ELEMENTS OF PHYSICAL BIOLOGY BT ALFRED J. LOTKA, M.A., D.Sc. "Voil& un homme qui a fait son miens pour _enniwer deux ou trois cents de ses concitoyens; mais son intention 6tait bonne: il n'y a pas de quoi dftniire PersepolM. Yoltovre BALTIMORE WILLIAMS & WLKINS COMPANY 1925 Composed and Printed at the WAVERLf PRESS BT THE VYlLLIAMS & WlLKINS Baltimore, Md., U. S. A. T COPTEIQHT 1925 WILLIAMS & WILKINS COMPANY Made in United States of America Published February, 1925 ALL BIGHTS KESERVIpD v^ DEDICATED TO THE MEMORY OF JOHN HENBY POYNTING PREFACE The preface is that part of a book which is written last, placed first, and read least. As I approach my concluding task I am moved to reflect why a preface should be written at all. This question, if followed into all the intricacies of which it holds potentiality, should apparently result in a composition new in literature, a Preface to the Preface. Such precedent should not be lightly established, for it suggests a vista of future degenerations after the pattern of Josiah Royce's infinite succession of maps, each containing within Itself its own replica on a reduced scale. But without going to such lengths as this, the philosophy of the preface may perhaps briefly be sum- marized to this effect, that It is the author's subjective introduction , to the more objective matter that should follow. Here he may, if this is deemed of any interest, say something regarding the circum- stances that gave origin to the w-ork, and the conditions under which it came into being. He may express his feelings as to its alleged pur- pose, and may follow custom by giving voice to pious wishes as to the function which the product of his presumptive mind may fulfill in an Universe in which no event, however trivial be it no more than the addition of one more book to the groaning library shelves is with- out distant reverberations. As to origin, the first plan of the work was laid about 1902, in the author's student days In Leipzig. The development of the topic is recorded, in outline, in various publications, of which the first appeared in 1907 in the American Journal of Science. Eeference to this and to its various sequels will be found in pertinent places in the text that follows. The last stage of the work, arrangement of the matter in collected form, and filling in the flesh about the skele- ton framework elaborated in the journal literature, was carried out at the Johns Hopkins University upon the invitation of the Depart- ment of Biometry and Vital Statistics. For the courtesies so extended to him the author wishes here to express his thanks, as well as for the interest shown in the progress of the work by Dr. Raymond Pearl and the members of the Department, notably Drs. W. T. Howard, L. J. Reed and J. R. Miner. Outside the walls of VU1 PREFACE this University I think with very particular appreciation of the never-failing succor in times of mathematical trouble, which I found at the hands of Prof. F. R. Sharpe of Cornell University; also of the patient assistance, upon more than one occasion, from Prof. W. B. Fite of Columbia University. And I gratefully recall en- couragement received from Dr. G. K. Burgess, Director of the Bureau of Standards, especially in the earlier stages of the work, when encouragement was most needed. Acknowledgment has been made in the text for numerous quota- tions. The somewhat extended excerpts from certain articles pub- lished in the Scientific Monthly call for special notice here, and I wish to express my thanks both to the author, Prof. G. W. Martin, and to the Editor of the Monthly, for permission to quote thus at length from its pages. I am similarly indebted to the Editor of Harpers Magazine for permission to reproduce here certain portions of an article from my pen, entitled "Biassed Evolution", which originally appeared in the May issue (1924) of that publication. Toward the publishers, Messrs. Williams and Wilkins and in particular Mr. C. C. Thomas, I have every occasion to entertain feelings of the most cordial appreciation. Through their courteous attentions the business of bookmaking was made a pleasure. My greatest debt is acknowledged in the dedication. Whatever merits this book possesses may well be credited to the influence and teaching of Poynting. There is little danger that its faults shall be charged to his account. As to the topic of the work it seems unnecessary to say many words here, inasmuch as a delineation of this has been made the subject of a special chapter on The Program of Physical Biology. Only this explanation it may be well to offer here, that, as proposed in Chapter V } the term Physical Biology has been employed to denote the broad application of physical principles and methods in the contemplation of biological systems, whereas Biophysics, in common parlance, relates rather to the special field of certain physical aspects of the life processes of the individual. With this terminology, Physical Biology would comprehend Biophysics within its scope. The writer cannot in reason expect to have produced a work without blemish. Even an approach to such absolute perfection is the rare privilege of a few. He would, however, be unjustified in addressing the reading public at all if he did not entertain the hope PEEFACE IX that, despite shortcomings, these pages may bring to the reader new assets, here and there a new piece for his mental furniture, now and again a new perspective, a new comprehensive outlook over a body of facts and relations in themselves perhaps familiar. The work has been largely one of systematization, and of develop- ment of method. Factual material has been introduced essentially for the purpose of illustrating the point of view to be set forth. There seems therefore hardly any occasion for apologetic explanations that anything of the nature of completeness in the presentation of perti- nent facts was in nowise armed at. Indeed, it must be obvious upon most casual reflection that such completeness, in a subject of the amplitude of that here taken in view, could be achieved only in a cyclopedic work of several tiers of volumes. Considerable care has been taken to cite in detail the sources con- sulted. It was felt that, on account of the wide dispersal of these citations over a broad field of scientific literature, few readers could be expected to be familiar with all the branches of pertinent library lore, and for this reason a collation of such references should have a value of its own, even apart from the text. At the same time the compilation of anything like a complete bibliography could not be undertaken on the present occasion. It is hoped that the mathematical mien of certain pages will not deter biologists and others, who may be disposed to look askance at symbols of the art, from acquiring an interest in other portions of the book. Biometricians will, presumably, not shrink on this score; to them, and to physicists, (whom I should greatly wish to number among my readers) I may perhaps confess that I have striven to infuse the mathematical spirit also into those pages on which symbols do not present themselves to the eye. For this I offer no apology. For the sake of space economy recapitulary paragraphs have, as a rule, not been given a place in the text. An exception has however been made in Chapters XX, XXXIII and XXIV, the last of which, in particular resumes and amplifies somewhat certain phases of the topics discussed in earlier chapters. The reader who may wish briefly to review the substance of his reading as he proceeds, should find suitable assistance in the rather detailed Analytical Synopsis of Chapters that has been placed immediately after the Table of Contents. And finally, a bird's eye survey of the general X PREFACE field covered in this work can be obtained by consulting the Tabular Synopsis at the end. Here, then, I make my exit from the prefatory stage and commend my work to the tender mercies of the reader; not without some trepi- dation, for I recall how Voltaire said of one: "II fit une philosophic comrne on fait un bon roman; tout parut vraiseniblable, et rien ne fut vrai;" and there comes to mind the language still plainer of du Maupassant "Depuis qu'ils ont appris a lire et a ecrire, la betise latente se degage." I trust that the reader's response to these pages may not be too fervent an Amen to the prayer of The Sceptical Chymisi "It is to be hoped that these men, finding that they can not longer write impertinently and absurdly .... will be reduced either to write nothing, or books that may teach us something . , . . ; and so, ceasing to trouble the world with riddles or im- pertineneies, we shall either by their books receive an advantage, or by their silence escape an inconvenience." ALFEED J. LOTSA. Johns Hopkins University, May, 1924- CONTENTS PART I. GENERAL PRINCIPLES CHAPTER I Regarding definitions 3 CHAPTER II Evolution Defined : The History of a System in the Course of Irreversible Transformation 20 CHAPTER III The Statistical Meaning of Irreversibility 30 CHAPTER IV Evolution Conceived as a Redistribution 41 CHAPTER V The Program of Physical Biology 49 PART n. KINETICS CHAPTER VI The Fundamental Equations of the Kinetics of Evolving Systems. Gen- eral Case 57 CHAPTER VII Special Case : Single Dependent Variable 64 CHAPTER VIII Special Cases : Two and Three Dependent Variables 77 CHAPTER IX Analysis of the Growth Function 100 CHAPTER X Further Analysis of the Growth Function 128 PART III. STATICS CHAPTER XI General Principles of Equilibrium 143 CHAPTER XII Chemical Equilibrium, as an Example of Evolution under a Known Law. . 152 xi Xll CONTENTS CHAPTER XIII Inter-species Equilibrium. . , 161 CHAPTER XIV Inter-species Equilibrium : Aquatic Life. 171 CHAPTER XV The Stage of the Life Drama 185 CHAPTER XVI The Circulation of the Elements. The Water Cycle 209' CHAPTER XVII The Carbon Dioxide Cycle 218 CHAPTER XVIII The Nitrogen Cycle 229 CHAPTER XIX The Phosphorus Cycle 246 CHAPTER XX Cycles : Conclusion and Summary 252 CHAPTER XXI Moving Equilibria. . . 259 CHAPTER XXII Displacement of Equilibrium 280 CHAPTER XXIII The Parameters of State 300 PART IV. DYNAMICS CHAPTER XXIV The Energy Transformers of Nature 325 CHAPTER XXV The Helation of the Transformer to Available Sources 336 CHAPTER XXVI The Correlating Apparatus 362 CHAPTER XXVII Extension of the Sensuous World Picture 371 CHAPTER XXVIII The Adjusters. 381 CONTENTS Xlll CHAPTER XXIX Consciousness 388 CHAPTER XXX The Function of Consciousness 394 CHAPTER XXXI The Origin of Consciousness in the Living Organism 402 CHAPTER XXXII Energy Relations of Consciousness 406 CHAPTER XXXIII Review of the Correlating Apparatus 410 CHAPTER XXXIV Conclusions ] Retrospect and Prospect 417 Synoptic chart of physical biology 435 LIST OF ILLUSTRATIONS FIG. 1. Graph of model process illustrating the statistical meaning of irreversibility. Reproduced from A. J. Lotka, Two Models in Statis- tical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 122 31 FIG. 2. Graph of second model process illustrating the statistical meaning of irreversibility. Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 124 33 FIG. 3. Frequency diagram for the deviations from the mean appearing in figure 2. Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 125 34 FIG 4. The law of population growth for the United States according to Pearl and Reed 68 FIG. 5. Growth of a population of Drosophila (fruit flies) undsr controlled experimental conditions, according to Pearl and Parker. 69 FIG. 6. Growth of a bacterial colony (B. dendroides). according to H. G. Thornton 71 FIG. 7. Growth of rat according to H. H. Donaldson and T. B. Robertson. 73 FIG. 8. Growth of sunflower seedlings according to H. S. Reed and R. H. Holland; computed curve by L. J. Reed 74 FIG. 9. Growth of sunflower seedlings. The same data as in figure 8, but plotted in logarithmic diagram 75 FIG. 10. Presumptive curve of growth of endemic malaria according to the Ross Equation. Reproduced from A. J. Lotka, Am. Jour. Hy- giene, January Supplement, 1923 84 FIG. 11. Course of parasitic invasion of insect species according to W. R. Thompson 85 FIG. 12. Increasing diffusion-in-time of successive generations in the progeny of a population element 86 FIG. 13. Course of parasitic invasion of insect species, according to Lotka; elementary treatment 90 FIG. 14. Course of parasitic invasion of insect species, according to Lotka; more exact treatment 91 FIG. 15. Some historical human survival curves, exhibiting an evolution- ary trend toward longer average duration of life 103 FiGi 16. Survival curves for the State of Massachusetts, for the three decades 1890-1900-1910. After Glover 104 FIG. 17. Survival curves for different countries. After Glover 105 FIG. 18. Survival curve plotted on logarithmic scale. II. S. 1910 107 FIG. 19. Logarithmic survival curves for man, Drosophila, and Proales de- cipiens. Plotted according to centiles of life-span. After R. Pearl. . 109 FIG. 20. Diagrams to illustrate proof of stability of normal age distribu- tion. Reproduced from A. J. Lotka, Proc. Natl. Acad. Sci., vol. 8, 1922, p. 339 , 113 xv ^ LIST OP ILLUSTRATIONS FIG. 21. "Stable" age distribution, as exemplified by the population of England and Wales in the decade 1871-1880 ........................... 114 FIG, 22. Diagram of relation between birth rate per head 6 and death rate per head d in population with stable age distribution ................ 117 FIG. 23. Diagrammatic illustration of influence of random and of selective slaughtering upon survival curve of biological species .............. 120 FIG. 24. Growth of favored type in mixed population of two phenotypes After J. B. S. Haldane ............................................. 124 FIG. 25. Effect of selection on population comprising two phenotypes with Mendelian inheritance. After J. B. S. Haldane ..................... 126 FIG. 26. Feed consumed, and increase in live weight of steers at several ages. After Moulton, Trowbridge and Haigh ....................... 133 io. 27. Some fundamental types of equilibrium, in a system with two dependent variables ............................ j 4 g FIG 28. Map of integral curves for the Ross malaria equations.' Repro- 1923 A ' J " L tka ' Am ' JoUr- Hygiene > Jan uary Supplement, FIG 29 Model of surface corresponding to figure 28 '."Reproduced from ff^ A ?' JoUr ' H yg iene > January Supplement, 1923 ......... 150 G. dO. Age distribution in population of molecules of two substances in monomolecular chemical equilibrium. Reproduced from A. J. Lotka, Am. Jour. Science, 1907, p. 208 ....... 155 n contents " of '' * of srasshoppe FIG 34. Seasonal food habits of the Meadow" Lark.' After H.c'.B^ant * ' ' ^ of FIG. 39. Key to figure 38 .............................................. 177 205 LIST OF ILLUSTRATIONS FIG. 45. Periodic classification of the elements, showing division into petrogenic elements and metallogenic elements. After H. S. Washington ^ 7 FIG. 46. Circulation of the elements in nature. The water cycle 215 FIG. 47. Circulation of the elements in nature. The carbon cycle 226 FIG. 48. Circulation of the elements in nature. The nitrogen cycle 230 OQTJ FIG. 49. Organic nitrogen circulation * 0i FIG. 50. The rise of the saltpeter industry 235 FIG. 51. The rise of the fixed nitrogen industry 24 FIG. 52. Circulation of the elements in nature. The phosphorus cycle. . . 247 FIG. 53. The Soxhlet extraction apparatus 253 FIG. 54. Circulation of the elements in nature. The sodium chloride cycle ; m FIG. 55. Uranium and its products of radioactive disintegration 263 FIG. 56. Equilibrium polygon for Radon (radium emanation) in contact with its disintegration products 266 FIG. 57. Relative abundance of the elements. Diagram according to W. D. Harkins, based on analysis of meteorites. (Jour. Am. Chem. Soc., 1916, p. 863) 271 FIG. 58. Equilibrium polygon for the human species and some of the spe- cies on which it depends for its food supply 277 FIG. 59. The Yellow Shark and some of his relatives, as an example of possibly orthogenetic development 296 FIG. 60. Law of urban concentration 308 FIG. 61. Relation between rate of reproduction in Drosophila (fruit fly) and density of mated population. After Pearl and Parker 309 FIG. 82. Relation between mean length of life and population density in Drosophila. After Pearl and Parker 310 FIG. 63. Hyperbolic curves obtained by plotting as ordinates the number of genera 1, 2, 3, . . . n species, and as abscissae the number n of such species. After J. G. Willis 314 FIG. 64. Relation between number and size of genera of all flowering plants, plotted logarithmically. After J. C. Willis 315 FIG. 65. Relation between number and size of Rubiaceae, plotted logarith- mically. After J. C. Willis 315 FIG. 66. Relation between number and size of genera of chrysomelid bee- tles, plotted logarithmically. After J. C. Willis 316 FIG. 67. Solubility of carbon dioxide in water, expressed in volumes of COj measured at normal temperature and pressure, per volume of water. After G. W. Martin 318 FIG. 68. The mill-wheel of life 334 FIG. 69. Mechanical walking beetle, exhibiting the several characteristic elements of the correlating apparatus 341 FIG. 70. The evolution of man's means of transportation 367 FIG. 71. Growth of American Railways 369 FIG. 72. How the future enters into the determination of the motion of the walking mechanical beetle, thus imitating purposive action (tele- ology) 382 LIST OF TABLES Table 1. The program of physical biology 53 Table 2. Growth of the population of the United States 67 Table 3. Growth of bacterial colony. After H. G. Thornton 70 Table 4. Growth of sunflower seedlings. After H. S. Reed and R. H. Holland 74 Table 5. Course of parasitic invasion of insect species, according to W. R. Thompson 86 Table 6. Tabular survey of different modes of interdependence of biologi- cal species 98 Table 7. Example of normal age distribution 113 Table 8. Relation between birth rate per head b and death rate per head d in a population with normal age distribution 117 Table 9. Growth of steers, and feed consumed 134 Table 10. Survey of methods of marine biological census 178 Table 11. Principal components of the earth's surface crust 185 Table 12. Distribution of gases in the atmosphere at different levels 188 Table 13. Constituents of the atmosphere, at earth's surface and in toto . . 190 Table 14. Composition of the Ocean 191 Table 15. Comparison of air and aquatic atmosphere 192 Table 16. Composition of lithosphere 194 Table 17. Composition of human body 197 Table 18. Composition of living organisms. After H. F. Osborne 198 Table 19. Comparison of composition of blood serum and sea water 201 Table 20. Moisture contents of foods 211 Table 21. Discharge of the World's rivers 214 Table 22. Progress of by-product coke ovens in the United States 233 Table 23. Growth of the saltpeter industry 237 Table 24. Uses of saltpeter 237 Table 25. Meteoric rise of the nitrogen fixation industry 242 Table 26. Supply of plant foods in the soil 257 Table 27. Rate of participation of the elements in the cycle of nature. . . 258 Table 28. Radioactive equilibrium of radium in contact with its disinte- gration products 265 Table 29. Radioactive equilibrium of radon (radium emanation) in con- tact with its disintegration products 268 Table 30. Geological time table. After C. Schuchert 269 Table 31. World's production of population and its accessories. After R. Pearl 278 Table 32. Association of high blood pressure with over-weight 295 Table 33. Distance travelled in one hour by different modes of conveyance 368 Table 34. Man's correlating apparatus, native and artificial contrasted. . 411 Table 35. Man's correlating apparatus (continued). The Adjusters 413 Table 36. Classification of the sciences 423 ANALYTICAL SYNOPSIS OF CHAPTERS PART I. GENERAL PRINCIPLE Chapter I. Regarding Definitions. Definitions are arbitrary, 3 But are governed by considerations of expediency, 3 The problem of expediency in framing definitions is not always a simple one, 3 Pseudo-problems arising from failure to recognize the arbitrary character of definitions: Hunting the Jabberwock, 4 Subjective discontinuities introduced by the senses, 5 Examples: Colors, e.g. blue, green; light and heat waves; distinction between animals and plants; between biological species; between living and non-living matter, 5 To such abrupt subjective divisions there may correspond no objec- tive discontinuity in nature, 5 Definitions in Biology, 5 Vitalism versus Mechanism; merely a question of terms, 7 Herbert Spencer's "proximate definition" of life, 7 Inadequacy of Spencer's definition, 7 Sir Edward Schaefer's standpoint, 8 Line of division recedes with increasing knowledge 8 Alleged characteristics of living matter: Growth from within 8 Chemical growth as distinguished from physical growth of crystals, 10 Growth from unsaturated solution, as distinguished from growth of crystals from super- saturated solution, 10 "Selective" growth, 10 Reproduction, 11 Vital Force, 13 Physical chemistry of structured systems, 13 Geometrical element lacking in physical chemistry of today, 13 Systems ordinarily considered are either structureless or of simple structure, 14 This absence of structural features in physico-chemical systems is due to subjective, not objective rea- sons, 14 Due in part to convenient arbitrary restrictions, 15 Relation between growth, environment and structure, 15 The laws of chemical dy- namics in structured systems will be the laws that govern the evolution of a system comprising living organisms, 16 Application to Biology: The organ- ism as a structured physico-chemical system, 16 The travelling environment, milieu inte"rieur, 17 Increasing independence of organism of its remote en- vironment, 18 The policy of resignation: Abandoning the attempt to define life, 18 Parallels in the history of science : Abandonment of the attempt to prove Euclid's twelfth postulate led to new systems of geometry, 18 Aban- donment of attempts to build perpetual motion machines was equivalent to recognizing the law of conservation of energy, 18 Abandonment of the at- tempt to detect the earth's motion through the ether is the foundation of the modern theory of relativity, 18 The ideal definition is quantitative, 19 Desirability of establishing a quantitative definition and conception of evo- lution, 19. Chapter II. Evolution Defined. Definition should conform as far as possi- ble to common usage of the term, 20 Analysis of common conception of evolu- tion, 20 Evolution is history, but not all history is evolution, 20 Systems in purely periodic motion would not be said to evolve. They repeat in endless xxi XXil ANALYTICAL SYNOPSIS OF CHAPTERS succession the same series of events. In an evolving system each day is unlike any other day, 21 Evolution not a mere changeful sequence, 21 Abortive attempts to formulate the direction of evolution, 21 These attempt defini- tion in terms of a single component, 22 Such definitions are foredoomed to failure, a successful definition must be framed in terms of the evolving system as a whole, 22 Evolution is the history of a system in the course of irreversible transformation, 24 Scope of this definition: What it excludes; what it includes 24 The line of division depends on the nature and extent of our knowledge regarding the system, 25 This is in harmony with the fact that problems of evolution are largely problems of probability, 25 All real trans- formations are irreversible, hence all real history is evolution, 26 What then is gained by the definition? 26 It indicates the direction of evolution as the direction of irreversible transformations, the direction of increasing entropy, 26 Example of pendulum. Irreversible feature introduced by fric- tional force, 27 Inertia-free or completely damped systems, 28 Accelerations vanish with velocities, 29 Velocities are single-valued functions of configura- tion, 29. Chapter III. The Statistical Meaning of Irreversibility., Apparent irrever- sibility (progressiveness in time) of certain theoretically periodic processes, 30 Their periodicity, with eventual return to initial state, never observed in practice, 30 Explanation of this discrepancy: Macroscopic model illustrative of irreversibility of gaseous diffusion, 30 Return to initial state possible, but exceedingly rare (highly improbable), 31 The model is competent to illustrate also the highly improbable event of return to initial state, 32 Dynamical theory indicates not only occasional but periodic return to initial state, 32 A second model illustrates this also, 32 Evolution aspassage from less probable to more probable states, 35 Inadequacy of this "principle" : it is indefinite in fail- ing to specify the characteristic with respect to which probability is reckoned; and it is incomplete in failing to draw attention to certain energy relations, 35 Irreversibility is relative, depending upon the means naturally available or arbitrarily permitted to operate upon the system, 35 Significance of this m organic world: Macroscopic irreversibility of diffusion processes in nature, 36 Need of a method of mathematical analysis to deal with cases intermedi- ate, in specified degree, between the following two extremes: (a) Wholly in- discriminate (pure chance) operation upon material in bulk, (b) Wholly determinate operation, with nothing left to chance, upon materials discrimi- nated and acted upon in detail, piece by piece, and circumstance by circum- stance, _36--This method must take account of degree of perfection of the mechanical and psychic equipment by which each organism reacts upon its environment, 37-Senses as a means of overcoming chance, 37-PhysicaI sig- nificance of our subjective sense of forward direction in time, which finds no expression in the differential equations of pure dynamics, 37-This subjective tune sense may be related to the influence of initial conditions in dynamics, 38-But the direction of evolution seems related rather to that directedness in Zdltl 18 f + r Ct T tiC f aperi dic r S6emingly a P eriodic ?**, 38- Inadequacy of thermodynamic method, 39-The linking of evolution with the ANALYTICAL SYNOPSIS OF CHAPTERS XX111 concepts of thermodynamics and statistical mechanics is instructive as sug- gesting a conception of the direction of evolution, the direction of increasing entropy, increasing probability, 39 This point of view, however, is inadequate for application to concrete cases of organic evolution, because data are fur- nished in terms unsuited to the methods of thermodynamics, 39 Neither are existing methods of statistical mechanics, as applied to molecules and the like, helpful; the instrument is ill adapted to the scale of the object, 39 New method needed, that shall accept its problems in terms of biological data, as thermodynamics accepts its problems in terms of physical data; a General Theory of State, an "Allgemeine Zustandslehre " 39. Chapter IV. Evolution Conceived as a Redistribution. Evolution viewed as a redistribution of matter among the components of a system, 41 System described by statement of mass of each component, and indication of value of certain parameters, 41 Analytical expression of history of system given by relations or equations established between the variables and parameters de- nning the state of the system, 41 Fundamental equations usually simplest in form of differential equations, 42 Particular form of equations of evolv- ing systems, 42 General form of equations of evolving system, 43 Equations as applied to life-bearing system, 43 Definition of the components arbitrary but conclusions relate to the components as defined, 44 Relation of evolu- tion, as here conceived, to the problem of the origin of species, 44 Inter-group and intra-group evolution, 44 Analytical indication of intra-group evolution, 45 Fundamental equations resemble in form the equations for an inertia-free or completely damped system, 47 Fundamental equations, as here given, may not cover all cases, but are at any rate of very wide scope, 47 Equations interpreted to include possible lag or lead effects, 47 Singular implications of lag and lead effects; possible relation to phenomena of memory and will, 48 Appearance of lag and lead effects in equations may, however, be spurious, 48. Chapter V. The Program of Physical Biology. Systematization and division of subject, 49 General mechanics of evolution, 49 Macro-mechanics and micro-mechanics, 50 Statistical mechanics as the connecting link, 50 Stoichiometry, the study of mass relations (material transformations), 50 Energetics or Dynamics, the study of the energy transformations, 50 Kinetics and Statics, 51 Equilibrium and steady states, 51 Moving equilibria, 51 Displacement of equilibrium; Le Chatelier's principle, 52 Sociological analogues of forces and "quasi-dynamics" (economics), 52 The term Physical Biology to be used to cover the territory indicated in this chapter, 52 Methods of obtaining data, 52 Chart of Program of Physical Biology, 53 Methods of elaborating data, 54. PART II. KINETICS Chapter VI. The Fundamental Equations of Kinetics of Evolving Systems. General case, 57 Some implications of the fundamental equations in their general form, 57 Equations of constraint, 58 Elimination of variables. Introduction of constants A, 58 Evolution with parameters P, Q and A constant, 58 Equilibria or steady states, 59 Number and character of XXIV ANALYTICAL SYNOPSIS OF CHAPTEES equilibria. Example: Fly population, 59 Fundamental equations trans- formed by introduction of excess of actual masses over equilibrium masses in place of the former, 60 Expansion in Taylor's series, and solution in exponen- tial series, 60 Characteristic equation for exponential coefficients, 60 Sig- nificance of sign and character of roots X of characteristic equation, 61 Sta- bility of equilibrium, 61 Mode of approach to equilibrium: Aperiodic and periodic (oscillatory) type, 61 Analytical confirmation and extension of a passage in Herbert Spencer's First Principles, 61 Zero or negative roots of the equilibrium equation: Unfit species, 62. Chapter VII. Fundamental Equations of Kinetics (Continued). Special Case: Single Dependent Variable. Law of population growth, 64 Popula- tion of United States, 66 Stability of equilibrium, 67 Experimental popula- tions, 69 Diminishing population, 70 Growth of individual organism, 71 Autocatakinesis, 76. Chapter VTII. Fundamental Equations of Kinetics (Continued). Special Cases: Two and Three Dependent Variables. Interdependence of species, 77 Several types of interdependence, 77 Analytical characteristics of those types, 78 Effect of these characteristics upon the nature of the solution of the equation of Kinetics for two variables, 79 Concrete examples, 5 Martini's equation for immunizing diseases, 79 A special case noted by Watson, 81 The Ross malaria equations, 81 Example in parasitology, 83 Thompson's treatment of the case, 83-Objections to this, 87-Treatment of case by general equation of kinetics, 88-Nature of the solution: it represents a periodic or oscillatory process, 90-Comparison with observed facts according to L O Howard^ 90 Annihilation of one species by another, 92-Case of three depen- dent variables, 94-Relation of this to a practical problem in sea fisheries, 95- Replaceable and irreplaceable components, 95-Limiting factors, 97-Lie- big s law, 97 Chart of types of interdependence of biological species 98 * 2!r K ' A * dysis of the Growth Unction. The form of thi function A+ ? T l aggregates 100-Demographic functions, 101-Survival M?^ '^ i T r , mal Ege diataibu *^ HO-Demogra P hic relations in normal population, 115-Rate of increase per generation, 118-Effect of selects slaughtering 119-Intelligence as a discriminating agency 20 (With MendeH J B. S Chapter X. Further Analysis of the Growth Function. Adjustment of the e teful produce, 132- nrnHnna 1QO T?~i j.- j ' -^ vvu^o uiv, muuiaiS ICept tor p^ss^s^it^s^ Sr P S-^;-- -:s-i; ANALYTICAL SYNOPSIS OF CHAPTEBS XXV from standpoint of domestic species is high efficiency from human standpoint, 135 Network of chains of interrelated species, 136 Transformation factors and their economic significance, 137. PART ill STATICS Chapter XI. General Principles of Equilibrium. Equilibria and station- ary states, 143 Scope of Statics, 143 Kinetic, dynamic, and energetic defini- tion of equilibrium, 143 Our chief interest here in stationary states not true equilibria, 145 General equilibrium condition, 145 Different types of equi- libria, 146 Illustration: Malaria equilibrium according to Ross, 147 Metas- table equilibrium, 151 Exceptional cases, 151. Chapter XII. Chemical Equilibrium, as an Example of Evolution under a Known Law. Case of simple balanced reaction, 152 Relation between "birth" rate and "death" rate of molecules, 153 The survival factor p (a) and life curve of molecules, 153 Reaction constant as a a force of mortality, 154 Chemical reaction as a case of survival of the fittest, 154 Analogy of chemical reaction to course of events in population of mixed biological species, 155 Presumptive mechanism of chemical reaction (Baly), 155 The fuga- ceous transitional state intermediate between chemical compounds (Schonbein), 156 Laws of thermodynamics, as determinants of the end state in an equi- librium reaction, are, in this case, the Law of Evolution of the system, 157- Natural laws conveniently expressed as maximum or minimum principles, 157- Law of organic evolution may be expected to take this form, 158 Law of chemical evolution is framed in terms of the system as a whole ; law of organic evolution must undoubtedly also be thus framed, 158 Law of equilibrium expressed in form of a minimum principle, 158. Chapter XIII. Inter-Species Equilibrium. Equilibrium condition in more particular form, 161 Numerical illustration, 162 Economic relationship of coefficients appearing in equilibrium condition, 163 Sources of information regarding biological equilibrium : Biological surveys, 164 Analysis of stomach, contents, 166 Intra-species equilibrium (with Mendelian inheritance), 170. Chapter XIV. Inter-species Equilibrium: Aquatic Life. Special occasion for demological studies of aquatic population, 171 Fishes as natural dragnets, 171 Aquatic food chains in relation to human food, 2 Importance of aqui- culture for the future, 172 Loss of fertilizer through modern methods of sewage disposal, 172 Partial restoration of such material to human food sup- ply through fisheries, 173 Different methods of census of marine population : dragnet; bottom samplers; recatching of marked catches; centrifuge; dilution culture, 173 The nannoplankton, 173 Work of Petersen in Kattegatt, 174 Quantitative estimates of principal groups of the marine population of Katte- gatt, and their interrelation, 175 Summary of methods of marine biological census (Table), 176 Food chains, 176 Necessity of occupying ourselves with the more remote links of our food chain, 177 "Feeding our foods, " a species of symbiosis, 180 Agriculture and aquiculture, as mining and as manufactur- ing industries, 180 Resources but recently tapped, and others still untouched, 180 Food chains in aquatic species, 180 Wastefulness of long food chains, XXVI ANALYTICAL SYNOPSIS OF CHAPTEES 181 Economic value of shell fisheries owing to simple food chain coverting vegetation directly into human food (oysters, etc.); 181 Primary, secondary and tertiarv foods, 171 Cycles and the circulation of the elements in Nature. 183. ' Chapter XV. The Stage of the Life Drama. The tripartite world, 185 Atmosphere, 185 How the earth holds her atmosphere, 185 Cosmic losses from the atmosphere negligible, 188 Cosmic accessions to the atmosphere, 192Hydrosphere, 192 Aquatic atmosphere, 193 Lithosphere, 193 Cosmic accessions to lithosphere, 195 Composition of earth's crust, 195 Relation to composition of organism, 197 Similarity in composition of sea water and blood serum, 201 Significance of this as regards aquatic origin of terrestrial fauna, 202 Chemical correlation in soil and in organism, 204 Accessibility of valuable earth constituents, 206 Accessibility not in any direct relation to abundance, 206 H. S. Washington's classification of petrogenic (rock-form- ing) and metallogenic (ore-forming) elements of the periodic table, 207 Dissipating and concentrating processes in Nature, 208. Chapter XVI. The Circulation of the Elements : The Water Cycle. Circu- lation of elements vaguely realized by ancients, 209 Water requirements of human body, 210Water requirements of plants; rainfall as limiting factor, 211 Sources of supply, 213 Quantitative estimates of several items in water cycle, 213 Water cycle diagram, 215 Fraction of total water circulation tak- ing part in life cycle, 216. Chapter XVII. The Carbon Dioxide Cycle. Combustion as an essential feature of the life process, 218 Analogy of flame arising from spark and life arising from germ, 218 Rarity of ignition except from pre-existing flame anal- ogous to apparent impossibility of life originating except from preexisting life, 218 Oxygen as an inorganic food, 219 Plant nature of man in his attitude toward this inorganic food, 219-The carbon cycle, 220-Carbon as the organic element, 220 Source and gate of entry into the organic carbon cycle, 220 Some estimates of quantities involved in the carbon cycle, 220 Interrelation of green plants and animals, 221 Carbonization of dead vegetable matter and its significance for the industrial civilization of the present era, 221The pres- ent an atypical epoch: man living on his capital, 222-Attempts to establish the balance sheet of the earth's carbon economy, 222 Absorption of C0 2 in weathering of rocks, 223-Formation of CO 2 by burning of coal, 224-The ocean as an equalizer regulating the C0 2 content of the air, 224-Unccrtainty as to net loss or gam of total C0 2 in atmosphere, 225 The oxygen cycle, 225-Oriin of atmospheric carbon dioxide and oxygen, 225-Carbon cycle diagram 226- Loss of oxygen from the atmosphere, 228. Chapter XVIII. The Nitrogen Cycle. Natural demand and supply 229- Seemmg abundance of nitrogen illusory, 229-Nitrogen cycle diagram,' 230- Cate of entry into nitrogen cycle, 232-Leak of nitrogen out of circulation 2ipLoss of nitrogen in combustion, distillation and coking of coal 233-Sk- mficance and rapid increase of by-product recovery type of coke ovens, 233- Other losses of combined nitrogen, 234-Accessory sources of combined nitro- nSJ f ^' 234 r Human ^erference in nitrogen cycle, 236-Ex- ploitation of guano deposits, 236-CMlean nitre beds, 236-Consumption of ANALYTICAL SYNOPSIS OF CHAPTERS XXV11 saltpeter as fertilizer and otherwise, 237 Origin of nitre beds, 238 Industrial nitrogen fixation processes, 239 Birkeland and Eyde nitric acid process. Cyanamide process. Haber ammonia process. Biicher cyanide process, 239 Combination of Haber process with Solvay soda process, 239 Ostwald oxidation process, 241 Meteoric rise of nitrogen fixation industry, 241 Its ethnological significance, 241 Economic and energetic significance of concen- tration, e.g. of supply of combined nitrogen, 243 Localized sources of concen- trated supplies as centers of attraction in man's economic activities, 244 Total circulation tends to increase, 245. Chapter XIX. The Phosphorus Cycle, Immobile elements, 246 Natural phosphorus supply of soils, 246 Phosphorus cycle diagram, 247 Leakage of phosphorus from circulation, 248 Phosphate rock and the migration of phos- phorus, 248 The role of fish and birds in the phosphorus cycle, 249 Soil losses of phosphorus, 250 Phosphate slag as fertilizer, 251. Chapter XX. Cycles: Conclusion and Summary. Circulation of chlorine and the alkalis, 252 The sea and the sun as a Soxhlet extractor, 252 Sodium chloride cycle diagram, 254 Differential behavior of sodium and potassium in the process of extraction, 255 Comparative rarity of potash in soils, 255 Effect of World War on potash market, 256 Circulation of sulphur, 256 Circulation of iron, 256 Summary of cycles, 257 Supply of plant food in soil, 257 Rate of participation of elements in cycles of nature, 258. Chapter XXI. Moving Equilibria. Principle of continuity, 259 Equation of slowly moving equilibrium; first approximation, 259 Higher approxima- tions, 260 Special case : pace set by slowest member in a chain, 261 Radio- active equilibrium, 261 Equilibrium polygon, 266 Extinction of unadapted species, 266 Example of inaccuracy of first approximation, 268 Radioactive chains as cosmic clocks, 268 Geological time table, 269 Origin of elements and ultimate genesis of organisms, 269 Relative abundance of the elements, 271 Terminal stages of the earth's evolution: Geophysics and geochemistry, 273 Joly's theory of periodic melting of the earth's crust, 275 Organic moving equilibria, 276 Equilibrium polygon, 277. Chapter XXII. Displacement of Equilibrium. Perfectly general case of influence of change in parameters will not here be considered, 280 Principle of Le Chatelier, 281 Some common misstatements of the principle, 282 Early essayed application to biology, 283 Conditions of validity of the prin- ciple, 284 Extension of scope of rigorous applicability, 286 Area and rent, 288 Discussion of displacement of equilibrium independently of Le Chatelier's principle, 289 Case I: Displacement of equilibrium between food and feeding species, 289 Case II: Change of circulation through moving cycles, 292 Some significant cases of instability, 294 Vicious circles, 294 Cumulative cycles simulating orthogenesis, 296 Benign cycles, 297. Chapter XXIII. The Parameters of State. Topographic parameters, 300 Intensity factor of energy, 303 Simpliest examples of topographic par- ameters : volume, area, 301 Complexity of topography in organic evolution, 301 Simplification of problem by "substituting ideal upon which it is possible to operate, for intractable reality", 302 Empirical study of biogeography and ecology, 302 Conjugate parameters, 303 The intensity law in organic and xxviii ANALYTICAL SYNOPSIS OF CHAPTERS economic systems, 303-Rent as a measure or index of population pressure 304 But population pressure exists independently of rent, e.g., in species other than man, 304-Distant analogy of law of population pressure to gas law, 305-Law of urban concentration, 306-Biological background of population pressure, 307 Influence of population density on rate of reproduction (Pearl and Parker), 308 Influence of population density on duration of life (Pearl and Parker), 309 Topographic parameters during period of diffusion, 311 Willis' theory of Age and Area, 311 Climatic parameters, 317 Their labora- tory investigation (Pearl and Parker), 319 Parameters of state and the analytical condition for equilibrium, 319 Thermodynamic analogy, 320 Inversion of typical problem of thermodynamics, 321 Systems of Quasi- Dynamics, 321. PART IV. DYNAMICS Chapter XXIV. The Energy Transformers of Nature. The fundamental equations of Kinetics do not exhibit any explicit reference to dynamical or energetic relations, 325 But certain of the components S are energy trans- formers, 325 Fundamental characteristics of energy transformers, 325 Cyclic working; output and efficiency, 326 Thermodynamic law of maximum output, 326 Reversible and irreversible processes, 327 Composite and coupled transformers, 327 Accumulators, 328 Chemical accumulators, 328 Growth, 328 Law of growth, 328 Anabions and catabions, 329 Systems of transformers, 329 Plant and animal as coupled transformer, 330 The World Engine, 331 Share of sun's energy that falls to different constituents of world, 331 Share falling to organic circulation, 331 Relation of transformer cycle to circulation of the elements, 334 Influence of limiting factors upon working of world engine, 334 Evolution of the World Engine, 335. Chapter XXV. Relation of the Transformer to Available Sources. Distrib- uted and localized sources of energy, 336 Random and aimed collisions, 337 Negative correlation, 337 The correlating apparatus, 338 Component ele- ments of correlating apparatus: Depictors, Receptors, Elaborators, Adjus- ters, 339 Receptor-effector circuit begins and ends in environment, 340 Significance of this, 340 Correlating apparatus not peculiar to living organ- isms, 340 Mechanical imitations of living beings (automatons) 341 Chess as a conventional model of the battlefield of life, 343 The biological contest considered in the light of the chess analogy, 343 Topographic map, centers of mobility and centers of influence as the elements of the game, 343 Zones of influence, 344 Collisions or encounters, 344 Zones of mobility, 345 Analyt- ical statement of problem of organic conflict, 345 The behavior schedxile, 346 Specific productivity, 347 Effect of change in zone pattern, (Intra- species evolution), 348 Biologic relation of economic value, 350 Effect of change in behavior schedule, 350 Rigid or automaton type and elastic or free-choice type of behavior schedule, 350 Relation between ideal and actual organism, 352 Effect of small departure from perfect adjustment, 353 Relation of economic value to physical energy, 354 Economic conver- sion factors of energy, 355 General or aggregate effect of individual ANALYTICAL SYNOPSIS OF CHAPTERS XXIX struggles for energy capture, 356 The law of evolution adumbrated as a law of maximum energy flux, 357- Statistical mechanics of a system of organ- isms, 358 Mean free path, 358 Frequency of collision and capture, 359 Influence of size of organism, 359 Curves of pursuit, 360 Random motion under a bias, 360 Use of models, 360. Chapter XXVI. The Correlating Apparatus, 362 Receptors, 363- Artificial receptors, 364 Significance of these in evolution of modern man, 364 Effec- tors, 366 Artificial effectors; industrial evolution, 367 Singular effects of in- dustrial evolution, 368. Chapter XXVII. Extension of the Sensuous World Picture. The Elabora- tors, 371 The scientific world picture : Systems of coordinates, 372 The ego as a coordinate reference frame, 372 The ego immaterial, 373 Interpenetra- tion of the egos, 374 Where is Mind?, 375 Fundamental premises and implicit assumptions, 376 Difficulty of shaking off preconceived premises, 377 Ob- stacle which this raises to understanding, 377 The communicators, 378 Orthogenesis in human evolution, 378 Orthogenesis does not suspend selec- tion, 380 Importance of curiosity in evolution, 380. Chapter XXVIII. The Adjusters. Mechanistic and teleological inter- pretation of adjusters, 381 Significance of the future in the operation of ad- justors, 382 The future that may be and the future that will be, 382 The doubtful cases, 383 Final causes or purposes not usually postulated when sufficient account can be given of events in terms of efficient causes (e.g., in terms of mechanistic explanation), 384 Adaptive adjustment of tastes, 385 Spencer's hedonistic principle, 386 Genuine utility for social service, 386. Chapter XXIX. Consciousness. Relation of consciousness to physical conditions, 388 Conditional relations, 388 A fundamental hypothesis ad- mitted, 389 Consciousness is closely bound up with life processes and struc- tures, 390 Consciousness dependent on metabolism. The personal element, 391 Consciousness possibly a general property of matter, 392. Chapter XXX. The Function of Consciousness. The contents of conscious- ness determined by past and present bodily states, 394 Operative relations of consciousness to physical conditions, 394 Not only knowledge but motive required in the working of the human organism, 394 Motivation in lower organisms appears fatalistic (simple tropisms), 395 Purposive action, 395 Dynamic psychology, instinctive drives to action, 396 Individual traits. In- stinct of workmanship and self-expression, 396 Influence of special aptitudes, 399 The industrial and the personal problem of satisfying instincts, 400. Chapter XXXI. The Origin of Consciousness in Living Organisms. Why has nature resorted to consciousness as means for effecting adaptive reactions of organisms? 402 The problem of psycho-physical parallelism, 402 The double aspect theory of consciousness, 403 Physical analogies, 403 Physico- chemical theory of consciousness as a state related to the transitional state of molecules in chemical reaction, 403 Argument of simplicity of structure may be invoked in favor of this theory, 404 Origin of consciousness, 404 Some elementary forms of consciousness perhaps a general property of all matter, 404 In that case not consciousness has been evolved, but only a par- ticular type of consciousness, a consciousness integrated around an ego, 404. XXX ANALYTICAL SYNOPSIS OF CHAPTEBS Chapter XXXII, Energy Relations of Consciousness. Seeming conflict between directive power of consciousness and determinate character of physi- cal events, 406 Possible explanations : First alternative possible inaccuracy of laws of dynamics, 406 Second alternative singular orbits with indetermi- nate motion, 407 Conception of Clerk Maxwell and of J. Boussinesq, 407 Third alternative possible influence of factors eliminated from equations of dynamics, 408. Chapter XXXIII. Review of the Correlating Apparatus. Tabular synopsis and classification of Depictors or Informants, 410 Epictors or Transformants, 411 Adjusters, 412 Internal adjusters, 412 Tabular synopsis of adjusters, 413 External adjustors, 414 Anatomical and physiological adjustment of social activities, 414 Economic adjustment of social activities, 415 Disad- vantages of latter form of adjustment, 415 Its advantages or necessity in the human community, 415. Chapter XXXIV. Conclusion; Retrospect and Prospect, The life struggle in the modern community, 417 New character of this struggle : competition is principally within the species, 417 Organization of motives lags behind indus- trial organization, 418 Philosophy as a necessary part of scientific enquiry, 418 Classification of Sciences in relation to Self and External World, 419 Bertrand Russell's viewpoint regarding object and subject, 420 This leads to a natural division and classification of the sciences, 421 The ego as at once a Knower and a Wilier; hence the reaction of knowledge upon the emotions, 421, 424 The poetry of science, 425 Significance of emotional reaction of knowl- edge for our future evolution, 425 Evolution of the Self may have overshot the mark, 426 Man's opportunity to influence the destiny of his species by his own initiative, 427 Desirability of concerted action, 427 Of constructive optimism, 428 Evolutionary value of nurture and tradition, 428 Fallacy of cynical deprecation of potency of nurture, 429 Extension of Spencer's hedonistic principle, 430 How it explains the appearance of design in nature, 430 Such design can be neither proved nor disproved, and may therefore be made an article of faith, 430 Man may embrace the World Purpose for his own, 430 Survival value of such attitude, 430 Orthogenesis in field of ethics, 430 Once more the reaction of knowledge upon the emotion, 432 Evolution is not achieved by struggling against cosmic forces, but by adaptation to them (Claude Bernard, Sir Charles Sherrington), 433 He that survives must be in some measure a collaborator with Nature, 433 Evolution towards fusion of personal will with natural law, 433. I GENERAL PRINCIPLES CHAPTER I REGARDING DEFINITIONS Truth comes out of error more readily than out of confusion. Bacon. A definition is a purely arbitrary thing. If I choose to define a triangle as a plane figure bounded by four sides and having four angles; and if, also, I define a quadrilateral as a plane figure bounded by three sides and having three angles, I shall run into no logical conflicts; my geometry need in no wise depart from that of Euclid; I shall need to make no changes in existing works on geometry, beyond that of substituting throughout the word triangle for the word quadrilateral, and vice versa. But while a definition is in this sense, from the point of view of logic, a purely arbitrary thing, while my definition of a triangle as a four-sided figure may be admissible, it is by no means expedient. Thus the definition of terms, which naturally forms one of the first steps in the systematic treatment of any subject, may present no particular problems of logic, but it does present certain problems of expediency. In the geometrical example cited, the unusual definitions given, though quite permissible, are inexpedient for simple etymological reasons. Such a choice of terms would be misleading, and, instead of assisting the memory, would impose upon it an unnecessary burden. In this case the application of the principle of expediency is obvious to the point of being grotesque, the example having purposely been chosen to illustrate the principle in drastic fashion. But the framing of definitions at times involves more subtle con- siderations of expediency, so subtle in fact, that they may be over- looked, or misunderstood, and a problem which is, in truth, a problem of definition, falsely masquerades as a problem of fact. Certain pseudo-problems of science have owed their origin to a failure to realize this circumstance. 1 1 On the other hand, some very fundamental advances of science are, upon critical examination, found to rest essentially upon the establishment of a 3 4 ELEMENTS OF PHYSICAL BIOLOGY The writer of the book of Genesis shows good judgment. Our legendary forebear, the originator of the first biological system of nomenclature, sees each creature first, and thereupon names it. We have not always been equally wise. Sometimes we have tried to invert the method; we have found or made a name, and then gaily set forth on an expedition to discover the thing that should answer to that name; we have hunted the Jabberwock. Forgetful of the wisdom of Mephistopheles : Derm eben wo Gedanken fehlen Da stellt ein Wort zur rechten Zeit sich ein we have given way to an inherent bias of the human ,mind described in characteristic fashion by H. G. Wells 2 : . . . . when we have a name we are predisposed and sometimes it is a very vicious predisposition to imagine forthwith something answering to the name If I say Wodget or Crump, you find yourself passing over the fact that these are nothings, .... and trying to think what sort of a thing a Wodget or a Crump may be. You find yourself insensibly, by subtle associations of sound and ideas, giving these blank terms attributes. 8 So the biologist of the past generation, finding in his native vocabulary the words animal and plant, forthwith proceeded in an effort to establish precise distinctions between animals and plants, never giving any thought, it would seem, to the fact that these names had already been parceled out generations ago, by "popular" consent, by unscientific persons without any regard to fine distinc- tions. There is clearly, here, the tacit assumption that because two distinct words are found in the vocabulary, therefore two correspond- ingly distinct things exist in nature. In point of fact, we know well enough (though we may not at all times have this knowledge clearly in the focus of our consciousness) that in nature many things form finely graded series, with extremes at the two ends, extremes to which judicious definition. A notable instance of this is the enunciation of the prin- ciple of the survival of the fittest, which is essentially of the nature of a defi- nition, since the fit is that which survives. Regarding the epistemological significance of definitions compare A. N. Whitehead and B. Russell, Principia Mathematica 1910, vol. 1, p. 12. * H. G. Wells, First and Last Things, 1908, p. 32. 8 "Gewohnlich glaubt der Mensch, wenn er nur Worte Es mtisse sich dabei auch etwas denken lassen." REGARDING DEFINITIONS 5 ^ our vocabulary has lent more or less definitely associated names, but with no definite line of demarcation between. Examples of this are \ innumerable. We speak of objects as being red, orange, yellow, green, blue, violet, etc. There is nothing in nature to correspond to such staccato classification of colors: the visible spectrum runs con- tinuously from a wavelength of about 8 x 10 ~* mm. (extreme red) to about 4 X 10~ 4 mm. (extreme violet). Cases therefore must neces- sarily arise when we are in doubt whether to call a thing blue, or green, for example; and such doubt can be resolved, if at all, only by arbitrary definition. The question is not "what is green, and what is blue," but, at best, "what shall we agree to call green, and what blue." It lies in the nature of the mechanism by which we enter into possession of our knowledge, that problems of definition of this kind *- arise. We are equipped with two separate and distinct senses, the one responding to electromagnetic waves ranging from about 4 x 10 ~ 4 to 8 x 10 ~ 4 mm., light waves; the other to somewhat longer waves otherwise of the same character, heat waves. Accordingly we have two separate terms in our language light and heat, to denote two phenomena which, objectively considered, are not separated by any line of division, but merge into one another by gradual transition. Here the question might be raised whether an electromagnetic wave of a length of 9 x 10 ~ 4 mm. is a light wave or a heat wave. The answer is obvious: Call it what you please, it is merely a question of arbitrary definition. We must beware of .... that false secondary power '* By which we multiply distinctions, then Deem that our puny boundaries are things That we perceive, and not that we have made. Wordsworth. Definitions in Biology. The attempt to establish a rigorous distinc- tion between "animals" and "plants" may be similarly regarded. Expediency demands that if these terms are appropriated for exact scientific use, their sense, when so used, shall, if possible, be reasonably near akin to the sense commonly associated with these words. The difficulties encountered in seeking to establish a satis- factory line of division between animals and plants were long regarded as difficulties in a problem of fact. It was thought that some biologi- * cal principle must be sought which divided animals from plants. 6 ELEMENTS OF PHYSICAL BIOLOGY The truth is, of course, that we may define "animals" and "plants" any way we please- as for instance by reserving the term, plant for an organism possessing cellulose' but whether such definition is "cor- rect" or "satisfactory" is not a question of biological fact, it is a question of expediency. It is not a question whether there is any definable difference between animals as a class and plants as a class, nor what this difference is, but whether it is expedient to retain for purposes of strict scientific classification the popular terms "animals" and "plants," which were not originally founded upon any rigorous examination of facts; and if so, where we should, by definition, draw the line of separation. When the problem is viewed in this way the difficulty of distin- guishing between animals and plants vanishes. In the case of the higher forms of life it is easy to establish biological distinctions that do not conflict with the popularly drawn lines of division. In the case of certain lowly forms of life popular distinctions cannot exist, since these forms are not known to the public except through biologi- cal publications. And the biological line of demarcation we can, by definition, draw arbitarily where we choose, or, better perhaps, we may say that the terms "animal," "plant," do not correspond to any fundamental objective distinction and, though conveniently applied to certain common forms of living matter, are entirely unnecessary 4 and only introduce difficulties of definition and classifica- tion when applied to certain simple organisms. What difference does it make whether we call Volvox a plant or an animal? Whether it is a plant or an animal is merely a matter of definition, not a question of biological fact. Somewhat similar remarks apply to the narrower divisions into which the biologist divides the world of living organisms. Disputes as to what constitutes a species are fruitless. "A species is a thing described as such." This is simply a matter of definition. If on grounds of expediency one definition is preferable to another, it may be well to urge its general adoption. But its adoption or rejection will neither add nor subtract one jot from our stock of ascertained facts. It is necessary to guard against the error of disputing about mere words. Not always does this error strut about in such blatant form as in the example quoted by Fechner: S. Sachs, in a book published 4 R. W. Glaser, Science, 1918, vol. 48, pp. 301-302: "We are justified at present in not classifying viruses either wit|i plants or animals." KEGARDING DEFINITIONS 7 in 1850, takes the astronomers to task for their presumptuous specula- tions: "How do they know that the star they call Uranus is Uranus?" If any one should think that in our day it is no longer necessary to guard against errors of this kind (though less gross, perhaps), let him consider such a question as this: Is not the perennial debate between vitalism and mechanism a quibble about words? Is not the whole situation summed up accurately in the words of L. J. Briggs: 5 "The mechanism of plant processes not at present explainable on a physico-chemical basis would be termed by the vitalistic school "vital," by the physico-chemical school "unknown"? And in searching for the essential characteristics of life, those that should finally and conclusively distinguish the living from the non- living, are we not just searching for the thing in nature that should correspond to a word in our vocabulary? Are we not hunting the Jabberwock? Definitions of Life. The difficulty of giving a precise meaning to the word life has been realized probably by everyone who has ever seriously attempted a definition. Herbert Spencer remarks: Classifications are subjective concepts, which have no absolute demarca- tions in Nature corresponding to them .... Consequently, when we attempt to define anything complex .... we can scarcely ever avoid including more than we intended, or leaving out something that should be taken in. Thus it happens that on seeking a definition of life, we have great difficulty in finding one that is neither more nor less than sufficient. Nevertheless he proceeds to establish his definition of life: "The continuous adjustment of internal relations to external relations." 8 It cannot be said that Spencer has been very happy in this choice of a definition or that he has been at all successful in avoiding the very pitfalls which he himself so clearly points out. For obviously many purely mechanical systems fall under this definition. It would, for example, include a windmill provided with a device automatically turning its arms into the most favorable plane according to the direction of the wind. 7 Indeed, in a sense it is true of every physical 6 L. J. Briggs, Jour. Washington Acad. Sci., 1917, vol. 7, p. 89; compare also E. M. East, Mankind at the Crossroads, 1923, p. 21. 6 Herbert Spencer, Principles of Biology, section 30. 7 Compare the following : "No one has yet succeeded in formulating a clean- cut definition of the limits of the reflex either at its lower or its higher extreme, and perhaps no one ever will; for the whole list of behavior types, from ma- chines to men, probably form a closely graded series." C. J. Herrick: The Evolution of Intelligence and Its Organs. Science, 1910, vol. 31, p. 18. 8 ELEMENTS OF PHYSICAL BIOLOGY system that it "adjusts its internal relations to external relations." For this statement simply implies that there is a tendency for the establishment of equilibrium between a selected portion of a physical system, and the remainder, the environment. Thus, for example, if the system 2H 2 + 2 is left to itself hi a suitable vessel at 1480C. 8 H 2 and one atmosphere pressure, the ratio ^rp: which we may term an "internal relation" of the system, assumes the value 0.0002. If now the external conditions of temperature and pressure are changed "FT to 2929C. and one atmosphere pressure, the internal relation rrr-z adjusts itself to the new external condition and acquires the value 0.11. With better judgment than Herbert Spencer, Sir Edward Schafer 9 frankly evades the definition of life. He remarks : The ordinary dictionary definition of life is "the state of living." Dastre, following Claude Bernard, defines it as "the sum total of the phenomena com- mon to all living beings." Both these definitions are, however, of the same character as Sidney Smith's definition of an Archdeacon as "a person who performs archidiaconal functions." I am not myself proposing to grapple with a task that has proved too great for the intellectual giants of philosophy, and I have the less inclination to do so because recent advances in knowledge have suggested the probability that the dividing line between animate and inanimate matter is less sharp than it has hitherto been regarded, so that the difficulty of finding an inclusive definition is correspondingly increased. It is, indeed, an elementary historical fact that, as knowledge has advanced, the scope embraced in the term "vital" processes has continually decreased, since Wohler took the first cut out of it in 1828 by the synthesis of a "vital product" (urea) in the laboratory; and the field of known physico-chemical processes going on in living organisms has correspondingly increased. For the rest, the most uncompromising vitalist does not deny that some, at least, of the processes going on in living matter are physico-chemical. Even so fundamentally biological a process as the stimulation of an ovum to development we have learnt to effect by purely physical means. Alleged Characteristics of Living Matter. On the other hand some of the features commonly ascribed to living matter as its peculiar 8 W. Nernst, Theoretische Chemie, 1913, p. 713. 9 E. A. Schaefer, Presidential Address at Dundee Meeting of Brit Assoc Adv. Sci. 1912. EEGAEDING DEFINITIONS and characteristic attributes seem Irrelevant to the point of triviality. "^his. remark applies particularly to the distinction sometimes claimed for living matter, that it grows "from within," as distinguished from crystals, which, in a suitable mother liquor, "grow from without." There may or may not be many and profound differences between a bacterial colony growing in a culture medium, on the one hand, and on the other hand a mass of crystals growing in a supersaturated solution. But whether the growth takes place from within or without is merely an accident of structure. If a droplet of chloroform is brought near to a glass particle coated with shellac, the drop flows around the particle, engulfs it, absorbs the shellac coating and finally rejects the "undigested" glass particle. 10 The droplet thus grows "from within." In point of fact "growth from within" is the rule and not the exception in chemical systems. For what do we mean by growth? We mean the increase of the mass of one component of a system at the expense of another. It is precisely the same thing as that which occupies the center of attention of the physical chemist, though he does not ordinarily call it growth. In fact, he does not find it neces- sary to give it any particular name, for, being accustomed to the use Ci'YYI of mathematical methods and symbols, he simply writes it -=- , rate Cit d /m\ of increase of mass with time, or, more often, T.\ I, rate of increase of concentration (mass/volume) with time. And in homogenous systems, at least, which (on account of their comparative ease of theoretical and experimental treatment) figure prominently in the physical chemistry of today, growth is necessarily from within. Some writers (J. Loeb, The Organism as a Whole, 1916, p. 28) have seen a characteristic feature, peculiar to living organisms, as distinguished, for example from crystals growing out of a solution, 10 "Let it be clearly understood that this illustration is here quoted, not as an example of life-like analogies in the world of non-living matter; nor as a veiled suggestion that such a drop of chloroform represents even a modest degree of success in the artificial imitation of life; nor yet again as an argument that the conduct of amoeba can today be fully accounted for on a physico- chemical basis; this example was cited merely to show that "growth from within" cannot be claimed as a distinguishing characteristic of living matter. For further discussion of so-called simulacra vitae see McClendon, Physical Chemistry of Vital Phenomena; Burns and Paton, Biophysics, 1921, p. 403. 10 ELEMENTS OF PHYSICAL BIOLOGY in the fact that the latter grow by a physical process, the former by chemical processes. Leaving aside the question as to whether there exists any fundamental distinction between physical and chemical processes, at most the point to which attention is drawn by these authors would class living organisms with chemical, as distinguished from physical systems, but would furnish no basis whatever for separating organisms in a class by themselves from other chemical systems. This is not saying that they are not in a class by them- selves, but only that the distinction suggested fails in effect. It has similarly been urged, as a distinction between the growth of a crystal and that of an organism, that the former will grow only in a supersaturated solution of its own substance, while the latter ex- tracts from an unsaturated solution the substance needed for its anabolism. This is really the same distinction in another form. It may distinguish the organism from the growing crystal, but leaves it m... one class with any chemically reacting system whatever, since in the case of the latter also there is "growth," i.e., formation of one or more products of reaction, in a system which need not be physically supersaturated in the narrow sense in which the crystallizing solution is. In a wider sense 11 the system may indeed be said to be super- saturated with regard to a chemical substance that is formed within it but in the same sense a system can probably be said to be super- saturated with regard to the substance of a bacterial colony growing therein. Neither can we subscribe to the view set forth by J. Loeb (The Organism as a Whole, 1906, p. 29), that the synthesis of specific materials from simple compounds of non-specific character distin- guishes living from non-living matter. In every chemical reaction specific materials are formed. In a mixture of hydrogen, chlorine, and nitrogen, the hydrogen and the chlorine unite, leaving the nitro- gen on one side unchanged. This is merely a brutally simple example of a universal fact. Chemical reaction is always selective. And if "complexity" is to be made the characteristic of life processes, then the question immediately arises, what degree of complexity is re- quired to place a given process in the category of life processes? "Namely in the sense that it is metastable, that is, its thermodynamic potential is not at a minimum. EEGARDING DEFINITIONS 11 Reproduction. Another characteristic that has been cited by some as exclusively peculiar to living organisms is the power of reproducing their kind. "How, says Driesch in effect, can a mechanism provide for its own reconstitution? No machine known to us is able to con- struct another like itself, nor can it repair its own parts." 12 Undue emphasis on this alleged distinction between living and non- living machines seems ill advised, for two reasons. In the first place, though it may be true that no man-made engine exists that performs the functions of self-repair and self-reproduction, no one has ever attempted, so far as I know, to demonstrate that no such engine can be built. Anyone who should be disposed to regard this objection as specious should reflect for a moment on the amazing development in technical arts within the last thirty or forty years. Half a century ago one might with equal justice have pronounced flight a funda- mental, essentially biological characteristic of birds, incapable of duplication by man-made engines. But in another, perhaps more significant respect, we must regard as misplaced the emphasis sometimes laid on the power of reproduc- tion in organisms, and its absence in human artefacts. It is based on an exaggerated conception of the part played by the parent in the making of the offspring. This probably has its origin in the instance of reproduction that to us is naturally of supreme interest, the repro- duction of man. As a mammal, the young human organism grows within the parent body, and seems to us to be in some way fashioned by the parent; this conception must be at the basis of The alleged ! distinction between organic reproduction and the incapacity of ; '. ; non-living engines to reproduce their kind, for without such con- ception the comparison would lack all parallel. Now, in point of fact, we need but call to mind the familiar hatching of a chick to realize that the part necessarily played by the parent in the formation of the young individual is really very restricted. The process in this case goes on, for the most part, in complete isolation from the parent. 13 12 H. C. Warren, Jour. Philos., Psychol. and Scientific Method, vol. 13, 1916, p. 36. 13 Compare E. G. Conklin, Heredity and Environment, 1918, pp. 99,45, 109: "The hen does not produce the egg, but the egg produces the hen and also other eggs We know that the child comes from the germ cells and not from the highly differentiated bodies of the par- ents, and furthermore that these cells are not made by the parents' bodies but IZ ELEMENTS OF .PJtilKlUAJLi As for the initiation of cell division of the ovum, we now know that, in some cases at least, this can be effected by ordinary physical means. Recent development in experimental embryology suggest a more rational view of this process of self-reproduction of the living engine, a view which strips it of at least some of its mystery, and which certainly takes from it any force it might otherwise have had as a basis for distinction between living and non-living matter. If, after the first division of the ovum of a frog, the two cells are separated, each will under suitable conditions develop into a separate and com- plete, normal organism. These two organisms A and B are, in fact twin brothers or sisters. No one would for a moment entertain the thought that in this case A reproduces B, or vice versa. Now suppose that in some way, after the first division, A alone grows into a com- plete mature organism, while the single cell B remains attached to it, say for six months. At the end of this time it is separated, and stimulated to start its growth into a frog. We would ordinarily describe this state of affairs by saying that A reproduced B as offspring, that B was the child of A. In point of fact it is merely a delayed twin brother or sister of its elder brother or sister A. u A had little or nothing to do with the production of B; the latter grew, very much in the same way as A grew in its own time. That nature has evolved, in surviving races, this method of delayed development, so as to stretch out the totality of living organisms in a long chain, a succession in tune, is of course a fact of most fundamental importance, the significance of which will deserve our profound contemplation. One of its consequences has been to render possible a practically infinite number of organisms, built from a finite and quite restricted amount of matter, the same substance being used over and over again, for it is literally true that we live on our forefathers. Had all these cells have arisen by the division of antecedent germ cells Parents da not transmit their characters to their offspring, but these germ cells in the course of long development give rise to adult characters similar to those of the parent." 14 The perhaps somewhat doubtfully authenticated cases of fetus in fetu, "those strange instances in which one might almost say that a man may be pregnant with his brother or sister," add a touch of realism to the discussion here presented. For further data on this singular subject see G. M. Gould and W. L. Pyle, Anomalies and Curiosities of Medicine, 1897, pp. 199 et seq. Compare also in this connection, the phenomenon of pedogenesis; see for example, G. H. Parker, Psyche, 1922, vol. 29, p. 127. , these organisms sought to grow simultaneously, their career would have been stopped by lack of material. , If anyone should object that these reflections leave out of account entirely the r61e of sex in reproduction, with all the complex phe- nomena of the fusion of gametes, the mingling of chromosomes, and biparental inheritance, the obvious reply is that these phenomena are now known to be less fundamental than they formerly appeared; that reproduction of an organism can very well take place without them; and that therefore they may at most serve to distinguish certain forms of life from non-living matter, but they cannot possibly be made the basis of a distinction between living matter in general and that which we commonly describe as non-h'ving. , Vital Force. If we have cause to hesitate in defining hie,_ still more is it the part of wisdom to be very conservative in the coming and use of such phrases as vital force, nerve energy, and the like. Shall we not do well to follow the biblical example, and wait, to name the animal, until it is physically present to our senses? Or, to pass from legend to the world of scientific fact, let us borrow, if we^ can, the method of the physicist: He discovers that a quantity f mv possesses certain important properties. Then, he proceeds to name it: Energy, in particular, kinetic energy. But biologists have been disposed sometimes to adopt the reverse procedure: they have named a vital force, a nerve energy, a mental energy, and what not, and now they entertain the pious hope that in due time they may discover these "things." That there is something radically at fault with such terms is evident from the fact that forces and energy are magnitudes, and "to define a magnitude and to say how it is measured are one and the same thing." 15 But who has ever told us how to measure vital force 18 and such like? Physical Chemistry of Structured Systems. In the physical chemistry of today structure, that is to say, geometrical configura- tion, plays a subordinate rdle. For obvious reasons the theory of chemical reaction in homogeneous, or in heterogeneous systems of comparatively simple form, is more approachable than that of systems which possess intricate structure, resulting in complicated mechanical 18 Nature, September 25, 1922, p. 405. 18 G. Bunge, in his Physiologic and Pathologic Chemistry, 1902, p. 1, re- marks : "I regard vital force as a convenient resting place where, to quote Kant, 'reason can repose on the pillow of obscure relations.'" Curiously enough this damning admission is made by an advocate of vitalism. 14 ELEMENTS OF PHYSICAL BIOLOGY interactions of their parts, in accompaniment of chemical reaction. In technical practise, too, reactions in homogeneous systems (solu- tion, gas) are common, and where there is heterogeneous structure, this is usually of a form very simple as compared with the complex biological structures. But this comparative absence, from physico-chemical discussion, of reference to structure, to geometrical features, is not due to any inherent characteristic property of chemical systems, as contrasted with the structurally complex organic systems: the reason for the simplicity is to be found in ourselves. It is not a physical phenom- enon of the thing observed, but a psychological phenomenon in the observer. Physical chemistry is still a comparatively young science, and naturally the simpler phenomena have been sought out for first attention. This is not because complex physico-chemical structures do not exist, nor even because they are unimportant. On the con- trary, it is to be expected that the future will bring important develop- ments in this direction, as followed, for example by Sir William Bayliss in his work Interfacial Forces in Physiology, The rate of formation, the rate of growth, of a chemical substance, is a definite function of its environment. In a structureless system the nature and state of this environment is defined in comparatively simple terms (e.g., by stating the concentration of each of the reacting substances) . But in a system possessing structure, the environment of a given portion of the system depends on the structure, the topography of the system, which, in general, will be variable with the time. In particu- lar, the structure may be such that a given substance or complex of substances carries its own immediate environment around with it. The rate of formation (growth) of that substance will then depend largely upon the mechanical properties of those portions of the system which accompany this substance or complex in its travels through the system. The complete discussion of a system of this kind may well fall outside the scope of present day physical chemistry, not because it is inherently foreign to that branch of science, but because no case of this kind, sufficiently simple to invite discussion on a mathematical and physico-chemical basis, has clearly presented itself. 17 ^Compare W. M. Bayliss, Physiology, 1915, p. XI, "All that we are justi- fied in stating is that, up to the present, no physico-chemical system has beea-**l|| EBGARDING DEFINITIONS JLO Yet there is absolutely nothing in such a case that in principle places it outside the pale of physico-chemical science. It is largely as the result of intentional selection of simple conditions that the systems with which the chemist ordinarily deals (outside of biological chemistry) are comparatively structureless. We can, in fact, even now lay down certain general observations with regard to structured physico-chemical systems. Let us consider a system of this kind in which local conditions are subject to variation from point to point and from instant to instant. We fix our attention on some one component which requires for its growth certain definite conditions of its immediate environment. If this component is associated with a structure whose geometrical and mechanical properties secure and maintain for it a comparatively constant suitable environment amid the changing conditions of the system, then that component will grow. Furthermore, the several components will compete with greater or less success for the material available for their growth, in proportion as their structure is more or less perfectly adapted to secure and maintain for them a suitable environment. The chemical dynamics of such a system, that is to say, the laws governing the distribution of matter among its several components, may evidently assume a fundamentally different character from that to which we are accustomed from our study of ordinary structureless systems. For in these latter the arrangement and rearrangement of matter within the system depends chiefly on chemical coefficients (affinity coefficients), and scarcely at all on geometrical features. In structured systems, on the other hand, there is the possibility that geometrical and mechanical features may play the dominant role. This possibility will present itself particularly in those systems which receive a continuous or periodic supply of free energy, for instance in the form of illumination. Here the advantage will go to those structures that are adapted to direct available energy into such channels as lead to the maintenance of the environment required for their growth. 1 * But a little reflection shows that this is precisely the princi- met with having the same properties as those known as vital; in other words, none have, as yet, been prepared of similar complexity and internal coordination. 18 It should be observed that nothing has been said of life in describing the system. The system may or may not comprise living organisms, the argument pie which governs survival in the struggle for existence among living organisms. Hence we may say: The laws of the chemical dynamics of a structured system of the kind^ described will be precisely those laws, or at least a very important section of those laws, which govern the evolution of a system comprising living organisms. For it is precisely structured systems of the kind considered above that are presented to us in living organisms growing in an "environ- ment." Application to Biology. The several organisms that make up the earth's living population, together with their environment, con- stitute one system, 19 which receives a d :1 y supply of available energy from the sun. Each Individual is composed of various chemical substances assem- bled into a definite structure and capable of growth, i.e., of accretion out of the environment by chemical reaction 1 provided a suitable medium or environment is offered. Moreover, each mobile organism carries with it a travelling en- vironment, suitable for the growth of its substance. It maintains this environment by virtue of the peculiar mechanical properties associated with its structure, whereby it is enabled to turn to this use, directly or indirectly, the available energy of the sun's light. And while the travelling environment may not be absolutely constant, remains the same. This suggests that a term, such as life, so vague that it defies definition, is perhaps not likely to play an important part in any exact argument; -we may, indeed, find it wholly unnecessary. It may, in time, in the literature of exact science, meet with the fate of the word cause: a term of rare and at best incidental occurrence in records of exact investigations. 19 This fact deserves emphasis. It is customary to discuss the "evolution of a species of organisms. 3 ' As we proceed we shall see many reasons why we should constantly take in view the evolution, as a whole, of the system [organ- ism plus environment]. It may appear at first sight as if this should prove a more complicated problem than the consideration of the evolution of a part only of the system. But it will become apparent, as we proceed, that the physical laws governing evolution in all probability take on a simpler form when referred to the system as a whole than to any portion thereof. It is not so much the organism or the species that evolves, but the entire system, species and environment. The two are inseparable. "The organism, as Uexkull teaches us, must be studied, not as a congeries of anatomical and physiological abstraction, but as a piece of machinery, at work among external conditions," 0, C. Glaser, Science, vol. 21, 1910, p. 303. it is more nearly so than the more remote portions of the system, and keeps within such limits of variation as are compatible with the survival of the organism or its species. A concrete illustration may help to make this point clear. Many aquatic forms of life are con- stantly bathed in a saline solution sea water. Their body fluids are accordingly in equilibrium with this environment. Variations in the salinity of their environment, if they exceed certain comparatively narrow bounds, are apt to be fatal to such organisms. The higher organisms have made themselves (largely) independent of their immediate environment. Their tissues are bathed from within by a fluid (the blood) which they carry around with them, a sort of "internal environment." 20 The degree of perfection with which this constancy of the internal or traveling environment, independently of the external environment, is developed, increases as we ascend the biologic scale. This is lucidly set forth, for example, by Claude Bernard: 21 Chez tous les 6tres vivants le milieu inte'rieur qui est un produit de 1'organ- isme, conserve les rapports ne'ce'ssaires d'Schange avec le milieu e'xte'rieur ; mais & mesure que 1'organisme devient plus parf ait, le milieu organique se sp< cifie et s'isole en quelque sorte de plus en plus du milieu ambiant. It is the peculiar structure and the mechanical properties of the organism that enable it to secure and maintain the required environ- ment (including the milieu interieur). The higher animals, in particular, are provided with an intricate apparatus, comprising many members, for securing food (internal environment) as well as for warding off hostile influences. 20 "Etant donne* que 1'eau de mer a un contact si intime avec les organismes de la mer et que non seulement elle les entoure de ses flots, mais qu'elle traverse leurs branchies et impregne en partie les corps des inverte"bre"s, il semble assez justind de la placer dans la m&ne cate"gorie que les autres liquides physio- logiques." S. Palitzsch, Comptes Rendus de Carlsberg, vol. 10, part 1, 1911, p. 93. Compare also the following: "Not only do the body fluids of the lower forms of marine life correspond exactly with sea water in their composition, but there are at least strong indications that the fluids of the highest animals are really descended from sea water .... the same substances are present in both cases, and in both cases sodium chloride largely predominates." L. J. Henderson, The Fitness of the Environment, 1913, pp. 187-188. See also ibid., pp. 116 and 153; H. F. Osborn, The Evolution and Origin of Life, 1917, p. 37 ; D'Arcy W. Thomp- son, Growth and Form, 1917, p. 127. 21 Introduction a l'6tude de la me'decine expe"rimentelle, 1885, p. 110. The increasing independence, as we ascend the biological; ~ jale, which the organism displays toward its more remote environment, is thus accompanied by a parallel increase in the perfection of the apparatus by which this independence is earned. Here again we may quote Claude Bernard : 22 A mesure que 1'ori s'eleve dans l'e"chelle des etres, ces appareils deviennent plus parfaits et plus eomplique's; ils tendent & affranchir completement 1'organisme des influences et des changements survenus dans le milieu ex- te"rieur. Chez les animaux inverte'bre's, au contraire, cette independence vis- a-vis du milieu exte"rieur n' est que relative. The Policy of Resignation : Its Parallel in Other Sciences. What- ever may be our ultimate conclusions, we may do well to adopt at least as a temporary expedient the policy of resignation; with Sir Edward Schafer we may abandon the attempt to define life. Perhaps, in doing this, we are following historical precedents : Geome- ters have had to resign themselves to the fact that Euclid's parallel axiom cannot be proved. But as the reward of this resignation came the new geometries of Bolyai, Lobatchewski and Riernann. Enlightened inventors have abandoned the attempt to build a perpetual motion machine; but again, resignation is rewarded with the recognition of a fundamental law, the law of conservation of energy. Physicists, following Einstein, have abandoned, for the time being at any rate, the attempt to determine experimentally the earth's absolute motion through space. The reward has been the theory of relativity, one of the greatest events in the history of science. The whole development of science, especially in recent years, is a record of tearing down barriers between separate fields of knowledge and investigation. Little harm, and perhaps much gain, can come from a frank avowal that we are unable to state clearly the difference ;' between living and non-living matter. This does not in any way ' commit us to the view that no such difference exists. l For the present, then, we shall adopt the position that the problem is essentially one of definition. The question is not so much "What is life," but rather, "What shall we agree to call life?" And the answer, for the present at any rate, seems to be that it is immaterial how we define life; that the progress of science and our understanding of natural phenomena is quite independent of such a definition. 22 Ibid. We shall, wherever convenient, continue to employ the terms life, living organism, merely as a matter of convenience. This use of the terms does not imply or presuppose any precise distinction between living and non-living matter; it merely rests upon the fact that in most cases ordinarily met there is essentially universal agreement as to whether a portion of matter is to be classed in the first or in the second category. We will adopt the policy of Sir William Bayliss : If asked to define life I should be inclined to do as Poinsot, the mathe- matician did, as related by Claude Bernard: "If anyone asked me to define time, I should reply: Do you know what it is that you speak of? If he said Yes, I should say, Very well, let us talk about it. If he said No, I should say, Very well, let us talk about something else." The ideal definition is, undoubtedly, the quantitative definition, one that tells us how to measure the thing defined; or, at the least, one that furnishes a basis for the quantitative treatment of the sub- ject to which it relates. We have already spoken of evolution. Most of what follows will relate directly or indirectly to evolution. It will be well here, while discussing definitions, to establish a defini- tion, a conception, of evolution that shall, as far as may be, have the quantitative stamp. CHAPTER II EVOLUTION DEFINED Nature must be considered as a whole if she is to be understood in detail. Bunge. As has been abundantly made plain, the choice of a definition is a matter of expediency. In adopting, for special use in exact science, a term already in general use, we must seek, so far as possible, to embody in our definition the fundamental and essential features of the concept denoted by the term as used popularly and by the best workers, thinkers and writers. 1 In so far as there is divergence in the use of the term, it may be well to frame the definition broadly, so as to cover a wide range of phenomena and lead to a compre- hensive view of natural events, corresponding to the essential unity of nature. In this way we shall be most likely to see the facts of nature arraying themselves in a natural order, and to achieve that economy of thought which is secured by a well devised system of classification. Facts which naturally belong together will, then, be found together, in our system, in the same or in neighboring pigeon- holes. Now if we seek to analyze what is in our minds when we speak of the evolution of a given system, we find' and on this probably all are agreed that the fundamental, the central thought, is that of the history of the system. But the concepts of the history and of the evolution of a system, though related, are not identical if they were, one word would suffice to denote the single concept. The popular and also the scientific conception of evolution contains as an essential feature the element of progress, of development . We would not ordinarily class as evolution the history of such a system as a swinging pendulum, or a celestial body circling in its orbit, in so far 1 Compare Bertrand Russell, Analysis of Mind, 1921, p. 197: "The use of the word comes first, and the meaning is to be distilled out of it by observation and analysis." "In each case the work consists chiefly in making explicit proc- esses which are instinctive," as J. W. N. Sullivan (Aspects of Science, 1923, p. 24) remarks apropos of certain other matters. 20 EVOLUTION DEFINED 21 as these motions are purely periodic or cyclic. In the history of such systems the element of progression in time, of development, is lacking. They repeat in endless succession the same series of events. The hand of the clock, like a symbol of perpetual youth, goes through its daily double cycle, making no distinction between yesterday, today and tomorrow. It is the calendar that reminds us we grow older year by year, the calendar that turns a new and different leaf each o!ay. "The book of Nature is the book of Fate. She turns the gigantic pages leaf after leaf, never returning one. . . . . " 2 But, to characterize the kind of history we speak of as evolution, it is not enough that each day be unlike every other; it is not merely that a system never passes twice through the same state; 3 not merely that a biological species never retraces its steps, 4 or that "when a race has lived its term it comes no more again." 5 Evolution not a Mere "Changeful Sequence." Such a statement as those cited in the preceding paragraph alone is insufficient to dis- tinguish evolution as a progress from merely a changeful sequence', it is insufficient to define the direction of evolution. 6 For if the world's events taken in historical order A, B, C . . . are a changeful sequence, the same is also true of the inverted series . . . C, B, A, Mere unlikeness of two days is insufficient to tell us which is antecedent to the other. To determine this we must know something regarding the character of the unlikeness. In a vague way this character is indicated by the term progress, which, as already remarked, is closely associated, in popular conception, with evolution. And the more rigorous scientific dis- ciplines of biology, too, leave us with a not very clearly defined idea of progression as one of the fundamental characteristics of those changes which are embraced by the term evolution. Such phrases as "the passage from lower to higher forms" which are often used to describe the direction of evolution, are vague, and 2 Emerson, Conduct of Life, Everyman's Library Edition, 1915, p. 157. Com- pare also Lee Wilson Dodd's lines: " . . . . Nor do the stars retrace their glistening snail marks of slow destiny." 3 J. Perrin, Traite" de Chimie Physique, 1903, vol. 1, p. 142. 4 Petronievics, Science Progress, 1919, p. 406. 6 Emerson, Conduct of Life, p. 158. 6 For this reason the characterization of the trend of evolution given by Petronievics, loc. cit., is inadequate. 22 ELEMENTS OP PHYSICAL BIOLOGY undoubtedly contain an anthropomorphic element. 7 At best they give every opportunity for divergence of opinion as to what con- stitutes a "higher form." If, on the other hand it is stated that evolution proceeds from simpler to more complex forms, or from less specialized to more specialized forms, then the direction of evolution is but poorly defined, for the rule is at best one with many exceptions. It should be particularly noted that all these efforts to specify the direction of evolution attempt to do so in terms of a single component of the evolving system. Such definitions of the direction of evolution are foredoomed to failure. It is the system as a whole that evolves, and we can hope to establish a definition of the direction of evolution only in terms of the system as a whole. Evidently, we must seek a more precise indication of the direction of evolution if our definition is to be truly expedient. We must analyze further the contents of our mind when it contemplates the concept of evolution. We return to our examples of the pendulum, or of the earth in its orbit. When frictional resistances are neglibible, or are disregarded, the periodic series of events in the system may be history, but seems hardly worthy of the name evolution. In actual fact the motion of the pendulum bob gradually dies down, owing to friction and other dissipative forces. The motion is not strictly periodic. The pendulum does not, actually, count out similar seconds, unidentified, but marks, by its greater amplitude, an earlier vibration as distinguished from a later. So also, the earth in its motion is slightly delayed by frictional forces introduced by the tides; it slows down a little as the centuries pass. The strictly periodic proc- ess is changed into one in which successive days differ by a trifle in length. The process has a definite direction in time. We feel justified in speaking of the system as "evolving." Now the thing to mark is that what has imparted to the process its directed character is frictional resistance, dissipative forces, typical irreversible effects, to speak in the language of the physicist. 7 "Evolution is thus almost synonymous with progress, though the latter term is usually confined to processes of development in the moral, as distin- guished from the physical world. Further, this idea, as Mr. Spencer remarks, has rather a subjective value in existence, as judged by our feelings" (Encycl' Brit., 9th edition, vol. 8, p. 751). Compare also Bertrand Russell Our Knowl- edge of the Eternal World, 1914, p. 12. "A process which led from amoeba to man appeared to the philosophers to be obviously a progress though whether the amoeba would agree with this opinion is not known." EVOLUTION DEFINED 23 Again, consider a typical example of what we are all agreed to speak of as evolution: the history of the earth and its living in- habitants. The readjustments, the re-adaptations of life-forms which have here taken place, were undoubtedly due in part to changes in external conditions, such as climate, geographic distribution of land and sea, etc. In part, also, such changes have gone on and are going on before our eyes independently of any external changes, and under approximately constant conditions. Organic evolution being a slow process, it takes a certain time, when equilibrium or near-equilibrium is disturbed, for a new equilibrium or near-equi- librium to become established. There is therefore a tendency for internal readjustments or changes to lag behind the external changes by which they are conditioned. As a special case, if an external change is followed by constant external conditions, internal changes may continue to proceed under constant external conditions. Now such internal changes in a material system, which lag behind the determining external changes, or which go on under constant external conditions, are typically irreversible processes.* 8 A. process is said to take place reversibly, if the direction of the change is reversed by a suitable alteration, however small, of the (generalized) force applied to produce the change. For example, if two equal weights are sus- pended from the ends of a string passed over a simple pulley, then, the weights being initially at rest, any weight however small, added on one side of the system will produce motion downward on that side, provided there is no friction at the pulley and no stiffness in the string. If, on the contrary, a weight, however small, is lifted off from the same side of the system, motion will be initiated in the opposite direction. Note that if there is friction at the pulley, these statements are no longer true. It will now require a weight of definite size, perhaps a decigram, or a milligram, to start or reverse the motion. In the first instance the change is reversible, in the second it is said to be irreversible. Note that in this example the circumstance that imparts to the process an irreversible character is the presence of friction, which causes the dissipation of energy, that is to say, its conversion into heat at the tempera- ture of the surroundings. Again, consider a vessel containing water at a temperature Ti in a room at temperature Tj. If TI is ever so slightly greater than T 2 , heat passes from the vessel to the surroundings, and vice versa. When, therefore, TI and T s are very nearly equal, the passage of heat from the vessel to the surroundings is essentially reversible. If there is a material difference between TI and T 8 , the heat transference is irreversible. For example, if the vessel is at 50C. and the room at 20C., heat will pass from the vessel to the room. And the direction of this heat transfer will remain unchanged if the temperature of the vessel is 24 ELEMENTS OF PHYSICAL BIOLOGY We are thus led, from two slightly different points of view to the following definition of evolution: Evolution is the history of a system undergoing irreversible changes. Scope of Definition, It is worth while at this point to consider briefly what kind of history this definition excludes and what it includes. It has already been noted that we have excluded certain purely mechanical systems of periodic habit, such as the frictionless pendulum and the planet circling in its orbit through empty space, in absence of tidal effects. 9 It is not the case, however, that all purely mechanical systems are excluded, that is to say, all systems in which all energy is either kinetic or potential (configurational), all forces either inertia forces or positional forces. If our knowledge of such a system is statistical in character, if we know only averages of certain of the variables defining the state of the system, it may happen that certain changes therein appear to us irreversible, and would accordingly be classed, by our definition, among processes of evolution. 10 This leads to the seemingly embarrassing conclusion that a process is or is not a process of evolution, according to the reduced 1, 2 or even 10. Not until the vessel is cooled by more than 30 will the stream of heat be reversed. The passage of heat in such case, from a body at one temperature to another at essentially lower temperature, is irreversible in this sense, the sense in which the term is employed by the physicist in dis- cussions of this kind. In the case in which internal readjustments lag behind changes in external conditions, there is necessarily a finite difference between the applied (gener- alized) force, and the opposing resistance. Such processes are, therefore, of necessity, irreversible. From the examples given, it will be seen that during a reversible change a system is at all times (very nearly) in equilibrium. It can therefore be said that a reversible change is one in which the system passes through a con- tinuous succession of equilibria. In fact, the change is strictly reversible only if the difference in the applied (generalized) force and the resistance is infini- tesimal, and the change is infinitely slow. 6 Such tidal effects act as brakes and destroy the exact periodicity of the motion. 1S The irreversibility also of those changes occurring in a system whose internal adjustments lag behind changes in the applied forces, may be appar- ent, and may disappear when detailed knowledge of the individual parts of the system takes the place of statistical data. The reason for this is that when the reaction or readjustment is expressed in a statistical way, an average of indi- vidual reactions may show a lag, although each individual reaction itself may be immediate. EVOLUTION DEFINED 25 nature and extent of our knowledge regarding the system. So, for example, the establishment of thermal equilibrium in a body of gas initially at non-uniform temperature is evolution if we merely know its total mass, composition, volume, pressure and initial temperature distribution. But should we be informed of the exact initial state of each molecule, then the process by which thermal equilibrium is established (if this does occur) would be classed, together with the journey of the earth in its orbit, among the cases excluded, as mere history, from our definition. This, upon reflection is neither as strange, nor as embarrassing as it may at first sight appear. For problems of evolution are in large measure problems of probabilities, statistical problems. Inciden- tally, this reflection disposes of the rather foolish objection sometimes raised against the theory of evolution, that it ascribes the course of events in an evolving system to chance. When we describe a phe- nomenon as being governed by chance, we do not, of course, mean that there are no definite causes (determining factors) at work; we merely state in these terms that the causes are complex and not known to us in detail. Practically there is no cause for embarrassment, since we never do know material systems in sufficient detail to compute their state at every instant from the initial state, except in terms of averages. In principle, however, it is necessary to make the admission that, in the last analysis, whether we class the history of a system as evolu- tion or not must depend on the extent and detail of our knowledge of that system. 11 It will thus be seen that the line of division between reversible (purely mechanical) and irreversible (dissipative) processes is not 11 To be quite exact, evolution, according to this, should be defined in terms of a point of view, say about as follows : Evolution is the history of a system, regarded as a progressive change or development, to which its unidirectional character is imparted by irreversible changes going on in the system. That a point of view is involved is also implied in the following definition given by Karl Pearson: "A causal description of the appearance of successive stages in the history of a system forms a theory of the evolution of that system. "If the theory be so satisfactory that it resumes in some simple statement the whole range of organic change, we term it the law of evolution," (Grammar of Science, 1900, p. 375). 26 ELEMENTS OF PHYSICAL BIOLOGY so very sharply drawn. Furthermore, the cases excluded are, in point of fact, ideal cases. Real processes are always irreversible. Hence, after all, history, real history, is always evolution, and, though f . in principle the two concepts may be distinct, in practice they "-(' coincide in scope. What then is gained by our definition of evolution? This is the gain: Having analyzed the submerged implications of the term evolution as commonly used, so as to bring them into the focus of our consciousness, and having recognized that evolution, so understood, is the history of a system in the course of irreversible transformation; we at once recognize also that the law of evolution is the law of irreversible transformations; that the direction of evolu- tion (which, we saw, had baffled description or definition in ordinary biological terms), is the direction of irreversible transformations. ' . And this direction the physicist can define or describe in exact terms. Jl For an isolated system, it is the direction of increasing entropy. 12 The law of evolution is, hi this sense, the second law of thermo- dynamics. 13 12 More generally, it is the direction of decreasing thermo dynamic poten- tial, this potential being variously defined, according to the conditions of transformation. 13 "The second law (of thermodynamics) is the law of evolution of the world accessible to our observation" (Chwolson, Lehrbuch der Physik, 1905, vol. iii, p. 499; Scientia, 1910, vol. iii, p. 51. the second law of the theory of energy is now generally regarded as essentially a statistical law. So viewed, the second law of energy becomes a principle stated wholly in terms of the theory of probability. It is the law that the physical world tends, in each of its parts, to pass from certain - -v less probable to certain more probable configurations of its moving particles. As thus stated the second principle .... becomes a law of evolution" (Josiah Royee, Science, 1914, vol. xxxix, p. 551.) "Tin systeme isole ne passe jamais deux f ois par le me'me e"tat. "Le second principe affirme un ordre nece"ssaire dans la succession de deux phenomenes, sans retour possible aux etats deja traverse's. C'est pourquoi j'ai cru expressif d' appeler ce principe un principe devolution. II se trouve qu'en proposant ce nom je suis fidele a la pens^e de Clausius, car le mot farpomft, d'oti ii a tire" entropie, signifie precisement Evolution." (J.Perrin, Trait6 de Chimie Physique, 1903, vol. 1, pp. 142-143.) "II est hautement improbable qu'un systeme isole" passe deux fois par le mgme e"tat; cela est d'autant plus improbable que la complication du systeme est plus grande, et pratiquement il serait insensS de se placer dans cette hypothec d'un retour a I'Stat initial." (J. Perrin, loc. cit., p. 146) . A EVOLUTION DEFINED 27 Simple Mechanical Example. It will be desirable, at this point, to consider, by the aid of a simple example, the manner in which some of the facts considered in the preceding pages find expression in the analytical formulation of the behavior of mechanical systems. Take the example of the simple pendulum. For small vibrations the restoring force, tending to draw back the bob to its lowest position, 3/ is easily shown to be mg -, where x is the horizontal displacement, 6 m the mass of the bob, and g the acceleration of gravity. This force is expended upon two items, first, in overcoming the inertia of the bob, and producing an acceleration a. The force so expended is cc measured by ma. Second, a part of the force mg - is expended in b overcoming the resistance of the air. If v is the velocity of the bob, this part of the force is measured (for ordinary velocities) by kv, where k is a constant depending on the shape of the bob, etc. We have then ma + kv = mg (1) /Jj* or, since the velocity v is the rate of change of x with time, i.e., =-> GlV fiir nD (i y and a is the rate of change of with time, i.e. a = = , at at ctt d?x dx x . . Now there are certain general characteristics to be observed in this equation, characteristics which are typical of the equations of motion of mechanical systems. The equation contains the first and second derivatives of x with regard to t and no higher derivatives, The first derivative is introduced by the frictional force, and dis- appears if this force is zero, Le., if the coefficient k in (2) is zero. The equation then takes the simpler form Now in this simplified form the equation has the following pecu- liarity: It is indifferent to the sign of t. For, in differentiating twice in succession with regard to t, the positive sign of the second deriva- 28 ELEMENTS OP PHYSICAL BIOLOGY tive is restored. This is the analytical symptom, as it were, of the reversibility of the process. 14 It should be noted that this peculiarity disappears at once if the frictional term k ~ is present, for a single differentiation with regard to-* yields a result with sign opposite to that of differentiation with regard to t. The presence of a fric- tional force, therefore, imparts to the process an irreversible character, it establishes a distinction between t and-i; it singles out one direc- tion in time as a peculiar direction, the forward direction, the direc- tion of progression. Now, in point of fact, in the equations of motion of all real systems j the frictional term (viscosity term) k j or its equivalent is present, d?x though it may be small as compared with the inertia term *-^, The reversible system, in which this term is wholly absent (zero) is an ideal case, it represents a limit towards which real systems may approach; an abstraction. Inertia-Free or Completely Damped Systems. There is another such ideal limiting case, another abstraction, which is of much interest because certain important classes of real systems approach it very closely. This is the case in which the inertia term m is negligible, so that in the case of the pendulum, di- for example, the equation representing the history of the system reduces to , dx x /, N k = -mg r (4) at I The history of such inertia-free systems is typically of the irreversible kind. They have, furthermore, a property illustrated by certain features in the equation (4) above : dx It will be observed that if cc is zero, then --=- also is zero, or, as we may put it, the velocity vanishes with the displacement from equilib- rium. Moreover, differentiation of (4) gives di* I dt u Compare H. PoincarS, Thermo dynamique, 1908, p. 441. EVOLUTION DEFINED 29 from which it is seen that the acceleration also vanishes with the velocity. 15 This implies that when the system is in its equilibrium position, it is also actually at r$si, unlike the pendulum, which swings twice through its equilibrium position in each vibration. The former property, the vanishing of the accelerations with the velocities, so that the equilibrium position is necessarily the position of rest, is characteristic of an important class of systems, including those with which we shall here be chiefly concerned. Another important characteristic of such systems, which is also exemplified by equation (4), is that the velocity is uniquely deter- mined for every value of x. This is not the case in the motion represented by (3). This latter equation gives, upon integration, --.C-* (6) dx so that for every value of x there are two possible values of . (M> W.e have, then, at the one extreme the "purely mechanical" system free from frictional (Viscosity) effects, and, in its most typical form, periodic in habit. As an intermediate link we have systems exhibiting both inertia and frictional effects. Their action may resemble that of a pendulum swinging in air; typically the history of such a system exhibits the phenomenon of damped oscillations, a periodicity over which there is superimposed the dying away of the motion. The damping is introduced by the frictional effects. At the other extreme we have inertia-free, or, as we might say, completely damped systems/ 6 typically irreversible in their history. The system and processes with which we shall largely be concerned here seem to belong essentially to this third type, as will be seen in the development of the theme. Compare E. Buckingham, Theory of Thermodynamics, 1900, p. 33. 16 This does not, however, preclude the possibility of oscillations. More will be said on this point later. IISC Lib B'lore 574 N25 2544 CHAPTER III THE STATISTICAL MEANING OF IRREVERSIBILITY Supposons que nous voulions placer un grain d'avoine au milieu d'un tas de hie": cella sera facile; supposons que nous voulions ensuite 1'y retrouver et 1'en retirer; nous ne pourrons pas y parvenir. Tous les phe*nom6nes irr6ver- sibles, d'apres certains physiciens, seraient construits sur ce mo dele. JET. Poincari. One point, to which, allusion has been made incidentally, calls for comment. Many processes which, viewed in the gross, present the appearance of typically dissipative, irreversible phenomena, have long been suspected, and have in recent years been fully demon- strated to be, in fact, of the reversible type, "purely mechanical" processes, the details of which are merely hidden from our view owing to the diminutive dimensions (and correspondingly immense number) of the units at play. So, for example, consider the case of two vessels A and B at equal pressures, communicating by a tube that can be closed by means of a turncock. Let the vessel A con- tain 1 gram of nitrogen gas, and let B contain 1 gram of oxygen gas, the communication between A and B being closed. It is a matter of common knowledge that if the stopcock is now opened, the gas from the A will flow over into the vessel B and vice versa, and in a short time an equilibrium is reached in which each vessel con- tains 0.5 gram of each gas. Now, in point of fact, the molecules of the gas behave (approximately) like a number of elastic spheres, their equations of motion contain no dissipative term, but are of the type (3) (Chapter II). We should therefore expect the sys- tem to exhibit periodic motion, we should expect that after a cer- tain lapse of time the initial condition should return, and that all the nitrogen should once more be contained in the vesel A, all the oxygen in B. In actuality, such a thing is never observed. How is this discrepancy to be explained? Let us replace the two vessels and the gas molecules by some simple analogues of dimensions readily accessible to our senses, and let us., watch a process analogous to the diffusion of the gas from one vessel into the other. We provide ourselves with two boxes 30 STATISTICAL MEANING OF IRREVERSIBILITY 31 or urns. 1 In one of these, A we place 50 black balls; in the other B, we place 50 white balls. We shuffle both boxes thoroughly, and then draw blindly a ball from A, and one from B, and we return them to opposite urns. We continue this as long as desired. The more lightly drawn curve in figure 1 shows the graphic record of an actual series of drafts of this kind. The stair-case-like line shows how in successive drafts the number of black balls in box A gradually diminished until at last there remained about 25, one- half of the original number, in box A. But note that there are fluctuations, sometimes the box contains 26, 27, 28, then again 3O 4O NUMBEff Cl 90 60 (DOUBLE.) Fia. 1. GRAPH OF MODEL PROCESS ILLUSTRATING THE STATISTICAL MEANING OF IRREVERSTBILITY The more lightly drawn curve records the number of black tickets remaining in urn A after successive drafts. The heavier curve records the previous and ensuing history of 50 tickets found in urn A at the end of the fiftieth draft, (Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 122.) 27, 26, etc. of the original balls. It is nowise impossible that, if we continue the drafts for a long time, some time or other all the original 50 black balls will be back in box A; but it is highly im- probable that this should happen within any reasonable time. Curi- ously enough, the urn model is competent to illustrate also this highly improbable course of events. For this purpose, instead of starting with 50 black and 50 white balls, we start with the balls, or in this case more conveniently tickets, all white, and numbered from 1 to 50 in urn A, and from 51 to 100 in urn B. After a suit- 1 A. J. Lotka, Am. Math. Monthly, March, 1924; Science, 1924, vol. 59, p. 532. 32 ELEMENTS OP PHYSICAL BIOLOGY able number of drafts, say 50 double drafts, in which a record is kept of all the numbers drawn, the urns are opened, and the tickets in the second urn are now blackened. The drawing is then con- tinued, for, say, another 50 drafts, recording each time the numbers drawn. The numbers on the tickets enable us to trace the previ- ous history of the 50 black tickets, before they were blackened. In an experiment actually carried out it was found that these 50 black tickets were originally distributed essentially evenly in the two urns. The curve representing the first 50 drafts is an ascending curve, the system passed, during this stage of the process, from more prob- able to less probable states, as shown in the first, ascending por- tion of the more heavily drawn curve in figure 1. In the second series of fifty drafts the curve descends in normal fashion, with increasing probability of the successive states of the system. It may seem like a contradiction of terms that what amounts prac- tically to an infinitely improbable series of drafts should be cap- able of actual realization at will. But if the series of drafts de- scribed were extended to great length in both directions, say one million drafts before blackening the tickets, and one million after, it would be seen that the peak on the curve is indeed a very ex- ceptional feature. It is a perfectly safe bet that in two million drafts not more than one such peak, going up to 50 black balls in one urn, would be encountered. The model described exemplifies among others the fact that in an exceedingly long lapse of time it may some time occur that the system will return to its original state. This is quite in accord with the laws of mechanics; in fact, as already noted, these laws actually demand that every mechanical system of finite dimensions must,, ultimately return to its initial state, and must do this not onee only, but in everlasting reiteration at regular intervals: the mo- tion is periodic. This property also is capable of illustration by a simple model, such as the following: Twenty-six pendulums of periods T = 0.5, 0.6, . . . 2.9, 3.0 seconds are started simultane- ously to the left from their equilibrium position, and are then al- lowed to oscillate undisturbed. Count is then made, at the end of every tenth of a second, of the number of pendulums on the left of ^ the median. In this way the staircase curve figure 2 was ob- tained (computation here taking the place of actual observation). It will be seen that in the brief fragment of a period covered by the STATISTICAL MEANING OF IKREVEBSIBILITY 33 record, this exhibits all the characteristics of a "passage from a less probable to a more probable distribution," though, in point of fact, we know that the system has a perfectly definite period of 7385 years. The appearance of chance in this wholly determinate mechanical process is brought into still greater prominence if we plot the deviations, from the mean, of the number of pendulums on the left of the median position, at successive counts. We thus obtain the points indicated by small circles in figure 3. These group themselves very obviously about a typical Gaussian curve of random distribution, namely one having a standard deviation of y2L', this curve has been drawn in the diagram, and, as will be seen, the agreement is good, considering the smallness of the sample FIG. 2. GBAPH OF SECOND MODEL PEOCESS ILLUSTRATING THE STATISTICAL MEANING OF IEEEVEESIBILITY Number of pendulums found on left of median position at successive epochs. (Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 124.) (412 observations, extending over 41.2 seconds, out of a total period of 7385 years). Thus for long stretches of time the periodicity of the motion of the system of pendulums is very effectively masked under an aspect of "chance." These simple models illustrate very clearly how the seeming con- flict between the periodicity of all mechanical motions and the ap- parently one-sided course of events, directed toward one definite end state, is resolved. The actual process of isothermal gaseous diffusion is, in fact, periodic, but with a period so long that humanly speaking, the return to the initial state never occurs at all. For all stretches of time that can have any real significance in human thought (and this includes the vast historical ranges of all geology and astronomy), it may therefore be said in a certain sense that evo- 34 ELEMENTS OF PHYSICAL BIOLOGY lution proceeds, in all but a vanisMngly small class of- exceptional cases, from less probable states (e.g., uneven distribution of the 50 black balls in the two urns) to more probable states, tending ul- timately toward a most probable state. This statement cannot however be allowed to pass without a word of caution. It is mean- -/ft 83 FIG. 3. FREQUENCY DIAGRAM FOE THE DEVIATIONS FROM THE MEAN APPEARING IN FIGURE 2 Abscissae represent deviations from the mean (13) in number of pendu- lums on left of median; ordinates represent corresponding frequencies, among the observations recorded in figure 2. (Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924 t> 125.) '*' STATISTICAL MEANING OF IEREVEESIBILITY 35 ingless unless the characteristic with regard to which probability is reckoned is explicitly or implicitly indicated. Probability is es- sentially a matter of classification. An improbable event is one that is a member of a small class, and whether it is so or not depends, clearly, on our system of classification. For this reason the broad statement which has sometimes been made, 2 that the direction of evolution is from less probable to more probable states, is not only inadequate, but is really meaningless. It is indefinite In failing to specify with regard to what characteristic probability is to be reck- oned; and it is incomplete in failing to call attention to the fund- amentally important connection between the particular probabilities in question and available energy. Another point, which has not hitherto perhaps received its de- served attention is clearly brought out by the two models de- scribed, namely, that irreversibility is a relative term. For, ob- viously, if we use our visual discrimination in selecting the balls drawn from the boxes (instead of drawing blindly), we can easily bring it about that in short order all the black balls are back in box A. Thus a process may be reversible or not, according to the means that are naturally available or arbitrarily permitted in oper- ating upon the system under consideration; somewhat as the tri- section of an angle is or is not an impossible geometric construc- tion, according as we are or are not forbidden the use of instru- ments other than ruler and compass. In the case of molecular aggregates this fact has long been duly appreciated, having been first pointed out by Clerk Maxwell, who remarked that a demon capable of dealing with individual molecules would be able to cheat the second law of thermodynamics. But it seems to have been pretty generally overlooked that the relative character of irreversi- bility has an important significance in certain natural processes taking place on a macroscopic scale. In point of fact this is a mat- ter whose importance in the world of living organisms can hardly be rated too high. For there are certain diffusive processes going on in nature which, from the standpoint of thermodynamics, are not of the irreversible type; but which might as well be, so far as any benefit derived from their reversibility by the organism (and, in particular, by man) is concerned. If I should be the fortunate 2 See for example, J. Royce, Science, 1914, vol. 34, p. 551. 36 ELEMENTS OF PHYSICAL BIOLOGY possessor of a pound of gold dust, and some malicious person should take it and scatter it far and wide, so that it became hope- lessly diluted with dust and refuse, it would be a small comfort to me to know that it were merely mechanically commingled with such foreign matters, that it had not irreversibly undergone solu- tion or chemical transformation, and that therefore it could theo- retically be recovered without the expenditure of work. In prac- tice its recovery might entail the expenditure of far more energy than if the gold were present in reasonably concentrated solution. The point is that in practice I am restricted to operations in bulk upon reasonably large quantities, in reasonably concentrated form, otherwise the theoretical ideal of recovery without work is very far from being attained. And with this restriction placed upon my operations, certain processes acquire an irreversibility which they do not possess apart from that restriction. The illustration of the pound of gold has the advantage of simplicity and cogency. But if by any chance it has conveyed the impression that only in pecu- liar and far-fetched cases does this kind of irreversibility enter into play, then it is an unfortunate example indeed, for nothing could be farther from the truth. The fact is that nature abounds in just such dissipative processes as the scattering to the four winds, to utter inutility, of materials of the highest importance to life; and one of the central problems which the organism has to solve in the struggle for existence, is the reconcentration, into his immediate environment and into his body, of valuable materials that have become scattered by agencies beyond his control. It is not the least of the triumphs which have made man the lord of creation, that he has learnt, beyond all comparison more effectively than any of his competitors, to carry out this process of reconcentration to satisfy his needs. So a fleet of ships, year by year, bear a burden of saltpeter from Chile to all civilized countries, to balance the losses from our depleted fields. So, in his most recent technical achieve- ment, man has learnt to draw from the air a supply that will con- tinue unfailing, long after the Chilean nitre beds are exhausted. The fact is, in dealing with the physics of such macroscopically irreversible effects, it will ultimately be necessary to develop a method of mathematical analysis that shall be competent to dis- tinguish and handle not only the extremes the case of a primitive organism that can deal only in the gross, without intelligent or STATISTICAL MEANING OF IKREVEBSIBILITY 37 other discrimination, with, the matter and situations presented to it; and an ideally perfect organism, that should expend just the minimum of effort, directed with absolute precision toward the attainment of its ends. A method must be devised that shall duly take account of, and use as a fundamental datum for its deduc- tions, the particular character, the particular degree of perfection of the mechanical and psychic equipment or organization by which each organism reacts more or less selectively upon its environment. We shall have occasion to refer to this matter again in greater detail in a~ later section. But here it is well to note that our two models are suggestive also with regard to this aspect of the subject. For it appears at first sight as if there were a fundamental differ- ence in character between the first and the second model, since it is essential for the operation of the urn model that the drawing be done blindly, so as to give chance a part in the process, as we would say; whereas the pendulum model we operate with our eyes open, apparently in full consciousness of what is going on. Chance seems to play no part here, the system is mechanically determinate. But there is a blindness which is not of the eye, and there is a vision that surpasses optical vision. The same struggle for ex- istence which has developed in man the organ of sight, to depict for him the external world, to furnish him with a map on which to base his plan of campaign, has also, in latter days, developed his internal vision, whereby he extends his world-picture beyond the powers of the bodily eye. It is immaterial by which process his map is drawn 'its function is the same; whether I peep into the urn and manipulate the drafts by the light of my eyes; or whether, in the light of my knowledge of mechanics, I adjust the pen- dulums to equal lengths and phases; or again, whether, in the more serious affairs of life I employ these same faculties to diverse ends, the effect is the same: In greater measure or less these organs and faculties emancipate me from the bonds of the fortuitous and make me a controller of events. Their function is to substitute choice for chance, to introduce aimed collisions in place of random encounters. Origin of Subjective Sense of Direction-in-Time. The failure of the differential equations of dynamics to discriminate between t and t raises the question as to the physical significance and origin of our subjective conviction of a fundamental difference between the forward and the backward direction in time, a con- 38 ELEMENTS OP PHYSICAL BIOLOGY viction that is intimately bound up with the concept of evolution, for, whatever may ultimately be found to be the law of evolution, it is plain that no trend of any kind can be denned or even de- scribed without reference to a favored direction in time. One view which suggests itself is that this conviction is our sub- jective appreciation of the trend from less probable to more prob- able states recognized in statistical mechanics. But this does not seem very satisfying, for we somehow feel that our conviction must rest on something more fundamental than this somewhat accidental circumstance, which, as the models described clearly show, is funda- mentally incompetent to distinguish between the forward and the backward direction in time. For the pr ^in figure 1, for example, may indifferently be traversed from lefTto right or vice versa, it presents the same general character in either sense. Another alternative is to suppose that the differential equations of dynamics, as formulated by us today, are either an incorrect, or else an incomplete statement of facts. The latter view is, indeed, upon reflection, found to have a certain warrant. For the differ- ential equations of motion alone do not fully determine the actual course of events; this depends further on the value of certain ar- bitrary constants of integration; or, to speak in terms of physical entities, upon the initial velocities of the particle of which the sys- tem is composed. Strictly speaking it is only when the initial velocities are zero, that the equations of motion, considered in their totality, are indifferent to the substitution t' = t. From this point of view our sense of the forward direction in time would ap- pear as our subjective appreciation of the fact that, once a material system has been started on a certain course, with certain initial velocities, there then remains no further freedom; its history must continue to unfold in the direction determined by the initial veloc- ities. It seems, however, that it is not with this perfectly general type of irreversibility of the course of events that we are chiefly con- cerned in the study of evolution. The concept of evolution, ac- cording to the analysis which has been made of it in preceding pages, applies principally, if not exclusively, to systems that out- wardly at least affect the aperiodic habit, systems that do not return periodically to their initial state, but show a definite trend, whereby yesterday and tomorrow are never alike, and differ more- STATISTICAL MEANING OF IEEEVEESIBILITY 39 over in some definite and characteristic fashion, even though we may not be fully competent, at the present epoch of science, to specify exactly wherein lies the characteristic difference. 3 Inadequacy of Thermo dynamic Method. Our reflections so far have linked the fundamental problem of the direction, the trend of evolution, with the disciplines of thermodynamics and statis- tical mechanics. From this point of view the direction of evolution is identified with the direction of the unfolding of irreversible proc- esses, the direction of increase of entropy (in thermodynamics) or of increasing probability (in statistical mechanics). A certain mental satisfaction may be derived from, this conclu- sion. It gives us, in principle at least, an answer to our question "Quo vadis?" But practically the answer is very inadequate. If the conclusions, the methods of thermodynamics, or of statistical mechanics, are to be applied to a concrete case, the data of the problem must be presented in a veiy particular form. So long as we deal with volumes, pressures, temperatures, etc., our thermo- dynamics serve us well. But the variables in terms of which we find it convenient to define the state of biological (life-bearing) systems are other than these. We may have little doubt that the principles of thermodynamics or of statistical mechanics do actu- ally control the processes occurring in systems in the course of organic evolution. But if we seek to make concrete application we find that the systems under consideration are far too complicated to yield fruitfully to thermodynamic reasoning; and such phases of statistical mechanics as relate to aggregation of atoms or mole- cules, seem no better adapted for the task. To attempt applica- tion of these methods to the prime problems of organic evolution is much like attempting to study the habits of an elephant by means of a microscope. It is not that the substance of the elephant is in- herently unfitted to be viewed with the microscope; the instrument is ill adapted to the scale of the object and the investigation. It would seem, then, that what is needed is an altogether new instrument; one that shall envisage the units of a biological popula- tion as the established statistical mechanics envisage molecules, atoms and electrons; that shall deal with such average effects as 3 Perhaps the objective interpretation of our subjective sense of direction in time must be sought in quantum mechanics. Cf. A. J. Lotka, loc. cit., p, 126, and, W. S. Franklin, Science, 1924, vol. 60, p. 258. 40 ELEMENTS OF PHYSICAL BIOLOGY population density, population pressure, and the like, after the man- ner in which thermodynamics deal with the average effects of gas concentration, gas pressures, etc.; that shall accept its problems in terms of common biological data, as thermodynamics accepts prob- lems stated in terms of physical data; and that shall give the answer to the problem in the terms in which it was presented. What is needed, in brief, is something of the nature of what has been termed "Allgemeine Zustandslehre," 4 a general method or Theory of State. It is somewhat along these lines that the system now to be sketched is conceived. A term introduced by J. R. Rydberg, quoted by C. Benedicks, Zeitschr. f. phys. Chemie, 1922, vol. 100, p. 42. CHAPTER IV EVOLUTION CONCEIVED AS A REDISTRIBUTION Toutes ces choses ne peuvent se determiner surement que par des mesures precises que nous chercherons plus tard; mais auparavant il fallait au moins sentir ie besoin de les chercher. /. B. Biot. It now behooves us to establish, with respect to the problem of evolution, a viewpoint, a perspective, a method of approach, which has hitherto received its_ principal development and application outside the boundaries of biological science. Such prior develop- ment and applications, however extraneous to our chief line of interest here, may well serve us in our present interrogations, since we shall be in a position to profit by the precedents established in methods, in conclusions, and, most particularly, in habit of thought. This perspective is that which contemplates an evolving system as an aggregation of numbered or measured components of several specified kinds, and which observes and enregisters the history of that system as a record of progressive changes taking place in the distribution, among those components, of the material of which the system is built up. It is thus that physical chemistry views the progressive changes in a system comprising several chemical species, that is to say elements, compounds, phases, etc. It describes the system by enumerating these components, by stating their character and ex- tent (mass); and by further indicating the values of certain quan- tities or parameters, such as volume or pressure, temperature, etc., which, together with the masses of the components, are found experimentally to be both necessary and sufficient, for the purposes in view, to define the state of the system. With the instantaneous state of the system thus defined, physical chemistry investigates by observation and by deductive reasoning (theory) the history, the evolution of the system, and gives analytical expression to that history, by establishing relations, or equations, between the vari- ables denning these states (after the manner set forth above), and the time. It is commonly found that these fundamental equations assume the simplest, the most perspicuous form, when they are written 41 42 ELEMENTS OF PHYSICAL BIOLOGY relative to rates of change of the state of the system, rather than relative to this state itself. That is to say, it is found that the expressions for the rate of increase in mass, the velocity of growth, of the several components, are simpler, more primitive in form, than the expressions giving directly the mass of each component as a function of the time. In the language of the calculus, the differ- ential equations display a certain simplicity in form, and are there- fore, in the handling of the theory at least, taken as the starting point, from which the equations relating to the progressive states themselves, as functions of the time, are then derived by integration. 1 So, for example, a simple system may be defined as comprising 4 grarn-rnoleeules of hydrogen, 2 gram-molecules of oxygen, and 100 gram-molecules of steam, at one atmosphere pressure, and at 1800C. The fundamental relation expressing the law of evolu- tion, the historical pattern, of the system, is in this case given by the law of mass action: . /{^ (_-! } v dt v 3 v 2 where v is the volume, mi is the mass of steam, m 2 the mass of hy- drogen, and m s , the mass of oxygen (all expressed in gram-mole- cules). The coefficients h, h, are functions of the temperature, or, for a given temperature, are characteristic constants of the re- action. We are not, here, interested in the particular form of the law of mass action. What does interest us is the general form of the equation (1). It states that the rate of increase in mass, the veloc- ity of growth of one component, steam (mass mi), is a function of the masses m*, nis, of the other components, as well as of the mass mi itself, and, besides, of the parameters v (volume) and T (tem- perature), the latter being contained in the coefficients fc x , /c 2 . This statement, in its more general form, is written, according to es- tablished notation 2 1 In experimental observation usually (though not always) the reverse attitude is adopted. 2 For the benefit of the non-mathematical reader it may here be explained that aquation (2) is merely a short-hand expression, so to speak, of the simple statement: The rate of increase of mass with time of the component Si is dt & junction of, or is determined by, the masses mi, m 2 , ma, of the components Si, S%, S s , as well as by the volume v and the temperature T. Precisely similar ia the construction to be placed on the equations (3). EVOLUTION CONCEIVED AS REDISTRIBUTION 43 dni-i = F (nn, m z , m s ; v, T) (2) at Now it is this habit of thought, expressed in equation (2) , that is to be transplanted into the contemplation of problems of evolu- tion in general, and organic evolution in particular; this point of view, this perspective, which regards evolution as a process of re- distribution of matter among the several components of a system, under specified conditions. 3 Having thus passed from the specific to the general from the case of physico-chemical systems to a general formulation -we now retrace our steps to the particular, but in a new direction. We now contemplate the kind of systems that form the object of study of the biological sciences. With the outlook gained in our preceding reflections we envisage the life-bearing system, in the progress of evolution, as an assembly of a number of components: Biological species; collections or ag- gregations of certain inorganic materials such as water, oxygen, carbon dioxide, nitrogen, free and in various combinations, phos- phorus, sulphur, etc. These components are placed in various relations of mutual in- teraction under specific conditions of area, topography, climate, etc. Under these conditions each may grow, decay, or maintain equi- librium. In general the rate of growth ~7T~of any one of these com- ponents will depend upon, will be a function of, the abundance in which it and each of the others is presented; this rate of growth will also be a function of the topography, 4 climate, etc. If these latter features are defined in terms of a set of parameters Pi P 2 . . . PJ, we may write, in the same sense as equation (2) 3 Compare F. B. Jevons, "Evolution," Macmillan, 1902, Chapter VI, p. 72. 4 Terrestial species have an essentially two-dimensional distribution, so that area functions here in a manner somewhat analogous to that in which volume enters into physico-chemical relations. Aquatic life, with its three- dimensional sphere of activity, is enacted in systems whose extension is described in terms of volume. More detailed topographic parameters may be required to define in sufficient completeness the configuration, the structure of these systems. The absence of such detail from the more familiar formu- lations of chemical dynamics is due to the purely accidental circumstance that the systems commonly dealt with in that branch of science are either homogeneous, or of comparatively simple heterogeneous structure. 44 ELEMENTS OF PHYSICAL BIOLOGY ~ = Fx (Z 1; Z,, . . . XnJ Pi, Pt, . . . Pi) CW j~v = Ft (Xi, Xz, . . . Xn', Pi, Pa, PI) (3) at j-y- = Fn (Xi, X^ . . . X R ; PI, Pj, . . . Pj) at In general there will be n such equations, one for each of the n components. The purport of these equations (3) remains uncertain so long as the components (e.g., biological species, etc.) Si Sz . . $ n are undefined. What definitions we may adopt for these is, in accord- ance with the principles discussed anteriorly, a matter for arbitrary disposition; though we may be guided in our choice by considera- tions of expediency. These may advise different policies from case to case, according to the particular phase of the problem taken in view. The conclusions reached will, of course, depend upon the particular definitions chosen. This is as it should be; the conclu- sions apply to the components as defined, and, in general, to no other. This seems clear enough, but if any further exposition is needed, it will be found In examples shortly to be considered. Intra-Group Evolution. While the precise definition of the com- ponents Si S 2 . . , SB may, and indeed must, be held over for determination as each separate phase of the general problem is singled out for treatment, yet there is one class of cases regarding which it is appropriate to set forth certain reflections at this juncture. It may have been observed that so far nothing has been said regarding one aspect of or/ganic evolution which, in the history of the subject in the past, and hi the minds and writings of its ex- ponents and students today, occupies a dominating position namely, the relation of evolution to the modification of species with the passage of time, and, in particular, to the origin of new species. Now a biological species, however defined, is not a homogenous group. It comprises portions (individuals) varying more or less widely with regard to numerous features, such as stature, weight, etc. If our description of the distribution of matter in the system is to be at all exhaustive, we shall need to know, not only the extent EVOLUTION CONCEIVED AS EEDISTEIBUTION 45 (total mass) of each species, but also its constitution, as expressed by the frequency, the relative abundance, of each statistical type within the species. In the case of man, for example, we may wish to know the fraction F (56) of the total population whose height is comprised within the limits 58 and 57 inches at a given instant. And in observing the evolution of the system of which this popula- tion forms part, we shall be interested, not only in the growth (or decay) of the population as a whole, but also in the rate of change of the abundance (frequency) of each statistical type. This phase of the problem does not differ essentially, in character, firom that first considered : it is essentially a question of distribution and changes in distribution of mass in the system among its several cornponent; only now we have fixed our attention on a different set of components, components defined in a different way, on a finer scale. In the first instance we had taken in view the dis- tribution and changes in distribution of the matter of the system among the several major groups or species; now we are considering the distribution within each such group. This division of the gen- eral problem of organic evolution into two aspects has certain prac- tical advantages, and it will be convenient to have names to desig- nate the two separate aspects or domains of evolution. We shall accordingly speak of inter-group evolution on the one hand, when referring to changes in the distribution of the matter of the system among several component groups; and we shall speak of intra-group evolution when referring to changes in the distribution of matter within the group, among its statistical types, however defined. It is possible to set up equations relating to intra-group evolu- tion, similar in general character to those set forth above relative to inter-group evolution. However, this phase of the problem is probably treated more satisfactorily in other ways, of which some examples will arise in due course. 5 It is nevertheless, desirable, to indicate in the system, of equa- tions (3) the incidence of intra-group evolution. Conveniently this may be done by writing the ith equation, for example, in more detailed form: = Fi (X 1} Z,, . . . X n ; Pi, P Sj . . . PI; Qi, Q, . . . Qk) (4) at 5 See Chapter IX, p. 122; Chapter XIII, p. 170; Chapter XXV, pp. 348 et seq. 46 ELEMENTS OF PHYSICAL BIOLOGY where Qi, Q 2) , . . Q k are parameters defining the character of several components (e.g., biological species) S; such definition may take the form of a set of characteristic frequency functions, or some other form. These parameters will, in general, be functions of the time, that is to say, each component may, in general, be variable in character through the occurrence of intra-group evolu- tion. Whether this variability is limited, or constrained to follow a certain course (orthogenesis), whether variations take place in continuously graded series or per saltum, or in any or all of these ways, are questions which will not be discussed at this point. At present all that need be said is that the origination of a new species in any of these ways falls within the scheme of our description of evolution as a change in the distribution of matter among the com- ponents of the system. 6 It may be remarked, in passing, that in general a complete def- inition of the system may require an infinite number of parameters P and Q; this does not necessarily cause any undue complication in practise, since in many cases certain of the parameters P, Q either remain constant, or change so slowly that, in discussing changes in the variables X, we may treat these parameters as if they were constant. The parameters Q, defining the character of the species, are in general functions of the time, as has already been remarked. In this respect organic evolution exhibits a very important distinc- tion as compared with chemical evolution, i.e., the evolution of a system in the course of chemical transformation. In such a sys- tem the character of the components is usually fixed once for all. Water is H 2 for all time, unlike a species of organism which is subject to change in character. One important result of this is that, so far as we know, organic evolution is a process without end, for there seems to be no limit to the variety of forms of living mat- ter, as there is no limit to the variety of geometric configurations and mechanical systems that can be formed from a given portion of matter. Chemical evolution, on the contrary, terminates, under constant condition, in a definite equilibrium, determined once for all, by those conditions. 8 This is not a new definition of evolution, it is a conception of evolu- tion wholly compatible with the definition that has been laid down in preceding pages. EVOLUTION CONCEIVED AS BEJHSTRIBTJTION 47 It will be observed that the fundamental equations (3) resemble in form the equation (4) of Chapter II, which was given as a typical example of the equation for an inertia-free or completely JV damped system. The velocities -37 are single-valued functions F of the variables X. It is the single-valued character of the func- tions F that gives the system its stamp as an inertia-free or completely damped system; a system in the course of typically irreversible transformation. It is not maintained that these equations cover all cases that may be brought within the purview of the present study; nor shall we, in all cases, be tied down to the scope of these equations. They are, however, of very broad scope, and, upon reflection, will be found to cover at least a large and significant portion of the field of our interests here. To one point, however, it may be well to draw attention. To read these equations in their broadest interpretation we must be prepared to consider cases in which the phenomenon of lag or lead enters. Perhaps the terms lag and lead require explanation. In some cases the course of events today depends on certain features in the state of the world at a previous date. So, for example, the number of persons of age 50 in the year 1924 depends (among other things) on the birthrate in the year 1874. Or, to quote another instance, the number of new cases of scarlet fever today depends on the number of infections a week ago. There is thus a lag in the appearance of the observable effect in the system. In other cases there may be a lead. The price of land on Church Street today may suffer an increase or perhaps decrease because it be- comes known that in a year's time a railway station is to be built nearby. Since effects of this kind must be contemplated as a possibility, we must be prepared to read our equations in the following sense: The rate of increase ~T7~of the mass of the component Si at the at present instant, is a function of the masses X i} Xz, etc., at some other instant of time, say Xi at (t pi), X 2 at (t p 2 ), etc., where some of the p's may be negative (corresponding to a lead). 48 ELEMENTS OF PHYSICAL BIOLOGY We must be prepared to consider our equations in this interpre- tation. Illustrative examples will not be offered here, as the math- ematical treatment of these cases is somewhat troublesome. The interested reader will find an illustration in the author's mono- graph on the Ross Malaria equations, Am. Jour. Hygiene, vol. 3, January Supplement, 1923. Only a reflection of general charac- ter shall find its place here. It is characteristic of systems whose history is defined by equations thus involving a lag, that, in general, the course of events at a v a instant is dependent upon the previ- ous history of the system over a certain finite range of time. The consequences of this feature are somewhat singular. If the world's events followed a system of equations of this kind, we might have two worlds, in every respect identical today, but each with a different past, and, in consequence each with a different future. And a similar reflection necessarily applies in the case of lead. This conclusion is perhaps not in harmony with a mechanistic conception of the universe. But the phenomena of memory and of will are of precisely such character as to introduce lags and leads into the world's equation, and we may be well advised to keep our minds open as to the possible effects of this circumstance upon the course of events. It should be observed, that the appearance of a lag or lead in our equation may be spurious. It may be due to a species of mathe- matical shorthand. It is easier to describe a person as having be- come infected with scarlet fever three days ago, than to describe precisely his present state today ensuant upon that infection. Hence we may prefer to, or, for lack of detailed knowledge we may be forced to write our equations in terms of (t p), although, if all the facts were known, we could, were we so disposed, write them in terms of t. And the same reflection applies to the appearance of a lead in our equations. Whether, with complete knowledge of all circumstances bearing on the situation, there should still remain a residuum of influences that could find expression only in terms of a lag or lead, this, perhaps, is fundamentally the nature of the problem of the influence of consciousness upon the course of the world's historv. CHAPTER V THE PROGRAM OF PHYSICAL BIOLOGY . . . . It will be the function of this new branch of science to investi- gate biological phenomena as regards their physical aspects, just as Physical Chomistry has treated the physical aspects of chemical phenomena. Because thiy field has not yet been systematically explored .... the individual data of Physical Biology appear, as yet, as more or less disconnected facts, or as regularities for which no proper place is found in the existing scheme of pre,Bent-day science; and the investigations of isolated problems in this field are as yet carried on as something of a scientific hobby by amateurs, with the result that they are guided by chance rather than by plan. . . . and are often totally lacking in any fundamental guiding principles or connecting theo.'y. As results gathered in this disconnected fashion accumulate, the need of their unification into a harmonious whole, into a distinct discipline of science, becomes more and more acutely felt. Such unification necessarily involves the working out of a viewpoint that shall make the several facts and relations fall in line naturally in an orderly system; in other words, what is needed is a labor of organisation. In the course of this, new and unforeseen problems will inevitably arise, and a fruitful field of scientific endeavor should thus be opened for the investigator. Porstmann. A first use to which we may with advantage put the results of the preceding analysis, is to systematise the subject here treated; there will thus be gained a general plan of work and a division of the topic into natural sections, upon which the arrangement of the succeeding chapters will, in the main, be based. Physical Biology, 1 as here conceived and discussed, is essentially a branch of the greater discipline of the General Mechanics oj Evolu- tion, the mechanics of systems undergoing irreversible changes in the distribution of matter among the several components of such system. 1 In introducing the term Physical Biology the writer would suggest that the term Biophysics be employed (as hitherto) to denote that branch of science which treats of the physics of individual life processes, as exhibited in the indi- vidual organism (e.g., conduction of an impulse along nerve or muscle); and that the term Physical Biology be reserved to denote the broader field of the application of physical principles in the study of life-bearing systems as a whole. Physical biology would, in this terminology, include biophysics as a subordinate province. For a summary statement of what might be termed the program of bio- physics see A. Forbes, Science, 1920, vol. 52, p. 331. 49 50 ELEMENTS OF PHYSICAL BIOLOGY It so happens that many of the components that play an tant role In nature, both organic and inorganic, are built large numbers of individuals, themselves very small as compe The with the aggregations which they form. Accordingly the stud^ono- S3 r stenis of this kind can be taken up in two separate aspects, nam x- 3, first with the attention centered upon the phenomena displace- by the component aggregates in bulk; we may speak of this as hose Bulk Mechanics or Macro-Mechanics of the evolving system. A^al, secondly, the study of such systems may be conducted with f evi- attention centered primarily upon the phenomena displayed The the individuals of which the aggregates are composed. This brand's of the subject may suitably be termed the Micro-Mechanics^wo the evolving system. It is evident that between these two branc'ent or aspects of the general discipline there is an inherent relat-ilar arising from the fact that the bulk effects observed are of the na of a statistical manifestation or resultant of the detail workiiistic the micro-individuals. The study of this inherent connected of accordingly, the special concern of a separate branch which w^ads speak of as Statistical Mechanics, This terminology is i]P our coincident with accepted usage, but in part must be unde n the to refer to an expansion of the subject beyond the bounds h covered, whereby its scope shall be extended so as to inched in statistical treatment of the dynamical problems presented &the- gregates of living organisms; that is to say, aggregates of ? be- transf ormers possessing certain significant special properties sribe Each of the branches of the mechanics of evolution enur'SAce so far naturally splits up into two subdivisions, according v be devote our attention to the material changes or to the energy ch all involved. By an extension of prior usage in physical chemistr^n may employ the term Stoichiometry to denote that branch of " science which concerns itself with the material transformations., with the relations between the masses of the components. The discussion of this branch presents itself as the more elementary task, and will therefore be taken up first; after this has been dis- posed of we shall be better prepared to discuss the second aspect, the Energetics or Dynamics of Evolution. Taking now a survey of the Stoichiometry of systems in evolu- tionary transformation, we can hardly find a better guide, for the organization of this subject, than the fundamental equations PROGRAM OF PHYSICAL BIOLOGY ^7 = F, (X, X,, . . . Xn; P, Q) at which we may speak of as the fundamental equations of the Kinetics of Evolution, since they furnish expressions for the velocities of trans- formation and exhibit the relations between these velocities and the masses of the several components, as well as the parameters P, Q. The very form of these equations suggests, as the first and most elementary problem, the treatment of the case of evolution under constant conditions, as defined by constant P's and Q's. This will, accordingly be the coarse here adopted, treating first the general case of n variables Xi, X 2 , . . . X n , and then some special cases in which the number of variables is restricted to 1, 2 and 3. The perfectly general case, of evolution under conditions of wholly unrestrained variability of the P's and Q's, is mathematically uninviting (though not wholly intractable), and is also of minor interest in practice. Little will therefore be said of this. Certain special types of changes in the P's and Q's (e.g., slow changes) will find a place in the next subdivision of the general subject, the Statics of evolving systems. This branch is, in a sense, a special division of the Kinetics of Evolution, namely that which concerns itself with systems in which the velocities of transformation are zero, so that there is Equilibrium, or, to be more exact, a Steady State. This, of course, implies, strictly speaking, constancy of the para- meters P, Q. But something very much like equilibrium presents itself in certain cases when these parameters change slowly. There may then arise what has been termed a Moving Equilibrium. In view of the important role which such moving equilibria play in nature, their discussion must form a part of the program of Physical Biology. A second special problem of evolution under changing conditions (changing parameters P) that lends itself to treatment with com- parative ease, is that which concerns itself with initial and end states (equilibria), as influenced by changing conditions, without demand- ing any information regarding the intermediate steps passed through by the evolving system. So, in physical chemistry, we may enquire what will be the effect, upon equilibrium, of a change in pressure or in temperature. Similar questions may be raised regarding 52 ELEMENTS OP PHYSICAL BIOLOGY equilibria or steady states in life-bearing systems, and the matter calls for at least passing notice. This leads to the consideration of the Principle of Le Chatelier and, as a natural sequel of the train of thought thus started, to the examination of relations which may exist between certain of the parameters P, somewhat as, in physical chemistry, significant relations exist between pressure and volume, for example. It is found that some, at any rate, of the parameters P present analogy to the intensity and the capacity factors of an energy; out of this have arisen in the past sporadic efforts to es- tablish systems of social dynamics and the like, which, however, have been based upon an acceptance as identity of what is only analogy. Those who have followed this road have been led, not so much perhaps to erroneous conclusions, as to blind alleys, to barren fields. True progress can be expected only by retracing one's steps from such tentative excursions and striking out in a new direction; forsaking the way of quasi-dynamics, and breaking a trail toward a system of true dynamics, both of the individual (micro-dynamics) and of the system as a whole (macro-dynamics). Of intra-species evolution, as expressed in changes in the para- meters Q that define the character of the several species, little will be said here. The reasons and justification for this step-fatherly treatment of so important an aspect of our topic have been set forth in the preceding text. A discussion of at least one phase of intra-species evolution, falling under the head of dynamics, will, however, be presented when dealing with that phase of the subject; and we shall briefly note, in due course, some aspects of intra- species evolution, as discussed more particularly by J. B. S. Haldane. These, in broad outline, are some of the principal land-marks in the territory ultimately to be covered by Physical Biology, and to be given a preliminary survey here. In concentrated form the lay of the land, as set forth above, is sketched in the diagram or scheme table 1, which should be found helpful both in presenting to the eye the salient features of the field of investigation, and als'i in furnishing a logical basis for the systematic arrangement of the subject in the ensuing chapters. It remains, in this chapter, to enumerate the methods by which Physical Biology may be expected to develop. For the gathering of data two methods are available: observation in natural condi- 1 .2 -S Q, 1H O " 54 ELEMENTS OP PHYSICAL BIOLOGY tions, and observation under experimental (laboratory) conditions. Examples of both these methods will be noted. For the elaboration of data, the establishment of regularities (Jaws/, there is available in this field, as everywhere in science, the method of induction, aided, if need be, by statistical technique. In this volume, however, emphasis will be laid upon deductive methods of mathematical analysis, as applied either to data fur- nished by observation, or to "unknown" quantities, blanks, as it were, in our formulae, ready for numerical substitution whenever concrete data become available. The principal subdivisions of our topic, and the relations between them, as outlined above, are summarised graphically in table 1. KINETICS CHAPTER VI THE FUNDAMENTAL EQUATIONS OF KINETICS OF EVOLVING SYSTEMS L'emploi des signea mathe'matiques est chose naturelle toutes les fois qu'il s'agit de discuter des relations entre des grandeurs. A. Cournot. General Case. We now proceed to the systematic study of the subject in accordance with the general plan laid down in the preceding survey of the Program of Physical Biology. According to this schedule (table l)we approach first of all the section of Macro- mechanics (that is to say, the mechanics of evolving systems regarded as bujlt up of component species in the gross), without carrying the analysis down to the finer details of individual organ- isms. And, of the field of Macromechanics, we shall here take up the section of Stoichiometry, that is to say, we shall for the present confine our attention to the relations between the masses involved, leaving aside for a later section the associated energy changes. Of the general field of Stoichiometry the first division to be taken up, according to our schedule, is the Kinetics of Evolution, and we shall here begin with a brief consideration of the Fundamental Equations in their general form = Fi (* X tl . . . X K ; P, Q) (1) at One phase of the general problem before us must evidently be the study (by observation, experiment, or any other method available) of the character or form of the functions F which tell us how the growth of each component is dependent upon the other components and the parameters P and Q. It might be supposed, indeed, that until this phase of the problem had received considera- tion, the system of equations (1), would be at best a barren expres- sion of facts. But this is a misconception. Without knowing anything regarding the precise form of these functions, a good deal of information of considerable interest can be derived from these equations; and before proceeding to the consideration of concrete 57 58 ELEMENTS OF PHYSICAL BIOLOGY cases, in which something is known regarding the particular form of the functions F, it is proper, at this point, to extract from the fun- damental equations (1) all the information that we can. This requires a little mathematical manipulation of compara- tively simple character, The variables X, the masses of the several components of the system, are not, in general, wholly independent. They are subject to certain constraining relations. For example, for any self-con- tained system, we shall have an equation of the form X, X a + . , .+X n = SZ = A = const. (2) expressing the constancy of the total mass of the system; and a similar equation holds separately for every chemical element. 1 These equations prohibit certain changes of mass in the system. They are analogous to equations introduced in mechanical problems^^,^ /- by the geometric constraints that limit possible displacements (as in the case of a ball rolling down an inclined surface, and pre- vented, by the resistance of the surface, from falling vertically). They are commonly spoken of as equations of constraint. Equations of constraint such as 2Z = A = const. will in general enable us to eliminate certain of the variables X, expressing them in terms of the other X's and of certain constants A. If, by the aid of m such equations of constraint, m variables have been eliminated, the system of equations (1) can be reduced to a simpler one, of identical form, but containing only (n m) variables ^.. ( and the same number of equations. .-. { In all that follows we shall assume that this has been done, that we now read equations (1) in this sense : the n variables Xi X% . . . . X n are those left over after eliminating as many of the X's as the equations of constraint permit. The functions F will in general, after this elimination, contain constants A introduced by the equa- tions of constraint. We shall now, except where otherwise stated, restrict our con- siderations to the case in which the parameters P and Q and A are either constant, or change so slowly that we may disregard f f /"' 1 Except in those rare cases in which radioactive or other atomic disintegra- /' ticms occur. / *_ FUNDAMENTAL EQUATIONS OF KINETICS 59 their variation. This means that we restrict ourselves to the con- sideration of simple inter-group evolution under constant condi- tions. With this understanding, the first point we may observe about the system of equations (1) is that they define certain conditions of equilibrium, or, to be more precise, of steady states. For such a state ensues whenever all the velocities vanish, that is to say, according to (1) whenever Fi = F, = . . . = F n = (3) We thus have n equations between the n variables Xi, X Zi . . . X n , which, in general, determine certain values (4) such that when masses of the several components have these values, they persist in these values; the system is at rest, as regards changes in the distribution of matter among its components Si, S 2 , . . . S n . In general there will be a number of such possible equilibria, some of which will be stable, some unstable. The determination of their number and character is a technical point in the theory of equations, for the general treatment of which the reader must be referred to the pertinent literature (see, for example, Picard, Traite" d' Analyse, vol. 1, 1891, pp. 83, 123; vol. 2, 1893, pp. 183, 193, 196, footnote) . To give a touch of concreteness to the discussion at this stage a very simple example may be given to illustrate how several equili- bria may occur in a life-bearing system. A perfectly screened dwelling may be kept indefinitely free from flies. This is a condi- tion of equilibrium, but of unstable equilibrium; for if only a few flies gain access, presently these will breed, and the room will become inhabited by a population of flies whose number will depend on the amount of food present, on the measures taken to combat the pest, etc. Unless these measures are v,ery active, the flies will not be wholly exterminated, but the population will attain some approxi- mately steady number (for a given season) . There are, then, in this case, two possible equilibria; one with a zero population of flies, the other with some positive number of fly population. 60 ELEMENTS OF PHYSICAL BIOLOGY The equilibria in nature, involving countless species, are of course much more complicated in character, but the general principle is the same; and we must expect that in general a variety of different equilibria are possible, some unstable and some stable. For the further discussion of the equations (1) it is now desirable to express these relations, not in terms of the masses X } but in terms of the excess of each mass Xi over its corresponding equili- brium value, xi = Xi-d (5) The equation (1) then takes the form , ~r- = / i at (6) the parameters P, Q being omitted, for the sake of brevity. Ex- panding the right hand member by Taylor's theorem we obtain the system of equations dt dt + + + . -f dt A general solution of this system of equation is (7) (8) where the (?'s are constants, of which n are arbitrary, and Xi, X 2 . . . X n are the n roots of the equation 2 for X Oil X 0,ii . . . dm - 2)(X) = (9) 2 This equation is commonly spoken of as the characteristic equation. For greater detail see A. J. Lotka, Proc. Am. Acad. Sci., vol. 55, 1920, pp. 137-153. FUNDAMENTAL EQUATIONS OF EJNETIC3 61 It is seen by inspection of the solution (8) that if all the X's are real and negative, x i} x 2 , . . , x n all approach zero as t increases toward infinity, since e~ m = 0. But according to (5)., as the re's approach zero, the X's approach their equilibrium valu_es C. In this case, obviously, equilibrium is stable, since, by allowing sufficient time to elapse, we can always make the system approach as near as we please to Xi = Ci, for all values of the subscript i. Precisely similar conclusions hold if some or all of the X's are com- plex, and the real parts of all the X's are negative. If all the roots X ar ( e real (and negative), 3 each term in the solution (8) diminishes continually and approaches zero asymptot- ically as t approaches infinity. If all the constants G are of the same sign, the sum. of the series also will, evidently have a similar type of approach to equilibrium. If some of the G's are positive, others negative, there may be a species of irregular oscillations, the mass X of the component rising sometimes above its equilibrium value (7, sometimes falling below it. Ultimately, however, the equilibrium is approached from one side only, s,ince ultimately tbe term containing the numerically smallest X will predominate over all other terms. If some of the roots X are complex, the solution will contain truly oscillatory terms, since the exponential function, for complex exponents, assumes the trigonometric form g ft _}_ i gi n ft) (10) In this case there will be regular oscillations about the equilibrium position; in general these oscillations will be damped, that is to say, their amplitude will diminish, so that equilibrium is approached more and more closely, but always with oscillation the equilibrium is approached from both sides at once, so to speak, the oscillations persisting forever, though on a diminishing and ultimately vanish- ing scajle. These conclusions are the analytical confirmation and extension of an inference drawn by Herbert Spencer 4 on qualitative grounds : 3 The analytical condition that the real parts of all the roots X shall be nega- tive is given by Hurwitz, Math. Ann., vol. 46, 1895, p. 521. See also Blondel, Ann. de Physique, 1919, pp. 117, 153. It might be noted here that a necessary though by no means sufficient condition, evidently is that the absolute term in D (X), that is to say D (0), shall be positive when n is even, and negative when n is odd; for this absolute term is equal to the product of all the roots X. * Herbert Spencer, First Principles, Chapter 22, Section 173. 62 ELEMENTS OF PHYSICAL BIOLOGY Every species of plant and animal is perpetually undergoing a rhythmical variation in number now from abundance of food and absence of enemies rising above its average, and then by a consequent scarcity of food and abun- dance of enemies being depressed below its average .... amid these oscillations produced by their conflict, lies that average number of the species at which its expansive tendency is in equilibrium with surrounding repressive tendencies. Nor can it be questioned that this balancing of the preservative and destructive forces which we see going on in every race must necessarily go on. Since increase of numbers cannot but continue until increase of mor- tality stops it: and decrease of number cannot but continue until it is either arrested by fertility or extinguishes the race entirely. It will be observed, however, that our analysis enables us to be considerably more specific in distinguishing several modes of ap- proach to equilibrium, and in indicating the particular conditions under which each occurs. A point that here deserves particular emphasis is that, in order to make these distinctions and indications, it is by no means necessary to have a complete knowledge, or even very extensive information regarding the functions F. Only the coefficients of the first (linear) terms in the Taylor expansion of these functions enters into the determinant D(X),, and only these need therefore be known to draw the requisite conclusions regarding the stability and mode of approach of equilibrium. Furthermore, Spencer's commentary relates specifically to the case of species related to each other in certain particular ways (food, enemies, etc.): the analysis here given is framed on perfectly general lines and covers any sort of interrelation, interdependence of the com- ponents Si, S . . . S n . Certain particular interrelations will be duly considered in the course of the further development of the theme. Here it may be well to draw attention once for all to the fact that there is nothing whatever to restrict the application of the principles and methods set forth to systems in organic evolu- tion. Indeed, a typical example to which these reflections apply is the case of a chain of elements in the course of radioactive trans- formation. 5 It will be seen later, in considering a concrete example, that the equilibrium equation (12) may yield zero or negative roots for C. A zero root is simply interpreted; if the equilibrium to which it relates is a stable one, this means that the species in question will 5 A. J. Lotka, Proc. Am. Acad. Sci., vol. 55, 1920, p. 148; Proc. Natl. Acad. Sci., vol. 7, 1921, p. 168; Phil. Mag., August, 1912, p. 353. FUNDAMENTAL EQUATIONS OF KINETICS 63 become extinct 1 '. It is unfit, is unadapted ultimately 7 to survive under the existing conditions 8 (as defined by the parameters P, etc.). A negative value of C may also signify that the species is incapable of survival under the existing conditions. Masses cannot assume negative values. As soon as any component passes through zero it ceases to function in the system, whose history is henceforth represented by a new set of equations in which this component does not appear. These conclusions must in one respect be accepted with a certain caution. Since there may be several equilibria, a species may be incapable of existence in the neighborhood of one such equilibrium, but might nevertheless succeed in maintaining itself in the neighbor- hood of another equilibrium. Whether such cases occur in prac- tise may be left an open question. Our analysis suggests this possibility. e This conclusion does not apply, of course, if the equilibrium with X = O is unstable, as in the example of a fly population cited above. 7 It maj*, however, persist for a time, and, it may be, for a long time. An instance in point is furnished, outside the field of organic evolution, by a chain of elements in radioactive transformation. Although here the ultimate equilibrium is one with a single survivor, namely the last link in the chain, yet for millions of years the several products exist side by side in constant ratio though in slowly diminishing amount. At the head of such a chain is always found a "parent substance" which is the longest-lived in the chain. This is not accident. It is easily shown that if at some prior period this substance was preceded by a more rapidly decaying pre-parent, this latter must have disappeared in the course of the ages. It is totally unfit, unadapted, even for a temporary equilibrium, under present conditions. (See A. J. Lotka, Phil. Mag., August, 1911, p. 354.) 8 It should hardly be necessary to point out that adaptation is purely relative. CHAPTER VII FUNDAMENTAL EQUATIONS OF KINETICS (CONTINUED)- SPECIAL CASE: SINGLE DEPENDENT VARIABLE If arithmetic, mensuration and weighing be taken away from any art, that which remains will not be much Plato. Law of Population Growth. It will add concreteness to the pres- ent exposition to consider at this point a numerical illustration. The simplest possible example of numerical application of the equations set forth in the preceding pages will be one in which there is only a single variable X. The fundamental system of equations then reduces to a single equation f=F(JT) (1) at A. case in point arises when for any reason one particular biological species or group grows actively, while conditions otherwise remain substantial!}" constant. This seems to be essentially what has occurred in certain human populations. It is true that, in their growth, they have carried along with them a complicated industrial system or group, comprising both living and non-living elements. We may, however, look upon the number of the human population itself as a sort of single index or measure of the growth of the group as a whole. 1 1 This is an example in which certain of the variables X are connected by equations of constraint. So, for example, the number of head of cattle Nc has for many years past, in the United States, been about six tenths of that of the human population, Nh so that we have an equation of constraint or, putting the average mass of cattle at 1000 pounds per head, that of a human being at 100 pounds (an average to include all ages) X e = Similar equations of constraint apply to the other species of domesticated animals and plants, so that the mass of each can be (approximately) expressed / in terms of the single variable Xh. 64 FUNDAMENTAL EQUATIONS OF KINETICS 65 Applying to this case the equation (1) and the general method set forth in the preceding pages, we are led to consider first of all the equi- librium equation ^ - F OT) = (2) at This equation, obviously, has a root at X = 0, for at least one female is required to start the growth of a population. Expanding by Taylor's theorem we shall therefore have for F a series lacking the absolute term, 2 thus = F (X) = aX + 6X> + cX 3 + . . . (3) dt Furthermore, the equation (2) will have at least one other root, since there must be some upper limit to the growth of the population. The simplest case satisfying this condition is that in which the right hand member of (3) terminates at the second degree term, i.e., ^f = oZ + &Z (4) at The characteristic equation 3 for X is in this case simply A - a = (5) and the solution of (4) therefore is X = (?*' + Sue 20 ' Substituting this in (4) and equating coefficients of homologous terms we find Gn =-<?! (7) a <?m = -!<?'x (8) a 2 so that (6) is a simple geometric series. Its sum is Z- f - (9) o a 1 -- f>at - p at _ 1 dX 2 Otherwise would not vanish, with X dt 3 Corresponding to equation (9), Chapter VI. 4 The series (6) is divergent for large values of t if a is positive. But the expression (9) for the sum of (6) remains a solution of (4). 66 ELEMENTS OF PHYSICAL BIOLOGY In formula (9) either Gi or the origin of time is arbitrary. A sim- plification can be effected by so adjusting the origin of time that (10) This fixes the value of the constant Gi (?:=-- (11) and the formula (9) becomes The equation (9) can also be written in another form. If we denote by X the value of X when t = o, and write x = X + 7, then (9) becomes a X = - - - (13) "'- 1 Population of United States. Formula (12) has been applied by Pearl and Reed 6 to the population growth of the United States. 7 The calculated curve for the number TV of the population fits the ob- served data over a long period of years (1790 to 1910) with remark- 5 This result can also be obtained by direct integration of (4) in finite form. The process given above has here been followed to illustrate the general method of the solution (S), (9), of Chapter VI. 6 P. F. Verhuist, Mem. Acad. Roy. Bruxelles, 1844, vol. 18, p. 1; 1846, vol. 20, p. 1; R. Pearl and L. J. Reed, Proc. Natl. Acad. ScL, vol. 6, 1920, p. 275; Scientific Monthly, 1921, p. 194; R. Pearl, The Biology of Death, 1922, p. 250. The last-mentioned work, especially, should be consulted for a detailed dis- cussion. For a general treatment of the population problem see also particu- larity E. M. East, Mankind at the Crossroads, 1923 (Scribners). 7 Measured, in this case, by the increase in the number N of persons. This Is evidently, in first approximation at any rate, proportional to the total mass X of the population. FUNDAMENTAL EQUATIONS OF KINETICS 67 able faithfulness, as will be seen from table 2 and the graph shown in figure 4. Numerically the formula (12) here takes the form N = 197,273,000 1+e- ,-0. 03134 1' (14) and the time t' (in years) is dated from April 1, 1914 (t 1 , being negative for dates anterior to this). This epoch is one of peculiar interest. It represents the turning point when the population passed from a progressively increasing to a progressively diminishing rate of growth. Incidentally it is interesting to note that if the population of the TABLE 2 Results of fitting United States population data 1790 to 1910 by equation (14) YEAH OBSBBVBD POPULATION CALCULATED POPU- LATION BY EQUATION (14) BBBOK 1790 3,929,000 3,929,000 1800 5,308,000 5,336,000 +28,000 1810 7,240,000 7,228,000 -12,000 1820 9,638,000 9,757,000 +119,000 1830 12,866,000 13,109,000 +243,000 1840 17,069,000 17,506,000 +437,000 1850 23,192,000 23,192,000 1860 31,443,000 30,412,000 -1,031,000 1870 38,558,000 39,372,000 +814,000 1880 50,156,000 50,177,000 +21,000 1890 62,948,000 62,769,000 -179,000 1900 75,995,000 76,870,000 +875,000 1910 91,972,000 91,972,000 United States continues to follow this growth curve in future years, it will reach a maximum of some 197 million souls, about double its present population, by the year 2060 or so. Such a forecast as this, based on a rather heroic extrapolation, and made in ignorance of the physical factors that impose the limit, must, of course, be accepted with reserve. Stability of Equilibrium. The equilibrium at X = 0, i.e., with total absence of human population, is evidently unstable, since A = a and a is an essentially positive quantity, its numerical value being, for the population of the United States, 0.0313395. The second equilibrium, corresponding to the saturation point, is evidently at 88 ELEMENTS OF PHYSICAL BIOLOGY X = r , and here it is easily found by the substitution x = X + that X = a, the equilibrium is stable. FIG. 4. THE LAW OF POPULATION GROWTH FOR THE UNITED STATES ACCORDING TO PEARL AND REED The lower S-shaped limb corresponds to the actual approach to equilib- rium from below. The upper limb represents the presumptive course of events for a diminishing population approaching equilibrium from above. FUNDAMENTAL EQUATIONS OF KINETICS Experimental Populations. Pearl has also fitted the same formula to the population of a number of countries, but the range covered by the available observation in these other cases is less extended, so that there is less opportunity for comparison with observed figures. Of particular interest is an application of the same formula, also by Pearl, to an experimental population of fruit flies (Drosophila). In this case practically the entire range of the S-shaped curve defined by equation (12) is realized, and a glance at the plot in figure 5 shows that the agreement of the observed figures (represented by small circles) and the calculated curve is exceedingly satisfactory. Still closer is the agreement in the case of bacterial cultures studied aso ss-s 125 JOO GftOWH Of DKOSOPHILA z Days. FIG. 5. GROWTH OF A POPULATION OF DROSOPHILA (FRUIT FLIES) TJNDEB CONTROLLED EXPERIMENTAL CONDITIONS, ACCORDING TO PEARL AND PARKER by H. G. Thornton 8 (Annals of Appl. Biology, 1922, p. 265), whose observations are set forth in table 3 and are shown by the small circles in figure 6 ; the theoretical curve to fit these points, as com- puted here in the laboratory, is shown in the fully drawn line. As will be seen the agreement is excellent. This is due in part to the fact that the figures plotted represent the means of a number of individual cultures. A veiy particular interest attaches to this example, inasmuch as it forms, as it were, a connecting link between the law of growth of a 8 Similar results have been obtained by A. G. McKendrick and M. Kesava Pai, Proc. Roy. Soc. Edin., vol. 31, 1911; for a study of the growth of yeast, see A. Slator, Trans. Chem. Soc., 1921, vol. 119, p. 126. 70 ELEMENTS OF PHYSICAL population, and the law of growth of the individual. A colony of unicellular organisms, regarded as a whole, is analogous to the body of a multicellular organism . Or, to put the matter the other way about, a man, for example, may be regarded as a population of cells. We need not, therefore, be greatly surprised, if the growth of the multi- cellular organism should be found to follow a law similar to that exhibited by populations. And in point of fact, as will be shown a little further on, this expectation is in not a few instances fulfilled. TABLE 3 ifrowth of bacterial colony, according to H. G. Thornton (Ann. Appl, Biol., p. 865) For graph see figure 6 AGE OF COLONY ABBA IN SQTTABH CENTIMBTEBB Observed Calculated* days 0.0 0.24 0.2511 0.5 0.7208 1.0 2.78 2.0324 1.5 5.4620 2.0 13.53 13.0761 2.5 25.1961 3.0 36.30 37.0479 3.5 44.2213 4.0 47.50 47.3930 4.5 48.5974 5.0 49.40 49.0231 *According to the equation y = 0.2524 e +0.005125 Diminishing Population. It may be noted here that in one respect the formula (12), while particularly simple in form, is of more restricted scope than (13). The former defines the characteristic S- shaped curve that appears in the graphs of actual populations shown in figure 4. But formula (13) gives a more complete definition of a curve composed of two limbs; one of these is the S-shaped curve already considered. The other is a steeply descending arc, shown in the upper portion of figure 4. This portion of the curve has not been 4 FUNDAMENTAL EQUATIONS OF KINETICS 71 realized in any recorded population. It represents the computed course of events if the population initially exceeds its equilibrium strength, i.e., if is positive. so 43 Pays. FIG. 6. GEOWTH OF A BACTEBIAL COLONY (B. DENDEOIDES) Observations by H. G. Thornton Growth of Individual Organism. Although, strictly speaking, the growth of the individual organism is a subject properly belonging to the field that has here been termed "micromechanics" (Chapter V), yet, in view of the very close analogy which has been found to exist, in certain cases, between the law of growth of a population, and that of the individual, it may be noted here that the formula (9) has also 72 ELEMENTS OF PHYSICAL BIOLOGY been applied by T. B, Robertson 9 and by Wo. Ostwald 10 and others, to the growth of the individual organism. An example of such appli- cation Is shown in figure 1, which exhibits the growth day by day, of male white rats according to observations by H. H. Donaldson" and computations by T. B . Robertson. The fully drawn curve repre- sents the calculated values of the mass of the rats (in grams) at dif- ferent ages. The circles indicate selected observed values, namely, those which diverge most widely from the computed values. It will be seen that up to about the one hundredth day the agreement is good. Above this there is no agreement worthy of the name. Another example,, and one in which the computed curve fits the observed values with remarkable agreement, is the growth (in height) of sunflower plants as studied by H. S. Reed and R. H. Holland. 12 The curve to fit these observations has been recomputed by the method of least squares by Dr. L. J. Reed, who has very kindly placed his results at the author's disposal. They are shown in figure 8. The observations on which they are based are shown in table 4. It will be seen that the fit is practically perfect through the whole range of observations. In practice we are not usually given the differential equation (4), but data corresponding to points on the integral curve (12). We then have the problem of determining from these points the charac- teristic constants of the curve. The detailed working out of theprob- 9 Archiv liir die Entwiekelungsmeehanik der Organismen, 1907, vol. 25, p. 4; 190S, vol. 26, p. 108; The Chemical Basis of Growth and Senescence, publ. Lippincott, 1923. T. B. Robertson also quotes A. Monnier, Publications of Irist. of Botany, Univ. Geneva, 1905, which in turn refers to Chodat as hav- ing recognized the analogy of organic to autocatalytic growth. The idea is rather an obvious one, which probably has occurred to many. The earliest reference noted by the writer is L. Errera, Revue de 1'Univeriste de Bruxelles, 1899-1900, May issue. Something very similar is found in Ostwald, Vorlesungen Ober Naturphilosophie, 1902, p. 342. These last-mentioned lectures were published in 1902. but were actually delivered somewhat prior to that date. The writer recalls that Ostwald referred to the matter in his lectures on Gen- eral Chemistry in 1902, and probably others will recall similar references on earlier occasions. 10 Die Zeitlichen Eigenschaften der Entwickelungsvorgange, Leipzig, 1908. 11 Boas Memorial Volume, 1906, p. 5; see also the same author's book The Rat, 1915. 12 H. S. Reed and R. H. Holland, Proc. Natl. Acad. Sci., vol. 5, 1919, pp. 135-144. FUNDAMENTAL EQUATIONS OP KINETICS 73 lem may well be left to the reader, after pointing out the following interesting property of the curve (12). Taking reciprocals and writing A for a/b, we have _ _ 3 a _ /o) A" log (A% -1) = - o(f-* ) (15) (16) oo O SO 100 (SO ZOO ZSQ 3CO Ays. in days. FIG. 7. GROWTH OF RAT ACCORDING TO H. H. DONALDSON AND T. B. ROBERTSON Circles indicate only those observations that diverge most widely from the calculated curve. where t is the time corresponding to the point of inflection of the S- curve. Thus if we plot A 1 against t on logarithmic curve paper, we shall obtain a straight line diagram. This is shown in figure 9 for the same data (growth of sunflower) which have already been 74 ELEMENTS OP PHYSICAL BIOLOGY 00 y- Z6/.I O 7 H 2/ 28 35 -?2 +9 ?& 63 TO 77" &<f Age in days. FIG. 8. GHOWTH OP SUNFLOWER SEEDLINGS ACCORDING TO H. S. REED AND R. H. HOLLAND; COMPUTED CURVE BY L. J. REED TABLE Growth in height of sunflower plants (H. S. Reed and R. H. Holland;* computed values by L. J". Reed) MEAN HEIGHT Observed Computed cm. cm. 7 17.93 21.89 14 36.36 37.74 21 67.76 62.08 28 98.10 95.42 35 131.00 134.56 42 169.50 172.99 49 205.50 204.64 56 228.30 227.16 63 247.10 241.56 70 250.50 250.14 77 253.80 255.05 84 254.50 257.79 Troc. Natl. Acad. Sci., vol. 5, 1919, p. 140. 10.0 7 63 7O 77 84- SI 38 Age. m day-s, FIG. 9. GEOWTH OF STJNFLOWEB SEEDLINGS The same data as in figure 8, but plotted in logarithmic diagram, with ordinates as indicated in the legend. 75 exhibited by another method in figure 8 . Taking now three equidis- tant points of time ^ t is and the corresponding ordinates &, 2, &, it is readily shown that ta-ft ^gLlLg; (17) ' (t + &)-2& TM-& where 3/ denotes the geometric mean of 1 and | 3 and while m denotes their arithmetic mean. Thus the constant A can be determined from three suitably chosen points on the curve (after smoothing graphically if need be). The other constants then follow easily. Autocatakinesis. Both Robertson and Wo. Ostwald .draw atten- tion to the similarity between the law of growth (12) and the law of formation of a chemical substance by autocatalysis under certain con- ditions. The analogy is interesting, but must not be taken too seriously, inasmuch as in the one case the rate of growth is deter- mined by ordinary chemical influence, in the other (organic growth) by a complicated combination of factors both of chemical and of physical character. Rather more to the point seems to be the sug- gestion made above in connection with the law of growth of bacte- rial colonies, that, the body of a multicellular organism being a "popu- lation" of cells, it is not altogether surprising that it should be found to follow the law of growth of a population. A suggestion of terminology by Wo. Ostwald is worth noting. He proposes that growth of any kind, in which the substance or structure itself acts as nucleus for the formation about it of further quantities of the same substance or structure, be broadly termed autocata- kinetic growth, the narrower terms autocatalysis of autocatalytic growth being reserved for that particular kind of autocatakinesis which is chemical in character. This is a useful suggestion, as it leaves the term autocatakinesis quite neutral and free from any impli- cation or restrictions as to the mechanism by which the growth takes place. CHAPTER VIII FUNDAMENTAL EQUATIONS OF KINETICS (CONTINUED) SPECIAL CASES: TWO AND THREE DEPENDENT VARIABLES Life is a system of relations rather than a positive and independent ex- istence G. A. Sola. Interdependence of Species. The case of two dependent vari- ables, X\, Xz, which we now approach, is of interest as the simplest example exhibiting the relation between interdependent species. This relation can take on a variety of forms. Most fundamental, perhaps, is that form of interdependence (1) in which one species Si serves as food to another species S Z) so that, in this sense, Si becomes transformed into S 2} thus Si > $2 All animal species, and many plants also, thus derive their sub- stance from other species on which they feed. And several dif- ferent types of this form of interdependence are observed. In the first type, (a) the organism $ 2 kills Si outright in the process of feeding upon it. We might term S 2 in such a case an episite of Si, in contradistinction from the second type, (&) in which 2 lives on Si without killing it outright, being parasitic upon Si. The host is in most cases more or less injured by the parasite, and all pathogenic organisms fall into this class. For this reason quantita- tive epidemiology appears as one of the special branches of the general subject under consideration here. A third type (c) of interdependence, is that in which S 2 is sapro- phagous or saprophytic, feeding upon the cadavers of Si after death from other causes; or, /S 2 may live on waste products discharged by Si. In contrast to episites and parasites, saprophagous species are presumably beneficial rather than injurious to the host species, since they function as scavengers. Still another type, (d), of inter- dependence is that of symbiosis, in which Si and *S 2 live in partnership which, as a rule, is in some degree mutually beneficial. Man and his domesticated animals and plants are obvious examples of this type. 77 7S ELEMENTS OF PHYSICAL BIOLOGY In addition to these types (Id) to (Id), another large group of cases (2) are those in which two or more species compete for a common food supply. In their general form the fundamental equations of kinetics for the case of two dependent variables are dt dl -j- AuXi -r AitXz + . + AtiXi -r AttKi T . (1) or, after the transformation (5), of Chapter VI = anXi -f- at dx s ~dt = CsjXj + QssZs 4- . . . (2) We may note, first of all, as a general rule, the following observa- tions regarding the coefficients a in the several types of interdepen- dence enumerated above: la. S 2 lives on Si by killing Si outright (episitic type). In this case, evidently S 2 unfavorably influences the growth of Si } while, on the contrary, Si is advantageous to the growth of &, so that 5 dZi Z 2 dt < 0, i.e., an < t ~ > O, i.e., an > oXi at (3) (4) lb. 82 parasitic upon Si. Here a iz < 0, 021 > 0, as in case la. Ic. S 2 saprophytic upon Si, or living upon waste products of Si. Here we may expect that a iz ^ 0, a 2 i > 0. Id. S 2 feeds on Si, but at the same time cultivates it in symbiosis. Here a iz > 0, a 2 i > 0. The characteristic equation for two variables is X X = (5) FUNDAMENTAL EQUATIONS OF KINETICS 79 or, In expanded form, X 2 (an + o^X + (andn 021^:2) = (6) X = I { (a u + a 22 ) =*= V(a u - a 22 ) 2 + 4a 2 iai 2 } (7 ) Certain general conclusions follow immediately. So, for example, if ai, ais are both of the same sign, as in the case of saprophytes and of symbiosis, the quantity under the radical is necessarily positive, and hence both roots for X are real. The oscillatory type of approach to equilibrium is here excluded. In the case of two species of the type (la) or (16), the episitic or parasitic type, on the contrary, the possibility of oscillations is indicated, under conditions where the equations (1) are applicable. CONCRETE EXAMPLES Martini's Equations for Immtinizing Diseases. Numerical appli- cations of the case of two dependent variables are not easily ob- tained. Of concrete examples in general terms (with unknown or very imperfectly known values of the constants involved) several are to be found in the literature. The simplest of these is a case for which the equations are given, without solution, by Martini in his Berechnungen und Beobachtungen zur Epidemiologie der Malaria (Gente, Hamburg, 1921, p. 70), namely, the case of the growth of an endemic disease of the type that confers acquired immunity upon persons who recover from it (e.g., measles, scarlet fever, etc.) Martini writes u = fraction of the population affected and infective i = fraction of the population not available for new infection (i.e., im- mune or already affected) (1 _ i ) = fraction of the population available for new infection p fraction of the population newly affected per unit of time q = fraction of the population of affected population that ceases to be so, per unit of time, by recovery or by death m = fraction of "unavailable" population that loses immunity or dies per unit of time a = infe.otivity (a proportionality factor) Martini puts the newly affected population, per unit of time, jointly proportional to the infective population u and to the population available for new infection, (1 i), so that p = a u(l t) 80 ELEMENTS OF PHYSICAL BIOLOGY Then, obviously, = a u(l - - qu = (a - q}u - a ui di I- (9) = a u(l i) mi = a u mi a ui di j The characteristic equation for X here reduces, near the origin, simply to l | (a _ q ] _ A ) (m - X) = (10) and the solution, near the origin, is (7, ^- mt 4- . . 1 (11) from which it is seen that the equilibrium near the origin is stable if and only if a < q There is a second equilibrium at in (a q) = I (12) Since, however, i can in reality assume only positive values, this equilibrium has a meaning only if a > q, i.e., just in the case in which the equilibrium at the origin is unstable. Hence we conclude that if a. < q the equilibrium at the origin is the only possible one, and is stable, so that the disease will die out. In the other alternative, a > q, the second equilibrium has a real meaning, and we can develop a solution in series - = ' Xlt ' X!t 1 (13) i - I = (?' Sll e Xj *+ (?' 2 , 2 e X2t + . . . where X 3 , X 2 are given by q- 1 A. J. Lotka, Nature, vol. Ill, 1923, p. 633. gi FUNDAMENTAL EQUATIONS OF KINETICS Oi We need not here consider the case (a - q) < 0, for, as pointed out above, in this case the second equilibrium is meaningless. But when a > q, it is seen from (14) that the real parts of the two roots will then in any case be negative, so that the equilibrium, if it exists at all, is stable. Furthermore, there will be two real and distinct, two real and coincident, or two complex roots, according as a- q 1 m ~> _ ft K) 2. -1 - \i"J a" a 4 q < In the last-mentioned case equilibrium will be approached by a series of oscillations above and below the final state of equilibrium, so that a series of "epidemic" waves will appear, a feature which has an obvious interest in connection with the type of disease here discussed. 2 The oscillations thus occasioned by the factors duly taken into account in this elementary analysis may in practise be enhanced by factors here neglected, such as varying virulence of the disease, exhaustion of susceptible population, seasonal influences, etc. The last-mentioned, however, will in general tend to produce a separate series of waves whose period will have no relation to that of the oscillations derived above. For the special case = 1, Prof. G. N. Watson has given a complete solution, which may be consulted in the original. 3 It is to be noted that this case cannot, according to the analysis here presented, lead to oscillations, since the condition for oscillations according to (15) reduces to ^" :o (16) and cannot be satisfied, as the square of a real quantity is necessarily positive. The Ross Malaria Equations. A system of equations has been established by Sir Ronald Ross 4 to represent, under certain 2 Compare J. Brownlee, Investigation into the Periodicity of Infectious Diseases, Public Health, vol. 25, 1915, p. 125. 3 G. N. Watson, Nature, vol. Ill, 1923, p. 808. 4 Sir Ronald Ross, The Prevention of Malaria, second edition, 1911, p. 679. This volume also contains a bibliography. For a detaile ddiscussion of the Ross malaria equations see A. J. Lotka, Am. Jour. Hygiene, January Supplement, 1923. 82 ELEMENTS OF PHYSICAL BIOLOGY conditions, the course of events in the spread of malaria in a human population by the bites of certain breeds of mosquitoes infected with the malaria parasite. These equations are of the same form as Martini's equations discussed above, and inasmuch as Ross's malaria equations have been very fully treated by the writer in a separate monograph, 5 their detailed study may here be omitted. It will suffice to reproduce from this monograph one of the curves representing the course of events, the presumptive growth of malaria in a human population, as defined by the differential equations of Sir Ronald Ross. -4 ~ t years FIG. 10. CUHVE OF GROWTH OF ENDEMIC MALARIA ACCORDING TO SIB RONALD Ross's EQUATIONS The upper (S-shaped) curve relates to the particular case in which the initial malaria rates in the human and the mosquito population stand in the ratio which they have at equilibrium or are both small. The lower curve represents the course of events when the initial malaria rate is 4.2 per cent in the human population, and 1.4 per cent in the mosquito population. (The zero of the time scale is arbitrary.) Ordinates are malaria rates (human) expressed as fraction of unity. (Reproduced from A. J. Lotka, Am. Jour, of Hygiene, January Supplement, 1923.) 5 A. J. Lotka, loc. cit. FUNDAMENTAL EQUATIONS OP KINETICS 83 It will be observed (fig. 10) that the curve showi consists of two parts, the one an Srshaped limb which, in point of fact, is very nearly identical in type with the Verhulst-Pearl population curve. The second limb ascends very steeply, almost vertically, and finally bends to join the S-shaped limb. The meaning of these curves is as follows: If a small nucleus of malarial infection is introduced into a (constant) population of human beings and mosquitoes, both being previously free from such infection, then the growth of malaria in the human population will follow the course represented by the S-shaped limb. It will be seen that the process is a rather leisurely one, its essential comple- tion occupying about ten years, according to Ross's figures. (Strictly speaking it is never quite complete in a finite time.) Seasonal effects are here disregarded. On the other hand, if at the start there is already present a certain malarial rate in the human population, and also in the mosquito population, then, in the case here depicted, there is for a time a very rapid increase of malaria in the human population, until, in the brief space of about two months, the S-shaped curve is reached. After that the course of events is the same as in the first case. In practise, in temperate climes, we can expect only short sections of the S-shaped curve to be realized, owing to the interruptions of the seasons. No data are available for a numerical comparison of these results with observed conditions. Close agreement is not to be expected, as the Ross equations refer to a rather highly idealised case, a constant population both of men and mosquitoes. The latter could be even distantly approached only in the tropics. There is room here for further analysis along more realistic lines. It must be admitted that this may lead to considerable mathematical difficulties. The case of periodic seasonal influences is perhaps the one that promises to yield most readily to mathematical treatment. The Ross malaria equations are a typical example of equations affected with a lag, owing to the period of incubation. For a detailed discussion of this feature the reader must be referred to the author's monograph published by the American Journal of Hygiene. An Example in Parasitology. An interesting and practically significant case of inter-group evolution, of conflict between two species, has been made the subject of an analytical study by W. R. 84 ELEMENTS OF PHYSICAL BIOLOGY Thompson. 6 He considers a host species^ numbering initially n in- dividuals, and a parasite species, initially p individuals. On a num- ber of simple assumptions, for which the reader must be referred to the original papers, he develops the following formula for the fraction a of the host population attacked, in the t ih generation, by parasites pa* a = (a* - a) n -p a 1 where a is the ratio of the "reproductive power" of the parasite to that of the host, the reproductive power being measured by the number of eggs deposited per female. It is assumed that only one egg is deposited in each host. Putting a 1 = e rt (18) and X (19) P P Thompson's formula becomes, after a simple transformation, -^r-i (20) which will be recognized, once more, as the equation of the law of simple autocatakinetic growth. According to the value of a and K several different cases may arise, whose graphs are shown in figure 11. It should be observed that K = 1 according as a= 1 (21) r = according as a= 1 (22) < > In the special case that a = 1 the formula (20) becomes indeterminate and a is then given by P n-p(t-l) a hyperbolic relation (see fig. 11, the last diagram). (23) W. R. Thompson, Comptes Rendus Acad. Sci., vol. 174, 1922, pp. 201, 1443, vol. 175, p. 65. See also R. A. Wardle and P. Buckle, The Principles of Insect Control, Longmans, Green & Co., 1924. FUNDAMENTAL EQUATIONS OF KINETICS 85 Thompson gives a number of numerical examples exhibited in table 5 . From these examples and from this formulae he concludes that the invasion of the host species by the parasite may at first Fia. 11. COTJESE OF PAKASITIC INVASION OF INSECT SPECIES ACCOBDINQ TO W. R, THOMPSON proceed only very slowly, and that nevertheless, after a certain time, the increase may become very rapid. From a practical stand- point this is important to observe, since it implies that the first effect of "sowing" the parasite among a species of insect hosts, ELEMENTS OF PHYSICAL BIOLOGY with a view to destroying them, may be quite discouraging, and that this must not be taken as indicative of ultimate failure. This TABLE 5 Percentage (IQOa) of host species attacked by parasite in t lh generation, according to W. R. Thompson p n a 1 2 3 4 5 6 7 8 9 10 10 1,000 1.5 1.5 2.28 3.50 5.40 8.64 14.2 24.6 50.0 100.0 10 10,000 1.5 0.15 0.23 0.33 0.51 0.77 1.16 1.76 2.7 4.15 6.5 10 100 1.0 10.0 11.1 12.5 14.35 16.6 20.0 25.0 33.3 50.0 100.0 10 200 1.0 5.0 5.25 5.55 5.88 6.25 6.66 7.14 7.7 8.3 9.1 p n a 11 12 13 14 15 16 17 18 19 20 10 10,000 1.5 9 6 18.6 34.2 78.0 10 200 1.0 10.0 11.1 12.5 14.3 16.6 20 25 33 50 100 p= initial number of parasite species. 7i=initial number of host species. parasite host a=ratio of reproductive powers The figures in the columns headed 1, 2, 3, etc., denote the values of 100 a in the 1st, 2d, 3d, etc., generation. 7i'm& in J>OO \ \ \ \ FIG. 12. INCREASING DIBTUSION-IN-TIME OF STJCCBSSIVH GENERATIONS IN THE PEOGENY OF A POPULATION ELEMENT observation is especially significant since in practise one must often be satisfied with the introduction of a relatively very small number of parasites. So, for example, in using the parasite species FUNDAMENTAL EQUATIONS OF KINETICS 87 Liparis di&par, commonly about one thousand individuals have been sown. Supposing that there were one thousand million hosts, and that the parasite reproduced twice as fast as the host, it would require, according to Thompson's calculations, about 19 generations to exterminate the host; and then, even to the sixteenth generation, only 10 per cent or less of the host species would be attacked, Thompson's formula is open to certain objections. Its derivation seems to involve the assumption that each generation of the parasite is coextensive in time with the corresponding generation of the host. Furthermore, the use of a generation as a sort of time unit is un- satisfactory, because a generation is a very diffuse thing, spread out over varying lengths of time. This is easily seen by considering the progeny of a batch of individuals all born at the same time t 0. If we call this the th generation, and if fli, ao are respectively the lower and the upper limit of the reproductive period, then it is clear that the next generation, the first, will extend over the interval of time from a L to a 2 , the second from 2a^ to 2a 2 , the n th from nai to na z , an interval which will ultimately become very large as n increases, as shown by successive intercepts between the two sloping lines in figure 12. Less objectionable, perhaps, is the fact that Thompson's formula is expressed in terms of the "rate of multiplication per generation" of the two species. This term is not as clear as it might be. From the context it appears that it refers to the number of eggs deposited per female. This number is closely and simply related to the ratio 12 of the total births in two successive generations. If an individual of age a reproduces, on an average /3 (a) individuals per unit of time, the ratio R is evidently given by f* = j Jo (a)p(a)do The relation of this R to the rate of increase r per head of the popula- tion is not altogether obvious and cannot be expressed in simple form. For a population with fixed age distribution, it will be shown in Chapter IX that r is given by = e - r a p( a )p( a )fa |"SO / = I @(a)p(d)da r I a^(a} Jo Jo + (25) 88 ELEMENTS OF PHYSICAL BIOLOGY Hence Jra aj3(a)p(a)da + . . . (26) In view of the doubtful features in Thompson's formula which have been indicated above, it appears desirable to attack his problem in quantitative parasitology from another angle. We may do this by following the general method which has here been set forth and exemplified. Treatment of the Problem by the Method of Kinetics. Let NI be the number of the host population, &i its birthrate per head, (the deposition of an egg being counted a birth), and di its death rate per head from causes other than invasion by the parasite. Let kNiN* be the death rate per head due to invasion by the parasite, in the host population, the coefficient k being, in general, a function of both jVi and JV 2 , the latter symbol designating the number of the parasite population. The birth of a parasite is contingent upon the laying of an egg in a host, and the ultimate killing of the host thereby. To simplify matters we will consider the case in which only one egg is hatched from any invaded host. If an egg is hatched from every host killed by the invasion, then the total birthrate in the parasite popula- tion is evidently kNiN*. If only a fraction k' of the eggs hatch, then the total birthrate in the parasite population is evidently kk' NiN, which we will denote briefly by KNiN 2 . Lastly, let the deathrate per head among the parasites be d z . Then we have, evidently dt (27) dt ~ i 2 - - - where r x has been written for (b { d t ). Regarding the function k, we shall now make the very broad assumption that it can be expanded as power series in NI and N z , thus k = a. + &Ni + yN-t + . . . (28) FUNDAMENTAL EQUATIONS OP KINETICS It will be convenient first of all to consider an approximation. 7 If the coefficients 3, 7, etc., are sufficiently small, we shall have, for values of A'i, X* not too large, essentially at (29) (30) Integrating, and putting k'a TI (^21 a. we obtain d 2 Iog(x + p) + n \og(y + q) - h'ax - ay = M (33) where M is an arbitrary constant of integration. Expanding by Taylor's theorem we find M' (34) By giving successively different values to the arbitrary constant M r a family of closed curves is obtained for the plot of (34) in rec- tangular coordinates, as indicated in figure 13. In the neighborhood of the origin, where terms of higher than second degree are negligible, (34) reduces simply to (- -^ = constant (35) P~ <f and the integral curves are small ellipses. 7 It will be observed that the "diagonal" terms in the linear part of (27), (29) are lacking, i.e., there is no linear term containing 2V 2 in the first equation of (27) and none containing NI in the second. This gives rise to an exceptional case to which the general solution given in Chapter VI is not applicable. 90 ELEMENTS OF PHYSICAL BIOLOGY The course of events represented by these curves is evidently a cyclic or periodic process, corresponding to a circulation around the closed curves. The period of oscillation, near the origin, 8 is given by T = 27r/Vr 1 d 2 (36) FIG. 13. COURSE OF PARASITIC INVASION op INSECT SPECIES, ACCOEDINQ TO LOTKA; ELEMENTARY TREATMENT This finding accords well with the observation made by L. 0. Howard : With all very Injurious lepidopterous larvae .... we constantly see a great fluctuation in numbers, the parasite rapidly increasing immediately after the increase of the host species, overtaking it numerically, and. reducing it to the bottom of another ascending period of development. 8 The purely periodic solutions have been discussed by the author in Proe. Natl. Acad. Sci., 1920, vol. 7, p. 410. The writer, however, at that time over- looked the existence also of the other types of solution, and also stated that the period of oscillation is independent of initial conditions. This is an error which he takes the present opportunity to correct. The expression given by him loc. cit. for the period of oscillation holds only in the neighborhood of x = y = 0. See also note 10 below. FU^TDAMEXTAL EQUATIONS OF KINETICS 91 The remarks of W. R. Thompson relative to this may also be quoted : Recent studies oa the utilization of entomophagous parasites seem to show that the role of these auxiliaries of man finds its maximum effective- n^== when the noxious insect has increased in numbers to the point of a ula-me, one or more of the factors of natural equilibrium having somehow failed ' to act as a check. The expansion of the noxious species then automatically produces an increase in the number of parasites; generation FIG. 14. COURSE OF PABASITIC INVASION OP INSECT SPECIES, ACCORDING TO LOTKA; MORE EXACT TREATMENT after generation this number increases at the expense of the host, until it first equals, and presently surpasses the number of the host species, and finally almost annihilates the host; but then comes a moment when the parasite pop- ulation, having grown to excess, largely disappears for the lack of food supply. 9 9 L. O. Howard, Bureau of Entomology, Technical Series No. 5, 1897, p. 48. It will be noted that the analysis given by W. R. Thompson fails to give any indication of this oscillatory process. 92 ELEMENTS OF PHYSICAL BIOLOGY In this elementary discussion the terms of second and higher degree in A r i and JV 2 in equations (27) have been neglected, as in (29). When they are taken into account it is found that, in general, the system of closed curves in figure 13 is replaced by a spiral winding about the second equilibrium point (see fig, 14). The process is still oscillatory, but, in general, it is the nature of a damped oscillation. For detail of this more exact treatment the reader must be referred to the original literature. 10 Annihilation of One Species by Another. The preceding example was suggested by W. H. Thompson's paper in the Comptes Rendus. Another case, leading to equations of the same form, had been previously treated by the writer, namely the following : A species S* feeds on a species Si, which, in turn feeds on some source presented in such large excess that the mass of this source may be considered constant during the period of time under consid- eration. Then we have the obvious relation (37) The first term in the right hand member, Mass of newly formed Si per unit of time, evidently must vanish with Xi } and we shall therefore write this term biXi, where 61 will, in general, be a func- tion of both Xi and X 2 . Similarly, and for the same reasons we shall write for the third term, excretory matter, etc., eliminated from Si per unit of time, diXi. Contracting these two terms into one we shall write for biXi diXi the single term riXi. The middle term, mass or Si destroyed per unit of time by 82, must evidently vanish with either Xi or X 2 separately. We shall ac- cordingly write this term kXiX 2 , so that the equation now appears Other dead Rate of in- crease of Xi per unit of time (Mass of newly 1 = \ formed Si per > [unit of time J Mass of Si destroyed by 2 per unit of time ,_, or excretory matter elimi- nated from Si per unit of time at (38) 10 H. Poincare, Journal de Mathem., 4th ser., vol. 1, p. 172. (See also E. Picard, Traite d' Analyse, vol. Ill, Chapter IX, p. 214); A. J. Lotka, Jour. Washington Acad. Sei., 1923, vol. 13, p. 152. FUNDAMENTAL EQUATIONS OF KINETICS 93 (39) For the species S which feeds on Si we have a corresponding relation 'Mass of newly] formed 2 per Mass of S-. I unit of time I destroyed or I (derived from] eliminated per S\ as food in- unit of time .gested) j which, for reasons precisely analogous to those set forth above with reference to equation (38), we shall write at (40) It will be seen that these two equations are identical in form with those discussed in the preceding example, and it is therefore un- necessary to repeat the analysis there given. But one point invites attention, and will naturally lead us on presently to the consideration of a case in three dependent variables, which offers certain features of special interest. Let us leave aside all other considerations, and restrict our enquiry here to the following question : Is it possible, under the circumstances represented by our equations, for the hostile species $ 2 to exter- minate completely the species Si upon which it feeds? This would, of course bring in its train also the extermination of S z . To answer this question we note that Xi (n- X, dX, is satisfied by the particular solution i = for all values of (41) (42) Hence if we plot the integral curves of (41) in rectangular coordinates, the axis of X% is itself one of the integral curves. Now if r x , k, K, and c? 2 are single-valued functions of X i} and X z , as it is reasonable to suppose that they are, then any point in the XiX 2 plane is traversed by only one integral curve, since the derivative -=? is uniquely ClX 2 determined by (41). It follows that no integral curve can cross the axis of X z and therefore Xi can never fall to the value zero, 94 ELEMENTS OF PHYSICAL BIOLOGY it can be zero only if it had that value from the beginning. The food species cannot, therefore be exterminated by the predatory species, under the conditions to which our equations refer. This argument fails, however, at the origin, where the derivative dXi/dXz assumes the indeterminate form 0/0. But in the neigh- borhood of the origin, where second degree terms are negligible, we have, essentially, dXi Xi TI dXz Xy dz from which it is seen that in the positive quadrant the integral curves always slope downward from left to right near the origin. They cannot, therefore, in the positive quadrant, cross the axis of Xz at the origin, any more than at any other point, and the con- clusion is now fully established, that the species Sz cannot exterminate Si under the conditions here considered. A word of caution, however, is perhaps in order. Although 8 2 cannot exterminate Si, it may so reduce the latter in numbers as to render it very vulnerable, and liable to extinction from those other influences which have deliberately been ignored in the development of the equations. Case of Three Dependable Variables. A singularly interesting conclusion is reached when we enquire what may be the effect of introducing, into the system discussed in the preceding section, a third species, which also serves as food for Sz, so that this now has two sources to draw upon, after the pattern One would perhaps naturally suppose that this alternative and additional source would in some measure protect Si from the attacks of Si. But we shall see presently that this is not necessarily the case, that, in fact, in certain circumstances, the introduction of the third species 83 may bring about the extermination of Si, from which, as we have seen, Si is naturally protected by the diminution of growth forced upon $2 when this species too greedily consumes its sole source of food. FUNDAMENTAL EQUATIONS OF KINETICS 95 The equations for this case are similar to those for the preceding, except that we must add a third equation for the species $3 ~ = r,X s - hXzX* (44) at and add a term to the equation for S 2 to represent the consumption of the species 83 as food by S 2 , so that this equation becomes dX z : JK.X i.X.2 -j- at + HX 3 - d 2 ) (45) We now have, near the axis of X s> _ . fAC\ J-tr V TJ"V~ ^ V*Q) and, if Xz is sufficiently large, the projection of the integral curves upon the XiX plane will slope upward from left to right in the positive quadrant. Such an integral curve may therefore cut through the X%X 3 plane, that is to say, the species Si may be reduced to zero. Similarly, near the axis of Xi _J = ! r J. (47) j~Y V K'Tf J ^ tt-A. 2 "- 2 ^Vwti. I ~~" ti2 and hence an integral curve may cut through the XsXi, plane, thus reducing X 3 to zero. This observation is of practical interest. It has been pointed out that in sea fisheries the accompanying presence of a common fish may cause the extermination of a rarer species which, were it present alone, would be protected by its very scarcity, since this would make fishing unprofitable. But the more abundant fish continues to render a balance of profit from the trawling operations, and thus the rarer species, so long as any of it remains, is gathered in with the same net that is cast primarily for common species. Replaceable and Irreplaceable or Indispensable Components. The last two examples present an illustration of a point which calls for brief comment. So long as the species Sz has only one source jy of food, S i} it is to be observed that becomes negative as soon dt 98 ELEMENTS OF PHYSICAL BIOLOGY as Xi is zero (see equation 40). In this sense Si is an essential component of the system, relatively to S, i.e., it is indispensable for the growth and even the mere continued existence of S. On the other hand, when two or more sources, such as Si and S s are provided, the vanishing of either Xi or X s singly does not bring j~y~ with it a negative value of . The components Si and $3 can more or less effectively replace, act as substitutes for, each other. When the feeding species ($ 2 in the example) is the human species, the facts indicated above find their expression in economic terms. It is an elementary fact of common knowledge that among the varied materials which the human race requires for its growth and sustenance are many that are more or less readily interchangeable. So, for example, a deficiency in the wheat crop may be in some degree met by increased supply of potatoes or other starchy food; or a deficiency in beef may be compensated by increased production in pork. On the other hand, there are certain requisites that are irreplaceable, and therefore absolutely essential. The most obvious example of such is our supply of oxygen. In terms of our general analysis these facts would be expressed somewhat as follows. If we survey the various components whose masses Xi,Xz . . . X n appear in the function FI = Fi (Xi, Xa, . . . -X" n ) at we may effect a classification by first of all dividing them into two classes, namely those that adversely affect the rate of growth of Xi, that is to say, those for which JL^< O dZj dt and those that promote the rate of growth of Xi, those for which dt Among the latter components, those favorable to the growth of X{, there is a special class distinguished by the following property : If X is the mass of a component of this special class, then dXi . ~j7 is invariably negative as soon as X^ is zero, no matter what FUNDAMENTAL EQUATIONS OP KINETICS ~ 97 may be the masses Xi, X , . . of the other components. Xk is indispensable for the growth and the sustenance of Xi] X& is an essential of irreplaceable component, relatively to Xi. The ground upon which we are here treading is evidently close to the biological basis of economics. A detailed analysis of -the relations involved belongs to the domain of the dynamics of life- bearing systems, and will in due course be considered in its natural place. Limiting Factors. In general the several components that promote the growth of the component of Si will be presented in varied abun- dance. If one essential component (or a group of components which jointly are essential) is presented in limited amount, any moderate increase or decrease in the ample supply of the other components will have little or no observable influence upon the rate" of growth Fi of Si. An essential component presented in limited supply thus acts as a check or brake, as a limiting factor, upon the growth of Si. The significance of such limiting factors seems to have been first pointed out by J. Liebig: 11 "Der Ertrag (des Bodens) ist von dem im minima in ihm enthaltenen Nahrstoff abhangig." And again : Fur die Wiederherstellung der Ertrage der durch die Cultur erschopften Felder durch Stallmistdiinguug ist die Zufuhr von alien den Nahrstoffen welche das Feld im tJberschuss enthalt, vollkommen gleichgiiltig, und es wirken nur diejenigen Bestandteile desselben gtinstig, durch. welche ein im Boden enstandener Mangel an einem oder zwei Nahrstoffen beseitigt wird. Limiting factors not only set certain bounds to the growth of the components to which they are thus related, but are competent also to give rise to the phenomenon of "moving equilibrium" the discussion of which is reserved for a later section dealing with equilibria generally, under the heading of Statics. This chapter may fittingly be concluded with table 6, which exhibits the principal modes of interdependence of biological species, and summarizes the analytical characters of these, as discussed above. 11 J. Liebig, Die Chemie in ihrer Anwendung auf Agricultur, 1876, p. 334. See also ibid., pp. 332, 333, 381, 382. i-l o O *:? ^5 5 **< o O "^ S _ o .2 N * S 05 " i "-< 'S i "o S* C! JS "o 5 S S "s - .E ^ 02 S ft '5 T3 ^ g 1 ^ ~ -Sfl ^ i < i-< S O *. ft ? OB r 5 4 S3 "2 S"- I ~C i ^3 a "P ', T - > .2 o bp S - d ' ^ "a '- 53 "g >* '43 y > ^ fc 'a ^ 3 1 m 1 fcb ? *? O O Q! S s - i - .S ic "x ^ 'S H s ? -p S -^ H C S2 ,? - Q -C T3 -J g ft _ -i-i .9 .2 T': .2 S v= ^3 > '42 Ml o a T3 g fcO - " HI as 'S -*S 1 S> & &o CQ - L.2 ^ ' W cu 6 fl ^ ^3 tb ' ft CO fl 5,'x 53 r-s i o , 03 O L.g ^ a L-i c3 Pn a .S o 3 o 8, ~f- H~ "a 3 +3 S a <* '% m % 1 d T) _i_ | I a & O u K Q [ !z z3 3 ~3 cj a ^ "a 2 o o a| 1 1 B a Ji o e i e s b u t ,u 5.1 1 S a o '3 *v o ition w lestruct S | S "$> "o 4 O |I 2 s o| 2 g 3 _g 1 1 1 i a "ft a s i-< ei /o M /o M /o /o II II i-t C5 M r-4 99 CHAPTER IX ANALYSIS OF THE GROWTH FUNCTION Elegant intellects which despise the theory of quantity are but half devel- oped. A. N. Whitehead. The Form of the Growth Function F. The fundamental equations (1) of Chapter VI express in a very general, and for that reason somewhat colorless way, the interdependence of the several compo- nents of evolving systems of the kind here under discussion. * In the special cases with one, two and three dependent varial^teg that have been presented as examples, the particular form affif the ^&^ concrete meaning of the functions F appearing in these equations -* has been illustrated for these specific instances, without any attempt at systematization from a general standpoint. It is desirable now to make a somewhat detailed analysis of the functions F in their more general aspect. A natural step to take is, first of all, to split up the function F into a positive and a negative term; that is to say, to express the rate of increase of the mass Xi of the component & as the balance, the surplus, of the mass Ui added to that component per unit of time, over the mass Vi eliminated therefrom per unit of time. Thus dXi -~r t .i\-r t (i) Growth of Aggregates. Among the components of systems of the kind in which we are here mainly interested, a particularly important type are those built up of a large number of essentially similar units or individuals. Such are the aggregates of molecules that constitute the components (chemical elements and compounds) of the systems with which physical chemistry is concerned; such, also, are the aggregates of individual organisms that constitute the biological species, the component population groups, of which are built up the systems in which organic evolution takes its course. In the case of an aggregate of this kind, if N is the number of individuals, and m their average mass per head, we have 100 ANALYSIS OF GROWTH FUNCTION 101 Xi = miNi (2) dXi dNi dnii . . = mi + N l - (3) If the average mass per individual, nii, is constant, the second term of the right hand member in (8) drops out, and we have simply, This holds strictly for aggregates of similar molecules, for example, whose masses are all equal and constant. It will often hold with close approximation (as will be set forth in greater detail shortly) for populations of living organisms of one species. dNi Now -J7 1 the rate of increase in numbers of the aggregate Si, naturally appears, after the manner indicated above, as the balance of the number of newly formed individuals Bi per unit of time, and the number Di eliminated per unit of time. When these symbols refer to a population of living organism, Bi is the (total) birth rate and D i the (total) death rate per unit of time. We have, then, f-^-A (5) at which it is often convenient to write at the lower case letters &, d denoting birth rate and death rate per head per unit of time. Combining (4) and (5) we have ^-(Ifc-DOmi (7) at Demographic Functions. The quantities B, D, or 6, d lend themselves to further analysis in terms of more fundamental char- acteristics of the aggregate. In a qualitative way everyone is familiar with the manner of the elimination of individuals from a population by death: some are carried off in infancy, some in child- 102 ELEMENTS OF PHYSICAL BIOLOGY hood, adolescence, and maturity, until the remnant is finally called in old age. Quantitatively this fact finds expression in an actu- arian's life table, or the corresponding life curve, of which some examples are shown in figures 15 to 17. Starting with some large number,, say a million, of newly born individuals, counted at birth, if we follow this sample batch of population through life, we find them thinning out, at first rather rapidly in infancy and child- hood (steep part of curve on left) : then more slowly (more gentle slope) in mid-life; and faster again as the natural term is approached. At any particular age a there are thus left, out of the original batch, a fraction p(a) of survivors. The fraction p(a), which we may speak of as the survival factor, 1 is a measure of the probability, at birth, that a random individual of the batch shall reach age a, under the conditions under which the data assembled in the life table were collected. The value of p(a) for every age of life depends, of course^" on the general conditions of life in the population. It therefore varies in different localities and at different epochs, as illustrated in figures 15, 16, 17. When it is desired to bring out the fact that p(a) depends on the time, it may be written p(a, f). But as a rule the change with time is not very rapid and we may often consider p(a) as a function of the age a alone. We shall do this in the present analysis^ which relates to a population in a selected locality under essentially constant conditions of life. If we denote by N a the survivors to age a, out of an original batch A r counted at birth, we have JV = N p(a) (8) We will -write dN a /1A\ \LO) ' or dN a d > r , j N a da da The coefficient ju a thus defined is termed the force of mortality at age a. From its definition it is clear that it measures the death rate per head in a population composed entirely of individuals of age a. 1 The function p (a) is that commonly denoted by l x in actuarial notation, p.nd tabulated in the principal column of a "Life Table," SSfSS] \ \ \ \ \ \ \ s.ooo 70,000 64,000 FIG. 15. SOME HISTORICAL HUMAN SURVIVAL CURVES, EXHIBITING AN EVOLU- TIONARY TREND TOWARD LONGBB AVERAGE DURATION OF LIFE The improvement is probably rather one of general hygiene than of man's physiological constitution. The earlier figures are of very doubtful accuracy. 103 104 ELEMENTS OF PHYSICAL BIOLOGY AGE IN YEARS 65,000 -45,000 40,000 35,000 30,000 25,000 20/500 15,000 IO.OOO. 5.000 x: X 4\ \ il 60 70 ASE IN YEARS 30,000 25,000 2O.OOO I5.OOO FIG. 16. STJEVIVAI, CUEVES FOB THE STATE OF MASSACHUSETTS, FOB THE THBEE DECADES 1890-1900-1910. AFTEB GLOYEB The data on -which these curves are based are naturally more reliable than those of the older life tables shown in figure 14, and exhibit very clearly the upward trend of the average length of human life in recent decades. ANALYSIS OF GROWTH FUNCTION 105 AEE M YEAJIS FIG. 17. SURVIVAL CURVES FOE DIFFERENT COUNTRIES, SHOWING INFLUENCE OP LOCAL CONDITIONS UPON LENGTH OF HUMAN LIFE. AFTER GLOVER A particularly simple case is that in which \i a is independent of a, the force of mortality is independent of the age, or is the same at all ages, We then have by integration of (10) and p(a) (11) (12) 106 ELEMENTS OF PHYSICAL BIOLOGY Such a simple life curve as this is not to be expected in a species of living organisms. It implies that the individual does not age, that his chance of living another year is just as good at ninety years of age as at fifty or at ten or at five; he can die, as it were, only by acci- dent; he is perpetually young. Survival curves of this form do occur and play a significant role in the aggregates of atoms and mole- cules which the chemist and the physicist make it their province to study. The atoms of an element in radioactive transforma- tion, for example, are picked off, one by one, according to a law of this form, and so are the molecules of a chemical compound decom- posing by a monomolecular reaction. If, in such a case, the plot the function p(a) is drawn on logarithmic paper, evidently a straight line is obtained, since log e p(a) = jua (13) (14) da The force of mortality is here seen as the (constant) slope of the curve representing log e p(a). In the more general case it is also often convenient to plot p(a) on a logarithmic scale. This has been done in figure 18 for the United States survival curve shown in figure 17. It is seen that the curve thus obtained is at first convex toward the axis of a, but soon becomes concave toward that axis and then remains so to the end. The significance of this is, of course, that the force of mortality is very high in infancy, decreases in early childhood, until it reaches a minimum about the twelfth year of life; and finally increases con- tinually to the end of the life span. The detailed analysis of the human survival curve is a matter of interest not only to the student of evolution, but also to the guard- ian of public health and to the insurer of human life, the actuarian. The essentially practical requirements of these has led to a highly developed technique, amounting virtuaUy to a separate branch of science, in the preparation and analysis of life tables. Of this phase of the subject no more needs to be said here, since there is a volumi- nous special literature available. Only the more strictly biological aspect of the matter is for us here of immediate interest, and of this more will be said later, in discussing the physical basis underlying the survival curve. ANALYSIS OF GROWTH FUNCTION 107 Survival Curve Data. Without having recourse to any refine- ments of mathematical analysis it is clear that a close relation exists between the survival curve of a given species and its rate of increase, its fate in the straggle for existence and for dominance. It might 10,000 /o K O -? 8 IS. 16 SO & S3 X 36 -fO 44 -fff .52 S6 60 64 68 7S 76 80 ff* 88 9 -36 /OO Ag<s in Years. FIG. 18. SURVIVAL CURVE PLOTTED ON LOGARITHMIC SCALE. UNITED STATES 1910 The slope of the curve thus plotted measures the Force of Mortality, This slope is at first very steep, decreases to about the twelfth year of life, when it reaches a minimum (point of inflection), and then increases again continuously to the end of life. be expected, therefore, that the study of such survival curves for different species of organisms should have formed an essential part of quantitative studies in evolution. As a matter of fact, however, data for survival curves of organisms in their natural environment 108 ELEMENTS OF PHYSICAL BIOLOGY are, for obvious reasons not easily obtained. It is difficult enough to make an accurate census of ages in a supposedly intelligent and responsible human community, where the cooperation of the Individ- ual can at least in a measure be engaged. A census of ages in the population of field, forest and stream must indeed be a severe test of the ingenuity of any investigator that should approach such an enterprise. Not that it is utterly hopeless. Isolated data can and have been gathered. Every layman knows that the age of trees, for example, can be estimated at least approximately by the number of rings in a cross section of the trunk. The age of certain fishes can be gaged from tell-tale marking on their scales. The age of certain birds can be told from their plumage. An estimate of the average age of herring gulls has been made, on this basis, by J. T. Nichols. 2 But data of this kind are at best scant. The problem is more manage- able in the case of domestic species, or of species in captivity. The most complete study of this character is probably the work of R. Pearl and S. L. Parker with colonies of fruit flies (Drosophila) grown in an experimental "universe," a glass bottle containing a suitable quantity of banana juice. 3 The survival curves thus ob- tained are singularly like those for the human species, when allow- ance is made for the difference in the total life span. The fruit fly's allotted days, in the most favorable case, number about ninety, or, say, one day for each year of human life. In figure 19 are shown, plotted on the same diagram, three survival curves; one of these is that of the human population of the United States in 1910; the second is the survival curve for one of the numerous varieties of fruit flies. Por only one other organism, Proales decipiens (a rotifer), a life table and curve has been prepared, on the basis of observations by B. Noyes. 4 The life curve has been computed by Pearl and Doe- ring 5 and compared with that of man, on the basis of the life span 2 J. T. Nichols, in The Condor, vol. 17, 1915, p. 181; for other data on the longevity of animals see C. S. Minot, The Problem of Age, Growth and Death, 1908, pp. 226, 266; A. Weismann, tJber die Dauer des Lebens, Jena, 1882. See also A. T. Masterman, Report on the scales of freshwater fish in relation to age determination. H. M. Stationery, Office 1924. 3 R. Pearl and S. L. Parker, American Naturalist, vol. 55, 1921, p. 503; vol. 56, 1922, p. 403. * B. Noyes, Jour. Exp. Zool., vol. 35, 1922, p. 225. * R. Pearl and C. R. Doering, Science, vol. 57, 1923, p. 211. ANALYSIS OF GHOWTH FUNCTION" 109 measured from the point at which, the force of mortality is a mimi- mura to the extreme end of life. It is the curves thus obtained that are shown in figure 19. As will be seen, they exhibit a close analogy. 30 4O SO 60 70 CENTILES OF LIFE SPAN 8O 90 fOQ FIG. 19. LOGARITHMIC SURVIVAL CURVES FOR MAN, DROSOPHILA, AND PBOALES DECIPIENS Plotted according to centiles of life-span; the human life curve is plotted with the twelfth year of life (point of inflection, see figure 18) as zero of the time scale. After R. Pearl. Influence of Age Distribution Upon Rate of Growth of Population. For our present purposes it is not sufficient to fix our attention on a group of individuals at the moment of their birth, and to observe the gradual dirninution of this group by deaths; we must take in 110 ELEMENTS OF PHYSICAL BIOLOGY view an actual population, comprising individuals of all ages, and we must enquire into the effect of deaths upon such a group as this. If the survival curve for such a population were of the simple form p(a) = e-v (12) with n, the force of mortality, independent of a, it is evident that the age distribution in the population would have no influence upon the total death rate. For if individuals of all ages are equally susceptible to death, it is evidently immaterial how the population is constituted as regards age. The death rate per head, in such a population would, in fact, be simply equal to the force of mortality. The Stable or "Normal" Age Distribution. But in the popula- tions with which the biologist and the vital statistitian deals, the force of mortality varies very decidedly with the age, and it might therefore be supposed that any discussion of the rate of increase of a population of organisms must fully take into account the age dis- tribution. This supposition, however, involves an assumption, namely, the assumption that the age distribution itself is variable. Now, in point of fact, the age distribution is indeed variable, but only within somewhat restricted limits. Certain age distributions will practically never occur, and if by arbitrary interference, or by a castastrophe of nature, some altogether unusual form were impressed upon the age distribution of an isolated population, the "irregu- larities'"'' would tend shortly to become smoothed over. There is, in fact a certain stable type of age distribution about which the act- ual age distribution varies, and toward which it returns if through any agency disturbed therefrom. 5 The form of this distribution, in an isolated population (i.e., with immigration and emigration negligible) is easily deduced, as follows: If we denote by N (f) the total number of the population at time t, and by N ($) c(\a} da that fraction of the total whose age is com- prised, at time t, within the limits a and a + da, it is evident that the individuals of this element of the population are the survivors of the B(t-a)da persons born at times (<-o). Hence if p(a) is the survival factor, we have s For a formal proof of this see F. R. Sharpe and A. J. Lotka, Phil. Mag., April, 1911, p. 435; A. J. Lotka, Proc. Natl. Acad., vol. 8, 1922, p. 339. ANALYSIS OF GKOWTH FUNCTION 111 B (t - o) p(o) da = tfft) c(o) da (15) If we denote by D (t) the total death rate at time t, we have, evi- dently '*' (17) We are supposing that we are dealing with a population in which the survival factor is constant, i.e., independent of t. If c(a) also is independent of tf, the integral in (17) is evidently a constant and we have . ) = const. = d (18) where d is the death rate per head. On the other hand, since c(w) is independent of the time, we have c(0) = const. = b (19) Brief reflection shows that the value of c(o) for zero age is the birth rate per head. Thus we have 6 d = r - const. (20) where r is the "natural rate of increase" of the population. We have further B(t - a} = bN(t - a) (21) = bN(tie~ ra (22) since the natural rate of increase r is constant. Thus by (16) c(a) = be- fa p(a) (28) The normal age distribution (23) is of fixed form in th'e sense that, once established, it perpetuates itself. More tlikn this it is also > stable > in the sense that, if disturbed by a temporary change m the conditions of life (e.g., war), it will spontaneously return upon restoration of normal conditions. 112 ELEMENTS OF PHYSICAL BIOLOGY A rigorous proof of this stability of the age distribution (23) cannot be briefly given, and for details of such proof the reader must be referred to the author's publications in the journal literature. 8 The general character of the proof may, however, be indicated. If the original population has any arbitrary age distribution, such as that represented by the more heavily drawn curve in figure 20 by judicious trimming we could reduce the population to the nor- mal distribution represented by the lower curve tangent to the heavy curve; or, we could, by filling in gaps, supplement the population to fit the upper tangent normal distribution curve. The trimmed down population, having the normal age distribution, would always retain it. The same is true of the supplemented population. The actual population will therefore always lie between the trimmed and the supplemented. Moreover, it can be shown that at intervals of about one hundred years (the span of life) new tangent curves can be drawn to the actual distribution curve, the new tangents lying between the old. Thus the two tangent curves ultimately approach until they coincide, and then, necessarily, the actual distribution curve lying between them coincides with them also. That is to say, the actual conforms with the normal age distribution. 7 It must be understood that the normal or stable form of age dis- tribution represents merely a broad type, toward which actual age distributions will tend. However, the approach seems to be at times very close, as is shown by the figures in table 7 giving the observed and the calculated age distribution for England and Wales in decennium 1871-1880. A graphic representation appears in figure 21, which shows the observational data, plotted (in dotted lines) as a stepped curve. The corresponding figures calculated by the formula (13) are plotted in two ways, namely, first as a stepped curve (drawn in full), for comparison with the observational data; and also as a continuous curve. The latter brings out a feature that may be noted in passing, namely the fact that the curve of age dis- tribution is very flat, roughly linear, over a wide range, from the 7 An exceedingly interesting effort of early date to demonstrate the ulti- mate approach to geometric increase of the birthrate, independently of initial conditions (e.g. starting with a single pair of parents) is to be found in L. Euler, Recherches g6n<rales sur la mortalite". See also E. T. Gumbel, Jahresber. Deutsch Math, verein., vol. 25, p. 251. ANALYSIS OF GROWTH FUNCTION 113 FIG. 20. DIAGRAMS TO ILLUSTRATE PROOF OP STABILITY OF NORMAL AGE DISTRIBUTION Reproduced from A. J. Lotka, Proc. Natl. Acad. Sci. ; vol. 8, 1922, p. 339 TABLE 7 Population of England and Wales 1871-1880, as an example of "normal" age distribution a- 1 , o MALES FEMALES PERSONS Calculated Observed Calculated Observed Calculated Observed 0-5 142 139 135 132 138 136 5-10 118 123 114 117 116 120 10-15 107 110 104 104 106 107 15-20 9S 99 95 95 96 97 20-25 88 87 86 91 87 89 25-35 150 144 148 149 149 147 35-45 117 112 117 115 117 113 45-55 84 84 87 87 86 86 55-65 57 59 63 61 60 59 65-75 29 31 36 35 32 33 75- co 11 12 13 15 12 13 1,001 1,000 998 1,001 999 1,001 114 ELEMENTS OP PHYSICAL BIOLOGY Si, .C3S 30 .028 .026 .OSf .022 420 ,0/8 .0/6 .O/-f MS .010 .008 .006 .OO$ .002 .OOO \ ^ .-=. 03373 &- 10 -30 -50 60 70 SO SO /<> a = age, FIG. 21. "STABLE" AGE DISTRIBUTION, AS EXEMPLIFIED BY THE POPULATION OF ENGLAND AND WALES IN THE DECADE 1871-1880 I IK ANALYSIS OF GHOWTH FUNCTION 1JLtJ fifth to the eightieth year of life. 5 This, of course, is a special fea- ture of a human population when the natural rate of increase r lies in the neighborhood of a certain value. Minor fluctuations of the age distribution will not greatly affect the birth rate and the death rate. Since actual populations approxi- mate the normal age distribution, i.e., that defined by equation (23), it seems permissible and it is certainly expedient to assume, in further discussion, that the normal age distribution is actually established; we may then virtually disregard the influence of the age distribu- tion upon the rate of increase of the component. This element is, as it were, automatically ruled out of further discussion, by the nat- ural establishment of the normal age distribution; 9 a circumstance which is in so far fortunate, as we are here interested in the relation of the rate of growth to the more fundamental biological character- istics; the age-distribution appearing merely as an adventitious element complicating the relation, without being essential to the fundamental characterization of the species. Demographic Relations in "Normal" Population, It is worth while to note briefly in passing that in the case of a population in normal age distribution, many demographic relations assume a simple form. So, for example, the relation between birth rate per head 6 and death rate per head d is here given by the formula 10 1/b = g-rp( ) d (25) Jo which, to second order approximation, is reducible to the simple form 1 - U U __ __ + j=*L (26) where L is the mean length of life, defined as Compare J. Brownlee, The Use of Death Rates as a Measure of Hygienic Conditions, Report to Medical Research Council, London, 1922, pp 36-37 8 Of. G. Eijkman, Onthr " J.--.-TTI .. .. ~ . >j.r- !, p. 269; 10 A. J. Lotka, Quart. Publ. Am. Statist. Assoc., 1918, p. 121; 1921, p. 998. 116 ELEMENTS OF PHYSICAL BIOLOGY Ceo p(a) da (27) while U is defined 11 by /"6co L ' = ap(a) da (28) <Jo L' U = Z"' (29) As an illustration it may be mentioned that in the saniv, population (England and Wales, 1871-1880) which as already been referred to, the relation (26) is found by actual computation to be, numerically 0.767 0.233 ___ + ___ = 41 _3 5 ( ma i es ) ^ 0.7369 0.2G31 . _ -- j -- . _ 44,62 (females) (31) Oi The observed values of 6 and d, together with those computed by the formulae (22), (23), are shown in table 8. It will be seen that the agreement is good. The relation (26) between b and d lends itself readily to graphic representation by means of a diagram involving only straight lines. This is shown in figure 22. The method is as follows. _ r Along OX mark off a length OP = L U Along OY mark off a length OQ == L Complete the rectangle QOPM. Suppose we are now given 6 - 0.0337 It is required to find d Along OX mark off 0V = 0.0337 Join VM and produce it to meet OY in W Read off at W on the scale of OY d = 0.0200 11 U is proper fraction. It then follows from (26) that L is intermediate , 1,1 between - and -; this circumstance has been noted by Bortkewitch, Mitt- lere-Lebensdauer. ANALYSIS OF GROWTH FUNCTION 117 A single equation, such as (25), connecting the two quantities b and d, is of course insufficient to determine their value. To do this requires a second independent relation. Such a relation Is furnished by the following considerations: The law of the fixed age distribution holds separately for each Thus, if we denote JV/ the total number of the female popula- sex. TABLE 8 (OBSEBVED) d (OBSERVED) d (COMPUTED) Males 0.036921 0.022612 0.02221 Fe n ioles 0.033727 020001 0.02000 b-,0337 .008 .016 b Bir1~h rafs pe.r head per annum. FIG. 22. DIAGRAM OF RELATION BETWEEN BIRTH RATE PER HEAD 6 AND DEATH RATE PEE HEAD d IN POPULATION WITH STABLE AGE DISTRIBUTION tion } by 6/ the number of female births per unit of time and per head of female population, and finally by pj- (a) the survival factor in the female population, then the number of females of an age between a and a + da is given by Njbje~ n p f (a) da. If these reproduce, on an average, 0/ (a,) females per unit of time, then the total number of female births per unit of time is evidently given by 118 ELEMENTS OF PHYSICAL BIOLOGY C a I t/oi ( 32 ) where a i} a are the lower and upper age limits of the female repro- duction period. But Nfy- = B f ; hence (a)8 f (a} da (34) This is the required second relation 12 between 6 and d. The birth rate and death rate in a population of fixed age distribution are thus completely determined as soon as the functions p f (a) and /?/ (a) are given. Aside from its direct interest as the second relation required to complete the determination of 5 and d in a population with fixed age distribution, equation (34) is also of value in enabling us to deduce certain secondary results, of which two will here be noted. Rate of Increase per Generation. To derive our first conclusion from (25), we expand e~ ra under the integral sign. We find 1 = /3(o) p(o) da - r aj8(a) p(a) da + ... 35 t/oj Jai A. little reflection shows that the first integral measures the ratio between the total number of births in two successive generations. Denoting this by R, and the second integral by S, we have 22 = 1 + rS - . . . (36) a formula wjhich connects the rate of increase per head per annum r and the rate of increase R in the total number of births in one generation, per generation. It will be seen that this relation is not one that could very well be foretold by any simple elementary or common sense principle. It is this relation that is implicitly involved in W. R. Thompson's treatment of a problem in parasitol- ogy cited in Chapter VIII (page 87) ; at least, we must know this relation if we are to interpret his results in terms of the usual tune unit, such as the year, whereas Thompson reckons in generations. 13 A. J. Lotka, Jour. Washington Acad. Sci., 1913, p. 293. ANALYSIS OF GROWTH FUNCTION 119 Effect of Selective Slaughtering. A second deduction from equa- tion ',34} follows from the fact that the integral extends only between the age limits of the reproductive period. In consequence of this the rate of increase, per head, of the population is independent of the form of the life curve (the survival factor) outside this period; thus the slaughtering of the "superannuated" members of a herd has no effect upon its rate of increase, though both death rate and birth rate are augmented (by equal amounts). 12 The economic significance of this, as a means for raising the productive capacity of the herd, considered as a food factory, is serf-evident. It is true, as previously noted, that generally the influence of age distribution upon the rate of growth of a species is eliminated from discussion by the spontaneous establishment of the fixed or normal distribution. But the subject referred to in the preceding paragraph clearly shows that there are exceptions to this general rule. We have here an instructive example in which the influence of the age distribution upon the course of events is so fundamental that a discussion of the case which should fail to cover this feature would be lacking in an essential particular. Indiscriminate killing of one species by another, as practised by the untutored savage or the dumb animals, has the effect of reducing the ordinates of the life curve of the food species more or less along its whole extent. This is illustrated in figure 23. We may suppose that the life curve of some species, wild cattle, for example, is that shown in a full line, curve I, and that curve II represents the life curve for the same species after the habitat of the species has been invaded by a tribe of primi- tive hunters, too thoughtless or too unintelligent to regulate their depredations in accordance with anything of the nature of game laws. Curve III is the type of curve that would be produced by selective killing guided by intelligent control, as distinguished from random lolling. The verticals erected at a\ and a 2 represent the limits of the period of reproduction for the cattle. The slaughtering, in the case of curve III, takes place exclusively after the end of the period of reproduction. This is an extreme case, which, in practice, for obvious reasons, would be only distantly approached. For both in game preservation and in animal husbandry on the farm many factors have to be considered; not only the quality of the meat of animals at different ages, but the cost of raising and maintenance. It is out of the question to restrict the slaughtering to the post- 120 ELEMENTS OF PHYSICAL BIOLOGY reproductive period. The unnecessary male may, in fact, be slaugh- tered as soon as the cost of his upkeep exceeds the corresponding gain hi marketable value of his carcass. Just what is the most advantage- ous tune for his slaughter is a question in agricultural economics which has been discussed, among others by A. Gouin and P. Andouard. 13 These authors compute that to bring up three calves, from weaning to the age of three and one-half years, requires about 33,000 kgm. of hay (or their equivalent); this quantity would suffice to bring FIG. 23. DIAGRAMMATIC ILLUSTRATION OF INFLUENCE OF RANDOM AND OP SF.LECTIVE SLAUGHTERING UPON SURVIVAL CURVE OF BIOLOGICAL SPECIES seven head of cattle to the end of their second year, which would give a gain, in meat, of 40 per cent, as compared with three head brought to the age of three and one-half years. Intelligence as a Discriminating Agency. It is particularly worth while to note how in this connection human intelligence exerts its influence upon the course of physical events by substituting systematic selection in the place of the more haphazard, more ran- dom actions characteristic of the mentally less developed species. 13 Bull. Soc. Natl. d' Agriculture, Paris, 1910, p. 695. ANALYSIS OF GROWTH FUNCTION 121 These latter must. In many ways, remain, dependent upon certain average manifestation, whereas man, in a corresponding situation, is able to discriminate and address himself directly to individual elements of which these averages are the expression in the gross. In the manifold interactions on a macroscopic scale that consti- tute the perennial struggle for existence on nature's battlefield, the lower organisms are in many respects situated in a position analogous to man's relation toward the ultra-microscopic elements of his environment, the molecules and atoms. These he can handle only in bulk, unable to avail himself of the distinctive features of this or that particular atom or molecule. Just as our senses and our bodily members are inadequate for the task of handling individual atoms, so the mental or other dis- criminating powers of the lower organism are often inadequate to lead it toward any other than a rather random selection, and by no means an optimum selection, among the somewhat varied oppor- tunities of self-service presented to it. However, though man does far excel the other creatures in this respect, the difference is, after all, one in degree and not in kind. Many, if not all organisms, possess in some degree the power of selec- tion, are in some measure independent of pure haphazard. This introduces an altogether peculiar complication into the dynamics of systems comprising living organisms, a complication of which the statistical dynamics of molecular physics are free. Not only is the living organism capable of performing, on a macroscopic scale, exploits analogous to those which in the world of molecules are permitted only to such figments of the imagination as Maxwell's demon; but this power of "cheating chance," as it were, is possessed in different degree by the several living organisms, and a dynamics of systems comprising living matter must necessarily take account not only of this faculty, but of the gradations in this faculty which have a large part in assigning to the several biological species their place in the scale of evolution. This will require the development of special methods. It is essential that we bear in mind constantly the ultimate aim of our reflections. The transformations of matter, the change in its distribution among the components of the physical system in the course of evolution, are the first to strike the eye, and are properly the first to receive our systematic considera- tion, just because they are of more obvious and elementary character. 122 ELEMENTS OF PHYSICAL BIOLOGY But our fundamental aim must ultimately be to gain an enlarged understanding of the dynamic relation involved, of the play of forces, the transformations of energy, Kinetics of Intra-Group Evolution. It is not intended to attempt here a systematic discussion in any sense exhaustive of the course of events in ultra-group evolution, that is to say, in the redistribution of matter among the several sub-types of which a biological species is composed. We shall however briefly note an enterprise, led more particularly by J. B. S. Haldane/ 4 to investigate the trend of evolution in a population in which selection is operating upon a character subject to Mendelian inheritance. Casel. Haldane considers first of all the simple case of a species of constant total population, consisting of two types (phenotypes) A and B that do not interbreed, but react upon each other merely through competition in the same environment. Haldane treats the case as follows : Let the nth generation, counted immediately after fertilization, 15 consist of types A and B in the ratio u a : 1, and let the coefficient of selection be k, i.e., let (1fc) of B survive (to breeding age) 16 for every one of A. Then the sur- vivors (to breeding age) of the nth generation, and hence the first numbers 17 of the (n-j-l)th, will be in the ratio 14 J. B. S. Haldane, Trans. Cambridge Phil. Soc., January, 1924, vol. xxiii, pp. 19-41. Part II, read July 14, 1924, has not yet reached the author. 15 The words in italics are here added to Haldane 's text, in accordance with the first paragraph of his paper, page 20. 18 The words in italics are here added to Haldane's text, in accordance with his paper, page 23, line 6. 17 The meaning of this term is not altogether clear. It seems to refer to the total births in the (n + l)th generation, or, in other words, to the (n+l)th generation, counted at birth. This does not seem altogether consistent with the rest of the argument, notably the wording referred to in footnote 15. On the other hand the terms referred to in footnote 16 are vague unless reproduc- tion takes place once and once only in the life of the individual, since in general there is not only one single age of reproduction, so that the fraction surviving to breed cannot be represented by any single number. It seems that it would be better to base the argument on the total number born in each generation, i.e., on the generation, counted at birth; and to regard the ratio R between the births in two successive generations as a function of /3 (a) and p (a), as set forth in Chapter VIII equation (24) and Chapter IX equations (35), (36). We would then start with equation (38) of page 123 as a fundamental assump- tion; beyond this the argument would remain essentially unchanged. ANALYSIS OF GROWTH FUNCTION 123 w n :l-A = Un + i:l (37) Ua + i "rr& (38) If u be the original ratio of A:B in the Oth generation, then u a = (1 - Ar)- n o (39) If we write yn. for the proportion of 2?'s to the total population of A and B in the nth generation, then (40) 1 -f u n 1 4- (1 k)~ n u * = y + (1 _ ^1, (1 _ y o) ^^ If we start with an equal number of births of A and B 2/o = i (43) 1 i i f~t 7.^1 n 1 -f- (I K) If h is very small, i.e., if selection is very slow, then approximately 1 (44) (45) fc n = log ^ (46) Vn In equation (45) we recognize once more the Verhulst-Pearl relation. It is seen that in the circumstances to which the discussion relates, the better adapted of the two types grows, along the typical S-shaped curve, at the ex- pense of the less well adapted, ultimately displacing it entirely. This is quite in accord with what we should expect, since the total population A + B is constant, that is to say, it just holding its own. Any constituent of the popu- lation that falls below the average in adaptation, must therefore be unable to hold its own, and diminishes continually. The curve representing graphically the change of the population increase of type A at the expense of B, which is ultimately displaced entirely is shown in figure 24 for the case k = 0.001, i.e., that 999 B's survive to reproduce, for every 1000 A's. In these circumstances 9184 generations are needed for the proportion of A'B to increase from 1 per cent to 99 per cent. 124 ELEMENTS OF PHYSICAL BIOLOGY Case 2. Selection of a Simple Mendelian Character, with intermingling of dominant and recessive type. Kaldane next takes the following case: Consider the case of a population containing two, one, or no "doses" of a completely dominant Mendelian factor A. Mating is at random and selection acts in equal degree in both sexes upon the character produced by the factor. Pearson 18 and Hardy 19 have shown that in a population mating at random, the square of the number of lieterozygotes is four times the product of the numbers of the two iiomozygous classes. Let u n A. : la be the proportion of the types of gametes produced in the (n 1) <A generation. Then in the nth gen- eration the initial proportion of the two classes of zygotes will be :2u n Aa:la (47) iGOr A -5COO +5000 FIG. 24. GROWTH OP FAVOHED TYPE IN MIXED POPULATION OF Two PHBNOTYPES Constant total population. Abscissae = number of generations; ordinates = percentage of favored type in total population. After J. B. S. Haldane. The proportion of recessives to the whole population is (48) Now only (1 k) of the recessives survive to breed, so that the survivors are in the proportion a Aa: (1 k) aa (49) 18 K. Pearson, Phil. Trans. Boy. Soc., 1904, A 203, p. 53. 18 G. H. Hardy. Science, 1908, vol. 28, p. 49. ANALYSIS OF GROWTH FUNCTION 125 The numbers of the next generation can most easily be calculated from the new gai:;etic ratio iir~ i. This is immediately obvious in the case of aquatic orga-uisfiia 7,iio shed their gametes into the water. If each zygote produces IV giiiietes v/aieh conjugate, the numbers of gametes of type A and a respectively, are. clearly. CVtin 2 T A r w) A and (Nu a + N { 1 - k } )o (50) so thut (51) 1 + n /C Xow if vre know the original proportion of recessives y , we start with a population u -AA:2uoAa:!Aa (52) where ^o = y ~* - 1 (53) and we can at once calculate 1 + MO & and thence u^ and so on. For complete selection we have k = 1 (recessives all die). Then n = n + M (56) t/ n = y (l+ W y J)-2 (57) If we start with a population containing J recessives, the second generation will contain the third ^, the nth - Thus 999 generations will be (n + IP required to reduce the proportion to 1: 1,000,000, and we need not wonder that recessive sports still occur in most of our domestic breed of animals. When selection is not very intense, we can proceed as follows : (58) (59) 126 ELEMENTS OP PHYSICAL BIOLOGY When k is small (60) 100 I K, 5 = u a u + log (61) (62) (63) 1 7 -5000 Generations. +5QOQ 10000 FIG. 25. EFFECT OP SELECTION ON POPTJLATION COMPRISING Two PHENOTYPES WITH MENBELIAN INHEEITANCE Upper curve, dominants favored; lower curve, recessives favored. Abscissae = generations; ordinates = percentage of population with the favored char- acter. After J. B. S. Haldane. If we start from a population containing 25 per cent recessives U Q = 1, kn = n + iogen - 1 (64) In figure 25 is shown the growth curve for the dominants when k 4- 0.001 (upper curve) and for the recessives when k = 0.001 (lower curve), i.e., the favored type has an advantage of 1 in 1000, as in figure 24. In each case 16,582 generations are required to increase the proportion of the favored type ANALYSIS OF GROWTH FUNCTION 127 front 1 per cent to 99 per cent, but dominants increase more rapidly than reces- sives when they are few, and more slowly when they are numerous. The change occurs mci=t rapidly when y a , the proportion of recessives to the total popu- Haldane deals also with several other cases, for the details of which the reader must be referred to the original. A continuation of Haldane's article is anticipated at the time of this writing. CHAPTER X FURTHER ANALYSIS OF THE GROWTH FUNCTION F We fat all creatures else to fat us, and we fat ourselves for maggots. Shakespeare, Adjustment of Birth Rate to Optimum. In the preceding dis- cussion of the equations (25) and (34) of chapter IX it has been supposed that the form not only of p(a), but also of /3(a), the rate of procreation at different ages, is fixed. It is only on this supposition that the two equations (25) and (34) of Chapter IX together deter- mine both 6 and d. In point of fact the function (3 (a) is undoubtedly, for most species of organsms, very elastic (much more so than p(a), the survival factor), and capable of adapting itself to varying cir- cumstances. This is especially so in the case of man, who exhibits in particularly high degree the rather astonishing phenomenon of a portion of matter whose growth is at least partially under the con- trol of a will in some manner associated with it. But, in the organic world at large alsoj there is presumably at least some tendency for the adjustment of the procreation factor so to take place as to make the rate of increase r a maximum under the existing conditions. Too high a procreation factor would lead to excessive sacrifices in progeny that could not be raised to maturity, and would increase the death rate more than the birth rate. On the other hand too small a procreation factor would obviously fail to give the maximum attainable rate of increase. Somewhere between the two extremes a certain optimum procreation factor will make r a maximum. From this point of view the two relations that effectively determine the actual values of the two variables 6 and d are i f*<o -= I e -ra( a ) d a (l) and r = maximum compatible with e~ ra {$(a} p(a) da 128 (2) ANALYSIS OF GROWTH FUNCTION 129 The view that the rate of procreation thus expands or contracts in sympathy with the expanding or contracting food supply (or economic conditions generally) has been developed in detail (for the human species) by R. Lascaux in his work La Production et la Population (1921), That some such adjustment occurs with many biological species is a very plausible, one might say an inevitable supposition; but the approach to the ideal optimum is probably often only very imperfect, if only because the nature of the case calls for a large "factor of safety."'' In any event it is true that the birth rate does not play so unquali- fiedly a dominant role in determining the rate of growth of a species as might appear on cursory reflection. The equation (5) of Chap- ter IX, expressing the rate of growth of the species in terms of the birth rate and death rate, while it renders correctly the quantitative relations to which it refers, is only a partial or one-sided representa- tion of the facts, and is even open to misinterpretation. Incau- tiously construed it might be taken to imply that growth of an aggre- gate of living organisms takes place by births of new individuals into the aggregate. This, of course, is not the case. The new material enters the aggregate in another way, namely in the form of food con- sumed by the existing organisms. Births and the preliminaries of procreation do not in themselves add anything to the aggregate, but are merely of directing or catalyzing influences, initiating growth, and guiding material into so many avenues of entrance (mouths) of the aggregate, provided that the requisite food supplies are presented, provided that the system is, in a sense, "supersaturated" with regard to the species seeking to grow therein. The final result may not depend very greatly on the number of births, somewhat as the final state of a crystallizing solution is independent of the number of crystal germs initially sown therein. It will be desirable to develop our analysis in such manner as to bring out the relations thus involved. Aggregates of Constant Units. In aggregates of a simpler kind, as presented to our view in ordinary physico-chemical systems, each individual unit retains its identical substance unchanged throughout its period of existence as such unit. So, for example, a molecule of water consists of two particular atoms of hydrogen united to one specific atom of oxygen; and these same atoms continue to exist together as the building stones of the molecule as long as this con- tinues in existence as a water molecule. 130 ELEMENTS OF PHYSICAL BIOLOGY In these circumstances the mass of each unit is obviously constant throughout its period of existence as such: and furthermore, addition to the component, or elimination therefrom, can take place only by the actual entry or departure of a complete unit. If, therefore, the mass of each unit is rtii, and if Bi new units are added to the aggre- gate per unit of time, while Dj are eliminated, we have, in this case, very simply, ^=(Bi->i)mi (3) at Aggregates of Variable Units. For aggregates of living organisms we can also write an equation identical in form with (3), as has already been noted in Chapter IX; Bi is in this case the total birth rate, DI the total death rate, and m^ the average mass per head of the living population. But this is an inadequate representation of the significant facts. The equation, thus written, glosses over certain important characteristics of living organisms. Unlike molecules in a system in the course of chemical transformation, each unit in an aggregate of living organisms does not retain its substance unchanged in identity or in total mass. In fact, each unit is itself an aggregate within the larger aggregate that constitutes the species or biological group, and for each individual unit (organism) separately we can write an equation analogous to equation (1) of Chapter IX where U'i is the total mass taken up (ingested) per unit of time by the unit organism, and Vi is the total mass eliminated therefrom per unit of time. So, for example, in the course of one year a boy ten years old and weighing 32.5 kgm. may consume about 600 kgm. of food (inclusive of water and oxygen), may eliminate about 599 kgm of wastes, and will grow in actual mass by about 1 kgm., so that we have ~ = 600 - 599 = 1 = O.OSmi (5) at The Stream of Substance Through, the Form of the Organism. It will be observed that a portion of the intake, but a portion only, is expended in adding to the total mass of the unit. The remainder ANALYSIS OF GROWTH FUNCTION 131 B,-17',-^ (6) at is expended without, apparently, 1 - any resulting increase in the total mass of the unit. This constant expenditure of substance, and the equally constant intake required to balance it, is a fundamental char- acteristic of the units here under discussion. In the adult, whose mass is (on an average) approximately constant, we have simply and the entire intake goes to meet the requirements of maintaining the mass of the unit at constant level. 2 Turning now from the consideration of the individual unit to that of the aggregate of N such units, evidently, if the average intake per unit of time per individual is U'i, and if the average elimination is Vi, then we shall have for the rate of increase of the total mass Xi of the aggregate. cLTi _ = NU'i- NV'i - Nm'idi (9) at where di is the death rate per head per unit of time and m'i is the aver- age mass of a unit (organism) at death. Two Types of Organisms: Economical and Lavish Birth Rate. The relative importance of the second and the third term in the right hand member of the equation (9) differs greatly in different biologi- cal species. At the one extreme we have a type of which perhaps the most characteristic representative is man. With a mean length of life of about fifty years, his body must be replaced about twice in a century to maintain the population equilibrium. If we assume (as a rough but sufficient approximation) that the average weight of man at death is 50 kgm., this means that the third term, 1 Indirectly a part of the excess Ri of the mass intake over the mass increase may contribute to that increase, namely by furnishing some of the energy re- quired for anabolism. But we are here discussing mass relations only. The energy relations are reserved for separate consideration later. - Except during gestation, if the mass of the fetus is reckoned in with that of the mother. 132 ELEMENTS OF PHYSICAL BIOLOGY vy 50 in a stationary population, is about ^p kgrn. for the entire popu- ou lation, or just about 1 kgm., or say 2 pounds per head per annum. To put it crudely, of the food consumed by each human individual in a year, 2 pounds go, on an average, to replace the bodies of his fellows departed that year. This, it will be seen, is an insignificant, almost wholly negligible fraction of the 1000 pounds 3 or more than he con- sumes, in all, in a year. Of the total food consumed by the human race, then, about 0.2 per cent 3 goes to replace the bodies eliminated by death. The remainder is for current maintenance of the living. And the total food consumption 3 may be of the order of 7 to 10 times the mass of the population per annum. 4 But, as already stated, man represents an extreme type, the ex- treme economy of life, with low death rate and correspondingly low birth rate. Of the opposite extreme, lavish, seemingly wasteful extravagance, examples are exceedingly common, though it may not be easy to give full quantitative detail. Among the most wasteful breeders are, no doubt many aquatic species, including fish, since their young are ill protected and become ready victims of other species. So a ling weighing 54 pounds was found to be carrying twenty-eight million eggs. 5 An oyster may have sixty million eggs. But some familiar land animals are prolific enough, even if they do fall far behind the standards just exemplified. The brown rat may have five or six litters averaging about eight or ten each, in a year. 6 Domestic Animals Kept for Produce. Accurate figures can be obtained in case of domestic animals. While these do not represent so extreme an example, a special interest attaches to them owing to their direct relation to human food economics. The most prolific among domestic animals is the pig. In reasonably good farm con- ditions a sow should average three litters in two years, each of seven farrows, of which five are successfully raised and marketed. Even with the high mortality artificially induced by man in his do- mestic stock the item of running expenditure in feed for mere main- tenance is far in excess of the replacement cost, that is to say, the feed 3 Exclusive of oxygen. 4 For quantitative data on growth in man see C. S. Minot, The Problem of Age, Growth and Death. 6 J. A. Thomson, The Wonders of Life, 1914, p. 130. s H. H. Donaldson, The Rat, 1915, p. 190. ANALYSIS OF GROWTH FUNCTION 133 stored up and finally utilized in the carcass of the slaughtered animal. From a detailed study of the vital economics of beef production made at the University of Missouri 7 figure 20 is reproduced here to show these relations. The convex curve shows the average growth per head in a group of steers fed with a ration regulated to secure a maxi- MOOO /DO SOO 30O -90O 00 600 700 goo SCO /OOO //OO /200 t30O HOO OOO FIG. 26. FEED CONSUMED, AND INCREASE IN LIVE WEIGHT OF STEERS AT SEVERAL AGES Dry matter consumed is represented on one-tenth the scale of the live weight. After Moulton, Trowbridge and Haigh. mum of growth, without storage of surplus fat; the approximately straight line mounting upward shows the steadily increasing inte- grated amount of feed consumed since birth. Only the dry weight of the feed is plotted, and the scale employed is ten times more 7 University of Missouri, College of Agriculture, Bulletins 43, 54, 55 (E. C. Trowbridge and L. D. Haigh). 134 ELEMENTS OP PHYSICAL BIOLOGY condensed than that used for the live weight else the second would rise too steeply as to lie for the most part far beyond the HnvT of the page. Thus is shown the great disproportion between the feed TABLE 9 Average Yearly Gains of Steers* PERIOD GROUP I GROUP II GROUP III days 30 to 360 360 to 720 to < 1,080 1,080 to ( to | Weight gained pounds 605.3 3,075.0 1.83 5.08 560.7 6,314.1 11.26 390.8 6,652.9 1.09 17.02 268.9 6,304.6 0.75 23.45 JL j CJ&Q / 22,346.6 1.30 12.24 pounds 409.7 1.24 4.71 320.8 3,619.3 0.89 11.28 126.9 3,629.2 0.35 28.60 270.6 4,259.9 0.75 15.74 11 oo n j J.ZJO . \J 3,438.0 0.80 11.91 pounds 241.6 1,265.0 0.73 5.24 235.6 2,420.1 0.65 10.27 121.2 2,811.7 0.34 23.20 311.2 3,562.3 0.86 11.45 911.6 0,059.1 0.65 11.03 Dry matter eaten Daily gain Dry matter per pound gain Weight gained Dry matter eaten. . . Daily gain Dry matter per pound gain. Weight gained Dry matter eaten.. . Daily gain Dry matter per pound gain Weight gained. . . Dry matter eaten Daily gain Dry matter per pound gain Weight gained. . . Dry matter eaten Daily gain Dry matter per pound gain Animals of group I were fed all they would eat. Srowth not body of the growing steer - and ceraM ; ' S &r M the interest of the P^uoer is con- cerned, in mere mamtenance, for the private satisfaction and benefit ANALYSIS OF GROWTH FUNCTION 135 of the animal, so to speak. The numerical data on which figure 26 is based are exhibited in table 9, together with the corresponding figures observed when the animals are somewhat underfed and over- fed respectively. The instances that have been cited man on the one hand, and the highly prolific species, both feral and captive, on the other are elo- quent illustrations of the elasticity of adaptation. Clearly, a species may hold its own, in the straggle for existence, either by the aid of well-developed protective devices resulting in a low death, rate, and requiring only a correspondingly low birth rate; or, a less well pro- tected species may balance a high death rate by an equally high birth rate. Which of these two methods would be chosen in the natural course of events is a question that it might be difficult to answer on any general a priori principle, so long as attention remained fixed on a single species. Perhaps one would have expected evolution to turn in a favor of the more economical method of meeting a low death rate with a low birth rate. In point of fact both types of organism the economical type (as judged by its own standard) with low death rate, and the wasteful with high death rate exist side by side in abun- dance. This is a good example to illustrate the purely relative charac- ter of fitness, and to remind us once more that we cannot expect any success in attempts to define the direction of evolution in terms of a single species. It is not the individual species, the individual com- ponents of the system, that evolve, but the system as a whole, com- prising all the species and their environment. The species of the economical type, with low death rate, are largely dependent for their subsistence on the presence of species of the opposite type; we must think here of a competition, not between individual species, but be- tween groups of species, groups consisting, in the simplest case, of two species each, a food species or prey, and a feeding or predatory species. Of two such groups, that one will, other things equal, have the advantage in the struggle, in which high productivity of the food species is accompanied by economy of life on the part of the feeding species. From the point of view of the hog, so to speak, the high mor- tality in the pen is a disastrous inefficiency and maladaptation, a misfortune to be borne, as best it may, with porcine philosophy 1 . From the point of view of the consumer on the other hand, this high mortality is, quite on the contrary, a measure of the efficiency, the eminent fitness of swine as producers of pork; and his only regret 136 ELEMENTS OF PHYSICAL BIOLOGY is that so much of the feed placed in the trough goes merely to carry on "what may be called the personal activities of the animals them- selves." 8 It is to be noted, however, that only a part of the material accountable as waste from the standpoint of the food species is gain for the feeding species. Deaths from disease are a pure loss to both species. Similar reflections, of course, apply, mutatis mutandis, to those cases in which the feeding species derives its nourishment from some cur- rent product of the life activity of the food species or host, instead of from its carcass. The most notable example of this in the food econ- omy of man is his exploitation of the milch cow, who is a far more efficient producer than the beef steer. The latter at best consumes over 6 pounds of nutriment for every pound of product. 9 According to the investigations of the National Research Council about 18 per cent of the energy of grain fed to cattle is recovered for human con- sumption in milk, but only about 3.5 per cent in beef. Similarly, crops on a given area will yield about four to five times as much pro- tein and energy when fed to dahy COW T S as when used for beef produc- tion. In providing mineral substances and vitamines the milk of cows contrasts even more favorably with the beef animal. The vitamines and calcium salts contained in hay and grain are stored in the muscular tissue only to a slight extent, but are in relative abun- dance in milk. 10 From the standpoint of the dairyman a thoroughbred prize cow, such as Glista Ernestine (a Holstein), which gave in one year 833 pounds of butter fat, and in one hundred days 10,000 pounds of milk, is a very model of efficiency, producing more than her own weight in milk each month. But from the point of view of the bo- vine species such record performances are gross inefficiency, approach- ing in some cases perilously near to total biological unfitness, for some of the record Jersey cows are probably unable, under the conditions of the stable at any rate, to raise their own calves the over-rich milk would probably kill the young animal. Network of Chains of Interrelated Species. The relation between man and the domesticated species of animals and plants on which he 8 I have here borrowed a felicitous phrase from an anonymous writer on the editorial page of the New York Times, February 10, 1921. 9 University of Minnesota, Agr. Exp. Station Bulletin 193, pp. 68, 69 (T. L. Haecker). 10 Jour. Franklin Inst., vol. 190, 1920, p. 155. ANALYSIS OF GROWTH FUNCTION 137 EO largely depends for food, in the present state of civilization, is only a particularly tangible, a particularly accessible example of an intri- cate network of relationships that connect more or less closely all living species. In this network each species or component is inter- laced, like a link in a meshed coat of mail, with other species, which in turn connect with still others, and so forth. In our effort to get some sort of mental grasp of the complicated interlocking of these elements we seize upon some one link, some one species or component, and we note, first of all, that whatever is eliminated from one com- ponent of a self-contained system must pass into one or more other components of the system. So, for example, the component Si may be a herd of cattle. The matter eliminated from this component goes in part as food to build up or sustain a human population; in part it goes as fertilizer on the fields to furnish nutriment for crops; still other parts are worked up into various industrial products, such as leather, glue, etc. We thus have, in schematic representation, Cattle \ Human population \ Leather s S S Fertilizer L \ \ s k On the other hand, the substance of the herd itself is recruited from certain other components of the system, grass, clover, corn, etc., so that we may further develop the scheme Clover Grass \ Corn / \ Cattle \ Human population \ \ Leather \ N Fertilizer Transformation Factors and Their Economic Significance. In general any one component thus appears as a link in a complicated chain or rather network of chains; the component S t , for example 138 ELEMENTS OP PHYSICAL BIOLOGY receives a certain fraction at, i of the mass VtXt eliminated per unit of time from the component St', it passes on to the component S& a certain fraction ik of the mass ViXi eliminated from Xi itself. / l~y The rate of growth n 2 of Xi is the balance of the sum Ui of all con- tributions it receives over and above the sum V\ of all the contri- butions which it makes to other components, thus dXi = mXi - viXi = Ui - Vi (10) at = ZatiVtXi - S/3 ik i>iXi (11) the first summation being extended over all those components St which contribute to Xi and the second over all those components 8k to which Si contributes. But we may also analyse the contributions to and from the com- ponent Si in another way. We may say that, of the total contribu- tions per unit of time UiXi to the mass Xi, a certain fraction yuUiXi is derived from Sf. Then (12) = v f Xf (13) 7 it and, substituting (13) in (10), Lastly, if the system is not self-contained, we must add a term Ji for "imports" per unit of time, and a term EI for "exports" per unit of time, that is to say, -^ = vtX f - ViXi + Ii-Ei (15) dt 7n When Si is the human species, the coefficients a, jS, 7 have an obvious economic significance. The restriction of this remark to the human species must not be taken to imply that there is in this feature some- thing wholly peculiar to man, but rather, that underlying our economic manifestations are biological phenomena which we share in common with other species; and that the laying bare and clearly formulating ANALYSIS OF GROWTH FUNCTION 139 of the relations thus involved In other words, the analysis of the bio- physical foundations of economics is one of the problems coming \ within the program of physical biology. Hints as to the direction in which we may or must look for light on this phase of our problem have now been noted upon several occasions. So it was observed that the components of a life-bearing system can be divided into two classes, relative to the component Si, namely, on the one hand those components S- } for which -^ ~^T was positive, or, as we may say, those useful to the species Si, those having for it a positive value, and. on the other hand, components 8% for which ^ 3i l <0, com- V.A ic at ponents harmful to Si, or having for it a negative value. Else- where we have noted the classification of components into replace- , able and indispensable components, a classification that at once recalls elementary economic reflections. These hints we note in passing. They may serve to put our minds in a state of preparedness for the more formal and decisive attack of the problem, to which we shall be led in the last division of our en- quiry, dealing with the dynamics of life-bearing systems. Jf STATICS CHAPTER XI GENERAL PRINCIPLES OF EQUILIBRIUM Repeatedly, in preceding chapters, occasion has arisen to refer to stationary states or equilibria. Inevitably, in the discussion of the kinetics of evolution, one is led to consider incidentally certain conditions and special cases in which the velocities of the changes in the evolving system are zero; when, that is to say, the system under discussion is in a steady or stationary state, in equilibrium. Viewed from this avenue of approach equilibrium presents itself as a special case of motion or change, namely motion or change with zero velocity. Indeed, something very like equilibrium occurs also with velocities that are merely small, not vanishingly small. In such case the phenomenon of moving equilibrium may present itself, as we shall have occasion to observe in greater detail in due course. Stationary states equilibria and near-equilibria play an im- portant role in nature, and it is desirable at this point to give them something more than incidental consideration; to sketch, at least in outline, their systematic study; to stake out, in the rough, that field which, in our survey of the Program of Physical Biology (Chapter V) was designated as the Statics of Evolution, and was systematized according to the schedule Statics "I Equilibria Moving equilibria Displacement of (steady states) equilibrium It will be convenient, in the development of the subject, to follow, in the main, the schedule thus set forth. Kinetic, Dynamic, and Energetic Conceptions of Equilbrium. While we shall, in this section, conceive a stationaiy state from the standpoint of kinetics, defining it as a state in which certain velocities vanish, it must be noted that there are also other concep- tions of equilibrium. Etymologically the word equilibrium is tied, in stricter usage, to a dynamic conception: Aequa libra, the poised balance, is symbolic of a state in which forces are balanced, in which the resultant force vanishes, 143 144 ELEMENTS OF PHYSICAL BIOLOGY A third conception of equilibrium, differing from the second, the dynamic, only in point of view, not in scope, is derived from a con- sideration of energy relations. A system in dynamic equilibrium is found to be characterized by the attainment of a minimum (or some- times a maximum) of certain functions having the dimensions of energy; a state in which the virtual work done in any very small displacement compatible with the constraints vanishes. 1 So, for example, a ball placed hi a hemispherical cup, is in equilibrium when its potential energy is a minimum compatible with the geometry of the system. More generally, equilibrium is, according to this view, defined as a state in which certain potentials have a minimum (or a maximum). Pedantic usage would demand that the term equilibrium be re- served for states satisfying the dynamic and energetic conditions of rest or invariability in time. It would deny the appellation equi- librium to certain states commonly so designated. Metabolic equi- librium, population equilibrium, and the like, are not true equilibria, in this narrower sense, but are steady states maintained with a con- stant expenditure, a constant dissipation, of energy. It is not neces- sary, however, at present, to lay any stress on this distinction. The occasional use of the word equilibrium in speaking of what is merely a steady state maintained with a continuous expenditure of free energy is not likely to cause any serious confusion; and we may as well take the usual liberties in the matter, whenever this course is dictated by convenience and does not offend against essential prin- ciples. Where express distinction becomes necessary, we may speak in specific terms of true equilibrium and quasi-equilibrium, respect- ively, to denote the two separate types included hi the generic term "stationary state" or "steady state." A complete treatment of the entire field of the statics of evolving systems should, to be entirely systematic, cover both types of stationary states. There are, however, two reasons for departing somewhat from such strictly systematic arrangement. The first is that the statics of true equilibria have been developed to a high degree in the discipline of thermodynamics, so that an exposition of the pertinent principles and conclusions would be little more than 1 Stability of equilibrium demands, further, that the work done on the system in any small, but finite, displacement, be positive, that the potential energy be a minimum (maximum being, in this case, excluded). GENERAL PRINCIPLES OF EQUILIBRIUM 145 a transcription into these pages of what can be found abundantly set forth elsewhere in the standard literature. However much one might be tempted, in the interest of a well rounded presentation, to sketch at this point at least an outline of the relevant chapters of thermo- dynamics, economy of space dictates the briefer expedient of referring the reader to the existing literature, so abundant that it seems super- fluous to mention titles. A second reason for passing lightly over true equilibria at this point, is that the steady states with which we are most frequently and most closely concerned in the field of organic evolution (our main topic here), are of the second class; not true, equilibria in the dynamic sense, equilibria in which all forces are balanced; but what we have termed above quasi-equilibria, states maintained constant or approximately so with a continual expenditure, a con- tinual dissipation or degradation of available energy. To such as these we shall give our chief attention, though in part our discussion will be framed broadly to cover indifferently either type of steady state. For the sake of example, too, reference will be made, on occasion, to systems evolving toward a true equilibrium; systems for which the law of evolution is capable of direct expression In com- paratively simple thermodynamical terms; systems which, by that very fact, are peculiarly adapted to serve as paradigms exhibiting the characteristic form of a law of evolution. General Equilibrium Condition. As has already been noted in- cidentally, the general condition for equilibrium, or, to be more precise, for a stationary state, is obtained by equating to zero the velocity of growth of each component of the system, thus = Fi(Xi, Xz, . . . X n ) (1) at F! = F z = . . . = F n = (2) This condition, in general, furnishes n independent equations, which determine one or more sets of values of the variables X, thus Xz ~ * ' (3) 146 ELEMENTS OP PHYSICAL BIOLOGY If the values C thus determined are real and positive 2 they evi- dently define an equilibrium or a steady state, the character (stability mode of approach) of which depends upon the nature of the roots X of a certain characteristic equation, as has been indicated on an earlier occasion. Different Types of Equilibrium. Graphic Representation. A particularly graphic representation of the different types of equilib- rium is obtained if, instead of seeking solutions of the fundamental equations (1) expressing Xi, X 2 . . . X n in terms of t, we on the contrary eliminate t from this system of equations. This is very readily effected by division, which leads to the new system dXi dXz dX n T! = ~F~* = ' ' ' " 71 (4) This system of equations defines a family of curves passing through the equilibrium points, which here appear as singular points. The situation is particularly transparent in the case of two variables Xi, X 2 , since this readily permits of plotting the integral curves in rectangular coordinates in the plane of the paper. We have already had occasion incidentally to employ this method of treatment in an example in Chapter VIII, in which the conflict between a host species and a parasite species was examined analytically. Without going into extensive technical details it is advisable now at least to enumer- ate and briefly describe the several types of equilibria and the topography, characteristic of each type, presented by the integral curves in and about a singular point. These types are somewhat numerous, even if we restrict ourselves to the case of two variables, and brevity is therefore imperative. Type 1. Roots Xi and X 2 real and negative. Equilibrium is stable; in- tegral curves run directly into singular point as in figure 27, A. Type 2. Roots Xi and X 2 real and positive. The topography is similar to that of type 1, but integral curves are traversed outward from singular point. Unstable equilibrium, figure 27, B. Type 3. Roots Xi and X 2 real and of opposite sign. Integral curves in general do not pass through singular points, but curve away from it. Unstable equi- librium, figure 27, C. 2 Masses cannot assume negative or imaginary values. Hence negative roots may fail to define equilibria; a similar statement holds regarding complex roots. GENERAL PKDSTCIPLES OF EQUILIBRIUM 147 Type 4. Roots Xi and X 2 complex, real parts negative. The integral curves are spirals winding into the origin, forever approaching it without ever reach- ing it. Stable equilibrium, figure 27, D, Type 5. Boots Xi and X 2 complex, real parts positive. The topography is similar to that of type 4, but integral curves are traversed outward from singular point. Unstable equilibrium, figure 27, E. Type 6. Roots Xi and X pure imaginaries. This gives rise to several dis- tinct subtypes. Subtype F. Integral curves are closed loops enclosing the origin. Process is purely periodic. Figure 27, F. Subtype G. Integral curves are spirals winding inward. Stable equilib- rium. This is the case treated in Chapter VIII, where a representative dia- gram will be found. Figure 27, G. Another subtype is similar to G but spiral winds outward. Unstable equilibrium. Subtype H. Integral curves are spirals winding about a closed loop. Types I AND J occur when \i = X 2 . Figure 27, 1, J. As an example illustrating the occurrence of two types of equilib- rium, two types of singular points, the topographic chart of the integral curves defined by the Ross equations for the spread of malaria under certain conditions is shown in figure 28. It will be seen that there are two singular points, one at the origin 0, unstable, of type C; the other at T, stable, of type A. This chart obviously suggests "stream lines" and a three dimensional model. Such a model (purely qualitative) is shown in figure 29. The feature of interest is that a singular point like 0, of type C, is represented by a col ("notch") in the landscape; whereas the stable equilibrium of type A is represented by a pit, as at the point T. While this model refers to a very particular case, it serves to bring out a noteworthy fact, namely, that there are necessarily certain regularities in the occurrence of the various types of equilibria. So, for example, it is clear that two pits of the character of the point T cannot occur without some other type of singular point between them, just as it is physically impossible for two mountains to rise from a landscape without some kind of a valley between. For a detailed study of this phase of the subject the reader must be re- ferred to the mathematical literature. 3 3 For further discussion of the various types of singular points that may occur the reader is referred to the mathematical literature, of which the following may be mentioned: E. Picard, Trait<5 d' Analyse, 1891, vol. 1, pp. 83, 123; 1893, vol. 2, pp. 183, 193, 196 (footnote); 1896, vol. 3, pp. 228, 238; v. Dyk, Sitzungsber Bayer. Akad. Wissensch. Miinchen, March 6, 1909,^AbhandL 15; Abhandlungen derKgl. Bayer. Akad. Wissensch., March, 1913, vol. 26,Abhandl. 10; Sitzungsber, 1891, p. 23; 1892, p. 101; H. Liebmann, Lehrbuch der Differ- ential gleichungen, 1901, pp. 101, 102, 134. ELEMENTS OF PHYSICAL BIOLOGY FIG. 27. SOME FUNDAMENTAL TYPES OP EQUILIBRIUM, IN A SYSTEM WITH Two DEPENDENT VAEIABLES GENERAL PRINCIPLES OF EQUILIBRIUM 149 FIG. 28. MAP OF INTEGRAL CURVES FOR THE Ross MALARIA EQUATIONS, AS AN EXAMPLE EXHIBITING Two SINGULAR POINTS, OP TYPE 1 AND 3 (SEE FIGURE 27, A AND C) The heavy lines are integral curves; the lighter lines are auxiliaries (isoclines) employed in constructing the graphic solution of the differential equations. (Reproduced from A. J. Lotka, Am. Jour. Hygiene, January Supplement, 1923.) 150 ELEMENTS OP PHYSICAL BIOLOGY . ,JH'/!' <>';'' X" \' A >' " "ft *M' f t' "'" s. '"-' ; GENERAL PRINCIPLES OF EQUILIBRIUM 151 Metastable Equilibrium. The graphic representations of the malaria equilibrium furnish the occasion for another remark of general character regarding certain equilibria. The circumstance that gives rise to the first malaria equilibrium, the one in which the malaria rate is zero, (point in figs. 28 and 29) is the autocatakinetic charac- ter of the growth, of a malaria endemic . This is a common character- istic of the growth of living systems; growth is initiated by a nucleus of the same species of matter that is added by the growth. Con- versely, in the entire absence of any nucleus of a particular species of living matter, growth of that species cannot take place, even though all other conditions for such growth may be satisfied, even though the system may he, as it were, supersaturated with regard to that species of matter. In these circumstances an equilibrium may be presented which is unstable in the sense that, upon the introduction of a suitable nucleus, growth immediately sets in. 4 Equilibria of this type, which are stable in the absence of a suitable "nucleus" but in which change is immediately initiated upon introduction of such a nucleus, have been termed "metastable" equilibria. Exceptional Cases: A brief reference must suffice regarding certain exceptional cases that may arise. So it may happen tbat one of the roots X of the characteristic equation vanishes. An example of this was encountered in dealing with the Ross malaria equations. It was found that as the number of mosquitoes per head of the human population approaches a certain critical value, two singular points approach each other, and finally fuse, giving one "double" point. 5 Another special case that may arise, and whose mention must here suffice, is that of multiple roots of the characteristic equation, the case in which two or more of the roots are equal. 6 4 In inorganic systems an analogous state of affairs is observed in super- saturated solutions or vapors which, are brought to crystallization or to con- densation by the introduction of a suitable nucleus. Dynamically the char- acteristic of a metastable equilibrium is that the thermodynamic potential of the system, though a minimum, is not an absolute minimum. 6 See A. J. Lotka, Am. Jour. Hygiene, 1923, vol. 3, January supplement, p. 12. 6 Compare H. Liebmann, loe. cit., pp. 102, 134; A. J. Lotka, Zeitschr. f. physikal. Chemie, 1912, vol. 80, p. 16. CHAPTER XII CHEMICAL EQUILIBRIUM AS AN EXAMPLE OF EVOLUTION UNDER A KNOWN LAW I wanted to remind the biologists that in the early stages of life what they are accustomed to speak of as natural selection passes over into what might be described as a mere physical selection of stabler compounds. K. Pearson. One of the simplest examples of equilibria in systems of the type that interests us here systems composed of several groups each consisting of numerous similar individuals as units is the equi- librium resulting from a pair of balanced or opposing chemical reactions. This case illustrates so well, in their simplest form, a number of typical traits of the phenomena here under discussion, that it will pay to give it brief consideration. We shall select for this purpose the simplest possible type of balanced chemical reaction at constant volume and temperature, namely a reaction which is monomolecular in both directions. A substance Si undergoes a transformation into S%, and Sz in turn is converted back into Si, one molecule alone taking part, in each case, in the transformation. If Xi and x z are the respective concentrations of Si and $2, we have, at a given temperature, by the law of mass action, the rate of decomposition of Si and Sz respectively. (Ztei) = - faxi (1) (D^) = - fax* (2) Or, since at constant volume concentrations x are proportional to numbers n of molecules (Dm) = - fciin (3) (Dnz) font (4) where fci, k 2 are coefficients (functions of the temperature) character- istic of the reaction. 152 CHEMICAL BQUILIBBIUM 153 The rate of increase of the substance S is the excess of its rate of formation 6 over its rate of decomposition, in strict analogy to the birth rate and death rate in a human population (fti-fcOm (5) at (6) at In a population of living organisms the material for the formation of new individuals must ultimately be derived from the bodies of those that have died. But the connection is a complicated one in- volving many steps. In the population of molecules here under consideration the relation between birth rate and death rate is of the simplest possible form. Each molecule of Si that "dies" becomes a molecule of 82, and vice versa. Thus equations (5), (6) assume the form and in equilibrium , , kiUi (7) at - = kiHi ^22 (8) at K\ If we fix our attention upon HI molecules of Si at the moment of their formation, we can apply to these particular molecules the equation (3), from which we have, by integration 1 * (10) similarly for S 2 = naCO)*-*** (ll) \ But 77:7 is the probability pi(a), at the moment of its formation, that a molecule of Si picked at random at such moment, will reach age o. The life curve for the molecules of Si is thus defined by Pi(a) = <r kia (12) 154 ELEMENTS OF PHYSICAL BIOLOGY and for Sz pj (a) = e-^ (13) while the mean lengths of life are ., /T = I Jo and in equilibrium ^ (16) ft 2 La the molecules are present in amounts proportional to their respective mean lengths of life, although, they are "born" in equal numbers, since fan* = AVZ.I. The significance of this is brought out in the diagram figure 30, in which the population of molecules is plotted "in age groups," for the substance Si and S 2 separately. Since the birth rate is the same for both populations, they begin at a common ordinate; but the curve for 82, the substance with greater k, greater force of mortality , lies entirely below that for Si; the areas of the two curves are, in fact, proportional to the mean lengths of life of the molecules of the corresponding substances. Thus, in the struggle for existence the stabler (fitter) molecules of Si have the advantage, being, on an average, longer-lived. There is thus an obvious analogy between the course of events in such a population of different species of molecules, on the one hand, and a mixed population of different species of organism on the other, an analogy which extends into details for the exposition of which space is lacking here. 1 The analogy is not a meaningless accidental circumstance, but depends on identity of type in the two cases. It can be said quite generally (so as to apply to either case) , that in a material system in which physical conditions vary from instant to instant and from point to point, certain individual constituents (molecules, organisms) may have a transitory existence as such, each lasting just so long as its conditions and those of its neighborhood continue within certain limits. Although the "life period" of each individual constituent may be thus limited, an aggregate of a number 1 For such details see A. J. Lotka, Am. Jour. Sci., vol. 24, 1907, pp. 199, 375. CHEMICAL EQUILIBRIUM 155 of such individuals may nevertheless have prolonged existence, pro- vided that the fluctuations in the conditions of the system, from point to point and from instant to instant, do not exceed certain limits, and that by some process or other new individuals are formed as the old are eliminated. Of the character of the fluctuations, and their relation to the "length of life" in the case of living organisms, more will be said in a later chapter. As to the circumstances, the fluctua- tions, that lead to the translation of a molecule in chemical reaction from one state into another, we may with advantage adopt a view- 3OOO FIG. 30. AGE DISTRIBUTION IN POPULATION OF MOLECULES OF Two SUB- STANCES IN MONOMOLECULAR CHEMICAL EQUILIBRIUM The birth rate per head, of both species, is the same as indicated by their common zero ordinate; but the species having the lesser force of mortality (reaction constant) predominates, as shown by the greater area under the corresponding curve. (Reproduced from A. J. Lotka, Am. Jour. Science, 1907, p. 208.) point set forth in some detail by the writer on an earlier occasion, 2 and expressed more recently by Professor Baly 3 in these terms : Every complete reaction consists of three separate stages, with each of which is associated its characteristic energy change. In general, molecules in the free state exist in a phase which is non-reactive, and in order to carry out any reaction it is first of all necessary to bring them into a reactive phase. This, which is the first stage of the reaction, requires that a definite amount 2 A. J. Lotka, Am. Jour. Sci., 1907, p. 213 et seq. 3 E. C. Baly, Photosynthesis. Nature, 1922, p. 344. j.uu ELEMENTS OP PHYSICAL BIOLOGY of energy should be supplied to each molecule, the amount necessary being the difference in energy contents of the initial phase and the particular phase necessary for the reaction in question. The second stage of the reaction is the atomic rearrangement whereby new molecules are produced, and it is this stage, and this stage alone, which is represented by the equation of the reaction. The third and final stage is the change in phase of the newly synthesised molecules, whereby they pass into their normal and non-reactive phases. These last two stages are both accompanied by an escape of energy. If the sum of the amounts of energy evolved in the second and third stages is greater than that absorbed in the first stage, the reaction is exothermic; whilst an endothermic reaction is one in which the energy necessary for the first stage is greater than the total amount evolved in the second and third stages. It should be remarked that the second stage is, apparently, passed through in an exceedingly brief space of time, so that at any instant only an imperceptibly small amount of substance exists in the transi- tional state. We are, in fact, almost wholly devoid of any informa- tion regarding matter in this state, and the words of Schonbein 4 hold true in almost their full force today: "Presumably, between the state in which two portions of matter exist after completion of chem- ical combination, and the state in which they previously existed separately, there is a series of transition states of which the chemistry of today knows nothing." Probably the only positive and direct experimental evidence we have of matter in this intermediate state between two compounds is furnished by the superlatively refined methods of Sir J. J. Thomson and Dr. F. W. Aston, which not only reveal but actually weigh such decapitated molecules as CHs, whose length of life is measured in ten-mil lionths of a second. 5 As to the agencies, the "fluctuations" that provide, every now and again, the requisite energy to carry a transforming molecule "over the crest of the hill," there is first the thermal agitation of the mole- cules, second the influence of incident light in photochemical reac- tions, and third the influence of catalysts, whose action probably depends on a flattening of the path over the hill crest, the point of 4 Jour. Prakt. Chem., vol. 55, I, p. 152. B Compare A. J. Lotta, loc. cit., p. 214. F. W. Aston, Science Progress, 1912; I. Langmuir, Jour. Am. Chem. Soc., 1920, vol. 42, p. 2190. Perhaps in this connection should be mentioned also recent studies on the duration of atoms in their several quantum states. See E. C. Tolman, Proc. Natl. Acad. Sci., 1924, vol. 10, p. 85; L. A. Turner, Phys. Rev., 1924, vol. 23, .> p. 464; F. M. Kannenstine, Astrophys. JL, 1924, vol. 59, p. 13. ^ CHEMICAL EQUILIBBIUM 157 departure and the final state remaining unchanged. For discussions of these technical details the reader must be referred to the literature, a few of the more recent publications being noted in a footnote below. 6 While the details of the manner of the "birth" and "death" of the molecules in chemical transformation are, as yet beyond the range of the observation of the physicist, the fundamental laws of ener- getics, which hold true generally, and independently of particular features of mechanism, are competent to give substantial information as to the end product, at any rate, of the evolution of such a system as considered in the simple example above. The final equilibrium must accord, as regards its dependence on temperature, pressure and other factors, with the second law of thermodynamics, which may thus be said to function as a law of evolution for a system of this kind. This is a point worth dwelling on a little at length, inasmuch as our knowledge of the form and character of the law of evolution for this special type of system may be expected to serve as a guide in the search for the laws of evolution in the more complicated systems, belonging to an essentially different type, which confront us in the study of organic evolution. The second law of thermodynamics can be expressed in various ways, but the form in which it serves our present purpose best is that which states that the system evolves toward a state in which certain functions (thermodynamic potentials) of the variables defining its condition are at a mrnimum, somewhat as a ball placed in a hemispherical bowl ultimately comes to rest in the position in which its (gravitational) potential is a minimum, namely, at the lowest point of the bowl. Mary laws of nature are conveniently 7 expressed in this form, as minimum (or maximum) 8 Regarding the r61e played by thermal agitation and by radiation see G. W. Todd and S. P. Owen, Phil. Mag., vol. 37, 1919, p. 224; I. Langmuir, Jour. Am. Chem. Soc., 1920, vol. 42, p. 2190; W. H. Rodebush, Jour. Am. Chem. Soc., vol. 45, 1923, p. 606; J. M. Lowry, Trans. Faraday Soc., vol. 17, 1922, p. 596; J. A. Christiansen, Zeitschr. phys. chem., vol. 103, 1922, p. 91. Regarding the influence of radiation see especially the publications of Professor Baly. See also J. Mellor, Chemical Statics and Dynamics, 1904, pp. 394, 414, 415; and Perrin and Hammick, Atoms, 1923, p. 168. The literature on catalysis is so extensive that no attempt is made here to give even a key to it. 7 Fundamentally this is a matter of convenience, and does not predicate anything narrowly characteristic of natural laws. The fact that the course of events is uniquely determined implies that the laws which determine that course can be expressed in the manner referred to. For a discussion of this question see J. Petzoldt, Maxima and Minima und Okonomie, Altenburg, 1891, pp. 17 et seq. 158 ELEMENTS OF PHYSICAL BIOLOGY laws, and it is to be expected that the law of evolution in life-bearing systems also, (where, as we shall see later, mechanism cannot be lightly waved aside into the convenient catch-all of the laws of thermodynamics), will be found to receive its most convenient expression in this form. In another respect the case of chemical evolution may confidently be expected to be found a good model in the treatment of the broader problem of evolution. It is to be noted that the law of chemical evolution is expressed in terms of the system as a whole. It is the thermodynamic potential of the entire system that approaches a minimum. Biologists have rather been in the habit of reflecting upon the evolution of individual species. This point of view does not bear the promise of success, if our aim is to find expression for the fundamental law of evolution. We shall probably fare better if we constantly recall that the physical object before us is an undivided system, that the divisions we make therein are more or less arbitrary importations, psychological rather than physical, and as such, are likely to introduce complications into the expression of natural laws operating upon the system as a whole. As regards the formulation of the laws of evolution in form of a maximum or minimum principle, it should be remarked that one such principle follows directly from the fundamental equations of kinetics as set forth in Chapter VI. If we multiply the first of these equations by Xi, the second by X 2 , and so on, we obtain at -2 . ~TT ^ ^1X1X2 + 022X2- +...-}- a2n.Xn.X2 + at (17) dt K n "" " nn- n. Hence by addition A -S & = 2Q(x h xt, . . . x n ) + . . . (18) where Q represents a quadratic form. The relation thus obtained is not of general utility in this form. However, by a linear substitution CHEMICAL EQUILIBRIUM 159 & = N lXl + N,X, + . . . + 2ST n a; n (19) the equation (18) can be transformed into ~ at (20) where Xi, X 2 , . . . X n are the n roots of the characteristic equation for X, the same X's that function as exponents in the series solution of the original system of equations. 8 Now it will be recalled that the condi- tion for stability at the origin is that all the real parts of the roots shall be negative. But in that case the quadratic form Q' is definite and negative. Hence the condition for stability at the origin can be expressed by saying that the quadratic form Q' must be definite and negative; or, by saying that Q' must have a minimum at the origin. And the law of evolution, near the origin, evidently is, according to (20), that S 2 continually decreases. (At points remote from the origin the terms of higher order, which have here been omitted, may cause increases in S.) The chief interest of the minimum principle here indicated lies in its analogy 9 to certain theorems in dynamics and thermodynamics, for which reference must be made to the literature, in particular to P. Duhem, Traite d'Energetique, 1911, vol. 1, pp. 460 et seq.; F. Michaud, Ann. de Phys., 1921, vol. 16, pp. 148 et seq. 8 For the sake of simplicity the argument has here been presented in the form in which it appears when all the roots X are distinct and real. For a detailed discussion of the conditions of stability when some of the roots X are multiple or complex see E. Goursat, Cours d' Analyse, 1915, vol. 3, pp. 31-43. 9 The analogy to the dynamical cases treated in the reference cited becomes particularly plain if we bear in mind that = when i =j= j = 2Xi when i j so that the quadratic form Q' can be written ' dQ' 5Q' 160 ELEMENTS OP PHYSICAL BIOLOGY where the bracketed exponent (2) denotes the symbolic square, in which '- } is> replaced by \JTC w\x W vjj -, and the product =-r- - >T~J i g replaced by -^ ^--. \ The condition that the form so defined shall be negative, is that the deter- minant shall be negative, and also all determinants derived from it by striking out the last p lines and the last p columns. In the present case the same condition, can be expressed in simpler form to the effect that Xi, \^, . . . \ a must all be negative. But it is worth while, in order to bring out the analogy, to note also the more complicated general form of the condition. CHAPTER XIII INTER-SPECIES EQUILIBRIUM Since the struggle for existence is chiefly a struggle for subsistence, a careful comparative account of the food of various competing species and genera at different places and seasons and at all ages of the individual .... can- not fail to throw much light upon the details, causes and effects of the struggle. Forbes. Equilibrium Condition in More Particular Form. The fundamen- tal relations of statics are derived immediately from the correspond- ing equations of kinetics by substituting in the latter the value zero for the several velocities. This has already been noted with regard to the equations of kinetics in their most general form. In somewhat more particular form, useful in common numerical applications, we have a condition for equilibrium derived from the system of equations (14) of Chapter X . X t - ViXi (1) dt 7if (2) 7 if X f JIM i (3) It should be noted that these formulae hold equally well if the masses are measured in ordinary units (e.g., pounds) or if they are measured in "head of population," with the proviso, of course, that the coefficients ui, Vi, are in each case expressed in corresponding units. The equation (1) expresses the fact that, for each component, the total inflow is just balanced by the total outflow, so that nowhere in the system is any accumulation of mass going on. This clearly implies that, unless there is complete equilibrium, the matter in the system must be in circulation, it must be going through one or more cycles. Such cycles are, indeed, very characteristic features in the scheme of nature. 161 162 ELEMENTS OF PHYSICAL BIOLOGY Numerical Illustration. We may, by the way of illustration, apply the formula (3) to the equilibrium between the several biological species comprised in a life-bearing system. For obvious reasons numerical data are most readily available for man and the species directly under his control. So, for example, we may let Xt. represent the mass (or number) of a human population, and Xt the mass (or number) of a population of sheep serving as food for that human population. In the United States in 1918 the consumption of mutton (or lamb) per head of the population per annum was 5.417 pounds. This is not strictly an equilibrium ration, since our population is increasing. However, the difference between this and the equilib- rium ration is probably small. In our example we will therefore put jitui = 5.417 pounds = 0.1096 sheep 1 (4) Again in the United States in 1918 the number of sheep slaughtered per annum was 23.22 per cent of the standing herd. Hence a fi t>i = 0.2322 (5) so that Xf = "^ Xi (6) f 0.2322 = 0.4718 Xi W In 1918 Xi = 103,587,955 head (8) Hence Xt = 48,873,000 head (^ i According to the Year Book of the Department of Agriculture, 1920, p. 759 the number of sheep slaughtered under Federal inspection m 1918 was 8 769,498. According to R. Pearl, The Nation's Food, 1920, p. 61, this repre- sented 77 per cent of all the sheep slaughtered in that year, so that the total number slaughtered was 1 1,370,000. The total dressed weight of these, accord- ing to the Year Book, p. 826, was 502,214,000 pounds, which makes the average of one Bhoop carcass 49.45 pounds. The standing herd of sheep in 1918, according to the Year Book, p. 701, was 48,C03,000 on farms, or, adding a correction for animals not on farms ! say 48,963,000. The percentage of animals slaughtered in a year out of the stand- ing herd, was therefore 23.2216 per cent. For a review of various estimates of the output of herds of cattle, sheep swine, etc., the reader ia referred to a paper by R. H. Rew in the Journal of the Royal Statistical Society, 1902, vol. 65, p. 666. INTER-SPECIES EQUILIBRIUM 163 The actual standing herd of sheep in 1918 was 48,963,000 head of which 90,000 head furnished mutton for export, the United States not being a self-contained system. It is to be noted that in this example the products 7jf U[ and oifi vi are more easily ascertained than the individual coefficients <y it, a, v. Inasmuch as these coefficients, in the formula (3), appear only in these products, it is not necessary, for the purposes of this example, to ascertain the values of the coefficients separately. The example of the equilibrium between a human population and the national herd of sheep, cited primarily for the purpose of illustrat- ing the equilibrium equation (3), incidentally brings out some other points that may be noted in passing. We meet here a pointed sug- gestion of economic factors entering into play in the processes which we are studying. For the coefficients a 7, u, v, have obvious economic relationship. So, for example, Vf, the proportion of sheep slaughtered per annum (in a stationary state of the system), is of the nature of interest on the standing herd, which latter, in turn, is of the nature of capital. The gross interest rate of 23.2 per cent is, of course, greatly diminished, in effect, by the extensive accessories, representing a further investment of capital, and by the general running expenses required, in addition to the mere herd, to produce, transport, and market the ware. On the other hand, certain secondary products (e.g., wool) add their quota to the returns on the invested capital. Again, the coefficient Aif, which measures what fraction of the total consumption by the component Si (human population) is derived from St (sheep), is clearly a factor of economic significance. Mutton .is typically a commodity of the replaceable type beef, pork, fish, etc., furnishing ready substitutes. In such case as this the factor 7 if will be elastic, capable of assuming, in ready response to slight changes in general conditions, a whole range of values from zero up. In the case of less readily replaceable commodities the coefficient 7 wih 1 be of more rigid habit. The full significance and the precise nature of the relation between the biological and the economic characteristics of a system must form the subject of special considerations to which we shall find ourselves led inevitably later, in our efforts to gain insight into the dynamics of life-bearing systems. At this juncture it may not be amiss to indicate in preparation of a viewpoint to be more fully developed later, that if economic factors force themselves upon our notice primarily in the 164 ELEMENTS OF PHYSICAL BIOLOGY consideration of systems comprising a human population, this is riot because the operation of economic stresses is peculiar to human aggregations, but only because these stresses find their ready numerical expression and measure in such communities; owing to the develop- ment of a system of social cooperation and division of labor, coupled with a very special mechanism of adjustment by "economic exchange/' which is peculiar to man. For though not a few other species, bees, ants, etc., display a social organization in some respects perhaps superior to ours, their organizations make use of other expedients than the transfer of ownership through a universal medium of exchange, in bringing about the allocation, to each indi- vidual, of his share in productive effort and in product. For obvious reasons our information regarding the interdependence of the several biological species and other components of our life- bearing system is most complete and most exact in so far as it relates to species under human cultivation, species that contribute, as pro- ducers, to our political economy. However, this does not mean that we arc wholly cut off from all information regarding the life balance of other species. Two sources, two methods of observation furnish us with data on this subject, namely, first, biological surveys, and second analyses of the stomachs of sample specimens. A third method would coiiHisli in establishing experimental systems comprising several HpocioH of organisms and making periodic censuses by direct count, after tho manner of the work of Pearl and Parker with a single tuxu'ioB (Drosophila). There is here an attractive field open for wViirch Perhaps the readiest, though not the most interesting approach to this problem would be the study of mixed bacterial crowbhH ay along the lines followed, for a single species, by H. G. Thornton, Annals of Applied Biology, vol. 9, 1922, p. 241. Biological Surveys. Biological surveys, supplemented by esti- mate* depending more or less on personal judgment, are aimed to rivo UB Jmo degree of quantitative description of our world by mves- oranisms. A the number and the variety of living organisms. biolORioal survey would enumerate the several general and ound in the locality examined, and would furthermore give mowuro of tho extent of each species, either in numbers or m some 1 "u tohte tonne. It would give us a species of "General Demol- v'' cor globe. Needless to say, in this matter we are very far lined perfection. The best that can be done is to INTER-SPECIES EQUILIBRIUM 165 give rather crude estimates, based, in the most favorable instances, on counts or observations made with some degree of care, but without pretense of great precision. As to the number of species, some interesting figures are given by J, A. Thomson. 2 On the small island of Britain alone 462 different birds have been observed; the total number of living species of birds he estimates at not less than ten thousand. Of vertebrates he quotes an estimate by H. Gadow. 3 The total number of recent species this author puts at 24,241, as follows: Mammals 2,702 Birds 9,818 Reptiles 3,441 Amphibians 925 Fishes 7,328 Primitive vertebrates 27 24,241 The vertebrate elite, however, forms but a small minority in the scheme of nature. It has been intimated that, if the present order of things should come to a term, the supremacy would, as likely as not, pass from the crowned vertebrate Homo sapiens to the now despised, presently perhaps to be feared, creeping thing, the insect. Indeed, it has been pointed out that were it not for the relentless internecine warfare which its members carry on among themselves, we should very soon find ourselves driven out of house, home and granary by the insect pest. Even as it is, though the largest insects barely exceed, individually, the size of some of the smallest verte- brates, yet, as D. Sharp remarks, "the larger part of the animal matter existing on the lands of the globe is in all probability locked up in the form of insects." He estimates th,e number of insect species that have been definitely named, at 250,000, and suggests that this is only about a tenth of the total. The number of plant species has been estimated at 200,000. Darwin records the finding of 20 species in a patch of turf four feet by three. As to the numerical strength of the several species, here again some telling figures are given by J. A. Thomson. 4 2 J. A. Thomson, The Wonder of Life, 1914, p. 11. 3 See also H. Gadow, The Wandering of Animals, 1913, p. 74. * J. A. Thomson, loe. cit., pp. 9-10. 166 ELEMENTS OF PHYSICAL BIOLOGY _ At the spring maximum of the Rotifer Synchoeta there may be about three millions to a square yard of lake. At the summer maximum of the slimy Alga Clathrocystis ceruginosa there may be 500 millions to the square yard; at the autumn maximum of a well-known diatom Melosira varians, which has a sum- mer maximum as well, there are about 7000 millions to the square yard, so that the waters of the lake form a veritable living soup In an ordi- nary sample from a warm part of the Atlantic and from a depth of 500 metres (which is the most densely populated as far as plants go), there are likely to be about 5,000 plant-cells to a liter; but there may be as many as a quarter of a million Elsewhere Thomson tells us that in the midst of a swarm of fish at spawning time in the Norwegian fjords a boat may be so lensely packed in among the mass of fish that an oar stuck upright into the swarm remains standing for an appreciable time after the hand relin- quishes its hold. 5 Examination of Stomach. Contents. The second method by which information has been gathered regarding the interdependence of biolog- ical species consists, as already stated, in examining the contents of trie stomachs of sample specimens. This method has been applied particularly to birds and fishes. The results of such an analysis of the feeding habits of the common crow and of the starling are strik- ingly brought to view in the accompanying charts figures 31 and 32, reproduced by courtesy of the Department of Agriculture from Bulle- tins 868 and 1102. Such a chart does not, of course, yield any direct information regarding the relative abundance of the several species upon which the crow feeds, but it does give us at least an indication of a resultant compounded of that relative abundance and a number of other factors, such as the preference or selective tastes of the crow, the greater or less degree of protective characters with which nature has endowed the various species exposed to attack, etc. The converse problem, also, has been investigated, namely the extent to which different species of birds participate in the destruction of one selected noxious insect. So, for example, Bulletin 107 (1914) of the Department of Agriculture lists 45 different species of birds that were found to have fed upon the alfalfa weevil. A similar study by H. C. Bryant 8 was carried out during a grasshopper outbreak in B For an account of bird censuses in the United States see M. T. Cooke, Bulletin No. 1165, Bureau of Biological Survey, U. S. Department of Agriculture. 6 University of California Publications in Zoology, 1912, no. 1, vol. 11. INTER-SPECIES EQUILIBRIUM 167 California, when "in the infected areas the grasshoppers were com- puted to number from 20 to 30 per square yard." Bryant's results are shown, in part, in tabular and diagrammatic form in figures 33 and 34 3 reproduced from his original publication. . 31. SEASONAL ANALYSIS OF THE STOMACH CONTENTS OF CROW ^ *' ; ; '^jfiftitAr.^' i "~\^^/&8SS8 V^&3PI FIG. 3:2. SEASONAL ANALYSIS OF THE STOMACH CONTENTS OF STARLING The same author has also given us a classic in his extended study "A Determination of the Economic Status of the Western Meadow Lark in California." This paper contains among other things a 1(58 .ELEMENTS OF PHYSICAL BIOLOGY detailed bibliography up to 1913, and a historical survey of the methods employed and the investigators who have labored in the : '' ' I" 1 :: :-"'.: :':: .. :: ? ; .V . ,; :, ' ;-:. ,;..'. . ..,'' ,';! V. ::..'::- ::::,.:.:.; 'r-'.; ! | . ., g , 'if r w Bullock Oriole- Cliff Swallow- 4,050 2.2S5 Western Kingbird- 1,260 Burrowing Owl-.- I.2SO California Shrike- 1,200 Anthony Green Heron- 1,050 Sundry- 593 TOTAL DESTRUCTION OF GRASSHOPPERS PEP SQUARE MILE DAILY, IS0.453 FlG. 33. COMPARATIVE DAILY DESTRUCTION OF GRASSHOPPERS BY SEVERAL SPECIES oir BIRDS After H. C Bryant ANIMAL FOOD VEGETABLE FOOD Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. TOTAL VM/M/////////A, Fia. 34. SEASONAL FOOD HABITS OP THE MEADOW LARK After H. C. Bryant field. The results obtained by Bryant exemplify the proverbial voracity of birds. Young birds require about one-half their own weight in food each day. In the course of a year the average meadow IXTEK-SPECIES EQUILIBHIUM 169 lark, weighing say, 4 ounces, consumes about the following quantities of food. pounds % ft FIG. 35. STOMACH CONTENTS OP A BREWER'S BLACKBIRD Since an adult bird weighs about 4 ounces, this means that it con- sumes on an average about 24 times its weight of food in a year. Dr. Bryant remarks: "If we consider that there is an average of one meadow lark to eveiy two acres of land available for cultivation (11 million acres) in the Sacramento and San Joaquin Valleys, and that each pair of birds raises an average of four young, it takes over 343| tons of insects each day to feed the young birds in the valley alone." His findings regarding the seasonal changes in the food of the meadow lark are summarized in a number of charts of which one is reproduced in figure 38. An illustration of the voracity of birds, and their destructiveness to insects is also seen in figure 35 (from Bulle- 170 ELEMENTS OP PHYSICAL BIOLOGY tin 107 of the Department of Agriculture) showing the stomach con- tents of a Brewer's Blackbird, for this bird was found to have gorged upon 374 larvae, 65 pupae and 3 adults of the alfalfa weevil. Obser- vations on the feeding habits of young English sparrows are recorded by E. R. Kalmbach in Bulletin 107 of the United States Department of Agriculture. This author remarks (p. 54) : From a series of five observations it appears that the parent English spar- rows visited their nest on an average about once every 5 minuted, or a little more than 11 trips an hour. The four adults captured had as food for tlieir young 2 kernels of wheat; 17 alfalfa weevil larvae; 1 ground beetle, 9 weevil larvae and a caterpillar; and 23 weevil larvae, respectively. Three other adults taken in the fields had food for nestlings in their bills. This amounted to IS weevil larvae and an aphid in the first, 5 larvae in the second, and 3 coccinellid larvae. 13 weevil larvae, and 2 pupae in the third. Though this is rather heterogeneous assortment, it would appear that 15 larvae of the weevil or their equivalent in bulk of other insects would be a fair estimate of an average amount of food brought in at each trip by adult birds. In fact, it is certain that the material brought in frequently greatly exceeded this amount. Allowing 15 larvae at each trip and 11 trips per hour, these birds would bring in 165 larvae per hour. Then, assuming that the young were being fed for 12 hours each day, a conservative estimate, we would have a total of 1980 larvae consumed by one brood in one day. Straw-thatched sheds containing upward of 100 nest holes, both old and new, are frequent, and it is not uncom- mon to find farmyards where this number of nests are occupied. There are also ample nesting sites about the other buildings and in the ever-present Lombardy poplar, cottonwood, or box elder. Such a colony of birds would devour a daily total of 198,000 larvae, or an equivalent bulk in other food. As the young birds remain in the nest for at least 10 days and are probably fed several days longer by the adults, they will have eaten food equivalent to the bulk of 1,980,000 larvae during their nestling life. Intra-Species Equilibrium. As has been remarked on a previous occasion, it is not intended, in this volume, to take up the discussion of the evolutionary changes within the confines of a species. Passing notice may, however, be given to the fact that the equilibrium within a Mendelian population has been discussed by G. H. Hardy 7 and by R. C. Punnett 8 and latterly by J. B. S. Haldane. 9 (See p. 122.) 7 G. H. Hardy, Science, 1908, vol. 28, p. 49. 8 R. C. Punnett, Mimicry in Butterflies. 8 J. B. S. Haldane, Cambridge Philosophical Society, 1923. CHAPTER XIV INTER-SPECIES EQUILIBRIUM AQUATIC LIFE Aquiculture is as susceptible to scientific treatment as agriculture; and the fisherman, who has been in the past too much the hunter, if not the devas- tating raider, must become in future the settled farmer of the sea, if his harvest is to be less precarious. W. A. Herdman. From what has already been set forth the direct economic im- portance of studies in general demology should be sufficiently clear, if this term be used to denote the quantitative study of the popula- tion of the several species of organisms living together in mutual interdependence through their food requirements, feeding habits, and in other ways. In no other field, perhaps, has the study, from this angle, and under this economic impetus, been so systematically undertaken, as in the biology of aquatic species. On the one hand the three- dimensional extension of the systems here involved (as distinguished from the essentially two-dimensional spread of land species over the earth's surface'), facilitates, in certain respects, the operation of sampling (by the use of the dragnet) and counting; on the other, the close relation of such investigations to the practical problems of our inland and our ocean fisheries has furnished alike the economic occasion and the financial support for work on an extended scale. The methods employed have by this time developed into a more or less standardized technique. The dragnet, already referred to, and the stomach and gills of fish, acting, as it were, as natural drag- nets, themselves caught within the collector's man-made dragnet, are among the principal accessories in this 'field of investigation. L. H. Tiffany recommends particularly the gizzard shad as a con- venient collector and sampler of Algae. 1 Contrasting these natural samplers with man-made contrivances he remarks: These living tow nets (i.e., gizzard shad) do not get caught on snags and roots, the string does not break, and the algal collection is very representa- tive of the body of water from which the fish were taken. It is only neces- 1 Science, 1922, vol. 56, p. 285. 171 172 ELEMENTS OF PHYSICAL BIOLOGY sary to catch the young fish and examine their stomachic and intestinal content to secure a proportionate sample of the plankton. Tiffany examined specimens from streams and ponds in Illinois, and also from Ohio. He points out that the gizzard shad fulfills an important rule as an intermediary link in the food chain: algae, shad, game-fishes, man. Thus, the gizzard shad is making useful for man the energy stored in plant forms which occupy no land areas, which do not interfere with the ordinary disposition or utilization of bodies of water ('except the occasional contam- ination of water for drinking purposes by some algae), which involve no labor of cultivation on the part of man, and which are of no value for direct humaa consumption. The world's population in the last hundred years has increased about 150 per cent. Along with this increase has had., to corne a corresponding increase in the world's food supply. One of the ways in which this necessity- has been met is the securing of new acres of soil in which to grow crops. It is easily seen, however, that there is a limit to new acreage. In the future, therefore, we may have to turn more of our attention to the cultivation of the waters for food supplies. We may have to develop an industry of aqui- culture as we have developed an industry of agriculture. The time is rapidly approaching when fish will be more highly prized as food and more extensively used than now. As that time comes, the cultivation of algae will be a first step toward greater fish production. A second step may be the introduction of fish like the gizzard shad into fish ponds and lakes to make more readily available the phytoplankton for fish food. In a summary survey of the Food Resources of the Sea, 2 G. W. Martin makes similar observations. He remarks: "So far as the actual cultivation of the sea's resources, as distinguished from their mere exploitation, is concerned, we have made only the feeblest beginnings." Somewhat in the same vein is W. A. Herdman's comment: "Aquiculture is as susceptible to scientific treatment as agriculture; and the fisherman who has been in the past too much the hunter, if not the devastating raider, must become in the future the settled farmer of the sea, if the harvest is to be less pre- carious." Viewing the matter from a slightly different angle W. F. Wells of the New York Conservation Commission draws attention to the fact that in our modern great cities, with the widespread adoption of the water system of sewage disposal, valuable fertilizer material is lost from its natural place in the fields. By enriching 2 Scientific Monthly, 1922, p. 456. INTER-SPECIES EQUILIBRIUM 173 the vegetation in the water and furnishing abundant life thereto, it may again be restored to the people as fish and shell fish in the place ^of beef and mutton. And the exchange is not such a bad bargain. For it has been found that in carp ponds, for example, the production of market ware was 95 pounds per acre, as contrasted with 73 pounds of beef per acre of farmland. 3 Perhaps one of the most telling illustrations of the economic importance of pisciculture is presented to us in the Alaska purchase: Within fifty years after the acquisition of our Northern province, it had yielded, in its salmon fisheries alone, seven and a half times its purchase price. The quantitative study of ocean life (on which must be based an intelligent system of marine aquiculture) may be said to elate from 1880, when Henson introduced the use of the dragnet. To this has since been added the use of bottom samplers or grabs; also an ingenious, if less trustworthy method of making a census of the marine population, which consists in catching a number of live fish, marking them, throwing them back in the water, and then noting the^ percentage of marked fish in the fishermen's catch during the period that follows. In this way it has been estimated, for example, that the North Sea contains about fifteen hundred million plaice, a figure about equal to the earth's human population. The field of utility of the dragnet and most of the other methods described is evidently limited to the larger denizens, such as are effectively held in the meshes of a net. Even a fine silk net will fail to hold ^a very numerous constituent of the population of the sea, a constituent that is highly important not only on account of its great extent, but also because of the role it plays in the food traffic of marine life. Lohmann showed (1911) how these fine organisms (the nannoplankton') can be collected by the use of a centrifuge. 4 Allen has developed a special dilution culture method of count for the nannoplankton organisms, which is modelled after the pattern of bacterial count technique. He showed sea water to contain 464,000 organisms per liter (exclusive of bacteria). Al- * It is true that beef is much superior in food value, pound for pound, but it is also much more costly to produce. It should also be noted that the yield from carp ponds is very high as compared with that of marine fisheries E J. Allen (Food from the Sea, 1917, quoted by W. F. Thompson, Scientific Monthly, 1922, p. 546) estimates the yield in the North Sea at 15 pounds per acre. 4 G. W. Martin, loc. cit., p. 461. 174 ELEMENTS OF PHYSICAL BIOLOGY lowing for systematic errors Allen thinks a population of one million organisms per liter (one per cubic millimeter) to be a conservative estimate. This is not excessive crowding, in view of the size of these organisms, as illustrated by the diagram, figure 36, reproduced from G. W. Martin's article. Before the extent and significance of this nannoplankton was realized, the amount of food required by the animals of the sea seemed so much in excess of the amounts \ FIG. 36. DENSITY OF MICROORGANISMS (EXCLUSIVE OF BACTERIA) IN SEA WATER About one organism, measuring 6 microns in diameter, is found per cubic millimeter of water. The relative dimensions are here shown. After G. W. Martin. revealed by the earlier methods of collection, as to give rise to the suggestion (Putter 1907-1909) that the nutrition of marine animals was on an entirely different plane from that of land animals, and that a large number of them, especially the smaller ones, absorbed dissolved organic matter directly from the water, without the mediation of plants. Putter's arguments have not been generally ac- cepted, and more recent studies have invalidated many of them. Never- theless, it is possible that something of this sort is more general than we realize. Mitchell (1917) reported an experiment strongly indicating that an oyster can utilize dextrose dissolved in sea water. The most elaborate attempts to calculate the production of the sea have been those of the Danish biologist Petersen and his associates. As a result of their studies, these workers have come to the conclusion that the plank- ton plays a very small part in the nutrition of the animals of the sea, and INTEK-SPECIES EQUILIBRIUM 175 176 ELEMENTS OF PHYSICAL BIOLOGY that the fundamental food of all marine forms in northern waters at any rate is the "dust fine detritus" of the sea bottom, derived primarily from the eel grass, Zostera. These investigators have studied in particular the conditions in the Kattegat, a rather shallow arm of the sea between Denmark and Sweden, about 150 miles in extreme breadth and 90 miles in extreme width. Their principal conclusions are exhibited in the diagram figure 37, and are as follows: It is assumed that about half of the total amount of Zostera annually produced in this area is washed elsewhere by the currents. The balance, estimated at 24,000,000 tons, serves as the basis for the animal life of the area. The useless animals, that is, those that are of no value to man and do not serve as food for fish, feeding directly on the Zostera, amount to about 5,000,000 tons. Useful animals, mainly those capable of serving as food for fish, are estimated at 1,000,000 tons. These are not all utilized by food fish, however. Starfish account for perhaps 200,000 tons; 500,000 tons are eaten by the larger gastropods and crustaceans, of which only a part are consumed by fish; while plaice and other flatfish consume about 50,000 tons producing 5000 tons of human food annually. Cod are much less economical, since they get their food at third hand, so to say, and each ton of the 6000 tons produced annually represents about one hundred times as much of the original synthesized organic food. On the other hand, the cod help to keep down the predatory gastropods and crustaceans (see figs. 38 and 39). The herring is the most important food fish feeding on the plankton, (mainly on copepods) and it in turn is eaten by the cod. Perhaps the most striking feature brought out by these figures is the comparatively trifling amount of human food finally produced from such a large amount of organic material. Summary of Methods. A tabular summary of some of the princi- pal methods by which data have been secured regarding relative and absolute frequency of organisms and species is given in table 10. Food Chains. It has already been remarked, in dealing with the general kinetics of the type of systems here under consideration, that each component of the system appears as a link in a chain or a network of chains, receiving contributions from components (sources') above, and discharging material into other components (sinks') below. Food chains, such as spoken of by Tiffany in the passage quoted on page 172, are a particular example of such chains of components. The study of food chains is one of the important tasks of the economic biologist. For we cannot afford to restrict our attention to the immediate source from which we draw our INTER-SPECIES EQUILIBRIUM 177 By courtesy of the Illustrated London News FIG. 38. THE VABIED DIET OF THE CODFISH 178 ELEMENTS OF PHYSICAL BIOLOGY -a .f CO a o ft he O o> ri o I--- S.2 !- ft 02 .5 INTER-SPECIES EQUILIBRIUM 179 supplies of food for the human population. The sources of these sources also demand attention. The problem forces itself upon our notice primarily (in the present state of society) in connection with agriculture. The fields cannot continue indefinitely 'to yield undiminished annual crops if the materials drawn from them are not in some way replenished. One important constituent needs no human intervention: carbon dioxide, owing to its gaseous form, automatically seeps in by diffusion as fast as it is absorbed by the FIG. 30. KEY TO FIGUKK 3 green plants. The same is true in some degree of free nitrogen , though the capacity of plants to assimilate this element is ^ strictly limited. 5 Water, also, is, in most agricultural areas, provided by the automatic meteorological processes of evaporation and conden- sation in rainfall. But as to certain other essential materials, notably combined nitrogen, phosphorus, potash and sulphur, the inherently immoUW constituents of the fertile soil, for these auto- * Science. November 24, 1922, p. 605. "Fiir sicli micht beweglich," Liebig, Die Chemie in ihrer Anwendung auf Agricultur, 1876, p. 382. 180 ELEMENTS OF PHYSICAL BIOLOGY matio replacement does not occur in sufficient measure ^ to satisfy the agricultural needs of the densely populated countries of this age. It becomes necessary for man to feed his food. Early man and primitive man, may reap where he has not sown. BuMong ago our tribe turned from the life of a nomad and hunter to tilling the soil and to animal husbandry. Thus was established a system of symbiosis with the links next above us in the food chain : the harvest ripening on the plain; and the cattle, grazing upon the pasture in summer, fed from the crib in winter. But early agricul- ture was essentially of the nature of a mining industry. It drew from the soil as from a bottomless well, without thought of a possible exhaustion of the source, or of any feasible replenishment to dimin- ishing resources. Except that, by a semi-automatic, semi-empirical process, natural fertilizer was allowed to restore to the soil at least a part of its strength to bring forth a crop. One more step forward and man graduated from mining farmer into manufacturing farmer. The field became a factory fed with raw materials in the form of saltpeter, potash salts and phosphate fertilizer, imported, if need be, from afar; and producing its output of agricultural flora arid fauna. Lastly, in our own generation, we have learned to divert into the life stream the sluggish element, so essential to life, so ill- named by the French -azote; to make ourselves independent of the saltpeter beds, to assure our future against a nitre famine by open- ing the inexhaustible mine of the atmosphere. It is a singular thing that this element, so accessible, so abundant, in which we arc literally bathed within and without, every instant of our life, should so long have remained foreign to our industrial economy. Strange circumstances, yet not without close parallel. For even now we are powerless to avail ourselves effectively of the golden flood of energy that daily pours upon us without limit from above while we turn earthward to dig laboriously for the plainly exhaustible supply of coal to supplement our limited bodily energies. Food Chains in Aquatic Species. The principle that long food chains are essentially wasteful finds particular application also in the practical problem of the economic and rational utilization of marine organisms for human sustenance. As Professor Martin observes, the most economical course would be to utilize marine vegetation directly as food for man and his domestic animals. The INTER-SPECIES EQUILIBRIUM 181 use of algae as food for the table can hardly be expected to become an item of any consequence. Its use as cattle fodder presents better prospects. But our chief reliance will no doubt continue to be on the assimilation of marine plants by fish. To quote again Professor Martin: 7 Since man prefers to harvest the plant life of the sea indirectly, those animals which feed directly on the plants are able to increase with less waste and at a more rapid rate, considered in total populations, than those which feed on other animals. Most of our food fish, for example, feed on smaller fish; these in turn feed upon small crustaceans and the latter eat the micro- scopic plants and detritus, so that in many instances the fish we eat are re- moved three or four steps, perhaps more, from the original food source. This is more significant than may seem apparent at first glance, since it involves an enormous waste. Before any organism can grow, the energy needed merely to live must be supplied, and by the time a crustacean is eaten by a minnow, or a minnow by a food fish, it will, on the average, have consumed a quantity of food several times its own weight. These facts are well brought out in the statistics of Petersen, and the diagram figure 37 based thereon. The edible shellfish, however oysters, clams, mussels and the like feed for the most part directly on the marine plants and this is one reason why the extension of the shell fisheries represents so much promise. Another advantage of this branch of aquiculture is also to be noted, namely, that most shellfish, like land crops, stay where they are planted. Even the scal- lop, which can swim about after a fashion, is restricted in its movement, and could readily be controlled. Oyster culture is already a great and im- portant industry, but it has not nearly approached its possibilities. Clam culture is still in an embryonic state, and scallop culture has yet merely been suggested. When some of the problems confronting the establishment of these industries have been solved, we may hope to have acquired addi- tional information concerning the ecology of the sea, which will help us in our approach to the more difficult problems of the future. Primary, Secondary and Tertiary Foods. It is often convenient to classify the foods for human consumption according' to their relative position in that portion of the food chain which is under human control. Commonly the meats of domestic animals are 7 For some further bibliographic indications regarding the food consumed by fishes see A. S. Pearce, Ecology, vol. 1924, p, 258. See also W. A, Herd- man, Founders of Oceanography, 1923; J. Johnstone, Introduction to Ocean- ography, 1923. 182 ELEMENTS OF PHYSICAL BIOLOGY classed as secondary foods, since the production of these meats takes plate in two steps, first the growing of the fodder (for the most part materials not comestible by man), and second the trans- formation of a part of this fodder, in the animal economy, into food adapted for human consumption. On the other hand the fisher- man's haul is for us primary food, growing feral, without human intervention. This classification must not be pressed. Where fields are supplied with fertilizer it might well be maintained that the cr^ps of wheat, potatoes, etc., commonly classed as primary, are secondary foods, while butchers' meats are tertiary. Even the catch of fish, and huntsman's quarry may not be strictly primary, in so far as game laws and regulations regarding the pollution of lakes and rivers represent some degree of symbiotic intervention on the part of man. Fine points apart, the distinction between the primary foods (crops and fish) and the secondary foods (butchers' meats) is economically most significant, for the consumption of fodder by farm animals is an item not merely comparable with human food consumption, but exceeding this latter manyfold. The fact, of course is, that farm animals are far from being economical and efficient converters of raw materials into food for human consumption. They repre- sent a luxury, a humoring of the tastes of men at the expense of their purses. With the present density of population we can afford the luxury. Presumably the future will see retrenchments, with pastures and cornfields converted to wheat. Still tastes arc not accidental thiogs. Allowing for vagaries and exceptions, the things we like are, on the whole, good for us and for the species. Whether man can maintain his present status with a materially abridged meat ration is perhaps an open question. Should the answer be in the negative, the conclusion would seem to be forced upon us that an overcrowding of the earth would react unfavorably upon the vigor of the race; quality would be sacrificed to quantity. How this might affect the ultimate fate of our species is a subject for speculation. The pessimist might take a cue from palaeontology, recalling that the extinction of a species seems to follow, not in- frequently, close upon its period of greatest development. The optimist, on the other hand, might perhaps extend the suggestion that when overcrowding does come, the ones to survive most surely, if not most abundantly, will be those whose superior qualities INTER-SPECIES EQUILIBRIUM 183 will enable them, in spite of intensified competition, to draw to themselves a sufficiency of the more desirable, though perhaps not absolutely essential articles of consumption. Natural selection would thus operate by the preferential survival of an aristocracy, while a submerged tenth would furnish a drain for the discharge of the unfit. Certainly, in the interests of the species, it were better that the inferior constituents be purged from the system than that they should drag down the general level to mediocrity and perhaps below the line of viability. But these are speculative reflections. Cycles. Food chains, were we able to trace them through their entire course, would undoubtedly be found to form closed cycles or a network of cycles. This is indeed a practical necessity for the continued performance of the processes or organic nature, processes that have gone on essentially unchanged in their general character, however modified in detail, for many millions of years. A few of the simpler food chains we may be able to follow with something approaching completeness through their cycle. For the most part, however, the system of interlocking cycles in nature is complex beyond all reasonable hope of detailed analysis in its entirety. If we are satisfied to omit innumerable details, we can trace, for each of the most important chemical elements 7 concerned, the broad outline of its cycle in nature. The elements and simple compounds principally concerned are Carbon (dioxide) COz Oxygen O 2 Nitrogen free N, NH 3 , nitrites and nitrates Water H 2 O Phosphorus (phosphates, etc. ) Brief consideration will presently be given to each of these cycles in turn. First, however, it will be well to review some of the es- sential facts regarding the occurrence of the chemical elements, generally, in nature. For the drama of life is like a puppet show in which stage, scenery, actors and all are made of the same stuff. The players, indeed, "have their exits and their entrances," but the exit is by way of translation into the substance of the stage; and each entrance is a transformation scene. So stage and players are T}ound together in the close partnership of an intimate comedy; and 184 ELEMENTS OF PHYSICAL BIOLOGY if we would catch the spirit of the piece, our attention must not all be absorbed in the characters alone, but must be extended also to the scene, of which they are born, on which they play their part, and with which, in a little while, they merge again. 8 8 Since the words above were written I have run across the following singu- larly apposite passage in- John Morley's Introduction to Wordsworth Col- lected Poetic Works: "Wordsworth's claim, his special gift, his lasting con- tribution, lies in the extraordinary stremxousness, sincerity and insight with which he first idealizes and glorifies the vast universe around us, and then makes of it, not a theatre on which men play their parts, but an animate presence, intermingling with our works, pouring its companionable spirit about us, and 'breathing grandeur upon the very humblest face of human life,'" CHAPTER XV THE STAGE OF THE LIFE DRAMA When the elements have been mingled in the fashion of a man, and come to the light of day, or in the fashion of the race of wild beasts or plants or birds, then men say that these come into being', and when they are separated, they call that in common parlance, death .... let not the error pre- vail over the mind that there is any other source of all the perishable crea- tures that appear in countless numbers. Empedocles. Our stage is a tripartite world: The heavens above, the waters of the sea, and the solid ground beneath our feet; the atmosphere, TABLE 11 Principal components of earth's surface VOLUME MILLIONS DENSITY MASS MILLIONS PUB CENT Atmosphere cu. miles short tons 5 82 X 10 16 0.03 Ocean 302 1.03 1.42 X 10 18 6.58 Solid crust 1,633 2.7 2 01 X 10 19 93.39 the hydrosphere and the lithosphere. The total mass of the earth is about 6.5 X 10 21 tons. But it is only the outer crust that interests us here, for the deeper layers have little or no part in terrestrial life. If we arbitrarily take a layer ten miles thick for the crust, the distri- bution of the material among the three main divisions is, according to F. W. Clarke, about as shown in table 11. The Atmosphere. There are two ways of confining a gas. The one most familiar in the laboratory and in products of human workman- ship generally, is to enclose the gas in a suitable envelope, such as a glass vessel, or the cylinder of an engine; to put something around the body of gas to be confined. The other way, nature's way on a large scale, is just the opposite, and consists in putting something into the gas, or putting the gas around something. It is so the earth holds her atmosphere by gravitational attraction. Her hold is not impartial. She draws closest to her the densest constituents, and 185 186 ELEMENTS OF PHYSICAL BIOLOGY gives longer leash to the lighter. The atmosphere, in consequence, is not a homogeneous body, but varies in composition with altitude. At the same time, owing to its elasticity, the air in its lower strata is compressed by the weight of the overlying atmosphere, so that 99 per cent of the whole is contained within a shell 30 km. (18| miles) thick. The remaining 1 per cent extends out into space without any Of CLOUDS * QROlMAW OUST J FIG. 40. CROSS SECTION OF THB ATMOSPHERE, SHOWING SOME FEATURES CHARACTERISTIC OF DIFFERENT ALTITUDES After W. J. Humphreys assignable limit, but at any rate to a height of some 300 km. (185 miles), as evidenced by the aurora. A graphic representation of the broad divisions of the atmosphere is shown in figure 40, adapted from Wegener and Humphreys. 1 A more detailed and exact statement of 1 Physikalische Zeitschrift, 1911, vol. 12, p. 172. See also W. J. Humph- reys' work, The Physics of the Air (Lippincott, 1920), pp. 68, 69. Also, the same author, Bulletin Mt. Wilson Weather Observatory, vol. 2, 1909, p. 67; STAGE OF THE LIFE DRAMA 187 O F CONSTANT TffMPf MTV fie. FIG. 41. COMPOSITION OF THE ATMOSPHERE AT DIFFERENT LEVELS After W. J. Humphreys Journ. Franklin Institute, vol. 175, 1913, p. 208, 212. A singular view has lately been put forward by L. Ve"gard. He identified certain green lines in the auroral spectrum with lines observed when solid nitrogen is rendered phosphorescent by x-rays. He concludes that the upper atmosphere contains solid nitrogen. (See Nature 1924 vol.113 p. 716.) Ve"gard's view has been op- posed by J. C. McLennan, Roy. Soc., June 19, 1924. 188 ELEMENTS OF PHYSICAL BIOLOGY its composition at different altitudes, up to 140 kin., is given in table 12 and illustrated graphically in figure 41, derived from Humphrey's work. A complete discussion of the r61e played by the atmosphere in the round of terrestrial life would amount to nothing less than a treatise on meteorology, such as forms no part of the present project. What TABLE 12 Percentage distribution of gases in the almosphere HEIGHT OASES PER CKNT OP TOTAL A.TMO- sniiDnE Argon Nitro- gen Water vapor Oxygen Carbon dioxide Hydro- gen Helium Total pressure km. mm. 140 0.01 99.15 0.84 0.0040 130 0.04 99.00 0.96 0.004C 120 0.19 98.74 1.07 0.0052 110 0.67 0.02 0.02 98.10 1.19 0.0059 100 2.95 0.05 0.11 95,58 1.31 0.0067 90 9.78 0.10 0.49 88.28 1.35 0.0081 80 32.18 0.17 1.85 64.70 1.10 0.0123 70 0.03 61.83 0.20 4.72 32.61 0.61 0.0274 60 0.03 81.22 0.15 7.09 10.68 0.23 0.0935 50 0.12 86.78 0.10 10.17 2.76 0.07 0.403 40 0.22 86.42 0.06 12.61 0.67 0.02 1.84 30 0.35 84.26 0.03 15.18 0.01 0.16 0.01 8.63 99.0 20 0.59 81.24 0.02 18.10 0.01 0.04 40.90 94.5 15 0.77 79.52 0.01 19.66 0.02 0.02 89.66 85.5 11 0.94 78.02 0.01 20.99 0.03 0.01 168.0 78.0 5 0.94 77.89 0.18 20.95 0.03 0.01 405.0 46.7 0.93 77.08 1.20 20.75 0.03 0.01 760.0 important factors weather and climate are in the business of provid- ing the sustenance of life is a matter of common knowledge, and a very particular concern of the farmer. Yet we must here be satis- fied with little more than a passing reference to meteorology, noting only a few elementary facts which bear directly upon the subject in hand, the circulation of the chemical elements in nature. Losses from the Atmosphere. A first question that suggests itself in this discussion of the economy of nature is this: Since the atmos- phere is "open at the top," so to speak, is there not a loss, a constant leakage of gas out into space? The answer to this question must be STAGE OF THE LIFE DRAMA 189 sought in terms of the molecular constitution of the gases of our atmosphere. A cubic centimeter of air contains (at 0C.) about 3.15 X 10 19 molecules. These are in continuous agitation, somewhat after the manner of a swarm of gnats, except that they flit about with speeds comparable with that of a rifle bullet (about 500 meters per second) rather than with the leisurely flight of an insect. At a tem- perature of 0C. a molecule of nitrogen has, on an average, a velocity of 492 meters per second. A molecule of hydrogen, under the same conditions, would have an average velocity of 1839 meters per second. It must be understood that these figures represent, in each case, a mean about which the velocities of individual molecules cluster, so that a certain proportion of them will fall below and others will exceed the figures stated. At the earth's surface the average distance travelled between two successive collisions is about ^iroTolTo cm - But in the upper ranges of the atmosphere conditions are very different. If we follow the estimates and computations of J. H. Jeans, we find that at an altitude of 3200 km. the atmospheric pressure is re- duced to about 1/10 14 of its value at sea level; but even at this low pressure there are still about 300,000 hydrogen molecules per cubic centimetre. (At this altitude all other gases except hydrogen are practically absent) . The mean free path between collisions is now 10,000 km. or about If times the earth's radius. In such circum- stances collisions between molecules are rare, and for the most part the molecules move freely through space in parabolic or elliptical orbits, and become virtually diminutive satellites of the earth. Since the day of Jules Verne's story From the Earth to the Moon it has been a matter of popular knowledge that a body projected from the earth with a velocity exceeding 7 miles per second will go off in a hyper- bolic orbit, never to return. This applies to the molecules of a gas. Any of them that may be travelling outward with such a speed in the region where collisions are so rare as to be negligible, will leave the earth for good and will thus be lost to our atmosphere. The rate of leakage from the atmosphere thus depends on the number of molecules per unit of time that acquire the limiting (outward) veloc- ity of 7 miles per second. This number, in turn, depends on the temperature in the region under consideration, a point regarding which our information is very uncertain. But an exact knowledge of this temperature is not needed to compute a major limit, a max- imum figure which the rate of escape certainly cannot exceed. It 190 ELEMENTS OF PHYSICAL BIOLOGY is thus found that under present conditions 2 the earth holds her atmosphere so effectively that there cannot be any appreciable leak even in many millions of years. Cosmic losses from the atmosphere then, are, for nil practical pur- poses; wholly negligible. Certain other subtractions from and acces- sions to the atmosphere we shall have occasion to note as we consider the circulation of the several elements. As a matter of fact the com- position of the atmosphere in the region in which living organism have their habitation is very nearly uniform 3 and constant, except TABLE 13 Constituents of the, atmosphere AT HAHTH For cout by volume 8 H imp AC XI 1'or cent by weight IN BNTinn ATMOHPKHmU Total amount Nitrogen 78.03 20.99 0.03 ' 0.94 0.01 0.0012 0.0004 75.45 23.21 0,05 1.30 0.0003 metric tons 3.9 X 1Q 15 1.2 X 10 15 2.2 X 10'* 6.2 X IQ U 1.3 X 10' 1 1.3 X 10 13 4.7 X 10i 0.4 X 10 0.3 X 10 9 1.2 X 10" Oxygen Carbon dioxide Argon Hydrogen , Water Neon Krypton Helium Xenon , Earth's surface = 1,97 X 10 8 square iniloH ~ 5.5 X 10 18 square foot. One square mile = 2.79 X 10 7 square fcofc. One metric ton = 2205 pounds. s It may be noted in passing that in the earth's pant history conditions may have been different. If at any time the temperature of the upper at- mosphere was about 750C., then there must have been a very distinct loss of hydrogen by leakage into space. The moon, and certain of the planets having a lesser gravitational pull or a higher temperature (Mercury), have probably lost in this way any atmosphere that they may have had. For a detailed discussion of this and other points in connection with the escape of gases from the atmosphere the reader may be referred to J. H. Jeans's Dynamical Theory of Gases, 1921, Chapter XV. See also K A. Milne, Trans. Cambr. Phil. Soc., vol. 22, 1923, p. 483; J. E. Jones, ibid., p. 535. 8 Except in the neighborhood of volcanoes, and in lesser degree, in or near large cities or manufacturing centers, where large amounts of waste gases may be discharged into the atmosphere. STAGE OF THE LIFE DRAMA 191 J c, K o2 -* ** &. B """^ r~^ rf( c*) C!^ rn ^^^ ^ d bO M CO 02 M ^ frl H !z; ^ S O W P! O H CQ u O OO * O CO (M t 1 1-- CO t> CD ** CM CO H O t~- O rff CO C-T O O P< 03 l> T-1 1 CO rH g |M K B H H fe CO co CO ^ h p | s F-t " nS ^ lO <D O, S 00 r-l & ** OS a tn n M 5 a o " ^ O !> O 1 1 O O rH ,-H rH B i 1 *- < -w o s X x x x x X H o K <M CS) <M O i* 1O i I OO r-t T-H 1C TH U3 O i^-ta O i-" ^ X X 19 8 Ss T"H p^ T-H rH ""* ra c3 Pt CD K o O S '" og t-s O p X X aT^eS So M S o -I ^ (i M r-( (M 'o *&** w d r- 1 O 1 ( w ^ o CD o T3 CO o ^ X X h. . t>> f^i & C3 OO CO 4J o C^ T^ J^J rt .a a ^ CO _rt . be -3 co a o > *3 ,t3 *^ 03 W "ra Q o rt HH r I <0 -s - ft to CO |SS s # si bfi fee o n-t O 1TJ ^^^11 1 fl TD "~* r ^ w o. w ^ S ,c) d pd /7 1 ^ O *H O CQ Q, Pi rQ rQ O rT to ft w * 43 ^ U .3 .3 fl 3-3 rt Water. . > rt =) p) s|2| ra r> a) % -r-i X ^ oj P O PE| O , . rl OQ 03 8 .C OQ B M- pj <o to 3 m a) 1-5 "2 3 p) rt .3 eg pi .s S % ?-2 ^ S'rS "o o i| i| r ^ o ,| a o>o)S^OPHSQ C5 pTrl [Xj * -i -h-H 192 ELEMENTS OF PHYSICAL BIOLOGY as regards its moisture content, and wo may accept for the composi- tion of the atmosphere at the earth's surface, moisture excluded, the figures shown in the first two columns of table 13. The third column shows the total amounts of the several constituents of the entire atmosphere, according to W. J. Humphreys (Monthly Weather Review, vol. 49, June, 1921, p. 341). Cosmic Accessions to the Atmosphere. Meteorites falling upon the earth from space bring with them certain o entities of entangled or occluded gases. While this contribution tt the atmosphere is at the present time, presumably, of negligible dimrnsions (see also page 195, Cosmic Accessions to the Lilhospfwrc), yet in the course of the TABLE 15 Comparison of air and aqiutlic atmosphere HIilA HII A. 8 13 A. A tic i'i<m IiITffllt* WAT I')U 15 "O. PHU Allt IIY VOM1MI8 W. VI' I'll I 15 "0. JIV AIK nr VVJQIOIrt 1 WAT Eli 15 "C. BY U'CHK* vofjirMii William.' cc. (1C. /XT ce.nl /><:r ct'.n,t 'IXiT Cllllt per cant o 209.9 6.8 21.0 lfi.0 23.2 12 8 N 780.3 11. ,'J 78.0 29,0 75.4 21 7 C0 2 0.3 21. 7f 0.03 50.0 05 05 6 * Measured at 0G. and 760 nun. Ug. f42.7ingm.; according to F. W. Clarke, Data of CicocluMnistry, Geological Survey Bulletin. 491, 1920, p. 142; G. Linck, KroiHlaufvorgiitige in dor Erd- gescMchte, Jena, 1912, p. 6. long procession of ages past this source may not have been wholly insignificant. Data on this question, are at best very uncertain, and a mere passing reference must suffice. 4 The Hydrosphere. In comparison with the ocean alt other aggre- gations of water upon the earth are insignificant in aimmnt. The bald statement of the total volume of the ocean 302 million cubic miles 'conveys but little to the mind. More impressive it is to recall that the average depth of the sea is 2$ miles, and that, even if these waters were spread over the whole earth, leaving no continents, the average depths would still be If miles. The average composition of the ocean is shown in table 14. Here again, the figure for the volume of total dissolved solids, 4.8 million cubic miles, is made more readily comprehensible by a graphic illus- tration. The salts of the ocean, made into one solid block, would * F. W. Clarke, Data of Geochemistry, U, S. Geological Survey Bulletin 695, 1920, pp. 58, 269, 282. STAGE OF THE LIFE DRAMA 193 cover the entire United States and Alaska to a depth, of Ixfr miles; or according to J. Joly, they would encrust the whole earth to a depth of 112 feet. 5 The Aquatic Atmosphere. Aquatic species perform their respira- tion in contact with an atmosphere of gases held in solution in the water that surrounds them. This atmosphere is very different both in concentration and also in composition from, that in which we live, as is apparent from table 15. A comparison of the several columns in table 15 is an object les- son on the adaptability of living organisms to varied conditions. The atmosphere in which fish and other marine animals live in com- fort would not only drown us with its principal constituent, water, but, even if this were removed, the residual gases would suffocate us for lack of oxygen; and if the deficiency in this gas were made up by the addition of the amount required to bring the percentage up to that to which we are accustomed, we would still be choked by the excessively high percentage of carbon dioxide. 6 The Lithosphere. Immense as the ocean appears to us, with its average depth of 2| miles, yet it constitutes less than vsV<r of the total mass of the earth, whose bulk is thus concentrated chiefly in lithosphere. Of the deeper layers of this lithosphere we have but scant and indirect knowledge. Earthquakes give evidence of some change in constitution about half way down to the center of the globe. Conditions and occurrences at such depth as this would seem, in the present state of our knowledge, to have little bearing upon the life at the surface. Other indirect evidence regarding the earth's inter- ior is derived as follows : The volume of the globe being known from triangulation, and its weight from direct determination with a bal- ance 7 or in other ways, the mean density of the earth is found to be 6 F. W. Clarke, ioc. cit., p. 24; Sci. Trans. Roy. Soc. Dublin, 1899, vol. 7, p. 30. 6 It must be admitted, however, that much of this COg in sea water is par- tially neutralized by alkali. 7 A concise survey of the principal determinations of the mass of the earth will be found in J. H. Poynting's little book (in the Cambridge Manuals series) The Earth (1913) It may add interest to the bald figures to note hero in passing that the density of the earth's crust (2.7) is not very widely different from that of the moon (3.46). This fact has a certain significance in connec- tion with Sir Charles Darwin's theory of the origin of the moon, according to which our satellite originally formed part of the earth and was thrown off by a species of tidal disruption. The bulk of the moon, then, would be formed of material derived from the outer layers of the earth. 194 ELEMENTS OF PHYSICAL BIOLOGY TABLE 16 Average composition of terrestrial matter* LITHO8PHERB (93 PERCENT) HYDBOSPHBBB (7 PEE CENT) AVEBAGK INCLUDING ATMOSPHERE Oxygen 49. 19 Silicon 25 . 71 Aluminum 7 . 50 Iron 4.68 Calcium 3.37 Sodium 2.61 Potassium 2 . 38 Magnesium 1 .94 Hydrogen ."': 0.872 Titanium 0.648 Chlorine .228 Bromine Phosphorus . 142 Carbon . 139 Manganese .108 Sulphur .093 Barium .075 Chromium .062 Zirconium .048 Vanadium .038 Strontium .032 Fluorine .030 Nickel 0.030 Nitrogen 0.030 Cerium, yttrium .019 Copper .010 Lithium 0.005 Zinc 0.004 Cobalt 0.003 Lead 0.002 Boron .001 Glucinum .001 100.000 85.79 0.05 -1-14 0.4 0.14 10.67 2.07 0.008 0.002 0.09 100.000 46.68 27.60 8.05 03 63 72 2.56 2.07 0.145 0.696 0.095 0.152 0.149 0.116 0.100 0.079 0.066 0.052 0.041 0.034 0.030 0.031 0.020 0.010 0.005 0.004 0.003 0.002 0.001 0.001 100.172 * Adapted from F. W. Clarke, Data of Geochemistry, 1921, p. 35 ; Clarke and Washington, Proc. Natl. Acad. ScL, 1922, vol. 8, p. 114. 5.5. But the mean density of the rocks outcropping at the surface 8 is only 2.7. Whatever may be the character of the earth's interior, 8 Compare also E. D. Williamson and L. H. Adams, Jour. Washington Acad. Sci., 1923, vol. 13, p. 413; H. S. Washington, ibid., p. 453. STAGE OF THE LIFE DRAMA 195 it is thus evident that it is composed of/lenser material than we find at the surface. In point of fact, comparison with meteorites, and other evidence, make it appear likely that the earth's interior is vir- tually a metallic regulus (chiefly iron), encrusted with a slag not un- like that which separates out and floats on the molten mass of metal in a blast furnace. Such, essentially, is the raw material of our land- scape, such our habitation, such the ground on which we tread and from which we draw the substance of our body. For the fertile soil in the plain is but the weathered variant of the granite strength of the hills, and we ourselves but a strangely metamorphosed portion of the world-stuff, the slag coating of a metal core. Cosmic Accessions to the Lithosphere. The earth receives a con- stant shower of meteorites from interstellar space, at an annual rate of some 20,000 tons. 9 This seems a large amount. As a matter of fact, spread over the surface of the globe (197 million square miles), the accumulated meteoric material of a thousand million years would make a layer only 1 inch thick, if its density were that of water, or a correspondingly thinner layer of denser material. Composition of the Earth's Crust. Table 16 adapted from F. W. Clarke's Data of Geochemistry and a more recent publication by Clarke and Washington, shows the average composition of the known terrestrial matter. The first column, in particular, gives the figures for the solid crust. This table exhibits a number of facts and rela- tions of interest. Perhaps the first significant circumstance that strikes the eye, in glancing at the table, is the very unequal deal with which nature has distributed matter among the ninety odd 10 known elements. One-half the lithosphere, and one-quarter of the atmosphere are made up of the element oxygen. The eight most abundant elements of the earth's crust (oxygen, silicon, aluminum, iron, calcium, sodium, potassium and magnesium) the only ones whose amounts are over 1 per cent constitute together over 98 per cent of the earth's crust. These, with hydrogen, titanium, carbon and chlorine twelve in all make up 99.5 per cent; thus leaving only one-half of one per 9 S. Arrhenius, Worlds in the Making, 1908, p. 108. See also Young, Astronomy, 1904, p. 475. 10 Uncountable isotopes aside. 196 ELEMENTS OF PHYSICAL BIOLOGY Ba Cr 0.08 tt LJL_ S 0-18 O.ll a d2\g.$8 IS Aly o.or Ca 3.ZZ Si Sff.80 o SO.OS. Compos. '/ Van of arf/>'o At 7.30 A/a Z.S8 a p r a?o r F Mn /y n 0.11 ato o.og s O.I1- Fe O.Ol 63.0 0.10 O.H Z.S Compost ft'an a'f Human Boc/y. FIG. 42. COMPARISON OF COMPOSITION OF EARTH'S CRUST AND HUMAN BODY Figures indicate percentages STAGE OF THE LIFE DRAMA 197 cent for all the other elements, among them some quite indispensible for our existing civilization. 11 Relation to Composition of the Organism. Of the elements spe- cifically significant for the living organism, only one, oxygen, is pres- ent in great abundance. Carbon, hydrogen and nitrogen, the prin- cipal "organic elements/' are among the less abundant constituents of the globe. On the whole it may be said the living organisms are composed of comparatively rare elements. We are, indeed, earth- born, but yet not altogether common clay. 12 This is well brought out TABLE 17 Average composition of human body POUNDS PERCENT Oxygen 97 20 63 03 Carbon 31 10 20.20 Hydrogen 15.20 9.90 Nitrogen 3 80 2 50 Calcium 3 80 2 50 Phosphorous 1 75 1.14 Chlorine 0.25 0.16 Fluorine 22 14 Sulphur 22 0.14 Potassium 0.18 0.11 Sodium. . . 16 0.10 Magnesium 0.11 0.07 Iron 01 01 Total 154.00 100.00 in the chart figure 42, which shows, side by side, the average composi- tion of the known terrestrial matter, and, in comparison, the approxi- mate composition of the human body. This latter is not exactly a representative sample of the totality of living matter (see tables 17 11 H. S. Washington, Jour. Franklin Institute (1920), vol. 190, p. 7. Tho figures have been slightly modified in accordance with the latest data of F. W. Clarke and H. S. Washington, Jour. Natl. Acad. ScL, 1922, vol. 8 p. 114. 12 Indeed, taken literally the expression "common clay," as applied to man, is an extreme case of poetic license; for aluminum and silicon the chief constituents of clay, and taking second and third place in rank of abun- dance among the components of the earth's crust, are both present only in tracea in the human body. 198 ELEMENTS OP PHYSICAL BIOLOGY |-|| ii d ^a l" W! n> *H ^ d rj O O CD T3 O O .r3 "o --s o d3 >> m Ct fl pC( X! Ss CS Kn C5 O _g o d bD O ao rv H OQ *"^ *^H ^ ""9 fli H ^H c _3 o f ftX 11 B slS | K|| a |j| B 1 I 'g g = 5 g ^^1 -P ,M J o tB-a O g M d Jj <u J3 > CD ^ 9 4 '3 a " is-s 11 iPiSi! 03 m ' : <tiwr i M ta ^TCQO) ^1 1 S & l S S.g i 8 S w-T.S jn o ,S "mpn ^ra'S'SpQ^'^^o !r* 9 -<j rS rJ-T 03 .u TT "-ua fl^Q 03 r~\ T^ S^cS "i h ' o "jip 5 aM & *-^ ri /^ W " ' *~> OJ M l_i '' -*-* * n .-H ^ rt ^^ *^ ri S t"^t rj Cl ""* CQ o3 CJ 'T^ d f-I f-i d DQ J-< rn m ft ^ 03 gfcl!>^c3gC^c3o3 CQ a CQ 8 8 q o .B _fl i ^-s ^^ i i c3 TJ ft o ^ ^S^a ^ TS < tfl a) ft o o o C a) Pt ft ^ # eo s t v> {* O bC fl 1^ . ft m a, -e f^ - '7 ' S's'S'^ jU^ .1 :! 8 '> K a ft -2^^ "^ **H "Srl^Irt e \^4 O !-< T3 * ^ O ^ ("5 cJ ^ 03 M rl cJ Cs H U j-j r ' ^Ca^ *~i ^fc.w u o 3 _g cfg cS Q.^d ^ ^ ^^gi g _g e j P< SJ rt -a S -g p s cf 1 ~w -g .t2 o 3 So 3 "o 'fl ^ ID 2 S-^ M d| *o "* p. K. bjQ t|_5 H ^j pjC] Q^ S> HJ 5* (P "** CL C "ti ~*-^ . rH CJ -|J O C^ C3 f ~ H (1 ^** ^ v^ ft d ^ s c Q* ' H -+-3'-c3KJrj ^ S) ""* ^ Q ^pos^t^on >^ 1 1 d s-5^-1 n e ^ 3 >> G fl W "3 g llliillilill 253 ^Ti g^S^ -5 ^g ft j ( <cj (^ pL| p] R ^ *H CO o -t-2 rt (D td o o iz M d 0) H 0] "^ w ^ f=5 <sj m ,2 P3 m |j S a s d R w a H J 1 1 'S | J t cj| QJ rj ,J W "tj 'P Sr h o A ^ bO P) ^j .a KJ t "H _j ~ n ^ oS O Wuo^ PH ,5 n M r ^ >H CO AH ^i O M s ta ft S3 (a ** ^ ^ * s B P o ^J M t> 1O !> "*?H CO B g ^ CD 00 T^ TH CM TjH l> CXD 1O JtS O H t>" CCl t^ Ci <7^| 00 O P. u ^00 U5 <M .H CO CO TH o ft 00 U5 3 o O O O rH o o o o >* o CO O W *-* O >-l CO O CXJ 2 & l-l t-4 H CO c<i o> # p ic CO CO CM "^ ^O ^ STAGE OF THE LIFE DRAMA 199 -P d -P rt .2 ri S) 03 'CQ -p -d d d ,J2 d d X5 03 1 13 d "d d o3 ft CO -< ft ^ GQ ". d o GQ ^ 13 rd o3 -2 *c8 a ft ft a 3 83 "d "d 03 O [o CQ CQ 13 d 13 1 '3 O 13 e3 03 bO d o "3 a d -a .3 13 J .2 ft CQ 'd 03 .1-1 o -p o CO CQ ^ -p d o d 03 3 bfi "p d g rS 03 CO 03 .3 CO 03 F-l .9 j 03 CQ 03 S bfl a M PH pn F-< PH elieved essential to all plants, but not demonstrated; found in ma- rine plants, especially Phseophyceas resent in many plants ; believed by some to be essential ; abundant in 03 -p .3 3 'o 03 ft CQ 03 88" bfl "3 d 1 a Phseophycese ound in all plants ; present in large quantities in the Diatomacece, both fresh-water and marine; in form of "silica" constitutes 0.5 ; to 7 per cent of the ash of ordinary marine alga3 pq PH fo 03 * O w rv.. rt, a .3 &-. d 'C o ;*:j o .S o OJ 6 .F-4 GQ co C33 ^o CM CM CO CD CO CO B CO 1C oo CM co ,3 s s &4H K^H CQ O O 03 r9 r-< fcfl bD ., d a "I ;-| "P* CQ O o "cl CQ CQ 03 ~ ^ 02 j 03 aj C3 t*} t^i JS -d -P CO FH Lj 1 Tn CO C3 03 JH CD ,d CD 03 l> > 2 d X) f 3 .3 o "K bfl .2 .3 -e o - S 6 r-S ra 'S 13 1 rS t> ^ CD d 03 03 -r-i ^ ^ 3 urj g -^ 13 t-T r-l .2 'd d 'd d ,O oJ d "S "IS "3 .2 3 g ^ O5 TH '-p -p~ .5 " '-P b d ff CQ c!i CQ -P M 03 CQ O C) 2 P* S a g d ^d 1 ra rv d 03 -d o d -3 M CQ 2 flj P.Q 3 _bO "S ft o H HH M hH o co a |.3.S JS .3 03 d CQ f^ o "r~j IT* _ o C3 S^ Mj |-Q ^? 8 CQ O FH c3 <^ c3 &> ^ o bO .3 I ! i 02 2 -5 O 03 CO 1 F4 JH 0? ^ [-^ 03 S !* a -p ~ 13 CQ co~ ^ CO ^ -p d d d r d S r2 ^ 3 -p ^ oT ^ o'3 5 M d 03 03 co "ft -^ ^. o3 " bD d a 3 a g M 03 r^ 03 FH J*3 F-< ft li'll 3 5 '| 9 "ft _d 03 "FH 3 03 go J O S 03 : aF-l ft H H H F-I d M-I d 03 v o Z ^ CQ 1 1 O R CO d d f g HH "o 03 d +3 o CQ -P "a 1 M d M s pq h "o w 03 1 CQ rt 03 03 *bJD S C-I .3 * EJ 03 .3 bfl '3 3 g 'C o ? O M S PQ S CO 1 CO j , t- aj i ( o O ^d CM Oi CO CM O O3 o o ft 03 Qj "S * o rM d ft C 03 CQ 200 ELEMENTS OF rriYBTCAL BTOLOftY 9 d IJJ Pi 4 <U M d b* 1 O m "S d CO a w m CD > m O) o af^9 ,2 d '~3 ^ C3 .J3 oQ W cj (LH 3 0-3 g m a ^ 3 d 3 'd 2 rt ^^ ^ i s a a s i a -9 '9 '3 g bs M '3 S f d d-5 h d d p CD ' H fl, aj ^ a s g s d s g g a g ,9,3 H H CD P- a ^3 o KI d 00 a M |H O bD 1 _d 3 *rH _pj w en OT en CQ M W r/ j r/3 r/] "3 +- 1 "P 4-j ~tn! 'tj ^f^ d d pi d d o d d 9 <3 JB rr) d 'S d d d 3 ^ d Q) OT ft ft p, ft ft p. ft p, ft ft g ft ^ ^: cu E- t 1 ^ CD CD Q ^cU CD pj [S CD 1* \f. CD d CD CD t)^ M *4H *4H ' "^ **t o l " ^ o d d co d TO r/3 ^ ^ , wg [/j C3 C3 d pi d d d d d M M M M M M d d d d 1 1 M M M fn op p "d o n CD , < m d O p) X5 .M [H d ^ p| a <r "*id CQ CO C) C3 PH ,--! [/i pcj f/j tS] CD ra H i_ P !- -1- .. -1 H ,.| ELEM: .a | a H a s .^, | a S 1 _g g -^ | ^sas^& g g -^^pqfqOO h5h-] r H H 4^J 0) P H .1 31 is X PH M N o j 5 B! # ,JS 2 E o fe CO M O 00 f.. rH M W M h-i co *o oo ^ CD CD H W g **tf O> >O CM * CM 6 d< o rH O O O ,-i rt o Ej H W CO b- ts. 1,^ O fh 00 CO l'~ i 3 rH CT) CO CO Cft ^O C^l CD CD O CO CO H ^ b- TH 1> r-H 00 CO t-- CD OO CO l~ <O "*l p CT l> CO rH kO CD O O CH 00 CD rH <M CM .a .a Ol V- CJ m S M. O O t CJ JD S ft cu q H cu . y w l>> 2 ti .in a -fl STAGE OF THE LIFE DKAMA 201 and 18) but will serve well enough for the present pupose. The chart brings out very pointedly the selective character of the organism's activity in gathering to itself the substance of its body. Thus car- bon, which both in function and in relative quantity figures so promi- nently in living matter, appears as an insignificant lifctle block in the chart of the earth's crust. A similar contrast, if not quite so extreme, is seen in the case of nitrogen. With aluminum and silicon the com- parison works the other way about very plentiful in the earth's crust, these elements are practically absent from the human body. 13 Taking the mother earth as a whole, and the organism as a whole, it certainly cannot be said that there is much evidence of "inheritance TABLE 19 Composition of the salts in sea water and in blood serum in per cent* SEA WATER BLOOD SBBDM Sodium Na 30.59 39 Magnesium Mg 3.79 0.4 Calcium Ca 1.20 1 Potassium K 1.11 2.7 Chlorine Cl 55.27 45 Sulphate ion S0 4 7.66 Carbonate ion C0 3 0.21 12 Bromine Br 0.19 Phosphoric acid . PaOfl 0.4 * Maccallum, Trans. Koy. Soc. Canada, 1908, II, p. 145. of parental characters" in their respective compositions. Still, resem- blance is not wholly lacking. This becomes evident if we compare, not the entire organism with the whole of the earth's crust, but the blood of a mammalian, for example, with the water of the ocean. This comparison is made in table 19 and figure 43. The likeness thus seen, imperfect as it is, can hardly be ascribed to accident. The fact is, 13 Silica furnishes, however, the skeletal support in a variety of living forms (radiolaria, sponges, plants). Cf. G. Bunge, Physiological and Path- ological Chemistry, 1902, p. 23-24. For a detailed discussion of the distri- bution of the chemical elements in organic nature see Vernadsky, Revue Ge"n. Sci. The reader interested in this phase of the subject should not fail to acquaint himself with this article, which came to the writer's attention too late to be given more than this passing note here. 202 ELEMENTS OF PHYSICAL BIOLOGY Cl 55.3 Na 30.6 SoJJds in 7.66 3.79 KU Br O.f3 O.O 0.4 1.0 2.7 0.4 Soh'ds in Blood Serum No 39 an a dry fi Cl 45 . 43. COMPARISON OF SEA WATER AND BLOOD SERUM STAGE OF THE LIFE DKAMA 203 as pointed out by Palitzsch, 14 that aquatic species are in such intimate contact, both at their body surface, and more particularly in their gills, with the surrounding water, that the latter might almost be considered continuous with their bloody fluids, so that sea water may justly be "placed in the same category as the other physiological fluids." "Frederiq 15 has shown that the amount of sodium chloride in the blood of Crustacea varies, and all but corresponds, with the density of the water in which the creature has been kept." More highly organized aquatic species have made themselves in greater degree independent of the salinity of their environment; 18 and finally, it would appear, when the marine ancestors of terrestrial vertebrates emerged from the sea and adventured life on dry land, they packed, as it were, a portion of their saline environment in their baggage, and took it along with them on their excursion as an essential part of their milieu interieur. And to this day, according to this view, we our- selves cany about with us in our arteries and veins, if not a portion of the actual ocean, at least a roughly approximate replica of its brine. For, as L. J. Henderson 17 remarks: "Not only do the body fluids of the lower forms of marine life correspond with sea water in their composition, but there are at least strong indications that the fluids of the highest animals are really descended from sea water." Some such indications may be seen in the reflections (conceived from a slightly different point of view) of G. Bunge, 18 "I am convinced that the remarkably high percentage of salt in vertebrate animals, as well as the desire to take salt with our food, can be satisfactorily explained only by the theory of evolution." In support of this consideration Bunge points out that in the weathering of rocks by the action of rain-water charged with carbonic acid, the sodium is dissolved and carried off as carbonate, while the potassium largely remains behind in combination with silica. The sodium carbonate being washed to 14 Comptes-rendus, Laboratoire de Carlsberg, 1911, vol. 10, part I, p. 93. See Chapter I, footnote 19. 16 Arch, de Zool. Exp. et Gen., 1885, Ser. 2, vol. 3, p. XXXV; see also D'Arcy Thompson, Growth and Form, 1917, p. 127. 16 Compare D'Arcy Thompson, loo. cit., p. 127-130; Claude Bernard, In- trod. a 1' etude de la me"decme exp., 1855, p. 110 as quoted on p. 17 (and foot- note 20) of Chapter I. 17 The Fitness of the Environment, 1913, p. 187. 18 Loc. cit., 1902, p. 101-103, 204 ELEMENTS OF PHYSICAL BIOLOGY the sea undergoes double decomposition with the alkali earth chlo- rides, these latter being deposited as carbonates (limestone and dolo- mite) while the sodium remains in solution as salt. Thus sea water is rich in sodium chloride and poor in potassium, while on dry land the balance is essentially reversed. 19 Plants and invertebrate ani- mals, Bunge points out, contain little sodium, unless they live in a highly saline habitat, in or near the sea, or on salt steppes. Yet the land vertebrates are all remarkably rich (comparatively) in salt, in spite of the scanty supply around them. Is not the large amount of sodium chloride found in the present inhab- itants of dry land another proof of the genealogical connection which we are forced to accept from morphological facts? There is no doubt that each of us in his individual development has come through a stage in which he still possessed the chorda dor satis and the branchial arches of his sea-dwel- ling ancestors. Why may not the high average of salt in our tissues be also inherited from them? Support for the supposition thus suggested is seen by Bunge in the fact that the younger a vertebrate is in its individual development, the more salt does it contain. Furthermore, cartilage contains the highest percentage of sodium of all the tissues of our body, and is also the tissue of greatest antiquity. The human skeleton is origi- nally composed of cartilage, which is replaced, for the most part, by bone as the individual matures. These are facts which lead most readily to the interpretation th? * the vertebrates living on dry land originally came from the sea, and aw still continuing to adapt themselves to their present surroundings, where they can get but little salt. We prolong this process of acclimation by taking advantage of the salt strata which have been left on the land by our primeval element, the salt flood. Chemical Correlation in Soil and in Organism. Our ancestral resemblance to the soil from which we spring is also exhibited by evidence converging from a different source. II. S. Washington, 19 This observation must be accepted with some caution. Compare Whitney, Science, 1922, vol. 56, p. 218. "Until we determine the actual loss, through chemical denudation, of silica, alumina, iron, potash and other electrolytes in the colloidal state, carried by rivers, we are in no position to even speculate as to whether erosion is a selective process which might change the chemical composition of the soil." STAGE OF THE LIFE DEAMA 205 in his studies on The Chemistry of the Earth's Crust, already cited, draws attention to the fact that in the rocks soda and iron tend to be associated together as a pair on the one hand, and potash and mag- nesia on the other. This is well brought out in the diagram figure 44, reproduced from Washington's memoir, in which it is seen that the points representing the analysis of a number of rock samples tend to group themselves about the diagonal of the square, indicating that FeO FlG. 44. COKBELATION IN THE OCCURRENCE OF Na, Fe AND ~K, Mg IN ROCKS A large number of analyses here plotted show a marked tendency to array themselves along the diagonal, showing that high percentage of Na is com- monly associated with high percentage of Fe; K and Mg follow a similar relation. After H. S. Washington. high content of soda goes together with high content of iron, but with low potash and low magnesia; and vice versa. The point of special interest to us here in the present connection is that to which Dr. Washington draws attention in the words: Curiously enough, the same correlation between these two pairs of ele- ments, soda and iron, and potassium and magnesium, seems to hold good in the organic world. This is apparently shown by the following facts: In 206 ELEMENTS OF PHYSICAL BIOLOGY autotrophic plant metabolism potash is an essential element, as is also mag- nesium, in that chlorophyll (which in the leaves acts as the carbon-trans- ferring substance) is a magnesium salt of a complex organic acid, while sodi- um and iron are generally toxic toward (at least the higher, gymnospermous and angiospermous) plants. On the other hand, sodium, rather than potas- sium, is the alkali metal essential to the higher animals, salt being a very necessary article of diet (in part because of its chlorine, and in part because of its sodium, content), and sodium chloride is present in the blood plasma; and at the same time, hemoglobin and its derivatives (which act as oxygen carriers, and are analogous to chlorophyll in plants) are iron salts of organic acids closely related to that of chlorophyll; while, similarly potassium and magnesium are more toxic toward the higher animals than are the other pair. This singular parallel must not, of course, be looked upon as an instance of resemblance due to anything of the nature of inheritance of ancestral traits in the biological sense. Rather must it be con- nected in our minds with the fact that systems composed of the same fundamental substances, will display certain analogies through inter- play, in them, of the same chemical affinities. Accessibility of Valuable Earth-Constituents. Such a comparison as has been made above of the relative abundance of the several ele- ments in the living organism and in the environment from which it draws its supplies, would be misleading if attention were not drawn to another factor aside from abundance, which enters strongly into play in the quest for the necessities of life. More important than mere abundance is accessibility. For, a substance may be present in comparatively large quantities, and yet be difficult to lay hold of, either on account of its wide dispersal in dilute form, or for other reasons. On the contrary, a comparatively rare substance may be procurable with relative ease, if it occurs segregated in concentrated or otherwise readily accessible form. Perhaps the most telling illus- trations of this are to be found in industry. The element copper for 2 example is found only to the extent of about -r per cent in the earth's crust. Yet it is one of the most important metals in the arts, and is not ordinarily thought of as particularly rare. This is because, in those regions where it does occur, it is found in concentrated form, either as native metal, or as rich ore. Other instances are readily cited. Tin, lead and zinc are all rarer than copper, and each rarer than its precursor, in the order named. Still rarer are silver, tungs- ten, gold, bromine and platinum, all of which find important use in STAGE OF THE LIFE DRAMA 207 ^ O r-l (-H r-t WrH M Si r-i CM CO 42803 22 19 iO CO t- 00 CJ 1SI n S 02 W O *z W ffl 02 o ,_, O CU -> J _O pa 5 R H -<1 ^ O p O M W fM 208 ELEMENTS OF PHYSICAL BIOLOGY the arts. But the mode of occurrence of these substances is such that they can be gathered or rained with comparative ease. It has been pointed out by H. S. Washington that the elements, as arranged in Mendeleeff's table, naturally fall into two groups divided by a zig- zag line, as shown in figure 45. Above this line are the rock elements or petrogenic elements, that enter into the principal rock-forming minerals (and also, the gases of the atmosphere). Below the line are the ore elements or metallogenic elements, which commonly occur in concentrated form as ores and as native metal. Now man's industrial activities are merely a highly specialized and greatly developed form of the general biological struggle for existence ; and this same feature of accessibility and of concentration in segregated supplies (ores and the like), which is a prime con- dition for the very existence of some of our industries, is also involved, in closely analogous manner, in the more primitive life processes. Our fields demand fertilizers bringing ammonia, nitrates, potash, phosphates, etc., in suitably concentrated form, if they are to bear a harvest commensurate with the needs of a modern community. And this again is merely an accentuated example of the still more primitive needs of the unsophisticated flora and fauna of virgin nature. Of scattering, dissipating processes there are plenty. Rain and snow wash most of what is soluble, and much that is not, into the rivers and^ut to sea. Our own activities in modern intensive agriculture briiig each year to the land a highly concentrated diet of fertilizers, which in the very act of cultivation are scattered and diluted many thousand times. And our modern sewage system is deliberately wasteful of vital substances, which it discharges into streams and out TO sea. All this dissipation must in some way be balanced if the regime is to continue. Thus the circulation of matter in nature must no/fc only provide for the mere presence of certain substances on which ^che maintenance of life depends, but it must furnish them in suitable ' concentration and, generally, in available form. It must, therefore, in many cases include, as a definite step, a segregating or concen- trating process 20 as well as simple motion through a cycle. We shall have occasion to note concrete illustrations of this in the separate consideration of the circulation of the several elements, to which we now proceed. 20 The significance of this from the point of view of energetics will engage our special consideration in a later section. CHAPTER XVI THE CIRCULATION OF THE ELEMENTS -THE WATEH CYCLE The great Sea-water finds its way Through long, long windings of the hills; And drinks up all the pretty rills And rivers large and strong: Then hurries back the road it came Returns on errand still the same; This did it when the earth was new; And this for evermore will do As long as earth shall last. Words worth. The ancients, totally blind as they were perforce to the fine details of material transformations revealed by the search light of modern chemistry, nevertheless recognized in its broad features the cycle of life, the circulation of the elements in nature. "Dust thou art, and unto dust shalt thou return," we read in an old book of wisdom. Heracleitus (536 to 470 B.C.), promulgator of the famous doctrine Tr&vra /Set has a more detailed, if not more accurate conception of the cycle of Nature, which he formulates in these terms: Earth S \ Fire Water Vapor (Air) The human mind was not yet schooled, then, to polish the facets of this rough gem, and bring out the sharp-edged truth as we see it today Solid \ Liquid Vapor 209 210 ELEMENTS OF PHYSICAL BIOLOGY But all honor to the minds that discerned through the mists of dawn the bold features of the landscape to be revealed in the sunlight of later day. Today we recognize not four elements, but over ninety, not counting those modern variants, the isotopes. And we follow in much detail not one cycle, but, as particularly pertinent to life, five major cycles the circulation of water, carbon, oxygen, nitrogen and phosphorus. This was noted already at the conclusion of Chapter XIV, but the discussion of the cycles was deferred to give space to a preliminary survey of the scene in which these cycles churn the planet's surface in their age-long duty. We are now prepared to take up the thread where we broke off; we turn our attention first to the water cycle. Water Requirements of Human Body. We do not ordinarily class water as a food, though we partake of it by the same channel .. | that gives entrance to the materials commonly so classed, and although the lack of water, if we are by any circumstance deprived of this substance, is felt even more acutely than an interruption in the adequate supply of food. The fundamental basis for this dis- tinction, its origin in the unsophisticated mind, is undoubtedly the fact that we have a separate sense of thirst, distinct from the signals of hunger and appetite originating from nutritive demands of the body. And this naive, unsophisticated distinction is entirely in accord with the reasoned analysis of the respective functions of water and of food in the narrower sense. It is undoubtedly just because of this difference in function that thirst and hunger have been developed as separately recognized sensations. Water acts merely as a vehicle; unlike the food, which undergoes extensive and complicated reac- tions within the economy, water leaves the body essentially as it enters it, unchanged chemically, though charged (in part) with sub- stances in solution. It would be a gross error, however, to suppose that water, because it functions thus in accessory capacity, and escapes the more intimate transmutations of metabolism, can be lightly regarded in making a survey of the participation of the several elements in the cycle of nature. It must be remembered that water constitutes as much as 60 per cent of the total mass of the human body, for example, and a still greater proportion of the substance of most of our food-stuffs, as shown in table 20. _ .,*f? WATER CYCLE 211 Thus an adult human being consumes per diem about 3 liters of water of which about 1 liter is contained in his solid food. In point of fact he consumes about 5 pounds of water for every pound of dry solid matter ingested. It is thus seen what an important item water is in the daily economy of the human organism. The excretion of water by the kidneys, lungs, intestine and skin is somewhat in excess of the intake. The excess of the outgo over the intake is formed in the body by the oxidation of hydrogen organically combined. 1 Water Requirements of Plants. Rainfall as a Limiting Factor. In the economy of most plants the traffic of water is of even greater importance 2 than in man and land animals generally, and TABLE 20 Moisture content of some common foods PER CENT PER CENT "RAAf (frpqM 43-65 40-71 66 11 87 27 12 35 Sugar granulated 78 81 63 94 86 1 Fish Cabbage Cucumbers Rntfpr Potatoes Tomatoes Strawberries Walnut, California White bread indirectly the moisture needs of plants are, of course, of funda- mental importance also to the animal population feeding on the vegetable growth of the soil, so that the water supply of a territory 1 L. J. Henderson (The Fitness of the Environment, p. 133) estimates that a man weighing 60-70 kgm. excretes daily: grams Water 2500-3500 Carbon dioxide 750- 900 All other substances 60- 125 and that water is, accordingly, three-fourths and carbon dioxide one-fifth of the total excreted. The proportion of water excreted from kidneys, skin, lungs and intestines, is given by Kirk (Physiology, p. 208) as 1:0.5:0.22:0.09, or 11.5:5.75:2.5:1. 2 L. J. Henderson, loc. cit., estimates that of the materials ingested by ordinary green plants, more than nine-tenths is water, and carbon dioxide at least five sixteenths of the remaining tenth. 212 ELEMENTS OF PHYSICAL BIOLOGY may function as the basic limiting factor of the total life which that territory is able to support. A study of these relations with partic- ular reference to the human population of the United States, has been made by W. J. McGee, who remarks : 3 Hellriegel in Germany and King in this country have shown that crop plants require for their growth_a quantity of water, measured by transpira- tion, averaging from 300 to 600 (with a mean of about 450) times the weight of the plants after drying; and common field experience indicates that, in addition to the moisture passing through the plants, the soil requires an even larger quantity to maintain a texture suitable for crop growth much of which passes away through evaporation and seepage. On this basis "the agricultural duty of water" in this country has been formulated as the pro- duction of one-thousandth part of its weight in average plant crop. Reck- oning human food and drink on this basis, and assuming that meats require (chiefly in the growth of plants used as feed for the animals) ten times the quantity of water represented in vegetal food, it appears that the adult who eats 200 pounds each of bread and beef in a year consumes something like 1 ton of water in drink and the equivalents of 400 tons in bread and 4000 tons in meat, or 4401 tons in all figures corresponding fairly with the results of intensive agriculture in arid districts. Accordingly, the "duty of water" considered in relation to human population may be stated roughly as the maintenance of a human life a year for each 5 acre-feet used effectively in agriculture. Now mainland United States (i.e., the chief body of our territory, ex- clusive of Alaska and the insular possessions) comprises something over 3,000,000 square miles, or somewhat less than 2,000,000,000 acres of land; yet the annual rainfall the sole original source of fresh water averages barely 2* feet (30 inches), or hardly 5,000,000,000 acre-feet. So while the land area, if peopled to the density of Belgium (over 640 per square mile,) would carry a population of 2,000,000,000, the water supply suffices for only 1,000,000,000. The conclusions of McGee may have to be modified in point of detail, and some of his figures may perhaps have to be revised; but the general principle underlying his reflections attract our attention. The moisture needs of the living population (all species included) are a large and fundamental item in biological economy; whether this item proves the ultimate limiting factor of population growth, as McGee suggests, is a question whose answer must be sought in 3 Science, 1911, vol. 34, p. 429; Yearbook of the Department of Agricul- ture, 1910, pp. 169-176; Bureau of Soils Bull. 71 (1911), pp. 7-14; World's Work, 1912, vol.. 23, p. 443. Compare also L. J. Briggs, The Water Require- ments of Plants, U. S. Department of Agric., Bureau of Plant Ind Bul- letins 284, 285. WATER CYCLE 213 terms of Liebig ? s Law of the Minimum; a dearth of other essentials may make itself felt before the limit of available moisture is reached. The Sources of Supply. Such, then, in broad outline, are the moisture needs of organic nature; as such they have existed, in greater or less degree, for millions of years, and have been satisfied, and will continue long to be satisfied, from a source essentially inexhaustible because constantly replenished by the return flow : The great reser- voir is the ocean, with its 302 million cubic miles of water, and an evaporating surface of over 144 million square miles. Annually there rise from this into the atmosphere about 63,300 cubic miles of water, to which some 22,800 more are added by evaporation from the land, making a total of 86,100 cubic miles. This figure also repre- sents the total precipitation in rain, snow, etc., but of the total about 56,700 fall back directly into the ocean, and only the balance, 29,400 is available for the needs of the land. It has been noted that the evaporation from the land is about 22,800 cubic miles. The difference between this and the precipitation on land, the balance of 6500 cubic miles, is the drainage from the land to the ocean by rivers. No attempt will be made to estimate what proportion of the precip- itation on land is derived from evaporation over the sea, and what proportion comes from the land itself. Some idea of the relation, however, can be formed from a consideration of the difference or the ratio of the rainfall within the area drained by rivers and the amount actually discharged by them into the sea. John Murray 4 has col- lected information on this subject, with the result shown in table 21, which covers 33 of the world's principal rivers. It will be seen that the total rainfall in the area of these rivers is 10,186 cubic miles, the discharge to the sea is 2182 cubic miles, leaving a difference of 8004 cubic miles unaccounted for. A certain portion of this perhaps represents seepage, but the bulk must correspond to water re- evaporated from the land and from inland waters. It will be ob- served that for the 33 rivers combined the proportion of the dis- charge to sea to the total rainfall is about one-fifth. For individual rivers the ratio varies widely, between the extreme of 0.58 (Rhone) and 0.027 (Nile). Naturally the climate very materially affects this figure. 4 Scottish Geographical Magazine, 1887, vol. 3, p. 76. 214 ELEMENTS OF PHYSICAL BIOLOGY There is also a circulation of waters of the sea in very large di- mensions. According to L. J. Henderson 5 the Gulf Stream in the Straits of Yucatan carries 200 million tons per second, 6 travelling TABLE 21 Showing the drainage area, annual rainfall, annual discharge, and ratio of discharge to rainfall, of S3 rivers in different parts of the world POSITION OP MOTJTH O BIVBB RIVER DRAINAGE AHBA ANNUAL RAINFALL MEAN ANNUAL DISCHARGE MEAN RATIO square miles cubic miles cubic miles 50-60N. Rhine 32,600 19,500 10,100 2,049 Oder 51,100 14,700 2,500 5,803 Niemen 36,450 10,355 3,783 2,737 Vistula 65,800 19,908 5,657 3,508 40-50N. St. Lawrence 565,200 338,967 87,312 3,904 Danube 320,300 198,736 67,511 2,746 Po 27,100 23,887 13,322 1,794 Volga 592, 300 152,384 43,736 3,484 Seine 23,250 10,266 5,469 1,877 Rhone 34,850 22,439 14,066 1,724 Dnieper 197,450 56,093 22, 195 2,527 Loire 42,600 18,210 7,810 2,332 Dniester 30,950 8,792 3,274 2,685 30-40N. Yang-tse-kiang 689,100 408,872 125,043 3,421 Hoang-ho 387,150 ' 117,711 28,591 4,159 Nile 1,293,050 892, 120 24,334 30,998 Pei-ho 65,000 22,354 1,650 13,551 20-30N. Mississippi 1,285,300 673,064 125,603 5,446 Rio Grande 232,300 113,655 12,676 8,966 Indus 360,050 104,416 26,032 4,886 Ganges 588,450 549,791 43,263 16,169 10-20N. Magdalena 92,900 116,746 59,451 1,964 Irawali 181,950 180,849 82,298 2,256 Kistna 81,300 61,025 14,776 4,137 Godavery 154,850 95,924 16,841 5,708 0-10N. Orinoco 429,700 603,397 122,242 4,936 Equator Amazon 2,229,900 2,833,830 527,951 8,009 0-10S. San Francisco 212,900 218,459 22, 197 9,842 Congo 1,540,800 1,213,044 419,291 3,213 20-30S. Orange 267, 150 50,913 21,875 2,327 30-40S. Olifant 14, 300 2,472 0,679 ^,639 De La Plata 994,900 904,687 188,740 091 Uruguay 151,000 130,890 32, 136 X073 Total 13,272,000 0,186,460 2,182,485 5 The Fitness of the Environment, 1913, p. 182. 8 Across any cross-section of the stream, presumably; the statement is not clear on this point. WATER CYCLE 215 with, a mean velocity of about 80 miles per day. This circulation in the ocean has of course no direct part in the water cycle of the organic world, but indirectly is most important on account of its climatic effects. m- \ r Oceo/% ///&./m. mill $f mi. figures \rifhout denomination a fie cubic FIG. 46. CIRCULATION OP THE ELEMENTS IN NATURE. THE WATEE CTCLEI Water Cycle Diagram. The principal figures relating to the circulation of water on the globe are exhibited in diagrammatic form in figure 46, which tells the story more effectively than words. It may here be added in explanation that the 3000 cubic miles of 216 ELEMENTS OF PHYSICAL BIOLOGY moisture contained in the air are practically restricted to the lower 6 miles or so of the atmosphere. For the rest, this moisture is un- evenly distributed, as everybody knows from personal weather observation. Water vapor and condensed water differs in this respect from the other permanently, gaseous constituents of the atmosphere. (See table 12 and figures 40, 41.) Fraction of Total Water Circulation Taking Part in Life Cycle. Only a fraction of the total circulation of water actually passes through the organic cycle. We may make an attempt as follows to obtain a rough idea of the order of magnitude of the fraction thus concerned. If the entire land surface were cultivated to produce crops at the rate adopted as standard by W. J. McGee, the growth produced (figured in dry weight) would be T ^u~ ff of the rainfall. Furthermore, this growth would evaporate, by transpiration, about 500 times its own weight of water, (this is assuming one crop per year). It would therefore evaporate just about one-half the annual rainfall. But the 22 800 total evaporation on the land is OQ'QQQ = tnrce fourths of the an- nual rainfall on land, the remaining fourth being drained to sea by rivers. Hence, of the water evaporated on land, one-half times four-thirds = two-thirds is evaporated by plants and thus takes direct part in the organic cycle. In comparison with the evapora- tion by plants, that from animals is undoubtedly negligible, espe- cially in view of the coarseness of our data. If we put the evapora- tion on land as one-fourth of the total evaporation, we finally arrive at the value one-sixth as that fraction of the total water in circu- lation, which takes actual part in the organic circulation. Desert areas cannot materially alter this estimate, since they con- tribute but little to either side of the account, both evaporation and life being meagre or absent. In some measure this remark also ap- plies to frigid wastes of the polar regions, where the low temperature makes for comparatively low evaporation (a factor counter-balanced, it is true, in some degree, by the extensive cover of ice and snow) . For temperate zones McGee's figure for cultivated fields is undoubt- edly too high to apply as an average for the entire land. In the trop- ics, on the other hand, it is perhaps not excessive. On the whole the fraction one-sixth computed as above is probably too high, but per- haps it serves to give us an idea of the order of magnitude involved. WATER CYCLE 217 Another estimate leading to a materially lower result, is obtained as follows: If we accept Engler's estimate that one-fiftieth of the atmospheric carbon dioxide, that is to say, 4.4 X 10 10 metric tons, takes part in the organic cycle; and we adopt the figure given by L. J. Henderson 2 that the water taken up by plants, is about 11 times the C0 2 which they absorb, we obtain, as a very rough estimate of the water engaged in the organic cycle, an amount of 5 X 10 U metric tons, or, in round numbers, 120 cubic miles. This, then, is about vfar of the total annual circulation of water, or about $%-$ of the total rainfall on land. CHAPTER XVII THE CARBON DIOXIDE CYCLE Behold how great a matter a little fire kindleth. St. James. If the lamp of life is a poetic symbol, it is an image essentially true to fact. Not only is life, in particular animal life, largely a combustion process: like the flame, life reaches out for fuel, and with the power gained, strains again for more. Like the flame it consumes, and it spreads. And as the fire sends out sparks, of which many die, but a few, falling upon favorable ground, flare up as a second generation, in reproduction of the parent flarne; so the living creature scatters its seed, some to die, but some also to live again the life of the parent. "But," someone perhaps will remark, "a fire may start without preexisting flame; whereas all life is itself begotten of life." Is this distinction really so fundamental? In nature undisturbed by man the starting of a fire spontaneously is a rare event; and that, after all, is the most that we can say posi- tively regarding the origination of life from the non-living it is either so rare or so unobtrusive 1 an event as to have escaped our observation. No doubt it took man many thousands of years to acquire the art of lighting a fire, may not in the lapse of time a second Prometheus arise to teach us also how to kindle the torch of life? Let us not delay his coming by closing our minds to the possibility. 2 1 Compare F. J. Allen's view, as presented by L. L. Woodruff in The Evolu- tion of the Earth and its Inhabitants, 1919, p. 102: "Life at this stage was of the humblest kind, since there were no definite organisms, only diffuse substances trading in energy, and between this stage and the evolution of cellular organisms an immense period elapsed." If this picture of the begin- ning of life is true to fact, the process was unobtrusive ; probably, if we were shown a specimen of such elementary "living" matter, we should not recognize it as such. All this is in accord with what has been said in an earlier ^ chapter regarding the definition of life. If we continue to use the word life, this is merely a matter of convenience and does not imply any departure from the point of view set forth in the opening chapters. See also p. 19. 218 CARBON DIOXIDE CYCLE 219 Be that as it may, the fundamental fact remains that slow com- bustion, oxidation as the chemist calls it, is a dominant feature in the physiology of animal life, and that the leading roles in this action are played by the elements carbon and oxygen. In our method of securing our supply of these elements there is a certain dyssymmetry. Carbon we eat in our meals; oxygen we breathe in in respiration. But in function the two elements stand in essentially symmetrical relation; the two together and impartially furnish us with the requisite energy for our life activities. ^ Thus we must regard oxygen as food as much as carbon. This fact deserves a passing note, since it is sometimes stated that assimila- tion of inorganic food is a characteristic of plants, as distinguished from animals. The statement rests on an arbitrary and wholly gratuitous exclusion of oxygen from our list of foods. It just so happens that there is one item on the animal's menu, namely, oxygen that is gaseous and is spread broadcast; does not therefore have to be hunted and captured. Toward this the animal assumes the same attitude which, presumably, plants adopt toward all their foods: he takes it in unconsiously. This touch of plant nature which we recognize in ourselves should serve to give us a sympathetic insight into the "psychology" of plants. At the same time it re- minds us once again of the esentially arbitrary character of the division of organisms into two classes, animals and plants. One and the same organism possesses both animal and plant characteris- tics, and this is true even of that most highly specialized of ^all animals, the human being. In view of the symmetry in function that exists between carbon and oxygen, and the inseparable relation 2 It is easy to strike a match, but this is merely a way of borrowing tho efforts of others, those who have made the match possible, and those who have manufactured it. How many city dwellers could, by their unaided ef- forts, start a fire where none was before? Anyone who, in an emergency, may have been forced to attempt the feat will appreciate how among tho ancients "the spark of fire was zealously guarded and soon invested with sacred attributes .... The chief function of the vestal virgins in Homo was to keep the perpetual fire; and in the Catholic church today with its never extinguished light we have the last survival of what was once a social custom " Another curious survival of this custom is quoted by E. A. Seligman (Principle of Economics, 1908, p. 69): "Whenever the location of gas works is changed the fire is transferred by a brand from tho old to the new building. Under no consideration would a new fire bo started." 220 ELEMENTS OP PHYSICAL BIOLOGY in which, they stand in the life processes of the organism, we shall consider jointly, in this chapter, the circulation of carbon on the one hand, and of oxygen on the other. The Carbon Cycle. A very particular interest attaches to the carbon cycle. Carbon is the organic element par excellence, whose absence from any chemical substance stamps this forthwith, by common if somewhat arbitrary consent, as inorganic: whose pres- ence affords the soil ? ,, \ season for the growth of what might be termed the tropical jungle in the domain of chemistry. For in the compounds of carbon nature seems to have run riot, in a revel of creative versatility, as if vying to set a record unapproached else- where in all the realm of chemistry, for number, variety and com- plexity of her children. Other elements 'oxygen, nitrogen, phos- phorus, sulphur, iron, indeed play a significant role in life processes; but the indispensable bond that ever links all other ingredients in organic unity is carbon. Furthermore, carbon is preeminently the energy carrier, the standard coin of the organic real, in which both the first cost of installation, of anabolic tissue building, and also the running cost of operation, of metabolism, is defrayed. Indescribably complex far beyond the understanding of the organic chemist of today as are the metamorphoses that carbon undergoes in the economy of the organism, its source and its gate of entry into the organic cycle are comparatively simple. Some two and a half million million tons (2.2 X 10 15 kgm.) of carbon dioxide, in the air, and perhaps twenty to twenty-five times this amount contained in the waters of ocean, lakes and rivers, these constitute the store from which all life ultimately draws its supply. Of this vast store, according to an estimate made by C. Engler, 3 about one-thousandth part actually takes part in the cycle of life. The total carbon locked up in living organisms, which would be a measure, in a way, of the spread of life on our globe, is difficult to gage even roughly. One hardly knows how much or how little significance to attach to an estimate by A. G. Hogbom 4 that the total quantity of carbon in all living matter is of about the same order as that contained in the atmosphere, namely, 6 X 10 11 metric 3 Linck, Kreislaufvorgaxige in der Erdgescliichte, 1912, p. 6; Engler, tJber Zerfalls-prozesse in der Natur. 4 Clarke, Data of Geochemistry, 1920, p. 49. CAKBON DIOXIDE CYCLE 221 tons; an amount which, spread over the entire surface of the globe, would cover it with a film of carbon 1 mm. thick, or with a film of living matter about | inch thick. The organic carbon cycle, reduced to its simplest terms, is a closed chain of three links. Atmosphere / Green Animals* Plants Green plants, under the influence of sun light, absorb C0 2 from the atmosphere and convert it, with elimination of oxygen, into the many and complex compounds of the plant substance. 5 Animals consume plants (directly or indirectly) as food, and in the course of the operations of their typically active lives (as compared with the typically passive, vegetative existence of the majority of plants) they reoxidize the carbon reduced in the photosynthetic plant processes, and return C0 2 to the atmosphere, thus completing the cycle. In actual fact this simple fundamental cycle is complicated by a number of influences. The decay of dead animals and plants adds a comparatively small item to the discharge of C0 2 into the atmos- phere. Plants are somewhat mor-e resistant to complete decay than animals, and one result of this is the accumulation of notable 6 To discuss here the chemistry of photosynthesis in plants would lead us too far afield. Very important advances have been made recently in this field of biochemistry. It must suffice here to refer to the original liter- ature of which the following articles may be mentioned: E. C. Baly, Photo- synthesis, Nature, March 16, 1922, p. 344. Report of discussion on photo- synthesis at the British Association Meeting, Nature, December 23, 1922, p. 856. I. M. Heilbron, The Photosynthesis of Plant Products, Nature, April 14, 1923, p. 502. 0. Baudisch, On the Formation of Organic Compounds from Inorganic by the Influence of Light, Science, April 20, 1923, p. 451. 0. Baudisch, The Influence of Light on Inorganic Matter and Life Processes, Jour. Industrial and Eng. Chemistry, May, 1923, p. 451. J. C. Bose, Effect of Infinitesimal Traces of Chemical Substances on Photosynthesis, Nature, July 21, 1922, p. 95. Baly, Heilbron and Parker, Photochemical Produc- tion of Formaldehyde, Nature, September 1, 1923, p. 323. J. H. Mathews, Trends in Photochemical Research, Jour. Ind. and Eng. Chem., September, 1923, p. 885. 222 ELEMENTS OF PHYSICAL BIOLOGY quantities of reduced carbon in the form of peat and coal. This process of fossilization is slow, and would not in itself, in any short period, materially affect the carbon cycle. It has, however, fur- /\ nished the occasion for a phenomenon which, judged in a cosmic , ' perspective, represents a purely ephemeral flare, such as must ulti- mately appear utterly insignificant in the geological calendar, if duration alone is considered; but which to us, the human race in the twentieth century is of altogether transcendent importance: The great industrial era is founded upon, and at the present day in- exorably dependent upon, the exploitation of the fossil fuel ac- cumulated in past geological ages. We have every reason to be optimistic; to believe that we shall be found, ultimately, to have taken at the flood this great tide in the affairs of men; and that we shall presently be carried on the crest of the wave into a safer harbor. There we shall view r\ with even mind the exhaustion of the fuel that took us into port, knowing that practically imperishable resources have in the mean- while been unlocked, abundantly sufficient for all our journeys to the end of time. But whatever may be the ultimate course of events, the present is an eminently atypical epoch. Economically we are living on our capital; biologically we are changing radically the complexion of our share in the carbon cycle by throwing into the atmosphere, from coal fires and metallurgical furnaces, ten times as much carbon dioxide as in the natural biological process of breathing. How large a single item this represents will be rea- lized when attention is drawn to the fact that these human agencies alone would, in the course of about five hundred years, double '-r the amount of carbon dioxide in the entire atmosphere, if no com- pensating influences entered into play. In point of fact the per- f centage of carbon dioxide in the atmosphere exhibits remarkable I constancy and there are several very large items, in addition to those already touched upon, both on the ingoing and outgoing side ' of the account. The case of man has been singled out for mention here merely because our knowledge of the human population and economy enables us to make a reasonably close estimate of his contributions. The quota supplied by the remaining animal species can hardly even be guessed at. But probably the greatest source of atmosphere C0 2 are volcanoes and mineral springs. Coto- paxi alone has been credited with an annual discharge of two million 4 tons of the gas. *- * [ CAEBON DIOXIDE CYCLE 223 On the debit side there is first of all the item of consumption by plants. E. H. Cook, 6 from very uncertain data, computes that leaf action alone more than compensates for the production of carbon dioxide, consuming about one-hundredth of the total at- mospheric oxygen in a year. Yost 7 computes that if the entire land area were planted with sun flowers, about 6.5 X 10 11 tons of CO 2 would be absorbed per annum. Forests would be considerably less efficient, and would take care of about 2.8 X 10 10 tons per an- num, or, say in round numbers one-hundredth of the atmospheric carbon dioxide. An older estimate, by Liebig, puts the annual output of the soil of Central Europe at 2.5 tons of dry organic matter per hectar, or, say 1 ton per acre. Allowing 40 per cent carbon in such organic matter we find for [the total annual production of carbon in plants 13,000 million (1.3 X 10 10 ) tons. This is about ten times the world's annual coal consumption, and about one-fiftieth of the total carbon in atmospheric carbon dioxide. Arrhenius points out that if all this carbon fixed by plants were deposited in peat bogs, the atmosphere would be depleted in half a century. But, of course, only a small proportion of the bodies of plants are thus "horded" and removed from the organic life cycle. A figure of perhaps greater interest, because based upon observa- tions of processes actually going on in a selected portion of the universe, is given by W. R. G. Atkins. From the observed change in the hydrogen ion concentration in the water of the English Channel, this author has calculated that 250 metric tons of organic carbon (figured as hexose) were produced per square kilometer between July and December. From similar observations made at Port Erin, Moore found a production of 300 metric tons per square kilometer during the six months that included the vernal maximum, of diatom production. 8 An important item among the withdrawals from the atmosphere is the absorption of carbon dioxide in the weathering of rocks, with replacement of silicates by carbonates. A. G. Hogborn reckons this item as about balancing the reproduction of carbon dioxide in the F. W. Clarke, loc. oit., p. 48; Phil. Mag., 5th ser., vol. 1882, p. 387. 7 Pflanzenphysiologie, 1913, p. 151. 8 Journal of Marine Biol. Assoc. October, 1922, vol. 12, no. 4, Nature, January 27, 1923, p. 132. 224 ELEMENTS OP PHYSICAL BIOLOGY combustion of coal. Its accumulated record is seen in the sedi- mentary rocks, which, according to F. W. Clarke, 9 contain 30,000 times as much as C0 2 as are today present in the atmosphere. 10 Sterry Hunt "illustrates the effect of weathering by the statement that the production from orthoclase of a layer of kaolin (china clay) 500 meters thick and completely enveloping the globe would con- sume 21 times the amount of C0 2 now present in the atmosphere." Chamberlin and others estimate that it would take about 10,000 years to consume the present amount of atmospheric C0 2 by the weathering of rocks. Loss of C0 2 by peat formation may be esti- mated at the same figure. The formation of COa by the burning of coal would, according to these estimates, cover the loss by weather- ing and peat formation combined, seven times over. Finally, there is the great reservoir of the ocea, in equilibrium with the atmosphere, and absorbing, under present conditions, some 18 to 25 times as much C0 2 as it leaves in the air above it. Thus the sea acts as a vast equalizer; of every ton of C02 thrown into the atmosphere by volcanoes, or by coal fires, for example, the ocean ultimately receives directly or indirectly, about 1900 pounds, only the balance of 100 pounds remaining in the atmosphere. It is thus seen that even extensive contributions from the lithosphore have but a slight effect upon the atmospheric store, and fluctuations are in this way ironed out and moderated. Arrhenius 11 points out, moreover, that at the present time the carbon dioxide content of the air over the ocean is on an average 10 per cent lower than over land. From this, and the fact that generation of C0 2 by coal (and probably also from volcanoes) has in late years been in- creasing, he concludes that the air is, at the present epoch, becom- ing richer in this gas. 8 Loc. cit., p. 48. See also C. Shuchert, The Evolution of the Earth and its Inhabitants (Yale Press, 1919), p. 52. 10 Hogbom's figure, quoted by Arrhenius (Worlds in the Making, 1908, p. 54), is 25,000 for limestones and dolomites. Am. Jour. ScL, 3d ser, 1880, vol. 19, p. 349. Clarke, loc. cit, p. 48. 11 Loc. cit., p. 54. Arrhenius' figure differs somewhat from the one given above. He supposes that the sea takes up five-sixths of the CO Z thrown into the air, i.e., 1 ton would yield up 1667 pounds to the ocean, leaving 333 pounds in the air. ft CARBON DIOXIDE CYCLE 225 As to which side of the account shows a net balance, in the carbon cycle, we have no certain knowledge. Arrhenius builds his con- ception of the future industrial development of our race on the expectation that the atmosphere is gaming in carbon dioxide, under the present regime of "evaporating" our coal mines, as it were into the air. On the other hand, if our atmospheric C0 2 is of volcanic origin, and the balance is maintained today with the aid of discharge from the lithosphere, then the ultimate extinction of the earth's plutonic fires would bring in its train the depletion of the atmosphere and secondarily the extinction of life. "The cessation of volcanism would signify the end of life on the globe." A similar position is taken by C. Schuchert, 12 who further remarks: We should add that if there were again as much life as there is at present, all the carbon of the atmosphere would be in the living plants and animals, and, if such a condition were possible death would come to them all .... Life and its abundance at any time are conditioned by the amount of this gas (COt) present in the atmosphere. This remark of Schuchert's is suggestive as illustrating in concrete manner the relative amounts of carbon concerned in the life balance. It is, perhaps, somewhat misleading in making no mention of the equalizing influence of the ocean which has been noted above. In point of fact, if all the C0 2 in the air were withdrawn, a nearly equal amount would rise from the ocean to take its place. A summary, in graphic form, of the principal relations in the carbon cycle noted in the preceding paragraphs, will be found in figure 47, which should aid in giving a comprehensive picture of the situation. < ; The Oxygen Cycle. The organic oxygen cycle is, of course,, directly related to the carbon cycle, although other features also enter into operation in regulating the oxygen balance of the atmos- phere. The complementary relation between animals (essentially oxidizors of carbon) and plants (essentially reducers of carbon diox- ide) is indeed a biological fact of fundamental importance at the present stage of evolution. But if we look back through the vista of ages, to the time before the advent of life, such as we^know it, our curiosity is aroused as to the origin of the atmospheric carbon i The Evolution of the Earth and its Inhabitants, 1919, p. 52. 226 ELEMENTS OP PHYSICAL BIOLOGY H O o NA B W w tJ W ft o o e ! t-3 P O M O (JABEON DIOXIDE CYCLE 227 dioxide and oxygen. Various views have been upheld on this subject. F. W. Clarke 13 remarks: It is likely that carbon dioxide has been added to the atmosphere by vol- canic agency, in some such manner as this: Primitive carbon, like the graph- ite found in meteorites, a,t temperature no greater than that of molten lava, reduced the magnetite of igneous rocks to metallic iron, such as is found in many basalts, and was itself thereby oxidized. Then, discharged into the atmosphere as dioxide,, it became subject to the familiar reactions which restored it to the lithowphere a,s coal or limestone. Arrhonius, 14 referring to Koehno's reflections on this subject, points out that the atmosphere contains about 1.2 X 10 1 ! 5 tons of oxygen, an amount which roughly* 6 corresponds with the mass of fossil coal in the sedimentary rocks. "The supposition appears natural, therefore, that all the oxygen of the air may have been formed at the expense of atmospheric carbon dioxide. Probably all the oxygen of the air owes its existence to plant life." F. W. Clarke resumes the views of a number of investigators as follows: 10 C. J. Koehne assumed that the primitive atmosphere contained no free oxygon, and he has boon followed by T. L. Plupson, 17 J. Lemberg, 18 J. Steven- son, 10 and Lord Kelvin." Lomberg and Kelvin, however, do not go to ex- tremes, but admit that possibly Homo free oxygen was present even in the earliest times. Lemberg argued that the primeval atmosphere contained chiefly hydrogen, nitrogen, volatile chlorides, and carbon compounds; the oxygen which is now free, being Mum united with carbon and iron. The 13 F. W. Clarke, loc. oil., p. 55. "Worlds iu the Making, 1908, p. 58. 111 The correspondence is rather distant if we accept Engler's estimate of the world's coal reserves, namely, !J X 10 1S tons, containing 75 per cent car- bon.. Hoc Lin ok, loo. oil;., p. 37, Kngler'H estimate is probably low. The World Altnanao, 1921, p. 201., given 7.5 X H) n! tons. This would correspond to 1.5 X 10 13 tons oxygon, as against 1.2 X ID 11 ' tons in the atmosphere. It is true that these ostinmteM of coal reserves cover only such coal as it would pay to mine. F. W. Clarke, loc. oil;., 1021, p. 50. 17 Chom. News, ISO.'J, vol. 07, p. i:?5. Also several notes in vols. 68, 69, and 70. For Koohno'n work wee PhipHou'n papers, 1893-1894. "Zeitsohr. Deutsoh, gool. (lenell, 1888, vol. 40, pp. 030-634. I'PhiloH. Mag., 1900, 5th ser., vol. 50, pp. 312-,'JOO; Oth ser., 1902, vol. 4 ; p. 435; 1905, vol. 9, p. MS; 1900, vol. 11, p. 220. * Ibid., 1899, 5th Her., vol. 47, pp. 85 89. 228 ELEMENTS OF PHYSICAL BIOLOGY liberation of oxygen began with the appearance of low forms of plant life, possibly reached a maximum in Carboniferous time, and has since diminished. Stevenson's argument is much more elaborate, and starts with an estimate of the uncombined carbon now existent in the sedimentary formations. In the deposition of that carbon, oxygen was liberated, and from data of this kind it is argued that the atmospheric supply of oxygen is steadily increasing, while that of carbon dioxide diminishes. The statement that no oxygen has been found in the gases extracted from rocks is also adduced in favor of the theory. First, an oxidized crust and no free oxygen in the air; then proc- esses of reduction coming into play ; and at last the appearance of lower forms of plants, which prepared the atmosphere to sustain animal life. The arguments are ingenious, but to my mind they exemplify the result of at- taching excessive importance to one set of phenomena alone. It is not clear that due account has been taken of the checks and balances which are ac- tually observed. At present the known losses of oxygen seem to exceed the gains. For example, C. H. Smyth 21 has estimated that the oxygen with- drawn from the air by the change of ferrous to ferric compounds, and so locked up in the sedimentary rocks, is equal to 68.8 per cent of the quan- tity now present in the atmosphere. G. Bunge 22 also supports the view that atmospheric oxygen is continually diminishing, becoming bound by the ferrous oxide resulting from the decomposition of silicates. Accumulation of carbon dioxide in the atmosphere, at any rate under present condi- tions, is also assumed by S. Arrhenius, as has already been remarked. 23 21 Jour. Geology, 1905, vol. 13, p. 319. 22 Text book of Physiological and Pathological Chemistry, 1902, pp. 16-17. 23 See Jour. Franklin Inst., 1920. vol. 190, pp. 114-121. CHAPTER XVIII THE NITROGEN CYCLE If the denuuul becomes inHiHteut enou K h, wo cannot doubt that methods will be dovimul which mil K iv us the desired results. To question that would bo to admit thai; man haw nearod the culmination of his evolutionary career and is preparing to bequeath the iwwtery of the earth to hia successor who- ever that may be.' (7. W. Martin, ' Natural Demand and Supply. The proportion of nitrogen to carbon in the human body IB 1 :3, in the atmosphere it is 5,500: 1. A human adult contains in IUH body about 42 pounds of nitrogen, and over 50 pounds of curium. Over every square foot of the earth's surface rifles a column containing Homo- .1500 pounds of nitrogen, and only about. 1/4 pound of carbon. The demand and the supply of these two element appear, therefore, at first sight, to be altogether out of all proportion favorable to nitrogen. Yet, in point of fact, the practical problem of Heeuring an adequate supply for the substance and expansion of life w incomparably more complex in the case of nitrogen than in the cawo of carbon. The reason for this somewhat remarkable, inversion in to bo noon in the fact that nitrogen is readily aecoHsibio an food for living organisms only when it occurs in certain chemical combinations, and nitrogen thus combined is far from plentiful. It has I een estimated by T. II. Norton that this available or "nomadic." nitrogen -i.e., that which taken part in the migration through the organic, cycle -amounts to only about two one-millionths of the total nif.rogen of the atmosphere, or, way, to about 8 X 10 9 tons. In fact, njUrogon w today probably the chief of those limiting factors' which, in accordance with Liobig's law of the minimum, establish the bounds for the extreme expansion of living matter upon the earth. At the name time the circumstance of the chemical idiosyncrasies of the element nitrogen introduces a certain com- plexity into the nitrogen cycle, which strikes the eye at a glance in the charts, figures -18 and 40, exhibiting the essentials of the nitrogen 1 Compare ("}. Bunge, I'liymolotfioal and Pathological Chemistry, 1902, p, 17; Grinmsll J<uu, Qu. Jour. lOconomioH, 1920, vol. 34, p. 394. 220 230 ELEMENTS OF PHYSICAL BIOLOGY NITEOGEN CYCLE 231 'Nitrogenous Vegetable. \tmosf>harit\ , Nftrogan / V ^A>^ J/IM'^***^ J x ! ll 1 I S : ^ *i / Nitrogenous Animal Compounds! Urea Ammonia Kl<!, '!!), OltOANIC NlTHOUIW CntOULATION ". -18 H!H>WH in hrondor outline the main feature of the oycki, while figure -ID cx!nl)i(,,s in greater detail especially those stages in the eyde UuU. am muni, iudijuately aKsociated with life agencies. 2 '' For dcituilH (in UUH ])linH<> of the milijocfc the reader must be i-oferred to Mio Hpocutl liUu'iil.iinv A. good Hiinunary, fairly detailed and complete, yot coticiHO, will IHI found in [{ llubor, Zur Sfcickstoff-Frage, Born 1908 pp. 1-10. 232 ELEMENTS OF PHYSICAL BIOLOGY Gate of Entry into Nitrogen Cycle, The natural gateway for the entry of atmospheric, elementary nitrogen, into the organic cycle is a narrow one. So far as at present known, only a limited class of organisms possess the faculty of "fixing" this element, that is, taking it in its gaseous state from the atmosphere and converting it into the condensed (liquid or solid) form, in which only it has com- mon acceptation as coin of the organic realm. The organisms known to take part in this natural process of nitrogen fixation are three, namely certain bacteria having their habitat in the soil; certain leguminous plants (peas, beans, clover, alfalfa), working in conjunc- tion, in symbiosis, with nitrogen-fixing bacteria lodged in tubercles upon their roots; and, thirdly, the wheat plant has recently been shown to possess the independent faculty of assimilating nitrogen from the air. This recent demonstration, 3 of course, suggests the ready question whether after all, a number of other plants may not be similarly endowed. This remains as a matter for further investiga- tion, but it is improbable that we shall have occasion to change materially our present impression, namely, that the natural avenues by which elementary nitrogen gains admission from the atmosphere into the cycle of life are rather narrowly restricted. As to other avenues opened up by the man, these will be considered presently. Leak of Nitrogen out of Circulation. While there is thus a narrowly restricted class of vegetable organism through which nitrogen trickles in a thin stream from the elementary supply in the atmosphere into the life cycle, the majority of plants derive the supply for their biological needs from ready formed nomadic nitro- gen in the soil, that is to say nitrogen combined in the form of ammonium salts, nitrites and nitrates. These substances are sub- ject to oxidation and reduction in the soil under the influence of various bacteria, as indicated in the chart figure 49. The result of these changes is that a certain fraction of the nomadic nitrogen is continually leaking out of the circulation and joins the general reservoir of free nitrogen in the atmosphere. This is only one of a number of items on the losing side of the balance sheet. Other items will be found in following up the details of the two charts already 3 C. B. Lipman and J. K". Taylor, Science, November 24, 1922, p. 605. These authors report that in their experiments wheat plants assimilated 13 to 21 per cent of their nitrogen content from the air. NITROGEN CYCLE 233 referred to. There is loss in the autumnal leaf fall of deciduous plants; in the decay of dead plants, or in their fossilization as peat, lignite and coal. Forest fires, and the burning of wood and coal; the distillation of coal for illuminating gas; the oxidation of coal in metallurgical furnaces, the coking of coal in ovens of the so-called beehive type; all these are operations in which a greater or less pro- portion of the combined nitrogen in the coal is liberated into the air in the free, unavailable form. In view of the great loss which this represents in the economy of our food resources, the highest impor- tance attaches to the modern drift away from the beehive coke oven to the by-product oven, in which the major part of the nitrogen in the coal is recovered as ammonia. TABLE 22 Fraction, of total output of coke in the United States, produced in by-product ovens YEAH PERCENT YEAH PER CENT YEAR PER CENT 1890 1910 17.1 1917 38.6 1900 5.3 1915 33.8 1918 46.0 1905 10.7 1916 35.0 1919 56.2 These by-product ovens were introduced in 1893. It is estimated that during the period from 1893 to 1910 alone, through the con- t Limed use of the old beehive type coke oven, over 9,300,000 tons of ammonium sulphate were wasted, representing, at the prices then prevailing, a value of 553 million dollars. In addition to this must be reckoned a further loss in the resulting field crops. Had all the nitrogen wasted in the beehive ovens been spread as fertilizer on the field, this would have increased the crops some 20 per cent. 4 An idea of the extent arid significance of the healthy modern drift to- wards replacement of the beehive by the recovery coke oven may be gathered from table 22, reproduced from an article by Grinnell Jones in the Quarterly Journal of Economics, 1920, vol. 34, p. 402. H. E. Fischer, writing in the Journal of the Franklin Institute, 1920, vol. 190, p. 191, remarks that if all the coal in the United States were used as coke, and the ammonia recovered in the process, this alone would 4 This and other data regarding nitrogen losses here set forth are drawn, largely, from an article by J. D. Pemiock in the Journal of Industrial and Engineering Chemistry, 1911. 234 ELEMENTS OF PHYSICAL BIOLOGY furnish one million tons of ammonia, which corresponds to about one-half the world's total production of nitrogen compounds. Returning to the chart (fig. 48) and continuing to trace the prog- ress of nitrogen in the organic cycle, we note next that combined nitrogen is absorbed from plants by animals in their food. It is rejected from the animal economy in part as excretory matter (manure, etc.), in part in the bodies of dead animals, in so far as these are not themselves consumed as food. A large item here, in the economics of the human community, is the refuse from slaughter houses. Certain portions of this are recovered for various uses (glue, leather, etc.). Some is made into fertilizer. Much of it goes to waste, and thus gives opportunity for another leak of nitrogen out of the life cycle to the elementary form in the atmosphere. It is difficult to form any estimate of the extent of this loss, but there can be no doubt that it represents a waste of hundreds of tons of nitrogen daily. Much also is lost from the other item of animal waste ma- terials, not a little of the loss being occasioned by modern methods of sewage disposal in large cities. Such methods represent, from the standpoint of agricultural economy, a luxury, which however, will be thought worth the price if the means are at hand to make good the loss from other sources. Those portions of animal refuse which are placed on the soil of crop-bearing fields and pastures return, at least in part, into the organic cycle. Accessory Sources of Combined Nitrogen. It is of course abso- lutely essential for the continuance of the life cycle that the losses of combined nitrogen which have been noted should in some way be compensated by equal or greater accessions to the total amount of nomadic nitrogen. One source of such compensating revenue has already been noted, namely the direct assimilation of elementary nitrogen from the atmosphere by a narrowly restricted class of plants. There are two other natural sources. Volcanoes and furnaroles belch notable quantities of ammonium chloride into the air; nitric acid is also formed by the action of lightning, while ammonia is produced by the passage of the silent electric discharge (aurora) through the atmosphere. Arrhenius 5 estimates that the amount of nitrogen annually bound in this way amounts to about 1.4 X 10 9 B S. Arrhenius, Worlds in the Making, 1908, p. 144. See also Haber, Zeitschr. f. Angev. Chemie, 1910, p. 685. NITEOGEN CYCLE 235 metric tons, or one part in 3 millions of the total atmospheric nitrogen. The products are washetl into the soil by the descending rain, to- gether with more or less of the same substances that have escaped into the atmosphere from the ground and are thus restored to the soil. Estimates which have been made of the quantities involved 3.6 I .8 O oo IB30 6O TO WOO /O SO 90 YEAR. FIG. 50. THE RISE OF THE SALTPETER INDUSTKY SO 30 are somewhat conflicting, but on the whole the gains of the soil in this way are held to exceed its losses. 6 The stages and agencies so far reviewed may be collectively desig- nated as those constituting the "natural nitrogen cycle," as dis- tinguished from a group now to be considered, which are charac- 6 F. W. Clarke, loc. cit., 1920, p. 52; Linck, Kreislaufvorgange, 1912, pp. 6-7. It has also been put forward (Sch.dn.bcin) that a certain amount of nitric acid is formed in the evaporation of moisture from the earth (Bunge Physiological Chemistry, 1902, p. 11). But this is doubted by Ostwald. (Grundlinien der Anorganischen Chemie, 1912, p. 384.) 236 ELEMENTS OF PHYSICAL BIOLOGY terized by human interference with the course of nature. It Is hardly necessary to point out that such a distinction between natural and artificial agencies is merely a convenient use of brief terms; the fact must never be lost sight of that man himself is very essentially part of nature, and that his development, whether physiological, psychological, sociological, economic, or what not, is part of the great process of nature. Human Interference in Nitrogen Cycle. Man's earliest con- scious, purposive intervention in the nitrogen cycle dates from antiquity, and primarily consisted merely in taking more or less pains, in an empirical way, that the nitrogenous waste material of animal economy be, as far as possible, restored to the soil. Perhaps the first recorded use of fertilizers not derived from current wastes of domestic animals is the exploitation of guano by the Incas, which dates from antiquity, and was brought to the notice of Europe by de la Vega in 1604. It seems to have aroused no interest until attention was again drawn to it two hundred years later by von Humboldt and by Justus Liebig, and large scale importations of guano into Europe began soon after this. A greater event in the history of agriculture was the opening up, in 1831, of the Chilean nitre beds. Without this source of saltpeter the modern develop- ment of intensive agriculture, and the consequent growth of popula- tion in all civilized countries, would have been at the least greatly hampered. The rapid rise of the saltpeter industry is clearly ex- hibited in table 23 and the corresponding graph figure 50. While our chief interest here is in the agricultural use of Chile saltpeter, its consumption in the industries is altogether too extensive to be passed by without mention. J. D. Pennock 7 gives the figures shown in table 24 for the relative amounts of saltpeter consumed in different uses. It should be observed that Pennock's figures relate to peace time conditions. Even so, nearly one-half the consumption is taken up in the manufacture of explosives. Nitrogen thus employed is, of course, lost to the life cycle. A certain loss also concurs in the re- fining of the caliche (native saltpeter), and in the production of nitric acid and sulphuric acid 8 therefrom. 7 J. D. Pennock, Jour. Industr. and Eng. Chem., 1911, p. 172. 8 When manufactured by the Chamber process. NITROGEN CYCLE 237 TABLE 23 Growth of the saltpeter industry SHORT TONS (I) (II) Sodium nitrato 1830 soo 1831 112 1850 22,800 22,350 1SSO 222,559 257,060 1890 1,050,119 998,970 1900 1,471,500 1,490,950 190(5 1,795,000 1,835,190 1911 2,482,000 2,624,250 1913 2,730,000 3,055,800 1914 2,422,000 2,715,050 19 If! 1,728,000 1,935,140 1916 2,868,000 3,210,008 1917 2,955,000 3,319,920 1918 2,821,000 3, 170, 106 (III) Nitrogon on basin of 15.05 por con I 17.5 3,498 40,230 156,339 233,334 287,207 410,095 478,230 424,900 302,850 502,370 519,570 496, 120 The figures in column I are the Chilean nitrate production, as given by Par- sons and Petit, Brokers. The figures in oolxims 11 and III, from 1831 to 19 11 inclusive, represent the World's consumption of saltpeter according to C!(5ni Civil, vol. 62, p. 192. The figures from 1913 to 1918 in oolums II and II I ropi'e- serit the total production of Chilean and Indian nitrate, according to Griunell Jones, Jour. Frankl. Inst., 1920, vol. 134, p. 398. The precipitous drop in the Chilean production in 1915 was due to a blockade established by the GermariH during the early stages of the, World War. See Fig. 50, in which circles indi- cate production, the drawn out curve consumption. TABLE 2t Distribution of Chili saltpeter consumption in the United States in 1010 among different uses In manufacture of fertilizer. In manufacture of dyestuffs. In general chemistry In glass In explosives In nitric acid In sulphuric acid Unaccounted for 13 12 10 4 41 9 100 238 ELEMENTS OF PHYSICAL BIOLOGY Origin of Nitre Beds. The origin of the nitre beds is uncertain. The presence of boron in the deposits, and the association of this element with ammonia in volcanic emanations, have been regarded by some as evidence that the saltpeter is of ultimately volcanic origin. Others have ascribed it to organic sources, such as altered guano deposits. But whatever be their origin, this is certain, that the salt- peter beds represent an accumulation of ages, and that the present rapid rate of consumption is out of all proportion with the rate of formation of the deposits. In other words, here, as in the case of coal, we are living on our capital, and must prepare ourselves for its impending exhaustion. 9 It is true that our other sources of combined nitrogen notably ammonia from coke ovens supplement our drafts upon the nitre beds, and thus help to defer the day of scarcity. But coal itself is a limited stock, and other sources of combined nitrogen seem quite inadequate for the needs which we have de- veloped under the stimulation of temporarily bountiful supplies. It is a peculiarity of living substance that in times of plenty it tends to grow beyond the bounds compatible with ultimate stability; it overshoots the mark so to speak; the curve along which it approaches its equilibrium is very apt to be humpbacked, or it may be oscil- latory. 10 The prospect of a period of actual diminution (not mere marking time) to follow upon a period of exuberant prosperity, is one that an organism gifted with foresight must look upon with disquietude. Such foresight may, then, lead an organism so gifted, to make efforts to provide for untoward future exigencies, either by laying by supplies, in times of plenty, for times of stress; or by devising means, if possible, to increase, by new measures, the supplies which, under the old regime, would presently fall short of require- ments. Man has not always made a display of brilliant foresight, but in this instance, in making ready for the exhaustion of the nitrate supply, he has taken time by the forelock, and all indications are that long before the emergency arises he will have made himself _ 9 The probable date of this exhaustion has been variously estimated. Little value can be attached to positive statements. More significant, per- haps, is the negative report made in 1913 to the Chilean Government by the Inspector General of Nitrate Deposits: "There is no fear of the Chilean nitrate deposits being exhausted for two hundred years" (Grinncll Jones loc. cit., p. 401). ' "Compare what has been said on this subject in Chapter XI. NITROGEN CYCLE 239 ready to meet it. For the last decade has seen the development, to full industrial capacity, of several processes for the fixation of atmos- pheric nitrogen, its conversion into compounds directly or in- directly adapted to enter the cycle of nomadic nitrogen. For details regarding these modern industrial developments the reader must be referred to the technical literature. 11 It must suffice to indicate here very briefly the nature of the several processes. 1. The Birkeland and Eyde Process, is essentially man's imitation of the production of nitric acid by lightning. Air is passed through an electric arc fanned out into a broad disc by a magnetic field. The process is commercially viable only where very cheap power is available, and has been developed mainly in Scandinavia, with the use of water power. 2. The Cyanamide (Frank and Caro) Process effects the ab- sorption of atmospheric nitrogen by calcium carbide in the electric furnace. The product can be employed directly as a fertilizer, or can be made to yield ammonia and other nitrogen compounds. 3. The Haber Process effects the synthesis of ammonia from nitrogen and hydrogen under pressure (100 to 200 atmospheres) in the presence of a catalyst. An allied process is that of Claude, which works at very high pressure (1000 atmospheres). ia 4. The Biicher Process, which has not yet passed beyond the experimental stage, yields cyanides; these can also, if the market warrants it, be made a source of ammonia. A highly significant development is the union of the Habcr ammonia process with the Solvay process, whereby the carbon dioxide obtained as a waste product in the manufacture of the hydrogen for ammonia synthesis, is utilized in the production of sodium carbonate; while, on the other hand, the formation of large quantities of the nearly worthless calcium chloride waste of the Solvay process, as ordinarily conducted, is avoided. Inasmuch us "soda ranks second only to sulphuric acid among all chemicals in magnitude of output (1,390,628 short tons in the United States in 1918) and fundamental importance, .... this (combination of the Haber and the Solvay processes) may well prove to be the most 11 Sec for example the articles by G. H. Fischer and Grinnell Jones already cited. 12 H. E. Fisher, Jour. Franklin lust., 1920, vol. 190, p. 201. 240 ELEMENTS OF PHYSICAL BIOLOGY 1,6 FIG. 51. THE RISE OF THE FIXED NITROGEN INDUSTRY NITROGEN CYCLE 241 significant development in industrial chemistry of the present decade." 13 5. The Ostwald Process, A subsidiary process bridging the gap from the product (ammonia) of the processes of the Haber and Frank and Garo types, to the market requirements of nitric, acid (nitrate and nitrites), is the Ostwald process for the catalytic oxida- tion of ammonia. The Meteoric Rise of Nitrogen Fixation Industries. In the period during arid immediately following the World War the situation in the nitrogen industries was abnormal, production being tempo- rarily activated to fever heat, inasmuch as nitrogen compounds are among the most indispensable of war materials. Tims it came about that the development of nitrogen fixation had for its imme- diate motive not so much the constructive spirit of the arts of peace, providing for the future needs of men, as the malice and forethought of the conspirators of war. The conflict left on our hands, upon the conclusion of the armistice, both completed and unfinished manufacturing plants in excess of immediate needs in times of peace. Legislative difficulties also hampered well-designed efforts to convert these plants to industrial use. These arc tem- porary conditions; although one may not be able to foresee exactly how, in detail, these industries will finally adjust themselves, there, can be little doubt that from now on synthetic nitrogen compounds will continue to be drawn in increasing amounts from the atmosphere into the life cycle. The phenomenal growth of this infant industry within the past ten or twelve years is forcibly brought out in the graph (fig. 51) and the corresponding table 25. It will bo observed that in 1909 only about 1 per cent of the world's needs in combined nitrogen were satisfied from the new-born industry. In .1.917 its contribution had swollen to 30 per cent, and by 1920 the capacity of existing plants was adequate to furnish 43 per cent of the world'n requirements. This extraordinary development is something much more than a fundamental new departure in industry. It represents nothing lens than the ushering in of a new ethnological era in the history of the human race, a new cosmic epoch. In the short span of a dozen years 'geologically speaking in an instant- man has initiated transformations literally comparable in magnitude with cosmic 13 Grinnell Jones, loc. cit., p. 414. 242 ELEMENTS OF PHYSICAL BIOLOGY " <, ^> (M CD <M o of w 3+J o co co o Pi "bfl wrd S co cSi ^ o O p| S P p, rH "JZJ " *"* ftg O3 O OO 'd 03 t* ^ Q O CM CD fJJ-2'43 Ol rH rl "-< >> O O O o *o ^ ^ o o co co P^ *O IH rH CO rH >Q S Ci" "3 *s> |ss ^ Tfl C3 >O IO % "ra s^ o> M rj ~j CO S t> o^ 7 1 5 Es 1 o CN 0) tj+3'3 o o o o o o O CO O ^ a -S g e *""* !* >o" oo" oo" *^ y _o Pn CN CO O rH "PJ S co co ^- rH Q .p~. PQ g g rH O o o> a ^ |1 ^O TP CD O ^ MH CD QJ C^'rt CD 1C) c*~i t., 4-* Cn ft S CD rH |i Uj~ w O ^ A CO O I'O *x~ I* \ S 0*3 <M" c<T co" oo" S 'P q 000 -M -rJ so -2 o o o 'a r3 -cJ T3 O TH O " r3 " Q) "9 M (H Tj O p] i^ o co a d *S> CO !> OS O O -iH o3 m^H l> C-l >O O CD CO 4- 1 rH r-i' rn '^ 'ff Pt *- o '"HJ >0 10 O 5 . u It t- 00 T}i ^ co CD T3 "3 O o 5? o "i * M ft Pi 1 o d ss H ^ a ^ a P) % Pk* 1 CO rH I^i bfl " S ^ t"* .* CQ o (D bfl O5 l-~ (^1 " CO o CO oo i co ."S "+H S a S2 rH Ol" Op 10" ^ r-, * CO in s! ** (N W i I tS "o CNJ la's >O ls- cj *K> . Cn> .p 2 co o O (j3 S ^ O "tf* c33 r3 -+J O ?* J^. ^J P 0+J IM" cjT ^H (M o g ^ % B 1 fc ^ CD & a S O 0:1 l> Q C4 "3 Os 1 II S S g "3 CN^ ^ 00*' rH O 1 ^ S s| M rt <*" <n "S g- n CO CO O *2 w OJ CO 50 s o 'O O CD" ,, 5 2 & ?3 <M >O ^ d ^ O CO . .V T"' " cu o PI "*^ J- c"S~'3^ ^ S'S ' t> __ 43 _=0 P O- ^X | p . J3 o. (3j K'HO *o d rt"" 1 ^9 p< rd . 'K (H J^fH,^- S W "3rg ? T3 rH Q P< K2 rH f^ W| m C^ O /H CU 5*3 K. e. jj jjs -f^> rn CU ^ -* r-t W ^_. *i-i r"^ i OUBCB OF SI! 5 a g - ,5, | ra a 'S g ft raO r i g 4 cuO-u '-S M "& +* O ert rH :; cu rH y rH W +3 ftj d **H W r^ ^ M S ^ > 03 L_J H ti -t-> H p _, CO Eflrd ffl '^1 1^- ill life 1 1 -t-J T) i: : H r^-j fj^^g es.oj 8co o M ^ rH"^ H O fQ 1 g *"* O S-i- S rH >, NITEOGBN CYCLE 243 processes. Accepting Arrhenius' liberal estimate of the total quan- tity of combined nitrogen washed down to the soil in the annual rain fall over all continents of the globe, namely 400 million tons, it is seen that the new industry even now is capable of furnishing a supplementary supply equal to one six-hundredth of this prodigious quantity. 14 Economic and Energetic Significance of Concentration. We are, of course, greatly more interested in this six-hundredth which is under our control 'and of which a due proportion falls, in con- sequence, upon our fields in concentrated form 'than in the very much larger quantity that nature scatters with sublime indifference on stony places and good soil alike. This is just one of those cases to which reference has already been made in a general way. It is not so much the quantity of the material provided by nature that counts, as its accessibility; and accessibility here means, among other things, suitable concentration. This question did not so obviously project itself into the discussion of the carbon cycle, because the __ natural source of carbon is atmospheric carbon dioxide, which, being a gas, in the very nature of things spreads evenly, and presents itself unsought, by a spontaneous process, at the mouth of the hungry plant. But the combined forms of nitrogen are for the most part solids or solutions, occurring in definitely localized amounts of greatly varying concentration. The labors forced upon us in our efforts to satisfy the nitrogen needs of our fields are, to be precise, not primarily work of production, but virtually work of concentration; or to be more exact, work of bringing about concentration at the particular locality where it is wanted by transportation if need be. It is only because we find k easier, in some instances, to produce than to concentrate existing supplies, that we elect the former expedient; just as we may prefer to feed a boiler with a fresh supply of water, rather than to u For land and sea together Arrhenius estimates 1500 million tons of nitro- gen in the rain fall. Note added in correcting 1 proof. Since the writing of this chapter there have become available Trade Infor- mation Bulletins No. 220 and 240 of the U. S. Department of Commerce, which give further and more recent data. In Bulletin 240 J. M. Braham estimates the capacity of the world's nitrogen fixation plants at 490,000 tons for 1923. The reader interested in this topic may also consult J. R. Partington, The Nitrogen Industry, publiBhcd by Constable, 1924. 244 ELEMENTS OF PHYSICAL BIOLOGY return to it the condensed exhaust from the engine. That the mere concentration of existing supplies should at all require the doing of physical work is a circumstance of particular interest, not only in its economic relations, 13 but also, and quite particularly, from the stand- point of energetics. This is a matter that will duly engage our attention in a later section, devoted to the energetics of the several processes that have here been considered in their purely material or stoiehiometi'ic aspect. Meanwhile it is interesting to observe that such localized sources of concentrated supplies as those presented in the Chilean nitre beds virtually function as centers of attraction toward which gravi- tates a stream of human beings 'Or their representatives in the form of ships and other conveyances arriving in search of cargo and going out laden with material. To an ultramundane observer who should survey the scene in suitable perspective, the activities around the nitre beds must appear very like the busy swarming of a colony of ants around the treasure trove of some silvan inhabitant departed this life; who, having completed his earthly career, is now yielding up, in the dissolution of death, such energies as still remain locked up in the carcass. Attractions such as this are, in a sense, merely apparent; they are the outward symptoms of a complicated chain of cause and effect 10 characteristic of the behavior of living organisms. Yet they are often so consistent in their action that it would not be unreasonable to essay a systematic treatment of the movements in a world comprising such centers of attraction and such moving pawns, on the basis of brute tropisms unanalyaed into their ultimate com- ponent agencies. Here it must suffice to have pointed out that our highly complex industrial system, our far-flung intricate network of lines of traffic by land and sea, is but a sublimated copy, on a heroic scale, of the hustle and bustle that is going on all around us in nature, in response to attractions, tropisms, determinants of the 15 A striking illustration of this is cited by Haber (Zeitschr. f Angew Chemie, 1910, I, p. 685). If the gold in sea water were extracted and" ap- portioned evenly to all the human inhabitants of the globe, we should all be millionaires three times over; yet it does not pay to as much as begin this extraction. _ 15 Commonly accompanied by that anticipatory inversion of the sequence in time, of effect and cause, which is the earmark of purposive action. NITHOGEN CYCLE 245 moves of an army of checkers over the mosaic of the earth's topography. 17 Total Circulation Tends to Increase. The study of the nitrogen cycle furnishes us with a first occasion to take note of a phenomenon the full significance of which will become apparent in dealing with the dynamics of evolving systems. It is to be observed that the general trend of man's effort, especially in this new epoch of nitrogen fixation, has been towards drawing into the organic circulation a greater amount of matter, enlarging the wheel of the mill of life, so to speak. There can be little doubt that this trend will continue and increase in the future, and that it is the expression of one aspect of a general law; 18 the other aspect of this law, and its full significance, must be reserved for later discussion, as already intimated. 17 For a discussion of the natural concentrating processes the reader may be referred to the following articles: A. C. Lane, Nature's Coneentraiors, Engineering and Mining Journal, 1897, vol. 03, p. 54-2. ,1. C. Russell, Con- centration as a Geological Principle, Bulletin Ccol. 8oc. An., 1907, vol. IS, pp. 1-28. E. Blackweldcr, The Geologic Role of Phosphorus, Aia. Jour. Sci., 1910, vol. 62, p. 285. W. Lindgren, Concentration and Circulation of the Elements from the Standpoint of Economic Geology, Economic Geology, 1923, vol. IS, pp. 419-44.2. 18 The phenomenon is, of course, closely related to that In which R. Lus- caux refers as "unc propriete particuliere do la vie: cclle do rextentiion" (La Production ct la Population 1921, p. 37). ftcc also A. J. Lotka, I'roc. Nat 'I Acad. Sci., 1922, p. 47. CHAPTER XIX THE PHOSPHORUS CYCLE There is no coming into being of aught that perishes, nor any end for it .... but only mingling, and separation of what has been mingled. Empedocles. Immobile Elements. The circulation of water, carbon dioxide and oxygen in nature is greatly assisted by the freely occurring processes of evaporation, condensation (rainfall) and diffusion. In the case of nitrogen these processes still give some aid, as in the dis- tribution broadcast of atmospheric nitrogen, and in the formation j. and precipitation of nitrogen "fixed" by lightning, etc. In phos- phorus we have, on the contrary, a typical example of the inherently immobile elements needful to the living organism to adopt Liebig's phrase (fur sich nicht beweglicli) . The successive steps in the con- centration, diffusion, and reconcentration of this element, as available for the substance of the organism, accordingly display a characteristic complexity. Some of the principal items and steps in the phos- phorus cycle are set forth in diagrammatic form in figure 52. Natural Phosphorus Supply of Soils. The virgin soil contains, in general, a certain natural supply of phosphates. So, for example, Van Hise 1 reports that the virgin soil of Ohio, Illinois and Wisconsin contained, to a depth of 8 inches, 2077 pounds of PaOg per acre. / After fifty-five years of cultivation this figure had sunk to 1813 pounds per acre, a loss of 36 per cent. These figures show the extreme im- portance of a conservation of our resources of phosphorus. The loss probably occurs partly through erosion by rain water charged with carbonic acid, which dissolves phosphates in the soil and in rocks, and ultimately "washes a certain proportion out to sea. To this unavoidable loss, however, is added a large item of preventable loss through our failure to ensure that the phosphorus of animal wastes be returned to the soil. So, for example, farmyard manure contains over three-fourths of the phosphorus in the feed and bedding supplied to the animals. It is therefore very essential that this be returned 1 Ann. Acad. Polit. Soc. Sci., 1909, vol. 33. . , 246 PHOSPHORUS CYCLE 247 o o W FM 248 ELEMENTS OF PHYSICAL BIOLOGY to the soil as completely as possible. Of the one-fourth of the phos- phorus fed to the animals, and not accounted for in the manure, a considerable fraction appears in the bodies of these animals, especially i in the bones. These, then, also should find their way back to the * soil, as they do to some extent through the practice of using bone meal, either as such, or after conversion into superphosphate, 2 as fertilizer. This practice is materially assisted by the modern methods of meat production on a large scale, with very complete utilization of by-products. Leakage of Phosphorus from Circulation. The human cadaver is in the great majority of cases returned to the soil in the regular course of events, although under conditions which, obviously do not render it very readily available for crop production. An adult contains about 1| pounds of phosphorus, or 3.4 pounds IW If we allow one-half of this for a "unit of population," i.e., 1.7 pounds 4 P 2 6 , with a death rate of 1.3 per cent per annum, we find that the ^ amount of P 2 5 annually committed to the cemeteries of the United States is about 1105 tons. This is about the equivalent of 3300 tons of phosphate rock, or about one-thousandth of the annual production of that material in the United States. But a very much larger amount of waste is occasioned from the human population by the practice of running the sewage from cities into rivers and thus to sea. Van Hise estimates that annually 400,000 tons P 2 5 or the equivalent of 1,200,000 tons of phosphate rock are thus ran to waste. He remarks: "The wide dispersal of the vast quantities of phosphorus which it took the process of nature an indefinite period to segregate, must cease. The loss is irrop- Jj arable." Much has been done in recent years to comply with this demand. Phosphate Rock and the Migration of Phosphorus. In the meantime, in our phosphorus economy also, as in the case of the combined nitrogen of the Chilean nitre beds, we are living on our capital. For, as our fields tend to become depleted of phosphorus under intensive agriculture, we restore some of the rarefied clement to the tired soil by drawing upon the accumulations of ages, in the form of phosphate rock. As a matter of fact, in this we are welding 2 By treatment with sulphuric acid, which renders the phosphorus more readily available to plants. These are technical details that cannot be on- tered into here. The reader must be referred to the pertinent agricultural A- and technological literature. PHOSPHOEUS CYCLE 249 the closing link in a very remarkable endless chain of nature. For the phosphates washed by the rivers 3 into the sea serve as food to the marine vegetation and indirectly to the fishes and other aquatic species. 4 These act in this case as concentrating agents, the bones and teeth of fish, and some shells of Crustacea and molluscs, being comparatively rich in phosphorus. 5 In its further migration the 3 For details the reader may be referred to an article by E. Blackwclder in the Am. Jour. Sci., 1916, vol. 62, p. 285, from which the following particu- larly pertinent passage may here be noted: "Of the vast quantity of dis- solved mineral matter annually delivered to the sea by the run-off, it is esti- mated that about 0.45 per cent consists of phosphorus pentoxide. Using the best available figures for the amount of water thus brought to the ocean annually, it is calculated that if the phosphatic material in the form of solid tri-calcium phosphate were loaded into standard railroad cars it would fill a train stretching continuously from Boston to Seattle and would be 7 to 12 times as great as the world's total production of phosphate rock in 1911. Nevertheless, so great is the volume of the oceans, and so vast the area of their floors, that if all this material were deposited in solid form uniformly over the bottom of the sea, it would build annually a layer less than 0.2 mm. thick. Of the phosphorus poured into the sea, so large a proportion is util- ized by living beings that the net working balance dissolved in oceanic water constantly averages less than 0.005 per cent, expressed as PzOs, or, in other words about 0.18 per cent of the dissolved salts. In this solution, phos- phorus seems to have reached the most dilute state in which it exists during the course of its complex migrations. Its subsequent transformations gen- erally tend to ever greater concentration, almost until the cycle is closed upon itself. 4 Compare also W. Lindgren, Concentration and Circulation of the Ele- ments from the Standpoint of Economic Geology: "In the sea water the blue- green algae concentrate phosphorus, certain molluslcs or crustaceans feed on the algae, and other meat-eating mollusks devour the vegetarians. Small fishes eat the molluslcs, large fishes eat the small, finally seals and birds swal- low the fishes, and so in about six transformations the phosphorus originally contained in the sea-water may come to rest in deposits of guano on desert islands or in accumulations of bones of vertebrate denizens of the sea" (Eco- nomic Geology, 1923, vol. 18, p. 431). 6 In this connection may be noted again a passage from Blackwelder's article, p. 289 (see footnote 3): "As phosphorus ascends in the evolutionary scale of animals, its concentration tends to increase, although irregularly. The protozoan, air dried, contains less than 0.6 per cent PaOc. According to Juday quantities of minute crustaceans from Lake Mendota contain in the air-dried condition 1.8 to 2.4 per cent of PaGc, or several times that of the protozoans. A Russian biochemist, Sempelovski, found in entire fresh specimens of a cartilaginous fish (the common skate) 0.91 per cent PaOs, whereas the average for eight Teleostean fishes with well-developed bones was about 1.5 per cent. Certain brachiopods, such as those of the family Lingulidae form shells of fibro-crystalline tricalcium phosphate probably either the mineral dahllite or staff elite." 250 ELEMENTS OF PHYSICAL BIOLOGY phosphorus here splits into two streams. On the one hand the hard parts of dead fish and other sea animals fall to the bottom and form a phosphatic deposit. So, for example, it is reported that in certain localities a single draft of the dredge has brought up 1500 shark's teeth from the sea bottom. These deposits become further enriched through the replacement of their calcium carbonate by calcium phosphate under the action of the sea water. 6 Subsequently some of the deposits so formed have been raised, in a crust upheaval, above the sea level, so as to form sedimentary strata from which we now derive some of our supplies of phosphate rock. The other division of the stream in the flow of phosphorus is per- haps one of the most remarkable examples of a cycle in the economy of nature. The fish of the sea are eaten by birds, who flock in great hordes and have their nesting places upon rocky islands and shores. 7 There an accumulation of immense amounts of guano has taken place in the course of centuries and ages. Of this guano some has been returned directly to the land by the agency of man. Other portions, of more ancient origin,, have undergone transformation, and have passed into fossil form by reaction with the rock base on which they were in the first instance deposited. This is the origin of a second class of (metamorphic) phosphate rock, which also wo mine and spread on our field, so that this loop in the chtiin also is closed. Soil Losses of Phosphorus. From this sketch of the migration of phosphorus in nature it is seen that qualitatively the path of tho element is a closed circulation. Unfortunately the cycle is quanti- tatively quite incomplete. Van Hise, quoting Whitson's invest! ga- tions, shows that, on a very conservative estimate, tho soil of the United States loses annually some 2 million tons of P 2 G , the equiva- lent of 6 million tons of phosphate rock. This is about double the total output of our phosphate quarries, and about four times our domestic consumption of that output. It is thus scon that tho situation with regard to our supply of phosphatic fertilizers is an exceedingly serious one it would perhaps not be out of place to say For a detailed discussion of some of these concentrating processes and related matters the reader is referred to F. W. Clarke, loc. cit., 1921, pp. 132 495, 502. See also Chapter XVIII, footnote 17. ^ 7 For an excellent illustrated account of the part played by the Guanay bird in this cycle see R. C. Murphy, Natl. Gcogr. Mag. Sept. 4, 1924, p. 279. PHOSPHORUS CYCLE 251 ao alarming one. For here we have no reserve which we may hope to find means of tapping in the future, such as is presented to us, in the case of nitrogen, by the inexhaustible supply in the atmosphere. Phosphorus is at best a comparatively rare element, constituting only about 0.14 per cent of the earth's crust (see table 16, Chapter XV). The general and alarming decrease in the crop yield per acre in various states so well described by Mr. James J. Hill, 8 is largely due to the depletion of the soil in phosphorus The work (at the Ohio Agricultural Experiment Station 9 ) upon different fertilizers shows that for the soils tested in their experiments phosphorus was the controlling element in producing an increase in the cereal crops. 10 And elsewhere 11 Van Ilise remarks : The average rock contains twenty times as much potassium as phosphorus. Therefore, looking toward the distant future, if we consider ratios, we may unhesitatingly assert that the problem of maintaining the fertility of the soil in phosphorus will be twenty times as difficult as for potassium; but this ratio by no means measures the real difference, for when a deposit con- tains a moderate percentage of a substance it may be possible to utilize it commercially, whereas, if the percentage falls below this amount it is with- out value. Phospbtatic Slag as Fertilizer. To the consideration of the migration of the element potassium thus referred to by Van Hise, we shall presently turn our attention. Before leaving the subject of the migration of phosphorus, one secondaiy source of phosphate fertilizer remains to be noted here, namely, the by-product (slag) obtained in certain processes of steel manufacture, in which the phosphorus contained in the ore (and very objectionable as a con- stituent in steel) becomes segregated, and is thus eliminated, in the slag. This latter, by suitable processes, is converted into a very serviceable fertilizer. 12 8 The Natural Wealth of the Land, and its Conservation. Paper given at the White House Conservation Conference, May 13, 1908. 9 Ohio State Agr. Coll. Bull. 141, 182. 10 Van Hise, loc. cit., p. 704. 11 Idem, loc. cit., p. 701. 12 For details regarding the use of phosphatic slags as fertilizer the reader may be referred to G. S. Eobertson, Basic Slag and Rock Phosphates, Cam- bridge University Press, 1922. CHAPTER XX CYCLES: CONCLUSION AND SUMMARY The saltness of the sea is due to the numerous springs of water, which in penetrating the earth, find salt mines, and dissolving parts of these carry them away with them to the ocean and to the other seas, from whence they are never lifted by the clouds that produce the rivers. Leonardo da Vinci. The Circulation of Chlorine and the Alkalis. It will be con- venient to consider jointly the migration of the elements chlorine, sodium, and potassium in nature, inasmuch as they are closely connected. There is a piece of laboratory apparatus known as the Soxhlet Extractor, of which the chemist makes use when he wants to pre- pare a solution, an extract, of one of the constituents of a mixture of substances. This apparatus, as represented in figure 53, operates on a simple principle. The solvent (water, ether, petroleum spirit, etc.) is placed in the flask A heated by a bimsen flame B, so as to drive vapors of the solvent up through the tube C to the top of a tubular vessel D containing the material M to be extracted. The whole apparatus is open at the top, but escape of vapor is prevented by a condenser tube E cooled by a water jacket F. The Vapor condensing in E drips down upon the material M, and dissolves out the substance to be extracted. The condensate accumulates in the vessel D and rises in the syphon tube I until it reaches the top of the syphon, whereupon it drains back into the flask A and is reevaporated, this cycle being repeated indefinitely as long as desired. Since the vapor of a liquid containing a non-volatile substance in solution is pure solvent, the action continues until practically all the soluble substance has been extracted. It is almost literally true that we pass our lives in the midst of a gigantic Soxhlet apparatus. The flask is the sea basin; the solvent is the water of the ocean, the rain, and our rivers and lakes. The material extracted is the earth's crust (rocks, soil, etc.). The place of the Bunsen burner is taken by the sun which raises water vapor from the ocean's surface, into clouds which drift over the land. 252 CYCLES: CONCLUSION AND SUMMARY 253 FIG. 53. THE SOXHLET EXTRACTION APPARATUS The action of this apparatus is closely analogous to the natural extrac- tion of soluble constituents from the earth's crust by the water in circu- lation through clouds, rain, rivers and the sea, under the influence of the sun's heat. 254 ELEMENTS OF PHYSICAL BIOLOGY \\ 1 fn| ? * ) . ^ c " V ,3 i K C i Afe /> Ocean _ . /4/w;^/ A& in Rivers" e <" 1.413 xtO'tfans ^ 1.584 * 10 Bfons ~ "~ '"""" &F.W.aafKe J Data of 6 cnemterry laau p /*? fer.se^ Should be consulted for- a and for minor correct/a* the basic figures given CHLOBIDE CYCLE Q ^ A*W j * Q CJ i 1 v *> v || j I 3 1^ M ^ k j> ^ tc J 'i^ 5l C * ^ H ^ *k CN i; w X Q S 5 EH (^ k S ^o"* ft. s. H s k U i i M 1 1 ! I f fc fe ^ V ^ o EH v * uioy ts H a * Q 1 Ci* c^ Hi S" 9 u 4 t H K W 1 B <u O A I j j j ^. O M *, A <a o 3 5 w 6 *w $ .ft (f '1 S <y ^* '^* Q) \ is C5 C \ ^ 1" ^ 1 *** II > <** ^t>M 1 fc < Si ^ <j <o i CYCLES: CONCLUSION AND SXJMMAEY 255 The cold upper atmosphere acts as a condenser (see fig. 54), beyond which no clouds can pass out above. Presently the moisture of the clouds is precipitated as rain over the face of the earth. It drains into rivers and lakes back into the sea, charged now with the soluble constituents of rock and soil. This operation has been going on over and over for ages, with the result that the greater part of the more soluble constituents of the rocks is by this time collected in the ocean, imparting to its water its characteristic salt taste. It is worthy of more than passing note that these relations, in all essentials, were recognized by that universal genius, Leonardo da Vinci, whose remarks on this subject appear at the head of this chapter. The principal soluble constituents of the earth's crust are the carbonates and chlorides of the alkali metals, sodium and potassium. These, then, mainly take part in the extraction process described. There is, however, an important difference in the behavior of sodium and potassium in this process. In the igneous rocks sodium and potassium are present in very nearly equal proportions (see table 16). Yet the ocean is very much richer in sodium than in potassium (see table 14). What becomes of the potassium? 1 It is a circum- stance highly significant for terrestrial life that potassium salts seem to be largely absorbed from their solutions on their passage through soil and clay. Thus the soil would retain a supply of the element so essential for plant growth, while the less vitally important sodium 'in other respects so similar to its next kin potassium has in large part passed on into the ocean. 2 Nevertheless, potassium is not present in most soils in profusion; under intensive agriculture the soil becomes impoverished in this constituent also, and recourse must be had to sources of potassium salts in concentrated form, the deposits left by the drying up of ancient seas, to make up the deficiency. In this field, also, the World War has materially affected the complexion of agricultural economics. Its influence has been twofold. In the first place, since 1 In this connection, and for details regarding the circulation of Na, K,Cl, and the related question of the age of the ocean, the reader may be referred to F. W. Clarke, Data of Geochemistry, pp. 136, 137, 145, et seq. 2 These suppositions must be viewed with a certain caution, as has already been pointed out; see p. 204, footnote 19. 256 ELEMENTS OF PHYSICAL BIOLOGY the only areas highly productive of potassium salts were contained within the domains controlled at that time by Germany, it became necessary for the Allies to find other sources of the needed element. One outcome of this was a temporary development of potash re- covery from the waste of cement works and other materials. Un- fortunately much of the commerce thus started had to be abandoned again when the customary sources of potash became once more available after the close of the war. The second effect of the war upon the potash situation arises from the political changes which it brought about. The Alsatian potash deposits, formerly controlled by the same monopoly as the German Stassfurt deposits, are now in French domains, and the monopoly is broken. 3 Some of the quantitative aspects of the migration of chlorine and sodium are shown in figure 54. The Circulation of Sulphur. Sulphur occurs in abundance in inorganic nature in the form of the sulphates of the alkalies and alkaline earths. Plants assimilate these compounds directly and then form proteids containing about 0.3 to 2 per cent of sul- phur. In this form it is assimilated by animals, who excrete the element chiefly in the form of sulphates. These, returned to the soil, complete the cycle. Tlie Circulation of Iron. The importance of iron in the economy of the living organism is out of proportion to the comparatively small amount of this element actually present in the body. Thus the body of a human adult holds only about 4.5 grams of iron, contained, for the most part, in the red blood corpuscles. The importance of this comparatively small amount of the metal arises out of the fact that it fulfills the essential function of an oxygen carrier, a catalyst as it were, mediating the transfer of oxygen from the air in the lungs to the tissues of the body through the blood stream. Similarly, in plants, the iron is contained chiefly in the chlorophyll, whose catalytic action is a fundamental condition for the assimilation of carbon dioxide from the air. This catalytic action of iron is attributable to the ease with which it passes from ferrous to the ferric condition and vice versa, and plays a significant 3 G. Jones, Q. Jour. Econ., 1920, vol. 34, p. 392, CYCLES: CONCLUSION AND SUMMARY 257 idle not only within the body of the organism, but also In the soil, where it hastens the oxidation of carbonaceous matter, thus render- ing carbon once more available for the organic cycle. 4 Summary of Cycles. In conclusion of this chapter a few remarks and tables regarding the circulation of the elements in general may be offered. In retrospect we may observe that alcharacteristic stamp is placed on certain of the elementary cycles by the form in which TABLE 26 Supply of plant foods in soil Number of years the supply of several elements would last, utilizing soil to a depth of 7 inches, and producing annually a crop of 100 bushels of com. Average compositon of soil assumed equal to that of 2110 samples of common rocks in the United States. (C. G. Hopkins, Annual Acad. Sci. 1909, vol. 33, p. 638). N g Soil 1 | K P Fe Ca Mg 50 700,000 2,600* 130 200,000 55,000 7,000 10 * Assuming stalks returned to the land. each element occurs or takes part in the circulation. Thus, the gases oxygen and carbon dioxide occur in nearly uniform distribu- tion, so that their migration is free from certain complications that arise in the case of the other elements. Water occupies a position intermediate between the gaseous 'elementsjand^those which like phosphorus, potash, etc., as solids, are subject to local segregation, and thus introduce problems of transportation in one form or another. As vapor, water drifts with air currents. But owing tn the phenom- enon of precipitation, water, unlike the permanent gases, is very unevenly distributed, the supply available for life processes being strictly a matter of climatic conditions. Thus, in desert regions water functions as the limiting factor of life. Nitrogen, although 4 G. Bunge, Physiological and Pathological Chemistry, 1902, p. 21. 258 ELEMENTS OF PHYSICAL BIOLOGY gaseous in the elementary state, is chiefly operative in combined forms, so that its distribution in available form, is also a locally varied phenomenon. All these facts have their influence not only on the primitive flora and fauna as a function of geographic site, but play also an important role in those secondary life phenomena which we com- monly describe as commerce and trade:. Two tables 26, and 27, are, finally appended, the one giving cer- tain data of interest regarding the Supply of Plant Foods in the TABLE 27 Rate of participation of the elements in cycles of nature METBIC TONS CIRCU- LATION BY WEIGHT ONE PART IN AUTHORITY* STANDARD OP HEFBBENCE Annual circulation Total in Standard of Reference A. 145 145 144 G. L. M. 434 H 2 O 1.4X10 14 4.4X10 10 3.5X10 10 1.4X10 9 1.3X10 18 2.2X10 13 1.2X10 15 3.9X10* 5 10,000 50 35,000 3,000,000 22, 59 4 7 7 7 Ocean Atmosphere CO. N * Authorities : A. = Arrhenius, Worlds in the Making, 1908 ; C. = Clarke, Data of Geochemistry, 1921; L = Linck, Kreislaufvorgange, 1912; M. = McGee, Science, 1911, vol. 134. The numbers in the column "Authority" refer to pages of the works cited. Soil, according to C. G. Hopkins; the other giving estimates of the Rate of Participation of the Elements in the Cycle of Nature. It seems hardly necessary to point out that the quantitative estimates cited in these chapters on the circulation of the elements in nature represent only very rough approximations, the best per- haps that can be attained in the present state of our knowledge. As F. W. Clarke 3 remarks, "Such estimates may have slight numeri- cal value, but they serve to show how vast and how important the processes under consideration are." Rough as the data are, they give us, presumably, at least an idea of the order of magnitudes involved. The least that can be claimed for them is, in the words of Clarke 5 once more: "In calculations of this sort there is a certain fascination, but their chief merit seems to lie in their suggestiveness." 6 F. W. Clarke, Data of Geochemistry, 1920, p. 48. CHAPTER XXI MOVING EQUILIBRIA Aus dieser Untersuchung wird kein Dualismua hervorgehen, sondern eine Wissenschaft, welche Organisches und Anorganisches umfasst, und die den beiden Gebieten gemeinsamen Thatsachen darstellt. E. Mack. In preceding pages we have considered as examples of biological "equilibria," states that quite obviously can be regarded only as rough approximations to equilibria or steady states; and we have not, so far, examined critically the justification for this attitude. It is desirable to give at least brief consideration to this matter. It is common custom, in dealing with the relatively simple systems studied in physical chemistry, to assume that a sufficiently slow change in one parameter (e.g., volume) defining the state of the system, brings in its train a succession of states each of which is essentially equi- librium. So, for example, if we slowly raise the piston in a gas-tight cylinder containing X grams of water and Y grams of water vapor at a temperature 6, it is commonly assumed that at every instant the quantities X, Y are such as correspond to equilibrium at the tempera- ture 8. The Principle of Continuity. The basis of the assumption referred to in the preceding paragraph is rarely if ever discussed. Obviously it is to be sought in the principle of continuity. If the parameter P is constant, at the value P , the variables X, Y, . . . defining the state of the system have certain values X , YQ, . . . We tacitly assume that if the parameter PO is nearly constant at the value P (that is to say, passes through P in very slow change) then the vari- ables X, Y will have nearly the value of Xo, Y , . . Or, in the notation of an earlier section, if ^-i^tt:,,*,, . . .P) (i) at and if, with P = P = constant ^ = p z = . . = p i = . . . = o (2) 260 ELEMENTS OF PHYSICAL BIOLOGY gives Xi = (?i = constant (3) then we assume that with P = P() (4) where P(t) is a slowly changing function of t, we shall have Xi = d(t) (5) where Ci(t") is a root of the system of equations Fi(i) = F z (ti = . . . = Fi(t) . . . = (6) (for all values of It should be noted that, strictly speaking, this involves a contra- diction. For if the velocities F are zero, the variable X cannot be changing. And, in point of fact, the result (5) represents a first approximation which is not in all cases free from significant error. Higher Approximation. It is possible, in certain cases, to proceed to second, third and higher approximations by successive steps, or, as will be shown, by a single formula. So, for example, for a system in two variables X, Y, we may write, first of all d -j = F^X, Y, i) (7) ^ = F(X, Y, t) (8) 'The first approximation here is F! = F z = (9) X = X 1 (t) (10) Y = Fi(<) (11) The second approximation we obtain by differentiating (10), (11), so as to obtain the derivatives Xi, YI, which, although not zero, are nevertheless small, according to our supposition of a slow change. Substituting these in (7), (8) we find Fi - Xi'(t) (12) F, = F,'(0 , (13) MOVING BQUILIBEIA 261 Hence X = Xt(t) (14) F = F 2 (*) (15) And so on, for successive higher approximations. But this process can be contracted into a compact expression. We have ^J = ?f 1 , ^Oj d JE *>1}*Y dt ~ di" bX dt bY dt ' ^! = 5f> _)_ ^J^I L ^. 2 ^I d d* d,Y di + dF d< ( If we substitute in (16) (17) f-X/> (18) dF Tt = YS0 (19) the right hand member must vanish, in view of (9), (10), (11) and we have +*'+*- from^which it is seen that X 2 (t), F 2 (0 can be expressed directly as the solution of the system of equations Similarly the (n + i) th approximation is found by writing _' = _ = o (24) Special Case. Returning to the general case of n variables, sup- pose that for all values of X l X z . . . differing appreciably from 262 ELEMENTS OF PHYSICAL BIOLOGY the equilibrium values, the velocities ] - ... are negligible (Z6 QJ(I dXr as compared with some one of them, say -j-. Then we can write, at practically X t , . . . X n ) = (25) F n (Xi, Xz, . , . X r , . . . X a ) where ^ r is excluded from the system (25) . This defines X 2 = C z (X r ) (26) etc. In addition to this we then have the equation TTLT * = F t (Ci, C*, . . . X t , . . . C n ) (27) at = F r (X r ) (28) Hence which is directly integrable ? ,v x (30) In such case as this, then, that particular change which is much slower than all the others, sets the pace and controls the whole process. It acts as a brake, as a limiting factor. RADIOACTIVE EQUILIBRIUM Moving equilibria play an important role in evolutionary processes of the most varied type, as emphasized almost ad nauseam by Herbert Spencer, 1 though some of the most typical and at the same time most fundamental examples were unknown to him. For, the most exact, quantitatively precise illustrations of moving equilibria are to be seen in the evolution of chemical elements by successive steps 1 First Principles, Chapter XXII. MOVING EQUILIBRIA 263 ggsiggaga^^ FIG. 55. URANIUM AND ITS PRODUCTS OF RADIOACTIVE DISINTEGRATION (U = uranium; To = ionium; Ra = radium; Rn = radon == radium emana- tion; Po = polonium; Pb = lead.) Each element in the chain is produced from its predecessor either by the emission of an alpha particle, i.e., a doubly charged helium atom, in which case the atomic number is decreased by two units; or by the emission of a beta particle, i.e., an electron, in which case the atomic number increases by one unit. 264 ELEMENTS OF PHYSICAL BIOLOGY of atomic disintegration, accompanied, in most cases known to us, by radioactive manifestations. It is not within the plan or compass of this work, to give a detailed account of what has by this time grown into an extensive special field of physical science. It must suffice to refer to the chart (fig. 55) of one of the typical series of radioactive transformation chains, and to state briefly the simple law of transfor- mation of such elements by spontaneous atomic disintegration : The amount of a substance transformed per unit of time is directly pro- portional to the amount of that substance present, so that if Si is the i th substance in a transformation chain and if we denote by Xt the mass of Si, then we have a system of equation "V T7" "\ T7" /fjrO -Ai-^-x-AA (32) where the coefficients X are constants, invariable under all condi- tions to which observation to the present date has extended. It will be seen that the system of equations (32) is a simple special case of the general form discussed in Chapter VI. Its solution 2 is of the form there indicated, but the simplicity of the differential equations is reflected in the integrals, which here appear as finite series, the expression for the mass of z th substance, being Xt = a,-,ie~ Xlt + ai,2e~ X4t 4- . . . + ai,t<T"V (33) The series contains only linear terms, and breaks off at the term in a#. If X A is (numerically) the least of the X's, then, evidently, after a sufficient lapse of time the term in X& outweighs all other terms, which thus become negligible, so that Xi ctik Y k = (34) The coefficients a are easily determined 3 as functions of the X's. It is thus found that ^ " (Am - X*) (X*+ 2 - X,) . . . Ow_! - X*) (X, - A*) if * > * (35) am - if i < k (36) 2 For a somewhat remarkable method of integration (by a multiple integral), see A. Debierne, Les ide"es modernes de la Matiere, 1913, p. 328. 8 See, for example, E. Rutherford, Radioactive Substances, 1913, pp. 422, 423. MOVING EQUILIBEIA 265 Thus, after a sufficient lapse of time, the substances Si, S 2 , . . . Sk are always present together in constant proportion, so that we have a moving equilibrium of a veiy simple type, illustrated graphi- cally in figure 56, in accordance with some of the constants given in tables 28 and 29. The most slowly decaying substance here acts as TABLE 28 Radioactive equilibrium of radium in contact with its disintegration products on the basis oj data in Jour. Am. Chem. Soc., 1923, vol. 45, pp. 872-873 AMOUNT IN EQUILIBRIUM l/X = MEAN LIFE Radium 1.00 ton 2,440 years* Radon 6 . 23 grams 5.55 days Radium A 3 37 mgm. 4 32 minutes Radium B 30.18 mgm. 38 7 minutes Radium C 21.91 rngm. 28 . 1 minutes Radium D 9.75 kgm. 23.8 years Radium E 8.08 grains 7 2 days Radium F| 220.01 grams 196 days *Half-decay period = = 190 years. fPolonium. the controlling, slowly variable parameter, and sets the pace with which all the subsequent members in the transformation chain keep step, so that the polygons representing the system in its successive stages are geometrically similar. (Compare fig. 58 on p. 277.) The second of the two expressions for the ratio a ifi: /a kk calls for brief comment. If i < /c, that is to say if the substance Si precedes, in the transformation chain, the substance Sk which has its lowest disintegration rate, then Si does not appear at all in the equilibrium. Hence when an aggregation of substances in radioactive equilibrium is found in nature, the substance at the head of the chain (the "par- ent substance") is always the one of slowest disintegration rate. But, obviously we cannot from this draw any conclusion as to whether or not it is itself a product of disintegration of a pre-parent of more rapid O i-% A decay .ate. Aay such pre-parei* OP PHYSICAL BIOLOGY as it were, have been CONTACT WITH ITS DISIN- M davs (half -decay period) torn a geometric series Tte different equations transformation "dily to deterge the . H. Mitchell, PhU. Mag., 23, p. 353. . , ^rairw of radioactive f^ ation> a nd there is rf OTCCess i v e approxima- . However, the first , P- ' A ' MOVING EQUILIBBIA 267 approximation is so simple, that it is commonly applied. Equating the right hand member of (32) to zero we find immediately -j = *L I (37) and by an obvious extension - ~* (38) Afc Xi It is easily shown that L i} the reciprocal of X;, is the "mean length of life" of an atom of the substance $,> We may write (38) ~ - ~ (39) which expresses the fact that, in first approximation, the amounts of several substances present together in radioactive equilibrium arc in the ratio of the respective mean lengths of life. 5 This result can also be read out of (35) if \\ h \, the least of the |X|'s, is negligible in comparison with all the other |X|'s, so that the denominator reduces to the product It is thus seen that the closeness of the first" approximation depends on the relative magnitude of X& and the remaining X's. In many chains of radioactive transformation the parent substance is very slow in its disintegration, and the first approximation (giving what Rutherford has termed the secular equilibrium) is exact within the limits of experimental error. But if one of the more rapidly decaying members is isolated and is then allowed to come into equilibrium with 5 This is a special case of a general law that if all the exponents \ arc real and negative, the final stages of the process of evolution are characterized by constancy in the ratios of the Variables .T. Compare p. 261, Special <7a.svj; also, Lotka, A. J., Proc. Am. Acad. Sci., vol. 55, 1920. Ifc should be noted, however, that in the general case, xt denotes not mass of Si, but CXCGHB of that mass over the equilibrium mass of Si. In the radioactive equilibrium there is no distinction between X and x, since the ultimate value of both is zero. For the second and higher approximation, applied to the radioactive equilibrium, the reader may be referred to A. J. Lotka, Proceedings Natl. Acad. Sci., 1921, vol. 7, p. 170. 268 ELEMENTS OF PHYSICAL BIOLOGY its own products of disintegration, the error of the first approxi- mation may become appreciable. 7 This is shown, for example in table 29, which exhibits the amounts of radon gas (radium emanation) and its several products of disintegration in radioactive equilibrium. It will be observed that in the case of radium C there is a discrepancy of about 1 per cent between the amount computed by first approxi- mation and the true amount. Radioactive Chains as Cosmic Clocks. It must be noted that all that has been said above regarding the amounts of the substances present in radioactive equilibrium does not apply to the last link in the chain, the end product. This does not, of course, take part in the equilibrium, but accumulates, if non-volatile, as in the case of lead, or, it may in part escape and be lost to observation, as in the case of helium. TABLE 29 Radioactive equilibrium of radon (radium emanation) in contact with its dis- integration -products, on basis of data in Jour. Am. Chem. Soc., 1923, vol. 45, pp. 872-873 X (PER SECOND) L = 1A EQUILIBRIUM AMOUNT Exact First approximate Radon. 2.085 X 10- 6 3.85 X 10 ~ 3 4.30 X 10-" 5.92 X 10-* 5.55 days* 4.32 minutes 38.7 minutes 28 . 1 minutes l.OOOkgm. 0.5416 grams 4.874 grams 3 . 552 grams l.OOOkgm. 0.5418 gram 4.849 grams 3 . 522 grams Radium A Radium B Radium C *Half~decay period = 0.69315 L = 3.85 days. If the amounts of parent substance and end product are large as compared with the amount of intermediates, the amount of any one end product formed is evidently simply proportional to the amount of parent substance lost by disintegration in a given time. The chain of substances in transformation behaves, in fact, much like a sandglass clock having a number of bulbs and from the accu- mulation in the end bulb we can obtain an indication of the age of the system, on the assumption that originally all was in the top bulb, that initially only the parent substance was present. The applica- tion of this principle to radioactive mineral deposits has given us a quantitative time scale in historical geology where before we had to 7 Compare E. Rutherford, Radioactive Substances, 1913, p. 430. MOVING EQUILIBRIA 269 rest satisfied with a crude chronology recognizing only order of pre- cedence, or at best dealing in exceedingly uncertain jjestimates of lapse of time. Thus the investigation of radioactivity, remote as it seems from the field of life phenomena, has nevertheless contributed to biology essential information regarding the time that has been avail- able for the evolution of the earth and its inhabitants. The estimate reached upon this basis is that the age of the radium-bearing rocks (uranium ore) examined is at least eight million years, and at most seventeen hundred million years old. For a resume" of various esti- mates of the age of the earth the reader may be referred to G. TABLE 30 Geologic time table After Sclmchert FROM BAItHBLI, 1917 ON tniANIUM BASIS Cenozoic per cent 5 per cent 4 Mesozoic 12 11 Paleozoic 28 30 Late Proterozoic 15 ) Early Proterozoic 15 J- 55 Archeozoic 25 Schuehert, The Evolution of the Earth and its Inhabitants, 1919' pp. 56 et seq.; 80; and to the Proceedings of the American Philo- sophical Society, 1922, vol. 61, pp. 247-288. See also E. Ruther- ford, Radioactive Substances, 1913. Schuchert's estimate is that "geologic time endured about eight hundred million years," dis- tributed among the several geological eras as indicated in table 30. The Origin of the Elements and the Ultimate Genesis of the Organism. The case of radioactive equilibrium has here been in- troduced primarily by the way of illustration, as probably the most typical example in nature of a moving equilibrium in a system in the course of evolution. But the matter is also of more material interest to us in our survey of the evolution of the earth as the abode of life. For, as has already been emphasized, we are not only on the earth but of it; we have thus a two-fold interest in the evolution of its substance first, as providing the stage upon which our life drama is set; and second, as furnishing the material of our bodies. Of these same ele- 270 ELEMENTS OF PHYSICAL BIOLOGY ments that make up the earth's crust we also are composed: their genesis is therefore also the first, remote chapter in the genesis of our own bodies. Through the discovery of radioactive chains of ele- ments we hold a clue regarding the fundamental influences that have determined the quantitative chemical composition of our world, and have thus appointed the measure of the supplies available for our needs. Those elements whose genesis is known to us came into being in perfectly definite proportions. Presumably the same is true also of those whose precise mode origin is still unknown. There is good evidence to support this view. We have at present no detailed quantitative knowledge of the laws which determine the value of the decay coefficients X of the radioactive elements, and which thus ulti- mately fix their relative abundance in equilibrium. But a significant qualitative relation has been pointed out by W. D. Harkins. 8 When the elements in a radioactive chain are arranged in order of their atomic numbers, 9 and are separated into two groups, those of odd and those of even number, it is found that each even-numbered ele- ment is more abundant than the adjacent odd-numbered elements. And, what is of particular significance, the law of relative abundance of odd and even-numbered elements extends also to those elements, regarding whose precise mode of origin we have not, as yet, that sure knowledge which is gained by direct observation within the four walls of the physical laboratory. (See fig. 57.) 8 Jour. Am. Chem. Soc., 1916, vol. 38, pp. 863, 869; 1923, vol. 45, pp. 1420- 1433. Compare also F. W. Aston, Nature, March 15, 1924, p. 394. For other regularities observed in the length of life of radioactive elements see K. Fajans, Radioaktivitat (Sainmlung Vieweg Heft 45) 1921. 9 The chemical elements, arranged in ascending order of atomic weights, beginning at hydrogen = 1, may be given ordinal numbers 1, 2, 3, etc., in- dicating their position in the series. These ordinal numbers have been found to have important relation to the atomic architecture. They have been termed the atomic numbers. The definition here given is not quite exact; in certain places allowance must be made, gaps left for unknown elements, and the several isotopes of one element receive the same atomic number, though differing in their atomic weights. A more precise definition is the following: The atomic number of an element represents the excess of posi- tive over negative charges in the constitution of the atomic nucleus. Each atomic number also represents the place occupied by the element in Men- deleef's table (Jour. Am. Chem. Soc., 1923, vol. 45, p. 868). For further in- formation the reader must be referred to the special literature; of compre- hensive works the following may be mentioned: Bragg, X-rays and Crys- tals; F. W. Aston, Isotopes. FIG. 57. RELATIVE ABUNDANCE OF THE ELEMENTS Each element of even atomic number is more plentiful than the adjacent elements of odd stomic numbers. Diagram according to W. D. Harking, based on analysis of meteorites. (Jour. Am. Chem. Soc., 1916, p. 863.) 271 272 ELEMENTS OF PHYSICAL BIOLOGY But the laboratory is not a prison, and the eye of the physicist is free to sweep the sky, where nature's great smelteries gleam at night. With the aid of the spectroscope he has studied the multitudes of the stars, and has recognized in them a number of distinct stages of evo- lution. Life's day is far too short to give the observer any oppor- tunity to study directly the evolutionary changes in any one star. But by piecing together the observations made upon the mixed popu- lation of stars of different ages, it has been possible to construct with considerable certainty the main stages in stellar evolution, just as the stages of human life could be gathered from a single observation of a mixed population comprising persons of all ages. The evidence points clearly that the elements, such as we know them, are the product of "the general brewing of material which occurs under the intense heat in the interior of the stars." Out of such foundry came our own abode, if we accept the well-considered views of Eddingtor.. '" "I do not say that the earth was a gaseous body when it first became recognizable as an independent planet, but I am convinced that its material was at one time merged in a completely gaseous sun/' And since we are of earth, ours also is the same origin. The hand that writes these words and the eye that reads them alike are composed of the selfsame atoms that came into being, ages and ages ago, in the young sun. Far, far more wonderful than any dream of old 10 A. S. Eddington, The Borderland of Astronomy and Geology, Nature, 1923, p. 18, also, The Interior of a Star, Supplt. to Nature, May 12, 1923. The T- der who wishes to acquaint himself in greater detail on this subject may refer to Eddingtons's -work Stellar Evolution. In the interest of unbiassed presentation it must be noted here that T. C. Chamberlin (The Origin of the Earth, 1916) has put forward a theory of the origin of the earth and the planets which is at variance with that sustained by Eddington. On the other hand it is also proper to mention a fundamental objection to theories of cosmogony of the type of that of Chamberlin and Moulton, which is based on the supposition that the luminous stars are formed by the collision of dead suns. "The distances separating the stars are enormous compared with their own dimensions. Sir Frank Dyson once used the illustration of twenty tennis balls distributed at random throughout the whole interior of the earth, to give a model of the density of distribution of the stars Taking a very liberal view of the kind of approach that can be held to con- stitute a collision, it is estimated that a star would suffer a collision about once in a hundred million million years" (Eddington). For a survey of the modern views on this subject see J. Barrell, The Evolution of the Earth (Yale University Press, 1919). MOVING EQUILIBBIA 273 mythology is the story of our creation. Thus was the birth of man prepared in the grey dawn of time ; thus the metal of his frame com- pounded in the flaming furnace of a star. .... TERMINAL STAGES OF THE EARTH'S EVOLUTION Geophysics and Geochemistry. For the last stages in the evolu- tion of the elements and their chemical combinations we do not look to the stars. We can study them at close quarters in the field and in the laboratory. In this way, with the application of physics and chemistry to general problems of geology, have grown up the sciences of geophysics and geochemisty. Indeed, it naturally might be supposed that on the terrestrial phases of inorganic evolution we should be altogether better informed than on those prior stages, far remote in time and space, which run their course in distant suns. But this is true, at best, only in restricted measure. It is a singular circumstance that, in some ways, we are better informed regarding the physics and chemistry of the stars, of which the nearest, outside the solar system, is twenty-five million million miles distant, than regard- ing that of our own planet. To say that the earth's surface layers accessible to our direct observation are comparable, scale for scale, to the shell of an egg, is to err on the side of liberality. The deepest burrow into the earth made by human agency, the mine shaft at Morro Velho, Brazil, 11 is 1-|- miles (6400 feet) deep or only about frsT of the earth's diameter. Direct observation can therefore give us at best only the most uncertain information regarding thecon- ditions at even moderate depths. Where the crust has been creased and thrown into folds, subsequent denundation may have exposed layers of some 50 or 60 miles aggregated thickness. 12 But, though this gives us an invaluable record of some of the most significant chapters in the earth's history, it adds little or nothing to our knowl- edge of conditions of temperature and pressure even at comparatively trivial depths, and regarding chemical composition also it gives us, after all mere surface indications. Indeed, our information on all these points is very largely of a negative character. We know from the earth's average density that the composition of the core must be very different from that of the shell. We know that the temperature u Sir Charles Parsons, Nature, February 19, 1920, p. 677. 12 G. Schuchert, The Evolution of the Earth, Yale University Press, 1919, p. 67. 274 ELEMENTS OF PHYSICAL BIOLOGY gradient observed in boreholes near the surface averages an increase of about 1F. for every 60 feet (1C. for every 35 meters) descent, but of the further course of temperature at greater depths we know with reasonable certainty only that it cannot continue at this gradient, which would give a phantastic temperature of 300,000F. (1 80,000C.) at the center. Lord Kelvin's calculations, based on the rate of cool- ing of the earth, and the more recent figures of the same character given by Van Orstrand, are rendered uncertain in their application owing to the presence of radium, in unknown amounts, evolving heat in its distiiitegration. Strutt has made the tentative estimate that the temperature rises uniformly to a depth of about 30 miles, and after that remains sensibly constant at 2700F. A little more definite is our information regarding the pressure in the earth's interior. A first approximation of its value is found by considering the earth as a fluid. It is thus found 13 that the pressure at the center would be three million atmospheres. In any case there can be no doubt whatever that the pressures reached vastly exceed anything at our command in the laboratory, where a pressure of twenty-four thousand atmospheres, employed by P. W. Bridgman in his researches, stands out as a record achievement, though it coiv responds to a depth of rock of only 56 miles. 14 It must be clear from what has been said above, that all conjectures as to the physical, chemical and subatomic transformations going on in the earth's interior are subject to a very large margin of uncertainty. In this connection it is well to recall the words of F. W. Clarke: 15 The chemistry of great pressures and concurrently high temperatures is entirely unknown, and its problems are not likely to be unravelled by any experiments within the range of our resources. The temperatures we can command, but the pressures are beyond our reach We may devise mathematical formulae to fit determinable conditions; but the moment we seek to apply them to the phenomena displayed at great depths, we are forced to employ the dangerous method of extrapolation, and our conclu- sions are not verified. 13 This supposition, in calculating pressures, probably does not err far from the truth. The researches of P. D. Adams (Journal of Geology, Feb- ruary, 1912), P. W. Bridgeman and others have shown that at depths of some 30 miles rocks probably give way like butter to the pressure of the layers above them. " W. D. Lambert, Jour. Washington Acad. Soi., 1920, p. 126. Sir Charles Parsons, loc. cit. 15 Data of Geochemistry, p. 271. MOVING EQUILIBRIA 275 In these circumstances we can feel but little confidence in inferences