"The values of the function of _x, ce__kx__, increase according to the terms of a geometrical progression as the variable x increases in arithmetical progression_....

"The most immediate application of a function in which the growth is proportional to the function itself is to the air. The decrease in the pressure of the air at the distance _h_ above the earth's surface is proportional to _h_.

"The expression _P = 760 e__-h/7990_ gives the numerical value of the pressure in millimeters of mercury for _h_ measured in meters. The negative exponent indicates that the pressure decreases as _h_ increases.

In inches as units of length of the mercury column, _h_ in feet,

_P = 29.92e__-h/26200_

This is known as Halley's law.

"The growth of bean plants within limited intervals and the growth of children, again between quite restricted limits, follow approximately the law of organic growth. Radium in decomposing follows the same law; the rate of decrease at any instant being proportional to the quant.i.ty. In the case of vibrating bodies, like a pendulum, the rate of decrease of the amplitude follows this law; similarly in the case of a noise dying down and in certain electrical phenomena, the rate of decrease is proportional at any instant to the value of the function at the instant....

"_The Curve of Healing of a Wound._-Closely allied to the formulas expressing the law of organic growth, _y = e__kt_, and the law of 'organic decay,' _y = e__-kt_, is a recently discovered law which connects algebraically by an equation and graphically by a curve, the surface-area of a wound, with time expressed in days, measured from the time when the wound is aseptic or sterile. When this aseptic condition is reached, by washing and flushing continually with antiseptic solutions, two observations at an interval commonly of four days give the 'index of the individual,' and this index, and the two measurements of area of the wound-surface, enable the physician-scientist to determine the normal progress of the wound-surface, the expected decrease in area, for this wound-surface of this individual. The area of the wound is traced carefully on transparent paper, and then computed by using a mathematical machine, called a planimeter, which measures areas.

"The areas of the wound are plotted as ordinates with the respective times of observation measured in days as abscissas. After each observation and computation of area the point so obtained is plotted to the same axes as the graph which gives the ideal or prophetic curve of healing.

"When the observed area is found markedly greater than that determined by the ideal curve, the indication is that there is still infection in the wound.... A rather surprising and unexplained situation occurs frequently when the wound-surface heals more rapidly than the ideal curve would indicate; in this event secondary ulcers develop which bring the curve back to normal....

"This application of mathematics to medicine is largely due to Dr. Alexis Carrel of the Rockefeller Inst.i.tute of Medical Research. He noted that the larger the wound-surface, the more rapidly it healed, and that the rate of healing seemed to be proportional to the area. This proportionality constant is not the same for all values of the surface or we would have an equation of the form,

_S = S__1__e__-kt_

in which _S_, is the area at the time that the wound is rendered sterile and observations to be plotted really begin....

"The data given are taken from the Journal of Experimental Medicine, reprints kindly furnished by Major George A. Stewart of the Rockefeller Inst.i.tute. The diagrams are reproduced from the issue of Feb. 1, 1918, pp.

171 and 172, article by Dr. T. Tuffier and R. Desmarres, Auxiliary Hospital 75, Paris....

[ A graph, showing the progress of healing of a surface wound of the right leg of a 31-year-old patient. It shows that as time pa.s.ses, the wounded area decreases. ]

Progress of healing of a surface wound of the right leg, patient's age 31 years.

"WAVE MOTION. General.-In nature there are two types of recurrent motion, somewhat closely connected mathematically, in which repet.i.tion of motion occurs at regular intervals.

"One type of this motion, in cycles as we may say, repeats the motion in one place, and is in a sense stationary. The tuning fork in motion moves through the same s.p.a.ce again and again; a similar movement is the motion of a vibrating string. Of this stationary type may be mentioned the heartbeats, the pulse, the respiration, the tides, and the rotation of a wheel about its axis.

"The second type of recurrent motion transmits or carries the vibratory impulse over an extent of s.p.a.ce as well as time. The waves of the sea are of this character. Sound waves, electrical vibrations or waves, and radiant energy vibrations are transmitted by a process similar to that by which the waves of the sea are carried.

"Both of these types of motion are representable mathematically by equations involving a sequence of trigonometric functions. To the fundamental and basic function involved, _y_ = sin _x_, we will direct our attention in the next section and to simple applications in other sections of this chapter....

"Sound Waves.-If a tuning fork for note lower C is set to vibrating, the free bar makes 129 complete, back-and-forth, vibrations in one second. By attaching a fine point to the end of the bar and moving under this bar at a uniform rate, as it vibrates, a smoke-blackened paper, a sinusoidal curve is traced on the paper. Our curve is traced by a bar vibrating 50 times in 1 second.

[ A graph, showing a wavy line. "Tuning fork vibrations recorded on smoked paper." ]

Tuning fork vibrations recorded on smoked paper.

"Corresponding to each movement of the vibrating rod there is a movement of the air. As the bar moves to the right it compresses the layer of air to its right and that _compression_ is immediately communicated to the layer of air to the right; as the bar moves back and to the left, the pressure on the adjacent air is released and a _rarefaction_ takes place.

In 1/50 of 1 second you have the air adjacent to the rod _compressed_, back to normal, and _rarefied_; during this time the neighboring air is affected and the compression is communicated a distance which is the _wave length_ of this given sound wave. In 1 second this disturbance is transmitted 1100 feet at 44 Fahrenheit. The wave length for this sound wave then is 1100/50 = 22 feet.

"The wave length is commonly designated by ?. If _V_ is the velocity, and _t_ the time of one vibration, ? = _Vt_."

[ A graph, showing five wavy lines, labeled "a", "ou", "r", "E", and "a".

"Vibration records produced by the voice: 'a' as in 'ate'; 'ou' as in 'about'; 'r' in 'relay'; 'e' in 'be'; and 'a' in 'father'. The tuning fork record, frequency 50 per second, gives the vibration frequencies." ]

"Vibration records produced by the voice: 'a' as in 'ate'; 'ou' as in 'about'; 'r' in 'relay'; 'e' in 'be'; and 'a' in 'father'. The tuning fork record, frequency 50 per second, gives the vibration frequencies."

This last drawing may help to visualize the fact in what manner wrong expressions and untrue teachings hamper the true progress of humanity.

Every word has its energy and produces some physico-chemical effects in the time-binding apparatus in accord with the idea which we a.s.sociate with the sound of the word. If we teach ideas which are untrue, then the physico-chemical effects produced are not proper-in other words the human mind does NOT WORK PROPERLY, that is, it does not work _naturally_ or _normally_ or true to the human dimension. There is every reason why the standards in our civilization are so low, because we have "poisoned," in a literal sense of the word, our minds with the physico-chemical effects of wrong ideas. This correct NATURAL APPROACH to the "Time-binding" energies will make it obvious how unmeasured is the importance of the manner in which we handle this subtle mechanism, as the poisoning with wrong ideas or with careless or incorrect words does not in any way differ in consequences from poisoning with any other stupor-producing or wrongly stimulating poison.

Monographs on Experimental Biology and Physiology

LOEB, J.: "Comparative Physiology of the Brain and Comparative Psychology." New York, 1900.

LOEB, J.: "Studies in General Physiology." Chicago, 1905.

LOEB, J.: "The Dynamics of Living Matter." New York, 1906.

LOEB, J.: "The Mechanistic Conception of Life." Chicago, 1912.

Selection from contents: I. The Mechanistic Conception of Life. II. The Significance of Tropisms for Psychology. III. Some Fundamental Facts and Conceptions concerning the Comparative Physiology of the Central Nervous System. IV. Pattern Adaptation of Fishes and the Mechanism of Vision. V.

On Some Facts and Principles of Physiological Morphology. VI. On the Nature of the Process of Fertilization. VII. On the Nature of Formative Stipulation (Artificial Parthenogenesis). VIII. The Prevention of the Death of the Egg through the Act of Fertilization. IX. The Role of Salts in the Preservation of Life. X. Experimental Study of the Influence of Environment on Animals.

LOEB, J.: The Organism as a Whole. G. P. Putnam's Sons. New York, 1916.

Selection from contents: I. Introductory Remarks. II. The Specific Difference between Living and Dead Matter and the Question of the Origin of Life. III. The Chemical Basis of Genus and Species: 1. The Incompatibility of Species not Closely Related. 2. The Chemical basis of Genus and Species and of Species Specificity. IV. Specificity in Fertilization. V. Artificial Parthenogenesis. VI. Determinism in the Formation of an Organism from an Egg. VII. Regeneration. VIII.

Determination of s.e.x, Secondary s.e.xual Characters and s.e.xual Instincts: 1.

The Cytological Basis of s.e.x Determination. 2. The Physiological Basis of s.e.x Determination. IX. Mendelian Heredity and its Mechanism. X. Animal Instincts and Tropisms. XI. The Influence of Environment. XII. Adaptation to Environment. XIII. Evolution. XIV. Death and Dissolution of the Organism.

LOEB, J.: "Forced Movements, Tropisms, and Animal Conduct." J. B.

Lippincott, Philadelphia, 1918.

Selection from contents: I. Introduction. II. The Symmetry Relations of the Animal Body as the Starting Point for the Theory of Animal Conduct.

III. Forced Movements. IV. Galvanotropism. V. Heliotropism. The Influence of One Source of Light. 1. General Facts. 2. Direct Proof of the Muscle Tension Theory of Heliotropism in Motile Animals. 3. Heliotropism of Unicellular Organisms. 4. Heliotropism of Sessile Animals. VI. An Artificial Heliotropic Machine. VII. Asymmetrical Animals. VIII. Two Sources of Light of Different Intensity. IX. The Validity of the Bunsen-Roscoe Law for the Heliotropic Reactions of Animals and Plants. X.

The Effect of Rapid Changes in Intensity of Light. XI. The Relative Heliotropic Efficiency of Light of Different Wave Lengths. XII. Change in the Sense of Heliotropism. XIII. Geotropism. XIV. Forced Movements Caused by Moving Retina Images: Rheotropism: Anemotropism. XV. Stereotropism.

XVI. Chemotropism. XVII. Thermotropism. XVIII. Instincts. XIX. Memory Images and Tropisms.

A list of _554 books on this subject_, in which any reader interested will find a vast storehouse of exact knowledge in this line. Author.

CONKLIN, EDWIN GRANT: "Heredity and Environment in the Development of Men." Princeton University Press, 1915.

Selection from contents: I. Facts and Factors of Development.

Introduction. A. Phenomena of Development. B. Factors of Development. II.

Cellular Basis of Heredity and Development. A. Introductory. B. The Germ Cells. C. The Mechanism of Heredity. D. The Mechanism of Development. III.

Phenomena of Inheritance. A. Observations on Inheritance. B. Statistical Study of Inheritance. C. Experimental Study of Inheritance. IV. Influence of Environment. A. Relative Importance of Heredity and Environment. B.

Experimental Modifications of Development. C. Functional Activity as a Factor of Development. D. Inheritance or Non-inheritance of Acquired Characters. E. Applications to Human Development: Euthenics. V. Control of Heredity: Eugenics. A. Domesticated Animals and Cultivated Plants. B.

Control of Human Heredity. VI. Genetics and Ethics.