Principles of General Physiology

In such kinds of processes, then, we have to deal, not with simple linear relationships, but with exponential or logarithmic ones. Other aspects of the question may be found in Newton's " Law of Cooling," one of the earliest cases 'to which the infinitesimal calculus was applied. Here the rate of cooling depends on the difference of temperature between the hot body and its surroundings, so that it steadily diminishes as the temperature difference becomes less ; in theory, equality of temperature is attained only after an infinite time, asymptotically, as it is called, after the straight lines to which such a curve as the hyperbola continually approaches without actually reaching ; this is due to the fact that each succeeding portion of the curve moves towards the asymptote a little less than the previous portion did. In such cases as loss of heat, or the rate

of a chemical reaction, one may look upon the driving force as becoming less and less. The velocity of chemical reactions will be dealt with in Chapter X. The increase of money lent at compound interest follows a similar law ; for this reason, the general law in which a function varies at a rate proportional to itself, an exponential function, was called by Kelvin, " the compound interest law." On this point, pp. 56-64 of Mellor's book (1909) will repay perusal.

The name "function" has just been used without explanation and it may be useful here to refer to some terms often met with in descriptions of phenomena from the mathematical standpoint. The volume of a given mass of a particular gas is different, according to the pressure to which it is exposed ; but it is always the same, other conditions being unchanged, when the same pn-ssimis applied. The volume of a gas is said to be a "function" of the pi insure. A function, then, is a quantity which changes according to some definite law when another quantity, of which it is said to be a function, changes. This is expressed in symbols : — r =f (p)t in the case of Boyle's law ; or, generally, y —f (x),

which means that, to every value of .r, there is a determinate value of y. x and y are CM! led " ftriable*." Any quantity which remains unchanged during a particular mathematical operation is called a "constant." When the value of one variable depends on that of the other, as in the example given, the first is called the " dependent mr table," the second, the " independent mriable." Which of the two is chosen as the independent variable is a matter of convenience. In cases involving time as one variable, it is usually taken as the independent variable, since its changes are the most uniform. When the values of y are simple arithmetical multiples or fractions of those of x, so that the graph is a straight line, y is said to be a " linear function" of x. When y varies as a power of x, it is said to be an "exponential Junction," and so on.

Speaking generally, the object of scientific research is to find out how one thing depends on another, in fact, what " function " the one is of the other. To return to our main theme, we find that the work done in compressing a gas isothermally from the volume v.2 to t\ is : — Further, since, by Boyle's law, pressures are inversely as volumes, we have : — and writing ct and c., for osmotic or molar concentrations of any two solutions ,-is being proportional to pl and/).,, we have a formula which gives the work done in concentrating a solution from the value c} to c.,, as in the case of the kidney when secreting urine of an osmotic pressure different from that of the blood, as will be seen later.

Or, again, if ct and c.2 represent the concentration of an ion in two solutions in contact with electrodes of the same substance, we have the electromotive force of the battery, due regard being taken as to the units in which R is expressed. We shall see later how this fact is made use of to determine the real acidity of a solution, and how it is related to the electrical -changes taking place in acti\<- organs. For further details as to this important law, the reader is referred to the work of Nernst (1911, pp. 51 and ~v2), and the essay of Benjamin Moore (1906, pp. 21, etc ).

The practical bearing of the logarithmic form of the equation may be seen in the case of a concentration battery in hydrogen ions, as used for determining the true acidity or alkalinity of a complex fluid like blood, for example. If the relative concentration of the hydrogen ions in the two solutions compared is, in one case, as 2 to 1, and, in another case, as 10 to 1, the electromotive force in the second case will not be five times that in the first, but in the ratio of log 10 to log i', that is, as 1 to 0-301, or about 3'3 times. Thus the actual E.M.F. of a battery, composed of a standard calomel electrode combined with a hydrogen electrode in one-tenth noimal hydrochloric acid, is 0-394 volt, while if one hundredth normal acid is taken, the value is 0-452 volt. It will be noted that the logarithmic form of the equation lessens the delicacy of the method.

This is the most appropriate place to refer to the view taken by some, that the introduction of mathematics into biological questions is mischievous. Huxley's (1902, p. 333) comparison of mathematics to a mill, which only gives out in another form what was put into it, is often quoted. At the same time we must not forget that this new form is much more useful than the original one. Plato remarks, "If arithmetic, mensuration and. weighing be taken away from any art, that which remains will not be much" ("Philebus," Jowett's translation, 1875, vol. iv. p. 104). Stephen Hales devoted himself to quantitative measurements in physiology and denned his point of view thus (1727, p. 2) : " Since we are assured that the all-wise Creator has observed the most exact proportions, of number, tveight, and measure, in the make of all things, the most likely way to get any insight into the nature of those parts of the creation, which come within our observation, must in all reason be to number, weigh and measure. And we have much encouragement to pursue this method of searching into the nature of things, from the great success that has attended any attempts of this kind." The Biblical passage referred to will be found in the beautiful 40th chapter of Isaiah, verse 12: "Who hath measured the waters in the hollow of his hancl, and meted out heaven with the span, and comprehended the dust of the earth in a measure, and weighed the mountains in scales, and the hills in a balance?"

If it be admitted that our physiological methods are limited to those of physics and chemistry, further remarks are unnecessary. The value of mathematics in physics is plain, to every one, and its value in chemistry becomes continually more obvious. As Arrhenius (1907, p. 7) points out, the expression of experimental results in a formula shows their relation to known laws in a way which is otherwise very difficult or impossible to attain. One is enabled to see whether all the factors have been taken into account and even an empirical formula may assist in deciding whether irregularities are due merely to experimental error or to some unsuspected real phenomenon in the process.

For example, the action of trypsin on a protein might be expected to follow the course of a unimolecular reaction (see Chapter X. ). Actually we find that the velocity constant calculated by the appropriate formula shows a continual diminution as the reaction proceeds. This fact leads us to look for the cause. In experiments on the influence of alkali we find that the activity of trj'psin is, within limits, in proportion to the degree of alkalinity of the digest. We naturally look for diminution of alkalinity in the course of tvypsin digestion and find that the production of amino-acids, especially the strongly acid di-carboxylic ones, is capable of producing a considerable change in the direction in question.

Possibly it may seem hard to add an extra burden to the already large equipment necessary for the physiological investigator. The reader will, no doubt, have been struck by the wide range of natural knowledge which has to be taken into account. At one moment we may be concerned with the movements of protoplasm in a vegetable cell, or the composition of the primeval ocean, and at the next, the work done in compressing a gas, the chemical properties of amino-acids, or the constitution of dyes.

In connection with the wide range of knowledge implied in the various problems with which physiology is concerned, it is interesting to remember that oxygen was discovered by a physiologist, Mayow, as we shall see in Chapter XXI., and many facts belonging to other sciences have also been brought to light in physiological investigations. " On the other side, we may note that the function of the heart was practically discovered by an artist, Leonardo ; the arterial pressure by a clergyman, Hales; the capillary circulation by a "bedell," Leeuwenhoek ; intravenous injection by an architect, Wren ; the nature of animal heat by a chemist, Lavoisier ; the function of the green plant by a clergyman, Priestly ; and so on.

A moderate amount of mathematics will probably have to suffice for most of us, enough to be able to understand and use the fundamental equations. But, since, as often insisted on already, vital phenomena are essentially changes, it will be obvious that the infinitesimal calculus, which deals with changing quantities, must be included, at least in its elements. It might indeed with advantage be allowed to take the place of much of the geometry and trigonometry

taught in our schools, as is well pointed out by Prof. Perry in his "Calculus for Engineers." As a brief introduction, the first chapter of Melloi's "Chemical Statics and Dynamics" may be recommended. The admirable book of Nernst and Schunflies, of which unfortunately no English translation exists, may follow, and then, perhaps, Mellor's " Higher Mathematics for Students of Chemistry and Physics." Experimental results can almost invariably be best expressed graphically, owing to the direct appeal to the eye.

The way in which algebraical foimuhe can be represented by geometrical figures, or vice versa, was discovered by Descartes and published in his famous "Geometric" in 1637. The co-ordinates when referred to two axes at «n angle to one another, are accordingly known as "Cartesian co-ordinates." This system, with the axes at right angles, is that most commonly used in representing experimental results in a graphic form. The fact should also be remembered that Descartes realised the import of his method as the commencement of "the expression by means of algebraical formula} of continuously varying quantities " (Playfair)

(From portrait in possession of the University of Edinburgh. From The Merchistonian, 1912-13.) In other words, the history of the differential calculus may be said to begin with him. His portrait will be found in Fig. 25. If the reader attempts to follow the reasoning given by Descartes himself, he will find it a difficult task. It seems as if the philosopher did not wish that his opponents, of whose mental capacity he had a very small opinion, should understand him too easily. Accessible editions of Descartes' works will be found given in the Bibliography at the end of the present work.

The experimenter, who uses a slide rule or a table of logarithms to diminish his arithmetical labours, should often feel grateful to the inventor of logarithms. This was Napier of Merchiston, whose portrait will be found in Fig. 2ti. Merchiston Tower is seen in Fig. '_'T. For most purposes, the short straight form of slide rule gives sufficient accuracy. If a greater number of significant places is required in the result, the spiral form of Fuller is very convenient in use. It is made by Stanley. The Handbook to the Exhibition at the Tercentenary of Napier, published by the Royal Society of Edinburgh in 1914, will be found useful in connection with the history and use of logarithms, as well as with other aids to calculation.

There is sometimes an unfounded prejudice against smoothed curves, but, if the data show any sort of regularity, the course of the phenomenon is more accurately shown by such a curve, since it eliminates accidental errors. It may be useful to describe the method, slightly mollified from that of Ostwald -Luther (1910, pp. 28-3(1), which I have found the mo-i convenient one for drawing curves for reproduction. The experimental values are first marked by -f at the intersection of the co-ordinates, given appropriate values, on squared paper; a curve is drawn as smoothly as possible by hand, using a pencil, through these

points. The paper is pasted on to a piece of moderately thick cardboard, which is then cut with scissors along the curve, so as to obtain a template. The movement of the hand in this operation is very regular, being sensitive to the least deviation from a regular course. Ostwald states that the co-ordination of hand and eye is sensitive to the second, or even third, differential coefficient. This template is used to draw a curve in pencil on Bristol l>oard, which curve is then inked in by means of a French curve or a flexible curve. (The best the " J. FL. B." made by Harling, Finsbury Pavement.) It will be plain that the lai^-r the scale, within limits, to which the curve is drawn, the better it will look when reduced for publication ; the slight inaccuracies in the use of the French curve will be invisible. The little work by Howard Duncan on "Practical Curve Tracing" (Longmans) will be useful.

A word of caution may be allowed. Although an equation may express in one line what would require pages of verbal description, it must not b« forgotten that it is, after all, but a kind of shorthand, and must never be permitted to serve in place of a clear conception of the process itself. The same thing may be said of structural formulae in chemistry, which are only a very convenient way of expressing certain facts in the play of molecular forces, whose nature is as yet unknown. This fact sometimes seems to be in danger of being forgotten, and "bonds" regarded as actual material threads holding alums together.

Structural formula; sometimes say too much even when regarded merely as records of experimental results ; in other ways they do not say enough. A. W. Stewart points out (Chemical World, December 1912, p. 415) that in the formula for acetic acid, if written thus : — there is experimental evidence that the three methyl hydrogen atoms are different from the hydroxyl one, but that there is no evidence for the existence of a CO group ; none of the reactions characteristic of its presence are given by acetic acid. In order to make the formula inform us of-the difference between the various hydrogen atoms, which is not diiectly indicated, we have to treat the groups CH3 and OH as wholes, saying that hydrogen is not the same when united with oxygen as when united with carbon. Moreover, carboxyl, as such, is not present in acetic acid ; when CO is united with OH, a new radical, COOH (carboxyl), is formed, which must itself be taken as a whole, so that the formula of acetic acid is more correctly written : —

These components of organic compounds behave, as it were, as elements, and, strictly speaking, to make structural formula: more complete in certain ways, it would be necessary to give each of these radicals a distinctive symbol. The essence of chemical combination is, of course, that the properties of elements are changed when united with others, as in the common illustration of mercuric iodide. The object of these remarks is merely to advocate more critical use of structural formula; than is apt to be made by a certain school of chemists, who appear to think that, if a formula can be made to indicate the possibility of a particular mode of combination, the fact is in itself proof that such a reaction actually occurs. G. H. Lewes (1864, p. 131) refers to the profound psychological mistake of holding " that whenever man can form clear ideas, not in themselves contradictor}', these ideas must of necessity represent truths of nature." This view was, at one time, very widely held, and even by so great a man as Descartes. For further discussion see Karl Pearson's book (1911, chapter viii.).

The question may properly be asked, What are the peculiarities that make organic chemistry a special domain and of especial importance in physiological science? The reason lies, as van't Hoff (1881, i. p. 34 ff., and ii. p. 240 ff.) points out, in the characteristic qualities of carbon itself. This author enumerates five items : — 1. The quadri valence renders possible an enormous number of derivatives of any one compound. 2. The capacity of carbon atoms of uniting with each other allows a great variety of modes of combination.

3. Its position in the periodic system, in the middle between positive and negative elements, gives it the power of uniting with the most different elements — hydrogen, nitrogen, oxygen, chlorine, etc. (see the table in Nernst's book, 1911, p. 180). Owing to this, it is readily capable of alternate oxidation and reduction, and thus of acting as a carrier of energy. 4. When three of its valencies are saturated, the fourth valency has a "positive" or "negative" character, according to the nature of the groups in the other three places. Thus while —

5. The slowness of reaction or inertia of the carbon compounds is of much significance in vital pITenomena. As an illustration, methyl sulphonic acid is much more stable than sulphurous acid, having a methyl group in place of hydrogen. Chemical reactions arrive at their point of equilibrium and stop dead at it without overshooting. They are, in fact, aperiodic, like processes in general taking place against resistance. This being so, a formula similar in form to that of Ohm's law in electricity must hold. Thus :—

Cl>emical force is a function of the free energy ; very little is definitely known as to chemical resistance, except that it is greatly diminished by rise of temperature. All chemical reactions are, therefore, increased in rate by rise of temperature. Some confusion is apt to arise with respect to endothermic reactions, on account of the effect of temperature on the equilibrium point, to be described presently. Endothennic reactions require to be supplied with energy from their surroundings, since the products have a greater store of potential energy than the bodies from which they are produced ; but it must not be forgotten that they progress of themselves. A chemical reaction takes place, in fact, when the intensity factor of the energy associated with the original mixture is greater than that of the final system (see Mellor's book, 1904, p. 25), whether the reaction be endo- or exothermic.

From the standpoint of the kinetic theory of heat, it is easy to see why all processes conditioned by rate of molecular movement are accelerated by rise of temperature. But, as Nernst points out (1911, p. 680), it is not so easy to see why the acceleration of chemical reactions is as great as it is. A rise of 10° C. usually doubles or trebles this rate (Law of van't Hoff), whereas " the velocity of molecular movement in gases, and in all probability in liquids also, is proportional to the square root of _the absolute temperature." So that, if it has a value of 100 at 20°, it will only increase to 101 '7 at 30°, instead of to 200; Goldschmidt (1909, p. 206), however, has shown that only those molecules react whose velocity exceeds a certain high value, so that the difficulty disappears.

Conclusions are sometimes drawn as to the nature of a particular process from the value of the temperature coefficient. This quantity varies so much, not only according to the position on the scale of temperature at which the reaction happens to take place, but also in individual cases, that, on this ground alone, caution must be exercised. For example, the saponification of ethyl butyrate by barium hydroxide between 50° and 60° has the low value for a chemical reaction of 1'33 for 10° (Trautz and Volkmann, 1908, p. 79), whereas diffusion, a physical process, has a value nearly as high, viz., 1"28 (Nernst, 1888, p. 624). Chick and Martin (1910, p. 415) find that the heat coagulation of haemoglobin has the extraordinarily high temperature coefficient of 13'8 for 10°, while that of albumin is even higher. It is of interest that P. von Schroeder (1903, p. 88) finds that gelatine solution, in a particular condition, has a viscosity at 21° represented by 13'76, whereas at 31° it is only 1 "42 ; that is about ten times less for 10° rise of temperature. As will be seen later, colloids of the type of gelatine play a large part in vital processes. The temperature coefficient of the rate of absorption of water by the seeds of barley has recently been shown by Adrian Brown and Worley (1912, pp. 546-553) to be of the order of that usually regarded as characteristic of chemical reactions. They also find that the rate is an exponential function of the temperature. This is, as Mellor points out (1904, p. 394), very rare for a physical process. The increase of the vapour pressure of a liquid is one of these rare cases, and, in fact, the value of the exponent in Brown and Worley's experiments is the same as that of the vapour pressure of water. The bearing of this fact on the effect of temperature on chemical reaction in general will be found in Chapter VIII.

The impossibility of forming conclusions as to the physical or chemical nature of a process from the temperature coefficient of its velocity is well shown by the work of Knowlton and Starling (1912, p. 206), on the effect of temperature on the rate of the heart-beat in the isolated heart-lung preparation. This rate is a linear function of the temperature, as shown by Fig. 28. In other words, a given rise of temperature produces the same increase at different points of the scale. But such a relationship is what we find in the simplest physical process, such as the expansion of a gas. Therefore, if the temperature coefficient is any index, the heart-beat is a purely physical process. This is obviously an absurd conclusion. We know that rise of temperature accelerates the chemical changes in the heart muscle, as evidenced by the increase in the oxygen consumption (Lovatt Evans, 1912, p. 231), and, in fact, it is very interesting to find that this increased

metabolism is directly proportional to the increase of rate, so that we have again a linear relation. It will be plain that, in such a case as that before us, one cannot speak of a " coefficient " in the strict sense. If such a number be calculated for any particular temperature, it will not apply to any other temperature. Consider indeed, for a moment, the complexity and variety of the forms of energy change involved in a muscular contraction— surface and volume energy, thermal, electrical and chemical energy. I think that it must be admitted that to attempt to draw conclusions from the temperature coefficient of the entire process does not seem likely to lead to results of much value. This remark, of course, applies to the activities of living protoplasm in general, as well as to muscle.