Chapter XII
THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.
We have hitherto considered the theory of probability only in its simple deductive employment, in which it enables us to determine from given conditions the probable character of events happening under those conditions. But as deductive reasoning when inversely applied constitutes the process of induction, so the calculation of probabilities may be inversely applied; from the known character of certain events we may argue backwards to the probability of a certain law or condition governing those events. Having satisfactorily accomplished this work, we may indeed calculate forwards to the probable character of future events happening under the same conditions; but this part of the process is a direct use of deductive reasoning (p. 226).
Now it is highly instructive to find that whether the theory of probability be deductively or inductively applied, the calculation is always performed according to the principles and rules of deduction. The probability that an event has a particular condition entirely depends upon the probability that if the condition existed the event would follow. If we take up a pack of common playing cards, and observe that they are arranged in perfect numerical order, we conclude beyond all reasonable doubt that they have been thus intentionally arranged by some person acquainted with the usual order of sequence. This conclusion is quite irresistible, and rightly so; for there are but two suppositions which we can make as to the reason of the cards being in that particular order:--
1. They may have been intentionally arranged by some one who would probably prefer the numerical order.
2. They may have fallen into that order by chance, that is, by some series of conditions which, being unknown to us, cannot be known to lead by preference to the particular order in question.
The latter supposition is by no means absurd, for any one order is as likely as any other when there is no preponderating tendency. But we can readily calculate by the doctrine of permutations the probability that fifty-two objects would fall by chance into any one particular order. Fifty-two objects can be arranged in 52 × 51 × ... × 3 × 2 × 1 or about 8066 × (10)^{64} possible orders, the number obtained requiring 68 places of figures for its full expression. Hence it is excessively unlikely that anyone should ever meet with a pack of cards arranged in perfect order by accident. If we do meet with a pack so arranged, we inevitably adopt the other supposition, that some person, having reasons for preferring that special order, has thus put them together.
We know that of the immense number of possible orders the numerical order is the most remarkable; it is useful as proving the perfect constitution of the pack, and it is the intentional result of certain games. At any rate, the probability that intention should produce that order is incomparably greater than the probability that chance should produce it; and as a certain pack exists in that order, we rightly prefer the supposition which most probably leads to the observed result.
By a similar mode of reasoning we every day arrive, and validly arrive, at conclusions approximating to certainty. Whenever we observe a perfect resemblance between two objects, as, for instance, two printed pages, two engravings, two coins, two foot-prints, we are warranted in asserting that they proceed from the same type, the same plate, the same pair of dies, or the same boot. And why? Because it is almost impossible that with different types, plates, dies, or boots some apparent distinction of form should not be produced. It is impossible for the hand of the most skilful artist to make two objects alike, so that mechanical repetition is the only probable explanation of exact similarity.
We can often establish with extreme probability that one document is copied from another. Suppose that each document contains 10,000 words, and that the same word is incorrectly spelt in each. There is then a probability of less than 1 in 10,000 that the same mistake should be made in each. If we meet with a second error occurring in each document, the probability is less than 1 in 10,000 × 9999, that two such coincidences should occur by chance, and the numbers grow with extreme rapidity for more numerous coincidences. We cannot make any precise calculations without taking into account the character of the errors committed, concerning the conditions of which we have no accurate means of estimating probabilities. Nevertheless, abundant evidence may thus be obtained as to the derivation of documents from each other. In the examination of many sets of logarithmic tables, six remarkable errors were found to be present in all but two, and it was proved that tables printed at Paris, Berlin, Florence, Avignon, and even in China, besides thirteen sets printed in England between the years 1633 and 1822, were derived directly or indirectly from some common source.[150] With a certain amount of labour, it is possible to establish beyond reasonable doubt the relationship or genealogy of any number of copies of one document, proceeding possibly from parent copies now lost. The relations between the manuscripts of the New Testament have been elaborately investigated in this manner, and the same work has been performed for many classical writings, especially by German scholars.
[150] Lardner, *Edinburgh Review*, July 1834, p. 277.
*Principle of the Inverse Method.*
The inverse application of the rules of probability entirely depends upon a proposition which may be thus stated, nearly in the words of Laplace.[151] *If an event can be produced by any one of a certain number of different causes, all equally probable à priori, the probabilities of the existence of these causes as inferred from the event, are proportional to the probabilities of the event as derived from these causes.* In other words, the most probable cause of an event which has happened is that which would most probably lead to the event supposing the cause to exist; but all other possible causes are also to be taken into account with probabilities proportional to the probability that the event would happen if the cause existed. Suppose, to fix our ideas clearly, that E is the event, and C_{1} C_{2} C_{3} are the three only conceivable causes. If C_{1} exist, the probability is *p*{1} that E would follow; if C{2} or C_{3} exist, the like probabilities are respectively *p*{2} and *p*{3}. Then as *p*{1} is to *p*{2}, so is the probability of C_{1} being the actual cause to the probability of C_{2} being it; and, similarly, as *p*{2} is to *p*{3}, so is the probability of C_{2} being the actual cause to the probability of C_{3} being it. By a simple mathematical process we arrive at the conclusion that the actual probability of C_{1} being the cause is
*p*{1}/(*p*{1} + *p*{2} + *p*{3});
[151] *Mémoires par divers Savans*, tom. vi.; quoted by Todhunter in his *History of the Theory of Probability*, p. 458.
and the similar probabilities of the existence of C_{2} and C_{3} are,
*p*{2}/(*p*{1} + *p*{2} + *p*{3}) and *p*{3}/(*p*{1} + *p*{2} + *p*{3}).
The sum of these three fractions amounts to unity, which correctly expresses the certainty that one cause or other must be in operation.
We may thus state the result in general language. *If it is certain that one or other of the supposed causes exists, the probability that any one does exist is the probability that if it exists the event happens, divided by the sum of all the similar probabilities.* There may seem to be an intricacy in this subject which may prove distasteful to some readers; but this intricacy is essential to the subject in hand. No one can possibly understand the principles of inductive reasoning, unless he will take the trouble to master the meaning of this rule, by which we recede from an event to the probability of each of its possible causes.
This rule or principle of the indirect method is that which common sense leads us to adopt almost instinctively, before we have any comprehension of the principle in its general form. It is easy to see, too, that it is the rule which will, out of a great multitude of cases, lead us most often to the truth, since the most probable cause of an event really means that cause which in the greatest number of cases produces the event. Donkin and Boole have given demonstrations of this principle, but the one most easy to comprehend is that of Poisson. He imagines each possible cause of an event to be represented by a distinct ballot-box, containing black and white balls, in such a ratio that the probability of a white ball being drawn is equal to that of the event happening. He further supposes that each box, as is possible, contains the same total number of balls, black and white; then, mixing all the contents of the boxes together, he shows that if a white ball be drawn from the aggregate ballot-box thus formed, the probability that it proceeded from any particular ballot-box is represented by the number of white balls in that particular box, divided by the total number of white balls in all the boxes. This result corresponds to that given by the principle in question.[152]
[152] Poisson, *Recherches sur la Probabilité des Jugements*, Paris, 1837, pp. 82, 83.
Thus, if there be three boxes, each containing ten balls in all, and respectively containing seven, four, and three white balls, then on mixing all the balls together we have fourteen white ones; and if we draw a white ball, that is if the event happens, the probability that it came out of the first box is 7/14; which is exactly equal to (7/10)/(7/10 + 4/10 + 3/10), the fraction given by the rule of the Inverse Method.
*Simple Applications of the Inverse Method.*
In many cases of scientific induction we may apply the principle of the inverse method in a simple manner. If only two, or at the most a few hypotheses, may be made as to the origin of certain phenomena, we may sometimes easily calculate the respective probabilities. It was thus that Bunsen and Kirchhoff established, with a probability little short of certainty, that iron exists in the sun. On comparing the spectra of sunlight and of the light proceeding from the incandescent vapour of iron, it became apparent that at least sixty bright lines in the spectrum of iron coincided with dark lines in the sun’s spectrum. Such coincidences could never be observed with certainty, because, even if the lines only closely approached, the instrumental imperfections of the spectroscope would make them apparently coincident, and if one line came within half a millimetre of another, on the map of the spectra, they could not be pronounced distinct. Now the average distance of the solar lines on Kirchhoff’s map is 2 mm., and if we throw down a line, as it were, by pure chance on such a map, the probability is about one-half that the new line will fall within 1/2 mm. on one side or the other of some one of the solar lines. To put it in another way, we may suppose that each solar line, either on account of its real breadth, or the defects of the instrument, possesses a breadth of 1/2 mm., and that each line in the iron spectrum has a like breadth. The probability then is just one-half that the centre of each iron line will come by chance within 1 mm. of the centre of a solar line, so as to appear to coincide with it. The probability of casual coincidence of each iron line with a solar line is in like manner 1/2. Coincidence in the case of each of the sixty iron lines is a very unlikely event if it arises casually, for it would have a probability of only (1/2)^{60} or less than 1 in a trillion. The odds, in short, are more than a million million millions to unity against such casual coincidence.[153] But on the other hypothesis, that iron exists in the sun, it is highly probable that such coincidences would be observed; it is immensely more probable that sixty coincidences would be observed if iron existed in the sun, than that they should arise from chance. Hence by our principle it is immensely probable that iron does exist in the sun.
[153] Kirchhoff’s *Researches on the Solar Spectrum*. First part, translated by Roscoe, pp. 18, 19.
All the other interesting results, given by the comparison of spectra, rest upon the same principle of probability. The almost complete coincidence between the spectra of solar, lunar, and planetary light renders it practically certain that the light is all of solar origin, and is reflected from the surfaces of the moon and planets, suffering only slight alteration from the atmospheres of some of the planets. A fresh confirmation of the truth of the Copernican theory is thus furnished.
Herschel proved in this way the connection between the direction of the oblique faces of quartz crystals, and the direction in which the same crystals rotate the plane of polarisation of light. For if it is found in a second crystal that the relation is the same as in the first, the probability of this happening by chance is 1/2; the probability that in another crystal also the direction will be the same is 1/4, and so on. The probability that in *n* + 1 crystals there would be casual agreement of direction is the nth power of 1/2. Thus, if in examining fourteen crystals the same relation of the two phenomena is discovered in each, the odds that it proceeds from uniform conditions are more than 8000 to 1.[154] Since the first observations on this subject were made in 1820, no exceptions have been observed, so that the probability of invariable connection is incalculably great.
[154] *Edinburgh Review*, No. 185, vol. xcii. July 1850, p. 32; Herschel’s *Essays*, p. 421; *Transactions of the Cambridge Philosophical Society*, vol. i. p. 43.
It is exceedingly probable that the ancient Egyptians had exactly recorded the eclipses occurring during long periods of time, for Diogenes Laertius mentions that 373 solar and 832 lunar eclipses had been observed, and the ratio between these numbers exactly expresses that which would hold true of the eclipses of any long period, of say 1200 or 1300 years, as estimated on astronomical grounds. It is evident that an agreement between small numbers, or customary numbers, such as seven, one hundred, a myriad, &c., is much more likely to happen from chance, and therefore gives much less presumption of dependence. If two ancient writers spoke of the sacrifice of oxen, they would in all probability describe it as a hecatomb, and there would be nothing remarkable in the coincidence. But it is impossible to point out any special reason why an old writer should select such numbers as 373 and 832, unless they had been the results of observation.
On similar grounds, we must inevitably believe in the human origin of the flint flakes so copiously discovered of late years. For though the accidental stroke of one stone against another may often produce flakes, such as are occasionally found on the sea-shore, yet when several flakes are found in close company, and each one bears evidence, not of a single blow only, but of several successive blows, all conducing to form a symmetrical knife-like form, the probability of a natural and accidental origin becomes incredibly small, and the contrary supposition, that they are the work of intelligent beings, approximately certain.[155]
[155] Evans’ *Ancient Stone Implements of Great Britain*. London, 1872 (Longmans).
*The Theory of Probability in Astronomy.*
The science of astronomy, occupied with the simple relations of distance, magnitude, and motion of the heavenly bodies, admits more easily than almost any other science of interesting conclusions founded on the theory of probability. More than a century ago, in 1767, Michell showed the extreme probability of bonds connecting together systems of stars. He was struck by the unexpected number of fixed stars which have companions close to them. Such a conjunction might happen casually by one star, although possibly at a great distance from the other, happening to lie on a straight line passing near the earth. But the probabilities are so greatly against such an optical union happening often in the expanse of the heavens, that Michell asserted the existence of some connection between most of the double stars. It has since been estimated by Struve, that the odds are 9570 to 1 against any two stars of not less than the seventh magnitude falling within the apparent distance of four seconds of each other by chance, and yet ninety-one such cases were known when the estimation was made, and many more cases have since been discovered. There were also four known triple stars, and yet the odds against the appearance of any one such conjunction are 173,524 to 1.[156] The conclusions of Michell have been entirely verified by the discovery that many double stars are connected by gravitation.
[156] Herschel, *Outlines of Astronomy*, 1849, p. 565; but Todhunter, in his *History of the Theory of Probability*, p. 335, states that the calculations do not agree with those published by Struve.
Michell also investigated the probability that the six brightest stars in the Pleiades should have come by accidents into such striking proximity. Estimating the number of stars of equal or greater brightness at 1500, be found the odds to be nearly 500,000 to 1 against casual conjunction. Extending the same kind of argument to other clusters, such as that of Præsepe, the nebula in the hilt of Perseus’ sword, he says:[157] “We may with the highest probability conclude, the odds against the contrary opinion being many million millions to one, that the stars are really collected together in clusters in some places, where they form a kind of system, while in others there are either few or none of them, to whatever cause this may be owing, whether to their mutual gravitation, or to some other law or appointment of the Creator.”
[157] *Philosophical Transactions*, 1767, vol. lvii. p. 431.
The calculations of Michell have been called in question by the late James D. Forbes,[158] and Mr. Todhunter vaguely countenances his objections,[159] otherwise I should not have thought them of much weight. Certainly Laplace accepts Michell’s views,[160] and if Michell be in error it is in the methods of calculation, not in the general validity of his reasoning and conclusions.
[158] *Philosophical Magazine*, 3rd Series, vol. xxxvii. p. 401, December 1850; also August 1849.
[159] *History*, &c., p. 334.
[160] *Essai Philosophique*, p. 57.
Similar calculations might no doubt be applied to the peculiar drifting motions which have been detected by Mr. R A. Proctor in some of the constellations.[161] The odds are very greatly against any numerous group of stars moving together in any one direction by chance. On like grounds, there can be no doubt that the sun has a considerable proper motion because on the average the fixed stars show a tendency to move apparently from one point of the heavens towards that diametrically opposite. The sun’s motion in the contrary direction would explain this tendency, otherwise we must believe that thousands of stars accidentally agree in their direction of motion, or are urged by some common force from which the sun is exempt. It may be said that the rotation of the earth is proved in like manner, because it is immensely more probable that one body would revolve than that the sun, moon, planets, comets, and the whole of the stars of the heavens should be whirled round the earth daily, with a uniform motion superadded to their own peculiar motions. This appears to be mainly the reason which led Gilbert, one of the earliest English Copernicans, and in every way an admirable physicist, to admit the rotation of the earth, while Francis Bacon denied it.
[161] *Proceedings of the Royal Society*; 20 January, 1870; *Philosophical Magazine*, 4th Series, vol. xxxix. p. 381.
In contemplating the planetary system, we are struck with the similarity in direction of nearly all its movements. Newton remarked upon the regularity and uniformity of these motions, and contrasted them with the eccentricity and irregularity of the cometary orbits.[162] Could we, in fact, look down upon the system from the northern side, we should see all the planets moving round from west to east, the satellites moving round their primaries, and the sun, planets, and satellites rotating in the same direction, with some exceptions on the verge of the system. In the time of Laplace eleven planets were known, and the directions of rotation were known for the sun, six planets, the satellites of Jupiter, Saturn’s ring, and one of his satellites. Thus there were altogether 43 motions all concurring, namely:--
Orbital motions of eleven planets 11 Orbital motions of eighteen satellites 18 Axial rotations 14 -- 43
[162] *Principia*, bk. ii. General scholium.
The probability that 43 motions independent of each other would coincide by chance is the 42nd power of 1/2, so that the odds are about 4,400,000,000,000 to 1 in favour of some common cause for the uniformity of direction. This probability, as Laplace observes,[163] is higher than that of many historical events which we undoubtingly believe. In the present day, the probability is much increased by the discovery of additional planets, and the rotation of other satellites, and it is only slightly weakened by the fact that some of the outlying satellites are exceptional in direction, there being considerable evidence of an accidental disturbance in the more distant parts of the system.
[163] *Essai Philosophique*, p. 55. Laplace appears to count the rings of Saturn as giving two independent movements.
Hardly less remarkable than the uniform direction of motion is the near approximation of the orbits of the planets to a common plane. Daniel Bernoulli roughly estimated the probability of such an agreement arising from accident as 1 ÷ (12)^{6} the greatest inclination of any orbit to the sun’s equator being 1-12th part of a quadrant. Laplace devoted to this subject some of his most ingenious investigations. He found the probability that the sum of the inclinations of the planetary orbits would not exceed by accident the actual amount (·914187 of a right angle for the ten planets known in 1801) to be (1/10)! (·914187)^{10} or about ·00000011235. This probability may be combined with that derived from the direction of motion, and it then becomes immensely probable that the constitution of the planetary system arose out of uniform conditions, or, as we say, from some common cause.[164]
[164] Lubbock, *Essay on Probability*, p. 14. De Morgan, *Encyc. Metrop.* art. *Probability*, p. 412. Todhunter’s *History of the Theory of Probability*, p. 543. Concerning the objections raised to these conclusions by Boole, see the *Philosophical Magazine*, 4th Series, vol. ii. p. 98. Boole’s *Laws of Thought*, pp. 364–375.
If the same kind of calculation be applied to the orbits of comets, the result is very different.[165] Of the orbits which have been determined 48·9 per cent. only are direct or in the same direction as the planetary motions.[166] Hence it becomes apparent that comets do not properly belong to the solar system, and it is probable that they are stray portions of nebulous matter which have accidentally become attached to the system by the attractive powers of the sun or Jupiter.
[165] Laplace, *Essai Philosophique*, pp. 55, 56.
[166] Chambers’ *Astronomy*, 2nd ed. pp. 346–49.
*The General Inverse Problem.*
In the instances described in the preceding sections, we have been occupied in receding from the occurrence of certain similar events to the probability that there must have been a condition or cause for such events. We have found that the theory of probability, although never yielding a certain result, often enables us to establish an hypothesis beyond the reach of reasonable doubt. There is, however, another method of applying the theory, which possesses for us even greater interest, because it illustrates, in the most complete manner, the theory of inference adopted in this work, which theory indeed it suggested. The problem to be solved is as follows:--
*An event having happened a certain number of times, and failed a certain number of times, required the probability that it will happen any given number of times in the future under the same circumstances.*
All the *larger* planets hitherto discovered move in one direction round the sun; what is the probability that, if a new planet exterior to Neptune be discovered, it will move in the same direction? All known permanent gases, except chlorine, are colourless; what is the probability that, if some new permanent gas should be discovered, it will be colourless? In the general solution of this problem, we wish to infer the future happening of any event from the number of times that it has already been observed to happen. Now, it is very instructive to find that there is no known process by which we can pass directly from the data to the conclusion. It is always requisite to recede from the data to the probability of some hypothesis, and to make that hypothesis the ground of our inference concerning future events. Mathematicians, in fact, make every hypothesis which is applicable to the question in hand; they then calculate, by the inverse method, the probability of every such hypothesis according to the data, and the probability that if each hypothesis be true, the required future event will happen. The total probability that the event will happen is the sum of the separate probabilities contributed by each distinct hypothesis.
To illustrate more precisely the method of solving the problem, it is desirable to adopt some concrete mode of representation, and the ballot-box, so often employed by mathematicians, will best serve our purpose. Let the happening of any event be represented by the drawing of a white ball from a ballot-box, while the failure of an event is represented by the drawing of a black ball. Now, in the inductive problem we are supposed to be ignorant of the contents of the ballot-box, and are required to ground all our inferences on our experience of those contents as shown in successive drawings. Rude common sense would guide us nearly to a true conclusion. Thus, if we had drawn twenty balls one after another, replacing the ball after each drawing, and the ball had in each case proved to be white, we should believe that there was a considerable preponderance of white balls in the urn, and a probability in favour of drawing a white ball on the next occasion. Though we had drawn white balls for thousands of times without fail, it would still be possible that some black balls lurked in the urn and would at last appear, so that our inferences could never be certain. On the other hand, if black balls came at intervals, we should expect that after a certain number of trials the black balls would appear again from time to time with somewhat the same frequency.
The mathematical solution of the question consists in little more than a close analysis of the mode in which our common sense proceeds. If twenty white balls have been drawn and no black ball, my common sense tells me that any hypothesis which makes the black balls in the urn considerable compared with the white ones is improbable; a preponderance of white balls is a more probable hypothesis, and as a deduction from this more probable hypothesis, I expect a recurrence of white balls. The mathematician merely reduces this process of thought to exact numbers. Taking, for instance, the hypothesis that there are 99 white and one black ball in the urn, he can calculate the probability that 20 white balls would be drawn in succession in those circumstances; he thus forms a definite estimate of the probability of this hypothesis, and knowing at the same time the probability of a white ball reappearing if such be the contents of the urn, he combines these probabilities, and obtains an exact estimate that a white ball will recur in consequence of this hypothesis. But as this hypothesis is only one out of many possible ones, since the ratio of white and black balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has to repeat the estimate for every such possible hypothesis. To make the method of solving the problem perfectly evident, I will describe in the next section a very simple case of the problem, originally devised for the purpose by Condorcet, which was also adopted by Lacroix,[167] and has passed into the works of De Morgan, Lubbock, and others.
[167] *Traité élémentaire du Calcul des Probabilités*, 3rd ed. (1833), p. 148.
*Simple Illustration of the Inverse Problem.*
Suppose it to be known that a ballot-box contains only four black or white balls, the ratio of black and white balls being unknown. Four drawings having been made with replacement, and a white ball having appeared on each occasion but one, it is required to determine the probability that a white ball will appear next time. Now the hypotheses which can be made as to the contents of the urn are very limited in number, and are at most the following five:--
4 white and 0 black balls 3 " " 1 " " 2 " " 2 " " 1 " " 3 " " 0 " " 4 " "
The actual occurrence of black and white balls in the drawings puts the first and last hypothesis out of the question, so that we have only three left to consider.
If the box contains three white and one black, the probability of drawing a white each time is 3/4, and a black 1/4; so that the compound event observed, namely, three white and one black, has the probability 3/4 × 3/4 × 3/4 × 1/4, by the rule already given (p. 204). But as it is indifferent in what order the balls are drawn, and the black ball might come first, second, third, or fourth, we must multiply by four, to obtain the probability of three white and one black in any order, thus getting 27/64.
Taking the next hypothesis of two white and two black balls in the urn, we obtain for the same probability the quantity 1/2 × 1/2 × 1/2 × 1/2 × 4, or 16/64, and from the third hypothesis of one white and three black we deduce likewise 1/4 × 1/4 × 1/4 × 3/4 × 4, or 3/64. According, then, as we adopt the first, second, or third hypothesis, the probability that the result actually noticed would follow is 27/64, 16/64, and 3/64. Now it is certain that one or other of these hypotheses must be the true one, and their absolute probabilities are proportional to the probabilities that the observed events would follow from them (pp. 242, 243). All we have to do, then, in order to obtain the absolute probability of each hypothesis, is to alter these fractions in a uniform ratio, so that their sum shall be unity, the expression of certainty. Now, since 27 + 16 + 3 = 46, this will be effected by dividing each fraction by 46, and multiplying by 64. Thus the probabilities of the first, second, and third hypotheses are respectively--
27/46, 16/46, 3/46.
The inductive part of the problem is completed, since we have found that the urn most likely contains three white and one black ball, and have assigned the exact probability of each possible supposition. But we are now in a position to resume deductive reasoning, and infer the probability that the next drawing will yield, say a white ball. For if the box contains three white and one black ball, the probability of drawing a white one is certainly 3/4; and as the probability of the box being so constituted is 27/46, the compound probability that the box will be so filled and will give a white ball at the next trial, is
27/46 × 3/4 or 81/184.
Again, the probability is 16/46 that the box contains two white and two black, and under those conditions the probability is 1/2 that a white ball will appear; hence the probability that a white ball will appear in consequence of that condition, is
16/46 × 1/2 or 32/184.
From the third supposition we get in like manner the probability
3/46 × 1/4 or 3/184.
Since one and not more than one hypothesis can be true, we may add together these separate probabilities, and we find that
81/184 + 32/184 + 3/184 or 116/184
is the complete probability that a white ball will be next drawn under the conditions and data supposed.
*General Solution of the Inverse Problem.*
In the instance of the inverse method described in the last section, the balls supposed to be in the ballot-box were few, for the purpose of simplifying the calculation. In order that our solution may apply to natural phenomena, we must render our hypotheses as little arbitrary as possible. Having no *à priori* knowledge of the conditions of the phenomena in question, there is no limit to the variety of hypotheses which might be suggested. Mathematicians have therefore had recourse to the most extensive suppositions which can be made, namely, that the ballot-box contains an infinite number of balls; they have then varied the proportion of white to black balls continuously, from the smallest to the greatest possible proportion, and estimated the aggregate probability which results from this comprehensive supposition.
To explain their procedure, let us imagine that, instead of an infinite number, the ballot-box contains a large finite number of balls, say 1000. Then the number of white balls might be 1 or 2 or 3 or 4, and so on, up to 999. Supposing that three white and one black ball have been drawn from the urn as before, there is a certain very small probability that this would have occurred in the case of a box containing one white and 999 black balls; there is also a small probability that from such a box the next ball would be white. Compound these probabilities, and we have the probability that the next ball really will be white, in consequence of the existence of that proportion of balls. If there be two white and 998 black balls in the box, the probability is greater and will increase until the balls are supposed to be in the proportion of those drawn. Now 999 different hypotheses are possible, and the calculation is to be made for each of these, and their aggregate taken as the final result. It is apparent that as the number of balls in the box is increased, the absolute probability of any one hypothesis concerning the exact proportion of balls is decreased, but the aggregate results of all the hypotheses will assume the character of a wider average.
When we take the step of supposing the balls within the urn to be infinite in number, the possible proportions of white and black balls also become infinite, and the probability of any one proportion actually existing is infinitely small. Hence the final result that the next ball drawn will be white is really the sum of an infinite number of infinitely small quantities. It might seem impossible to calculate out a problem having an infinite number of hypotheses, but the wonderful resources of the integral calculus enable this to be done with far greater facility than if we supposed any large finite number of balls, and then actually computed the results. I will not attempt to describe the processes by which Laplace finally accomplished the complete solution of the problem. They are to be found described in several English works, especially De Morgan’s *Treatise on Probabilities*, in the *Encyclopædia Metropolitana*, and Mr. Todhunter’s *History of the Theory of Probability*. The abbreviating power of mathematical analysis was never more strikingly shown. But I may add that though the integral calculus is employed as a means of summing infinitely numerous results, we in no way abandon the principles of combinations already treated. We calculate the values of infinitely numerous factorials, not, however, obtaining their actual products, which would lead to an infinite number of figures, but obtaining the final answer to the problem by devices which can only be comprehended after study of the integral calculus.
It must be allowed that the hypothesis adopted by Laplace is in some degree arbitrary, so that there was some opening for the doubt which Boole has cast upon it.[168] But it may be replied, (1) that the supposition of an infinite number of balls treated in the manner of Laplace is less arbitrary and more comprehensive than any other that can be suggested. (2) The result does not differ much from that which would be obtained on the hypothesis of any large finite number of balls. (3) The supposition leads to a series of simple formulas which can be applied with ease in many cases, and which bear all the appearance of truth so far as it can be independently judged by a sound and practiced understanding.
[168] *Laws of Thought*, pp. 368–375.
*Rules of the Inverse Method.*
By the solution of the problem, as described in the last section, we obtain the following series of simple rules.
1. *To find the probability that an event which has not hitherto been observed to fail will happen once more, divide the number of times the event has been observed increased by one, by the same number increased by two.*
If there have been *m* occasions on which a certain event might have been observed to happen, and it has happened on all those occasions, then the probability that it will happen on the next occasion of the same kind (*m* + 1)/(*m* + 2). For instance, we may say that there are nine places in the planetary system where planets might exist obeying Bode’s law of distance, and in every place there is a planet obeying the law more or less exactly, although no reason is known for the coincidence. Hence the probability that the next planet beyond Neptune will conform to the law is 10/11.
2. *To find the, probability that an event which has not hitherto failed will not fail for a certain number of new occasions, divide the number of times the event has happened increased by one, by the same number increased by one and the number of times it is to happen.*
An event having happened *m* times without fail, the probability that it will happen *n* more times is (*m* + 1)/(*m* + *n* + 1). Thus the probability that three new planets would obey Bode’s law is 10/13; but it must be allowed that this, as well as the previous result, would be much weakened by the fact that Neptune can barely be said to obey the law.
*3. An event having happened and failed a certain number of times, to find the probability that it will happen the next time, divide the number of times the event has happened increased by one, by the whole number of times the event has happened or failed increased by two.*
If an event has happened *m* times and failed *n* times, the probability that it will happen on the next occasion is (*m* + 1)/(*m* + *n* + 2). Thus, if we assume that of the elements discovered up to the year 1873, 50 are metallic and 14 non-metallic, then the probability that the next element discovered will be metallic is 51/66. Again, since of 37 metals which have been sufficiently examined only four, namely, sodium, potassium, lanthanum, and lithium, are of less density than water, the probability that the next metal examined or discovered will be less dense than water is (4 + 1)/(37 + 2) or 5/39.
We may state the results of the method in a more general manner thus,[169]--If under given circumstances certain events A, B, C, &c., have happened respectively *m*, *n*, *p*, &c., times, and one or other of these events must happen, then the probabilities of these events are proportional to *m* + 1, *n* + 1, *p* + 1, &c., so that the probability of A will be (*m* + 1)/(*m* + 1 + *n* + 1 + *p* + 1 + &c.) But if new events may happen in addition to those which have been observed, we must assign unity for the probability of such new event. The odds then become 1 for a new event, *m* + 1 for A, *n* + 1 for B, and so on, and the absolute probability of A is (*m* + 1)/(1 + *m* + 1 + *n* + 1 + &c.)
[169] De Morgan’s *Essay on Probabilities*, Cabinet Cyclopædia, p. 67.
It is interesting to trace out the variations of probability according to these rules. The first time a casual event happens it is 2 to 1 that it will happen again; if it does happen it is 3 to 1 that it will happen a third time; and on successive occasions of the like kind the odds become 4, 5, 6, &c., to 1. The odds of course will be discriminated from the probabilities which are successively 2/3, 3/4, 4/5, &c. Thus on the first occasion on which a person sees a shark, and notices that it is accompanied by a little pilot fish, the odds are 2 to 1, or the probability 2/3, that the next shark will be so accompanied.
When an event has happened a very great number of times, its happening once again approaches nearly to certainty. If we suppose the sun to have risen one thousand million times, the probability that it will rise again, on the ground of this knowledge merely, is (1,000,000,000 + 1)/(1,000,000,000 + 1 + 1). But then the probability that it will continue to rise for as long a period in the future is only (1,000,000,000 + 1)/(2,000,000,000 + 1), or almost exactly 1/2. The probability that it will continue so rising a thousand times as long is only about 1/1001. The lesson which we may draw from these figures is quite that which we should adopt on other grounds, namely, that experience never affords certain knowledge, and that it is exceedingly improbable that events will always happen as we observe them. Inferences pushed far beyond their data soon lose any considerable probability. De Morgan has said,[170] “No finite experience whatsoever can justify us in saying that the future shall coincide with the past in all time to come, or that there is any probability for such a conclusion.” On the other hand, we gain the assurance that experience sufficiently extended and prolonged will give us the knowledge of future events with an unlimited degree of probability, provided indeed that those events are not subject to arbitrary interference.
[170] *Essay on Probabilities*, p. 128.
It must be clearly understood that these probabilities are only such as arise from the mere happening of the events, irrespective of any knowledge derived from other sources concerning those events or the general laws of nature. All our knowledge of nature is indeed founded in like manner upon observation, and is therefore only probable. The law of gravitation itself is only probably true. But when a number of different facts, observed under the most diverse circumstances, are found to be harmonized under a supposed law of nature, the probability of the law approximates closely to certainty. Each science rests upon so many observed facts, and derives so much support from analogies or connections with other sciences, that there are comparatively few cases where our judgment of the probability of an event depends entirely upon a few antecedent events, disconnected from the general body of physical science.
Events, again, may often exhibit a regularity of succession or preponderance of character, which the simple formula will not take into account. For instance, the majority of the elements recently discovered are metals, so that the probability of the next discovery being that of a metal, is doubtless greater than we calculated (p. 258). At the more distant parts of the planetary system, there are symptoms of disturbance which would prevent our placing much reliance on any inference from the prevailing order of the known planets to those undiscovered ones which may possibly exist at great distances. These and all like complications in no way invalidate the theoretic truth of the formulas, but render their sound application much more difficult.
Erroneous objections have been raised to the theory of probability, on the ground that we ought not to trust to our *à priori* conceptions of what is likely to happen, but should always endeavour to obtain precise experimental data to guide us.[171] This course, however, is perfectly in accordance with the theory, which is our best and only guide, whatever data we possess. We ought to be always applying the inverse method of probabilities so as to take into account all additional information. When we throw up a coin for the first time, we are probably quite ignorant whether it tends more to fall head or tail upwards, and we must therefore assume the probability of each event as 1/2. But if it shows head in the first throw, we now have very slight experimental evidence in favour of a tendency to show head. The chance of two heads is now slightly greater than 1/4, which it appeared to be at first,[172] and as we go on throwing the coin time after time, the probability of head appearing next time constantly varies in a slight degree according to the character of our previous experience. As Laplace remarks, we ought always to have regard to such considerations in common life. Events when closely scrutinized will hardly ever prove to be quite independent, and the slightest preponderance one way or the other is some evidence of connection, and in the absence of better evidence should be taken into account.
[171] J. S. Mill, *System of Logic*, 5th edition, bk. iii. chap. xviii. § 3.
[172] Todhunter’s *History*, pp. 472, 598.
The grand object of seeking to estimate the probability of future events from past experience, seems to have been entertained by James Bernoulli and De Moivre, at least such was the opinion of Condorcet; and Bernoulli may be said to have solved one case of the problem.[173] The English writers Bayes and Price are, however, undoubtedly the first who put forward any distinct rules on the subject.[174] Condorcet and several other eminent mathematicians advanced the mathematical theory of the subject; but it was reserved to the immortal Laplace to bring to the subject the full power of his genius, and carry the solution of the problem almost to perfection. It is instructive to observe that a theory which arose from petty games of chance, the rules and the very names of which are forgotten, gradually advanced, until it embraced the most sublime problems of science, and finally undertook to measure the value and certainty of all our inductions.
[173] Todhunter’s *History*, pp. 378, 379.
[174] *Philosophical Transactions*, [1763], vol. liii. p. 370, and [1764], vol. liv. p. 296. Todhunter, pp. 294–300.
*Fortuitous Coincidences.*
We should have studied the theory of probability to very little purpose, if we thought that it would furnish us with an infallible guide. The theory itself points out the approximate certainty, that we shall sometimes be deceived by extraordinary fortuitous coincidences. There is no run of luck so extreme that it may not happen, and it may happen to us, or in our time, as well as to other persons or in other times. We may be forced by correct calculation to refer such coincidences to a necessary cause, and yet we may be deceived. All that the calculus of probability pretends to give, is *the result in the long run*, as it is called, and this really means in *an infinity of cases*. During any finite experience, however long, chances may be against us. Nevertheless the theory is the best guide we can have. If we always think and act according to its well-interpreted indications, we shall have the best chance of escaping error; and if all persons, throughout all time to come, obey the theory in like manner, they will undoubtedly thereby reap the greatest advantage.
No rule can be given for discriminating between coincidences which are casual and those which are the effects of law. By a fortuitous or casual coincidence, we mean an agreement between events, which nevertheless arise from wholly independent and different causes or conditions, and which will not always so agree. It is a fortuitous coincidence, if a penny thrown up repeatedly in various ways always falls on the same side; but it would not be fortuitous if there were any similarity in the motions of the hand, and the height of the throw, so as to cause or tend to cause a uniform result. Now among the infinitely numerous events, objects, or relations in the universe, it is quite likely that we shall occasionally notice casual coincidences. There are seven intervals in the octave, and there is nothing very improbable in the colours of the spectrum happening to be apparently divisible into the same or similar series of seven intervals. It is hardly yet decided whether this apparent coincidence, with which Newton was much struck, is well founded or not,[175] but the question will probably be decided in the negative.
[175] Newton’s *Opticks*, Bk. I., Part ii. Prop. 3; *Nature*, vol. i. p. 286.
It is certainly a casual coincidence which the ancients noticed between the seven vowels, the seven strings of the lyre, the seven Pleiades, and the seven chiefs at Thebes.[176] The accidents connected with the number seven have misled the human intellect throughout the historical period. Pythagoras imagined a connection between the seven planets and the seven intervals of the monochord. The alchemists were never tired of drawing inferences from the coincidence in numbers of the seven planets and the seven metals, not to speak of the seven days of the week.
[176] Aristotle’s *Metaphysics*, xiii. 6. 3.
A singular circumstance was pointed out concerning the dimensions of the earth, sun, and moon; the sun’s diameter was almost exactly 110 times as great as the earth’s diameter, while in almost exactly the same ratio the mean distance of the earth was greater than the sun’s diameter, and the mean distance of the moon from the earth was greater than the moon’s diameter. The agreement was so close that it might have proved more than casual, but its fortuitous character is now sufficiently shown by the fact, that the coincidence ceases to be remarkable when we adopt the amended dimensions of the planetary system.
A considerable number of the elements have atomic weights, which are apparently exact multiples of that of hydrogen. If this be not a law to be ultimately extended to all the elements, as supposed by Prout, it is a most remarkable coincidence. But, as I have observed, we have no means of absolutely discriminating accidental coincidences from those which imply a deep producing cause. A coincidence must either be very strong in itself, or it must be corroborated by some explanation or connection with other laws of nature. Little attention was ever given to the coincidence concerning the dimensions of the sun, earth, and moon, because it was not very strong in itself, and had no apparent connection with the principles of physical astronomy. Prout’s Law bears more probability because it would bring the constitution of the elements themselves in close connection with the atomic theory, representing them as built up out of a simpler substance.
In historical and social matters, coincidences are frequently pointed out which are due to chance, although there is always a strong popular tendency to regard them as the work of design, or as having some hidden meaning. If to 1794, the number of the year in which Robespierre fell, we add the sum of its digits, the result is 1815, the year in which Napoleon fell; the repetition of the process gives 1830 the year in which Charles the Tenth abdicated. Again, the French Chamber of Deputies, in 1830, consisted of 402 members, of whom 221 formed the party called “La queue de Robespierre,” while the remainder, 181 in number, were named “Les honnêtes gens.” If we give to each letter a numerical value corresponding to its place in the alphabet, it will be found that the sum of the values of the letters in each name exactly indicates the number of the party.
A number of such coincidences, often of a very curious character, might be adduced, and the probability against the occurrence of each is enormously great. They must be attributed to chance, because they cannot be shown to have the slightest connection with the general laws of nature; but persons are often found to be greatly influenced by such coincidences, regarding them as evidence of fatality, that is of a system of causation governing human affairs independently of the ordinary laws of nature. Let it be remembered that there are an infinite number of opportunities in life for some strange coincidence to present itself, so that it is quite to be expected that remarkable conjunctions will sometimes happen.
In all matters of judicial evidence, we must bear in mind the probable occurrence from time to time of unaccountable coincidences. The Roman jurists refused for this reason to invalidate a testamentary deed, the witnesses of which had sealed it with the same seal. For witnesses independently using their own seals might be found to possess identical ones by accident.[177] It is well known that circumstantial evidence of apparently overwhelming completeness will sometimes lead to a mistaken judgment, and as absolute certainty is never really attainable, every court must act upon probabilities of a high amount, and in a certain small proportion of cases they must almost of necessity condemn the innocent victims of a remarkable conjuncture of circumstances.[178] Popular judgments usually turn upon probabilities of far less amount, as when the palace of Nicomedia, and even the bedchamber of Diocletian, having been on fire twice within fifteen days, the people entirely refused to believe that it could be the result of accident. The Romans believed that there was fatality connected with the name of Sextus.
“Semper sub Sextis perdita Roma fuit.”
[177] Possunt autem omnes testes et uno annulo signare testamentum Quid enim si septem annuli una sculptura fuerint, secundum quod Pomponio visum est?--*Justinian*, ii. tit. x. 5.
[178] See Wills on *Circumstantial Evidence*, p. 148.
The utmost precautions will not provide against all contingencies. To avoid errors in important calculations, it is usual to have them repeated by different computers; but a case is on record in which three computers made exactly the same calculations of the place of a star, and yet all did it wrong in precisely the same manner, for no apparent reason.[179]
[179] *Memoirs of the Royal Astronomical Society*, vol. iv. p. 290, quoted by Lardner, *Edinburgh Review*, July 1834, p. 278.
*Summary of the Theory of Inductive Inference.*
The theory of inductive inference stated in this and the previous chapters, was suggested by the study of the Inverse Method of Probability, but it also bears much resemblance to the so-called Deductive Method described by Mill, in his celebrated *System of Logic*. Mill’s views concerning the Deductive Method, probably form the most original and valuable part of his treatise, and I should have ascribed the doctrine entirely to him, had I not found that the opinions put forward in other parts of his work are entirely inconsistent with the theory here upheld. As this subject is the most important and difficult one with which we have to deal, I will try to remedy the imperfect manner in which I have treated it, by giving a recapitulation of the views adopted.
All inductive reasoning is but the inverse application of deductive reasoning. Being in possession of certain particular facts or events expressed in propositions, we imagine some more general proposition expressing the existence of a law or cause; and, deducing the particular results of that supposed general proposition, we observe whether they agree with the facts in question. Hypothesis is thus always employed, consciously or unconsciously. The sole conditions to which we need conform in framing any hypothesis is, that we both have and exercise the power of inferring deductively from the hypothesis to the particular results, which are to be compared with the known facts. Thus there are but three steps in the process of induction:--
(1) Framing some hypothesis as to the character of the general law.
(2) Deducing consequences from that law.
(3) Observing whether the consequences agree with the particular facts under consideration.
In very simple cases of inverse reasoning, hypothesis may seem altogether needless. To take numbers again as a convenient illustration, I have only to look at the series,
1, 2, 4, 8, 16, 32, &c.,
to know at once that the general law is that of geometrical progression; I need no successive trial of various hypotheses, because I am familiar with the series, and have long since learnt from what general formula it proceeds. In the same way a mathematician becomes acquainted with the integrals of a number of common formulas, so that he need not go through any process of discovery. But it is none the less true that whenever previous reasoning does not furnish the knowledge, hypotheses must be framed and tried (p. 124).
There naturally arise two cases, according as the nature of the subject admits of certain or only probable deductive reasoning. Certainty, indeed, is but a singular case of probability, and the general principles of procedure are always the same. Nevertheless, when certainty of inference is possible, the process is simplified. Of several mutually inconsistent hypotheses, the results of which can be certainly compared with fact, but one hypothesis can ultimately be entertained. Thus in the inverse logical problem, two logically distinct conditions could not yield the same series of possible combinations. Accordingly, in the case of two terms we had to choose one of six different kinds of propositions (p. 136), and in the case of three terms, our choice lay among 192 possible distinct hypotheses (p. 140). Natural laws, however, are often quantitative in character, and the possible hypotheses are then infinite in variety.
When deduction is certain, comparison with fact is needed only to assure ourselves that we have rightly selected the hypothetical conditions. The law establishes itself, and no number of particular verifications can add to its probability. Having once deduced from the principles of algebra that the difference of the squares of two numbers is equal to the product of their sum and difference, no number of particular trials of its truth will render it more certain. On the other hand, no finite number of particular verifications of a supposed law will render that law certain. In short, certainty belongs only to the deductive process, and to the teachings of direct intuition; and as the conditions of nature are not given by intuition, we can only be certain that we have got a correct hypothesis when, out of a limited number conceivably possible, we select that one which alone agrees with the facts to be explained.
In geometry and kindred branches of mathematics, deductive reasoning is conspicuously certain, and it would often seem as if the consideration of a single diagram yields us certain knowledge of a general proposition. But in reality all this certainty is of a purely hypothetical character. Doubtless if we could ascertain that a supposed circle was a true and perfect circle, we could be certain concerning a multitude of its geometrical properties. But geometrical figures are physical objects, and the senses can never assure us as to their exact forms. The figures really treated in Euclid’s *Elements* are imaginary, and we never can verify in practice the conclusions which we draw with certainty in inference; questions of degree and probability enter.
Passing now to subjects in which deduction is only probable, it ceases to be possible to adopt one hypothesis to the exclusion of the others. We must entertain at the same time all conceivable hypotheses, and regard each with the degree of esteem proportionate to its probability. We go through the same steps as before.
(1) We frame an hypothesis.
(2) We deduce the probability of various series of possible consequences.
(3) We compare the consequences with the particular facts, and observe the probability that such facts would happen under the hypothesis.
The above processes must be performed for every conceivable hypothesis, and then the absolute probability of each will be yielded by the principle of the inverse method (p. 242). As in the case of certainty we accept that hypothesis which certainly gives the required results, so now we accept as most probable that hypothesis which most probably gives the results; but we are obliged to entertain at the same time all other hypotheses with degrees of probability proportionate to the probabilities that they would give the same results.
So far we have treated only of the process by which we pass from special facts to general laws, that inverse application of deduction which constitutes induction. But the direct employment of deduction is often combined with the inverse. No sooner have we established a general law, than the mind rapidly draws particular consequences from it. In geometry we may almost seem to infer that *because* one equilateral triangle is equiangular, therefore another is so. In reality it is not because one is that another is, but because all are. The geometrical conditions are perfectly general, and by what is sometimes called *parity of reasoning* whatever is true of one equilateral triangle, so far as it is equilateral, is true of all equilateral triangles.
Similarly, in all other cases of inductive inference, where we seem to pass from some particular instances to a new instance, we go through the same process. We form an hypothesis as to the logical conditions under which the given instances might occur; we calculate inversely the probability of that hypothesis, and compounding this with the probability that a new instance would proceed from the same conditions, we gain the absolute probability of occurrence of the new instance in virtue of this hypothesis. But as several, or many, or even an infinite number of mutually inconsistent hypotheses may be possible, we must repeat the calculation for each such conceivable hypothesis, and then the complete probability of the future instance will be the sum of the separate probabilities. The complication of this process is often very much reduced in practice, owing to the fact that one hypothesis may be almost certainly true, and other hypotheses, though conceivable, may be so improbable as to be neglected without appreciable error.
When we possess no knowledge whatever of the conditions from which the events proceed, we may be unable to form any probable hypotheses as to their mode of origin. We have now to fall back upon the general solution of the problem effected by Laplace, which consists in admitting on an equal footing every conceivable ratio of favourable and unfavourable chances for the production of the event, and then accepting the aggregate result as the best which can be obtained. This solution is only to be accepted in the absence of all better means, but like other results of the calculus of probability, it comes to our aid where knowledge is at an end and ignorance begins, and it prevents us from over-estimating the knowledge we possess. The general results of the solution are in accordance with common sense, namely, that the more often an event has happened the more probable, as a general rule, is its subsequent recurrence. With the extension of experience this probability increases, but at the same time the probability is slight that events will long continue to happen as they have previously happened.
We have now pursued the theory of inductive inference, as far as can be done with regard to simple logical or numerical relations. The laws of nature deal with time and space, which are infinitely divisible. As we passed from pure logic to numerical logic, so we must now pass from questions of discontinuous, to questions of continuous quantity, encountering fresh considerations of much difficulty. Before, therefore, we consider how the great inductions and generalisations of physical science illustrate the views of inductive reasoning just explained, we must break off for a time, and review the means which we possess of measuring and comparing magnitudes of time, space, mass, force, momentum, energy, and the various manifestations of energy in motion, heat, electricity, chemical change, and the other phenomena of nature.
XIII. THE EXACT MEASUREMENT OF PHENOMENA
As physical science advances, it becomes more and more accurately quantitative. Questions of simple logical fact after a time resolve themselves into questions of degree, time, distance, or weight. Forces hardly suspected to exist by one generation, are clearly recognised by the next, and precisely measured by the third generation. But one condition of this rapid advance is the invention of suitable instruments of measurement. We need what Francis Bacon called *Instantiæ citantes*, or *evocantes*, methods of rendering minute phenomena perceptible to the senses; and we also require *Instantiæ radii* or *curriculi*, that is measuring instruments. Accordingly, the introduction of a new instrument often forms an epoch in the history of science. As Davy said, “Nothing tends so much to the advancement of knowledge as the application of a new instrument. The native intellectual powers of men in different times are not so much the causes of the different success of their labours, as the peculiar nature of the means and artificial resources in their possession.”
In the absence indeed of advanced theory and analytical power, a very precise instrument would be useless. Measuring apparatus and mathematical theory should advance *pari passu*, and with just such precision as the theorist can anticipate results, the experimentalist should be able to compare them with experience. The scrupulously accurate observations of Flamsteed were the proper complement to the intense mathematical powers of Newton.
Every branch of knowledge commences with quantitative notions of a very rude character. After we have far progressed, it is often amusing to look back into the infancy of the science, and contrast present with past methods. At Greenwich Observatory in the present day, the hundredth part of a second is not thought an inconsiderable portion of time. The ancient Chaldæans recorded an eclipse to the nearest hour, and the early Alexandrian astronomers thought it superfluous to distinguish between the edge and centre of the sun. By the introduction of the astrolabe, Ptolemy and the later Alexandrian astronomers could determine the places of the heavenly bodies within about ten minutes of arc. Little progress then ensued for thirteen centuries, until Tycho Brahe made the first great step towards accuracy, not only by employing better instruments, but even more by ceasing to regard an instrument as correct. Tycho, in fact, determined the errors of his instruments, and corrected his observations. He also took notice of the effects of atmospheric refraction, and succeeded in attaining an accuracy often sixty times as great as that of Ptolemy. Yet Tycho and Hevelius often erred several minutes in the determination of a star’s place, and it was a great achievement of Rœmer and Flamsteed to reduce this error to seconds. Bradley, the modern Hipparchus, carried on the improvement, his errors in right ascension, according to Bessel, being under one second of time, and those of declination under four seconds of arc. In the present day the average error of a single observation is probably reduced to the half or quarter of what it was in Bradley’s time; and further extreme accuracy is attained by the multiplication of observations, and their skilful combination according to the theory of error. Some of the more important constants, for instance that of nutation, have been determined within the tenth part of a second of space.[180]
[180] Baily, *British Association Catalogue of Stars*, pp. 7, 23.
It would be a matter of great interest to trace out the dependence of this progress upon the introduction of new instruments. The astrolabe of Ptolemy, the telescope of Galileo, the pendulum of Galileo and Huyghens, the micrometer of Horrocks, and the telescopic sights and micrometer of Gascoygne and Picard, Rœmer’s transit instrument, Newton’s and Hadley’s quadrant, Dollond’s achromatic lenses, Harrison’s chronometer, and Ramsden’s dividing engine--such were some of the principal additions to astronomical apparatus. The result is, that we now take note of quantities, 300,000 or 400,000 times as small as in the time of the Chaldæans.
It would be interesting again to compare the scrupulous accuracy of a modern trigonometrical survey with Eratosthenes’ rude but ingenious guess at the difference of latitude between Alexandria and Syene--or with Norwood’s measurement of a degree of latitude in 1635. “Sometimes I measured, sometimes I paced,” said Norwood; “and I believe I am within a scantling of the truth.” Such was the germ of those elaborate geodesical measurements which have made the dimensions of the globe known to us within a few hundred yards.
In other branches of science, the invention of an instrument has usually marked, if it has not made, an epoch. The science of heat might be said to commence with the construction of the thermometer, and it has recently been advanced by the introduction of the thermo-electric pile. Chemistry has been created chiefly by the careful use of the balance, which forms a unique instance of an instrument remaining substantially in the form in which it was first applied to scientific purposes by Archimedes. The balance never has been and probably never can be improved, except in details of construction. The torsion balance, introduced by Coulomb towards the end of last century, has rapidly become essential in many branches of investigation. In the hands of Cavendish and Baily, it gave a determination of the earth’s density; applied in the galvanometer, it gave a delicate measure of electrical forces, and is indispensable in the thermo-electric pile. This balance is made by simply suspending any light rod by a thin wire or thread attached to the middle point. And we owe to it almost all the more delicate investigations in the theories of heat, electricity, and magnetism.
Though we can now take note of the millionth of an inch in space, and the millionth of a second in time, we must not overlook the fact that in other operations of science we are yet in the position of the Chaldæans. Not many years have elapsed since the magnitudes of the stars, meaning the amounts of light they send to the observer’s eye, were guessed at in the rudest manner, and the astronomer adjudged a star to this or that order of magnitude by a rough comparison with other stars of the same order. To Sir John Herschel we owe an attempt to introduce a uniform method of measurement and expression, bearing some relation to the real photometric magnitudes of the stars.[181] Previous to the researches of Bunsen and Roscoe on the chemical action of light, we were devoid of any mode of measuring the energy of light; even now the methods are tedious, and it is not clear that they give the energy of light so much as one of its special effects. Many natural phenomena have hardly yet been made the subject of measurement at all, such as the intensity of sound, the phenomena of taste and smell, the magnitude of atoms, the temperature of the electric spark or of the sun’s photosphere.
[181] *Outlines of Astronomy*, 4th ed. sect. 781, p. 522. *Results of Observations at the Cape of Good Hope*, &c., p. 37.
To suppose, then, that quantitative science treats only of exactly measurable quantities, is a gross if it be a common mistake. Whenever we are treating of an event which either happens altogether or does not happen at all, we are engaged with a non-quantitative phenomenon, a matter of fact, not of degree; but whenever a thing may be greater or less, or twice or thrice as great as another, whenever, in short, ratio enters even in the rudest manner, there science will have a quantitative character. There can be little doubt, indeed, that every science as it progresses will become gradually more and more quantitative. Numerical precision is the soul of science, as Herschel said, and as all natural objects exist in space, and involve molecular movements, measurable in velocity and extent, there is no apparent limit to the ultimate extension of quantitative science. But the reader must not for a moment suppose that, because we depend more and more upon mathematical methods, we leave logical methods behind us. Number, as I have endeavoured to show, is logical in its origin, and quantity is but a development of number, or analogous thereto.
*Division of the Subject.*
The general subject of quantitative investigation will have to be divided into several parts. We shall firstly consider the means at our disposal for measuring phenomena, and thus rendering them more or less amenable to mathematical treatment. This task will involve an analysis of the principles on which accurate methods of measurement are founded, forming the subject of the remainder of the present chapter. As measurement, however, only yields ratios, we have in the next chapter to consider the establishment of unit magnitudes, in terms of which our results may be expressed. As every phenomenon is usually the sum of several distinct quantities depending upon different causes, we have next to investigate in Chapter XV. the methods by which we may disentangle complicated effects, and refer each part of the joint effect to its separate cause.
It yet remains for us in subsequent chapters to treat of quantitative induction, properly so called. We must follow out the inverse logical method, as it presents itself in problems of a far higher degree of difficulty than those which treat of objects related in a simple logical manner, and incapable of merging into each other by addition and subtraction.
*Continuous Quantity.*
The phenomena of nature are for the most part manifested in quantities which increase or decrease continuously. When we inquire into the precise meaning of continuous quantity, we find that it can only be described as that which is divisible without limit. We can divide a millimetre into ten, or a hundred, or a thousand, or ten thousand parts, and mentally at any rate we can carry on the division *ad infinitum*. Any finite space, then, must be conceived as made up of an infinite number of parts each infinitely small. We cannot entertain the simplest geometrical notions without allowing this. The conception of a square involves the conception of a side and diagonal, which, as Euclid beautifully proves in the 117th proposition of his tenth book, have no common measure,[182] meaning no finite common measure. Incommensurable quantities are, in fact, those which have for their only common measure an infinitely small quantity. It is somewhat startling to find, too, that in theory incommensurable quantities will be infinitely more frequent than commensurable. Let any two lines be drawn haphazard; it is infinitely unlikely that they will be commensurable, so that the commensurable quantities, which we are supposed to deal with in practice, are but singular cases among an infinitely greater number of incommensurable cases.
[182] See De Morgan, *Study of Mathematics*, in U.K.S. Library, p. 81.
Practically, however, we treat all quantities as made up of the least quantities which our senses, assisted by the best measuring instruments, can perceive. So long as microscopes were uninvented, it was sufficient to regard an inch as made up of a thousand thousandths of an inch; now we must treat it as composed of a million millionths. We might apparently avoid all mention of infinitely small quantities, by never carrying our approximations beyond quantities which the senses can appreciate. In geometry, as thus treated, we should never assert two quantities to be equal, but only to be *apparently* equal. Legendre really adopts this mode of treatment in the twentieth proposition of the first book of his Geometry; and it is practically adopted throughout the physical sciences, as we shall afterwards see. But though our fingers, and senses, and instruments must stop somewhere, there is no reason why the mind should not go on. We can see that a proof which is only carried through a few steps in fact, might be carried on without limit, and it is this consciousness of no stopping-place, which renders Euclid’s proof of his 117th proposition so impressive. Try how we will to circumvent the matter, we cannot really avoid the consideration of the infinitely small and the infinitely great. The same methods of approximation which seem confined to the finite, mentally extend themselves to the infinite.
One result of these considerations is, that we cannot possibly adjust two quantities in absolute equality. The suspension of Mahomet’s coffin between two precisely equal magnets is theoretically conceivable but practically impossible. The story of the *Merchant of Venice* turns upon the infinite improbability that an exact quantity of flesh could be cut. Unstable equilibrium cannot exist in nature, for it is that which is destroyed by an infinitely small displacement. It might be possible to balance an egg on its end practically, because no egg has a surface of perfect curvature. Suppose the egg shell to be perfectly smooth, and the feat would become impossible.
*The Fallacious Indications of the Senses.*
I may briefly remind the reader how little we can trust to our unassisted senses in estimating the degree or magnitude of any phenomenon. The eye cannot correctly estimate the comparative brightness of two luminous bodies which differ much in brilliancy; for we know that the iris is constantly adjusting itself to the intensity of the light received, and thus admits more or less light according to circumstances. The moon which shines with almost dazzling brightness by night, is pale and nearly imperceptible while the eye is yet affected by the vastly more powerful light of day. Much has been recorded concerning the comparative brightness of the zodiacal light at different times, but it would be difficult to prove that these changes are not due to the varying darkness at the time, or the different acuteness of the observer’s eye. For a like reason it is exceedingly difficult to establish the existence of any change in the form or comparative brightness of nebulæ; the appearance of a nebula greatly depends upon the keenness of sight of the observer, or the accidental condition of freshness or fatigue of his eye. The same is true of lunar observations; and even the use of the best telescope fails to remove this difficulty. In judging of colours, again, we must remember that light of any given colour tends to dull the sensibility of the eye for light of the same colour.
Nor is the eye when unassisted by instruments a much better judge of magnitude. Our estimates of the size of minute bright points, such as the fixed stars, are completely falsified by the effects of irradiation. Tycho calculated from the apparent size of the star-discs, that no one of the principal fixed stars could be contained within the area of the earth’s orbit. Apart, however, from irradiation or other distinct causes of error our visual estimates of sizes and shapes are often astonishingly incorrect. Artists almost invariably draw distant mountains in ludicrous disproportion to nearer objects, as a comparison of a sketch with a photograph at once shows. The extraordinary apparent difference of size of the sun or moon, according as it is high in the heavens or near the horizon, should be sufficient to make us cautious in accepting the plainest indications of our senses, unassisted by instrumental measurement. As to statements concerning the height of the aurora and the distance of meteors, they are to be utterly distrusted. When Captain Parry says that a ray of the aurora shot suddenly downwards between him and the land which was only 3,000 yards distant, we must consider him subject to an illusion of sense.[183]
[183] Loomis, *On the Aurora Borealis*. Smithsonian Transactions, quoting Parry’s Third Voyage, p. 61.
It is true that errors of observation are more often errors of judgment than of sense. That which is actually seen must be so far truly seen; and if we correctly interpret the meaning of the phenomenon, there would be no error at all. But the weakness of the bare senses as measuring instruments, arises from the fact that they import varying conditions of unknown amount, and we cannot make the requisite corrections and allowances as in the case of a solid and invariable instrument.
Bacon has excellently stated the insufficiency of the senses for estimating the magnitudes of objects, or detecting the degrees in which phenomena present themselves. “Things escape the senses,” he says, “because the object is not sufficient in quantity to strike the sense: as all minute bodies; because the percussion of the object is too great to be endured by the senses: as the form of the sun when looking directly at it in mid-day; because the time is not proportionate to actuate the sense: as the motion of a bullet in the air, or the quick circular motion of a firebrand, which are too fast, or the hour-hand of a common clock, which is too slow; from the distance of the object as to place: as the size of the celestial bodies, and the size and nature of all distant bodies; from prepossession by another object: as one powerful smell renders other smells in the same room imperceptible; from the interruption of interposing bodies: as the internal parts of animals; and because the object is unfit to make an impression upon the sense: as the air or the invisible and untangible spirit which is included in every living body.”
*Complexity of Quantitative Questions.*
One remark which we may well make in entering upon quantitative questions, has regard to the great variety and extent of phenomena presented to our notice. So long as we deal only with a simply logical question, that question is merely, Does a certain event happen? or, Does a certain object exist? No sooner do we regard the event or object as capable of more and less, than the question branches out into many. We must now ask, How much is it compared with its cause? Does it change when the amount of the cause changes? If so, does it change in the same or opposite direction? Is the change in simple proportion to that of the cause? If not, what more complex law of connection holds true? This law determined satisfactorily in one series of circumstances may be varied under new conditions, and the most complex relations of several quantities may ultimately be established.
In every question of physical science there is thus a series of steps the first one or two of which are usually made with ease while the succeeding ones demand more and more careful measurement. We cannot lay down any invariable series of questions which must be asked from nature. The exact character of the questions will vary according to the nature of the case, but they will usually be of an evident kind, and we may readily illustrate them by examples. Suppose that we are investigating the solution of some salt in water. The first is a purely logical question: Is there solution, or is there not? Assuming the answer to be in the affirmative, we next inquire, Does the solubility vary with the temperature, or not? In all probability some variation will exist, and we must have an answer to the further question, Does the quantity dissolved increase, or does it diminish with the temperature? In by far the greatest number of cases salts and substances of all kinds dissolve more freely the higher the temperature of the water; but there are a few salts, such as calcium sulphate, which follow the opposite rule. A considerable number of salts resemble sodium sulphate in becoming more soluble up to a certain temperature, and then varying in the opposite direction. We next require to assign the amount of variation as compared with that of the temperature, assuming at first that the increase of solubility is proportional to the increase of temperature. Common salt is an instance of very slight variation, and potassium nitrate of very considerable increase with temperature. Accurate observations will probably show, however, that the simple law of proportionate variation is only approximately true, and some more complicated law involving the second, third, or higher powers of the temperature may ultimately be established. All these investigations have to be carried out for each salt separately, since no distinct principles by which we may infer from one substance to another have yet been detected. There is still an indefinite field for further research open; for the solubility of salts will probably vary with the pressure under which the medium is placed; the presence of other salts already dissolved may have effects yet unknown. The researches already effected as regards the solvent power of water must be repeated with alcohol, ether, carbon bisulphide, and other media, so that unless general laws can be detected, this one phenomenon of solution can never be exhaustively treated. The same kind of questions recur as regards the solution or absorption of gases in liquids, the pressure as well as the temperature having then a most decided effect, and Professor Roscoe’s researches on the subject present an excellent example of the successive determination of various complicated laws.[184]
[184] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790.
There is hardly a branch of physical science in which similar complications are not ultimately encountered. In the case of gravity, indeed, we arrive at the final law, that the force is the same for all kinds of matter, and varies only with the distance of action. But in other subjects the laws, if simple in their ultimate nature, are disguised and complicated in their apparent results. Thus the effect of heat in expanding solids, and the reverse effect of forcible extension or compression upon the temperature of a body, will vary from one substance to another, will vary as the temperature is already higher or lower, and, will probably follow a highly complex law, which in some cases gives negative or exceptional results. In crystalline substances the same researches have to be repeated in each distinct axial direction.
In the sciences of pure observation, such as those of astronomy, meteorology, and terrestrial magnetism, we meet with many interesting series of quantitative determinations. The so-called fixed stars, as Giordano Bruno divined, are not really fixed, and may be more truly described as vast wandering orbs, each pursuing its own path through space. We must then determine separately for each star the following questions:--
1. Does it move?
2. In what direction?
3. At what velocity?
4. Is this velocity variable or uniform?
5. If variable, according to what law?
6. Is the direction uniform?
7. If not, what is the form of the apparent path?
8. Does it approach or recede? 9. What is the form of the real path?
The successive answers to such questions in the case of certain binary stars, have afforded a proof that the motions are due to a central force coinciding in law with gravity, and doubtless identical with it. In other cases the motions are usually so small that it is exceedingly difficult to distinguish them with certainty. And the time is yet far off when any general results as regards stellar motions can be established.
The variation in the brightness of stars opens an unlimited field for curious observation. There is not a star in the heavens concerning which we might not have to determine:--
1. Does it vary in brightness?
2. Is the brightness increasing or decreasing?
3. Is the variation uniform?
4. If not, according to what law does it vary?
In a majority of cases the change will probably be found to have a periodic character, in which case several other questions will arise, such as--
5. What is the length of the period?
6. Are there minor periods?
7. What is the law of variation within the period?
8. Is there any change in the amount of variation?
9. If so, is it a secular, *i.e.* a continually growing change, or does it give evidence of a greater period?
Already the periodic changes of a certain number of stars have been determined with accuracy, and the lengths of the periods vary from less than three days up to intervals of time at least 250 times as great. Periods within periods have also been detected.
There is, perhaps, no subject in which more complicated quantitative conditions have to be determined than terrestrial magnetism. Since the time when the declination of the compass was first noticed, as some suppose by Columbus, we have had successive discoveries from time to time of the progressive change of declination from century to century; of the periodic character of this change; of the difference of the declination in various parts of the earth’s surface; of the varying laws of the change of declination; of the dip or inclination of the needle, and the corresponding laws of its periodic changes; the horizontal and perpendicular intensities have also been the subject of exact measurement, and have been found to vary with place and time, like the directions of the needle; daily and yearly periodic changes have also been detected, and all the elements are found to be subject to occasional storms or abnormal perturbations, in which the eleven year period, now known to be common to many planetary relations, is apparent. The complete solution of these motions of the compass needle involves nothing less than a determination of its position and oscillations in every part of the world at any epoch, the like determination for another epoch, and so on, time after time, until the periods of all changes are ascertained. This one subject offers to men of science an almost inexhaustible field for interesting quantitative research, in which we shall doubtless at some future time discover the operation of causes now most mysterious and unaccountable.
*The Methods of Accurate Measurement.*
In studying the modes by which physicists have accomplished very exact measurements, we find that they are very various, but that they may perhaps be reduced under the following three classes:--
1. The increase or decrease, in some determinate ratio, of the quantity to be measured, so as to bring it within the scope of our senses, and to equate it with the standard unit, or some determinate multiple or sub-multiple of this unit.
2. The discovery of some natural conjunction of events which will enable us to compare directly the multiples of the quantity with those of the unit, or a quantity related in a definite ratio to that unit.
3. Indirect measurement, which gives us not the quantity itself, but some other quantity connected with it by known mathematical relations.
*Conditions of Accurate Measurement.*
Several conditions are requisite in order that a measurement may be made with great accuracy, and that the results may be closely accordant when several independent measurements are made.
In the first place the magnitude must be exactly defined by sharp terminations, or precise marks of inconsiderable thickness. When a boundary is vague and graduated, like the penumbra in a lunar eclipse, it is impossible to say where the end really is, and different people will come to different results. We may sometimes overcome this difficulty to a certain extent, by observations repeated in a special manner, as we shall afterwards see; but when possible, we should choose opportunities for measurement when precise definition is easy. The moment of occultation of a star by the moon can be observed with great accuracy, because the star disappears with perfect suddenness; but there are other astronomical conjunctions, eclipses, transits, &c., which occupy a certain length of time in happening, and thus open the way to differences of opinion. It would be impossible to observe with precision the movements of a body possessing no definite points of reference. The colours of the complete spectrum shade into each other so continuously that exact determinations of refractive indices would have been impossible, had we not the dark lines of the solar spectrum as precise points for measurement, or various kinds of homogeneous light, such as that of sodium, possessing a nearly uniform length of vibration.
In the second place, we cannot measure accurately unless we have the means of multiplying or dividing a quantity without considerable error, so that we may correctly equate one magnitude with the multiple or submultiple of the other. In some cases we operate upon the quantity to be measured, and bring it into accurate coincidence with the actual standard, as when in photometry we vary the distance of our luminous body, until its illuminating power at a certain point is equal to that of a standard lamp. In other cases we repeat the unit until it equals the object, as in surveying land, or determining a weight by the balance. The requisites of accuracy now are:--(1) That we can repeat unit after unit of exactly equal magnitude; (2) That these can be joined together so that the aggregate shall really be the sum of the parts. The same conditions apply to subdivision, which may be regarded as a multiplication of subordinate units. In order to measure to the thousandth of an inch, we must be able to add thousandth after thousandth without error in the magnitude of these spaces, or in their conjunction.
*Measuring Instruments.*
To consider the mechanical construction of scientific instruments, is no part of my purpose in this book. I wish to point out merely the general purpose of such instruments, and the methods adopted to carry out that purpose with great precision. In the first place we must distinguish between the instrument which effects a comparison between two quantities, and the standard magnitude which often forms one of the quantities compared. The astronomer’s clock, for instance, is no standard of the efflux of time; it serves but to subdivide, with approximate accuracy, the interval of successive passages of a star across the meridian, which it may effect perhaps to the tenth part of a second, or 1/864000 part of the whole. The moving globe itself is the real standard clock, and the transit instrument the finger of the clock, while the stars are the hour, minute, and second marks, none the less accurate because they are disposed at unequal intervals. The photometer is a simple instrument, by which we compare the relative intensity of rays of light falling upon a given spot. The galvanometer shows the comparative intensity of electric currents passing through a wire. The calorimeter gauges the quantity of heat passing from a given object. But no such instruments furnish the standard unit in terms of which our results are to be expressed. In one peculiar case alone does the same instrument combine the unit of measurement and the means of comparison. A theodolite, mural circle, sextant, or other instrument for the measurement of angular magnitudes has no need of an additional physical unit; for the circle itself, or complete revolution, is the natural unit to which all greater or lesser amounts of angular magnitude are referred.
The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. In every measurement we equate some multiple or submultiple of one quantity, with some multiple or submultiple of another, and equality is always the fact which we ascertain by the senses. By the eye, the ear, or the touch, we judge whether there is a discrepancy or not between two lights, two sounds, two intervals of time, two bars of metal. Often indeed we substitute one sense for the other, as when the efflux of time is judged by the marks upon a moving slip of paper, so that equal intervals of time are represented by equal lengths. There is a tendency to reduce all comparisons to the comparison of space magnitudes, but in every case one of the senses must be the ultimate judge of coincidence or non-coincidence.
Since the equation to be established may exist between any multiples or submultiples of the quantities compared, there naturally arise several different modes of comparison adapted to different cases. Let *p* be the magnitude to be measured, and *q* that in terms of which it is to be expressed. Then we wish to find such numbers *x* and *y*, that the equation *p = (x/y)q* may be true. This equation may be presented in four forms, namely:--
First Form. Second Form. Third Form. Fourth Form. *p = (x/y)q* *p(y/x) = q* *py = qx* *p/x = q/y*
Each of these modes of expressing the same equation corresponds to one mode of effecting a measurement.
When the standard quantity is greater than that to be measured, we often adopt the first mode, and subdivide the unit until we get a magnitude equal to that measured. The angles observed in surveying, in astronomy, or in goniometry are usually smaller than a whole revolution, and the measuring circle is divided by the use of the screw and microscope, until we obtain an angle undistinguishable from that observed. The dimensions of minute objects are determined by subdividing the inch or centimetre, the screw micrometer being the most accurate means of subdivision. Ordinary temperatures are estimated by division of the standard interval between the freezing and boiling points of water, as marked on a thermometer tube.
In a still greater number of cases, perhaps, we multiply the standard unit until we get a magnitude equal to that to be measured. Ordinary measurement by a foot rule, a surveyor’s chain, or the excessively careful measurements of the base line of a trigonometrical survey by standard bars, are sufficient instances of this procedure.
In the second case, where *p(y/x) = q*, we multiply or divide a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it. As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiplying them without introducing error. Thus the expansion of a metallic bar when heated from 0°C to 100° may be multiplied by a train of levers or cog wheels. In the common thermometer the expansion of the mercury, though slight, is rendered very apparent, and easily measurable by the fineness of the tube, and many other cases might be quoted. There are some phenomena, on the contrary, which are too great or rapid to come within the easy range of our senses, and our task is then the opposite one of diminution. Galileo found it difficult to measure the velocity of a falling body, owing to the considerable velocity acquired in a single second. He adopted the elegant device, therefore, of lessening the rapidity by letting the body roll down an inclined plane, which enables us to reduce the accelerating force in any required ratio. The same purpose is effected in the well-known experiments performed on Attwood’s machine, and the measurement of gravity by the pendulum really depends on the same principle applied in a far more advantageous manner. Wheatstone invented a beautiful method of galvanometry for strong currents, which consists in drawing off from the main current a certain determinate portion, which is equated by the galvanometer to a standard current. In short, he measures not the current itself but a known fraction of it.
In many electrical and other experiments, we wish to measure the movements of a needle or other body, which are not only very slight in themselves, but the manifestations of exceedingly small forces. We cannot even approach a delicately balanced needle without disturbing it. Under these circumstances the only mode of proceeding with accuracy, is to attach a very small mirror to the moving body, and employ a ray of light reflected from the mirror as an index of its movements. The ray may be considered quite incapable of affecting the body, and yet by allowing the ray to pass to a sufficient distance, the motions of the mirror may be increased to almost any extent. A ray of light is in fact a perfectly weightless finger or index of indefinite length, with the additional advantage that the angular deviation is by the law of reflection double that of the mirror. This method was introduced by Gauss, and is now of great importance; but in Wollaston’s reflecting goniometer a ray of light had previously been employed as an index. Lavoisier and Laplace had also used a telescope in connection with the pyrometer.
It is a great advantage in some instruments that they can be readily made to manifest a phenomenon in a greater or less degree, by a very slight change in the construction. Thus either by enlarging the bulb or contracting the tube of the thermometer, we can make it give more conspicuous indications of change of temperature. The ordinary barometer, on the other hand, always gives the variations of pressure on one scale. The torsion balance is remarkable for the extreme delicacy which may be attained by increasing the length and lightness of the rod, and the length and thinness of the supporting thread. Forces so minute as the attraction of gravitation between two balls, or the magnetic and diamagnetic attraction of common liquids and gases, may thus be made apparent, and even measured. The common chemical balance, too, is capable theoretically of unlimited sensibility.
The third mode of measurement, which may be called the Method of Repetition, is of such great importance and interest that we must consider it in a separate section. It consists in multiplying both magnitudes to be compared until some multiple of the first is found to coincide very nearly with some multiple of the second. If the multiplication can be effected to an unlimited extent, without the introduction of countervailing errors, the accuracy with which the required ratio can be determined is unlimited, and we thus account for the extraordinary precision with which intervals of time in astronomy are compared together.
The fourth mode of measurement, in which we equate submultiples of two magnitudes, is comparatively seldom employed, because it does not conduce to accuracy. In the photometer, perhaps, we may be said to use it; we compare the intensity of two sources of light, by placing them both at such distances from a given surface, that the light falling on the surface is tolerable to the eye, and equally intense from each source. Since the intensity of light varies inversely as the square of the distance, the relative intensities of the luminous bodies are proportional to the squares of their distances. The equal intensity of two rays of similarly coloured light may be most accurately ascertained in the mode suggested by Arago, namely, by causing the rays to pass in opposite directions through two nearly flat lenses pressed together. There is an exact equation between the intensities of the beams when Newton’s rings disappear, the ring created by one ray being exactly the complement of that created by the other.
*The Method of Repetition.*
The ratio of two quantities can be determined with unlimited accuracy, if we can multiply both the object of measurement and the standard unit without error, and then observe what multiple of the one coincides or nearly coincides with some multiple of the other. Although perfect coincidence can never be really attained, the error thus arising may be indefinitely reduced. For if the equation *py* = *qx* be uncertain to the amount *e*, so that *py* = *qx* ± *e*, then we have *p* = *q(x/y)* ± *e/y* , and as we are supposed to be able to make *y* as great as we like without increasing the error *e*, it follows that we can make *e* ÷ *y* as small as we like, and thus approximate within an inconsiderable quantity to the required ratio *x* ÷ *y*.
This method of repetition is naturally employed whenever quantities can be repeated, or repeat themselves without error of juxtaposition, which is especially the case with the motions of the earth and heavenly bodies. In determining the length of the sidereal day, we determine the ratio between the earth’s revolution round the sun, and its rotation on its own axis. We might ascertain the ratio by observing the successive passages of a star across the zenith, and comparing the interval by a good clock with that between two passages of the sun, the difference being due to the angular movement of the earth round the sun. In such observations we should have an error of a considerable part of a second at each observation, in addition to the irregularities of the clock. But the revolutions of the earth repeat themselves day after day, and year after year, without the slightest interval between the end of one period and the beginning of another. The operation of multiplication is perfectly performed for us by nature. If, then, we can find an observation of the passage of a star across the meridian a hundred years ago, that is of the interval of time between the passage of the sun and the star, the instrumental errors in measuring this interval by a clock and telescope may be greater than in the present day, but will be divided by about 36,524 days, and rendered excessively small. It is thus that astronomers have been able to ascertain the ratio of the mean solar to the sidereal day to the 8th place of decimals (1·00273791 to 1), or to the hundred millionth part, probably the most accurate result of measurement in the whole range of science.
The antiquity of this mode of comparison is almost as great as that of astronomy itself. Hipparchus made the first clear application of it, when he compared his own observations with those of Aristarchus, made 145 years previously, and thus ascertained the length of the year. This calculation may in fact be regarded as the earliest attempt at an exact determination of the constants of nature. The method is the main resource of astronomers; Tycho, for instance, detected the slow diminution of the obliquity of the earth’s axis, by the comparison of observations at long intervals. Living astronomers use the method as much as earlier ones; but so superior in accuracy are all observations taken during the last hundred years to all previous ones, that it is often found preferable to take a shorter interval, rather than incur the risk of greater instrumental errors in the earlier observations.
It is obvious that many of the slower changes of the heavenly bodies must require the lapse of large intervals of time to render their amount perceptible. Hipparchus could not possibly have discovered the smaller inequalities of the heavenly motions, because there were no previous observations of sufficient age or exactness to exhibit them. And just as the observations of Hipparchus formed the starting-point for subsequent comparisons, so a large part of the labour of present astronomers is directed to recording the present state of the heavens so exactly, that future generations of astronomers may detect changes, which cannot possibly become known in the present age.
The principle of repetition was very ingeniously employed in an instrument first proposed by Mayer in 1767, and carried into practice in the Repeating Circle of Borda. The exact measurement of angles is indispensable, not only in astronomy but also in trigonometrical surveys, and the highest skill in the mechanical execution of the graduated circle and telescope will not prevent terminal errors of considerable amount. If instead of one telescope, the circle be provided with two similar telescopes, these may be alternately directed to two distant points, say the marks in a trigonometrical survey, so that the circle shall be turned through any multiple of the angle subtended by those marks, before the amount of the angular revolution is read off upon the graduated circle. Theoretically speaking, all error arising from imperfect graduation might thus be indefinitely reduced, being divided by the number of repetitions. In practice, the advantage of the invention is not found to be very great, probably because a certain error is introduced at each observation in the changing and fixing of the telescopes. It is moreover inapplicable to moving objects like the heavenly bodies, so that its use is confined to important trigonometrical surveys.
The pendulum is the most perfect of all instruments, chiefly because it admits of almost endless repetition. Since the force of gravity never ceases, one swing of the pendulum is no sooner ended than the other is begun, so that the juxtaposition of successive units is absolutely perfect. Provided that the oscillations be equal, one thousand oscillations will occupy exactly one thousand times as great an interval of time as one oscillation. Not only is the subdivision of time entirely dependent on this fact, but in the accurate measurement of gravity, and many other important determinations, it is of the greatest service. In the deepest mine, we could not observe the rapidity of fall of a body for more than a quarter of a minute, and the measurement of its velocity would be difficult, and subject to uncertain errors from resistance of air, &c. In the pendulum, we have a body which can be kept rising and falling for many hours, in a medium entirely under our command or if desirable in a vacuum. Moreover, the comparative force of gravity at different points, at the top and bottom of a mine for instance, can be determined with wonderful precision, by comparing the oscillations of two exactly similar pendulums, with the aid of electric clock signals.
To ascertain the comparative times of vibration of two pendulums, it is only requisite to swing them one in front of the other, to record by a clock the moment when they coincide in swing, so that one hides the other, and then count the number of vibrations until they again come to coincidence. If one pendulum makes *m* vibrations and the other *n*, we at once have our equation *pn* = *qm*; which gives the length of vibration of either pendulum in terms of the other. This method of coincidence, embodying the principle of repetition in perfection, was employed with wonderful skill by Sir George Airy, in his experiments on the Density of the Earth at the Harton Colliery, the pendulums above and below being compared with clocks, which again were compared with each other by electric signals. So exceedingly accurate was this method of observation, as carried out by Sir George Airy, that he was able to measure a total difference in the vibrations at the top and bottom of the shaft, amounting to only 2·24 seconds in the twenty-four hours, with an error of less than one hundredth part of a second, or one part in 8,640,000 of the whole day.[185]
[185] *Philosophical Transactions*, (1856) vol. 146, Part i. p. 297.
The principle of repetition has been elegantly applied in observing the motion of waves in water. If the canal in which the experiments are made be short, say twenty feet long, the waves will pass through it so rapidly that an observation of one length, as practised by Walker, will be subject to much terminal error, even when the observer is very skilful. But it is a result of the undulatory theory that a wave is unaltered, and loses no time by complete reflection, so that it may be allowed to travel backwards and forwards in the same canal, and its motion, say through sixty lengths, or 1200 feet, may be observed with the same accuracy as in a canal 1200 feet long, with the advantage of greater uniformity in the condition of the canal and water.[186] It is always desirable, if possible, to bring an experiment into a small compass, so that it may be well under command, and yet we may often by repetition enjoy at the same time the advantage of extensive trial.
[186] Airy, *On Tides and Waves*, Encyclopædia Metropolitana, p. 345. Scott Russell, *British Association Report*, 1837, p. 432.
One reason of the great accuracy of weighing with a good balance is the fact, that weights placed in the same scale are naturally added together without the slightest error. There is no difficulty in the precise juxtaposition of two grams, but the juxtaposition of two metre measures can only be effected with tolerable accuracy, by the use of microscopes and many precautions. Hence, the extreme trouble and cost attaching to the exact measurement of a base line for a survey, the risk of error entering at every juxtaposition of the measuring bars, and indefatigable attention to all the requisite precautions being necessary throughout the operation.
*Measurements by Natural Coincidence.*
In certain cases a peculiar conjunction of circumstances enables us to dispense more or less with instrumental aids, and to obtain very exact numerical results in the simplest manner. The mere fact, for instance, that no human being has ever seen a different face of the moon from that familiar to us, conclusively proves that the period of rotation of the moon on its own axis is equal to that of its revolution round the earth. Not only have we the repetition of these movements during 1000 or 2000 years at least, but we have observations made for us at very remote periods, free from instrumental error, no instrument being needed. We learn that the seventh satellite of Saturn is subject to a similar law, because its light undergoes a variation in each revolution, owing to the existence of some dark tract of land; now this failure of light always occurs while it is in the same position relative to Saturn, clearly proving the equality of the axial and revolutional periods, as Huygens perceived.[187] A like peculiarity in the motions of Jupiter’s fourth satellite was similarly detected by Maraldi in 1713.
[187] *Hugenii Cosmotheoros*, pp. 117, 118. Laplace’s *Système*, translated, vol. i. p. 67.
Remarkable conjunctions of the planets may sometimes allow us to compare their periods of revolution, through great intervals of time, with much accuracy. Laplace in explaining the long inequality in the motions of Jupiter and Saturn, was assisted by a conjunction of these planets, observed at Cairo, towards the close of the eleventh century. Laplace calculated that such a conjunction must have happened on the 31st of October, A.D. 1087; and the discordance between the distances of the planets as recorded, and as assigned by theory, was less than one-fifth part of the apparent diameter of the sun. This difference being less than the probable error of the early record, the theory was confirmed as far as facts were available.[188]
[188] Grant’s *History of Physical Astronomy*, p. 129.
Ancient astronomers often showed the highest ingenuity in turning any opportunities of measurement which occurred to good account. Eratosthenes, as early as 250 B.C., happening to hear that the sun at Syene, in Upper Egypt, was visible at the summer solstice at the bottom of a well, proving that it was in the zenith, proposed to determine the dimensions of the earth, by measuring the length of the shadow of a rod at Alexandria on the same day of the year. He thus learnt in a rude manner the difference of latitude between Alexandria and Syene and finding it to be about one fiftieth part of the whole circumference, he ascertained the dimensions of the earth within about one sixth part of the truth. The use of wells in astronomical observation appears to have been occasionally practised in comparatively recent times as by Flamsteed in 1679.[189] The Alexandrian astronomers employed the moon as an instrument of measurement in several sagacious modes. When the moon is exactly half full, the moon, sun, and earth, are at the angles of a right-angled triangle. Aristarchus measured at such a time the moon’s elongation from the sun, which gave him the two other angles of the triangle, and enabled him to judge of the comparative distances of the moon and sun from the earth. His result, though very rude, was far more accurate than any notions previously entertained, and enabled him to form some estimate of the comparative magnitudes of the bodies. Eclipses of the moon were very useful to Hipparchus in ascertaining the longitude of the stars, which are invisible when the sun is above the horizon. For the moon when eclipsed must be 180° distant from the sun; hence it is only requisite to measure the distance of a fixed star in longitude from the eclipsed moon to obtain with ease its angular distance from the sun.
[189] Baily’s *Account of Flamsteed*, p. lix.
In later times the eclipses of Jupiter have served to measure an angle; for at the middle moment of the eclipse the satellite must be in the same straight line with the planet and sun, so that we can learn from the known laws of movement of the satellite the longitude of Jupiter as seen from the sun. If at the same time we measure the elongation or apparent angular distance of Jupiter from the sun, as seen from the earth, we have all the angles of the triangle between Jupiter, the sun, and the earth, and can calculate the comparative magnitudes of the sides of the triangle by trigonometry.
The transits of Venus over the sun’s face are other natural events which give most accurate measurements of the sun’s parallax, or apparent difference of position as seen from distant points of the earth’s surface. The sun forms a kind of background on which the place of the planet is marked, and serves as a measuring instrument free from all the errors of construction which affect human instruments. The rotation of the earth, too, by variously affecting the apparent velocity of ingress or egress of Venus, as seen from different places, discloses the amount of the parallax. It has been sufficiently shown that by rightly choosing the moments of observation, the planetary bodies may often be made to reveal their relative distance, to measure their own position, to record their own movements with a high degree of accuracy. With the improvement of astronomical instruments, such conjunctions become less necessary to the progress of the science, but it will always remain advantageous to choose those moments for observation when instrumental errors enter with the least effect.
In other sciences, exact quantitative laws can occasionally be obtained without instrumental measurement, as when we learn the exactly equal velocity of sounds of different pitch, by observing that a peal of bells or a musical performance is heard harmoniously at any distance to which the sound penetrates; this could not be the case, as Newton remarked, if one sound overtook the other. One of the most important principles of the atomic theory, was proved by implication before the use of the balance was introduced into chemistry. Wenzel observed, before 1777, that when two neutral substances decompose each other, the resulting salts are also neutral. In mixing sodium sulphate and barium nitrate, we obtain insoluble barium sulphate and neutral sodium nitrate. This result could not follow unless the nitric acid, requisite to saturate one atom of sodium, were exactly equal to that required by one atom of barium, so that an exchange could take place without leaving either acid or base in excess.
An important principle of mechanics may also be established by a simple acoustical observation. When a rod or tongue of metal fixed at one end is set in vibration, the pitch of the sound may be observed to be exactly the same, whether the vibrations be small or great; hence the oscillations are isochronous, or equally rapid, independently of their magnitude. On the ground of theory, it can be shown that such a result only happens when the flexure is proportional to the deflecting force. Thus the simple observation that the pitch of the sound of a harmonium, for instance, does not change with its loudness establishes an exact law of nature.[190]
[190] Jamin, *Cours de Physique*, vol. i. p. 152.
A closely similar instance is found in the proof that the intensity of light or heat rays varies inversely as the square of the distance increases. For the apparent magnitude certainly varies according to this law; hence, if the intensity of light varied according to any other law, the brightness of an object would be different at different distances, which is not observed to be the case. Melloni applied the same kind of reasoning, in a somewhat different form, to the radiation of heat-rays.
*Modes of Indirect Measurement.*
Some of the most conspicuously beautiful experiments in the whole range of science, have been devised for the purpose of indirectly measuring quantities, which in their extreme greatness or smallness surpass the powers of sense. All that we need to do, is to discover some other conveniently measurable phenomenon, which is related in a known ratio or according to a known law, however complicated, with that to be measured. Having once obtained experimental data, there is no further difficulty beyond that of arithmetic or algebraic calculation.
Gold is reduced by the gold-beater to leaves so thin, that the most powerful microscope would not detect any measurable thickness. If we laid several hundred leaves upon each other to multiply the thickness, we should still have no more than 1/100th of an inch at the most to measure, and the errors arising in the superposition and measurement would be considerable. But we can readily obtain an exact result through the connected amount of weight. Faraday weighed 2000 leaves of gold, each 3-3/8 inch square, and found them equal to 384 grains. From the known specific gravity of gold it was easy to calculate that the average thickness of the leaves was 1/282,000 of an inch.[191]
[191] Faraday, *Chemical Researches*, p. 393.
We must ascribe to Newton the honour of leading the way in methods of minute measurement. He did not call waves of light by their right name, and did not understand their nature; yet he measured their length, though it did not exceed the 2,000,000th part of a metre or the one fifty-thousandth part of an inch. He pressed together two lenses of large but known radii. It was easy to calculate the interval between the lenses at any point, by measuring the distance from the central point of contact. Now, with homogeneous rays the successive rings of light and darkness mark the points at which the interval between the lenses is equal to one half, or any multiple of half a vibration of the light, so that the length of the vibration became known. In a similar manner many phenomena of interference of rays of light admit of the measurement of the wave lengths. Fringes of interference arise from rays of light which cross each other at a small angle, and an excessively minute difference in the lengths of the waves makes a very perceptible difference in the position of the point at which two rays will interfere and produce darkness.
Fizeau has recently employed Newton’s rings to measure small amounts of motion. By merely counting the number of rings of sodium monochromatic light passing a certain point where two glass plates are in close proximity, he is able to ascertain with the greatest accuracy and ease the change of distance between these glasses, produced, for instance, by the expansion of a metallic bar, connected with one of the glass plates.[192]
[192] *Proceedings of the Royal Society*, 30th November, 1866.
Nothing excites more admiration than the mode in which scientific observers can occasionally measure quantities, which seem beyond the bounds of human observation. We know the *average* depth of the Pacific Ocean to be 14,190 feet, not by actual sounding, which would be impracticable in sufficient detail, but by noticing the rate of transmission of earthquake waves from the South American to the opposite coasts, the rate of movement being connected by theory with the depth of the water.[193] In the same way the average depth of the Atlantic Ocean is inferred to be no less than 22,157 feet, from the velocity of the ordinary tidal waves. A tidal wave again gives beautiful evidence of an effect of the law of gravity, which we could never in any other way detect. Newton estimated that the moon’s force in moving the ocean is only one part in 2,871,400 of the whole force of gravity, so that even the pendulum, used with the utmost skill, would fail to render it apparent. Yet, the immense extent of the ocean allows the accumulation of the effect into a very palpable amount; and from the comparative heights of the lunar and solar tides, Newton roughly estimated the comparative forces of the moon’s and sun’s gravity at the earth.[194]
[193] Herschel, *Physical Geography*, § 40.
[194] *Principia*, bk. iii. Prop. 37, *Corollaries*, 2 and 3. Motte’s translation, vol. ii. p. 310.
A few years ago it might have seemed impossible that we should ever measure the velocity with which a star approaches or recedes from the earth, since the apparent position of the star is thereby unaltered. But the spectroscope now enables us to detect and even measure such motions with considerable accuracy, by the alteration which it causes in the apparent rapidity of vibration, and consequently in the refrangibility of rays of light of definite colour. And while our estimates of the lateral movements of stars depend upon our very uncertain knowledge of their distances, the spectroscope gives the motions of approach and recess irrespective of other motions excepting that of the earth. It gives in short the motions of approach and recess of the stars relatively to the earth.[195]
[195] Roscoe’s *Spectrum Analysis*, 1st ed. p. 296.
The rapidity of vibration for each musical tone, having been accurately determined by comparison with the Syren (p. 10), we can use sounds as indirect indications of rapid vibrations. It is now known that the contraction of a muscle arises from the periodical contractions of each separate fibre, and from a faint sound or susurrus which accompanies the action of a muscle, it is inferred that each contraction lasts for about one 300th part of a second. Minute quantities of radiant heat are now always measured indirectly by the electricity which they produce when falling upon a thermopile. The extreme delicacy of the method seems to be due to the power of multiplication at several points in the apparatus. The number of elements or junctions of different metals in the thermopile can be increased so that the tension of the electric current derived from the same intensity of radiation is multiplied; the effect of the current upon the magnetic needle can be multiplied within certain bounds, by passing the current many times round it in a coil; the excursions of the needle can be increased by rendering it astatic and increasing the delicacy of its suspension; lastly, the angular divergence can be observed, with any required accuracy, by the use of an attached mirror and distant scale viewed through a telescope (p. 287). Such is the delicacy of this method of measuring heat, that Dr. Joule succeeded in making a thermopile which would indicate a difference of 0°·000114 Cent.[196]
[196] *Philosophical Transactions* (1859), vol. cxlix. p. 94.
A striking case of indirect measurement is furnished by the revolving mirror of Wheatstone and Foucault, whereby a minute interval of time is estimated in the form of an angular deviation. Wheatstone viewed an electric spark in a mirror rotating so rapidly, that if the duration of the spark had been more than one 72,000th part of a second, the point of light would have appeared elongated to an angular extent of one-half degree. In the spark, as drawn directly from a Leyden jar, no elongation was apparent, so that the duration of the spark was immeasurably small; but when the discharge took place through a bad conductor, the elongation of the spark denoted a sensible duration.[197] In the hands of Foucault the rotating mirror gave a measure of the time occupied by light in passing through a few metres of space.
[197] Watts’ *Dictionary of Chemistry*, vol. ii. p. 393.
*Comparative Use of Measuring Instruments.*
In almost every case a measuring instrument serves, and should serve only as a means of comparison between two or more magnitudes. As a general rule, we should not attempt to make the divisions of the measuring scale exact multiples or submultiples of the unit, but, regarding them as arbitrary marks, should determine their values by comparison with the standard itself. The perpendicular wires in the field of a transit telescope, are fixed at nearly equal but arbitrary distances, and those distances are afterwards determined, as first suggested by Malvasia, by watching the passage of star after star across them, and noting the intervals of time by the clock. Owing to the perfectly regular motion of the earth, these time intervals give exact determinations of the angular intervals. In the same way, the angular value of each turn of the screw micrometer attached to a telescope, can be easily and accurately ascertained.
When a thermopile is used to observe radiant heat, it would be almost impossible to calculate on *à priori* grounds what is the value of each division of the galvanometer circle, and still more difficult to construct a galvanometer, so that each division should have a given value. But this is quite unnecessary, because by placing the thermopile before a body of known dimensions, at a known distance, with a known temperature and radiating power, we measure a known amount of radiant heat, and inversely measure the value of the indications of the thermopile. In a similar way Dr. Joule ascertained the actual temperature produced by the compression of bars of metal. For having inserted a small thermopile composed of a single junction of copper and iron wire, and noted the deflections of the galvanometer, he had only to dip the bars into water of different temperatures, until he produced a like deflection, in order to ascertain the temperature developed by pressure.[198]
[198] *Philosophical Transactions* (1859), vol. cxlix. p. 119, &c.
In some cases we are obliged to accept a very carefully constructed instrument as a standard, as in the case of a standard barometer or thermometer. But it is then best to treat all inferior instruments comparatively only, and determine the values of their scales by comparison with the assumed standard.
*Systematic Performance of Measurements.*
When a large number of accurate measurements have to be effected, it is usually desirable to make a certain number of determinations with scrupulous care, and afterwards use them as points of reference for the remaining determinations. In the trigonometrical survey of a country, the principal triangulation fixes the relative positions and distances of a few points with rigid accuracy. A minor triangulation refers every prominent hill or village to one of the principal points, and then the details are filled in by reference to the secondary points. The survey of the heavens is effected in a like manner. The ancient astronomers compared the right ascensions of a few principal stars with the moon, and thus ascertained their positions with regard to the sun; the minor stars were afterwards referred to the principal stars. Tycho followed the same method, except that he used the more slowly moving planet Venus instead of the moon. Flamsteed was in the habit of using about seven stars, favourably situated at points all round the heavens. In his early observations the distances of the other stars from these standard points were determined by the use of the quadrant.[199] Even since the introduction of the transit telescope and the mural circle, tables of standard stars are formed at Greenwich, the positions being determined with all possible accuracy, so that they can be employed for purposes of reference by astronomers.
[199] Baily’s *Account of Flamsteed*, pp. 378–380.
In ascertaining the specific gravities of substances, all gases are referred to atmospheric air at a given temperature and pressure; all liquids and solids are referred to water. We require to compare the densities of water and air with great care, and the comparative densities of any two substances whatever can then be ascertained.
In comparing a very great with a very small magnitude, it is usually desirable to break up the process into several steps, using intermediate terms of comparison. We should never think of measuring the distance from London to Edinburgh by laying down measuring rods, throughout the whole length. A base of several miles is selected on level ground, and compared on the one hand with the standard yard, and on the other with the distance of London and Edinburgh, or any other two points, by trigonometrical survey. Again, it would be exceedingly difficult to compare the light of a star with that of the sun, which would be about thirty thousand million times greater; but Herschel[200] effected the comparison by using the full moon as an intermediate unit. Wollaston ascertained that the sun gave 801,072 times as much light as the full moon, and Herschel determined that the light of the latter exceeded that of α Centauri 27,408 times, so that we find the ratio between the light of the sun and star to be that of about 22,000,000,000 to 1.
[200] Herschel’s *Astronomy*, § 817, 4th. ed. p. 553.
*The Pendulum.*
By far the most perfect and beautiful of all instruments of measurement is the pendulum. Consisting merely of a heavy body suspended freely at an invariable distance from a fixed point, it is most simple in construction; yet all the highest problems of physical measurement depend upon its careful use. Its excessive value arises from two circumstances.
(1) The method of repetition is eminently applicable to it, as already described (p. 290).
(2) Unlike other instruments, it connects together three different quantities, those of space, time, and force.
In most works on natural philosophy it is shown, that when the oscillations of the pendulum are infinitely small, the square of the time occupied by an oscillation is directly proportional to the length of the pendulum, and indirectly proportional to the force affecting it, of whatever kind. The whole theory of the pendulum is contained in the formula, first given by Huygens in his *Horologium Oscillatorium*.
Time of oscillation = 3·14159 × √(length of pendulum/force).
The quantity 3·14159 is the constant ratio of the circumference and radius of a circle, and is of course known with accuracy. Hence, any two of the three quantities concerned being given, the third may be found; or any two being maintained invariable, the third will be invariable. Thus a pendulum of invariable length suspended at the same place, where the force of gravity may be considered constant, furnishes a measure of time. The same invariable pendulum being made to vibrate at different points of the earth’s surface, and the times of vibration being astronomically determined, the force of gravity becomes accurately known. Finally, with a known force of gravity, and time of vibration ascertained by reference to the stars, the length is determinate.
All astronomical observations depend upon the first manner of using the pendulum, namely, in the astronomical clock. In the second employment it has been almost equally indispensable. The primary principle that gravity is equal in all matter was proved by Newton’s and Gauss’ pendulum experiments. The torsion pendulum of Michell, Cavendish, and Baily, depending upon exactly the same principles as the ordinary pendulum, gave the density of the earth, one of the foremost natural constants. Kater and Sabine, by pendulum observations in different parts of the earth, ascertained the variation of gravity, whence comes a determination of the earth’s ellipticity. The laws of electric and magnetic attraction have also been determined by the method of vibrations, which is in constant use in the measurement of the horizontal force of terrestrial magnetism.
We must not confuse with the ordinary use of the pendulum its application by Newton, to show the absence of internal friction against space,[201] or to ascertain the laws of motion and elasticity.[202] In these cases the extent of vibration is the quantity measured, and the principles of the instrument are different.
[201] *Principia*, bk. ii. Sect. 6. Prop. 31. Motte’s Translation, vol. ii. p. 107.
[202] Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i. p. 33.
*Attainable Accuracy of Measurement.*
It is a matter of some interest to compare the degrees of accuracy which can be attained in the measurement of different kinds of magnitude. Few measurements of any kind are exact to more than six significant figures,[203] but it is seldom that such accuracy can be hoped for. Time is the magnitude which seems to be capable of the most exact estimation, owing to the properties of the pendulum, and the principle of repetition described in previous sections. As regards short intervals of time, it has already been stated that Sir George Airy was able to estimate one part in 8,640,000, an exactness, as he truly remarks, “almost beyond conception.”[204] The ratio between the mean solar and the sidereal day is known to be about one part in one hundred millions, or to the eighth place of decimals, (p. 289).
[203] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 333.
[204] *Philosophical Transactions*, (1856), vol. cxlvi. pp. 330, 331.
Determinations of weight seem to come next in exactness, owing to the fact that repetition without error is applicable to them. An ordinary good balance should show about one part in 500,000 of the load. The finest balance employed by M. Stas, turned with one part in 825,000 of the load.[205] But balances have certainly been constructed to show one part in a million,[206] and Ramsden is said to have constructed a balance for the Royal Society, to indicate one part in seven millions, though this is hardly credible. Professor Clerk Maxwell takes it for granted that one part in five millions can be detected, but we ought to discriminate between what a balance can do when first constructed, and when in continuous use.
[205] *First Annual Report of the Mint*, p. 106.
[206] Jevons, in Watts’ *Dictionary of Chemistry*, vol. i. p. 483.Return to passage
Determinations of length, unless performed with extraordinary care, are open to much error in the junction of the measuring bars. Even in measuring the base line of a trigonometrical survey, the accuracy generally attained is only that of about one part in 60,000, or an inch in the mile; but it is said that in four measurements of a base line carried out very recently at Cape Comorin, the greatest error was 0·077 inch in 1·68 mile, or one part in 1,382,400, an almost incredible degree of accuracy. Sir J. Whitworth has shown that touch is even a more delicate mode of measuring lengths than sight, and by means of a splendidly executed screw, and a small cube of iron placed between two flat-ended iron bars, so as to be suspended when touching them, he can detect a change of dimension in a bar, amounting to no more than one-millionth of an inch.[207]
[207] British Association, Glasgow, 1856. *Address of the President of the Mechanical Section*.Return to passage
XIV. UNITS AND STANDARDS OF MEASUREMENT
As we have seen, instruments of measurement are only means of comparison between one magnitude and another, and as a general rule we must assume some one arbitrary magnitude, in terms of which all results of measurement are to be expressed. Mere ratios between any series of objects will never tell us their absolute magnitudes; we must have at least one ratio for each, and we must have one absolute magnitude. The number of ratios *n* are expressible in *n* equations, which will contain at least *n* + 1 quantities, so that if we employ them to make known *n* magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some concrete standard, some actual object, which if once lost and irrecoverable, all our measures lose their absolute meaning. This concrete standard is in all cases arbitrary in point of theory, and its selection a question of practical convenience.
There are two kinds of magnitude, indeed, which do not need to be expressed in terms of arbitrary concrete units, since they pre-suppose the existence of natural standard units. One case is that of abstract number itself, which needs no special unit, because any object which exists or is thought of as separate from other objects (p. 157) furnishes us with a unit, and is the only standard required.
Angular magnitude is the second case in which we have a natural unit of reference, namely the whole revolution or *perigon*, as it has been called by Mr. Sandeman.[208] It is a necessary result of the uniform properties of space, that all complete revolutions are equal to each other, so that we need not select any one revolution, but can always refer anew to space itself. Whether we take the whole perigon, its half, or its quarter, is really immaterial; Euclid took the right angle, because the Greek geometers had never generalised their notions of angular magnitude sufficiently to treat angles of all magnitudes, or of unlimited *quantity of revolution*. Euclid defines a right angle as half that made by a line with its own continuation, which is of course equal to half a revolution, but which was not treated as an angle by him. In mathematical analysis a different fraction of the perigon is taken, namely, such a fraction that the arc or portion of the circumference included within it is equal to the radius of the circle. In this point of view angular magnitude is an abstract ratio, namely, the ratio between the length of arc subtended and the length of the radius. The geometrical unit is then necessarily the angle corresponding to the ratio unity. This angle is equal to about 57°, 17′, 44″·8, or decimally 57°·295779513... .[209] It was called by De Morgan the *arcual unit*, but a more convenient name for common use would be *radian*, as suggested by Professor Everett. Though this standard angle is naturally employed in mathematical analysis, and any other unit would introduce great complexity, we must not look upon it as a distinct unit, since its amount is connected with that of the half perigon, by the natural constant 3·14159... usually denoted by the letter π.
[208] *Pelicotetics, or the Science of Quantity; an Elementary Treatise on Algebra, and its groundwork Arithmetic.* By Archibald Sandeman, M. A. Cambridge (Deighton, Bell, and Co.), 1868, p. 304.
[209] De Morgan’s *Trigonometry and Double Algebra*, p. 5.
When we pass to other species of quantity, the choice of unit is found to be entirely arbitrary. There is absolutely no mode of defining a length, but by selecting some physical object exhibiting that length between certain obvious points--as, for instance, the extremities of a bar, or marks made upon its surface.
*Standard Unit of Time.*
Time is the great independent variable of all change--that which itself flows on uninterruptedly, and brings the variety which we call motion and life. When we reflect upon its intimate nature, Time, like every other element of existence, proves to be an inscrutable mystery. We can only say with St. Augustin, to one who asks us what is time, “I know when you do not ask me.” The mind of man will ask what can never be answered, but one result of a true and rigorous logical philosophy must be to convince us that scientific explanation can only take place between phenomena which have something in common, and that when we get down to primary notions, like those of time and space, the mind must meet a point of mystery beyond which it cannot penetrate. A definition of time must not be looked for; if we say with Hobbes,[210] that it is “the phantasm of before and after in motion,” or with Aristotle that it is “the number of motion according to former and latter,” we obviously gain nothing, because the notion of time is involved in the expressions *before and after*, *former and latter*. Time is undoubtedly one of those primary notions which can only be defined physically, or by observation of phenomena which proceed in time.
[210] *English Works of Thos. Hobbes*, Edit. by Molesworth, vol. i. p. 95.
If we have not advanced a step beyond Augustin’s acute reflections on this subject,[211] it is curious to observe the wonderful advances which have been made in the practical measurement of its efflux. In earlier centuries the rude sun-dial or the rising of a conspicuous star gave points of reference, while the flow of water from the clepsydra, the burning of a candle, or, in the monastic ages, even the continuous chanting of psalms, were the means of roughly subdividing periods, and marking the hours of the day and night.[212] The sun and stars still furnish the standard of time, but means of accurate subdivision have become requisite, and this has been furnished by the pendulum and the chronograph. By the pendulum we can accurately divide the day into seconds of time. By the chronograph we can subdivide the second into a hundred, a thousand, or even a million parts. Wheatstone measured the duration of an electric spark, and found it to be no more than one 115,200th part of a second, while more recently Captain Noble has been able to appreciate intervals of time not exceeding the millionth part of a second.
[211] *Confessions*, bk. xi. chapters 20–28.
[212] Sir G. C. Lewis gives many curious particulars concerning the measurement of time in his *Astronomy of the Ancients*, pp. 241, &c.
When we come to inquire precisely what phenomenon it is that we thus so minutely measure, we meet insurmountable difficulties. Newton distinguished time according as it was *absolute* or *apparent* time, in the following words:--“Absolute, true, and mathematical time, of itself and from its own nature, flows equably without regard to anything external, and by another name is called *duration*; relative, apparent and common time, is some sensible and external measure of duration by the means of motion.”[213] Though we are perhaps obliged to assume the existence of a uniformly increasing quantity which we call time, yet we cannot feel or know abstract and absolute time. Duration must be made manifest to us by the recurrence of some phenomenon. The succession of our own thoughts is no doubt the first and simplest measure of time, but a very rude one, because in some persons and circumstances the thoughts evidently flow with much greater rapidity than in other persons and circumstances. In the absence of all other phenomena, the interval between one thought and another would necessarily become the unit of time, but the most cursory observations show that there are changes in the outward world much better fitted by their constancy to measure time than the change of thoughts within us.
[213] *Principia*, bk. i. *Scholium to Definitions*. Translated by Motte, vol. i. p. 9. See also p. 11.
The earth, as I have already said, is the real clock of the astronomer, and is practically assumed as invariable in its movements. But on what ground is it so assumed? According to the first law of motion, every body perseveres in its state of rest or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. Rotatory motion is subject to a like condition, namely, that it perseveres uniformly unless disturbed by extrinsic forces. Now uniform motion means motion through equal spaces in equal times, so that if we have a body entirely free from all resistance or perturbation, and can measure equal spaces of its path, we have a perfect measure of time. But let it be remembered that this law has never been absolutely proved by experience; for we cannot point to any body, and say that it is wholly unresisted or undisturbed; and even if we had such a body, we should need some independent standard of time to ascertain whether its motion was really uniform. As it is in moving bodies that we find the best standard of time, we cannot use them to prove the uniformity of their own movements, which would amount to a *petitio principii*. Our experience comes to this, that when we examine and compare the movements of bodies which seem to us nearly free from disturbance, we find them giving nearly harmonious measures of time. If any one body which seems to us to move uniformly is not doing so, but is subject to fits and starts unknown to us, because we have no absolute standard of time, then all other bodies must be subject to the same arbitrary fits and starts, otherwise there would be discrepancy disclosing the irregularities. Just as in comparing together a number of chronometers, we should soon detect bad ones by their going irregularly, as compared with the others, so in nature we detect disturbed movement by its discrepancy from that of other bodies which we believe to be undisturbed, and which agree nearly among themselves. But inasmuch as the measure of motion involves time, and the measure of time involves motion, there must be ultimately an assumption. We may define equal times, as times during which a moving body under the influence of no force describes equal spaces;[214] but all we can say in support of this definition is, that it leads us into no known difficulties, and that to the best of our experience one freely moving body gives the same results as any other.
[214] Rankine, *Philosophical Magazine*, Feb. 1867, vol. xxxiii. p. 91.
When we inquire where the freely moving body is, no perfectly satisfactory answer can be given. Practically the rotating globe is sufficiently accurate, and Thomson and Tait say: “Equal times are times during which the earth turns through equal angles.”[215] No long time has passed since astronomers thought it impossible to detect any inequality in its movement. Poisson was supposed to have proved that a change in the length of the sidereal day amounting to one ten-millionth part in 2,500 years was incompatible with an ancient eclipse recorded by the Chaldæans, and similar calculations were made by Laplace. But it is now known that these calculations were somewhat in error, and that the dissipation of energy arising out of the friction of tidal waves, and the radiation of the heat into space, has slightly decreased the rapidity of the earth’s rotatory motion. The sidereal day is now longer by one part in 2,700,000, than it was in 720 B.C. Even before this discovery, it was known that invariability of rotation depended upon the perfect maintenance of the earth’s internal heat, which is requisite in order that the earth’s dimensions shall be unaltered. Now the earth being superior in temperature to empty space, must cool more or less rapidly, so that it cannot furnish an absolute measure of time. Similar objections could be raised to all other rotating bodies within our cognisance.
[215] *Treatise on Natural Philosophy*, vol. i. p. 179.
The moon’s motion round the earth, and the earth’s motion round the sun, form the next best measure of time. They are subject, indeed, to disturbance from other planets, but it is believed that these perturbations must in the course of time run through their rhythmical courses, leaving the mean distances unaffected, and consequently, by the third Law of Kepler, the periodic times unchanged. But there is more reason than not to believe that the earth encounters a slight resistance in passing through space, like that which is so apparent in Encke’s comet. There may also be dissipation of energy in the electrical relations of the earth to the sun, possibly identical with that which is manifested in the retardation of comets.[216] It is probably an untrue assumption then, that the earth’s orbit remains quite invariable. It is just possible that some other body may be found in the course of time to furnish a better standard of time than the earth in its annual motion. The greatly superior mass of Jupiter and its satellites, and their greater distance from the sun, may render the electrical dissipation of energy less considerable than in the case of the earth. But the choice of the best measure will always be an open one, and whatever moving body we choose may ultimately be shown to be subject to disturbing forces.
[216] *Proceedings of the Manchester Philosophical Society*, 28th Nov. 1871, vol. xi. p. 33.
The pendulum, although so admirable an instrument for subdivision of time, fails as a standard; for though the same pendulum affected by the same force of gravity performs equal vibrations in equal times, yet the slightest change in the form or weight of the pendulum, the least corrosion of any part, or the most minute displacement of the point of suspension, falsifies the results, and there enter many other difficult questions of temperature, friction, resistance, length of vibration, &c.
Thomson and Tait are of opinion[217] that the ultimate standard of chronometry must be founded on the physical properties of some body of more constant character than the earth; for instance, a carefully arranged metallic spring, hermetically sealed in an exhausted glass vessel. But it is hard to see how we can be sure that the dimensions and elasticity of a piece of wrought metal will remain perfectly unchanged for the few millions of years contemplated by them. A nearly perfect gas, like hydrogen, is perhaps the only kind of substance in the unchanged elasticity of which we could have confidence. Moreover, it is difficult to perceive how the undulations of such a spring could be observed with the requisite accuracy. More recently Professor Clerk Maxwell has made the novel suggestion, discussed in a subsequent section, that undulations of light *in vacuo* would form the most universal standard of reference, both as regards time and space. According to this system the unit of time would be the time occupied by one vibration of the particular kind of light whose wave length is taken as the unit of length.
[217] *The Elements of Natural Philosophy*, part i. p. 119.
*The Unit of Space and the Bar Standard.*
Next in importance after the measurement of time is that of space. Time comes first in theory, because phenomena, our internal thoughts for instance, may change in time without regard to space. As to the phenomena of outward nature, they tend more and more to resolve themselves into motions of molecules, and motion cannot be conceived or measured without reference both to time and space.
Turning now to space measurement, we find it almost equally difficult to fix and define once and for ever, a unit magnitude. There are three different modes in which it has been proposed to attempt the perpetuation of a standard length.
(1) By constructing an actual specimen of the standard yard or metre, in the form of a bar.
(2) By assuming the globe itself to be the ultimate standard of magnitude, the practical unit being a submultiple of some dimension of the globe.
(3) By adopting the length of the simple seconds pendulum, as a standard of reference.
At first sight it might seem that there was no great difficulty in this matter, and that any one of these methods might serve well enough; but the more minutely we inquire into the details, the more hopeless appears to be the attempt to establish an invariable standard. We must in the first place point out a principle not of an obvious character, namely, that *the standard length must be defined by one single object*.[218] To make two bars of exactly the same length, or even two bars bearing a perfectly defined ratio to each other, is beyond the power of human art. If two copies of the standard metre be made and declared equally correct, future investigators will certainly discover some discrepancy between them, proving of course that they cannot both be the standard, and giving cause for dispute as to what magnitude should then be taken as correct.
[218] See Harris’ *Essay upon Money and Coins*, part. ii. [1758] p. 127.
If one invariable bar could be constructed and maintained as the absolute standard, no such inconvenience could arise. Each successive generation as it acquired higher powers of measurement, would detect errors in the copies of the standard, but the standard itself would be unimpeached, and would, as it were, become by degrees more and more accurately known. Unfortunately to construct and preserve a metre or yard is also a task which is either impossible, or what comes nearly to the same thing, cannot be shown to be possible. Passing over the practical difficulty of defining the ends of the standard length with complete accuracy, whether by dots or lines on the surface, or by the terminal points of the bar, we have no means of proving that substances remain of invariable dimensions. Just as we cannot tell whether the rotation of the earth is uniform, except by comparing it with other moving bodies, believed to be more uniform in motion, so we cannot detect the change of length in a bar, except by comparing it with some other bar supposed to be invariable. But how are we to know which is the invariable bar? It is certain that many rigid and apparently invariable substances do change in dimensions. The bulb of a thermometer certainly contracts by age, besides undergoing rapid changes of dimensions when warmed or cooled through 100° Cent. Can we be sure that even the most solid metallic bars do not slightly contract by age, or undergo variations in their structure by change of temperature. Fizeau was induced to try whether a quartz crystal, subjected to several hundred alternations of temperature, would be modified in its physical properties, and he was unable to detect any change in the coefficient of expansion.[219] It does not follow, however, that, because no apparent change was discovered in a quartz crystal, newly-constructed bars of metal would undergo no change.
[219] *Philosophical Magazine*, (1868), 4th Series, vol. xxxvi. p. 32.
The best principle, as it seems to me, upon which the perpetuation of a standard of length can be rested, is that, if a variation of length occurs, it will in all probability be of different amount in different substances. If then a great number of standard metres were constructed of all kinds of different metals and alloys; hard rocks, such as granite, serpentine, slate, quartz, limestone; artificial substances, such as porcelain, glass, &c., &c., careful comparison would show from time to time the comparative variations of length of these different substances. The most variable substances would be the most divergent, and the standard would be furnished by the mean length of those which agreed most closely with each other just as uniform motion is that of those bodies which agree most closely in indicating the efflux of time.
*The Terrestrial Standard.*
The second method assumes that the globe itself is a body of invariable dimensions and the founders of the metrical system selected the ten-millionth part of the distance from the equator to the pole as the definition of the metre. The first imperfection in such a method is that the earth is certainly not invariable in size; for we know that it is superior in temperature to surrounding space, and must be slowly cooling and contracting. There is much reason to believe that all earthquakes, volcanoes, mountain elevations, and changes of sea level are evidences of this contraction as asserted by Mr. Mallet.[220] But such is the vast bulk of the earth and the duration of its past existence, that this contraction is perhaps less rapid in proportion than that of any bar or other material standard which we can construct.
[220] *Proceedings of the Royal Society*, 20th June, 1872, vol. xx. p. 438.
The second and chief difficulty of this method arises from the vast size of the earth, which prevents us from making any comparison with the ultimate standard, except by a trigonometrical survey of a most elaborate and costly kind. The French physicists, who first proposed the method, attempted to obviate this inconvenience by carrying out the survey once for all, and then constructing a standard metre, which should be exactly the one ten millionth part of the distance from the pole to the equator. But since all measuring operations are merely approximate, it was impossible that this operation could be perfectly achieved. Accordingly, it was shown in 1838 that the supposed French metre was erroneous to the considerable extent of one part in 5527. It then became necessary either to alter the length of the assumed metre, or to abandon its supposed relation to the earth’s dimensions. The French Government and the International Metrical Commission have for obvious reasons decided in favour of the latter course, and have thus reverted to the first method of defining the metre by a given bar. As from time to time the ratio between this assumed standard metre and the quadrant of the earth becomes more accurately known, we have better means of restoring that metre by reference to the globe if required. But until lost, destroyed, or for some clear reason discredited, the bar metre and not the globe is the standard. Thomson and Tait remark that any of the more accurate measurements of the English trigonometrical survey might in like manner be employed to restore our standard yard, in terms of which the results are recorded.
*The Pendulum Standard.*
The third method of defining a standard length, by reference to the seconds pendulum, was first proposed by Huyghens, and was at one time adopted by the English Government. From the principle of the pendulum (p. 302) it clearly appears that if the time of oscillation and the force actuating the pendulum be the same, the length of the pendulum must be the same. We do not get rid of theoretical difficulties, for we must assume the attraction of gravity at some point of the earth’s surface, say London, to be unchanged from time to time, and the sidereal day to be invariable, neither assumption being absolutely correct so far as we can judge. The pendulum, in short, is only an indirect means of making one physical quantity of space depend upon two other physical quantities of time and force.
The practical difficulties are, however, of a far more serious character than the theoretical ones. The length of a pendulum is not the ordinary length of the instrument, which might be greatly varied without affecting the duration of a vibration, but the distance from the centre of suspension to the centre of oscillation. There are no direct means of determining this latter centre, which depends upon the average momentum of all the particles of the pendulum as regards the centre of suspension. Huyghens discovered that the centres of suspension and oscillation are interchangeable, and Kater pointed out that if a pendulum vibrates with exactly the same rapidity when suspended from two different points, the distance between these points is the true length of the equivalent simple pendulum.[221] But the practical difficulties in employing Kater’s reversible pendulum are considerable, and questions regarding the disturbance of the air, the force of gravity, or even the interference of electrical attractions have to be entertained. It has been shown that all the experiments made under the authority of Government for determining the ratio between the standard yard and the seconds pendulum, were vitiated by an error in the corrections for the resisting, adherent, or buoyant power of the air in which the pendulums were swung. Even if such corrections were rendered unnecessary by operating in a vacuum, other difficult questions remain.[222] Gauss’ mode of comparing the vibrations of a wire pendulum when suspended at two different lengths is open to equal or greater practical difficulties. Thus it is found that the pendulum standard cannot compete in accuracy and certainty with the simple bar standard, and the method would only be useful as an accessory mode of restoring the bar standard if at any time again destroyed.
[221] Kater’s *Treatise on Mechanics*, Cabinet Cyclopædia, p. 154.
[222] Grant’s *History of Physical Astronomy*, p. 156.
*Unit of Density.*
Before we can measure the phenomena of nature, we require a third independent unit, which shall enable us to define the quantity of matter occupying any given space. All the changes of nature, as we shall see, are probably so many manifestations of energy; but energy requires some substratum or material machinery of molecules, in and by which it may be manifested. Observation shows that, as regards force, there may be two modes of variation of matter. As Newton says in the first definition of the Principia, “the quantity of matter is the measure of the same, arising from its density and bulk conjunctly.” Thus the force required to set a body in motion varies both according to the bulk of the matter, and also according to its quality. Two cubic inches of iron of uniform quality, will require twice as much force as one cubic inch to produce a certain velocity in a given time; but one cubic inch of gold will require more force than one cubic inch of iron. There is then some new measurable quality in matter apart from its bulk, which we may call *density*, and which is, strictly speaking, indicated by its capacity to resist and absorb the action of force. For the unit of density we may assume that of any substance which is uniform in quality, and can readily be referred to from time to time. Pure water at any definite temperature, for instance that of snow melting under inappreciable pressure, furnishes an invariable standard of density, and by comparing equal bulks of various substances with a like bulk of ice-cold water, as regards the velocity produced in a unit of time by the same force, we should ascertain the densities of those substances as expressed in that of water. Practically the force of gravity is used to measure density; for a beautiful experiment with the pendulum, performed by Newton and repeated by Gauss, shows that all kinds of matter gravitate equally. Two portions of matter then which are in equilibrium in the balance, may be assumed to possess equal inertia, and their densities will therefore be inversely as their cubic dimensions.
*Unit of Mass.*
Multiplying the number of units of density of a portion of matter, by the number of units of space occupied by it, we arrive at the quantity of matter, or, as it is usually called, the *unit of mass*, as indicated by the inertia and gravity it possesses. To proceed in the most simple manner, the unit of mass ought to be that of a cubic unit of matter of the standard density; but the founders of the metrical system took as their unit of mass, the cubic centimetre of water, at the temperature of maximum density (about 4° Cent.). They called this unit of mass the *gramme*, and constructed standard specimens of the kilogram, which might be readily referred to by all who required to employ accurate weights. Unfortunately the determination of the bulk of a given weight of water at a certain temperature is an operation involving many difficulties, and it cannot be performed in the present day with a greater exactness than that of about one part in 5000, the results of careful observers being sometimes found to differ as much as one part in 1000.[223]
[223] Clerk Maxwell’s *Theory of Heat*, p. 79.
Weights, on the other hand, can be compared with each other to at least one part in a million. Hence if different specimens of the kilogram be prepared by direct weighing against water, they will not agree closely with each other; the two principal standard kilograms agree neither with each other, nor with their definition. According to Professor Miller the so-called Kilogramme des Archives weighs 15432·34874 grains, while the kilogram deposited at the Ministry of the Interior in Paris, as the standard for commercial purposes, weighs 15432·344 grains. Since a standard weight constructed of platinum, or platinum and iridium, can be preserved free from any appreciable alteration, and since it can be very accurately compared with other weights, we shall ultimately attain the greatest exactness in our measurements of mass, by assuming some single kilogram as a *provisional standard*, leaving the determination of its actual mass in units of space and density for future investigation. This is what is practically done at the present day, and thus a unit of mass takes the place of the unit of density, both in the French and English systems. The English pound is defined by a certain lump of platinum, preserved at Westminster, and is an arbitrary mass, chosen merely that it may agree as nearly as possible with old English pounds. The gallon, the old English unit of cubic measurement, is defined by the condition that it shall contain exactly ten pounds weight of water at 62° Fahr.; and although it is stated that it has the capacity of about 277·274 cubic inches, this ratio between the cubic and linear systems of measurement is not legally enacted, but left open to investigation. While the French metric system as originally designed was theoretically perfect, it does not differ practically in this point from the English system.
*Natural System of Standards.*
Quite recently Professor Clerk Maxwell has suggested that the vibrations of light and the atoms of matter might conceivably be employed as the ultimate standards of length, time, and mass. We should thus arrive at a *natural system of standards*, which, though possessing no present practical importance, has considerable theoretical interest. “In the present state of science,” he says, “the most universal standard of length which we could assume would be the wave-length in vacuum of a particular kind of light, emitted by some widely diffused substance such as sodium, which has well-defined lines in its spectrum. Such a standard would be independent of any changes in the dimensions of the earth, and should be adopted by those who expect their writings to be more permanent than that body.”[224] In the same way we should get a universal standard unit of time, independent of all questions about the motion of material bodies, by taking as the unit the periodic time of vibration of that particular kind of light whose wave-length is the unit of length. It would follow that with these units of length and time the unit of velocity would coincide with the velocity of light in empty space. As regards the unit of mass, Professor Maxwell, humorously as I should think, remarks that if we expect soon to be able to determine the mass of a single molecule of some standard substance, we may wait for this determination before fixing a universal standard of mass.
[224] *Treatise on Electricity and Magnetism*, vol. i. p. 3.
In a theoretical point of view there can be no reasonable doubt that vibrations of light are, as far as we can tell, the most fixed in magnitude of all phenomena. There is as usual no certainty in the matter, for the properties of the basis of light may vary to some extent in different parts of space. But no differences could ever be established in the velocity of light in different parts of the solar system, and the spectra of the stars show that the times of vibration there do not differ perceptibly from those in this part of the universe. Thus all presumption is in favour of the absolute constancy of the vibrations of light--absolute, that is, so far as regards any means of investigation we are likely to possess. Nearly the same considerations apply to the atomic weight as the standard of mass. It is impossible to prove that all atoms of the same substance are of equal mass, and some physicists think that they differ, so that the fixity of combining proportions may be due only to the approximate constancy of the mean of countless millions of discrepant weights. But in any case the detection of difference is probably beyond our powers. In a theoretical point of view, then, the magnitudes suggested by Professor Maxwell seem to be the most fixed ones of which we have any knowledge, so that they necessarily become the natural units.
In a practical point of view, as Professor Maxwell would be the first to point out, they are of little or no value, because in the present state of science we cannot measure a vibration or weigh an atom with any approach to the accuracy which is attainable in the comparison of standard metres and kilograms. The velocity of light is not known probably within a thousandth part, and as we progress in the knowledge of light, so we shall progress in the accurate fixation of other standards. All that can be said then, is that it is very desirable to determine the wave-lengths and periods of the principal lines of the solar spectrum, and the absolute atomic weights of the elements, with all attainable accuracy, in terms of our existing standards. The numbers thus obtained would admit of the reproduction of our standards in some future age of the world to a corresponding degree of accuracy, were there need of such reference; but so far as we can see at present, there is no considerable probability that this mode of reproduction would ever be the best mode.
*Subsidiary Units.*
Having once established the standard units of time, space, and density or mass, we might employ them for the expression of all quantities of such nature. But it is often convenient in particular branches of science to use multiples or submultiples of the original units, for the expression of quantities in a simple manner. We use the mile rather than the yard when treating of the magnitude of the globe, and the mean distance of the earth and sun is not too large a unit when we have to describe the distances of the stars. On the other hand, when we are occupied with microscopic objects, the inch, the line or the millimetre, become the most convenient terms of expression.
It is allowable for a scientific man to introduce a new unit in any branch of knowledge, provided that it assists precise expression, and is carefully brought into relation with the primary units. Thus Professor A. W. Williamson has proposed as a convenient unit of volume in chemical science, an absolute volume equal to about 11·2 litres representing the bulk of one gram of hydrogen gas at standard temperature and pressure, or the *equivalent* weight of any other gas, such as 16 grams of oxygen, 14 grams of nitrogen, &c.; in short, the bulk of that quantity of any one of those gases which weighs as many grams as there are units in the number expressing its atomic weight.[225] Hofmann has proposed a new unit of weight for chemists, called a *crith*, to be defined by the weight of one litre of hydrogen gas at 0° C. and 0°·76 mm., weighing about 0·0896 gram.[226] Both of these units must be regarded as purely subordinate units, ultimately defined by reference to the primary units, and not involving any new assumption.
[225] *Chemistry for Students*, by A. W. Williamson. Clarendon Press Series, 2nd ed. Preface p. vi.
[226] *Introduction to Chemistry*, p. 131.
*Derived Units.*
The standard units of time, space, and mass having been once fixed, many kinds of magnitude are naturally measured by units derived from them. From the metre, the unit of linear magnitude follows in the most obvious manner the centiare or square metre, the unit of superficial magnitude, and the litre that is the cube of the tenth part of a metre, the unit of capacity or volume. Velocity of motion is expressed by the ratio of the space passed over, when the motion is uniform, to the time occupied; hence the unit of velocity is that of a body which passes over a unit of space in a unit of time. In physical science the unit of velocity might be taken as one metre per second. Momentum is measured by the mass moving, regard being paid both to the amount of matter and the velocity at which it is moving. Hence the unit of momentum will be that of a unit volume of matter of the unit density moving with the unit velocity, or in the French system, a cubic centimetre of water of the maximum density moving one metre per second.
An accelerating force is measured by the ratio of the momentum generated to the time occupied, the force being supposed to act uniformly. The unit of force will therefore be that which generates a unit of momentum in a unit of time, or which causes, in the French system, one cubic centimetre of water at maximum density to acquire in one second a velocity of one metre per second. The force of gravity is the most familiar kind of force, and as, when acting unimpeded upon any substance, it produces in a second a velocity of 9·80868 . . metres per second in Paris, it follows that the absolute unit of force is about the tenth part of the force of gravity. If we employ British weights and measures, the absolute unit of force is represented by the gravity of about half an ounce, since the force of gravity of any portion of matter acting upon that matter during one second, produces a final velocity of 32·1889 feet per second or about 32 units of velocity. Although from its perpetual action and approximate uniformity we find in gravity the most convenient force for reference, and thus habitually employ it to estimate quantities of matter, we must remember that it is only one of many instances of force. Strictly speaking, we should express weight in terms of force, but practically we express other forces in terms of weight.
We still require the unit of energy, a more complex notion. The momentum of a body expresses the quantity of motion which belongs or would belong to the aggregate of the particles; but when we consider how this motion is related to the action of a force producing or removing it, we find that the effect of a force is proportional to the mass multiplied by the square of the velocity and it is convenient to take half this product as the expression required. But it is shown in books upon dynamics that it will be exactly the same thing if we define energy by a force acting through a space. The natural unit of energy will then be that which overcomes a unit of force acting through a unit of space; when we lift one kilogram through one metre, against gravity, we therefore accomplish 9·80868 . . units of work, that is, we turn so many units of potential energy existing in the muscles, into potential energy of gravitation. In lifting one pound through one foot there is in like manner a conversion of 32·1889 units of energy. Accordingly the unit of energy will be in the English system, that required to lift one pound through about the thirty-second part of a foot; in terms of metric units, it will be that required to lift a kilogram through about one tenth part of a metre.
Every person is at liberty to measure and record quantities in terms of any unit which he likes. He may use the yard for linear measurement and the litre for cubic measurement, only there will then be a complicated relation between his different results. The system of derived units which we have been briefly considering, is that which gives the most simple and natural relations between quantitative expressions of different kinds, and therefore conduces to ease of comprehension and saving of laborious calculation.
It would evidently be a source of great convenience if scientific men could agree upon some single system of units, original and derived, in terms of which all quantities could be expressed. Statements would thus be rendered easily comparable, a large part of scientific literature would be made intelligible to all, and the saving of mental labour would be immense. It seems to be generally allowed, too, that the metric system of weights and measures presents the best basis for the ultimate system; it is thoroughly established in Western Europe; it is legalised in England; it is already commonly employed by scientific men; it is in itself the most simple and scientific of systems. There is every reason then why the metric system should be accepted at least in its main features.
*Provisional Units.*
Ultimately, as we can hardly doubt, all phenomena will be recognised as so many manifestations of energy; and, being expressed in terms of the unit of energy, will be referable to the primary units of space, time, and density. To effect this reduction, however, in any particular case, we must not only be able to compare different quantities of the phenomenon, but to trace the whole series of steps by which it is connected with the primary notions. We can readily observe that the intensity of one source of light is greater than that of another; and, knowing that the intensity of light decreases as the square of the distance increases, we can easily determine their comparative brilliance. Hence we can express the intensity of light falling upon any surface, if we have a unit in which to make the expression. Light is undoubtedly one form of energy, and the unit ought therefore to be the unit of energy. But at present it is quite impossible to say how much energy there is in any particular amount of light. The question then arises,--Are we to defer the measurement of light until we can assign its relation to other forms of energy? If we answer Yes, it is equivalent to saying that the science of light must stand still perhaps for a generation; and not only this science but many others. The true course evidently is to select, as the provisional unit of light, some light of convenient intensity, which can be reproduced from time to time in the same intensity, and which is defined by physical circumstances. All the phenomena of light may be experimentally investigated relatively to this unit, for instance that obtained after much labour by Bunsen and Roscoe.[227] In after years it will become a matter of inquiry what is the energy exerted in such unit of light; but it may be long before the relation is exactly determined.
[227] *Philosophical Transactions* (1859), vol. cxlix. p. 884, &c.
A provisional unit, then, means one which is assumed and physically defined in a safe and reproducible manner, in order that particular quantities may be compared *inter se* more accurately than they can yet be referred to the primary units. In reality the great majority of our measurements are expressed in terms of such provisionally independent units, and even the unit of mass, as we have seen, ought to be considered as provisional.
The unit of heat ought to be simply the unit of energy, already described. But a weight can be measured to the one-millionth part, and temperature to less than the thousandth part of a degree Fahrenheit, and to less therefore than the five-hundred thousandth part of the absolute temperature, whereas the mechanical equivalent of heat is probably not known to the thousandth part. Hence the need of a provisional unit of heat, which is often taken as that requisite to raise one gram of water through one degree Centigrade, that is from 0° to 1°. This quantity of heat is capable of approximate expression in terms of time, space, and mass; for by the natural constant, determined by Dr. Joule, and called the mechanical equivalent of heat, we know that the assumed unit of heat is equal to the energy of 423·55 gram-metres, or that energy which will raise the mass of 423·55 grams through one metre against 9·8... absolute units of force. Heat may also be expressed in terms of the quantity of ice at 0° Cent., which it is capable of converting into water under inappreciable pressure.
*Theory of Dimensions.*
In order to understand the relations between the quantities dealt with in physical science, it is necessary to pay attention to the Theory of Dimensions, first clearly stated by Joseph Fourier,[228] but in later years developed by several physicists. This theory investigates the manner in which each derived unit depends upon or involves one or more of the fundamental units. The number of units in a rectangular area is found by multiplying together the numbers of units in the sides; thus the unit of length enters twice into the unit of area, which is therefore said to have two dimensions with respect to length. Denoting length by *L*, we may say that the dimensions of area are *L* × *L* or *L*^{2}. It is obvious in the same way that the dimensions of volume or bulk will be *L*^{3}.
[228] *Théorie Analytique de la Chaleur*, Paris; 1822, §§ 157–162.
The number of units of mass in a body is found by multiplying the number of units of volume, by those of density. Hence mass is of three dimensions as regards length, and one as regards density. Calling density *D*, the dimensions of mass are *L*^{3}*D*. As already explained, however, it is usual to substitute an arbitrary provisional unit of mass, symbolised by *M*; according to the view here taken we may say that the dimensions of *M* are *L*^{3}*D*.
Introducing time, denoted by *T*, it is easy to see that the dimensions of velocity will be *L/T* or *LT*^{-1}, because the number of units in the velocity of a body is found by *dividing* the units of length passed over by the units of time occupied in passing. The acceleration of a body is measured by the increase of velocity in relation to the time, that is, we must divide the units of velocity gained by the units of time occupied in gaining it; hence its dimensions will be *LT*^{-2}. Momentum is the product of mass and velocity, so that its dimensions are *MLT*^{-1}. The effect of a force is measured by the acceleration produced in a unit of mass in a unit of time; hence the dimensions of force are *MLT*^{-2}. Work done is proportional to the force acting and to the space through which it acts; so that it has the dimensions of force with that of length added, giving *ML*^{2}*T*^{-2}.
It should be particularly noticed that angular magnitude has no dimensions at all, being measured by the ratio of the arc to the radius (p. 305). Thus we have the dimensions *LL*^{-1} or *L*^{0}. This agrees with the statement previously made, that no arbitrary unit of angular magnitude is needed. Similarly, all pure numbers expressing ratios only, such as sines and other trigonometrical functions, logarithms, exponents, &c., are devoid of dimensions. They are absolute numbers necessarily expressed in terms of unity itself, and are quite unaffected by the selection of the arbitrary physical units. Angular magnitude, however, enters into other quantities, such as angular velocity, which has the dimensions 1/*T* or *T*^{-1}, the units of angle being divided by the units of time occupied. The dimensions of angular acceleration are denoted by *T*^{-2}.
The quantities treated in the theories of heat and electricity are numerous and complicated as regards their dimensions. Thermal capacity has the dimensions *ML*^{-3}, thermal conductivity, *ML*^{-1}*T*^{-1}. In Magnetism the dimensions of the strength of pole are *M*^{1/2}*L*^{3/2}*T*^{-1}, the dimensions of field-intensity are *M*^{1/2}*L*^{-1/2}*T*^{-1}, and the intensity of magnetisation has the same dimensions. In the science of electricity physicists have to deal with numerous kinds of quantity, and their dimensions are different too in the electro-static and the electro-magnetic systems. Thus electro-motive force has the dimensions *M*^{1/2}*L*^{1/2}*T*^{-1}, in the former, and *M*^{1/2}*L*^{3/2}*T*^{-2} in the latter system. Capacity simply depends upon length in electro-statics, but upon *L*^{-1}*T*^{2} in electro-magnetics. It is worthy of particular notice that electrical quantities have simple dimensions when expressed in terms of density instead of mass. The instances now given are sufficient to show the difficulty of conceiving and following out the relations of the quantities treated in physical science without a systematic method of calculating and exhibiting their dimensions. It is only in quite recent years that clear ideas about these quantities have been attained. Half a century ago probably no one but Fourier could have explained what he meant by temperature or capacity for heat. The notion of measuring electricity had hardly been entertained.
Besides affording us a clear view of the complex relations of physical quantities, this theory is specially useful in two ways. Firstly, it affords a test of the correctness of mathematical reasoning. According to the *Principle of Homogeneity*, all the quantities *added* together, and equated in any equation, must have the same dimensions. Hence if, on estimating the dimensions of the terms in any equation, they be not homogeneous, some blunder must have been committed. It is impossible to add a force to a velocity, or a mass to a momentum. Even if the numerical values of the two members of a non-homogeneous equation were equal, this would be accidental, and any alteration in the physical units would produce inequality and disclose the falsity of the law expressed in the equation.
Secondly, the theory of units enables us readily and infallibly to deduce the change in the numerical expression of any physical quantity, produced by a change in the fundamental units. It is of course obvious that in order to represent the same absolute quantity, a number must vary inversely as the magnitude of the units which are numbered. The yard expressed in feet is 3; taking the inch as the unit instead of the foot it becomes 36. Every quantity into which the dimension length enters positively must be altered in like manner. Changing the unit from the foot to the inch, numerical expressions of volume must be multiplied by 12 × 12 × 12. When a dimension enters negatively the opposite rule will hold. If for the minute we substitute the second as unit of time, then we must divide all numbers expressing angular velocities by 60, and numbers expressing angular acceleration by 60 × 60. The rule is that a numerical expression varies inversely as the magnitude of the unit as regards each whole dimension entering positively, and it varies directly as the magnitude of the unit for each whole dimension entering negatively. In the case of fractional exponents, the proper root of the ratio of change has to be taken.
The study of this subject may be continued in Professor J. D. Everett’s “Illustrations of the Centimetre-gramme-second System of Units,” published by Taylor and Francis, 1875; in Professor Maxwell’s “Theory of Heat;” or Professor Fleeming Jenkin’s “Text Book of Electricity.”
*Natural Constants.*
Having acquired accurate measuring instruments, and decided upon the units in which the results shall be expressed, there remains the question, What use shall be made of our powers of measurement? Our principal object must be to discover general quantitative laws of nature; but a very large amount of preliminary labour is employed in the accurate determination of the dimensions of existing objects, and the numerical relations between diverse forces and phenomena. Step by step every part of the material universe is surveyed and brought into known relations with other parts. Each manifestation of energy is correlated with each other kind of manifestation. Professor Tyndall has described the care with which such operations are conducted.[229]
[229] Tyndall’s *Sound*, 1st ed. p. 26.
“Those who are unacquainted with the details of scientific investigation, have no idea of the amount of labour expended on the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat. There is a morality brought to bear upon such matters which, in point of severity, is probably without a parallel in any other domain of intellectual action.”
Every new natural constant which is recorded brings many fresh inferences within our power. For if *n* be the number of such constants known, then 1/2 (*n*^{2}--*n*) is the number of ratios which are within our powers of calculation, and this increases with the square of *n*. We thus gradually piece together a map of nature, in which the lines of inference from one phenomenon to another rapidly grow in complexity, and the powers of scientific prediction are correspondingly augmented.
Babbage[230] proposed the formation of a collection of the constant numbers of nature, a work which has at last been taken in hand by the Smithsonian Institution.[231] It is true that a complete collection of such numbers would be almost co-extensive with scientific literature, since almost all the numbers occurring in works on chemistry, mineralogy, physics, astronomy, &c., would have to be included. Still a handy volume giving all the more important numbers and their logarithms, referred when requisite to the different units in common use, would be very useful. A small collection of constant numbers will be found at the end of Babbage’s, Hutton’s, and many other tables of logarithms, and a somewhat larger collection is given in Templeton’s *Millwright and Engineer’s Pocket Companion*.
[230] British Association, Cambridge, 1833. Report, pp. 484–490.
[231] *Smithsonian Miscellaneous Collections*, vol. xii., the Constants of Nature, part. i. Specific gravities compiled by F. W. Clarke, 8vo. Washington, 1873.
Our present object will be to classify these constant numbers roughly, according to their comparative generality and importance, under the following heads:--
(1) Mathematical constants. (2) Physical constants. (3) Astronomical constants. (4) Terrestrial numbers. (5) Organic numbers. (6) Social numbers.
*Mathematical Constants.*
At the head of the list of natural constants must come those which express the necessary relations of numbers to each other. The ordinary Multiplication Table is the most familiar and the most important of such series of constants, and is, theoretically speaking, infinite in extent. Next we must place the Arithmetical Triangle, the significance of which has already been pointed out (p. 182). Tables of logarithms also contain vast series of natural constants, arising out of the relations of pure numbers. At the base of all logarithmic theory is the mysterious natural constant commonly denoted by *e*, or ε, being equal to the infinite series 1 + 1/1 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4 +...., and thus consisting of the sum of the ratios between the numbers of permutations and combinations of 0, 1, 2, 3, 4, &c. things. Tables of prime numbers and of the factors of composite numbers must not be forgotten.
Another vast and in fact infinite series of numerical constants contains those connected with the measurement of angles, and embodied in trigonometrical tables, whether as natural or logarithmic sines, cosines, and tangents. It should never be forgotten that though these numbers find their chief employment in connection with trigonometry, or the measurement of the sides of a right-angled triangle, yet the numbers themselves arise out of numerical relations bearing no special relation to space. Foremost among trigonometrical constants is the well known number π, usually employed as expressing the ratio of the circumference and the diameter of a circle; from π follows the value of the arcual or natural unit of angular value as expressed in ordinary degrees (p. 306).
Among other mathematical constants not uncommonly used may be mentioned tables of factorials (p. 179), tables of Bernoulli’s numbers, tables of the error function,[232] which latter are indispensable not only in the theory of probability but also in several other branches of science.
[232] J. W. L. Glaisher, *Philosophical Magazine*, 4th Series, vol. xlii. p. 421.
It should be clearly understood that the mathematical constants and tables of reference already in our possession, although very extensive, are only an infinitely small part of what might be formed. With the progress of science the tabulation of new functions will be continually demanded, and it is worthy of consideration whether public money should not be available to reward the severe, long continued, and generally thankless labour which must be gone through in calculating tables. Such labours are a benefit to the whole human race as long as it shall exist, though there are few who can appreciate the extent of this benefit. A most interesting and excellent description of many mathematical tables will be found in De Morgan’s article on *Tables*, in the *English Cyclopædia*, Division of Arts and Sciences, vol. vii. p. 976. An almost exhaustive critical catalogue of extant tables is being published by a Committee of the British Association, two portions, drawn up chiefly by Mr. J. W. L. Glaisher and Professor Cayley, having appeared in the Reports of the Association for 1873 and 1875.
*Physical Constants.*
The second class of constants contains those which refer to the actual constitution of matter. For the most part they depend upon the peculiarities of the chemical substance in question, but we may begin with those which are of the most general character. In a first sub-class we may place the velocity of light or heat undulations, the numbers expressing the relation between the lengths of the undulations, and the rapidity of the undulations, these numbers depending only on the properties of the ethereal medium, and being probably the same in all parts of the universe. The theory of heat gives rise to several numbers of the highest importance, especially Joule’s mechanical equivalent of heat, the absolute zero of temperature, the mean temperature of empty space, &c.
Taking into account the diverse properties of the elements we must have tables of the atomic weights, the specific heats, the specific gravities, the refractive powers, not only of the elements, but their almost infinitely numerous compounds. The properties of hardness, elasticity, viscosity, expansion by heat, conducting powers for heat and electricity, must also be determined in immense detail. There are, however, certain of these numbers which stand out prominently because they serve as intermediate units or terms of comparison. Such are, for instance, the absolute coefficients of expansion of air, water and mercury, the temperature of the maximum density of water, the latent heats of water and steam, the boiling-point of water under standard pressure, the melting and boiling-points of mercury, and so forth.
*Astronomical Constants.*
The third great class consists of numbers possessing far less generality because they refer not to the properties of matter, but to the special forms and distances in which matter has been disposed in the part of the universe open to our examination. We have, first of all, to define the magnitude and form of the earth, its mean density, the constant of aberration of light expressing the relation between the earth’s mean velocity in space and the velocity of light. From the earth, as our observatory, we then proceed to lay down the mean distances of the sun, and of the planets from the same centre; all the elements of the planetary orbits, the magnitudes, densities, masses, periods of axial rotation of the several planets are by degrees determined with growing accuracy. The same labours must be gone through for the satellites. Catalogues of comets with the elements of their orbits, as far as ascertainable, must not be omitted.
From the earth’s orbit as a new base of observations, we next proceed to survey the heavens and lay down the apparent positions, magnitudes, motions, distances, periods of variation, &c. of the stars. All catalogues of stars from those of Hipparchus and Tycho, are full of numbers expressing rudely the conformation of the visible universe. But there is obviously no limit to the labours of astronomers; not only are millions of distant stars awaiting their first measurements, but those already registered require endless scrutiny as regards their movements in the three dimensions of space, their periods of revolution, their changes of brilliance and colour. It is obvious that though astronomical numbers are conventionally called *constant*, they are probably in all cases subject to more or less rapid variation.
*Terrestrial Numbers.*
Our knowledge of the globe we inhabit involves many numerical determinations, which have little or no connection with astronomical theory. The extreme heights of the principal mountains, the mean elevations of continents, the mean or extreme depths of the oceans, the specific gravities of rocks, the temperature of mines, the host of numbers expressing the meteorological or magnetic conditions of every part of the surface, must fall into this class. Many such numbers are not to be called constant, being subject to periodic or secular changes, but they are hardly more variable in fact than some which in astronomical science are set down as constant. In many cases quantities which seem most variable may go through rhythmical changes resulting in a nearly uniform average, and it is only in the long progress of physical investigation that we can hope to discriminate successfully between those elemental numbers which are fixed and those which vary. In the latter case the law of variation becomes the constant relation which is the object of our search.
*Organic Numbers.*
The forms and properties of brute nature having been sufficiently defined by the previous classes of numbers, the organic world, both vegetable and animal, remains outstanding, and offers a higher series of phenomena for our investigation. All exact knowledge relating to the forms and sizes of living things, their numbers, the quantities of various compounds which they consume, contain, or excrete, their muscular or nervous energy, &c. must be placed apart in a class by themselves. All such numbers are doubtless more or less subject to variation, and but in a minor degree capable of exact determination. Man, so far as he is an animal, and as regards his physical form, must also be treated in this class.
*Social Numbers.*
Little allusion need be made in this work to the fact that man in his economic, sanitary, intellectual, æsthetic, or moral relations may become the subject of sciences, the highest and most useful of all sciences. Every one who is engaged in statistical inquiry must acknowledge the possibility of natural laws governing such statistical facts. Hence we must allot a distinct place to numerical information relating to the numbers, ages, physical and sanitary condition, mortality, &c., of different peoples, in short, to vital statistics. Economic statistics, comprehending the quantities of commodities produced, existing, exchanged and consumed, constitute another extensive body of science. In the progress of time exact investigation may possibly subdue regions of phenomena which at present defy all scientific treatment. That scientific method can ever exhaust the phenomena of the human mind is incredible.