such a law; and I then ascertain its truth, by proving deductively from the rules of decimal numeration, that any number ending in five must be made up of multiples of five, and must therefore be itself a multiple.

To make this more plain, let the reader now examine the numbers

7, 17, 37, 47, 67, 97.

They all end in 7 instead of 5, and though not at equal intervals, the intervals are the same as in the previous case. After consideration, the reader will perceive that these numbers all agree in being prime numbers, or multiples of unity only. May we then infer that the next, or any other number ending in 7, is a prime number? Clearly not, for on trial we find that 27, 57, 117 are not primes. Six instances, then, treated empirically, lead us to a true and universal law in one case, and mislead us in another case. We ought, in fact, to have no confidence in any law until we have treated it deductively, and have shown that from the conditions supposed the results expected must ensue. No one can show from the principles of number, that numbers ending in 7 should be primes.

From the history of the theory of numbers some good examples of false induction can be adduced. Taking the following series of prime numbers,

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c., it will be found that they all agree in being values of the general expression x2 + x + 41, putting for a in succession the values, 0, 1, 2, 3, 4, &c. We seem always to obtain a prime number, and the induction is apparently strong, to the effect that this expression always will give primes. Yet a few more trials disprove this false conclusion. Put x 40, and we obtain 40 × 40 + 40 + 4!, or 41 x 41. Such a failure could never have happened, had we shown any deductive reason why a2 + x + 41 should give primes.

There can be no doubt that what here happens with forty instances, might happen with forty thousand or forty million instances. An apparent law never once failing up to a certain point may then suddenly break down, so that inductive reasoning, as it has been described by some writers, can give no sure knowledge of what is to come. Babbage pointed out, in his Ninth Bridgewater

Treatise, that a machine could be constructed to give a perfectly regular series of numbers through a vast series of steps, and yet to break the law of progression suddenly at any required point. No number of particular cases as particulars enables us to pass by inference to any new case. It is hardly needful to inquire here what can be inferred from an infinite series of facts, because they are never practically within our power; but we may unhesitatingly accept the conclusion, that no finite number of instances can ever prove a general law, or can give us certain knowledge of even one other instance.

General mathematical theorems have indeed been discovered by the observation of particular cases, and may again be so discovered. We have Newton's own statement, to the effect that he was thus led to the all-important Binomial Theorem, the basis of the whole structure of mathematical analysis. Speaking of a certain series of terms, expressing the area of a circle or hyperbola, he says: "I reflected that the denominators were in arithmetical progression; so that only the numerical co-efficients of the numerators remained to be investigated. But these, in the alternate areas, were the figures of the powers of the number eleven, namely 11o, 111, 112, 113, 114; that is, in the first I; in the second I, I; in the third 1, 2, 1; in the fourth 1, 3, 3, I; in the fifth 1, 4, 6, 4, 1.1 I inquired, therefore, in what manner all the remaining figures could be found from the first two; and I found that if the first figure be called m, all the rest could be found by the continual multiplication of the terms of the formula

It is pretty evident, from this most interesting statement, that Newton, having simply observed the succession of the numbers, tried various formulæ until he found one which agreed with them all. He was so little satisfied with this process, however, that he verified particular results of his new theorem by comparison with the results of common

1 These are the figurate numbers considered in pages 183, 187, &c. 2 Commercium Epistolicum. Epistola ad Oldenburgum, Oct. 24, 1676. Horsley's Works of Newton, vol. iv. p. 541. See De Morgan in Penny Cyclopædia, art. " Binomial Theorem," p. 412.

multiplication, and the rule for the extraction of the square root. Newton, in fact, gave no demonstration of his theorem; and the greatest mathematicians of the last century, James Bernoulli, Maclaurin, Landen, Euler, Lagrange, &c., occupied themselves with discovering a conclusive method of deductive proof.

There can be no doubt that in geometry also discoveries have been suggested by direct observation. Many of the now trivial propositions of Euclid's Elements were probably thus discovered, by the ancient Greek geometers; and we have pretty clear evidence of this in the Commentaries of Proclus. Galileo was the first to examine the remarkable properties of the cycloid, the curve described by a point in the circumference of a wheel rolling on a plane. By direct observation he ascertained that the area of the curve is apparently three times that of the generating circle or wheel, but he was unable to prove this exactly, or to verify it by strict geometrical reasoning. Sir George Airy has recorded a curious case, in which he fell accidentally by trial on a new geometrical property of the sphere.2 But discovery in such cases means nothing more than suggestion, and it is always by pure deduction that the general law is really established. As Proclus puts it, we must pass from sense to consideration.

Given, for instance, the series of figures in the accompanying diagram, measurement will show that the curved lines approximate to semicircles, and the rectilinear figures to right-angled triangles. These figures may seem to suggest to the mind the general law that angles inscribed

1 Bk. ii. chap. iv.

2 Philosophical Transactions (1866), vol. 146, p. 334

in semicircles are right angles; but no number of instances, and no possible accuracy of measurement would really establish the truth of that general law. Availing ourselves of the suggestion furnished by the figures, we can only investigate deductively the consequences which flow from the definition of a circle, until we discover among them the property of containing right angles. Persons have thought that they had discovered a method of trisecting angles by plane geometrical construction, because a certain complex arrangement of lines and circles had appeared to trisect an angle in every case tried by them, and they inferred, by a supposed act of induction, that it would succeed in all other cases. De Morgan has recorded a proposed mode of trisecting the angle which could not be discriminated by the senses from a true general solution, except when it was applied to very obtuse angles.1 In all such cases, it has always turned out either that the angle was not trisected at all, or that only certain particular angles could be thus trisected. The trisectors were misled by some apparent or special coincidence, and only deductive proof could establish the truth and generality of the result. In this particular case, deductive proof shows that the problem attempted is impossible, and that angles generally cannot be trisected by common geometrical methods.

Geometrical Reasoning.

This view of the matter is strongly supported by the further consideration of geometrical reasoning. No skill and care could ever enable us to verify absolutely any one geometrical proposition. Rousseau, in his Emile, tells us that we should teach a child geometry by causing him to measure and compare figures by superposition. While a child was yet incapable of general reasoning, this would doubtless be an instructive exercise; but it never could teach geometry, nor prove the truth of any one proposition. All our figures are rude approximations, and they may happen to seem unequal when they should be equal, and equal when they should be unequal. Moreover figures may from chance be equal in case after case, and

1 Budget of Paradoxes, p. 257.

SO.

yet there may be no general reason why they should be The results of deductive geometrical reasoning are absolutely certain, and are either exactly true or capable of being carried to any required degree of approximation. In a perfect triangle, the angles must be equal to one halfrevolution precisely; even an infinitesimal divergence would be impossible; and I believe with equal confidence, that however many are the angles of a figure, provided there are no re-entrant angles, the sum of the angles will be precisely and absolutely equal to twice as many rightangles as the figure has sides, less by four right-angles. In such cases, the deductive proof is absolute and complete; empirical verification can at the most guard against accidental oversights.

There is a second class of geometrical truths which can only be proved by approximation; but, as the mind sees no reason why that approximation should not always go on, we arrive at complete conviction. We thus learn that the surface of a sphere is equal exactly to two-thirds of the whole surface of the circumscribing cylinder, or to four times the area of the generating circle. The area of a parabola is exactly two-thirds of that of the circumscribing parallelogram. The area of the cycloid is exactly three times that of the generating circle. These are truths that we could never ascertain, nor even verify by observation; for any finite amount of difference, less than what the senses can discern, would falsify them.

There are geometrical relations again which we cannot assign exactly, but can carry to any desirable degree of approximation. The ratio of the circumference to the diameter of a circle is that of 3.14159265358979323846.... to I, and the approximation may be carried to any extent by the expenditure of sufficient labour. Mr. W. Shanks has given the value of this natural constant, known as π, to the extent of 707 places of decimals. Some years since, I amused myself by trying how near I could get to this ratio, by the careful use of compasses, and I did not come nearer than I part in 540. We might imagine measurements so accurately executed as to give us eight or ten places correctly. But the power of the hands and

1 Proceedings of the Royal Society (1872-3), vol. xxi. p. 319.