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 x 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 + 41, or 41 × 41. Such a failure could never have happened, had we shown any deductive reason why x2 + 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 110, 111, 112, 113, 114; that is, in the first 1; in the second 1, 1; in the third 1, 2, 1; in the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.[138] 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
m - 0/1 × m - 1/2 × m - 2/3 × m - 3/4 × &c.”[139]
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 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.[140] 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.[141] 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.
