The question first of all arises, How may we describe this series of numbers? What is uniformly true of them? The reader cannot fail to perceive at the first glance that they all end in five, and the problem is, from the properties of these six numbers, to infer the properties of the next number ending in five. If we test their properties by the process of perfect induction, we soon perceive that they have another common property, namely that of being *divisible by five without remainder*. May we then assert that the next number ending in five is also divisible by five, and, if so, upon what grounds? Or extending the question, Is every number ending in five divisible by five? Does it follow that because six numbers obey a supposed law, therefore 376,685,975 or any other number, however large, obeys the law? I answer *certainly not*. The law in question is undoubtedly true; but its truth is not proved by any finite number of examples. All that these six numbers can do is to suggest to my mind the possible existence of 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 *x*^{2} + *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 *x*^{2} + *x* + 41 should give primes.