based upon this that the simplest ratios are the most common; this is verified in the formulæ of analysis and is found again in natural phenomena, in crystallization, and in chemical combinations. This simplicity of ratios will not appear astonishing if we consider that all the effects of nature are only mathematical results i of a small number of immutable laws.
Yet induction, in leading to the discovery of the general principles of the sciences, does not suffice to establish them absolutely. It is always necessary to confirm them by demonstrations or by decisive experiences; for the history of the sciences shows us that induction has sometimes led to inexact results. I shall cite, for example, a theorem of Fermat in regard to primary numbers. This great geometrician, who had meditated profoundly upon this theorem, sought a formula which, containing only primary numbers, gave directly a primary number greater than any other number assignable. Induction led him to think that two, raised to a power which was itself a power of two, formed with unity a primary number. Thus, two raised to the square plus one, forms the primary number five; two raised to the second power of two, or sixteen, forms with one the primary number seventeen. He found that this was still true for the eighth and the sixteenth power of two augmented by unity; and this induction, based upon several arithmetical considerations, caused him to regard this result as general. However, he avowed that he had not demonstrated it. Indeed, Euler recognized that this does not hold for the thirty-second power of two, which, augmented by unity, gives 4,294,967,297, a number divisible by 641.
