EXPONENT, a modern term used in algebra to indicate the number of times the expression to which it refers is to be taken as a factor. For example, in the monomial the exponent of x is 2 and it states how many times x is taken as a factor; that is, means 3xx. Although the idea of power of a number is very old, and the use of the exponent (but in a form unlike the present one) is found as early as the i4th century, the actual form first appears in Descartes's La Geometrie (1637). The early concept of exponent was broadened as algebra developed, and gradually meanings were found for such forms as a°, at, aNI and so on, the exponent being any kind of number whatso ever. As new types of exponent were suggested, the question arose as to what meanings were to be assigned to them so that the fundamental laws of positive integral exponents should be retained. This desideratum led to defining a° in such a way that the law ama" = am+" should remain valid; that is, a°a" must con tinue to be a°+", or a". Evidently this required that a° should be equal to 1, excepting when a = o. Similarly, al came to mean -I a, al to mean and to mean 1/a". These facts had long been known, but without our modern exponential symbols. In like manner meanings came to be assigned to a\ 2 and even to Indeed, one of the most interesting relations, and most fruitful, is that which asserts that e"t = —I, where e is the base of the natural system of logarithms, r is the incommensurable ratio of the cir cumference of a circle to the diameter, i= — 1, and — 1 is the negative unit. It is by means of this relation, to which the name of Euler is attached, but which is easily derived from the relation Oi=log (cos 4'+i sin 4), due to Cotes (1722), that it can be shown that every number has an infinite number of logarithms, only one being real. Even before the time of Cotes it appears that De Moivre (1 707) knew a similar relation. The first algebraist to explain fully the significance of negative and fractional ex ponents was John Wallis (1655), his work being supplemented soon after (1669) by Newton. (D. E. S.)