In order to see clearly the meaning of this result, let us suppose that we have to divide unity or 1 1 1 1 successively by the numbers 1, 10' 100' 1000' 1 &c. we will have the quotient 1, 10, 100, 10000' 1000, 10000, &c. continually and indefinitely increasing, because the divisors are continually and indefinitely decreasing; but these divisors tend towards zero, which they cannot attain, although they approach to it continually, or that the difference becomes less and less; and at the same time the value of the fraction increases continually, and tends to that which corresponds to the divisor zero or 0: and it is as much impossible that the fraction in its successive augmentations, attains as it is that the 1 1 0' 1 denominator in its successive diminutions arrives at zero. Thus is the last term or limit of the increasing values of the fraction at this period, it has received all its augmentations: 0 Ꮎ is not therefore a number, it is the superior limit of numbers; such is the notion that we must have of this result which the analysts call for abbrevia 1 0 tion, infinity, and which is denoted by the character co, (Art. 35). It is frequently given as ananswer to an impossible question, (which will be noticed in a subsequent part of the Work,) and in fact, it is very proper to announce this circumstance, since that we cannot assign the number denoted by this sign. It may still be remarked, that if we would take but the first six terms of the series, we must close the developement by the corresponding remainder divided by this divisor, which gives, this equality, absurd in appearance, proves that six terms at least do not hinder the series from being indefinitely continued. And in fact, if after having taken away six terms from this series, it would cease to be infinite, or become terminated, in restoring to it these six terms, it should be composed of a definite or assignable number of terms, which it is not. Therefore the surplus of the series must have the same sum as the total. We can yet say that, inasmuch as it is not a magnitude, can receive no augmentation, so that 1+1+1+, &c. +must remain equal to 1 0 1 Hence, we might conclude that a finite quantity added to, or subtracted from infinity, makes no alteration. Thus, a∞. However, it may be necessary in this place to observe, that, although an infinity cannot be increased, or decreased, by the addition, or subtraction, of finite quantities; still, it may be increased or decreased, by multiplication or division; in the same manner as any other quantity; Thus, if be equal to infinity, thrice, and so on. 2 will be the double of it, See EULER'S Algebra, Vol. I. 1 3 10 100 to be the fractions, in which the denominators are Now, as 1 divided by any assignable quantity, however great it may be, can never arrive completely at 0, consequently the fractions in their successive augmentations can never arrive at infinity, except that unity or 1, be divided by a quantity infinitely great; that is to say, it must be divided by infinity; hence we may conclude that is in reality equal 1 ∞ 166. It may not be improper to take notice in this place of other properties of nought and infinity. I. That nought added to or subtracted from any quantity, makes it neither greater nor less; that is, a+0=a, and a-0-a. II. Also, if nought be multiplied or divided by any quantity, both the product and quotient will be nought; because any number of times 0, or any part of 0, is 0: that is, Oxa, or a X0=0, and 0. α III. From the last property, it likewise follows, that nought divided by nought, is a finite quantity, of some kind or other. For since 0 Xa, or 0= Oxa, it is evident from the ordinary rules of divi sion, that IV. Farther, if nought be multiplied by infinity, the product will be some finite quantity. For since 1 α or∞ ; therefore, OX∞ =α. 0 0 167. It may be also remarked, that nought multiplied by 0 produces 0; that is, 0X0=0. For, since 0 Xa=0, whatever quantity a may be, then, supposing a=0, 0X0=0. From this we might infer, according to the rules 0 of division, that the value of=0, or that nought 0 divided by nought is nought, in this particular case: Also, that 0, raised to any power, is 0; that is, 0m 0 0=0; it follows that but if in am Om 0 am a" (Art. 86), we suppose a=0, which may be allowed, since a designates any number, we have 0o 0 0 If we really effect the division of 0 by 0, we could put for the quotient any number whatever, since any number, multiplied by zero, gives for the product zero, which is here the dividend. 0 This expression, 0°, appears therefore to admit of an infinity of numerical values; and yet such a result as can, in many cases, admit of a finite and determined value. It is thus, for example, that the Κατ fraction in the hypothesis of a=0, becomes 2 an KX0 0 0 Ο But, if at first we write this fraction under the form Kam", and that we put a=0, we find that it becomes K×0m-", which is 0 for m>n; in case of m<n, or m=n-d, we shall have (Art. 86), K K = ; Od 0 which is equal to infinity, as has been already observed; finally, for m=n, we can divide above and below by am, and the fraction is reduced to K, which is a finite quantity. 168. If we suppose, in the fraction (Art. 164), a=2, we find 1 1-2 =-1=1+2+4+8+16+32+64+, &c., which at first sight will appear absurd. But it must be remarked, that if we wish to stop at any term of the above series, we cannot do so without joining the fraction which remains. Suppose, for example, we were to stop at 64; after having written 1+2 +4+8+16+32+64, we must join the fraction 128 1-2' 128 or, or -123; we shall therefore have for the complete quotient 127-128, that is in fact 1. Here, however far the fractional term may be extended, its numerical value, which is negative, will always surpass, by a unit, that of the integral part, so that this is totally destroyed; and as in the hypothesis of a>1, we shall always subtract more than what we will add, we shall never meet with 169. These are the considerations which are necessary when we assume for a numbers greater than unity; but if we now suppose a less than 1, the whole becomes more intelligible; for example, let |