rules that have been given, nor from continuing it as far as we please, and we shall thus not fail to find the true quotient, though under different forms. 161. To prove this, let us actually divide a° or 1, by 1-a, thus ; Now, by considering the first of these formula, which is 1 If we follow the same process with regard to the second expression, that is to say, if we reduce the integral part 1+ a to the same denominator, 1-a, we shall have the fraction 1-a2 a2 to which if we add we shall have 1 α 1-a2+a2 1- -a 1 a In the third formula of the quotient, the integers 1+a+a2 1-a3 reduced to the denominator 1— a make and if we add Therefore each of these formulæ is in fact the value of 162. This being the case, we may continue the series as far as we please, without being under the necessity of performing any more calculations; by observing, in the first place, that each of these formulæ is composed of an integral part which is the sum of the successive powers of a, beginning with a 1 inclusively; Secondly, of a fraction which has always for the denominator 1-a, and for the numerator the letter a, with an exponent greater, by unity, than that of the same letter in the last term of the integral part. This constant formation of the successive formulæ, is what Analysts call a law. And the manner of deducing general laws by the consideration of certain particular cases, is usually called induction; which, though not a strict method of proof, says LAPLACE, has been the source of almost all the discoveries that have hitherto been made, both in analysis and physics, of which all the phenomena are the mathematical results of a small number of invariable laws. It is thus that NEWTON, by following the law of the numeral coefficients, in the square, the cube, the fourth power, &c. of a binomial, arrived soon at the general law, that is to say, at the general formula, that bears his name, and which will be demonstrated in one of the following Chapters: This Geometer has carefully added, that in following this mode of investigation, we must not generalize too bastily; as it often happens, that a law, which appears to take place in the first part of a process, is not found to hold good throughout. Thus, in the simple instance of reducing to a decimal, its equivalent va 531251 lue is 17174949, &c., of which the real, repeating period is 49, and not 17, as might, at first, be imagined. 163. From what has been observed with regard to the successive quotients, (Art. 161), we can, in general, put an+1 l-a' n being a whole positive number, which augmented by unity, gives the place of the term. In fact, making n=3, an becomes a3, which is the fourth term of the quotient; for n=4, an becomes a1, which is the fifth term. But as nothing hinders us from removing indefinitely the fractional term which terminates the series, that is, of adding always a term to the integral part; so that we might still go on without end; for which reason it may be said that the proposed fraction has been resolved into an infinite series; which is, 1+a+a2+a® +aa+a+ao+a1+a®+ao+a1o+a"+a12+, &c. to infinity: and there are sufficient grounds to maintain that the value of this infinite series is the same as that of the fraction Or that, 1 =1+a+a2+a3+aa+ ; &c. 1 1. -α 164. What has been just observed may at first appear strange; but the consideration of some particular cases will make it easily understood. Let us suppose, in the first place, a=1; the general quotient above will become a particular quotient corresponding In order to see clearly the meaning of this result, let us suppose that we have to divide unity or 1 successively by the quotient, 1, 10, 100, 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 denominator in its 1 Ο successive diminutions arrives at zero. Thus 10 is the last 1 0 term or limit of the increasing values of the fraction: at this period, it has received all its augmentations: is not therefore a number, it is the superior limit of numbers; such is the notion that we must have of this result which 0' 1 the analysts call for abbreviation, infinity, and which is denoted by the character ∞, (Art. 35). It is frequently given as an answer 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 0 1 Hence, we might conclude that a finite quantity added to, or subtracted from infinity, makes no alteration. Thus, ±α=∞. ∞ 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 1 same manner as any other quantity; Thus, if be equal to 2 3 0 infinity, will be the double of it, thrice, and so on. See 0 tions, in which the denominators are 1, 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 to nothing, or 1 ∞ 165. 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, 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=0, or 0=0Xa, it is evident from the ordinary rules of division, that 1 α IV. Farther, if nought be multiplied by infinity, the product will be some finite quantity. For since or co; 0 0 therefore, OX∞ =α. 166. It may be also remarked, that nought multiplied by O produces 0; that is, 0X0=0. 'For, since 0Xa=0, whatever quantity a may be, then, supposing a=0, 0X0=0. From this we might infer, according to the rules of division, that the value of 0 0, or that nought divided by nought is nought, in this particular case. Also, that 0, raised to any power, is 0; that is, 0m=0; it Om 0 am follows that = ; but if in am-m (Art. 86), we sup Om 0 am pose a=Q, 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 oe This expression, 0°, appears therefore to admit of an infinity of numerical values; and yet such a result as many cases, admit of a finite and determined value. It is thus, can, in |