14 INFINITESIMAL CALCULUS however small, and which can be diminished indefinitely, so as to approach as near as we please to zero, without ever absolutely attaining to it. For instance, the difference between the area of a circle and that of an inscribed regular polygon can, by increasing the number of sides of the polygon, be made less than any assigned area, however small ; but, no matter how largo the number of sides may be, this difference can never become absolutely zero. It would be easy to give other illustrations of the sense in which the word infinitesimal is employed in analysis. 11. Again, if a be regarded as an infinitesimal of the first order, a 2 , being infinitely small in comparison with a, is regarded as an infinitesimal of the second order. In like manner a 3 , a 4 , ... a", may be regarded as representing infinitesimals of the third, fourth . . . ?ith orders, respectively. Again, two infinitesimals a, are said to be of the same order if a Q the fraction - tends to a finite limit. If tends to a finite limit, a a" /3 is called an infinitesimal of the nth order in comparison with a. 12. To avoid misconception, it should be borno in mind that infinitesimals are not regarded as being actual quantities in the ordinary acceptation of the words, or as capable of exact representa tion. They are introduced for the purpose of abridgment and simplification of our reasonings, and are an ultimate phase of mag nitude when it is conceived by the mind as capable of diminution below any assigned quantity, however small. Such magnitudes are in all cases, as styled by Carnot, auxiliary quantities, introduced for the purpose of facilitating our investigations, but they should disappear from our final results. We shall illustrate this statement by the example of drawing a tangent to a curve, in which problem the method of infinitesimals may be said to have originated. We introduce the infinitesimals dx and dy, for the purpose of finding their ultimate ratio, i.e., in order to determine the limit of -J-. Now this limit is in all cases dx a function of x and y, the coordinates of the point of contact, and cannot contain in it either dx or dy, since they must be taken as evanescent quantities when we proceed to the limit. Likewise in all other applications of infinitesimals in the differen tial calculus, we endeavour to find the ultimate ratio of two indefinitely small quantities, or infinitesimals ; and it is unnecessary to attach any precise meaning to such infinitesimals during the course of our investigations, further than to regard them as variable quantities, which become evanescent when we proceed to our final results. In employing infinitesimals in such cases, we proceed on the principle that the limit of the ratio of two infinitesimals, a and ft, is the same as that of o and & , provided the limit of => 1, and limit a 0> of = 1. This is evident since, in all cases, we have a a a /3 a result which must hold in the limit. In consequence of this principle, before proceeding to the limit, we may neglect an infinitesimal of any order in comparison with one of a lower order. For instance, in seeking the ultimate ratio of 7 to 5, where 7 = A x cc + A 2 a 2 + A 3 a 3 + &c. + A n a n + &C. , 8 = Bi/3 + B 2 j8 2 + B 3 3 + &c. + B,, jB" + &c. , in which A 1( A 2 , A 3 . . . Bj, B.,, &c., are finite, and independent of the infinitesimals o and , we may neglect a 2 , a 3 , /3 3 , 3 . . . in com parison with a and ft, and we get the limit of ^- = i x limit of . o Bj & In general, if a, # be infinitesimals of the same order, their ratio has a finite magnitude; and if y=/ 1 (a, )9), 8=/ 2 (a,/3), then in finding the limiting value of -?. we take the terms of the lowest 5 order in o and in /j and / 2> neglecting all infinitesimals of higher orders ; substituting in the result the limiting value of , we obtain the required limit for -2- . 8 13. Again, if Oi + e^ . . . +etn represents the sum of a number of infinitely small quantities which approaches a finite limit when n is indefinitely increased, and if fa, 0. 2 . . . , be another system of infinitely small quantities, such that O! a, a, where 6 1} e 2 . . . f n become evanescent in the limit, then, when n is indefinitely increased, the limit of the sum of fa, /8 2 . . . is equal to that of a 1} a. 2 . . .a,,. This is evident from the elementary algebraic O i O i Q principle that the ratio 53 _" H C s between the greatest and Oj + Oo . . . + OM the least values of the fractions -- 1 , --- . . . ; and it accordingly aj a 2 a,t has unity for its limit, under the supposed conditions. For example, suppose any magnitude divided into a number of parts, and that each is capable of subdivision into two parts, one of which can be simply found, and the other not so. Let A + a be the first part, of which A is of the former species, and a of the latter. In like manner let B + &, C + c, &c., be the other parts. Then the re quired magnitude is represented by A+B+C+ &c. +a + b + c + &c. Now suppose that when a sufficiently great number of parts is taken we can make a, b, c, &c. , as small as we please in comparison with A, B, C, &c., then a + b + c+ &c., can be made as small as we please wdth respect to A-f B + C+ &c. ; consequently by continuing the process indefinitely, the limit of the sum of A + B + C+ &c., is equal to the required magnitude, without the necessity of paying any attention to the remaining parts. This latter may be regarded as the fundamental principle of the integral calculus, and the former, given in 12, as that of the differential. 14. In consequence of metaphysical objections to the employment of infinitesimals, many writers on the calculus have confined them selves exclusively to the method of limits or limiting ratios, and by so doing have in many cases involved themselves in long and cum brous demonstrations of theorems which follow with great facility by the adoption of infinitesimals. In reality the difference between the method of infinitesimals and that of limits (when exclusively adopted) is, that in the latter method it is usual to retain evanescent quantities of higher orders until the end of the calculation, and then to neglect them. On the other hand, such quantities are neglected from the commencement in the infinitesimal method, from the con viction that they cannot affect the final result, as they must neces sarily disappear when we proceed to the limit. A very little re flexion will show that the result obtained in both cases must be the same. Moreover such quantities are neglected, not, as Leibnitz stated, because they are infinitely small in comparison with those that are retained, which would produce an infinitely small error, but because they must be neglected to obtain a rigorous result ; since such result must be definite and determinate, and consequently in dependent of these variable indefinitely small quantities. It may be added that the precise principles of the infinitesimal calculus, like those of any other science, cannot be thoroughly apprehended except by those who have already studied the science, and made some progress in the application of its principles. 15. The preceding statements may also be regarded in connexion with the different meanings of the terms "zero" and "an evanes cent quantity." There is but one process in arithmetic which yields an absolute zero, namely subtraction, thus - = 0. But from no other arithmetical process does zero arise, except by the same train of ideas as leads us to the use of the word infinite. Wo cannot, for example, obtain the quotient zero by dividing one finite magnitude by another. We can make the result as small as AVO please, but not absolutely zero. When, therefore, we consider an equation made by addition or subtraction of terms, the absolute zero may be used without reservation, thus 2x + a = b, and 2x + a - b = may be substituted for each other without any particular examina tion of the symbol 0. But in any other case we consider zero or as the limit towards which we approach by a series of diminutions, none of which is final. Thus when we see that we can nearly arrive at a certain conclusion by attributing a small value to a particular magnitude, that wo can more nearly attain this con clusion by attributing to it a smaller value, and so on without limit, i.e., that we can approximate to this conclusion as nearly as we please by the use of a value as small as we please, but that we never attain it as long as the magnitude has any finite amount, then such conclusion is said, for abbreviation, to be absolutely true when the magnitude is nothing or zero. These considera tions will help to explain the sense in which Euler was correct when he stated that dx and dy in the calculus must be regarded each as zero. 16. We now return to the consideration of the method of finding the derived functions or differential coefficients of the different forms of functions of x. Before doing so, however, it will be necessary to establish two or three general principles. We commence with the differentiation of a product. Let y = uv, where u and v are functions of a; ; then Ay = (u + Au)(v + Av) - uv Ay _ Ax Ax Av , . Au
- + (v + AV)
Ax proceeding to the limit, this becomes d(uv] dv du dx dx dx In like manner, if y = umv, we get dy du dv dw = vw -j- + wu j- + U-D-J- dx dx dx dx
