1 1+a is a positive quantity less than 1 ; let 1 + x = where 1+2 z is a small positive quantity less than 1, and becomes infinitesimal simultaneously with ; then which, when z is infinitesimal, becomes e by the former part of the Lemma. Hence we conclude that when a is a small positive or nega 1 tive quantity, approximating to 0, (1+x) approximates to the value e; that is, differs from e by a quantity less than any assignable quantity, when a diminishes without limit; that is, when a is an infinitesimal, (1 + x) * = e. COROLLARY I. Hence, taking Napierian logarithms of both sides of the last equation, 1 loge (1+x)=log, e = 1; .. log. (1+x) = x, when x is an infinitesimal. COR. II. And taking logarithms to the base a, we have (11) COR. III. As the above determination of the value of e is independent of any previously-expanded logarithmic series, it is well to shew how other results follow from it. cancelling the units, dividing through by y, making y infinitesimal, and equating the finite terms, we have 22.] LEMMA II. To prove that tan x, x, and sin x may be used indifferently for each other, when a is an infinitesimal arc, and the radius is finite. Let AP (see fig. 2) be the arc of a circle whose radius = 1, and let the arc AP = x, x being the circular measure of the angle PCA; let AT be the tangent and PM the sine of the arc. At P draw a tangent to the circle, viz. PT', and draw the other lines as in the figure; then, since TPT is a right angle, TT' is greater than PT', .. ATAT+T'P; and because two sides of a triangle are greater than the third, RT' + T'S > RS; à fortiori AT > AR+RS+SP: and similarly, if tangents are drawn to the arc at points between A and Q and Q and P, it may be shewn that the sum of all the lines similar to SR is less than AT; but the limit of all such lines is the circular arc; therefore AT is greater than the arc. Again, the chord AP is greater than PM, which is the sine of x, and PQ+QA is greater than AP; therefore PQ+QA > PM: and, drawing other chords from A and P to intermediate points on the arcs, it may be shewn that the sum of such chords is greater than the chord AP, and therefore, à fortiori, than PM; and as the arc AP is the limit of all such chords, it follows that it is greater than the sine PM. Therefore the arc is less than its tangent and greater than its sine. Again, bearing in mind that cos x by its definition = 1, when a is an infinitesimal, we have sin x tan x = cos x = 1, when x is an infinitesimal; whence it follows that sina tan x, when x is an infinitesimal; and since x is, as above shewn, always intermediate to these, it is clear that all three are equal to, and therefore may be used indifferently for, each other. Hence, when x is an infinitesimal, sin xx tan x; which proposition is frequently expressed in the form, (13) The limiting ratio of the sine, the arc, and the tangent is that of equality. This result is also involved in the former Lemma. The exponential value of the sine gives us, To evaluate e2-1-1, when x is an infinitesimal; when z is an infinitesimal, by Cor. I, Lemma I. Hence e2x√—1_1 = 2x √−1 ; COR. I. When x is an infinitesimal, chord x = x. that is, the chord of an infinitesimal arc is equal to the arc; which is the VIIth Lemma of the first Section of Book I of Newton's Principia. COR. II. Hence also, if x is infinitesimal, -1 sin-1x = x = tan-1x: (14) that is, the sine and the tangent may be used indifferently for the arc, when the arc is infinitesimal. 23.] Although it may be beside our defined path, yet it is worth while to shew, that the arc and the sine are equal, when the arc is infinitesimal, only by omitting terms of a higher order, and which must be neglected in accordance with Theorem VI, Art. 9. Assuming the validity of the trigonometrical proof of the series, sin x = x X3 x5 ... if x is an infinitesimal, x3, x5, ... must be neglected as they are algebraically added to x, and we have it follows by reason of Theorem VI, Article 9, that, if x is infinitesimal, x2 versin x = ; and therefore, if the arc is an infinitesimal of the first order, the versed-sine of it is an infinitesimal of the second order. The geometrical proof of this truth is so manifest, that it is unnecessary to do more than to suggest it to the student. SECTION 3.-Examples of the Infinitesimal method. 24.] The principles above explained are sufficient for a variety of problems; examples of which are subjoined, to give the student an insight into the kind of processes which he has to perform. Ex. 1. To differentiate x2, that is, to determine the change of x2 due to an infinitesimal change of value of x. omitting the term (dx)2, which can have no value, because it is an infinitesimal of the second order, and added to one of the first order. dy Hence = 2x, and therefore if, in accordance with the dx notation of derived functions, f(x) = x2, d.f(x) = f'(x) = 2x; which result, and others of a similar kind, as will be shewn hereafter, justifies the presumption of Article 18, that the ratio of the infinitesimal variation of f(x) to that of x is a finite quantity. The above process may thus be explained geometrically : Let a represent the straight line AP, fig. 3; and suppose AP to be increased by PQ, which is represented by ax; then from the figure it is plain that the square is increased by the rectangles DB, BQ, and the square BR, the values of which are x × ▲x, X×▲X, (^x)2; whence Ay = A.x2 = 2x ▲x+(x)2. Now let PQ be infinitesimal, that is, let ar become dx, whereby also ay becomes dy, and we have dy =d.x22x dx + (dx)2; but (da) must be omitted for the following reason: x dx sym |