. Differential and integral calculus, an introductory course for colleges and engineering schools. ng limits, we have equations (3) as before. Equations (4) are not accurately true: the true equations are(1) and (2) above. Moreover, the use of the inaccurate equations(4) is permissible, not because, when the triangle PBQ is very small,As cannot be distinguished by the eye from a right line, and equations(4) seem then very close approximations to the truth, but because, byvirtue of our theorem of infinitesimals, equations (4) lead to resultsthat are absolutely true. We may now go a step farther
. Differential and integral calculus, an introductory course for colleges and engineering schools. ng limits, we have equations (3) as before. Equations (4) are not accurately true: the true equations are(1) and (2) above. Moreover, the use of the inaccurate equations(4) is permissible, not because, when the triangle PBQ is very small,As cannot be distinguished by the eye from a right line, and equations(4) seem then very close approximations to the truth, but because, byvirtue of our theorem of infinitesimals, equations (4) lead to resultsthat are absolutely true. We may now go a step farther in the substitution of one infin- itesimal for another. For it can be shown that lim df{x) = 1 (let the student give the proof), and therefore the increment of a function can be replaced by its dif-ferential in that kind of problemto which our theorem of infinitesi-mals applies. Hence in the firstfigure of this article Ay may bereplaced by dy and As by ds. Thatis, the triangle PBT may be re-garded as coincident with the tri-angle PBQ. Then from the accompanying figure we may writedown at once. ds = Vdx2 + dx ds cos 6, dy • o dy sin d, -j-dx tan0. From the foregoing illustrations it is obvious that there is advan-tage in putting our theorem of infinitesimals into the followinggeometrical form: In any geometrical problem which involves the taking of the limitof the ratio of the lengths of infinitesimal lines, either straight orcurved, any such line may be replaced by any other infinitesimal line(straight or curved), provided the limit of their ratio is unity. 103 THE METHOD OF INFINITESIMALS 147 Let us apply this principle to obtain the formula? of Art. the figure of that article it is plain that PB = psinA0, PR = pA0, lim PB ,. p sin A 6hm ^r- = 1. PR p&d Therefore PB may be replaced by PR. Again, r>^ a , /, a m ^^ a BQ „ 1—cosA0 A#BQ= Ap + p(l-cosA0), RQ = AP, ^| = 1 + p ^ — whence lim RQ 1, and BQ may be replaced by RQ. We already know that c may bereplace
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