Isaac Newton

“ For DB may be taken as AD3: in which case no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. And by a like reasoning, if DB be made successfully as AD4, AD5, AD6, AD7, &c., we shall have a series of angles of contact, proceeding in infinitum, wherein every succeeding term is infinitely less than the preceding. ”

“ If in comparing indetermined quantities of different sorts with another, any one is said to be as any other directly or inversly: the meaning is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. and if any one is said to be as any other two or more directly or inversly: the meaning is, that the first is augmented or diminished in the ratio's in which the others, or the reciprocals of the others, are augmented or diminshed. ”

“ Let the curve Abc be similar to the curve ABC, and draw the right line Ag so as to touch both curves in A, and cut the ordinates DB, EC, db, ec, in F, G, f, g. Then supposing the length Ae to remain the same, let the points B and C meet the point A; and the angle cAg vanishing, the curvilinear areas Abd, Ace will coincide with the rectilinear areas Afd, Age; and therefore (by Lem 5) will be one to other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas ”

“ But by the same argument it may be alledged, that a body arriving at a certain place, and there slopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity ”

“ Pl. 2. Fig. 1. The same things being supposed, I say that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality. Pl. 2. Fig. 1.
For while the point B approaches the point A, consider always AB and AD as produc'd to the remote point b and d; and parallel to the secant BD draw bd; and let the arc Acb be always similar to the arc ACB. Then, supposing the points A and B to coincide, the angle dAb will vanish, by the preceding lemma ”

“ For demonstrations are more contracted by the method of indivisibles: but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratio's of nascent and evanescent quantities, that is, to the limits of those sums and ratio's ”