34. If it is said that fluxions may be expounded or expressed by finite lines proportional to them; which finite lines, as they may be distinctly conceived and known and reasoned upon, so they may be substituted for the fluxions, and their mutual relations or proportions be considered as the proportions of fluxions - by which means the doctrine becomes clear and useful. I answer that if, in order to arrive at these finite lines proportional to the fluxions, there be certain steps made use of which are obscure and inconceivable, be those finite lines themselves ever so clearly conceived, it must nevertheless be acknowledged that your proceeding is not clear nor your method scientific. For instance, it is supposed that AB being the abscissa, BC the ordinate, and VCH a tangent of the curve AC , Bb or CE the increment of the abscissa, Ec the increment of the ordinate, which produced meets VH in the point T and Cc the increment of the curve. The right line Cc being produced to K , there are formed three small triangles, the rectilinear CEc , the mixtilinear CEc , and the rectilinear triangle CET . It is evident that these three triangles are different from each other, the rectilinear CEc being less than the mixtilinear CEc , whose sides are the three increments above mentioned, and this still less than the triangle CET . It is supposed that the ordinate bc moves into the place BC , so that the point c is coincident with the point C ; and the right line CK , and consequently the curve Cc , is coincident with the tangent CH . In which case the mixtilinear evanescent triangle CEc will, in its last form, be similar to the triangle CET : and its evanescent sides CE , Ec and Cc , will be proportional to CE , ET and CT , the sides of the triangle CET . And therefore it is concluded that the fluxions of the lines AB , BC , and AC , being in the last ratio of their evanescent increments, are proportional to the sides of the triangle CET , or, which is all one, of the triangle VBC similar thereunto. [`Introd. ad Quadraturam Curvarum.'] It is particularly remarked and insisted on by the great author, that the points C and c must not be distant one from another, by any the least interval whatsoever: but that, in order to find the ultimate proportions of the lines CE , Ec , and Cc ( i.e. the proportions of the fluxions or velocities) expressed by the finite sides of the triangle VBC , the points C and c must be accurately coincident, i.e. one and the same. A point therefore is considered as a triangle, or a triangle is supposed to be formed in a point. Which to conceive seems quite impossible. Yet some there are who, though they shrink at all other mysteries, make no difficulty of their own, who strain at a gnat and swallow a camel.
35. I know not whether it be worth while to observe, that possibly some men may hope to operate by symbols and suppositions, in such sort as to avoid the use of fluxions, momentums, and infinitesimals, after the following manner. Suppose x to be one abscissa of a curve, and z another abscissa of the same curve. Suppose also that the respective areas are xxx and zzz : and that z - x is the increment of the abscissa, and zzz - xxx the increment of the area, without considering how great or how small those increments may be. Divide now zzz - xxx by z - x , and the quotient will be zz + zx + xx : and, supposing that z and x are equal, the same quotient will be 3 xx , which in that case is the ordinate, which therefore may be thus obtained independently of fluxions and infinitesimals. But herein is a direct fallacy:
for in the first place, it is supposed that the abscissae z and x are unequal, without such supposition no one step could have been made; and in the second place, it is supposed they are equal; which is a manifest inconsistency, and amounts to the same thing that hath been before considered. [Sect. 15.] And there is indeed reason to apprehend that all attempts for setting the abstruse and fine geometry on a right foundation, and avoiding the doctrine of velocities, momentums, &c.
will be found impracticable, till such time as the object and the end of geometry are better understood than hitherto they seem to have been. The great author of the method of fluxions felt this difficulty, and therefore he gave in to those nice abstractions and geometrical metaphysics without which he saw nothing could be done on the received principles: and what in the way of demonstration he hath done with them the reader will judge.
It must, indeed, be acknowledged that he used fluxions, like the scaffold of a building, as things to be laid aside or got rid of as soon as finite lines were found proportional to them. But then these finite exponents are found by the help of fluxions. Whatever therefore is got by such exponents and proportions is to be ascribed to fluxions: which must therefore be previously understood. And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
