Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/201

 and I will give you an answer. Tell me therefore, how much do you think sufficeth to make that motion swifter than this?

I will say for example, that if that motion by the tangent were a million of times swifter than this by the secant, the pen, yea, and the stone also would come to be extruded.

You say so, and say that which is false, onely for want, not of Logick, Physicks, or Metaphysicks, but of Geometry; for if you did but understand its first elements, you would know, that from the centre of a circle a right line may be drawn to meet the tangent, which intersecteth it in such a manner, that the part of the tangent between the contact and the secant, may be one, two, or three millions of times greater than that part of the secant which lieth between the tangent and the circumference, and that the neerer and neerer the secant shall be to the contact, this proportion shall grow greater and greater in infinitum; so that it need not be feared, though the vertigo be swift, and the motion downwards slow, that the pen or other lighter matter can begin to rise upwards, for that the inclination downwards always exceedeth the velocity of the projection.

I do not perfectly apprehend this businesse.

I will give you a most universal yet very easie demonstration thereof. Let a proportion be given between BA [in Fig. 3.] and C: And let BA be greater than C at pleasure. And let there be described a circle, whose centre is D. From which it is required to draw a secant, in such manner, that the tangent may be in proportion to the said secant, as BA to C. Let AI be supposed a third proportional to BA and C. And as BI is to IA, so let the diameter FE be to EG; and from the point G, let there be drawn the tangent GH. I say that all this is done as was required; and as BA is to C, so is HG to GE. And in regard that as BI is to IA, so is FE to EG; therefore by composition, as BA is to AI; so shall FG be to GE. And because C is the middle proportion between BA and AI; and GH is a middle term between FG and GE; therefore, as BA is to C, so shall FG be to GH; that is HG to GE, which was to be demonstrated.

I apprehend this demonstration; yet neverthelesse, I am not left wholly without haesitation; for I find certain confused scruples role to and again in my mind, which like thick and dark clouds, permit me not to discern the cleernesse and necessity of the conclusion with that perspicuity, which is usual in Mathematical Demonstrations. And that which I stick at is this. It is true that the spaces between the tangent and the circumference do gradually diminish in infinitum towards the contact; but it is also true on the contrary, that the propension of the moveable to