ABSTRACT
PASSING by all prefatorial and circumstantial Speeches, according to my accustomed Manner (yet most candid Auditors, not without congratulating your present Health, and wishing it a long Continuance to you hereafter) I apply myself to the Business, and resume my too long intermitted Task. But, with your leave, permit me to stick close to my former Method, and finish out what remains of my Undertaking, concerning the Preparatives to the Mathematics, which I hope to effect with this Term; that after we have stuck so long upon the Shore, and cruised along the Coasts of these Sciences, we may at length be carried out into the Main Sea. Though in Truth the Subjects we are now undertaking to discuss, whether we respect their Subtilty or Utility, do seem not to be far from the very inmost and deepest Part of the Mathematics. For Reason or Proportion is first to be considered, then Analogy or Proportionality; about which the Whole of Mathematics is some Way or other employed either as its Objects or Instruments of Invention and Demonstration; and therefore the divine a Plato in his most divine Work (I mean his b Timœus) thinks it to be the Chain and Bond of the Mathematics, or rather of all Disciplines. Nor does the most illustrious Author of the famous Discourse concerning the Method of using Reason, differ from hence. Because, says c he, I observed that those (viz. particular Sciences called the Mathematics) although conversant about different Objects, did yet all agree in this, in examining nothing but certain Relations and Proportions, which are found in them, I thought that these Proportions only are to be considered. I add Eratosthenes, who pronounces, That all Things in the Mathematics consist of the Proportion of certain Quantities. Therefore the Consideration of Proportions pervades and in a manner contains the Whole of these Disciplines: So that while we are contemplating the Nature of Proportions we ought not to think we are touching the outward Skin, but are entring their very Pith and Marrow. The Theory of the determinate Situation or Position of Magnitudes will succeed this, whereby the various Similitude and manifold Diversity of Figures appears, together with their Proportion; next, of the Species or Specifical Agreements and Differences of the Magnitudes consequent to these; lastly, of the Motion whereby Magnitudes are generated: I say, it will remain to consider these. Thus after we have examined the most, at least the principal of all the Properties of Magnitudes, which are laid down by Mathematicians or implied in their Hypotheses, returning into the same Track from which we first departed, we will produce certain Particulars worthy to be observed concerning the Hypotheses themselves and other Principals. Which Business being dispatched, we will apply ourselves to that which, I remember well, was heretofore promised, viz. to explain in the Method of Mathematical Invention. And I should be very glad if we could be able to finish this proposed Undertaking within the Course of the present Term. These Things being first spoken of Let us first begin to discourse of Reasion or Proportion.
