Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

XIV.

After this way we design to go through the following Scheme. 1. We shall deduce several propositions of Euclid, Archi∣medes, and Apollonius from our definitions, and the genera∣tions of Magnitudes therein proposed; as Corollaries necessarily flowing from them, and confirm'd only by an immediate and simple consequence. 2. We shall demonstrate their chief Theo∣rems (for the sake of which they were forced to Demon∣strate several others before hand, the knowledge whereof for their own sake was not so necessary or valuable) without a∣ny long series of antecedent Propositions, or Foreign princi∣ples, from a few direct and intrinsick Principles of their own. Whence 3. It will follow, that after this Method we shall propose things, and treat of them; in a more natural Or∣der, and first of all deliver those which are most universal and common to all quantities, and then descend to those which in a more special manner regard Magnitude, and distribute and dispose all according to certain general distinct Classes of the things to be treated of, and their affections. Hence also 4. We deduce from those universal Theorems, by way of Corollary, the Precepts of vulgar Arithmetick, and specious Computation, which afterwards we make use of in particular Demonstra∣tions after a very short and compendious way; and, for this very reason, some Learned men of the present Age are of opi∣nion, that the Ancients often fell into that tedious and intri∣cate prolixity in their Demonstrations, because they would not acknowledge the great affinity there was between Arithme∣tick

and Geometry, taking particular care not to introduce the Terms and Operations of Arithmetick into Geometry; tho' at the same time they never scrupled to transfer the names of Plan, Square, Cube, and such like to numbers. 5. Lastly, Having first Demonstrated the first and Fundamental Theo∣rems of Elementary Geometry, we may safely build on them the Praxes of all kinds of Mathematical Arts, that are most useful and requisite to several Exigencies of human Life, as, first, Trigonometry both Plain and Spherical, the Constructi∣on and use of the Tables of Sines and Tangents. 2. The Construction of Logarithms, and a compendious application of them to Trigonometry: And in the 3. and last place, the fundamental Precepts of Algebra, or the Analytic Art; by the help whereof the learner may at length arrive to the higher and more recluse parts of Geometry, and become master thereof: Not to mention several Geometrical and A∣rithmetical Problems, which we have all along derived from several of our Theorems, by way of Corollary, which it may be some other time, may make an Appendix of this work.