5'. Ideam Mathematicam non nisi obiter inspexi, jamq; tantum memini nihilame in illa reperiffe a quo multum dissentirem, & valde probaste quod primoloco omnis supellex Mathematica ibi enumeretur, & postea ipse Mathematicus tes.* ’f'bfido.tanquam «uVoifWK ex leiplo contentus describatur. In eundem enim fere -i-, f. 144.sensiim duo soleo in Mathesi distinguere, Historiam scilicet & Scientiam : PerHistoriam intelligo illud omne quod jam inventum est, atque in libris contine-tur : Per Scientiam vero, Peritiam Quxstiones omnes resolvendi, atque adeoinveniendi propria Industria illud omne quod ab humano ingenio in ea Scien-tia potest inveniri, quam qui habet, non sane multum aliena desiderat, atqueadeo valde proprie ttvT&pwis appellatur. Valde autem optandum foret, utilla Historia Mathematica quae in multis Voluminibus sparsa, nondum integra& perfecta est, in unum Librum tota colligeretur : Neque ad hoc ulli sum-ptus in perquirendis aut coemendis libris eflent faciendi: Cum enim Authoresalii ex aliis multa exscripserint, nihil ullibi extat, quod non in quavis medio-criter instructa Bibliotheca alicubi reperiatur; nec tam diligentia opus essetad omnia colligenda, quam judicio ad superflua rejicienda, & Scientia ad eaquae nondum inventa sunt supplenda. Atqui si talis Liber extarer, facile ex eounusquisque omnem Historiam Mathematicam, atque etiam aliquam partemScientiae addisceret. Si quis autem omnia quae ad ejus Praxin pertinent, ha-bere vellet, ut Instrumenta, Machinas, Automata, &c. nx ille si Rex esset,orbis Terrarum impensis omnibus ad hoc necessariis sufficere nunquam posset.

Neque vere etiam illis opus habet, sed fatis est si omnium norit descriptionem,adeo ut ea, cum ustis exiget, vel ipse facere, vel per Artifices fieri curarepossit.

IITTTIE Propositions which I (hallendeavour to demonstrate independently some of Eu-1 from all others, fhall be these; the z id, and 47th of the Firlt Bookj, cliä', Propositi.moft of the Sccond and Flfth Bool y; the ist and irith of the Si.xrb ; with their Co-rolkries. In order to demonstrate the zid; I siippose it known what is meant ly from the re ff,by an Angle, Triangle, Circle, External Angle, Parallels, and that the Mea- Mr - Asl, >fure of an Angle is the Arch of a Circle intercepted between its Sides; lirat fj g 7 , 2 ’a Right Angle is measiir’d by a Quadrant, and two Right Angles by a Semi-circle. I sey then, that in the Triangle ABC, the External Angle BCE, %. I.is equal to the two opposite Intemal ones ABC, BAC; For let a Circlebe drawn, C being the Center, and B C the Radius, and let C D be drawnparallel to A B, those two Lines being always equidistant, will both have theseme Inclination to any third Line falling upon them; that is, (by the Defini-tion of Angle) they will malte Equal Angles-with it j For if any Part ofCD (for Instance) did incline more to B Q than did AB, upon that veryaccount they would not be Parallel ; it follows therefore that the AnglesABC, B CD, are Equal: Also BAC —D CE, becaufe A E falis upontwo Parallels ; but the External Angle B C E =r B C D DCE, whichwere before prov’d to be Equal to ABC, BAC. g. E. D. Hence maybe inferr’d as a Corollary, That the three Angles of every Triangle are Equalto two Right ones ; for the Angles A CB -f BCE, are measiir’d by a

Semieircle,