ELEMENTS; The whole Fifteen Books compen- By Mr. ISAAC BARROW. Fellow of Tri- And Translated out of the Latin. Καθαρμοὶ ψυχῆς λογικῆς εἰσιν αἱ μαθηματικοί στις μοι. Hierocl. LONDON, Printed by R. DANIEL, for WILLIAM THE AUTHOR'S PREFACE. Norder to the Reader's fatisfaction concerning the Book put into his hand, I 'am to advertise him of fome few things, and that according to the nature of the Work, briefly; as followeth. My Undertaking aimed principally at two Ends. The firft of which was to conjoin the greatest Compen dioufnefs of Demonftration with as much Per fpicuity as the quality of the fubje&t would admit that fo the Volume might bear no bigger bulk then would render it conveniently portable. Which I have fo farr attained, that though poffibly fome other perfon might with greater curiofity.yet (I prefume) none could with more concife neffe have demonftrated moft propofitions; efpecially, fince I have altered nothing in the number and order of the Propofitions, nor taken the li berty to leave out any one of Euclide's as leffe neceffary, or to reduce certain of the eafieft into the Claffis of Axiomes. Which notwithstanding fome have done; as that moft accurate Geometrician Andr.Tacquet, whom I mention the rather, because I esteem it ingenuous to acknowledge fome things taken from him. And, indeed, Ifhould have attempted nothing after his moft elegant Edition, bad it not pleafed that learned Perfon to publish (...) 2 only only Eight of Euclide's Books illuftrated by his paines, either flighting or undervaluing the other Seven as leffe relating to the Elements of Geometry. But I had a different Purpose from the beginning; not to compofe Elements of Geometry any-wife at my difcretion, but to demonftrate Euclide himself,and all of him, and that with all poffible brevity. For as for Foure of his Books, the Seventh, Eighth, Ninth and Tenth, although they do not fo neerly pertain to the Elements of Plane and Solid Geometry, as the Six Firft & the Two fubfequent; yet no man that ha's arriv'd to any measure of skill in Geometry is ignorant how exceedingly ufefull they are in Geometricall matters, afwell in regard of the very neer alliance between Arithmetick and Geometrie, as for the knowledge of Commenfurable and Incommen furable Magnitudes which is highly important to the understanding both of Plane and Solid Fi gures. And the noble Theory of the Five Regular Bodies, contained in the Three Laft Books,could not be omitted without prejudice & injury ; fince our Author of thefe Elements,being a Sectator of Plato's Schole, is reported to have compiled the whole Syfteme only in reference to that Contemplation; which Produs attefteth in these words, Ὅθεν δὴ καὶ τῆς Γυμπάσης εοιχειώσεως τέλος προςήσατο τω καλεμένων Πλατωνικών χημάτων (ύςασιν. Moreover, I was eafily induc'd to believe, that it would be acces ptable to all Lovers of thefe Sciences to have the Intire work of Euclide by them,as it is ufually cited and recommended by all men, Wherefore I deter min'd to leave out no Book or Propofition of thofe which are found in Peter Herigon, whofe footsteps I became neceffitated to follow closely by having refolved to make ufe for the most part of the Schemes of his Book, upon a forefight that my fpeedy departure out of England, would not allow me time to defcribe New, although I fome. times defired fo to doe. Upon the fame account alfo I purposed to ufe generally no other then Euclide's own Demonftrations, contracted into a more fuccin& form, faving perchance in the Second and Thirteenth, & fparingly in the Seventh, Eighth, and Ninth Books, where it feem'd convenient to vary fomething from him. So that it may be reafonably hoped that in this Particular our own Defign and the Wishes of the Studious are in fome manner fatisfi'd. The other End aimed at,was in favour of Their defires who more affect Symbolical then Verbal Demonftrations.In which kind,feeing most of our own Nation are accustomed to the Notes of Mr. Oughtred, Iefteemed it more convenient to make ufe of them principally throughout. For no man hitherto that I know of, faving only Peter Herigon, ha's attempted to fet forth and interpret Euclide according to this way. The Method of which moft learned Perfon, though in many other refpects very excellent, and exactly accommodated to his peculiar purpose, feem'd to me notwith ftanding doubly defective. First,in that, whereas of feverall Propofitions brought to the proving of fome one Theoreme or Probleme the Latter do's |