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The whole Fifteen Books compen-

diously Demonstrated
By M'. ISAAC BARROW. Fellow of Tri-

nity Colledge in CAMBRIDGE.
Und Translated out of the Latin.

Καθαρμοί ψυχής λογικής είσιν αι μαθηματικα σπιςήμου. Ηierocl.

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Printed by R. DANIEL , for WILLIAM
NEALAND Bookseller in Cambridge ; and
are to be fold there, and at the Crown in

Duck-Lane 1660,

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Norder to the Reader's fatisfa&ion con.

cerning the Book put into his hand, I am to advertise him of some few

things, and that according to the nature of the Work, briefly; as followeth. My Undertaking aimed principally at two Ends. The first of which was to conjoin the greatest Compen. diousness of Demonstration with as much Perfpicuity as the quality of the subject would ad. mit ; that fo the Volume might bear no bigger bulk then would render it conveniently portable: Which I have fo farr attained, that though posli. bly some other perfon" might with greater

curiosity yet (I presume) none could with more concisenefíe bave demonstrated most propositions; espe. cially , 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 lessene. cessary, or to reduce certain of the easiest into the Claflis of Axiomes. Which notwithstanding some have done ; as that most accurate Geometrician

Andr. Tacquet, whom I mention the rather, because I esteem it ingenuous to acknowledge some things taken from him. And, indeed, I should have at. tempted nothing after his most elegant Edition, bad it not plealed that learned Person to publish

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only Eight of Euclide's Books illustrated by his paines , either flighting or undervaluing the other Seven as lesse relating to the Elements of Geometry. But I had a different Purpose from the beginnings not to compose Elements of Geometry any-wise at my discretion, but to demonstrate Euclide bimself, and all of him, and that with all poslible brevity.For as for Foure ofhis Books, the Seventh, Eighth, Ninth and Tenth , although they do not so neerly pertain to the Elements of Plane and Solid Geometry, as the Six Firft & the Two subsequent; yet no man that ha's arriv’d to any measure of skill in Geometry is ignorant how exceedingly usefull they are in Geometricall matters , aswell in regard of the very neer alliance between Arithmetick and Geometrie , as for the knowledge of Commensurable and Incommen, furable Magnitudes which is highly important to the understanding both of Plane and Solid Fie gures. And the noble Theory of the Five Regular Bodies, contained in che Three Last Books could not be omitted without prejudice & injury; since our Author of these Elements,being a Sedator of Plato's Schole, is reported to have compiled the whole Systeme only in reference to that Contemplation ; which Proclus attefteth in these words , Όθεν δη και της ζυμπί σης στοιχειώσεως, τέλG- αθθισή σατο τω καλεμένων Πλατωνικών χημάτων (ύσασιν. Moreover, Iwas easily induc'd to believe, that it would be acce. ptable to all Lovers of these Sciences to have the Intire work of Euclide by them as it is usually cited and recommended by all men. Wherefore I deter, min’d to leave out no Book or Proposition of those which are found in Peter Herigon , whose footsteps I became neceflitated to follow closely by baving resolved to make use for the most part of the Schemes of his Book, upon a forefight that my speedy departure out of England, would not allow me time to describe New, although I some. times defired fo to doe. Upon the same account also I purposed to use generally no other then Euclide's own Demonstrations, contraded into a more fuccin& form, saving perchance in the Se cond and Thirteenth, & fparingly in the Seventh, Eighth, and Ninth Books, where it seem'd convenient to vary something from him. So that it may be reasonably, hoped that in this particular our own Design and the Wishes of the Studious are in some manner fatisfid.

The orber End aimed at was in favour of Their desires who more affe& Symbolical then Verbal Demonstrations.In which kind, feeing most of our Own Nation are accustomed to the Notes of Mr. Ougbtred, I esteemed it more convenient to make use of them principally throughout. For no man bitherto that I know of, faving only Peter Herigon, ha's attempted to set forth and interpret Euclide according to this way. The Method of which most learned Person", though in many other refpe&s very excellent, and exa&tly accommodated to bis peculiar purpose , seem'd to me notwith. handing doubly defective. First,in that, whereas of severall Propofitions brought to the proving of fome one Theoreme or Probleme the Latter


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