The whole Fifteen BOOKS
Compendiously Demonstrated.

Of the Sphere and Cylinder, investi-
gated by the Method of Indivisibles.

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By ISAAC BARROW, D. D. Late Master

Trinity College in Cambridge.

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with Marinus's Preface.

And a Brief TREATISE of

Kalupeos funns aozenñs exciv cu manna ning tushuria

LONDON: Printed and Sold by W. Redmayne

in Jewen-street, R. Mount on Tower-bil, and J. and
B. Sprint in Little-britain. 1714.

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F pou are desirous, Courteous Reader to know what I have perform'd in this Edition of the Elements of Euclide, I all here explain it to you in fhort,

according to the nature of the Work. I have endeqvar'd to attain two ends chiefly , the first, to be very perfpicuous, and at the same time fo very brief, that. the Bçok may not (well to

. Juch a' Bulk, as may be carry about one in which I think I have Lucceeded, unless is my absence the Printer's careshould fruftrate my Design. Some of a brighter Genits, and endued with greater Skill, may have demonstrated moje of these Propofitions with more nicery, but perbaps none with more succin&tness than I kave; especially since 1 alter'd nothing in the number and order of the Autbor's Propositions; por presum'd either to take ibe liberty of reje&ting, u lefs.necelary, any of them, or of reducing Jame of the easier sert into the rank of Axioms, as fem veral bave done į and among others, that most expert Geometrician A. Tacquetus C. (whom I the more willingly name, because I think it is but civil to acknowledge that I have imitated him in some points) after whose molt acurate Edition I had no Thoughts of attempting any thing of this nature, till I considera that ibis moft learned Man thought fit to publish only eight of Euclide's Books, which he took the pains to explain and embellish, having in a manner rejected and undervalued tbe as her leven, as lefs appertaining to the Elements of Geometry. Ixt, my Province was


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quite different, not that of writing the Elements of Geometry, after wbat method foever I pleas’d, but of demonstrating, in a few words as possible i cou'd, the whole Works of Euclide. As to four of the Books, viz. the seventb, eigbeb, nintb, and tentb, altho' they don't fo nearly appertain to the Elements of plain and solid Geometry, as tbe fix precedent and the two subsequent, yet none of the more skilful Geometricians can be so ignorant as not to know that they are very mseful for Geometrical matters, not only by rean son of the mighty near affinity that is between Arithmetick and Geometry, but also for the knowledge of both measurable and

unmeasurable Magnitudes, fo excom seeding necellary for the Doctrine of borb plain and for lid figures. Now the noble Contemplation of the five Regular Bodies that is contain'd in the three laft Books, cannot without great Injustice be pretermitted, since that for the sake thereof our sorteiatas, being a Philosopher of the Platonic Seet, is said to have compos'd ibis universal System of Elements; as Proclus lib. 2. witnelseth in these Words, Osv sej This owurdons στοιχειώσεως τέλΘ προσήσαιο την 8 καλυμβόων πλαtarinowy gmpestor cuseer. Befades, I easily perswaded my self to think, that it would not be unacceptable to any Lover of these Sciences to bave in bis Posession the whole Euclidean Work, as it is commonly. cited and celebrated by all Men. Wherefore I refolu'd to omit no Book or Proposition of those 'ibat are found in P. Herigonius's Edition, whose Steps I was oblig’d closely to follow, by reason I took a Resolution to make use of most of the Schemes of the said Book, very well foreseeing that time would not allow me to form


new ones, tbofometimes I chose rather to do it. For she same reason I was willing to use for sbe most part Euclide's own Demonftrations, baving only expresid them in a more succinct Form, unless perbaps in the second, thirteenth, and very few in the seventb, eigbeby and nintb Book, in which it seem'd not worth my while to deviate in any particular from bim. There- fore I am not without good bopes that as to this part I bave in some measure Latisfied both my own Intentions, and the Defire of the Studious. As for some certain Problems and Theorems that are added in the Scholions (or frost Expofitions) either appertaining (by reason of their frequent Use) to the nature of these Elements, or conducing to the readj Demonftration of those things that follow, or which do intimate the reasons of some principal Rules of practical Geometry, reducing them to their original Fountains, these I say, will not, I bope, make the Book swell to a Size beyond the de figoid Proportion.

The other Butt, which I levelld at, is to content obe Defores of those who are delighted more with Symo bolical than verbal Demonstrations. In which kind, whereas most among us are accustom'd to the Symbols of Gulielmus Oughtredus, I therefore thought beft to make use, for the most part, of bis.. None hitherto (as I know of) bas attempted to interpret and publisha Euclide after this manner, except P. Herigonius; whole Merbod (tho indeed most excellent in many, things, and very well accommodated for the particular purpose of that most ingenious Man) yet feems in my Opinion to labour under a double Defect. First, in regard shar, alıbo' of two or more Propositions, produs


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