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ELEMENTS
The whole Fifteen BOOKS
Compendioully Demonstrated.

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

By ISAAC BARROW, D. D. Late Master

of Trinity College in Cambridge.

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LONDON: Printed and Sold by W. Redmayne

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

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F you are desirous, Courteous Reader to know what

I have perform'd in this Edition of the Elements of. Euclide, I shall bere explain it to you in short,

according to the nature of the Work. I have endeavor’d to attain two ends chiefly; the first, to be very perfpicuous, and at the same time fo very brief, that. the Book may not (well to Juch a Bulk, as may be troublefume to carry about one, in which I think I have Succeeded, unless is my absence the Printer's care should

in fruftrate my Design. Some of a brighter Genites, and endued with greater Skill, may have demonstrated most of these Propofitions with more nicery, but perhaps none with more succin&tness than I have; especially since I alter'd ncthing in the number and order of the Author's Propositions; por presum'd either rejecting, as lefs.necesary, any of them, or of reducing Some of the easier fart into the rank of Axioms, as (everal 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 aca knowledge that I have imitated him in fome Points) after whose molt acurate Edition I had no Thoughts of attempting any thing of this nature, till I consider'd that ibis moft learned Man thought fit to publish only eight af Euclide's Books, which he took the pains to explain and embellisla, baving in a manner reje&ted and undervalued the ar her leven, as less appertaining to the Elements of Geometry. Iut my Province was

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quite different, not that of writing tbe Eles ments of Geometry, after wbat met bod

foever I pleas’d, but of demonstrating, in as 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 so 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 useful for Geometrical matters, not only by reason 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 necessary for the Doctrine of bocb plain and for lid figures. Now the noble Contemplation of the five Regular Bodies that is contain'd in the tbree laft Books, cannot without great Injustice be pretermitted, since that for the sake thereof our sorteftatie, being a Philosopher of the Platonic Seet, is said to have compos'd obis universal System of Elements; as Proclus lib. 2. witnesseth in these Words, "Oder sig sj rñs owurdous soravanotas Tiro agorsávalo thoshte nan syfów miagarina gemuetar ouseosv. Besides, 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. Wberefore I refolu'd to omit no Book or Proposition of those that are found in P. Herigonius's Edition, whose Steps I was obligd 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

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new ones, tho' sometimes I chose rather to do it. For she same reason I was willing to use for the most part Euclide's own Demonstrations, baving only expresid them in a more fuccin& Form, unless perlaps in the second, thirteentb, and very few in the seventb, eigbeby and ninth Book, in which it seem'd not worth my while to deviate in any particular from bim. Therefore I am not without good hopes that as to this part I bave in some measure satisfied both my own Intentions, and the Defire of the Studious. As for some certain Problems and Theorems that are added in the Scholions (or fhoort Expofitions) either appertaining (by reason of their frequent 'Use) to the nature of these Elements, or conducing to tbe 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 (well 80 m Size beyond the dea ligri'd Proportion.

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

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