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The whole Fifteen BOOKS
Compendiously Demonstrated,


Of the Sphere and Cylinder, invefti-
gated by the Method of Indivifibles.

By ISAAC BARROW, D. D. Late Mafter
of Trinity College in Cambridge.

To which is added in this Edition,

with Marinus's Preface.

And a Brief TREATISE of


Καθαρμοί ψυχῆς λογικῆς εἰσὶν οἱ μαθηματικοί όλιςήμοι.

LONDON: Printed and Sold by W. Redmayne
in Fewen-ftreet, R. Mount on Tower-bill, and F. and
B. Sprint in Little-britain. 1714.


7-30-26 13266




To the READER.


Fyou are defirous, Courteous Reader,to know what I have perform'd in this Edition of the Elements of Euclide, I shall here 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 fame time fo very brief, that. the Book may not fowell to fuch a Bulk, as may troublefome to carry about one, in which I think I have fucceeded, unless in my abfence the Printer's care should fruftrate my Defign. Some of a brighter Genius, and endued with greater Skill, may have demonftrated most of thefe Propofitions with more nicety, but perhaps none with more fuccinctness than I have; especially fince I' alter'd nothing in the number and order of the Author's ·Propofitions; nor prefum'd either to take the liberty of rejecting, as lefs necellary, any of them, or of reducing Some of the easier fart into the rank of Axioms, as (everal bave done 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 fome Points) after whose most acurate Edition I had no Thoughts of attempting any thing of this nature, till I confider'd that this mast 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 the other feven, as less appertaining to the Elements of Geometry, But my Province was origi

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originally quite different, not that of writing the Elements of Geometry, after what method foever I pleas'd, but of demonftrating, in as few words as poffible 1 cou'd, the whole Works of Euclide. As to four of the Books, viz. the seventh, eighth, ninth, and tenth, altho' they don't fo nearly appertain to the Elements of plain and folid Geometry, as the fix precedent and the two fubfequent, yet none of the more skilful Geometricians can be fo ignorant as not to know that they are very useful for Geometrical matters, not only by reas fon of the mighty near affinity that is between Arithmetick and Geometry, but also for the knowledge of both measurable and unmeasurable Magnitudes, fo exseeding neceffary for the Doctrine of both plain and folid 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, fince that for the fake thereof our sorgerne, being a Philofopher of the Platonic Sect, is faid to have compos'd this univerfal System of Elements; as Proclus lib. 2. witneffeth in thefe Words, "Ov ♪ rñe ovμrdons τοιχειώσεως τέλο προςήσατο τὴν τω καλεμθύων πλα τωνικών χημάτων σύςασιν. Befides, I eaftly perfuaded my felf to think, that it would not be unacceptable to any Lover of thefe Sciences to bave in his Poffeffion the whole Euclidean Work, as it is commonly cited and celebrated by all Men. Wherefore I refolv'd to omit no Book or Propofition of those that are found in P. Herigonius's Edition, whofe Steps I was oblig'd clofely to follow, by reason I took a Refolution to make ufe of most of the Schemes of the faid Book, very well forefeeing that time would not allow me to form


new ones, tho' fometimes I chose rather to do it." For the fame reafon I was willing to use for the most part Euclide's own Demonftrations, having only exprefs'd them in a more fuccinct Form, unless perhaps in the Second, thirteenth, and very few in the seventh, eighth, and ninth Book, in which it feem'd not worth my while to deviate in any particular from him. Therefore I am not without good hopes that as to this part I bave in fome measure fatisfied both my own Intentions, and the Defire of the Studious. As for fome certain Problems and Theorems that are added in the Scholions (or Short Expofitions) either appertaining (by reason of their frequent Ufe) to the nature of thefe Elements, or conducing to the ready Demonftration of those things that follow, or which do intimate the reafons of fome principal Rules of practical Geometry, reducing them to their original Fountains, thefe I fay, will not, I bope, make the Book fwell to a Size beyond the defign'd Proportion.

The other Butt, which I levell'd at, is to content the Defires of thofe who are delighted more with fymbolical than verbal Demonstrations. In which kind, whereas most among us are accuftom'd to the Symbols of Gulielmus Oughtredus, I therefore thought beft to make ufe, for the most part, of his. None hitherto (as I know of) has attempted to interpret and publif Euclide after this manner, except P. Herigonius; whofe Method (tho' indeed most excellent in many things, and very well accommodated for the particular purpofe of that most ingenious Man) yet feems in my Opinion to labour under a double Defect. First, in regard that, altho' of two or more Propofitions, produ


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