Notwithstanding this, Barrow's Demonftrations are fo very short, and are involv'd in fo many Notes and Symboles, that they are render'd obfcure and difficult to one not vers'd in Geometry. There are many Propofitions which appear confpicuous in reading Euclid himself, are made knotty and fcarcely intelligible to Learners by this Algebraical way of Demonftrations as is, for Example, Prop. 13. Book 1 And the Demonftrations which he lays down in Book 2. are fill more difficult : Euclid himself has done much better, in Shewing their Evidence by the Contemplations of Figures, as in Geometry Should always be done. The Elements of all Sciences ought to be handled after the most Simple Method, and not to be involved in Symboles, Notes, or obfcure Principles, taken elsewhere.

As Barrow's Elements are too short, fo are thofe of Clavius too prolix, abounding in fuperfluous Scholiums and Comments: For in my Opinion, Euclid is not so obfcure as to want fuch a number of Notes, neither do I doubt but a Learner will find Euclid him felf, eafier than any of his Com

memtators. As too much Brevity in Geometrical Demonftrations begets Obfcurity, So too much Prolixity produces Tediousness and Confufion.

On thefe Accounts principally, it was that I undertook to publish the first fix Books of Euclid, with the 11th and 12th, according to Commandinus's Edition; the reft I forbore, because those first mention'd are fufficient for understanding of moft parts of the Mathematicks now studied.

Farther, for the Ufe of those who are defirous to apply the Elements of Geometry to fes in Life, we have added a Compendium of Plain and Spherical Trigonometry, by means whereof Geometrical Magnitudes are measured, and their Dimensions expressed in Numbers.

J. KEIL

Mr. CUN N's

PREFACE,

Shewing the USEFULNESS and EXCELLENCY of this WORK.

R.KEIL, in his Preface, hath fufficiently declar'd

D how much eafier, plainer, and eleganter, the Elements of Geometry written by Euclid are, than those writ

ten by others; and that the Elements themfelves, are fitter for a Learner, than

thofe

those publish'd by fuch as have pretended to Comment on, Symbolize, or Tranfpofe any of his Demonftrations of fuch Propofitions as they intended to treat of. Then how must a Geometrician be amaz'd, when he meets witha Tract* of the Ift, 2d, 3d, 4th, 5th, 6th, 11th,and 12th Books of the Elements, in which are omitted the Demonstrations of all the Propofitions of that most noble universal Mathefis, the 5th; on which the 6th, 11th, and 12th fo much depend, that the Demonstration of not fo much as one Propofition in them can be obtain'd without thofe in the 5th,

The 7th, 8th, and 9th Books treat of fuch Properties of Numbers which are neceffary for the Demonftrations of the 10th, which treats of Incommenfurables; and the 13th, 14th, and 15th, of the five Platonick Bodies. But though the Doctrine of Incommenfurables, because expounded in one and the fame Plane, as the first fix Elements were, clam'd by a Right of Order, to be handled before Planes interfected by Planes, or the more compounded Doctrine of Solids; and the Properties of Numbers were neceffary to the Reasoning about Incommenfurables:

Vide the laft Edition of the English Tacquet.

Yet

Yet because only one Propofition of these four Books, viz. the 1ft of the 10th is quoted in the 11th and 12th Books; and that only once, viz. in the Demonstration of the 2d of the 12th, and that 1ft Propofition of the 10th, is fupplied by a Lemma in the 12th: And because the 7th, 8th, 9th, roth, 13th, 14th, 15th Books have not been thought (by our greatest Masters) neceffary to be read by fuch as defign to make natural Philofophy their Study, or by fuch as would apply Geometry to practical Affairs, Dr. Keil in his Edition, gave us only thefe eight Books, viz. the firft fix, and the 11th and 12th.

And as he found there was wanting a Treatife of thefe Parts of the Elements, as they were written by Euclid himself; he publish'd his Edition without omitting any of Euclid's Demonftrations, except two; one of which was a fecond Demonftration of the 9th Propofition of the third Book; the other a Demonftration of that Property of Proportionals call'd Converfion, (contain'd in a Corollary to the 19th Propofition of the 5th Book,) where inftead of Euclid's Demonftration, which is univerfal, moft Authors have given us only particular ones of their own. The first of these which was omitted is here

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