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Notwithstanding this, Barrow's Dei monftrations are so very short, and are involv'd in so many Notes and Symboles, that they are render'd obscure and difficult to one not vers'd in Geometry. There are many Propositions which appear conspicuous in reading Euclid himself, are made knotty and scarcely intelligible to Learners by this Algebraical way of Demonstrations as is, for Example, Prop. 13. Book i. And the Demonstrations 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 fimple Method, and not to be involved in Symboles, Notes, or obfcure Principles, taken elfewhere.

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As Barrow's Elements are too short, so are those of Clavius too prolix, abounding in fuperfluous Scholiums and Comments: For in my opinion, Euclid is not so obfcure as to want such a number of Notes, neither do. I doubt but a Learner will find Euclid himself, casier than any of his Com.

mentators

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momtators. As too much Brevity in Geometrical Demonstrations begets Obfcurity, so too much

Prolixity produces Tediousness and Confusion.

On these 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 sufficient for understanding of most parts of the Mathematicks now studied,

Farther, for the Use of thofe who are desirous to apply the Elements of Geometry to Yfes in Life, we have added a Compendium of Plain aud Spherical Trigonometry, by means whereof Geometrical Magnitudes are measured, and their Dimensions expressed in Numbers.

J. KEIL.

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R.KEIL, in his Preface,

hath sufficiently declar'd D

how much easier, plainer, and eleganter, the Elements of Geometry written by

Euclid are, than those written by others; and that the Elements themselves, are fitter for a Learner, than

those

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those publish’dby such as have pretended to Comment on, Symbolize, or Transpose any of his Demonstrations of such Propositions as they intended to treat of. Then how must. a Geometrician be amaz'd, when he meets with a Tract* of the ist, 2d, 3d, 4th, 5th, 6th, 11th and 12th Books of the Elements, in which are omitted the Demonstrations of all the Propositions of.that most noble universal Mathesis, the 5th; on which the 6th, 11th, and izth so much depend, that the Demonstration of not so much as one Proposition in them can be obtain'd without those in the 5th,

The 7th, 8th, and 9th Books treat of fuch Properties of Numbers, which are necessary for the Demonftrations of the Ioth, which treats of Incommensurables; and the 13th, 14th, and 15th, of the five Platonick Bodies. But though the Doctrine of Incommensurables, because expounded in one and the same 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 necessary to the Reasoning about Incommensuraħles:

Yet

* Vide the laft Edition of the English Tacquet.

Yet because only one Proposition of these four Books, viz. the ift of the Icth is

quoted in the 11th and 12th Books; and that only once, viz. in the Demonftration of the 2d of the 12th, and that ist Proposition of the roth, is supplied by a Lemma in the 12th: And because the 7th, 8th, 9th, Toth, 13th, 14th, 15th Books have not been thought (by our greatest Masters) neceffary to be read by such as design to make natural Philosophy their Study, or by such as would apply Geometry to practical Affairs, Dr. Keil in his Edition, gave us only these eight Books, viz. the first six, and the 11th and 12th.

And as he found there was wanting a Treatise of these parts of the Elements, as they were written by Euclid himfelf; he publish'd his Edition without omitting any of Euclid's Demonstrations, except two; one of which was a second Demonstration of the 9th Propofition of the third Book; the other a Demonstration of thar Property of Proportionals call'd Conversion, (contain’d in a Corollary to the 19th Propofition of the 5th Book,) where instead of Euclid's Demonstration, which is universal, most Authors have given us only particular ones of their own. The first of these which was omitted is here

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