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plexed Ideas, than to the Demonstrations themselves. And, however some may find Fault with the Disposition and Order of his Elements, yet, notwithstanding, I do not find any Method, in all the Writings of this Kind, more proper and easy for Learners than that of Euclid.

It is not my Business here to answer, feparately, every one of these Cavillers; but it will easily appear to any one, moderately versed in these Elements, that they rather frew their own Idleness, than any real Faults in Euclid. Nay, I dare venture to say, there is not one of these new Systems, wherein there are not more Faults, nay, groffer Paralogisms, than they have been able even to imagine in Euclid.

After so many unsuccessful Endeavours in the Reformation of Geometry, some very good Geometricians, not daring to make new Elements, have deservedly preferred Euclid to all others; and have accordingly made it their Business to publish those of Euclid. But they, for what Reason I know not, have entirely omitted some Propositions, and have altered the Demonstrations of others, for worse. Among whom are chiefly Tacquet and Dechales, both of which have unhappily rejected some elegant Propoßtions in the Elements (which ought to have been retained), as imagining them trifling and uselefs; juch, for Example, as Prop. 27, 28, and 29, of the fixth Book, and some others, whose Uses they might not know. Farther,

where

wherever they use Demonstrations of their own, instead of Euclid's, in those DemonArations, they are faulty in their Reasoning, and deviate very much from the Conciseness of the Antients.

In the fifth Book, they have wholly rejeEled Euclid's Demonstrations, and have given a Definition of Proportion different from Euclid's, and which comprehends but one of the two Species of Proportion, taking in only commensurable Quantities. Which great Fault, no Logician or Geometrician would ever have pardoned, had not those Authors done laudable Things in their other Mathematical Writings. Indeed, this Fault of theirs is common to all Modern Writers of Elements, who all split on the same Rock; and, to sew their skill, blame Euclid, for what, on the contrary, he ought to be commended; I mean, the Definition of proportional Quantities, wherein he fizews an easy Property of those Quantities, taking in both commensurable and incommensurable ones, and from which all the other Properties of Proportionals do easily follow.

Some Geometricians, for footh, want a Demonstration of this Property in Euclid; and undertake to supply ibe Deficiency by one of their own. 'Here, again, they shew their Skill in Logic, in requiring a Demonstration for the Definition of a Term; that Definition of Euclid being such as determines those Quantities Proportionals, which have the Conditions Specified in the said De

finition.

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finition. And why might not the Author of the Elements give what Names be thought fit to Quantities, having such Requistes ? Surely be might use his own Liberty, and accordingly has called them Proportionals.

But it may be proper here to examine the Method whereby they endeavour to demonstrate that Property: Which is by first af Juming a certain Affection, agreeing only to one Kind of Proportionals, viz. Commenfurables ; and thence, by a long Circuit, and a perplexed Series of Conclusons, do deduce that universal Property of Proportionals which Euclid affirms ; a Procedure foreign enough to the just Methods and Rules of Reasoning. They would certainly have done much better, if they had first laid down. that universal Property by Euclid, and thence have deduced that particular Property agreeing to only one Species of Proportionals. But, rejecting this Method, they have taken the Liberty of adding their Demonstration to this Definition of the fifth Book. Those who have a mind to see a farther Defence of Euclid, may consult the Matbematical Lectures of the learned Dr. Barrow.

As I have happened to mention this great Geometrician, I must not pass by the Elements published by him, wherein, generally, he has retained the Constructions and Demonftrations of Euclid himself, not having omisted so much as one Proposition. Hence, bis Demonstrations become more strong and nervous, bis Constructions more neat and 5

elegant,

elegant, and the Genius of the antient Geometricians more conspicuous, than is ufually found in other Books of this Kind. To tbis he has added everal Corollaries and Scholia, which serve not only to shorten the DemonAtration of what follows, but are likewise of Use in other Matters.

Notwithstanding this, Barrow's Demonstrations are so very mort, and are involved in so many Notes and Symbols, that they are rendered obfcure and difficult to one not ver fed in Geometry. There, many Propofitions, which appear conspicuous in reading Èuclid himself, are made knotty, and scarcely intelligible to Learners, by his Algebraical Way of Demonstration ; as is, for Example

, Prop. 13. Book I. And the Demonstrations which he lays down in Book II. are still more difficult : Euclid bimself has done much better, in Newing their Evidence by the Contemplations of Figures, as in Geometry mould always be done. The Elements of all Sciences ought to be handled after the most fimple Method, and not to be involved in Symbols, Notes, or obscure Principles, taken elsewhere.

As Barrow's Elements are too short, fo are those of Clavius too prolix, abounding in fuperfluous Scholiums and Comments: For, in my opinion, Euclid is not so obscure as to want such a Number of Notes, neither do I doubt, but a Learner will find Euclid much easier than any of his Commentators. As too much Brevity in Geometricial Demon

strations

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ftrations begets Obscurity, fo too much Prolixity produces Tediousness and Confufon.

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 rest I forbore, because those, first-mentioned, are Jufficient for understanding of most Parts of the Mathematics now studied.

Farther, for the Ŭfe of those who are defirous to apply the Elements of Geometry to Uses in Life, we have added a Compendium of Plane and Spherical Trigonometry; by Means whereof, Geometrical Magnitudes are measured, and their Dimension expressed in Numbers.

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