perplexed Ideas, than to the Demonftrations themfelves, And however fome may find Fault with the Difpofition and Order of his Elements, yet notwithstanding I do not find any Method, in all the Writings of this kind, more proper and eafy for Learners than that of Euclid.

It is not my Bufinefs here to answer feparately every one of these Cavillers; but it will eafily appear to any one, moderately verfed in thefe Elements, that they rather fhew their own Idleness than any real Faults in Euclid. Nay, I dare venture to fay, there is not one of these New Syftems, wherein there are not more Faults, nay, groffer Paralogifms, than they have been able even to imagine in Euclid.

After fo many unfuccessful Endeavours, in the Reformation of Geometry, fome very good Geometricians, not daring to make new Elements, have defervedly preferr'd Euclid to all others; and have accordingly made it their Bufinefs to publish thofe of Euclid. But they, for what Reafon I know not, have entirely omitted fome Propofitions, and have altered the Demonftrations of others for worse. Among whom are chiefly Tacquet and Dechalles, both of which have unhappily rejected fome elegant Propofitions in the Elements (which ought to bave been retained) as imagining them trifling and ufelefs; fuch, for Example, as Prop. 27, 28, and 29, of the Sixth Book, and fome others, whoje Ufes they might not know. Farther, wherever they ufe Demonftrations of their

own,

own, inftead of Euclid's, in thofe Demonftrations they are faulty in their Reasoning, and deviate very much from the Conciseness of the

Antients.

In the fifth Book, they have wholly rejected Euclid's Demonftrations, 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 commenfurable Quantities. Which great Fault no Logician or Geometrician would have ever pardoned, had not thofe 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 fame Rock; and to fhew their Skill, blame Euclid, for what, on the contrary, he ought to be commended; I mean the Definition of Proportional Quantities, wherein he fhews an eafy Property of thofe Quantities, taking in both Commenfurable and Incommenfurable ones, and from which all the other Properties of Proportionals do eafily follow.

Some Geometricians, forfooth, want a Demonftration of this Property in Euclid; and undertake to supply the Deficiency by one of their own. Here, again, they fhew their Skill in Logick, in requiring a Demonftration for the Definition of a Term; that Definition of Euclid being fuch as determines thofe Quantities Proportionals which have the Conditions fpecified in the faid Definition. And why might A 3

not

not the Author of the Elements give what Names he thought fit to Quantities having fuch Requifites? Surely he might ufe his own Liberty, and accordingly has called them Proporti

onals.

But it may be proper here to examine the Method whereby they endeavour to demonftrate that Property: Which is by firft affuming a certain Affection, agreeing only to one kind of Proportionals, viz. Commenfurables; and thence, by a long Circuit, and a perplexed Series of Conclufions, do deduce that univerfal Property of Proportionals which Euclid affirms; a Procedure foreign enough to the juft Methods and Rules of Reafoning. They would certainly have done much better, if they had first laid down that univerfal Property affigned 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 Demonftration to this Definition of the fifth Book. Thofe who have a Mind to fee a further Defence of Euclid, may confult the Mathematical 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 Conftructions and Demonftrations of Euclid himself, not having omitted fo much as one Propofition. Hence, his Demonftrations become more strong and nervous, bis Con

ftructions

ftructions more neat and elegant, and the Genius of the ancient Geometricians more confpicuous, than is ufually found in other Books of this kind. To this he has added feveral Corollaries and Scholias, which ferve not only to Shorten the Demonftrations of what follows, but are likewife of use in other Matters.

Book

Notwithstanding this, Barrow's Demonftrations are fo very fhort, and are involved in fo many Notes and Symbols, that they are rendered obfcure and difficult to one not verfed in Geometry. There, many Propofitions which appear confpicuous in reading Euclid himself, are made knotty and fcarcely intelligible to Learners by this Algebraical Way of Demonftration, as is, for Example, Prop. 13. I. And the Demonftrations which he lays down in Book II. are ftill more difficult: Euclid bimfelf has done much better, in fhewing their Evidence by the Contemplations of Figures, as in Geometry fhould always be done. The Elements of all Sciences ought to be handled after the moft fimple Method, and not to be involv'd in Symbols, Notes, or obfcure Principles, taken elsewhere.

As Barrow's Elements are too fhort, so are thofe of Clavius too prolix, abounding in fuperfluous Scholiums and Comments: For in my Opinion, Euclid is not fo obfcure as to want fuch a lumber of Notes, neither do I doubt but a Learner will find Euclid himself easier than any of his Commentators. As too much Bre= A 4

with

vity in Geometrical Demonftrations begets Obfcurity, fo too much Prolixity produces Tedioufnefs 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 mentioned are fufficient for understanding of most Parts of the Mathematicks now ftudied.

Farther, for the Ufe of those who are defirous to apply the Elements of Geometry to Ufes in Life, we have added a Compendium of Plain and Spherical Trigonometry, by means whereof Geometrical Magnitudes are meafur'd, and their Dimenfions expreffed in Numbers.

J. KEIL