plexed Ideas, than to the Demonftrations themfelves. And, however fome may find Fault with the Difpofition and Order of his Elements, yet, notwithflanding, 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 anfwer, feparately, every one of thefe 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 thefe new Systems, 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 preferred Euclid to all others; and have accordingly made it their Bufinefs to publish those of Euclid. But they, for what Reafon I know not, have entirely omitted fome Propofitions, and have altered the Demonftrations of others, for worfe. Among whom are chiefly Tacquet and Dechales, both of which have unhappily rejected fome elegant Propofitions in the Elements (which ought to have been retained), as imagining them trifling and uselefs; fuch, for Example, as Prop. 27, 28, and 29, of the fixth Book, and fome others, whofe Ufes they might not know. Farther,

where

wherever they ufe Demonftrations of their own, instead of Euclid's, in thofe DemonAtrations, they are faulty in their Reasoning, and deviate very much from the Concifeness 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 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 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 incommensurable 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 fupply the Deficiency by one of their own. Here, again, they fhew their Skill in Logic, in requiring a DemonAtration for the Definition of a Term; that Definition of Euclid being fuch as determines thofe Quantities Proportionals, which have the Conditions Specified in the faid De

A 3

finition.

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

But it may be proper here to examine the Method whereby they endeavour to demonAtrate that Property: Which is by firft af fuming 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 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. Those who have a mind to fee a farther 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 DemonArations of Euclid himself, not having omitted fo much as one Propofition. Hence, bis Demonftrations become more ftrong and nervous, his Conftructions more neat and

5

elegant,

elegant, and the Genius of the antient Geometricians more confpicuous, than is ufually found in other Books of this Kind. To this be has added everal Corollaries and Scholia, which ferve not only to fhorten the Demonftration of what follows, but are likewife of Ufe in other Matters.

Notwithstanding this, Barrow's Demonftrations are fo very short, and are involved in fo many Notes and Symbols, that they are rendered obfcure and difficult to one not verfed in Geometry. There, many Propofi tions, which appear confpicuous in reading Euclid bimfelf, are made knotty, and scarcely intelligible to Learners, by his Algebraical Way of Demonftration; as is, for Example, Prop. 13. Book I. And the Demonftrations which he lays down in Book II. are ftill more difficult: Euclid himself has done much better, in fhewing 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 moft fimple Method, and not to be involved in Symbols, 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 fso obscure as to want fuch 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

ftrations 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 rest I forbore, becaufe those, first-mentioned, are fufficient for understanding of moft Parts of the Mathematics 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 Plane and Spherical Trigonometry; by Means whereof, Geometrical Magnitudes are meafured, and their Dimenfion expressed in Numbers.

J. KEILL.