Elements are ill disposed; and that they have found out innumerable Falfities in them, which had lain bid to their Times.

But by their Leave, I make bold to affirm, that they Carp at Euclid undeServedly: For his Definitions are destinct and clear, as being taken from first Principles, and our most easy and simple Conceptions; and his Demonstrations elegant, perspicuous and concife, carrying with them fuch Evidence, and so much Strength of Reason, that I am easily induced to believe the Obscurity, Sciolifts fo often accused Euclid with, is rather to be attributed to their own perplex'd 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 Separately every one of these Cavellers; but it will easily appear to any one, moderately versed in these Elements, that

they

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they rather shew 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, grosser Paralogisms, than they have been able even to imagine in Euclid.

After so many unsuccessful Endeavours. in the Reformations of Geometry, fome very good Geometicians, not daring to make new Elements, have deservedly preferr'd Euclid to all others; and have accordingly made it their Business to pub lish those of Euclid. But they, for what Reason I know not, have entirely omitted fome Propositions, and have altered the Demonstrations of others for worse. Among whom are chiefly Tacquet and Dechalles, both of which have un. happily rejected some elegant Propositions in the Elements, (which ought to have been retain'd,) as imagining them trifling and useless; such, for Example, as Prop. 27, 28, and 29. of the Sixth Book, and fome others, whose Uses they might not know. Farther, wherever they use Demonstrations of their own, inStead

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ftead of Euclid's, in those Demonstrations 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 Demonstrations, and have given a Definition of Proportion different from Euclid's; and which comprebends but one of the two Species of Proportion, taking in only commensurable Quantities. Which great Fault no Logician or Geometrician would have ever 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 Shew their Skill, blame Euclid, for what, on the contrary, be ought to be commended; I mean, the Definition of Proportional Quantities, wherein be Shews an easy Property of those Quantities, taking in both Commenfurable and Incommensurable ones, and from which, all the other Properties of Propor tionals do easily follow.

Some Geometricians, forfooth, want a

Demonstration of this Froperty in Euclid; and and undertake to supply the Deficiency by one of their own. Here, again, they shew their Skill in Logick, in requiring a Demonstration for the Definition of a Term; that Definition of Euclid being fuch as determines those Quantities Proportionals which have the Conditions specified in the faid Definition. And why might not the Author of the Elements give what Names he thought fit to Quantities having fuch Requisites; furely be might use his own Liberty, and accordingly has called them Proportio

nals,

But it may be proper here to examine the Method whereby they endeavour to Demonstrate that Property: Which is by first affuming a certain Affection, agreeing only to one kind of Proportionals, viz. Commenfurables; and thence, by a long Circuit, and a perplex'd Series of Conclusions, do deduce that univerfal 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 univerfal A 3 Property Property assign'd 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 further Defence of Euclid, may confult the Mathematical Lectures of the learn'd Dr. Barrow.

As I have happened to mention this great Geometrician, I must not pass by the Elements publish'd by him, wherein generally he has retain'd the Constructions and Demonstrations of Euclid himself, not having omitted so much as one Propofition, Hence, bis Demonstrations became more Strong and nervous, bis Construction more neat and elegant, and the Genius of the antient Geometricians more confpicuous, than is usually found in other Books of this kind. To this he has added, feveral Corollaries and Scholias, which ferve not only to shorten the Demonstrations of what follows, but are likewise of use in other Matters.

Not

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