Elements are ill difpofed; 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 deftinct and clear, as being taken from first Principles, and our most easy and fimple Conceptions; and his Demonftrations elegant, perfpicuous and concife, carrying with them fuch Evidence, and fo much Strength of Reason, that I am eafily induced to believe the Obfcurity, Sciolifts fo often accufed Euclid with, is rather to be attributed to their own perplex'd Ideas, than to the Demonftrations themselves. And however Some 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 easy for Learners than that of Euclid. It is not my Bufinefs here to Answer Separately every one of thefe Cavellers ; but it will easily appear to any one, moderately verfed in thefe Elements, that they they rather shew their own Idleness, than any real Faults in Euclid. Nay, I dare venture to fay, there is not one of the fe New Syftems, wherein there are not more Faults, nay, groffer Paralogifms, than they have been able even to imagine in Euclid. After fo many un fuccessful Endeavours,. in the Reformations of Geometry, fome very good Geometicians, not daring to make new Elements, have defervedly preferr'd Euclid to all others; and hav accordingly made it their Bufinefs to pub lifh thofe of Euclid. But they, for what Reason I know not, have entirely omitted fome Propofitions, and have altered the Demonftrations of others for worfe. Among whom are chiefly Tacquet and Dechalles, both of which have un. happily rejected fome elegant Propofitions in the Elements, (which ought to have been retain❜d,) as imagining them trifling and ufelefs; fuch, for Example, as Prop. 27, 28, and 29. of the Sixth Book, and fome others, whose Uses they might not know. Farther, wherever they ufe Demonftrations of their own, in A 2 Stead ftead of Euclid's, in thofe Demonftrations. 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 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 those Quantities, taking in both Commenfurable and Incommenfurable ores, and from which, all the other Properties of Propor tionals do easily follow. Some Geometricians, for footh, want a Demonftration of this Froperty in Euclid; and and undertake to fupply the Deficiency by one of their own. Here, again, they shew 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 Specified in the faid Definition. And why might not the Author of the Elements give what Names he thought fit to Quantities having fuch Requifites; furely he 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 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 perplex'd Series of Conclufions, 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 firft laid down that univerfal A 3 Property Property affign'd by Euclid, and thence have deduc'd 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 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 publifh'd by him, wherein generally he has retain'd the Conftructions and Demonftrations of Euclid him felf, not having omitted fo much as one Propofition, Hence, his Demonftrations became more Strong and nervous, his Conftruction more neat and elegant, and the Genius of the antient 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 fhorten the Demonftrations of what follows, but are likewise of use in other Matters, Not |