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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 destinet and clear, as being taken from first Principles, and our most easy and simple Conceptions ; and his Demonstrations elegant, perfpicuous and concise , carrying with them such Evidence and so much Strength of Reason, that I am easily induced to believe the Obscurity, Sciolists fa 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

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, grofferi Paralogisms, than they have been able even to imagine in Euclid.

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After so many unsuccessful Endeavours, in the Reformations of Geometry, fome very good Geometicians, not daring to make nere Elements, have deservedly preferrd Euclid to all others ; and have accordingly made it their Business to pub. lish those of Euclid. But they for what Reafon I know not, have entirely omitted fome Propositions, and have als tered the Demonstrations of other's for worlê. 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, äs Prop. 27, 28, and 29. of the Sixth Book, and some others, whose Uses they might not know. Farther, wherever they use Demonstrations of their own, ina

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stead

Read 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 same Rock; and to Shew their Skill, blame Euclid, for what, on the contrary, he ought to be commended ; I mean, the Definition of Proportional Quantities, where. in he Shews an ealy Property of those Quantities, taking in both Commensurable and Incommensurable oxes, and from which, all the other Properties of Proportionals do easily follow.

Some Geometricians, for footh, 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 such as determines those Quantities Proportionals which have the Conditions specified in the said Definition. And why might not the Author of the Elements give what Names he thought fit to Quantities having such Requisites; surely he might use his own Liberty, and accordingly has called them Proportia nals,

But it

may be proper here to examine the Method whereby they endeavour to Demonstrate that Property : Which is by first assuming a certain" Affection, agreeing only to one kind of Proportionals, viz. Commensurables; and thence, by a long Circuit, and a perplex'd Series of Conclusions, do deduce that universal Property of Proportionals which Euclid affirms; a Procedure foreign enough to the just Me thods and Rules of Reasoning. They would certainly have done much better, if they had first laid down that universal

Property

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Property aligu'd by Euclid, and thence have dedriced that particular Property agreeing to onlyoneSpecies of Propartionals. 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 consult the Mathematical Lectures of the learn’d Dr. Barrow.

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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; his Construction more neat and elegant, and the Genius of the antient Geometricians more conspicuous, than is ufually found in other Books of this kind. To this he has added, several Co , rollaries 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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