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Dr. K E IL L's


A YOUNG Mathematician


be surprised to see the old obfolete Elements of Euclid appear afresh in Print ; and that, too, after so many new Elements of Geometry as bave been lately published ; especially those who gave us the Elements of Geometry, in a new Manner, would bave us believe they have detected a great many Faults in Euclid. These acute Philosophers pretend to have discovered, that Euclid's Definitions were not conspicuous enough; that his Demonstrations are scarcely evident ; that his whole Elements are illdisposed; and that they have found out innumerable Falfties in them, which had lain bid to their Times.

But, by their Leave, I make bold to afirm, that they carp at Euclid undefervo edly: for bis Definitions are distinɛt and clear, as being taken from the first Principles, and our molt easy and fimple Conceptions ; and bis Demonstrations elegant, perspicuous, and concise, carrying with them such Evidence, and so much Strength of Reafon, that I am easily induced to believe, that the Obscurity Sciolists fo often accuse Euclid with, is rather to be attributed to their own per


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plexed Ideas, than to the Demonstrations themselves. And bowever some may find Fault with the Disposition and Order of bis 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 Cavilers; but it will easily appear to any one moderately versed in these Elements, that they rather Shew their own Idleness tban 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 Pan ralogisms, than they have been able even to imagine in Euclid.

After so many unsuccessful Endeavours in the Reformation of Geometry, some very good Geometricians, not daring to make new Elements, have deservedly preferred Euclid to all others, and have accordingly made it their Business to publish those of Euclid. But they, for what Reason I know not, have entirely omitted some Propofitions, and have altered the Demonstrations of others, for worse. Among whom are chiefly Tacquet and Dechales, both of which have unhappily rejected some elegant Propositions in the Elements (which ought to have been retained), as imagining them trifling and uselefs; such, for Example, as Prop. 27, 28, and 29, of the nith Book, and some others, whose Uses they might not know. Farther,


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wherever they use Demonstrations of their own, instead of Euclid's, in those Demonftrations, they are faulty in their Reasoning, and deviate very much from the Conscifeness of the Antients.

In the fifth Book, they have wholly rejected Euclid's Demonstrations, and bave given a Definition of Proportion different from Euclid's, and which comprehends but one of the two Species of Proportion, taking in only commensurable Quantities. Which great Fault, no Logician or Geometrician would ever have pardoned, had not tboje Author's 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 sew their skill, blame Euclid, for what, on the contrary, be ought to be commended ; I mean, the Definition of proportional Quantities, wherein beshews an easy Property of those Quantities, taking in both commensurable and incommenfurable ones, and from which all the other Properties of Proportionals do easily follow.

Some Geometricians, forfooth, want a Demonstration of this Property in Euclid; and undertake to supply the Deficiency by one of their own. Here, again, they shew their Skill in Logic, in requiring a Demonftration 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 De


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finition. And why might not the Author of
the Elements give what Names be thought
fit to Quantities, having such Requisites ?
Surely he might use his own Liberty, and ac-
cordingly has called them Proportionals.
But it


proper here to examine the Method whereby they endeavour to demonstrate that Property : Which is by first afjuming a certain Affection, agreeing only to one Kind of Proportionals, viz. Commenfurables; and thence, by a long Circuit, and a perplexed Series of Conclusions, do deduce that universal Property of Proportionals which Euclid afirms; a Procedure foreign enough to the just Methods and Rules Reasoning. They would certainly have done much better, if they had first laid down that universal 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 Demonstration to this Definition of the fifth Book. Those wbo have a mind to see a farther Defence of Euclid, may consult the Mathematical Lečtures of the learned Dr. Barrow.

As I bave happened to mention this great Geometrician, I must not pass by the Elements published by him, wherein, generally, he has retained the Constructions and Demonftrations of Euclid himself, not having omitted so much as one Proposition. Hence, bis Demonjirations become more frong and nervous, bis Constructions more neat and


elegant, and the Genius of the antient Geometricians more conspicuous, than is usually found in other Books of this Kind. To this be bas added several Corollaries and Scholia, which ferve not only to shorten the Demonftration of what follows, but are likewise of Use in other Matters.

Notwithstanding this, Barrow's Demonstrations are so very short, and are involved in so many Notes and Symbols, that they are rendered cbscure and dificult to one not versed in Geometry. There, many Propoptions which appear conspicuous in reading Euclid bimself, are made knotty, and scarcely intelligible to Learners, by bis Algebraical Way of Demonstration; as is, for Example, Prop. 13. Book I. And tbe Demonstrations which he lays down in Book II. are still more difficult : Euclid himself has done much better, in Mewing 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 most fimple Method, and not to be involved in Symbols, Notes, or obscure Principles, taken elsewhere.

As Barrow's Elements are too fort, so are those of Clavius too prolix, abounding in superfluous Scholiums and Comments : For, in my Opinion, Euclid is not so obscure as to want such 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 Gemetrical Demon


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