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perplexed Ideas, than to the Demonstration themselves, And however some may find Fau 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 Cavillers; but it will easily appear to any one, moderately versed in thejė Elements, that they rather sew 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, groffer Paralogisms, 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 neró Elements, have deservedly preferr’d Euclid to all others; and have accordingly made it their Bufiness to publish those of Euclid. But they, for what Reason I know not, have entirely omitted fome Propositions, and have altered the

Demonstrations of others for worfe. Among whom are chiefly Tacquet and Dechalles, both of which have unhappily rejected some elegant Propositions in the Elements (which ought to have been retained) as imagining them trifling and useless ; such, for Example, as Prop. 27, 28, and 29, of the Sixth Book, and some others, whojé Uses they might not know. Farther, wherever they use Demonstrations of their

own,

own, instead 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 commenfurable quantities. Which great Fault no Logician or Geometrician would have ever pardoned, had not those Authors done laudable Things in their otser Mathematical Writings. Indeed, this Fault of theirs is common to all Modern Writers of Elements, who all split on the same Rock; and to flew their Skill, blame Euclid, for what, on the contrary, be ought to be commended ; I mean the Definition of Proportional Quantities, wherein he shews 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, forsooth, want a Demonstration of this Property in Euclid; and undertake to supply the Deficiency by one of their own. Here, again, they show 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 said Definition. And why might A 3

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not the Author of the Elements give what Names be thought

fit to Quantities having such Requistes ? Surely be 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 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 perplexed Series of Conclusions, do deduce that universal Property

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 universal Property assigned 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 consult 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 be has retained the Constructions and Demonstrations of Euclid himself

, not having omitted so much as one Proposition. Hence, bis Demonstrations become more strong and nervous, bis Con

structions

.

structions more neat and elegant, and the Géa rius of the ancient Geometricians more confpicuous, than is usually found in other Books of this kind. To this be has added several Coroltaries and Scholias, which ferve not only to shorten the Demonstrations of what follows, but are likewise of use in other Matters.

Notwithstanding this, Barrow's Demonstrations are so very short, and are involved in fa many Notes and Symbols

, that they are rendered obscure and difficult to onė not versed in Geometry. There, many Propofitions which appear conspicuous in reading Euclid himself, are made knotty and scarcely intelligible to Learners by this Algebraical Way of Demonftration, as is, for Example, Prop. 13. Book 1. And the Demonstrations which he lays down in Book II. are still more difficult : Euclid bimself has done much better, in Newing their Evidence by the Contemplations of Figures, as in Geometry should always be done. The Élements of all Sçiences ought to be handled

after the moji simple Method, and not to be involv'd in Symbols, Notes, or obscure Principles, taken elsewhere.

As Barrow's Elements are too short, so arė those of Clavius too prolix, abounding in superfluous Scholiums and Comments: For in my Opinion, Euclid is not so obfcure as to want fuch a lumber of Notes, neither do I doubt but a Learner will find Euclid himself eaper than any of his Commentators. As too much Brez

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vity in Geometrical Demonstrations begets Obscurity, so too much Prolixity produces Tediousness and Confusion.

On these 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, because those first mentioned are sufficient for understanding of most Parts of the Mathematicks now studied.

Farther, for the Use of those who are defirous to apply the Elements of Geometry to Uses in Life, we have added a Compendium of Plain and Spherical Trigonometry, by means whereof Geometrical Magnitudes are meafur’d, and their Dimensions expressed in Numbers.

J. KEIL

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