Dr. KEIL L's


AYOUNG Mathematician may be furprised to see the old obfolete Elements of Euclid appear afresh in Print; and that, too, after fo many new Elements of Geometry as have been lately published; efpecially thofe who gave us the Elements of Geometry, in a new Manner, would have us believe they have detected a great many Faults in Euclid. Thefe acute Philofophers pretend to have difcovered, that Euclid's Definitions were not confpicuous enough; that his Demonftrations are scarcely evident; that his whole Elements are illdifpofed; and that they have found out innumerable Falfities in them, which had lain hid to their Times.

But, by their Leave, I make bold to affirm, that they carp at Euclid undefervedly: for his Definitions are diftinct and clear, as being taken from the first Principles, and our most eafy 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, that the Obscurity Sciolifts fo often accufe Euclid with, is rather to be attributed to their own per

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plexed Ideas, than to the Demonftrations themselves. And however Jome may find Fault with the Difpofition and Order of bis Elements, yet notwithstanding, I do not find Method in all the Writings of this Kind, more proper and eafy for Learners than that of Euclid.


It is not my Bufinefs here to answer, feparately every one of thefe Cavilers; but it wwill eafily appear to any one moderately verfed in thefe Elements, that 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, groffer Paralogifms, than they have been able even to imagine in Euclid.

After fo many unfuccessful Endeavours in the Reformation of Geometry, fome very good Geometricians, not daring to make new Elements, have defervedly preferred Euclid to all others, and have accordingly made it their Business to publish those of Euclid. But they, for what Reafon I know not, have entirely omitted fome Propofitions, and have altered the Demonftrations of others, for worfe. Among whom are chiefly Tacquet and Dechales, both of which have unhappily rejected fome elegant Propofitions in the Elements (which ought to have been retained), as imagining them trifling and ufelefs; fuch, for Example, as Prop. 27, 28, and 29, of the 11th Book, and fome others, whofe Ufes they might not know. Farther,


wherever they ufe Demonftrations of their own, instead of Euclid's, in thofe Demonftrations, they are faulty in their Reasoning, and deviate very much from the Confcifenes 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 ever have pardoned, had not thofe Authors done laudable Things in their other Mathematical Writings. Indeed, this Fault of theirs is common to all Modern Writers

Elements, who all Split on the famé 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 thofe Quantities, taking in both commenfurable and incommenfurable ones, and from which all the other Properties of Proportionals do eafily follow.

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

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finition. And why might not the Author of the Elements give what Names he thought fit to Quantities, having fuch Requifites? 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 demonftrate 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 Conclufions, do deduce that univerfal Property of Proportionals which Euclid affirms; a Procedure foreign enough to the juft Methods and Rules of Reajoning. They would certainly have done much better, if they had first laid down that univerfal 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 Demonftration to this Definition of the fifth Book. Those who have a mind to fee a farther Defence of Euclid, may confult the Mathematical Lectures 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 Conftructions and DemonHrations of Euclid himself, not having omitted fo much as one Propofition. Hence, bis Demonftrations become more Arong and nervous, bis Conftructions 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 be has added feveral Corollaries and Scholia, which ferve not only to shorten the Demonftration of what follows, but are likewife of Ufe in other Matters.


Notwithstanding this, Barrow's Demonftrations are fo very short, and are involved in fo many Notes and Symbols, that they are rendered cbfcure and difficult to one verfed in Geometry. There, many Propofitions which appear confpicuous in reading Euclid bimfelf, are made knotty, and scarcely intelligible to Learners, by his Algebraical Way of Demonftration; as is, for Example, Prop. 13. Book I. And the Demonftrations which he lays down in Book II. are fill more difficult: Euclid himself has done much better, in fhewing 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 obfcure Principles, taken elsewhere.

As Barrow's Elements are too short, fo are thofe of Clavius too prolix, abounding in fuperfluous Scholiums and Comments : For, in my Opinion, Euclid is not fo obfcure as to want fuch 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 DemonA 4


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