Dr. K E IL L’s





YOUNG Mathematician may be

surprised to see the old obsolete 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 have 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 Falsties 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 distinɛt and clear, as being taken from the first Principles, and our most easy and fimple Conceptions ; and his Demonstrations elegant, perspicuous, and concise, carrying with them such Evidence, and so much Strength of Reason, that I am eafly induced to believe, that the Obscurity Sciolists fo often accuse Euclid with, is rather to be attributed to their own perA 2

plexed plexed Ideas, than to the Demonstrations themselves. And however 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 pew 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 Pa. 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 bave accordingly made it their Business to publish those of Euclid. But they, for what Reason I know not, have entirely omitted some Propositions, 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 uith 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 have 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 those 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 thew 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 DemonAtration 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 bere 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. Commensurables ; 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 of 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 have happened to mention this great Geometrician, I must not pass by the Elemients published by him, wherein, generally, he has retained the Constructions and Demonfrations of Euclid bimself, not having omitted so much as one Proposition. Hence, bis Demonjirations become more Arong and nervous, bis Constructions more neat and


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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 forten the DemonAtration of what follows, but are likewise of Ufe in other Matters.

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

As Barrow's Elements are too short, 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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