Sidebilder
PDF
ePub

fufficiently valuable for the purposes they were intended by Euclid himself, do not fo nearly belong to the elements of plane and folid geometry, as the first fix, eleventh, and twelfth books.

Accordingly these eight books alone by most of the moderns have been looked upon as fufficient Elements of that plane and folid geometry, in ufe and fashion amongst us.

If it should be asked, what occafion there was for publishing afresh another English translation of thefe eight books, when there are fo many already, especially after the feveral English impreffions of Commandine's Euclid; I anfwer, this tranflation is from the best and most correct edition of Euclid' himself by Dr. Gregory, who himself says, in the preface, that his own Latin tranflation is vastly more correct than Commandine's; that Dr. Hudson carefully compared the copy with the best original Greek manufcripts, given by Sir Henry Saville for the use of the univerfity of Oxford; and diligently revised the sheets over and over before they were printed off, as they came from the prefs.

The figures are annexed to the feveral propofitions, and not at a distance. Those of the fifth book are more diftinct and better adapted to the generality of the propofitions, and oftentimes better fuited to the comprehenfion of the demonftrations. The notes I have added, do explain and clear up fome difficulties and obfcurities that may occur to learners, and easier demonftrate fome propofitions, and clear Euclid from fome feeming faults and overfights; for I am not entirely of opinion with Dr. Keil, who fays, in his preface to Commandine's Euclid, that Euclid himself is clearer than any of his commentators. Indeed tho' Clavius's Commentary upon Euclid, as the Dr. rightly obferves, is in general certainly too tedious and prolix, yet many of his obfervations are useful, and make Euclid

the

[ocr errors][ocr errors]

the eafier. I never in my life knew a learner that did underftand Euclid's fifth definition of the fifth book, without a further explanation.

I have alfo added feveral propofitions to this edition, containing many valuable, useful, and elegant theorems and problems, which, with those of Euclid himself, do render the whole more compleat elements of common geometry and fome of thefe are new, at least to me; fuch as Prop. ii. at page 54. Prop. vii. at page 64. the Scholium at page 97. the Scholium at page 100. Prop. viii. at page 204. Prop. ix. at page 205. the Scholium at page 106. and the propofition. Prop. ix. at page 158. Prop. x. at page 160. Prop. xi. at page 161. Prop. xii. at page 163. Prop. xiii. at page 164. Prop. vi. page 197. the theorem at the end of page 273. Prop. viii, page 298. and feveral others.

I hope the errors of the prefs are but few, an honeft able friend of mine, very skilful in geometry, having affifted me in correcting the fheets as they came from the prefs. The demonstrations of the lemma at page 102, of Prop. xxiii of the fixth book. at page 273. and of Prop. ii, of the additions to the fixth book, at page 291, are his.

*Mr. William Payne, a teacher of mathematics.

Once

[ocr errors][merged small]

A

T the End of this Second Edition are added feveral Things not in the former, tending to free thefe Elements from Error, and clear them yet more from the real or feeming Blemishes that may have happened, or be thought to have happened, either from Euclid himself, which I am loath to fuppofe, or any body elfe.What is herein contained of the greatest Note, is an Obfervation of mine, which but lately occurred to me, on the Fifth Definition of the Fifth Book about Magnitudes having the fame Ratio, viz. that this Definition does really extend to commenfurable Magnitudes only, and not to incommenfurable ones; although it has been generally thought, by all the modern Writers I have ever seen, to take in both commenfurable and incommenfurable Magnitudes, moft People thinking it was for this Purpose that Definition itself was invented, and put down. -Now that Part of the Fifth Definition of the Fifth Book (see the Definition itself in its Place) which fays, "If the Multiple of the First Magni

tude be equal to the Multiple of the Second, "the Multiple of the Third will be equal to the Multiple

66

66 Multiple of the Fourth," Cannot exist when the Magnitudes are incommenfurable; because when the First and Second, and the Third and Fourth Terms of Two equal Ratios, or Four Proportionals are incommenfurable, no Number of Times the Firft can be equal to any Number whatsoever of Times the Second, nor any Number of Times the Third, equal to any Number whatsoever of Times the Fourth; for otherwise incommenfurable Magnitudes would be to one another as one Number is to another, which Euclid has demonftrated to be impoffible, at Prop. 7 of his 10th Book. Therefore it is evident this Fifth Definition is not a good one, as containing an impoffible Condition in the Cafe of incommenfurable Magnitudes.-Within this Day or two I have been induced to think Euclid himself (if he was the Author of the Fifth Book) knew this Fifth Definition of the Fifth Book extended only to commenfurable Magnitudes; becaufe in his Tenth Book he calls commenfurable Magnitudes Rationals, and incommenfurable Ones Irrationals, i. e. the former fuch as have a Ratio, or may be compared together according to Quantity, and the latter thofe that cannot be compared together according to Quantity, as not having a Ratio according to his Notion and the Meaning of the Word in his Third Definition of the Fifth Book, notwithstanding the Fourth Definition of the fame Book fays, "Magnitudes have a Ratio, the Leffer "of which can be multiplied fo as to exceed the "other;" which I think was put down rather to fhew that a Line and Superficies, or a Solid, &c. have no Ratio at all to one another, being quite incomparable according to Quantity.-Moreover, I do not know how juftly the Author of the Fifth Book can apply his Propofitions of it to fhew the Proportionality of Lines, and plain Figures in the

Sixth Book; and of Lines, Surfaces, and Solids in the Eleventh and Twelfth Books, being Magnitudes of different Kinds. Because I must think all his Magnitudes in the Propofitions of the Fifth Book are agreeable to the Fourth Definition of it, and therefore they are all of the fame Kind, viz. Lines, Superficies, or Solids, &c.-If this be not granted, I am certain the Twelfth, Fourteenth, Sixteenth, and Twenty-fifth Propofitions of the Fifth Book will not hold good when the Magnitudes in those Propofitions are of different Kinds. -I therefore hold it more probable to fuppofe the original Author of the Fifth Book defignedly reprefented in this Book all Magnitudes of different Kinds by right Lines; and whatever held true of thefe, was to be taken as fuch in any Magnitudes of different Kinds represented by these right Lines. I fay it is more probable to suppose this, than to make that Author guilty of putting down the Four Propofitions abovementioned, that cannot pass without being mended by the Addition of the Words, all of the fame Kind.-I am, moreover, certain that taking right Lines for the Reprefentatives of Magnitudes of different Kinds, either Curve Lines, Superficies, or Solids, &c. the whole Doctrine of the Proportionality of Magnitudes contained in the Fifth Book, may be clearly, briefly, and easily, ftated and demonftrated, whether the Magnitudes be commenfurable or incommenfurable, by a clear Definition of Proportionality very different from the Fifth of the Fifth Book, and by help of a few of the Propofitions of the First and Third Books.-But this by the bye.-I do not think it neceffary to fay more here about thefe Notes, &c. which I have added to this Second Edition at the End; they are but short, and the geometrical Reader himself will foon peruse them and judge

better

« ForrigeFortsett »