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fufficiently valuable for the purposes they were intended by Euclid himself, do not so nearly belong to the elements of plane and solid geometry, as the first fix, eleventh, and twelfth books.
Accordingly these eight books alone by most of the moderns have been looked upon as sufficient Elements of that plane and solid geometry, in use and fashion amongst us.
If it should be asked, what occafion there was for publishing afresh another English translation of these eight books, when there are so many already, especially after the several English impressions of Commandine's Euclid; I answer, this translation is from the best and most correct edition of Euclid himself by Dr. Gregory, who himself says, in the preface, that his own Latin tranlation is vastly more correct than Commandine's; that Dr. Hudson carefully compared the copy with the best original Greek manuscripts, given by Sir Henry Saville for the use of the university of Oxford ; and diligently revised the sheets over and over before they were printed off, as they came from the press.
The figures are annexed to the several propofitions, and not at a distance. Those of the fifth book are more distinct and better adapted to the generality of the propositions, and oftentimes better fuited to the comprehension of the demonstrations. The notes I have added, do explain and clear up fome difficulties and obscurities that may occur to learners, and easier demonstrate some propositions, and clear Euclid from some seeming faults and oversights; for I am not entirely of opinion with Dr. Keil, who says, 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 observes, is in general certainly too tedious and prolix, yet many of his observations are useful, and make Euclid
the easier. 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 also added several propositions to this edition, containing many valuable, useful, and elegant theorems and problems, which; with those of Euclid himself, do render che whole more compleat elements of common geometry and fome of these are new, at least to me; such 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 påge 106. and the proposition. 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 2731 Prop. viii, page 298. and several others.
I hope the errors of the press are but few, an honeft able friend of mine, very skilful in geometry, having aslifted me in correcting the sheets as they came from the press. The demonftrations 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 sixth book, at page 291, are his.
Mr. William Payne, a teacher of mathematics.
Once more to the R E A D E R.
T the End of this Second Edition are added
several Things not in the former, tending to free
these Elements from Error, and clear them yec more from the real or seeming Blemishes that may have happened, or be thought to have happened, either from Euclid himself, which I am loath to
fuppose, or any body else. -- 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 same Ratio, viz. that this Definition does really extend to commensurable Magnitudes only, and not to incommensurable ones ; although it has been generally thought, by all the modern Writers I have ever seen, to take in both commensurable and incommensurable 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 says, “ If the Multiple of the First Magni*** cude be equal to the Multiple of the Second, che Maltiple of the Third will be equal to the
“ Multiple of the Fourth,” cannot exist when the Magnitudes are incommensurable ; because when the First and Second, and the Third and Fourthi Terms of Two equal Ratios, or Four Proportionals are incommensurable, no Number of Times the First 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 incommensurable Magnitudes would be to one another as one Number is to another, which Euclid has demonstrated to be impossible, at Prop. z of his roth Book." Therefore it is evident this Fifth Definition is not a good one, as containing an impossible Condition in the Case of incommensurable 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 commensurable Magnitudes; because in his Tenth Book he calls commenfurable Magnitudes Rationals, and incommensurable Onės Irrationals, i. e. the former such as have a Ratio, or may be compared together according to Quantity, and the latter those 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 same Book says, “ Magnitudes have a Ratio, the Leffer “ of which can be multiplied so as to exceed the " other;" which I think was put down rather to shew 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 Propositions of it to shew 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 same Kind, viz. Lines, Superficies, or Solids, &c.—If this be not granted, I am certain the Twelfth, Fourteenth, Sixteenth, and Twenty-fifth Propositions of the Fifth Book will not hold good when the Magnitudes in those Propositions are of different kinds. -I therefore hold' it more probable to suppose the original Author of the Fifth Book designedly represented in this Book all Magnitudes of different Kinds by right Lines; and whatever held true of these, was to be taken as such in any Magnitudes of different kinds represented by these right Lines. I say it is more probable to suppose this, than to make that Author guilty of putting down the Four Propositions abovementioned, that cannot pass without being mended by the Addition of the Words, all of the same Kind.--I am, moreover, certain that taking right Lines for the Representatives 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, stated and demonstrated, whether the Magnitudes be commensurable or incommensurable, by a clear Definition of Proportionality very different from the Fifth of the Fifth Book, and by help of a few of the Propositions of the First and Third Books.-But this by the bye. I do not think it necessary to say more here about these Notes, &c. whích I have added to this Second Edition at the End; they are but short, and the geometrical Reader himself will soon peruse them and judge