But enough of this in general ; let us come nearer our present work, and more particularly observe, that we Britons have had amongst us scarcely any who have ventured to write Elements of Geometry of their own; altho' this nation has produced both the greatest mathematicians, and the greatest number of them too. We have generally liked Euclid's Elements beft, and know them too well to be at the trouble of compiling new Elements of Geometry (as many of the French, and other foreigners have done) and thereby changing better for worse. We think it far more eligible to do nothing at all, than vainly busy ourselves in matters introducing confusion and error into the geometrical world, and, by avoiding falfe praise, have likewise been free from evident difgrace, and real dishonour. For the method and order of Euclid's Elements of Geometry can neither be mended nor altered, but for the worse. And the demonstrations are mostly so very accurate, nervous, and elegant, as not to be exceeded, if equalled, by any geometrical writer whatsoever, either ancient or modern. His method may be defended for ever, and his demonftrations will be approved by all men of found judgment to the end of the world. His first principles or axioms are few, simple, and clear, taken from our primitive and natural conceptions of things, and such as every one can easily apprehend, and no man in his senses can deny. And his method is such, that nothing is taken as true, unless it be demonstrated, and nothing is demonstrated, but from what went before. And the demonstrations, as I said before, are in the main so perfect and compleat, that the most severe, critic could never find a real fault in them. The greatest masters of reasoning have always been captivated and charmed by their beauty and elegance, and, as one may fay, they ravish the reader's affent, and force an absolute com 44 mand mand over the mind that dares encounter them. But other element-writers, whether fuch as have alter'd Euclid's order; such as have given other demonstrations of some of his propositions ; such as have left out some of the most simple propositions, or ranked them amongst the class of axioms, are generally faulty in some respect or other; their own demonstrations are oftentimes imperfect, or sophiftical, and sometimes obscured and vitiated by the mixture of algebra and algebraical signs. In a word, the Elements of Euclid far exceed them all, especially as to method and elegance of demonftration, As to the fifteen books of Euclid's Elements, it is true there are some of more importance and use than others in the geometry requisite to the necesfary mechanic arts, and useful reputable fciences, now exercifed and cultivated amongst the several nations of Europe. The seventh, eighth, and ninth books are pretty short elements of arithmetic, and not of geometry ; altho' these, with the rest, are altogether usually called Euclid's Elements of Geometry. The tenth book, being the Elements of the doctrine of incommenfurability, is indeed fine; but its length, and apparent inutility in any. of the favourite mathematical studies of these ages, makes it very unpalatable, and much neglected. The same may almost be said of the 13b, 14th, and 15th books, containing a theory of the five regu- . lar solids, or Platonic bodies, as they are called. What divinity the antients found in these bodies I cannot at all imagine ; surely there must have been something very extraordinary in them, for Euclid is expresly related by Proclus to have compiled the whole system of his Elements only for the sake of the doctrine of the five regular solids. However it must be owned, that these books, tho* elegant in themselves, and, it may be prefum'd, sufficiently I 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 thefe 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 tranNation 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 proposi. tions, 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 suited to the comprehension of the demonstrations. The notes I have added, do explain and clear up some 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 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; fuch as Prop. ii. at page 54. Prop. vii. at page 64. the Scholium at page 97. the Scholium at page 100. Prop: viü. at page 204. Prop. ix. at page 205. the Scholium at page 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 273. 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 assisted me in correcting the sheets as they came from the press. 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 3 Once more to the READER. A 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 Observation 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 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 says, “ If the Multiple of the First Magni* cude be equal to the Multiple of the Second, che Multiple of the Third will be equal to the Multiple |