ftrations begets Obfcurity, fo too much Prolixity produces Tedioufnefs and Confufion. On thefe Accounts, principally, it was, that I undertook to publish the first fix Books of Euclid, with the 11th and 12tb, according to Commandinus's Edition; the reft I forbore, because those, firft-mentioned, are fufficient for understanding of moft Parts of the Mathematics now ftudied. Farther, for the Ufe of those who are defirous to apply the Elements of Geometry to Ufesin Life, we have added a Compendium of Plane and Spherical Trigonometry; by Means whereof, Geometrical Magnitudes are meafured, and their Dimenfions expressed in Numbers. J. KEILL. PREFACE, SHEWING, The USEFULNESS and EXCELLENCY of this W OR K. D R. KEILL, in his Preface, hath fufficiently declared how much eafier, plainer, and more elegant, the Elements of Geometry written by Euclid are, than those written by others; and that the Elements themselves' are fitter for a Learner, than those published by fuch as have pretended to comment on, fymbolize, or tranfpofe, any of his Demonftrations of fuch Propofitions as they intend to treat of. Then how muft a Geometrician be amazed, when he meets with a Tract * of the 1st, 2d, 3d, 4th, 5th, 6th, 11th, and 12th Books of the Elements, in which are omitted the Demonftrations of all the Propofitions of that moft noble univerfal Mathefis, the 5th; on which the 6th, 11th, and 12th, fo much depend, that the Demonftration of not fo much as one Propofition, in them, can be obtained without thofe in the fifth? *Vide the laft Edition of the English Tacquet. The The 7th, 8th, and 9th Books treat of fuch Properties of Numbers which are neceffary for the Demonftration of the 10th, which treats of Incommenfurables; and the 13th, 14th, and 15th, of the five Platonic Bodies. But though the Doctrine of Incommenfurables, because expounded in one and the fame Plane, as the first fix. Elements were, claimed, by a Right Or-. der to be handled before Planes interfected by Planes, or the more compounded Doctrine of Solids; and the Properties of Numbers were neceffary to the Reasoning about Incommenfurables; yet, because only one Propofition of these four Books, viz. the ift of the 10th, is quoted in the 11th and 12th Books; and that only twice, viz. in the Demonftration of the 2d and 16th of the 12th; and that ift Propofition of the 19th is fupplied by a Lemma in the 12th; and because the 7th, 8th, 9th, 10th, 13th 14th, and 15th Books have not been thought (by our greatest Masters) neceffary to be read by fuch as defign to make Natural Philofophy their Study, or by ffuch as would apply Geometry to practical Affairs; Dr. Keill, in his Edition, gave us only these eight Books, viz. the firft fix, and the 11th and 12th. And as he found there was wanting a Treatife of thefe Parts of the Elements, as they were written by Euclid himself; he he published his Edition without omitting any of Euclid's Demonftrations, except two; one of which was a fecond Demonftration of the 9th Propofition of the third Book; and the other a Demonstration of that Property of Proportionals called Converfions (contained in a Corollary to the 19th Propofition of the fifth Book ;) where, instead of Euclid's Demonstration, which is univerfal, most Authors have given us only particular ones of their own. The first of these, which was omitted, is here fupplied: And that which was corrupted is here restored*. And fince several Perfons, to whom the Elements of Geometry are of vast Use, either are not fo fufficiently skilled in, or perhaps have not Leifure, or are not willing to take the Trouble, to read the Latin; and fince this Treatife was not before in English, nor any other which may properly be faid to contain the Demonftrations laid down by Euclid himself: I do not doubt but the Publication of this Edition will be acceptable as well as ferviceable. Such Errors, either typographical, or in the Schemes which were taken Notice of in the Latin Edition, are corrected in this. * Vide Page 55, 107, of Euclid's Works, published by Dr. Gregory. Ac As to the Trigonometrical Tract, annexed to these Elements, I find our Author, as well as Dr. Harris, Mr. Cafwell, Mr. Heynes, and others of the Trigonomemetrical Writers, is mistaken in some of the Solutions. ་ 1 That the common Solution of the 12th Cafe of Oblique Spherics is falfe, I have demonftrated, and given a true one. Page 318. See In the Solution of our 9th and 10th Cafes, by our Authors called the ift and 2d, where are given and fought oppofite Parts, not only the afore-mentioned Authors, but all others that I have met with, have told us, that the Solutions are ambiguous; which Doctrine is, indeed, fometimes true, but fometimes falfe: For fometimes the Quæfitum is doubtful, and sometimes not; and when it is not doubtful, it is fometimes greater than 90 Degrees, and fometimes lefs: And fure I fhall commit no Crime, if I affirm, that no Solution can be given without a juft Diftinction of thefe Varieties. For the Solution of thefe Cafes, fee Pages 320, 321. In the Solution of the 3d and 7th Cafes, in other Authors reckoned the 3d and 4th, where there are given two Sides, and an Angle oppofite to one of them, to find the 3d Side, or the Angle oppofite to it; all the Writers |