PREFACE, SHEWING, The USEFULNESS and EXCELLENCY of this WOR K. D R. KEIL L, in his Preface, hath fufficiently declared how much eafier, plainer, and more elegant, the Elements of Geometry written by Euclid are, than thofe written by others; and that the Elements themselves are fitter for á Learner, than those published by such as have pretended to comment on, fymbolize, or tranfpofe, any of his Demonstrations of fuch Propofitions as they intend to treat of. Then how muft a Geometrician be amazed, when he meets with a Tract* of the ift, 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 Demonstration 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 Order, 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 fuch 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 these 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 Demonstration of the 9th Propofition of the third Book; and the other a Demonftration of that Property of Proportionals called Converfions (contained in a Corollary to the 19th Propofition of the fifth Book ;) where, inftead of Euclid's Demonftration, which is univerfal, moft Authors have given us only particular ones of their own. The firft of thefe, which was omitted, is here fupplied: And that which was corrupted is here restored *. And fince feveral Perfons, to whom the Elements of Geometry are of vast Use, either are not so fufficiently fkilled in, or perhaps have not Leifure, or are not willing to take the Trouble, to read the Latin; and fince this Treatife was not be fore 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. As 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 fome of the Solutions. 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 1ft 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 fometimes not; and when it is not doubtful, it is fometimes greater than 90 Degrees, and fometimes lefs: And fure 1 fhall commit no Crime, if I affirm, that no Solution can be given without a just Distinction of these Varieties. For the Solution of these Cafes, fee Pages 320, 321. In the Solution of our 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 Writers of Trigonometry, that I have met with, who have undertaken the Solutions of these two, as well as the two following Cafes, by letting fall a Perpendicular, which is undoubtedly the shortest and best Method for finding either of these Quæfita, have told us, that the Sum {Difference } of the Vertical Angles, or Bases, fhall be the } not be known, unless the Species of that unknown Angle, which is opposite to a given Side, be first known. Here they leave us firft to calculate that unknown Angle, before we shall know whether we are to take the Sum or the Difference of the vertical Angles or Bafes for the fought Angle or Bafe: And in the Calculation of that Angle have left us in the Dark as to its Species; as appears by the Obfervations on the two preceding Cafes. The Truth is, the Quafitum here, as well as in the two former Cafes, is fometimes doubtful, and fometimes not; when doubtful, fometimes each Answer is lefs than 90 Degrees, fometimes each is greater; but fometimes one lefs, and the other greater, as in the two last-mentioned Cafes. When it is not doubtful, the Qua |