of this WORK,


R. KEIL L, in his Preface, hath sufficiently declared how much

easier, 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 such as have pretended to comment on, symbolize, or transpose, any of his Demonstrations of such Propositions as they intend to treat of. Then how must a Geometrician be amazed, when he meets with a Tract * of the ist, 2d, 3d, 4th, 5th, 6th, 11th, and 12th Books of the Elements, in which are omitted the Demonstrations of all the Propofitions of that most noble universal Mathesis, the 5th; on which the 6th, Ilth, and 12th, so much depend, that the Demonstration of not so much as one Proposition, in them, can be obtained without those in the fifth !

* Vide the last Edition of the English Tacquet.


The 7th, 8th, and 9th Books, treat of such Properties of Numbers which are necessary for the Demonstration of the roth, which treats of Incommensurables; and the 13th, 14th, and 15th, of the five Platonic Bodies. But though the Doctrine of Incommensurables, because expounded in one and the fame Plane, as the first fix Elements were, claimed, by a Right Order, to be handled before Planes intersected by Planes, or the more compounded Doctrine of Solids; and the Properties of Numbers were necessary to the Reasoning about Incommensurables; yet, because only one Proposition of these four Books, viz. the ist of the 10th, is quoted in the 11th and 12th Books; and that only twice, viz. in the Demonstration of the 2d and 16th of the 12th ; and that ift Proposition of the 19th is supplied 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) necessary to be read by fuch as design to make Natural Philosophy their Study, or by such as would apply Geometry to practical Affairs; Dr. Keill, in his Edition, gave us only these eight Books, viz. the firft fix, and the rith and 12th.

And as he found there was wanting a Treatise of these parts of the Elements, as they were written by Euclid himself;


he published his Edition without omitting any of Euclid's Demonstrations, except two; one of which was a second Demonftration of the gth Proposition of the third Book ; and the other a Demonstration of that Property of Proportionals called Conversions (contained in a Corollary to the 19th Proposition of the fifth Book ;) where, instead of Euclid's Demonftration, which is universal, most Authors have given us only particular ones of their own. The first of these, which was omitted, is here supplied: And that which was corrupted is here restored *

And fince several Persons, to whom the Elements of Geometry are of vast Use, either are not so sufficiently skilled in, or perhaps have not Leisure, or are not willing to take the Trouble, to read the Latin; and since this Treatise was not before in English, nor any other which may properly be said to contain the Demonstrations laid down by Euclid himself, I do not doubt but the Publication of this Edition will be acceptable, as well as serviceable.

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 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.

That the common Solution of the 12th Case of Oblique Spherics is false, I have demonstrated, and given a true one. See Page 318.

In the Solution of our gth and 1 oth Cafes, by our Authors called the ist and 2d, where are given and sought opposite 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, sometimes true, but sometimes false: For sometimes the Quæfitum is doubtful, and sometimes not; and when it is not doubtful, it is sometimes greater than 90 Degrees, and sometimes less : And sure I shall 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, see Pages 320, 321.

In the Solution of our 3d and 7th Cases, in other Authors reckoned the 3d and 4th, where there are given two Sides, and an Angle opposite to one of them, to find the 3d Side, or the Angle opposite to it; all the


Writers of Trigonometry, that I have met with, who have undertaken the Solutions of these two, as well as the two following Cases, by letting fail aPerpendicular, which is undoubtedly the shortest and best Method for finding either of these Quæßta,

Sum have told us, that the

of the

Difference Vertical Angles, or Bases, shall be the sought Angle or Side, according as the



within not be known, unless the Species of that unknown Angle, which is opposite to a given Side, be first known.

Here they leave us first to calculate that unknown Angle, before we shall know whether we are to take the Sum or the Difference of the vertical Angles or Bases for the sought Angle or Base: And in the Calculation of that Angle have left us in the Dark as to its Species ; as appears by the Observations on the two preceding Cases.

The Truth is, the quahtum here, as well as in the two former Cafes, is fometimes doubtful, and sometimes not; when doubtful, sometimes each Answer is less than 90 Degrees, sometimes each is greater ; but sometimes one less, and the other

greater, as in the two last-mentioned Cases. When it is not doubtful, the Qua

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