Mr. CUN N's

PREFACE,

Shewing the USEFULNESS and EXCELLENCY of this WORK.

D

R. KEIL, 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, Symbolize, or Tranfpofe any of his Demonftrations of fuch Propofitions as they intended to treat of. Then how must a Geometrician be amazed, when he meets with a Tract of the 1ft, 2d, 3d, 4th, 5th, 6th, 11th and 12th Books of the Elements, in which are omitted the Demonftrations of all the Propofitions of that most noble univerfal Mathefis, the 5th; on which the 6th, 11th, and 12th fo much depend, that the Demonstration of not fo much as one Propofition in them can be obtained without those in the 5th?

* Vide the laft Edition of the Englife Tacquet.

The

The 7th, 8th, and 9th Books treat of fuch Properties of Numbers which are neceffary for the Demonftrations of the roth, which treats of Incommenfurables; and the 13th, 14th, and 15th, of the five Platonick 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 of 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 Reafoning 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 once, viz. in the Demonftration of the 2d of the 12th, and that Ift Propofition of the 10th, is fupplied by a Lemma in the 12th: And because the 7th, 8th, 9th, 10th, 13th, 14th, 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. Keil 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 Trea tife of these Parts of the Elements, as they were written by Euclid himfelf; he published his Edition without omitting any of Euclid's Demonstrations, except two; one of which was a fecond Demonftration of the 9th Propofition of the third Book; and the other a Demonftration of that Property of Proportionals callled Converfion, (contained in a Corollary to the

19th Propofition of the 5th Book,) where inftead of Euclid's Demonftration, which is univerfal, moft Authors have given us only particular ones of their own. The firft of these 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 fo 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 before in English, nor any other which may properly be faid to contain the Demonstrations laid down by Euclid himfelf; I do not doubt but the Publication of this Edition will be acceptable, as well as ferviceable.

1

Such Errors, either typographical, or in the Schemes, which were taken Notice of in the Latin Edition, are corrected in this.

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 Trigonometrical Writers, is mistaken in fome of the Solutions.

That the common Solution of the 12th Cafe of Oblique Sphericks is falfe, I have demonftrated, and given a true one. See Page

319.

*Vide Page 55, 107, of Euclid's Works, published by Dr. Gregory.

In

In the Solution of our 9th and 10th Cafes, by other Authors called the 1ft and 2d, where are given and fought oppofite Parts, not only the aforementioned 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 these Varieties. For the Solution of these Cases fee my Directions at Pages 321, 322.

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 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 Difference}

of the Vertical Angles, or Bafes, fhall be the fought Angle or Side, according as the Perpenwhich cannot be

dicular falls {without;

}

known, unless the Species of that unknown Angle, which is opposite to a given Side, be firft known.

Here

Here they leave us firft to calculate that unknown Angle, before we fhall 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 my Obfervations on the two preceding Cafes.

The Truth is, the Quafitum here, as well as in the two former Cafes, is fometimes doubtful, and sometimes not; when doubtful, fometimes each Anfwer is lefs than 90 Degrees, fometimes each is greater; but fometimes one lefs, and the other greater, as in the two laft mentioned Cafes. When it is not doubtful, the Quafitum is fometimes greater than 90 Degrees, and fometimes lefs. All which Diftinctions may be made without another Operation, or the Knowledge of the Species of that unknown Angle, oppofite to a given Side; or which is the fame thing, the falling of the Perpendicular within or without. For which see my Directions at Pages 324, 325.

In the Solution of our 1ft and 5th Cafes, called in other Authors, the 5th and 6th; where there are given two Angles, and a Side opposite to one of them, to find the 3d Angle, or the Side oppofite to it; they have told us, Sum that the Difference of the Vertical Angles,

or Bafes, according as the Perpendicular falls

{within shall be the fought Angle or Side;

without

and that it is known whether the Perpendicu

lar