Mr. C U N N's


Shewing the USEFULNESS and Ex



R. KEIL, in his Preface, hath sufficiently declared how much easier, plainer, and more elegant, the Ele

ments 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 intended 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 Propositions of that most noble universal Mathesis, the 5th ; on which the 6th, 11th, and 12th so much depend, that the Demonstration of not so much as one Proposition in them can be obtained without those in the 5th?

# Vide the last Edition of the Englise Tacquct.


The 7th, 8th, and oth Books treat of such Properties of Numbers which are necessary for the Demonstrations of the joth, which treats of Incommensurables ; and the 13th, 14th, and 15th, of the five Platonick Bodies. But though the Doctrine of Incommensurables, because expounded in one and the same 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 necessary to the Reasoning about Incommensurables": Yet because only one Proposition of these four Books, viz. the ift of the roth is quoted in the 11th and 12th Books ; and that only once, viz. in the Demonstration of the ad of the 12th, and that Ist Proposition of the 10th, is supplied by a Lemma in the 12th : And because the 7th, 8th, 9th, roth, 13th, 14th, 15th Books have not been (thought by our greatest Masters) neceffary to be read by such as design to make natural Philosophy their Scudy, or by such as would apply Geometry to practical Affairs, Dr. Keil in his Edition, gave us only these eight Books, viz. the first six, 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 Demonstration of the gth Proposition of the third Book ; and the other a Demonftration of that Property of Proportionals callled Conversion, (contained in a Corollary to the



19th Proposition of the 5th Book,) where instead of Euclid's Demonstration, 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 since several Persons to whom the Ele. ments 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 servi. ceable.

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 some of the Solutions.

That the common Solution of the 12th Case of Oblique Sphericks is false, I have demonstrated, and given a true one. 319.

Sec Page

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

Dr. Gregory

In the Solution of our gth and 10th Cafes, by other Authors called the ist and 2d, where are given and fought opposite 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, sometimes true, but sometimes false : For sometimes the Quæsitum 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 Cases see my Directions at Pages 321, 322.

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 Cafes, by letting fall a Perpendicular, which is undoubtedly the shortest and best Method for finding either of these

the of the Vertical Angles, or Bases, shall be the fought Angle or Side, according as the Perpenwithin

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

dicular falls { without ;


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 fought Angle or Base : And in the Calculation of that Angle have left us in the dark as to its Species ; as appears by my Observations on the two preceding Cases.

The Truth is, the Quæsitum here, as well as in the two former Cases, is sometimes doubtful, and sometimes not; when doubtful, fometimes each Answer is less than go 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 Quæsitum is sometimes greater than 90 Degrees, and sometimes less. All which Distinctions may be made without another Operation, or the Knowledge of the Species of that unknown Angle, opposite to a given Side ; or which is the same thing, the falling of the Perpendicular within or without. For which see my Directions at Pages 324, 325.

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In the Solution of our ist and 5th Cases, 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 opposite to it ; they have told us,

Difference} of the Vertical Angles, or Bases, according as the Perpendicular falls { without shall be the sought Angle or Side; and that it is known whether the Perpendicu


that the s Sum

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