supplied : And that which was corrupted is here restor'd *.

And since several Persons to whom the Elements of Geometry are of yast Use, either are not so fufficiently Skilld 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.

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 Cafe of Oblique Sphericks is false, I have demonstrated, and given a true one. See Page 319.

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

In

In the Solution of our 9th and roth Cases, by other Authors called the ift and ad, where are given and fought oppofite Parts, not only the aforemention'd 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 juft Distinction of these Varieties. For the Solution of these Cafes see my Directions at Pages 321, 322.

In the Solution of our 3d and 7th Cases, in other Authors reckon'd 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 Thortestand best Method for finding either of these Quæfita, have told us, that the

Sum

Sum * Difference} of the Vertical Angles, or Bases, shall be the sought Angle or Side, according as the perpendicular falls

} which cannot be known, unwithout;} less the Species of that unknown Angle, which is opposite to a given Side, be first

{within

:

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 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 Cafes.

The Truth is the Quafitum here, as well as in the two former Cases, is fometimes doubtful, and sometimes not; when doubtful, fometimes each Answer is less than 90 Degrees, sometimes each is greater; but sometimes one less, and the other greater, as in the two fast mention'd Cafes. When it is not doubtful, the Quæfitum is sometimes greater than 90 Degrees, and and sometimes less. All which Distinctions may be made without another Operation, or the Knowledge of the Species of that

un

unknown Angle, opposite to a given Side; or which is the fame thing, the falling of the Perpendiculari within or without. For which fee my Directions at Pages 324,

325.

In the Solution of our ift 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

opposite to it; they have told us, that the Sum

{Sifference } of the Vertical Angles, or

Bases, according as the Perpendicular falls swithin 3 shall be the fought Angle or .

without Side; and that it is known whether the Perpendicular falls within or without, by the Affection of the given Angles.

Here they seem to have spoken as tho’ the Quæsitum was always determin'd, and never ambiguous ; for they have here determined whether the Perpendicular falls within or without, and thereby whether they are to take the Sum or the Difference of the Vertical Angles or Bases, for the fought Angle or Side.

But, notwithstanding these imaginary Determinations, I affirm; tliat the Que

fitum here, as in the two Cases last mentioned, is sometimes ambiguous, and sometimes not; and that too, whether the Perpendicular falls within, or whether it falls without. See my Solutions of these two Cafes in Page 323.

The Determination of the 3d Case of Oblique Plane Triangles. See in Page 325

SAM. CUNN.

EVC LID's