supplied: And that which was corrupted is here restor'd*. And fince several Persons to whom the Elements of Geometry are of vaft Use, either are not so sufficiently Skill'd in, or perhaps have not Leisure, or are not willing to take the Trouble to read the Latin; and fince this Treatise was not before 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 asserviceable, 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 demonftrated, and given a true one. See Page 319. 1 T *** Vide Page 55, 107, of Euclid's Works, publish'd by Dr. Gregory. In 1 In the Solution of our 9th and 10th Cafes, by other Authors called the ift and 2d, 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 just 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 Cafes, in other Authors reckon'd 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 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 Quafita, have told us, that the Sum Sum Difference of the Vertical Angles, or Bases, shall be the fought Angle or Side, according as the Perpendicular falls within without which cannot be known, unless the Species of that unknown Angle, which is oppofite 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 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 Cafes, is fometimes doubtful, and sometimes not; when doubtful, fometimes each Answer is less than 90 Degrees, fometimes each is greater; but sometimes one less, and the other greater, as in the two last mention'd Cafes. When it is not doubtful, the Quafitum is sometimes greater than 90 Degrees, and and fometimes less. All which Distinctions may be made without another Operation, or the Knowledge of the Species of that un unknown Angle, oppofite to a given Side; or which is the fame thing, the falling of the Perpendicular within or without. For which fee my Directions at Pages 324, 325. In the Solution of our ist 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 Sunference of the Vertical Angles, or Bases, according as the Perpendicular falls within 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 feem to have spoken as tho the Quafitum 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; that the Quafitum . fitum here, as in the two Cafes 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 Cafe of Oblique Plane Triangles. See in Page 325. 1 SAM. CUNN. : EUCLID's |