2 4th Prop. the Triangles SRT, EN V, will have the Base ST, equal to the Base E V; the Angle T, to the Angle N VE, and the Angle S, to the Angle NEV. But ST, (which is equal to E V,) by Supposition, is equal to the Measure of the Angle HGD; to which Measure X E is also equal. Therefore, EV is equal to XE; and consequently, by the 7th Prop. the Angle EVX is equal to the Angle EXV; and the Angle EX V (whose Measure, as hath been shewn above, is equal to GD) is equal to the Angle T, (or NVE,) fince by Suppofition, the Measure of this is also equal to GD. Therefore the Angle E V X is equal to the Angle E VN, and so both right ones; and confequently ĚX V a right one also. Therefore, by the ad Cor. to the ad Prop. EV and EX are both Quadrants. But if E V be a Quadrant, and at right Angles to NX, then E, by zd Prop. and its Coroll. is the Pole of NX; and so EN a Quadrant also, and the Angle ENV a right one. Therefore, if the sides of a Triangle (NEV, or its Equal) RST, are equal to the Measures of the Angles of some other Triangle G HD, and the Meafures of the Angles of the former, equal to the Sides of the latter; two sides of such a Triangle RST, or GHD, must be Quadrants, and two Angles of each right ones. Therefore, if a Triangle RST be constructed whose Sides are equal to the Measures of the Angles of another Triangle GHD: the Measures of the Angles of the Triangle RST, shall not be equal to the Šides of the Triangle GHD, unless in the one Case beforementioned. Therefore the Measures of the Angles of the Triangle GHD, used as the Sides of a Triangle in the nth Case, will not give us a Side of GHD, but the Meafure of an Angle of the Triangle RST, unlefs in the one afore-mention'd Case; which was to be demonstrated. But to find a Side G D of spherical Triangle GHD, whose Angles are all given, produce MN, that Side of the supplemental Triangle, which is equal to the Supplement of the Measure of GHD, the Angle opposite to the Side fought, and MX, either of the other Sides till they meet as in E. And there, as hath been before thewn, the Sides E X, EN, of the Tri angle angle E XN, are exactly equal to the Measures of the Angles HGD, GHD, of the Triangle GHD; and of the Angles E XN, ENX, of the Triangle E XN, the Measures are equal to GD, HD. But the Side XN is equal to the Supplement of the Measure of the Angle G DH. And of the Angle XEN, the Meafure is equal to the Supplement of GH. Therefore the SOLUTION is thus : Change one of the Angles GDH, adjacent to the Side fought into its Supplement ; and then work with the Measures of the Angles as tho they were Sides, and the Result will be GD, the Side fought. The preceding Fault, as well as the Omiffions hereafter mention'd, are not peculiar to our Author ; but may be found in Dr. Harris, Mr. Caswell, Mr. Heynes, and many other Trigonometrical Writers. In the Solution of our 8th and oth Cases, they have told us, that the Quæfita are ambiguous; which sometimes, indeed, is true, but sometimes also false: Therefore, as I conceive it, they ought to have laid down Rules, by Help of which we might discover when the Quæfita are ambiguous, and when not. This Oversight may be corrected by the following Directions : Wherein, because every Sine çorresponds to two Arches, to one less than a Quadrant, and to another, which is the Supplement of the former to a Semicircle, (a true Distinction of which of these are to be used, being necessary to be known, before a proper Solution can be given to fuch Problems as these are,) I shall beg Leave, for Brevity Sake, to call the leffer Arch the acute Value, and the greater the obtuse ; whether the Sine be of an Angle or a Side. In the tenth Case, there are given two Angles B, D, and BC, a: Side opposite ta one of those Angles D, to find DC the Side opposite to the other. O the acute Value of DC, and also to its obtuse one, add BC; and if each of these Sums are greater Cacute Sgreater Šthan a Semicircle, when the Sum of the Angles B, D, is greater than two Right Angles ; both the Values of DC may be admitted, and then is ambiguous : But when only one of those Sums is { pereater than a Semicircle , only one Value of S obtuse DC, and one Angle B, Opposite to DC one of other. O the acute Value of D, and also to its obtufe Value, add B; and if each of these Sums is {reater than two Right Angles, when the Sum of the Sides is greater than a Semicircle, both the Values of D may be admitted, and consequently, D is ambiguous : But when only one of those Sums is greater than two Right Angles, only one Value less of D is true , viz. the {abite e and then not ambiguous. one ; Nor are we better used in the first Propofition; for tho' it is determined by the given Angles, whether the Perpendicular falls within or without the Triangles, yet in each of those Varieties, the Quefita will be sometimes ambiguous, and sometimes not. 1n In this first Proposition there are given two Angles B, D, and BC, a Side opposite to D, one of them, to find C the third Angle. T 1. Let the Perpendicular fall within ; that is, let the given Angles be of the fame Species. tuse one, add the Angle BCA; and if each of these Sums is less than two Right Angles, then either the acute Value of DCA, or its obtuse one added to BCA, gives a Value of BCD; which, therefore, is ambiguous. And when only one of these Sums is less than two Right Angles, the acute Value of DCA, added to BCA, gives the only Value of BCD; which then is not ambiguous; tho' in both Varieties the Perpendicular fell within. 2. Let the Perpendicular fall without; that is, let the given Angles be of different Species. WHEN the obtufe Value of the Angle DCA is less than the Angle BCA, the Angle BCD may be had by subtracting either Value of DCA from BCA; and then B CD is ambiguous. But when the obtuse Value of DCA is not less than BCA, the acute Value of DCA, taken from BCA, gives the single Value of BCD; which, therefore, is not ambiguous; tho' in both Varieties the perpendicular fell without. In the fifth Case we lie under the fame Misfortune, where there are given, as in the first, the Angles B, D, and the Side BC, to find BD the Sidelying between those given Angles. 1. When the Perpendicular falls within ; that is, when the given Angles are of the same Species. O the acute Value of DA, and so also to its is less than a Semicircle, then either the acute Value of D A, or its obtuse one, added to B A, gives the Value Value of BD; which thence is ambiguous. And when only one of these Şums is less than a Semicircle, the acute Value of D A, added to B A, gives the only Value of BD; which then is not ambiguous; tho' in both Varieties the Perpendicular fell within. 2. When the Perpendicular falls without; that is, when the given Angles are of different Species. WHEN the obtuse Value of DA is less than BA, BD will be had by subtracting either Value of D A from BA; and then B D is ambiguous. But when the obtuse Value of D A is not less than BA, the acute Value of D A, taken from B A leaves the only Value of BD; which, therefore, is not ambiguous; tho' in both Varieties the Perpendicular fell without. In the third, we have the Same Omission; where there are given two Sides BC, CD, and B an Angle opposite to CD one of them, to find the tbird Side BD. IRST, we may observe, that the Species of DA is always known ; for it is of theiferent} Af fection with the Angle B, when DC is {greater } than a Quadrant. And, If AD be less than AB, and also the Sum of AD and A B less than a Semicircle ; then AD, either added to, or subtracted from AB, will give the Value of BD, which, therefore, is ambiguous. But if AD be not less than AB, or if their Sum be not less than a Semicircle ; then their Sum in the former, and their Difference in the latter Variety, shall give one single Value of BD, and then is not ambiguous. 1 Tbe |