The following REMARK by SAMUEL CUNN.

"HAT this is true but in a particular Case, viz. THA when two of the Angles of the Triangle are Right ones, and two of the Sides Quadrants, may be thus demonftrated. For if poffible, let fome Triangle RST, Fig to Prop. 14th be fuch, that its Sides RS, ST, TR, be equal to the Measures of GHD, HGD, GDH, the Angles of a Triangle GHD; and alfo, that the Measures of RST, STR, TRS, the Angles of the Triangle RST be equal to GH, GD, HD, the Sides of the Triangle GHD. And produce MX, MN, two Sides of the fupplemental Triangle to Semicircles, and they will meet fomewhere; fuppofe at E; and there will be conftructed thereby the Triangle NEX, of which XE (the Supplement of X M, which, by the 14th Prop. was the Supplement of the Measure of the Angle HGD) is equal to the Measure it felf of the fame Angle HGD: And in like manner, NE (the Supplement of NM, which, by the 14th Prop. was the Supplement of the Measure the Angle GHD) is equal to the Measure it self of the fame Angle GHD. But the third Side X N, is not the Measure of the third Angle GDH, but its Supplement, by the 14th Prop. Moreover, of the Angle EXN (whofe Supplement is NX M) the Measure, by the 14th Prop. is equal to GD; and of the Angle X NE, (whofe Supplement is MNX) the Measure, by the 14th Prop. is equal to HD. But of the third NEX, (which is equal to NMX) the Measure is not equal to GH, but its Supplement.

Now make NV=RT=BK, the Measure of the Angle GDH, and draw the great Circle E V. And fince RS, by Suppofition, is equal to the Meafure of the Angle GHD, which is equal to EN; and fince the Measure of the Angle SRT, is by Suppofition, equal to DH, which is alfo equal to the Measure of the Angle XNE; the Angle X NE is equal to the Angle R. Then confequently, by the

4th Prop. the Triangles SRT, EN V, will have the Bafe ST, equal to the Bafe EV; the Angle T, to the Angle NVE, and the Angle S, to the Angle NEV. But ST, (which is equal to E V,) by Suppofition, is equal to the Measure of the Angle HGD; to which Measure X E is alfo equal. Therefore, EV is equal to XE; and confequently, by the 7th Prop. the Angle EVX is equal to the Angle EX V; and the Angle EXV (whofe Measure, as hath been fhewn above, is equal to GD) is equal to the Angle T, (or NVÉ,) fince by Suppofition, the Measure of this is alfo equal to GD. Therefore the Angle EV X is equal to the Angle EVN, and fo both right ones; and confequently EX V a right one alfo. Therefore, by the 2d Cor. to the 2d Prop. EV and EX are both Quadrants.

But if EV be à Quadrant, and at right Angles to NX, then E, by zd Prop. and its Coroll. is the Pole of NX; and fo 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 fome other Triangle GHD, and the Measures of the Angles of the former, equal to the Sides of the latter; two Sides of fuch a Triangle RST, or GHD, must be Quadrants, and two Angles of each right ones.

Therefore, if a Triangle RST be conftructed whose Sides are equal to the Meafures of the Angles of another Triangle GHD: the Measures of the Angles of the Triangle RST, fhall not be equal to the Sides of the Triangle GHD, unless in the one Cafe beforementioned. Therefore the Measures of the Angles of the Triangle GHD, used as the Sides of a Triangle in the 11th Cafe, will not give us a Side of GHD, but the Measure of an Angle of the Triangle RST, unlefs in the one afore-mention'd Cafe; which was to be demonftrated.

But to find a Side GD of spherical Triangle G HD, whofe Angles are all given, produce MN, that Side of the fupplemental Triangle, which is equal to the Supplement of the Measure of GHD, the Angle oppofite to the Side fought, and MX, either of the other Sides till they meet as in E. And there, as hath been before fhewn, the Sides EX, EN, of the Tri

angle

angle EX N, 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 GDH. And of the Angle XEN, the Meafure is equal to the Supplement of GH.

Therefore the SOLUTION is thus:

Change one of the Angles GD H, 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. Cafwell, Mr. Heynes, and many other Trigonometrical Writers.

In the Solution of our 8th and 9th Cafes, they have told us, that the Quæfita are ambiguous; which fometimes, indeed, is true, but sometimes alfo falfe: Therefore, as I conceive it, they ought to have laid down Rules, by Help of which we might difcover when the Quafita are ambiguous, and when not.

This Overfight may be corrected by the following Directions: Wherein, because every Sine çorrefponds to two Arches, to one lefs than a Quadrant, and to another, which is the Supplement of the former to a Semicircle, (a true Diftinction of which of thefe are to be used, being neceffary to be known, before a proper Solution can be given to fuch Problems as these are,) I fhall 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 Cafe, there are given two Angles B, D, and BC, a Side oppofite to one of those Angles D, to find DC the Side oppofite to the other.

the acute Value of DC, and also to its obtufe one, add BC; and if each of these Sums are greater

Sgreater
lefs S

than 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 thofe Sums is

{greater than a Semicircle, only one Value of DC can be true, viz. the obtufe

is not ambiguous.

acute

}

one; and then

In the ninth Cafe, there are given two Sides BC, DC, and one Angle B, oppofite to DC one of thofe Sides, to find D the Angle oppofite to the other.

O the acute Value of D, and also to its obtuse

{greater than two Right Angles, when the Sum }

2 lefs

of the Sides is greater than a Semicircle, both the 2 lefs

Values of D may be admitted, and confequently, D is ambiguous But when only one of thofe Sums is greater than two Right Angles, only one Value lefthan

{

of D is true, viz. the

ambiguous.

Jobtufe

acute

}

one; and then not

Nor are we better used in the firft 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 Quæfita will be fometimes ambiguous, and fometimes

not.

In

In this firft Propofition there are given two Angles B, D, and BC, a Side oppofite to D, one of them, to find C the third Angle.

1. Let the Perpendicular fall within; that is, let the given Angles be of the fame Species.

To the acute Value of DCA, and alfo to its ob

tufe one, add the Angle B CA; and if each of thefe Sums is lefs than two Right Angles, then either the acute Value of DCA, or its obtufe one added to BCA, gives a Value of B CD; which, therefore, is ambiguous. And when only one of these Sums is less than two Right Angles, the acute Value of DCA, added to B CA, gives the only Value of B CD; 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 lefs than the Angle BCA, the Angle BCD may be had by fubtracting either Value of DCA from BCA; and then BCD is ambiguous. But when the obtufe Value of DCA is not less than BCA, the acute Value of DC A, taken from BCA, gives the fingle Value of B CD; which, therefore, is not ambiguous; tho' in both Varieties the Perpendicular fell without.

In the fifth Cafe 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 Side lying between those given Angles.

1. When the Perpendicular falls within; that is, when the given Angles are of the fame Species. "O the acute Value of DA, and fo alfo to its obtufe one, add B A; and if each of these Sums is lefs than a Semicircle, then either the acute Value of DA, or its obtufe one, added to BA, gives the

T