PROP. XXV. THEOR.

If in an oblique anged spherical triangle (ABC), two right angled spherical triangles (ADC and BDC) be formed, by letting fall a perpendicular (CD), on any side (AB) considered as the base, from the opposite angle, and the perpendicular (CD) be assumed as the middle part in each of the right angled triangles; the rectangle under the cosines of the remote parts in one of the right angled triangles, is equal to that under the cosines of the remote parts in the other; and the rectangle under the tangents of the adjacent parts in one, is equal to that under the tangents of the adjacent parts in the other.

For the rectangles under the cosines of the remote parts in the triangle ADC, which are the complements of A and AC, and in the triangle BDC, which are the complements of CBD and BC, are each of them equal to the rectangle under radius and the sine of the middle part CD (23 Sph. Tr.), and therefore to

each other.

And the rectangle under the tangents of the adjacent parts in the triangle ADC, which are AD and the complement of ACD, and in the triangle BDC, which are BD and the complement of BCD, are each of them equal to the rectangle under radius and the sine of the middle part CD (24. Sph. Tr.), and therefore to each other.

Cor. The tangents of the segments of the base [AD and DB] are to each other, as the tangents of the vertical angles [ACD and BCD].

For, by this proposition, the rectangle under the tangent of AD and cotangent of ACD is equal to the rectangle under the tangent of DB and cotangent of BCD; therefore the tangent of AD is to the tangent of DB, as the cotangent of BCD is to the cotangent of ACD (16, 6 Eu.), or, the tangents of any two arches

or angles being reciprocally as their cotangents (Cor. 5 Def. Pl. Tr.), as the tangent of ACD is to the tangent of BCD.

PROP. XXVI. THEOR.

The same things being supposed, except that a middle part be so assumed in each of the right angled triangles, that the perpendicular may have a similar situation with respect to both these middle parts, and of course may be an extreme part of the same name, in both the right angled triangles; the sines of the middle parts are to each other, as the cosines of the remote, or as the tangents of the adjacent parts, which are peculiar to each of the right angled triangles, according as the perpendicular becomes a remote or an adjacent part.

Case 1. When the perpendicular CD, see fig. to prec. prop., is a remote part.

The rectangle under radius and the sine of the part assumed as a middle part in the triangle ADC, is equal to the rectangle under the cosines of CD and the other remote part in the same triangle (23. Sph. Tr.); and the like being true with respect to the triangle BDC; the rectangle under radius and the sine of the middle part in the triangle ADC, is to the rectangle under radius and the sine of the middle part in the triangle BDC, as rectangle under the cosines of CD and the other remote part in the triangle ADC, is to the rectangle under the cosines of CD and the other remote part in the triangle BDC (Cor. 1. 7. 5 Eu.); whence, the two first terms of these proportionals having a common side, namely, radius, and also the two latter, namely, the cosine of CD, by omitting these common sides, the sines of the middle parts are to each other, as the cosines of the remote parts which are peculiar to the triangles ADCand BDC (1. 6 and 11. 5 Eu).

Case 2. When the perpendicular CD is an adjacent part.

The rectangle under radius and the sine of the part assumed as a middle part in the triangle ADC, is equal to the rectangle under the tangents of CD and the other adjacent part in the same triangle (24 Sph. Tr.); and the like being true with respect to the triangle BDC; the rectangle under radius and the sine of the middle part in the triangle ADC, is to the rectangle under radius and the sine of the middle part in the triangle BDC, as the rectangle under the tangents of CD and the other adjacent part in the triangle ADC, is to the rectangle under the tangents of

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CD and the other adjacent part in the triangle BDC (Cor. 1.7.5 Eu.); whence, the two first terms of these proportionals having a common side, namely, radius, and also the two latter, namely, the tangent of CD, by omitting these common sides, the sines of the middle parts are to each other, as the tangents of the adjacent parts which are peculiar to the triangles ADC and BDC (1. 6 and 11. 5 Eu).

Cor. 1. If in a right angled spherical triangle [ABC, see fig. to prop. 25], a perpendicular [CD] be let fall on any side [AB] considered as the base, from the opposite angle; the cosines of angles at the base [A and ABC] are to each other, as the sines of the vertical angles [ACD and BCD].

In the right angled triangles ADC and BDC, the complements of A and CBD being assumed as middle parts, CD and the complements of the angles ACD and BCD are remote parts; therefore, by case 1 of this proposition, the cosine of A is to the cosine of CBD, or, by Cor. 1 Def. Pl. Tr., in fig. 2, of CBA, as the sine of ACD is to the sine of BCD.

Cor. 2. The same thing being supposed; the cosines of the sides [AC and BC] are to each other, as the cosines of the segments of the base [AD and BD].

In the same right angled triangles, assuming the complements of AC and CB as middle parts, CD and the segments of the base AD and BD are remote parts; therefore, by case 1 of this proposition, the cosines of AC and CB are to each other, as the cosines of AD and BD.

Cor. 3. The same thing being supposed; the sines of the segments of the base [AD and BD] are to each other, reciprocally as the tangents of the angles at the base [A and ABC].

In the same right angled triangles, assuming AD and BD as middle parts, CD and the complements of the angles A. and CBD become adjacent parts; therefore, by case 2 of this proposition, the sines of AD and BD are to each other, as the cotangents of the angles A and CBD, or by Cor. 1 Def. Pl. Tr., in fig. 2, CBA, or, the tangents of any two angles being inversely as their cotangents (Cor. 5 Def. Pl. Tr.), reciprocally as the tangents of the same angles.

Cor 4. The same thing being supposed; the cosines of the vertical angles [ACD and BCD] are to each other, reciprocally as the tangents of the sides [AC and BC].

In the same right angled triangles, the complements of the angles ACB and BCD being assumed as middle parts, CD and the complements of AC and CB are adjacent parts; therefore, by case 2 of this proposition, the cosines of ACB and BCD are

to each other, as the cotangent of AC and CB, or, the tangents of any two arches being inversely as their cotangents (Cor. 5 Def Pl. Tr.), reciprocally as the tangents of the same arches.

PROP. XXVII. THEOR.

The sines of the sides of spherical triangles, are to each other, as the sines of the opposite angles.

In the case of right angled spherical triangles, the proposition is manifest from 21 Sph. Tr., 16. 6 Eu. and Cor. 2 Def. Pl. Tr. But if the spherical triangle be oblique angled, as ABC, see the figures to proposition 25, the sines of any two sides, as AC and BC, are to each other, as the sines of the opposite angles CBA and CАВ.

Let fall on the third side AB, produced if necessary, the perpendicular CD, which perpendicular being assumed as a middle. part in each of the right angled triangles ADC and BDC, the complements of AC and of the angle A become the remote parts in the former triangle, and the complements of BC and the angle DBC, the remote parts in the latter; therefore the rectangle under the sines of AC and the angle A, is equal to the rectangle under the sines of BC and the angle DBC (25 Sph. Tr.); therefore the sine of AC is to the sine of BC, as the sine of the angle DBC, or, which is equal in fig. 2 (Cor. 1 Def. Pl. Tr.), of ABC, is to the sine of the angle A (16. 6 Eu).

PROP. XXVIII. THEOR.

If on any side of a spherical triangle, considered at its base, a perpendicular be let fall from the opposite angle; the rectangle under the tangents of the half sum and half difference of the legs, is equal to the rectangle under the tangents of the half sum and half difference of the segments of the base, between its extremes and the perpendicular.

Let ABC be a spherical triangle, and CD a perpendicular let fall on the base AB from the opposite angle ACB; the rectangle under the tangents of the half sum and half difference of AC and BC, is equal to the rectangle under the tangents of the half sum and half difference of AD and BD.

For the cosine of AC is to the cosine of BC, as the cosine of AD is to the cosine BD (Cor. 2. 26 Sph. Tr.); therefore, by comparing the sums and differences of the terms, the sum of the cosinęs of AC and BC is to their difference, as the sum of the cosines of AD and BD is to their difference (17. 18 and 22. 5 Eu.); but the sum of the cosines of any two arches is to their difference, as the cotangent of half their sum is to the tangent of half their difference (5 Pl. Tr.); therefore, substituting the latter ratios for the former, the cotangent of half the sum of AC and BC is to the tangent of half their difference, as the cotangent of half the sum of AD and BD is to the tangent of half their difference; and, forming rectangles from the two first terms, with a common side of the tangent of the half sum of AC and BC, and from the two last, with a common side of the tangent of the half sum of AD and BD, the rectangle under the tangent and cotangent of the half sum of AC and BC, is to the rectangle under the tangents of the half sum and half difference of AC and BC, as the rectangle under the tangent and cotangent of the half sum of AD and BD, is to the rectangle