COR. If two right-angled spheric triangles have one common angle, the sines of their hypotenuses are as the sines of their legs, opposite to this angle.

Since the planes GQP, GBP, are perpendicular to each other, if the radius be made equal to unity, the values, specified in the following table, may be easily obtained for all the parts of

the triangle; it being recollected, that tang. =

RR COS.

sin.

COS.

, and cot.=

5. ZBCA

BPxGQ QPXBG BPxGQ
BQXGP
GPXBQ QPXBG

Now, to show how these several expressions are demonstrated, it is sufficient to give merely the demonstration of the first line. For this purpose assume any line GR, and regard it as the radius of the Tables; let fall the perpendicular RS on GB. Then it is evident, that RS is the sine of the arc BC, and GS its cosine. In the similar triangles GRS, GQB, as QG: QB:: GR=1

: RS sin. BC= ; and QG GB GR=1

cos. BC= ; whence is deduced tang. BC =

QB
QG

GB
QG

BG

cot. BC

BQ

The other expressions are demonstrated in the

same manner.

THEOREM II.

As radius is to the cosine of either angle, so is the tangent of the hypotenuse to the tangent of the leg adjacent to this angle.

That is, R: cos. B: tang. BC tang. AB;

Or, R cos. C: tang. BC tang. AC.

Cor. If two right-angled spheric triangles have one common leg, the tangents of their hypotenuses are in, the inverse ratio of the cosines of the angles adjacent to this leg.

THEOREM III.

As radius is to the cosine of one of the legs, so is the cosine of the other leg to that of the hypotenuse.

That is, R: cos. AB :: cos. AC :

cos. BC; Or, R cos. AC:: cos. AB cos. BC..

The expressions for the cotangents, being obtainable by merely inverting those for the tangents, are not inserted in the table.

The truth of the first six Theorems is proved by substituting the particular values of the terms of the proportions to be demonstrated, and then comparing the product of the extremes with that of the means. For the two products will always be found to be exactly equal.

Thus, Theorem I. R sin. BC :: sin. B: sin. AC.

NOTE. By this Theorem, the expressions for the sine, esine, and tangent of the angle BCA were obtained.

1

COR. If two right-angled spheric triangles have one common leg, the cosines of their hypotenuses are as the cosines of their other legs.

THEOREM IV.

As radius is to the sine of either angle, so is the cosine of the adjacent leg to the cosine of the other angle.

That is, R sin. B:: cos. AB : cos. C;
Or, R sin. C:: cos. AC : cos. B.

COR. If two right-angled spheric triangles have one common leg, the cosines of the angles, opposite to this leg, are to each other as the sines of the adjacent angles.

THEOREM V.

As radius is to the sine of one of the legs, so is the tangent of its adjacent angle to the tangent of the other leg.

That is, R sin. AB :: tang. B: tang. AC;

Or, R sin. AC

:

tang. C: tang. AB.

COR. If two right-angled spheric triangles have one common leg, the sines of their other legs are reciprocally as the tangents of the angles at these legs.

COR. 2. If they have one common angle, the tangents of their legs, opposite to this angle, are as the sines of the legs adjacent to it.

THEOREM VI.

As radius is to the cotangent of one of the angles, so is the cotangent of the other angle to the cosine of the hypotenuse; or, which is the same, radius is to the cosine of the hypotenuse, as the tangent of one angle is to the cotangent of the other.

That is, R cot. B: cot. C :

cos. BC;

Or, R cos. BC :: tang. B cot. C: tang. C cot. B.

NOTE. These six Theorems are sufficient for the solutions of all the cases of right-angled spheric triangles.

THEOREM VII.

The product of radius and the sine of the middle part is equal to the product of the tangents* of the adjacent extremes, or to that of the cosines of the opposite extremes.f

NOTE. This Theorem is general, and equally applicable to every case of Rectangular Spheric Trigonometry. It is NAPIER'S Method of the five circular parts, and is sometimes called the Catholic Proposition.

It will assist the memory in recollecting this proposition to observe, that the second letters in tangents and cosines are respectively the same with the first letters of adjacent and opposite.

+ DEMONSTRATION. This Theorem may be demonstrated by substituting for each particular term the value, specified in the small table belonging to the demonstration of the first Theorem. Thus, if we assume AB, in the figure belonging to the definitions, for the middle part, AC and B are the adjacent extremes, and BC and C the opposite extremes; then, according to this Theor. R × sin. AB=tang. AC × cot. B = sin. BC x sin. C. The expressions of the values of these quantities being

taken from the aforesaid table, we have Rx

=

BP QPXBP GP GPXQP

BQ×BP×GQ.
GQXBQXGP'

each of which expressions is evidently the

THEOREM VIII.

The angles at the hypotenuse are always of the same affection with their opposite sides; and the hypotenuse is less or greater than a quadrant, according as the legs are of the same or different affection.*

NOTE.

The converse of this Theorem is true in all its parts. And when the hypotenuse is exactly a quadrant, one or each of the legs is 90°.

same, when reduced to its lowest terms.

may any other case be proved.

in all cases. Q. E. D.

* DEMONSTRATION.

If the

In the same manner Therefore the Theorem is true

F

sides AB and AC be each 90°; then, since the angle A is right by supposition, the points B and C will be the poles of the arcs AC and AB; and consequently the angles B and C will be right ones, being measured by arcs, which are supposed to be 90°; that is, they will be of the same affection with their opposite sides.

B

a

To prove, that, when the legs AB and AC are both acute, their opposite angles will be so also; let the side AC be produced to F, or till it be 90°. Then, as the point F will be the pole of the arc AB, the angle B will be also 90°; and consequently the angle ABC, which is less than the angle ABF, will be acute. We may prove in the same manner, that the angle at C is acute, when the opposite side AB is acute.

It is equally evident, that the angles B and C, in the triangle BaC, right-angled at a, are obtuse, when the opposite sides aB, aC, are obtuse.

If one of the sides Ab be obtuse, and the other side AC acute, as in the right-angled triangle bAC, the angle at C will also be obtuse, and that at b acute. For, having taken the arc_AG=90°