Now, in the triangle BCD,

BČ< BD + DC, by last Prop. .. AB+ AC+ BC < AB+ BD + AC+ CD, ABD + ACD,

< circumference of great circle. Note. In elementary geometry the only spherical triangles considered are those in which each side is less than a semicircumference, and each angle less than two right angles.

Should a spherical triangle be taken without these restrictions, it will be found that the residue of the surface of the sphere will be a triangle having portions of the same circumferences as boundaries with the given triangle, and falling within the restrictions; when all the parts of this latter triangle are known, the parts of the other may be derived from them by subtracting the known angles and the known sides from 180 or 360 degrees.

Triangles not limited by the restrictions above mentioned, therefore, being dependent upon those which are thus limited, the first class may be rejected, and our attention, as it has been in the preceding theorems, confined to the second.

PROP. IX.

Two spherical triangles are either identical or symmetrical, 1°. When they have two sides and the included angle of the one equal to the same in the other; 2°. When they have a side and two adjacent angles; 3°. When they have three sides respectively equal.

These follow from the cor- A responding theorems in trihedral angles (Prop. 3, and exer. 3, 4, 5), but may be proved by superposition of the given triangles, the one upon the other, or its symmetrical triangle, as at th. 1, 2, &c., in Plane Geometry.

B

F

D

E

The preceding diagram exhibits symmetrical tri

angles, viz., EDF and EDF', or ABC and EDF'. The one could not be superposed upon the other, for, on turning it over so as to bring the equal parts opposite to each other, the convexities of the two surfaces would be turned toward each other, and could touch in but one point.

PROP. X.

Symmetrical triangles are nevertheless equal in surface, which may be proved as follows:

Let ABC, DEF be two

symmetrical triangles, in

which AB-DE, AC = DF, BC= EF.

Let G be the pole of the small circle passing through B the three points A, B, C, and H the pole of the small circle passing through the

three D, E, F. Join G with A, B, C, and H with D, E, F by arcs of great circles. The triangles AGB, BGC, AGC, HDE, HFE, HDF are all isosceles, and the corresponding ones in the two diagrams admit of superposition, because, in turning them over to bring the convexities of their surfaces the same way, equal sides are not turned away from each other, and this arises from the triangles being isosceles. The three triangles of the one diagram being respectively identical, therefore, with the three of the other, we have AGB + BGC-AGC-DHE + EHF — DHF, or A BC= DEF. Q. E. D.

PROP. XI.

An isosceles spherical triangle has its two angles equal, and conversely.

PROP. XII.

In any spherical triangle, the greater side is opposite the greater angle, and conversely.

These may be proved precisely as in plane triangles.

PROP. XIII.

Two spherical triangles (unlike two plane triangles in this respect) are equal when the three angles of the one are equal to the three angles of the other, each to each.

For the polar triangles of the two given triangles will have equal sides (Prop. 6), and, consequently, equal angles (Prop. 9). Hence the given triangles will have equal sides.

NOTE.

The equal triangles in question in the preceding theorems need not be supposed on the same sphere, if their sides and angles are given in degrees and fractions of a degree. Indeed, there would be much advantage gained by discarding spherical triangles from geometry except for purposes of mensuration on the surface of the sphere, and using trihedral angles in their place, especially in the application to Astron omy, which, as a science of observation, depends entirely on angular measurements.

PROP. XIV.

The sum of the angles of a spherical triangle is greater than two and less than six right angles.

For each angle is less than two (note to Prop. 8), hence the sum of the three is less than six right angles. Again, each angle being measured by a semicircumference, minus the side opposite in the polar triangle, the sum of the three angles will be three semicircumferences, minus the sum of the three sides of the polar triangle, but the latter is is than a circumference (Prop. 8); hence the measure of the sum of the three angles will be greater than one semicircumference or two right angles.

Note. In birectangular spherical triangle, two of the sides are quadrants; and in a trirectangula, tri

angle all three of the sides are quadrants. This latter triangle is sometimes taken as the unit of measure on the surface of the sphere. As there are four such triangles in each hemisphere, the whole surface of the sphere would be expressed by the number 8.

PROP. XV.

The surface of a lune is to the whole surface of the sphere as the angle of the lune is to four right angles, or as the arc which measures the angle of the lune is to a circumference.

It is evident, from a mere inspection of the diagram, that the lune ABDC is the same aliquot part of the whole surface of the sphere that the arc BC is of a whole circumference, or that B the angle BAC, measured by this arc, is of four right angles. The demonstration may be made more full by dividing the triangle ABC into a number

C

A

D

of equal triangles, having their common vertex at A and their bases equal portions of BC,* and dividing, also, the hemisphere into triangles of the same size, and thus showing that the ratio of the triangle ABC to the hemisphere is the same as the ratio of BC to the whole circumference, because both are in the ratio of the same two numbers, viz., the number of triangles in ABC to the number in the hemisphere, or the number of bases in each.

Schol. The angle of the lune is to four as twice this angle is to eight. Hence, if the whole sphere be expressed by 8, the lune will be expressed by 2A.

These triangles will be equal because their sides are equal.

PROP. XVI.

The two opposite spherical triangles on a hemisphere are together equal to a lune having the same angle.

Let the two triangles ABC, ADE be on the same hemisphere, having their common vertex at A. Then will their sum be equal to the lune ABFC.

For the triangle BFC may be proved equilateral with the triangle ADE, and, therefore, of the same surface. Q. E. D.

PROP. XVII.

B

C

A

F

E

The measure of a spherical triangle is the excess of

the sum of its angles above two right angles.

Let ABC be a spherical trian

gle; its measure will be A+B+

H

G

gether, we obtain evidently the whole hemisphere, which is expressed by four, together with twice the triangle ABC.

.. 4 + 2 ▲ ABC=2A + 2B + 2C ;

ABC A+ B+ C-2.

=

Q. E. D.

Corol. 1. The spherical triangle is equivalent to a lune whose angle is half the above expression.

Corol. 2. Two spherical triangles are of equal surface when the sum of their angles is the same, and vice versa.

*

A, B, and C must be here understood as expressed not in degrees, &c., but in fractions of a right angle.