AC + CB + AB < (AC+CD) + (BD+AB); ... < ACD + ABD; .. < (π + π); :. < 2 π. COR. Hence, the sum of the sides of a polygon ACEFGA (the sides being arcs of great circles) is less than 2 π. For, by the Proposition, AD + DH+AH is < 2π; or, AC + (CD + DE) + EF + (FH + HG) + AG < 2 ᅲ, but CD + DE CE, and, FH+HG > FG; : a fortiori, AC + CE + EF + FG + AG < 2 π. PROP. IV. If CD be drawn from the centre C, perpendicularly to the plane of the circle ANMB, then, D is the pole of the circle ANMB and of all small circles, such as anmb, that are parallel to it. For, since DC is perpendicular to the plane of ANMB, it is perpendicular to all lines in it, as CN, CM, &c. (See Euclid, Book XI. Def. 3). Hence, DCA, DCN, DCM, each = 90°; consequently, since DC is common, and CA, CN, CM, are equal, the hypothenuses, which are the chords of the arcs DA, DN, DM, are equal; ... the arcs DA, DN, DM are equal to one another and to 90°, and ... D, (by Def. 3.) is the pole of ANMB. Again, in the small circle anmb, ca, cn, cm, are equal, as are the angles Dca, Dcn, Dcm; ... as before the chords of Da, Dn, Dm, are equal, and the arcs Da, Dn, Dm;..Dis equally distant from every point in the circumference of anmb, and therefore is its pole. Cor. 1. By Definition 6, the spherical angle AMD is equal to the inclination of the planes AMBC, DCM, and therefore is a right angle. COR. 2. Hence, to find the pole of a great circle, draw, from the point M, a great circle perpendicular to AM, and take MD=90°; then D is the pole: and, reversely, if D be the pole, DMN, DNM are right angles, and DN, DM are quadrants. COR. 3. Hence, to describe a great circle, of which D is the pole, take DN, DM, each = 90°; then let a plane pass through N, M, and C, and its intersection with the surface of the sphere is the circle required. COR. 4. Since NDM may be of any magnitude from 0 to 180°, and since the angles DNM, DMN, are, each = 90°, the sum of the three angles of a triangle, as DNM, may be any angle between 180° and 360°, which are the two limits. COR. 5. The radius of the small circle a nmbis Cn, which is the sin Dn, or, cos. Nn; and, if the great circle ANMB be divided into any number of equal parts, each equal to NM, the small circle an mb will be divided into the same number of equal parts, each part being equal tonm; but, the magnitude of nm will be to the magnitude of NM, as the circumference of anmb to that of ANMB, consequently, as the radius on to the radius CN; or, as sin. Dn to sin. DN; as sin. Dn to radius; or, as cos. N n to radius. PROP. V. If a plane TDI is perpendicular to CD, it is a tangent to the sphere. For take any point 7, join DT, then CDT is a right angle; ... CT is greater than CD; ... Tis without the surface, and since this is true of every point in the plane, except the point D, the plane TDI (by Def. 4.) is a tangent to the sphere. PROP. VI. If DT, Dt be drawn, in the planes DCA, DCN respectively, tangents to the arcs Da, Dn, at the point D, the angle TDt is equal to the angle ADN, both of which angles are measured by the arc AN. R For, DT, Dt are perpendicular to DC, which is the intersection of the planes DCA, DCN; ... by Euclid, Book XI. Def. 6. TD t measures the inclination of the planes, and therefore is equal to the angle ADN; but TDt, the inclination of the planes, = ACN, of which AN is the measure; ... AN is the measure of TD t and of ADN. Cor. If two arcs of great circles intersect each other, their vertical angles are equal. PROP. VII. If from the points A, B, C, of the triangle ABC, as poles, be described, the arcs EF, FD, DE, forming a triangle DEF; then, reciprocally, the points D, E, F are poles of the arcs BC, AC, AB. Since A is the pole of EF, the arc of a great circle drawn from A, to any point in EF, and therefore to a point as E, is a quadrant; similarly, since C is the pole of DE, the arc of a great circle from C, to any point in DE, and therefore to E, is a quadrant. Hence, E is distant from two points A, C, in an arc AC, by the quantity of a quadrant; ... by Cor. 2, Prop. 4, E is the pole of AC; and similarly, F is the pole of AB, and D of BC. PROP. VIII. The former construction remaining, the measures of the angles at A, B, C, are the supplements of the sides opposite, that is, of EF, DF, DE; and, reciprocally, the measures of the angles at D, E, F, are the supplements of the sides BC, AC, AB. For, the measure of the angle at A (see Prop. 6.) is GH. Now, GH = EH – EG = EH - (FE – FG) = EH + FG – FE = 90° + 90° - FE = 180° - FE = (p. 12.) the supplement of FE. Again, the measure of the angle at B, or KI=FK – FI=FK-(DF-DI) = FK + DI – DF=90°+90°- DF = 180° – DF = supp'. DF; and similarly, LM, the measure of the angle at C, is the supplement of DE. Secondly, the measure of the angle at D, or MI=MC+CI= MC+IB – BC = 90° + 90°-CB = 180° CB = supp1. CB; and similarly, LH, KG, the measures of the angles at E and F, are the supplements of AC and AB. From its properties, the triangle DEF has been called, by English Geometers, the Supplemental Triangle; and from the mode of its description, by the French, the Polar Triangle. Cor. 1. If any angle as B, =90°, then DF=180°-90°=90°, or is a quadrant: and if B and C each = 90°, DF, DE, each is equal to a quadrant. COR. 2. If (Cor. 4. Prop. 4.), the angles at B and C each = 90°, and the angle at A is nearly 180° = 180° x, x being a very small angle, then the side of the supplemental triangle opposite to A is equal to x; and the sum of the three sides of the supplemental triangle = semicircle + x, = 180° + x. PROP. IX. The sum of the three angles of a spherical triangle is > 2 right angles < 6 right angles. For, the angles of the triangle + sides of the supplemental triangle = 180° + 180° + 180° = 6 × 90°; ... since the sides of the supplemental triangle must have some magnitude, the angles of the triangle must be less than 6×90°: again, by Prop. 3, p. 127, the sides of any triangle; and ... the sides of the supplemental triangle, are less than 4 x 90°, =4×90-x; suppose, consequently, the angles of the triangle = 6×90° - (4.90°−x) = 2 × 90° + X COR. 1. Hence a spherical triangle may have two or three right angles, two or three obtuse angles. COR. 2. If the angles at B and C are right angles, AB, AC are quadrants, and A is the pole of BC: if also the angle at A is a right angle, the triangle ABC coincides with the supplemental or polar triangle, and the triangle ABC is contained eight times in the surface of the sphere. PROP. X. The angles at the base of an isosceles spherical triangle are equal. In the triangle ACB, let AC=BC, draw the tangents AS, BS, which are equal and which cut their secant OS in the common point S. Draw also from A and B two tangents AT, BT, which, by Euclid, Book III. Prop. 37. are equal. Hence, in the triangles SAT, SBT; SA, AT, ST, are respectively equal to SB, BT, ST; .. by Euclid, Book I. Prop. 8, the angle SAT = the angle SBT, and... by Prop. 6, page 129, the spherical angle at A = the spherical angle at B. To prove the reverse proposition, that is, to prove if the angles are equal, the sides are equal; take the supplemental triangle; then, |