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(1.) SPHERICAL Trigonometry means in strictness, the measurement of the various triangles which can be drawn upon a sphere. In the modern form of the science, it must be called the investigation of the relations which exist between the several parts of a solid angle.
A solid angle is a name improperly given to the figure made by three straight lines which meet in a point, and no two of which are in the same plane; as O A, O B, and O C.
In a solid angle there are six things to considernamely, the three lines O A, OB, and OC; and the three planes AO B, BOC, COA. The lines make three rectilinear angles, A O B, BOC, COA; and the planes make three dihedral angles; each of which may be determined, as usual, by drawing from a point, in the line of intersection of two planes, perpendiculars to that line in the two planes. (Geometry, book iv. prop. 17.)
(2.) In future, we shall abbreviate references as follows:
(G. iv. 17.) refers to Treatise on Geometry, book iv. prop. 17.
Trigonometry, Article 20. (St. 50.) Study of Mathematics, page 50. (3.) We have felt it unnecessary to supply a Geometrical Treatise on the Sphere; but we should strongly recommend the student to read that in the ninth book of the Treatise on Geometry. Clear ideas will never be acquired without considerable familiarity with the properties of the sphere; to acquire which, the minimum of spherical geometry frequently attached to Treatises on this subject is insufficient. We shall take it for granted that the student has some acquaintance with this part of solid Geometry.
(4.) We should also recommend the beginner to provide himself with such an apparatus * as here drawn; that is, a wooden sphere of three or four inches in diameter, with a hemispherical cup of the same radius, in which the sphere may be placed in any position. Any great circle (or circle which halves the sphere) may then be drawn through two given points, by bringing these points down to the edge of the cup, and drawing a pencil round the edge.
(5.) In this treatise we do not consider any solid angles except those
* Messrs. Watkins and Hill, of Charing Cross, are in the habit of constructing this apparatus.
in which the rectilinear angles are severally less than two right angles; in which case the dihedral angles will also be severally less than two right angles. And no side or angle is ever considered as negative.
(6.) If a sphere be drawn, whose centre is at the point O of the solid angle, a portion of the surface will be intercepted by the planes AOB, BOC, COA; which portion is called a spherical triangle. The arcs CA, A B, B C, are its sides; the angles made by the circles (that is, by tangents drawn to the circles) are its angles: and these six parts may be used to represent the six parts of the solid angle. For the arc AC bears to C B the same pro- p portion as the angle A O C to the angle C O B, the radii being the same. And since tangents drawn at A, to the circles which meet in that point, are perpendicular to O A in the planes of these circles, the angle made by the tangents is that made by the planes CO A, A OB. But it would do equally well to draw the perpendiculars from any other point P in O A.
(7.) A spherical triangle is made only by those circles which pass through the centre of the sphere, or by great circles. All triangles made by other circles are not considered.
(8.) When we talk of the side of a spherical triangle, we mean, therefore, the angle which that side subtends, at the centre of the sphere; and, as it is an angle which we are speaking of in reality, we are liable to the apparent confusion of calling a line an angle. Thus, we may say that the side of a spherical triangle is a right angle; meaning thereby that it is a quadrant of a circle, and subtends a right angle at the centre of the sphere.
(9.) But when we speak of the angles of a spherical triangle, we always mean the dihedral angles of a solid angle. Thus, a right-angled spherical triangle is cut off by three planes passing through the centre of a sphere, two of which planes are at right angles.
(10.) There is no direct affinity between the terms sides and angles, in a plane and in a spherical triangle. In a plane triangle the sides and angles are different species of magnitudes; in a spherical triangle they are the same. In a plane triangle, a side cannot be expressed in terms of angles only; in a spherical triangle, the sides (8) can be found when the angles are known. A plane triangle cannot have more than one right angle; a spherical triangle may have three; and so on.
(11.) The sides of a spherical triangle are usually denoted by the letters a, b, c; the opposite angles by A, B, C. In the solid angle, the dihedral angle, made by the planes meeting in OC, is called opposite to the rectilinear angle BOA; and the dihedral angle, formed at OC, is said to be contained by the rectilinear angles A O Č, CO B.
(12.) The angles are usually measured in degrees, minutes, and seconds, so long as they occur only in sines, cosines, &c.; but when used independently, they are measured as in (Tr. 7.) (St. 90.); that is, the number chosen to represent the angle is
The arc subtended
ON RIGHT-ANGLED SPHERICAL TRIANGLes.
(13.) Let A B C be a right-angled spherical triangle; or let the planes BOC, COA, be at right angles. From any point
Q in OB, draw QR in plane BO C, perpendicular to plane COA (G. iv. 18); from R draw R P, perpendicular to AO, and join QP, which (G. iv. 4.) is perpendicular to OA, and QR (G. iv. Def. 1.) is perpendicular to PR. Hence, QOPR is a triangular pyramid, having none but right-angled triangles; one angle in each of which right-angled triangles is one of the parts of the spherical triangle-namely:
QOR, the side a (8).
ROP, the side b.
POQ, the hypothenuse c.
QPR, the angle A (9), because QP and RP are perpendicular to O A. Hence the general fractional relation
where K, L, and M, are any quantities whatever, gives the following;
These formulæ will be readily seen from the definitions (Tr. 13.), and from what has just been stated, to be identical relations existing among the ratios of the sides of the pyramid PQOR. Each of the first three has another like it, similarly deduced from the other side: we shall range them as follows; in which a side and its opposite angle are in the same type, whether Roman or Italic, in the same formula. We add two other formulæ, in which two angles enter, which we shall presently deduce.
(15.) The two new formula R, and R, are demonstrated as follows. Multiply together the two in R,, already demonstrated, which gives
Again, multiply together the first in R, and R. crosswise, which gives
sin. a tan. c cos. B
tan. a sin. c sin. A
(16.) The six formulæ in (14), by repetition ten, solve all cases of right-angled triangles. For since, in every one of them, is a distinct combination of three, taken from among
a, b, A, B, c;"
and since, from out of five quantities, only ten distinct combinations of three can be taken, it is manifest that each of the five enters, with every other two, in one or other of the formula; so that any two being given, any other can be found. For example:
(17.) There are certain mnemonical formulæ, called Napier's rules of circular parts, which are generally explained. We do not give them, because we are convinced that they only create confusion, instead of assisting the memory We recommend the beginner to learn the six formulæ in words, as given, and not to proceed further until he can apply them readily. To assist him, we give the parts of a right-angled triangle, together with the logarithms of the sines, cosines, and tangents of each, from which he can immediately see whether he is able to verify any formulæ.
We carry this opinion to the extent of thinking that they have been, to many, a serious impediment to a ready knowledge of applications to plane astronomy.