Raising of powers? Extraction of roots? How is the logarithmic sine, tangent, &c., of any arc found when consisting of degrees and minutes only? How the log. sine, tangent, and secant, when of seconds also? How the cosine, cotangent, and cosecant in the last case? How are the logarithmic secant and cosecant computed from the logarithmic sine and cosine? What is the arithmetical complement of a logarithm? For logarithms entering in what way into formulas are arith. compa. used? What must be rejected from the logarithmic sum for each ar. comp. used? QUESTIONS ON THE CIRCLES OF THE CELESTIAL SPHERE. What is the axis of the earth? The axis of the heavens? What is the celestial equator? The ecliptic? What are the equinoxes? What are declination circles? What is the meridian of a place? What is the hour angle of a heavenly body? What is the horizon of a place? What are the poles of the horizon called? What are vertical circles? Which is called the prime vertical ? What is the declination of a heavenly body? The right ascension? What is the celestial latitude? Longitude? What is the altitude of a heavenly body? The azimuth? What are the co-ordinates of a heavenly body? How many sets are there? Which are obtained from observation? Which of the observed co-ordinates are preferable, and why? Ans. R. A, and D, because they are the same for every place on the earth, whereas altitude and azimuth are different for every place. QUESTIONS ON THE TRANSIT INSTRUMENT. What are the different kinds of time? What is apparent solar time? Mean solar time? What is siderial time? How much longer is a mean solar than a siderial day? How is a mean solar interval of time converted into a siderial interval, and the contrary? What instrument is employed for observing the time of transit of stars over the meridian? Of what parts does the transit instrument consist? How many and what are the adjustments? How is the instrument collimated? How is the striding level adjusted? How is the instrument adjusted to the meridian? When the instrument is completely adjusted, in what plane does the line of collimation move? How is an observation made with the transit instrument? How is the equatorial interval of the wires determined? How the interval between any wire and the middle wire for a particular star? What is the use of knowing the intervals of the wires? How does the probable error of observation compare with the number of wires observed upon? What is the formula for the inclination of the supporting axis? What the formula for the correction of the time of meridian transit for level error? What the formula for determining error in azimuth or deviation from the meridian? What for consequent error in the time of meridian transit ? How is the equatorial error of collimation found? How from this the collimation error for any star? What is used with the transit instrument? If the clock keep true siderial time, what does the time of meridian transit show? If the right ascension of the object observed is known by catalogue, what does the difference between this and the time of meridian transit show? How is siderial time converted into solar and the converse ?* *The above questions will suffice to show the nature of those which should be put upon the subsequent parts of the work with which we shall not take up further space. APPENDIX II. ON UNLIMITED SPHERICAL TRIANGLES AND THEIR SOLUTION.* OF THE VARIOUS TRIANGLES FORMED BY THE SAME THREE POINTS ON THE SPHere. 1. If any two points, A and B, be taken upon the surface of the sphere, the arc of a great circle joining them may be considered to be either the arc A B (< 180°), or 360 -A B; or if we do not limit the arcs to values less than a circumference, we may consider it to have an indefinite number of values expressed generally by the formula 2 na, a denoting that value which is less than or a semicircumference, and n any whole number or zero. 2. If two arcs of great circles intersect in a point A, the angle which they form may be considered to be either the angle a (< 90°), or 180°-A, or 1800+a, or 360° -A; or, taking the most general view of angular magnitude, the angle will have an indefinite number of values expressed by the formula mA, A denoting the value which is less than, and m any whole number or zero. 3. If, therefore, any three points, A, B, C, be taken on the surface of the sphere, and great circles, made to pass through each pair, we shall have an infinite series of triangles whose sides will be generally expressed by a, b, c, denoting the arcs less than joining the pairs of points BC, AC, A B, respectively; A, B, C, the angles less than formed at those points by the intersection of these arcs; and n and m, any whole numbers, or zero. 4. It is evident, however, that we cannot assume that any three values of the sides from the series (1), combined with any three values of the angles from (2), will form a spherical triangle. Some general relations of the parts composing a triangle must first be established, from which corresponding values of n and m in (1) and (2) * Introduced by Gauss. Notwithstanding the elegance and generality thus given, to the solutions of many astronomical problems, nothing is to be found on this subject in our trigonometrical works. The present paper is from Prof. Chauvenet of the U. S. Naval Acad. The explanatory notes are the author's. may be deduced. Although these general relations are well known, it may not be out of place to add here a concise demonstration of them. Let the point c,* one of the angular points of the spherical triangle ABC, be referred by rectangular co-ordinates to three planes, one of which, the plane of x y, is the plane of the great circle A B ; let the axis of x be the diameter of the sphere passing through B, and let the origin be the centre of the sphere. The formulas of transformation from these co-ordinates to polar co-ordinates, the origin being the same, the polar axis being the axis of x, and the fixed plane the plane of x y, are where B denotes the angle which the plane passing through the polar axis and the point c makes with the fixed plane; R, the radius-vector in this plane, or distance of the point c from the origin; and a the angle which this radius-vector makes with the polar axis. B is an angle of the spherical triangle, and a is the side opposite the angle a; and, according to the principles of analytical geometry, в and a may be altogether unlimited, due regard being had to the signs of their trigonometric functions, and to those of x, y, and z. Let us now transform from these rectangular co-ordinates to others also rectangu lar, the origin and the plane of x y remaining the same, but the axis of x in the new system passing through the point A, and therefore making with the first axis the angle c, c also expressing the side of the triangle opposite the angle c. The known formulas of transformation become Ꮓ *The rest of Art. 4 implies some knowledge of Analytical Geometry. It may be readily understood, however, by the mere student of trigonometry from the annexed diagram, with the following explanations. R in formulas (3) is equal to o c in the diagram. The projection of R or of oc, on oв which is called the axis of x, that is to say the distance between the foot of a perpendicular from c on o в and the point o is the value of x in the formulas, the projection of o c a line called the axis of y, drawn from o in the plane A OB perpendicular to o B, is the value of y in the formulas, and on A B the projection of R or oc on oz perpendicular to the plane A O в, is the value of z in the formulas. The first of formulas (3) is now obvious enough; in the second R sin a evidently expresses the value of a perpendicular from c to OB in the plane c o B, and this perpendicular multiplied by the cosine of the angle which it makes with its projection on the plane A o в equal to the angle of the two planes or the angle B, expresses the length of the projection on the plane A O B, which is evidently equal to the projection of R on the axis of Y; or multiplied by the sin в expresses the height of c above the plane A O B, which is equal to the projection of R on oz the axis of z. N. B. The axes of x, y, and z are at right angles each to the plane of the other two. So also are those of x', y', and z'. The student will readily deduce formulas (4) by the rules of Plane Trigonometry. |