Page. CHAP. I. Preliminary definitions and principles . . 1. IV. Plane trigonometry considered analytically 36 V. Application of plane trigonometry to the determination of heights and distances. .. 61 VI. Spherical trigonometry, in six sections - Seel. 1. Fundamental principles, general pro- perties, and formulae . . . . . 79 Sect. 2. Right angled spherical triangles . 86 Sect. 3. Oblique angled ditto . . . . . 89 Sect. 4. Analogies of Napier - . . 94 Sect. 5. Mnemonics of spherical trigonometry 98 Sect. 6. Areas of spherical triangles and po- Sect. 2. Orthographic projection . . . 122 Sect. 3. Stereographic projection . . . 125 IX. Principles of dialling . . . . . . . . 134 X. Astronomical problems . . . . . . 151 XI. Minute variations of triangles, with a table of differential equations . . . . . . 187 14. Let a line be drawn from C to the point of contact A, in the first figure. 29, line 8, for answer read answers 37, line 2, for and is read it is 41, line 3, for . read: 156, Er. 1. Add, In the Connaissance des Tems, pour l'an 1817, there is given a table by M. Burckhardt to shorten the computation of this problem. 167, line 9 from bottom, for limited, read limits 212, last line but one, for +2 V #p, read of 2 V #p, TRIGONOMETRY. CHAPTER I. 1.THE word Trigonometry signifies the measure of triangles. But in and enlarged sense we comprehend under this name, the science by which we are enabled to determine the positions and dimensions of the different parts of space, by means of the previous knowledge of some of those parts. 2. If we conceive any different points whatever, posited in space, to be joined one to another by right lines, there will be presented to our consideration three things: 1st. The lengths of those lines. 2dly. The angles they respectively form. 3dly. The angles formed respectively, by planes in which those lines are, or may be imagined to be, comprehended. On the comparison of these three objects depends the solution of the various questions which can be proposed, as to the measure of extension, and of its parts. 3. The intersections of three or more lines in one and’ the same plane, constitute angles limited by right lines, and plane or rectilinear triangles, or polygons susceptible of being resolved into triangles. And the intersections of three or more planes, form plane angles and triedral or polyedral surfaces, the determination of the magnitudes and relations of which is facilitated by reR ference to the surface of a sphere. Hence mathematicians have been led to attempt the solution of two general problems. I. Knowing three of the six things, whether angles or sides, which enter the constitution of a rectilinear triangle, to determine the other three; when it is possible. II, Knowing three of the six things which compose a triangle formed on the surface of a sphere, by the inter sections of three planes, which also meet in the centre of that sphere, to determine the other three, when possible. The resolution of the first of these general problems appertains to plane or rectilinear trigonometry: that of the second, to spherical trigonometry. 4. Lines and angles, being magnitudes of different kinds, do not admit of comparison. It becomes necessary, therefore, to have recourse to quantities of an inter mediate kind, akin to the one, yet having an obvious, dependence upon the other, and serving as a common vinculum. Such are the lineo-angular quantities denominated sines, tangents, &c. which we are about to define. They are lines, but lines which admit of being measured only by parts of an assigned line, the radius of a certain circle; and lines which at the same time depend altogether, for their value, upon arcs of that circle, which arcs are, themselves, adequate measures of the angles included between the radii which limit such arcs. 5. By means of this happy invention of intermediate quantities, the business of trigonometry is greatly facilitated. For, by imagining a perpendicular let fall from the vertical angle of an oblique angled plane triangle upon the base, or base prolonged, it will at once be manifest that the resolution of triangles generally, may be referred to that of right angled triangles. Thus, as suming for a term of comparison, the hypothenuse of a . right angled triangle, equal to unity, for example, computing the bases and perpendiculars of all possible right angled triangles having the assigned hypothenuse, and arranging them in different columns of a table, the magnitudes of the different parts of any proposed triangle, would become determinable upon the known principles of similar triangles. Such a table as this, would, as will soon be seen, be no other than a table of natural sines. PLANE TRIGONOMETRY. 6. Plane trigonometry is that branch of mathematics, by which we learn how to determine or compute three of the six parts of a plane, or rectilinear triangle, from the other three ; when that is possible. This limitation is necessary, although there is only one case in which it can occur, namely, that in which the three angles of a rectilinear triangle are given. For, it is plain from Euc. vi. 4, that while the three angles of a triangle remain the same, the sides, though retaining the same mutual relation, may be greater or less, in all conceivable proportions. 7. Lemma I. Let AcB be a rectilinear angle: if, about the point c as a centre, and with any distance, or radius, CA, a circle be described, intersecting cA, CB, the right lines that include the angle AcB, in A and B; the angle AcB will be D to four right angles, as the arc AB to the whole circumference of the circle ADF.E. Produce Ac to meet the circle in F, and through the centre c draw pe, another diameter, to meet the circle in D, E. Then, Euc. vi. 33, ang. AcB : right ang. AcD :: arc As : arc AD, and, Euc. v.4, cor. quadrupling the consequents, we have, angle AcB: 4 right angles: arc AB:4AD, that is, to the whole circumference. |