formulas which are often employed in trigonometrical calculations for reducing two terms to a single one. XXIV. From the first four formulas of Art XXIII. and the first -sin q= sin p+sin q_sin} (p+q) __tang } (p+q) cos p+cos q sin p+ sin q = cos q-cos p sin (p-q) sin P sin (p-q)_tang (p-q) cos p+cos q cos (p-q) R R sin p+sin q_2sin sin (p+q) cos ↓ (p−q) ___ cos } (p—q) (p+q)2sin (p+q) cos (p+q) cos (p+q) = = sin p-sin q_2sin † (p−q) cos (p+q) __ sin } (p— sin (p+q) 2sin(p+q) cos (p+q) sin (p+q) Formulas which are the expression of so many theorems. From the first, it follows that the sum of the sines of two arcs is to the difference of these sines, as the tangent of half the sum of the arcs is to the tangent of half their difference. XXV. In order likewise to develop some formulas relative to tangents, let us consider the expression tang (a+b)= R sin (a+b), in which by substituting the values cos (a+b) of sin (a+b) and cos (a+b), we shall find tang (a+b)= R (sin a cos b+sin b cos a) cos a cos b-sin b sin a substitute these values, dividing all the terms by cos a cos b; we shall have tang (a+b)= R2 (tang a+tang b) ; which is the value of the tangent of the sum of two arcs, expressed by the tangents of each of these arcs. For the tangent of their difference, we should in like manner find tang (a-b) R2 (tang a-tang b) R2+tang a tang b. Suppose b=a; for the duplication of the arcs, we shall have the formula R2-tanga Suppose b=2a; for their triplication, we shall have the for mula XXVI. Scholium. The radius R being entirely arbitrary, is generally taken equal to 1, in which case it does not appear in the trigonometrical formulas. For example the expression for the tangent of twice an arc when R=1, becomes, If we have an analytical formula calculated to the radius of 1, and wish to apply it to another circle in which the radius is R, we must multiply each term by such a power of R as will make all the terms homogeneous: that is, so that each shall contain the same number of literal factors. CONSTRUCTION AND DESCRIPTION OF THE TABLES. XXVII. If the radius of a circle is taken equal to 1, and the lengths of the lines representing the sines, cosines, tangents, cotangents, &c. for every minute of the quadrant be calculated, and written in a table, this would be a table of natural sines, cosines, &c. XXVIII. If such a table were known, it would be easy to calculate a table of sines, &c. to any other radius; since, in different circles, the sines, cosines, &c. of arcs containing the same number of degrees, are to each other as their radii. XXIX. If the trigonometrical lines themselves were used, it would be necessary, in the calculations, to perform the operations of multiplication and division. To avoid so tedious a method of calculation, we use the logarithms of the sines, cosines, &c.; so that the tables in common use show the values of the logarithms of the sines, cosines, tangents, cotangents, &c. for each degree and minute of the quadrant, calculated to a given radius. This radius is 10,000,000,000, and consequently its logarithm is 10. XXX. Let us glance for a moment at one of the methods of calculating a table of natural sines. The radius of a circle being 1, the semi-circumference is known to be 3.14159265358979. This being divided successively, by 180 and 60, or at once by 10800, gives .0002908882086657, for the arc of 1 minute. Of so small an arc the sine, chord, and arc, differ almost imperceptibly from the ratio of equality; so that the first ten of the preceding figures, that is, .0002908882 may be regarded as the sine of 1'; and in fact the sine given. in the tables which run to seven places of figures is .0002909. By Art. XVI. we have for any arc, cos=√(1--sin3). This theorem gives, in the present case, cos l'=.9999999577. Then by Art. XXII. we shall have 2 cos l'x sin l'-sin 0'sin 2′=.0005817764 2 cos 1'x sin 2'-sin 1'-sin 3'=.0008726646 2 cos 1'x sin 3'-sin 2′-sin 4'.0011635526 Thus may the work be continued to any extent, the whole difficulty consisting in the multiplication of each successive result by the quantity 2 cos 1'-1.9999999154. Or, the sines of 1' and 2' being determined, the work might be continued thus (Art. XXI.): sin 1': sin 2'-sin 1':: sin 2' + sin 1': sin 3 In like manner, the computer might proceed for the sines of degrees, &c. thus: sin 1° sin 2°-sin 1° :: sin 2° + sin 1° : sin 3° : sin 2° sin 3°-sin 1° :: sin 3° + sin 1° : sin 4° : sin 3°: sin 4°-sin 1° :: sin 4° + sin 1° : sin 5° Above 45° the process may be considerably simplified by the theorem for the tangents of the sums and differences of arcs. For, when the radius is unity, the tangent of 45° is also unity, and tan (a+b) will be denoted thus: And this, again, may be still further simplified in practice. The secants and cosecants may be found from the cosines and sines. TABLE OF LOGARITHMS. XXXI. If the logarithms of all the numbers between 1 and any given number, be calculated and arranged in a tabular form, such table is called a table of logarithms. The table annexed shows the logarithms of all numbers between 1 and 10,000. The first column, on the left of each page of the table, is the column of numbers, and is designated by the letter N ; the decimal part of the logarithms of these numbers is placed directly opposite them, and on the same horizontal line. The characteristic of the logarithm, or the part which stands to the left of the decimal point, is always known, being 1 less than the places of integer figures in the given number, and therefore it is not written in the table of logarithms. Thus, for all numbers between 1 and 10, the characteristic is 0: for numbers between 10 and 100 it is 1, between 100 and 1000 it is 2, &c. PROBLEM. To find from the table the logarithm of any number. CASE I. When the number is less than 100. Look on the first page of the table of logarithms, along the columns of numbers under N, until the number is found; the number directly opposite it, in the column designated Log,, is the logarithm sought. CASE II. When the number is greater than 100, and less than 10,000. Find, in the column of numbers, the three first figures of the given number. Then, pass across the page, in a horizontal line, into the columns marked 0, 1, 2, 3, 4, &c., until you come to the column which is designated by the fourth figure of the given number: to the four figures so found, two figures taken from the column marked 0, are to be prefixed. If the four figures found, stand opposite to a row of six figures in the column marked 0, the two figures from this column, which are to be prefixed to the four before found, are the first two on the left hand; but, if the four figures stand opposite a line of only four figures, you are then to ascend the column, till you come to the line of six figures: the two figures at the left hand are to be prefixed, and then the decimal part of the logarithm is obtained. To this, the characteristic of the logarithm is to be prefixed, which is always one less than the places of integer figures in the given number. Thus, the logarithm of 1122 is 3.049993. In several of the columns, designated 0, 1, 2, 3, &c., small dots are found. Where this occurs, a cipher must be written for each of these dots, and the two figures which are to be prefixed, from the first column, are then found in the horizontai line directly below. Thus, the log. of 2188 is 3.340047, the two dots being changed into two ciphers, and the 34 from the column 0, prefixed. The two figures from the colum 0, must also be taken from the line below, if any dots shall have been passed over, in passing along the horizontal line: thus, the logarithm of 3098 is 3.491081, the 49 from the column 0 being taken from the line 310. |