$31. Proportion by Logarithms. Since the fourth term of a proportion is found by dividing the product of the second and third terms by the first, we obviously have this RULE. From the sum of the logarithms of the second and third terms, subtract the logarithm of the first term. What is the fourth term of a proportion of which the first, second, and third terms are respectively 0-0146, 45, and 1.07? log. 4.5 = 0·653213 log. 1·07 = 0·029384 0.682597 subtract log. 0.01462.164353 log. of fourth term = 2.518244 Consequently the fourth term = 329·794, nearly. ARITHMETICAL COMPLEMENT. § 32. The subtraction of a logarithm from the sum of two or more logarithms is usually made to depend upon addition, by making use of its Arithmetical Complement, which is the difference between 10 and the given logarithm, and is readily taken from the table, by beginning at the left hand and subtracting each figure from 9, except the last significant figure on the right, which must be taken from 10. Now it is obvious, that if instead of subtracting a given number from another, we first subtract the number from 10, and then add the result, we shall, after rejecting 10, obtain the true difference. Hence to work a proportion by logarithms, we may use this second RULE. Take the arithmetical complement of the logarithm of the first term, and the logarithms of the second and third terms, and from the sum of the three reject 10 from the characteristic. EXAMPLES. 1. What is the fourth term of the proportion 6·5:30·5 :: 1·25? 0-768297, which sum, after rejecting 10, is the log. of 5.86539, the fourth term. 2. What is the fourth term in the proportion 3456 : 10·5 :: 6543 ? ar. co. log. 3456 = 6·461426 log. 10.5 = 1.021189 log. 6543 = 3.815777 fourth term = 19.8788, log. = 1.298392 3. In the proportion 1.23 : 2.34 :: 3·45, what is the fourth term? Ans. 6·56342, nearly. TABLE III. NATURAL SINES AND TANGENTS. $33. It is necessary to explain the arrangement of this table before explaining that of Table II., since the values of the latter have been obtained from those of the former. Table II., being logarithmic values, is with propriety placed immediately after the table of logarithms. Table III. contains the natural sines and tangents to every minute of the first quadrant, and consequently those of the cosines and cotangents. The first 9 pages are devoted to sines and cosines, the remainder of this table gives the tangents and cotangents. When the number of degrees does not exceed 45, they are to be found at the top of the page, and the additional minutes are given in the first or left-hand column of the page. But if the degrees exceed 45, they are to be found at the bottom of the page, and the additional minutes are then given in the last, or right-hand column of the page. It will be seen by examining this table that the number of degrees at the top of any page added to the number at the bottom of the same, gives 89, to which adding the 60 intermediate Ininutes makes just 90 degrees. Thus, on page 70 we have 34° at the top and 55° at the bottom, the sum of which is 89°. Since the cosine and cotangent (§ 8) of an angle is the same as the sine and tangent respectively of its complement, it follows that by this arrangement of the table those columns which are marked sine and tangent at the top of the page will be marked cosine and cotangent respectively, at the bottom of the page. As the sine and cosine of any angle cannot exceed the radius, which is a unit, it follows that their values are always expressed in this table by a decimal. The tangent and cotangent may in some cases exceed a unit, and will then be expressed by a decimal joined to an integral part. Thus, turning to this table, we find sin. 17° 13'=0.29599; cos. 17° 13'=0·95519. TABLE II. LOGARITHMIC SINES AND TANGENTS. 8 34. This table is constructed by taking the logarithms of the values given in Table III., and increasing each logarithm by 10. The arrangement is the same as given in the Table of Natural Sines, Tangents, &c.; that is, the degrees are to be found at the top of the page and minutes in the first column, when the angle is less than 45°. When the angle is greater than 45°, the degrees are to be found at the bottom of the page and the minutes in the last column. As in the last table, the column marked sine at top is cosine at the bottom; the column marked tangent at the top is cotangent at the bottom, and conversely. Obtuse angles are also provided for in this table: when found at the top of the page they are at the right-hand side, and the minutes are then to be found in the right-hand column. But when found at the bottom of the page they are at the left-hand side, and the minutes are then to be found in the left-hand column. As most cases of Trigonometry are solved by the assistance of proportions, which are most readily wrought by the help of logarithms, it has been found convenient to prepare beforehand the logarithms of the sines and tangents of all angles expressed by degrees and minutes within the limit of the first quadrant. Since the sines, as well as tangents, of many arcs are less than a unit, the characteristic of their logarithms will be negative. By adding 10 to each logarithm, the characteristics will all be positive, and we shall not be as liable to commit errors in using them, as we should if some had positive characteristics, and others negative characteristics. §35. To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given angle. If the angle is less than 45°, we seek the degrees at the top of the page and the minutes in the first column on the left; then passing across the page horizontally until we come to the column. having the appropriate heading of sine, cosine, tangent, or cotangent, as the case may require, we shall find the logarithm sought. Thus, on page 53, we find sin. 35° 53'-9-767999; cos. 35° 53' 9.908599. If the angle is greater than 45°, we seek for the degrees at the bottom of the page, and the minutes on the right-hand side of the page; then, as before, passing horizontally across the page till we reach the column having at the bottom the appropriate designation of sine, cosine, tangent, or cotangent, as the case may require, and the number thus given will be the one sought. If the angle is obtuse, we must make use of its supplement (§ 13). § 36. When the angle is expressed in degrees, minutes, and seconds, we proceed as follows: Let it be required, for example, to find the logarithmic sine of 13° 14' 17". By what has already been said, we readily find Hence, 537 ÷ 60 = 8·95 = diff. for 1′′, and 895 = diff. for 100”. If we suppose the logarithmic sines to increase in the same ratio as the increase of their corresponding arcs, which supposition is very nearly correct, we shall be able to find the difference corresponding to 17" by multiplying 8.95 by 17, which gives 152.15. sin. 13° 14′ = 9.359678 152 add. 9.359830 = sine of 13° 14′ 17′′. The column D, immediately at the right of the column of sines, gives the differences of the logarithmic sines for 100". Thus, in the tables opposite the sine of 13° 14' in the column D, we find 895, which we have already shown to be the increase for 100". In the same way, the column D, immediately after the column of cosines, gives the difference of the logarithmic cosines for 100". The column D, between the tangent and cotangent columns, answers for both, giving also the difference for 100". Hence, to find the logarithmic sine of an arc given in degrees, minutes, and seconds, we must first seek the value corresponding to the degrees and minutes; then multiply the corresponding tabular number in column D by the number of seconds, cutting off two places to the right for decimals; add the product to the value already found, and the result will be the logarithmic sine sought. Thus, find the sine of 37° 31′ 13′′. Sin. 37° 31'9-784612. Tabular number D=274, which multiplied by 13, the number of seconds, gives 274× 13=3562; pointing off two decimals, we have 35.62, or 36 nearly, which added gives |