inner TABLES OF LOGARITHMS OF NUMBERS AND OF SINES AND TANGENTS FOR EVERY TEN SECONDS OF THE QUADRANT, WITH OTHER USEFUL TABLES. BY ELIAS LOOMIS, LL.D., PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN THE UNIVERSITY OF THE CITY OF Entered, according to Act of Congress, in the year one thousand eight hund ed and forty-eight, by in the Clerk's Office of the District Court of the Southern District of New York. PREFACE. THE accompanying tables were designed to afford the means of performing trigonometrical computations with facility and precision. The tables chiefly used in this country for purposes of education extend to six decimal places, like those in the present collection; but the precision which they are designed to furnish is only attained by a serious expenditure of labor. In the Table of Logarithms of Numbers they do not furnish the correction for a fifth figure in the natural number, and the labor of computing this correction is such that I always prefer the use of Hutton's Tables, extending to seven places, even in computations to which six-place logarithms are abundantly competent. In the pres ent collection, the correction for a fifth figure of the natural number is introduced at the bottom of each page, and the table is thus rendered nearly as useful as one of the common kind extending to 100,000. The whole has been carefully compared with standard authors, and nearly a dozen errors have thus been detected in the common tables. The principal table in this collection is that of Logarithmic Sines and Tangents. The common tables in this country extend only to minutes, with differences to 100". If, in a trigonometrical computation, angles are only required to the nearest minute, tables to five places are quite sufficient; but if the computation is to be carried to seconds, these can only be obtained from the common tables by a great expenditure of time and labor. In the present collection, the sines and tangents are furnished to every ten seconds of the quadrant, and at the bottom of each page is given the correction for any number of seconds less than ten, so that the precision of seconds can be obtained with almost the same facility as that of minutes with the tables in common use. Moreover, near the limits of the quadrant, by means of an auxiliary table, sines and tangents are readily obtained, even for a fraction of a second. The method of arrangement of the sines and tangents was suggested by a table in Mackay's Longitude; but the errors of that table, amounting to several thousand, have been corrected by a careful comparison with the work of Ursinus. By comparison with the same standard, more than two hundred errors (chiefly in the final figures) have been detected in the tables in common use. The Table of Natural Sines and Tangents is of less use than the loga rithmic; nevertheless, it is often important for reference, particularly in analytical geometry and the calculus; and it is useful as a stepping sines and tangents. The Traverse Table commonly usea in this country furnishes the latitude and departure to every quarter degree of the quadrant, for distances from 1 to 100, and occupies ninety pages. The accompanying table occupies but six pages, and yields ten times greater precision. The Table of Meridional Parts extends to tenths of a mile, and great care has been taken to insure its accuracy. For this purpose, I have compared all the similar tables within my reach, and among them have found two which appeared to have been computed independently. Between them there were detected 674 discrepancies in the final figures. These cases were all recomputed, and 78 errors were detected in the Jest copy compared. It is probable that the numbers in this table are not in every instance true to the nearest tenth of a mile; but it is be lieved that the remaining errors are few in number, as well as minute. This table is confidently pronounced more accurate than any similar one with which I have been able to compare it. The Table of Corrections to Middle Latitude was computed entirely anew. The corresponding table in common use, which was originally computed by Workman, contains more than four hundred errors, several of them amounting to two minutes. On the whole, it is believed that the accompanying tables will be found more convenient to the computer than any tables of six decimal places hitherto published in this country; and that they will be pronounced sufficiently extensive for all purposes of academic and collegi. ate instruction, as well as for practical mechanics and surveyors. EXPLANATION, OF THE TABLES. TABLE OF LOGARITHMS OF NUMBERS, pp. 1-20. LOGARITHMS are numbers contrived to diminish the labor of Multiplica. tion and Division by substituting in their stead Addition and Subtrac tion. All numbers are regarded as powers of some one number, which is called the base of the system; and the exponent of that power of the base which is equal to a given number, is called the logarithm of that number. The base of the common system of logarithms (called, from their inventor, Briggs' logarithms) is the number 10. Hence all numbers are to be regarded as powers of 10. Thus, since whence it appears that, in Briggs' system, the logarithm of every number between 1 and 10 is some number between 0 and 1, i. e., is a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2, i. e., is 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e., is 2 plus a fraction, and so on. The preceding principles may be extended to fractions by means of negative exponents. Thus, since 101=0.1, -1 is the logarithm of 0.1 101=0.01, 66 in Briggs' system; 66 -2 0.01 66 66 Hence it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1, or may be represented by -1 plus a fraction; the logarithm of every number between 0.1 and .01 is some number between |