Prob. 10. Required the course and distance from New York, latitude 40° 43′ N., longitude 74° 0′ W., to San Francisco, latitude 37° 48′ N., longitude 122° 28′ W., on the shortest route. Ans. The course is S. 78° 16′ W. Distance, 2229.8 nautical miles. Prob. 11. Required the course and distance from San Francisco, latitude 37° 48′ N., longitude 122° 28′ W., to Jeddo, in latitude 35° 40′ N., longitude 139° 40′ E., on the shortest Ans. The course is N. 56° 41′ W. Distance, 4461.9 nautical miles. Prob. 12. Required the course and distance from San Francisco to Batavia in Java, latitude 6° 9′ S., longitude 106° 53′ E., on the shortest route. route. Ans. The course is N. 67° 30′ W. Prob. 13. Required the course and distance from San Francisco to Port Jackson, latitude 33° 51′ S., longitude 151° 14′ E., on the shortest route. Ans. The course is S. 59° 50′ W. Prob. 14. Required the course and distance from San Francisco to Otaheite, latitude 17° 29′ S., longitude 149° 29′ W., o the shortest route. on Ans. The course is S. 30° 31′ W. Distance, 3514.8 nautical miles. Prob. 15. Required the course and distance from San Francisco to Valparaiso, latitude 33° 2′ S., longitude 71° 41 W., on the shortest route. Ans. The course is S. 55° 9′ E. Distance, 5108.5 nautical miles. Prob. 16. Suppose two ports, one in north latitude 30°, and the other in north latitude 40°, the difference of longitude between them being 50°. Required the bearing and distance of each of those ports from an island that lies in south latitude 18°. and which is equally distant from both of the said ports. Ans. Bearing from first port, S. 40° 52′ 9′′E. The distance, 59° 23′ 19′′-3563.3 nautical miles. THE END. N 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 NEW YORK, AND AUTHOR OF “a courSE OF MATHEMATICS.” SEVENTH EDITION. NEW YORK: HARPER & BROTHERS, PUBLISHERS, 329 & 331 PEARL STREET, FRANKLIN SQUARE. 1859 Entered, according to Act of Congress, in the year one thousand eight hund ed and forty-eight, by HARPER & BROTHERS, 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 ist 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 steppingstone to assist the beginner in comprehending the nature of logarithmic |