3.141592653589 ::: 360° × 60: 3437·74677, &c. consequently m = 2.3025, &c. x 3437-74677, &c. x log. tang.comp.lat. =7915·7044679 × log.; Q.E.D. tang.comp.lat.' Note. It would be improper to make the value of r in the latter part of the expression=3437.7467, &c., unless the table" of logarithmical tangents was calculated to the same radius. EXAMPLE. Let it be required to find the meridional parts correspondent to 5 degrees. 90° 5 2185 Log. radius=10° Half comp.lat.42.30 the tang. by Ulacq's tables, 9.9620524617 0379475383 this multiplied by 7915.7044679 produces 300 38149846, &c. exactly agreeing with the answer before found by the secants. In the same manner you will find the meridional parts of 10° to be 603-069579, &c., the logarithmical tangent of 40° being 9.9238135302. (C) The following table comprehends the meridional parts answering to every five degrees of latitude. The first column. contains the degrees, the second the meridional parts copied from Wright's original treatise, and the third the meridional parts truly calculated by this problem.-As our tables of meridional parts are seldom carried to decimals, the reader will readily perceive that Wright's tables may be used without sensible error 6971-5485 5965-9179 6970-3390 10 603-0475 603.0696 40 2622-7559 2622-6902 70 5966-6811 15 910-4325 910-460645 3030-1271 3029-9392|75 20 1225-1292|1225·1390|50 3474-6045 8474-4720 80 8377-3416 8375-1970 25 1549-9878 1549-9952 55 3968-1879 3967.9661 85 10769.6200 10764 6210 30 1888-3768 1888-3754|| 604527-7106 4527.3677||89| 16317-5324 16299-5563 (D) A table of meridional parts being formed, by any of the preceding methods, to every degree and minute of the quadrant, a Mercator's chart, may be readily constructed therefrom. EXAMPLE. Let it be required to construct a Mercator's chart con containing 60 degrees of east longitude, and extending from the equator to 40 degrees of north latitude? I. Draw wE to represent the equator (Plate VI. Fig. 24.), on which set off wE=3600 miles (the number of miles contained in 60 degrees, the extent of the chart,) from any scale of equal parts. II. Divide the line we into 6 equal parts in the points 10, 20, 30, 40, &c., each part containing 10 degrees; and each of these parts may be subdivided into single degrees, half degrees, &c. if necessary. Then straight lines drawn through the points w, 10, 20, 30, &c. perpendicular to wE, will represent the meridians passing through every 10 degrees. III. From the same scale of equal parts take 2622.7 (the meridional parts answering to 40 degrees) and set them off from w to and from E to L, and join Ll, which will represent the parallel of 40 degrees of latitude, and determine the extent of the chart northward. IV. From the scale of equal parts take 603 the meridional parts answering to 10 degrees and set them off from w to a, and from E to 10, and join A, 10, this will represent the parallel of 10 degrees. In the same manner w, 20,=1225; and w, 30,1888; and draw the parallels of latitude; or make ▲, 20, =1225-603; 20, 30,=1888-1225; &c. If the chart does not begin at the equator, it is evident that the meridional parts correspondent to the least latitude contained in it, must be subtracted from the meridional parts of each point of greater latitude. PROPOSITION II. (E) Radius, is to the tangent of the course; as the meridional difference of latitude, is to the difference of longitude. In the annexed figure BC is the proper difference of latitude, DC the meridional difference of latitude, the angle ACB the course, AC the distance sailed, AB the departure, and ED the difference of longitude. Then EDC is a rightangled triangle, and similar to ABC (W. 363). Now DC radius :: ED: tangent of ACB; therefore radius: tangent of ACB :: DC: ED. PROPOSITION III. (F) The number 00012633114, &c. Is to the natural tangent of the course; As the difference between the logarithmical tangents of half the complements of the latitudes sailed from, and bound to, Is to the difference of longitudes between these places, în geographical miles. Let the less latitude be represented by l, and the greater by L; then the meridional parts correspondent to /= tang.comp. × 7915-7044, &c. (B. 365.) and by the same rule the meridional parts correspondent to L= tang.comp. L. tang comp. 1. the meridional difference of latitude in miles. Now (by E. 368.) r log. tang. comp. L. But, log. is evidently tang.comp. 1. =log. tang.comp. l. log. tang. comp. L; hence rad. tang. course:: (log. tang. comp. l. log, tang. comp. L.) x 7915 7044, &c. : diff. longitude. Divide the antecedents by 7915-7044, &c. then rad. 7915-7044, &c. tang. course :: (log. tang. comp.l.log.tang : (log.tang. rad. comp. L.): diff. longitude, but, 7915-7044, &c. 1 7915-7044, &c. 000126331143874, &c., hence it follows that ⚫000126331143874, &c. : tang. course :: (log. tang. comp. l. log. tang.comp. L): diff. longitude. (G) COROLLARY. Find the logarithmical tangent of half the complement of the latitude sailed from, the logarithmical tangent of half the complement of the latitude bound to, multiply their difference by 10000, and find the common logarithm of the product. THEN, The logarithmical tangent of the angle 51°.38.9".14""= 10.1015103, Is to the logarithm found above; As the logarithm tangent of the course, Is to the logarithm of the difference of longitude in miles. For, 00012633114, &c. : the natural tang. course :: (log. tang.comp. log. tang. comp. L.): diff. longitude in miles. Multiply the antecedents by 10000, then 1.2633114, &c. : the natural tang.course:: (log.tang. comp. log. tang.comp. L.) x 10000: diff. longitude, but 1 2633114, &c. is the natural tangent of 51°.38'.9".14", the logarithm of which is 10-1015103; hence by taking the logarithms of all the terms in the last proportion, we deduce the above rule. (H) SCHOLIUM. The second proposition (E. 368.) and the following, which is immediately deduced therefrom, (viz. Meridional difference of latitude, is to difference of longitude; as radius is to the tangent of the course,) will solve all the cases that can occur in Mercator's sailing by the help of a table of meridional parts. And the corollary (G. 369.) will solve the whole with the assistance of a table of logarithmical tangents (independent of a table of meridional parts) by varying the proportion. THE END OF NAVIGATION. I. A table of the Logarithms of Numbers, from an unit to ten thousand. II. A table of natural sines to every degree and minute of the quadrant. III. A table of Logarithmical sines and tangents to every degree and minute of the quadrant. IV. A table of the Refraction in altitude, of the heavenly bodies. sea. V. A table of the depression, or dip, of the horizon of the VI. A table of the sun's parallax in altitude. VII. A table of the augmentation of the moon's semi-dia meter. VIII. A table of the right ascensions and declinations of thirty-six principal fixed stars, corrected to the beginning of the year 1813. TABLE I. LOGARITHMS OF NUMBERS FROM 1 TO 10,000. No. Log. No. Log. No. Log. No. | Log. No. | Log. 234 5 6889 84510 90′ 09 26 41497 46 66276 00000 21 32222 41 61278 61 78533 81 90849 82 91381 83 91908 64 80618 65 81291 66 81954 86 93450 9 95424 29 46240 49 69020 69 83885 89 94939 |