XXV. For determining the Latitude by the Pole Star......... 239 XXX. To change Mean Solar into Sidereal Time.................... XXXI. To change Sidereal into Mean Solar Time......... XXXII. To convert Mean Time into Parts of the Equator..... 250 97 XXXIII. Lengths of Circular Arcs....... XXXIV. to XLVIII. For computing the Corrections of the 250 97 Fixed Stars...... 250 98 XLIX. Mean Obliquity of the Ecliptic......... LIV. Right Ascensions and Declinations of Stars for 1830... LV. Decimal Numbers for each Day in the Year......... LVI. Sun's R. A. for 1828.... 253 104 253 106 254 107 LIX. Correction of Longitude by Chronometers..... 255 109 ... LXI. To convert Space into Time....... 257 111 LXIII. Useful Numbers in Calculation.............. 257 111 LXXI. Contraction of the Semidiameter of the Sun and Moon 261 116 LXXII. Solar Nutation of a Star in R. A. and Dec........... During the time this work has been in the press, some improvements occurred to Page 5 Delete lastly, line 5 from bottom. Add Cor. log (N+1) = Log (N − 1) + 100, in tables to six places of decimals. 2M N nearly when N exceeds Page 24 for Exmaples read Examples. 22 30 32 40 41 51 53 55 - - 30 read 50, in Table for Face of Hexagon. Add. Fosse of Ravelin is Schuchburgh read Schuckburgh, line 38. volumes read volume, line 41. Supply difference, line 28. Heineker read Heineken. 16 (t+t') read 15 (t+t'), line 28. Add. The following is a very easy formula for computing heights by Noon read Meridian. - 100 Supply to the rule. The dip from Table XI. must be subtracted · KPL read KPZ, line 13, and for 20. 7hread 8h, line 2 from bottom. read 20.8, line 28. 2.8 Supply Ans. to ex. 2, 48° 30' 26" W. same evening read morning, 8th September, 1825. 113 ――0′.9 in col. of diff. read -1'.9, for -2′.8 read-3'.8 at 50° alt. 51° 54′ read 52° 54′, line 12, and for 52′ read 53′, line 2 from bottom. Add to note. By omitting log. radius, and adding the correction from 28°6 inches read 29.6, line 4. 28m read 8", line 6. Supply latitude 34° 37' S., ex. 2.1 Add. Should the log. μ from the Berlin Ephemeris, be used in place 23° 53' 30" read 23° 52′ 36′′.5, middle of page. Delete second part of foot-note, and read, Captain Kater's Floating - C lat. 55° 40′ 5′′ N. long. 2° 27′ 38′′ W. not 37′ read 55° 39′ 54′′ N. Deal 1.000460. C. Kater. Add to ex. 9, the angle of elevation being 32° 30′ compressed read expressed, line 35. Delete, true, line 11, from foot-note. It may be added to the note, that, if the expansion of moist air be Page 239 0.002083 for dry air, the very small errors of this table would be still diminished. page 85 read 110, last line in note. 242 1' 33" read 0′ 33′′, 2d diff., and for 38" read 33", 3d diff. 245 38° read 33o, line 4 from bottom. 245 and 246, Tables XXVIII. and XXIX. may be carried beyond similar manner. 247 for June 21 read June 16, line 3, under the table. Tables XXVII. and XXVIII. read XVII. and XVIII. read r', line 2 from bottom. E read C, line 18 from top. — (— a) read (s—a) in some copies. Page 103. Nov. 1, Table XLIX. for 0".360 read 0".370, and supply accents to 108. annual diminution, and these numbers. 106. Dec. 18, for 0.862, read 0.962, and since the numbers are decimals of 110. Time of high water at London, for 2h 50m read 2h 15. Vienna, for log 17° read 16°. 111. Table LXI., for 1", &c. read l' in the middle column. In Table LXIX. increase the days of the month by unity, that is, for 235. Tables XIII. and XIV. will serve to correct altitudes observed at land with an artificial horizon by increasing the numbers in XIII. under 16 feet by 4', and diminishing those in XIV. under 16 feetby 4', and then applying the sum and remainder, according to the title of each table, to the altitude observed, the result will be the true altitude. It may be remarked, that Professor Bessel of Konigsberg has found, very lately, that the usual corrections, Section V., page 200, &c., employed to reduce pendulum experiments to a vacuum, will require some modification, which future investigations can alone determine. PREPARING FOR PUBLICATION, BY THE SAME AUTHOR. I. A KEY to this Work; containing Solutions to all the Problems, and Exemplifications of all the Formula, to render their Application easy and familiar. For the use of Teachers, and of private Students who may not have the assistance of a Master. To be printed uniformly with the Tables. II. A Short but Comprehensive TREATISE on MATHEMATICAL and ASTRONOMICAL INSTRUMENTS; in which the Principles of those most generally useful will be clearly explained, and their Application to Practice fully illustrated. One volume 8vo. This work will form a second volume to the Author's Mathematical and Astronomical Tables, which may be considered the first, and the two taken together will constitute a useful introduction to the proper management of Mathematical and Astronomical Instruments, and to the most approved methods of obtaining the results generally derived from them by observation and calculation. INTRODUCTION. PART I. OF LOGARITHMIC AND TRIGONOMETRICAL TABLES. SECTION I. Of the Properties of Logarithms. 1. LOGARITHMS are a series of numbers, originally invented by Baron Napier, for the purpose of facilitating arithmetical calculations. This end is attained by their enabling us to perform the operations of multiplication by addition, of division by subtraction, of involution by multiplication, and of the extraction of roots by division.* 2. It is evident that any two series of numbers, the one being in arithmetical and the other in geometrical progression, possess these properties: thus, for example, let the Ar. series be 0 1 2 3 4 1 10 100 1000 10,000 100,000} .&c. Now, if we add any two numbers in the arithmetical series, such as 2 and 3, which are equal to 5, and multiply the corresponding numbers under them, 100 and 1000, we have 100,000, the number immediately under 5, which was obtained by the addition of 2 to 3. Hence, then, it is clear that, if tables of this kind, sufficiently extensive, were formed, by a reference to them, the operation of multiplication could be performed by means of addition. In like manner, we perform division by subtraction; for, if from 5 we take 3, the remainder is 2, under which we get 100; that is, 100,000, the number under 5, divided by 1000, that under 3, gives 100 as a quotient. Roots are readily determined in a similar way; thus, 4, in the arithmetical series, divided by 2, gives 2, under which, in the geometrical series, is 100; that is, the second, or square root of 10,000 the number under 4, is 100, the number under 2, and so on. Napier called the first series the logarithms of the corresponding numbers in the second. 3. Since the two series may be assumed at pleasure, we may have as many different systems of logarithms as we choose. 4. The series in art. 2 being adapted to the common denary scale of arithmetic, is, on the whole, the most convenient for general purposes, though other systems have, in particular cases, their peculiar advantages. On considering these series, it appears that the logarithm of 1 is * The identity of this process with that performed upon the exponents of quantities in the corresponding operations of algebra, will be obvious to those who have acquired the rudiments of that branch of mathematics. |