the 1st edition of the Tables Portatives de Logarithmes of Callet, in small 8vo., printed at Paris 1783 (n); all of which are adapted to the sexagesimal division of the circle, used by Vlacq and most of the later compilers. Besides these, several other tables, of a different kind, have been lately published by the French; in which the quadrant is divided, according to their new system of measures, into 100 degrees, the degree into 100 minutes, and the minute into 100 seconds; the principal of which are the 2d edition of the Tables Portatives of Callet, beautifully printed in stereotype, at Paris, by Didot, 8vo. 1795, with great additions and improvements; the Trigonometrical Tables of Borda, in 4to. an. ix, revised and enriched with various new precepts and formulæ by Delambre; and the tables lately published at Berlin, by Hobert and Ideler, which are also adapted to the decimal division of the circle, and are highly praised for their accuracy by the French computors. Among the various tables, however, of the sexagenary kind, none have been more esteemed for their usefulness and accuracy than those of Gardiner, printed in 4to. at London, in 1742; which contain the logarithms of all numbers from 1 to 102100, and the logarithmic sines and tangents for every ten seconds of the quadrant, to 7 places of decimals, with several other (n) This neat portable work, which is now become extremely scarce, contains all the tables in Gardiner's 4to vol. hereafter mentioned, with several additions and improvements; and is, by far, the most useful and convenient performance of the kind that has yet been offered to the public. b necessary tables. A new edition of which work was also printed at Avignon, in France, in 1770, under the care of Pezenas, who added to it the sines and tangents of every single second, for the first 4 degrees, and a small table of hyperbolic logarithms, taken from Simpson's Fluxions. But of all the trigonometrical tables hitherto published, the most extensive and best adapted for obtaining accurate results, in many delicate astronomical and geodetical observations, are those of Taylor, printed in large 4to. at London, 1792; which contain the logarithms of the first common numbers from 1 1260, to eight places of decimals; the logarithms of all numbers from 1 to 101000 to 7 places; and the logarithmic sines and tangents of every second in the drant, to 7 places; as also a preface, and various precepts for the explanation and use of the tables, which, from the author's dying before the last sheet of his work was printed off, were supplied by Dr. Maskelyne, the astronomer royal. qua It may here also be observed, that besides the common tables hitherto mentioned, which contain the logarithms of numbers in their usual order, others, of a different kind, have been constructed, for the more readily finding the number corresponding to any given logarithm; of which the principal one, of any considerable extent, is the Antilogarithmic Canon of Dodson, published at London, in 1742; which contains the numbers corresponding to every logarithm, from 1 to 100000, to eleven places of figures, with their differences and proportional parts; and, though little 2. Extend from 47° 48′ (the complement of BC) to 42° (the complement of A) on the sines, and that extent will reach, on the same line, from 90° to 64° 35′ the Z B. 3. Extend from 48° (ZA) to 42° 12′ (BC) on the sines, and that extent will reach, on the same line, from 90° to 64° 40', the side AB (z). 2. In the right-angled spherical triangle ABC, The leg BC 11° 30′ AC 27° 54' or 152° 6' B 69° 22′ or 110° 38′ AB 30° O'or 150° 0' spherical triangle ABC, Required the other parts. AC 75° 25′ or 104° 35' (x) In following the three general rules, which have been given for right-angled spherical triangles, the proportion used in the calculation is not always that which is adapted to the instrumental solution; but one may be easily reduced to the other by proper substitutions. Thus, since rad: tan BC :: cot A: sin ac by the first forдов mula, if tan A be put in the place of its equal, the cot a, the pro portion will become tan a : tan BC :: rad: sin a C, which is that applied to the instrument; and, if thought necessary, will equally serve for the numeral solution. It may also be observed, that, in the instrumental computation, when the extent on the tangents reaches beyond the line, it must be set as far back as it reaches over; the method of doing which may be seen in the solution of case iv. following; where it is more fully described. iven 4. In the right-angled B Required the other parts. spherical triangle ABC, B C 62° 28′ or 117°32′ 4A75° 53' or 104° 7' A B 66° 7′or 113°53′ Note. Several of the problems, given in this part of the work, may often be more conveniently resolved by means of some of the formulæ in the table of cases, page 146 et seq., where the limits of the data, and other circumstances, are more particularly pointed out. CASE II. When a leg and its adjacent angle are given, to find the rest. 1. To find the other leg. : As cot given sin adjacent, or given leg :: rad: tan other leg. Which leg is like its opposite 4. 2. To find the other 4. As rad sin given :: cos adjacent, or given leg: : cos other 4. Which is like its opposite leg. 3. To find the hypothenuse. As tan given leg: cos adjacent, or given :: rad: cot hyp. Which hyp. is less than 90° if the given leg and are like; but greater than 90° if they are unlike. EXAMPLES. 1. In the right-angled spherical triangle, ABC, having the leg AC 54° 43′, and its adjacent angle a 48°, to find the rest. AFFECTIONS OF RIGHT-ANGLED SPHERICAL TRIANGLES. 1. The legs are of the same kind as their opposite angles; and conversely. 2. The hypothenuse is less or greater than 90°, according as a leg and its adjacent angle, or the two legs, or the two angles, are like or unlike. 3. A leg is less or greater than 90°, according as its adjacent angle and the hypothenuse, or the other leg and the hypothenuse, are like or unlike. 4. An angle is acute or obtuse, according as its adjacent leg and the hypothenuse, or the other angle and the hypothenuse, are like or unlike. OTHER PROPERTIES OF RIGHT-ANGLED SPHERICAL TRIANGLES. 1. If the hypothenuse be 90°, one of the legs and its opposite angle will be each 90°; and the other leg and angle will be measured by the same number of degrees. 2. And if a leg, or an angle, be 90°, the opposite angle, or leg, and the hypothenuse, will be each 90°;. and the other leg and angle will be measured by the same number of degrees. 3. If a leg be less than the hypothenuse, their sum will be less than 180°; and if it be greater than the hypothenuse, their sum will be greater than 180° (x). (x) Properties similar to this, and the following one, are given by Cagnoli, p. 244, Traité de Trig., and by Maskelyne, in his Introduction to Taylor's Logarithms; but are so expressed, that, ir G |