heir applicumericalecis fartherced, and the lained, varrotta proceeds to give the solutions of the various cases of right and oblique angled plane triangles, explains the utility of different solutions to the same case, and next investigates formulæ for the sines and cosines of multiple arcs. Some cu. rious and well-known properties of chords, equations, &c. by Vieta, Waring, De Moivre, and Cotes, are then exhibited; expressions for the powers of sines and cosines of arcs, and for the tangents of multiple arcs, are given; the construction of the trigonometrical canon is explained, various formulæ of verification are deduced, and the utility of trigonometrical formulæ is farther shewn, in the solution of particular numèrical equations of different orders, and in their application to some curious inquiries in physical astro-* nomy. These particulars occupy the first 85 pages. The second part is devoted to Spherical Trigonometry. Mr. Woodhouse presents definitions and the chief proposi. tigns in spherical geometry. He then finds the areas of spherical triangles and polygons, and investigates the principal formulæ of solution for right-angled spherical triangles. We have next a proof of Napier's rules for the “ circular parts", as they are termed ; remarks on quadrantal triangles, and on the affections of sides and angles. These are fol. lowed by the solution of oblique spherical triangles, by the explanation of Napier's "analogies”, and by the solution of the most useful problems that would occur in a large tri. gonometrical survey : here we have modes of reduction of observed to horizontal angles, the use of expressions for the area of a spherical triangle, remarks on the “ spherical ex-* cess”, Legendre's theorem for solving spherical triangles that are nearly plane, and the reduction of spherical angles to angles contained by the chords. 1 The last 44 pages are occupied by an appendix. Here the properties of logarithms are explained, and series for computing them given : the advantage of Briggs's over the Neperean system is shewn : tables of proportional parts are explained: expressions for cosines and sines of multiple arcs, series for sines and cosines of arcs, and for logarithmic sinese and cosines, are given : sines and tangents are computed by different methods, and demonstrations are exhibited of Legendre's formulæ of reduction, and of his theorem for soly. ing spherical triangles as though they were rectilinear. Having thus described the contents of Mr. Woodhouse's treatise, we shall make one or two favourable and useful ex., tracts, before we give our opinion of its general merits. For the 4th case of obliqué angled plane triangles, where the sides a, b, c, are given to find the angles A, B, C, our author form: vives four solutions. In the first, putting i.(a + b + c) = S, one of the angles, A for example is found by the following 6 .. . - log. sin. A - 10:=log ., I flog, $ + log. (5- a) + log. (8 5) + log, (S –.c}. y? Jog. b. --- log. c. By the second method, which depends upon an expression for half one of the angles, we have in 2 log. sin. - '20 4- log: (s 6) + log. (S - ¢) - log. b. --- log. c. . The third method gives . .. 2 log, cos. = 20 + log. S + log: (8 --.a) — log. 6— log.c.. In the fourth method. " 2 log. tan. = 20 + log: (8 — ), + log: (8 + c)' – log. S ii.i log: (S-a). -This variety of solutions, of course, is not new; it is fol. lowed; however, by some remarks which are worth the atten-, tion of the student. . . * Each of the preceding methods is adapted to logarithmic computation, and each, in an analytical point of view is a complete solution. One soor lution would have been sufficient, and one alone been given, if the same applied with equal convenience and equal numerical accuracy to all in, , stances; but the fact is otherwise. If an example were proposed in which the angle A should be nearly 90', as C is in the former example: the log, și n. A might be deduced from the first solution ; but from such value of the log. sin. A, the angle A cannot be determined with any precision : for instance, if the numerical value of log. sin. A should be 9 9999998, A might equal (by the tables) 89° 56' 19", or 89' 57'sil, or any angle intermediate of these two angles : the reason of this is, the very small variation of the sine of an angle nearly equal to 90', which is plain from the inspection of the geometrical figure, or which analytically may be thus shewn : let A be an arc nearly=90°, and let it be increased by a. certain quantity, 11' for instance, theri sin. (A +1")=sin. A cos. 1" tocos. A. sin. ?". : sin. A = sin. A ..sin. (A +11)-sin.A=sin.A(cos. )"-1)+cos. A. sin. " or since cos. l'=l nearly, and sin. 1"=0 nearly. sin. (A+1")-sin. A varies as cos. A nearly, varies... as a very small quantity, when A is nearly=909. • It must not, however, be unnoticed, that the want of precision in the determination of the angle is partly owing to the construction of the lo.. garithmic and trigonometrical Tables. The cables referred to, and in common use, are computed to seven places of figures; but if we had ta. bles computed to a greater number of places, to double the number, for instance, then the logarithmic sines of all angles between 89° 56' 18". and 89° 57' 9'', in such tables, would not be expressed, as they are in tables now in use, by the same figures : and in such circumstances, we should obtain conclusions very little remote from the truth; but then such tables would be extremely incommodious for use, would, in all common cases, give results to a degree of accuracy quite superfluous and useless ; and besides, such tables in the extreme cases which we have mentioned, are not essentially necessary : since in those extreme cases their use can be superseded, by abandoning the first method of solution, and recurring cither to the 2d, 3d, or 4th method. . • When the angle (A) sought then is nearly = 90°, the first method must not be used, but one of the latter methods, in which either the sine, cosine or tangent of half the angle is determined ; and in such an extreme case, it is a matter of indifference whether instead of the first method, we substitute the 2d, or 3d, or 4th ; but in other cases, it is not a matter of indifference: for since, as it has been shewn, the variation or increment of the sine is as the cosine, and of the cosine as the sine, these two variations are equal at 45*, but beyond 459. up to 90', that of the sine is less, that of the cosine greater, and the contrary happens between 45° and 0 ; consequently we have this rule : if the angle sought be <90°, use the second method ; if -... .................. >90°, use the third method. The 4th method may be used and commodiously, for all values of the angles sought from O up to angles nearly=180°: when, however, the angle (A) is nearly=180°, tan. which is nearly tan. 90°, is very large and its variations, (which are as the square of the secant) are also very - large and irregular. If, therefore, we use Sherwin's Tables, which are computed for every minute only of the quadrant, the logarithms corresponding to the seconds, taken out by proportional parts, will not be exact : for in working by proportional parts, it is supposed that if the difference between the logarithmic tangents of 2 arcs differing by 60 seconds be d, that the difference between the logarithmic tangents of the first arc, and of another arc, that differs -from it only by a seconds is to id: now this is not true for arcs nearly 90°; and an example will most simply shew it: by Sherwin's Tables : i. lo. ... log. tan. 89° 30=12.0591416 log. tan. 890 29=120449004....... log. diff. corresponding to 60" = 142412 diff. corresponding to 30" 71206......................[47 '. :.. by Rule, log. tan, 89° 29' 30" ([2]+[4]) = 12.0520210 whereas true log. tan. 89. 29' 30'', by Taylor's log. =12.0519626 "log. tan. 89. 50=12 5362727 log. tan. 89. 49'=12.4948797 "' log. diff. corr8 to 601 4 13930 ... diff. corr8 to 8" 41393 .. . , " " .. by Rule log: tan, 89. 49' 61512.4990190 whereas true log. tán. 89, 49' 6', by Taylor's Logo 12.498845. ( la these instances the log. tangeot, determined by the proportional, 2.0591416 · again, 12000! parts, is too large, which it plainly must be, for the logarithmic in- . : J... diff. corresponding to 30=12635. . .. by Rule log. tan. 44• 29' 30'=9.99229335 v. and the true log. tan. by Taylor's Tables=9. 9922934 • It appears then, from the assigned reason, and by the instances given, : that an angle nearly 90° cannot exactly be found from its logarithmic tangent. The determination of the angle by means of proportional parts will be wrong in seconds.by Sherwin's tables ; and will be wrong in the parts of seconds by Taylor's tables ; and in computing the values - of angles, two inconveniences may occur, either when the successive - logarithmic numbers are too nearly alike, as in the case of sines of-an- . gles nearly 90°, or too widely different, as in the case of the tangents of angles nearly equal to 90° : and it is the business, of the Analyst to provide formulæ, by which these inconveniences may be remedied or avoided.' It has appeared, that the 4th method of solution ought not to be vused, wh is nearly=900 : it must not be used also when A is a very small angle, for very small angles cannot be exactly found from their logarithmic sines and tangents ; not exactly in seconds, : by Sherwin's tables, nor exactly, in parts of seconds, by Taylor's tables; and therefore, as great exactness may be required, and is commonly required in those cases, in which a very small angle is to be determined, the tables are not then to be used : but a peculiar computation, of which, withoat demoosiration, Dr. Maskelyne has given the rule in his Introduction to Taylor's Logarithms, p. 17 and 22. ^ This rule and similar rales will be stated and demonstrated in a subsequent part of this work, when the analytical series for the side and tangent of an arc-are de. duced.' We should be pleased to lay before the reader two or three of Mr. Woodhouse's solutions of problems in Physical Astronomy : bilt these, though they are rather too concise -for the uninitiated student, are too long for our limits ; especially as we wish to extract our autbor's observations on the advantages of Briggs's over Napier's system of logarithmns, advantages which we do not recollect having seen so fully stated by any other author.. • Napier's system, in which, (m1) - I ( e l)? + }(e-13 - &c. I fe= base] is apparently, so very simple, that there must exist some substantial reason for the adoption of Briggs's. Now, in this fattet system, the logarithm of 10 is 1, the logarithms of 100 or 10, of 1000 or 10?, &c, are 2, 3, &c. respectively '; consequently, the logarithm (L) of a number N being known, the logarithms of all num N bers corresponding to N X 10m or ion can be expressed by an al teration of the simplest kind in L: thus, if the logarithm of 2.7342 be 4368144, the logarithms of the numbers 27. 311, 273. 41, 2734. 1, 27341, 273410, are 1.4368144, 2.4368114, 3.4368144, 4.4368144, 5.4368144, that is, these latter logarithms are formed from the first by merely prefixing to the decimal, 1, 2, 3, 4, 5, which are called characteristics, and which characteristics are always numbers one less than the number of the figures of the integers in the numbers whose loga.' rithms are required : the reason is this, 27.341 = 10 x 2 7:41:.: "log. 27-541 = log.' 10 t. log. 2.7341- '1.4368144 2734.1 1000 x 273411.. log..2734/ 1 log: 1000-4 fog 27341 = 3.1369141 and generally log, 10 X N =log. 10n + log. Nam +L aod, similarly, it is plain, that the logarithms of I to, · May 12.73411. 2.7341.44; 27341 2.7341 . .. the pr i so t ning Site main, that is, of. - de bongs vir 10 :1916 400 W 1000.. 10,000 ': : : 28' 27341, 5X.027341,'*.0027341, 00027341, ** ; must be the logarithm of 2.7941, or .4368144, subtracting, respectively, the numbers 1, 2, 3, 4, which subtraction, it is usual thus to indicate 14368144, M: 2.4368144, 3.4368144, 4.4368144 The logarithm of a number (N), then, being inserted in the tables, it is needless to insert the logarithms of those numbers that can be formed by multiplying or dividing N by 10 and powers of 10: , Hence we are enabled to contract the size of Logarithmic Tables : and this advantage is peculiarly connected with the decimal system of nota. tion. If there had been in common use scales of notation, the roots of which were 9, or 7, or 3:[then the inost convenient systems of logarithms would have been those, the bases of which, are, 9, 7, 3, respectively; for in such cases, having computed the logarithm of any number N, we could immediately, by means of the characteristics, assign the logarithm of any number represented by 91 * Nora [root ='9}, which numbers would, analogously to the present method, be denoted by merely altering the place of the point or comma that separates integers from fractions : The root then in the scale of. notation ought to determine the choice of the base in a system of logarithms : we may construct logarithms with a base = 3, and then, having computed the logarithm L of a number N, * N . the logarithms of all numbers corresponding to $" *iN, and would |