Note, The logarithms found according to this method, in numbers between 10000 and 20000, arë true to 8 or 9 places of figures: thofe of numbers between 20000 and 50000 err only in the 9t or 10th place; and those of above 50000 are true to 10 places, at least. th Having explained the manner of conftructing a table of logarithms, and that by various methods, I now come to fhew the ufe of fuch a table in the bufinefs of trigonometry. First, in the rightangled plane triangle ABC, let there be given the hypothenufe AC = 17910 feet, and the angle A = 35° 20′; to find the perpendicular BC and the bafe AB. : A Here, because radius ; line 35° 20′:: 17910:BC (by Theor.2.p.6.) we have BC —fine 35° 20′×17910 radius therefore, because the addition and fubtraction of logarithms anfwers to the multiplication and divifion of the natural numbers (See p. 38, 39.) we have log. BC=log. fine 35° 20′ + log. 17910— log. radius. But, by the tables of artificial, or logarithmic, fines*, the log. fine of 35° 20′ will appear to be 9,7621775; to which add 4,2530956, the log. of 17910, and from the fum (14,0152731) take 10, the log. of radius, and there refults 4,0152731 the log. of BC; which, in the tables, anfwers to 10358, the length of BC required. A table of artificial fines is nothing more than a table of the logarithms of the numbers expreffing the natural fines, to the radius 10050000000; whofe logarithm is 10. E 2 Again, B Again, for AB, it will be, as radius: fine of C (54° 40′): ; AC (17910) : AB (by Theorem 2.) Whence, by adding the logarithms of the fecond and third terms together and fubtracting that of the first (as above), we have AB 14611. See the operation, Which 14° added to 66', the half fum of the angles C and B, gives the greater C = 80°; and fubtracted therefrom, leaves the leffer B = 52°. 4 Lastly, Lastly, in the rightangled fpherical triangle ABC, let there be given the hypothenufe AC = 60°, and the angle A= 23°29'; to find the bafe and perpendicular. Then B (by Theor. 1. p. 25.) the operation will be as follows: Having exhibited the manner of refolving all the common cafes of plane and fpherical triangles, both by logarithms and otherwife; I fhall here fubjoin a few propofitions for the folution of the more difficult cafes which fometimes occur; when, inftead of the fides and angles themselves, their fums, or differences, &c. are given. PROPOSITION I. The fine, co-fine, or verfed fine of an arch being given, to find the fine and co-fine, &c. of half that arch. 1 E 3 From B E Q D C F From the two extremes of the diameter AB, let the chords AE and BE be drawn, and let the radius CQ_bifect A AE, perpendicularly, in D (Vid. 1. 3.); then I will AD be the fine, and CD the co-fine, of the ACE. angle ACD, or But 4AD AE2 (by Cor. 1. to 6. 2.) = AB x AF (by Cor. to 19. 4.) 2ACxAF; whence AD2 AC × AF: alfo 4CD BE ABX BF = 2AC BF; whence CDAC × BF. From which it appears, that the Square of the fine of half any arch, or angle, is equal to a rectangle under half the radius and the verfed fine of the whole; and that the fquare of its co-fine is equal to a rectangle under half the radius and the verjed fine of the fupplement of the whole arch, or angle. PROP. II. The fines and co-fines of two arches being given, to find the fines, and the co-fines, of the fum and difference of thefe arches. Let AC and CD (=BC) be the two proposed arches ; ler CF and OF be the fine and cofine of the greater AC, and let mD (Bm) and Om, be thofe of the leffer CD (or BC): moreover, let DG and OG be the fine and co-fine of the fum AD; and BE and OE, thofe of the difference AB. Draw min parallel to CF, meeting AO in n; also draw mʊ and 9 and BH parallel to AQ, meeting GD in v and H: then it is plain, because Dm Bm, that Dv is = Hv, and munG= En; and that the triangles OCF, Omn and mDv are fimilar; whence we have the following proportions, OC: Om::CF:mn OC: OF:: Dm: Dv OC:OF::Om: On OC:CF:: Dm: my whence mn x OC Omx CF. Dvx OC Dm×OF. Now, by adding the two firft of these equations together, we have mn+ Dux OC (DG x OC) = Om x CF + Dm x OF; whence DG is known. Moreover, by taking the latter from the former, we get mn - Dv x OC (BE x OC)= Om x CFDmx OF; whence BE is known. In like manner, by adding the third and fourth equations together, we have On+mv x OC (OEx OC) = Om X OF + Dm x CF; and, by fubtracting the latter from the former, we have On-my x OC (OG × OC)=Om × OF Dm x CF; whence OE and OG are alfo known. 2. E. I. COROLLARY I. Hence, if the fines of two arches be denoted by S and s; their co-fines by C and c; and radius by R; then will the fine of their fum= Sc + sc R SC-SC the fine of their difference= the co-fine of their fum= R the co-fine of their difference = R Cc+ Ss: R |