Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . I. 44. cor.) As rad ; j tans:. PV : :tang. GC + tang. MD : cof. GPV or MPV. P R O B. CLII. 04. In the fpherkal triangle ABC, we have given the an-gles ADC, CDB, BDA, about the point D ; andthe triangle ABC itfelf\ to find the diftances AO,BO, CO. Let j, c be the fine and cofine of A ; p% qzzfineand coi. B, , , , , £=cof. AB, »=cof. ADB, x9 -z;:zfi
Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . I. 44. cor.) As rad ; j tans:. PV : :tang. GC + tang. MD : cof. GPV or MPV. P R O B. CLII. 04. In the fpherkal triangle ABC, we have given the an-gles ADC, CDB, BDA, about the point D ; andthe triangle ABC itfelf\ to find the diftances AO,BO, CO. Let j, c be the fine and cofine of A ; p% qzzfineand coi. B, , , , , £=cof. AB, »=cof. ADB, x9 -z;:zfine andcof. DAB; yi z zz fine and cof. DBA. Then (Trig. I. 6.) sv—, and pz— , and in the triangles CAD, BAD, h : j dsv-dcx d : : sv—ex : —-— 3= \ and a : /: : pz — . fP^fV = ; therefore adsv — adcx =a hfpz — bfqy. In the triangle ADB (Cafe 10. Trig.) bxy—vzzzn*But v zz y/i — a*, 2 zz y/i —^ thereforeadss/i — xx — adcx zz hfps/1 —yy — bxy ;— \/i — xx X \/1 —yy — %» From thefe two equations, the roots may beeafieft found by problem xcv; otherwife it willafcend to a high equation : or if you pleafe youmay proceed by rule 5, proh. xcii. PROB,. ^fyL Sed. VIII. PROBLEMS. 443 P R O B. CLIII. FJg- Given the difference of the azimuths of three knownJlars -, to find their altitudes. Let Z be the zenith; A, B, C, the ftars. Since places are given, the triangle ABC will be gi-ven. Put j, c zz fine and cof. 4-ABC, a zz S/AB;£, nzztzoL AC, q = cof. AZC\ \ZB% ^; xyyzzfine and cof. - •, Then jy+^z=^z^, and sy— And in the triangles ABZ, CBZ, d : a : : z : — — d , and b : e : : v : — = S»CZ; and in the tri- b angle AZC (Trig. III. 32.) *~ + ST^Tz X *S 1—vv zz n\ and tranfpofing, V^i—zz K y qazve , r yi—vvzzn—jrr> and squaring, 1—»z — 2nqae . qqaaee , , , ot + vraz = nn ~r vz + T777 v * > and bd bbdd tranfpofmg, 1 — nn zz zz + vv —
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