igo 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1675. 



which is n, by the right lines a, b, c, &c. (whose lengths are required) making 

 with the rectilineal diagonal RZ the angles «, |3, y, &c. Then RAa = 



^A, RAb = ^A, RAC=^A, &c. Let ARZ =/), and AZR = ^. Having 

 given therefore the legs r, z, with the contained angle A, and thence the sum 

 of the remaining angles p -\- q, the rest will be founds p obtuse, and q acute : 

 for z + r : z - r : : tang. t±l : tang. ^; and^ + ^ = j^. Then, 



knowing the angles p and - A, and thence the remaining angle a, with the in- 

 cluded side r; the side a will thence be found ; viz. sin. a : r : : sin. p : a. And, 

 in like manner, having known p and - A; p and - A; p and - A; &c., 

 there will be had b; c; d; &c. 



For example — Let r = 1 ; / = 0.2; z = 1.2; A = lO'. Therefore p-\- qz=. 



179° 50' ; and ^-—^ = 89° 55'. Then, as z + r = 2.2 : z — r = 0.2 : : 

 tang. ^^—^ = 687.5488693 : 62.5044427 = tang. ^-~; to which answers the 



angle 89° 5' O" M'" nearly. Therefore ^^ + ^^ =/) = 179° o' o" \f" 



nearly; the sine of which, or of 0° 59' 59" 43'", is 0.017451 1. Then, cutting 

 the angle A into 10 parts, each of them of l', there will then be found «, b^ c, r/, 

 €>f, Si K ii thus, as 



Sin. » (0° 58' 59" 43'") 0.017l603 : r = 1 



Sin. /3 (0 57 59 43 ) 0.0168694 : r = 1 



Sin. y (0 56 59 43 ) 0.0165780 : r = 1 



Sin. ^ (0 55 59 43 ) 0.0162877 : &c. 



Sin. i (0 54 59 43 ) 0.0159969 : 



Sin. C (0 53 59 43 ) 0.0157060 ; 



Sin. » (0 52 59 43 ) 0.0154152 : 



Sin. 6 (0 51 59 43 ) 0.0151243 : 



Sin. * (0 50 59 43 ) 0.0148335 : 



S'm.p = 0.0174511 

 Sin. p = 0.0174511 

 Sin. ^ = 0.0174511 

 &c. 



1.00000 = r 

 1.01694 = a 

 1.03448 = b 

 1.05264 = c 

 1.07144 = d 

 1.09091 = e 

 1.11110 =/ 

 1.13206 =g 

 1.15383 = h 

 1.17647 = i 

 1.20000 = z 



Difs. 

 1694 

 1754 

 I816 

 1880 

 1547 

 2019 

 2096 

 1177 

 2264 

 2353 



The Calculation otherwise. 



Let r = 1 ; / = 0.1 ; z = 1.1 ; A = lO'. Therefore /j + q = 179° 50', and 

 ^-^1=89° 55', whose tangent is 687-5488693. Then, as 2.1:0.1:: 



687.5488698 : 32.7404223-1- = tang. 88° 15' i" bf"\- = tang. ^-^. There- 



fore 



p + q 



+ 



lz=p = 178° 10' 1^/57' 



2 ' 2 



49' 58" 2'%, the sine of which is 0.03] 9827. Therefore, 



the supplement of which is 1^ 



