ORIGINATING THE CONSTRUCTION OF A HYDROSTATIC QUADRANT. 183 



CGEHI20, 



and according to our notation, it is shown that 



c B zr x ; 

 consequently, by subtraction, we obtain 



B G ~ x, and B G E rz: 20 x. 



But by the nature of the circle and the principles of Plane Trigo- 

 nometry, it is manifest, that BD is the versed sine of the arc BC ; 

 B H the versed sine of the arc B G, and B F the versed sine of the arc B G E ; 

 therefore, by re-establishing the respective symbols, we shall obtain 

 F H =r d =. vers. (20 x) vers. (0 a 1 ), 

 and by a similar subtraction, we get 



D H zz I =: vers.z vers. (0 x). 



Let both sides of this equation be multiplied by m, the co-efficient 

 of 8 in equation (140), and we shall have 



mS m {vers. a; vers. (0 x)} ; 

 consequently, by comparison, we obtain 

 vers. (20 x} vers<(0 x) ~ m{ vers.x vers. (0 x)}. 

 Now, in order to simplify the reduction of this equation, it will be 

 proper to substitute for the several versed sines of which it is com- 

 posed, their corresponding values in terms of the radius and cosines ; 

 for, by such a substitution we obtain 



cos. (0 x) cos. (20 x) = m {cos. (0 x) cos. a-} ; 



but by the arithmetic of sines, we have 

 cos. (0 #)i=:cos.0cos.a7 -f- sin. sin. a:, and cos. (20 ^)zzcos.20 



cos. x -)-. sin. 20 sinfjgj 



consequently, by substitution, we get 



cos. cos. x -f- sin.0sin.a: cos. 20 cos. x sin. 20 sin. z:zzwi{cos.0 



cos. a? -j- sin.0 sin. a: cos. a;} ; 



and from this, by transposition, we have 



mcos.x cos.20cos.a:-j-sin.20sm.,r-}-( 7W l)(cos.0cos.o;-4-sin.0sin.a:). 



Let all the terms of this equation be divided by cos. x, and we shall 

 obtain 



w=icos.20 -f- sin. 20 tan .x -}- (m l)(cos.0-|- sin.0 tan. a:) ; * 

 therefore, by separating and transposing the terms, we get 



* It is demonstrated by the writers cm Analytical Trigonometry, that the sine 

 divided by the cosine to the same radius, is equal to the tangent; hence we have 



sin.r 



tan. a zr . 



cos..r 



