340 PHILOSOPHICAL TRANSACTIONS. - [aNNO J753. 



degrees and parts of degrees as are intended to be put on the scale, from the 

 logarithm versed sine of ] 80" ; then the remainder taken from the foresaid scale 

 of equal parts, and laid successively from the termination of this line, will give the 

 several divisions sought. 



Hence it appears, that the least versed sine, which can be introduced within 

 the length of a double radius, falls between 10" and 20°, where the index changes 

 from 1 to 2 ; which will happen about 11° 28'. 



If a table of logarithm versed sines to 180° are wanting, they are easily made 

 by the following rule : Take the logarithm sine of 30° from twice the logarithm 

 sine of (n) any number of degrees ; the remainder is the logarithm versed sine of 

 (2n, or) twice those degrees." For it is a well-known goniometrical property, 

 that the sine of any arc (a), is a mean proportional between radius (r) and half 

 the versed sine of twice that arc. 



Therefore, putting v for the versed sine, and ,s for the sine ; 



thenu 2a = (' — = ^^a X =ssa x tV=) "^^ X i; radius being 10. 



Or the log. z; 2a = 2 log. s\ — log. 5. 



But when radius is 10, the sme of 30° is 5. 



Therefore the log. i'2a = 2 log. *a — log. sine of 30°. 



Most of the writers on this subject give the following rule for laying down the 

 divisions of this line : From the line of logarithmic sines, take the distance be- 

 tween 90° and any arc ; that distance being twice repeated, from the termi- 

 nation of the line of versed sines, will give the division for twice the complement 

 of that arc." Thus the distance between 90° and 20° on the sines twice repeated, 

 gives the versed sine of 140°; or twice 70°, the complement of 20°. For the 

 divisions, to be laid on this line, are the difTerences between the logarithm versed 

 sine of 180°, and the logarithm versed sines of the successive arcs. 



Now the difference between the logarithm versed sines of 1 80°, and of any arc 

 2a, is log. ver., sine 180 — 2 log. sin, a -|- log. sin. of 30°. 



Or, 10,30103 4- 9,69897 — twice log. sin. of a. 



Or, 20,00000 — twice logarithm sine of a. 



Or the arithmetical complement of twice logarithm sine of a. That Is, the 

 difference between the logarithm versed sine of 180°, and the logarithm versed 

 sine of any arc, is equal to double the arithmetical complement of the logarithm 

 sine of half that arc, rejecting the indices. 



But, as these differences give the divisions to the supplements of the real versed 

 sines ; therefore the arithmetical complement of the logarithm sine of any arc 

 being doubled, will give the distance of the division for the supplement of twice 

 that arc on the line of versed sines. 



Thus, for 70°, the logarithm sine is 9397299 



