OF PLANE TRIGONOMETRY. 
248 
,, i. t\/~i , , \ n I ^ Om x CF f Dm x FO 
that DC (=m»-bDy) will be zz 
Uv 
j „ tIS Ow / CF— Dm/FO r 
and BE (= mn — »H) = : from 
Uv 
whence it appears, that the sine (DC) of the sum (AD) 
of any two arches (AC and CD) is equal to the sum of 
the rectangles of the sine of the one into the co-sine 
of the other, alternately, divided by the radius; and 
that the sine (BE) of their difference (AB) is equal to 
the difference of the same rectangles, divided also by the 
radius. 
corol. 2. 
Moreover, seeing, DG f BE (2m») is — 
U w 
and DG — BE ( = DH±2D?;) —7^—-—* from the 
former of these, we have DG zz — BE, and 
from the latter, DG = F - + BE ; which, ex 
pressed in words, gives the following Theorems. 
Theor. l. Jf the sine o f the mean of three eqttidifferent 
arches (supposing the radius unityj be multiplied by 
twice the co-sine of the common difference, and the sine 
of either extreme be subtracted from the product, the re 
mainder will be the sine of the other extreme. 
Theor. 2. Or, i f the co-sine of the mean he multiplied 
ly twice the sine of the common difference,and the product 
be added to or subtracted from the sine of one of the 
extremes, the sum or remainder will be the sine of 
the other extreme. 
These two theorems are of excellent use in the con 
struction of the trigonometrical canon: for, supposing 
Vhe sine and co-sine of an arch of 1 minute to be found, 
by Prop. 6 and 1, and to be denoted by p and q, respec 
tively; then the sine of 2 minutes being given from 
Prop.4, the sine of3 minutes will from hence he known, 
being r 27 x sine 2' — sine 1' [by Theor. 1) or := 2p 
x co-sine of 2' F sine of 1' [by Theor. 2.) After the same