123 
Dr . Maskelyne on a new Property, &c. 
the first or third quadrant of the circle being considered as 
positive, and those belonging to the second and fourth qua- 
drant, being of a contrary direction, as negative ; in like 
manner as the sines in the first semi-circle are considered as 
positive, and in the second semi-circle as negative ; and the 
cosines in the first and fourth quadrant are considered as 
positive, and in the second and third quadrants as negative ; 
they lying, in the second case, on the contrary side of the 
diameter passing through the point of ninety degrees, to what 
they do in the former. Hence it easily follows, that the tan- 
gent of any arch and of its supplement to the whole circum- 
ference, or 360 degrees, are equal and contrary to one another, 
or the one negative of the other. 
Let t, u , w, be put for the tangents of the three arches A, 
B, C respectively, and r for the radius, and o for the whole 
circumference. Then A -f- B -f- C == o , and C = o — A-f- B. 
By trigonometry, t, A-f- B = and the tang. C = tang, 
(o — A -j- B ) — — tang. A -J~ B, by what has been said above. 
Therefore t, A -4- 1, B 4- t, C or t 4- u 4- w = t -j- u — — 
1 1 1 1 r z — t u 
= tu x — » —i— t u ; but t and u are the expressions for the tan- 
gents of A and B respectively, and — * s ^ ie ex pression 
for the tangent of C, or for w. Therefore, r' 1 x t -j- u -f- w, or 
the square of the radius multiplied into the sum of the three 
tangents of A, B, and C —tuw,or the product of the tangents. 
Q. E. D. 
