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IX. 0 « a nezv Property of the Tangents of three Arches trisect- 
ing the Circumference of a Circle, by Nevil Maskelyne, D. D. 
F.R. S. and Astronomer Royal. 
Read February 18, 1808. 
JVIr. William Garrard having shewn me a curious property 
of the tangents of the three angles of a plane triangle, or in 
other words, of the tangents of three arches trisecting a semi- 
circle, in a paper which I have communicated to this Society, 
I was led to consider whether a similar property might not 
belong to the tangents of three arches trisecting the whole 
circumference ; and, on examination, found it be so. 
Let the circumference of a circle be divided any how into 
three arches A, B, C ; that is, let A B -j- C be equal to the 
whole circumference. I say, the square of the radius mul- 
tiplied into the sum of the tangents of the three arches A, B, C, 
is equal to the product of the tangents multiplied together. I 
shall demonstrate this by symbolical calculation, now com- 
monly called (especially by foreign mathematicians) analytic 
calculation. 
It may be proper to premise, that the signification of the 
symbolical expressions of the tangents of an arc, whether with 
respect to geometry or numbers, are to be understood ac- 
cording to their position as lying on one side, or the other 
side of the radius, passing through the point of commence- 
ment of the arc of the circle ; those tangents which belong to 
