340 NEW PROPERTY OF TANGENTS. 



IV. 



On a neti> Properfij of the Tangents of three Arches trisect" 

 ing the Circumference of a Circle, By Nevil Maske- 

 LYNE, D. D. F.R.S. and Astronomer Royal*. 



j8am of three IVIR. William Garrard having shown mc a curious pro- 

 tangents of perty of the tangents of the three angles of a plane trian- 

 three arches tri- \ . , , ^ , « . , 



secting a circle, gle, or in other words, of the tangents of three arches tn- 



multiplied by gecting ascmicircle, in a paper which I hare communicated 



radius, equal to 



their product, to this Society, I was led to consider, whether a similar pro- 

 perty might not belong to the tangents of three arches tri- 

 secting the whole circumference; and, on examination, 

 found it to be so. 



Let the circumference of a circle be divided any how into 

 three arches A, B, C; that is, let A -{- B + 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. 



Preliminary It maybe 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 

 according 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 the first or third quadrant of the circle being considered as 

 positive, atid 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 semicircle are considered as 

 positive, and in the second semicircle as negative ; and the 

 cosines in the first and fourth quadrant are considered as po- 

 sitive, 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 

 tht$y do in the former. Hence it easily followsj that the tan» 



ire mark. 



Ibid, p, 122, 



gent 



