Mr. Siiarpe's Table of the Swi's Semi-diameter, 259 



But since the tangent varies faster than the arc, and the 

 arc varies as its complement, the cotang. varies faster than 

 the arc ; that is, increases faster than the arc decreases. Thus 

 we have in the present case, since 



^ < -^ and cot (-^) > cot (^) 



Whence the ratios of the areas, or, 



, cot( — ^ 



since the last factor is a fraction less than -^ gives us the 

 second side of the equation a fraction less than unity ; or we 



have — < 1. 



That is ai > a; or we deduce the conclusion that under 

 the same perimeter, the polygon with the greatest number of 

 sides has the greatest area. 



XXXVI. Tahleof the Sun's Semi-diameter. i?j/ S. Sharpe, 

 Esq. F.G.S. 



To the Editors of the Philosophical Magazine and Annals. 



Gentlemen, 



THE planetary perturbations which alter the earth's radius 

 vector are so small, as never, I believe, in the case of the 

 sun's semi-diameter, to amount to the tenth of a second, and 

 hence in that element are neglected. And a table of the sun's 

 semi-diameter, in which the argument is the mean anomaly, 

 is nearly perpetual. And in the same way a table, in which the 

 argument is the day of the mouth, will serve for any number 

 of years, if we do but make a small correction each year, de-^ 

 pendent on the mean anomaly at the beginning of the year. 

 Such a Table I have here constructed out ot lables I. and 

 XXIV. of Vince's Astronomy, vol. 3rd. And although the 

 sun's semi-diameter is always given in the Nautical Almanac, 

 2 L 2 sUll 



