VOL. LXV,] PHILOSOPHICAL TRANSACTIONS. 653 



equal to the radius ; and the other part, lying without the circle, is equal to the 

 radius minus twice the portion lying between the side of the square, and the cir- 

 cumference of the circle ; or is equal to that part of the radius that lies between 

 the centre and the side of the square minus the remainder of the radius ; that 

 is CL is equal to the radius, and li = kg — '2mg ; or li = km — mg. fg being 

 parallel to bc, and consequently perpendicular to ic, must divide the chord lc 

 in two equal parts ; so that mc being equal to ke, lc must be equal to 2ke ; 

 but KE (by Eucl. 1. xiii, pr. 12, cor. 2. Clav.) is equal to ed ; therefore lc = kd 

 the radius. The side of the square ic, being equal to bc, is likewise equal to 

 NM ; but LC being equal to kg, the remaining part li must be equal to nk — mg ; 

 or to KM — MG. a.E.D. ~ ^ 



XXIX. On Polygons of the Greatest and Least Area, or Perimeter, inscribed 

 in a Circle, or circumscribed about a Circle. By S. Horsley, LL.D., Sec. 

 R.S. p. 301. Translated from the Latin. 



Theorem 1 . If a right line touch a circular arc intercepted by two tangents ; 

 then its segments intercepted by its point of contact and those tangents, will be 

 equal or unequal, according as the arc is equally or unequally divided by the 

 point of contact. And the greater or less segments of the arc (when unequally 

 divided) and of the right line, lie on the same side of the dividing point. — Thus, 

 if the right line bd, fig. I and 2, pi. 13, in the point e touch the circular arc 

 AEC, intercepted by the two tangents ab, cd. Then the right line bd will be 

 equally or unequally divided in the point e, accordingly as the arc aec is equally 

 or unequally divided at the same point e. So that, when ae = ce, then 

 BE = DE, as in fig. 1 ; but when ae is greater than ce, then be is greater than 

 DE, as in fig. 2. 



Theorem 1. Of all the right lines which touch a circular arc, and meet two 

 other tangents at the extremities of the arc ; that is the least which touches the 

 arc in its middle point. — Thus, of all the lines, ac, gh, touching the arc bed, 

 fig. 3, and intercepted by the two tangents bag, dhc, that ac is the least 

 which touches the arc in its middle point e. 



Theorem 3. Of all the polygons, of a given number of sides, and circum- 

 scribing a given circle, the equiangular one has the least perimeter. 



Theorem 4. Of all the polygons, having a given number of sides, and cir- 

 cumscribing a given circle, the equiangular one has the least' area. 



Theorem 5 and 6. Of all polygons, having a given number of sides, and in- 

 scribed in a given circle, the equilateral one has the greatest perimeter and area. 

 All which theorems Dr. Horsley demonstrates with his usual geometrical 

 rigour. 



