380 0?i the Measure of Polygons. 



group is doubtless represented in the West by the upper beds of 

 blue limestone and marlite of Ohio, Indiana, Kentucky and Ten- 

 nessee, and the lower part of the magnesian lead-bearing rock of 

 Iowa and Wisconsin. The English equivalent must be some 

 part of the Caradoc sandstone. 



From the great change in the organic remains at the termina- 

 tion of this group, the New York geologists consider this the 

 line of division between the lower and higher portion of the 

 New York system. D. D. O. 



(To be continued.) 



Art. XTII. — On the Measure of Polygons ;-\)j Rev. George C. 

 Whitlock, Professor of Mathematics and Experimental Sci- 

 ence in the Genesee Wesleyan Seminary. 



Let a, b, c, . . .j, k, I, (fig. 1,) be the sides taken in order of 

 any polygon P. Divide the polygon into triangles, (A, b), (A, c), 

 (A, d), . . . (A, k), by diagonals drawn from pig. l. 



the junction A, of the first and last sides, to 

 the extremities of the other sides. From A 

 let fall the perpendicular p upon the produc- 

 tion of any side c : we then have 



2tri. (A, c)=cp. 

 But p is evidently the sum of the projections 

 of the sides a, b, preceding c, upon a line per- 

 pendicular toe; therefore jo = acos.(a,j9)-l-6cos.(6, jp)=asin.(a, c) 

 -}-6sin.(6, c) ; whence, 2tri. (A, c) = acsin.(a, c)-{-bcsm.{b, c). 



From what precedes it is obvious that the double areas of the 

 triangles constituting the polygon will be 

 2(A, b)=absm.{a, b). 

 2(A, c) = acsin.(a, c) + 6csin.6c. 

 2(A, d)=adsin.{a, d)-{-bdsm.{b, c?)-f-cc?sin.(c, d). 



&-C. •ifcc. &c. 



2{A,k)=aksm.{a,k)+bksm.{b,k) + cksin.{c,k) -{- . . . ■}-jksin,(j,ky, 

 .'.by addition there results for the double polygon, 

 2P = a6sin.(a,6) + acsin.(a,c)-l-ac?sin.(a,c?)4- • . . -{-aksin.{a,k) 

 -\-bcs\n.[b,c)-\-bdsin.{b,d)-\- . . . -^bksm.{b,k) 

 -\-cdsm.{c,d)-}- . . . -f cA;sin.(c,A;) 

 -{- &c. 



-\-jksm.U,k)' 



