Dr. Deane on the Discovery of Fossil Footmarks. 381 



Therefore, the double area of any polygon is equal to the sum 

 of the products of its sides, save one, taken two and two, multi- 

 plied into the sines of the angles embraced by the sides forming 

 the products severally. 



This beautiful theorem is essentially that given by Hutton 

 under the article Polygonometry. It is remarkable, that obvi- 

 ously useful as it is, it is not to be found in any other of our 

 popular works on mathematics. The above demonstration is 

 more analytical and vastly more simple than that given by Hut- 

 ton. 



An interesting form for the mensuration of the regular polygon 

 will immediately result from the preceding by put- 

 ting 



a — b = c=...=j—k, and 



(«, b)~e, (a, c) = 2e, ...(«, k)=^(7i—l)e, &c., 

 71 denoting the number of sides, we have 



2P= ['sin.e-l-sin.2e+sin.3e-[- • • • -f-sin.(w-l)e 

 +sin.e+sin.2e-f- . . . +sin.(w-2)e 

 -fsin.e-l- . . . -l-sin.(?2-3)e 

 -f &c. 



+ sin.e 

 or 2P = [(?i — l)sin.e+(?i — 2)sin.2e-{- . . . -\-sm.(n — l)e] . a^. 



Fig. 2. 



Art. XIV. — On the Discovery of Fossil Footmarks ; by James 



Deane, M. D. 



A definite settlement of the priority of claims to the discovery 

 of footprints in the sandstone of Connecticut River, is due to the 

 parties preferring these claims, and to the cause itself. 



It is remarkable that these striking impressions, abounding in 

 almost every sandstone quarry, should have escaped observation 

 so long. It is true they have been noticed by many old quarry- 

 men in the service of the canal companies, and by others, but 

 without the slightest comprehension of their origin or charac- 

 ter. Although they were formerly seen, they were nevertheless 

 as much unknown to the learned world as when concealed in the 

 depths of the earth. The eye of science had not seen, nor the 



Vol. xLvii, No. 2.— July-Sept. 1844. 49 



