1864.] 415 f^®*- 



obituary notice of Mr. Quinsy was received from Mr. E. 

 Everett, dated Boston, August 2Ttb, 1864. On motion Mr. 

 Everett was excused. 



Photographs for the Album were received from Dr. W. S. 

 "VV. Ruschenberger, Professor W. Chauvenet, Dr. L. Stro- 

 meyer, and Dr. Isaac Hays. 



Letters of acknowledgment were received from the Asiatic 

 Society of Bengal, Calcutta, October 3d, 1863; the Corpora- 

 tion of Harvard College, August 22d, and Captain Gilliss, 

 Washington, August 20th, 1864. 



Letters of envoi were received from the Smithsonian Insti- 

 tution, and Mr. J. W. Irwin, of New York City. 



Donations for the Library were received from Dr. Stro- 

 meyer ; the Annales des Mines ; the London Society of 

 Antiquaries; Harvard College; Silliman's Journal; the 

 Brooklyn Mercantile Library Association; Messrs. Blan- 

 chard k Lea, and Mr. Eli K. Price, of Philadelphia. 



Mr. Lea made a communication of a discussion of " Prime 

 Right- Angled Triangles and \/2," from a private letter ad- 

 dressed to him from Dr. James Lewis, of Mohawk, New 

 York. 



Prime Right-Angled Triangles, and v/2. 



In any R. A. Triangle, let H = hypothenuse, P i= perpendieu- 

 lar, B = base. 



Then H^=P^+B^; whence H^— P^=B^ H^— P^ is the pro- 

 duct of two factors, H+P (=a) and H — P (=b). Accordingly, 

 H^— p==(H4-P)x(H— P)=ab=B^ and v/ab=B. 



a + b = (H+P)-f(H— P) =2 H and^A=H. 



a_b = (H-t.P)— (H— P) =2 P and-^^=:P. 



The radical sign before ab implies that the terms a and b are 

 squares; the fractional expressions ^— - and ^^ implies that those 



terms should be multiplied by 2. Substituting for the terms a and b, 

 others that meet these indications, viz., a=2N- and b=2S', the sides 

 of R. A. Triangles have the following general expression : Hz=N^+ 

 S^, P=N2— 8^=, B=2NS. If the terras N and S be any whole num- 

 bers, their expansion as indicated in the formula, will evolve the sides 

 of Prime Right-Angled Triangles (prime, in the sense that the tri- 



