On the Analysis of Square JS'umbers. 



87 



the differences in level between them. It also gives the distances of 

 the same points from Philadelphia and from each other. The dis- 

 tances are measured by the route of the public works, and the levels 

 are those of the canal or rail road at the points named. 



The mode of using the table will be evident on examination. 



Table of elevation and 

 distances on the main 

 line of the Pennsyl- 

 vania Internal Im- 

 provements. 



Philadelphia. . . 



Mine ridge summit. 

 Columbia. . . 

 Middletown. 

 Harrisburg. 

 Duncan's Island. 

 Huntingdon. 

 Hollidaysburg. 

 Blair's Gap Summit 

 Johnstown. ' 

 Blairsville. . 

 Freeport. 

 Pittsburg. . . 



It was my intention to have added to this some account of the 

 Branch canals, but it would increase the length of my communication 

 too much, and must be deferred for the present. They are of great 

 extent and importance. 



Art. X. — On the Analysis of Square JVumbers, by A. D. Wheeler, 

 Instructor of the Latin Grammar School, Salem, Mass. 



The following properties, appertaining to square numbers, are cu- 

 rious, and of frequent use in Analytical investigations. Several of 

 them have been demonstrated by Fermat and Euler, but the demon- 

 strations here given are in my opinion, more simple and direct. 



1. If A and B contain each, the sum of two square numbers, 

 their product A'B will also, contain the sum of two square numbers. 

 Demonstration. hetA=a^+b-, and B^c^-ft/^, then, 

 A-'B = {a^ -\-b^y{c" i-d^) = {ac+bd)'' -{-{ad- bey [A] ; or, 

 A-B={a^ +b^y{c' +d')={ac-bdy -\-{ad+bcyiB.] Q.E.D. 



