On the Analysis of Square Numbers. 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. 
—_ 2 6° 
Table of elevation and E 3 | & 
distances on the main| g | & el tél ¢ Bn 
ne of the Pennsyl-| 5 |; 2) . Die) 6, oa) &) eis 2 
vania rnal a =z 2 & B 3 21 a 5 Bd Boho kee 
provements. Lax g =| 2) § = sles i a 3 
| 8, S18| s| S| Ss) Slelalele ls 
SISlSiSizjalaitia ls lalela 
feet, 
Philadelphia. . . . 560) 237) 290|312|3321604/928/2327|1151| 904) 761) 680) 
miles * ; 
Mine ridge summit. . 52 323|270/248/228| 44\368/1767| 591) 344) 201) 1 
Columbia. . . 82] 30 53 gc 95|367|691,2090, 914) 667; 524) 443 
Middletown. . . . | 101} 49) 19 22 fs 314'638/2037| 861) 614) 471| 390 
Harrisbur. 110} 58}.28} 9 201302 616|2015} 839 449 
Duncan’s Island 125) 73 24) 15 272\596/1995| 819) 572) 4 
untingdon. . . . | 215/163; 324|1723| 547! 300! 157 
Hollidaysburg. 254|202|172)153|144/129| 39; 1399] 223). 24| 167) 248 
Blair’ Aes 2 Summit . 264 alt. 163 beset 19| 10 1176|1423'1566/1647 
obnstow: 291 See whe 1/166) 76) 37 mi| 2 su Paid 471 
Blairsville. fh ets aay ate .* 148) 224 
Freeport. . . . . | 866/314 4 OBS See ‘ena ei 81 
Pittsburg. . . . . | 395/343 tah 131! 104] 74| 29) 
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. a 
Art. X.—On the Analysis of Square Numbers, by A.D. Wurrter, 
Instructor of the Latin Grammar School, Salem, Mass. 
Tue following enipshion appertaining to square auinbert, 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 AB will also, contain the sum of two square 
Demonstration. Let A=a?+-b?, and B=ec?+d’, then, 
A-B=(a?-+5?)-(c? +d?)=(ae+bd)?+(ad—be)*[A]; or, . 
A'B=(a? +5? )-(c? +d?)=(ac—bd)? +(ad+bc)* [B.] Q.E.D. 
