185 
of Edinburgh, Session 1863-64. 
of two cubes never can be a cube ; and yet one after another of the 
most eminent mathematicians have tried, and, though foiled, have 
again and again essayed the proof of Fermat’s negation. So it 
may also appear to be a matter of indifference whether or not we 
can construct a four-sided figure which may have its four sides, 
and also its two diagonals, all integer multiples of the linear unit ; 
and yet such inquiries- present to the mathematician attractions 
sufficiently powerful to balance those of more practical investi- 
gations. 
Nor is the labour bestowed on the cultivation of such subjects 
altogether or in any degree lost, since the various branches of 
science are so interwoven, that we cannot improve -our acquaintance 
with one without augmenting our knowledge of those allied to it. 
The first part of the paper is occupied with the subject of the 
orthagonal trigon, and is a collection of previously known proposi- 
tions, the novelty, if any, being in the arrangement. 
The second powers of numbers form the only exception to Fer- 
mat’s Theorem ; the sum of two squares may be a square number ; 
that is to say, the equation 
-j- h- ~ 
is pqssible in integer numbers ; or, in other words, the altitude, the 
base, and the hypotenuse of a right-angled trigon may all be ex- 
pressed in integers. 
Among the remarkable properties of these Pythagorean numbers, 
as they are often called, are that, when in the lowest terms, one or 
other of the two sides is divisible by 3 ; that one or other of the 
two sides is divisible by 4 ; and that one of the three is divisible 
by 5. These three propositions are all exemplified in the well- 
known solution, 
It is also a very singular property, that the hypotenusal number 
can never be a multiple of 7, of 11, of 19, or, in general, of any 
prime number of the form 4n~ 1, unless the other also be so; and 
thus that no prime number of that form can ever be a divisor of 
the hypotenuse when the trigon is in its lowest terms. 
And as a companion, we have this other property, that every 
prime number of the form 4n-f- 1, and every product of such prime 
factors, may be the hypotenuse of a right-angled trigon. 
