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XLV. — On the Theory of CommensuraUes. By Edward Sang, Esq. 



(Read 7th March 1864.) 



The general proposition in the theory of commensurables is to determine the 

 conditions under which lines, surfaces, or solidities, connected with prescribed 

 figures or forms, may have their ratios expressible by integer numbers. 



The attention of geometers must have been drawn to this subject by the con- 

 templation of incommensurable lines : the altitude of an equilateral trigon is 

 incommensurable with the base ; the diagonal of a square incommensurable with 

 the side, and so on. And, when the two sides of a right angle are expressed by 

 two numbers, the hypotenuse is, in the great multitude of cases, incommensurable 

 with the sides : thus, if the sides be 5 and 7 inches respectively, the length of the 

 subtense cannot be accurately expressed either in integers or fractions. How- 

 ever, when the sides are 3 and 4 units, the hypotenuse is exactly 5 of the same 

 units. 



This circumstance seems to have turned the inquiries of geometers at a very 

 early period to the discovery of those cases in which the three sides of a right- 

 angled trigon. are all commensurable; and the computation of Pythagorean 

 numbers was to them a very interesting problem. 



Many problems of a similar nature may be proposed : thus we may require 

 that the three sides of a trigon, and the line bisecting one of the angles, be all 

 commensurable ; that the four sides and the two diagonals of a tetragon be all 

 expressed by integer numbers, and so on. 



The methods of solving such problems may be said to constitute the theory of 

 commensurables. 



Section 1. — On the Bight-Angled Trigon. 



1. If A and B represent the sides of a right-angled trigon, of which the hypo- 

 tenuse is C, we must have the equation 



and the question is to discover integer values of A, B, and C, which may satisfy 

 this equation. 



When it happens that two of these three numbers have a common divisor, the 

 third number must have the same divisor ; for if A and B had the common divi- 

 sor n, so that A — na, B = nb, A^ -|- B^ that is C^ would be divisible by n^; in other 

 " words, C would be divisible by n, and thus the case A, B, C might be reduced by 

 division to the lower numbers a, b, c. 



And conversely, when we have obtained one solution, as 3, 4, 5, we can thence 



VOL. XXIII. PART III. 9 H 



