THE ‘THEOREM OF PYTHAGORAS’ 
147 
ions for 
13), was 
le Sata- 
) eighth 
i-Sulba- 
y means 
rational 
rational 
. as the 
actually 
There is 
j fact in 
js based 
i of two 
bs. But 
her the 
cable to 
rational 
entioned 
Baudha- 
ihad no 
forming 
refers to 
[.ructions 
riangles. 
mixed in 
i is even 
a square 
3.4.34' 
irpose of 
of course 
bion that 
anything 
approxi- 
is of the 
}uare, he 
equal to 
I, to con- 
y found. 
Thus the theorem is enunciated and used as if it were of 
general application; there is, however, no sign of any general 
proof ; there is nothing in fact to show that the assumption of 
its universal truth was founded on anything better than an 
imperfect induction from a certain number of cases, discovered 
empirically, of triangles with sides in the ratios of whole 
numbers in which the property (1) that the square on the 
longest side is equal to the sum of the squares on the other 
two was found to be always accompanied by the property 
(2) that the latter two sides include a right angle. But, even 
if the Indians had actually attained to a scientific proof of 
the general theorem, there is no evidence or probability that 
the Greeks obtained it from India; the subject was doubtless 
developed quite independently in the two countries. 
The next question is, how was the theorem proved by 
Pythagoras or the Pythagoreans ? Vitruvius says that 
Pythagoras first discovered the triangle (3, 4, 5), and doubtless 
the theorem was first suggested by the discovery that this 
triangle is right-angled; but this discovery probably came 
to Greece from Egypt. Then a very simple construction 
would show that the theorem is true of an isosceles right- 
angled triangle. Two possible lines are suggested on which 
the general proof may have been developed. One is that of 
decomposing square and rectangular areas into squares, rect 
angles and triangles, and piecing them together again after 
the manner of End., Book II; the isosceles right-angled 
triangle gives the most obvious case of this method. The 
other line is one depending upon proportions; and we have 
good reason for supposing that Pythagoras developed a theory 
of proportion. That theory was applicable to commensurable 
magnitudes only; but this would not be any obstacle to the 
use of the method so long as the existence of the incom 
mensurable or irrational remained undiscovered. From 
Proclus’s remark that, while he admired those who first 
noticed the truth of the theorem, he admired Euclid still 
more for his most clear proof of it and for the irrefutable 
demonstration of the extension of the theorem in Book VI, 
it is natural to conclude that Euclid’s proof in I. 47 was new, 
though this is not quite certain. Now VI. 31 could be proved 
at once by using I. 47 along with VI. 22; but Euclid proves 
L 2