146 
PYTHAGOREAN GEOMETRY 
older than the book itself ; thus one of the constructions for 
right angles, using cords of lengths 15, 36, 39 (= 5, 12, 13), was 
known at the time of the Tdittimya Samhitd and the Sata- 
patha Brdhmana, still older works belonging to the eighth 
century b. c. at latest. A feature of the Apastamba-Sulha- 
Sütra is the construction of right angles in this way by means 
of cords of lengths equal to the three sides of certain rational 
right-angled triangles (or, as Apastamba calls them, rational 
rectangles, i.e. those in which the diagonals as wmll as the 
sides are rational). The rational right-angled triangles actually 
used are (3, 4, 5), (5, 12, 13), (8, 15, 17), (12, 35, 37). There is 
a proposition stating the theorem of Eucl. I. 47 as a fact in 
general terms, but without proof, and there are rules based 
upon it for constructing a square equal to (l) the sum of two 
given squares and (2) the difference of two squares. But 
certain considerations suggest doubts as to whether the 
proposition had been established by any proof applicable to 
all cases. Thus Apastamba mentions only seven rational 
right-angled triangles, really reducible to the above-mentioned 
four (one other, 7, 24, 25, appears, it is true, in the Bâudhâ- 
yana S. S., supposed to be older than Apastamba) ; he had no 
general rule such as that attributed to Pythagoras for forming 
any number of rational right-angled triangles; he refers to 
his seven in the words ‘ so many recognizable constructions 
are there ’, implying that he knew of no other such triangles. 
On the other hand, the truth of the theorem was recognized in 
the case of the isosceles right-angled triangle ; there is even 
a construction for V2, or the length of the diagonal of a square 
with side unity, which is 
constructed as 
K 
11 1 N 
+ 3.4 ~ 3.4.34' 
of the side, and is then used with the side for the purpose of 
drawing the square on the side: the length taken is of course 
an approximation to /2 derived from the consideration that 
2.12 2 = 288 = 17 2 — 1 ; but the author does not say anything 
which suggests any knowledge on his part that the approxi 
mate value is not exact. Having drawn by means of the 
approximate value of the diagonal an inaccurate square, he 
proceeds to use it to construct a square with area equal to 
three times the original square, or, in other words, to con 
struct V3, which is therefore only approximately found.