A History of Mathematics From Mesopotamia to Modernity

128 A History ofMathematics

then the product ofEAwithABtogether with the square onBFare equal to the square onAF. But

the product ofEAwithABis known, and the square onBFis known. Hence the square onAFis

known, and soAFalso is known, and if from it is subtractedBF, which also is known, there is left

the knownAB, that is the root. If we multiply it by another equal to itself, we find the squareABCD

is known. This is what it was required to show.

Appendix C. From al-K ̄ash ̄i,The Calculator’s Key, book 4, chapter 7

On the measurement of bodies with regular faces.

...

There are seven bodies. [Al-K ̄ash ̄i considers not only the usual five but two semiregular solids

(see Fig. 9) which have their faces all regular, and regularly arranged, but not all the same.]

Thefirstcontains four faces, which are equilateral triangles in the sphere, that is, it is the body

bounded by four equal equilateral triangles. It appears as a pyramid with a triangular base, and

is made up of four pyramids, whose bases are its faces, and whose vertices are at the centre. The

measurement of this is as follows: take the square on the diameter of the circumscribed sphere,

and find the root of two thirds of it, and also the root of half the square on the diameter, and the

first will be the side of the base, and the second the height of the triangular side. If we multiply

one of them by half the other, we find the area of one side. If we multiply this by two ninths of the

diameter of the sphere, we find the volume.

Another way. We multiply the diameter once by 0 48 59 23 15 41 fifths, and we obtain its side,

and another time by 0 42 25 35 3 53 fifths, and we obtain the height of the triangle. And do the

rest as before.

[The main relations =

√

2

3 ·d, of the side of the tetrahedron to the diameter of the sphere,

is to be found in Euclid XIII.13 and so ‘common knowledge’ among the savants at Samarkand;

which is presumably why al-K ̄ash ̄i feels there is no need to prove it. As has been said, his book is

an exposition of methods, not of proofs, although from his other works we know that he could

produce serious proofs when needed. As for the actual figures, in sexagesimals to ‘fifths’ (1/ 605 ,

or roughly 1.2× 10 −^9 ), they follow from the standard method, which he has set out earlier, for

extracting square roots; the first number is

√

2

3 and the second

√

1

2. It is interesting to compare

the second figure with the Babylonian version on the Yale tablet (Chapter 1, Fig. 6), which has the

Fig. 9Al-K ̄ash ̄i’s seven regular solids (the five ‘platonic’ solids of Chapter 2, and two ‘semiregular’ ones).