A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE COLLECTION. BOOK VIII 439
Then he proves that, if we join A B, A B is equal to the length
of the side of the hexagon required.
Produce BC to D so that BD = BA, and join DA. ABD
is then equilateral.
Since EB is a tangent to the segment, AE.EC — EB 2 or
AE: EB = EB : EC, and the triangles EAB, EBC are similar.
Therefore BA 2 : BC 2 = AE 2 : EB* = AE'.EC = 9 : 4
and BC = %BA = §52), so that £6' = 2 CD.
But Ci^= 2C.4 ; therefore AC:CF= DC:CB, and 47), BF
are parallel.
Therefore Itf 7 : AD
= BC.CD = 2 : 1, so that
BF=2AD = 2AB.
Also £FBC= A BDA = 60°, so that ZARF= 120°, and
the triangle J.I?i^is therefore equal and similar to the required
triangle NLO.
Construction of toothed ivheels and indented screws.
The rest of the Book is devoted to the construction (1) of
toothed wheels with a given number of teeth equal to those of
a given wheel, (2) of a cylindrical helix, the cochlias, indented
so* as to work on a toothed wheel. The text is evidently
defective, and at the end an interpolator has inserted extracts
about the mechanical powers from Heron's Mechanics.
ALGEBRA:
XX
DIOPHANTUS OF ALEXANDRIA
Beginnings learnt
from Egypt.
In algebra, as in geometry, the Greeks learnt the beginnings
from the Egyptians. Familiarity on the part of the Greeks
with Egyptian methods of calculation is well attested. (1)
These methods are found in operation in the Heronian writings
and collections. (2) Psellus in the letter published by Tannery
in his edition of Diophantus speaks of the method of arith-
'
metical calculations used by the Egyptians, by which problems
in analysis are handled '
; he adds details, doubtless taken
from Anatolius, of the technical terms used for different kinds
of numbers, including the powers of
the unknown quantity.
(3) The scholiast to Plato's Charmides 165 E says that 'parts
of XoyiarTiKtj, the science of calculation, are the so-called Greek
and Egyptian methods in multiplications and divisions, and
the additions and subtractions of fractions '. (4) Plato himself
in the Laws 819 A-c says that free-born boys should, as is the
practice in Egypt, learn, side by side with reading, simple
mathematical calculations adapted to their age, which should
be put into a form such as to combine amusement with
instruction : problems about the distribution of, say, apples or
garlands, the calculation of mixtures, and other questions
arising in military or civil life.
r
'
Hau '-calculations.
The Egyptian calculations here in point (apart from their
method of writing and calculating in fractions, which, with
the exception of §, were always decomposed and written
as the sum of a diminishing series of aliquot parts or submultiples)
are the /iau-calculations. Hau, meaning a heap, is
the term denoting the unknown quantity, and the calculations