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

, . a
ANTHEMIUS 543
the focus to the intersection of two tangents bisects the angle
between the straight lines joining the focus to the two points
of contact respectively.
In the third portion of the fragment Anthemius proves that
parallel rays can be reflected
to one single point from a parabolic
mirror of which the point is the focus. The directrix is
used in the construction, which follows, mutatis mutandis, the
same course as the above construction in the case of the ellipse.
As to the supposition of Heiberg that Anthemius may also
be the author of the Fragmentum mathematicum Bobiense, see
above (p. 203).
. The Papyrus of Akhmvm.
Next in chronological order must apparently be placed
the Papyrus of Akhmlm, a manual of calculation written
in Greek, which was found in the metropolis of Akhmim,
the ancient Panopolis, and is now in the Musee du
Gizeh. It was edited by J. Baillet l in 1892. According
to the editor, it was written between the sixth and
ninth centuries by a Christian. It is interesting because
it preserves the Egyptian method of reckoning, with proper
fractions written as the sum of primary fractions or submultiples,
a method which survived alongside the Greek and
was employed, and even exclusively taught, in the East. The
advantage of this papyrus, as compared with Ahmes's, is that
we can gather the formulae used for the decomposition of
ordinary proper fractions into sums of submultiples. The
formulae for decomposing a proper fraction into the sum of
two submultiples may be shown thus
1 1
0) t=-t-t. +
be b + c
7
b + c
c . b .
a
a
_ 2 11 3 1 1 18 11
Examples — = — »
=
? =
F
11 666 110 7077 323 34 38
, , a 1 1
am
be b + mc , b + mc 1
c. b.
a
•
1
Memoires publies par les membres de la Mission archeologique frangaise
au Caire, vol. ix, part 1, pp. 1-89.
544 COMMENTATORS AND BYZANTINES
E
X
*
1 1
J^
1^ 3
176 ~" /16 + 3 . 11\ + /16 + 3. 11\ 1~ = 77 + 112'
U (—7^-)
16 (-^)3
3 1 1 11
and again — = H =
6
112 /16 + 2.7\ /16 + 2.7\1 70 80
a 1 1
1
/3) ==
7 (-3—)
16 (— r~)i
.
'
cdf cd + df cd + df
f
a
a
Example.
28 28 1 111
+
1320 10.12.11" 120 + 132 120+132 90 99
'
28~
'
28
The object is, of course, to choose the factors of the denominator,
and the multiplier m in
(2), in such a way as to make
the two denominators on the right-hand side integral.
When the fraction has to be decomposed into a sum of three
or more submultiples, we take out an obvious submultiple
first, then if necessary a second, until one of the formulae
will separate what remains into two submultiples. Or we
take out a part which is
not a submultiple but which can be
divided into two submultiples by one of the formulae.
For example, to decompose -^j^. The factors of 61 6 are 8.7 7
or 7 . 88. lake out gg, and ^T e
= gg 6 T6 - = 8 8 7 7 = 88 77 11 5
and T
2
T = eV A by formula (1), so that ^ = -£? TV is A
Take ^V The factors of 6460 are 85.76 or 95.68. Take
2 3 9
out q-
1^, and 6 4 6 o
= st ihtwo
• Again take out 93-, and we have
ws 9*5 eif or •
is 9V is
r^ie actual problem here is to find
which latter expression reduces to
3^3 rd of HJ^to eV,
20 • ^39.
The sort of problems solved in the book are (1) the division
of a number into parts in the proportion of certain given
numbers, (2) the solution of simple equations such as this:
From a certain treasure we take away j^th, then from the
remainder y7 th of that remainder, and we find 150 units left;
what was the treasure? \ =.R.