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

THE PAPYRUS OF AKHMIM. PSELLUS 545
(3) subtractions such as: From § subtract iVttiVAhis
1111111. Ill 1 1 1 An«wpr JL JL
40 44 5 55 60 66 7 7 7 88 9 95 100 110* auBVVC1
) 10 50'
The book ends with long tables of results obtained (1) by
multiplying successive numbers, tens, hundreds and thousands
up to 10,000 by §, -§, i, J, |, &c, up to ^, (2) by multiplying
all the successive numbers 1, 2, 3 ... n by -> where n is successively
11, 12, ... and 20; the results are all arranged as the
sums of integers and submultiples.
The Geodaesia of a Byzantine author formerly called, without
any authority, Heron the Younger ' ' was translated into
Latin by Barocius in 1572, and the Greek text was published
with a French translation by Vincent. 1 The place of the
author's observations was the hippodrome at Constantinople,
and the date apparently about 938. The treatise was modelled
on Heron of Alexandria, especially the Dioptra, while some
measurements of areas and volumes are taken from the
Metrica.
Michael Psellus lived in the latter part of the eleventh
century, since his latest work bears the date 1092. Though
he was called ' first of philosophers ', it cannot be said that
what survives of his mathematics suits this title. Xylander
edited in 1556 the Greek text, with a Latin translation, of
a book purporting to be by Psellus on the four mathematical
sciences, arithmetic, music, geometry and astronomy, but it
evident that it cannot be entirely Psellus's own work, since
the astronomical portion is dated 1008. The arithmetic contains
no more than the names and classification of numbers
and ratios. The geometry has the extraordinary remark that,
while opinions differed as to how to find the area of a circle,
the method which found most favour was to take the area as
the geometric mean between the inscribed and circumscribed
squares; this gives tt =