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

PROCLUS. MARINUS 537
straight lines in a figure formed by taking a triangle with
sides 3, 4, 5 as ABG, then drawing
BD from the right angle B perpendicular
to AG, and lastly drawingperpendiculars
BE, BF to AB, BC.
A diagram in the text with the
lengths of the segments shown alongside
them in the usual numerical
notation shows that Paterius obtained from the data AB = 3,
BG = 4, CA = 5 the following
£Z) = |S/ lf'=2^ [=2|]
42>=«S«V = lHA [=lf]
w = /3 s *y = 2f
j i T 5 [=2|f]
^ = .^^=11^* [=i«]
BE=aSy' »? v' = 1H A A [
= Iff]
This is an example of the Egyptian method of stating fractions
preceding by some three or four centuries the exposition
of the same method in the papyrus of Akhmim.
Marinus of Neapolis, the pupil and biographer of Proclus,
wrote a commentary or rather introduction to the Data of
Euclid. 1 It is mainly taken up with a discussion of the
question ri to deSo/ievov, what is meant by given 1 There
were apparently many different definitions of the term given
by earlier and later authorities. Of those who tried to define
it
in the simplest way by means of a single differentia, three
are mentioned by name. Apollonius in his work on vevaei?
and his ' general treatise ' (presumably that on elementary
geometry) described the given as assigned or fixed (reTay-
\xkvov), Diodorus called it kno%vn (yvdopifiov); others regarded
it as rational {p-qrov) and Ptolemy is classed with these, rather
oddly, because ' he called those things given the measure of
which is given either exactly or approximately'. Others
1
See Heiberg and Menge's Euclid, vol. vi, pp. 234-56.
538 COMMENTATORS AND BYZANTINES
combined two of these ideas and called it assigned or fixed
and procurable or capable of being found (nopifioit); others
'
fixed and known and a third ', class known and procurable ' '.
These various views are then discussed at length.
Domninus of Larissa, a pupil of Syrianus at the same time
as -Proclus, wrote a Manual of Introductory Arithmetic eyx^piSiov
dpiOfj.r]TiKrjs elaayooyfjs, which was edited by Boissonade
1
and is the subject of two articles by Tannery, 2 who also left
a translation of it, with prolegomena, which has since been
published. 3 It is a sketch of the elements of the theory of
numbers, very concise and well arranged, and is interesting
because it indicates a serious attempt at a reaction against the
Introductio arithmetica of Nicomachus and a return to the
doctrine of Euclid. Besides Euclid, Nicomachus and Theon
of Smyrna, Domninus seems to have used another source,
now lost, which was also drawn upon by Iamblichus. At the
end of this work Domninus foreshadows a more complete
treatise oh the theory of numbers under the title Elements of
Arithmetic (dpiO/jirjTiKr} a-Toi^e'iadcns;), but whether this was
ever written or not we do not know. Another tract
attributed to Domninus ttco? eari \6yov e/c Xoyov afeXelv
(how a ratio can be taken out of a ratio) has been published
with a translation by Ruelle 4 ; if it is not by Domninus, it
probably belongs to the same period.
A most honourable place in our history must be reserved
for Simplicius, who has been rightly called ' the excellent
Simplicius, the Aristotle-commentator, to whom the world can
never be grateful enough for the preservation of the fragments
of Parmenides, Empedocles, Anaxagoras, Melissus,
Theophrastus and others' (v. Wilamowitz-Mollendorff). He
lived in the first half of the sixth century and was a pupil,
first of Ammonius of Alexandria, and then of Damascius,
the last head of the Platonic school at Athens.
When in the
year 529 the Emperor Justinian, in his zeal to eradicate
paganism, issued an edict forbidding the teaching of philo-
1
Anecdota Graeca, vol. iv, pp. 413-29.
2
Me'moires scientifiques, vol. ii, nos. 35, 40.
3
Revue des etudes grecques, 1906, pp. 359-82; Memoires scientifiques^
vol. iii, pp. 256-81.
4
Revue de Philologie, 1883, p. 83 sq.