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

THE COLLECTION. BOOKS VII, VIII 427
but the genuine proof is indicated, although it breaks off
before it is quite complete.
Since
or AB :
But, by hypothesis,
therefore
AF:FB = BG:GC,
AB:FB = BC:GC t
BC = FB :
GO.
AB:BC=BG:GC;
BF = BG.
From this point the proof apparently proceeded by analysis.
'
Suppose it done ' (y^yoverco), i.e. suppose the proposition true,
and BED perpendicular to AC.
Then, by similarity of triangles, AD :
DB = AB : BC
;
therefore AF'.FB—AD.DB, and consequently the angle
ADB is bisected by DF.
Similarly the angle BDC is bisected by DG.
Therefore each of the angles BDF, BDG is half a right
angle, and consequently the angle FDG is a right angle.
Therefore B, G, D, F are concyclic ; and, since the angles
FDB, BDG are equal,
FB = BG.
This is of course the result above proved.
Evidently the interpolator tried
to clinch the argument by
proving that the angle BDA could not be anything but a right
angle.
Book
VIII.
Book VIII of the Collection is mainly on mechanics, although
it contains, in addition, some propositions of purely geometrical
interest.
428 PAPPUS OF ALEXANDRIA
Historical 'preface.
It begins with an interesting preface on the claim of
theoretical mechanics, as distinct from the merely practical
or industrial, to be regarded as a mathematical subject.
Archimedes, Philon, Heron of Alexandria are referred to as
the principal exponents of the science, while Carpus of Antioch
is also mentioned as having applied geometry to ' certain
(practical) arts'.
The date of Carpus is uncertain, though it is probable that
he came after Geminus; the most likely date seems to be the
first or second century A. D.
Simplicius gives the authority of
Iamblichus for the statement that Carpus squared the circle
by means of a certain curve, which he simply called a curve
generated by a double motion. 1 '
Proclus calls him Carpus the
writer on mechanics (o firjxaviKos) and quotes from ',
a work of
his on Astronomy some remarks about the relation between
problems and theorems and the 'priority in order' of the
former. 2 Proclus also mentions him as having held that an
angle belongs to the category of quantity (ttoo-ou), since it
represents a sort of ' distance ' between the two lines forming
it, this distance being ' extended one way ' {k %v SiecrTcos)
though in a different sense from that in which a line represents
extension one way, so that Carpus's view appeared to be the
'
greatest possible paradox ' 3
;
Carpus seems in reality to have
been anticipating the modern view of an angle as representing
divergence rather than distance, and to have meant by eft ef
in one sense (rotationally), as distinct from one way or in one
dimension (linearly).
Pappus tells us that Heron distinguished the logical, i.e.
theoretical, part of mechanics from the practical or manual
(xtLpovpyiKov), the former being made up of geometry, arithmetic,
astronomy and physics, the latter of work in metal,
architecture, carpentering and painting ; the man who had
been trained from his youth up in the sciences aforesaid as well
as practised in
the said arts would naturally prove the best
architect and inventor of mechanical devices, but, as it is
difficult
or impossible for the same person to do both the necessary
1
Simplicius on Arist. Categ., p. 192, Kalbfleisch.
2
y
Proclus on Eucl. I, pp. 241-3. lb., pp. 125. 25-126. 6.