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

198 SUCCESSORS OF THE GREAT GEOMETERS
quantity, it was only an abbreviation for
the word dpiOfios,
XV
THE SUCCESSORS OF THE GREAT GEOMETERS
With Archimedes and Apollonius Greek geometry reached
its culminating point. There remained details to be filled
in, and no doubt in a work such as, for instance, the Conies
geometers of the requisite calibre could have found propositions
containing the germ of
independent development.
theories which were capable of
But, speaking generally, the further
progress of geometry on general lines was practically
barred by the restrictions of method and form which were
inseparable from the classical Greek geometry. True, it was
open to geometers to discover and investigate curves of a
higher order than conies,
such as spirals, conchoids, and the
like. But the Greeks could not get very far even on these
lines in the absence of some system of coordinates and without
freer means of manipulation such as are afforded by modern
algebra, in contrast to the geometrical algebra, which could
only deal with equations connecting lines, areas, and volumes,
but involving no higher dimensions than three, except in so
far as the use of proportions allowed a very partial exemption
from this limitation. The theoretical methods available
enabled quadratic, cubic and bi-quadratic equations or their
equivalents to be solved. But all the solutions were geometrical
;
in other words, quantities could only be represented by
lines, areas and volumes, or ratios between them. There was
nothing corresponding to operations with general algebraical
quantities irrespective of what they represented. There were
no symbols for such quantities. In particular, the irrational
was discovered in the form of incommensurable lines ; hence
irrationals came to be represented by straight lines as they
are in Euclid, Book X, and the Greeks had no other way of
representing them. It followed that a product of two irrationals
could only be represented by a rectangle, and so on.
Even when Diophantus came to use a symbol for an unknown
with the meaning of an undetermined multitude of units ' ',
not a general quantity. The restriction then of the algebra
employed by geometers to the geometrical form of algebra
operated as an insuperable obstacle to any really new departure
in theoretical geometry.
It might be thought that tbere was room for further extensions
in the region of solid geometry. But the fundamental
principles of solid geometry had also been laid down in Euclid,
Books XI-XIII ; the theoretical geometry of the sphere had
been fully treated in the ancient spkaeric ; and any further
application of solid geometry, or of loci in three dimensions,
was hampered by the same restrictions of method which
hindered the further progress of plane geometry.
Theoretical geometry being thus practically at the end of
its resources, it was natural that mathematicians, seeking for
an opening, should turn to the applications of geometry. One
obvious branch remaining to be worked out was the geometry
of measurement, or mensuration in its widest sense, which of
course had to wait on pure theory and to be based on its
results. One species of mensuration was immediately required
for astronomy, namely the measurement of triangles, especially
spherical triangles ; in other words, trigonometry plane and
spherical. Another species of mensuration was that in which
an example had already been set by Archimedes, namely the
measurement of areas and volumes of different shapes, and
arithmetical approximations to their true values in cases
where they involved surds or the ratio (it) between the
circumference of a circle and its diameter ;
the object of such
mensuration was largely practical. Of these two kinds of
mensuration, the first (trigonometry) is represented by Hipparchus,
Menelaus and Ptolemy ;
the second by Heron of
Alexandria.
chapters ;
These mathematicians, will be dealt with in later
this chapter will be devoted to the successors of the
great geometers who worked on the same lines as the latter.
Unfortunately we have only very meagre information as to
what these geometers actually accomplished in keeping up the
tradition.
No geometrical works by them have come down
to us in their entirety, and we are dependent on isolated
extracts or scraps of information furnished by commen-