The Legacy of Greece: Essays By: Gilbert Murray, W. R. Inge, J. Burnet, Sir T. L. Heath, D'arcy W. Thompson, Charles Singer, R. W. Livingston, A. Toynbee, A. E. Zimmern, Percy Gardner, Sir Reginald Blomfield
Philosophy
The Legacy of Greece: Essays By: Gilbert Murray, W. R. Inge, J. Burnet, Sir T. L. Heath, D'arcy W. Thompson, Charles Singer, R. W. Livingston, A. Toynbee, A. E. Zimmern, Percy Gardner, Sir Reginald Blomfield
Greece -- Civilization
The other treatises included in Pappus's account are (1) On _Determinate
Section_; (2) _Contacts_ or _Tangencies_, Book II of which is entirely
devoted to the problem of drawing a circle to touch three given circles
(Apollonius's solution can, with the aid of Pappus's auxiliary
propositions, be satisfactorily restored); (3) _Plane Loci_, i. e. loci
which are straight lines or circles; (4) Νευσεις {Neuseis},
_Inclinationes_ (the general problem called a νευσις {neusis} being to
insert between two lines, straight or curved, a straight line of given
length _verging_ to a given point, i. e. so that, if produced, it passes
through the point, Apollonius restricted himself to cases which could be
solved by 'plane' methods, i. e. by the straight line and circle only).
Apollonius is also said to have written (5) a _Comparison of the
dodecahedron with the icosahedron_ (inscribed in the same sphere), in
which he proved that their surfaces are in the same ratio as their
volumes; (6) _On the cochlias_ or cylindrical helix; (7) a 'General
Treatise', which apparently dealt with the fundamental assumptions, &c.,
of elementary geometry; (8) a work on _unordered irrationals_, i. e.
irrationals of more complicated form than those of Eucl. Book X; (9) _On
the burning-mirror_, dealing with spherical mirrors and probably with
mirrors of parabolic section also; (10) ωκυτοκιον {ôkytokion} ('quick
delivery'). In the last-named work Apollonius found an approximation to
π {p} closer than that in Archimedes's _Measurement of a Circle_; and
possibly the book also contained Apollonius's exposition of his notation
for large numbers according to 'tetrads' (successive powers of the
myriad).
In astronomy Apollonius is said to have made special researches
regarding the moon, and to have been called ε {e} (Epsilon) because the
form of that letter is associated with the moon. He was also a master of
the theory of epicycles and eccentrics.
With Archimedes and Apollonius Greek geometry reached its culminating
point; indeed, without some more elastic notation and machinery such as
algebra provides, geometry was practically at the end of its resources.
For some time, however, there were capable geometers who kept up the
tradition, filling in details, devising alternative solutions of
problems, or discovering new curves for use or investigation.
Public-domain text, read in full here on John Shaqi.
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