THE WORKS OF ARCHIMEDES.
The range of Archimedes's writings will be gathered from the list of his
various treatises. An extraordinarily large proportion of their contents
represents entirely new discoveries of his own. He was no compiler or
writer of text-books, and in this respect he differs from Euclid and
Apollonius, whose work largely consisted in systematising and
generalising the methods used and the results obtained by earlier
geometers. There is in Archimedes no mere working-up of existing
material; his objective is always something new, some definite addition
to the sum of knowledge. Confirmation of this is found in the
introductory letters prefixed to most of his treatises. In them we see
the directness, simplicity and humanity of the man. There is full and
generous recognition of the work of predecessors and contemporaries; his
estimate of the relation of his own discoveries to theirs is obviously
just and free from any shade of egoism. His manner is to state what
particular discoveries made by his predecessors had suggested to him the
possibility of extending them in new directions; thus he says that, in
connexion with the efforts of earlier geometers to square the circle, it
occurred to him that no one had tried to square a parabolic segment; he
accordingly attempted the problem and finally solved it. Similarly he
describes his discoveries about the volumes and surfaces of spheres and
cylinders as supplementing the theorems of Eudoxus about the pyramid,
the cone and the cylinder. He does not hesitate to say that certain
problems baffled him for a long time; in one place he positively
insists, for the purpose of pointing a moral, on specifying two
propositions which he had enunciated but which on further investigation
proved to be wrong.
The ordinary MSS. of the Greek text of Archimedes give his works in the
following order:--
1. _On the Sphere and Cylinder_ (two books).
2. _Measurement of a Circle._
3. _On Conoids and Spheroids._
4. _On Spirals._
5. _On Plane Equilibriums_ (two books).
6. _The Sandreckoner._
7. _Quadrature of a Parabola._
A most important addition to this list has been made in recent years
through an extraordinary piece of good fortune. In 1906 J. L. Heiberg,
the most recent editor of the text of Archimedes, discovered a
palimpsest of mathematical content in the "Jerusalemic Library" of one
Papadopoulos Kerameus at Constantinople. This proved to contain writings
of Archimedes copied in a good hand of the tenth century. An attempt had
been made (fortunately with only partial success) to wash out the old
writing, and then the parchment was used again to write a Euchologion
upon. However, on most of the leaves the earlier writing remains more or
less legible. The important fact about the MS. is that it contains,
besides substantial portions of the treatises previously known, (1) a
considerable portion of the work, in two books, _On Floating Bodies_,
which was formerly supposed to have been lost in Greek and only to have
survived in the translation by Wilhelm of Morbeke, and (2) most precious
of all, the greater part of the book called _The Method, treating of
Mechanical Problems_ and addressed to Eratosthenes. The important
treatise so happily recovered is now included in Heiberg's new (second)
edition of the Greek text of Archimedes (Teubner, 1910-15), and some
account of it will be given in the next chapter.
The order in which the treatises appear in the MSS. was not the order of
composition; but from the various prefaces and from internal evidence
generally we are able to establish the following as being approximately
the chronological sequence:--
1. _On Plane Equilibriums_, I.
2. _Quadrature of a Parabola._
3. _On Plane Equilibriums_, II.
4. _The Method._
5. _On the Sphere and Cylinder_, I, II.
6. _On Spirals._
7. _On Conoids and Spheroids._
8. _On Floating Bodies_, I, II.
9. _Measurement of a Circle._
10. _The Sandreckoner._
In addition to the above we have a collection of geometrical
propositions which has reached us through the Arabic with the title
"Liber assumptorum Archimedis". They were not written by Archimedes in
their present form, but were probably collected by some later Greek
writer for the purpose of illustrating some ancient work. It is,
however, quite likely that some of the propositions, which are
remarkably elegant, were of Archimedean origin, notably those concerning
the geometrical figures made with three and four semicircles
respectively and called (from their shape) (1) the _shoemaker's knife_
and (2) the _Salinon_ or _salt-cellar_, and another theorem which bears
on the trisection of an angle.
An interesting fact which we now know from Arabian sources is that the
formula for the area of any triangle in terms of its sides which we
write in the form
[Delta] = [root]{s(s - a)(s - b)(s - c)},
and which was supposed to be Heron's because Heron gives the geometrical
proof of it, was really due to Archimedes.
Archimedes is further credited with the authorship of the famous
Cattle-Problem enunciated in a Greek epigram edited by Lessing in 1773.
According to its heading the problem was communicated by Archimedes to
the mathematicians at Alexandria in a letter to Eratosthenes; and a
scholium to Plato's _Charmides_ speaks of the problem "called by
Archimedes the Cattle-Problem". It is an extraordinarily difficult
problem in indeterminate analysis, the solution of which involves
enormous figures.
Of lost works of Archimedes the following can be identified:--
1. Investigations relating to _polyhedra_ are referred to by Pappus,
who, after speaking of the five regular solids, gives a description of
thirteen other polyhedra discovered by Archimedes which are
semi-regular, being contained by polygons equilateral and equiangular
but not similar. One at least of these semi-regular solids was, however,
already known to Plato.
2. A book of arithmetical content entitled _Principles_ dealt, as we
learn from Archimedes himself, with the _naming of numbers_, and
expounded a system of expressing large numbers which could not be
written in the ordinary Greek notation. In setting out the same system
in the _Sandreckoner_ (see Chapter V. below), Archimedes explains that
he does so for the benefit of those who had not seen the earlier work.
3. _On Balances_ (or perhaps _levers_). Pappus says that in this work
Archimedes proved that "greater circles overpower lesser circles when
they rotate about the same centre".
4. A book _On Centres of Gravity_ is alluded to by Simplicius. It is
not, however, certain that this and the last-mentioned work were
separate treatises, Possibly Book I. _On Plane Equilibriums_ may have
been part of a larger work (called perhaps _Elements of Mechanics_), and
_On Balances_ may have been an alternative title. The title _On Centres
of Gravity_ may be a loose way of referring to the same treatise.
5. _Catoptrica_, an optical work from which Theon of Alexandria quotes a
remark about refraction.
6. _On Sphere-making_, a mechanical work on the construction of a sphere
to represent the motions of the heavenly bodies (cf. pp. 5-6 above).
Arabian writers attribute yet further works to Archimedes, (1) On the
circle, (2) On a heptagon in a circle, (3) On circles touching one
another, (4) On parallel lines, (5) On triangles, (6) On the properties
of right-angled triangles, (7) a book of _Data_; but we have no
confirmation of these statements.