Chapter 3 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 M?rbeke, 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

Δ = √{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.

* * *