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4.5
In this review I shall offer an overview of the works of Archimedes with special attention to the considerations that seem to have motivated the directions of his research.Archimedes's determination of the volume (and secondarily the surface area) of a sphere may be considered the undisputed crown jewel of his works. Indeed, Archimedes himself seems to have thought so judging by the story that he requested it inscribed on his tombstone (xviii). Like an axiom needs no demonstration, this beautiful result needs no motivation.But no sooner has Archimedes proved this result than he turns, in the same treatise, to matters of much less evident value. Thus he solves numerous baroque problems about areas and volumes of various sections of spheres such as "Given two segments of spheres, to find a third segment of a sphere similar to one of the given segments and having its surface equal to that of the other" (82) and other things along similar lines. What is the purpose of this? What use could one ever make of such a problem?One might say: Archimedes is here working out the complete theory of spherical areas and volumes for future reference. The main result is beautiful in its own right, but these technical further propositions form a "toolbox" of results that subsequent mathematicians will find useful in technical contexts. If this is so (which I doubt) then it may be observed that previous mathematicians were apparently not inclined to do something similar when they worked out the theory of the cone. For Archimedes cites previous propositions on volumes of cones without proof (22), yet he has to work out the rather basic result of the surface area of frustum from scratch (21).Another possible motivation for these technical propositions emerges from Archimedes's preface to another work, where he says of a list of problems along these lines that: "there are two included among them which are impossible of realisation [and which may serve as a warning] how those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible" (151). Thus a sprinkling of technical propositions such as those mentioned above can serve a useful purpose even if they are of no actual use to any mathematician; the use being to expose poseur mathematicians by a kind of acid test. Indeed this seems to fit with Archimedes's preface to the original treatise where he appears to think that there is hardly anyone capable of judging his work: "these discoveries of mine ... ought to have been published while Conon was still alive, for I should conceive that he would best have been able to grasp them and to pronounce upon them the appropriate verdict" (2). (Incidentally, perhaps this lack of competent peers is also part of the reason why Archimedes chose to withdraw to a recluse life in Syracuse rather than remaining in Alexandria where he studied (xvi).)In the Measurement of a Circle, Archimedes offers the two-dimensional analog of the above result: "The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle" (91).Why did Euclid not prove this rather basic theorem about the areas of circles? The answer is obvious: because the triangle in question cannot be constructed by ruler and compass. Thus the problem cannot be considered truly solved. Indeed, Archimedes himself seems to agree with this when he writes: "Some of the earlier geometers tried to prove it possible to find a rectilineal area equal to a given circle ..., assuming lemmas not easily conceded, so that it was recognised by most people that the problem was not solved" (233). It seems clear from this quotation that the mere result that the area of a circle is equal to such-and-such a triangle was well known, since people proved it by *some* means. The problem, rather, was with the method used in those proofs. In other words, the problem was, as Archimedes says, to *find* the rectilineal area in question, meaning to construct it, not to prove that it was in fact equal to the circle.It seems clear that this is the reason Archimedes studies his spiral, for if it is admitted that the spiral and its tangents can be drawn then a construction of the triangle in question follows immediately (171), though probably Archimedes would have been the first to admit that these spiral constructions are just one more of those "lemmas not easily conceded" that he spoke of above. One indication that the spiral is meant to fit into the tradition of ruler-and-compass constructions is that Archimedes defines it in terms of motions and derives the locus-definition (essentially r=theta) as a theorem (155) even though it is the latter rather than the former that is needed in the subsequent proofs.It is often assumed that the ruler-and-compass paradigm was favoured over a numerical conception of mathematics because things like the irrationality of square roots pose problems for the latter but not the former. Perhaps it is understandable then that when the ruler-and-compass paradigm proved poorly suited for dealing with the quadrature of the circle, the numerical conception of geometry was revived. This could perhaps explain why Archimedes takes pains to establish bounds for pi (93), and perhaps by extension and analogy his interest in determining how many grains of sand fit in the universe (221-232), and maybe by one more tenuous step of association his interest in numerical problems generally, as in the cattle-problem (318-326) and apparently another lost work (xxxvi).Archimedes worked out axiomatic theories of statics and hydrostatics, showing that the geometrical method is remarkably and perhaps surprisingly well-suited to deal with these physical matters. In this it seems he had "no fore-runners" (xl). Some might look for his motivations in his work on warfare machines (xvi-xvii), but I would suggest that it actually followed quite naturally from his interest in volumes. As Archimedes says (2), his work on the volume of the sphere builds on Eudoxus's result that the volume of a cone is one third of the corresponding cylinder. Seeing as plane geometry is always "operationalised" in terms of ruler-and-compass constructions, one might look for a way to "operationalise" this result. Thus the use of a lever or balance suggests itself: a cylinder in one bowl balances with three cones in the other, or one cone at three times the distance. Alternatively, three cones or one cylinder dropped into a container of water make the surface rise by equal amounts. To put it another way, Greek results about volumes and areas are always comparative rather than absolute: they don't say it's equal to such-and-such a number or formula, but to such-and-such other figure. This comparative aspect goes hand in hand with the idea of a physical balance in equilibrium, again suggesting that Archimedes's work grew out of geometry.Archimedes's work on statics is also the basis for his "Method" of finding areas and volumes by (hypothetically) balancing them on levers. As he says: "Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge." (Method appendix, 13)Notably the volume of a sphere and the area of a parabolic segment are found in this way. Both are proved rigorously using the method of exhaustion in other works. In those works, the volume of the sphere is investigated by approximating a circle with a regular polygon and rotating it, so that the volume of the sphere is approximated by conical frustums; and the parabolic area is approximated by triangles stacked upon triangle fitted closer and closer to the perimeter. Both of these methods of approximation are natural and straightforward. It seems, therefore, that the only reason they were not feasible before the Method was that it was too difficult to brute-force sum these pieces (which basically amounts to simplifying intricate algebraic expressions) without knowing in advance what answer should be.In reality, however, the "mechanical" aspect of the method has rather little to do with it. More importantly the Method consists in investigating the shapes slice by slice. True, each slice is then balanced with some suitable slice of a comparison figure with some suitable choice of lever arm, but this is little more than a convenient and flexible way of expressing an algebraic equality. The greater contrast with the rigorous proofs is the slicing of shapes in to "lines," which basically means investigating the figures by their algebraic equations, since the line in question has height y=f(x).More generally, Archimedes seems to have taken a constant interest in conic sections simply for the fact that they are the most tractable "next" figures beyond circles. He constantly generalises all of his results about circles and spheres to conics, for example investigating their areas (in the parabolic case), rotational volumes, centers of gravity, and even the behaviour of a paraboloid submerged in water. None of these results seem particularly useful or natural except as a way of generating more mathematics from existing ideas. When they were first introduced conic sections surely had to be motivated and serve a purpose for solving established problems, but clearly at the time of Archimedes they were accepted as part of the standard repertoire of curves and it was seen as natural to "cut one's teeth" on the conics by generalising to them anything one did for circles.
