Mathematics in Aristotle

Thomas Heath

• Publisher: Routledge
• Year: 2015
• ISBN:
  9781317380597
• Edition: 1
• Topic: Philosophie
• Subtopic: Geschichte & Theorie der Philosophie

Originally published in 1949. This meticulously researched book presents a comprehensive outline and discussion of Aristotle's mathematics with the author's translations of the greek. To Aristotle, mathematics was one of the three theoretical sciences, the others being theology and the philosophy of nature (physics). Arranged thematically, this book considers his thinking in relation to the other sciences and looks into such specifics as squaring of the circle, syllogism, parallels, incommensurability of the diagonal, angles, universal proof, gnomons, infinity, agelessness of the universe, surface of water, meteorology, metaphysics and mechanics such as levers, rudders, wedges, wheels and inertia. The last few short chapters address 'problems' that Aristotle posed but couldn't answer, related ethics issues and a summary of some short treatises that only briefly touch on mathematics.

VI

PHYSICS

(a) Quadrature of the circle: Hippocrates, Bryson, Antiphon

Soph. El, 11. 171^b12–18; 171^b34–172^a7

Phys. I. 2. 185^a14–17

The translation of the first two passages has been given above.¹ Both fall to be considered in connexion with the passage from the Physics, which has also been mentioned.²

The first passage says that the so-called geometrical Fallacies are not ‘eristic’ because they come within the art of geometry, ‘much less any such fallacy as relates to a true proposition, as e.g. that of Hippocrates or the squaring by means of lunes’. But the method of Bryson is sophistical because it uses principles which are not confined to geometry, but of wider application.

The second passage repeats that ‘the quadrature of the circle by means of lunes’ is not eristic, whereas Bryson’s method is sophistical or eristic ‘because it is addressed to the mass of people, who do not know what is possible and what impossible in each science, for it will fit any; and the same thing is true of Antiphon’s quadrature’ (ἢ ὡς Άντιϕῶν έτετραγώνιζεν).

Phys. I. 2. 185^a14–17

‘The exponent of any science is not called upon to solve every kind of difficulty that may be raised, but only such as arise through false deductions from the principles of the science: with others than these he need not concern himself. For example, it is for the geometer to expose the quadrature by means of segments, but it is not the business of the geometer to refute the argument of Antiphon.’

We have first to consider the allusion to Hippocrates³ οὐδέ γ’ εἲ τί έστι ψευδογράϕημα περὶ ἀληθές, οἷον τὸ ‘Ιπποκράτους ἢ ὁ τετραγωνισμός ὁ διά των μηνίσκων. The word understood with τό must be ψευδογράϕημα, ‘the false argument of Hippocrates or the quadrature by means of lunes’. It was a false argument ‘relating to something true’, by which we must presumably understand an argument which, though supposedly fallacious, aimed at establishing a true proposition, in this case a real solution of the quadrature-problem.

Two questions arise here. When Aristotle speaks of ‘the fallacy of Hippocrates or the quadrature by means of lunes’, does he refer to one and the same fallacy, or to two different ones? And what is ‘the quadrature by means of segments’? Is it the same as ‘the quadrature by means of lunes’? On this Simplicius¹ writes shrewdly as follows: ‘As regards “the squaring of the circle by means of segments” which Aristotle reflected on as containing a fallacy, there are three possibilities, (1) that it hints at the quadrature by means of lunes (Alexander was right in expressing the doubt implied by his words “if it is the same as the quadrature by means of lunes”), (2) that it refers, not to the proofs of Hippocrates, but to some others, one of which Alexander actually reproduced, or (3) that it is intended to reflect on the squaring by Hippocrates of the circle plus the lune, which Hippocrates did in fact prove by means of segments, namely the three 〈in the greater circle and the six〉 in the lesser circle. Perhaps this proof might more properly be called “The proof by means of segments” than “the (proof) by means of lunes”.… On this third hypothesis the fallacy would lie in the fact that the sum of the circle and the lune is squared and not the circle alone.’

Now the remarkable fact is that the genuine quadratures of lunes by Hippocrates contain no fallacies whatever. There are only five types of lunes such as those of Hippocrates which can be squared by ‘plane’ methods in geometry, i.e. by the use of the straight line and circle only; Hippocrates discovered three of the five and squared them: a very brilliant achievement. The squaring of the other two was left for Martin Johan Wallenius of Abo (1766) and the great Euler (1772) respectively.

Equally clever was the squaring of the sum of a circle and a lune by Hippocrates.²

Since the ‘quadrature by means of segments’ was such that, in the opinion of Aristotle, the geometer need not even consider it, it cannot have been the last-mentioned case. It must therefore have been some quadrature by means of lunes which contained a fallacy, probably the fallacious quadrature given by Alexander.³ The explanation is no doubt that Aristotle, as well as Alexander, was not well informed on the subject of the quadratures of lunes that were genuinely attributable to Hippocrates.

We are more fortunate in that we have authentic information on the subject. We owe this to the admirable Simplicius, who gives in his commentary on the passage of the Physics a long extract from the History of Geometry by Eudemus, the pupil of Aristotle, which is lost except for extracts preserved by Simplicius, Proclus, and others. The particular extract by Simplicius is one of the most precious documents that we possess about the history of geometry before Euclid’s time.

Simplicius, after reproducing Alexander’s version of the quadratures of lunes, observes that, so far as Hippocrates is concerned, we must allow that Eudemus was in a better position to know the facts since he was nearer the times, being a pupil of Aristotle. Simplicius says accordingly that he will quote Eudemus word for word (κατὰ λ̓∊ξιν) except for a few additions referring to Euclid’s Elements which are necessitated by the summary (‘memorandum-like’) style of Eudemus. The separation of Eudemus’ text from Simplicius’ additions can be made with fair certainty.¹

Antiphon

The objection taken by Aristotle to Antiphon’s attempted quadrature of the circle was clearly that it was based on other than geometrical principles,² i.e. on something which conflicted with the admitted principles of geometry. Simplicius gives us the tradition about Antiphon’s procedure. He began with an equilateral triangle or a square inscribed in a circle; then in each of the segments of the circle cut off by the sides of the triangle or square, he inscribed an isosceles triangle, thereby obtaining an inscribed regular polygon with double the number of sides. Continuing this process with that polygon, and so on, he thought ‘that in this way the area of the circle would be used up, and we should some time have a polygon the sides of which, owing to their smallness, would coincide with the circumference of the circle. And, as we can construct a square equal to any polygon … we shall be able to make a square equal to a circle.’³

No doubt Eudemus was right in supposing that the geometrical principle infringed by Antiphon was the truth that geometrical magnitudes are divisible without limit, so that Antiphon’s process would never end, and he could never arrive at a regular polygon which is equal in area to the circle.⁴ Antiphon was, in fact, confronted with the precise difficulty which Zeno and Democritus had stated with such force. But Antiphon deserves an honourable place in the history of geometry for having originated the idea of exhausting the area of a circle by means of inscribed regular polygons with a continually increasing number of sides, an idea upon which Eudoxus based his great and fruitful ‘method of exhaustion’ for finding the areas and volumes of curvilinear figures. Even Eudoxus did not solve the difficulty raised by Zeno (which does not seem even today to be finally disposed of); but he circumvented it in such a way as to enable geometry to advance without hindrance by showing that it is sufficient for the purpose to prove, for example, in the case of the circle that, if Antiphon’s procedure be continued far enough, the sum of the small segments left over between the sides of the inscribed polygon and the circumference of the circle can be made less than any assigned area.¹ Eudoxus did not, of course, any more than Antiphon, actually ‘square the circle’, i.e. construct, or show how to construct, a square equal in area to a given circle; for this is beyond the power of ‘plane’ geometry. But, by his method, one area or solid content involving π, the ratio of the circumference of a circle to its diameter, can be found in terms of another area or solid content also involving π. Thus Eudoxus proved by his method that circles are to one another as the squares on their diameters, spheres are to one another as the cubes on their diameters, and a cone is equal to one-third of the cylinder on the same base and of equal height; and he found the actual content of any pyramid by proving that it is one-third of that of a prism on the same base and of equal height. These propositions duly appear in Euclid.²

(b) Things known to us and things prior in the order of nature. Definition of circle

Physics I. 1. 184^a10–^b12

‘In all studies in which there are principles or causes or elements, it is from an acquaintance with these that knowledge, that is, scientific knowledge, is acquired. For we consider that we know a thing when we are acquainted with the first causes and the first principles down to the elements. Obviously therefore, in the science of Nature too, we must first try to determine what belongs to the principles. The natural course leads from what is better known and clearer to us to what is clearer and better known in the order of nature; for it is not the same things that are known to us and known in the absolute sense respectively. Hence it is necessary to advance in this way from what is less clear in the order of nature but clearer to us to what is clearer and better known in the order of nature. Now the things which are at first obvious and clear to us are more of the nature of confused aggregates; it is only afterwards that, as we analyse them, the elements and the principles become known to us. Hence we must proceed from the universal [here, the concrete whole] to t...