• Antiphon's Squaring of the Circle©

by Henry Mendell (Cal. State U., L.A.)

In the text in Aristotle discussed by Simplicius, Aristotle claims that he does not have to refute Parmenides' view that what is is just one and unchangeable.  In a book on nature, he does not have to concern himself with hypotheses which reject nature altogether.  He then draws a contrast between two attempts to square the circle, one through segments, and one by Antiphon.  The mathematician needs to concern himself with a refutation of squaring by segments, but does not need to be concerned with refuting Antiphon's, which rejects mathematical principles.  Elsewhere, in Met. K 1 (assuming Aristotle to be the author), he appears to hold that such a refutation belongs to first philosophy.  Simplicius identifies the squaring through segments with the construction of lunules by Hippocrates of Chios, as suggested by Aristotle, Sophistical Refutations 171^b15-16.
 

• Characters of the Drama

Simplicius wrote his commentary on the Physics sometime around the 540's.  Simplicius, On Aristotle's Physics I 185^a14, pp. 53.27-55.24 is the principal source for Antiphon.  Besides Aristotle's comments, there are two other discussions of Antiphon that preserve some substantial issues on the squaring of the circle.

John Philoponus, On Aristotle's Physics I 185^a16, pp. pp. 31.1-32.3 makes only one trivial point not found in Simplicius, that the angles get 'smaller' as the sides increase, where, I assume, the angles are two sides forming an angle.  Philoponus was a contemporary and school-mate of Simplicius (Simplicius has little fondness for him), but wrote his commentary before Simplicius.

Themistius lived in the 4th century C.E.  He has little taste for expounding mathematical arguments.  Yet his account, Paraphrase of Aristotle's Physics 3.30-4.8 is somewhat different from that of Simplicius.

There can be little doubt that the conceptions Antiphon uses were related to mathematical research into circle squaring and 5th century B.C.E. arguments for the theorem that circles are as the squares of their diameters, which appears in Euclid, Elements XII 2 using the method of exhaustion.

In opposition to the thesis which states that being is one and unchangeable he sets down the thesis that all or some things by nature change, the evidence for which is manifestly obvious.  To prevents someone from saying, "You've supposed what is sought as already agreed on," the reason for the hypothesis, he proposes instead what's true before refuting what's false.  For it is not appropriate to refute every false claim since at the same time it is easy and not at all difficult to grasp the consequences of an absurd hypothesis.  But this is only for those things which someone presents falsely in a discussion from principles, and where they aren't, it isn't appropriate.  Those who say that it is one and unchangeable preserve neither any principle nor nature.  And so there is nothing absurd in supposing things  whose evidence is manifestly obvious before resolving the contrary arguments, since there is no need to resolve everything.  Now he shows the difference between falsehoods which are useful to resolve and those which are not with reference to certain false-diagrams (or arguments) in geometry.

Among the many people who looked for a squaring of the circle (i.e., the setting out a square equal to a circle), both Antiphon and Hippocrates thought they found it and were deceived.  In fact, it is not the job of the geometer to resolve the falsehood in Antiphon's since it does not set out from geometrical principles, as we shall learn.  But it is her job to resolve Hippocrates', since he was deceived but preserved geometrical principles.  For it is only necessary to resolve those arguments which observed the appropriate principles of the method but in this way make mistakes in their reasoning.  One should not resolve those arguments through which they are led astray and which destroy the principles.

 

After drawing a circle, Antiphon inscribes in some polygonal area of those that can be inscribed.  Let the inscribed figure, arbitrarily, be a square.

 

Then he bisects each side of the square and draws a perpendicular from the cut to the arcs.  Clearly, each perpendicular bisects its own segment of the circle.  Then he draws lines joining the cuts to the corners of the lines of the square, so that there are now four triangles on the straight lines, but the whole inscribed figure is an octagon.

And so, keeping to the same method, he bisects each of the sides of the octagon, leads perpendiculars from the cut to the circumference and draws lines joining the points at which the constructed perpendiculars touch the arcs to the limits of the divided lines, he made a 16-gon as the inscribed figure.

And by the same argument he cuts the sides of the inscribed 16-gon and draws lines joining the points, and so doubles the inscribed polygon.  By repeatedly doing this so that when the plain is exhausted a certain polygon is inscribed in this way in the circle, whose sides, because of their smallness, coincide with the circumference of the circle.

But we are able to set out a square equal to a given polygon, as we learned in the Elements.  Since the polygon was supposed as equal to the circle and as coinciding with it, we will have set forth a square equal to the circle. And it is clear that the procedure has arisen contrary to geometrical principles, although not in the way that Alexander says, "The geometer hypothesizes as a principle that the circle is tangent to a straight line at a point, but Antipon destroys this."  For the geometer does not hypothesize this, but he proves it in the third book.  And so it is better to state a principle that it is impossible for a straight line to coincide with an arc, but a line outside touches the circle at one point, while the line inside touches it at merely two and not more.  And so the tangent at the point comes about.  However, it is not the case that by repeated cutting the plane between the line and the circumference of the circle one will exhaust it, nor that it will ever overtake the circumference of the circle.  But if it does overtake it, a geometrical principle will be destroyed, namely the one that says that magnitudes are infinitely divisible.  Eudemus too says that this is the principle destroyed by Antiphon.

Simplicius actually states three candidates for principles 'destroyed' by Antiphon.

1. Alexander:   The circle is tangent to a straight line at a point.
2. Simplicius (and Philoponus as well, see below):  It is impossible for a straight line to coincide with an arc.
3. Eudemus:  Magnitudes are infinitely divisible.

• John Philophonus, On Aristotle's Physics I 185^a16, pp. 31.1-32.3

Hippocrates of Chios was a merchant who came across a pirate ship and lost everything.  He came to Athens to fill out a writ against the pirates.  Since he was staying for a long time in Athens because of the writ, he wandered into a group of philosophers.  He developed so much geometrical ability that he attempted to find the squaring of the circle.  And he did not find it, although after he squared the lunule, he falsely thought he would square a circle from it..  For he thought that he deduce the squaring of the circle from the squaring of the lunule.  Now Antiphon also attempted to square the circle, but without preserving geometrical principles.  He made his attempt as follows.  If, he says, I make a circle and draw in it a square, and I bisect the segments of the circle which arise from the square, and then lead straight lines from the each cut respectively to the endpoints of the segment, I make an octagon figure.  And if we again bisect the segments containing the angles, and again lead straight lines respectively from the cuts to the endpoints of the segments, we will make a polygonal figure.  And so if we do this more times, a most polygonal figure will having very small angles (sic), and the straight lines enclosing the them, because of their smallness will coincide with the circle.  And so since it is possible to square every given rectilinear figure, if I square this polygon, since it coincides with the circle, I will have also squared the circle.  And so this man destroys the geometrical principles.  For it is a geometrical principle that a straight line never coincides with any arc, but this man gives out that because of its smallness, a certain line coincides with a certain arc.  Now Hippocrates sets out from geometrical principles and squares a certain moon-like segment of the circle.  He concludes the rest poorly, in that he wants to deduce from this the squaring of the lunule as well.  However, Antiphon destroys geometrical principles, namely that a straight line never coincides with an arc, and in this way concludes the rest.  And so (Aristotle) says that it is the job of the geometer to refute the squaring of the circle due to Hippocrates which is false, since HIppocrates preserves the geometrical principles, while the geometer will not resolve the squaring of Antiphon, since he concludes in this way, given the geometrical principles are destroyed.

• Themistius, Paraphrase of Aristotle's Physics I, pp. 3.30-4.8

For the geometer must resolve those false diagrams which preserve geometrical hypotheses, but she should set aside those which fight with them.  For example, two people attempted to square a circle, Hippocrates of Chios and Antiphon.  One should resolve the squaring of Hippocrates.  For he reasons fallaciously while preserving the principles since that man merely squares the lunule which is drawn on the side of the square inscribed in the semicircle.... the geometer would not at all ahve to state a demonstration against Antiphon.  He inscribes an equilateral triangle in the circle and on each of the sides he constructs another isosceles triangle at the circumference of the cirlce.  He keeps doing this in successions and thinks that the straight line of the last triangle will coincide with the circumference.  But this is the case when he destroys the hypothesis which the geometer assume, unlimited division.

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