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Aristotle mentions Bryson's squaring of the circle in the Posterior Analytics and in the Sophistical Refutations.  We have many different versions of what he did, and it may not be possible to recover his argument with certainty.

We may look at Bryson as a comment on the nature of Greek mathematics.  What Greeks such as Aristotle considered wrong with his argument tells us how Greek mathematicians conceived of mathematical constructions.  Some such as T.L. Heath have seen Bryson's approach as a precusor of Archimedes' compression method.  This seems optimistic.  However, the approach does treat the size of the figure in relation to outside and inside figures.

Bryson the Sophist lived in the mid-4th century B.C.E.  From Aristotle and some remarks elsewhere we know a little about him.  His father was Herodorus (Aristotle, History of Animals 563a7, 615a9-10) and he came from Heracleia.1  In the Rhetoric 1405b9, Aristotle says that he held that no one speaks with a foul mouth.  The spurious Letter 13 (360C5) attributed to Plato also indicates that Plato was close to Polyxenus, a student of Bryson.

Aristotle mentions Bryson six times in his extant writings, of which three pertain to his squaring of the circle.

John Philoponus, On Aristotle's Aristotle's Analytics 75b37, pp. 111.3-114.17 is our primary source for attempts to elaborate on Aristotle's meagre remarks.  It is quite clear that he has no direct evidence of what Bryson said and is in a position to comment on him that is merely marginally better than our own based on the evidence he presents.  He cites Alexander of Aphrodisias, Proclus, and the philosopher (his teacher, Ammonius, who was also a student of Proclus).

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.

Michael of Ephesus lived in the early 12th century, but his commentary on the Sophistical Refutations, also attributed to Alexander of Aphrodisias, shows learning.  However, his comments on Bryson, On Sophistical Refutations p. 76.11-24, 88.29-90.25, 92.21-93.2, may not improve on Philoponus.

As with Antiphon, there can be little doubt that the conceptions Bryson 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.  However, it seems plausible that his argument is conceptually very different from Antiphon's.  In fact, this is a requirement of any viable interpretation.

1  Aristotle (On the Generation of Animals 757a4-5) mentions Herodorus from Heracleia, whose Account of Heracles survives in a few fragments, while Athenaeus (Deipnosophistai xi 118) quotes Theopompus of Chios in an attack on Plato's works as mentioning Bryson of Heracleia.  The Suda (sigma, entry 829.31-36) identifies Bryson of Heracleia as introducing contentious dialectic with Eucleides of Megara.  Of course, this assumes that these are all the same.  Diogenes Laertius also mentions Bryson as someone who did not write a book (I 16.6), Bryson of Achaia with whom, according to Hippobotus, Crates may have studied (VI 85.2-3, and Bryson, the son(?) of Stilpo (younger contemporary of Aristotle) with whom, according to Alexander the historian, Pyrrho studied ((IX 61.3-4).  The text in Diogenes Laertius perhaps actually said that Pyrrho studied with "Bryson or Stilpo."  Could there have been two or even three philosophical Brysons in the 4th century?  The name is not common.


The goal is to find a square equal to the given circle.  In each interpretation we start by inscribing a polygon and circumscribing a circle.

1. The circle is smaller than every circumscribed polygon and larger than every inscribed polygon.

2a.  Between every circumscribed polygon and inscribed polygon there is a polygon.

3a.  [Hence, there is a polygon between every inscribed and circumscribed polygon; let it be A]

4a.  If A and the circle are larger than and less than the same, then A = the circle.

Attribution:  Alexander, according to Philoponus.  Philoponus reports an objection of Proclus.

Comment 1:  Proclus, according to Philoponus, argues that this argument amounts to claiming that the cirlce and polygon A coincide, which would make it the case that a rectilinear figure and a circumference coincide, the assumption of Antiphon.

However, to use Proclus' objection, one needs a convergence argument, e.g. there is a single limit of the inner and outer approximations.

Comment 2:  The argument is invalid.  Let the circle be C.

xy((Polygon(x) & Polygon(y) & x< C < y) => z(Polygon(z) & x < z < y))
does not entail
z(Polygon(z) & xy((Polygon(x) & Polygon(y) & x< C < y) => x < z < y))

 
 
1. The circle is smaller than every circumscribed polygon and larger than every inscribed polygon.

2.  Where there is a larger and a smaller, there is an equal (cf. Plato Parmenides 161D)

Hence,
3.  There is a polygon equal to the circle.

Attribution:  Proclus, according to Philoponus.  Philoponus raises his own objections.

However, note that Philoponus' objection, if applicable to the argument, would imply that the squaring of the circle is impossible in the sense that there is no square equal to the circle.

Comment 1:  Most modern readers tend to accept Proclus' interpretation.  However, it must be conceded that there is no evidence that Proclus knows more than Alexander.

Comment 2:  The core of this argument is an existential claim, not a construction.  Is this what Aristotle means?


Since it is obvious that it is not possible to demonstrate each thing except from the principles of each thing, if what is proved [to belong] belongs qua that, this is not knowing[if it is proved from true, undemonstrable, and immediate principles.  For it is possible to prove in this way, just as Bryson did the squaring.  For these sort of people prove it according to something common, which also belongs to something else.  Hence, the arguments fit with other things not of the same genus.  And so he will not know qua that, but accidentally.  For the demonstration would then fit into another genus.
  For false-diagrams are not contentious (their fallacious deductions do not come under the craft), at least not if it is a false-diagram concerned with truth, as in the case of the false-diagram of Hippocrates or the squaring through lunules.  But the way in which Bryson squared the circle, if the circle even gets squared, since it is not directed at the fact of the matter, is sophistical on account of this.  Thus the apparent syllogism concerning these things is a contentious argument, and the deduction which apparently aims at the fact of the matter, even if it is a deduction, is and contentious.  For it apparently aims tat the fact of the matter, so as to be deceptive and wrong. The contentious argument is in some way to the dialectical as the argument with false-diagrams is to the geometrical, since it argues fallaciously from the same things as the dialectical, while the producer of false-diagrams does the same with the geometer.  But he is not contentious, since he argues with false-diagrams from principles and conclusions which come under his craft, while it is clear that the argument which comes under dialectic in other matters will be contentious.  For example, the squaring through lunules is not contentious, but the argument of Bryson is contentious.   And it is not possible for it to cross over (to another genus) but must be solely directed towards geometry, since it is from its own principles, but the other is directed at the many who do not know that is possible and impossible in each case.  Or as Antiphon squared it.  Or if someone should deny that it is better to take a walk after dinner because of the argument of Zeno, the argument is not medical.  For it is common.  And so, if the contentious holds in every way to the dialectical as the producer of false-diagrams is to the geometer, it would not be contentious concerning those things.  As it is, the dialectical is not concerned with some determinate genus, and it isn't aimed at proving anything, and it isn't like universal arguments either.


p. 75b37  "Since it is obvious that it is not possible to demonstrate each thing except from the principles of each thing, if what is proved [to belong] belongs qua that, this is not knowing...

Furthermore he adds this to the things proved about knowledge, that getting true and immediate premises is not adequate for producing a demonstration.  It is also necessary that these be appropriate for what is supposed to be demonstrated.  for if I speak in this way, "Every stone is colored, ever colored is a body, therefore every stone is a body," I have taken true premises, but also immediate ones (for I have no need of a middle term for the demonstration other than that ever stone is colored or that every colored is a body)..  However, the middle term is not appropriate for what is supposed to be demonstrated.  For being colored belongs to many other things.  But it is necessary, as has often been said, that the demonstration arise from principles appropriate to each thing, i.e., so that the middle term be appropriate for the extreme terms and not be common to anything else.  Thus, he says, it is necessary for the premises to be taken not merely from true and immediate statements, but also from those appropriate for the conclusion.  For in this way, he says, it is even possible to prove the squaring of Bryson from certain more common principles and not principles appropriate for what is proposed.  Now Aristotle says this much about the squaring of Bryson, but Alexander says that Bryson attempted to square the circle in this manner.  The circle, he says, is larger than every inscribed rectilinear figure, but is less than every circumscribed one (the figure drawn inside the circle is said to be 'inscribed', while one drawn outside is said to be 'circumscribed').  But also the rectilinear figure drawn between the circumscribed and inscribed figure is smaller than the circumscribed and larger than the inscribed figure.  Things larger and smaller than the same are equal to one another.  Therefore, the circle is equal to the rectilinear figure drawn between the inscribed figure and the circumscribed figure.  But we can to construct a square equal to every given rectilinear figure.  Therefore it is possible to produce a square equal to the circle.  Thus Alexander.

The philosopher used to say that Proclus, his teacher, railed against the interpretation of Alexander, since if Bryson had squared the circle in this way, he would concurred with the squaring of Antiphon.  For the rectilinear figure drawn between the inscribed and circumscribe figure will coincide with the circumference of the circle (Antiphon did this- up to where it coincided), a straight line to an arc, which is impossible.  There was discussion of this in the Physics.  And so Aristotle would not have put the squaring of Bryson as be different from that of Antiphon, if Bryson really had squared it in this way.  I say, says Proclus, that even the axiom is false.  For it is not true that the those which are larger and less than the same are equal to one another.  10 is larger than 8 but less than 12.  In fact even 9 is likewise less then 12 but larger than 8.  And I don't suppose that 10 and 9 are equal, since they are both larger than and less than the same, 12 and 8.  Therefore, it is not the case, even if the circle and the rectilinear figure drawn between inscribed and circumscribed figure [are larger and smaller] than the same, that because of this they are equal to one another, unless someone should claim the view of Antiphon, just what was mentioned, that the rectilinear figure drawn between the inscribed and circumscribed coincides with the circle, which is impossible.  For a straight line will never coincide with an arc.  And so Proclus used to say that Bryson squared the circle in this way. "The circle," he says, "is larger than every inscribed rectilinear figure and smaller than every circumscribed figure.  That for which there is a larger and a smaller, there is also an equal.  There is a rectilinear figure larger and one smaller than the circle.  Therefore there is also an equal to it."  And it is possible to speak against the claims of Proclus, since if Bryson had constructed the squaring of the circle in this way, he would not at all have constructed it, but one would have begged the question.  For those who square the circle did not seek this either:  if it is possible for a square to be equal to the circle, except that as believing that it is possible that attempted in this way to generate a square equal to the circle.  But he proved that it is possible for there to be a square equal to the circle, if in fact even this is admitted.  However, he did not construct a square equal to the circle and he did not teach how this might come about either, which is what those who would square the circle seek to do.  And Aristotle spoke as if the circle were squared by Bryson, just not geometrically.  Thus, the interpretation of Proclus does not appears to be suitable either.

If someone should thus also agree that Bryson did a construction, it is possible to speak against him since the argument will be true in the case of things in the same genus, that for which there is a larger and a smaller, there is also an equal, this is no longer true of things in dissimilar genera.  This is proved then by the geometer:
 
On semicircle GDB, the line AG drawn perpendicular from the endpoint of the diameter GB falls completely outside the circle, with two angles forming  from the circumference and the diameter  and from the line drawn perpendicular and the circumference.  I say that of these angles, the outer AGD and the inner DGB, the outer is smaller than every acute rectilinear angle and the inner is larger than every rectilinear acute angle.
N.b. when we have proved that they are larger and smaller than the same acute rectilinear angle, we will not be able to find an equal angle, since the magnitudes are of dissimilar genera, which they call horn-like.  And now the paradox:  since the outer angle can increase ad infinitum and the inner can be diminished, and again the inner can be increased ad infinitum and the outer diminished, neither will the outer increasing ad infinitum ever become equal to an acute rectilinear angle, but will always be smaller than every one, nor will the inner angle increasing ad infinitum every be equal to a right angle.  We increase the outer angle by drawing smaller circles.  For example, suppose I divide GB diameter at point E and bisect line GE at point Z and draw a circle with center Z and diameter ZG, whose semicircle is GHE.  Then the outer angle AGH increases, and again is no less smaller than every acute angle for the mentioned reason.
This theorem was proved by the geometer for every circle.  In the same manner again, I cut the diameter of the inner circle and draw a smaller circle and doing this ad infinitum, I repeatedly increase the outer angle but decrease the inner.  And neither will the outer angle every become equal to a rectilinear acute angle nor will the inner, but the outer will always be less while the inner will always be larger.  In this way I increase the outer and decrease the inner.  Again I increase the inner and decrease the outer by circumscribing larger circles in this way.  For I extend diameter GB in a straight line to E, and with center B and distance BG I draw a circle whose semicircle is GZE, and it is clear that semicircle GZE falls within straight-line AG since it has been proved that the line drawn perpendicular from the endpoint of the diameter in every case falls outside the circle.
It is clear from this that no portion of the outer semicircle GZE touches any potion of the inner semicircle GDB.  For if it is tangent, let there be joined lines HB and HQ from the point at which they touch, arbitrarily H, to the centers of the circles B and Q.  And so since point Q is the center of the inner semicircle, QH is equal to QG.  Again since B is the center of the outer semicircle GZE, BH is equal to BG.  But BQ and GQ are equal to QH.  Therefore HB is equal to BQ and to QH.  Therefore the two sides HQ and BQ of triangle HQB are equal to one side HB, which is impossible.  Therefore no part of the outer circle touches any part of the inner.  Therefore the outer circle cuts angle AGH.  And in this same way, by drawing outer circles, I decrease the outer angle ad infinitum and increase the inner.  And the inner increasing angle will never be equal to the right angle but will always be larger than every rectilinear right angle.  If then it has been proved that it is possible for something to be larger and smaller than the same, it still won't be equal due to the dissimilarity of the magnitudes.  Therefore, Bryson poorly assumed that if the circumscribed rectilinear figure is larger than the circle and the inscribed figure is smaller, therefore the figure between the inscribed and circumscribed figure is equal.  For the magnitudes in these cases are dissimilar, I mean the rectilinear with the circle, so that they will not be equal.



And so what do we mean by the claim that there are certain common axioms, e.g., equals taken from equals, and one must deny or affirm in every case?  Or is it that [demonstrations] do not make use of these as common, but each assigns them to an appendix of its own appropriate material:  for geometry it would be 'from equal magnitudes', for arithmetic it would be 'from numbers', and similarly with contradictions.  For each weaves together  and states the genus with which it works.  For example, odd exists or doesn't.  Therefore the fact that the premises are true, nor that they are undemonstrable and immediate, is not by itself adequate for demonstration, but it is necessary that they also be appropriate and not fit into many genera.  For this reason, someone might deny the squaring of Bryson as geometrical demonstration.  For it makes use of an axiom that is true but common.  This is:  things than which the same things are larger and smaller are equal to one another.  For the axiom is true not merely in the case of magnitudes, but also in the case of number and time and many other things.

Now what is it that Bryson gets for believing that he has squared the circle.  It doesn't pertain to the present argument-suppose it is for love of learning.  The circle, he says, is greater than every inscribed polygon and smaller than every circumscribed polygon.  Similarly with the polygon drawn between the polygons inscribed and the circumscribed about the circle.  The circle and this polygon are greater and smaller than the same, so that they are equal to one another due to the stated axiom.  But this axiom is true, if you wish, as well as immediate and undemonstrable.  But the deduction does not have as its principle one appropriate for you in proving it, since you do not take it qua circle, nor qua magnitude either.  Similarly too for deductions from accidents, since those get assumptions which are detached from the underlying nature, but someone who truly and not accidentally knows knows from the principles and from the per se attributes, for which it is necessary that the middle term and the premise holding the deduction together either be from the same and single genus in every case or from a related genus not far off, just as we said harmonic intervals hold to numbers.
 

p. 76.11-24
True refutations are unlimited in number, he says, as false ones are unlimited.  And he says here that false refutations are not the sophistical ones, but those which are thought to be, but aren't, from the appropriate principles of the science, about which they make the argument.  Thus, since the true refutations are unlimited, the false would also be infinite.  And his statements are clear.  The expression 'such as according to geometry the geometrical (refutation)'
was stated elliptically.  The full statement might be:
the geometrical refutation which is false according to geometry is the one which arises contrary to the principles of geometry, such as the squaring of Bryson.
Not merely does he say that whichever ones do not set out from the principles of the science under which the problem occurs  are thought to be false refutations, but also those who set out from the appropriate principles by arguing fallaciously in some way, such as are the "False Diagrams" of Euclid.

p. 90.1-25
For this reason "false diagrams" are not contentious, since they set out from the appropriate principles of geometry.  The drawings do not arise as necessary.  And it was stated in the first book of the Topics concerning these matters.  For he says there that someone who draws 'false diagrams' (101a15-17) "produces the fallacious argument by either drawing the semicircles in a way that one oughtn't or by drawing certain lines but not as they may be drawn."  And so 'false diagrams' are not contentious, since they argue fallaciously from appropriate premises.  Not even if the false diagram is something concerning truth will it be contentious, such as the squaring of the circle of Hippocrates and the one through lunules of Antiphon (sic).  For they are not contentious since they observe the appropriate principles of geometry.  But the squaring of the circle of Bryson is contentious and sophistical, since it does not proceed from the appropriate principles of geometry but from certain more common ones.  For circumscribing a square about the circle and inscribing another within and between the two squares another between, and then saying that the circle between the two squares as well as the square between the two squares are smaller than the outer square and larger than the inner, while those larger than the same things and smaller than the same are equal, and, therefore, the circle and the square are equal, these are from common principles, but also from false ones; from common since it applies even in the case of numbers, times, places, and other common things, and false, since eight and nine are smaller than ten and larger than seven, and, nevertheless, they are not equal.  But also someone who concludes that virtues are teachable from the claim that teachable things are things which were not earlier in us later come to be in us, but virtues , which were not earlier in us, later come to be in us, therefore virtues are teachable, is contentious and sophistical.  For he does not conclude what is proposed from principles appropriate to the matter.  For this will also fit the claim in the case of many other things

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