by Corry Shores

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Democritus’ Paradox of the Cone,

in Plutarch’s discussion of the Stoic doctrines of the elements,

in his *De communibus notitiis adversus Stoicos*

Summary from Sections 37-39:

According to common conceptions, two bodies cannot occupy the same location. But the Stoics have a confused understanding of the world and its parts. For example, they say that if you mix one cup of wine with two cups of water, the ‘one’ of the wine becomes ‘two’, because it extends throughout the whole of the two cups of wine. [In the Stoic theory of mixtures, there is a type of mixture called ‘total blending’, and the wine example illustrates it. For more, see Sellars’ explanation of the Stoic theory of mixtures. One odd thing about this kind of mixture is that the water and wine on the one hand form a third thing, but on the other hand the parts maintain their qualities, and can be extracted. So insofar as the wine retains its qualities and can be extracted, it is one. But insofar as it has thoroughly integrated with the water to make a third body, it is two.] The one is also three, because all the parts now total three parts. It is also four, because by being mixed with the two other parts, the original one is now two. So we have the original wine, which is now two, and the original water, which is also two, now making four.

Now this fine subtilty is a consequence of their putting bodies into a body, and so likewise is the unintelligibleness of the manner how one is contained in the other. For it is of necessity that, of bodies passing one into another by mixture, the one should not contain and the other be contained, nor the one receive and the other be received within; for this would not be a mixture, but a contiguity and touching of the superficies, the one entering in, and the other enclosing it without, and the rest of the parts remaining unmixed and pure, and so it would be merely many different things. But there being a necessity, according to their axiom of mixture, that the things which are mixed should be mingled one within the other, and that the same things should together be contained [p. 412] by being within, and by receiving contain the other, and that neither of them could possibly exist again as it was before, it comes to pass that both the subjects of the mixture mutually penetrate each other, and that there is not any part of either remaining separate, but that they are necessarily all filled with each other.

[Plutarch Section 37]

Another strange result of the Stoic theory of mixture is that one drop of wine in the sea mixes throughout the sea’s entirety. “And this Chrysippus admits, saying immediately in his First Book of Natural Questions, that there is nothing to hinder one drop of wine from being mixed with the whole sea. And that we may not wonder at this, he says that this one drop will by mixtion extend through the whole world; than which I know not any thing that can appear more absurd.” [Plutarch Section 37] As we will see, this idea that the drop extends to the whole world would be a main source of problems for the Stoic theory of mixtures, in Plutarch’s assessment.

Thus a further strange conclusion is that bodies have no ends to them, and we extend infinitely. [The drop of wine unites with the water which mixes with the air which falls and is absorbed in the soil and perhaps in this way everything is mixed with everything else] So also we cannot say that one body is larger than another, if both extend to infinity. Thus inequality becomes impossible [since nothing has determinate bounds. All things permeate with all other things]. But if there is no inequality, then there is also no roughness or unevenness of bodies. “**unevenness is the inequality of the same superficies with itself, and roughness is an unevenness joined with hardness**.” But in the Stoic theory, bodies do not terminate in a last part [because they extend into larger and larger mixtures] [and thus for example a surface cannot be said to be rough, because it does not actually have determinate boundaries of any kind, jagged or otherwise.] [Plutarch Section 38]

Plutarch then quotes the Stoic Chryssippus. [Here it seems that Chryssipus is explaining the composition of bodies. If he says our body for example is divisible into smaller parts, the question is, do they terminate? If so, why cannot those parts be divided? Or, if it goes on to infinity, would that not mean there is no fundamental composition to bodies, since there are no basic parts? It would be divisions all the way down with no basic substance. So Chrysippus might want to argue that the basic parts are neither finite nor infinite, even if he cannot specify what they might instead be.]

Chrysippus says: ‘If we are asked, if we have any parts, and how many, and of what and how many parts they consist, we are to use a distinction, making it a position that the whole body is compacted of the head, trunk, and legs, as if that were all which is enquired and doubted of. But if they extend their interrogation to the last parts, no such thing is to be undertaken, but we are to say that they consist not of any certain parts, nor yet of so many, nor of infinite, nor of finite.’

[Plutarch Section 38]

Plutarch notes this is a contradiction. Chrysippus wants it both ways, and in the end undermines the possibility of his claim having any truth value.

For if there were any medium between finite and infinite, as the indifferent is between good and evil, he should, by telling us what that is, have solved the difficulty. But if—as that which is not equal is presently understood to be unequal, and that which is not mortal to be immortal—we also understand that which is not finite to be immediately infinite, to say that a body consists of parts neither finite nor infinite is, in my opinion, the same thing as to affirm that an argument is compacted of positions neither true nor false.... [Plutarch Section 38]

Plutarch says that Chryssippus elaborates on this impossible status of infinite and finite with the example of the sides of a pyramid. [The distinction he makes will be clearer in the following Democritus paradox. But for now…] A pyramid is made of triangles who sides incline toward the juncture at the top. [I am not certain, but perhaps we are to think of two sets of triangles forming the pyranid. There are the sides which form the faces of the pyramid, and there are the stacking horizontal layers of triangles, assuming that it has a triangular base. At any rate, in light of the following example, Plutarch seems to be saying that if we were to consider the pyramid as being made of stacking layers, that one layer to the next can either be equal in size to the lower or unequal in size.] Chrysippus says that [these layers] are unequal [meaning the next one up is slightly smaller] and yet no one exceeds [its neighboring layer.] But this again leads into a contradiction.

To this he with a certain youthful rashness adds, that in a pyramid consisting of triangles, the sides inclining to the juncture are unequal, and yet do not exceed one another in that they are greater. Thus does he keep the common notions. For if there is any thing greater and not exceeding, there will be also something less and not deficient, and so also something unequal which neither exceeds nor is deficient; that is, there will be an unequal thing equal, a greater not greater, and a less not less. [Plutarch Section 39]

Plutarch proceeds then to the example of Democritus’ paradox of the cone’s composition. If the circles making up the cone were equal, the sides would be even and smooth, but the shape would be a cylinder and not a cone. If instead the cone’s circles were unequal, the figure would come to a point, but its sides would be uneven, making its surface rough and giving it the appearance of a staircase. [See my animate diagram below:]

[Moving diagram by Corry Shores, made with Open Office Draw and Unfreez]

Plutarch seems to indicate that Chrysippus’ solution is to say that the cone is unequal, but the circles composing it are neither equal nor unequal. Plutarch notes that this means we have an unequal body that is made of parts which are not unequal, and this seems absurd. [Plutarch following reasoning is not clear to me, but I provide it below. He does at least seem to be saying that] if a body is unequal, that could only mean that its parts are unequal.

Here, that he may convince Democritus of ignorance, he says, that the superficies are neither equal or unequal, but that the bodies are unequal, because

the superficies are neither equal nor unequal.[p. 415] Indeed to assert this for a law, that bodies are unequal while the superficies are not unequal, is the part of a man who takes to himself a wonderful liberty of writing whatever comes into his head. For reason and manifest evidence, on the contrary, give us to understand, that the superficies of unequal bodies are unequal, and that the bigger the body is, the greater also is the superficies, unless the excess, by which it is the greater, is void of a superficies. For if the superficies of the greater bodies do not exceed those of the less, but sooner fail, a part of that body which has an end will be without an end and infinite. For if he says that he is compelled to this, . . . For those rabbeted incisions, which he suspects in a cone, are made by the inequality of the body, and not of the superficies. It is ridiculous therefore to take the superficies out of the account, and after all to leave the inequality in the bodies themselves. But to persist still in this matter, what is more repugnant to sense than the imagining of such things? For if we admit that one superficies is neither equal nor unequal to another, we may say also of magnitude and of number, that one is neither equal nor unequal to another; and this, not having any thing that we can call or think to be a neuter or medium between equal and unequal.

[Plutarch Section 39]

Plutarch continues to say if this reasoning applies in this case, then why not for other types of figures?

Besides, if there are superficies neither equal nor unequal, what hinders but there may be also circles neither equal nor unequal? For indeed these superficies of conic sections are circles. And if circles, why may not also their diameters be neither equal nor unequal? And if so, why not also angles, triangles, parallelograms, parallelepipeds, and bodies? For if the longitudes are neither equal nor unequal to one another, so will the weight, percussion, and bodies be neither equal nor unequal. How then dare these men inveigh against those who introduce vacuities, and suppose that there are some indivisible atoms, and who say that motion and rest [p. 416] are not inconsistent with each other, when themselves affirm such axioms as these to be false: If any things are not equal to one another, they are unequal to one another; and the same things are not equal and unequal to one another? [Plutarch Section 39]

Let’s look at how Plutarch formulates the issue in the following sentence, because it will help us integrate this puzzle with the history of calculus. He says:

But when he says that

there is something greater and yet not exceeding, it were worth the while to ask, whether these things quadrate with one another. For if they quadrate, how is either the greater? And if they do notquadrate, how can it be but the one must exceed and the other fall short? For if neither of these be, the other both will and will not quadrate with the greater. [Plutarch Section 39]

Here the wording is, one figure is greater than another yet does not exceed it (in magnitude). If we think of each partition of the cone as narrowing to the infinitely small, then one such circular partition is larger than one of its immediate neighbors, however, it does not exceed or go beyond the smaller one by any finite amount. In that sense one partition can both be larger than its neighbor without extending beyond it. [See further discussion by Katz and Sherry.] Also notice Plutarch refers to quadrating the figures. Perhaps this is like finding the quadrature of a figure. If so, that would be interesting again with respect to the history of calculus. Boyer describes Archimedes’ method of find the area of a parabola segment using a method of exhaustion, in Archimedes’ text “Quadrature of the Parabola.” Let me quote from an old source that explains quadrature and its place in the history of calculus:

QUADRATURE, in geometry, denotes the reducing a figure to a figure. Thus, the finding of a square which shall contain just as much surface or area as a circle, an ellipsis, a triangle, &c. The quadrature, especially among the ancient mathematicians, was a great postulatum. The quadrature of rectilinear figures is easily found, for it is merely finding their areas or surfaces, i.e., their squares; for the squares of equal areas are easily found by only extracting the roots of the areas thus found. The quadrature of the curvilinear spaces is of more difficult investigation; and in this respect extremely little was done by the ancients, except

the finding the quadrature of the parabola by Archimedes. In 1957, Sir Paul Neil, Lord Brouncker, and Sir Christopher Wren, geometrically demonstrated the equality of some curvilinear spaces to rectilinear spaces; and soon after the like was proved, both at home and abroad of other curves, and it was afterwards brought under an analytical calculus; the first specimen of which was given to the public in 1688 by Mercator, in a demonstration of Lord Brounker’s quadrature of the hyperbola, Dr Wallis’s reduction of a fraction into an infinite series of division. Sir Isaac Newton, however, had before discovered a method of attaining the quantity of all quadruple curves analytically by his fluxions before 1668. It is disputed whether Sir Christopher Wren or Mr Huygens first discovered the quadrature of any determinate cycloidal space. Mr Leibniz afterwards found that of another space; and in 1669 Bernoulii discovered the quadrature of an infinity of cycloidal spaces both segments and sectors &c. [Note, previous word ‘infinity’ illegible in my source, and might be incorrect.] [Encyclopædia Perthensispage 526]

In the Boyer text, he describes how for the parabolic segment, Archimedes inscribes a large triangle, then other within the remaining space, and others still.

(From Boyer, p.52)

By summing the areas of the series of triangles, we can find the total area.

[So perhaps if we were to quadrate one of the circles of the cone, we would obtain a polygon with infinitely many sides and whose area differs from the circle’s only by an infinitesimal amount. In that sense, it would be less than the circle but no part of the circle would exceed the polygon’s boundaries by any finite amount. It does not seem that Plutarch is comparing the size of the circle with its quadrature, but rather the quatradatures of two neighboring circles. However, it is interesting that he evokes the method of exhaustion. Again, see Katz and Sherry for an excellent treatment of this cone example and Archimedes’ method of exhaustion.

Plutarch. *De communibus notitiis adversus Stoicos*. From* The Perseus Digital Library*

which cites the material thusly:

Plutarch. *Plutarch's Morals*. Translated from the Greek by several hands. Corrected and revised by. William W. Goodwin, PH. D. Boston. Little, Brown, and Company. Cambridge. Press Of John Wilson and son. 1874. 4.

Particularly the pages:

Section 37:

http://data.perseus.org/citations/urn:cts:greekLit:tlg0007.tlg138.perseus-eng1:37

Section 38:

http://data.perseus.org/citations/urn:cts:greekLit:tlg0007.tlg138.perseus-eng1:38

Section 39:

http://data.perseus.org/citations/urn:cts:greekLit:tlg0007.tlg138.perseus-eng1:39

Boyer, Carl.* The History of the Calculus and Its Conceptual Development (The Concept of the Calculus).* New York: Dover, 1949.

*Encyclopædia Perthensis Edinburgh: John Brown, Ancor Close, 1816*

**Viewed on Google Books:**

http://books.google.com.tr/books?id=SGvfGB5X_ncC&printsec=frontcover#v=onepage&q&f=false

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