# The three unsolved problems of antiquity

In Sidelights on the Cardan-Tartaglia Controversy (Apr., 1938) by Martin A. Nordgaard in the National Mathematics Magazine, Vol. 12, No. 7, pp. 327-364, it is written on the first page

The solution of the cubic had presented itself to the human mind as an intellectual problem already in the fifth century B. C.; it became a scientific need in Archimedes' calculation on floating bodies in the third century B. C.; it confronted the Arab astronomers in the Middle Ages. And now it was solved! The first of "the three unsolved problems of antiquity" to be solved.

But which are the other two of "the three unsolved problems of antiquity"? What people usually seem mean when they refer to "the three unsolved problems of antiquity" are the trisecting of the angle, the doubling of the cube (or the Delian problem) and the squaring of the circle (just use Google to verify). Evidently Nordgaard refers to these as one. The other problem could then very well be the question of the necessity of Euclid's fifth postulate.

Is there any other place were this terminology is used in this manner, or is it just an imagination of Nordgaard? Are my suspicions correct? Or does he refer to different problems? Was the solution to the cubic indeed such a "big deal" to the ancients?

• Why do you think ("evidently", even) that the quote regards the three classical problem as one? Solving cubics won't help you square a circle. It is related to angle trisection, but mostly by the fact that Cardano's formula requires one to take complex cube roots, which assumes that we can already trisect angles! (Likewise it assumes that we can take real cube roots, so Cardano is really no help with either doubling a cube or trisecting an angle). – Henning Makholm Apr 29 '16 at 22:09
• About the big deal: I am not versed in that matter, but it seems natural to me that after solving the linear and quadratic equations with some ease, the Ancients attacked the cubic and hit a wall, which made it an attractive challenge. – Yves Daoust Apr 29 '16 at 22:12
• @Jori: I still don't see how you conclude that the author treats the construction problems at one problem. Your comment just repeats that conclusion, but makes me none the wiser why you conclude that. It seems to me to be a total non-sequitur to say that you think the author considers them to be one problem because "they are extremely well-known" -- what does being well known have to do with your conclusion that such-and-such author cannot count to three? Throwing around words like "evidently" is not going to actually make such an unexplained jump to strange conclusions evident. – Henning Makholm Apr 30 '16 at 10:24
• The much more straightforward interpretation is that when this author refers to three classical problems, what he means is actually the three classical problems: (1) doubling the cube, (2) trisecting an angle, (3) squaring the circle, counting them as three because there are three of them. Somehow the author must have gotten into his head that one of these problems -- probably cube-doubling -- would be considered "solved" by having an algebraic formula for solutions to general cubics using roots, even though this is not the really the same as the classical problem. – Henning Makholm Apr 30 '16 at 10:29
• It is not difficult to see how such a misunderstanding can come about, because someone who thinks in terms of numbers (like we all tend to do these days) can easily end up thinking the real content of the cube-doubling problem is to solve the equation $x^3=2a^3$. When one hears that Cardano found solutions to all cubics in general, that can sound a lot like cube-doubling was therefore resolved as a special case of this. (Only this isn't really true because Cardano's method assumes one can already take cube roots, that is, it does not simplify equations of the form $x^3=\text{constant}$). – Henning Makholm Apr 30 '16 at 10:34

## 2 Answers

Here I provide some text passages which rather indicate that the three great/unsolved problems of antiquity

• duplication of the cube (Delian problem)

• trisection of the angle

• squaring the circle

were not considered as three connected problems by the ancients.

These three problems are discussed in B.L. van der Waerdens Science Awakening Vol. 1. Most of the following information is from chapter VII, the Alexandrian Era (330 - 200 B.C.)

duplication of the cube:

• (p. 124) ... The pure cubic $x^3=V$, in geometric form, poses the problem of contructing a cube of given volume. The ancients were concerned with this problem as well; a special case is the famous duplication of the cube, ...

• (p.136) ... Hippocrates (ca. 470 - ca. 410 B.C.) must also have occupied himself with the duplication of the cube, more exactly with the problem of constructing a cuve whose volume has a given ration (e.g. $1:2$) to that of a given cube.

It is an established tradition (see Eutocius in Archimedis Opera III, 104) that Hippocrates was the first to recognize that the problem would be solved if one could construct two mean proportionals $x$ and $y$ between two given line segments $a$ and $b$ whose ratio is the given ratio, \begin{align*} a:x=x:y=y:b\tag{1} \end{align*} For, as we have developed before, if (1) holds then the cube $a^3$ would have to $x^3$ the same ratio as $a$ has to $b$. ...

Section The duplication of the cube

... I shall discuss the history of the problem in some detail, known to us chiefly from Eutocius' commentary on Archimedes. A few of the main points will be touched upon now. We have already seen that Hippocrates of Chios reduced the general problem of increasing the volume of a cube in a given ratio to that of constructing two mean proportionals between two given line segments. Archytas of Taras (428 - 347 B.C.) discovered an extremely ingenious three-dimensional solution of this problem. Later, solutions were given by Eudoxus, Menaechmus, Plato (?), Eratosthenes, Nicomedes, Appolonius, Heron, Philon of Byzantium, Diocles, Sporus and Pappus. Obviously great importance was attributed to this problem.

In the supplement to the Laws of Plato, known as the Epinomis, counted as one of Plato's works but not published until after his death, the problem of the two mean proportionals is indeed considered as the problem of solid geometry.



About Archytas of Taras:

• ... The versatility of this remarkable South-Italian Dorian is unparalleled even in that era. By means of an extremely ingenious space construction, he obtained a solution of the famous Delian problem, the duplication of the cube.

On page 150,151 we can read about the really ingenious construction. See here the corresponding graphic Figure (1) which is also shown in v.d. Waerdens book. Later on, when v.d. Waerden discusses the possible origins of the sources we can read about two legends.

... Fortunately, a quotation in Theon of Smyrna gives the source of the second legend

In his work entitled Platonics, Eratosthenes relates that, when God announced to the Delians through an oracle that, in order to be liberated from the pest, they would have to make an altar, twice a great as the existing one, the architects were much embarassed in trying to find out how a solid cube could be made twice as great as another one. They went to consult Plato, who told them that the god had not given the oracle because he needed a doubled altar, but that it had been declared to censure the Greeks for their indifference to mathematics and their lack of respect for geometry.

The same story, in almost the same words, is found in Plutarch (de Ei apud Delphos, 386 E). Elsewhere (de genio Socratis 579 CD), he adds that Plato referred the Delphians to Eudoxus and to Helicon of Cyzicus for the solution of the problem. ...

Trisection of the angle:

We start with Nicomedes (280 - 210 B.C.)

• (p. 235) ... Chronologically, Nicomedes comes between Eratosthenes and Apollonius, since he criticised Eratosthenes' duplication of the cube, while Appolonius refers somewhere to the cochloid of Nicomedes. Apparently he belonged to the same group of Alexandrian mathematicians as Eratosthenes and Apollonius.

... Pappus states that Nicomedes described an instrument, with which the curve can be drawn. ...

... Nicomedes took great pride in the discovery of this curve. He applied his neusis-construction to two famous problems, viz. the trisection of the angle and the duplication of the cube. ... The underlying idea of the trisection will be seen to be the same as that of the trisection of Archimedes.

But Nicomedes only uses a neusis between two straight lines, while Archimedes calls for a neusis between a circle and a straight line. Pappus shows that the neusis of Nicomedes can also be obtained by the intersection of a circle and a hyperbola. Nicomedes used the intersection of his cochloid with a straight line, every bit as simple, since a cochloid is defined and drawn more easily than a hyperbola.

It's interesting to note how the ancients considered the use of different techniques and means to solve geometric problems.

About constructions by means of compasses and straight edge by the ancients

(p. 264) The idea, sometimes expressed, that the Greeks only permitted constructions by means of compasses and straight edge, is inadmissible. It is contradicted by the numerous constructions, which have been handed down, for the duplication of the cube and the trisection of the angle.

It is true however, that such constructions were considered to be more elementary, and somewhere Pappus says that, whenever a construction is possible by means of straight edge, more advanced means should not be used. See in this connection A.D. Steele, Quellen und Studien, B 3, p.287.

Some hundred years later Pappus of Alexandria (290 - 350 A.D.) did a great unifying work

• (p. 286) ... The reader will gradually have gotten some idea of his principal work, the Mathematical Collection, ... In this work, Pappus brought together everything in the work of his predecessors which interested him: on higher plane curves, on the quadrature of the circle, the duplication of the cube, the trisection of the angle, the method of analysis, etc.

Wherever explanations of or supplements to the works of the great geometers seemed to him necessary, he formulated them as lemmas. This gives us a great deal of varied information about what was contained in the lost works of Euclid and of Apollonius. But furthermore, Pappus supplemented and extended in several points the work of his predecessors.

The third of the problems:

Squaring the circle:

• (p. 130) ... It has already been mentioned that, while he was in prison, Anaxagoras, (510 - 428 B.C.) occupied himself with the quadrature of the circle.

Altogether this problem was very popular towards the end ot the 5th century. The comic poet Aristophanes even made a joke about it. In The Birds, he introduces the astronomer Meton, who says: ... Meton's quadrature was only a jest.

But, although not accurate, the quadrature of the circle of the sophist Antiphon was intended to be serious. ... He inscribes some polygon in the circle, perhaps a triangle or a square. By bisecting the arcs, he obtains a polygon of double the number of sides, a hexagon or an octagon.

He believed that, by continuing in this way, he would ultimately obtain a polygon, whose sides, on account of their small size, would coincide with the circumference. Since for any polygon, one can construct a square of equal area, and since the polygon that has been obtained is to be considered as equal to the circle, one can construct a square as equal to the circle, he concluded that the circle can also be squared.

• (p.131) ... The most famous among the geometers of the 5th century, Hippocrates of Chios (not to be confused with the still more famous physician Hippocrates of Cos), also occupied himself with the quadrature of the circle.

... We can get some idea of the way in which he attacked this problem from a statement which Simplicius, the most learned and most reliable among the commentators of Aristotle, gets from his teacher Alexander of Aphrodisias. According to this Alexander, Hippocrates began with an isosceles right triangle and proved that the sum of the areas of the two lunules, formed by the semicircles on the right sides and by the semi-circle circumscribed about the triangle, is equal to the area of the triangle. ...

Another contributions from Dinostratus and Nicomedes

• (p. 191) ... In the large compendium of Pappus, which must have been written in the time of the emperor Diocletian (284 - 305), it is mentioned that Dinostratus and Nicomedes used for the quadrature of the circle a curve, which for that reason was called the quadratrix. ...

Conclusion:

• The impression when looking at this information is that different ingenious geometers have attacked these problems with different techniques. Sometimes as we could see by Nicomedes one technique, the Neusis construction could be used to attack two of the three problems. But there is no evidence, that the ancients have considered the three problems as related.

• It seems to me that nowadays for our convenience these problems are put together, since these are famous examples of what can't be solved by means of compasses and straight edge alone. A typical example is chapter 10 of this introduction to Galois theory by Richard Koch.

• I would like to point to an instructive answer of a related question by DavidSpeyer who transforms the problem to a more familiar one, namely to the standard five function calculator: $+$, $-$, $\times$, $\div$, $\sqrt{ \ }$. So the question was: using these keys, and the ability to type in integers or recopy numbers that you have computed before, what can you compute? Can you compute $\sqrt{2}$, or $\cos (20^{\circ})$?.

• A typical convenient representation for us nowadays is presented by Jennifer Doung and Daniel Szara in The 3 unsolved problems of Greek Mathematics.

• @Jori: Thanks a lot for granting the bounty! :-) Btw: I have also sources from Oscar Becker a mathematician and historian, well-known in the german community for his "Grundlagen der Mathematik in geschichtlicher Entwicklung" about foundations in mathematics. In his book "Das mathematische Denken der Antike" (roughly: Mathematical Thinking in Ancient Times) he presents aspects similar to B.L. v.d. Waerden which also supports the conclusion section. – Markus Scheuer May 9 '16 at 8:33

It took me a few minutes to figure this out but I think this is an issue of deceptive terminology. Mathscinet lists only 6 articles by Martin Norgaard who seems to have been more of a historian than mathematician. I think there is a subtle mathematical point here that he did not appreciate. The 17th century solution of the cubic does not answer the question about the cube (as pointed out in the comments) even though the terminology is similar.

The solution of the cubic equation could not have been a concern of the ancients because they had no way of expressing it, lacking the algebraic notation that was only developed later. The ancients dealt with conic sections which can be expressed and studied geometrically.

If you are interested in examples of mathematical incompetence on the part of certain historians of mathematics you could consult such examples given in this 62-page study of Euler.

• Nordgaard's 1922 Ph.D. Dissertation (under David Eugene Smith), A historical survey of algebraic methods of approximating the roots of numerical higher equations up to the year 1819, is freely available on the internet. However, in looking through it I did not see anything helpful in answering Jori's question. My guess is that Nordgaard meant the three ancient construction problems and was a little careless in his exposition (in the way that both Henning Makholm and Mikhail Katz have pointed out). – Dave L. Renfro May 3 '16 at 17:09
• @DaveL.Renfro, did he talk about Simon Stevin's solution of a cubic by divide-and-conquer? – Mikhail Katz May 3 '16 at 17:39
• I just looked (been busy with something else since I posted my comment), and there is a little discussion of Simon Stevin on pp. 21-22. By the way, Nordgaard's ordering of topics is chronological, and this made the Simon Stevin discussion easy to locate. Humm...looking again, I see that Stevin is mentioned in the table of contents. For some reason I missed this when I looked a few moments ago, which is why I had to use the chronological ordering of topics. – Dave L. Renfro May 3 '16 at 18:27
• The ancients definitely did talk about cubic equations, even if they lacked the algebraic notation for it. – Doug M May 3 '16 at 21:55
• @DougM, that depends on what you mean by "the ancients". Simon Stevin is ancient even by the standards of infinitesimal calculus, and he definitely talked about cubic equations, which he presented as a proportion to fit the contemporary style. Could you be more specific? – Mikhail Katz May 4 '16 at 7:16