what was the illness of trying to square a circle

There are three classical bug in Greek mathematics which were extremely influential in the development of geometry. These problems were those of squaring the circle, doubling the cube and trisecting an bending. Although these are closely linked, nosotros cull to examine them in split up articles. The present article studies what has become the most famous for these bug, namely the problem of squaring the circle or the quadrature of the circle as it is sometimes called.

Ane of the fascinations of this problem is that information technology has been of interest throughout the whole of the history of mathematics. From the oldest mathematical documents known up to the mathematics of today the problem and related problems concerning π have interested both professional mathematicians and amateur mathematicians.

One of the oldest surviving mathematical writings is the Rhind papyrus, named after the Scottish Egyptologist A Henry Rhind who purchased it in Luxor in 1858. Information technology is a curl about 6 metres long and $\large\frac{ane}{3}\normalsize$ of a metre broad and was written around 1650 BC by the scribe Ahmes who copied a document which is 200 years older. This gives date for the original papyrus of virtually 1850 BC simply some experts believe that the Rhind papyrus is based on a piece of work going dorsum to 3400 BC.

In the Rhind papyrus Ahmes gives a dominion to construct a square of area nigh equal to that of a circle. The rule is to cut $\large\frac{1}{9}\normalsize$ off the circle's bore and to construct a square on the remainder. Although this is non really a geometrical construction as such it does show that the problem of constructing a square of area equal to that of a circle goes back to the beginnings of mathematics. This is quite a good approximation, corresponding to a value of 3.1605, rather than 3.14159, for π.

The problem of squaring the circle in the form which we think of it today originated in Greek mathematics and it is non e'er properly understood. The problem was, given a circumvolve, to construct geometrically a foursquare equal in area to the given circle. The methods one was immune to utilize to do this construction were not entirely clear, for really the range of methods used in geometry by the Greeks was enlarged through attempts to solve this and other classical bug. Pappus, writing in his work Mathematical drove at the cease of the menses of Greek development of geometry, distinguishes three types of methods used by the ancient Greeks (see for example [ 5 ] ):-

There are, nosotros say, three types of problem in geometry, the so-chosen 'plane', 'solid', and 'linear' bug. Those that can be solved with straight line and circle are properly called 'plane' bug, for the lines by which such problems are solved have their origin in a plane. Those problems that are solved by the utilise of one or more sections of the cone are called 'solid' bug. For it is necessary in the construction to use surfaces of solid figures, that is to say, cones. There remain the third type, the so-called 'linear' trouble. For the construction in these cases curves other than those already mentioned are required, curves having a more varied and forced origin and arising from more irregular surfaces and from complex motions.

Now we usually think of the trouble of squaring the circle to be a problem which has to be solved using a ruler and compass. This is really asking whether squaring the circumvolve is a 'aeroplane' trouble in the terminology of Pappus given above (we shall often refer to a 'plane solution' rather than use the more cumbersome 'solutions using ruler and compass"). The ancient Greeks, however, did non restrict themselves to attempting to observe a aeroplane solution (which we now know to be impossible), but rather adult a keen variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method.

The first mathematician who is on record as having attempted to square the circle is Anaxagoras. Plutarch, in his piece of work On Exile which was written in the first century AD, says [ 4 ] :-

There is no place that tin can take away the happiness of a man, nor still his virtue or wisdom. Anaxagoras, indeed, wrote on the squaring of the circle while in prison.

Now the trouble must have become quite popular shortly after this, non just amongst a small number of mathematicians, just quite widely, since there is a reference to information technology in a play Birds written past Aristopenes in most 414 BC. 2 characters are speaking, Meton is the astronomer (see D Barrett (trs.), Aristophanes, Birds (London, 1978) or [ four ] for a shorter quote):-

Meton: I propose to survey the air for you: information technology will have to be marked out in acres.

Peisthetaerus: Proficient lord, who practice you lot think you are?

Meton: Who am I? Why Meton. THE Meton. Famous throughout the Hellenic earth - you must have heard of my hydraulic clock at Colonus?

Peisthetaerus (eyeing Meton's instruments): And what are these for?

Meton: Ah! These are my special rods for measuring the air. Yous come across, the air is shaped - how shall I put it? - like a sort of extinguisher: so all I have to do is to attach this flexible rod at the upper extremity, take the compasses, insert the signal here, and - yous see what I mean?

Peisthetaerus: No.

Meton: Well I at present apply the straight rod - so - thus squaring the circle: and there you are. In the eye you take your market place: direct streets leading into it, from hither, from here, from here. Very much the same principle, really, as the rays of a star: the star itself is circular, but sends out directly rays in every direction.

Peisthetaerus: Brilliant - the man's a Thales.

At present from this time the expression 'circle-squarers' came into usage and it was applied to someone who attempts the impossible. Indeed the Greeks invented a special discussion which meant 'to busy oneself with the quadrature'. For references to squaring the circle to enter a popular play and to enter the Greek vocabulary in this way, there must have been much action between the piece of work of Anaxagoras and the writing of the play. Indeed we know of the piece of work of a number of mathematicians on this trouble during this menstruum: Oenopides, Antiphon, Bryson, Hippocrates, and Hippias.

Oenopides is idea by Heath to exist the person who required a plane solution to geometry problems. Proclus attributes ii theorems to Oenopides , namely to draw a perpendicular to a line from a given point not on the line, and to construct from a given point on a given line, a line at a given angle to the given line. Heath believes that the significance of these elementary results was that Oenopides gear up out for the offset time the explicit 'aeroplane' or 'ruler and compass' type of structure. Heath writes [ 2 ] :-

... [Oenopides] may take been the get-go to lay down the brake of the means permissible in constructions with ruler and compasses which became a canon of Greek geometry for all plane constructions...

At that place is no record of any attempt by Oenopides to square the circle past plane methods. In fact it is a rather remarkable fact that the Greeks did not produce fallacious 'proofs' that the circle could exist squared by plane methods. The few claims for such false proofs rather seem to result from less able mathematicians declining to understand exactly what some of the more brilliant contributions to the problem were intended to show. Sadly afterwards mathematicians did not follow the practiced example shown past the ancient Greeks and indeed many claimed incorrectly to accept discovered a 'ruler and compass' proof. Amateur mathematicians, greatly attracted to the classical bug, have produced (and notwithstanding go along to produce) thousands of false proofs.

Antiphon and Bryson both produced arguments relating to squaring the circle which were to prove of import in the hereafter development of mathematics. Retort inscribed a square in a circle, then a regular polygon with viii sides, so one with 16 sides and he continued the process continually doubling the number of sides. It appears that Bryson improved the statement of Retort by not only inscribing polygons in a circumvolve but also circumscribed polygons. Themistius states [ 1 ] :-

... that Bryson declared the circle to exist greater than all inscribed, and less than all circumscribed polygons.

Hippocrates was the beginning to actually use a plane construction to notice a square with area equal to a figure with circular sides. He squared sure lunes, and also the sum of a lune and a circumvolve. Now although he squared certain lunes, he had non shown that every lune can be squared. In particular the lune that he squared in his plane structure of a square of area equal to that of a sure lune and a circumvolve was i he could non square by plane methods. Of course this lune cannot be squared by plane methods otherwise Hippocrates would have squared the circle. Although some, such as Aristotle, seemed to neglect to understand the logic of Hippocrates argument, there seems little dubiety that Hippocrates was perfectly aware that his methods had failed to square the circle. Examples of Hippocrates' methods of squaring lunes are given in his biography in this annal.

Hippias and Dinostratus are associated with the method of squaring the circumvolve using a quadratrix. The curve it thought to exist the invention of Hippias while its awarding to squaring the circle appears to exist due to Dinostratus. The structure of this curve with a diagram is given in the biography of Hippias in this archive. Now this bend certainly solves the problem of squaring the circle simply, as given by Hippias, the bend is synthetic past mechanical ways given by a uniform motion of a line in a time equal to the rotating radius of a circle. The structure was rightly criticised as requiring a noesis of the ratio of a line and an arc of a circle, so one assumed equally known the holding required to square the circle in the start place. It is clear that Dinostratus never claimed that the quadratrix gave a plane method to square the circle. Nicomedes many years afterwards also used the quadratrix to square the circle.

Aristotle did non seem to capeesh the contributions of those who had attempted to square the circle. He wrote in his work Physics :-

The exponent of whatever scientific discipline is not called upon to solve every kind of difficulty that may exist raised, but only such as ascend through imitation deductions from the principles of the scientific discipline: with others than these he need non business organisation himself. For example, it is for the geometer to expose the quadrature past means of segments, but it is not the concern of the geometer to refute the arguments of Retort.

In this quote "quadrature past means of segments" refers to Hippocrates quadrature of lunes which Aristotle mistakenly thinks was intended equally a proof that the circle tin can be squared past plane methods. Antiphon's methods come up in for even more criticism from Aristotle, just all credit to Antiphon whose methods contained of import ideas which would pb eventually to integration. Aristotle also wrote in similar terms in Sophistical refutations again probably having had handed down to him an incorrect estimation of what Retort and Bryson had attempted to show:-

The method by which Bryson tried to square the circle, were it ever and so much squared thereby, is yet fabricated sophistical past the fact that information technology has no relation to the matter in mitt. ... The squaring of the circle by means of lunes is not eristic, merely the quadrature of Bryson is eristic. The reasoning used by the erstwhile cannot be practical to any subject other than geometry alone, whereas Bryson's argument is directed to the mass of people who do non know what is possible and what is impossible in each section, for it will fit whatsoever. And the same is true of Antiphon's quadrature.

Next nosotros should consider the contributions of Archimedes to the problem of squaring the circle. Now Archimedes is famed for his introduction of the screw curve, but why did he introduced this bend? The authors of [ seven ] suggest three reasons:-

Is it for purely geometric reasons because he studied this curve every bit a ways of calculating π, and squaring the circumvolve? Is it because of his astronomical interests, trying to calculate geometrically the spiral movements of the planets? Or is it finally through the interest of a mechanical listen in a bend which results from the combination of ii regular uniform movements, one in a directly line the other in a circle? these three reasons are evident at ane and the aforementioned time...

Archimedes gives the following definition of the spiral in his work On spirals (run into [ v ] for example):-

If a direct line drawn in a plane revolves uniformly any number of times most a fixed extremity until it returns to its original position, and if, at the same time equally the line revolves, a signal moves uniformly along the straight line beginning at the fixed extremity, the point will draw a spiral in the airplane.

To square the circle Archimedes gives the following construction. Allow $P$ be the betoken on the spiral when it has completed one turn. Let the tangent at $P$ cutting the line perpendicular to $OP$ at $T$ . Then Archimedes proves in Proposition 19 of On spirals that $OT$ is the length of the circumference of the circumvolve with radius $OP$ . Now it may non exist clear that this is solved the problem of squaring the circle but Archimedes had already proved every bit the first proposition of Measurement of the circumvolve that the surface area of a circle is equal to a right-angled triangle having the ii shorter sides equal to the radius of the circumvolve and the circumference of the circle. And then the area of the circle with radius $OP$ is equal to the area of the triangle $OPT$ .

Both Apollonius and Carpus used curves to square the circle merely it is non articulate exactly what these curves were. The one used by Apollonius is called past Iamblichus 'sister of the cochloid' and this has led to various guesses as to what the curve might have been. Once more the bend used past Carpus of Antioch is chosen the 'curve of double movement' which Paul Tannery argued was the cycloid.

Now nosotros leave the ancient Greek period and look at later developments but the first comment we should make is that the Greeks were certainly not the only ones to be interested in squaring the circle at this time. Mathematicians in India were interested in the problem (see for case [ eleven ] ) while in Prc mathematicians such equally Liu Hsiao of the Han Dynasty showed himself to exist one of the prominent of those attempting to square the circumvolve in around 25 AD.

Some time later the Arab mathematicians were, like the Greeks, fascinated by the problem. In [ 6 ] the piece of work of al-Haytham on squaring the circle is discussed. Now al-Haytham aimed to convince people that squaring the circle was possible by a plane construction simply since his promised treatise on the topic never appeared he must at to the lowest degree have realised that he could not solve the problem.

Not long after the work of al-Haytham, Franco of Liège in 1050 wrote a treatise De quadratura circuli Ⓣ on squaring the circle. The text is reproduced in [ viii ] and [ 9 ] and in it Franco examines 3 earlier methods based on the assumption that π is $\large\frac{25}{eight}\normalsize , \large\frac{49}{16}\normalsize$ or iv. Franco states (reasonably enough) that these are fake, then gives his own construction which is based on the supposition that π is $\large\frac{22}{seven}\normalsize$ . Although this treatise is of great historical involvement, it does evidence how European mathematics at the time was far behind the ancient Greeks in depth of understanding.

Moving forrad to well-nigh 1450, Cusa attempted to evidence that the circle could be squared past a plane construction. Although his method of averaging sure inscribed and confining polygons is quite beguiling, it is one of the first serious attempts in 'modern' Europe to solve the problem. Again information technology is worth commenting that the ancient Greeks basically knew that the circle could non be squared past plane methods, although they stood no chance of proving it. Regiomontanus, who brought a new impetus to European mathematics, was quick to point out the fault in Cusa'south arguments.

The mechanical methods of the Greeks certainly appealed to Leonardo who thought nearly mathematics in a very mechanical way. He devised several new mechanical methods to square the circle. Many mathematicians in the sixteenth century studied the trouble, including Oronce Fine and Giambattista della Porta. The 'proof' by Fine was shown to be incorrect by Pedro Nunes shortly after he produced information technology. The ancestry of the differential and integral calculus led to an increased involvement in squaring the circle, only the new era of mathematics still produced fallacious 'proofs' of airplane methods to foursquare the circle. One such false proof, given by Saint-Vincent in a book published in 1647, was based on an early on type of integration. The problem was however providing much impetus for mathematical development.

James Gregory developed a deep understanding of space sequences and convergence. He practical these ideas to the sequences of areas of the inscribed and circumscribed polygons of a circle and tried to employ the method to prove that there was no airplane structure for squaring the circle. His proof essentially attempted to prove that π was transcendental, that is non the root of a rational polynomial equation. Although he was correct in what he tried to evidence, his proof was certainly not correct. Nonetheless, others such every bit Huygens, believed that π was algebraic, that is that information technology is the root of a rational polynomial equation.

There was all the same an involvement in obtaining methods to square the circle which were not airplane methods. For example Johann Bernoulli gave a method of squaring the circle through the formation of evolvents and this method is described in detail in [ 12 ] .

The historian of mathematics, Montucla, made squaring the circle the topic of his first historical work published in 1754. This was written at a time long before the problem was finally resolved, then is necessarily very outdated. The piece of work is, however, a archetype and even so well worth reading.

A major footstep forrad in proving that the circle could not exist squared using ruler and compasses occurred in 1761 when Lambert proved that π was irrational. This was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers tin can be synthetic with ruler and compass. It but led to a greater flood of amateur solutions to the problem of squaring the circle and in 1775 the Paris Académie des Sciences passed a resolution which meant that no further attempted solutions submitted to them would be examined. A few years afterwards the Royal Society in London besides banned consideration of whatsoever further 'proofs' of squaring the circumvolve equally large numbers of amateur mathematicians tried to achieve fame by presenting the Society with a solution. This conclusion of the Purple Society was described by De Morgan nearly 100 years later as the official accident to circumvolve-squarers.

The popularity of the trouble continued and there are many agreeable stories told by De Morgan on this topic in his book Budget of Paradoxes which was edited and published by his wife in 1872, the year after his death. De Morgan suggests that St Vitus be fabricated the patron saint of circle-squarers. This is a reference to St Vitus' dance, a wild leaping trip the light fantastic in which people screamed and shouted and which led to a kind of mass hysteria. De Morgan also suggested the term 'morbus cyclometricus' as beingness the 'circle squaring disease'. Clearly De Morgan found himself having to attempt to persuade these circumvolve-squarers that their methods were incorrect, yet many stubbornly held to their views despite the best efforts of the professional mathematicians. For example a certain Mr James Smith wrote several books attempting to evidence that $\pi = \large\frac{25}{eight}\normalsize$ . Of course Mr Smith was able to deduce from this that the circle could exist squared simply neither Hamilton, De Morgan nor others could convince him of his errors.

The final solution to the trouble of whether the circle could be squared using ruler and compass methods came in 1880 when Lindemann proved that π was transcendental, that is information technology is not the root of any polynomial equation with rational coefficients. The transcendentality of π finally proves that there is no ruler and compass construction to square the circle.

One might imagine that this would be the terminate of interest in the problem of squaring the circumvolve, merely this was certainly non the instance. It neither prevented the stream of publications claiming that π had some unproblematic rational value, nor did information technology prevent the stream of publications of quite correct constructions to approximately square the circle with ruler and compass. Equally an case of the onetime type of claim, the New York Tribune published a letter in 1892 in which the author claimed to take rediscovered a secret going back to Nicomedes which proved that π = iii.2. Maybe more surprising is the fact that there were many who were totally convinced by this letter of the alphabet and firmly believed thereafter that π = 3.two.

Amid the correct guess constructions to square the circle was one by Hobson in 1913. This was a fairly accurate structure which was based on constructing the approximate value of 3.14164079... for π instead of iii.14159265.... . More remarkable, however, was the ruler and compass constructions published by Ramanujan. In the Journal of the Indian Mathematical Society in 1913 in a paper named Squaring the circle Ramanujan gave a construction which was equivalent to giving the gauge value of $\large\frac{355}{113}\normalsize$ for π, which differs from correct value only in the seventh decimal identify. He ended the newspaper with the post-obit:-

Notation.- If the area of the circle be 140,000 square miles, then [the side of the foursquare] is greater than the true length past virtually an inch.

Among other constructions given by Ramanujan in 1914 (Gauge geometrical constructions for π, Quarterly Journal of Mathematics XLV (1914), 350-374) was a ruler and compass construction which was equivalent to taking the strange yet remarkable approximate value for π to be $(ix^{2} + 19^{2}/22)^{1/4}$ . Now this is 3.1415926525826461253.... which differs from π only in the ninth decimal place (π = iii.1415926535897932385...). For a circumvolve of diameter 8000 miles, the error in the length of the side of the square constructed was merely a fraction of an inch.

Additional Resources (prove)

Written by J J O'Connor and E F Robertson
Last Update Apr 1999

holmesfinestower.blogspot.com

Source: https://mathshistory.st-andrews.ac.uk/HistTopics/Squaring_the_circle/

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