Talk:Squaring the circle/Archive 1

Archive 1

Archive 2

The illustration on this page is very misleading at best, since it shows a square that obviously has a larger area than that of the circle, and appears to have sides of length as big as the diameter of the circle. Obviously lengths of the sides should be conspicuously shorter than the diameter of the circle. Michael Hardy 19:39 Mar 25, 2003 (UTC)

probably best to superimpose the two figures too. -- Tarquin 19:41 Mar 25, 2003 (UTC)

You are welcome to replace the figure with one of you own choice... -- Jörgen Nixdorf

Done. Do you like it ? Theresa knott 21:25 Mar 25, 2003 (UTC)

I just finished wrestling with some graphics software to create a new one. I turns out to be very similar to the one you've put there; the square and circle are concentric; mine is in black-and-white. Your circle looks a bit wider than its length, however. Can you adjust it or shall I replace it? Michael Hardy 02:06 Mar 26, 2003 (UTC)

By all means upload, I can colour your diagram in later if wanted. What do people think - is colour or b/w better. Theresa knott 08:47 Mar 26, 2003 (UTC)

Michael - you've uploaded the image with a different name. If you upload it with the same name so that it overwrites the old file you do two things.

1. you save having to edit the page to insert the new picture name - less work.
2. More importantly you don't orphan the old image that then has to be deleted by someone. Theresa knott 10:00 Mar 26, 2003 (UTC)

But it seemed prudent not to discard the old one yet, since there might be discussion of the relative merits of the two images. Michael Hardy 01:00 Mar 27, 2003 (UTC)

When you replace an image, the old version is not lost. It can be found on the image description page, and can, in fact be reverted.{just like you can revert a page to a prior version}. All you need to is add some text to the description page asking that the old image not be deleted for the time being as we are not sure which version we want to settle on Theresa knott 08:41 Mar 27, 2003 (UTC)

I am very pleased with the new image, please discard the old one. -- Jörgen Nixdorf

---

Great article--I just moved it to the more common name, as that's the Wikipedia naming convention. --The Cunctator

quadrature

The page dealing with quadrature states that it is impossible to square the circle with only a rul and compass, because only algerbraic numbers can be found in this way and pi is a non algerbraic number. I have squared the circle by that method and not being a mathematician, but a philosopher, I do know why it cannot be done. Any one like to explain, please. Try Google, 'quadrature how to square the circle' and then any one who claims the circle cannot be squared in this manner, please tell me I am wrong and why. Michael Rack 05/08/03.

you're wrong. next! -- Tarquin 10:44, 5 Aug 2003 (UTC)

You just explained yourself why what you did isn't correct. You can only create algebraic numbers by ruler and compasses. Pi is not algebraic, it's transcendental. Dysprosia 10:45, 5 Aug 2003 (UTC)

you're wrong. Nixdorf

found the link: http://www.geocities.com/mikerack_uk/ To me, those numbers have an accuracy that far exceeds the demands of all but a very few applications or procedures. -- that sentence alone shows you do not understand what mathematics is. -- Tarquin 21:20, 5 Aug 2003 (UTC)

Fractals?

If I remember correctly, there was an article in New Scientist several years ago about a mathematician who squared the circle by dividing it into a finite number of fractal-edged pieces. I have looked on Google, but I can't seem to find any more info about this. Has anyone heard of it? Axl 21:48, 14 Dec 2004 (UTC)

The mathematician was Laczkovich. See Tarski's circle-squaring problem. There really should have been a link to this article, so I'll add one now. --Zundark 22:05, 14 Dec 2004 (UTC)

Thanks, Zundark. :-) Axl 17:02, 15 Dec 2004 (UTC)

finding suitable category to express the idea of pointless scientific efforts

Discussion originally spread out on users’ talk pages

I don't think "Pathological science" is an appropriate category for squaring the circle. Squaring the circle is a mathematics problem proposed by ancient Greek geometers, not proved to be impossible until 1882. That is a substantial part of what the article says. Although it also mentions briefly that there are crackpots who ignore the discovery made in 1882, that is not a major part of what the article is about, nor should it be, in my view. Michael Hardy 19:52, 20 May 2005 (UTC)

Written at the same time and re-edited now:
I don't like the name "Category:Pathological science", either, but this was the closest i could find. If you have a better category that expresses the idea of many "free spirits" who've invested years in a misguided scientific endeavour, i'd be happy if you could assign it to that one. This article certainly should not only be under Category:pi! — Sebastian (talk) 20:06, 2005 May 20 (UTC)

I see no reason why Category:pi is not appropriate. It seems very appropriate to me. Michael Hardy 20:11, 20 May 2005 (UTC)

Of course it is! But it shouldn’t be the only one. Squaring the circle is the prime example for the text marked green above. We should provide a way to link it. How about if we rename that whacky category name to some other name that is less controversial? — Sebastian (talk) 20:24, 2005 May 20 (UTC)

Kennedy's assassination may be the prime example of a conspiracy of space aliens propounded by a certain author, about whom a Wikipedia article exists. But that doesn't mean that author or that theory should even be mentioned in the Wikipedia article about the Kennedy assassination. The fact that crackpots keep working on something proved impossible in 1882 is of at most minor importance in this article. Maybe this article could link to an article about those crackpots, and in that other article "pathological science" could be a category. "Pi" is probably the most important of the categories in which this article should be included. Doubtless there are others. "Patholigical science" and "pointless scientific efforts" should not be what this article is about. Michael Hardy 20:35, 20 May 2005 (UTC)

I would agree with you if someone had assigned the Pi article to PointlessScientificEfforts. However, that is not the case.

This is what we’re discussing:

$SquaringTheCircle\hookrightarrow PointlessScientificEfforts$.

You’re asserting that this is equivalent to this:

${KennedysAssassination}\hookrightarrow {AlienConspiracy}$.

This is incongruent. The correct equivalence would be:

$KennedysAssassination\hookleftarrow KennedyWasAssasinatedByAliensHypothesis\hookrightarrow AlienConspiracy$.

$Pi\hookleftarrow SquaringTheCircle\hookrightarrow PointlessScientificEfforts$.

HTH, Sebastian (talk) 22:09, 2005 May 20 (UTC)

Why? In effect you're comparing the hypothesis that Kennedy was assasinated by space aliens to the topic of squaring the circle. But the problem of squaring the circle is a legitimate mathematical topic, and that is not controversial. Michael Hardy 22:48, 20 May 2005 (UTC)

Now dont make me angry. Who brought up the "example of a conspiracy of space aliens"? I've gone the extra mile your way to indulge your eccentric example, and this is what i get in return! Please stop putting your own BS into other people's mouth. — Sebastian (talk) 23:38, 2005 May 20 (UTC)

I don't know what I've done that would make you angry. I'm attempting to discuss this. I am indeed asserting what you wrote:

$SquaringTheCircle\hookrightarrow PointlessScientificEfforts$.

is equivalent to this:

${KennedysAssassination}\hookrightarrow {AlienConspiracy}$.

legitimate mathematical topics

Squaring the circle is a legitimate mathematical topic. That is not controversial. The mathematical problem and the proof of impossibility published in 1882 are what this article is about. This article is not and should not be about crackpots working on squaring the circle after the publication, in 1882, of the proof of impossibility. Michael Hardy 03:35, 21 May 2005 (UTC)

The set of circle squaring mathematical problems includes both those with rules which allow perfectly valid solutions and those with rules which do not. Rather than imply that only one set, which has conditions that make the solution impossible and the results irrational, is legitimate for research, why not allow that as in the primary source Egyptian examples in the article, workable solutions are also of interest, particularly those which allow rulers to be used with unit fractions as calculators. The set of Egyptian mathematical problems includes both arithmetic and geometric series, plane, solid and analytic geometry, equations of at least the first and second degree, roots and powers, and what looks like matrix algebra using tables of continued fractions in the form of sliding logarythmic scales. The continued fractions part is of interest because while the math is essentially binary the tables of equivalents are fascinatingly subtle and sophisticated. For just one example the use of variable coordinates can at the same time generate both circle and spiral. Using rulers with varying increments and a set of tools which include things like the remen which is defined as the diagonal of a triangle formed by two other sets of coordinates, and could by extension be so defined in any number of dimensions, even the relatively simple concepts like entasis spiral off into uncharted areas.Rktect 11:50, September 10, 2005 (UTC)

Agree with Michael. Squaring the circle is one very important problem the ancients tried really hard but never got to solve. It was proved to be impossible to solve only relatively recently, and some people are still trying to do it (now that Fermat's last theorem is proved, this seems to be the only thing left. :) Oleg Alexandrov 04:29, 21 May 2005 (UTC)

... as such, it does not deserve to be considered a pathological science, no more than Fermat's last theorem can. Oleg Alexandrov 05:11, 21 May 2005 (UTC)

I am sorry about the confusion the category name caused. Please read what i wrote above. Have a good day! — Sebastian (talk) 06:47, 2005 May 21 (UTC)

This discussion continued on Wikipedia_talk:WikiProject_Mathematics#Talk:Squaring_the_circle

Dispute

The dispute is with regards to this edit by User:rktect. Dissecting his contribution sentence by sentence:

In the last millennium many mathematicians thought of the classical problems of Greek antiquity as irrational.

Exactly who are these mathematicians who thought that this compass-and-rule problem was irrational? I believe many mathematicians though it interesting or intriguing. And some spent significant time at it. The entire crux of the problem is the compass-and-rule limitation, not solving the problem in a practical sense. (And if this sentence is meant as a clumsy way of expressing that the problem involves a transcendental number, then the article already covers this in a much better manner).

Attempting to square a circle was viewed by many as proved impossible because since Pi is irrational the problem involves a transcendental number and is thus irrational or foolish to pursue. I agree the entire crux of the problem is the set of rules, but that bit of information is often presented below the fold. -- Comment by Rktect

The argument that they could be rigorously proven impossible in three dimensions was all that they found interesting even though Greeks such as Eratosthenes and Plato had built machines to solve the problem.

They certainly spent time on analyzing the problem, but what exactly machines are we talking about here?

curve drawing machines -- Comment by Rktect

Those who could readily solve the problem in four dimensions by allowing an arc to be defined by a compass mounted on a board which was moving in time and space were on no interest to mathematicians because the problem is not mathematically challenging in more than three dimensions.

This sentence does not make sense at all. I get the feeling the editor is confusing a compass/rule limitations with a limitation to 2D. If you forego compass/rule, there are various solutions, even in 2D. The talk of 4D is nonsense: it is like claiming that the act of drawing circle with a compass is a 4D thing because the compass is in 3D and it takes time, so that makes it 4D.

No matter how you solve it you have to break some of the rules. Since many of the rules have been added after the original ruler and compass conditions were stated that doesn't seem unfair. -- Comment by Rktect

In recent years the analytic geometry of the Egyptians unit fraction algorythms have become interesting. The mathematics underlying the solutions of the Egyptians who first attacked the Problem in the Rhind Papyrus are once again being studied because of their implications for continued fractions.

Who on Earth would claim that the mathemathical problems of the Rhind papyrus are about the classical squaring the circle by compass/rule limitations. That would be highly controversial. "You can see more about the Rhind papyrus in the History topic article Egyptian papyri.

"In the Rhind papyrus Ahmes gives a rule to construct a square of area nearly equal to that of a circle. The rule is to cut 1/9 off the circle's diameter and to construct a square on the remainder. Although this is not 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 (256/81) 3.1605, rather than 3.14159, for pi"

[rhind]

[rhind] -- Comments by Rktect

There is a construction by an Egyptian architect which dates back to the 3rd millennium BC which solves the problem by the use of two different coordinate systems.

We are certainly eager to hear exactly who came to this marvelous conclusion? No less than User:rktect himself, perhaps?

Egyptian circle Somers Clarke and R. Englebach "Ancient Egyptian Construction and Architecture" illustrations 53 and 54 dated to the 3rd millenium BC -- Comment by Rktect

The problem really dates back to the invention of geometry and has occupied mathematicians for millennia. It was not until 1882 that the impossibility was proven rigorously, though even the ancient geometers had a very good practical and intuitive grasp of its intractability. It should be noted that it is the limitation to just compass and straightedge that makes the problem difficult. If the straightedge is allowed to be a ruler or if other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area or to trisect an angle or double a cube.

What's this? Here is a section makes perfect sense, and is worthy Wikipedia material. What's up? It turns out this part of rktects edit is in fact a copy of the first section of this article. Is it only me, or are other editors of the opinion that it is all right to have a section appear twice in an article, since they are leaving this section intact?

right|320px|The square and the circle have the same area

This is an drawing by rktect, perhaps based on the link below:

Egyptian analytic geometry

My drawing was done from the information in the book, but the link does have the advantage of showing that the artifact exists. -- Comment by Rktect

This is a link to an amateur Egyptolog, who, it seems, based on a fraction of something claimed to be a drawing for a structure in Saqqara, extrapolates very far reaching conclusions. But even this webpage does not go as far as suggesting that the Egyptians were solving the Squaring the circle problem, as far as I can see. -- Egil 01:30, 12 September 2005 (UTC)

I used the link because it has the picture which can also be found in the book cited above, the mathematics goes its own way. The point I wished to make was that by using a ruler and compass, this architect from the 3rd millenium BC had managed a very close aproximation of a circle. Its often thought that using a ruler divided into 7 parts makes it easier to come up with a ratio like 3 1/7 but that isn't what is happening either here or in the rhind papyrus. -- Comment by Rktect

I agree with Egil. In particular:

• To say that "In the last millennium many mathematicians thought of the classical problems of Greek antiquity as irrational" makes it sound as if it's been a whole millennium since it was proved not solvable, and "irrational" is very definitely the wrong word; no mathematician thinks it is "irrational". I think rktekt was in fact clumsily trying to say that mathematicians have now found that the straightedge-and-compass challenge cannot be met and regard as irrational those who still insist on believing otherwise. But those latter people are not particularly important to this article.

• One should be very specific about these alleged machines, and one should make it clear that their existence does not in any way detract from Lindemann's impossibility proof (if indeed there are such machines -- reference please!).

Mechanical linkage for the conchoid of Nicomede Moving square of Newton These "instruments" are a among a large catagory of drafting aids that culminate in the line and circle drawing algorythms of modern cad programs that are capable of such interesting constructions as nurbs and splined polylines. -- Comment by Rktect

• The claims that some related problems are solvable are FAR to vaguely worded. They should be made clear enough so that a reasonable mathematician would know what is being said.

" Since you like this phrasing, and I do as well, lets use it again "If the straightedge is allowed to be a ruler or if other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area or to trisect an angle or double a cube" -- Comment by Rktect

Michael Hardy 23:34, 12 September 2005 (UTC)

Rktect 00:51, September 13, 2005 (UTC)

[[This article has now been reverted 3 times in the last hour by user drini who seems not to read discussion pages. The Egyptian architect at Saqarra and the rhind papyrus are well known examples of Egyptian circle squaring Rktect 02:56, September 13, 2005 (UTC)

Another of Rktect LIES against me.. Looking at page history: 3 consecutive edits with removal of content (discussed below) with diff [1] and then a second removal per same reasons at [2]. I haven't reverted 3 times (and for invoking 3RR against me, I'd have to revert 4 times, not 3 within 24 hours). This has been added to the RFAr as evidence of false claims in bad faith. -- (☺drini♫|☎) 05:05, 13 September 2005 (UTC)

Comments to the above: In its current state, after rktect having dumped text whereever it pleases him (and perhaps expecting others to clean up after him, I don't know), the above discussion is now pretty incomprehensible for outsiders, and has thus become largely valueless. (The starting point is here).

That said, nothing of the above seems to support the claims of rktect. If one in this article wants to discuss the problem of finding an approximation of the area of a circle as a precursor of the squaring the circle problem (which may, or may not, be relevant for this analytical problem), then the Rhind papyrus could be mentioned in the history section.

Wrt. the claims of 3 revertions by drini, these are quite obviously wrong. Drini made 3 edits, which is something quite different. -- Egil 07:42, 13 September 2005 (UTC)

Egyptian history with the problem

Both the circle squaring problems of the Rhind papyrus (involving a circle with a circumference of one mile which results in a a plane geometry algrythm used in other solid geometry problems) and the Saqarra architect's sketch are well known legitimite predecursors to the Greek interest in the problem. Incredulity simply is an inadequate reason to butt heads with the facts. Cites and references are given. If you go to ABE books you can buy and get the reference material shipped

cheap and fast so there is little excuse for your chosen methodology of simply not doing your homework and saying "nothing of the above seems to support the claims of rktect" is simply false.Rktect 10:33, 13 September 2005 (UTC)

Since much of the ground work to frame the problems was demonstrably done by Ahmes in the Hyksos period,it seems wrong to attribute it all to the Greeks just because a millenia or so later Thales, Solon, Plato and Pythagoras stopped by to visit and discovered the Egyptians had a nice collection of materials that deserved a Library and some commentaries. Its true that later Greek geometers reformulated all the Egyptians ideas into theorems which attempted new more restrictive rules and formulations of the general case, but its simply wrong not to give the Egyptians the credit for breaking ground on the problem. Rktect 10:33, 13 September 2005 (UTC)

The Saqqara architects sketch does some things that are obvious and some things that are less obvious and its dating is unchallenged. First it attempts to measure an arc in terms of its unit coordinates.Rktect 10:33, 13 September 2005 (UTC)

What's less obvious is that like the unit rise and run seked problems of the pyramids at Saqqara, it has different x and y units for its coordinates. What that means is that although each axis is internally integral the relation between them is not. One unit interval is 6/7 of the other. Rktect 10:33, 13 September 2005 (UTC)

Is there a prohibition in the problem against this substitution of coordinates for a straightedge?

There certainly was not at the time the Egyptians first created it along with the related trisection of the angle and the doubling of the cube. Many people have noted that the British Imperial System doubles its volumes. One person recently commented what's the big deal, its a binary system? What I think is a big deal is that the sides of the cubes containing the volumes are Egyptian units of measure. If you think that is original research you can find the volumes given in cubic inches in Klein's "The World of Measurements" p 46. The volume of 4 bushels is 8877.2 cu inches. the side of a cube containing that volume is 20.7" = 525.9 mm = 1 Egyptian royal cubit. Its half is the strike and its half is the bushel.Rktect 10:33, 13 September 2005 (UTC)

Since the point at which the compass used to swing the arc can be seen in the drawing its pretty clear its a portion of a circle. What the architect then does is record the height given in royal cubits (units of 7 palms or 28 fingers), palms and fingers at unit runs or intervals of an ordinary cubit (units of 6 palms 24 fingers). Rktect 10:33, 13 September 2005 (UTC)

I really enjoyed reading your intuitive insights to this riddle from antiquity! All this talk about 'compass-and-rule limitations' and 'the irrational prevelance of pi as a number (arithmetic or not)' is tiresome! My friends, the answer is in the pendentive! It is the closest method of visually seeing the process of 'squaring the circle'. Unfortunately, it is an architectural achievement that speaks volumes over your discussions!

drini

According to Drini's earlier comments to Zoe, substantive edits constitute reverts. That's aside from his notation of them, and the fact that there is supposed to be discussion first not after a change and that the attitudes evidenced are rather unwiki. It's neither lying nor inappropriate to remind someone that they are pushing a limit.Rktect 10:33, 13 September 2005 (UTC)

I've pointed my changes below, and the claim was that if consecutive edits reconstruct a previous version in the same way a revert would, then yes, it is considered. But I checked the previous version and the first 3 edits were not a revert, only the 4th, but even if you stretch that in your mind, it would account to 2 reverts, not breaking 3RR as you falsely accused me. -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

revision needed

Much of the information contained in the article is incorrect. A square with an area identical to the area of a given circle cannot exist in the real world. It is irrelevant what instrument is used to construct the square. The problem does not "become trivial". The impossiblity results from the fact that pi is a transcendental number. No real world object with length which is a multiple of pi can exist, even though the length can approximate a multiple of pi as closely as we would like. To construct a square with the same area as a given circle would require that the length of the sides of the square be a multiple of pi. 128.238.44.102 16:50, 12 September 2005 (UTC)

The comment above is a misunderstanding. The problem of squaring the circle is not about physical objects with circular shapges; it's about an abstract mathematical object: Euclidean space. And your assertion that physical objects cannot have ratios of lengths that are transcendental numbers I find very dubious. Michael Hardy 23:17, 12 September 2005 (UTC)

Reasoning behind removing the disputed content

A revision was indeed needed. I removed a whole chunk, and I'll explain why.

The ancient Greeks, however, did not restrict themselves to attempting to find a plane solution (which we now know to be impossible), but rather developed a great variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method. Greek circle squarers. Trisecting the angle. Doubling the cube.

First, The geometrical problem was about abstract (as in no physical) forms. This is important, the classical problem was to determine the side of a square with the same area as a circle with straigh-edged ruler and ocmpass. Any other mean is not about "squaring the circle" in the classic greek sense.

Then as usual, our friend rktect supplies links that supposedly backup his claims (but which actually don't). But let's check the links on the first links page (the greek circle squares):

• Anaxagoras: While in prison he tried to solve the problem of squaring the circle, that is constructing with ruler and compasses a square with area equal to that of a given circle. This is the first record of this problem being studied and this problem, and other similar problems, were to play a major role in the development of Greek mathematics.
• Antiphon: Antiphon made an early and important contribution to mathematics when he made an attempt to square the circle. In doing so he became the first to propose a method of exhaustion although it is not entirely clear how well he understood his own proposal. He proposed successively doubling the number of sides of a regular polygon inscribed in a circle so that the difference in areas would eventually become exhausted.
• Apolloniuis: No mention of circle squaring on the content.
• Archimedes: As above
• Bryson: Aristotles criticises him (...) for his method of squaring the circle. We do know some details of this methods of squaring the circle and, despite the criticisms of Aristotle, it was an important step forward in the development of mathematics. Aristotle's criticism appears to have been based on the fact that Bryson's proof used general principles rather than on geometric ones, but it is somewhat unclear exactly what Aristotle meant by this.
• Dinostratus It is usually claimed that Dinostratus used the quadratrix, discovered by Hippias, to solve the problem of squaring the circle.

but a few lines below:

Proclus, who claims to be quoting from Eudemus, writes

Nicomedes trisected any rectilinear angle by means of the conchoidal curves, of which he had handed down the origin, order, and properties, being himself the discoverer of their special characteristic. Others have done the same thing by means of the quadratrices of Hippias and Nicomedes.

This makes somewhat less convincing the claim that Dinostratus used the quadratrix, discovered by Hippias, to square the circle since Eudemus does not even mention Dinostratus.

Then, we should look at Hippias for the first mention of a curve used:

• Hippias: Hippias's only contribution to mathematics seems to be the quadratrix which may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts.
• Hippocrates: In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii.
• Sporus: Sporus worked mainly on the classical problems of squaring the circle and duplicating the cube. His solution of the problem of duplicating the cube is similar to that of Diocles
• Oepides: no mention of method
• Nicomedes: Nicomedes also used the quadratrix, discovered by Hippias, to solve the problem of squaring the circle.
• Diocles: no mention in the content.

Thus so far, only one special curve is mentioned, and no mechanical device is hinted. There is no evidence that quadratix was "invented specially for the purpose", so I removed that paragraph (unless Rktect can provide better sources for evidence). HOWEVER, the quadratrix bit is really important, so to get a better NPOV, it'd be much better to create a section on how the quadratrix relates and is used for gettinga "solution". Continuing with the disputed material:

Although squaring the circle by straightedge and compass could be rigorously proven impossible in three dimensions

Previous sentence makes no sense as it was not a space geometry problem with a plane problem. It is inded impossible to solve in Euclidean plane.

Greeks such as Eratosthenes and Plato had built machines to solve the problem of doubling the cube which is mathematically related to squaring the circle.

Evidence?

In recent years the analytic geometry of the Egyptians unit fraction algorythms have become interesting.

To our friend Rktect, obviously. No such thing as Egyptian analytic geometry.

The mathematics underlying the solutions of the Egyptians who first attacked the problem in the Rhind Papyrus are once again being studied because of their implications for continued fractions.

Not in the mathematics world. Continuous fractions have deep relationships with number theory, not geometry. The sentence "The mathematics underlying the solutions" is ambiguous and need to be stated more accurately. For instance, Egyptians knew thata triangle 3-4-5 is a right triangle, but that doens't mean that they knew the underlying concepts that make pythagora's theorem to work. Finally, the sentence:

There is a construction by an Egyptian architect which dates back to the 3rd millennium BC which solves the problem by the use of two different coordinate systems.

is nothing more than rktect's unsubstantiated claim pushing his believings on egyptian world (see his contribution history). Those are the reasons for the removal of the content. -- (☺drini♫|☎) 03:31, 13 September 2005 (UTC)

User drini is unfamiliar with the history of the problem before the Greeks redefined its constraints. The history section in the form before its deletion by drini pushed the problem back in time to its original Egyptian form. To neglect this leaves us wondering why Greeks such as Thales, Solon, Plato and Pythagoras should have been so influenced by their visits to Egypt as to engage in the discussion in the first place. Its also of interest that even the Greeks do not at first have the constraint that a ruler may not be used and this is reflected in the phrase "Ruler and compass construction" Rktect 13:59, 13 September 2005 (UTC)

I'm quite familiar with mathematuical history. I'm only stating that There is a construction by an Egyptian architect which dates back to the 3rd millennium BC which solves the problem by the use of two different coordinate systems. is unsubstantiated and I doubt it's completely true since nor greeks nore egyptians used coordinate systems. egyptians did NOT knew analytic geometry (and before you answer "google has 98000 links" as claimed on [3], try searching "egyptian analytic geometry" with quotes) -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

The links pages mention a number of Greeks who engaged in the related problems of squaring a circle, trisecting an angle and doubling a cube. Drini allows Anaxagoras, Aristotle, Dinostratus, Hippias, Hippocrates, Nicomedes, Sporus but denies Apolloniuis, Archimedes, Oenopides, Diocles.Rktect 13:59, 13 September 2005 (UTC)

I DID went and check the content on the page "greek circle squarers" that you gave, and while I acknowledge they tried to solve the circle squaring, the information posetd above is the only given in the pages you show. So *at least those apges* aren't evidence of having use a great variety of methods nor they mention the mechanical devices you talk about. I'm very pleased that this time you used a very trustworthy source (groups.dcs.st-and.ac.uk) but again you fell into your old habit of giving urls pretending they support claims stronger than they actually do. I'm not denying Archimedes, Diocles, etc, I'm only pointint that the evidence you gave, makes no mention about what they were trying.

"Both Apollonius and Carpus used curves to square the circle but it is not clear exactly what these curves were. The one used by Apollonius is called by Iamblichus 'sister of the cochloid' and this has led to various guesses as to what the curve might have been. Again the curve used by Carpus of Antioch is called the 'curve of double motion' which Paul Tannery argued was the cycloid."Rktect 13:59, 13 September 2005 (UTC)

"To square the circle Archimedes gives the following construction. Let P be the point on the spiral when it has completed one turn. Let the tangent at P cut 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 circle with radius OP. Now it may not be clear that this is solved the problem of squaring the circle but Archimedes had already proved as the first proposition of Measurement of the circle that the area of a circle is equal to a right-angled triangle having the two shorter sides equal to the radius of the circle and the circumference of the circle. So the area of the circle with radius OP is equal to the area of the triangle OPT."Rktect 13:59, 13 September 2005 (UTC)

The previous is a very interesting example that if often discussed. But that didn't solve the problem in the classic sense. -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

I'm not saying that it solved the problem "in the classical sense" I'm saying that before the conditions of the problem were changed there were pre-classical attempts at solution which are worth looking at. Rktect 07:48, 17 September 2005 (UTC)

" Oenopides is thought by Heath to be the person who required a plane solution to geometry problems. Proclus attributes two 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 set out for the first time the explicit 'plane' or 'ruler and compass' type of construction. Heath writes [2]:- ... [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions t the ruler and compasses which became a canon of Greek geometry for all plane constructions... There is no record of any attempt by Oenopides to square the circle by plane methods."Rktect 13:59, 13 September 2005 (UTC)

There is no record of any attempt by Oenopides to square the circle by plane methods. Notice also that the comoments from above are not to be found int he links that rktect presents as evidence. -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

Drini apparently misses the fact that the three classical problems of Greek antiquity change over time both in definition and methodology and thus have a history which it is perfectly valid to research. In addition there is good evidence that they pre-exist the Greeks in the form of the rhind papyrus problems and other problems posed by Ahmes.Rktect 13:59, 13 September 2005 (UTC)

The three classical problems of the greek did not change. That's why aythey're referred as the three classical problems of the Greek. What changed were the solution paths attempted. -- (☺drini♫|☎) 15:58, 13 September 2005 (UTC)

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. It is a scroll about 6 metres long and 1/3 of a metre wide 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 about 1850 BC but some experts believe that the Rhind papyrus is based on a work going back to 3400 BC. (This work is the sketch of the saqarra architect.) You can see more about the Rhind papyrus in the History topic article Egyptian papyri. In the Rhind papyrus Ahmes gives a rule to construct a square of area nearly equal to that of a circle. The rule is to cut 1/9 off the circle's diameter and to construct a square on the remainder. Although this is not 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 p.Rktect 13:59, 13 September 2005 (UTC)

If we're tot talk about pi's approximations, Egyptians weren't neither the only nor the first before greeks. -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

This is not original research or something that I just made up. That drini apparently doesn't know much about the topic may just be due to his being poorly informed on the history of mathematical problems and the discovery of various curves, and the machines used to generate them. Rktect 13:59, 13 September 2005 (UTC)

This is Rktect's ad hominem attack against me. It happens that I'm very knowledgeable about geometry, mathematics in general and its history. -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

An ad hominem argument, also known as argumentum ad hominem (Latin, literally "argument to the man"), is a logical fallacy that involves replying to an argument or assertion by addressing the person presenting the argument or assertion rather than the argument itself.

Of particular interest to me is the Egyptian architects analysis of a section of arc using the technology already developed in his time for determining a slope by unit rise and run. There is nothing particularly earth shattering about the method. He drew an arc with a compass on a standard Egyptian inscription grid and then he measured it to get its proportions. My understanding is that the description of an arc in terms of its x and y coordinates would be considered analytic geometry by some, particularly if as the Egyptians then generalized or reused the algorythm or formula in other problems. What I particularly liked about the problem was that he chose to use different intervals for his x and y coordinates so that while both remain integrals in their own axis they are not necessarily related as integrals Rktect 13:59, 13 September 2005 (UTC)

Yes, but that's your understanding, it's a subjective claim. It depends on both the interpretation of t he analysis. Notice that you cannot readily draw an arc with a compass other than a circle arc. So, what specific arc are you talking about? Can you describe (using modern mathematics) the equation of such curve? Can you prove it? -- (☺drini♫|☎) 15:58, 13 September 2005 (UTC)

The basic parameters that define a circle are the center coordinates, (xc,yc) and the radius r. We can express the equation of a circle in several forms using either Cartesian or polar coordinate parameters. A standard form for the circle equation is the Pythagorean theorem (x-xc^2) +(y-yc^2)=r^2.This equation could be used to draw a circle by stepping along the x axis in unit steps from xc-r to xc+r and checking the coresponding y values at each position as y = yc +/- sqrt r^2-(x-xc)^2.

I know circle's equation. I only want a single line as an answer: what was the equation of the curve the egyptians used t osquare the circle? -- (☺drini♫|☎) 05:57, 17 September 2005 (UTC)

I'm proposing they used an algorythm. Rather than a formula this routine actually transforms the circle routine into a discriminating function. Rewrite the normal equation for a circle as f(x,y)=x^2+y^2+r^2. When f(x,y)=0 we are on our circle. When it is greater than 0, we are outside our circle, and a negative result means we are inside the circle.

That is not an algorithm, by the way. But again you hit the nail. It's you who are proposing such claims. It's not backed up. Waht's the equation of the curve egyptians used to square circle? I need not the circle's equation, I need the equation of the auxiliary curve. Still, no evidence given. -- (☺drini♫|☎) 23:51, 23 September 2005 (UTC)

Actually its a common algorythm that you as a mathematician should recognize. What I find interesting about it is that its both really old and newly reintroduced as a line and circle drawing algorythm for cad work. Donald Hearn / M. Pauline Baker "Computer Graphics" Prentice Hall, 1986 ISBN 0-13-165382-2 Chapter 3 "Output Primatives" p 67 Bresenham's Circle Algorithm Rktect 02:07, 24 September 2005 (UTC)

No, that's not an algorithm. I recognize the set of conditions, but that's miles away from giving a clue or making progress towards squaring a circle. --

(☺drini♫|☎) 02:38, 24 September 2005 (UTC)

It is an algorithm, one that you should be familiar with as a mathematician. It makes it possible to describe a circle as a set of rectangular coordinates. The image in the upper right hand corner of the article uses it. The rhind papyrus takes the next step and squares a circle giving the area of a circle as equivalent to the area of a square. At this stage in the progression toward a general solution the question is still being formulated rather than the answer. Most people are familiar with the concept that in order to get the right answers you need to ask the right questions. Asking the right questions is in many cases harder than answering the questions once they have been properly stated. The work of finding a solution to the problem of squaring a circle really amounts to stating a set of conditions that is properly constrained to allow a general solution. Rktect 12:42, 24 September 2005 (UTC)

An improvement in the methodology takes advantage of the symetry of circles to reduce the multiplication and square root calculations of the Pythagorean theorem and the multiplications and trigonometric calculations of the parametric equations as much as possible to integer arithmetic. Integer positions along a circular path can be obtained by determining which of two points is nearer the circle (parabole) at each step.

details? I'm a mathematician, don't be afraid to throw in specific details, I can handle them. -

- (☺drini♫|☎) 05:57, 17 September 2005 (UTC)

Look at the sketch you keep deleting, see if you can find the unit intervals on the x axis and the measured coordinates on the y axis. Rktect 07:48, 17 September 2005 (UTC)

What the Pythagoreans called this was ellipsis, parabole or hyperbole according as the rectangle placed on a line segment fell short of the line segment, coincided with it or exceeded it. That is in fact what the Egyptian architect does. For a circle centered at the coordinate origin (xc=0 and yc=0)unit steps are taken in the x direction starting from x=0 and ending when x=y. The starting coordinate is thus (0,r). At a position (xi,yi) which has been choisen closer to the circle path the next point is then either (xi+1,yi) or (xi+1,yi-1). The actual y on the circle path is determined as y^2=r^2-(xi+1)^2. The Saqarra Architect uses as his unit step an ordinary cubit. He then writes the coresponding y values at each step and gives his results in royal cubits, palms and fingers. A measure of the difference in coordinate positions can be defined in terms of the square of the y values as d1=y^2i-y^2=y^2i-r^2+(xiI1)^2 and d2=y^2-(yi-1)^2=r^2-(xi+1)^2-(yi-1)^2. He now sets up a parameter for determining the next coordinate position as the difference between d1 and d2=2(xi+1)^2+y^2i+(yi-1)^2-2r^2. if p is negative he selects the point at position yi, otherwise he selects the point at yi-1. The test for the next point holds whether the actual path passes above yi or below yi-1

Rktect 00:56, 17 September 2005 (UTC)

what pythagoreans called an ellipsis was not the same as the ycalled an ellipse. And no, the difference isn't about a rectangle. They're defined as geometric locus where certain quantity (a distance, a sum or its difference) remains constant. your mathematics make no sense sir.

The etymology of the words ellipse, parabola and hyperbola can be traced back to Appolonius who comes up with the names for curves, ellipse, parabola, and hyperbola cut through a cone from the Pythagorean names for areas, ellipsis, parabole or hyperbole according as the rectangle placed on a line segment fell short of the line segment, coincided with it or exceeded it. Rktect 07:48, 17 September 2005 (UTC)

" Analytic Geometry. The study of the geometry of figures by algebraic representation and manipulation of equations describing their positions, configurations, and separations. Analytic geometry is also called coordinate geometry since the objects are described as -tuples of points (where in the plane and 3 in space) in some coordinate system." Rktect 13:59, 13 September 2005 (UTC)

[Analytic Geometry]

[squaring the circle]

Bottom line I would be interested to know specifically what claim is made that drini thinks is not backed up?

Yes I do know what analytic geometry is. I stand foir my claim that Egytptians did not know it. You claimed at Talk:Unit fraction that googling for egyptian analytic geometry gave over 90 thousand links, but I got zero links when looking for the phrase "egyptian analytic geometry". Here's some food for thre tought: if egyptians did knew analytic geometry, why is that greeks did not pick it up from the contact with egyptians (your main claim seems to be that greek math is in large part egyptian math). -- (☺drini♫|☎) 15:50, 13 September 2005 (UTC)

I thought you agreed there that I made my point and could rest my case? Rktect 01:32, 16 September 2005 (UTC)

Academic Greeks began coming to Egypt c 600 BC. Three centuries later they began to reformulate the old Egyptian problems into a more general case with more constraints. Looking at the Greek proofs out of context is sort of like looking at an iterative proof without the first iteration. If you look at analytic geometry outside of our own familiar notation the most important element is the use of coordinates. An Egyptian inscription grid is a coordinate system. In any work on Analytic Geometry which is the study of geometric properties by means of algebraic statements and methods, definitions begin with the distance betwen two points and the direction of a line expressed by a slope. Egyptian Sekeds problems use a coordinate system with the concept of slope. Definitions continue with Line segments and the Equation of a line. The Egyptians used Equations of the first and second degree, arithmetric and geometric series, and formulas and algorythms for the solution to problems using unit fractions do constitute geometric analysis. Analytic Geometry definitions include detached coefficient methods for finding the intersection of a pair of lines. Many people think of Egyptian arithmetic as cumbersome or primitive because they don't understand unit fractions but there is a real beauty there in the establishement of cannons of proportion that give rise to all nature of curves. In the Rhind papyrus there are problems dealing with the first five powers of seven, as well as squaring the circle. In the Egyptian Mathematical Leather Roll, a formula for a truncated pyramid illustrating the use of V = 1/3h(a2+ab+b2). In the seked problems there are a number of pythagorean triples, sets of positive integers like (3,4,5) that constitute the sides of right triangles, using the same sexigesimal algorythm a = 2uv, b= u2-v2, c= u2+v2 found in a secant table for angles 45 to 31 degrees. The Egyptians also first discovered amicable numbers. (see if you can think of a well known Egyptian example of the pair 220 with 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 summing 284 with 1, 2, 4, 71 and 142 summing 220). Egyptian inscription grids have proportional increments such that if 2^n-1 is prime then 2^n-1(2^n-1)is an increment, ie; 6 dj = 4 1/2" = fist, 28 dj = 21" = royal cubit, 496 dj = 31 feet = 124 palms. The pre Pythagorean unit of the remen is related to the foot and the quarter as three palms = 1 quarter, four palms = 1 foot, 5 palms = 1 remen. 6 palms = 1 cubit, 8 palms = 1 nibw, 10 palms = 1 double remen ie; x^2 +y^2=Z^2, x^3+y^3+z3=w^3 for x=3, y=4, z=5,w=6. Egyptian Horus eye fractions double the cube Eratosthenes measures the earth in Egyptian stadia, Appolonius comes up with the names for curves, ellipse, parabola, and hyperbola cut through a cone from the Pythagorean names for areas, ellipsis, parabole or hyperbole according as the rectangle placed on a line segment fell short of the line segment, coincided with it or exceeded it.If we conside the curve referred to by a Cartesian coordinate system having its x and y axis along a line AB and its perpendicular AR and if we designate the coordinates of arc P by xand y then the curve is an ellipse if y^2<px, a parabola if y^2 = px and a hyperbola if y^2>px. In the cases of the ellipse and hyperbola y^2= px +/- px^2/d where d is the length of the diameter through vertex A. That's just for starters. Rktect 01:32, 16 September 2005 (UTC)

More on Bresenham

Sorry to start a new section, but I am a bit confused. Can I deduct from the above that the opinion of rktect is that the ancient Egyptians were familiar with Bresenhams algorithm, and that the Egyptians (or their architects) used it to solve the squaring of the circle problem in an analytical manner? Just curious. -- Egil 13:33, 24 September 2005 (UTC)

Ancient math often does have implications for modern problems. The ancient Egyptians were familiar with the use of inscription grids which allowed them to draw a slope at a unit rise and run and in a similar manner to describe a circle. Modern mathematicians also find such graphs useful. You can read the rest of the comments to find out about the Egyptians method for finding the area of a circle that was a good aproximation to the area of a square. Rather than saying that the Egyptians were familiar with Bresenham's circle algorithm, or that he was familiar with the Rhind papyrus (which is more likely), its probably best to allow that both worked independant solutions to a similar problem. What Bresenham's algorithm leads to is first polylines and then nurbs and splines. I'm sure drini is familiar with them and can explain to you how the math works. Rktect 01:08, 25 September 2005 (UTC)

I take it that What Bresenham's algorithm leads to is first polylines and then nurbs and splines. is a theory of yours based on the same sort of insight in computer graphics algorithms that you have in subjects like ancient Egypt? -- Egil 07:58, 25 September 2005 (UTC)

You should credit Donald Hearn/M. Pauline Baker. Although in both cases the knowledge is based on solid research its not my original research. Donald Hearn/M. Pauline Baker "Computer Graphics" Prentice Hall, 1986 ISBN 0-13-165382-2 Chapter 3 Output Primatives p 76 Exercise 3.2 "Extend Bresenham's line algorithm to generate lines with any slope. Implement the polyline command using this algorithim as a routine that displays the set of straight lines between the n output points. For n = 1 the routine displays a single point." Rktect 17:16, 25 September 2005 (UTC)

The "source" you are citing is simply a textbook in computer graphics, where the student is asked to expand an implementation of Bresenhams line drawing algorithm so that it can handle polylines. In fact, I'm afraid your statement "What Bresenham's algorithm leads to is first polylines and then nurbs and splines." is just as meaningless as saying that "What the espresso machine leads to is coffee plants.". But thanks for the insight. -- Egil 06:29, 27 September 2005 (UTC)

Approximations using rationals

The section Modern approximations has a hall of fame: Ramanujan, one C. D. Olds, Martin Gardner, and one B. Bold [is that WP:BOLD?], who allegedly all gave geometric constructions for 355/113. Is that worth remarking? Who are these Olds and Bolds, and why is it remarkable that anyone (including no-one less than Ramanujan) gave such a construction unless there is something notable about the construction specifically -- in which case we should read here what that is. Any ancient Greek geometer could have constructed 355/113, they only did not know it was a good numerical approximation. I can construct 21053343141/6701487259 = 3.14159265358979323846238... . If I manage to get that published in the Journal of Incredibly Tedious Mathematics, will I be induced in the Wikipedia Hall of Fame? Lambiam^Talk 10:59, 17 April 2006 (UTC)

I have seen some of these constructions; they are fairly elegant, but would be tedious to describe without a diagram (most of them depend on 113 = 7² + 8²). I would leave this in, and possibly ask someone with the tools to make a diagram or two. Septentrionalis 14:27, 17 April 2006 (UTC)

I have tried to expand the introduction to the section to make it clearer what it is about. But the relevance entries are really impossible to judge without having some measure of how complex each of the constructions is. Could someone in the know please provide an indication of this? Henning Makholm 09:46, 28 May 2006 (UTC)

Archimedean spirals

If other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area.

The first part of the sentence (now removed), dealt with the use of rulers, which will solve the problem of trisecting the angle, but not this one. I think this sentence has the same confusion, since the Archimendean spiral can, trivially, be used to trisect, but it is much harder to see how it solves transcendental problems. Septentrionalis 14:22, 17 April 2006 (UTC)

Hey you are missing the boat the real task is r2= pi (r2) the circumference of a circle to area of any square. I am not a math wizz but a craftsmen who wanted a dome over a square padio robb ha ha i did it .it looks great. —Preceding unsigned comment added by 4.159.77.135 (talk • contribs) 2006-05-06 02:35:01