Talk:Unit fraction

Explanation of revert

The revert I made it's because the content added by Rktect is a POV attempt to push his beliefs on egyptian world. Namely, that egyptians did knew how to square the circle.

[[Image:egyptian circle.jpg|right|400px|The square and the circle have the same area]]

Such image implictly states that egyptians were squaring circles, which accepted knowledge states as false.

"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. The Moscow and Akhmim Wooden

tablets date to 2,000 BC, and show clear links to the RMP. 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, because all of Egyptian arithmetic was based on remainders.

The point is that attempting to square the circle is not the same as actually doing it. And if by approximations of pi we are talking, egyptians aren't neither the only nor the first before greeks. 132.248.196.25 15:25, 13 September 2005 (UTC)

[rhind papyrus]

user drini may not be very well informed on mathematics and the history of curves, but there is certainly no reason to suggest this is anything but the accepted mainstream position regarding Egyptian mathematics. It is well documented by Gillings in "Mathematics in the time of the Pharoahs" . -- Rktect

Ok, that paragraph is nothing more than attempting an Ad hominem attack. I'm more informed about mathematics, geometry and history of mathematics than you think. 132.248.196.25 15:25, 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.

The point is that you are unfamiliar with the topic, which is well enough known to be recognized by most mathematicians who are familiar with geometry and the history of mathematics, and yet you want to take it upon yourself the right to deny others the opportunity to have access to the information. Rktect 17:50, 13 September 2005 (UTC)

No I'm not. I'm a mathematician and I'm familiar with geometry and its history. But I won't further reply to your ad hominem attacks, but we can discuss other things. -- (☺drini♫|☎) 19:00, 13 September 2005 (UTC)

There is a construction by an Egyptian architect which dates back to the 3rd millennium BC which attempts to solve the problem by the use of two different coordinate systems. It should be noted that it is the limitation to just compass and straightedge that makes the problem difficult.

The ancient Egyptians and later the Greeks, did not restrict themselves to attempting to find a plane solution but rather developed a great variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method." Many of these devices became common drafting aids in the offices of architects. Some of the dimensioned drawings of ancient Egyptian architects can be found in Somers Clarke and R. Englebacj "Ancient Egyptian Construction and Architecture", Dover 1990.

The discussion about the curves used by greeks on circle squaring are to be found at Talk:Squaring the circle where the three links mentioned below are analyzed and showing that they do not support a claim as strong as Rktect pretends

[circle squarering]

[Trisection]

[and .ac.uk/~history/HistTopics/Doubling_the_cube.html Doubling a cube]

The Greeks began visiting Egypt and picking up useful bits of mathematics in the time of Thales. Solon, Plato, Pythagorus and Herodotus all visited Egypt and over a period of three centuries wrote about their religion, philosophy, mathematics, science, history and geography. For another three centuries in the Ptolomaic period, a great library was established at Alexandria where Greeks and Romans from all over the Mediterranian world came to study and to copy and extrapolate on the Egyptians ideas. Well into the Medeival period Egyptian unit fractions were the standard mathematical notation. To just ignore that millenia of exploration of Egyptian mathematics like it didn't exist and to just attribute all of their ideas to the Greeks and Romans who wrote about them latter seems more than a bit ethocentric. -- Rktect

I wasn't claiming that greek mathematics did not benefits from contsact with Egypt, but it is stablished knowledge that greeks did advance the field greatly from their predecessors, which were only calculational into a methodic and abstrac-concept based corpus. However the lciam that egyptian fractions were the standard mathematical notation well into medieval ages is plain false. To just ignore the millenia of mathematical developments from other later civilizations like they didn't exist and to just attribute the developments to the egyptians who made first attempts seems more than a bit ethocentric. 132.248.196.25 15:25, 13 September 2005 (UTC)

The problem really dates back to the invention of geometry and has occupied mathematicians for millennia. Ancient geometers had a very good practical and intuitive grasp of the problems complexity.

No, the circle squaring problem does not dates as back as the earliest geometry problems, which were calculational, not constructional.

drini may wish to take that up with the other cited authors who disagree, but the earliest circle squaring problem I know of, the one refered to above, is indeed Egyptian and dated to its earliest dynasties. As to whether its calculational as opposed to constructional, a sketch of an arc is drawn, apparently with a compass and then its x and y coordinates are measured, there being no prohibition against the use of a ruler in the Egyptian history of the problem.

My point is that origins of geometry date way back further. 132.248.196.25 15:25, 13 September 2005 (UTC)

Could you supply a date? The Old Babylonian Plimpton type problems would be c 1,750 BC whereas the saqarra sketch is c 3,000 BC. If you know of any older geometry you have the advantage on me. Rktect 17:24, 14 September 2005 (UTC)

Still using Egyptian unit fractions for their calculations, Greeks such as Eratosthenes and Plato built machines to solve the problem of doubling the cube which is mathematically related to squaring the circle.

No, greeks did not have a decimal system, but they weren't limited to using Egyptian fractions, for they could use rations that had other demonimators than 1. Indeed, any positive rational number can be represented as a finite sum of unit fractions, but in practice this is not as easily doable without good arithmetic system: 2/17 decomposes as egyptian fractions as 1/9 + 1/153, and denominators can become ugly even for small fractions. Furthermore, the fact that greeks did use all rational numbers in their calculations is illustrated by the fact that they believed ANY number was indeed as rational number (ergo the shock with sqrt(2))

1.) The Greeks did have a decimal system, many of their measures are decimal multiples.

2.) They could use whatever notation they wanted but the fact is the use of Egyptian unit fractions is common in Greek mathematics

3.) The Egyptians had a good aritmetic system

4.) If you actually study unit fractions you will find that the Egyptians could and did find unit fraction equivalents using rulers as calculators, and there is nothing ugly about their mathematics.

5.) [the contra argument] [next in thread]

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.

See comments in Talk:Squaring the circle. BAsically, it's only to Rktect and a small handful of people that believe that such think as egyptian analytic geometry existed.

Drini apparently found that it isn't just me (92,800 google hits for Egyptian analytic geometry)

Actually, googling for "egyptian analytic geometry" with quotes gives zero results.

"Engels, Hermann. Quadrature of the circle in ancient Egypt. Historia Math. 4 (1977), 137--140. (Reviewer: L. Guggenbuhl.) SC: 01A15, MR: 56 #5124. Explains the Egyptian formula for the area of a circle in terms of the practices of Egyptian stone masons. In order to form a relief, the stone masons covered their designs with a grid. The hypothesized construction involves an error which would confirm the now commonly held view that the ancient Egyptians did not properly understand the Pythagorean theorem. Closely related topics: Ancient Egypt, The Circle, Coordinates, and Pythagorean Triangles and Triples. "

[analytic geometry]

As phrased, it sounds that it's indeed topic subject to research by many mathematicians and scientists, whereas it is not. Therefore, I reverted [1]. -- (☺drini♫|☎) 03:49, 13 September 2005 (UTC)

Mathematics is either a topic drini doesn't know much about or or a topic he can't be intellectually honest enough about to admit where he is wrong. Rktect 14:54, 13 September 2005 (UTC)

Again his ad hominem attacks. I'm very knowledgeable about mathematics, specially geometry, its history and philosophy. 132.248.196.25 15:25, 13 September 2005 (UTC)

Since I wasn't logged in, comments from 132.248.196.25 are mine: -- (☺drini♫|☎) 15:45, 13 September 2005 (UTC)

So then the explanation would be that you knew all about the unit fractions in the Rhind papyrus and the Egyptians analytic geometry and quadratures as cited above but denied them anyway? Just for the heck of it why don't you tell us about some of the work you have done with unit fractions? Rktect 17:24, 13 September 2005 (UTC)

Here is a well known problem for you show off with drini.

     35     1      1       1
    ---  =  -  +  --  +  -----  .
    179     7     19     23807

Find such an expression for 3/179. Rktect 17:50, 13 September 2005 (UTC)

1/60 + 1/10740 , so what? -- (☺drini♫|☎) 19:00, 13 September 2005 (UTC)

It is known that any fraction with an odd denominator is a sum of reciprocals of distinct odd integers

an odd Egyptian puzzle can I rest my case?

Rktect 23:29, 13 September 2005 (UTC)

Yes it is known. I knew that. It's a well stablished theorem. But that's not how greeks did math. -- (☺drini♫|☎) 02:37, 14 September 2005 (UTC)

Actually it is, Egyptian unit fractions continued to be used through the middle ages.

On the external links

Can you explain the relavance of angle trisection, doubling the cube to unit fractions? Links on "External links" section should be relevant to the topic discussed. Otherwise, no reason to keep them. -- (☺drini♫|☎) 02:40, 14 September 2005 (UTC)

You are aware that the Egyptians used unit fractions to explore the squaring of the circle, you may be less aware that they also explored the doubling of the cube (Horus eye fractions) and the trisection of angles. The Egyptians used both algrythms or formulas and or tables to find the 2/3 of any number. They did this so automatically that there are 18 problems in the Rhind where in order to find the 1/3 of any number they found the 2/3 first. They also included a number of seked and pesu problems, arithmetic and geometric series, equations of the first and second degree and many other interesting precursors to Greek mathematics. The Egyptians divided the heavens into 360 angular degrees and then recombined them into dekans. Then they divided the dekans into hours of the day and night and divided the hours into minutes. Finally they related units of time to units of space and made their measures of distance to agree.

"The problem of trisecting a line segment is no more difficult than finding its n-th part for an arbitrary n. However, the general problem of trisecting and angle (i.e., trisecting an arbitrary angle) is not solvable in a finite number of steps." The Egyptians didn't care about the general problem and they used rulers rather than straight edges.

" :Firstly, some angles are trisectable with a ruler and a compass. An obvious example is supplied by the three quarters of the straight angle - 67.5o. (The angle itself is constructible as it is obtained by two consecutive angle bisections. Its third is obtained along the way.) Angles of 30o (draw a right triangle with a side 1 and hypotenuse 2) and 45o (bisect the right angle) are both constructible. Therefore, the latter also admits a classical trisection."

Indeed some angles are trisectable. Any integer angle which is a multiple of 3 is constructible, and those are the only ones. You can find my proof at PlanetMath: constructible angles with integer values in degrees. Therefore, since any angles of the form 3k and 9k are constructible, you can trisect the second by constrcting the first. -- (☺drini♫|☎) 13:44, 14 September 2005 (UTC)

Secondly, consider the geometric series 1/4 + 1/16 + 1/64 + ... that adds up to 1/3. Once we know how to bisect an angle, we may also find its 2-n-th part, for any n. In particular, one may construct 1/4, 1/16, ... of any angle and, in principle, find its third after an infinite number of steps. This solution is universal but requires a forbidden (infinite) number of steps.

When the Ancient Greeks restricted the allowed operations to using a straightedge and a compass. and thus specifically forbade using a ruler for the sake of measurement that was a substantive change to the Egyptian methodology much of which was based on the use of rulers and measurement. Many of the early Greek attempts still procede by neusis which is the Egyptian method.

"Hippocrates (470-410 B.C.) of Chios, Famous for his work on quadrature of circular lunes and the arrangement of theorems in a logical manner, later used by Euclid in his Elements, also left the first known construction for the trisection of an angle".

"As the trisection of Archimedes, this one, too, is not done by straightedge and compass, which we know is impossible. (Such constructions that employ tools beyond straightedge and compass are known as neusis constructions.)"

angle trisection

All of the early Greek mathematicians trace Greek mathematics back to the Egyptian as they do their philosophy religion and science. Many of those Greeks who like Thales, Solon, Plato , Pythagoras who spent 20 years of more studying in Egypt and got at least halfway through the curiculum required for matriculation became famous mathematicians in the Greek world. Rktect 10:26, 14 September 2005 (UTC)

So, at most you're claiming that information on unit fractions may be interesting when one studies angle trisection. Therefore, if anything, the external links should be on trisection angle entry, not the other way. -- (☺drini♫|☎) 13:34, 14 September 2005 (UTC)

I'm claiming that information on unit fractions is interesting period. It is relevant to any mathematics done by the Greeks because they used the Egyptian unit fractions in their calculations. It is also interesting to mathematicians who are using it today to do some cutting edge math. Its interesting to me as an architect because it directly relates to canons of proportion. I also like the doubling of cubes (measures of grain and beer) first incorporated in the Horus eye fractions and Rhind papyrus pesu (rations) problems and preserved in the British Imperial System.Rktect 17:52, 14 September 2005 (UTC)

Yes, but the point is, information relevnt to the topic (unit fractions) is what goes into externa links. So you assert that unit fractions are related to some attempst for trisecting angle, so perhaps adding links from unit fraction pages into trisectin angles entry would be more appropiate that adding tristectin angle links into unit fractions (since a priori and for a visitor who has no previous knowledge of the relationship, a link for angle trisection is out of context) -- (☺drini♫|☎) 17:40, 14 September 2005 (UTC)

All that information is relevant to unit fractions, how they were used in the past, who used them, what they are being used for today. When I read an article I'm not bothered if there is a link I personally don't need as long as all the links I do need are there Rktect 17:52, 14 September 2005 (UTC)

The sum of every unit fraction

I think that the sum of every unit fraction i.e. 1/1+1/2+1/3+1/4+1/5... is about 11.66757818, encase anyone wants to mention that in the article. Robo37 (talk) 10:39, 14 June 2009 (UTC)

No, it grows to infinity as you include more fractions. See harmonic series. —David Eppstein (talk) 17:42, 14 June 2009 (UTC)

GA Review

This review is transcluded from Talk:Unit fraction/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Brachy0008 (talk · contribs) 11:22, 28 March 2023 (UTC)

GA review (see here for what the criteria are, and here for what they are not)

---

1. Is it well written?

    1a. The prose is clear and concise, and understandable to an appropriately broad audience; spelling and grammar are correct

1. It is reasonably well written.

   a. (prose, spelling, and grammar):

   The prose is clear and concise, and also introduces the readers to the terms and equations related to unit fractions.

    1b. It complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation: 

1. b. (MoS for lead, layout, word choice, fiction, and lists):

   The layout is ok, no words to watch and list incorporations.

    2. Is it verifiable with no original research?
    2a. It contains a list of all references (sources of information), presented in accordance with the layout style guideline: 
    References do not need to be consistently formatted or bibliographically complete, but they should contain enough information for you to be able to identify the source. Dead links should not be bare URLs.

1. It is factually accurate and verifiable.

   a. (reference section):

   Reflist is neat

    2b. All in-line citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons – science-based articles should follow the scientific citation guidelines:
    Verify that the 5 types of statements listed above are supported by inline citations. Check if there are any unreliable sources (see WP:RSP), including self-published sources and user-generated content. 

1. b. (citations to reliable sources):

   Spotchecks find no unreliable or deprecated sources

    2c. It contains no original research:
    Check at least some of the cited sources to see if they verify the article text. The article should not synthesize material from multiple sources to reach a conclusion not stated by any of them.

1. c. (OR):

   Spotchecked it already. No original research

    2d. It contains no copyright violations or plagiarism:
    While verifying citations, check if any text has been copied or closely paraphrased into the article. Earwig's tool can help check for plagiarism of online sources (https://tools.wmflabs.org/copyvios), but it should not replace manual checks as it cannot fully detect close paraphrasing. 

1. d. (copyvio and plagiarism):

   So far so good

    3. Is it broad in its coverage?
    3a. It addresses the main aspects of the topic:
    Ensure coverage of the main aspects. Note that to meet criteria 3a and 3b, comprehensiveness is not required: "broad in its coverage" allows shorter articles, articles that do not cover every fact or detail, and overviews of large topics.

1. It is broad in its coverage.

   a. (major aspects):

   Generally covers things the reader can know about unit fractions.

    3b. It stays focused on the topic without going into unnecessary detail (see summary style):
    Check for undue emphasis on tangents or minor details. Lengthy sections on subtopics should be spun off into their own articles, leaving summaries in their place.

1. b. (focused):

   Does not go out of the point related to Unit Fractions.

    4. Is it neutral?
    It represents viewpoints fairly and without editorial bias, giving due weight to each:
    Each viewpoint should be weighted in proportion to its prevalence in reliable sources on the topic. Check if minority views are given undue weight in terms of the depth of detail, prominence of placement or word choice. Ensure that the article describes disputes without engaging in them. 

1. It follows the neutral point of view policy.

   Fair representation without bias:

   No bias found

    5. Is it stable?
    It does not change significantly from day to day because of an ongoing edit war or content dispute:
    Scan the article's history and talk page for edit wars and content disputes. Good-faith improvements (such as copyediting), changes made in response to the review, proposals to merge or split content, and reversions of vandalism do not apply here. Stability is based on the article's current state, not any potential for instability in the future.

1. It is stable.

   No edit wars, etc.:

   No edit warring so far, no potential for instability

    6. Is it illustrated, if possible, by media such as images, video, or audio?
    6a. Media are tagged with their copyright status, and valid non-free use rationales are provided for non-free content:
    Click on media to check for valid copyright tags and, if not freely-licensed, a valid non-free use rationale. If the article has no media, but there is a readily available, relevant image with an acceptable license, it should be included. Otherwise, mark 6a and 6b as passed since media are not required for GA status.

1. It is illustrated by images and other media, where possible and appropriate.

   a. (images are tagged and non-free content have non-free use rationales):

   All media are either own work or licensed by Creative Commons

    6b. Media are relevant to the topic, and have suitable captions:
    Check that media are not primarily decorative and have relevance to the topic. Captions should be succinct and informative.

1. b. (appropriate use with suitable captions):

   Images are properly tagged

    Overall: Pass, fail or on hold
    If the article meets all 6 criteria, mark it as a pass. If it is only partially compliant or non-compliant, you may place the review on hold to allow time for issues to be fixed (generally 7 days). Often the nomination is brought up to standard during the review. If so, note this and close the review as a pass. If the nomination does not meet the criteria, close it as a fail. --> 

1. Overall:

   Pass/fail:

   Since no kind soul would volunteer their 2nd opinion, I’ll have to do the rest, and now I’m happy to pass you.

(Criteria marked are unassessed)

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by Theleekycauldron (talk) 00:57, 13 April 2023 (UTC)

(

• Comment or view
• Article history

)

• ... that when adding or subtracting two unit fractions, the sum isn't usually a unit fraction? Source: Betz, William (1957), Algebra for Today, First Year, Ginn, p. 370
  • Reviewed:

Improved to Good Article status by David Eppstein (talk). Nominated by Brachy0008 (talk) at 01:16, 30 March 2023 (UTC). Post-promotion hook changes for this nom will be logged at Template talk:Did you know nominations/Unit fraction; consider watching this nomination, if it is successful, until the hook appears on the Main Page.

• Wouldn't it be better to say "the result" rather than "the sum"? The result of a subtraction is called the difference and not the sum. --Moscow Connection (talk) 02:02, 30 March 2023 (UTC)
• I'd like to propose an alternate hook that I think is very catchy:

ALT1: ... that Gottfried Wilhelm Leibniz, James Gregory, and an unknown Indian mathematician all found a way to apprximate pi using unit fractions?

The source is here, also referenced in the article. This hook has 188 characters. TheLonelyPather (talk) 18:33, 2 April 2023 (UTC)

I'm very sympathetic to the goal of making a hook that is not just a statement of mathematical fact, but somehow relates more broadly to the world. But this one is not a statement that is actually in the article. The article itself just links to a series formula. The series formula that I assume you mean is for π/4, not π itself. The article says nothing about Leibniz beyond repeating his name in the name for the formula. It says nothing about Gregory. It says nothing about "an unknown Indian mathematician", nor for that matter about known Indian mathematician Madhava. Nor should it; that would be an unnecessary level of detail for this article. The phrasing, out of chronological order, emphasizing the European contribution and dismissing the Indian contribution as unknown, is unnecessarily unflammatory. And the article says nothing about this formula being intended as a way to approximate π. Without these claims, I don't think this hook is usable. Instead, how about:

Pizza divided into 1/8 fractions

• ALT2: .... that exercises in fair division of food (pictured) are often used to teach unit fractions? Source

—David Eppstein (talk) 19:06, 2 April 2023 (UTC)

• Understood. I agree that the hook is a far stretch in the "approximate pi" part, and so I have retracted the hook.

To respond to your concerns, the order of the mathematicians is introduced not by me, but by the source pdf: "The series (2) was obtained independently by Gottfried Wilhelm Leibniz [...] James Gregory [...] and an Indian Mathematician of the fifteenth century whose identity is not definitely known." (pp. 291) TheLonelyPather (talk) 20:08, 2 April 2023 (UTC)

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: - Offline/paywalled citation accepted in good faith Interesting:

Image: Image is freely licensed, used in the article, and clear at 100px.
Freely licensed: Used in article: Clear at 100px:

QPQ: None required.

Overall: Approve ALT2. BorgQueen (talk) 23:03, 4 April 2023 (UTC)

History of unit fractions

@Brachy0008 and David Eppstein: just a query about whether this article should have detail on the history of unit fractions and more information on their use within the context of Egyptian fractions... I think there's a lot of interesting information here, including the fact that fractions such as 1/3 predated things like 2/3 in the human development of maths, and would think this would be part of the "broad coverage" necessary for a GA tick. Cheeers  — Amakuru (talk) 09:29, 25 April 2023 (UTC)

Feel free to be bold and add it in. Brachy08 (Talk)(Contribs) 10:17, 25 April 2023 (UTC)

This is one of those topics that it's difficult to say much about the history of, because it belongs to the prehistory of mathematics rather than the history itself. But if you can find enough sourced material on the history as a general topic, rather than only on some specific subtopic like Egyptian fractions (which use unit fractions but are not unit fractions themselves), then I'm definitely open to including this material here. —David Eppstein (talk) 15:32, 25 April 2023 (UTC)

predated things like 2/3 in the human development of maths – this seems like an overly strong assertion. Ancient people definitely had some concept of “two thirds”, but didn’t express that using the modern fraction concept per se. In ancient Mesopotamia there were very complicated systems of units going back to like 5000 BC. ––jacobolus (t) 16:15, 25 April 2023 (UTC)

Well, they "predate things like 2/3 in the human development of maths" in the sense that each individual human, as a child, typically learns about concepts like 1/3 earlier than concepts like 2/3. But that's not about history, and it's already covered in the article, in its "Fair division and mathematics education" section. It is entirely believable that unit fractions predate other kinds of fractions in ancient mathematical history as well, but saying such a thing here would require finding reliable published scholarly work about it. —David Eppstein (talk) 00:05, 26 April 2023 (UTC)

To be honest, my understanding of this area isn't really very advanced so I may have been barking up the wrong tree with this. I actually heard the idea that early people had 1/3 but not 2/3 from a Vsauce youtube video, which is obviously not a rigorous academic source . Anyway, if there isn't a lot of history associated with this concept specifically, then all is good, thanks for the responses.  — Amakuru (talk) 10:18, 26 April 2023 (UTC)

@Amakuru I am not an expert in this either, but in searching around I found the hypothetical explanation in Mott's book Natural Knowledge in Preclassical Antiquity to be pretty interesting, starting from p. 36. –jacobolus (t) 00:54, 27 April 2023 (UTC)