Talk:Regula falsi

I think, the results from "Improved Algorithms of Illinois-Type for the Numerical Solution of Nonlinear Equations", Ford, 1995, should be integrated in the article. Especially his new methods, which are better than Anderson and Björck, and some of the results from his experiments. Gabriel (talk) 14:00, 25 April 2018 (UTC)Reply

WP:Be bold! --Qwfp (talk) 20:15, 25 April 2018 (UTC)Reply

I removed the text because I found it impossible to justify. Calculus is not needed to arrive at the idea of secant lines. The linear problems mentioned in the Nine Chapters do not use secant lines; in fact, I would argue they do not use the false position method either but linear interpolation. Finally, I could not find any evidence in History of calculus and Moscow and Rhind Mathematical Papyri that the Egyptians developed calculus, except for the false statement that one needs calculus to calculate the volume of a frustum. -- Jitse Niesen 18:18, 4 Feb 2005 (UTC)

To the editor with IP address 209.150.67.45: Could you please explain (or give references explaining) how the ancient Egyptians used the false position method? The example in Egyptian mathematics suggests that given an equation ax = b, they try one value, say y, compute ay and compare the result with b to get the solution. I am trying to understand the differences with the method of double false position as used by the Chinese. Thanks, Jitse Niesen 10:24, 4 Apr 2005 (UTC)

How does the method described in this article fit in with the "method of false position" described at [1] and [2]? Do we need to say something here about other uses of the terminology? Tom Harrison ^Talk 20:53, 6 May 2006 (UTC)Reply

Yes, it should be mentioned, perhaps via a disambiguation page. I think that the methods are related, but the rule of false position described on the web pages you mention is considerably less sophisticated. I haven't found a good reference on this though. -- Jitse Niesen (talk) 12:15, 7 May 2006 (UTC)Reply

Another external link for when the Rule of False Position page is created: *Rule of False Position at Convergence --Nic bor 18:23, 3 July 2007 (UTC)Reply

False position as used in Ancient Egypt is described in sources such as

* Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0871692320

* Katz, Victor J. (editor),Imhausen, Annette et.al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007 ISBN 978-0691114859

* Gillings, Richard J., Mathematics in the Time of the Pharaohs, Dover Publications, 1982 reprint (1972) ISBN 0-486-24315X

From their description the method of false position (also called false assumption) as used in Ancient Egypt is not iterative. It really does not follow the general outline of the technique from numerical analysis as outlined on this article. I have only seen examples where the method is used to solve linear and simple quadratic equations. It may be a very simple version of the modern technique, but I really like Jitse Niesen's suggestion that this method be given it's own page and that this one links to it via disambiguation. I will probably write something up over the weekend (time permitting). Regards --AnnekeBart (talk) 13:34, 9 October 2010 (UTC)Reply

 

is anyone interested in integrating this into the article? i find it looks a bit nicer than the old one... —The preceding unsigned comment was added by 212.117.72.126 (talk • contribs) 17:06, 31 May 2006 (UTC)Reply

212.117.72.126: It is "do it yourself" around here. Why not add it yourself, if you think that it is good. But I would not remove the old diagram, if I were you. JRSpriggs 04:52, 1 June 2006 (UTC) P.S. You should give copyright information (who created it and confirmation that he/she gave permission) even for diagrams posted on talk pages like this one.Reply

Thanks for the example, shown in the animated diagram, in which Regula Falsi works in a situation that would be particularly unfavorable to other popular methods.

--108.132.202.166 (talk) 18:41, 10 December 2015 (UTC)MichaelOssipoffReply

Regarding the example code: since the compiler will convert the floating-point division by a constant into a floating-point multiplication by the (precomputed) reciprocal of that constant at compile time, there is no need to make the code less clear in order to gain speed: the machine code generated will be the same in either case. -- The Anome 12:05, 11 January 2007 (UTC)Reply

Even faster than multiplying by one half, is shifting one bit to the right. Which optimization the compiler will also do for us, if it is any good (and knows what the constant is). More important still is that the code should be clear to the reader. JRSpriggs 05:35, 12 January 2007 (UTC)Reply

Since only variables with the type double is divided by 2 in the C code, it would wreak havoc on the result to shift it in order to achieve a division by two. Shifting is only equivelant to division and multiplication (depending on the direction of the shift) when dealing with integers -- never floats. —Preceding unsigned comment added by 77.212.20.61 (talk) 03:35, 15 January 2009 (UTC)Reply

Regarding the example code: the stopping criteria seems to be bad. Since the bracket width |b-a| will not converge to zero, it is most likely that n iterations will be spent when much fewer will suffice to meet the tolerance e. And it is ( fabs(s) + fabs(t) ), not fabs(s+t), i.e., imagine s+t=0. — Preceding unsigned comment added by 5.186.124.99 (talk) 15:39, 24 May 2022 (UTC)Reply

in the last equation, :${\displaystyle f(b_{k})+{\frac {f(b_{k})-f(a_{k})}{b_{k}-a_{k}}}(c_{k}-b_{k})=0.}$ 

shouldn't ${\displaystyle c_{k}}$  and <b_k></math> be switched like this?

${\displaystyle f(b_{k})+{\frac {f(b_{k})-f(a_{k})}{b_{k}-a_{k}}}(b_{k}-c_{k})=0.}$  —Preceding unsigned comment added by Ceazare (talk • contribs) 20:37, 18 March 2010 (UTC)Reply

Difference Between False Position And Bi-Section Method for Finding real roots —Preceding unsigned comment added by Ankurpatel12 (talk • contribs) 06:52, 6 June 2010 (UTC)Reply

It would be helpful if the article could explain why the method bears such a poetic title. Tkuvho (talk) 14:46, 3 March 2011 (UTC)Reply

You mean "False Position"? That's easy: "False Position is a really ridiculous mis-translation of "Regula Falsi".

Why is it called "Regula Falsi"? In Latin, Regula Falsi means "Rule of Falsehood", or "Ruler of Falsehood", or maybe (by extension from Ruler), "Straight Line of Falsehood".

Actually, in Latin, Regula can refer to straight things other than a ruler, and so "Straight Line of Falsehood", is the best guess for Regula Falsi's meaning. ...because Regula Falsi is based on a false, or only approximate, assumption that the function is linear.

--MichaelOssipoff (talk) 17:27, 10 December 2015 (UTC)MichaelOssipoffReply

After I posted that answer, it was pointed out to me, at this page, that the name "False Positions" was in use before the name "Regula Falsi" was coined. ...so "False Position" wasn't intended as a translation of "Regula Falsi". So I'd delete my answer above, except that someone said that we shouldn't do deletion at the talk-page.

So I'll just acknowledge that False Position isn't an intended translation of Regula Falsi. ...And that was my main objection to the "False Position" name.

So why was Regula Falsi given as unspecific a name as "False Positions"? At that time, there weren't as many other numerical equation-solving methods to distinguish it from, so there wasn't a need for the name to specify details or a distinction. Regula Falsi distinguished itself from Bisection by its effort to estimate, by a linearity-assumption, the position of the solution.

Of course the the name "Bracketing Secant" would be more descriptive, but it's also appropriate for an ancient method to be referred to by its early names. — Preceding unsigned comment added by MichaelOssipoff (talk • contribs) 12:10, 15 December 2015 (UTC)Reply

--MichaelOssipoff (talk) 12:11, 15 December 2015 (UTC)MichaelOssipoffReply

In the section "Illinois algorithm" it says: "While it is a misunderstanding to think that the method of false position is a good method, it is equally a mistake to think that it is unsalvageable." This statement seems subjective and not appropriate for an encyclopedia. Usually one compares one method to another and points out pros and cons objectively. The author should at least elaborate why he/she considers this method so bad. Italo Tasso (talk) 21:32, 8 March 2012 (UTC

Yes, the fact that something can be improved on doesn't mean that it isn't already "good".

Regula Falsi, even without improvement, always converges, and usually considerably faster than Bisection.

Yes there are situations that can slow Regula Falsi down, even to a prohibitive degree. But often those situations are ones that would prevent Newton's method or Secant from converging at all.

Some improvements on ordinary Regula Falsi include the Illinois version, Pegasus, and the Anderson-Bjőrk version.

Those methods usually avoid the slowdowns that can happen to Regula Falsi. In general they do a good job of avoiding the problems that difficult equations can cause--something that is a problem for all root-finding, equation-solving methods.

Illinois is pretty good for such a simple improvement. In Galdino's "A family of Regula Falsi Methods", he reports numerical tests that he did. In his tests, Anderson-Bjőrk was the clear winner, for simple roots.

For multiple roots, no method improved significantly on Bisection, and the only ones that even did as well as Bisection were three new ones proposed by Galdino. But, even they didn't improve a lot on Bisection. ...And, as I said above, Anderson-Bjőrk was the clear winner with simple roots--a kind of problem in which the best methods were a lot faster than Bisection.

--MichaelOssipoff (talk) 17:27, 10 December 2015 (UTC)MichaelOssipoffReply

On 19 March 2006, Deeptrivia added this sentence: "The oldest surviving documents demonstrating knowledge and proficiency in the false position method is the Indian mathematical text Vaychali Ganit..." (since then, the spelling was changed from Vaychali to Vaishali). No source for this statement was provided at the time, nor has any been supplied since that date. The statement needs to be reliably sourced, or else it should be removed. Rks22 (talk) 00:12, 9 June 2012 (UTC)Reply

http://books.google.com/books?id=AG2XBCmxYcUC&pg=PA157#v=onepage&q&f=false

http://muslimheritage.com/topics/default.cfm?ArticleID=993

Dun, Liu, 2002. "A Homecoming Stranger: Footsteps of the Double-False-Position Method." In: From China to Paris. 2000 Years Transmission of Mathematical Ideas. Edited by Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts & Benno van Dalen. Stuttgart: Steiner. [Abstract: Appearing first in the Nine Chapters on Mathematical Procedures (ca. 50 AD), the Double-False-Position Method spread from China into Central Asia in the Middle Ages and became known as the "Khitan algorithm" [hisâb al-khata'ayn] among Arabic mathematicians. Leonardo Fibonacci (1170?-1250) devoted a separate chapter to this method in his Liber Abaci (1202). When the Jesuits introduced Western mathematical knowledge into China in the early 17th century, they claimed that the Double-False-Position Method was a new technique invented by Western mathematicians and could not be found in the "old text" of the Nine Chapters. This is because ancient Chinese mathematical books had become extremely rare at that time. Therefore when the Double-False-Position Method appeared in the Tongwen suanzhi (1613) and Xijinglu (ca. 1610), it was said that "a stranger came from overseas"].

07:47, 8 February 2014 (UTC)

Around December 8th or 9th, I added text to the article, pointing out that "False Position" is a ridiculous mis-translation of Regula Falsi.

I told why:

In Latin, adjectives must agree with their noun in gender, number and case. ("case" refers to how the noun is used-subject, object, etc.)

"Regula" is feminine, nominative singular. "Falsi" is either masculine nominative plural, or masculine or neuter genitive singular.

Even if you say that maybe "Regula" is masculine with feminine form, that still leaves the blatant disagreement in case, number, or both.

"Falsi", as an adjective, doesn't agree with "Regula" as a noun.

Obviously "Falsi" is a noun.

"Regula" means "Rule" (in the sense of a regulation), or "Ruler" (straight-edge). But "Regula" was also used to refer to straight things other than a ruler.

"Falsus", or "Falsum", as a noun, means "Falsehood".

The best suggestion for the meaning of Regula Falsi is:

Straight Line of Falsehood

(because "Regula" can refer to straight things other than a ruler).

That's because Regula Falsi indeed uses a false assumption that the function is linear.

As I said, I added that text to the article.

Someone removed it within about 2 days. Why? Because you found an "authoritative source" that you could cite for the "False Position" translation? I don't care how authoritative your source was. Regula Falsi couldn't be Latin for False Position.

Making the embarrassment worse still, you have "False Position Method" as the article's title.

The person who deleted my comments about the ridiculous "False Position" translation said that the accuracy of the name isn't relevant to the fact that it's used. ...but it's relevant to the matter of whether it makes any sense or should be used.

And, if we're going to make up a name that differs in meaning from the classical name, then at least make up something that has some descriptive value. Every iterative approximation uses and gives "false-positions". The name "Regula Falsi" specifies something that is straight-linear and a falsehood. That's a good description of the straight line used in Regula Falsi. — Preceding unsigned comment added by 108.132.202.166 (talk) 21:54, 10 December 2015 (UTC) Michael OssipoffReply

An article like is probably not watched by many active editors so it may take quite a long time before someone who understands the topic and the terminology sees the changes or discussions (I am not that person). The edit on 8 December 2015 which I reverted was diff. Regardless of its accuracy, that text is not appropriate because the article makes no claim about the terms "false position method" and "regula falsi" other than they are both used to refer to the technique discussed. I agree that an encyclopedic article would benefit from a brief discussion of the origin of the name, but it would need a reliable source. I asked for assistance at WT:WikiProject Mathematics#False position method. Johnuniq (talk) 23:37, 10 December 2015 (UTC)Reply

I reply:

It would make a lot more sense to ask a Latinist whether Regula Falsi means False Position.

Regardless of whether the article makes a claim about the meaning of Regula Falsi in Latin, you're using a nonsensical translation as the article's title. As I said, that use, as the title, worsens the embarrassment of your mis-translation.

And no, you don't just say that both terms are used. You use "False Position" as the article's title, and call the method by that name, throughout the article.

--MichaelOssipoff (talk) 00:07, 11 December 2015 (UTC)MichaelOssipoffReply

While the Latin scholar may be correct about this point, the revert was necessary as this is WP:SYN and not relevant to this article for the simple reason that the term false position was translated into Latin and not from it.

You haven't shown that.

"Regula Falsi" is not a valid Latin translation of "False Position". So, how do you support a claim that Regula Falsi is an attempted Latin translation of False Position?

Suppose, hypothetically, that an Italian expression meaning "False Position" were in use in 1591, and that, a century later, it was replaced by Regula Falsi, a Latin term.

Does that make Regula Falsi a Latin translation of (the Italian for) False Position?

Of course not. All that can be said is that Regula Falsi replaced False Position (or, rather, the Italian expression with that meaning). There's no reason to believe that Regula Falsi was intended as a translation for False Position. In fact, the fact that Regula Falsi is not a valid Latin translation of False Position, weighs against such a claim.

--108.132.202.166 (talk) 13:22, 11 December 2015 (UTC)MichaelOssipoffReply

(D. E. Smith, History of Mathematics, Vol. 2, pg. 437, Dover) Regula Falsi appears in 1690, but Fibonacci, Pacioli and Tartaglia all used the method before then, referring to it as "El cataym" (from the Arabic sources).

But that doesn't favor "False Position" unless "El cataym" means "False Position". Does it?

--108.132.202.166 (talk) 13:22, 11 December 2015 (UTC)MichaelOssipoffReply

Pagani in 1591 says, "Delle Regole del Cattaino ouero false positioni".

1. One person saying it doesn't constitute widespread usage.

2. In any case, for that to lend any support to "False Position", two things would have to be shown:

...a) That "Delle Regole del Cattaino ouero false positioni" (or some part of it) means False Position, in Italian.

...b) That Pagani was referring to the Regula Falsi method when he said that.

I must apologize for not looking up an Italian translation before starting this posting--I intended to. Of course I'll do so after I post this.

As for a): The "-i" ending on a noun, in Italian, is a plural ending. On a plural noun like "positioni", derived from a Latin 3rd declension noun, the Italian "-i" plural would be used for either gender.

The "-e" ending on Regole corresponds to the Latin "-ae", a feminine singular noun form. "Regole" is the Italian for the Latin "Regulae". Rules or rulers, or some sort of things that are straight.

The "-e" ending on "False" suggests the feminine plural ending on a feminine noun, or on a feminine plural form of an adjective (one of those adjectives derived from a 1st-&-2nd declension Latin adjective using the nominative singular endings of "-us" (masculine), "-a" (feminine), and "-um" (neuter) ). In Latin, "Falsus" is a 1st-&-2nd declension adjective. "False" would be the Italian form of the Latin "Falsae", the feminine plural nominative. Italian doesn't specify case.

So, "false positioni" would mean "False Positions". The necessary agreement in number and gender, between adjective and noun, would then be fulfilled, in Italian.

So, isn't that phrase saying something like this?:

"Of the rules (or rulers or straight-lines) of Cattaino (probably the name of a person or a place)[something] false positions."

I have no idea what that [something] ("ouero")is. I briefly tried to look it up, and the online Italian translation program said that it had detected Galacian, and translated "ouero" as "egg".

If the Italian phrase you quoted is a sentence, then it needs a verb. If "ouero" were the verb, it would be 1st person singular present-tense.

Anyway, "false positioni" is definitely plural.

Again, I apologize for not looking up a translation of your Italian quote before posting this. I'll look it up afterwards though.

"Delle" is a contraction of a preposition and a definite article. "Di le". "Of the" (plural).

What if "ouero" is a verb, the 1st person singular present of a verb "ouerare", and what if "ouerare" means "like" or "prefer"?

"Ouero" doesn't sound a lot like the Spanish "quiero", but, if it's needed for "ouero" to be a verb, then that suggests itself as a possibility.

If Pagani was saying "Of the rules of Cattaino, I like (or prefer) False Positions.", then that would support your suggestion regarding what Pagani was saying.

It isn't at all obvious that "false positioni" is referring to an equation-solving method. As I said, it means "False Positions", and that's so non-specific that it could refer to any rule involving positions that are false. There's no particular reason to believe that it refers to the Regula Falsi method.

But even if it were established that something meaning "False Positions" were being used to name that method, in 1591, but it was replaced by "Regula Falsi" (which is not a Latin translation of "false positioni") by 1691, then, with Regula Falsi becoming the widespread term, that wouldn't support False Positions as the more legitimate term.

I'll try again to look up a translation for the Italian phrase that you quoted from Pagani.

--108.132.202.166 (talk) 13:22, 11 December 2015 (UTC)MichaelOssipoffReply

This and many other examples are quoted in Smith. Bill Cherowitzo (talk) 00:18, 11 December 2015 (UTC)Reply

My initial post about this was removed, presumably because I posted it at the wrong place (at the top of the page, above the official page-heading).

So then, let me post it here, in the usual way of adding a section to the "talk page":

Around December 8th or 9th, I added text to the article, pointing out that "False Position" is a ridiculous mis-translation of Regula Falsi.

I told why:

In Latin, adjectives must agree with their noun in gender, number and case. ("case" refers to how the noun is used--subject, object, etc.)

"Regula" is feminine, nominative singular. "Falsi" is either masculine nominative plural, or masculine or neuter genitive singular.

Even if you say that maybe "Regula" is masculine with feminine form, that still leaves the blatant disagreement in case, number, or both.

"Falsi", as an adjective, doesn't agree with "Regula" as a noun.

Obviously "Falsi" is a noun.

"Regula" means "Rule" (in the sense of a regulation), or "Ruler" (straight-edge). But "Regula" was also used to refer to straight things other than a ruler.

"Falsus", or "Falsum", as a noun, means "Falsehood".

The best suggestion for the meaning of Regula Falsi is:

Straight Line of Falsehood

(because "Regula" can refer to straight things other than a ruler).

That's because Regula Falsi indeed uses a false assumption that the function is linear.

As I said, I added that text to the article.

Someone removed it within about 2 days. Why? Because you found an "authoritative source" that made the same error? That obedient copying policy is what propagates an error forever (if allowed to).

Can you cite a Latinist who says that Regula Falsi means False Position? Read a page or two about Latin grammar. Or ask any highschool (secondary school) freshman 1st-year Latin student. Falsi as an adjective doesn't agree with Regula as a noun. Falsi couldn't be an adjective, as I pointed out above. "False Position" is, as I said, a ridiculous mis-translation of Regula Falsi.

--MichaelOssipoff (talk) 00:00, 11 December 2015 (UTC)MichaelOssipoffReply

See above section. Bill Cherowitzo (talk) 00:20, 11 December 2015 (UTC)Reply

To add to the above discussion: it doesn't matter whether it's a bad translation; what matters is that "false position" is the standard English-language name for this method. See WP:COMMONNAME. —David Eppstein (talk) 00:22, 11 December 2015 (UTC)Reply

So it doesn't matter that "False Position" is useless and meaningless as a descriptive name..that it completely fails to distinguish Regula Falsi from any of the other iterative approximation methods? Every iterative approximation method gives and uses "false position"s.

--MichaelOssipoff (talk) 00:36, 11 December 2015 (UTC)MichaelOssipoffReply

So lobby the textbook authors to change their terminology. But we have to follow them here, not lead them. See WP:SOAPBOX. —David Eppstein (talk) 00:48, 11 December 2015 (UTC)Reply

I concede that. But Regula Falsi and False Position are both popular and well-established names of the method. You'd be following established usage, and not leading, by using either one of those names.

So that policy of sticking with established usage doesn't choose between the names "Regula Falsi" and "False Position".

One of those names (Regula Falsi) is more descriptive than the other, and has widespread usage going back to 1690. The other (False Position) is evidently just a miss-guessed English translation of Regula Falsi. (I address the objections to that claim in my reply in the other section on this topic, on this page).

I'm not versed in Latin; you could well be right about a bad translation. But for the purposes of Wikipedia, the quality of the translation may be irrelevant. Wikipedia articles are descriptive of a topic, not prescriptive. They summarize the topic as it is described in reliable sources. False position and regula falsi are often treated as names for the same algorithm or class of algorithms.

Yes, I don't deny that, and I concede that Wikipedia's policy is not to promote new usages. But, as you said, both names are widely-used and well-established. That means that Wikipedia isn't compelled to use one instead of the other.

So, which one should be chosen?: The one that's more descriptive ("Straight Line of Falsehood"), with widespread usage back to 1690, or the one that is less descriptive and evidently a miss-guessed translation of the other?

You're free to choose. Or, I should say, we are free to choose. So (and it wouldn't violate Wikipedia policy at all) let's use, in the title and within the article, the name that is better, with regard to descriptiveness and legitimate origin.

Choose "False Position" if you think that it's better in some way (but say why it's better), not because Wikipedia policy requires it--it doesn't.

--108.132.202.166 (talk) 11:49, 11 December 2015 (UTC)MichaelOssipoffReply

Look in any book on numerical analysis; I just verified it in my copy of Numerical Recipes. Hence, even if the translation is bad, using the commonly accepted names is the right approach for a Wikipedia article.

Quite so. Regula Falsi and False Position are both commonly-accepted names. Using either would be the right approach, in accord with Wikipedia policy. So let's use whichever name is better. ...whichever name is more descriptive and has longer widespread usage and better etymological legitimacy.

Just a suggestion.

--108.132.202.166 (talk) 11:49, 11 December 2015 (UTC)MichaelOssipoffReply

In fact looking at some historical sources, it is apparent that there are a lot confusions, misinterpretations, and mistakes concerning the varieties of false position and secant methods. Our job is describe the history and current practice, not prescribe a canonical algorithm or better translation. If there is a reliable historical source that discusses the translation, I wouldn't be opposed to adding a sentence about the translation. Otherwise, it is best to leave out of the article personal linguistic findings on the topic. --Mark viking (talk) 01:09, 11 December 2015 (UTC)Reply

I'm adding this post to this last section, where it's more likely to be noticed.

Looking at some books on the Internet, yes it does seem as if "False Position" or "False Positions", whether in English or Italian, was a widespread name for the Regula Falsi method. ...even before the 1691 date which someone here gave for the first appearance of the "Regula Falsi" name.

So yes, both names have equal historical legitimacy, and maybe "False Position(s)" was used earlier (but "Regula Falsi" is not a translation of it).

If "Regula Falsi" means "Rule of Falsehood" (in the sense of a suggestion for how to do something, like solve equations), then both terms have about equal specificity.

But if "Regula Falsi" meant "Straight Line of Falsehood", then that has better specificity than "False Position(s)".

That's why I prefer "Regula Falsi", because it might specify more, in its meaning.

...and maybe because it sounds better to use a Latin term, rather than an English term that obviously doesn't mean anything.

For those reasons, and because an ancient method deserves to be named in an ancient language (or at least a Classical language), the name "Regula Falsi" is more aesthetically-appealing.

Anyway, of course either will do, equally historically-legitimately. So I won't pursue the name-legitimacy issue.

I claim that "Regula Falsi" would be better, for the reasons that I've just given, above, but I won't push for a change about it.

--108.132.202.166 (talk) 15:46, 11 December 2015 (UTC)MichaelOssipoffReply

I tried again, but couldn't find a meaning for "ourero" at the online Italian translation pages. Maybe it isn't used in modern Italian.

But the other words of the sentence tell us that Pagiani is saying that, of the rules of Cattiano, he likes, dislikes, prefers, recommends, uses, or avoids False Positions.

...or some such verb.

That was just a wild guess, relating it to quiero. I realize that there isn't enough resemblence to even justify it as a guess. But, for him to choose it to mention it, it's more likely that he mentioned it because he liked it, that because he disliked it.

Another possibility: The English word "Operate" suggests a Latin word "Operare". Spanish has "Obrero" for worker, and we have the Latin expresion "Opus Magnum", and the word "Opera" (the plural of "Opus").

So maybe, with a 17th century word related to those, Pagiani was saying that he uses False Positions.

Anyway, as I was saying, I won't make an issue about the name of that method.

...other than to just add this:

So they were calling it false positioni (false positions) before Regula Falsi was was coined. But Latin was the internatinal language then. So the Latin word would have been the international word, and likely the more official one.

...suggesting that Regula Falsi had more official status than False Positions.

Just saying...

--MichaelOssipoff (talk) 00:02, 12 December 2015 (UTC)MichaelOssipoffReply

I've just added that new section, at the beginning of the article. I didn't delete any previous work. I just added my introduction to the beginning. My addition introduces equation-solving; the generalities of methods for numerical equation-solving, and the need for them; the 2-point bracketing methods in general; Bisection; and Regula Falsi (False Position).

I put my added sections first in the article because I felt that the simpler, more basic, more familiarly worded, everyday-language introduction and explanations should come first in the article, in keeping with wikipedia guidelines to that effect.

I claim that my introduction & explanation of those topics better abides by the wikipedia guidelines specifying that articles start out simple, with things that the reader will already know about, speaking in language that the reader will understand. In these article sections that I added, the purpose was to say these things in a way that the most readers will understand. That's in the spirit of wikipedia's guidelines on such matters, and that was my purpose, in adding those sections at the beginning of the article.

I understand that the status-quo has a lot of weight at wikipedia. But I ask that, before you delete my added sections, you take a look at the simple, basic, familiarly-worded, clear, direct, concise, and sound introductions and explanations that I offer.

I didn't want to delete anyone else's work. Because the previous, already-existing article emphasized historical meanings for False Position, and its earlier forms, and my sections don't, I added the section-heading "Discussion from Historical Context" at the beginning of the already-existing article.

If anyone is inclined to delete my additions, then can we discuss it here first? I posted this announcement and discussion of my edits, and all I ask is that we have some discussion here before anyone starts deleting my added sections.

--MichaelOssipoff (talk) 01:16, 22 December 2015 (UTC)MichaelOssipoffReply

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I made the correction to the expression

${\displaystyle x={\frac {b_{1}x_{2}-b_{2}x_{1}}{b_{1}-b_{2}}},}$ 

and I still don't see how the original can be correct given that we're trying to find the x satisfying

${\displaystyle f(x)=b}$ .

After all, our formula for x should depend on b. This expression would be correct if b = 0, and I was merely correcting it to the more general case.

You are mistaking the generic expression f(x) = b for actual data. This method predates algebraic manipulation to find x, but in order to describe it to a modern audience it is necessary to use algebraic notation. The method involves two guesses x₁ and x₂ and the values of the function at these two points b₁ and b₂, respectively. The linear equation you are trying to solve is ax + b = 0 (I am using the symbol b again to point out a possible confusion), so in particular ax₁ + b = b₁ and ax₂ + b = b₂. From these values, a better approximation for x is obtained (exact if the equation is linear). If you carry out the algebra involved, you will see that the b's are cancelled out and do not appear in the final version. --Bill Cherowitzo (talk) 19:56, 9 February 2019 (UTC)Reply

I'm not mistaking anything, though perhaps the page itself is. The page assumes the following conditions:

${\displaystyle f(x)=b}$ 

${\displaystyle f(x_{1})=b_{1}}$ 

${\displaystyle f(x_{2})=b_{2}}$ 

Given these conditions, the correct approximation for x is

${\displaystyle x\approx {\frac {(b_{1}-b)x_{2}-(b_{2}-b)x_{1}}{b_{1}-b_{2}}}}$ .

Feel free to change one of the original conditions rather than the approximation for x, but something on the page must be changed for everything there to agree. -- JoshuaGreen1981 (talk) 21:00, 9 February 2019 (UTC)Reply

Perhaps I was being a bit too glib. Here is the complete derivation and I have changed a condition to avoid an ambiguity as to what the function actually is (which I still believe is the cause of this problem).

f(x) = ax + c = 0 is the equation to be solved.

f(x₁) = ax₁ + c = b₁

f(x₂) = ax₂ + c = b₂

Solve these two equations for a, the slope of the line, to obtain

a = ⁠b₁ − b₂/x₁ − x₂⁠.

The solution of the equation of interest is x = ⁠−c/a⁠.

Using either of the last two equations solve for c, say,

c = b₁ − ax₁.

Now substitute these values for a and c into the solution and simplify to obtain the result

x = ⁠b₁x₂ − b₂x₁/b₁ −b₂⁠

I hope that this convinces you.--Bill Cherowitzo (talk) 05:05, 10 February 2019 (UTC)Reply

I agree that the page is now correct, as you've replaced the previous

${\displaystyle f(x)=b}$ 

with

${\displaystyle f(x)=ax+c=0}$ 

which essentially sets b = 0 in my version. I disagree that you've merely "avoid[ed] an ambiguity as to what the function actually is" as, in fact, you've fundamentally changed the equation that we're trying to solve. We're now looking for a root, solving f(x) = 0 rather than the more general f(x) = b that was there before. -- JoshuaGreen1981 (talk) 13:35, 10 February 2019 (UTC)Reply

My apologies. Since I knew the result was correct I didn't look far enough backwards to see where the real problem was. My editing style is to keep as much of the original work as possible and here that got me into trouble. I took it for granted that the introductory material was correct and didn't see the flaw in the assumptions. --Bill Cherowitzo (talk) 00:25, 11 February 2019 (UTC)Reply

That's understandable. Apology accepted. -- JoshuaGreen1981 (talk) 03:08, 12 February 2019 (UTC)Reply

The term "method of false position" has consistently been more common than "false position method" or "rule of false position" during the 20-21st centuries, according to Google ngrams.

This article should be moved to Method of false position.

Comments? --Macrakis (talk) 03:26, 19 July 2019 (UTC)Reply

Personally I would prefer to move it to regula falsi. --Trovatore (talk) 04:10, 19 July 2019 (UTC)Reply

Silly me, I didn't plot that on ngrams. And sure enough, regula falsi turns out to be more common than the alternatives! I will start by putting that in first position in the lead. --Macrakis (talk) 05:47, 19 July 2019 (UTC)Reply

@Macrakis: do search results on “false position method” include false-position, with hyphen? Incnis Mrsi (talk) 16:37, 19 July 2019 (UTC)Reply

@Incnis Mrsi: Yes, they do, but if you look at the "false+position+method" search results (I suggest setting page size to 100), you'll find that the variant with the hyphen is rare, even if it is arguably more correct grammatically. --Macrakis (talk) 17:05, 19 July 2019 (UTC)Reply

In the article it is claimed (section "History") that "Regula Falsi appears as the Latinized version of Rule of False as early as 1690".

There is at least one book of Gemma Frisius, in which the terms appears already in 1545:

1. https://gallica.bnf.fr/ark:/12148/bpt6k58333155/f63.double.r=.shift#

So the whole section on how the term evolved should probably be reworked. — Preceding unsigned comment added by P-flowgie (talk • contribs) 13:56, 27 October 2020 (UTC)Reply

Added a famous root finding problem from The Nine Chapters.

• Liu Hui et al. The Nine Chapters on the Mathematical Art Copyright (C) 263 Academy of Wei & Jin Dynasties
• Shen Kangshen, The Nine Chapters on the Mathematical Art, Oxford University Press 1999. Copyrighted in the United States until 2091

--205.175.118.10 (talk) 04:29, 1 December 2021 (UTC)Reply