Talk:Exsecant

Ratios of sides instead of devices

cosine(C) = (a² + b² - c²) / (2ab);

versine(C) = 1 - cos(C);

haversine(C) = sin²(A/2),

all of this can be done as a series of fractions; it helps if one knows how to extract roots, but that is a simple matter.

exsecant(A) = versine(A) secant(A).

First principles are the key to understanding ancient technologies.

-- AptitudeDesign (talk) 10:15, 31 May 2014 (UTC)

Potential to-dos

• Flesh out article by adding more historical sources and usage examples.
• Try to find historical origins of names secans exterior, external secant, exsecant, coexsecant, excosecant and their abbreviations.
• Try to find synonyms in non-English speaking countries in order to track down usage distribution. (For example, there don't seem to be synonyms for exsecant and excosecant in the German language.)
• Possible expansion into complex domain
• Add more mathematical properties

--Matthiaspaul (talk) 12:23, 6 December 2015 (UTC)

excosecant vs coexsecant

I've reverted the change to "coexsecant". While it's not everything, a quick Google search returns ~77000 results for excosecant, and ~240 results for coexsecant. Even if coexsecant was used first, we should be using the term that's used more today (rare as it is). Plus, here's a book from 1909 that uses "excosecant", so it's not as recent as you're claiming. I'm going to go through and revert any related changes on other pages, but please keep the discussion here. --Deacon Vorbis (talk) 12:49, 9 August 2017 (UTC)

As you can see from my other edits and comments above I was (am!) actively searching for historical sources for excosecant for quite some while without turning up anything except for lots(!) of recent sources (all dated since about 2006). So, it very much looked as if Wikipedia would have spread another neologism.

If your ref above actually states excosecant in 1909, that would be a great find (and I would have happily reverted myself without increasing my reversion counter), because I find excosecant far more logical and linguistically pleasing than coexsecant. However, the link above does not point to an online version of the book, probably one of the reasons why I didn't find it. Do you have a copy of that very book in front of you, or how do you know, that it actually says excosecant? Can you, perhaps, even provide a quote? --Matthiaspaul (talk) 22:12, 9 August 2017 (UTC)

a use of coexsecant

I was recently given a book written in 1891 and published the following year that has four extra trigonometric functions, other than the basic six. The (shortened) name of the book is "Architect's and Builder's Pocket-Book." (I should also mention that mine is the fourth edition.) These were versine (vers), coversine (covers), exsecant (exsec), and coexsecant (coexsec). Of course, since it is mainly for architects (it says that on a page I don't remember the number of, and I'm not going to dig through about 1,600 pages to figure it out), it does not offer much insight on what any of these were used for. As you could probably also tell, it did call it coexsecant instead of excosecant. I will provide a link to this book below.

https://archive.org/details/architectsbuilders00kiddrich/mode/2up

Please message me if this link does not work. There's also a square root method I've never seen on page four. RandomSerbia (talk) 00:25, 4 August 2023 (UTC)

It's page 99: https://archive.org/details/architectsbuilders00kiddrich/page/99/mode/1up

Here are some more sources with coexsecant:

• https://archive.org/details/surveyingmanual0000loui/page/108/mode/1up
• https://archive.org/details/bwb_P8-ABB-831/page/78/mode/1up
• https://archive.org/details/civilengineerspo00frye/page/n40/mode/1up
• https://archive.org/details/planetrigonometr00boharich/page/235/mode/1up
• https://archive.org/details/in.ernet.dli.2015.204961/page/n322/mode/1up
• https://archive.org/details/surveyingmanual1993unse/page/n509/mode/1up
• https://archive.org/details/fieldbook00holm/page/n126/mode/1up
• https://archive.org/details/johnsonstables0000unse/page/37/mode/1up
• https://archive.org/details/railwayrightofw00fryegoog/page/n59/mode/1up
• https://archive.org/details/sevenplacenatura0000vari/page/142/mode/1up
• https://archive.org/details/surveyingintrodu0000lega/page/416/mode/1up
• https://archive.org/details/theorypracticeof02john/page/625/mode/1up
• https://archive.org/details/practicaloilgeol0000dors/page/437/mode/1up
• https://archive.org/details/inspectionofmate00byrn/page/n536/mode/1up
• https://archive.org/details/analyticgeometry0000unse_h3p4/page/199/mode/1up

–jacobolus (t) 00:52, 4 August 2023 (UTC)

Are the sources for secans exterior being mischaracterized?

I don't read Latin, but in looking at all of these linked sources, it looks to me like the term secans exterior is just being used for any segment of a secant line exterior to the circle with one endpoint on the circle. That would be a different thing than the trigonometric function. Are there any secondary sources discussing this, or any contemporary (17th–18th century) English sources that might be a bit easier to read? Is there a source where the trigonometric context per se is clearly established? –jacobolus (t) 10:19, 27 March 2024 (UTC)

Ping @Matthiaspaul. It looks like you added the discussion of secans exterior in special:diff/794236108. Do you know if there are any relevant sources which are expressly trigonometric? (Can you read Latin?) If not, we should probably say explicitly that this was not about a trigonometric quantity per se. –jacobolus (t) 19:32, 28 March 2024 (UTC)

While we're at it, the claim that the "exsecant function" was used by Galileo seems clearly false. As far as I can tell Galileo was referring to particular line segments, not to any function of angle measure. I think the historical sections here need to be rewritten for clarity to avoid anachronism and to avoid misleading readers. The trigonometric use of exsecant, as far as I can tell from searching around, dates from the mid 19th century. –jacobolus (t) 19:45, 28 March 2024 (UTC)

I am also concerned that many of the more recent "sources" cited here, used as support for names, abbreviations, formulas, etc., are random self-published web pages, which should really not be taken as reliable (for example, van Vlijmen, Gottschalk, van den Doel, Calvert, and frankly Weisstein isn't a very good source either). In my opinion these should all be removed, possibly along with the claims they ostensibly support, and at the very most one or two could go in the "external links" section. –jacobolus (t) 20:02, 28 March 2024 (UTC)

Is there clear explicit support anywhere for the claim that the exsecant function was used in each of "fields such as surveying, railway engineering, civil engineering, astronomy, and spherical trigonometry"? The main sources I can see are just about planning/measuring railway curves (this is railway engineering, and could be considered part of surveying or civil engineering), mostly downstream of Haslett who was a civil engineer working for a railroad. It might be more accurate to say that Haslett developed an exsecant table and used it in preference to alternative trigonometric functions, was mildly influential from ~1850–1900 mainly confined to the field of railway design/layout in the USA, and that thereafter it was occasionally inserted into general trigonometry resources (general textbooks, reference books) but that it never really caught on in a broader context, and is now mainly a historical footnote. Then the article could focus on specifically what methods/tools Haslett or his followers developed/recommended and justifying why they were useful in the context of railway engineering, which would likely be much more interesting to readers the pile of miscellaneous anachronistic (original research) formulas which currently make up this article. –jacobolus (t) 20:17, 28 March 2024 (UTC)

I'm thinking of changing the lead section to say something like:

The external secant function (exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function:

$\operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.$

Like the versine function, $\operatorname {vers} \theta =1-\cos \theta ,$ exsecant is practically useful in calculations involving small angles, for which the secant is very nearly ⁠$1$⁠, which compromises precision when calculating with the secant function directly. For example, the common logarithm of the secant of 1° is 0.000066, with the first 5 digits wasted on zeros, while the logarithm of the exsecant of 1° is −3.817220 all of whose digits are meaningful.

The exsecant function was introduced in the mid 19th century by American civil engineer Charles Haslett for use in designing and measuring circular sections of railroad track, and by the end of the 19th century was commonly briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. While it has occasionally found other applications, today it is obscure and mainly of historical interest.

As a line segment, an external secant of a circle has one endpoint on the circumference, and then extends radially. This usage is older, dating to the 17th century.

Thoughts? –jacobolus (t) 06:32, 29 March 2024 (UTC)

I can't figure out what is supposed to be relevant in these sources. I can't find a mention of the exsecant.

• Boyer, Carl Benjamin (1969) [1959]. "5: Commentary on the Paper of E. J. Dijksterhuis (The Origins of Classical Mechanics from Aristotle to Newton)". In Clagett, Marshall (ed.). Critical Problems in the History of Science (3 ed.). Madison, WI: University of Wisconsin Press. pp. 185–190. ISBN 0-299-01874-1.
• Hawking, Stephen William, ed. (2002). On the Shoulders of Giants: The Great Works of Physics and Astronomy. Philadelphia: Running Press. ISBN 0-7624-1698-X.

–jacobolus (t) 19:51, 30 March 2024 (UTC)

Alright, continuing my conversation with myself here, I've now rewritten the article to give fuller context to readers and hopefully no longer be misleading/incorrect. I also took out most of the miscellaneous identities copied from the "review exercises" section of a random trigonometry book, which don't seem to be particularly important per se. I cut the parts about coexsecant down to a brief mention as I can't find a single example where this was ever used for any practical purpose. And I reduced the symbols and terminology down to the dominant version, removing some variants that were only found on now-dead self-published web pages. I'm sure the article could still be better (e.g. it could use a diagram of a railroad "simple curve"), but I think it's in okay shape for now. –jacobolus (t) 01:07, 1 April 2024 (UTC)