Talk:Brahmagupta interpolation formula

cleanup
The article says:


 * The Brahmagupta interpolation formula is:
 * $$r \sin\theta = \frac{\triangle\theta}{h} \left[\left(\frac{D_{p+1} + D_p}{2}\right) + \frac{\triangle\theta}{h}\left(\frac{D_{p+1} - D_p}{2}\right)\right]. $$
 * $$r \sin\theta = \frac{\triangle\theta}{h} \left[\left(\frac{D_{p+1} + D_p}{2}\right) + \frac{\triangle\theta}{h}\left(\frac{D_{p+1} - D_p}{2}\right)\right]. $$

&theta; is self-explanatory, but what is Dp and what is h? Obviously the article needs to say what those are. And how is the quantity called r on the left side related to anything on the right side? It seems as if you'd have to be so confused in order to leave out things like that that it's hard to imagine how you could be able to write the formula at all. I don't know if I'll ever understand this kind of behavior. Michael Hardy 21:48, 1 June 2007 (UTC)


 * Certainly this edit helps clarify things, but it would seem that the question about r remains. Michael Hardy (talk) 17:25, 7 May 2008 (UTC)


 * .... and also, what is h? Michael Hardy (talk) 20:27, 10 September 2008 (UTC)

The formula becomes a reasonable interpolation formula if one assumes that "r" is a transcription error for "&Delta;", with &Delta;sin(&Theta;) =def sin(&Theta; + &Delta;&Theta;) - sin(&Theta;), and that h is the length of the intervals over which differences are being taken, with Dp =def sin(&Theta; + h) - sin(&Theta;) and Dp+1 =def sin(&Theta; + 2 h) - sin(&Theta; + h).

But we shouldn't have to guess. It would be worth checking the reference to see if its account is as incoherent as the article's currently is. If so, we need to find a better one. &mdash;David Wilson (talk · cont) 17:30, 11 September 2008 (UTC)


 * If h is the interval over which differences are taken, then what is &Delta;&theta;? Michael Hardy (talk) 17:55, 11 September 2008 (UTC)


 * I was surmising that &Delta;&theta; would be the difference between the value, &theta; + &Delta;&theta;, of the argument at which the interpolated value of the sine function is to be calculated and the lowest of the three values, &theta;, &theta; + h and &theta; + 2h, of the argument at which the value of the sine function is already given. I have now checked the reference and found that this is not quite correct. The formula, for an arbitrary function f, should read:


 * $$ f(\theta + h + \triangle\theta) - f(\theta + h ) \approx $$
 * $$\frac{\triangle\theta}{h} \left[\left(\frac{D_{p+1} + D_p}{2}\right) + \frac{\triangle\theta}{h}\left(\frac{D_{p+1} - D_p}{2}\right)\right]$$,


 * where h > 0, -h &le; &Delta;&theta; &le; h, Dp =def f( &theta; + h ) &minus; f(&theta;), and Dp+1 =def f( &theta; + 2h ) &minus; f( &theta; + h ).


 * I was also wrong about "r" being a transcription error for "&Delta;". The reference only gives the right hand side of the formula, and then only in the context of a specific numerical example, with  f(x) = 150 sin(x), &theta; = 45&deg;, h = 15&deg; and &Delta;&theta; = 7&deg;. That is, the reference uses the formula to interpolate the value of  150 sin 67 &deg; from already tabulated values of 150 sin(&theta;) at &theta; = 45&deg;, &theta; = 60&deg; and &theta; = 75&deg;. The r sin &theta; which currently appears on the left side of the equation in the article is an erroneous presumption of what the right side of the formula means on the part of the editor who transcribed it.


 * All this raises a delicate point of policy. While I am sure my interpretation of the reference is correct, it is nevertheless an interpretation, which required the application of previously acquired mathematical knowledge on my part to supply the correct left hand side of the approximation, and to extend the reference's specific numerical example correctly to the general case. Thus, technically at least, this violates Wikipedia's policy of no original research and should not be used in the article. However, the reference does say that Brahmagupta's formla is equivalent to the Newton-Stirling formula to second-order differences, and gives an expression for that formula, so we can use that without violating the no original research policy.  I shall make the correction tomorrow.
 * &mdash;David Wilson (talk · cont) 18:06, 12 September 2008 (UTC)

I checked out the book yesterday and will look closely. Michael Hardy (talk) 18:38, 12 September 2008 (UTC)


 * I have now made the change proposed above.
 * The previous version of the article gave the impression, not supported by the indicated pages of the cited reference, that Newton and Stirling were aware of Brahmagupta's formula, and used it as a starting point to develop their more general formula. I have taken the opportunity to eliminate that as unsupported original research. Of course, it should be reinstated if support for it can be found in the cited reference or another reliable source.
 * Is the formula sufficiently notable to warrant its own article? It seems to me that a section or subsection of the article on Brahmagupta would be a much more appropriate place for it. I will therefore propose that it be merged into that article.
 * &mdash;David Wilson (talk · cont) 03:19, 13 September 2008 (UTC)

Please. Look at my most recent edits. Your way of using various mathematical notations suffers from a number of weaknesses. In particular, you twice used a hyphen instead of a minus sign in non-TeX notation, you inexplicably ended the math environment and then immediately started it again, you wrote "\ +" instead of " + {}" in TeX, and you didn't use sufficient spacing in non-TeX notation. Michael Hardy (talk) 06:44, 13 September 2008 (UTC)


 * Thank you for the tips, and for cleaning up the text. The explanation for my breaking the math environment in the displayed formula is that I have a very old computer with a narrow screen. Thus, with the typical width I have my browser set at, the single-line formula overflows the right edge of the page by a considerable margin.  Ideally, it would be nice to be able to write the TeX so that the formula displays on a single line if the browser  is wide enough, but splits in an appropriate place and displays on two lines with appropriate spacing when it isn't. Unfortunately, I can't think of a way to make it do this.
 * &mdash;David Wilson (talk · cont) 08:30, 13 September 2008 (UTC)

Proposal to merge into Brahmagupta article
Even if the formula which is the subject of this article is sufficiently notable to justify a stand-alone article on it, it seems very likely that the article will never contain enough material to become anything more than a stub. On the other hand, it strongly satisfies two of the four criteria listed as being good reasons for merging one article into another in the Wikipedia help page on mergers. It also partially satisfies the fourth criterion, though not as strongly as it does the second and third.
 * Overlap. Given the level of detail already supplied in the Mathematics section of the article on Brahmagupta, the information given in this article is clearly very appropriate for inclusion in that section.
 * Text. The text of the article is currently very short, and, as noted above, is unlikely to grow much longer.
 * Context. While the information in the article on Brahmagupta is not necessary to understand the article on the interpolation formula, it nevertheless provides a solid context into which the information on the interpolation formula fits nicely. This information clearly bears a relation as closely similar to the article on Brahmagupta as that which all the information on his other mathematical contributions do.

Consequently, I propose that this article be merged into the subsection on trigonometry in the article on Brahmagupta. &mdash;David Wilson (talk · cont) 07:40, 13 September 2008 (UTC)
 * This is about a formula, while that article is about a person. The Brahmagupta article contains so much content that the details about the formula could ideally be retained and expanded in this article without merging them both. ­ Kris (talk) 20:38, 30 November 2008 (UTC)


 * But as far as I can tell there is nothing particularly notable about the formula, except for the fact that it is one of Brahmagupta's many discoveries. Also, the Brahmagupta article doesn't seem to me to be all that long by Wikipedia standards.  A printed copy of the article contains only 8 pages of readable prose, and by the method recommended in the guideline on Article Size for estimating the amount of readable prose in an article, it contains only 19kb, well below the sizes of 10 pages or 30kb which the guideline suggests as the lower limits at which one should consider splitting off otherwise relevant content. Even including all the article content that doesn't count as readable prose, the total size is only 11 pages, or 35kb.


 * There also doesn't seem to me to be much scope for expanding this article on Brahmagupta's formula, which is one of the main reasons why I suggested that it be merged. In what way do you think it could be expanded?
 * &mdash;David Wilson (talk · cont) 13:17, 1 December 2008 (UTC)