# Talk:Snell's law

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Consistency problem:

In the top explanatory figure n2>n1 is explicitly stated. In section "Total internal reflection and critical angle", however, the following must hold true (see also the corresponding figure): n1>n2. As in the text there is no indication of this inversion, this leads to confusion. We should either make the relationship consistent throughout, or use different names for the refractive indices in the different examples.

145.64.134.221 (talk) 11:53, 1 October 2009 (UTC)

It would be nice to spell Snel's name correctly (i.e., with a single "l"). His name is "Snel" in his native language, or "Snellius" in Latin. The common spelling "Snell" is a solecism committed by people who know neither Dutch nor Latin.

Perhaps it would also be wise to point out that Snel was not the *original* discoverer; the law was first found by Thomas Harriot, about 1600 -- two decades before Snel's work.

http://mintaka.sdsu.edu/GF/explain/optics/discovery.html

An experimental paragraph added. See a short article by Kwan, Dudley and Lantz about this in Physics World in 2003 or 2002. This article says that Thomas Harriot (Hariot may be his preferred spelling) was actually not the first.

• I've improved the history section with information from that article, and incorporated the references in it. Hope that's clearer. I'm not sure what to do with the external link, which contradicts those articles (says Harriot was first). -- Bob Mellish 17:36, 30 September 2005 (UTC)

Snell may be a solecism, but it's English, and it's the English name of the law, even if not how the man wrote his own name. Given how fluid spelling could be in the 17th century, it could be that Snell spelt his name in several different ways. Can someone check this?

## Image wrong?

The image shows that the normal is the boundary between the media, rather than being horizontal to the boundary. Also, ${\displaystyle \theta _{1}}$ and ${\displaystyle \theta _{2}}$ are wrongly labelled as a result. The image disagrees with the text. See ScienceWorld [1] for how the image should be corrected.

The image is misleading and confusing, and should be rectified as soon as possible; however, as I lack experience with graphics, I request someone to upload a corrected version as soon as possible.

• Um, no. The image is correct, and correctly labelled. The image in the article has the two media on the left and right sides of the diagram; the interface runs vertically. The image at Scienceworld has the two media at the top and bottom of their diagram; their interface runs horizontally. Try rotating one of the diagrams by 90°, and you'll see how they match. -- Bob Mellish 18:25, 26 September 2005 (UTC)

## Snell's law spelling

I agree it is Snel and not Snell's law. Also in English. The 'discoverer' (not getting in the historical issue here, just the spelling bit) of the quantitative law of refraction was 'Willebrord Snel van Royen', thus one l, and there is no fluidity to this spelling as far as I know. The two-l-spelling has nothing to do with 'Snel' spelled differently in the English language but with incorrect de-latinazation of Snellius.

FWIW, in the Netherlands, up to the present, the person has always been known as Snellius, not Snel. I think, most Dutchmen who have heard about him would be surprised to learn that he wasn't born as Snellius. Iterator12n Talk 05:29, 11 July 2007 (UTC)
Also FWIW, the Dutch encyclopedia Winkler Prins, under "Snellius", shows "Snell van Royen" as the original family name - with two l's. Finally, with Snellius goes "Willebrordus", not "Willebrord." This is all you're going to get from me on the name of Snellius! Iterator12n Talk 20:07, 14 July 2007 (UTC)

His name was Willebrod Snel van Royen. The sine law carries his name which is snel and not snell. As mentioned above the double l is due to incorrect de-latinazation.

## Other formulae

I've seen this formula used quite a lot in some establishments as an expansion of the original law for wavelength...

${\displaystyle {\frac {\lambda _{0}}{\lambda _{1}}}={\frac {v_{o}}{v_{m}}}={\frac {c/n_{1}}{c/n_{2}}}={\frac {n_{2}}{n_{1}}}}$

to... ${\displaystyle \lambda _{1}sin\theta _{1}=\lambda _{2}sin\theta _{2}}$

James S 00:12, 11 December 2006 (UTC)

Those ratios of velocities, wavelengths, indices of refraction, etc. are fine, but they are not Snell's law. Snell's law needs to have the sines of the angles in it. We seem to have forgotten to state the law near the top of the article anywhere. I'm working on it... Dicklyon 20:04, 22 December 2006 (UTC)
OK, I fixed the lead, putting the law equation and illustration into it; and I added a book page of history. I don't really understand the point of what someone was trying to do with these relations in the Explanation section, so I'll leave it for now. But as I said, without the angles, or some measurement proportional to their sines, it's not Snell's law. Dicklyon 20:40, 22 December 2006 (UTC)

## Merge from Angle of refraction?

I propose to merge Angle of refraction into Snell's law, since it covers exactly the same material. Please support, oppose, or otherwise comment here. Dicklyon 08:36, 23 December 2006 (UTC)

support The Photon 03:46, 24 December 2006 (UTC)
Expecting no objection, I went ahead and incorporated a few bits from there that we didn't have here, and converted it to a redirect. Dicklyon 05:30, 24 December 2006 (UTC)

## Reverting JCraw's extensive uncommented changes

JCRaw, that's a lot of changes to do all at once without even any change comments. You've de-linked the references from what they refer to, and made them into a very hard-to-maintain form (because the numbers don't track automatically). And you've introduced non-words (e.g. constance) and grammatical errors into the lead. I haven't reviewed most of the changes yet, but on these bases alone I'm going to revert, and we can make the changes you want more slowly and carefully, giving other editors a chance to collaborate on them, please. Dicklyon 17:00, 4 January 2007 (UTC)

JSpudeman, the way you've put it back is really no better. You still have the hard-to-maintain ref style, grammatical errors in the lead, and unclear what point you're trying to make there. The statement about "it's [sic] original form" is probably wrong, since the constant ratio of sines was articulated before velocities or indices of refraction were known. Dicklyon 23:24, 4 January 2007 (UTC)

Point noted; i know of the application Fermat's principle to Snell's Law, but i was unaware of the history linking them together. However -- what was the original formula that was used before the inclusion of least-time? It would be interesting to know how the original formula was developed. On that note, do you own that book? I presume that from your reference to it's contents that you do? If so, why not reference it? James S 23:23, 7 January 2007 (UTC)

Which book are you referring to? I have Huygens. The others I mostly just find on books.google.com. Dicklyon 01:14, 8 January 2007 (UTC)

As a slight offshoot to the topic here, perhaps the introduction should be more explanatory:

In optics and physics, Snell's law (also known as Descartes' Law or the law of refraction) is a formula that relates the angles where a ray of light crosses a boundary between different media, such as air and glass.

Although it's fine, it doesn't quite explain the reason for the relation of the angles, or what is being related other than "angles" (i.e incidence/refraction).

The law was determined before a reason for it could be found, if by reason you mean the underlying physical basis. When Descartes and Fermat articulated the law as being based on a "principle of least time," they still didn't have an underlying physical reason to explain that principle. Huygens explained it with his wave theory, but it took another 120 years and rediscovery of that approach for that reason to begin to be accepted. In the mean time, Snell's law served geometric optics admirably, even without any "reason" behind it. So, I think the reason can come later, or can remain divorced from the law. In fact, all that's being related is angles. The law just says the ratio of sines is constant, just like Ibn Sahl said with a geometric relationship in his construction. Dicklyon 01:14, 8 January 2007 (UTC)

Similarly, i was just taking a glance at the edit history and noticed that apart from a slight change in grammar, some of the explanations were removed. Again, i'm staying well away from this one, but i'm wondering why that is exactly. Although practically the same information is there, it makes it more difficult for those who are reading the article as an impartial/non-informed user, i think, to pick up on the article.

The explanations that ended up in footnotes are removed until someone takes the time to incorporate them better. The "ref" mechanism was already in use, keeping a list of numbered refs in sync with numbers in the text, when it was taken over to use for footnotes instead, leaving the numbered references with no automatic way to stay synchronized. That's why I reverted it. If there's stuff in there that was useful, why not help incorporate it? Dicklyon 01:14, 8 January 2007 (UTC)

I'll leave it in your hands, as my edits would undoubtedly be reverted ;-) James S 23:36, 7 January 2007 (UTC)

Not if you don't hijack the "ref" mechansim again ;^} Dicklyon 01:14, 8 January 2007 (UTC)
JCraw, I looked over your notes again, and I'm having trouble getting the point of them, or why you added them. It looks like you have a couple of useful references with derivations and applications, but what you said about them in the footnotes was difficult to understand. Please join the discussion here to tell us what issue you are trying to address. Dicklyon 06:51, 8 January 2007 (UTC)

Where did the idea that it was ever known as "Descartes's Law" come from? I have never heard of it under that name. Out of 5 different books on E&M/Optics(Frankel, Feynmann, Jackson, Ditchburn, Stratton) handy for me to check, not one uses this term.

It the alternate name is really that rare, it only adds clutter to mention it in the Wikipedia article. It adds no useful information.

On a side note: "optics and physics" is redundant, since optics by definition is a branch of physics. Slighly more informative would have been "optics and wave theory", since that makes it clearer that Snell's Law applies to radio waves as well. But it would have made even more sense (following Jackson's hints) to rephrase the whole sentence as:

In optics and wave theory, Snell's law (also known as Descartes' Law or the law of refraction) is a kinematical formula that relates the angle of incidence and that of refraction where a ray of light crosses a boundary between transparent media with differing optical characteristics, such as air and glass.

68.166.188.143 (talk) —Preceding undated comment was added at 18:39, 1 September 2008 (UTC)

## The original form of the law

Here's an 1803 book that explains that Snel did the same thing that Ibn Sahl had done. Nowhere does the velocity of propagation or the index of refraction enter into his observation that the ratio of sines is a constant for a given pair of media. Later, when it was realized that light speed varies in different media, it was realized that the law of sines was in agreement with a principle of least time, or Fermat's principle; that's where velocity and index started to come into the equation, via their ratio. Let's not get the cart before the horse on this. Dicklyon 01:42, 5 January 2007 (UTC)

## Applicability to sound waves

Snell's law is not just limited to propagation of light but can also be used to explain the propagation of sound waves across different medium where the speed of sound changes. The article fails to mention this. I think this law can be used for explaining other kind of wave propagation too, though I am not sure. But at least we sound waves should be mentioned in the introduction of the article. -- Myth (Talk) 05:50, 16 April 2007 (UTC)

It says "light or other waves, passing through a boundary between two different isotropic media". Is that not sufficient to encompass sound? Dicklyon 06:05, 16 April 2007 (UTC)
I agree, but the article focuses heavily on propagation of light waves and the importance is not evident in relation to other kinds of waves. A reader not aware of the applicability of the law in other cases, won't realize it easily. -- Myth (Talk) 06:21, 16 April 2007 (UTC)
That's because light is the usual application. Sound diffracts so much that Snell's law is seldom a useful approximation to sound wave behavior. If you have other applications, or relevant sources, please do bring them up. Dicklyon 14:58, 16 April 2007 (UTC)
Snell's law is very important in the study of underwater sound. See, for example: Robert J. Urick, "Principles of Underwater Sound (2nd edition)." New York: McGraw-Hill, 1975, p.116. Ephesians 2:10 (talk) 23:10, 8 September 2009 (UTC)

## How to reach consensus

See WP:BRD. If you make a change (e.g. adding "by scientists") and someone reverts it, bring it up on the talk page if you care; don't just make it again. Dicklyon 05:26, 1 August 2007 (UTC)

## Proposal to rotate the image

The image on the right (Image:Snells law.svg) is quite nice, but I think in the literature the interface between media is usually horizontal (like between air and water). I have rotated the image, see Image:Snells law2.svg (also on the lower right). I propose to replace the original with this rotated version. Comments? Oleg Alexandrov (talk) 00:58, 25 December 2007 (UTC)

In optics, rays are most often traced from left to right. But I agree it looks good going top to bottom. Dicklyon (talk) 01:59, 25 December 2007 (UTC)
Done. Rotated, the image shows the details better while taking less real estate. Oleg Alexandrov (talk) 04:54, 1 January 2008 (UTC)

## Explanation for section removal

I removed a section from the article, for the following reasons:

1. The informational content of the section is minimal, it repeats what is already stated both in the article and the diagram
2. Then name of the section is misleading, that section is not about calculating refractive indeces
3. The analogy with the car going from highway to the mud causing it to change angle is I think misleading, it is not as if a ray is a pair of parallel waves and one of them reaches the seconc medium first and forces the second wave to turn. Besides, is mud denser than asphalt?

Comments? Oleg Alexandrov (talk) 04:34, 10 January 2008 (UTC)

## Formula confusion

I'm confused about the formula. If the angle of incidence is 0, then how does the formula work? Or does it not work for maxima and minima of the sine wave? STYROFOAM☭1994TALK 22:58, 29 January 2008 (UTC)

The equation ${\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}$ is satisfied with θ1 = 0 and θ2 = 0 for any indices of refraction. That is, if the ray comes in perpendicular to the surface, it stays that way. Dicklyon (talk) 00:09, 30 January 2008 (UTC)

## Vector Form Inconsistency

The "vector form" example calculates cos(theta_1) incorrectly. The dot product of -l and n is clearly positive, but the example shows that the result is negative. When I calculate the reflected angle, I only get the correct answer if I define cos(theta_1) as the dot product of l and n (instead of -l and n).

The article does not give a source for the formula. Does anyone have a source (and therefore a way to verify the correct formula)? —Preceding unsigned comment added by 71.235.123.252 (talk) 03:03, 30 January 2008 (UTC)

I had difficulty with this formula in 3D vector mode:

Note: ${\displaystyle \mathbf {n} \cdot (-\mathbf {l} )}$ must be positive. Otherwise, use

${\displaystyle \mathbf {v} _{\mathrm {refract} }=\left({\frac {n_{1}}{n_{2}}}\right)\mathbf {l} -\left(\cos \theta _{2}+{\frac {n_{1}}{n_{2}}}\cos \theta _{1}\right)\mathbf {n} .}$

I got results to agree with Breault ASAP raytrace results by modifying the first formula:

${\displaystyle \mathbf {v} _{\mathrm {refract} }=\left({\frac {n_{1}}{n_{2}}}\right)\mathbf {l} +\left({\frac {n_{1}}{n_{2}}}\cos \theta _{1}-\cos \theta _{2}\right)\mathbf {n} }$

to the more general:

${\displaystyle \mathbf {v} _{\mathrm {refract} }=\left({\frac {n_{1}}{n_{2}}}\right)\mathbf {l} +\left({\frac {n_{1}}{n_{2}}}\cos \theta _{1}-\left(sign\left(\cos \theta _{1}\right)\right)\cos \theta _{2}\right)\mathbf {n} }$

This worked for both cases.

Palmeroo (talk) 17:34, 28 February 2008 (UTC)

Are there any reasons why this hasn't been corrected?

Anyway, I've updated it, so please check if it is correct now. Anders Ytterström (talk) 20:26, 2 May 2008 (UTC)

Is the formula still wrong? it disagrees with this webpage: http://www.nationmaster.com/encyclopedia/Snell's-law and it doesnt even make sense... how can refraction depend on the position of the light? It has nothing to do with lighting... Someone has to fix this. 128.100.32.72 (talk) 00:19, 30 November 2008 (UTC)

That web page is just an old copy of this article. If there's an actual source that disagrees, that would be more interesting. The light vector is just the direction of travel of a light ray to be analyzed; don't think of its source as a lighting source, just where a ray is coming from. Dicklyon (talk) 00:58, 30 November 2008 (UTC)

Does anyone know why Ibuwan edited the second formula for v_refract on Feb 14, 2012? I believe this to be wrong; the formula worked before, (with two additions) and is broken now (with two subtractions). (171.66.163.219)

I changed the formula for a better understanding for symmetry. it used to read ${\displaystyle +\left(-A+B\right)}$ and i changed it to${\displaystyle -\left(A-B\right)}$so both are actually equal. It would be interesting to see it work with only two additions, because i carefully verified my result. (not for this article, but for my work) so please feel free to correct if im wrong. Ibuwan (talk) 01:37, 8 May 2012 (UTC)

## Metamaterials?

Um, Snell's law has been disproven/broken/shattered for the better part of a year now... when is this article going to be updated to reflect that? Light can be bent at left angles using metamaterials, and the light traveling "backwards" in a vacuum exceeds the "normal" limit of light speed (the only thing (other than tachyons)) that travels faster than light is still light... just light traveling backwards). 68.185.167.117 (talk) 15:02, 26 November 2008 (UTC)

That work has not bothered Snell's law at all, but if you'd like to add a bit about it, citing a good source, that might be useful. Dicklyon (talk) 16:44, 4 December 2008 (UTC)

## Al Haythem, A.I.Sabra and the Sine Law of Refraction

I have removed a sentence from the main text in which it was claimed that Al Haythem had known the sine law of refraction and which gave A.I. Sabra's Theories of Light From Descartes to Newton as source for this claim. This is not true Al Haythem did not have knowledge of the sine law of refraction and this is clear from the following passage from Sabra(page97):

Now let us suppose that Ibn al-Haythem moved one step further and assumed the increase... (there follow a series of mathematical deductions)... In other words the sines of the angles of incidence and refraction are in a constant ratio, which is the geometrical statement of the law of refraction .(...) He did not, however, take that step...

Sabra then goes on to argue that this might have been the route taken by Descarte who knew Al Haythem's work in discorvering the law of refraction. What we have here is a hypothetical argument concerning the possible route of discovery taken by Descarte and not the claim that Al Haythem had discovered the law himself, which he had not.Thony C. (talk) 13:12, 4 December 2008 (UTC)

## Snell's Law Obsolete?

This was from the Faster-Than-Light article:

This is influenced by man-made metamaterials, which allow light to be bent backwards; the discovery of these shattered the now defunct Snell's Law, an old "law of physics".

Shouldn't something to that effect be added to this article, then? SineSwiper (talk) 01:52, 18 December 2008 (UTC)

Negative-index metamaterials still obey Snell's law, just with a negative index of refraction. — Steven G. Johnson (talk) 04:41, 5 August 2009 (UTC)

## Finding the speed of light

I think the formula for finding the actual speed of light (${\displaystyle n=c/v}$) should be added into the section. This is the only appropriate place on Wikipedia to put it on. However, I cannot find a reliable source to cite. If someone can find this, I think it will help a great many people. Fireedud (talk) 18:14, 3 January 2009 (UTC)

I just found it on Refractive index, but I think it should also be added here. Fireedud (talk) 18:21, 3 January 2009 (UTC)

## "Today's Featured Picture" for 9/23/2009 incorrect !!

The image, in the article at "Explanation#Derivations", is incorrect in the lower part, below the interface. It shows the wavefronts as becoming hyperbolic, so that the portions at large distances from the central axis asymptotically become straight lines. However, the refracted rays below the interface would then not appear to diverge from a point, and that is not the case in reality. The point source position is shifted (upwards, in the figure), but the light source seen from below the interface still appears as a point. These means the wavefronts must continue to diverge from a point, and thus remain segments of a sphere, with only the center of the sphere being shifted. The figure needs to be corrected by someone with a facility in graphics animation. A few other editors should verify and confirm my conclusion, but it is really quite obvious, and easily apparent visually so that it needs to be fixed quickly. Thanks, Wwheaton (talk) 15:53, 23 September 2009 (UTC)

I believe it is correct. The refracted rays will NOT appear to come from a point, as is well known; there will be spherical and chromatic aberrations, as is the case when putting a prism behind a lens that's designed to image through air. The rays far from the source will approach the critical angle. Dicklyon (talk) 16:02, 23 September 2009 (UTC)
I'm going to have to disagree with you as well. According to Snell's Law, ${\displaystyle \theta _{2}=\arcsin \left({\frac {n_{1}}{n_{2}}}\sin \theta _{1}\right)}$. Note that n1 (air) = 1 and n2 (water) = 1.33 (and thus n2/n1 = 0.75), and that ${\displaystyle \sin \theta _{1}=\sin \left(\arctan {\frac {x}{h}}\right)}$, where x is the horizontal distance from the central axis and h is the height of the source over the water. For my purposes, I'll just say h=2 meters (the height and the units are arbitrary and unimportant). So we get that ${\displaystyle \theta _{2}=\arcsin \left(0.75\sin \left(\arctan {\frac {x}{2}}\right)\right)}$. Due to some lovely trigonometry, ${\displaystyle \theta _{2}}$ is not just the angle of refraction from vertical; it's also the angle of the underwater wave (which is perpendicular to the angles of refraction) from horizontal (i.e. the angle of inclination of the wave). So, we can plot ${\displaystyle \theta _{2}}$, as defined above, in WolframAlpha. You'll see that ${\displaystyle \theta _{2}}$ changes close to zero (the central axis), but that it quickly approaches an asymptote on both the positive an negative ends. (Near-)constant ${\displaystyle \theta _{2}}$ (angle of inclination) far from the central axis means (near-)straight line. That's what the image depicts, so I think the image is correct. Tell me if you think I made an error in my calculations and explanation; they were hastily done. -- tariqabjotu 20:38, 23 September 2009 (UTC)
Your argument seems plausible to me at the moment, though I need to verify it when I have more time. The fact that objects under water appear sharp and unblurred led me to believe that the refraction of a plane surface does not affect the spherical character of the waveform of light from a point source, but I see now that the small-angles approximation may indeed account for this, and the existence of critical refraction supports your argument. Thanks for (probably) correcting me, and apologies for the false alarm. Wwheaton (talk) 14:25, 25 September 2009 (UTC)
If you look at objects underwater, or embedded in glass or plastic, from an angle, they actually appear with significant blur and color fringing. See my patent on how such aberrations can be approximately corrected, in the case of an optical system designed to focus in air being used through glass. Dicklyon (talk) 03:46, 26 September 2009 (UTC)

## Is it really a law?

About 20 years ago, I remember being told in college that Snell's Law is not a law because

1. It can be derived (which has since made me wonder about the nature of laws)
2. It is an approximation

If there's any truth to this, I think it would be good to include. (If there's wide misconception about this, I think it'd be even better to attend to it) Googling for '"snell's law" "not a law"' has been inconclusive. AngusCA (talk) 23:35, 13 June 2010 (UTC)

Without a source, there's not much you can do. Lots of things are called laws; I wonder where those criteria come from. Dicklyon (talk) 05:25, 15 June 2010 (UTC)

There is no categorical definition as to what constitutes a law in physics and Snell's Law is known as Snell's Law and that's that. As to your objections what has being derived got to do with it? All of the mathematical laws of physics are derived from raw empirical experimental data. Finally in what sense is Snell's Law an approximation? Please elucidate?Thony C. (talk) 13:23, 15 June 2010 (UTC)

I can't. I haven't seen that teacher in 20 years. I don't even remember his name, which is fortunate, because he was a jackass. But he seemed to know what he was talking about. AngusCA (talk) 02:39, 16 June 2010 (UTC)

Actually is a rule derived from Fermat principle.This principle does not explains underwater refraction if the point Q is at the bottom of the sea and sunlight cannot reach it.Be carefull about math:it is not a trick!You cannot put Q on a "ray" because P & Q define the ray!It is exactly the Q point whitch defines the ray. — Preceding unsigned comment added by El662009 (talkcontribs) 10:28, 8 January 2015 (UTC)

## Wolf's paper

Did anybody get a chance to check Wolf's paper to see what it says about history of mathematics? Tkuvho (talk) 11:08, 3 June 2011 (UTC)

## Second Picture is Incorrect

The second picture in the article(the one that shows the incident, refracted and reflected wave) has an inaccuracy in the Refracted Wave. The inaccuracy occurs where the angle of reflection is labeled. I have already made an image that I believe contains more relevant information to the less informed and does not contain the inaccuracy in the only two angles that are relevant to Snell's Law and will be uploading it as the successor shortly. — Preceding unsigned comment added by Chapin J (talkcontribs) 21:07, 26 September 2013 (UTC)

## Vector form

Every few months, we get an uncommented edit or two to change a sign or two in the "Vector form" section. I tried to check the source but the relevant pages aren't showing up in gbs. The ref came in in this uncommented edit, which didn't change the equations, so I don't really know if the equations came from there, or were consistent with the ref at that time, or what. Someone should check, and put a comment in the wikisource about staying true to the ref. Or work it out and make sure it's right. Part of the problem is that the direction of the surface normal is not defined, so people may be trying it different ways. An appropriate figure would help. Dicklyon (talk) 18:33, 26 January 2014 (UTC)

I reworked the presentation considerably, making sure it is understandable and self-consistent. If someone has the source and wants to make sure I didn't say anything that it doesn't supprot, please let us know so we can adjust. Dicklyon (talk) 19:35, 26 January 2014 (UTC)

## Elaborate on Ibn Sahl's authorship of the law

It is claimed in this article (and in the article on Ibn Sahl) that Ibn Sahl was the first to understand the law of refraction correctly, namely in the following way (text under image):

Image 1: For any given angle ${\displaystyle \theta _{\mathrm {i} }}$ of the incoming ray with the surface normal, one is able to construct a refracted ray (at an angle ${\displaystyle \theta _{\mathrm {r} }}$ with the surface normal) by keeping the ratio of the two hypothenuses ${\displaystyle b/a}$ constant. The constant is characteristic of the interface, its value being the reciprocal of the nowadays known relative refractive index ${\displaystyle n_{\mathrm {rel} }=n_{\mathrm {i} }/n_{\mathrm {r} }}$ .
Image 2: An excerpt from Ibn Sahl's manuscript

I am not convinced by the material I have found on the internet so far. Here are a few reasons:

1. The articles on Wikipedia provide insufficient justification. Image 2 is presented in each of them, yet no translation of the surrounding arabic text is given. It seems very important to me, what the text is about. If the text is explaining the construction of a refracted ray on the plano-convex lens in the sketch (lower right) with the aid of the two right triangles (upper left), then that is concrete evidence of Sahl's understanding of the law. However, if he is talking about something unrelated (if the two right triangles have no corelation with refraction), then that is quite a substantial reason to doubt his understanding. If the person who posted this image to support the claim of Sahl's authorship knew of the meaning of the text and did not provide a translation because it has no relation to refraction, I would consider that very dishonest.
2. There is only one article containing original research on the subject and it is hard to come by. That is the article by Roshdi Rashed, A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses, a free version of which I have not been able to find. All of the other cited references (and all literature I have been able to find) do not provide any justification for Sahl's authorship. They simply assume it, citing Rashed's article. If somebody has a full english version of Rashed's article, please give me a link. It's the only reason I'm still on the fence about this and haven't completely dismissed the claim.
3. Step 0: I have renamed the points with Latin letters so it's easier for me to reference them. I have also colored the plano-convex lens gray.
On image 2, I see no construction of the refracted ray on the plano-convex lens itself. I would expect Ibn Sahl to use the two right triangles as an aid in constructing the refracted ray. Perhaps he did and it's just not immediately evident from the sketch, let's investigate. It's claimed that one of the notches on the optical axis is supposed to be the focal point. Quote from the article about Ibn Sahl: "The curvature of the convex part of the lens brings all rays parallel to the horizontal axis (and approaching the lens from the right) to a focal point on the axis at the left." The lens is a thick spherical lens and so does not have a well defined focal point, but let's not hold this against the explanation. Let's instead assume that one of the notches on the optical axis is where the parallel ray from the right, refracted at the point marked on the upper right part of the lens' surface intersects the optical axis. I have tried reconstructing the refracted ray using vector graphics (Inkscape). Here's what I did:
Refracted ray reconstruction
Step 1: I have tried fitting a yellow semicircle to the semicircle going through points A, B and F as a test. This is the best fit I could achieve. It's not great, also the center is in point N, which is not marked on the original picture. Could be that the aspect ratio of the original is off. Let's forget about this for now.
Step 2: We'll try to construct a refracted ray through point R following instructions from image 1. First, we need to find the surface normal in R. For this we fit a red circle to the semicircle going through A, P, B. The best fit is achieved with the center in M (not marked on the original). Clearly, the line MP is now the surface normal of the imaginary right half of the lens in P and thus also of the real lens in R.
Step 3: We now take the blue line XY, rotate it (preserving length) and fit it through M and O, so that one end is in M. We mark the other end with L. Now we need to move the green line XZ with one end in L and fit the other end to the red surface normal MP we found in the previous step. This gives us the direction of the refracted ray LN.
Step 4: Finally, we draw a line parallel to LN through R (our refracted ray). We also draw a blue incoming ray through R. As we can see, the ray is not refracted much and so does not go through any of the origally marked points. This is perhaps not surprising, since the relative refractive index calculated from the ratio of the blue (XY) and green (XZ) lines is only about 1.26. For example, the relative refractive index for a glass-air interface is about 1.5.

I interpret this as a negative result for the corelation between the original image and Snell's law. A very positive result would be if the refracted ray went through one of the points marked on the optical axis. I didn't want to give up yet, so I changed the aspect ratio of the original image (suspecting that the original was wrong) so that the blue line XY was a perfect fit between C and H. Specifically, I squished the image horizontally to 78% of its original width. I think H was intended (in the original) to be the point on the optical axis, closest to P (thus directly above P). The procedure in the following pictures is the same as above so I will only comment on important differences.

Refracted ray reconstruction with a modified aspect ratio
Step 1: The test fit of a yellow circle is now much better. Also, the center of the yellow circle is now actually in point O.
Step 2: The center of the red circle now falls in point C. This is promising. The surface normal is the line CP.
Step 3: The blue line of the incoming ray now stretches from H (a point present in the original image) to C. This is not surprising, since the image was resized based on this criterion.
Step 4: However, the ray is now refracted even less, the relative refractive index being around 1.18.

Again, a negative result. You can try to tinker with these .svg files yourself, all you need is Inkscape. Somebody please refute or confirm my doubts. I am writing the history section of the article on light on sl.wikipedia and I want to present the truth, not speculation.

Marko Petek 00:05, 15 July 2015 (UTC) — Preceding unsigned comment added by Marko Petek (talkcontribs)

Here's the actual supporting passage:

"The hyperbola as a conic section: The law of refraction.

Ibn Sahl first considers refraction on a plane surface. Defining ${\displaystyle GF}$ as the plane surface of a piece of crystal of homogenous transparency, he emphasizes a relation that is the reciprocal of the refractive index ${\displaystyle n}$ of this crystal in relation to air.19

'Let ${\displaystyle DC}$ be a light ray in the crystal, which is refracted [see figure] in the air along ${\displaystyle CE}$. The perpendicular to the plane surface ${\displaystyle GF}$ at ${\displaystyle G}$ intersects line ${\displaystyle CD}$ at ${\displaystyle H}$ and the refracted ray at ${\displaystyle E}$.'

The ratio ${\displaystyle CE/CH<1}$, which Ibn Sahl uses throughout his study, is the reciprocal of ${\displaystyle n}$:

'Let ${\displaystyle i_{1}}$ and ${\displaystyle i_{2}}$ be the angles formed by ${\displaystyle CD}$ and ${\displaystyle CE}$, respectively, with the normal; we have

${\displaystyle {\frac {CE}{CH}}={\frac {CE}{CG}}\cdot {\frac {CG}{CH}}={\frac {\sin i_{1}}{\sin i_{2}}}={\frac {1}{n}}}$.

Let ${\displaystyle I}$ be a point on segment ${\displaystyle CH}$ such that ${\displaystyle CI=CE}$, and let point ${\displaystyle J}$ be the middle of ${\displaystyle IH}$. We have ${\displaystyle CI/CH=1/n}$. Therefore ${\displaystyle C,I,J,H}$ characterize the crystal for any refraction.'

This result of considerable importance, encountered here for the first time, enabled Ibn Sahl to utilize the law of inverse return in the case of refraction, which is essential for the study of biconvex lenses, as we shall see later.

19[Tehran manuscript of ibn Sahl's treatise, pages] 5-9."

— Rashed (1990), p. 478
According to Rashed, the plate with the Arabic text comes from page 7 of the manuscript, and the geometric references are to the triangle on that page. The sketches themselves, I admit, are too vague to be considered on their own. Nevertheless, if the translation that Rashed gives is faithful to the original work (especially the last sentence, suggesting that the relative location of those collinear points are a material property of crystals that determine an invariant quantity for refraction experiments), I think it's fair to say that ibn Sahl gets priority for the law of refraction. Conformancenut347 (talk) 19:01, 29 July 2016 (UTC)