# Newton's $\sin(r)/\sin(i)=n$ [closed]

$$\frac{\sin(r)}{\sin(i)}=n$$

In which works and how did Newton get it? Hence inverting the value. $$n$$ is the refractive index of water, thus light speed increased in water according to this.

It's wrong but I want to know how Newton got sines and not cosines or tangents there.

• How Newton got it is a question for hsm.se. If you want an arbitrary proof, it can be derived from e.g. Fermat's principle. (This way is fun.)
– J.G.
Jul 18 '21 at 16:52
• @J.G. can u then shift it ti hsm.se, also if there is a space in the name how can I refer using @ Jul 18 '21 at 16:57
• i saw the springs example earlier but insight into newton would be great. Jul 18 '21 at 16:58
• Unfortunately, I think only a moderator can move this question.
– J.G.
Jul 18 '21 at 17:01
• I’m voting to close this question because it belongs on a site dedicated to history of science. Jul 18 '21 at 20:24

As requested, here is a very common derivation to the formula above, aka Snell's law. The core idea behind this derivation is that light always takes the shortest path (Fermat's principle). Here is how it goes:

The time light takes is given by:

$$t=\frac{\sqrt{a^2+x^2}}{v_1} + \frac{\sqrt{b^2+(d-x)^2}}{v_2}.$$

This time must be minimised, so $$\frac{dt}{dx}=0$$.

Differentiating the above equation gives us $$\frac{dt}{dx}=0\longrightarrow\frac{x}{v_1\sqrt{a^2+x^2}} - \frac{d-x}{v_2\sqrt{b^2+(d-x)^2}}=0;$$

$$\frac{x}{v_1\sqrt{a^2+x^2}}= \frac{d-x}{v_2\sqrt{b^2+(d-x)^2}}.$$

Also,

$$\sin(i)= \frac{x}{\sqrt{a^2+x^2}}\;\mathrm{and}\; \sin(r) = \frac{d-x}{\sqrt{b^2+(d-x)^2}}.$$

Thus, $$\frac{\sin(i)}{v_1}=\frac{\sin(r)}{v_2}.$$ Voila! Snell's Law.

• The question does not ask for a derivation. It asks for the historical works where it first appeared. Jul 18 '21 at 20:23
• @Brick Ig he was asking me "how Newton got sines and not cosines or tangents there" So essentially a derivation Jul 18 '21 at 20:24
• "In which works and how did Newton get it?" Jul 18 '21 at 20:25
• Ok if he asks that i am not sure, as i am only familiar with the physics of it. But the comments on his question indicated that he needed a derivation. Jul 18 '21 at 20:28
• See the comment on the question by the moderator. Your answer is blocking this question from being migrated to a history-oriented site. Jul 19 '21 at 19:19