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When was the first time we see a square in an equation describing some physical effect?

Well, there is the area of circle (or square of course ...), but a circle area is not - in this question definition at least - a physical phenomenon. An example of a physical phenomenon would be the lever, but we see no square there, rather a simple multiplication of a quantity with another quantity: $M_a*D_a = M_b*D_b$. The equation: $E_v=\frac{mv^2}{2}$ on the other hand, is a qualified example where we square a quantity. But there are, I suppose, much earlier examples of this. What are they?

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    $\begingroup$ How is the length of a lever's leg a "physical phenomenon", but side length of a square land parcel not a "physical phenomenon"? Would the inverse square law or Kepler's third law, where the squares of periods are as the cubes of major axes, qualify? $\endgroup$
    – Conifold
    Dec 6, 2021 at 0:00
  • $\begingroup$ (First part) @Conifold, Well, I have no good answer for this question. But I do sense they can be physiological difference between them as this stays in the realm of Geometry. If the community finds this question off let it be so. But maybe it would have been better to reject circle/square area by philosophizing and saying that area as such is really multiplication of different sizes that coincide in square. (and in circle let multiply the circumference by R and divide by 2 -- as if infinite number of congruent triangles where to measure the area). $\endgroup$
    – d_e
    Dec 6, 2021 at 0:17
  • $\begingroup$ (Second part) @Conifold, Certainly Kepler would qualify. Actually, this is the real context of my question as I saw Kepler "flirting" with $distance^2$ already in his Astronomy Nova and I found this to be not so trivial. I wanted to see if Kepler had some other examples from which he could see that in nature. $\endgroup$
    – d_e
    Dec 6, 2021 at 0:22
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    $\begingroup$ He saw that in optics, and analogized gravity and magnetism to light:"as much of the form would be in a wide and farther-away sphere of this sort as is in the narrow and nearer sphere. And since the ratio of convex spheres is the ratio of the squares of their diameters : therefore the form will be made weaker in unequal spheres in the ratio of the square of its distance", Epitome of Copernican Astronomy. $\endgroup$
    – Conifold
    Dec 7, 2021 at 21:41
  • $\begingroup$ At least two millennia ago, the Babylonians were using squared numbers in the calculation of time and their calendar. mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_mathematics $\endgroup$
    – DJohnson
    Dec 12, 2021 at 17:30

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I'm not sure if the following satisfy your notion of physical phenomenon. I suggest Babylonian "System B" for planet motions. According to this system planets move around zodiac with constantly increasing speed for some time and then they move with constantly decreasing speed. That is, they approximate planet speed as a piecewise linear function. Then they need to square time to calculate the distance the planet travel. That is, they used formula $d=\frac{at^2}{2}$ for motion with constant acceleration.

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This is not a full answer but information I find useful throwing in. As @Conifold suggested in a comment, we saw a usage of square physical quantity in Kepler's third law, and he indeed might drew inspiration from his work Optics on the spreading of light:

Just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely. For according to [propositions] 6 & 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there. (Source: Wikipedia).

It should be noted however that in his Astronomia Nova Kepler, though he draws some similarities between the emission of light and gravity, does not recognize the inverse square distance law with respect to gravity; rather he establish there the "distance-law" (which later became the "area law" though not entirely equivalent) which missing the square. Kepler indeed compares gravity with Magnetism -- but even in the case of Magnetism no one up to his time - so it seems - has formulated any "distance-square" relation.

According to Wikipedia the first to suggest a case of inverse square law, was the French astronomer Ismaël Bullialdus.

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Whilst geometry is most often thought of as a mathematical discipline, it is also inherently physical. That's why geometry is called what it is -- the measurement of the earth -- and by this they mean lengths, angles, areas and volumes and the addition and division of such.

Hence I would suggest the area of a square was the first known physical phenomenon to exhibit squaring. It's also probably why we call it a square.

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