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As we know, even Archimedes did soon some experimental calculations.

My question were, who calculated first time the exact formulas ($V=\frac{4\pi}{3}r^3$, $A=4\pi r^2$)?

As I know, these formulas require the higher understanding of differential calculus, thus I think it happened after Newton and Leibnitz. But who did them?

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  • $\begingroup$ We have some reason to believe that the Greeks used sand to demonstrate geometric proofs. I would be surprised if the ancient Greeks and Egyptians didn't use sand to "stumble upon" the formulae and later fill in the proofs to match the prediction by sand. What do I mean? Make a spherical container. Make a cylindrical container that would very closely circumscribe the spherical one. Fill the sphere up with sand. Dump the sand into the cylindrical container. Shake it up. Eye ball it and see it fills about 2/3 of it up. Prove it. $\endgroup$
    – user6918
    Jun 17, 2018 at 8:18
  • $\begingroup$ @Robert Afaik the Greeks were very axiomatical in their ideas, they hadn't ever accept an experimental measurement without an axiomatical proof. This was being done by the Agyptians and the Babylonians. And here came their next major problem, that not knowing irrationals they could not discover $\pi$. Even their ideas about rational numbers was the proportion of natural numbers. For example, from the size and the diagonal of the square they held, they are not "co-measurable", i.e. they knew that $\sqrt 2$ is not rational but they has seen it that it is not a number. $\endgroup$
    – peterh
    Jun 17, 2018 at 13:45
  • $\begingroup$ I'm not saying they would have stopped short of a proof and accepted merely just experimental evidence. But you can "get the formula for $\pi$" from experimental evidence, in the sense that you can get the relevant theorems about proportions that would nowadays be expressed with $\pi$. Take three different cut-out circles covered in a thin layer of sand and then "brush" that sand into three different squares equally thin, and you could eyeball that the proportion of areas between any two of those squares is the same. $\endgroup$
    – user6918
    Jun 17, 2018 at 16:00

2 Answers 2

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This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to history, is that only Euler presented volume formulas in this form in his textbooks after 1737.

The principal step was no doubt made by Archimedes in On Sphere and Cylinder, where he proved rigorously that $V_S:V_C=2:3$, where $V_S$ is the volume of a sphere and $V_C$ is the volume of the circumscribed cylinder (he also gave proportion for the surface area). "Obviously", $V_C=\pi r^2\times2r$, so we get the modern formula, right? Although this is a popular way of projecting modern concepts onto history it sells short the ingenuity of ancient Greeks and the difficulties they managed to overcome, not to mention the work of countless others who brought about our mathematical paradise. Here is one problem: how does one assign a number to a volume, or even to a length for that matter? Today we use real numbers and integration theory, but ancient Greeks had none of that. Their ingenious solution was to make do without both. Geometric magnitudes (volumes, areas, lengths) weren't assigned numbers at all, they were related to other like magnitudes as ratios. These ratios weren't numbers, they could be compared but not added, and only occasionally they could be expressed as ratios of whole numbers, the only numbers proper. This is why Archimedes expressed the "formula" the way he did.

But to get to our modern formula even in the ratio form like $V_S:r^3=4\pi:3$ there is another problem standing in the way. This proportion relates the volume of the sphere not to a cylinder, but to the cube on its radius. Problem is, $4\pi:3$ is not a ratio of whole numbers, unlike $2:3$. Of course, Archimedes didn't know that for sure, but Pythagoreans already got into hot water by assuming it for the side and the diagonal of a square, and later proving otherwise. So Archimedes, like geometers before and after him, did not write it that way, and they did not write $A=\pi r^2$ or $A:r^2=\pi$ for the area of a circle either. Not in equations and not in words. Yet again, Greeks rose to the occasion despite the absence of our modern machinery. The theory of proportion, an ingenious invention of Eudoxus of Cnidus presented in Book V of Euclid's Elements, allowed them to make sense of estimates like $s:t<A:r^2<p:q$ with whole numbers $p,q,s,t$. Archimedes and many of his successors did prove many such estimates without invoking mysterious entities, which remained undefined for almost two millenia hence, and without the modern idea that unbeknownst to them those ratios were "approximating" $\pi$.

If this last step seems trivial to us today let me point out that in 17th century Cavalieri and Roberval were still presenting their volumes and areas as ratios to other simpler volumes and areas, and recall the history of zero, which was not understood or used as a number for centuries after Babylonians and Alexandrian astronomers were using a symbol for it as a placeholder. With $\pi$ there wasn't even a symbol. Only at the end of middle ages some Arabs and Europeans started thinking of irrationals as some kind of numbers, while giving them telling nicknames, like "deafmute numbers". And this was for irrationals like $1+\sqrt{5}$ or $\sqrt[3]{2}$ given by algebraic formulas.

It appears that the first person to contemplate that $\pi$ and $e$ were also "some kind of numbers" in print was James Gregory in The True Squaring of the Circle and of the Hyperbola published in 1667. He was also the first to suggest a possibility that quadrature of the circle was unsolvable with straightedge and compass, although his argument for it was flawed. Even then it took time for the idea to percolate until William Jones in 1706 was bold enough to assign a symbol to the new "number", our modern $\pi$, while still saying "the exact proportion between the diameter and the circumference can never be expressed in numbers". Integration theory was sufficiently developed by then to be comfortable with volumes and areas as numbers as well, so Jones could dispose with the ratios and write $A=\pi r^2$. And when Euler adopted the symbol 30 years later, and made it famous, he could finally write $V_S=\frac43\pi r^3$.

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    $\begingroup$ This is one of the best answers that I've yet seen on hsm.SE. Thank you! $\endgroup$
    – dotancohen
    Jan 27, 2015 at 13:36
  • $\begingroup$ Your link to Andy Engelward's page "history of zero" is broken. $\endgroup$
    – KCd
    Dec 27, 2023 at 20:41
  • $\begingroup$ @KCd He had a pdf of the MacTutor's page, I relinked directly to them. $\endgroup$
    – Conifold
    Dec 28, 2023 at 5:20
  • $\begingroup$ If I understand you correctly, one of your main points is that $\pi$ is irrational and hence could not have been considered a number by the greeks, so they could not have written or proved the equation $V=\frac{4}{3}\pi r^3$. But if Archimedes considered the proportion $V_S : V_C$ and treated it as a mathematical object and Pythagoras considered "side : diagonal of square", did they not also consider "circumference : diameter of circle" as an object of study (which could be taken as a definition of $\pi$ from modern perspective)? $\endgroup$ Dec 28, 2023 at 21:00
  • $\begingroup$ And could they then not (in principle) have stated and proved something like $V_S:V_{cu} = C : 6D$ where $V_{cu}$ is the volume of a circumscribed cube and $C$ is the circumference while $D$ the diameter of a circle? This only involves proportions of geometric objects and natural numbers, and is easily seen to be equivalent to the equation $V=\frac{4}{3}\pi r^3$. $\endgroup$ Dec 28, 2023 at 21:01
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Archimedes calculated the exact formulas (in the way that the ancient Greeks gave formulas) in his book On the Sphere and Cylinder. This was not "experimental": He gave a full geometric proof, rigorous for its time period.

He considered this his greatest work. He asked that a diagram representing his proof be inscribed on his tomb. This was apparently done as at least one visitor to later Syracuse reported seeing the diagram.

Some people think that Archimedes discovered calculus and found the formulas in that way, but hid his discovery. Newton did much the same thing later, using Calculus to discover much about gravity but using geometric proofs when he wrote about gravity. Newton finally revealed his own work in Calculus when forced by Leibniz. Newton realized that Calculus would be controversial. Perhaps Archimedes did as well.

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    $\begingroup$ @Peter Horvath - see Reviel Netz (editor), The Works of Archimedes : Volume 1 The Two Books On the Sphere and the Cylinder (2004), page 148. $\endgroup$ Jan 24, 2015 at 16:23
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    $\begingroup$ Archimedes did NOT have "the ideas basically equivalent to our formulas" because that basically requires treating $\pi$ as a number. The difference is far beyond using words for equations. And people who think that "Archimedes discovered calculus" are conspiracy theorists. We have Archimedes's letter to Eratothenes, recently rediscovered in a palimpsest but known since 1897, where he describes exactly how he discovered his "formulas" cut-the-knot.org/pythagoras/Archimedes.shtml. It was based on a mechanical analogy and "indivisibles". $\endgroup$
    – Conifold
    Jan 25, 2015 at 1:43
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    $\begingroup$ I did not like the expression "for its time period". Are Archimedes arguments not rigorous for our time period? $\endgroup$ Jan 25, 2015 at 2:17
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    $\begingroup$ @Conifold: I disagree. One does not need "measure theory" to define and find the volume of the ball. And the "real numbers" were perfectly dealt with by Archimedes. When I was a teenager, they taught this in high school, completely rigorously, without any "measure theory" and following Archimedes. $\endgroup$ Jan 25, 2015 at 14:00
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    $\begingroup$ Although I accepted the answer, my main problem with this whole Archimedes thing, that he was an ancient greek. And the ancient greeks didn't know the irrational numbers as $\pi$ (even for the non-integers they used ratios of integers, actually in their mind exclusively integer numbers existed). I think, maybe Archimedes was able to find some interesting relations (i.e. 2:3) between the sphere and the cylinder, maybe could give an algorithm to the volume and surface of a sphere, but I don't think he had used the $pi$ in any of his formulas. $\endgroup$
    – peterh
    Jan 26, 2015 at 2:49

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