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> It was only 282 years ago that Euler presented in his textbooks the exact formula for the volume of a sphere It's strange that Fermat's Library stated this s

> It was only 282 years ago that Euler presented in his textbooks the exact formula for the volume of a sphere It's strange that Fermat's Library stated this since the formula for volume of sphere was discovered by Archimedes more than 2000 years ago. [1]: https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder

ColinWright 7y ago
Yes and no. My understanding is that Archimedes didn't, and never could have, produced the actual formula, because that way of writing formulas didn't exist. That's discussed in the sub-thread here: https://twitter.com/daveinstpaul/status/1213469023461138432 https://twitter.com/daveinstpaul/status/1213469023461138432 Additionally, the result was probably known before Archimedes, but again, writing that formula in that form probably didn't happen until Euler.
- quietbritishjim 7y ago
  That tweet is merely about the way the formula was expressed, not about its mathematical content: > Archimedes discovered the formula many centuries before Euler, although he could not express it in modern algebraic notation. If Archimedes was able to express the formula in natural language (e.g. "take the radius of the hypersphere, multiply it by itself as many times as there are dimensions, ..." etc.) in a way that was unambiguously understood by other mathematicians (or natural philosophers) of the time, then that formula has definitely been discovered. Writing it in modern notation with one-character symbols for addition, multiplication (typically zero symbols!) etc. is nowhere near required for it to count as being discovered.
- Cybiote 7y ago
  Yeah, the subtle difference is that discovering a geometric concept and being able to express it in a compact and flexible notation are not quite the same. It's very easy to take for granted the sheer amount of innovation and ideas that went into being able to express that formula just because they are grade school topics today. The concept and acceptance of irrational numbers, the relationship between geometry and algebra, the use of symbols in algebra and even assigning symbols for special real numbers took centuries and a steady stream of creative leaps of genius. An excellent and detailed explanation I highly recommend can be found here: https://hsm.stackexchange.com/a/1872 https://hsm.stackexchange.com/a/1872.

ColinWright 7y ago

Yes and no. My understanding is that Archimedes didn't, and never could have, produced the actual formula, because that way of writing formulas didn't exist. That's discussed in the sub-thread here: https://twitter.com/daveinstpaul/status/1213469023461138432 https://twitter.com/daveinstpaul/status/1213469023461138432 Additionally, the result was probably known before Archimedes, but again, writing that formula in that form probably didn't happen until Euler.

quietbritishjim 7y ago
That tweet is merely about the way the formula was expressed, not about its mathematical content: > Archimedes discovered the formula many centuries before Euler, although he could not express it in modern algebraic notation. If Archimedes was able to express the formula in natural language (e.g. "take the radius of the hypersphere, multiply it by itself as many times as there are dimensions, ..." etc.) in a way that was unambiguously understood by other mathematicians (or natural philosophers) of the time, then that formula has definitely been discovered. Writing it in modern notation with one-character symbols for addition, multiplication (typically zero symbols!) etc. is nowhere near required for it to count as being discovered.
Cybiote 7y ago
Yeah, the subtle difference is that discovering a geometric concept and being able to express it in a compact and flexible notation are not quite the same. It's very easy to take for granted the sheer amount of innovation and ideas that went into being able to express that formula just because they are grade school topics today. The concept and acceptance of irrational numbers, the relationship between geometry and algebra, the use of symbols in algebra and even assigning symbols for special real numbers took centuries and a steady stream of creative leaps of genius. An excellent and detailed explanation I highly recommend can be found here: https://hsm.stackexchange.com/a/1872 https://hsm.stackexchange.com/a/1872.