Calculus 5.2c - Infinitesimals - Archimedes

[@gpinon4346]

Fantastic! Just fascinating...

[@annaclarafenyo8185]

Archimedes did not call this the method of exhaustion, the method of exhaustion is due to Eudoxus and it's equivalent to epsilon delta. Archimedes used infinitesimals in the method of mechanical theorems, by balancing things using the law of the lever, as explained on Wikipedia.

[@yanntal954]

Thank you for pointing this out!

[@gabitheancient7664]

no no, he used levers to make intuition, in his "The Method" he explains that using mechanics is an amazing way of gaining some knowledge on mathematical objects before making any proof, but his proofs are basically bullet proof to this day, like his proof of the area of a circle

[@annaclarafenyo8185]

@@gabitheancient7664 The major difference is that today we can make his lever proofs just as bulletproof as his rigorous constructions. It's a pity he didn't change mathematical practice to admit these arguments, although I am sure he would feel right at home in a modern calculus class.

[@gabitheancient7664]

@@annaclarafenyo8185 can we? but I mean his lever arguments really do seem non-rigorous, I'm pretty sure you can't prove his center of mass and lever rules, I can't even think how you could make these ideas rigorous and usable to prove stuff I'm open to be proven wrong tho this seems interesting

[@annaclarafenyo8185]

@@gabitheancient7664 His lever arguments are standard integral calculus. He is taking "dx" width slices of one shape and comparing them to "dx" width slices of another shape at distance x on the same lever. His argument amounts today to the definition of center of mass, and symmetry arguments for shapes with a symmetry around the center of mass, both of which we make rigorous by defining integrals. It takes effort to define integrals rigorously, Riemann and Lebesgue did it though.

[@stabiljka]

Thank you. That's fascinating story.

[@ThomasDeLello]

It shouldn't be so hard to get one's head around the concept of a limit of a function yielding a transcendant constant, in this case the slope of a line. In Archamedes' example, a tiny arc.

[@Vonzi0000]

Can't you just think of infinitesimals as being limits that approch zero (but never quite reach it)? And dy/dx being a quotient of such limits?

[@Vonzi0000]

@derekowens Ok I'm looking forward to that. Good luck to your son at the game!

[@namdevchintapalli]

Wonderful

[@samatarMohamed]

Shit imagine if mathemeticiams had this! Who knows how this could change our lives.

[@goshiluvarchie]

Lol, damn monks.

[@Thewerwolf]

no one knows what archimedes look like

[@jimkeller3868]

Leave it to the religion to destroy and obscure works of genius.

[@derekowens]

But remember that Newton himself was very devout. James Gleik's short biography on Newton is a very accessible and enjoyable work, highly recommended.

[@annaclarafenyo8185]

They also preserved the same works. Parchment was very expensive back then.