AHC and WI: Earlier Calculus

How early could Calculus* have plausibly arisen in (or been introduced to) Western Europe with a relatively simple PoD? And what would the effects of it being invented around that time be?

Bear in mind -- I know nothing about advanced mathematics, aside from what I get on Wikipedia...

*an understanding of both derivatives and integrals, or at least OTLs fundamental theorom
 
Some of the fundamentals were invented in the 14th century in India; interesting POD if they were further developed. Also, the ancient Greeks figured out the areas of circles using the method of exhaustion, but didn't seem to be able to reconcile infinitesimals with smoothness, which is sort of an intellectual prerequisite for calculus.
 
Yeah, supposedly the greeks had got close to developing it. Were finding the areas and volumes of shapes by summing up small slices.

You may also want to advance physics. Maybe have a heliocentric model become favored a lot earlier. Then has people try to describe the motions of the heavens, they develop more sophisticated math, eventually coming up with calculus.

Practically, I have no idea what this would do, if anything. I can't think of any really practical applications of calculus before people need thermodynamics, ie an industrial revolution. And if you want to study elasticity to say build bigger cannons.
 
In many ways calculus is a special case of geometry. However the two hardest concepts you must absorb before you can make the jump from geometry to calculus are zero and infinity. However these are all intellectual challenges and there are no reasons why calculus couldn't be developed a lot earlier

The more interesting question is how late could calculus be developed. You could easily get to the 20th Century without it's development as their are other mathematical tools that can be employed to solve problems
 
Some of the fundamentals were invented in the 14th century in India; interesting POD if they were further developed. Also, the ancient Greeks figured out the areas of circles using the method of exhaustion, but didn't seem to be able to reconcile infinitesimals with smoothness, which is sort of an intellectual prerequisite for calculus.

Archimedes was most of the way there.

Part of the problem is that Newton and Leibnitz used wonky math. "Infinitesmals" can be made to work
Wiki said:
After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s. Robinson's approach, called non-standard analysis, uses technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus.
but that requires LOADS of modern math to make work rigorously.

Newton basically handwaved it and said 'ah, it works, dunnit?'

It wasn't until the 1800s that Calculus became anything like rigorous.

But, ja, most societies would be happy with Newton's level of rigor and calculus could certainly have been invented in Roman times, say.
 
Another big thing that really helps is modern elementary algebra--that is, the divorcing of a value from it's numerical representation. The fact that most ancient societies used letter-based numbering systems didn't help with that.

danderson said:
I can't think of any really practical applications of calculus before people need thermodynamics, ie an industrial revolution. And if you want to study elasticity to say build bigger cannons.

Well, mechanics. Mechanics lives and dies on calculus, no matter what form you use. The big application there would be improved calenders; better mechanics means that you can more accurately calculate planetary orbits, and therefore increase the accuracy of your calenderical system. Mostly in terms of being able to predict the equinoxes better and such. Of course, given astrology, that sort of thing has a certain religious aspect as well. It would be interesting if (as sort-of happened OTL, but more so) calculus is developed by priests of some religion that heavily involves astrology, and used religiously.
 
Well, mechanics. Mechanics lives and dies on calculus, no matter what form you use. The big application there would be improved calenders; better mechanics means that you can more accurately calculate planetary orbits, and therefore increase the accuracy of your calenderical system. Mostly in terms of being able to predict the equinoxes better and such. Of course, given astrology, that sort of thing has a certain religious aspect as well. It would be interesting if (as sort-of happened OTL, but more so) calculus is developed by priests of some religion that heavily involves astrology, and used religiously.

Yeah, I was thinking more in terms of how does it help life in general. Sure you can have a more accurate calender, but aside from a few people once you have a pretty good idea of when to plant the harvest you are set.

I did over look the astrology aspect of it. That may have more of an impact. And yeah, equinoxes were more important pre-Christianity.
 
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