Question

How ancient get the formulas for circumference of a circle and semi-cubical parabola without calculus?Can I derive them without calculus!

Answer 1

The circumference of a circle formula doesn't come from calculus in any way, it's a direct consequence of the definition of pi. I think the ancients were able to discover these things because they were actually using calculus in a sense, it just depends on what you mean when you say "calculus".

Answer 2

I mean by calculus what we you nowadays.
and I knew that the length of semi-cubical parabola and circle was known before sir Newton,so I am looking for their methodology to get 8r.

Answer 3

I think Archimedes drew a regular polygon on the inside and outside of the circle and argued the circumference was bigger than the perimeter of the inscribed polygon but less than the circumscribed polygon. Then he let the sides of the polygon approach infinity

Answer 4

approach infinity means calculus right, but my book said that it was determined without calculus and also emi-cubical parabola

Answer 5

@Catch.me

Note: Approaching infinity, decreasing without bound are phrases not restricted to Calculus.

Archimedes used a sandy beach for his drawings. No technology, no formal Calculus.

Info at the link describes how Archimedes found the circumference of a circle.

http://scienceforkids.kidipede.com/math/geometry/circumferenceproof.htm

Answer 6

Yeah I think the best way to think of it is that calculus always existed, it's just it wasn't always known to people and labelled in a formal way. So it's not like nobody could use it, it was just different.

Answer 7

What about semi-cubical parabola? How did they know it converges ?

Answer 8

You'll need to google up how William Neile cranked that out without "formal" calculus.

The Semi-cubical Parabola is probably the first curve in our NCB collection whose history is more fascinating than its mathematics.

http://curvebank.calstatela.edu/semicubicpar/semicubicpar.htm

@Catch.me