# Archimedes and his constant **URL:** \ **Category:** Factual Questions\ **Created:** [December 2, 2004, 5:38pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721 "2004-12-02T17:38:02Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 2, 2004, 5:38pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/1 "2004-12-02T17:38:02Z")
In short, how did Archimedes prove that the ratio of the circumference of a circle to its diameter is constant? It’s very easy to do with analytic geometry and calculus, but he didn’t have those tools available to him. ---
**Author:** ![ccwaterback](https://avatars.discourse-cdn.com/v4/letter/c/df705f/32.png) [@ccwaterback](https://boards.straightdope.com/u/ccwaterback)\ **Post date:** [December 2, 2004, 5:51pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/2 "2004-12-02T17:51:53Z")
From : [http://www.ualr.edu/~lasmoller/pi.html](http://www.ualr.edu/~lasmoller/pi.html) The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 \< Pi \< 22/7. Archimedes’s results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. Beginning with a hexagon, he worked all the way up to a ploygon with 96 sides! ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 2, 2004, 5:53pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/3 "2004-12-02T17:53:55Z")
So he didn’t know that it was a constant? When was that first discovered? ---
**Author:** ![aahala](https://avatars.discourse-cdn.com/v4/letter/a/a88e4f/32.png) [@aahala](https://boards.straightdope.com/u/aahala)\ **Post date:** [December 2, 2004, 5:58pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/4 "2004-12-02T17:58:48Z")
Archimedes contribution wasn’t the proof of the constanct, the knowledge there was a constant preceeded him. It was his discovery, shown above, how one could repeat a process indefinately to get an ever increasingly accurate value for pi. ---
**Author:** ![ccwaterback](https://avatars.discourse-cdn.com/v4/letter/c/df705f/32.png) [@ccwaterback](https://boards.straightdope.com/u/ccwaterback)\ **Post date:** [December 2, 2004, 6:00pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/5 "2004-12-02T18:00:18Z")
So who did prove it was a constant? ---
**Author:** ![Kimstu](https://avatars.discourse-cdn.com/v4/letter/k/ecd19e/32.png) [@Kimstu](https://boards.straightdope.com/u/Kimstu)\ **Post date:** [December 2, 2004, 6:05pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/6 "2004-12-02T18:05:43Z")
Yes, certainly it was known in pre-Archimedean times that the ratio of the circumference of a circle to its diameter is a constant. Eudoxus of Cnidus in the fourth century BCE [proved](http://www.maths.uwa.edu.au/~schultz/3M3/L6Euclid.html) that the area of a circle is a constant times the square of the diameter; the constant relationship of the circumference to the diameter was also well-known. I think I recall hearing a couple years ago that someone had found an Old-Babylonian (second millennium BCE) cuneiform tablet with an approximate value of pi on it. Ancient Egyptian and Chinese texts also preserve (cruder) values of pi. The knowledge that the circumference had the same ratio to the diameter in every circle was old, old news to Archimedes. I don’t know where the first known proof of it is, though. Everything you ever wanted to know about how Archimedes proved his result is [here](http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 2, 2004, 9:54pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/7 "2004-12-02T21:54:35Z")
Everything you wanted to know about [pi](http://www.straightdope.com/mailbag/mpiorigin.html), but were afraid to ask. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 2, 2004, 10:21pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/8 "2004-12-02T22:21:31Z")
**bibliophage** ’s article is nice and all that, but I’m really interested in the proofs of three facts using no calculus: 1. That the area of a circle is a constant multiple of the square of its radius (got that one, thanks to **Kimstu** ). 2. That the circumference of a circle is a constant multiple of the radius. 3. And that those two constants are the same. ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [December 2, 2004, 10:48pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/9 "2004-12-02T22:48:59Z")
Well, one of Archimedes’ theorems was that the area of a circle is half the radius multiplied by the circumference. More specifically, in Heath’s translation, Proposition 1 of _Measurement of a Circle_ is: > [@](#): > > The area of any circle is equal to a right-angled triangle in which one of the sides is equal to the radius, and the other to the circumference, of the circle. Following from the earlier result cited by **Kimstu** , one can then define pi via _C=2 pi r_. If _A = rC/2_ by the above theorem, then _A = pi r[sup]2[/sup]_. At a glance, Archimedes’ proof of the theorem doesn’t seem to involve even an anticipation of calculus. ---
**Author:** ![Josh\_dePlume](https://avatars.discourse-cdn.com/v4/letter/j/bbe5ce/32.png) [@Josh\_dePlume](https://boards.straightdope.com/u/Josh_dePlume)\ **Post date:** [December 2, 2004, 11:38pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/10 "2004-12-02T23:38:40Z")
> [@ultrafilter](#): > > **bibliophage** ’s article is nice and all that, but I’m really interested in the proofs of three facts using no calculus: > > 1. That the area of a circle is a constant multiple of the square of its radius (got that one, thanks to **Kimstu** ). > > 2. That the circumference of a circle is a constant multiple of the radius. What’s the proof that the circumference of a square is a constant multiple of its diagonal, or the analogous thing for other polygons? Once you’ve got that, the circle thing seems pretty clear (I know that’s not a formal proof). > [@](#): > > 1. And that those two constants are the same. You just slice the circle up into equal-sized wedges and arrange them in a rectangle-like thing. As the number of wedges gets large, this looks more and more like a rectangle with a width of _r_ and a length of half the circumference. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 3, 2004, 6:13am UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/11 "2004-12-03T06:13:49Z")
> [@Josh\_dePlume](#): > > You just slice the circle up into equal-sized wedges and arrange them in a rectangle-like thing. As the number of wedges gets large, this looks more and more like a rectangle with a width of _r_ and a length of half the circumference. Take the limit, eh? That’s calculus. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 3, 2004, 7:32am UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/12 "2004-12-03T07:32:41Z")
> [@](#): > > At a glance, Archimedes’ proof of the theorem doesn’t seem to involve even an anticipation of calculus. Actually, it does. Archimedes didn’t have the concept of a limit pre-defined for him, but he used it nonetheless in this and several others of his proofs. Basically, he had to re-derive the concept each time he used it, which had to be cumbersome for him. Basically, his proof amounted to “Given any value less than pi_r[sup]2[/sup], I can inscribe a polygon with area greater than that value, so the area of the circle is not less than pi_r[sup]2[/sup]. And given any value greater than pi_r[sup]2[/sup], I can circumcribe a polygon with area less than that value, so the area of the circle is not greater than pi_r[sup]2[/sup]. Since the area of the circle is neither greater than nor less than pi_r[sup]2[/sup], it must be equal to pi_r[sup]2[/sup].”. ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [December 3, 2004, 1:43pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/13 "2004-12-03T13:43:25Z")
> [@Chronos](#): > > Actually, it does. At a second glance, agreed. ---
**Author:** ![Josh\_dePlume](https://avatars.discourse-cdn.com/v4/letter/j/bbe5ce/32.png) [@Josh\_dePlume](https://boards.straightdope.com/u/Josh_dePlume)\ **Post date:** [December 3, 2004, 1:54pm UTC](https://boards.straightdope.com/t/archimedes-and-his-constant/277721/14 "2004-12-03T13:54:46Z")
> [@ultrafilter](#): > > Take the limit, eh? That’s calculus. Call it what you like. It doesn’t involve an integral (which is what comes to mind when I think of finding areas using “calculus”). It doesn’t even involve taking the limit of a specified expression. One needs no exposure to calculus to understand it. I think people were doing this sort of thing long before the advent of calculus as we know it. If you want to say that they were _really_ doing calculus but didn’t know it, fine, but be prepared for someone to say that they were _really_ doing something else even more advanced (I don’t know, maybe measure theory).