User dave richeson - MathOverflow

Who first proved that the value of C/d is independent of the choice of circle?

2011-08-12T20:20:54Z

I have an elementary question about the history of $\pi$. I thought the answer would be easy to find. But, to the contrary, after quite a bit of searching and after consulting math historians, I have been unable to find a satisfactory answer.

Who first proved that $C/d$ is independent of the choice of circle ($C$ and $d$ are the circumference and diameter, respectively)?

Or equivalently:

Who first proved that given two circles with circumferences $C_1$ and $C_2$ and diameters $d_1$ and $d_2$, that $C_1/C_2=d_1/d_2$? (Or, as I imagine Euclid would have written it: the circumferences of circles are to one another as their diameters.)

Most accounts of the history of $\pi$ spend a lot of time talking about how this fact has been "known" for a long time (giving Egyptian, Babylonian, biblical, etc. approximations to the value). But they never say who first proved it. I expected it to be in Euclid's Elements, but was surprised to find that it isn't. Can I take that to mean that it hadn't been proved by then? I would be very surprised if the proof was known to Euclid and he had not included it in Elements.

Note: Euclid does contain Eudoxus's proposition that $A_1/A_2=d_1^2/d_2^2$, where the $A_i$ are the areas of the two circles (Elements XII.2: Circles are to one another as the squares on their diameters.). This implies that the value of $A/d^2$ is independent of the choice of circle.

If we jump ahead a few years from Euclid we find the fact that $C/d$ is constant given implicitly in Archimedes's Measurement of the Circle. First of all, he finds bounds for $C/d$ (it being between $223/71$ and $22/7$). So presumably he knew that it was a constant. But also, it follows logically from his result that $A=rC/2$, where $r$ is the radius of the circle (Archimedes says that the area of a circle is equal to the area of a triangle with height $r$ and base $C$): if we take Eudoxus's proposition as saying $A=kd^2$ (for some constant $k$) and Archimedes's result as $A=dC/4$, then setting them equal we get $kd^2=dC/4$, or equivalently $C/d=4k$ (i.e., $k=\pi/4$).

So, my question is: who first prove this fact? Was it Archimedes? I've read that the version of the Measurement of the Circle that we have may be only a part of what Archimedes actually wrote. Do people conjecture that it was proved and stated explicitly in the missing part of this document?

This all seems very mysterious to me. I would be a little surprised to discover that the answer to this question is lost to history since it is such a major mathematical result (but maybe that is so). I would be surprised if it took until Archimedes to get a proof of this; if it was "known" empirically for the entire Greek period (which I assume it was), one would imagine that a rigorous proof would be highly sought after. One imagines a proof would have been within Eudoxus's reach. Finally, whether the answer the answer to the question is known or not known, I have been very surprised that no one has written about this fact (or at least not that I've found).

Answer by Dave Richeson for h-Cobordism Theorem and Dynamical Systems

2011-08-24T20:15:48Z

You might want to look at John Franks's 1982 CBMS monograph (#49) Homology and Dynamical Systems. The $h$-cobordism theorem appears on page 13. It is used in connection with gradient-like flows on manifolds of dimension greater than 5.

Answer by Dave Richeson for What are the lengths that can be constructed with straightedge but without compass?

2011-04-08T16:19:02Z

The short answer is that nothing is constructible. As is standard, we begin with the two points $(0,0)$ and $(1,0)$. Then we can draw a line between them, and that's it. We can't draw any more lines, and hence we can't construct any new points. The Euclidean rules say that we are only allowed to draw a new line if we are joining two already-constructed points, and a point can only be constructed if it is the intersection of two lines (or, irrelevant to this discussion, two circles or a line and a circle).

However, suppose you begin with a finite collection of points $(x_1,y_1),\ldots, (x_n,y_n)$. Let $C$ be the set of points constructible from this set using only a straightedge (unmarked). If a point $(x,y)$ is in $C$, then $x$ and $y$ can be formed from $x_1,\ldots,x_n,y_1,\ldots,y_n$ using the operations $+$, $-$, $\cdot$, and $\div$ (since new points are created as intersections between lines). However the converse of this is not true (as the two-point example shows). I suppose you could say more about what $C$ looks like, but it would probably be messy.

Answer by Dave Richeson for Papers that debunk common myths in the history of mathematics

2010-06-03T01:47:12Z

There is a nice article by Brian Hayes in the American Scientist (May-June 2006), "Gauss' Day of Reckoning," in which he looks at the history of the so-called "baby Gauss" story (that Gauss amazes his teacher by summing the first 100 positive integers).

There is a famous anecdote about Euler embarrassing Diderot in Catherine the Great's court. He claimed to have mathematical proof of the existence of God, when in fact he just stated mathematical nonsense (which Diderot did not understand): "Monseur, $(a+b^n)/n=x$ donc Dieu existe; répondez!" B. H. Brown tracked down the source of this myth in "The Euler-Diderot Anecdote" (Amer. Math. Monthly, Vol 49, 1942, reprinted in William Dunham's The genius of Euler: reflections on his life and work (2007))

Finally, there is a mathematical urban legend that I thought was surely was false, but is apparently true (according to Snopes). This is the story about the student who comes late to class and sees the homework written on the board. After a lot of effort he solves the problems. Only later did he discover that they were not homework, but open problems. It turns out that the student in the story was George Dantzig. Snopes cites a 1986 interview with Dantzig from the College Mathematics Journal.

Comment by Dave Richeson

2011-08-13T17:18:35Z

Thanks for your comments and the link to this article. Very interesting and helpful. It had occurred to me that understanding the question may involve knowing how they defined arc length. According to the commentary here aleph0.clarku.edu/~djoyce/java/elements/bookI/… Euclid viewed arcs of circles as magnitudes (like line segments), but they could only be compared to arcs of circles with the same radius. This had changed by the time of Archimedes because, as you wrote, he compares the length of the circumference to the perimeters of the inscribed and circumscribed polygons.

Comment by Dave Richeson

2011-08-12T20:59:04Z

Yes, that's typical of most accounts of the history of $\pi$. But again, I think they're referring to the empirical fact that it is a constant slightly larger than 3. I doubt the Greeks would have been satisfied with that. I'd think they'd demand a rigorous proof.