Revisions - MathOverflow

Return to Answer

added 328 characters in body

view source

edited Oct 1 2011 at 2:33

roy smith
4,871●34●28

I realized this summer that this allows a computation of the volume of the 4 dimensional ball. I.e. this ball results from revolving half a 3 ball, hence can be calculated by revolving a cylinder and subtracting the volume of revolving a cone. Since Archimedes knew the center of gravity of both those solids he knew this.

added 183 characters in body; added 164 characters in body

view source

edited Jan 13 2011 at 4:51

roy smith
4,871●34●28

As a remark on criterion 2 of the original question, I find it is not at all necessary to read all of a paper by a master to get some insight. One word in Euclid enlightened me, and before the translation came out, I had already gained most of my understanding of Riemann's argument for RR just from reading the headings of the paragraphs. I learned a proof of RR for plane curves from reading only the introduction to a paper of Fulton. A single sentence of Archimedes, that a sphere is a cone with vertex at the center and base equal to the surface, makes it clear the volume is 1/3 the surface area. Moreover this shows the same ratio holds for a bicylinder, whereas the area of a bicylinder is considered so difficult we do not even ask it of calculus students. So one should not be discouraged by the difficulty of reading it all of a masters' paper, although of course it wouldn't hurt.

added 926 characters in body

view source

edited Jan 13 2011 at 2:38

roy smith
4,871●34●28

A related question occurs in many cases since the classical arguments of the "ancients" are preserved but only in classical texts such as Van der Waerden in algebra, and newer books have found slicker methods to avoid them. E.g. the method of LaGrange resolvents is useful in Galois theory for proving an extension of prime degree in characteristic zero is radical. There are faster less precise methods of showing this such as Artin/Dedekind's method of independence of characters, but the older method is useful when trying to use Galois theory to actually write down solution formulas of cubics and quartics. Thus today we often have an intermediate choice of reading modern expositions which reproduce the methods of the creators, or ones that avoid them, sometimes losing information. (This is discussed in the math 843-2 algebra notes on my web page, where, being a novice, I give all competing methods of proof.)

added 119 characters in body

view source

edited Jan 12 2011 at 23:20

roy smith
4,871●34●28

added 444 characters in body

view source

edited Jan 12 2011 at 22:23

roy smith
4,871●34●28

added 551 characters in body

view source

edited Jan 12 2011 at 21:33

roy smith
4,871●34●28

added 612 characters in body

view source

edited Jan 12 2011 at 20:57

roy smith
4,871●34●28

added 360 characters in body

view source

edited Jan 12 2011 at 20:11

roy smith
4,871●34●28

[made Community Wiki]

view source

answered Jan 12 2011 at 20:04

roy smith
4,871●34●28

I agree 100% with Igor and Andrew L., on the benefit of reading the creator's version of the same thing available from later expositors. I have gained mathematical insights from reading Euclid, Archimedes, Riemann, Gauss, Hurwitz, Wirtinger, as well as moderns like Zariski.... on topics I already thought I understood.

Just Euclid's use of the word "measures" for "divides" finally made clear to me the elementary argument that the largest number dividing 2 integers is also the smallest positive number one can measure using both of them. This is clear thinking of (commensurable) measuring sticks, since by translating it is obvious the set of lengths that one can so measure are equally spaced, hence the smallest one would measure them all.

I was unaware also that Euclid's characterization of a tangent line to a circle was not just that it is perpendicular to the radius, but is the only line meeting the circle locally once and such that changing its angle ever so little produces a second intersection, i.e. Newton's definition of a tangent line. It is said Newton read Euclid just before giving his own definition.

I did not realize until reading Archimedes that the "Cavalieri principle" follows just from the definition of the Riemann integral, without needing the fundamental theorem of calculus. hence one can conclude all the usual volume formulas for pyramids, cones, spheres, even the bicylinder, just by starting from the decomposition of a cube into three right pyramids, applying Cavalieri to vary the angle of the pyramid, then approximating and using Cavalieri. It is an embarrassment to me that I had thought the volume of a bicylinder a more difficult calculus problem that that for a sphere, when it follows immediately from comparing horizontal slices of a double square based pyramid inscribed in a cube. I.e. by Cavalieri and the Pythagorean theorem, the volume of a sphere is the difference between the volumes of a cylinder and an inscribed double cone. The same argument shows the volume of a bicylinder is the difference between the volumes of a cube and an inscribed double square based pyramid. This led to an intuitive understanding of the simple relation between the volumes of certain inscribed figures that I then noticed had been recently studied by Tom Apostol.

Having read everywhere that Hurwitz' theorem was that the maximum number of automorphisms of a Riemann surface of genus g is 84(g-1), I had a difficult proof that the maximum number in genus 5 is 192, using Jacobians, Prym varieties, and classifications of representations of planar groups, until Macbeath referred me to Hurwitz' original paper where a complete list of the possible orders was easily given: 84(g-1), 48(g-1),....I subsequently explained this easy argument to some famous mathematical figures. Sometime later a more complicated such example for which Macbeath himself was usually credited was found also to occur in the 19th century literature.

Having studied Riemann surfaces all my life, but unable to read German well, I thought I had acquired some grasp of the Riemann Roch theorem, in particular I thought Riemann had given only an inequality l(D) ≥ 1-g + deg(D). When the translation from Kendrick press became available, I learned he had written down a linear map whose kernel computed l(D), and the estimate derived from the fundamental theorem of linear algebra. The full equality also follows, but only if one can compute the cokernel as well. That cokernel of course was already shown by him to be what we now call H^1(D). Hence Riemann's original theorem was the so called "index" version of RR. Since he expressed his map in terms of path integrals, it was natural to evaluate those integrals by residue calculus as Roch did. This is explained in my answer to "why is Riemann Roch [not precisely] an index problem?" Although there are many fine modern expositions of Riemann Roch, the most insightful perhaps being that in the chapter on Riemann surfaces in Griffiths and Harris, I had not seen how simple it was until reading Riemann.

Reading Wirtinger's 19th century paper on theta functiions, even though unfortunately for me only available in the original German, I learned that when a smooth Riemann surface acquires a singularity, the elementary holomorphic differential with a non zero period around that vanishing cycle, becomes meromorphic, and that period becomes the residue at the singuklar point. At last this explains clearly why one defines "dualizing differentials" as one does, in algebraic geometry.

Once as grad student in Auslander's algebraic geometry class, I vowed to try out Abel's advice and read the master Zariski's paper on the concept of a simple point. I was very discouraged when several hours passed and I had managed only a few pages. Upon returning to class, Auslander began to pepper us with questions about regular local rings. I found out how much I had learned when I answered them all easily until he literally told me to be quiet, since I obviously knew the subject cold. (To be honest, I did not know the very next question he posed, but I was off the hook.)

In my answer to a question about where to learn sheaf cohomology I have given an example of insight only contained in Serre's original paper.

As you can see these are elementary examples hence from a fairly naive and uneducated person, myself, who has not at all plumbed the depth of many original papers. But these few forays have definitely convinced me there is a benefit that cannot be gained elsewhere, as these exposures can transform the understanding of ordinary mortals closer to that of more knowledgeable persons, at least in a narrow vein. So while it might be thought that only the strongest mathematicians can attempt these papers, my advice would be that reading such masters may be even more helpful to us average students.