Do you read the masters?

Question

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a principle that people give lip service to because it sounds good, but which is honored in the breach more than in the observance? If not, which masterworks have you found to be most enlightening?

To keep the question focused, let me lay down some ground rules.

1. List only papers/books from the 19th century or earlier. I recognize that this is an arbitrary cutoff but I want to draw a line somewhere.

2. It must be something that you personally have read in its entirety (or almost in its entirety). I'm not really interested in secondhand evidence ("So-and-so says that X is a must-read").

3. You must have acquired important mathematical insights (not just historical insights) from the paper/book that you feel that you would never have acquired had you restricted your reading to 20th-century and 21st-century literature. It's not enough, for my purposes, that you found the paper/book "interesting" but not really essential. If possible, briefly describe these insights in your response.

[Edit: In response to a comment that suggested that I have set the bar impossibly high, let me violate one of my own ground rules and point to this discussion on the $n$-Category Cafe that gives some secondhand examples. That discussion should also help to clarify what I am asking for more examples of.]

Answer 1

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. I.e. it follows just from the definition of a volume as a limit of approximating slices, and was known to Archimedes. 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.

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.

Having read everywhere that Hurwitz's 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: $84(g-1), 48(g-1),\ldots$ 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 $\ell(D) ≥ 1-g + \deg(D).$ When the translation from Kendrick press became available, I learned he had written down a linear map whose kernel computed $\ell(D),$ 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.

Perhaps this is only historical knowledge, but reading Riemann one sees that he also knew completely how to prove (index) Riemann Roch for algebraic plane curves, without appealing to the questionable Dirichlet principle, hence the usual impression that a rigorous proof had to await later arguments of Clebsch, Hilbert, or Brill and Noether, is incorrect.

Reading Wirtinger’s 19th century paper on theta functions, 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 singular 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.

The sense of wonder and awe one gets upon reading people like Riemann or Euler, is also quite wonderful. Any student who has struggled to compute the sum of the even powers of the reciprocals of natural numbers $1/n^{2k},$ will be amazed at Euler's facile accomplishment of this for many values of $k.$ Calculus students estimating $\pi$ by the usual series to 3 or 4 places will also be impressed at his scores of correct digits. On the other hand, anyone using a modern computer can detect an actual error in his expansion of $\pi,$ I forget where, in the 214th place? but an error which was already noticed long ago.

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.

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 all of a masters' paper, although of course it wouldn't hurt.

A remark on the definition of master, versus creator. There are cases where a later master re - examines an earlier work and adds to it, and in these cases it seems valuable to read both versions. In addition to examples given above of Newton generalizing Euclid and Mumford using Hilbert, perhaps Mumford's demonstration of the power of Grothendieck's Riemann Roch theorem in calculating inavriants of moduli space of curves is relevant.

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 844-2 algebra notes on my web page, where, being a novice, I give all competing methods of proof.)

Answer 2

In algebraic number theory, the existence of a Frobenius element at any prime $p$ in a Galois extension $K/{\mathbf Q}$ is crucial. That is, for any prime ideal $\mathfrak p$ lying over $p$ in $K$ there is some $\sigma \in {\rm Gal}(K/{\mathbf Q})$ that looks like the $p$-th power map mod $\mathfrak p$: $$ \sigma(\alpha) \equiv \alpha^p \bmod \mathfrak p $$ for all $\alpha$ in the integers of $K$. (This can be jazzed up to the relative case, but I'll keep the base field as $\mathbf Q$ for simplicity here.)

In any introductory algebraic number theory book I have seen which shows the existence of $\sigma$, first the decomposition field is introduced in order to make a reduction to the case where the base field is the decomposition field. But if you look at the original proof by Frobenius (1896) it is different, using multivariable polynomials in an interesting way and there is no decomposition field. The argument fits in one page; see https://kconrad.math.uconn.edu/blurbs/gradnumthy/frobeniuspf.pdf, where I consider a fairly general setup using the method of Frobenius. This nice proof by Frobenius has been completely forgotten, even though it handles the general case. (Frobenius himself worked with base field ${\mathbf Q}$.)

What is the mathematical insight here? That you can prove this theorem without having to mention decomposition fields (which also makes it easier for students new to the subject to follow the proof). I found this essential when teaching a course on algebraic number theory since it meant I did not have to introduce decomposition fields in the lectures at all; they could safely be left to homework assignments, if I so chose. The proof is also a nice illustration of the usefulness of multivariable polynomials, especially considering that a lot of basic algebraic number theory only requires polynomials in one variable.

Answer 3

Riemann's original paper Über die Anzahl der Primzahlen unter einer gegebenen Grösse (On the Number of Primes Less Than a Given Magnitude), 1859, is definitely a master well worth reading. In just 8 or so pages he shows how useful the zeta function is for questions about the primes, proves the functional equation, the explicit formula, and makes several deep and far-reaching conjectures (all proven except one infamous example).

This is the paper which (arguably) began the extremely fruitful method of applying complex analysis to number theoretic questions. It lacks details in some places, but it contains a lot of invaluable motivation and exposition.

It certainly helped me to understand why complex analysis is so useful, and how one might discover these connections for himself.

EDIT: Just so you have no excuse, here's a link to an English translation: https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf (Remember that he writes $tt$ for $t^2$ and $\Pi(s-1)$ for $\Gamma(s)$).

Answer 4

"Read the masters" should not be taken as blanket advice, because some masters are much easier to read, or more congenial to modern mathematicians, than others. Some 19th century works that I have learned from are:

1. Dirichlet/Dedekind. Dirichlet's Lectures on Number Theory, edited and supplemented by Dedekind, are very clear and inspiring. They cover everything from the basics up to Dirichlet's own breakthroughs on class numbers and primes in arithmetic progressions.

2. Dedekind's Theory of Algebraic Integers. Dedekind wrote this because he was disappointed with the initial response to his theory of ideals. He goes to great pains to motivate the theory from the problem of unique prime factorization (using the now standard example of $\mathbb{Z}[\sqrt{-5}]$).

3. Poincare's papers on automorphic functions. Whether or not you want to know about automorphic functions, these papers are a great introduction to hyperbolic geometry, fuchsian groups, and Kleinian groups. Like Dedekind, Poincare writes very clearly and simply.

Disclaimer. These are all books that I translated, so naturally I think they are good. If you can read them in the original language they are probably even better.

I should add that I came to these books after being disappointed with certain 20th century books, which seemed to me too terse, unmotivated, and abstract. If you haven't had this experience, then you probably won't enjoy 19th century books.

Answer 5

There is more than one reason to read "masters". One such reason is field-specific and can be phrased as "read the latest work right before a scientific revolution" (standard example is the large body of work by Cayley, Sylvester, Gordan, etc., in the pre-Hilbert classical invariant theory). Often such results are more powerful in very specific cases of interest.

Another practical reason to read "masters" is to avoid embarrassment. Lots of (mostly minor) results are not mentioned in later treatises, so a number of people rediscover these results because they are either too lazy to read, or simply assume that "masters" couldn't have possibly be so smart to figure out these results back then... When going through the references in writing this survey, I read all 80 pages of J.J. Sylvester, A constructive theory of partitions, arranged in three acts, an interact and an exodion, Amer. J. Math. 5 (1882), 251–330. As a result, I discovered that a number of recent results were already proved there, sometimes by leaders in the field (let me not name them here - see the survey).

Answer 6

I interpret "read the masters" as advice to learn a theory from those who created it. Often they are mathematicians of higher caliber than those who follow, and hence they offer unique insights missed by later expositions; insights missed either because they were never understood, or as they are considered common knowledge, the "new normal". Examples I have in mind are Thurston's "Notes", Wall's and Browder's books on surgery theory, Gromov's "Hyperbolic groups".

Answer 7

While I feel it's certainly worthwhile to read the masters (by which, I mean the initial works that created entire fields of mathematics by their founders), my reasoning is somewhat different then most. Reading the masters is really more for conceptual depth than actual mathematical enlightenment.

There's a myth surrounding Abel's dictum that stems from the unreadability of the masters like Gauss as a measure of their nearly inhuman brilliance. This is a fallacy. The reason the masters are so difficult to read is because we are catching them with their pants down in the act of creation, i.e. they are groping towards the right notation and terminology, but aren't quite there yet. For example, it's pretty clear Riemann in his doctoral lecture was trying to explain the need for higher dimensional spaces that went beyond familiar three dimensional space ("multiply extended quantities") which preserved all the familiar properties of the usual Euclidean spaces, i.e. Kleinian transformations and calculus in local neighborhoods. The problem was without either linear algebra or the fundamentals of topology, it was next to impossible to express this idea clearly and precisely. He just ends up babbling on about what's needed. But all the same, Riemann recognized what was needed even if how to express it correctly was beyond his ability.

A more recent and readily available example will clarify this further: One of my favorite books is Hassler Whitney's Geometric Integration Theory. I have friends in differential geometry who tell me it's a dinosaur, that his proof of the de Rham theorem is incredibly coarse and tedious. Yes, it is — but it has the advantage of being a DIRECT proof from the construction of simplexes on the boundary of an embedded manifold. I love the book because although Whitney's ideas were old fashioned, they were incredibly powerful IDEAS that allow us to tackle the subject concretely and with an amazing amount of insight. THAT'S what we get from reading the masters — their insight and depth of understanding that allows us to see beyond the machinery into why things are defined as they are.

Answer 8

Once I needed a good exposition of Newton's polygon (for my lectures). It is hard to find in modern books. I asked a great expert in my department (an expert in both algebraic geometry and in classical literature, Shreeram Abhyankar): Where to find a good exposition of Newton's polygon, with examples? His answer was: In Mathematical Papers of Newton. Then he added: In Chrystal's Algebra (1879). This is a British high school algebra textbook. I read both, and I am very satisfied, and used this in my lectures.

Answer 9

I certainly have read a lot of classics, and have learned a lot (mostly about historical developments) even from Euclid's elements. When it comes to research mathematics, I'd at least like to mention that Weil got the idea for the Weil conjectures from reading Gauss's articles on biquadratic residues.

As an example more in line with the question let me add that, in a letter to Goldbach dated April 15, 1749, Euler mentions that he has found, after quite some effort, a parametrized solution of the equation $xyz(x+y+z) = a$. Elkies found a way of deriving Euler's solution and found a simpler one, using methods from modern algebraic geometry. How Euler found his solution is still open.

Answer 10

I am no big shot in reading mathematical papers, but there are a few examples where I think "Reading the masters" is really worthwhile:

1. I read the theory of irrationals from Hardy's Pure Mathematics and found mention of Dedekind's original and concise version "Stetigkeit und Irrationale Zahlen". I read the English translation of this paper and I must say it is better than anything we can find in modern books on real analysis simply because it is easier to comprehend and appreciate. I somehow find this approach to irrationals the simplest requiring least mathematical machinery like you can teach this to a 15 year old kid with just the working knowledge of rationals.

2. Next I read lot of proofs of Jacobi's Triple product identity from wikipedia and other sources, but none of them matches the simplicity of the one given in Fundamenta Nova of Jacobi. He just multiplies the factors on one side and gets the infinite series on the other side. Plain simple multiplication like 2x3 = 6. In fact his whole theory of Elliptic functions as presented in Fundamenta Nova is based on integral transformation theory and is much easier to introduce than the modern modular form approach.

3. Another example is Lambert's proof of irrationality of pi based on the continued fraction expansion of tan(x). The real gem is not the irrationality of pi, but the beautiful formula for tan(x) (much more beautiful than Taylor's series for sin(x) and cos(x)). Compared to his proof, the proof of irrationality of pi by Ivan Niven is quite short and simple to grasp, but is highly non-obvious.

4. Best of all examples is the theory of periods in Disquisitiones Arithmeticae by Gauss. The entire proof of construction of regular polygons based on theory of periods is not to be found in modern texts at least in the form accessible to first year undergraduates. This theory and its application to constructions of polygons is so exciting and awe inspiring. No parallel in modern papers.

In my view the modern authors have made it a habit to work in too abstract terms so that only a post-graduate student of mathematics can understand the papers. There are few exceptions definitely and these are the ones I read online, but as a majority the books/papers on maths are increasingly becoming inaccessible to anyone other than a mathematics researcher.

Answer 11

Although your rule #1 would bar the following, I don't think anybody would disagree with me if I call Grothendieck a master. I cannot say I have read the whole of EGA, but I did go through most of volume 1 and a big chunk of volume 2, and got a lot out of it. The clarity of exposition is superb. If only it had examples...

Answer 12

I enjoyed reading Gauss's Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fläche auf einer andern gegebnen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird. This work was written in 1822 and deals with conformal mappings between the globe and the plane. I dare say I actually learned some mathematics from the experience too.

Answer 13

Whether or not Alfredo Capelli's papers about the Capelli identity fit the rubric of "old, classic texts by great mathematicians" is open to debate. What is true is that they offer a very clear and surprisingly modern perspective on the "first fundamental theorem of invariant theory for the general linear group" (the terminology is due to Hermann Weyl, who used Capelli's method in his book "Classical groups", but in an oblique and essentially incomprehensible way). In particular, Capelli introduced the universal enveloping algebra of $\mathfrak{gl}_n$ and its center and computed the action of the special central elements that he constructed on the polynomial algebra over matrices, deriving the Gordan – Capelli decomposition (or ($GL_n, GL_m$)-duality). Roger Howe, beginning in the late 1980s, had produced the only faithful modern account of Capelli's approach that I was aware of at the time that I read Capelli's papers. Of course, since it was Roger who introduced me to this area, I read his 20th century exposition first!

P.S. From the wording of the question, I get an impression that the rules have been rigged in order to confirm the favored hypothesis. I have several more worthy examples of "read the masters", but I feel as if I would need to argue the case more than I care to.

Answer 14

Boole, George (1854), An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.

Answer 15

Bolyai: Appendix, the first account of non-Euclidean geometry by one of its inventors. If nothing else, the beauty, clarity and brevity of exposition alone make this a must-read.

Answer 16

I have read Gauss Disquisitiones Arithemeticae (Tr AA Clarke SJ) and Euclid's elements (tr & commentary by Heath) - but both well after engaging with the fields they covered. To an extent the subject had moved on, and later insights had provided better definitions, postulates, generalisations, primitive concepts and theorems. But it did become clear in the reading what had motivated further development. The sheer ingenuity required by Euclid (reread over Christmas) to do arithmetic is a joy to behold, even though we would do it very differently now, and rather tedious on a second reading, Some of the foundational subtleties are also glossed over or rather taken for granted in more modern treatments. With both there is also a kind of pedagogical simplicity by which you arrive suddenly and without apparent effort at a significant result - I guess this is what constitutes true mastery.

Answer 17

I agree with the slogan, but completely disagree with reading original or historical papers. Quality is dictated from objective criteria, which we can try to formalize but - I think - everyone here recognize a quality paper when he reads it. What I personally like to find in a good exposition are:

• Meaningful insights - a way to imagine the subject and develop an intuition;
• Formal elegance and clarity - such insights become manifestly rigorous once the framework is set up;
• A clever idea or a nice picture to bring home - this can be a non trivial way of applying some tools in a context, or a wonderful trick;
• How things fit into a bigger picture.

The thing is: very often masters do give meaningful insights and are able to perform clever tricks, but they are not necessarily able to be clear or explain how things fit in a bigger picture; this depends on the ability to give a clear exposition, and the knowledge of the subject at the time in which the paper was written. Maybe a bigger picture just *didn't * exist, or it was at least significantly smaller than now. This applies in particular to papers from 19th, which I consider to be only of historical relevance. That's why I am thankful to the writers that digest the rough and precious material that "high level" mathematicians produce and then try to explain to a wider audience.

However, empirically, a lot of authors I love are masters. For example, I love the style of Deligne and Lurie, because you can taste the depth of what they write, no matter how much technical the things get. But how many of the algebraic geometers here have formed their intuition on the 13 volumes of EGA? Grothendieck was my inspiration for a long time, and he is the founder of modern algebraic geometry; it's just that things now are better understood and can be explained in a simpler way.

Answer 18

From the 1991 book by Bryant et. al, Exterior Differential Systems:

This book grew through our efforts to work through and appreciate Partie II of Cartan's "Oeuvres Completes" (Cartan [1953]), which we found to be full of interesting ideas and details. Hopefully our presentation will help the study of the original work, which we cannot replace. In fact, for readers who have gone through most of this book we propose the following problem as a final examination: Give a report on his famous five-variable paper, "Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre" (Cartan [1910]).

I read Cartan's five-variable paper when I was on sabbatical, and I learned a lot. It took most of my sabbatical time. I got a clearer idea of how to deal with the tension between explicit computation and abstraction in differential geometry (no one knows how to compute in coordinates the differential forms and differential invariants that arise in that paper), how to look for moduli of geometric structures, and how in practice to find geometric structures with symmetries.

Answer 19

It may still be useful for beginning students who know some basic algebra and some basic geometry to read Gauss' Disquisitiones Arithmeticae (1801). Gauss wrote out everything in components, when dealing with matrices and even vectors. Knowing more concise modern notion has the advantage of making one feel some (probably false) sense of accomplishment of being able to see further. I was doing just this a while ago, and noted many $2 \times 2$ determinants and realized that he was giving algorithms for inverting the Plucker embedding of the Grassmannian map $$ \alpha_{n,k} : (\mathbb{Z}^n)^k \rightarrow G_{n,k}(\mathbb{Z}) \subset \mathbb{Z}^{{n \choose k}},$$ in special cases over the integers.

I then read a small part of Bhargava's interpretation of Gauss's composition of definite binary quadratic forms in terms of $2 \times 2 \times 2$ cube and then observed that the algorithmic step for Gauss's composition is just inverting $\alpha_{4,2}$.

Given two positive definite primitive binary quadratic forms $Q_2=(a_2,b_2,c_2),Q_3=(a_3,b_3,c_3)$ with the same discriminant $b_i^2-4a_ic_i$. We form the vector $$ X=\begin{pmatrix} X_{12} \cr X_{13} \cr X_{14} \cr X_{23}\cr X_{24} \cr X_{34} \end{pmatrix} := \begin{pmatrix} -a_3 \cr -a_2 \cr (b_2+b_3)/2 \cr (b_2-b_3)/2 \cr -c_2 \cr -c_3 \end{pmatrix}. $$ and observes that $X$ satisfies the Plucker condition $X_{12}X_{34}-X_{13}X_{24}+X_{14}X_{23}=0$ for $G_{4,2}$ because $Q_2,Q_3$ have the same discriminant. This means we can find vector $x, y \in \mathbb{Z}^4$ (unique upto $SL_2(\mathbb{Z})$ transform) such that $x \wedge y=X$, ie. $X_{ij}=x_iy_j-x_jy_i$. If we now labelled vertices on the front and back faces of a $2 \times 2 \times 2$ cube $A$ by $$ M_1=\begin{pmatrix} x_1 & x_2 \cr x_3 & x_4 \end{pmatrix},\;\; N_1=\begin{pmatrix} y_1 & y_2 \cr y_3 & y_4 \end{pmatrix}. $$ We then have the corresponding left/right, top/bottom faces $$M_2=\begin{pmatrix} x_1 & x_3 \cr y_1 & y_3 \end{pmatrix}, N_2=\begin{pmatrix} x_2 & x_4 \cr y_2 & y_4 \end{pmatrix}$$ $$M_3=\begin{pmatrix} x_1 & y_1 \cr x_2 & y_2 \end{pmatrix}, N_3=\begin{pmatrix} x_3 & y_3 \cr x_4 & y_4 \end{pmatrix}, $$ and $Q_j^A(x,y)=-\det(M_jx-N_jy),$ we will satisfies $Q_j^A(x,y)=Q_j(x,y),j=2,3$ so that by Bhargava, $$ Q_1^A(x,y)=(-\det M_1, \det \begin{pmatrix} x_1 & x_2 \cr y_3 & y_4 \end{pmatrix}+ \det \begin{pmatrix} y_1 & y_2 \cr x_3 & x_4 \end{pmatrix}, -\det N_1), $$ (the middle term being the sum of determinants of the diagonal slices) is Gauss' composition of $Q_2$ and $Q_3$. Gauss's definition of composition is very complicated and it is very satisfying that we can give it a very simple and basic geometric meaning.

Answer 20

1] Carl Ludwig Siegel's Topics in Complex Function Theory 3 volumes 2] Kunihiko Kodaira Complex Manifolds and Deformation of Complex Structures

Answer 21

I once used the UK Interlibrary-loan service to borrow Fibonacci's Liber Abaci. This was not so much for the mathematical content, but because I was guiding students writing an essay on the influence of Islamic culture, and I wanted to get a bit closer to the sources.

Answer 22

Charles Sanders Peirce — Beginning with volumes 3 and 4 of his Collected Papers and covering pretty much everything he wrote on logic and mathematics that I could get my hands on.

Answer 23

1. Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

2. Aristotle, “On Interpretation”, Harold P. Cooke (trans.), pp. 111–179 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

3. Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Animadversion 1

One of the reasons to “Read the Masters!” is that you almost always learn how different their actual intellectual contexts, motivations, and reasoning were from what you tend to find in the reports of $2$nd, $3$rd, and $n$th hand sources.

In the case of Aristotle, one of the first shocks — that I still distinctly remember — was discovering that he was a far less binary, dichotomous, or dualistic thinker than all my previous readings and teachers had told me. This has a bearing that goes far beyond the purely historical interest to the substantive issue of how deductive reasoning proper relates to what was later described as "inductive" and "abductive" inference.

Animadversion 2

I call it “mathematics” when I see hints of form that inform and rule the appearances in view. The test of a “practically essential” source, ancient or modern, is much like the test of a chemical catalyst — it is not that we'd never get the desired product by any other reaction pathway, but that we'd be highly unlikely to get it anywhere near as easily in our lifetime. It is very often the forms that permeate our current airs of knowledge that we, like the proverbial fish in water, can hardly see for all their pervasion.

Animadversion 3

Another reason to study our mathematical organon in embryo is that it makes it easier to see the early integuments and initial embeddings of topics that grow detached and remote from each other as they develop. By way of example, here's a draft of an essay I started on the precursors of category theory.