Many more books

I recognize myself in these notes by Chris Tiee on tensor analysis—the fever of acquisition, the uses of comparison, and the stray insight that makes the whole book worth it (as the one track does for the whole album):

“One way to learn a lot of mathematics is by reading the first chapters of many books.”—Paul R. Halmos

Ah, the dreaded discussion of texts for tensor analysis. I am addicted to collecting math books (I also often have 10 books checked out from the library simultaneously) and reading the first 20 pages of them. Very occasionally I make it through farther than that. Despite the proliferation of bad tensor analysis texts (some would say all of them are bad), I have to admit I have gleaned everything I have learned about Tensor Analysis from reading these books, collecting the knowledge into a gigantic patchwork. The fact is, each text actually has a gem of insight or two that is not presented in any others. There is much overlap in the bad parts, and some in the good parts too, but of course, it’s always hard to consult so many references, since I often forget the transformation laws on those overlaps. . . not to mention also the transformation laws that tell how the notation changes—regarding this, we have the following

A.1. Joke. Differential geometry is the study of those things invariant under change of notation.

Another problem is also that it’s very hard to strike a balance between being intuitive in the explanations of what these things are—and hence being vague—and also developing a precise, formal theory that is the real deal—hence being obfuscatory.

The quote from Halmos isn’t quite right. It’s actually, “I wish I had read the first ten pages of many more books—a splendid mathematical education can be acquired that way.” As for tensors, a newer book rich in intuition is Dwight Neuenschwander’s Tensor Calculus for Physics.

Visual X

Needham’s book is the example par excellence of treating modern analytical material in a more classical, geometric vein; some of these books approach it in that regard, while others just include nice pictures or illustrations. I’m sure I’m forgetting about many texts with beautiful figures; I’ll add to the list as I’m reminded of them.

Tristan Needham, Visual Complex Analysis

Nathan Carter, Visual Group Theory

Martin Weissman, An Illustrated Theory of Numbers

Elias Wegert, Visual Complex Functions

Siegmund Brandt and Hans Dieter Dahmen, The Picture Book of Quantum Mechanics

James Callahan, Advanced Calculus: A Geometric View

H. M. Schey, Div, Grad, Curl and All That

Michio Kuga, Galois’ Dream: Group Theory and Differential Equations

Charles Misner, Kip Thorne, and John Archibald Wheeler, Gravitation

Michael Spivak, A Comprehensive Introduction to Differential Geometry

David Hilbert and Stephan Cohn-Vossen, Geometry and the Imagination

Admirers so few and so languid

Samuel Taylor Coleridge, age 18, to his brother George, with a very green poem setting Euclidean reasoning to verse:

Dear Brother,

I have often been surprising that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the case; viz. that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert. To assist Reason by the stimulus of Imagination is the design of the following production. In the execution of it much may be objectionable. The verse (particularly in the introduction of the ode) may be accused of unwarrantable liberties, but they are liberties equally homogeneal with the exactness of Mathematical disquisition, and the boldness of Pindaric daring. I have three strong champions to defend me against the attacks of Criticism; the Novelty, the Difficulty, and the Utility of the work. I may justly plume myself, that I first have drawn the nymph Mathesis from the visionary caves of abstracted Idea, and caused her to unite with Harmony. The first-born of this Union I now present to you; with interested motived indeed—as I expect to receive in return the more valuable offspring of your Muse.

This is now—this was erst,
Proposition the first—and Problem the first.


On a given finite line
which must no way incline;
To describe an equi—
—lateral Tri—
—A, N, G, E, L, E.
Now let A. B.
Be the given line
Which must no way incline;
The great Mathematician
Makes the Requisition,
That we describe an Equi—
—lateral Tri—
—angle on it:
Aid us Reason—aid us Wit!


From the centre A. at the distance A. B.
Describe the circle B. C. D.
At the distance B. A. from B. the centre
The round A. C. E. to describe boldly venture.
(Third postulate see.)
And from the point C.
In which the circles make a pother
Cutting and slashing one another,
Bid the straight lines a journeying go.
C. A. C. B. those lines will show
To the points, which by A. B. are reckon’d,
And postulate the second
For authority ye know.
A. B. C.
Triumphant shall be
An Equilateral Triangle,
Not Peter Pindar carp, nor Zoilus can wrangle.


Because the point A. is the centre
Of the circular B. C. D.
And because the point B. is the centre
Of the circular A. C. E.
A. C. to A. B. and  B. C. to B. A.
Harmoniously equal must forever stay;
Then C. A. and B. C.
Both extend the kind hand
To the basis A. B,
Unambitiously join’d in Equality’s Band.
But to the same powers, when two powers are equal
My mind forebodes the sequel;
My mind does some celestial impulse teach,
And equalizes each to each.
Thus C. A. with B. C. strikes the same sure alliance.
That C. A. and B. C. had with A. B. before
And in mutual affiance
None attempting to soar
Above another,
The unanimous three
C. A. and B. C. and A. B.
All are equal, each to his brother,
Preserving the balance of power so true:
Ah! the like would the proud Autocratix do!
At taxes impending not Britain would tremble,
Nor Prussia struggle her fear to dissemble;
Nor the Mah’met-sprung wight
The great Mussulman
Would stain his Divan
With Urine the soft-flowing daughter of Fright.


But rein your stallion in, too daring Nine!
Should Empires bloat the scientific line?
Or with dishevell’d hair all madly do ye run
For transport that your task is done?
For done it is—the cause is tried!
And Proposition, gentle maid,
Who soothly ask’d stern Demonstration’s aid,
Has prov’d her right, and A. B. C.
Of angles three
Is shown to be of equal side;
And now our weary stead to rest in fine,
‘Tis raised upon A. B. the straight, the given line.