Very many thanks to all who have written in pointing out errors in the Princeton Companion. I am told that it won’t be too long before the next printing, so I have finally been forced to collect together the errata in a systematic way. I am copying the list I have just sent to Princeton University Press so that if anyone finds an error now they can easily check whether it has already been spotted. (I was asked if I would do this some time ago — now at last I have.) The list appears after the break. From now on if people point out further errors I will add them to the list, with some indication of whether they have yet been corrected.
The secondary purpose of this post is to suggest that you should wait a bit if you are thinking of buying the book. I don’t want to hit sales too hard, but I’m guessing that not all PCM buyers are avid readers of this blog so I might as well reward those who are. Of course, you could take the attitude that the error-strewn version is a collector’s item: if so, hurry while stocks last.
The errors that have so far been spotted.
I will use the notation A.B to stand for page A column B.
An erratum with three asterisks in front of it denotes one that is problematic because it is more than just a minor change, or will affect pagebreaks.
Back flap: I suggest “illustrating a logarithmic spiral” instead of the phrase about the Fibonacci sequence.
14.1 Displayed statement (4′) should read
(4′) For all , lifelong happiness is at least as good as .
(I hope that still fits on one line.)
14.2 line -17
The reference to sentence (6) should be to sentence (9).
24.2 First displayed equation should read
27.2 line 18
… it is a simple exercise to show that every homomorphism that is not identically zero is automatically an isomorphism …
30.1 line 7
… such that is proportional to …
The v here should be bold face.
33.2 line -5
It should say “rate of change of ” rather than just “rate of change “.
40.1 Sixth line of section 6.4
It should be n-dimensional rather than d-dimensional.
42.2 line -12
It should say ““.
Two lines later it should say ““.
In line -2 it should again say ““.
43.1 line 13
Change “with a “point at infinity”” to “with a “line at infinity””
***43.1 line -6
Ideally, we would omit everything from “(This is quite hard to imagine” to “but a projective plane.)” since what I wrote in those brackets is unfortunately wrong. But omitting it will affect many pagebreaks. I can’t immediately think of any text that could replace what we have at present. We may need to discuss this.
43.2 line -1
The very last equation of the column should be ““.
57.2 Middle of column, it should say
… natural properties (for instance for all functions , , and ) …
62.1 In the paragraph headed (i), it should say
… which leads to the formula .
replace “” by “”
63.2 At the end of section 6.2 it should read
… was proved by Huxley in 2003 (the latest in a long line of successive improvements), is that is at most for some constant .
64.2 At the very end of section 6.3 it should say
… as many as .
In other words, ms need to be ns.
75.1 line -21
The number 181 needs to be added to the list of numbers here (in the appropriate place).
121.1 line 4
This displayed equation should read
124.1 line -7
Replace “in every bounded infinite set” by “for every bounded infinite set”.
131.1 line -13
It should be DBCF rather than DBFC here.
169.1 Middle of column.
… if one maps the real number to the point , then …
That is, the final “squared” comes inside the bracket.
171.1 End of penultimate paragraph.
This list is missing some numbers. It should go
and then continue as before. If it’s a problem to add a line here, then we could end the list with “4, -4, …” instead of ending at “5, -5, …”
***172.1 line -20
A new half-sentence is needed here. It should say
… for any elements and ; this involution is required to satisfy the –identity .
176.2 Middle of column.
Where it says “where and .” it should say “where and .”
180.2 line 5
Instead of “” it should say ““.
Earlier in the same line it would be better to write “” instead of ““.
183.2 line 16
… written either as 0.1 or as 0.02222…
(instead of 0.22222…)
184.2 lines 12-14
It should read
… sense: if is less than , then the -dimensional Hausdorff measure of is infinite, while if is greater than , then it is 0.
187.2 line 7
It should be “octahedron” rather than “octagon”.
194.2 line -8
The final should be (so that it matches the in the exponent).
***202.2 End of column
I would like to add the following sentence if possible.
… coefficients quickly. A method for doing this was discovered by Cooley and Tukey in 1965 (though it turned out that Gauss had anticipated them over 150 years earlier).
223.1 line -16
In the display here there is a stray comma in the last expression, which should be . (NB the “e” should still be a Roman e — it’s just the comma that needs removing.)
228.1 line 3
It should end with . I.e., it’s half the square root of 3 and not the square root of half of three.
232.1 bottom of column
It should read
***238.1 line 1
The factor 6 here is not explained. Unfortunately, it may take more serious rewriting to clarify this. (It’s not exactly wrong, but comes from a slightly different form of the original equation.) So I think we’ll have to leave it for now.
However, it should say
… equation . If we add …
240.2 line 8
It should say
… is the matrix defined by .
249.2 line 11
It should be
I.e., there’s a missing q in the bracket.
266.1 line -21
It should say “,” instead
272.2 line 4
It should say “generated by the polynomial .”
277.1 line -4
Reword as follows.
… a system of numbers where we introduce not just one square root of -1 but three, called i, j, and k (together with their negatives). Once one knows …
***278.2 Middle of column, item (ii).
Ideally we should extend the sentence as follows:
… is a real number, and similarly for multiplication on the right.
If the extra line causes problems, then we can perhaps live without this, but it will bother some readers.
297.1 line 21
It should say
… spherical harmonics are eigenvectors of the Laplacian, but they …
310.2 Second paragraph of section 2.1
This should read
Suppose, then, that we are given two points P and Q in the plane. We take as our class of admissible functions all smooth, real-valued functions , defined on some interval , such that and . The length …
314.2 line -5
It should say “when one does include the”
323.2 display in line -9
Where it says it should say
It should say “published in 1952” rather than “published in 1934”.
325.1 line -11
It should say “It is easy to deduce that is itself a prime in the …”
NB the “i” should be in the same font as the other ones round here when this correction is made.
327.1 lines -8, -6, -5
The three s that appear here should be s.
339.1 line 3
It should say “arithmetic progression” rather than merely “arithmetic” here.
353.2 line 12
“Pollardand” should be “Pollard and”.
355.1 line -16
“Pollardfor” should be “Pollard for” (!)
424.2 line 18
Change “the first three” to “three of the”
425.2 Middle of column.
Change to (it comes at the end of a line).
***444.1 line 18
Change “non-trivially” to “”properly discontinuously””, so it should read
… if a group acts “properly discontinuously” as a set of isometries on …
A better change, but one that would affect linebreaks and pagination, would be
… if a group acts properly discontinuously (this means that for any compact set there are only finitely many such that and intersect) as a set of isometries on …
445.1 line 19
Replace “smooth compact manifold” by “complete Riemannian manifold”.
Replace “Given any closed path” by “Given any contractible closed path”.
460.1 line 4
It should say
… matrices that satisfy …
496.1 line -19
A few small changes needed here. It should say
However, if we apply ten times to and , then their respective eleventh digits have shifted leftwards and become the first digits of and . These two …
523.2 line -21
Replace “slide rules” by “log tables”.
531.2 line 9
It should say image( ker() rather than
the other way round.
555.1 line -7 it should say
… is by constructing a (preferably nice) bijection …
558.1 line 22
It should read “When there is exactly … ”
568.1 line -13
It should read
… set of integers using …
575.2 line -5
It should say “Two very obvious conditions” here.
595.1 lines 16-17
.. at least . Therefore, the probability … is at most , which is …
645.1 last line
The right-hand side should read
This one will need careful formatting.
695.1 line -6
A mathematical mistake here. It should say
We define the index of the fixed point to be the number of turns made by the vector from to , counting this negatively if these turns are in the opposite direction to the way that goes around . (This definition is problematic if for some , but again …
701.2 line 7
It should say “no proof of the formula , where is the …”
706.1 lines 9 and 11
The two capital Fs should be small Fs. That is, we want
“a function ” and we want f(x,y) inside the integral.
708.1 line -16
It should say “tile the plane if and only if the algorithm
fails to halt.”
708.2 line 13
It should say “if and only if the algorithm halted at .”
708.2 line -3
It should say “in the form .”
714.2 line 1
Change to .
***lines 2-5 This parenthetical remark should be removed, though it will change the pagination.
756.1 Middle of column.
For consistency with the rest of the book, we should say “going on to find seven more proofs over the years.” However, we should not make this change unless Jeremy Gray is happy with it.
It would be better to entitle this article “Ernst Eduard Kummer”
The first word of the column should be spelt “Theologie.”
813.1 line 5
Omit “closed”, so that it reads “formally real fields”.
832.1 At the end of the first full paragraph, add words so that it reads
… can take any value greater than 1.
The description of CBC and OFB modes is oversimplified. The CBC description gives the impression that two successive cleartext blocks are added before encryption (so a 128-bit repeat in cleartext would result in a 64-bit repeat in ciphertext, similar to the ECB problem described in the previous paragraph)
when actually each cleartext block is added to the previous ciphertext block before encryption. In turn this reads like the description given of OFB mode; OFB does not feed the input cleartext through the block cipher at all, but starts with an initialization vector and repeatedly enciphers that to generate
a keystream, that is then added mod 2 to the cleartext to generate a ciphertext.
I think this is going to be too complicated to change this time, but I’ll keep a record of it.
909.1 line 14
“Becky has five siblings” should read “Desta has five siblings”.
“” (in the second paragraph) should read ““.
In the display the should be removed inside the “exp”, and the “” in the square root on the bottom should be ““.
Four lines later, there is again a stray , and again the “” on the bottom needs to be a ““. And this time, we also need to change the “” to a ““.
984.1 line -2
“quatre-vingt” should be “quatre-vingts”
In equation (7), “” should read ““.
1014.2 The remark in brackets that looks as though it explains Robinson’s Non-Standard Analysis has nothing to do with it and should appear after the following entry on the Langlands program.