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| This page includes both errata and comments (mine and readers'), all referring to the hardback version of Unknown Quantity, first published in May 2006. I owe a number of the most perceptive comments and corrections to Fernando Q. Gouvea, and have credited him in each case as "FQG." Prof. Harold M. Edwards of the Courant Institute also had numerous comments, credited as "HME". Points I shall think actually worth correcting in any future edition of the book are marked with an asterisk in the leftmost column. (Though these asterisks may come and go as I check things out or just change my mind; and whether there are future editions is in the hands of readers and my publisher!) Other points are just interesting comments of various kinds.
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| Page | Line(s) | Error or Comment | |
| * | 2 | 18 | "Jacobean" should be "Jacobian." These varieties were named for Carl Jacobi. |
| 10 | 6 etc. | [FQG]: I don't know how (or whether) it might fit into your outline, but it's interesting to note that the Greeks seem to have adapted quite comfortably to the discovery of incommensurability. Their solution was to distinguish 'numbers' from 'magnitudes,' the latter being a more general notion. Line segments were not numbers for them, and they didn't think in terms of 'length' (a number that we attach to a line segment); instead they treated segments (and areas, and angles, and other things, most notably ratios) as a different kind of magnitude. | |
| * | 24 | 18 | [HME]: Plimpton 322 is at Columbia, not, as I say, at Yale. |
| 26 | 1 etc. | [FQG]: Here you could have used to advantage [Jens] Hoyrup's work. Among other things, he has developed a whole new way of translating the mathematical terms in problems like this one. | |
| * | 27 | 16 etc. | I think the indentation and small font went on too long here, confusing the problem with my comments on it. From line 16 on we should have reverted to normal-sized font an no indentation. |
| 28 | 12 etc. | [FQG]: Here I started to get a
little worried about the lack of any distinction, in your text, between
'problems' and 'equations.' This is a big theme of
Reviel Netz's work, and I
think he's onto something. Basically, he highlights that the ancient texts
are about solving problems, and not about solving equations. One of the
distinctions is this: when solving problems, one wants an elegant, neat
solution (think of geometric constructions); when solving equations, one
wants a uniform method, so that the goal is to make the solution as standard as possible. In this framework, the Old Babs are closer to equations, though they still need to dress them up as problems. My reading of what is going on in their texts, actually, is that they hit on a method, and then set out to create problems that their method could solve. Your idea about why they were interested in these problems is ingenious, but I don't believe it. I think Hoyrup is closer to the truth: the scribes needed ways to show off their virtuosity. |
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| 30 | 9 | [FQG]: I think describing Ahmes's methods as 'trial and error' is a mistake. False position is a method! Plus, there are various other methods deployed there. | |
| 32 | 4 etc. | [FQG]: Mathematics in ancient Mesopotamia changed more than you say. In fact, the virtuosic stuff with quadratic equations is sharply limited in time, and sort of disappeared for long periods, eclipsed by astronomy and other worries. See Hoyrup's article on this. | |
| 32 | 21 etc. | [FQG]: The 'usual story' about the
shock of the discovery of incommensurability has been rather brutally
challenged recently. I usually recommend David Fowler's The Mathematics of Plato's Academy (second edition), with the proviso that Fowler's destruction of the 'usual story' is more convincing than his proposed reconstruction. |
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| * | 35 | 3 | "Less than half" should be "more than half": We have 6 of Diophantus's 13 books in Greek and four more in Arabic. |
| 36 | 22 | [FQG]: Heath conjectured that the 'terminal sigma' in Diophantus is actually an abbreviation for arithmos, much as his other 'symbols' are also abbreviations of dynamos, kybos, etc. I think most people accept this. This means we should think of Diophantus's notation as a sort of syncopated algebra, analogous to the Italian style in the 1500s. | |
| 38 | 25 | Any linear substitution would have led to a quadratic equation here. Diophantus's trick is to pick one that gives rational solutions. Perhaps I didn't make this sufficiently clear. | |
| * | 40 | 11 | "He knew the rule of signs..." FQG notes: "I'm skeptical about Diophantus and 'negative numbers'. The law of signs you quote refers (quite clearly, in context) to multiplying subtracted quantities, not negative numbers. That is, he is explaining how to multiply (a+b) by (a-c)." |
| 41 | 10 | [FQG]: The reason why I wouldn't refer to Diophantus as the 'father of algebra' is that he had so little influence. All the way to Euler and Fermat, he kept getting rediscovered (you note Bombelli's rediscovery; there were others), but once rediscovered he seemed to immediately get forgotten again. | |
| 43 | 2 | "...somewhat unfair..." Since we actually *got* al-jabr from the Arabic mathematicians, not (see above) from Diophantus, I guess it is not so unfair. | |
| * | 44 | 26 | [FQG]: Hypatia is not the first female name in the history of mathematics. The first is mentioned in Pappus: her name is Pandrosian. She is not mentioned as often as Hypatia, I think because Pappus doesn't seem to have thought much of her abilities... |
| * | 48 | 26, 33 | "Algorithmi" should be "Algorismi," and "algorithmists" should be "algorismists." |
| 55 | 6 | [FQG]: I think Victor [Katz] is right, and that Khayyam could indeed solve all types of cubics. Of course, only some of them required conic sections to solve; the others he reduced to quadratic and linear equations. | |
| 62 | Table | Several readers couldn't follow this. I wish I'd had space for an example. The equation x3-15x+22=0 works well. | |
| 67 | 9 | [FQG]: It's "got tagged." Fibonacci was never known by that name in his time; the name seems to have been invented in the 19th century. | |
| 70 | 13 | [FQG]: "The great obstacle..." The great obstacle is that this stuff is *hard*, much harder than we all seem to think. One needs to learn to think numerically and algebraically. Remember that the Greek tradition still had enormous authority at this time, and the number/magnitude distinction sometimes got in the way. The importance of Leonardo of Pisa has a lot to do with his ability to build a bridge between the Greek way and a more numerical way. | |
| 73 | 15 | [FQG]: Cardano recommends, in one of his books on astrology, that one should always predict long life, peace, and a successful reign when casting a king's horoscope. He was not an impractical man, in his own crazy way... | |
| * | 79 | 6 | [FQG]: As far as I can tell by reading Cardano, he never did manage to understand the casus irreducibilis. He certainly did *not* use complex numbers. The only instance I can find is the one you quote, involving a quadratic, where he felt himself in surer ground. I think this is really not there before Bombelli. |
| * | 93 | 31 | Thomas Harriot, not "John." |
| 102 | 6 | [HME]: "Newton's Theorem" normally refers, I think, to the formula on page 12 of my Galois Theory. [A formula, which I won't attempt to duplicate here, relating the sums of k-th powers of the variables to the elementary symmetric polynomials in them.--JD] I believe, as I say in that book, that Newton was fully aware of what you call "Newton's Theorem," but that that only became clear when Whiteside published Newton's private writings. But I would like to be wrong about this, if you have a citation. | |
| 108 | Last | As a footnote to my remarks about the
Fundamental Theorem of Algebra, mathematician Joe Shipman offers the following:
"It
was nice to see you credit my old teacher Mike Artin so much, but the book
doesn't do justice to his dad Emil Artin, who is more responsible than
anyone else for the way the subject is taught today, and whose great work on
Formally Real fields and Real Closed fields should have been mentioned (I
understand why you left out his development of Class Field Theory). |
|
| * | 112 | last line | "k = 6" should be "k = 5" |
| * | 113 | 21 | The second item in the last row of that table is shown as minus omega-squared. It should just be minus omega. |
| 143 | 15 | [FQG]: In your reading, did you
find any hint about when 'linear associative algebras' (Peirce's name for
them) became just 'algebras', or of when 'linear algebra' came to mean the name of the subject dealing with vector spaces and linear transformations? I'm curious about that. [JD: No, I didn't. Any information gratefully received.] |
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| 159 | 25 | [HME]: The phrase "the zero vector
has factors" is not well chosen. For that matter, though, the usual
phrase "zero divisors" is not well chosen either. Zero times anything is zero, so in this sense everything is a factor, or a divisor, of zero. |
|
| * | 165 | 33 | "quartet" should be "triplet." |
| 216 | 17 | [FQG]: When I got to the end of the second paragraph here, 'And notice one thing...' something crystallized in my mind that had been nagging me for a while. After years of teaching this stuff to undergraduates, I can see that in this sentence you're assuming too much of the reader. Were I saying this to my students, I'd have to say 'since the definition of normal hinges on a comparison between multiplying from the left and multiplying from the right, it follows that if a group is commutative...' Otherwise, they just wouldn't see it. Perhaps you read mathematician-speak for a bit too long, and may have fallen into our bad habits here and there. It's hard for me to imagine a 'general reader' following you at some of these points. Alas, I didn't mark down all of them. [JD: Nothing made me tear my hair as much as trying to find an accessible definition of "normal subgroup." The only consolation I can find here is that, to the best of my knowledge, no other pop-math author has been able to find one either. One mathematician, not FQG, said to me that until you have internalized the concept of homomorphism, normal subgroups don't really make sense. I'm afraid that may be right.] | |
| 220 | 8 | [FQG]: You have D2 wrong here. To get a 2-gon, draw a circle and mark two dots on diametrically opposed points. (If you like, distort it a bit, but maintain symmetry between the top and bottom halves.) Then there are four symmetries (180 degree rotation, flip vertical, flip horizontal, identity), so this is a copy of the Klein four-group. [JD: Hmph. I think this is a stretch.] | |
| 233 | 25 | [HME]: "Dedekind's definitions do
not look very modern, but he was on the right track." Seems to equate
"modern" with "the right track," which I object to. [JD: !] |
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| * | 234 | 29 | [HME]: Hilbert was never at Erlangen. Is it possible that Hilbert directed his thesis from afar? [JD: No. I wrote "Hilbert" when I should have written "Noether." Lasker took some advice from Hilbert, that's all.] |
| * | 238 | 14 | [HME]: "Ring-fields" is not the correct translation of Ringbereichen. The usual translation of Bereich is "domain". |
| * | 257 | 31 | (x2 + y2) should be (x2 - y2) |
| * | 291 | 24 | [FQG]: Z5 is certainly complete! (It's the completion of Z with respect to the 5-adic topology.) [JD: Drat, of course FQG is right. Though he is too much of a gentleman to say so, he has explained this, and everything else you want to know about p-adic numbers, in his excellent book on the topic. Several readers--including one Amazon reviewer--said I lost them with p-adic numbers. Buy Fernando's book!] |
| * | 296 | 30 | [FQG]: You have "Sao Paulo" wrong every time here. You got the hard part right: Sao with the tilde on the a. But it's Paulo, in Portuguese, not Paolo, as in Italian. (That's where I was born, and I studied at the same university there. Amazingly, I could see no trace of the influence of Weil and Zariski... except maybe for an older professor who had a 'thing' for projective geometry. I once asked an older professor there, and she said she thought no one could understand what they were talking about... There were some traces of Grothendieck's visit in the 1950s, mostly to be seen in several folks interested in topological vector spaces and functional analysis.) |
| 308 | 8 | [HME]: I felt you allowed the
possibility that the emperor has no clothes to come through in the
Grothendieck story. I would love to know whether Grothendieck will be remembered a century from now as a great bearer of light or as a strange and incomprehensible 20th century phenomenon. Carl Ludwig Siegel told me that Andre Weil had told him in a letter that "The work of Grothendieck is a swamp and I don't want to spend my last years in that swamp." (From memory, ergo a highly unreliable quote.) [JD: As I said on p.314, I do not understand Grothendieck's work myself, and so can have no useful opinion.] |
|
| 320 | 22 | [FQG]: As a confirmed Augustinian, I'm curious about which side it goes on... [JD: Er, the left... I think.] | |
| * | 322 | Note 7 | [FQG]: I don't think the proof you outline was given by Euclid. [JD: Good grief! This is a thing I have idly assumed all my life; but looking into Book 10 of the Elements, I see it's not there. Grrrr.] |
| * | 324 | Line 12 | "Since [san] stands for 7..." That san should be a zeta. |
| 324 | Note 16 | [FQG]: "recycled to show thousands" suggests a more positional system than it was; they needed to be marked in a particular way to mean thousands. | |
| 326 | Note 28 | A couple of readers queried "ethny," wondering if it shouldn't be "ethnicity." I have been using "ethny" since reading Pierre L. van den Berghe's The Ethnic Phenomenon. Van den Berghe is a respectable scholar, and as far as I'm concerned, two syllables trumps four. | |
| * | 327 | 20 | "(x-3)" should be "(x+3)." |
| * | 360 | Edwards | There should be an index entry for the long quote from Prof. Edwards on pp.258-9 |