The Should I Become a Mathematician? Thread
|Oct4-12, 01:58 AM||#3299|
The Should I Become a Mathematician? Thread
freshman calc: Courant/Hardy/Landau - Foundations of Analysis
how long were people using Hardy? I got the impression both Courant and Hardy did well into the early and mid 60s, though i think hardy faded a bit quicker. Esp with so many people trying to replace both with all the newer 60s texts.
What did you think of Hardy and Landau?
Hardy seems like a pretty rough ride for anyone taking math after the Space Race.
I think anyone reading it would go through it at a glacial pace, and i wonder if anyone finished the damn thing...
Landau looks cool, totally minimalist, and puzzling as Babylonian cuniform....
sophomore calculus: i forget what book,
oh damn, that's the best part...
Was it more of Courant? and was there another vector book?
were any of these possibly on the reading list, or recommended by the teachers?
1952 Kaplan - Advanced Calculus - Addison-Wesley
1955 AE Taylor - Ginn
1957 Apostol [I'd think you'd remember that one]
1959 Nickerson Spencer and Steenrod - van Nostrand
1961 Olmstead - Appleton-Crofts
1964 Protter and Morrey - Addison-Wesley [all these would probably be after you took your degree/classes]
1964 Smirnov - Addison-Wesley
1965 Buck - McGraw-Hill [actually that's probably the second edition, there was probably a first edition 1957-1963ish]
1965 Fleming - Addison-Wesley
1967? Spivak - WA Benjamin?
1968 Loomis and Sternberg - Addison-Wesley- free pdfs at his website
1970 Rossi - Addison-Wesley [oh oh another Brandeis person]
[I'm not sure if missed anyone from 1955-1980s there, but if there's any famous forgotten text from the 50s 60s 70s, tell me someone]
[oh hell tell me about the terrible ones too!]
my feeling there wasnt really anything out in the 70s... just Thomas and Finney clones and 15% of the books just mentioned...
I get the feeling that Apostol and Buck soaked up most of the sales at the high end, and Thomas and Finney for the rest]
when did the first Spivak come out? wasnt that like in 1967 I assume you read it after your degree, and the other book he did i think was 1965 on manifolds..
[or did you zoom through it after your degree and before grad school]
I always found it interesting where i'd struggle with a mainstream book and then eons later, find it more approachable [or find the easy and hard books on the same subject more approachable]
I used to think that you liked Loomis before, but it was more 'something you went through' but wouldnt really recommend... [when you clarified things a while later]
sophomore algebra - linear algebra and matrix theory - nering
fundamental concepts of higher algebra - aa albert
What did you think of Nering?
I assume that was a fixed up edition of albert's 1930's abstract algebra books
[Modern Abstract Algebra - Chicago 1937]
[Introduction to Algebraic Theories - Chicago 1941 - more an introduction to the other book]
Linear Algebra didnt really seem to take off till the 50s/60s, or bits of it in a Calculus III part of the text....
[or they dropped it being called Theory of Equations like using that famous Uspensky book and made it way easier and modern looking in the mid 60s]
[maybe it was all the mainframes doing Linear Programming that got it popular in the schools]
1951 Wade - The Algebra of vectors and matrices - Addison-Wesley
1952 Perlis - Theory of Matrices - Addison-Wesley
1952 Stoll - LInear Algebra and Matrix Theory - McGraw-Hill
1964 Bickley-Thompson - Matrices and their Meaning - van Nostrand 1964
- complex variables, text by Henri Cartan
so the pures went cartan and the applied went to churchill? [or did anyone do the easiest thing and read churchill first?]
Kaplan did a big Addison-Wesley on Complex too in 1953...
advanced calculus: official text: calculus of several variables by wendell fleming, but the lectures followed more closely the book Foundations of modern analysis by Jean Dieudonne
Did you take adv calculus at two different times, or was fleming out that early?
[I got the impression that Courant and Spivak and Fleming were the best of the texts from the good ole days from you]
senior: real analysis taught by lynn loomis, no textbook, it covered abstract measure theory as in the book of Halmos
Halmos came out in 1950 and probably the closest in style is Bruckner.
I remember seeing a strange set of analysis books at Simon Fraser, they used Goldberg [Wiley 1976] and Bruckner [Prentice-Hall 1996]
Goldberg looked stiff, but i heard it's pretty traditional and a touch gentler as far as dry analysis books go, but it's sure a rare one, musta been popular in the mid 70s and with the MAA and got tossed into obscurity when Rudin got pushed more and more...
[I still find Binmore or Colin Clark [The Theoretical Side of Calculus] as the two easier books out there]
and didnt Marsden write a pretty gentle and wordy Analysis text? It seemed the book to read before tackling Hardy]
algebraic topology taught by raoul bott, text: algebraic topology by spanier. most people today recommend the book by allen hatcher.
How did you find Bott's texts? [Bott and Tu]
Spanier..... well i was going to say, amazon, but i peeked and it's from the chicago list of books...
[Spanier is the maximally unreadable book on algebraic topology. It's bursting with an unbelievable amount of material, all stated in the greatest possible generality and naturality, with the least possible motivation and explanation. But it's awe-inspiring, and every so often forms a useful reference. I'm glad I have it, but most people regret ever opening it.]
I want to remind that i did not learn much from this somewhat harsh and user unfriendly first exposure to mathematics
people say that Caltech's course probably 'teaches' more, but if you throw teaching out the window, Harvard is the most difficult one...
I found these notes 'somewhere' and it had to deal with Rudin's textbook ...
[Harvard 55ab takes about 50 hrs a week of study]
[Thoughts on the flaws of Harvard 55]
[After having chosen Caltech over Princeton and Harvard to pursue a math major, I feel strongly that the math department's main feeder course here - Math 5 - is by far the strongest of the various courses at top universities which are taken by the strongest math students. It's main virtue is that it is long enough (a year) to do something serious, and that it does it in a thorough methodical way, building up steadily to huge, important theorems that you actually understand fully by the time you get to them.]
[I know that the 'stronger than the others' claim is true for sure in comparison to Princeton, since I actually took their math major feeder courses when I was a high school senior. (Problems there: teaching quality haphazard, too-advanced material rushed through so that even the brightest students are lost, though Jordan Ellenberg's Math 214 was a well-known and beautiful exception - but he's not there anymore.) And yes, I think Math 5 here is stronger even than Harvard's Math 55. While Harvard's famous course covers a lot of esoteric and advanced topics, it does so with very little unity and requires overwhelming amounts of outsdie reading so that even the best students miss 30% or so of the ideas.]
[After a year and a half at Caltech, I knew everything that a Math 55 graduate knew, but various comments I've heard make it pretty clear that most of them come out with a "scattered" feeling - they've been exposed to a lot but don't have a particularly unified picture. Math 5 keeps to a more manageable area and explores it more deeply, and so one comes away with some very tangible and coherent knowledge.]
[Those are my feelings on the subject.]
[Caltech Math108a - used Rudin and Carothers and Elias Stein Complex Book - 2 real+1complex]
[the combo of the three is better than Harvard 55]
[Loomis and Sternberg's book used to be used for Harvard 55ab]
[I think this book is inappropriate for use as an undergraduate textbook. Its use at the introductory graduate level is defensible, but I see no reason to choose this book when better ones are available. Apostol's Analysis book is at a similar level but has much richer discussion and is more comprehensive. For a book slightly more elementary than that, I would recommend Taylor and Mann. Like I said above--as a sequel to this or similar books, I think the Rudin "Real and Complex Analysis" book is absolutely wonderful. This book does have one purpose for which I found it to be very well-suited: it is useful to work through, perhaps only once, to review the subject and solidify your understanding of the material. But its value as such does not warrant purchasing it at the obscene price.]
- almost all algebra books seem to fail this test to me
[high school or abstract?]
c. Beachy and Blair
those 7 i think are the easiest ones on my list, and the first two are probably 'well-known'
how did you find Paul Cohn's books [1970s-1990s]
[i think one of his introductory books was fixed up considerably with the newer editions]
not sure what to think though, since it's not used that much in any of the syllabuses out there [or anymore]
- along with books like rudin's analysis.
- Many people recommend Dummitt and Foote and it does have many good qualities but I have several criticisms of it.
What texts are you somewhat [or completely] sour on?
It's rare to actually hear people criticize a popular book, or classic [in whole or part]
Heck, the first time i saw Apostol's texts i said, man, none of this is really necessary... but i was impressed at how huge the books were, and thought man it would be one hell of a school that used these as 60 weeks of 'an introduction to calculus'....
but i'm sure if one tackled a mini calculus course or had a book to read in parallel, it would be much better. But as a first and only textbook, oh i shuddered, but i definately spent a good 30 minutes at it in the 1980s saying, wow this is surreal, it's the hardest calculus book i seen.
much later on, i added it to my 'shopping list'
I added three books to the list too..
Nering is a new one...
Mackey's complex text
your stories definately do get better the more we hear them mathwonk!
|Oct4-12, 12:07 PM||#3300|
well heres one more story about my sophomore calc book and why i don't remember the name. after getting a D- in freshman honors calc part 2 from john tate (a course i had only attended once a month, during my slow decline before eventually getting kicked out for a year), when i returned in the fall i had to take non honors sophomore calc, taught as it happens also by tate.
tate was a great prof, but in the non honors course he had to use the book chosen by the departmental calculus committee instead of picking his own. So it was one of those routine mediocre books they use at places that are not harvard, reasonable but not too challenging (Taylor?). the course was ridiculously easy in comparison to the previous year's course, and although i did not work or attend much and seldom handed in hw, i was still passing as i recall.
one day in discussing the implicit function theorem in class on a day when i was there, tate read disgustedly from the book's treatment: "the proof of this result is beyond the scope of this book". He slammed the book on the desk and said loudly "well it's not beyond the scope of this course!" and went over to the board.
Then he stopped, looked back at the offending book lying on the desk, strode quickly back, grabbed the book and slammed it into the trash can with both hands.
Then at the end of the class, he went back, calmly retrieved the book from the trash and assigned homework from it.
|Oct4-12, 12:12 PM||#3301|
|Oct4-12, 01:00 PM||#3302|
another reason for not remembering the name of the sophomore calc book may be that i did not own a copy and just borrowed one to read the day before the test. i thought that was cool, then.
|Oct4-12, 06:41 PM||#3303|
I got mixed feelings about tossing proofs or epsilons into a first calculus course, and i thnk the new math did 'kill off' the Syl Thompsons, JE Thompsons, and the easy to read, easy to understand calculus texts common till the late 40s/early 50s [Granville Longley Smith as well, which i liked browsing in the library, when texts were built so you could read it all, and follow it all]
And well, there should be a point made where honours calculus and regular calculus has to do some trade-offs, a math teacher does need to know what is essential and what is 'merely details'.
[Bueche made that point in his introduction to his College Physics text where sometimes you *need* to push the essential ideas and do it well sometimes].
But, it's hard to say, how good/awful the book is, for some book is challenging enough, which could be the *audience* of the book... Remember that in the majority of cases the math or physics course is just a 'feeder' for engineering or basic requirements for some 'other course'. It's not math for mathematicians or physics for physicists... though i think actually it might be nicer in some cases for people to jump through the hoop twice, with an easy book and then a super detailed book.
There's a lot of Taylor's but i don't think it was AE Taylor...
Sherwood and Taylor did their prentice-hall book in the 40s and it was definately in the top 10 books for the 1945-1950 period.
the early 40s is when the last edition of Horace Lamb's Calculus book, which was probably THE long winded calculus text paired with Hardy's Pure Mathematics, and the late 40s is when the last tweak of Longley Smith came out after 50 plus years of handholding.... [it was a popular one for teaching in the US Military too]
and then Taylor and Mann did Advanced Calculus in 1955 and was/is still going in a third edition into the 1980s.....
Taylor and Mann [1ed 1955 2ed 1972? 3ed 1983]
[Excellent Clarity of Presentation]
[This book has a clarity unparalleled among books covering similar topics. While it contains an extensive amount of prose, it is still fairly compact: the book explains each result, the motivation for it, and points out possible pitfalls and considerations. Examples are well-chosen, proofs are easily followed. The order of the book is a bit chaotic, but it's written in such a way that it is easy to skip around in it.]
[My only complaint about this book is that I wish it covered a bit more material. This book might not go quite as far as some people might want, especially for a two-semester sequence or for courses at the graduate level.]
[I would recommend this book to anyone who already knows calculus and wants to learn (the more rigorous topic of) analysis on their own, or anyone selecting a textbook for an undergraduate advanced calculus course. This book also makes a good reference, and I was happy to permanently add it to my collection. For a more advanced book covering topics beyond those covered in this book, I would recommend Apostol's analysis book.]
[Worth every penny]
[This is the advanced calculus text I used at University of Washington while getting my BS in mathematics. I loved it then, and I've just purchased another copy to use for review. It's extremely well written. If you're looking for a good second year calculus text, this one's it.]
[I am no expert in the area of Mathematical Analysis, but I am an avid reader of any book that pertains the subject. I found this book in my schools mathematics lounge and could not resist reading it from cover to cover. This book is of the quality of such authors as Buck, Widder, Courant, and Rudin. As another reviewer has noted, this book is definitely worth every penny. It is not dry or to pedantic as some of the other afore mentioned authors, yet it is not simple and lacking in content. Of course like any quality Advanced Calculus book it requires the reader to have mathematical maturity as well as patience and the drive to self-explore the concepts. If one cannot follow simple examples and from those examples formulate their own, they may want to review the very basics of mathematics or consider a different major. I would highly recommend this book to advanced undergraduates or beginning gradutes students as a reference book or for self study.]
Anyhoo it is surprising that Har would use in the early 60s a mainstream calculus text that wanted a minimum of proofs....
Actually here's a good question, what would be the ideal textbook and supplementary texts that you'd pick Mathwonk for 1960 Harvard, for honours and mainstream calculus?
[I thought of the question when i thought, gee i wonder if Thomas would be a way better choice for the non-honours class than the 'unknown textbook']
a. Franklin - McGraw-Hill 1953
b. Thomas - Addison-Wesley [2ed 1953 3ed 59-61ish] [before it was Thomas and Finney]
a. Courant Blackie/Interscience 1938
b. Kaplan [for Calculus 3/4] Addison-Wesley 1952
[maybe Taylor for the second class]
[maybe Apostol for both classes]
I just wonder if back then you'd find Thomas too easy, and Apostol too challenging...
i found it interesting that there wasnt too much choice till the New math days really when good and bad textbooks on calculus [and high school and second year] just exploded
Courant was used from the depression till the Space Race and was still pretty strong 60-65 for books... and then the creepier gold courant/fritz john book came out, which was neater and weirder, basically courant bowed down to the new math pressures [heh] and well most people like it, with mixed feelings, but almost *always* prefer the original
I think he started the second edition unneccessary textbook change hype *grin*
|Oct4-12, 09:01 PM||#3304|
for me it was not so much the book, as when i started to take learning seriously, but some books like spivak went out of their way to reach me before i knew how to study. i.e. no book is too hard for a serous student, but some books reach out to the clueless.
|Oct5-12, 12:18 AM||#3305|
- for me it was not so much the book, as when i started to take learning seriously, but some books like spivak went out of their way to reach me before i knew how to study. i.e. no book is too hard for a serous student, but some books reach out to the clueless.
I got frustrated with lukewarm books [common in the 70s and 80s] and i often looked for easy books, that were far clearer, far easier, and well, there was also the appeal for the super detailed, super lengthy hardcore books too.
I think doing things differently and adding tons of topics not found in any other textbook is why i started liking those things...
I still get the impression that one needs a good mix of old and new textbooks, and for me about 25% new 75% older is a cool balance.
I was looking at the easy calculus books [syl thompson/JE Thompson/Sherman Stein/calculus for electronics 60s McGraw-Hill/ and things like courant and john.. and hardy... [apostol then was way too formal and scary for me then]
same thing liking old physics texts, Feynman and the Berkeley Series, Resnick, Kleppner, Symon, Reif, PSSC]
i thought it was interesting that the books were EASIER in the old days
yet they were HARDER too...
like they didnt forget what's so sorely needed for people to get up to speed, and slowly learn how to study properly....
But i think the newer textbooks are superior with way more examples [Schaum's outlines were there for a reason!] and sometimes way more problem sets.
I just thought that there was a time where the best easy math/science books and the best/harder textbooks were just passed off as unsuitable by the curriculum because they didnt *easily* fit....
and well, i see nothing at all wrong with textbooks written for people who got problems crawling.... or courses at higher institutions that teach people from zero math or zero physics [and do it well]... as well as making courses hard to fail if you 'follow the teacher's recommendations'....
otherwise, nothing at all wrong with repeating a class 7 times till you get it right, and go to the next rung of the ladder [I think there's something ungood in the fabric of schools of, if you didnt get a B, get out and try another career].... not a good tendency at all.
- no book is too hard for a serous student
especially true if you know how to tackle it, and eliminate any teacher or exam or grade stresses...
I found it so liberating to know that true self-accomplishment came from trying to tackle just one chapter as best as one can, and to keep plugging the hours into it, if it takes 8-15 hours, unlock the secrets of all the examples, reread the text carefully, and well enjoy the text once you're soaking in 98% of it, and try to see that the problems are meant to be totally taken as a whole, and it should all be workable with the 30 pages studied....
too many people fall into a trap of accomplishments by 'passing a whole course' or
'passing an exam'
and i think that's really a good way of not getting the most out of a text, the accompliment is mastering just one chapter....
doing 2 chapters [knowing it inside and out] and not touching the rest of the book says more to me, than taking 3 courses and getting 57%....
and i think i know both of those extremes well in my earlier days
i think there was a slow transition from lectures to textbooks from the 1910s to the 1960s.... a good example is a lot of the early quantum stuff, there wasnt a textbook for a while, and for years it was lectures and readings of papers, and sometimes 3 people and a teacher trying it out...
and as the decades flowed, the textbooks got easier in some ways, and there's a lot of interesting stuff out there, now...
I think textbooks are really highly polished lecture notes...
but remember there's lots of geniuses in math or physics, who didnt rely much on teachers or the curriculum to start off their box of tools. They didnt wait till Algebra 11 or Physics 11 or Resnick or Courant.... they soaked in a few textbooks and library books on their own, and then at a higher stage, fell into place into following the 'syllabus and curriculum'....
I think all the hope is placing a ton of effort into the lower stuff... and to make people do more than 97% of the others....
it feels like 3% of the people who did algebra, will get into a calculus text...
or 97% of people in first year physics people won't take a course in intermediate mechanics...
and i think we stopped making things 'friendly yet DEEP' at the elementary levels too, where i think the 1960 and 1965 PSSC system worked, and then the curriculum killed it for being too weird, too deep and spending months before you learned 'mechanics', and well, how many high schools or colleges or unis teach calculus with Syl Thompson's Calculus Made Easy?
I think that would be a great class for people, for credit or no credit at all. And it might toss people the courage to get into good books like Spivak.
i think we need lots of easy classes for algebra and physics for the clueless, hell in only a few weeks or months you can slowly show them how to study things deeply too.. but the biggest impediments i think are, the hoops and ladders to get a good solid background in algebra, or basic physics these days, if not also the financial strains of society that keep growing, and unis going from nearly free, to nearly impossible things to pay for.
My mantra is teach a student to only be 'serious' about learning *one chapter*
i think it's way easier than mastering 'one course'....
|Oct5-12, 12:01 PM||#3306|
i agree one can only learn one topic at a time. i try not to worry about how few books i have read completely, and focus only on how many individual topics i understand thoroughly. E.g. i think I understand the (classical) Riemann Roch theorem pretty well now and am finally beginning to grasp the Riemann singularities theorem.
oh yes, and if you think about why courses like math 55 at harvard are sop hard, you have to think about who they are aimed at. A friend of mine's son took that and flourished in it. But he prepared by taking not just a full and challenging math major sequence at UGA while he was in high school, but also took and starred in a number of graduate courses too, all before entering college and attempting math 55.
So this successful student was essentially at the advanced graduate level before taking what is listed as a second year advanced calculus class. Oh yes I believe he also took and did well in the (college level) Putnam exam while a high school student.
So don't believe what it says in the catalog about some of these courses and wonder why you and I were not able to deal with them, when all we had was the actual stated prerequisites. I bombed in 3rd year college french too after 2 years of high school french, in a class in which every other student had taken 4 or more years of french, some had taken 8, and at least one had lived in france. One of my friends who tried to take first year italian was the son of an Italian employee at the Italian consulate, and they spoke italian as the primary language at home.
|Oct5-12, 05:14 PM||#3307|
- i agree one can only learn one topic at a time. i try not to worry about how few books i have read completely, and focus only on how many individual topics i understand thoroughly.
When you get higher up, yeah, you go from books to concepts....
Well, some courses are there to teach you, and sometimes try for coherence in letting most all of it to soak in....
And the other courses who are for people who are self-taught who bring their own advanced box of tools, and there's little unity and *no one* soaks in more than 70% of the material. But if you like esoteric cannonballs fired at you, fine... I'd rather just get the reading list and some structured outline of 'what to read when' and do it way way more slowly...
Not to mention, i wonder how the course changed through the decades with the outside readings, and such... The good side is people are exposed to a 'lot', but it's a rush job...
seems like in the glory days of the 60s, you just had Loomis...
[well with Fleming and Dieudonne too]
now they throw Axler and Rudin at you, and add bits of
c. Counterexamples in Analysis
d. Korner's Fourier Analysis
and caltech does similar throwing at you
c. Elias Stein Complex Book [not well liked at all]
[people do like Carothers and Burn, both outta cambridge in the 80s and 90s...]
notes on carothers:
[I do agree with him on that the book is very informal in the exposition and is chatty. I feel that this might be very distracting for those who do not wish to be specialists in analysis, or to those who are seeing analysis for the first time. However, for someone who has finished, say Baby Rudin, this book IS AMAZING. His chatty 'foreshadowing' is the best part, since by now you are trying to see the 'big picture'. In this respect, the chattiness tells of the shortcomings of the previous theory and points one to the right questions to ask. ]
[When I first began using this book, I felt uncomfortable, since the tone of the author was so casual and might I say unprofessional.]
Axler people said that it was the closest thing in style, like if Spivak did a textbook on linear algebra [not sure if that's true or not]
most seem to think axler is better than average but not superb, but it's easy to read for an abstract linear book and good for self-study.
I just think to myself is that all Har 55 is, basically cramming Hardy's Pure Mathematics and Hoffman and Kunze asap into someone who wants to read 7 other books at the same time [and not the most friendly or approchable supplementary readings either]
I think 750 hours could be stretched out, so no one drops out... and well Binmore's book starts off easy enough and tells you a pretty good list of what to read in his three books and when to tackle Royden.
- So don't believe what it says in the catalog about some of these courses and wonder why you and I were not able to deal with them, when all we had was the actual stated prerequisites.
what i would like to see is someone who's done a syllabus from the 1920s-1970s for all the big schools... some of the schools in the 50s actually would print the name of the textbook used in the calendar for a class...
i got a lot of neat insights looking at all the AJP Transcripts of famous physics people and teachers and listening to what textbooks they had in school or what they taught from..
found out Slater who was popular for writing first and second year textbooks in physics in the 30s and 40s, got his math from
EB Wilson - Advanced Calculus - Ginn 1912
all 566 pages of that.
and that's probably the oldest textbook of *any* use to people today....
mind you, sometimes that stuff is fragmentary
I think he used Osgood's mechanics, which is like Macmillian 1937, so maybe that's what he taught from before writing one in the 40s [Slater and Frank] which basically got pushed out by Synge and Griffith and later Symon.
[i found out Synge and Griffith was used in the 40s and 50s at Cornell because my copy i picked up in the used bookstore said PHY xxx Cornell 1950 in it] which is about the closest i got to Bethe or Feynman....
Slater used Abraham and Becker for Electromagnetism [1932 translation] and i still wonder why the Part II in German didnt get translated as well...
Slater also used James Jeans - The Mathematical Theory of Electricity and Magnetism - Cambridge 1925 5ed - for his EM classes
Leighton who worked with Feynman on the lectures went through Smythe's Static and Dynamic Electricity - McGraw-Hill 1939
so it's a neat thing to see a fragmentary picture of what people used in uni back then, or taught from...
Still not sure what feynman used for his high school or calculus physics, but it was probably what he could 'find', and he was still jumping from Math to Electrical Engineering to finally Physics as the happiest balance between theory and applied...
|Oct7-12, 09:48 AM||#3308|
If one were interested in taking a look at "Elements of Algebra" by Euler, what translation/version would you recommend?
|Oct7-12, 10:58 PM||#3309|
since you seem to read english, i suggest this english version:
|Oct8-12, 12:48 AM||#3310|
best write up on Euler's book is
Euler's Elements of Algebra
Leonhard Euler, edited by Chris Sangwin
paperback - 276 pages (2006)
Tarquin Books $22
[The style is engaging; the structure and language is clear, and the explanations logical. The approach is surprisingly modern and does not suffer either from being nearly 250 years old, or from being an edited version of a "charming" English translation from the 19th century. In fact, this English text comes from an 1822 English translation of a French translation of the original German. That such writing can still be called clear and readable is something of a miracle, and testament to Euler's original clarity and readability. This edition has excised various later accretions such as editors' footnotes and introductions, as well as an entire chapter added by Lagrange, material which may be reproduced if a reprint of Part II of Euler's work is ever attempted.]
[For me, the mystery of this old school textbook, which doesn't hold your hand and so seems to lead you rapidly through a ton of material, is that so much is conveyed in a spare, clean style. In fact, I expect that more material is covered than in more modern textbooks which spend an age going over and over material, and yet books like Elements seem less hurried than modern ones.]
[For example, Euler's definition of the integers seems to exclude zero. Later, he gives good reason to suppose that there is an infinity of numbers between two integers, but he couldn't know of the different "sizes" of those infinities which Georg Cantor discovered, and which a brief note might bring alive. He also anticipates the great utility of imaginary numbers. An index would also increase the usability of the book, especially for those interested in the history and development of mathematical concepts.]
[Overall, the book is to be highly recommended. The broad range of elementary topics means the book can and should be referred to often. The structure, readability, and standard of explanations lead to a rapid and rewarding learning experience, while the elegance of the prose is frankly a joy to read. The book soothes ageless anxiety caused by learning the mysteries of logarithms and imaginary numbers and yet does not shy away from addressing practical problems, even how to calculate interest — a footnote on the dangers of credit cards would go well here.]
I'm not yet sold on it, anyone wanna twist my arm?
A few years ago Springer in 3 vols did his calculus text, finally translated in English, seemed interesting enough off amazon for me to dump it in my 'neat' list.... people seemed to like it browsing at what was essentially the first textbook on calculus...
[hold on let me drag it out]
32 Foundations of Differential Calculus - Leonhard Euler - Springer - $70
[The First calculus texts]
[more intuition than formalism]
33 Introduction to Analysis of the Infinite: Book I (Books 1 + 2) - Leonard Euler - Springer - $105
34 Introduction to Analysis of the Infinite: Book II - Leonard Euler - Springer - $90
If you got $275 kicking around... but it's probably a better and weirder read than new copies of Stewart or Thomas and Finney.
|Oct8-12, 01:59 AM||#3311|
Heres a page I found some time ago. I can't comment on the quality of translations, but it contains lots of old works of math translated to english for free. Like Eulers "calculus" books.
|Oct14-12, 03:57 PM||#3312|
|Oct14-12, 08:04 PM||#3313|
i am puzzled. the copy of euler i have linked contains hundreds of exercises.
|Oct15-12, 12:01 AM||#3314|
- However, if one's purpose is to study and learn mathematical substance/skills from the book, it seems to lack of exercises. Do you perhaps know of any supplementary text/exercises to the book?
I think you're putting a modern question to a rather old book... Some people actually didnt like textbooks that tossed in a ton of problems, thinking the easy ones are just 'confidence builders' and this stuff are merely 'drills'... Yet the trend in the 50s and 60s and 70s were to put out new editions of textbooks with 30% more problems in the newer editions... [which what happenned with Resnick and Bueche's physics texts going into the 70's.]
There's a reason Schaum's outlines were popular...
and if you like problems there's always Chrystal's Textbook of algebra
Yet there's two good essays out there
Albrecht Heefer has some neat comments about the book:
[now remember Euler's book is from 1770]
"In his selection of problems in the Algebra, Euler shows himself familiar with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. An extensive historical database with algebraic problems, immediately reveals Euler’s use of the Stifel’s edition of Rudolﬀ ’s Coss for his repository of problems. This work, published in 1525 in Strassburg, was the ﬁrst German book entirely devoted to algebra."
"Stifel used many problems from Rudolﬀ in his Arithmetica Integra of 1544 and found the work too important not to publish his own annotated edition. The ﬁrst volume of Euler’s Algebra on determinate equations contains 59 numbered problems. Two thirds of these can be directly matched with the problems from Rudolﬀ."
"The third chapter dealing with linear equations in one unknown has 21 problems. They clearly show how Euler successively selected suitable examples from Rudolﬀ’s book. The problems are put in practically the same order as Rudolﬀ’s. They include well-known problems from recreational mathematics, ...the legacy problems, two cups and a cover, alligation, division and over- taking problems. The fourth chapter deals with linear problems in more than one unknown, including the mule and a-s-s problem, doubling each other’s money and men who buy a horse."
"The ﬁfth chapter is on the pure quadratic equation with ﬁve problems all taken from Rudolﬀ. The sixth has ten problems on the mixed quadratic equation, of which nine are taken from Rudolﬀ. Chapter eight, on the extraction of roots of binomials, has ﬁve problems, none from Rudolﬀ. Finally, the chapter of the pure cubic has ﬁve problems, two from Rudolﬀ and on the complete cubic there are six problems, of which four are from Stifel’s addition. Cardano’s solution to the cubic equation was published in 1545, between the two editions of the Coss. While Euler also treats logarithms and complex numbers, no problems on this subject are included."
"Having determined the source for Euler’s problems, the question remains why he went back almost 250 years. The motive could be sentimental. In the Russian Euler archives at St-Petersburg a manuscript is preserved containing a short autobiography dictated by Euler to his son Johann Albrecht on the ﬁrst of December, 1767. He states that his father Paulus taught him the basics of mathematics using the Stifel edition of Christoﬀ Rudolﬀ’s Coss. The young Euler practiced mathematics for several years using this book, studying over four hundred algebra problems. When he decided to write an elementary textbook, Euler conceived his Algebra as a self study book, much as he used Rudolﬀ’s Coss, the educational value of which Euler amply recognized."
"Arithmetic books before the 16th century use a great many recipes to solve a wide variety of problems. With the emergence of symbolic algebra in the second half of the 16th century, many of these recipes became superﬂuous and the corresponding problems losttheir appeal. Several types of problems disappeared from arithmetic and algebra books for the next two centuries. The algebra textbooks of the eighteenth century abandoned the constructive role of problems in producing algebraic theorems. Problems were used only to illustrate theory and practice the formulation of problems into the algebraic language. The new rhetoric of problems in algebra textbooks explains why Euler found in Rudolﬀ ’s Coss a suitable repository of examples."
"A typical example of this type of problems is a legacy problem, which emerged during the late Middle Ages and is found in Fibonacci’s Liber Abbaci. It is a riddle about a dying man who distributes gold pieces to an unknown number of children, each receiving the same amount. With i children, each child gets ai plus (1/n)^th of the rest. The question is how many children there are and what the original sum is."
"After Euler, many of the textbooks on elementary algebra of the 19th century include this and other problems from Rudolﬀ as excercises. In this way, Euler’s Algebra functioned as a gateway for the revival of Renaissance recreational problems."
Christoff Rudolff’s influence
"In his selection of problems in the Algebra, shows himself familiar with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. Taking up the task of tracing the sources of these problems I found a strong similarity with the books by Valentin Mennher de Kempten. Originating from Kempten, in the south of Germany, Mennher was a reckoning master living in Antwerp. He published several books on arithmetic and algebra in French. His Arithmetique seconde, first published in 1556, has a large section with problems which are very similar to these of Euler’s Algebra. A close comparison shows that many problems from Euler could be reformulations of the text and values of Mennher’s problems. A German translation was published in Antwerp in 1560 for the German market. Possibly it circulated in Berlin where Euler might have been charmed by its pedagogical qualities. Still, why would Euler base his examples on a two-centuries old book from Antwerp, with so many alternatives at his disposal?"
"Lacking the crucial motive, I looked at later publications for the missing link. The eighteenth-century algebra treatise which matches Euler’s Algebra best is A Treatise of Algebra by Thomas Simpson (1745). This book was also indented as an elementary work in algebra, treating the basic operations on polynomials. It also has a large section on the resolution of equations as well as a chapter on indeterminate problems. Simpson’s book became highly succesful as ten editions were released in the UK from 1745 to 1826 and at least three editions in the US from 1809. However, there are only about twenty problems which can directly be matched between Simpson’s and Euler’s books. In fact, Simpson’s problems show a better correlation with Mennher than with Euler."
"Recently, a digital version of Stifel’s edition of Rudolff’s Coss has become available. A fist glance reveal immediately evident that Euler used this book for his repository of problems. The original edition was the first German book entirely devoted to algebra."
"It was published in 1525 in Strassburg under the title 'Behend vnnd Hubsch Rechnung durch die kunstreichen regeln Algebre so gemeincklich die Coss genennt werden'. Stifel used many problems from Rudolff in his own Arithematica Integra of 1544 but found the work too important not to publish his own annotated edition in 1553, 'Die Coss Christoffs Rudolffs mit schonen Exempeln der Coss'."
"Having determined the source for Euler’s problems, the question remains about his motive for going back almost 250 years. The motive could be sentimental. In the Russian Euler archives at St-Petersburg is preserved a manuscript containing a short autobiography dictated by Euler to his son Johann Albrecht on the first of December, 1767 (Fellmann 1995). He states that his father Paulus Euler taught him the basics of mathematics with the use of the Stifel edition of Christoff Rudolff’s Coss (Stifel, 1553). The young Euler practiced mathematics for several years using this book, studying over four hundred algebra problems."
"When he decided to write an elementary textbook on algebra, he must have had in mind the first mathematics book he owned. The book was to be used for self study, in the same way that he had used Rudolff’s book. As the many examples from Rudolff had helped Euler to practice his algebraic skills, so would he also include many aufgaben related to the resolution of equations. So while the motivation to use a sixteenth-century book may have been partly sentimental, the recognized educational value of algebraic problem solving was an important contributing factor."
"Given that Euler’s Algebra is separated from Rudolff’s Coss by more than two centuries of algebraic practice, the structure of both works is rather close."
"Rudolff treats the same subjects but his organization reflects more the tradition of medieval algorisms. For each of the different species, whole numbers, fractions, etc, he first gives the numeration and then discusses the possible operations which he calls algorithms. The rest of Rudolff’s book consists of eight sections on the eight rules of algebra. These correspond with linear equations, the six Arab types of quadratic equations and the cubic equation with only the cube term. A division into eight equations is a
simplification of the 24 types given by Johannes Widman (Codex Leipzig 1470). As the subdivision of quadratic equations in separate rules disappeared in the early seventeenth century, Euler’s arrangement is different. He has separate sections on linear problems in one unknown, linear equations in multiple unknowns, the pure quadratic equation, the mixed quadratic, the pure cubic and the complete cubic equation."
"The third chapter dealing with linear equations in one unknown has 21 problems. They clearly show how Euler sequentionally selected suitable examples from Rudolff’s book. The problems are practically in the same order as in Rudolff (1553)."
"The fifth chapter is on the pure quadratic with five problems all taken from Rudolff. The sixth has ten problems on the mixed quadratic equation, of which nine are taken from Rudolff. Chapter eight, on the extraction of roots of binomials, has five problems, none from Rudolff."
"Finally, the chapter of the pure cubic has five problems, two from Rudolff and on the complete cubic there are six problems, of which four are from Stifel’s addition. While Euler also treats logarithms and complex numbers, he included no problems on this subject."
"The English edition of John Hewlett adds 51 ‘problems for practice’. It is not clear where they originate from, as they do not appear in the French edition (Euler 1774). It seems doubtful that the bible translator Hewlett (1811) added the problems himself. In any case, they were not selected by Euler."
There ya go...
|Oct15-12, 12:13 AM||#3315|
I think this part is one of the more interesting parts in the Euler paper, showing the origins of some of the problems and how we approached them....
Phases in rhetoric development of treatises on algebra - The medieval tradition
"One of the first Latin problem collections found in the Western world is attributed to Alcuin of York under the title Propositiones ad Acuendos Juvenes or Problems to Sharpen the Youth. The text dates from around 800 and consist of 53 numbered problems with their solution. As an example let us look at problem 16 on Propositio de duobus hominibus boves ducentibus, appearing twice in the
Two men were leading oxen along a road, and one said to the other: “Give me two oxen, and I’ll have as many as you have.” Then the other said: “Now you give me two oxen, and I’ll have double the number you have.” How many oxen were there, and how many did each have?
Solution. The one who asked for two oxen to be given him had 4, and the one who was asked had 8. The latter gave two oxen to the one who requested them, and each then had 6. The one who had first received now gave back two oxen to the other who had 6 and so now had 8 which is twice 4, and the other was left with 4 which is half 8.
"The rhetorical structure of these problems is that of a dialogue between a master and his students and is typical for the function of quaestiones since antiquity. Rhyme and cadence in riddles and stories provided mnemonic aids and facilitated the oral tradition of problem solving. Many of the older problems are put in verse. Some best known examples are 'Going to St-Yves' using the geometric progression 7 + 7^2 + 7^3 + 7^4, (Tropfke 1980). We know also many problems in rhyme from Greek epigrams19 such as Archimedes cattle problem (Hillion and Lenstra, 1999), the a-s-s and mule problem from Euclid (Singmaster, 1999) and age problems (Tropfke 1980). During the Middle Ages complete algorisms were written this way, taking over 500 verses (Karpinski and Waters, 1928; Waters, 1929). Even without rhyme, problems were cast into a specific cadence to make it easier to learn by heart. The 53 problems of Alcuin clearly show a character of declamation, specific for the medieval system of learning by rote. Medieval students were required to calculate the solution to problems mentally and to memorize rules and examples. The solution depends on precepts, easy to remember rules for solving similar problems, and adds no explanation."
"The structure of a problem as a dialogue between master and student is also explicitly present in early Hindu mathematical writings. These treatises consist of long series of verses in which a master challenges a student with problems. An example from the Ganitasarasangraha of Mahavīra is as follows:
(Padmavathamma and Rangacarya 2000, stanza 80 1/2):
'Here, (in this problem,) 120 gold pieces are divided among 4 servants in the proportional parts of 1/2 , 1/3 , 1/4 and 1/6. O arithmetician, tell me quickly what they obtained.'
The student is addressed as friend, arithmetician or learned man and is defied in solving difficult problems. In one instance, Brahmagupta states in his Brahmasphutasiddhanta of 628 AD that (Colebrooke 1817):
He, who tells the number of [elapsed] days from the number of days added to past revolutions, or to the residue of them, or to the total of these, or from their sum, is a person versed in the pulverizer.
Thus someone who is able to solve this problem on lunar revolutions, should have memorized the verses describing the Kuttaka or pulverizer method for solving indeterminate problems. Literally stated, the memorization of the rules formulated in stanzas by the master is a precondition for problem solving. Hindu algebra is based on the reformulation of problems to a format for which a memorized rule can be applied. The rhetorical function of problems in medieval, as well as Hindu texts, is to provide templates for problem solving which can be applied in similar circumstances.
Aint that cool?
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