Should I Become a Mathematician?

In summary, to become a mathematician, you should read books by the greatest mathematicians, try to solve as many problems as possible, and understand how proofs are made and what ideas are used over and over.
  • #3,291


- i want to be a mathematician, but i don't hve the natural talent for it..

In your last course, did you do 'every problem' in each chapter?I used to be skeptical, of the advice i once got, but you can likely get a B almost all the time if you burn 10-15 hours a week on the problems...

and without any talent i think you could crank through and pass 80% of any math class for a degree, if not more...

heck if you burn 300 hours on a complete textbook, maybe you'll create a 'toolbox' for talent... -----

skill is what comes with practice, start small, wring out 101% out of one chapter of your math or physics book... flip a coin and try the next chapter later on...don't rush a textbook, and don't cheat yourself not doing 90% of the problems. If you can read the whole book, do all the problems, all it takes is a enormous amount of time...

but what you do learn will be pretty damn solid.
 
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  • #3,292


"What was Mike Spivak's inspiration for his calculus and diff geometry books?"

We would need to ask Mike this, but he went to Harvard where they used Courant and Hardy when I was there. Also when I taught from his book I noticed some of the proofs were similar to ones used in Courant, so my personal conjecture then was that Courant was at least one inspiration for his Calculus.

We really should ask him this question though.

The diff geom book is much more ambitious and comprehensive, and from reading it, it seems to be inspired by the original sources it references, such as Gauss and Riemann, perhaps Weil for the local proof of Poincare duality. I have not read many of the more modern sources, but they seem to include Cartan,... (Mike was also Milnor's student.)
 
  • #3,293


- We would need to ask Mike this, but he went to Harvard where they used Courant and Hardy when I was there. Also when I taught from his book I noticed some of the proofs were similar to ones used in Courant

Hardy would be a rough ride if you had to go through it quickly...then again any analysis text is...if you need to push through half in 12-15 weeks

I always wondered what people used from the middle 60s onwards, when courant was still going strong, was Apostol an instant classic when it came up like in 57-58 or did it take years for it to catch on?

It would be great if you could recall all the textbooks you went though during your undergrad years, i know you trickled bits and pieces here through the years...of the main ones...- The diff geom book is much more ambitious and comprehensive, and from reading it, it seems to be inspired by the original sources it references, such as Gauss and Riemann, perhaps Weil

actually that was the most interesting part of the Brandeis links you tossed me about 8 of the 22 or so faculty had homepages with texts, and it was the differential geometry courses that most impressed me, and i actually had one more textbook to add to my list...

damn, Wulf something an oxford text...and it was interesting to see how they would use three books in tandem, and i only knew of two of the books being used 'together'...

so it was nice to see what books people can read at roughly the same time when taking a first or second class...

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a. Brandeis in some classes used Erdmann's book on Lie Algebras, since all you need to tackle that textbook is linear algebra. And *then* then go into Humphreys and Fulton.

[though some feel that to tackle humphreys it's best to read a. Herstein b. Hoffman and Kunze, which are both considerably harder and would take a lot of time]

b. Brandeis used Lang's Real and Functional Analysis text with Zimmer at the same time

c. Brandeis for a second class in Real Analysis did a. Lang's Real and Functional Analysis b. Rudin's Real and Complex Green book c. Real Analysis by Kolmogorov and Fomin - Dover

d. surprised they used the 70s early 80s Mardsen and Tromba for vector calculus, which is generally well disliked, but if you put a ton of effort into it or supplement it, it's better. To the plus it's got detailed explanations, doesn't shove definitions at you, has meaningful illustrations, but it can be a confusing text and the cause of many many headaches. And it's not the smoothest for self-study either... [my guess is people read thomas and finney and marsden and tromba together]

e. Brandeis uses Fraleigh and Gallian for Abstract Algebra [my notes show it's an easier hop to start with Fraleigh, then read Artin and then read Dummitt]

f. Brandeis for Topology uses Hatcher-Greenberg-Munkres together, likely after the main Munkres text]

g. Brandeis also used Hocking for Topology, though i think for the course only do chapter 1 2 and then hop to 5. It's a clear book and good for self-study i hear.

h. Rolfsen's book on Knots and Links [Harvard would use cromwell as a main text, and then supplement it with Livingstone/Gilbert/[and more lightly]/Burde/Rolfsen/Kauwach]

i. Brandeis for Diff Geo I goes a. Spivak b. Warner c. Milnor d. Bott and Tu

j. Brandeis for Diff Geo II goes a. doCarmo [Riemann Geometry] b. more Spivak c. Milnor and Stasheff d. Roe

k. [yet other courses for Diff Geo II use doCarmo and use Petersen's Riemannian book and Lee's Riemannian book with Warner]

l. If you're reading Warner's book Foundations of Differential Manifolds and Lie Groups, Wulf Rossmann's Oxford book is great to read with it. [and well ideally you'd need to read a. Lee b. the other Lee book c. a bit of Messay d. a bit of Boothby e. a bit of Warner] So basically one new book on the list when one tackles Lee's Introduction to Topological Manifolds and Lee's Introduction to Smooth Manifolds. Assuming i actually finish a topology and differential geometry book that is...

m. Brandeis also liked Pressley's Differential Geometry book which is an easy read, much like Erdmann's on Lie Algebra...

n. one odd thing for Diff Geo was a. Gallot-Hullin-Lafontaine b. Spivak c. Milnor. I would assume for a second course...

o. they are big fans of D'angelo's proof book which seems friendlier than most of the others, which probably helps people later on so they don't go to pieces with Analysis...

p. Falcon's Fractal Geometry book, which is one of the best ones out there was used in some second year grad school course on 'Hausdorf Dimension'...

q. Markov Chains - they use Norris with Lawler as two of the main texts sometimes [though they seem to like adding miserable textbooks on financial economic stuff on the reading list too, since it's trendy or a good career option] Mind you, Hoel's books and Lawler look like real gems [I'm all for the great 70s books still in print by Howard Mifflin in Statistics and Stochastic processes,like Hoel's without any need for being in the 27th edition either, though they are pricey, they are still IN PRINT, and they are easy gentle reads.] [yup Mifflin and Hoel did a great job in the early 70s comming out with an easy book on probability, an easy book on statistics and an easy book on stochastic processes, all in print with no need for useless updating either to look new. They are fine as they are. I think Stanford in the 70s was big on those three textbooks]

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Anyhoo, that's some of my notes, and some of the stuff you helped me out with by suggesting the Brandeis teacher homepage links... much appreciated!
 
  • #3,294


RJinkies said:
***This is RJinkies quote to Mathwonk, I cut the rest off***

It would be great if you could recall all the textbooks you went though during your undergrad years, i know you trickled bits and pieces here through the years...of the main ones...

I would also be really interested. Though I'm just learning the basics now on my slow pace, but it would be really interesting resource.

Infact, it would be great if you someday would like to write more of a guide to learning mathematics. Like from elementary to end of undergraduate studies. Like about the order of topics (e.g. when would be useful to study topology and other things like that..), and which are good books for people with different talents. And books for those of us who really have no talent at all but study it anyway for the fun of it..

Of course it would be a lot of work and I already appreciate all the help you have given. This whole thread was really interesting to read but the info is little scattered around.. : )
 
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  • #3,295
when i was at harvard the instructors gave book recommendations, but never followed them. the books were just for your outside reading. the lecturers composed their own version of the material. to be honest most of the time the lectures were significantly better than anything in the best books, but not always as detailed.

freshman calc: john tate, recommended texts: Courant, and G.H Hardy: Pure Mathematics, and Foundations of Analysis by E. Landau.

sophomore algebra: garrett birkhoff, text: survey of modern algebra by birkhoff and maclane.

sophomore calculus: i forget what book, maybe Taylor, but the book was apparently chosen by a committee and the professor was contemptuous of at least some of it, e.g. lack of proof of implicit functiion theorem.

sophomore diff eq. herman gluck; text: earl coddington. the best part of this course occurred at the end, when prof gluck departed from the routine stuff in the book and presented a beautiful proof of the existence theorem for solutions of first order ode's, by the contraction lemma for complete metric spaces.

sophomore algebra, instructor newcomb greenleaf, texts: linear algebra and matrix theory, by evar nering; fundamental concepts of higher algebra by a. adrian Albert, galois theory by emil artin. also notes by Andrew Gleason on linear algebra available from the dept.

complex variables, text by Henri Cartan. except when taught by Ahlfors, who used his own book.

junior:
advanced calculus: lynn loomis, official text: calculus of several variables by wendell fleming, but the lectures followed more closely the book Foundations of modern analysis by Jean Dieudonne'; including sturm liouville theory, supplemented by lectures on content theory and a lovely presentation of vector geometry via the group of motions in intrinsic euclidean geometry. much of this course is now recorded in the book by Loomis and Sternberg.

another reference text for this course was advanced calculus by spencer, steenrod, and nickerson,.

a very useful course on introductory analysis taught by george mackey with no text. a good book now on related material is his text on complex analysis: Lectures on the theory of functions of a complex variable.

senior:
real analysis taught by lynn loomis, no textbook, it covered abstract measure theory as in the book of Halmos, and some Banach algebras and stone weierstrass theorem. much too abstract to be really useful. some of my friends who understand the material now recommend the book by zygmund and wheeden at least for the integration theory.

algebraic topology taught by raoul bott, text: algebraic topology by spanier. most people today recommend the book by allen hatcher.

I want to remind that i did not learn much from this somewhat harsh and user unfriendly first exposure to significant mathematics. but i admit those who worked harder did.

very few of my courses in graduate school even recommended any texts at all, everything was in the lectures. the only books even referred to in any grad school course were the hand written notes on algebraic topology and differential forms by friedlander, griffiths, and morgan, the book course in arithmetic by serre, and some seminars read the books on several complex variables by gunning and rossi, and by hormander, and the lectures on riemann surfaces by gunning, and the book Topology from the differentiable viewpoint, by John Milnor.when i began recovering my math career after my first unsuccessful attempt at mastering it from some of the books and courses mentioned above, i learned more from spivak's calculus and calculus on manifolds, and frederick greenleaf's book on one complex variable. i also read lectures on algebraic topology by marvin greenberg, and books by chinn and steenrod, and william massey. i also liked hurewicz' book on ode, and hurewicz and wallman on dimension theory, kelley on general topology, and lang's analysis I. modern algebra by van der waerden was also frequently helpful but not always, as was algebra by lang.

I personally find it hard to find many algebra books that are really user friendly, but there is one significant exception, the book Algebra by Michael Artin is quite wonderful. Many people recommend Dummitt and Foote and it does have many good qualities, but I have some criticisms of it.

I do not like books that are written to show off how clever the author is rather than to make the material look easy and clear, and many algebra books seem to fail this test to me, along with books like rudin's analysis.

Virtually everyone likes Geometry of Algebraic Curves, vols I and II, for that more specialized subject, carefully written over 30 years by, (in the interest of full disclosure, my good friends), E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris.
 
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  • #3,296


@mathwonk:

I was wondering, does one need to get into a "super" university like Harvard to learn a lot of math, or what?
 
  • #3,297
no. the advantages of harvard are a deservedly good reputation and lots of money, hence they attract a strong interesting student body, outstanding professors, high quality living quarters, a prestigious reputation for the school and its degree, expensive equipment and facilities, opportunities available in boston, horrible winters...oops that's a negative.

harvard does not necessarily offer better advice to people struggling, or more personal attention.

one learns by hard work on material that has been clearly presented. if you actually read the books listed above that my harvard instructors recommended but that i did not read, you will obviously be miles ahead of me and many other harvard students.
 
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  • #3,298


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...

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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...

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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 wouldn't really recommend... [when you clarified things a while later]

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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


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- 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...

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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]


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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]

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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.]

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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 ...

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[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.]


and...

[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]

and

[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.]

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- almost all algebra books seem to fail this test to me

[high school or abstract?]

a. Gallian
b. Fraleigh
c. Beachy and Blair
d. Allenby
e. Saracino
f. Pinter
g. Childs

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]

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- 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 definitely 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'


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I added three books to the list too..

Nering is a new one...
Mackey's complex text
and Arbarello...

your stories definitely do get better the more we hear them mathwonk!
much appreciated
 
  • #3,299


well here's 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.
 
  • #3,300


mathwonk said:
well here's 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.

This needs to be said.
 
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  • #3,301


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.
 
  • #3,302


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 definitely 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...

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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.]
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[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.]
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[Wonderfully Masterful]
[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.]

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Anyhoo it is surprising that Har would use in the early 60s a mainstream calculus text that wanted a minimum of proofs...

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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']

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I'm thinking

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*
 
  • #3,303
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 serious student, but some books reach out to the clueless.
 
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  • #3,304


- 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'...
 
  • #3,305
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 so 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.
 
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  • #3,306


- 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
a. Rudin
b. Carothers
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...
 
  • #3,307


If one were interested in taking a look at "Elements of Algebra" by Euler, what translation/version would you recommend?
 
  • #3,308


since you seem to read english, i suggest this english version:

'http://archive.org/details/elementsalgebra00lagrgoog
 
  • #3,309


hi Dowland
hi mathwonk

best write up on Euler's book is

http://plus.maths.org/content/eulers-elements-algebra

------

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 want to 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.
 
  • #3,310


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.

http://www.17centurymaths.com/
 
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  • #3,311


mathwonk said:
since you seem to read english, i suggest this english version:

'http://archive.org/details/elementsalgebra00lagrgoog[/QUOTE]
I like Euler's writing style and his exposition of the subject. I also think Elements of Algebra contains a lot of interesting stuff that standard textbooks in Algebra does not contain. 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?
 
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  • #3,312


i am puzzled. the copy of euler i have linked contains hundreds of exercises.
 
  • #3,313


- 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

http://logica.ugent.be/albrecht/thesis/AlgebraRhetoric.pdf
http://logica.ugent.be/albrecht/thesis/EulerProblems.pdf

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 successful 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 you go...
 
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  • #3,314


Euler II
---------

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
Patrologia Latina"

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.

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Aint that cool?
 
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  • #3,315


Euler III
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Phases in rhetoric development of treatises on algebra - The abacus tradition

"While the medieval tradition of riddles or problems with standard recipes was carried through to sixteenth-century arithmetic books, a new tradition of algebraic problem solving emerged in Renaissance Italy. The Catalogue by Warren van Egmond (1980) provides ample evidence of a continuous thriving of algebraic practice from the fourteenth till the sixteenth century."

"Over two hundred manuscripts provides an insight in the practice of teaching the basics of arithmetic and algebra to sons of merchants in the abacus schools of major towns in Renaissance Italy. The more skilled of these abacus masters drafted treatises on algebraic problem solving in the vernacular."

"These consist typically of a short introduction on the basic operations on polynomials and the rules for solving problems (resolving equations). The larger part of these treatises is devoted to the algebraic solution of problems. We can state that the algebraic practice of the abacus tradition is the rhetorical formulation of problems using an unknown. The solution typically depends on the reformulation of the problems in terms of the hypothetical unknown. The right choice of the unknown is half of the solution to the problem. Once the several unknown quantities are expressed in the rhetorical unknown, the analytic method consists of manipulating the polynomials and applying the rules of algebra (resolution of equations) to the point of the resolution of a value for the unknown."

As an example of the rhetoric of algebraic problem solving let us look at the major abacus master of the fourteenth century, Antonio de’ Mazzinghi (Problem 9, Arrighi 1967):

Italian: Truova 2 numeri che, multiplichato l’uno per l’altro, faccino 8 e i loro quadrati sieno 27

English: Find two numbers which, multiplying one with the other gives 8, and [adding] their squares gives 27. After the problem text is given, the solution typically starts with the hypothetical definition of an unknown: “Suppose that the first quantity is one cosa”. The skill of abacus master and the elegance of the problem-solving method depends mostly on the clever choice of the unknown. Maestro Antonio not only was skilful in this, he also was the very first to introduce multiple unknowns for solving difficult problems in an elegant way.

"Ma per aguagliamenti dell’algibra anchora possiamo fare; e questo è che porremo che lla prima quantità sia una chosa meno la radice d’alchuna quantità, l’altra sia una chosa più la radice d’alchuna quantità. Ora multiplicherai la prima quantità in sè et la seconda quantità in sè et agugnerai insieme et araj 2 censi et una quantità non chonosciuta, la quale quantità non chonosciuta è quel che è da 2 censj infino in 27, che v’è 27 meno 2 censj, dove la multiplichatione di quella quantità è 13 1/2 meno i censo."

------

Instead of using the cosa for one of the numbers, or two unknowns for the two numbers, Maestro Antonio here uses

x-sqr(y) and x+sqr(y).

Squaring these two numbers gives

x^2-2x*sqr(y)+y and x^2+2x*sqr(y)+y respectivelyAdding them together results in 2x^2+2y, which is equal to 27.

The auxillary unknown thus is 13 1/2 - x^2.

-------

"This text fragment from the end of the fourteenth century is exemplary for the abacus tradition. Algebraic practice consists of analytical problem solving. The rhetorical structure depends on the reformulation of the given problem in terms of the cosa and applying the analytical method to arrive at a value for the unknown. The unknown quantities can then easily be determined. A test subsitituting the values of the quantities in the original problem provides proof of the validity of the solution."

[More painful word problems anyone?]
 
  • #3,316


Euler IV

The Beginning of Algebraic Theory: from Pacioli to Cardano

"By the end of the fifteenth century we observe a change in the rhetorical structure of algebra treatises. While the solution to problems still remains the major focus of the texts, authors pay more attention to the introductory part. While a typical abacus text on algebra was limited to thirty or forty carta, the new treatises easily fill hundred folio’s. Two trends contribute to more comprehensive approach: the use of the algorism as a rhetorical basis for an
introductory theory and the extraction of general principles from practice."

The amalgamation of the algorism with the abacus text

"The algorism, as grown from the first Latin translations of Arab adoptions of Hindu reckoning, describes the Hindu-Arabic numerals and the basic operations of addition, subtraction, multiplication and division. In later texts we also find doubling and root extraction as separate operations. These operations are applied to natural numbers, fractions and occasionally also sexadecimal numbers. Through the influence of Boetian arithmetic, some algorisms also include sections on proportions and progressions. Whereas we find this structure also in abacus texts on arithmetic, the treatises on algebra have a different character."

"The introductory part extends on early Arab algebra with the six rules for solving quadratic problems, lengthened by some derived rules. By the end of the fifteenth century algebraic treatises also incorporate the basic operations on arithmetic and broaden the discussion on whole numbers and fractions with irrational binomials and cossic numbers. We witness this evolution in Italy as well as in Germany. The culmination of this evolution is reflected in the Practica Arithmeticae of Cardano (1539). Cardano begins his book with the numeration of whole numbers, fractions, and surds (irrational numbers) as in the algorisms. He then adds de numeratione denominationum placing
expressions in an unknown in the same league with other numbers, which is completely new."

"In doing so he shows that the expansion of the number concept has progressed to the point of accepting polynomial expressions as one of the four basic types of numbers. He further discusses the basic operations in separate chapters and applies each operation to the four types. Also, he applies root extraction to powers of an unknown in the same way as done for whole numbers (chapter 21). He continues by constructing aggregates of cossic numbers with whole numbers, fractions or surds (chapter 33 to 36).

As an example of the aggregation of cossic numbers with surds, he shows how

sqr(3) multiplied with 4x^2+5x gives sqr(48x^4+120x^3+75x^2).

Though Cardano was not the first, his Practica Arithmeticae is a prime example of the adoption of the algorism for the rhetorical structure of the new textbooks on algebra, and functioned as a model for later authors. Cossic numbers were in this way fully integrated with the numeration of the species of number and presented as the culmination of the application of the operations of arithmetic.

---------

Extracting general principles from algebraic practice

"For a second trend in the amplification of an introductory theory in algebraic treatises we can turn to Pacioli. It has long been suspected that Pacioli based his Summa de arithmetica geometria proportioni et proportionalita of 1494 on several manuscripts from the abacus tradition."

"These claims have been substantiated during the past decades for large parts of the Geometry. Ettore Picutti has shown that “all the ‘geometria’ of the Summa, from the beginning on page 59v. (119 folios), is the transcription of the first 241 folios of the Codex Palatino 577”, (cited in Simi and Rigatelli 1993). Margaret Daly Davis (1977) has
shown that 27 of the problems on regular bodies in Pacioli’s Summa are reproduced from Pierro’s Trattao d’abaco almost literally. Franci and Rigatelli (1985) claim that a detailed study of the sources of the Summa would yield many surprises. Yet, for the part dealing with algebra, no hard evidence for plagiarism has been given. While studying the history of problems involving numbers in geometric progression (GP), I found that a complete section of the Summa is based on the Trattato di Fioretti of Maestro Antonio. Interestingly, this provides us with a rare insight in Pacioli’s restructuring of old texts, and as such, in the shift in rhetorics of algebra books."Pacioli: Famme de 13 tre parti continue proportionali che multiplicata la prima in laltre dui, la seconda in
laltre dui, la terça in laltre dui, e queste multiplicationi gionti asiemi facino 78.

Maestro Antonio: Fa’ di 19, 3 parti nella proportionalità chontinua che, multiplichato la prima chontro all’altre 2 e
lla sechonda parte multiplichato all’altre 2 e lla terza parte multiplichante all’altre 2, e quelle 3 somme agunte insieme faccino 228. Adimandasi qualj sono le dette parti.In modern notation, the general structure of the problem is as follows:

x/y = y/z

x+y+z=a

x(y+z)+y(x+z)+z(x+y)=b

Maestro Antonio is the first to treat this problem and uses values a=19 and b=228. Expanding the products and summing the terms gives:

2xy+2xz+2yz=228, but as y^2=xz

we can write this also as

2xy+2y^2+2yz=228, or 2y(x+y+z)=228

Given the sum of 19 for the three terms, this results in 6 for the middle term. Antonio then proceeds to find the other terms with the procedure of dividing a number into two extremes such that their product is equal to the square of the middle term. Pacioli solves the problem in exactly the same way. However, the rhetorical structure is quite different. Maestro Antonio performs an algebraic derivation on a particular case. Instead, Pacioli justifies the same step as an application of a more general principle, defined as a general key...

"The restructuring of material and the shift in rhetoric is in itself an important aspect in the development of sixteenth-century textbooks on algebra. Pacioli raised the testimonies of algebraic problem solving from the abacus masters to the next level of scientific discourse, the textbook. When composing the Summa, Pacioli had almost twenty years of experience in teaching mathematics at universities all over Italy. His restructuring of abacus problem solving methods is undoubtedly inspired by this teaching experience. Cardano’s Practica Arithmeticae continues to build on this evolution and the two works together will shape the structure of future treatises on algebra."

-----

[now doesn't that look like part of the algebra book puzzler that mathwonk tossed at us this summer?]
[With that x(y+z)+y(x+z)+z(x+y)=b fragment!]
 
  • #3,317


Euler V
--------

Algebra as a model for method and demonstration

"The two decades following Cardano’s Practica Arithmeticae were the most productive in the development towards a symbolic algebra. Cardano (1545) himself secured his fame by publishing the rules for solving the cubic equation in his Ars Magna and introduced operations with two equations. In Germany, Michael Stifel (1544) produce his Arithmetica Integra which serves as a model of clarity and method for many authors during the following two centuries."

"Stifel also provided significant improvements in algebraic symbolism, which have been essential during the sixteenth century. He was followed by a Johannes Scheubel (1550) who included an influential introduction to algebra in his edition of the first six books on Euclid’s Elements. This introduction was published separately in the subsequent year in Paris as the Algebrae compendiosa (Scheubel, 1551) and reissued two more times. In France, Jacques Peletier (1554) published the first French work entirely devoted to algebra, heralding a new wave of French algebraists after the neglected Chuquet (1484) and de la Roche (1520)."

"Johannes Buteo (1559) built further on Cardano, Stifel and Peletier to develop a method for solving simultaneous linear equations, later perfected by Guillaume Gosselin (1577). In 1560, an anonymous short Latin work on algebra was published in Paris. It appeared to be of the hand of Petrus Ramus and was later edited and republished by Schoner (1586, 1592). The work depended on Scheubel’s book to such a measure that Ramus refrained from publishing it under his own name. In Flanders, Valentin Mennher published a series of books between 1550 and 1565, showing great skill in the application of algebra for solving practical problems."

"England saw the publication of the first book treating algebra by Robert Recorde (1557). This Whetstone of witte was based on the German books of Stifel and more importantly Scheubel. It introduced the equation sign as a result of the completion of the concept of an equation. It would take too long to review all these books. Only some general trends and changes in the rhetorical structure of the sixteenth-century algebra textbook will be discussed."

"Giovanna Cifoletti (1993) is one of the few who wrote on the rhetoric of algebra and specifically on this period. She attributes a high importance to Peletier’s restructuring of the algebra textbook. However, we have shown that
the merger of the algorism with the practical treatises of the abacus tradition was initiated by the end of the fifteenth century, culminating in Cardano (1539)."

"This trend cannot be attributed to Peletier, as proposed by Cifoletti. On the other hand, Peletier was an active participant in the humanist reform program which aimed not only at language and literature but also at science publications. His works on arithmetic (1549), algebra (1554) and geometry (1557) make explicit references to this program and reflections on the rhetoric of mathematics teaching. Cifoletti (1993) demonstrates how Peletier intentionally evokes the context of the author as the classical Orator in order to approach a textbook from the point of view of rhetoric. He rebukes on the demonstration of mathematical facts by his predecessors, explicitly referring to Stifel and Cardano. His ideal model for mathematical demonstration is exemplified by the rules of logic represented under the form of a syllogism. In his introduction to Euclid’s Elements he considers the application of syllogisms in mathematical proof as analogous with that of an lawyer at the court house, the rules of rhetoric:

"Que si quelqu’un recherche curieusement, pourquoi en la démonstration des propositions ne se fait voir la forme du syllogisme, mais seulement y apparoissent quelques membres concis du syllogisme, que celui là sache, que ce seroit contre la dignité de la science, si quand on la traite à bon escient, il falloit suivre ric à ric les formules observées aux écoles. Car l’advocat, quand il va au barreau, il ne met pas sur ses doigts ce que le Professeur en rhétorique lui a dicté: mais il s’étudie tant qu’il peut, encore qu’il soit fort bien recours des preceptes de rhétorique, de faire entendre qu’il ne pense rien moins qu’à la rhétorique."

[it's interesting how you can skim through it pretty easily seeing three words stick out: advocat/syllogisme/rhetorique]

-------

So, how did Peletier apply his understanding of rhetoric in his Algebre? Cifoletti (1993) points at the contamination of the rhetorical notion of quaestio and the algebraic notion of problems, initiated by Ramus and Peletier, and fully apparent in the Regulae of Descartes. She goes as far as to identify the algebraic equation with the rhetorical quaestio (Cifoletti 1993):

"But I also think that from the point of view of the history of algebra, so crucial for later theoreticians of Method, 'quaestio' has played a fundamental role because it has allowed consideration of the process of putting mathematical matters into the form of equations in a rhetorical mode.'

"In Cicero’s writings, the quaestio is an important part of rhetorical theory. He distinguishes between the 'quaestio finita', related to time and people, and the 'quaestio infinita', as a question which is not constrained. The quaestio finita is also called causa, and the alternative name for quaestio infinita is propositum, related to the aristotelian notion of thesis. Cicero discerns the two types of propositum, the first of which is propositum cognitionis, theoretical, and the second is propositum actions, practical. Both these types of quaestio infinita have their role in algebra as the art addresses both theoretical and practical problems."

--------

"I believe the rhetorical function of algebra recognized by the authors cited above, is contained more in the development of algebraic symbolism, than in the changing role of quaestio. I have argued elsewhere that the period between Cardano (1539) and Buteo (1559) has been crucial for the development of the concept of the symbolic equation."

"The improved symbolism of Viète, and symbols in general, are the result, rather than the start, of symbolic reasoning. It is precisely Cardano, Stifel, Peletier and Buteo who shaped the concept of the symbolic equation by defining the combinatorial operations which are possible on an equation. The process of representing a problem in a symbolic mode and applying the rules of algebra to arrive at a certain solution, have reinforced the belief in a mathesis universalis. Such a universal mathesis allows us not only to address numerical problems but possibly to solve all problems which we can formulate."

"The thought originates within the Ramist tradition as part of a broader philosophical discussion on the function and method of mathematics, but the term turns up first in the writings of Adriaan Van Roomen (1597). The idea will flourish in the seventeenth century with Descartes and Leibniz. A mathesis universalis is inseparably connected with the newly invented symbolism. As Archimedes only needed the right lever to be able to lift the world, so did the new algebraist only need to formulate a problem in the right symbolism to solve it. Nullum non problema solvere, or “leave no problem unsolved” as Viète would zealously write at the end of the century. Much has been written on the precise interpretation of Descartes’ use of the term. The changing rhetoric of algebra textbooks at the second half of the sixteenth century gives support to the interpretation of Chikara Sasaki, in which mathesis universalis can be considered as algebra applied as a model for the normative discipline of arriving at certain knowledge. This is the function Descartes describes in Rule IV of his Regulae. Later, Wallis (1657) uses Mathesis Universalis as the title for his treatise on algebra and includes a large historical section discussing the uses of symbols in different languages and cultures. As a consequence, the study of algebra delivers us also a tool for reasoning in general."

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  • #3,318


Euler VI
---------

The generalization of problems to propositions

"For the modern reader of sixteenth-century algebras, it is difficult to understand why it took so long before algebraic problems became formulated in more general terms. Many of the textbooks of mid-sixteenth century contained hundreds of problems often of similar types intentionally dispersed over the pages. It is evident that someone who can solve the general case, can all individual problems belonging to that case. What is more, the need for generality was duly recognized. For example Cardano (1545) writes “We have used this variety of examples so that you may understand that the same can be done in other cases” (Witmer 1968)."

"There is a specific historical reason for the lack of generality. By 1560, the algebraic symbolism was developed to a point where multiple equations of higher degree could be simultaneously formulated without ambiguities."

"One crucial aspect was missing: the tools for the generalization of the values of the coefficients. This required the generalization of the concept of an equation to a general structure which can be approached under different circumstances. It was Viète who initiated the shift from the solution of problems to the study of the structure of equations and transformations of equations."

"Let us look at one example as an illustration of the importance of the new symbolism for coefficients. In the In Artem Analyticem Isagoge, Viète (1591) studies several problems with numbers in GP, as did Cardano and Stifel before him. The latter two construct equations in order to solve specific instances of problems with numbers in GP. On the other hand, Viète is interested in the relationship between the properties of numbers in GP and the structure of the quadratic and cubic equation. He investigates the circumstances in which one can be transformed into the other."

...

"Before Viète this crucial property of this quadratic equation could not be represented. Viète therefore introduced the use of the vowels A, E, I, O and U to represent unknowns, and the use of consonants for the constants and coefficients of an equation."

"However, others after Viète show the inclination to reformulate classic problems in more general terms. Christopher Clavius, the great reformer of mathematics teaching, published his own Algebra in Rome in 1608.
Unexpectedly, he ignores most of the achievements and improvements in symbolism of the second half of the sixteenth century and goes back to Stifel’s Arithmetica Integra as a model for structure and for most of his large problem collection."

"This method of generalization is completed by Jacques de Billy (1643) who treats no less than 270 problems on numbers in GP in his Nova Geometriae Clavis Algebra. For each problem, he gives a general formulation, a construction method, an algebraic derivation and a general canon. de Billy abandons the terms ‘problem’ and quaestio and instead uses propositio. The general formulation of problems thus allows him to dispose of problems altogether and move to general propositions which constitute a new body of mathematical theory."

"The generalization of problems thus achieved more than one had hoped for. Not only did it provide a solution method to all problems of that type. Also it constituted a body of mathematical knowledge that could be referred to in a rhetorical exposition, strengthening its persuasive power."

-----------

An attempt at an axiomatic theory

"The method of de Billy, of generalizing problems and turning their solution into canons which are universally applicable and to be used in the derivation of other propositions, was taken over by a new wave of algebraists in England. Despite of the fact that he published only a concise introduction to algebra in French (1637) and the Latin treatise on numbers in GP (1643), de Billy was well appreciated in England."

"The books were not issued again in France. In England however, William Leybourn (1660) added a translation of de Billy’s 'Abrégé des préceptes d'algèbre' as the fourth part of his Arithmetic, first published in 1657. This popular work was reprinted several times up to the eighteenth century. But it is de Billy’s other work which influenced the rhetoric of English algebra textbooks in the later half of the seventeenth century. In England, the need for rigor in the demonstration of algebraic reasoning was felt more directly. The prime model for truthful reasoning was, without doubt, Euclidian geometry, constructing theorems which follow from axioms by deductive reasoning."

"Before the seventeenth century, algebra was considered a practice, performed by those skilled in the art. It required experience and knowledge of many rules, which had their own name such as the regula alligationis. The idea of a universal mathesis rendered knowledge of such rules superfluous."

"Algebra was basically not different from geometry or arithmetic (Wallis 1657). Algebra starts from simple facts which can be formulated as axioms. All other knowledge about algebraic theorems can be derived from these axioms by deduction. John Wallis introduced the term axioms in relation to algebra in an early work, called Mathesis Universalis, included in his Operum mathematicorum (1657). With specific reference to Euclid’s Elements, he gives nine Axiomata, called communes notationes, referring to the function of symbolic rewriting.

1 Due eidem sunt aequalia, sunt et inter se aequalia
if A = C and B = C then A = C

2 Si aequalibus aequalia addantur, tota sunt
if A = B then A + C = B + C

6 Quae eiusdem sunt dupliciae sunt inter se aequalia
2A = A + A

7 Quae eiusdem sunt dimidia, sunt aequalia inter se
A/2 = A – A/2

etc etc...

"Some years later, John Kersey (1673, Book IV) expanded on this and formulated 29 axioms “or common notions, upon which the force of inferences or conclusions, about the equality, majority and minority of quantities compared to one another, doth chiefly depend”. Although using many more axioms, he basically reformulates those from Wallis. The method of constructing theorems or canons and the belief in the infallibility of the chain of reasoning becomes apparent from Kersey’s explication of the difference between the analytic and the synthetic approach in the introduction (Kersey 1673):

'Algebra which first assumes the quantity sought, whether it be a number or a line in a question, as if it were known, and then, with the help of one or more quantities given, proceeds by undeniable consequences, until that quantity which at first was but assumed or supposed to be known, is found to some quantity certainly known, and is therefore known also.'

'Which analytical way of reasoning produceth in conclusion, either a theorem declaring some property, proportion or equality, justly inferred from things given or granted in a proposition, or else a canon directing infallibly how that may be found out or done which is desired; and discovers demonstrations of the certainty of the resulting theorem or canon, in the synthetical method, or way of composition, by steps of the analysis, or resolution.'

"The quote is an excellent example to illustrate how the rhetoric of the algebra textbooks in the second half of the seventeenth century adopts of the Euclidian style of demonstration."

------

"The attempt to grasp the foundations of algebraic reasoning in basic axioms, was pursued until the early eighteenth century. Before Euler in Germany, the most influential writer of textbooks on mathematics was Christian Wolff (1713-1715). His Elementa matheseos universae was originally issued in two volumes. The first one treats the traditional disciplines arithmetic, algebra, geometry and trigonometry. A later addition added a wide variety of practical mathematics, from optics and astronomy to fortification and pyrotechnics. With the Basel edition this standard textbook was enlarged to five volumes, reprinted and adapted several times in the eighteenth century (Wolff, 1732). Immanuel Kant owned a copy of the first edition and was intimitaly acquainted with Wolff’s work (Warda 1922).The book had an important influence on Kant’s conception of the synthetic a priori in his Critique of Pure Reason (Shabel, 2003). Especially Kant’s view on the role of algebra in symbolic construction, as based on the manipulation of geometrically constructible objects, is strongly influenced by the way Wolff conceived algebra."

The part on algebra in the Elementa was also published separately in a Compendium (Wolff, 1742) and translated into English (Wolff, 1739) and German. Wolff starts his Compendium with an introduction to the methodo mathematica describing the axiomatic method. In the introduction to arithmetic, preceding the algebra, he gives eight axioms “on which the general way of calculation is founded”, corresponding with these of Wallis (1657). He adds (Wolff, 1739):

'The delivering of these may seem superfluous, but it will be found that they are of great help to the understanding of Algebra, giving a clear idea of the way of reasoning that is used therein.'

"While the axioms define the basic properties of quantities and, as such, belong to the realm of arithmetic, they are considered functional for the study of algebra. Wolff deals with many problems, always formulated in the general way, leading to a general solution and illustrated by a numerical example. The solution is often presented as a theorem."

"While the axiomatic approach was abandoned in the most common textbooks after Euler, the attempts by Wallis, Kersey and Wolff extented into the nineteenth century through some lesser-know works. Perkins (1842) lists ‘four axioms used in solving equations’. Ingrid Hupp (1998) studied a tradition of three university professors teaching mathematics at the university of Wurzburg. Franz Huberti (1762), Franz Trentel (1774) and Andreas Metz (1804) all continued Wolff’s approach to express the essentials of algebra and arithmetic by axioms."

"Their motivation may have been more didactical than in pursuance of a mathesis universalis. The axiomatic method brings rigor, clarity and brevity to the mathematical discipline, all too much inundated by numerous individual rules and recipes. Metz uses these properties of the axiomatic structure of algebra explictly as an argument to include it in an elementary textbook on arithmetic (Metz 1804)."

"While Wolff defined axioms but never used them in his Algebra, Huberti and to a larger extent Trentel and Metz, occasionally apply the axioms in derivations (Hupp 1998)."

"Though the axiomatic method, found in algebra textbooks until the early nineteenth century does not match the standards of mathematical logic emerging in the late nineteenth century, the axiomatic model of Euclidian geometry is used rhetorically to arrive at “undeniable consequences”. The purpose of algebra moves from the solution of numerical problems to the construction of a body of certain mathematical knowledge formulated by means of theorems and derived by rigorous deduction. Importantly, problems are the main instrument in this rhetorical transition. The whole body of knowledge, in the form of theorems, is derived from generalized problems. The changing role of problems has facilitated the rhetorical transition of algebra textbooks."

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  • #3,319


Euler VIII
---------

The generalization of problems to propositions

"For the modern reader of sixteenth-century algebras, it is difficult to understand why it took so long before algebraic problems became formulated in more general terms. Many of the textbooks of mid-sixteenth century contained hundreds of problems often of similar types intentionally dispersed over the pages. It is evident that someone who can solve the general case, can all individual problems belonging to that case. What is more, the need for generality was duly recognized. For example Cardano (1545) writes “We have used this variety of examples so that you may understand that the same can be done in other cases” (Witmer 1968)."

"There is a specific historical reason for the lack of generality. By 1560, the algebraic symbolism was developed to a point where multiple equations of higher degree could be simultaneously formulated without ambiguities."

"One crucial aspect was missing: the tools for the generalization of the values of the coefficients. This required the generalization of the concept of an equation to a general structure which can be approached under different circumstances. It was Viète who initiated the shift from the solution of problems to the study of the structure of equations and transformations of equations."

"Let us look at one example as an illustration of the importance of the new symbolism for coefficients. In the In Artem Analyticem Isagoge, Viète (1591) studies several problems with numbers in GP, as did Cardano and Stifel before him. The latter two construct equations in order to solve specific instances of problems with numbers in GP. On the other hand, Viète is interested in the relationship between the properties of numbers in GP and the structure of the quadratic and cubic equation. He investigates the circumstances in which one can be transformed into the other."

...

"Before Viète this crucial property of this quadratic equation could not be represented. Viète therefore introduced the use of the vowels A, E, I, O and U to represent unknowns, and the use of consonants for the constants and coefficients of an equation."

"However, others after Viète show the inclination to reformulate classic problems in more general terms. Christopher Clavius, the great reformer of mathematics teaching, published his own Algebra in Rome in 1608.
Unexpectedly, he ignores most of the achievements and improvements in symbolism of the second half of the sixteenth century and goes back to Stifel’s Arithmetica Integra as a model for structure and for most of his large problem collection."

"This method of generalization is completed by Jacques de Billy (1643) who treats no less than 270 problems on numbers in GP in his Nova Geometriae Clavis Algebra. For each problem, he gives a general formulation, a construction method, an algebraic derivation and a general canon. de Billy abandons the terms ‘problem’ and quaestio and instead uses propositio. The general formulation of problems thus allows him to dispose of problems altogether and move to general propositions which constitute a new body of mathematical theory."

"The generalization of problems thus achieved more than one had hoped for. Not only did it provide a solution method to all problems of that type. Also it constituted a body of mathematical knowledge that could be referred to in a rhetorical exposition, strengthening its persuasive power."

-----------

An attempt at an axiomatic theory

"The method of de Billy, of generalizing problems and turning their solution into canons which are universally applicable and to be used in the derivation of other propositions, was taken over by a new wave of algebraists in England. Despite of the fact that he published only a concise introduction to algebra in French (1637) and the Latin treatise on numbers in GP (1643), de Billy was well appreciated in England."

"The books were not issued again in France. In England however, William Leybourn (1660) added a translation of de Billy’s 'Abrégé des préceptes d'algèbre' as the fourth part of his Arithmetic, first published in 1657. This popular work was reprinted several times up to the eighteenth century. But it is de Billy’s other work which influenced the rhetoric of English algebra textbooks in the later half of the seventeenth century. In England, the need for rigor in the demonstration of algebraic reasoning was felt more directly. The prime model for truthful reasoning was, without doubt, Euclidian geometry, constructing theorems which follow from axioms by deductive reasoning."

"Before the seventeenth century, algebra was considered a practice, performed by those skilled in the art. It required experience and knowledge of many rules, which had their own name such as the regula alligationis. The idea of a universal mathesis rendered knowledge of such rules superfluous."

"Algebra was basically not different from geometry or arithmetic (Wallis 1657). Algebra starts from simple facts which can be formulated as axioms. All other knowledge about algebraic theorems can be derived from these axioms by deduction. John Wallis introduced the term axioms in relation to algebra in an early work, called Mathesis Universalis, included in his Operum mathematicorum (1657). With specific reference to Euclid’s Elements, he gives nine Axiomata, called communes notationes, referring to the function of symbolic rewriting.

1 Due eidem sunt aequalia, sunt et inter se aequalia
if A = C and B = C then A = C

2 Si aequalibus aequalia addantur, tota sunt
if A = B then A + C = B + C

6 Quae eiusdem sunt dupliciae sunt inter se aequalia
2A = A + A

7 Quae eiusdem sunt dimidia, sunt aequalia inter se
A/2 = A – A/2

etc etc...

"Some years later, John Kersey (1673, Book IV) expanded on this and formulated 29 axioms “or common notions, upon which the force of inferences or conclusions, about the equality, majority and minority of quantities compared to one another, doth chiefly depend”. Although using many more axioms, he basically reformulates those from Wallis. The method of constructing theorems or canons and the belief in the infallibility of the chain of reasoning becomes apparent from Kersey’s explication of the difference between the analytic and the synthetic approach in the introduction (Kersey 1673):

'Algebra which first assumes the quantity sought, whether it be a number or a line in a question, as if it were known, and then, with the help of one or more quantities given, proceeds by undeniable consequences, until that quantity which at first was but assumed or supposed to be known, is found to some quantity certainly known, and is therefore known also.'

'Which analytical way of reasoning produceth in conclusion, either a theorem declaring some property, proportion or equality, justly inferred from things given or granted in a proposition, or else a canon directing infallibly how that may be found out or done which is desired; and discovers demonstrations of the certainty of the resulting theorem or canon, in the synthetical method, or way of composition, by steps of the analysis, or resolution.'

"The quote is an excellent example to illustrate how the rhetoric of the algebra textbooks in the second half of the seventeenth century adopts of the Euclidian style of demonstration."

------

"The attempt to grasp the foundations of algebraic reasoning in basic axioms, was pursued until the early eighteenth century. Before Euler in Germany, the most influential writer of textbooks on mathematics was Christian Wolff (1713-1715). His Elementa matheseos universae was originally issued in two volumes. The first one treats the traditional disciplines arithmetic, algebra, geometry and trigonometry. A later addition added a wide variety of practical mathematics, from optics and astronomy to fortification and pyrotechnics. With the Basel edition this standard textbook was enlarged to five volumes, reprinted and adapted several times in the eighteenth century (Wolff, 1732). Immanuel Kant owned a copy of the first edition and was intimitaly acquainted with Wolff’s work (Warda 1922).The book had an important influence on Kant’s conception of the synthetic a priori in his Critique of Pure Reason (Shabel, 2003). Especially Kant’s view on the role of algebra in symbolic construction, as based on the manipulation of geometrically constructible objects, is strongly influenced by the way Wolff conceived algebra."

The part on algebra in the Elementa was also published separately in a Compendium (Wolff, 1742) and translated into English (Wolff, 1739) and German. Wolff starts his Compendium with an introduction to the methodo mathematica describing the axiomatic method. In the introduction to arithmetic, preceding the algebra, he gives eight axioms “on which the general way of calculation is founded”, corresponding with these of Wallis (1657). He adds (Wolff, 1739):

'The delivering of these may seem superfluous, but it will be found that they are of great help to the understanding of Algebra, giving a clear idea of the way of reasoning that is used therein.'

"While the axioms define the basic properties of quantities and, as such, belong to the realm of arithmetic, they are considered functional for the study of algebra. Wolff deals with many problems, always formulated in the general way, leading to a general solution and illustrated by a numerical example. The solution is often presented as a theorem."

"While the axiomatic approach was abandoned in the most common textbooks after Euler, the attempts by Wallis, Kersey and Wolff extented into the nineteenth century through some lesser-know works. Perkins (1842) lists ‘four axioms used in solving equations’. Ingrid Hupp (1998) studied a tradition of three university professors teaching mathematics at the university of Wurzburg. Franz Huberti (1762), Franz Trentel (1774) and Andreas Metz (1804) all continued Wolff’s approach to express the essentials of algebra and arithmetic by axioms."

"Their motivation may have been more didactical than in pursuance of a mathesis universalis. The axiomatic method brings rigor, clarity and brevity to the mathematical discipline, all too much inundated by numerous individual rules and recipes. Metz uses these properties of the axiomatic structure of algebra explictly as an argument to include it in an elementary textbook on arithmetic (Metz 1804)."

"While Wolff defined axioms but never used them in his Algebra, Huberti and to a larger extent Trentel and Metz, occasionally apply the axioms in derivations (Hupp 1998)."

"Though the axiomatic method, found in algebra textbooks until the early nineteenth century does not match the standards of mathematical logic emerging in the late nineteenth century, the axiomatic model of Euclidian geometry is used rhetorically to arrive at “undeniable consequences”. The purpose of algebra moves from the solution of numerical problems to the construction of a body of certain mathematical knowledge formulated by means of theorems and derived by rigorous deduction. Importantly, problems are the main instrument in this rhetorical transition. The whole body of knowledge, in the form of theorems, is derived from generalized problems. The changing role of problems has facilitated the rhetorical transition of algebra textbooks."

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  • #3,320


Euler IX
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Practicing the algebraic language

"Taking the body of algebraic knowledge for granted, the rhetoric of problems in algebra textbooks shifts again during the eighteenth century. Newton’s Arithmetica universalis is a good example. From the inventory of his library we know which books he owned on algebra and arithmetic (Harrisson 1978). The two copies of Oughtred’s Clavis (1652, 1667) and the standard work of Kersey (1673) appear to be the most influential on the Arithmetica universalis. Helena Pycior describes how John Collins persistently tried to find and publish an algebra textbook in English suitable for use at universities (Pycior 1997). The only existing algebra in 1660 was Oughtred (1652) and this abstruse Latin work was not considered appropriate to expound on the algebraic achievements of the seventeenth century. Looking at foreign textbooks Collins found the Algebra of Gerard Kinckhuysen (1661) best suited for the task."

"He had the book translated into Latin and asked Newton in 1669 to write a commentary. Although Newton was very critical of Kinckhuysen, especially on the lack of generality in problem solutions, he would use several of his problems in his own Arithmetica universalis published three decades later. Newton’s introduction on the difference between the synthetic and analytic method echoes that from Kersey, cited above. He also follows Oughtred’s Clavis in the view of algebra as leading to universal thruth. Everything derived through algebra can be considered a theorem."

"Although Newton recognizes the universality of the method, he does not use axioms with respect to algebra, as done by Kersey. Also, problems have a very different role in Newton’s Arithmetica. In Kersey’s Algebra the theorems are formulated as the result of problem solving. Newton uses far less problems than in algebra textbooks before him and they serve no function in the construction of a body of theory. The sixteen numbered problems on arithmetic are given as an illustration and for practicing the algebraic language:

'Let the learner proceed to exercise or put in practice these operations, by bringing
problems to aequations and lastly, let him learn or contemplate the nature and
resolution of aequations.'

The function of problems in Newton’s textbook is thus a complete shift from previous works on algebra. Also, the nature of the problems is different. Newton includes problems which were not seen again since the first half of the sixteenth century. Take for example the following simple arithmetical problem
(Newton 1720, 71):

'Problem IV: A person being willing to distribute some money among some
beggars, wanted eight Pence to give three Pence a piece to them; he therefore
gave to each two Pence, and had three Pence remaining over and above. To find
the number of beggars.'

"Using x for the number of beggars, the sum of money equals 3x – 8 when giving three each or 2x + 3 when giving two each. Both these expression are equal, so x = 11. The generalization of this problem to a theorem would be trivial and is not the function of problems in Newton’s Arithmetica. These problems only serve the purpose of practicing the art of “translating out of the English, or any other tongue it is proposed in, into the algebraical language, that is, into characters fit to denote our conceptions of the relations of quantities” (Newton 1720). In fact, the changing function of problems allowed Newton to incorporate this problem again in a textbook. This problem, better known in the formulation of handing out figs to children, was popular during the Middle Ages and the Renaissance. It probably originated from Hindu sources and was traditionally solved by a recipe, as formulated in the Bija-Ganita of Bhaskaracarya (c. 1150, Colebrook 1817).With the general form

ax+b=cx-d=y

it can be solved as

y=(ad+bc)/(a-c)

as well by x=(b+d)/(c-a).

"Both solutions appear as separate recipes in Medieval sources. These problems functioned as vehicles for the transmission of arithmetical recipes before the advent of algebra. It is one of Widman’s many rules called regula augmenti et decrementi (Widman 1489). The problems appeared in the sixteenth century for the last time in Mennher (1550). After that, such simple problems were not interesting enough to be included in the program of the French algebraists of constructing a body of mathematical theory from algebraic problem solving. With the changing rhetoric of problems in the eighteenth century, simple problems reaffirm their function, now for exercising and practicing the new symbolism. Formulating simple problems in algebraic equations is a required deftness for eighteenth-century men of science. Algebra has turned into a language which learned men cannot afford to neglect. Problems happen to be the primary tools in textbooks to acquire the necessary skills in symbolic algebra."

The changed role of problems became the new standard in eighteenth-century textbooks. Thomas Simpson adopted the rhetoric of problems as practice in his popular Treatise of Algebra. He included a large number of recreational and practical problems popular during the Renaissance. The purpose of the many word problems is to practice the process of abstraction and to identify the essential algebraic structure of problems (Simpson 1809):

'This being done, and the several quantities therein concerned being denoted by proper symbols, let the true sense and meaning of the question be translated from the verbal to a symbolic form of expression; and the conditions, thus expressed in algebraic terms, will, if it be properly limited, give as many equations as are necessary to its solution.'

Simpson gives 75 determinate problems in the section The Application of Algebra to the Resolution of Numerical Problems. Several of these were not seen anymore in algebra textbooks of the previous century. An example is the lazy worker problem, which was very popular during the fifteenth century (“Der faule Arbeiter”, Tropfke 1980). A man receives a pence for every day he works and has to return b pence for every day he fails to turn up. At the end of a period of c days he is left with value d. How many days did he work? This simple problem leads to two linear equations in two unknowns:

x+y=c
ax-by=d

with solutions

x=(bc+d)/(a+b)

and

y=(ac-d)/(a+b)


"The early formulations of the problem often had d = 0 and applied the recipe of dividing the product bc by the sum a + b, without any explanation, let alone an algebraic derivation (e.g. Borghi 1484). It disappeared from algebra books by 1560 because it did not function within the rhetoric of that time."

While books on algebra in the sixteenth and seventeenth century were the testimonies of mathematical scholarship, new algebraic methods, from the late seventeenth century onwards, were more and more divulged in scientific periodicals as the Acta Eruditorum in Leipzig, the Philosophical Transactions in London and the Histoire de l'Académie royale des sciences in Paris. With some expections, as Cramer (1750), the algebra books of the eighteenth century are primarily intended as textbooks, as part of the mathematics curriculum. Simpson (1740) is an early example. He reintroduces simple problems much as the lazy worker again, mainly to practice the translation and interpretation of word problems. It is within this new rhetoric that we have to situate Euler’s Algebra. What Euler did not state himself, was made clear by the publisher (Euler, 1822, xxiii):

'We present the lovers of Algebra a work, of which a Russian translation appeared two years ago. The object of the celebrated author was to compose an Elementary Treatise, by which the beginner, without any other assistance might make himself complete master of Algebra.'

"The rhetoric of problems is emphasized over and over again throughout the book: ‘To illustrate this method by examples’ (Euler 1822, §609, p. 207), and ‘in order to illustrate what has been said by an example’ (§726, p. 256). Euler’s book was the most successful of all algebra textbooks ever. By appropriating the problems from the antique book of Rudolff his father used for teaching him mathematics, Euler appealed to a large audience. His lucid accounts, such as the explanation why the quadratic equation has two roots (Euler 1822, 244-248), are illustrated with practical and recreational problems to practice the translation into algebraic language."

-----

Conclusion

"The examination of algebra textbooks from the point of view of the changing rhetoric of problems provides us with some interesting insights. Different ways of presenting problems have played a crucial role in the transformation of early abacus manuscripts on algebra into the typical eighteenth-century textbook. While algebra consisted originally of problem solving only, an expansion through the amalgamation of medieval algorisms with abacus texts was the first step towards the modern textbook."

"Pacioli’s appropriation of abacus texts in his Summa initiated an important restructuring of algebraic derivations into a theoretical introduction and its application in problem solving."

"The extension of the number concept and the treatment of operations on irrational binomials and polynomials by Cardano set a new standard for algebra textbooks by his Practica Arithmeticae. Humanists such as Ramus and Peletier were inspired by the developments within rhetoric to restructure algebra books and paid more attention to the art of demonstration in algebraic derivations."

"The emergence of symbolic algebra in the mid-sixteenth century contributed to the idea of a mathesis universalis, as a normative discipline for arriving at certain knowledge. By the end of the sixteenth century the change of focus to the study of the structure of equations led to a more general formulation of problems. The solutions to general problems yielded theorems, propositions and canons, which constituted an extensive body of algebraic knowledge."

"The rhetoric of seventeenth-century textbooks adopted the Euclidian style of demonstration to provide more rigor in demonstration. The algebra textbooks of the eighteenth century abandoned the constructive role of problems in producing mathematical knowledge. Instead, problems were used only for illustration and for practicing the algebraic language. Recreational problems from the Renaissance, which disappeared from books for almost two centuries, acquired the new function of exercises in transforming problems into equations."

"Euler’s Algebra is the textbook intended for self-study par excellence, which revives many older problems. This new established role of problems in algebra textbooks explains why Euler found in Rudolff’s Coss a suitable repository of examples."

FIN

---------
---------

I decided to quote the best 15% of Albrecht's paper since it actually had some interesting things to show the influence of Hindu and Arabic puzzles and problems and al-gore-isms and al-gorithms, and how all the italians and french reworked the problems and symbols and how it led from word problems to formalistic symbolisms, and shades of Euclid, and it flopped from Newton to Euler.

Mind you, nothing is more cool than Edna Kramer's 1970 huge math history book [with a creepy black and green dustjacket] [her husband was the guy big into babylonian stuff so there's some good ancient mathematics there clearly explained] and Morris Kline's Mathematics the Loss of Certainty (1980), were my two cool books i bought new and used...The things you read up on a sunday night 10pm to midnite...
gee thanks
 
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  • #3,321


mathwonk said:
i am puzzled. the copy of euler i have linked contains hundreds of exercises.
Well, maybe there are enough exercises then. I guess I just got a little suspicious of the fact that there are no "Questions for Practice" after several topics. I probably have to take a deeper look into the material.

RJinkies, thanks for posting. A related topic might be the approach some writers of mathematical learning materials have today; problem based material rather than a rigorous exposition of the subject by the writer himself/herself. For instance, "Polynomials" by Barbaue (which is a book I have looked into and maybe aspire to get some time in the future to learn from) is an algebra book that basically gives you a bunch of exercises/problems, and it seems to me that the idea is that you more or less are expected to discover and conceive the general theory yourself. I also think that "Algebra" by Gelfand has a similar approach.
 
  • #3,322


Well remember that Cauchy was someone who looked down upon diagrams being in textbooks, and he was all for analytical rigour.

- A related topic might be the approach some writers of mathematical learning materials have today; problem based material rather than a rigorous exposition of the subject by the writer himself/herself.

On the web there's an interesting quote or two from Chrystal's book...

[George Chrystal is perhaps best known for his book on algebra. The first volume of the book, whose full title is Algebra : An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges, was published in 1886. The authors of this History of Mathematics Archive are particularly proud to be members of the Department of the University of St Andrews once led by Chrystal and they attempt to follow his example as a fine teacher of mathematics.]

"As examples of the special features of this book, I may ask the attention of teachers to chapters iv. and v. With respect to the opening chapter, which the beginner will doubtless find the hardest in the book, I should mention that it was written as a suggestion to the teacher how to connect the general laws of Algebra with the former experience of the pupil."

"In writing, this chapter I had to remember that I was engaged in writing, not a book on the philosophical nature of the first principles of Algebra, but the first chapter of a book on their consequences. Another peculiarity of the work is the large amount of illustrative matter, which I thought necessary to prevent the vagueness which dims the learner's vision of pure theory; this has swollen the book to dimensions and corresponding price that require some apology."

"The chapters on the theory of the complex variable and on the equivalence of systems of equations, the free use of graphical illustrations, and the elementary discussion of problems on maxima and minima, although new features in an English text-book, stand so little in need of apology with the scientific public that I offer none."

"With respect to the very large number of Exercises, I should mention that they have been given for the convenience of the teacher, in order that he might have, year by year, in using the book, a sufficient variety to prevent mere rote-work on the part of his pupils. I should much deprecate the idea that anyone pupil is to work all the exercises at the first or at any reading. We do too much of that kind of work in this country."

I think i bought a new Chelsea in the late 80s/early 90s for about 40 dollars as two black bricks, and in the early 90s saw some grubby Dovers that i passed on because they were too well pawed through and i wanted a tighter binding for like 12 dollars each] Not sure why they didnt keep both editions in print, i assume chelsea started printing it in the 70s after Do let theirs go out of print in the 60s. is Chelsea still going strong and are they reprinting anything at all these days? A fair bit was pretty intimidating, but I had about a dozen titles, Altschiller-Court's Geometry, Hausdorff's Set theory, Chrystal and a few others, MacDuffee's Theory of Matrices, most of the stuff was too hard for me, but i saw a deal if i bought them all in a lump.

----------

now i'd like to hear about if anyone knew of any algebra or calculus books that just had dynamite problem sets or a ton of problems...

I always thought well about the Schaum's Calculus and Advanced Calculus
[thought i wonder if they borrowed from 40s 50s calculus textbook examples]
Franklin's Calculus, and Sherman Stein's Calculus
maybe Harley Flanders...

and Dolciani and Munem's Algebra books...

those stood out for me a little bit...
and there was some Springer book of worked out calculus problems too
which if i remember in the 80s or 90s was just obscenely priced as a paperback, and should be a hardback only... considering how much use it would get for students.
- For instance, "Polynomials" by Barbaue (which is a book I have looked into and maybe aspire to get some time in the future to learn from) is an algebra book that basically gives you a bunch of exercises/problems, and it seems to me that the idea is that you more or less are expected to discover and conceive the general theory yourself.

Ed Barbeau is professor emeritus of mathematics at the University of Toronto.

27 University of Toronto, Canada
[#42 World Ranking Physics]
[#43 World Ranking Mathematics]
[#26 World Ranking Chemistry]
[#19 World Ranking Engineering Techology]

And that's a Springer textbook of his...
He is currently associate editor in charge of the Fallacies, Flaws and Flimflam column in the College Mathematics Journal.

Institutions - University of Western Ontario, University of Toronto

Alma mater - University of Toronto, University of Newcastle-upon-Tyne

---------

I added his book on my algebra list, actually...

1 Polynomials - E.J. Barbeau
[outstanding treatment of polynomials with lots of examples, it doesn't require you to know any thing beyond the average high school math]

Heck you and the others might enjoy my list, since it's got Euler, Barbeau and Gelfand on it, and well I'm not sure i would recommend them all to others, but for me, i found these interesting books to look into

-----------
Algebra
1 Polynomials - E.J. Barbeau
2 Basic Notions of Algebra - I.R. Shafarevich
3 Trigonometry for the Practical Man - J.E. Thompson
4 Algebra for the Practical Man - J.E. Thompson
5 Algebra - I. M. Gelfand - Birkhauser 2003
6 Trigonometry*by I.M. Gelfand - Birkhauser
7 Functions and Graphs - I. M. Gelfand - Dover
8 The Method of Coordinates - I. M. Gelfand - 84 pages - Birkhauser 1990/Dover
9 Algebra, Functions and Graphs - I. M. Gelfand - Birkhauser
10 Sequences, Combinations, Limits - S. I. Gelfand/Gerver/Kirillov/Konstantinov - 160 pages - Dover 1969/2002 [originally 1969]
11 Introductory Mathematics: Algebra and Analysis - Geoffrey C. Smith - Second Corrected Edition - Springer 1998 - 216 pages
12 Trigonometric Delights - Eli Maor - Princeton 1998
13 G. Wentworth and D. E. Smith - Plane Trigonometry and Tables - Fourth Edition - Ginn and Company 1943
14 J. J. Corliss and W. V. Berglund - Plane Trigonometry - Houghton Mifflin, 1950
15 Fundamental Concepts of Algebra - Bruce E. Meserve - Addison-Wesley 1953/Dover
16 Bronshtein and Semendyayev, A Guide Book To Mathematics - Zurich: Harri Deutsch 1973
17 J. B. Rosenbach and E. A. Whitman - College Algebra - Third Edition - Ginn & Co 1949
18 Algebra - 2 Volumes - Welchons and Krickenberger - Ginn 1953
19 Mathematics For High School - Elementary Functions Teacher's Commentary - SMSG - Yale 1961
20 Mathematics fo High School - First Course in Algebra Part I Student's Text - SMSG - Yale
21 Concepts of Algebra - Donald R. Clarkson - SMSG V111 - Yale 1961
22 Introduction to Matrix Algebra - Student's Text - Unit 23 - SMSG - Yale
23 Elementary Algebra - Student Textbook- Harold R. Jacobs - VHPS/WH Freeman 1979 - 876 pages
24 Vision in Elementary Mathematics - W. W. Sawyer
25 Algebra and Geometry: Japanese Grade 11 (Mathematical World, V. 10) - Kunihiko Kodaira
26 Basic Analysis: Japanese Grade 11 (Mathematical World, V. 11) - Kunihiko Kodaira
27 Mathematics 1: Japanese Grade 10 (Mathematical World, V. 8) - Kunihiko Kodaira
28 Mathematics 2: Japanese Grade 11 (Mathematical World) - Kunihiko Kodaira
29 Algebra I: Expressions, Equations, and Applications - Paul A. Foerster
30 Basic Mathematics - Serge Lang
31 Precalculus with unit circle trigonometry [no information]
32 Algebra: Structure and Method: Book I and Book II - Dolciani, Berman, and Wooton - Houghton-Mifflin 1963
33 Mathematics 6: An Award Winning Textbook from Russia - Enn Nurk and Aksel Telgmaa - 1987
34 Introductory Algebra - Tenth Edition - (Bittinger Developmental Mathematics Series) (Paperback) - Marvin L. Bittinger - Pearson/Addison-Wesley 2006 - 864 pages
35 Intermediate Algebra - Tenth Edition - (Bittinger Developmental Mathematics Series) (Paperback) - Marvin L. Bittinger - Addison-Wesley 2006 - 960 pages
36 College Algebra - Marvin L. Bittinger - Addison-Wesley 2000
37 Trigonometry: Triangles and Functions - Keedy and Bittinger - Addison-Wesley
38 Mathematics Dictionary - Fourth Edition - Robert Clarke James and Glenn James - Van Nostrand
39 Precalculus Mathematics in a Nutshell - George F. Simmons - 120 pages
40 Hall and Knight - Elementary Algebra - Second Edition - 1896 - 516 pages
41 Hall and Knight - Higher Algebra - Third Edition - 1889 - 557 pages
42 Modern Algebra, a Logical Approach - Helen R. Pearson and Frank B. Allen - Ginn 1964
[extras from parke and 2 additions i put in]
a. Peacock 1842
b. Hall and Knight I 2ed 1896 Macmillan
c. Hall and Knight II 3ed 1889 MacMillan
d. Chrystal 6ed 1900 A&C Black/Dover/Chelsea
e. Fine 1904 Ginn
f. Knebelman and Thomas 1942 Prentice-Hall
g. Ferrar I 1945 Oxford
h. Albert 1946 McGraw-Hill
i. Ferrar II 1948 Oxford
j. Welchons and Krickenberger 1953 Ginn
k. Dolciani Houghton-Mifflin 1963
l. Allen and Pearson 1964 Ginn
43 Euler's Elements of Algebra - Leonhard Euler/edited by Chris Sangwin - Tarquin Books 2006 - 276 pages
[44] Algebra and Trigonometry - Munem - Third Edition early 80s
-----------

I'm not sure if i remember anything about Fundamental Concepts of Algebra - Bruce E. Meserve, if it was approachable or more abstract algebra, or an uneasy mixture of both...

- I also think that "Algebra" by Gelfand has a similar approach.

Here's my notes on that book

------

5 Algebra - I. M. Gelfand - Birkhauser 2003
[University of Chicago uses it]
[California State University, Hayward uses it]

[Cheap, challenging, and excellent preparation for further mathematics]

[set of four books:
a. Algebra
b. Trigonometry
c. Algebra, Functions and Graphs
d. The Method of Coordinates]

[Splendid and illuminating algebra text]

[This text, which is intended to supplement a high school algebra course, is a concise and remarkably clear treatment of algebra that delves into topics not covered in the standard high school curriculum. The numerous exercises are well-chosen and often quite challenging.]

[The text begins with the laws of arithmetic and algebra. The authors then cover polynomials, the binomial expansion, rational expressions, arithmetic and geometric progressions, sums of terms in arithmetic and geometric progressions, polynomial equations and inequalities, roots and rational exponents, and inequalities relating the arithmetic, geometric, harmonic, and quadratic (root-mean-square) means. The book closes with an elegant proof of the Cauchy-Schwarz inequality.]

[Topics are chosen with higher mathematics in mind. In addition to gaining facility with algebraic manipulation, the reader will also gain insights that will help her or him in more advanced courses.]

[The exercises, which are numerous, often involve searching for patterns that will enable the reader to tackle the problem at hand. Many of the exercises are quite challenging because they require some ingenuity. Some of the exercises are followed by complete solutions. These are instructive to read because the authors present alternate solutions that offer additional insights into the problem.]

[I also highly recommend the other texts in the Gelfand School Outreach Program. They include The Method of Coordinates, Functions and Graphs (Dover Books on Mathematics), and Trigonometry. Also, to gain additional insights into the inequalities at the end of this text, the reader may wish to consult an Introduction to Inequalities (New Mathematical Library) by Edwin Beckenbach and Richard Bellman.]

[A few novelties]

[This is a good, intelligent introduction]
-------

Actually one thing to remember...

Foundations of Differential Calculus - Leonhard Euler - Springer - $70
is one of the first calculus texts, so i wonder how spooky the problems are inside...

and when and what the next textbook was to replace it...

the oldest texts that might be still useful today might be
a. Horace Lamb - An Elementary Course in Infinitessimal Calculus 3ed 1919 [corrected 3ed 1944] - I got an early 50s copy of that one, and it seems likely that was paired with Hardy's Pure Mathematics

b. . Granville Longley Smith - 1904/1946 last update
in 1904 and 1911 it was just Granville and Smith
Granville was at Gettsyburg College
and Smith, and later Longely were both at Yale.

c. Sylvanius P Thompson - Calculus made Easy - 1914 and last tweaked in the 40s, i think...
I think the macmillian edition in paperback was great with the blue white and black artwork and the chalkboard graphic, and the 40s 50s Granville are thin and small and sturdy too [i think that was before the last tidying up] and Martin Gardner from Scientific American seemed to do a totally unnecessary new edition, which i think was more a forward saying how much he liked the book and then modernizing the english in it.

I'm not really sure why people criticized the book so much at the time, I thought it was great. I know he was something like an electrical engineer in the 1890s or 1900s and was in the English Roentgen society if i recall right.

but a lot of the books before Granville's time where a bit sloppy with function talk and limit talk when Weiderstrauss and Riemann were working all that crap out. Euler i think was the intuitive sort of guy for textbooks, and Cauchy the hard *** formalist.

...
 
  • #3,323


Thank you for that Algebra-list! I have worked through the algebra part in Basic Mathematics by Lang and I am halfway through Algebra by Gelfand. However, I feel that I lack som basic skills that I want to have from elementary algebra, such as standard techniques of factoring polynomials amongst other things. I also want to go into logs, cubic equations, etc. One more book on algebra after I'm finished with Gelfand is my plan...

My attention is mainly drawn to "Algebra for the Practical Man", "Fundamental Concepts of Algebra" by Bruce E Meserve (looked it up on Amazon and it seems interesting!), Euler's book and "Polynomials" by Barbeau. What a smorgasbord.
 
  • #3,324


- Thank you for that Algebra-list!

Lets not forget the MAA list, which one day i'ld like to see the out of print books dropped [slipped back in], or the newer additions...Allendoerfer, C.B. and Oakley, C.O. Principles of Mathematics New York, NY: McGraw-Hill, 1963.
Webber, G. and Brown, J. Basic Concepts of Mathematics Reading, MA: Addison-Wesley, 1963.
Allendoerfer, C.B. and Oakley, C.O. Fundamentals of Freshman Mathematics, New York, NY: McGraw-Hill, 1965. Second Edition.
Ayre, H.G.; Stephens, R.; and Mock, G.D. Analytic Geometry: Two and Three Dimensions, New York, NY: Van Nostrand Reinhold, 1967. Second Edition.
Auslander, Louis. What Are Numbers? Glenview, IL: Scott Foresman, 1969.
* Martin, Edward, ed. Elements of Mathematics, Book B: Problem Book St.~Louis, MO: CEMREL-CSMP, 1975.
* Usiskin, Zalman. Advanced Algebra with Transformations and Applications River Forest, IL: Laidlaw Brothers, 1976.
Larson, Loren C. Algebra and Trigonometry Refresher for Calculus Students New York, NY: W.H. Freeman, 1979.
* Rising, Gerald R., ed. Unified Mathematics, Boston, MA: Houghton Mifflin, 1981. 3 Vols.
Devlin, Keith J. Sets, Functions, and Logic: Basic Concepts of University Mathematics New York, NY: Chapman and Hall, 1981.
Simmons, George F. Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry Los Altos, CA: William Kaufmann, 1981.
** Demana, Franklin D. and Leitzel, Joan R. Transition to College Mathematics Reading, MA: Addison-Wesley, 1984.
Coxford, Arthur F. and Payne, Joseph N. Advanced Mathematics: A Preparation for Calculus San Diego, CA: Harcourt Brace Jovanovich, 1984.
Seymour, Dale. Visual Patterns in Pascal's Triangle Palo Alto, CA: Dale Seymour, 1986.
* Foerster, Paul A. Precalculus with Trigonometry: Functions and Applications Reading, MA: Addison-Wesley, 1987.
** Leithold, Louis. Before Calculus: Functions, Graphs, and Analytic Geometry, New York, NY: Harper and Row, 1985, 1989. Second Edition.
* Swokowski, Earl W. Algebra and Trigonometry with Analytic Geometry, Boston, MA: PWS-Kent, 1989. Seventh Edition.
Cohen, David. Precalculus, St.~Paul, MN: West, 1984, 1989. Third Edition.
Grossman, Stanley I. Algebra and Trigonometry Philadelphia, PA: Saunders College, 1989.
*** Demana, Franklin D. and Waits, Bert K. Precalculus Mathematics---A Graphing Approach Reading, MA: Addison-Wesley, 1990.
* Kaufmann, Jerome E. College Algebra and Trigonometry, Boston, MA: PWS-Kent, 1987, 1990. Second Edition.
* Lewis, Philip G. Approaching Precalculus Mathematics Discretely: Explorations in a Computer Environment Cambridge, MA: MIT Press, 1990.
*** Demana, Franklin D., et al. Graphing Calculator and Computer Graphing Laboratory Manual, Reading, MA: Addison-Wesley, 1991. Second Edition.
* Sobel, Max A. and Lerner, Norbert. Algebra and Trigonometry: A Pre-Calculus Approach, Englewood Cliffs, NJ: Prentice Hall, 1983, 1991. Third Edition.

-----

I'm not totally satisfied with finding enough information of books from 1954-1980 yet, and I am sort of surprised at the ghost-town of old titles liked or recommended by people from the 1960s and 70s, especially considering the huge changes going on with the New Math.

I always wonder if the Dolcianis, Swokowskis, Thomas and Finneys and Stewarts choke out 95% of the other textbooks, when schools and curriculum freaks adopt something like lemmings and get into the New Math or Computer fads and shake up the math curriculum where it needs the LEAST shaking up...

to say nothing about textbook authors that get headlocked by their editors saying, oh you need limits or Newton's method, or stuff all the other textbooks contain, if you want to be 'adopted' by the people who choose the curriculum...

that sort of stuff would kill calculus made easy, or Feynman's lectures or anything 'too different' or 'too easy' or 'too much odd stuff'It almost makes me feel like there was a huge failure because there's really not a lot of the older texts fondly remembered. the MAA tries that but only goes back 'so far' and they often got a weird fetish for the computer fad books or a few that are a bit overboard on the formalism [and are sometimes disliked a fair deal as being not a great first introduction to the subject]

Mind you, I'm starting to feel that almost *all schooling* from grade 8 to second year uni, might best be done as a library project with no tests and exams, and just hand kids duotangs with reading lists and tell them
a. what book
b. what chapter

from like a choice of a dozen books...

you get a gold star for reading the chapter, and 3 gold stars for doing all the problems

and set up films each week for math chem physics...

I mean MIT has like a 24 hour a day physics channel on the tube, where you can watch one of the main guys, go through all the problems and stuff, it's like a cross between a 1964 CHEM 35mm film/PSSC film/Schaums Outline in one.

and there was in the 50s and 60s Encyclopedia Brittanica films that for obscene amounts of money you could buy films of a teacher at the blackboard, and the lab work, and you could do a whole physics 11 and 12 course for schools where they couldn't afford a teacher or lab equipment in some dinky little towns, and other courses too...

all you needed was a 35mm film projector and a shelf for 120 canisters of film

that stuff would be awesome in every library...if not every home...

[heck I'm still trying to search for the list of the CHEM films for schools, which was that textbook edited by Seaborg about 1964]
which was like the chemistry PSSC, and the text is a great read most of the time, though getting the algebraic skills seemed a bit weak.]
------


- I have worked through the algebra part in Basic Mathematics by Lang and I am halfway through Algebra by Gelfand.

What are the minuses and pluses you see with Lang or Gelfand?

- However, I feel that I lack som basic skills that I want to have from elementary algebra, such as standard techniques of factoring polynomials amongst other things.

I liked Dolciani 1964 and Munem 1982 pretty much, Munem being faster and smoother for getting the info out. You needed to work a bit harder with Dolciani i think, but i think that the earlier the chapter you started reading, the easier her book was. It's pretty effortless if you started at the beginning, but pretty difficult if you had half a course from another crappy textbook and then got thrown into the middle of Dolciani.

I feel that happens as well with Thomas and Finney, a lot of schools preferred it for Calculus III, and you get used to the homework and studying far more if you started right at the bottom.

-------

- My attention is mainly drawn to "Algebra for the Practical Man"

algebra trig and calculus are the trilogy of the 5 books to browse...

you might look at the schaum's outlines too, i think there was COLLEGE ALGEBRA which i thought was a nice browse, it was one of those darker green ones with the crinkle quilt paper... i totally lost my enthusiasm for the series when they started to use flat shiny paper and then those nasty white things with the rainbow crayon scribble eyesores.

they were always great as the BLACK and TAN books
or the BLUE and PINK and GREEN quilty books
 
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  • #3,325
for old books, (around 1900), i like goursat's course in analysis, 3 volumes, recommended by the brilliant russian mathematician, arnol'd.
 
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