Other Should I Become a Mathematician?

dustbin · Jul 7, 2012

@miglo:

Check out REU's (Research Experience for Undergraduates) to find opportunities for research. There are several sites that have lists and other resources.

@homeomorphic:

I do not find the material itself difficult... I am just having to work very hard at gaining comfort in proofs (both reading and writing). Prior to a few months ago I had never even seen or worked one out (save for my trig teacher proving the quadratic formula). I am particularly terrible at brevity... I just finished a proof using Rolle's Theorem which was two paragraphs long. I compared it with another answer to the problem which was only 3 sentences. At least my answer was correct

I just need more practice and experience.

RJinkies · Jul 7, 2012

- for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles linear algebra and diff equations )
- I am guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is.

It depends where, usually the lower is

calculus i ii iii iv
linear
diff eqs
intro to analysisbut some linear and diff equations can be considered upper if they deal with some analysis and extra stuff.

sometimes you can see the regular linear being second year [sometimes the 200 levels], yet the honours classes can be the upper division [at the 300 levels]

same with diff eqs, regular classes could be 200 level, and honours at the 300 level.

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with analysis, some like to bunch it with the calculus classes, others as a separate course of half a year or a whole year, and then your next class will be a 300 level/upper division onegeometry classes can be upper or lower too depending how intense, and some schools that do really lite courses on abstract algebra can be lower division.

often they'll use linear algebra as a prerequisite, though I am not sure it should really matter that much.

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For my money, all one needs to really focus on is
a. calculus i ii iii iv
b. basic analysis and more analysis and then probably more analysis

all the rest is filler...

and neat if you're doing good,

and a drag if it's painful... [where you're skipping too fast and not going deep enough]

heck with a super duper calculus text, and two supplementary texts maybe you got (a) and (b) both

as for proof and rigour, you can face that at any stage, first year extra hard calculus texts, second year linear with tons of abstract spaces and forcing you to generalize/do proofs, or you can get hit hard with analysis classes or abstract algebra with it...

A lot can really depend on your choice of textbooks...

and a really limited and inflexible curriculum i think is why you get people who face these 'hard' things somewhere up the ladder, and sometimes the higher the costs of education [and textbooks] the curriculum gets worse by being more bare bones...

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if i had my way, a uni would be a library where you get a duotang for all the math texts, year 1 2 3 4, and the reading lists [and options]

and a duotang for the physics texts, year 1 2 3 4, and the reading lists...

exams? what exams, doing one chapter and all the problems, is your damn exam lol

spend 40 hours on a chapter, and no not pass go, till the time clocks says 40 hours...

might take 8 years to get your degree, but it'd be like 'speed learn' in the Prisoner, 100% Entry, 100% Pass.

homeomorphic · Jul 7, 2012

I do not find the material itself difficult... I am just having to work very hard at gaining comfort in proofs (both reading and writing). Prior to a few months ago I had never even seen or worked one out (save for my trig teacher proving the quadratic formula). I am particularly terrible at brevity... I just finished a proof using Rolle's Theorem which was two paragraphs long. I compared it with another answer to the problem which was only 3 sentences. At least my answer was correct I just need more practice and experience.

The interesting thing about brevity is that one of my profs said he was really impressed with my brevity, yet I never gave any thought to it. I think it's probably because my thought process is very conceptual. In trying to understand something deeply, you usually want the simplest explanation possible. When you make that rigorous, often, the proof ends up being short. That doesn't always happen, but it's my theory as to why my proofs tended to be shorter than most people's. I tend not to plow through stuff using technical brute force. It could also be that after a while, I knew which steps I could skip when I wrote the proof down, since they were clear enough.

miglo · Jul 7, 2012

dustbin said:

@miglo:

Check out REU's (Research Experience for Undergraduates) to find opportunities for research. There are several sites that have lists and other resources.

so even though I am at a community college i can still apply for REU's? i always thought that only applied to undergraduates at universities.

dustbin · Jul 9, 2012

Miglo, some are only for undergraduates at universities. I have found some that are open to all undergrad students. When you look, just look through the requirements and such info about the application. Some do not specify, which I assume means they are open to all undergrads...

mathwonk · Jul 17, 2012

here is a problem from an 1895 high school algebra book, Treatise on Algebra, by Charles Smith:

{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. any takers? (show work.)

micromass · Jul 17, 2012

mathwonk said:

here is a problem from an 1895 high school algebra book, Treatise on Algebra, by Charles Smith:

{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. any takers? (show work.)

That was fun! Everybody should definitely try this one.

Here's my solution:

Let

\frac{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)}{a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}

be our expression. We multiply numerator and denominator by abc to get

\frac{a^3 (c - b) + b^3 (a - c) + c^3 (b - a)}{a^2(c - b) + b^2(a - c) + c^2(b - a)}

Rearranging gives us

\frac{a^3 (c - b) - a(c^3-b^3) +bc(c^2-b^2)}{a^2(c - b) - a(c^2-b^2) +bc(c-b)}

We can eliminate c-b from numerator and denominator to get

\frac{a^3 - a(c^2+bc+b^2) +bc(c+b)}{a^2 - a(c+b)+ bc}

Rearranging again gives us

\frac{a(a^2 - c^2)-bc(a-c)-b^2(a-c)}{a(a-c)- b(a-c)}

Eliminating a-c and we get

\frac{a(a+c)-bc-b^2}{a- b}

Rearranging again and we get

\frac{(a^2-b^2)+c(a-b)}{a- b}

Eliminating a-b yields

a+b+c

AnTiFreeze3 · Jul 17, 2012

I got the same thing as well, Micro.

I would show my work, but I did it on some graph paper that I had nearby.

That was fun to do though. I solved mine a little differently than you, so I might take of picture of my work and show it that way. I haven't taken the time to get Matlab or LaTeX or anything like that yet, so I don't want to just type in all of my math and have it be a huge, ugly mess.

RJinkies · Jul 17, 2012

looked at a little bit of the problem, the pattern is interesting:ax+by+cz
------------
a+b+c

not making it messy, now that's a challenge...

AnTiFreeze3 · Jul 17, 2012

micromass said:

...

Rearranging again and we get

\frac{(a^2-b^2)+c(a-b)}{a- b}

Eliminating a-b yields

a+b+c

I'm not sure that I understand this step (even though it looks very simple). I did it a different way, so the way I got to that answer was different. What I'm not seeing is how you have a-b in the denominator, yet three separate occasions of a-b in the numerator (a² - b²; and a-b), yet when you essentially cancel them out, you are somehow left with a + b + c.

In my mind, when you cancel out the a-b on the bottom with any of the three pairs of a-b on top, you are either left with:

(a-b) +c(a-b), or

(a² - b²) + c

What do I seem to be missing, or not understanding?

micromass · Jul 17, 2012

AnTiFreeze3 said:

I'm not sure that I understand this step (even though it looks very simple). I did it a different way, so the way I got to that answer was different. What I'm not seeing is how you have a-b in the denominator, yet three separate occasions of a-b in the numerator (a² - b²; and a-b), yet when you essentially cancel them out, you are somehow left with a + b + c.

In my mind, when you cancel out the a-b on the bottom with any of the three pairs of a-b on top, you are either left with:

(a-b) +c(a-b), or

(a² - b²) + c

What do I seem to be missing, or not understanding?

You know that a^2-b^2=(a-b)(a+b)

So

\frac{(a^2-b^2)+c(a-b)}{(a-b)}=\frac{(a-b)((a+b)+c)}{a-b}=a+b+c

AnTiFreeze3 · Jul 17, 2012

I'm not going to try to right this with brevity like Micro, but instead I want to explain my thought process, because I feel that I may have done something incorrect.

My solution:

{a²(1/b - 1/c) + b²(1/c - 1/a) + c²(1/a - 1/b)}
____________________________________________________________________

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)

I noticed from the start that the binomials would cancel out, so long as I was able to manipulate the problem and get them next to each other, so I didn't see a reason to get rid of the fractions, since I knew they would cancel out anyways with their respective opposites. I then simplified a, b, and c to get rid of any multiplication in the denominator, and I used the commutative property to rearrange the denominator:

{ a(1/b - 1/c) + b(1/c -1/a) + c(1/a - 1/b)}
___________________________________

{(1/a - 1/a + 1/b - 1/b + 1/c - 1/c)}

The denominator then cancels out to equal 1, so I am left with:

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}

This next step is where I have broken math. I recognized what the answer should be, but I think that I may have cheated in order to get to that final result. As a result, I did this:

{(a + b + c)(1/a - 1/a + 1/b -1/b + 1/c - 1/c)

Then, similarly as before, the fractions cancel each other out, so I was ultimately left with this:

a + b + c

I didn't peak at Micro's answer, and actually came to the correct answer myself. Regardless of that, I still feel as if that last step isn't allowed. Is it even possible to solve it correctly using the process that I used?

EDIT:

I messed it up in the first step, which is why I ended up in a situation where I couldn't correctly solve it.

jbunniii · Jul 17, 2012

AnTiFreeze3 said:

I'm not going to try to right this with brevity like Micro, but instead I want to explain my thought process, because I feel that I may have done something incorrect.

My solution:

{a²(1/b - 1/c) + b²(1/c - 1/a) + c²(1/a - 1/b)}
____________________________________________________________________

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)

I noticed from the start that the binomials would cancel out, so long as I was able to manipulate the problem and get them next to each other, so I didn't see a reason to get rid of the fractions, since I knew they would cancel out anyways with their respective opposites. I then simplified a, b, and c to get rid of any multiplication in the denominator, and I used the commutative property to rearrange the denominator:

{ a(1/b - 1/c) + b(1/c -1/a) + c(1/a - 1/b)}
___________________________________

{(1/a - 1/a + 1/b - 1/b + 1/c - 1/c)}

Can you explain what you did to get to this step? It is clearly not correct, because:

The denominator then cancels out to equal 1

Actually, it cancels to 0, not 1.

AnTiFreeze3 · Jul 17, 2012

jbunniii said:

Can you explain what you did to get to this step? It is clearly not correct, because:
Actually, it cancels to 0, not 1.

I already mentioned that I messed up the first step, and that that is what threw off my whole solution. Thanks though.

EDIT: Although, if you are curious as to what was going through my mind, I embarrassingly forgot that I needed to simplify it before I could just eliminate a, b, and c. The rest of my problems stemmed from that.

I think it was coincidental that my answer ended up being a + b + c, even after making two big mistakes. Or maybe it wasn't coincidental, and I have just inadvertently invented a new form of Algebra where you break rules until you get the answer.

mathwonk · Jul 17, 2012

Very impressive micromass and Antifreeze! nice solutions!

micromass and Antifreeze are very strong, but we can also make progress using some basic principles to help us.

Here is a hint for other possible solutions: Generalized factor theorem: if f is an irreducible polynomial, and if f = 0 implies g = 0, then f divides g. (This is a basic result in “algebraic geometry”, and generalizes the basic result that x-r is a factor if r is a root.).)

For instance, suppose a-b = 0, then what about a^3(c-b) + b^3(a-c) + c^3(b-a), does it vanish too? Then what?

Now how did we guess to try a-b=0? Recall the "rational root theorem"? It says you look for roots of form X-r by trying factors r of the "constant term.

As miromass observed, we can rewrite the top of the fraction after simplifying,

as a^3(c-b) - a(c^3-b^3) + bc(c^2-b^2). Think of this as a polynomial in a.

thus the constant term has prime factors ±b,±c, ±(c-b),±(c+b).

(also other products of these factors, possibly.)

So we should try setting a equal to those factors. e.g. a=b iff a-b = 0.

Dowland · Jul 29, 2012

I don't want to start a new topic for this question, so i post it here:

How important is (euclidean) geometry in the higher (that is at the university) mathematics education? I'm currently in high school and feel that I've barely touched the subject, only simple computations with area, proportions, and some volume problems, together with a few "angle games".

I'm thinking of maybe getting the following book: https://www.amazon.com/dp/0201508672/?tag=pfamazon01-20

But maybe it's all too much, and not so important? I've enjoyed the little euclidean geometry I've done, but if I don't have very much use of it in the basic calculus and linear algebra courses, I'll probably skip it (for now).

Thoughts on that?

(Sorry for possible language errors, english is not my native, hope it's all readable

)

mathwonk · Jul 29, 2012

its important, that's a good book: here' a cheaper one:
http://www.abebooks.com/servlet/Sea...rtby=17&sts=t&tn=elementary+geometry&x=74&y=9

and here's the all time great original version from euclid:

https://www.amazon.com/s/ref=nb_sb_...-keywords=euclid+green+lion&tag=pfamazon01-20and an excellent companion volume:

http://www.abebooks.com/servlet/SearchResults?kn=hartshorne+euclid&sts=t&x=52&y=12

dustbin · Jul 29, 2012

Mathwonk, after finishing Elementary Geometry from an Advanced Standpoint and Principles of Mathematics, what would you suggest next? I'm about halfway through A&O and Chartrand's proof book, which I should have finished up relatively soon, since most of my time has been devoted to my summer calculus class.

mathwonk · Jul 29, 2012

the natural continuation would be a strong calculus book like spivak or apostol. since you are already taking calculus that makes sense only if your course is at a lower level.

other basic topics are topology and abstract algebra.

dustbin · Jul 29, 2012

Great, that will be my plan then! I just wanted to make sure there wasn't some other basic text I should work through after these. My calculus course is taught from Stewart and is almost purely computational, which is at a significantly lower level. I have done some supplementary work/reading from Apostol, but it does not line up 100% with my course in a manner that I can concurrently work through Apostol, though. I may be taking an honors, proof-based intro to linear algebra course this fall, though.

RJinkies · Jul 29, 2012

Hi dowland...

Edwin Moise's book Elementary Geometry from an Advanced Standpoint is one of the classics of the 60s like Coxeter's Introduction to Geometry. Both are lively and fun texts, yet they both go pretty deep. Moise doesn't make it dry and boring, and he does help out with proofs as well.

I'd say from the late 60s, geometry isn't really essential for a degree anymore, but if you wanted one text for a whole year to tackle, it was Coxeter, or maybe Moise as a second choice as the one and only 'offering'...

one interesting book was Altschiller-Court.
I think it's Modern Pure Solid Geometry from 1935, which has some of the weirder problems around. Dover has reprinted two of his books, and well the 1935 one was a 60s 70s Chelsea reprint...

The Dover reprints are:
a. College Geometry
b. Mathematics in Fun and in Earnest (recreational mathematical)

and moise should be remembered for writing a good calculus book as well as a good geometry book, as well.

-------

Hi dustbin

- Mathwonk, after finishing Elementary Geometry from and Advanced Standpoint and Principles of Mathematics, what would you suggest next?

a. Some of the New Mathematical Library titles from the 60s and 70s on geometry are good elementary and not so elementary books to collect. Originally started about 1961 by Random House and then reprinted by the MAA from about 1975-now. Sure wish they didnt update them, I think the cryptology one got a new look and more material, but i like the 1960s look of the series... It's about 40-46 books now. and 5 of the books are on geometry, two by coxeter.

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other books:

b. Introduction to Geometry - Coxeter - Wiley 1960?/1969 Second Edition.
c. Fundamental Concepts of Geometry - Addison-Wesley/Dover - Bruce E. Meserve
[touches n some topology at the end]
d. A Course in Modern Geometries - Judith N. Cederberg - Springer
e. The Four Pillars of Geometry - John Stillwell - Springer
f. Lines and Curves: A Practical Geometry Handbook - Victor Gutenmacher - Birkhauser 2004
g. Geometry - Michele Audin - Springer [not an elementary textbook]
[if you took Differential Geometry with DoCarmo and Spivak [and Coxeter] then you can safely run through this book]
h. Geometry: Euclid and Beyond - Robin Hartshorne - Springer
[after the 1960s, two authors that stand out in geometry are Jacobs and Hartshorne]
i. Geometry for the Classroom - C.Herbert Clemens - Springer
[mathwonk uses clemens and hartshorne together as a substitution for Jacobs]
j. Modern Geometries - James R. Smart [5 editions of this one]
[a difficult text in places unless you took geometry in the 1960s]
[mathwonk's written a few things about this book]
k. Geometry: A Metric Approach with Models - Richard Millman and George Parker - Springer 1981/1991
[mathwonk's written about this one as well - it can get technical getting into things Euclid overlooked]
[MAA tosses this a 1 star recommendation - Geometry: Surveys]
l. Foundations of projective geometry: Lecture notes - Robin Hartshorne
m. The Foundations of Geometry and the Non-Euclidean Plane - G.E. Martin - Springer 1975
[clear and complete, explained beautifully]
[MAA - 1 star recommendation - Geometry: Euclidean and Non-Euclidean Geometry]
n. Transformation Geometry: An Introduction to Symmetry - George E. Martin - Springer 1982
[MAA - 1 star recommendation - Geometry: Polyhedra, Tilings, Symmetry]
o. Geometry - David A. Brannan and Esplen and Gray - Cambridge 1999
[one needs a first course in geometry before tackling this one]
[modern British approach - often used with Rees - Notes on Geometry - Springer]
p. Notes on Geometry - Elmer G. Rees - Springer 1983
[brannan and rees are sometimes used together]
q. Elementary Geometry - John Roe - Oxford 1993
[clean simple introduction to Euclidean Geometry and Differential Geomtry]
[people use Stillwell and Roe together]
[accessible if you already read one easy geometry textbook]
r. Lectures on Analytic and Projective Geometry - Dirk J. Struik - Addison-Wesley 1953/Dover 2011
[mentioned in the classic Parke III - under: Geometry: Analytic Geometry]
s. Beyond Geometry: Classic Papers from Riemann to Einstein - Peter Pesic - Dover
[Very interesting]
t. Geometries and Groups - V. V. Nikulin - Springer 1987
u. Geometry: Seeing, Doing, Understanding - First Edition and Third Edition - Harold R. Jacobs - WH Freeman - an 800 page monster
[mathwonk liked the first and second editions more of Jacobs, the third edition was an easier textbook, and the opinions are still mixed if the book is better or worse off]
[Jacobs did a kickass Elementary Algebra book - WH Freeman 1979 with an Escher cover, as well as Geometry:Seeing,Doing, Understanding. As well as the awesome and friendly text - Mathematics: A Human Endeavor]
[one flaw with Jacobs is that you don't really get taught proofs and that's probably best done with the more elementry but *rigorous* text - Geometry by Moise and Floyd
v. Geometry - Moise and Floyd
w. Euclidean and Non-Euclidean Geometries: Development and History - Marvin J. Greenberg
[half the book is accessible to most folks]

There you go...

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For the truly hardcore and insane you could do your own Harvard 130 - Classical Geometry course on your own in six textbooks:
a. Ryan - Euclidean and Non-Euclidean Geometry, an Analytic Approach
[short text]
b. Yaglom - A Simple Non-Euclidean Geometry and its Physical Basis
[flawed masterpiece]
c. M.K. Bennett - Affine and projective geometry
[great reference
d. Meschkowski - Noneuclidean Geometry
[short book]
e. David Hilbert - Foundations of Geometry
[looks elementary but is very subtle]
f. Euclid - The Elements
[perhaps you heard of this one]
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All my notes from the catacombs...

mathwonk · Jul 30, 2012

I especially like nikulin (and shafarevich) geometry and groups. a followup to that is a book on geometry of surfaces by John Stilllwell. Another good provocative book is Experiencing Geometry by David Henderson and sometimes Daina Taimina.

Dowland · Jul 30, 2012

@ mathwonk, RJinkies

Hi guys, thanks for the responses. Out of pure curiosity, what's so important about euclidean geometry? The mentioned book seems to go very deep, and I suspect there's much unnecessary drilling with profoundly derived techniques, if you know what I mean.

BTW: When I come to think about it, Serge Langs book "Basic Mathematics" includes a part called "Intuitive geometry", which I suspect includes much euclidean geometry. I have ordered the book mainly to learn some algebra, trigonometry, etc. Maybe the geometry the book covers is quite sufficient for now? Do you have any experience with the book? In that case, would you say that the geometry included in the book is enough to have in your luggage when entering the world of university mathematics?

micromass · Jul 30, 2012

I guess I kind of disagree with the previous posters. I don't find Euclidean geometry important enough to read an indepth book on it.

Sure, Euclidean geometry is very beautiful and trains people to use logic and proofs. As such, it is valuable. But I feel that most theorems in Euclidean geometry are not used very much in university classes. For example, if you draw angle bisectors in all the angles of a triangle, then the bisectors will intersect in one point. This is a remarkably beautiful theorem. But I have never used it in my entire college education.

However, geometry is still important. And with geometry, I mean here: coordinate geometry. Knowing about equations of lines and planes, inner products, vectors, etc. That is extremely useful stuff in college education. Also, trigonometry is extremely useful. If I were you, I would focus on these two subjects.

Basic mathematics by Lang certainly covers all of these things. So I guess it is good enough. Lang also has a geometry book though that covers more stuff (and that probably covers it in more detail).

Dowland · Jul 30, 2012

Thanks, micromass.

By "geometry" above, I was loosely referring to "euclidean geometry". Do you know how well that's covered in Lang's book?

dustbin · Jul 30, 2012

Out of curiosity...
I often hear people say that Spivak and Apostol's Calculus texts are basically introductory analysis texts. What is the difference between Spivak/Apostol and books that are specifically titled along the lines of Introductory Analysis or Introduction to Anaysis (such as Rosenlicht)?

RJinkies · Jul 30, 2012

hi Mathwonk

good to know nikuklin flows into Stillwell's Geometry of Surfaces book

i got some interesting notes/quotes for that one and i actually plopped it in book 17 under topology *grin*

Notes:
[This is the book that made me a mathematician.]
[Interesting advanced undergraduate course]
[It is an attractive mixture of topology, algebra and a smidgen of analysis.]
[The main theme here is the deep connections with complex function theory.]

------

The preceding book was Stillwell, which because of the comments and the MAA rating, is on my list of old junky books to buy one day...

-----
16 Classical Topology and Combinatorial Group Theory - John Stillwell - [Springer 1980?]
[This book is great! No book on this list coincides with my own mathematical esthetics like this one: I checked this book out this summer while I was doing research on surface topology and read it cover to cover: you'll see how geometry relates to topology relates to group theory. I wish this was my first algebraic topology book, because it's full of exciting theorems about surfaces, three-manifolds, knots, simple loops, geodesics - in other words, it's rippling with geometric/topological content intead of commutative diagrams. Let me also recommend Stillwell's book Geometry of surfaces, along the same lines.]
[an excellent guide]
[Chapter 1 is very intriguing and contains lots of ideas.]
[Chapters 2-5 were a bit slowed down by foundational issues, but now in chapters 6-8 it's all topology all the time.]
[There are many ways to destroy the soul of topology. Stillwell says in the preface: "In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams."]
[Stillwell protects us from such dangers by his emphasis on low dimensions, his insistence on the fundamental group as the best unifying tool, visualisation and illustrations, and his great respect for primary sources. The latter is reflected in frequent references and in the commented, chronological bibliography, which is very useful.]
[MAA - 1 star recommendation] - Topology: Algebraic Topology
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-------

a. Henderson is new, what book/s do you recommend before tackling it?

and

b. which book/s by Taimina would you suggest, and what's texts are good before attempting it?

-------

RJinkies · Jul 30, 2012

hi dowland...

- Serge Lang "Basic Mathematics" includes a part called "Intuitive geometry", which I suspect includes much euclidean geometry. I have ordered the book mainly to learn some algebra, trigonometry, etc. Maybe the geometry the book covers is quite sufficient for now? Do you have any experience with the book? In that case, would you say that the geometry included in the book is enough to have in your luggage when entering the world of university mathematics?

Here's my notes on the book... You could say that the book was Lang's way of saying, read this before you take math in uni-cursity.

-----
Notes:
Basic Mathematics - Serge Lang
[Do you have any gaps in your High School mathematics? Teaches basic math in an abstract way, such as by congruence.]
[Preparation for college mathematics from a mathematician's standpoint]
[Serge Lang's text presents the topics that he feels students should understand before commencing their study of college mathematics. As such, working through this text is a good way for you to supplement what you learned in high school with material that will aid you in studying mathematics in college. Therefore, I particularly recommend it for prospective mathematics majors.]
[The material in the text is well motivated and clearly presented. While Lang explains how to perform routine calculations, he focuses on the underlying structure of the mathematics. The material is developed logically and results are proved throughout the text. However, the presentation of the material is marred by numerous errors, most, but not all, of which are typographical.]
[The problems range from routine calculations to proofs. Many of the problems are challenging and some require considerable ingenuity to solve. Answers to some of the exercises are presented in the back of the text. I should warn you that if you are used to artificial textbook problems in which the correct solution is a "nice" number, you will find that is not the case here. Also, it is useful to read through the problem sets before you begin solving them so that you can do related problems at the same time.]
[The first section of the book covers algebra. Properties of the integers, rational numbers, and real numbers are examined and compared. There is also more routine material on linear equations, systems of linear equations, powers and roots, inequalities, and quadratic equations.]
[A brief discussion of logic precedes a section on geometry. Basic assumptions about distance, angles, and right triangles are used as a starting point rather than Euclid's postulates. This leads to a discussion of isometries, including reflections, translations, and rotations. Area is discussed in terms of dilations. The treatment here is different from that in the high school text Geometry which Lang wrote with Gene Murrow. I found the material on isometries quite interesting. Be aware that the notation and some of the terminology in this section is not standard.]
[The third section of the book covers coordinate geometry. Distance is interpreted in terms of coordinates. This leads to a discussion of circles. Transformations are reinterpreted using coordinates. Segments, rays, and lines are presented using parametric equations. A chapter on trigonometry covers standard topics, but also includes a section on rotations. The section concludes with a chapter on conic sections. Of particular interest is a proof that all Pythagorean triples can be generated from points on the unit circle with rational coordinates.]
[The final section of miscellaneous topics addresses functions, more generalized mappings, complex numbers, proofs by mathematical induction, summations, geometric series, and determinants. The text concludes by demonstrating how determinants can be used to solve systems of linear equations.]
[The eminent mathematicians I. M. Gelfand and Kunihiko Kodaira have also contributed to books intended for high school students. Those of you planning to study mathematics in college would benefit from working through their texts as well.]
------

So I'd place Lang with the half dozen Gelfland books [usually white and green], and the 40+ NML New Mathematical LIbrary books from the 1960s-date...

----
for contrast

Introduction to Geometry - Second Edition - Coxeter - Wiley 1969 - 485 pages

[Coxeter's introduction is a classic text. It is not a systematic account but contains a lot of material you won't easily find in one book.]
[A sweeping book on geometry by a modern master. Part IV is on differential geometry; part III includes a chapter on hyperbolic geometry.]
[This is the best book I've seen covering geometry at this level. Coxeter was known as an apostle of visualization in geometry; many other books that cover this material just give you page after page of symbols with no diagrams. He motivates all the topics well, and lays out the big picture for the reader rather than just presenting a compendium of facts. This is a survey of a huge field, but he does a great job of focusing on the most important results. As other reviewers have noted, this book is not "introductory" in the sense of high school geometry; it's introductory in the sense of being the kind of book a college math major would use in his/her first upper-division geometry course. It doesn't presuppose a great deal of mathematical knowledge, but it probably isn't a book that one could appreciate without having already developed quite a high level of mathematical maturity.]

------

I would say that want to plop into physics with differential geometry, or you like MC Escher, the four Wenninger books on building Polyhedron Models out of paper/cardboard, or want to get the Tinkertoy for professionals ZomeTools/Zomeworks...this is the book for you, and if it's too spooky, it's pretty to look at and read a few cool fragments...

I think it's the single best all in one, only book you need for 5% of folks... especially in it's day.

I think the comments for his dinky NML Book applies to his other works as well...

Geometry Revisited (New Mathematical Library)*- H. S. M. Coxeter
[Very useful for solving challenging problems in geometry]
[it has a pleasantly non-brain-dead quality to it. There are interesting geometric facts that you probably haven't seen before in here.]
[NML 19]

Now you know what the problem of geometry is, oops...
-------

I just go on how the first 3 pages speak to me, weird pictures and recommended readings [before tackling the book, or after you finish a chapter or the whole book]..

------
------hi micromass

- I guess I kind of disagree with the previous posters. I don't find Euclidean geometry important enough to read an indepth book on it.

Which is why, it's probably only tackled now as a third year optional course for 5% of math majors.Though, mathwonk makes a good case for one book:
Geometry for the Classroom - C.Herbert Clemens
"Clemens has written a very spare, absolutely elementary, and yet substantive treatment of the most important fundamental and useful parts of euclidean geometry. He has also sketched the other main geometries "sphereworld" and "hyperbolicland" in his eminently understandable yet authoritative style."
"If you need a book that starts from scratch, quickly reviews the basic intuition of elementary geometry, then passes to constructions, and only then to the idea of proofs, take a look at this little work by a world expert geometer who is deeply commited to teaching and improving teaching throughout the world."

-----

- Sure, Euclidean geometry is very beautiful and trains people to use logic and proofs. As such, it is valuable.

maybe that was the goal in the 1890-1960s, but i think the worst thing about geometry is to *use* it as a service course for set theory and logic and proofs!

one could argue that topology is a service course, being the caboose on the Analysis Train.

- But I feel that most theorems in Euclidean geometry are not used very much in university classes.

I think as the sciences broke down their 'absolutes' [Darwin and Einstein in part] it takes a while for that to hit mathematics [Godel]. Euclid in victorian england was a sacred cow, and well when you fight one force that, so much in high school geometry seems 'obvious' and a thorough treatment seems dry and almost drain dead, it sure don't help. And Neither does it when some books 'fill in the gaps' Euclid omitted, and if they weren't that *obvious* for 2000 years, you can be it's too *subtle* for students!

Some of that is addressed in Morris Kline's "Mathematics: The Loss of Certainity", or quirky mystic-philosophers like JG Bennett in his 'The Dramatic Universe' where he takes an interesting stand on uncertainity and 'hazard' being a fundamental factor in life, and he embraces 'Absolute Relativism'

But as the Icktorian world got shaken up with Euclid not being a rock solid foundation anymore, with a decline in geometry circa 1914 [maybe that was educational reform with public schools], and the Failure of the New Math [Kline's Johnny can't Add] with rigour before vigour [and tossing many post Sputnik high school teachers with a stroke, with all the weird formalism, with that anxiety filtering down into the students], you saw geometry disappear.

i think it's sort of neat that it disapppeared from grade 11 math and snuck it's way into a rarely used part of Third Year Math.

I only remember the barest of geometry in grade 5 and grade 9 and no more, cept for 25 people a year getting it in one class in grade 11.

I do wonder if it's a good 'side' course for differential geometry though
thoughts anyone?

Heck i always wondered why there weren't topology courses for people without analysis
maybe:
A Topological Picturebook - George K. Francis - Springer
Intuitive Concepts in Elementary Topology - BH Arnold - 1962

RJinkies · Jul 30, 2012

hi dustbin

- I often hear people say that Spivak and Apostol's Calculus texts are basically introductory analysis texts. What is the difference between Spivak/Apostol and books that are specifically titled along the lines of Introductory Analysis or Introduction to Anaysis (such as Rosenlicht)?

Calculus just blurs into analysis, hardy and rudin are lumped with advanced calculus like courant and kaplan. When it's elementary calculus that's the 'vigour' and when you apply the spiral approach come back to it, with advanced calculus, you add the 'rigour'. Books like Courant and Apostol just start off with a bang with both. And it helps if you started with something like syl thompson or je thompson [calculus made easy/calculus for the practical man]

And if you used older terminology, like in the early 50s, rudin could also be classed as 'functions of a real variable'

and rosenlicht is no different than rudin.

which is well analysis and you could say it turns into real analysis and real variables too...

what they are doing is overhauling what the number system is, and shaping your intuition about what functions are and what variables are... and suppossedly...magically one day you end up with a box of tools where don't fuss with trivial issues... and well one *hopes* that soon after you stop fearing mathematics, can can more often get to the heart of the problem.

So, with that emphasis of Nathan Grier Parke [guide to the literature of mathematics Dover 1957] where analysis goes... you can suppossedly save a ton of hours with those 'trivial issues' which went on a century or two before in mathematics...

and well with all that analysis you an be 'rigorous' with Fourier Series, and 'rigorous' with probability theory too. And well you can do stretchy rubber sheet geometry too, whoops topology.

dustbin · Aug 1, 2012

Thanks for your insights RJinkies. I appreciate all of your vault notes!

RJinkies said:

And well you can do stretchy rubber sheet geometry too, whoops topology.

Lol.

Nano-Passion · Aug 1, 2012

RJinkies said:

hi dustbin

- I often hear people say that Spivak and Apostol's Calculus texts are basically introductory analysis texts. What is the difference between Spivak/Apostol and books that are specifically titled along the lines of Introductory Analysis or Introduction to Anaysis (such as Rosenlicht)?

Calculus just blurs into analysis, hardy and rudin are lumped with advanced calculus like courant and kaplan. When it's elementary calculus that's the 'vigour' and when you apply the spiral approach come back to it, with advanced calculus, you add the 'rigour'. Books like Courant and Apostol just start off with a bang with both. And it helps if you started with something like syl thompson or je thompson [calculus made easy/calculus for the practical man]

And if you used older terminology, like in the early 50s, rudin could also be classed as 'functions of a real variable'

and rosenlicht is no different than rudin.

which is well analysis and you could say it turns into real analysis and real variables too...

what they are doing is overhauling what the number system is, and shaping your intuition about what functions are and what variables are... and suppossedly...magically one day you end up with a box of tools where don't fuss with trivial issues... and well one *hopes* that soon after you stop fearing mathematics, can can more often get to the heart of the problem.

So, with that emphasis of Nathan Grier Parke [guide to the literature of mathematics Dover 1957] where analysis goes... you can suppossedly save a ton of hours with those 'trivial issues' which went on a century or two before in mathematics...

and well with all that analysis you an be 'rigorous' with Fourier Series, and 'rigorous' with probability theory too. And well you can do stretchy rubber sheet geometry too, whoops topology.

Just to be completely sure, are you saying that it is better to spend your time on Parke's book rather than Spivak & Apostol?

RJinkies · Aug 1, 2012

Well, do remember that the first book on my geometry list was Jacobs because of mathwonk's comments about the different editions... and my own frustrations with books that weren't too hard or too brain dead proofy, or out of touch...

people still think the 60s dolciani geometry book with 2 other authors is a bit sterile, but a lot of the 60s books for the schools were that way...

yet it was odd how the MIT PSSC physics group was pre sputnik, and the Yale SMSG Experimental Math Thing was post sputnik.

Dolciani's algebra book in 1964 , and the Wooten/Dolciani Analysis book for high schools in the 60s [and the other geometry book, though lots of others too], were basically the flowers that bloomed from the Yale thing [probably the origin of the New Math] yet the experimental paperbacks were considered pretty damn good though not polished...

but then again the new math crashed and burned, and i think computers in schools or CAI crashed and burned too, and the whole calculators yes or no for math exams debate now.. or the rotten books that are with 30% missing and it's web content or CD roms usually missing from the books if bought used...

but the neat thing, is that half the books that are useful are old, and half the books are new, so I still think that there should be way more than chelsea, or Dover out there getting all the math and physics out there. Heck, I still wonder why McGraw Hill just doesn't crank out their classes and let them stay in print endlessly. Wiley did that with their classics but sadly as those crappy thick black paperbacks with courant and the rest...

If you can't do it as good as Dover don't do it lol
But in the long term, 95% of what people will read will be public domain...

----

Me i just wanted to make a coherent booklist for my own uses, and well when the ole Physics Faq by Vijay Fafat came out [that 1994-2005 list of books] I wanted to fill in my own books and do something similar for math.. while still struggling to find books i found thoroughly cool.

It's a *lot* harder* for math books, but that's the great thing about this place, finding out what people like, and well making the path easier...

for me, i think a math degree is just
3 calculus texts and 3 books on analysis... for 80% of it...

and if you want supplementary reading multiply by x3 x4 x5 books, so you got a library of your own...

for math you got your
high school with dolciani
and you got your calculus, with the easy and hard books - with the goal of enough there to study Vector Calculus for 15-30 weeks on your own]

and then getting up to analysis, with maybe 3 texts on it.
[Binmore/Bartle/Rudin/Apostol/Royden]

physics you got your [60s PSSC-Zacharias High School]
[Halliday-Resnick and Wolfson] for first year
[the whole 5 books of the Berkley Course Mech/EM/Waves/Quantum/Stat Mech]
[Symon and Kleppner Kolenkow for mech]
[Butkov for Mathematical Physics]
[the three books by Griffin - EM-Quantum-Particles]

[if you can get into Purcell's EM book by Berkley and Griffy's EM and QM texts, who needs anything else, you're halfway there]

and well with math, i guess it's getting to
algebra - dolciani seems to be the easiest way for mastering grade 10 11 12
vector calculus
a course on Diff Eqs
one book/two books on analysis

and the crown is one book on topology and one plastic man comic book

oddly, i had to find out about Halliday, Symon, , Purcell, Griffin, Syl Thompson, JE Thompson, Courant, Binmore all on my own

and how i think why math and physics for high school still didnt top 1965 with the Great Society Era where Dolciani and Zacharias aint been bettered]

and don't think you *need* the ratrace of the school system, or exams, as long as you know that you can pump 200 hours into a textbook, reading *all* of it, and doing *all* the problems, and putting in 8 hours a chapter [as a guideline], heck 5-10 hours lol
you don't need no teachers, or a piece of paper...

but if you want to be a shooting star and get paid, sure, do that too. Just don't let curriculum or time be your enemy.

I felt liberated when i felt that a better benchmark is self-study and completing *one* chapter, and don't get into any traps about exams, pressure, and cramming... the sooner you self-learn the more you'll get out the experience.

and it doesn't matter if you read one chapter or the whole book, or how far up the ladder you go. Just *enjoy* understanding how nature works, and be curious for life, ...being happy can be a fast paced thing, or a slow paced thing...

just be happy...

RJinkies · Aug 1, 2012

- Just to be completely sure, are you saying that it is better to spend your time on Parke's book rather than Spivak & Apostol?

oh no... Parke is just a good source for what books were considered useful for a bunch of catagories in science [mostly math/physics, some chemistry/engineering/electronics] and it didnt touch anything after Sputnik. Parke was an applied mathematician with his own laboratory and consulting firm and did a book in the 40s for McGraw-Hill with about 2500 books, and then in 1956 came out with a second edition for Do-er which was double the size with about 5000 books.
And about half of the textbooks were his own personal library for his business...

and he had people run to MIT for card catalogues and in his spare? time he came out with a pretty useful guidebook for knowing what the cool books were 1900-1955.

He goes into interesting ideas about parallel reading and how to tackle new subjects you know little about, and helpful stuff like that.

Apostol's book came out a year after Parke... and Rudin and Hardy are probably the only books people would recognize anymore...

--------

Here's a sample of Parke, though i rearranged things in chronological order...

Guide to the Literature of Mathematics and Physics - Nathan Grier Parke III 1956
Physics - Chronological - Title
- - - - - - - - - - - - - - - - - - - - - - - - -
Ganot 10 - 18ed - Elementary Treatise on Physics, Experimental and Applied [Wood, New York] - 1225 pages
Duncan 20 - 2ed - A Text-book of Physics for the Use of Students of Science and Engineering [Macmillan, London] [revised in 1948] - 1063 pages
Gerlach 28 - Matter, Electricity, Energy [Van Nostrand, New York] - 427 pages
Poynting 28 - 9ed - A Textbook of Physics: Heat [Griffin, London] - 354 pages
Poynting 29 - 12ed - A Textbook of Physics: Properties of Matter [Griffin, London] [revised with new title in 1947] - 228 pages
-----
Franklin 30 - General Physics [Franklin & Charles, Lancaster PA] - 705 pages
Pohl 30 - Physical Principles of Electricity and Magnetism [Blackie, London] - 250 pages
Jauncey 32 - 1ed - Modern Physics: a Second Course [Van Nostrand, New York] [revised with new title in 1948] - 568 pages
Pohl 32 - Physical Principles of Mechanics and Acoustics [Blackie, London] [only revised in German in 1953 - 12ed Springer] - 338 pages
Eldridge 34 - The Physical Basis of Things [McGraw-Hill, New York] - 407 pages
Grimsehl 32-35 - A Textbook of Physics [5 volumes] [Blackie, London]
Knowlton 35 - 2ed - Physics for College Students [McGraw-Hill, New York] - 623 pages
Duff 37 - 8ed - Physics [Blakiston, Philadelphia] - 715 pages
Frank 39 - 2ed - Introduction to Mechanics and Heat [McGraw-Hill, New York] - 384 pages
Hausman 39 - 2ed - Physics [Van Nostrand, New York] [revised in 1948] - 756 pages
Smyth 39 - Matter, Motion and Electricity: a Modern Approach to General Physics [McGraw-Hill, New York] - 648 pages
-----
Frank 40 - Introduction to Electricity and Optics [McGraw-Hill, New York] [revised in 1950] - 398 pages
Lindsay 40 - General Physics for Students of Science [Wiley, New York] - 534 pages
Champion 39-42 - Properties of Matter [5 volumes] [Blackie, London]
Richtmyer 42 - 3ed - Introduction to Modern Physics [McGraw-Hill, New York] [revised in 1947 and 1955] - 723 pages
Stranathan 42 - The Particles of Modern Physics [Blakiston, Philadelphia] - 571 pages
Lemon 43 - Analytical Experimental Physics [University of Chicago] - 584 pages
Nedelsky 45 - The Physical Sciences [McGraw-Hill, New York] - 335 pages
Semat 45 - Fundamentals of Physics [Farrar, New York] [revised and with a new publisher in 1951] - 593 pages
Sears 44-46 Principles of Physics [3 volumes] [Addison-Wesley, Cambridge MA]
Semat 46 - 2ed - Introduction to Modern Physics [Farrar, New York] - 384 pages
Poynting 47 - 14ed - University Textbook of Physics: Volume I - Properties of Matter [Griffin, London] [Volume II Sound 10ed 1949 - See Acoustics]
Richtmyer 47 - 4ed - Introduction to Modern Physics [McGraw-Hill, New York] [revised in 1955] [36 extra pages in 4ed from the 1942 edition] - 759 pages
Smith 47 - 3ed - Intermediate Physics [Arnold, London] - 1033 pages
Duncan 48 - 2ed [revision of the 1920 2ed] - A Text-book of Physics for the Use of Students of Science and Engineering [Macmillan, London] - 1063 pages
Hausman 48 - 3ed - Physics [Van Nostrand, New York] [37 extra pages in 3ed from the 1939 edition] - 793 pages
Jauncey 48 - 3ed - Modern Physics: a Second Course in College Physics [Van Nostrand, New York] - 561 pages
Sears 49 - University Physics [Addison-Wesley, Cambridge MA] - 848 pages
Semat 49 - Physics in the Modern World [Rinehart, New York] - 434 pages
-----
Crowther 50 - 5ed - A Manual for Physics [Oxford University Press] - 594 pages
Frank 50 - 2ed - Introduction to Electricity and Optics [McGraw-Hill, New York]
Nelkon 50 - Light and Sound [Heinemann, London] - 342 pages
Shortley 50 - Physics: Fundamental Principles for Students of Science and Engineering [2 volumes] [Prentice-Hall, New York]
Starling 50 - Physics [Longmans, New York] - 1301 pages
Semat 51 - 2ed Fundamentals of Physics [Rinehart, New York] [256 extra pages in 2ed from the 1945 edition] - 849 pages
US Bureau of Naval Personnel 51- Physics for Electronics Technicians [US Government Printing Office, Washington] - 378 pages
Bitter 52 - Currents, Fields and Particles [Technology Press, Cambridge MA]
Boulind 52 - Heat and Light [Murray, London] - 368 pages
Champion 52 - Properties of Matter [Blackie, London] - 328 pages
Furry 52 - Physics for Science and Engineering Students [Blakiston, Philadelphia] - 694 pages
Marcus 52 - Physics for Modern Times [Prentice-Hall, New York] - 762 pages
Pilborough 52 - Foundations of Engineering Science [Blackie, London] - 468 pages
Sears 52 - 2ed - College Physics [Addison-Wesley, Cambridge MA] - 912 pages
Stead 52 - 8ed - Elementary Physics, for Medical, First-Year University Science Students and General Use in the Schools [Churchill, London] - 578 pages
Winans 52 - Introductory General Physics [Ginn, Boston] - 765 pages
Margenau 53 - 2ed - Physics: Principles and Applications [McGraw-Hill, New York] - 814 pages
Rogers 53 - 3ed - Physics for Medical Students [Melbourne University Press] - 405 pages
White 53 - 2ed - Modern College Physics [Van Nostrand, New York] - 823 pages
Ballard 54 - Physics Principles [Van Nostrand, New York] - 743 pages
Brown 54 - 2ed - Physics: The Story of Energy [Heath, Boston] - 596 pages
Burns 54 - Physics, A Basic Science [Van Nostrand, New York] - 546 pages
Frye 54 - Essentials of Applied Physics [Prentice-Hall, New York] - 369 pages
Kimball 54 - 6ed - College Textbook of Physics [Holt, New York] - 942 pages
Kronig 54 - Textbooks of Physics/Leerboek der Natuurkunde [in English - translation of Third Dutch edition] [Pergammon, London] - 855 pages
Richtmyer 55 - 5ed - Introduction to Modern Physics [McGraw-Hill, New York]
White 55 - 2ed - Practical Physics [McGraw-Hill, New York] - 484 pagesCalculus: Elementary - Chronological - Title
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Osgood 22 - Introduction to the Calculus [Macmillian] - 449 pages
-----
Dresden 40 - Introduction to the Calculus [Holt] - 428 pages
Dull 41 - 2ed - Mathematics for Engineers [McGraw-Hill] - 780 pages
Gale 41 - Elementary Functions and Applications [Holt] - 409 pages
Bacon 42 - Differential and Integral Calculus [McGraw-Hill] - 771 pages
Klaf 44 - Calculus Refresher for Technical Men [McGraw-Hill] [Dover 1956] - 431 pages
Lamb 44 - 3ed corrected - An Elementary Course of Infinitesimal Calculus [Cambridge] - 530 pages
Oakley 44 - An Outline of the Calculus [Barnes and Noble] [1944 outline of current texts] - 221 pages
------
Granville 46 - Elements of Calculus [Ginn] - 549 pages
Randolph 46 - Analytic Geometry and Calculus [Macmillian] - 642 pages
Sherwood 46 - revised edition - Calculus [Prentice-Hall] - 568 pages
Thompson 46 - Calculus for the Practical Man [Van Nostrand] - 342 pages
Douglass 47 - Calculus and its Applications [Prentice-Hall] - 568 pages
Murnaghan 47 - Differential and Integral Calculus: Functions of One Variable [Remsen Press] - 502 pages
Goodstein 48 - A Text-Book of Mathematical Analysis: the Uniform Calculus and its Applications [Oxford] - 475 pages
Boyer 49 - The Concepts of the Calculus: a Critical and Historical Discussion of the Derivative and the Integral [Hafner] - 346 pages
Kells 49 - 2ed - Calculus [Prentice-Hall] - 508 pages
Miller 49 - Analytic Geometry and Calculus: a Unified Treatment [Wiley] - 658 pages
Smail 49 - Calculus [Appleton-Century-Crofts] - 592 pages
Toeplitz - 49 - Die Entwicklung der Infinitesimalrechnung: eine Einleitung in die Infinitesimalrechnung nach der genetischen Methode [Springer, Berlin] - [Translated 1963 - The Calculus: A Genetic Approach - reissued 1981 University of Chicago with new introduction]
------
Michell 50 - The Elements of Mathematical Analysis [2 volumes] [Macmillian] - 1087 pages
Peterson 50 - Elements of Calculus [Harper] - 369 pages
Urner 50 - Elements of Mathematical Analysis [Ginn] - 561 pages
Fort 51 - Calculus [Heath] - 560 pages
Palmer 52 - Practical Calculus [McGraw-Hill] - 470 pages
Randolph 52 - Calculus [Macmillian] - 483 pages
Siddons 52 - A New Calculus [could be multivolume] [Cambridge]
Franklin 53 - Differential and Integral Calculus [McGraw-Hill] - 641 pages
Smail 53 - Analytic Geometry and Calculus [Appleton-Century-Crofts] - 644 pages
Thomas 53 - 2ed - Calculus and Analytic Geometry [Addison-Wesley] - 731 pages
Wylie 53 - Calculus [McGraw-Hill] - 565 pages
Love 54 - 5ed - Differential and Integral Calculus [Macmillian] - 526 pages
Merriman 54 - Calculus: An Introduction to Analysis, and a Tool for the Sciences [Holt] - 625 pagesCalculus: Advanced - Chronological - Title
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Wilson 12 - Advanced Calculus [Ginn] - 566 pages
Bliss 13 - Fundamental Existence Theorems [American Mathematical Society] - 107 pages
Hardy 16 - 2e - The Integration of Functions of a Single Variable [Cambridge] - 67 pages
Goursat 04-17 - A Course in Mathematical Analysis [Ginn]
-----
Edwards 22 - The Integral Calculus [2 volumes] [Chelsea] - 1922 pages
Osgood 25 - Advanced Calculus [Macmillian] - 530 pages
Fine 27 - Calculus [Macmillian] - 421 pages
Landau 30 - Foundations of Analysis [Chelsea] [published in German 1930 translated 1951] - 134 pages
Landau 34 [English 51] - Einfuhrung in die Differentialrechnung und Integralrechnung [Noordhoff, Groningen] - 368 pages [Chelsea translated this 1951]
Woods 34 - Advanced Calculus: a Course Arranged with Special Reference to the Needs of Students of Applied Mathematics [Ginn] - 397 pages
Chaundy 35 - The Differential Calculus [Oxford] - 459 pages
Courant 38 - Differential and Integral Calculus [2 volumes] [Interscience/Blackie/Nordemann]
Burrington 39 - Higher Mathematics with Applications to Science and Engineering [McGraw-Hill] - 844 pages
Gillespie 39 - Integration [Oliver and Boyd] - 126 pages
-----
Franklin 40 - A Treatise on Advanced Calculus [Wiley] - 595 pages
Stewart 40 - Advanced Calculus [Methuen]
Sokolnikoff 41 - 2ed - Advanced Calculus [McGraw-Hill] - 587 pages
Franklin 44 - Methods of Advanced Calculus [McGraw-Hill] - 486 pages
Hardy 45 - 8e - A Course of Pure Mathematics [Cambridge] - 500 pages
Widder 47 - Advanced Calculus [Prentice-Hall] - 432 pages
-----
Gillespie 51- Partial Differentiation [Oliver and Boyd/Interscience] - 105 pages
Kaplan 51 - 2e - Advanced Calculus for Engineers and Physicists [Ann Arbor] - 338 pages
Wylie 51 - Advanced Engineering Mathematics [McGraw-Hill] - 640 pages
Hardy 52 - 10e - A Course of Pure Mathematics [Cambridge] - 509 pages
Kaplan 52 - Advanced Calculus [Addison-Wesley] - 679 pages
Rudin 52 - Principles of Mathematical Analysis [McGraw-Hill] - 227 pagesbasically just a record of what Parke thought were the best books of the era, and still useful. 50 years later, i'd say that many arent going to be that interesting to a modern student, but if you're interested in new supplementary textbooks and old supplementary textbooks, it's good to know what was in the forefront through the decades...

just because i like Sokolnikoff's 1941 calculus book doesn't mean 95% of others will!
[i know i liked it for being easy and nicer than a 1981 Thomas and Finney]

and i think a lot of people would cringe at half of those physics books since PSSC and Halliday and Resnick...
but some people cringe at Hardy too thinking Rudin is way better...

- Spivak & Apostol?

I'd say that with Rudin Spivak Bartle Binmore Apostol, who needs Parke...

If you like supplementary textbooks, it's just nice that Parke offers his Siskel and Ebert Thumbs up to about 5000 books. Stuff like Topology and Analysis are in some ways another world... but if you're someone who likes 30 books on one subject, he's worth knowing if your local library doesn't satisfy you.

Nano-Passion · Aug 1, 2012

RJinkies said:

- Just to be completely sure, are you saying that it is better to spend your time on Parke's book rather than Spivak & Apostol?

oh no... Parke is just a good source for what books were considered useful for a bunch of catagories in science [mostly math/physics, some chemistry/engineering/electronics] and it didnt touch anything after Sputnik. Parke was an applied mathematician with his own laboratory and consulting firm and did a book in the 40s for McGraw-Hill with about 2500 books, and then in 1956 came out with a second edition for Do-er which was double the size with about 5000 books.
And about half of the textbooks were his own personal library for his business...

and he had people run to MIT for card catalogues and in his spare? time he came out with a pretty useful guidebook for knowing what the cool books were 1900-1955.

He goes into interesting ideas about parallel reading and how to tackle new subjects you know little about, and helpful stuff like that.

Apostol's book came out a year after Parke... and Rudin and Hardy are probably the only books people would recognize anymore...

--------

Here's a sample of Parke, though i rearranged things in chronological order...

Guide to the Literature of Mathematics and Physics - Nathan Grier Parke III 1956
Physics - Chronological - Title
- - - - - - - - - - - - - - - - - - - - - - - - -
Ganot 10 - 18ed - Elementary Treatise on Physics, Experimental and Applied [Wood, New York] - 1225 pages
Duncan 20 - 2ed - A Text-book of Physics for the Use of Students of Science and Engineering [Macmillan, London] [revised in 1948] - 1063 pages
Gerlach 28 - Matter, Electricity, Energy [Van Nostrand, New York] - 427 pages
Poynting 28 - 9ed - A Textbook of Physics: Heat [Griffin, London] - 354 pages
Poynting 29 - 12ed - A Textbook of Physics: Properties of Matter [Griffin, London] [revised with new title in 1947] - 228 pages
-----
Franklin 30 - General Physics [Franklin & Charles, Lancaster PA] - 705 pages
Pohl 30 - Physical Principles of Electricity and Magnetism [Blackie, London] - 250 pages
Jauncey 32 - 1ed - Modern Physics: a Second Course [Van Nostrand, New York] [revised with new title in 1948] - 568 pages
Pohl 32 - Physical Principles of Mechanics and Acoustics [Blackie, London] [only revised in German in 1953 - 12ed Springer] - 338 pages
Eldridge 34 - The Physical Basis of Things [McGraw-Hill, New York] - 407 pages
Grimsehl 32-35 - A Textbook of Physics [5 volumes] [Blackie, London]
Knowlton 35 - 2ed - Physics for College Students [McGraw-Hill, New York] - 623 pages
Duff 37 - 8ed - Physics [Blakiston, Philadelphia] - 715 pages
Frank 39 - 2ed - Introduction to Mechanics and Heat [McGraw-Hill, New York] - 384 pages
Hausman 39 - 2ed - Physics [Van Nostrand, New York] [revised in 1948] - 756 pages
Smyth 39 - Matter, Motion and Electricity: a Modern Approach to General Physics [McGraw-Hill, New York] - 648 pages
-----
Frank 40 - Introduction to Electricity and Optics [McGraw-Hill, New York] [revised in 1950] - 398 pages
Lindsay 40 - General Physics for Students of Science [Wiley, New York] - 534 pages
Champion 39-42 - Properties of Matter [5 volumes] [Blackie, London]
Richtmyer 42 - 3ed - Introduction to Modern Physics [McGraw-Hill, New York] [revised in 1947 and 1955] - 723 pages
Stranathan 42 - The Particles of Modern Physics [Blakiston, Philadelphia] - 571 pages
Lemon 43 - Analytical Experimental Physics [University of Chicago] - 584 pages
Nedelsky 45 - The Physical Sciences [McGraw-Hill, New York] - 335 pages
Semat 45 - Fundamentals of Physics [Farrar, New York] [revised and with a new publisher in 1951] - 593 pages
Sears 44-46 Principles of Physics [3 volumes] [Addison-Wesley, Cambridge MA]
Semat 46 - 2ed - Introduction to Modern Physics [Farrar, New York] - 384 pages
Poynting 47 - 14ed - University Textbook of Physics: Volume I - Properties of Matter [Griffin, London] [Volume II Sound 10ed 1949 - See Acoustics]
Richtmyer 47 - 4ed - Introduction to Modern Physics [McGraw-Hill, New York] [revised in 1955] [36 extra pages in 4ed from the 1942 edition] - 759 pages
Smith 47 - 3ed - Intermediate Physics [Arnold, London] - 1033 pages
Duncan 48 - 2ed [revision of the 1920 2ed] - A Text-book of Physics for the Use of Students of Science and Engineering [Macmillan, London] - 1063 pages
Hausman 48 - 3ed - Physics [Van Nostrand, New York] [37 extra pages in 3ed from the 1939 edition] - 793 pages
Jauncey 48 - 3ed - Modern Physics: a Second Course in College Physics [Van Nostrand, New York] - 561 pages
Sears 49 - University Physics [Addison-Wesley, Cambridge MA] - 848 pages
Semat 49 - Physics in the Modern World [Rinehart, New York] - 434 pages
-----
Crowther 50 - 5ed - A Manual for Physics [Oxford University Press] - 594 pages
Frank 50 - 2ed - Introduction to Electricity and Optics [McGraw-Hill, New York]
Nelkon 50 - Light and Sound [Heinemann, London] - 342 pages
Shortley 50 - Physics: Fundamental Principles for Students of Science and Engineering [2 volumes] [Prentice-Hall, New York]
Starling 50 - Physics [Longmans, New York] - 1301 pages
Semat 51 - 2ed Fundamentals of Physics [Rinehart, New York] [256 extra pages in 2ed from the 1945 edition] - 849 pages
US Bureau of Naval Personnel 51- Physics for Electronics Technicians [US Government Printing Office, Washington] - 378 pages
Bitter 52 - Currents, Fields and Particles [Technology Press, Cambridge MA]
Boulind 52 - Heat and Light [Murray, London] - 368 pages
Champion 52 - Properties of Matter [Blackie, London] - 328 pages
Furry 52 - Physics for Science and Engineering Students [Blakiston, Philadelphia] - 694 pages
Marcus 52 - Physics for Modern Times [Prentice-Hall, New York] - 762 pages
Pilborough 52 - Foundations of Engineering Science [Blackie, London] - 468 pages
Sears 52 - 2ed - College Physics [Addison-Wesley, Cambridge MA] - 912 pages
Stead 52 - 8ed - Elementary Physics, for Medical, First-Year University Science Students and General Use in the Schools [Churchill, London] - 578 pages
Winans 52 - Introductory General Physics [Ginn, Boston] - 765 pages
Margenau 53 - 2ed - Physics: Principles and Applications [McGraw-Hill, New York] - 814 pages
Rogers 53 - 3ed - Physics for Medical Students [Melbourne University Press] - 405 pages
White 53 - 2ed - Modern College Physics [Van Nostrand, New York] - 823 pages
Ballard 54 - Physics Principles [Van Nostrand, New York] - 743 pages
Brown 54 - 2ed - Physics: The Story of Energy [Heath, Boston] - 596 pages
Burns 54 - Physics, A Basic Science [Van Nostrand, New York] - 546 pages
Frye 54 - Essentials of Applied Physics [Prentice-Hall, New York] - 369 pages
Kimball 54 - 6ed - College Textbook of Physics [Holt, New York] - 942 pages
Kronig 54 - Textbooks of Physics/Leerboek der Natuurkunde [in English - translation of Third Dutch edition] [Pergammon, London] - 855 pages
Richtmyer 55 - 5ed - Introduction to Modern Physics [McGraw-Hill, New York]
White 55 - 2ed - Practical Physics [McGraw-Hill, New York] - 484 pagesCalculus: Elementary - Chronological - Title
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Osgood 22 - Introduction to the Calculus [Macmillian] - 449 pages
-----
Dresden 40 - Introduction to the Calculus [Holt] - 428 pages
Dull 41 - 2ed - Mathematics for Engineers [McGraw-Hill] - 780 pages
Gale 41 - Elementary Functions and Applications [Holt] - 409 pages
Bacon 42 - Differential and Integral Calculus [McGraw-Hill] - 771 pages
Klaf 44 - Calculus Refresher for Technical Men [McGraw-Hill] [Dover 1956] - 431 pages
Lamb 44 - 3ed corrected - An Elementary Course of Infinitesimal Calculus [Cambridge] - 530 pages
Oakley 44 - An Outline of the Calculus [Barnes and Noble] [1944 outline of current texts] - 221 pages
------
Granville 46 - Elements of Calculus [Ginn] - 549 pages
Randolph 46 - Analytic Geometry and Calculus [Macmillian] - 642 pages
Sherwood 46 - revised edition - Calculus [Prentice-Hall] - 568 pages
Thompson 46 - Calculus for the Practical Man [Van Nostrand] - 342 pages
Douglass 47 - Calculus and its Applications [Prentice-Hall] - 568 pages
Murnaghan 47 - Differential and Integral Calculus: Functions of One Variable [Remsen Press] - 502 pages
Goodstein 48 - A Text-Book of Mathematical Analysis: the Uniform Calculus and its Applications [Oxford] - 475 pages
Boyer 49 - The Concepts of the Calculus: a Critical and Historical Discussion of the Derivative and the Integral [Hafner] - 346 pages
Kells 49 - 2ed - Calculus [Prentice-Hall] - 508 pages
Miller 49 - Analytic Geometry and Calculus: a Unified Treatment [Wiley] - 658 pages
Smail 49 - Calculus [Appleton-Century-Crofts] - 592 pages
Toeplitz - 49 - Die Entwicklung der Infinitesimalrechnung: eine Einleitung in die Infinitesimalrechnung nach der genetischen Methode [Springer, Berlin] - [Translated 1963 - The Calculus: A Genetic Approach - reissued 1981 University of Chicago with new introduction]
------
Michell 50 - The Elements of Mathematical Analysis [2 volumes] [Macmillian] - 1087 pages
Peterson 50 - Elements of Calculus [Harper] - 369 pages
Urner 50 - Elements of Mathematical Analysis [Ginn] - 561 pages
Fort 51 - Calculus [Heath] - 560 pages
Palmer 52 - Practical Calculus [McGraw-Hill] - 470 pages
Randolph 52 - Calculus [Macmillian] - 483 pages
Siddons 52 - A New Calculus [could be multivolume] [Cambridge]
Franklin 53 - Differential and Integral Calculus [McGraw-Hill] - 641 pages
Smail 53 - Analytic Geometry and Calculus [Appleton-Century-Crofts] - 644 pages
Thomas 53 - 2ed - Calculus and Analytic Geometry [Addison-Wesley] - 731 pages
Wylie 53 - Calculus [McGraw-Hill] - 565 pages
Love 54 - 5ed - Differential and Integral Calculus [Macmillian] - 526 pages
Merriman 54 - Calculus: An Introduction to Analysis, and a Tool for the Sciences [Holt] - 625 pagesCalculus: Advanced - Chronological - Title
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Wilson 12 - Advanced Calculus [Ginn] - 566 pages
Bliss 13 - Fundamental Existence Theorems [American Mathematical Society] - 107 pages
Hardy 16 - 2e - The Integration of Functions of a Single Variable [Cambridge] - 67 pages
Goursat 04-17 - A Course in Mathematical Analysis [Ginn]
-----
Edwards 22 - The Integral Calculus [2 volumes] [Chelsea] - 1922 pages
Osgood 25 - Advanced Calculus [Macmillian] - 530 pages
Fine 27 - Calculus [Macmillian] - 421 pages
Landau 30 - Foundations of Analysis [Chelsea] [published in German 1930 translated 1951] - 134 pages
Landau 34 [English 51] - Einfuhrung in die Differentialrechnung und Integralrechnung [Noordhoff, Groningen] - 368 pages [Chelsea translated this 1951]
Woods 34 - Advanced Calculus: a Course Arranged with Special Reference to the Needs of Students of Applied Mathematics [Ginn] - 397 pages
Chaundy 35 - The Differential Calculus [Oxford] - 459 pages
Courant 38 - Differential and Integral Calculus [2 volumes] [Interscience/Blackie/Nordemann]
Burrington 39 - Higher Mathematics with Applications to Science and Engineering [McGraw-Hill] - 844 pages
Gillespie 39 - Integration [Oliver and Boyd] - 126 pages
-----
Franklin 40 - A Treatise on Advanced Calculus [Wiley] - 595 pages
Stewart 40 - Advanced Calculus [Methuen]
Sokolnikoff 41 - 2ed - Advanced Calculus [McGraw-Hill] - 587 pages
Franklin 44 - Methods of Advanced Calculus [McGraw-Hill] - 486 pages
Hardy 45 - 8e - A Course of Pure Mathematics [Cambridge] - 500 pages
Widder 47 - Advanced Calculus [Prentice-Hall] - 432 pages
-----
Gillespie 51- Partial Differentiation [Oliver and Boyd/Interscience] - 105 pages
Kaplan 51 - 2e - Advanced Calculus for Engineers and Physicists [Ann Arbor] - 338 pages
Wylie 51 - Advanced Engineering Mathematics [McGraw-Hill] - 640 pages
Hardy 52 - 10e - A Course of Pure Mathematics [Cambridge] - 509 pages
Kaplan 52 - Advanced Calculus [Addison-Wesley] - 679 pages
Rudin 52 - Principles of Mathematical Analysis [McGraw-Hill] - 227 pagesbasically just a record of what Parke thought were the best books of the era, and still useful. 50 years later, i'd say that many arent going to be that interesting to a modern student, but if you're interested in new supplementary textbooks and old supplementary textbooks, it's good to know what was in the forefront through the decades...

just because i like Sokolnikoff's 1941 calculus book doesn't mean 95% of others will!
[i know i liked it for being easy and nicer than a 1981 Thomas and Finney]

and i think a lot of people would cringe at half of those physics books since PSSC and Halliday and Resnick...
but some people cringe at Hardy too thinking Rudin is way better...

- Spivak & Apostol?

I'd say that with Rudin Spivak Bartle Binmore Apostol, who needs Parke...

If you like supplementary textbooks, it's just nice that Parke offers his Siskel and Ebert Thumbs up to about 5000 books. Stuff like Topology and Analysis are in some ways another world... but if you're someone who likes 30 books on one subject, he's worth knowing if your local library doesn't satisfy you.

Ohh, okay thanks. And man, that was quite a list!

I'll likely pick Apostol or Spivak and an intro to proof book, I want to make sure I can master my first proof class later on.

Cod · Aug 1, 2012

All this talk of old texts has me wondering, have any great mathematicians of today written a text that people will be talking about in 50+ years?

RJinkies · Aug 1, 2012

Hi Cod...

- All this talk of old texts has me wondering, have any great mathematicians of today written a text that people will be talking about in 50+ years?

It's exactly what 50% of people wonder about all books...

That's what i liked about going through Parke, seeing what textbooks in english were still considered good for the 50s, and how many were on the tip of my tongue, or recommended by others today...

for Physics, Duncan and Starling was used back in the 20s, it came out during the middle or end of WWI, and yet it was used like at the University of British Columbia like in 1951 and i think still in 1955.

Poynting with JJ Thompson did the quintisential late victorian english Physics text i think it was in the late 1880s, and it was cranked out till after WWII. The high water mark being 1928 and 1947, well if you trust Parke lol It's a significantly difficult book, and i think i had the blue and orangey-red Dovers from the 1960s and let's just say that it's quite a struggle, it was like the Halliday and Resnick of 1960, but not terribly friendly, but if you were patient enough there's a ton of stuff there. But that's one of the things with the texts, they sometimes get to be an easier and smoother read with time. Though that might not be so true with calculus...

Notice you see Sears with his trilogy in 1944 and then his main text about 1952 with Sears and Zemansky being the classic till the 1980s and then mutating into Freedman and Young... as they passed away...

I remember when i was just about starting calculus, and i found a used yellow striped copy of Tipler (81?) and Furry from 1952, and whew Furry was pretty difficult, incredibly dense and probably a horror for anyone with a weak background in high school physics or starting from zero... but if your algebra and intro calculus was pretty awesome, and you manage to last the first 40 pages, the book was dense, solid and certainly crammed full of neat stuff. But you had to work at reading it, and it's something to go through after you read a more modern and gentler book. But definitely a solid book though a bit unfriendly...

As for Sears i think he'll still be liked 30 40 50 years from now... just like Halliday and Resnick. I had a half of the grey edition of part II [an odd printing that one in the early 60s] , and 20 years later the Orange part of Orange/Blue 1960 edition...

I wasnt aware of HR being famous then, but i said man why don't they use this book today, it was hard but awesome reading and i liked the 1960s graphics of all the shaded particles... Then i found out, with something like PSSC and then easing into HR, it was as closer as you got to the royal road or the Ivy League...

And well, i still think the 1960 ed of Halliday is great, and so is the 1986 Third Edition of Fundamentals of Physics, I don't really subscribe to the fact that the Fundamentals text is all that much dumbed down, maybe the earlier one was a decade before, but i found that what was mostly chopped was the historical stuff, and some of the thinking experiments before the problem sets [i recall one that was a water filled hollow sphere as a pendulum, and you wonder does it swing the same, or slow down or speed up or what]...

I thought of that problem as a hot water tank sitting on a swing and you let it leak and 'film it'... I'm not sure of the answer still to this day but i think that the initial and the final swing is the same, and i think there's some part of it as it drains will be speedier and another part of it slower... Not sure if that's the part of the swing or the return swing, or just if it's more than half full/less than half full... but it was quite the discussion i had with someone who toyed with the problem off and on for months..

But anyhoo, I think that 90% of the fundamentals of Physics text is the same, some of the most difficult stuff was pared down, to get the page count down and the history gutted, which makes me feel it wasnt all that necessary a thing... but i think the fundamental changes were just that they made the text easier and clearer, not dumbed down at all! And all the editions before 1990 i think rule...

As for math, I think Granville, Longley and Smith, was pretty neat as in they didnt bother with any formalism or analysis at all, the book was easy and it's where i learned that Jacobi created in the 1850s the del sign for partial differentiation. Something that 99% of other texts don't tell you. Most of the book is the same old 1904 edition, and still a great read and probably the easiest text of the day... Same goes for the much stomped on and much praised Sylvanius P Thompson's calculus made easy. I'm still not sure why it was disliked or why Parke didnt include it. Mathwonk found it useful when he was taking first year calculus at Harvard, and it's what others recommended.

It only reinforces Parke's spiral approach, read the baby book, then the hard one! Parke mentions the books Feynman used like JE Thompson Calculus for the Practical Man, and the rather blah Love text, which was like the Thomas and Finney of it's day...

It came out in 1921 and like Granville-Longley-Smith, CE Love with Rainville were just the early guys on the block, and it lasted till a 1962 6th edition, before going poof.

And if you wanted busy and long winded and difficult, you could go the British route with what probably complemented Hardy - Horace Lamb's calculus book from Cambridge. [Third Edition was 1919] and still used in the 40s and early 50s...

and with JE Thompson was Farley Woods which Feynman used. Woods is probably hyped too much and some of the theory is long winded, but there's lots of applied math gunk that the main books didnt touch. But any Advanced Calculus book with enough of a page count, would match it. Being under 400 pages, you get like almost 300 pages more in Kaplan...

With calculus, i'd say that Granville, Franklin, and Thomas and JE Thompson were awesome. And Thomas was probably best in the late 60s or early 70s, and peaked probably about the 7th edition in 1986. [that's the bluey one] I'm not really impressed with the later editions, and i think mathwonk said the 9th edition was the last one before it got botched up. I probably like the 60s edition, a 1972 ish 4th alt edition, one of the early 70s ones, and that 1986 ish 7th one...
not too fond of the early 80s one or 90s editions...

and Courant and Kaplan and half of those advanced books will be peachy decades from now..

A *lot* depends on how well prepared you are, for tackling the older books, sometimes the first chapter is the hardest one because sometimes your previous math course wasnt that 'hot' or the older textbooks were more thorough, and you learned more, with less frills.

I'd say that most of the old books are great, but they might not be as easy a read, but often a good number of them are *way* easier to read. I still think half of the books of Parke's are still good, and maybe 85%, if you're a masochist, or like reading 4 calculus books end to end, before saying 'no more' lol

always wondered what parke would pick after 1955...

for calculus.. maybe
55 AE Taylor
57 Apostol
61 Olmstead
64 Protter and Morrey
64 Smirnov
68 Loomis and Sternberg

what i think is cool is that Parke really doesn't touch the easy books on calculus before 1940...
he thinks a baby book on calculus and zoom into Courant, nothing else needs to be said.. though i question that sort of crappy Barnes and Noble Outline of Calculus by Oakley, i think both Thompsons or Granville are a billion times better.

------
Parke on page 143

"Granville, Smith and Longley is used by the US Armed Forces Institute. Franklin is a vetran writer and his calculus is certainly first-rate. Murnaghan is a first-rate applied mathematician and his calculus is written from a rather novel point of view. However, our personal inclination is to get as much as possible out of the Barnes and Noble Outline of the Calculus, and proceed as early as possible to the serious study of Courant's Differential and Integral Calculus, cited under the advanced texts. Courant will give the student the best possible balance between vigor and rigor.'

i think what kills the old books and the good books is curriculum. Feynman got pushed out because it didnt fit, same goes for the Berkeley Physics Course. Sadly book 1 by Kittel on mechanics isn't talked about much, and neither is the swedish guy who did book 4 on quantum. All you hear is endless praise for purcell's EM book where all of them are awesome. 5 books was just too much for people, some would do book 1 and 2 for first year and then cram the other three in second year.

I think Halliday and Resnick's old edition suffered, and PSSC suffered more, by the time the 1971 Third edition came out, people were rearranging the order and killing the elegant beauty of the 1960 and 1965 writing... basically the whole PSSC high school course was killed because of time pressure, teachers wanted to get to mechanics right away and all the conceptual layering meant you lost the build up of 150 pages or so before you get there...so people only used the mechanics and EM part and junked the other 50%... and I'm not so sure the last edition was the best, it's interesting, but all i end up doing is miss the 1965 edition more...

And there's not enough praise for the schaums outlines or the weirder REA books. Calculus and chemistry and physics and vector calculus and intermedia mechanics are all nicely done in those books. I'd rather use a schaums outline than 40% of the new texts out there. At least there's no bulls,er crap with Schaums outlines. Shame they changed the look, i liked the tan and blacks or the quilty blue/pink/greens with the white border, now they look like they're from hell and no more fun to collect. REA has nice plain covers and now there's hideous sherlock holmes artwork...

and the awesome 60s style IBM Selectic like fonts make it neat, though the schaums are way way way more nicely typeset.

Apostol and Courant, Spivak's calculus, college math, and algebra based physics i think all suffer with the curriculum and end up like feynman's lectures, liked by 20% of the teachers who sadly say, oh that would be too much reading, or it's too hard, or I'm not the head of the department who chooses the books...

the best thing about some of the better schools, esp for a course in Quantum Mechanics is they dump like 4-12 textbooks on you, and that's your whole bookshelf for QM I II III, and you're suppossed to jump around... and well assumed to eat sleep and breathe the course with 4 texts and 7 supplementary texts and burn 2 hours a day on it lol

...

Again i think the best path is 50% new books and 50% old books, and well 85% of the old ones are still great.

You just got to know which 50% of the old books and new books stink.

Pick the books that *speak* to you, or pick the one with the freaky diagrams and weird **** that no other text tries to accomplish. Look at Feynman, Look at Wheeler, look at Courant, they got stuff in there no other books have. They might not be popular anymore, but there is a definate minority cult out there.

If you can handle books without full colour pictures, and 1700 words on a page not 300 words a page, the old books, rule lol

dkotschessaa · Aug 8, 2012

Silly question - when do you folks read all these books (Like Apostol) if they are not part of the classes you are already taking? Are you doing this while you are taking classes or after you've gotten through the traditional sequence?

I do lots of extra reading and studying, but nothing quite this heavy yet.

-Dave K

RJinkies · Aug 8, 2012

I just think people collect the books, before or after their classes [if they do the classsic]

One just finds the books that speak to you in the library, or if you're lucky, you find someone to talk to or a list somewhere. [a lot easier with the internet in some ways]

If you're aware of the curriculum and know what the general syllabus is for the courses, you just go on a lifelong easter egg hunt and find what 'fits' your style.It's one thing to browse and another thing to 'study' the books, but don't ignore the joy of browsing and searching, it's all a part of getting your own unique box of tools.

sometimes the curriculum helps and often it hurts...

i remember there wasnt any good algebra books at home, but for calculus there was the quirky and likeable Sherman K Stein's book [1969 and then a few 70s editions] and JE Thompson's calculus book from the early 30s. But i probably would hath been better off if i read Stein and Thompson rather than waiting around for an actual class in calculus, in hindsight...

But i was buying Symon and Kleppner for physics without a damn calculus physics problem in my life, and those books 'spoke' to me.

Courant i heard about, and didnt see a copy till after i took calculus. Though i saw the creepy gold dustjacket of the first part of the 1963 Courant and John edition, which was definitely a 'weird' one...

I think how the curriculum goes against you, is i still think the best ways of learning some things are by taking a course twice, with an easy text and then a harder one after. There is something sort of magical about seeing how clear and straightforward something like Calculus for Electronics can be, and often you get a better working box of tools with that outside of the classroom, than *inside* one with a regular text.

Things like the Berkeley Physics Course and Feynman's Lectures didnt take off, though i tend to think of Griffith's books now as a new form of that [now that he's written the other two texts], and surprisingly they are now a solid part of the mainstreain curriculum.

----------

I got a good question...

a. Why did Stewart's calculus textbook take off so successfully? and is anyone out there bold enough to toss some detailed minuses, and detailed pluses to the text?
[and like Thomas and Finney the earlier books were better...]

b. I thought Flanders book on Calculus was something close to taking off as a popular text in the late 80s - WH Freeman
[it's a glossy white one, and the first edition was white and red cloth
[he was much more famous for the differential forms calculus book way way earlier]

and what were some of the famous calculus textbooks, when Apostol/Spivak and Thomas-Finney weren't used in the 50s 60s 70s... I thought it odd how Thomas and Finney gradually turned into a second year only textbook and dropped for most with first year calculus...

AnTiFreeze3 · Aug 9, 2012

Here's a fun problem to solve that I did a little while back:

Let x=\frac{1-t^2}{1+t^2}
and y=\frac{2t}{1+t^2}

Show that x^2+y^2=1

micromass · Aug 9, 2012

Let t=\tan(x/2)

dkotschessaa · Aug 9, 2012

RJinkies said:

I got a good question...

a. Why did Stewart's calculus textbook take off so successfully? and is anyone out there bold enough to toss some detailed minuses, and detailed pluses to the text?
[and like Thomas and Finney the earlier books were better...]

I'd like to know the answer to that too, though I think I can contribute somewhat to the discussion. I think the strongest point of Stewart's book may also be what makes it so disliked by many students - namely, that is extremely concise.

Most students complain that it is "hard to read" and that "it doesn't have enough examples." I think the philosophy of the text is to keep students away from the "plug and chug" method of heading straight for the homework problems, looking for examples that are similar, and re-arranging the necessary formulas.

The explanations are actually very good but have to be read very carefully and "unpacked." Sometimes there are VERY IMPORTANT details that are relegated to a small, fine print marginal sentences. When I took notes out of this book I would often re-write what was contained in a single paragraph to something (for my understanding) that would fill a whole page of notes.

I looked at earlier editions of the book and it seems to have gotten thinner and thinner as the new editions came out. Stewart is putting supplements online, but most of the students I studied with weren't even aware of this (even though it is advertised in the book.)

-Dave K

A. Bahat · Aug 9, 2012

That problem is much more fun in reverse! (Finding a rational parametrization of the unit circle, that is.) Give this one a try:
parametrize the curve y²=x³+ax+b

RJinkies · Aug 9, 2012

a. Why did Stewart's calculus textbook take off so successfully?

dkotschessaa - I'd like to know the answer to that too, though I think I can contribute somewhat to the discussion. I think the strongest point of Stewart's book may also be what makes it so disliked by many students - namely, that is extremely concise.

I hear people like and dislike Larson/Edwards aka Larson/Hosteller/Edwards] which goes from being a junk book to crystal-clear at times depending on who's opinion and what edition.

On LE/LHR:
[like a typical intro calc book - it's not rigorous enough, has too much brute force, too many applications, not enough mathematics, not enough creativity.]
[I have many of the same criticisms of this book as I do of the Stewart, although I do think this book does a slightly better job in the very beginning]
[This book does provide the concepts and theory critical to an understanding of calculus. Unfortunately, it is in a wordy, technical, abstract, and thoroughly annoying format...This book gives you plenty of abstract proofs that look like bull@!#t, but falls far short of my engineering book in encouraging an understanding of calculus. The truth is, this book gives you hundreds of formulas to memorize, instead of a relative few like my engineering book that can cover every problem.]

LHR/LE went through 10 editions, and suffered a *lot* in the 4ed from 1993 with the horrible idea of using computers and graphic calculators and other stuff. Thomas-Finney at least in the 80s just plopped in all the freaky 3D graphs and didnt need you to play with software or odd CAI stuff] But i wonder if the later editions got better and dropped those fads... and the ratings went up. It's interesting since people think the minuses of stewart's book apply here too.

-----
I think the secret to Stewart getting cult status by some is due to his influence of Polya with trying to show students how to actually solve the problems. [Something i forgot was hidden in my notes lol]

Here are some of the gems people said about Stewart [and other texts]

a. [i recommend the Second edition of stewart. it went downhill after that. - Mathwonk]

b. [A few people I know have trudged all the way through Apostle's tome, and found they had to skip over entire sections reverting to stewarts book to tell them what the hell is going on intuitively.]

c. [Presentation of Applications Confuse Students - 2 out of 5]
[This book was used at my undergraduate and graduate institutions; I am currently forced to teach out of it. I don't understand why it's considered such a great book. I have seen many students confused by it, and I find it mediocre as a reference text.]
[It is my belief that calculus should be presented in a simple and pure way so that students can master the fundamentals, and then (simple) applications be presented later. Instead, this book introduces fairly complex and "ugly" applications right from the start.]
[The net effect is that students using this book often fail to master the fundamentals of the subject, and find calculus overwhelming and confusing.]
[The book's covering of advanced topics is better than the earlier chapters, but there are far better calculus books out there, and I would not under any circumstances recommend this one.]

d. [Stewart does not sugarcoat or resort to gimmicks or superficiality in order to make the material learnable]

e. [I am teaching honors calc this fall and cannot find a good book. I do not mean Spivak or Apostol, those are too hard for my "honors" course. There only seem to be really weak books for non honors, or really hard books for super honors courses, Any good plain old intermediate honors books out there? I don't want to be difficult but I also dislike heavy books, and space wasted on technology, or bundled CD's. I want clear explanations, some rigor, and a logical sequence of ideas, intelligently written. I have considered the old Courant, but it looks a bit unattractive on the page for todays kids. I once liked Stewart, and Thomas Finney, but subsequent editions have been dumbed down. - Mathwonk]

f. [stewart is a joke compared to spivak. i.e. stewart (2nd edition) is a good non honors book. spivak is an excellent super honors book,(not just regular honors). - Mathwonk]

-------Two promising books, which few know about, but i like a fair deal, is something, I'd like others to chime in about...

Leithhold [circa 1968]
and
a. 43 The Calculus 7 aka TC7 (Hardcover) - Louis Leithold - Harpercollins 1995 - 1216 pages
[extremely approachable text - well liked for Third World Engineering types]
[dates before 1968]b. 44 Calculus and Analytic Geometry - Second Edition - Sherman K. Stein - McGraw Hill 1973
[aka Calculus in the First Three Dimensions - 1967 First Edition]
[Sherman Stein - PhD Columbia 1953]
[Taught at the University of California, Davis - retired 1993]
[a gem to have]
[brilliant method]
[This book is literally the best basic calculus text you can possibly get.]
[Reading this book really gave me an true understanding of basic calculus.]
[Stein offers several suggestions on how to solve certain problems. It is a shame - this book does not attract the amount of attention it deserves.]
[I did get stuck a couple times]
[when I need to refresh some calculus and geometry techniques, this book is really the best to sharpen my intuition and understanding of calculus.]
[I wish all math books were like this]
[If math is not your strongest skill and you need to learn some higher calculus this book will be your excellent companion helping you to gain the insight and intuition you need.]
[may disappoint the reader who is looking for rigor]
[perfect book to gain insight in calculus]
[unique calculus book with a physical bent - tech book guy - los angeles]
[This book starts out with integration and presents the main ideas in a very concrete fashion. Although many books would have the student think otherwise - the techniques of calculus were developed to solve concrete physical problems and model natural phenomena. This book does a good job of helping the student realize this.]
[Another thing I like about this book is that it actually assumes the reader knows pre-calculus mathematics rather than trying to review everything. The inclusion of pre-requisite material is what usually drives calculus books into phone book size.]

[First Edition] - 1967 [or 1968?] [could be called Calculus in the First Three Dimensions - 613 pages]
[Second Edition] - 1973 - the one i like
[Third Edition] - 1982
[Fourth Edition] - 1987
[Fifth Edition] - 1992

------
Stein was the first calculus book in my house! It was tossed at me about 1974-1975 when i was in elementary school. Some stuff was easy but in places, i did feel stonewalled. But it was probably due to youth and not enough algebra, or just finding it at the time, clear clear clear uh oh impossible [hide the book for a whole year]

Other books I'd like an opinion on was

d. Calculus - Harley Flanders
[first edition - 70s/80s?]
[second edition 1985 WH Freeman]
All i know about him is he got a BA in 1946 at Chicago
50s - Faculty Berkeley
50s fellowship at Caltech
60s - Purdue
Tel Aviv University 1970-1977
Ann Arbor Michigan 1985-1997

people like his first textbook Differential Forms from 1964 and Dover has it out now, but it's not an elementary textbook.

[I did a mistake a few days ago thinking it was Flanders, but the book was Edwards]

e. Advanced Calculus: A Differential Forms Approach - Harold M. Edwards - 1969/now Birkhauser 1994
[I'm just going to come out and say it: this book is the best treatment of multivariable calculus that I've seen. Unlike the usual multivariable textbook, this book gives lucid, clear, and elegant explanations and proofs for nearly all principles introduced, i.e. the method of Lagrange Multipliers. The author never keeps you guessing; he starts low and builds up quickly and brilliantly.]
[An inviting, unusual, high-level introduction to vector calculus, based solidly on differential forms. Superb exposition: informal but sophisticated, down-to-earth but general, geometrically and physically intuitive but mathematically rigorous, entertaining but serious. Remarkably diverse applications, physical and mathematical.]
[In fact this book looks decidedly 19th century in places. This is the opposite to a book by Lang, Dieudonne or Rudin. To be fair the author has gone to great lengths to motivate the mathematics and for this reason it may well be very popular with engineers and physicists.]

I thought it was Flanders was the Differential Forms text that people wet their pants about, but it was Edwards...

----

The main thing is that i thought Flanders for the 80s had a book that though weird in places [he obsessed about getting students to draw crappy diagrams a lot] it seemed like a book that almost pushed Thomas and Finney out of place. [then again WH Freeman used to be like Addison-Wesley, almost anything they printed was awesome]
Seemed like a solid unusual book...

--------

another one the MAA liked, but it's probably more for the later books than the initial books was

f. Jerrold E. Marsden and Alan Weinstein - Calculus I, II, III - New York, NY: Springer-Verlag, 1985. Second Edition - three book set

I think Marsden might hath popped the book out in the late late 70s. and for a while it was used when they both taught at Berkeley.

------

g. 72 Allendoerfer, C.B. and Oakley, C.O. Principles of Mathematics - McGraw-Hill 1963
[probably Second Edition 1963]
Mathwonk used it and liked it...but there's a lot of proofs...

----
h. Thomas and Finney...

[The prose is clear and tight. The figures are fantastic. Great examples. Great discussion of the mean value theorem. The discussion of limits is rigorous but not overly so. The 4th edition went overboard on rigor as that was vogue in the early 1970's. Subsequent editions became heavy phone book size calculus texts. Of all the editions of this text this is the one to get. Although there are some other good older calc texts out there this is the cream of the crop.]

[First Edition] 1951
[Second Edition] 1956 - 731 pages [maybe 1953] - Parke III recommendation
[Third Edition] 1961
[Fourth Edition] 1968 - [this went overboard on New Math Rigor]
[Third Alternative Edition] 1972 - 1025 pages
[i can't figure out the alt editions, anyone know??]
[Fifth Edition] 1979
[Sixth Edition] 1984 - a bit of a let down
[Seventh Edition] 1988 - a nice edition

i hear the 8th and 9th are okay - 1991 and 1995
but the 10th-12 editions from 2001 to 2010 arent as good now

seemed like an okay book of the 50s and 60s
and it was up and down in places in the 70s 80s...

but if anyone wants to add any forgotten books 1955-1980 that Thomas-Finney or Apostol or Apostol.. didnt steamroller into obscurity, do tell.

Leithhold, Stein and Flanders were i think three that stood out.. and seem more fun to browse than Stewart. But i think my guess is that Polya's influence is what got Stewart his 10 million house in Toronto lol...Some say

i. Calculus - RT Smith and RB Minton
is better than Stewart...

but i think there were problems with the first edition with proof reading and the binding, and McGraw-Hill offered replacement texts. Nothing like mistakes or falling apart books to ruin a promising beginning...
Not sure when the first edition came out or if it's useable
but the second edition was 2002 McGraw-Hill...

and the third edition is 2007 with like 7000 problems and 1000 examples...

[sounds like a schuams outline with handholding, how can you go wrong!]

anyhoo, some say Smith and Minton beats Stewart, Larson and Anton, so it's worth a look...

------

dk - students complain that it is "hard to read" and that "it doesn't have enough examples." I think the philosophy of the text is to keep students away from the "plug and chug" method of heading straight for the homework problems, looking for examples that are similar, and re-arranging the necessary formulas.

dk - The explanations are actually very good but have to be read very carefully and "unpacked." Sometimes there are VERY IMPORTANT details that are relegated to a small, fine print marginal sentences. When I took notes out of this book I would often re-write what was contained in a single paragraph to something (for my understanding) that would fill a whole page of notes.

dk - I looked at earlier editions of the book and it seems to have gotten thinner and thinner as the new editions came out. Stewart is putting supplements online, but most of the students I studied with weren't even aware of this (even though it is advertised in the book.)

A lot of the older textbooks were short on examples till the 1970s. Hard to read can sometimes show a huge flaw, or sophisication, so it's a hard one to judge. Kittel's solid state physics texts are infamous for being hard to read, but if you're extremely slow and careful you see the method to his madness...

I think *any* online or CD supplements are a long term death strategy for authors... the worst book with that was
the Quantum Physics book by Gasiorowicz...

[probably the bext textbook combo for QM was
[Cohen-Tannoudji/Gasiorowicz/Griffiths/Liboff/Merzbacher/Sakurai/ Ohanian/Shankar/Feynman/Bransden/Dicke/Schiff]
if you had the right *edition* of Gasio...

Gasio in the First [1974] and Second Edition [1995] was 500 pages
and then the Third Edition was 350 pages
with just 30% of the book taken out, and then plopped online
which i think is almost a criminal thing to do, on top of a ton of mistakes with crappy proofreading...

It was a no nonsense book, but sadly one where like a lot of calculus texts, it's awesome after you took a class but as a text to learn from, it usually seems like knocking your head against a wall. Gasio's book was liked because it was in many ways a replacement for Schiff's 1960s text...

[I truly truly hated this book when I was using it as an undergraduate. It’s thin, explains things with extreme economy of words, and the problems are quite difficult in comparison to the depth of explanation in the chapters. It assumes a very decent mathematical background in linear algebra/Hilbert spaces. That said, now that I understand the material it’s a great reference. I think I would have really liked this book if my math background had been stronger, and it’s still a good source of brushing-up on a few basic topics while taking grad school QM.]

[I have pretty strong math skills and most of the time I have no clue how or why he does things. The text is written very math and equation oriented. There is little to no explanation as to why or what the author is trying to show, he just runs through the equations, section by section. He overly uses terms like "We know that" or "Its clear that" as a means of explanation, and the reader is stuck wondering why something is done.]

[He lays out the concept, manipulates the equation in a few brief steps, and leading to the final equation. Entire sections can be covered in a few sentences.]

[In defense of the text, there is a focus on the physically interesting material, while extraneous mathematical stuff has been skipped. However, the text is too hard for an introduction, but skips too much material to be a comprehensive guide. Perhaps as the second or third quantum mechanics book on your shelf, this book will do, but not as the first. For the mathematically inclined look to Sakurai. For a very readable if non-standard approach see The Feynman Lectures. Or, for a lighter introduction see Griffiths.]

----

[Terrible book. Half it is put online, and it's absolutely gaunt compared to other more comprehensive texts. Completely glosses over many fundamental derivations. Avoid at all costs. - Quaoar]

[It does not teach QM conceptually, instead it just states stuff, giving no reason for why things are done as they are. Overall, this book is terrible - bad for undergrads who will learn little, and horrible for grads who won't learn principles. The sooner this book goes out of print, the better.]

-----

So the best of books can be *ruined* by online gimmicks, or cd rom supplements, or computer crap tie-ins, or graphic calculator or TI-55 button mashing/mathematica problems... I got more respect for a mathbook with APL symbols really...

If a publisher can't cram all the weird stuff into a text or needs to resort to animations, run to your nearest copy of any Sylvanius P thompson from 1914. Heck at least Thomas and Finney in 1988 could cram all the pretty computer pictures into the book without increasing the page count or taking stuff out... and there are still books that could do wonders with black and white or only occasional diagrams.

Again, Courant doesn't put stuff online, or use funny colour photographs or play around with side margins, and people still think it's pretty close to ideal and hard to top, though not a cakewalk...

after a decade all those computer gimmick textbooks, or online gimmick texts end up on the junkpile, not liked anyone cept for the xmas bonfire... Half of the worth of these books are as a reference *later* on... and if we need gimmicks, i'd prefer the 1960s 35-mm film to go with my math or physics text lol

another nightmare were those integrated first year math-first year physics textbooks, all in one... [and the mechanical uni-curse physics textbook seemed like a fad too, i think caltech was the only place that still used it in some classes, where i still think the PSSC films *said* more]

It's strange how any truly creative calculus text, dropped topics that were filler, and included unusual stuff, and sadly the textbooks that get adopted are ones that include the kitchen-sink approach [and not in a good way] out of fear that if something about Newton's method or the new-math delta-episilon stuff is left out, 80% of curriculum writers just don't adopt the textbook.

I don't know if Feynman, PSSC, the Berkeley Physics Series, or Spivak or Courant would fly today with the strangulating feel of a bland curriculum, or the Banesh Hoffman 'tyranny of testing' that pushes out the quirky texts.

egyptman · Aug 13, 2012

hi.i.love.this.website...>>>but.iam.weak.at.math.:(
im.from.egypt.im.at
3.prepschool.
please
i
wanna
help.:)

ovael · Sep 8, 2012

Mathwonk, could you tell about differences between first and second edition of Allendoerfer's
and Oakley's Principles of Mathematics?

I am currently studying more basic material, and I plan to study that book next but I just don't know
yet which edition to get. I think you recommended the first (1955) in somewhere, but the second edition seems to be almost 100 pages longer, so I was wandering if it contains some useful additions?

I don't really have money to get both editions right now, so I will probably get the first edition if there are no recommendations to do otherwise.

Thanks if you can answer, and also to anyone else who might know. Mathwonk has just been recommending that book a lot.

RJinkies · Sep 8, 2012

Allendeorfer and Oakley - Principles of Mathematics

There's a Third Edition 1969 McGraw-Hill

and some of the books might be with title changes so the diffences could be minor or substantial...
in the day it was likely the same book tweaked for college students...Allendoerfer & Oakley - Principles of Mathematics - McGraw-Hill 1ed 1955 - 540? pages
Allendoerfer & Oakley - Principles of Mathematics - McGraw-Hill 3ed 1969 - 705? pages
Allendoerfer & Oakley - Fundamentals of Freshman Mathematics. McGraw-Hill 1959
Allendoerfer & Oakley - Fundamentals of College Algebra. McGraw-Hill 1967

Allendoerfer - Calculus of Several Variables and Differentiable Manifolds - Macmillan 1974

[he died in 1974]

and i think he was pretty much a fixture at the Uni of Washington, in Seattle from 1951 onwards, being one of the many big cheeses with the New Math...

------'Noting the trend to abstraction in New Math, Morris Kline says "abstraction is not the first stage but the last stage in a mathematical development." ...'

----------

There's been some famous algebra books:
Chrystal i got in the chelsea edition...Peacock - A Treatise on Algebra 1842
Hall and Knight - Elementary Algebra 2ed 1896 - 516 pages
Hall and Knight - Higher Algebra 3ed 1889 MacMillian - 557 pages
George Chrystal - Textbook of Algebra - A&B Black, London 1900/Dover/Chelsea - 1235 pages
Fine - College Algebra - Ginn 1904 - 595 pages
Knebelman and Thomas - Principles of College Algebra - Prentice-Hall 1942 - 380 pages
Ferrar - Higher Algebra - Oxford 1945 - 222 pages
Ferrar - Higher Algebra: A Sequel - Oxford 1948 - 320 pages
Albert - College Algebra - McGraw-Hill 1946 - 278 pages
Welchons and Krickenberger - Algebra - Ginn 1953

you could add after Parke's choices [oddly he mixed up modern algebra with it like Merserve's Fundamental Concepts of Algebra and stuff]...

well you could add

Allendorfer 1955 [and all the other texts he did]
Dolciani 1964

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

I'm actually interested in any texts people liked from 1955-1980 actually, since there's a lot of 50s 60s texts that slip through the radar...

-------

one of the stranger ones was Hayden's 1960s book for Allyn and Bacon, talk about being a freaky advanced concepts supplementary text for Honours high school people...

it's got extremely extremely few examples, lots of New Math, and, challenging and scary on most every page. I got one of the two books in the set...

[Algebra Two - Dunstan Hayden and Gay Fischer - Allyn and Bacon 1965 - 454 pages]

IT was ideally meant for three semesters in most cases...

Where I'm assuming they meant 4 quarters for the year, 3 quarters on this text, and one final quarter where they teach the basics of probability and statistics or calculus.

---------

RJinkies · Sep 8, 2012

Here's a really good summary of Allendorger off Amazon:-------
quite a good book on the theory of equations Nov 21 2010
By Bruce D. Wilner - Published on Amazon.com

I used A&O in an experimental high school class in 1976. The book provides thorough, strong, and unique coverage of assorted fun topics in the theory of functions--synthetic division, Descartes's rule of signs, fundamental theorem of algebra, rational root theorem, and such--as well as (as I recall, dredging up thirty-year-old memories as best I can) good stuff on sequences and series. The pedagogy is a bit dated, which is why I withhold the fifth star. But A&O is an enjoyable book that covers lots of stuff one won't find elsewhere. You might also enjoy Hall & Knight, but, like so many British "texts," they don't teach--they just present and assume that you'll follow completely. Even the great Bruce David Wilner gets put off by this approach very occasionally . . .
---------

dustbin · Sep 8, 2012

ovael said:

Mathwonk, could you tell about differences between first and second edition of Allendeorfer's
and Oakleys' Principles of Mathematics?

I am currently studying more basic material, and I plan to study that book next but I just don't know
yet which edition to get. I think you recommended the first (1955) in somewhere, but the second edition seems to be almost 100 pages longer, so I was wandering if it contains some useful additions?

I don't really have money to get both editions right now, so I will probably get the first edition if there are no recommendations to do otherwise.

Thanks if you can answer, and also to anyone else who might know. Mathwonk has just been recommending that book a lot.

I do not know if this is still the case, but I read a while back that mathwonk prefers the first edition. I am almost done working through the first edition. At first, I did not like it a whole lot, but as I've worked through it and gotten used to the writing/style, I've come to like it quite a bit, actually. But I cannot give a comparison of the editions :-(

mathwonk · Sep 8, 2012

Yes, in general I always recommend first editions. In this case however I have not seen any other edition since my experience too is simply from having used it in high school in an experimental class in 1959.

As a general rule, the first edition is the authors' own best shot at exactly what they want to say and do, hence it is the best. Usually later editions exist only because the publisher wants to be able to sell more copies and not have to compete with cheaper used copies. So they pressure the author to change it somehow to make suckers, oops, I mean students, buy the newer pricier one.

I myself cannot think of a single book for which I would prefer a later edition. It is tempting to want that extra chapter, but truthfully I seldom even get through all of the shorter version, and if I do, I almost surely do not need the extra stuff in the later version.

Lets put it this way, if you wait until finishing the first edition before worrying about needing the second or third, you will almost never get to that point of having worries.

It is true as mentioned above that some old books are written in a more serious style, i.e. some modern books are written more for students who cannot actually read as well as used to be assumed, so they use smaller words and so on.

However the later editions are not usually much worse than the first one, and then I would be guided by price, although there are a few exceptions as noted below.

Some successful calculus books introduce easier problem sets in later editions to broaden their audience, or actually delete useful material, in favor of including more easier material.

E.g. the 2nd edition of van der waerden's algebra book (the first one available in english) omitted the material on well ordering and restricted to the case of countable fields, so as to include other material on valuations which is less interesting to me personally.

Then the 3rd or 4th edition restored the well ordering and added a chapter on the algebraic riemann roch theorem and topological algebra, neither of great interest to me. To make room for those, it dropped the chapter on elimination theory, which I find quite interesting especially today with the rise of computational methods.

Thus as always, it is helpful to take a look at the books in a library and see which appeals to you.
Since I have not studied this elementary material for some 50 years, it is also quite possible that some more recent books I have not heard of are more palatable and useful. So browse around on the library shelf near this book for others as well.

ovael · Sep 8, 2012

Thanks for answers RJinkies, dustbin and mathwonk.

As I don't have any experience posting forums and didn't yet get the hang of quoting, I will reply
for you here.

RJinkies, that's interesting that Allendoerfer was proponent of New Math. I glanced over his wikipedia page but missed that completely. And I am also interested in old mathbooks, as they seem to be better than what's available nowadays.

By the way if someone is reading this thread and doesn't yet know you can get many books RJinkies mentioned free from archive.org. Like Hall and Knights algebra books.

For example here is Leonhard Eulers Elements of Algebra:

http://archive.org/details/elementsofalgebr00euleDustbin, that's great that you have nearly completed the book. Could you tell a bit more about the experience? Like what kind of math backround you had and do you perhaps now prefer the style it has to other books you have studied previously? Did you think the problems were hard/interesting?Mathwonk, yes i will not worry about those minor differences, I was more curious, as I thought previously that I could get both the first and second edition. We have a system here (Finland) that you can order books from other cities/universities libraries and I was told it's (nearly) free but apparently it isn't.

And sadly we don't have that extensive collection of mathbooks in libraries, not at least elementary books. Local university library has luckily some, and I got my current study materials from there. They are finnish school books from the 1960's, which were first published in the 1940's.
And they are way better than the books I had in high school.. Funny or sad, depends how you look at it. And even they have been "watered down" a little due to curriculum changes from the 40's editions.

That really prompted me to ask about the Allendoerfers book, since this whole trouble of finding decent mathbooks and the whole general state of math education is really quite frustrating.

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