Other Should I Become a Mathematician?

RJinkies · Oct 5, 2012

- for me it was not so much the book, as when i started to take learning seriously, but some books like spivak went out of their way to reach me before i knew how to study. i.e. no book is too hard for a serous student, but some books reach out to the clueless.

I got frustrated with lukewarm books [common in the 70s and 80s] and i often looked for easy books, that were far clearer, far easier, and well, there was also the appeal for the super detailed, super lengthy hardcore books too.

I think doing things differently and adding tons of topics not found in any other textbook is why i started liking those things...

I still get the impression that one needs a good mix of old and new textbooks, and for me about 25% new 75% older is a cool balance.

I was looking at the easy calculus books [syl thompson/JE Thompson/Sherman Stein/calculus for electronics 60s McGraw-Hill/ and things like courant and john.. and hardy... [apostol then was way too formal and scary for me then]

same thing liking old physics texts, Feynman and the Berkeley Series, Resnick, Kleppner, Symon, Reif, PSSC]

i thought it was interesting that the books were EASIER in the old days
yet they were HARDER too...

like they didnt forget what's so sorely needed for people to get up to speed, and slowly learn how to study properly...

But i think the newer textbooks are superior with way more examples [Schaum's outlines were there for a reason!] and sometimes way more problem sets.

I just thought that there was a time where the best easy math/science books and the best/harder textbooks were just passed off as unsuitable by the curriculum because they didnt *easily* fit...

and well, i see nothing at all wrong with textbooks written for people who got problems crawling... or courses at higher institutions that teach people from zero math or zero physics [and do it well]... as well as making courses hard to fail if you 'follow the teacher's recommendations'...
otherwise, nothing at all wrong with repeating a class 7 times till you get it right, and go to the next rung of the ladder [I think there's something ungood in the fabric of schools of, if you didnt get a B, get out and try another career]... not a good tendency at all.

- no book is too hard for a serous student

especially true if you know how to tackle it, and eliminate any teacher or exam or grade stresses...

I found it so liberating to know that true self-accomplishment came from trying to tackle just one chapter as best as one can, and to keep plugging the hours into it, if it takes 8-15 hours, unlock the secrets of all the examples, reread the text carefully, and well enjoy the text once you're soaking in 98% of it, and try to see that the problems are meant to be totally taken as a whole, and it should all be workable with the 30 pages studied...

too many people fall into a trap of accomplishments by 'passing a whole course' or
'passing an exam'

and i think that's really a good way of not getting the most out of a text, the accompliment is mastering just one chapter...

doing 2 chapters [knowing it inside and out] and not touching the rest of the book says more to me, than taking 3 courses and getting 57%...

and i think i know both of those extremes well in my earlier days

i think there was a slow transition from lectures to textbooks from the 1910s to the 1960s... a good example is a lot of the early quantum stuff, there wasnt a textbook for a while, and for years it was lectures and readings of papers, and sometimes 3 people and a teacher trying it out...

and as the decades flowed, the textbooks got easier in some ways, and there's a lot of interesting stuff out there, now...

I think textbooks are really highly polished lecture notes...

but remember there's lots of geniuses in math or physics, who didnt rely much on teachers or the curriculum to start off their box of tools. They didnt wait till Algebra 11 or Physics 11 or Resnick or Courant... they soaked in a few textbooks and library books on their own, and then at a higher stage, fell into place into following the 'syllabus and curriculum'...

I think all the hope is placing a ton of effort into the lower stuff... and to make people do more than 97% of the others...

it feels like 3% of the people who did algebra, will get into a calculus text...

or 97% of people in first year physics people won't take a course in intermediate mechanics...

and i think we stopped making things 'friendly yet DEEP' at the elementary levels too, where i think the 1960 and 1965 PSSC system worked, and then the curriculum killed it for being too weird, too deep and spending months before you learned 'mechanics', and well, how many high schools or colleges or unis teach calculus with Syl Thompson's Calculus Made Easy?

I think that would be a great class for people, for credit or no credit at all. And it might toss people the courage to get into good books like Spivak.

i think we need lots of easy classes for algebra and physics for the clueless, hell in only a few weeks or months you can slowly show them how to study things deeply too.. but the biggest impediments i think are, the hoops and ladders to get a good solid background in algebra, or basic physics these days, if not also the financial strains of society that keep growing, and unis going from nearly free, to nearly impossible things to pay for.

My mantra is teach a student to only be 'serious' about learning *one chapter*

i think it's way easier than mastering 'one course'...

mathwonk · Oct 5, 2012

i agree one can only learn one topic at a time. i try not to worry about how few books i have read completely, and focus only on how many individual topics i understand thoroughly. E.g. i think I understand the (classical) Riemann Roch theorem pretty well now and am finally beginning to grasp the Riemann singularities theorem.

oh yes, and if you think about why courses like math 55 at harvard are so hard, you have to think about who they are aimed at. A friend of mine's son took that and flourished in it. But he prepared by taking not just a full and challenging math major sequence at UGA while he was in high school, but also took and starred in a number of graduate courses too, all before entering college and attempting math 55.

So this successful student was essentially at the advanced graduate level before taking what is listed as a second year advanced calculus class. Oh yes I believe he also took and did well in the (college level) Putnam exam while a high school student.

So don't believe what it says in the catalog about some of these courses and wonder why you and I were not able to deal with them, when all we had was the actual stated prerequisites. I bombed in 3rd year college french too after 2 years of high school french, in a class in which every other student had taken 4 or more years of french, some had taken 8, and at least one had lived in france. One of my friends who tried to take first year italian was the son of an Italian employee at the Italian consulate, and they spoke italian as the primary language at home.

RJinkies · Oct 5, 2012

- i agree one can only learn one topic at a time. i try not to worry about how few books i have read completely, and focus only on how many individual topics i understand thoroughly.

When you get higher up, yeah, you go from books to concepts...------

Well, some courses are there to teach you, and sometimes try for coherence in letting most all of it to soak in...

And the other courses who are for people who are self-taught who bring their own advanced box of tools, and there's little unity and *no one* soaks in more than 70% of the material. But if you like esoteric cannonballs fired at you, fine... I'd rather just get the reading list and some structured outline of 'what to read when' and do it way way more slowly...

Not to mention, i wonder how the course changed through the decades with the outside readings, and such... The good side is people are exposed to a 'lot', but it's a rush job...

seems like in the glory days of the 60s, you just had Loomis...
[well with Fleming and Dieudonne too]

now they throw Axler and Rudin at you, and add bits of
c. Counterexamples in Analysis
d. Korner's Fourier Analysis

and caltech does similar throwing at you
a. Rudin
b. Carothers
c. Elias Stein Complex Book [not well liked at all]

[people do like Carothers and Burn, both outta cambridge in the 80s and 90s...]

notes on carothers:
[I do agree with him on that the book is very informal in the exposition and is chatty. I feel that this might be very distracting for those who do not wish to be specialists in analysis, or to those who are seeing analysis for the first time. However, for someone who has finished, say Baby Rudin, this book IS AMAZING. His chatty 'foreshadowing' is the best part, since by now you are trying to see the 'big picture'. In this respect, the chattiness tells of the shortcomings of the previous theory and points one to the right questions to ask. ]
[When I first began using this book, I felt uncomfortable, since the tone of the author was so casual and might I say unprofessional.]

Axler people said that it was the closest thing in style, like if Spivak did a textbook on linear algebra [not sure if that's true or not]
most seem to think axler is better than average but not superb, but it's easy to read for an abstract linear book and good for self-study.

I just think to myself is that all Har 55 is, basically cramming Hardy's Pure Mathematics and Hoffman and Kunze asap into someone who wants to read 7 other books at the same time [and not the most friendly or approchable supplementary readings either]

I think 750 hours could be stretched out, so no one drops out... and well Binmore's book starts off easy enough and tells you a pretty good list of what to read in his three books and when to tackle Royden.

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- So don't believe what it says in the catalog about some of these courses and wonder why you and I were not able to deal with them, when all we had was the actual stated prerequisites.

what i would like to see is someone who's done a syllabus from the 1920s-1970s for all the big schools... some of the schools in the 50s actually would print the name of the textbook used in the calendar for a class...

i got a lot of neat insights looking at all the AJP Transcripts of famous physics people and teachers and listening to what textbooks they had in school or what they taught from..

found out Slater who was popular for writing first and second year textbooks in physics in the 30s and 40s, got his math from

EB Wilson - Advanced Calculus - Ginn 1912
all 566 pages of that.

and that's probably the oldest textbook of *any* use to people today...

mind you, sometimes that stuff is fragmentary

I think he used Osgood's mechanics, which is like Macmillian 1937, so maybe that's what he taught from before writing one in the 40s [Slater and Frank] which basically got pushed out by Synge and Griffith and later Symon.

[i found out Synge and Griffith was used in the 40s and 50s at Cornell because my copy i picked up in the used bookstore said PHY xxx Cornell 1950 in it] which is about the closest i got to Bethe or Feynman...

Slater used Abraham and Becker for Electromagnetism [1932 translation] and i still wonder why the Part II in German didnt get translated as well...

Slater also used James Jeans - The Mathematical Theory of Electricity and Magnetism - Cambridge 1925 5ed - for his EM classes

Leighton who worked with Feynman on the lectures went through Smythe's Static and Dynamic Electricity - McGraw-Hill 1939

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so it's a neat thing to see a fragmentary picture of what people used in uni back then, or taught from...

Still not sure what feynman used for his high school or calculus physics, but it was probably what he could 'find', and he was still jumping from Math to Electrical Engineering to finally Physics as the happiest balance between theory and applied...

Dowland · Oct 7, 2012

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

mathwonk · Oct 7, 2012

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

'http://archive.org/details/elementsalgebra00lagrgoog

RJinkies · Oct 8, 2012

hi Dowland
hi mathwonk

best write up on Euler's book is

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

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Euler's Elements of Algebra
Leonhard Euler, edited by Chris Sangwin
paperback - 276 pages (2006)
Tarquin Books $22[The style is engaging; the structure and language is clear, and the explanations logical. The approach is surprisingly modern and does not suffer either from being nearly 250 years old, or from being an edited version of a "charming" English translation from the 19th century. In fact, this English text comes from an 1822 English translation of a French translation of the original German. That such writing can still be called clear and readable is something of a miracle, and testament to Euler's original clarity and readability. This edition has excised various later accretions such as editors' footnotes and introductions, as well as an entire chapter added by Lagrange, material which may be reproduced if a reprint of Part II of Euler's work is ever attempted.]

[For me, the mystery of this old school textbook, which doesn't hold your hand and so seems to lead you rapidly through a ton of material, is that so much is conveyed in a spare, clean style. In fact, I expect that more material is covered than in more modern textbooks which spend an age going over and over material, and yet books like Elements seem less hurried than modern ones.]

[For example, Euler's definition of the integers seems to exclude zero. Later, he gives good reason to suppose that there is an infinity of numbers between two integers, but he couldn't know of the different "sizes" of those infinities which Georg Cantor discovered, and which a brief note might bring alive. He also anticipates the great utility of imaginary numbers. An index would also increase the usability of the book, especially for those interested in the history and development of mathematical concepts.]

[Overall, the book is to be highly recommended. The broad range of elementary topics means the book can and should be referred to often. The structure, readability, and standard of explanations lead to a rapid and rewarding learning experience, while the elegance of the prose is frankly a joy to read. The book soothes ageless anxiety caused by learning the mysteries of logarithms and imaginary numbers and yet does not shy away from addressing practical problems, even how to calculate interest — a footnote on the dangers of credit cards would go well here.]

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I'm not yet sold on it, anyone want to twist my arm?

A few years ago Springer in 3 vols did his calculus text, finally translated in English, seemed interesting enough off amazon for me to dump it in my 'neat' list... people seemed to like it browsing at what was essentially the first textbook on calculus...

[hold on let me drag it out]

32 Foundations of Differential Calculus - Leonhard Euler - Springer - $70
[The First calculus texts]
[more intuition than formalism]

33 Introduction to Analysis of the Infinite: Book I (Books 1 + 2) - Leonard Euler - Springer - $105

34 Introduction to Analysis of the Infinite: Book II - Leonard Euler - Springer - $90

If you got $275 kicking around... but it's probably a better and weirder read than new copies of Stewart or Thomas and Finney.

ovael · Oct 8, 2012

Heres a page I found some time ago. I can't comment on the quality of translations, but it contains lots of old works of math translated to english for free. Like Eulers "calculus" books.

http://www.17centurymaths.com/

Dowland · Oct 14, 2012

mathwonk said:

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

'http://archive.org/details/elementsalgebra00lagrgoog[/QUOTE]
I like Euler's writing style and his exposition of the subject. I also think Elements of Algebra contains a lot of interesting stuff that standard textbooks in Algebra does not contain. However, if one's purpose is to study and learn mathematical substance/skills from the book, it seems to lack of exercises. Do you perhaps know of any supplementary text/exercises to the book?

mathwonk · Oct 14, 2012

i am puzzled. the copy of euler i have linked contains hundreds of exercises.

RJinkies · Oct 15, 2012

- However, if one's purpose is to study and learn mathematical substance/skills from the book, it seems to lack of exercises. Do you perhaps know of any supplementary text/exercises to the book?

I think you're putting a modern question to a rather old book... Some people actually didnt like textbooks that tossed in a ton of problems, thinking the easy ones are just 'confidence builders' and this stuff are merely 'drills'... Yet the trend in the 50s and 60s and 70s were to put out new editions of textbooks with 30% more problems in the newer editions... [which what happenned with Resnick and Bueche's physics texts going into the 70's.]

There's a reason Schaum's outlines were popular...

and if you like problems there's always Chrystal's Textbook of algebra

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Yet there's two good essays out there

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

Albrecht Heefer has some neat comments about the book:

[now remember Euler's book is from 1770]

"In his selection of problems in the Algebra, Euler shows himself familiar with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. An extensive historical database with algebraic problems, immediately reveals Euler’s use of the Stifel’s edition of Rudolﬀ ’s Coss for his repository of problems. This work, published in 1525 in Strassburg, was the ﬁrst German book entirely devoted to algebra."

"Stifel used many problems from Rudolﬀ in his Arithmetica Integra of 1544 and found the work too important not to publish his own annotated edition. The ﬁrst volume of Euler’s Algebra on determinate equations contains 59 numbered problems. Two thirds of these can be directly matched with the problems from Rudolﬀ."

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"The third chapter dealing with linear equations in one unknown has 21 problems. They clearly show how Euler successively selected suitable examples from Rudolﬀ’s book. The problems are put in practically the same order as Rudolﬀ’s. They include well-known problems from recreational mathematics, ...the legacy problems, two cups and a cover, alligation, division and over- taking problems. The fourth chapter deals with linear problems in more than one unknown, including the mule and a-s-s problem, doubling each other’s money and men who buy a horse."

"The ﬁfth chapter is on the pure quadratic equation with ﬁve problems all taken from Rudolﬀ. The sixth has ten problems on the mixed quadratic equation, of which nine are taken from Rudolﬀ. Chapter eight, on the extraction of roots of binomials, has ﬁve problems, none from Rudolﬀ. Finally, the chapter of the pure cubic has ﬁve problems, two from Rudolﬀ and on the complete cubic there are six problems, of which four are from Stifel’s addition. Cardano’s solution to the cubic equation was published in 1545, between the two editions of the Coss. While Euler also treats logarithms and complex numbers, no problems on this subject are included."

"Having determined the source for Euler’s problems, the question remains why he went back almost 250 years. The motive could be sentimental. In the Russian Euler archives at St-Petersburg a manuscript is preserved containing a short autobiography dictated by Euler to his son Johann Albrecht on the ﬁrst of December, 1767. He states that his father Paulus taught him the basics of mathematics using the Stifel edition of Christoﬀ Rudolﬀ’s Coss. The young Euler practiced mathematics for several years using this book, studying over four hundred algebra problems. When he decided to write an elementary textbook, Euler conceived his Algebra as a self study book, much as he used Rudolﬀ’s Coss, the educational value of which Euler amply recognized."

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"Arithmetic books before the 16th century use a great many recipes to solve a wide variety of problems. With the emergence of symbolic algebra in the second half of the 16th century, many of these recipes became superﬂuous and the corresponding problems losttheir appeal. Several types of problems disappeared from arithmetic and algebra books for the next two centuries. The algebra textbooks of the eighteenth century abandoned the constructive role of problems in producing algebraic theorems. Problems were used only to illustrate theory and practice the formulation of problems into the algebraic language. The new rhetoric of problems in algebra textbooks explains why Euler found in Rudolﬀ ’s Coss a suitable repository of examples."

"A typical example of this type of problems is a legacy problem, which emerged during the late Middle Ages and is found in Fibonacci’s Liber Abbaci. It is a riddle about a dying man who distributes gold pieces to an unknown number of children, each receiving the same amount. With i children, each child gets ai plus (1/n)^th of the rest. The question is how many children there are and what the original sum is."

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"After Euler, many of the textbooks on elementary algebra of the 19th century include this and other problems from Rudolﬀ as excercises. In this way, Euler’s Algebra functioned as a gateway for the revival of Renaissance recreational problems."

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Christoff Rudolff’s influence

"In his selection of problems in the Algebra, shows himself familiar with the typical recreational and practical problems of Renaissance and sixteenth-century algebra books. Taking up the task of tracing the sources of these problems I found a strong similarity with the books by Valentin Mennher de Kempten. Originating from Kempten, in the south of Germany, Mennher was a reckoning master living in Antwerp. He published several books on arithmetic and algebra in French. His Arithmetique seconde, first published in 1556, has a large section with problems which are very similar to these of Euler’s Algebra. A close comparison shows that many problems from Euler could be reformulations of the text and values of Mennher’s problems. A German translation was published in Antwerp in 1560 for the German market. Possibly it circulated in Berlin where Euler might have been charmed by its pedagogical qualities. Still, why would Euler base his examples on a two-centuries old book from Antwerp, with so many alternatives at his disposal?"

"Lacking the crucial motive, I looked at later publications for the missing link. The eighteenth-century algebra treatise which matches Euler’s Algebra best is A Treatise of Algebra by Thomas Simpson (1745). This book was also indented as an elementary work in algebra, treating the basic operations on polynomials. It also has a large section on the resolution of equations as well as a chapter on indeterminate problems. Simpson’s book became highly successful as ten editions were released in the UK from 1745 to 1826 and at least three editions in the US from 1809. However, there are only about twenty problems which can directly be matched between Simpson’s and Euler’s books. In fact, Simpson’s problems show a better correlation with Mennher than with Euler."

"Recently, a digital version of Stifel’s edition of Rudolff’s Coss has become available. A fist glance reveal immediately evident that Euler used this book for his repository of problems. The original edition was the first German book entirely devoted to algebra."

"It was published in 1525 in Strassburg under the title 'Behend vnnd Hubsch Rechnung durch die kunstreichen regeln Algebre so gemeincklich die Coss genennt werden'. Stifel used many problems from Rudolff in his own Arithematica Integra of 1544 but found the work too important not to publish his own annotated edition in 1553, 'Die Coss Christoffs Rudolffs mit schonen Exempeln der Coss'."

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"Having determined the source for Euler’s problems, the question remains about his motive for going back almost 250 years. The motive could be sentimental. In the Russian Euler archives at St-Petersburg is preserved a manuscript containing a short autobiography dictated by Euler to his son Johann Albrecht on the first of December, 1767 (Fellmann 1995). He states that his father Paulus Euler taught him the basics of mathematics with the use of the Stifel edition of Christoff Rudolff’s Coss (Stifel, 1553). The young Euler practiced mathematics for several years using this book, studying over four hundred algebra problems."

"When he decided to write an elementary textbook on algebra, he must have had in mind the first mathematics book he owned. The book was to be used for self study, in the same way that he had used Rudolff’s book. As the many examples from Rudolff had helped Euler to practice his algebraic skills, so would he also include many aufgaben related to the resolution of equations. So while the motivation to use a sixteenth-century book may have been partly sentimental, the recognized educational value of algebraic problem solving was an important contributing factor."

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"Given that Euler’s Algebra is separated from Rudolff’s Coss by more than two centuries of algebraic practice, the structure of both works is rather close."

"Rudolff treats the same subjects but his organization reflects more the tradition of medieval algorisms. For each of the different species, whole numbers, fractions, etc, he first gives the numeration and then discusses the possible operations which he calls algorithms. The rest of Rudolff’s book consists of eight sections on the eight rules of algebra. These correspond with linear equations, the six Arab types of quadratic equations and the cubic equation with only the cube term. A division into eight equations is a
simplification of the 24 types given by Johannes Widman (Codex Leipzig 1470). As the subdivision of quadratic equations in separate rules disappeared in the early seventeenth century, Euler’s arrangement is different. He has separate sections on linear problems in one unknown, linear equations in multiple unknowns, the pure quadratic equation, the mixed quadratic, the pure cubic and the complete cubic equation."

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"The third chapter dealing with linear equations in one unknown has 21 problems. They clearly show how Euler sequentionally selected suitable examples from Rudolff’s book. The problems are practically in the same order as in Rudolff (1553)."

"The fifth chapter is on the pure quadratic with five problems all taken from Rudolff. The sixth has ten problems on the mixed quadratic equation, of which nine are taken from Rudolff. Chapter eight, on the extraction of roots of binomials, has five problems, none from Rudolff."

"Finally, the chapter of the pure cubic has five problems, two from Rudolff and on the complete cubic there are six problems, of which four are from Stifel’s addition. While Euler also treats logarithms and complex numbers, he included no problems on this subject."

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"The English edition of John Hewlett adds 51 ‘problems for practice’. It is not clear where they originate from, as they do not appear in the French edition (Euler 1774). It seems doubtful that the bible translator Hewlett (1811) added the problems himself. In any case, they were not selected by Euler."

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There you go...

RJinkies · Oct 15, 2012

Euler II
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I think this part is one of the more interesting parts in the Euler paper, showing the origins of some of the problems and how we approached them...

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Phases in rhetoric development of treatises on algebra - The medieval tradition

"One of the first Latin problem collections found in the Western world is attributed to Alcuin of York under the title Propositiones ad Acuendos Juvenes or Problems to Sharpen the Youth. The text dates from around 800 and consist of 53 numbered problems with their solution. As an example let us look at problem 16 on Propositio de duobus hominibus boves ducentibus, appearing twice in the
Patrologia Latina"

Two men were leading oxen along a road, and one said to the other: “Give me two oxen, and I’ll have as many as you have.” Then the other said: “Now you give me two oxen, and I’ll have double the number you have.” How many oxen were there, and how many did each have?

Solution. The one who asked for two oxen to be given him had 4, and the one who was asked had 8. The latter gave two oxen to the one who requested them, and each then had 6. The one who had first received now gave back two oxen to the other who had 6 and so now had 8 which is twice 4, and the other was left with 4 which is half 8.

"The rhetorical structure of these problems is that of a dialogue between a master and his students and is typical for the function of quaestiones since antiquity. Rhyme and cadence in riddles and stories provided mnemonic aids and facilitated the oral tradition of problem solving. Many of the older problems are put in verse. Some best known examples are 'Going to St-Yves' using the geometric progression 7 + 7^2 + 7^3 + 7^4, (Tropfke 1980). We know also many problems in rhyme from Greek epigrams19 such as Archimedes cattle problem (Hillion and Lenstra, 1999), the a-s-s and mule problem from Euclid (Singmaster, 1999) and age problems (Tropfke 1980). During the Middle Ages complete algorisms were written this way, taking over 500 verses (Karpinski and Waters, 1928; Waters, 1929). Even without rhyme, problems were cast into a specific cadence to make it easier to learn by heart. The 53 problems of Alcuin clearly show a character of declamation, specific for the medieval system of learning by rote. Medieval students were required to calculate the solution to problems mentally and to memorize rules and examples. The solution depends on precepts, easy to remember rules for solving similar problems, and adds no explanation."

"The structure of a problem as a dialogue between master and student is also explicitly present in early Hindu mathematical writings. These treatises consist of long series of verses in which a master challenges a student with problems. An example from the Ganitasarasangraha of Mahavīra is as follows:

(Padmavathamma and Rangacarya 2000, stanza 80 1/2):

'Here, (in this problem,) 120 gold pieces are divided among 4 servants in the proportional parts of 1/2 , 1/3 , 1/4 and 1/6. O arithmetician, tell me quickly what they obtained.'

The student is addressed as friend, arithmetician or learned man and is defied in solving difficult problems. In one instance, Brahmagupta states in his Brahmasphutasiddhanta of 628 AD that (Colebrooke 1817):

He, who tells the number of [elapsed] days from the number of days added to past revolutions, or to the residue of them, or to the total of these, or from their sum, is a person versed in the pulverizer.

Thus someone who is able to solve this problem on lunar revolutions, should have memorized the verses describing the Kuttaka or pulverizer method for solving indeterminate problems. Literally stated, the memorization of the rules formulated in stanzas by the master is a precondition for problem solving. Hindu algebra is based on the reformulation of problems to a format for which a memorized rule can be applied. The rhetorical function of problems in medieval, as well as Hindu texts, is to provide templates for problem solving which can be applied in similar circumstances.

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Aint that cool?

RJinkies · Oct 15, 2012

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

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

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

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

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

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

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

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

------

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

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

Squaring these two numbers gives

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

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

-------

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

[More painful word problems anyone?]

RJinkies · Oct 15, 2012

Euler IV

The Beginning of Algebraic Theory: from Pacioli to Cardano

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

The amalgamation of the algorism with the abacus text

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

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

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

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

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

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

---------

Extracting general principles from algebraic practice

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

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

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

x/y = y/z

x+y+z=a

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

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

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

we can write this also as

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

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

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

-----

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

RJinkies · Oct 15, 2012

Euler V
--------

Algebra as a model for method and demonstration

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

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

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

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

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

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

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

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

-------

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

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

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

--------

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

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

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

-------

RJinkies · Oct 15, 2012

Euler VI
---------

The generalization of problems to propositions

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

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

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

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

...

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

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

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

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

-----------

An attempt at an axiomatic theory

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

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

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

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

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

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

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

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

etc etc...

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

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

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

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

------

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

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

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

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

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

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

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

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

-------

RJinkies · Oct 15, 2012

Euler VIII
---------

The generalization of problems to propositions

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

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

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

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

...

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

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

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

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

-----------

An attempt at an axiomatic theory

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

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

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

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

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

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

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

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

etc etc...

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

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

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

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

------

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

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

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

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

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

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

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

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

-------

RJinkies · Oct 15, 2012

Euler IX
---------

Practicing the algebraic language

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

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

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

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

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

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

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

ax+b=cx-d=y

it can be solved as

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

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

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

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

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

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

x+y=c
ax-by=d

with solutions

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

and

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

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

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

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

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

-----

Conclusion

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

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

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

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

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

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

FIN

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

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

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

Dowland · Oct 15, 2012

mathwonk said:

i am puzzled. the copy of euler i have linked contains hundreds of exercises.

Well, maybe there are enough exercises then. I guess I just got a little suspicious of the fact that there are no "Questions for Practice" after several topics. I probably have to take a deeper look into the material.

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

RJinkies · Oct 16, 2012

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

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

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

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

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

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

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

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

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

----------

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

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

and Dolciani and Munem's Algebra books...

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

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

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

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

Institutions - University of Western Ontario, University of Toronto

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

---------

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

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

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

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

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

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

Here's my notes on that book

------

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

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

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

[Splendid and illuminating algebra text]

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

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

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

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

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

[A few novelties]

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

Actually one thing to remember...

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

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

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

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

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

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

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

...

Dowland · Oct 17, 2012

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

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

RJinkies · Oct 17, 2012

- Thank you for that Algebra-list!

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

-----

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

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

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

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

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

from like a choice of a dozen books...

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

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

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

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

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

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

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

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

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

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

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

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

-------

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

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

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

they were always great as the BLACK and TAN books
or the BLUE and PINK and GREEN quilty books

mathwonk · Oct 17, 2012

for old books, (around 1900), i like goursat's course in analysis, 3 volumes, recommended by the brilliant russian mathematician, arnol'd.

RJinkies · Oct 18, 2012

mathwonk - for old books, (around 1900), i like goursat's course in analysis, 3 volumes, recommended by that cranky brilliant russian mathematician, arnol'd.

Not that easy to find at one time, since libraries usually got tons of the french copies and there were 1910's translations too by Ginn and sold em for a few decades, and dover finally did a Phoenix hardcover of them. Hardy was really a huge fan of Goursat and that was a influence for him.

I remember seeing stuff nicely stated, but i wondered just how good one's french would need to be to tackle that and what level someone to be tackling it, with no analysis, with some analysis etc etc... but i do remember out of the blue little hunks of set theory would be tossed in with wonderfully crisp and strange fonts and then there was talk about a Jordan curve, and only later with Parke i said, oh it's in english, funny how the uni library didnt have a copy of that.

-----

did you like any of the 60s and 70s calculus texts out there, aside from the one's you mentioned and used? I remember seeing a lot of odd experimental 60s texts, maybe 70% of them seemed stiff with a lot of bland robotic New math formalism, which felt like all the set theory and analysis parts of Dolciani hatched on you. the 1960s New American Landau...

there were a lot of other texts out there than thomas/finney and Stewart... thinks like Campbell and Dierker or Harley Flanders gets lost in the cracks of out print books

I thought Campbell/Dierker [late 70s] was dull and i think Marsden and others did a good book in the 70s and 80s for calculus, maybe not the most gentle though. Flanders i liked but i was bothered a great deal with his suggestions for students to do all these sloppy freehand like scribbles and stuff

-----

Parke does mention goursat for advanced calculus
[things like Goursat/Hardy/Franklin's other book/Rudin are really analysis courses but closely bundled here]

Calculus: 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
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 pages

[I got Woods, Courant, Franklin [his analysis book, the calculus one is in the basic calculus list], Sokolnikoff [i thought it was way easier than Thomas and Finney and gentle], Hardy, and i think i got either Widder or Kaplan, but not both, maybe]

[I seen Edwards, and Landau and Rudin but didnt find any copies when i was collecting them]

what else would you [or anyone else] name drop from the 50s 60s 70s, that you haven't mentioned before that you think might be almost as neat as Courant, Kaplan or Widder?

I used to think that pre 1970 most any McGraw-Hill text was mainstream in the schools, and well anything from the 40s to now Addison-Wesley never did a bad textbook.

I always wondered why Ginn, Macmillian and also Blakiston would seem to be receding from the 50s to now from the textbook market, usually Ginn and Macmillian always wanted to cater to the old style high school textbooks, and i think as competition grew esp after sputnik, they both shrank, but people still use a number of their classic texts for reading...

lets add the MAA elementary calc books:

MAA: Elementary Calculus

1968
Levi, Howard. Polynomials, Power Series, and Calculus New York, NY: Van Nostrand Reinhold, 1968.
*** Thomas, George B., Jr. and Finney, Ross L. Calculus and Analytic Geometry, Reading, MA: Addison-Wesley, 1968, 1987. Seventh Edition.

1972
Dorn, William S.; Bitter, Gary G.; and Hector, David L. Computer Applications for Calculus Boston, MA: Prindle, Weber and Schmidt, 1972.

1975
Swokowski, Earl W. Calculus, Boston, MA: PWS-Kent, 1975, 1991. Fifth Edition.

1976
* Keisler, H. Jerome. Foundations of Infinitesimal Calculus Boston, MA: Prindle, Weber and Schmidt, 1976.
* Keisler, H. Jerome. Elementary Calculus, Boston, MA: Prindle, Weber and Schmidt, 1976, 1986. Second Edition.
* Lax, Peter; Burstein, Samuel; and Lax, Anneli. Calculus with Applications and Computing New York, NY: Springer-Verlag, 1976.

1977
* Goldstein, Larry J.; Lay, David C.; and Schneider, David I. Calculus and Its Applications, Englewood Cliffs, NJ: Prentice Hall, 1977, 1990. Fifth Edition.
* Kline, Morris. Calculus: An Intuitive and Physical Approach, New York, NY: John Wiley, 1977. Second Edition.

1979
Henle, James M. and Kleinberg, Eugene M. Infinitesimal Calculus Cambridge, MA: MIT Press, 1979.
** Priestley, William M. Calculus: An Historical Approach New York, NY: Springer-Verlag, 1979.

1980
* Anton, Howard. Calculus with Analytic Geometry, New York, NY: John Wiley, 1980, 1988. Third Edition.
Bittinger, Marvin L. Calculus: A Modeling Approach, Reading, MA: Addison-Wesley, 1980, 1988. Fourth Edition.
** Spivak, Michael D. Calculus, Boston, MA: Publish or Perish, 1980. Second Edition.

1982
** Stein, Sherman K. Calculus and Analytic Geometry, New York, NY: McGraw-Hill, 1982, 1987. Fourth Edition.

1985
Ash, Carol and Ash, Robert B. The Calculus Tutoring Book Los Angeles, CA: IEEE Computer Society, 1985.
* Hamming, Richard W. Methods of Mathematics Applied to Calculus, Probability, and Statistics Englewood Cliffs, NJ: Prentice Hall, 1985.
* Marsden, Jerrold E. and Weinstein, Alan. Calculus, New York, NY: Springer-Verlag, 1985. Second Edition.
* Simmons, George F. Calculus with Analytic Geometry New York, NY: McGraw-Hill, 1985.

1988
Grossman, Stanley I. Calculus, San Diego, CA: Harcourt Brace Jovanovich, 1988. Fourth Edition.

1989
Berry, John; Norcliffe, Allan; and Humble, Stephen. Introductory Mathematics Through Science Applications New York, NY: Cambridge University Press, 1989.

1990
* Finney, Ross L. and Thomas, George B., Jr. Calculus Reading, MA: Addison-Wesley, 1990.
Fraleigh, John B. Calculus with Analytic Geometry, Reading, MA: Addison-Wesley, 1990. Third Edition.
Seeley, Robert T. Calculus San Diego, CA: Harcourt Brace Jovanovich, 1990.
Small, Donald B. and Hosack, John M. Explorations in Calculus with a Computer Algebra System New York, NY: McGraw-Hill, 1990.
Small, Donald B. and Hosack, John M. Calculus: An Integrated Approach New York, NY: McGraw-Hill, 1990.

1991
Feroe, John and Steinhorn, Charles. Single Variable Calculus with Discrete Mathematics San Diego, CA: Harcourt Brace Jovanovich, 1991.
** Strang, Gilbert. Calculus Wellesley, MA: Wellesley-Cambridge Press, 1991.

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

and the higher up MAA calculus books:

MAA: Advanced Calculus

1937
*** Courant, Richard. Differential and Integral Calculus, New York, NY: Interscience, 1937. 2 Vols.

1952
* Kaplan, Wilfred. Advanced Calculus, Reading, MA: Addison-Wesley, 1952, 1984. Third Edition.

1956
* Knopp, Konrad. Infinite Sequences and Series Mineola, NY: Dover, 1956.

1959
*** Hardy, G.H. A Course of Pure Mathematics New York, NY: Cambridge University Press, 1959.

1962
Hildebrand, Francis B. Advanced Calculus for Applications, Englewood Cliffs, NJ: Prentice Hall, 1962, 1976. Second Edition.

1965
Bromwich, Thomas J. l'Anson. An Introduction to the Theory of Infinite Series New York, NY: Macmillan, 1965.
** Buck, R. Creighton. Advanced Calculus, New York, NY: McGraw-Hill, 1965, 1978. Third Edition.
Landau, Edmund G.H. Differential and Integral Calculus New York, NY: Chelsea, 1965.
*** Spivak, Michael D. Calculus on Manifolds Reading, MA: W.A. Benjamin, 1965.

1967
*** Apostol, Tom M. Calculus, New York, NY: John Wiley, 1967, 1969. 2 Vols., Second Edition.

1969
Cronin-Scanlon, Jane. Advanced Calculus: A Start in Analysis, Lexington, MA: D.C. Heath, 1969. Revised Edition.
Fulks, Watson. Advanced Calculus, New York, NY: John Wiley, 1969, 1978. Third Edition.

1972
Williamson, Richard E.; Crowell, Richard H.; and Trotter, Hale F. Calculus of Vector Functions, Englewood Cliffs, NJ: Prentice Hall, 1972. Third Edition.

1973
** Schey, H.M. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus New York, NY: W.W. Norton, 1973.

1974
Sagan, Hans. Advanced Calculus of Real-Valued Functions of a Real Variable and Vector-Valued Functions of a Vector Variable Boston, MA: Houghton Mifflin, 1974.

1976
** Marsden, Jerrold E. and Tromba, Anthony J. Vector Calculus, New York, NY: W.H. Freeman, 1976, 1988. Third Edition.

1980
Amazigo, John C. and Rubenfeld, Lester A. Advanced Calculus and Its Applications to the Engineering and Physical Sciences New York, NY: John Wiley, 1980.

1982
* Simmonds, James G. A Brief on Tensor Analysis New York, NY: Springer-Verlag, 1982.

1983
Taylor, Angus E. and Mann, W. Robert. Advanced Calculus, New York, NY: John Wiley, 1983. Third Edition.

1984
Price, G. Baley. Multivariable Analysis New York, NY: Springer-Verlag, 1984.

1985
* Marsden, Jerrold E. and Weinstein, Alan. Calculus III, New York, NY: Springer-Verlag, 1985. Second Edition.

1986
Grossman, Stanley I. Multivariable Calculus, Linear Algebra, and Differential Equations, New York, NY: Academic Press, 1986. Second Edition.

1987
* Widder, David V. Advanced Calculus, Mineola, NY: Dover, 1987. Second Edition.

1988
Bamberg, Paul and Sternberg, Shlomo. A Course in Mathematics For Students of Physics, New York, NY: Cambridge University Press, 1988, 1990. 2~Vols.
Magnus, Jan R. and Neudecker, Heinz. Matrix Differential Calculus with Applications in Statistics and Econometrics New York, NY: John Wiley, 1988.

1989
** Courant, Richard and John, Fritz. Introduction to Calculus and Analysis, New York, NY: Springer-Verlag, 1989. 2 Vols. [this should be like 1965-1966]

1990
* Knopp, Konrad. Theory and Application of Infinite Series Mineola, NY: Dover, 1990.
Loomis, Lynn H. and Sternberg, Shlomo. Advanced Calculus, Boston, MA: Jones and Bartlett, 1990. Revised Edition.

1991
* Bressoud, David M. Second Year Calculus New York, NY: Springer-Verlag, 1991.

--------

there has to be some mid-late 50s , 60s 70s 80s books slipped through the cracks with the MAA list...

again, i always wondered why they cut out the older titles, their recommendations would keep things in print, and i guess there is always a bias for something 'new' to keep 'some' people busy lol

[and i would definitely think that the dates and judging of the different editions would have been liked too]

----------

mathwonk · Oct 18, 2012

all the books in those days were good. even the routine books were good, like kiokemeister; thomas; smith, granville and stewart?,

then there were good books that never caught on like the one by lipman bers, and quirky but interesting ones like harley flanders, or m.e.munroe.

strong classic style books included buck, widder, and so on.

the three excellent new 1960's books were by apostol, spivak, and joseph kitchen.

flaky, over the top books included spencer steenrod and nickerson, which i have never heard of anyone learning from.

another one is loomis and sternberg.

really good new advanced calc books include wendell fleming, spivak, apostol.

but really one only needs one good book to read.my goursat is translated into english, by hedrick i think, indeed pub. by Ginn, 1904(vol 1), 1916 (vol2), 1917(vol3).

RJinkies · Oct 18, 2012

- all the books in those days were good. even the routine books were good, like kiokemeister; thomas; smith, granville and stewart?

So true... i just remember thinking that all the 60s textbooks for physics and math seemed to offer just all that 'extra' detail. And one felt that a lot of books or texts where they cram something into 'one semester' should be two semesters with books like that. [Which was true of Symon for mechanics]

A slower pace, as well as starting with basics and handholding for the first 100 pages, but by the time you're at the end of the book, you feel like you were pushed into something like the top elite of Feynman's lectures or Hardy..

I think it was easier say before 1968, or possibly in other things before Sputnik where you almost got algebra and calculus in your first year uni, because the schools didnt trust the high schools to teach alike... As well as showing you 'how to study'...The older books are harder, occasionally clunky, and sometimes a more leisurely pace, or 30% to 300% times the content of most 70s 80s 90s texts that are all watered down.

but if you can get through them, or part way through... [one chapter is a lifetime of accomplishment - is one of my mantras] i think you come out with a pretty neat outlook and a unique box of tools...

and that goes for a schaums outline as well. There's something there about just seeing example upon example, almost like 20 fold what a teacher could explain on a blackboard and if you're lazy, or you're really out to bash yourself out knowing some of them inside out... you can definitely get something different out of the experience.

I think schaums outlines threw out a need for 'more problems' and 'worked out problems' which is basically what a teacher does at the damn blackboard. Which in fact makes something like a schaum's outline way way more something to treasure than taking a class and seeing the lectures.

ideally you got a great text, great supplementary texts, infinite hours to burn reading, and doing problems, enough patience to see that xx hours a week will get results out of a textbook, a great teacher and hopefully something like a schaum's outline.

But i think a lot of physics books and math books in the late 60s started to wake up and by the 70s you saw books adding more problems and then more problems, and still more worked out problems, to address what they lacked, and what schaum's offered.

I heard a rumour that schaum's outlines were like collections of problems from all the major texts of the day, and if you had like all three of the major textbooks in physics and did ALL the problems and then went though all the schaums outlines, you'd say, hey, that's nothing new, oh wait, this problem looks identical to what i just worked out... etc etc

----

one reason the texts were so good, people realized they had one shot to make it, and proof read it, so there weren't any mistakes or typos... now people just seem to rush it, and new editions seem like half hearted rewrites that often create more errors and errata.

some of the great books, the new edition was something with 4 new chapters at the end, and 99% of the text wasn't tampered with, just new problems.

occasionally people will totally tighten up one or two chapters, or merge them... or break one big chapter into two.

but i liked textbooks where there wasnt a lot of editions, or dramatic changes through the editions...

one of the bothersome things, was something like an electromagnetics text, where, each edition had radically different problems, and if you had like 3 or 4 of the major editions, you got the 'whole' story. Where you don't get that with Halliday and Resnick or Symon or Courant. They dump the whole BRICK on you.

I remember looking at the 60s Courant and John, and thinking, how incredibly hard this gold brick is, but yet there's something really attractive here with it going over the top, and going into stuff that no other calculus text does, but it feels like 8 times the effort to get through a chapter.

Yet, it took years before i found a courant, and i found a 50s blackie edition of that interscience classic. a few years later i found a second copy but it was only one of the two books, orphaned, and i still didnt see courant and john outside of a library...

and they feel totally different in the beginning...kiokemeister - don't know that one

johnson and kiokemeister - 1974 Allyn and Bacon
is that the one?
brown 1978 - 6th edition
goes back to 1960 at least...

lipman bers

228 Calculus - Lipman Bers [and Frank Karal] - Second Edition - Holt Rinehart and Winston 1976
[crystal clear explanations]
[I had come across this book in the university library. Before that I had been getting excellent marks in Calculus by mechanically going thru the rules in my mind. This book changed all that and gave me a proper perspective on the discipline. The explanations are clear and this book is eminently suitable for self study.]
[Recommend this book whole-heartedly at least for the first and second years of calculus. This was about twenty-five years ago! But it's just as relevant now.]
[Barry says Bers was a Russian mathematician emigre to the U.S. who was familiar with Russian textbooks.]
[First Edition] ?
[Second Edition] 1976

m.e.munroe - don't know much about that one

joseph kitchen

88 Calculus of one variable - (Addison-Wesley series in mathematics) - Joseph W Kitchen - Addison-Wesley 1968 - 785 pages
[a superb honors level book - Mathwonk]
[This book is on the level of the ones by spivak, courant, and apostol, and is very modern]
[mathwonk - a nice book - but if the price gets astronomical on the used market then it is absurd to buy it, and that almost anyone of the other good books will give you an enormous amount of education - Fleming or Dieudonne or Courant for cheap is better than an expensive copy of kitchen in the short term]

------

outstanding commentary mathwonk...

any other quirky calculus books, from strange disasters to forgotten failures that were pretty cool?

i had the feeling that from 1963 to 1975, you could tell what book was going to be like because of the crazy graphic design...

mathwonk · Oct 18, 2012

http://www.abebooks.com/servlet/SearchResults?an=m.e.munroe&tn=calculusanother quirky but good one: lectures on freshman calculus by cruse and granberg.

i was asked to referee this in 1970, but did not realize the referee report was not supposed to contain any criticism. this is an excellent book, but in my picky way i pointed out the errors in it. i was very upset that the published version did not correct any of the errors i observed. this made me cynical and think that the referee report as a scam.

it did not go far in the market, although it had my favorite quality of carefully and beautifully motivating each main concept. maybe if i had written them a better review it would still be available.

https://www.amazon.com/dp/B002PCY21O/?tag=pfamazon01-20the thing i remember from bers' book was the remark: "calculus is essentially about solving differential equations." i didn't know that, and i appreciated being told that. to me as a student, d.e. was just an annoyingly unenjoyable course i had to take, with tedious solution techniques to memorize and little interesting theory. i didn't realize it is possible to appreciate problems, that have no easy solutions. the problems had not been clearly stated to me. i.e. a d.e. is a vector field. that has beautiful geometry.

RJinkies · Oct 19, 2012

Part I

so the messy part of cruse and granberg was something to do with Decartes' method of tangents on parabolas and how you're only suppossed to cut the curve once?
Was the problem with with complicated functions the tangent might be hitting multiple points with higher degree curves or somethings? [at least that's the gist of the complaint on amazon]. Didnt other textbooks use that method, and fall into similar traps? or it was just something that worked for some equations and not higher degree equations where it could be strange or messy.

[watch out it's sin(x)/x]

- Any thoughts on:
a. Calculus for the Practical Man - JE Thompson [1931/fixed up 1948 - and obsessed with rates and flows]
b. Quick Calculus - Kleppner
[wasnt that late 60s or early 70s, i don't remember if it came out before or after Introduction to Mechanics... but i thought it was a great gesture, all you need with high school algebra is my 'other book' to read 'my other book]

i was pretty skeptical of the need help with calculus textbooks but two textbooks after the 80s seem to be quite good

c. The Calculus Lifesaver: All the Tools You Need to Excel at Calculus - Adrian Banner - Princeton 2007 - 752 pages
[Banner's style is informal, engaging and distinctly non-intimidating, and he takes pains to not skip any steps in discussing a problem. Because of its unique approach, The Calculus Lifesaver is a welcome addition to the arsenal of calculus teaching aids. - MAA]
[I used Adrian Banner's The Calculus Lifesaver as the sole textbook for an intensive, three-week summer Calculus I course for high-school students. I chose this book for several reasons, among them its conversational expository style, its wealth of worked-out examples, and its price. This book is designed to supplement any standard calculus textbook, thus my students will be able to use it again when they take later calculus courses. The students in my class came from diverse backgrounds, ranging from those who had already seen much of the material to others who were struggling with basic algebra. They all uniformly praised the book for being one of the clearest mathematics texts they have ever read, and because it reviews the required prerequisite material. The numerous worked-out examples are an ideal supplement to the lectures. The only difficulty in using this book as a primary text is the lack of additional exercises in the text. However, there are so many sites and sources for calculus problems that this was not a problem. I would definitely use this book again. - Steven J. Miller, Brown University]
[some wonder about the lack of reinforcement]
[not the best for clarity]
[not always easy to follow]
[for volumes with shell and disks - far more complicated than main textbooks and still leaves out a lot of explanation]

d. How to Ace Calculus

and then there is
e. Calculus - Elliot Gootman - Barron's Educational Series 1997 - 342 pages
[said to be much better than the dummies book]
[and for some more useful than the how to ace book]
[how they do it - 'Once you master about twenty basic procedures, the rest becomes far more approachable. I recommend this book highly to those frustrated with standard textbooks or simply wishing to understand the basics of how calculus works.']

f. The Humongous Book of Calculus Problems: For People Who Don't Speak Math - W. Michael Kelley - Alpha 2007 - 576 pages
[Kelley does a great job of stripping away the gobbledygook and delivering you the nuts and bolts of calculus ON PAR with the "hardcore texts". There are many of those "hardcore" books, and they just don't teach well. What this author has done is to teach you how to solve the problems as well as the underlying logic.]

two older super-obscure classics that i found fascinating opinions on:

g. Calculus - Fadell - early 60s
[considered one of the neater post Sputnik calculus books]
[The best may be the book by Fadell also written in the early 60's which has some fantastic figures and a very unique treatment of calculus]

h. Differential and integral calculus - James Ronald Fraser Kent
[verbose older calc text]
[I picked up a used copy of this text based on the five star review that was given. I think this book proves that all of first year calculus can be covered in a compact book. It assumes the reader has mastered pre-calc math and does not waste time covering much of the pre-req material. However, this book still packs a maximum density of information given its size.]
[The Book is less than half the size of the prototype modern calc text.]
[The text is very wordy and broken down into compact subsections. At points, I felt the author could have done a better job explaining certain topics wi th less words and a few more equations. The figures are also not as good as in other older texts like Fobes or early editions of Thomas. However, this book is still much better than most of the calc books in print today. All in all a very decent older text that is worth digging up if you are into calculus pedagogy.]

-----

eek

i. Calculus and Pizza: A Cookbook for the Hungry Mind - Clifford A. Pickover - Wiley 2003 - 208 pages
[A must see for 9th and 10th grade high school students]
[one word this book is: Enthusiasm]
[This book is the simple solution to every young student avoiding complications in calculus later in life. I was given this book early on during basic algebra (which I wasn't great at). When I finished reading this book I didn't claim to know calculus: I skimmed the first couple chapters over and over. But, I had an idea of what people meant when they said "Calculus."]
[America's public educational systems lack the rigor that is required by its universities and colleges because students are not getting "very basic" ideas early on. This book is a definitive solution. Reading parts of this book in 9th or 10th grade can give students time to let the fundamental simplicity of calculus percolate, something that cannot be rushed in a semester.]
[Students don't need trigonometry, or advanced algebra. They need insight early on. If you're searching for a calculus book because you're having trouble with it now, do your younger friends a favor and recommend this book. It could mean the difference between success and failure when they transition from Precalculus to calculus. This book should be treated the same way astronomy and science survey books are written to inspire interest in young people. Move over earth, life, and health sciences and make some room for Calculus and Pizza - food for the hungry mind.]
[This book served to demistify the entire basics of the calculus for me. Without it, I'd still be wondering about the derivative, or about limits or integrals. On the other hand, it contains about 5% of the stuff in a real calc book, which is why I'm glad I've got both. Even today I refer back to this when the definitions Swokowski gives me are too obscure to understand.]
[If you have trouble understanding calculus, buy this, not a copy of Schaum's outlines. This will open you up to fundamental concepts, and once you have those down, reading any normal calculus text will be a breeze.]
[A really fun read, and you learn Calculus too]
[From the first couple of pages I felt as though I had been thrown in the deep end of the pool in order to learn how to swim. I was anticipating a more accessible book and I was disappointed. The examples of tomato sauce mold, rocket launched meatballs and giant pepperoni (don't ask) didn't serve to ground calculus in the real world for me. Again, maybe a terrific text for people that already have a grounding in the subject, but hardly as comforting as the title would lead you to believe.]

[Yeah, I'm probably the first person to bring up a book called calculus and pizza, but if it is a book that can teach someone calculus 4 years before most people encounter it in school, that's a good thing]

[I recall some book in a 1970 Edmund Scientic Catalogue that had some package or book [i think it was like a book with extra demo materials like cardboard cutouts or something] and the blurb was about how elementary school children could be taught ideas that are in calculus, and i thought that this pizza book is doing similar stuff, and well books that do this sort of thing are rarer than hen's teeth]

-----

the best newer textbook [yeah another Addison-Wesley book, how creepy is that... as i said they always put out good stuff]

j. Multivariable Calculus - William L. Briggs and Lyle Cochran - Addison Wesley 2010 - 656 pages
[used at UCLA]
[Most readable calculus book I've yet to come across]
[I was re-taking multivariable calculus this past semester (as kind of a filler class at the community college. I just had some general ed. class to take, so I thought I'd try calc III again and see if I would actually learn anything about vector calculus this time around). We were loaned out the paperback Multivariable edition of the Briggs/Cochran calculus book. One down-side of these copies - the ink smudged way too easily. But that's really not a factor in my four-star rating, I promise. ]
[I've managed to take long enough getting through school (as I mostly just take evening and online classes, what with working during the day) that I've used three different calculus books - Stewart, Thomas and now Briggs. Also, a friend and I are kinda math/physics junkies so we both have fairly extensive collections of Dover books and other various textbooks. Point being, I've come across a lot of different calculus books.]
[And this one has just become my favorite. It never feels dumbed-down (like Stewart did), and it's significantly more readable than Thomas calculus (which does Ok at times, then falls apart at other times). If you've happened to used the Knight physics textbook recently, the Briggs/Cochran book is similar in flavor - conversational yet extremely thorough. It still requires focused reading and plenty of practice, but at least the book won't be an obstacle to learning - as is the case with so many other textbooks in the math/physics world, I find.]
[Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher’s voice beyond the classroom. That voice–evident in the narrative, the figures, and the questions interspersed in the narrative–is a master teacher leading readers to deeper levels of understanding. The authors appeal to readers’ geometric intuition to introduce fundamental concepts and lay the foundation for the more rigorous development that follows. Comprehensive exercise sets have received praise for their creativity, quality, and scope.]

[Though I was a little skeptical about a first edition, my skepticism faded quickly after reading through the beginning of the book, particularly limits. Very, very good explanations and examples that thoroughly prepare the reader for the upcoming exercises. The definitions are great, and the graphics are very well laid out and explained. All in all, though I haven't read through the etire book yet, I have read enough Calculus books to know a good one from a bad one. This being a very good one.]
[Will never be as popular as Stewart's Calculus, and it is far from being a serious, self-respecting Calculus book - such as the one written by Apostol. Not a good text-book for students in Science and Engineering who need to have a better understanding of Calculus and applications, based on more serious Engineering and Physics-born examples, with more serious computations and proofs!]
[This book is actually pretty good, good for self study. But if you want a really good book, I would recommend Ron Larson's Calculus book instead.]
[The book would be great for a high school student who is trying out Calculus, but is not good at Math at all. It may be good for the Liberal Art student pursuing multidisciplinary studies: that is, a mixed salad of Humanities, Education, Social Sciences and Life Sciences, spiced up with some Calculus just for the sake of sounding like a true intellectual!]

[it's got some moody blue and black artwork on it too]

-----

k. Jerrold E. Marsden and Alan Weinstein - Calculus I, II, III - Springer-Verlag [came out in the late 70s or early 80s] second edition is 1985 is all i know about it.. and it was used at Berkeley, since i think Marsden is there and cranks out 3-4 textbooks through the decades...

I heard extremely little about it, any ideas on when it first came out, and how the different editions are, by anyone out there?
Im sure people didnt like 3 orangey yellow textbooks with 3 study guides and then you possibly get pushed into marsden's vector calculus textbook afterwards...

heck here is a neglected text these days from the 60s

l. What Is Calculus About? (New Mathematical Library) - W. W. Sawyer
[someone should talk about one of the first NML books, i thought they were one of the greatest ideas around, a huge series of books to supplement you from high school on up]
[i think the closest anyone came to something sort of like that might be the oxford chemistry series that had all these strange silver and back thin 80s paperbacks which were like 50-70 titles i think...]

[physics only had the anchor science series for teenagers, and man those arent easy to see, but you could always see a few in the bookstores of the 70s, usually the electronics book or some of the history books] It looked like so much promise in the 60s and it petered out in the 70s with the PSSC texts [or likely nixon gutting the libraries and education funding stuff that got pushed 1960-1968]

-----

another newish one that looks good

m. Calculus: The Elements - Michael Comenetz
[Best Textbook on Calculus - Concise & Fun to Read & Comprehensive]
[It's no doubt that Stewart's book is the most popular textbook on Calculus. It's comprehensive and standard. However, it's a pain to read through every page and do all the exercises.]
[In that regard, I've found Michael Comenetz 'Calculus: the Elements' most suitable for students without a solid background who intend to major in physics, math, chemistry, and engineering. Comenetz' book is not only comprehensive but very concisely written. Problems are well chosen - unlike Stewart's that has repetitive/similar problems all over the textbook. Yet, my advice would be 'keep Stewart's as a reference while learn from Comenetz's' This, based on my own experience, is the most effective to achieve high scores in tests and excellent grades.]

RJinkies · Oct 19, 2012

Part II

off topic but a 'friendly' book as in the rudin path to math texts

n. Advanced Calculus: A Friendly Approach - Witold A.J. Kosmala - Prentice-Hall - 700 pages - 1998
[I have copies of Rudin, Apostol, Bear, Fulks, and Protter, but this book beats them all as an introductory text. If you are looking for a self-study text, or if you want a reference companion to help you understand Rudin or Apostol, try this book first. You won't be disappointed.]
[The author of this book has used " a friendly approach " to present the stuff so that readers will actively be engaged in learning with less strain. This has not in a sense simplified the difficult elements of Calculus but bringing along the readers to think and reason while studying the subject.]
[Designed to be readable and intimidation-free, this advanced calculus book presents material that flows logically allowing readers to grasp concepts and proofs. Providing in-depth discussion of topics, the book also features common errors to encourage caution and easy recall of errors. It also presents many proofs in great detail and those which should not provide difficulty are either short or simply outlined. Throughout the book, there are a number of important and useful features, such as cross-referenced functions, expressions, and ideas; footnotes which place mathematical development in historical perspective; an index of symbols; and definitions and theorems which are clearly stated and well marked. An important reference for every professional who uses advanced math.]

For the last huff, jump in anyone...

o. Calculus With Analytic Geometry - 9th edition 2008 now...
[Ron Larson and Edwards] or [Larson, Hosteller and Edwards] - DC Heath and Brooks/Cole

people think the highest and lowest of this textbook, though it's been through a hell of a lot of editions, and i think in the 80s it flaked out with some computer gunk and then went back to basics...

the comments are all over the place *grin*[this isn't Edwards and Penney]
[liked by Alexander Shaumyan - New Haven, CT]
[easy to follow]
[it doesn't really explain things adequately]
[it skips too many steps in the examples]
[some think it's got a nice format and easy to follow]
[too software fixated with frills and fluff and fad though]
[Excellent treatise of 3-semester calculus. A classic]
[Decent text but by no means excellent - 3 out of 5 rating]
[if people complain this book makes calculus too simple, so what? If you are struggling and can't do the easy stuff, then how on Earth are you going to start doing the hard stuff later on?]
[i get the feeling this book isn't better than Sherman Stein's or Thomas and Finney really]
[starts off simple, but then goes into too many shallow applications, with skimpy second year stuff]
[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, for example, when introducing the limit, and also in that it leaves out some of the extraneous and confusing attempts at applications in the first chapter. I still think the book contains too many confusing applications from the second chapter onward. I do think the book would be improved by having a completely separate section covering the definition of the limit, however.]
[I like the prose in the examples. I like the presentation of some of the material from multivariable calculus. But again, this book is 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. This book doesn't cultivate the awe and wonder that should be present when a student learns calculus.]
[There is no text, in my opinion, more suited towards use in any introductory Calculus series, but this text is also ideal for self-study. The theory is presented in crystal clear fashion, and then multiple examples are given in order of increasing complexity.]
[just another junk book]
[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. I used this book for calculus 1 and 2. However, unlike my classmates, I learned all the material from an engineering math book (kenneth stroud, engineering mathematics).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. Most importantly, I can create these formulas if I need to, because I actually UNDERSTAND what is going on. By the way, I got an A+ in both courses, and I never bothered to learn the epsilon delta crap.]

i ain't got much of a timeline on the book but i got this much
[First Edition]
[Second Edition]
[Third Edition] [started to use computer generated graphs - ugh]
[Fourth Edition] 1993 [started to use computers and graphing calculators - ugh]
[Fifth Edition] [started to use a CD Rom - ugh]
[Sixth Edition] 1998 - 1316 pages [started to do stuff online - ugh]
[Seventh Edition]
[Eighth Edition] 2005 - 1328 pages - Brooks/Cole
[Ninth Edition] 2009 - 1328 pages - now just Larson and Edwards

oh one more

p1 and p2. Lang's simple and non scary calculus text, came out in like 1964 for a basic course, and through the changes in curriculum people found that it's still useful today...

p1. A First Course in Calculus - First edition - Lang - Springer 1964 - 264 pages
[reissued in the past decade as - Short Calculus - yeah the first edition is back]

p2. A First Course in Calculus - fifth edition - Lang - Springer 1998 -752 pages
[the bloated new editions]

the comments:
[simple, but not unsophisticated]
[As a high school teacher, I used this text with great success several times for both AP Calculus BC and AP Calculus AB courses. It is my favorite calculus text to teach from, because it is very user-friendly and the material is presented in such an eloquent way. There are no gratuitous color pictures of people parachuting out of airplanes here. Opening this book is like entering a temple: all is quiet and serene. Epsilon-delta is banished to an appendix, where (in my opinion) it belongs, but all of the proofs are there, and they're presented in a simple (but not unsophisticated) way, with a minimum of unnecessary jargon or obtuse notation. He doesn't belabor the concept of "limit"; most calculus books beat this intuitively obvious concept into the ground. Even though it doesn't cover all of the topics on the AP syllabus, I would rather supplement and use this text rather than any other. - B. Jacobs]
[Calculus for beginning college students]
[I needed to bring my high school calculus up to speed for first year physics studies and found this to be the only book which covered the necessary ground. The material is presented in a thorough manner with the great majority of topics shown with proofs. The book is very well organized and there are abundant worked examples. Some problems are offered which deal with matters not covered in the text, but usually there is a worked example given among the answers. Lang deals with the material in a clear fashion so that the subject matter is usually not difficult to follow.]
[On the negative side I can say that there is no human touch between the covers. His sole attempt at humor is an item following a list of problems in which he notes "relax". In the foreword he exhibits his firm belief that many freshmen arrive unprepared for college calculus, which may be true. But nowhere in the book is there a note of encouragement, so it cannot be described as reader friendly. Finally the index is pathetic - just three pages for a book of 624 pages, so that finding things can be frustrating.]
[Effectively conveys key concepts and skills]
[Serge Lang's text does an effective job of teaching you the skills you need to solve challenging calculus problems, while teaching you to think mathematically. The text is principally concerned with how to solve calculus problems. Key concepts are explained clearly. Methods of solution are effectively demonstrated through examples. The challenging exercises reinforce the concepts, while enabling you to develop the skills required to solve hard problems. Answers to the majority of exercises (not just the odd-numbered ones) are provided in a hundred page appendix, making this text suitable for self-study. In some sections, such as related rates and max-min problems, Lang provides many fully worked out solutions.]
[As effectively as Lang conveys the key concepts and teaches you how to solve problems, he does not neglect the subject's logical development. Topics are introduced only after their logical foundations have been laid. Results are derived. Theorems are proved when Lang feels that they will add to the reader's understanding. Through his exposition and his grouping of logically related exercises, Lang teaches the reader how a mathematician thinks about the subject.]
[The book is divided into five sections: review of basic material, differentiation and elementary functions, integration, Taylor's formula and series, and functions of several variables. The heart of the course is the middle three sections.]
[Most of the topics covered in the review of basic material should be familiar to most readers. However, it is still worth reading since there are challenging problems, properties of the absolute value function are derived from defining the absolute of a number as the square root of the square of the number, conic sections and dilations may be unfamiliar to some readers, and Lang views the material through the prism of a mathematician who knows what concepts are important for understanding higher mathematics.]
[Lang introduces the derivative as the slope of a curve in order to motivate the introduction of the idea of a limit. Next, Lang teaches you techniques of differentiation and shows you how to use them solve applications such as related rate problems. After a detailed discussion of the sine and cosine functions, Lang introduces the Mean Value Theorem and illustrates how it can be used for curve sketching and solving for maxima or minima. Lang covers properties of inverse functions before concluding the section by defining the natural logarithm of x as the area under the curve y = 1/x between 1 and x and defining the exponential function f(x) = e^x as its inverse.]
[The integral is introduced as the area under a curve, with the natural logarithm taken as the motivating example. Lang explains the relationship between integration and differentiation before introducing techniques of integration and their applications. Integration with respect to polar and parametric coordinates is introduced to expand the range of applications. The exercises introduce additional tricks that enable you to solve integrals that do not succumb to the basic techniques. A table of integrals is included on the inside of the book's front and back covers.]
[Lang's demonstrates the power of differential and integral calculus through his discussion of approximation of functions through their Taylor polynomials. This chapter should also give you an idea of how your calculator calculates square roots and the values of trigonometric, exponential, and logarithmic functions. The behavior of series, including convergence and divergence tests, concludes the material on single variable calculus.]
[The material on functions of several variables in the final section of the book is covered in somewhat greater detail in Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics). Since the corresponding chapters in that text include additional sections on the cross product, repeated partial derivatives, and further techniques in partial differentiation and an expanded section on functions depending only on their distance from the origin, I chose to read these chapters in Lang's multi-variable calculus text. The material that is included here, on vectors, differentiation of vectors, and partial differentiation, should give the reader a solid foundation for a course in multi-variable calculus.]
[I have some caveats. There are numerous errors, including some in the answer key. Some terminology is nonstandard, notably the use of bending up (down) for concave up (down), or missing, limiting the text's usefulness as a reference. In the chapter on Taylor polynomials, when Lang requests an answer accurate to n decimal places, what he really means is that the error in the answer should be less than 1/10^n, which is not the same thing. At one point, Lang claims that the Extreme Value Theorem, which he leaves unnamed, is obvious. I turned to the more rigorous texts Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition) by Tom M. Apostol and Calculus by Michael Spivak, where I discovered proofs covering one and half pages of text of the Extreme Value Theorem and a preliminary result on which it depends that Lang does not state until an appendix much later in the book. Perhaps Lang meant the Extreme Value Theorem is intuitive. While I found much of the text to be clear, I sometimes found myself turning to Apostol's text for clarification when I read Lang's proofs.]
[Despite my reservations, I think this text is well worth reading. Reading the text and working through the exercises gives you a good understanding of the key concepts and techniques in calculus, enables you to develop strong problem solving skills, prepares you well for more advanced mathematics courses, and gives you a sense of how mathematicians think about the subject. ]
[a book that focuses on the foundation without trying to do too much and it does that very well. self-contained and easy-to-follow, this book promotes understanding of the basics]

mostly recent stuff i packrat into my books for calculus... but i figure that almost any of these books should be worth discussing here, by anyone who's got a copy, used a copy, browsed it in the library, or utterly hates the book...

--------

another thing to talk about, what were the MOST popular textbooks out there 50s or earlier to today?

Thomas and Finney seemed popular [i wonder if that's because it was just enough to make engineering happy, as well as the math majors and the people who just need calculus once]
[I heard the alt editions were better, and what were those, it sounded like all the unreadable fluff and proofs were yanked out, but those only came out in the 60s or 70s, and the alt editions i think had unique numbering]

and i do recall

[i also think the writing of the 9th edition is actually clearer than in thomas original book - mathwonk]

I'm not sure of story, but wasnt the second edition pushed out really quickly for thomas, and I'm wondering if the first edition had problems, or just so much more was written but not fully completed for the first edition, and well, when the book took off, he said, i finally finished the last few chapters which i needed a few more years to finish up... etc etc
stewart i think started to get popular about 1990 or so..what was always surreal is how some older bookstores would just carry stacks and stacks of the 1967-1974 textbooks for calculus, which were all the mainstream, don't take too many chances, write for all audiences, and keep all that formalism, don't make the book too easy, don't make it too eccentric, don't stick in any material if the other top 7 sellers don't include it... and no one would buy them at 10 dollars and you'd see 15 years of dust on them...yet they would be great books for 2 dollars for the store to dump on people who want 'supplementary reading'

i always thought that the super easy books were far better, and the super difficult ones... the books in the middle just were compromised far too much, and lacked any vision...

any why is that no syllabus around tosses a schaum's outline for calculus or physics on the list?

mathwonk · Oct 19, 2012

my perception of a calculus book is partly influenced by when it came out relative to my math education. kleppner i believe was a harvard physics prof who wrote his book after i had taken a spivak style course from john tate at harvard (as spivak also may have), so did not interest me at the time.

gootman is one of my favorite books for struggling students and i have a copy signed by gootman, my long time colleague and a master teacher and analyst.

i liked lang's calculus books and learned how easy and simple riemann integration is from them.i loved comenetz's book, and wrote the initial rave review of that book. unfortunately i gave away my review copy as a prize to a good student. I attach my (edited) review, no longer available on the publisher's website: (see below)
unfortunately for the buyer, the price has increased from under $40 to over $125. Perhaps that is one reason my review has been removed, since it originally contained a grateful comment about the price.i loved the first edition of edwards and penney, two wonderful scholars and teachers and friends of mine, but to my taste the book did not improve through several editions apparently designed to enlarge its audience at the behest of the publisher. it seemed to serve as the model for stewart's book.

schaum's outline series was wonderful in the old days, extensive and good problems, plus brief and useful theoretical summaries; but more recently when i tried to use it in a course, it seemed greatly reduced in quality and usefulness somehow, no longer worth it.

the elementary error in cruse and granberg is the fact that the fermat criterion for a tangent line is not that the polynomial which vanishes where the line meets the curve should have only one root, but that it should have a double root at the given point.

this is easy to check for polynomials where one can always divide (x-a) out from
[f(x)-f(a)]/(x-a) because of the first forced root, and after doing so, simply set x=a to see if there is a second root. the result is as usual that the slope of the tangent line to y = x^n at x=a is na^(n-1).in fact i have experimented using this method to teach derivatives to undergrads, for polynomials. of course more analysis is required for transcendental functions like sin, e^x.i wrote out this result in complete detail for the author and publishers when they commissioned me to review the book prior to publication, but they ignored it. perhaps the authors did not understand it either, but i suspected at the time, the book was already ready to go to print and thy did not care to know its flaws.

i have written this method up completely with examples in the class notes attached to post #6 of this thread:

https://www.physicsforums.com/showthread.php?t=441018

dustbin · Oct 19, 2012

In reponse to RJinkies partI above:

My calculus course is using Briggs this semester. I think it is a pretty good book... but I feel the exercises are too easy. The explanations are good, though. Definitely better than what I've read of Stewart. Actually, my favorite "popular" calc book is Thomas, I think. There are tons of exercises (100+ per section typically); some of which I've found are also in Apostol and Spivak (decent selection of proof problems). However, certainly not as good as Apostol, Spivak, Courant...

I read a while ago a suggestion for Calculus by Kitchen (forget first name) from mathwonk... I happened to see it in a university library today. Looked like a nice book that covers a lot of material most other books do not.

tinylights · Oct 20, 2012

Hi, I wanted to introduce myself. :)

I have recently discovered that math is my calling, and am studying it at a small 2-year college before transferring out next Fall to pursue my BS. I'm taking Calc 1 right now with a Stewart textbook (though due to the earnest recommendations for it all over this site I have ordered Spivak's Calculus as well) and am doing well, though there is a definite change in difficulty level between Pre-Calculus math and Calculus. It's actually quite exciting to me because I remember finding myself so bored in other classes when I could easily predict where my teachers were going with every idea, and the course I am in now is a lot closer to my pace.

Out of curiosity, does anyone know what the best colleges/universities in Florida are for a solid math education? I live nearby UCF so it is my most likely option, but I want to consider others so as to avoid my grad school speaking at me in a new language. And I've heard of a lot of people having issues with UCF's massive enrollment, primarily that of never getting a chance to connect with your professors.

Secondly, I've looked at a lot of grad school programs and they recommend acquiring reading fluency of mathematical texts in French, German or Russian. Which one(s) are most useful to learn, in your experience?

mathwonk · Oct 20, 2012

I "read" French, German, and Russian, well enough to pass a grad school math proficiency test, but only French well enough to actually read a math paper fairly easily.

As far as Russian goes, so few English speakers read it that most big Russian journals are routinely translated into English.

I staggered through a few sections of Riemann's papers in German but even those are at last available in English.

I always thought I could read Serre's clear papers in French, but boy the English version of Algebraic groups and class fields is much easier to get something out of.So while it is recommended to learn these languages, at least french, and less so german, most of us get by quite well in english, occasionally having to struggle through an original language with a dictionary. but even to do that you need to know the basics of the language.

i.e. learn what languages you can, but be aware that you will be able to read almost everything written fairly recently in english. original languages are needed especially for reading some important works from the 19th century and early 20th cent.

e.g. with my weak german, i still have not read the great paper on linear series on algebraic curves, treated purely algebraically, including an early algebraic proof of the riemann roch theorem, by brill and noether.

it was kind of entertaining trying to struggle through a russian textbook on vector spaces (vyektornye prostranstva) when i kept running across the same words (ochevidno shto and silno) over and over, which turned out to mean "obviously" and "clearly"!

symbolipoint · Nov 4, 2012

MATHWONK,

you described your career progress a few times, but not remembering exactly, could you tell us: Did you study anything (Mathematics) while you were a meat-lugger, not in school? Or did you just work your labor job without studying your subject?

mathwonk · Nov 4, 2012

thats a little like asking country joe mcdonald what he remembers about the 60's, and he answers "nothing".

this is not a thread for discussing politics, but that was a great distraction. those were years when we were fighting in vietnam. it was hard to focus on just preparing for a narrow scientific career. the one advance i made in those years was by assisting/grading in honors calculus, i had to read spivak's calculus book, and learned a lot of calc i should have known much earlier.

converting1 · Nov 10, 2012

has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)? Reading up on mathematicians it seems as though everyone makes great work in their early twenties then just die down

micromass · Nov 10, 2012

converting1 said:

has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)? Reading up on mathematicians it seems as though everyone makes great work in their early twenties then just die down

http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list

converting1 · Nov 10, 2012

micromass said:

http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list

Guess it's never too late then! Thought I had little time left seeing as I'm 17,

thanks

homeomorphic · Nov 10, 2012

has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)?

30 isn't that old. Actually, very few mathematicians today even get to the point where they can make any significant contributions UNTIL they are about that age. The average PhD age is like 27 or 28, and my impression is that postdocs were this extra thing that they had to stick in because a PhD isn't really enough to become a mathematician anymore. So, by the time you are done just getting started, you're that old.

eliya · Nov 10, 2012

30 isn't that old to start or to finish. A lot of mathematicians ``made impact" beyond their 30's. Andrew Wiles, for instance, missed the Fields Medal by a few months.

As a general rule though, don't think about making an impact. Every mathematician who's active and writing papers is changing mathematics, of course, to different extents. To paraphrase Robion Kirby, don't worry about the significance of your mathematical results, worry about being the best mathematician you can be, and the rest will follow.

chiro · Nov 11, 2012

Take a look at George Polya, who started late relative to a lot of others (consider also that mathematics has exploded since 100 years ago) and didn't start studying mathematics:

http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/polya-george.pdf

Born in 1987, got the doctorate in 1912 so got the doctorate at the age of 25 (but please put that into context for mathematics especially probability at that time, and I am not denigrating Polya when I make these statements).

mathwonk · Nov 11, 2012

whether or not you will do important math is not determined by your age, surname, gender, or anything else. It is based on your desire. go for it.

dkotschessaa · Nov 12, 2012

Just turned 36. Still an undergrad. Not giving up. :)

Mariogs379 · Nov 27, 2012

@mathwonk,

regardless of what I do re: staying in NYC vs. Brandeis program, I'm going to take some math in the spring semester. Seems like it makes sense, for continuity's sake, to take real analysis II.

Was also thinking Algebra I. Thoughts?

giorgiv19 · Nov 30, 2012

Well, I have read some of the posts about textbook recommendations and want to offer an insight of my own:
Normal calculus textbooks? Don't bother. Don't read them, they do more damage, than good. The best thing to do is pick up a Russian Analysis textbook, like Fihtengolz, Zorich or Kydriatsev. They all come in 3 volumes.
Also no textbook is good without exercises. For this the best one by far is Demidoviche's "A Collection of Problems in Analysis".
The other essential thing for mathematics is linear algebra and analytic geometry. Serge Lange has very good book in linear algebra.
But the most important thing is not just studying at a university. You should look for open seminars. These seminars will give you much greater knowledge, than any course ever would.

mathwonk · Dec 1, 2012

thank you for these views which differ from many usually found here, and supplement them nicely!

NegativeDept · Dec 7, 2012

mathwonk said:

By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me. Please correct me on this point, since nothing this general is ever true.

I raise my hand with magnitude ##r \in (0,\tfrac{1}{2}]##. I'm a physics PhD student with a math undergrad degree. My thesis is on quantum decoherence, but it consists entirely of equations, simulations, theorems, and other people's data. When asked, I identify as either "applied mathematician" or "theoretical physicist."

Arnol'd, who is a MUCH better mathematician than me, says math is "a branch of physics, that branch where experiments are cheap." At this late date in my career I am trying to learn from him, and have begun pursuing this hint. I have greatly enjoyed teaching differential equations this year in particular, and have found that the silly structure theorems I learned in linear algebra, have as their real use an application to solving linear systems of ode's. I intend to revise my linear algebra notes now to point this out.

I agree! I just wrote a linear-systems-of-ODEs numerical software package which uses silly theorems of linear algebra to beat the hell out of RK4. (The catch: linear systems only. If you're interested, look up "Magnus expansion.") I'm sure my advisor, who has published huge amounts of Arnol'd-related stuff, would also applaud your effort. I suspect we're both working on one of his big long-term goals: show scientists and engineers that Sophus Lie's view of ODEs can be really practical and useful.

Hercuflea · Dec 9, 2012

Is it possible to receive an applied math Ph.D, but do your dissertation in some other area of science or engineering? I am asking because I want to get a solid foundation on some mathematics courses (functional analysis, advanced and numerical linear algebra, ODE's, PDE's, hilbert spaces, several complex variables) at the graduate level, but I would not really have a chance to take all of these courses if I did an engineering Ph.D. However It seems like it would be the best of both worlds if I could go for an applied math Ph.D. and do my dissertation in nuclear fusion which is ultimately my intended research interest, whilst being able to get the solid mathematical background.

Do you know if this is a common thing to do in applied math programs?

QuantumP7 · Dec 12, 2012

I just got What Is Mathematics: An Elementary Approach to Ideas and Methods, Second Edition. It's by Richard Courant, Herbert Robbins and revised by Ian Stewart. I'm REALLY looking forward to solidifying my knowledge of the really basic parts of mathematics. Hopefully, it'll answer some questions I have about the fundamental concepts.

Cod · Dec 13, 2012

QuantumP7 said:

I just got What Is Mathematics: An Elementary Approach to Ideas and Methods, Second Edition. It's by Richard Courant, Herbert Robbins and revised by Ian Stewart. I'm REALLY looking forward to solidifying my knowledge of the really basic parts of mathematics. Hopefully, it'll answer some questions I have about the fundamental concepts.

A great book you just got. The beauty of it is, its not a book that must be used in order. You can skip around as you see fit in order to meet your goals.

QuantumP7 · Dec 14, 2012

Cod said:

A great book you just got. The beauty of it is, its not a book that must be used in order. You can skip around as you see fit in order to meet your goals.

Thanks! I'm really loving this book so far!

Other Should I Become a Mathematician?

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