Should I Become a Mathematician?

[ktm]

Invictious, what you said doesn't make sense. Teaching is a very small burden for professors at research universities. My math professor teaches one class at a time and I believe my physics professor teaches two classes at a time. This amounts to 2-5 hours in the classroom per week. I'm sure you would spend just as much time per week doing necessary chores. And for such a small burden, a tenured professor averages around $70-90k+ a year, and up to mid 100's at top universities (Wikipedia). Plus, they can't get fired. So you see, a tenured professor has a very comfortable position financially and a very small workload.

 

[d_leet]

Invictious, what you said doesn't make sense. Teaching is a very small burden for professors at research universities. My math professor teaches one class at a time and I believe my physics professor teaches two classes at a time. This amounts to 2-5 hours in the classroom per week. I'm sure you would spend just as much time per week doing necessary chores. And for such a small burden, a tenured professor averages around $100k+ a year, and $160k+ at a top university. Plus, they can't get fired. So you see, a tenured professor has a very comfortable position financially and a very small workload.

I would imagine it is a lot more work to teach a class than to just put in the 2-5 hours that are actually in the classroom.

 

[ktm]

A little, but not that much more. A professor is usually responsible for the lectures, creating the homework, creating the tests/quizzes, and the final grades.

The homework can just be assigned out of a textbook, though it doesn't take much time to come up with problems at the class level. The tests/quizzes would probably have to change from year to year, but coming up with the problems shouldn't take that long and it's not an everyday thing. The final grades also shouldn't take that long -- the TA's could compute it actually. And then there's office hours, which could amount to 2 hours a week.

So I would estimate a minimum rate of 5 hours per week per class. This is much better than the 40 hour a week norm. And even if I'm off in my estimate, it would still be much better than the 40 hour a week norm. Of course, a lot of professors do more than this because they enjoy teaching.

Also, the TAs usually do the grading of homeworks, tests, and quizzes, which a big proportion of the work involved in teaching a class, especially a big class.

 

[mathwonk]

ktm, i take the maximum exception to your remarks. you can only say this if you have never taught yourself. trying to do a good job of teaching a class that meets 3 times a week, is a large order, especially if it has 30-40 students or more. and teaching two of them essentially takes all your time.

there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...

if you try to give your students the experience of making presentations, there is also tutoring them in the material in advance, wroting notes for them, scheduling opportunities to hear them practice the presentation,...

I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

then there are committees which meet regularly and endless paperwork.

all three jobs, teaching, research, and administration, are potentially infinite. you must always make choices and compromises to do them all minimally, much less to do them all well.

then we are not counting trying to have some time for family, not to mention a private life of ones own. one spends literally years with no time even to go out to dinner or read a book.

i once made a pact never to sleep over at the office, no matter how much work i had, in order to at least see my home every night. that semester, i once came home at 5am, slept 45 minutes total, and went back to work at 6:30am.

i also once worked 36 hours straight at the office, trying to go through over 700 job applications.

respectfully, you do not have a clue what you are talking about.

there is a huge difference between teaching a class or two, and having a semester off to do research full time. try it sometime.

of course there are people who do not care about doing a good job, and spend little time or energy on their teaching, but they are very rare in my experience. the hardest thing is to keep your research alive in the face of all these demands on your time.

oh yes, and i do not have a TA in either of my classes this semester, not even for grading, much less for office hours, or lecturing.

in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.

then i had to also write and grade the prelim. that was only one of two courses that semester. teaching is a huge time sink. having a year or even a semester off for research is a tremendous boost to ones productivity.

forgive me for unloading on you, but i am tired, i have taught already 7 classes the first three days of this week, and have scheduled 3 more review classes for friday. listening to know nothings say how easy it is, is just too much to take right now.

i wanted to present a seminar on my recent research this week but had no time to prepare it properly.

of course different departments are different, and biological sciences professors e.g. have much lower teaching responsibilities. But English profs may have more.

 

[pivoxa15]

Which institutions if any offer tenure for full time research (no teaching whatsoever) in pure maths (I know there are more for more applied research but all pure maths academics do teaching in my uni)?

How hard is it to get these positions?

 

[JasonRox]

ktm, i take the maximum exception to your remarks. you can only say this if you have never taught yourself. trying to do a good job of teaching a class that meets 3 times a week, is a large order, especially if it has 30-40 students or more. and teaching two of them essentially takes all your time.

there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...

if you try to give your students the experience of making presentations, there is also tutoring them in the material in advance, wroting notes for them, scheduling opportunities to hear them practice the presentation,...

I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

then there are committees which meet regularly and endless paperwork.

all three jobs, teaching, research, and administration, are potentially infinite. you must always make choices and compromises to do them all minimally, much less to do them all well.

then we are not counting trying to have some time for family, not to mention a private life of ones own. one spends literally years with no time even to go out to dinner or read a book.

i once made a pact never to sleep over at the office, no matter how much work i had, in order to at eklast see my home every night. that semester, i once came home at 5am, slept 45 minutes total, and went back to work at 6:30am.

i also once worked 36 hours straight at the office, trying to go through over 700 job applications.

respectfully, you do not have a clue what you are talking about.

there is a huge difference between teaching a class or two, and having a semester off to do research full time. try it sometime.

of course there are people who do not care about doing a good job, and spend little time or energy on their teaching, but they are very rare in my experience. the hardest thing is to keep your research alive in the face of all these demands on your time.

oh yes, and i do not have a TA in either of my classes this semester, not even for grading, much less for office hours, or lecturing.

in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.

then i had to also write and grade the prelim. that was only one of two courses that semester. teaching is a huge time sink. having a year or even a semester off for research is a tremendous boost to ones productivity.

Marking sucks ass. Some people just don't know how brutal it is to mark, especially papers from clueless first year students!

My question now is... aren't you better off working 9 to 5pm so you just go to work and do what you want after work without hassles? Because it seems like you have to put a great deal of your plans out of the way just to be a professor.

 

[JasonRox]

Which institutions if any offer tenure for full time research (no teaching whatsoever) in pure maths (I know there are more for more applied research but all pure maths academics do teaching in my uni)?

How hard is it to get these positions?

Win a Field's Medal and maybe a school will consider you to do only research every now and then.

 

[muppet]

Win a Field's Medal and maybe a school will consider you to do only research every now and then.

And even then you'll be lucky. Hawking is head of his department at Cambridge, has considerable prestige and research value, has the best possible reason not to teach given his medical condition, and still has to supervise a couple of PhDs. Admittedly not a field medallist... but are you THAT good?

 

[cristo]

And even then you'll be lucky. Hawking is head of his department at Cambridge, has considerable prestige and research value, has the best possible reason not to teach given his medical condition, and still has to supervise a couple of PhDs. Admittedly not a field medallist... but are you THAT good?

Hawking isn't head of DAMTP; a guy called Peter Haynes is. Besides, supervision is not the same as teaching duties; generally researchers like having research students. If anything, it boosts their paper count!

 

[eastside00_99]

ktm, i take the maximum exception to your remarks. you can only say this if you have never taught yourself. trying to do a good job of teaching a class that meets 3 times a week, is a large order, especially if it has 30-40 students or more. and teaching two of them essentially takes all your time.

there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...

if you try to give your students the experience of making presentations, there is also tutoring them in the material in advance, wroting notes for them, scheduling opportunities to hear them practice the presentation,...

I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

then there are committees which meet regularly and endless paperwork.

all three jobs, teaching, research, and administration, are potentially infinite. you must always make choices and compromises to do them all minimally, much less to do them all well.

then we are not counting trying to have some time for family, not to mention a private life of ones own. one spends literally years with no time even to go out to dinner or read a book.

i once made a pact never to sleep over at the office, no matter how much work i had, in order to at eklast see my home every night. that semester, i once came home at 5am, slept 45 minutes total, and went back to work at 6:30am.

i also once worked 36 hours straight at the office, trying to go through over 700 job applications.

respectfully, you do not have a clue what you are talking about.

there is a huge difference between teaching a class or two, and having a semester off to do research full time. try it sometime.

of course there are people who do not care about doing a good job, and spend little time or energy on their teaching, but they are very rare in my experience. the hardest thing is to keep your research alive in the face of all these demands on your time.

oh yes, and i do not have a TA in either of my classes this semester, not even for grading, much less for office hours, or lecturing.

in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.

then i had to also write and grade the prelim. that was only one of two courses that semester. teaching is a huge time sink. having a year or even a semester off for research is a tremendous boost to ones productivity.

forgive me for unloading on you, but i am tired, i have taught already 7 classes the first three days of this week, and have scheduled 3 more review classes for friday. listening to know nothings say how easy it is, is just too much to take right now.

i wanted to present a seminar on my recent research this week but had no time to prepare it properly.

of course different departments are different, and biological sciences professors e.g. have much lower teaching responsibilities. But English profs may have more.

That sounds brutal.

 

[ktm]

mathwonk,

First of all, I admit I was speculating (while appearing too sure of myself), as I am not a professor myself and so I'm not in a very good position to make such estimates. Your opinion is probably much more valid than my speculation. I'm sorry for speculating, and for most likely saying something very inaccurate.

Second of all, I recognize your analogy of teaching students and teaching yourself, but it's no basis for saying that I've never taught myself anything and that I'm a "know-nothing". I've taught myself a lot and don't consider myself a "know-nothing".

Thirdly, I was only considering teaching, and I didn't mean to include doctoral students, job applications, committees, and department meetings. Regardless, I admit I was still speculating, and that I still probably said something very inaccurate.

Anyways, it's good that this dispelled an incorrect notion of mine.

 

[cristo]

Marking sucks ass. Some people just don't know how brutal it is to mark, especially papers from clueless first year students!

Tell me about it! I never knew how much work lecturers actually put into teaching until I started as a TA this semester, and has to attend tutorials and mark scripts. As mathwonk says, I can see how at least half a week can easily be taken up with teaching one class.

 

[Jekertee]

The decision is yours.
If it wrong then you learn

I used to decide to leave Italy. It WAS my decision. No one has ever taught me to do taht anyway

 

[mathwonk]

i'm sorry ktm, as i said i just got home and I'm tired and i unloaded my frustrations on you. please forgive me.

being a professor is certainly easier than most jobs. it beats the meat lugging job i used to do. the heaviest piece of meat i ever lifted there was 305 pounds. now i complain if a calc book weighs 5 pounds.

 

[musicheck]

I'm currently a freshman in college (at Yale) and am very interested in a major in math (or math/physics) and quite possibly grad school beyond (though it is very premature for me to make such a decision). The math class I am currently taking (Year-long Vector Calc+Linear Algebra with textbook "Multivariable Mathematics" by Mathwonk's colleague Ted Shifrin) is interesting but feels a bit light. I studied the first 7 chapters of Baby Rudin (and a bit of linear algebra too) last summer, which was a much more invigorating experience. In my spare time, what would be the best way to learn more math? Is studying putnam exam type questions a good idea? Should I read textbooks on my own? Should I put more time into my (slightly dull) math class, or read about more advanced related materials? Should I simply study harder for my math class, and if so how? I've heard it said that foundations in linear algebra are essential in more advanced math, so would a linear algebra book like Hoffman and Kunze be a good idea for enrichment?

I feel like I'm really passionate about math right now and not 100% sure what step to take next.

 

[JasonRox]

now i complain if a calc book weighs 5 pounds.

Oh god, I hate that.

I love the textbooks you get later on. Much smaller. I go to school carrying my textbook with some paper stuffed inside and a pencil in my pocket. I hate carrying school bags and junk.

 

[k3N70n]

...there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...
...
I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.
...
i also once worked 36 hours straight at the office, trying to go through over 700 job applications.
...
in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.
...

Wow! I had no idea it was so much work. I have a new respect for profs.

 

[DeadWolfe]

When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

That seems inappropriately intrusive.

 

[symbolipoint]

That seems inappropriately intrusive.

That is a very pessimistic interpretation. Mathwonk's concern is for helping the student to achieve academic success or maintain a level of progress. Further, in some educational systems, maintaining classroom attendance is a very critical struggle which if not successful, result in closure of the class section; and so the effort to reach non-attending students is an essential part of the teaching position.

 

[mathwonk]

let me add a bit of comment:

it is true as a colleague of mine told me some years ago,. that a professors job is not that hard, if he does not want it to be. i.. we have a great deal of latitude in what we choose to do, and if we become tired and disillusioned, we may choose to do very little.

as an example, today I had a reading day, i.e. a day off. but i offered my students a day of review, if they so chose. i drove 130 miles round trip to be there for over 4 hours of review, but only 8 of my 24 students bothered to show up.
these were mostly of course the very strongest of the class, who did not need the review at all.

i could have spent the day doing my own research, and would have been more rewarded for that, but i tried to help the students who were struggling to pass the course, but to little effect since most of those did not even come in for help.

in such a situation, you may understand that many of us choose not to bother next time, but to do what benefits ourselves, since so few students even care to take advantage of our gift of time.

so a professor's job is indeed easy if he choses not to do a good job of it, but it is quite challenging if he chooses to do his best.

it is not even that simple, since if he chooses to teach the same courses over and over he can do a good job with very little time spent preparing. but this semester for example, i am teaching two courses i have never taught before, so both take great amounts of preparation. not everyone offers to do this.

 

[mathwonk]

back to math

i have finished teaching my geometry course, and ended with some beautiful calculations due to archimedes that i used to do using calculus. my comments follow:

Reflections after teaching geometry (5200) and integral calculus (2260) fall 2007.

It seems to me the method of “calculus” is properly the use of the fundamental theorem, i.e. antiderivatives, to calculate limits that were already used by the ancients to express areas and volumes. On the other hand, “Riemann sums” were used by Archimedes quite effectively for shapes described by quadratic and linear equations, and his use of the technique can help illuminate calculus methods for the student.

In particular:
1) Defining area as a limit of inscribed rectangles and triangles, and volume as a limit of pyramids and prisms, was known to the ancient Greeks.

2) One easily deduces directly the “Cavalieri principle”, additivity, and well definedness of areas and volumes for triangles, circles, pyramids, prisms, cylinders, cones, and spheres, and the principles of volumes by washers and by shells.

3) Hence, if b = base length, B = base area, C = circumference, r = radius, h = height, C(r) = circle of radius r, one gets the classical formulas: A(rectangle) = bh, A(triangle) = (1/2)bh, A(circle) = (1/2)Cr, V(cone) = (1/3)Bh, V(pyramid)=(1/3)Bh, V(sphere) = (1/3)Sr, S = 4A(great circle) = (2/3)A(cylinder circumscribing sphere) , and Pappus’ formulas: V(torus of major radius b, minor radius a) = C(b)A(circle of radius a), A(torus) = C(a).C(b), A(cone frustum) = slant height.C(average.). One also obtains area and volume formulas for spherical segments and sectors.

4) None of those computations becomes easier by the fundmental theorem of calculus. In fact the volume of a spherical sector or segment, or torus, are harder and omitted from our book along with the formulas of Pappus. The area of a torus by parametrized areas of surfaces is done without mentioning Pappus’ theorem. They also do not mention the fact analogous to that for spheres (hence potentially clear to Archimedes), that the surface area of a torus is the derivative of its volume w.r.t. the minor radius.

5) I suggest that a calculus course, especially for engineers, should build upon and enhance the results of elementary geometry, i.e. preserve old insights when they remain best, and show the student how to use calculus to obtain results which were out of reach of older methods. It should show the power of calculus to solve problems that are suited to it, and how to recognize which ones those are. The advantage of antiderivatives should be revealed via its use in computing limits of truly difficult Riemann sums, such as the arclength of a parabola, possibly unknown to Archimedes.

6) The topic that truly goes beyond methods used by the Greeks is power series, and it should be connected thoroughly with the other topics in the course. Numerical estimates like the trapezoid or Simpson rules should be illuminated and refined via Taylor series. Elliptic and other previously exotic integrals deserve the same treatment, since power series render them as “elementary” as the trig. and exponential functions. Finally they should be used to augment the discussion of separable differential equations.

 

[mathwonk]

to explain my policy of contacting students, you should know that 4-5 absences results in being dropped from the course, usually with a WF. hence after 2-3 absences I have 2 options, 1) either wait until the student has missed 2 more classes and drop them with a WF, or contact them to allow them to explain what is going on, welcome them back, and offer to help them catch up.

often the student is embarrassed after a few misses and needs the ice to be broken before they feel comfortable returning.

in over 30 years, i cannot recall any student preferring i use option 1.

 

[quasar987]

Nice post (1177).. thank you.

 

[mathwonk]

thanks. i need to go back now and look at apostol's book, as he begins with area, and integration, as opposed to diiferentiation. this is the historically correct order of the subject. indeed it now seems to me that differentiation should enter the topic of integration only as a method of calculation, after the fundamental ideas are clear.

 

[mathwonk]

ok now i think i get it. archimedes had the cavalieri principle, that two solids have the same volume if all plane slices of them by planes parallel to a given plane have equal areas.

that let's him equate volumes of different figures. then using the additivity principle for volumes and areas, and the equality of volumes of congruent figures, he can deduce the volumes of figures, first that are fractions of known ones, e.g. that pyramids have 1/3 the volume of their circumscribed cylinder.

but he can only work out cases where the new solid can be compared with one that is known by the decomposition method.

the advance offered by the fundamental theorem of calculus is that not only can the cavalieri principle be used to compare volumes but also to calculate them. i.e. now it can be said that the area of a plane slice is actually the derivative of the moving volume function. i.e. the function whose value is the portion of the volume lying on one side of the slicing plane.

so now we do not need to have another known figure with slices of the same area as our new figure, we can directly compute the volume of our new figure from knowing the areas of those slices by antidifferentiation. so this is a real advance.

but it still makes little sense to hide from the student, the fact that archimedes could already do many of the most natural and interesting problems of this type, without antidifferentiation. and it also makes little sense not to present the pappus principle, and ask students to use calculus in a clumsy way, making the derivation of old results actually harder with calculus than with the methods of the ancient greeks.

the two points of view can illuminate each other, if taught together.

 

[muppet]

I've found my geometry courses make little or no mention of the basic Euclidean results that teenagers do at school. From a student's perspective mathwonk, where that kind of thing comes from would be nice to know! It'd also be useful for my physics problems- you never quite know if your approximation that a certain quantity is constant over an infinitesmal change in a variable is right or not until you get your final answer!

 

[mathwonk]

well i have spent my whole life in math and am only now learning the basics of greek geometry, so i recommend reading euclid, with the guide of Hartshorne's book, geometry: euclid and beyond. also some searching on the internet for Archimedes' ideas is useful.

look for archimedes, the method.

 

[pivoxa15]

well i have spent my whole life in math and am only now learning the basics of greek geometry, so i recommend reading euclid, with the guide of hartshorne's book, geometry: euclid and beyond. also some searching on the internet for ARCHIMEDES IDEAS IS USEFUL.

look for archimedes, the method.

How can you be a professional algebraic geometer but only just started to learn classical geometry? Certainly an algebraic topologist would know classical topology?

Is algebraic geometry more a field of algebra but algebraic topology more a field of topology?

What is the core principles in algebraic geometry? i.e The core principles in algebraic topology would be homotopy theory including the fundamental group? Using algebraic methods to answer topology questions such as knowing when two spaces are not homeomorphic rather then when is homeomorphic.

 

[PowerIso]

I'm going to go on a limb to say that Monkwonk knew classical geometry to one degree or another, but it's only now that he is really learning it the way it was meant to be learn.

 

[JasonRox]

I'm going to go on a limb to say that Monkwonk knew classical geometry to one degree or another, but it's only now that he is really learning it the way it was meant to be learn.

I don't believe there is a "meant to be learn", but merely learning it in the classical sense.

Just like no one knows calculus in the classical sense. It's rather normal to not know the classical methods.

 

[mathwonk]

amazing isn't it? I have been so focused in my little narrow field of specialization that i have not learned the history of my subject. Now i am very psyched to learn my subject more thoroughly. i feel i have now a much better appreciation of how calculus developed. now i want to learn how algebraic geometry developed from scratch to grothendieck, and hopefully even the current stuff that my young people are doing now.

i.e. to be honest, after 40 years, i still seem not to know much about math. but i have hopes.

a suggestion: try to read the great ones: euclid, archimedes, gauss, riemann, ...

if they are hard to understand read interpreters, but make your goal understanding those great workers.

i.e. read a calc writer like me or stewart, or edwards / penney, or whoever, but make your goal to use their help to understand the greats.

 

[JasonRox]

Finding prints of the original copy is probably hard to find!

 

[Phred101.2]

It's a language that you have to learn. I see it like having to learn how to play different kinds of instruments, and understanding a musical score. Reading a paper about quantum processing and density-of-states and you don't quite understand the symbols, but you understand entropy, and harmonic states or resonances. It's how you look at what it really is. Learning a language other than the one you speak normally is quite possible (some can learn several), it's all down to how interesting you think it is.

Also it helps if you try to explain what you think you know about math to someone, if they ask questions about what you tell them, i.e. you test your own knowledge, or you explain it to yourself, at the same time you try to explain it to anyone else.

 

[mathwonk]

well we have good English translations of copies of copies of the originals. There is a cheap Dover edition of Heath's translation of an anonymous copy of the "Valla" copy of Archimedes' works, and a project at Stanford is currently engaged in making the last few pages visible.

We have a fine version from Green Lion press of Euclid, also translated by Heath, and Hartshorne has written a great companion volume, Geometry: Euclid and beyond.

Gauss' Disquisitiones is available in English translation, and so is Riemann. now if someone would just translate Galois.

 

[mathwonk]

http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html

here is the extent to which i knew classical geometry: in high school it was my favorite subject and i won the mid state HS geometry contest. i also won the statewide "comprehensive" math contest including "solid geometry".

then in college i studied mostly calculus, real analysis, spectral analysis, banach spaces, and some n dimensional theory of content due to jordan, and a little abstract measure theory. no differential geometry, no classical geometry. so this was mostly very abstract stuff designed to give the extremely rigorous underpinning to, and precise formulation of, analysis. there was little or no attempt to actually teach us more "content". the one fascinating course that went into interesting geometric material was Bott's algebraic topology.

then in grad school i studied mostly commutative algebra, complex analysis, algebraic topology and algebraic geometry.

in my job i have taught foundations of euclidean geometry and transformation geometry, but never really going back through all the classical results, rather focusing on the subtleties of the foundations, the interplay between models and axioms, questions of consistency, completeness,...

until this year. i decided that i would have better success if i repeated the topics from the actual high school course my students would be asked to teach, even though i assumed i myself knew this stuff cold from high school. i got a big surprize: i did not know the old basic stuff at all well, i.e. i did not really remember the proofs, just the statements, and there were some statements i did not know.

so i started getting a big boost myself, and i started understanding it better, as if for the first time. i had never studied euclid himself, and did not realize that he proved his first 27 or so propositions without the parallel postulate, so i had no good grasp of just where that postulate is needed. i had never read saccheri, and did not know his "weak" or "neutral" version of the exterior angle theorem, or that without the euclidean parallel postulate one does not have rectangles.

i also had never examined the elementary material with a sophisticated eye, so i did not realize either how the theory of similarity is connected with but independent of that of area, nor how tricky it is to make area well defined. i did not understand that there are three different statements of pythagoras: a^2 = b^2 = c^2 ( a statement about numbers), or equality of the areas of two figures (which requires a proof that area makes sense), or as a statement that two different figures can be dissected into congruent smaller figures).

these three statements are often regarded as equivalent, but to prove them requires different arguments. the numerical version for examples follows from the theory of similarity, while euclids own version, the decomposition version, requires a proof of transitivity of the relation of "equidecomposable" that euclid took for granted. if you open the beautiful book "geometry revisited" e.g. by coxeter and greitzer, you see immediately a simple proof of ceva's theorem but one requiring the existence of an area function, not at all a trivial fact, and one most euclidean geometry courses omit to prove. so i proved it in my class (although quite briefly) using hilbert's lovely argument (millman and parker get moise's argument wrong).

i did not realize that a theory of limits is needed to discuss similar triangles whose side ratio is irrational. this is why euclid considered similarity more sophisticated than equidecomposability, and placed it later, whereas moderns like me and it turns out Hilbert, would invert the order of these concepts. i began to notice that almost every fact provable using area has another proof using similarity, like ceva's theorem.

and i did not know Hilbert's 3rd problem so well, that even if two polyhedra do have equal volume, it may be impossible to decompose them into congruent smaller figures, and why not. i did not have time to present this problem's solution by Dehn, but it is in Hartshorne.

i also did not realize that archimedes had used the concept of centers of gravity to compute volumes, as shown on the websites above. finally, after presenting these concepts, i began to see how some of them generalize to 4 dimensions and higher. e.g. archimedes arguments also show that in any dimension n, the volume of an n - ball is R/n times its surface area. this generalizes the fact that area of a circle is R/2 times circumference, and volume of a sphere is R/3 times surface area. (I presented this 4 dimensional stuff on the last day of class).

This was archimedes own explanation of how he proceeded from computing volume of a sphere to knowing its area as i learned yesterday reading "the method". i also learned that archimedes and euclid's proofs of equality of quantities requiring limits was precisely the one now encoded in epsilon - delta definitions, namely to show that the difference between the two quantities is less than any assigned positive amount. i always thought this discrete concept of limits due to the 19th century analysts like Cauchy. actually they just put it into their own words.

i also learned that Archimedes proved his volume insights by taking limits of sums of approximations. whether all commentators agree or not to call these "Riemann sums", there they are for all to see in Archimedes. e.g. he computed areas and circumferences of circles as limits of areas and perimeters of polygons, which were done by adding up triangles. and he did it from both within and without, computing the error between the two approximations and showing it went down by more than half each time, hence "approaching zero" as we would say.

as i see now, he had the Cavalieri principle as far as being able to prove that parallel slices of area do determine volume, and could use it to compare volumes of cones cylinders and spheres. our advance today is being able to take the function of area of slices, and integrate it to get the volume function. thus we can not only compare volumes by this principle, but compute them. this is the advance over archimedes which is made possible by the FTC.

what books point this out? if none, then where can one learn it if not from the source?