Ok, I am trying to see the solution for a cubic in terms of "completing a cube". It doesn't seem to be quite that, but there is away to see it geometrically, analogous to the picture of completing a square. It seems to be more matter of decomposing a figure as a cube. I.e. a (u+v) cube normally decomposes into 8 pieces, a u cube a v cube, 3 u^2v rectangkes, and 3 uv^2 rectangles. But we can also clump together one u^2v rectangle and one uv^2 rectangle on each face, and get 5 pieces, the two cubes, and 3 identical rectangles that measure u by v by (u+v). I have drawn this roughly on a piece of paper and it looks ok. So if we have a cubic equation like X^3 = pX +q, we try to see it as decomposed like that. I.e. we try to find two numbers u,v such that q = u^3 + v^3 is the sum of the two cubes, and such that also p = 3uv. Then letting X = u+v, we have a (u+v) cube on the left side of the equation, and on the right side it is decomposed as 3 u by v by X (= u+v) rectangular solids, and the two cubes u^3 + v^3 = q. Then X = u+v solves the equation. Moreover finding u and v means solving a quadratic, because all we have to find is their cubes u^3 and v^3, and we know both their sum, since u^3+v^3 = q, and their product, since 27u^3v^3 = p^3, so u^3v^3 = p^3/27. And solving quadratics is a matter of finding two numbers whose sum and product is known, as emphasized by Euler. Well ok, it is not nearly as clear as the square case, but it did take over 1,000 more years to solve! In the quadratic case, we are given rs and r+s and we want r and s. In the cubic case we are given r+s+t, rst, and rs +rt + st, and we want r,s and t. Amazing how much harder it is. Still in some sense the quadratic case is solved by looking for r,s as a sum of two square roots, and the cubic case is solved by looking for r,s,t as the sum of two cube roots. I am not sure if the quartic case is solved as a sum of two 4th roots, but euler seems to solve it as a sum of 3 square roots, which three numbers are themselves obtained as roots of a cubic.

Good for you! In about 1970 when I lived in Harvard square, I read that the top student in that year's graduating class had never had formal institutional schooling until entering Harvard. I think he had lived in Hong Kong and been home schooled, perhaps by diplomat parents. Well google never ceases to amaze - 43 years later, it may have been Peking, and this seems to be his cv: http://www.yale.edu/...cottBoorman.pdf

swimmermom3. yes i agree! hearing you say it, i realize it is just a matter of making two choices: 1) where do we put the origin? and 2) how long is our chosen unit? I.e. where do we zero it out, and what is the scale? (sliding and stretching)

ah yes, even without following the link yet, that already stimulates my visual imagination, just hearing you say to view it geometrically. I.e. then we look at X^2 + bX as the sum of a square and a rectangle, and then we are led to split the rectangle bX into two rectangles both with one side X and one side bX/2, and fit them into a figure that looks like a big square, but minus a little square with both siudes equal to, well b/2 I guess. So we make a big (X + b/2) square out of a little X square, two rectangles (both X by b/2) and a little b/2 square! wonderful! now I am going to look at the link... yes!! exactly so!! This is so much clearer than the way I was taught. Oh yes, and now I see the connection with Euclid's Book II, Prop 14, since your way clearly shows that to complete the square we add in a small square, so the original quantity X(X+b ) = X^2 + bX, has been expressed as a difference of two squares, the (X + b/2) square, minus the b/2 square. cool!

kiana, I think you are right, and my problem was partly a lack of familiarity with completing the square. Now that I think about it, this technique pops up in Euclid in a geometric context. that might have helped me as I always prefer to visualize things geometrically. The idea there is that one knows from Pythagoras how to take square roots of things which are expressed as the difference of 2 squares. I,.e. if we want the square root of A^2-B^2 we just construct a right triangle with hypotenuse A and side B, and then the third side C satisfies C^2 = A^2 - B^2, by Pythagoras. Then the clever remark Euclid makes is that any product at all can be expressed as the difference of 2 squares. I.e. if we have AB, where B>A, let C = (A+B)/2 and D = (B-A)/2. Then C+D = B, and C-D = A. Thus AB = C^2-D^2. This is exactly completing the square. I.e. if we have X^2 + bX, we regard it as a product X(X+b ), and perform the previous construction getting C = [X+X+b]/2 = X + b/2, and D = [X -X -b]/2 = -b/2. Then C+D = (X+b/2) - b/2 = X and C-D = (X + b/2) + b/2 = X+b. Moreover, the product X(X+b ) is now expressed as C^2-D^2 = (X+b/2)^2 - (b/2)^2, exactly what we get by completing the square. Euclid uses this technique in Prop. 11 and Prop 14, Book II where he solves the special quadratic equation X^2 +cX = c^2, and then takes arbitrary square roots, purely geometrically. In Euclid, "multiplication" means forming a rectangle, so taking a square root means finding a square with the same area as a given rectangle. This shows another benefit of the suggestion made elsewhere by regentrude of combining geometry and algebra. Indeed trying to do Book II of Euclid without using any algebra, as Euclid does, and where the algebra is done entirely geometrically, is a bit clunky and complicated, to me. E.g. you get stuff like: "If a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line." instead of (a+b )^2 = a^2 + b(2a+b ).

yes, my high school book also did it by completing the square, but that didn't seem well motivated to me, so I had trouble with it. I didn't see the "why" in that approach. Now years later I sort of see it as follows: we know how to solve X^2 = d, by taking a square root, so we just want to see if we can reduce our equation X^2 -bX + c = 0, to that form. Knowing that the coefficient b is the sum of the roots, we can make that sum zero by subtracting b/2 from both roots. So we make a transformation Y = X-b/2. or X = Y+b/2. This makes X^2 -bX + c = (Y+b/2)^2 - b(Y+b/2) + c = Y^2 +bY +b^2/4 -bY -b^2/2 +c = Y^2 +c -b^2/4 = 0, so we have Y^2 = b^2/4 -c or Y = ± (1/2)sqrt(b^2-4c). Thus X = Y+b/2 = b/2 ± (1/2) sqrt(b^2-4c). Of course the quick method is just to say lets complete the square using the first two terms, and just add and subtract b^2/4 getting X^2 -bX + b^2/4 -b^2/4 + c, so (X-b/2)^2 = b^2/4 -c, but my mind just wants to ask "why did we do that?" or "how did you think of doing it"? I also like the fact that the other method explains, to me at least, why the formula looks as it does, where the 1/2 and the 4 come from. Not understanding this, I just had to memorize it (I still have my high school text, and the quadratic formula is written in the front cover in my hand writing, since I struggled to recall it at first). I also was not very good with fractions, and it was hard for me to simplify (X-b/2)^2 = b^2/4 -c, and get X = [b ± sqrt(b^2-4c]/2. So I just kind of turned off to that derivation. I.e. it was just a calculation, with no motivation, and I have always had trouble following a computation when I don't understand why it is being carried out. Another problem with it, is it is of virtually no use when trying to go further and solve cubics. I.e. as far as I understand, one does not solve cubics by completing the cube. So I always want to know "what have I learned from this solution? where can I use it again?" What insight does it give me? Of course that is highly personal and other people may gain insight from methods that leave me puzzled, as is the case with completing the square. But to solve a cubic, one does proceed by first eliminating the X^2 term, in the same way as above for quadratics, (but one cannot also eliminate the X term). And then one reduces to solving a quadratic, using the idea that a quadratic is equivalent to being given the sum and product of two numbers, and finding the numbers. Or maybe in a way one does complete the cube. I.e. to solve a cubic like X^3 = pX + q, one tries to find u,v, such that q = u^3 +v^3 and p = 3uv. since then X^3 =pX+q becomes X^3 = (3uv)X + (u^3+v^3) = (3uv)(u+v) + (u^3+v^3) = (u+v)^3. Hmmmmm... its not exactly the cube of X-something, but it is a cube of u+v = X....... E.g. to solve X^3 = 30X + 133, one comes up with u=2, v=5, so u^3+v^3 = 125 + 8 = 133, and 3uv = 3(2)(5) = 30. Then X = 2+5 = 7, solves the equation. To me this illustrates why learning is challenging, because we can have these irrational mental blocks against grasping the idea behind something, while other people may not have any at all for that same thing. That's why I like personal learning, so the instructor can share with me their way of seeing something. Then maybe I can see it that way too.

there are a few standard books for the most mathematically oriented students and courses, spivak, apostol, courant, and kitchen. essentially all other books such as stewart are aimed at a completely different and larger audience. stewart 2nd edition, was one of my favorite books for the average calculus audience that i normally taught. i thought it was excellent! If you want a recommendation for a student, I would suggest working through Stewart first, and then working through Spivak afterwards. stewart will teach the methods, facts and techniques, and then spivak will explain why those things are true, as well as supplement some facts that were omitted in stewart. in general, i advocate using any source that helps. there is no benefit to using a high level source one does not learn from. i myself was a student in a spivak type course and was pretty much blown away by it. only years later did i reach that level. (smile)

there is no comparison between stewart and spivak. they are both good books, but mike's book is aimed at future mathematicians, and stewart's book is aimed at the larger market of average good calculus students. stewart was a good choice for our average calculus course at UGA. Mike's book was the standard choice for super honors calculus at UGA and Harvard. i.e spivak is far more theoretical and complete, and the problems are much harder. as a rough guide, choose spivak if your child's sat math scores are over 700. oops now i need to read the link you posted. ok i read it. that seems to be written by a young high school student as a course requirement. it is roughly correct but pretty minimal.

I have known Mike since 1965. He began publishing his own books a long time ago to become more independent. So this is likely either him, or a company that he pays to do his publishing. He does publish other people's works. But he has never yet ventured into the high school market.

These are excellent ideas. I have forwarded them to Harold. That "pop" publisher is probably Mike Spivak himself.

I don't know Harold's exact plans, but I understand some search for a publisher is underway. I'll stay in contact. yes I do believe Freeman is ceasing their entire participation in high school textbook publishing.

In answer to a question posed on TWTM, Harold Jacobs wrote back and said he had recently learned that after 45 years working with him, Freeman is going out of the high school textbook business due to increasing competition and increasingly specialized local school district criteria for adopting books. I.e. apparently it is getting harder and harder to produce even a math book that will meet the criteria for adoption of enough different places to make money. Freeman is thus ceasing publication of all three of Harold's books and returning their rights to him. He may seek a new publisher, possibly as soon as next month. As of now the used sales are very strong and his share of the market including those is actually higher than when new copies were available. To me, the fact that some of the best elementary math books on the market are not viable because of administrative considerations, makes for a sad day. I.e. it seems that math books are now often chosen or rejected for reasons that have little or nothing to do with either mathematics or pedagogy. I still own a copy of the geometry book by Harold, but unfortunately I parted company with my last copy of his algebra book when I moved out of my office upon retirement, but my granddaughter does have one I bought for her. Hopefully ancillaries for the books will still be available at Ask Dr. Callahan, to serve home schoolers. http://www.askdrcallahan.com/ I hope this information is useful.

yes! When I use the word "know:" it never means "memorize mindlessly". So it did not dawn on me that there was any other way to "know" this other than the Pythagorean theorem.

I think it would be fine. There are many different courses called calculus. In high school there are AB and BC and so on, and they seem only to mean that the BC is a little harder than the AB. When i got these students in college they had almost never seen any proofs or any of the reasoning that I expected and so AB, BC or whatever they all needed pretty much to start from scratch. I could essentially not tell the difference between students with high school AP background and those with no calculus at all. Indeed I would much prefer my students had a good solid precalculus preparation than to have calculus as would most other professors I know.\ I measured these AP courses by the tests they inspired. The AB tests I saw had no proofs at all. The BC tests had maybe one proof question at the end of the test. But at Stanford, my son's class had 100% proofs on the tests. Where does one prepare for such a class? Certainly not in a typical AP class. What we look for even in a state college calculus class is a familiarity with algebra and geometry and some trig. We would like our students to know that r is a root of a polynomial if and only if (X-r) is a factor. And that the only rational roots r/s in lowest terms, of a polynomial with integer coefficients has r dividing the constant term and s dividing the lead term. We would also,like our students to know the sine and cosine of angles of 30, 60, 45, and 90 degrees, We essentially never get any of this. High school courses are taught by high school teachers, usually with at most a math major in college. College courses are often taught by professional research mathematicians, and as such are on an entirely different level. But you never know, as the proliferation of AP classes in high school has flooded college with so many poorly prepared students that many college courses have also been dumbed down to the level that will accommodate them. My apologies for this rant as your students are probably in the small percentage of those who do have everything we would like but who go away to the better colleges and we never see them at state college. But it never hurts to take as many classes as challenge you and teach you, no matter what their names are. if you want to see how many different varieties of calculus there are, look at books like Calculus made easy, by Silvanus P. Thompson, Calculus and analytic geometry by George B. Thomas, Lectures on Freshman Calculus, by Cruse and Granberg, Calculus by Michael Spivak, Calculus by Tom Apostol, Differential and Integral Calculus by Richard Courant, Calculus by Joseph Kitchen, or if you want to know what advanced freshmen took at Harvard in 1965, look at Advanced Calculus by Loomis and Sternberg....... This ludicrously abstract book was used in a class that actually had mostly freshmen enrolled. http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf

swellmomma, I apologize for muddying the water. My own experience in learning math has been that there is no simple formula, but if one wants to learn the subject one just persists.

swellmomma: Did you see the link for a free online copy of euler in post 38? I had never heard of it either, and I was a professional mathematician and math professor for 35 years. In retirement I get to read stuff I like and not just run in the rat race of publishing more papers on obscure topics. As a high school student I thought quadratic equations was about remembering that x = (1/2a)(-b ± sqrt(b^2-4ac)) and had trouble understanding where that came from. As an old man I read euler and learned that all we need to find the roots r,s of a quadratic is to know their sum (r+s) and their difference (r-s), and then we can add and subtract these and get 2r = (r+s)+(r-s), and 2s = (r+s) - (r-s). Then we divide by two to get r and s. So that last part is where the (1/2) comes from in the formula, and also the ± sign in the formula. In fact in the formula, -b/a = (r+s), and sqrt(b^2-4ac)/a = (r-s) I.e. in a simplified equation like X^2 -bX + c = 0, the sum r+s is just b, and the product rs is c. So since we know that (r+s)^2 - 4rs = (r-s)^2, we can compute the square of the difference as (r-s)^2 = b^2-4c, and that's where the 4 came from! So now all the numbers in the formula make sense. the 1/2, the 4, the sqrt... Why didn't anyone ever tell me this as a child? Well, maybe it's because I read Welchons and Krickenberger instead of Euler. (And Welchons et al is a lot better than most choices out there today.) In fact I learned by reading another great elementary algebra book, by LaGrange, that this clear conceptual explanation of solving quadratics is in the book of Diophantus, written almost 2,000 years ago! And this may not appeal to everyone but I love the charming problems in Euler, like: "Two country girls took their cheeses to market and sold them. Afterwards one said to the other, if I had sold yours at my price I should have received 10 shillings more,...." This might seem odd, but after 35 years as a mathematics researcher with a PhD, I am now relearning high school algebra by reading Euler and LaGrange. Why? Because I want to understand it. For the same reason I am still interested in learning physics from scratch. It has dawned on me I should probably red Newton. I still sigh at the folly of lightening my load 35 years ago by selling my unread copy of the Principia for ten cents! to a used book buyer. I am also learning that things that high school students used to learn, like how to solve cubic equations, was dropped out of the curriculum many years ago, but i don't know why, since euler makes it look so easy. One reason was the move toward universal high school education. For that to work, the curriculum had to be dumbed down a lot. Before 1900, when at most maybe 10% of people went to high school, the level was much higher. Remember those PBS shows about Abigail Adams and her husband the president and their accomplishments with a high school education or less? Now that we have more universal college education, the same thing has happened there. College is now high school, graduate school is college.......

"why do we have to sleep?" As a grad student I became a vegetarian partly in order to spend less time sleeping and digesting heavy food. I also skipped lunch for the same reason, and to save money. I tried to sleep only 6-7 hours per night, and ran 2-4 miles each morning to...... sorry, these details are not interesting, but the moral is, we do a lot to provide for our families. you guys inspire me anew. And this is definitely not advice,.. I am laughing and crying when I recall this. Hang in there!

That;s an easy one, "yes!". And you are a gift to me!

a very interesting and thought provoking post.

Here is a link (hopefully) for the beautiful Hartshorne essay: http://www.ams.org/n...-hartshorne.pdf One basic point is that Euclid does not assume one understands "real numbers" before doing geometry as do almost all other modern geometry books. That allows him to first build a familiarity with the geometric concepts of lines and triangles and planes and areas, before introducing numbers to measure them. This is the historic development of the subject. Real numbers grew out of a need to quantify the geometric notion of similarity in Euclid, which appears there (Book VI) using a sequence of rational approximations. In fact the notions of similarity and area are equivalent, but that is not made clear in some modern treatments. E.g. in the excellent book on geometry by AOPS, area of a triangle is defined to be (1/2) the product of the length of the base by the height. Unfortunately a triangle has three potential sides to be considered as the base and there is no way to prefer one over the others, so one needs to know that every choice leads to the same result. To prove this fact uses the principle of similarity. That principle however is proved in AOPS later in the book, as Euclid did, and relies in turn on the concept of area. So the reasoning there seems to be circular. Euclid had given a different more geometric development of area. I.e. in AOPS, two triangles with the same base and "in the same parallel lines" (hence same height) have the same area by definition, but Euclid shows instead that they can be decomposed into congruent pieces. This does not mean AOPS is not a wonderful book. But what it is attempting to do I think, is build problem solving skills, and convey good mastery of geometric facts, rather than give a completely theoretically sound logical development of the subject. It does a wonderful job at achieving its goals, but it helps i think to know what those goals are. Not only is writing ones own materials from scratch mostly unfeasible, as regentrude said, but even choosing from among the available sources seems challenging to me. I would like to comment on some of the different books out there, since there is not just "geometry", but many flavors of that topic and many presentations and focal points for it. I would suggest that the primary goal of AOPS is as its name implies, learning to solve problems. So it teaches creative thinking and it also imparts a lot of sound information. I believe the series was conceived and executed by someone who enjoyed math contests and tries to prepare students for them.. I would say the strength of Harold Jacobs' books is that they teach that math can be fun, they explain the concepts clearly, and the concepts are related to everyday situations with cartons and humor. This may be a book many kids will "take a shine to" and agree to read alone. As the Jacobs geometry book progressed from first to third edition, it became less reasoning oriented, with less logic content, and more factual. I.e. more facts were taken for granted and fewer were deduced by reasoning from prior material. The Saxon books seem to have as strong point good retention of what is taught, and lots of repetitive drill on fundamental manipulative skills. They are said to raise scores on standardized tests. Deep explanations and fun may be lacking. The book of Euclid provides a foundation for geometry via logical reasoning, the historical development of real numbers, the combination of ideas from geometry and algebra, and the beginnings of reasoning by limiting processes, which leads to calculus. Even the deeper results of geometry in Euclid, like constructing a pentagon, were omitted from my high school book. I hope this helps choose ones own sources. Of course they should always be previewed oneself, preferably at leisure in a library, for free.

The recommended version of Euclid is the beautiful Green Lion edition. at about $15 paper. A recommended companion volume is Hartshorne's Geometry: Euclid and Beyond, about $50, or my much less useful but free epsilon camp notes on my web site at the UGA math dept, under retired faculty. Hartshorne's book is also chock full of advanced material that will last for years and years. Chapter one alone suffices for a first course from Euclid. I do not believe at all that my recommendations of Euler and Euclid will work for every particular case, and I welcome advice from those with more experience as to what uses they may have. They have worked with the epsilon camp kids, in an intense 2 week setting, with highly motivated and "gifted" math types. I am just trying to throw them out there and get more content choices in the mix, for what that may be worth. These suggestions are aimed at long range value of the material learned, as being of high quality, but the everyday crux is what works in the real world of teaching it. Even if these sources are better as second courses, for those already knowing the material, they need to be known for that to be possible. It may be that in practice the usual suspects are the most workable, but maybe these could be useful as review and a test of how well the main ideas have been grasped. for those interested in trying Euclid, there is an excellent essay on Teaching according to Euclid, by Robin Hartshorne, available on the website of the Notices of the AMS.

If Euler does not suit, then for modern day books, written by a pedagogical masters, and very child friendly, I suggest the two books by Harold Jacobs, Elementary Algebra, and Geometry (preferably first edition). These books make math fun, and still teach solid material. There are used copies available on abebooks.com for $15.50 for geometry and about $32 for algebra.

Here is a free version of Euler: http://archive.org/details/elementsalgebra00lagrgoog One can see on pages 244-245 a clear explanation of the meaning of quadratic equations. It is very simple. There is one basic fact, that a quadratic equation factors into a product of two factors, each determined by one root: X^2 -bC +c = (X-r)(X-s) where r,s are the two riots of X^2 -bX + c = 0. there are two immediate consequences: 1) the equation X^2 -bX + c = 0 can be true only if the equation (X-r)(X-s) = 0 is true, which happens if and only if at least one factor in the product is zero, i.e. if and only if X=r or X=s. Thus such an equation can have only two solutions. 2) Multiplying out, gives X^2 -bX + c = (X-r)(X-s) = X^2 -rX-sX+rs = X^2 -(r+s)X + rs. Hence by comparing coefficients, we must have b = r+s and c = rs; i.e. the coefficients of the equation itself always determine both the sum and the product of the two solutions sought for. There are also many clever worked examples and sample word problems. Still it is not cheap, my Cambridge Univ Press edition is about $50. And it is in old style English from 1822 or so, and may require adult assistance for even a bright child. The advantage is that it is was written by an absolute master of the subject, with a view to instructing complete beginners. It is said that Euler aimed it at his butler, an intelligent man innocent of mathematics. It is essentially unheard of today for great mathematicians to write elementary books, not even calculus books, much less high school algebra. But please just take a look at the explanation on pages 244-245, and maybe the absolute begining introduction on pages 1-2, as to what "quantities are:" and how to introduce numbers into measurement of arbitrary quantities, to see how this book differs from all others. If this appeals, I suggest further preview of it before investing any significant sum of money. And the Amazon reviews warn against some editions such as that from Tarquin. It may be that Euler is not practical as a standalone book for home school. But I hope it could be useful as a supplement for child or parent, since it does explain what the material means. In this role the free versions online may suffice. I think I can guarantee that a child who masters the version in Euler will understand much more than one taught from traditional modern texts. E./g. Euler teaches how to use ones knowledge of solving quadratics to also solve cubics, a few pages later. This is not taught in any modern high school algebra books that I know of, but Euler makes it look relatively easy. E.g. If u^3+v^3 = q and 3uv = p, then X = u+v solves X^3 = pX+q. (You can "easily" check this by multiplication.) E.g. since 1^3 + 3^3 = 28, and 3(1)(3) = 9, then X = 1+3 = 4 solves X^3 = 9X + 28. Moreover, given the coefficients p and q, to find u,v, that satisfy u^3+v^3 = q and 3uv = p, is easy, since we only need u^3+v^3 = q and uv = p/3, or u^3v^3 = p^3/27. Then since we know both the sum and product of u^3,v^3, we can find them by solving a quadratic! Namely just solve t^2 - qt + p^3/27, for t = u^3, v^3, and take cube roots. In the example above, p=9 and q = 28, so we solve t^2 -28t + 27 = 0, for t=1,27, and take cube roots to get u,v = 1,3.

katilac, my lengthy answer to recommended books just got lost. I will try again. Basically, it is Euler's Elements of Algebra, and Euclid's Elements (for geometry), but there were lots of useful examples and some caveats included.

@regentrude: as to the math background for learning physics, I would leave that flexible. I.e. I would hope that there is a basic core of physics that does not require much math, but if you recommend one should know a certain amount of math first, I would like to know that as well. Are there basic concepts of physics one can understand without much math? If so, I would want to start there. Or if not, which physics concepts really require which math methods to grasp? Here is an example: In teaching calculus in school I struggled to teach its applications to concepts like "work" for which I had little intuitive grasp. I wanted to "see" or "feel" work somehow. It helped to think of it as change in potential energy, or the damage that a big rock would do if dropped after being lifted a certain height. I finally got help from connecting it with volume, i.e. linking physics with geometry helps me. E.g. in physics I believe there is the principle that the work done by moving a wire or plate of uniform density is proportional to the mass times the distance traveled by the center of mass. This parallels an ancient principle in geometry (Pappus' theorem) that area or volume generated by revolving that figure is proportional to the distance traveled by the center of mass. It finally dawned on me that since physicists also understand the work done by moving a solid, so too one could think of the 4 dimensional volume generated by revolving a solid, (around a plane in 4 space I guess), as proportional to the work done by raising that solid against gravity. I wrote this up in the epsilon camp notes, showing how Archimedes could have calculated the volume of a 4 dimensional ball, just by computing the work done by moving a half ball a certain distance against gravity and multiplying by 2Ï€. So the physicists' concepts seemed to me more comprehensive than the mathematicians', since their concepts involving motion and time allow one to contemplate 4 dimensions. Thus even Archimedes knew enough physics to compute 4 dimensional volumes, if those had made sense to him. Until I realized this, I had struggled with the trig identities needed to do this volume integral. So as a mathematician I am very envious of the intuition physicists have that gives them a leg up in understanding both the world and the math we use to study it.