I now have 66 prints prepared for the World Science Fiction Convention art show print shop. I will share the print shop with dozens of other artists. The show runs from 2 to 5 September in Boston. It's part of the convention and thus is only open to paying guests, I'm sorry to say. The images in this upcoming show of mine date from 1995 through 2002 and all of them can be seen on my Website. I feel rather annoyed and even unhappy that I don't have any new original art to display at the convention, but with job, mathematics, and commercial commissions, I just haven't had the time to make new art. I hope that the older pieces will still be appreciated (and bought) in my old territory of Boston, where I grew up and spent my first years in the science fiction fan community. I am far less active among the fans than I used to be, but it is still good to go to a Worldcon and see friends from all around the country, and even some from overseas.

Math with Paul Klee

I will be in New England for a week after the convention, visiting my parents' home in the "north woods." I expect to add some entries to this Weblog while up there. Of course, I'll have my math with me. I put 1958 back on the shelf, and I put Barron's algebra review text (with the "Ruritanians") back too. I have never had any luck with the Barron's texts, as you remember from my trigonometry studies. I can only assume that these books are made for use in a class with a teacher. Also, the algebra text expects the student to know simple computer programming, which I certainly don't. There are assignments at the end of each problem set where the student must program some computer to do basic mathematical computation. I know nothing of computer programming, at least not yet. I am still working on learning how to use the slide rule.

The book I'm taking with me is my underused "White Paul Klee" book. It is a businesslike but mild-mannered text for first-year college students. It has thick, smooth white paper, a blocky shape, and weighs a lot, so it isn't a casual briefcase type of book. The real title of "White Paul Klee" is: INTERMEDIATE ALGEBRA:GRAPHS AND FUNCTIONS by Roland E Larson, Robert P. Hostetler, and the evocatively named Carolyn F. Neptune, whose name suggests a mathematical mermaid. A search on Amazon shows pages of math text books written by the prolific Roland "Ron" Larson. I suppose someone's gotta write these things. The current edition of this book is listed on Amazon for a whopping $105.56. I got the ten-year-old edition, in mint condition, from a discount mail-order book list for about $8. I definitely "did the math" there.

The Paul Klee on the cover is titled "Pyramide, 1930.138." It is one of Klee's geometric watercolors. Paul Klee is one of my favorite artists, especially when he works in the geometric style you can see if you scroll down on the link page to Klee's neat picture with the arrows in it. I imitate his art, and that of his friend Wassily Kandinsky constantly. I want art that looks like math. Art that snaps shut like well-crafted equations. Art that solves problems.

I'll be reviewing logarithms, sequences, and progressions with this book over the next few weeks, when I get the time. The book has plenty of clear exposition, worked-through examples, problem sets, and "real life situation" problems with applied math. It is patient and does not rush the student. The headings and many accents are in colored printing, but it isn't at all garish. Best of all, it has no fantasy characters in it. It also has no logarithm table in the back. Just press the button, Wassily, it's 1994.

Posted at 2:39 am | link

I have not had much time to do math this week due to pressures at my day job and preparations for the World Science Fiction Convention. I'll be showing prints there, of many of the pieces that you can see on my main website. Unfortunately, I won't be showing any new or non-printed "original" work. I simply have not had time to make any.

I have had to cut about 60 mats and mounts, the cardboard frames that hold prints. This is one of the most tedious jobs in commercial art. Not only do I have to cut them using a mat-cutting knife (I have no room for a professional mat-cutting device), I have to place the signed and numbered print on the mounting board, place the mat over it in just the right position, fasten the print to its mounting board, then fasten the mat to the print's mounting board to produce a sturdy package. And even then my work is not done. I put documentation and a written (printed) note about the picture on the back of each print. Finally, each print needs a tag for tracking and pricing at the print shop of the convention art show.

Even with all this slicing and sticking, I have done a little math. I can't live without doing at least some of it. "Positive addiction," you know. I've been reviewing my recently learned logarithms with the two more modern math texts I mentioned an entry or so ago, namely Barron's and "White Paul Klee." Barron's is much more practical than 1958. Interestingly, it still has logarithm tables, though it recommends the use of a calculator. Barron's problem sets are not totally abstract, but are related to widely used logarithmic-scale measurements such as decibels (for sound) and the magnitudes of stars (brightness as seen by observers on Earth). 1958 had some of these, too, but not many.

When I read the Barron's text introducing logarithms, for a minute I thought that I was reading about some other type of math altogether, something that I had completely failed to learn. This was a frightening minute. It was just like one of those nightmares you have where you are sitting to take a final exam and realized not only that you have not studied, but you don't know the material at all. But reading on in the Barron's text, I realized that they were just putting the elements of logarithms and exponential functions in a different order, and that the stuff I was familiar with was right there.

I get that review panic feeling sometimes when I look back into an algebra or trigonometry text and realize that I would not be able to pass a test on a subject that I slaved and struggled over just a year ago. The antidote for this panic, of course, is to do more reviewing, until I am re-familiarized with what I need to know. For instance, I want to revisit series and progressions, along with their sums and limits, which I last worked on in early 2003. I will also need to review my beginning studies in classical mechanics, such as accelerations, vectors, and Newton's laws.

Interestingly, I don't get panic when I look back at things like quadratic equations, polynomials, or congruent triangles. I have been through these things so many times that I feel at ease with them. I'd like to have that feeling with many more things in both math and physics. Only revisiting and review can give me that facility. It's the way my path goes; sometimes I have to go back in order to go ahead.

Posted at 2:04 am | link

I'm bidding goodbye to 1958 for now. After I had finished my weeks of logrolling, the chapter sections on logarithmic scales and exponential graphs were too archaic even for me. The slide rule was the best you could get back then, logarithmic scale and all. I realized, perusing this relic, that though slide rules are cool and retro is metro, (huh?) it began to be counterproductive for me not to work with calculator and computer. Whatever I will be doing with my math and science in the future, it will inevitably involve these technologies, not a slide rule. It may eventually be important for me to learn computer programming. Don't ask me why; I'm supposed to be a middle-aged dilettante here.

I have two other books to teach me logarithmic scales and exponential functions. One is from 1994, a college text which I picked up from a very discount book catalog, and it has been useful to me in the past. I'll call this one the "White Paul Klee" book, since its white cover is centered with a very nice geometric abstraction by Paul Klee. The other is yet another Barron's high school study guide, which I used to re-learn algebra some years ago. This Barron's, from 1996, features the same pathetic little "Ruritanian" fantasy characters who attempted to teach me trigonometry last year. I don't think I'll stay too long with them, but the book is at least bigger (larger print) and more readable than 1958. And both these books have plenty of problem sets for me to work on.

Problems problems problems. As I said before, how can I know I've learned something unless I've solved problems with it? I spent hours and hours doing the logarithm problems in 1958. Every time I solved one, my reward was to get another, which was more complex and convoluted than the previous one. This progression of increasingly difficult problems is such a basic part of doing math that I believe that this must reflect the ultimate underlying structure of our lives on earth. You start on simple problems, then they get harder and harder and harder as we grow older. I have never found a problem set where the authors, in order to have fun or trip up the unsuspecting student, place an easy, simple problem towards the end of the list.

This is why I don't look too far ahead in my studies. Often times my Friendly Mathematicians or Scientists, in their eagerness to share what they know and to teach me, offer me something which is just a little too far ahead of what I am currently doing. Or more than a little too far ahead. It seems simple to them, but it freaks me out. I seize up, like a computer asked to do too many things at once. The same thing happens to me when I look too far ahead in one of my math or physics books and magazines. Arggh! I'll never be able to learn that!

This is where my age and lack of "math/science talent" shows. If I were younger, much younger, like those brilliant teenagers I read about so often in science/math writing, I would be leaping about my math program like an Olympic gymnast on the tumbling floor. I would do my problems swiftly and impatiently, and grasp concepts in a flash. I'd be far ahead of where I am now, given the same amount of time spent.

But that's not what my math path is like. I'm just slow. I assimilate new information and ideas slowly, one bit at a time, like an old, very slow modem. I hack my way through problems one at a time, like the Capricornian goat picking its way up the slope rather than the Sagittarian arrow speeding to its goal. (Disclaimer: I don't believe in astrology, I'm only using this as a metaphor, and I am neither a Capricorn nor a Sagittarian.)

I try to tell my preceptors this, and they don't quite understand. Maybe this is because most of them are younger than I am, and talented professionals in their fields. Why don't I just move on to more complex mathematics and more physics? I answer that I go so slow because I want to make sure I know what has gone before, and be familiar with it, before I move on to that frightening new thing.

This works for me, but it just doesn't work very fast. Confidence is always an issue here. Only when I am no longer spooked by seeing strange numbers and symbols in front of me will I know that it is time to move on to new strange numbers and symbols. Until that time, I will continue my low-baud-rate ways,
one
problem
at
a
time.

Posted at 2:20 am | link

Tonight, like a whole lot of other people, I watched the opening ceremonies of the Athens Olympic Games on TV. The theatrical part was fabulous for an old classicist like me, since I grew up studying ancient Greek language and culture. The mixture of stylized historic costumes, slow pace, acrobatics, big suspended shapes, Mahler music, and splashing water was very Symbolist and "European" and probably bewildered many Americans, despite the well-meaning commentary attempting to explain it. There was only one pop singer ("Bjork," from Iceland). It was quite un-Hollywood, no lineups of chorus girls or chrome pickup trucks or Oriental fire dancers. Surprisingly Serious, for a big international pageant.

I am an unashamed fan of big pageantry. I just love it. Maybe it's an artist thing; I would imagine it is not a scientist or mathematician thing. I love costumes, dancing, special effects, light-shows, and fireworks, as long as I watch them and don't have to actually do them myself. So I had a great time watching the Olympic show tonight, complete with the neo-Zoroastrian lighting of the futuristic torch, which when erect (yes) resembles a gas burn-off tower at an oil refinery.

I also loved the parade of athletes, from all those countries I've only barely heard of (Sao Tome et Principe? Timor Deste?). More costumes! The Central Asian countries had particularly wonderful flag-bearers. Each Olympics I watch for the Mongolian group because they usually feature a burly wrestler as flag-bearer, wearing a fur loincloth, fur boots and hat, and a big wide cape, kind of like Conan the Barbarian. But since the ceremony this year took place in 90-degree heat, the Mongolians, as the commentators said, "left their fur at home."

The cameras lovingly dwelt on these thousands of young athletes, each one of them more beautiful, glowing, and super-fit than the next. Bright, ephemeral flowers of humanity, here for one glorious moment in the summer sunlight. Most of them will never win anything, but will do their best and have two weeks of fun. Those who are ranked and pressured competitors may have less fun, but they are more inspiring to me.

I don't care much about the issues of performance-enhancing drugs, or cheating, or politics. It's the uplifting ideal of intense discipline, struggle and victory (or honorable defeat) which I believe in. Back in 2000, though, I couldn't bear to watch the Olympics. When I saw the young athletes and their life-consuming, goal-oriented striving, I felt intense envy, leading to depression. I felt as though despite making lots of nice art, I had achieved nothing, and I was in my late forties. I was wasting my time, with no goal in sight. I would never be one of those enchanted youths, and never had been. I had nothing to strive for.

But that year I visited Fermilab, and experienced the inspiration which gave me my own goal to aim for. I know that I will not win medals or prizes, never enter the "math Olympics" (a contest for the brainier kind of flowerlike youth) and I won't ever make any new scientific discoveries. But learning math and physics has become my own personal goal, my own private Olympic quest. My victories are small, and won't win me any fame or fortune. But they mean something to me. My Olympic torch shines in the energies of particle accelerators and quasars. And so I can watch the athletic Olympics again with joy, knowing that I too can strive for the Pythagorean laurels.

Posted at 2:56 am | link

Pyracantha to logarithm tables: I give up. Despite an hour-and-a-half long phone conversation with one of my Friendly Mathematicians, in which he with infinite patience led me through one of those fractional-exponent evaluation problems, I simply cannot do one right. I follow the process. I search and interpolate. I divide, multiply, and reciprocate. I find the antilogarithm. I compute the results. Still wrong. I simply cannot stop making simple computation and arithmetic mistakes, even if I correctly follow the logarithm-negative-positive-reciprocal-antilogarithm process. Somewhere in all that hand calculation (no electronic means used) I make a mistake; one decimal point off, one number wrong, and then the whole thing is off. Or I fail to reverse one negative-positive logarithm number. Something always goes wrong.

I could try again, and again, and again, and again, and I still wouldn't get them right, even if I did a hundred of them. It's pathetic not only that I can't do them, but that I've tried so hard and done so many, still hoping to finally get it right.

This is what math for me in middle and high school was like. Endless, meaningless numbers, in complicated processes I hardly understood, trying to get things right and failing over and over again, exhausted and filled with shame.

I finally asked the Friendly Mathematician: How important for my future nonexistent physics career is it to correctly compute these things using only tables and working the math out by hand? "We do all these things with calculators and computers now," he answered."Other than understanding how logarithms arise and how they have been used to do computation, it's irrelevant and obsolete."

I tried to imagine a young boy, already immersed in math with the hope of going on to a career in physics or engineering. Would he be doing all this work? "No, he would have already finished his homework using the calculator and would be playing a video game."
I don't have that option. Ladies of a Certain Age do not play video games. But I do have a calculator. This logarithm game has been enough for me. I did not pass 1958. I will not help defeat the Commies with my scientific expertise. But I have to remember it's 2004 now. Time to proceed to exponential equations. Logging out for now.

Posted at 2:52 am | link

It's still 1958, and I am tasked to find a given number, less than 1, to the negative 1/3 power, using logarithms and the table, all other computations by hand. No pocket calculator, they haven't been invented yet. How should I proceed? I try to follow the example in the book.

First, state your negative exponent as a reciprocal fraction, that is, one over the number whose exponent is now positive, but still fractional. Then find the logarithm of the number. State the number in scientific notation, which will give me the "characteristic" that is the integer part of the logarithm. In this case it's negative 2. I find the mantissa (four-digit decimal part of the logarithm) in the table by interpolating between two entries on the chart. The logarithm is negative, which is a no-no, you can't work with that in this system. You have to re-state your logarithm as the difference between two positive numbers, for instance a negative 2 must be stated as "8 minus 10."

Now you want to divide your logarithm by three, since it's a fractional exponent, one third. You can either do this by upping your two positive numbers so that they can be divided by three and retain their original subtractive relationship coming out to negative 2. Or you can do more calculations and find the negative logarithm that is the reciprocal of the one you have, by subtracting the smaller from the larger and giving this remainder a negative sign. Then you divide that by three. Either way, it's supposed to get you a fractional exponent. Or a root, if that's what you're after. They're the same thing, sort of.

When you have the logarithm divided by three, then you must reverse it to its reciprocal again and find its antilogarithm, which is supposed to be the final number you were looking for in the first place. For any other number in this computation, perform the same process of logarithmizing, reciprocation, and antilogarithmizing, and then do the other computations necessary to get your totally pointless result.

This business takes me about 15 to 20 minutes for each problem. If I should peek back into the twenty-first century and use my little pocket calculator, it takes about a second.

I have been working on these problems for days now. Hours go by as I find logs in the table, reverse them, multiply or divide them, re-reverse them, negative or positive, reciprocal or original, and then find the anti-log in the table. The slide rule isn't good enough, it only gives about three significant digits, and the book asks for four. I have done one problem after another. The third root of (0.8210)² minus the fifth root of 2.927. Or (2.138)³ multiplied by (43.10)^-2. The hours go by and things get heavier and more involved and more soaked with the fog of logs.

And no matter how many of these things I do, I haven't gotten a single one of them right. There's always something I've missed. I miss a negative/positive sign. I miss a reversed or reciprocal logarithm. I miss a digit in hand calculation. I miss altogether, having inattentively switched or added some four-digit number with something else it wasn't supposed to be with. My eyes blur from looking so long at the small print in the tables.

I keep hoping that sooner or later I will master the twists and turns and reverses and finally get one right. For all I know, this type of calculation is necessary practice for my working with far more complex and twisty-turny negative-positive-negative problems later on. Or it could be a waste of time. Perhaps there is some virtue to mastering a technique which has not been practiced since the advent of far more efficient mechanical calculators. At this point, I just don't know. Maybe one of my Friendly Mathematicians will tell me what this is supposed to do.

Some physics formulas have logarithms in them, so I know I will need this somewhere. But how much logging do I have to do with the wooden tables? I know some high-minded types who insist that just because a computer does the work now does not excuse you from learning the original way it was done. But at this point it feels like it will be years before I get one of these problems right using the tables and the reciprocals. And I have such a long, long way ahead of me.

Posted at 2:38 am | link

I've been doing lots of logarithm problems, the old-fashioned way using the log tables and sometimes the slide rule. It takes an entire page of scratch paper to work out just one of them, since there might be four or five different numbers each one with its own logarithm which has to be added, subtracted, or multiplied, then turned back into its anti-logarithm, i.e. the number which would be the answer to the whole computation. I am constantly having to "reverse" negative logarithms into the difference of two positive numbers or vice versa, after I find their mantissa on the log table and interpolate it.

I get most of these problems wrong. Maybe all of them wrong. When I trace my many steps back through the log table and the reversing of negatives, it always comes down to one arithmetic mistake, somewhere. An 8 instead of a 9. A negative sign instead of a positive. And then I re-do the calculations, and check them not only against the answer that appears in the 1958 book (at least for the odd-numbered problems) but the answer that my calculator gives me. Wrong again. Write a frightening red X over the wrong answer and start all over again.

OK, I'm a little obsessive. Maybe more than a little. I close doors that are left open. I want things to be neat and lined up. I perceive the world as a maddening mess of details, all of them clamoring for my attention at once. And maybe I'm not very good at ordinary arithmetic. But I want to know that I am doing things right. I want to master these archaic mathematical tools, once I have taken them up, even though calculators and computers now have probably irrevocably taken the place of hand-computing by tables and slide rule. Yet I am defeated, again and again.

I experience a kind of anxiety as I progress in mathematics. Ahead of me now are places I've never been, mathematics I've never, ever worked before, full of arcane Greek letters and forbidding symbols. It looks to me like magic, like Things I Am Not Meant To Know. If I fail in the small workings, how can I attain the Great Work? I remember that this quest is something I have taken on myself to do. No one is making me do it. There is no usefulness to it, no engineering or scientific career waiting for me if I do well and persevere. I am not a fresh young thing dreaming of doing research and making new world-shaking scientific discoveries.

Nevertheless, there may come a time when I actually will need to know what
-0.8619 to the minus fourth power is. (With only a log table and no calculators or computers available, which is the least probable part of this possibility.) I guess I'll do some more of these logarithm problems. Sooner or later I'll get one right.

Posted at 2:37 am | link

Toiling through these long-winded logarithm problems, I find it interesting that the results I get by hand or slide rule differ by one or even two significant digits from the results that the electronic calculator gives. In order to get calculator-like results "by hand," I have to do extra sub-calculations to match up the mantissa numbers with the desired subject numbers. The process of multiplying and dividing long numbers by adding or subtracting their logarithms seems so time-consuming and involved that I wonder whether it is really a more convenient method at all. The negative logarithms are especially confusing.

The mechanical devices like slide rules and calculators are so much of an advance on the pencil-pushing use of tables and interpolations that it reminds me of something I once heard concerning the history of science and technology. Only when mathematicians and scientists could gain a reasonable level of accuracy in calculating things like logarithms, could science and engineering progress forward. The measurements that lead to scientific discoveries or technical inventions, despite all those banks of computing girls doing the work by hand, need far more accuracy than mistake-prone hand calculation can do. And of course nowadays, the level of measurement and accuracy is so incredibly high that only electronic devices of vast and ponderous power can crank out the results.

One of my Friendly Scientists assures me that I have actually done more physics than I thought, and a quick review through one of my physics texts would send me on my way into the next "semester's" work. But it's still a long way from where I am, to the time when I will be formally introduced, that is, mathematically introduced, to atoms and the world of subatomic particles, let alone energy and space-time. I am still in the eighteenth century scientifically and mathematically, though in the nineteen-fifties technologically. To use a musical analogy, I'm listening to a peculiar combination of Bach and Elvis.

Posted at 2:43 am | link

Midnight at the Electron. The air is as warm and heavy as cream. Two layers of clouds pass over the Blue Moon of July: one lower down, swift and tumbling and fluffy, the other higher up, variegated, sharp, and iridescent in the full moon's light. Every so often heat lightning, reflecting from distant storms, flickers on the clouds. The trees are full of insect song and katydid chatter. Brown bats dash through the air, swooping in and out of their nests inside my building's roof. In the clear black sky-spaces between the clouds, I can see the bright stars of the Great Summer Triangle, directly overhead: Vega, Deneb, and Altair. Moon, lightning, stars, the photons of August.

Fear of Physics

I'm multiplying and dividing long numbers by using logarithms. Or trying to, at any rate. I must get the logarithm of each number by going to the table and interpolating between the log readings until I match the fourth significant digit. Then I must add or subtract the logarithms, making sure I take into account the negative logarithms which are to be expressed by the difference of positive numbers. Then I must find the anti-logarithm of the result, again interpolating into the log tables using the not-so-handy charts of "proportional parts." This is a long process. The tables are in small print and the first number of each mantissa is not printed with it, but set off to the left side, so it's easy to lose track of what the first digit might be. Since I am adding and subtracting by hand, I often make clerical mistakes. And even when I'm finished after at least fifteen minutes of calculation, the number I get does not match exactly the one that my calculator squirts out in a half a second. It is usually one significant digit different, even when you round off the longer number to a shorter one.

Clerical accuracy, Miss Pyracantha. How will you find your place among the "computer girls" in the astrophysics lab if you make so many mistakes? Here, sez I, let me show you this device I sneaked in from the twenty-first century. It's a pocket calculator.

It's been almost four years since my epiphany at Fermilab. I started seriously studying mathematics in early 2001. I've worked on arithmetic, basic and intermediate algebra, geometry, trigonometry, and now logarithms. But what about physics? After all, this is what I resolved to learn. Yet in these years I have had only a little of what would truly be called "physics." Sure, I've done a large amount of mathematics, especially for someone who had no ability and little knowledge of it when I started. And I remember a quote from some famous physicist (can't remember who) when he was asked what you needed to learn to study physics: "Mathematics, mathematics, and more mathematics."

Plenty of math, then, but not much physics. What I've done has been fragmentary. I've read about Newton's laws, and can quote them to you, but without truly understanding what an "equal and opposite reaction" is despite many patient attempts to explain it to me. (How can a wall "push back?") I've done distance, rate, and time problems. I've read about pseudoforces and momentum and velocity. I've done a fair number of vector problems, but if I were faced with one right now I would panic and I'd have to go back and read about it in my book again before I could solve it. I've learned what the acceleration of gravity is, what acceleration is, and how you calculate it given the data on how fast something is going, how fast it was going, and how long it's been going. Many fragments, but no continuity.

I have so far searched in vain for a text or website which would connect all these things together and make sense to me. Since I have never taken a physics course, I have "no prior knowledge of the subject" which puts me in a class with elementary-school children. The materials for my level of physics understanding are for kids. They are colorful, toy-like Web animations or virtual worlds full of cartoon characters and earnest multicultural children doing simple fun experiments that I never did in school. I guess no one thinks that any middle-aged person would ever start learning physics from the ground up, any more than a middle-aged person would take up some physically taxing sport like gymnastics or rugby when he/she had never done it before.

Some of my Friendly Scientists still suggest that there are "physics for poets" or "physics for non-concentrators" books and courses which have no equations or math so they would teach me and not trouble me with any of that "hard" stuff. Well, dudes, you just don't understand. I WANT hard stuff. I don't want charming little animated pink cartoons, but dry text, black and white diagrams, and whole pages of mathematical problems to solve. I want that victorious rush of solving a problem correctly, not a warm fuzzy glow of soft learning. How can I know whether I have really understood something if I don't solve problems in it?

I've got a couple of old college physics texts that one of my Friendly Mathematicians gave me, but they quickly vault into calculus, where I have not yet ventured. Same with other college physics texts. Lately I have come across a fairly good physics learning site, The Physics Classroom which is aimed at high school kids but seems to have a bit more dignity than some of the other learning sites. It has problems to solve, but it still has cartoons. The trouble with learning from websites is, obviously, that I have to be at my computer to use them. I have plenty of paper books purporting to teach me basic physics but somehow I haven't been able to work with them. I figure that this must be my fault, either through laziness or lack of concentration or what physicist and writer Lawrence Krauss called, in one of his book titles, "Fear of Physics." (I've read that book, but should read it again.) There's even a "Fear of Physics" site which I once frequented. I may have learned a bit from it, and I appreciated the problems it was able to give me, but I didn't like its childish graphics and kid-oriented text. I keep looking, but perhaps I'm not looking in the right place, or I'm too picky. Or perhaps what I'm looking for doesn't exist. I am ashamed of myself. I've had almost four years to learn basic physics, and I haven't gotten beyond middle school. What is all this mathematical toil for, if not to do physics the way the big boys do it?

Posted at 1:26 am | link