"Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." ---L. Euler (Preface Introduction to Analysis of the Infinite (1748)). "Algebra is conducive to symbolic reasoning." ....PSSM, p.345 "The problem with wrong proofs to correct statements is that it is hard to give a counterexample." ---H. Lenstra "It is an equally awful truth that four and four makes eight, whether you reckon the thing out in eight onions or eight angels, eight bricks or eight bishops, eight minor poets or eight pigs." ---G.K. Chesterton "And yet, almost half of our 17-year-olds do not have the basic understanding of math needed to qualify for a production associate's job at a modern auto plant." ---Secretary of Education Margaret Spellings in testimony to Senate Committee on Health, Education, Labor and Pensions, 9 February 2006 From Bridge magazine in Britain: Q: "Do you have to be a brilliant mathematician to play bridge?" A: "No. The truth of the matter is that bridge is much more about inference, deduction, and judgement, than it is about mathematics." "Moore and Russell were having a philosophical discussion in Hall. Russell suddenly said: `You don't line me, Moore, do you?' Moore replied `No'. This point disposed of, the discussion proceeded as before." "The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic." ---G. W. Leibniz ``If things are nice, there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do. '' ---Richard Askey "It is curious how `no mathematics without total understanding' sometimes advocated for one stage contrasts with `mathematics without any understanding' advocated for another. Nowadays at the primary level there is a great insistence on understanding preceding memory in the teaching of `tables', whereas at the secondary level pupils are taught certain geometric facts without being expected to either understand the proofs or even be aware that there are logical connections between the facts. For example, many GCSE candidates will have memorized the angle properties of circles but will have never seen their derivation from properties of triangles. As argued earlier, it is precisely in the logical deduction of one geometrical fact from others that the mathematics and the educational value lie. The situation is rather like a parent disciplining a child, who gives detailed explanations when the child is a toddler and arbitrary decisions with no explanation when the child is a teenager." --- From "Is the proof in the practice" by Tony Barnard, which appeared in Mathematics in School, September, 1989. "You are capable, competent, creative, careful. Prove it." --Fortune from Cookie "The Mathematicks, though not absolutely necessary to a Devine in the way of his Profession, yet has a great Influence upon his Studies; it gives him a Habit of Thinking abstractedly upon every Subject; enduces him with with Patience to investigate the most knotty Problems, for the sole Pleasure of finding out the Truth; and is useful in explaining most other Sciences." ---The London Tradesman, published in 1747, "for the Information of PARENTS, and Instruction of YOUTH in their Choice of Business." It has to do with preparation for, "the first learned science, divinity." Son: Dad, can you help me with my algebra homework? Dad: No, it wouldn't be right. Son: Sure, but you could try. "I know that two and two make four -- and should be glad to prove it too if I could -- though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure." --Lord Byron 1788-1824 "Long separated by cruel fate, the star-crossed lovers raced across the grassy field toward each other like two freight trains, one having left Cleveland at 6:36 p.m. traveling at 55 mph, the other from Topeka at 4:19 p.m. at 35 mph." "How happy the lot of the mathematician! He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. No cashier writes a letter to the press complaining about the incomprehensibility of Modern Mathematics and comparing it unfavorably with the good old days when mathematicians were content to paper irregularly shaped rooms and fill bathtubs without closing the waste pipe." ---W. H. Auden 1907-1973 "When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian functions. How did this happen? In mathematics, as in other sciences, the same process can be observed again and again. First new questions arise, for internal or external reasons, and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and there fore require less foreknowledge - even if it only a matter of axiomatics, or set theory, or some such thing." ---Felix Klein (1849-1925) "If I explained my investigations as broadly as Fourier's book is written it would take me just a quarter of the time and several huge volumes." -- K. F. Gauss (1777-1855) "I don't doubt that [Sacramento City School Superintendent] Jim Sweeney loves children and had dedicated his life's career to improving education. The school district has done some wonderful things ... but (on state tests) half the students are still below the 50th percentile. That's a problem." ---Lauren Hammond, member of Sacramento City Council The best way to get a good idea is to get a lot of ideas. ---Linus Pauling "We've all heard that a million monkeys banging on a million typewriters would eventually reproduce the entire works of Shakespeare. Now thanks to the Internet we know that this is not true" ---Robert Silensky "It is of course easier to supply the proof when we have previously acquired some knowledge of the question by the method, than it is to find it without any previous knowledge." --Archimedes (in his *Method* letter to Eratosthenes) "If 90% of the ideas you generate aren't absolutely worthless, then you're not generating enough ideas". --Mike Artin "Archimedes will be remembered when Aeschylus is forgotten because langages die but mathematical ideas do not." --G.H. Hardy (*MA*, p. 81) "Beauty is the first test; there is no permanent place in the world for ugly mathematics." --G.H. Hardy (*MA*, p. 8) "Mathematics is not yet ready for such problems." --Paul Erdo"s talking about the 3n+1 (Collatz) Problem. There are 10^11 stars in the galaxy. That used to be a huge number. But it's only a hundred billion. It's less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers. --Richard Feynman (1918-1988) The education sargeant (commissar in the Soviet version) gives a lecture in the course of which he says that water boils at 90 degrees. One of the soldiers corrects him, "100 degrees" "No, 90, sit down." A month later the sargeant gives another lecture. An honest fellow, he starts by saying "Your comrade was right last month. Water does boil at 100 degrees. What confused me was that the right angle boils at 90 degrees." The irrationals exist in such variety, indeed, that no notation whatever is capable of providing a separate name for each of them. --Willard Van Orman Quine [*Mathematical Logic, Revised Edition*, page 273 (Harper Torchbook)] [12 + 144 + 20 + 3 * sqrt(4)]/7 + 5 * 11 = 9^2 + 0 A dozen, a gross and a score Plus three times the square root of four When divided by seven (Plus five times 11) Is exactly nine squared and no more 1,264,853,971.2758463 One thousand two hundred and sixty Four million eight hundred and fifty Three thousand nine hun- Dred and seventy one Point two seven five eight four six three Leigh Mercer, in Martin Gardner's *The Unexpected Hanging*, 1969, p238. "It is a profoundly erroneous truism, repeated by all copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle--they are strictly limited in number, they require fresh horses, and must only be made at decisive moments."-- A.N. Whitehead ``The ultimate goal of mathematics is to eliminate any need for intelligent thought.'' --A. N. Whitehead Most of the difficulties students have learning calculus can be traced to weak algebra skills. -- Leonhard Euler "Mathematicians can never leave well enough alone -- what if this, what if that? .. You never get to the end of anything." ---Sarah Flannery's Mom in the book "In Code, A Mathematical Journey" There are 10 types of people in the world: those who understand binary, and those who don't. I graduated from Douglass College without distinction. I was in the top 98% of my class and damn glad to be there. I slept in the library and daydreamed my way through history lecture. I failed math twice, never fully grasping probability theory. I mean, first off, who cares if you pick a black ball or a white ball out of the bag? And second, if you're bent over about the color, don't leave it to chance. Look in the damn bag and pick the color you want. ---*Hard Eight*, p. 218. (A Stephanie Plum novel) "When aiming for the common denominator, be prepared for the occasional division by zero." *Colleague to Mac Lane*: "I am sorry, but I will not be able to come to your lecture. What will you say in the lecture?" *MacLane*: "If I could say it in 5 minutes I would not be wasting an hour of everybody else's time!" "...at about the age of sixteen, I was offered a choice, which, in retrospect, I can see that I was not mature enough, at the time, to make wisely. The choice was between starting on the calculus and, alternatively, giving up mathematics altogether and spending the time saved from it on reading Greek and Latin literature more widely. I chose to give up mathematics, and I have lived to regret this keenly after it has become too late to repair my mistake. The calculus, even a taste of it, would have given me an important and illuminating additional outlook on the Universe, whereas, by the time at which the choice was presented to me, I had already gone far enough in Latin and Greek to have been able to go farther with them unaided. So the choice I made was the wrong one, yet it was natural that I should choose as I did. I was not good at mathematics; I did not like the stuff....Looking back, I feel sure that I ought not to have been offered the choice; the rudiments, at le! ast, of the calculus ought to have been compulsory for me. One ought, after all, to be initiated into the life of the world in which one is going to have to live. I was going to have to live in the Western World...and the calculus, like the full-rigged sailing ship, is one of the characteristic expressions of the modern Western genius." --Arnold Toynbee (*Experiences*, Oxford University Press, 1969, pp. 12-13) "He was an arithmetician, not a mathematician. None of the music, humor, or mysticism of higher mathematics ever entered his head." --John Steinbeck (*The Moon is Down*) "Rationality of thought imposes a limit on a person's concept of his relation to the cosmos." ---John Forbes Nash, mathematician If A equals success, then the formula is A = X + Y + Z. X is work. Y is play. Z is keep your mouth shut. -- Albert Einstein "For fifteen days I struggled to prove that no functions analogous to those I have since called Fuchsian functions could exist; I was then very ignorant. Every day I sat down at my work table where I spent an hour or two; I tried a great number of combinations and arrived at no result. One evening, contrary to my custom, I took black coffee; I could not go to sleep; ideas swarmed up in clouds; I sensed them clashing until, to put it so, a pair would hook together to form a stable combination. By morning I had established the existence of a class of Fuchsian functions, those derived from the hypergeometric series. I had only to write up the results which took me a few hours." ---Henri Poincare "The Russian students are terrible. We teach them something, they do not understand. We teach it again, they do not understand. We teach the same thing a third time, now we understand, they still do not understand." ---I. M. Gelfand "One should always be able to say, instead of 'points, lines, and planes", 'tables, chairs, and beer mugs'." David Hilbert "I heard the sweet flute of the Pied Piper that Hilbert was, seducing so many rats to follow him into the deep river of mathematics." ---Hermann Weyl, describing the prose in Hilbert's preface to Hilbert's *Zahlbericht* (*Report on the Theory of Numbers*) "The winner is the first player to get the most points." --Card game rules Higgeldy-Piggeldy Summertime swimming, where Topless and bikinied Females are found. Watching them all is A mathematician, who Says his concern is The least upper bound. ---From a President Emeritus of the MAA "I think there's no way they should have to teach [math] now. We have computers. We no longer need to know why 3x = (2y)/4." ---Talk-show host Rosie O'Donnell, arguing that there's no place for math in the curriculum. "English you speak 50 million Englishmen speak. English I speak 200 million Russians speak" --Besicovich Finally, I will ask young readers who may get hold of this book not to try to get through it strenuously; don't try to get through it in a few weeks or even months. Take a year or more over it. There is a lot in this book; and I don't think it is any exaggeration to say that its contents would make a generous slice in anyone's mathematical education. One cannot expect to cram this amount of knowledge into one's brain in a few weeks or months without detriment to it. What I have learnt over 52 years cannot hurt my young readers to spend 52 weeks in learning! ---Albert Eagle *The Elliptic Functions as They Should Be* (Cambridge, Gateway & Porter Ltd., 1958) ``Counting pairs is the oldest trick in combinatorics... Every time we count pairs, we learn something from it''-- Gil Kalai (``Combinatorics and Convexity'', lecture) The road to wisdom/Well, it's plain/And simple to express: Err/And err/And err again/But less/and less/and less. -- Piet Hein "Lines that are parallel meet at Infinity!" Euclid repeatedly, heatedly, urged. Until he died, and so reached that vicinity; in it he found that the damned things diverged. ---Piet Hein "The primary purpose of the DATA statement is to give names to constants; instead of referring to pi as 3.141592653589793 at every appearance, the variable PI can be given that value with a DATA statement and used instead of the longer form of the constant. This also simplifies modifying the program, should the value of pi change." ---FORTRAN manual for Xerox Computers "I never did experiment, I never did; I just fiddled around. I made radios and gadgets. I fiddled around. Gradually, through books and manuals I began to discover there were formulas applicable to electricity in relating the current and resistance, and so on. One day, looking at the formulas in some book or other, I discovered a formula for the frequency of a resonant circuit which was 2 pi sqrt(LC) where L is the inductance and C the capacitance of the circuit. And there was pi, and where was the circle? You laugh but I was very serious then. pi was a thing with circles, and here is pi coming out of an electric circuit, where [it stood for] the circle. Do you who laughed know how that pi comes about? I have to love the thing. I have to look for it. I have to think about it. And then I realized, of course, that the coils are made in circles. About a half year later, I found another book which gave the inductance of round coils and square coils, and there were other pi's in these formulas. I began to think about it again, and I realized that the pi did not come from the circular coils. I understand it better now; but in my heart I still don't quite know where that circle is, where that pi comes from..." ---Richard Feynman, addressing the National Teachers' Association (April 1966) "God does not care about our mathematical difficulties. He integrates empirically." ------ Albert Einstein "The whole of science is nothing more than a refinement of everyday thinking." --- Albert Einstein "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality." ---Albert Einstein "Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever." --- Albert Einstein "Much of mathematics means translating problems into abstract representations and converting numerical solutions into understanding. It's something that neither calculator nor computer program can do. It's what each of us struggles with whenever we enter the world of numbers...It's why we'll forever need arithmetic, algebra, and calculus. And it's why computers don't belong in math class." ---Clifford Stoll Next day Brendan raises his hand. Dotty gives him the little smile. Sir, what use is Euclid and all the lines when the Germans are bombing everything that stands? The little smile is gone. Ah, Brendan. Ah, Quigley. Oh, boys, oh, boys. He lays his stick on the desk and stands on the platform with his eyes closed. What use is Euclid? he says. Use? Without Euclid the Messerschmitt could never have taken to the sky. Without Euclid the Spitfire could not dart from cloud to cloud. Euclid brings us grace and beauty and elegance. What does he bring us, boys? Grace, sir. And? Beauty, sir. And? Elegance, sir. Euclid is complete in himself and divine in application. Do you understand that, boys? We do, sir. I doubt it boys, I doubt it. To love Euclid is to be alone in this world. He opens his eyes and sighs and you can see the eyes are a little watery. *Angela's Ashes* Frank Mc Court Irish-Americans - Biography ISBN 0-684-87435-0 Scribner, 1996 Euclid alone has looked on beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese Gabble and hiss, but heroes seek release From dusty bondage into luminous air. O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Euclid alone Has looked on Beauty bare. Fortunate they Who, though once only and then but far away, Have heard her massive sandal set on stone. ---Edna St. Vincent Millay In 1952, E. Teller (the father of the H-bomb) was asked: "Will the thermonuclear device work?" He replied, "I don't know." "But if you didn't know 5 years ago, haven't you made any progress since then?" Oh, yes," replied Teller, "Now I don't know on much better grounds." "Discovery is seeing what everyone has seen and thinking what nobody has thought" -- Albert Szent-Gyogyi I'm glad I'm educated I think it's simply grand To know so many facts and stuff That I don't understand -- S.O. Barkerd One physicist said of mathematics that if mathematics hadn't been invented "physics as a science would have been set back maybe a week." Upon which the mathematician to whom he was speaking said, "You mean the week when the Lord created the heavens and the earth?" Bitte vergiss alles, was Du auf der Schule gelernt hast; denn Du hast es nicht gelernt. ---Edmund Landau Don't let schooling get in the way of your education. ---Mark Twain The job of the teacher is to translate the textbook into the vernacular. ---Paul Halmos, 1990 Anyone who is about to teach the undergraduate curriculum should come down to earth by looking through the Schaum's outline series before burdening students with those well-printed, many colored, highly advertised hardcover volumes that are pathetically passed off as text books. ---Gian-Carlo Rota 1997 "On my income tax 1040 it says 'Check this box if you are blind.' I wanted to put a check mark about three inches away." --Tom Lehrer, 4/4/90 If a `religion' is defined to be a system of ideas that contains unprovable statements, then Go"del taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. ---John Barrow "I can calculate the motions of heavenly bodies, but not the madness of people." ---Sir Isaac Newton, 1720, upon hearing that he had lost L20,000 in the Great South Sea Bubble "His book is a marvel of laboured detail. No expositor could take more pains with his reader, space being held of no moment if clearness had to be secured. As might be expected, therefore, all that is really worth preserving of his work is but a small fraction of the 264 pages which he occupies in exposition." --Muir on Desnanot's work on determinants "Young man, in mathematics you don't understand things; you just get used to them". --John von Neumann The way to do research is to attack the facts at the point of greatest astonishment. -- Celia Green Politics is for the moment. An equation is for eternity. --- Albert Einstein But what are all these mysteries to me, Whose thoughts are full of indices and surds. 2 X + 7 X + 53 = 11 / 3. -- Lewis Carrol Friedrich Engels, writing to Karl Marx, 18 August 1881: ....Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and was pleased to find that I did not need them. I complement you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes with the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls. . . You need not fear that any mathematician has preceded you here... I've said that the passing grade in mathematics should be 100 percent. Suppose it's 70 percent; then you go on to the next year, and again you only understand 70 percent of the 70 percent that you did understand, or 49 percent. The following year, you understand 70 percent of that, or 35 percent, and so on. So the student gives up and says, "I'm just incapable of understanding!-- Linus Pauling "All of science is mathematics: Physics is the noisy part of mathematics, Chemistry is the smelly part of mathematics, Biology is the slimy part of mathematics." ---Charles S. Slichter (U.Wisc. Appl. math) "If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics; thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." ---Bertrand Russell It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that. ---G. H. Hardy The purpose of computing is insight, not numbers. --- Richard Hamming Life is complex, has both real and imaginary parts. "There is, strictly, no such thing as mathematical proof: we can, in the lasty analysis, do nothing but point; proofs are what Littlewood and I call 'gas'; rhetorical flourishes designed to affect psychology, pictures on board in the lecture, devices to stimulate the imagination of pupils." ---G. H. Hardy, Mathematical Proof. Mind 38, 1929, pp. 1-25. But all the brooding, all the searching was for nothing; finally a few days ago, I succeeded. But not by long searches but by the sheer grace of God, I may say, like lightning strikes, the riddle was solved; I myself would not be able to find the connection between what I knew previously, with what I used for my last attempt and with what finally succeeded. ---K.F. Gauss Becoming sufficiently familiar with something is a substitute for understanding it. --- John H. Conway "Any historian of mathematics conscious of the perils and pitfalls of Whig history quickly discovers that the translation of past mathematics into modern symbolism and terminology represents the greatest danger of all. The symbols and terms of modern mathematics are the bearers of its concepts and methods. Their application to historical material always involves the risk of imposing on that material, a content it does not in fact possess." ---Michael S. Mahoney "The good Christian should beware of mathematicians, and all those who make empty prophecies.The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell. " ---St. Augustine (354-430) DeGenesi ad Litteram, Book II, xviii, 37 "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist. The City of God." ---St. Augustine (354-430) "I am a carnivorous fish swimming in two waters, the cold water of art and the hot water of science." ---Salvador Dali. Freeman Dyson, upon being told by senior physicists after the market collapse of the 1970's that "economic factors were not relevant to the decision to enter physics." responded by saying something like "Terrific, If that's true, a 5% voluntary pay cut by senior scientists should be enough to prop up the market for entering physicists." Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing. ---Alfred Adler, from "Mathematics and Creativity," the New Yorker Magazine, Feb. 19, 1972. The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell. -- Saint Augustine "...today's top mathematicians are in fact a group of exciting, dynamic, and glamorous individuals." -- Dave Barry And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? ---George Berkeley ...by a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull. ---R.P. Boas When analyzing arbitrary "if / then" rules, many other researchers found that fewer than one-quarter of those tested offer logically correct answers...when analyzing a "social contract" ...70 to 90 percent of volunteers accurately pick out cheaters on a Wasou test. ---Bruce Bower, from "Roots of Reason," Service News, Vol 145, January 29, 1994. Statistics are no substitute for judgment. ---Henry Clay, quoted in "Flaws and Fallacies in Statistics", by Campbell. ...scientific power is like inherited wealth: attained without discipline. You read what others have done, and you take the next step. You can do it very young...there is no mastery, old scientists are ignored. There is no humility before nature...Its a form of inherited wealth. And you know what assholes congenitally rich people are. ---Ian Malcolm (Michael Crichton), from Jurrasic Park. Failure is instructive. The person who really thinks learns quite as much from his failures as from his successes. ---John Dewey, quoted in Contemporary Abstract Algebra, by J. Gallian. We only think when confronted with a problem. ---John Dewey, quoted in Harper's Quotes. How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? Sir Arthur Conan Doyle Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore. Albert Einstein, quoted in The World of Mathematics, by J.R. Newman. Whatever your difficulties in mathematics, I can assure you mine are far greater. Albert Einstein As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein, quoted in A Mathematical Sampler: Topics for Liberal Arts, by W.P. Berlinghoff and K.E. Grant. Science is the attempt to make the chaotic diversity of our sense-experiments correspond to a logically uniform system of thought. Albert Einstein Pure mathematics is, in its way, the poetry of logical ideas. Albert Einstein "It is in fact nothing short of a miracle that the modern methods of instruction have not entirely strangled the holy curiosity of inquiry; for this delicate little plant, aside from stimulation, stands mainly in need of freedom; without this it goes to wrack and ruin without fail." ---Albert Einstein