Calculus

"It is supposed that the results cannot be truly accurate, because certain quantities are neglected; thus, a finite number of infinitesimals is neglected if it is added to or subtracted from a finite quantity; and similarly a finite quantity must be and is neglected when it is added to or subtracted from an infinity. And it is supposed that an error is hereby introduced, which vitiates and destroys the accuracy of the whole work, and that the results are at last true only because a compensation is made. Now let the nature of our Calculus be clearly understood; it is of itself a part of the science of number; its subject matter is continuous number; and according as its results and theorems in number are exact or not, so is its exactness to be tested. It is capable of application to other sciences; those of Geometry, Motion, &c.; but its truth is not to be tried by the results of these subjects, because their matter may not be so conformable to that of our Calculus, but that discrepancy may be between them. In Infinitesimal Calculus may properly so called, our symbols are symbols of number only: we make our own materials; our infinitesimals and infinities are created by us; and are subject to certain conditions which we choose to impose on them; we make them subject to certain laws; and so long as they are employed consistently with these conditions and these laws, and in accordance with the rules of correct , the conclusions which they lead to are strictly correct. There is no error: the neglecting of infinitesimals is a necessary stop in our process, and therefore it is that I have used the language of necessity: I have said in Theorem VI that a finite sum of infinitesimals must be neglected, not may be neglected, when it is added to or subtracted from a finite quantity: were this not done, infinitesimals would not be what they are, and our rules for the discovery of them would be other than they are. When however we apply to other subject matter, say of space or of motion, the conception of infinitesimals, it may be that this particular subject-matter does not admit of the continuous infinitesimal change with that exactness which the conception of numerical infinitesimal growth is capable of; it may be that there is a discrepancy, and consequently an error; for which afterwards compensation has to be made. Thus, for instance, we may in our conception of the infinitesimal Calculus as applied to Geometry assume the line joining two consecutive points in a circle to be straight, and represent it by a symbol which denotes a straight line; whereas from the geometrical definition of a circle we know that the curvature of a circle is continuous, and that the line joining two points of it, however near together they are, cannot be straight; and thus our symbols, though representatives of such straight lines, only approximately represent them. In this case doubtless there may be an error; an error not in the work of the calculus; that is true and exact; but because the geometrical quantities are not adequately expressed by the symbols; but when by means of integration we pass from the infinitesimal element to the finite function, then the finite function becomes the exact and adequate representative of the geometrical quantity, and a compensation has taken place in the act of passing from the infinitesimal element to the finite function. On investigation it will, I venture to think, be found that the exactness of the Calculus has been impugned on these and similar grounds; and therefore that it has been unfairly impugned: let it be tried on its own principles; on them I venture to say it will stand the attack. It creates its own materials, and is subject to its own laws; let it not be condemned because other materials, which you try to bring within its grasp, refuse to submit to these laws."

"Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define."

"Of possible quadruple algebras the one that had seemed to him by far the most beautiful and remarkable was practically identical with quaternions, and that he thought it most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels."

"If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing."

"As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience)."

"It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics."

"While still in high school and having just learned the geometric significance of the first and second derivatives of a function, I asked my father what significance there might be to derivatives of higher order than two. Naturally, he mentioned that the fourth derivative of the deflection curve of a beam would be proportional to the intensity of the load distribution responsible for this deflection. I learned a good deal more about this subject during my first year at the university as a student in my father's mechanics course."

"One of the most important theorems in calculus is the Mean Value Theorem (MVT), which is used to prove many theorems of both differential and integral calculus, as well as other subjects such as numerical analysis. MVT is said to be the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance. The proof of the Mean-Value Theorem is based on a special case of it known as Rolle’s Theorem."

"The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be."

"The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself... had ideas, upon this topic which can only be described as extremely crude. He appears to have held that... the Calculus is only approximate, but is justified practically by the fact that the errors... are less than those of observation. When he was thinking of , his belief in the actual hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which... infinite division was supposed to lead. And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules. At other times... he definitely rejects infinitesimals as philosophically valid; but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor's sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. But Newton was... ignorant of the fact that his Lemmas depend upon the modern theory of continuity; moreover, the appeal to time and change, which appears in the word , and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuity had been given. Whether Leibniz avoided this error, seems highly doubtful... in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations... are as yet little known among philosophers..."

"In another way, calculus is fundamentally naive, almost childish in its optimism. Experience teaches us that change can be sudden, discontinuous, and wrenching. Calculus draws its power by refusing to see that. It insists on a world without accidents, where one thing leads logically to another. Give me the initial conditions and the law of motion, and with calculus I can predict the future—or better yet, reconstruct the past. I wish I could do that now."

"Looking at numbers as groups of rocks may seem unusual, but actually it's as old as math itself. The word "calculate" reflects that legacy—it comes from the Latin word calculus, meaning a pebble used for counting. To enjoy working with numbers you don't have to be Einstein (German for "one stone"), but it might help to have rocks in your head."

"The greatest achievements of lie in that twilight realm where math meets reality. Indeed, the story of James Clerk Maxwell and his equations offers one of the eeriest instances of the . Somehow, by shuffling a few symbols, Maxwell discovered what light is."

"Calculus is the study of limits. In simple terms, a limit allows us to look at what happens to the dependent variable as the independent variable gets close to something."

"The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus."

"Newton, of course, was the inventor of so his place in the tale is quite special. Even the loss of two years worth of formulae wasn’t enough to dislodge him from his dogged path …"

"But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes."

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. ...Fermat applied his method ...and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same,ssumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"J.M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Paul Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"An untold amount of intellectual energy has been expended on the quadrature of the circle, yet no conquest has been made by direct assault. The circle-squarers have existed in crowds ever since the period of Archimedes. After innumerable failures to solve the problem at a time, even when investigators possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, while those who still persisted were completely ignorant of its history and generally misunderstood the conditions of the problem. ...But progress was made on this problem by approaching it from a different direction and by newly discovered paths. Lambert proved in 1761 that the ratio of the circumference of a circle to its diameter is incommensurable. Some years ago, Lindemann demonstrated that this ratio is also transcendental and that the quadrature of the circle, by means of the ruler and compass only, is impossible. He thus showed by actual proof that which keen minded mathematicians had long suspected; namely, that the great army of circle-squarers have, for two thousand years, been assaulting a fortification which is as indestructible as the firmament of heaven."

"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"The chief object of the present work is, as its title indicates, to furnish to the student examples by which to illustrate the processes of the Differential and Integral Calculus. In this respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation. But as Professor Peacock informed me that he had not leisure to superintend the publication of a second edition of his "Examples" which had been long out of print, I thought that I should do a service to students by preparing a work on a similar plan, but with such modifications as seemed called for by the increased cultivation of Analysis in this University."

"As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer..."

"[Human and physical events are] equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough."

"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers."

"God does not care about our mathematical difficulties — he integrates empirically."

"A Calculus Carol Oh, Calculus; Oh, Calculus, How tough are both your branches. Oh, Calculus; Oh, Calculus, To pass what are my chances? Derivatives I cannot take, At integrals my fingers shake. Oh, Calculus; Oh, Calculus, How tough are both your branches. Oh, Calculus; Oh, Calculus, Your theorems I can't master. Oh, Calculus; Oh, Calculus, My Proofs are a disaster. You pull a trick out of the air, Or find a reason, God knows where. Oh, Calculus; Oh, Calculus, Your theorems I can't master. Oh, Calculus; Oh, Calculus, My limit I am reaching. Oh, Calculus; Oh, Calculus, For mercy I'm beseeching. My grades do not approach a B, They're just an epsilon from D. Oh, Calculus; Oh, Calculus, My limit I am reaching."

"The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment. The sciences must proceed to first causes, above all mathematics where one cannot assume, as in physics, principles that are unknown to us. For there is in mathematics, so to speak, only what we have placed there... If, however, mathematics always has some essential obscurity that one cannot dissipate, it will lie, uniquely, I think, in the direction of the infinite; it is in that direction that mathematics touches on physics, on the innermost nature of bodies about which we know little."

"In general the position as regards all such new calculi is this — That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able — without the unconscious inspiration of genius which no one can command — to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"Imagine Aristotle revivified and visiting Manhattan. Nothing in our social, political, economic, artistic, sexual or religious life would mystify him, but he would be staggered by our technology. Its products—skyscrapers, cars, airplanes, television, pocket calculators—would have been impossible without calculus."

"I have no fault to find with those who teach geometry. That science is the only one which has not produced sects; it is founded on analysis and on synthesis and on the calculus; it does not occupy itself with the probable truth; moreover it has the same method in every country."

"It was certainly Isaac Newton who first devised a new infinitesimal calculus and elaborated it into a widely extensible algorithm, whose potentialities he fully understood; of equal certainty, the differential and integral calculus, the fount of great developments flowing continuously from 1684 to the present day, was created independently by Gottfried Wilhelm Leibniz. Whatever we feel of the relations between these two men, we cannot but admire their analogous creative achievements with as much impartiality as our emotions will admit."

"Love was actually more like calculus or physics. What was the half-life of love? Did it have cosigns and slopes, or quarks that morphed from wave to particle faster than you could say, please don’t leave?”"

"J.M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. [Paul] Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

"That Leibniz here (in the field of calculus) wholly uninfluenced by others, gained his crucial insights unaided, is beyond all doubt."

"The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason."

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"But just as much as it is easy to find the differential [derivative] of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not."

"It is here, at this very moment when the first utterly trivial differential equation is solved, that the secret form of words is revealed that makes modern science possible."

"Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Fermat, Wallis, and Barrow. It was Newton's good luck to come at a time when everything was ripe for the discovery, and his ability enabled him to construct almost at once a complete calculus."

"In February 1994, [Mary] Tai authored a paper in the journal Diabetes Care entitled "A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves." …What Tai "discovered," even being so bold as to term it Tai's Model, is integral calculus. ...the trapezoidal rule for calculating the area below a curve, which seems to have been known to Newton. And yet Tai's article passed through the editors and has received well over one hundred citations in the scientific literature. A number of letters written in response to Tai in a later issue... pointed out that this technique is well-known and available in many introductory textbooks. ...in many situations knowledge can spread far slower than we might realize."

"The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experimentalist, and the statesman."

"I don't have a word for how it smelled. Like calculus, perhaps?"

"(2) \int which is merely a long S... may be called, if you like, "the sum of." Thus \int dx means the sum of all the little bits of x or \int dt means the sum of all the little bits of t. Ordinary mathematicians call this symbol "the integral of.""

"(1) d... merely means "a little bit of." Thus dx means a little bit of x or du means a little bit of u. ...you will find that these little bits or elements may be considered to be indefinitely small."

"The preliminary terror, which chokes off most [students]... from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common sense terms—of the two principal symbols that are used in calculating."

"Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can."

"Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks."

"Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics—and they are mostly clever fools—seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way."