Calculus

"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."

"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."

"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."

"(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."

"(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.""

"Now any fool can see that if x is considered as made up of a lot of little bits, each of which is called dx, if you add them all up together you get the sum of all the dxs (which is the same thing as the whole of x). The word "integral" simply means "the whole.""

"When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow. That's all."

"We shall have... to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness."

"1 minute is a very small quantity of time compared with a whole week, Indeed, our forefathers considered it small as compared with an hour, and called it "one minùte," meaning a minute fraction—namely one sixtieth—of an hour. When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth's days, they called "second minùtes" (i.e. small quantities of the second order of minuteness). Nowadays we call these... "seconds." But few people know why they are so called."

"If, for the purpose of time, 1/60 be called a small fraction, then 1/60 of 1/60 (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness."

"The smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders, if only we take the small quantity of the first order small enough in itself."

"It must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred."

"Now in the calculus we write dx for a little bit of x. These things such as dx, and du, and dy, are called "differentials," the differential of x, or of u, or of y, as the case may be. [You read them as dee-eks or dee-you or dee-wy.] If dx be a small bit of x, and relatively small of itself, it does not follow that such quantities as x\cdot dx or x^2 dx or a^x dx are negligible. But dx\times dx would be negligible, being a small quantity of the second order."

"Let us think of x as a quantity that can grow by a small amount so as to become x + dx, where dx is the small increment added by growth. The square of this is x^2 + 2x\cdot dx + dx^2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2."

"The witty Dean Swift once wrote: "So Nat'ralists observe, a Flea "Hath smaller Fleas that on him prey. "And these have smaller Fleas to bite 'em, "And so proceed ad infimitum. An ox might worry about a flea of ordinary size—a small creature of the first order of smallness. But he would probably not trouble himself about a flea's flea, being of the second order of smallness, it would be negligible. Even a gross of fleas' fleas would not be of much account to the ox."

"Let x and y be respectively the base and the height of a right-angled triangle (Fig. 4), of which the slope... is fixed at 30º. If we suppose this triangle to expand and yet keep its angles the same as at first, then, when the base grows so as to become x + dx the height becomes y + dy. Here, increasing x results in an increase of y. The little triangle, the height of which is dy and the base of which is dx is similar to the original triangle; and it is obvious that the value of the ratio dy/dx is the same as that of the ratio y/x."

"We call the ratio dy/dx "the differential coefficient of y with respect to x." It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ratio dy/dx."

"The process of finding the value of dy/dx is called "differentiating." But, remember, what is wanted is the value of this ratio when both dy and dx are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute."

"Let us begin with the simple expression y = x2 . ..Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. ...the enlarged y will be equal to the square of the enlarged x. Writing this down we have y + dy = (x + dx)2. Doing the squaring [see Ch.2, Fig.2 above] we get y + dy = x^2 + 2x\cdot dx + (dx)^2. ...dx2 will mean a little bit of a little bit of x; that is... a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have: y + dy = x^2 + 2x\cdot dx. Now y = x2; so let us subtract this from the equation and we have left dy = 2x\cdot dx. Dividing across by dx, we find dy/dx = 2x."

"Try differentiating y = x3. We let y grow to y + dy while x grows to x + dx. Then we have y + dy = (x + dx)3. ...[By a similar argument as above] dy/dx = 3x^2. ...Try differentiating y = x4. Starting as before by letting both y and x grow a bit, we have: y + dy = (x + dx)4. ...[By a similar argument as above] dy/dx = 4x^3."

"Let us collect the results to see if we can infer any general rule..."

"Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of x by 1 (for example in the last case reducing x4 to x3), and at the same time multiplying by a number (the same number in fact which originally appeared as the power). Now, when you have once seen this, you might easily conjecture how the others will run. You would expect that differentiating x5 would give 5x4 or differentiating x6 would give 6x5."

"Following out logically our observation, we should conclude that if we want to deal with any higher power,—call it n—we could tackle it in the same way. Let y = x'n, then we should expect to find that dy/dx = nx(n-1). For example let n = 8 then y = x8; and differentiating it would give dy/dx = 8x7."

"Thirdly, among the dreadful things they will say about "So Easy" is this: that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not? You don't forbid the use of a watch to every person who does not know how to make one? You don't object to the musician playing on a violin that he has not himself constructed. You don't teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus."

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

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"[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?”"

"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..."

"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 calculus of utility aims at supplying the ordinary wants of man at the least cost of labour."

"In this work I have attempted to treat economy as a calculus of pleasure and pain, and have sketched out, almost irrespective of previous opinions, the form which the science, as it seems to me, must ultimately take."

"In the history of mathematics and science, few conflicts have attained the notoriety of the Newton/Leibniz dispute. … A carefully reconstructed chronology reveals that Newton had formulated the essentials of his calculus by 1666, years before Leibniz had attained the mathematical knowledge necessary to develop his own point of view on the calculus. … There is much that can never be known about such a feud. This feud is peculiar in that it erupted late, and was both sparked and carried on to a large degree by the followers of the men involved. There are scientific reasons for it (the divergences in their interpretation of "the calculus" itself; personal reasons (a history of suspicion, not only between the two principles but between each of them and other rivals; nationalism, never a negligible factor; and the bitterness associated with disputes on related matters, notably the ongoing rivalry between the Newtonian and Cartesian theories of gravity. At a personal level, Newton's pride, suspicious character, and reluctance to publish collided with Leibniz' naive optimism, arrogance, and his belief in "systems" as more valuable than inspiration, in a long-delayed but virulent explosion."

"The theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it."

"Among all of the mathematical disciplines the theory of differential equations is the most important … It furnishes the explanation of all those elementary manifestations of nature which involve time."

"Who has not been amazed to learn that the function y = ex, like a phoenix rising from its own ashes, is its own derivative?"

"In England, where it originated, the calculus fared less well. ...by siding completely with Newton in the priority dispute, they cut themselves off from developments on the Continent. They stubbornly stuck to Newton's dot notation of fluxions, failing to see the advantages of Leibniz's differential notation. As a result, over the next hundred years, while mathematics fluorished in Europe as never before, England did not produce a single first-rate mathematician. When the period of stagnation finally ended around 1830, it was not in analysis but in algebra that the new generation of English mathematicians made their greatest mark."

"Accordingly, we find Euler and Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations."

"This is a tricky domain because, unlike simple arithmetic, to solve a calculus problem - and in particular to perform integration - you have to be smart about which integration technique should be used: integration by partial fractions, integration by parts, and so on."

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.""

"In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Bionomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly."

"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."

"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."

"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."

"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."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"It will be sufficient if, when we speak of infinitely great (or more strictly unlimited), or of infinitely small quantities (i.e., the very least of those within our knowledge) it is understood that we mean quantities that are indefinitely great or indefinitely small, i.e., as great as you please, or as small as you please, so that the error that any one may assign may be less than a certain assigned quantity. Also, since in general it will appear that, when any small error is assigned, it can be shown that it should be less, it follows that the error is absolutely nothing; an almost exactly similar kind of argument is used in different places by Euclid, Theodosius and others; and this seemed to them to be a wonderful thing, although it could not be denied that it was perfectly true that, from the very thing that was assumed as an error, it could be inferred that the error was non-existent. Thus by infinitely great and infinitely small, we understand something indefinitely great, or something indefinitely small, so that each conducts itself as a sort of class, and not merely as the last thing of a class. If any one wishes to understand these as the ultimate things, or as truly infinite, it can be done, and that too without falling back upon a controversy about the reality of extensions, or of infinite continuums in general, or of the infinitely small, ay, even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of the calculation, just as the algebraists retain imaginary roots with great profit. For they contain a handy means of reckoning, as can manifestly be verified in every case in a rigorous manner by the method already stated. But it seems right to show this a little more clearly, in order that it may be confirmed that the algorithm, as it is called, of our differential calculus, set forth by me in the year 1684, is quite reasonable."

"To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus."

"In the beginning of the year 1665 I found the method of approximating series and the rule for reducing any dignity [power] of any binomial to such a series [ i.e. the binomial theorem ]. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions [i.e. the elements of differential calculus], and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions [i.e. integral calculus], and in the same year I began to think of gravity extending to the orb of the moon … and having thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly. All this was in the two years of 1665 and 1666, for in those years I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since."

"[S]cientific 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..."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language —the language of vectors —to describe those quantities. This language is also used in engineering, the other sciences, and even in common speech."

"Differential geometry originally sneaked into theoretical physics through Einstein's theory of general relativity."

"現代幾何学においても、特性類の重要性は減ずるどころかますます大きくなっている。しかも単なるコホモロジー類としてだけでではなく、それを表わす微分形式自 身の詳細な解析がなされるなど、より深い役割を果たすようになってきている。たとえば、de RhamコホモロジーとRiemann計量の関わりを記述する理論である調和積 分論を、ずっと大きな枠組みのなかで一般化するという、壮大な試みが進行中である。そしてこれらの新しい展開のなかで、微分形式は、たとえてみれば生物にとっ ての水や空気のような役割を果たしているといっても過言ではないだろう。"

"Nothing in Descartes' work led directly to Leibniz's calculus, but Descartes' discoveries in mathematics were certainly forerunners of the calculus. We know that in 1661... Newton read books about Descartes' mathematics. ...without Descartes' unification of algebra and geometry it would have been impossible to describe graphs using mathematical equations, and hence, except perhaps as a pure theory, the calculus would be completely devoid of meaning."

"In Sorbière's day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisibles and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes's acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and banned the concept.. Even as late as the 1730s... George Berkeley mocked mathematicians for their use of infinitesimals... Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitesimally small. These included, in addition to Wallis: Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton."

"On the one side were ranged the forces of hierarchy and order—Jesuits, Hobbesians, French Royal Courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative "liberalizers" such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come."

"[Joseph-Louis Lagrange's] lectures on differential calculus form the basis of his Theorie des fonctions analytiques which was published in 1797. ...its object is to substitute for the differential calculus a group of theorems based upon the development of algebraic functions in series. A somewhat similar method had been previously used by John Landen in his Residual Analysis... Lagrange believed that he could... get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. ...Another treatise in the same lines was his Leçons sur le calcul des fonctions, issued in 1804. These works may be considered as the starting-point for the researches of Cauchy, Jacobi, and Weierstrass."

"Nothing is easier... than to fit a deceptively smooth curve to the discontinuities of mathematical invention. Everything then appears as an orderly progression... with Cavalieri, for instance, indistinguishable from Newton in the neighborhood of the calculus, or Lagrange from Fourier in that of trigonometric series, or Bhaskara from Lagrange in the region of Fermat's equation. Professional historians may sometimes be inclined to overemphasize the smoothness of the curve; professional mathematicians, mindful of the dominant part played in geometry by the singularities of curves, attend to the discontinuities. ...That such differences should exist is no disaster. Dissent is good for the souls of all concerned."

"Descartes' method of finding tangents and normals... was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. ...Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus."

"Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first."

"The fundamental definitions of the calculus, those of the derivative and integral, are now so clearly stated in textbooks on the subject... that it is easy to forget the difficulty with which these basic concepts have been developed."

"The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition."

"Just as the problem of defining instantaneous velocities in terms of the approximation of average velocities was to lead to the definition of the derivative, so that of defining lengths, areas, and volumes of curvilinear configurations was to eventuate in the formation of the definite integral. ...This definition then invokes, apart from the ordinary operations of arithmetic, only the concept of the limit of an infinite sequence of terms, precisely as does that of the derivative. The realization of this fact, however, followed only after many centuries of investigation by mathematicians."

"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, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"The Calculus of Variations owed its origin to the attempt to solve a very interesting and rather narrow class of problems in Maxima and Minima, in which it is required to find the form of a function such that the definite integral of an expression involving that function and its derivative shall be a maximum or a minimum."

"Methods of drawing tangents were invented by Roberval and Fermat... Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations."

"Every great epoch in the progress of science is preceded by a period of preparation and prevision. The invention of the differential and integral calculus is said to mark a "crisis" in the history of mathematics. The conceptions brought into action at that great time had been long in preparation. The fluxional idea occurs among the schoolmen—among Galileo, Roberval, Napier, Barrow, and others. The differences or differentials of Leibniz are found in crude form among Cavalieri, Barrow, and others. The undeveloped notion of limits is contained in the ancient method of exhaustion; limits are found in the writings of Gregory St. Vincent and many others. The history of the conceptions which led up to the invention of the calculus is so extensive that a good-sized volume could be written thereon."

"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. 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."

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of s. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"In the method of exhaustion, Archimedes possessed all the elements essential to an infinitesimal analysis. ...the idea of limit as conceived by Archimedes was adequate for the development of the calculus of Newton and Leibnitz and... it remained practically unchanged until the days of Weierstrass and Cantor. ...the principle ...consists in "trapping" the variable magnitude between two others, as between two jaws of a vise. Thus, in the case of the periphery of a circle... Archimedes grips the circumference between two sets of regular polygons of an increasing number of sides... one set is circumscribed... and the other is inscribed. ...By this method he also found the area under a parabolic arch..."

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. ...It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858."

"If a cone is cut by surfaces parallel to the base, then how are the sections, equal or unequal? If they are unequal then the cone would have the shape of a staircase; but if they were equal, then all sections will be equal, and the cone will look like a cylinder, made up of equal circles; but this is entirely nonsensical."

"...nor have I found occasion to depart from the plan... the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange... had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits."

"I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold..."

"When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or ∞ or 1⁄0 for its limit."

"Kepler imagined a given geometrical figure to be decomposed into infinitesimal figures, whose areas or volumes he added up in some ad hoc way to obtain the area or volume... Cavalieri proceeded by setting up a one-to-one correspondence between the indivisible elements of two geometrical figures. If corresponding indivisibles of the two figures had a certain (constant) ratio, he concluded that the areas of volumes of one of the figures had the same ratio. Typically, the area or volume of one of the figures was known in advance, so this gave the other. ... Kepler thought of a geometrical figure as being composed of indivisibles of the same dimension [as the original figure]... from some process of successive subdivision... However, Cavalieri generally considered a geometrical figure to be composed of an indefinitely large number of indivisibles of lower dimension. ...an area as consisting of ...line segments, and a volume as consisting of... plane sections... Rigor, he wrote in the Exercitationes, is the affair of philosophy rather than mathematics."

"Newton regarded the curve f(x,y) = 0 as the locus of the intersection of two moving lines, one vertical and the other horizontal. The x and y coordinates of the moving point are then functions of the time t, specifying the locations of the vertical and horizontal lines... The motion is then the composition of a horizontal motion with velocity vector having length \dot{x} and a vertical motion with velocity vector having length \dot{y}. ...the velocity vector is the parallelogram sum of these ...It follows that the slope of the tangent line to the curve is \frac{\dot{y}}{\dot{x}}."

"Shortly after his arrival in Paris in 1672, [ Leibniz ] noticed an interesting fact about the sum of differences of consecutive terms of a of numbers. Given the sequencea_0, a_1, a_2, ..., a_nconsider the sequenced_1, d_2, ..., d_nof differences d_i = a - a_i. Thend_1 + d_2 +... + d_n = (a_1 - a_0) + (a_2 - a_1) + ... (a_n - a_{n-1})= a_n - a_0. Thus the sum of the consecutive differences equals the difference of the first and last terms of the original sequence. ... His result on sums of differences also suggested... the possibility of summing an infinite series of numbers. ... If, in addition, \lim_{n\to \infty} a_n = 0[ -\sum_{n=1}^\infty d_n= a_0 ]"

"Pascal's aritmentic triangle and Leibniz' harmonic triangle enjoy a certain inverse relationship... These considerations implanted in Leibniz' mind a vivid conception that was to play a dominant role in his development of the calculus—the notion of an inverse relationship between the operation of taking differences and that of forming sums of the elements of a sequence."

"In the first two thirds of the seventeenth century mathematicians solved calculus-type problems, but they lacked a general framework in which to place them. This was provided by Newton and Leibniz. Specifically, they a. invented the general concepts of and —though not in the form we see them today... b. recognized differentiation and integration an inverse operations. Although several mathematicians... noted the relation... in specific cases... the clear and explicit recognition, in its complete generality, of... the belongs to Newton and Leibniz. c. devised a notation and developed algorithms to make calculus a powerful computational instrument. d. extended the range and applicability of the methods... While in the past those methods were applied mainly to polynomials, often of low degree, they were now applicable to "all" functions, algebraic and transcendental."

"The subject of s was forced upon the Greek mathematicians so soon as they came to close grips with the problem of the quadrature of the circle. Antiphon the Sophist was the first to [inscribe] a series of successive regular polygons in a circle, each of which had double as many sides as the preceding, and he asserted that, by continuing this process, we should at length exhaust the circle: [according to Simplicius, on Aristotle, Physics] 'he thought that in this way the area of the circle would sometime be used up and a polygon would be inscribed in the circle the sides of which on account of their smallness would coincide with the circumference.' Aristotle roundly said that this was a fallacy... Antiphon's argument.. as early as the time of Antiphon himself (a contemporary of Socrates) had been subjected to a destructive criticism expressed with unsurpassable piquancy and force. No wonder that the subsequent course of Greek geometry was profoundly affected by the arguments of Zeno on motion. Aristotle... called them 'fallacies', without being able to refute them. The mathematicians, however, knew better, and, realizing that Zeno's arguments were fatal to infinitesimals, they saw that they could only avoid the difficulties connected with them by once for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or diminishing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please. If they used infinitesimals at all, it was only as a tentative means of discovering propositions; they proved them afterwards by rigorous geometrical methods. An illustration of this is furnished by the Method of Archimedes. ...Archimedes finds (a) the areas of curves, and (b) the volumes of solids, by treating them respectively as the sums of an infinite number (a) of parallel lines, i.e. infinitely narrow strips, and (b) of parallel planes, i.e. infinitely thin laminae; but he plainly declares that this method is only useful for discovering results and does not furnish a proof of them, but that to establish them scientifically a geometrical proof by the , with its double ' is still necessary."

"The history of modern mathematics is to an astonishing degree the history of the calculus. This calculus was the first great achievement of mathematics since the Greeks and it dominated mathematical exploration for centuries. The questions it answered and... raised lay at the heart of man's understanding of not only geometry and number, but also space and time and mathematical truth. It began with the surprising unification of two rather different geometrical problems, and almost immediately its ideas bore fruit in dozens of seemingly unrelated areas. The methods it developed gave the physical sciences an impetus without parallel in history, for through them natural science was born, and without them physics could not have progressed much further than the mystical vortices of Descartes."

"In the beginning there were two calculi, the differential and the integral. The first had been developed to determine the slopes of tangents to... curves, the second to determine... areas... bounded by curves. Algebra, geometry, and trigonometry were simply insufficient to solve general problems of this sort, and prior to the late seventeenth century mathematicians could at best handle only special cases."

"The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating \pi by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of \pi to any assigned degree of approximation. The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression\frac{\pi}{2} = \frac {2}{1}\cdot\frac {2}{3}\cdot\frac {4}{3}\cdot\frac {4}{5}\cdot\frac {6}{5}\cdot\frac {6}{7}\cdot\frac {8}{7}\cdot\frac {8}{9}\cdotsfor \pi as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value \frac{\pi}{2} according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for \frac{\pi}{8} the area of a semi-circle of diameter 1 as the definite integral \int\limits_{0}^{1}\sqrt{x-x^2}dx. The expression has the advantage over that of Vieta that the operations required are all rational ones."

"As to Cavalierian methods: one deceives oneself if one accepts their use as a demonstration, but they are useful as a means of discovery preceding a demonstration. ...Nevertheless, that which comes first and which matters most is the way in which the discovery has been made. It is this knowledge which gives most satisfaction and which one requires from the discoverers. It seems, therefore, preferable to supply the idea through which the result first came to light and through which it will be most readily understood. We will thereby save ourselves much labour and writing and the others the reading; it is necessary to bear in mind that mathematicians will never have enough time to read all the discoveries in Geometry (a quantity which is increasing day to day and seems likely in this scientific age to develop to enormous proportions) if they continue to be presented in a rigorous form, according to the manner of the ancients."

"This history of the development of calculus is significant because it illustrates the way in which mathematics progresses. Ideas are first grasped intuitively and extensively explored before they become fully clarified and precisely formulated even in the minds of the best mathematicians. Gradually the ideas are refined and given polish and rigor which one encounters in textbook presentations. In the instance of the calculus, mathematicians recognized the crudeness of their ideas and some even doubted the soundness of the concepts. Yet they not only applied them to physical problems, but used the calculus to evolve new branches of mathematics... They had the confidence to proceed so far along uncertain ground because their methods yielded correct results. Indeed, it is fortunate that mathematics and physics were so intimately related in the seventeenth and eighteenth centuries—so much so that they were hardly distinguishable—for the physical strength supported the weak logic of mathematics. Of course, mathematicians were selling their birthright, the surety of the results obtained by strict deductive reasoning from sound foundations, for the sake of scientific progress, but it is understandable that the mathematicians succumbed to the lure."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"That method [of infinitesimals] has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratio of two quantities, as long as they remain finite, that ratio offers the to mind no clear and precise idea, as soon as its terms become, the one and the other, nothing at the same time."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"It will be useful to write \int for\, omn., so that \int l = omn. l, or the sum of the l's... I propose to return to former considerations. Given l and its relation to x, to find \int l. Now this comes from the contrary calculus, that is to say if \int l = ya. Let us assume that l = ya/d, or as \int increases, so d will diminish the dimensions. But \int means a sum, and d a difference. From the given y, we can always find ya/d or l, or the difference of the y's. Hence one equation may be changed into the other..."

"My method is but a corollary of a general theory of transformations, by the help of which any given figure whatever, by whatever equation it may be accurately stated, is reduced to another analytically equivalent figure... Furthermore, the general method of transformations itself seems to me proper to be counted among the most powerful methods of analysis, for not merely does it serve for infinite series and approximations, but also for geometric solutions and endless other things that are scarcely manageable otherwise... The basis of the transformation is this: that a given figure, with innumerable lines [ordinates] drawn in any way (provided they are drawn according to some rule or law), may be resolved into parts, and that the parts—or others equal to them—when reassembled in another position or another form compose another figure, equivalent to the former or of the same area even if the shape is quite different; whence in many ways the quadratures can be attained... These steps are such that they occur at once to anyone who proceeds methodically under the guidance of Nature herself; and they contain the true method of indivisibles as most generally conceived and, as far as I know, not hitherto expounded with sufficient generality. For not merely parallel and convergent straight lines, but any other lines also, straight or curved, that are constructed by a general law can be applied to the resolution; but he who has grasped the universality of the method will judge how great and how abstruse are the results that can thence be obtained: For it is certain that all squarings hitherto known, whether absolute or hypothetical, are but limited specimens of this."

"The prime occasion from which arose my discovery of the method of the Characteristic Triangle, and other things of the same sort, happened at a time when I had studied geometry for not more than six months. Huygens, as soon as he had published his book on the pendulum, gave me a copy of it; and at that time I was quite ignorant of Cartesian algebra and also of the method of indivisibles, indeed I did not know the correct definition of the . For, when by chance I spoke of it to Huygens, I let him know that I thought that a straight line drawn through the center of gravity always cut a figure into two equal parts... Huygens laughed when he heard this, and told me that nothing was further from the truth. So I, excited by this stimulus, began to apply myself to the study of the more intricate geometry, although... I had not at that time really studied the Elements. But I found in practice that one could get on without a knowledge of the Elements, if only one was master of a few propositions. Huygens, who thought me a better geometer than I was, gave me to read the letters of Pascal, published under the name of Dettonville; and from these I gathered the method of indivisibles and centers of gravity, that is to say the well-known methods of Cavalieri and Guldinus."

"When M. Huygens lent me the "Letters of Dettonville" (or Pascal), I examined by chance his demonstration of the measurement of the spherical surface, and in it I found an idea that the author had altogether missed... Huygens was surprised when I told him of this theorem, and confessed to me that it was the very same as he had made use of for the surface of the parabolic . Now, as that made me aware of the use of what I call the "characteristic triangle" CFG, formed from the elements of the coordinates and the curve, I thus found as it were in the twinkling of an eyelid nearly all the theorems that I afterward found in the works of Barrow and Gregory. Up to that time, I was not sufficiently versed in the calculus [analytic geometry] of Descartes, and as yet did not make use of equations to express the nature of curved lines; but, on the advice of Huygens, I set to work at it, and I was far from sorry that I did so: for it gave me the means almost immediately of finding my differential calculus. This was as follows. I had for some time previously taken a pleasure in finding the sums of series of numbers, and for this I had made use of the well-known theorem, that, in a series decreasing to infinity, the first term is equal to the sum of all the differences. From this I had obtained what I call the "harmonic triangle," as opposed to the "arithmetical triangle" of Pascal; for M. Pascal had shown how one might obtain the sums of the figurate numbers, which arise when finding sums and sums of sums of the natural scale of arithmetical numbers. I on the other hand found that the fractions having figurate numbers for their denominators are the differences and the differences of the differences, etc., of the natural harmonic scale (that is, the fractions 1/1, 1/2, 1/3, 1/4, etc.), and that thus one could give the sums of the series of figurate fractions1/1 + 1/3 + 1/6 + 1/10 + etc, 1/1 + 1/4 + 1/10 + 1/20 + etc. Recognizing from this the great utility of differences and seeing that by the calculus of M. Descartes the ordinates of the curve could be expressed numerically, I saw that to find quadratures or the sums of the ordinates was the same thing as to find an ordinate (that of the ), of which the difference is proportional to the given ordinate. I also recognized almost immediately that to find tangents is nothing else but to find differences (differentier), and that to find quadratures is nothing else but to find sums, provided that one supposes that the differences are incomparably small. I saw also that of necessity the differential magnitudes could be freed from (se trouvent hors de) the fraction and the root-symbol (vinculum), and that thus tangents could be found without getting into difficulties over (se mettre en peine) irrationals and fractions. And there you have the story of the origin of my method."

"In the famous dispute regarding the invention of the infinitesimal calculus, while not denying... the priority of Newton... some... regard Leibnitz's introduction of the integral symbol \int as alone a sufficient substantiation of his claims to originality and independence, so far as the power of the new science was concerned."

"Many of the greatest discoveries of science, — for example, those of Galileo, Huygens, and Newton,—were made without the mechanism which afterwards becomes so indispensable for their development and application. Galileo's reasoning anent the summation of the impulses imparted to a falling stone is virtual integration; and Newton's mechanical discoveries were made by the man who invented, but evidently did not use to that end, the doctrine of s."

"Since the operations of computing in numbers and with variables are closely similar... I am amazed that it occurred to no one (if you except N. Mercator with his quadrature of the hyperbola) to fit the doctrine recently established for decimal numbers in similar fashion to variables, especially since the way is then open to more striking consequences. For since this doctrine in species has the same relationship to Algebra that the doctrine in decimal numbers has to common Arithmetic, its operations of Addition, Subtraction, Multiplication, Division and Root extraction may be easily learnt from the latter's."

"In a correspondence in which I was engaged with the very learned geometrician Mr. Leibnitz ten years ago, having informed him, that I was acquainted with a method of determining the maxima and minima, drawing tangents, and doing other similar things, which succeeded equally in rational equations and radical quantities, and having concealed this method by transposing the letters of the words, which signified: an equation containing any number of flowing quantities being given, to find the fluxions, and inversely: that celebrated gentleman answered, that he had found a similar method; and this, which he communicated to me, differed from mine only in the enunciation and notation, and in the idea of the generation of quantities."

"\frac {dy}{dx} = \frac {\omega^2x}{g}...The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form \frac {dy}{dx}...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...\frac {dy}{dx} is the limit of a ratio of dy to dx... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio... and multiply by dx to achieve the separation of variables. We get {dy} = \frac {\omega^2x}{g}xdx"

"The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself—who, one would have supposed, should have been competent to give a correct account of his own invention—had ideas, upon this topic which can only be described as extremely crude. He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation. When he was thinking of , his belief in the actual infinitesimal 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, in his philosophy, 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, it is true, 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, of course, entirely 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 fluxion, 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; it is at any rate certain that, 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, as is natural, are as yet little known among philosopher..."

"In connection with the study of curves Fermat proceeded to apply the idea of infinitesimals to the questions of quadrature and of maxima and minima as well as to the drawing of tangents. In this he seems to have anticipated the work of Cavalieri, but the date of his discovery is unknown."

"Leibniz's thirtieth year and his last in the City of Light was his annus mirabulus. ...The year of miracles began in late August 1675 with the arrival of Walther Ehrenfried von Tschirnhaus. ...The two young Germans became instant best friends, achieving a degree of intimacy rarely matched in the course of Leibniz's life. ... In the Hôtel des Romains, the two expatriots promptly engaged in mathematical parleys. ...the papers preserved in Leibniz's files are crisscrossed with the scribbled handwriting of both men. It was around this time that Leibniz passed the threshold of the calculus. In a note from October 29, 1675, two months after Tschirnhaus's arrival, Leibniz for the first time used the symbol ∫ to stand for integration, replacing the earlier "omn" (for "omnes" [all]). Two weeks later, on November 11, he used dx for the first time to represent the "differential of x." Leibniz now believed himself to be in sole possession of the general method we call calculus. At some point he shuffled his new equations over to Tschirnhaus ...[who] dismissed it all as mere playing with symbols."

"[Leibniz] introduced the sign, ∫, in his De geometria... and proved the , that integration is the inverse of differentiation. The result was known to Newton and even, in geometric form, to Newton's teacher Barrow, but it became more transparent in Leibniz's formalism. For Leibniz, ∫ meant "sum," and \int f(x) dx was literally a sum of terms f(x) dx, representing infinitesimal areas of height f(x) and width dx. The difference operator d yields the last term f(x) dx in the sum, and dividing by the infinitesimal dx yields f(x). So voila!\frac{d}{dx}\int f(x) dx = f(x)"

"Algebra made an enormous difference to geometry. Whereas Archimedes had to make an ingenious new approach to each new figure... calculus dealt with a great variety of figures in the same way, via their equations. That was the whole point. Calculus was a method of calculating results, rather than proving them. If pressed, mathematicians could justify their calculations by the method of exhaustion, but it seemed impractical if not unnecessary... Huygens was probably the only major mathematician who stuck to the 'methods of the ancients.' The methods of calculus were so much more powerful and efficient that rigour became secondary. ...By the middle of the eighteenth century, calculus had solved almost all the problems of classical geometry, and new ones the ancients had not dreamed of. It had also revealed the secrets of the heavens, explaining the motions of the moons and planets with uncanny precision."

"The "exhaustion method" (the term "exhaust" appears first in , 1647) was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... by reducing problems... to... formal logic only. ...This indirect method... the standard Greek and Renaissance mode of strict proof in area and volume computation was quite rigorous, ...It had the disadvantage that the result... must be known in advance, so that the mathematician finds it first by another less rigorous and more tentative method. ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...several mathematicians before Newton, notably Kepler, used essentially the same conceptions... our modern limit conceptions have made it possible to build this... into a theory as rigorous as... "exhaustion"... The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. ...it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"The origins of calculus are clearly empirical. Kepler's first attempts at integration were formulated as "dolichometry"—measurement of kegs—that is, volumetry for bodies with curved surfaces. This is... post-Euclidean geometry, and... nonaxiomatic, empirical geometry. Of this, Kepler was fully aware. The main effort and... discoveries, those of Newton and Leibniz, were of an explicitly physical origin. Newton invented the calculus "of fluxions" essentially for the purpose of mechanics—in fact... calculus and mechanics were developed by him more or less together. The first formulations of the calculus were not even mathematically rigorous. An inexact, semiphysical formulation was the only one available for over a hundred and fifty years after Newton! And yet, some of the most important advances of analysis took place during this period... ! Some of the leading mathematical spirits... were clearly not rigorous, like Euler; but others, in the main, were, like Gauss or Jacobi. The development was as confused and ambiguous as can be, and its relation to empiricism was certainly not according to our present (or Euclid's) ideas of abstraction and rigor. Yet... that period produced mathematics as first class as ever existed! And even after the reign of rigor was... re-established with Cauchy, a... relapse into semiphysical methods took place with Riemann."

"Riemann gave a rigorous definition of the integral by enclosing it between... the "lower sum"... the sum of the areas of the rectangles below the curve, and the "upper sum"... the sum of rectangles of somewhat greater height, which cover the area. The treatise on conoids and spheroids shows that Archimedes was familiar with this method of inclusion and... used it for the determination of volumes. But... one cannot say that he was familiar with the concept of the integral. His integrals always remained tied to a definite geometric interpretation, as volumes or as areas of plane figures. We have no evidence that he understood that one single concept is the foundation of all these geometric interpretations... he bases his rigorous proofs on totally different methods... Nevertheless, his rigorous determination of areas and volumes make Archimedes the precursor of the modern integral calculus."

"On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"Are there indivisible lines? And, generally, is there a simple unit in every class of quanta? §1. Some people maintain this thesis on the following grounds:— (i) If we recognize the validity of the predicates 'big' and 'great', we must equally recognize the validity of their opposites 'little' and 'small'. Now that which admits practically an infinite number of divisions, is 'big' not 'little' . Hence, the 'little' quantum and the 'small' quantum will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates 'little' and 'small' apply. (ii) Again, if there is an Idea of line, and if the Idea is first of the things called by its name:—then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible. And on the same principle, the Ideal Square, the Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts: for otherwise it will result that there are elements prior to each of them. (iii) Again, if Body consists of elements, and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought. (iv) Again, Zeno's argument proves that there must be simple magnitudes. For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any 'stretch' which is not simple, motion—unless there be simple magnitudes—involves that the moving body touches successively one-by-one an infinite number of points in a finite time: which is impossible. But even if the body which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the 'stretch' which it traverses in an equal time: and since the movement of thought is quickest of all movements:—it follows that thought too will come successively into contact with an infinity of objects in a finite time. And since 'thought's coming into contact with objects one-by-one' is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible there must be such a thing as an indivisible line. ..."

"Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out."

"You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used..."

"This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals."

"Around 1650 I came across the mathematical writings of Torricelli, where among other things, he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met."

"It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. ...the art of making discoveries should be extended by considering noteworthy examples of it."

"Among the most renowned discoveries of the times must be considered that of a new kind of mathematical analysis, known by the name of the differential calculus; and of this... the origin and the method of the discovery are not yet known to the world at large."

"Its author invented it nearly forty years ago, and nine years later (nearly thirty years ago) published it in a concise form; and from that time it has... been a method of general employment; while many splendid discoveries have been made by its assistance... so that it would seem that a new aspect has been given to mathematical knowledge arising out of its discovery."

"Now there never existed any uncertainty as to the name of the true inventor, until recently, in 1712, certain upstarts... acted with considerable shrewdness, in that they put off starting the dispute until those who knew the circumstances, Huygens, Wallis, Tschirnhaus, and others, on whose testimony they could have been refuted, were all dead."

"They have changed the whole point of the issue, for... they have set forth their opinion... as to give a dubious credit to Leibniz, they have said very little about the calculus; instead every other page is made up of what they call infinite series. Such things were first given as discoveries by of Holstein who obtained them by the process of division, and Newton gave the more general form by extraction of roots binomial expansion by the interpolation method of Wallis]. This is certainly a useful discovery, for by it arithmetical approximations are reduced to an analytical reckoning; but it has nothing at all to do with the differential calculus. Moreover, even in this they make use of fallacious reasoning; for whenever this rival works out a quadrature by the addition of the parts by which a figure is gradually increased, at once they hail it as the use of the differential calculus... By the selfsame argument, Kepler (in his Stereometria Doliorum), Cavalieri, Fermat, Huygens, and Wallis used the differential calculus; and indeed, of those who dealt with "indivisibles" or the "infinitely small," who did not use it? But Huygens, who as a matter of fact had some knowledge of the method of fluxions as far as they are known and used, had the fairness to acknowledge that a new light was shed upon geometry by this calculus, and that knowledge of things beyond the province of that science was wonderfully advanced by its use."

"On his return from England to France in the year 1673... at the instigation of Huygens he began to work at Cartesian analysis (which afore-time had been beyond him), and in order to obtain an insight into the geometry of quadratures he consulted the Synopsis Geometriae of Honoratus Fabri, Gregory St. Vincent, and a little book by Dettonville (i.e., Pascal [letters to M. de Carcavi]). Later on from one example given by Dettonville, a light suddenly burst upon him, which strange to say Pascal himself had not perceived in it. For when he proves the theorem of Archimedes for measuring the surface of a sphere or parts of it, he used a method in which the whole surface of the solid formed by a rotation round any axis can be reduced to an equivalent plane figure. From it our young friend made out for himself the following general theorem. Portions of a straight line normal to a curve, intercepted between the curve and an axis, when taken in order and applied at right angles to the axis give rise to a figure equivalent to the moment of the curve about the axis. When he showed this to Huygens the latter praised him highly and confessed to him that by the help of this very theorem he had found the surface of parabolic s and others of the same sort, stated without proof many years before in his work on the pendulum clock. Our young friend, stimulated by this and pondering on the fertility of this point of view, since previously he had considered infinitely small things such as the intervals between the ordinates in the method of Cavalieri and such only, studied the triangle... which he called the Characteristic Triangle..."

"To find the area of a given figure, another figure is sought such that its subnormals are respectively equal to the ordinates of the given figure, and then this second figure is the of the given one; and thus from this extremely elegant consideration we obtain the reduction of the areas of surfaces described by rotation to plane quadratures, as well as the rectification of curves; at the same time we can reduce these quadratures of figures to an inverse problem of tangents. From these results, our young friend [Leibniz] wrote down a large collection of theorems (among which in truth there were many that were lacking in elegance) of two kinds. For in some of them only definite magnitudes were dealt with, after the manner not only of Cavalieri, Fermat, Honoratus Fabri, but also of Gregory St. Vincent, Guldinus, and Dettonville; others truly depended on infinitely small magnitudes, and advanced to a much greater extent. But later our young friend did not not trouble to go on with these matters, when he noticed that the same method had been brought into use and perfected by not only Huygens, Wallis, van Huraet, and Neil, but also by James Gregory and Barrow."

"The Method of Fluxions is the general Key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature. And as it is that which hath enabled them so remarkably to outgo the Ancients in discovering Theorems and solving Problems, the exercise and application thereof is become the main, if not sole, employment of all those who in this Age pass for profound Geometers. But whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgment, and that of every candid Reader."

"It is said, that the minutest Errors are not to be neglected in Mathematics: that the Fluxions are Celerities, not proportional to the finite Increments though ever so small; but only to the Moments or nascent Increments, whereof the Proportion alone, and not the Magnitude, is considered. And of the aforesaid Fluxions there be other Fluxions, which Fluxions of Fluxions are called second Fluxions. And the Fluxions of these second Fluxions are called third Fluxions; and soon, fourth, fifth, sixth, &c. ad infinitum. Now as our Sense is strained and puzzled with the perception of Objects extremely minute, even so the Imagination, which Faculty derives from Sense, is very much strained and puzzled to frame clear Ideas of the least Particles of time, or the least Increments generated therein: and much more so to comprehend the Moments, or those Increments of the flowing Quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite Particles."

"And it seems still more difficult, to conceive the abstracted Velocities of such nascent imperfect Entities. But the Velocities of the Velocities, the second, third, fourth and fifth Velocities, &c. exceed, if I mistake not, all Humane Understanding. The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered; the Objects, at first fleeting and minute, soon vanishing out of sight."

"[T]o conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust."

"[O]ur modem Analysts are not content to consider only the Differences of finite Quantities: they also consider the Differences of those Differences, and the Differences of the Differences of the first Differences. And so on ad infinitum. That is, they consider Quantities infinitely less than the least discernible Quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding Infinitesimals, and so on without end or limit."

"Insomuch that we are to admit an infinite succession of Infinitesimals... in an infinite Progression towards nothing, which you still approach and never arrive at."

"All these Points, I fay, are supposed and believed by... Men who pretend to believe no further than they can see. ...But he who can digest a second or third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity. ...{W]ith what appearance of Reason shall any Man presume to say, that Mysteries may not be Objects of Faith, at the fame time that he himself admits such obscure Mysteries to be the Object of Science?"

"[T]he modern Mathematicians... scruple not to say, that by the help of these new Analytics they can penetrate into Infinity itself: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they express it) or an Infinity of Infinites."

"But, notwithstanding all these Assertions and Pretensions, it may be justly questioned whether, as other Men in other Inquiries are often deceived by Words or Terms, so they likewise are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species."

"But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves... we shall discover much Emptiness, Darkness, and Confusion; nay, if I mistake not, direct Impossibilities and Contradictions."

"The ancients drew tangents to the conic sections, and to the other geometrical curves of their invention, by particular methods, derived in each case from the individual properties of the curve in question. Archimedes determined in a similar manner the tangents of the spiral, a mechanical curve. Among the moderns, des Cartes, Fermat, Roberval, Barrow, Sluze, and others, had invented uniform methods, of more or less simplicity, for drawing tangents to geometrical curves, which was a great step: but it was previously necessary, that the equations of the curves should be freed from radical quantities, if they contained any; and this operation sometimes required immense, if not absolutely impracticable calculations. The tangent of the , a modern mechanical curve, had been determined only by some artifices founded on it's nature, and from which we could derive no light in other cases. A general method, applicable indifferently to curves of all kinds, geometrical or mechanical, without the necessity of making their radical quantities disappear in any case, remained to be discovered. This sublime discovery, the first step in the method of fluxions, was published by Leibnitz in the Leipsic Transactions for the month of October, 1684. The ever memorable paper that contained it is entitled: 'A New Method for Maxima and Minima, and likewise for Tangents, which is affected neither by Fractions nor irrational Quantities; and a peculiar Kind of Calculus for them.' In this we find the method of differencing all kinds of quantities, rational, fractional, or radical, and the application of these calculi to a very complicated case, which points out the mode for all cases. The author afterward resolves a problem de maximis et minimis, the object of which is to find the path, in which an atom of light must traverse two different mediums, in order to pass from one point to another with most facility. The result of the solution is, that the sines of the angles of incidence and refraction must be to each other in the inverse ratio of the resistances of the two mediums. Lastly he applies his new calculus to a problem, which Beaune had formerly proposed to des Cartes, from whom he obtained only an imperfect solution of it. ...Leibnitz showed in a couple of lines the required curve to be ...the common logarithmic curve."

"In two small tracts on the quadratures of curves, which appeared in 1685, Leibniz] published the first ideas of the calculus summatorius, or inverse method of fluxions. These are farther developed in another tract, entitled, 'Of recondite Geometry, and the Analysis of Indivisibles, and Infinites,' published the following year. In this Leibnitz gives the fundamental rule of the integral calculus; and explains in what the problems of the inverse method of tangents consist, which have since been varied in so many ways. ...and he observes generally, that all the problems of quadratures, before given by geometricians, might be resolved without any difficulty by his method."

"While Leibnitz was in possession of all these treasures, Newton had yet published nothing, from which the world could learn, that he on his part had arrived at similar results. But toward the end of the year 1686, his Philosophiæ naturalis Principia mathematica issued from the press: a vast and profound work ...the key of the most difficult problems resolved in it is the method of fluxions, or analysis of infinites, but exhibited in a form which disguised it, and rendered the author difficult to follow. Accordingly at first it had not all the success it deserved: it was charged with obscurity, with demonstrations derived from sources too remote, and an affected use of the synthetic method of the ancients, while analysis would much better have made known the spirit and progress of the invention. ...mathematicians did Newton the justice to acknowledge, that, at the period when his book was published, he was master of the method of fluxions to a high degree, at least with respect to that part which concerns the quadratures of curves."

"Two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli."

"Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters; and the periodical publications became a kind of learned amphitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de l'Hopital combated with bloodless weapons; the honour of France being ably supported by the marquis for several years."

"The following problem, proposed by John Bernoulli, in 1693, contributed greatly to the progress of the methods for summing up differences. To find a curve such that the tangents terminating at the axis shall be in a given ratio with the parts of the axis comprised between the curve and these tangents. This was resolved by Huygens, Leibniz, James Bernoulli, and the marquis de l'Hopital. On this occasion Huygens passed on the new methods an encomium so much the more honourable, as this great man, having made several sublime discoveries without them, might have been dispensed from proclaiming their advantages. He confessed, that he beheld 'with surprise and admiration the extent and fertility of this art; that, wherever he turned his eyes, it presented new uses to his view; and that it's progress would be as unbounded as it's speculations.' How unfortunate, that science was bereft of him at an age, when with this new instrument he might still have rendered it so many important services!"

"We find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother; and exhorts him to moderate his pretensions; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy; and to acknowledge, that, 'as quantities in geometry increase by degrees, so every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived."

"In 1696 a great number of works appeared which gave a new turn to the analysis of infinites. ...and above all the celebrated work of the marquis de l'Hopital, entitled: 'The Analysis of Infinites, for the understanding of curve Lines,'... Such a work had long been a desideratum. 'Hitherto,' says Fontenelle, in his eulogy on the marquis, 'the new geometry had been only a kind of mystery, a cabbalistic science, confined to five or six persons. Frequently solutions were given in the public journals, while the method, by which they had been obtained, was concealed: and even when it was exhibited, it was but a faint gleam of the science breaking out from those clouds, which quickly closed upon it again. The public, or, to speak more properly, the small number of those who aspired to the higher geometry, were struck with useless admiration, by which they were not enlightened; and means were found to obtain their applause, while the information, with which it should have been repaid, was withheld.' The work of the marquis de l'Hopital, completely unveiling the science of the differential calculus, was received with universal encomiums, and still retains it's place among the classical works on the subject. But the time was not yet arrived for treating in the same manner the inverse method of fluxions, which is immense in it's detail, and which, notwithstanding the great progress it has made, is still far from being entirely completed. Leibnitz promised a work, which, under the title of Scientia Infiniti, was to comprise both the direct and inverse methods of fluxions: but this, which would have been of great utility at that time, never appeared."

"The marquis de l'Hopital had given in his work on the analysis of Infinites a very ingenious rule... No person thought proper to dispute his title to this while he lived; but about a month after his death, John Bernoulli, remarking that this rule was incomplete, made a necessary addition to it, and thence took occasion to declare himself it's author. Several of the marquis de l'Hopital's friends complained loudly... Instead of retracting his assertion, John Bernoulli went much farther; and by degrees he claimed as his own every thing of most importance in the Analysis of Infinites. The reader will indulge me in a brief examination of his pretensions. In 1692 John Bernoulli came to Paris. He was received with great distinction by the marquis de l'Hopital, who soon after carried him to his country seat at Ourques in Touraine, where they spent four months in studying together the new geometry. Every attention, and every substantial mark of acknowledgment, were lavished on the learned foreigner. Soon after, the marquis de l'Hopital found himself enabled, by persevering and excessive labour which totally ruined his health, to solve the grand problems, that were proposed to each other by the geometricians of the time. From the year 1693 he made one in the lists of mathematical science, in which he distinguished himself till his death. At this period he was ranked among the first geometricians of Europe; and it is particularly to be observed, that John Bernoulli was one of his most zealous panegyrists. Perhaps he was exalted too high during his lifetime: but the accusation brought against him by John Bernoulli after his death forms too weighty a counterpoise, and justice ought to restore the true balance. ... The extracts of letters, which John Bernoulli has brought forward, are far from proving what he has asserted. ...It is true we find from them, that John Bernoulli had composed lessons in geometry for the marquis de l'Hopital, but by no means that these lessons were the Analysis of Infinites... We see too in these extracts, that the marquis, while at work on his book, solicited from John Bernoulli, with the confidence of friendship, explanations relative to certain questions, which are treated in it... Amid all these uncertainties, it is most equitable and prudent, to adhere to the general declaration made by the marquis in his preface, that he was greatly indebted to John Bernoulli [aux lumiéres de J. B.]; and to presume, that if he had any obligations to him of a particular nature, he would not have ventured to mask them in the expressions of vague and general acknowledgment. If... any one should think proper to credit John Bernoulli on his bare word, when he gives himself out for the author of the Analysis of infinites, the code of morality... will never absolve him, for having disturbed the ashes of a generous benefactor, in order to gratify a paltry love of self, so much the less excusable, as he possessed sufficient scientific wealth besides."

"Toward the end of the year 1704, Newton gave to the World in one volume his Optics in english, an enumeration of lines of the third order, and a treatise on the quadrature of curves, both in latin. ...the treatise on quadratures, belongs to the new geometry. The particular object of this treatise is the resolution of differential formulæ of the first order, or of a single variable quantity; on which depends the precise, or at least the approximate, quadrature of curves. With great address Newton forms series, by means of which he refers the resolution of certain complicated formulæ to those of more simple ones; and these series, suffering an interruption in certain cases, then give the fluents in finite terms. The development of this theory affords a long chain of very elegant propositions, where among other curious problems we remark the method of resolving rational fractions, which was at that time difficult, particularly when the roots are equal. Such an important and happy beginning makes us regret, that the author has given only the first principles of the analysis of differential equations. It is true he teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem. Newton's opponents have argued from his treatise on quadratures, that, when this work appeared, the author was perfectly acquainted only with that branch of the inverse method of fluxions which relates to quadratures, and not with the resolution of differential equations. Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the world in english. In 1740 it was translated into french, and a preface was prefixed to it, in which the merits of Leibnitz are depreciated so excessively, and in such a decided tone as might impose on some readers, if the writer of this preface Buffon] had not sufficiently blunted his own criticisms, by betraying how little knowledge of the subject he possessed."

"Nicholas Facio de Duillier... thought proper to say, in a little tract 'on the curve of swiftest descent, and the solid of least resistance,' which appeared in 1699, that Newton was the first inventor of the new calculus... and that he left to others the task of determining what Leibnitz, the second inventor, had borrowed from the english geometrician. Leibnitz, justly feeling himself hurt by this priority of invention ascribed to Newton, and the consequence maliciously insinuated, answered with great moderation, that Facio no doubt spoke solely on his own authority; that he could not believe it was with Newton's approbation; that he would not enter into any dispute with that celebrated man, for whom he had the profoundest veneration, as he had shown on all occasions; that when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia, that neither had borrowed any thing from the other; that when he published his differential calculus in 1684, he had been master of it about eight years; that about the same time, it was true, Newton had informed him, but without any explanation, of his knowing how to draw tangents by a general method, which was not impeded by irrational quantities; but that he could not judge whether this method were the differential calculus, since Huygens, who at that time was unacquainted with this calculus, equally affirmed himself to be in possession of a method, which had the same advantages; that the first work of an english writer, in which the differential calculus was explained in a positive manner, was the preface to Wallis's Algebra, not published till 1693; that, relying on all these circumstances, he appealed entirely to the testimony and candour of Newton, &c."

"In 1708, Keil... renewed the same accusation. ...Keil returned to the charge; and in 1711, in a letter to sir , secretary to the Royal Society, he was not contented with saying, that Newton was the first inventor; but plainly intimated, that Leibnitz, after having taken his method from Newton's writings, had appropriated it to himself, merely employing a different notation; which was charging him in other words with plagiarism. Leibnitz, indignant at such an accusation, complained loudly to the Royal Society; and openly required it to suppress the clamours of an inconsiderate man, who attacked his fame and his honour. The Royal Society appointed a committee, to examine all the writings that related to this question, and in 1712 it published these writings, with the report of the committee, under the following title: Commercium epistolicum de Analysi promota. Without being absolutely affirmative, the conclusion of the report is, that Keil had not calumniated Leibnitz. The work was dispersed over all Europe with profusion. Newton was at that time president of the Royal Society, where he enjoyed the highest respect and most ample power..."

"Newton, gifted by nature with superiour intellect, and born at a time when Harriot, Wren, Wallis, Barrow, and others, had already rendered the mathematical sciences flourishing in England, enjoyed likewise the advantage of receiving lessons from Barrow in his early youth at Cambridge. The whole bent of early youth was toward studies of this kind, and the success he obtained was prodigious. ... Leibnitz, who was four years younger, found but moderate assistance in his studies in Germany. He formed himself alone. His vast and devouring genius, aided by an extraordinary memory, took in every branch of human knowledge; literature, history, poetry, the law of nations, the mathematical sciences, natural philosophy, &c. This multiplicity of pursuits necessarily checked the rapidity of his progress in each; and accordingly he did not appear as a great mathematician till seven or eight years after Newton. Both these great men were in possession of the new analysis long before they made it known to the world. If priority of publication determined priority of discovery, Leibnitz would have completely gained his cause: but this is not sufficient..."

"If Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of december 1672, Leibnitz equally invented it on his part, without borrowing any thing from his rival. These two great men by the strength of their genius arrived at the same discovery through different paths: one, by considering fluxions as the simple relations of quantities, which rise or vanish at the same instant; the other, by reflecting, that, in a series of quantities which increase or decrease, the difference between two consecutive terms may become infinitely small, that is to say, less than any determinable finite magnitude. This opinion, at present universally received except in England, was that of Newton himself, when he first published his Principia... At that time the truth was near it's source, and not yet altered by the passions. In vain did Newton afterward change his language, led away by the flattery of his countrymen and disciples; in vain did he pretend, that the glory of a discovery belongs entirely to the first inventor, and that second inventors ought not to be admitted to share it. ...two men, who separately make the same important discovery, have an equal claim to admiration; and... he who first makes it public, has the first claim to the public gratitude."

"The design of stripping Leibnitz, and making him pass for a plagiary, was carried so far in England, that during the height of the dispute it was said... that the differential calculus of Leibnitz was nothing more than the method of Barrow. What are you thinking of, answered Leibnitz, to bring such a charge against me? ...If the differential calculus were really the method of Barrow (which you well know it is not) who would most deserve to be called a plagiary? Mr. Newton, who was the pupil and friend of Barrow, and had opportunities of gathering from his conversation ideas, which are not in his works? or I, who could be instructed only by his works, and never had any acquaintance with the author?"

"John Bernoulli who... learned the analysis of infinites from the writings of Leibnitz, ... advances not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany."

"The death of Leibnitz, which happened in 1716, it may be supposed, should have put an end to the dispute: but the english, pursuing even the manes of that great man, published in 1726 an edition of the Principia in which the scholium relating to Leibnitz was omitted. This was confessing his discovery in a very authentic and awkward manner. Must they not be aware, that the chimerical design of annihilating the testimony, which an honourable emulation had formerly rendered to truth, would be ascribed to national prejudice, or to a sentiment even still more unjust?"

"In later times there have been geometricians, who... have objected... that the metaphysics of his method were obscure, or even defective; that there are no quantities infinitely small; and that there remain doubts concerning the accuracy of a method, into which such quantities are introduced. But Leibnitz might answer: ...I have no need of the existence of infinitely small quantities: it is enough for my purpose, as I have said in several of my works, that my differences are less than any finite quantity you please to assign; and that consequently the errour, which may result from my supposition, is less than any determinable errour, which is the same as absolutely nothing. The manner in which Archimedes demonstrates the proportion of the sphere to the cylinder, has a similar principle for it's basis. ...The metaphysics of my calculation, therefore, are perfectly conformable to those of the method of exhaustion of the ancients, the certainty of which has never been questioned by any one."

"It would seem from Fermat's correspondence with Descartes as if he had thought out the principles of analytical geometry for himself before reading Descartes' Discours, and had realized that from the equation of a curve (or as he calls it the "specific property") all its properties could be deduced. His extant papers on this subject deal however only with the application of infinitesimals to geometry; it seems probable that these papers are a revision of his original manuscripts (which he destroyed) and were written about 1663, but he was certainly in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629. Kepler had already remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this to a few examples. Thus to find the maximum value of x(a - x) he took a consecutive value of x, namely x - e where e is very small, and put x(a - x) = (x - e) (a - x + e). Simplifying and ultimately putting e = 0 he got x = \frac{1}{2}a. This value of x makes the given expression a maximum. [This] is the principle of Fermat's method, but his analysis is more involved."

"[Fermat] obtained the to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e; but there is nothing to indicate that he was aware that the process was general, and though in the course of his work he used the principle, it is probable that he never separated it, so to speak, from the symbols of the particular problem he was considering. The first definite statement of the method was due to Barrow and was published in 1669."

"In 1669 [Isaac Barrow] issued his Lectiones opticæ et geometricæ: this, which is his only important work, was republished with a few minor alterations in 1674. A complete edition of all Barrow's lectures was edited for Trinity College by W. Whewell, Cambridge, 1860. It is said in the preface to the Lectiones opticæ et geometricæ that Newton revised and corrected these lectures adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. ... The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves. Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it [the tangent line] was known; hence if the length of the MT could be found (thus determining the point T) then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn he got a small triangle PQR (which he called the differential triangle because, its sides PR and PQ were the differences of the abscissas and ordinates of P and Q) so thatTM : MP = QR : RP.To find QR : RP he supposed that x,y were the coordinates of P and x - e, y - a those of Q. ...Using the equation of the curve and neglecting the squares and higher powers of e and a as compared with their first powers he obtained e : a The ratio a/e was subsequently (in accordance with a suggestion made by de Sluze) termed the angular coefficient of the tangent at the point. Barrow applied this method to the following curves (i) x^2 (x^2 + y^2) = r^2y^2; (ii) x^3 + y^3 = r^3; (iii) x^3 + y^3 = rxy, called la galande; (iv) y = (r - x) tan\frac{\pi x}{2r}, the quadratrix; and (v) y = r \tan \frac{\pi\,x}{2r}. ...take as an illustration the simpler case of the parabola y^2 = px. Using the notation given above we have for the point P, y^2 = px; and for the point Q, (y - a)^2 = p(x - e). Subtracting we get 2ay - a^2 = pe. But if a is an infinitesimal quantity, a^2 must be infinitely smaller and may therefore be neglected: hence e : a = 2y : p. Therefore TM : y = e : a = 2y : p. That is TM = \frac{2y^2}{p} = 2x. This is exactly the procedure of the differential calculus, except that we there have a rule by which we can get the ratio \frac{a}{e} or dy \over dx directly without the labour of going through a calculation similar to the above for every separate case."

"The most notable of Wallis' mathematical works] was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and shewed that x^{-n} stood for the reciprocal of x^n and that x^\frac{p}{q} stood for the q^{th} root of x^p. He next proceeded to find by the method of indivisibles the area enclosed between the curve y = x^m, the axis of x, and any ordinate x = h; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio 1:m + 1. He apparently assumed that the same result would also be true for the curve y = ax^m, where a is any constant. In this result m may be any number positive or negative, and he considered in particular the case of the parabola in which m = 2, and that of the hyperbola in which m = -1: in the latter case his interpretation of the result is incorrect. He then shewed that similar results might be written down for any curve of the form y = \sum{ax^m}; so that if the ordinate y of a curve could be expanded in powers of the abscissa x, its quadrature could be determined. Thus he said that if the equation of a curve was y = x^0 + x^1 + x^2 +... its area would be y = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 +... He then applied this to the quadrature of the curves y = (1 - x^2)^0, y = (1 - x^2)^1, y = (1 - x^2)^2, y = (1 - x^2)^3, &c. taken between the limits x = 0 and x = 1: and shewed that the areas are respectively1,\quad \frac{2}{3},\quad \frac{8}{15},\quad \frac{16}{35},\quad \&c."

"[Wallis] next considered curves of the form y = x^\frac{1}{m} and established the theorem that the area bounded by the curve, the axis of x, and the ordinate x = 1 is to the area of the rectangle on the same base and of the same altitude as m:m + 1. This is equivalent to finding the value of \int_{0}^{1}x^\frac{1}{m}dx. He illustrated this by the parabola in which m = 2. He stated but did not prove the corresponding result for a curve of the form y = x^\frac{p}{q}."

"As [Wallis] was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."

"In 1659 Wallis published a tract on s in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of s: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the x^3 = ay^2. This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified."

"The writings of Wallis published between 1655 and 1665 revealed and explained to all students the principles of those new methods which distinguish modern from classical mathematics. His reputation has been somewhat overshadowed by that of Newton, but his work was absolutely first class in quality. Under his influence a brilliant mathematical school arose at Oxford. In particular I may mention Wren, Hooke, and Halley as among the most eminent of his pupils. But the movement was shortlived, and there were no successors of equal ability to take up their work."

"[Isaac Barrow's] lectures delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones mathematicae: these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones opticae et geometricae, which is his most important work. ...The geometrical lectures contain some new ways of determining the areas and tangents of curves. The latter is solved by a rule exactly analogous to the procedure of the differential calculus, except that a separate determination of what is really a had to be made for every curve to which it was applied. Thus he took the equation of the curve between the coordinates x and y, gave x a very small decrement e and found the consequent decrement of y, which he represented by a. The limit of the ratio a/e when the squares of a and e were neglected was defined as the angular coefficient of the tangent at the point, and completely determined the tangent there."

"Barrow's lectures failed to attract any considerable audiences, and on that account he felt conscientious scruples about retaining his chair. Accordingly in 1669 he resigned it to his pupil Newton, whose abilities he had been one of the earliest to detect and encourage."

"[Isaac Newton's] subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures."

"[Isaac Newton] took his BA degree in 1664. There is a manuscript of his written in the following year, and dated May 28, 1665, which is the earliest documentary proof of his discovery of fluxions. It was about the same time that he discovered the ."

"Leibnitz did not reply to this letter till June 21, 1677. In his answer he explains his method of drawing tangents to curves, which he says proceeds "not by fluxions of lines but by the differences of numbers"; and he introduces his notation of dx and dy for the infinitesimal differences between the coordinates of two consecutive points on a curve. He also gives a solution of the problem to find a curve whose subtangent is constant, which shews that he could integrate."

"The two letters to Wallis in which Newton] explained his method of fluxions and fluents were written in 1692, and were published in 1693. Towards the close of 1692 and throughout the two following years Newton had a long illness, suffering from insomnia and general nervous irritability. He never quite regained his elasticity of mind, and though after his recovery he shewed the same power in solving any question propounded to him, he ceased thenceforward to do original work on his own initiative, and it was difficult to stir him to activity."

"In 1704 Newton] published his Optics, containing an account of his emission theory of light. To this book two appendices were added; one on cubic curves, and the other on the quadrature of curves and his theory of fluxions. Both of these were old manuscripts which had long been known to his friends at Cambridge, but had been previously unpublished."

"The second appendix to Newton's] Optics was entitled De quadratura curvarum. Most of it had been communicated to Barrow in 1666, and was probably familiar to Newton's pupils and friends from about 1667 onwards. It consists of two parts. The bulk of the first part had been included in the letter to Leibnitz of Oct. 24, 1676. This part contains the earliest use of literal indices, and the first printed statement of the : these are however introduced incidentally. The main object of this part is to give rules for developing a function of a in a series in ascending powers of x; so as to enable mathematicians to effect the quadrature of any curve in which the ordinate y can be expressed as an explicit function of the abscissa x. Wallis had shewn how this quadrature could be found when y was given as a sum of a number of powers of x and Newton here extends this by shewing how any function can be expressed as an infinite series in that way. ...Newton is generally careful to state whether the series are convergent. In this way he effects the quadrature of the curves y = \frac{a^2}{b + x},\quad y = (a^2 \pm x^2)^\frac{1}{2},\quad y = (x - x^2)^\frac{1}{2},\quad y = (\frac{1 + ax^2}{1 - bx^2})^\frac{1}{2}, but the results are of course expressed as infinite series. He then proceeds to curves whose ordinate is given as an implicit function of the abscissa; and he gives a method by which y can be expressed as an infinite series in ascending powers of x, but the application of the rule to any curve demands in general such complicated numerical calculations as to render it of little value. He concludes this part by shewing that the rectification of a curve can be effected in a somewhat similar way. His process is equivalent to finding the integral with regard to x of (1 + \dot{y}^2)^\frac{1}{2} in the form of an infinite series. This part should be read in connection with his Analysis by infinite series published in 1711, and his Methodus differentialis published in 1736. Some additional theorems are there given, and in the latter of these works he discusses his method of . The principle is this. If y = \theta(x) is a function of x and if when x is successively put equal to a1, a2,... the values of y are known and are b1, b2,.. then a parabola whose equation is y = p + qx + rx^2 +\cdots can be drawn through the points (a_1,b_1), (a_2,b_2),\cdots and the ordinate of this parabola may be taken as an approximation to the ordinate of the curve. The degree of the parabola will of course be one less than the number of given points. Newton points out that in this way the areas of any curves can be approximately determined. The second part of this second appendix contains a description of his method of fluxions and is condensed from his manuscript..."

"It is probable that no mathematician has ever equalled Newton in his command of the processes of classical geometry. But his adoption of it for purposes of demonstration appears to have arisen from the fact that the infinitesimal calculus was then unknown to most of his readers, and had he used it to demonstrate results which were in themselves opposed to the prevalent philosophy of the time the controversy would have first turned on the validity of the methods employed. Newton therefore cast the demonstrations of the Principia into a geometrical shape which, if somewhat longer, could at any rate be made intelligible to all mathematical students and of which the methods were above suspicion. ...in Newton's time and for nearly a century afterwards the differential and fluxional calculus were not fully developed and did not possess the same superiority over the method he adopted which they do now. The effect of his confining himself rigorously to classical geometry and elementary algebra, and of his refusal to make any use even of analytical geometry and of trigonometry is that the Principia is written in a language which is archaic (even if not unfamiliar) to us. The subject of optics lends itself more readily to a geometrical treatment, and thus his demonstrations of theorems in that subject are not very different to those still used. The adoption of geometrical methods in the Principia for purposes of demonstration does not indicate a preference on Newton's part for geometry over analysis as an instrument of research, for it is now known that Newton used the fluxional calculus in the first instance in finding some of the theorems (especially those towards the end of book I. and in book II.), and then gave geometrical proofs of his results. This translation of numerous theorems of great complexity into the language of the geometry of Archimedes and Apollonius is I suppose one of the most wonderful intellectual feats which was ever performed."

"The fluxional calculus is one form of the infinitesimal calculus expressed in a certain notation just as the differential calculus is another aspect of the same calculus expressed in a different notation. Newton assumed that all geometrical magnitudes might be conceived as generated by continuous motion: thus a line may be considered as generated by the motion of a point, a surface by that of a line, a solid by that of a surface, a plane angle by the rotation of a line, and so on. The quantity thus generated was defined by him as the fluent or flowing quantity. The velocity of the moving magnitude was defined as the fluxion of the fluent."

"At one time, while purchasing wine, [Johannes Kepler] was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the , which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria. Other points of mathematical interest in Kepler's works... [include] a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear... The Stereometria led Cavalieri... to the consideration of infinitely small quantities."

"... a pupil of Galileo and professor at Bologna, is celebrated for his Geometria indivisibilibus continuorum nova quadam ratione promota 1635. This work expounds his method of Indivisibles, which occupies an intermediate place between the of the Greeks and the methods of Newton and Leibniz. Indivisibles were discussed by Aristotle and the scholastic philosophers. They commanded the attention of Galileo. Cavalieri does not define the term. He borrows the concept from the scholastic philosophy of Bradwardine and Thomas Aquinas, in which a point is the indivisible of a line, a line the indivisible of a surface, etc. Each indivisible is capable of generating the next higher continuum by motion; a moving point generates a line, etc. The relative magnitude of two solids or surfaces could then be found simply by the summation of series of planes or lines. For example... he concludes that the pyramid or cone is respectively 1/3 of a prism or cylinder of equal base and altitude... By the Method of Indivisibles, Cavalieri solved the majority of the problems proposed by Kepler. Though expeditious and yielding correct results, Cavalieri's method lacks a scientific foundation. If a line has absolutely no width, then the addition of no number, however great, of lines can ever yield an area; if a plane has no thickness whatever, then even an infinite number of planes cannot form a solid. Though unphilosophical, Cavalieri's method was used for fifty years as a sort of integral calculus. It yielded solutions to some difficult problems. [Paul] Guldin made a severe attack on Cavalieri... [who] published in 1647... a treatise entitled Exercitationes geometriece sex in which he replied to the objections of his opponent and attempted to give a clearer explanation of his method. ...A revised edition of the Geometria appeared in 1653."

"There is an important curve not known to the ancients which now began to be studied with great zeal. Roberval gave it the name of" ," Pascal the name of "roulette," Galileo the name of "." The invention of this curve seems to be due to Charles Bouvelles who...in 1501 refers to this curve in connection with the problem of the . Galileo valued it for the graceful form it would give to arches in architecture. He ascertained its area by weighing paper figures of the cycloid against that of the generating circle and found thereby the first area to be nearly... thrice the latter. A mathematical determination was made by his pupil ... By the Method of Indivisibles he demonstrated its area to be triple that of the revolving circle and published his solution. This same quadrature had been effected a few years earlier (about 1636) by Roberval in France, but his solution was not known to the Italians. ... another prominent pupil of Galileo, determined the tangent to the cycloid. This was accomplished in France by Descartes and Fermat."

"In France, where geometry began to be cultivated with greatest success, Roberval, Fermat, Pascal, employed the and made new improvements in it. Giles Persone de Roberval... claimed for himself the invention of the Method... Roberval and Pascal improved the rational basis of the Method of Indivisibles, by considering an area as made up of an indefinite number of rectangles instead of lines, and a solid as composed of indefinitely small solids instead of surfaces. Roberval applied the method to the finding of areas, volumes, and centres of gravity. He effected the quadrature of a parabola... [and] cycloid. Roberval is best known for his method of drawing tangents, which, however, was invented at the same time if not earlier by Torricelli. Torricelli's appeared in 1644 under the title Opera geometrica. Roberval gives the fuller exposition of it."

"Roberval's method of drawing tangents is allied to Newton's principle of fluxions. Archimedes conceived his spiral to be generated by a double motion. This idea Roberval extended to all curves. Plane curves, as for instance the conic sections, may be generated by a point acted upon by two forces, and are the resultant of two motions. If at any point of the curve the resultant be resolved into its components, then the diagonal of the parallelogram determined by them is the tangent to the curve at that point. The greatest difficulty connected with this ingenious method consisted in resolving the resultant into components having the proper lengths and directions. Roberval did not always succeed in doing this, yet his new idea was a great step in advance. He broke off from the ancient definition of a tangent as a straight line having only one point in common with a curve,—a definition which by the methods then available was not adapted to bring out the properties of tangents to curves of higher degrees, nor even of curves of the second degree and the parts they may be made to play in the generation of the curves. The subject of tangents received special attention also from Fermat, Descartes, and Barrow, and reached its highest development after the invention of the differential calculus. Fermat and Descartes defined tangents as secants whose two points of intersection with the curve coincide. Barrow considered a curve a polygon and called one of its sides produced, a tangent."

"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems.""

"The labors of L. Euler, J. Lagrange, and P. S. Laplace lay in higher analysis, and this they developed to a wonderful degree. By them analysis came to be completely severed from geometry. During the preceding period the effort of mathematicians not only in England, but, to some extent, even on the continent, had been directed toward the solution of problems clothed in geometric garb, and the results of calculation were usually reduced to geometric form. A change now took place. Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected an elaborate structure. There are few great ideas pursued by succeeding analysts which were not suggested by L. Euler, or of which he did not share the honor of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, J. Lagrange developed the infinitesimal calculus and put analytical mechanics into the form in which we now know it. P. S. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labors of Newton, gave a full analytical discussion of the solar system. ... Comparing the growth of analysis at this time with the growth during the time of K. F. Gauss, A. L. Cauchy, and recent mathematicians, we observe an important difference. During the former period we witness mainly a development with reference to form. Placing almost implicit confidence in results of calculation, mathematicians did not always pause to discover rigorous proofs, and were thus led to general propositions, some of which have since been found to be true in only special cases. ...But in recent times there has been added to the dexterity in the formal treatment of problems, a much needed rigor of demonstration. A good example of this increased rigor is seen in the present use of infinite series as compared to that of Euler, and of Lagrange in his earlier works. ... The ostracism of geometry, brought about by the master-minds of this period, could not last permanently. Indeed, a new geometric school sprang into existence in France before the close of this period."

"There have been four general steps in the development of what we commonly call the calculus... The first is found among the Greeks. In passing from commensurable to incommensurable magnitudes their mathematicians had recourse to the , whereby, for example, they "exhausted" the area between a circle and an inscribed regular polygon, as in the work of Antiphon (c. 430 B.C.) The second general step... taken two thousand years later,... the method of s... began to attract attention in the first half of the 17th century, particularly in the works of Kepler (1616) and Cavalieri (1635), and was used to some extent by Newton and Leibniz. The third method is that of fluxions and is the one due to Newton (c. 1665). It is this form of the calculus that is usually understood when the invention of the science is referred to him. The fourth method, that of limits, is also due to Newton, and is the one now generally followed."

"The Greeks developed the about the 5th century B.C. Zeno of Elea (c. 450 B.C.) was one of the first to introduce problems that led to a consideration of magnitudes. He argued that motion was impossible, for this reason:"

"(c. 440 B.C.) may possibly have been a pupil of Zeno's. Very little is known of his life and we are not at all certain of the time in which he lived, but Diogenes Laertius (2nd century) speaks of him as a teacher of Democritus (c. 400 B.C.). He and Democritus are generally considered as the founders of that atomistic school, which taught that magnitudes are composed of individual elements in finite numbers. It was this philosophy that led Aristotle (c. 430 B.C.) to write a book in indivisible lines."

"Antiphon (c. 430) is one of the earliest writers whose use of the is fairly well known to us. In a fragment of Eudemus (c. 335 B.C.)... we have the following description:"

"(370 B.C.) is probably the one who placed the theory of exhaustion on a scientific basis. ...[In] Book V of Euclid's Elements (the book on proportion)... it is thought that the fundamental principles laid down are his. The fourth definition... is: "Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another," and this includes the relation of a finite magnitude to a magnitude of the same kind which is either infinitely great or infinitely small. ...According to Archimedes, this method had already been applied by Democritus (c. 400 B.C.) to the mensuration of both the cone and the cylinder."

"It is known that Hippocrates of Chios (c. 460 B.C.) proved that circles are to one another as the squares of their diameters, and it seems probable that he also used the ... Archimedes tells us that the "earlier geometers" had proved that spheres have to one another the triplicate ratio of their diameters, so that the method was probably used by others as well."

"It is to Archimedes... that we owe the nearest approach to actual integration to be found among the Greeks. His first noteworthy advance... was concerned with his proof that the area of a parabolic segment is four thirds of the triangle with the same base and vertex, or two thirds of the circumscribed parallelogram. This was shown by continually inscribing in each segment between the parabola and the inscribed figure a triangle with the same base and... height as the segment. If A is the area of the original inscribed triangle, the process... leads to the summation of the seriesA + \frac{1}{4}A + (\frac{1}{4})^2A + (\frac{1}{4})^3A+...or...A[1 + \frac{1}{4} + (\frac{1}{4})^2 + (\frac{1}{4})^3+...]so that he really finds the area by integration and recognizes, but does not assert, that(\frac{1}{4})^n \to 0~\text{as}~n \to \infty,this being the earliest example that has come down to us of the summation of an infinite series. ... In his treatment of solids bounded by curved surfaces he arrives at conclusions which we should now describe by the following formulas: Surface of a sphere,4\pi a^2 \cdot \frac{1}{2} \int\limits_{0}^{\pi} \sin\theta d\theta = 4\pi a^2.Surface of a spherical segment,\pi a^2 \int\limits_{0}^{a} 2\sin\theta d\theta = 2\pi a^2 (1-\cos\alpha).Volume of a segment of a hyperboloid of revolution,\int\limits_{0}^{b} (ax + x^2) dx =b^2(\frac{1}{2}a + \frac{1}{3}b).Volume of a segment of a spheroid,\int\limits_{0}^{b} x^2 dx = \frac{1}{3}b^3.Area of a spiral, \frac{\pi}{a} \int\limits_{0}^{a} x^2 dx = \frac{1}{3} \pi a^2.Area of a parabolic segment, \frac{1}{A^2} \int\limits_{0}^{A} \bigtriangleup^2 d\bigtriangleup = \frac{1}{3} A."

"Among the more noteworthy attempts at integration in modern times were those of Kepler (1609). In his notable work on planetary motion he asserted that a planet describes equal focal sectors of ellipses in equal times. This... demands some method for finding the areas of such sectors, and the one invented by Kepler was called by him the... "sum of the radii," a rude kind of integration. He also became interested in the problem of gaging, and published a work on this... and on general mensuration as set forth by Archimedes. ...[Kepler's] was a scientific study of the measurement of solids in general. ...composed "as it were" (veluti) of infinitely many infinitely small cones or infinitely thin disks, the summation of which becomes the problem of later integration."

"Kepler's attempts at integration... led Cavalieri to develop his method of indivisibles... which may also have been suggested to him by Aristotle's tract De lineis insecabilibus [On indivisible lines]... It may also have been suggested by one of the fragments of Xenocrates (c. 350 B.C.)... who wrote upon indivisible lines. ... Cavalieri... seems to have looked upon a solid as made up practically of superposed surfaces, a surface as made up of lines, and a line as made up of points, these component parts being the ultimate possible elements in the decomposition of the magnitude. He then proceeded to find lengths, areas, and volumes of the summation of these "indivisibles," that is, by the summation of an infinite number of s. Such a conception of magnitude cannot be satisfactory to any scientific mind, but it formed a kind of intuitive step in the development of the method of integration and undoubtedly stimulated men like Leibniz to exert their powers to place the theory upon a scientific foundation. ... Cavalieri was able to solve various elementary problems in the mensuration of lengths, areas, and volumes, and also to give a fairly satisfactory proof of the theorem of Pappus with respect to the volume generated by the revolution of a plane figure about an axis."

"The problem of tangents, the basic principle of the theory of maxima and minima, may be said to go back to Pappus (c. 300). It appears indirectly in the Middle Ages, for Oresme (c. 1360) knew that the point of maximum or minimum of a curve is the point at which the ordinate is changing most slowly. It was Fermat, however, who first stated substantially the law as we recognize it today, communicating (1638) to Descartes a method which is essentially the same as the one used at present, that of equating [the ] f^\prime(y) to zero. Similar methods were used by René de Sluze (1652) for tangents, and by Hudde (1658) for maxima and minima."

"The first British publication of great significance bearing upon the calculus is that of John Wallis, issued in 1655. It is entitled Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilineorum Quadraturum, aliaque difficiliora Matheseos Problemata, and is dedicated to Oughtred. By a method similar to Cavalieri the author effects the quadrature of certain surfaces, the cubature of certain solids, and the rectification of certain curves. He speaks of a triangle, for example, "as if" (quasi) made up of an infinite number of parallel lines in arithmetic proportion, of a paraboloid "as if" made up of an infinite number of parallel lines, and of a spiral as an aggregate of an infinite number of arcs of similar sectors, applying to each the theory of the summation of an infinite series. ...he expresses his indebtedness to such writers as Torricelli and Cavalieri. He speaks of the work of such British contemporaries as Seth Ward and Christopher Wren, who were interested in this relatively new method, and, indeed, his dedication to Oughtred is the best contemporary specimen that we have of the history of the movement just before Newton's period of activity. All this, however, was still in the field of integration, the first steps dating... from the time of the Greeks."

"What is considered by us as the process of differentiation was known to quite an extent to Barrow (1663). In his Lectiones opticae et geometricae he gave a method of tangents in which, in the annexed figure, Q approaches P, as in our present theory, the result being an indefinitely small (indefinite parvum) arc. The triangle PRQ was long known as "Barrow's differential triangle," a name which, however, was not due to him. ...this method, and the figure... must have had a notable influence upon the mathematics of his time."

"Isaac Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow's work; obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow's book that he purchased in 1673. The above is the ultimate conclusion that I have arrived at as the result of six months' close study of a single book, my first essay in historical research. By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a product, a quotient, or a power of a function; and also a recognition and demonstration of the fact that differentiation and integration are inverse operations.""

"The case of Newton is to my mind clear enough. Barrow was familiar with the paraboliforms, and tangents and areas connected with them, in from 1655 to 1660 at the very latest; hence he could at this time differentiate and integrate by his own method any rational positive power of a variable, and thus also a sum of such powers. He further developed it in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he communicated to Newton the great secret of his geometrical constructions, as far as it is humanly possible to judge from a collection of tiny scraps of circumstantial evidence; and it was probably this that set Newton to work on an attempt to express everything as a sum of powers of the variable. During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per Æquations. This, though composed in 1666, was not published until 1711."

"Leibniz bought a copy of Barrow's work in 1673, and was able "to communicate a candid account of his calculus to Newton" in 1677. In this connection, in the face of Leibniz' persistent denial that he received any assistance whatever from Barrow's book, we must bear well in mind Leibniz' twofold idea of the "calculus": (i) the freeing of the matter from geometry, (ii) the adoption of a convenient notation. Hence, be his denial a mere quibble or a candid statement without any thought of the idea of what the "calculus" really is, it is perfectly certain that on these two points at any rate he derived not the slightest assistance from Barrow's work; for the first of them would be dead against Barrow's practice and instinct, and of the second Barrow had no knowledge whatever. These points have made the calculus the powerful instrument that it is, and for this the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. in modern phraseology, the differentiation and integration of any function whatever (such as were known in Barrow's time), must be ascribed to Barrow."

"The beginnings of the Infinitesimal Calculus, in its two main divisions, arose from determinations of areas and volumes, and the finding of tangents to plane curves. The ancients attacked the problems in a strictly geometrical manner, making use of the "s." In modern phraseology, they found "upper and lower limits," as closely equal as possible, between which the quantity to be determined must lie; or, more strictly perhaps, they showed that, if the quantity could be approached from two "sides," on the one side it was always greater than a certain thing, and on the other it was always less; hence it must be finally equal to this thing. This was the method by means of which Archimedes proved most of his discoveries. But there seems to have been some distrust of the method, for we find in many cases that the discoveries are proved by a ', such as one is familiar with in Euclid. To Apollonius we are indebted for a great many of the properties, and to Archimedes for the measurement, of the conic sections and the solids formed from them by their rotation about an axis."

"The first great advance, after the ancients, came in the beginning of the seventeenth century. Galileo (1564-1642) would appear to have led the way, by the introduction of the theory of composition of motions into mechanics; he also was one of the first to use s in geometry, and from the fact that he uses what is equivalent to "virtual velocities" it is to be inferred that the idea of time as the independent variable is due to him."

"Kepler (1571-1630) was the first to introduce the idea of infinity into geometry and to note that the increment of a variable was evanescent for values of the variable in the immediate neighbourhood of a maximum or minimum; in 1613, an abundant vintage drew his attention to the defective methods in use for estimating the... contents of vessels, and his essay on the subject (Nova Stereometria Doliorum [Vinariorum]) entitles him to rank amongst those who made the discovery of the infinitesimal calculus possible."

"In 1635 Cavalieri published a theory of "indivisibles," in which he considered a line as made up of an infinite number of points, a superficies as composed of a succession of lines, and a solid as a succession of superficies, thus laying the foundation for the "aggregations" of Barrow. Roberval seems to have been the first, or at the least an independent, inventor of the method; but he lost credit for it, because he did not publish it, preferring to keep the method to himself for his own use; this seems to have been quite a usual thing amongst learned men of that time, due perhaps to a certain professional jealousy. The method was severely criticized by contemporaries, especially by Guldin, but Pascal (1623-1662) showed that the method of indivisibles was as rigorous as the method of exhaustions, in fact that they were practically identical. In all probability the progress of mathematical thought is much indebted to this defence by Pascal. Since this method is exactly analogous to the ordinary method of integration, Cavalieri and Roberval have more than a little claim to be regarded as the inventors of at least the one branch of the calculus; if it were not for the fact that they only applied it to special cases, and seem to have been unable to generalize it owing to cumbrous algebraical notation, or to have failed to perceive the inner meaning of the method when concealed under a geometrical form. Pascal himself applied the method with great success, but also to special cases only; such as his work on the ."

"The next step was of a more analytical nature; by the method of indivisibles, Wallis (1616-1703) reduced the determination of many areas and volumes to the calculation of the value of the series (0^m + 1^m + 2^m +... n^m / (n + 1)n^m, i.e. the ratio of the mean of all the terms to the last term, for integral values of n; and later he extended his method, by a theory of interpolation, to fractional values of n. Thus the idea of the Integral Calculus was in a fairly advanced stage in the days immediately antecedent to Barrow."

"What Cavalieri and Roberval did for the integral calculus, Descartes (1596-1650) accomplished for the differential branch by his work on the application of algebra to geometry. Cartesian coordinates made possible the extension of investigations on the drawing of tangents to special curves to the more general problem for curves of any kind. To this must be added the fact that he habitually used the index notation; for this had a very great deal to do with the possibility of Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make any very great progress on his own account in the matter of the drawing of tangents, owing to what I cannot help but call an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three:—"

"Fermat (1590-1663) adopted Kepler's notion of the increment of the variable becoming evanescent near a maximum or minimum value, and upon it based his method of drawing tangents. Fermat's method of finding the maximum or minimum value of a function involved the differentiation of any explicit algebraic function, in the form that appears in any beginner's text book of today (though Fermat does not seem to have the "function" idea); that is, the maximum or minimum values of f(x) are the roots of f'(x) = 0, where f'(x) is the limiting value of [f(x+h) - f(x)]/h; only Fermat uses the letter e or E instead of h."

"Here then we have all the essentials for the calculus; but only for explicit integral algebraic functions, needing the binomial expansion of Newton, or a general method of rationalization which did not impose too great algebraic difficulties, for their further development; also, on the authority of Poisson, Fermat is placed out of court, in that he also only applied his method to certain special cases. Following the lead of Roberval, Newton subsequently used the third definition of a tangent, and the idea of time as the independent variable, although this was only to insure that one at least of his working variables should increase uniformly. This uniform increase of the independent variable would seem to have been usual for mathematicians of the period and to have persisted for some time; for later we find with Leibniz and the Bernoullis that d(dy/dx) = (d2y/dx2)dx. Barrow also used time as the independent variable in order that, like Newton, he might insure that one of his variables, a moving point or line or superficies, should proceed uniformly; ...Barrow... chose his own definition of a tangent, the second of those given above; and to this choice is due in great measure his advance over his predecessors. For his areas and volumes he followed the idea of Cavalieri and Roberval."

"Thus we see that in the time of Barrow, Newton, and Leibniz the ground had been surveyed, and in many directions levelled; all the material was at hand, and it only wanted the master mind to "finish the job." This was possible in two directions, by geometry or by analysis; each method wanted a master mind of a totally different type, and the men were forthcoming. For geometry, Barrow; for analysis, Newton and Leibniz with his inspiration in the matter of the application of the simple and convenient notation of his calculus of finite differences to infinitesimals and to geometry. With all due honour to these three mathematical giants, however, I venture to assert that their discoveries would have been well-nigh impossible to them if they had lived a hundred years earlier; with the possible exception of Barrow, who, being a geometer, was more dependent on the ancients and less on the moderns of his time than were the two analysts, they would have been sadly hampered but for the preliminary work of Descartes and the others I have mentioned (and some I have not—such as Oughtred), but especially Descartes."

"Almost all of fluid dynamics follows from a differential equation called the Navier-Stokes equation. But this general equation has not, in practice, led to solutions of real problems of any complexity. In this sense, the curve of a baseball is not understood; the Navier-Stokes equation applied to a base ball has not been solved."

"I ought also to mention Jacobi]'s papers on Abelian transcendants; his investigations on the theory of numbers... his important memoirs on the theory of differential equations, both ordinary and partial; his development of the ; and his contributions to the problem of three bodies, and other particular dynamical problems. Most of the results of the researches last named are included in his Vorlesungen über Dynamik."

"The meaning of the differential equation now follows:\frac{df(t)}{dt} = Af(t)expresses the claim that the rate of change in f(t)... is proportional at t to f(t) itself. And this makes sense. How fast a colony of bacteria will grow is contingent on the... number of bacteria on hand and the relative percentage of bacteria engaged in reproduction. ... Equations are... acts of specification in the dark; something answers to some condition. ...Specification in the dark corresponds to the...process by which a sentence in which a pronoun figures—He smokes—acquires the stamp of specificity when the antecedent... is dramatically or diffidently revealed—Winston Churchill, say, or a lapsed smoker seeking an errant cigarette in a bathroom. The differential equation describing uniform growth admits a simple but utterly general solution by means of the exponential functionf(t) =ke^{At}.The number e is an irrational number lying on the leeward side of the margin between 2 and 3 and playing, like \pi, a strange and essentially inscrutable role throughout all of mathematics; takes e to a power... in this case... specified by A and t. The constant k has an interpretation as the problem's initial value... some... (weight or mass) of bacteria. ... as time scrolls backward or forward in the... imagination, ke^{At} provides a running account of growth or decay... This is in itself remarkable, the temporal control achieved by what are after all are just symbols, quite unlike anything else in language or its lore or law, but when successful, specification in the dark achieves an analysis of experience that goes beyond any specific prediction to embrace a universe of possibilities loitering discreetly behind the scenes."

"[W]e find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances... This memoir concluded with an invitation to mathematicians, to sum up a very general differential equation, of great use in analysis. The solution which James Bernoulli had found of this problem, as well as those which Leibnitz and John Bernoulli gave of it, were published in the Leipsic Transactions."

"Newton] teaches us to take the s, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem."

"Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the world in english."

"The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843."

"We should speak of a dialectics of the calculus... the problem element in so far as... distinguished from the properly mathematical element of solutions. Following Lautman... a problem has three aspects: its difference in kind from solutions; its transcendence in relation to the solutions... and its immanence in the solutions which cover it, the problem being the better resolved the more it is determined. Thus the ideal connections constitutive of the problematic ([Platonic] dialectical) Idea are incarnated in the real relations which are constituted by mathematical theories and carried over into problems in the form of solutions... like the discontinuities compatible with differential equations."

"In general, a differential equation arises whenever you have a quantity subject to change. ...Strictly speaking, the changing quantity should be one that changes continuously. ...However, change in many real life situations consists of a large number of individual discrete changes, that are miniscule compared with the overall scale of the problem, and in such cases there is no harm in simple assuming that the whole changes continuously."

"Maxwell's equations had abstract mathematical qualities which were profoundly new and important. Maxwell's theory was formulated in terms of a new style of mathematical concept, a extending throughout space and time and obeying coupled partial differential equations of peculiar symmetry. ...If they had taken Maxwell's equations to heart as Euler took Newton's, they would have discovered, among other things, Einstein's theory of special relativity, the theory of s and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis. A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwell's equations naturally lead."

"If the idea of physical reality had ceased to be purely atomic, it still remained for the time being purely mechanistic; people still tried to explain all events as the motion of inert masses; indeed no other way of looking at things seemed conceivable. Then came the great change, which will be associated for all time with the names of Faraday, Clerk Maxwell, and Hertz. The lion's share in this revolution fell to Clerk Maxwell. He showed that the whole of what was then known about light and electro-magnetic phenomena was expressed in his well known double system of differential equations, in which the electric and magnetic fields appear as the dependent variables. Maxwell did, indeed try to explain, or justify, these equations by intellectual constructions. But... the equations alone appeared as the essential thing and the strength of the fields as the ultimate entities, not to be reduced to anything else."

"Mr Gregory devoted to it a chapter of his work, and noticed particularly some of the more remarkable applications of definite integrals to the expression of the solutions of partial differential equations. It is not improbable that in another edition he would have developed this subject at somewhat greater length. He had long been an admirer of Fourier’s great work on heat, to which this part of mathematics owes so much; and once, while turning over its pages, remarked to the writer,—“ All these things seem to me to be a kind of mathematical paradise.""

"Any progress in the theory of partial differential equations must also bring about a progress in Mechanics."

"A wide variety of economic problems lead to differential, difference, and integral equations. Ordinary differential equations appear in models of economic dynamics. Integral equations appear in dynamic programming problems and asset pricing models. Discrete-time dynamic problems lead to difference equations."

"A of the Riccati type is derived for the of the optimal filtering error. The solution of this ' equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the . Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The duality principle relating estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of s."

"Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity."

"The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived... A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations."

"In the interval between the discovery of and that of the general differential equations of Elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibrations of bars and plates, and the stability of columns."

"At the end of the year 1820 the fruit of all the ingenuity expended on elastic problems might be summed up as—an inadequate theory of flexure, an erroneous theory of torsion, an unproved theory of the vibrations of bars and plates, and the definition of . But such an estimate would give a very wrong impression of the value of the older researches. The recognition of the distinction between shear and extension was a preliminary to a general theory of strain; the recognition of forces across the elements of a section of a beam, producing a resultant, was a step towards a theory of stress; the use of differential equations for the deflexion of a bent beam and the vibrations of bare and plates, was a foreshadowing of the employment of differential equations of displacement; the Newtonian conception of the constitution of bodies, combined with , offered means for the formation of such equations; and the generalization of the principle of in the Mécanique Analytique threw open a broad path to discovery in this as in every other branch of mathematical physics."

"Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. Happily here is a commodity of which a little may be made to go a long way. ...the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks."

"The differential equation of the first order \frac {dy}{dx} = f(x,y) ...prescribes the slope \frac {dy}{dx} at each point of the plane (or at each point of a certain region of the plane we call the field"). ...a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines. ...It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction.""

"When Born and Heisenberg and the Göttingen theoretical physicists] first discovered they were having, of course, the same kind of trouble that everybody else had in trying to solve problems and to manipulate and to really do things with matrices. So they had gone to Hilbert for help and Hilbert said the only time he had ever had anything to do with matrices was when they came up as a sort of by-product of the eigenvalues of the boundary-value problem of a differential equation. So if you look for the differential equation which has these matrices you can probably do more with that. They had thought it was a goofy idea and that Hilbert didn’t know what he was talking about. So he was having a lot of fun pointing out to them that they could have discovered Schrödinger’s wave mechanics six month earlier if they had paid a little more attention to him."

"Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations."

"[D]ifferential equations... represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both. ...[S]ince Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns. ...[T]theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram... Loosely translated... "It is useful to solve differential equations.""

"Science is a differential equation. Religion is a ."

"Simulators set up the required system of interdependences, usually between electrical potentials or voltages as variables, by means of valve-amplifiers and electrical networks. Since the voltage across a capacitance is proportional to the integral of a current, that across an inductance to the first derivative of a current, and that across a resistor to the current itself, it is possible to arrange a network of electrical elements, with amplifiers and feeds-back where necessary, so that a given linear differential equation is caused to relate an ’output’ voltage to an ’input’ voltage. Thus a given linear system of interdependences can be simulated, either directly or in any convenient transformation. If non-linear relationships are required there is no universally applicable simple device, but there do exist a great variety of non-linear elements with non-linear characteristics that are known and to some extent; adjustable. These include non-linear resistors... and the characteristic curves of thermionic valves, of rectifiers and discharge vessels and of magnetic materials. Limits may be set by the use of neon tubes that become conducting when a certain voltage is exceeded, or by relays, and so on"

"The methods of the Bernoullis and of Taylor, were held, at the time of their invention, to be most complete and exact. Several imperfections, however, belong to them. They do not apply to problems involving three or more properties; nor do they extend to cases involving differentials of a higher order than the first: for instance, they will not solve the problem, in which a curve is required, that with its radius of curvature and evolute shall contain the least area. Secondly, they do not extend to cases, in which the analytical expression contains, besides x, y, and their differentials, integral expressions; for instance, they will not solve the second case proposed in James Bernoulli's Programma.. if the Isoperimetrical condition be excluded; for then the arc s, an integral, since it =\int \!dx \sqrt(1+\frac{dy^2}{dx^2}), is not given. Thirdly, they do not extend to cases, in which the differential function, expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever."

"Malliavin’s work inspired many new results in stochastic analysis. Examples include filtering theorems (Michel ...), a deeper understanding of the Skorohod integral and the development of an anticipating stochastic calculus (Nualart and Paradox ...), an extension of Clark’s formula (Ocone ...), Bismut’s probabilistic analysis of the small-time asymptotics of the heat kernel of the Dirac operator on a Riemannian manifold ... and his subsequent proof of the associated index theorem ..., and a sharp hypoellipticity theorem for Hörmander operators with hypersurfaces of infinite type (Bell and Mohammed ...)."

"Denis R. Bell:"

"The mathematical theory now known as Malliavin calculus was first introduced by Paul Malliavin .. as an infinite-dimensional integration by parts technique. The purpose of this calculus was to prove the results about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motiion."

"We present an approach that allows one to introduce a Malliavin type calculus for functionals of general Lévy processes and to obtain sufficient conditions for the absolute continuity of solutions of stochastic differential equations with jumps (we do not pose any assumptions about regularity of the intensity of the jumps). Our investigations are motivated by a pioneering idea due to Bismut ... and developed further by many authors. The idea is to extend the Malliavin approach to regularity of Wiener functionals to more general probability spaces by introducing a smooth structure in these spaces in terms of a “differentiation rule”, integration-by-parts formula, and by further applications of the stochastic calculus of variations to smooth functionals with nondegenerate derivatives."

"The Malliavin calculus refers to a part of Probability theory which can loosely be described as a type of calculus of variations for Brownian motion. It is intimately concerned with the interplay between Markov processes with continuous paths (i.e., diffusions) and partial differential equations. ... What Malliavin did was to provide a probabilistic proof of Hörmander's theorem by constructing a kind of calculus of variations for Brownian motion. This in turn gave probabilistic proofs of the smoothness of the transition densities. This has the advantage of giving probabilistic insight and intuition into what is seen as a fundamental probabilistic result; it has the disadvantage of giving a longer and perhaps harder proof of Hörmander's theorem than is available in the PDE literature ... However Malliavin's methods (credit should also be given to those whose work he built upon such as Gross, Kree, Kuo, Eels, Elworthy, .,. ) are profound, and they are already having ramifications in other areas of probability."