calculus

"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."

"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."

"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."