History of calculus

"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."