Search Results for Calculus


Biographies

  1. Mikusinski biography
    • The participants of Wazewski's seminar, in 1943, were the first persons to come in contact with a new theory which is now very well known in the world of mathematics as the Mikusinski operational calculus.
    • The paper 'Hypernumbers' represents the first version of the Mikusinski operational calculus (improved afterwards by the use of the Titchmarsh theorem) and contains already the main ideas of this theory.
    • He submitted a collection of his papers under the title A new approach to the operational calculus for a degree similar to the present D.Sc.
    • Rudolf Hilfer, Yury Luchko and Zivorad Tomovski write in [Fractional Calculus and Applied Analysis 12 (3) (2009), 299-318.',7)">7]:- .
    • In the 1950's, Jan Mikusinski proposed a new approach to develop an operational calculus for the operator of differentiation (see J Mikusinski, Operational Calculus (Pergamon Press, New York, 1959)).
    • The Mikusinski operational calculus was successfully used in ordinary differential equations, integral equations, partial differential equations and in the theory of special functions.
    • Arthur Erdelyi, reviewing the Polish version of Mikusinski's book The Calculus of Operators (1953), writes:- .
    • By and large, freshman and sophomore calculus are not quite sufficient for reading this book, a good course in advanced calculus is more than sufficient.
    • Henry Schaerf, reviewing the second Polish edition of Mikusinski's The Calculus of Operators (published in 1957), writes [Bull.
    • In several papers the author has published a theory containing a direct justification of the Heaviside Calculus as opposed to the various well known indirect methods using functional transforms.
    • The Professor delivered for them a series of lectures on operational calculus.
    • The topics of the seminar were closely connected with the mathematical interests of Professor Jan Mikusinski and included the operational calculus, generalized functions, convergence structures and integration theory.
    • We should mention a number of Mikusinski's books in addition to The Calculus of Operators (1953) which we discussed above.
    • The sequential approach (1973); The Bochner integral (1978); Operational calculus.
    • I (1983) and (with Thomas K Boehme) Operational calculus.
    • Based on the lifetime work of leading teacher and researcher Jan Mikusinski, this classroom-tested book provides a thorough grounding in mathematical analysis, calculus and mathematical proofing.

  2. Bernoulli Johann biography
    • Jacob was lecturing on experimental physics at the University of Basel when Johann entered the university and it soon became clear that Johann's time was mostly devoted to studying Leibniz's papers on the calculus with his brother Jacob.
    • Johann's first publication was on the process of fermentation in 1690, certainly not a mathematical topic but in 1691 Johann went to Geneva where he lectured on the differential calculus.
    • De l'Hopital was delighted to discover that Johann Bernoulli understood the new calculus methods that Leibniz had just published and he asked Johann to teach him these methods.
    • After Bernoulli returned to Basel he still continued his calculus lessons by correspondence, and this did not come cheap for de l'Hopital who paid Bernoulli half a professor's salary for the instruction.
    • However it did assure de l'Hopital of a place in the history of mathematics since he published the first calculus book Analyse des infiniment petits pour l'intelligence des lignes courbes (1696) which was based on the lessons that Johann Bernoulli sent to him.
    • The well known de l'Hopital's rule is contained in this calculus book and it is therefore a result of Johann Bernoulli.
    • After de l'Hopital's death in 1704 Bernoulli protested strongly that he was the author of de l'Hopital's calculus book.
    • In 1692, while in Paris, he met Varignon and this resulted in a strong friendship and also Varignon learned much about applications of the calculus from Johann Bernoulli over the many years which they corresponded.
    • Johann's solution to this problem was less satisfactory than that of Jacob but, when Johann returned to the problem in 1718 having read a work by Taylor, he produced an elegant solution which was to form a foundation for the calculus of variations.
    • He strongly supported Leibniz and added weight to the argument by showing the power of his calculus in solving certain problems which Newton had failed to solve with his methods.
    • Although Bernoulli was essentially correct in his support of the superior calculus methods of Leibniz, he also supported Descartes' vortex theory over Newton's theory of gravitation and here he was certainly incorrect.
    • As a study of the historical records has justified Johann's claims to be the author of de l'Hopital's calculus book, so it has shown that his claims to have published Hydraulica before his son wrote Hydrodynamica are false.
    • History Topics: The rise of Calculus .

  3. Varignon biography
    • In 1687 Varignon published Projet d'une nouvelle mechanique which studied composition of forces using Leibniz's differential calculus in the study of mechanics.
    • From the earliest of his publications such as Projet d'une nouvelle mechanique in 1687, it was clear that he understood the value of Leibniz's calculus.
    • This was surprisingly soon after Leibniz's two articles on the new differential calculus were published in the Acta Eruditorum in October 1684 and June 1686.
    • Although Varignon made no major mathematical contributions, he developed analytic dynamics by adapting Leibniz's calculus to the inertial mechanics of Newton's Principia being one of the first French scholars to recognise the power and importance of the calculus.
    • Varignon put aside these philosophical worries and began to rework large sections of the Principia into the Leibniz's approach to the differential and integral calculus.
    • Among his other work was a publication in 1699 on applications of the differential calculus to fluid flow and to water clocks.
    • In 1702 he applied the calculus to clocks driven by a spring.
    • Again this work is developed using Leibniz's approach to the calculus.
    • Varignon played a major role in defending the calculus from attacks.
    • For example in 1700 Rolle argued against the calculus both on the grounds that it was without sound foundation and that it led to errors.
    • Varignon argued before the Academy of Sciences that Rolle's arguments which suggested that the calculus led to errors was wrong.

  4. Leibniz biography
    • It was during this period in Paris that Leibniz developed the basic features of his version of the calculus.
    • In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy.
    • In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.
    • by Leibniz's approach but the formalism was to prove vital in the latter development of the calculus.
    • In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus..
    • In 1686 Leibniz published, in Acta Eruditorum, a paper dealing with the integral calculus with the first appearance in print of the ∫n notation.
    • Much of the mathematical activity of Leibniz's last years involved the priority dispute over the invention of the calculus.
    • Leibniz demanded a retraction saying that he had never heard of the calculus of fluxions until he had read the works of Wallis.
    • whence Leibniz derived the principles of that calculus or at least could have derived them.
    • However, when Newton wrote to him directly, Leibniz did reply and gave a detailed description of his discovery of the differential calculus.
    • History Topics: The rise of the calculus .

  5. Franklin biography
    • However, he is best known for textbooks he published on calculus, differential equations, complex variable and Fourier series.
    • In particular he wrote Differential equations for electrical engineers (1933), Treatise on advanced calculus (1940), The four color problem (1941), Methods of advanced calculus (1944), Fourier methods (1949), Differential and integral calculus (1953), Functions of a complex variable (1958) and Compact calculus (1963).
    • Let us here quote from Richard Courant's review of A Treatise on Advanced Calculus [Science 94 (2448) (1941), 518.
    • This book is an extraordinarily satisfactory addition to the literature of advanced calculus.
    • This text is indeed a treatise which covers completely the infinitesimal calculus and includes much prerequisite algebra and analysis (and most other concepts) that are needed for geometric and physical applications.
    • Four years after publishing this rigorous text, Franklin published another text Methods of advanced calculus on similar material but now with very different students in mind.
    • the new volume may be regarded as one of the best textbooks now available for any advanced calculus course which is intended to be a terminal course in mathematics for engineers, physicists and the like.
    • As an editor of 'The Journal of Mathematics and Physics' and the author of studies in algebra and calculus you have made important contributions to scholarship.

  6. Lagrange biography
    • By the end of 1754 he had made some important discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations (which mathematicians were beginning to study but which did not receive the name 'calculus of variations' before Euler called it that in 1766).
    • In 1756 Lagrange sent Euler results that he had obtained on applying the calculus of variations to mechanics.
    • He published his beautiful results on the calculus of variations, and a short work on the calculus of probabilities.
    • His work in Berlin covered many topics: astronomy, the stability of the solar system, mechanics, dynamics, fluid mechanics, probability, and the foundations of the calculus.
    • Lagrange published two volumes of his calculus lectures.
    • the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.
    • Not everyone found Lagrange's approach to the calculus the best however, for example de Prony wrote in 1835:- .
    • Lagrange's foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine engineering, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools.
    • History Topics: The rise of the calculus .

  7. Euler biography
    • The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics.
    • Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.
    • He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics (extending the book by Robins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publication Letters to a Princess of Germany (3 vols., 1768-72).
    • He made decisive and formative contributions to geometry, calculus and number theory.
    • He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis.
    • This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously.
    • In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus of finite differences.
    • The calculus of variations is another area in which Euler made fundamental discoveries.
    • published in 1740 began the proper study of the calculus of variations.
    • History Topics: The rise of the calculus .

  8. Picone biography
    • While in Catania, he published two books: Teoria introduttiva delle equazioni differenziali ordinarie e calcolo delle variazioni (Introductory Theory of ordinary differential equations and calculus of variations) (1922) and Lezioni di Analisi infinitesimale (Lessons on Infinitesimal Analysis) (1923).
    • They worked in the "Institute for Calculus" which had been founded by Picone in Naples in 1927 with financial assistance from the Banco di Napoli.
    • With admirable single mindedness and great political skill Picone succeeded in getting appointed to a chair of mathematical analysis in Rome, left Naples University together with his brilliant assistant Miranda and, besides teaching the kind of mathematics acceptable to his illustrious colleagues, obtained a small grant from the newly established Italian National Research Council to start in a small apartment in Rome in a new section of town the high-sounding "Institute for the Applications of the Calculus." .
    • Picone was the director of the Institute for the Applications of the Calculus in Rome from its founding until 31 July 1960 when he retired from his chair at the university and was made professor emeritus.
    • Yet by 1935 Picone's vision and persistence had given the institute, by now the National Institute for the Applications of Calculus, the entire top floor of the new palace of the National Research Council and enough money to pay (miserly) a staff of thirty.
    • The work undertaken by the Institute included functional analysis, partial differentiation, integral equations, calculus of variations, special functions, probability theory, rational mechanics and mathematical physics.
    • Some of his most important books which Picone published during his years in Rome are: Appunti di Analisi superiore (1940), which studies harmonic functions, Fourier, Laplace and Legendre series and the equations of mathematical physics; Lezioni di Analisi funzionale (1946), which concerns the calculus of variations; Teoria moderna dell'integrazione delle funzioni (1946), containing a detailed discussion of the r-dimensional Stieltjes integrals; (with Tullio Viola) Lezioni sulla teoria moderna dell'integrazione (1952), which is basically the previous work by Picone with three extra chapters by Viola; and (with Gaetano Fichera) Trattato di Analisi matematica (Vol 1, 1954, Vol 2, 1955), which puts into a treatise Picone's way of teaching calculus particularly slanted towards the applications studied at the Institute for Applied Calculus.
    • Over the last eight years in the chair he had concentrated his research on a classical approach to the integrals of the calculus of variations.

  9. Hermann biography
    • Leibniz had published his ideas on the differential calculus in 1884 and, two years later, his ideas on the integral calculus.
    • Bernhardt Nieuwentijt (1654-1718) was a Dutch philosopher and mathematician who was highly critical of Leibniz's differential and integral calculus and, in 1696, he published Considerationes secundae in which [Studia Leibnitiana 21 (1 ) (1989), 69-86.',16)">16]:- .
    • In 1701 Hermann became a member of the Berlin Academy of Science, his election being very much due to support from Leibniz who was delighted to see the clarity with which he had defended the infinitesimal calculus.
    • As soon as he was in Italy he began to make contacts with other Italian scientists such as Bernardino Zendrini (1679-1747) [Jacob Hermann and the diffusion of the Leibnizian calculus in Italy (Leo S Olschki, Florence, 1997).',3)">3]:- .
    • One of the ways in which Hermann made the Leibnizian calculus known in Italy while he was in Padua was through frequent exchanges, by letter and in person, with Italian mathematicians, scientists, diplomats and scholars.
    • He lectured on mechanics in November 1708 and in December of that year he wrote to Grandi giving him a detailed explanation of how to use Leibniz's calculus to deduce the differential equation of the logarithm function.
    • Hermann, who also kept in frequent contact with Johann Bernoulli, published five articles on the inverse problems of central forces between 1710 and 1713 [Jacob Hermann and the diffusion of the Leibnizian calculus in Italy (Leo S Olschki, Florence, 1997).',3)">3]:- .
    • However, his work with other scientists was almost entirely devoted to understanding, developing and applying methods in the infinitesimal calculus.
    • However, his knowledge of calculus is evident in the way in which he deals with infinitesimals.

  10. Begle biography
    • Ed and I first taught Algebra to Army ASTP students, and the Calculus to Yale freshmen, spending hours discussing the problems of that teaching.
    • Out of this grew Begle's elementary calculus text, which was unique at the time in that it contained serious mathematics written not for colleagues but for the students themselves ..
    • This textbook was Introductory Calculus, with Analytic Geometry which Begle published in 1954.
    • This text differs from most others in this field in that it treats calculus as a branch of mathematics rather than as a mere adjunct of the physical and engineering sciences.
    • We start with a list of axioms and show how the theorems of the calculus are derived from these axioms.
    • Our aim in presenting calculus in this fashion is to give the student more of an understanding of the basic concepts of the subject than is usually done in an introductory course.
    • With new calculus books appearing each year, it is the rare instructor who has the time, the inclination, or the opportunity to examine all of them.
    • Equally rare is the book which makes a real contribution to understanding of calculus at the first year level.
    • In the reviewer's opinion Begle's book does make such a contribution and should be examined carefully by all instructors of beginning calculus.

  11. Tonelli biography
    • Over these years his publication record is truly remarkable with eight papers appearing in 1925, including important ones on the calculus of variations, and eleven papers in 1926.
    • His 1940 paper L'analisi funzionale nel calcolo delle variazioni is interesting in that it gives an insight into Tonelli's thinking on one of the topics for which he is most famed, namely the calculus of variations.
    • This paper is an exposition of the reasons that led the author to develop his approach to the calculus of variations and of the results obtained by himself and others.
    • He discusses at some length the classical calculus of variations before showing how the functional calculus was brought to bear on the calculus of variations by means of the concept of semi-continuity.
    • Some of the more important results of his work are mentioned in passing, and the essay closes with a few remarks on extensions of the calculus of variations to abstract spaces.
    • The second volume Calcolo delle variazioni, published in 1961, contains 27 articles, namely, all those dealing with the calculus of variations which appeared between 1911 and 1924.
    • The third volume Calcolo delle variazioni, published in 1962, contains all papers on the calculus of variations that appeared after 1926, including a posthumous paper of 1950.

  12. Church biography
    • He created the λ-calculus in the 1930's which today is an invaluable tool for computer scientists.
    • attempt[s] to show that Church's great discovery was lambda calculus and that his remaining contributions were mainly inspired afterthoughts in the sense that most of his contributions, as well as some of his pupils', derive from that initial achievement.
    • It is effectively a rewritten and polished version of lectures Church gave in Princeton in 1936 on the λ-calculus.
    • This, of course, is in contrast with the propositional calculus which has a decision procedure based on truth tables.
    • The subject matter is more or less classical, namely, the propositional algebra and the functional calculus of first order, to which is added a chapter summarizing without proofs certain features of functional calculi of higher order.
    • For the expert the chief interest in the tract is that it makes readily accessible careful detailed formulation and proofs of certain standard theorems, for example, the deduction theorem, the reduction to truth tables, the substitution rule for the functional calculus, Godel's completeness theorem, etc.
    • Chapters I and II are concerned with the propositional calculus, discussing tautologies and the decision problem, duality, consistency and completeness, and independence of the axioms and rules of inference.
    • The first order functional calculus is studied in Chapters III and IV, while Chapter V deals mainly with second order functional calculi.
    • Church bases his form of the theory of types on his λ-calculus.

  13. Bliss biography
    • However mathematics was his real love and, in 1898, he began his doctoral studies working on the calculus of variations.
    • His interest in the calculus of variations came through two sources, firstly from lecture notes of Weierstrass's 1879 course, of which he had a copy, and secondly from the inspiring lectures by Bolza which Bliss attended.
    • Two further papers by him on the calculus of variations appeared in 1904, both in the Transactions of the American Mathematical Society.
    • They were An existence theorem for a differential equation of the second order, with an application to the calculus of variations and Sufficient condition for a minimum with respect to one-sided variations.
    • There he very effectively applied methods from the calculus of variations to solve problems relating to correcting missile trajectories for the effects of wind, changes in air density, rotation of the Earth and other perturbations.
    • Bliss's main work was on the calculus of variations and he produced a major book, Lectures on the Calculus of Variations , on the topic in 1946.
    • This is a sound, thorough and up-to-date text on the single integral problems of the calculus of variations, based on courses given by the author at the University of Chicago.
    • The theory here presented marks the culmination of the modern phase of development of the calculus of variations, begun by Weierstrass and continued by Hilbert, Bolza and Bliss.

  14. Federer biography
    • It had the notable feature of lower semicontinuity, which is crucial in the calculus of variations.
    • In 1961, in collaboration with Bjarni Jonsson, Federer published the undergraduate text Analytic Geometry and Calculus.
    • This text presents beginning calculus using a set theoretic approach and strongly emphasizes the theory of the calculus.
    • Included in 'Analytic Geometry and Calculus' are the ordered pairs definition of relation and function; in equalities; absolute values; and the epsilon delta approach to limits and the wrapping function in trigonometry.
    • The book presents a semi-rigorous approach to the calculus but this does not mean that the intuitive basis has been neglected.
    • These advances have given us deeper perception of the analytic and topological foundations of geometry, and have provided new direction to the calculus of variations.
    • Its appearance in 1969 was timely, as it brought together earlier studies of geometric Hausdorff-type measures, work on rectifiability of sets and measures of general dimension, and the fast developing theory of geometric higher-dimensional calculus of variations.

  15. Bernoulli Jacob biography
    • Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and the two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus..
    • It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were the first to try to understand and apply Leibniz's theories.
    • We shall now examine some of the major contributions made by Jacob Bernoulli at an important stage in the development of mathematics following Leibniz's work on the calculus.
    • Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning.
    • Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability.
    • History Topics: The rise of calculus .

  16. Miranda biography
    • After graduating, Miranda became an assistant in infinitesimal calculus, supporting the chair of mathematics held by Picone.
    • They worked in the "Institute for Calculus" which had been founded by Picone in Naples in 1927.
    • With admirable single mindedness and great political skill Picone succeeded in getting appointed to a chair of mathematical analysis in Rome, left Naples University together with his brilliant assistant Miranda and, besides teaching the kind of mathematics acceptable to his illustrious colleagues, obtained a small grant from the newly established Italian National Research Council to start in a small apartment in Rome in a new section of town the high-sounding "Institute for the Applications of the Calculus." .
    • Miranda continued to work as part of the National Institute for the Applications of the Calculus (INAC) in Rome.
    • He continued in this role in 1936-37 but, during 1937, he won the competition for the chair of Algebra and Infinitesimal Calculus at the University of Genoa.
    • This was largely due to the efforts of Picone who made Miranda a consultant for the National Institute for the Applications of the Calculus (INAC) and argued that the Institute was vital for military research and therefore its staff (including consultants such as Miranda) must remain in post and not be drafted into the army.
    • He had indicated in a report in 1938 that the Institute was undertaking important theoretical work, and here he emphasised Miranda's work on the calculus of variations and eigenvalue problems.
    • Proceedings of international Meeting dedicated to the memory of Professor Carlo Miranda (Naples, 1983).',1)">1] and divides these contributions into the following areas: (a) Integral equations, series expansions, summation methods; (b) Harmonic mappings, potential theory, holomorphic functions; (c) Calculus of variations, differential forms, elliptic systems; (d) Numerical analysis; (e) Propagation problems; (f) Differential geometry in the large; (g) General theory for elliptic equations; and (h) Functional transformations.

  17. Arbogast biography
    • His entry was to bring him fame and an important place in the history of the development of the calculus.
    • Also in 1789 he submitted a major report on the differential and integral calculus to the Academie des Sciences in Paris which was never published.
    • I then foresaw the birth of the first inkling of the ideas and methods which, when developed and extended, formed the substance of the calculus of derivatives.
    • In 1794 he was appointed Professor of Calculus at the Ecole Centrale (soon to become the Ecole Polytechnique) but he taught at the Ecole Preparatoire.
    • As well as introducing discontinuous functions, as we discussed above, he conceived the calculus as operational symbols.
    • Arbogast was friendly with Francois Francais and together they worked on the calculus of derivations and the operational calculus.
    • He continued Arbogast's work on the operational calculus and presented a memoir on this topic, in particular applying the methods to study projectiles in a resistant medium, to the Academie des Sciences in 1804.

  18. Berkeley biography
    • Berkeley is best known in the world of mathematics for his attack on the logical foundation of the calculus as developed by Newton.
    • In his tract The analyst: or a discourse addressed to an infidel mathematician, published in 1734, he tried to argue that although the calculus led to true results its foundations were no more secure than those of religion.
    • He declared that the calculus involved a logical fallacy of a shift in the hypothesis.
    • Berkeley's criticisms were well founded and important in that they focused the attention of mathematicians on a logical clarification of the calculus.
    • By reviewing Berkeley's lifetime and the content of the "Analysts", we conclude that his critique was correct and that it impelled the improvement of the foundations of calculus objectively.
    • Many of the other references which we give also discuss Berkeley's attack on the calculus; see [George Berkeley 1685-1753 (Basel, 1989).',5)">5], [Berkeley\'s philosophy of mathematics (Chicago, 1993).',11)">11], [Etudes sur l\'histoire du calcul infinitesimal, Rev.
    • De Moivre, Taylor, Maclaurin, Lagrange, Jacob Bernoulli and Johann Bernoulli all made attempts to bring the rigorous arguments of the Greeks into the calculus.
    • History Topics: The rise of calculus .

  19. Ferrand biography
    • Returning to Ferrand's career, she was promoted to full professor at Caen in 1948 and, later in the same year, she was appointed to the chair of calculus and higher geometry at the University of Lille, filling the chair left vacant when Bertrand Gambier (1879-1954) retired.
    • She published her textbook on advanced calculus Cours d'analyse (1968-70) in three volumes.
    • Volume I covered multivariable differential calculus, with a little differential geometry.
    • Volume III covered multivariable integral calculus, further topics in functions of a complex variable, Fourier series and ordinary differential equations.
    • The second volume, published in 1974, covered analysis (multivariable differential calculus and one-variable integral calculus) while the third volume, published in the following year covered geometry with applications to mechanics.
    • The fourth and final volume was published in 1974 and covered ordinary differential equations, multivariable integral calculus and holomorphic functions.

  20. Doob biography
    • His favourite course during his first year had been calculus and its applications so, since he had always enjoyed mathematics, there was an obvious direction in which to take his studies.
    • He was taught calculus by Osgood and this led to his taking the third year calculus course at the same time.
    • [Osgood] taught my sophomore calculus course, using his own textbook.
    • After a few weeks of his class I appealed to my adviser Marshall Stone to get me into a calculus section with a more lively teacher.
    • Of course Stone did not waste sympathy on a student who complained that a teacher got on his nerves, and he advised me that if I found sophomore calculus too slow I should take junior calculus at the same time! ..

  21. Mayer Adolph biography
    • It was Richelot who advised Mayer to undertake research on the calculus of variation, and he followed this advice working on this topic for the rest of his life.
    • He began teaching at Leipzig University in 1867 giving two courses in that year, Analytic Mechanics and the Calculus of variations, as well as teaching a Mathematical Exercises class.
    • In the following years he taught Differential and Integral Calculus, Theory of Definite Integrals, Some chapters from mechanics and the calculus of variations, Higher Algebra, Differential Equation of Mechanics and the Calculus of Variations, Analytic Geometry, and many more courses of a similar type.
    • Mayer worked on differential equations, the calculus of variations and mechanics.
    • In the winter semester of session 1907-08 he had to cancel his lecture course on the Calculus of Variations because of a stabbing pain in his chest, especially at night.

  22. Carre biography
    • This stimulated Carre's interest in mathematics and, from this time on, he began working hard on writing a calculus text.
    • This book was, in particular, devoted to applications of the integral calculus.
    • The greatest achievement we have made until now is probably the Differential Calculus, which consists as one knows in descending to quantities with infinitely small differences to discover their nature, and the nature of those variables of which they are differences.
    • Early authors of calculus texts discussed the problem and computed the value of the centre of oscillation for several solids.
    • In Une methode pour Ia mesure des surfaces, the first French textbook on the integral calculus, Carre made a mistake in calculating the integral for the moment of inertia of a cone suspended from its vertex, a mistake that led to an incorrect expression for the centre of oscillation of the cone.
    • Lenore Feigenbaum explains that the story of Carre's mistake and the subsequent propagation of his error in eighteenth-century calculus textbooks [From ancient omens to statistical mechanics, Acta Hist.
    • is instructive in several regards: first, in showing how some of the methods of the calculus were interpreted and absorbed during the early 18th century; second, in shedding light on the nature of the textbook industry of the time; and finally, in providing us with a modicum of historical sympathy when we find our own students making the same kind of mistakes .

  23. Scholz biography
    • This is a treatise on the classical (two-valued) propositional algebra and the predicate calculus of first order based on it.
    • The author develops both the algebra and the calculus from this point of view; the deductive, axiomatic standpoint is not considered at all for the algebra, and is relegated to a secondary role for the calculus.
    • For the algebra this amounts to a treatment from the matrix point of view; but for the calculus it entails that the line between constructive and nonconstructive results is not as sharply drawn as one would expect in a modern work.
    • This monograph treats the propositional calculus, the predicate calculus of first order with and without identity ..
    • There is no clear indication which branches of mathematics satisfy this condition; it seems that abstract algebras and other axiomatic theories do, intuitive geometry and numerical arithmetic, called a 'kind' of calculus ..

  24. Bukreev biography
    • By the end of the 1890s Bukreev's research interests had moved somewhat and he began to undertake research into differential geometry; in 1900 he published A Course on Applications of Differential and Integral Calculus to Geometry.
    • After 1900 he became interested in the theory of series, publishing papers such as Notes on the theory of series and he also worked on the Calculus of Variations.
    • He taught courses on analysis, differential and integral calculus and their applications to geometry, the theory of integration of differential equations, the theory of series, algebra, and other topics.
    • In 1934, he published An Introduction to the Calculus of Variations.
    • The Calculus of Variations is central to the physical and mathematical sciences.
    • One who is studying the Calculus of Variations not only repeats and learns infinitesimal analysis, but also understands that this analysis is a powerful tool to address the many issues that are of a purely practical nature.
    • Boris Yakovlevic worked on the theory of functions of a complex variable, on mathematical analysis, on algebra, on the calculus of variations, and on differential geometry.

  25. Genocchi biography
    • In 1862 he moved chairs again, but remaining in Turin, to the Introduction to the Calculus and the following year to Infinitesimal Calculus.
    • The main research topics which Genocchi worked on were number theory, series and the integral calculus.
    • In 1884 Differential Calculus and Fundamentals of Integral Calculus was published under Genocchi's name.
    • He was many times urged to publish his calculus course, but he never did.
    • I used notes made by his students at his lessons, comparing them point by point with all the principal calculus texts, as well as with original memoirs..

  26. Van der Pol biography
    • even in mathematics, his papers covered number theory, special functions, operational calculus and nonlinear differential equations.
    • ',2)">2] lists van der Pol's main contributions under the headings: propagation of radio waves; non-linear circuits: relaxation oscillations; transient phenomena, and operational calculus.
    • Let us look first at the last of these topics on which Bremmer and van der Pol collaborated in writing the classic text Operational Calculus: Based on the Two-Sided Laplace Integral.
    • This book is intended as a treatise on the application of the operational calculus in its modern form to mathematics, physics, and engineering.
    • They seem to feel that on the one hand modern applications of the operational calculus require theorems under rather general conditions, and on the other hand the readers they have in mind are not interested in, or not able to follow, proofs under such general conditions.
    • This 1950 book was not the first joint work of van der Pol and Bremmer on the operational calculus, for example they published two papers with the title Modern operational calculus based on the two-sided Laplace integral in 1948.

  27. Rolle biography
    • Some basic principles of the calculus and the theory of equations can definitely be traced to their origin as incidental propositions of the method.
    • Rolle's theorem, an important proposition of the calculus, also owes its origin to the method.
    • It might be assumed from what we have just written about Rolle's work that he was developing the infinitesimal calculus.
    • This would be a serious error, for Rolle described the infinitesimal calculus as a collection of ingenious fallacies and he believed that the methods could lead to errors.
    • The burden of his critique rested on two arguments, one stressing the inadequacy and the lack of logical rigour of the fundamental concepts and principles of the new calculus, the other pretending to show (with the aid of cleverly selected examples) that the new calculus led to error, insofar as it did not yield the same results obtained in the classical, algebraically inspired methods of Fermat and, more especially, Hudde.

  28. Cesari biography
    • show applications of these concepts in the transformation of double integrals, calculus of variations, and Lebesgue area theory.
    • His course on calculus of variations and seminar fostered my interest in that area.
    • These include concepts of bounded variation, absolute continuity, and generalized Jacobians for continuous mappings T, and the principal theorems relating to these concepts; tangential properties of general continuous surfaces; the theory of a general Weierstrass-type double integral; extremely general forms of the Gauss-Green and Stokes theorems; theorems on homotopy and retraction for general continuous surfaces; and far-reaching results in calculus of variations for double integrals.
    • One of Cesari's abiding interests was the study of problems in the calculus of variations, and he also did a great deal of work in optimal control.
    • Aside from Young's book [L C Young, Lectures on the calculus of variations and optimal control theory (1969)], Cesari's text is perhaps the only one attempting to bridge the gap between the calculus of variations and optimal control theory.

  29. Fox Ralph biography
    • He also mentioned in his talk to the International Congress of Mathematicians his ideas on how to study group presentations using the "free differential calculus." Five papers on this topic appeared over the following ten years: Free differential calculus.
    • Derivation in the free group ring (1953); Free differential calculus.
    • The isomorphism problem of groups (1954); Free differential calculus.
    • Subgroups (1956); (with Roger Lyndon and Kuo-Tsai Chen) Free differential calculus.
    • The quotient groups of the lower central series (1958); and Free differential calculus.

  30. Ito biography
    • At that time, few mathematicians regarded probability theory as an authentic mathematical field, in the same strict sense that they regarded differential and integral calculus.
    • With clear definition of real numbers formulated at the end of the19th century, differential and integral calculus had developed into an authentic mathematical system.
    • Calculation using the "Ito calculus" is common not only to scientists in physics, population genetics, stochastic control theory, and other natural sciences, but also to mathematical finance in economics.
    • In fact, experts in financial affairs refer to Ito calculus as "Ito's formula." Dr.
    • A recent monograph entitled Ito's Stochastic Calculus and Probability Theory (1996), dedicated to Ito on the occasion of his eightieth birthday, contains papers which deal with recent developments of Ito's ideas:- .
    • Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater.

  31. Bolza biography
    • He attended Weierstrass's 1879 lecture course on the calculus of variations which was to have a lasting effect on the direction that Bolza's mathematical interests would take.
    • However, he worked on the calculus of variations from 1901.
    • Papers which appeared in the Transactions of the American Mathematical Society over the next few years were: New proof of a theorem of Osgood's in the calculus of variations (1901); Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so-called isoperimetric problems (1902); Weierstrass' theorem and Kneser's theorem on transversals for the most general case of an extremum of a simple definite integral (1906); and Existence proof for a field of extremals tangent to a given curve (1907).
    • His text Lectures on the Calculus of Variations published by the University of Chicago Press in 1904, became a classic in its field and was republished in 1961.
    • Immediately after his return to Germany Bolza continued teaching and research, in particular on function theory, integral equations and the calculus of variations.
    • Oskar Bolza's Calculus of Variations .

  32. Forder biography
    • He reported that it was at present possible for a student to get First Class Honours and be completely unaware of the existence of the whole of modern mathematics, and by modern mathematics he meant not the mathematics of this century but the last! Such was the inertia of the University of New Zealand system, coupled with the political need to cater for large numbers of part-time and exempt students, that it took him until 1936 to get the theory of complex variables into the course, and until 1938 to get calculus into the syllabus for Pure Mathematics I.
    • These are: The Foundations of Euclidean Geometry (1927), A School Geometry (1930), Higher Course Geometry (1931), The Calculus of Extension (1941), Geometry (1950), and Coordinates in Geometry (1953).
    • The Calculus of Extension (1941) is in many ways the most interesting of all Forder's books.
    • The book is the first modern textbook of Grassmann's calculus of extension.
    • Forder's 'The calculus of extension' not only presents a wealth of specific applications of the subject to geometry, that are either new or not readily obtainable elsewhere; but in addition furnishes an admirable and fresh exposition of the 'Ausdehnungslehre'.
    • Books that contain a wealth of material are never easy to read through, and it is my conviction that The calculus of extension provides the best exposition of the fundamental processes of the 'Ausdehnungslehre' and the most inclusive treatment of the geometrical applications available at present.

  33. Foulis biography
    • Wayman once confided to me that he had made an observation when Dave was one of his Advanced Calculus students in Miami; he said, "Dave is someone who does not know how to make a bad proof".
    • These are Fundamental Concepts of Mathematics (1962), (with Mustafa A Munem) Calculus (1978), (with Mustafa A Munem) Calculus: With Analytic Geometry (1984), (with Mustafa A Munem) After Calculus: Algebra (1988), (with Mustafa A Munem) After Calculus: Analysis (1989), (with Mustafa A Munem) Algebra and Trigonometry with Applications (1991), and (with Mustafa A Munem) College Algebra with Applications (1991).
    • Hyla wrote the solutions manual to Dave's first Calculus book, providing solutions to approximately 5000 problems, an impressive feat before the availability of graphing calculators.

  34. Maclaurin biography
    • In 1742 Maclaurin published his 2 volume Treatise of fluxions, the first systematic exposition of Newton's methods written as a reply to Berkeley's attack on the calculus for its lack of rigorous foundations.
    • Grabiner gives five areas of influence of Maclaurin's treatise: his treatment of the fundamental theorem of the calculus; his work on maxima and minima; the attraction of ellipsoids; elliptic integrals; and the Euler-Maclaurin summation formula.
    • Maclaurin appealed to the geometrical methods of the ancient Greeks and to Archimedes' method of exhaustion in attempting to put Newton's calculus on a rigorous footing.
    • The Treatise of fluxions is not simply a work designed to put the calculus on a rigorous basis, for Maclaurin gave many applications of calculus in the work.
    • History Topics: The rise of calculus .

  35. Gregory biography
    • teaching profoundly influenced Gregory, particularly in providing the twin keys to the calculus, the method of tangents (differentiation) and of quadratures (integration).
    • the first attempt to write a systematic text-book on what we should call the calculus.
    • By the time that Gregory published this work Newton had formed his ideas of the calculus so probably had not been influenced by Gregory.
    • Essentially Newton and Gregory were working out the basic ideas of the calculus at the same time, as, of course, were other mathematicians.
    • For his reluctance to publish his "several universal methods in geometry and analysis" when he heard through Collins of Newton's own advances in calculus and infinite series, he postumously paid a heavy price ..
    • H W Turnbull's Scottish Contribution to the Calculus .

  36. Manfredi Gabriele biography
    • However, he became interested in mathematics when his brother Eustachio was learning differential calculus from Giovanni Domenico Guglielmini (1655-1710).
    • He read mathematical works by Gottfried Leibniz, Johann Bernoulli and Jacob Bernoulli as well as studying Guillaume de l'Hopital's differential calculus text Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (1696).
    • Through contacts with members of this group, Manfredi was put in touch with Pierre Varignon who was working in Paris on applications of the differential and integral calculus.
    • This correspondence with Grandi was particularly important since it marks the beginning of innovative research on the calculus being undertaken in Italy.
    • While Manfredi was in Rome, Grandi published his important work on the differential calculus Quadratura circoli et hyperbolae per infinitas hyperbolas et parabolas quadrabiles geometrice exhibita (1703).
    • This challenge was issued as part of the ongoing argument over the claims made for the superiority of Newton's or Leibniz's approach to the calculus.

  37. Pascal Ernesto biography
    • It was in that year that he entered the competition for the chair of infinitesimal calculus at the University of Pavia.
    • The German translation of the "Calcolo delle variazioni" published by Ernesto Pascal in 1897 gives to American mathematicians in convenient form the best book on the calculus of variations that has, to our knowledge, appeared up to the present time.
    • A valuable feature of the work will certainly be found to be the very excellent and apparently complete bibliography given in connection with brief accounts of the development of the calculus of variations.
    • That such an end is in the calculus of variations especially difficult to attain appears from the fact that the proofs are not always precise and that the author prefers often to tell us that the work given is not rigorous rather than to attempt to make it so.
    • It is certain that through the profound changes which the critical spirit has made in the foundations of the calculus, even a course intended for those for whom mathematics is a means rather than an aim, cannot but use the new results which have been reached .
    • it would therefore exhibit a shortsighted view and little esteem for the ability of the future engineer, to believe that it would be sufficient for them, at least if they can, to learn to operate the calculus in about the way in which a workman knows how to operate a machine made by others, and of which he does not know the inner connections.

  38. Ohm Martin biography
    • His treatise The theory of maxima and minima (1825) may be regarded as a successor to Dirksen's treatise on the calculus of variations published in 1823.
    • Todhunter writes [A History of the Calculus of Variations (Macmillan and Co, Cambridge, 1861).',2)">2]:- .
    • The first 84 pages contain an Introduction, in which the author collects the propositions in algebra and the differential and integral calculus, which are especially used in the ordinary theory of maxima and minima, and in the calculus of variations.
    • The portion of the book extending over pages 87-127 is called Calculus of Variations.
    • The pages 131-208 contain the theory of maxima and minima, which is given in ordinary treatises on the Differential Calculus.

  39. Strong biography
    • It was natural that American mathematicians of this period should be influenced by British rather than Continental mathematics and this largely disadvantaged the Americans since English mathematics was still too strongly influenced by Newton's approach to the calculus.
    • Strong, however, was more influenced by the approach of the Scottish mathematicians who advocated the Continental approach to the calculus.
    • He advocated the Continental approach to the calculus in his article Fluxions which appeared in Encyclopaedia Britannica in 1810 and in his article Fluxions for the Edinburgh Encyclopaedia which was published in 1815 he used Leibniz's differential notation.
    • He was therefore the first to write an English treatise on the calculus using differential notation and it was around this time that Strong began to build up a library of Continental texts.
    • The use of Leibniz's approach, as developed by Laplace and Lagrange, was used by Strong in his papers from about 1825 onwards so he participated actively in the introduction of the continental approach to differential and integral calculus into America.
    • The were A treatise on elementary and high algebra (1959) and A treatise on the differential and integral calculus (1869), the second appearing in print shortly after his death.

  40. Heaviside biography
    • His operational calculus, developed between 1880 and 1887, caused much controversy however.
    • He introduced his operational calculus to enable him to solve the ordinary differential equations which came out of the theory of electrical circuits.
    • Although highly successful in obtaining the answer, the correctness of Heaviside's calculus was not proved until Bromwich's work.
    • Burnside rejected one of Heaviside's papers on the operational calculus, which he had submitted to the Proceedings of the Royal Society , on the grounds that it:- .
    • Whittaker rated Heaviside's operational calculus as one of the three most important discoveries of the late 19th Century.

  41. Hudde biography
    • He gives his rule as follows [The Origins of the Infinitesimal Calculus (Dover, 2003).',3)">3]:- .
    • He gives the following instructions [The Origins of the Infinitesimal Calculus (Dover, 2003).',3)">3]:- .
    • A fuller account of Hudde's rule was given in a letter he wrote dated 21 November 1659 which was not published at the time but, was published during the Newton-Leibniz controversy on who deserved priority for discovering the calculus, Hudde's letter was published as part of the evidence.
    • The manuscripts must have had an important influence on Leibniz's introduction of the calculus.
    • History Topics: The rise of the calculus .

  42. Simpson biography
    • This was a high-quality textbook devoted to the calculus of fluxions, the Newtonian version of the infinitesimal calculus.
    • The topic was advanced -- it was no trivial exercise to write such a book in the 1730s, when the calculus was mastered by only a few mathematicians in Europe.
    • The method of approximating the roots did not use the differential calculus.
    • His two volume work The Doctrine and Application of Fluxions in 1750 contains work of Cotes and is considered by many to be the best work on Newton's version of the calculus published in the 18th century.

  43. Babbage biography
    • It is a little difficult to understand how Woodhouse's Principles of Analytic Calculation was such an excellent book from which to learn the methods of Leibniz, yet Woodhouse was teaching Newton's calculus at Cambridge without any reference to Leibniz's methods.
    • Babbage tried to buy Lacroix's book on the differential and integral calculus but this did not prove easy in this period of war with Napoleon.
    • I then drew up the sketch of a society to be instituted for translating the small work of Lacroix on the Differential and Integral Calculus.
    • They gave a history of the calculus, and of the Newton, Leibniz controversy they wrote:- .
    • These are the English translation of Lacroix's Sur le calcul differentiel et integral published in 1816 and a book of examples on the calculus which they published in 1820.

  44. Schiffer biography
    • During these years he did excellent research on the Calculus of Variations.
    • The 'Calculus of Variations' - formulating and solving problems in terms of a quantity to be maximized or minimized and analysing the properties of such extremal solutions - had already been and remains an established, highly developed, and highly effective area of mathematical analysis and its applications.
    • Their joint publications began in 1949 and included papers such as The coefficient regions of schlicht functions (1949) (which also had A C Schaeffer as a co-author), The coefficient problem for multiply-connected domains (1950), A variational calculus for Riemann surfaces (1951) and Some remarks on variational methods applicable to multiply connected domains (1952).
    • Our ultimate aim is to help develop this sixth sense in as wide a readership as possible; we have restricted technical mathematics to minimum, just algebra and some calculus.
    • Students and teachers at both high school and college level will be pleased, perhaps surprised, at how readily they may follow this development with the aid of only a modest background in differential calculus.

  45. Goldstine biography
    • Appointed an assistant to Gilbert Bliss at Chicago in 1936 working with him on the calculus of variations for three years before moving to the University of Michigan in 1939 as an Instructor.
    • The first publication of Goldstine's was Minimum problems in the functional calculus which was based on results in his thesis.
    • In 1942 he published two papers, The modular space determined by a positive function (with R W Barnard) and The calculus of variations in abstract spaces.
    • A history of the calculus of variations from the 17th through the 19th century was published in 1980.
    • The exposition is good and the author has clarified the argument of, and has indicated the connection between, many of the fundamental works to which he refers with the result that the book gives a better understanding of the calculus of variations than do many modern texts.

  46. Walsh biography
    • He became convinced that the differential calculus was a delusion; that Sir Isaac Newton was a shallow sciolist, if not an impostor; and that the universities and academics of Europe were engaged in the interested support of a system of error.
    • The printed tracts and papers are for the most part occupied with the announcement of some discovery which was designed to supersede the differential calculus in its application to problems respecting curves.
    • The method in question consisted in transferring the origin of coordinates to a point upon the curve, developing the ordinate y in terms of the abscissa x, and making use of the coefficients of the expansion just in the same way as the ordinary principles of the differential calculus would direct us to do.
    • Memoir on the Calculus of Variations, showing its total unreality.
    • Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of July 7th, 1844; exactly twenty-two years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.

  47. Davies biography
    • At Swansea, Davies had worked under Dienes who advised him to work on the absolute differential calculus.
    • The leading expert on the absolute differential calculus was Tullio Levi-Civita who lectured in Rome, so in August 1926, following Dienes' advice, Davies travelled to Rome.
    • His steady stream of publications is testimonial to his authority in the fields of Riemannian geometry and the calculus of variations.
    • His steady stream of publications in differential geometry and the calculus of variations attests to his authority in this field.
    • In papers such as On the invariant theory of contact transformations (1953) Davies studies invariant theory of contact transformations by using tensor calculus.

  48. Fox biography
    • He wrote only one book An introduction to the calculus of variations (1950, 2nd edition 1963, reprinted 1987).
    • On the flyleaf of this book it is claimed that "In this work the Calculus of Variations is developed both for its intrinsic interest and because of its wide and powerful applications to modern Mathematical Physics." It appears, however, that the author's main preoccupation is with the applications and that his interest in the Calculus of Variations derives from its applicability to physical problems rather than its intrinsic discipline.
    • The book provides an excellent introduction to the subject for those primarily concerned with having available techniques and rules of procedure for tackling concrete problems, and a most exhaustive treatment of the classical problems of the Calculus of Variations (e.g.
    • However, for the student of pure mathematics, it does not seem to replace Hadamard's 'Lecons sur le Calcul des Variations' or Bliss's 'Calculus of Variations' (whose mathematical content broadly coincides with its own).

  49. Taylor biography
    • Also in 1712 Taylor was appointed to the committee set up to adjudicate on whether the claim of Newton or of Leibniz to have invented the calculus was correct.
    • Returning to the paper, it is a mechanics paper which rests heavily on Newton's approach to the differential calculus.
    • Taylor added to mathematics a new branch now called the "calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion.
    • The importance of Taylor's Theorem remained unrecognised until 1772 when Lagrange proclaimed it the basic principle of the differential calculus.
    • History Topics: The rise of calculus .

  50. Murnaghan biography
    • this makes it a pleasure and a duty of the mathematician to adapt his powerful methods to the needs of the physicist and especially to explain these methods in a manner intelligible to anyone well-grounded in algebra and calculus.
    • It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus.
    • After he retired from his position in Brazil and returned to the United States he published The calculus of variations and The Laplace transformation , both in 1962.
    • 103A (1), (2003), 101-112.',2)">2] Lewis gives a delightful quote from Russell Baker (who went on to become a prize-winning journalist) who attended Murnaghan's calculus lectures in 1942:- .

  51. Fontaine des Bertins biography
    • His papers are rather confused, and ignorant of the work of others, but do contain some very original ideas in the calculus of variations, differential equations and the theory of equations.
    • The methods which he developed to solve these problems led to the calculus of variations.
    • 38 (3) (1981), 251-290.',5)">5] shows how Fontaine progressed from a calculus of variations to a calculus of several variables.
    • 11 (1) (1984), 22-38.',3)">3] Greenberg considers Fontaine's work and that of his contemporaries who are usually given credit for laying the foundations for the calculus of several variables.

  52. Cauchy biography
    • His text Cours d'analyse in 1821 was designed for students at Ecole Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible.
    • He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitesimal while in 1829 in Lecons sur le Calcul Differentiel he defined for the first time a complex function of a complex variable.
    • his two theories of elasticity and his investigations on the theory of light, research which required that he develop whole new mathematical techniques such as Fourier transforms, diagonalisation of matrices, and the calculus of residues.
    • Cauchy's Calculus .
    • History Topics: The rise of calculus .

  53. Reinhardt biography
    • In 1934 he published the book Methodische Einfuhrung in die Hohere Mathematik which contained a novel approach to teaching calculus.
    • On account of this belief Professor Reinhardt has presented us with a book in which integral calculus is developed first, completely independent of differential calculus; in fact the latter appears simply as the inverse operation of the former.
    • Although it represents an introduction to the calculus, it cannot be compared with our ordinary calculus texts; the American youth would find difficulty in studying it before his junior or senior year as it contains many topics which appear here in courses on the theory of functions of a real variable.

  54. Kluvanek biography
    • The book, covered Differential and Integral calculus, Analytic geometry, Differential equations and Complex variable [S Tkacik, J Guncaga, P Valihora and M Gerec (eds.), Igor Kluvanek: Prispevky zo seminara venovaneho nedozitym 75.
    • To illustrate Kluvanek's views on teaching mathematics, let us quote from his paper What is wrong with calculus? written in 1988:- .
    • Since its invention, the foundations of the differential and integral calculus were clarified and also the techniques were improved.
    • And that gives us a key for the understanding of the deficiencies in the teaching of 'calculus'.
    • The processes of clarification, simplification and improvement stopped at a certain stage and/or were confined to some aspects of the differential and integral calculus.

  55. Kochin biography
    • He wrote the textbook 'Vector Calculus and the Principles of Tensor Calculus', many editions of which have been published, and finally, together with Ilia Afanasevich Kibel (1904-1970) and Nikolai Vladimirovich Roze, wrote the outstanding two-volume course 'Theoretical Fluid Mechanics', which has been, and continues to be, used to teach many generations of Russian mechanics.
    • We note that the first edition of Vector Calculus and the Principles of Tensor Calculus was titled Vector Calculus and published in 1927.

  56. Osgood biography
    • Osgood's main work was on the convergence of sequences of continuous functions, solutions of differential equations, the calculus of variations and space filling curves.
    • Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).
    • Other classic texts included Introduction to Infinite Series (1897), A First Course in the Differential and Integral Calculus (1909), Topics in the theory of functions of several complex variables published by the American Mathematical Society in 1914, Plane and Solid Analytic Geometry (with W C Graustein, 1921), Advanced Calculus (1925), and Mechanics (1937).

  57. Banachiewicz biography
    • In 1925 Banachiewicz introduced the Krakowian calculus which involved a column-by-column multiplication of matrices.
    • To simplify matrix computations, Banachiewicz introduced the Krakowian calculus, which greatly simplified and improved computations on calculating machines, which was his principal goal.
    • The Krakowian calculus was used by Banachiewicz in many computational techniques that he developed for astronomical purposes, mainly aimed at orbit calculations.
    • One of Banachiewicz's great achievements in theoretical astronomy was the simplification [using the Krakowian calculus] of Olbers' method of determining parabolic orbits.

  58. Korkin biography
    • Korkin also discussed problems on the calculus of variation in his essay.
    • At this time he left the First Cadet School and taught trigonometry, analytic geometry and integral calculus at the University.
    • In addition to teaching at the University, he also began teaching calculus at the Nikolaevskaya Naval Academy.
    • He continued to lecture at the university until the year of his death 1908, and he continued to teach calculus at the Naval Academy until 1900.

  59. Bordoni biography
    • In the following year he was appointed to the Chair of Higher Calculus, Geodesy and Hydrometrics.
    • We note that Higher Calculus (actually called Sublime Calculus) essentially meant Mathematical Analysis.
    • At Pavia, Bordoni joined the other Chairs in Mathematics, namely Marchesi in Architecture, Lotteri in Preliminary Calculus, and Gratognini in Mechanics, Statics, Hydrodynamics, Hydraulics (Applied Mathematics).

  60. MacColl biography
    • For example The Calculus of Equivalent Statements was a series of eight papers published in the Proceedings of the London Mathematical Society between 1877 and 1898.
    • I discovered my Calculus of Limits, or as I then called it, my 'Calculus of Equivalent Statements and Integration Limits', I regarded it at first as a purely mathematical system restricted to purely mathematical questions ..
    • When I found that my method could be applied to purely logical questions unconnected with the integral calculus or with probability, I sent a second and a third paper to the '[London] Mathematical Society', which were both accepted, and also a paper to 'Mind' (published January 1880).

  61. Boole biography
    • The first advanced mathematics book he read was Lacroix's Differential and integral calculus.
    • Boole began to give Mary informal mathematics lessons on the differential calculus.
    • Boole also worked on differential equations, the influential Treatise on Differential Equations appeared in 1859, the calculus of finite differences, Treatise on the Calculus of Finite Differences (1860), and general methods in probability.

  62. Hesse biography
    • He examines Jacobi's 1837 result on the calculus of variations and Hesse's reformulation in 1857:- .
    • Jacobi's result does not have much visibility in current texts on the calculus of variations and even less so Hesse's.
    • Indeed, Hesse shifts from an algorithmic approach to the calculus of variations to an emphasis on its analytical character.
    • This was the line of research adopted in the method of fields of extremals, which characterizes progress in the calculus of variations in the late nineteenth century.

  63. Stampacchia biography
    • For three years he produced outstanding examination results in a wide range of courses such as Tutorial Sessions in Analysis and in Geometry, Calculus of Variations, Theory of Functions, and Ordinary Differential Equations.
    • From the time Stampacchia took up his appointment in Naples, his research output was impressive consisting mainly of papers on differential equations and the calculus of variations.
    • In 1948-49 he was awarded a National research Council scholarship to enable him to undertake research on the calculus of variations and functional analytic methods.
    • The years that Stampacchia spent in Pisa and Naples characterize the formation of his personality as an analyst: he was a passionate specialist in calculus of variations and in the theory of partial differential equations, a practitioner and an inspirer of research works of considerable depth and originality of thought.

  64. Schroder biography
    • He was the first to use the term 'propositional calculus' and seems to be the first to use the term 'mathematical logic'.
    • It offers the first exposition of abstract lattice theory, the first exposition of Dedekind's theory of chains after Dedekind, the most comprehensive development of the cakculau of relations, and a treatment of the foundations of mathematics in relation calculus that Lowenheim in 1940 still thought was as reasonable as set theory.
    • Schroder developed Peirce's relative calculus much further and much more systematically than did Peirce.
    • He understood that there are notions such as countability that are beyond relative calculus (and also beyond first-order predicate logic).

  65. Woodhouse biography
    • Woodhouse was interested in the theoretical foundations of the calculus, the importance of notation, the nature of imaginary numbers and other similar topics.
    • He wrote an three papers in the Philosophical Transactions of the Royal Society in 1801, 1802 and an important book Principles of Analytic Calculation in 1803 which attempted to put the calculus on a rigorous algebraic foundation using a formal series expansions method similar to that developed by Lagrange [Dictionary of National Biography (Oxford, 2004).',2)">2]:- .
    • He demanded that analysis in general and the calculus specifically be placed upon a purely algebraic footing free of geometric and physical encumbrances such as limits or infinitesimals.
    • Woodhouse's other works include A Treatise on Plane and Spherical Trigonometry (1809), A Treatise on Isoperimetrical Problems and the Calculus of Variations (1810), Treatise on Astronomy (1812) and a work on gravitation published in 1818.

  66. Levi-Civita biography
    • He wrote a dissertation, which was supervised by Ricci-Curbastro, on absolute invariants but this also marks the beginning of his use of the tensor calculus [Historia Math.
    • He is best known, however, for his work on the absolute differential calculus and with its applications to the theory of relativity.
    • In 1886 he published a famous paper in which he developed the calculus of tensors, following on the work of Christoffel, including covariant differentiation.
    • Levi-Civita: Absolute Differential Calculus .

  67. Snyder biography
    • suggested that the department of mathematics introduce a regular course in descriptive geometry as an alternative for those Arts students who ought to have some mathematics, but who found the calculus too difficult or too unattractive to be studied with profit.
    • He published (with James McMahon) Treatise on Differential Calculus (1898), (with John I Hutchinson) Differential and Integral Calculus (1902), (with John H Tanner) Plane and Solid Geometry (1911), (with John I Hutchinson) Elementary Textbook on the Calculus (1912), and (with Charles H Sisam) Analytic Geometry of Space (1914).

  68. Lang biography
    • Your 'Calculus for undergraduates' went through many editions in the seventies and eighties, and your 'Algebra' textbook is a standard reference in the field.
    • Perhaps Lang's most used undergraduate text is A first course in calculus which he first published in 1964.
    • Foreword to Lang's First Course in calculus .
    • Serge Lang - A first course in calculus .

  69. Curry biography
    • After giving a very clear exposition of the fundamentals of combinatory logic, showing its close relationship to the λ-calculus developed by Church, Curry went on to describe his recent work.
    • He presented in his address: a critique of non-formal theories; the notion of a formal system (illustrated by Dickson's postulates for a group); the notion of a calculus; discussion of a metatheory; the definition of mathematics; and acceptability of a formal system, discussing criticisms from intuitionists and formalists.
    • He published The Heaviside operational calculus in 1943.
    • The authors of [To H B Curry : essays on combinatory logic, lambda calculus and formalism (London-New York, 1980), vii-xi.',3)">3] make some nice comments about Curry and his wife:- .

  70. Lax Peter biography
    • In 1972 Lax, together with his wife Annelli Lax and Samuel Burstein, wrote Calculus with applications and computing.
    • The calculus material in this book is fairly standard (except that it is oriented towards applications) but the computing flavour is unorthodox, successful and highly recommended.
    • [Anneli and my] calculus book was enormously unsuccessful, in spite of containing many excellent ideas.
    • A calculus book has to be fine-tuned, and I didn't have the patience for it.

  71. McClintock biography
    • Among the honours which McClintock received, many were for his mathematical work on the Calculus of Enlargement [American National Biography 14 (Oxford, 1999), 876-877.',7)">7]:- .
    • in his "An Essay on the Calculus of Enlargement" (1879) ..
    • he sought to develop a unified theory of the calculus of finite differences and the differential calculus.

  72. Douglas biography
    • In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space (1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28).
    • In particular the award was for three papers all published in 1939: Green's function and the problem of Plateau and The most general form of the problem of Plateau published in the American Journal of Mathematics and Solution of the inverse problem of the calculus of variations published in the Proceedings of the National Academy of Sciences.
    • The third paper does not give the compete proof for the solution of the inverse problem of the calculus of variations but is an announcement of the result.
    • Another five papers by Douglas appeared in 1940: Theorems in the inverse problem of the calculus of variations; Geometry of polygons in the complex plane; On linear polygon transformations; A converse theorem concerning the diametral locus of an algebraic curve and A new special form of the linear element of a surface.

  73. Lupas biography
    • At this time it consisted of two departments, the Department of Approximation and Calculus, and the Computer Department.
    • Professor at the Department of Mathematics of the University "Lucian Blaga" in Sibiu, Romania, he was a specialist in Approximation Theory, Classical Analysis, Inequalities, Convexity, Numerical Analysis, Special Functions, Finite Operatorial Calculus (Umbral Calculus) and q-Calculus.

  74. Schouten biography
    • He became not only one of the founders of the "Ricci calculus" but also an efficient organiser (he was a founder of the Mathematical Centre at Amsterdam in 1946) and an astute investor.
    • He published a monograph On the Determination of the Principle Laws of Statistical Astronomy in 1918 and his classic work on the Ricci calculus Der Ricci-Kalkul : Eine Einfuhrung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie in 1924.
    • We mentioned above Schouten's Ricci-calculus.
    • Since the publication of the author's 'Der Ricci-Kalkul' [1924] the subject of tensor calculus as applied to differential geometry has grown very considerably, and the present edition is more than a mere translation of the first.

  75. Peano biography
    • In his second year he was taught calculus by Angelo Genocchi and descriptive geometry by Giuseppe Bruno.
    • This book Course in Infinitesimal Calculus although based on Genocchi's lectures was edited by Peano and indeed it has much in it written by Peano himself.
    • In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic.
    • When the calculus volume of the Formulario was published Peano, as he had indicated, began to use it for his teaching.

  76. Schubert biography
    • Using methods of Chasles, with Schroder's logical calculus as a model, he set up a system to solve such problems, he called it the principal of conservation of the number.
    • Schubert's achievement was to combine this procedure, which he called "the principle of conservation of number", with the Chasles correspondence principle, thus establishing the foundation of a calculus.
    • With the aid of this calculus, which he modelled on Ernst Schroder's logical calculus, Schubert was able to solve many problems systematically.

  77. Burkill biography
    • The groundwork in analysis and calculus with which the reader is assumed to be acquainted is, roughly, what is in Hardy's "A course of pure mathematics "(1908).
    • well-written text is designed as an introductory course in real and complex analysis for students familiar with elementary calculus and linear algebra.
    • The book covers: sets and functions, metric spaces, continuous functions on metric spaces, real and complex limits and series, uniform convergence, Riemann-Stieltjes integration, multivariable differential and integral calculus, Fourier series, Cauchy's theorem, Laurent expansions, residue calculus, infinite products, the factor theorem of Weierstrass, asymptotic expansions, and applications to special functions in particular the gamma function.

  78. Tao biography
    • sitting in the far corner of a room reading a hardback book with the title 'Calculus'.
    • Clements discovered that Terry knew the definition of a group and could solve graph sketching problems using the differential calculus.
    • This two-volume introduction to real analysis is intended for honours undergraduates, who have already been exposed to calculus.
    • The material starts at the very beginning - the construction of the number systems and set theory, and then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several-variable calculus and Fourier analysis, and finally the Lebesgue integral.

  79. De L'Hopital biography
    • Bernoulli at this time was 24 years old and he had just arrived in Paris after giving lectures on the latest development in mathematics, namely Leibniz's differential calculus.
    • In 1696 L'Hopital's famous book Analyse des infiniment petits pour l'intelligence des lignes courbes was published; it was the first text-book to be written on the differential calculus.
    • I must here in justice own (as Mr Leibniz himself has done in 'Journal des Scavans' for August 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis ..
    • It was used for a long time, with new editions produced until 1781, and it was also a model for the next generation of calculus books.

  80. Bellavitis biography
    • The geometrical calculus which he developed (in his own words):- .
    • What he introduced was a barycentric calculus more general than that of Mobius.
    • Later on, in 1858, Bellavitis included the system of quaternions into his geometric calculus.
    • He developed very personal critical observations about the calculus of probabilities and the theory of errors.

  81. Sidler biography
    • In his house, both in Zug and in Unterstrass, he had a little observatory, and Graf reports in [Mitteilungen der Naturforschenden Gesellschaft in Bern, 1907, 230-256.',2)">2] that 'he always took Kastner's Foundations of Mathematics or Lacroix's Introduction to Differential and Integral Calculus along to meetings of the National Court'.
    • Moreover, he also took on P F Servient's lectures on differential and integral calculus in French.
    • Moreover, it is written in a way that he is not required to have any particular knowledge of infinitesimal calculus, such as gamma functions.
    • Whilst he mainly lectured on analytic geometry, infinitesimal calculus, theory of functions and number theory, Sidler primarily gave lectures on theoretical astronomy and synthetic geometry.

  82. Serrin biography
    • Also in 1959 he published the monograph Mathematical principles of classical fluid mechanics in the Handbuch der Physik as well as papers on the calculus of variations such as the two major papers On a fundamental theorem of the calculus of variations and A new definition of the integral for non-parametric problems in the calculus of variations which appeared one following the other in Acta Mathematica.
    • Mathematical analysis, calculus of variations and geometry were not sufficient for James Serrin.

  83. Ampere biography
    • His method involves the use of infinitesimals but since Ampere had not studied the calculus the paper was not found worthy of publication.
    • Shortly after writing the article Ampere began to read d'Alembert's article on the differential calculus in the Encyclopedie and realised that he must learn more mathematics.
    • After taking a few lessons in the differential and integral calculus from a monk in Lyon, Ampere began to study works by Euler and Bernoulli.
    • This work was followed by one on the calculus of variations in 1803.

  84. Razmadze biography
    • Razmadze wrote the first textbooks in Georgian on analysis and integral calculus.
    • 214 (1980), 21-28; 229.',2)">2] Razmadze's Introduction to differential calculus a little-known textbook published in Russian in 1923 is described.
    • His work was on the calculus of variations, continuing work by Weierstrass and Hilbert.
    • The fundamental lemma of the calculus of variations is named after him.

  85. Newton biography
    • While Newton remained at home he laid the foundations for differential and integral calculus, several years before its independent discovery by Leibniz.
    • However the last portion of his life was not an easy one, dominated in many ways with the controversy with Leibniz over which of them had invented the calculus.
    • In this capacity he appointed an "impartial" committee to decide whether he or Leibniz was the inventor of the calculus.
    • History Topics: The rise of the calculus .

  86. Li Shanlan biography
    • With Alexander Wylie, Li translated Elements of Analytical Geometry and of the Differential and Integral Calculus which had been written by Elias Loomis and published in New York in 1851.
    • Their Chinese translation was published in 1859 and became the first book to introduce Newton's calculus into China.
    • By their fourth year students were studying the differential and integral calculus.
    • The works of the great astronomer Guo Shoujing concerning the inequalities of the solar and lunar motion, Wang Lai's iterated sums, Dong Fangli's cyclotomical computations, and lastly the summation of series which appear in the algebra and the differential calculus of the Westerners constitute the major part of this chapter.

  87. Fabri biography
    • He also studied magnetism, optics and calculus.
    • In calculus he was closer to Newton than to Cavalieri but his notation was cumbersome.
    • His work on the calculus appeared in his major mathematical publication Opusculum geometricum de linea sinuum et cycloide (1659).
    • Fabri had a major influence on the development of the calculus through Leibniz.

  88. Saint-Vincent biography
    • During his years in Louvain, Saint-Vincent worked on mathematics and developed methods which were important in setting the scene for the invention of the differential and integral calculus.
    • He treats the hyperbola summing the area under the curve using a sequence of ordinates in geometric progression [The Origins of the Infinitesimal Calculus (Pergamon Press, Oxford, 1969).',2)">2]:- .
    • Basically these procedures are closely related to the development of the integral calculus and the numerical methods which were subsequently developed for the calculation of logarithms form a fascinating study.
    • Margaret Baron writes [The Origins of the Infinitesimal Calculus (Pergamon Press, Oxford, 1969).',2)">2]:- .

  89. Ricci-Curbastro biography
    • He changed area somewhat to undertake research in differential geometry and was the inventor of the absolute differential calculus between 1884 and 1894.
    • The method he used to demonstrate [the invariance of the quadratics] led him to the technique of absolute differential calculus, which he discussed in its entirety in four publications written between 1888 and 1892.
    • The algorithm of absolute differential calculus, the instrument materiel of the methods ..
    • Ricci-Curbastro's absolute differential calculus became the foundation of tensor analysis and was used by Einstein in his theory of general relativity.

  90. Schwartz biography
    • The theory of distributions is a considerable broadening of the differential and integral calculus.
    • Heaviside and Dirac had generalised the calculus with specific applications in mind.
    • I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ..
    • Later work by Schwartz on stochastic differential calculus is described by him in the survey article [Analyse Mathematique et Applications (Gauthier-Villars, Montrouge, 1988), 445-463.',45)">45], see also [Mathematical analysis and applications A (Academic Press, New York-London, 1981), 1-25.',44)">44].

  91. Saint-Venant biography
    • Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845.
    • Saint-Venant used this vector calculus in his lectures at the Institut Agronomique, which were published in 1851 as "Principes de mecanique fondes sur la cinematique".
    • In this book Saint-Venant, a convinced atomist, presented forces as divorced from the metaphysical concept of cause and from the physiological concept of muscular effort, both of which, in his opinion, obscured force as a kinematic concept accessible to the calculus.
    • Although his atomistic conceptions did not prevail, his use of the vector calculus was adopted in the French school system.

  92. Young biography
    • Perhaps his most important contribution was to the calculus of several variables.
    • He set out this theory beautifully in his treatise The fundamental theorems of the differential calculus (1910).
    • All advanced calculus books now use his approach to functions of several complex variables.
    • W H Young: Differential Calculus .

  93. Caratheodory biography
    • He worked on the calculus of variations and was much influenced by both Hilbert and Klein.
    • Caratheodory made significant contributions to the calculus of variations, the theory of point set measure, and the theory of functions of a real variable.
    • He added important results to the relationship between first order partial differential equations and the calculus of variations.
    • Caratheodory wrote many fine books including Lectures on Real Functions (1918), Conformal representation (1932), Calculus of Variations and Partial Differential Equations (1935), Geometric Optics (1937), Real functions Vol.

  94. Cunha biography
    • Da Cunha wrote a 21 part encyclopaedia of mathematics Principios Mathematicos which he began to publish in parts from 1782 (it was published as a complete work in 1790) which contained a rigorous exposition of mathematics, in particular a rigorous exposition of the calculus.
    • The book contained the elements of geometry and algebra in addition to the calculus.
    • This book is characterised by the attempts at rigor, especially in the calculus.
    • 26 (1) (1973), 3-22.',11)">11], claims that da Cunha should rank with Bolzano, Cauchy, Abel and others for his contributions to the principles of the calculus.

  95. Glenie biography
    • In fact, even before being sent to North America, he had discovered what he called the antecedental calculus in 1774.
    • The was an attempt to base Newton's fluxional calculus on the binomial theorem rather than on the concept of motion.
    • In 1778 the Royal Society published Glenie's paper on the antecedental calculus.
    • Glenie still retained his interest in mathematics and he published his ideas in a book entitled the Antecedental Calculus (1793, 1794).

  96. Whitehead biography
    • The chief examples of such systems are Hamilton's Quaternions, Grassmann's Calculus of Extension, and Boole's Symbolic Algebra.
    • The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every providence of thought, or 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.

  97. Grave biography
    • At St Petersburg he taught analytic geometry, algebra, calculus I and a special course on the theory of surfaces.
    • He also began publishing books based on his lecture courses, for example: Course of analytical geometry (1893); and A course in differential calculus (1895).
    • He also studied the history of algebraic analysis and published two volumes of his Treatise on Algebraic Calculus (Russian) in 1938 and 1939.

  98. Boggio biography
    • In 1903 he was appointed to teach mathematical physics at the University of Pavia and as an assistant to Giuseppe Peano to teach calculus at the University of Turin.
    • A text which Boggio wrote on the differential calculus with geometrical applications, published in 1921, was reviewed by his colleague Peano who says the books use of vector methods:- .
    • In session 1949-50 he taught Infinitesimal Calculus but by this time he was officially retired and taught as an assistant to the chair.

  99. Nalli biography
    • She was ranked third in the competition for the chair of calculus at the University of Modena in 1922 but then ranked first in the competition for the chair of calculus at the University of Pavia in the following year.
    • However, the greatest indication of the reply Nalli must have received is seen from the fact that at this time she changed her research topic and after this worked on tensor calculus, the topic for which Levi-Civita is famed.

  100. Poleni biography
    • In fact Padua had been fortunate to have had Jakob Hermann teaching there from 1707 to 1713 and he had introduced the methods of Leibniz' differential and integral calculus to the university.
    • Poleni was familiar with the new differential and integral calculus and with the natural philosophy of Newton.
    • The two mathematicians discussed: the application of calculus to some controversial issues in dynamics; hydraulics; and administrative matters related to the regulation of waters in the Venetian region.

  101. Paoli biography
    • His research was on analytic geometry, calculus, partial derivatives, and differential equations.
    • The third part was further divided into two sections, the first containing the differential calculus, the second being devoted to methods related to the integral calculus.

  102. Jones biography
    • It included the differential calculus, infinite series, and is also famed since the symbol π is used in it with its modern meaning.
    • Jones then served on the Royal Society committee set up in 1712 to decide who had invented the infinitesimal calculus, Newton or Leibniz.
    • As an appendix to this work Jones added Newton's Tractatus de quadratura curvarum which was a shortened version of the work on analytical calculus which Newton had written in 1691.

  103. Bonferroni biography
    • It was a highly appropriate conference for Bonferroni to attend since, for the first time, a section was introduced covering Statistics, Mathematical Economy, Calculus of Probability, and Actuarial Science.
    • The author establishes above all a symbolic calculus which enables the expression in a rapid and uniform manner of the various probabilities of survival and death amongst a group of assured, expressed as a function of a particular type assumed as primary.
    • This calculus does not require the hypothesis that the assured lives should be independent, as is usual in treatments of this problem.

  104. Tacquet biography
    • Of course this idea is a early form of what would become clear when the calculus was invented, namely that the derivative and integral were inverse to each other.
    • This book had a considerable effect on Pascal and was important in setting the scene for the invention of the calculus.
    • The importance of Tacquet's work is not so much in the actual results he proved, for he was not greatly inventive in this respect, but rather for the clarity of his writings and the fact that in many ways his approach was important in preparing the way for Newton and Leibniz's integral and differential calculus.

  105. Blackwell biography
    • The most interesting thing I remember from calculus was Newton's method for solving equations.
    • That was the only thing in calculus I really liked.
    • It was a course on real analysis, based on Hardy's Pure Mathematics, rather than the calculus which really turned him on to a career in mathematics [Mathematical People (Boston, 1985), 18-32.',2)">2]:- .

  106. Stern biography
    • During his 55 years at the University of Gottingen, Stern lectured on a wide variety of topics, including algebraic analysis, analytic geometry, differential and integral calculus, variational calculus, mechanics, popular astronomy and, of course, number theory.
    • In any case, Riemann received a profound knowledge of the state of the art of analysis as taught in Germany at that time from Stern's lectures on calculus.

  107. Scheffers biography
    • This calculus is a geometer's calculus.
    • This book provides an unusually complete course in calculus for the sincere student.

  108. Dedekind biography
    • There he was to receive a good understanding of basic mathematics studying differential and integral calculus, analytic geometry and the foundations of analysis.
    • His other courses covered material such as the differential and integral calculus, of which he already had a good understanding.
    • In fact it was while he was thinking how to teach differential and integral calculus, the first time that he had taught the topic, that the idea of a Dedekind cut came to him.

  109. Grandi biography
    • In fact Grandi taught methods of the infinitesimal calculus from 1702 in private lessons he was giving; he was the first to do so in Italy.
    • But I have here and there also inserted the dx, dy typical of differential calculus, and their methods of being differentiated and added.
    • Thus had I been able to introduce them in my previous pamphlets too! But then, the secrets of that method had been inaccessible to me, while now, their usefulness and fecundity having been proven, why not insert them among the other methods I am familiar with? Also, the significance of the symbols is very clear, because it only signifies an infinitely small difference between the same x and y, and you will easily find the very rules of calculus if you observe and peruse this tract carefully - unless you want to recourse to the illustrious De L'Hopital who explains them in a more complete way in his tract on the infinitely small.

  110. Pieri biography
    • He was taught 'Calculus' in 1881-82 by Ulisse Dini who also taught him 'Higher analysis' in the following year.
    • During the next years they worked to enhance the university's offerings in mathematics, creating schools of fundamentals of algebra, of ornamental design and architecture, and of infinitesimal calculus in 1911.
    • In 1911 Pieri became interested the vector calculus through the work of Cesare Burali-Forti and Roberto Marcolongo.

  111. Kline biography
    • In 1967 he published a two-volume calculus text Calculus, An intuitive and Physical Approach which teaches calculus through physical problems but also tries to develop the students intuition by approaching problem solving as a beginner might, making false starts and changing tack.

  112. Baiada biography
    • In 1940 he received the Merlani Award for "contributions relating to the calculus of variations".
    • Although not formally qualified as a lecturer at this stage, nevertheless he taught the analysis courses, namely the theory of functions, calculus, and rational mechanics.
    • We have mentioned some of Baiada's publications above but we note that his output totals 60 scientific publications on a wide range of different fields in analysis: ordinary and partial differential equations, Fourier series and the series expansion of orthonormal functions, topology, real analysis, functional analysis, calculus of variations, measure and integration, optimisation, and the theory of functions.

  113. Hamming biography
    • In the calculus book we are currently using on my campus, I found no single problem whose answer I felt the student would care about! The problems in the text have the dignity of solving a crossword puzzle - hard to be sure, but the result is of no significance in life.
    • His attempt to move to a new way of teaching calculus is exhibited in his book Methods of mathematics applied to calculus, probability, and statistics (1985).

  114. Darmois biography
    • The 1914-1918 war, having oriented me towards ballistics and artillery problems, and then towards location by sound and the problems of measuring and of wave propagation, had deeply inflected my spirit towards mathematical physics and the calculus of probability.
    • The decision I took in Nancy in 1923, to connect the teaching and investigations of the calculus of probability with several applications to statistics, stemmed from the desire of constituting in France a school of theoretical and practical statistics.
    • In 1949 he succeeded Frechet when appointed Professor of the Calculus of Probabilities and Mathematical Physics at the Sorbonne.

  115. Frechet biography
    • The thesis concerns 'functional operations' and 'functional calculus' and is developed from ideas due to Hadamard and Volterra.
    • The functional calculus of his thesis is then the systematic study of functional operations.
    • Frechet also made important contributions to statistics, probability and calculus.

  116. Kneser biography
    • The second main area of his work was the calculus of variations.
    • He published Lehrbruch der Variationsrechnung (Textbook of the calculus of variations) (1900) and he gave the topic many of the terms in common use today including 'extremal' for a resolution curve, 'field' for a family of extremals, 'transversal' and 'strong' and 'weak' extremals [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • But above all, the decisive advances towards the solution of the so-called Mayer Problem, recently introduced to the calculus of variations, are due to Kneser.

  117. Arnold biography
    • Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics.
    • Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universities Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).
    • The face of modern mathematics would be unrecognisable without his work in dynamical systems, classical and celestial mechanics, singularity theory, topology, real and complex algebraic geometry, symplectic and contact geometry, hydrodynamics, variation calculus, differential geometry, potential theory, mathematical physics, superposition theory, etc.

  118. Clairaut biography
    • Alexis used Euclid's Elements while learning to read and by the age of nine he had mastered the excellent mathematics textbook of Guisnee Application de l'algebre a la geometrie which provided a good introduction to the differential and integral calculus as well as analytical geometry.
    • He wrote the paper Sur quelques questions de maximis et minimis in 1733 on the calculus of variations, written in the style of Johann Bernoulli and, in the same year, he published on the geodesics of quadrics of rotation again studying a topic to which Johann Bernoulli had contributed.
    • In 1739 and 1740 he published further work on the integral calculus, proving the existence of integrating factors for solving first order differential equations (a topic which also interested Johann Bernoulli, Reyneau and Euler).

  119. Young Laurence biography
    • In 1969 Young published Lectures on the calculus of variations and optimal control theory.
    • Before leaving Lectures on the calculus of variations and optimal control theory let us note that Hestenes gives this assessment:- .
    • The book is an important contribution to the calculus of variations and optimal control theory.

  120. Threlfall biography
    • He was promoted to a tenured extraordinary professor of analytical geometry, calculus of variations, analysis and function theory on 1 April 1936.
    • This book is a first-class introduction to the topological part of Marston Morse's book, 'The Calculus of Variations in the Large' ..
    • This theory is applied in detail to the study of a typical problem in the calculus of variations in the large.

  121. Couturat biography
    • Couturat argued that all of these generalisations had at first encountered strong opposition, but had become accepted in the end because they were suitable for representing new magnitudes and they allowed a calculus of operations which was impossible before their introduction.
    • Leibniz had proposed a calculus of reason which would allow the mind to think directly of things themselves.
    • He wanted this calculus of reason to be supported by a logical universal language.

  122. Craig biography
    • and the "Acta Eruditorum" of Leipzig ranked him among the originators of the calculus (after Leibniz, but before Newton).
    • However, he was involved in a dispute with Jacob Bernoulli over the calculus and he also had a dispute with Tschirnhaus.
    • It is worth noting that despite using Leibniz's notation for the calculus in his earlier works, in this later one Craig used Newton's fluxion notation.

  123. Goursat biography
    • Edouard Goursat's three-volume 'A Course in Mathematical Analysis' remains a classic study and a thorough treatment of the fundamentals of calculus.
    • As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition.
    • Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.

  124. Bottasso biography
    • the simplicity and the quickness of vector calculus in the approach to different problems for which Cartesian methods are too difficult.
    • He used the vector calculus in studying problems in geometry, mechanics and physics.
    • Bottasso gave lectures to the high school teachers on numerical calculus in March 1915.

  125. Briot biography
    • He taught engineering and surveying in the year he moved back to Paris, then he taught a calculus course in 1853 and, two years later, courses on mechanics and astronomy.
    • Briot, however, developed a sophisticated mathematical theory to study these properties, and although his work has no great importance to physics, the analysis he had to develop during his working out of the theory led to significant results in the integral calculus and also in the theory of elliptic and abelian functions.
    • He wrote textbooks which covered most of the topics from a mathematics course: arithmetic, algebra, calculus, geometry, analytic geometry, and mechanics.

  126. McAfee biography
    • I taught a course in differential and integral calculus in one summer to all of the engineers who were entering Rutgers that fall.
    • I was surprised, some of these engineers didn't have the slightest idea about calculus.
    • Finally, at the end of the class there were only about four people I would have passed in that course in calculus and there were about twenty-nine or thirty in the class.

  127. Weierstrass biography
    • In 1860/61 he lectured on the Integral calculus.
    • Calculus of variations or applications of elliptic functions.
    • Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.

  128. Peacock biography
    • While undergraduates Peacock, Herschel and Babbage planned to bring reforms to Cambridge and, in 1815, they formed the Analytical Society whose aims were to bring the advanced continental methods of calculus to Cambridge.
    • In 1816 the Analytical Society produced a translation of a book of Lacroix in the differential and integral calculus.
    • Peacock published Collection of Examples of the Application of the Differential and Integral Calculus in 1820, a publication which sold well and helped further the aims of the Analytical Society.

  129. Zeno of Elea biography
    • So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing.
    • Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus.
    • History Topics: The rise of the calculus .

  130. Smith Karen biography
    • At high school she studied the usual mathematical topics to prepare her for university entrance, namely geometry, algebra, pre-calculus, and then Calculus (AB).
    • She entered Princeton University and in her first year took a calculus class with Charles Fefferman.

  131. Verhulst biography
    • At this time Verhulst worked on the theory of numbers, and, influenced by Quetelet, he became interested in the calculus of probability and social statistics.
    • On Quetelet's recommendation, in 1834 Verhulst was appointed as a Repetiteur at the Academy to teach calculus.
    • There he gave courses on astronomy, celestial mechanics, the differential and integral calculus, the theory of probability, geometry and trigonometry.

  132. Radon biography
    • He was awarded a doctorate in 1910 for a dissertation on the calculus of variations carried out under Gustav von Escherich's supervision.
    • Radon applied the calculus of variations to differential geometry which led to applications in number theory.
    • It was while he was studying applications of the calculus of variations to differential geometry that he discovered curves which are now named Radon curves.

  133. Saurin biography
    • He became friends with de L'Hopital, Malebranche and Varignon but, by 1702, he was in dispute with Rolle over the calculus.
    • Saurin made contributions to the calculus, wrote on Jacob Bernoulli's problem of quickest descent and Huygens' theory of the pendulum.
    • Rather, firmly committed to the new infinitesimal calculus, he explored the limits and possibilities of its methods and defended it against criticism based on lack of understanding.

  134. Mansion biography
    • Schaar was not quite fifty years old when he died in April 1867 leaving the Chair of Differential and Integral Calculus and Higher Analysis vacant.
    • In 1892 Mansion succeeded Emmanuel-Joseph Boudin when he was appointed to the Chair of the Calculus of Probabilities at Ghent.
    • Author of many works on mathematical analysis, the calculus of probabilities, non-Euclidean geometry, the history and philosophy of science, [Mansion] held a prominent place in the Belgian scientific world.

  135. Allan Graham biography
    • In this survey article the author outlines those parts of holomorphic function theory (in particular, the holomorphic functional calculus) that have been applied in the study of commutative Banach algebras, specifically excluding topics that are peculiar to uniform algebras, where the applications have been most extensive.
    • The main object of the exposition is the construction of the holomorphic functional calculus in several variables and the application of this calculus to the Silov idempotent theorem, the local maximum modulus theorem and the Arens-Royden theorem.

  136. Salem biography
    • Another direction in which [Salem] did a lot was applications of the calculus of probability to Fourier series and, curiously enough, this has connection with problems of uniqueness.
    • Moreover, it seem that, far from being incidental, as it might have appeared some 30 or so years ago, the calculus of probability is becoming a standard method of attacking problems of trigonometric series.
    • Going through the papers of [Salem] one sees this growing role of the calculus of probability.

  137. Nelson biography
    • Her teaching first year calculus was highly successful [Algebra Universalis 26 (1989), 259-266.
    • she had a clear vision of the importance of the first year calculus course and a serious concern for the right way of teaching it, which necessarily made her a demanding instructor.
    • Given this superb attitude and success at teaching first year calculus it is ironical that colleagues on the faculty at McMaster argued against giving her a position in the Department because they believed that she could not handle large first year classes.

  138. Vitali biography
    • There he was strongly influenced by Luigi Bianchi, who taught him analytic geometry, and Ulisse Dini who taught him infinitesimal calculus.
    • The fact itself of his having met with one of the creators of the new directions of integral calculus, Lebesgue, is proof that he stayed within the mainstream of this research, in which such results naturally presented themselves.
    • In his last years he worked on a new absolute differential calculus and a geometry of Hilbert spaces.

  139. Spencer biography
    • In 1959 Spencer, along with H K Nickerson and N E Steenrod, pubished the textbook Advanced calculus.
    • The contents of this remarkable book have served as notes for a special course in advanced calculus at Princeton University.
    • The book stems from a widespread dissatisfaction with the method of presentation of the subject matter of the traditional course in advanced calculus and succeeds in showing that the traditional subject matter can form an integral part of modern mathematics.

  140. Morse biography
    • It builds on the classical results in the calculus developed by Hilbert and his students.
    • Morse's major works include Calculus of variations in the large (1934), Functional topology and abstract variational theory (1938), Topological methods in the theory of functions of a complex variable (1947) and Lectures on analysis in the large (1947).
    • In 1933 the American Mathematical Society awarded him the Bocher Prize for his memoir The foundations of a theory of the calculus of variations in the large in m-space published in Transactions of the American Mathematical Society in 1929 (which he shared with Norbert Wiener).

  141. Cramer biography
    • That he made little use of Euler's work is supported by the rather surprising fact that throughout his book Cramer makes essentially no use of the infinitesimal calculus in either Leibniz' or Newton's form, although he deals with such topics as tangents, maxima and minima, and curvature, and cites Maclaurin and Taylor in footnotes.
    • One conjectures that he never accepted or mastered the calculus.
    • The suggestion that Cramer never mastered the calculus must be considered doubtful, particularly given the high regard that he was held in by Johann Bernoulli.

  142. Gibson biography
    • He published several excellent textbooks, for example An Elementary Treatise on the Calculus, An Elementary Treatise on Graphs, and A First Course in Calculus, based on Graphic Methods.
    • George Gibson: Calculus .

  143. Fergola biography
    • His interests were broad and he studied in depth the developments in the differential and integral calculus and their application to physical problems.
    • He taught advanced mathematics at his school and he also undertook research producing works on applications of calculus such as Risoluzione di alcuni problemi ottici (1780) and La ver misura delle volte a spira (1783).
    • A vigorous argument between supporters of the synthetic method and those of the analytic method broke out in 1810 when Ottavio Colecchi (1773-1847), who taught differential and integral calculus at the Scuola di Applicazione in Naples, criticised Fergola for putting too much emphasis on pure geometry and not enough emphasis on the new methods of analysis.

  144. Kepler biography
    • He also did important work in optics (1604, 1611), discovered two new regular polyhedra (1619), gave the first mathematical treatment of close packing of equal spheres (leading to an explanation of the shape of the cells of a honeycomb, 1611), gave the first proof of how logarithms worked (1624), and devised a method of finding the volumes of solids of revolution that (with hindsight!) can be seen as contributing to the development of calculus (1615, 1616).
    • 1598 - 1647) and is part of the ancestry of the infinitesimal calculus.
    • History Topics: The rise of the calculus .

  145. Hay biography
    • She published Axiomatization of the infinite-valued predicate calculus in the Journal of Symbolic Logic in 1963 in which she gave a set of nine axiom schemes and two rules for the predicate calculus based on the infinite-valued sentential calculus of Lukasiewicz.

  146. Hayes Charles biography
    • The book is the first English text on Newton's method of fluxions, or, to phrase it in more modern terms, the first English calculus text.
    • The book is a very full treatise, about three times the size of de l'Hopital's famous calculus book.
    • We noted above that Hayes dedicated his calculus text to the Director of the Royal African Company.

  147. Keill biography
    • He claimed that Leibniz had plagiarised Newton's invention of the calculus and he served as Newton's avowed champion.
    • The part played by Keill in the controversy over who invented the calculus is fully brought out in [Philosophers at war : the quarrel between Newton and Leibniz (1980).',3)">3].
    • He wrote two articles explaining Newton's error and suggested that they demonstrated that Newton could not have invented the calculus independently of Leibniz since he was incapable of it.

  148. Bessel-Hagen biography
    • He submitted his thesis on the calculus of variations Uber eine Art singularer Punkte der einfachen Variationsproblerme in der Ebene and was awarded the degree in 1920.
    • Caratheodory thought Bessel-Hagen's disertation the first important advance in the theory of discontinuous solutions for problems in the calculus of variations since his own work in 1905.

  149. Christoffel biography
    • He wrote important papers which contributed to the development of the tensor calculus of C G Ricci-Curbastro and Tullio Levi-Civita.
    • Indeed this influence is clearly seen since this allowed Ricci-Curbastro and Levi-Civita to develop a coordinate free differential calculus which Einstein, with the help of Grossmann, turned into the tensor analysis mathematical foundation of general relativity.

  150. Bernoulli Daniel biography
    • Daniel, like his father, really wanted to study mathematics and during the time he studied philosophy at Basel, he was learning the methods of the calculus from his father and his older brother Nicolaus(II) Bernoulli.
    • Another important aspect of Daniel Bernoulli's work that proved important in the development of mathematical physics was his acceptance of many of Newton's theories and his use of these together with the tolls coming from the more powerful calculus of Leibniz.

  151. Landen biography
    • He sent his results on making the differential calculus into a purely algebraic theory to the Royal Society.
    • Lacroix discussed this work in Traite Du calcul differentiel where he stated that Landen was the first to present calculus in a purely algebraic setting.

  152. Mosteller biography
    • In his second year he took courses on calculus, mechanics, French, quantitative analysis, and physical measurements.
    • Using only high school algebra and no calculus, they develop the subject from probability, permutations and combinations, through set theory to frequency distributions and statistical inference.

  153. Robbins biography
    • He entered Harvard University in 1931, when only sixteen tears old, and took courses on literature but he also took a calculus course [Mathematical People (Birkhauser, Boston-Basel-Stuttgart, 1985), 283-297.',4)">4] or [The College Mathematics Journal 15 (1) (1984), 2-24.',5)">5]:- .
    • Having just entered Harvard with practically no high school mathematics, I knew calculus would be useful if I ever wanted to study any of the sciences.

  154. McShane biography
    • McShane is famous for his work in the calculus of variations, Moore-Smith theory of limits, the theory of the integral, stochastic differential equations, and ballistics.
    • In 1974, the year he retired and was made Professor Emeritus at Virginia, McShane published Stochastic calculus and stochastic models which again reflected his work on the mathematical setting for quantum mechanics.

  155. Barbier biography
    • He also wrote on probability and calculus.
    • These were entirely on mathematical topics and he made worthwhile contributions to the study of polyhedra, integral calculus and number theory.

  156. Torricelli biography
    • Collections of paradoxes which arose through inappropriate use of the new calculus were found in his manuscripts and show the depth of his understanding.
    • History Topics: The rise of the calculus .

  157. Hayes biography
    • During her years at Wellesley College, Hayes wrote a number of fine textbooks: Lessons on Higher Algebra (1891); Elementary Trigonometry (1896); Algebra for High Schools and Colleges (1897); and Calculus with Applications, An Introduction to the Mathematical Treatment of Science (1900).
    • My courses with her included Calculus, Celestial Mechanics, Logic, and Astronomy.

  158. Wallis biography
    • Wallis contributed substantially to the origins of calculus and was the most influential English mathematician before Newton.
    • He studied the works of Kepler, Cavalieri, Roberval, Torricelli and Descartes, and then introduced ideas of the calculus going beyond that of these authors.

  159. Graffe biography
    • While in Gottingen, Graffe wrote a prize winning dissertation Die Geschichte der Variationsrechnung vom Ursprung der Differential und Integralrechnung bis auf die heutige Zeit zu schreiben (The history of the calculus of variations from the origin of differential and integral calculus to the present time) which he submitted to the Faculty of Philosophy on 4 June 1825.

  160. Poretsky biography
    • He developed an original system of axioms of logical calculus and proposed a very convenient mode of determining all the conclusions that are deducible from a given logical premise, and of determining all possible logical hypotheses from which given conclusions may be deduced.
    • He applied his logic calculus to the theory of probability.

  161. Haar biography
    • Between 1917 and 1919 he worked on the variational calculus, proving Haar's Lemma, and applying his results to problems like Plateau's problem [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • A multitude of papers by others show the influence this lemma exerted on the whole area of variational calculus.

  162. Fubini biography
    • In addition to the areas of analysis detailed above, he worked on the calculus of variations where he studied reducing Weierstrass's integral to a Lebesgue integral and also he worked on the expression of surface integrals in terms of two simple integrations.
    • His most important work was on differential projective geometry where he used the absolute differential calculus.

  163. Mengoli biography
    • By examining the limits of sums, products and quotients of variable quantities, Mengoli was setting up the basic rules if the calculus thirty years before Newton and Leibniz.
    • History Topics: The rise of the calculus .

  164. Konigsberger biography
    • They had spent the evenings reading books on differential calculus and analytical geometry, working at a table in a small sitting room which they shared.
    • My lectures on differential and integral calculus, mechanics, etc.

  165. Clebsch biography
    • Even before his appointment at Karlsruhe there had been signs of Clebsch moving towards pure mathematics with his work on the calculus of variations.
    • Pure mathematics became Clebsch's main research topic when he began to study the calculus of variations and partial differential equations.

  166. Padoa biography
    • He returned to the University of Turin for session 1894-95 where he attended two courses given by Giuseppe Peano, one on the infinitesimal calculus and the other on higher geometry which made particular study of the geometric contributions of Hermann Grassmann.
    • However, Padoa also gave many lecture courses at Naval Academy in Genoa including 'Algebraic Analysis' (1911-12 and 1913-14), 'Infinitesimal Calculus' (1912-13) and 'Further Mathematics' (1914-1930).

  167. Chazy biography
    • The purpose of this book, which is the development of a course taught at the Faculty of Sciences of Paris in 1927, is to expose as clearly as possible the theory of relativity in dealing with celestial mechanics, taking as a starting point the knowledge of a student who has attended a few lessons on differential and integral calculus, and mechanics.
    • (1) The calculus of variations.

  168. Silva biography
    • In these papers Silva investigates analytic functionals, the theory of distributions, vector-valued distributions, ultradistributions, the operational calculus, and differential calculus in locally convex spaces.

  169. Brisson biography
    • The main idea in these reports was the application of the functional calculus, through symbols, to the solution of certain kinds of linear differential equations and of linear equations with finite differences.
    • Cauchy followed the way opened by Brisson, who thus became one of those who developed the methods of functional calculus.

  170. Mollweide biography
    • From these books he taught himself calculus and then progressed to the study of algebra.
    • The mathematics involved in the construction requires mainly high school algebra and trigonometry with only a bit of calculus (which can be plausibly avoided if one so desires).

  171. Pascal biography
    • He applied Cavalieri's calculus of indivisibles to the problem of the area of any segment of the cycloid and the centre of gravity of any segment.
    • It has been suggested that it was his too concrete turn of mind that prevented his discovering the infinitesimal calculus, and in some of the Provinciales the mysterious relations of human beings with God are treated as if they were a geometrical problem.

  172. Julia biography
    • In 1931 he was appointed to the Chair of Differential and Integral Calculus, then in 1937 he was appointed to the Chair of Geometry and Algebra at the Ecole Polytechnique when Maurice d'Ocagne retired.
    • Volume 4 is again in four parts: (i) Functional calculus and integral equations; (ii) Quasianalyticity; (iii) Various techniques of analysis; and (iv) Works concerning Hilbert space.

  173. Marczewski biography
    • At the University of Warsaw he was taught by Kuratowski who was teaching the first year calculus course that Marczewski attended.
    • during the first tutorials in that subject [calculus] he attracted my attention by his extraordinary ingenuity.

  174. Bolzano biography
    • (Pure Analytical Proof) (1817), which contain an attempt to free calculus from the concept of the infinitesimal.
    • Other topics covered here include various approaches to the calculus (including the method of exhaustion), and grounds for asserting the certainty of mathematics.

  175. Gompertz biography
    • The following year he read a paper to the Society which applied the differential calculus to the calculation of life expectancy.
    • Gompertz applied the calculus to actuarial questions and he is best remembered for Gompertz's Law of Mortality.

  176. Carslaw biography
    • One was An introduction to the infinitesimal calculus published in 1905.
    • This put Heaviside's operational calculus on a rigorous footing following the approach proposed by Gustav Doetsch.

  177. Wallace biography
    • Perhaps his most significant contribution, however, was the fact that he advocated the Continental approach to the calculus and was one of the first in Britain to do so.
    • In this Encyclopaedia Britannica article Wallace uses Newton's notation, but in his article Fluxions for the Edinburgh Encyclopaedia which was published in 1815 he used Leibniz's differential notation and was therefore the first to write an English treatise on the calculus using differential notation.

  178. Hall biography
    • In it, in addition to its main aims of developing the theory of regular p-groups, Hall introduces the commutator calculus, commutator collection, and the connection between p-groups and Lie rings.
    • This discussion is kept concise by the use of an elegant calculus of closure operations on group properties.

  179. Huygens biography
    • On his return to Holland Huygens wrote a small work De Ratiociniis in Ludo Aleae on the calculus of probabilities, the first printed work on the subject.
    • History Topics: The rise of the calculus .

  180. Gateaux biography
    • As Gateaux only mentions that he followed two of Volterra's lectures in Rome (one of Mathematical Physics, the other about applications of functional calculus to Mechanics), it is probable that the delay refers to Volterra's political involvements as Senator.
    • has left very advanced researches on functional calculus (his thesis was composed to a great extent, and partly exposed in notes to the Academy), researches for which M Volterra and myself have a big consideration.

  181. Vinti biography
    • Among his most important publications during his years in Perugia, we mention On the Weierstrass integrals of the calculus of variations over BV varieties: recent results of the mathematical seminar in Perugia (1989) and Problems associated with the theory of finitely additive measures: some recent results of the Scuola Matematica Perugina (1990).
    • The scientific interests of Calogero Vinti covered several areas of Mathematical Analysis, from Calculus of Variations to Differential Equations, from Approximation Theory to Real Analysis and Measure Theory.

  182. Folie biography
    • In 1857 he was appointed to teach courses in Algebra, Calculus and Analytic Geometry at the University of Liege.
    • It was a course he continued to give for a number of years but, in 1868, he gave up teaching the courses in Algebra, Calculus and Analytic Geometry.

  183. Lebesgue biography
    • This generalisation of the Riemann integral revolutionised the integral calculus.
    • He also made major contributions in other areas of mathematics, including topology, potential theory, the Dirichlet problem, the calculus of variations, set theory, the theory of surface area and dimension theory.

  184. Kantorovich biography
    • He published his first book in 1933, coauthored with Vladimir Ivanovich Krylov (1902-1994) and Vladimir Ivanovich Smirnov, entitled Calculus of variations.
    • These other areas include functional analysis and numerical analysis and within these topics he published papers on the theory of functions, the theory of complex variables, approximation theory in which he was particularly interested in using Bernstein polynomials, the calculus of variations, methods of finding approximate solutions to partial differential equations, and descriptive set theory.

  185. Picken biography
    • First let us list a few of the papers that Picken published in The Mathematical Gazette: Ratio and proportion (January 1920); The complete angle and geometrical generality (December 1922); Some general principles of analytical geometry (July 1923); The complete angle (October 1923); The notation of the calculus (October 1923); Parallelism and similarity (October 1924); The approach to the logarithmic and exponential functions (December 1926); and The approach to the calculus (October 1927).

  186. Castelnuovo biography
    • An interest in the history of mathematics is evident from the interesting history book Le origini del calcolo infinitesimale nell'era moderna (1938) which he wrote on the calculus up to the time of Newton and Leibniz.
    • Calculus was not brought into the school syllabus in Italy as early as in many other countries and the concept of a function only brought in around 1910 after Castelnuovo's efforts.

  187. Blum biography
    • Calculus uses real numbers rather than counting numbers because it's measuring things in the real world.
    • Continuity is the mathematics of calculus and physics but there's never been a theory of computation that deals with this continuum.

  188. Egorov biography
    • In 1923 Egorov published the book Principles of the calculus of variations.
    • After he was appointed as a professor, he taught courses on differential geometry, the integration of differential equations, integral equations, the calculus of variations, number theory, and the theory of surfaces.

  189. Nevanlinna biography
    • After World War II, during his association with Zurich, his mathematical interests moved to the calculus of variations and applications to physics.
    • For example he wrote the paper Calculus of variation and partial differential equations (1967).

  190. Olds biography
    • He was also teaching me calculus, and he had a huge voice, was very good humoured, and had astonishingly big teeth.
    • Most regular work emphasized calculus or geometry; the discrete mathematics I learned with the outside readings was what was required.

  191. Bowditch biography
    • Bowditch had begun to learn algebra in 1787 and two years later he began to study the differential and integral calculus.
    • He learnt calculus so that he might study Newton's Principia and in 1790 he learnt Latin which was also necessary to enable him to read Newton's famous work.

  192. Agnesi biography
    • With Rampinelli's help Agnesi studied Reyneau's calculus text Analyse demontree (1708).
    • Rampinelli encouraged Agnesi to write a book on differential calculus.

  193. Malebranche biography
    • The two had many meetings when they discussed ideas both of philosophy and of mathematics and, in particular, Leibniz conveyed many of his ideas about his new calculus to Malebranche.
    • Malebranche's other work includes research into the nature of light and colour, studies in the infinitesimal calculus and work on vision.

  194. Gillman biography
    • He took courses in French and Analytic Geometry to start with but, in his second year took Differential Calculus and another French course.
    • Next he took courses on Integral Calculus, Matrix Theory, and Differential Equations.

  195. Schoen biography
    • Schoen, 40, continues his research in differential geometry, nonlinear partial differential equations and the calculus of variations.
    • He serves on the editorial boards of: the Journal of Differential Geometry, Communications in Analysis and Geometry, Communications in Partial Differential Equations, Calculus of Variations and Partial Differential Equations, and Communications in Contemporary Mathematics.

  196. Condorcet biography
    • During this period he produced several important works, including one in 1772 on the integral calculus which was described by Lagrange as:- .
    • Also in 1786 he again worked on his ideas for the differential and integral calculus, giving a new treatment of infinitesimals.

  197. Buffon biography
    • He corresponded with Gabriel Cramer on mechanics, geometry, probability, number theory and the differential and integral calculus.
    • He next published Memoire sur le jeu de franc-carreau which introduced differential and integral calculus into probability theory.

  198. Albanese biography
    • During the time Albanese was studying at the Scola Normale Superiore, Ulisse Dini was the director of the School, but he also gave lectures on infinitesimal calculus which Albanese attended.
    • However, Dini died in 1918, and following his death Albanese became an assistant to Onorato Nicoletti (1872-1929), an expert in the theory of Hermitian forms who took over Dini's courses on infinitesimal calculus after his death.

  199. Lerch biography
    • Some of his work is fundamental in modern operator calculus.
    • He is remembered today for his solution of integral equations in operator calculus and for the 'Lerch formula' for the derivative of Kummer's trigonometric expansion for log G(v).

  200. Holmboe biography
    • Holmboe published Stereometrie (Stereometry) in 1833, Plan og sfaerisk Trigonometrie (Plane and Spherical Trigonometry) in 1834, and an advanced calculus book Laerebog i den hoiere mathematik in 1849.
    • advanced calculus text was evidently influenced by Abel's research.

  201. Edgeworth biography
    • This work, really on economics, looks at the Economical Calculus and the Utilitarian Calculus.

  202. Cohen biography
    • was only nine years old when his sister Sylvia checked out a book about calculus from a New York library for him.
    • Librarians were reluctant to let her have the book, much less for her younger brother, arguing that even some college professors didn't understand calculus.

  203. Cercignani biography
    • Before the age of eighteen I had learned the basics of analysis through consulting encyclopaedias and by constructing missing proofs on my own (some quite correct, others only fanciful) in the Calculus of Variations, in Rational Mechanics, in Tensor Calculus, and I thoroughly read articles on Einstein's special and general theory of relativity.

  204. Spence biography
    • In 1808 Spence was again in London, and during the several months that he lived there, he published An Essay on the various Orders of logarithmic Transcendents; with an Inquiry into their Applications to the Integral Calculus, and the Summation of Series.
    • Spence published his last work "Outlines of a theory of algebraical equations: deduced from the principles of Harriot, and extended to the fluxional or differential calculus" in 1814.

  205. Pierpont biography
    • The student of mathematics, on entering the graduate school of American universities, often has no inconsiderable knowledge of the methods and processes of the calculus.
    • The problem therefore arises to examine more carefully the conditions under which the theorems and processes of the calculus are correct, and to extend as far as possible or useful the limits of their applicability.

  206. Archimedes biography
    • gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.
    • History Topics: The rise of the calculus .

  207. Adamson biography
    • He proposes to talk instead of "a function f whose domain is the set of real numbers", claiming that on this basis he can develop the calculus without the use of variables at all; his book 'Calculus: A modern approach' is a triumphant vindication of this claim.

  208. Fantappie biography
    • For these the author establishes the usual concepts of the differential calculus such as the derivative, with the customary properties and rules and the series expansion of such a function.
    • However, these results contain a Cauchy integral formula for the many-variable case, which permits an operational calculus of a somewhat restricted sort.

  209. Savage biography
    • His grades began to improve: C in analytic geometry; B in calculus; B in differential equations; A in Raymond Wilder's foundations of mathematics; and A in Raymond Wilder's point set topology course.
    • While at the Institute he solved an open problem in the calculus of variations suggested to him in discussions with John von Neumann and Marston Morse.

  210. Ackermann biography
    • Ackermann was also the main contributor to the development of the logical system known as the epsilon calculus, originally due to Hilbert.
    • The system is formalized in an applied first-order calculus with identity using a binary predicate e (membership) and a singulary predicate M (being a set).

  211. Bass biography
    • One of the courses in which Bass enrolled in his first year of study was a calculus course lectured by Emil Artin and tutored by Serge Lang and John Tate.
    • I had no reason to believe that this course was substantially different from any other college calculus course.

  212. Pincherle biography
    • At Bologna, Pincherle became a colleague of Luigi Donati (1846-1932), who was appointed to Bologna in 1877, and Cesare Arzela who had been appointed to the chair of Infinitesimal Calculus in 1880.
    • The lectures on the infinitesimal calculus which I gave to the press at the end of 1915, not without fear and trembling, have won favour beyond all expectations with the mathematical public ..

  213. Mahler biography
    • Already, from the summer vacation of 1917, he began teaching himself logarithms (the arithmetic properties of which turned out to be one of his abiding interests in transcendental number theory) plane and spherical trigonometry, analytic geometry and calculus.
    • At Frankfurt, supported financially by his parents and several members of the Krefeld Jewish community, he attended lectures by Max Dehn on topology, Ernst Hellinger on elliptic functions, Carl Siegel on calculus and Otto Szasz [Biographical Memoirs of Fellows of the Royal Society of London 39 (1994), 265-279.',2)">2]:- .

  214. Du Bois-Reymond biography
    • Du Bois-Reymond's work is almost exclusively on calculus, in particular partial differential equations and functions of a real variable.
    • However he had already developed a theory of infinitesimals in Uber die Paradoxen des Infinitar-Calculs ("On the paradoxes of the infinitary calculus") in 1877.

  215. Almgren biography
    • The reasons for this are partly indicated, for readers with only an advanced calculus background, in terms of examples, illustrated by a series of rather beautiful diagrams in colour.
    • Frederick Justin Almgren, Jr, one of the world's leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia.

  216. Wilkins Ernest biography
    • He then continued with his doctoral studies at Chicago and submitted his dissertation Multiple Integral Problems in Parametric Form in the Calculus of Variations which led to his being awarded a Ph.D.
    • In 1944 four of his papers appeared: On the growth of solutions of linear differential equations; Definitely self-conjugate adjoint integral equations; Multiple integral problems in parametric form in the calculus of variations; and A note on skewness and kurtosis.

  217. Bagnera biography
    • In 1901 he entered the competition for the extraordinary professorship of infinitesimal calculus at the University of Messina and was appointed to the post.
    • He was still holding the chair of infinitesimal calculus at the University of Messina at this time but he had travelled to Palermo to spend the Christmas vacation there.

  218. Coulomb biography
    • Perhaps the most significant fact about this memoir from a mathematical point of view is Coulomb's use of the calculus of variations to solve engineering problems.
    • A reason, perhaps, for the relative neglect of this portion of Coulomb's work was that he sought to demonstrate the use of variational calculus in formulating methods of approach to fundamental problems in structural mechanics rather than to give numerical solutions to specific problems.

  219. Golab biography
    • In 1956 Golab published the book Tensor calculus (Polish).
    • This is a careful book, in the classical style and the usual best traditions of the Polish school, on the Tensor Calculus, written from a geometrical point of view, and intended for students of Physics and Engineering as well as those of Mathematics.

  220. Schwarz biography
    • Schwarz attended Weierstrass's lectures on The integral calculus in 1861 and the notes that Schwarz took at these lectures still exist.
    • While in Berlin, Schwarz worked on minimal surfaces (surfaces of least area), a characteristic problem of the calculus of variations.

  221. Lowenheim biography
    • Lowenheim analysed and improved upon the customary methods of solving equations in the calculus of classes or domains (that is, set theory in its Peirce-Schroder [Charles Peirce and Ernst Schroder] setting) and proved what is now known as Lowenheim's general development theorem for functions of functions.
    • This appears in the paper Uber Moglichkeiten im Relativkalkul (On possibilities in the calculus of relatives) published in 1915.

  222. Moisil biography
    • He was appointed as Professor of Differential and Integral Calculus on 1 November 1936 at Iasi, then as Professor of Calculus in 1939.

  223. De Beaune biography
    • This statement, which is not easily justified except in the language of differential and integral calculus, was fifty years ahead of scientific developments and - by itself - reveals Debeaune's singular ability to translate physical questions into the abstract language of mathematical analysis, despite the inadequacies of the operative means of his time.
    • History Topics: The rise of the calculus .

  224. Kotelnikov biography
    • The thesis he presented for the Master's Degree was The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics.
    • Kotelnikov obtained his doctorate in 1899 for a thesis The Projective Theory of Vectors which generalised the vector calculus to the non-euclidean spaces of Lobachevsky and Riemann.

  225. Volterra biography
    • In 1890 Volterra showed by means of his functional calculus that the theory of Hamilton and Jacobi for the integration of the differential equations of dynamics could be extended to other problems of mathematical physics.
    • The theory of functionals as a generalization of the idea of a function of several independent variables was developed by Volterra in a series of papers published since 1887 and was inspired by the problems of the calculus of variations.

  226. Zermelo biography
    • His doctorate was completed in 1894 when the University of Berlin awarded him the degree for a dissertation Untersuchungen zur Variationsrechnung which followed the Weierstrass approach to the calculus of variations.
    • Immediately following the award of the degree he was appointed as a lecturer at Gottingen on the strength of his contributions to statistical mechanics as well as to the calculus of variations.

  227. Weatherburn biography
    • Gibbs and Heaviside had been early exponents of the vector calculus while its chief opponents had been Tait.
    • He published An Introduction to Riemannian Geometry and the Tensor Calculus in 1938 and it was reissued in 1966.

  228. Spottiswoode biography
    • In 1861, the year of his marriage, Spottiswoode published On typical mountain ranges: an application of the calculus of probabilities to physical geography, which attempted to use statistical methods to determine whether the mountain ranges of Asia had been formed by one or several causes.
    • His article in Journal of the Royal Asiatic Society of Great Britain and Ireland (1860) discussed the fact that that most of the principles of the differential calculus were known in ancient Indian mathematics before the period of Bhaskara II in the 12th century.

  229. Suschkevich biography
    • For example, he attended the two courses Differential calculus and Integral calculus by Hermann Schwarz, Number theory by Frobenius, and Algebra by Schur.

  230. Vagner biography
    • He published a major 70 page paper General affine and central projective geometry of a hypersurface in a central affine space and its application to the geometrical theory of Caratheodory's transformations in the calculus of variations (Russian) in 1952.
    • Vagner published the book Geometria del calcolo delle variazioni in Italian in 1965 in which he gave a systematic treatment of his own approach to the geometry of the calculus of variations, which he developed during the years 1942-1952.

  231. Colson biography
    • It was the first book on calculus written by a woman, and Colson intended the translation to make calculus more accessible to women.

  232. Mason biography
    • Mason's mathematical research interests lay in differential equations, the calculus of variations and electromagnetic theory.
    • He published seven papers in the Transactions of the American Mathematical Society between 1904 and 1910: Green's theorem and Green's functions for certain systems of differential equations (1904), The doubly periodic solutions of Poisson's equation in two independent variables (1905), A problem of the calculus of variations in which the integrand is discontinuous (1906), On the boundary value problems of linear ordinary differential equations of second order (1906), The expansion of a function in terms of normal functions (1907); The properties of curves in space which minimize a definite integral (1908) and Fields of extremals in space (1910).

  233. Gregory Duncan biography
    • Two other important works by Duncan Gregory are Examples of the Processes of the Differential and Integral Calculus and A Treatise on the Application of Analysis to Solid Geometry.
    • The first became an important text at Cambridge which, by this time, had accepted Peacock, Herschel and Babbage's Analytical Society reforms, and continental methods of calculus were being taught in Cambridge.

  234. Gutzmer biography
    • Highly gifted as a teacher, showing infectious enthusiasm for his topic, Gutzmer taught courses on a wide variety of topics including differential and integral calculus, and analytic geometry at lower level.
    • Among the advanced courses he taught we list: ordinary differential equations, analytic mechanics, calculus of variations, number theory, higher algebra, function theory and the theory of algebraic curves.

  235. Cavalieri biography
    • This contains the method of indivisibles which became a factor in the development of the integral calculus.
    • History Topics: The rise of calculus .

  236. Eisenhart biography
    • Important contributions to it were made by Bianchi, Beltrami, Christoffel, Schur, Voss, and others, and Ricci-Curbastro coordinated and extended the theory with the use of tensor analysis and his absolute calculus.
    • In 1933 Eisenhart published Continuous Groups of Transformations which continues the work of his earlier books looking at Lie's theory using the methods of the tensor calculus and differential geometry.

  237. Lesniewski biography
    • The three major logical systems which Lesniewski developed were: Protothetic, a theory of propositions and propositional functors, similar in power to a theory of propositional types, providing an extended propositional calculus with quantified functional variables; Ontology, which is an axiomatised theory of common names based on protothetic which may be characterised as a cross between traditional term logic and modern type theory, containing, besides singular terms, also empty and plural terms and a host of other interesting features; and Mereology, which is an axiomatic extension of ontology for a theory of classes quite different from set theory providing a formal theory of part and whole similar to the calculus of individuals.

  238. Berge biography
    • The symbolic calculus which he discussed in this major paper is a combination of generating functions and Laplace transforms.
    • He then applied this symbolic calculus to combinatorial analysis, Bernoulli numbers, difference equations, differential equations and summability factors.

  239. Poisson biography
    • Lagrange and Laplace recognised Fermat as the inventor of the differential and integral calculus; he was French after all while neither Leibniz nor Newton were! Poisson, however, wrote in 1831:- .
    • This [differential and integral] calculus consists in a collection of rules ..

  240. Bortolotti biography
    • From 1892 he undertook postgraduate studies at Paris then, in 1893, he was appointed to the University of Rome and taught in Rome until 1900 when he became professor of infinitesimal calculus at Modena.
    • Bortolotti studied topology at first but later went in the direction of analysis considering the calculus of finite differences, continued fractions, convergence of infinite algorithms, summation of series, the asymptotic behaviour of series and improper integrals.

  241. Valiron biography
    • He was named professor of general mathematics in 1941 then, later in the same year, named professor of differential and integral calculus at the Faculty of Science.
    • Fundamental theorems and the calculus of residues; XIV.

  242. Herschel biography
    • We should also say that the Analytical Society was not the first move towards Continental mathematics in England, for Woodhouse who was one of Herschel's lecturers at Cambridge, had written a fine book which took the Leibniz approach to the calculus rather than Newton's approach.
    • Herschel, together with Peacock, translated Lacroix's Traite du calcul differentiel et du calcul integral which examined these different approaches to the calculus.

  243. Gardner biography
    • We certainly do not want to even list the titles of over sixty works so we will give a selection: Logic Machines and Diagrams (1958); The Annotated Alice (1960); Relativity for the Million (1962); The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds (1964); Mathematical Carnival: A New Round-up of Tantalizers and Puzzles from "Scientific American" (1975); The Incredible Dr Matrix (1976); Aha! Insight (1978); Science: Good, Bad, and Bogus (1981); Aha! Gotcha: Paradoxes to Puzzle and Delight (1982); The Whys of a Philosophical Scrivener (1983); Codes, Ciphers and Secret Writing (1984); Entertaining Mathematical Puzzles (1986); Time Travel and Other Mathematical Bewilderments (1987); Perplexing Puzzles and Tantalizing Teasers (1988); Fractal Music, Hypercards and More (1991); My Best Mathematical and Logic Puzzles (1994); Classic Brainteasers (1995); Calculus Made Easy (1998); A Gardner's Workout: Training the Mind and Entertaining the Spirit (2001); Mathematical Puzzle Tales (2001); and Bamboozlers (2008).
    • (4) I devised a novel way to diagram the propositional calculus.

  244. Chebyshev biography
    • Brashman was particularly interested in mechanics but his interests were wide ranging and, in addition to courses on mechanical engineering and hydraulics, he taught his students the theory of integration of algebraic functions and the calculus of probability.
    • [I] found an occasion each day to talk with this geometer concerning [applications of calculus to number theory] as well as other questions on pure and applied analysis.

  245. Hilton biography
    • Hilton had also published Differential calculus, a 56-page text in the Library of Mathematics series.
    • This book is intended to provide the university student in the physical sciences with information about the differential calculus which he is likely to need ..

  246. Arzela biography
    • Arzela only taught at Palermo for two years for, having entered a competition for a professorship at the University of Bologna, he was appointed as professor of Infinitesimal Calculus in 1880.
    • He also wrote 'Complementi di algebra elementare' (1896) and (with G Ingrami) 'Aritmetica razionale' (1894) for the secondary school audience in addition to the university-level text, 'Lezioni di calcolo infinitesimale' (1901-06), which encompasses the lectures on infinitesimal calculus given at the University of Bologna beginning in the academic year 1880-1881.

  247. Joachimsthal biography
    • At the University of Berlin Joachimsthal taught courses on analytic geometry and calculus, giving more advanced courses on the theory of surfaces, the calculus of variations, statics and analytic mechanics.

  248. Knopp biography
    • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.

  249. Cherubino biography
    • In 1913 Cherubino was appointed as an assistant to the Chair of Infinitesimal Calculus at the University of Naples but he continued to teach in secondary schools.
    • He gave courses on infinitesimal calculus at the School of Architecture in the academic year 1927-28.

  250. Bogolyubov biography
    • In 1928 he successfully defended his thesis The Application of the Direct Methods of the Calculus of Variations to Investigation of Irregular Cases of a Simplest Problem.
    • The works of his first period, some of which were carried out by him jointly with his teacher N M Krylov, deal with direct methods of the calculus of variations, to the theory of nearly-periodic functions and approximate solutions of boundary-value differential equations.

  251. Borel biography
    • In [Enseignement mathematique 11 (1965), 1-95.',8)">8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
    • In addition to many textbooks, Borel published more than fifty papers between 1905 and 1950 on the calculus of probability.

  252. Faedo biography
    • The topic of the thesis was the Calculus of Variations and he developed ideas on the "direct method" introduced by Tonelli in the 1920s.
    • We have already seen that he made contributions to a wide variety of areas such as the calculus of variations, the theory of linear ordinary differential equations, the theory of partial differential equations, measure theory, the Laplace transform for functions of several variables, questions relating to existence for linear equations in Banach spaces, and foundational problems such as his work on Zermelo's principle in infinite-dimensional function spaces.

  253. Privalov biography
    • Later textbook were: Fourier series (1930); Course of differential calculus (1934); Course of integral calculus (1934); Integral equations (1935); Foundation of the analysis of infinitesimals, textbook for self-education (1935); and Elements of the theory of elliptic functions (1939).

  254. Francais Francois biography
    • Francais was friendly with Arbogast and together they worked on the calculus of derivations.
    • After Arbogast died in 1803, Francais inherited his mathematical papers and continued to work on the calculus of derivations.

  255. Kramp biography
    • His treatise Elements d'arithmetique universelle (Cologne, 1808) attempted to fuse ideas in the calculus of variations introduced by Louis Arbogast with basic combinatorial techniques [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • He thus strove to create an intimate union of differential calculus and ordinary algebra, as had Lagrange in his last works.

  256. Birnbaum biography
    • After arriving in Gottingen, Edmund Landau became his advisor, and he attended several lecture courses: differential equations given by Courant; calculus of variations given by Courant; power series given by Landau; higher geometry given by Herglotz; probability calculus given by Bernays; analysis of infinitely many variables given by Wegner; and attended the mathematical seminar directed by Courant and Herglotz.

  257. Valerio biography
    • 25 (2) (1983), 227-249.',7)">7] by Divizia aims to show how, in De centro gravitatis, Valerio anticipated the concept of limit and the methods of the integral calculus to calculate areas and volumes.
    • History Topics: The rise of the calculus .

  258. Fichera biography
    • However, by the time Fichera was studying with her, Nalli had become interested in tensorial calculus.
    • In pure mathematics Gaetano Fichera achieved considerable results in the following fields: mixed boundary value problems of elliptic equations; generalized potential of a simple layer; second order elliptic-parabolic equations; well posed problems; weak solutions; semicontinuity of quasi-regular integrals of the calculus of variations; two-sided approximation of the eigenvalues of a certain type of positive operators and computation of their multiplicity; uniform approximation of a complex function f(z); extension and generalization of the theory for potentials of simple and double layer; specification of the necessary and sufficient conditions for the passage to the limit under integral sign for an arbitrary set; analytic functions of several complex variables; solution of the Dirichlet problem for a holomorphic function in a bounded domain with a connected boundary, without the strong conditions assumed by Francesco Severi in a former study; construction of a general abstract axiomatic theory of differential forms; convergence proof of an approximating method in numerical analysis and explicit bounds for the error.

  259. Uhlenbeck Karen biography
    • I started out my mathematics career by working on Palais' modern formulation of a very useful classical theory, the calculus of variations.
    • She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ).

  260. Courant biography
    • Richard Courant - Differential and Integral calculus - German edition .
    • Richard Courant - Differential and Integral calculus - English edition .

  261. Faa di Bruno biography
    • It can also be found in books on partitions [George E Andrews, 'The theory of partitions' (1976)], mathematical statistics, matrix theory, calculus of finite differences, computer science [Donald E Knuth, 'The Art of Computer Programming' (1968)], symmetric functions, and miscellaneous mathematical techniques.
    • In 1871 he was put in charge of teaching calculus and analytic geometry and he was appointed as an extraordinary professor of higher analysis in 1876.

  262. Hahn biography
    • Hahn's first results were contributions to the classical calculus of variations.
    • He also made important contributions to the calculus of variations, mostly between 1903 and 1913, developing ideas of Weierstrass.

  263. Turnbull biography
    • As a result of careful scrutiny it has been established that Gregory made several remarkable and unsuspected discoveries, particularly in the calculus and the theory of numbers, which he never published.
    • H W Turnbull's Scottish Contribution to the Calculus .

  264. Barrow biography
    • They contain the important work on tangents which was to form the starting point of Newton's work on the calculus.
    • History Topics: The rise of the calculus .

  265. Humbert Pierre biography
    • His main mathematical work from the mid 1930s onwards was in developing the symbolic calculus.
    • He also wrote on applications of the symbolic calculus to mathematical physics.

  266. Lacroix biography
    • Not only did Monge use his influence to obtain this position for Lacroix, but he also acted as his mathematical advisor, recommending that he undertake research on partial differential equations and the calculus of variations.
    • He also became a professor at the Ecole Polytechnique in 1809, filling Lagrange's chair, and a professor of differential calculus at the Faculte des Sciences in 1810.

  267. Dirksen biography
    • Analytische Darstellung der Variations-Rechnung, mit Anwendung derselben auf die Bestimmung des Grossten und Kleinsten (1823) is one of the earliest comprehensive accounts of the calculus of variations.
    • This work was highly praised in reviews at the time and recommended as necessary reading by anyone wishing to delve deeply into the calculus of variations.

  268. Riccati Vincenzo biography
    • Riccati and Saladini also considered the principle of the substitution of infinitesimals in the 'Institutiones analyticae', together with the application of the series of integral calculus and the rules of integration for certain classes of circular and hyperbolic functions.
    • Their work may thus be considered to be the first extensive treatise on integral calculus ..

  269. Wilson Edwin biography
    • In 1912 Wilson published the first American advanced calculus text [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • a comprehensive text on advanced calculus that was the first really modern book of its kind in the United States.

  270. Stieltjes biography
    • I have been offered, some days ago, a professorship in analysis (differential and integral calculus) at the University of Groningen.
    • In the same year Stieltjes was appointed to the University of Toulouse, being appointed to a chair of differential and integral calculus in Toulouse in 1889.

  271. Peirce B O biography
    • Together they taught calculus and its applications, constructing a two year course which Peirce and Byerly taught starting in alternate years.
    • Byerly's text Elements of the integral calculus was first published in 1881, while Peirce's text Elements of the theory of the Newtonian potential function was first published in 1886.

  272. Good biography
    • He continued to read mathematics books, for example when he was older he read Joseph Edward's Differential Calculus with Applications and Numerous Examples and G H Hardy's Pure Mathematics.
    • The author, in this book, is interested in how the results of those manipulations in the probability calculus can be used in practical circumstances.

  273. Lichtenstein biography
    • He did pioneering work in potential theory, integral equations, calculus of variations, differential equations and hydrodynamics.
    • [Lichtenstein] made important contributions to the theory of partial differential equations, and the calculus of variations.

  274. Toeplitz biography
    • For example he wrote an excellent book on the history of the calculus The Calculus: A Genetic Approach (1963).

  275. Somov biography
    • Among his works (all written in Russian) were Analytic theory of the undulatory motion of the ether (1847), Foundations of the theory of elliptical functions (1850), Course in differential calculus (1852), Analytic geometry (1857), Elementary algebra (1860), Descriptive geometry (1862) and the two volume treatise Rational mechanics (1872-74).
    • He was the author of important works in the field of theoretical mechanics, theory of hinged mechanisms, synthesis of mechanisms, and screw and vectorial calculus.

  276. Norlund biography
    • This is the first book to develop the theory of the difference calculus from the function-theoretic point of view and to include a significant part of the recent researches having to do with the analytic and asymptotic character of the solutions of linear difference equations.
    • The author presents a connected account of what appears to him to be the most important and the best developed domains of the difference calculus.

  277. Sommerfeld biography
    • In his note he proposed a special way of denoting vectors, vector calculus and the electromagnetic magnitudes, which became obligatory for all contributors.
    • Its aim was to create a unified vector symbolism and calculus.

  278. Einstein biography
    • He studied mathematics, in particular the calculus, beginning around 1891.
    • About 1912, Einstein began a new phase of his gravitational research, with the help of his mathematician friend Marcel Grossmann, by expressing his work in terms of the tensor calculus of Tullio Levi-Civita and Gregorio Ricci-Curbastro.

  279. Bougainville biography
    • This extended de l'Hopital's book, written more that half a century earlier, to cover the integral calculus and also updated the differential calculus providing an up-to-date text.

  280. Williams biography
    • My first teacher in the calculus was Professor Williams.
    • Kind though he was, he also had high standards, and he failed 75 percent of the calculus class one year! .

  281. Bisacre biography
    • In 1921 Bisacre's Applied calculus; an introductory textbook was published by Blackie and Son.
    • F F P Bisacre - Applied calculus .

  282. Kiselev biography
    • Here are the most important of Kiselev's textbooks: Systematic arithmetic course for secondary schools (1884); Elementary Algebra (1888); Elementary Geometry (1892-1893); Additional topics in algebra (a course of the 7th grade real schools) (1893); Quick arithmetic for urban schools (1895); A Brief algebra for girls' schools and seminaries (1896); An elementary physics for secondary schools with a number of exercises and problems (1902); Physics (two volumes) (1908); Elements of the differential and integral calculus (1908); The initial study of derivatives for the 7th grade real schools (1911); Graphical representation of some of the features discussed in elementary algebra (1911); On topics in elementary geometry, which are usually solved by means of limits (1916); Brief algebra (1917); Brief arithmetic for urban county schools (1918); Irrational numbers, considered as infinite non-periodic fractions (1923); and Elements of algebra and analysis (2 volumes) (1930-1931).
    • However, books on other subjects such as Physics, which went through 13 editions, and Elements of the differential and integral calculus which also went through multiple editions were also popular but never came near to matching the incredible popularity of Systematic arithmetic course for secondary schools, Elementary Algebra and Elementary Geometry.

  283. Teichmuller biography
    • I am not concerned with making difficulties for you as a Jew, but only with protecting - above all - German students of the second semester from being taught differential and integral calculus by a teacher of a race quite foreign to them.
    • But I know that many academic courses, especially the differential and integral calculus, have at the same time educative value, inducting the pupil not only to a conceptual world but also to a different frame of mind.

  284. Thabit biography
    • played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.
    • Thabit's work on parabolas and paraboliods is of particular importance since it is one of the steps taken towards the discovery of the integral calculus.

  285. Serret biography
    • Two years later he was appointed to the chair of differential and integral calculus at the Sorbonne.
    • Jean-Claude Bouquet took over Serret's lecture courses at the Faculty of Science in 1871 and was appointed professor of differential and integral calculus in 1874.
    • Serret also published papers on number theory, calculus, the theory of functions, group theory, mechanics, differential equations and astronomy.

  286. Lhuilier biography
    • The Academy sought the best article on the theory of the mathematical infinity and they designed the competition to encourage mathematicians to seek a sound basis for the new differential calculus.
    • The standard concepts and notation for derivatives, and the standard elementary theorems on limits which appear in an undergraduate calculus text today appear in a remarkably similar form in Lhuilier's prize winning essay.

  287. Bachelier biography
    • One of his courses was Probability calculus with applications to financial operations and analogies with certain questions from physics.
    • convergence in distribution), martingales and Ito stochastic calculus.

  288. Scherk biography
    • At Halle, Scherk taught a wide range of courses such as: analytic geometry of lines and the conic sections; analytic geometry of lines and surfaces of the first and second degree; algebra and algebraic geometry; plane and spherical trigonometry; integral calculus; and differential calculus and its application to algebra, analysis and geometry.

  289. Cimmino biography
    • Professor Gianfranco Cimmino can be considered, among other things, as one of the founders of the Istituto per le Applicazioni del Calcolo, to which he lent his continuing and productive assistance during the embryonic stages of the Institute itself, in Naples, in the laboratory annexed to that university's Calculus chair, from July 1928 to October 1932.
    • However, he also made important contributions to many other areas of mathematics, for example the calculus of variations; differential geometry; conformal and quasi-conformal mappings; topological vector spaces; and the theory of distributions.

  290. Stokes biography
    • a student was to become acquainted with the differential and integral calculus and to go on to statics, dynamics, conic sections and the first three sections of Newton's "Principia"..
    • In those days boys coming to the University had not in general read so far in mathematics as is the custom at present; and I had not begun the differential calculus when I entered the College, and had only recently read analytical sections.

  291. Paman biography
    • As this states, Paman had written his work to answer the criticisms of the calculus presented by Berkeley in The Analyst published in 1734.
    • Paman managed to give foundations to the calculus using these concepts, and he needed all of them.

  292. Levy Paul biography
    • This involved extending the calculus of functions of a real variable to spaces where the points are curves, surfaces, sequences or functions.
    • notions of calculus of probabilities and the role of Gaussian law in the theory of errors.

  293. Bayes biography
    • Bayes also wrote an article An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst (1736) attacking Berkeley for his attack on the logical foundations of the calculus.
    • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.

  294. Viviani biography
    • However, he lived at a time when the calculus was beginning to be used to prove geometric results.
    • The two got on well during this meeting but after Leibniz published Solutio problematis a Galilaeo propositi de linea catenaria in 1692, Viviani became unhappy about the use of the differential and integral calculus which he believed to be nothing but a kind of game that could only solve its own problems.

  295. Feynman biography
    • He studied a lot of mathematics in his own time including trigonometry, differential and integral calculus, and complex numbers long before he met these topics in his formal education.

  296. Durell biography
    • Among the books he wrote around this time were: Readable relativity (1926), A Concise Geometry (1928), Matriculation Algebra (1929), Arithmetic (1929), Advanced Trigonometry (1930), A shorter geometry (1931), The Teaching of Elementary Algebra (1931), Elementary Calculus (1934), A School Mechanics (1935), and General Arithmetic (1936).

  297. Weyl biography
    • There I attended his lectures on the Elie Cartan calculus of differential forms and their application to electromagnetism - eloquent, simple, full of insights.

  298. Democritus biography
    • History Topics: The rise of the calculus .

  299. Lueroth biography
    • This mechanics book makes heavy use of the vector calculus, being the first one to do so.

  300. Kramer biography
    • Her examination of Omar Khayyam and algebra, Newton and calculus, Fermat and probability, Lewis Carroll and logic and Einstein and relativity provides an intriguing book for non-mathematicians and a valuable reference source for teachers and students.

  301. Titeica biography
    • Returning to Romania, Titeica was appointed as an assistant professor at the University of Bucharest where he taught the course on differential and integral calculus.

  302. Descartes biography
    • History Topics: The rise of the calculus .

  303. Tannery Jules biography
    • From 1884 he was an adviser of studies at the Ecole Normale and, from 1903, Professor of differential and integral calculus at the Faculty of Science in Paris.

  304. Lamb biography
    • Lamb wrote books in addition to those mentioned above, including Infinitesimal Calculus (1897), Dynamical Theory of Sound (1910), and Higher Mechanics (1920).

  305. Chisholm Young biography
    • She continued to work on mathematical research and, between 1914 and 1916, she published work on the foundations of calculus under her own name.

  306. Carmichael biography
    • Also in 1927, in collaboration with James Henry Weaver (1883-1942), Carmichael published The Calculus.

  307. Bartik biography
    • Returning to Northwest Missouri State Teachers College for her junior year, she found that nobody else was majoring in mathematics and, in some of the courses, for example Calculus and Astronomy, she was one of only two students, the other student being a young man from Peru.

  308. Fraser biography
    • In fact Fraser had been intending to address the meeting of the Society on Friday 14 November 1890 on the topic of the history of the controversy concerning the differential calculus.

  309. Calderon biography
    • In particular Calderon wanted to describe a calculus for elliptic differential operators and, from this beginning in the 1950s, the theory of pseudodifferential operators grew in the 1960s.

  310. Kovalevskaya biography
    • Studying the wallpaper was Sofia's introduction to calculus.

  311. Ostrogradski biography
    • His papers at this time show the influence of the mathematicians in Paris and he wrote on physics and the integral calculus.

  312. Janovskaja biography
    • The history of mathematics was another topic which attracted Janovskaja and she published work on Egyptian mathematics On the theory of Egyptian fractions (1947), Zeno of Elea's paradoxes, Rolle's criticisms of the calculus in Michel Rolle as a critic of the infinitesimal analysis (1947), Descartes's geometry (see below), and Lobachevsky's work on non-euclidean geometry in papers such as The leading ideas of N I Lobachevsky - a combat weapon against idealism in mathematics (1950), On the philosophy of N I Lobachevsky (1950), and On the Weltanschauung of N I Lobachevsky (1951).

  313. Guldin biography
    • Clavius was, however, a classical mathematician teaching only Euclid's geometric methods and Guldin would also take this classical approach and oppose the newer ideas of the calculus which were beginning to appear around this time.

  314. Manfredi biography
    • Guglielmini taught Manfredi the differential calculus and he soon became interested in hydraulics, but also taught himself astronomy.

  315. Doeblin biography
    • It contains pieces of what we now call stochastic calculus, including a version of Ito's formula.

  316. Sluze biography
    • One has to say that this work on tangents makes de Sluze a major figure in the development of the calculus.

  317. Schonflies biography
    • He wrote textbooks on descriptive geometry and analytic geometry, and a calculus textbook Einfuhrung in die mathematische Behandlung der Naturwissenschaft (1895) written jointly with Walter Nernst.

  318. Stark biography
    • Besides items usually dealt with in textbooks of elementary analytic geometry the reader finds here the introduction to synthetic projective geometry, to the theory of matrices (and determinants) with the usual applications and to the (three-dimensional) elementary vector calculus.

  319. Pick biography
    • His mathematical work was extremely broad and his 67 papers range across many topics such as linear algebra, invariant theory, integral calculus, potential theory, functional analysis, and geometry.

  320. Tschirnhaus biography
    • He also sets out some of the fundamental principles of the calculus that he has developed.

  321. Kurschak biography
    • Another topic which Kurschak investigated was the differential equations of the calculus of variations.

  322. Kalman biography
    • The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations.

  323. Tucker Albert biography
    • He published papers such as: An abstract approach to manifolds (1933); On tensor invariance in the calculus of variation (1934); Non-Riemannian subspaces (1935); Cell spaces (1936); On chain-mappings carried by cell-mappings (1939); and The algebraic structure of complexes (1939).

  324. Feigl biography
    • At this time Feigl was working with Schmidt on a book on differential and integral calculus.

  325. Cartan biography
    • His appointment in 1909 was as a lecturer at the Sorbonne but three years later he was appointed to the Chair of Differential and Integral Calculus in Paris.

  326. Pars biography
    • Thereafter he published little, although he did publish two valuable textbooks, Introduction to dynamics in 1953 and Calculus of variations in 1962, until he published a monumental 650 page work Treatise on analytical dynamics in 1965.

  327. La Hire biography
    • Although his rejection of the infinitesimal calculus may have rendered a part of his mathematical work sterile, his early works in projective, analytic, and applied geometry place him among the best of the followers of Desargues and Descartes.

  328. Ruffini biography
    • Among his teachers of mathematics at Modena were Luigi Fantini, who taught Ruffini geometry, and Paolo Cassiani, who taught him calculus.

  329. Roberval biography
    • History Topics: The rise of Calculus .

  330. Greenhill biography
    • Among these books were Differential in integral calculus (1886), The applications of elliptic functions (1892), Treatise on hydrostatics (1894), Notes on dynamics (1908), Theory of stream lines with applications to an aeroplane (1910), Dynamics of mechanical flight (1912), and Gyroscopic theory (1914).

  331. Mendelsohn biography
    • Two of Mendelsohn's papers An algorithmic solution for a word problem in group theory (1964) and (with Clark T Benson) A calculus for a certain class of word problems in groups (1966) were particularly important in launching this strand of my own research career - thank you Nathan! .

  332. Von Staudt biography
    • By his calculus of 'throws' he gave an outline of the modern algebraisation of axiomatic theory.

  333. Halmos biography
    • I was a routine calculus student - I think I got B's.

  334. Cosserat biography
    • In 1896 he became professor of differential and integral calculus there, replacing Thomas Stieltjes who had died on 31 December 1894, and, from that time on, he divided his work between the Faculty of Science and the Observatory.

  335. Milne-Thomson biography
    • Of course the main mathematical tool used in constructing tables was the method of finite differences and in 1933 Milne-Thomson published his first textbook, The Calculus of Finite Differences, a text in which he set out to explain to students the techniques which he used in table making.

  336. Takagi biography
    • At Tokyo University Takagi took courses on calculus and analytic geometry.

  337. Moore Robert biography
    • Robert received a good education at a private high school in Dallas, and before he entered university he had learnt university level calculus by studying the university textbooks.

  338. Delaunay biography
    • For his doctoral dissertation Delaunay undertook research on the calculus of variations and was awarded his doctorate for his thesis De la distinction des maxima et des minima dans les questions qui dependent de la methode des variations in 1841.

  339. Godel biography
    • He completed his doctoral dissertation under Hahn's supervision in 1929 submitting a thesis proving the completeness of the first order functional calculus.

  340. Feigenbaum biography
    • He had already taught himself to play the piano when he was about 12 years old, but at high school he taught himself calculus.

  341. Word problems
    • Independently of Godel, Alonzo Church was developing the λ-calculus designed to clarify the foundations of mathematics, in particular the meaning of variables.

  342. Word problems
    • Independently of Godel, Alonzo Church was developing the λ-calculus designed to clarify the foundations of mathematics, in particular the meaning of variables.

  343. Belanger biography
    • So the Ecole Centrale des Arts et Manufactures devised a teaching plan, attempting to satisfy the condition without sacrificing in proofs the tight logic without which mathematics becomes an often misleading semi-science, not compromising clarity by excessive brevity, but by choosing those parts of analytic geometry and the infinitesimal calculus which every engineer must know, and especially those which are necessary for the study of mechanics viewed from the point of view of its practical application to industrial work.

  344. Weyr Eduard biography
    • In 1900 Weyr was asked by a Committee of the Union of Czech Mathematicians to write a calculus textbook.
    • He published Differential calculus in 1902.

  345. Black Fischer biography
    • Inter alia, Bachelier, had shown in his thesis [The Random Character of Stock Market Prices, MIT Press, Cambridge, Massachusetts (contains the translation from French of Bachelier\'s doctoral thesis and contains Sprenkle\'s, 1961 paper).',88)">88] the close connection between random walks and the Fourier heat equation, something that was expanded on by Kac, in 1951, [Ito\'s stochastic calculus and probability theory, Tokyo, ix-xiv.

  346. Goodstein biography
    • which presented a novel approach to elementary differential and integral calculus ..

  347. Mises biography
    • He classified his own work, not long before his death, into eight areas: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy.

  348. Gopel biography
    • At Pisa he was taught algebra, differential calculus, statics, analytical mechanics, theoretical physics and experimental physics.

  349. Penrose biography
    • This volume covered two-spinor calculus and relativistic fields while the second volume covering spinor and twistor methods in space-time geometry appeared two years later.

  350. Whyburn biography
    • Robert Moore had been appointed as associate professor at the University of Texas in 1920 and it was Moore who taught Whyburn calculus early in his university studies.

  351. Plessner biography
    • In 1921 Plessner went to Gottingen where he took courses on Dirichlet series and Galois theory by Edmund Landau; algebraic number fields by Emmy Noether; and calculus of variations by Courant.

  352. Kirby biography
    • The techniques and objects emphasized here are framed links and the Kirby calculus, spin structures, and immersion methods.

  353. Pitt biography
    • This compact account develops the theory as it applies to abstract spaces, describes its importance to differential and integral calculus, and shows how the theory can be applied to geometry, harmonic analysis, and probability.

  354. Castel biography
    • In particular he taught infinitesimal calculus and mechanics at the Lycee.

  355. Schmidt Harry biography
    • The first two chapters give a clear exposition of the elements of vector algebra and calculus with some physical applications.

  356. Fefferman biography
    • It is claimed that he had mastered the calculus before the age of twelve.

  357. Angeli biography
    • History Topics: The rise of the calculus .

  358. Dijkstra biography
    • A joint work with Carel S Scholten, Predicate calculus and program semantics, was published in 1990.

  359. Duhem biography
    • It is to this reading, to these exchanges of views, that I owe the greater part of my later works, almost all of which deal with the calculus of variations, the theory of Hugoniot, hyperbolic partial differential equations, Huygens' principle.

  360. Hammersley biography
    • This covered plenty of Euclidean geometry (including such topics as the nine-point circle) and algebra (Newton's identities for roots of polynomials) and trigonometry (identities governing angles of a triangle, circumcircle, incircle, etc), but no calculus.

  361. Siegel biography
    • By 1928 Siegel was teaching 143 students in the differential and integral calculus course, and had to put in many hours work correcting students exercises.

  362. Cotes biography
    • His substantial advances in the theory of logarithms, the integral calculus, in numerical methods particularly interpolation and table construction of integrals for eighteen classes of algebraic functions led Newton to say:- .

  363. Krylov Nikolai biography
    • With his collaborator and former student N N Bogolyubov, he published On Rayleigh's principle in the theory of differential equations of mathematical physics and on Euler's method in calculus of variations (1927-8) and On the quasiperiodic solutions of the equations of the nonlinear mechanics.

  364. Frege biography
    • He lectured on all branches of mathematics, in particular analytic geometry, calculus, differential equations, and mechanics, although his mathematical publications outside the field of logic are few.

  365. D'Adhemar biography
    • He taught differential and integral calculus.

  366. Shimura biography
    • He taught linear algebra and calculus and continued to undertake research publishing articles A note on the normalization-theorem of an integral domain (1954) and Reduction of algebraic varieties with respect to a discrete valuation of the basic field (1955).

  367. Pringsheim biography
    • He also suggested that the paradoxes of the infinitary calculus arose from transferring properties of real numbers to infinite-dimensional domains where they fail, and agreed with Cantor that any use of infinitesimals in analysis would necessarily lead to inconsistencies.

  368. Vashchenko biography
    • He published The Symbolic Calculus and its Application to the Integration of Linear Differential Equations in 1862.

  369. Branges biography
    • George Thomas was writing a text on the calculus and analytic geometry which was tested on the incoming freshman class.

  370. Hopf Eberhard biography
    • His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis.

  371. Grauert biography
    • I (Differential- and integral calculus Vol.

  372. Chen biography
    • He published Integration in free groups (1951), Commutator calculus and link invariants (1952), Isotopy invariants of links (1952), and A group ring method for finitely generated groups (1954).

  373. Matiyasevich biography
    • The first five lead to the negative solution of Hilbert's Tenth Problem; the remaining chapters are devoted to various applications of the method used by the author, which is, in a sense, more important than the solution itself: it has applications to Hilbert's eighth problem, decision problems in number theory, Diophantine complexity, decision problems in calculus, and Diophantine games.

  374. Mihoc biography
    • Only two of the 26 are singled authored: Treatise on actuarial mathematics (Romanian) (1943) and An Introduction to the calculus of probability (Romanian) (1954).

  375. Artin Michael biography
    • This is a remarkable text designed for highly motivated undergraduates having some preparation in linear algebra and some other post-calculus mathematics.

  376. Lukasiewicz biography
    • He worked on mathematical logic, wrote essays on the principle of non-contradiction and the excluded middle around 1910, developed a three value propositional calculus (1917) and worked on many valued logics.

  377. Le Cam biography
    • He continued to take University of Paris courses in calculus and rational mechanics but needed a third course to obtain a diploma.

  378. Hedrick biography
    • This strengthen his interests in differential equations, the calculus of variations, and functions of a real variable which he would work on for the rest of his life.

  379. Kellogg biography
    • He wrote a number of articles on the teaching of mechanics, and published a textbook, written jointly with Hedrick, Applications of the calculus to mechanics (1909).

  380. Gelfond biography
    • Many of his contributions to approximation and interpolation theories are recounted in Ischislenie konechnykh raznostey (The calculus of finite differences) (1952).

  381. Thomae biography
    • After the Seven Weeks' War (as this short war is called) Thomae returned to Gottingen and gave a lectures on determinants and on the differential and integral calculus.

  382. Birman biography
    • in mathematics from Barnard College in 1948 but had been put off continuing to study mathematics by courses such as calculus which left her feeling she did not understand the subject.

  383. Bezout biography
    • In particular Harvard University adopted them as calculus textbooks.

  384. Efimov biography
    • gives a clear elementary treatment of the calculus of exterior differential forms in Rn, leading up to a proof of Stokes's theorem.

  385. Pfaff biography
    • constituted the starting point of a basic theory of integration of partial differential equations which, through the work of Jacobi, Lie, and others, has developed into a modern Cartan calculus of extreme differential forms.

  386. Schneider biography
    • From a position of almost no students in 1920, by 1928 there were 143 students taking the first semester differential and integral calculus course.

  387. Severi biography
    • There he began teaching a variety of courses from calculus to higher geometry.

  388. De Giorgi biography
    • Influenced by methods which Caccioppoli had developed, De Giorgi went on to develop new techniques in geometric measure theory and he applied his results to the calculus of variations proving his regularity theorem for almost all minimal surfaces.

  389. MacDuffee biography
    • The students are familiar with the process from integral calculus and some of them are astonished when you point out that the universality of the method had not been proved to them.

  390. MacRobert biography
    • The treatment is at the level of a course in advanced calculus; accordingly no contour integration methods are used, and all variables and parameters are real, except in the last two chapters in which the variable is complex.

  391. Ribenboim biography
    • In July of that year he returned from France to Rio de Janeiro where he was appointed to teach calculus at Escola Tecnica do Exercito and also appointed to teach analytic functions in the Centro Brasileiro de Pesquisas Fisicas.

  392. Pople biography
    • He read about the differential and integral calculus, teaching himself how to solve differential equations.

  393. Lovasz biography
    • In recent years there has been a plethora of textbooks on discrete mathematics, designed as a counter-balance to the over-emphasis on calculus in colleges and universities.

  394. Kober biography
    • He received the degree for his thesis Konjugierte kinetische Brennpunkte which made important contributions to the calculus of variations.

  395. Ahlfors biography
    • the high school curriculum did not include any calculus, but I finally managed to learn some on my own, thanks to clandestine visits to my father's engineering library.

  396. Kahler biography
    • [For my thesis I] received much advice from Kahler, from whom I learned the subject of exterior differential calculus and what is now known as the Cartan-Kahler theory.

  397. Cramer Harald biography
    • In this classic of statistical mathematical theory, Harald Cramer joins the two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probabilists transformed the classical calculus of probability into a rigorous and pure mathematical theory.

  398. Levi Beppo biography
    • In content it has much in common with American works on "advanced calculus," differing from them in the greater attention given to algebra and in the commendably comprehensive points of view.

  399. Du Val biography
    • He published on the de Sitter model of the universe and Grassmann's tensor calculus.

  400. Pairman biography
    • Miss Pairman also attended a short course I give on Hamiltonian Quaternions with Physical Applications, and there I was impressed with her capacity for appreciating the theoretical foundations of the calculus.

  401. Antoine biography
    • Antoine published a textbook based on courses he had taught on the differential and integral calculus; the 234 page volume Calcul differentiel et calcul integral: Calcul integral (1948) and the 194 page volume Calcul differentiel et calcul integral: Calcul differentiel (1949).

  402. Skolem biography
    • It states that if a theory within first-order predicate calculus has a model then it has a countable model.

  403. Lagny biography
    • One should note that although methods based on the differential calculus were being developed at this time, neither Lagny not Halley used these new ideas.

  404. Libri biography
    • Arago, the perpetual secretary of the Academy, helped him obtain a further appointment at the College de France in 1833 where he taught, and in the following year he was elected as assistant professor in the calculus of probabilities at the Sorbonne.

  405. Hilbert biography
    • Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.

  406. Hecht biography
    • His later texts covered topics such as quadratic and cubic equations, differential and integral calculus, and arithmetic and geometry.

  407. Casorati biography
    • In 1862 he was appointed as an ordinary professor at Pavia and in the following year he succeeded Mainardi to the chair of infinitesimal calculus.

  408. Peschl biography
    • The titles of the chapters are: Algebra and geometry of complex numbers; Fundamental topological concepts, sets, sequences of complex numbers and infinite series; Functions, real and complex differentiability and holomorphy; Integral theorems and their consequences; Winding number and curves homologous to zero; Taylor development of holomorphic functions; Elementary transcendental functions; Laurent series, isolated singularities and residue calculus; Holomorphic and meromorphic functions obtained by limiting processes; Analytic continuation; and Conformal mappings.

  409. Ladyzhenskaya biography
    • Not only did she love to discuss mathematics with her father but she also studied calculus with him as an equal.

  410. Ferrar biography
    • He published Mathematics for science (1965), Calculus for beginners (1967) and Advanced mathematics for science (1969) all when over the age of 70.

  411. Piaggio biography
    • In Nature Piaggio published articles such as The operational calculus (1943) and The significance and development of Hamilton's quaternions (1943).

  412. Laurent Hermann biography
    • It is divided into two parts, of two and five volumes respectively, on the differential and integral calculus, and included not only the standard treatment of the derivative and the integral and their applications to geometry but also substantial sections on the theory of functions, determinants and elliptic functions.

  413. Griffiths biography
    • in 1978) Topics in algebraic and analytic geometry (1974); Entire holomorphic mappings in one and several complex variables (1976); Principles of algebraic geometry (1978); An introduction to the theory of special divisors on algebraic curves (1980); (with John W Morgan) Rational homotopy theory and differential forms (1981); Exterior differential systems and the calculus of variations (1983); (with Gary R Jensen) Differential systems and isometric embeddings (1987); Introduction to algebraic curves (1989); and (with Mark Green, a doctoral student of Griffiths' who was awarded his Ph.D.

  414. Whittaker biography
    • His many lecture courses on this topic were collected into a book which he published in 1924 The Calculus of Observations: a treatise on numerical mathematics.

  415. Renyi biography
    • He also produced a number of outstanding books including The calculus of probabilities (Hungarian) (1954).

  416. Tichy biography
    • As a result, the author discusses a wide range of living philosophical issues in the latter half of his book: Church's logic of sense and denotation, Montague's intensional logic, Gentzen's sequent calculus, and Hilbert's formal axiomatics, to name a few.

  417. Robinson biography
    • Thus, there exist extensions of the field of real numbers that possess all the properties of the system of real numbers that are formulated in the lower predicate calculus in terms of some given set of relations.

  418. Menger biography
    • I am lecturing on the calculus of variations.
    • Also during the war years Menger's contribution to the war effort was teaching calculus to Naval cadets as part of the V-12 Navy College Training Program which ran from 1942 to 1944.
    • This led to his interest in mathematical education and, during the 1950s and 1960s, he wrote articles on mathematical education and published books with new ideas on teaching calculus, geometry and other branches of mathematics.
    • Menger on the Calculus of Variations .

  419. Tapia biography
    • He used Newton-like iterations to solve the generalized Euler-Lagrange equation of the calculus of variations.

  420. Brash biography
    • In Brash's case he joined in December 1912 although while his application for membership was being processed he had read a paper to the Society entitled Two general results in the differential calculus at the meeting on Friday 14 June 1912.

  421. Dubreil-Jacotin biography
    • she became a full professor at Poitiers where in 1955 she held the chair in Differential and Integral Calculus.

  422. Erdelyi biography
    • In addition to the five volumes which arose from the Bateman project mentioned above, Erdelyi wrote two other texts of major importance Asymptotic expansions (1955) and Operational calculus and generalised functions (1962).

  423. Nikodym biography
    • Some of his other books were: Introduction to differential calculus, (Warsaw, 1936) (jointly with his wife), Theory of tensors with applications to geometry and mathematical physics, I, (Warsaw, 1938), Differential Equations, (Poznan, 1949).

  424. Sato biography
    • Sato's theory of hyperfunctions allows much freer calculations than does classical calculus.

  425. Drach biography
    • that the theory of groups is inseparable from the study of the transcendental quantities of the integral calculus.

  426. Smullyan biography
    • He had never completed sufficient courses to merit the award, but to make up the number Chicago credited him with a calculus course which he had never taken but was teaching.

  427. Flajolet biography
    • The approach to quantitative problems of discrete mathematics provided by analytic combinatorics can be viewed as an operational calculus for combinatorics organized around three components.

  428. Aubin biography
    • It covers topics every working mathematician (or theoretical physicist) ought to know: tensorial, differential and integral calculus on smooth manifolds, and basic Riemannian geometry.

  429. Walsh Joseph biography
    • The topics he taught, rotating them from year to year, included calculus, algebra, mechanics, differential equations, complex variable, probability, number theory, potential theory, approximation theory, and function theory.

  430. Mascheroni biography
    • Despite the error in the calculation, Mascheroni's work shows a deep understanding of Euler's calculus.

  431. Blaschke biography
    • Although Weierstrass had supplied the missing proofs using the calculus of variations, this did not satisfy Blaschke who gave proofs in the style of Steiner in Kreis und Kugel.

  432. De Morgan biography
    • (Over the years he was to write 712 articles for the Penny Cyclopedia.) The Penny Cyclopedia was published by the Society for the Diffusion of Useful Knowledge, set up by the same reformers who founded London University, and that Society also published a famous work by De Morgan The Differential and Integral Calculus.

  433. Kirchhoff biography
    • Problems remained, however, which Kirchhoff solved using variational calculus.

  434. Hsiung biography
    • It begins with a review on point-set topology, multi-dimensional calculus and linear algebra.

  435. Schroeter biography
    • Schroter quickly learnt the foundations of the differential calculus and he decided that he wanted to apply mathematics.

  436. Schlafli biography
    • Although he was only fifteen years old when he entered the Gymnasium, Schlafli was already studying the differential calculus using Kastner's famous book Mathematische Anfangsgrunde der Analysis des Unendlichen.

  437. Weldon biography
    • Realising that his mathematical skills were somewhat less than he wished, Weldon read widely studying, in particular, the leading works by the French mathematicians on the calculus of probability.

  438. Grunsky biography
    • He then introduces the calculus of alternating multilinear forms and gives a proof of Stokes's theorem for manifolds.

  439. Liu Hui biography
    • There is also evidence that he is beginning to understand concepts associated with early work on the differential and integral calculus.

  440. Singer biography
    • He is perhaps the only American mathematician to hold a Distinguished University Professorship who regularly teaches ordinary (as opposed to Honours) first semester calculus.

  441. Szekeres biography
    • He had taken a calculus course while an undergraduate at the Technological University of Budapest but this was the only mathematics course he had officially studied.

  442. Jeffery Ralph biography
    • He took a degree in economics at Acadia University, but while studying for this degree he took two mathematics courses, one in calculus and one in analytic geometry.

  443. Roch biography
    • These included: Differential and Integral Calculus; Analytic Geometry; and Elliptic and Abelian Functions.

  444. Cantelli biography
    • In the same year he published La tendenza a un limite nel senso del calcolo delle probabilita (Convergence to a limit in the sense of the calculus of probabilities).

  445. Picard Emile biography
    • In 1885 Picard was appointed to the chair of differential calculus at the Sorbonne in Paris when the chair fell vacant on the death of Claude Bouquet.

  446. Jordan biography
    • However, the courses at the Ecole Polytechnique were supposed to train students to become civil and military engineers and this does not seem to be the approach which one would take trying to teach applications of the calculus to engineers.

  447. Possel biography
    • He held the chairs of rational mechanics, then differential and integral calculus.

  448. Black biography
    • [Black suggests] that intelligent choice might well be thought of as the exercise of an informal, practical art, rather than the application of a mathematical calculus.

  449. Lorentz biography
    • He was also the author of a textbook of the differential and integral calculus; "Visible and Invisible Movements", 1901; and "Clerk Maxwell's Electromagnetic Theory", 1924.

  450. Todhunter biography
    • In addition to the fellowship of the Royal Society he served on its Council in 1874, the same year in which he was awarded the Adams Prize for his work Researches on the calculus of variations.

  451. Schubert Hans biography
    • At Halle Schubert taught a variety of different courses such as differential and integral calculus, partial differential equation, and integral equations.

  452. Vranceanu biography
    • Other topics he studied include the absolute differential calculus of congruences, analytical mechanics, partial differential equations of the second order, non-holonomic unitary theory, conformal connection spaces, metrics in spherical and projective spaces, Lie groups, global differential geometry, discrete groups of affine connection spaces, locally Euclidean connection spaces, Riemannian spaces of constant connection, differentiable varieties, embedding of lens spaces into Euclidean space, tangent vectors of spheres and exotic spheres, the equivalence method, non-linear connection spaces, and the geometry of mechanical systems.

  453. Martin Lajos biography
    • Szily applied the calculus of variations in his computations but gave up the tiresome examination of the second variation.

  454. Privat de Molieres biography
    • Privat de Molieres published Lecons de mathematiques (1726), a work on the principles of algebra and calculus.

  455. Tarski biography
    • During this period, in 1941, he published an important paper calculus of relations.

  456. Meyer Paul-Andre biography
    • The book consists of three parts [Elements of Probability Calculus; Martingale theory; Analytical tools in Potential theory] which are "connected by a pattern of analogies rather than by explicit logical relations." But ..

  457. Herstein biography
    • Among the methods and problems discussed in some detail are a derivation of the Slutsky equation via the calculus, a problem in Welfare Economics treated by the theory of convex sets, matrix theory as applied to international trade, and a game-theoretical approach to the personnel assignment problem.

  458. Hamilton biography
    • I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions [the calculus] was for the close of the seventeenth.

  459. Zylinski biography
    • Besides items usually dealt with in textbooks of elementary analytic geometry the reader finds here the introduction to synthetic projective geometry, to the theory of matrices (and determinants) with the usual applications and to the (three-dimensional) elementary vector calculus.

  460. Hooke biography
    • History Topics: The rise of the calculus .

  461. Post biography
    • thesis, in which he proved the completeness and consistency of the propositional calculus described in the Principia Mathematica by introducing the truth table method.

  462. Ledermann biography
    • Other books which Ledermann has written for undergraduates include Complex numbers (1960), Integral calculus (1964), Multiple integrals (1966), Introduction to group theory (1973), and Introduction to group characters (1977).

  463. Rademacher biography
    • His initial mathematical interests were in the theory of real functions which he was taught by Caratheodory who also taught him the calculus of variations.

  464. Monge biography
    • The four memoirs that Monge submitted to the Academie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics.

  465. Lexell biography
    • He also gave a proof which was not based on using the calculus of variations.

  466. Lewy biography
    • he published a series of fundamental papers on partial differential equations and the calculus of variations.

  467. Cosserat Francois biography
    • This work was carried out in collaboration with his brother, Eugene Cosserat, who was, at the time the collaboration began, professor of differential and integral calculus at the University of Toulouse.

  468. Fine Henry biography
    • Among the elementary texts he wrote are Number system of algebra treated theoretically and historically (1891), A college algebra (1905), Coordinate geometry (1909), and Calculus (1927).

  469. Specker biography
    • In his first term at ETH he attended linear algebra lectures by Michel Plancherel and calculus lectures by W Saxer.

  470. Wright Sewall biography
    • The next five years spent at Lombard College saw him move away from chemistry and instead he studied mathematics, reaching differential and integral calculus, and surveying in classes taught by his father.

  471. Sokhotsky biography
    • Furthermore, Sokhotskii was the first to apply the calculus of residues to Legendre polynomials.

  472. Carleman biography
    • Carleman wrote also a Textbook in differential and integral calculus together with geometrical and mechanical applications, Stockholm 1928 (2nd ed.

  473. Bethe biography
    • By the age of fourteen he had taught himself calculus.

  474. Uhlenbeck biography
    • In his last couple of years at school his physics teacher strongly encouraged him, gave him texts on the differential and integral calculus and suggested that he read undergraduate texts on mathematics and physics.

  475. Rasiowa biography
    • Her thesis, presented in 1950, was on algebra and logic Algebraic treatment of the functional calculus of Lewis and Heyting and these topics would be the main areas of her research throughout her life.

  476. Russell biography
    • His contributions relating to mathematics include his discovery of Russell's paradox, his defence of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus.

  477. Berzolari biography
    • He graduated with his laurea in 1884 and continued to work at Pavia as an assistant to Casorati, who held the chair of infinitesimal calculus.

  478. Kontsevich biography
    • Calculus of differential forms, symplectic forms, Hamiltonian vector fields and Poisson brackets in noncommutative geometry are sketched.

  479. Morgan William biography
    • ',8)">8]) to the Royal Society, he showed (without any use of the integral calculus) how the present value of assurances contingent on the survival of one life beyond another should be calculated.

  480. Wiltheiss biography
    • Wiltheiss predominantly lectured to students who were beginning their studies, on topics such as differential and integral calculus, geometry, and algebra.

  481. Taniyama biography
    • In a long suicide note he left, he took great care to describe exactly where he had reached in the calculus and linear algebra courses he was teaching and to apologise to his colleagues for the trouble his death would cause them.

  482. Mackenzie biography
    • At Honours level: Natural Philosophy, Mathematics, Final Natural Philosophy, Final Mathematics, Calculus, General Analysis, Heat, Electricity I and II, General Physics, Higher Algebra and Geometry.

  483. Whiteside biography
    • Before the first volume appeared, however, Whiteside published the first of two volumes The mathematical works of Isaac Newton which contained English translations of Newton's published mathematical works on the calculus.

  484. Frank biography
    • In mathematics he worked on the calculus of variations, Fourier series, function spaces, Hamiltonian geometrical optics, Schrodinger wave mechanics, and relativity.

  485. Borda biography
    • Borda made good use of the differential calculus and of experimental methods to unify areas of physics.

  486. Laplace biography
    • Laplace's first paper which was to appear in print was one on the integral calculus which he translated into Latin and published at Leipzig in the Nova acta eruditorum in 1771.

  487. D'Alembert biography
    • In May 1741 d'Alembert was admitted to the Paris Academy of Science, on the strength of these and papers on the integral calculus.

  488. Rouche biography
    • He also wrote several textbooks including Traite de geometrie elementaire (written jointly with Ch De Comberousse) (1874), Elements de Statique Graphique (1889), Coupe des pierres : precedee des principes du trait de stereotomie (written jointly with Charles Brisse) (1893), and Analyse infinitesimale a l'usage des ingenieurs (1900-02) which was a calculus text written for engineers.

  489. Guinand biography
    • Guinand worked on summation formulae and prime numbers, the Riemann zeta function, general Fourier type transformations, geometry and some papers on a variety of topics such as computing, air navigation, calculus of variations, the binomial theorem, determinants and special functions.

  490. Rogosinski biography
    • There was little science and the mathematics course contained no calculus but plenty of geometry.

  491. Euclid biography
    • The standard of rigour was to become a goal for the inventors of the calculus centuries later.

  492. Sylvester biography
    • He published important papers in 1852 and 1853, namely On the principle of the calculus of forms and On the theory of syzygetic relations and two rational integer functions.

  493. Thomson James biography
    • While in Belfast he published A Treatise on Arithmetic in Theory and Practice (1819), Trigonometry, Plane and Spherical (1820), Introduction to Modern Geography (1827), and The Differential and Integral Calculus (1831).

  494. Bendixson biography
    • From 1892 until 1899 he taught at the Royal Technological Institute in Stockholm and he also taught calculus and algebra at Stockholm University.

  495. Kemeny biography
    • It was designed because he was unhappy that mathematics (entirely calculus in first year courses at that time) was [Notices Amer.

  496. Crofton biography
    • Crofton wrote most of his papers on pure mathematics, publishing on geometry and the operator calculus.

  497. Boys biography
    • The school had no mathematics department so Boys learnt mathematics from books including Todhunter's Integral Calculus.

  498. Rado Richard biography
    • Together with Erdős he developed the partition calculus.

  499. De Moivre biography
    • In 1710 de Moivre was appointed to the Commission set up by the Royal Society to review the rival claims of Newton and Leibniz to be the discovers of the calculus.

  500. Bott biography
    • My first teacher in the calculus was Professor Williams.

  501. Sintsov biography
    • Cantor quotes these equations as examples of equations in three variables which can be solved by the method of differential calculus due to Niels Henrik Abel.

  502. Kirillov biography
    • The author also acquaints the reader with the notion of a von Neumann algebra and the idea of "supermathematics", the calculus of anti-commuting variables.

  503. Hutton biography
    • The second volume contains Newton's approach to the differential and integral calculus.

  504. Prodi biography
    • It follows essentially the same scheme (differential calculus in Banach spaces, local inversion theorems, global inversion theorems, semilinear Dirichlet problems, bifurcation and applications).

  505. Halley biography
    • He supported Newton in his controversy with Leibniz over who invented the calculus, serving as secretary of a committee set up by the Royal Society to resolve the dispute.

  506. Nirenberg biography
    • The work of Louis Nirenberg has enormously influenced all areas of mathematics linked one way or another with partial differential equations: real and complex analysis, calculus of variations, differential geometry, continuum and fluid mechanics.

  507. Le Paige biography
    • Some of these papers were on topics he had worked on before he settled on geometry as his main interest, for example there are papers on continued fractions, differential equations, the difference calculus, and Bernoulli numbers.

  508. Schrodinger biography
    • In mathematics he was taught calculus and algebra by Franz Mertens, function theory, differential equations and mathematical statistics by Wilhelm Wirtinger (whom he found uninspiring as a lecturer).

  509. Wang Yuan biography
    • However he did write a number of books such as: (with Hua Loo Keng) Applications of number theory to numerical analysis (1978); Goldbach Conjecture (1984); (with Hua Loo Keng) Popularising mathematical methods in the People's Republic of China (1989); Diophantine equations and inequalities in algebraic number fields (1991); (with Fang Kai-Tai) Number theoretic methods in statistics (1994); Hua Loo Keng (1995); and (with Fong Yuen) Calculus (1997).

  510. Eisenstein biography
    • He began by learning the differential and integral calculus from the works of Euler and Lagrange.

  511. Lobachevsky biography
    • Despite this heavy administrative load, Lobachevsky continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics.

  512. Samelson biography
    • World War II was still taking place and he was involved teaching in the Army's Specialized Training Program where soldiers attended for a six-week crash course in calculus.

  513. Bouquet biography
    • In 1874 Bouquet was appointed professor of differential and integral calculus at the Sorbonne, succeeding Serret who had retired due to ill health.

  514. Carlyle biography
    • In 1817 he tried to understand the Continental approach to the calculus by reading Wallace's article Fluxions which was published in the Edinburgh Encyclopaedia in 1815 and used Leibniz's differential notation.

  515. Landau Lev biography
    • Although calculus was not part of the school syllabus, Lev had studied this own topic his own and, later in life, would say that he could not remember a time when he was not proficient at differentiation and integration.

  516. Finkel biography
    • Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions with Notes and Explanations by B F Finkel.

  517. Heuraet biography
    • This work was typical of that being carried out by van Schooten's research group and this work was important since attempts to discover properties of curves of this type led to methods which eventually gave rise to the differential and integral calculus.

  518. Witt biography
    • Oswald Teichmuller and Ludwig Schmid were also members of the seminar, and Schmid collaborated with Witt on ideas which would lead to the Witt vector calculus.

  519. Netto biography
    • There he taught courses on advanced algebra, the calculus of variations, mechanics, Fourier series, and synthetic geometry.

  520. Gregory David biography
    • David Gregory certainly supported Newton strongly in the Newton - Leibniz controversy arguing, as did Gregory's friend Wallis, that Leibniz had learnt of the calculus through a letter from Collins.

  521. Gromov biography
    • In 1985 Gromov was a plenary speaker at the British Mathematical Colloquium in Cambridge when he lectured on Differential geometry with and without infinitesimal calculus: anatomy of curvature.

  522. Germain biography
    • She had not derived her hypothesis from principles of physics, nor could she have done so at the time because she had not had training in analysis and the calculus of variations.

  523. Boyle biography
    • History Topics: The rise of the calculus .

  524. Bruno biography
    • From 1852 to 1858 he taught algebra and geometry, while from 1860 to 1862 he taught differential and integral calculus.

  525. Zeuthen biography
    • As we mentioned above, he developed the enumerative calculus, proposed by Chasles, for counting the number of curves touching a given set of curves.

  526. Levin biography
    • In addition to undergraduate courses of calculus, theory of functions of a complex variable and functional analysis, he taught advanced courses on entire functions, quasi-analytic classes, almost periodic functions, harmonic analysis and approximation theory, and Banach algebras.

  527. Dodgson biography
    • I mean to have read by next time, Integral Calculus, Optics (and theory of light), Astronomy, and higher Dynamics.

  528. Love biography
    • The treatment throughout is severely analytical, but it took form too early to incorporate the tensor calculus.

  529. Antiphon biography
    • History Topics: The rise of the calculus .

  530. Saunderson biography
    • Although Saunderson never wrote up any of his other courses for publication, he did leave a large amount of material on his teaching of the differential and integral calculus.

  531. Al-Karaji biography
    • theory of algebraic calculus ..

  532. Lefebure biography
    • In May 1843 Lacroix died and, in July of the same year, Lefebure succeeded him in the chair of differential and integral calculus at the Faculty of Science in Paris.

  533. Bartels biography
    • Bartels took up his post at professor of mathematics at Kazan in 1808 and, during the following twelve years, he lectured on the History of Mathematics, Higher Arithmetic, Differential and Integral Calculus, Analytical Geometry and Trigonometry, Spherical Trigonometry, Analytical Mechanics and Astronomy.

  534. Schmetterer biography
    • Because his family was poor, they could not afford to buy him mathematics books, which were expensive, but he still managed to find and read books on algebra and calculus.

  535. Ghizzetti biography
    • He published Sull'uso della trasformazione di Laplace nello studio dei circuiti elettrici (1937), and La trasformazione di Laplace e il calcolo simbolico degli elettrotecnici (1941) which provided an expository representation of the symbolic calculus of electric circuit theory based on the Laplace transformation.

  536. Roberts biography
    • He also wrote on the calculus of operations, interpolation etc.

  537. Hadamard biography
    • Jacques Hadamard on "Who discovered the calculus" .

  538. Rokhlin biography
    • Rokhlin's interest in mathematics is so strong that he independently studied the beginnings of calculus, analytical geometry and higher algebra.

  539. Osipovsky biography
    • His most famous work was the three-volume handbook A Course of Mathematics (1801-1823) which covered function theory, differential equations, and the calculus of variations.

  540. Moser Jurgen biography
    • A course of lectures that Moser have at ETH in the spring of 1988 became the basis for Selected chapters in the calculus of variations (2003).

  541. Schmid biography
    • In fact Schmid and Witt collaborated on developing what today is called the 'Witt vector calculus'.

  542. Fermat biography
    • History Topics: The rise of Calculus .

  543. Tilly biography
    • Tilly must have carried on with his methods of teaching calculus despite these warnings and as a consequence he was dismissed from his post and forced into early retirement in August 1900.

  544. Nash biography
    • Soon, however, his growing interest in mathematics had him take courses on tensor calculus and relativity.

  545. Robertson biography
    • His contributions to differential geometry came in papers such as: The absolute differential calculus of a non-Pythagorean non-Riemannian space (1924); Transformation of Einstein space (1925); Dynamical space-times which contain a conformal Euclidean 3-space (1927); Note on projective coordinates (1928); (with H Weyl) On a problem in the theory of groups arising in the foundations of differential geometry (1929); Hypertensors (1930); and Groups of motion in space admitting absolute parallelism (1932).

  546. Budan de Boislaurent biography
    • He did not appeal to the theory of finite differences or to the calculus of these coefficients, preferring to give them "by means of simple additions and subtractions".

  547. Wangerin biography
    • He taught many courses at the University of Halle including: linear partial differential equations; calculus of variations; theory of elliptical functions; synthetic geometry; hydrostatics and capillarity theory; theory of space curves and surfaces; analytic mechanics; potential theory and spherical harmonics; celestial mechanics; the theory of the map projections; hydrodynamics; and the partial differential equations of mathematical physics.

  548. De Vries biography
    • That his life was now more stable is shown by the fact that, in 1912, de Vries published two further papers on his 'calculus rationis' in the Proceedings of the Royal Netherlands Academy of Sciences.

  549. Fenyo biography
    • The book opens with a discussion of elementary set theory, Lebesgue integration and Stieltjes integration and then goes on to the first major topic, the operator calculus, following the ideas of Mikusinski and others.

  550. Bilimovic biography
    • The main characteristic of his scientific opus is that he did not address only problems of one narrow scientific field, but Bilimovic also studied the problems of theory of curves and surface, rational mechanics, celestial mechanics and geophysics, nonanalytical functions and vector calculus.

  551. Fiorentini biography
    • There are three parts, dealing with logic (i.e., first order predicate calculus), set theory and algebraic structures, respectively.

  552. Reyneau biography
    • Reyneau struggled to assimilate the differential and integral calculus participating in debates provoked by Rolle on these topics.

  553. Shnirelman biography
    • L A Lyusternik became a friend and important collaborator with Shnirelman and together they made significant contributions to topological methods in the calculus of variations in a series of paper written jointly between 1927 and 1929.

  554. Riccati biography
    • In the second volume, when dealing with Integral Calculus, the reader will find a new method for polynomials; this is due to the famous Count Jacopo Riccati, a personality of unique merit in all sciences, and well known to the literate world.

  555. Boussinesq biography
    • Indeed he succeeded in the following year when he was appointed Professor of Differential and Integral Calculus at the Faculty of Science in Lille.

  556. Machin biography
    • Keill repeated his accusations in a letter to Leibniz saying that two letters from Newton, sent to Leibniz through Oldenburg, must have given him the principles of the calculus.

  557. Thompson Robert biography
    • He discussed: quantitative prediction; high and low roads; the numerical range; similarity invariants of principal submatrices; commutators; the triangle inequality; the facial structure of the unit ball; the Gershgorin circle theorem; matrices, graphs, inertia, number theory; power embeddings and dilatations; the Schubert calculus; the spectrum of a sum of Hermitian matrices; the Hadamard-Schur product; the exponential function; the exponential function and commutativity; integral quadratic forms; the matrix-valued numerical range; inequalities with subtracted terms; and further uses of the computer.

  558. Liouville biography
    • One of the first topics he studied, which developed from his early work on electromagnetism, was a new topic, now called the fractional calculus.

  559. Hoeffding biography
    • Hoeffding took this course but also studied, among other, advanced calculus with Erhard Schmidt and number theory with Alfred Brauer (who he considered the best of all his lecturers).

  560. Lindelof biography
    • In addition to the 1905 work referred to above which is largely on his own research, he wrote the textbook Differential and integral calculus and their applications which was published in four volumes between 1920 and 1946.

  561. Boutroux biography
    • Following this he was appointed as a lecturer in mathematics at the University of Montpellier before becoming professor of integral calculus at Poitiers in 1908.

  562. Mineur biography
    • The work of his thesis was published in 1925 as was the paper Theorie analytique des groupes continus finis and, three years later, he published a paper on the absolute differential calculus Calcul differentiel absolu.

  563. Sleszynski biography
    • Introduction to mathematical logic, complete proof, mathematical proof, exposition of the theory of propositions, the Boolean calculus, Grassmann's logic, Schroder's algebra, Poretsky's seven laws, Peano's doctrine, Burali-Forti's doctrine - these are some of the themes pursued in this work, from which I personally have learned a great deal and thanks to which I have got a clear idea of many an unclear thing.

  564. Iyanaga biography
    • He helped Takagi with his calculus course in 1935-36, the last time Takagi gave it before he retired.

  565. Lakatos biography
    • The point is not merely to rethink the reasoning of Cauchy, not merely to use the mathematical insight available from Robinson's non-standard analysis to re-evaluate our attitude towards the whole history of the calculus and the notion of the infinitesimal.

  566. Janiszewski biography
    • There he taught courses on analytic functions and functional calculus.

  567. Mackey biography
    • I defined all terms - even those of calculus and elementary algebra - and proved all theorems except those which I decided to leave to the students as exercises.

  568. Dieudonne biography
    • For example, the differential calculus is developed in terms of linear approximation to functions on an open subset of a Banach space to a Banach space.

  569. Grieve biography
    • An elementary knowledge of Calculus is assumed; but alternative methods are given, usually in the examples at the end of a chapter.

  570. Whittaker John biography
    • Whittaker attended Fettes from 1918 until 1920 during which time he was introduced to calculus.

  571. Klein biography
    • The essential change recommended was the introduction in secondary schools of the rudiments of differential and integral calculus and the function concept.

  572. Seifert biography
    • they published their second joint book in 1938 which was the monograph Variational calculus in the large which was a text on Morse theory.

  573. Pollaczek biography
    • The circumstances that this theory employs uniquely analytic methods and dispenses with all resources of classical Probability Calculus probably accounts for the fact that hitherto my methods have been employed by nobody save myself.

  574. Conforto biography
    • In addition to his duties as an assistant in algebraic geometry, Conforto also worked at Mauro Picone's National Institute for the Applications of Calculus.

  575. Graham biography
    • That book was 'Calculus' by Granville, Smith and Longley.

  576. Shtokalo biography
    • Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics.
    • In 1960 he published On the operational calculus (Russian), giving the following English summary:- .
    • This paper contains a brief outline of investigations in the field of the calculus of operations, indicating the main trends and principal stages of development, as well as a substantiation of the methods of the calculus of operations.
    • His 303-page book Operational calculus (Russian) was published in 1972.
    • This book is an introduction to operational calculus, which is based on the theory of the Laplace transformation.
    • Teacher in primary, intermediate, and advanced schools, scholar, organiser and director of scientific endeavours, researcher in the area of the theory of differential equations, in the domain of operational calculus, in the area of the history of mathematics, organiser of the publication of works which are classics of indigenous science - such is a far from complete enumeration of the contributions of Iosif Zakharovich Shtokalo to the development of domestic science and culture.

  577. FitzGerald biography
    • telegraphy owes a great deal to Euclid and other pure geometers, to the Greek and Arabian mathematicians who invented our scale of numeration and algebra, to Galileo and Newton who founded dynamics, to Newton and Leibniz who invented the calculus, to Volta who discovered the galvanic coil, to Oersted who discovered the magnetic actions of currents, to Ampere who found out the laws of their action, to Ohm who discovered the law of resistance of wires, to Wheatstone, to Faraday, to Lord Kelvin, to Clerk Maxwell, to Hertz.

  578. Eudoxus biography
    • History Topics: The rise of the calculus .

  579. Planck biography
    • In this case the quantum of action must play a fundamental role in physics, and here was something completely new, never heard of before, which seemed to require us to basically revise all our physical thinking, built as this was, from the time of the establishment of the infinitesimal calculus by Leibniz and Newton, on accepting the continuity of all causative connections.

  580. Sasaki biography
    • Although in earlier years there were no mathematics texts in Japanese, by the time Sasaki attended High School there were Japanese texts on algebra, analytic geometry, trigonometry and calculus, all of which he studied.

  581. Glaisher biography
    • By pure mathematics I do not mean the ordinary processes of algebra, differential and integral calculus etc., which every worker in the so-called mathematical sciences should have at his command.

  582. Bromwich biography
    • In a series of papers he put Heaviside's calculus on a rigorous basis treating the operators as contour integrals.

  583. Pless biography
    • When Vera was about 12 years old she was taught calculus by a family friend who was a graduate student at the University of Chicago.

  584. Bachmann Friedrich biography
    • This remarkable book is essentially an elaboration of an idea of G Thomsen (The treatment of elementary geometry by a group-calculus, Math.

  585. Landsberg biography
    • In particular he studied the role of these curves in the calculus of variations and in mechanics.

  586. Cesaro biography
    • Cesaro's interest in mathematical physics is also evident in two very successful calculus texts which he wrote.

  587. Billy biography
    • Adiectus est calculus, aliquot eclipseon solis & lunae, quae proxime per totam Europam videbuntur.

  588. Forsythe biography
    • In this last mentioned paper he introduced generalizations of the weak law of large numbers and the central limit theorem of the calculus of probability.

  589. Seki biography
    • Secrecy surrounded the schools in Japan so it is hard to determine the contributions made by Seki, but he is also credited with major discoveries in the calculus which he passed on to his pupils.

  590. Leucippus biography
    • History Topics: The rise of the calculus .

  591. Bernstein Felix biography
    • There was no graduate program in those days at all and poor old Felix Bernstein had to teach college algebra; the highest course he taught I guess was first and second year calculus, nothing further.

  592. Allardice biography
    • For example at the meeting held on Friday 14 March 1884 he read a paper on the geometry of the spherical surface; at the meeting on Friday 8 January 1886 he discussed a problem of symmetry in an algebraical function; on 11 February 1887 he communicated a note on a theorem in algebra; on 11 January 1889 he contributed a note on a formula in quaternions; on 13 December 1889 he discussed some theorems in the theory of numbers; on 13 November 1891 his paper Barycentric Calculus of Mobius was read by John Alison; on 14 December 1901 his paper Four Circles Touching a Common Circle was communicated to the meeting by Mr George Duthie; and on 13 January 1911 his paper On the envelope of the directrices of a system of similar conics through three points was communicated by E D Williamson.

  593. Tait biography
    • It was the physical insight which Hamilton's quaternion differential calculus then gave which impressed Tait and he began to work hard developing a physical theory.

  594. Jeffreys biography
    • In pure mathematics he studied operational methods (where he improved on Heaviside's operational calculus and Laplace transforms), cartesian tensors and asymptotic approximations.

  595. Neumann Hanna biography
    • Her introduction to higher mathematics was a course given by Georg Feigl and, in addition, she was taught analytic and projective geometry by Ludwig Bieberbach, differential and integral calculus by Erhard Schmidt, and number theory by Issai Schur.

  596. Richardson biography
    • In addition to his 1922 book, Richardson published about 30 papers on the mathematics of the weather and in these he made contributions to the calculus and to the theory of diffusion, in particular eddy-diffusion in the atmosphere.

  597. Johnson biography
    • In the area of logic he published papers such as The logical calculus (1892) and Analysis of thinking (1918), both of which appeared in Mind.

  598. Fiske biography
    • I had attended only a few lectures by Cayley on 'The calculus of the extraordinaires' when, slipping on the ice, he suffered a fracture of the leg, which brought the lectures to an end.

  599. Steiner biography
    • He attended lectures at the Universities of Heidelberg on combinatorial analysis, differential and integral calculus and algebra.

  600. Robins biography
    • This was written to support the differential calculus against attacks by George Berkeley and James Jurin.

  601. Rudin Walter biography
    • It will serve as a good text for courses at this level and, for those prepared to "dig," it would be a good choice for independent study of the fundamental ideas underlying calculus.

  602. Forsyth biography
    • After his 1893 treatise he published many other texts, the most important of which are Lectures on the differential geometry of curves and surfaces (1912), Lectures introductory to the theory of functions of two complex variables (1914), Calculus of variations (1927), Geometry of four dimensions which was in two volumes and published in 1930, and Intrinsic geometry of ideal space also in two volumes, published in 1935.

  603. Hall Marshall biography
    • He entered Yale as an undergraduate and there he took many advanced courses in topics such as calculus of variations and algebraic numbers.

  604. Sturm biography
    • For around ten years he gave excellent lectures but his wish to give his students the best possible courses meant that he gave a great deal of his time to preparing his lecture courses on differential and integral calculus and on rational mechanics.

  605. Olech biography
    • It includes a number of other branches of mathematics, and his works are classified by Mathematical Reviews as including, among others, the area of linear and multilinear algebra, measure and integration theory, calculus of variations, convex and discrete geometry, operations research and general systems theory.

  606. Warga biography
    • Warga's extensive exposition will be of value to mathematicians interested in functional analysis, calculus of variations, optimal control and systems science.

  607. Tietz biography
    • The analytical calculus; 5.

  608. MacLane biography
    • He defended his thesis Abbreviated Proofs in the Logical Calculus, with Weyl as examiner, on 19 July 1933 and quickly returned to the United States.

  609. Khinchin biography
    • The book was designed to be used to supplement a standard course on the calculus and gives a careful treatment of some of the basic notions of mathematical analysis.

  610. Ramsey biography
    • The combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate calculus.

  611. Bombelli biography
    • He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave an expression involving the square roots of negative numbers.

  612. Gorenstein biography
    • He taught himself calculus at the age of 12 years.

  613. Heisenberg biography
    • In fact his mathematical abilities were such that in 1917 he tutored a family friend who was at university in calculus.

  614. Keynes biography
    • The mathematical calculus is astonishingly powerful, considering the very restricted premises which form its foundation..

  615. Stueckelberg biography
    • constitutes the first complete and easily generalizable instance of a manifest relativistically invariant perturbative calculus.

  616. Krull biography
    • his earlier studies, but also dealt with other fields of mathematics: group theory, calculus of variations, differential equations, Hilbert spaces.

  617. Kolmogorov biography
    • This was published jointly with Khinchin and contains the 'three series' theorem as well as results on inequalities of partial sums of random variables which would become the basis for martingale inequalities and the stochastic calculus.

  618. Dahlquist biography
    • The prerequisites are slight (calculus and linear algebra and preferably some acquaintance with computer programming) so that some of the finer theoretical points (those at which numerical analysis becomes applied functional analysis, for example) are outside the scope of the book.

  619. Rellich biography
    • He had gone to Hamburg in the previous year where he lectured on differential and integral calculus.

  620. Pearson biography
    • and in general to convert statistics in this country from being the playing field of dilettanti and controversialists into a serious branch of science, which no man could attempt to use effectively without adequate training, and more than he could attempt to use the differential calculus, being ignorant of mathematics.

  621. Kalmar biography
    • He worked on mathematical logic solving certain cases of the decision problem for the first order predicate calculus, simplified results of Bernays, and worked on ideas of Post, Godel and Church.

  622. Rudolph biography
    • Calculus reform with an eye to helping minorities.

  623. Diaconis biography
    • I didn't know calculus, or at least not enough.

  624. Householder biography
    • by the University of Chicago in 1937 for a thesis on the calculus of variations.

  625. Frenet biography
    • Frenet's exercise book on the calculus Recueil d'exercises sur le calcul infinitesimal, first published in 1856, ran to seven editions, the seventh being published in 1917 [Dictionary of Scientific Biography (New York 1970-1990).

  626. Simson biography
    • For Simson the best vehicle for presenting a mathematical argument was geometry and, although he was familiar with the recent developments in algebra and the infinitesimal calculus, he preferred to express himself in geometrical terms wherever possible.

  627. Mittag-Leffler biography
    • Mittag-Leffler made numerous contributions to mathematical analysis particularly in areas concerned with limits and including calculus, analytic geometry and probability theory.

  628. Eddington biography
    • The level to which the school was able to take Arthur was, however, not very advanced and his good grounding in mathematics stopped short of reaching the differential and integral calculus.

  629. Bolyai biography
    • By the time Bolyai was 13, he had mastered the calculus and other forms of analytical mechanics, his father continuing to give him instruction.

  630. Thom biography
    • Thom's theory is an attempt to describe, in a way that is impossible using differential calculus, those situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes.

  631. Ulam biography
    • I was sixteen when I really learned calculus all by myself from a book by Kowalevski, a German not to be confused with Sonia Kovalevskaya ..

  632. Grossmann biography
    • It was Grossmann who pointed out to him the relevance to general relativity of the tensor calculus which had been proposed by Elwin Bruno Christoffel in 1864, and developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita around 1901.

  633. Canard biography
    • Citizen Canard, although a mathematics teacher, is ignorant of or forgets the elements of the calculus of functions ..

  634. Ceva Giovanni biography
    • In Geometria motus, opusculum geometricum in gratiam aquarum excogitatum (1692) he, to some extent, anticipated the infinitesimal calculus in his study of curves such as parabolas and hyperbolas using infinitesimal methods of the type introduced by Bonaventura Cavalieri.

  635. Hankel biography
    • Beginning with a revised statement of George Peacock's principle of permanence of formal laws, he developed complex numbers as well as such higher algebraic systems as Mobius' barycentric calculus, some of Hermann Grassmann's algebras, and W R Hamilton's quaternions.

  636. Eisenbud biography
    • V I Arnold once referred to [the] celebrated formula of Eisenbud-Levine, which links calculus, algebra and geometry, as a "paradigm" more than a theorem that provides a local manifestation of interesting global invariants and that "would please Poincare and Hilbert (also Euler, Cauchy and Kronecker, to name just those classical mathematicians, whose works went in the same direction)." Given this early work, it was natural for David's attention to turn to the study of singularities and their topology.

  637. Polozii biography
    • Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by G Polozii of Kiev, who introduced a new notion of p-analytic functions, defined the notion of derivative and integral for these functions, developed their calculus, obtained a generalised Cauchy formula, and devised a new approximation method for solution of problems in elasticity and filtration.


History Topics

  1. Calculus history
    • A history of the calculus .
    • The main ideas which underpin the calculus developed over a very long period of time indeed.
    • In fact, because of this work, Lagrange stated clearly that he considers Fermat to be the inventor of the calculus.
      Go directly to this paragraph
    • Huygens was a major influence on Leibniz and so played a considerable part in producing a more satisfactory approach to the calculus.
      Go directly to this paragraph
    • In fact, although Barrow never explicitly stated the fundamental theorem of the calculus, he was working towards the result and Newton was to continue with this direction and state the Fundamental Theorem of the Calculus explicitly.
    • This was a work which was not published at the time but seen by many mathematicians and had a major influence on the direction the calculus was to take.
    • He also calculated areas by antidifferentiation and this work contains the first clear statement of the Fundamental Theorem of the Calculus.
    • On returning to Paris Leibniz did some very fine work on the calculus, thinking of the foundations very differently from Newton.
      Go directly to this paragraph
    • For Newton the calculus was geometrical while Leibniz took it towards analysis.
    • His results on the integral calculus were published in 1684 and 1686 under the name 'calculus summatorius', the name integral calculus was suggested by Jacob Bernoulli in 1690.
      Go directly to this paragraph
    • After Newton and Leibniz the development of the calculus was continued by Jacob Bernoulli and Johann Bernoulli.
      Go directly to this paragraph
    • However when Berkeley published his Analyst in 1734 attacking the lack of rigour in the calculus and disputing the logic on which it was based much effort was made to tighten the reasoning.
      Go directly to this paragraph
    • Maclaurin attempted to put the calculus on a rigorous geometrical basis but the really satisfactory basis for the calculus had to wait for the work of Cauchy in the 19th Century.
      Go directly to this paragraph
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_rise_of_calculus.html .

  2. Calculus history references
    • References for: A history of the calculus .
    • M E Baron, The origins of the infinitesimal calculus (New York, 1987).
    • C B Boyer, The History of the Calculus and Its Conceptual Development (New York, 1959).
    • C H Edwards, The Historical Development of the Calculus (New York, 1979).
    • J O Fleckenstein, The line of descent of the infinitesimal calculus in the history of ideas, Arch.
    • E Giusti, A comparison of infinitesimal calculus in Leibniz and Newton (Italian), Rend.
    • N Guicciardini, Three traditions in the calculus : Newton, Leibniz and Lagrange, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 308-317.
    • N Guicciardini, The Development of Newtonian Calculus in Britain, 1700-1800 (Cambridge, 1989).
    • T Guitard, On an episode in the history of the integral calculus, Historia Mathematica 14 (2) (1987), 215-219.
    • P Kitcher, Fluxions, limits, and infinite littlenesse : A study of Newton's presentation of the calculus, Isis 64 (221) (1973), 33-49.
    • A Nikolic, Space and time in the apparatus of infinitesimal calculus, Zb.
    • L Pepe, The infinitesimal calculus in Italy at the beginning of the 18th century (Italian), Boll.
    • A Rosenthal, The history of calculus, The American Mathematical Monthly 58 (1951), 75-86.
    • A B Shtykan, On the question of the origin of the differential and integral calculus (Russian), Voprosy Istor.
    • O Toeplitz, The Calculus: A Genetic Approach (1963).
    • J Vernet, The infinitesimal calculus and Spanish mathematics of the 18th century (Spanish), Arch.
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_rise_of_calculus.html] .

  3. Calculus history references
    • References for: A history of the calculus .
    • M E Baron, The origins of the infinitesimal calculus (New York, 1987).
    • C B Boyer, The History of the Calculus and Its Conceptual Development (New York, 1959).
    • C H Edwards, The Historical Development of the Calculus (New York, 1979).
    • J O Fleckenstein, The line of descent of the infinitesimal calculus in the history of ideas, Arch.
    • E Giusti, A comparison of infinitesimal calculus in Leibniz and Newton (Italian), Rend.
    • N Guicciardini, Three traditions in the calculus : Newton, Leibniz and Lagrange, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 308-317.
    • N Guicciardini, The Development of Newtonian Calculus in Britain, 1700-1800 (Cambridge, 1989).
    • T Guitard, On an episode in the history of the integral calculus, Historia Mathematica 14 (2) (1987), 215-219.
    • P Kitcher, Fluxions, limits, and infinite littlenesse : A study of Newton's presentation of the calculus, Isis 64 (221) (1973), 33-49.
    • A Nikolic, Space and time in the apparatus of infinitesimal calculus, Zb.
    • L Pepe, The infinitesimal calculus in Italy at the beginning of the 18th century (Italian), Boll.
    • A Rosenthal, The history of calculus, The American Mathematical Monthly 58 (1951), 75-86.
    • A B Shtykan, On the question of the origin of the differential and integral calculus (Russian), Voprosy Istor.
    • O Toeplitz, The Calculus: A Genetic Approach (1963).
    • J Vernet, The infinitesimal calculus and Spanish mathematics of the 18th century (Spanish), Arch.
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_rise_of_calculus.html .

  4. History overview
    • Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
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    • Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability.
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    • However the calculus was to be the topic of most significance to evolve in the 17th Century.
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    • Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature.
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    • However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th Century rather than that of Newton.
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    • Leibniz's influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application.
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    • The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry.
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    • The best known is probably the notation for the calculus used by Leibniz and Newton.
    • Leibniz's notation lead more easily to extending the ideas of the calculus, while Newton's notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered.
    • For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered.
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    • Neither did since Newton certainly learnt the calculus from his teacher Barrow.
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    • Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.
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  5. Brachistochrone problem
    • It is not surprising, given the dispute over the calculus, that Johann Bernoulli had included these words in his challenge:- .
    • It was an unpleasant incident, but one of great value to mathematics for the problems being argued about led directly to the founding of the calculus of variations.
    • It gave an analytic method to attach calculus of variations type problems.
    • The first problem of this type [calculus of variations] which mathematicians solved was that of the brachistochrone, or the curve of fastest descent, which Johann Bernoulli proposed towards the end of the last century.
    • Since, however, the rules were not sufficiently general, the famous Euler undertook the task of reducing all such investigations to a general method which he gave in the work "Essay on a new method of determining the maxima and minima of indefinite integral formulas"; an original work in which the profound science of the calculus shines through.
    • a method which only requires a straightforward use of the principles if the differential and integral calculus; but I must strongly emphasise that since my method requires that a quantity be allowed to vary in two different ways, so as not to confuse these different variations, I have introduced a new symbol δ into my calculations.

  6. Bourbaki 1
    • The year is 1934 and for weeks Cartan has been asking Weil how he would teach different aspects of the differential and integral calculus.
    • Weil, like Cartan, is unhappy with the recommended text, Goursat's Traite d'Analyse, and has been suggesting to him better ways to introduce various concepts in the calculus.
    • to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis.
    • A large number of subcommittees were formed, given the size of the group, and these were to cover the following topics: algebra, analytic functions, integration theory, differential equations, existence theorems for differential equations, partial differential equations, differentials and differential forms, calculus of variations, special functions, geometry, Fourier series, and representations of functions.
    • It proved impossible to retain the classical division into analysis, differential calculus, geometry, algebra, number theory, etc.

  7. Brachistochrone problem references
    • T Koetsier, The story of the creation of the calculus of variations : the contributions of Jakob Bernoulli, Johann Bernoulli and Leonhard Euler (Dutch), in 1985 holiday course : calculus of variations (Amsterdam, 1985), 1-25.
    • K Pedersen and K M Pedersen, The early history of the calculus of variations (Danish), Nordisk Mat.
    • J Peiffer, Le probleme de la brachystochrone a travers les relations de Jean I Bernoulli avec L'Hopital et Varignon, in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli, Basel, 1987, Studia Leibnitiana Sonderheft 17 (Wiesbaden, 1989).

  8. function concept
    • It was a concept whose introduction was particularly well timed as far as Johann Bernoulli was concerned for he was looking at problems in the calculus of variations where functions occur as solutions.
    • ',28)">28] for more information about how the author considers the calculus of variations to be the mathematical theory which developed most intimately in connection with the concept of a function.
    • exponentials, logarithms, and others which integral calculus supplies in abundance.
    • This might have been a huge breakthrough but after giving this wide definition, Euler then devoted the book to the development of the differential calculus using only analytic functions.

  9. Brachistochrone problem references
    • T Koetsier, The story of the creation of the calculus of variations : the contributions of Jakob Bernoulli, Johann Bernoulli and Leonhard Euler (Dutch), in 1985 holiday course : calculus of variations (Amsterdam, 1985), 1-25.
    • K Pedersen and K M Pedersen, The early history of the calculus of variations (Danish), Nordisk Mat.
    • J Peiffer, Le probleme de la brachystochrone a travers les relations de Jean I Bernoulli avec L'Hopital et Varignon, in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli, Basel, 1987, Studia Leibnitiana Sonderheft 17 (Wiesbaden, 1989).

  10. Brachistochrone problem references
    • T Koetsier, The story of the creation of the calculus of variations : the contributions of Jakob Bernoulli, Johann Bernoulli and Leonhard Euler (Dutch), in 1985 holiday course : calculus of variations (Amsterdam, 1985), 1-25.
    • K Pedersen and K M Pedersen, The early history of the calculus of variations (Danish), Nordisk Mat.
    • J Peiffer, Le probleme de la brachystochrone a travers les relations de Jean I Bernoulli avec L'Hopital et Varignon, in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli, Basel, 1987, Studia Leibnitiana Sonderheft 17 (Wiesbaden, 1989).

  11. Abstract linear spaces
    • Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space.
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    • He credits Leibniz, Mobius's 1827 work, Grassmann's 1844 work and Hamilton's work on quaternions as providing ideas which led him to his formal calculus.
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    • It gives the basic calculus of set operation introducing the modern notation ∩, ∪, belongs for intersection, union and an element of.

  12. Bolzano publications.html
    • Contains his thoughts on Euclidean geometry, manipulations of series, functions and foundations of calculus, and topics in mechanics.
    • Covers topics such as geometry, calculus, and mechanics frequently making philosophical commnts.

  13. Classical time
    • Yet basic mathematics takes time as understood and develops the calculus around a particle whose position at time t is given by x(t), its velocity is dx/dt , the derivative of x(t) with respect to time, and its acceleration is the second derivative.
    • The calculus was his theory of fluxions, relating motion to this universal flux of time.

  14. Infinity
    • Leibniz's development of the calculus was built on ideas of the infinitely small which has been studied for a long time.
    • The Church had failed to silence Bruno despite putting him to death, it had failed to silence Galileo despite putting him under house arrest and it would not stop progress towards the differential and integral calculus by banning the teaching of indivisibles.

  15. Planetary motion
    • There are several ways of carrying this out, which unfortunately involve either sophisticated calculus or fairly heavy algebra.
    • Then we apply result (11) above to the first term, and, as one possibility for the second term, introduce a formula for change of variable which is found in some calculus textbooks: .

  16. EMS History
    • There he obtains a knowledge of Synthetic and Analytical Conics, the elements of the Differential Calculus, and, it may be, of the Integral Calculus as well.

  17. references
    • (1983) A stochastic calculus model of Continuous Trading: Complete Markets, Stochastic Processes and their Applications, 15, 313-316.
    • (1996), in Ikeda N, Watanabe S, Fukushima M and Kunita H (eds.), It™'s stochastic calculus and probability theory, Tokyo, ix-xiv.

  18. Neptune and Pluto references
    • F Morgan, Calculus, planets, and general relativity, SIAM Rev.

  19. General relativity references
    • F Morgan, Calculus, planets, and general relativity, SIAM Rev.

  20. Bolzano's manuscripts references
    • K Macak, Bernard Bolzano and the calculus of probabilities (Czech), Mathematics in the 19th century (Czech), Vyskov, 1994 (Prometheus, Prague, 1996), 39-55.

  21. Classical time references
    • A Nikoli'c, Space and time in the apparatus of infinitesimal calculus, Zb.

  22. Real numbers 2 references
    • J V Grabiner, The origins of Cauchy's rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).

  23. Trigonometric functions references
    • V J Katz, The calculus of the trigonometric functions, Historia Mathematica 14 (4) (1987), 311-324.

  24. Neptune and Pluto references
    • F Morgan, Calculus, planets, and general relativity, SIAM Rev.

  25. General relativity references
    • F Morgan, Calculus, planets, and general relativity, SIAM Rev.

  26. Bolzano's manuscripts references
    • K Macak, Bernard Bolzano and the calculus of probabilities (Czech), Mathematics in the 19th century (Czech), Vyskov, 1994 (Prometheus, Prague, 1996), 39-55.

  27. Classical time references
    • A Nikoli'c, Space and time in the apparatus of infinitesimal calculus, Zb.

  28. Real numbers 2 references
    • J V Grabiner, The origins of Cauchy's rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).

  29. Trigonometric functions references
    • V J Katz, The calculus of the trigonometric functions, Historia Mathematica 14 (4) (1987), 311-324.

  30. The number e
    • It would be fair to say that Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium.

  31. Topology history
    • Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated.
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  32. General relativity
    • In 1913 Einstein and Grossmann published a joint paper where the tensor calculus of Ricci and Levi-Civita is employed to make further advances.
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  33. Indian mathematics
    • The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835.

  34. Real numbers 2
    • Grabiner writes [The origins of Cauchy\'s rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).',2)" onmouseover="window.status='Click to see reference';return true">2]:- .

  35. Word problems
    • Independently of Godel, Alonzo Church was developing the λ-calculus designed to clarify the foundations of mathematics, in particular the meaning of variables.

  36. Quantum mechanics history

  37. Squaring the circle
    • The beginnings of the differential and integral calculus led to an increased interest in squaring the circle, but the new era of mathematics still produced fallacious 'proofs' of plane methods to square the circle.

  38. U of St Andrews History
    • Collins sent Gregory Barrow's book and, within a month of receiving Barrow's book, Gregory was sending Collins results which would give him an excellent claim to be a co-inventor of the calculus.
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Famous Curves

  1. Cycloid
    • It was one of the earliest variational problems and its investigation was the starting point for the development of the calculus of variations.

  2. Kampyle
    • His work contains elements of the calculus with a rigorous study of the method of exhaustion.

  3. Hyperbolic
    • He was one of the first French scholars to recognise the value of the calculus.


Societies etc

  1. Wolf Prize
    • for his fundamental contributions to pure and applied probability theory, especially the creation of the stochastic differential and integral calculus.
    • for his innovating ideas and fundamental achievements in partial differential equations and calculus of variations.

  2. AMS Bôcher Prize
    • for his memoir "The foundations of a theory of the calculus of variations in the large in m-space".
    • for his memoirs "Green's function and the problem of Plateau", "The most general form of the problem of Plateau", and "Solution of the inverse problem of the calculus of variations".

  3. Turin Mathematical Society
    • The second volume was published in the summer of 1762 and again included important contributions from Lagrange such as a work on the calculus of variations and another paper on the propagation of sound.
    • The third volume of the Melanges de Turin, containing a paper by Lagrange on the integral calculus, appeared in 1766.

  4. Fermat Prize
    • for several important contributions to the theory of variational calculus, which have consequences in Physics and Geometry.
    • for his impressive contributions to the Calculus of Variations and Geometric Measure Theory, and their link with partial differential equations.

  5. BMC 2005
    • Ball, J MOpen problems in the calculus of variations and elasticity .

  6. Bulgarian Academy of Sciences
    • For example the Institute of Mathematics and Mechanics of the Bulgarian Academy of Sciences organised a conference on Generalized Functions and Operational Calculus which took place in Varna, from 29 September to 6 October 1975.

  7. International Congress Speaker
    • Marston Morse, The Calculus of Variations in the Large.

  8. Collatz Prize
    • for his highly original and profound contributions to applied mathematics, calculus of variations and nonlinear partial differential equations, the mechanics of continua, and mathematical material sciences.

  9. BMC 1985
    • Gromov, M Differential geometry with and without infinitesimal calculus: anatomy of curvature .

  10. BMC 2006
    • Arendt, WPerturbation of generators, cosine functions and functional calculus .

  11. BMC 1991
    • Ray, N Sequences of polynomials in topology, combinatorics and formal calculus .

  12. BMC 2008
    • Thalmeier, ABrownian motion of Jordan curves and stochastic calculus on the diffeomorphism group of the circle .


References

  1. References for Leibniz
    • E J Aiton, The application of the infinitesimal calculus to some physical problems by Leibniz and his friends, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 133-143.
    • G V Coyne, Newton's controversy with Leibniz over the invention of the calculus, in Newton and the new direction in science (Vatican City, 1988), 109-115.
    • E Giusti, A comparison of infinitesimal calculus in Leibniz and Newton (Italian), Rend.
    • H Grant, Leibniz - beyond the calculus, Math.
    • J E Hofmann, G W Leibniz (14.11.1716) - der Erfinder des Calculus.
    • J E Hofmann, G W Leibniz (14.11.1716) - der Erfinder des Calculus.
    • S H Hollingdale, Leibniz and the first publication of the calculus in 1684, Bull.
    • M Horvath, On the attempts made by Leibniz to justify his calculus, Studia Leibnitiana 18 (1) (1986), 60-71.
    • V M Kir'yanova, The ideas of the symbolic calculus of Leibniz and Euler (Russian), in Questions on the history of mathematical natural science (Kiev, 1979), 91-96.
    • E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, in Conference on the History of Mathematics (Rende, 1991), 293-313.
    • G Roncaglia, Modality in Leibniz' essays on logical calculus of April 1679, Studia Leibnitiana 20 (1) (1988), 43-62.
    • K D Stiegler, Zur Entstehung und Begrundung des Newtonschen calculus fluxionum und des Leibnizschen calculus differentialis, Philos.
    • C Swoyer, Leibniz's calculus of real addition, Studia Leibnitiana 26 (1) (1994), 1-30.
    • A P Yushkevich, Leibniz and the foundations of the infinitesimal calculus (Russian), Uspekhi Matem.

  2. References for Franklin
    • E A Cameron, Review: Differential and Integral Calculus by Philip Franklin, Amer.
    • R Courant, Review: A Treatise on Advanced Calculus by Philip Franklin, Science 94 (2448) (1941), 518.
    • R D Doner, Review: Methods of Advanced Calculus by Philip Franklin, National Mathematics Magazine 20 (2) (1945), 105-106.
    • E D Helliger, Review: A Treatise on Advanced Calculus by Philip Franklin, National Mathematics Magazine 16 (7) (1942), 361-362.
    • R L Jeffery, Review: A Treatise on Advanced Calculus by Philip Franklin, Amer.
    • N D Kazarinoff, Review: Compact Calculus by Philip Franklin, Amer.
    • M Marden, Review: Methods of Advanced Calculus by Philip Franklin, Amer.
    • D A Quadling, Review: A Treatise on Advanced Calculus (Dover reprint of 1940 edition) by Philip Franklin, The Mathematical Gazette 50 (372) (1966), 191.
    • M E Shanks, Review: Differential and Integral Calculus by Philip Franklin, Science 118 (3067) (1953), 422.
    • D V Widder, Review: Methods of Advanced Calculus by Philip Franklin, Science 101 (2612) (1945), 64-65.

  3. References for Newton
    • E J Aiton, The contributions of Isaac Newton, Johann Bernoulli and Jakob Hermann to the inverse problem of central forces, in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli (Wiesbaden, 1989), 48-58.
    • T R Bingham, Newton and the development of the calculus, Pi Mu Epsilon J.
    • I B Cohen, Isaac Newton, the calculus of variations, and the design of ships, in For Dirk Struik (Dordrecht, 1974), 169-187.
    • G V Coyne, Newton's controversy with Leibniz over the invention of the calculus, in Newton and the new direction in science (Vatican City, 1988), 109-115.
    • E Giusti, A comparison of infinitesimal calculus in Leibniz and Newton (Italian), Rend.
    • R C Hovis, What can the history of mathematics learn from philosophy? A case study in Newton's presentation of the calculus, Philos.
    • P Kitcher, Fluxions, limits, and infinite littlenesse : A study of Newton's presentation of the calculus, Isis 64 (221) (1973), 33-49.

  4. References for Lagrange
    • C G Fraser, Isoperimetric problems in the variational calculus of Euler and Lagrange, Historia Math.
    • C G Fraser, Joseph Louis Lagrange's algebraic vision of the calculus, Historia Math.
    • C G Fraser, J L Lagrange's changing approach to the foundations of the calculus of variations, Arch.
    • C G Fraser, Isoperimetric problems in the calculus of variations of Euler and Lagrange (Spanish), Mathesis 8 (1) (1992), 31-53.
    • J V Grabiner, The calculus as algebra, the calculus as geometry : Lagrange, Maclaurin, and their legacy, in Vita mathematica (Washington, DC, 1996), 131-143.

  5. References for Mikusinski
    • P C Chatwin, Review: Operational Calculus, Vol.
    • J W Dettman, Review: Operational Calculus, Vol.
    • R Hilfer, Y Luchko and Z Tomovski, Operational method for the solution of fractional differential equations with generalised Riemann-Liouville fractional derivatives, Fractional Calculus and Applied Analysis 12 (3) (2009), 299-318.
    • H M Schaerf, Review: Rachunek Operatorow (Operational calculus) by Jan Mikusinski, Bull.
    • R G Woolley, Review: Operational Calculus by Jan Mikusinski, Chromatographia 18 (6) (1984), 329.

  6. References for Euler
    • C G Fraser, The origins of Euler's variational calculus, Arch.
    • C G Fraser, Isoperimetric problems in the variational calculus of Euler and Lagrange, Historia Math.
    • E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, Conference on the History of Mathematics (Rende, 1991), 293-313.
    • J Lutzen, Euler's vision of a general partial differential calculus for a generalized kind of function, Math.

  7. References for Curry
    • J P Seldin and J R Hindley (eds.), To H B Curry : essays on combinatory logic, lambda calculus and formalism (London-New York, 1980).
    • A short biography of Haskell B Curry, in To H B Curry : essays on combinatory logic, lambda calculus and formalism (London-New York, 1980), vii-xi.
    • Bibliography of Haskell B Curry, in To H B Curry : essays on combinatory logic, lambda calculus and formalism (London-New York, 1980), xiii-xx.
    • J P Seldin, Curry's program, in To H B Curry : essays on combinatory logic, lambda calculus and formalism (London-New York, 1980), 3-33.

  8. References for Cauchy
    • J V Grabiner, The Origins of Cauchy's Rigorous Calculus (Cambridge, Massachusetts, 1981).
    • J M Dubbey, Cauchy's contribution to the establishment of the calculus, Ann.
    • J V Grabiner, Who gave you the epsilon? Cauchy and the origins of rigorous calculus, Amer.
    • T Koetsier, Cauchy's rigorous calculus : a revolution in Kuhn's sense?, Nieuw Arch.

  9. References for Maclaurin
    • G Giorello, The 'fine structure' of mathematical revolutions : metaphysics, legitimacy, and rigour, The case of the calculus from Newton to Berkeley and Maclaurin, in Revolutions in mathematics (New York, 1992), 134-168.
    • J V Grabiner, Was Newton's calculus a dead end? The continental influence of Maclaurin's treatise of fluxions, Amer.
    • J V Grabiner, The calculus as algebra, the calculus as geometry : Lagrange, Maclaurin, and their legacy, in Vita mathematica (Washington, DC, 1996), 131-143.

  10. References for Schouten
    • D J Struik, J A Schouten and the tensor calculus, Nieuw Arch.
    • D J Struik, Schouten, Levi-Civita and the Emergence of Tensor Calculus, in David Rowe and John McCleary (eds.), History of Modern Mathematics Vol.
    • A G Walker, Review: Ricci-calculus (2nd edition) by J A Schouten, The Mathematical Gazette 40 (333) (1956), 225-226.

  11. References for Berkeley
    • The case of the calculus from Newton to Berkeley and Maclaurin, in Revolutions in mathematics (New York, 1992), 134-168.
    • I Grattan-Guinness, Berkeley's criticism of the calculus as a study in the theory of limits, Janus 56 (1969), 215-227.
    • X X Ren, Berkeley and his critique to the early calculus (Chinese), J.

  12. References for Kline
    • R W Cowan, Review: Calculus Part One by Morris Kline, Mathematics Magazine 40 (5) (1967), 277.
    • R W Cowan, Review: Calculus Part Two by Morris Kline, Mathematics Magazine 40 (5) (1967), 277.
    • W G Kellaway, Review: Calculus by Morris Kline, The Mathematical Gazette 52 (380) (1968), 171-172.

  13. References for Hermann
    • S Mazzone and C S Roero, Jacob Hermann and the diffusion of the Leibnizian calculus in Italy (Leo S Olschki, Florence, 1997).
    • E J Aiton, The contributions of Isaac Newton, Johann Bernoulli and Jakob Hermann to the inverse problem of central forces, in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli (Wiesbaden, 1989), 48-58.
    • R H Vermij, Bernard Nieuwentijt and the Leibnizian Calculus, Studia Leibnitiana 21 (1 ) (1989), 69-86.

  14. References for Fontaine des Bertins
    • J L Greenberg, Alexis Fontaine's route to the calculus of several variables, Historia Math.
    • J L Greenberg, Alexis Fontaine's integration of ordinary differential equations and the origins of the calculus of several variables, Ann.
    • J L Greenberg, Alexis Fontaine's 'fluxio- differential method' and the origins of the calculus of several variables, Ann.

  15. References for Laplace
    • M A B Deakin, Corrigendum: 'Operational calculus and the Laplace transform', Austral.
    • M A B Deakin, Operational calculus and the Laplace transform, Austral.
    • I Schneider, Laplace and thereafter : the status of probability calculus in the nineteenth century, in The probabilistic revolution 1 (Cambridge, MA-London, 1987), 191-214.

  16. References for Heaviside
    • P A Kullstam, Heaviside's Operational Calculus: Oliver's Revenge, IEEE Transactions on Education 34 (2) (1991), 155-156.
    • J Lutzen, Heaviside's operational calculus and the attempts to rigorise it, Archive for History of Exact Sciences 21 (1979), 161-200.
    • S S Petrova, Heaviside and the Development of the Symbolic Calculus, Archive for History of Exact Sciences 37 (1987), 1-23.

  17. References for Grassmann
    • F D Kramar, The geometric Grassmann calculus (Russian), in Proc.
    • K Reich, Rudolf Mehmke, an outstanding propagator of Grassmann's vector calculus, in From Past to Future: Grassmann's Work in Context (Basel, 2010), 209-220 .

  18. References for Black Fischer
    • (1983) A stochastic calculus model of Continuous Trading: Complete Markets, Stochastic Processes and their Applications, 15, 313-316.
    • (1996), in Ikeda N, Watanabe S, Fukushima M and Kunita H (eds.), Ito's stochastic calculus and probability theory, Tokyo, ix-xiv.

  19. References for Wallace
    • A D D Craik, Calculus and analysis in early 19th-century Britain : the work of William Wallace, Historia Math.
    • M Panteki, William Wallace and the Introduction of Continental Calculus to Britain : A Letter to George Peacock, Historia Mathematica 14 (1987), 119-132.

  20. References for Ito
    • N Ikeda, S Watanabe, M Fukushima and H Kunita (eds.), Ito's stochastic calculus and probability theory (Tokyo, 1996).
    • Kiyosi Ito, in N Ikeda, S Watanabe, M Fukushima and H Kunita (eds.), Ito's stochastic calculus and probability theory (Tokyo, 1996), ix-xiv.

  21. References for Viete
    • I G Bashmakova and E I Slavutin, F Viete's calculus of triangles, and the study of Diophantine equations (Russian), Istor.-Mat.
    • S S Glushkov, An interpretation of Viete's 'Calculus of triangles' as a precursor of the algebra of complex numbers, Historia Math.

  22. References for De L'Hopital
    • I Grattan-Guinness and H J M Bos, From the Calculus to Set Theory 1630-1910: An Introductory History (Princeton University Press, 2000).
    • J Peiffer, Le probleme de la brachystochrone a travers les relations de Jean I Bernoulli avec L'Hopital et Varignon, in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli (Wiesbaden, 1989), 59-81.

  23. References for Rolle
    • P Sergescu, An episode in the struggle for the triumph of differential calculus; the Rolle-Sourin polemic 1702-1705 (Romanian) (Bucharest, 1942).
    • P Mancosu, The metaphysics of the calculus: a foundational debate in the Paris Academy of Sciences, 1700-1706, Historia Math.

  24. References for Saint-Vincent
    • M E Baron, The Origins of the Infinitesimal Calculus (Pergamon Press, Oxford, 1969).
    • A Meskens, Gregory of Saint Vincent : A Pioneer of the Calculus, The Mathematical Gazette 78 (483) (1994), 315-319.

  25. References for Federer
    • A Lytle, Review: Analytic Geometry and Calculus, by Herbert Federer and Bjarni Jonsson, The Mathematics Teacher 55 (4) (1962), 296.
    • L Wahlstrom, Review: Analytic Geometry and Calculus, by Herbert Federer and Bjarni Jonsson, Amer.

  26. References for Fontenelle
    • M Blay, Du fondement du calcul differentiel au fondement de la science du mouvement dans les 'Elements de la geometrie de l'infini' de Fontenelle,in Der Ausbau des Calculus durch Leibniz und die Bruder Bernoulli (Wiesbaden, 1989), 99-122.

  27. References for Zu Chongzhi
    • D Dennis, V Kreinovich and S M Rump, Intervals and the origins of calculus, Reliab.

  28. References for Levi-Civita
    • D J Struik, Schouten, Levi-Civita, and the emergence of tensor calculus, in The history of modern mathematics, Vol.

  29. References for Carnot
    • C C Gillispie, Lazare Carnot Savant : a monograph treating Carnot's scientific work, with facsimile reproduction of his unpublished writings on mechanics and on the calculus, and an essay concerning the latter by A P Youschkevitch (Princeton, 1971).

  30. References for Fermat
    • Von Fermat und Descartes bis zur Erfindung des Calculus und bis zum Ausbau der neuen Methoden (Berlin, 1957).

  31. References for Pontryagin
    • E J McShane, The Calculus of Variations from the beginning through Optimal Control Theory, SIAM Journal on Control and Optimization 27 (5) (1989), 916-939.

  32. References for Pascal
    • P Dupont, The foundations of the calculus of probabilities in Blaise Pascal (Italian), Atti Accad.

  33. References for Huygens
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  59. References for Hesse
    • C G Fraser, Jacobi's result (1837) in the calculus of variations and its reformulation by Otto Hesse (1857), in A study in the changing interpretation of mathematical theorems.

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    • R Zach, The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program, Synthese 137 (1-2) (2003), 211-259.

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    • S S Petrova and O E Mitryaeva, Some results of James Gregory on the integral calculus (Russian), Istor.-Mat.

  65. References for Servois
    • L A Ljusternik and S S Petrova, From the history of symbolic calculus (Russian), Istor.-Mat.

  66. References for Jacobi
    • C G Fraser, Jacobi's result (1837) in the calculus of variations and its reformulation by Otto Hesse (1857) : A study in the changing interpretation of mathematical theorems, in History of mathematics and education : ideas and experiences (Gottingen, 1996), 149-172.

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    • R W Hamming, Calculus and Discrete Mathematics, The College Mathematics Journal 15 (5) (1984), 388-389.

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    • M E Baron, The Origins of the Infinitesimal Calculus (Dover, 2003).

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  71. References for Ohm Martin
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    • M E Flashman, Historical motivation for a calculus course : Barrow's theorem, in Vita mathematica (Washington, DC, 1996), 309-315.

  73. References for Ricci-Curbastro
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Additional material

  1. Menger on the Calculus of Variations
    • Menger on the Calculus of Variations .
    • Karl Menger wrote an article What Is Calculus of Variations and What Are Its Applications? This interesting article, written by an outstanding mathematician and expert in the field, gives insights into both the history of the topic and also into understanding Menger's views on mathematics:- .
    • The calculus of variations belongs to those parts of mathematics whose details it is difficult to explain to a non-mathematician.
    • The first human being to solve a problem of calculus of variations seems to have been Queen Dido of Carthage.
    • The branch of mathematics which establishes a rigorous proof of this statement is the calculus of variations.
    • Some problems concerning maxima and minima are studied in differential calculus, taught in college.
    • In differential calculus we deal thus with maxima and minima of so-called functions of points, i.e., of numbers associated with points; in calculus of variations, however, with maxima and minima of so-called functions of curves, that is, of numbers associated with curves or of numbers associated with still more complicated geometric entities, like surfaces.
    • There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature.
    • It is obvious that for this reason the mathematical theory of economics is to a large extent application of calculus of variations.
    • In the mathematical theory of the maximum and minimum problems in calculus of variations, different methods are employed.
    • In order to find such criteria a considered curve is a little varied, and it is from this method that the name "calculus of variations" for the whole branch of mathematics is derived.
    • This second method of calculus of variations was initiated by the German mathematician Hilbert at the beginning of the century.
    • Another way of calculus of variations was started in this country.
    • While the minimum and maximum problems of calculus of variations correspond to the problem in the ordinary calculus of finding peaks and pits of a surface, the minimax problems correspond to the problem of finding the saddle points of the surface (the passes of a mountain).
    • One of the greatest advances of calculus of variations in recent times has been the development of a complete and systematic theory of stationary curves due to Marston Morse (Institute of Advanced Study).
    • There are many technical details of calculus of variations which are hardly available to a non-mathematician.
    • Certainly this is true in the case of calculus of variations: If the cars, the locomotives, the planes, etc., produced today are different in form from what they used to be fifteen years ago, then a good deal of this change is due to calculus of variations.
    • But if we wish to discover the form which guarantees the least resistance, then we need calculus of variations.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Calculus_of_Variations.html .

  2. Richard Courant: 'Differential and Integral calculus' German edition
    • Richard Courant: Differential and Integral calculus German edition .
    • In 1934 Richard Courant published an English edition of his German text Differential and Integral calculus.
    • Differential and Integral calculus by Richard Courant .
    • Although there is no lack of textbooks on the differential and integral calculus, the beginner will have difficulty in finding a book that leads him straight to the heart of the subject and gives him the power to apply it intelligently.
    • The reader will notice especially the complete break away from the out-of-date tradition of treating the differential calculus and the integral calculus separately.
    • This separation, a mere result of historical accident, with no good foundation either in theory or in practical convenience in teaching, hinders the student from grasping the central point of the calculus, namely, the connection between definite integral, indefinite integral, and derivative.
    • With the backing of Felix Klein and others, the simultaneous treatment of differential calculus and integral calculus has steadily gained ground in lecture courses.
    • This first volume deals mainly with the integral and differential calculus for functions of one variable; a second volume will be devoted to functions of several variables and some other extensions of the calculus.
    • The beginner should note that I have avoided blocking the entrance to the concrete facts of the differential and integral calculus by discussions of fundamental matters, for which be is not yet ready.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Courant_calculus_german.html .

  3. George Gibson: 'Calculus
    • George Gibson: Calculus .
    • In 1901 George Gibson published An Elementary Treatise on the Calculus with Illustrations from Geometry, Mechanics and Physics with Macmillan and Co., Limited, St Martin's Street, London.
    • It is sometimes alleged that a thorough knowledge of the derivatives and integrals of the simpler powers, of the exponential and the logarithmic functions, and perhaps of the sine and the cosine, is quite sufficient preparation in the Calculus for the engineer.
    • It may be possible to state and illustrate in a few lessons a sufficient amount of the special results of the Calculus to enable a student to follow with some intelligence.
    • the more elementary treatment of mechanical and physical problems; but, though such a meagre course in the Calculus may not be without value, it is quite inadequate, both in kind and in quantity, m a preparation for the serious study of such practical subjects as Alternate Current Theory, Thermodynamics, Hydrodynamics, and the theory of Elasticity, and to a student so prepared much of the recent literature in Physics and Chemistry would be a sealed book.
    • Subsequent specialisation makes it the more, not the less, necessary that the mathematical training in the earlier stages should be the same whether the student afterwards devotes himself to pure mathematics or to the more practical branches of science, especially as the processes of thought involved in any serious study of mechanical, physical, or chemical phenomena have much in common with those developed in the study of the Calculus.
    • The early text-books on the Calculus, such as Maclaurin's or Simpson's, were not written for pure mathematicians alone, but drew their illustrations largely from Natural Philosophy; the later text-books, probably in consequence of the ever-widening range of Physics, gradually dropped physical applications, and even tended to become treatises on Higher Geometry.
    • In the present position of mathematical science, however, it is just as much out of place to make an elementary work on the Calculus a text-book of Higher Geometry as.
    • What may be reasonably required of an elementary work on the Calculus is that it should prepare the student for immediately applying its principles and processes in any department of his studies in which the Calculus is generally used.
    • As regards Chemistry, a sound knowledge of the Calculus is of special importance, since it is the properties of functions -of more than one variable that are predominant in chemical investigations; the lately published book of Van Laar, Lehrbuch der Mathematischen Chemie, is a sign of the times that cannot be mistaken.
    • After considerable hesitation I have included in my plan the elements of Coordinate Geometry, so far as these were likely to be of real service in elucidating fundamental principles or important applications; but for many applications of the Calculus an extensive acquaintance with Coordinate Geometry is not necessary, and I hope that a sufficiently clear account of its principles has been given to meet the practical needs of many students.
    • Another innovation is the chapter on the Theory of Equations; the innovation seems to be justified, not merely as an arithmetical illustration of the Calculus, but also by the practical importance of the subject, and by the absence of elementary works that treat of transcendental equations.
    • The somewhat lengthy discussion of the conceptions of a rate and a limit I have found in practice to be the simplest method of enabling a student to grapple with the special difficulties of the Calculus in its applications to mechanical or physical problems; when these notions have been thoroughly grasped, subsequent progress is more certain and rapid.
    • them; and with the object of encouraging the student to put himself through the drill that is absolutely necessary for the acquisition of facility and confidence in applying the Calculus, I have freely given hints towards the solution of the more important examples.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Gibson_Calculus.html .

  4. Charles Bossut on Leibniz and Newton
    • We present below a version of Bossut's account of the Leibniz-Newton controversy over their priority in inventing the calculus.
    • Urged on the one hand by the English, and on the other by personal resentment against Leibniz, from whom he professed not to have received the marks of esteem he conceived to be his due, he 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 said this for the sake of truth and his own conscience; and that he left to others the task of determining what Leibniz, the second inventor, had borrowed from the English geometrician.
    • Leibniz, 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 when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia that neither had borrowed anything from the other; that, when he published his differential calculus in 1884, 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 work of an English writer, in which the 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, etc.
    • But whether it were or not, I shall proceed to demonstrate that Leibniz either had no knowledge of these two pieces before he discovered his differential calculus, or derived no information from them.
    • All these researches are ingenious and seem to have at least a remote relation to the calculus of differences.
    • In fact the theory of series was already far advanced in England at that time; and though Leibniz had likewise penetrated deeply into it, he always acknowledged that the English, and Newton in particular, had preceded and surpassed him in that branch of analysis: but this is not the differential calculus, and the English have shown too evident a partiality in their endeavours to connect these two objects together, .
    • Let us hear and examine the history which Leibniz gives of his discovery of the differential calculus.
    • He relates that, on combining his old remarks on the differences of numbers with his recent meditations on geometry, he hit upon this calculus about the year 1676; that he made astonishing applications of it to geometry; that being obliged to return to Hanover about the same time he could not entirely follow the thread of his meditations; that endeavouring nevertheless to bring forward his new discovery, he went by the way of England and Holland; that he stayed some days in London where he became acquainted with Collins who showed him several letters from Gregory, Newton, and other mathematicians, which turned chiefly on series.
    • According to this account, it would appear that Leibniz, wishing to spread abroad his new discovery, must then have made known in England the differential calculus.
    • But if the account given by Leibniz be just, or if his memory did not deceive him, when he said he was in possession of the differential calculus before his second visit to England, no doubt some private reason then occurred to induce him to keep his discovery secret, contrary to the design he had first formed of bringing it forward: for in this very letter Collins mentions another from Leibniz to Oldenburg written from Amsterdam the 28th of November 1676 in which Leibniz proposes the construction of tables of formulas tending to improve the method of Sluze, instead of explaining the differential calculus or at least pointing it out as much more expeditious and more convenient.
    • The English therefore are justified in saying that Leibniz, when he passed through London in 1676, did not teach them the differential calculus: but they ought to acknowledge that the same letter conclusively proves that he likewise learnt nothing from them on the subject.
    • Then he explains openly and without mystery that of the differential calculus, affirming that he had long employed it for drawing the tangents of curve lines.
    • The design of stripping Leibniz and making him pass for a plagiary was carried so far in England that during the height of the dispute it was said (and Newton himself was not ashamed to support the objection) that the differential calculus of Leibniz was nothing more than the method of Barrow.
    • What are you thinking of, answered Leibniz, to bring such a charge against me? Will you have the differential calculus to be nothing but the method of Barrow when I claim it? and at the same time say it was invented by Mr Newton when you wish to rob me of it? Can you be so blinded by passion as not to perceive this manifest contradiction? 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 a pupil and friend of Barrow and had the 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? .
    • Johann Bernoulli, who in concert with his brother learned the analysis of infinities from the writings of Leibniz, opposed to the Commercium epistolicum a letter where he 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 though all the journals of Holland and Germany.
    • His reasons are in substance, first, the Commercium epistolicum exhibits no vestige of Newton's having employed dotted letters to denote fluxions in the writings alleged; secondly, in the Principia, where the author had so frequently occasion for employing this calculus and giving it's algorithm, he has not done it; he proceeds everywhere by means of lines and figures without any determinate analysis, and simply in the manner of Huygens, Roberval, Cavalleri, etc,: thirdy, the dotted letters first began to appear in the third volume of Wallis's Works, several years after the differential calculus was everywhere known; fourthly, the true method of differencing differences, or of taking the fluxions of fluxions, was unknown to Newton, since even in his treatise on quadratures, not published till 1704, the rule he gives at the end for determining the fluxions of all orders by considering these fluxions as the terms of the power of a binomial formed of a variable quantity, and it's first fluxion, and treating the first fluxion as constant, is false except simply for the term which answers to the first fluxion: fifthly, at the same period of 1704 Newton was not versed in the integral calculus of differential equations which Leibniz and the two Bernoullis had already carried so far; otherwise he would not have failed to treat this part of the analysis of infinities, the most difficult, and at least as worthy of being promulgated and carried to perfection as the quadratures on which he enlarged so much.
    • To this letter the English answered that the notation did not constitute the method: that the principles of the calculus of fluxions were contained in Newton's great work and in his letters: that the rule in the treatise on quadratures for finding the fluxions of all orders was true, suppressing the denominators of the terms of the series and gave by consequence quantities proportional to the true fluxions.
    • But Leibniz might answer: 'I have proposed the existence of infinitely small quantities only as subsidiary, or as a simple hypothesis, serving to abridge the calculus and reasonings on which it is founded.
    • He has the advantage over Newton of having invented and carried to a great length the integral calculus of differential equations.
    • The one has left us a greater mass of geometrical truths: the other more accelerated the progress of science in his time by the simple and commodious notation of his calculus, the applications he made of it himself, or enabled others of the learned to make, the encouragements he gave them, and the new paths he was continually opening to their meditations.
    • To conclude, whatever length of time the completion of the Principia may have required, we ought not to forget that this work did not appear till two or three years after Leibniz had published his differential calculus, and some sketches of the integral.

  5. Oskar Bolza: 'Calculus of Variations
    • Oskar Bolza: Calculus of Variations .
    • In 1904 the University of Chicago Press published Oskar Bolza's Lectures on the Calculus of Variations.
    • The principal steps in the progress of the Calculus of Variations during the last thirty years may be characterized as follows: .
    • This was - in advance of great importance for all geometrical applications of the Calculus of Variations; for the older method implied - for geometrical problems - a rather artificial restriction.
    • These discoveries mark a turning-point in the history of the Calculus of Variations.
    • Chiefly under the influence of Weierstrass's theory a vigorous activity in the Calculus of Variations has set in during the last few years, which has led - apart from extensions and simplifications of Weierstrass's theory - to the following two essentially new developments: .
    • Hilbert's a priori existence proof for an extremum of a definite integral - a discovery of far-reaching importance, not only for the Calculus of Variations, but also for the theory of differential equations and the theory of functions.
    • And the present volume is, in substance, a reproduction of these lectures, with such additions and modifications as seemed to me desirable in order that the book could serve as a treatise on that part of the Calculus of Variations to which the discussion is here confined, viz., the case in which the function under the integral sign depends upon a plane curve and involves no higher derivatives than the first.
    • For a rigorous treatment of the Calculus of Variations the principal theorems of the modern theory of functions of a real variable are indispensable; these I had therefore to presuppose, the more so as I deviate from Weierstrass and Kneser in not assuming the function under the integral sign to be analytic.
    • My principal source of information concerning Weierstrass's theory has been the course of lectures on the Calculus of Variations of the Summer Semester, 1879, which I had the good fortune to attend as a student in the University of Berlin.
    • Besides, I have had at my disposal sets of notes of the courses of 1877 (by Mr G Schulz) and of 1882 (a copy of the set of notes in the "Lesezimmer" at Gottingen for which I am indebted to Professor Tanner), a copy of a few pages of the course of 1872 (from notes taken by Mr Ott), and finally a set of notes (for which I am indebted to Dr J C Fields) of a course of lectures on the Calculus of Variations by Professor H A Schwarz (1898-99).
    • I regret very much that I have not been able to make use of the articles on the Calculus of Variations in the Encyclopaedie der mathematischen Wissenschaften by Adolf Kneser, Zermelo, and Hahn.
    • For the same reason no reference could be made to Hancock's Lectures on the Calculus of Variations.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Bolza_calculus.html .

  6. Serge Lang: 'A first course in calculus
    • Serge Lang: A first course in calculus .
    • Serge Lang wrote A first course in calculus in 1964.
    • In the Foreword to the book he explains something of his ideas about teaching calculus:- .
    • A first course in calculus by Serge Lang .
    • The purpose of a first course in Calculus is to teach the student the basic notions of derivative and integral, and the basic techniques and applications which accompany them.
    • At present in the United States, the trend is to introduce Calculus in high schools, and I agree that the material covered in the present book should ultimately be the standard fare of the last two years of secondary schools.
    • I have made no great innovations in the exposition of calculus.
    • As for the question: Why write one more calculus book? I would answer: Because practically all existing ones are too long (500 to 600 pages) and one loses sight of the over-all ideas, sacrificed for the sake of topics which have hung on through habits, bad habits, I would say.

  7. Richard Courant: 'Differential and Integral calculus' English edition
    • Richard Courant: Differential and Integral calculus English edition .
    • In 1934 Richard Courant published an English edition of his German text Differential and Integral calculus.
    • Differential and Integral calculus by Richard Courant .
    • When American colleagues urged me to publish an English edition of my lectures on the differential and integral calculus, I at first hesitated.
    • I felt that owing to the difference between the methods of teaching the calculus in Germany and in Britain and America a simple translation was out of the question, and that fundamental changes would be required in order to meet the needs of English-speaking students.
    • Thus the first volume contains the material for a course in elementary calculus, while the subject-matter of the second volume is more advanced.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Courant_calculus_english.html .

  8. Levi-Civita: 'Absolute Differential Calculus
    • Levi-Civita: Absolute Differential Calculus .
    • In 1925 Levi-Civita published Lezioni di calcolo differenziale assoluto and, two years later an English translation appeared entitled The Absolute Differential Calculus (Calculus of Tensors).
    • CALCULUS .
    • (Calculus of Tensors) .
    • Two new chapters have been added, which are intended to exhibit the fundamental principles of Einstein's General Theory of Relativity (including, of course, as a limiting case, the so-called Special or Restricted Theory) as an application of the Absolute Calculus.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Levi-Civita_Calculus.html .

  9. W H Young: 'Differential Calculus
    • W H Young: Differential Calculus .
    • The fundamental theorems of the differential calculus by W H Young was No 11 in the series and published in 1910.
    • THE FUNDAMENTAL THEOREMS OF THE DIFFERENTIAL CALCULUS .
    • The Differential Calculus is concerned with those continuous functions that possess differential coefficients and with these differential coefficients themselves.
    • No more knowledge of the language or concepts of this theory will however here be required than a serious mathematical student may now be supposed to have gained before completing his Degree course, and, with this exception, the present account of the fundamental theorems of the Differential Calculus will, it is hoped, be found to be complete in itself For the rest a brief account is given in Appendix III of the definitions and results from the Theory of Sets of Points actually employed in the Tract.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Young_calculus.html .

  10. Hadamard on the calculus
    • Jacques Hadamard on "Who discovered the calculus" .
    • In An Essay on the Psychology of Invention in the Mathematical Field (Princeton University Press, 1945), Jacques Hadamard looks at the question: "Who discovered the Infinitesimal Calculus." We present here a version of his argument.
    • What is certain is that Gregory was the first to publish a proof of this fundamental theorem of the calculus.
    • Does this mean, as many are inclined to think, that he invented the Differential Calculus? In one sense we must answer "yes," for we see him applying his method to various problems, and even pointing out that the method could he applied to similar ones.
    • But the Differential Calculus is not the whole Infinitesimal Calculus.
    • There is a second branch, the Integral Calculus, the fundamental operation of which is the valuation of plane curvilinear areas; and this implies a discovery which lay deep and had been entirely unsuspected, viz.
    • Thus Oresme, Keper, and Fermat failed to discover the Differential Calculus because they did not pursue their initial and fruitful ideas.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Hadamard_calculus.html .

  11. H W Turnbull: 'Scottish Contribution to the Calculus
    • H W Turnbull: Scottish Contribution to the Calculus .
    • H W Turnbull gave a lecture in "History" on The Scottish Contribution to the Early History of the Calculus.
    • THE SCOTTISH CONTRIBUTION TO THE EARLY HISTORY OF THE CALCULUS .
    • Recent examination of original and mostly unpublished manuscripts and letters has thrown new light upon the early developments of the Calculus by James Gregory (1638-1675) and Isaac Newton (1642-1727).

  12. Eulogy to Euler by Fuss
    • Euler turned his attention to these different subjects, he perfected integral calculus; was the inventor of a new type of calculus of sines; he simplified analytical operations; with the help of these powerful tools and the astonishing facility with which he knew to manipulate the most intractable expressions, he found a new way to spread light onto all the parts of the mathematical sciences.
    • The great revolution that the discovery of differential and integral calculus had provided for in all of the branches of the mathematical sciences, did not neglect to change Mechanics entirely.
    • It is there that we find remarkably the full measure of the theory of curves: tautochrones, brachistochrone, trajectories and the very deep research in integral calculus, on the nature of numbers, concerning series, the motion of heavenly bodies, the attraction of spheroid-elliptical bodies and on an infinity of subjects of which one hundredth part would suffice in making the reputation of anyone else.
    • Euler revisited this important topic and in 1744 published a complete treatise on isoperimetrics where we can say that he mined the riches of this sublime analysis and he established the first basis for the calculus of variations, when he considered curves which differ in infinitely small distances for a determined curve.
    • He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author, and the recurrent series, he provides for in the second part the general theory of curves with their divisions and sub-divisions and in a supplement the theory of solids and their surfaces while showing how their measurement leads to the equations with three variables and he ends finally this important work by developing the idea of curves with double curvature which provides for the consideration of the intersection of curved lined surfaces.
    • In succession to this Introductio came about his lessons in differential calculus as well as those on integral calculus published by our Academy which Mr.
    • The foremost quality of the former work which travels concerning the part of infinitesimal calculus already perfected by its inventors Newton and Leibniz and the Bernoulli, is consistent with the point of view where Mr.
    • Euler had already envisaged the true principles in the systematic order in which he has exposed them and with the methodology which exists and the clarity with which he has shown the utility of the calculus in relation to the doctrine of series and to the theory of the maxima and the minima.
    • The first steps concerning the origins of integral calculus are lost in differential calculus but are far from the perfection that the latter attained.
    • His glory is due to the fact that he pushed back the limits of this sublime calculus far beyond the reach of the primary discoverers.
    • The third volume of his Integral Calculus contains a new form of calculus which has enriched infinitesimal analysis: the calculus of variations.
    • Euler at the Berlin Academy and he was eventually able to disengage the calculus of variations from its geometrical origins and made it into a problem of analysis and made it possible to resolve problems through this new genre of calculus that Mr.
    • Euler has since perfected he named it the calculus of variations since the relations between the variable quantities is in itself variable.
    • During the same time that the Academy was publishing this work its presses were occupied in printing the Letters to a Princess of Germany, Integral Calculus, Elements of Algebra, the calculation concerning the comet of 1769, the sun's eclipse and the passage of Venus, all in the same year, the new lunar theory and that of navigation, not withstanding the huge number of memoires which are found in the Commentarii of this period.
    • Since Daniel Bernoulli's Hydrodynamica, the perfection of calculus which in Mr.

  13. Eulogy to Euler by Fuss
    • Euler turned his attention to these different subjects, he perfected integral calculus; was the inventor of a new type of calculus of sines; he simplified analytical operations; with the help of these powerful tools and the astonishing facility with which he knew to manipulate the most intractable expressions, he found a new way to spread light onto all the parts of the mathematical sciences.
    • The great revolution that the discovery of differential and integral calculus had provided for in all of the branches of the mathematical sciences, did not neglect to change Mechanics entirely.
    • It is there that we find remarkably the full measure of the theory of curves: tautochrones, brachistochrone, trajectories and the very deep research in integral calculus, on the nature of numbers, concerning series, the motion of heavenly bodies, the attraction of spheroid-elliptical bodies and on an infinity of subjects of which one hundredth part would suffice in making the reputation of anyone else.
    • Euler revisited this important topic and in 1744 published a complete treatise on isoperimetrics where we can say that he mined the riches of this sublime analysis and he established the first basis for the calculus of variations, when he considered curves which differ in infinitely small distances for a determined curve.
    • He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author, and the recurrent series, he provides for in the second part the general theory of curves with their divisions and sub-divisions and in a supplement the theory of solids and their surfaces while showing how their measurement leads to the equations with three variables and he ends finally this important work by developing the idea of curves with double curvature which provides for the consideration of the intersection of curved lined surfaces.
    • In succession to this Introductio came about his lessons in differential calculus as well as those on integral calculus published by our Academy which Mr.
    • The foremost quality of the former work which travels concerning the part of infinitesimal calculus already perfected by its inventors Newton and Leibniz and the Bernoulli, is consistent with the point of view where Mr.
    • Euler had already envisaged the true principles in the systematic order in which he has exposed them and with the methodology which exists and the clarity with which he has shown the utility of the calculus in relation to the doctrine of series and to the theory of the maxima and the minima.
    • The first steps concerning the origins of integral calculus are lost in differential calculus but are far from the perfection that the latter attained.
    • His glory is due to the fact that he pushed back the limits of this sublime calculus far beyond the reach of the primary discoverers.
    • The third volume of his Integral Calculus contains a new form of calculus which has enriched infinitesimal analysis: the calculus of variations.
    • Euler at the Berlin Academy and he was eventually able to disengage the calculus of variations from its geometrical origins and made it into a problem of analysis and made it possible to resolve problems through this new genre of calculus that Mr.
    • Euler has since perfected he named it the calculus of variations since the relations between the variable quantities is in itself variable.
    • During the same time that the Academy was publishing this work its presses were occupied in printing the Letters to a Princess of Germany, Integral Calculus, Elements of Algebra, the calculation concerning the comet of 1769, the sun's eclipse and the passage of Venus, all in the same year, the new lunar theory and that of navigation, not withstanding the huge number of memoires which are found in the Commentarii of this period.
    • Since Daniel Bernoulli's Hydrodynamica, the perfection of calculus which in Mr.

  14. F F P Bisacre - Applied calculus
    • F F P Bisacre - Applied calculus .
    • In 1921 Frederick Francis Percival Bisacre's Applied calculus; an introductory textbook was published by Blackie and Son.
    • It is not in general the policy of the Bulletin to review elementary textbooks on the calculus.
    • Its avowed intention is to provide an introductory course in the calculus for the use of students in the natural and applied science whose knowledge of mathematics is slight.
    • Its interest is further enhanced by biographical notes and portraits of mathematicians who have contributed to the development of the calculus and to the physical theories covered.
    • It is doubtful whether the book could be advantageously used in courses on the calculus in this country and yet it is not at all clear that some schools of applied science or technology might not find it distinctly available in their work.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Bisacre_Applied_Calculus.html .

  15. L'Hôpital: 'Analyse des infiniment petits' Preface
    • This was the first text-book to be written on the differential calculus and it is interesting to examine the Preface of the work in which de L'Hopital gives historical comments as well as describing the contents of the work: .
    • Barrow's work did not stop there: he also invented a kind of calculus based on this method, but this calculus, like that of Descartes, could only be used once all fractions and roots had been removed.
    • Barrow's calculus was replaced by that of M Leibniz, an accomplished geometer who started his own work where Barrow and others had ended theirs.
    • His calculus led him into domains hitherto unknown and the discoveries he made amazed the most brilliant mathematicians of Europe.
    • The Bernoullis were the first to recognise the elegance of Leibniz's method, and they in turn developed his calculus to a degree which enabled them to solve problems which had previously seemed too difficult to attempt.
    • This calculus is of immense scope: it can be used for the curves which occur in mechanics, transcendental curves such as the catenary, as well as for purely geometrical curves, squares or other roots do not cause any difficulty (and may even be an advantage), any number of variables may be considered, and it is equally easy to compare infinitely small quantities of any type.
    • The first describes the principles of the calculus of differences [the differential calculus].
    • The third shows how the calculus is used in problems connected with maxima and minima.
    • The eighth section describes how the calculus is used to find the curves which touch an infinite number of given straight lines or curves.
    • The tenth section describes a new way of using the differential calculus for geometrical curves: from which we can derive the method used by M Descartes and M Hudde, which is applicable only to this kind of curve.
    • I had intended to include an additional section which was to have described the marvellous use to which the calculus may be put in physics, what accuracy can thereby be obtained, and to show how useful the calculus would be in mechanics.
    • All this is only the first part of M Leibniz's work on calculus, which consists of working down from integral quantities to consider the infinitely small differences between them and comparing these infinitely small differences with each other, whatever their type: this part is called Differential Calculus.
    • The other part of M Leibniz's work is called the Integral Calculus, and consists of working up from these infinitely small quantities to the quantities of totals of which they are the differences: that is, it consists of finding their sums.
    • But M Leibniz wrote to me to say that he himself was engaged upon describing the integral calculus in a treatise he calls De Scientia infiniti, and I did not wish to deprive the public of such a work, which will deal with all the most interesting consequences of this inverse method of tangents, showing how it can be used to find the lengths of curves, to find the area they enclose, to find the volumes and surfaces of their solids of revolution, to find centres of gravity etc.
    • M Leibniz himself acknowledges his debt to M Newton, who, as it appears in his excellent work Philosophiae naturalis principis mathematica of 1687, had already invented a technque very like that of the differential calculus, which he uses throughout his book.
    • But M Leibniz's use of the characteristic makes his calculus much simpler and quicker, and sometimes also proves very helpful.

  16. Cauchy's Calculus
    • Cauchy's Calculus .
    • In it he attempted to make calculus rigorous and to do this he felt that he had to remove algebra as an approach to calculus.
    • Cauchy's approach to the calculus: .
    • C H Edwards, The historical development of the calculus (Springer, New York, 1979).
    • J W Grabiner, The origins of Cauchy's rigorous calculus (MIT Press, 1981).
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Cauchy_Calculus.html .

  17. Franklin's textbooks
    • A Treatise on Advanced Calculus (1940).
    • Not only are the traditional subjects of a book on advanced calculus covered, but also many more advanced topics are included.
    • This book is a valuable contribution to the field of advanced calculus.
    • This book is an extraordinarily satisfactory addition to the literature of advanced calculus.
    • This text is indeed a treatise which covers completely the infinitesimal calculus and includes much prerequisite algebra and analysis (and most other concepts) that are needed for geometric and physical applications.
    • Of course, this text is not for beginners; the reader must be acquainted with the elementary calculus and he must have some proficiency in its technique in order to follow and appreciate the author's developments and precise deductions.
    • Methods of advanced calculus (1944).
    • first to refresh and improve the reader's technique in applying elementary calculus; second, to present those methods of advanced calculus which are most needed in applied mathematics.
    • As is well-known, this textbook is the second written by Professor Franklin on the subject of Advanced Calculus, the first having been published in 1940 by John Wiley and Sons under the title of "A treatise on advanced calculus." The two books are in some respects different as to content, but in all respects different as to point of view.
    • the new volume may be regarded as one of the best textbooks now available for any advanced calculus course which is intended to be a terminal course in mathematics for engineers, physicists and the like.
    • Differential and Integral Calculus (1953).
    • This is a soundly written standard text for a first course in the calculus.
    • This is a substantial text in elementary calculus.
    • Much appeal is made to geometric intuition, which is right and proper in a beginning course in calculus.
    • Compact calculus (1963).
    • The choice of title is accurate: the print is large and uncrowded on pages of medium size, the tersely phrased text is divided into sections of about one page, the total number of pages is small for the ground covered, which is differential and integral calculus of one variable, infinite series, and partial derivatives and multiple integrals ..
    • A Treatise on Advanced Calculus (Dover reprint of 1940 edition) (1965).

  18. Kline's books
    • the keen inquirer who wants to know why it is that mathematics is the key to our knowledge of the physical world but has no equipment beyond elementary mathematics (and there are many such people in the world today) will get a surprisingly long way in Kline's hands, and so may be inspired to learn some calculus from this book and pursue its applications elsewhere.
    • Important questions of sciences such as earth measure, cosmology, gravitation, and electromagnetism, and of arts, such as perspective drawing and musical composition, are described in detail sufficient to motivate discussions of mathematical notions they generate: geometry, algebra, trigonometry, calculus, and so on.
    • Calculus, An intuitive and Physical Approach (1967).
    • "In my opinion, a rigorous first course in the calculus is ill advised for numerous reasons.
    • However, the calculus divorced from applications is meaningless." Again: "In this book real problems are used to motivate the mathematics, and the latter, once developed, is applied to genuine physical problems - the magnificent, impressive, and even beautiful problems tendered by nature." It is in this "physical approach" that this text has a very great deal to offer.
    • It is refreshing to find that books are still being written on elementary calculus in which the subject is approached from an intuitive and heuristic standpoint rather than attempting to inculcate in the beginning student the deadly rigorous viewpoint of the sophisticated mathematician.
    • For the purpose for which they were designed these calculus books by Morris Kline would be splendid texts for a beginning student who needs a book that is readable, interesting, and not too demanding in mathematical rigour.

  19. Levi-Civita: 'Lezioni di calcolo differenziale assoluto
    • In 1925 Levi-Civita published Lezioni di calcolo differenziale assoluto and, two years later an English translation appeared entitled The Absolute Differential Calculus (Calculus of Tensors).
    • Riemann's general metric and a formula of Christoffel constitute the premises of the absolute differential calculus.
    • There is a chapter on the foundations of the absolute calculus, with special reference to the transformation of the equations of dynamics, in Wright's Tract, Invariants of Quadratic Differential Forms (Cambridge University Press, 1908); apart from this, while special researches based on the use of this method were, continued after 1901 by a limited number of mathematicians, yet general attention was not again directed to it until the great renaissance of natural philosophy, due to Einstein, which found in the absolute differential calculus the necessary instrument vii for formulating the new ideas mathematically and for the subsequent numerical work.
    • In an earlier memoir Einstein had given a new exposition of those elements and formulae of the absolute calculus which more specifically served his purposes.
    • A similar standpoint was subsequently adopted by the most distinguished workers in the field of general relativity, in particular by Weyl, Laue, Eddington, and Birkhoff, all of whom made conspicuous original contributions, both of idea and of method, to the physical theories, in addition to useful and elegant developments of the tensor calculus.
    • Similar statements can be made for Carmichael, Marcolongo, Kopff, Becquerel - to mention, from the vast literature on the subject, only the books I have myself had occasion to consult - while de Donder has avoided the notation of the absolute calculus and used instead the theory of integral invariants.
    • In recent years there have been some general treatises devoted to the absolute calculus; for instance, those of Juvet, Marais, and Galbrun.
    • Lastly, there is another calculus, in a new order of ideas, not less comprehensive and perhaps even more general, invented by Schouten, and developed with the collaboration of Struik.

  20. Finkel's Solution Book
    • Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions.
    • Many of the Formulae in Mensuration have been obtained by the aid of the Calculus, the operations alone being indicated.
    • This feature of the work will not detract from its merits for those persons who do not understand the Calculus; for those who do understand the Calculus it will afford an excellent drill to work out the steps taken in obtaining the formulae.
    • But the Calculus has been used for the sake of presenting the beauty and accuracy of that powerful instrument of mathematics.
    • Indeed, until quite recently, there were very few books on Arithmetic, Algebra, Geometry or Calculus that were not mere copies of the works written a century ago, and in this way the method, the spirit, the errors and the solecisms of the past two hundred years were preserved and handed down to the present generation.
    • The most of the text-books on Arithmetic, Algebra, Geometry, and the Calculus, written within the last five years, are evidence of this progress." .
    • - Invariants; Differential and Integral Calculus; Modern Methods in Geometry.
    • - Galois's Theory of Equations; Lie's Theory as applied to Differential Equations; Riemann's Theory of Functions; The Calculus of Variations; Functions Defined by Linear Differential Equations; The Theory of Numbers; The Theory of Planetary Motions; Theory of Surfaces; Linear Associative Algebra; the Algebra of Logic; the Plasticity of the Earth; Elasticity; and the Elliptic and the Abelian Transcendants.

  21. MacDuffee Addresses
    • Objectives in Calculus .
    • C C MacDuffee, Objectives in Calculus, Amer.
    • What should be the objectives in a beginning course in the calculus? That is a question which many college teachers ask themselves, and to which it is difficult to frame an answer.
    • Calculus is the course for which the student has long been preparing through college algebra, trigonometry and analytics, and for many a student it is the last mathematics course which he will ever take.
    • What could be more natural than a combination course of basic physics and calculus? This course would probably have to be spread over two years if it were to contain a complete course in both physics and calculus.
    • Can you think of a better background for scientists of the present age? Regardless of the framework in which it is taught, the first course in calculus must be handled with a fine sense of balance.
    • I think it should be a course in which advanced algebra and the rudiments of analytic geometry are integrated, and which contains a few of the essential ideas of the calculus in the second semester.
    • The question which we have come here to discuss is the proper content of a course or courses in algebra for a student who has just completed elementary calculus.

  22. De Montmort: 'Essai d'Analyse
    • This visit gave further impetus to his study of mathematics, and he came back to Paris to pursue his studies in algebra, geometry and the new calculus, which he found "thorny".
    • [One may speculate here about the new calculus.
    • Possibly Montmort had a contact with a pupil of Jacob - it is thought he did not meet Nicolaus Bernoulli until 1709 - and this contact inspired him to pursue the new calculus with its fascinating sidelines of the summation of infinite series and the manipulation of binomial coefficients.
    • The fact that he wrote at all is probably a fortunate one for the probability calculus.
    • This is possibly the first exponential limit in the calculus of probability, but having achieved it Montmort can't make much use of it.
    • Usually he carefully took the middle road in argument, as may be seen in his careful neutrality at the time of the calculus controversy.
    • The generalisations of the various topics discussed in the first edition are interesting, without adding anything particularly new to the probability calculus, although the various methods for the summation of series show the skill of the Bernoullis in that part of algebra.
    • With the publication of this second edition Montmort seems to have given up researches on the probability calculus.
    • It may have been that the short history which he wrote about the theory of probability (or possibly the calculus controversy) piqued his curiosity, but he wrote to Nicolaus (August 20th, 1713):- .

  23. David Hilbert: 'Mathematical Problems
    • The calculus of variations owes its origin to this problem of Johann Bernoulli and to similar problems.
    • So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations.
    • The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential - to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.
    • But the most striking example for my statement is the calculus of variations.
    • Weierstrass showed us the way to a new and sure foundation of the calculus of variations.
    • By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of the calculus of variations.
    • Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigour a difficult theorem on the continuity of functions or the existence of points of condensation? Who could dispense with the figure of the triangle, the circle with its centre, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences? .

  24. Von Neumann: 'The Mathematician
    • The second example is calculus - or rather all of analysis, which sprang from it.
    • The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance.
    • The origins of calculus are clearly empirical.
    • Newton invented the calculus "of fluxions" essentially for the purposes of mechanics - in fact, the two disciplines, calculus and mechanics, were developed by him more or less together.
    • The first formulations of the calculus were not even mathematically rigorous.
    • Second, that the empirical origin of mathematics is strongly supported by instances like our two earlier examples (geometry and calculus), irrespective of what the best interpretation of the controversy about the "foundations" may be.

  25. ELOGIUM OF EULER
    • Euler's oeuvre changed the face of integral calculus as the ripest discovery which man has ever possessed.
    • There are a great number of particular methods based on different principles which are spread throughout his works and brought together in his Treatise of Integral Calculus.
    • Here is an opportunity to mention another body of calculus which belongs almost in its entirety to Mr.
    • Taylor was made into an important branch of integral calculus by assigning a simple and workable notation which was found to apply successfully to the theory of series.
    • Euler regarded his new analysis as a science which one day would be useful in the progress of integral calculus.
    • The vibrating string problem and all those that belong to the theory of sound or the laws of oscillations in air had been subjected to analysis by these new methods which in turn enriched the calculus of partial differentials equations.

  26. Olds' teaching articles
    • The second semester began today and, with it came the challenge - a class of "repeaters" in Differential Calculus.
    • Not a new challenge for, each year, throughout the collegiate world, boys "flunk" Calculus and then have to repeat it.
    • Somehow it seemed that they didn't care to get too close to Calculus.
    • This provided the proper opening for the statement that we (the class) were going to discuss Calculus together just as though we knew nothing about it - and we did just that.
    • The purpose of this note is to outline the manner in which the heuristic method has been used to develop the concept of Curvature with a class in Calculus.
    • Furthermore, there is some suspicion that many teachers, college, high-school, grade, do not know arithmetic, algebra, calculus.

  27. Euler Elogium.html.html
    • Euler's oeuvre changed the face of integral calculus as the ripest discovery which man has ever possessed.
    • There are a great number of particular methods based on different principles which are spread throughout his works and brought together in his Treatise of Integral Calculus.
    • Here is an opportunity to mention another body of calculus which belongs almost in its entirety to Mr.
    • Taylor was made into an important branch of integral calculus by assigning a simple and workable notation which was found to apply successfully to the theory of series.
    • Euler regarded his new analysis as a science which one day would be useful in the progress of integral calculus.
    • The vibrating string problem and all those that belong to the theory of sound or the laws of oscillations in air had been subjected to analysis by these new methods which in turn enriched the calculus of partial differentials equations.

  28. Ernesto Pascal's books
    • The German translation of the "Calcolo delle variazioni" published by Ernesto Pascal in 1897 gives to American mathematicians in convenient form the best book on the calculus of variations that has, to our knowledge, appeared up to the present time.
    • A valuable feature of the work will certainly be found to be the very excellent and apparently complete bibliography given in connection with brief accounts of the development of the calculus of variations.
    • That such an end is in the calculus of variations especially difficult to attain appears from the fact that the proofs are not always precise and that the author prefers often to tell us that the work given is not rigorous rather than to attempt to make it so.
    • In his preface, written May 1917, Pascal says " It is certain that through the profound changes which the critical spirit has made in the foundations of the calculus, even a course intended for those for whom mathematics is a means rather than an aim, cannot but use the new results which have been reached .
    • it would therefore exhibit a shortsighted view and little esteem for the ability of the future engineer, to believe that it would be sufficient for them, at least if they can, to learn to operate the calculus in about the way in which a workman knows how to operate a machine made by others, and of which he does not know the inner connections." .

  29. Mathematics in Aberdeen.html

  30. Durell and Robson: 'Advanced Trigonometry
    • There have been such radical changes in method and outlook that it has become necessary to treat large sections of some of the standard books merely as (moderately) convenient collections of examples and to supply the bookwork in the form of notes; especially is this true of Algebra, Trigonometry, and the Calculus.
    • Methods of the Calculus are freely used in courses of Algebra and Trigonometry, while matter which used to find a place in the Algebra text-book is now included more conveniently elsewhere.
    • For many years past leading mathematicians have advocated a definition which transfers the chapter on the theory of logarithms from the Algebra to the Calculus text-book, and makes it the basis from which the exponential function is discussed, thus reversing the order commonly followed.
    • By tradition the theory of the exponential and logarithmic functions of a complex variable is included in books on Advanced Trigonometry and this is a very reasonable arrangement ; it seems equally desirable to include also the theory of the corresponding functions of a real variable instead of relegating it to the Calculus book.
    • The authors are planning text-books parallel to the present volume on Advanced Algebra and Calculus, written from a similar point of view.

  31. Mathematics in Aberdeen
    • The subjects are Geometrical Conic Sections, Spherical Trigonometry, Analytical Geometry, and the Differential and Integral Calculus.
    • The text-books recommended are Drew's "Conic Sections," Salmon's or Todhunter's "Conic Sections," Todhunter's "Spherical Trigonometry," and Todhunter's or Williamson's "Differential Calculus," and "Integral Calculus".
    • Geometry, Algebra, Trigonometry (Plane and Spherical), Conic Sections, Theory of Equations, Analytical Geometry of Two and Three Dimensions, and Differential and Integral Calculus, including the Solution of Differential Equations.
    • The examination in the Integral Calculus will be confined to the following subjects:- " Integration, application to lengths and areas of curves, volumes of solids, and to questions of mean value; definition and chief properties of the Gamma functions." .

  32. Gillespie: 'Integration
    • The first four chapters of this book are devoted to an elementary account of integration and demand from the student only a slight knowledge of the differential calculus.
    • For a student who desires a working knowledge of the integral calculus Chapters I to IV and Chapter VI cover the most important parts of the ground.
    • The integral calculus may be said to have been begun by the Greek mathematicians who strove to evaluate the area of a circle.
    • This method is essentially that of the integral calculus.
    • In the language of the calculus we say that the area of the circle is the limit for n tending to infinity of the area of the regular n-sided inscribed or circumscribed polygon.

  33. E W Hobson: 'Mathematical Education
    • That is the very important question as to the possibility of making a rudimentary treatment of the ideas and processes of the Calculus part of the normal course of Mathematics in the higher classes of schools.
    • In the hands of a really skilful teacher, the purely formal element in the treatment of the Calculus could be reduced to very small dimensions; all the leading notions and processes could be sufficiently illustrated by means of functions of the very simplest types.
    • The Calculus, as embodying and utilizing the fundamental notion of a "limit," is the gate to a Mathematical world of incomparably greater dimensions than the one in which the student has moved during the earlier part of his course.
    • I do not propose to indicate now, even in outline, a schedule of those parts of the Calculus which would be suitable as part of a general education.

  34. Andrew Forsyth addresses the British Association in 1905, Part 2
    • In particular, the infinitesimal calculus in its various branches (including, that is to say, what we call the differential calculus, the integral calculus, and differential equations) received the development that now is familiar to all who have occasion to work in the subject.
    • When this calculus was developed, it was applied to a variety of subjects; the applications, indeed, not merely influenced, but immediately directed, the development of the mathematics.

  35. John Walsh's delusions
    • There is no differential calculus, no Taylor's theorem, no calculus of variations, &c.
    • He sought, in his Theory of Partial Functions, to substitute "partial equations" for the differential calculus.
    • In his diary there is an entry: "Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of 7 July 1844; exactly 22 years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.] .

  36. Todd: 'Basic Numerical Mathematics
    • It is probably more efficient to present such material after a strong grasp of (at least) linear algebra and calculus has already been attained - but at this stage those not specializing in numerical mathematics are often interested in getting more deeply into their, chosen field than in developing skills for later use.
    • An alternative approach is to incorporate the numerical aspects of linear algebra and calculus as these subjects are being developed.
    • For instance, in a 6-quarter course in Calculus and Linear Algebra, the material in Volume 1 can be handled in the third quarter and that in Volume 2 in the fifth or sixth quarter.
    • In the first of these, subtitled "Numerical Analysis", we assume that the fundamental ideas of calculus of one variable have been absorbed: in particular, the ideas of convergence and continuity.

  37. André Weil: 'Algebraic Geometry
    • VII gives a translation of the main results of intersection-theory into a new language, particularly well adapted to applications, the "calculus of cycles".
    • An algebraic calculus of cycles can then be developed, closely analogous to the Algebra of homology-classes constructed by modern topologists: the main difference between the two is that, while the latter deals with classes, the former operates with the cycles themselves, but is unable, because of this, to have an intersection-product defined without any restrictive assumption.
    • This, as will be seen, entails, in the practical handling of the calculus, a certain amount of inconvenience, which probably could be avoided, as the analogy suggests, by substituting, for the cycles, classes of cycles modulo a suitable concept of equivalence.
    • VIII gives, on the basis of the results of the preceding chapters, a detailed treatment of the theory of divisors on a variety, both for its own sake and in order to provide the reader with some examples of the use of our calculus.

  38. Edmund Landau: 'Foundations of Analysis' Prefaces
    • The full title, including subtitles, was Foundations of Analysis: the Arithmetic of Whole, Rational, Irrational and Complex Number: A Supplement to Text-Books on the Differential and Integral Calculus.
    • Differential and Integral Calculus .
    • V), my daughters have been studying (chemistry) for several semesters, think they have learned differential and integral calculus in school, and yet even today don't know why .
    • I will refrain from speaking at length about the fact that often even Dedekind's fundamental theorem (or the equivalent theorem in the development of the real numbers by means of fundamental sequences) is not included in the basic material; so that such matters as the mean-value theorem of the differential calculus, the corollary of the mean-value theorem to the effect that a function having a zero derivative in some interval is constant in that interval, or, say, the theorem that a monotonically decreasing bounded sequence of numbers converges to a limit, are given without any proof or, worse yet, with a supposed proof which in reality is no proof at all.

  39. The Tercentenary of the birth of James Gregory
    • He lodged in the house of his fellow countryman, Dr Caddenhead, the Professor of Philosophy in that city: it was there that Gregory was brought into contact with the great Italian school of geometers, particularly that of Cavalieri whose method of indivisibles led, through Gregory, to the integral calculus.
    • Barrow and Newton had discovered the differential calculus, but within a month of receiving Barrow's book Gregory poured out such a volley of equations in his next letter that Collins was convinced beyond a doubt that Gregory had made the same discovery too.
    • These rough notes, written, who knows? in this very room, are the silent but inevitable witness giving Gregory the right to take his place with Barrow, Newton and Leibniz as a principal discoverer of the differential calculus: indeed in this one aspect of the subject he attained a result which neither of the others are known to have found.

  40. Donald C Spencer's publications
    • M Schiffer and D C Spencer, A variational calculus for Riemann surfaces, Ann.
    • P R Garabedian and D C Spencer, A complex tensor calculus for Kahler manifolds, Acta Math.
    • H K Nickerson, D C Spencer, and N E Steenrod, Advanced Calculus (Van Nostrand, Princeton, New Jersey, 1959).

  41. Thomas Bromwich: 'Infinite Series
    • In the remainder of the book free use is made of the notation and principles of the Differential and Integral Calculus; I have for some time been convinced that beginners should not attempt to study Infinite Series in any detail until after they have mastered the differentiation and integration, of the simpler functions, and the geometrical meaning of these operations.
    • The use of the Calculus has enabled me to shorten and simplify the discussion of various theorems (for instance, Arts.
    • [While my book has been in the press, three books have appeared, each of which contains some account of this theory: Gibson's Calculus (ch.

  42. Charles Bossut on Leibniz and Newton Part 2
    • The author gives the name increments or decrements of variable quantities to the differences, whether finite or infinitely small, their calculus, either direct or inverse, belongs to the Leibnizian analysis or the method of fluxions; and Dr Taylor resolves a great number of problems of this kind.
    • But when the differences are finite the method of finding the relations they bear to the quantities that produce them forms a new kind of calculus, the first principles of which were given by Dr Taylor; and in this respect the book has the merit of originality.
    • The two excellent papers which he published on this subject in the Memoirs of the Academy of Sciences for 717 and 1728 may be considered as the first methodical and luminous elementary treatise on the integral calculus with finite differences that ever appeared.

  43. Mathematics in St Andrews
    • The subjects are:- Co-ordinate Geometry, Differential and Integral Calculus.
    • Text-Book's.- Todhunter's Differential Calculus, and Williamson's Integral Calculus.

  44. Felix Klein on intuition
    • The naive intuition, on the other hand, was especially active during the period of the genesis of differential and integral calculus.
    • This idea of building up science purely on the basis of axioms has since been carried still further by Peano, in his logical calculus ..
    • Here a practical difficulty presents itself in the teaching, let us say, the elements of the calculus.

  45. Eddington: 'Mathematical Theory of Relativity' Introduction
    • It might well seem impossible to realise so comprehensive an outlook; but we shall find that the mathematical calculus of tensors does represent and deal with world-conditions precisely in this way.
    • For this reason the somewhat difficult tensor calculus is not to be regarded as an evil necessity in this subject, which ought if possible to be replaced by simpler analytical devices; our knowledge of conditions in the external world, as it comes to us through observation and experiment, is precisely of the kind which can be expressed by a tensor and not otherwise.
    • And, just as in arithmetic we can deal freely with a billion objects without trying to visualise the enormous collection; so the tensor calculus enables us to deal with the world-condition in the totality of its aspects without attempting to picture it.

  46. Muir on research in Scotland
    • There he obtains a knowledge of Synthetic and Analytical Conics, the elements of the Differential Calculus, and, it may be, of the Integral Calculus as well.
    • The subjects of the courses will be somewhat like the following, which constitute an actual case (1) Differential and Integral Calculus; (2) Definite Integrals; (3) Elliptic Functions; (4) Differential Equations; (5) the Function Theory of Weierstrass; (6) Theory of Equations; (7) Determinants; (8) Curved Surfaces and Curves of Double Curvature: and if the visitor goes back the following session he will not fail to find several courses that are new.

  47. Gibson obituary.html
    • The effect of his contact with students whose main interest lay in the applications of Mathematics is seen in his Treatise on the Calculus, published in 1901.
    • The elements of the Calculus and Analytical Geometry were included in the syllabus of work for the pass degree; the curriculum for honours was broadened and the standard raised, each student being required to read some branch of the subject not treated in the class lectures and to profess it for examination.
    • Until his death, on 1st April 1930, he was busily engaged in the preparation of a book on Advanced Calculus.

  48. Three Sadleirian Professors
    • Among the topics on which he lectured during his tenure of the Sadleirian chair may be mentioned Differential Geometry and the Calculus of Variations.
    • His lectures on the Calculus of Variations were the earliest in Cambridge to expound the Weierstrass theory: these were embodied in a treatise published in 1927 which extended the whole range of the subject and included much new research.
    • Most of Professor Hobson's researches have been connected with the theory of functions of real variables, but he has also dealt with Legendre's and Bessel's functions, integral equations, potential theory, the conduction of heat, and the calculus of variations.

  49. Gheorghe Mihoc's books
    • An Introduction to the calculus of probability (Romanian) (1954).
    • (with O Onicescu and C T Ionescu Tulcea) The calculus of probability and its applications (Romanian) (1956).
    • The fourth part treats index numbers, the fifth is devoted to sampling theory, and the sixth part to modern methods of statistical calculus.

  50. Mathematics in Edinburgh
    • On Mondays, Wednesdays, and Fridays - Higher Algebra, Analytical Geometry, Differential and Integral Calculus, Calculus of Finite Differences.
    • Differential and Integral Calculus.- Williamson's Treatises.

  51. EMS obituary
    • His early work included contributions to the theory of numbers, to the theory of functions of two complex variables, to partial differential equations, and, most notably of all, to the absolute differential calculus.
    • Originally it was a technique rather than a separate branch of mathematics, providing as it did a way of writing theorems of differential geometry and the calculus in a form at once concise and general, and it was not until after the development of relativity, followed shortly afterwards by Levi-Civita's definition of parallelism in Riemannian geometry, that it assumed the full place it now holds as one of the main branches of modern mathematics.
    • Lezioni di calcolo differenziale assoluto, 1925 (English translation with additional chapters, The absolute differential calculus, 1927).

  52. Muir on research in Scotland
    • There he obtains a knowledge of Synthetic and Analytical Conics, the elements of the Differential Calculus, and, it may be, of the Integral Calculus as well.
    • The subjects of the courses will be somewhat like the following, which constitute an actual case (1) Differential and Integral Calculus; (2) Definite Integrals; (3) Elliptic Functions; (4) Differential Equations; (5) the Function Theory of Weierstrass; (6) Theory of Equations; (7) Determinants; (8) Curved Surfaces and Curves of Double Curvature: and if the visitor goes back the following session he will not fail to find several courses that are new.

  53. Tullio Levi-Civita

  54. Levi-Civita.html
    • His early work included contributions to the theory of numbers, to the theory of functions of two complex variables, to partial differential equations, and, most notably of all, to the absolute differential calculus.
    • Originally it was a technique rather than a separate branch of mathematics, providing as it did a way of writing theorems of differential geometry and the calculus in a form at once concise and general, and it was not until after the development of relativity, followed shortly afterwards by Levi-Civita's definition of parallelism in Riemannian geometry, that it assumed the full place it now holds as one of the main branches of modern mathematics.
    • Lezioni di calcolo differenziale assoluto, 1925 (English translation with additional chapters, The absolute differential calculus, 1927).

  55. The Edinburgh Mathematical Society: the first hundred years
    • Fundamental notions of the differential calculus (A Y Fraser).
    • There he obtains a knowledge of Synthetic and Analytical Conics, the elements of the Differential Calculus, and, it may be, of the Integral Calculus as well.

  56. Andrew Forsyth addresses the British Association in 1905
    • The Italian mathematicians, of whom Cavalieri is the least forgotten, were developing Greek methods of quadrature by a transformed principle of indivisibles; but the infinitesimal calculus was not really in sight, for Newton and Leibniz were yet unborn.
    • That century also saw the discovery of the fluxional calculus by Newton, and of the differential calculus by Leibniz.

  57. Association 1904 Part 2.html

  58. Whittaker EMS Obituary.html
    • The fifth of the standard works by Whittaker and in this case prepared with the aid of his colleague G Robinson, is entitled The calculus of observations; a treatise on numerical mathematics and grew out of his lectures in the mathematical laboratory.
    • His interest in Relativity manifested itself also at the undergraduate level, for the Honours course entitled Higher Algebra and Geometry contained neither Algebra nor Geometry in the ordinary sense of these terms but comprised Tensor Calculus with Riemannian Geometry and its generalisations.

  59. G C McVittie papers
    • J L Synge writes: The author employs the technique of tensor calculus to transform the equations of classical hydrodynamics to moving curvilinear coordinates.
    • H P Robertson writes: The first three of the five chapters present a rapid survey of our knowledge of extra-galactic nebulae, of the tensor calculus and of the principles of the general theory of relativity.

  60. The Works of Sir John Leslie
    • His University course, as displayed in his University text books, by neglecting, or subordinating computation, algebra, coordinate geometry, differential and integral calculus, is enough to startle a mathematician.
    • He remarks quaintly: "The superior elegance and perspicuity with which the geometrical process unfolds the properties of these higher curves, may show that the fluxionary calculus should be more sparingly employed, if not reserved for the solution of problems of a more arduous nature." After that it comes quite as a shock to meet mere differential equations masquerading in such elegant geometrical company, but these are seen to be rank outsiders, members of the nouveaux riches.

  61. Weyl on Hilbert
    • Paul Bernays's publications cover a variety of areas of mathematical science: the representation of positive integers by binary quadratic forms (dissertation supervised by Edmund Landau), elementary theory of landau's function of Picard's theorem (habilitation thesis in Zurich), Legendre's condition in the calculus of variations, one-dimensional gas as an example of an ergodic system, axiomatic treatment of Russell's propositional calculus (habilitation thesis, Gottingen, not printed).

  62. American Mathematical Society Colloquium
    • The Simplest Type of Problems in the Calculus of Variations.
    • Lectures on the Calculus of Variations, Chicago, 1904.

  63. William Lowell Putnam Mathematical Competition
    • Problems were set on calculus, analytic geometry, and differential equations and West Point Cadets were victorious.
    • The questions will be taken from the fields of calculus (elementary and advanced) with applications to geometry and mechanics not involving techniques beyond the usual applications, higher algebra (determinants and the theory of equations), elementary differential equations and geometry (advanced plane and solid analytic geometry).

  64. Halmos: creative art
    • of inertia by calculus.
    • Similarly many, perhaps most, whose interests are in the mathematics of today, earn their bread and butter by teaching arithmetic, trigonometry, or calculus.

  65. R L Wilder: 'Cultural Basis of Mathematics II
    • The influence of hydrodynamics on function theory, of Kantianism and of surveying on geometry, of electromagnetism on differential equations, of Cartesianism on mechanics, and of scholasticism on the calculus could only be indicated [in his book]; - yet an understanding of the course and content of mathematics can be reached only if all these determining factors are taken into consideration." In his third chapter Struik gives a revealing account of the rise of Hellenistic mathematics, relating it to the cultural conditions then prevailing.
    • What had become stagnant came to life-analytic geometry appeared, calculus-and the flood was on.

  66. E C Titchmarsh on Counting
    • The book covers Counting, Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath (see this link).

  67. EMS 1938 Colloquium
    • For example he had evolved a theory of the calculus that we taught in schools today, except for the notation.
    • and proved that he should rank with Barrow, Newton and Leibniz as a founder of the Calculus.

  68. Mathematics at Aberdeen 4
    • From 1848 the first three books of Euclid and elementary algebra became a prerequisite for the first mathematical class, enabling Cruickshank to introduce Leibniz' differential and integral calculus, which had replaced fluxions, in his second class.
    • The syllabus for this senior class included Spherical Trigonometry, Conic Sections, Analytical Geometry and Differential and Integral Calculus.

  69. Whittaker RSE Prize
    • To the great benefit of students he had put together from time to time, in the more permanent form of a book, his systematic thoughts upon a particular subject, whether analysis or dynamics or optics or the calculus of observations, a phase of his activities which was the expression of his profound interest in teaching, and which, on the personal side, had made him an inspiring force both as a teacher and as a leader of mathematical research.
    • His recent contributions to much-needed developments in the application of the relativity calculus were the latest instance in point.

  70. Kuratowski: 'Introduction to Set Theory
    • In geometry we consider sets whose elements are points, in arithmetic we consider sets whose elements are numbers, in the calculus of variations we deal with sets of functions or curves; on the other hand, in the theory of sets we are concerned with the general properties of sets independently of the nature of the elements which comprise these sets.
    • In Chapters I and III we have given the main facts from this subject concerning the calculus of propositions, propositional functions and quantifiers.

  71. Planetary motion tackled kinematically
    • There are several ways of carrying this out, which unfortunately involve either sophisticated calculus or fairly heavy algebra.
    • Then we apply result (11) above to the first term, and, as one possibility for the second term, introduce a formula for change of variable which is found in some calculus textbooks: .

  72. Jacques Hadamard's failures
    • Now, investigating that would have led me to the principle of the so-called "Absolute Differential Calculus," the discovery of which belongs to Ricci and Levi Civita.
    • Absolute differential calculus is closely connected with the theory of relativity; and in this connection, I must confess that, having observed that the equation of propagation of light is invariant under a set of transformations (what is now known as Lorentz's group) by which space and time are combined together, I added that "such transformations are obviously devoid of physical meaning." Now, these transformations, supposedly without any physical meaning, are the base of Einstein's theory.

  73. Ernest Hobson addresses the British Association in 1910, Part 3
    • Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.
    • The great and increasing importance of a knowledge of the differential and integral calculus for students of engineering and other branches of physical science has led to the publication during the last few years of a considerable number of text-books on this subject intended for the use of such students.

  74. Einar Hille: 'Analytic Function Theory
    • The systematic study of holomorphic functions occupies the last three chapters, devoted to complex integration, representation theorems, and the calculus of residues.
    • A student who intends to use this book should have had a good course in advanced calculus.

  75. Science at St Andrews
    • In Italy Gregory, inspired by the recent advances of the Italian and French schools, made his first discoveries in the differential and integral calculus, probably quite unaware that Barrow and Newton were doing the like at Cambridge.
    • This period of discovery culminated for Gregory in the central expansion theorems of interpolation and the differential calculus, the former of which he announced in a letter to Collins, November 1670, and the latter of which he exemplified in the following February by half a dozen examples and again a year later by the solution of Kepler's problem - on determining the theoretical position in its orbit of a planet at a given time - which Gregory solved by invoking the properties of the cycloid and repeated differentiation.

  76. Edmund Landau: 'Foundations of Analysis' Contents
    • The full title, including subtitles, was Foundations of Analysis: the Arithmetic of Whole, Rational, Irrational and Complex Number: A Supplement to Text-Books on the Differential and Integral Calculus.
    • Differential and Integral Calculus .

  77. Rudio's Euler talk
    • He dedicated two particular major works to these investigations: his Introduction to Infinitesimal Calculus and his Manual for Differential and Integral Calculus.

  78. Malcev: 'Foundations of Linear Algebra' Introduction
    • In the middle of the last century, investigations of non-commutative algebras (Hamilton), led to the development of a matrix calculus (Cayley and Sylvester), which played a major role in the subsequent growth of linear algebra.
    • At about the same time the development of differential geometry for many -dimensional spaces and of the theory of transformations of algebraic forms of higher powers led to the creation of the tensor calculus, upon which was built the theory of relativity.

  79. Horace Lamb addresses the British Association in 1904, Part 2
    • We have discussions on the principles of mechanics, on the foundations of geometry, on the logic of the most rudimentary arithmetical processes, as well as of the more artificial operations of the Calculus.
    • The pure mathematician, for his part, will freely testify to the influence which it has exercised in the development of most branches of Analysis; for example, we owe to it all the leading ideas of the Calculus.

  80. Carathéodory: 'Conformal representation
    • In the proof of this theorem, which forms the foundation of the whole theory, he assumes as obvious that a certain problem in the calculus of variations possesses a solution, and this assumption, as Weierstrass (1815-1897) first pointed out, invalidates his proof Quite simple, analytic, and in every way regular problems in the calculus of variations axe now known which do not always possess solutions.

  81. Bolzano's publications
    • Contains his thoughts on Euclidean geometry, manipulations of series, functions and foundations of calculus, and topics in mechanics.
    • Covers topics such as geometry, calculus, and mechanics frequently making philosophical commnts.

  82. Mathematics in Glasgow
    • Subjects: Trigonometry, Geometrical and Analytical Conics, Differential and Elements of Integral Calculus.
    • Subjects: Integral Calculus, Spherical Trigonometry, Geometry of Three Dimensions, Differential Equations, Finite Differences.

  83. Cohen on mathematics
    • The misconceptions which the average student brings to the Calculus course often causes him to see it merely as a set of rules for handling special problems.
    • For its discoverers, Newton and Leibniz, however, the essential element of the calculus was a new point of view rather than special problems.

  84. Cajori: 'A history of mathematics' Introduction
    • 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.
    • Even the value of mathematical training is called in question, quote the inscription over the entrance into the academy of Plato, the philosopher: "Let no one who is unacquainted with geometry enter here." Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, Leibniz, and Lagrange played in creating that science.

  85. E C Titchmarsh: 'Aftermath
    • The book covers Counting (see this link), Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath.

  86. Semple and Kneebone: 'Algebraic Projective Geometry
    • We have accordingly taken for granted acquaintance with the elements of linear algebra and the calculus of matrices, and, except in one instance, we have not gone into the proofs of purely algebraic theorems.

  87. Rota's lecture on 'Mathematical Snapshots
    • By an approximation process such as one finds in an advanced calculus textbook, one shows that that these axioms imply that the volume of a sphere Sr of radius r is given by the known formula v(Sr) = 4/3 π r3.

  88. Turnbull lectures on Colin Maclaurin, Part 2
    • Here Maclaurin expounds the calculus of variations by his geometrical and fluxional method, after first alluding to the early discoveries of Newton and James Bernoulli, on the solid of least resistance and the line of swiftest descent.

  89. Mathematics at Aberdeen 2
    • The Universities were now ready to go forward in the eighteenth century with the great advances in Mathematics resulting from the introduction of Analytical Geometry and Calculus.

  90. H W Turnbull's LMS Obituary by Ledermann
    • His enthusiasm for invariants, and in particular for the "symbolical calculus" of Clebsch and Aronhold, remained with him throughout his life.

  91. Paul Levy and René Gateaux
    • This is defined on locally convex topological vector spaces and generalises the idea of a directional derivative from differential calculus.

  92. John Maynard Keynes: 'Newton, the Man
    • He regarded the universe as a cryptogram set by the Almighty - just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibniz.

  93. L E Dickson: 'Linear algebras
    • The more general paper by Wedderburn is based upon a rather abstract calculus of complexes, comparable with the theory of abstract groups (§ 61).

  94. James Clerk Maxwell on the nature of Saturn's rings
    • The question may be made to depend upon the conditions of a maximum or a minimum of a function of many variables, but the theory of the tests for distinguishing maxima from minima by the Calculus of Variations becomes so intricate when applied to functions of several variables, that I think it doubtful whether the physical or the abstract problem will be first solved.

  95. EMS 1938 Colloquium 2.html
    • For example he had evolved a theory of the calculus that we taught in schools today, except for the notation.

  96. Percy MacMahon addresses the British Association in 1901, Part 2
    • Jacob Bernoulli, in his Ars Conjectandi, 1713, established the fundamental principles of the Calculus of Probabilities.

  97. George Temple's Inaugural Lecture II
    • In gas dynamics Schwartz's theory of distributions seems destined to play an important part, but there are other techniques which cannot be neglected, such as the theory of non-linear differential equations and the calculus of variations.

  98. Somerville's Booklist
    • LacroixDifferential and integral calculus .

  99. Mathematics at Aberdeen 3
    • Interested students could go on to the Professor's optional third class of Advanced Algebra, Quadrature and Fluxions (Newton's approach to Calculus), with parts of Newton's Principles of Philosophy.

  100. EMS 1938 Colloquium 4.html
    • and proved that he should rank with Barrow, Newton and Leibniz as a founder of the Calculus.

  101. Marie-Louise Dubreil-Jacotin
    • In 1939 she became an associate professor at Lyon until in 1943 she became a full professor at Poitiers where in 1955 she held the chair in Differential and Integral Calculus.

  102. Max Planck: 'Quantum Theory
    • Or the radiation theory is founded on actual physical ideas, and then the quantum of action must play a fundamental role in physics, and proclaims itself as something quite new and hitherto unheard of, forcing us to recast our physical ideas, which, since the foundation of the infinitesimal calculus by Leibniz and Newton, were built on the assumption of continuity of all causal relations.

  103. Oliver Heaviside and Newton Abbott
    • his own operational calculus now successfully applied in different branches of pure .

  104. Truesdell's books
    • I have presented the theory in such a way that this text could be made (and indeed it already has been made) the basis of a first course in thermodynamics for gifted and thoughtful undergraduates, with the proviso, nowadays difficult of fulfilment, that they master the elements of differential and integral calculus, not merely its lingo.

  105. Berge books
    • This book is essentially self-contained, requiring but a casual knowledge of linear algebra and advanced calculus.

  106. Sneddon: 'Special functions
    • For that reason the methods it employs should be intelligible to anyone who has completed a first course in calculus and has a slight acquaintance with the theory of differential equations.

  107. Percy MacMahon addresses the British Association in 1901
    • In particular, it has been stated that the study of the fundamental principles of the infinitesimal calculus may profitably be deferred indefinitely so long as the student is able to differentiate and integrate a few of the simplest functions that are met with in pure and applied physics.

  108. Lehrer Songs
    • We must fight for an enlightened calculus, .

  109. Henry Baker addresses the British Association in 1913, Part 2
    • Our ordinary integral calculus is well-nigh powerless when the result of integration is not expressible by algebraic or logarithmic functions.

  110. Planck's quanta.html
    • In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was, since the establishment of the infinitesimal calculus by Leibniz and Newton, upon the acceptance of the continuity of all causative connections.

  111. James Clerk Maxwell on the nature of Saturn's rings
    • The question may be made to depend upon the conditions of a maximum or a minimum of a function of many variables, but the theory of the tests for distinguishing maxima from minima by the Calculus of Variations becomes so intricate when applied to functions of several variables, that I think it doubtful whether the physical or the abstract problem will be first solved.

  112. Carl B Boyer
    • Coordinate geometry, the function concept, and the calculus had arisen by the seventeenth century; yet it was the Introductio which in 1748 fashioned these into the third member of the triumvirate - comprising geometry, algebra, and analysis.

  113. Dickson: 'Theory of Equations
    • The use of elementary calculus allows a clear treatment and a complete solution of the problem, "given an equation to locate its real roots," while the methods of Chap.

  114. G H Hardy: 'Integration of functions
    • This pamphlet is intended to be read as a supplement to the accounts of 'Indefinite Integration' given in text-books on the Integral Calculus.

  115. Cochran: 'Sampling Techniques' Preface
    • As an indication of the level at which the book is directed, the minimum mathematical equipment necessary for an easy understanding of the proofs is a knowledge of calculus as far as the determination of maxima and minima (using Lagrange's multipliers where required), plus a familiarity with elementary algebra, and especially with the use of summation signs.

  116. EMS obituary
    • Our ordinary integral calculus is well-nigh powerless when the result of integration is not expressible by algebraic or logarithmic functions.

  117. Sneddon: 'Special functions
    • For that reason the methods it employs should be intelligible to anyone who has completed a first course in calculus and has a slight acquaintance with the theory of differential equations.

  118. Ernest Hobson addresses the British Association in 1910, Part 2
    • Since the time of Newton and Leibniz there has been almost unceasing discussion as to the proper foundations for the so-called infinitesimal calculus.

  119. C Chevalley on Herbrand's thought

  120. Bernstein on teaching
    • Apart from arithmetic and the elementary parts of algebra and geometry, the rudiments of differential and integral calculus and analytical geometry would satisfy these requirements.

  121. EMS obituary
    • Occasionally he had pupils who gave a third or even a fourth year's attendance, for which he would accept no fee, and he took particular pleasure in initiating them into the mysteries of the calculus." One might be tempted to hint that such a wide range of subjects would be only superficially covered, but I know, from frequent conversations with Dr Mackay, that in his case at least the work was very thoroughly done; to the end of his days he could repeat verbatim many of the enunciations in geometry and in spherical geometry as he had learned them under Dr Miller.

  122. William and Grace Young: 'Sets of Points
    • In subjects as wide apart as Projective Geometry, Theory of Functions of a Complex Variable, the Expansions of Astronomy, Calculus of Variations, Differential Equations, mistakes have in fact been made by mathematicians of standing, which even a slender grasp of the Theory of Sets of Points would have enabled them to avoid.

  123. Gregory tercentenary
    • "These rough notes, written, who knows (?) in this very room, are the silent, but inevitable witness giving Gregory the right to take his place with Barrow, Newton, and Leibniz as a principal discoverer of the differential calculus"; indeed, in this one aspect of the subject he attained a result which neither of the others is known to have found.

  124. C Chevalley: 'On Herbrand's thought
    • They are not of a different nature from those which engendered the birth of the infinitesimal calculus, which one wished to exclude for philosophical reasons because one wanted to see a 'real' object in the differential.

  125. Gibson History 7 - Robert Simson
    • quantities as generated by motion and Mr Cotes's view of them as the sums of ratiunculae; and to demonstrate Newton's lemmas concerning the limits of ratios and then to give the elements of the fluxionary calculus; and to finish off his course with a select set of propositions in Optics, gnomonics and central forces.

  126. Henry Baker addresses the British Association in 1913
    • Calculus of Variations.

  127. Gibson History 13 - Postscript
    • (ii) The other phase of this one-sidedness is the long neglect of the calculus except in its purely Newtonian form.

  128. Caius Iacob: 'Applied mathematics and mechanics
    • Applications of integral calculus to the determination of the centres of gravity and moments of inertia of some geometric bodies.

  129. Library of Mathematics

  130. W H and G C Young
    • For 41/2 more years the flood of my parents' papers still continued, unabated and uninterrupted: 1 by my mother alone, 73 by my father or joint, as well as my father's Cambridge tract on the fundamental theorems of the differential Calculus.

  131. Carl Runge: 'Graphical Methods
    • The Graphical Methods of the Differential and Integral Calculus.

  132. Carl B Boyer: 'Foremost Modern Textbook
    • Coordinate geometry, the function concept, and the calculus had arisen by the seventeenth century; yet it was the Introductio which in 1748 fashioned these into the third member of the triumvirate - comprising geometry, algebra, and analysis.

  133. L R Ford - Differential Equations
    • No use is made of Cauchy's method of the calculus of limits and just a mention is included of the Cauchy-Lipschitz method.

  134. Gibson History 2 - Mathematics in the schools
    • The oldest of these academies is Perth Academy, founded in 1760, and it began with a very ambitious programme in mathematics, viz., the higher branches of arithmetic; mathematical, physical and political geography; algebra, including the theory of equations, and the differential calculus; geometry, consisting of the first six books of Euclid; plane and spherical trigonometry; mensuration of surfaces and solids; navigation, fortification; analytical geometry and conic sections, natural philosophy, consisting of statics, dynamics, hydrostatics, pneumatics, optics and astronomy.

  135. Gibson History 8 - James Stirling
    • The work by which Stirling is best known is his Methodus Differentialis - not a treatise on the Differential Calculus as that term is now understood but rather on what we call Finite Differences, though that name is inadequate.

  136. Chrystal.html
    • Horse-play fled before the Differential Calculus in spectacles.

  137. The Edinburgh Mathematical Society: the first hundred years (1883-1983) Part 2
    • He was the author of books on calculus which were notable for their rigour at a time when this was the exception rather than the rule, but his main interest was the history of mathematics.

  138. A Napierian logarithm before Napier
    • In the ordinary histories of mathematics there are very few suggestions about the way in which John Napier conceived the idea of his great discovery, truly one of the most beautiful made by man, not only As supplying a new method for saving time and trouble in tedious calculations, but also as forming one of the most important steps towards the discovery of the infinitesimal calculus.

  139. Combinatorial group theory
    • The subjects of Nielsen Transformations (Chapter 3), Free and Amalgamated Products (Chapter 4), and Commutator Calculus (Chapter 5) are treated here in a more detailed fashion than in the works of Kurosh and of Hall.

  140. Edwin Elliot's Algebra of Quantics
    • In such additions as I have made to my mention of that method I have as a rule adopted, not the usual symbolism, but a modification of it which stands in direct relationship to the calculus of differential operations.

  141. Gibson History 5 - James Gregory
    • Gregory's developments in the Geometria and the Exercitationes show plainly that what we now call the Differential and Integral Calculus was near at hand, but while the great range of results and the ingenuity of the demonstrations are worthy of recognition the vital connection between differentiation and integration is not yet stated as is necessary for the advance that followed from Barrow's developments (Lect.

  142. Finkel and The American Mathematical Monthly
    • 'The American Mathematical Monthly' will, therefore, devote a due portion of its space to the solution of problems, whether they be the easy problems in Arithmetic, or the difficult problems in the Calculus, Mechanics, Probability, or Modern Higher Mathematics.

  143. Carol R Karp: 'Languages with expressions of infinite length
    • Techniques for proving completeness theorems in logic and representation theorems for Boolean algebras combined to yield a completeness theorem: Valid formulas of denumerable length in which only finitely many variables can be quantified at a time are provable in a system very much like the ordinary first-order predicate calculus.

  144. H Weyl: 'Theory of groups and quantum mechanics'Preface to Second Edition
    • Transcendental methods, which are in group theory based on the calculus of group characteristics, have the advantage of offering a rapid view of the subject as a whole, but true understanding, of the relationships is to be obtained only by following an explicit elementary development.

  145. Mathematicians and Music 3
    • In the eighteenth century when calculus had become a tool, there was a notable series of theoretical discussions of vibrating strings.

  146. Kuratowski: 'Introduction to Topology
    • The latter has various applications in differential and algebraic geometry, the calculus of variations, and in other branches of analysis.

  147. Professor Chrystal
    • Horse-play fled before the Differential Calculus in spectacles.

  148. Hans Hahn: 'The crisis in intuition
    • [This] directly affects the foundations of differential calculus as developed by Newton (who started with the concept of velocity) and Leibniz (who started the so-called tangent problem) ..

  149. Gibson History 11 - John Playfair, Sir John Leslie
    • Even the disastrous controversy over the rise of the Calculus is handled with a freedom from prejudice that is a sure guarantee of the genuine scientific spirit; it is, I think (with the possible exception of Maclaurin's Fluxions), the first direct statement in English of the essential elements in the case that is free from a decidedly national bias.

  150. Gibson History 12 - Minor figures, Arithmetic Books
    • At the same time it is, I think, the case that advance was being made and that the level of attainment was fairly high; up and down the country, schools were to be found whose mathematical curriculum included conic sections and elementary calculus, and the presence of such schools, even though not in every town, indicates a provision of teachers and an outlook for their pupils that could only be met by the Universities.

  151. Bartlett's reviews
    • The treatment in a limited space has required a concentrated style, and the reader must have a working knowledge of calculus and applied statistics.

  152. Max Planck and the quanta of energy
    • In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was, since the establishment of the infinitesimal calculus by Leibniz and Newton, upon the acceptance of the continuity of all causative connections.

  153. H Weyl: 'Theory of groups and quantum mechanics' Introduction
    • It is somewhat distressing that the theory of linear algebras must again and again be developed from the beginning, for the fundamental concepts of this branch of mathematics crop up everywhere in mathematics and physics, and a knowledge of them should be as widely disseminated as the elements of differential calculus.

  154. MacDuffee's books
    • (2) The student will benefit in encountering the "arbitrarily small epsilon" notation of his calculus course in several places such as in proofs of Budan's and Sturm's theorems.

  155. Gibson History 10 - Matthew Stewart, John Stewart, William Trail
    • Like his master Simson he was jealous of the encroachments that Algebra was making on Geometry, and it was his constant aim to reduce to the level of ordinary Geometry problems that were supposed to require the higher calculus.

  156. Edwin Elliot: 'Algebra of Quantics
    • In such additions as I have made to my mention of that method I have as a rule adopted, not the usual symbolism, but a modification of it which stands in direct relationship to the calculus of differential operations.


Quotations

  1. Quotations by Leibniz
    • Quoted in G Simmons Calculus Gems (New York 1992).
    • Quoted in G Simmons Calculus Gems (New York 1992).
    • Therefore, I have attacted [the problem of the catenary] which I had hitherto not attempted, and with my key [the differential calculus] happily opened its secret.

  2. Quotations by Arago
    • The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experimentalist, and the statesman.
    • It is the calculus of probabilities, which, after having suggested the best arrangements of the tables of populations and mortality, teaches us to deduce from those numbers, in useful character; it is the calculus of probabilities which alone can regulate justly the premiums to be paid for assurances; the reserve funds for the disbursements of pensions, annuities, discounts, etc.

  3. Quotations by Euler
    • These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people.
    • Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus.
    • Quoted in G Simmons Calculus Gems (New York 1992).

  4. A quotation by Toeplitz
    • Regarding all these basic topics in infinitesimal calculus which we teach today as canonical requisites ..
    • The calculus : A genetic approach (Chicago, 1963).

  5. Quotations by Abel
    • Quoted in G F Simmons, Calculus Gems (New York 1992).
    • Quoted in G F Simmons, Calculus Gems (New York 1992).

  6. Quotations by Ball
    • Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Pascal, 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.

  7. Quotations by Laplace
    • Quoted in G Simmons Calculus Gems (New York 1992).
    • The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account.

  8. Quotations by Gauss
    • Quoted in G Simmons Calculus Gems (New York 1992).
    • Quoted in G Simmons Calculus Gems (New York 1992).

  9. Quotations by Poincare
    • Quoted in G Simmons Calculus Gems (New York 1992).
    • 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.

  10. A quotation by Hirst
    • The night was chill, I dropped too suddenly from Differential Calculus into ladies' society, and could not give myself freely to the change.
    • Calculus, and forgot the ladies..

  11. Quotations by Russell
    • Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.
    • Quoted in G Simmons Calculus Gems (New York 1992).

  12. Quotations by Dedekind
    • As professor in the Polytechnic School in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic.

  13. Quotations by Klein
    • Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.

  14. Quotations by Archimedes
    • Quoted in G Simmons Calculus Gems (New York 1992).

  15. Quotations by Einstein
    • Quoted in G Simmons Calculus Gems (New York 1992).

  16. Quotations by Hobbes
    • Quoted in G Simmons Calculus Gems (New York 1992).

  17. Quotations by Newton
    • Quoted in G Simmons Calculus Gems (New York 1992).

  18. Quotations by Hermite
    • Quoted in G F Simmons Calculus Gems (New York 1992).

  19. Quotations by D'Alembert
    • [advice to those who questioned the calculus] .

  20. Quotations by Bernoulli Johann
    • Simmons, Calculus Gems, New York: McGraw Hill, 1992, p.

  21. A quotation by Chebyshev
    • Quoted in G Simmons, Calculus Gems (New York 1992).

  22. A quotation by Lacroix
    • Calculus (1799) .

  23. A quotation by Somerville
    • Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity.

  24. A quotation by Tietze
    • Quoted in G Simmons Calculus Gems (New York 1992).

  25. Quotations by Pascal
    • G Simmons Calculus Gems (New York 1992).

  26. Quotations by Gardner
    • Simmons Calculus Gems, 1992.

  27. Quotations by Osgood
    • The calculus is the greatest aid we have to the application of physical truth in the broadest sense of the word.

  28. Quotations by De Prony
    • [Lagrange's foundations of the calculus] is assuredly a very interesting part of what one might call purely philosophical study; but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine, geodesy, and the different branches of the science of the engineer, the consideration of the infinitely small leads to the aim in a manner [which is] more felicitous, more prompt, and more immediately adapted to the nature of the questions, and this is why the Leibnizian method has, in general, prevailed in French schools.

  29. Quotations by Recorde
    • Quoted in G Simmons Calculus Gems (New York 1992).

  30. A quotation by Arnold
    • Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

  31. Quotations by Descartes
    • Quoted in G Simmons Calculus Gems (New York 1992).

  32. Quotations by Legendre
    • tables (values of trignometric functions), constructed by means of new techniques based principally on the calculus of differences, are one of the most beautiful monuments ever erected to science.


Chronology

  1. Mathematical Chronology
    • Thabit ibn Qurra makes important mathematical discoveries such as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.
    • His methods are early uses of the calculus.
    • This work is an early contribution to the differential calculus.
    • Newton discovers the binomial theorem and begins work on the differential calculus.
    • James Gregory publishes Geometriae pars universalis which is the first attempt to write a calculus textbook.
    • Barrow publishes Lectiones Geometricae which contains his important work on tangents which forms the starting point of Newton's work on the calculus.
    • Leibniz publishes details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus.
    • Brook Taylor publishes Methodus incrementorum directa et inversa (Direct and Indirect Methods of Incrementation), an important contribution to the calculus.
    • Jacob Bernoulli's work on the calculus of variations is published after his death.
    • It contains a thorough treatment of the calculus applied to logarithmic and circular functions.
    • He argues that although the calculus led to true results its foundations were no more secure than those of religion.
    • D'Alembert publishes Memoire sur le calcul integral (Memoir on Integral Calculus).
    • Maclaurin publishes Treatise on Fluxions which aims to provide a rigorous foundation for the calculus by appealing to the methods of Greek geometry.
    • It is the first systematic exposition of Newton's methods written in reply to Berkeley's attack on the calculus for its lack of rigorous foundations.
    • Agnesi writes Instituzioni analitiche ad uso della giovent italiana which is an Italian teaching text on the differential calculus.
    • This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously.
    • D'Alembert studies the "three-body problem" and applies calculus to celestial mechanics.
    • Lagrange makes important discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations.
    • Euler publishes Institutiones calculi differentialis which begins with a study of the calculus of finite differences.
    • It applies calculus to study the orbits of celestial bodies and examines the stability of the Solar System.
    • Bolzano publishes Rein analytischer Beweis (Pure Analytical Proof) which contain an attempt to free calculus from the concept of the infinitesimal.
    • Designed for students at the Ecole Polytechnique it was concerned with developing the basic theorems of the calculus as rigorously as possible.
    • The idea comes to him while he is thinking how to teach differential and integral calculus.
    • Bertrand publishes Treatise on Differential and Integral Calculus.
    • Ricci-Curbastro begins work on the absolute differential calculus.
    • Levi-Civita publishes a paper developing the calculus of tensors.
    • Church invents "lambda calculus" which today is an invaluable tool for computer scientists.

  2. Chronology for 1740 to 1760
    • Maclaurin publishes Treatise on Fluxions which aims to provide a rigorous foundation for the calculus by appealing to the methods of Greek geometry.
    • It is the first systematic exposition of Newton's methods written in reply to Berkeley's attack on the calculus for its lack of rigorous foundations.
    • Agnesi writes Instituzioni analitiche ad uso della giovent italiana which is an Italian teaching text on the differential calculus.
    • This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously.
    • D'Alembert studies the "three-body problem" and applies calculus to celestial mechanics.
    • Lagrange makes important discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations.
    • Euler publishes Institutiones calculi differentialis which begins with a study of the calculus of finite differences.

  3. Chronology for 1650 to 1675
    • Newton discovers the binomial theorem and begins work on the differential calculus.
    • James Gregory publishes Geometriae pars universalis which is the first attempt to write a calculus textbook.
    • Barrow publishes Lectiones Geometricae which contains his important work on tangents which forms the starting point of Newton's work on the calculus.

  4. Chronology for 1720 to 1740
    • It contains a thorough treatment of the calculus applied to logarithmic and circular functions.
    • He argues that although the calculus led to true results its foundations were no more secure than those of religion.
    • D'Alembert publishes Memoire sur le calcul integral (Memoir on Integral Calculus).

  5. Chronology for 1880 to 1890
    • Ricci-Curbastro begins work on the absolute differential calculus.
    • Levi-Civita publishes a paper developing the calculus of tensors.

  6. Chronology for 1700 to 1720
    • Brook Taylor publishes Methodus incrementorum directa et inversa (Direct and Indirect Methods of Incrementation), an important contribution to the calculus.
    • Jacob Bernoulli's work on the calculus of variations is published after his death.

  7. Chronology for 1930 to 1940
    • Church invents "lambda calculus" which today is an invaluable tool for computer scientists.

  8. Chronology for 1625 to 1650
    • This work is an early contribution to the differential calculus.

  9. Chronology for 1780 to 1800
    • It applies calculus to study the orbits of celestial bodies and examines the stability of the Solar System.

  10. Chronology for 1820 to 1830
    • Designed for students at the Ecole Polytechnique it was concerned with developing the basic theorems of the calculus as rigorously as possible.

  11. Chronology for 1850 to 1860
    • The idea comes to him while he is thinking how to teach differential and integral calculus.

  12. Chronology for 1600 to 1625
    • His methods are early uses of the calculus.

  13. Chronology for 1860 to 1870
    • Bertrand publishes Treatise on Differential and Integral Calculus.

  14. Chronology for 1675 to 1700
    • Leibniz publishes details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus.

  15. Chronology for 500 to 900
    • Thabit ibn Qurra makes important mathematical discoveries such as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.

  16. Chronology for 1810 to 1820
    • Bolzano publishes Rein analytischer Beweis (Pure Analytical Proof) which contain an attempt to free calculus from the concept of the infinitesimal.


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JOC/BS August 2001