Newton v. Leibniz calculus controversy
Encyclopedia
The calculus controversy was an argument between 17th-century mathematician
s Isaac Newton
and Gottfried Leibniz
(begun or fomented in part by their disciples and associates – see Development of the quarrel below) over who had first invented calculus
. It is a question that had been the cause of a major intellectual controversy over who first discovered calculus, one that began simmering in 1699 and broke out in full force in 1711.
Newton claimed to have begun working on a form of the calculus (which he called "the method of fluxions and fluents
") in 1666, at the age of 23, but did not publish it except as a minor annotation in the back of one of his publications decades later. (A relevant Newton manuscript of October 1666 is now published among his mathematical papers..) Gottfried Leibniz
began working on his variant of the calculus in 1674, and in 1684 published his first paper employing it. L'Hopital
published a text on Leibniz's calculus in 1696 (in which he expressed recognition about Newton's Principia
of 1687, that Newton's work was "nearly all about this calculus".) Meanwhile, Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the Principia of 1687, did not explain his eventual fluxional notation for the calculus in print until 1693 (in part) and 1704 (in full).
, Newton, and others, over whether Leibniz had discovered calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's. Newton manipulated the quarrel. The most remarkable aspect of this barren struggle was that no participant doubted for a moment that Newton had already developed his method of fluxions when Leibniz began working on the differential calculus. Yet there was seemingly no proof beyond Newton's word. He had published a calculation of a tangent with the note: "This is only a special case of a general method whereby I can calculate curves and determine maxima, minima, and centers of gravity." How this was done he explained to a pupil a full 20 years later, when Leibniz's articles were already well-read. Newton's manuscripts came to light only after his death.
The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials, or, as noted above, it was also expressed by Newton in geometrical form, as in the 'Principia' of 1687. Newton employed fluxions as early as 1666, but did not publish an account of his notation until 1693. The earliest use of differentials in Leibniz's notebooks may be traced to 1675. He employed this notation in a 1677 letter to Newton. The differential notation also appeared in Leibniz's memoir of 1684.
The claim that Leibniz invented the calculus independently of Newton rests on the fact that Leibniz
According to Leibniz's detractors, to rebut this case it is necessary to show that he (I) saw some of Newton's papers on the subject in or before 1675 or at least 1677, and (II) obtained the fundamental ideas of the calculus from those papers. They see the fact that Leibniz's claim went unchallenged for some years as immaterial.
No attempt was made to rebut #4, which was not known at the time, but which provides very strong evidence that Leibniz came to the calculus independently from Newton. For instance Leibniz came first to integration
, which he saw as a generalization of the summation of infinite series, whereas Newton began from derivatives. However, to view the development of calculus as entirely independent between the work of Newton and Leibniz misses the point that both had some knowledge of the methods of the other, and in fact worked together on some aspects, in particular power series, as is shown in a letter to Henry Oldenburg
dated October 24, 1676 where he remarks that Leibniz had developed a number of methods, one of which was new to him. Both Leibniz and Newton could see by this exchange of letters that the other was far along towards the calculus (Leibniz in particular mentions it) but only Leibniz was prodded thereby into publication.
That Leibniz saw some of Newton's manuscripts had always been likely. In 1849, C. J. Gerhardt, while going through Leibniz's manuscripts, found extracts from Newton's De Analysi per Equationes Numero Terminorum Infinitas (published in 1704 as part of the De Quadratura Curvarum but also previously circulated among mathematicians starting with Newton giving a copy to Isaac Barrow
in 1669 and Barrow sending it to John Collins
) in Leibniz's handwriting, the existence of which had been previously unsuspected, along with notes re-expressing the content of these extracts in Leibniz's differential notation. Hence when these extracts were made becomes all-important. It is known that a copy of Newton's manuscript had been sent to Tschirnhaus in May 1675, a time when he and Leibniz were collaborating; it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, when Leibniz discussed analysis by infinite series with Collins and Oldenburg. It is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which one or both of them surely possessed. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Shortly before his death, Leibniz admitted in a letter to Abbot Antonio Conti, that in 1676 Collins had shown him some of Newton's papers, but Leibniz also implied that they were of little or no value. Presumably he was referring to Newton's letters of 13 June and 24 October 1676, and to the letter of 10 December 1672, on the method of tangent
s, extracts from which accompanied the letter of 13 June.
Whether Leibniz made use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which no direct evidence is available at present. It is, however, worth noting that the unpublished Portsmouth Papers show that when Newton went carefully (but with an obvious bias) into the whole dispute in 1711, he picked out this manuscript as the one which had probably somehow fallen into Leibniz's hands. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704; hence Newton's conjecture was not published. But Gerhardt's discovery of a copy made by Leibniz tends to confirm its accuracy. Those who question Leibniz's good faith allege that to a man of his ability, the manuscript, especially if supplemented by the letter of 10 December 1672, sufficed to give him a clue as to the methods of the calculus. Since Newton's work at issue did employ the fluxional notation, anyone building on that work would have to invent a notation, but some deny this.
At first, there was no reason to suspect Leibniz's good faith. In 1699 Nicolas Fatio de Duillier
had accused Leibniz of plagiarizing
Newton, but Fatio was not a person of consequence. It was not until the 1704 publication of an anonymous review of Newton's tract on quadrature
, a review implying that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician doubted that Leibniz had invented the calculus independently of Newton. With respect to the review of Newton's quadrature work, all admit that there was no justification or authority for the statements made therein, which were rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubts emerged. Had Leibniz derived the fundamental idea of the calculus from Newton? The case against Leibniz, as it appeared to Newton's friends, was summed up in the Commercium Epistolicum of 1712, which referenced all allegations. That document was thoroughly machined by Newton.
No such summary (with facts, dates, and references) of the case for Leibniz was issued by his friends; but Johann Bernoulli
attempted to indirectly weaken the evidence by attacking the personal character of Newton in a letter dated 7 June 1713. When pressed for an explanation, Bernoulli most solemnly denied having written the letter. In accepting the denial, Newton added in a private letter to Bernoulli the following remarks, Newton's claimed reasons for why he took part in the controversy. "I have never," he said, "grasped at fame among foreign nations, but I am very desirous to preserve my character for honesty, which the author of that epistle, as if by the authority of a great judge, had endeavoured to wrest from me. Now that I am old, I have little pleasure in mathematical studies, and I have never tried to propagate my opinions over the world, but I have rather taken care not to involve myself in disputes on account of them."
Leibniz explained his silence as follows, in a letter to Conti dated 9 April 1716:
While Leibniz's death put a temporary stop to the controversy, the debate persisted for many years.
To Newton's staunch supporters this was a case of Leibniz's word against a number of contrary, suspicious details. His unacknowledged possession of a copy of part of one of Newton's manuscripts may be explicable; but it appears that on more than one occasion, Leibniz deliberately altered or added to important documents (e.g., the letter of June 7, 1713, in the Charta Volans, and that of April 8, 1716, in the Acta Eruditorum
), before publishing them, and falsified a date on a manuscript (1675 being altered to 1673). All this casts doubt on his testimony.
Several points should be noted. Considering Leibniz's intellectual prowess, as demonstrated by his other accomplishments, he had more than the requisite ability to invent the calculus. What he is alleged to have received was a number of suggestions rather than an account of the calculus; it is possible that since he did not publish his results of 1677 until 1684 and since the differential notation was his invention, Leibniz may have minimized, 30 years later, any benefit he may have enjoyed from reading Newton's work in manuscript. Moreover, he may have seen the question of who originated the calculus as immaterial when set against the expressive power of his notation.
In any event, a bias favoring Newton tainted the whole affair from the outset. The Royal Society set up a committee to pronounce on the priority dispute, in response to a letter it had received from Leibniz. That committee never asked Leibniz to give his version of the events. The report of the committee, finding in favor of Newton, was written by Newton himself and published as "Commercium Epistolicum" (mentioned above) early in 1713. But Leibniz did not see it until the autumn of 1714.
The prevailing opinion in the 18th century was against Leibniz (in Britain, not in the German-speaking world). Today the consensus is that Leibniz and Newton independently invented and described the calculus in Europe in the 17th century.
One author has identified the dispute as being about "profoundly different" methods:
On the other hand, other authors have emphasized the equivalences and mutual translatability of the methods: here N Guicciardini (2003) appears to confirm L'Hopital (1696) (already cited):
Analysi per Equationes Numero Terminorum Infinitas, English translation, free e-Book
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
(begun or fomented in part by their disciples and associates – see Development of the quarrel below) over who had first invented calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
. It is a question that had been the cause of a major intellectual controversy over who first discovered calculus, one that began simmering in 1699 and broke out in full force in 1711.
Newton claimed to have begun working on a form of the calculus (which he called "the method of fluxions and fluents
Method of Fluxions
Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
") in 1666, at the age of 23, but did not publish it except as a minor annotation in the back of one of his publications decades later. (A relevant Newton manuscript of October 1666 is now published among his mathematical papers..) Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
began working on his variant of the calculus in 1674, and in 1684 published his first paper employing it. L'Hopital
Guillaume de l'Hôpital
Guillaume François Antoine, Marquis de l'Hôpital was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞...
published a text on Leibniz's calculus in 1696 (in which he expressed recognition about Newton's Principia
Principia
Principia could refer to:*Philosophiae Naturalis Principia Mathematica, Isaac Newton's three-volume work containing explanations of his laws of motion and his law of universal gravitation*Principia , a stem-group coralline alga...
of 1687, that Newton's work was "nearly all about this calculus".) Meanwhile, Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the Principia of 1687, did not explain his eventual fluxional notation for the calculus in print until 1693 (in part) and 1704 (in full).
The quarrel
The last years of Leibniz's life, 1709–1716, were embittered by a long controversy with John KeillJohn Keill
John Keill was born in Edinburgh, Scotland, and was primarilya mathematician and important disciple of Isaac Newton. He studied at Edinburgh University, under David Gregory, and obtained his bachelors degree in 1692 with a distinction in physics and mathematics...
, Newton, and others, over whether Leibniz had discovered calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's. Newton manipulated the quarrel. The most remarkable aspect of this barren struggle was that no participant doubted for a moment that Newton had already developed his method of fluxions when Leibniz began working on the differential calculus. Yet there was seemingly no proof beyond Newton's word. He had published a calculation of a tangent with the note: "This is only a special case of a general method whereby I can calculate curves and determine maxima, minima, and centers of gravity." How this was done he explained to a pupil a full 20 years later, when Leibniz's articles were already well-read. Newton's manuscripts came to light only after his death.
The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials, or, as noted above, it was also expressed by Newton in geometrical form, as in the 'Principia' of 1687. Newton employed fluxions as early as 1666, but did not publish an account of his notation until 1693. The earliest use of differentials in Leibniz's notebooks may be traced to 1675. He employed this notation in a 1677 letter to Newton. The differential notation also appeared in Leibniz's memoir of 1684.
The claim that Leibniz invented the calculus independently of Newton rests on the fact that Leibniz
- Published a description of his method some years before Newton printed anything on fluxions;
- Always alluded to the discovery as being his own invention. Moreover, this statement went unchallenged some years;
- Rightly enjoyed the strong presumption that he acted in good faith;
- Demonstrated in his private papers his development of the ideas of calculus in a manner independent of the path taken by Newton.
According to Leibniz's detractors, to rebut this case it is necessary to show that he (I) saw some of Newton's papers on the subject in or before 1675 or at least 1677, and (II) obtained the fundamental ideas of the calculus from those papers. They see the fact that Leibniz's claim went unchallenged for some years as immaterial.
No attempt was made to rebut #4, which was not known at the time, but which provides very strong evidence that Leibniz came to the calculus independently from Newton. For instance Leibniz came first to integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, which he saw as a generalization of the summation of infinite series, whereas Newton began from derivatives. However, to view the development of calculus as entirely independent between the work of Newton and Leibniz misses the point that both had some knowledge of the methods of the other, and in fact worked together on some aspects, in particular power series, as is shown in a letter to Henry Oldenburg
Henry Oldenburg
Henry Oldenburg was a German theologian known as a diplomat and a natural philosopher. He was one of the foremost intelligencers of Europe of the seventeenth century, with a network of correspondents to rival those of Fabri de Peiresc, Marin Mersenne and Ismaël Boulliau...
dated October 24, 1676 where he remarks that Leibniz had developed a number of methods, one of which was new to him. Both Leibniz and Newton could see by this exchange of letters that the other was far along towards the calculus (Leibniz in particular mentions it) but only Leibniz was prodded thereby into publication.
That Leibniz saw some of Newton's manuscripts had always been likely. In 1849, C. J. Gerhardt, while going through Leibniz's manuscripts, found extracts from Newton's De Analysi per Equationes Numero Terminorum Infinitas (published in 1704 as part of the De Quadratura Curvarum but also previously circulated among mathematicians starting with Newton giving a copy to Isaac Barrow
Isaac Barrow
Isaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...
in 1669 and Barrow sending it to John Collins
John Collins (mathematician)
John Collins was an English mathematician. He is most known for his extensive correspondence with leading scientists and mathematicians such as Giovanni Alfonso Borelli, Gottfried Leibniz, Isaac Newton, and John Wallis...
) in Leibniz's handwriting, the existence of which had been previously unsuspected, along with notes re-expressing the content of these extracts in Leibniz's differential notation. Hence when these extracts were made becomes all-important. It is known that a copy of Newton's manuscript had been sent to Tschirnhaus in May 1675, a time when he and Leibniz were collaborating; it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, when Leibniz discussed analysis by infinite series with Collins and Oldenburg. It is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which one or both of them surely possessed. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Shortly before his death, Leibniz admitted in a letter to Abbot Antonio Conti, that in 1676 Collins had shown him some of Newton's papers, but Leibniz also implied that they were of little or no value. Presumably he was referring to Newton's letters of 13 June and 24 October 1676, and to the letter of 10 December 1672, on the method of tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
s, extracts from which accompanied the letter of 13 June.
Whether Leibniz made use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which no direct evidence is available at present. It is, however, worth noting that the unpublished Portsmouth Papers show that when Newton went carefully (but with an obvious bias) into the whole dispute in 1711, he picked out this manuscript as the one which had probably somehow fallen into Leibniz's hands. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704; hence Newton's conjecture was not published. But Gerhardt's discovery of a copy made by Leibniz tends to confirm its accuracy. Those who question Leibniz's good faith allege that to a man of his ability, the manuscript, especially if supplemented by the letter of 10 December 1672, sufficed to give him a clue as to the methods of the calculus. Since Newton's work at issue did employ the fluxional notation, anyone building on that work would have to invent a notation, but some deny this.
Development of the quarrel
The quarrel was a retrospective affair. In 1696, already some years later than the events that became the subject of the quarrel, the position still looked potentially peaceful: Newton and Leibniz had each made limited acknowledgements of the other's work, and L'Hopital's 1696 book about the calculus from a Leibnizian point of view had also acknowledged Newton's published work of the 1680s as 'nearly all about this calculus' ('presque tout de ce calcul'), while expressing preference for the convenience of Leibniz's notation.At first, there was no reason to suspect Leibniz's good faith. In 1699 Nicolas Fatio de Duillier
Nicolas Fatio de Duillier
Nicolas Fatio de Duillier was a Swiss mathematician known for his work on the zodiacal light problem, for his very close relationship with Isaac Newton, for his role in the Newton v. Leibniz calculus controversy, and for originating the "push" or "shadow" theory of gravitation...
had accused Leibniz of plagiarizing
Plagiarism
Plagiarism is defined in dictionaries as the "wrongful appropriation," "close imitation," or "purloining and publication" of another author's "language, thoughts, ideas, or expressions," and the representation of them as one's own original work, but the notion remains problematic with nebulous...
Newton, but Fatio was not a person of consequence. It was not until the 1704 publication of an anonymous review of Newton's tract on quadrature
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
, a review implying that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician doubted that Leibniz had invented the calculus independently of Newton. With respect to the review of Newton's quadrature work, all admit that there was no justification or authority for the statements made therein, which were rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubts emerged. Had Leibniz derived the fundamental idea of the calculus from Newton? The case against Leibniz, as it appeared to Newton's friends, was summed up in the Commercium Epistolicum of 1712, which referenced all allegations. That document was thoroughly machined by Newton.
No such summary (with facts, dates, and references) of the case for Leibniz was issued by his friends; but Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
attempted to indirectly weaken the evidence by attacking the personal character of Newton in a letter dated 7 June 1713. When pressed for an explanation, Bernoulli most solemnly denied having written the letter. In accepting the denial, Newton added in a private letter to Bernoulli the following remarks, Newton's claimed reasons for why he took part in the controversy. "I have never," he said, "grasped at fame among foreign nations, but I am very desirous to preserve my character for honesty, which the author of that epistle, as if by the authority of a great judge, had endeavoured to wrest from me. Now that I am old, I have little pleasure in mathematical studies, and I have never tried to propagate my opinions over the world, but I have rather taken care not to involve myself in disputes on account of them."
Leibniz explained his silence as follows, in a letter to Conti dated 9 April 1716:
Pour répondre de point en point à l'ouvrage publié contre moi, il falloit entrer dans un grand détail de quantité de minutiés passées il y a trente à quarante ans, dont je ne me souvenois guère: il me falloit chercher mes vieilles lettres, dont plusiers se sont perdus, outre que le plus souvent je n'ai point gardé les minutes des miennes: et les autres sont ensevelies dans un grand tas de papiers, que je ne pouvois débrouiller qu'avec du temps et de la patience; mais je n'en avois guère le loisir, étant chargé présentement d'occupations d'une toute autre nature.
[In order to respond point by point to all the work published against me, I would have to go into much minutiae that occurred thirty, forty years ago, of which I remember little: I would have to search my old letters, of which many are lost. Moreover, in most cases I did not keep a copy, and when I did, the copy is buried in a great heap of papers, which I could sort through only with time and patience. I have enjoyed little leisure, being so weighted down of late with occupations of a totally different nature.]
While Leibniz's death put a temporary stop to the controversy, the debate persisted for many years.
To Newton's staunch supporters this was a case of Leibniz's word against a number of contrary, suspicious details. His unacknowledged possession of a copy of part of one of Newton's manuscripts may be explicable; but it appears that on more than one occasion, Leibniz deliberately altered or added to important documents (e.g., the letter of June 7, 1713, in the Charta Volans, and that of April 8, 1716, in the Acta Eruditorum
Acta Eruditorum
Acta Eruditorum was the first scientific journal of the German lands, published from 1682 to 1782....
), before publishing them, and falsified a date on a manuscript (1675 being altered to 1673). All this casts doubt on his testimony.
Several points should be noted. Considering Leibniz's intellectual prowess, as demonstrated by his other accomplishments, he had more than the requisite ability to invent the calculus. What he is alleged to have received was a number of suggestions rather than an account of the calculus; it is possible that since he did not publish his results of 1677 until 1684 and since the differential notation was his invention, Leibniz may have minimized, 30 years later, any benefit he may have enjoyed from reading Newton's work in manuscript. Moreover, he may have seen the question of who originated the calculus as immaterial when set against the expressive power of his notation.
In any event, a bias favoring Newton tainted the whole affair from the outset. The Royal Society set up a committee to pronounce on the priority dispute, in response to a letter it had received from Leibniz. That committee never asked Leibniz to give his version of the events. The report of the committee, finding in favor of Newton, was written by Newton himself and published as "Commercium Epistolicum" (mentioned above) early in 1713. But Leibniz did not see it until the autumn of 1714.
The prevailing opinion in the 18th century was against Leibniz (in Britain, not in the German-speaking world). Today the consensus is that Leibniz and Newton independently invented and described the calculus in Europe in the 17th century.
It was certainly Isaac Newton who first devised a new infinitesimal calculus and elaborated it into a widely extensible algorithm, whose potentialities he fully understood; of equal certainty, the differential and integral calculus, the fount of great developments flowing continuously from 1684 to the present day, was created independently by Gottfried Leibniz. (Hall 1980: 1)
One author has identified the dispute as being about "profoundly different" methods:
Despite... points of resemblance, the methods [of Newton and Leibniz] are profoundly different, so making the priority row a nonsense. (Grattan-Guinness 1997: 247)
On the other hand, other authors have emphasized the equivalences and mutual translatability of the methods: here N Guicciardini (2003) appears to confirm L'Hopital (1696) (already cited):
... the Newtonian and Leibnizian schools shared a common mathematical method. They adopted two algorithms, the analytical method of fluxions, and the differential and integral calculus, which were translatable one into the other. (Guicciardini 2003, at page 250)
See also
- Isaac NewtonIsaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
- Gottfried LeibnizGottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
- Possibility of transmission of Kerala School results to Europe
- History of calculusHistory of calculusCalculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was...
- List of scientific priority disputes
Analysi per Equationes Numero Terminorum Infinitas, English translation, free e-Book
Sources
- Ivor Grattan-GuinnessIvor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
, 1997. The Norton History of the Mathematical Sciences. W W Norton. A thorough scholarly discussion. - Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Gottfried Leibniz. Cambridge Uni. Press.
- W. W. Rouse BallW. W. Rouse Ball-External links:*...
, 1908. A Short Account of the History of Mathematics, 4th ed. Dated. - Kandaswamy, Anand. The Newton/Leibniz Conflict in Context.
- Stephen HawkingStephen HawkingStephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
, 1988. A Brief History of TimeA Brief History of TimeA Brief History of Time is a popular science book written by renown physicist Stephen Hawking and first published by the Bantam Dell Publishing Group in 1988. It became a best-seller and has sold more than 10 million copies...
From the Big Bang to Black Holes. Bantam BooksBantam BooksBantam Books is an American publishing house owned entirely by Random House, the German media corporation subsidiary of Bertelsmann; it is an imprint of the Random House Publishing Group. It was formed in 1945 by Walter B. Pitkin, Jr., Sidney B. Kramer, and Ian and Betty Ballantine...