MATH 5400, History of Mathematics Lecture 7: 17Th to Early 18Th Centuries

MATH 5400, History of Mathematics Lecture 7: 17th to early 18th centuries

Professor: Peter Gibson

[email protected] http://people.math.yorku.ca/pcgibson/math5400

November 10, 2016 Overview and historical context

The 17th century was marked by imperial expansion and rivalry. The Russian empire expanded to encompass vast territories, governed by the Romanov imperial family (which ruled from 1584-1917). In particular, Peter the Great (r. 1682-1725) consolidated both territory and political power. In western Europe there was rivalry between the French Bourbon kings and the Hapsburg rulers of Spain and Austria.

The Dutch Republic broke away from Spanish control at the end of the Thirty Years War (1618-1648). There was an expansion of naval power among European states, accompanying increased overseas trade and colonisation of the Americas.

The slave trade ﬂourshed during this time.

P. Gibson (YorkU) Math 5400 2 / 29 An estimate 12M tons of sugar was transported to Europe in the century 1690-1790, during which time an estimated 12M African slaves were forced into labour in European colonies.

Politically, the prevailing model was one of absolutist, dynastic monarchy, as typiﬁed by Louis XIV (r. 1643-1715) in France or Peter the Great in Russia. (The situation in England was an exception, with the civil war having resulted in power sharing between the monarchy and the nobility.)

The balance of power shifted away from Spain and Portugal, toward Britain and France.

Overall, mercantalism was the dominant economic model, involving extraction of wealth from colonial territory in the Americas and elsewhere, and transportation of goods to Europe.

P. Gibson (YorkU) Math 5400 3 / 29 Overall, mercantalism was the dominant economic model, involving extraction of wealth from colonial territory in the Americas and elsewhere, and transportation of goods to Europe. An estimate 12M tons of sugar was transported to Europe in the century 1690-1790, during which time an estimated 12M African slaves were forced into labour in European colonies.

The balance of power shifted away from Spain and Portugal, toward Britain and France.

P. Gibson (YorkU) Math 5400 3 / 29 Mathematics in 17th century

Just as political power gradually shifted toward Britain and France, so too did intellectual achievement.

Some mathematicians active in this period, in addition to Kepler, Galileo and Descartes: Girard Desargues (1591-1661) Blaise Pascal (1623-1662) Pierre de Fermat (1601-1665) Evangelista Torricelli (1608-1647) Christiaan Huygens (1629-1695) John Wallis (1616-1703) Isaac Barrow (1630-1677)

P. Gibson (YorkU) Math 5400 4 / 29 Fermat was a lawyer and member of the parlement of Toulouse. He communicated with Blaise Pascal, Marin Mersenne, Ren´eDescartes, John Wallis, Gilles de Roberval, as part of a long distance community of scholars that would come to be known as the Republic of Letters.

P. Gibson (YorkU) Math 5400 5 / 29 while Arnol’d asserts that Barrow understood the fundamental theorem of calculus.

What is clear, is that mathematics and physics were profoundly impacted by one person (who is traditionally credited with co-development of diﬀerential and integral calculus).

There were new developments in (synthetic) geometry, in probability, and in the calculation of areas of regions and lengths of curves.

A considerable amount of work centred on inﬁnitesimals, about which there was much dispute.

Laplace credits Fermat with the discovery of diﬀerential calculus,

P. Gibson (YorkU) Math 5400 6 / 29 What is clear, is that mathematics and physics were profoundly impacted by one person (who is traditionally credited with co-development of diﬀerential and integral calculus).

There were new developments in (synthetic) geometry, in probability, and in the calculation of areas of regions and lengths of curves.

A considerable amount of work centred on inﬁnitesimals, about which there was much dispute.

Laplace credits Fermat with the discovery of diﬀerential calculus,while Arnol’d asserts that Barrow understood the fundamental theorem of calculus.

P. Gibson (YorkU) Math 5400 6 / 29 There were new developments in (synthetic) geometry, in probability, and in the calculation of areas of regions and lengths of curves.

A considerable amount of work centred on inﬁnitesimals, about which there was much dispute.

Laplace credits Fermat with the discovery of diﬀerential calculus,while Arnol’d asserts that Barrow understood the fundamental theorem of calculus.

What is clear, is that mathematics and physics were profoundly impacted by one person (who is traditionally credited with co-development of diﬀerential and integral calculus).

P. Gibson (YorkU) Math 5400 6 / 29 -Franois-Marie Arouet to Maupertuis

Newton est notre Cristophe Colomb. Il nous a mene dans un nouveau monde, et je voudrais bien y voyager...

Newton is our Cristopher Colombus. He has led us to a new world, and I would very much like to go there...

P. Gibson (YorkU) Math 5400 7 / 29 -Franois-Marie Arouet to Maupertuis

Newton est notre Cristophe Colomb. Il nous a mene dans un nouveau monde, et je voudrais bien y voyager...

Newton is our Cristopher Colombus. He has led us to a new world, and I would very much like to go there...

P. Gibson (YorkU) Math 5400 7 / 29 Newton est notre Cristophe Colomb. Il nous a mene dans un nouveau monde, et je voudrais bien y voyager...

Newton is our Cristopher Colombus. He has led us to a new world, and I would very much like to go there... -Franois-Marie Arouet to Maupertuis

P. Gibson (YorkU) Math 5400 7 / 29 Newton in 1689

P. Gibson (YorkU) Math 5400 8 / 29 Some of his works: Philosophiæ Naturalis Principia Mathematica (1687) Opticks (1704) Method of Fluxions (1736)

Newton lived from 1642-1727. His self-proclaimed annus mirabilis was 1666.

P. Gibson (YorkU) Math 5400 9 / 29 Philosophiæ Naturalis Principia Mathematica (1687) Opticks (1704) Method of Fluxions (1736)

Newton lived from 1642-1727. His self-proclaimed annus mirabilis was 1666. Some of his works:

P. Gibson (YorkU) Math 5400 9 / 29 Opticks (1704) Method of Fluxions (1736)

Newton lived from 1642-1727. His self-proclaimed annus mirabilis was 1666. Some of his works: Philosophiæ Naturalis Principia Mathematica (1687)

P. Gibson (YorkU) Math 5400 9 / 29 Method of Fluxions (1736)

Newton lived from 1642-1727. His self-proclaimed annus mirabilis was 1666. Some of his works: Philosophiæ Naturalis Principia Mathematica (1687) Opticks (1704)

P. Gibson (YorkU) Math 5400 9 / 29 Newton lived from 1642-1727. His self-proclaimed annus mirabilis was 1666. Some of his works: Philosophiæ Naturalis Principia Mathematica (1687) Opticks (1704) Method of Fluxions (1736)

P. Gibson (YorkU) Math 5400 9 / 29 Less known is his career as Warden and Master of the Royal Mint. Also, Newton worked extensively on alchemy and theology. How can this be reconciled with his scientiﬁc achievement?

Newton’s mathematical and scientiﬁc contributions are well known.

P. Gibson (YorkU) Math 5400 10 / 29 Also, Newton worked extensively on alchemy and theology. How can this be reconciled with his scientiﬁc achievement?

Newton’s mathematical and scientiﬁc contributions are well known. Less known is his career as Warden and Master of the Royal Mint.

P. Gibson (YorkU) Math 5400 10 / 29 How can this be reconciled with his scientiﬁc achievement?

Newton’s mathematical and scientiﬁc contributions are well known. Less known is his career as Warden and Master of the Royal Mint. Also, Newton worked extensively on alchemy and theology.

P. Gibson (YorkU) Math 5400 10 / 29 Newton’s mathematical and scientiﬁc contributions are well known. Less known is his career as Warden and Master of the Royal Mint. Also, Newton worked extensively on alchemy and theology. How can this be reconciled with his scientiﬁc achievement?

P. Gibson (YorkU) Math 5400 10 / 29 the Great Plague (1664-1666) the Great Fire of London (1666)

Two important events took place in London during Newton’s most productive years:

P. Gibson (YorkU) Math 5400 11 / 29 Two important events took place in London during Newton’s most productive years: the Great Plague (1664-1666) the Great Fire of London (1666)

P. Gibson (YorkU) Math 5400 11 / 29 P. Gibson (YorkU) Math 5400 12 / 29 P. Gibson (YorkU) Math 5400 13 / 29 P. Gibson (YorkU) Math 5400 14 / 29 Year Population of London 1100 10-20,000 1300 20-50,000 1350 25-50,000 1500 50-100,000 1600 200,000 1650 350,000 1700 550,000 1750 700,000 1801 959,300 1831 1,655,000

P. Gibson (YorkU) Math 5400 15 / 29 Newton entered Trinity College at Cambridge University in 1661.

P. Gibson (YorkU) Math 5400 16 / 29 In 1669 he bacame Lucasian Professor of Mathematics, taking the place of Isaac Barrow.

In the late 1680s he became famous, with the publication of his Principia.

In 1695 he left Cambridge to become Warden of His Majesty’s Mint.

During the years 1664-1666 Newton (having recently received his ﬁrst degree) was forced to the family farm in Woolsthorpe because of the bubonic plague, which is estimated to have killed 100 000 people.

P. Gibson (YorkU) Math 5400 17 / 29 In the late 1680s he became famous, with the publication of his Principia.

In 1695 he left Cambridge to become Warden of His Majesty’s Mint.

In 1669 he bacame Lucasian Professor of Mathematics, taking the place of Isaac Barrow.

P. Gibson (YorkU) Math 5400 17 / 29 In 1695 he left Cambridge to become Warden of His Majesty’s Mint.

In 1669 he bacame Lucasian Professor of Mathematics, taking the place of Isaac Barrow.

In the late 1680s he became famous, with the publication of his Principia.

P. Gibson (YorkU) Math 5400 17 / 29 During the years 1664-1666 Newton (having recently received his ﬁrst degree) was forced to the family farm in Woolsthorpe because of the bubonic plague, which is estimated to have killed 100 000 people.

In 1669 he bacame Lucasian Professor of Mathematics, taking the place of Isaac Barrow.

In the late 1680s he became famous, with the publication of his Principia.

In 1695 he left Cambridge to become Warden of His Majesty’s Mint.

P. Gibson (YorkU) Math 5400 17 / 29 Huygens & Barrow, Newton & Hooke, by Vladimir Arnol’d Newton and the Counterfeiter, Thomas Levenson

There are many, many secondary sources on Newton.

Among these are: Never at Rest, a biography by Richard Westfalls

P. Gibson (YorkU) Math 5400 18 / 29 Newton and the Counterfeiter, Thomas Levenson

There are many, many secondary sources on Newton.

Among these are: Never at Rest, a biography by Richard Westfalls Huygens & Barrow, Newton & Hooke, by Vladimir Arnol’d

P. Gibson (YorkU) Math 5400 18 / 29 There are many, many secondary sources on Newton.

Among these are: Never at Rest, a biography by Richard Westfalls Huygens & Barrow, Newton & Hooke, by Vladimir Arnol’d Newton and the Counterfeiter, Thomas Levenson

P. Gibson (YorkU) Math 5400 18 / 29 An important figure in the transmission of Newton’s work was Emilie du Châtelet.Her translation of Newton’s Principia into French is still the definitive version.

P. Gibson (YorkU) Math 5400 19 / 29 Gottfried Wilhelm Leibniz (1646-1716)

Leibniz had a classical education in Leipzig, supplemented by his (late) father’s library 1661, entered the University of Leipzig, obtaining a general bachelor’s degree, then a bachelor’s degree in law 1667, granted a doctorate in law at University of Altdorf served as secretary to the Nuremburg alchemical society employed by Baron von Boineburg in Frankfurt; worked for the Elector of Mainz travelled to Paris and England studied under Cristiaan Huygens; communicated with Oldenberg of the Royal Society

P. Gibson (YorkU) Math 5400 20 / 29 Leibniz, con’t

Z 1675, published a manuscript with the notation f (x) dx communicated with Newton by letter 1676, recruited by the Duke of Hanover, Johann Friedrich 1680, worked for the new Duke, Ernst August failed project in the Harz mines 1684 published on inﬁnitesimal calculus in Acta Eruditorum 1689, elected to the Accademia had > 600 correspondents in his lifetime (unjustly?) discredited by the Royal Society

P. Gibson (YorkU) Math 5400 21 / 29 the Royal Society; the Acta Eruditorum.

The dissemination of mathematics in the late 17th and early 18th centuries rested largely on personal communication. Two incipient new forums that emerged as important at this time were:

P. Gibson (YorkU) Math 5400 22 / 29 the Acta Eruditorum.

The dissemination of mathematics in the late 17th and early 18th centuries rested largely on personal communication. Two incipient new forums that emerged as important at this time were: the Royal Society;

P. Gibson (YorkU) Math 5400 22 / 29 The dissemination of mathematics in the late 17th and early 18th centuries rested largely on personal communication. Two incipient new forums that emerged as important at this time were: the Royal Society; the Acta Eruditorum.

P. Gibson (YorkU) Math 5400 22 / 29 Leibniz was an external member of the Royal Society

P. Gibson (YorkU) Math 5400 23 / 29 P. Gibson (YorkU) Math 5400 24 / 29 A number of mathematicians who took up the mantle had the same name: Bernoulli.

The work of Newton and Leibniz provided new tools which were taken up by others and applied to a range of previously intractable problems.

P. Gibson (YorkU) Math 5400 25 / 29 The work of Newton and Leibniz provided new tools which were taken up by others and applied to a range of previously intractable problems.

A number of mathematicians who took up the mantle had the same name: Bernoulli.

P. Gibson (YorkU) Math 5400 25 / 29 Bernoulli family tree http://www-history.mcs.st-and.ac.uk/Biographies/Bernoulli_Johann...

P. Gibson (YorkU) Math 5400 26 / 29

1 of 1 16-11-08 5:02 PM The brachistochrone problem, posed in the Acta Eruditorum by Johann Bernoulli, was solved by Newton, Leibniz and Jakob Bernoulli.

Later the journal backed Leibniz in his bitterly-contested rivalry with Newton.

P. Gibson (YorkU) Math 5400 27 / 29 Scientiﬁc legacies

Newton and Leibniz share several commonalities: personal histories have some common features alchemy, theology work on what we now refer to as calculus

P. Gibson (YorkU) Math 5400 28 / 29 Newton laid the cornerstone for mathematical physics while Leibniz had many highly original ideas, many of his projects failed or were never completed Newton was a master experimentalist and expositor Leibniz was engaged in diplomatic and legal eﬀorts early in his career The eighteenth century, the Age of Enlightenment, was strongly inﬂuenced by the popular conception of the mathematics and physics of Newton.

But their legacies are very diﬀerent:

P. Gibson (YorkU) Math 5400 29 / 29 The eighteenth century, the Age of Enlightenment, was strongly inﬂuenced by the popular conception of the mathematics and physics of Newton.

But their legacies are very diﬀerent: Newton laid the cornerstone for mathematical physics while Leibniz had many highly original ideas, many of his projects failed or were never completed Newton was a master experimentalist and expositor Leibniz was engaged in diplomatic and legal eﬀorts early in his career

P. Gibson (YorkU) Math 5400 29 / 29 But their legacies are very different: Newton laid the cornerstone for mathematical physics while Leibniz had many highly original ideas, many of his projects failed or were never completed Newton was a master experimentalist and expositor Leibniz was engaged in diplomatic and legal efforts early in his career The eighteenth century, the Age of Enlightenment, was strongly influenced by the popular conception of the mathematics and physics of Newton.

P. Gibson (YorkU) Math 5400 29 / 29