Mathematics in the Late 19th Century

The 19th century saw an explosion of applied mathematics as the industrial revolution took hold throughout Europe.  An example of this is the development of the École Polytechnique in Paris.  Galois had actually applied to the École Polytechnique in 1828, but failed his entrance exam. Many students from The École Polytechnique went on to one of the more specialized schools of engineering – School of Bridges and Roads, School of Mines, School of Engineering, and the School of Artillery.  This was the birth of the many specialized fields of engineering (electrical, mechanical, civil, aeronautical, chemical) that all rely on advanced mathematics, much of which was developed during the 18th and 19th century.

Some important results in applied mathematics from the 19th century are listed below:

Joseph Fourier researched and developed theories explaining the behavior of heat transfer.

The Scottish physicist and mathematician James Clerk Maxwell produced important results in electromagnetic theory.

The German physicist Gustav Kirchhoff also did important research with electricity, as well as heat and light.

The French mathematicians Augustin-Jean Fresnel and Augustin-Louis Cauchy both did important work researching the behavior of light and optics.

In the mid-19th century, Cauchy and a German mathematician named Karl Weierstrass developed a logical explanation/justification for the methods of calculus that Newton and Leibniz had developed 150 years before.  The issue with the calculus was that, although it worked well enough, it rested on a concept of “infinitesimals.”

These are quantities that are so small that sometimes they are treated like numbers greater than zero, but other times they are treated as equivalent to zero (because they’re so small).  Cauchy and Weierstrass developed a method that is quite useful, particularly in applications.

In applied mathematics, small rounding errors in initial calculations can sometimes produce much larger errors as those calculations are carried forward through a series of equations.  Cauchy and Weierstrass’ methods provided a way to limit these errors ahead of time by putting constraints on the initial values.

These ideas of infinitesimals had been considered by the Greek philosophers, most notably by Zeno.

Towards the end of the 19th century, a mathematician named Georg Cantor began to research the ideas of infinity.  The ideas of infinite sets (like the set of whole numbers – {1,2,3,4,…..}) had not been deeply considered by mathematicians until Cantor.

Cantor’s ideas rely heavily on the concept of a one-to-one correspondence.  This is a very useful idea for a number of reasons.  If you go to a party and everyone throws their coats on the bed, it can be difficult for everyone to find their coat at the end of the night.  At a restaurant, which must handle many coats every night for a large group of strangers, it makes sense to use a one-to-one correspondence to keep track of things.

Each person turns in their coat and receives a number.  The coat hanger is coded with the same number as the chip that the patron receives.  This creates a one-to-one correspondence between the chips and the coats on their hangers.  As each person leaves, it becomes much easier to match up each person with their coat.

Cantor’s application of the one-to-one correspondence says that any set that can be put into a one-to-one correspondence with the whole numbers (also called natural numbers) is the same size as the set of whole numbers.  This leads to some interesting results.

The set of even numbers {2,4,6,8,…..} is a subset of the set whole numbers {1,2,3,4,……}.  It would seem that the even numbers should be smaller than the whole numbers because all the odd numbers are missing.  However, in creating a one-to-one correspondence between the two sets, Cantor said that the two sets are the same size.  His one-to-one correspondence was this: each whole number “N” would be matched up with the even number “2N.”  Because both sets are infinite, we never run out of numbers in either set, so they are the same size.

Cantor called this size “aleph null”

Aleph is the first letter of the Hebrew alphabet, so Cantor used it to represent the first infinite size.  Once he had done this, he began to wonder if there were infinities that are larger than aleph null.

His conclusion, given in the now-famous “diagonal argument” is that the set of real numbers is larger than the set of whole numbers.  The real numbers are best thought of as the number line where each point on the line represents a number and each number corresponds to a point on the line.  Cantor showed that if you tried to create a one-to-one correspondence between the real numbers and the whole numbers, there would be at least one number missing from the list of real numbers, meaning that there are more of them than can be set into a one-to-one correspondence with the whole numbers.

These ideas can be very confusing and paradoxical.  Things may seem true one day and false the next.  I actually experienced this when I was in graduate school doing work for a class that was based on some of Cantor’s ideas.  In working on a problem, I would think that I had solved it.  Then, an hour later, I would think I was wrong and go back to fix it.  Then an hour later, I would think that my first answer was right and change it again.

Cantor’s ideas were not received with much enthusiasm by other mathematicians.  He found it difficult to have his papers published in important mathematical journals and other mathematicians refused to even consider the implications of his ideas.  Because of this professional stress and the difficulty of the ideas themselves, Cantor began to suffer from bouts of depression and was institutionalized for brief periods in 1884, 1899, and 1903.  Beginning in 1904, Cantor’s depression became chronic and he took time off during many of winter semesters leading up to his retirement in 1913.

Towards the end of his life, younger mathematicians began to be more interested in Cantor’s ideas and in 1911, he was invited to St. Andrew’s, Scotland for the 500th anniversary of the University there.

When World War I began, conditions in Germany became very difficult and food became scarce.  Cantor’s health suffered as a result.  He entered a sanatorium for the last time in June of 1917 and died of a heart attack in January of 1918.

Mathematics in the Early 19th Century

C. F. Gauss

Carl Friedrich Gauss (1777-1855) was a German mathematician whose talents were comparable to those of Euler.  Much of Gauss’ reputation was based on his monumental 1801 work called Disquisitiones Arithmeticae.  This book was a combination of original work by Gauss together with his organizing and summarizing of the work of several other older mathematicians – primarily Euler, Legendre and Lagrange.

Many of the ideas which Gauss addressed in his book had to do with “residues” or remainders.  Although these concepts were seen for a long time as being among the most abstract areas of mathematics with no practical applications, today they form the basis for digital cryptography which controls most of the computer security systems in the world.

Gauss was a mathematical prodigy whose talent for mathematics was recognized early on in his life.  A well-known story describes an incident that occurred when Gauss was seven years old.  His teacher asked the class to add up the numbers from 1-100 so that the teacher could get some work done without being disturbed.  A few minutes later, Gauss produced the answer: 1+2+3+4+….+98+99+100=5050.

Gauss demonstrated that if you added the smallest numbers to the largest numbers in pairs:  1+100=101, 2+99=101, 3+98=101…50+51=101, each pair added up to 101.  There were 50 of these pairs so Gauss calculated 50*101=5050.  Gauss eventually received a stipend from the local Duke to support his studies.  He attended the University of Göttingen from 1795-1798, but left without a degree in the autumn of 1798.  He later received his doctorate from the University of Helmstedt in 1799.

After publishing Disquisitiones Arithmeticae, Gauss became interested in astronomy.  The Italian astronomer Piazzi had discovered a small planetoid (today considered an asteroid) between Mars and Jupiter in 1801, but was only able to observe it for a short time until it disappeared behind the sun.  Many astronomers worked to calculate where it would reappear.  Gauss’ prediction was quite different from all the others, and, as it turned out, was correct.

In 1807, the Duke of Brunswick, who had supported Gauss’ studies was killed fighting for the Prussian army.  Gauss returned to Göttingen to oversee the establishment of an observatory there.  He published his second book in 1809, this time focusing on astronomy and continued working on the new observatory which opened in 1816.

The period from 1817-1830 was personally difficult for Gauss.  While he was taking care of his ailing mother, his second wife wanted the family to move to Berlin although Gauss was happier in Göttingen.  In 1831, Wilhelm Weber arrived in Göttingen as the new physics professor and he and Gauss worked together frequently.

Gauss had been researching the magnetic phenomena of the earth for several years and, together with Weber, used this knowledge of magnetism and the relatively new field of electricity to establish a primitive working telegraph machine in 1833.  It wasn’t practical and other scientists soon surpassed his discoveries in using the telegraph, but Gauss was usually pleased to see his ideas taken and expanded by others.

Weber left Göttingen in 1837, and Gauss’ production of new research diminished.  Though he still corresponded with other mathematicians and oversaw the the doctoral research of two of the mid-19th centuries greatest mathematical minds in Richard Dedekind and Bernhard Riemann, he was essentially an “elder statesman” of mathematics for the rest of his life.  Gauss died in his sleep in February 1855.

Evariste Galois

Evariste Galois (1811-1832) was a French mathematician who is known primarily for his proof that equations of the fifth degree or higher could not be solved by formula the way that quadratic, cubic and quartic equations could.

Several weeks ago, I assigned a reading on Italian mathematicians of the 16th century.  This reading dealt mainly with the intrigues surrounding the formula for solving cubic equations.  Eventually Ludovico Ferrari found a solution for the quartic equation, but all later mathematicians were stymied in their attempts to find a formula to solve 5th degree equations.  As time went by, many mathematicians suspected that it was not possible to solve quintic equations by a formula, but if this was the case, then they would have to prove that it was not possible.

Galois was primarily a math student who worked frequently on outside materials beyond what was required by his courses.  He often studied the works of Lagrange and Legendre, the famous French mathematicians of the mid-18th century.

In 1830, while Galois was attending the Ecole Normale in Paris, there was a revolution in France and King Charles X fled the country.  In the ensuing political disputes, Galois was expelled from the school, joined the anti-royalist Artillery of the National Guard and was arrested several times when the new King Louis-Phillipe outlawed the Artillery of the National Guard.  His connection with the Artillery of the National Guard resulted in his imprisonment on and off throughout the years 1830-32.

Galois was involved in a duel with Perscheux d’Herbinville in May 1832, which was apparently instigated by a disagreement over Galois’ affections for a woman who was the daughter of a doctor who served in one of the prisons Galois had spent time in.  Galois spent the night before the duel working on a math paper.  He was shot during the duel and died of his wounds the following day.

Galois’ papers were collected by his brother and his friend Chevalier and sent out to many of the best mathematicians of the time.  The French mathematician Liouville eventually recognized the importance of Galois’ work and announced to Paris Academyof the Sciences in 1843 that Galois had proven that it was impossible to solve equations of the fifth degree by formula.

Galois’ proof relied on what are today called Permutations and Combinations.  These concepts are concerned with the number of ways things can be selected from a group and are a part of what is known as Discrete Mathematics, which is very useful in computer science.

Galois’ ideas were also the genesis of an almost entirely new branch of mathematics called Abstract Algebra which focuses on the structure of number systems rather than the methods of solving equations which is the focus of Classical Algebra.

Swiss Mathematicians of the 18th Century

Leonhard Euler (1707-1783, pronounced Oiler) is recognized as being one of the greatest mathematicians who ever lived.  But, as is true in many fields (not just mathematics), Euler had the benefit of working closely with other talented mathematicians and was able to learn from them and build on their work – in Euler’s case, the Bernoulli family provided substantial support in his education and career.

The Bernoullis

Nicolaus Bernoulli (1623-1708) was a spice merchant in Basel, Switzerland whose family had been driven from what is today Belgium due to political and religious unrest in that region.  The northern part of the Netherlands was Protestant and the southern portion (what is today Belgium) was Catholic.  The Bernoulli family were Protestants (Calvinist) who lived in the Southern Netherlands, so when war broke out between Spain and the Netherlands, the Bernoullis fled, eventually ending up in Switzerland.

Two of Nicolaus Bernoulli’s ten children became famous mathematicians – Jacob and Johann.  Jacob taught at the University of Basel when his brother Johann (who was twelve years younger) attended school there.  Jacob traveled to France, the Netherlands and England before returning to Basel  in 1683 to teach at the University until his death in 1705.

Johann attended the University of Basel while his brother was teaching there and later spent extensive time in Paris (where he tutored l’Hopital) and Germany before returning to Basel in 1705.  He was unaware that his brother had recently passed away, but found out the news while en route to Basel.  He took over his brother’s teaching duties at the University of Basel and remained there until his death in 1748.

Johann supported Leibniz’s version of the calculus and was able to solve difficult problems using Leibniz’s methods that Newton had been unable to solve using his own methods.

Leonhard Euler

Leonhard Euler’s father Paul Euler had been a roommate of Johann Bernoulli when they were students together at the University of Basel.  As a young boy, Leonhard Euler was sent to Basel to attend school and lived there with his grandmother.  He entered the University of Basel in 1720 and received his Master’s in Philosophy in 1723.  However, during this time he was encouraged to read mathematics on his own by Johann Bernoulli, who also offered private tutoring sessions to answer any questions Euler had.

Euler began to study theology in 1723, but he and Johann Bernoulli were able to convince his father that he was better suited to study mathematics and so he switched his focus of study to mathematics.  Euler completed his mathematics degree in Basel in 1726, while also completing substantial studies in medicine and physiology.

Johann Bernoulli’s sons Daniel and Nicolaus Bernoulli (grandson of the original Nicolaus Bernoulli) had taken positions as professors of mathematics at the St. Petersburg Academy in Russia in 1725, but Nicolaus died unexpectedly in 1726 at the age of 31.  Euler was recommended to replace him in St. Petersburg and arrived there in May 1727.

Euler initially served as a medical officer in the Russian Navy until 1730 when he was appointed to a position as professor of physics.  In 1733, he took over the senior chair in mathematics from Daniel Bernoulli.  This improved academic position allowed Euler to marry in 1734.  His new wife was named Katharina Gsell

the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet.

Besides pure mathematics, Euler researched many applied fields including cartography, science education, magnetism, fire engines, ship building and navigation, artillery and ballistics, calculation of planetary orbits, the motion of the moon, optics, acoustics, the wave theory of light, hydraulics and music.

Euler remained in St. Petersburg until 1741, when he accepted an offer from Frederick the Great of Prussia to join in the founding of the Academy of Sciences in Berlin.  While in Berlin, Euler continued his research and also oversaw many applied projects for the Prussian government:

Euler undertook an unbelievable amount of work for the Academy [1]:-

… he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal … At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.

This was not the limit of his duties by any means. He served on the committee of the Academy dealing with the library and of scientific publications. He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal.

Despite this, Euler and Frederick the Great did not have a very good relationship and Euler returned to St. Petersburg in 1766.  Shortly after his return to St. Petersburg, Euler became almost completely blind.  Because of his incredible memory and calculating ability he continued to work and produce valuable research despite his blindness.

Euler’s wife Katharina died in 1773 and he married her half-sister in 1776.  He continued to live and work in St. Petersburg until his death in 1783.

Other mathematicians were amazed by Euler’s abilities.  The French mathematician Laplace was quoted as saying:

Read Euler – he is the master of us all.

Another French mathematician named Arago was quoted as saying:

He calculated without any apparent effort – as other men breathe, or as eagles sustain themselves in the air.

Euler can be seen as a pivot point that catapulted mathematics from the early work of Descartes, Huygens and Newton to the more modern mathematics of Gauss, Fourier and Galois.

Newton and Leibniz

Isaac Newton

Isaac Newton was born in 1642/3 in Woolsthorpe, England.  The doubled year of birth is related to the gradual adoption of the Gregorian calendar throughout Europe.  The Catholic church had developed the Gregorian calendar to correct the inefficiencies of the Roman (or Julian) calendar which had been in use for a little over 1500 years at that point.  The Roman calendar had leap years every four years, which is slightly too many days.  If you add too many days to a calendar, you’ll find that the calendar says it’s February 29th when it’s really March 1st, or after 1500 years, the calendar says that it’s the end of December 1642 when it’s really the beginning of January 1643.  The Catholic church and various astronomers noticed that Easter was moving later and later into the spring.  As early as the 8th century, scholars had noticed that astronomical events such as the full moon, the equinoxes and solstices were occurring earlier than their scheduled date on the calendar.  Great Britain adopted the Gregorian system in 1752, when September 2, 1752 was followed on the calendar by September 14, 1752.  But enough about that.

Newton’s father had died several months before he was born.  His mother remarried and Newton had an uneasy relationship with his extended step-family.  After completing an education at the local grammar school, Newton entered Trinity College, Cambridge in 1661.  The mathematician De Moivre said that Newton’s interest in mathematics was piqued by his attempt to read a book on astrology.  He couldn’t understand the math that was used in astrology and set out to learn trigonometry, then figured out that he needed to study geometry to understand that.  This led him to study Euclid’s and Des Cartes’ work on geometry as well as a number of other texts including John Wallis‘ work on algebra.

In 1665, Cambridge University shut down due to an outbreak of plague in England.  Newton returned to his family home in Woolsthorpe, Lincolnshire and began his independent investigations in mathematics and physics.  Newton experimented extensively with light and optics, and began to develop his ideas about gravity and his methods of calculus.

The ‘method of fluxions’, as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions. Newton’s De Methodis Serierum et Fluxionum was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736.

from the St. Andrews University biography of Newton

Newton’s habit of not publishing or sharing his work is part of what led to the controversy over who had first developed the methods of calculus.  It is pretty well accepted that Newton developed his methods first, and then Leibniz developed his own (different) methods independently about a decade later.

After the university reopened in 1667, Newton returned to Cambridge as a professor and was promoted to Lucasian Professor of Mathematics in 1670.  His determination that white light was composed of different colored frequencies of light was a dramatic breakthrough in the study of optics.  Newton remained at Cambridge throughout the 1680s and 1690s, being elected to represent the area in Parliament in 1689.  Since this involved his spending much time in London, he became more involved in government and eventually accepted a position at the Royal Mint in 1696 and took over as Master of the Royal Mint in 1699.

Newton published the “Principia Mathematica” in 1687 and his work on “Opticks” was published in 1704.

Gottfried Leibniz

Leibniz was born Leipzig, Germany in 1646.  His father was a Professor of Moral Philosophy at the University of Leipzig, but died when Leibniz was six years old.  Leibniz’s mother let young Gottfried read freely in his father’s library as a child and he consequently mastered the Latin language and many advanced ideas at a young age.

Leibniz entered the University of Leipzig in 1661 at age fourteen.  During the early 1660s, Leibniz earned a Master’s in Philosophy and a Bachelor’s degree in Law, although the University of Leipzig turned him down for a Doctorate of Law.  He eventually earned a Doctorate in Law at the University of Altdorf in 1667.

Shortly after this he met and began to work for Baron Johann Christian von Boineburg, working mainly in Nuremberg.  In 1672, Leibniz traveled to Paris on a diplomatic mission at the Baron’s request.  There, he met Christiaan Huygens, a well known Dutch clock-maker and mathematician.  Leibniz began to study math and physics with Huygens and continued his work in these areas throughout the 1670s.  He traveled to London in 1673 and met several members of the Royal Society, a group of early scientific pioneers in England.  Isaac Newton was associated with this group as well (although he did not meet Leibniz during his visit) and served as its president from 1703 until his death in 1727.

During 1675-6 Leibniz developed his notation for the methods of calculus.  His notation for integrating (finding the area under a curve) “∫  f (x) dx” and differentiating (finding the slope of a tangent line)  d(xⁿ) = nx^n-1 are still commonly used today.  In 1676, Leibniz returned to Germany to manage the library for the Duke of Hanover and, although he traveled frequently, remained there until his death in 1716.

Leibniz’ notation for calculus was considered to be easier to use and understand and, consequently, mathematicians on the European continent began to advance more quickly in their discoveries than the British mathematicians.

Eighteenth century British scientists made many important discoveries in the development of the steam engine, calculating the speed of light, predicting the behavior of comets, finding a method to determine longitude, and understanding the behavior of gasses.  However, British mathematics did not truly rejoin the mainstream of the European continent until about 1820, when British mathematicians began to accept and work with Leibniz’ notation and the subsequent 100 years of research that had followed it.

The Development of the Calculus

After the advent of Descartes’ analytic geometry, mathematicians had a lot of algebraic representations of geometric curves to analyze.  Victor Katz, in his textbook A History of Mathematics, lays out the quantities that interested mathematicians in relation to curves in the plane – arc length, the tangent line (and subtangent), the normal line (and subnormal), the area of a region with a curve for a boundary, and the volume of a solid of revolution based on a figure in the plane.

These various quantities can be very helpful in analyzing many problems in physics and other applied disciplines.  The slope of the tangent line to a curve can tell you at what rate a quantity is changing.

In this diagram the blue line is the tangent and the red line is the secant.  The idea of Calculus is to move the point Q closer and closer to the point P and use our knowledge of the behavior of the secant to understand the behavior of the tangent.  Notice that as the point Q moves closer to point P, the secant line more and more closely approximates the tangent line.

Finding the line tangent to a curve at a particular point also allows us to find what are called extrema – or maximum and minimum values.  This is useful in physics to find the maximum height of something shot into the air.  It can also be useful in finding the minimum cost in manufacturing or minimum travel time.  Visually, these max and min values look like this:

I’ll put up one more visual, which shows a way to use calculus to find the volume of an object called a solid of revolution:

These problems were all attacked individually with various mathematicians having varying degrees of success in solving particular problems.  Johannes Kepler, Bonaventura Cavalieri, Evangalista Torricelli, Pierre Fermat, Gilles de Roberval, Rene Descartes, Isaac Barrow and James Gregory all did important work that would today be considered Calculus, but none of them were able to generalize their results.  This is why Isaac Newton and Gottfried Leibniz generally receive credit for developing the techniques of Calculus independently of each other.  They both saw the connections among these seemingly different problems and ideas and they organized them into a coherent mathematical system.

I’ll discuss the biographies of Newton and Leibniz in the next post.

French Mathematics of the 17th century

Francois Viète (1540-1603)

Francois Viète was the son of a lawyer in 16th century France.  He is credited with devising a scheme* in which unknown quantities in algebra would be represented by letters that are vowels and constant quantities would be represented by letters that are consonants.  At the time, the Arabic algebra that had been transferred to Europe over the previous 500 years was based on prose writing – everything was described in words.  After Viète’s initial use of letters for unknowns and constants, René Descartes later began to use letters near the end of the alphabet for unknowns (x, y, z) and letters from the beginning of the alphabet for constants (a, b, c).  This practice continues today.

In 1593, the Dutch ambassador to France said to French King Henry IV that a well-known Dutch mathematician had posed a problem that was beyond the capabilities of ANY French mathematician.  Henry IV passed the problem along to Viète and Viète was able to solve it.

Viète began a correspondence with Roomen, the Dutch mathematician who had posed the problem originally and became one of the first internationally recognized French mathematicians.  He worked mainly in trigonometry, astronomy and the theory of equations.

*This link is a paper written by a college student at Rutgers University in New Jersey.  Papers on other subjects by other students in the same course can be found here.

Marin Mersenne (1588-1648)

Marin Mersenne was a French monk best known for his research into prime numbers.  He also did important research into the musical behavior of a vibrating string, showing that the frequency of the vibration was related to the length, tension, cross section and density of the material.

Mersenne primes are prime numbers of the form , where p is a prime number itself.  For example

which is prime

and so on.

Mersenne was also interested in the work that Copernicus had done on the movement of the heavenly bodies and despite the fact that, as a monk, he was closely tied to the Catholic church, he promoted the heliocentric theory in the 1600′s.

Mersenne was also known as a friend, collaborator and correspondent of many of his contemporaries.  Fermat, Pascal, Descartes, Huygens, Galileo, and Torricelli all corresponded with Mersenne and the exchange of ideas among these scientists promoted the understanding of music, weather and the solar system.

René Descartes (1596-1650)

René Descartes is probably best known for two things.  One is the conclusion “I think therefore I am” (Cogito ergo sum in Latin and Je pense donc je suis in French) and the other is the geometric coordinate system generally known as the Cartesian plane.

Descartes joined the army of Prince Maurice of Nassau in 1619 and was in Bavaria (southern Germany) and Bohemia (Czech Republic) during the beginning of the Thirty Years War.

The importance of the Cartesian Plane is difficult for us to understand today because it is a concept that we are taught at a young age.  Locating objects on a grid by their horizontal and vertical coordinates is so deeply embedded in our culture that it is difficult to imagine a time when it did not exist.

Before Descartes’ grid system took hold, there was Geometry:

and there was Algebra:

(Click on photo for larger view)

…and they were separate fields of endeavor.  The idea that a geometric shape like a parabola could be described by an algebraic formula that expressed the relationship between the curve’s horizontal and vertical components really is a ground-breaking advance.  It is so ground-breaking that once it happened, people began to forget that it hadn’t always been that way.

Once this new method for describing curves was developed, the question of finding the area under a curve was addressed.  This is the general problem of Integral Calculus.  Descartes (among others) saw that, given a polynomial curve , the area under the curve could be found by applying the formula

These were the rudimentary beginnings of the development of the Calculus that would be devised by Isaac Newton and Gottfried Leibniz in the ensuing years.

Fermat (1601-1665)

Pierre Fermat is also mostly remembered for two important ideas – Fermat’s Last Theorem and Fermat’s Little Theorem.  Fermat’s Last Theorem is a simple elegant statement – that Pythagorean Triples are the only whole number triples possible in an equation of the form .

Pythagorean Triples are interesting groups of numbers that satisfy the Pythagorean relationship .  Triples such as {3,4,5} {6,8,10} {8,15,17} {7, 24, 25} can be found that satisfy the equation.  But – Fermat’s Last Theorem says that if the in the original equation is any number higher than two, then there are no whole number solutions.

It’s true – but very difficult to prove.  Mathematicians tried for 350 years or so to prove this theorem before it was finally accomplished by Andrew Wiles in 1995.

By the way, you can generate Pythagorean Triples using the following formulas:

Pick two numbers and , with

Fermat’s Little Theorem is a useful and interesting piece of number theory that says that any prime number divides evenly into the number , where is any number that doesn’t share any factors with .

Blaise Pascal (1623-1662)

Blaise Pascal was the son of Etienne Pascal, who was a lawyer and amateur mathematician.  Etienne Pascal knew Marin Mersenne and often visited him at his Paris monastery, and when Blaise was a teenager he sometimes accompanied his father on these visits.

Pascal’s first published paper was a work on the conic sections.  He also did research on the composition of the atmosphere and noticed that the atmospheric pressure decreased as the elevation increased.  This led him to believe that beyond the atmosphere there existed a vacuum in which there was no atmospheric pressure.

René Descartes visited Pascal in 1647 and they argued about the existence of a vacuum beyond the atmosphere.  Descartes felt that this was impossible and criticized Pascal, saying that he must have a vacuum in his head.

Pascal is known for the structure of Pascal’s Triangle, which is a series of relationships that had previously been discovered by mathematicians in China and Persia.

Here is Pascal’s version:

Here is the Chinese version:

Here is a version that we often see in textbooks:

Each successive level is created by adding the two numbers above it, so in the 6th row {1,5,10,10,5,1} the 10 is created by adding the 4 and the 6 from the row above it.  These number patterns are actually quite useful in a wide variety of situations.

In raising a binomial to a power like , the coefficients of each term are the same as the numbers from the 6th row:

These numbers are also related to Discrete Mathematics and Combinatorics which describes how many ways there are to choose something from a series of possibilities.

There was a lot of great mathematics happening in Italy, England, Holland and Germany during the 17th century, but this collection of French mathematicians spanning nearly 100 years produced a tremendous amount of very important mathematical ideas.

The English, Germans and Swiss would make great contributions to mathematics in the 18th century with Newton, Leibniz, the Bernoullis, Euler and others, while the French would still contribute with the works of Laplace, Lagrange and Legendre.

Applied Mathematics and the Heavenly Bodies

Before we leave the time period of the early 16th century, I think it would be helpful to recognize three very important astronomer/mathematicians of that time -

Nicolaus Copernicus (1473-1543)

Tycho Brahe (1546-1601)

Johannes Kepler (1571-1630)

Nicolaus Copernicus

Copernicus was a Polish astronomer and scientist who first proposed that the earth revolved around the sun.  It was at about this time in the early 1500s that people started to notice that the Roman calendar that had been in use for some 1500 years was starting to be unreliable.  That is, the date of the equinox was coming earlier and earlier in the year and people could see by the angle of the sun and other methods that the calendar was wrong.  This problem was eventually corrected with the advent of the Gregorian Calendar (also here).

Copernicus was not directly involved in the debate surrounding the Calendar, but he did decide over the course of his life that the earth revolved around the sun.  This contradicted not only Ptolemy’s ideas from the 2nd century, but also the Catholic church’s assertion that the earth was the center of the universe.

In 1543, as Copernicus was dying, his ideas were published as the book De revolutionibus orbium coelestium (On the revolution of the the celestial orbs).  In the ensuing decades, his ideas were not well-received.  Because of the Protestant reformation, the Catholic church in Rome was very concerned about its loss of authority and wasn’t interested in any “non-traditional” theories about the earth, sun and moon.  In addition, the Protestants themselves believed for the most part that all knowledge should be based on scripture.

Although Copernicus was correct in his assertion that the earth revolved around the sun, he incorrectly believed that the orbits of the heavenly bodies were circular and this created defects in any calculations made from using his system.  The correction for this would come later.

Tycho Brahe

Tycho Brahe was a Danish nobleman who worked as an astronomer for the Holy Roman Emperor Rudolph II in Prague.  His belief was that the sun and moon revolved around the earth, but that the other planets revolved around the sun.  He saw some benefit in the Copernican model but couldn’t prove that it was true because of the mistaken belief that the orbits were circular and the difficulty in measuring a predicted phenomena of stellar parallax.  The first accurate measurement of stellar parallax was made in 1838 by Friedrich Bessel.  In 1600, Brahe hired the German astronomer Johannes Kepler to work in the royal observatory in Prague.

Johannes Kepler

The collaboration between Kepler and Brahe was short-lived as Brahe died in 1601.  Using Brahe’s data from the previous 38 years as well as his own observations between 1600-1605, Kepler finally realized a way to reconcile his observed data with the various theories that had been proposed.

He realized that the planets orbited the sun in elliptical orbits.  His book Nova Astronomia was published in 1609.  Ten years later, in 1619, Kepler published Harmonices Mundi or The Harmony of the World in which he expounded his three laws of planetary motion.