Blaise Pascal
Kevin Kappenman
Blaise Pascal’s life
Blaise Pascal was born on June nineteenth in the year 1623. He was born in a town called Clermont-Ferrand,
in France. His father, Etienne Pascal was a judge at the tax court in Clermont-Ferrand, and his mother died
a few years after Blaise Pascal’s birth. Blaise was the third of four children that the family had and was the
only son.
After his mother’s death Etienne Pascal moved the family to Paris, France where Blaise was educated at home.
The studies that Pascal pursued specifically omitted the study of math. Etienne believed that the study of
mathematics would prove so fascinating that Pascal would be unable to focus on any of his other classical
studies, and so Pascal was forbidden from learning mathematics. Pascal pursued the study of literature and
the humanities as instructed by his tutors, but of course the elusive subject sparked Blaise’s attention and he
began to study geometry, and discover geometrical concepts on his own.
It has been said that Pascal began drawing geometric figures during his free time. He drew these figures with
such accuracy that he was able to begin to study them. It is said that Pascal was able to determine that
the sum of all the angles in a triangle added up to two right angles all on his own. He did this by folding
the triangle in such a way that the edges lined up and the sum was clear. When Etienne saw how well the
boy had taken to mathematics he gave Pascal a copy of Euclid’s Elements to study and allowed Pascal to
attend meetings of the mathematical society of which his father was part, Academie libre. It was through these
meetings that Pascal came into contact with the latest findings in mathematics. His study of mathematics
continued to grow and would culminate to new heights throughout his life.
In 1640 the family moved to Rouen where Etienne had received a new position as a tax collector. It was here
in Rouen that Blaise began a spiritual revolution. When Pascal’s father injured his leg he was cared for by a
pair of Jansenist converts. In the process of caring for Etienne their religion impacted Blaise who proceeded
to become devoutly religious. The Janesites offered a religion different from the Jesuit practices that were
common in the area. The Janesites lived with the Pascal family for almost a year while Etienne recovered
from his broken leg.
In the year 1654 Pascal went through an experience that he believed to be a call from the Heavens to abandon
the trivialities of the world. While he was riding a horse drawn carriage the horses ran wild toward the edge
of a bridge, he was only saved when the train, connecting the horses to the carriage, broke. He moved to Port
Royal and began to study and contemplate religion. It was his religious and philosophical works for which he
is said to be best known. The first such work is Les Provinciales which were a blow against the lax morality of
the Jesuits. These works would be read by many different religious sects. A future Pope, Innocent XI corrected
many of the points that Pascal had brought up within this work to help return to the Catholic religion to a
more focused position. Pascal also began working on the Apologie, but this work remained unfinished upon
his death. The work was published under the title Pensees. This work was compilation of fragments and notes
that Pascal had written that told of the failures of mankind, as well as the infinite glory and importance of
Christ.
Although Blaise Pascal is often thought of in terms of Physics and Mathematics, it is his works in religion that
are often overlooked. Blaise Pascal died in 1662. He was only thirty nine years old at the time of his death.
The cause of death for Blaise Pascal is a bit of a mystery. It was assumed to be tuberculosis until an autopsy
was performed after his death. The autopsy revealed problems with his stomach and a lesion in his brain. The
problems with his stomach seem to be stomach cancer, although that was not a diagnosis given in 1662, and
the lesion in his brain could be responsible for his chronic headaches. The cause of death generally accepted
is tuberculosis, but with these other physical organ problems more issues may have caused his death.
Blaise Pascal’s works in Science and Mathematics
Pascal’s fascination with mathematics began due to the fact that his father forbid the subject from being
taught at a young age. He discovered some of the basics in geometry on his own, such as the sum of the
interior angles of a triangle. He began seriously working with mathematics soon after his father introduced
him to the group, Academie libre. In front of the group of mathematicians Pascal began to present some of
his early mathematical works. One of his early theorems involved his mystical hexagon the idea behind his
hexagon is that if a hexagon is inscribed within a circle then the three intersection points of opposite sides all
lie on a single line. He named this line the Pascal line. Around this same time Pascal began developing his
work with conic sections.
Pascal’s work on conic sections was written in the essay Essai pour les Coniques, essay on conics. In this essay
his work with the mystical hexagon could be found. The work also contained hundreds of propositions on
conics. One of these ideas involved a quadrilateral inscribed in a conic, and a straight line cutting the sides
taken in order in the points A,B,C,D , and then the conic P and Q, this means that
P A.P C : P B.P D = QA.QC : QB.QD
This formed an interesting new way of looking at the relation between a quadrilateral and a curved conic
section. The source of his information for this work came from studying Apollonius and his successors. By
building upon the developments of the past Pascal developed his own new ideas. This work containing his
ideas served as a big step in projective geometry, taking a three dimensional figure and placing it on a two
dimensional plane. This work was published in 1640, when Pascal was only seventeen years old. It was around
this same time that Pascal began to study Torricellis experiment.
This experiment involved overturning a tube of mercury into a bowl that was also filled with mercury. The
mercury would fall to a certain point in the tube and then stop. This was used as a way to estimate the weight
of the atmosphere. The vacuum created by the falling mercury would pull the mercury up the tube to varying
heights based upon the atmospheric pressure. Pascal continued this experiment with many variations. He
performed this experiment on a mountain overlooking his birthplace of Clermont-Ferrand, as well as in Paris
to determine that the level of Mercury in the tube changes along with changes in elevation. This led to Traite
du vide (Treatise on a Vacuum) being published.
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The equation developed from Torricelli’s experiment is as follows:
Pressure = (Density of Mercury) · (Height of Mercury) · (Earth’s gravitational constant)
This allowed for atmospheric pressure to be calculated, Earth’s gravitational constant has been found to be
9.8. From his work in physics with the atmospheric pressure on the bowl of Mercury Pascal also came to
develop the syringe and a hydraulic press which used a principle now known as Pascal’s Law. This law states
that pressure applied to a confined liquid is transmitted through the liquid regardless of the area of the applied
force.
(Force on one side)/(Area of applied force) = (Force on other side)/(Area of applied force)
In essence this means that any force applied to a liquid in a confined space can be transferred through the
liquid to a new direction. These works were an impressive step in physics.
When Pascal’s father became a tax collector in Rouen he began working on a very interesting machine. It
was an early form of the calculator, capable of simple addition, subtraction, multiplication, and division. He
called the machine a Pascaline. The Pascaline had eight movable dials that represented numerical digits such
as ones, tens and hundreds. Pascal attempted to sell models of his Pascaline, but it was not well received and
quickly went out of production.
In 1650 Pascal began work with what he hoped would be a perpetual motion machine. The idea behind the
machine was that it would produce more energy than was consumed in setting the machine in motion. Such
a device would have solved the world’s energy problems before they started. The attempts to build such a
device failed. However this did lead to Pascal’s wheel, which he named the roulette machine. Roulette being
the French word for little wheel. The Roulette gambling wheel was developed in the eighteenth century and
was loosely based upon Pascal’s version.
In 1654 Pascal began his work with his famous triangle. It is an easy tabular method for finding the binary
coefficients of a multiple of binary terms. Each number in each subsequent line, down the triangle, is the sum
of the two numbers to its right and left in the row above it. A binomial is an algebraic expression that is
fairly straight forward, it involves two numbers, call them a and b, these numbers are then raised to a power
as follows.
(a + b)n
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This is equivalent to (a+b) (a+b) (a+b) .... These terms would be multiplied out to n terms.
For example if the phrase (a + b)3 were given this would be equivalent to
(a + b)(a + b)(a + b)
When multiplied out this multiplication would have eight individual terms.
a3 + a2 b + a2 b + a2 b + ab2 + ab2 + ab2 + b3
However some of these are like terms and could be reduced to a series of coefficients.
a3 + 3a2 b + 3ab2 + b3
It is the process of finding these coefficients that Pascal’s Triangle simplifies.
The first few lines of this triangle are as follows.
As can be seen the sum of the numbers in each line corresponds with the number of individual terms found
from taking a binomial to a power. The first line corresponds with the exponent 0 then 1, 2,3,4 and so on.
This allows for binomials to be expanded easily.
Pascal also found a few patterns within the triangle. The diagonals of the triangle provide some interesting
points. The first diagonal down and to the right is simply a series of ones. The second diagonal down and
to the right is the series of natural numbers. The third diagonal down and to the right contains the series of
triangular numbers. The fourth diagonal down and to the right contains the pyramidal triangular numbers.
It is possible to find others series and partial patterns within the triangle as well.
Pascal also used this triangle in the theory of combinations. This theory of combinations involves having a set
number of items, call the number of items m, and taking a set number of them at a time, call it n. In such a
way that he stated as:
((n + 1)(n + 2)(n + 3)(n + 4)(n + 5)...m)/(m − n)!
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Pascal’s triangle also has an application to the study of probability that will be discussed in a later section as
he collaborated on this theory.
Pascal’s last work was with the cycloid curve. This curve is defined as a curve traced by a single point on
the circumference of a circle as it rolls in a straight line. This curve was first studied by Galileo who thought
the curve was beautiful enough to build new bridges in the shape of. Future mathematicians would study
the cycloid. Roberval found the area of the cycloid. Descartes and Fermat worked to find the tangents to
this curve. Several problems remained unsolved in regards to the curve, its surface area, volume, and center
of mass, when revolved around its axis, its base, and the tangent line at its vertex, were all questions that
mathematicians wondered. These questions were solved by Pascal in 1658. Pascal used a method of indivisibles,
which is highly similar to integral calculus that could be applied today. It was claimed by D’Alembert, that
these researches for a link between Archimedean geometry and infinitesimal calculus that Newton developed
later in the seventeenth century.
Collaboration with other scholars
The mathematical theory of probability did not have a good deal of support until the works of Pascal and
Fermat came into the picture. The correspondence between Fermat and Pascal laid the groundworks for the
Theories in Probabilities that these two would develop in regards to a game of gambling. Together the two
contemporaries came up with very similar results. The theory developed as a way for two individuals to walk
away from a game of on which money was bet before the game was over. The money cannot all go to one
person who is in the lead because that person has not yet won the game. While the man who is losing cannot
receive all of his money back, because he is behind. This problem has an interesting application in how the
pot of money should be split between two individuals playing this game of chance.
This problem called The Problem of Points is simplest when described in a winner take all sort of game. One
game used for example is that of flipping a coin. The first of two players to reach a set number of heads or
tails wins a pot of money, let’s call the pot 100. Let’s call the target the men are trying to reach ten. Let’s
say that one player has reached eight points, and the other is at seven points. The game would end after four
tosses of the coin, either the first player would receive the two points that he needs or the second player would
reach the three points which he needs. Since it is entirely possible for either of these events to come to pass
the money cannot all go to one individual. Instead the results of the possible coin flips can be written out
exhaustively.
HHHH HHHT HHTH HHTT TTTT
HTHH HTHT HTTH HTTT TTTH
THHH THHT THTH THTT TTHH TTHT
This means that there are a total of sixteen ways for the game to end with the coin tosses. Of these sixteen
eleven favor the player who had eight points to begin with while only five favor the player who had seven
points. This means that the pot of francs should be split according to the following for the first and second
players respectively.
100 · (11/16) =
100 · (5/16) =
5
Pascal looked for a method of generalizing the problem instead of listing all of the possibilities exhaustively
as that would become tedious. He realized that there are only two outcomes for the above game, making the
outcomes similar to a binary expression:
(T + H)4
By going to the fifth row of Pascal’s triangle the coefficient numbers can be found to be: 1 4 6 4 1
From this it can be seen that the sum of all the numbers in the fifth row of the triangle is 16, the total number
of ways the coin game could have ended. The sum of the first three terms is 11 which is the total number of
ways the first player could have won, and the sum of the last three terms is 5 which was the number of was that
the second player could have won. By using binary terms and coefficients the process of exhaustively finding
outcomes was reduced. This was a way of formalizing the idea of equally probable outcomes. The probability
of something occurring could be simplified down to, the sum of the positive outcomes that could occur divided
by the total number of outcomes that could occur. This allowed for the use of fractions when calculating the
probability of events, and gave rise to the process of finding the probability of other events than just coin flips.
There is the probability of rolling the same number twice on a six sided die given by multiplication.
1/6 · 1/6 = 1/36
And the process of finding the probability of rolling an even number on a six sided die given by addition.
1/6 + 1/6 + 1/6 = 1/2
Although Pascal and Fermat did much of the work required for early probabilities they were not the first to
publish any such work. A work on probabilities was first published by Christiaan Huygens in 1657. His works
were largely based on the ideas of Fermat and Pascal. Pascal later used his work on probabilities within one
of his religious works, Pensees, in referencing that though the likelihood of finding eternal happiness is small,
eternal happiness must be quantitatively worthwhile to make up for the small odds in achieving it.
Historical events that marked Pascal’s life.
Pascal lived in France from the year 1623 till his death in 1662. During this time France was ravaged by a
series of uprisings and was engaged in international wars. These wars and unrest within the country led to
some interesting changes within the power structure of France at this time.
The wars that France was involved in included the Thirty years war, and the Franco Spanish war. The Thirty
years war began in the year 1618 as a conflict between Bohemian nobleman and the Austrian Hapsburgs.
France remained out of the war until the year 1635. France entered the war by declaring war on the neighboring
country of Spain. By extension this meant that France had declared war on Spain’s allies throughout the Holy
Roman Empire.
By declaring war against the Holy Roman Empire, France took a step toward siding with the protestant powers
in Europe. Cardinal Richelieu was the individual who chose for France to enter the war against Spain, with
his death in 1642 and Louis XIII’s death in 1643, and the heir to the throne being only five years old, power
was in a questionable state.
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From this arose Cardinal Mazarin, and Anne of Austria who was Louis XIII’s wife. This change of power
occurred after France was already eight years into its war with the Holy Roman Empire. France was forced
to raise taxes and tax the common people to pay for the war effort. The Pascal family was respected enough
to remain above the common people and as part of the governmental powers. It was around this time that
Pascal’s father was appointed to be a tax collector. The taxes that were being levied against the people of
France enraged some of these groups. Cardinal Mazarin felt that the thirty years war was near an end and
that when France emerged victorious it would receive monetary compensations for the war effort from the
losing countries, but in the short run the people of France were angry.
Anne of Austria and Cardinal Mazarin were soon to find themselves facing a series of uprisings and times of
civil unrest within France. It seemed very similar to the revolution occurring in England at the same time, but
never rose to that level of unrest, at least not in this century. As time came for Mazarin and Anne to put forth
a new tax before the judges of the Paris courts the Parliament called a secret meeting of these same judges.
The meeting was called the Chambre Saint Louis and this marked the beginning of the uprisings, known as
the Fronde. Street demonstrations were organized by the Chamber that showed strong support for the judges
by the people of France.
People participating in the Fronde aimed their anger toward Mazarin, due to his foreign status, and his high
taxes. They claimed he was using ”his position to enrich himself and ruin the country.” There is no mention
of the Pascal family’s responses to these events occurring in their vicinity. Anne of Austria, also a foreigner,
continued to stand by Mazarin throughout this time, protecting the future of France, and the heir to the
throne. The summoning of the Chamber Saint Louis was an act of defiance to the Royal Authority within
France. It did not culminate into a revolution as the Revolution in the eighteen hundreds would do. This was
due to weak support for the cause of the revolution. The revolution was sparked due to taxes for a war that
would soon draw to a close.
In the year 1648 the Peace of Westphalia ended the Thirty Years war. France was among the victors and
would become a dominant force within the continent. Conflict with Spain continued after the end of the thirty
years war, in the form of the Franco Spanish war. This war would continue into 1659. These wars continued
to draw the poor ranks of young men to the fighting fronts where money could often be found. The war was
not fought by a unified French army, but instead it was done by private companies who were not bound by a
central mode of operation. By using private companies such as this the French Royals could focus on the civil
unrest at home.
Many noblemen continued to support Mazarin and Anne of Austria throughout the Fronde. When it became
clear in 1649 that the Royals could not subdue the unrest in Paris Mazarin, Anne, and Louis XIV fled the
capital to go to the nearby city of Saint- Germain.
The parliamentary Fronde gave way to the Fronde of the princes in 1648. Provinces revolted against the royal
authority and went to war with Mazarin’s forces eventually forcing Mazarin to flee France. In the year 1651
Louis XIV was officially recognized as King at the age of thirteen.
Although Louis XIV did not assume full control of the Royal powers until Mazarin’s death in 1661 when he
did he took several drastic measures. Louis XIV moved the royal Palace out of Paris to Versailles, and made
sure that his absolute authority would be respected. This absolute authority would persist into the eighteenth
century, when a unified French rebellion would eventually topple the monarchy.
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Significant historical events around the world during Pascal’s life
In the early seventeenth century Galileo studied Jupiter’s moons and Tycho Brahe took large quantities of
mathematical notes on the positions of planets in the sky. Brahe’s assistant, Johannes Kepler, began exploring
the idea of planetary motion which was aided by the development of the logarithm. The astronomists of this
time made great progress in figuring out how the planets interacted with one another. In 1633 Galileo was
tried before the inquisition and sentenced to house arrest by the church for his wild claims that the Sun was
the center of the galaxy.
Japan became an isolationist country in 1633 and would stay this way for a few centuries. This allowed other
civilizations to develop technologies that Japan did not acquire until much later in its development as a nation.
The Wars of the Three Kingdoms began in 1651, these wars were civil wars throughout Scotland, Ireland,
and England. These wars were fought between the Puritans and the Catholics for a religious and political
revolution. The civil unrest began when the monarchy of Charles II was reestablished. The English Parliament
eventually came out victorious in this series of wars establishing England as a constitutional monarchy. Charles
II was executed in the process of making the Parliament supreme over the Kings of England.
The Peace of Westphalia that ended the Thirty years war also spelled the demise of the Holy Roman Empire
and Spain as world powers. The reparations that were owed at the end of the war sent both countries into a
state of financial disrepair.
In 1660 the Commonwealth of England ended. This meant that the Parliament was once again subject to
the authority of a monarchy. At the same time the Royal Society of London for the Improvement of Natural
Knowledge was founded. This organization would serve as a center of academic achievement in the coming
decades.
Significant mathematical progress during the Pascal’s lifetime
Mathematics took some large steps within the seventeenth century. The first of these large steps was taken
by John Napier who developed the logarithm. He published his work on logarithms in the book Mirifici
Logarithmorum Canonis Descriptio. The book contained a number of pages and tables related to logarithms
and natural logarithms. The book also holds some theories in regards to spherical trigonometry which is
usually referred to as Napier’s Rule of Circular Parts.
When Henry Brigs visited Napier in 1615 he suggested a logarithm based on base ten numbers instead of
those based on the constant e. Napier calculated tables of these logarithms as well and they aided the fields
of physics and astronomy by greatly reducing the calculations necessary. They sped up the process of writing
in scientific notation immensely. And in the year 1622 William Oughted would introduce a logarithmic slide
rule, an instrument that allowed for the swift calculation of logarithms, that would be used for the next three
hundred years.
Napier also improved upon the decimal notation which was originally written by Simon Stevin. The decimal
notation that Napier created helped to popularize the use of the decimal point.
Around this same time an individual named Marin Mersenne developed an interesting theory in regards to
prime numbers. His work claimed that there exist Mersenne primes, which are numbers one less than a power
of 2.
3 = (22 − 1), 7 = (23 − 1), 31 = (25 − 1), 127 = (27 − 1)...
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When the list of Primes that Mersenne had listed was later checked it was found that he had omitted several
primes from his list, as well as including some which were not prime at all. Although interesting that some
prime numbers could be found in this way it did not prove to be a viable method of finding prime numbers.
Rene Descartes next entered the scene of mathematics to revolutionize algebraic notation. He continued
upon the works of other mathematicians who had begun to use variables to represent numbers, and set the
groundwork for the common notation used today which includes using the lowercase a, b , and c for known
quantities and the lowercase x and y for unknown variables. He introduced this method of notation within his
book titled La Geometrie.
It was within La Geometrie that Descartes first began to propose the idea that any point can be described in
two dimensions. He did this by placing the point on a plane and defining it by two numbers. These numbers
came from the point’s position relative to a pair of axes. This came as a large step in projectional geometry
which allowed for figures to be placed onto a two dimensional plane. The points that Descarte found he
named Cartesian Coordinates. Equations could readily be represented on a pair of perpendicular axes. The x
coordinate would come first followed by the y coordinate so that a line such as y=3x would have points: (0,0)
(1,3) (2,6) (3,9) and so on.
This also allowed for curves to be plotted such as: x2 , x3 , and even circles y 2 + x2 = r2 .
This work is often referred to as analytic geometry or Cartesian geometry, and allowed for the conversion
between geometry and algebra to take place. A pair of equations could now be solved either algebraically or
graphically. To solve the equations graphically one would find the points at which the two lines intersect. This
interaction between algebra and geometry would allow for the development of calculus by Leibniz and Newton
later in the century.
Descartes is also attributed with developing the rule of signs. This was used to determine the number of
positive or negative real roots that a polynomial would contain. The rule of signs does not actually give the
solution to the polynomial, but it gives information on the number of roots it possesses.
The number of positive roots that a polynomial has is either equal to, or an integer number less than the
number of times p(x) changes sign.
x5 − 3x4 − 2x3 + 6x2 − 2
This example changes signs 3 times and therefore there is the potential for 3 positive roots.
The number of negative roots that a polynomial contains is either equal to, or an integer number less than the
number of times p(-x) changes sign.
(−x)5 − 3(−x)4 − 2(−x)3 + 6(−x)2 − 2 = −x5 − 3x4 + 2x3 + 6x2 − 2
Because the sign changes twice this means that there is the potential for two negative roots to the above
equation.
Descartes also made the use of superscript notation such as 2 · 2 · 2 · 2 = 24 popular.
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The French mathematician Fermat conducted work with number theory. Most of his ideas within number
theory were sent in correspondences with friends, often with little mathematical proof to support his claims.
Fermat came up with an interesting theorem in relation to prime numbers. It says that any number divisible
by four which leaves a remainder of one can be rewritten as the sum of two squares.
(4n + 1) = p
such that
13 = 3 · 4 + 1 = 32 + 22
This theorem can be used to test for prime numbers, and is often used in encryption methods today. Fermat
also identified a subset of the natural numbers which he called Fermat numbers. They are found by taking
two to a power of two and adding one such two to the two to the n plus one equals a Fermat number. The
first five in the series come out to be 21 + 1 = 3 22 + 1 = 5 24 + 1 = 17 28 + 1 = 257 and 21 6 + 1 = 65537.
These numbers are interesting due to the fact that the first five are all prime numbers. The Fermat numbers
of higher values of n than five are not prime numbers.
Fermat is best known for something that he did not even prove. His Last Theorem which was a conjecture
scribbled in the margin of his copy of Arithmetica. His Last theorem states that no three positive integers a,
b, and c can satisfy the following equation when n is greater than two:
an + bn = cn
There are many examples of when this theorem is true for values of two, the Pythagorean Triples are all
examples. Fermat claimed to have a proof to this concept, and that the margin of Arithmetica was too small
to hold the proof. According to documents of Fermat’s found after his death he had only partial proven this
theorem for the example of n=4. This theorem was eventually proven by Andrew Wiles in 1995. The final
proof used complex mathematics that was far outside of the reach of Fermat, leading to the conclusion that
he did not actually have a proof for this theorem.
Fermat did some work with a precursor to calculus. While experimenting with finding the centers of gravity for
a variety of figures Fermat developed a way for finding the maxima, minima, and tangents to curves, which are
essential processes in differential calculus. His work with Fermat on probability was also a large mathematical
concept to emerge from Fermat.
An oxford professor laid more of the groundwork for the coming development of calculus in his work Atithmetica
Infinitum. The professor’s name was John Wallis. He applied the operation of mathematics to the system
of the tides. He also developed the modern notation that is used for infinity. He was also responsible of
originating the idea of a number line and extending the notation of powers to include negative integers. These
negative integers to a power were referred to as continued fractions.
A mathematician to whom many accolades belong of the seventeenth century was Sir Isaac Newton. One of the
first ever scientists to be knighted for his progress in physics that can be found in his work the Principia. He
worked very diligently for two years during the Great Plague on his works in a theory of light, quantification of
a gravitational constant, and he developed his most important mathematical work, ifinitesimal calculus. His
theory of Calculus improved upon the works of his fellow mathematicians, John Wallis and Isaac Barrow. He
also used work from mathematicians who were not English such as Descartes and Fermat with their methods
of graphing curves. The use of calculus allowed Newton to study motion and change in the world through a
plot of the motion of objects.
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The first problem with which Newton mused was that it was easy to calculate the average slope of a curve
at a point, but it was always changing. There was no method at that time to find the slope of a curve at a
single point. Newton developed a method for finding the derivative of a function which gives the slope of a
function, f(x), at any point of x. He called the derivative, the instantaneous rate of change at a given point, a
fluxion. He called the changing values of x and y the fluents. Newton developed a series of rules in relation to
differentiation. These rules generalized the derivatives of functions so that they could be easily calculated. His
rules referred to power functions, exponential functions, logarithmic functions, and trigonometric functions.
By developing these general rules Newton was able to find the fluxion of many mathematical graphs which
then allowed him to find the slope of the tangent to the graph at a single point, by using a value of x.
The opposite of the derivative, or the method of fluents, is integration. Newton was able to define this as the
area under the curve. Newton did not publish his work immediately, he instead waited until 1693. By this
time another mathematician named Gottfried Leibniz had already published a work on calculus in the year
1684. The Royal Society accused Leibniz of plagiarizing Newton’s work, but in the end it was determined that
the two methods were developed independent of one another.
Newton is also credited with his work with binomials. He developed the binomial theorem which describes the
expansion for algebraic expressions. He used fractional exponents and coordinate geometry to find solutions
to Algebraic equations that only allowed for integer variables, known as Diophantine equations. He was able
to develop a method for finding the zeros and roots of a polynomial function.
Leibniz, as mentioned before, developed a system of infinitesimal calculus on his own He also chose to use
matrices as a method for solving equations, and he developed a calculating machine that used the binary
system. The Royal Society gave credit of the first publication of calculus to Leibniz while they gave credit for
the first discovery of calculus to Newton. The notation that Leibniz used was more standardized and is still
the method used for working with calculus today.
Leibniz work with matrices allowed linear equations to be expressed as a matrix. These matrices could then
be added, subtracted, multiplied, or divided with relative ease. Later mathematicians would further Leibniz’s
work with the matrix. Leibniz’s calculating machine was a forerunner of the computer. It operated on a
binary system of ones and zeroes. The system corresponds each digit and number to a series of on and off
sequences within the ones and zeroes. Leibniz believed that the mind should not be preoccupied with simple
mathematics that a machine could perform.
Connections between history and the development of mathematics
Mathematics develops in two ways, passively without a defined purpose, and actively, with a purpose in mind.
Throughout the seventeenth century mathematics took large leaps in both of these ways. The development of
logarithms by Napier was done passively and did not have a great deal of focus as to the application of the
logarithms, based upon the scientific needs of the seventeenth century he adapted his logarithms to base ten
evaluations. This development of mathematics occurred due to scientific progress.
The passive development of number theory, probability, and the Cartesian plane all eventually led to the
development of calculus. Calculus developed as the sum of the mathematical work that came from the mathematician’s work who laid the groundwork. The ideas of Newton and Leibniz were incredibly forward thinking,
but they relied upon the previous work of the mathematicians of the centuries. Newton’s development of
calculus had a direct correlation to history. While studying at Cambridge the plague broke out and Newton
retreated to a family farm to await the passing of the plague. During this time he was able to focus on his
work in physics which in turn led to his developments in calculus. Although he developed the calculus on his
own, the plague forced him the focus on his studies while away at the farm.
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The foundation of the Royal Society of London for the Improvement of Natural Knowledge also played a
crucial role in the development of mathematics. Similar to the Academie Libre of which Pascal was a part, the
Royal Society provided a central location for the mathematicians of the era to share ideas. Newton’s ideas on
Calculus were passed around social circles within the Royal Society for several years before their publication.
This allowed for works to be evaluated, and collaborated on before their presentation to the world at large.
The developments of mathematics within the seventeenth century included many passive ideas that allowed
for more tangible progress to be made throughout the scientific and mathematic community of the century.
Remarks
The seventeenth century is filled with mathematical innovations. Without the development of the logarithm,
Cartesian coordinates, calculus, and many other mathematical concepts the world of mathematics would look
very different today. The work of Pascal is important to the ideas of probability and binomial expansion
which serve as common applications in mathematics today. If you were to remove the developments of the
seventeenth century from the mathematics textbooks, the theories of mathematics may not have developed in
the same way, or at all. The brilliant minds that worked on mathematics throughout the century performed
impressive feats with numbers.
The theory of probability and variable rates of change are such common subjects today that it would be
difficult to imagine the world without them. Many inventions and scientific marvels could not have come to
pass without calculus. No man would have reached the moon, without variable rates of change. Statistics are
all around us in the world today and these are just two examples of the mathematical works that arose from
this century.
Pascal is noted for his work in physics, philosophy, and religion. The Pascal unit for pressure was named after
him, as well as a computer language that was developed by Nicklaus Wirth. This level of recognition years
later is evidence of the impact that pascal had on the development of the fields of physics and mathematics.
References
1. http://www.biography.com/people/blaise-pascal-9434176
2. http://www.storyofmathematics.com/17th leibniz.html
3. http://www.storyofmathematics.com/17th newton.html
4. http://www.storyofmathematics.com/17th.html
5. http://www.bshm.ac.uk/
6. http://fclass.vaniercollege.qc.ca/web/mathematics/about/history.htm
7. http://www.sparknotes.com/history/european/scientificrevolution/section4.html
8. http://www.storyofmathematics.com/17th fermat.html
9. http://www.storyofmathematics.com/17th descartes.html
10. http://www.storyofmathematics.com/17th pascal.html
11. http://libro.uca.edu/payne1/payne15.htm
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12. http://www.uky.edu/ popkin/540syl2007/540Fronde.htm
13. Ball, Rouse: A Short Account of the History of Mathematics. Np. 1908
14. https://www.famousscientists.org/blaise-pascal/
15. http://www.britannica.com/print/articl/445406
16. https://math.berkley.edu/ robin/Pascal/
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