Pascal and the Arithmetic Triangle

Carrie Tapp

20498109

Math 680: History of Mathematics

August 4, 2013

Abstract

Over his short, fragile life, Blaise Pascal significantly contributed to mathematics, specifically the

field of probability. A mathematical prodigy, teenaged Pascal made novel geometric observations in

his first paper Essai pour les coniques, including the mystic hexagon theorem. To prevent his father

from doing monotonous calculations, Pascal invented an adding and multiplying machine, called the

Pascaline. He discovered that the height of a column of mercury varied with altitude and in 1971,

the pascal (Pa) became the SI measurement for pressure equal to one newton of force per square

meter.

Along with Fermat and the prompting questions of the gambler De Mere, Pascal began to delve

into probability theory, answering several dilemmas in a straightforward, simple manner. Perhaps

Pascal’s most famous mathematical endeavor was the study of the arithmetic triangle. Known to

many today as “Pascal’s Triangle”, the arrangement of binomial coefficients led Pascal to nineteen

never before published properties. The Traite du Triangle Arithmetique contained equalities and

proportions that Pascal had discovered from the triangle. It is in this work that Pascal also laid the

foundation for mathematical induction, a commonly used proof technique today.

After a spiritual realization at the age of 31, Pascal abandoned the study of mathematics and

devoted almost all of the rest of his life to religion. He wrote several works under the pseudonym

Louis de Montalte that have become cornerstones of French religious literature. The only reversion

to the field of math was a short eight-day study of the cycloid. With so many contributions in his

short life, Burton (2011) speculates that “perhaps if Pascal had not died so young, or if he had not

abandoned mathematics for the religious life, he would have had the honor accorded Newton and

Leibniz for the discovery of the calculus” (p. 454).

i

Contents

1 The Place 1

1.1 Internal Wars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Foreign Wars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Person 5

2.1 1623-1639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 1640-1651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 1651-1654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 De Mere’s Problem of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 1655-1662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The Problem: The Arithmetic Triangle 12

3.1 History of the Arithmetic Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Pascal’s Nineteen Consequences of the Arithmetic Triangle . . . . . . . . . . . . . . 15

3.2.1 Consequence I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2 Consequence II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.3 Consequence III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.4 Consequence IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.5 Consequence V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.6 Consequence VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.7 Consequence VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

ii

CONTENTS iii

3.2.8 Consequence VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.9 Consequence IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.10 Consequence X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.11 Consequence XI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.12 Consequence XII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.13 Consequence XIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.14 Consequence XIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.15 Consequence XV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.16 Consequence XVI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.17 Consequence XVII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.18 Consequence XVIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.19 Consequence XIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 1

The Place

This chapter illustrates the turmoil that was occuring around Blaise Pascal in his native country

and the importance of religion to French culture, which would have a pronounced effect on Pascal in

his later years. France was a major European power during Pascal’s lifetime and although winning

many foreign conquests, the financial burden on the homeland was tremendous and after years

of poor crops and heavy taxation, the people demanded the rulers pay attention to their domestic

issues. France was at the centre of a renewed religious focus and its upper class became progressively

more educated and refined. Pascal was raised in the upper-middle class lifestyle and hence received

instruction in arts, sciences, and religion.

1.1 Internal Wars

Seventeenth-century France was a bustling place, with Paris at the centre of the intellectual and

social scene. Louis XIII and then Louis XIV were in power, although many of the decisions in Pas-

cal’s time were made by the Cardinals Richelieu and Mazarin (Banker, 2009). Cardinal Richelieu

encouraged manufacturing industries, created a navy, established colonies on other continents, and

secured French territory through military conquests, but he was still not a well-liked man (France,

2013). His ambitions tripled government expenses between 1620 and 1640, requiring drastic tax

increases. He weakened the nobles’ influence and reduced the Parlement’s power, laying the foun-

dation for Louis XIV absolutist ideals. He also attempted to ban the common practice of duelling

1

CHAPTER 1. THE PLACE 2

(Banker, 2009).

When power was transferred to Queen Anne of Austria and her Italian Cardinal Mazarin in 1643,

they continued in the same tyrranical fashion as Richelieu (the Fronde, 2013). An attempt to raise

taxes again started a series of French civil wars called the Fronde in 1648. The Parlement tried

to put a limit on the monarchy through twenty-seven different demands, including reducing taxes,

Parlement ’s approval of all new taxes, and an end to arbitrary imprisonment (the Fronde, 2013).

Pascal was part of the officer class that led the revolt, but he was certain that uprisings would

only make matters worse for the poor, and so he opposed the fighting (Rogers, 2003, p.11). Due

to the war with Spain, on July 31, 1648, Mazarin and his government decided to approve many

of the Parlement ’s requests so that the foreign war could be the focus. When a French victory

was announced the next month, Queen Anne and Mazarin had two parlementaires arrested, only

to to be forced to release them two days later as a result of a Parisian uprising against the act.

Tension continued to rise until war broke out in January, 1649 (the Fronde, 2013). With the gov-

ernment still wanting to focus on foreign conquests as well as provincial disturbances spurred by

crop failures, famine, poverty, and tax increases, they negotiated the Peace of Reuil on April 1, 1649.

A second phase, The Fronde of the Princes, led by King Louis XIV’s disenchanted cousin, The Great

Conde, occurred between January, 1650 and September, 1653 (the Fronde, 2013). The Great Conde

was a military leader who had sided with the government in the first Fronde. Realizing that he

was not going to gain political power, he decided to lead a second round of uprisings. After Conde

was arrested on January 18th, 1650, his friends led more provincial uprisings and were able to bring

about Conde’s release and Mazarin’s dismissal (the Fronde, 2013). Queen Anne strategically caused

a divide between the Frondeurs and despite aid from Spain, Conde lost the support of the bour-

geoisie and never did receive Parlement ’s backing. Conde left his Parisian barricade on October

13th, 1652, the young King Louis XIV returned eight days later, and Mazarin followed victoriously

on February 3rd, 1653 (the Fronde, 2013). The Fronde not only revealed the lack of leadership of the

Parlement and selfishness of the nobility, it also eliminated their role as a ruling party with the King.

CHAPTER 1. THE PLACE 3

There would not be another challenge of the monarchy’s power until The French Revolution of 1789.

The Fronde had a lasting effect on King Louis XIV. Having witnessed all the turmoil as a boy, he

never forgave Paris, nobility, or common-folk. With the end of the Fronde, Louis the Grand Monarch

came in to absolute power as a teenager; he would continue down this extravagant, arrogant path

during the rest of his reign, going so far as to proclaim “L’etat, c’est moi,” which in English is “I

am the state” (France, 2013).

1.2 Foreign Wars

France was at war with Spain for much of Pascal’s life. France had entered into the Thirty Years’

War in 1635 by declaring war on Spain. The war officially ended in 1648 with the Peace of West-

phalia, but the conflict between the Spanish and the French would continue under the premise of

the Franco-Spanish war, which also ended with the French being victorious in 1659 (France, 2013).

Under Louis XIII, armies were not the rigorous groups they are known as today. The men wore

whatever uniform they pleased and were not a regimented group. Many French noblemen who were

not the firstborns that would inherit their familial property joined the army above the common

folk and were fast tracked to become officers. Each commander was able to set their own rules and

procedures for their privately-financed company. Soldiers would often float between companies and

were more faithful to their commander than to their country (Banker, 2009). However, when Louis

XIV took control, the men were all to be dressed alike and stand in straight lines (Blashfield, 1890).

1.3 Culture

Salons served as the gathering places of the educated who dicussed literature, language, music, sci-

ences, and sometimes just gossip. Father Marin Mersenne hosted a mathematical salon that Etienne

and Blaise Pascal attended. Baroque music, especially the operas of Claudio Monteverdi, were in

CHAPTER 1. THE PLACE 4

vogue (Burkholder, J.P. & Palisca, C.V., 2006, p. 397). Demonstrating the French commitment to

the arts, young boys were castrated before puberty in order to preserve their higher vocal range.

Making a castrato was a risk for parents as they could never guarantee that the youngster’s vocal

ability would translate to talent as an adult (Burkholder, J.P. & Palisca, C.V., 2006, p. 433). Those

that did end up as beautiful singers with a brilliant sound and powerful voices were treated as

celebrities. This tradition continued for a few centuries until the last surviving castrato, Alessandro

Moreschi (1858 -1922), gave his last concert. Today, the parts are sung by women, male contraltos,

or transposed down an octave and sung by tenors or baritones.

Chapter 2

The Person

This chapter explains the influences and factors that led Pascal to his mathematical discoveries, as

well as his religious devotion. Much of what is known about Pascal comes from his letters to family,

friends, and acquaintances and what others have written about him. His father Etienne Pascal and

Father Marin Mersenne would pave the way for his mathematical success, providing avenues of both

study and presenting his findings. Pascal, a Jansenist, devoted most of the last eight years of his life

to his religion after a dream in which he believed God spoke to him. Many mathematicians speak of

what Pascal’s brilliant mathematical mind could have discovered (including infinitesimal calculus) if

he had not devoted so much of his short life to religion. (Rogers, 2003, p. 17). However, disciples of

the church may have wished Pascal had denounced his scientific endeavours much sooner to support

their beliefs.

2.1 1623-1639

Blaise Pascal (1623-1662) was the third born into an upper middle class French family. His father,

Etienne, a local judge and hobby mathematician, took over the education of Blaise and his two

daughters after his wife died in 1625. Blaise was a nervous, sickly child (Burton, 2011, p. 447), who,

thanks to his father’s unorthodox views on education, never attended school and was allowed to

sleep in. The original education plan was to prohibit mathematics education until his son reached

fifteen (Burton, 2011, p.447). Legend has it that a 12-year old Blaise made the conjecture that the

5

CHAPTER 2. THE PERSON 6

sum of the angles in a triangle is 180◦and when Etienne caught him trying to prove it, Etienne

revised his strategy and provided Blaise with Euclid’s Elements (Burton, 2011, p. 447).

Salons in 17th century France often focused on literature and the arts, but Father Marin Mersenne

created one with a scientific concentration. The “academy” (Burton, 2011, p. 448) held weekly

meetings, which both Etienne and Blaise Pascal regularly attended. At 16, Pascal brought forward

to the group Essai pour les coniques, which in one page contained many theorems, including the

mystic hexagon theorem [Figure 2.1]. This theorem says that “if a hexagon is inscribed in a circle,

the three points of intersection of pairs of oposite sides lie on a straight line” (Burton, 2011, p. 448).

Pascal’s work on conics, published in 1640, led to at least four hundred propositions, including how

Figure 2.1: Mystic Hexagon. Burton, 2011, page 448.

to construct a tangent to a conic at any given point (Burton, 2011, p. 449). Most mathematicians

heralded the young genius’ work, but Descartes unenthusiastically replied to Mersenne in a letter

that he could not “pretend to be interested in the work of a boy” (Burton, 2011, p. 449).

2.2 1640-1651

In 1639, Etienne Pascal was appointed by the government to manage the tax records in the province

of Normandy straight. This job required many hours of work doing repetitive calculations. Blaise,

wanting to help his father, realized that since precise calculations were necessary, some sort of

clockwork machine would be useful. Ater two years, the Pascaline was a shoe box sized machine

that used eight dials to add numbers with a maximum of eight digits (Bellis, 2013). Unfortunately,

CHAPTER 2. THE PERSON 7

the excessive friction of the gears and the high selling price of 100 livres limited the Pascaline to

being a curiousity, rather than a useful calculator [Figure 2.2].

Figure 2.2: The Pascaline. Bellis, 2013.

In 1646, Etienne Pascal fell and broke a leg (Rogers, 2003, p. 9). Two local bone-setters, the

Dechamps brothers moved in to the Pascal’s Rouen home to help heal the leg. As Jansenist dis-

ciples, they had a more profound affect on the family than they did on Etienne’s health. After

providing spiritual guidance literature to Blaise, the brothers converted him and then the rest of his

family to a more rigorous from of devotion than was their current practice. The effect on Jacqueline

Pascal was so tremendous that she was ready to become a nun. Inhibited by her father for the time

being, she would join the convent after his passing.

During the height of the Fronde, the Pascals fled Paris to live in Clermont with the eldest daughter,

Gilberte, and her husband, Florin Perier for a year (Rogers, 2003, p.12). Pascal would return to

their home in 1662 to die while under the care of his doting sister.

2.3 1651-1654

When Etienne Pascal died in 1651, Blaise was left to his own devices, and in Burton’s (2011)

words, entered into a “profane period” (p. 450). This was the time when he conducted pressure

experiments with gases and liquids, corresponded with Fermat to create the basis for probability

theory, and made numerous discoveries from the Arithmetic Triangle. Pascal’s “zeal for scientific

activity” (Burton, 2011, p. 450) made him an intellectual celebrity in the French society. The

Arithmetic Triangle is the focus of the final chapter of this paper, but to show some of Pascal’s

CHAPTER 2. THE PERSON 8

intuitive thinking, the Problem of Points is included here.

2.3.1 De Mere’s Problem of Points

Gambler and French noble Antoine Gombaud, Chevalier de Mere, posed to Pascal some gaming

questions in 1654, which led to the now famous interchange of letters between Pierre de Fermat of

Toulouse and Blaise Pascal. One of the things De Mere wanted to know was how many rolls of

two dice it would take to have a greater than 50% chance of rolling a double six. By De Mere’s

mathematical calculations, he would need 24 rolls. However, his experimental trials had always

proven to require 25 rolls. His thinking involved first finding the probability with rolling one die.

There is a 16 chance of rolling a 6 and a 5

6 chance of not rolling a 6. Throwing the same die twice

means there is a 56 ∗

56 =

(56

)2chance of not rolling a 6 on either of the rolls. Following the same

train of thought, there is a(56

)nof not rolling a 6 in n turns. Rolling a 6 at least once would have a

probability of 1−(56

)n. In order to have a better than 50% chance of rolling at least one 6, n must

be 4 or greater. From this, De Mere knew that he could safely bet on rolling at least one 6 if he had

four rolls and more often than not, he would make money.

When throwing two different coloured dice, there are 36 distinct possibilities. De Mere used the

“ancient, but mistaken gambling rule” (Burton, 2011, p.456) of n required trials to have a better

than even chance of success on one die divided by N options on one die multiplied by the total

outcomes of rolling two dice. Hence 46 ∗ 36 = 24 rolls.

Pascal was quickly able to sort out De Mere’s mathematical contradiction and get his experimental

answer of 25 rolls. When throwing two dice, there is one way to roll a double six and 35 ways to

not roll a double six. When throwing n pairs of dice, there is a(3536

)nprobability of not throwing

a double six and thus a 1 −(3536

)nof throwing a double six. Substituting in n = 24 provides a

probability of 0.4914 while n = 25 yields the desired better than even probability of 0.5055.

CHAPTER 2. THE PERSON 9

2.4 1655-1662

As much as Blaise Pascal had immersed himself in mathematics and science in the previous three

years, the subsequent years would be devoted to religion. On November 23, 1654, Pascal had a

near-death experience that he regarded as a sign from God to “consecrate his talents to the Chris-

tian faith” (Burton, 2011, p. 450). Pascal believed God spoke to him in a two hour vision, and

after regaining full consciousness, wrote down his revelation. This document, known as “Pascal’s

Talisman” was sewn into the lining of his clothing, near his heart, and found after he died (Latiste,

1911). When Pascal’s niece, Marguerite Perier’s eye abcess was cured by a relic of the Holy Thorn,

he became even more pious (Rogers, 2003, p. 17). Latiste (1911) exemplified Pascal’s devotion when

describing that he “wore a cincture of nails which he drove into his flesh at the slightest thought of

vanity.”

Pascal, a Jansenist, defended his religion in a series of pamphlets, Les Lettres Provinciales, against

what he felt to be the hypocritical Jesuits. In order to avoid being caught committing the crime of

ridiculing religion, the pamphlets were published under the pseudonym Louis de Montalte, which

was chosen specifically to enrage King Louis XIV. In the beginning Pascal adopted “the persona of

a concerned but bemoused outsider, who sets out to explain to a friend in the provinces what is

really going on” (Rogers, 2003, p. 15). The first letters spoke highly of Antoine Arnauld, a Jansenist

leader in Port-Royal. After Arnauld’s expulsion, the next letters mock the Jesuit’s eagerness for

converts at the expense of proper Christian ethics. The final letters answer the Jesuit’s rebuttals

to the earlier letters and contnue to defend his earlier accusations. Pascal had begun to drop his

persona in these final letters which perhaps led to their abrupt cessation. Twelve thousand copies of

the Provincial Letters made their way into the hands of educated people in every corner of France

(Burton, 2011, p. 451). Considering that the University of Ottawa (2013) estimates the literate

portion of the population was 5% and the population was around twenty million, there was about

one copy for every 83 educated people.

Pascal began to write an apology for the Christian Religion, but was unable to finish it before he

CHAPTER 2. THE PERSON 10

died. Leaving behind a collection of apparently random jot notes, the thoughts were later organized

and published as Pensees de M. Pascal sur la Religion. Pascal wrote quips based on math, language,

art, humanity, nature, and then the bulk of the work focused on religion. Like the Provincial Letters,

Pascal clearly felt his religion was the superior. “The Church has three kinds of enemies: the Jews,

who have never been of her body; the heretics, who have withdrawn from it; and the evil Christians,

who rend her from within.” (Pascal, trans. 2013). Pascal even applied his mathematical skills to

religion, creating what is now known as “Pascal’s Wager,” which assigns odds to whether to believe

in God or not.

If I wager for and God is - infinite gain;

If I wager for and God is not - no loss.

If I wager against and God is - infinite loss;

If I wager against and God is not - neither loss nor gain (Latiste, 1911).

As wagering against God gave one the possibility of losing everything, according to Pascal’s logic,

one was much better off gambling and either infinitely gaining or losing nothing by believing in God.

In 1658, Pascal made one more short reversion into the field of mathematics. According to Burton

(2011), while suffering insomnia at the hands of a toothache, Pascal began to take his mind off of the

pain by considering the cycloid curve, the path traced by a point on the circumference of a rotating

wheel [Figure 2.3]. When the pain began to subside, Pascal interpreted it as divine permission to

continue exploring the “Helen of Geometry” (Burton, 2011, p. 452). In eight days, he solved sev-

eral of the problems associated with the cycloid, including finding the area under one arch and the

volume of the shape created when the curve is rotated around an axis. The results were published

under Amos Dettonville, an anagram of Louis de Montalte. Burton (2011) speculates that Pascal

did not use his own name as he did not want society to know he had reentered the scientific field

that he had previously abandoned in favour of sacred pursuits (p. 453).

CHAPTER 2. THE PERSON 11

Figure 2.3: The Cycloid. Burton, page 452.

J.J. O’Connor and E.F. Robertson (1996) claim that after a life of poor health, Pascal died at the

age of 39, when a malignant growth spread from his stomach to his brain.

Chapter 3

The Problem: The Arithmetic

Triangle

This chapter delves into the history of the arithmetic triangle and the nineteen consequences that

Pascal observed from it. Pascal laid the foundations for proof via mathematical induction in verifi-

cation of the twelfth consequence. As Pascal discovered so many significant features of the triangle,

today, it often appears under the name of “Pascal’s Triangle.”

3.1 History of the Arithmetic Triangle

Although commonly called “Pascal’s Triangle,” Blaise Pascal was not the first to discover or publish

this triangular arrangement of binomial coefficients. Jovanovic (2004) suggests that Pascal’s Tri-

angle made its way to Europe from China via Arabia. He identifies that Yang Hui in China knew

of the first six rows of the triangle as early as 1261, although he attributes his knowledge of it to

the eleventh century mathematician Jia Xian. Chu Shih-Chien published eight rows in 1303 in The

Precious Mirror of the Four Elements (Burton, 2011, p. 457). Omar Khayyam, the Arab author of

Rubaiyat, also knew of the triangle in the 11th century and the Persian Al-Tusi created a triangle

of twelve rows in the 13th century.

The first European arrangement was published in 1527 by Peter Apian on the title page of Rechnung

12

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 13

(Burton, 2011, p. 457). In 1544, Michael Stifel published seventeen lines in a vertical formation in

Arithmetic Integra noting that more lines could easily be created [Figure 3.1].

Figure 3.1: 1544: Michael Stifel’s version. Jovanovic, 2004.

According to Jovanovic (2004), in 1591, Francois Viete began assigning Latin names to the columns,

which would continue to be used in the next century by Fermat, Pascal’s correspondent. The En-

glish translation for the column with numbers 3, 6, 10, 15, . . . is triangular numbers, as with any of

these numbers of points, an equilateral triangle can be formed [Figure 3.2]. Using a similar figurate

reasoning, the next columns are “pyramidales”, “triangulo-trianguli”, and “triangulo-pyramidales.”

Figure 3.2: Triangular Numbers. Weisstein, 2013.

In 1636, Father Marin Mersenne, the close friend of Etienne and Blaise Pascal, published Harmon-

icorum Libri XII, with the following table included [Figure 3.3].

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 14

Figure 3.3: 1636: Mersenne’s Table. Jovanovic, 2004.

It is very likely that Pascal saw Father Mersenne’s table and it may have sparked his interest in

exploring it further. Although many others had published the table before Pascal, his Traite du

Triangle Arithmetique, printed in 1654 and distributed in 1665, was groundbreaking enough to

forever link his name with the triangle. Pascal created a labelling system of parallel (horizontal)

and perpendicular (vertical) ranks. Cells along the same positively sloped diagonal were in the same

base. The nth base, Pascal recognized, was the coefficients for the binomial expansion of (x+ y)n−1.

The negatively sloped diagonal was an axis of symmetry. Cells in the same base, which are the same

distance from the axis of symmetry, were called reciprocals. Pascal’s rule for creating the triangle

was that each cell is equal to the sum of the cell above it in its perpendicular rank and the cell to

the left of it in its parallel rank. The letter Z in the upper left corner was called the generator and

although 1 is the most frequently used value, any other number could be used to create the triangle

[Figure 3.4].

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 15

Figure 3.4: 1654: Pascal’s Triangle. Burton, 2011, p. 458.

From the triangle, Pascal observed 19 properties, which he called consequences when he published

his treatise. The current day notation for binomial coeffiecients of(nr

)was not to become convention

until 1827 and n! was introduced in 1808 by Christian Kramp (Burton, 2011, p. 459-460). Thus,

although Pascal would not have written the following consequences in this modern day notation, for

sake of understanding, it will be used in some explanations. For example, the generating rule that

creates the triangle by adding the cells immediately above and to the left of the desired cell would

be represented as(nr

)+(

nr+1

)=(n+1r+1

)(Brown, 2011).

3.2 Pascal’s Nineteen Consequences of the Arithmetic Triangle

3.2.1 Consequence I

In every arithmetic triangle, all the cells of the first parallel rank and of the first perpendicular rank

are equal to the generator (Pengelley, n.d.). As each of the cells in the first parallel and perpendicular

ranks have no cells that precede them other than the generator, they must be equal to the generator

plus zero, hence just equal to the generator.

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 16

3.2.2 Consequence II

In every arithmetic triangle, each cell is equal to the sum of all the cells of the preceding parallel

rank, going left from its own perpendicular rank to the first, inclusive (Pengellely, n.d.).

Using the basic definition that creates the triangle and Consequence I, Consequence II is easily

proven. Using the notation of Pascal’s triangle as above, it would need to be shown that some cell,

for example θ, equals the sum of the row above it: G +σ + π.

By the basic definition, θ = ψ + π and ψ = G + σ, so by substitution θ = G + σ + π as desired.

This process of substituting the definition several times works for all cells of the triangle. In today’s

notation,(nr

)=(n−1r−1

)+(n−2r−2

)+ . . . +

(n−r−1

0

). This consequence is often called the hockey stick

pattern, due to its shape (Bogomolny, 2013).

3.2.3 Consequence III

In every arithmetical triangle, each cell is equal to the sum of all the cells of the preceding perpen-

dicular rank, going up from it’s own parallel rank to the first, inclusive (Pengelley, n.d.).

This proof would be symmetrical to Consequence II and in today’s notation would appear as(nr

)=(n−1r−1

)+(n−2r−1

)+ . . .+

(r−1r−1

). This consequence is also called the hockey stick pattern.

3.2.4 Consequence IV

In every arithmetical triangle, each cell exceeds by the generator the sum of all the cells in the

rectangle above and to the left of it (Pengelley, n.d.).

The proof of this uses the same substitution and definition process as the proof for Consequences II

and III. For example,(42

)=(21

)+(20

)+(11

)+(10

).

3.2.5 Consequence V

In every arithmetical triangle, each cell is equal to its reciprocal (Pengelley, n.d.).

Each of the extremes are equal to the generator and thus must be equal to each other. Using the

basic definition of the triangle, cell F = E + C and cell ω = R + C. Now it needs to be shown

that the reciprocals R = E. By the definition, R = λ + θ and E = B + D. As λ and D are both

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 17

extremes, they are equal to both the generator and each other. Wanting to show θ = B, by the

definition, θ = π + ψ and B = ψ + A. Again both A and π are equal to the generator and each

other. Finally, the equations both contain the same variable, ψ. Thus, θ = B, R=E, and F = ω.

This substitution process could be used for any given cell. In today’s notation, this consequence is

represented as(nr

)=(

nn−r

), a commonly used rule when studying combinations.

3.2.6 Consequence VI

In every arithmetic triangle, the nth parallel rank has equal cells in the same order to the nth

perpendicular rank (Xavier University, n.d.).

As they are composed of reciprocals, by Consequence V, the parallel and perpendicular ranks must

be equal.

3.2.7 Consequence VII

In every arithmetic triangle, the sum of the cells of any positive sloping base is double the preceding

base (Xavier University, n.d.).

As the extremes will both be equal to the generator in both bases, it is only the inner cells that

need to be considered. For example, compare the base Aψπ to DBθλ. It needs to be shown that

D + B + θ + λ = 2 ∗ (A + ψ + π). By the definition, B = A + ψ and θ = ψ + π and the extremes

A = D and π = λ. Substituting in for D+B+ θ+λ provides A+A+ψ+ψ+π+π = 2(A+ψ+π)

as desired. Again, this proof could be extended for any desired base.

3.2.8 Consequence VIII

In every arithmetic triangle, the sum of the cells of each base is equal to Z ∗ 2n−1 where n is the nth

base and Z is the generator (Pengelley, n.d.).

By Consequence VII, each base is double the previous, giving rise to Consequence VIII. In modern

day notation, 2n =(n0

)+(n1

)+(n2

)+ . . .+

(nn

)(Burton, 2011, p. 460).

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 18

3.2.9 Consequence IX

In every arithmetic triangle, each base diminished by the generator is equal to the sum of all the

preceding cells (Xavier University, n.d.).

Call the sum of the first base Z, where Z is the value of the generator. By Consequence VII, the

sum of the second base is 2Z, the third is 4Z, and the fourth is 8Z. It is easy to see that the sum

of the first three bases is 7Z, which is one generator less than the fourth base’s sum of 8Z. As the

triangle is based on the double progression, this consequence will always hold true, no matter how

many bases in.

3.2.10 Consequence X

In every arithmetic triangle, the sum of as many contiguous (adjacent) cells from the same base,

beginning with a generator at an extremity, is equal to as many cells as the preceding base plus as

many cells of the preceding base minus one (Xavier University, n.d.).

For example, summing the first three cells of the fifth base, H + E + C would equal the first three

cells of the fourth base, plus the first two cells of the fourth base, i.e. D+B+ θ+D+B. Using the

generator value of 1, H +E +C = 1 + 4 + 6 = 11 and D+B + θ+D+B = 1 + 3 + 3 + 1 + 3 = 11.

The proof also comes from the definition, H+E+C = D+D+B+B+θ = (D+B+θ)+(D+B).

Again, the beginning contiguous cells could be from any base and the proof would hold.

3.2.11 Consequence XI

Each cell on the line of symmetry (also called a “cell of the divide”) is double the cell preceding it

in both the parallel and perpendicular ranks (Xavier University, n.d.).

The horizontal and vertical cells that precede the cells of the divide are reciprocals and by Conse-

quence V are even. By the definition that creates the triangle, those two equal reciprocals are added

to created the cells of the divide, and thus the cells of the divide are double the reciprocals that

precede it.

The first eleven consequences have given equalities. The next seven are proportion based, all stem-

ming from Consequence XII.

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 19

3.2.12 Consequence XII

In every arithmetic triangle, if two cells are contiguous in the same base, the upper (superior) is to

the lower (inferior) as the number of cells from the upper to the top of the base is to the number of

cells from the lower to the bottom ofthe base, inclusive (Burton, 2011, p.461).

(n

r+1

)(nr

) =n− rr + 1

(Burton, 2011, p. 460).

This consequence is of historical importance because its proof begins to use mathematical induction.

It is clear to see how Pascal would have come up with this rigorous proof technique as many of the

other consequences end with the concept of “this will hold true for which ever base/rank/cell is

used.” Using the given notation, if E and C were chosen as the contiguous cells, then it needs to be

shown E : C :: 2 : 3 as there are two cells below and including E (H, E) and three cells above and

including C (C, R, µ).

The proof begins by proving the base case by looking at the second base, φ : σ :: 1 : 1. As both φ

and σ equal the generator, they are equal and have a one to one proportion.

Next, Pascal made the realization that if a proportion is found in one base, it will be found in the

next base. Using the inductive hypothesis to prove the next stage allows the rule to be continued

on to infinity. So assuming the proportions for the fourth base (i.e. D : B :: 1 : 3, B : θ :: 2 : 2, and

θ : λ :: 3 : 1), the proportion for the fifth base (E : C :: 2 : 3) needs to be shown. As E = D + B

and C = B + θ by the definition, E : B :: D+B : B :: 1+3 : 3 :: 4 : 3 and C : B :: B + θ :: 2 + 2 :

2 :: 4 : 2. As E : B :: 4 : 3 and B : C :: 2 : 4, combining these provides E : C :: 2 : 3 as desired. �

3.2.13 Consequence XIII

In every arithmetic triangle, with two contiguous cells in the same perpendicular rank, the inferior

is to the superior as the the base of the superior is to the parallel rank of the superior (Xavier

University, n.d.).

For example, F : C :: 5 : 3 as C is in the 5th base and 3rd parallel rank. By the definition, F = E

+ C and from the last proof it is known that E : C :: 2 :3. Thus F : C :: E + C : C :: 2 + 3 : 3 ::

5 : 3 as desired.

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 20

3.2.14 Consequence XIV

In every arithemetic triangle, two contiguous cells in the same parallel rank, the further left cell is

to the further right cell as the base of the fiurther right cell is to the perpendicular rank of the further

right cell (Xavier University, n.d.).

With a similar proof as the last consequence, it needs to be shown that F : E :: 5 : 2 as E is in the

5th base and 2nd perpendicular rank. By the definition, F = E + C and E : C :: 2 : 3. Thus F : E

:: E + C : E :: 2 + 3 : 2 :: 5 : 2 as hoped.

3.2.15 Consequence XV

In every arithmetic triangle, the sum of the cells of a parallel rank is to the furthest right cell in that

parallel rank as the base of that cell is to parallel rank of that cell (Xavier University, n.d.).

For example, using the triangle that only goes to the fourth base, i.e. GDλ and the second parallel

rank, φ + ψ + θ : θ :: 4 : 2. Using the definition a few times, φ + ψ + θ = C . From Consequence

XIII, C : θ :: 4 :2 as desired. Again using induction, this proof could be also built on all other

parallel ranks and bases.

3.2.16 Consequence XVI

In every arithmetic triangle, the sum of any parallel rank is to sum of the rank above it (inferior

rank) as the number of the inferior rank is to the number of cells it contains (Xavier University,

n.d.).

For example, consider the triangle that extends to the fifth base, GHµ, and the third parallel rank,

which will include ABC, compared to its inferior rank of φψθR. So A+B+C : φ+ψ+ θ+R :: 2 : 2.

By Consequence II, φ+ ψ + θ+R = ω and A+B +C = F . By Consequence V, ω = F , giving the

1 :1 or 2 : 2 ratio as predicted by Consequence XVI. Consequence XII could also have been used in

this last step, providing the same answer of ω : F :: 3 : 3.

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 21

3.2.17 Consequence XVII

In every arithmetic triangle, any cell that is added to all cells of its perpendicular rank, is to the

same cell added to all cells of its parallel rank, as the number of cells in the perpendicular rank is to

the number of cells in the parallel rank (Xavier University, n.d.).

Starting with cell B, the perpendicular rank includes three cells B, ψ, and σ, to a sum of 6, and

the parallel rank includes two cells B and A, to a sum of 4. As predicted by the consequence, 6 : 4

:: 3 : 2. For a proof, by Consequence III, B + ψ + σ = C and by Consequence II, A + B= E. By

Consequence XII, C : E :: 3 : 2 as desired.

3.2.18 Consequence XVIII

In every arithmetic triangle, when two parallel ranks equally distant from the extremities, the sum

of the one is to the other as the number of cells of the one is to the number of cells of the other

(Xavier University, n.d.).

For example, choose the triangle that extends to the fifth base, GHµ. Parallel ranks 2 and 4 are

equally distant by a value of one from the extremities. In the second base, there are four cells

and when the generator is 1, φ + ψ + θ + R = 10. In the 4th base, there are two cells and

D + E = 5.Therefore the ratio is 10 : 5 :: 4 : 2.

The proof lies in the fact that by Consequence VI, the sum of the nth parallel rank is equal to the

sum of the nth perpendicular rank. In the above example, n = 2. As the perpendicular rank will

intersect at the outermost cell with the other parallel rank, by Consequence XVII, Consequence

XVIII is proven true.

3.2.19 Consequence XIX

In every arithmetic triangle, when two cells of the divide are contiguous, the larger is to the quadruple

the smaller, as the number of the base of the smaller is to the number of the base of the smaller

increased by one (Xavier University, n.d.).

As an example, consider cells ψ = 2 in the third base, and C = 6 i.e. 6 : 4*2 = 8 :: 3 : 4.

By Consequence XI, each cell of the divide is double the cell above it or to the left of it. Considering

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 22

the above example, ψ = 2∗σ and 2ψ = 4σ. So 4σ : ψ :: 2 : 1, and by Consequence XIV, θ : ψ :: 3 : 2.

Combining these two proportions, C : 4ψ :: 3 : 4.

3.3 Mathematical Induction

Mathematical Induction is a common method for proving statements involving the integers. It

requires the statement to be true for some integer n (often called the base case and letting n = 1)

and if assumed true for n, then proving it is also true for n+1. This rigorous method is often likened

to climbing a ladder: if one can get on the first rung and each rung has a rung above it, one could

climb the ladder infinitely far. Hence, when proving a statement via the principle of mathematical

induction, it holds true for all positive integers. Burton (2011) illustrates the concept with;

An infinite row of dominoes, all standing on edge and arranged in such a way that when

one falls it knocks down the next in line. if either no domino is pushed over (if there is

no basis for the induction) or the spacing between them is too large (the induction step

fails), then the complete line will not fall (p. 461).

Pascal was the first to use this process in the proof of Consequence XII in his Triangle Arithmetique,

although Francesco Maurolico had described “reasoning by recurrence” in the sixteenth century

(Burton, 2011, p. 463). A modern inductive proof of Consequence XII would look like the following:

Proof. By induction, this proof will show that(nr)

( nr+1)

= r+1n−r

Base Case: n = 1, r = 0 (10

)(11

) =1

1=

0 + 1

1− 0

Inductive hypothesis: ∀n, r ∈ Z+, n > r, assume(nr)

( nr+1)

= r+1n−r . It now needs to be shown that

(n+1r

)(n+1r+1

) =r + 1

n+ 1− r

CHAPTER 3. THE PROBLEM: THE ARITHMETIC TRIANGLE 23

By the generating rule for the triangle,

LHS =

(nr

)+(

nr−1

)(n

r+1

)+(nr

)Factoring

(nr

)out of both the numerator and denominator,

=

1 +

[( nr−1)(nr)

][

( nr+1)(nr)

]+ 1

Using the inductive hypothesis for each of the brackets,

=1 +

[r

n−(r−1)

][n−rr+1

]+ 1

Simplifying,

=n−r+1+rn−r+1

n−r+r+1r+1

=r + 1

n+ 1− r

By the Principal of Mathematical Induction, ∀n ∈ Z+, n > r,(nr)

( nr+1)

= r+1n−r .

Conclusion

Although Pascal’s involvement with the field of mathematics was relatively short, it made great

strides and advancements that have helped probability theory and other areas become what they

are today. He discovered that the sum of the angles of a triangle is 180◦at a young age, convincing

his father to let him study Euclid’s Elements. From this research, Pascal was provided with a spring

board into conics, leading to his first published work, Essai pour les coniques. During his “profane

period” (Burton, 2011, p. 450), Pascal conducted scientific experiments with pressure and explored

the Arithmetic Triangle. His correspondence with Fermat led to many solutions for probability

problems. After an epiphany, Pascal’s narrow-minded devotion to the Jansenist faith allowed him

create two classic French literature pieces that are still referenced in the twenty-first century: Les

Lettres Provinciales and Pensees. In his abbreviated lifespan, with an even briefer time spent on

math, Pascal was able to answer many queries and notice new patterns that others had overlooked.

24

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