Blaise Pascal and the Pascal Triangle


Pascal’s triangle, based on the expansion of any binomial expression (x+y)ⁿ, can be used for several elements within a polytope in the modern world. Some examples include significant ties to number theories and the Fibonacci sequence. In addition, pascal triangle numbers can be the coefficients of perfect squares in polynomials. For example, the coefficient of its first-row expansion is (x+y)⁰=1, its second-row expansion is (x+y)¹=1x+1y, and its third-row expansion is representative of (x+y)², which equals 1x²+2xy+1y². To this day, the binomial theorem is taught in algebra for determining the permutations and combinations of probabilities. As an early pioneer of the binomial theorem, Blaise Pascal was one of the most influential mathematicians and philosophers during the 17th century; he developed probability theory with other mathematicians and statisticians, theorized the conic sections, and discovered the pascal triangle. Although his involvement in ethics, politics, and existentialism, along with his constant defense of catholicism made him a controversial figure, his contribution to Pascalines (calculating machines named after him) established him as one of the two inventors of the mechanical calculator. Pascal also conducted the first Torricellian, demonstrating that air pressure varies from higher to lower altitudes, where later, he established the hydraulic press. Perhaps he could have accomplished much more in his 39 years, but radical religious beliefs in Jansenism made this possibility short-lived. Soon after the major publication on probability theory, the notion of expected value was introduced. This soon led to his invention of the roulette table, alongside other perpetual motion machines. However, after this theory, Pascal started focusing on catholicism and existentialism in general, where he published the Pascal Wager in the justified belief of God and virtuous life. Pascal’s most regarded theological work, Pensées (\”Thoughts\”), along with his theorems and inventions, is what cemented him as one of the most influential persons in the 17th century.


The following article looks at the accomplishments of Blaise Pascal and his inventions as arguably the most influential mathematical innovations in the 17th century. In light of the 398th anniversary of his birth, this article will also discuss his impact on modern sciences and his involvement in the latter half of his life.

Blaise Pascal was born on June 19th, 1623 in Clermont-Ferrand, which was in France’s Auvergne region, to Etienne and Antoinette Pascal. From a young age, Pascal showed a prodigious aptitude for mathematics and science, to the extent in which people mistook his work for his father’s. His essay on Essai pour les coniques demonstrated the relation points of three intersections if a hexagon is inscribed in a circle. Pascal was only 16 when he wrote the theorem, which was so surprising that Descartes and Mersenne both suggested it was a product of his father. The Essay on Conic Sections transferred a 3-D object onto a 2-D field, which was novel in geometry at the time. Since his father was a tax collector, 18-years-old Pascal decided to construct a mechanical calculator to ease the calculations of taxes owed and paid. The Pascaline (as it was named after him), at that time, was only a status symbol since it was expensive to obtain. Later, it was transferred into our modern calculators when computer engineering became prevalent. The Pascaline had multiple movable parts, including dials, which made it hard for people to use. Although it was the first of its kind, the Pascaline had multiple glitches. Thus, even with 50 prototypes, the Pascaline was not as popular as the abacus.

At the height of his life, Pascal discovered Pascal’s Triangle and its various applications in mathematics and statistics, including its ability to solve combinations. In addition, the modern probability theory was the first reasoning on an expected value (which is used when two people have equal chances of winning each round). Later, he focused on theological beliefs and published multiple religious works documenting his belief in Jansenism.


The pascal triangle is arguably the most useful discovery by mathematicians and statisticians alike. Through the triangle, Pascal was able to derive the binomial theorem and solve complex equations containing variable coefficients. From Pascal’s triangle, we can simply derive exponents of 11, squares between the second diagonal and the sum of the number next to it, and the Fibonacci sequence (by adding diagonally from the top right side of the triangle). The in-depth approaches in Pascal’s Triangle include the probability of any combination. For example, tossing a coin four times will not always land the same combinations, the results skewing towards the center of Pascal’s triangle. There is one combination that will give four heads, and there are four combinations that give three heads and one tail. At the center of this “board” is six combinations, showing two heads and two tails as the most likely combination. Meanwhile, the right side combination contains four combinations that will give one head and three tails and only one combination with all tails. This represents the fifth row in Pascal’s triangle, “1,4,6,4,1”. The triangle also shows how many combinations of objects there are possible. These represent the “choose problems”, where you calculate the number of ways you can choose from, and in Pascal’s Triangle, all the numbers are written in combinations. The Quincunx, which was invented as a direct result of Pascal’s Triangle, represents a modern-day Plinko Table, where the possibilities of the ball bouncing to the bottom of the triangle is where their probable position will be. For example, the fifth row (1,4,6,4,1) would have a ⅜ chance of dropping into one of the middle slots. Consequently, Pascal’s triangle can simplify many situations involving complex probability or counting, making this discovery critical in the face of the math world.


The problem of points, also known as the division of stakes, is a problem in probability theory. Pascal had the first explicit reasoning on the expected value of the game. The Problem of Points derives from a game of change where two players have an equal chance of winning each round. The players contribute 50-50 to the prize and agree that the first player who wins a certain number of rounds will collect the entire prize. The problem now concerns the possibility of external situations before the game has ended, which ultimately led to questions on how one divides the pot fairly. Pascal and Fermat provided a solution to the problem in 1654 when Chevalier de Mere posed it to Blaise Pascal. What Pascal and Fermat sought did not necessarily depend on the history of the interrupted game but the percentage of the games they have won. For example, if a player has a lead of 9-4 in a game to 10, they have the same chances of eventually winning as a player with an 18-8 lead in a game to 20. Fermat reasoned the game would have won in r + s – 1 additional round, where r and s are the amount of points player 1 and player 2 need to win respectively. Therefore, the total round would have 2ʳᐩᔆ⁻¹ possible continuations. Although he was right, Pascal elaborated on his argument and considered it impractical if r + s – 1 is more than 10. Supposed the players had been able to play one more round before being interrupted, and the judges already decided how to fairly divide the stakes after one more round, this way, the judges would only need to calculate two possibilities with a fair division of the stakes, and they can split the difference between two future divisions evenly. Pascal was the first mathematician to introduce expected value in a direct application where Pascal’s step-by-step rule is significantly quicker than Fermat’s predicted outcome rule.


Pascal’s encounter with ethics, politics, and Catholicism transpired after an accident when he was 22. In 1646, Etienne, Blaise\’s father, suffered serious injuries from a fall and fractured his hip which left him bed-bound. Since the Pascal family never fully accepted local Jesuit ideas, this accident led to a change in the family\’s religious beliefs. After the accident, Etienne received medical care from the Deschamp brothers, and the Pascals came to live with them for three months. The two brothers were followers of Jansenism, which is a special sect of the Catholic Church. Their influence, believed to be related to Etienne\’s health trauma, led the whole family to convert. Pascal became passionately pious, and Sister Jacqueline became a nun.

Although this encounter with Jansenist theology is sometimes referred to as Pascal\’s first conversion, it is unlikely that he made the final decision on the insignificance of the mathematics and science work that marked his dramatic changes in the 1650s from being involved in mathematics and science to catholicism and politics.

1) Ethics and Politics

Pascal’s views of politics made him a controversial person in front of many. In his ‘Lettres provincials, he described the morality of Jesuits as ‘perniciously lax’. Pascal classified many human behaviors as obviously immoral behaviors—such as homicide, which is not justifiable as self-defense—and described them as contradicting \”natural light\”, \”common sense\”, or \”natural law\”

Pascal’s philosophical thinking on morality vaguely revealed his view on the law of God. According to God\’s law and its elements that survive the common beliefs of people all over the world, certain behaviors are inherently good or bad. Our moral obligations include not only the most famous examples, such as the obligation to avoid voluntary homicide, but also the “obligation to give alms from the surplus fairly to meet the general needs of the poor” per Cayetano’s words. Pascal agrees with Cayetano “as the rich have rights over their surplus and they control what is distributed to other stewards and to choose from those in need”.

Pascal\’s political theory is also governed by his description of human happiness. According to Pensees\’s 90th segment, \”desire and power are the sources of all our behavior; happiness leads to voluntary behavior, and violence leads to involuntary behavior\” (II, 570). Although the state of nature could determine human behavior before the fall of Adam, human relationships are now destroyed by desire and the power of one person over another. Being forced to obey those who exercise political power against us can be interpreted as punishment for our guilty country. This pessimistic explanation of political power and its possible abuse coincides with Luther and Calvin\’s explanation.

2) Catholicism and Existentialism

Pascal believed in the catholic faith all his life, and since he was interested in scientific and mathematical problems long after his commitment to Jansenism, it seems unfair to call his later years a betrayal of his scientific principles. Pascal was a lifelong Catholic whose personal conversion from tepid to unconditional belief was achieved not through rational arguments, but a life-changing mystical experience.

His greatest works don\’t seem to be solely masterpieces of French prose, however sterling defenses of the Christian faith. In his ‘Lettres provincials, he attacked the Jesuits and defended the Jansenists\’ demand for a return to morality and Augustine’s belief in divine grace.

Pascal is often included within the ranks of “existentialist” philosophers, but many argue he is undeserving of that title. If a shaping attribute of philosophical doctrine is an endorsement of Sartre’s maxim that “existence precedes essence,” then Pascal doesn’t qualify.


1) Fluid Dynamics

Pascal’s law, also called Pascal’s principle, is a statement claiming that if a fluid rests in a closed container, a pressure change in one part of the container will be transmitted (without loss of energy) to the other wall of the container. The pressure is equal to the force divided by the area on which it is compressed against. According to Pascal’s law, in which he demonstrated a simple diagram between the original force and expected force, if the second piston has an area 10 times the first, the force on the second piston is 10 times greater while pressure remains the same. Using Pascal’s principles, applications such as the hydraulic press were invented and used in inventions like the hydraulic brakes. Another point to this law was how Pascal discovered the pressure at the point where the fluid rests is the same from all directions, meaning that the pressure would be the same on all sides within the specific point. Taken from Pascal’s law, it states, “…when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container.” This law and the experiments shown below are key developments in the modern field of hydrodynamics. According to Pascal’s principle, the original pressure (P1) exerted on (A1) will produce an equal pressure (P2) on (A2). Again, because A2 has 10 times the area of A1, it will produce a force that is 10 times greater than the original force. Therefore, through Pascal’s principle, a small force exerted on a hydraulic press can apply 10 times the force used to lift a relatively heavy object, a car in this case.

2) Vacuum Stimulation

The second half of Pascal’s life was haunted by many grievances, putting him off of his scientific endeavors. However, after his conversion to Jansenism, Pascal reabsorbed himself into his scientific interests, beginning by testing the theories of Galileo and Evangelista Torricelli (a physicist who discovered the principle of the barometer). He amplified and reproduced experiments on atmospheric pressure through constructing mercury barometers and measuring air pressure, both of which took place at high altitudes (in Paris and on top of a mountain overlooking Clermont-Ferrand). These tests paved his studies in hydrodynamics and hydrostatics, where Pascal invented the syringe and created the hydraulic press through vacuum testing. His publications on the problems of the vacuum added to his reputable knowledge, where he composed treatises on the equilibrium of liquid solutions. These experiments led to potential papers on the weight and density of air and his book: Traité de l’équilibre des liqueurs et de la pesanteur de la masse de l’air (The Physical Treatises of Pascal, 1937). In his experimentation, he compared barometric observations made at different altitudes. From his experiments, he deduced the existence of a vacuum in the real world, which subsequently, provoked many controversies on the possibilities of the vacuum. His theory of “the vacuum within a vacuum” suggested vertical columns of air, its weight, and the absolute vacuum to be derived from barometric observations. Discoveries by Pascal’s vacuum stimulation led to the introduction of pump and seal equipment today.


During Pascals\’ last years, many of his family and close friends passed away, leading to stronger convictions in the Christian faith. Pascal wrote summaries of his religious conversion in a brief document titled “Memorial”. After 1654, he terminated his mathematical discussions and planned to dedicate his time to the other-worldly attitude of his devotion. The last few years of his short life were dominated by religious controversy, continual illness, and loneliness. During this time, he began to collect ideas in defense of the Catholic faith. He constructed a radical theological position to prove it was impossible to support genuine religious faith by reason, contradicting the popular opinion that religious faith was a gift from God. In late 1659, Pascal fell ill, where he rejected the administrations of his doctors by saying “Sickness is the natural state of Christians.” His last major achievement was inventing the first bus line, or carrosses a cinq sols, moving passengers within Paris with multiple seats and specified routes. Blaise Pascal passed away on August 18th, 1662, where he went into multiple convulsions and died the next morning. His last words were “May God never abandon me,” and he was buried in Saint-Étienne-du-Mont cemetery. After Pascal died, collaborators posthumously published his unfinished version of Entretien avec M. de Sacy and Pensées (Thoughts), his most famous theological book focusing on the thoughts he had when converting. Reportedly, he could only find consolation for his misery in his final years through religion since his extreme ill-health and loneliness remained after he converted to Jansenism.


Pascal’s Triangle has been documented as one of the most important discoveries in binomial sequences and combinations. Blaise Pascal would go on to find many theories regarding physics and math that can account for modern inventions like the mechanical calculator and the carriage. Given math was not as prevalent in the 17th century as today, Blaise Pascal’s contributions to mathematics were astronomical since it consisted of many modern theorems and laws. In conclusion, Pascal’s works on mathematics, physics, and religion will always be remembered, even centuries after his death.


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Jerphagon, Lucien. “Blaise Pascal.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 2021,

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