* The earliest extant Chinese illustration of ' Pascal's Triangle ' is from Yang Hui's book Xiangjie Jiuzhang Suanfa, published in this year, although knowledge of Pascal's Triangle existed in China by at least 1100.

Yang Hui was also the first person in history to discover and prove " Pascal's Triangle ", along with its binomial proof ( although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD ).

The earliest extant Chinese illustration of ' Pascal's Triangle ' is from Yang's book Xiangjie Jiuzhang Suanfa ( 详解九章算法 ) of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using " Yang Hui's Triangle " was invented by mathematician Jia Xian who expounded it around 1100 AD, about 500 years before Pascal.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

Pascal's work was so precocious that Descartes was convinced that Pascal's father had written it.

Gottfried Leibniz ( 1646 – 1716 ), building on Pascal's work, became one of the most prolific inventors in the field of mechanical calculators ; he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator.

Pascal's work built upon reports of the mercury tube experiment which had reached Paris via Marin Mersenne in 1644.

Pascal's religious conversion led him into a life of asceticism, and the Pensées was in many ways his life's work.

The concept ( but not the term ) " Pascal's Wager " stems from a portion of this work.

* Blaise Pascal's Lettres Provinciales is burned as a heretical work on the orders of King Louis XIV of France.

Extrapolating from these findings, Dr. Franz Hammer, another one of Kepler's biographers, challenged in 1957 the fact that Pascal had invented the mechanical calculator by announcing to the world that the drawings of this previously unknown calculating clock, predating Pascal's work by twenty years, had been found in these two letters.

Apian's work included in mathematics — in 1527 he published a variation of Pascal's triangle, and in 1534 a table of sines — as well as astronomy.

This work did not keep him from publishing a paper on Pascal's theorem in 1906.

Brilliantly written by Pascal, the Provincial Letters would not have been possible without the work of theologians from Port-Royal ; indeed, most of the arguments Pascal deployed were already to be found in Arnauld's Théologie morale des Jésuites, something which led the Jesuit Nicolas Caussin to reply to Pascal's perceived libel.

Pascal's use of wit, humor, and mockery in attacking existing institutions made his work extremely popular.

The notation was introduced by Andreas von Ettingshausen in 1826, although the numbers were already known centuries before that ( see Pascal's triangle ).

The arithmetical triangle — a graphical diagram showing relationships among the binomial coefficients — was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

In 2008, the system was brought up to a new level and the resulting language termed " Pascaline " ( after Pascal's calculator ).

He also sent to France his famous " formulary ", that was to be signed by all the clergy as a means of detecting and extirpating Jansenism and which inflamed public opinion, leading to Blaise Pascal's defense of Jansenism.

This concept was first formulated in a slightly extended form by French mathematician and philosopher Blaise Pascal in 1647 and became known as Pascal's Law.

SNOBOL4's programmer-defined data type facility was advanced at the time — it is similar to the earlier Cobol's and the later Pascal's records.

Some people believe that although Torricelli's experiment was crucial, it was Blaise Pascal's experiments that proved the top space really contained vacuum.

Pascal's Traité du triangle arithmétique ( Treatise on Arithmetical Triangle ) was published posthumously in 1665.

The triangle was later named after Pascal by Pierre Raymond de Montmort ( 1708 ) who called it " Table de M. Pascal pour les combinaisons " ( French: Table of Mr. Pascal for combinations ) and Abraham de Moivre ( 1730 ) who called it " Triangulum Arithmeticum PASCALIANUM " ( Latin: Pascal's Arithmetic Triangle ), which became the modern Western name.

Blaise Pascal's Écrits sur la grâce, based on what Michel Serres has called his " anamorphotic method ," attempted to conciliate the contradictory positions of Molinists and Calvinists by stating that both were partially right: Molinists, who claimed God's choice concerning a person's sin and salvation was a posteriori and contingent, while Calvinists claimed that it was a priori and necessary.

It was not until the beginning of the 20th century that scholars began to understand Pascal's intention.

( See, also, the monumental edition of his Oeuvres complètes ( 1964 – 1991 ), which is known as the Tercentenary Edition and was realized by Jean Mesnard ; this edition reviews the dating, history, and critical bibliography of each of Pascal's texts.

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri ( 1000 ) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

Pascal's Wager ( also known as Pascal's Gambit ) is an argument in apologetic philosophy which was devised by the seventeenth-century French philosopher, mathematician, and physicist, Blaise Pascal.

The most natural setting for Pascal's theorem is in a projective plane since all lines meet and no exceptions need be made for parallel lines.

File: Blaise pascal. jpg | Blaise Pascal ( 1623-1662 ): experimented with fluids, formulated Pascal's law in the 1650s stating that the pressure applied to a fluid taken in a closed container is transmitted with equal force throughout the container, proved that air has weight and that air pressure can produce a vacuum, namesake of the unit of pressure: the pascal ( Pa )

The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.

where the coefficients a < sub > i </ sub > in this expansion are precisely the numbers on row n of Pascal's triangle.

When divided by 2 < sup > n </ sup >, the nth row of Pascal's triangle becomes the binomial distribution in the symmetric case where p = 1 / 2.

This is the experiment where a vacuum is created in a mercury tube: Pascal's experiment had his brother-in-law carry a barometer to the top of the Puy-de-Dôme.

If the coefficients of Pascal's triangle are labeled C ( i, j ) and the coefficients of the Tetrahedron are labeled C ( n, i, j ), where n is the layer of the Tetrahedron, i is the row, and j is the column, then the relation can be expressed symbolically as:

The film ends as the other balloons in Paris come to Pascal's aid and take him on a cluster balloon ride over the city as the narrator states that other balloons took Pascal up to a magical world where balloons and children can be friends forever.

Each Pascal's m-simplex is a semi-infinite object, which consists of a semi-infinite series ( n ≥ 0 ) of finite ( m − 1 )- simplices, where m is the number of terms of a polynomial and n is a power the polynomial is raised to.

Rains became the first actor to receive a million dollar salary, playing Julius Caesar in Gabriel Pascal's lavish and unsuccessful version of Shaw's Caesar and Cleopatra ( 1945 ), made in Britain.

He also had a single prominent scene as a commander of Roman centurions in the film described at the time as the most expensive ( and financially ruinous ) British film enterprise ever made, Gabriel Pascal's production of George Bernard Shaw's Caesar and Cleopatra, starring Vivien Leigh and Claude Rains.

He made use of principles from previous mechanical calculators like the stepped reckoner of Leibniz and Pascal's calculator.