Blaise Pascal was one of those few geniuses whose name is familiar with a wide range of scholars and students; Mathematicians recognise him through Pascal’s Triangle, as well as for pioneering the probability theory, physicists through his pioneering works on vacuum and hydrostatics, computer engineers for inventing the first mechanical calculating machine, students of philosophy and theology know him through the famous Pascal’s Wager, which itself form a part of one of the most renowned work, “Pensées” published after his death.
Blaise Pascal was born on June 19, 1623 in Clermont-Ferrand, France, in an aristocratic family. He had two sisters, Jacqueline and Gilberte. He lost his mother Antoinette Begon at the age of three. His father, Etienne Pascal, trained as a lawyer was one of the highest ranking officials in King Henri III’s service. He served as a tax assessor, then as a senior financial magistrate. He was an accomplished humanist with fluent Greek and Latin. He was also one of the leading mathematicians of his age and was a member of Mersenne’s[1] academy.
Pascal was a child prodigy and his father recognised it. Pascal was also a sick child. Indeed, throughout his life, Pascal used to suffer from various ailments. Pascal’s sister Gilberte wrote the first biographic sketch of Pascal; La Vie de M. Pascal (The Life of M. Pascal) which was published in 1684 and we learned that from the age of 23, there was never a day when Pascal did not suffer from pain. Partly because of Pascal’s health and partly because he had certain ideas about education, Etienne Pascal decided to take the responsibility of educating his young sibling in his own hand (rather unusual at that time). He sold his job (in those days, you could do that), left Clermon-Ferrand to settle in Paris. Knowing the strong scientific bent of Pascal, Eteinne thought that it would be prudent to acquaint the child first with languages and then with more engrossing mathematics. He removed all the books of mathematics from the household. The act naturally aroused child’s curiosity more and he begged his father to tell him about mathematics. Etienne told that mathematics is to find the right relations between different kinds of figures. What follows then is interesting. According to his sister Gilberte;
“And being alone in a room where he (Pascal) was accustomed to amuse himself, he took a piece of charcoal and drew figures upon the boards, trying, for example, to make a circle perfectly round, a triangle of which the sides and angles were equal, and similar figures. He succeeded in his task, and then endeavoured to determine the proportions of the figures, although so careful had his father been in hiding from him all knowledge of the kind, that he did not even know the names of the figures. He made names for himself, then definitions, then axioms; and in this way had pushed his researches as far as the thirty-second proposition[2] of the first book of Euclid.”
When Etienne learned about his sibling’s achievement, he relented and furnished Pascal with a copy of Euclid’s Elements[3]. Pascal progressed rapidly and when he was 13 years old, the proud father introduced his son to Marsenne’s academy. There, even before he was 16, Pascal read his first mathematical paper on Conic sections, “Essai pour les Coniques (Essay on Conics)”. He proved what is now known as Pascal’s theorem (or Hexagrammum mysticum theorem) that if a hexagon is inscribed in a conic section (say, ellipse, parabola or hyperparabola etc.), then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line. It was rather remarkable achievement for a boy of 16 because the geometry was not the conventional metrical geometry, rather projective geometry. In metrical geometry, magnitude or value has a role to play, but not in projective geometry. In Pascal’s theorem, magnitude has no role to play, whatever be the magnitude of the sides or angles of the hexagon, Pascal’s theorem remain valid. When Mersenne informed Rene Descartes[4] about Pascal’s achievement, Descartes was incredulous as well as astonished. Indeed, he accused Etienee of writing the paper in his son’s name.
Around this time misfortune fell upon Pascal’s family. Etienee Pascal had invested most of his wealth in Government bonds. Thirty years’ war[5] left the French government impoverished. Under duress, government devalued the bonds. The angered the bond-bearers protested under the leadership of Eietenee Pascal, and Etienee who had at one time served as an adviser to Cardinal Richelieu[6], was threatened with imprisonment and went underground. Eventually, he was restored to Cardinal’s good graces by the intervention of his daughter Jacqueline. Jacqueline was an accomplished poet and artist. Cardinal Richelieu, a well-known art lover and patron of the theatre was impressed by Jacqueline’s dramatic skills and Jacqueline appealed to him for her father’s pardon. The appeal was made in the form of poem which was preserved,
“O marvel not, Armand, the great, the wise,
If I have failed to please thine ear, thine eyes;
If I have failed to please thine ear, thine eyes;
My sorrowing spirit, torn by countless fears,
Each sound forbiddeth save the voice of tears.
With power to please thee wouldst though me inspire?
Recall from exile now my hapless sire.”
Cardinal also appointed Etienee the chief tax administrator for the city of Rouen and the whole the family moved to Rouen. The duty of the tax administrator was heavy and to lighten father’s burden, young Pascal made a calculating machine, the second such machine to be made[7]. Pascal worked for the machine for three years. With a local craftsman, he made 50 pieces of his calculating machine (called Pascaline). One copy of his machine, he send to Christina, the Queen of Sweden, considered to be most educated woman of 17th century Europe. Pascal’s fame soared high with his calculating machine. However, Pascaline was not a commercially success. The French currency system was rather complex then. There were 20 sols in a livre and 12 deniers in a sol. With 240 division of livre, Pascal had to solve much harder technical problem than he would have had if the division had been 100 and the mechanical calculator was prohibitively costly. In 1970’s Swiss computer scientist Niklaus Wirth designed several programming languages. To honour Pascal’s contribution to mechanical computation, he named one language as “PASCAL.”
Around this time, Marin Mersenne returned back from Italy and informed Pascal about Torricelli’s experiment[8]. Pascal accepted Torricelli’s explanation that the space above the mercury column is devoid of any matter and is vacuum. He carried out a series of experiments for verifying the explanation and banishing for ever the scholastic nonsense of Nature’s abhorrence of a vacuum. He understood that if the weight of air was the cause for sustaining mercury column up to a certain height in a Torricellian tube, then that height will vary at different elevations. He was then at Rouen where higher levels are too insignificant to draw a definite conclusion. 10 Kilometer from Clermont-Ferrand was Puy de Dome, a large lava dome of about 3000 feet height. He asked his brother-in law in Clermond-Ferrand to carry out the experiment. On 19 September 1648, the experiment was carried out in front of a large audience. Two Torricelli tubes were prepared and the mercury column was marked at 26 inches 3 and ½ lines. One was kept below and the other was taken up to the summit. At the summit, the mercury column stood at 23 inches and 2 line. The experiment was repeated several times with the same results. Such profound was the results of the experiments that adherents of Aristotelian doctrine of nature abhor vacuum could not argue more but to accept the result. Aristotelian dictum that “nature abhors vacuum” was banished forever. Atomists won a decisive victory. To prove that the mass of air presses by its weight all bodies which it surrounds and that it is elastic and compressible, he performed a simple but spectacular experiment. A half inflated balloon was taken to the top of Puy de Dome. It gradually inflated itself as it ascended, and at the top of the dome, it was fully inflated. It deflated again when brought down at the bottom. In one experiment he made two unequal apertures in a closed vessel filled with fluid. He then applied two pistons to these apertures, pressed by forces proportional to the respective apertures. The fluid remained in equilibrium. From his studies he formulated the famous law of hydrostatics (known as Pascal’s law): “A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid.” The experiments also led to the invention of the hydraulic press, the mechanical machine for multiplying forces. According to the noted historian of science Sir David Brewster[9] that discovery alone would have immortalised Pascal. In 1795, the English inventor and locksmith, Joseph Bramah, patented the hydraulic press and introduced to the market.
One of the most notable achievement of Pascal was the theory of probability, the branch of mathematics concerned with the analysis of random phenomena. Pascal and the French mathematician Pierre de Fermat (1601-1665) independently created this theory that brought under the rule of law and order the apparently lawlessness of pure chance. Probability theory enormously enhanced human knowledge as chance or randomness appears to be most dominating phenomena of nature.
Around 1653-1654, Pascal got interested in analysis of random phenomena. A professional gambler Chevalier de Mére introduced Pascal to the problem of points (also called problem of stake). The problem was as follows:
Say, two players are engaged in a fair game of chance and it is agreed that whoever wins a certain point (say 10 points) would win a stake of certain amount. The question is if the game is interrupted before the decisive score, how the stake amount will be divided. It was an old problem. Italian mathematician Luca Pacioli (1447–1517) and Niccolò Tartaglia (1499/1500-1557) tried to solve the problem but could not get the right answer. Pascal solved the problem and independently of him the French mathematician Pierre de Fermat (1601-1665) also solved the problem. The trick was to focus not on the completed games, but on the games to be completed. Say, a fair game of chance was interrupted when the score was 6:8 and decisive score was 10. How the stake will be divided? One can convince oneself that the game would have been decisive if 5 more rounds were played. One can write down 2^5=32 possible outcomes and divide the stake in appropriate ratios. Pascal gave a simple method using the arithmetic triangle known by his name; Pascal’s triangle.
The triangle is easy to write: number at n-th row and m-th column is;
N(n,m)=N(n-1,m-1)+N(n-1,m+1)
with N(0,0)=1.
- Figure 1: Pascal’s triangle upto 5th row.
From the 5th row for the score of 8:6 in a 10 point game, the appropriate division of the stake can be calculated as,
(1+5):(10+10+5+1) or 3:13.
Even though the arithmetic triangle is known as Pascal’s triangle, Pascal didn’t invent it. Much before Pascal, the triangle was known in ancient China. Chinese mathematician Chu Shih Chieh, in his 1303 treatise “The Precious Mirror of the Four Elements” depicted the triangle and indicated its use in the binomial expansion. You see, the triangle arranges the coefficients of the binomial expansion:
(a+b)^n, n=0,1,2,3….
(a+b)^0=1
(a+b)^1=a+b
(a+b)^2=a^2+2ab+b^2
(a+b)^3=a^3+3a^2b+3ab^2+b^3
(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4
etc.
Possibly, the triangle was known even in India. Indians mathematician Pingala (5th century), Varahamihir (6th century), Halayudh (10th century) knew of the binomial expansion and its use in combinatorial problem. The arithmetical triangle was also known to the 11th-century Persian mathematician Omar Khayyam. Nowadays, Omar Khahyam is more known as a poet. His fame as poet grew after Edward Fitzerald translated his poems under the title: “Rubaiyat of Omar Khayyam” which caught the fancy of the Western World, but he was an accomplished mathematician and astronomer. In certain parts of the World, the triangle is known as Khayyam-Pascal triangle. In 1653, Pascal discussed in details several features of the arithmetic triangle in his exposition “Traite du triangle arithmetique” and possibly for this reason, his name is associated with the triangle.
Pascal mostly abandoned scientific study after 1654. He briefly returned to mathematics for eight days in 1658. Of many ailments of Pascal, one was bad teeth. One night, when he was awake due to tooth ache, he found that thinking about mathematical problems gave him relief. He worked on the problem of cycloid.[10] For eight consecutive days he worked and succeeded in solving many of the problems connected with the cycloid. His scientific career ended with his work on cycloid, but, he had another invention of a different kind: introduction of the first public conveyances─ the omnibus service in Paris. Horse drawn coaches, with six-eight passengers ran on select routes of Paris. The Carosses à Cinq Sous, or Five-Penny Coaches, were popular at first, but the concept was ahead of time. France was still a feudal country and idea that a noble and a common man will share the same carriage was not tolerated. A decree was issued by the King restricting its use to the Nobility only (who needed it the least) and in due time the service was out of business.
No account of Pascal is complete without mentioning his contributions to religious philosophy. Pascal was Catholic but strong scientific mind made him rather indifferent to the religion. Two incidences, one in early 1646 and the other in late 1654, changed his views on religion. In 1646, Pascal’s father, Etienee slipped on ice and seriously injured himself. During his confinement he was treated by two brothers. The brothers followed Jansenism[11] and they introduced the family to Jansenism. Pascal’s sister Jacqueline was most influenced and wanted to join Port-Royal, the theological centre for Jansenism and become a nun. However, Etienne Pascal strongly opposed the idea. Only after her father’s death in 1651, Jacqueline could join Port-Royal and become a nun (Pascal initially resisted, possibly on amount of the dowry Jacqueline wanted to bestow upon Port-Royal but relented later.) Pascal was no less influenced by the brothers and for the first time in his life, was drawn towards religion. In the language of his sister,
“Providence led him to the study of such pious writings while he was not even twenty four years of age; and God so enlightened him by this course of reading, that he came to realise that the Christian religion obliges us to live only for God, and to have no other object besides Him. So clear and necessary appeared this truth to him, that he gave up for a time all his researches, renounced all other knowledge, and applied himself alone to the ‘one thing needful’ spoken of by our Lord.”
The 1646 event is now known as Pascal’s first conversion. In his new found zeal, Pascal even wrongly accused a monk of heresy, but later realizing his mistake withdraw the accusation. After his first conversion, Pascal did not abandoned his scientific study. Indeed, during 1647-48 he performed most of the experiments on vacuum and hydrostatics. On November 23, 1654 Pascal had his second conversion. On that day Pascal had a miraculous escape from an accident when his horses bolted and he was thrown into the roadway. That night he had a spiritual experience. He saw the horses plunging over the precipice and abyss open beside him—the abyss of eternity and from the abyss there appeared a globe of fire encircled with the Cross. He wrote down the experience on a piece of parchment and for the rest of his life he carried parchment sewed into his coat. Following his death the parchment was recovered, on it was written;
“Fire.
God of Abraham, God of Isaac, God of Jacob, not of the philosophers and of savants. Certitude. Certitude. Sentiment. Joy. Peace. God of Jesus Christ, My God and Your God. Thy God will be my God—Oblivion of the world and of all save God. He is found by the ways taught in the Gospel. Grandeus of the human soul. Just father, the world hath not known thee, but I have known thee. Joy, joy, joy, tears of joy. I have separated myself from him—They have forsaken Me, the fountain of living water. My God, will you forsake me?—Oh, may I not be separated from him eternally! ‘This is life eternal, that they know Thee the only true God, and Him whom you hast sent, J.-C. Jesus Christ– Jesus Christ—I have separated from Him; I have fled, renounced, crucified Him. Oh that I may never be separated from Him~ He is only held fast by the ways taught in the Gospel.”
One can’t make much from this obscure piece of writing. Undoubtedly, the accident had shaken him and in his weak physical and mental condition, he must have taken it as a divine signal to follow the path of God. Whatever may be the case, from that day onward, Pascal led an ascetic life. He even wore a girdle of iron next to his skin, the sharp points of which he pressed closely when he thought himself in danger of experiencing any worldly pleasure. He devoted his remaining life in the service of God. Between January 1656 and March 1657, he produced a remarkable piece of writing, the Provincial Letters. It was a series of eighteen letters (and a fragment of a nineteenth) written in defence of his Jansenist friend Antoine Arnauld who was accused of heresy by the Jesuits. Pascal used the pseudonym of Louis de Mantalte. In the letters, Pascal defended Jansenism and attacked the Jesuits for their moral laxity, casuistry (Jesuit’s method of resolving resolve moral problems by extracting or extending theoretical rules from particular instances and applying these rules to new instances). Pascal knew how to make discussions on a subject like theology accessible to a wider readership. Perfect blend of humour, mockery and satire made the letters hugely popular. The letters also influenced the style of French writings. Voltaire[12] and Jean-Jacques Rousseau[13]‘s writings were greatly influenced by it. According to Voltaire, Pascal’s Provincial Letters stabilised the French Language, setting an enduring example of linguistic precision and accuracy.
Pascal’s last and most renowned work, “Pensèes (Thoughts)” was published posthumously. It is considered to be an exemplary piece on religious (Christian) apologetics. After his religious conversion, Pascal led an ascetic life. He even wore a girdle of iron next to his skin, the sharp points of which he pressed closely when he thought himself in danger of experiencing any worldly pleasure. Seventeenth century France tried to reason God away; “Who needs God? Man can make it on his own.” Descartes and Voltaire, among others, tried to fashion a world view ruled completely by reason. Pascal found reason inadequate. In Pensées he wrote, “We know truth, not only by reason, but also by the heart, and it is in this last way that we know first principles.” For him faith is indispensable for a mortal man. Sometime around 1660, He planned to write a Christian Apologetics[14] to convert the worldly sceptics to Christianity. If he could have completed, it would have been a remarkable work. However, he could not complete the work due to his failing health. Whenever possible, he started to jot down on fragments of paper, his thoughts on God, Religion, human behaviour. The writings were never intended for publication, rather for further elaboration by the author. However, after his death, those fragments were collected, edited and published as “Pensées.” Since the pieces were not numbered or dated, editors did their best to arrange them in an order, but one obtains an impression of chaos, even though Pascal’s brilliance is evident there. In Pensées also, we find his famous “wager,” Pascal’s argument for believing in God. Some parts of his “wager” from Pensées are reproduced below,
- …Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.—”That is very fine. Yes, I must wager; but I may perhaps wager too much.”—Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; wherever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.
Pascal loathed indifference. In all his writings, Pascal wanted his readers to take a stance. In his wager, he thus urges, “You must wager, It is not optional.” He appeals to men’s rationality to wager for existence of God. In wager Pascal made use of his probability theory. Wager was also the first formal use of decision theory, theory that deals with methods for determining the optimal course of action when a number of alternatives are available and their consequences cannot be forecast with certainty. In decision theory, a given action is associated with a set of possible outcomes and each outcome has a certain value or “utility” (gain minus loss); the “expectation” for each outcome is equal to its utility multiplied by the probability of its happening. One then compute the expectation for a given action as the sum of the expectations for each possible associated outcome. The course of action having the maximum expectation is the rational one to follow.
According to Pascal, existence or non-existence of God is beyond human comprehension. In Pensées he wrote;
- It is incomprehensible that God should exist, and it is incomprehensible that He should not exist; that the soul should be joined to the body, and that we should have no soul; that the world should be created, and that it should not be created, etc.; that original sin should be, and that it should not be.
He then balanced loss and gain in believing and disbelieving God. There are four possible cases,
- God exists and you believe in God’s existence:
Gain: eternal happiness or infinite reward,
Loss: finite, in the form of abstinence from worldly pleasure.
Outcome: gain-loss= positive infinite - God exists and you do not believe in God’s existence:
Gain: finite, in the form of worldly pleasure.
Loss: infinite, eternal happiness or salvation denied.
Outcome: gain-loss= negative infinite. - God does not exists and you believe in God’s existence:
Gain: Nil.
Loss: finite, in the form of abstinence from worldly pleasure.
Outcome: gain-loss=negative finite. - God does not exists and you do not believe in God’s existence:
Gain: finite, in the form of worldly pleasure.
Loss: Nil.
Outcome: gain-loss= positive finite.
The four possible cases can be written as a matrix (pay off matrix):
| God exists | God does not exists | |
| You believe in God | positive infinite | negative finite |
| You don’t believe in God | negative infinite | positive finite |
In Pensées Pascal wrote;
- It is incomprehensible that God should exist, and it is incomprehensible that He should not exist; that the soul should be joined to the body, and that we should have no soul; that the world should be created, and that it should not be created, etc.; that original sin should be, and that it should not be.
Since existence or non-existence of God is beyond human comprehension, one can assign equal probabilities ½ for existence or non-existence of God. Using the above table, the expectation for believing in God = (positive infinity x ½) + (negative finite x ½) = positive infinity; and the expectation for not believing = (negative infinity x ½) + (positive finite value x ½) = negative infinity. Hence it is rational to believe in God.
Pascal lived a short life. In June 1662, his already feeble health rapidly declined. He had given up most of his wealth and even his own house to a poor family and moved to his sister Gilberte’s home. He wished to die in the hospital among the poor, but doctors feared to move him as his health deteriorated precariously. On 19 August, 1662, he breadth his last breath. Even though Pascal’s life was short, he achieved more than any mortal can aspire for. In today’s world, Pascal’s religious writings are of historical interest only. But he will remain immortal for his contributions in mathematics and physical science.
In preparing the article, I have freely used the following references.
1. Blaise Pascal: Mathematician, Physicist and Thinker about God, By D. Adamson, St. Martin Press, USA 1995.
2. The Cambridge companion to Pascal, Nicholas Hammond, Cambridge University Press, 2003.
3. Pascal, John Tulloch, William Blackwood and Sons, London, 1878.
4. Oxford World’s Classics; Pensèes and other writings, Blaise Pascal translated by Honor Levy,, Oxford University Press, 1995.
5. The provincial letters by Pascal, Ed. John De Soyres, George Bell and Sons, London, 1880.
[1] French Priest and mathematician Marin Mersenne (8 September 1588-1 September 1648) is best known for Mersenne prime numbers, those which can be written in the form Mn = 2n − 1 for some integer n. He also did seminal work on music theory and often referred to as the “father of acoustics”. He discovered the Mersenne Law, the law governing the vibrating strings: frequency is proportional to the square root of the tension and inversely proportional to length and square root of mass per unit length of the string. In addition to his scientific contributions, he is remembered for his role as a correspondent, a publiciser and a disseminator of science and scientific work. Mersenne was well known in the scientific community in Europe. Among his close acquaintances were Galileo Galilei, Rene Descartes, Blaise Pascal, Etinne Pascal (Blaise Pasal’s father), Christian Huygens, Pierre de Fermat and others. He also used travel frequently and was in communication with nearly all the scientists, mathematicians of his time. Through him, new ideas spread over the Europe, for example, Mersenne was instrumental in making Galileo Galilie known outside Italy. In 1635 Mersenne formed the informal, private Académie Parisienne also known as Mersenne’s Academy (the precursor to the French Academy of Sciences), where many of the leading mathematicians and natural philosophers of France shared their research. No wonder, he was called “the center of the world of science and mathematics” during the first half of the 1600s.”
[2] The thirty second proposition in Euclid’s element was the famous proposition that sum of three internal angles of a triangle is equal to sum of two right angles.
[3] Euclid of Alexandria was a Greek mathematician who lived around 300 BCE. He is best known famous for his treatise, Elements. Elements is considered to be one of the most important books in the history of mathematics. From its publication in around 300 BC, until 19th century, it served as the main textbook for teaching mathematics (especially geometry).
[4] Rene Descartes was a French mathematician and philosopher, who famously said, “Cogito ergo sum (I think, therefor I am). See https://asischaudhuri.wordpress.com/2016/09/30/what-is-truth-part-1/ for more details on Descartes.
[5] The Thirty Years’ War was one of the longest and devastating was fought between various European states in the name of religion (Protestant vs. Catholic). It was one of the deadliest war resulting into eight million casualties.
[6] Cardinal Richelieu was a French Clergyman and statesman. In 1624 King Louis XIII appointed him prime minister of France, a post he held until his death in 1642. An able administrator, he reformed the army, crushed any rebellions, raised money by any methods required and supervised a foreign policy to pull back France from verge of collapse to make it the greatest power in Europe.
[7] In 1623, approximately, 20 years predating Pascal’s machine, Wilhelm Schikard, a German professor of Hebrew and Astronomy designed the world’s first mechanical calculator. It is uncertain whether a machine was actually built.
[8] To learn about Torricelli’s experiment see; https://asischaudhuri.wordpress.com/2017/08/19/evangelista-torricelli-and-discovery-of-barometer/
[9] David Brewster was a Scottish physicist and a prominent figure in the popularization of science. He invented the Kaleidoscope.
[10] A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. Galileo suggested its use as an arch for bridges. For mechanical reason, cycloid arch is superior to any other construction and it was intensely studied curve in 16th and 17th century.
[11] Jansenism was a 17th-century Catholic movement, initiated primarily in France from the posthumously published work of the Dutch Theologian, Cornelius Jansen (1585-1638). Janesist repudiated free will, accepted predestination, and taught that divine grace, rather than good works, was the key to salvation. The theological centre of the movement was the convent of Port-Royal-des-Champs Abbey. Even though Janesist’s prophesied to follow Augustine of Hippo’s teachings, the orthodox Catholics, symbolised by Jesuits (members of Society of Jesus, the religious congregation of Catholic Church) disliked and distrusted them. In 1653, Pope condemned five cardinal doctrines of Jansenism as heresy (strongly at variance with established believe). The papal condemnation seriously hampered the movement and over time Jansenism completely lost to orthodox Catholicism.
[12] Voltaire (21 November 1694-30 May 1778) was a French writer and philosopher, famous for his wit and satirical writings. A prolific writer, he advocated civil liberties, freedom of speech, and freedom of religion. He was critical of the Catholic Church and wanted separation of the State and Church. His writings and philosophy greatly influenced France as well as the entire Europe in eighteenth century.
[13] Jean-Jacques Rousseau (28 June 1712 – 2 July 1778) was a French philosopher and writer who greatly influenced France and across Europe, during the Enlightenment period in eighteenth century.
[14] In Greek, ‘apology’ is a speech in one’s own defence. Christian Apologetics is a field of Christian theology that defends Christianity against objections by presenting, historical, reasoned and evidential bases for Christianity.
Cooking Cosmos 