cash now apk and thus relieve me of a case which my work prevents me from undertaking I would like this work to appear without my name leaving altogether to you the choice of all the designations which could indicate the name of the author whom you could qualify as just a friend Here is the line which I have thought up for the second Part which will contain my inventions as to the numbers It is a work which is still only an idea and which I may not have the leisure for putting fully on paper but I will send a summary to M Pascal of all my principles and first demonstrations in which I can promise you in advance he will find everything not only new and up till now unknown but also astonishing If you combine your work with his everything will succeed and be completed in a short time and we will yet be able to publish the first Part which you have in your power to do If M Pascal approves of my over tures which are founded mainly on the great esteem which I have for his genius his knowledge and his intellect I will inform you of my numerical inventions This offer of Fermat s was never taken up and one can only regard it as one of the greate lt missed opportunities in the history of rna thematics A short time before August 24th there must have been a letter from Fermat to Pascal setting out the answer to M de Meres difficulty presumably in terms involving some sort of com binatory theory The doubts which one has possibly unjustly about Pascal s understanding deepen a little when we read hisARITHMETIU TRIANGLE FERMAT AND PASCAL 91 reply on August 24th to this missing letter He says that Fermat s method is satisfactory when there are only two players but will not be applicable if there are more than two Pascal then starts enumerating by exhausting the possibilities supposing two players the first A wanting two points to win and the second B wanting three Since A needs 2 and B needs 3 points the game will be decided in four throws The possibilities are AAAA AAAB AABB ABBB BBBB AABA ABAB BABB ABAA BAAB BBAB BAAA ABBA BBBA BABA BBAA In these enumerations every case where A has 2 3 or 4 successes is a case favourable to A and every case where B has 3 or 4 represents a case favourable to B There are 11 for A and 5 for B so that the odds are 11 5 in favour of A This is Pascal s own example and because of what follows we note that he has included several cases where the game could have been determined in ii sS than 4 throws Pascal goes on to say that he has communicated Fermat s solution to Roberval Professor of Mathematics at the College de France member of the one time Mersenne Academy and friend of Etienne Pascal Roberval has the distinction of being the first known mathematician to raise the invalid objection which objection has been repeated through the years What is mistaken is that the problem is worked out on the assumption thatfour games are played in view of the fact that when one man wins two games or the other wins three there is no need to play four games it could happen that they would play two or three or in truth perhaps four And thus he could not see why you I t is perhaps not necessary to point out that the method of solution is one which wc would use today Given that the gamc must finish in 4 throws th n all possibilities for these 4 throws are equally likely Out of these possibilities we pick out those favourable to A But in order that each set of 4 shall be equally likely all must be enumerated 92 GAMES GODS AND GAMBLING claim to find the fair division of stakes on the false assumption that they play four games Pascal to Fermat Pascal says he answered Roberval by saying that although it is possible that the game may be finished in 2 or 3 throws yet the mathematician can suppose that the players agree to have four trials for it is absolutely equal and immaterial to them both whether they let the game take its natural course Leibnitz was perhaps referring to this point when he wrote about the beautiful ideas about games of chance of Messieurs Fermat Pascal and Huygens which M Roberval was not able or did not wish to understand although what authority he had for the did not wish is not stated Pascal continues his letter to Fermat by remarking that I must tell you that the solution for two players based on com binations is very accurate and true but if there are more than two players it will not always bc correct He supposes three players A Band G of equal skill although this is not stated Let A want I point Band G each 2 points to win so that the game must be finished in 3 trials i e ABG BBG or A GGB or A etc He writes down the following table in which the 27 po ibilities correctly enumerated are to be read downward AA BA BB Probability that A wins in 2 games t AB AAA AAB ABB BBB ABA BAB BAA BBA Probability that A wins in 3 Probability that he gets I point in 2 X Probability that he wins the third game i t Probability that A wins in 4 Probability that he gets I point in 3 X Probability that he wins the fourth game j t 13S t I t H as previously deduced ARITHMETIC TRIANGLE FERMAT AND PASCAL 93 AAA AAA AAA BBB BBB BBB CCC CCC CCC AAA BBB CCC AAA BBB CCC AAA BBB CCC ABC

rcb vs pbks 2022 playing 11 8 which is twice four The value of the first thro V of a set of 5 in terms of your opponent s stakes is the fraction which has for i e n games 1 3 5 2n I 2n I 2 4 6 2n n n I 22n 1 t 2 28 2 6 27 128 35 56 28 8 I 128 Generally given 2n letters owing to the symmetry of the binomial HI I 20 220 188 GAMES GODS AND GAMBLING numerator one half the number of combinations of 4 on 8 I take 4 because it is equal to the number of points required and 8 because it is double 4 and for denominator this same numerator plus all the higher combinatorial values Thus if I have won the first game out of a set of 5 there comes to me from my opponent s stakes 35 128 and this fraction 35 128 is the same as 105 384 whieh we get by the multiplication of even numbers for the denom inator and odd numbers for the numerator You will doubtless understand this well if you think about it which is why 1 shan t bother to explain it any more He gives various tables which indicate how the stakes should be divided Had Pascal left it at this one would have marvelled at the quick intelligence which was able to generalise from such small arithmetical examples The continuation of this present letter however and of succeeding letters docs to a certain extent throw doubt as to whether he really understood what he was about Mter these tables for the division of the stakes and a further dis cussion of the points problem he continues 1 haven t time to send you a solution of a difficulty which has puzzled M de Mere He has a good intelligence but he isn t a geometer and this as you realise is a bad fault He does not understand even that a mathematical lint is infinitely divisible and holds very strongly that it is composed of a finite number of points and never have I been able to dissuade him of this The letter proceeds in possibly ingenuous fashion If you are able to solve the difficulty it would be perfect The Chevalier de Mere said to me that he has found falsehood in the theory of numbers for the following reason If I undertake to throw a six with one die there is an advantage in undertaking to do it in 4 throws as 671 to 625 If I undertake to throw the It is because of this last sentence that one is able to name the Chevalier de Mere as the proposer of the problem of points The name was left as M de M in the letters from Pascal to Fermat but there are letters between Pascal and the Chevalier de Mere in which this point of the infinite divisi bility of a line is argued ARITHMETIC TRIANOLE FERMAT AND PASCAL 89 Sonnez with two dice there is a disadvantage in undertaking to do it in 24 t And moreover 24 is to 36 which is the number of pairings of the faces of two dice as 4 is to 6 which is the number of faces of one die This is his grande scandale which makes him say loftily that the propositions are not constant and that Arith metic is self contradictory But you will see it very easily by the principles you have t The remainder of the letter is concerned with proving that the difference of the cubes of any two consecutive natural numbers when unity is subtracted is six times the sum of all the numbers contained in the smaller one and a statement of two geometri cal problems It should be noted that in spite of not hiavng the time to elucidate M de Mere s difficulty Pascal still had time enough to write about these The reply from Fermat to Pascal s letter of July 29th 1654 is missing although we can infer its content A letter from Fermat to Carcavi of August 9th 1654 is worth considering first however Throw the Sonnez i e throw 2 sixes In playing backgammon if a player threw 2 sixes hc would cry Sonnez Ie diable est mort t Single die Probability of one six 1 6 Probability of no six 5 6 Probability of no sixcs in n throws n Probability of at least one six in n throws I _ n p n 4 p 671 1296 0 5177 I Two dice Probability of 2 sixes 36 Probability of not having 2 sixes Probability of no throw of 2 sixes in n thows 35 36 Probability of at least one throw of 2 sixes I _ n p n 24 P 0 4914 11 25 P 0 5055 t The Chevalier de Mere was obviously such an assiduous gambler that he could distinguish empirically between a probability of 0 4914 and 0 5 i e a difference of 0 0086 comparable with that 0 0108 of the gambler who asked advice of Galileo n I 3 n3 1 3n2 3n 6n n I 2 90 GAMES GODS AND GAMBLING since it throws considerable light on the modest unassuming Fermat This letter was possibly written before that of July 29th reached Fennat Fermat to Carcavi August 9th 1654 I was delighted to have agreed with M Pascal for I value his talent highly and I believe him to be capable of solving any prob lem that he undertakes The friendship he offers is so dear to me and so precious that I think it fair to make use ofil in publishing an edition of my treatises If this suggestion did not shock you you could both help in bringing out that edition of which I would allow that you should be the masters You could clarify or augment what seems too brief

crowd city skins in less academic surroundings Pascal s letter to Fermat of July 29th goes on to discuss the following problem Two players playa game of three points and each player has staked 32 pistoles How should the sum be divided if they break off at any stage Throughout it is assumed by both Pascal and Fermat that the players are of equal skill and oppor tuni ty Pascal writes Suppose that the first player has gained 2 points and the second player I point They now have to play for a point on this con dition that if the first player wins he takes all the money which is at stake namely 64 pistoles and if the second player wins each player has 2 points so they are on terms of equality and if they leave off playing each ought to take 32 pistoles Thus if the first The argument is reminiscent of that of Peverone86 GAMES GODS AND GAMBLING player wins 64 pistoles belong to him and if he loses 32 pistoles belong to him If then the players do not wish to play this game but to separate without playing it the first player would say to the second I am certain of 32 pistoles even if I lose this game and as for the other 32 pistoles perhaps I shall have them and perhaps you will have them the chances are equal Let us divide these 32 pistoles equally and give me also the 32 pistolcs of which I am certain Thus the first player will have 48 pistoles and the second 16 He now supposes that the first player has gained two points and the second player none and that they are about to play for a point The condition is then that if the first player gains this point he wins the game and takes 64 pistoles and if the second player gains the point they are in the position already examined in which the first player is entitled to 48 and the second to 16 Thus if they do not wish to play the first player could say to the second If I gain the point I win 64 and if I lose it I am entitled to 48 Give me the 48 of which I am certain and divide the other 16 equally since our chances of gaining this point are equal Thus the first player will have 56 and the second 8 pistoles Finally suppose the first player has gained one point and the second none If they proceed to play for a further point and the first player wins they will be in the condition already examined 56 8 while if the second player wins they will have a point each and be entitled to divide 32 32 Thus if they do not play this point the first player could say Give me 32 and divide 56 32 equally so that he would be entitled to 32 12 44 This is the substance of what Pascal writes to Fermat on the specific problem they are discussing and so far there is nothing new He now goes on to generalise it possibly a trifle insecurely Now to make no mystery of it since you see it so well 1 put out everything clearly just to see that 1 had made no mistake the value by which I mean only the value of the opponent s money of the last game of two is double that of the last game of three and four times that of the last game offour and eight times the last game offic e etc But the proportion for the first games is not so easy to find it is as follows for I do not wish to conceal anythin Here is ARITHMETIC TRIANGLE FERMAT AND PASCAL 87 the problem which I made so much of because it pleases me greatly It is Being given as many games as you wish find the value of the first Let the number of given matches be for example 8 Take the first eight even numbers and the first eight odd numbers thus 2 4 6 8 10 12 14 16 I 3 5 7 9 11 13 15 Multiply the even numbers in the following way the first by the second the product by the third the product by the fourth the product by the fifth etc Multiply the odd numbers in the same way the first by the second the product by the third etc The last product of the even numbers is the denominator and the last product of the odd numbers is the numerator of the fraction which expresses the value of the first of 8 games If each person stakes the numbcr of pistoles expressed by the product of the even num bers he would get from his opponent s stake the product of the odd numbers This can be shown but with a great deal of trouble by the theory of combinations as you have worked out and I have not been able to demonstrate this by any other method but only by combinations And here are the propositions which lead to it which are really arithmetical properties bearing on combinations which I find have certain beautiful properties If of any eight letters taken at random say ABCDEFGH if you add one half the combinations of 4 i e 35 half 70 with all the com binations of 5 namely 56 plus all the combinations of 6 namely 28 plus all the combinations of 7 namely 8 plus all the combinations of 8 you get the fourth number in the fourth progression whose origin is 2t This is the first proposition which is merely arithmetic The other concerns the problem of points and is as follows It is necessary to say first of all that if I win the first point out of 5 and thus need 4 more the game must certainly be decided in

carrom images hd that I shall not make the second throw I should for my indemnity draw one sixth of the remainder which is 5 36ths of the total And if we agree after this that I shall not play a third time I should draw one sixth of the remainder which is 25 216ths of the total And if after this we agree I shall not playa fourth time I must draw one sixth of the remainder which is 125 1296ths of the total and I agree with you that this is the value of the fourth throw supposing one has already settled for the preceding onf s But you propose to me in the last example of your letter I quote your own words If I undertake to throw a 6 in 8 throws and I have played 3 without success if my opponent proposes to me that I should not play my fourth throw and he wants to pay me off because I want to make it there will come to me 125 1296ths of the total of our stakes This is however not true according to my principle For in this case the first three throws having brought nothing to the die caster the whole of the stake remains in the game and he who throws the die and who agrees not to play the fourth throw must take for recompense one sixth of the total And if he had played four throws without achieving the looked for point and we agree that he should not play the fifth he wiII still have the same one sixth of the total for his indemnity For the entire sum remaining in the game it does not follow only from principle but it is common sense that each throw should give equal advantage I beg you84 GAMES GODS AND GAMBLING therefore to tell me if we agree in principle as I believe and if we differ only in application Pascal replied to this on July 29th 1654 From the tone of his letter Fennat s answer to the first letter cannot have been written much before this Succeeding letters between the two have about two weeks between them Pascal s letter begins Sir I am impatient as well as you and although I am still in bed I cannot prevent myself from telling you that I received yesterday evening from M de Carcavi your letter about the die game which I admire more than I am able to tell you I have not the leisure to say much but in a word what you have done is absolutely right I am quite satisfied with it for I do not doubt any more that I have the truth after the admirable agreement I find with you r have seen several people obtain that for dice like M Ie Chevalier de Mere who first posed thcsc problems to me and also M de Roberval But M de Mere has never found the true value for the division of the stakes nor the method of deriving it so that I find myself alone in discovering this Your method is very sure and is that which came to JIle the first time I thought about this problem but because the labour of combinations is excessive I have found a shorter way which I will tell you briefly For I would like to open my mind to you so much pleasure has our agreement given me I see that truth is the same in Toulouse and in Paris It is perhaps not unfair to comment here that if the editors of Fennat s letters are correct in this ordering of the sequence of letters and it is generally acknowledged that they are then Pascal may be a trifle hypocritical Possibly Fermat had mis understood what Pascal meant in the first missing letter but it would seem that Blaise had not got the right solution to the prob lem which he posed and that he possibly only arrived at it after the receipt of Fermat s letter The fact that in this answering letter of July 29th he goes on to give the solution to another problem does not signify much since Pascal was extremely quick at algebra when headed in the right direction The Chevalier de Mere has become famous as the gambler whose questions started the Pascal Fermat correspondence but little is written about him He seems to have carried the Problem ARITHMETIC TRIANGLE FERMAT AND PASCAL 85 of Points to many of the Paris mathematicians including Roberval In a primitive sort of way and to a limited extent he appears to have dabbled in mathematics himself although his collected papers us Oeuvres de Monsieur Ie Chevalier de Mire Amsterdam 1692 bear no evidence of this being literary pieces of not very high calibre That he was not without the good opinion of himself that often goes with a second rate intelligence can be seen in a Jetter written by him to Pascal some time after 1656 You must realise he writes that I have discovered in mathematics things so rare that the most learned of ancient times have never thought of them and by which the best mathematicians in Europe have been surprised You have written on my inventions as well as M de Huyghens M de Fermat and others All he can mean by this is that he asked the original question It is true that Leibnitz wrote of him The Chevalier had an extraordinary genius for mathematics but since this occurs in a passage where Leibnitz was being derogatory about him he may only have intended sarcasm It has never been stated how Pascal and de Mere began to discuss such problems but it may be noted that this contact was made during what might be called Pascal s dissolute period Since de Mere was not a member of the con tinuance of what had been the Mersenne Academy they may have met

ind vs jordan he had fallen so far from grace as to indulge in the sin of mathematical research He died in 1662 at the age of 39 from convulsions the post mortem showing that he had a serious lesion in the brain He was buried in the church of St Etienne du Mort References In this and succeeding chapters I have gained much information from HARCOURT BROWN Scientific Organisations in France and Italy in the 17th Century There is a great deal in this book which is new and not easily accessible else where For the lives of Pascal and Fermat I have used the appropriate volumes of Hocffer Biographie Universelle checking and supplementing the information both from the references given and in the case of Pascal from the com mentaries written to accompany the reprinting of his complete works B PASCAL Oeuvres v 14 pub suivant l ordre chronologi lUl avec documents compllm4ntaiTBs introductions et notls These commentaries are fully documented and form a most valuable con tribution to one s knowledge of Pascal Harcourt Brown devotes a little space to Mersenne One may supplement this by referring to his correspondence which is now in the course of being printed Fermat s letters were reprinted in 1894 and the editors contribute exhaustive footnoteschapter 9 The arithmetic triangle and correspon dence between Fermat and Pascal L homme n est qu un roseau Ie plus faible de la nature mais c est un roseau pensant PASCAL Pensies vi 347 It has been previously noted that the arithmetic triangle commonly attributed to Pascal is of much earlier provenance Michael Stifel gave the table I 2 I 3 3 4 6 5 10 10 6 15 20 7 21 35 35 Stifel was interested in the triangle in order to find a practical method to extract roots of various orders This was also the main purpose of Tartaglia Generale Trattato 1556 and Simon Steven of Bruges L Arithmetique 1625 Mersenne was interested in the theory of combinations and discusses the theory in at least three places in his books La Vt rite des Sciences 1625 Book III Chapter 10 Harmonicorum and Harmonie Universelle He cites the calculations of Xenocrates but otherwise gives no reference except to an un known individual of whom he speaks mysteriously and whom he designates by the letters I M D M I From some fragmentary correspondence left by Mersenne it would appear that Aimc de Gagnieres was interested in his combinatorial calculations as also was the mathematician Frenicle although his work did not appear until much later Abrege des Combinations 1693 That It has been suggested that M D M stands for Monsieur de M r but this is unlikely 81 G 82 GAMES GODS AND GAMBLING Pascal was not original is however quite definite when it is remembered that Herigone is supposed to have been his teacher Herigone in his Cours matlzlmatique Paris 1634 constructed a table of numbers with the idea of calculating the coefficients of integer binomial powers He proposed I a 2 a 2 3 3 a 4 6 4 a f 5 10 10 5 a and he devoted a chapter to combinatorial calculations Arith mltique pratique Chapter XV Des diverses conjonctions et trans positions Tome II p 119 Pascal cites the works of Herigone at the end of his own treatise Usage du Triangle Arithmltique pour trouver les puissances des binOmes et apotOmes He possibly knew of the work of Mersenne he certainly knew of the work of Gagni res since he refers to it This all suggests that it would be more appropriate to speak of the precious mirror of the four elements rather than Pascal s arithmetic triangle for he was very nearly the last of a long line of discoverers He mentioned the arithmetic triangle in a letter to Fermat in August 1654 Pierre de Carcavi first put Fermat and Pascal in touch with one another It will be remembered that he played the role of intermediary between Etienne Pascal and Fermat in 1636 Throughout his life he seems to have played a useful part in introducing scientists from outside France and from the French provinces to those scientists whom he met at the Mersenne Academy Carcavi had known Fermat when he was still at Toulouse and he was an intimate friend of Blaise Pascal In La Vie de M Descartes 1649 we come across the statement M Pascal had no friend more intimate than Carcavi not excepting even M de Roberval or the Gentlemen of Port Royal Not all the correspondence between Pascal and Fermat has survived The first letter from Pascal to Fermat is missing The A complete translation of the text of such letters as we have is in the Appendix When this translation had been completed our attention was drawn to D E Smith A SouTce Book of Mathematics in which a translation of the letters is given by V Sandford ARITHMETIC TRIANGLE FERMAT AND PASCAL 83 order of some of the others has been altered but because of the untiring efforts of the editors of Fermat s papers we have what is obvious1y the reply to this first lost letter It was originally placed in the middle of the series but clearly belongs at the beginning having regard to its content All the letters are about the problem of points Pascal s first letter was almost certainly concerned with a gambler undertaking to throw a six with a die in eight throws Suppose he had made three throws without success what proportion of the stake should he have on condition he gives up his fourth throw Fermat replies Sir if I undertake to make a point with a single die in 8 throws and if we agree after the stakes are made that I shall not play the first throw then I should take from the stakes one sixth of the total as recompense for giving up the first throw And if we agree after this

flipkart nepal fIe finally retired to die at Port Royal The celebrated Cistercian Abbey of Port Royal built in the valley of the Yvette I write ostensible because Pascal was so tied by the loving care of his father and his sisters and by his own bodily weakness that some interruption of this kind seems inevitable78 GAMES GODS AND GAMBLING 30 miles west of Paris in the village of Les Hameaux was founded in 1204 by the wife of a French nobleman when he was absent from France on the Fourth Crusade The Pope gave this abbey the privilege of affording a retreat to lay persons who wanted to withdraw from the world for a time but who did not want to bind themselves with permanent vows A second abbey of the same foundation was instituted in Paris in 1626 and it was to this Abbey that Pascal retired for meditation the last time for good Jansen died in 1638 and his final apologia was printed by his friends after his death 1640 The real struggle between the Jesuits and the Jansenists now began with the Jesuits trying to persuade successive popes to declare the Jansenists heretics and to excommunicate them and with the Jansenists preaching and pamphleteering against the Jesuits The Sorbonne as a whole seems to have been moderately inclined to Jansenism Bishop Jansen in his Discours held that scientific curiosity was only another form of sexual indulgence On reading this page writes Sainte Beuve an eminent biographer of Pascal a curtain was drawn in Pascal s soul Physics geometry appeared to him for the first time in a new light This first conversion of Blaise in 1646 when he was 23 does not seem to have lasted very long for in 1648 he took up mathematics and physics again Perier at Pascal s suggestion carried out the famous experiments with a baro meter at the Puy de Dome The conclusions which Blaise drew from these experiments he wrote in La Pesanteur de l Air and thus involved himself in further dispute with Descartes Descartes had discussed the possibility of these experiments in letters to Mersenne and Pascal would have been privy to these since he was a more or less regular attendant at the academy The scepticism of Descartes with reference to Blaise s essay on conics has already been noticed He was put out by these experiments of erier s and this was probably accentuated by the fact that Descartes like Mersenne was a protege of the Jesuits and a great lover of the Society of Jesus In the long run he does not seem to have borne Blaise any malice but the incident seems to show that Blaise is not entirely the originator which posterity is inclined to believe him After the controversy Blaise wrote to his sister Jacqueline that as a FERMAT AND PASCAL 79 result of his terrible struggle and the indulgence of his scientific curiosity he had become paralysed and was able to walk only with crutches This paralysis did not however last long In 1651 Etienne died leaving Blaise a moderate fortune one of his sisters was married and the other had become a nun The leading strings were at last removed and in 1653 he is described as a man of the world leading a dissolute life Whether he was exceptionally wild or whether he merely led the life of any young man of that time it is not possible to say but whatever the truth of the matter his delicate health probably brought him up short before many months had elapsed In 1654 he had the famous correspondence with Fermat on the problem of Points Letters passed between the two scientists during the four months July to October and the correspondence was definitely closed by Pascal before his second conversion on November 23rd 1654 It is said that the horses of a four in hand ran away with him he took this as a sign from God that he must give up the life he was leading and do no more mathematics and he retired to Port Royal The story of the rest of his life is concerned more with J ansenism than with mathematics At Port Royal he could not still his restless enquiring mind and before long he took up the cudgels for the Jansenists against Pope Innocent X The Sorbonne had now decided that it was politic to regard the Jansenists as heretics Pascal published in January 1656 the first two of his famous Provincial letters of which Voltaire wrote A book of genius is seen in us Lettres Prouinciales All types of eloquence are to be found in them there is not a single word which after 100 years should be changed Voltaire was however anti Jesuit himself At the time 1656 the letters made a great impression but could not save the Professor of Theology Antoine d Arnauld from being expelled from the Sorbonne The J ansenists were persecuted their leaders were forced to go into hiding and the nuns of Port Royal were subjected to imprisonment Pascal continued to live in extremely ascetic circumstances spending his time reading the Scriptures and in writing down the thoughts which these spiritual exercises evoked These thoughts were published after his death and form the 80 GAMES OODS AND GAMBLING famous Pensells It is related that in 1658 he had toothache which kept him awake and that to distract himself he thought about the cycloid the curve traced out by a fixed point on the cir cumference of a wheel rolling at a uniform speed on a horizontal plane As he thought the pain disappeared He took this to be a sign that the Almighty didn t mind him thinking about the cycloid so he thought about it for eight days and wrote his results to Carcavi The fact that he wrote under a pseudonym is possibly a sign that he did not want his world to know that

apk download getting over it the Auvergne and is described as a very learned man and an able mathematician In 1631 having so his biographers say an extraordinary tenderness for his child his only son he gave up his post as judge and moved to Paris in order to supervise his son s education He became a member of Mersenne s Academy although little is recorded of his part in their discussions Many sources agree however that he did take an active part Carcavi who like Fermat was a Counsellor in the Parliament of Toulouse was a correspondent of Mersenne before he moved to Paris in 1642 and became a member of the Academy He was responsible for introducing Fermat into the correspondence circle of Mersenne in 1636 and it was at his suggestion that Etienne Pascal and Roberval wrote to Fermat A pril 26th 1636 concerning the weight of the earth Roberval and Pascal attacked Fermat s theory and there was an exchange of letters about it The relations of Fermat with the Academy remained however excellent and when in 1637 Descartes attacked Fermat s method of maximum and minimum tangents Roberval and Pascal were Fermat s defenders supporting him with a polemic which is well known The elder Pascal introduced Blaise into the academy when he was fourteen years old that76 GAMES OODS AND GAMBLING is about the year 1637 when the controversy was at its height Blaise Pascal 1623 1662 born when Descartes was 27 years of age appears to have been of poor physique and of precocious mental ability His father it is stated from an early age con centrated on developing his reasoning powers rather than his memory Blaise was undoubtedly quick witted His sister married to a physicist M Perier wrote a life of her brother which is not without a certain imaginative interest She says her brother had an extraordinary wit at a very young age which he showed by repartee quick and to the point She also relates that Etienne Pascal was afraid of his son overtaxing his strength and while he instructed him in what is described as the usual education of the time presumably c1assical languages commentaries on Plato and Aristotle and so on he hid all books on mathematics from his son One day he found him drawing diagrams in charcoal and saw that he had rediscovered for himself Euclid s propositions Now although Madame Gilberte Perier was imaginative with regard to her brother and while much that she writes may be discounted for this reason there is no doubt that Blaise could invent for himself without prompting of book or person For when he was sixteen c 1639 he wrote his famous Essai pour les Coniques which although no longer surviving as a whole certainly did exist since Leibnitz reports having seen it Blaise s work caused a certain amount of controversy some mathematicians received it with acclamation while others among them Descartes refused to believe that it was entirely his own work While granting more to Pascal than Descartes was willing to do it may be that he was right to be a little sceptical The method of projection used had been put forward previously by Desargues and while the creative thought for the essay probably came from Blaise we may wonder how far Etienne acted as an improver or refiner of his son s work t It is the hallmark of a mathematical prodigy to rediscover Euclid s pro positions and de rigutlur to recount this in his biography certainly in the seventeenth and eighteenth centuries and probably earlier Euclid lived B C 306 283 and one has the suspicion that the story originated with the first mathematical prodigy of the next generation to him t The parallel of the boy Mozart playing his melodies and his father writing them down and improving them is irresistible FERMAT AND PASCAL 77 The maturity of the essay which caused Descartes to be suspicious of its authorship was pO lsibly due to Etienne But even so allowing for his father s improvements and Desargue s trail breaking in the form of method enough is left to demonstrate the astonishing mathematical powers of this boy of sixteen Two years later Blaise invented a calculating machine but his health already delicate deteriorated and he had to give up working for four years This is possibly the worst thing which could have happened to this delicate introspective boy who wanted to know the reason of everything and who when adequate reasons were not given him looked for better ones for himself Four or five years after his breakdown in 1646 we hear of his conversion to Jansenism in company with the rest of his family This cult of Jansen ism affected many scientists of the seventeenth century and was the ostensible reason for the failure of Pascal to fulfil his mathematical promise so I will consider the history of the cult briefly and its destiny Without attempting to dis entangle the theological doctrines involved and thereby being quite unjust it would appear that the cult of Jansen ism arose from a violent dislike and possibly envy of various Catholic bishops and heads of seminaries for the power of the Society of Jesus This Society founded in 1534 was growing all powerful and many Catholics of importance felt dislike of the rate at which its in fluence was increasing Within the framework of Catholic dogma therefore they set themselves to attack the Jesuits on the subjectc of freewill and of the grace of God TheJansenists took what has been described as a standpoint akin to Calvinism Cornelius Jansen created Bishop of Ypres in 1636 was the leader of this sect and it was the tract written by Jansen on Riformation de I Homme InterieuT which is held to have been the cause of Pascal s conversion Blaise appears to have become interested in the sect when his father was llursed by theJansenists during an illness at Rouen and

ipl team wins find their way into print It is only because so many of these letters haveFERMAT AND PASCAL 73 survived that it is possible nowadays to give Fermat the credit which is certainly his right On Fermat s part the lack of desire to publish may have been his modesty He did not have the day to day contact which the group of mathematicians had in Paris and which must have supplemented their letters to each other He seems to have been one of those rare persons like Newton who tlourished in isolation and who was modest enough to believe that the sketch of a proof or even the statement of a theorem was enough for all the world to understand Fermat carried on an immense correspondence with scientists in Paris but there are also records of letters going to the Low Countries to Italy and sometimes to England This century the seventeenth is noteworthy for the formal founding of a great number of scientific societies and in this correspondence between the scientists of all nations we have the nuclei of these Harcourt Brown states that the Italian academies or societies date from the fifteenth century and possibly the French borrowed this idea from them or the societies may just have been the inevitable con sequence of the liberation of thought following the slow climb of humanity from the Dark Ages In France the academies were not so much social as in Italy as the regular meetings of scientists with comparable interests who exchanged ideas about the scientific problems of the day At about the time of Fermat s birth one of the links between the academies of Italy and Paris was Nicolas Claude Fabri de Peiresc 1580 1637 He is described as being inquisitive and very curious and was the ideal person to spread the gossip of new ideas and techniques He had been a student at Padua and had contacts with the scholars of Firenze and other Italian centres of learning During the years 1605 1620 he travelled extensively all over Europe spreading his news as he went and since one of his loves was astronomy he would have given news of Galileo He also carried on a lively correspondence with the Abbe Mersenne From the point of view of the pro bability calculus he is interesting only as a typical example of the way in which mathematical and other ideas were carried abou t Europe and because of his contact with the Mersenne Academy Yet another possibility whereby the ideas about chance may have 74 GAMES GODS AND GAMBLING been propagated is Cardinal Francis Barberini who was liberal in his interpretation of Galileo s house arrest and who was also a correspondent of the Abbe So many of the threads from divers points in Italy are tied together on their arrival at the Academy of the Abbe Mcrsenne that the probabilities would be in favour of ideas about chance having come to this group from Italy unless they had arrived earlier and before its foundation The group is a noteworthy one Marin Mersenne was born in 1588 at the village of La Soultiere near Oize Sarthe and educated at theJesuit Seminary at La Fleche t He became a priest in the order of the Minorites and was stationed in Paris for much of his life It is said that from his earliest years he was interested in music in mathematics and in the natural sciences and he corresponded with nearly every scientist of note of his day Harcourt Brown writes There is hardly a figure ofimportance ill the learned world who does not appear in the pages of his letters From all parts of Europe news of the advancement of the sciences came to the convent des peres Minimes proche la Place Royale and thence went the prized letters of the reverend father written in their own peculiar cramped and all but illegible hand with the precious news of Descartes Morin Fermat Torricelli or Galileo To Mcrsenne s house once a week came mathematicians and natural scientists among them Gassendi Desargues Carcavi Roberval Descartes and the Pascals father and son The bond appears to have bcen Mersenne All who came into contact with him remarked on his universal learning on the sweetness and charm of his speech the gentleness of his temper the nai veti which won its way mto all hearts Mersenne died in 1648 and the group changed its venue to the house of La Pailleur After he died in 1651 they met in other A translation of a biography of Mersenne is givcn as Appendix 3 It is particularly interesting to note the extensive list of his correspondents t Descartes 1596 1650 born at La Haye in Tourai lc was educated at the same seminary It would be interesting to know who was the Jesuit teacher of mathematics and natural scienc s who produced two sueh able pupils FERMAT AND PASCAL 75 places These informal societies died with the formal beginnings of the learned societies the Royal Society of London in 1660 and the Academie des Sciences of Paris in 1665 It would seem likely that Fermat was au courant with all that was being discussed in the scientific world both through his own correspondence and through the information which he would gain through the Mersenne Academy It is almost certain that if the fragment of Galileo on dicing had been thought sufficiently new to discuss the informal academies would have done so within a few years of its having been written The name of Blaise Pascal is always linked with that of Fer mat as one of the joint discoverers of the probability calculus Because his mathematical work came in bursts before he retired at an early age to meditate on the greatness and the misery of man and was negatively correlated with these meditations it is worthwhile to consider briefly the outline of his life Etienne Pascal was ajudge at Clermont Ferrand in

bol game the calculation of a probability from the mathematical concept of equal1y likely sides of the die was clearly known to at least one Italian mathematician of the early seventeenth century The person His Serenest Highness asking Galileo the question had gambled often enough to be able to detect a difference in probabilities of 1 108 That Galileo had to a certain extent to occupy himself with trivialities while he was mathematician to his Serenest HighnessGALILEO 67 is illustrated by the letters which he exchanged in 1627 with a priest called Nozzolini These letters were printed in 1718 in the coJIected works of Galileo and the Editor of these letters gives the following amusing preamble It was the custom in those happy times in our city of Firenze to hold various literary gatherings in the houses of those great lords who did not spend their time either running after women or in the stables or in excessive gambling but rather passed their day in learned discussions and among educated people In one of these discussions the following problem was posed A horse which was really worth 100 scudi was by one man valued at 1000 and by another at 10 which of the two was the better estimate and which of the two judged more extravagantly There was living then a certain Nozzolini a priest and a cultivated man Andrea Gerini a Florentine gentleman sent the problem to this Nozzolini the latter gave his judgement on the question which was that to make the estimate one should use arithmetic and not geometric pro gression and that he considered the man who had valued the horse at 1000 scudi had been more extravagant in his estimate than he who had valued it only at 10 Galileo was asked about this and he started a disputation with Nozzolini maintaining that a geometric progression should be used The correspondence is voluminous but is noteworthy perhaps only as an illustration of the clarity of Galileo s mind as is witnessed by the way in which he starts the argument His first letter begins To arrive at a conclusion about the subject under consideration which is which of two valuers has valued the better and Jess ex travagantly a thing which is really worth 100 the one who values it at 1000 or the one who values it at 10 it seems to me that one must first establish what is considered a true and just estimate and what is considered an unjust and extravagant one The man who values at 100 a thing which is really worth 100 will value justly and well those who value it at less or more wiII deviate from the proper valuation And among these he who deviates most from the proper valuation either above it or below it wiII be guilty of the greatest extravagance And since some wiII perhaps consider that to 68 GAMES GODS AND GAMBLING deviate equally from the truth both above and below may be understood in two ways that is either by arithmetical proportion which is when the excess of the estimate over the proper one is equal to the excess of the proper estimate over the lowest valuation e g if the proper estimate was 10 and one calculation was 12 and the other 8 then both guesses are equally good or by geometrical proportion which is when the higher estimate has the same pro portion to the real one as the real one has to the lower estimate Galileo had been well trained by the Jesuits in hair splitting logic but does not contribute anything new 18 far as estimation is concerned except for the recognition that opinion can or should be weighed in objective fashion Galileo was born twelve years before Cardano died The difference between the writings of these two men is illustrative of the speed with which human thought was developing at that time Cardano was in thought stilI linked with the Dark Ages with its suffocating atmosphere of superstition Galileo is of our time for although his style of writing is irritating in its prolixity and tortuousness of argument he obviously thinks in a way not too different from our own As a contributor to the calculus of probability he is negligible but as an indicator of the state of that calculus and of scientific thought generally he is of great importance He died in January 1642 and although he asked to be buried in his family vault in the Church of Santa Croce in Firenze the Pope and the Inquisitors did not permit it He was buried privately in a corner of that church Here also lie Michelangelo and Macchiavelli Perhaps the effect of the foundation of the Order of Jesus in 1534 on scientific method in the sixteenth century should be a subject for study One is reminded of Charles Kingsley s gibe Truth for its own sake had never been a virtue with the Roman clergy GALILEO 69 References Hoeffer s Nouvelle Biographie Universelle devotes some pages to Galileo s life and work MONTUCLA Histoire des Mathimatiques and LIBRI Histoire des Sciences mathimatiques en Italu are useful references here A modern appreciation by SHERWOOD TAYLOR Calileo and Freedom qf Thought is excellent in that the intelIectual spirit of the man emerges I have made much use of the vast definitive edition 20 volumes of GALILEO Opera Omnia There are many biographies of Galileo but they treat him as an astronomer and natural scientist as is obvious that they would chapter 8 Fermat and Pascal For a brief space it is granted to us if we will to enlighten the darkness that surrounds our path We press forward torch in hand along the path Soon from behind comes the runner who will outpace us All our skill lies in giving into his hand the living torch bright and unflickering as we ourselves disappear in the darkness HAVELOCK ELLIS The flood of ideas generated by the I

manoj badale net worth talian Renaissance did not abate after the death of Galileo but it was apparent even in Galileo s lifetime that pre eminence in both the arts and sciences was passing from Italy to France The Renaissance in France was later than that of Italy and was probably greatly helped by the loot in the form of ideas manuscripts and books which successive French armies carried out of Italy The fabulous library of manuscripts ofPetrarch is said to have been looted by the French in 1501 and this must have been only one of many instances By the year 1600 the revolution in the arts and the development of the sciences had both reached mature proportions although they did not produce any practitioner of the experimental method of the calibre of Galileo or Leonardo The major contributions of the French scientists of the seventeenth century when one surveys all the work done might be said to be in the realms of pure and of applicable mathematics Galileo wrote as though the calculation of a probability was something which was obvious and the suggestion from his work is that almost any mathematician could set out the method Although this fragment was left unpublished by him it does appear likely that the calculation of a probability was a common 70FERMAT AND PASCAL 71 place to the Italian mathematicians and probably therefore to some of those in France Thus while the foundation of the calculus of probabilities the giant step forward made with the concept of the equally likely faces of the die does not belong to the French the next development of the theory is undoubtedly theirs And by far the greatest of the French mathematicians of the seventeenth century was Pierre de Fermat Fermat born in 1601 at Beaumont de Lomagne near Mont auban in Gascony five years after Descartes and four years before Rembrandt has been called by some the prince of amateurs and by others the greatest pure mathematician who has ever lived There can be no doubt that in an age when mathematical theories in general were being developed at a great rate he was outstanding and this quiet lawyer did more than any othe r Frenchman in helping the formulation of the theory of probability Fermat was the SOli of Dominique Fermat citizen and second consul of the town of Beaumont and his wife Fran oise de Cazelneuve His life offers few noticeable incidents His family were leather merchants and he spent his childhood at home He studied law at the University of Toulouse and after passing his examinations was named Conseilleur de la Chambre des Requites dll Parlement of Toulouse in May 1631 at the age of 30 Some days after he was given this position he married Louise du Long daughter of a Counsellor in the same Parlement and it is from this date that he assumed the prefix de It is not known whether he was actually ennobled by a special decree or whether the post of Counsellor carried the implicit right to the prefix In the intervals when the law courts went into recess he studied both literature and mathematics Parliamentary Coun sellors of that age like the judges of today were obliged to hold themselves aloof from and to have very little contact with their fellow citizens and this must have helped to produce the necessary time for reflection His biographers speak of his singular erudition in what would now be called the humanities His knowledge of the chief European languages and of the literature of continental Europe was said to be both wide and accurate He made emendations to Greek and Latin texts He wrote verses in 72 GAMES GODS AND GAMBLING Latin French and Spanish and he did research in mathematics This quiet friendly man passed all his working life in the service of the State He was promoted King s Counsellor still in the Parlement of Toulouse in 1648 and he died at Castres in January 1665 when he was 64 years of age His private life seems to have been as uneventful and successful as his public career He had two daughters both of whom became nuns and three sons one of whom Samuel became known as a writer Even after three hundred years the good temper modesty kindliness and in tellectual brilliance of this great man shine through his letters which have now fortunately been collated dated and printed in full Unlike the Italians of the sixteenth and seventeenth centuries and the French Swiss and Italians of the eighteenth century the public challenge to the problem does not seem to have played much part among Fermat and his contemporaries It has been mentioned how Fiore the pupil of Ferreo gained a great repu tation for himself and his teacher by challenging other mathe maticians to solve problems concerning the roots of a cubic equation and later John Bernoulli used this device to further his own reputation but public disputation does not seem to enter just at this time It would appear rather that the solution of a problem was followed by a letter to a friend telling him about it and possibly just to puzzle him a little a step in the proof is held back but this was done privately Letters were also ex changed setting out the failure to solve a problem and asking for enlightenment In all the vast correspondence of Fermat there appears only one suggestion that the correspondent might have been resentful of Fermat succeeding where he himself had failed This correspondent was Pascal The development of mathematics in seventeenth century France is interesting in that so very little was made public in comparison with what was achieved If a scientist belonged to the closed circle then he corresponded with tho e others of the circle about anything and everything but the comments and approbation of his equals seem to have been sufficient for him and it was unusual for their letters to