Among the contemporaries of Descartes none displayed greater natural genius than Pascal, but his mathematical reputation rests more on what he might have done than on what he actually effected, as during a considerable part of his life he deemed it his duty to devote his whole time to religious exercises.
Blaise Pascal was born at Clermont on June 19, 1623, and died at Paris on Aug. 19, 1662. His father, a local judge at Clermont, and himself of some scientific reputation, moved to Paris in 1631, partly to prosecute his own scientific studies, partly to carry on the education of his only son, who had already displayed exceptional ability. Pascal was kept at home in order to ensure his not being overworked, and with the same object it was directed that his education should be at first confined to the study of languages, and should not include any mathematics. This naturally excited the boy's curiosity, and one day, being then twelve years old, he asked in what geometry consisted. His tutor replied that it was the science of constructing exact figures and of determining the proportions between their different parts. Pascal, stimulated no doubt by the injunction against reading it, gave up his play-time to this new study, and in a few weeks had discovered for himself many properties of figures, and in particular the proposition that the sum of the angles of a triangle is equal to two right angles. I have read somewhere, but I cannot lay my hand on the authority, that his proof merely consisted in turning the angular points of a triangular piece of paper over so as to meet in the centre of the inscribed circle: a similar demonstration can be got by turning the angular points over so as to meet at the foot of the perpendicular drawn from the biggest angle to the opposite side. His father, struck by this display of ability, gave him a copy of Euclid's Elements , a book which Pascal read with avidity and soon mastered.
At the age of fourteen he was admitted to the weekly meetings of Roberval, Mersenne, Mydorge, and other French geometricians; from which, ultimately, the French Academy sprung. At sixteen Pascal wrote an essay on conic sections; and in 1641, at the age of eighteen, he constructed the first arithmetical machine, an instrument which, eight years later, he further improved. His correspondence with Fermat about this time shews that he was then turning his attention to analytical geometry and physics. He repeated Torricelli's experiments, by which the pressure of the atmosphere could be estimated as a weight, and he confirmed his theory of the cause of barometrical variations by obtaining at the same instant readings at different altitudes on the hill of Puy-de-Dôme.
In 1650, when in the midst of these researches, Pascal suddenly abandoned his favourite pursuits to study religion, or, as he says in his Pensées , "contemplate the greatness and the misery of man"; and about the same time he persuaded the younger of his two sisters to enter the Port Royal society.
In 1653 he had to administer his father's estate. He now took up his old life again, and made several experiments on the pressure exerted by gases and liquids; it was also about this period that he invented the arithmetical triangle, and together with Fermat created the calculus of probabilities. He was meditating marriage when an accident again turned the current of his thoughts to a religious life. He was driving a four-in-hand on November 23, 1654, when the horses ran away; the two leaders dashed over the parapet of the bridge at Neuilly, and Pascal was saved only by the traces breaking. Always somewhat of a mystic, he considered this a special summons to abandon the world. He wrote an account of the accident on a small piece of parchment, which for the rest of his life he wore next to his heart, to perpetually remind him of his covenant; and shortly moved to Port Royal, where he continued to live until his death in 1662. Constitutionally delicate, he had injured his health by his incessant study; from the age of seventeen or eighteen he suffered from insomnia and acute dyspepsia, and at the time of his death was physically worn out.
His famous Provincial Letters directed against the Jesuits, and his Pensées , were written towards the close of his life, and are the first example of that finished form which is characteristic of the best French literature. The only mathematical work that he produced after retiring to Port Royal was the essay on the cycloid in 1658. He was suffering from sleeplessness and toothache when the idea occurred to him, and to his surprise his teeth immediately ceased to ache. Regarding this as a divine intimation to proceed with the problem, he worked incessantly for eight days at it, and completed a tolerably full account of the geometry of the cycloid.
I now proceed to consider his mathematical works in rather greater detail.
His early essay on the geometry of conics, written in 1639, but not published till 1779, seems to have been founded on the teaching of Desargues. Two of the results are important as well as interesting. The first of these is the theorem known now as "Pascal's Theorem," namely, that if a hexagon be inscribed in a conic, the points of intersection of the opposite sides will lie in a straight line. The second, which is really due to Desargues, is that if a quadrilateral be inscribed in a conic, and a straight line be drawn cutting the sides taken in order in the points A , B , C , and D , and the conic in P and Q , then
PA . PC : PB . PD = QA . QC : QB . QD .
Pascal employed his arithmetical triangle in 1653, but no account of his method was printed till 1665. The triangle is constructed as in the figure below, each horizontal line being formed form the one above it by making every number in it equal to the sum of those above and to the left of it in the row immediately above it; ex. gr. the fourth number in the fourth line, namely, 20, is equal to 1 + 3 + 6 + 10.
The numbers in each line are what are now called figurate numbers. Those in the first line are called numbers of the first order; those in the second line, natural numbers or numbers of the second order; those in the third line, numbers of the third order, and so on. It is easily shewn that the mth number in the nth row is (m+n-2)! / (m-1)!(n-1)!
Pascal's arithmetical triangle, to any required order, is got by drawing a diagonal downwards from right to left as in the figure. The numbers in any diagonal give the coefficients of the expansion of a binomial; for example, the figures in the fifth diagonal, namely 1, 4, 6, 4, 1, are the coefficients of the expansion ( a + b ) 4 . Pascal used the triangle partly for this purpose, and partly to find the numbers of combinations of m things taken n at a time, which he stated, correctly, to be (n+1)(n+2)(n+3) ... m / (m-n)!
Perhaps as a mathematician Pascal is best known in connection with his correspondence with Fermat in 1654 in which he laid down the principles of the theory of probabilities. This correspondence arose from a problem proposed by a gamester, the Chevalier de Méré, to Pascal, who communicated it to Fermat. The problem was this. Two players of equal skill want to leave the table before finishing their game. Their scores and the number of points which constitute the game being given, it is desired to find in what proportion they should divide the stakes. Pascal and Fermat agreed on the answer, but gave different proofs. The following is a translation of Pascal's solution. That of Fermat is given later.
The following is my method for determining the share of each player when, for example, two players play a game of three points and each player has staked 32 pistoles.
Suppose that the first player has gained two points, and the second player one point; they have now to play for a point on this condition, that, if the first player gain, he takes all the money which is at stake, namely, 64 pistoles; while, if the second player gain, each player has two points, so that there are on terms of equality, and, if they leave off playing, each ought to take 32 pistoles. Thus if the first player gain, then 64 pistoles belong to him, and if he lose, then 32 pistoles belong to him. If therefore 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 will have them and perhaps you will have them; the chances are equal. Let us then divide these 32 pistoles equally, and give me also the 32 pistoles of which I am certain." Thus the first player will have 48 pistoles and the second 16 pistoles.
Next, suppose 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 then is that, if the first player gain this point, he secures the game and takes the 64 pistoles, and, if the second player gain this point, then the players will be in the situation already examined, in which the first player is entitled to 48 pistoles and the second to 16 pistoles. Thus if they do not wish to play, the first player would say to the second, "If I gain the point I gain 64 pistoles; if I lose it, I am entitled to 48 pistoles. Give me then the 48 pistoles of which I am certain, and divide the other 16 equally, since our chances of gaining the point are equal." Thus the first player will have 56 pistoles and the second player 8 pistoles.
Finally, suppose that the first player has gained one point and the second player none. If they proceed to play for a point, the condition is that, if the first player gain it, the players will be in the situation first examined, in which the first player is entitled to 56 pistoles; if the first player lose the point, each player has then a point, and each is entitled to 32 pistoles. Thus, if they do not wish to play, the first player would say to the second, "Give me the 32 pistoles of which I am certain, and divide the remainder of the 56 pistoles equally, that is divide 24 pistoles equally." Thus the first player will have the sum of 32 and 12 pistoles, that is, 44 pistoles, and consequently the second will have 20 pistoles.
Pascal proceeds next to consider the similar problems when the game is won by whoever first obtains m + n points, and one player has m while the other has n points. The answer is obtained using the arithmetical triangle. The general solution (in which the skill of the players is unequal) is given in many modern text-books on algebra, and agrees with Pascal's result, though of course the notation of the latter is different and less convenient.
Pascal made an illegitimate use of the new theory in the seventh chapter of his Pensées . In effect, he puts his argument that, as the value of eternal happiness must be infinite, then, even if the probability of a religious life ensuring eternal happiness be very small, still the expectation (which is measured by the product of the two) must be of sufficient magnitude to make it worth while to be religious. The argument, if worth anything, would apply equally to any religion which promised eternal happiness to those who accepted its doctrines. If any conclusion may be drawn from the statement, it is the undersirability of applying mathematics to questions of morality of which some of the data are necessarily outside the range of an exact science. It is only fair to add that no one had more contempt than Pascal for those who changes their opinions according to the prospect of material benefit, and this isolated passage is at variance with the spirit of his writings.
The last mathematical work of Pascal was that on the cycloid in 1658. The cycloid is the curve traced out by a point on the circumference of a circular hoop which rolls along a straight line. Galileo, in 1630, had called attention to this curve, the shape of which is particularly graceful, and had suggested that the arches of bridges should be built in this form. Four years later, in 1634, Roberval found the area of the cycloid; Descartes thought little of this solution and defied him to find its tangents, the same challenge being also sent to Fermat who at once solved the problem. Several questions connected with the curve, and with the surface and volume generated by its revolution about its axis, base, or the tangent at its vertex, were then proposed by various mathematicians. These and some analogous question, as well as the positions of the centres of the mass of the solids formed, were solved by Pascal in 1658, and the results were issued as a challenge to the world, Wallis succeeded in solving all the questions except those connected with the centre of mass. Pascal's own solutions were effected by the method of indivisibles, and are similar to those which a modern mathematician would give by the aid of the integral calculus. He obtained by summation what are equivalent to the integrals of sin φ , sin 2 φ , and φ sin φ , one limit being either 0 or 1/2π. He also investigated the geometry of the Archimedean spiral. These researches, according to D'Alembert, form a connecting link between the geometry of Archimedes and the infinitesimal calculus of Newton.
介紹布萊茲·帕斯卡及其遺產
布萊茲·帕斯卡是一位傑出的法國數學家、物理學家、發明家、作家和哲學家，生活在 17 世紀。帕斯卡於 1623 年出生，從小就展現出非凡的天賦，尤其是在數學和科學方面。儘管他早年就展現出巨大的潛力並做出了重大貢獻，但他一生的大部分時間都致力於宗教反思和哲學。他的作品對許多領域產生了持久的影響，包括幾何學、概率論，甚至文學。
背景和歷史背景
帕斯卡生活在科學和宗教經常交匯，有時甚至相互衝突的時代。17 世紀是一個偉大科學發現的時期，笛卡爾和伽利略等人物挑戰了舊觀念，為現代科學奠定了基礎。帕斯卡是這場知識革命的一部分，但也深入參與了宗教思想，特別是通過他與皇家港的詹森主義運動的聯繫，該運動是一個強調虔誠和道德嚴謹的宗教團體。
他的父親本身是一位法官和科學家，在培養帕斯卡的早期教育方面發揮了關鍵作用。有趣的是，帕斯卡最初被禁止學習數學，這反而激發了他的好奇心，並使他自己發現了許多數學真理。這種早期的自主學習為他後來的成就奠定了基礎。
帕斯卡的主要貢獻
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數學和幾何學： 帕斯卡在射影幾何學方面取得了重要發現，包括關於圓錐曲線內接六邊形的帕斯卡定理。他還發明了著名的帕斯卡三角形，這是一個有助於計算組合和二項式展開的工具，至今仍在學校教授。
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概率論： 與皮埃爾·德·費馬一起，帕斯卡為現代概率論奠定了基礎。他們的通信始於一個賭博問題，並演變成一個正式的數學理論，幫助我們理解機會和風險——這是統計學、經濟學和決策的關鍵領域。
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物理學和發明： 帕斯卡進行了關於大氣壓力的實驗，證實並擴展了托里拆利早期的工作。他發明了最早的機械計算器之一，這是邁向現代計算的第一步。
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哲學和文學： 帕斯卡的《思想錄》是一部關於人性、宗教和哲學的反思集。他著名的“賭注”論證使用概率來推理對上帝的信仰，展示了他獨特的科學推理和靈性的結合。
對學生的意義和重要性
帕斯卡的生活和工作為學生和年輕讀者提供了豐富的教訓：
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好奇心和自主學習： 帕斯卡早期對限制的反抗以及他的自主學習表明了好奇心和獨立學習的力量。學生們可以學到，有時質疑規則並自己探索會帶來偉大的發現。
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跨學科思維： 帕斯卡將數學、科學和哲學結合起來，表明知識是相互關聯的。這鼓勵學生們保持開放的心態，並了解不同的學科如何相互啟發和豐富。
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平衡激情與反思： 帕斯卡從科學轉向宗教沉思，提醒我們平衡智力追求與個人價值觀和反思的重要性。
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韌性和目標： 儘管帕斯卡面臨健康問題和個人挑戰，但他仍然繼續工作並深入思考重要問題。他的毅力對任何面臨困難的人都具有啟發性。
在日常生活和學習中的應用
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數學和解決問題： 理解帕斯卡三角形和概率可以提高學生的邏輯思維和決策能力，這在遊戲、學習和涉及風險的現實生活中很有用。
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倫理學和批判性思維： 帕斯卡的哲學反思鼓勵學生思考大問題——關於幸福、目標和信仰——這可以幫助他們發展自己的價值觀和道德推理。
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科學探究： 帕斯卡的實驗表明了觀察、測試和好奇心在科學中的重要性，鼓勵學生探索和質疑他們周圍的世界。
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文學欣賞： 帕斯卡在他的信件和《思想錄》中清晰優雅的寫作風格是展示如何簡單而美麗地表達複雜思想的一個很好的例子，這是一項在所有交流中都很有價值的技能。
如何培養帕斯卡的精神
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保持好奇心： 像帕斯卡一樣，不要害怕探索新主題，即使是那些看起來很難或被禁止的主題。
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跨學科思考： 嘗試將不同領域的想法聯繫起來——數學、科學、文學、哲學——以獲得更深入的理解。
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反思你的信仰： 像帕斯卡晚年一樣，花時間思考對你來說重要的事情。
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堅持不懈： 挑戰和挫折是學習的一部分；堅持不懈地努力實現你的目標。
結論
布萊茲·帕斯卡的故事不僅僅是關於數學或科學；它是一個年輕的頭腦在好奇心、發現、信仰和反思中旅程的故事。對於今天的學生來說，他的一生教會了我們智力激情、道德質疑和韌性的價值。通過學習帕斯卡，年輕的學習者可以培養幫助他們在學業上取得成功並成長為思想深刻、全面發展的個體的技能和態度。
