It was almost a matter of course that the English should at first have adopted the notation of Newton in the infinitesimal calculus in preference to that of Leibnitz and consequently the English school would in any case have developed on somewhat different lines to that on the continent, where a knowledge of the infinitesimal calculus was derived solely from Leibnitz and the Bernoullis. But this separation into two distinct schools became very marked owing to the action of Leibnitz and John Bernoulli, which was naturally resented by Newton's friends; and so for forty or fifty years, to the disadvantage of both sides, the quarrel raged. The leading members of the English school were Cotes, Demoivre, Ditton, David Gregory, Halley, Maclaurin, Simpson, and Taylor. I may, however, again remind my readers that as we approach modern times the number of capable mathematicians in Britain, France, Germany and Italy becomes very considerable, but that in a popular sketch like this book it is only the leading men whom I propose to mention.
To David Gregory, Halley and Ditton I need devote but few words.
David Gregory
David Gregory, the nephew of the James Gregory mentioned above, born at Aberdeen on June 24, 1661, and died at Maidenhead on Oct. 10, 1708, was appointed professor at Edinburgh in 1684, and in 1691 was on Newton's recommendation elected Savilian professor at Oxford. His chief works are one on geometry, issued in 1684; one on optics, published in 1695, which contains [p. 98] the earliest suggestion of the possibility of making an achromatic combinations of lenses; and one on the Newtonian geometry, physics, and astronomy, issued in 1702.
Halley
Edmund Halley, born in London in 1656, and died at Greenwich in 1742, was educated at St. Paul's School, London, and Queen's College, Oxford, in 1703 succeeded Wallis as Savilian professor, and subsequently in 1720 was appointed astronomer-royal in succession to Flamsteed, whose Historia Coelestis Britannica he edited; the first and imperfect edition was issued in 1712. Halley's name will be recollected for the generous manner in which he secured the immediate publication of Newton's Principia in 1687. Most of his original work was on astronomy and allied subjects, and lies outside the limits of this book; it may be, however, said that the work is of excellent quality, and both Lalande and Mairan speak of it in the highest terms. Halley conjecturally restored the eighth and lost book of the conics of Apollonius, and in 1710 brought out a magnificent edition of the whole work; he also edited the works of Serenus, those of Menelaus, and some of the minor works of Apollonius. He was in his turn succeeded at Greenwich as astronomer-royal by Bradley.
Ditton
Humphry Ditton was born at Salisbury on May 29, 1675, and died in London in 1715 at Christ's Hospital, where he was mathematical master. He does not seem to have paid much attention to mathematics until he came to London about 1705, and his early death was a distinct loss to English science. He published in 1706 a text book on fluxions; this and another similar work by William Jones, which was issued in 1711, occupied in England much the same place as l'Hospital's treatise did in France. In 1709 Ditton issued an algebra, and in 1712 a treatise on perspective. He also wrote numerous papers in the Philosophical Transactions . He was the earliest writer to attempt to explain the phenomenon of capillarity on mathematical principles; and he invented a method for finding the longditude, which has been since used on various occasions.
Cotes
Roger Cotes was born near Leicester on July 10, 1682, and died at Cambridge on June 5, 1716. He was educated at Trinity College, Cambridge, of which society he was a fellow, and in 1706 was elected to the newly-created Plumian chair of astronomy in the university of Cambridge. From 1709 to 1713 his time was mainly occupied in editing the second edition of the Principia . The remark of Newton that if only Cotes had lived "we might have known something" indicates the opinion of his abilities held by most of his contemporaries.
Cotes's writings were collected and published in 1722 under the titles Harmonia Mensurarum and Opera Miscellanea . His lectures on hydrostatics were published in 1738. A large part of the Harmonia Mensurarum is given up to the decomposition and integration of rational algebraical expressions. That part which deals with the theory of partial fractions was left unfinished, but was completed by de Moivre. Cotes's theorem in trigonometry, which depends on forming the quadratic factors of x n - 1, is well known. The proposition that "if from a fixed point O a line be drawn cutting a curve in Q 1 , Q 2 , ... , Q n , and a point P be taken on the line so that the reciprocal of OP is the arithmetic mean of the reciprocals of OQ 1 , OQ 2 , ... , OQ n , then the locus of P will be a straight line" is also due to Cotes. The title of the book was derived from the latter theorem. The Opera Miscellanea contains a paper on the method for determining the most probable result from a number of observations. This was the earliest attempt to frame a theory of errors. It also contains essays on Newton's Methodus Differentialis , on the construction of tables by the method of differences, on the descent of a body under gravity, on the cycloidal pendulum, and on projectiles.
de Moivre
Abraham de Moivre was born at Vitry on May 26, 1667, and died in London on November 27, 1754. His parents came to England when he was a boy, and his education and friends were alike English. His interest in the higher mathematics is said to have originated in his coming by chance across a copy of Newton's Principia . From the éloge on him delivered in 1754 before the French Academy it would seem that his work as a teacher of mathematics had led him to the house of the Earl of Devonshire at the instant when Newton, who had asked permission to present a copy of his work to the earl, was coming out. Taking up the book, and charmed by the far-reaching conclusions and the apparent simplicity of the reasoning, de Moivre thought nothing would be easier than to master the subject, but to his surprise found that to follow the argument overtaxed his powers. He, however, bought a copy, and as he had but little leisure he tore out the pages in order to carry one or two of them loose in his pocket so that he could study them in the intervals of his work as a teacher. Subsequently he joined the Royal Society, and became intimately connected with Newton, Halley, and other mathematicians of the English school. The manner of his death has a certain interest for psychologists. Shortly before it he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer each day than the preceding one. The day after he had thus reached a total of something over twenty-three hours he slept up to the limit of twenty-four hours, and then died in his sleep.
He is best known for having, together with Lambert, created that part of trigonometry which deals with imaginary quantities. Two theorems on this part of the subject are still connected with his name, namely, that which asserts that sin nx + i cos nx is one of the values of (sin x + i cos x ) n , and that which gives the various quadratic factors of x 2 n - 2 px n + 1. His chief works, other than numerous papers in the Philosophical Transactions , were The Doctrine of Chances , published in 1718, and the Miscellanea Analytica , published in 1730. In the former the theory of recurring series was first given, and the theory of partial fractions which Cotes's premature death had left unfinished was completed, while the rule for finding the probability of a compound event was enunciated. The latter book, besides the trigonometrical propositions mentioned above, contains some theorems in astronomy, but they are treated as problems in analysis.
Stewart
Maclaurin was succeeded in his chair at Edinburgh by his pupil Matthew Stewart, born at Rothesay in 1717 and died at Edinburgh on January 23, 1785, a mathematician of considerable power, to whom I allude in passing, for his theorems on the problem of three bodies, and for his discussion, treated by transversals and involution, of the properties of the circle and straight line.
背景和歷史背景
這篇文章讓讀者得以一窺17世紀末和18世紀初英國數學和天文學的發展。它描述了英國和歐洲大陸的微積分學派之間的競爭，重點關注艾薩克·牛頓和戈特弗里德·威廉·萊布尼茨的追隨者。這場競爭塑造了數十年來的數學思想。文中提到的人物——大衛·格雷戈里、埃德蒙·哈雷、漢弗萊·迪頓、羅傑·科茨、亞伯拉罕·德·莫弗和馬修·斯圖爾特——是這一時期科學和數學進步的關鍵貢獻者。
關於作者和數學家
- 艾薩克·牛頓（1643–1727）：雖然這裡沒有詳細介紹，但牛頓的著作奠定了所討論的大部分微積分和物理學的基礎。他的《自然哲學的數學原理》徹底改變了對運動和引力的理解。
- 大衛·格雷戈里（1661–1708）：一位數學家和天文學家，他為幾何學和光學做出了貢獻，包括關於消色差透鏡的早期想法。
- 埃德蒙·哈雷（1656–1742）：以哈雷彗星聞名，他也有助於出版牛頓的著作，並為天文學做出了重大貢獻。
- 漢弗萊·迪頓（1675–1715）：以其關於流數（微積分）和代數的教科書而聞名，並在毛細現象和經度計算方面進行了早期研究。
- 羅傑·科茨（1682–1716）：牛頓《原理》第二版的編輯，以其在三角學中的重要定理和關於誤差理論的早期研究而聞名。
- 亞伯拉罕·德·莫弗（1667–1754）：概率論和涉及虛數的三角學的先驅，他的工作為統計學奠定了基礎。
- 馬修·斯圖爾特（1717–1785）：以其關於三體問題和幾何性質的研究而聞名。
詳細闡釋和意義
這組傳記和成就突出了科學進步的合作性和有時的競爭性。英國學派對牛頓記號的偏好影響了英國微積分的發展方向，而歐洲大陸的數學家則遵循萊布尼茨的方法。這些學派之間的爭論減緩了進程，但也提高了數學的嚴謹性。
提到的著作——從幾何學和光學到天文學和概率——表明了這些領域之間的相互聯繫。例如，哈雷的努力確保了牛頓的革命性思想傳播給了更廣泛的受眾，而德·莫弗的概率論已成為當今統計學和風險分析的基礎。
給學生的經驗教訓和見解
- 毅力和好奇心：這些數學家中有許多都面臨著挑戰，從知識爭論到個人困境。他們在研究複雜問題方面的毅力教會了學生奉獻的價值。
- 跨學科學習：幾何學、物理學、天文學和概率的融合說明了聯繫不同領域以解決問題的重要性。
- 合作與尊重：儘管存在競爭，但這些學者經常在彼此的基礎上進行研究。這表明了尊重他人的貢獻和共同努力的重要性。
- 創新和批判性思維：德·莫弗對虛數和概率的處理方式表明了創新思維如何開啟新的知識領域。
- 實際應用：從通過計算經度來改善導航到理解毛細現象，這些發現都具有真實世界的影響，鼓勵學生看到數學和科學的相關性。
在日常生活中應用這些經驗教訓
- 在學習中：效仿這些數學家的好奇心和紀律性，逐步解決具有挑戰性的科目，並試圖理解它們的實際應用。
- 在社交場合：欣賞不同的觀點並互相尊重地合作，認識到進步往往來自共同的努力。
- 在解決問題中：使用跨學科的方法——結合來自不同領域的知識來尋找創造性的解決方案。
- 在個人成長中：培養耐心和毅力，知道掌握需要時間和努力。
從這些故事中培養積極的特質
學生可以學會重視終身學習，迎接挑戰，並保持知識上的謙遜。這些故事鼓勵擁抱新想法，即使很困難，並強調為更大的知識共同體做出貢獻的重要性。
通過研究這些歷史人物及其著作，學生不僅可以獲得數學知識，還可以獲得在科學、技術、工程和數學（STEM）領域追求自己道路的靈感，從而培養一種重視好奇心、合作和毅力的思維方式。
