Isaac Barrow was born in London in 1630, and died at Cambridge in 1677. He went to school first at Charterhouse (where he was so troublesome that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to Felstead. He completed his education at Trinity College, Cambridge; after taking his degree in 1648, he was elected to a fellowship in 1649; he then resided for a few years in college, but in 1655 he was driven out by the persecution of the Independents. He spent the next four years in the East of Europe, and after many adventures returned to England in 1659. He was ordained the next year, and appointed to the professorship of Greek at Cambridge. In 1662 he was made professor of geometry at Gresham College, and in 1663 was selected as the first occupier of the Lucasian chair at Cambridge. He resigned the latter to his pupil Newton in 1669, whose superior abilities he recognized and frankly acknowledged. For the remainder of his life he devoted himself to the study of divinity. He was appointed master of Trinity College in 1672, and held the post until his death.
He is described as "low in stature, lean, and of a pale complexion," slovenly in his dress, and an inveterate smoker. He was noted for his strength and courage, and once when travelling in the East he saved the ship by his own prowess from capture by pirates. A ready and caustic wit made him a favourite of Charles II., and induced the courtiers to respect even if they did not appreciate him. He wrote with a sustained and somewhat stately eloquence, and with his blameless life and scrupulous conscientiousness was an impressive personage of the time.
His earliest work was a complete edition of the Elements of Euclid, which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of the Data. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones Opticae et Geometricae. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of the Conics of Apollonius, and of the extant works of Archimedes and Theodosius.
In the optical lectures many problems connected with the reflexion and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflexion or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow.
The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, Hudde and Sluze were working on the lines suggested by Fermat towards the methods of the differential calculus.
Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangent MT could be found (thus determining the point T), then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn, he got a small triangle PQR (which he called the differential triangle, because its sides PR and PQ were the differences of the abscissae and ordinates of P and Q), so that
TM : MP = QR : RP.
To find QR : RP he supposed that x, y were the co-ordinates of P, and x - e, y - a those of Q (Barrow actually used p for x and m for y, but I alter these to agree with modern practice). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point. Barrow applied this method to the curves (i) x² (x² + y²) = r²y²;(ii) x³ + y³ = r³; (iii) x³ + y³ = rxy, called la galande; (iv) y = (r - x) tan πx/2r, the quadratrix; and (v) y = r tan πx/2r. It will be sufficient here if I take as an illustration the simpler case of the parabola y² = px. Using the notation given above, we have for the point P, y² = px; and for the point Q, (y - a)² = p(x - e). Subtracting we get 2ay - a² = pe. But, if a be an infinitesimal quantity, a² must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe. Hence 2ay = pe, that is, e : a = 2y : p. Therefore TP : y = e : a = 2y : p. Hence TM = 2y²/p = 2x. This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.
背景介紹與作者介紹
艾薩克·巴羅是17世紀一位傑出的數學家和神學家。他於1630年出生於倫敦，生活在科學發現和思想發生巨大變革的時代。巴羅不僅是一位學者，還是一位勇敢的冒險家，以其力量和智慧而聞名。儘管他年輕時有些不羈，但他成為了那個時代最傑出的數學家之一，在劍橋大學擔任著享有盛譽的學術職位。巴羅的研究為微積分奠定了重要的基礎，而微積分後來由他的著名學生艾薩克·牛頓進一步發展。
巴羅的人生故事教導我們關於毅力、求知慾以及認識和鼓勵他人才能的重要性。他決定辭去享有盛譽的盧卡斯教授席位，讓給牛頓，這表明了他謙遜的品格以及致力於推進知識發展，超越個人野心的決心。
詳細解釋和意義
以上文本描述了巴羅對數學的貢獻，特別是在幾何學和光學方面。他研究了涉及光反射和折射的問題，解釋了圖像如何通過透鏡形成——這仍然是當今物理學和光學的基礎課題。他在幾何學方面的講授引入了尋找曲線切線的新方法，這是微積分中的一個關鍵概念。
巴羅對切線的處理涉及我們現在所稱的“微分三角形”，這是一個巧妙的幾何工具，幫助他在微積分正式發展之前近似曲線的斜率。這種方法是通往牛頓和萊布尼茨後來將其形式化的強大技術的墊腳石。巴羅對圓錐曲線的研究以及他對阿波羅尼奧斯和阿基米德等古代數學家的評論，也有助於保存和推進古典數學知識。
學生可以學到什麼
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數學的歷史背景： 了解巴羅的研究有助於學生欣賞數學思想是如何隨著時間推移而演變的。它表明，微積分這個經常被認為很難的學科，是由許多思想家一步一步發展起來的，他們建立在彼此的發現之上。
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跨學科學習： 巴羅將他在數學、神學和哲學方面的興趣結合起來，展示了廣泛教育的價值以及不同學科之間的相互聯繫。
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批判性思維和解決問題： 巴羅分析曲線和光的方法教導學生創造性地和邏輯地處理問題，將複雜的想法分解成更簡單的部分。
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謙遜和指導： 巴羅對牛頓卓越才能的認可以及他願意退居二線，是關於謙遜和支持他人成長的重要性的有力一課。
在生活和學習中的應用
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在學校： 學生可以效仿巴羅的例子，對學習保持好奇心和毅力，尤其是在數學和科學等具有挑戰性的科目中。了解概念背後的歷史可以使它們變得更有趣，更容易理解。
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在社交場合： 巴羅的故事鼓勵尊重他人的才能和指導的重要性。幫助同學並認識到他們的優勢可以建立一個積極的學習社區。
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在個人成長中： 巴羅的人生表明，早期的錯誤或困難並不能定義一個人的未來。通過努力和奉獻，任何人都可以取得偉大的成就。
如何從巴羅的故事中培養積極的特質
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好奇心： 像巴羅探索光和曲線的本質一樣，永遠提出問題並試圖理解事物是如何運作的。
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勇氣： 不要害怕面對挑戰或新的體驗，就像巴羅在旅行和反抗海盜時所做的那樣。
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謙遜： 認識到其他人有更好的想法，並願意向他們學習。
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責任心： 像巴羅在學術和個人生活中所做的那樣，努力保持誠信和謹慎的工作。
反思與欣賞
閱讀關於艾薩克·巴羅的故事可以激勵學生重視知識和品格。他兼具智慧和個人勇氣，使他不僅成為有抱負的科學家和數學家的榜樣，也成為任何希望成長為一個有思想、有禮貌和勇敢的個體的人的榜樣。通過學習他的人生和工作，年輕的學習者可以看到對學習的奉獻和對他人的友善如何齊頭並進，從而對世界產生有意義的影響。
