John Wallis was born at Ashford on November 22, 1616, and died at Oxford on October 28, 1703. He was educated at Felstead school, and one day in his holidays, when fifteen years old, he happened to see a book of arithmetic in the hands of his brother; struck with curiosity at the odd signs and symbols in it he borrowed the book, and in a fortnight, with his brother's help, had mastered the subject. As it was intended that he should be a doctor, he was sent to Emmanuel College, Cambridge, while there he kept an "act" on the doctrine of the circulation of the blood; that was said to have been the first occasion in Europe on which this theory was publicly maintained in a disputation. His interests, however, centred on mathematics.
He was elected to a fellowship at Queens' College, Cambridge, and subsequently took orders, but on the whole adhered to the Puritan party, to whom he rendered great assistance in deciphering the royalist despatches. He, however, joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I., by which he incurred the lasting hostility of the Independents. In spite of their opposition he was appointed in 1649 to the Savilian chair of geometry at Oxford, where he lived until his death on October 28, 1703. Besides his mathematical works he wrote on theology, logic, and philosophy, and was the first to devise a system for teaching deaf-mutes. I confine myself to a few notes on his more important mathematical writings. They are notable partly for the introduction of the use of infinite series as an ordinary part of analysis, and partly for the fact that they revealed and explained to all students the principles of the new methods of analysis introduced by his contemporaries and immediate predecessors.
In 1655 Wallis published a treatise on conic sections in which they were defined analytically. I have already mentioned that the Géométrie of Descartes is both difficult and obscure, and to many of his contemporaries, to whom the method was new, it must have been incomprehensible. This work did something to make the method intelligible to all mathematicians: it is the earliest book in which these curves are considered and defined as curves of the second degree.
The most important of Wallis's works was his Arithmetica Infinitorum, which was publishd in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and greatly extended, but their logical exposition is open to criticism. It at once became the standard book on the subject, and is constantly referred to by subsequent writers. It is prefaced by a short tract on conic sections. He commences by proving the law of indices; shews that x 0 , x -1 , x -2 ... represents 1, 1/ x , 1/ x ²...; that x 1/2 represents the square root of x , that x 2/3 represents the cube root of x 2 , and generally that x -n represents the reciprocal of x n , and that x p/q represents the q th root of x p .
Leaving the numerous algebraical applications of this discovery he next proceeds to find, by the method of indivisibles, the area enclosed between the curve y = x m , the axis of x , and any ordinate x = h ; and he proves that the ratio of this area to that of the parallelogram on the same base and of the same altitude is equal to the ratio 1 : m + 1. He apparently assumed that the same result would be true also for the curve y = a x m , where a is any constant, and m any number positive or negative; but he only discusses the case of the parabola in which m = 2, and that of the hyperbola in which m = -1: in the latter case his interpretation of the result is incorrect. He then shews that similar results might be written down for any curve of the form
and hence that, if the ordinate y of a curve can be expanded in powers of the abscissa, x , its quadrature can be determined: thus he says that if the equation of the curve were y = x 0 + x 1 + x 2 + ⋅⋅⋅ , its area would be x + 1/2 x 2 + 1/3 x 3 + ⋅⋅⋅. He then applies this to the quadrature of the curves y = ( x - x 2 ) 0 , y = ( x - x 2 ) 1 , y = ( x - x 2 ) 2 , y = ( x - x 2 ) 3 , etc. taken between the limits x = 0 and x = 1; and shews that the areas are respectively 1, 1/6, 1/30, 1/140, etc. He next considers curves of the form y = x -m and establishes the theorem that the area bounded by the curve, the axis of x , and the ordinate x = 1, is to the area of the rectangle on the same base and of the same altitude as m : m + 1. This is equivalent to finding the value of x 1/m dx . He illustrates this by the parabola in which m = 2. He states, but does not prove, the corresponding result for a curve of the form y = x p/q .
Wallis shewed considerable ingenuity in reducing the equations of curves to the forms given above, but, as he was unacquainted with the binomial theorem, he could not effect the quadrature of the circle, whose equation is y = ( x - x 2 ) 1/2 , since he was unable to expand this in powers of x . He laid down, however, the principle of interpolation. Thus, as the ordinate of the circle y = ( x - x 2 ) 1/2 is the geometrical mean between the ordinates of the curves y = ( x - x 2 ) 0 and y = ( x - x 2 ) 1 , it might be suppose that, as an approximation, the area of the semicircle ( x - x 2 ) 1/2 dx which is 1/8π might be taken as the geometrical mean between the values of
that is, 1 and 1/6; this is equivalent to taking 4√2/3 or 3.26... as the value of π. But, Wallis argued, we have in fact a series 1, 1/6, 1/30, 1/140, ... and therefore the term interpolated between 1 and 1/6 ought to be chosen so as to obey the law of this series. This, by an elaborate method, which I need not describe in detail, leads to a value for the interpolated term which is equivalent to taking
The mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct analysis.
In this work also the formation and properties of continued fractions are discussed, the subject having been brought into prominence by Brouncker's use of these fractions.
A few years later, in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves; and gave a solution of the problem to rectify the semi-cubical parabola x ³ = ay ² which had been discovered in 1657 by his pupil William Neil. Since all attempts to rectify the ellipse and hyperbola had been (necessarily) ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Torricelli, and was the first curved line (other than the circle) whose length was determined by mathematics, but the extension by Neil and Wallis to an algebraical curve was novel. The cycloid was the next curve rectified; this was done by Wren in 1658.
Early in 1658 a similar discovery, independent of that of Neil, was made by van Heuraët, and this was published by van Schooten in his edition of Descartes's Geometria in 1659. Van Heuraët's method is as follows. He supposes the curve to be referred to rectangular axes; if this be so, and if ( x,y ) be the coordinates of any point on it, and n be the length of the normal, and if another point whose co-ordinates are ( x , η ) be taken such that η : h = n : y , where h is a constant; then, if ds be the element of the length of the required curve, we have by similar triangles ds : dx = n : y . Therefore h ds = η dx . Hence, if the area of the locus of the point ( x , η ) can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve y ³ = ax ² but added that the rectification of the parabola y ² = ax is impossible since it requires the quadrature of the hyperbola. The solutions given by Neil and Wallis are somewhat similar to that given by van Heuraët, though no general rule is enunciated, and the analysis is clumsy. A third method was suggested by Fermat in 1660, but it is inelegant and laborious.
The theory of the collision of bodies was propounded by the Royal Society in 1668 for the consideration of mathematicians. Wallis, Wren, and Huygens sent correct and similar solutions, all depending on what is now called the conservation of momentum; but, while Wren and Huygens confined their theory to perfectly elastic bodies, Wallis considered also imperfectly elastic bodies. This was followed in 1669 by a work on statics (centres of gravity), and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject.
In 1685 Wallis published an Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. This algebra is noteworthy as containing the first systematic use of formulae. A given magnitude is here represented by the numerical ratio which it bears to the unit of the same kind of magnitude: thus, when Wallis wants to compare two lengths he regards each as containing so many units of length. This perhaps will be made clearer if I say that the relation between the space described in any time by a particle moving with a uniform velocity would be denoted by Wallis by the formula s = vt , where s is the number representing the ratio of the space described to the unit of length; while the previous writers would have denoted the same relation by stating what is equivalent to the proposition s 1 : s 2 = v 1 t 1 : v 2 t 2 . It is curious to note that Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity. The latter opinion may be tenable and not inconsistent with the former, but it is hardly a more simple one.
背景介绍和作者介绍
约翰·沃利斯是一位杰出的数学家和学者，于1616年出生于英格兰阿什福德。最初，沃利斯被期望从事医学，但他的好奇心引导他走向了数学，在那里他做出了开创性的贡献。沃利斯毕业于剑桥大学伊曼纽尔学院，他深深地参与了他那个时代的思想和政治潮流，包括清教徒运动和英国内战。他的职业生涯在牛津大学蓬勃发展，在那里他担任了享有盛誉的萨维尔几何学教授超过50年。除了数学，沃利斯还对神学、哲学，甚至聋哑人的教育做出了贡献，展现了他广泛的才智。
沃利斯工作的详细解释
沃利斯最著名的著作《无穷算术》（1656年）彻底改变了数学家们处理无穷级数和微积分的方式。在这篇论文中，他扩展了笛卡尔和卡瓦列里等早期数学家的方法，系统地使用无穷级数来解决涉及曲线下面积的问题，这是积分学的先驱。
他引入了使用分数和负指数的概念，解释了类似(x^{1/2})的表达式代表平方根，而(x^{-1})代表倒数。这是代数符号和理解方面的一个重大进步。
沃利斯还解决了寻找曲线下面积（求积）的挑战。使用不可分量法，他计算了类似(y = x^m)的曲线下的面积，证明了当今微积分中具有基础性的关系。尽管由于当时缺乏二项式定理，他在处理某些曲线（如圆）时遇到了困难，但他开发了插值方法来逼近诸如(\pi)之类的值。
他对连分数和诸如摆线和半立方抛物线之类的曲线的求长（测量长度）的研究进一步推动了数学分析。沃利斯对物体碰撞的研究引入了动量守恒的早期思想，对物理学产生了深远的影响。
意义和含义
沃利斯的贡献为微积分的发展奠定了重要的基础，微积分是牛顿和莱布尼茨不久后正式确立的数学分支。他的方法使数学家们能够严格地处理无穷过程，以新的方式连接代数和几何。
对于学生来说，沃利斯的故事说明了好奇心和毅力如何带来突破。尽管资源有限，并且存在一些不正确的假设，但他富有创新精神的思维推动了知识的边界。他愿意参与政治和宗教争议也表明了在为社会做出贡献的同时坚持自己原则的重要性。
给学生的教训和启发
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好奇心带来发现： 沃利斯的旅程始于对算术符号的简单好奇。学生们应该拥抱他们的问题并深入探索主题。
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克服挑战的毅力： 沃利斯面临智力上的困难和政治上的反对，但他仍然致力于他的工作。这教会了在学习和生活中的韧性。
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跨学科思维： 沃利斯的工作涵盖了数学、神学、逻辑和教育。学生们可以学习连接不同领域的价值，以丰富理解。
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清晰沟通的重要性： 沃利斯帮助其他人理解了复杂的思想，表明清晰地分享知识与发现一样重要。
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伦理和社会参与： 沃利斯的政治参与提醒学生，学者也可以是积极的公民，影响他们的社区。
如何在生活中应用这些教训
- 在学习中： 以好奇心对待新主题，不要害怕提问或寻求帮助，就像沃利斯对他的兄弟所做的那样。
- 在解决问题中： 使用创造性的方法，并对近似和逐步改进持开放态度，反映沃利斯的插值技术。
- 在社交场合中： 坚定地坚持你的价值观，同时尊重他人的观点，学习沃利斯平衡的政治立场。
- 在个人成长中： 培养韧性，将挫折视为学习和成长的机会，受到沃利斯毅力的启发。
培养积极的态度和行为
学生们可以通过以下方式培养沃利斯精神：
- 在学习中练习耐心和奉献精神。
- 探索跨学科主题以建立广泛的知识。
- 在讨论和辩论中保持尊重的态度。
- 帮助他人理解困难的概念，培养协作学习环境。
- 将数学和逻辑思维应用于日常问题。
结论
约翰·沃利斯的生活和工作为好奇心的力量、数学思想的演变以及学者在社会中的作用提供了丰富的见解。他的遗产鼓励学生们充满激情地追求知识，创造性地思考，并为他们周围的世界做出有意义的贡献。
