Almost contemporaneously with the publication in 1637 of Descartes' geometry, the principles of the integral calculus, so far as they are concerned with summation, were being worked out in Italy. This was effected by what was called the principle of indivisibles, and was the invention of Cavalieri. It was applied by him and his contemporaries to numerous problems connected with the quadrature of curves and surfaces, the determination on volumes, and the positions of centres of mass. It served the same purpose as the tedious method of exhaustions used by the Greeks; in principle the methods are the same, but the notation of indivisibles is more concise and convenient. It was, in its turn, superceded at the beginning of the eighteenth century by the integral calculus.
Bonaventura Cavalieri was born at Milan in 1598, and died at Bologna on November 27, 1647. He became a Jesuit at an early age; on the recommendation of the Order he was in 1629 made professor of mathematics at Bologna; and he continued to occupy the chair there until his death. I have already mentioned Cavalieri's name in connection with the introduction of the use of logarithms into Italy, and have alluded to his discovery of the expression for the area of a spherical triangle in terms of the spherical excess. He was one of the most influential mathematicians of his time, but his subsequent reputation rests mainly on his invention of the principle of indivisibles.
The principle of indivisibles had been used by Kepler in 1604 and 1615 in a somewhat crude form. It was first stated by Cavalieri in 1629, but he did not publish his results till 1635. In his early enunciation of the principle in 1635 Cavalieri asserted that a line was made up of an infinite number of points (each without magnitude), a surface of infinite number of lines (each without breadth), and a volume of an infinite number of surfaces (each without thickness). To meet the objections of Guldinus and others, the statement was recast, and in its final form as used by the mathematicians of the seventeenth century it was published in Cavalieri's Exercitationes Geometricae in 1647; the third exercise is devoted to a defence of the theory. This book contains the earliest demonstration of the properties of Pappus. Cavalieri's works on indivisibles were reissued with his later corrections in 1653.
The method of indivisibles rests, in effect, on the assumption that any magnitude may be divided into an infinite number of small quantities which can be made to bear any required ratios ( ex. gr. equality) one to the other. The analysis given by Cavalieri is hardly worth quoting except as being one of the first steps taken towards the formation of an infinitesimal calculus. One example will suffice. Suppose it be required to find the area of a right-angled triangle. Let the base be made up of, or contain n points (or indivisibles), and similarly let the other side contain na points, then the ordinates at the successive points of the base will contain a , 2 a ..., na points. Therefore the number of points in the area is a + 2 a + ... + na ; the sum of which is 1/2 n 2 a + 1/2 na . Since n is very large, we may neglect 1/2 na for it is inconsiderable compared with 1/2 n 2 a . Hence the area is equal to 1/2( na ) n , that is, 1/2 x altitude x base. There is no difficulty in criticizing such a proof, but, although the form in which it is presented is indefensible, the substance of it is correct.
It would be misleading to give the above as the only specimen of the method of indivisibles, and I therefore quote another example, taken from a later writer, which will fairly illustrate the use of the method when modified and corrected by the method of limits.
Let it be required to find the area outside a parabola APC and bounded by the curve, the tangent at A , and a line DC parallel to AB the diameter at A . Complete the parallelogram ABCD . Divide AD into n equal parts, let AM contain r of them, and let MN be the ( r + 1)th part. Draw MP and NQ parallel to AB , and draw PR parallel to AD . Then when n becomes indefinitely large, the curvilinear area APCD will be the the limit of the sum of all parallelograms like PN . Now
area PN : area BD = MP . MN : DC . AD .
But by the properties of the parabola
MP : DC = AM 2 : AD 2 = r 2 : n 2 ,
and MN : AD = 1 : n . Hence MP . MN : DC . AD = r 2 : n 3 . Therefore area PN : area BD = r 2 : n 3 . Therefore, ultimately,
area APCD : area BD = 1 2 + 2 2 + ... + (n-1) 2 : n 3 = 1/6 n (n-1)(2n-1) : n 3
which, in the limit, = 1 : 3.
It is perhaps worth noticing that Cavalieri and his successors always used the method to find the ratio of two areas, volumes, or magnitudes of the same kind and dimensions, that is, they never thought of an area as containing so many units of area. The idea of comparing a magnitude with a unit of the same kind seems to have been due to Wallis.
It is evident that in its direct form the method is applicable to only a few curves. Cavalieri proved that, if m be a positive integer, then the limit, when n is infinite, of (1 m + 2 m + ... + n m )/ n m+1 is 1/( m +1), which is equivalent to saying that he found the integral of x to x m from x = 0 to x = 1; he also discussed the quadrature of the hyperbola.
背景和作者介绍
本文介绍了博纳文图拉·卡瓦列里的开创性工作，他是17世纪早期数学史上的重要人物。卡瓦列里于1598年出生于米兰，是一位耶稣会牧师，也是博洛尼亚大学的数学教授。他的工作为积分学奠定了基础，积分学是数学的一个分支，处理对无限多个小量求和以求得面积、体积和其他量的问题。卡瓦列里的不可分量原理是一个革命性的思想，它帮助数学家们超越了古希腊的穷竭法，为计算面积和体积提供了一种更简单、更灵活的方法。
理解不可分量原理
卡瓦列里的原理指出，一条线由无限多个点组成，一个面由无限多条线组成，一个体积由无限多个面组成。这个想法乍听起来可能很抽象，甚至令人困惑，但它是现代微积分中积分概念的关键一步。通过将形状想象成由无限薄的切片或点组成，卡瓦列里可以通过比较不同形状之间的这些切片来计算面积和体积。
例如，为了找到一个直角三角形的面积，卡瓦列里将底边想象成由许多点组成，而高则包含成比例的点数。通过对这些点求和，他得出了三角形面积的熟悉公式：底边乘以高的一半。虽然他的方法缺乏我们今天所期望的严谨性，但其基本思想是正确的，并为未来的数学家铺平了道路。
意义和影响
卡瓦列里的工作意义重大，因为它引入了一种关于几何和测量的全新思维方式，这种方式比以前的方法更直观，更不笨拙。他的不可分量原理预示了后来牛顿和莱布尼茨开发的积分学。这种方法使数学家能够解决以前非常困难或不可能处理的涉及曲线和表面的问题。
他的工作也影响了对抛物线、球体和双曲线的研究，扩大了对这些形状及其性质的理解。卡瓦列里的方法帮助弥合了几何学和代数之间的差距，从而产生了当今科学和工程中使用的强大数学工具。
学生可以学到什么
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数学创造力和创新： 卡瓦列里的故事表明，新想法通常建立在旧想法的基础上。他采用了古希腊的穷竭法，并通过将形状想象成由不可分割的部分组成来改进它。这教导学生创造性思维的价值，并从新的角度看待问题。
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微积分的基础： 虽然微积分可能看起来很复杂，但卡瓦列里的原理提供了一个简单的介绍，即对无限多个小部分求和以找到整体的概念。理解这一原理有助于学生欣赏微积分的起源和重要性。
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历史背景： 了解卡瓦列里有助于学生了解数学是如何随着时间的推移而发展的，以及不同的文化是如何为知识做出贡献的。它也表明了科学和宗教是如何共存的，因为卡瓦列里是一位耶稣会牧师和数学家。
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解决问题： 给出的例子，例如找到抛物线下的面积，说明了数学推理如何解决实际问题。学生可以学习应用逻辑步骤并使用近似值来处理复杂的问题。
在生活和学习中应用这些课程
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在学校： 学生可以使用卡瓦列里的原理作为理解微积分课程中积分的垫脚石。它鼓励将复杂的问题分解成更小、更易于管理的部分，这在任何科目中都是一项有用的技能。
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在日常生活中： 将小部分相加来理解整体的想法可以应用于预算、烹饪或计划项目。例如，通过将任务分成更小的部分来管理时间，这反映了不可分量的方法。
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在社交场合： 卡瓦列里对信仰和科学的奉献表明了平衡生活的不同方面和尊重不同知识领域的重要性。学生可以学会欣赏多种观点并在不同学科之间进行协作。
从卡瓦列里的工作中培养积极的特质
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好奇心和开放心态： 卡瓦列里探索新想法的意愿鼓励学生保持好奇心并乐于学习，即使概念看起来很困难或不熟悉。
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毅力： 他的工作最初受到批评，并且没有被完全接受，但他继续完善他的想法。这教导了在面对挑战时坚持不懈的价值。
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分析性思维： 不可分量的方法需要仔细的分析和逻辑推理，这些技能在学术和日常决策中都很有价值。
结论
博纳文图拉·卡瓦列里的不可分量原理不仅仅是一种数学技术；它是一个关于创新、毅力和人类理解演变的故事。对于学生来说，它提供了一个了解微积分起源和不同思考方式的力量的视角。通过研究他的工作，年轻的学习者可以深入了解解决问题、科学史以及将创造力与逻辑相结合的重要性。这些课程超出了数学的范围，鼓励一种好奇、坚持和分析的心态——这些品质将在他们生活的各个领域为他们提供帮助。
