An interdisciplinary approach to education is nowadays considered an important aspect to improve the critical skills of the learners so that they can guess how several aspects of the human knowledge are interconnected. A key aspect of interdisciplinary education is represented by the use of the history of a certain subject within the teaching of the subject itself. This is particularly conspicuous in science education. For, the appropriate and not superficial introduction of historical elements within science education allows the pupils to discover the human aspects of science, the problems behind the creation and development of many concepts, which are often presented only in a formal manner. As a matter of fact, after having grasped that some problems are difficult and after having understood and appreciated the efforts that in the course of history the scientists have carried out to solve such problems, the learners will accept the necessity of a formalization and will not consider such a formalization as a sort of abstract doctrine imposed by the teachers for unspecified reasons. Furthermore, history of science has profound relations with history, mathematics, science, philosophy, physique and technique, so that it is a typical interdisciplinary subject which can be exploited in an educative context. The considerations which follow concern the calculations of areas and volumes of curvilinear and mixtilinear figures. At a first glance, one might think that the problem is connected only to mathematics, to its history and its education, but, in fact, this is not the case, because it encompasses the whole of exact sciences education insofar as it is related to the concept of integral, which is one the mathematical basic notions used in physics and in all the mathematized sections of other sciences. In general integral calculus is taught at the last but one or the last year of scientific or technical gymnasia/ high schools. In most cases the concept of definite integral is introduced in the form of Riemann integral or Cauchy integral. In both cases an enough heavy formal apparatus is necessary. Obviously the relation between the concept of definite integral and the calculations of plane areas is explained and the fundamental theorem of integral calculus is proved. The whole of this approach is correct, but is there a more intuitive way which allows the learners to understand why the resort to a formal apparatus is appropriate and which, at the same time, make them guess the background of problems which induced the mathematicians to work on the precise definition of the concept of integral? This way exists and it passes through history of mathematics. The first and fundamental phase of our proposed itinerary which connects mathematics and science education with mathematics and science history takes place in Siracusa, the Sicilian city where in the third century B.C. the greatest mathematician of antiquity and one of the greatest ever worked: Archimedes. Euclid, some decades before Archimedes, had written his Elements, the fundamental text of Greek geometry, where, in part collecting a series of knowledge already existing in Greek mathematics, in part adding his new results, had offered an impressive picture of Greek geometry. The width of the subject dealt with by Euclid as well as his rigorous approach made the Elements an unavoidable reference point for all the mathematicians active after Euclid. The problem of the areas of rectilinear flat figures is faced in the first and, after the introduction of the concept of proportion (fifth book), in the sixth book of the Elements. Euclid is, in substance, able to solve the problem reducing the calculation of rectilinear flat areas to the calculation of the areas of triangles. However, with regard to curvilinear areas and volumes, things are by far more difficult. All the results obtained by Euclid are referred to in the 12th book: in proposition 2, Euclid proved that the circles are as the squares of their diameters; in proposition 10 that the volume of a cone is equal to 1/3 of that of a cylinder having its same base and height; in proposition 11 that cones and cylinders having the same heights are as the bases; in proposition 12 that similar cones and cylinders are as the cubes of their bases’ diameters; finally, in proposition 18 that the volume of a sphere is proportional to the cube of its

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使用阿基米德方法计算面积和体积：一些教学考虑

如今，跨学科的教育方法被认为是提高学习者关键技能的一个重要方面，以便他们能够猜测人类知识的几个方面是如何相互关联的。跨学科教育的一个关键方面是在学科本身的教学中使用某个学科的历史。这在科学教育中尤为突出。因为，在科学教育中适当而不是肤浅地介绍历史元素，可以让学生发现科学的人文方面，以及许多概念的创造和发展背后的问题，这些概念通常只以正式的方式呈现。事实上，在认识到有些问题是困难的，在理解和欣赏了科学家在历史进程中为解决这些问题所做的努力之后，学习者会接受形式化的必要性，而不会将这种形式化视为一种形式化的方法。教师出于未指明的原因强加的某种抽象教义。此外，科学史与历史、数学、科学、哲学、体质和技术有着深刻的联系，是一门典型的可以在教育背景下开发的交叉学科。下面的考虑涉及曲线和混合线图形的面积和体积的计算。乍一看，人们可能认为这个问题只与数学、它的历史和它的教育有关，但实际上，情况并非如此，因为它涵盖了整个精确科学教育，因为它与积分概念有关，积分是物理学和其他科学的所有数学部分中使用的数学基本概念之一。一般来说，积分是在科学或技术体育馆/高中的最后一年或最后一年教授的。在大多数情况下，定积分的概念是以黎曼积分或柯西积分的形式引入的。在这两种情况下，都需要足够重的正式设备。明显地解释了定积分的概念与平面面积计算之间的关系，证明了积分学的基本定理。整个方法是正确的，但是有没有一种更直观的方法可以让学习者理解为什么诉诸正式的仪器是合适的，同时让他们猜测问题的背景，从而促使数学家对概念进行精确定义积分？这种方式存在并且它穿越了数学史。我们提议的将数学和科学教育与数学和科学史联系起来的行程的第一个和基本阶段发生在西西里城市锡拉库萨，公元前三世纪最伟大的古代数学家和有史以来最伟大的数学家之一：阿基米德。欧几里得比阿基米德早几十年写了他的几何学基础著作《几何学》，其中部分地收集了希腊数学中已经存在的一系列知识，部分加上他的新结果，提供了令人印象深刻的希腊几何图景。欧几里德处理的主题的宽度以及他的严谨方法使元素成为欧几里德之后活跃的所有数学家不可避免的参考点。直线平面图形的面积问题在第一部以及在引入比例概念（第五本书）之后，在《元素》的第六本书中都遇到过。Euclid 本质上能够解决将直线平面区域的计算减少到三角形区域的计算的问题。然而，关于曲线区域和体积，事情要困难得多。欧几里得得到的所有结果都在第12本书中提到：在命题2中，欧几里得证明了圆是它们直径的平方；在命题 10 中，圆锥的体积等于底和高相同的圆柱的体积的 1/3；在命题 11 中，具有相同高度的圆锥和圆柱是底；在命题 12 中，相似的圆锥和圆柱是它们底直径的立方；最后，在命题 18 中，球体的体积与其球体的立方成正比