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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

THE CALCULATIONS OF AREAS AND VOLUMES USING THE METHOD OF ARCHIMEDES: SOME DIDACTIC CONSIDERATIONS