27 Nov 2008

Cavalieri's Indivisibles

by Corry Shores
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Cavalieri concieved of a surface as made up of an indefinite number of equidistant parallel lines and of a solid as composed of parallel equidistant planes, these elements being designated the indivisibles of the surface and of the volume respectively.
(Boyer 117c).

Cavalieri says the number of these indivisbles must be "indefinitely great," but he remained "agnostic" as to the nature of infinity (Hegel distinguishes Cavalieri's notion of indivisibles with the vanaishing divisibles of Newton).

Cavalieri says that if one were to rotate the set of indivisibles (parallel lines) making up a planar shape, then the area of the new planar piece will equal the original. Likewise if we slid the parallel planar sections of a solid, we would obtain another solid with the same area:

This last result can be strikingly illustrated by taking a vertical stack of cards and then pushing the sides of the stack into curved surfaces; the volume of the disarranged stack is the same as that of the original stack.
(Eves 387d).

The reason, then, that Cavalieri did not relate these indivisibles to the notion of infinity was because he focused more on the correspondaence between the indivisibles of the two configurations, rather than upon the totality of indivisibles within a single area or volume.

But still the applications of Cavalieri's indivisibles imply some notion of infinity, as for example when demonstrating that a parallelogram is made up of the sum of the double of the lines of either of its internal triangles:

The parallelogram is divided into two triangles by a diagonal line. Then, one forms the smaller triangles in the corners by marking off equal lines, BC and EF, then drawing lines BM and and HE parallel to CD. These lines are equal because their internal angles must be equal. Thus for every parallel line in the one triangle, there is an equal counterpart in the other, hence the double the sum of all the indivisibles in one triangle equals the sum of all lines in the whole parallelogram (Boyer 118b-119).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.

Eves, Jamie. An Introduction to the History of Mathematics, with Cultural Connections. London: Brooks/Cole - Thomson Learning, 1990.

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