Bonaventura Cavalieri was born in Milan in 1598, became a Jesuat (not a Jesuit, as is frequently incorrectly stated) at the age of fifteen, studied under Galileo, and served as a professor of mathematics at the age of forty-nine. He was one of the most influential mathematicians of his time and wrote a number of works on mathematics, optics, and astronomy. He was largely responsible for the early introduction of logarithms into Italy. But his greatest contribution to mathematics was a treatise, Geometria indivisibilibus, published in its first form in 1635, devoted to the precalculus method of indivisibles. Though the method can be traced back to Democritus (circa 450B.C.) and Archimedes (circa 287-212 B.C.), very likely it was Keplerís attempts to find certain areas and volumes that directly motivated Cavalieri.
Cavalieriís treatise is verbose and not clearly written, and it is difficult to know precisely what is to understood by an ďindivisibleĒ. It seems that an indivisible of a given planar piece is a chord of that piece, and an indivisible of a given solid is a plane section of that solid. A planar piece is considered as made up of an infinite set of parallel chords and a solid as made up of an infinite set of parallel plane sections.
Now, Cavalieri argued, if we slide each member of the set of parallel chords of some given planar piece along its own axis, so that the end points of the chords still trace a continuous boundary, then the area of the new planar piece so formed is the same as that of the original planar piece, inasmuch as the two pieces are made up of the same chords. A similar sliding of the members of a set of parallel planar sections of a given solid will yield another solid having the same volume as the original one. This last result can be strikingly illustrated by taking a vertical stack of cards and then pushing the sides of the stack into curving surfaces; the volume of the disarranged stack is the same as that of the original stack. These results, slightly generalized, give the so-called Cavalieri principles:
1. If two planar pieces
are included between a pair of parallel lines, and if the lengths of the
two segments cut by them on any line parallel to the including lines are
always in a given ratio, then the areas of the two planar pieces are also
in this ratio.
2. If two solids are included between a pair of parallel planes, and if the areas of the two sections cut by them on any plane parallel to the including planes are always in a given ratio, then the volumes of the two solids are also in this ratio.
Cavalieriís principles constitute a valuable tool in the computation of areas and volumes, and their intuitive bases can easily be made rigorous with the modern integral calculus. Accepting these principles as intuitively apparent, one can solve many problems in mensuration that normally require the more advance techniques of the calculus.