Six Geometrical Exercises (Exercitationes Geometricae Sex)

I.

In the New Geometry of Indivisibles, the very indivisibles of the continuum are employed as the principal instrument for comparing the measurement of figures, whether plane or solid.

II.

We establish a twofold path for this. The first is exhibited by the first six books of the said Geometry (even though the first contains rather the lemmatic doctrine common to both paths, while it considers principally the properties of similar figures, cylinders, and cones, which definitions 3, 4, 10, and 11 explain); the second, however, is contained in the seventh book. Therefore, we shall not incongruously call both the method of indivisibles: namely, the former, the first, and the latter, the second.

Book VII?
containing?
the method?

III.

In both methods, for the measurement of plane figures, straight lines are employed that are parallel to one certain designated line (which is called their rule), capable of being described by the mind in infinite number within the figures themselves, and terminating at those two [lines] which touch the same figures from opposite sides, and are called, in the 8th definition of the first book, their opposite tangents, of which the one is accustomed to be taken as the rule of the remaining parallels. For the measurement of solids, however, planes are employed that are equidistant to one certain designated plane (which is called their rule), capable of being described by the mind in infinite number within the solids themselves, and terminating at those two planes which touch the solids themselves from opposite sides, and are called, in the 8th definition of the 2nd book, their opposite tangent planes, of which one is accustomed to be taken as the rule of the remaining equidistant planes.

IV.

Hence, it is manifest that plane figures are to be conceived by us after the likeness of a fabric woven with parallel threads: but solids, after the likeness of books which are heaped together by parallel leaves.