Six Geometrical Exercises (Exercitationes Geometricae Sex)

...discussio discussion) it was necessary for the reader to be taught the subject of the dispute beforehand, so that, being sufficiently capable of the same, he might subsequently be able to render a legitimate judgment as to whether any utility is to be hoped for by Geometers from this, or if this new kind of demonstration is to be rejected entirely as pseudogeometric. Certainly, if we attentively consider the fruit collected hitherto by the help of Indivisibles—especially (to pass over in silence the insignificance of my own inventions in the aforementioned Geometry) through the industry and skill of the outstanding and most keen Geometer and Mathematician to the Serene Grand Duke of Tuscany, Evangelista Torricelli—those things will not seem altogether to be neglected. For it is now sufficiently evident to the whole Republic of Letters, in his Geometric Works, how easily he obtained the solution of the most difficult problems through these. As, for instance, in the multiple method for squaring the Parabola through the Indivisibles themselves, and in the dimension of the cycloidal space, which was made so passable through these that a problem viewed by most Geometers as of the greatest difficulty has turned out to be very easy. Indeed, he also subjected to measurement a body (which he himself calls the acute hyperbolic solid) of infinite length—unusual in geometry, yet admirable—as well as a cochlea spiral/screw, though through those curved Indivisibles (by which he himself has wonderfully expanded the bounds of this method). But so that this utility might shine forth more clearly, I shall also add some things invented by me in the said Geometry after the third Exercise, so that all these things, kind reader, may be submitted to your judgment. Now, however, follow the Notes on the second book of the said Geometry, so that its doctrine, in which the prior method of Indivisibles chiefly consists, may become easier to grasp.