The Great Art of Artillery, Part One

...as there is no method (I say truly Geometric) using a common pair of compasses and a ruler, as is usually done in enlarging planes, to double, triple, or infinitely increase a cube: which can by no means be done without an accurate and true discovery of the two mean proportionals. This most noble and useful problem in mechanical matters has been attempted to be resolved and demonstrated by many Geometers, both ancient and modern, as if it were planar and linear (since there are those who include it among the problems of solids), by means of certain ingenious lines, some mixed, others simple, having their origin in the plane, such as straight and circular lines. Of this number, Nicomedes Ancient Greek mathematician used the conchoid a specific curve line, Diocles Ancient Greek mathematician the cissoid ivy-shaped curve, Menaechmus Ancient Greek mathematician the conic sections, and others the parabola. Eratosthenes, Sporus, and Plato, however, used straight and circular lines. Then also Pappus, Hero, Apollonius of Perga, Philo of Byzantium, Orontius, Villalpandus, Clavius, and many others have endeavored to perform this in various ways. But what have they achieved in this step? It is not for me to pass judgment on such great men, who have deserved so much from the Mathematical Republic, or to weigh their works in a more curious scale. It is nevertheless most heralded among many well-versed in Geometry that there is no method by means of planes to exactly multiply a cube, a fact that I observe even those who have labored so greatly in this work have acknowledged. Yet these discoveries and studious efforts of theirs are by no means to be rejected or held as false; rather, let us use them until a happier age suggests better and more perfect ones. From these, I here insert a single method, which seems somewhat more excellent and more Geometric than the others for increasing a Cube, or for finding two mean proportionals in a continuous series between two other given ones, which I consider sufficient for correctly handling Pyrotechnic matters. Let two mean proportionals be found in a continuous series between the two straight lines named above, namely A B and A D. First, let them be placed at right angles to each other, and let a parallelogram A B C D be constructed upon them, and let A B and A D be produced to infinity. Then, having drawn the diagonals B D and A C, let them be placed at the intersection H. Then let a ruler be applied to point C, cutting the produced lines A B and A D in such a way at points E and F that H F and H E are equal. This having been accomplished, D E and B F will be the two mean continuously proportional lines between the given A B and A D; for they will be such that as C D (which is A B) is to D E, so B C (which is A D) is to B F.

I willingly omit other methods, many of which can be seen both among the authors cited above and in Mario Bettino’s Aerarium Philosophiae Mathematicae Treasury of Mathematical Philosophy, recently published in Bologna. In it, he also attempts to show in every way that both the ancient Geometers and certain modern ones, whose names he lists, have found a true, genuine, and perfect method in all respects for finding two mean proportionals between two other given ones, and have demonstrated it geometrically, so that nothing more can be desired by anyone in this step. But let it also be pleasing to hear his words on this matter; for thus he has it in Scholium 7, Proposition 15, Book 6:

"Therefore, what we once pronounced in Apianus 3, Problem 1, concerning the conchoid of Nicomedes, as if doubtful and fearful, we here expressly profess. To show that the better part of Geometric Philosophy concerning solids is truly solid, we affirm that two mean proportionals have long since been discovered geometrically and demonstratively. For, to omit the discoveries of the remaining Ancients, and to indicate at least one, traces of which are also among us, the two means discovered by the method of Nicomedes with the aid of the conchoid line possess that Geometric certainty than which no greater can be desired in any Geometric demonstration of any problem, etc." And a little further down: "For which reasons, no ground for doubting remains."