Various Questions and Inventions

our quadrant. And if at the said elevation of the 6th point it will have less effect, it would have much less at any other lower elevation. But if by chance the distance from said fortress to the artillery pieces that were on the plain were 600 paces, that is diametrically, and from the same to those that were on the summit of the mountain were 150 paces, in such a case I say that the said colubrine will have a much greater effect on the said wall standing on the plain (or foot of the mountain at the elevation of the said 6th point) than it would standing on the summit of the mountain. This is because, standing on the plain, the balls fired from it will come to strike the said wall about 200 paces before the end of its procedure in a straight line, and those fired from the summit of the mountain would come to strike only at 50 paces before the end of its travel in a straight line. And because the difference of said effects, that is, from the 50 paces to the 200 (which strike before their sensible declination), is about 150 paces, and therefore the said colubrine, not only at the elevation of the sixth point of our quadrant but also at the elevation of the fifth point, will have a greater effect. But of this, I do not wish to labor to make a demonstration, because I know that it would become tiresome to you. Therefore, if in such a great height (as we have supposed in this last case) the said colubrine would have a greater effect (standing on the plain at the elevation of the 6th and also of the 5th point) than it would standing on the summit of the mountain, much more evidently would such an effect follow in the first case proposed by Your Excellency, in which it was supposed that the mountain and also the fortress were equally high, only 60 paces, and the distance of the roots of the two mountains, or their peaks, to be 100 paces. Whence the diametrical or diagonal line, that is, the distance of said fortress to the place at the root of the mountain where it is supposed the artillery stands on the plain, by the penultimate proposition of the first book of Euclid the Greek geometer, will be about 116 paces (leaving aside fractions). And therefore, the balls fired from our said colubrine, standing on the summit of the mountain, would come to strike the said wall about 140 paces before the end of their travel in a straight line, and those fired from the same standing on the plain at the elevation of the 6th point would come to strike the said wall about 684 paces before the end of their travel in a straight line. And because such a difference is very great, that is, from 140 paces to 684 paces that strike before the end of their travel in a straight line, it is a thing evident and clear in this case that not only at the elevation of the 6th point will the said colubrine standing at the foot of the mountain have a greater effect on said fortress than it would standing on the summit, but also at the elevation of whatever point it may be raised, which is the proposition.

Duke: You have resolved this inquiry for me quite well.

THIRD INQUIRY MADE BY THE SAME

Duke: But in your arguing you have brought me into another greater difficulty, or doubt, because if you remember well, you have said that the ball, once it has left the piece, never travels any part of its motion in a straight line, except by firing it straight upward toward the sky.

N. Niccolò Tartaglia: Or straight downward toward the center of the world?