OF THE

FORTIFICATIONS

BY BUONAIUTO LORINI

NOBLE FLORENTINE.

FIRST BOOK.

OF THE FIRST PRINCIPLES OF GEOMETRY. CH. I.

Geometry necessary in all operations.

SINCE Geometry is not only useful but necessary, as it is the foundation of all our operations, it must be highly esteemed. Especially since, by means of it, we must begin everything proposed to be treated in the following Books. Without such means, it would be impossible to be able to execute, or even to well understand, anything. Since even shoemakers and the executors of the lowest exercises are necessitated to form and understand the outlines with which they must represent those surfaces and forms that they wish to make. And even more is this expected of those who desire to dedicate themselves to real and much worthier works, such as Fortifications, where the conservation of States and the defense of peoples are treated, especially against the infidel and barbarian peoples, our common enemies. Because from this science depend the clearest and easiest demonstrations that will need to be made. With them, one can not only represent in reality all things created by nature, but also those that we wish to find with the value and artifice of our talent; or to those to add, or diminish, and judge their perfections or imperfections, as if they were made real. Without such means, it would be impossible to be able to teach, or to show anything in its being, as will be diffusely said when we treat of drawing, which goes with the same science. And having to reason of such a principle, we will show the bodies formed by simple lines, however, in accordance with what we judge can serve for the understanding of that which will have to be treated in the present Work, as this principle of the Mathematical sciences is the principal cause of arriving at all the greater and more hidden understandings of nature. And therefore we will treat here only of three things: that is, of the point, of the line, and of the surface.

DEFINITION I.

And first of the point, noted with A. I say that although it is the beginning and the end of all lines, not for this is any part of thickness, width, or depth included in it, but as an indivisible thing it must always be considered, because it serves for nothing other than as a simple limit for the divisions or for the compartments of the bodies.

A diagram showing a single point labeled with the letter A.

DEFINITION II.

The line is a continuous extension from one point to another, and as was said, without width, thickness, or depth, with which all figures that are formed by the idea are circumscribed. Nor should any part of matter or body be considered in it, but always imagine it as simple length BC, so that it attends to nothing other than to represent those forms that one wishes to make.

A diagram showing a horizontal straight line labeled with letters C at the left end and B at the right end.

DEFINITION III.

Parallel lines are two or more, as is seen by these two letters FD, GE, drawn equally distant from one another, so that going into infinity by length, they can never join together.

A diagram showing two horizontal parallel lines; the top line is labeled D and F, the bottom line is labeled G and E.