PROPOSITION VII.

A geometric lathe, unknown elsewhere, which uses a single gauge to reduce and turn cylinders and cones into any desired oval shape with its instruments, applicable to any material that can be turned.

The elegance and subtlety of this lathe are such that they cannot be praised enough, for through an understanding of it, one can construct others so that not only ovals, but any other shape whatsoever can be turned. This method of turning is not to be despised, for along with the pleasure it brings, it offers great utility to artisans. By this means, they will complete a piece of work sooner and better than with the help of many tools specific to their trade. To achieve this, one must understand its parts, which are first (as in common lathes) the feet and the frame, the moving heads, the treadle original: "marche"; the foot pedal used to power the lathe, and the pole original: "perche"; a flexible wooden lath that acts as a return spring. The specific details of the heads are as we describe them. From the end of the feet toward the north original: "septẽtrion"; in these diagrams, cardinal directions often refer to positions on the machine's frame come two fixed heads: one on the East, the other on the West. In the eastern head is a square hole where a sufficiently long square piece of wood passes, which serves to move the points that hold the work-piece closer or further away, as well as the middle heads which are mobile. In the western head (as also in the eastern) is the end of a spindle-handle, which turns in the round hole located in this head. The two moving heads are bored through to receive the axle of the points that turn, upon which the work-piece is placed. Now, on the sections of the axle of the points located between a fixed head and a moving one, there appear two orbs original: "orbes"; circular discs or cam-plates like those used on an oval-drawing compass. The entire subtlety of the lathe depends on these; their centers are on the same line as the center of the work-piece, and they are made so they can be freely adjusted according to the shape one wishes to give the oval, just as with a compass where there are two circles like these that serve the same purpose: for they guide the tool. These orbs must be set identically to one another using the toothed quadrants original: "quarts de cercles dentez"; geared arcs used for precise adjustment that are visible there. As for the fact that the ends of the axle carrying them are elbowed original: "coudez"; bent or cranked, this is done to give more force to the movement; otherwise, it would be the same if they were straight. Furthermore, the two moving heads are slotted the width of a board, which enters them freely and moves up and down. That board is perforated, and has in the middle a serpentine slot original: "fente en mode de serpent"; a winding or S-curved opening to place the iron tool according to the will of the craftsman (for the board guides the instrument, while the hand only supports it). The board rests upon the round discs which, during the movement, cause it to rise and fall, so that the iron only touches the work according to how the perforated board holding the tool rises or falls. Thus, it is evident, following the description of the oval, that the circles lose their roundness in relation to the axle which they approach from two sides. Two orbs are placed here so that one can make any oval at will, as otherwise one would need infinitely many templates. Furthermore, regarding what is seen toward the north, these are the tools and the axle with the work-piece. It should be noted here that if instead of circles one places other figures, such as those described by the second compass, one will turn those figures in the same way, as the logic does not change.

PROPOSITION VIII.

Another new type of lathe derived from the preceding one, for hollowing and working cups and goblets into oval shapes from any material that can endure iron tools.

Just as it is pleasant to turn ovals or other shapes as declared previously, it is also agreeable to hollow them out; for usually one hardly goes without the other, especially when it is a question of some beautiful vases, for which lords are curious and for whom these things are made. To achieve this, this lathe is presented, derived from the other, having the same base and similar heads, except that one is elbowed: namely the one coming from the eastern foot, so that the craftsman can more easily lean upon it. The rest is apparent to the eye. Among other things, the orb which is between the two moving heads with its axle should, for greater safety, be accompanied by another, as shown in the figure above the lathe. As for the rest, it can be adjusted as one pleases, always following the path of the invention, and observing that in