ARITHMETIC, GEOMETRY, DRAWING, ETC

Page 6

**44.** This instrument is no more than a simple application of the screw to dividing straight lines, but it will suffice to enable the reader to understand the principles on which the more complicated instruments are based.

A plate, P, (fig. 35), supports a bracket *a*, in which a screw, similar to the one described in paragraph **30**, is engaged by means of a collet; it rotates, being supported between this bracket and the small bearing *b*, that receives the pivot at the end of the screw.

The screw is fitted carefully into a nut *n*, which is rigidly attached to the small plate *h*; this carries a fine marker, movable on an axis, and terminating with a chisel edge or a fine diamond point, according as the instrument is to be used for engraving metal or glass; or it may be provided with a fine pencil if the object is merely to make subdivisions on a drawing.

This being understood, it will be evident that, if a rule or rod of any form be fixed by screws or otherwise between *f* and *g*, it can be graduated by means of the marker, the screw being made to advance; the millimeter screw can be used for dividing into millimeters and fractions direct, or, with a little calculation, into fractions of an inch. Each complete rotation of the head means a displacement of the marker by a millimeter; a half turn will be half a millimeter, etc.

**45. First method.** Having fixed a sheet of drawing paper on a smooth board, draw a line M N, (fig. 36), longer than the rule which is required to be divided. Then wth a compass or graduated scale, mark off a series of equidistant points, commencing at N, equal in number to the required series on the rule, and let M be the last division. With the center N and radius N M describe the circular arc *p v*, and with M as a center and the same radius, describe a second arc *r s* intersecting the first at O. Join O with M and N. Assume *a c* to be the rule that is to be divided into equal parts; slide it on the paper parallel to M N until the extremities , *a* and *c*, coincide with the lines O M, O N, and are equidistant from O. This position can be easily found by the aid of a compass wit one of its centers at O. Now fix the rule in position with sealing -was, or by some other means, and, with a firm upright pin, center the brass rule R at O, so that it can rotate round this centeron the pin as a pivot. It now only remains to trace a series of lines O*k*, O*b*, O*d*, etc., with the rule, to the division pointsof the line M N. The line *a c* is thus divided into as many equal parts as the line M N. The graduations will be all the more exact according as the divisions of the line M N are longer.

**46. Second method.** By side of the chuck of a wheel-cutting engine, arrange a horizontal slide *y f*, (fig. 37), that can travel easily in a direction perpendicular to *a* T. A watch fusee-chain, or a very flexible spring, is fixed by one end to the chuck, and by the other to the slide at *d*. The chain or spring is kept stretched by a weight which tends to draw the slide *f* towards *d*.

The rule to be graduated, *a*, is now fixed on the slide, and an initial division is marked on it with a pointed rotating cutter in the position usually occupied by the wheel-cutter, or else by striking a small pointed or flat-edged chisel arranged for the purpose, in such a manner as not to be liable to derangement.

Rotate the table through a definite distance; the rule a will advance throug the same distance; mark the second division; then having moved the division-plate through a distance equal to its first displacement, mark the third graduation, and so on. Suppose, for example, that it be required to make 30 divisions on the rule between *f* and *d*; select on the plate the circle corresponding to twice or thrice this amount, so that the radius of the chuck may not be relatively too short, and that the chain or spring may not act at a disadvantage; take the number 60 for example.

The two marks a *d* and *g* on the spring indicate the length that corresponds to the straight line to be divided.

The chuck is placed in the lath and reduced in diameter until the half circumference is exactly equal to the distance between these two marks on the spring, which thus fal on a diameter, *i g*, of the chuck.

The spring having been fixed by its two extremities, the slide with the rule attached is placed in position, so that the mark *g* is on the line *a* T; it will be evident from the figure that each displacement of the division-plate through one-sixtieth of its circumference will cause *a* to advance through one-thirtieth of the space between *f* and *d*

*Remarks.*- Knowing the relation of a diameter to the circumference (as 1 : 3.1416), we can determine the diameter of the chuck at once by calculation.

Its form should be a true cylinder, and it is well to place guides that will prevent the spring or chain from assuming a helical position.

The side that carries the rule should be strictly perpendicular to *a* T; and the portion of the spring that is not coiled on the chuck should always be parallel to this slide.

The chuck and spring must be quite clean and smooth, and the latter should be very pliable. A greater weight will be needed to keep the spring stretched than will suffice for a chain, and it must be increased as the strength of spring is greater.

The slide, *y f*, may simply travel over a horizontal surface between pins planted in two parallel lines. But it would be preferable to adopt some other method, for instance, to make this piece F, (fig. 37), travel with a little friction along a perfectly true cylindrical rod.

**47. Third method.** This is merely an application of the arrangement mentioned in paragraph **44**. The lathe can be employed fpr marking off a series of equidistant points in a straight line. Knowing the pitch of the slide-rest screw, determine the distance apart in, say, millimeters, of the required divisions, and fix the rule perfectly flat on the face-plate, which must be rendered immovableby any convenient means. Then mark the first point with the drill-stock. Advance the screwby the amount previously determined upon and mark the second point. After withdrawing the drill, again advance by the same amount and mark the third point, etc. Always be careful, before making the first mark, that the screw has already traveled some distance in the direction it will continue to move, so as to avoid backlash, or loss of time.

Submitted by: Samuel Kirk (##)