In dividing diagonal scales, the beginnings and endings of the inclining lines are marked off with the dividing-knife from a pattern of equal parts, and afterwards those lines are traced by means of a straight edge, similar to the blade of the square. The knife being dropped into one of the marks, the ruler is brought into contact with it, and very nearly up to the other mark ; the thumb of the left hand is placed exactly over the ruler at the former, and the knife being dropped into the latter, the ruler is brought into con tact with it, being turned under the thumb as on a centre. Instead of drawing the diagonal lines by hand, some use the bevel ; and as by this the whole is done at once from the pattern, it would seem to he the more methodical way, but it requires a very nice adjustment to make the bevel agree with the square which had been previously used to draw the perpendicular lines. But which ever method is used, to do it with sufficient exactness is one of the most difficult operations of this department of the art.
In the sector, and every other instrument, where the lines, which bound the length of the dividing strokes, are not parallel to the edge, so as to he traced with the gauge, the straight-edge and dividing-knife are used : The begin ning and ending of the lines are set off in their proper po sitions by the spring-dividers : a tool so important in every branch of the art, that a description of it might have sooner claimed a place.
This is a kind of compasses formed altogether of fine steel. It consists of a circular bow and two legs, all in one piece, as represented in full dimensions by Fig. 5. Plate CCLXXVIII. The bow is strong and well tempered, and, without danger of breaking, allows a motion of the points from the distance of about an inch to their contact. This is their range for use, but the elasticity is not exhausted until the points are separated nearly twice as far. At about three-fifths, reckoning from the bow, the adjusting screw has its place : It is at right angles with the legs when the distance between the points is small, because short lengths are most commonly taken. The screw, throughout its whole length, has a fine deep thread of about sixty turns to an inch ; is fastened to one of the legs by a pin passing through it, and upon which it turns as on a centre, in order to obey the circular spring of the bow ; and the other leg is perforated, in order that, at every distance of the points, it may pass freely through it. A nut, the female screw of which is nicely fitted to the former, by its action overcomes the expansive force of the bow, and regulates the distance between the points : But this nut does not come in contact with the leg ; there is interposed between them a saddle-piece, which exterior to the screw is made conical, and this is received by a hollow conical part in the nut, which it exactly fits. The saddle-piece next the leg is
formed into a knife-edge, or sharp angle, which, resting upon the sharp bottom of a notch in the leg, keeps the for mer from turning round with the nut, at the same time that it allows the angular change, as the distance of the points is varied. By this contrivance, freedom of action of the screw is preserved, and the possibility of its change of place, and consequent alteration of the distance between the points, prevented. The legs are bored in the direction of their length, to a depth of about three quarters of an inch ; they have each, near their extremities, a part which projects outwards. These parts, as well as the length bored, are cut open on the outer edge by a saw. A screw, in each of the projecting parts, passes freely through one half, but acting in the other, brings the parts nearer toge ther, and furnishes the means or securing the points firmly in their places. The points themselves are cylinders, ex actly filling the bore of the legs, and their ends are worked to the requisite sharpness; and, in order that they may measure the shortest possible distance, are brought very near the inner extremity of the diameter. At the very point, however, they should be round, and in every direc tion the sides must make equal angles with the perpen dicular ; for, were they not so, a distance set off with them, it is obvious, would be altered by pressure.
In using this instrument, the fore-finger is pressed upon the bow, the thumb and middle finger keeping it upright, while the other fingers prop the hand ; but where a dis tance is to be set off many times in succession, the dividers are to be twirled round in the same direction, making a dot at every half turn. This is the manner of handling the tool for common purposes ; but for the accurate bisection of a distance, they must not he touched by any other part than the leg near the point, which is lodged in one extremity, while with the other a faint arc is described : the same thing being done from the other end of the distance, the middle point is secured by making a dot with a fine conical poin tril. In every use of the dividers, a magnifying glass is to be held in the left hand.