GRADUATION, the art of dividing mathematical, astronomical, and other instruments. The simplest problem in graduation is the dividing of a straight line, such as an ordi nary,scale or rule. This is commonly done by copying from a standard scale, for which purpose a dividing square and a suitable knife for cutting the diVisions are used. The dividing square is'a hard steel straight-edge, with a shoulder at right angles like a car penter's square. This is made to slide along the standard scale, and halt at each required division, when a corresponding one is ent.upon the rule, etc., by using the steel straight-edge as a ,guide to the knife. The original graduation of a straight line into equal divisions, as in making a first standard scale, etc., is performed either on the principle of bisection or stepping. In bisecting, the points of a beam-compass (see CO31 PAss) are adjusted to nearly half the length of the line to be divided; one point is then placed at one end of the line, and a faint arc struck towards the middle: this is repeated at the other end; the small distance between these arcs is then carefully bisected with the aid of a fine pointer and magnifier, which gives an accurate half of the line. The half thus obtained is again bisected in like manner, and these quarters bisected again, and so on until the required subdivision is attained. Stepping is performed with delicately pointed springi-dividers, which are set at once as nearly_as possible to the op ening of the small division required; then the points are made to step on, leaving at each step a very fine dot; and when it is found that the last dot either falls short of or overpasses the end of the line, the opening is adjusted accordingly, until perfect accuracy is obtained. Thus, if a line were divided into a thousand parts, and each division were Di o th too long or too short, the error would amount to a whole division at the end of a s thousand steps. The method of bisection is practically the most accurate, and has been adopted by Graham. Bird, Ramsdell, Troughton, and other eminent artists in original graduation. Ourved lines are divided on this principle. The chord of an arc of 60° is equal to the radius; therefore, the opening of the compasses required for striking the circle gives this arc at once to start with. • An arc 90°, or a quadrant, is obtained by
bisecting 60° and adding the half. By continual bisection of 60°, the finer graduations are produced. The amount of care, patience, skill, and delicacy of touch required in the original graduation of important astronomical instruments, is such, that not above one or two men in a generation have been found competent to the task, and these have become almost as famous as the astronomers who have successfully used the instru meets. it would be out of place here to point out in detail the minute precautions and methods 'of correction that are adopted in this most delicate manipulation; but, as an example,• we may mention the fact, that Graham, when dividing the mural quadrant for the Greenwich observatory, measured his larger chords from a scale made for the pur pose; but before laying these down on the quadrant, he left tire scale, beam•compasses, and quadrant to stand for a whole night, in order to acquire exactly the same tempera ture, and that neglect of this precaution would have involved a notable amount of error. The necessity of such extreme accuracy will be understood when we consider the appli cation that is made of these divisions. When, for example, the mariner determines his latitude by taking the meridian altitude of the sun, the graduated arc of the limb of the sextant or quadrant he uses represents, practically, the curved surface of the globe, and the drror is magnified just to the same extent as the radius of the earth exceeds that of the divided arc of the instrument. Supposing this arc to be part of a circle of 60 inches' circumference, each degree will occupy VI of an inch. An error of an inch in the division would thus mislead the mariner to an extent of more than four statute miles as regards his position on the waters. But such a ship's quadrant is but a coarse and rude instrument compared with astronomical instruments for measuring celestial angular distances by means of a divided arc; in these, an error of a thousandth part of an inch would be regarded as one of serious magnitude.