verse-making, the full title being Gradus ad Parnassum, a step to Parnassus, the abode of the Muses.

GRADE, GRADIENT, GRADUAL, GRADUATE, GRADUS, etc.: see under GRADATION.

GRAD'UAL, in the Liturgy of the Roman and other Western Churches: that portion of the mass between the epistle and the gospel, consisting of a few verses of Holy Scripture, usually from the Psalms. It was called original. ly Responsum,' or 'Cantus Responsorius;' but, probably for distinction from other portions of the service called by the same name, its present appellation has been substituted. The name 'gradual' is derived from the place at which this response was chanted, which was either the ambo or chanting pulpit approached by 'steps' (gradus), or the 'steps' themselves, whether of the ambo or of the altar. Originally, as we find from St. Augustine, the G. response consisted of an entire psalm, and in the mass or the first Sunday in Lent the entire xci. (xc. in the Vulgate) psalm is chanted. In the Ambrosian Liturgy, the G. is partly from the Old and partly from the New Testament. Strictly, only the first sentence was called the G.; the remainder was called the 'verse.' The G. in the Roman Liturgy, is always followed by the alleluia,' except in penitential time, when a slow and mournful chant, called the 'Tract,' is substituted: see GRADUALE.

GRADUALE, grăd-ū-ā'lē: the music of the Gradual q.v.) in the Roman Liturgy. It is performed during mass after the epistle is read, and it is said to have been used from the earliest times to allow the officiating priest time to take his place on the steps of the reading-desk, or on the steps of the gospel side of the altar. The music is acrording to the character of the words, and may be either aria, duet, or chorus. The composition must not be long, as the priest has little ceremony to go through during its performance. The best specimens of the graduale are Haydn's Insane et Vana Curae; Salve Regina, or Mozart's Misericordias Domini; Sancta Maria; De Profundis, etc.

GRADUAL PSALMS, or PSALMS OF THE STEPS,' or 'SONGS OF DEGREES': in both the Hebrew and the Christian service-books, the fifteen psalms, cxx.-cxxxiv. (cxix.-cxxxiii. in the Vulgate). The origin of the name is uncertain. The rabbins trace it to a fabulous incident connected with the building of the second temple; others explain it as an allusion to the 15 steps by which (Ezekiel xl. 22-26) the temple was reached; others, again, regard these psalms as containing a prophetic allusion to the return from captivity, which, in the language of the Jews, was a going up,' psalm cxxxiv. being the full outburst of exultation at the accomplishment of that great object of hope and longing. These psalms, in the Roman Church, form part of the office of each Wednesday during Lent.

GRADUATION: art of dividing mathematical, astronomical, and other instruments. The simplest problem in G. is the dividing of a straight line, such as an ordinary

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 carpenter's square. This is made to slide along the standard scale, and halt at each required division, when a corresponding one is cut 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 on the principle either of bisection or of stepping. In bisecting, the points of a beam-compass (see COMPASS) are adjusted to nearly half the length of the line to be divided; one point is then plaeed at one end of the line, and a faint arc struck toward 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 spring-dividers, which are set at once as nearly as possible to the opening 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 1000 too long or too short, the error would amount to a whole division at the end of a thousand steps. The method of bisection is prac tically the most accurate, and has been adopted by eminent artists in original graduation. Curved 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 G. 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 instruments. Graham, when dividing the mural quadrant for the Greenwich Observatory, measured his larger chords from a scale made for the purpose; but before laying these down on the quadrant, he left the scale, beam-compasses, and quadrant to stand for a whole night, in order to acquire exactly the same temperature: neglect of this precaution would have involved a notable amount of error. The necessity of such extreme accuracy arises from the application 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 that he uses represents, practically, the

Graduation.-Dividing-engine: A, A, Circle, ratched into teeth at lower edge,

C; E, axis of circle; TT', Tangent-screw; T', A handle for turning it; T, A disk shaped head; D, Carriage; P, P, P, P, Pillars; K, A handle.

Vol. 13.

curved surface of the globe, and the error 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 of an inch. An error of 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 of serious magnitude. The methods of original graduation above described are not practically adopted except for the largest and most important astronomical or geodesical instruments. Ordinary instruments are graduated by dividing plates or engines which copy and adapt a set of already existing divisions. The dividing plate for common purposes, such as dividing compass rings, etc., is a divided circle with a steel straight-edge, made movable on the axis or arbor of the plate in such a manner that its edge during every part of its revolution shall fall in the exact line from centre to circumference. The ring, protractor, or other instrument to he divided, is clamped upon the plate with its centre exactly coinciding with that of the plate, and the straightedge is moved round, and made to halt at the required divisions on the circumference of the dividing-plate, and by using the steel straight-edge as a guide, corresponding divisions are marked off on the concentric arc of the instrument to be divided. The dividing-engine is a very complex machine, requiring the greatest accuracy and care in its construction; so much so, that the possession of a good one may sometimes afford a good income, with a moderate amount of labor in using it. Among the most celebrated dividing-engines may be mentioned those of Ramsden, Troughton, Simms, and Ross. Their principal parts consists of a large circle divided with extreme care by original graduation. This wheel is racked on its edge with teeth as equal and accurate as the divisions; a very carefully constructed endless screw works in these teeth, and is moved through any given number of revolutions, or any measured fraction of a revolution, by means of a treadle or other suitable power, thus making the requisite steps for each division; another part of the machine cutting a fine line at the moment of the halt of each step. These divisions are cut upon an arc of silver, gold, or platinum, which is soldered or inlaid upon the limb of the instrument, the precious metals being used, on account of the oxidation to which common metals are liable.

GRÆ CIA, MAG'NA. See MAGNA GRÆCIA.

GRAVIUS, grā'vē-ûs (or GRÄFE, grä'feh, or GREFFE, rgeffeh), JOHANN GEORG: 1632, Jan. 29-1703, Jan. 11; b. Naumberg, Saxony: classical scholar. He was educated at the Gymnasium of Pforta, and began studying law in the Univ. of Leipsic. where, in his 18th year, he