GRACES. 



GRADUATION. 



Band ch*nicu-r. but leave the mind without real faith 

 which U necessary for Miration, being; granted only to such individuals 

 a* hare bean elected to everlasting life. [ELECTION.] There i> also 

 difference of opinion among theologian! on what it called irmi*tU>le 

 orn<Y, many considering "that grace may be resisted and rendered 

 ineffectual by the perverse will of obstinate sinners ; * while others 

 believe " that it is never on the whole finally rejected, an on to fail 

 working faith m those who are the happy subjects of it." 



GRACES, GRATIS, or CHAIMTES, in ancient mythology, are 

 represented an three young and handsome sinters, the attendants of 

 Venus. Their names were Aglaia, Euphrosyne, and Thalia. Homer 

 mentions but one Chans. The Lacedicmoniana had only two, whom 

 they called Kleta or Klyta, and Phaenne, and a temple in honour of 

 them existed in the time of Pausanias between Sparta and Amyclic 

 (iii. IS ; ix. 35). Hwiod, however, speaks of three. Some poets name 

 I*uithea as one of the Graces. The idea of the Graces was, acn .nlhu- 

 to some, originally a symbolic personification : Aglaia represented the 

 harmony and splendour of the creation, Euphrosyne represented cheer- 

 fulness and mirth, and Thalia feasts and dances. In short, they were 

 an esthetic conception of all that is beautiful and attractive in the 

 physical as well as in the social world. Some called them the daugh- 

 ters of Zeus' by Hera, Eurynome, Harmonia, or Lethe ; others of 

 Dionysus and Aphrodite ; others, again, of AppUo and Enanthe. Their 

 worship is said to have originated in Bowtia. They were originally 

 represented as lightly clothed, but in latter times the sculptors made 

 them entirely naked : they are usually figured holding each others 

 hands or embracing, and bearing ears of com or flower*. They were 

 invoked to preside at the festive board, at nuptials, at births, kc. 

 Their images were multiplied on an infinite number of Kculpturcs, 

 paintings, gems, and votive inscriptions were affixed to them. Groups 

 of the three Graces have been found, forming one of the most pleasing 

 representations of ancient art ; and modern sculptors, Canova among 

 the rest, have sought to rival the ancients in reproducing the same 

 subject. (Millin ; Hirt ; Mullcr.) 



GRADIENTS. [RAILWAY.] 



GRADUATE. [ARTS, DEGREES is.] 



GRADUATION is the name commonly applied to the art of dividing 

 mathematical and astronomical instruments. The nature of this work 

 will not admit of a detailed account of the various methods and 

 machines used in different branches of the art ; we shall only give an 

 outline of the different processes, with reference to the standard 

 authorities, and add a few suggestions for the consideration of the 

 astronomer and artist. 



Graduation, or, as the workmen more generally style it, dividing, is 

 performed in two ways, by mating a copy of a system of <li 

 already existing, or by ariyinal dividing. The straight scales and rules 

 which are in common use are divided thus : The original pattern, and 

 the scale on which the copy is to be laid, are placed side by side ; a 

 straight edge, with a shoulder at right angles, like a carpenter's square, 

 is made to slide along the original, stopping at each division, when a 

 corresponding stroke is cut by the dividing knife on the copy. With 

 care and practice, this method admits of considerable accuracy. By 

 making the straight edge turn on the centre of a divided circle, the 

 divisions of that circje .may be copied upon any concentric circle. 

 Common protractors are thus divided, and scales upon circular limbs. 

 The original circle, which may have several orders of divisions for 

 different purposes, is called a dividing plate. 



The above method requires a itandard, which must be divided 

 originally. This will be noticed hereafter. 



Small theodolites and ordinary circular instruments must have been 

 thus divided, previous to the invention by Ramsden of his >'. 

 fnifine. The errors were of course large, and Mayer proposed to get 

 rid of them by his principle of repetition [REPEATING CIRCLE] ; but 

 Bamsden'a discovery of a machine for rapid and accurate dividing was 

 better adapted to ordinary purposes. The general principle of Ramsden's 

 dividing machine may perhaps be understood by the following descrip- 

 tion : A horizontal circle of four feet diameter turns upon a vertical 

 axis ; the outer edge is ratchcd, or notched, by on endless screw, one 

 revolution of which carries the circle round 10' ; the pressure of the 

 foot upon a treadle turns the screw forward, and there is a series of 

 very ingenious contrivances which enable the divider to turn the screw 

 through any portion of its revolution at each descent of the treadle, 

 and which restore the position of the part*, when the foot is taken off, 

 without allowing any return motion to the screw. The circle to be 

 divided is fixed upon the dividing engine, and made concentric with it, 

 and a division cut after each pressure of the foot The Board of 

 Longitude gave Ramsden a reward of 3001. for the invention of thin 

 machine, and 3151. for the machine itself, leaving it, during pleasure, 

 in his possession, on condition that he would divide sextants at 6. and 

 octant* at .'!*.. for other mathematical instrument makers. Machine* 

 of a similar kind, with some alterations and improvements, have since 

 beA constructed by John Troughton, Edward Troughton, and others, 



are still employed in all instruments which are not large 

 enough, or not sufficiently valuable, to require original dividing. 



Ramsden invented a machine for dividing straight lines, in which he 

 used a screw as his original. In the form proposed by Ramsden the 

 machine has not been deemed of any value, since a long screw can 

 never be made so accurate as a scale divided by bisections. Mr. Bryan 



Donkiu ha* contrived a machine where a screw is indeed the scale, but 

 where the errors of the screw are corrected by additional mechanism. 

 We do not think that this machine has ever been figured or described, 

 but scales have been divided, and screws cut by it of extraordinary 

 accuracy. 



Diviiling engines have been constructed somewhat differently by 

 nboch and others in Germany, and by Gambey in Paris. Much 

 t' 'I !. German division is excellent, and probably superior to any 

 English engine-dividing. It is understood to be performed by copying. 

 A large circle having been divided originally with great care, the copy 

 is placed upon it, and concentric with it A microscope is fixed inde- 

 pendently over the divided circle, the divisions are brought in suc- 

 cession under the wires of the micrometer, and a line is cut in the 

 copy after each bisection. This process is much more tedious than 

 ili'' Kuglish engine-dividing, but it admits of the greatest accuracy 

 wh( -n the workman is careful and expert It is a defect in the Ei>glih 

 engine that the circle to be divided must be detached from iU centre 

 and framing, and that when refrained there is frequently a sensible 

 eccentricity, that U, the centre of the divided circle is not in the axis 

 of rotation. This does not, however, cause any error if two or more 

 opposite readings be used. It is a worse fault, that if the inxtrnnii 'lit 

 be badly framed the circle may become distorted when the lustrum, nt 

 is again put together. Uut when the divided limb is only a part of a 

 circle, as in the sextant, any error of excentricity is of serious i: 

 ance, and this error may be very sensible after the utmost care of the 

 artist. [SEXTANT.] Gambey has constructed a dividing engine, by 

 which the instrument is divided upon its own centre, but we cannot 

 here describe the contrivance, and are not aware that it has been 

 published. 



It has not, we believe, been ascertained what average amount of 

 error is to be feared in an English circle, engine-divided. We h.ive 

 not been able to learn a more important point, whether 

 the same engine are fac-similes. If they are, it would lie < 

 mine the error of one copy, and to apply correcting pieces to the stop 

 of the dividing screw. This point is worth the attention of th< 

 for if the engine does always give the same result, the correction would 

 neither be troublesome nor expensive ; and if it does not, nor can be 

 made so to do, the German mode of copying must be followed where 

 great accuracy is necessary. 



In what has preceded, the existence of a standard is presumed ; we will 

 IMW give a brief ^md necessarily imperfect sketch of the art of original 

 dividing, Before the invention of the telescope, almost any mode of 

 division must have been sufficiently accurate. In a circle of three feet 

 radius, 1' is rather more than O'Ol inches, a quantity the half or third 

 of which is readily seen and still more easily felt, so that the observa- 

 tions of Tycho and Hevelius might very well have been exact to 1"" if 

 their greatest errors had arisen from the erroneous division of their 

 instruments. The earliest essays in dividing which we are aware of 

 are those of Hooke and Roomer. Hooke proposed to cut the edge of 

 his quadrant by an endless screw, just as in Kamsden's engine, nnd to 

 use the revolutions and parts of the screw as a (liviniun. This was 

 done in Flamsteed's sector, constructed by Tompion, probably under 

 Hooke's control, and also in his. mural arc, but both the limbs were 

 also divided into degrees by diagonal lines, &c. ; and in the ' Historia 

 Celestis,' the revolutions and parts of the screw are set down, a* well 

 as the divisions. It is found in practice that such a mode of dividing 

 is liable to verj considerable errors unless checked and corrected by 

 independent divisions. Roeiner, when he had constructed his transit- 

 circle, directed his pupil Horrebow to describe a number of <<>!< 

 circles on the limb, very near each other, and then to divide them into 

 equal parts by stepping along each with a pair of compasses opened to 

 a space of about 10'. All that he required was to have the dots round 

 and the spaces equal ; the actual value of each n\<acc was to be ascer- 

 tained by finding how many were contained in the arc of 90, between 

 the pole and the equator. Horrebow informs us that only one of these 

 series of divisions was executed which turned out exact enough to 

 satisfy Roomer. The objection to this division is the same as to 

 Hooke's endless screw, that there is no check upon an accumulation of 

 small errors ; still it is probable that Hoeiner's circle was the best 

 .!.. I. -d instrument then in existence, and the idea of determining the 

 value of the parts by observation is worthy of its author. We have 

 no account of the way in which either Klamsteed's sector or his nun .1 

 arc was was divided All we know is that the latter instrument was 

 divided by the " skilful hand of Abraham Sharp," then Flamsteed's 

 assistant. 



The art of dividing assumed a now form under the celebrated 

 Graham, the father of all good clock, watch and in-tnmi<'iit making in 

 this country, and the worthy associate of Bradley. He pointed out 

 the fundamental principle of original graduation, that you can divide 

 a given line accurately into two parts, but not into three or five equal 

 parts. The dividing tool employed by Graham was the beam-compass, 

 a straight rod of wood or metal, on which perpendicular jwints ct 

 are fixed. Now if a line or arc is to be bisected, the points of the 

 beam-compass are placed utar/i/ at the distance of half the line, or the 

 chord of half the arc between the dots. One point is placed in one 

 dot, and a faint arc is struck with the other point towards the distant 

 dot, and this operation is repeated with the second dot as a i 

 The two faint arcs will cither include a small space, or leave a small 



