Chapter II: Sphere of Government (3)

GRACE, WILLIAM GILBERT (1848- ), English cricketer, was born at Downend, Gloucestershire, on the 18th of July 1848. He found himself in an atmosphere charged with cricket, his father (Henry Mills Grace) and his uncle (Alfred Pocock) being as enthusiastic over the game as his elder brothers, Henry, Alfred and Edward Mills; indeed, in E. M. Grace the family name first became famous. A younger brother, George Frederick, also added to the cricket reputation of the family. "W. G." witnessed his first great match when he was hardly six years old, the occasion being a game between W. Clarke's All-England Eleven and twenty-two of West Gloucestershire. He was endowed by nature with a splendid physique as well as with powers of self-restraint and determination. At the acme of his career he stood full 6 ft. 2 in., being powerfully proportioned, loose yet strong of limb. A non-smoker, and very moderate in all matters, he kept himself in condition all the year round, shooting, hunting or running with the beagles as soon as the cricket season was over. He was also a fine runner, 440 yds. over 20 hurdles being his best distance; and it may be quoted as proof of his stamina that on the 30th of July 1866 he scored 224 not out for England _v._ Surrey, and two days later won a race in the National and Olympian Association meeting at the Crystal Palace. The title of "champion" was well earned by one who for thirty-six years (1865-1900 inclusive) was actively engaged in first-class cricket. In each of these years he was invited to represent the Gentlemen in their matches against the Players, and, when an Australian eleven visited England, to play for the mother country. As late as 1899 he played in the first of the five international contests; in 1900 he played against the players at the Oval, scoring 58 and 3. At fifty-three he scored nearly 1300 runs in first-class cricket, made 100 runs and over on three different occasions and could claim an average of 42 runs. Moreover, his greatest triumphs were achieved when only the very best cricket grounds received serious attention; when, as some consider, bowling was maintained at a higher standard and when all hits had to be run out. He, with his two brothers, E. M. and G. F., assisted by some fine amateurs, made Gloucestershire in one season a first-class county; and it was he who first enabled the amateurs of England to meet the paid players on equal terms and to beat them. There was hardly a "record" connected with the game which did not stand to his credit. Grace was one of the finest fieldsmen in England, in his earlier days generally taking long-leg and cover-point, in later times generally standing point. He was, at his best, a fine thrower, fast runner and safe "catch." As a bowler he was long in the first flight, originally bowling fast, but in later times adopting a slower and more tricky style, frequently very effective. By profession he was a medical man. In later years he became secretary and manager of the London County Cricket Club. He was married in 1873 to Miss Agnes Day, and one of his sons played for two years in the Cambridge eleven. He was the recipient of two national testimonials: the first, amounting to L1500, being presented to him in the form of a clock and a cheque at Lord's ground by Lord Charles Russell on the 22nd of July 1879; the second, collected by the M.C.C., the county of Gloucestershire, the _Daily Telegraph_ and the _Sportsman_, amounted to about L10,000, and was presented to him in 1896. He visited Australia in 1873-1874 (captain), and in 1891-1892 with Lord Sheffield's Eleven (captain); the United States and Canada in 1872, with R. A. Fitzgerald's team.

Dr Grace played his first great match in 1863, when, being only
fifteen years of age, he scored 32 against the All-England Eleven and
the bowling of Jackson, Tarrant and Tinley; but the scores which first
made his name prominent were made in 1864, viz. 170 and 56 not out for
the South Wales Club against the Gentlemen of Sussex. It was in 1865
that he first took an active part in first-class cricket, being then 6
ft. in height, and 11 stone in weight, and playing twice for the
Gentlemen _v._ the Players, but his selection was mainly due to his
bowling powers, the best exposition of which was his aggregate of 13
wickets for 84 runs for the Gentlemen of the South _v._ the Players of
the South. His highest score was 400 not out, made in July 1876
against twenty-two of Grimsby; but on three occasions he was twice
dismissed without scoring in matches against odds, a fate that never
befell him in important cricket. In first-class matches his highest
score was 344, made for the M.C.C. v. Kent at Canterbury, in August
1876; two days later he made 177 for Gloucestershire _v._ Notts, and
two days after this 318 not out for Gloucestershire _v._ Yorkshire,
the two last-named opposing counties being possessed of exceptionally
strong bowling; thus in three consecutive innings Grace scored 839
runs, and was only got out twice. His 344 was the third highest
individual score made in a big match in England up to the end of 1901.
He also scored 301 for Gloucestershire _v._ Sussex at Bristol, in
August 1896. He made over 200 runs on ten occasions, the most notable
perhaps being in 1871, when he performed the feat twice, each time in
benefit matches, and each time in the second innings, having been each
time got out in the first over of the first innings. He scored over
100 runs on 121 occasions, the hundredth score being 288, made at
Bristol for Gloucestershire _v._ Somersetshire in 1895. He made every
figure from 0 to 100, on one occasion "closing" the innings when he
had made 93, the only total he had never made between these limits. In
1871 he made ten "centuries," ranging from 268 to 116. In the matches
between the Gentlemen and Players he scored "three figures" fifteen
times, and at every place where these matches have been played. He
made over 100 in each of his "first appearances" at Oxford and
Cambridge. Three times he made over 100 in each innings of the same
match, viz. at Canterbury, in 1868, for South v. North of the Thames,
130 and 102 not out; at Clifton, in 1887, for Gloucestershire _v._
Kent, 101 and 103 not out; and at Clifton, in 1888, for
Gloucestershire _v._ Yorkshire, 148 and 153. In 1869, playing at the
Oval for the Gentlemen of the South _v._ the Players of the South,
Grace and B. B. Cooper put on 283 runs for the first wicket, Grace
scoring 180 and Cooper 101. In 1886 Grace and Scotton put on 170 runs
for the first wicket of England _v._ Australia; this occurred at the
Oval in August, and Grace's total score was 170. In consecutive
innings against the Players from 1871 to 1873 he scored 217, 77 and
112, 117, 163, 158 and 70. He only twice scored over 100 in a big
match in Australia, nor did he ever make 200 at Lord's, his highest
being 196 for the M.C.C. _v._ Cambridge University in 1894. His
highest aggregates were 2739 (1871), 2622 (1876), 2346 (1895), 2139
(1873), 2135 (1896) and 2062 (1887). He scored three successive
centuries in first-class cricket in 1871, 1872, 1873, 1874 and 1876.
Playing against Kent at Gravesend in 1895, he was batting, bowling or
fielding during the whole time the game was in progress, his scores
being 257 and 73 not out. He scored over 1000 runs and took over 100
wickets in seven different seasons, viz. in 1874, 1665 runs and 129
wickets; in 1875, 1498 runs, 192 wickets; in 1876, 2622 runs, 124
wickets; in 1877, 1474 runs, 179 wickets; in 1878, 1151 runs, 153
wickets; in 1885, 1688 runs, 118 wickets; in 1886, 1846 runs, 122
wickets. He never captured 200 wickets in a season, his highest record
being 192 in 1875. Playing against Oxford University in 1886, he took
all the wickets in the first innings, at a cost of 49 runs. In 1895 he
not only made his hundredth century, but actually scored 1000 runs in
the month of May alone, his chief scores in that month being 103, 288,
256, 73 and 169, he being then forty-seven years old. He also made
during that year scores of 125, 119, 118, 104 and 103 not out, his
aggregate for the year being 2346 and his average 51; his innings of
118 was made against the Players (at Lord's), the chief bowlers being
Richardson, Mold, Peel and Attewell; he scored level with his partner,
A. E. Stoddart (his junior by fifteen years), the pair making 151
before a wicket fell, Grace making in all 118 out of 241. This may
fairly be considered one of his most wonderful years. In 1898 the
match between Gentlemen _v._ Players was, as a special compliment,
arranged by the M.C.C. committee to take place on his birthday, and he
celebrated the event by scoring 43 and 31 not out, though handicapped
by lameness and an injured hand. In twenty-six different seasons he
scored over 1000 runs, in three of these years being the only man to
do so and five times being one out of two.

During the thirty-six years up to and including 1900 he scored nearly
51,000 runs, with an average of 43; and in bowling he took more than
2800 wickets, at an average cost of about 20 runs per wicket. He made
his highest aggregate (2739 runs) and had his highest average (78) in
1871; his average for the decade 1868-1877 was 57 runs. His style as a
batsman was more commanding than graceful, but as to its soundness and
efficacy there were never two opinions; the severest criticism ever
passed upon his powers was to the effect that he did not play slow
bowling quite as well as fast. (W. J. F.)

GRACE (Fr. _grace_, Lat. _gratia_, from _gratus_, beloved, pleasing; formed from the root _cra-_, Gr. [Greek: chas-] cf. [Greek: chairo, charma, charis]), a word of many shades of meaning, but always connoting the idea of favour, whether that in which one stands to others or that which one shows to others. The _New English Dictionary_ groups the meanings of the word under three main heads: (1) Pleasing quality, gracefulness, (2) favour, goodwill, (3) gratitude, thanks.

It is in the second general sense of "favour bestowed" that the word has its most important connotations. In this sense it means something given by superior authority as a concession made of favour and goodwill, not as an obligation or of right. Thus, a concession may be made by a sovereign or other public authority "by way of grace." Previous to the Revolution of 1688 such concessions on the part of the crown were known in constitutional law as "Graces." "Letters of Grace" (_gratiae, gratiosa rescripta_) is the name given to papal rescripts granting special privileges, indulgences, exemptions and the like. In the language of the universities the word still survives in a shadow of this sense. The word "grace" was originally a dispensation granted by the congregation of the university, or by one of the faculties, from some statutable conditions required for a degree. In the English universities these conditions ceased to be enforced, and the "grace" thus became an essential preliminary to any degree; so that the word has acquired the meaning of (_a_) the licence granted by congregation to take a degree, (_b_) other decrees of the governing body (originally dispensations from statutes), all such degrees being called "graces" at Cambridge, (_c_) the permission which a candidate for a degree must obtain from his college or hall.

To this general sense of exceptional favour belong the uses of the word in such phrases as "do me this grace," "to be in some one's good graces" and certain meanings of "the grace of God." The style "by the grace of God," borne by the king of Great Britain and Ireland among other sovereigns, though, as implying the principle of "legitimacy," it has been since the Revolution sometimes qualified on the continent by the addition of "and the will of the people," means in effect no more than the "by Divine Providence," which is the style borne by archbishops. To the same general sense of exceptional favour belong the phrases implying the concession of a right to delay in fulfilling certain obligations, e.g. "a fortnight's grace." In law the "days of grace" are the period allowed for the payment of a bill of exchange, after the term for which it has been drawn (in England three days), or for the payment of an insurance premium, &c. In religious language the "Day of Grace" is the period still open to the sinner in which to repent. In the sense of clemency or mercy, too, "grace" is still, though rarely used: "an Act of Grace" is a formal pardon or a free and general pardon granted by act of parliament. Since to grant favours is the prerogative of the great, "Your Grace," "His Grace," &c., became dutiful paraphrases for the simple "you" and "he." Formerly used in the royal address ("the King's Grace," &c.), the style is in England now confined to dukes and archbishops, though the style of "his most gracious majesty" is still used. In Germany the equivalent, _Euer Gnaden_, is the style of princes who are not _Durchlaucht_ (i.e. Serene Highness), and is often used as a polite address to any superior.

In the language of theology, though in the English Bible the word is used in several of the above senses, "grace" (Gr. [Greek: charis]) has special meanings. Above all, it signifies the spontaneous, unmerited activity of the Divine Love in the salvation of sinners, and the Divine influence operating in man for his regeneration and sanctification. Those thus regenerated and sanctified are said to be in a "state of grace." In the New Testament grace is the forgiving mercy of God, as opposed to any human merit (Rom. xi. 6; Eph. ii. 5; Col. i. 6, &c.); it is applied also to certain gifts of God freely bestowed, e.g. miracles, tongues, &c. (Rom. xv. 15; 1 Cor. xv. 10; Eph. iii. 8, &c.), to the Christian virtues, gifts of God also, e.g. charity, holiness, &c. (2 Cor. viii. 7; 2 Pet. iii. 18). It is also used of the Gospel generally, as opposed to the Law (John i. 17; Rom. vi. 14; 1 Pet. v. 12, &c.); connected with this is the use of the term "year of grace" for a year of the Christian era.

The word "grace" is the central subject of three great theological controversies: (1) that of the nature of human depravity and regeneration (see PELAGIUS), (2) that of the relation between grace and free-will (see CALVIN, JOHN, and ARMINIUS, JACOBUS), (3) that of the "means of grace" between Catholics and Protestants, i.e. whether the efficacy of the sacraments as channels of the Divine grace is _ex opere operato_ or dependent on the faith of the recipient.

In the third general sense, of thanks for favours bestowed, "grace" survives as the name for the thanksgiving before or after meals. The word was originally used in the plural, and "to do, give, render, yield graces" was said, in the general sense of the French _rendre graces_ or Latin _gratias agere_, of any giving thanks. The close, and finally exclusive, association of the phrase "to say grace" with thanksgiving at meals was possibly due to the formula "Gratias Deo agamus" ("let us give thanks to God") with which the ceremony began in monastic refectories. The custom of saying grace, which obtained in pre-Christian times among the Jews, Greeks and Romans, and was adopted universally by Christian peoples, is probably less widespread in private houses than it used to be. It is, however, still maintained at public dinners and also in schools, colleges and institutions generally. Such graces are generally in Latin and of great antiquity: they are sometimes short, e.g. "Laus Deo," "Benedictus benedicat," and sometimes, as at the Oxford and Cambridge colleges, of considerable length. In some countries grace has sunk to a polite formula; in Germany, e.g. it is usual before and after meals to bow to one's neighbours and say "Gesegnete Malzeit!" (May your meal be blessed), a phrase often reduced in practice to "Malzeit" simply.

GRACES, THE, (Gr. [Greek: Charites], Lat. _Gratiae_), in Greek mythology, the personification of grace and charm, both in nature and in moral action. The transition from a single goddess, Charis, to a number or group of Charites, is marked in Homer. In the _Iliad_ one Charis is the wife of Hephaestus, another the promised wife of Sleep, while the plural Charites often occurs. The Charites are usually described as three in number--Aglaia (brightness), Euphrosyne (joyfulness), Thalia (bloom)--daughters of Zeus and Hera (or Eurynome, daughter of Oceanus), or of Helios and Aegle; in Sparta, however, only two were known, Cleta (noise) and Phaenna (light), as at Athens Auxo (increase) and Hegemone (queen). They are the friends of the Muses, with whom they live on Mount Olympus, and the companions of Aphrodite, of Peitho, the goddess of persuasion, and of Hermes, the god of eloquence, to each of whom charm is an indispensable adjunct. The need of their assistance to the artist is indicated by the union of Hephaestus and Charis. The most ancient seat of their cult was Orchomenus in Boeotia, where their oldest images, in the form of stones fallen from heaven, were set up in their temple. Their worship was said to have been instituted by Eteocles, whose three daughters fell into a well while dancing in their honour. At Orchomenus nightly dances took place, and the festival Charitesia, accompanied by musical contests, was celebrated; in Paros their worship was celebrated without music or garlands, since it was there that Minos, while sacrificing to the Charites, received the news of the death of his son Androgeus; at Messene they were revered together with the Eumenides; at Athens, their rites, kept secret from the profane, were held at the entrance to the Acropolis. It was by Auxo, Hegemone and Agraulos, the daughter of Cecrops, that young Athenians, on first receiving their spear and shield, took the oath to defend their country. In works of art the Charites were represented in early times as beautiful maidens of slender form, hand in hand or embracing one another and wearing drapery; later, the conception predominated of three naked figures gracefully intertwined. Their attributes were the myrtle, the rose and musical instruments. In Rome the Graces were never the objects of special religious reverence, but were described and represented by poets and artists in accordance with Greek models.

See F. H. Krause, _Musen, Gratien, Horen, und Nymphen_ (1871), and the
articles by Stoll and Furtwangler in Roscher's _Lexikon der
Mythologie_, and by S. Gsell in Daremberg and Saglio's _Dictionnaire
des antiquites_, with the bibliography.

GRACIAN Y MORALES, BALTASAR (1601-1658), Spanish prose writer, was born at Calatayud (Aragon) on the 8th of January 1601. Little is known of his personal history except that on May 14, 1619, he entered the Society of Jesus, and that ultimately he became rector of the Jesuit college at Tarazona, where he died on the 6th of December, 1658. His principal works are _El Heroe_ (1630), which describes in apophthegmatic phrases the qualities of the ideal man; the _Arte de ingenio, tratado de la Agudeza_ (1642), republished six years afterwards under the title of _Agudeza, y arte de ingenio_ (1648), a system of rhetoric in which the principles of _conceptismo_ as opposed to culteranismo are inculcated; _El Discreto_ (1645), a delineation of the typical courtier; _El Oraculo manual y arte de prudencia_ (1647), a system of rules for the conduct of life; and _El Criticon_ (1651-1653-1657), an ingenious philosophical allegory of human existence. The only publication which bears Gracian's name is _El Comulgatorio_ (1655); his more important books were issued under the pseudonym of Lorenzo Gracian (possibly a brother of the writer) or under the anagram of Gracian de Marlones. Gracian was punished for publishing without his superior's permission _El Criticon_ (in which Defoe is alleged to have found the germ of _Robinson Crusoe_); but no objection was taken to its substance. He has been excessively praised by Schopenhauer, whose appreciation of the author induced him to translate the _Oraculo manual_, and he has been unduly depreciated by Ticknor and others. He is an acute thinker and observer, misled by his systematic misanthropy and by his fantastic literary theories.

See Karl Borinski, _Baltasar Gracian und die Hoflitteratur in
Deutschland_ (Halle, 1894); Benedetto Croce, _I Trattatisti italiani
del "concettismo" e Baltasar Gracian_ (Napoli, 1899); Narciso Jose
Linan y Heredia, _Baltasar Gracian_ (Madrid, 1902). Schopenhauer and
Joseph Jacobs have respectively translated the _Oraculo manual_ into
German and English.

GRACKLE (Lat. _Gracculus_ or _Graculus_), a word much used in ornithology, generally in a vague sense, though restricted to members of the families _Sturnidae_ belonging to the Old World and _Icteridae_ belonging to the New. Of the former those to which it has been most commonly applied are the species known as mynas, mainas, and minors of India and the adjacent countries, and especially the _Gracula religiosa_ of Linnaeus, who, according to Jerdon and others, was probably led to confer this epithet upon it by confounding it with the _Sturnus_ or _Acridotheres tristis_,[1] which is regarded by the Hindus as sacred to Ram Deo, one of their deities, while the true _Gracula religiosa_ does not seem to be anywhere held in veneration. This last is about 10 in. in length, clothed in a plumage of glossy black, with purple and green reflections, and a conspicuous patch of white on the quill-feathers of the wings. The bill is orange and the legs yellow, but the bird's most characteristic feature is afforded by the curious wattles of bright yellow, which, beginning behind the eyes, run backwards in form of a lappet on each side, and then return in a narrow stripe to the top of the head. Beneath each eye also is a bare patch of the same colour. This species is common in southern India, and is represented farther to the north, in Ceylon, Burma, and some of the Malay Islands by cognate forms. They are all frugivorous, and, being easily tamed and learning to pronounce words very distinctly, are favourite cage-birds.[2]

In America the name Grackle has been applied to several species of the genera _Scolecophagus_ and _Quiscalus_, though these are more commonly called in the United States and Canada "blackbirds," and some of them "boat-tails." They all belong to the family _Icteridae_. The best known of these are the rusty grackle, _S. ferrugineus_, which is found in almost the whole of North America, and _Q. purpureus_, the purple grackle or crow-blackbird, of more limited range, for though abundant in most parts to the east of the Rocky Mountains, it seems not to appear on the Pacific side. There is also Brewer's or the blue-headed grackle, _S. cyanocephalus_, which has a more western range, not occurring to the eastward of Kansas and Minnesota. A fourth species, _Q. major_, inhabits the Atlantic States as far north as North Carolina. All these birds are of exceedingly omnivorous habit, and though destroying large numbers of pernicious insects are in many places held in bad repute from the mischief they do to the corn-crops. (A. N.)

FOOTNOTES:

[1] By some writers the birds of the genera _Acridotheres_ and
_Temenuchus_ are considered to be the true mynas, and the species of
_Gracula_ are called "hill mynas" by way of distinction.

[2] For a valuable monograph on the various species of _Gracula_ and
its allies see Professor Schlegel's "Bijdrage tot de Kennis von het
Geschlacht Beo'" (_Nederlandsch Tijdschrift voor de Dierkunde_ i.
1-9).

GRADISCA, a town of Austria, in the province of Gorz and Gradisca, 10 m. S.W. of Gorz by rail. Pop. (1900) 3843, mostly Italians. It is situated on the right bank of the Isonzo and was formerly a strongly fortified place. Its principal industry is silk spinning. Gradisca originally formed part of the margraviate of Friuli, came under the patriarchate of Aquileia in 1028, and in 1420 to Venice. Between 1471 and 1481 Gradisca was fortified by the Venetians, but in 1511 they surrendered it to the emperor Maximilian I. In 1647 Gradisca and its territory, including Aquileia and forty-three smaller places, were erected into a separate countship in favour of Johann Anton von Eggenberg, duke of Krumau. On the extinction of his line in 1717, it reverted to Austria, and was completely incorporated with Gorz in 1754. The name was revived by the constitution of 1861, which established the crownland of Gorz and Gradisca.

GRADO, a town of northern Spain, in the province of Oviedo; 11 m. W. by N. of the city of Oviedo, on the river Cubia, a left-hand tributary of the Nalon. Pop. (1900) 17,125. Grado is built in the midst of a mountainous, well-wooded and fertile region. It has some trade in timber, live stock, cider and agricultural produce. The nearest railway station is that of the Fabrica de Trubia, a royal cannon-foundry and small-arms factory, 5 m. S.E.

GRADUAL (Med. Lat. _gradualis_, of or belonging to steps or degrees; _gradus_, step), advancing or taking place by degrees or step by step; hence used of a slow progress or a gentle declivity or slope, opposed to steep or precipitous. As a substantive, "gradual" (Med. Lat. _graduale_ or _gradale_) is used of a service book or antiphonal of the Roman Catholic Church containing certain antiphons, called "graduals," sung at the service of the Mass after the reading or singing of the Epistle. This antiphon received the name either because it was sung on the steps of the altar or while the deacon was mounting the steps of the ambo for the reading or singing of the Gospel. For the so-called Gradual Psalms, cxx.-cxxxiv., the "songs of degrees," LXX. [Greek: ode ana bathmon], see PSALMS, BOOK OF.

GRADUATE (Med. Lat. _graduare_, to admit to an academical degree, _gradus_), in Great Britain a verb now only used in the academical sense intransitively, i.e. "to take or proceed to a university degree," and figuratively of acquiring knowledge of, or proficiency in, anything. The original transitive sense of "to confer or admit to a degree" is, however, still preserved in America, where the word is, moreover, not strictly confined to university degrees, but is used also of those successfully completing a course of study at any educational establishment. As a substantive, a "graduate" (Med. Lat. _graduatus_) is one who has taken a degree in a university. Those who have matriculated at a university, but not yet taken a degree, are known as "undergraduates." The word "student," used of undergraduates e.g. in Scottish universities, is never applied generally to those of the English and Irish universities. At Oxford the only "students" are the "senior students" (i.e. fellows) and "junior students" (i.e. undergraduates on the foundation, or "scholars") of Christ Church. The verb "to graduate" is also used of dividing anything into degrees or parts in accordance with a given scale. For the scientific application see GRADUATION below. It may also mean "to arrange in gradations" or "to adjust or apportion according to a given scale." Thus by "a graduated income-tax" is meant the system by which the percentage paid differs according to the amount of income on a pre-arranged scale.

GRADUATION (see also GRADUATE), the art of dividing straight scales, circular arcs or whole circumferences into any required number of equal parts. It is the most important and difficult part of the work of the mathematical instrument maker, and is required in the construction of most physical, astronomical, nautical and surveying instruments.

The art was first practised by clockmakers for cutting the teeth of their wheels at regular intervals; but so long as it was confined to them no particular delicacy or accurate nicety in its performance was required. This only arose when astronomy began to be seriously studied, and the exact position of the heavenly bodies to be determined, which created the necessity for strictly accurate means of measuring linear and angular magnitudes. Then it was seen that graduation was an art which required special talents and training, and the best artists gave great attention to the perfecting of astronomical instruments. Of these may be named Abraham Sharp (1651-1742), John Bird (1709-1776), John Smeaton (1724-1792), Jesse Ramsden (1735-1800), John Troughton, Edward Troughton (1753-1835), William Simms (1793-1860) and Andrew Ross.

The first graduated instrument must have been done by the hand and eye alone, whether it was in the form of a straight-edge with equal divisions, or a screw or a divided plate; but, once in the possession of one such divided instrument, it was a comparatively easy matter to employ it as a standard. Hence graduation divides itself into two distinct branches, _original graduation_ and _copying_, which latter may be done either by the hand or by a machine called a dividing engine. Graduation may therefore be treated under the three heads of _original graduation_, _copying_ and _machine graduation_.

_Original Graduation._--In regard to the graduation of straight scales elementary geometry provides the means of dividing a straight line into any number of equal parts by the method of continual bisection; but the practical realization of the geometrical construction is so difficult as to render the method untrustworthy. This method, which employs the common diagonal scale, was used in dividing a quadrant of 3 ft. radius, which belonged to Napier of Merchiston, and which only read to minutes--a result, according to Thomson and Tait (_Nat. Phil._), "giving no greater accuracy than is now attainable by the pocket sextants of Troughton and Simms, the radius of whose arc is little more than an inch."

The original graduation of a straight line is done either by the
method of continual bisection or by stepping. In continual bisection
the entire length of the line is first laid down. Then, as nearly as
possible, half that distance is taken in the beam-compass and marked
off by faint arcs from each end of the line. Should these marks
coincide the exact middle point of the line is obtained. If not, as
will almost always be the case, the distance between the marks is
carefully bisected by hand with the aid of a magnifying glass. The
same process is again applied to the halves thus obtained, and so on
in succession, dividing the line into parts represented by 2, 4, 8,
16, &c. till the desired divisions are reached. In the method of
stepping the smallest division required is first taken, as accurately
as possible, by spring dividers, and that distance is then laid off,
by successive steps, from one end of the line. In this method, any
error at starting will be multiplied at each division by the number of
that division. Errors so made are usually adjusted by the dots being
put either back or forward a little by means of the dividing punch
guided by a magnifying glass. This is an extremely tedious process, as
the dots, when so altered several times, are apt to get insufferably
large and shapeless.

The division of circular arcs is essentially the same in principle as the graduation of straight lines.

The first example of note is the 8-ft. mural circle which was
graduated by George Graham (1673-1751) for Greenwich Observatory in
1725. In this two concentric arcs of radii 96.85 and 95.8 in.
respectively were first described by the beam-compass. On the inner of
these the arc of 90 deg. was to be divided into degrees and 12th parts
of a degree, while the same on the outer was to be divided into 96
equal parts and these again into 16th parts. The reason for adopting
the latter was that, 96 and 16 being both powers of 2, the divisions
could be got at by continual bisection alone, which, in Graham's
opinion, who first employed it, is the only accurate method, and would
thus serve as a check upon the accuracy of the divisions of the outer
arc. With the same distance on the beam-compass as was used to
describe the inner arc, laid off from 0 deg., the point 60 deg. was at
once determined. With the points 0 deg. and 60 deg. as centres
successively, and a distance on the beam-compass very nearly bisecting
the arc of 60 deg., two slight marks were made on the arc; the
distance between these marks was divided by the hand aided by a lens,
and this gave the point 30 deg. The chord of 60 deg. laid off from the
point 30 deg. gave the point 90 deg., and the quadrant was now divided
into three equal parts. Each of these parts was similarly bisected,
and the resulting divisions again trisected, giving 18 parts of 5 deg.
each. Each of these quinquesected gave degrees, the 12th parts of
which were arrived at by bisecting and trisecting as before. The outer
arc was divided by continual bisection alone, and a table was
constructed by which the readings of the one arc could be converted
into those of the other. After the dots indicating the required
divisions were obtained, either straight strokes all directed towards
the centre were drawn through them by the dividing knife, or sometimes
small arcs were drawn through them by the beam-compass having its
fixed point somewhere on the line which was a tangent to the
quadrantal arc at the point where a division was to be marked.

The next important example of graduation was done by Bird in 1767. His
quadrant, which was also 8-ft. radius, was divided into degrees and
12th parts of a degree. He employed the method of continual bisection
aided by chords taken from an exact scale of equal parts, which could
read to .001 of an inch, and which he had previously graduated by
continual bisections. With the beam-compass an arc of radius 95.938
in. was first drawn. From this radius the chords of 30 deg., 15 deg.,
10 deg. 20', 4 deg. 40[min] and 42 deg. 40' were computed, and each of
them by means of the scale of equal parts laid off on a separate
beam-compass to be ready. The radius laid off from 0 deg. gave the
point 60 deg.; by the chord of 30 deg. the arc of 60 deg. was
bisected; from the point 30 deg. the radius laid off gave the point 90
deg.; the chord of 15 deg. laid off backwards from 90 deg. gave the
point 75 deg.; from 75 deg. was laid off forwards the chord of 10 deg.
20'; and from 90 deg. was laid off backwards the chord of 4 deg. 40';
and these were found to coincide in the point 85 deg. 20'. Now 85 deg.
20' being = 5' X 1024 = 5' X 2^10, the final divisions of 85 deg. 20'
were found by continual bisections. For the remainder of the quadrant
beyond 85 deg. 20', containing 56 divisions of 5' each, the chord of
64 such divisions was laid off from the point 85 deg. 40', and the
corresponding arc divided by continual bisections as before. There was
thus a severe check upon the accuracy of the points already found,
viz. 15 deg., 30 deg., 60 deg., 75 deg., 90 deg., which, however, were
found to coincide with the corresponding points obtained by continual
bisections. The short lines through the dots were drawn in the way
already mentioned.

The next eminent artists in original graduation are the brothers John
and Edward Troughton. The former was the first to devise a means of
graduating the quadrant by continual bisection without the aid of such
a scale of equal parts as was used by Bird. His method was as follows:
The radius of the quadrant laid off from 0 deg. gave the point 60 deg.
This arc bisected and the half laid off from 60 deg. gave the point 90
deg. The arc between 60 deg. and 90 deg. bisected gave 75 deg.; the
arc between 75 deg. and 90 deg. bisected gave the point 82 deg. 30',
and the arc between 82 deg. 30' and 90 deg. bisected gave the point 86
deg. 15'. Further, the arc between 82 deg. 30' and 86 deg. 15'
trisected, and two-thirds of it taken beyond 82 deg. 30', gave the
point 85 deg., while the arc between 85 deg. and 86 deg. 15' also
trisected, and one-third part laid off beyond 85 deg., gave the point
85 deg. 25'. Lastly, the arc between 85 deg. and 85 deg. 25' being
quinquesected, and four-fifths taken beyond 85 deg., gave 85 deg. 20',
which as before is = 5' X 2^10, and so can be finally divided by
continual bisection.

The method of original graduation discovered by Edward Troughton is
fully described in the _Philosophical Transactions_ for 1809, as
employed by himself to divide a meridian circle of 4 ft. radius. The
circle was first accurately turned both on its face and its inner and
outer edges. A roller was next provided, of such diameter that it
revolved 16 times on its own axis while made to roll once round the
outer edge of the circle. This roller, made movable on pivots, was
attached to a frame-work, which could be slid freely, yet tightly,
along the circle, the roller meanwhile revolving, by means of
frictional contact, on the outer edge. The roller was also, after
having been properly adjusted as to size, divided as accurately as
possible into 16 equal parts by lines parallel to its axis. While the
frame carrying the roller was moved once round along the circle, the
points of contact of the roller-divisions with the circle were
accurately observed by two microscopes attached to the frame, one of
which (which we shall call H) commanded the ring on the circle near
its edge, which was to receive the divisions and the other viewed the
roller-divisions. The points of contact thus ascertained were marked
with faint dots, and the meridian circle thereby divided into 256 very
nearly equal parts.

The next part of the operation was to find out and tabulate the errors
of these dots, which are called _apparent_ errors, in consequence of
the error of each dot being ascertained on the supposition that its
neighbours are all correct. For this purpose two microscopes (which we
shall call A and B) were taken, with cross wires and micrometer
adjustments, consisting of a screw and head divided into 100
divisions, 50 of which read in the one and 50 in the opposite
direction. These microscopes were fixed so that their cross-wires
respectively bisected the dots 0 and 128, which were supposed to be
diametrically opposite. The circle was now turned half-way round on
its axis, so that dot 128 coincided with the wire of A, and, should
dot 0 be found to coincide with B, then the two dots were 180 deg.
apart. If not, the cross wire of B was moved till it coincided with
dot 0, and the number of divisions of the micrometer head noted. Half
this number gave clearly the error of dot 128, and it was tabulated +
or - according as the arcual distance between 0 and 128 was found to
exceed or fall short of the remaining part of the circumference. The
microscope B was now shifted, A remaining opposite dot 0 as before,
till its wire bisected dot 64, and, by giving the circle one quarter
of a turn on its axis, the difference of the arcs between dots 0 and
64 and between 64 and 128 was obtained. The half of this difference
gave the apparent error of dot 64, which was tabulated with its proper
sign. With the microscope A still in the same position the error of
dot 192 was obtained, and in the same way by shifting B to dot 32 the
errors of dots 32, 96, 160 and 224 were successively ascertained. In
this way the apparent errors of all the 256 dots were tabulated.

From this table of apparent errors a table of _real_ errors was drawn
up by employing the following formula:--

1/2(x(a) + x(c)) + z = the real error of dot b,

where x(a) is the real error of dot a, x(c) the real error of dot c,
and z the apparent error of dot b midway between a and c. Having got
the real errors of any two dots, the table of apparent errors gives
the means of finding the real errors of all the other dots.

The last part of Troughton's process was to employ them to cut the
final divisions of the circle, which were to be spaces of 5' each. Now
the mean interval between any two dots is 360 deg./256 = 5' X 16-7/8,
and hence, in the final division, this interval must be divided into
16-7/8 equal parts. To accomplish this a small instrument, called a
subdividing sector, was provided. It was formed of thin brass and had
a radius about four times that of the roller, but made adjustable as
to length. The sector was placed concentrically on the axis, and
rested on the upper end of the roller. It turned by frictional
adhesion along with the roller, but was sufficiently loose to allow of
its being moved back by hand to any position without affecting the
roller. While the roller passes over an angular space equal to the
mean interval between two dots, any point of the sector must pass over
16 times that interval, that is to say, over an angle represented by
360 deg. X 16/256 = 22 deg. 30'. This interval was therefore divided
by 16-7/8, and a space equal to 16 of the parts taken. This was laid
off on the arc of the sector and divided into 16 equal parts, each
equal to 1 deg. 20'; and, to provide for the necessary 7/8ths of a
division, there was laid off at each end of the sector, and beyond the
16 equal parts, two of these parts each subdivided into 8 equal parts.
A microscope with cross wires, which we shall call I, was placed on
the main frame, so as to command a view of the sector divisions, just
as the microscope H viewed the final divisions of the circle. Before
the first or zero mark was cut, the zero of the sector was brought
under I and then the division cut at the point on the circle indicated
by H, which also coincided with the dot 0. The frame was then slipped
along the circle by the slow screw motion provided for the purpose,
till the first sector-division, by the action of the roller, was
brought under I. The second mark was then cut on the circle at the
point indicated by H. That the marks thus obtained are 5' apart is
evident when we reflect that the distance between them must be 1/16th
of a division on the section which by construction is 1 deg. 20'. In
this way the first 16 divisions were cut; but before cutting the 17th
it was necessary to adjust the micrometer wires of H to the real error
of dot 1, as indicated by the table, and bring back the sector, not to
zero, but to 1/8th short of zero. Starting from this position the
divisions between dots 1 and 2 were filled in, and then H was adjusted
to the real error of dot 2, and the sector brought back to its proper
division before commencing the third course. Proceeding in this manner
through the whole circle, the microscope H was finally found with its
wire at zero, and the sector with its 16th division under its
microscope indicating that the circle had been accurately divided.

_Copying._--In graduation by copying the pattern must be either an accurately divided straight scale, or an accurately divided circle, commonly called a _dividing plate_.

In copying a straight scale the pattern and scale to be divided, usually called the work, are first fixed side by side, with their upper faces in the same plane. The dividing square, which closely resembles an ordinary joiner's square, is then laid across both, and the point of the dividing knife dropped into the zero division of the pattern. The square is now moved up close to the point of the knife; and, while it is held firmly in this position by the left hand, the first division on the work is made by drawing the knife along the edge of the square with the right hand.

It frequently happens that the divisions required on a scale are either greater or less than those on the pattern. To meet this case, and still use the same pattern, the work must be fixed at a certain angle of inclination with the pattern. This angle is found in the following way. Take the exact ratio of a division on the pattern to the required division on the scale. Call this ratio [alpha]. Then, if the required divisions are longer than those of the pattern, the angle is cos^-1 [alpha], but, if shorter, the angle is sec^-1 [alpha]. In the former case two operations are required before the divisions are cut: first, the square is laid on the pattern, and the corresponding divisions merely notched very faintly on the edge of the work; and, secondly, the square is applied to the work and the final divisions drawn opposite each faint notch. In the second case, that is, when the angle is sec^-1 [alpha], the dividing square is applied to the work, and the divisions cut when the edge of the square coincides with the end of each division on the pattern.

In copying circles use is made of the dividing plate. This is a circular plate of brass, of 36 in. or more in diameter, carefully graduated near its outer edge. It is turned quite flat, and has a steel pin fixed in its centre, and at right angles to its plane. For guiding the dividing knife an instrument called an index is employed. This is a straight bar of thin steel of length equal to the radius of the plate. A piece of metal, having a V notch with its angle a right angle, is riveted to one end of the bar in such a position that the vertex of the notch is exactly in a line with the edge of the steel bar. In this way, when the index is laid on the plate, with the notch grasping the central pin, the straight edge of the steel bar lies exactly along a radius. The work to be graduated is laid flat on the dividing plate, and fixed by two clamps in a position exactly concentric with it. The index is now laid on, with its edge coinciding with any required division on the dividing plate, and the corresponding division on the work is cut by drawing the dividing knife along the straight edge of the index.

_Machine Graduation._--The first dividing engine was probably that of Henry Hindley of York, constructed in 1740, and chiefly used by him for cutting the teeth of clock wheels. This was followed shortly after by an engine devised by the duc de Chaulnes; but the first notable engine was that made by Ramsden, of which an account was published by the Board of Longitude in 1777. He was rewarded by that board with a sum of L300, and a further sum of L315 was given to him on condition that he would divide, at a certain fixed rate, the instruments of other makers. The essential principles of Ramsden's machine have been repeated in almost all succeeding engines for dividing circles.

Ramsden's machine consisted of a large brass prate 45 in. in diameter,
carefully turned and movable on a vertical axis. The edge of the plate
was ratched with 2160 teeth, into which a tangent screw worked, by
means of which the plate could be made to turn through any required
angle. Thus six turns of the screw moved the plate through 1 deg., and
1/60th of a turn through 1/360th of a degree. On the axis of the
tangent screw was placed a cylinder having a spiral groove cut on its
surface. A ratchet-wheel containing 60 teeth was attached to this
cylinder, and was so arranged that, when the cylinder moved in one
direction, it carried the tangent screw with it, and so turned the
plate, but when it moved in the opposite direction, it left the
tangent screw, and with it the plate, stationary. Round the spiral
groove of the cylinder a catgut band was wound, one end of which was
attached to a treadle and the other to a counterpoise weight. When the
treadle was depressed the tangent screw turned round, and when the
pressure was removed it returned, in obedience to the weight, to its
former position without affecting the screw. Provision was also made
whereby certain stops could be placed in the way of the screw, which
only allowed it the requisite amount of turning. The work to be
divided was firmly fixed on the plate, and made concentric with it.
The divisions were cut, while the screw was stationary, by means of a
dividing knife attached to a swing frame, which allowed it to have
only a radial motion. In this way the artist could divide very rapidly
by alternately depressing the treadle and working the dividing knife.

Ramsden also constructed a linear dividing engine on essentially the same principle. If we imagine the rim of the circular plate with its notches stretched out into a straight line and made movable in a straight slot, the screw, treadle, &c., remaining as before, we get a very good idea of the linear engine.

In 1793 Edward Troughton finished a circular dividing engine, of which the plate was smaller than in Ramsden's, and which differed considerably in simplifying matters of detail. The plate was originally divided by Troughton's own method, already described, and the divisions so obtained were employed to ratch the edge of the plate for receiving the tangent screw with great accuracy. Andrew Ross (_Trans. Soc. Arts_, 1830-1831) constructed a dividing machine which differs considerably from those of Ramsden and Troughton.

The essential point of difference is that, in Ross's engine, the
tangent screw does not turn the engine plate; that is done by an
independent apparatus, and the function of the tangent screw is only
to stop the plate after it has passed through the required angular
interval between two divisions on the work to be graduated. Round the
circumference of the plate are fixed 48 projections which just look as
if the circumference had been divided into as many deep and somewhat
peculiarly shaped notches or teeth. Through each of these teeth a hole
is bored parallel to the plane of the plate and also to a tangent to
its circumference. Into these holes are screwed steel screws with
capstan heads and flat ends. The tangent screw consists only of a
single turn of a large square thread which works in the teeth or
notches of the plate. This thread is pierced by 90 equally distant
holes, all parallel to the axis of the screw, and at the same distance
from it. Into each of these holes is inserted a steel screw exactly
similar to those in the teeth, but with its end rounded. It is the
rounded and flat ends of these sets of screws coming together that
stop the engine plate at the desired position, and the exact point can
be nicely adjusted by suitably turning the screws.

A description is given of a dividing engine made by William Simms in the _Memoirs of the Astronomical Society_, 1843. Simms became convinced that to copy upon smaller circles the divisions which had been put upon a large plate with very great accuracy was not only more expeditious but more exact than original graduation. His machine involved essentially the same principle as Troughton's. The accompanying figure is taken by permission.