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Fig. 15.

ing of two pieces, one on each side of the rib, notched to t and the beam, and fastened by bolts and straps.

Fig. 18.

level of the top of the walls.

In small spans this may be done by the simple arrangement called the carpenter's bost (A, Fig. 16), in which a firm union is effected between the beam and the rafters without the use of nails or pins. Such a roof can only press injuriously on the walls by the rafters sinking into a concave form, which however their lower ends are very liable to do. In such a case additional strength may be obtained by inserting a longitudinal truss, as in B, Fig. 16, where c represents the end of the truss,

Fig. 16. B

which should be firmly built into the gables. d and e are side views of two longitudinal trusses suitable for such a situation, the first being stiffened by an arch of iron notched into the short vertical pieces, and the second formed of timber only. Similar trusses are occasionally introduced under the purlins. Roofs without ties may be greatly strengthened by the use of parabolic curves of iron, notched into the rafters of each inclined face, and abutting on the wallplates, which in such a case are firmly bolted together. The timbers of such a roof may be framed together in planes, each having a distinct ridge-piece, and the ridges being screwed or otherwise firmly connected together. The curves may be cast in short segments, as they are compressed when in use, it being merely necessary to provide that the joints should always abut on a rafter. Tredgold, in his 'Elementary Principles of Carpentry,' recommends the use of similar curves, of either wood or iron, in the trusses of an ordinary roof, by which the derangement often arising from the shrinking of the king-posts and queen-posts may be avoided. In this case the curves take the place of the principal rafters, and, if made of wood, may be constructed of short straight pieces, arranged as shown in Fig. 17, and held together by bolts or wooden keys. When curved tim

Fig. 17.

ber can be obtained it is to be preferred, as it reduces the number of joints. For small roofs timbers may be bent into the required form, as it is found that a piece of wood the thickness of which does not exceed th part of its length, may be bent into a curve rising one-eighth of its span without impairing its elasticity. Two such pieces may be laid together, and bent by twisting a rope attached to their ends, as is done in tightening the frame of a bow or pit saw; and, being bolted together while curved, they will spring back but little when the rope is relaxed. Another mode of forming such a rib is to take a piece of wood whose thickness is about one-sixtieth of its length, and cutting along the middle with a thin saw from each end, leaving about eight feet in the centre solid. The beam may then be bent, and bolted or pinned together as before described. In either case the rib should be bent about onefourth more than it is intended to remain, to allow for springing back. A parabolic curve is the form most recommended; but a circular arc, rising half the height of the roof, will answer the purpose. Fig. 18 represents the truss of a truncated roof strengthened by a curved rib, the Suspending pieces being, when the rib is formed in the manfirst described, placed at each joint, and each consist

One of the advantages of this mode of construction that the tie-beams may be suspended from any number points, which is important in large spans, where the bes have to be formed of several pieces scarfed togethe [SCARFING.] Diagonal braces, though unnecessary w parabolic curves, may be added to meet accidental stra as shown by the dotted lines in the cut. This principle a construction, with an arc composed of several pieces of t ber, was followed in one of the largest roofs ever built, the erected in 1791 over a riding-house at Moscow. The s of this roof, which has been said to be the most extens in the world, is stated by Tredgold at 235 feet, the s being about 19°, and the external dimensions of the bu ing 1920 by 310 feet. He states that it had sunk so mai that it was proposed to add a second curve for addition strength.

A simple and economical roof, invented by Mr. A. H Holdsworth, and rewarded by the Society of Arts in 1892 is supported by curved ribs of timber applied in a differes: manner. A detailed description is given in the 38th volume of the Society's Transactions,' but Fig. 19 will sufficiently explain the principle of its construction a is a beat

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serving as a tie-beam, and also to support the upper floor! the building; b b are curved ribs, formed in a similar ner to those just described, the lower ends of which ar firmly secured to the tie-beam a. The principal rafters red on these ribs, and their lower ends bear upon short timbe resting on the walls, these pieces being fastened by str iron straps to the curved ribs, to counteract the outwar thrust of the rafters. By this arrangement the whole of the interior of the roof, which is usually encumbered with king posts, queen-posts, braces, &c., is rendered available for u ful purposes, in addition to which it effects a consideraka saving of timber.

Wrought-iron straps of various forms are very usef when judiciously applied, in strengthening the joints of a roof. They should be fixed with regard to the unavoidab tendency of the timbers to shrinking, so that while ther may, in some cases, counteract or lessen its effect, they may so far yield to it as to prevent a strain which should co upon a timber, being entirely thrown, by its alteration form, upon the strap. Tie-beams are often suspended the trussing-posts by means of straps, so arranged as allow the beam to be keyed up to its true position in as of the roof sinking. When this is not the case, the ties a sometimes drawn up into a slightly convex or cambered form, to meet the same contingency. Height may to gained inside a building by disposing the timbers as in t 20, the want of a continuous tie-beam being compensated

Fig. 20.

for by an iron strap to unite the ties to the bottom of the king-post at a; but it is evident that the safety of the plas must depend wholly on the straps, which alone counterat the outward thrust of the rafters.

In roofing a church with a nave and side aisles, the continuity of the tie-beams may be dispensed with, intermediate support being obtained from columns. It is however necessary to guard carefully against any lateral strain to the columns.

Many of the high-pitched roofs of old Gothic churches and halls are very ingeniously contrived, but they often throw great pressure on the walls, owing to the absence or elevated position of the ties; thereby rendering very solid walls and buttresses necessary. The Norman roof is an ingenious contrivance for the construction of roofs of large span with small pieces of wood. Fig. 21 shows this arrangement, in which all the rafters abut on joggled king-posts, of which there are several, their relative position being maintained by diagonal braces. The timbers of this kind

Fig. 21.

of roof are often left visible, being so carved as to have an ornamental effect. Such a roof may be made to exert very little injurious pressure on the walls.

When the space covered in is of an irregular shape, it is best to arrange the inclined planes of the roof in a similar imanner to those of a rectangular building, leaving a level platform in the centre, corresponding to the plan of the inclosed space. Where the space covered is circular, elliptical, or polygonal, although the construction of the roof may appear more complicated to the eye, it is, in fact, simpler and easier than that of a quadrangular building, the strain of the roof being more equally distributed. The nearer a roof approaches to a circle in plan, the stronger it will be, the parts deriving that mutual support from each other which forms the distinguishing character of the dome. Domes of wood, of great size, have been made without trussing, simply by forming the timbers into curved ribs abutting on the wall plates, which then form a circle, and kept in their proper positions by horizontal circles framed with them at intervals. As the ribs approach the upper part of the dome, the intervals between them diminish in width, to allow for which every second or third rib is discontinued at intervals, the ends of the ribs thus discontinued being received by the horizontal circles, which may be compared to purlins, the ribs taking the place of rafters. The wooden dome formerly existing at the Halle aux Blés, at Paris, was a remarkably bold example of this kind, being 200 feet in diameter, and having a large opening in the centre. It was built at the suggestion of M. Moulineau, and, having been destroyed by fire, bas been replaced by a similar structure of iron, but of smaller dimensions.

works on carpentry, allusion can here be made to only one other. It is an admirably simple plan for making a very flat roof, described in the 37th volume of the 'Transactions' of the Society of Arts, in a communication from the inventor, Mr. Smart. The beams or rafters are cut, with a circular saw, as shown at a, fig. 23, while b represents their form when in use, a wedge being inserted between the ends of the parts that are elevated into a sloping position. These may be raised to an angle of 10° or 12°, and will bear a great

Fig. 23.

a

weight, as they cannot be depressed without thrusting off the ends of the beam, or breaking the lower part of it by tension. This is called, by the inventor, the bow and string rafter, and was used by him to support a roof at the Ordnance Wharf, Westminster Bridge. Strong laths were nailed upon the rafters, and on these a platform of bricks was laid in cement, the whole being covered with tiles also bedded and pointed with cement, and twice coated with hot linseed oil. The cost of this roof is stated to be not more than half that of lead. For a further notice of the experiments of the inventor of this simple truss, see TRUSSING.

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In the valuable practical works of Nicholson, Tredgold, &c. the methods of calculating the strength necessary in the various parts of a roof may be found; and in the Principles of Carpentry,' by the latter author, tables are given of the dimensions suitable for different spans. The table here quoted refers to a roof similar to fig. 7; the trusses being not more than ten feet apart, and the pitch at an angle of about 27° with the horizon, for a covering of slates. The scantlings are suited for yellow fir, and must be some what increased for timber of inferior quality.

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For the strength of different materials, under various circumstances, the reader may consult MATERIALS, STRENGTH OF, vol. xv., p. 8. As a general remark, it may be observed that oak, when exposed to tension, is weaker than fir, and is therefore less adapted for ties. Being however less comWhen the roof approaches the circular form, but not pressible, it is usually preferred for rafters, straining pieces, sufficiently to have the character of a dome, it may be con- and struts; but Tredgold observes that its greater tendency sidered as consisting of several trusses resembling those of to warping in summer renders it less fit for rafters and an ordinary roof, but so contrived as to intersect each other purlins than foreign fir. Cast-iron is not much used, except in the centre; the king-post being common to all the in fire-proof roofs, and each piece requires to be well tested. Fig. 22, representing a design for a polygonal Wrought-iron is very useful for straps and fastenings, and roof, from Nicholson, may illustrate this, and exemplify also for ties and trussing-posts; but care is always necessary also some of the applications of iron straps: a shows the to guard against imperfections, which are more likely to form of the strap by which the ties are secured to the king-pass unobserved than in wood. Wherever iron is applied, post; the post having as many faces, and the strap as many arms, as there are trusses in the roof.

trusses.

Fig. 22.

*

Though the number of contrivances for the construction of roofs is very great, as may be seen by reference to various

provision should be made for its expansion and contraction, and it is desirable to protect it from oxidation by painting. Though iron is far stronger for its size than any kind of timber, it is neither so strong nor so cheap as yellow fir, weight for weight.

The joints in the frame-work of a timber roof are of various kinds, according to the nature of the strain they have to resist. They should be formed with great care, and with due regard to such probable changes of form as all constructions of timbers are liable to from shrinking and warping. Cocking or cogging is the name given to that kind of joining in which one piece of timber, in a state of tension, is so attached to another that it cannot be drawn away without one piece breaking. Figs. 24 and 25 represent two methods. of cocking the ends of tie-beams on the wall-plates, giving a plan and elevation of each. In both figures a represents the beam, and b the wall-plate. In the first plan, which was formerly much practised, the contraction of the dove-tailed end of the beam would allow it to be

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and curved or angular surfaces generally; and also for! roofs, or such as have too little slope for slating. Lea the most common material for such purposes, though. per, iron, tinned iron, and recently zinc, are also u Lead terraces or flats are commonly laid on boarding plaster. The joints are sometimes soldered, but the a approved method is to roll or wrap the edges into c other, making allowance for expansion and contraction fall of a quarter of an inch in a foot is sufficient for surf covered with sheet metal.

Cements of various kinds have been applied to the! mation of roofs, and in some cases with success, the they have often been found to crack, and thereby ben permeable to water. Mixtures of tar with lime, sand, gra ashes, &c., have been recommended; and asphalte has applied to this purpose, apparently with great advant Compositions of tar, resin, and similar substances, sp upon sheets of coarse paper, have also been used. (Nicholson's Architectural Dictionary, Practical B:.'é &c., &c.; Tredgold's Principles of Carpentry; Robe Mechanical Philosophy.)

drawn considerably out of its place, and would therefore permit the walls to spread; but in the second the amount of contraction is diminished, owing to the small width of the rectangular tongue that enters the tie-beam, while its position is such as to prevent the beam being drawn out of its place beyond the actual extent of the contraction of the tongue. The shrinking of the joggles of king-posts and queen-posts is often productive of serious derangement, a circumstance greatly in favour of the substitution of iron for wood for such parts, especially in large roofs. This inconvenience is sometimes avoided by making the upper ends of the principal rafters abut immediately upon each other, as ROOK (Corvus frugilegus, Linn.), This well-known represented in fig. 12. A similar arrangement is made, in garious and familiar bird (for it seems to affect the neighb some cases, where wooden king-posts are used, the king-hood of man, and even not to be scared by the post and rafters being strapped together with iron. The atmosphere of great towns) is the Cornacchia nera am." sinking of a roof, particularly if it be of low pitch, is very nacchione of the Italians; Graye, Grolle, Freux, and Fa injurious to the mortise-and-tenon joints of the struts and one of the French Corneille Moissoneuse of Br rafters, by throwing the strain on the shoulders of the Schwartze Krähe of the Germans; Roka of the Swe tenons in such a way as to break off the tenons or splinter Rook of the modern British, and Ydfran of the ank the wood. To guard against such injuries, it has been pro- British. posed by M. Perronet, a French engineer, instead of making the tenons and joggles square, to form them into circular arcs, the centres being at the opposite end of the strut or rafter. This plan appears worthy of general adoption, as it allows the joints to accommodate themselves to changes of form without injury. All the timbers of a roof are usually fitted and framed together on the ground, and taken to pieces again before being elevated to the building.

Allusion has been made in a previous column to the various materials used for the covering of roofs, with reference to the different degrees of inclination suitable for them. Thatched roofs have been considered by some to maintain the most equable temperature in the buildings covered by them, keeping out alike the extreme heat of summer and cold of winter. They are objectionable on account of their harbouring vermin, being easily damaged by wind, and dangerously combustible. The frequent reparrs required make thatch also an expensive material. Besides straw, reeds and heath are sometimes used for thatching, and possess the advantage of greater durability. Tiles admit heat and moisture more than good slates. Pantiles, having no holes for nailing through, are simply hung, by ledges, upon laths nailed to the rafters. Plain tiles, laid in mortar, and over-lapping, so as to be double thickness everywhere, make a very good though heavy covering. Tiles of a peculiar form, called hip-tiles, are used for cover ing salient angles; and gutter-tiles, which are similar to them, but placed with the concave side upwards, in the valleys or receding angles. Slates are laid in various ways. They are sometimes nailed down on a close boarding; or, if large, on battens, or pieces of wood from two and a half to three inches wide, and three-quarters of an inch to an inch thick, which are nailed to the rafters at intervals regulated by the length of the slates. Lozenge-shaped slating is occasionally used, and has an ornamental appearance, but is easily injured, as there is but one nail through each slate. It is always laid on boarding. For what is called patent slating the best large slates are selected, and fixed without either boarding or battening, the common rafters being placed at such a width as to come under the joints. The slates are screwed down, the courses over-lapping about two auches. The meeting joints are covered by fillets of slate about three inches wide, set in putty, and screwed down; and the hips and ridges are sometimes covered in the same manner, though it is best in all such cases to use lead. Patent slating when well executed, is water-tight with as low a slope as one in six. In some districts lamina of stone are used in leu of hates or tiles. Shingles, which are like sates, but made of wood, were formerly much used in coverang pyramidal sterputs, and nroofs of steep pitch. They are plan used as the Lined Sites, and are usually laid on

of manner to common slates.

Sheets of metal are very convenient for covering domes,

Belon and Caius, the latter of whom names the R Spermologus, seu Frugilega, appear to be of opinion th it is the σπεpμoλóyog of Aristotle (Hist. Anim., viii. 3). I! doubtless, as Pennant observes, the Corvus of Virgil, has happily described a flock of them

E pastu decedens agmine magno.' (Georg, lib. i., v. 381.) Geographical Distribution.-The Rook is spread over t greater part of Europe; but nowhere does it seem to more abundant than in Great Britain and Ireland. Wood and cultivated districts are its favourite haunts. The fart north the observer goes in Scotland, the fewer rooks does see. In Orkney and Shetland there are none, nor are the any in Guernsey and Jersey. They do not appear to be r merous in Denmark, nor in the southern districts of Swed there. In Italy the rook is common and permanent: i. nor in Russia and northern Asia, though they may be Europe. In France it is also common, and the follo it appears to be migratory over a part of the continer: quatrain appears under the cut of it in the Portraits d

seaux:

Jamais le Freux ne hante le rivage,

Et ne se paist que de grains et de vers,
Il est oyseau commun, gros et pervers,
Qui vole en trouppe, et crie à l'avantage.'

It occurs between the Black and Caspian seas; and Von Siebold and M. Bürger note it among the Eunbirds seen by them in Japan.

Food, Habits, &c.-Grain, and insects especially, form food of the Rook, and there can be no doubt that it an repays the farmer for the seed which it takes, by its ass ity in clearing his land of wire-worms and the larvæ e cock-chaffer (Melolontha vulgaris). These last are ca Rook-worms in many places, and the birds may be sect lowing the plough-tail to gather them up as the share poses them. In the end of May and beginning of 1 when the young are able to fly and go abroad with! parents, they may often be seen among the bright leaves of the horse-chestnut and other trees bending branches with their weight as they assemble to pick of cock-chaffers in the winged state. Where these birds b been inconsiderately destroyed, on account of the supp damage which they had done, a total failure of the crop le made the farmer glad to try to get them back again stick-built nest contains four or five pale greenish et blotched with dark greenish brown: these are seme palmed upon the undiscerning for Plover's eggs, ba't easily distinguished from them. Not that a rook's egg ** any means bad; though far inferior in every respect t other. The male is most attentive to the female whẩe s is setting, and feeds her assiduously; both are very i trious in supplying their young, and the skin under tongue may at this season be often seen dilated into of pouch by the collected food. During the building s

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they have great squabbles among themselves about their nests. An account of one of their battles with the Herons for the possession of a disputed territory is alluded to in the article HERONS [vol. xii. p. 167]. They frequently visit their nest-trees in the autumn on their way to roost in some distant wood, and come to them for the purpose of repairing their nest, and setting about the business of incubation early in March.

state of the nostrils and anterior part of the head, both of which are entirely destitute of feathers. But he notices another specimen, in the possession of Mr. R. Wood of Manchester, which has the mandibles greatly elongated and much curved. Now,' says Mr. Blackwall, it is evident that the bird possessing a bill thus formed could not thrust it into the ground in search of worms and larvæ of insects, as the rook is known to do habitually; and accordingly the plumage at the base of the bill of this individual, and the bristly feathers which cover its nostrils, are very conspicuous, not having sustained the slightest injury. The opinion entertained by many persons that the naked condition of the nostrils and anterior part of the head is an original peculiarity in the rook, is thus satisfactorily proved to be incorin these particulars, is sufficiently conclusive on this point; but the possibility of an entire species being endowed with an instinct destructive of a usual portion of its organization was probably never contemplated by these observers; it is not surprising therefore that the inference deduced from a partial view of the subject should be erroneous.'

The Rook is not without the power of mimicry granted so largely to the greater part of the true crows, is docile, capable of learning amusing tricks, and becomes much attached to the kind hand that feeds it. It has been heard to imitate the note of a jackdaw (Hewitson) and the barking of dogs so perfectly that if the mimic had been out of sight, no ear could have discovered the deception. (Macgil-rect; indeed, the fact that young rooks exhibit no deficiency livray.)

Varieties.-White, pied, and cream-coloured. A gentleman,' says the charming author of the History of Selborne, 'had two milk-white rooks in one nest. A booby of a carter, finding them before they were able to fly, threw them down and destroyed them, to the regret of the owner, who would have been glad to have preserved such a curiosity in the rookery. I saw the birds myself nailed against the end of a barn, and was surprised to find that their bills, legs, feet, and claws were milk-white.' These perhaps were perfect albinos, and might so have continued; but instances are not wanting where the original light colour deepens into the usual sable with age. Mr. Yarrell quotes Mr. Hunt of Norwich, who states that a gentleman of his acquaintance had in 1816 a young rook of a light ash-colour, most beautifully mottled all over with black, and the quill and tail feathers elegantly barred; but when the bird moulted it became a jet black rook, and in this state was suffered to join its sable brethren in the fields. Mr. Yarrell remarks that this agrees with his own observations, and he adds that accidental varieties will generally be found to be comparatively small and weak birds; as these young birds increase in age and gain constitutional power, the secretions, he observes, become perfect, and the plumage assumes its natural colours, whilst the assumption of white feathers, by old birds, is probably the effect of the converse operation of the physiological law. (British Birds.)

Head and Foot of Rook.

It has been, and indeed still is with some, a question whether the loss of the feathers at the base of the beak in the young rook upon the first moult, is or is not a specific distinction, or merely the result of denudation from plungmg the bill into the ground in search of prey. It must be borne in mind that some foreign birds resemble the rook in this particular. Mr. John Blackwall's observations (Researches in Zoology) touching this matter are full of interest. He refers to a rook preserved in the Manchester Museum, which has its mandibles crossed near their extremities, but so slightly that the malformation could not have interfered materially with the mode of procuring food usually resorted to by rooks, as is clearly shown by the denuded

ROOKE, SIR GEORGE, ADMIRAL, the eldest son of Sir William Rooke, was born at his father's seat, the priory of St. Lawrence near Canterbury, in the year 1650. He entered the navy as a volunteer, and at the age of thirty had attained the rank of post-captain. In 1689 he was sent out as commodore with a squadron to the coast of Ireland, where his services were such as to induce William III. to promote him to the rank of rear-admiral of the red. He soon afterwards bore a part in the indecisive action between the earl of Torrington's fleet and that of the French admiral Tourville, off Beachy Head.

In 1692 Rooke was advanced to the rank of vice-admiral of the blue, and greatly distinguished himself in the battle off Cape La Hogue (properly La Hague) between the French fleet and the combined English and Dutch fleets under admiral Russell, May 19, 1692; but a part of the French fleet having escaped into La Hogue, and being hauled up so high that the English ships of the line could not reach them, Rooke volunteered on the following day to attack them with the boats of his squadron. This service he performed at night under cover of a fire from his frigates and smaller vessels; and so well was his plan contrived, and so unexpectedly and suddenly executed, that though six French three-deckers were burnt that night and seven other ships of the line on the following morning, the loss of the English only amounted to ten men. For this exploit Rooke was rewarded with the rank of vice-admiral of the red, a pension of 1000l. a year, and the honour of knighthood.

After the peace of Ryswick in 1697, Sir George Rooke was elected member of parliament for Portsmouth; and though he was attached to the Tory party, then in opposition to the government, queen Anne, on her accession in 1702, appointed him vice-admiral and lieutenant of the admiralty, and also lieutenant of the fleets and seas of this kingdom,' having previously constituted her royal consort prince George of Denmark generalissimo of her forces by land and

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sea.

The war of the succession had now commenced, and an attack upon Cadiz was resolved upon, the land-forces being under the command of the duke of Ormond, and the combined English and Dutch fleets under Rooke. The attack was begun, but, in consequence of the opposition of the prince of Hesse, was not persevered in. Having received intelligence however that the Plate fleet, under convoy of a French squadron, had taken shelter in the port of Vigo, the duke and Sir George resolved to proceed there. The duke stormed the town with 3000 men, while the fleet took and destroyed seventeen ships; six galleons being taken by the English and five by the Dutch, who burnt five others. The value of the specie and goods taken was estimated at five millions of dollars.

Sir George Rooke having been joined by Sir Cloudesley Shovel, with a large reinforcement from England, they resolved to make an attack upon Gibraltar. On the 21st of July, 1704, the prince of Hesse, with 1800 marines, was landed on the isthmus, while the ships commenced a cannonade upon the fortress, which, having been kept up for about six hours, the Spaniards began to fly from the batteries. The boats were then manned and armed, and the seamen succeeded in making themselves masters of the great platform, which they retained till the following day, when a reinforcement of seamen enabled them to carry an

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other strong battery, which put them in possession of most [ equation is called a root of that equation. Thus, 2, 1,.of the enemy's cannon. The governor then accepted the (-3) and 1-√ (−3) are the roots, and all the rocks offered terms of capitulation, and the fortress surrendered. the equation On the 9th of August, 1704. Rooke fell in with the French fleet under the Comte de Toulouse, who had recently put to sea from Toulon, with fifty-two ships of the line and twentyfour galleys. The French admiral endeavoured to get away, though, according to Rooke's statement, he had a superiority of 600 guns, but on the 13th of August Rooke brought him to action off Malaga. The battle began in the forenoon, and ended with the day, when the French went off to leeward, and, the weather being hazy, escaped. This was a hard-fought battle. The French lost upwards of 3000 men, and the English upwards of 2000.

Sir George Rooke on his return to England was received by queen Anne at Windsor with great distinction, but find ing that the government was hostile to him, he resigned his employments, gave up his seat in parliament, and passed the rest of his life at his seat of St. Lawrence, where he died on the 24th of January, 1709, aged fifty-eight, and was buried in the cathedral of Canterbury. He was thrice married. (Campbell's Lives of the Admirals; Locker's Gallery of Greenwich Hospital.)

ROOKER, MICHAEL ANGELO, an artist of considerable merit as a landscape-painter and engraver, was born in London about the year 1743. His father, Edward Rooker, also a skilful designer and engraver, who excelled in landscapes and architectural views, appears to have been a singular character, having for some time acted as a harlequin at Drury-Lane Theatre. Michael Angelo was taught engraving by his father, and executed the head-pieces to the 'Oxford Almanack' for several years, from his own drawings. In landscape-drawing, which is said to have been his favourite occupation, he was instructed by Paul Sandby, whose style he imitated. His manner is not powerful, but his drawings display much taste and feeling. For several years Rooker painted the scenes for the Haymarket Theatre. He was one of the earliest associates of the Royal Academy, and died on the last day of February, 1801, at the age of fifty-seven or fifty-eight.

since they are the only algebraical formulæ and aritht cal numbers which satisfy it. On this general use of term root, see THEORY OF EQUATIONS and INVOLUTION. The more common use of the term root is as follows. seventh root of 8 is the incommensurable fraction vi seventh power is 8, or the solution of the equation = There are altogether seven such solutions, one only arit: tical, the others of the form a+b√(−1); the meth obtaining the arithmetical solution has already been diserin the article INVOLUTION; the importance of the Sq. Roor will justify its consideration in an article apart reserve for the present article the method of finding using any root (in the common sense) of any algebr. quantity, a necessary completion of the article NEGA AND IMPOSSIBLE QUANTITIES.

Every algebraical result is of the form a+b√(-?widest, or may be reduced to that form. Here a anmeant to be real algebraical quantities, that is, redu to positive or negative whole numbers or fractions, mensurable or incommensurable. Thus, if b=0, we bare% simple real quantity a; if a=0, we have the simple a sible quantity b(-1). It is indifferent, as to the pres article, in what light the impossible quantity √(-1) se sidered; whether, as in NEGATIVE AND IMPOSSIBLE QUAS TITIES, upon that extended system of definitions . makes it explicable and rational, or upon the more com system in which it is used without such explanation: we are now merely considering all algebraic formula: results, subject to certain laws by which their use is to regulated, and without any reference to interpretati. When we desire to consider only the arithmetical root of. arithmetical quantity, we shall use the symbols √, V, V, &: but the exponential fractions .,, &c. will denote: one of the algebraical roots of a formula. Thus 16 me simply 4; but (16) is an ambiguous symbol standing? ROOS, PHILIP PETER, a painter commonly called either +4 or -4. And when we have an equation Rosa da Tivoli, from his long residence at that place, was presents an ambiguous formula equated to an unambig born at Frankfort in 1655. He was instructed in art by one, we mean that the unambiguous side of the equalit his father, who was in the service of the landgrave of Hesse, one of the values of the ambiguous one: in this sense! = by which prince Philip was sent to Italy, and allowed a pen-(-1+(-3)). When we use the simple arithme sion during the period of his study. On arriving at Rome, symbol before an algebraical quantity, as in √(−3) = he applied himself assiduously to painting, and acquired a most astonishing facility of hand; indeed, such was his ramerely mean to signify that the two values of (-3) are pidity of execution, that C. le Blond, who was at the same tinguished into +√ (−3) and − √ (−3). time at Rome, declares that Roos copied in chalk the arch Assume r cos. ea, r sin. 0=b, which gives Let us now take a quantity of the form a+b

of Titus within half an hour, and that with a considerable degree of finish. He devoted his talents chiefly to painting animals, which he designed mostly from nature. To facilitate his studies he established himself at Tivoli, where he kept a kind of menagerie for the purpose of drawing from the life with correctness such animals as he required for his pictures. His other subjects generally represent pastoral scenes, with herdsmen and cattle, and works of a similar nature, some of which are executed as large as life. His groups are composed with great judgment; and the landscapes in his backgrounds, his skies and distances, are treated with surpassing truth, and executed in a masterly style. Yet, although he painted with great facility, his productions betray no appearance of negligence or inattention; they are free, without being deficient in finish. His pictures, according to Lanzi, are to be found in the galleries of Vienna, Dresden, and other capital cities of Germany, besides an immense number in Italy and many in England, though we have no specimen by his hand in the National Gallery. He was a member of most of the principal, academies of Europe. He is said by Huber to have etched a few plates of pastoral subjects, which are very scarce. M. Périès, in the Biographie Universelle,' mentions three pictures by this master which were in the Musée Napoléon, but which were returned to Vienna, whence they had been taken, in 1815. These are a view of the Cascade of Tivoli, a picture of animals, and a wolf devouring a sheep, the .andscape in which latter work was painted by Tempesta. (Pilkington's Dictionary, by Fuseli; Lanzi, Storia Pttiorica, ii. 174; Biographie Universelle.)

ROOT. The mathematical use of this term has gradually been extended, until it may be defined as follows: every value of an unknown quantity which satisfies a given

b

r = ± √ (b2+a3) tan. 0=

Let us choose for r, which is called the modulus of '2 expression, the positive value (b+a2). We can tr always make the angle 0 give the equation

a+b√ (−1) =r cos. 0+r sin ✪ √ (− 1) ................ (1) identically true. If a and b be both positive, ✪ must l between 0 and a right angle, or between 0 and 4 [ANGLI if a be positive and b negative, e must lie between 2: if b be positive and a negative, e must lie betwee and : and if both be negative, e must lie between and t Thus reducing angles to degrees and minutes,

2+3√(−1)=√13 (cos. 56° 19' + sin. 56° 19′. N ( -2+3(-1)=√13 cos. 123° 41'+sin. 123° 41′ √ (2-3(-1)=13 (cos. 303° 41'+sin. 303° 41′ (-2—3√(−1)=√13 (cos. 236° 19'+sin. 236° 19' (-. Generally, if a and b be positive, and if, returning to theoretical mode of measuring angles, be that ar which lies between 0 and 4, and has b:a for its tange we must use for a+b (-1), π-0 for a+b√-i 2-0 for a−b√(−1), and +0 for a−b √(−1).

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Again, since 0+2k has the same sine and cosine ** when k is any whole number, positive or negative, take so as to satisfy (1), we find that the following is satisfied:

a+b√ (−1) =r {cos. (0+2kπ) + sin. (0+2k′′). √ (−! (2)

for all integer values of k positive or negative, but not fr any fractional value of k whatsoever. This and various of results of common trigonometry should be familiar to eve student who attempts the present subject.

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