in 



KOOF. 



ROOT. 



178 



Kor the strength of different material*, under various circumstances, 

 the reader may consult MATERIALS. STIIKXI:TII or. As a general 

 remark, it ny be observed that oak, when exposed to tension, in 

 weaker than fir, and U therefore lea* adapted for tie*. Being, however, 

 )e cotnprnMble, it U usually preferred for rafters, straining pieces, 

 and ttniU ; but Tredgold observes that its greater tendency to warping 

 in cummer renders it ten fit for rafters and purlins than foreign fir. 

 Cast-iron U not much used in the framing of wooden roofs, excepting 

 for show, king-port heads, bolt-heads, and collars at the feet of 

 struts and straining pieces. Wrought-iruu is very useful for straps and 

 fsstmingi. and also for ties and tnuing-posU ; but care is always 

 Decenary to guard against imperfections, which are more likely to pass 

 unobserved than in wood. Wherever iron is applied, provision should 

 be made for it* expansion and contraction, and it is desirable to 

 protect it from oxidation by painting. Though iron is far stronger ior 

 lU MM- than any kind of timber, it is neither so strong nor so cheap as 

 yellow fir, terigltt for might, provided the spans of the roofs are 

 moderate. 



The jnints 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 timber are liable to from shrink- 

 ing nii'l warping. Coctiiu/ or cogging is the name given to that kind of 

 joining in which one piece of timber, hi a state of tension, is so attached 

 to another that it cannot be drawn away without one piece breaking. 

 F<I.I. 32 and S3 represent two methods of cocking the ends of tie- 



Fig. 33. 



beams on the wall-plates, giving a plan and elevation of each. In 

 both figures a represents the beam, and It the wall-plate. In the first plan, 

 which was formerly much practised, the contraction of the dovetailed 

 < n.| "1" the beam would allow it to be 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 produc- 

 tive 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 repre- 

 sented in /;/. 12. A similar arrangement is made, in some coses, 

 where wooden king-posts are used, the king-post and rafters being 

 strapped together with iron. The sinking of a roof, particularly if it 

 be of low pitch, is very injurious to the mortise-and-tenon joints of the 

 strut* and rafters, by throwing the strain on the shoulders of the 

 tenons in such a way as to break off the tenons or splinter the wood. 

 To guard against such injuries, it has been proposed by M. Perronet, a 

 Frrnch 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 mate- 

 rials used for the covering of roofs, with reference to the different 

 degrees of inclination suitable for them. Thatched roofs have been 

 usutMiirad by some to mintjin the most equable temperature in the 

 1'iiildings 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 danger- 

 ously combustible. The frequent repairs required make thatch also 

 an expensive material. Besides straw, reeds and heath are sometimes 

 used for Unteh'ng, and possess the advantage of greater durability. 

 Tiles admit heat and moisture more than good slates. Pantiles, having 

 no hole* fnr nailing through, are simply hung, by ledges, upon laths 



nailed to the raf ters. 1 'lain tiles, laid in mortar, and over-lapping, so 

 as to be of double thickness everywhere, make a very good though 

 heavy covering. Tiles of a peculiar form, called hip-tiles, are used for 

 covering salient angles ; and gutter-tile*, which are similar to them, 

 but placed with the concave side upwards, iu the valleys or receding 

 angles. Slates are laid iu various ways. They are sometimes nailed 

 down on a close boarding; or, if large, on batteni, 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 interval* 

 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 rafter being placed at such a wjdth as to come under the 

 joints. The slates are screwed down, the courses over-lapping about 

 two inches. 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 iu all such cases to use lead. Patent slating, when well executed, 

 is water-tight with as low a slope as one in six. In souie districts 

 lamiiKc of stone are used in lieu of slates or tiles. Shingles, which ,ir<; 

 like slates, but made of wood, were formerly much used in covering 

 pyramidal steeples, and in roofs of steep pitch. They are still used in 

 the United States, and are usually laid on boarding, in a similar 

 manner to common slates. 



Sheets of metal are very convenient for covering domes, and curved 

 or angular surfaces generally ; and also for flat roofs, or such as have 

 too little slope for slating. Lead is the most common material for 

 such purposes, though copper, iron, tinned iron, and recently zinc, are 

 also used. Lead terraces or flats are commonly laid on boarding or 

 plaster. The joints are sometimes soldered, but the most approved 

 method is to roll or wrap the edges into each other, making allowance 

 for expansion and contraction. A fall of a quarter of an inch in a foot 

 is sufficient for surfaces covered with sheet metal. 



Cements of various kinds have been applied to the formation of 

 roofs, and in some cases with success, though they have often been 

 found to crack, and thereby become permeable to water. Mixtures of 

 tar with lime, sand, gravel, ashes, &c., have been recommended ; and 

 osphalte has been applied to this purpose, apparently with great 

 advantage. Compositions of tar, resin, and similar substances, spread 

 upon sheets of coarse paper, have also been used. 



(Nicholson, Architectural Dictionary, Practical Builder, &c., &c. ; 

 Tredgold, Principle* of Carpentry; Robison, Mechanical P/tilnsiij,/,,/ ; 

 Rondelet, I' Art de bdttir.) 



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 equation is called a root of 

 that equation. Thus, 2, 1, 1 + V( 3 ) and 1 \?(-3) are the roots, 

 and all the roots, of the equation 



since they are the only algebraical formulae and arithmetical numbers 

 which satisfy it. On this general use of the term root, see THEORY OF 

 EQUATIONS and INVOLUTION. 



The more common use of the term root is as follows : the seventh 

 root of 8 is the incommensurable fraction whose seventh power is 8, or 

 the solution of the equation 2^=8. There are altogether seven such 

 solutions, one only arithmetical, the others of the form a + b\'( l); 

 the method of obtaining the arithmetical solution has already been 

 discussed in the article INVOLUTION ; the importance of the SO.CARK 

 HOOT will justify its consideration in an article apart. We reserve for 

 the present article the method of finding and using any root (in the 

 common sense) of any algebraical quantity. 



Every algebraical result is of the form a + 6V( 1) at widest, or 

 may be reduced to that form. Here a and 6 are meant to be real 

 algebraical quantities, that is, reducible to positive or negative whole 

 numbers or fractions, commensurable or incommensurable. Thus, if 

 5 = 0, we have the simple real quantity a; if a = 0, we have the simple 

 impossible quantity 4V( !) It is indifferent, as to the present article, 

 in what light the impossible quantity */( 1) is considered ; whether 

 [ AI.C.I IIIIA | upon that extended system of definitions which makes 

 it explicable and rational, or upon the more common system iu which it 

 is used without such explanation : for we are now merely considering 

 all algebraic formula; as results, subject to certain laws by which 

 their use is to be regulated, and without any reference to inter- 

 pretation. When we desire to consider only the arithmetical root of 

 an arithmetical quantity, we shall use the symbols V, V. (/, *c-, but 

 the exponential fractions |, J, J, &c., will denote any one of the alge- 

 braical roots of a formula. Thus V16 means simply 4 ; but (16)' is 

 an ambiguous symbol standing for either +4 or 4. And when we 

 have an equation which presents an ambiguous formula equated to 

 an unambiguous one, we mean that the unambiguous sido of the 

 equation is one of the values of the ambiguous one : iu this sense 

 (!) = i( 1 + V( 3) ). When we use the simple arithmetical symbol 

 V before an algebraical quantity, as in V( 3), we merely mean to 

 signify that the two values of ( 3)' are distinguished into -f v'( - 3) 

 and ^f( - 3). 



