^^ 0^ V ^ 5> -^ ^^•''-^ -^^ C?\^i:i'''/^ co^v^lll,'"/-^^ m , '-P ^ <*' ■^z.d« '"■% . ^^^L^tte^'^^^^ ^\^^ms^'< -t . - .^^^ I 5 <' i, I' V il ^. PARTS OF AN ORDER. 1 5 22. — Tlie Shaft: the upright part of a column standing upon the base and crowned with the capital, is from shafto, to dig — in the manner of a well, whose inside is not unlike the form of a column. 23. — The Capital: from kephale or caput, the head, is the uppermost and crowning part of the column. 24. — The Architrave : from arcJii, chief or principal, and trabs, a beam, is that part of the entablature which lies in immediate connection with the column. 25. — The Frieze: irom. fibron, a fringe or border, is that part of the entablature which is immediately above the architrave and beneath the cornice. It was called by some of the ancients zopJwriis, because it was usually enriched with sculptured animals. 26. — The Cornice: from corona, a crown, is the upper and projecting part of the entablature — being also the upper- most and crowning part of the whole order. 27- — The Pccliinent : above the entablature, is the tri- angular portion which is formed by the mclined edges of the roof at the end of the building. In Gothic architecture, the pediment is called a gable. 28. — The Tyinpaiium : is the perpendicular triangular surface which is enclosed by the cornice of the pediment. 29. — The Attic: is a small order, consisting of pilasters and entablature, raised above a larger order, instead of a pediment. An attic story is the upper story, its windows being usually square. 30. — Proportioii§ in an Order. — An order has its several members proportioned to one another by a scale of 60 equal parts, which are called minutes. If the height of buildings were always the same, the scale of equal parts would be a hxed quantity — an exact number of feet and inches. But as buildings are erected of different heights, the column and 1 6 ARCHITECTURE, its accompaniments are required to be of different dimen- sions. To ascertain the scale of equal parts, it is necessary to know the height to which the Avhole order is to be erected. This must be divided by 'the number of diameters which is directed for the order under consideration. Then the quotient obtained by such division is the length of the scale of equal parts — and is, also, the diameter of the column next above the base. For instance, in the Grecian Doric order the whole height, including column and entablature, is 8 diameters. Suppose now it were desirable to construct an example of this order, forty feet high. Then 40 feet divided by 8 gives 5 feet for the length of the scale ; and this being divided by 60, the scale is completed. The up- right columns of figures, marked H and P, by the side of the drawings illustrating the orders, designate the height and the projection of the members. The projection of each member is reckoned from a line passing through the axis of the column, and extending above it to the top of the entab- lature. The figures represent minutes, or 6oths, of the major diameter of the shaft of the column. 31. — Grecian Styles.— The original method of building among the Greeks was in what is called the Doric order : to this were afterwards added the Ionic and the Corinthian. These three were the only styles known among them. Each is distinguished from the other two by not only a peculiar- ity of some one or more of its principal parts, but also by a particular destination. The character of the Doric is robust, manly, and Herculean-like ; that of the Ionic is more deli- cate, feminine, matronly ; while that of the Corinthian is extremely delicate, youthful, and virgin-like. However they may differ in their general character, they are alike famous for grace and diguii}', elegance and grandeur, to a high degree of perfection. 32. — The l>oric Order: {Fig^, 2,) is so ancient that its origin is unknown — although some have pretended to have discovered it. But the most general opinion is, that it is an improvement upon the original wooden buildings of the ^^iiiiiiiiiiiE:: aiiiiiiiii FA^XIFUL ORICIX OF THE DORIC. 17 Grecians. These no doubt were very rude, and perhaps not unhke the following figure. Fig. I. — Supposed Origin of Doric Temple. The trunks of trees, set perpendicularly to support the roof, may be taken for columns ; the tree laid upon the tops of the perpendicular ones, the architrave ; the ends of the cross-beams which rest upon the architrave, the triglyphs ; the tree laid on the cross-beams as a support for the ends of the rafters, the bed-moulding of the cornice ; the ends of the rafters which project beyond the bed-moulding, the mutules; and perhaps the projection of the roof in front, to screen the entrance from the weather, gave origin to the portico. The peculiarities of the Doric order are the triglyphs — those parts of the frieze which have perpendicular channels cut in their surface ; the absence of a base to the column — as also of fillets between the flutings of the column ; and the plainness of the capital. The triglyphs should be so dis- posed that the width of the m.etopes — the space between the triglyphs — shall be equal to their height. 33. — The Intcrcolumniation : or space between the col- umns, is regulated by placing the centres of the columns under the centres of the triglyphs — except at the angle ot the building ; where, as may be seen in Fig. 2, one edge of i8 ARCHITECTURE. -i-LJ— i_n-T-rT^ T .n n-n-i-uT jn_n_nLJ.^-i-j_i r^ /^^ 28 3 1 30 30 J LJ U 2n 22J s; n ^-c^j vli L-J lJ LJ 1 I f n — ' Fig, 2. — Grecian Doric. k PECULIARITIES OF THE DORIC. 1 9 the triglyph must be over the centre of the column."^' Where the columns are so disposed that one of them stands beneath every other triglyph, the arrangement is called vtono-triglyph and is most common. When a column is placed beneath every third triglyph, the arrangement is called diastyle ; and when beneath every fourth, arceostyle. This last style is the worst, and is seldom adopted. 34.— The Doric Order: is suitable for buildings that are destined for national purposes, for banking-houses, etc. Its appearance, though massive and grand, is nevertheless rich and graceful. The Patent Office at Washington, and the Treasury at New York, are good specimens of this order. 35- — The Ionic Order. {Fig. 3.) — The Doric Avas for some time the only order in use among the Greeks. They gave their attention to the cultivation of it, until perfection seems to have been attained. Their temples were the prin- * Grecian Doric Order. When the -width to be occupied by the whole front is limited, to determine the diameter of the column. The relation between the parts may be expressed thus : _ 60 <7 ^ ~ ^7(^+ c) + (60 — c) Where a equals the width in feet occupied by the columns, and their inter- columniations taken collectively, measured at the base ; b equals the width of the metope, in minutes ; c equals the width of the triglyphs in minutes ; d equals the number of metopes, and x equals the diameter in feet. Example. — A front of six columns — hexastyle — 61 feet wide ; the frieze having one triglyph over each intcrcolumniation, or mono-triglyph. In this case, there being five intercolumniations and two metopes over each, therefore there are 5 x 2 = 10 metopes. Let the metope equal 42 minutes and the triglyph equal 28. Then a = 61 ; (^ = 42 ; <: = 28 ; and d =z io\ and the formula above becomes 60 X 61 60 X 61 3660 — — =5 feet = the diameter 10(42 + 28) + (60 — 28) 10 X 70 + 32 732 required. Example. — An octastyle front, 8 columns, 1S4 feet wide, three metopes over each intercolumniation, 21 in all, and the metope and triglyph 42 and 28, as before. Then 60 X 184 11040 -. , , J. , J ^ — — ;; a\ 72 ^ — • = 7-35t5 0!Z feet = the diameter required. 21 (42 + 28) + (60 — 28) 1502 ' -'-'i^o-f n 20 ARCHITECTURE. cipal objects vipon which their skill in the art was displayed ; and as the Doric order seems to have been well fitted, by its massive proportions, to represent the character of their male deities rather than the female, there seems to have been a necessity for another style which should be emble- matical of feminine graces, and with which they might decorate such temples as were dedicated to the goddesses. Hence the origin of the Ionic order. This was invented, according to historians, by Hermogenes of Alabanda ; and he being a native of Caria, then in the possession of the lonians, the order was called the Ionic. The distinguishing features of this order are the volutes or spirals of the capital ; and the dentils among the bed- mouldings of the cornice: although in some instances dentils are wanting. The volutes are said to have been designed as a representation of curls of hair on the head of a matron, of whom the whole column is taken as a sem- blance. The Ionic order is appropriate for churches, colleges, seminaries, libraries, all edifices dedicated to literature and the arts, and all places of peace and tranquillity. The front of the Custom-House, New York City, is a good specimen of this order. 36. — The Intcrcolumniation : of this and the other orders — both Roman and Grecian, with the exception of the Doric — are distinguished as follows. When the interval is one and a half diameters, it is cdXi^^ pycno style, or columns thick-set; when two diameters, systyle ; when two and a quarter diameters, t'?/j/j/^/ when three diameters, <^/rt'j/>/^ / and when more than three diameters, arcEostyle, or columns thin-set. In all the orders, when there are four columns in one row, the arrangement is called tetrastyle ; when there are six in a row, Jiexasiyle ; and when eight, octastyle. 37. — To De§cribc the Ionic Volute.— Draw a perpen- dicular from ^ to ^ {Fig. 4), and make a s equal to 20 min. or to -4- of the whole height, a c ; draw s 2X right angles to s a, and equal to i^ min. ; upon o, with 2\ min. for radius. 'ROPORTIONS OF GRECIAN IONIC. Fig. 3. — Grecian Ionic. '22 ARCHITECTURE. describe the eye of the volute ; about o, the centre of the eye, draw the square, r t i 2, with sides equal to half the diameter of the eye, viz. 2^ min., and divide it into 144 equal parts, as shown at Fig. 5. The several centres in rotation are at the angles formed by the heavy lines, as figured, i, 2, 3, 4, 5, 6, etc. The position of these angles is determined by commencing at the point, i, and making each heavy line one part less in length than the preceding one. No. i is the Fig. 4. — Ionic Volute. THE IONIC VOLUTE. 23 centre lor the arc a b {Fig, 4 ;) 2 is the centre for the arc b c ; and so on to the last. The inside spiral line is to be described from the centres, x, x, x, etc. {Fig. 5), being the centre of the first small square towards the middle of the eye from the centre for the outside arc. The breadth of the fillet at a j is to be made equal to 2-^^ min. This is for a spiral of three revolutions ; but one of any number of revolutions, as 4 or 6, may be drawn, by dividing of {Fig. 5) into a cor- responding number of equal parts. Then divide the part nearest the centre, «»»r-5<>t^o;K)^!»'»»<>-v^^ •i.-^?5>'>?»-'^M Fig. 10. — Roman Corinthian. 32 ARCHITECTURE. ^. M. 41'% MkMUMJ^J^Uj^^M 26^ .?.q;;: USicfinUioTis; I^inciples. — The upper sur- face is called the cxtrados, and the inner, the intrados. The spring is where the intrados meets the abutments. The span is the distance between the abutments. The wedge- shaped stones which form an arch are sometimes called voussoirs, the uppermost being- the keystone. The part of a pier from which an arch springs is called the impost, and the curve formed by the upper side of the voussoirs, the arcJii- volt. It is necessary that the walls, abutments, and piers on which arches are supported should be so firm as to resist the lateral thrust, as well as vertical pressure, of the arch. It will at once be seen that the lateral or sidewa}^ pressure of an arch is very considerable, when we recollect that every stone, or portion of the arch, is a wedge, a part of whose force acts to separate the abutments. For want of attention to this circumstance, important mistakes have been commit- ted, the strength of buildings materially impaired, and their ruin accelerated. In some cases, the Avant of lateral firmness in the walls is compensated by a bar of iron stretched across the span of the arch, and connecting the abutments, like the tie-beam of a roof. This is the case in the cathedral of Milan and some other Gothic buildings. 66. — All Arcade: or continuation of arches, needs only that the outer supports of the terminal arches should be strong enough to resist horizontal pressure. In the inter- mediate arches, the lateral fbrce of each arch is counter- acted by the opposing lateral force of the one contiguous to it. In bridges, however, where individual arches are liable to be destroyed by accident, it is desirable that each of the piers should possess sufficient horizontal strength to resist the lateral pressure of the adjoining arches. 67. — The Vault : is the lateral continuation of an arch, serving to cover an area or passage, and bearing the same relation to the arch that the wall does to the column, A simple vault is constructed on the principles of the arch, and distributes its pressure equally along the walls or abutments, A complex or groined \^\x\i is made by two vaults intersect- ing each other, in which case the pressure is thrown upon VARIOUS CONSTRUCTIONS OF THE DOME. 53 springing- points, and is greatly increased at those points. The groined vault is common in Gothic architecture. 68. — The Dome : sometimes called cupola, \s, a concave covering to a building, or part of it, and may be either a segment of a sphere, of a spheroid, or of any similar figure. When built of stone, it is a very strong kind of structure, even more so than the arch, since the tendency of each part to fall is counteracted, not only by those above and below it, but also by those on each side. It is only necessary that the constituent pieces should have a common form, and that this form should be somewhat like the frustum of a pyra- mid, so that, when placed in its situation, its four angles may point toward the centre, or axis, of the dome. During the erection of a dome, it is not necessar}^ that it should be sup- ported by a centring, until complete, as is done in the arch. Each circle of stones, when laid, is capable of supporting itself without aid from those above it. It follows that the dome may be left open at top, without a keystone, and yet be perfectly secure in this respect, being the reverse of the arch. The dome of the Pantheon, at Rome, has been always open at top, and 3'et has stood unimpaired for nearly 2000 years. The upper circle of stones, though apparently the weakest, is nevertheless often made to support the addi- tional weight of a lantern or tower above it. In several of the largest cathedrals, there are two domes, one within the other, which contribute their joint support to the lantern, which rests upon the top. In these buildings, the dome rests upon a circular wall, which is supported, in its turn, by arches upon massive pillars or piers. This construction is called building upon pcndcntivcs, and gives open space and room for passage beneath the dome. The remarks which have been made in regard to the abutments of the arch apply equally to the walls immediately supporting a dome. They must be of sufficient thickness and solidity to resist the lateral pressure of the dome, which is very great. The walls of the Roman Pantheon are of great depth and solid- ity. In order that a dome in itself should be perfectly secure, its lower parts must not be- too nearly vertical, since, 54 ARCHITECTURE. ill this case, they partake of the nature of perpendicular walls, and are acted upon by the spreading force of the parts above them. The dome of St. Paul's Church, in Lon- don, and some others of similar construction, are bound with chains or hoops of iron, to prevent them from spreading at bottom. Domes Avhich are made of wood depend, in part, for their strength on their internal carpentry. The Halle du Bled, in Paris, had originally a wooden dome more than 200 feet in diameter, and only one foot in thickness. This has since been replaced by a dome of iron. (See Art- 235.) 69. — The Roof: is the most common and cheap method of covering buildings, to protect them from rain and other effects of the weather. It is sometimes flat, but more fre- quently oblique, in its shape. The flat or platform roof is the least advantageous for shedding rain, and is seldom used in northern countries. The pent roof, consisting of two oblique sides meeting at top, is the most common form. These roofs are made steepest in cold climates, where they are liable to be loaded with snow. Where the four sides of the roof are all oblique, it is denominated a hipped roof, and where there are two portions to the I'oof, of different ob- liquity, it is a curb, or mansard roof. In modern times, roofs are made almost exclusively of wood, though frequently covered with incombustible materials. The internal struc- ture or carpentry of roofs is a subject of considerable me- chanical contrivance. The roof is supported by rafters, w^hich abut on the walls on each side, like the extremities of an arch. If no other timbers existed except the rafters, they would exert a strong lateral pressure on the walls, tending to separate and overthrow them. To counteract this lateral force, a tie-beam, as it is called, extends across, receiving the ends of the rafters, and protecting the wall from their horizontal thrust. To prevent the tie-beam from sagging^ or bending downward with its own weight, a king- post is erected from this beam, to the upper angle of the rafters, serving to connect the whole, and to suspend the weight of the beam. This is called trussing. Queen-posts MANNER OF CONSTRUCTING ROOFS. 55 are sometimes added, parallel to the king-post, in large roofs! also various other connecting timbers. In Gothic buildings, where the vaults do not admit of the use of a tie-beam, the rafters are prevented from spreading, as in an arch, by the strength, of the buttresses. In comparing the lateral pressure of a high roof with that of a low one, the length of the tie-beam being the same, it will be seen that a high roof, from its containing most materials, may produce the greatest pressure, as far as weight is concerned. On the other hand, if the weight of both be equal, then the low roof will exert the greater press- ure ; and this will increase in proportion to the distance of the point at which perpendiculars, drawn from the end of each rafter, w^ould meet. In roofs, as well as in wooden domes and bridges, the materials are subjected to an inter- nal strain, to resist which the cohesive strength of the ma- terial is relied on. On this account, beams should, when possible, be of one piece. Where this cannot be effected, two or more beams are connected together by splicing. Spliced beams are never so strong as whole ones, yet they may be made to approach the same strength, by affixing lat- eral pieces, or by making the ends overlay each other, and connecting them with bolts and straps of iron. The ten- dency to separate is also resisted, by letting the two pieces into each other by the process called scarfing. Mortices, in- tended to truss or suspend one piece by another, should be formed upon similar principles. Roofs in the United States, after beino^ ^boarded, receive a secondary covering of shingles. When intended to bo incombustible, they arc covered with slates or earthen tiles, or with sheets of lead, copper, or tinned iron. Slates are preferable to tiles, being lighter, and absorbing less moisture. Metallic sheets are chiefly used for flat roofs, wooden domes, and curved and angular surfaces, which require a flexible material to cover them, or have not a sufficient pitch to shed the rain from slates or shingles. Various artificial composi- tions are occasionally used to cover roofs, the most common of which are mixtures of tar with lime, and sometimes with sand and gravel. — Ency. Ain. (See Art. 202.) SECTION II.— CONSTRUCTION. Art. 70. — Construction X:§sentia]. — Construction is that part of the Science of Building which treats of the Laws of Pressure and the strength of materials. To the architect and builder a knowledge of it is absolutely essential. It de- serves a larger place in a volume of this kind than is gene- rally allotted to it. Something, indeed, has been said upon the styles and principles, by which the best arrangements may be ascertained ; yet, besides this, there is much to be learned. For however precise or workmanlike the several parts may be made, what Avill it avail, should the system of iraming, from deficient material, or an erroneous position of its timbers, fail to sustain even its own weight ? Hence the necessity for a knowledge of the laws of pressure and the strength of materials. These being once understood, Ave can with confidence determine the best position and dimen- sions for the several pieces which compose a floor or a roof, a partition or a bridge. As systems of framing are more or less exposed to heavy weights and strains, and, in case of failure, cause not only a loss of labor and material, but fre- quently that of life itself, it is very important that the mate- rials employed be of the proper quantity and quality to serve their destination. And, on the other hand, any superfluous material is not only useless, but a positive injury, as it is an unnecessary load upon the points of support. It is neces- sar}', therefore, to know the least quantity of material that will suffice for strength. Not the least common fault in framing is that of using an excess of material. Economy, at least, would seem to require that this evil be abated. Before proceeding to consider the principles upon which a system of framing should be constructed, let us attend to a few of the elementary laws in MccJianicSy Avhich Avill be found to be of great value in determining those principles. INTERIOR OF THE CATIIJ:DRAL. SIENNA. DIRI-XT AND OBLIQUE SUPPORTS. 57 71. — I^a\v§ of Pressure. — (i.) A heavy body always ex- erts a pressure equal to its own weight in a vertical direc- tion. Example: Suppose an iron ball weighing loo lbs. be supported upon the top of a perpendicular post {Fig. 22-A) ; then the pressure exerted upon that post will be equal to the weight of the ball, viz., 100 lbs. (2.) But if two inclined posts {Fig. 22-B) be substituted for the perpendicular sup- port, the united pressures upon these posts will be more than equal to the weight, and will be in proportion to their position. The farther apart their feet are spread the greater will be the pressure, and ince vci'sa. Hence tremendous strains may be exerted by a comparatively small weight. And it follows, therefore, that a piece of timber intended for a strut or post should be so placed that its axis may coincide, as nearly as possible, with the direction of the pressure. The direction of the pressure of the weight W {Fig. 22-B) is in the vertical line b d\ and the weight W would fall in that line if the two posts were removed ; hence the best position for a support for the weight would be in A. that line. But as it rarely occurs in systems of framing that weights can be supported by any single resistance, they requiring generally two or more supports (as in the case of a roof supported by its rafters), it becomes important, there- fore, to know the exact amount of pressure any certain weight is capable of exertmg upon oblique supports. Now, it has been ascertained that the three lines of a triangle, drawn parallel with the direction of three concurring forces in equilibrium, are in proportion respectively to these 58 CONSTRUCTION. forces. For example, in Fig. 22-By we have a representation of three forces concurring in a point, which forces are in equiUbrium and at rest ; thus, the weight W is one force, and the resistances exerted by the two pieces of timber are the other two forces. The direction in which the first force acts is vertical — downwards ; the direction of the two other forces is in the axis of each piece of timber respectively. These three forces all tend towards the point b. Draw the axes a b and b c oi the two supports ; make b d vertical, and from d draw d e and d f parallel w4th the axes b c and b a repectively. Then the triangle b d c has its lines parallel respectively with the direction of the three forces ; thus, bd\s in the direction of the weight PV,dc paral- lel with the axis of the timber i^,and ^ ^ is in the direction of the timber C. In accordance with the principle above stated, the lengths of the sides of the triangle b d e are in propor- tion respectively to the three forces aforesaid ; thus — As the length of the line b d Is to the number of pounds in the weight IV, So is the length of the line b e To the number of pounds' pressure resisted by the timber C. Again — As the length of the line /; d Is to the number of pounds in the weight JF, So is the length of the line d c To the number of pounds' pressure resisted by the timber D. And again — As the length of the line b e Is to the pounds' pressure resisted by C, So is the length of the Ime d c To the pounds' pressure resisted by D. These proportions are more briefly stated thus — \st. bd\ W \\b e\ P, P being used as a symbol to represent the number of pounds' pressure resisted by the timber C. PARALLELOGRAM OF FORCES. 59 O representing the number of pounds' pressure resisted by the timber D. Zd, be:P::de:Q. 72. — Parallelog^ratn of Forces. — This relation between lines and pressures is applicable in ascertaining the pres- sures induced by known weights throughout any system of framing. The parallelogram b e d f is called the Parallelo- gram of Forces ; the two lines be and bf being called the components, and the line b d the resultant. Where it is re- quired to find the components from a given resultant [Fig. 22-B)y the fourth line d f nt^d not be drawn, for the triangle b d e gives the desired result. But when the resultant is to be ascertained from given components {Fig. 28), it is more convenient to draw the fourth line. 73b — The He§o!utBoii of Forces: is the findinsf of two or more forces which, acting in different directions, shall exactly balance the pressure of any given single force. To make a practical application of this, let it be required to ascertain the oblique pressure inFig. 22-B. In this figure the line bd measures half an inch (0'5 inch), and the line be three tenths of an inch (0-3 inch). Now if the weight W be supposed to be 1200 pounds, then the first stated propor- tion above, b d '. JV : : b e : P, becomes 0-5:1 200 : : o • 3 : P. And since the product of the means divided by one of the extremes gives the other extreme, this proportion n:ay be put in the form of an eqtiation, thus — 1 200 X o • 3 _ p Performing the arithmetical operation here indicated — that is, multiplying together the two quantities above the line, and dividing the product by the quantity under the line — the 6o CONSTRUCTION. quotient will be equal to the quantity represented by P, viz., the pressure resisted by the timber C. Thus — 1 200 0.3 0.5)360-0 720 = P. The strain upon the timber C is, therefore, equal to 720 pounds; and since, in this case (the two timbers being- in- clined equally from the vertical), the line ^ ^is equal to the line b c, therefore the strain upon the other timber D is also 720 pounds. Fig. 23. 74. — Iiiclinatioia of Supports Uaiequal. — In Fig. 23 the pressures in the two supports are unequal. The supports are also unequal in length. The length of the supports, however, does not alter the amount of pressure from the concentrated load supported ; but generally long timbers are not so capable of resistance as shorter ones. For, not being so stiff, they bend more readily, and, since the com- pression is in proportion to the length, they therefore shorten more. To ascertain the pressures in Fig. 23, let the weight suspended from b dh^ equal to two and three quarter tons (2-75 tons). The line b d measures five and a half tenths of an inch (0-55 inch), and the line b c half an inch (0-5 inch). Therefore, the proportion b d \ W w b c '. P becomes o • 5 5 : 2 • 75 : : o- 5 : P, JTRAIN IN PROPORTION TO INCLINATION. 6l and i:7Lx_o^ ^ ^, 0-55 2-75 0.5 I 10 275 275 The strain upon the timber A is, therefore, equal to two and a half tons. Again, the line r d measures four tenths of an inch (0-4 inch) ; therefore, the proportion b d : W :: e d : Q becomes o- 55 : 2-75 : : 0-4 : (7, and 2- •75 X o-4_ -~Q: 0.55 2-75 0-4 •55) I • 100(2 = I 10 e. The strain upon the timber B is. therefore, equal to two tons. 75. — The Strains Exceed the ^Weights. — Thus the united pressures upon the two inclined supports always exceed the weight. In the last case, 2 J tons exert a pressure of 2^ and 2 tons, equal together to 4I- tons ; and in the former case, 1200 pounds exert a pressure of twice 720 pounds, equal to 1440 pounds. The smaller the angle of inclination to the horizontal, the greater will be the pressure upon the sup- ports. So, in the frame of a roof, the strain upon the rafters decreases gradually with the increase of the angle of incli- nation to the horizon, the length of the rafter remaining the same. 62 CONSTRUCTION. This is true in comparing one system of framing with another ; but in a system where the concentrated weight to be supported is not in the middle (see Fig. 23), and, in con- sequence, the supports are not inclined equally, the strain will be greatest upon that support which has the greatest inclination to the horizon. 76. — Minimum TlBriB§t of RafteE'§. — Ordinarily, as in roofs, the load is not concentrated, it being that of the fram- ing itself. Here the amount of the load will be in proportion to the length of the rafter, and this will increase with the mcrease of the angle of inclination, the span remaining the same. So it is seen that in enlarging the angle of inclina- tion to the horizon in order to lessen the oblique thrust, the load is increased in consequence of the elongation of the rafter, thus increasing the oblique thrust. Hence there is an economical angle of inclination. A rafter Avill have the least oblique thrust when its angle of inclination to the horizon is 35° 16' nearly. This angle is attained very nearly when the rafter rises 8J inches per foot, or when the height B C {Fig. 32) is to the base A C as 8^ is to 12, or as 0-7071 is to I • o. 77. — Practical MctSiod of Determining Strain§. — A com- parison of pressures in timbers, according to their position, may be readily made by drawing various designs of framing and estimating the several strains in accordance with the parallelogram of forces, always drawing the triangle b d e so that the three lines shall be parallel with the three forces or pressures respectively. The length of the lines forming this triangle is unimportant, but it will be found more con- venient if the line drawn parallel with the known force is made to contain as many inches as the known force contains pounds, or ai many tenths of an inch as pounds, or as many inches as :o is, or tenths of an inch as tons; or, in general, as many divisions of any convenient scale as there are units of weight or pressure in the known force. If drawn in this manner, then the number of divisions of the same scale found in the other two lines of the triangle will equal the units of pressure or weight of the other two forces respect- HORIZONTAL THRUST OF RAFTERS. 63 ively, and the pressures sought will be ascertained simply by applying the scale to the lines of the triangle. For example, in Fig. 23, the vertical line b d, of the tri- angle, measures fifty-live hundredths of an inch (0-55 inch); the line be, fifty hundredths (0-50 inch); and the line e d, forty (0-40 inch). Now, if it be supposed that the vertical pressure, or the weight suspended below b d, is equal to 55 pounds, then the pressure on A will equal 50 pounds, and that on B will equal 40 pounds ; for. by the proportion above stated, b d: JV:: b e : P, 55 : 55 :: 50: 50; and so of the other pressure. If a scale cannot be had of equal proportions with the forces, the arithmetical process will be shortened somewhat by making the line of the triangle that represents the knozvn weight equal to unity of a decimally divided scale, then the other lines will be measured in tenths or hundredths; and in the numerical statement of the proportions between the lines and forces, the first term being unity, the fourth term will be ascertained simply by multiplying the second and third terms together. For example, if the three lines are i, 0-7, and 1-3, and the known weight is 6 tons, then b d \ IV :: b e : P becomes I : 6 :: o-y : P = 4'2, equals four and two tenths tons. Again — b d : W :\ c d : Q becomes I :6:: 1.3 : e = 7.8, equals seven and eight tenths tons. 78.— Horizontal Tlirust.— In Fig. 24, the weight Impresses the struts in the direction of their length ; their feet, 71 11, therefore, tend to move in the direction no, and would so move Avere they not opposed by a sufficient resistance from the blocks, A and A. If a piece of each block be cut off at 64 CONSTRUCTION. the horizontal line, a ;/, the feet of the struts would slide away from each other along- that line, in the direction na; but if, instead of these, two pieces were cut off at the verti- cal line, nb, then the struts would descend vertically. To estimate the horizontal and the vertical pressures exerted by the struts, let n o be made equal (upon any scale of equal parts) to the number of tons with which the strut is pressed ; Fig. 24. construct the parallelogram of forces by drawing c parallel to an, and of parallel to bn; then nf (by the same scale) shows the number of tons pressure that is exerted by the strut in the direction n a, and ;/ e shows the amount exerted in the direction n b. By constructing designs similar to this, giving various and dissimilar positions to the struts, and then estimating the pressures, it will be found in every case that the horizontal pressure of one strut is exactly equal to that of the other, however much one strut may be inclined more than the other; and also, that the united vertical pressure of the two struts is exactly equal to the weight W, (In this calculation the weight of the timbers has not been taken into consideration, simply to avoid complication to the learner. In practice it is requisite to include the weight of the framing with the load upon the framing.) Suppose that the two struts, B and B {Fig. 24), were rafters of a roof, and that instead of the blocks, A and A, the walls of a building were the supports: then, to prevent TIES DESIRABLE IN ROOFS. 65 the walls from being thrown over by the thrust of B and B^ it would be desirable to remove the horizontal pressure. This may be done by uniting- the feet of the rafters with a Fig. 25. rope, iron rod, or piece of timber, as in Fig. 25. This figure is similar to the truss of a roof. The horizontal strains on the tie-beam, tending to pull it asunder in the direction of its length, may be measured at the foot of the rafter, as was shown at^ Fig. 24 ; but it can be more readily and as accu- rately measured by drawing from /and e horizontal lines to the vertical line, h d, meeting it in and o ; then/^ will be the horizontal thrust at B^ and co at A ; these will be found to equal one another. When the rafters of a roof are thus connected, all tendency to thrust out the walls horizontally is removed, the only pressure on them is in a vertical direc- tion, being equal to the weight of the roof and whatever it has to support. This pressure is beneficial rather than otherwise, as a roof having trusses thus formed, and the trusses well braced to each other, tends to steady the walls. 79.— PosilioBB of Supports.— /^zVy. 26 and 27 exhibit two methods of supporting the equal weights, W and IV. Let it be required to measure and compare the strains produced on the pieces, A B and A C. Construct the parallelogram of forces, ebfd, according to Art. 71. Then b f \\\\\ show the 66 CONSTRUCTION. Strain on A B, and b e the strain on A C, By comparing the figures, bd being equal in each, it will be seen that the strains in Fig. 26 are about three times as great as those in Fig. 27. Fig. 27 ; the position of the pieces, A B and A C, in Fig 27, is therefore far preferable. Fig. 28. 80i— The Composition of Forces : consists in ascertain- ing the direction and amount of one force which shall be just capable of balancing two or more given forces, acting in different directions. This is only the reverse of the resolu- STRAINS INVOLVED IN CRANE. 67 tion of forces ; and the two are founded on one and the same principle, and may be solved in the same manner. For example, let A and B {Fig. 28) be two pieces of timben pressed in the direction of their length towards b — A by a force equal to 6 tons weight, and B 9 tons. To find the direction and amount of pressure they would unitedly exert, draw the lines b e and ^/ in a line with the axes of the timbers, and make be equal to the pressure exerted by ^, viz., 9 ; also make ^/ equal to the pressure on A^ viz., 6, and complete the parallelogram of forces ebfd; then bd, the diagonal of the parallelogram, will be the direction, and its length, 9-25, will be the amount, of the united pressures of A and of B, The line b d\'s> termed the resultant of the two forces b f and be, U A and B are to be supported by one post, C, the best position for that post will be in the direc- tion of the diagonal bd; and it will require to be sufficiently strong to support the united pressures of A and of B, which are equal to 9-25 or 9^ tons. Fig. 29. 81.— Another Example.— Let Fig. 29 represent a piece of framing commonly called a crane, which is used for hoist- ing heavy weights by means of the rope, B bf, which passes over a pulley at b. This, though similar to Figs. 26 and 27, is, however, still materially different. In those figures, the strain is in one direction only, viz., from b to d\ but in this there are two strains, from A to B and from A to IV. The strain in the direction A B Is evidently equal to that in the 68 CONSTRUCTION. direction A JV. To ascertain the best position for the strut A Cy make d e equal to hf, and complete the parallelogram of forces cbfd; then draw the diagonal b d, and it will be the position required. Should the foot, C, of the strut be placed either higher or lower, the strain on A C would be in- creased. In constructing cranes, it is advisable, in order that the piece B A may be under a gentle pressure, to place the foot of the strut a trifle lower than where the diagonal /;rt^ would indicate, but never higher. 7^ HvJ Fic.. 30. W^vJy 82.— Ties aii> CIS m ^- S H H > Pounds Pounds Pounds per inch. per inch. per inch. 0-613 9500 840 2250 0-762 1 1 700 1 160 2800 0-774 8000 1250 2650 0-369 7S50 540 650 0-388 6650 480 800 0-423 5700 370 800 0-397 3400 800 0-491 6700 1250 0-517 5850 .... 3100 0-574 8450 2700 0-877 13750 4100 0-494 9050 2500 0-421 7800 2100 0-837 1 1 600 .... 5700 0-439 4900 .... 1700 0-916 moo 6S0O 1-282 1 2 100 .... 7700 r. Georgia Pine Locust White Oak Spruce White Pine Hemlock White Wood Chestnut Ash Maple Hickory Cherry Black Walnut Mahogany (St. Domingo) (Bay Wood) .. Live Oak Lignum Vitae 900 1 1 20 1060 260 320 320 320 500 1240 1080 1640 1000 840 2280 680 2720 3080 The resistance of timber of the same name varies much ; depending as it obviously must on the soil in which it grew, on its age before and after cutting, on the time of year when cut, and on the manner in which it has been kept since it was cut. And of wood from the same tree much depends upon its location, whether at the butt or towards the limbs, and whether at the heart or of the sap, or at a point mid- way from the centre to the circumference of the tree. The 80 CONSTRUCTION. pieces submitted to experiment were of ordinary good quality, such as would be deemed proper to be used in framing. The prisms crushed were generally small, about 2 inches long, and from i inch to i^ inches square ; some were Avider one way than the other, but all containing in area of cross section from i to 2 inches. The weight given in the table is the average weight per superficial inch. Of the first six woods named, there were nine specimens of each tested ; of the others, generally three specimens. The results for the first six woods named are taken from the author's work on Transverse Strams, published by John Wiley & Sons, New York. The results for these six woods, as well as those for all the others named in the table, were obtained by experiments carefully made by the author. The first six woods named Avere tested in 1874 and 1876, and upon a testing machine, in which the power is transmitted to the pieces tested, by levers acting upon knife-edges. For a description of this machine, see Transverse Strains, Art. 704. The woods named in the table, other than the first six, Avere tested some twenty years since, and upon a hydraulic press, which, owing to friction, gave results too low. The results, as thus ascertained, were given to the public in the 7th edition of this work, in 1857. I^^ the present edi- tion, the figures in Table L, for these woods, are those Avhich have resulted by adding to the results given by the hydraulic press a certain quantity thought to be requisite to compensate for the loss by friction. Thus corrected, the figures in the table may be taken as sufficiently near approx- imations for use in the rules, — although not so trustworthy as the results given for the first six woods named, as these were obtained upon a superior testing machine, as above stated. In the preceding table, the second column contains the specific gravity of the several kinds of wood, showing their comparative density. The weight in pounds of a cubic foot of any kind of wood or other material is equal to its specific gravity multiplied by 62-5, this number being the weight in pounds of a cubic foot of water. The third column EXPLANATION OF TABLE L 8 1 contains the weight in pounds required to crush a prism having a base of one inch square ; the pressure applied to the fibres longitudinally. In practice, it is usual never to load material exposed to compression with more than one fourth of the crushing weight, and generally with from one sixth to one tenth only. The fourth column contains the weight in pounds which, applied in line with the length of the fibres, is required to force off a part of the piece, causing the iibres to separate by sliding, the surface separated being one inch square. The fifth column contains the weight in pounds required to crush the piece when the pressure is applied to the fibres transversely, the piece being one inch thick, and the surface crushed being one inch square, and depressed one twentieth of an inch deep. The sixth column contains the value of P in the rules; P being the weight in pounds, ap- plied to the iibres transversely, which is required to make a sensible impression one inch square on the side of the piece, this being the greatest weight that would be proper for a post to be loaded with per inch surface of bearing, resting on the side of the kind of wood set opposite in the table. A greater weight would, in proportion to the excess, crush the side of the wood under the post, and proportionably derange the framing, if not cause a total failure. It will be observed that the measure of this resistance is useful in limiting the load on a post according to the kind of material contained, not in the/(?5/, but in the timber upon which the post presses. 95. — Rcsistaatcc to Tension. — The resistance of materials to the force of stretching, as exemplified in the case of a rope from which a weight is suspended, is termed the resist- ance to tension. In fibrous materials, this force will be differ- ent in the same specimen, in accordance with the direction in which the force acts, whether in the direction of the length of the fibres or at right angles to the direction of their length. It has been found that, in hard Avoods, the resistance in the former direction is about eisfht to ten times what it is in the latter ; and in soft woods, straight grained, such as white pine, the resistance is from sixteen to twenty times. A knowledge of the resistance in the direction of the fibres is the most useful in practice. 82 CONSTRUCTION. In the following table arc recorded the results of ex- periments made to test this resistance in some of the woods in common use, and also in iron, cast and wrought. Each specimen of the woods was turned cylindrical, and about 2 inches diameter, and then the middle part reduced to about f of an inch diameter, at the middle of the reduced part, and thence gradually increased toward each end, where it was considerably larger at its junction with the enlarged end. The results, in the case of the iron and of the first six woods named, are taken from the author's work, Trans- verse Strains, Table XX. Experiments were made upon the other three woods named by a hydraulic press, some twenty years since, and the results were first published in the 7th edition of this work, in 1857. These results, owing to friction, wxre too low. Adding to them what is supposed to be the loss by friction of the machine, the results thus corrected are what are given for these three woods in the following table, and may be taken as fair approximations, but are not so trustworthy as the figures given for the other six woods and for the metals. Table II. — Resistanxe to Tension. Material. Georgia Pine Locust White Oak Spruce White Pine Hemlock Hickory. . . Maple Ash Cast Iron, American ) from English \ to Wrought Iron, American \ from " English \ to Specific Gravity. 0-65 0.794 o 774 o 432 0-458 0402 0-75I 0694 o-6oS 6-944 7 584 7 600 7-792 Pounds re- quired to rup- ture one inch bquare. 16OCO 24800 I95CO 19500 I20C0 8700 26OCO 20000 15000 27000 17000 Coooo 50COO The figures in the table denote the ultimate capability of a bar one inch square, or the weight in pounds required to VALUES C)i'^ MATERIALS FOR CROSS-STRAINS. 83 produce rupture. Just what portion of this should be taken as the safe capabiHty will depend upon the nature of the strain to which the material is to be exposed. In practice it is found that, through defects in workmanship, the attach- ments may be so made as to cause the strain to act along one side of the piece, instead of through its axis; and that in this case fracture will be produced with one third of the strain that can be sustained through the axis. Due to this and other contingencies, it is usual to subject materials exposed to tensile strain with only from one sixth to one tenth of the breaking weight. 96, — Resistance to Transverse Strains. — In the follow- ing table are recorded the results of experiments made to test the capability of the various materials named to resist the effects of transverse strain. The figures are taken from the author's work. Transverse Strains^ before referred to. Table III. — Transverse Strains. Material. Georgia Pine Locust White Oak . Spruce White Pine Hemlock White Wood Chestnut Ash Maple Hickory Cherry Black Walnut Mahogany (St. Domingo) (Bay Wood).. Cast Iron, American .... " English Wrought Iron, American " English . . Steel, in Bars Blue Stone Flagging . . . Sand Stone Brick, common " pressed Marble, East Chester . . . B. Resistance to Rupture. 850 1200 650 500 450 600 480 900 1 100 1050 650 650 850 2500 2100 2600 1900 6000 200 59 33 37 147 F. Resistance to Flexure. 5900 5050 3100 3500 2900 2800 3450 2550 4000 5150 3850 2850 - 3900 3600 4750 50000 40000 62000 60000 70000 Extension of Fibres. •00109 •0015 •00086 • 00098 ■0014 •00095 • 00096 •00103 ■001 I I 0014 •0013 •001563 •00104 ■OOII6 00109 Margin for Safety. 1-84 2 -20 3-39 2-23 I-7I 35 52 54 82 12 91 03 57 16 28 84 CONSTRUCTION. The figures in the second column, headed B, denote the weight in pounds required to break a luiit of the material named when suspended from the middle, the piece being supported at each end. The unit of material is a bar one inch square and one foot long between the bearings. The third column, headed F, contains the values of the several mate- rials named as to their resistance to flexure, as explained in Arts. 302-305, Transverse Strains. These values of F, as constants, are used in the rules. The fourth column, headed Cy contains the values of the several materials named, denot- ing the elasticity of the fibres, as explained in Art. 312, Transverse Strains. These values of r, as constants, are used in the rules. The fifth column, headed a, contains for the several ma- terials named the ratio of the resistance to flexure as com- pared with that to rupture, and which, as constants used in the rules, indicate the margin of safety to be given for each kind of material. The figures given in each case show the smallest possible A'alue that may be safely given to a, the factor of safety. In practice it is generally taken higher than the amount given in the table. For example, in the table the value of B, the constant for rupture b}^ transverse strain for spruce, is 550. Now, if the dimensions of a spruce beam to carry a given weight be computed by the rules, using the constant B, at 550, the beam will be of such a size that the given weight will just break it. But if, in the computation, instead of taking the full value of B, only a part of it be taken, then the beam will not break immediately ; and if the part taken be so small that its effect upon the fibres shall not be sufficient to strain them beyond their limit of elasticity, the beam will be capable of sustaining the weight for an indefinite period ; in this case the beam will be loaded by what is termed the safe weight. Or, since the value oi a for spruce is 2-23 in the table, if, in- stead of taking B at 550, its full value, only the quotient arising from a division of B by a be taken — or 550 divided by 2-23, which equals 246-6 — then the beam will be of suffi- cient size to carry the load safely. Therefore, while the con- THE VAKIOUS CLASSES OF PRESSURES. 85 stant B is to be used for a breaking- weight, for a safe load P the quotient of — is to be used* But, again, if a be taken at a its value as given in the table, the computed beam v/ili be loaded up to its limit of safety. So loaded that, if the load be increased only in a small degree, the limit of safety will be passed, and the beam liable, in time, to fail by rupture. Therefore, as the values of a, in the table, are the smallest possible, it is prudent in practice always to take a larger than the table value. For example, the table value of a for spruce is 2-23, but in practice let it be taken at 3 or 4. 97. — Resistance to Compression. — The resistance of ma- terials to the force of compression may be considered in several ways. Posts having their heights less than ten times their least sides will crush before bending ; these belong- to one class : another class is that which com- prises all posts the height of which is equal to ten times their least sides, or more than ten times ; these will bend before crushing. Then there remains to be considered the manner in which the pressure is applied : whether in line with the fibres, or transversely to them ; and, again, whether the pressure tends to crush the fibres, or simply to force off a part of the piece by splitting. The various pressures may be comprised in the four classes following, namely: 1st. — When the pressure is applied to the fibres trans- versely. 2d.— When the pressure is applied to the fibres longi- tudinally, and so as to split off the part pressed against, causing the fibres to separate by sliding. 3d. — When the pressure is applied to the fibres longi- tudinally, and on short pieces. 4th. — When the pressure is applied to the fibres longi- tudinall}^, and on long pieces. These four classes will now be considered in their reg- ular order. 86 CONSTRUCTION. 98. — Compression Transversely to the Fibres. — In this first class of compression, experiment has shown that the resistance is in proportion to the number of fibres pressed, that is, in proportion to the area. For example, if 5000 pounds is required to crush a prism with a base i inch square, it will require 20,000 pounds to crush a prism having a base of 2 by 2 inches, equal to 4 inches area ; because 4 times 5000 equals 20000. Therefore, if any given surface pressed be multiplied by the pressure per inch which the kind of material pressed ma}^ be safely trusted with, the product will be the total pressure which may with safety be put upon the given sur- face. Now, the capability for this kind of resistance is given in column P, in Table I., for each kind of material named in the table. Therefore, to find the limit of weight, proceed as follows: 99. — The liiinit of IVeiglit. — To ascertain what weight a post may be loaded with, so as not to crush the surface against which it presses, we have — Rule 1. — Multiply the area of the post in inches by the value of P, Table I., and the product will be the weight re- quired in pounds ; or — W^AP. (i.) Example, — A post, 8 by 10 inches, stands upon a white- pine girder; the area equals 8 x 10 = 80 inches. This being multiplied by 320, the value of /^, Table I., set opposite white pine, the product, 25600, is the required weight in pounds. 100. — Area of Post.— To ascertain what area a post must have in order to prevent the post, loaded with a given weight, from crushing the surface against which it presses, we have — Rule II. — Divide the given weight in pounds by the value of P, Table I., and the quotient will be the area required in inches ; or — RESISTANCE TO RUPTURE UV SLIDING. 8/ Example. — A post standing on a Georgia-pine girder is loaded with 100,000 pounds: what must be its area? The weight, 1 00000, divided by 900, the value of P, Table I., set opposite Georgia pine, the quotient, in • 11, is the required area in inches. The post may be 10 by 11-^, or 10 by 11 mches; or if square each side will be 10-54 inches, or w^^ inches diameter if round. 101.— Riaptiirc by Sliding. — In this the second class of rupture by compression, it has been ascertained by ex- periment that the resistance is in proportion to the area of the surface separated without regard to the form of the sur- face. Now, in Table I., column //, we have the ultimate resistance to this strain of the several materials named. But to obtain the safe load per inch, the ultimate resist- ance of the table is to be divided by a factor of safety, of such value as circumstances may seem to require. Gener- ally this factor may be taken at 3. Then to obtain the safe, load for any given case, we have but to multiply the given surface by the ultimate resistance, and divide by the factor of safety ; therefore, proceed as follows : 102- — TJae Limit of WclgliJ. — To ascertain what weight may be sustained safely by the resistance of a given area of surface, when the weight tends to split off the part pressed against by causing, in case of fracture, one surface to slide on the other, we have — Rule III. — Multiply the area of the surface by the value of H, in Table I. divide by the factor of safety, and the quotient w^ill be the weight required in pounds ; or — IF = 4^ (3.) a Example. — The foot of a rafter is framed into the end of its tie-beam, so that the uncut substance of the tie-beam is I 5 inches long from the end of the tie-beam to the joint of the rafter ; the tie-beam is of white pine, and is 6 inches thick: what amount of horizontal thrust will this end of the tie-beam sustain, without danger of having the end o| 88 CONSTRUCTION. the tie-beam split off? Here the area of surface that sus- tains the pressure is 6 by 15 inches, equal to 90 inches. This multiplied by 480, the value of //, set opposite to white pine, Table I., and divided by 3, as a factor of safety, gives a quotient of 14400, and this is the required weight in pounds. 103- — Area of Surface— To ascertain the area of surface that is required to sustain a given Aveight safely, when the weight tends to split off the part pressed against, by causing, in case of fracture, one surface to slide on the other, we have — Rule \Y . — Divide the given weight in pounds by the value of Hy Table I. ; multiply the quotient by the factor of safety, and the product will be the Required area in inches ; or — ^ = -77- ^-^ Example. — The load on a rafter causes a horizontal thrust at its foot of 40,000 pounds, tending to split off the end of the tie-beam : what must be the length of the tie-beam be- yond the line where the foot of the rafter is framed into it, the tie-beam being of Georgia pine, and 9 inches thick ? The weight, or horizontal thrust, 40000, divided by 840, the value of //, Table I., set opposite Georgia pine, gives a quo- tient of 47-619, and this multiplied by 3, as a factor of safety, gives a product of 142-857. This, the area of surface in inches, divided by 9, the breadth of the surface strained (equal to the thickness of the tie-beam), the quotient, 15.87, is the length in inches from the end of the tie-beam to the rafter joint, say 16 inches. 104.— Tenons and Splices.— A knowledge of this kind of resistance of materials is useful, also, in ascertaining the length of framed tenons, so as to prevent the pin, or key, with which they are fastened from tearing out ; and, also, in cases where tie-beams, or other timber under a tensile strain, CRUSiriNG STRENGTH OF STOUT POSTS. 89 are spliced, this rule gives the length of the jog-gle at each end of the splice. 105. — §toui P©§ts.— These comprise the third class of ob- jects subject to compression (Art. 97), and include all posts which are less than ten diameters high. The resistance to compression, in this class, is ascertained to be directly in pro- portion to the area of cross-section of the post. Now in Table I., column C, the ultimate resistance to crushing is given for the several kinds of materials named ; from Avhich the safe resistance per inch may be obtained by dividing it by a proper factor of safety. Having the safe resistance per inch, the resistance of any given post may be determined b}^ multiplying it by the area of the cross-section of the post. Therefore, proceed as follows : 106. — TBie L.Qinit of WefgBU. — To find the weight that can be safely sustained by a post, when the height of the post is less than ten times the diameter if round, or ten times the thickness if rectangular, and the direction of the pressure coinciding with the axis, we have — Rule V. — Multiply the area of the cross-section of the post in inches by the value of C, in Table I. ; divide the pro- duct by the factor of safety, and the quotient will be the re- quired weight in pounds ; or — W^ ™- ■ (5-) Example. — A Georgia-pine post is 6 feet high, and in cross-section 8 X 12 inches: what weight will it safely sus- tain? The height of this post, 12 x 6=: 72 inches, which is less than 10 x 8 (the size of the narrowest side) =: 80 inches; it therefore belongs to the class coming under this rule. The area = 8 x 12 = 96 inches ; this multiplied by 9500, the value of C, in the table, set opposite Georgia pine, and divided by 6, as a factor of safety, the quotient, 152000, is the weight in pounds required. It will be observed that the weight would be the same for a Georgia-pine post of any height less than 90 CONSTRUCTION. ID times 8 inches = 80 inches == 6 feet 8 inches, provided its breadth and thickness remain the same, 12 and 8 inches. (07.— Area of Post. — To find the area of the cross-sec- tion of a post to sustain a given weight safely, the height of the post being less than ten times the diameter if round, or ten times the least side if rectangular, the pressure coinciding with the axis, Ave have — Rule VI. — Divide the given weight in pounds by the value of C, in Table I. ; multiply the quotient by the factor of safety, and the product will be the required area in inches ; or — A = 'IJ'-. (6.) Example. — A weight of 40,000 pounds is to be sustained by a white-pine post 4 feet high : what must be its area of section to sustain the weight safely ? Here, 40000 divided by 6650, the value of C, in Table 1., set opposite white pine, and the quotient multiplied by 6, as a factor of safety, the pro- duct is 36 ; this, therefore, is the required area, and such a post may be 6 x 6 inches. To find the least side, so that it shall not be less than one tenth of the height, divide the height, reduced to inches, by 10, and make the least side to exceed this quotient. The area divided by the least side so determined will give the wide side. If, however, by this process, the first side found should prove to be the greatest, then the size of the post is to be found by Rule IX., X., or XI. 108. — Area of Round Posts. — In case the post is to be round, its diameter may be found by reference to the Table of Circles in the Appendix, in the column of diameters, op- posite to the area of the post to be found in the column of areas, or opposite to the next nearest area. For example, suppose the required area, as just found by the example under Rule VI., is 36 : by reference to the column of areas, 35-78 is the nearest to 36, and the diameter set opposite is CRUSHING STRENGTH OF SLENDER POSTS. 9 1 6.75, which is a trifle too small. The post may therefore be, sa}^, 6J inches diameter. 109. — SBender Posts. — When the height of a post is less than ten times its diameter, the resistance of the "post to crushing is approximately in proportion to its area of cross- section. But when the height is equal to or more than ten diameters, the resistance per square inch is diminished. The resistance diminishes as the height is increased, the diameter remaining the same (^Transverse Strains, Art. 643). The strength of a slender post, consists in a combination of the resistances of the material to bending and to crushing, and is represented in the following rule : 110. — The Liinilt of liVeiglij. — To ascertain the weight that can be sustained safely by a post the height of Avhich IS at least ten times its least side if rectangular, or ten times its diameter if round, the direction of the pressure coincid- ing with the axis, we have — RiileYW, — Divide the height of the post in inches by the diameter, or least side, in inches ; multiply the quotient by itself, or take its square ; multiply the square by the value of e, in Table III., set opposite the kind of material of which the po^ is made ; to the product add the half of itself ; to the sum add unity (or one) ; multiply this sum by the factor of safety, and reserve the product for use, as below. Now multiply the area of cross-section of the post in inches by the value of C, in Table I., set opposite the material of the post, and divide the product by the above reserved product; the quotient will be the required weight in pounds ; or — J^-Vr^n (7-) Example : A Round Post. — What weight may be safely placed upon a post of Georgia pine 10 inches diameter and 10 feet high, the pressure coinciding with the axis of the post ? The height of the post, (10 x 12 =) 120 inches, divided by 10, its diameter, gives a quotient of 12; this multiplied 92 CONSTRUCTION'. by itself gives 144, its square; and this by -00109, the value of c for Georgia pine, in Table III., gives • 15696 ; to whicli adding its half, the sum is 0-23544; to which adding unity, the sum is 1-23544 ; and this multiplied by 7, as a factor of safety, the product is 8 -648, the reserved divisor. Now the area of the post is (see Table of Areas of Circles, in the Ap- pendix, opposite its diameter, 10)78-54; this multiplied by 9500, the value of C for Georgia pine, in Table I., gives a product of 746130; which divided by 8-648, the above re- served divisor, gives a quotient of 86278, the required weight in pounds. AnotJicr Example : A Rectangular Post. — What weight may be safely placed upon a white-pine post 10 x 12 inches, and 15 feet high, the pressure coinciding with the axis of the post? Proceeding according to the rule, we find the height of the post to be 180 inches, which divided by 10, the least side of the post, gives 18 ; this multiplied by itself gives 324^ its square ; which multiplied by -0014, the value of e for white pine, in Table III., gives -4536; to which adding its half, the sum is -6804; to which adding unity, the sum is I -6804 ; and this multiplied by 8, as a factor of safety, the pro- duct is 13-4432, the reserved divisor. Now the area of the post, (10 X 12 =) 120 inches, multiplied by 6650, the value of C for white pine, in Table I., gives a product of 798,000, and this divided by 13-4432, the above reserved divisor, the quo- tient, 59360, is the required Aveight in pounds. Ml. — Diameter of tlic Post: Avlien Round. — To ascertain the size of a round post to sustain safely a given weight, when the height of the post is at least ten times the diameter ; the direction of the pressure coinciding with the axis of the post; we have — RuleWW. — Multiply the given weight by the factor of safety, and divide the product by i • 5708 times the value of C for the material of the post, found in Table I. ; reserve the quotient, calling its value G. Now multiply 432 times the value of e for the material of the post, found in Table III., by the square of the height in feet, and by the above quo- tient G\ to the product add the square of G\ extract the SIZE OF POST FOR GIVEN WEIOHT. 93 square root of the sum, and to it add the value of (j \ then the square root of this sum will be the required diameter; or — 1.5708 6 ^ ' •' =\' |. /432 G c r' + G' ^- G (9.) Example. — What should be the diameter of a locust post 10 feet high to sustain safely 40,000 pounds, the pressure coinciding with the axis? Proceeding by the rule, the given weight multiplied by 6, taken as a factor of safety, equals 240000. Dividing this by 1-5708 times 11700, the value of 6" for locust, in Table I., the quotient, 13-06, is the value of G, the square of which is 170-53. Now, the value of e for locust, in Table III., is -0015. This multiplied by 432, by 100, the square of the height, and by the above value of G, gives a product of 846-2 ; w^hich added to 170-53, the above square of G, gives the sum of 1016-73. To 31-89, the square root of this, add the above value of G\ then ^-j, the square root of this sum, is the required diameter of the post. The post therefore requires to be 6-^, say 6|- inches diameter. 112. — Side of tUe Post: ^vliesi Square.— To ascertain the side of a square post to sustain safely a given weight, w^hen the height of the post is at least ten times the side ; the pres- sure coinciding with the axis ; we have — Rule IX. — Multiply the given Aveight by the factor of safety, and divide the product by twice the value of C for the material of the post, found in Table I. ; reserve the quo- tient, calling its value G. Now multiply 432 times the value of e for the material of the post, found in Table III., by the square of the height in feet, and by the above quotient G\ to the product add the square of G ; extract the square root of the sum, and to it add the value of G ; then the square root of this sum will be the required side of the post ; or — - G=--^. (,o.) 94 CONSTRUCTION. S =i/ . /I^gVI^TJ' + G. ^^ ^•) J//432 Example. — What should be the side of a Georgia-pine square post 15 feet high to sustain safely 50,000 pounds, the pressure coinciding with the axis of the post? Proceeding by the rule, 50,000 pounds multiplied by 6, as a factor of safety, gives 300000 ; this divided by 2 x 9500 (the value of (7)= 19000, the quotient, 15-789, is the value of G. The value of € for Georgia pine is -00109; the square of the height is 225 ; then, 432 times -00109 by 225 and by 15-789 (the above value of G) gives a product of 1672 - 86 ; the square of 6^ equals 249-31 ; this added to 1672-86 gives a sum of 1922- 17, the square root of which is 43-843 ; which added to 15-789, the value of G, gives 59-632, the square root of which is 7-722, the required side of the post. The post, therefore, requires to be, say, 7f inches square. 113. — Tliickiaess of a Rectangular Post.— This may be definitely ascertained when the proportion which the thick- ness shall bear to the breadth shall have been previously determined. For example, when the proportion is as 6 to 8, then I J times 6 equals 8, and the proportion is as 1 to i^; again, when the proportion is as 8 to 10, then i^ times 8 equals 10, and in this case the proportion is as i to ij. Let the latter figure of the ratio i to ij, i to i J, etc., be called 11, or so that the proportion shall be as i to n, then — To ascertain the thickness of a post to sustain safely a given Aveight, when the height is at least ten times the thick- ness ; the action of the weight coinciding with the axis ; Vv'c have — Ride X. — Multiply the given weight by the factor of safety, and divide the product by twice the value of C for the material of the post, found in Table I., multiplied by ;/, as above explained ; reserve the quotient, calling it G. Now multiply 432 times the value of e for the material of the post, found in Table III., by the square of the height in feet, and by the above quotient G ; to the product add the square of G ; extract the square root of the sum, and to it add the value liREAUTH OF POST VOR GIVEN THICKNESS. 95 of G\ then the square root of this sum will be the required thickness of the post ; or — 0=1'^: (-) 2 C n = /v 432 G c r ^ G' -V G. (13.) Example. — What should be the thickness of a white-pine rectangular post 20 feet high to sustain safely 30,000 pounds, the pressure coinciding with the axis, and the proportion between the thickness and breadth to be as 10 to 12, or as i to I -2 ? Proceeding according to the rule, we have the pro- duct of 30000, the given weight, by 6, as a factor of safety, equals 180000 ; this divided by twice Cx n, or 2 x 6650 x i -2, (=15960) gives a quotient of 11-278, the value of G. Then, we have r = -0014, the square of the height equals 400; therefore, 432 X -0014 x 400 x 1 1 -278 = 2728-43. To this add- ing 127-2, the square of G^ we have 2855 63, the square root of which is 53-438; and this added to G gives 64.-^16, the square root of which is 8-045, the required thickness of the post. Now, since the thickness is in proportion to the breadth as i to 1-2, therefore 8-045 x 1-2 = 9-654, the re- quired width. The post, therefore, may be made 8x9! inches. 114.— Breadth of a Rcctang^ular Post.— When the thick- ness of a post is fixed, and the breadth required ; then, to ascertain the breadth of a rectangular post to sustain safely a given weight, the direction of the pressure of which coin- cides with the axis of the post, we have — Rule XI. — Divide the height in inches by the given thick- ness, and multiply the quotient by itself, or take its square ; multiply this square by the value of ^ for the material of the post, found in Table 111. ; to the product add its half, and to the sum add unity ; multiply this sum by the given weight, and by the factor of safety ; divide the product b}^ the pro- duct of the given thickness multiplied by the value of C for 96 CONSTRUCTION. the material of the post, found in Tabic I., and the quotient will be the required breadth ; or — Example. — What should be the breadth of a spruce post 18 feet high and 6 inches thick to sustain safely 25,000 pounds, the pressure coinciding with the axis of the post? According to the rule, 216 (= 12 x 18), the height in inches, divided by 6, the given thickness, gives a quotient of 36, the square of which is 1296; the value of ^' for spruce is -00098 ; this multiplied by 1296, the above square, equals i -27 ; which increased by -635, its half, amounts to 1-905 ; this increased by unity, the sum is 2-905 ; which multiplied by the given weight, and by the factor of safety, gives a product of 435749; and this divided by 6 (the given thickness) times 7850 (the value of (7 for spruce) = 47100, gives a quotient of 9-2516, the required breadth of the post. The post, therefore, re- quires to be 6 X 9|- inches. Observe that when the breadth obtained by the rule is less than the given thickness, the result shows that the con- ditions of the case arc incompatible with the rule, and that a new computation must be made ; taking now for the breadth what was before understood to be the thickness, and proceeding in this case, by Rule X., to find the thickness. 115. — Re§istancc lo Ten§ion.— In Art. 95 are recorded the results of experiments made to test the resistance of vari- ous materials to tensile strain, showing in each case the ca- pability to such resistance per square inch of sectional area. The action of materials in resisting a tensile strain is quite simple ; their resistance is found to be directly as their sec- tional area. Hence — 116. — The Umit of Wcijj^ht.— To ascertain the weight or pressure that may be safely applied to a beam or rod as a tensile strain, we have — Ride XII, — Multiply the area of the cross-section of the beam or rod in inches by the value of T, Table II.; divide AREA OF BEAM FOR TEx\SILE STRAIN. 97 the product by the factor of safety, and the quotient will be the required weight in pounds ; or — W=-^^. (.5-) The cross-section here intended is that taken at the small- est part of the beam or rod. A beam, in framing, is usually cut with mortices ; the area will probably be smallest at the severest cutting ; the area used in the rule must be that of the uncut fibres only. Example. — The tie-beam of a roof-truss is of white pine, 6 X 10 inches; the cutting for the foot of the rafter reduces the uncut area to 40 inches : what amount of horizontal thrust from the foot of the rafter will this tie-beam safely sustain ? Here 40 times 12000, the value of T, equals 480000; this divided by 6, as a factor of safety, gives 80000, the required weight in pounds. 1(7. — ISectional Area. — To ascertain the sectional area of a beam or rod that will sustain a given weight safely, when applied as a tensile strain, we have — Ride XIII. — Multiply the given weight in pounds by the factor of safety ; divide the product by the value of T, Table II., and the quotient will be the area required in inches; or — This is the area of uncut fibres. If the piece is to be cut for mortices, or for any other purpose, then for this an adequate addition is to be made to the result found by the rule. Example. — A rafter produces a thrust horizontally of 80,000 pounds ; the tie-beam is to be of oak : what must be the area of the cross-section of the tie-beam in order to sustain the rafter safely ? The given weight, 80000, multiplied by 10, as a factor of safety, gives 800000; this divided by 19500, the value of T, Table II., the quotient, 41, is the area of uncut fibres. This should have usually one half of its amount 98 CONSTRUCTION. added to it as an allowance for cutting; therefore, 41+21 = 62. The tie-beam may be 6 x 10^ inches. Another Example. — A tie-rod of American refined wrought iron is required to sustain safely 36,000 pounds: what should be its area of cross-section ? Taking 7 as the factor of safety, 7 x 36000 = 252000; and this divided by 60000, the value of T, Table II., gives a quotient of 4-2 inches, the required area of the rod. I!8. — TTeiglit of the Suspending Piece Included.— Pieces subjected to a tensile strain are frequently suspended verti- cally. In this case, at the upper end, the strain is due not only to the w^eight attached at the lower end, but also to the weight of the rod itself. Usually, in timber, this is small in comparison with the load, and may be neglected ; although in very long timbers, and where accuracy is decid- edly essential, as, also, when the rod is of iron, it may form a part of the rule. Taking the effect of the weight of the beam into account, the relation existing between the Aveights and the beam requires that the rule for the weight should be as follows : Rule XIV. — Divide the value of T for the material of the beam or rod, Table II., by the factor of safety ; from the quotient subtract 0-434 times the specific gravity of the ma- terial in the beam or rod multiplied by the length of the beam or rod in feet ; multiply the remainder by the area of cross-section in inches, and the product will be the required weight in pounds ; or — W: A \~j -0-434/^). (i;-) N. B. — This rule is based upon the condition that the sus- pending piece be not cut by mortices or in any other way. Example. — What weight may be safely sustained by a white-pine rod 4x6 inches, 40 feet long, suspended verti- cally? For white pine the value of T is 12000; this divid- ed by 8, as a factor of safety, gives 1500; from which sub^ tracting 0-434 times 0-458 (the specific gravity of white pine. Table II.) multiplied by 40, the length in feet, the remainder RESISTANXE TO TRANSVERSE STRAINS. 99 is 1492-049; which multiplied by 24 (=4x6, the area of cross-section) equals 35,761 pounds, the required weight to be carried. The weight which the rule would give, neglecting the weight of the rod, would have been 36000; ordinarily, so slight a difference would be quite unimportant. 1(9. — Area of Suspending^ Piece. — To ascertain the area of a suspended rod to sustain safely a given weight, when the weight of the suspending piece is regarded, we have — Rule XV. — Multiply 0-434 times the specific gravity of the suspending piece by the length in feet ; deduct the pro- duct from the quotient arising from a division of the value of T, Table II., by the factor of safety, and with the remain- der divide the given weight in pounds ; the quotient will be the required area in inches ; or — A ^ T , ' (i8.> — -0-434/^ ^ ^ a N.B. — This rule is based upon the condition that the rod be not injured in anywise by cutting. Example. — What should be the area of a bar of English cast iron 20 feet long to sustain safely, suspended from its lower end, a weight of 5000 pounds ? Taking the factor of safety at 7-0, and the specific gravity also at 7, and the value of T, Table II., at 17000, we have the product of 0-434 X 7.0 X 20 = 60-76; then 17000 divided by 7 gives a quotient of 2428-57; from which deducting the above 60-76, there remains 2367-81 ; dividing 5000, the given weight, by this remainder, we have the quotient, 2 • 1 1, which is the required area in inches. RESISTANCE TO TRANSVERSE STRAINS. 120. — Transverse Strains: Rupture. — A load placed upon a beam, laid horizontally or inclined, will bend it, and, if the weight be proportionally large, will break it. The power in the material that resists this bending or breaking is termed the resistance to cross-strains, or transverse strains. lOO CONSTRUCTION. While in posts or struts the material is compressed or short- ened, and in ties and suspending pieces it is extended or lengthened, in beams subjected to cross-strains the material is both compressed and extended. (See Art. 91.) When the beam is bent the fibres on the concave side are compressed, while those on the convex side are extended. The line where these two portions of the beam meet — that is, the portion compressed and the portion extended — the hori- zontal line of juncture, is termed the neutral line or plane. It is so called because at this line the fibres are neither com- pressed nor extended, and hence are under no strain what- ever. The location of this line or plane is not far from the middle of the depth of the beam, when the strain is not suf- ficient to injure the elasticity of the material ; but it re- moves towards the concave or convex side of the beam as the strain is increased, until, at the period of rupture, its distance from the top of the beam is in proportion to its dis- tance from the bottom of the beam as the tensile strength of the material is to its compressive strength. (21. — Liocatioii of Mortices.— In order that the diminution of the strength of a beam by framing be as small as possible, all mortices should be located at or near the middle of the depth. There is a prevalent idea with some, who are aware that the upper fibres of a beam are compressed Avhen sub- ject to cross-strains, that it is not injurious to cut these top fibres, provided that the cutting be for the insertion of an- other piece of timber — as in the case of gaining the ends of beams into the side of a girder. They suppose that the piece filled in will as effectually resist the compression as the part removed would have done, had it not been taken out. Now, besides the effect of shrinkage, which of itself is quite suf- ficient to prevent the proper resistance to the strain, there is the mechanical difficulty of fitting the joints perfectly throughout ; and, also, a great loss in the power of resist- ance, as the material is so much less capable of resistance when pressed at right angles to the direction of the fibres than when directly with them, as the results of the experi- ments in the tables show. STRENGTH OF BEAMS FOR CROSS-STRAINS. lOI 122- — Transverse Strains : Relation of Weight lo I>i- mcnsions. — The strength of various materials, in their re- sistance to cross-strains, is given in Table III., Art. 96. The second column of the table contains the results of experi- ments made to test their resistance to rupture. In the case of each material, the figures given and represented by B indicate the pounds at the middle required to break a tc7tit of the material, or a piece i inch square and i foot long between the bearings upon which the piece rests. To be able to use these indices of strength, in the computation of the strength of large beams, it is requisite, first, to establish the relation between the unit of material and the larger beam. Now, it may be easily comprehended that the strength of beams will be in proportion to their breadth ; that is, Avhen the length and depth remain the same, the strength will be directly as the breadth ; for it is evident that a beam 2 inches broad will bear twice as much as one which is only I inch broad, or that one which is 6 inches broad will bear three times as much as one which is 2 inches broad. Thij; establishes the relation of the weight to the breadth. With the depth, however, the relation is different ; the strength is greater than simply in proportion to the depth. If the boards cut from a squared piece of timber be piled up in the order in which they came from the timber, and be loaded with a heavy weight at the middle, the boards will deflect or sag much more than they would have done in the timber before sawing. The greater strength of the material when in a solid piece of timber is due to the cohesion of the fibres at the line of separation, by which the several boards, before sawing, are prevented from sliding upon each other, and thus the resistance to compression and tension is made to contribute to the strength. This resistance is found to be in proportion to the depth. Thus the strength due to the depth is, first, that which arises from the quantity of the material (the greater the depth, the more the material), which is in proportion to the depth ; then, that which en- sues from the cohesion of the fibres in such a manner as to prevent sliding ; this is also as the depth. Combining the two, we have, as the total result, the resistance in proportion I02 CONSTRUCTION. to the square of the depth. The relation between the weight and the length is such that the longer the beam is, the less it will resist ; a beam which is 20 feet long will sus- tain only half as much as one which is 10 feet long ; the breadth and depth each being the same in the two beams. From this it results that the resistance is inversely in pro- portion to the length. To obtain, therefore, the relation between the strength of the unit of material and that of a larger beam, we have these facts, namely : the strength of the unit is the value of j5, as recorded in Table III. ; and the strength of the larger beam, represented by W, the Aveight required to break it, is the product of the breadth into the square of the depth, divided by the length ; or, while for the unit we have the ratio — B\ I, we have for the larger beam the ratio — Therefore, putting these ratios in an expressed proportion, we have — ^: I :: W\^-^l From which (the product of the means equalling the pro- duct of the extremes ; see Art, 373) we have — lV=-j-. (19.) In which JV represents the pounds required to break a beam, when acting at the middle between the two supports upon which the beam is laid ; of which beam ^ represents the breadth and <^ the depth, both in inches, and / the length in feet between the supports ; and B is from Table III., and represents the pounds required to break a unit of material like that contained in the larger beam. IJMIT OF WEIGHT AT MIDDLE. IO3 (23- — Safe IVeight : Load at Hiddle. — The relation established, in the last article, between the weight and the dimensions is that which exists at the moment of rupture. The rule (19.) derived therefrom is not, therefore, directly practicable for computing the dimensions of beams for buildings. From it, however, one may readily be deduced which shall be practicable. In the fifth column of Table III. are given the least values of ^, the factor of safety, explained in Art. 96. Now, if in place of By the symbol for the break- ing weight, the quotient of B divided by a be substituted, then the rule at once becomes practicable ; the results now being in consonance with the requirements for materials used in buildings. Thus, with this modification, we have — W — . — -— • (20.) a I Therefore, to ascertain the weight which a beam may be safely loaded with at the centre, we have — Rule XVI.— Multiply the value of B, Table III., for the kind of material in the beam by the breadth and by the square of the depth of the beam in inches ; divide the pro- duct by the product of the factor of safety into the length of the beam between bearings in feet, and the quotient will be the weight in pounds that the beam will safely sustain at the middle of its length. Example. — What weight in pounds can be suspended safely from the middle of a Georgia-pine beam 4x lo inches, and 20 feet long between the bearings ? For Georgia pine the value of B, in Table III., is 850, and the least value of a is 1-84. For reasons given in Art. 96, let a be taken as high as 4; then, in this case, the value of b is 4, and that of d is 10, while that of/ is 20. Therefore, proceeding by -the rule, 850 X 4 X 10' = 340000 ; this divided by 4 x 20 (= 80) gives a quotient of 4250 pounds, the required weight. Observe that, had the value of a been taken at 3, instead of 4, the result by the rule would have been a load of 5667 pounds, instead of 4250, and the larger amount w^ould be none too much for a safe load upon such a beam ; although, I04 CONSTRUCTION. with it, the deflection would be one third greater than with the lesser load. The value of a should always be assigned higher than the figures of the table, which show it at its least value ; but just how much higher must depend upon the firmness required and the conditions of each particular case. (24-. — Breadth of Beam witli Safe Load. — By a simple transposition of the factors in equation (20.\ we obtain — a rule for the breadth of the beam. Therefore, to ascertain what should be the breadth of a beam of given depth and length to safely sustain at the middle a given weight, we have — ■ RuL' XVII. — Multiply the given Aveight in pounds by the factor of safety, and by the length in feet, and divide the product by the square of the depth multiplied by the value of B for the material in the beam, in Table III. ; the quotient will be the required breadth. Example. — What should be the breadth of a white-pine beam 8 inches deep and lo feet long between bearings to sustain safely 2400 pounds at the middle? For white pine the value of B, in Table III., is 500. Taking the value of a at 4, and proceeding by the rule, we have 2400 x4x 10 = 96000 ; this divided by i^^ x 500 =) 32000 gives a quotient of 3, the required breadth of the beam. 826. — Beptli of Beam Avith §afe Load. — A transposition of the factors in equation (21.), and marking it for extraction of the square root, gives — B 2i rule for the depth of a beam. Therefore, to ascertain what should be the depth of a beam of given breadth and length to safely sustain a given weight at the middle, we have — WEKillT AT ANY POINT. I05 Ride XVIII. — Multiply the given weight by the factor of safety, and by the length in feet; divide the product by the product of the breadth into the value of B for the kind of wood, Table III. ; then the square root of the quotient will be the required depth. Example. — What should be the depth of a spruce beam 5 inches broad and 10 feet long between bearings to sustain safely, at middle, 4500 pounds ? The value of B from the table is 550; taking a at 4, and proceeding by the rule, we have 4500 X 4x15 = 2700CO ; this divided by (550 x 5 ==) 2750 gives a quotient of 98-18, the square root of which is 9-909, the required depth of the beam. The beam should be 5 x 10 inches. 126- — Safe Load at any Point. — When the load is at the middle of a beam it exerts the greatest possible strain ; at any other point the strain would be less. The strain de- creases gradually as it approaches one of the bearings, and when arrived at the bearing its effect upon the beam as a cross-strain is zero. The effect of a weight upon a beam is in proportion to its distance from one of the bearings, mul- tiplied by the portion of the load borne by that bearing. The load upon a beam is divided upon the two bearings, a3 shown at Art. 88. The weight which is required to rup- ture a beam is in proportion to the breadth and square of the depth, b d', as before shown, and also in proportion to the length divided by 4 times the rectangle of the two parts into which the load divides the lenorth, or- ^ (see Fi^^. 35). This, when the load is at the middle, may be put as = ---, a result coincidinjr with the relation before 4xi/x-^/ / given in Art. 122, viz. : "The resistance is inversely in pro- portion to the length." The total resistance, therefore, put- ting the two statements together, is in proportion to . There are, therefore, the^e two ratios, viz., JV: — - — - and 4 7/1 n B : \^ from which vjc have the proportion — I06 CONSTRUCTION. /.'::::/F:^^, 4 m 11 from which we have — 1V= . (23.) 4 ;;/ 71 r> This is the relation at the point of rupture, and when — is ^ ^ a used instead of B, the expression gives the safe weight. Therefore — 7F=_^Mlf (24.) 4 a 7/1 71 is an expression for the safe Aveight. Now, to ascertain the weight which may be safely borne by a beam at any point in its length, we have — Rule XIX. — Multiply the breadth by the square of the depth, by the length in feet, and by the value of B for the material of the beam, in Table III. ; divide the product by the product of four times the factor of safety into the rec- tangle of the two parts into which the centre of gravity of the weight divides the beam, and the quotient will be the required weight in pounds. Exai7tph\ — What weight may be safely sustained at 3 feet from one end of a Georgia-pine beam which is 4 x 10 inches, and 20 feet long? The value of B for Georgia pine, in Table III., is 850 ; therefore, by the rule, 4 x lo"^ x 20 x 850 = 6800000. Taking the factor of safety at 4, we have 4x4x3x17=816. Using this as a divisor with which to divide the former product, we have as a quotient 8333 pounds, the required weight. 127. — Breadtli or Depth: Load at any Point. — By a proper transposition of the factors of (24.) we obtain — , .2 4 W a 171 71 . . hd =^y-, 125.) an expression showing the product of the breadth into the square of the depth ; hence, to ascertain the breadth or DISTRIBUTED WEIGHT. 10/ depth of a beam to sustain safely a given weight located at any point on the beam, wc have — Rule XX. — Multiply four times the given weight by the factor of safety, and by the rectangle of the two parts into which the load divides the length ; divide the product by the product of the length into the value of B for the mate- rial of the beam, found in Table III., and the quotient will be equal to the product of the breadth into the square of the depth. Now, to obtain the breadth, divide this product by the square of the depth, and the quotient will be the required breadth. But if, instead of the breadth, the depth be de- sired, divide the said product by the breadth ; then the square root of the quotient Avill be the required depth. Example. — What should be the breadth (the depth being 8) of a white-pine beam 12 feet long to safely custain 3500 pounds at 3 feet from one end ? Also, what should be its depth when the breadth is 3 inches? By the rule, taking the factor of safety at 4, 4 x 3500 x 4 x 3 x 9 = 15 12000. The value of B for white pine, in Table III., is 500 ; there- fore, 500 X 12 = 6000; with this as divisor, dividing 15 12000, the quotient is 252. Now, to obtain the breadth when the depth is 8, 252 divided by (8x8=) 64 gives a quotient of 3.9375, the required breadth; or the beam may be, say, 4x8. Again, when the breadth is 3 inches, we have for the quotient of 252 divided by 3 = 84, and the square root of 84 is 9- 165, or 9jr inches. For this case, therefore, the beam should be, say, 3 X 9J inches. 120. — ^Veiglit UnBformly Di§tribme4l. — When the load is spread out uniformly over the length of a beam, the beam will require just twice the weight to break it that would be required if the weight were concentrated at the centre. Therefore, we have JF= — , where U represents the dis- tributed load. Substituting this value of W in equation (20.), we have — U_BbjP 2 a I ' or — £/=---_-. (36.) I08 CONSTRUCTION. Therefore, to ascertain the weight which may be safely sus- tained, when uniformly distributed over the length of a beam, wc have — Rule XXL — Multiply twice the breadth by the square of the depth, and by the value of B for the material of the beam, in Table III., and divide the product by, the product of the length in feet by the factor of safety, and the quotient will be the required weight in pounds. Example. — What weight uniformly distributed may be safely sustained upon a hemlock beam 4x9 inches, and 20 feet long? The value of B for hemlock, in Table III., is 450 ; therefore, by the rule, 2 x 4 x 9' x 450 = 291 600. Tak- ing the factor of safety at 4, we have 4 x 20 = 80, the pro- duct by which the former product is to be divided. This division produces a quotient of 3645, the required weight. 129.— Breadth or Oeptli : Load Uniformly Distributed.— By a proper transposition of factors in (26.), we obtain — , ,2 Ual , . bd^ = ~^^, (27.) an expression giving the value of the breadth into the square of the depth. From this, therefore, to ascertain the breadth or the depth of a beam to sustain safely a given weight uni- formly distributed over the length of a beam, we have — Rule XXII. — Multiply the given w- eight by the factor of safety, and by the length ; divide the product by the pro- duct of twice the value of B for the material of the beam, in Table III., and the quotient will be equal to the breadth into the square of the depth. Now, to find the breadth, divide the said quotient by the square of the depth ; but if, instead of the breadth, the depth be required, then divide said quotient by the breadth, and the square root of this quotient will be the required depth. Example. — What should be the size of a white-pine beam 20 feet long to sustain safely 10,000 pounds uniformly distributed over its length? The value of ^ for white pine, in Table III., is 500. Let the factor of safety be taken at 4. Then, by the rule, loooo x 4 x 20 = 800000; this divided by (2 x 500 =) WEIGHT PER BEAM IN FLOORS. 1 09 a quotient of 800. Now, if the depth be fixed at 12, then the said quotient, 8oo, divided by (12 x 12=) 1.44 gives 5-y, the required breadth of beam ; and the beam may be, say, 5} x 12. Again, if the breadth is fixed, say, at 6, and the depth is required, then the said quotient, See, divided by 6 gives 1 33 J, the square root of which, 11 • 55, is the required depth. The beam in this case should therefore be, say, 6 X lif inches. 130. — ILoad per Foot Superficial. — When several beams are laid in a tier, placed at equal distances apart, as in a tier of floor-beams, it is desirable to know what should be their size in order to sustain a load eqiially distributed over the floor. ,^.--^ If the distance apart at which they are placed, measured from the centres of the beams, be multiplied by the length of the beams between bearings, the product will equal the area of the floor sustained by one beam ; and if this area be multiplied by the weight upon a superficial foot of the floor, the product will equal the total load uniformly distributed over the length of the beam ; or, if c be put to represent the distance apart between the centres of the beams in feet, and /represent the length in feet of the beam between bearings, and/ equal the pounds per superficial foot on the floor, then the product of these, or <://, will represent the uni- formly distributed load on a beam ; but this load was before represented by U {Art. 128); therefore, we have cfl— U^ and they may be substituted for it in (26.) and (27.). Thus we have — 2 B or- bd-"^ -'^~-. (28.) 2B Therefore, to ascertain the size of floor-beams to sustain safely a given load per superficial foot, we have — 7?///^ XXIII. — Multiply the given weight per superficial foot by the factor of safety, by the distance between the no CONSTRUCTION. centres of the beams in feet, and by the square of the length in feet; divide the product by twice the value of B for the material of the beams, in Table III., and the quotient will be equal to the breadth into the square of the depth. Now, to obtain the breadth, divide said quotient by the square of the depth, and this quotient will be the required breadth. But if, instead of the breadth, the depth be required, divide the aforesaid quotient by the breadth ; then the square root of this quotient will be the required depth. Example. — What should be the size of white-pine floor- beams 20 feet long, placed 16 inches from centres, to sustain safely 90 pounds per superficial foot, including the weight of the materials of construction — the beams, flooring, plas- tering, etc.? The value of B for white pine is 500; the factor of safety may be put at 5. Then, by the rule, we have 90 X 5 X 11- X 20^ = 240000. This divided by (2 x 500 = ) 1000 gives 240. Now, for the breadth, if the depth be fixed at 9 inches, then 240 divided by (9^ = ) 81 gives a quotient of 2-963. The beams therefore should be, say, 3x9. But if the breadth be hxed, say, at 2-5 inches, then 240 divided by 2-5 gives a quotient of 96, the square root of which is 9-8 nearly. The beams in this case would require therefore to be, say, 2|- x 10 inches. N. B. — It is well to observe that the question decided by Rule XXII. is simply that of strength only. Floor-beams computed by it will be quite safe against rupture, but they will in most cases deflect much more than Avould be consist- ent with their good appearance. Floor-beams should be computed by the rules which include the effect of deflection. (See Art, 152.) 131. — L.evcrs : I^oad at One End. — The beams so far con- sidered as being exposed to transverse strains have been supposed to be supported at each end. When a piece is held firmly at one end only, and loaded at the other, it is termed a lever ; and the load which a piece so held and loaded will sustain is equal to one fourth that which the same piece would sustain if it were supported at each end and loaded at the middle. Or, the strain in a beam sup- LEVERS TO SUSTAIN GIVEN WEIGHTS. I I I ported at each end caused by a given weight located at the middle is equal to that in a lever of the same breadth and depth, when the length of the latter is equal to one half that of the beam, and the load at its end is equal to one half of that at the middle of the beam. Or, when P represents the load at the end of the lever, and n its length, then W =^ 2 P, and l—2n. Substituting these values of [Fand / in equa- tion (20.), we have — .P^ Bbd' 2 an p^ Bbd' from which- /?/, //'^ (29.) ^an Hence, to ascertain the weight which may be safely sus- tained at the end of a lever, we have — Rule XXIV.— Multiply the breadth of the lever by the square of its depth, and by the value of B for the material of the lever, in Table III. ; divide the product by the pro- duct of four times the length in feet into the factor of safety, and the quotient will be the required weight in pounds. Example. — What weight can be safely sustained at the end of a maple lever of which the breadth is 2 inches, the depth is 4 inches, and the length is 6 feet ? The value of B for maple, in Table III., is iioo; therefore, by the rule, 2 X 4^" X 1 100 = 35200. And, taking the factor of safety at 5, 4x5x6= 120, and 35200 divided by 120 gives a quotient of 293 •33» or 293^- pounds. N. B. — When a lever is loaded with a weight uniformly distributed over its length, it will sustain just twice the load which can be sustained at the end. 132. — Ll§tributecl Ijoad. — The deflection shown in equation (40.) is just six times that shown in equation (33.). The result by (33.) mul- tiplied by 6 will equal the result by (40.); therefore, we have — Rule XXXV.— Proceed as directed in Rule XXVIII.; the result thereby obtained multiplied by 6 will give the required deflection. Example. — To what depth will 500 pounds deflect a 3 x 10 inch white-pine lever 10 feet long, the weight uniformly distributed over the lever? Here, by Rule XXVIII., we obtain the result 0-05747 ; this multiplied by 6 gives 0-3448, the required deflection. 148. — Deflection of Ijever§ : IVeiglit when Uniformly Distributed. — By a transposition of factors in (40.), we ob- tain — TT Fb d' d / ■ This is equal to one sixth that of equation (31.) ; therefore, we have — Rule XXXVI. -Proceed as directed in Rule XXVI.; the quotient thereby obtained divide by 6, and the quotient thus obtained will be the required weight. Example. — What weight will be required to deflect a 4x5 inch spruce lever i inch, the weight uniformly dis- tributed over its length ? Proceeding as directed in Rule XXVI., the result thereby obtained is 1750; this divided by 6 gives 291 f, the required weight in pounds. 149. — Deflection of Le§cribed. — Floors are most generally con, structed siitgle; that is, simply a series of parallel beams, each Fig. 39. Spanning the width of the building, as seen at Fig. 39. Oc- Fig. 40. casionally floors are constructed double, as at Fig. 40 ; and sometimes framed, as at Fig. 41 ; but these methods are RULES APPLIED TO FLOORS. 25 seldom practised, inasmuch as either of these requires more timber than the single floor. Where lathing and plastering is attached to the floor-beams to form a ceiling below, the springing of the beams, by customary use, is liable to crack the plastering. To obviate this in good dweUings, the double and framed floors have been resorted to, but more in former times than now, as the cross-furring (a series of narrow strips of board or plank nailed transversely to the underside of Fig. 41. the beams to receive the lathing for the plastering) serves a like purpose very nearly as well. as- (62. — Floor-Beams. — The size of floor-beams can be certained by the preceding rules for the stiffness of materials. These rules give the required dimensions for the various kinds of material in common use. The rules may be some- what abridged for ordinary use, if some of the quantities represented in the formula be made constant within certain limits. For example, if the load per foot superficial upon the floor be fixed, and the deflection, then these, together with the constant represented by F, may be reduced to one 126 CONSTRUCTION. constant. For dwellings, the load per foot may be taken at 70 pounds, the weight proper to be allowed for a crowd of people on their feet. {Transverse Strains, Art. 114.) To this add 20 for the weight of the material of which the floor is composed, and the sum, 90, is the value of/, or the weight per foot superficial for dwellings. Then <://== U {Art. 130). The rate of deflection allowable for this load may be fixed at 0-03 inch per foot of the length, or (5^ = 0-03 /. Substitut- ing these values in equation (35.), we obtain — I -6 Fx 'Oi I ~ 1-6 X '01 F '~ F or — bd'=l^cl. (43.) Putting/ to represent — -p—, we have — i875_ F bd^=jcl\ (44.) T R7C Now, by reducing — ^ — , for the six woods in common use, the value of 7 for each is found as follows: Georgia Pine j = 0-32 Locust J = 0-37 White Oak / = o-6 Spruce J = o • 54 White Pine / = 0-65 Hemlock / =z c-Sy Equation (44.) is a rule for the floor-beams of dwellings ; it may be used also to obtain the dimensions of beams for stores for all ordinary business • for it will require from 3 to 5 times the weight used in this rule, or from 200 to 400 (average 300) pounds to increase the deflection to the limit of elasticity in beams of the usual depths and lengths. For light stores, therefore, loaded, say, to 150 pounds per foot, the beams would be safe, but the deflection would be in- CONSTANTS FOR USE IN THE RULES. 12/ creased to o-o6 per foot. When so great a deflection as this would not be objectionable to the eye, then this rule (44.) will serve for the beams of light stores. But for first-class stores, taking the rate of deflection at -04 per foot, and fixing the weight per superficial foot at 275 pounds, includ- ing the weight of the material of which the floor is con- structed, and letting k represent the constant, then — bd' = kc l\ (45.) and for — Georgia Pine k = 0-73 Locust k = 0-85 White Oak ^ = i .38 Spruce k = 1-48 White Pine k = 1-23 Hemlock k = 1-53 153. — Floor-Beams for Dwellings. — To find the dimen- sions of floor-beams for dwellings, when the rate of deflection is 0-03 inch per foot, or for ordinary stores when the load is about 150 pounds per foot, and the deflection caused by this weight is within the limits of the elasticity of the material, we have the following rule : Rule XXXVIIL— Multiply the cube of the length by the distance apart between the beams (from centres), both in feet, and multiply the product by the value of 7* {Art. 152) for the material of the beam, and the product will equal the product of the breadth into the cube of the depth. Now, to find the breadth, divide this product b}^ the cube of the depth in inches, and the quotient will be the breadth in inches. But if the depth is sought, divide the said product by the breadth in inches, and the cube root of the quotient will be the depth in inches ; or if the breadth and depth are to be in proportion as r is to unity, r representing any re- quired decimal, then divide the aforesaid product by the value of r, and extract the square root of the quotient, and the square root of this square root will be the depth re- quired in inches, and the depth multiplied by the value of r will be the breadth in inches. 128 CONSTRUCTION. Example, —In a dwelling Or ordinary stOre, what must be the breadth of the beams, when placed 15 inches from centres, to support a floor covering a span of 16 feet, the depth being 1 1 inches, the beams of white oak ? By the rule, 4096, the cube of the length, by \\, the distance from centres, and by o-6, the value oi j for white oak, equals 3072. This divided by 1331, the cube of the depth, equals 2-31 inches, or 2^ inches, the required breadth. But if, in- stead of the breadth, the depth be required, the breadth being fixed at 3 inches, then the product, 3072, as above, di- vided by 3, the breadth, equals 1024 ; the cube root of this is io-o8, or, say, 10 inches nearly. But if the breadth and depth are to be in proportion, say, as 0-3 to i-o, then the aforesaid product, 3072, divided by 0-3, the value of r, equals 10240, the square root of which is ioi-2, and the square root of this is io-o6, the required depth. This multiplied by 0-3, the value of r, equals 3-02, the re- quired breadth ; the beam is therefore to be, say, 3 x 10 inches. I54-. — Floor-Beams for First-Class Stores. — To find the breadth and depth of the beams for a floor of a first-class store sufficient to sustain 250 pounds per foot superficial (exclusive of the weight of the material in the floor), with a deflection of 0-04 inch per foot of the length, we have — Rule XXXIX.— The same as XXXVIII., with the ex- ception that the value of /^ {Art. 152) is to be used instead of the value of j. Example. — The beams of the floor of a first-class store are to be of Georgia pine, with a clear bearing between the walls of 18 feet, and placed 14 inches from centres: what must be the breadth when the depth is 11 inches? By the rule, 5832, the cube of the length, and i^, the distance from centres, and 0-73, the value of k for Georgia pine, all multi- plied together equal 4966-92 ; and this product divided by 1 33 1, the cube of the depth, equals 3-732, the required breadth, or 3f inches. But if, instead of the breadth, the depth be required : what must be the depth when the breadth is 3 inches? DISTANCE APART OF FLOOR-BEAMS. 1 29 The said product, 4966-92, divided by 3, the breadth, equals 1655-64, and the cube root of this, 11-83, o^> say, 12 inches, is the depth required. But if the breadth and depth are to be in a given pro- portion, say 0-35 to i-o, the 4966-92 aforesaid divided by 0-35, the value of r, equals 14191, the square root of which is 119-13, and the square root of this square root is 10-91, or, say, 11 inches, the required depth. And 10-91 multiplied by 0-35, the value of r, equals 3-82, the required breadth, say 3^ inches. 155, — Floor -Beams: Distance from Centres. — It is sometimes desirable, when the breadth and depth of the beams are fixed, or when the beams have been sawed and are now ready for use, to know the distance from cen- tres at which such beams should be placed in order that the floor be sufficiently stiff. By a transposition of the factors in equation (44.), we obtain — jl rr.' (46.) In like manner, equation (45.) produces — These, in words at length, are as follows : Rule XL. — Multiply the cube of the depth by the breadth, both in inches, and divide the product by the cube of the length in feet multiplied by the value of 7', for dwellings and for ordinary stores, or by k, for first-class stores, and the quotient will be the distance apart from centres in feet. Example. — A span of 17 feet, in a dwelling, is to be cov- ered by white-pine beams 3x12 inches: at what distance apart from centres should they be placed? By the rule, 1728, the cube of the depth, multiplied by 3, the breadth, equals 5184. The cube of 17 is 4913 ; this by 0-65, the value of j for white pine, equals 3193-45. The aforesaid 5184 divided by this 3193-45 equals 1-6233 feet, or, say, 20 inches. 130 CONSTRUCTION. 156. — Framed Openings for Cliimneys and Stairs. — Where chimneys, flues, stairs, etc., occur to interrupt the bearing, the beams are framed into a piece, b {Fig. 42), called a header. The beams, a a, into which the header is framed are called trimmers or carriage-beams. These framed beams require to be made thicker than the common beams. The header must be strong enough to sustain one half of the weight that is sustained upon the /^//-beams, c c (the wall at the opposite end or another header there sustaining the other half), and the trimmers must each sustain one half of the weight sustained by the header in addition to the weight it supports as a common beam. It is usual in practice to make Fig. 42. these framed beams one inch thicker than the common beams for dwellings, and two inches thicker for heavy stores. This practice in ordinary cases answers very well, but in extreme cases these dimensions are not proper. Rules applicable generally must be deduced from the conditions of the case — the load to be sustained and the strength of the material. (57. — Breadtli of Headers. — The load sustained by a header is equally distributed, and is equal to the superficial area of the floor supported by the header multiplied by the load on every superficial foot of the floor. This is equal to the length of the header multiplied by half the length of the tail-beams, and by the load per superficial foot. Putting g I DIMENSIONS OF HEADERS. I3I for the length of the header, n for the length of the tail- beams, and / for the load per superficial foot ; U, the uni- formly distributed load carried by the header, will equal J f n g. By substituting for U, in equation (35.), this value of it, we obtain — i'6F6 The symbols g and / here both represent the same thing, the length of the header ; combining these, and for d putting its value ^r, we obtain— 1'2 Fr To allow for the weakening of the header by the mor- tices for the tail-beams (which should be cut as near the middle of the depth of the header as practicable), the depth should be taken at, say, one inch less than the actual depth. With this modification, we obtain — i = _^^^3. (48.) If /be taken at 90, and r at 0-03, we have, by reducing — Fid- if ^^^^ which is a rule for the breadth of headers for dwellings and for ordinary stores. This, in words, is as follows : Rule XLI. — Multiply 937-5 times the length of the tail- beams by the cube of the length of the header, both in feet. The product divided by the cube of one less than the depth multiplied by the value of F, Table III., will equal the breadth of the header in inches for dwellings or ordinary stores. Example. — A header of white pine, for a dwelling, is 10 feet long, and sustains tail-beams 20 feet long ; its depth is 12 inches: what must be its breadth? By the rule, 937-5 X20X 10^=18750000. This divided by (12- r)^x 2900= 132 CONSTRUCTION. 3859900, equals 4-858, say 5 inches, the required breadth. Y ox first-class stores, f ^\\o\x\^ be taken at 275, and r at 0-04. With these values the constants in equation (48.) reduce to 2148-4375, or, say, 2150. This gives— 7. 2150 «^3 r . a rule for the breadth of a header for first-class stores. It is the same as that for dwellings, except that the constant 2150 is to be used in place of 937-5. Taking the same ex- ample, and using the constant 2150 instead of 937-5, we obtain 1 1 • 14 as the required breadth of the header for a first- class store. Modifying the question by using Georgia pine instead of white pine, we obtain 5 -476 as the required thick- ness, say 5^ inches. 158. — Breadth of Carriage-Beams. — A carriage-beam or trimmer, in addition to its load as a common beam, carries one half of the load on the header, which, as has been seen in the last article, is equal to one half of the superficial area of the floor supported by the tail-beams multiplied by the weight per superficial foot of the load upon the floor ; therefore, when the length of the header in feet is repre- sented by g, and the length of the tail-beams by n, w equals ^ X - x/, equals \ f g n.^ For a load not at middle, we have (25.) — ^ '^ = —bT- * The load from the header, instead of he'ing ^ f g n, is, more accuratel)", ifn{g — c)\ because the surface of floor carried by the header is only that which occurs between the surfaces carried by the carriage-beams, each of which carries so much of the floor as extends half way to the first tail-beam Q from it, or the distance - ; therefore, the width of the surface carried equals the length of the header less [ 2 x - = J r, or ^ — / tuting- for a, the factor of safety, ^ , , its value in terms of resistance to flexure {Transverse Strains, (154.)), we have — , ,, ^WBlinn 4. Win 71 a - or- In this expression, W is a concentrated weight at the dis- tances VI and n from the two ends of the beam. Taking the load upon a carriage-beam due to the load from the header, as above found, and substituting it for JF, we obtain — ^ ^^3 _ 4 X \fgn m n ^ fgm n ' ^ Fr Ft This is the expression required for the concentrated load. To this is to be added the uniformly distributed load upon the carriage-beam ; this is given in equation (35.). Substi- tuting for U of this equation its value, f c /, gives — i'6F6 . Fr Combining these two equations, we have for the total load — ,^==/k!^i±if.a (51.) Fr If, in this equation, /be taken at 90, and r at 0-03, these reduce to 3000 ; therefore, with this value of -, we have — 3000{gmn' + %cr) . . d = -^rjj . i52.) This rule for the breadth of carriage-beams with one header, for dwellings and for ordinary stores, is put in words as follows : 134 CONSTRUCTION. Rule XLII. — Multiply the length of the framed opening by its breadth, and by the square of the length of the tail- beams ; to this product add | of the cube of the length into the distance of the common beams from centres — all in feet ; divide 3000 times the sum by the cube of the depth in inches multiplied by the value of /'"for the material of the beam., in Table HI., and the quotient will be the breadth in inches. Example, — In a tier of 3 x 10 inch beams, placed 14 inches from centres, what should be the breadth of a Georgia-pine carriage-beam 20 feet long, carrying a header 12 feet long, having tail-beams 15 feet long? Here the framed opening is 5x12 feet. Therefore, according to the rule, 12x5x15^== 13500; to which add(|x 20^x-if =) 5833!-; the sum is 19333J, and this by 3000= 58000000. The value of F for Geoi-gia pine, in Table III., is 5900; the cube of the depth is 1000; the product of these two is 5900000 ; therefore, dividing the above 58000000 by 5900000 gives a quotient of 9.83, the required breadth in inches. If, in equation (51.), / be taken at 275, and r at 0-04, then - becomes 6875, and the equation becomes — b = j-di ' (53-) a rule for the breadth of carriage-beams iov first-elass stores ; the same as that for dwellings, except that the constant is 6875 instead of 3000. 159. — Breadth of Carriage-Beams Carrying Two Sets of Tail-Beams. — A rule for this is the same as that for a car- riage-beam carrying one set of tail-beams, if to it there be added the effect of the second set of tail-beams. Equation (51.) with the addition named becomes — in which ;/ is the length of one set of tail-beams, and s the lensrth of the other set ; and ;;/ + n = I. CARRIAGE-BEAMS WITH TWO HEADERS. 1 35 If / be taken at. 90, and r at 0-03, these two reduee to 3000, and we have — b^- -j^, > 155.) a rule for the breadth of a carriage-beam carrying two sets of headers, for dwelHngs and for ordinary stores. It may be stated in words as follows : Rule XLl II.— Multiply the length of the longer set of tail-beams by the difference between this length and the length of the carriage-beam, and to the product add the square of the length of the shorter set of tail-beams ; mul- tiply the sum by the length of the longer set of tail-beams, and by the length of the header ; to this product add f of the product of the cube of the length of the carriage- beam into the distance apart from centres of the common beams ; multiply this sum by 3000 ; divide this product by the product of the cube of the depth in inches into the value of F ior the material of the carriage-beam, in Table III., and the quotient will be the required breadth. Example. — In a tier of 3 x 12 inch beams, placed 14 inches from centres, what should be the breadth of a spruce car- riage-beam 20 feet long in the clear of the bearings, carry- ing two sets of tail-beams, one of them 9 feet long, the other 5 feet ; the headers being 15 feet long ? The difference between the longer set of tail-beams and the carriage-beam is (20 — 9 =) II feet. Therefore, by the rule, 9x11 + 5'' = 124; then (124x9x15=) i6740 + (f X 20^x|f =) 5833^ = 22573i; then 22573^x3000 = 67720000. Now the value of F for spruce. Table III., is 3500; this by 12^, the cube of the depth, equals 6048000 ; by this dividing the aforesaid 67720000, we obtain a quotient of 11-197, the required breadth of the carriage-beam. If, in equation (54.), f be taken at 275, and r at 0-04, these reduce to 6875, and we obtain — ^ ^ 6875 \_gn{1nn-Vs'')^-lc P'\^ - ^^^^ a rule for the breadth of carriage -beams carrying two sets 136 CONSTRUCTION. of tail-beams, in the floors of first-class stores. This is like the rule for dwellings, except that the constant is 6875 in- stead of 3000. (60. — Breadth of Carriage -Beam wUh Well -Hole at Middle. — When the framed opening between the two sets of tail-beams occurs at the middle, or when the lengths of the two sets of tail-beams are equal, then equation (54.) reduces to and if /be taken at 90, and r at 0'03, these reduce to 3000, and we have — , 3000 /(^;/ 2 + 1^^ ^^^^ a rule for the breadth of a carriage-beam carrying two sets of tail-beams of equal length, in the floor of a dwelling or of an ordinary store ; and which in words is as follows : Rule XLIV.— Multiply the length of the header by the square of the length of the tail-beams, and to the product add |- of the product of the square of the length of the car- riage-beam by the distance apart from centres of the com- mon beams ; multiply the sum by 3000 times the length of the carriage-beam ; divide the product by the product of the cube of the depth into the value of F for the material of the carriage-beam, in Table III., and the quotient will be the required breadth. Example. — In a tier of 3 x 12 inch beams, placed 12 inches from centres, what must be the thickness of a hemlock car- riage-beam 20 feet long, carrying two sets of tail-beams, each 8 feet long, with headers 10 feet long? By the rule, 10 X 8' + f X I X 20' = 890 ; 890 X 3000 X 20 = 53400000. Now, the value of F, in Table III., for hemlock is 2800 ; this by the cube of the depth, 1728, equals 4838400; by this dividing the former product, 53400000, and the quotient, 11-0367, is the required breadth of the carriage-beam. CROSS-BRIDGING. 37 If, in equation (57.), /be taken at 275, and rat 0.04, these will reduce to 6875, and we shall have — b=^ (59-) a result the same as in equation (58.), except that the constant is 6875 instead of 3000. Equation (59.) is a rule for the breadth of carriage-beams carrying two sets of tail-beams of equal length, in the floor of a iirst-class store. In words at length, it is the same as Rule XLIV., except that the con- stant 6875 is to be used in place of 3000. (61.— Cro§§-Briclgiiis, or Herrings-Bone Bridi^ing.— The diagonal struts set between floor-beams, as in Fig. 43, are known as cross-bridging, or herring- bone bridging. By connecting the beams thus at intervals, say, of from 5 to 8 feet, the stiffness of the floor is greatly increased. The absolute strength of a tier of beams to resist a weight uniformly distributed over the Avhole tier is augmented but lit- tle by cross-bridging ; but the power of any one beam in the tier to re- sist a concentrated load upon it, as a heavy article of fur- niture or an iron safe, is greatly increased by the cross- bridging; for this device, by connecting the loaded beam with the adjacent beams on each side, causes these beams to assist in carrying the load. To secure the full benefit of the diagonal struts, it is very important that the beams be well secured from separating laterally, by having strips, such as cross-furring, firmly nailed to the under edges of the beams. The tie thus made, together with that of the floor-plank on the top edges, will prevent the thrust of the struts from sep- arating the beams. 162.— Bridging: Value to Resist Concentrated Lioad§. — A rule for determining the additional load which any one beam connected by bridging will be capable of sustaining, by the assistance derived from the other beams, through the Fig. 43. 138 CONSTRUCTION. bridging, may be found in Chapter XVI II., Transverse Strains. This rule may be stated thus : R = ^^^(i + 2^ + 3 V 4^4- eie) ; (60.) in which R is the increased resistance, equal to the addi- tional load which may be put upon the loaded beam ; c is the distance from centres in feet at which the beams in the tier are placed ; /is the load in pounds per superficial foot upon the floor ; / is the length of the beams in feet ; and ei is the depth of the beams in inches. The squares within the bracket are to be extended to as many places as there are beams on each side which contribute assistance through the bridging. The rule given in the work referred to, for ascer- taining the number of spaces between the beams, is — or, the depth of the beam in inches divided by the square of the distance from centres, in feet, at which the beams are placed will give the number of spaces between the beams which contribute on each side in sustaining the concentrated load. The nearest whole number, minus unity, will equal the required number of beams. The value of <: for beams in floors of dwellings is given in equation (46.), and lor those in first-class stores in equation (47.). By a modification of equation (34.), putting e f I for U, we have — cfl^ r ' and— c = y-p , (62.) \'6Fbd^r r.. or— c =r — (63.) These equations give general rules for the value of c. INCREASED LOAD BY CROSS-BRIDGING. 1 39 Now, the rule, in words at length, for the resistance offered by the adjoining beams to a weight concentrated upon one of the beams sustained by cross-bridging to the others, is — Rule XLV". — Divide the depth of the beam in inches by the square of the distance apart from centres in feet at which the floor-beams are placed ; from the quotient deduct unity, and call the whole number nearest to the remainder the First Result. Take the sum of the squares of the con- secutive numbers from unity to as many places as shall equal the above first result ; multiply this sum by 5 times the length in feet, by the load per foot superficial upon the floor, and by the fifth power of the distance apart from centres in feet at Avhich the beams are placed ; divide the product by 4 times the square of the depth in inches, and the quotient will be the weight in pounds required. Example. — In a tier of 3 x 12 inch floor-beams 20 feet long, placed in a dwelling 16 inches from centres and well bridged, what load maybe uniformly distributed upon one of the beams, additional to the load which that beam is capable of sustaining safely when unassisted by bridging? Here, according to the rule, 12 divided by (iJ-4-ii- = ) i-J equals 6f ; 6|— I = 5f, the nearest whole number to which is 6, the first result. The sum of the square of the first 6 numbers equals (i + 2' + 3^ + 4'+ 5' + 6' =) i +4 + 9 + 16 + 25 + 36 = 91. Therefore, 91 x 5 x 20 x 90 x (|)' ~ 345 1266.'^ The square of the depth (12 x 12 = ) 144x4 = 576; by this dividing the above 3451266, we have the quotient 5991-78, say 5992 pounds, the required weight. This is the additiojial load which may be placed upon the beam. At 90 pounds per superficial foot, the common load on each beam, we have * The value of r, 16 inches, equals | feet. The fifth power of this, or (5)*, is obtained by involving both numerator and denominator to the fifth power, and dividing the fifth power of the former by the fifth power of the latter ; for (1)5 = z_. For the numerator we have 4x4x4x4x4=1024, and for the de- 3^ nominator 3x3x3x3x3 = 243. The former divided by the latter gives as a quotient 4-214, the value of (|)^ The process of involving a number to a high power, or the reverse operation of extracting high roots, may be performed by logarithms with great facility. (See Art. 427.) I40 CONSTRUCTION. 90 X 20 X 4. = 2400 as the common load. To this add 5992, the load sustained through the bridging by the other beams, and the sum, 8392 pounds, will be the total load which may be safely sustained, uniformly distributed, upon one beam — nearly 3J times the common load. 163.— Oirclcrs.— When the distance between the walls of a building is greater than that which would be the limit for the length of ordinary single beams, it becomes requisite to introduce one or more additional supports. Where sup- ports are needed for a floor and partitions are not desirable, it is usual to use a large piece of timber called a girder, sus- tained by posts set at intervals of from 8 to 15 feet; or, when posts are objectionable, a framed construction called a framed girder {Art, 196); or an iron box called a tubular iron girder {Art. 182). When a simple timber girder is used it is advisable, if it be large, to divide it vertically from end to end and reverse the two pieces, exposing the heart of the timber to the air in order that it may dry quickly, and also to detect decay at the heart. When the halves are bolted together, thin slips of wood should be inserted between them at the several points at which they are bolted, in order to leave sufficient space for the air to circulate freely in the space thus formed between them. This tends to prevent decay, which \vill be found first at such parts as are not exactly tight, nor yet far enough apart to permit the escape of moisture. When girders are required for a long bear- ing, it is usual to truss them ; that is, to insert between the halves two pieces of oak which are inclined towards each other, and which meet at the centre of the length of the girder like the rafters of a roof-truss, though nearly if not quite concealed within the girder. This and many similar methods, though extensively practised, are generally worse than useless ; since it has been ascertained that, in nearly all such cases, the operation has positively ivcakcncd the girder. A girder may be strengthened by mechanical contrivance, when its depth is required to be greater than any one piece of timber will allow. Fig. 44 shows a very simple yet invalu- able method of doing this. The two pieces of which the gir- CONSTRUCTION OF GIRDERS. I4I der is composed are bolted or pinned together, having keys inserted between to prevent the pieces from sliding. The keys should be of hard wood, well seasoned. The two pieces should be about equal in depth, in order that the joint between them may be in the neutral line. (See Arts. 120, 121.) The thickness of the keys should be about half their breadth, and the amount of their united thickness should be equal to a trifle over the depth and one third of the depth of the girder. Instead of bolts orpins, iron hoops are sometimes used ; and when they can be procured, they are far preferable. In this case, the girder is diminished at the ends, and the hoops driven from each end towards the middle. A girder may be spliced if timber of a sufficient length cannot be obtained ; though not at or near the mid- Fig. 44. die, if it can be avoided. (See Art. 87.) Girders should rest from 9 to 12 inches on each wall, and a space should be left for the air to circulate around the ends, that the damp- ness may evaporate. 164-- — Girders: Dimensions. — The size of a girder, for any special case, may be determined by equations (21.), (22.), (25.), (27.), and (28.), to resist rupture ; and to resist deflection, by equations (32.) and (35.)- For girders in dwellings, equa- tion (44,) may be used. In this case, the value of c is to be taken equal to the width of floor supported by the girder, which is equal to the sum of the distances half way to the wall or next bearing on each side. When there is but one 142 CONSTRUCTION. girder between the two walls, the value of c is equal to half the distance between the walls. The rule for girders for dtvellijtgs, in words, is — Rule XLVI. — Multiply the cube of the length of the gir- der by the sum of the distances from the girder half way to the next bearing on each side, and by the value of J for the material of the girder, in Art. 152; the product will equal the product of the breadth of the girder into the cube of the depth. To obtain the breadth, divide this product by the cube of the depth ; the quotient will be the breadth. To obtain the depth, divide the said product by the breadth ; the cube root of the quotient will be the depth. If the breadth and depth are to be in a given proportion, say as r ', i-o, then divide the aforesaid quotient by the value of r ; take the square root of the quotient ; then the square root of this square root will be the depth, and the depth multi- plied by the value of r will be the breadth. Example. — In the floor of a dwelling, what should be the size of a Georgia-pine girder 14 feet long between posts, placed at 10 feet from one wall and 20 feet from the other? The value of c here is J^o, 4. ^0. _ 3_o. — j^^ -phe value oi j for Georgia pine {Art. 152) is 0-32. By the rule, 14^ x 15 x 0-32 = 13171 .2. Now, to find the breadth when the depth is 12 inches ; 13171 -2 divided by the cube of 12, or by 1728, gives a quotient of 7-622, or 7f, the required breadth. Again, to find the depth, when the breadth is 8 inches : 13171-2 divided by 8 gives 1646-4, the cube root of which is II -808, or, say, \\\ inches, the required depth. But if neither breadth nor depth have been previously determined, except as to their proportion, say as 0-7 to i-o, then 13171-2 divided by 0-7 gives 18816, of which the square root is 137- 171, and of this the square root is 11 -712, or, say, I if inches, the required depth. For the breadth, we have 1 1 -712 by 0-7 equals 8-198, or, say, 8 J, the required breadth. Thus the girder is required to be 7f x 12, 8 x iif, or8ixiif inches. This example is one in a dwelling or ordinary store ; ior first-class stores the rule for girders is the same as the last, except that the value of k is to be taken instead of/, in Art. 152. FIRE-PROOF TIMBER FLOORS. I43 165. — Solid Timber Floors. — Floors constructed with rolled-iron beams and brick arches are proof against fire only to a limited degree ; for experience has shown that the heat, in an extensive conflagration, is sufficiently intense to deprive the iron of its rigidity, and consequently of its strength. Singular as it may seem, it is nevertheless true that wood, under certain circumstances, has a greater fire- resisting quality than iron. Floors of timber constructed, as is usual, with the beams set apart, have but little power to resist fire, but if the spaces between the beams be filled up solid with other beams, which thus close the openings against the passage of the flames, and the under surface be coated with plastering mortar containing a large portion ot plaster of Paris, and finished smooth, then this w^ooden floor will resist the action of fire longer than a floor of iron beams and brick arches. The wooden beams should be se- cured to each other by dowels or spikes. 166. — Solid Timber Floors for Dwellings and Assem- bly-Rooms. — From Transverse Strains, Art. 702, we have — (82+j^/V^ 0-576/^ ' which may be modified so as to take this form : 0-576^^ ^ ^^ which is a rule for the depth or thickness of solid timber floors for dwellings, assembly-rooms, or office buildings, and in which y and // are constants depending upon the mate- rial ; thus, for — Georgia Pine jF = 4, and // = 0-3x4 Spruce / = 2-5", " // = 0-365 White Pine y = 2 J, " /i = 0-389 Hemlock J = 2, " // = o-39 The rule may be stated in words thus : Ru/e XLVII. — Multiply the length by the value of y, 144 CONSTRUCTION. and by the value of //, as above given ; to the product add 82 ; multiply the sum by the cube of the length ; divide this product by 0.576 times the value of F, in Table III. ; then the cube root of the quotient will be the required depth in inches. ^^^;;/// (69.) This is a rule by which to ascertain the size of a rolled-iron beam to sustain a given weight at middle with a given de- flection, and, in words at length, is as follows: Rule XLVIII. — Multiply the weight in pounds by the cube of the length in feet ; divide the product by 744000 times the deflection in inches, and the quotient will be the DEFLECTION OF IRON BEAMS. I47 moment of inertia of the required beam, and may be found, or the next nearest number, in Table IV. in column headed /. Opposite to the number thus found, to the left, will be found the name, depth, and weight per yard of the required beam. Example. — Which of the beams of Table IV. would be proper to carry 10,000 pounds at the middle with a deflection of one inch, the length between bearings being 20 feet ? Here we have, substituting for the symbols their values — W l^ 10000x20^ 80000000 7440DO 6 744000 X I 744000 / :j / ' or, the momentof inertia of the required beam is 107-527, the nearest to which, in the table, is 107-793, pertaining to the Phoenix 9-inch, 84-pound beam. This, then, is the required beam. 170. — Rolled-Iron Beams: Deflection ^vhen IVeiglit is at Hiddle. — By a transposition of symbols in equation (69.), we have — _ Wl^ ~ 744000 /' (70-) or a rule for the deflection of rolled-iron beams when the weight is at the middle. This, in words, is — Ride XLIX. — Multiply the weight in pounds b}- the cube of the length in feet ; divide the product by 744000 times the value of / for the given beam, and the quotient will be the required deflection in inches. Example. — What will be the deflection of a Phoenix 9- inch, 70-pound beam 20 feet long, loaded at the middle with 7500 pounds? The value of / for this beam, in Table IV., is 92 • 207 ; therefore, substituting for the symbols their val- ues, and proceeding by the rule, we have — W P 7500x20^ _ ^ d = = • — — = 0.87461 ; 744000 / 744000 x 92 • 207 or, the deflection will be, say, f of an inch. 148 CONSTRUCTION. 171. — Rolled -Iron Beams: Tl^eiglit when at middle.— A transposition of factors in equation (70.) gives — _ 744000 / S P ' (71.) This is a rule for the weight at middle, and, in words, is — Ru/e L. — Multiply 744000 times the value of / by the deflection in inches ; divide the product by the cube of the length, and the quotient will be the required weight in pounds. Example. — What weight at the middle of a Buffalo 9-inch, 90-pound beam will deflect it one inch, the length between bearings being 20 feet ? The value of / for this beam, in Table IV., is 109- 117; therefore — 744000/ (J 74400QXI09-II7 XI ,^^^„ p. P 20^ ^/ ' or, the required weight is, say, 10,148 pounds. (72. — Rolled -Iron Beam§ : Weiglit at any Point. — The equation for a load at any point is ( Transverse Strains, Art. 485)- 186000 IS ~ I in n ' (72.) in which wand n represent the two parts in feet into which the point where the load rests divides the length. This, in words, is as follows : Rule LI. — Multiply 186000 times the value of / by the deflection in inches ; divide the product b}^ the product of the length into the rectangle formed by the two parts into which the point where the load rests divides the length ; the quotient will be the required weight in pounds. Example. — What weight is required, located at 10 feet from one end, to deflect i^ inches a Paterson 12^-inch, 125- pound beam 25 feet long between bearings ? The value of /for this beam, in Table IV., is 292-05 ; /;/ = 10, and n — I — 7;/ = 25 — 10 -^ 15 ; therefore — VARYING WEIGHTS ON IRON BEAMS. I49 i86ooo/(^ 186000x292-05x1.5 [[/ = ■ = • ^--^ = 21728-52; / 7n u 25 X 10 X 15 ' -^ ' or, the required weight is, say, 21,730 pounds. 173,— Rolled-Iron Beams: Dimensions; Weig^ht at any Point.— By transposition of factors in equation (72.), we ob- tain — _ W I m n ^ ~ 186000^- (73-) This may be expressed in words as follows : Rule LII. — Multiply the weight by the length, and by the rectangle of the two parts into which the point where the weight rests divides the length ; divide the product by 186000 times the deflection, and the quotient will be the value of /, which (or its next nearest number) may be found in Table IV., opposite to which will be found the required beam. Example. — What beam 10 feet long will be required to carry 5000 pounds at 3 feet from one end Avith a deflection of 0-4 inch? Here we have m equal 3, and 71 equal 7; therefore — _ W I m n 5000 X 10x3x7 ~ 1860006' ~ 186000x0-4 ~ 4* i- The value of / is 14- 113, the nearest number to which in the table, is 14-317, the moment of inertia of the Phoenix 5- inch, 36-pound beam ; this, therefore, is the beam required. 174. — Rolled-Iron Beams: Dimensions; Weiglit Uniform- ly Distributed. — Since %U — W{Art. 138), equation (69.) may be modified by the substitution of this value of W^ when we obtain — / which reduces to — JJLIl 744000 s' 1 160400 o I50 CONSTRUCTION. a rule for the dimensions of a beam for a uniformly distrib- uted load, which, in words, is as follows : Rule LI 1 1. — Multiply the uniformly distributed load by the cube of the length ; divide the product by 1 190400 times the deflection, and the quotient will be the value of /, corre- sponding to which, or to its next nearest number will be found in Table IV. the required beam. Example. — What beam 10 feet long is required to sus- tain an equally distributed load of 14,000 pounds with a de- flection of half an inch ? For this we have — 14000 X 10^ I = -—^ : = 23 • 52. 1190400x0-5 ^ -^ This is the moment of inertia of the required beam ; nearly the same as 23-761, in Table IV., the value of / for a Tren- ton 6-inch, 40-pound beam, which will serve as the re- quired beam. 175. — Rolled-Iron Beams : I>efleeUoii ; WeiglitUiiiforinJy Distributed. — A transposition of the factors in equation (74.) gives — 1 190400/' (75-) a rule for the deflection of a uniformly loaded beam, and which may be put in these words, namely : Rule LIV. — Multiply the uniformly distributed load by the cube of the length ; divide the product by 1 190400 times the value of /, Table* IV., and the quotient will be the re- quired deflection. Example. — To what depth will 14,000 pounds, uniformly distributed, deflect a Buffalo io|^-inch, 90-pound beam 20 feet long ? The value of / for this beam, as per the table, is 151-436; therefore — 14000 X 20^ ^ d — -pr — 0-6213 ; 1 190400 X 151 -436 or, the required deflection is, say, f of an inch. IRON FLOOR-BEAMS FOR DWELLINGS. 151 176. — Rolled -Iron Beams: ^Velght ^irhen Uniforitily distributed. — Equation (75.), by a transposition of factors, gives — £/= "904og _^g, (76.) a rule for the weight uniformly distributed, and which may be worded thus : Rule LV. — Multiply 1190400 times the value of /, Table IV., by the deflection ; divide the product by the cube of the length, and the quotient will be the required weight. Example. — What weight uniformly distributed upon a Buffalo loj-inch, 105 -pound beam 25 feet long between bearings will deflect it f of an inch ? The value of / for this beam, as per Table IV., is 175-645 ; therefore — _. 1190400 X 175-645 X f . or, the required weight is, say, 10,036 pounds. (77. — Rolled-Iron Reams: Floors of Dwellings or A§- sembly- Rooms. — From Transverse Strains^ Art. 500, we have — '- P 420' ^^^^ a rule for the distance from centres of rolled-iron beams in floors of dwellings, assembly-rooms, or offices, where the spaces between the beams are hlled in with brick arches and concrete. In the equation, c is the distance apart from cen- tres in feet, and y is the weight per yard of the beam. This, in words, is thus expressed : Ride LVI. — Divide 255 times the value of / by the cube of the length ; from the quotient deduct one 420th part of the weight of the beam per yard, and the remainder will be the required distance apart from centres. Example. — What should be the distance apart from cen- 152 CONSTRUCTION. tres of Buffalo i2:}-inch, 125 -pound beams 25 feet long between bearings, in the floor of an assembly-room? For these beams, in Table IV., / equals 286.019, and y— 125; therefore — _ 255 X 286-019 125 ^ ~ 25' 420' 72934-8 125 ^^o 15625 420 y H- 0/ . or, the required distance from centres is, say, 4 feet 4^^ inches. (78, — Rolled-Iron Beam§: Floors of First-€la§s Storc§. — From Transverse Strains, Art. 504, we have — 148-8/ / a rule for the distance from centres of rolled-iron beams in the floor of a first-class store ; the spaces between the beams being filled with brick arches and concrete. This rule may be put in words as follows : Rule\N\\. — Divide 148-8 times the value of / by the cube of the length ; from the quotient deduct one 960th part of the weight of the beam per yard, and the remainder will be the distance apart of the beams from centres in feet. Example. — W]i?± should be the distance apart from cen- tres of Buffalo 12^-inch, 180-pound beams 20 feet long between bearings, in the floor of a first-class store? For these beams the value of /, Table IV., is 418-945, and the value of J/ is 180; therefore — 148-8 X 418-945 180 ^ 20^ 960 or, the required distance from centres is, say, 7 feet 7J inches. TIE-RODS FOR IRON BEAMS. 1 53 (79. — Floor-Arches: Oeneral Considerations. — In fill- ing the spaces between the iron beams of a floor, the arches should be constructed with hard whole brick of good shape, laid upon the supporting centre in contact with each other, and the joints thoroughly filled with cement grout, and keyed with slate. Made in this manner, the arches need not be over four inches thick at the crown for spans extending to 7 or 8 feet, and 8 inches thick at the springing, where they should be started upon a proper skew-back. The rise of the arch should not be less than i^ inches for each foot of the span. (80. — Floor - Arches ; Tie -Rods: Dxvcllings. — From Transverse Strains, Art. 507, we have — <^=i 4/0.0198 <: i", (79.) which is a rule for the diameter in inches of a tie-rod for an arch in the floor of a bank, office building, or assembly- room ; in which d is the diameter in inches of the rod, s is the span of the arch, and c is the distance apart between the rods (s and c both in feet). This rule requires that the arch rise \\ inches per foot of the span, and that the brick-work and the superimposed load each weigh 70 pounds, or to- gether 140 pounds. This rule, in words, is as follows: Rule LVIII. — Multiply the span of the arch by the dis- tance apart at which the rods are placed, and by the decimal 0-0198 ; the square root of the product will be the diameter of the required rod. Example. — What should be the diameter of the wrought- iron ties of brick arches of 5 feet span, in a bank or hall of assembly, where the ties are 8 feet apart ? For this we have — d = ^0-0198 X 8 X 5 = i^-792 = 0-89 ; or, the diameter of the required rods should be, say, | of an inch. (8(. — Floor-Arches; Tie-Rods: First-Class IStores. — From the same source as in last article, we have — ^ = 4/0 • 045 27 c s, (80.) 154 CONSTRUCTION. which is a rule for the size of tie-rods for the brick arches of the floors of first-class stores, where the arches have a rise of i|- inches for each foot of the span, and where the weight of the brick arch and concrete* is not over 70 pounds per superficial foot of the floor, and the loading does not exceed 250 pounds per superficial foot. As the rule is the same as the one in the preceding article, except the deci- mal, a recital of the rule, in words, is not here needed. To obtain the required diameter, proceed as directed in Rule LVIIL, using the decimal 0-04527 instead of the one there given. TUBULAR IRON GIRDERS. (82. — TiBbiilar Iron Girders: B>e§criptioii. — The use of wooden beams for floors is limited to spans of about 25 feet. When greater spans than this are to be covered, some expe- dient must be resorted to by which intermediate bearings for the floor- beams may be provided. Wooden girders may be used, but these need to be supported b}^ posts at intervals of from 10 to 15 feet, unless the girders are trussed, or made up of top and bottom chords, struts, and ties. And even this is objectionable, owing to the height such a piece of framing requires, and which encumbers the otherwise free space of the hall. A substitute for the framed girder has been found in the tubular iron girder, as in Fig. 46, made of rolled plate iron and angle irons, riveted. They require to be stiffened by an occasional upright T iron along eachside, and a cross-head at least at each bearing. 183. — Tubular Iron GSrders: Area of Flanges ; Load at Middle. — In wrought-iron tubular girders it is usual to make the top and bottom flanges of equal thickness. From Transverse Strains, Art. 551, we have — Fig. 46. TUBULAR IRON GIRDERS. 155 a rule for the area of the bottom flange ; in which a' equals the area of the flange in inches, W the weight in pounds at the middle, / the length and d the depth of the girder, both in feet, and k the safe load in pounds per inch with which the metal may be loaded, and which is usually taken at 9000. The rule may be stated thus : Riile LIX. — Multiply the weight by the length : divide the product by 4 times the depth into the value of k, and the quotient will be the required area of the bottom flange. Example. — In a girder 40 feet long and 3 feet high, to carry 75,000 pounds at the middle, what area of metal is required in the bottom flange, putting k at 9000 ? For this we have, by the rule — A,d k 4 X 3 X 9000 or, the area required is 27-I inches. This is the amount of uncut metal. An allowance is required for that which will be cut by rivet-holes. This is usually an addition of one sixth. 184. — Tubular Iron Oirders : Area of Flanges; Load at aiiy Point.— The equation suitable for this {Transverse Strains^ Art. 553) is — a'-w-^^^' (82.) ^ -'^ dkl ' ^ ^ in which in and n are the distances respectively from the lo- cation of the load to the two ends of the girder. The other symbols are the same as in the last article. This rule may be thus stated : Rule LX. — Multiply the weight by the values of m and of n\ divide the product by the product of the depth into the length and into the value of k, and the quotient will be the required area of the bottom flange. Example. — In a girder 50 feet long between bearings and 156 CONSTRUCTION. 3|- feet high, what area of metal is required in the bottom flange to sustain 50,000 pounds at 20 feet from one end, when k equals 9000^ By the rule, we have — ,^^ 171 n 50000 X 20 X xo d k I 3^ X 9000 X 50 or, each flange requires 19 inches of solid metal uncut for rivets. 185. — Tubular Iron Crirders : Area of Flanges; Load Uniformly I>i§trlbuted. — The equation appropriate here is {Trafisverse Strains^ Art. 555) — This is a rule by which to obtain the area of cross-sec- tion of the bottom flange at any point in the length of the girder, the load uniformly distributed ; in and n being the respective distances from the point measured to the two ends of the girder, and U representing the uniformly dis- tributed load in pounds. This, in words, is described as follows : Rtile LXI. — Divide the weight by the product of twice the depth into the length and into the value of k ; then the quotient multiplied by the values of in and of n will be the required area of the bottom flange at the point measured, the distance of which from the ends equals ;;/ and ;/. Example. — In a girder 50 feet long and ih feet high, to carry a uniformly distributed load of 120,000 pounds, what area of cross-section is required in the bottom flange, at the middle and at intervals of 5 feet thence, to each support ; k being taken at 9000? Here we have, first — ^^ in n 120000 mil a — U — T-r-r = ' :. " — 0-038095 in n. 2dkl 2 X 3|- X 9000 X 50 ^ ^^ Now, when m — u— 25, we have the middle point ; then — a' = 0-038095 m n = 0-038095 x 25 x 25 = 23-81 ; SHEARING STRAIN. 15/ or, the area of the bottom flange at mid-length is 23-81 inches. When in — 20, then n — 30, and — a' = 0- 038095 X 20 X 30 = 22 • 86 ; or, the required area, at 5 feet either way from the middle, is 22^ inches. When 7/1= 15, then 71 = 35, and — a = 0-038095 X 15 X 35 = 20-0 ; or, at 10 feet either way from the middle, the required area is 20 inches. When 771 = 10, then 7t =40, and — a = 0-038095 X 10x40 = 15-24 ; or, at 15 feet either way from the middle, the required area is I5:|- inches. When 77i = 5, then 71 = 4^, and — ^' = 0-038095 X 5x45 r= 8-57; or, at 20 feet each side of the middle, the required area is 8f inches. The area of cross-section found in every case is that of the uncut fibres ; to this is to be added as much as will be cut by the rivets. This is usually about one sixth of the area given by the rule. The top flange is to be made equal in area to the bottom flange. The flanges are unvarying in Avidth from end to end, the variation of area being obtained by varying the thickness of the flanges, and this being at- tained by building the flange in lamina, or plates ; but these should not be less than a quarter of an inch thick. There should be added to the length of the girder, in the clear, about one tenth of its length for supports on the walls : thus, a girder 30 feet long requires 3 feet added for supports, or 18 inches on each wall. 186.— Tutoular Iron Girders: Shearing Strain,— The top and bottom flanges are provided of sufficient size to i-esist 158 CONSTRUCTION. the transverse strain ; the two upright plates, technically termed the web, need, therefore, to be thick enough to resist only the shearing strain. This, upon a beam uniformly loaded, is at the middle theoretically nothing, but from thence it increases regularly towards eac«h support, where it equals half the whole weight. For example, the girder of Art. 185, 50 feet long between supports, carries 120,000 pounds uniformly distributed over its length. In this case the shearing strain at the wall at each end is the half of 120,000 pounds, or 60,000 pounds; at 5 feet from the wall it is -^^ or \ less, or 48,000 pounds ; at 10 feet from the wall it is f less, or 36,000 pounds ; at 1 5 feet it is 24,000 ; at 20 feet it is 12,000; and at 25 feet or the middle, it is nothing. 187- — Tubular Iron Oirder§: Thickness of Web.— The equation appropriate for this is — '=^'-' (^^-^ in which t is the thickness of the web (equal to the sum of the thicknesses of the two side plates), d is the height of the plate {t and d both in inches), G is' the shearing strain, and k' is the effective resistance of wrought iron to shearing per inch of cross-section. This may be put in words as follows • Ride LXII. — Divide the shearing strain by the product of the depth in inches into the value of k' , and the quotient will be the thickness of the web, or of the two side plates taken together. Example, — What is the required thickness of web in a girder 50 feet between bearings, side plates 38 inches high between top and bottom flanges, and to carry 120,000 pounds, uniformly distributed ? Here, putting the shearing resistance of the plates at 7000 pounds per inch, we have — /. = dk 38x7000 266000 The shearing strain at the supports, as in last article, is 60000 ; therefore, we have for this point — LIGHT IRON GIRDERS. 1 59 60000 When G = 48000, then t = —^ = 0-225. 266000 48000 _ / — -^ = o-i8; 266000 and when G = 36000, then — 36000 / = -^p- — 0-135. 266000 ^^ Those nearer the middle of the girder are still less than these ; and these are all below the practicable thickness, which is half an inch for the two plates. The plates ought not in practice ever to be made less than a quarter of an inch thick. 188. — Tubular Iron Girders, for FIoor§ of Di¥ellin§^§, Assembly-Rooms, and Offlee Buildings.— When the floors of these buildings are constructed with rolled-iron beams and brick arches, then the following (Art. 568, Transverse Strains) is the appropriate equation for the area of cross-section of the bottom flange of the girder : , .'=(,4o+^)^_-^x-^^; (85.) in which a' is in inches, and c, c\ d, /, ?;/, and n are in feet. Also, a' is the area required ; y is the weight per yard of the rolled-iron beam of the fioor ; c, their distances from centres ; c\ the distance from centres at which the tubular girders are placed, or the breadth of floor carried by one girder ; d, the depth of the girder; k, the effective resistance of the metal per inch in the fianges of the girder ; and m and n are the distances respectively from the two ends of the girder to the point at which the area of cross-section of the bottom flange is required. The rule ma}^ be thus described : Ride LXIII. — Divide the weight per yard of the rolled- iron beams by 3 times their distance from centres; to the quotient add 140 and reserve the sum ; deduct the length in feet from 700, and with the remainder as a divisor divide 700 ; multiply the quotient by the above reserved sum, and l6o ' CONSTRUCTION. by the value of c' ; divide the product by the product of twice the depth into the value of k, and the quotient multi- plied by the values of m and of n will be the required area of cross-section of the bottom flange at the point in the length distant from the two ends equal to 7n and n respec- tively. Example. — In a floor of 9-inch, 70-pound beams, 4 feet from centres, what ought to be the area of the bottom flange of a tubular girder 40 feet long between bearings, 2 feet 8 inches deep, and placed 17 feet from the walls or from other girders ; the area of the flange to be ascertained at every 5 feet of the length ; the value of k to be put at 9000? Here y =70, c — 4, c' — ly, I— 40, and d^ 2%, Therefore, by the rule — / ( 70 \ 700 17 a =i\ 140 + I X 5 X m n • V ^ 3x4/ 700 - 40 2 X 2f X 9000 ' a' — 145 -Sjx I -0606 X 0-0003 54if X ;;2;2; . a' = 0-05478 V1 11. The values of ;;/ and n are — At the middle ;;2 = 20 ; ;/ = 20 5 feet from middle 7/2 = 15; ;/ = 25 10 '' '^ " ;;2=io; ;/ = 30 15 '' " " m— 5 ; ;^ = 35 These give — At the middle ^' = 0-05478 x 20 x 20 = 21 -91 ^' 5 feet from middle... .a' = 0-05478 x 15 x 25 = 20- 54 "10 " " '' ....^' = 0-05478x10x30=16-43 "15 '' '' - ...../ = 0-05478 X 5x35= 9-59 These are the areas of uncut fibres at the points named, in the lower flange ; the upper flange requires the same sizes. 189. — Tubular Iron Oirder§, for Floors of First-Class Stores.— The equation proper for this is {Transverse Strains, Art. 570)— CAST IRON COMPARED WITH WROUGHT. l6l a' ^ ('320 + -^') - 700 c in n (86.) a rule the same in form as that of the previous article ; hence it needs no particular exemplification. Rule LXIII. of last article may be used for this case, simply by using the constant 320 in place of that of 149. CAST-IRON GIRDERS. (90. — Cast-iron Girders: Inferior. — Rolled-iron beams have been so extensively introduced within a few years as to have superseded almost entirely the form^erly much used cast-iron beam or girder. The tensile strength of cast iron is far inferior to that of wrought iron. This inferiority and the contingencies to which the metal is subject in casting render it very untrustworthy ; it should not be used where rolled-iron beams can be procured. A very substantial gir- der to carry a brick wall is made by placing two or more rolled-iron beams side by side, and securing them together by bolts at mid-height of the web ; placing thimbles or sep- arators at each bolt. As there may be cases, however, in which cast-iron girders will be used, a few rules for them will here be given. 191.— Cast-iron Girder: Load at Middle. — The form of cross-section given to this girder usually is as shown in Fig. 47. In the cross-section, the bottom flange is made to contain in area four times as much as the top flange. The strength will be in proportion to the area of the bottom flange, and to the height or depth of the girder at middle. Hence, to obtain the greater strength from a given amount of material, it is requisite to make the upright part, or the web, rather thin ; yet, in order to prevent injurious strains in the casting while it is cooling, the parts should be nearly equal in thickness. The thickness of the three parts — web, l62 CONSTRUCTION. top flange, and bottom flange — may be made in proportion as 5, 6, and 8. For a weight at middle, the form of the web should be that of a triangle ; the top flange forming two straight lines declining from the centre each way to the bottom flange at the ends, like the rafters of a roof to its tie-beam. From Transverse Strains, Art. 583, we have — W a l 4850 .) a rule of like form with that of the last article ; therefore, Rule LXIV. may be used for this case, simply by substitut- ing 9700 for 4850, 193.— Cast-iron Bowstring^ Girder.— An arched girder, such as that in Fig. 48, is technically termed a '' bowstring girder." The curved part is a cast-iron beam of T form in section, and the horizon- tal line is a wrought-iron tie-rod attached to the ends of the arch. This girder has but little to commend it, and is by no means worthy the confi- dence placed in it by builders, with many of Avhom it is quite popular. The brick arch usually turned over it is adequate to sustain the entire compressive force induced from the load (the brick wall built above it), and it thereby supersedes the necessity for the iron arch, which is a useless expense. The tie-rod is the only useful part of the bowstring girder, but it is usually made too small, and not infrequently is seriously injured by the needless strain to which it is subjected when it is '* shrunk in" to the sockets in the ends of the arch. The bow- string girder, therefore, should never be used. 1 94-.— Substitute for the Bowstring Girder. — As the cast- iron arch of a bowstring girder serves only to resist com- 164 CONSTRUCTION. pression, its place can as well be filled by an arch of brick, footed on a pair of cast-iron skew-backs ; and these held in position by a pair of wrought-iron tie-rods, as shown in Fig. 49. This system of construction is preferable to the bowstring girder, in that the tie-rods are not liable to injur)' by '' shrinking in," and the cost is less. From Transverse StraiiiSy Art. 596, we have — D /- U I 9425 d (89.) Fig. 49. an equation in which D is the diameter in inches of each of the two tie-rods of the brick arch ; U is the load in pounds uniformly distributed over the arch ; / is the span of the arch in feet ; and d, in inches, is its versed sine, or its height measured from the centre of the tie-rod to the centre of the thickness or height of the arch at middle. This equation may be put in words as folloAvs : Ride LXV. — Multiply the weight by the length ; divide the product by 9425 times the depth, and the square root of the quotient will be the diameter of each rod. Example. — What should be the diameter of each of the pair of tie-rods required to sustain a brick arch 20 feet span from centres, with a versed sine or height at middle of 30 inches, to carry a brick wall 12 inches thick and 30 feet high, weighing 100 pounds per cubic foot? The load upon this arch will be for so much of the wall as will occur over the opening, which will be about one foot less than the span of the arch, or 20 — i = 19 feet. Therefore, the load will equal 19 x 30X i x 100 r= 57,000 pounds ; and hence, U = 57000, / — 20, d = 30, and, by the rule — n a/ 57000 X 20 ,/ — „ —V — ■ = y 4-0318 = 2-008; 9425 X 30 or, the diameter of each rod is required to be 2 inches. STRAINS REPRESENTED GRAPHICALLY. 65 FRAMED GIRDERS. (95- — Graphic Representation of Strains. — In the first part of this section, commencing at Art. 71, the method was developed of ascertaining the strains in the various parts of a frame by the parallelogram or triangle of forces. The method, so far as there explained, is adequate to solve sim- ple cases ; but when more than three pieces of a frame con- verge in one point, the task by that method becomes difficult. This difficulty, however, disappears when recourse is had to the method known as that of *' Reciprocal Figures, Frames, and Diagrams of Forces," proposed by Professor I. Clerk Maxwell in 1867. This is an extension of the method by the triangle of forces, and may be illustrated as follows : Fig. 50. Let the lines in Fig. 50 represent, in direction and amount, four converging forces in equilibrium in any frame, as, for example, the truss of a roof; let the lines in Fig. 51 be drawn parallel to those in Fig. 50, in the manner fol- lowing, namely : Let the line A B be drawn parallel with the line of Fig. 50 which is between the corresponding letters A and B, and let it be of corresponding length ; from B draw the line B C parallel with the line of Fig. 50 which is be- tween the letters B and C, and of corresponding length ; then from C draw C Z>, and from A draw A D, respectively parallel with the lines of Fig. 50 designated by the corre- sponding letters, and extend them till they intersect at D. The lengths of these two lines, the last two drawn, are de- termined by the point D where they intersect ; their lengths, therefore, need not be previously known. The lengths of the lines in Fig. 5 1 are respectively in proportion to the 1 66 CONSTRUCTION. several strains in Fig. 50, provided these strains are in equilibrium. Fig. 51 is termed a closed polygon of forces. A system of such polygons, one for each point, in the frame where forces converge, so constructed that no line repre- senting a force shall be repeated, is termed a diagram of forces. This diagram of forces is a reciprocal of the frame from which it is drawn, its lines and angles being the same. The facility of tracing the forces in the diagram of forces depends materially upon the system of lettering here shown, and which was proposed by Mr. Bow, in his excellent work on the Ecofiouiics of Construction. In this system each line of the frame is designated by the two letters which it separates ; thus the line between A and B is called line A B ; that between C and D is called line CD ; and so of others ; and in the diagram the corresponding lines are called by the same letters, but here the letters designating the line are, as usual, at the ends of the line. Any point in a frame where forces converge is designated by the several letters which cluster around it; as, for example, in Fig. 50, the point of convergence there shown is designated as point A B C D. This invaluable method of defining graphically the strains in the various pieces composing a frame, such as a girder or roof-truss, is remarkably simple, and is of general application. Its utility will now be exemplified in its appli- cation to framed girders, and afterwards to roof-trusses. 196.— Framed Girders. — Girders of solid timber are use- ful for the support of floors only where posts are admissible as supports, at intervals of from 8 to 15 feet. For unob- structed long spans it becomes requisite to construct a frame to serve as a girder (Arts. 163, 182). A frame of this kind requires two horizontal pieces, a top and a bottom chord, and a system of struts and suspension-pieces by which the top and bottom chords are held in position, and the strains from the load are transmitted to the bearings at the ends of the girders. Various methods of arranging these struts and ties have been proposed. One of the most simple and effective is shown in Fig. 52, forming a series of isos- celes triangles. The proportion between the length and height of a girder is important as an element of economy d^—-', (90.) RULES FOR FRAMED GIRDERS. 1 6/ both of space and cost. When circumstances do not control in limiting the height, it may be determined by this equation from Transverse Strains, Art. 624 — (1 75+^) /. 2400 ' in which d is the depth or height between the axes of the top and bottom chords, and / is the length between the cen- tres of bearings at the supports {d and / both in feet). This equation in words is as follows : Rule LXVI. — To the length add 175 ; multiply the sum by the length ; divide the product by 2400, and the quotient will be the required height between the axes of the top and bottom chords. Exa7nple.—\Yh2it should be the depth of a girder which is 40 feet long between the centres of action at the supports ? For this the rule gives — ^^ (i7S+40)x40 ^ 2400 J D A ^ or, the proper depth for economy of material is 3 feet and 7 inches. The number of bays, panels, or triangles into which the bottom chord may be divided is a matter of some considera- tion. Usually girders frorn — 20 to 59 feet long should have 5 bays. 59 *' 85 '' " " " 6 '' 85 " 107 '' '' '' "7 " 107 '' 127 " " '' '' 8 " 127 " 146 '' " " " 9 " (97-— rramed Girder and Diag^ram of Forces; — Let Fig: 52 represent a framed girder of six bays of, say, ii feet each, or of a total length of 66 feet. The lines shown are the axial lines, or the imaginary lines passing through the axes of the several pieces composing the frame. The six arrows indicate the six pressures into which the equally distributed load is supposed to be divided. Each of these is at the apex of a triangle, the base of which lies along the lower chord. i68 CONSTRUCTION. The spaces between the arrows are lettered ; so, also, the space between the last arrow at either end and the point of support has a letter, and so has each triangle, and there is one for the space beneath the lower chord. These letters are to be used in describing the diagram of forces, as was explained in Art, 195. The diagram of forces {Fig. 53) for this girder-frame is drawn as follows, namely : Upon a verti- 52. IP cal line A iVmark the points A, O, P, Q, R, S, and N, at equal distances, to represent the six equal vertical pressures indi- cated by the arrows in Fig. 52. The equal distances A (9, OP, etc., may be made of any convenient size; but it will HGrF Fig. 53. serve to facilitate the measurement of the forces in the dia- gram if they are made by a scale of equal parts, and the number of parts given to each division be made equal to the number of tons of 2000 pounds each which is contained in the pressure indicated by each arrow. On this vertical line the distance A O represents the load at the apex of the tri- angle B, or the points OCB {Art. 195); the distance OP GRAPHICAL DIAGRAMS OF FORCES. 1 69 represents the weight at the second arrow, or at the point O P'E D C, and so of the rest. If the weights upon the points in the upper chord had been unequal, then the divi- sion of the vertical line A N would have had to be corre- spondingly unequal, each division being laid off by the scale, to accord with the weight represented by each. The line of loads, A N, being adjusted, the other lines are drawn from it {Art. 195), so as to make a closed polygon for the forces converging at each point of the frame. Fig, 52 — commenc- ing with the point A B Ty Fzg. $2, where there are three forces, namely, the force acting through the inclined strut A By the horizontal force in B T, and the vertical reaction A T at the point of support. This last is equal to half the entire load, or equal to the pressure indicated by the three arrows, A O, O P, and P Q, and is represented in Fig. 53 by A Q or A T. From the point Q draw a horizontal line Q B ; this is parallel with the force B T oi Fig. 52, in the lower chord. From the point A draw A B parallel with the strut A B oi Fig. 52. This line intersects the line B T in B and closes the polygon A B TA ; the point B defines the length of the lines A B and B T, and these lines measured by the scale by Avhich the line of loads was constructed give the required pressures in the corresponding lines, A B and B T, oi Fig. 52. Taking next the point A B C O, where four forces meet, of which we already have two, namely, the force in the strut A B and the load A O — from the point O draw the hori- zontal line O C ; this is parallel to the horizontal force O C of Fig. 52. Now from B draw BC parallel with the suspen- sion-piece B C oi Fig. 52. This line intersects O C\n (7, and the point C limits the lines O C and B C and closes the poly- gon A B C O A, the four sides of which are respectively in proportion to the four forces converging at the point A B CO oi Fig. 52, and when measured by the scale by which the line of loads was constructed give the required strains re- spectively in each. Taking next the point B C B T, where four forces converge, of which we already have two, B C and B T— from B extend the horizontal line T B to D \ from C draw CD parallel with C D oi Fig. 52, and extend it to in- tersect TD in Dy and thus close the polygon. T B C D T. I/O CONSTRUCTION. The lines in a part of this polygon coincide — those from B to T\ this is because the two strains B 7" and D T, Fig. 52, He in the same horizontal line. Again, taking the point OCD EP, where five forces meet, three of which, O P, O C, and CD, we already have — draw from D the line D E parallel with D E oi Fig. 52, and from Pthe line PE horizontally or parallel with PE of Fig. 52. These two lines intersect at E and close the polygon P O C D E P, the sides of which meas- ure the forces converging in the ^omt P O C D E , Fig. 52. Next in order is the point DEFT, Fig. 52, where four forces meet, two of which, TD and D E, are known. From ^ draw ^/^ parallel with E F in Fig. 52; and from T, TF parallel with T F'vcv Fig. 52 ; these two lines meet in F and close the polygon TD E F T, the sides of which measure the required strains in the hues converging at the point DEFT, Fig, 52. Taking next the point PEFG Q, Fig. 52, where live forces meet, of which we already have three, QP, PE,Sind E F— from jpdraw a line parallel with F G oi Fig. 52, and from Q a line parallel with Q G oi Fig. 52. These two intersect at G and complete the polygon QPE EG Q, the lines of which measure the forces converging at PEFG Q in Fig. 52. In this last polygon, a peculiarity seems to indicate an error: the line EG has no length ; it begins and ends at the same point ; or, rather, the polygon is complete without it. This is easily understood when it is considered that the two lines /^G^ and G H do not contribute any strength towards sustaining the loads PQ and QR, and in so far as these weights are concerned they might be dispensed with, and the space occupied by the three triangles F, G, and H left free, and be designated by only one letter instead of three. Thus it appears that there are only four instead of five forces at the point PEFG Q, and that the four are represented by the lines of the polygon QPE FQ. The peculiarity above explained arises from considering loads only on the top chord : the analysis of the case is cor- rect as worked from the premises given ; but in practice there is always more or less load on the bottom chord at the middle, which should be considered. This will be included in a case proposed in the next article. One half of the dia- LOAD ON BOTH CHORDS. 171 gram of forces is now complete. The other half being ex- actly the same, except that it is in reversed order, need not here be drawn. 198. — Framed Oirder§ : L.oad on Both Chords. — Let Fig. 54 represent the axial lines of a girder carrying an Fig. 54- equally distributed load on each chord, represented by the arrows and balls shown in the figure. Let each bay measure 10 feet, or the length ot the girder be 50 feet, and its height / B / D X\ V ^ / /-» \ /\ X y\ " / \ / \ / n / \ /\ / \/ \/ \^ / \/ \ ■ /\ y\ \y \ \ / \ F \ \/ y H "^ Fig. 55. be 4|- feet. The diagram of forces (Fig. 55) for this girder is obtained thus : The plan of the girder, Fig. 54, requires to be lettered as shown ; having one letter within each panel and outside the frame, and one between every two weights or strains. Then, in Fig. 55, mark the vertical X\x\q KV ^V L,M,N,0, 172 CONSTRUCTION. and P, dividing' it by scale into equal parts, corresponding with the weights on the top chord represented by the ar- rows. For example, if the load at each arrow equals 6J tons, make K L, L M, M N, etc., each equal to 6\ parts of the scale. Then K P \n\\\. equal the total load on the top flange. Make the distance P V equal to the sum of the loads on the bottom chord. Then iT F equals the total load on the gir- der. Bisect KV\xi U\ then K U or U Fequals half the total load ; consequently, equals the reaction of the bearing at K or P of Fig. 54. Now, to obtain the polygon of forces converging at K A U, Fig. 54, we have one of these forces, K U, or the re- action of the bearing at KA f/, equal to K U, Fig. 55. From U draw UA parallel with UA of Fig. 54, and from K draw KA parallel with the strut KA^ Fig. 54, and intersecting the line UA at ^, a point which marks the limit of K A and UA, and closes the polygon KA UK, the sides of which are in proportion respectively to the three strains which converge at the point A UK, Fig. 54. For example, since the line K U by scale measures the vertical reaction, K U, of the bearing at A UK, Fig. 54, therefore the line K A oi the diagram of forces by the same scale measures the strain in the strut K A, Fig. 54, and the line ^ ^ of the diagram by the same scale measures the strain in the bottom chord at A U, Fig. 54. For the strains converging at K A B L, Fig. 54, of which two, K A and K L, are already known, we draw from A the line A B parallel with the line A B, Fig. 54, and from L draw L B parallel with L B, Fig. 54, meeting A B Sit B, a point which limits the two lines and closes the polygon KA B L K, the lines of which are in proportion respectively to the strains converging at the point KA B L, Fig. 54, as before explained. Of the five strains converging at U A B C T, we already have three — T U, UA,3,nd AB; to obtain the other two, make UQ equal to PV, equal to the total load upon the lower flange ; divide U Q into four equal parts, QR,RS, ST, and T U, corresponding with the four weights on the lower chord, and represented by the four balls. Fig. 54. Now, from T, the point marking the first of these divisions, draw T C parallel with T C, Fig. 54, and from B draw B C paral- VARIOUS STRAINS IN FRAMED GIRDERS. 1 73 lei with the strut B C, Fig, 54, meeting TC in Cj a point which limits the lines B C and TC and closes the polygon T UA BC T, the sides of which are in proportion respectively to the strains converging in the point T UA B C 7", Fig. 54. Of the five forces converging at MLB CD, we already have three — ML, LB, and B C; to obtain the other two, from M draw ML) parallel with ML>, Fig. 54, and from C draw CLJ parallel with CL>, Fig. 54, meeting ML) at D, a point Hmit- ing the lines M L> and CD and closing the polygon MLB CD M, the sides of which are in proportion to the strains converging at the point MLB CD, Fig. 54. Of the five forces converging at the point 5 T C D E, three — 5 T, T Cj and CD — are known ; to obtain the other two, from 5 draw SE parallel with SE, Fig. 54, and from D draw D E^ parallel with the strut D E, Fig. 54, meeting the line S E in E, a point limiting the two lines S E and D E and closing the polygon 5 TC D ES, the sides of which are in proportion to the strains converging at vS TCD E, Fig. 54. One half of the strains in Fig. 54 are now shown in its diagram of forces, Fig. 55 ; and since the two halves of the girder are symmetrical, the forces in one half corresponding to those in the other, hence the Hues of the diagram for one half of the forces may be used for the corresponding forces of the other half. 199. — Framed Girders: Dimensions of Parts. — The parts of a framed girder are the two horizontal chords (top and bottom) and the diagonals — the struts and ties. The top chord is in a state of compression, while the bottom chord experiences a tensile strain. Those of the diagonal pieces which have a direction from the top to the bottom chord, and from the middle towards one of the bearings of the girder, as KA, B C, or D E, Fig. 54, are struts, and are sub- jected to compression. The diagonal pieces which have a direction from the bottom to the top chord, and from the middle towards one of the supports, as A B or C D, Fig. 54, are ties, and are subjected to extension, {Art. 83). The amount of strain in each piece in a framed girder having been ascertained in a diagram of forces, as shown in Arts. 197 and 198, the dimensions of each piece may be obtained 174 CONSTRUCTION. by rules already given. The dimensions of the pieces in a state of compression are to be ascertained by the rules for posts in Arts. 107 to 114, and those in a state of tension by Arts. 117 to 119 (see Arts. 226 to 229). Care is required, in obtaining the size of the lower chord, to allow for the joints which necessarily occur in long ties, for the reason that tim- ber is not readily obtained sufficiently long without splicing. Usually, in cases where the length of the girder is too great to obtain a bottom chord in one piece, the chord is made up of vertical lamina, and in as long lengths as practicable, and secured with bolts. A chord thus made will usually require about twice the material ; or, its sectional area of cross-sec- tion will require to be twice the size of a chord which is in one whole piece ; and in this chord it is usual to put the fac- tor of safety at from 8 to 10. The diagonal ties are usually made of wrought iron, and it is well to secure the struts, especially the end ones, with iron stirrups and bolts. And, to prevent the evil effects of shrinkage, it is well to provide iron bearings extending through. the depth of each chord, so shaped that the struts and rods may have their bearings upon it, instead of upon the wood. PARTITIONS. 200. — PartUloii§. — Such partitions as are required for the divisions in ordinary houses are usually formed by tim- ber of small size, termed stitds or joists. These are placed upright at 12 or 16 inches from centres, and well nailed. Upon these studs lath are nailed, and these are covered with plastering. The strength of the plastering depends in a great measure upon the cHnch formed by the mortar which has been pressed through between the lath. That this clinch may be interfered with in the least possible degree, it is proper that the edges of the partition-joists which are presented to receive the lath should be as narrow as prac- ticable ; those which are necessarily large should be reduced by chamfering the corners. The derangements in floors, plastering, and doors which too frequently disfigure the interior of pretentious houses with gaping cracks in the FRAMED TARTITIONS. 75 plastering and in the door-casings are due in nearly all cases to defective partitions, and to the shrinkage of floor-timbers. A plastered partition is too heavy to be trusted upon an ordi- nary tier of beamsj unless so braced as to prevent its weight from pressing upon the beams. This precaution becomes es- pecially important when, in addition to its own weight, the partition serves as a girder to carry the weight of the floor- beams next above it. In order to reduce to the smallest practicable degree the derangements named, it is important that the studs in a partition should be trussed or braced so as to throw the weight upon firmly sustained points in the construction beneath, and that the timber in both partitions and floors should be well seasoned and carefully framed. To avoid the settlement due to the shrinkage of a tier of beams, it is important, nia partition standing over one in the story below or over a girder, that the studs pass between the beams to the plate of the lower partition, or to the girder ; and, to be able to do this, it is also important to ar- range the partitions of the several stories vertically over each other. All principal partitions should be of brick, especially such as are required to assist in sustaining the floors of the building. r~-l 1 ^ ; 1 -^// 1 1 ' 1 i ' 1 1 l! 1( ^ ' !' i 1 V 1 1 1 1 ^ J U Fig. 56. 201. — Examples of Partitions. — Fig. 56 represents a par- tition having a door in the middle. Its construction is simple but effective. Fig. 57 shows the manner of constructing a 176 CONSTRUCTION. partition having doors near the ends. The truss is formed above the door-heads, and the lower parts are suspended from it. The posts a and b are halved, and nailed to the tie c d and the sill ef. The braces in a trussed partition iS^.-^ Fig. 57. should be placed so as to form, as near as possible, an angle of 40 degrees with the horizon. The braces in a partition should be so placed as to discharge the weight upon the P^. R^ \-Y/ ^, r ^ 1^ y Fig. 58. points of support. All oblique pieces that fail to do this should be omitted. When the principal timbers of a partition require to be large for the purpose of greater strength, it is a good plan WEIGHT UPON PARTITIONS. 1 77 to omit the upright filling-in pieces, and in their stead to place a few horizontal pieces, as in Fig. 58, in order that upon these and the principal timbers upright battens may be nailed at the proper distances for lathing. A partition thus constructed requires a little more space than others ; but it has the advantage of insuring greater stability to the plas- tering, and also of preventing to a good degree the conver- sation of one room from being overheard in the adjoining one. Ordinary partitions are constructed with 3x4, 3x5, or 4 X 6 inch joists, for the principal pieces, and with 2x4, 2x5, or 2x6 filling-in studs, well strutted at intervals of about 5 feet. When a partition is required to support, in addition to its own weight, that of a floor or some other burden resting upon it, the dimensions of the timbers should be ascertained, by applying the principles which regulate the laws of pressure and those of the resistance of timber, as explained in the first part of this section, and in Arts. 196 to 199 for framed girders. The following data may assist in calculating the amount of pressure upon partitions : White-pine timber weighs from 22 to 32 pounds per cubic foot, varying in accordance with the amount of seasoning it has had. Assuming it to weigh 30 pounds, the weight of the beams and floor-plank in every superficial foot of the flooring w^ill be — 6 pounds when the beams are 3x8 inches, and placed 20 inches from centres 7i " " " 3 X 10 " " iS 9 " " " 3 X 12 " " 16 " " II " " " 3x12 " " 12 " " 13 " " " 4x12 " " 12 " " 13 " " " 4x14 *' " 14 In addition to the beams and plank, there is generally the plastering of the ceiling of the apartments beneath, and some- times the deafening. Plastering may be assumed to weigh 9 pounds per superficial foot, and deafening 1 1 pounds. Hemlock weighs about the same as white pine. A parti- tion of 3x4 joists of hemlock, set 12 inches from centres, therefore, will w^eigh about 2\ pounds per foot superficial and when plastered on both sides, 20J- pounds. 178 CONSTRUCTION. ROOFS. 202. — Roofs. — In ancient Norman and Gothic buildings, the walls and buttresses Avere erected so massive and firm that it was customary to construct their roofs without a tie- beam, the walls being abundantly capable of resisting the lateral pressure exerted by the rafters. But in modern buildings, usually the walls are so slightly built as to be in- capable of resisting much if any oblique pressure ; hence the necessity of care in constructing the roof so as to avoid oblique and lateral strains. The roof so constructed, instead of tending to separate the walls, will bind and steady them. Fig. 59. Fig. 60. Fig. 61. Fig. 62. Fig. 63. Fig. 64. Fig. 65. Fig. 66. Fig. 67. 203. — Comparison of Roof-Trii§sc§. — Desipfus for roof- trusses, illustrating various principles of roof construction, are herewith presented. The designs at Figs. 59 to 63 are distinguished from those at Figs. 64 to 6y by having a horizontal tie-beam. In the latter group, and in all designs similarly destitute of the horizontal tie at the foot of the rafters, the strains are much greater than in those having the tie, unless the truss be pro- VARIOUS FORMS OF ROOF-TRUSSES. 179 tected by exterior resistance, such as may be afforded by competent buttresses. To the uninitiated it may appear preferable, in Fig. 64, to extend the inchned ties to the rafters, as shown by the dotted lines. But this would not be beneficial ; on the con- trary, it would be injurious. The point of the rafter where the tie would be attached is near the middle of its length, and consequently is a point the least capable of resisting- transverse strains. The weight of the roofing itself tends to bend the rafter ; and the inclined tie, were it attached to the rafter, would, by its tension, have a tendency to increase this bending. As a necessary consequence, the feet of the rafters would separate, and the ridge descend. In Fig. 65 the inclined ties are extended to the rafters ; but here the horizontal strut or straining beam, located at the points of contact between the ties and rafters, counteracts the bending tendency of the rafters and renders these points stable. In this design, therefore, and only in such designs, it is permissible to extend the ties through to the rafters. Even here it is not advisable to do so, because of the in- creased strain produced. (See Figs. 77 and 79.) The design in Fig, 64, 66, or 6j is to be preferred to that in Fig. 65. 204-.— rorce Diag^ram : L or, transposmg — 2 /i : - : : R : H = — ^,- 2 4/1 THE STRAINS SHOWN GEOMETRICALLY. I99 where // equals the horizontal thrust in the tie-beam. To obtain R, the weight of the roof, multiply M, the load per foot, as in equation (96.), by s, the span, and by c, the dis- tance from centres at which the trusses are placed ; or — R = M c s. With this value of R substituted for it, we have — Mcsl , and — (99.) .=^,f, (.00, in which Y equals the strain in the axis of the rafter, and H the strain in the tie-beam. These are the greatest strains in the rafter and tie-beam. At certain parts of these pieces the strains are less, as will be shown in the next article. 224. — Slraiii§ in Roof-Timbers Shown Geometrically. — The pressure in each timber may be obtained as shown in Fig. 84, where A B represents the axis of the tie-beam, A C the axis of the rafter, D E and FB the axes of the braces, and DG^ F E, and C B the axes of the suspension-rods. In this design for a truss, the distance A B is divided into three equal parts, and the rods located at the two points of division, G and E. By this arrangement the rafter A Cis supported at equidistant points, B and F. The point D supports the rafter for a distance extending half-way to A and half-way to F, and the point F sustains half-way to D and half-way to C. Also, the point C sustains half-way to F, and, on the other rafter, half-way to the corresponding point to F. And because these points of support are located at equal distances apart, there- fore the load on each is the same in amount. On D G make Da equal by any decimally divided scale to the number of hundreds of pounds in the load on Z>, and draw the parallel- ogram abDc. Then, by the same scale, Db represents {Art. 71) the pressure in the axis of the rafter by the load at 200 CONSTRUCTION, Fig. 84. STRAINS IN A TRUSS. 201 D\ also, Z^^ the pressure in the brace /)^. Draw <:<^ hori- zontal ; then D d'ls the vertical pressure exerted by the brace D E Sit E. The point /^sustains, besides the common load represented by Da, also the vertical pressure exerted by the brace DE\ therefore, make Fe equal to the sum oi D a and Dd, and draw the parallelogram F gef. Then Fg, meas- ured by the scale, is the pressure in the axis of the rafter caused by the load at F, and Ff is the load in the axis of the brace FB. Draw /// horizontal ; then Fk is the vertical pressure exerted by the brace F B2X B. The point C, besides the common load represented by D a, sustains the vertical pressure Fh caused by the brace FB, and a like amount from the corresponding brace on the opposite side. There- fore, make Cj equal to the sum oi D a and twice F h, and draw j k parallel to the opposite rafter. Then Ck is the pressure in the axis of the rafter at C. This is not the only pressure in the rafter, although it is the total pressure at its head C. At the point Z', besides the pressure (^/', there is F g. i\t the point Z>, besides these two pressures, there is the pressure D b. At the foot, at A, there is still an addi- tional pressure ; for while the point D sustains the load half- way to F and half-way to A, the point A sustains the load half-way to D. This load is, in this case, just half the load at D. Therefore draw A vi vertical, and equal, by the scale, to half of Da, Extend CA to/; draw ml horizontal. Then ^ / is the pressure in the rafter at A caused by the weight of the roof from A half-way to D. Now the total of the pressures in the rafter is equal to the sum oi A 1+ D b + /^^ added to C k. Therefore make kn equal to the sum of A l+Db + Fg, and draw no parallel with the opposite raf- ter, and nj horizontal. Then Co, measured by the same scale, will be found equal to the total weight of the roof on both sides of C B. Since Da represents s, the portion of the weight borne by the point D, therefore Co, representing the whole weight of the roof, should equal six times Da, as it does, because D supports just one sixth of the whole load. Since C n is the total oblique thrust in the axis of the rafter at its foot, therefore nj is the horizontal thrust in the tie- beam at A, 202 CONSTRUCTION. 225.— Application of tlie Geometrical System of Strains.— The strains in a roof-truss can be ascertained geometrically, as shown in Art. 224. To make a practical application of the results, in any particular case, it is requisite first to as- certain the load at the head of each brace, as represented by the line D a, Fig. 84. The load corresponding to any part of the roof is equal to the product of the superficial area of that particular part (measured horizontally) multiplied by the weight per square foot of the roof. Or, when M equals the weight per square foot, c the distance from centres at which the trusses are placed, and 71 the horizontal distance between the heads of the braces, then the total load at the head of a brace is represented by — N=Mcn. (loi.) The value of M is given in general terms in equation (96.). To show its actual value, let it be required to find the weight per square foot upon a root 52 feet span and 13 feet high at middle ; or {Fig. 84), where A B equals half the space, or 26 feet, and CB 13 feet, then/i C, the length of the rafter, will be 26-069, nearly. And where the weight of covering per square foot, on the inclination, is 12 pounds, the force of the wind against a vertical plane is 30 pounds ; the weight of snow per foot horizontal is 20 pounds ; the Aveight of the plastering forming the ceiling at the tie-beam is 9 pounds ; and the load in the roof is nothing ; — with these quantities substituted, equation (96.) becomes — M — 2q-o6qx 12 { --+ 0-0154) + 30 — ^-^ — + 20 -f 9 +0 ; \52 / 29-069' M = (29-069 X 12 X 0-05386) + (30 X 0-2) + 20 + 9; M = 18-788 + 6 + 29 = 53-788; or, say, 53-8 pounds. Then if c, the distance from centres between trusses, is 10 feet, and Ji, the distance between braces, is one third of A B, Fig. 84, or y = 8f , the total load at the head of a brace will be, as per equation (loi.) — i\^= 53-8 X 10 X 8| = 4663; TABLE OF STRAINS. 203 or, say, 4650 pounds. Now, by any decimally divided scale, make D a, Fig. 84, equal to 46^^ parts of the scale ; this being the number of hundreds of pounds contained in the weight at D, as above. Then, by the same scale, the several lines in the figure drawn as before shown will be found to repre- sent respectively the weights here set opposite to them, as follows : Dd^^da=^he=^ 23^, and represents 2325 pounds; Da=zdc = hf=Fh^46^ '' 4650 Dc = Db = Al = Fg=^2 " 5200 " Fe = Da +'£> d = 6gi '' 6975 F/ = 65! '^ 6575 0" = 3^«=i39j- " 13950 CK= 3 Z>^ = 156 '' 15600 C 11 — C k^ Fg^ D b-\- A l=i\2 '' 31200 " Cn^ Ck^lDb^ 6 Db=2Ck = 312 *' 31200 " Nj = C — 6 Da = 6x 46J = 279 '' 27900 '' It should be observed here that the equality of the lines 7ij and Co is a coincidence dependent upon the relation which in this particular case the line CB happens to bear to the line A B ; A B being equal to twice C B. And so of some other lines in the figure. If the inclination of the roof were made greater or less, the equality of the lines referred to would disappear. It should also be observed that the strains above found are not quite exact ; they are, however, correct to within a fraction of a hundred pounds, which is a suffi- ciently near approximation for the purpose intended. From the results obtained above, we ascertain that the strain in the rafter, from F to C, is represented by C K, and is equal to 15,600 pounds ; while the strain at the foot of the rafter, from A to />, is represented by C71, and equals 3 1,200 pounds, or double that which is at the head of the rafter. We ascer- tain, also, that the maximum strain in the tie-beam, repre- sented by nj\ is 27,900 pounds; that that in the brace £>E, represented by F> c, is 5200 pounds; and that that in the brace F B, represented by F/,is 6575 pounds. The strain 204 CONSTRUCTION. in the vertical rod D G is theoretically nothing. There is, however, a small strain in it, for it has to carry a part of the tie-beam and so much of the ceiling as depends for support upon that part. But the manner of locating the weights, adopted in this article, does not recognize any load located at the point G. This is an objection to this system, but it is not material. For a recognition of weights at the tie-beam, see Arts. 205 to 211. The load at G may be found by obtaining the product of the surface carried into the weight per foot of the ceiling; or, say, 10 en = 10 x lox 8f = 867 pounds. The load to be carried by the rod FE is shown at D d=^ ke^ which above is found to be 2325 pounds. To this is to be added 867 pounds for the ceiling at E, as before found for the ceiling at G; or, together, 3192 pounds. The central rod CB has to carry the two loads brought to B by the two braces footed there ; and also the weight of the ceiling sup- ported by B. The vertical strain from the brace F B is rep- resented at F/i, and equals 4650 pounds ; therefore, the total load on CB is 4650 + 4650 + 867 = 10,167 pounds. 226. — Roof-Timbers: the Tie-Beam. — The roof-timbers comprised in the truss shown in Fi^: 84 are the rafters, tie-beam, two braces, and three rods. Of these, taking first the tie-beam, we have a piece subject to tension and some- times to cross-strain (see Art. 682, Transverse Strains), In this case the tensile strain only need be considered. For this a rule is given in Art. 117. In this rule, if the factor of safety be taken at 20, the result will be sufficiently large to allow for necessary cuttings at the joints. Therefore, if the beam be of Georgia pine, equation (16.), Art. 117, becomes — . 27000 X 20 - ^=^7655^- =341; or, say, 35 inches. This is ample to resist the tensile strain ; but, to resist the transverse strains to which such a long piece of timber is subjected in the hands of the workman, it would be proper to make it, say, 6x9. STRAIN UPON THE RAFTER. 205 227. — The Rafter. — A rafter, like a post, is subject to a compressive force, and is liable to fail in three ways, name- ly : by flexure, by being crushed, or by crushing the material against which it presses. To render it entirely safe, there- fore, it is requisite to ascertain the requirements for resisting failure in each of these three ways. Of these it will be convenient to consider, first, that of the Kability to being crushed. The rule for this is found in Art, 107. Let the rafter be of Georgia pine, then the value of C, Table I., will be 9500. The strain in the rafter {Art. 225) is 31,200 pounds. Now, taking the value of a, the fac- tor of safety, at lo, we have, by Rule VI. {Art. 107.) — 31200 X 10 A — • = 32-737; 9500 J /J/' or, 33 inches area of cross-section. This is the size of the rafter at its smallest section ; for example, at any one of the joints where it is customary to reduce the area by cutting for the struts and rods. Again : Let the liability of the rafter to flexure be now considered. For this we have a rule in Art. 114. The length of the rafter between unsupported points is nearly 9! feet, or 9f X 12 = 1 16 inches. Let the thickness of the rafter be taken at 6 inches. Then, by Rule ILX. {Art. 114), we have — h — ^^^(^ + 1^^') _ 31200 X 10(1+ f X .00109 X r') C t ~ 9500 X 6 / 116 , ^=7==-5-=-i9i; 19*' = 373-8. Then, f x .00109 x 373-8 = 0-611127 adding unity = i • I .611127 Substituting this, we have — _ 31200 X IPX 1 .611127 _ 502671 .624 ^ ~ 9500 x"6 ~~^o^^~"'"^'^^9; 206 CONSTRUCTION. or, to resist flexure the breadth is required to be 8-82, or, say, 9 inches ; or, the rafter is to be 6 xq inches at the foot. The strain in the rafter at the upper end is only half that at the foot ; the area of cross-section, therefore, at the head need not be more than half that which is required at the foot ; but it is usual to make it there about f of the size at the foot. In this case it would be, therefore, 6x6 inches at the upper end. Lastly, the requirement to resist crushing the surfaces against which the rafter presses is to be considered. The fibres of timber yield much more readily when pressed together by a force acting at ri£-/it angles to the di- rection of their length than when it acts in a line with their length. The value of timber subjected to pressure in these two ways is shown in Arts. 94, 98. In Table I., the value per square inch of the first stated resistance is expressed by P, C and the ultimate resistance of the other by — . The value ■^ a of timber per square inch to safely resist crushing may be ex- C pressed by — , in which a is the factor of safety. Timber pressed in an oblique direction will resist a force exceeding C that expressed by P, and less than that expressed by — . When the angle of inclination at which the force acts is just C 45°, then the force will be an average between P and — . And for any angle of inclination, the force will vary inverse- ly as the angle ; approaching P as the angle is enlarged, C but approaching — as the angle is diminished. It will be C equal to — when the angle becomes zero, and equal P when the angle becomes 90°. The resistance of timber per square inch to an oblique force is therefore expressed by — 90 (5-P); (102.) RESISTANCE OF SURFACES. 20/ where ^° equals the complement of the angle of inclination. In a roof, ^° is the acute angle formed by the rafter with a vertical line. If no convenient instrument be at hand to measure the angle, describe an arc upon the plan of the truss — thus : with C B {Fig. 84) for radius, describe the arc B g, and get the length of this arc in feet by stepping it off with a pair of dividers. Then — -- = 0-631 f; 90 // where k equals the length of the arc, and h equals B Cy the height of the roof. Therefore — k fC il/=:P+o.63||(^--p) (103.) equals the value of timber per square inch in a tie-beam, C and P being obtained from Table I., Art. 94. When C for the kind of wood in the tie-beam exceeds C set opposite the kind of wood in the rafter, then the latter is to be used in the rules instead of the former. The value of M, equation (103.), is the resistance per square inch of the surface pressed at the foot of the rafter. The resistance of the entire surface will therefore he MA, where A equals the area of the joint. Then, when the re- sistance equals the strain, we will have — MA=S = A[p+o.6i%l{^-p) from which we have — A ^-^^•^3f|g-^' (104.) in which 5 is the strain to be resisted. Now, the end of the rafter must be of sufficient size to afford a joint the area of which will not be less than that expressed by A in equation (104.). For example, the strain to which the rafter, Fig. 84, is subject at its foot is ascertained to be(^r/. 225) 31,200 pounds. For Georgia pine, the material of the tie-beam, P — goo {Art. 94, Table /.), and (7 = 9500. 208 CONSTRUCTION. The length of the arc Bo- is about 14-4 feet; the height B Cis 13 feet. Let a, the factor of safety, be taken at 10, then we have (104.) — 900 + (o-63t X -W ) (910 _ 900) _ 3T200 _ ~ 900 + (O • 705 X 50) ~ ^'^ * ^ ' or, the superficial area of the bearing at the joint required to prevent crushing the tie-beam is 33^^ inches. The results of the computations show that the rafter is required to be 6 inches thick, 9 inches wide at the foot, and 6 inches wide at the top. It is also ascertained that, in cut- ting for the bearing for the struts and boring for the sus- pension-rods, it is required that there shall be at least 33 inches area of cross-section left intact ; and, farther, that the area of the surface of the joint against the tie-beam should not be less than 33^ inches. 228. — The Braces. — Each brace is subject to compres- sion, and is liable to fail if too small, in the same manner as the rafter. Its size is to be ascertained, therefore, in the manner described for the rafter ; which need not be here repeated, except, perhaps, as to the liability to fail by flexure ; for in this case we have the breadth given, and need to find the thick- ness. The breadth of the- brace is fixed b}^ the thickness of the rafter, for it is usual to have the tw^o pieces flush with each other. Rule XI. {ArL 114) is to be used, but with this difference, namely : instead of the thickness, use the breadth as one of the factors in the divisor. Thus — t = '-.-^ -^. (105.) In working this rule, it is required, in order to get the value of r, the ratio between the height and thickness, to assume the thickness before it is ascertained ; and after com- putation, if the result shows that the assumed value was not a near approximation, a second trial will have to be made. Usually the first trial will be sufficient. STRAIN UPON BRACES. 209 For example, the brace D E \s about 9I feet or 1 16 inches long. As the strain in it is only 5200 pounds, the thickness will probably be not over 3 inches. Assuming it at this, we / t Therefore, we have — have r=-= J-p = 38I ; the square of which is about 1495. I X 0-00109 X 1495 = 2-4445 add unity = i. 3-4445 The equation reduces, therefore, to this — _ 5200x10x3 -4445 _ ^ ^- 5^^^^6 -3-1424, or, the required thickness of the brace is 34- inches, or the brace should be, say, 3J x 6 inches. In this case the result is so near the assumed value, a second trial is not needed. For the second brace, we have the length equal to about 12J feet or 147 inches ; and the strain equal to 6575 pounds {Art. 225). The ratio, therefore, may be obtained by assum- ing the thickness, say, at 4. With this, we have — / r =- =i|^= 36-75 ; the square of which is I350t*^. With this value of r" — I X -00109 X i350y^^ = 2-2081 add unity = i. Then- 3 -2081 6575x10x3.2081 ^^55^ =3-7006. Comparing this result with the assumed value of t = 4, we find the difference so great as to require a second trial. As the value of r was taken too low, the result obtained is correspondingly low. The true value is somewhere between 3 • 7 and 4. Assume it now, say, at 3-9. With this value, we have — / 147 ^ = ^= —- = 37-692 ; the square of which is 1420-7. 210 CONSTRUCTION. With this value of r'— f X -00109 X 1420-7 = 2-32282 add unity =: i • 3-32282 Then — _6575 X 10 X 3-32282 ^~ 95"6o76 =3-833. This result is a trifle less than the assumed value, 3-9. The true value is between these, and probably is about 3-86. This is quite near enough for use. This brace, therefore, is required to be 3 -86 x 6 inches, or, say, 4x6 inches. 229. — The Suspen§ion-Rod§.— These are usually made of wrought iron. This metal, when of excellent quality, may be safely trusted with 12,000 pounds per inch sectional area. But it is usual, for good work, to compute the area at only 9000 pounds per inch, and, as ordinarily made, these rods ought not to be loaded with more than 7000 pounds. The strain divided by this value per inch of the metal will give the sectional area of cross-section. For example, the strain in the rod jD G, Fig, 84, is 867 pounds {Art, 225); therefore — A 867 A — — ^=:0-i24; 7000 ^ or, the sectional area required is only an eighth of an inch. By reference to the table of areas of circles in the Appen- dix, the diameter of a rod containing the required area, as above, will be found to be a little less than half an inch. A rod half an inch in diameter will therefore be of .ample strength. For appearance's sake, however, no rod in a truss should be less than | of an inch in diameter. The rod FE has to resist a strain of 3192 pounds. For this, then, we have — ^=^^-^ = 0.456. 7000 ^^ A reference to the table of areas shows that a rod contain- ROOF-BEAMS. 211 ing this area would be a little more than f of an inch in di- ameter ; it would be of ample strength, say, at ^ of an inch in diameter. The rod C B, at the centre, has to carry a strain of 10,167 pounds. For this, then, we have — 10167 A — — 1-452. 7000 ^^ A reference to the table of areas shows that this rod should be i^ inches in diameter. 230. — Roof-Beams, Jack-RaHers, and Purlins. — These timbers are subject to loads nearly uniformly distributed, and their dimensions may be obtained by Rule XXX., equa- tion {i^'.), Art. 140. In this equation, U — cf I {Art. 152). Substituting this value for U, and r I for (J, equation (35.) be- comes — \'K)Fr and putting for r the rate of deflection, .04, we have — bd^ — -" ^ ^ , (106.) a formula convenient for roof-timbers. Example. — In a roof where the roofing is to be supported on white-pine roof-beams 10 feet long, placed 2\ feet from centres, and where the load per foot superficial is to be 40 pounds, including wind and snow : Avhat should be the di- mensions of the roof-beams? By equation (106.) — , ,, 40 X 2^- X 10^ bd'=.^—T-- =538.8. 0.064x2900 ■'^ Now if b, the breadth, be fixed, say, at 3, then — .= = ^^=.796; ^=: 5-64 nearly. 212 CONSTRUCTION. The roof-beams, therefore, require to be 3 x 5I, or, say, 3x6. All pieces of timber subject to cross-strains will sustain safely much greater strains when extended in one piece over two, three, or more distances between bearings ; therefore, roof-beams, jack-rafters, and purlins should, if possible, be made in as long lengths as practicable ; the roof-beams and purlins laid on, not framed into, the principal rafters, and extended over at least two spaces, the joints alternating on the trusses ; and likewise the jack-rafters laid on the purlins in long lengths. 231. — Five Examples of Roof^: are shown at Figs. 85, 86, 87, 88, and 89. In Fig. 85, a is an iron suspension-rod, b, b are braces. In Fig. 86, a, a, and b are iron rods, and d, d, c, c are braces. In Fig. Zj, a, b are iron rods, d, d braces, and c the straining beam. In Fig. 85. Fig. 88, a, a, b, b are iron rods, e, e, d, d are braces, and r is a straining beam. In Fig. 89, pur- lins are located at P P, etc. ; the inclined beam that lies upon them is the jack-rafter ; the post at the ridge is the king- TRUSS WITH BUILT-RIB. 213 post, the others are queen-posts. In this design the tie-beam is increased in height along the middle by a strengthening piece {Art. 163), for the purpose of sustaining additional weight placed in the room form- ed in the truss {Art. 216). Fi£: 90 shows a method of constructing a truss having a built-rib in the place of prin- cipal rafters. The proper form for the curve is that of the par- abola {Art. 560). This curve, when as fiat as is described in the figure, approximates so close- ly to that of the circle that the latter may be used in its stead. The height, a b^ is just half of a c, the curve to pass through the middle of the rib. The rib is composed of two series of abutting pieces, bolted together. These pieces should be as long as the dimensions of the timber will admit, in order that there may be but few joints. The sus- pending pieces are in halves, notched and bolted to the tie- beam and rib, and a purlin is framed upon the upper end of each. A truss of this construc- tion needs, for ordinary roofs, no diagonal braces between the suspending pieces, but if extra strength is required the braces may be added. The best place for the suspending pieces is at the joints of the rib. A rib of this kind will be sufficiently strong if the area of its section contain about one fourth more timber than is required for that of a rafter for a roof of the same size. The proportion of the depth to the thick- ness should be about as 10 to 7. 214 CONSTRUCTION. 232. — Roof-Truss witli Elevated Tie-Beam. — Designs such as are shown in Fig. 91 have the tie elevated for the ac- commodation of an arch in the ceiUng. This and all similar designs are seriously objectionable, and should always be avoided ; as the smaii height gained by the omission of the tie-beam can never compensate for the powerful lateral strains which are exerted by the oblique position of the supports, tending to separate the walls. Where an arch is required in the ceiling, the best plan is to carry up the walls as high as the top of the arch. Then, by using a horizontal tie-beam, the oblique strains will be entirely re- HIP-ROOFS. 215 moved. It is well known that many a public building has been all but ruined by the settling of the roof, consequent upon a defective plan in the formation of the truss in this respect. It is very necessary, therefore, that the horizontal Fig. 91. tie-beam be used, except where the walls are made so strong and firm by buttresses, or other support, as to prevent a possibility of their separating. (See Art. 212.) 233. — Hip-Roofs: Lines and BeTil§. — The lines a b and be, in Fig. 92, represent the walls at the angle of a building; b c\s> the seat of the hip-rafter, and g f oi a jack or cripple rafter. Draw e h at right angles to be, and make it equal 2l6 CONSTRUCTION. to the rise of the roof ; join b and h, and h b will be the length of the hip-rafter. Through e draw di at right angles to <^^; upon b, with the radius bh, describe the arc hi^ cutting di in i\ join b and /, and extend gfto meet b i iny; then gj will be the length of the jack-rafter. The length of each jack-rafter is found in the same manner — by extend- ing its seat to cut the line b i. From / draw fk at right angles to fg, also// at right angles to be-, make/y^ equal to// by the arc Ik, or make g k equal to gj by the arc j k ; then the angle at / will be the top-bevil of the jack-rafters, and the one at k will be the down-bevil.^ 234. — Tlie Backings of the Hip-Rafter. — At any con- venient place in <^ ^ {Fig. 92), as 0, draw m n at right angles to be) from 0, tangical to b h, describe a semicircle, cutting b e in s ; join m and s and n and s\ then these lines will form at s the proper angle for bevilling the top of the hip-rafter. DOMES.f 235. — Domes. — The usual form for domes is that of the sphere ; the base circular. When the interior dome does not Fig. 93. rise too high, a horizontal tie may be thrown across, by which any degree of strength required may be obtained. * The lengths and bevils of rafters for rooUva/leys can also be found by the above process. f See also Art 68. CONSTRUCTION OF DOMES. 217 Fig. 93 shows a section, and Fig. 94 the plan, of a dome of this kind, a b being the tie-beam in both. Two trusses of this kind {Fig. 93), parallel to each other, are to be placed one on each side of the opening in the top of the dome. Upon these the whole framework is to depend for support, Fig. 94. and their strength must be calculated accordingly. (See Arts, yo to 80 and 214 to 222.) If the dome is large and of importance, two other trusses may be introduced at right angles to the foregoing, the tie-beams being preserved in Fig. 95. one continuous length by framing them high enough to pass over the others. 236. — Ribbed Dome. — When the interior must be kept free, then the framing may be composed of a succession of ribs standing upon a continuous circular curb of timber, as 2l8 CONSTRUCTION. seen at Figs. 95 and 96 — the latter being a plan and the former a section. This curb must be well secured, as it serves in the place of a tie-beam to resist the lateral thrust of the ribs. In small domes these ribs may be easily cut from wide plank ; but where an extensive structure is required, they must be built in two thicknesses so as to break joints^ in the same manner as is described for a roof at Art. 231. They should be placed at about two feet apart at the base, and strutted as at a in Fig. 95. Fig. 96. The scantling of each thickness of the rib may be as fol- lows : For domes of 24 feet diameter, i x 8 inches, a u u 26 '' '' i^x 10 '' U ^Q u u 2 ^ j^ " 90 '' '' 2ix 13 " u jQg u u 2 ^ j^ 237. — Dome : Curve of Equilibrium. — The surfaces of a dome may be finished to any curve that may be desired, but the framing should be constructed of such form that the curve of cqiiilibriiini shall be sure to pass through the middle of the depth of the framing. The nature of this curve is such that, if an arch or dome be constructed in accordance with it, no one part of the structure will be less capable than another of resisting the strains and pressures to which the whole fabric may be exposed. The curve of equilibrium for an arched vault or a roof, w^here the load is equally diffused CURVE OF EQUILIBRIUM. 219 over the whole surface, is that of a parabola (ArL46o); for a dome having no lantern, tower, or cupola above it, a cubic parabola {Fig. 97) ; and for one having a tower, etc., above it, a curve approaching that of an hyperbola must be adopted, as the greatest strength is required at its upper parts. If the curve of a dome be circular (as in the vertical section, Fig,g^), the pressure will have a tendency to burst the dome outwards at about one third of its height. Therefore, when this form is used in the construction of an extensive dome, an iron band should be placed around the framework at that height ; and whatever may be the form of the curve, a band or tie of some kind is necessary around or across the base. J^""^ '^ oy^ X y \ \ "A i 1 /[ 1 ' / ' ' <^ J i h g / e d Fig. 97. If the framing be of a form less convex than the curve of equilibrium, the weight will have a tendency to crush the ribs inwards, but this pressure may be effectually overcome by strutting between the ribs ; and hence it is important that the struts be so placed as to form continuous horizontal circles. 238. — Cubic Parabola Computed. — Let a b {Fig. 97) be the base, and b c the height. Bisect a b at d, and divide a d into 100 equal parts ; of these give dc 26, cf i8J,/^ hJ,^/^ 12J-, h i lof, ij 9I and the balance, 8|, Vo j a ; divide b c into 8 equal parts, and from the points of division draw lines parallel to a /^ to meet perpendiculars from the several points 220 CONSTRUCTION. of division in a b, at the points o, o, o, etc. Then a curve traced through these points will be the one required. 239. — Small Domes over Stairways : are frequently made elliptical in both plan and section ; and as no two of the ribs in one quarter of the dome are alike in form, a method for obtaining the curves may be useful. Fig. 99. To find the curves for the ribs of an elliptical dome, let abed {Fig. 98) be the plan of a dome, and ef the seat of one of the ribs. Then take e f ior the transverse axis and twice the rise, og, of the dome for the conjugate, and de- COVERING OF DOMES. 221 scribe (according to Arts. 548, 549, etc.) the semi-ellipse eg-/, which will be the curve required for the rib eg-/. The other ribs are found in the same manner. 24-0. — Covering for a Spherical Dome. — To find the shape, let A [Fig. 99) be the plan, and B the section, of a given dome. From a draw ^ ^ at right angles \.o ab \ find the stretch-out {Art. 524) of ob, and make dc equal to it; divide the arc b and the line d c each into a like number of equal parts, as 5 (a large number will insure greater accuracy than a small one) ; upon c, through the several points of division in cd, describe the arcs do, i e i, 2/2, etc. ; make do equal to half the width of one of the boards, and draw os parallel to ^ ^ ; join s and a, and from the points of division in the arc ob drop perpendiculars, meeting asintjk/; from these points draw ^ 4,7*3, etc., parallel to <^^; make do,e i, etc., on the lower side of ac, equal to do, e i, etc., on the upper side ; trace a curve through the points 0, 1,2, 3, 4, c, on each side of dc ; then ^ <; ^ will be the proper shape for the board. By dividing the circumference of the base A into equal parts, and making the bottom, do, of the board of a size equal to one of those parts, every board may be made of the same size. In the same manner as the above, the shape of the covering for sections of another form may be found, such as an ogee, cove, etc. To find the curve of the boards when laid in horizontal courses, \^t A B C {Fig. 100) be the section of a given dome, 222 CONSTRUCTION. and Z>^ its axis. Divide BC into as many parts as there are to be courses of boards, in the points i, 2, 3, etc. ; through I and 2 draw a line to meet the axis extended at a ; then a will be the centre for describing the edges of the board F. Through 3 and 2 draw 3 b ; then b will be the centre for de- FlG. lOI, scribing F. Through 4 and 3 draw A,d\ then ^ will be the centre for G. B is the centre for the arc i 0. If this method is taken to find the centres for the boards at the base of the dome, they would occur so distant as to make it impracti- cable ; the following method is preferable for this purpose : G being the last board obtained by the above method, ex- tend the curve of its inner edge until it meets the axis, D B, in e ; from 3, through e, draw 3 /, meeting the arc y^ ^ in /; join / and 4, / and 5 , and /and 6, cutting the axis, D B, in s, n, and in ; from 4, 5, and 6 draw lines parallel to A (7 and cutting the axis in c, p, and r ; make c 4 {jFig. loi) equal to ^ 4 in the pre- vious figure, and cs equal to c s also in the previous figure ; then describe the inner edge of the board //", according to Art. 516; the outer edge can be obtained by gauging from the inner edge. In like manner proceed to obtain the next DESIGNS OF BRIDGES. 223 board — taking/ 5 for. half the chord, and pn for the height of the segment. Should the segment be too large to be de- scribed easily, reduce it by finding intermediate points in the curve, as at ^r/. 515. 24 L — Polygonal Dome: Form of Angle-Rib. — To ob- tain the shape of this rib, let A G H {Fig. 102) be the plan of a given dome, and C D ^ vertical section taken at the line ef. From i, 2, 3, etc., in the arc CD draw ordinates, paral- lel to A D, to meet fG ; from the points of intersection on fG draw ordinates at right angles to f G \ make ^ i equal to I, s 2 equal to 2, etc. ; then GfB, obtained in this way, will be the angle-rib required. The best position for the sheathing-boards for a dome of this kind is horizontal, but if they are required to be bent from the base to the vertex, their shape may be found in a similar manner to that shown at Fig. 99. BRIDGES. Fig. 103 242. — Bridges. — Of plans for the construction of bridges, perhaps the following are the most useful. Fig. 103 shows a method of constructing wooden bridges where the banks of the river are high enough to permit the use of the tie- beam, ab. The upright pieces, c d, are notched and bolted on in pairs, for the support of the tie-beam. A bridge ot this construction exerts no lateral pressure upon the abut- ments. This method may be employed even where the banks of the river are low, by letting the timbers for the roadway rest immediately upon the tie-beam. In this case the irame- work above will serve the purpose of a railing. 224 CONSTRUCTION. Fig. 104 exhibits a wooden bridge without a tie-beam. Where staunch buttresses can be obtained this method may be recommended ; but if there is any doubt ot their stabihty, it should not be attempted, as it is evident that such a sys- tem of framing is capable of a tremendous lateral thrust. Fig. 104. 24-3. — Bridges: Built-Rib. — Fig. 105 represents a bridge with a built-rib (see Art. 231) as a chief support. The curve of equihbrium will not differ much from that of a parabola ; this, therefore, may be used — especially if the rib is made Fig. 105. gradually a little stronger as it approaches the buttresses. As it is desirable that a bridge be kept low, the following table is given to show the least rise that may be given to the rib. Span in Feet. Least Rise in Feet Span in Feet. Least Rise in Feet Span in Feet. Least Rise in Feet 30 0-5 120 7 280 24 40 0-8 140 8 300 28 50 1-4 160 10 320 32 60 2 180 II 350 39 70 2* 200 12 380 47 80 3 220 14 400 53 90 4 240 17 TOO 5 260 20 DIMENSIONS OF THE BUILT-RIB. 22'5 The rise should never be made less than this, but in all cases greater if practicable ; as a small rise requires a greater quantity of timber to make the bridge equally strong. The greatest uniform weight with which a bridge is likely to be loaded is, probably, that of a dense crowd of people. This may be estimated at 70 pounds per square foot, and the fram- ing and gravelled roadway at 230 pounds more ; which amounts to 300 pounds on a square foot. The following rule, based upon this estimate, may be useful in determining the area of the ribs. Rule LXVI I.— Multiply the width of the bridge by the square of half the span, both in feet, and divide this pro- duct by the rise in feet multiplied by the number of ribs ; the quotient multiplied by the decimal o-ooii will give the area of each rib in feet. When the roadway is only planked, use the decimal 0-0007 instead of o-ooii. Example. — What should be the area of the ribs for a bridge of 200 feet span, to rise 15 feet and be 30 feet wide, with three curved ribs? The half of the span is 100, and its square is loooo ; this multiplied by 30 gives 300000, and 15 multiplied by 3 gives 45; then 300000 divided by 45 gives 6666|, which multiplied by o-ooii gives 7-333 feet or 1056 inches for the area of each rib. Such a rib may be 24 inches thick by 44 inches deep, and composed of 6 pieces, 2 in width and 3 in depth. The above rule gives the area of a rib that would be requisite to support the greatest possible uniform load. But in large bridges, a variable load, such as a heavy wagon, is capable of exerting much greater strains ; in such cases, therefore, the rib should be made larger.^* In constructing these ribs, if the span be not over 50 feet, each rib may be made in two or three thicknesses of timber (three thicknesses is preferable), of convenient lengths bolted together; but in larger spans, where the rib will be such as to render it difficult to procure timber of sufficient breadth, they may be constructed by bending the pieces to the proper curve and bolting them together. In this case, where tim- * See Tred^old's Carpentry by Hurst, Arts. 174 to 177. 226 CONSTRUCTION. ber of sufficient length to span the opening cannot be ob- tained, and scarfing is necessary, such joints naust be made as will resist both tension and compression (see Fig. 1 14). To ascertain the greatest depth for the pieces which compose the rib, so that the process of bending may not injure their elasticity, multiply the radius of curvature in feet by the decimal 0-05, and the product will be the depth in inches. £;r(^;;2//^.— Suppose the curve of the rib to be described with a radius of 100 feet, then what should be the depth? The radius in feet, 100, multiplied by 0-05 gives a product of 5 inches. White pine or oak timber 5 inches thick would freely bend to the above curve ; and if the required depth of such a rib be 20 inches, it would have to be composed of at least 4 pieces. Pitch pine is not quite so elastic as white pine or oak— its thickness may be found by using the deci- mal 0-046 instead of 0-05. Fig. 106. 244-- — Bridges: Framed Rib. — In spans of over 250 feet, "d. framed rib, as in Fig. 106, would be preferable to the foregoing. Of this, the upper and the lower edges are formed as just described, by bending the timber to the proper curve. The pieces that tend to the centre of the curve, called radials,.7XYQ notched and bolted on in pairs, and the cross-braces are halved together in the middle, and abut end to end between the radials. The distance between the ribs of a bridge should not exceed about 8 feet. The roadway should be supported by vertical standards bolted to the ribs THE ROADWAY AND ABUTiMENTS. 22/ at about every lo to 15 feet. At the place where they rest on the ribs, a double, horizontal tie should be notched and bolted on the back of the ribs, and also another on the un- derside ; and diagonal braces should be framed between the standards, over the space between the ribs, to prevent lat- eral motion. The timbers for the roadway may be as lig-ht as their situation will admit, as all useless timber is only an unnecessary load upon the arch. 245. — Bridges: Roadway. — If a roadway be 18 feet wide, tv/o carriages can pass without inconvenience. Its width, therefore, should be either 9, 18, 27, or 36 feet, ac- cording- to the amount of travel. The width of the foot-> path should be two feet for every person. When a stream of water has a rapid current, as few piers as practicable should be allowed to obstruct its course ; otherwise the bridge will be liable to be swept away by freshets. When the span is not over 300 feet, and the banks of the river are of sufficient height to admit of it, only one arch should be employed. The rise of the arch is limited by the form of the roadwa^v, and by the height of the banks of the river (see Art. 243). The rise of the roadway should not exceed one in 24 feet, but as the framing settles about one in 72, the roadway should be framed to rise one in 18, that it may be one in 24 after settling. The commencement of the arch at the abutments — the spring, as it is termed — should not be* below high-water mark ; and the bridge should be placed at right angles with the course of the current. 246. — Bridges: Abiitmeiats. — The best material for the abutments and piers of a bridge is stone ; and no other should be used. The following rule is to determine the ex- tent of the abutments, they being rectangular, and built with stone weighing 120 pounds to a cubic foot. Rule LXVHI— Multiply the square of the height of the- abutment by 160, and divide this product by the weight of a square foot of the arch, and by the rise of the arch ; add unity to the quotient, and extract the square root. Dimin- ish the square root by unity, and multiply the root sj dimin- 228 CONSTRUCTION. ished by half the span of the arch, and by the weight of a square foot of the arch. Divide the last product by 120 times the height of the abutment, and the quotient will be the thickness of the abutment. Example. — Let the height of the abutment from the base to the springing of the arch be 20 feet, half the span 100 feet, the weight of a square foot of the arch, including the great- est possible load upon it, 300 pounds, and the rise of the arch 18 feet: what should be its thickness? The square of the height of the abutment, 400, multiplied by 160 gives 64000, and 300 by 18 gives 5400; 64000 divided by 5400 gives a quotient of 11-852; one added to this makes 12-852, the square root of which is 3-6 ; this, less one is 2-6 ; this mul- tiplied by 100 gives 260, and this again by 300 gives 78000; this divided by 120 times the height of the abutment, 2400, gives 32 feet 6 inches, the thickness required. . The dimensions of a pier will be found by the same rule ; for, although the thrust of an arch may be balanced by an adjoining arch when the bridge is finished, and while it re- mains uninjured, yet, daring the erection, and in the event of one arch being destroyed, the pier should be capable of sustaining the entire thrust of the other. Piers are sometimes constructed of timber their princi- pal strength depending on piles driven into the earth ; but such piers should never be adopted where it is possible to avoid them ; for, being alternately wet and dry, they decay much sooner than the upper parts of the bridge. Spruce and elm are considered good for piles. Where the height from the bottom of the river to the roadway is great, it is a good plan to cut them off at a little below low-water mark, cap them with a horizontal tie, and upon this erect the posts for the support of the roadway. This method cuts off the pan ihat Is continually wet from that which is only occa- sionally so, and thus affords an opportunity for replacing the upper part. The pieces which are immersed will last a great length of time, especially when of elm ; for it is a Well-established fact that timber is less durable when subject to alternate dryness and moisture than when it is either con- tinually Avet or continually dry. It has been ascertained that CENTRING FOR BRIDGES. 229 the piles under London Bridge, after having been driven about 600 years, were not materially decayed. These piles are chiefly of elm, and wholly immersed. ; 247. — Centres for Stone Bridges. — Fig. 107 is a design for a centre for a stone bridge where intermediate supports, as piles driven into the bed of the river, are practicable. Its timbers are so distributed as to sustain the weight of the arch-stones as they are being laid, without destroying the original form of the centre ; and also to prevent its destruc- tion or settlement, should any of the piles be swept away. The most usual error in badly-constructed centres is that the timbers are disposed so as to cause the framing to rise at the crown during the laying of the arch-stones up the sides. To remedy this evil, some have loaded the crown with heavy stones ; but a centre properly constructed will need no such precaution. i Experiments have shown that an arch-stone does not press upon the centring until its bed is inclined to the horizon at an angle of from 30 to 45 degrees, according to the hardness of the stone, and whether it is laid in mortar or not. For general purposes, the point at which the pressure com- mences may be considered to be at that joint which forms an angle of 32 degrees with the horizon. At this point the pressure is inconsiderable, but gradually increases towards the crown. The following table gives the portion of the weight of the arch-stones that presses upon the framing at the various angles of inclination formed by the bed of the 230 CONSTRUCTION. Stone with the horizon. The pressure perpendicular to the curve is equal to the weight of the arch-stone multiplied by the decimal — •o, when the angle of mclination is 32 degrees. 04 34 08 36 12' 38 17 40 21 ' * 42 25 44 29 46 33 48 37 50 4 52 44 54 48 56 52 58 54 60 Fig. 108. From this it is seen that at the inclination of 44 degrees the (pressure equals one quarter the weight of the stone ; at 57 degrees, half the weight; and when a vertical line, as ad {Fig. 108), passing through the centre of gravity of the arch-stone, does not fall within its bed, c d, the pressure may be con- sidered equal to the whole weight of the stone. This will be the case at about 60 degrees, when the depth of the stone is double its breadth. The direction of these pressures is considered in a line with the radius of the curve. The weight upon a centre being known, the pressure may be estimated and the timber calculated accordingly. But it must be remembered that the whole weight is never placed upon the framing at once — as seems to have been the idea had in view by the designers of some centres. In building the arch, it should be commenced at each buttress at the same time (as is generally the case), and each side should progress equally towards the crown. In designing the fram- CENTRE FOR A STONE BRIDGE. ^51 ing, the effect produced by each successive layer of stone should be considered. The pressure of the stones upon one side should, by the arrangement of the struts, be counter- poised by that of the stones upon the other side. Over a river whose stream is rapid, or where it is neces- sary to preserve an uninterrupted passage for the purposes of navigation, the centre must be constructed without in- termediate supports, and without a continued horizontal tie at the base ; such a centre is shown at Fig. 109. In laying the stones from the base up to a and c, the pieces bd and bd act as ties to prevent an}^ rising at b. After this, while the stones are being laid from a and from c to b, they act as struts; the piece f g is added for additional security. Upon this plan, with some variation to suit circumstances, Fig. 109. centres may be constructed for any span usual in stone- bridge building. In bridge centres, the principal timbers should abut, and not be intercepted by a suspension or radial piece between. These should be in halves, notched on each side and bolted. The timbers should intersect as little as possible, for the more joints the greater is the settling ; and halving them together is a bad practice, as it destroys nearly one half the strength of the timber. Ties should be introduced across, especially where many timbers meet ; and as the centre is to serve but a temporary purpose, the whole should be de- signed with a view to employ the timber afterwards for other uses. For this reason, all unnecessary cutting should be avoided. 232 CONSTRUCTION. Centres should be sufficiently strong to preserve a staunch and steady form during the whole process of build- ing; for any shaking or trembling will have a tendency to prevent the mortar or cement from setting. For this pur- pose, also, the centre should be lowered a trifle immedi- ately after the key-stone is laid, in order that the stones may take their bearing before the mortar is set ; otherwise the joints will open on the underside. The trusses, in centring, are placed at the distance of from 4 to 6 feet apart, accord- ing to their strength and the weight of the arch. Between every two trusses diagonal braces should be introduced to prevent lateral motion. In order that the centre may be easily lowered, the frames, or trusses, should be placed upon wedge-formed sills, as is shown at d {Fig. 109). These are contrived so as to admit of the settling of the frame by driving the wedge d with a maul, or, in large centres, with a piece of timber mounted as a battering-ram. The operation of lowering a centre should be very slowly performed, in order that the parts of the arch may take their bearing uniformly. The wedge pieces, instead of being placed parallel with the truss, are sometimes made sufficiently long and laid through the arch, in a direction at right angles to that shown at Fig. 109. This method obviates the necessity of stationing men beneath the arch during the process of lowering ; and was originally adopted with success soon after the occurrence of an accident, in lowering a centre, by which nine men were killed. To give some idea of the manner of estimating the pres- sures, in order to select timber of the proper scantling, cal- culate the pressure {Art. 247) ot the arch-stones from i to b (Fig. 109), and suppose half this pressure concentrated at a, and acting in the direction a f. Then, by the parallelogram of forces {Art. 71), the strain in the several pieces compos- ing the frame bda may be computed. Again, calculate the pressure of that portion of the arch included between a and c, and consider half of it collected at b, and acting in a vertical direction ; then, by the parallelogram of forces, the pressure on the beams bd and bdrnd^y be found. Add the JOINTS OF THE ARCII-STONES. 233 pressure of that portion of the arch which is included be- tween i and b to half the weight of the centre, and consider this amount concentrated at d, and acting in a vertical direc- tion ; then, by constructing the parallelogram of forces, the pressure upon dj may be ascertained. The strains having been obtained, the dimensions of the several pieces in the frames b a d "^xidi bed may be found by computation, as directed in the case of roof-trusses, from Arts. 226 to 229. The tie-beams b d, b d, if made of suffi- cient size to resist the compressive strain acting upon them from the load at b, will be more than large enough to resist the tensile strain upon them during the laying of the first part of the arch-stones below a and c. 248.— ArcSi-Stones : Joiiit§. — In an arch, the arch-stones are so shaped that the joints between them are perpendicu- lar to the curve of the arch, or to its tangent at the point at which the joint intersects the curve. In a circular arch, the Fig. 1 10. joints tend toward the centre of the circle ; in an elliptical arch, the joints may be found by the following process: To find the direction of the joints for an elliptical arch ; / ^ // f A/^ e \ kfY^ J " Fig. III. a joint being wanted at a {Fig. 1 10), draw lines from that point to the foci,/ and // bisect the angle /^/ with the line ab ; then a b will be the direction of the joint. 234 CONSTRUCTION. To find the direction of the joints for a parabolic arch : a joint being wanted at a {Fig. 1 1 1), draw a c ^t right angles to the axis eg\ make eg equal to c c, and join a and g\ draw ah at right angles to ag\ then ah will be the direction of the joint. The direction of the joint from b is found in the same manner. The lines ^^and b f are tan- gents to the curve at those points respectively ; and any number of joints in the curve may be obtained by first ascertaining the tangents, and then drawing lines at right angles to them. {ScQArt. 462.) JOINTS. 249. — Timljer JJoants. — The joint shown in Fig. 112 is simple and strong ; but the strength consists wholly in the bolts, and in the friction of the parts produced by screwing the pieces firmly together. Should the timber shrink to ^. Fig. 112. even a small degree, the strength would depend altogether on the bolts. It would be made much stronger by indent- ing the pieces together, as at the upper edge of the tie-beam C -CZ]- -■E> Cr- FiG. 113. in Fig. 113, or by placing keys in the joints, as at the lower edge in the same figure. This process, however, weakens the beam in proportion to the depth of the indents. 1 Fig. 114 Fig. 1 14 shows a method of scarfing, or splicing, a tie- beam without bolts. The keys are to be of well-seasoned, hard wood, and, if possible, very cross-grained. The addi- THE SPLICING OF TIMBER. 235 tion of bolts would make this a very strong splice, or even white-oak pins would add materially to its strength. Fi^. 115 shows about as strong a splice, perhaps, as can well be made. It is to be recommended for its simplicity; as, on account of there being no oblique joints in it, it can be readily and accurately executed. A complicated joint is the worst that can be adopted ; still, some have proposed joints that seem to have little else besides complication to recommend them. In proportioning the parts of these scarfs, the depths of all the indents taken together should be equal to one third of the depth of the beam. In oak, ash or elm, the whole length of the scarf should be six times the depth, or thick- ness, of the beam, when there are no bolts ; but, if bolts in- stead of indents are used, then three times the breadth ; and when both methods are combined, twice the depth of the beam. The length of the scarf in pine and similar soft woods, depending wholly on indents, should be about 12 times the thickness, or depth, of the beam ; when depend- ing wholly on bolts, 6 times the breadth ; and when both methods are combined, 4 times the depth. Fig. 116. Sometimes beams have to be pieced that are required to resist cross-strains— such as a girder, or the tie-beam of a roof when supporting the ceiling. In such beams, the fibres of the wood in the upper part are compressed ; and therefore a simple butt joint at that place (as in F{^. 116) is far preferable to any other. In such case, an oblique joint is the very worst. The under side of the beam being in a state of tension, it must be indented or bolted, or both ; and an iron plate under the heads of the bolts gives a great addition of strengfth. 236 CONSTRUCTION. Scarfing requires accuracy and care, as all the indents should bear equally; otherwise, one being strained more than another, there would be a tendency to sphnter off the parts. Hence the simplest form that will attain the object is by far the best. In all beams that are compressed end- wise, abutting joints, formed at right angles to the direction of their length, are at once the simplest and the best. For a temporary purpose, Fig. 112 would do very well ; it would be improved, however, by having a piece bolted on all four sides. Fig. 113, and indeed each of the others, since they have no oblique joints, would resist compression well. In framing one beam into another for bearing purposes, such as a floor-beam into a trimmer, the best place to make the mortise in the trimmer is in the neutral line {Arts. 120, 121), which is in the middle of its depth. Some have thought that, as the fibres of the upper edge are compressed, r-j Fig. 117. a mortise might be made there, and the tenon driven in tight enough to make the parts as capable of resisting the compression as they would be without it ; and they have therefore concluded that plan to be the best. This could not be the case, even if the tenon would not shrink ; for a joint between two pieces cannot possibly be made to resist compression so well as a solid piece without joints. The proper place, therefore, for the mortise is at the middle of the depth of the beam ; but the best place for the tenon, in the floor-beam, is at its bottom edge. For the nearer this is placed to the upper edge, the greater is the liability for it to splinter off; if the joint is formed, therefore, as at Fig. 117, it will combine all the advantages that can be obtained. Dou- ble tenons are objectionable, because the piece framed into is needlessly weakened, and the tenons are seldom so accu- rately made as to bear equally. For this reason, unless the tusk THE FRAMING IN A ROOF-TRUSS. 237 at a in the figure fits exactly, so as to bear equally with the tenon, it had better be omitted. And in sawing the shoulders care should be taken not to saw into the tenon in the least, as it would wound the beam in the place least able to bear it. Thus it will be seen that framing weakens both pieces, more or less. It should, therefore, be avoided as much as possible , and where it is practicable one piece should rest upon the other, rather than be framed into it. This re- mark applies to the bearing of floor-beams on a girder, to the purlins and jack-rafters of a roof, etc. In a framed truss for a roof, bridge, partition, etc., the joints should be so constructed as to direct the pressures through the axes of the several pieces, and also to avoid every tendency of the parts to slide. To attain this object, Fig. 118. Fig. iig. Fig. 120. the abuttinsf surface on the end of a strut should be at right angles to the direction of the pressure ; as at the joint shown in Fig. 118 for the foot of a rafter (see Art. 86), in Fig. 119 for the head of a rafter, and in Fig. 120 for the foot of a strut or brace. The joint at Fig. 118 is not cut completely across the tie-beam, but a narrow lip is left standing in the middle, and a corresponding indent is made in the rafter, to prevent the parts from separating sideways. The abutting surface should be made as large as the attainment of other necessary objects will admit. The iron strap is added to prevent the rafter sliding out, should the end of the tie- beam, by decay or otherwise, splinter off. In making the joint shown at Fig. 119, it should be left a little open at a, so as to bring the parts to a fair bearing at the settling of the truss, which must necessarily take place from the shrink- ing of the king-post and other parts. If the joint is made fair at first, when the truss settles it will cause it to open at 238 CONSTRUCTION. the under side of the rafter, thus throwing the whole pres- sure upon the sharp edge at a. This will cause an indenta- tion in the king-post, by which the truss will be made to settle further ; and this pressure not being in the axis of the rafter, it will be greatly increased, thereby rendering the rafter liable to split and break. 1^-^ Fig. 121. Fig. 122. If the rafters and struts were made to abut end to end, as in Figs. I2i, 122 and 123, and the king or queen post notched on in halves and bolted, the ill effects of shrinking would be avoided. This method has been practised with success in some of the most celebrated bridges and roofs in Europe ; and, were its use adopted in this country, the un- seemly sight of a hogged ridge would seldom be met with. Fig. 124. Fig. 125. A plate of cast-iron between the abutting surfaces will equalize the pressure. Fig. 124 is a proper joint for a collar-beam in a small roof: the principle shown here should characterize all tie- joints. The dovetail joint, although extensively practised in the above and similar cases, is the very worst that can be employed. The shrinking of the timber, if only to a small WHITE-OAK PINS AND IRON STRAPS. 239 degree, permits the tie to withdraw — as is shown at Fig. 125. The dotted Une shows the position of the tie after it has shrunk. Locust and white-oak pins are great additions to the strength of a joint. In many cases they would supply the place of iron bolts ; and, on account of their small cost, they should be used in preference wherever the strength of iron is not requisite. In small framing, good cut nails are of great service at the joints ; but they should not be trusted to bear any considerable pressure, as they are apt to be brittle. Iron straps are seldom necessary, as all the joinings in carpentry may be made without them. They can be used to advantage, however, at the foot of suspending-pieces, and for the rafter at the end of the tie-beam. In roofs for ordinary purposes, the iron straps for suspending-pieces may be as follows: When the longest unsupported part of the tie-beam is— 10 feet, the strap may be i inch wide by -f^ thick. 20 "• " " 2 " " i " In fastening a strap, its hold on the suspending-piece will be much increased by turning its ends into the wood. Iron straps should be protected from rust; for thin plates of iron decay very soon, especially when exposed to dampness. For this purpose, as soon as the strap is made let it be heated to about a blue heat, and, while it is hot, pour over its entire surface raw linseed oil, or rub it with beeswax. Either of these will give it a coating which dampness will not penetrate. SECTION III.— STAIRS. 250. — Stairs : General I&equlrement§. — The STAIRS is that commodious arrangement of steps in a building by which access is obtained from one story to another. Their position, form, and finish, when determined with discrimi- nating taste, add greatly to the comfort and elegance of a structure. As regards their position, the first object should be to have them near the middle of the building, in order that they may afford an equally easy access .to all the rooms and passages. Next in importance is light ; to obtain which they would seem to be best situated near an outer wall, in which windows might be constructed for the purpose ; yet a skylight, or opening in the roof, would not only provide light, and so secure a central position for the stairs, but may be made, also, to assist materially as an ornament to the building, and, what is of more importance, afford an oppor- tunity for better ventilation. All stairs, especially those of the most important build- ings, should be erected of stone or some equally durable and fire-resisting material, that the means of egress from a burn- ing building may not be too rapidly destroyed. Winding stairs, or those in which the direction is gradu- ally changed by means of winders, or steps which taper in width, are interesting by reason of the greater skill required in their construction ; but are objectionable, for the reason that children are exposed to accident by their liability to fall when passing over the narrow ends of the steps. Stairs of this kind should be tolerated only where there is not suffi- cient space for those with flyers, or steps of parallel width. Stairs in one long continuous flight are also objection- able. Platforms or landings should be introduced at inter- vals, so that any one flight may not contain more than about twelve or fifteen steps. The width of stairs should be in accordance with the im- KHORSABAD.-ASSYRIAN TEMPLE, RESTORED.^ ..^ THE GRADE OF STAIRS. 241 portance of the building in which they are placed, varying from 3 to 12 feet. Where two persons are expected to pass each other conveniently the least width admissible is 3 feet. Still, in crowded cities, where land is valuable, the space allowed for passages is correspondingly small, and in these stairs are sometimes made as narrow as 2^ feet. From 3 to 4 feet is a suitable width for a good dwelling ; while 5 feet will be found ample for stairs in buildings occu- pied by many people ; and from 8 to 12 feet is sufficient for the width of stairs in halls of assembly. To avoid tripping or stumbling, care should be exer- cised, in the planning of a stairs, to secure an even grade. To this end, the iioshig, or outer edge, of each step should be exactly in line with all the other nosings. In stairs com- posed of both flyers and winders, precaution in this regard is especially needed. In such stairs, the steps — fiyers and winders alike — should be of one width on the line along which a person would naturally walk when having his hand upon the rail. This tread-X\\\(t, consequently, would be paral- lel with the hand-rail, and is usually taken at a distance of from 18 to 20 inches from the centre of it. In the plan of the stairs this tread-line should be drawn and divided into equal parts, each part being the tread, or width of a flyer from the face of one riser to the face of the next. 251.— The Orade of Stairs — The extra exertion required in ascending a staircase over that for walking on level ground is due to the weight which a person at each step is required to lift ; that is, the weight of his own body. Hence the difficulty of ascent will be in proportion to the height of each step, or to the rise, as it is termed. To facilitate the operation of going up stairs, therefore, the risers should be low. The grade of a stairs, or its angle of ascent, depends not only upon the height of the riser, but also upon the width of the step ; and this has a certain relation to the riser ; for the width of a step should be in proportion to the smallness of the ano:le of ascent. The distance from the top of one riser to the top of the next is the distance travelled at each step taken, and this dis- 242 STAIRS. tance should vary as the grade of the stairs ; for a person who in cUmbing a ladder, or a nearly vertical stairs, can travel only 12 inches, or less, at a step, will be able with equal or greater facility to travel at least twice this distance on level ground. The distance travelled, therefore, should be in proportion inversely to the angle of ascent ; or, the di- mensions of riser and step should be reciprocal : a low rise should have a wide step, and a high rise a narrow step, 252. — Fitcli-Koard : Relation of Ri§e to Tread. — Among the various devices for determining the relation of the rise to the tread, or net width of step, one is to make the sum of the two equal to 18 inches. For example, for a rise of 6 inches the tread should be 12, for 7 inches the tread should be 1 1 ; or — 6 + 12 = 18 8 + 10 = 18 61 + III = 18 8i + 9i= 18 7 + II = 18 9 + 9 = 18 7^+ ioi= 18 9J + 8^= 18 This rule is simple, but the results in extreme cases are not satisfactory. If the ascent of a stairs be gradual and easy, the length from the top of one rise to that of another, or the hypothenuse of the pitch-board, may be proportionally long ; but if the stairs be steep, the length must be shorter. There is a French method, introduced by Blondel in his Coiirs d' Architecture. It is referred to in G wilt's Encyclo- pedia, Art. 2813. This method is based upon the assumed distance of 24 inches as being a convenient step upon level ground, and upon 12 inches as the most convenient height tt) rise when the ascent is vertical. These are French inches, old system. The 24 inches French equals about 25-^^ inches English. With these distances as base and perpendicular, a right- angled triangle is formed, which is used as a scale upon which the proportions of a pitch-board are found. For example, let a line be drawn from any point in the hypothe- nuse of this triangle to the right angle of the triangle ; then this line will equal the length of the pitch-board, along the PROPORTIONS OF THE PITCH-BOARD. 243 rake, for a stairs having a grade equal to the angle formed by this line and the base-line of the scale. In the absence of the triangular scale, the lengths of the pitch-boards, as found by this rule, may be computed by this expression— ^=25-j-V-2/^; (107.) in which /^equals the tread, or base of the pitch-board, and // the riser, or its perpendicular height. For example, let // = 6 ; then — This result is greater than would be proper in some cases. The length of the hypothenuse of the pitch-board should be proportional not only to the angle of ascent {Art. 251), but also to the strength and height of the class of people who are to use the stairs. Tall and strong persons will take longer steps than short and feeble people. The hypothe- nuse of the pitch-board should be made in proportion to the distance taken at a step on level ground by the persons who are to use the stairs. If people are divided into two classes, one composed of robust workmen and the other of delicate women and in- firm men, then there may be two scales formed for the pitch- boards of stairs — one to be used for shops and factories, and the other for dwellings. The distance on level ground trav- elled per step, by men, varies from about 26 to 32 inches, or on an average 28 inches. The height to which men are accustomed to rise on ladders is from 12 to 16 inches at each step, or on the average 14 inches. With these dimensions, therefore, of 14 and 28 inches, a scale may be formed for pitch-boards for stairs, in buildings to be used exclusively by robust workmen. And with 12 and 24 inches another scale may be formed for pitch-boards for stairs, in buildings to be used by women and feeble people. These two scales are both shown in F/o-. 126. They are made thus : Let C A B he ii right angle. Make A B equal to 28 inches, and A C equal to 14; then join B and 244 STAIRS. C. At right angles to C B, from A, draw A F; then with A F for radius describe the arc F G. Then a line, as y^ AT or A L, drawn from A at any angle with A B and limited by the line G F B v^iW give the length of the hypothenuse of the pitch-board, for shop stairs of a grade equal to the angle which said line makes with A B. From K, perpendicular to A B, draw K N\ then K N will be the proper riser for a pitch-board of which ^ A^is the tread. So, likewise, L M will be the appropriate riser for the tread A M. The arc F G is introduced to limit the rake-line of pitch-boards occur- ring between F and C, in order to avoid making them longer than the one at F. The scale for the stairs for dwellings is made in the same manner ; A D — 2\ inches being the base, A E =. \2 inches the rise, and J H D the line limiting the rake-lines of pitch-boards. To compute the length of risers an,d treads, we have for the scale for shops, for those occurring between i^and B — r:= 1(28-0: (io8.) / = 28 — 2 r ; (109.) and for those between F and G, we have — (108, A.) (109, A.) r= 1/156.8 - /^ ^= |/i56.8-r^ For the scale for dweUings, we have, for those occurring between H and D — r =: 1(24-/); (108, B.) / = 24 — 2 r; (109, B.) STAIRS FOR SHOPS AND FOR DWELLINGS. and for those between H and J, we have — 245 r^ 1/115.2-/^ /= 4/115.2 — r'; (108, C.) (109, C.) where, in each equation, r represents the riser, and t the tread, or net step. By these formulae, the following tables have been com- puted : Stairs for Shops. Rise. Tread. Ratio— Rise to Tread. 24- 22- 2T- 20- 19. 18. 17-20 16 -60 i6- 15-50 15- 14-60 1420 14- 13-60 to 33 12 7' 6- 5- 4-22 3-6o 3-19 .91 Rise. Tread. Ratio— Ris2 to Tread, 7 -40 13-20 I to 1-78 7-60 12-80 1-68 7-80 12-40 1-59 8- 12- I -50 8 -20 II-6 1-41 8-50 II- 1-29 8-80 10-40 I -18 9- 10- I- II 9-30 9-40 I-OI 9-60 8-80 0-92 10 • 8- o-8o 10-50 7- 0-67 II- 6- 0-55 11-50 4-95 0-43 12- 3-58 I" 0-30 Stairs for Dwellings. Rise. Tread. Ratio— Rise to Tread. Rise. Tread. Ratio— Rise to Tread. 2- 20- I to 10- 7-40 9 20 I to 1-24 3 18 I " 6- 7 50 9 I " 1-20 3 50 17 I " 4-86 7 60 8 80 I " 1. 16 4 16 I '• 4- 7 70 8 60 I " I-I2 4 50 15 I " 3-33 7 80 8 40 I " 1-08 5 14 I •' 2-80 7 90 8 20 I " I -04 5 40 13 20 I " 2-44 8 8 I " I- 5 70 12 60 I " 2-21 8 10 7 80 I " 0-96 6 12 I " 2- 8 30 7 40 I " 0-89 6 25 II 50 I «' 1-84 8 50 7 I " 0-82 6 50 II I " 1-69 8 75 6 50 I " 0-74 6 75 10 50 I " 1-56 9 6 I " 0-67 7 10 I " 1-43 9 30 5 40 I " 0-58 7 10 9 80 I '• 1-38 9 60 4 80 I " 0-50 7 20 9 60 I " 1-33 10 3 90 I " 0-39 7-30 9-40 I " 1-29 10-50 2-20 I " 0-2I These tables will be useful in determining questions in- 246 STAIRS. volving the proportion between the rise and tread of a pitch-board. For stairs in which the run is limited, to determine the number of risers which would give an easy ascent : Divide the riui by the Jicight^ and find in the proper table, above, the ratio nearest to the quotient^ and in a line with this ratio, in the second column to the left, will be found the corre- sponding riser. With this divide the rise in inches ; the quo- tient^ or the nearest zvhole number thereto, will be the required number of risers in the stairs. Example. — For the stairs in a dwelling, let the rise be 12' 8", or 152 inches. Let the run between the extreme risers be 17' 2". To this, for the purpose of obtaining the correct angle of ascent, by having an equal number of risers and treads, add, for one more tread, say 10 inches, its probable width; thus making the total run 18 feet, or 216 inches. Thus we have for the run 216, and for the rise 152. Divid- ing the former by the latter gives i -42 nearly. In the table of stairs for dwellings, the ratio nearest to this is i -43, and in the line to the left, in the second column, is 7, the approxi- mate size of riser appropriate to this case. Dividing the rise, 152 inches, by this 7, we have 2 if as the quotient. This is nearer to 22 than to 21 ; therefore, the number of risers required is 22. When the number of risers is determined, then the rise divided by this number will give the height of each riser ; thus, in the above case, the rise is 152 inches. This divided by 22 gives 6-909 inches for the height of the riser. When the height of the riser is known, then, if the run is unlimited, the width of tread will be found in the proper table above. For example, if the riser is 7 inches or nearly that, then in the table of stairs for dwellings, in the next column to the right, and opposite 7 in the column of risers, is found 10, the approximate width of tread. By the use of equation (109, B.), the width may be had exactly according to the scale. For example, equation (109, B.) with 6-91 for the riser, becomes— / — 24 — 2 X 6-91 = 10- 18, or about io^\ inches. TO CONSTRUCT THE PITCH-BOARD. 247 When the run is limited and the number of risers is known, then the width of tread is obtained by dividing the run by the number of treads. There are always of treads one less than there are of risers, in each flight. 253. — i>imensions of the Pitch-Board. — The first thing in commencing to build a stairs is to make the /zV<://-board ; this is done in the following manner: Obtain very accurate- ly, in feet and inches, the rise, or perpendicular height, of the story in which the stairs are to be placed. This must be taken from the top of the lower floor to the top of the upper floor. Then, to obtain the number of rises and treads and their size, proceed as directed in Art. 252. Having obtained these, the pitch-board may be made in the following man- ner: Upon a piece of well-seasoned board about |- of an inch thick, having one edge jointed straight and square, lay the corner of a steel square, as shown at Fig. 127. Make a b equal to the riser, and b c equal to the tread ; mark along the edges with a knife, and cut by the ^^'^' ^^7- marks, making the edges of the pitch - board perfectly square. The grain of the wood should run in the direction indicated in the figure, because, in case of shrinkage, the rise and the tread will be equally affected by it. When a pitch-board is first made, the dimensions of the riser and tread should be preserved in figures, in order that, in case of shrinkage or damage otherwise, a second may be made. 254. — The String of a Stairs. — The space required for timber and plastering under the steps is about 5 inches for ordinary stairs, or 6 inches if furred ; set a gauge, there- fore, at 5 or 6 inches, as the case requires, and run it on the lower edge of the plank, Tis ab {Fig. 128). Commencing at one end, lay the longest side of the pitch-board against the gauge-mark, a b, as at c, and draw by the edges the lines for the first rise and tread ; then place it successively as at d, e, 248 STAIRS. and y, until the required number of risers shall be laid down. To insure accuracy, it is well to ascertain the theoretical raking length of the pitch-board by computation, as in note to Art. 536, by getting the square root of the sum of the squares of the rise and run, and using this by which to divide the line ab into equal parts. Fig. 128, 255. — Step and Ri§er Connection. — Fig. 129 represents a section of step and riser, joined after the most approved method. In this, a represents the end of a block about 2 UJ. Fig, 129. inches long, two or three of which, in the length of the step, are glued in the corner. The cove at b is planed up square, glued in, and stuck or moulded after the glue is set. PLATFORM STAIRS. 256. — Platform Stair§ : the Cylinder. — A platform stairs ascends from one story to another in two or more flights, having platforms or landings between for resting and to change their direction. This kind of stairs, being simple, is CYLINDER OF PLATFORM STAIRS. 249 easily constructed, and at the same time is to be preferred to those with ivinders, for the convenience it affords in use {Art. 250). The cylinder may be of any diameter desirable, from a few inches to 3 or more feet, but it is generally small, about 6 inches. It may be worked out of one solid piece, but a better way is to glue together 3 pieces, as in Fig. 130 ; in which the pieces a, b, and c compose the cylinder, and d and e represent parts of the strings. The strings, " ^^^' ^'^^' after being glued to the cylinder, are secured with screws. The joining at and is the most proper for that kind of joint. Fig. 131, 257. — Form of I^ower Edge of Cylinder. — Find the stretch-out, de {Fig. 131), of the face of the cylinder, a be, 250 STAIRS. according to Art. 524; from d and e draw <^/ and eg at right angles to de\ draw kg parallel to de, and make kf and ^? each equal to one riser; from i and / draw ij and fk parallel to kg] place the tread of the pitch-board at these last lines, and draw by the lower edge the lines kh and il\ parallel to these draw m n and op, at the requisite distance for the dimensions of the string ; from s, the centre of the plan, draw sq parallel to df-, divide h q and q g each into two equal parts, as at v and w ; from v and w draw v n and w parallel to f d\ join n and 0, cutting qs\x\r\ then the angles 7t7ir and r^'^', being eased off according to Art. 521, will give the proper curve for the bottom edge of the cylinder. A centre may be found upon which to describe these curves, thus : from ti draw u x at right angles to m n ; from r draw rx at right angles to 7to\ then x will be the centre for the curve ti r. The centre for the curve r t may be found in a similar manner. Centres from which to strike these curves are usually quite unnecessary ; an experienced workman will readily form the curves guided alone by his practised eye. &. Fig. 132. 258. — Position of the Balusters. — Place the centre of the first baluster, b {Fig. 132), half its diameter from the face of the riser, cd, and one third its diameter from the end of the step, €d\ and place the centre of the other baluster, a, half the tread from the centre of the first. A line through the centre of the rail will occur vertically over the centres of the balusters. The usual length of the balusters is 2 feet 5 inches and 2 feet 9 inches respectively, for the short and long balusters. Their length may be greater than is here indicated, but, for safety, should never be less. The differ- ence in length between the short and long balusters is equal to one half the height of a riser. CONSTRUCTION OF WINDING STAIRS. 251 259. — ^Viiiding Stairs: have the steps narrower atone end than at the other. In some stairs there are steps of parallel width incorporated with the tapering steps ; in this case the former are called ^j^ers, and the latter winders. 260. — Itcgiilar Windings Stair§. — In Fig. 133, abed rep- resents the inner surface of the wall enclosing- the space allotted to the stairs, a e the length of the steps, and e f g h the cylinder, or face of the front-string. The line ^ ^ is given as the face of the first riser, and the point / for the limit of Fig. 133. the last. Make e z^ equal to 18 inches, and upon ^, with i for radius, describe the arc i j \ obtain the number of risers and of treads required to ascend to the floor at j\ according to Art. 252, and divide the. arc ij into the same number of equal parts as there are to be treads ; through the points of division, i, 2, 3, etc., and from the wall-string to the front- string, draw lines tending to the centre, o ; then these lines will represent the face of each riser, and determine the form and width of the steps. Allow the necessary projection for the nosing beyond a e, which should be equal to the thick- 252 STAIRS. ness of the step, and then a e I k will be the dimensions for each step. Make a pitch-board for the wall-string having a k for the tread, and the rise as previously ascertained ; with this lay out on a thicknessed plank the several risers and treads, as at Fig. 128, gauging from the upper edge of the string for the line at which to set the pitch-board. Upon the back of the string, with a i^-inch dado plane, make a succession of grooves \\ inches apart, and parallel with the lines for the risers on the face. These grooves must be cut along the whole length of the plank, and deep enough to admit of the plank's bending around the curve abed. Then construct a drum, or cylinder, of any com- mon kind of stuff, made to fit a curve with a radius the thickness of the string less than oa ; upon this the string must be bent, and the grooves filled with strips of wood, called keys^ which must be very nicely fitted and glued in. After it has dried, a board thin enough to bend around on the out- side of the string must be glued on from one end to the other, and nailed with clout-nails. In doing this, be careful not to nail into any place opposite to where a riser or step is to enter on the face. After the string has been on the drum a sufficient time for the glue to set, take it off, and cut the mortices for the steps and risers on the face at the lines previously made ; which may be done by boring with a centre-bit half through the string, and nicely chiselling to the line. The drum need not be made to extend over the whole space occupied by the stairs, but merely so far as requisite to receive one piece of the wall-string at a time ; for it is evident that more than one Avill be required. The front-string may be constructed m the same manner ; taking e I instead of a k for the tread of the pitch-board, dadoing it with a smaller dado plane, and bending it on a drum of the proper size. 261. ^Winding §tair§ : Shape and Po§Uioii of Tiiiiber§. — The dotted lines in Fig. 133 show the position of the timbers as regards the plan ; the shape of each is obtained as follows: In Fig. 134, the line i a is equal to a riser, less the thickness of the floor, and the lines 2 7^, 3 n, a.o,'^ p, and 6 q are each TIMBERS FOR WINDING STAIRS. 253 equal to one riser. The line a 2 is equal to a in in Fig. 133, the line m 3 to m n in that figure, etc. In drawing this fiofure, commence at a, and make the lines a i and « 2 of the length above specified, and draw them at right angles to each other ; draw 2 in at right angles to a 2, and ;;/ 3 at right angles to in 2, and make 2 in and in 3 of the lengths as above specified ; and so proceed to the end. Then through the points i, 2, 3, 4, 5, and 6 trace the line i b ; upon the points i, 2, 3, 4, etc., with the size of the timber for radius, describe arcs as shown in the figure, and by these the lower line may be traced parallel to the upper. This will give the proper shape for the timber, a b, in Fig. 133 ; and that of the others may be found in a similar manner. In ordinary cases, the shape of one face of the timber will be sufficient, for a good workman can easily hew it to its proper level by that ; but where great accuracy is desirable, a pattern for the other side may be found in the same man- FiG. 134. ner as for the first. In many cases, the timbers beneath cir- cular stairs are put up after the stairs are erected, and with- out previously giving them the required form ; the work- man in shaping them being guided by the form marked out by the lower edge of the risers. 262. — IVinding Stairs urith Flyers : Grade of Front- Strini;. — In stairs of this kind, if the winders are confined to the quarter circle, the transition from the winders to the flyers is too abrupt for convenience, as well as in appear- ance. To remove this unsightly bend in the rail and string, it is usual to take in among the winders one or more of the flyers, and thus graduate the width of the winders to that of the flyers. But this is not always done so as to secure the best results. By the method now to be shown, both rail and strings will be gracefully graded. In Fig. 135, a b repre- sents the line of the facia along the floor of the upper story, 254 STAIRS. bee the face of the cylinder, and c d the face of the front- string. Make ^<^ equal to \ of the diameter of the baluster, and parallel to a b, b e c, and c d draw the centre-line of the rail, fg, g h i, and ij\ make gk and ^/ each equal to half the width of the rail, and through k and /, parallel to the centre- line, draw lines for the convex and the concave sides of the rail ; tangical to the convex side of the rail, and parallel to k in, draw n o ; obtain the stretch-out, q r, of the semicircle, k p lUy according to Art. 524 ; extend a b to t, and k m to s ; make c s equal to the length of the steps, and i u equal to 18 inches, and parallel to m p describe the arcs s t and u 6 ; from t draw t w, tending to the centre of the cylinder ; from 6, and on the line 6 u x, run off the regular tread, as at 5, 4, 3, 2, I, and v; make 71 x equal to half the arc u 6, and make the point of division nearest to x, as v, the limit of the par- allel steps, or flyers ; make r equal to m z ; from draw «^* at right angles to n 0, and equal to one riser; from a'^ draw a'^ s parallel to 710, and equal to one tread; from s, through 0, draw s b^. Then from zv draw w ^^ at right angles to 71 0, and set up on the line w c^^ the same number of risers that the floor. A, is above the first winder, B, as at i, 2, 3, 4, 5, and 6 ; through 5 (on the arc 6 u) draw d'^ e^, tending to the centre of the cylinder; from e" draw e^" f- at right angles to no, and through 5 (on the line w <:") draw g"^ f^ parallel to it o\ through 6 (on the line w c") and /^ draw the line h'^ b'^ \ make 6 <;^ equal to half a riser, and from c^ and 6 draw c^- i'^ and 67^ parallel to 71 o] make h^ P equal to h'^ f^] from P draw P k^ at right angles to i'^ /i^, and from /^ draw f^k^ at right angles to f"JP\ upon k'^, \vith k'^ f^ for radius, de- scribe the arc f^P\ make b^ P equal to ^'^/^, and ease off the angle at /^^ by the curve /-^ P. In the figure, the curve is described from a centre, but as this might be imprac- ticable in a full-size plan, the curve may be obtained accord- * In the references fl^ 6', etc., a new form is introduced for the first time. During the time taken to refer to the figure, the memory of the form of these may pass from the mind, while that of the sound alone remains; they may then be mistaken for a2, b 2, etc. This can be avoided in reading by giving them a sound corresponding to their meaning, which is a second^ b secotid^ etc. MOULDS FOR QUARTER-CIRCLE STAIRS. 255 ing to Art. 521. Then from i, 2, 3, and 4 (on the line zv c-) draw lines parallel to 71 0, meeting the curve in in^, ;/^ 0'^, and /^; from these points draw lines at right angles to no, and meeting it in ;l^^ ^'^ ^^ and /^; from x^ and r'^ draw «3 hz hi (fz Fig. 135. lines tending to it -, and meeting the convex side of the rail in y^ and ^^; make ;;/ 7^- equal to r s^, and 7/1 w^ equal to rt^; from J/-, -a v^^ and w", through 4, 3, 2, and i,draw lines meeting the line of the wall-string in a^, d^, c^, and d^ ; from 256 STAIRS. ^^ where the centre-line of the rail crosses the line of the floor, draw e^ f'^ at right angles to ;z 0, and from/^ through 6, draw/^-; then the heavy hnes /V^'^^^^ 7^ «^ ^^ ^^ v'^c^, w"- d^, and z y will be the lines for the risers, which, being extended to the line of the front-string, b e c d, will give the dimensions of the winders and the grading of the front-string, as was required. HAND-RAILING. 263. — Hand-Railing for Stairs. — A piece of hand-rail- ing intended for the curved part of a stairs, when properly shaped, has a twisted form, deviating widely from plane sur- faces. If laid upon a table it may easily be rocked to and fro, and can be made to coincide with the surface of the table in only three points. And yet it is usual to cut such twisted pieces from ordinary parallel-faced plank ; and to cut the plank in form according to a face-vaoxxld, previously formed from given dimensions obtained from the plan of the stairs. The shape of the finished wreath differs so widely from the piece when first cut from the plank as to make it appear to a novice a matter of exceeding difficulty, if not an impossibility, to design a face-mould which shall cover accu- rately the form of the completed wreath. But he will find, as he progresses in a study of the subject, that it is not only a possibility, but that the science has been reduced to such a system that all necessary moulds may be obtained with great facility. To attain to this proficiency, however, re- quires close attention and continued persistent study, yet no more than this important science deserves. The young car- penter may entertain a less worthy ambition than that of desiring to be able to form from planks of black-walnut or mahogany those pieces of hand-railing which, when secured together with rail-screws, shall, on applying them over the stairs for which they are intended, be found to fit their places exactly, and to form graceful curves at the cylinders. That railing which requires to be placed upon the stairs before cutting the joints, or which requires the curves or butt-joints to be refitted after leaving the shop, is discredit- PRINCIPLES OF HAND-RAILING. 257 able to the workman who makes it. No true mechanic will be content until he shall be proved able to form the curves and cut the joints in the shop, and so accurately that no altera- tion shall be needed when the railing is brought to its place on the stairs. The science of hand-railing requires some knowledge of descriptive geometry — that branch of geometry which has for its object the solution of problems involving three dimensions by means of intersecting planes. The method of obtaining the lengths and bevils of hip and valley rafters, etc., as in Art. 233, is a practical example of descrip- tive geometry. The lines and angles to be developed in problems of hand-railing are to be obtained by methods dependent upon like principles. 264. — Hand-Railing^: Definitions; Planc§ and Solids. — Preliminary to an exposition of the method for drawing the face-moulds of a hand-rail wreath, certain terms used in descriptive geometry need to be defined. Among the tools used by a carpenter are those well-known implements called planes, such as the jack-plane, fore-plane, smoothing-plane, etc. These enable the workman to straighten and smooth the faces of boards and plank, and to dress them out of iviiid, or so that their surfaces shall be true and unwinding. The term plane, as used in descriptive geometry, however, refers not to the implement aforesaid, but to the unwinding surface formed by these implements. A plane in geometry is defined to be such a surface that if any two points in it be joined by a straight line, this line will be in contact with the surface at every point in its length. With like results lines may be drawn in all possible directions upon such a sur- face. This can be done only upon an unwinding surface ; therefore, a plane is an unwinding surface. Planes are understood to be unlimited in their extent, and to pass freely through other planes encountered. The science of stair-building has to do with prisms and cylinders, examples of Avhich are shoAvn in Figs. 136, 137, and 138. A right prism {Figs. 136 and 137) is a solid standing upon a horizontal plane, and with faces each of which is a plane. Two of these faces — top and bottom — are horizontal 258 STAIRS. and are equal polygons, having their corresponding sides parallel. The other faces of the prism are parallelograms, each of which is a vertical plane. When the vertical sides of a prism are of equal width, and in number increased indefi- nitely, the two polygonal faces of the prism do not differ essentially from circles, and thence the prism becomes a cylinder. Thus a right cylinder may be defined to be a prism, with circles for the horizontal faces {Fig. 138). Fig, 136. Fig. 137. Fig. 138. 265. — Hand - Railings : Preliminary €on$lderation§. — If within the well-hole, or stair-opening, of a circular stairs a solid cylinder be constructed of such diameter as shall fill the well-hole completely, touching the hand-railing at all points, and then if the top of this cjdinder be cut off on a line with the top of the hand-railing, the upper end of the cylinder would present a winding surface. But if, instead of cutting the cylinder as suggested, it be cut by several planes, each of which shall extend so as to cover only one of the wreaths of the railing, and be so inclined as to touch its top in three points, then the form of each of these planes, at its intersec- tion with the vertical sides of the cylinder, would present the shape of the concave edge of the face-mould for that particular piece of hand - railing covered by the plane. Again, if a hollow cylinder be constructed so as to be in contact with the outer edge of the hand-railing throughout its length, and this cylinder be also cut by the aforesaid FACE-MOULDS FOR HAND-RAILS. 259 planes, then each of said planes at its intersection with this latter cylinder would present the form of the convex edge of the said face-mould. A plank of proper thickness may now have marked upon it the shape of this face-mould, and the piece covered by the face-mould, when cut from the plank, will evidently contain a wreath like that over which the face-mould was formed, and which, by cutting away the surplus material above and below, may be gradually wrought into the graceful form of the required wreath. By the considerations here presented some general idea may be had of the method pursued, by which the form of a face-mould for hand-railing is obtained. A little reflection upon what has been advanced will show that the problem to be solved is to pass a plane obliquely through a cylinder at certain given points, and find its shape at its intersection with the vertical surface of the cylinder. Peter Nicholson was the first to show how this might be done, and for the invention was rewarded, by a scientific society of London, with a gold medal. Other writers have suggested some sHght improvements on Nicholson's methods. The method to which preference is now given, for its simplicity ot work- ing and certainty of results, is that which deals with the tangents to the curves, instead of with the curves themselves; so we do not pass a plane through a cylinder, but through a prism the vertical sides of which are tangent to the cylinder, and contain the controlling tangents of the face-moulds. The task, therefore, is confined principally to finding the tangents upon the face-mould. This accomplished, the rest is easy, as will be seen. The method by which is found the form of the top of a prism cut by an oblique plane will now be shown. 266. — A Prism Cwt toy an ©toHqiic Plane. — A prism is shown in perspective at Fig. 139, cut by an oblique plane. The points abed are the angles of the horizontal base, and abg, bef, edef, and adeg are the vertical sides; while ^f^S is the top, the form of which is to be shown. 267. — Form of Top of Prism — In Fig. 139 the form of the top of the prism is shown as it appears in perspective, 26o STAIRS. not in its real shape ; this is now to be developed. In Fig. 140, let the sauare « ^ <; <^ represent by scale the actual form Fig. 139. and size of the base, abed, of the prism shown in Fig, 139. Make c, and dd^ respectively equal to the actual heights at .4 Fig. 140. cf and de^ Fig. 139 ; the lines dd^ and ee^ being set up per- pendicular to the line de. Extend the lines de and d^ e^ until ILLUSTRATION BY PLANES. 26 1 they meet in Jl ; join b and //. Now this line b h is the inter- section of two planes : one, the base, or horizontal plane upon which the prism stands ; the other, the cutting plane, or the plane which, passing obliquely through the prism, cuts it so as to produce, by intersecting the vertical sides of the prism, the form b feg, Fig. 139. To show that b h is the line of intersection of these two planes, let the paper on which the triangle dhd^ is drawn (designated by the letter E) be lifted by the point d^ and revolved on the line dh until d^ stands vertically over d, and c^ over c\ then B will be a plane standing on the line dh, vertical to the base-plane A . The point h being in the line cd extended, and the line ^<^ being in the base-plane ^, there- fore h is in the base-plane A. Now the line d^c^ represents the line ef oi Fig, 139, and is therefore in the cutting plane ; consequently the point //, being also in the line d^c^ ex- tended, is also in the cutting plane. By reference to Fig. 139 it will be seen that the point b is in both the cutting and base planes ; we must therefore conclude that, since the two points b and h are in both the cutting and base planes, a line joining these two points must be the intersection of these two planes. The determination of the line of intersection of the base and cutting planes is very important, as it is a control- ling line ; as will be seen in defining the lines upon which the form of the face-mould depends. Care should therefore be taken that the method of obtaining it be clearly under- stood. It will be observed that the intersecting line bh, being in the horizontal plane A, is therefore a horizontal line. Also, that this horizontal line b h being a line in the cutting plane, therefore all lines upon the cutting plane which are drawn parallel to b h must also be horizontal lines. The import- ance of this will shortly be seen. Through a, perpendicular to b Jl, draw the line b^, d^, and parallel with this line draw ddj^jj] on d as centre describe the arc d^d^^^/, draw^/^^^^^v parallel with dd^^, and extend the latter to d^^, ; on d^, as centre describe the arc d^ d^^^ ; join b,, and d^^^. We now have three vertical planes which are to be brought into position around the base-plane A, as follows: Revolve B 262 STAIRS. upon dh^ E upon dd^^, and C upon b^^ d^^, each until it stands perpendicular to the plane A. Then the points d^ and d^^^^ will coincide and be vertically over d\ the points d^^^ and d^ will coincide and stand vertically over d^^ ; and c^ will cover c. These vertical planes will enclose a wedge-shaped figure, lying with one face, b^^d^^dh, horizontal and coincident with the base-plane A, and three vertical faces, b^^ d^^ d^^^, dd^^ d^ d^^^^, and hdd^. By drawing the figure upon a piece of stout paper, cutting it out at the outer edges, making creases in the lines hd, dd^^, ^u^/n then folding the three planes B, E, and C at right angles to A, the relation of the lines will be readily seen. Now, to obtain the form of the top or cover to the wedge-shaped figure, perpendicular to b^^ d^^^ draw b^^ h^ and d^^^e\ on b^^ as centre describe the arc hJi^ ; make d^^^e equal to d^^d; join e and h^. Now the form of the top of the wedge-shaped figure is shown within the bounds d^^^ b^^ h^ c. By revolving this plane D on the line b^^ d^.^ until it is at a right angle to the plane C^ and this while the latter is supposed to be vertical to the plane A, it will be perceived that this movement will place the plane D on top of the wedge-shaped figure, and in such a manner as that the point e Avill coincide with d^^^^ d^, and the point h^ will fall upon and be coincident with the point h, and the lines of the cover will coincide with the corresponding lines of the top edges of the sides of the figure; for example, the fine b^^d^^/is common to the top and the side C] the line d^^^e equals d^^ dy which equals d^ d^^^^ ; therefore, the line d^^^ e will coincide with ^v<^//// of the side E\ the line eh^ will coincide with dji of the side B ; and the line b^^ h, will coincide with the line b^Ji. Thus the figure D bounded by b^^d^^^eh^ will exactly fit as a cover to the wedge-shaped figure. Upon this cover Ave may now develop the form of the top of the prism. PreUminary thereto, however, it will be observed, as was before remarked, that lines upon the cutting plane which are parallel to the intersecting line b^^ h^ are horizontal ; and each, therefore, must be of the same length as the line in the base-plane A vertically beneath it. For example, the line d^^^ e^ is a line in the cutting plane D, parallel with the line b^^ h^ in the same plane, and this line b^^ h^ will (when the EXPLANATION OF THE DIAGRAMS. 263 cutting plane D is revolved into its proper position) be co- incident with the intersecting Hne b^^ h ; therefore, the line d^^^e is a Une in the cutting plane D, drawn parallel with the intersecting line b^^ Ji. Now this line d^^^ c, when in position, will be coincident with the line d^^^^d^., which lies vertically over the line d^^ d of the base-plane A ; its length, therefore, is equal to that of the latter. In like manner it may be shown that the length of any line on the plane D parallel to b^^ hiy is equal in length to the corresponding line upon the plane A vertically beneath it. Therefore, to obtain the form of the top of the prism, we proceed as follows : Perpendicular to b^^ d^ draw c c^^^ and aa,ij\ perpendicular to b^^d^^^ draw ^^^^/ and equal to<:^^<:; on b^j as centre describe the arc b b^ ; join b^ a^.^, b^f, and a^^^ e. Now we have here in plane D the form of the top of the prism, as shown in the figure bounded by the lines a^j^b,fc. This will be readily seen when the plane D is revolved into position. Then the point a^^^ will be vertically over a ; the point e coincident with d^ d^^^^ and vertically over <^; the point / coincident with c^ and vertically over c ; while b^ will coin- cide with b of the base-plane A. The figure a^^^ b^fe, therefore, represents correctly both in form and size the top of the prism as it is shown in per- spective at bfeg. Fig. 139. The line ef, Fig. 140, is equal to the line d^ c^, and so of the other lines bounding the edges of the figure. The cutting plane bfeg, Fig. 139, may be taken to repre- sent the surface of the plank from which the wreath of hand- railing is to be cut ; the w^reath curving around from b to e^ as shown in Fig. 141, the lines b g and g e being tangent to the curve in the cutting plane ; while a b and a d are tan- gents to the curve on the base plane, or plane of the cylin- der. The location of the cutting plane, however, is usually not at the upper surface of the plank, but midway between the upper and under surfaces. The tangents in the plane are found to be more conveniently located here for deter- mining the position of the butt-joints. For a moulded rail two curved lines, each with a pair of tangents, are required upon the cutting plane, one for the outer edge of the rail, 264 STAIRS. and the other for the inner edge ; but for a round rail only one curve with its tangents is required, as that from ^ to ^ in Fig. 141, which is taken to represent the curved line run- ning through the centre of the cross-section of the rail. As an easy application of the principles regarding the prism, just developed, an example will now be given. 268. — Face-Hould for Hand-Railings of Platform Stairs. — Lety /^ and / vi, Fig. 142, represent the central or axial lines of the hand-rails of the two flights, one above, the other be- low the platform ; and let the semicircleyV/ be the central line of the rail around the cylinder at the platform, the risers at the platform being located at/ and /. Vertically over the platform risers draw ^^^ ; make gr^ equal to a riser of the lower flight, and r^g^ and ss^ each equal to a riser of the upper flight. Draw g^s and gk^ horizontal and equal each to a tread of each flight respectively. Through r^ draw k, a^^y and through g^ draw Si t^. Vertically over d draw a^ t^. Horizontally draw a^^ a^^^, and t^ t^^. It is usual to extend the wreath of the cylinder so as to include a part of the straight rail — such a part as convenience may require. Let the straight part here to be included ex- tend from / to ^ on the plan. Vertically over b draw b^ c^^^y and horizontally draw b^ w^^ ; at any point on b^ w^^ locate w^^, and make w^^w^ equal to //, and bisect it in iv ; erect the perpendiculars w^ a^^^y w d^^^, and w^^ v ; join t^^ and a^^^^ ; from ^v// horizontally draw d^^^ d^^ ; parallel with r^ k^ draw ^v/ Cii.r W^ ^^^ have the plan and elevations of the prism. FACE-MOULD FOR PLATFORM STAIRS. 265 containing at its angles the tangents required for the wreath extending from ^o ^ on the plan. The elevation i^ is a view of the cylinder looking in the direction dc. Fig. 142. Comparing Fig. 142 with Fig. 141, the line h^zv^, is the trace, upon a vertical plane, of the horizontal plane abed 266 STAIRS. of Fig. 141, or is the ground-line from which the heights of the prism are to be taken. The triangle <^;, b^ a^^ is represented in Fig. 141 at ab g, and the inclined hne b^ a^^ is the tangent of the rail of the lower flight, and is represented in Fig. 141 at b g ; while a^^^^ t^^ is the tangent of the railing around the cylinder, and the half of it is represented in Fig. 141 at ge. The height b^c^^^^ is shown in Fig. 141 at cf, while the height w d^^^, or a^ d^^, is shown in Fig. 141 at <^^. The vertical planes B F C may now be constructed about the prism as in />^. 140, proceeding thus: Make cc^ equal to b^ Cjjji, and dd^ equal to a^ d^^ ; through c^ draw d^ h ; through b draw // b^^ ; perpendicular to // b^^ through a draw b^^ dy ; from <^ parallel with /^^^ then transferred to Fig. 145, and placed at right angles to ^ the plan of a line pass- ing through the centre of the rail around the cylinder as before, and je is a vertical section of two steps starting from the floor , hg. Bisect eJi in d, and through d draw df HAND-RAIL AT RAKE AND LEVEL. 273 parallel to hg\ bisect f n in /, and from / draw It parallel to nj \ from n draw nt at right angles to jn ; on the line ob make ot equal to n t. Then, to obtain a mould for the twist going up the flight, proceed as at Fig. 145 ; making ^^ in that figure equal to en in Fig. 150, and the other lines of a length and position such as is indicated by the letters of reference in each figure. To obtain the mould for the level Fig. 150. rail, extend bo (Fig. 150) to i ; make o i equal to //, and join / and ^ ; make^2(/^^> 151) equal to ci Tit Fig. 150; through Fig. 151. c draw c d ?it right angles to ci\ make dc equal to df at Fig. 150, and complete the parallelogram odci\ then pro- ceed as in the previous cases to find the mould. 273. — Application of Face-Moulds to Plank. — All the moulds obtained by the preceding examples have been for round rails. For these, the mould may be applied to a plank of the same thickness as the rail is intended to be, and the 2/4 STAIRS. plank sawed square through, the joints being cut square from the face of the plank. A twist thus cut and truly rounded will hang in a proper position over the plan, and present a perfect and graceful wreath. 274. — Face-Moulds for Moulded Rails upon Platform Stairs. — In Fig. 152, ^ ^^ is the plan of a line passing through Fig. 152. the centre of the rail around the cylinder, as before, and the lines above it are a vertical section of steps, risers, and plat- form, with the lines for the rail obtained as in Fig. 144. Set half the width of the rail from b to f and from b to r, and from / and r draw fe and r d parallel to ca. At Fig. 153 the centre-lines of the rail jc and c I are obtained as in the ^previous examples, making Jc equal jn of Fig. 152, ct KACE-MOCLD APPLIED TO PLANK. 2/5 rou-^l ct of Fi-. 152, and tl equal si of Fig. 152. Make r ^■ Jd ; each eV.al to .. at mg. 152. and draw the lines ... and kg parallel to .y ; make /. and /r equal to « c and « rf at Fk .£ and draw rf« and eq parallel to /.; also, through j driwL- parallel to Ic; then, in the parallelograms «.;/r. and ..;., f^nd the elliptic curves, dm and ^S.-^-o.^^U. Art\^x, and they will define the curves. The fine dp, being drawn through / perpendicular to /r, defines the joint which is to be cut square through the plank. 275.— Application of Face-MouKU to Planli.— In Fig. ,S- make a drawing, from ./ to //, of the cross-section ol the hand-rail, and tangent to the lower corner draw the nie g li The distance between the lines 7> and g h is the thickness of the plank from which the rail is to be cut. Lay the face- mould upon the plank, mark its shape upon the plank, and Fig. 153- saw it square through. To proceed strictly in accordance with the requirements of the principles upon which the lace- mould is formed, the cutting ought to be made vertically through the plank, the latter being in the position which it would occupy when upon the stairs. Formerly it was the custom to cut it thus, with its long raking lines. But, owing to the great labor and inconvenience of this method, efforts were made to secure an easier process. By investigation it was found that it was possible, without change in the face- mould, to cut the plank square through and still obtain the correct figure for the raiUng, and this method is the one now usually pursued. Not only is the labor of sawing much re- duced by this change ; but to the workman it is an entire re- lief, as he now, after marking the form of the wreath upon the plank, sends it to a steam saw-mill, and, at a small cost, has it !76 STAIRS. cut out with an upright scroll-saw. When thus cut out in the square, the upper surface of the plank is to be faced up true and unwinding, and the outer edge jointed straight and square from the face. Then a figure of the cross-section of the hand-railing is to be carefully drawn on the ends of the squared block as shown in Figs. 154 and 155, and w^hich are regulated so as to be correctly in position, as follows. First, as to the end Ji of the straight part hj\ In Fig. 154, X^tab c dh^ an end view of the squared block, of which a cfd is the shape of the end of the straight part. Let the point g be the centre of this end of the straight part ; through g draw upon the end a e f d the line j k, so that the angle bjk shall be equal to the angle kt c, Fig. 152. This is the angle at which the plank is required to be canted, revolving it on Fig. 154. Fig. 155. the axis of the straight part of the rail. Through g draw the line n h parallel with a h. Upon a thin sheet of metal (zinc is preferable) mark carefully the exact figure of the cross-section of the rail, drawing a vertical line through its centre, cut away the surplus metal, then, with this template as a pattern, mark upon the end a efd, Fig. 154, the figure of the rail as show^n, the vertical line upon the template being made to coincide with the Wnej'k. From n and h draw the vertical lines h in and In parallel with j k. Now, as to the other end of the square block : Let b cfe. Fig. 155, represent the block, of which bcvn is the form of the end at the curved part, and o its centre. Through dra^v/^, so that the angle e pq shall be equal to the angle jnb, Fig. 152. Also, through draw d h parallel with e b\ CUTTING THE TWIST-RAIL. 277 from ^and Ji draw the vertical lines Ji rand ds parallel with pq. Place the template on bcvn, the end of the block, so that the vertical line throusfh its centre shall coincide with pq; mark its form, then from j, at mid-thickness, draw iv y parallel with / q. In applying the mould, let Fig. 156 represent the upper face of the squared block, .^ x with the face-mould lying ^ upon it. With the distance a /, Fig. 154, and by the edge a x, mark a gauge-line upon the upper face of the Fig. 156. squared block. Set the outer edge of the lace-mould to coin- cide with this gauge-line. Let the end of the face-mould be set at zv, e iv being equal to e zv, Fig. 155; then mark the block by the edge of the face-mould. Now turn the block over and apply the face-mould to the underside, as in Fig. 157. With the distance d i?i, Fig. 154, and by the outer edge of the block, mark a gauge- line from m, Fig. 157. Set the inner edge of the face- mould to this gauge-line, and slide it endwise till the distance em shall equal ezu, Fig. 155, then mark the block by the edges of the face-mould. The over wood may now be re- moved as indicated by the vertical lines at the sides of the cross-section marked on each end of the block (see also Fig. 167) : the direction of the cutting at the curves must be verti- cal ; the inner curve will require a round-faced plane. A com- parison of the several figures referred to, with the directions given, together with a little reflection, will manifest the reasons for the method here given for applying the face- mould. Especially so when it is remembered that the face- mould was obtained not for the top of the rail, but for the rail at the mid-thickness of the block. So, therefore, in the application to the upper surface of the block, the face-mould is slid up the rake far enough to put the mould in position vertically over its true position at mid-thickness ; and on the •m Fig. 157. 2/8 STAIRS. contrary, in applying the face-mould to the underside of the plank, it is slid down until it is vertically beneath its true position at the mid-thickness of the block. When the vertical faces are completed,, the over wood above and below the wreath is to be removed. In doing this, the form at the ends, as given by the template, is a suf- ficient guide there. Between these the upper and under surfaces are to be warped from one end to the other, so as to form a graceful curve. With a little practice an intelli- gent mechanic will be able to work these surfaces with facility. The form of cross-section produced by this opera- tion is that of a parallelogram, tangent to the top, bottom, and two sides of the rail ; and which at and near the ends of the block is not quite full. The next operation is that of working the moulding at the sides and on top, first by re- bates at the sides, then chamfering, and finally moulding the curves. Templates to fit the rail, one at the sides, another on top, are useful as checks against cutting away too much of the wood. The joints are all to be worked square through the plank in the line drawn perpendicular to the tangent, as shown in F^g' 153- 276. — Hand-Railing for Circular Stairs. — Let it be re- quired to furnish the face-moulds for a circular stairs similar to that shown in Fig. 133. Preliminary to making the face-moulds it is requisite to make a plan, or horizontal projection of the stairs, and on this to locate the projections of the tangents and develop their vertical projections. For this purpose let bcdefg, Fig. 158, be the horizontal projection of the centre of the rail, and the lines numbered from 1 to 19 be the risers. At any point, a, on an extension of the line of the first riser locate the centre of the newel. On <^ as a centre describe the two circles ; the larger one equal in diameter to the diame- ter of the newel-cap, the inner one distant from the outer one equal to half the width of the rail. Let the first joint in the hand-rail be located at b, at the fourth riser ; through b draw h k tangent to the circle. Select a point, //, on this PLAN OF CIRCULAR STAIRS. 279 tangent which shall be equally distant from b and from the inner circle of the newel-cap, measured on a line tending to a ; join h and a^ and from a point, ^, on the line b describe Fig. 158. the curve from b to the point of the mitre of the newel-cap, the curve being tangent, at this point, to the line a h. Select positions for the other joints in the hand-rail as at c, d, c, and /. 280 STAIRS. Through these draw lines tangent to the circle.* Then the horizontal projection of the tangents will be the lines Jik, kl, I in, 1/1 11, and np. Now, if a vertical plane stand upon each of these lines, these planes would form a prism not quite complete standing upon the base-plane, A. Upon these ver- tical planes, C, D, E, F, G, and H, lines may be drawn which at each joint shall be tangent to the central line of the rail. These are the tangents now to be sought. Perpendicular to the tangents at b, c, d, etc., draw the lines bb^,cc^,dd^,ee^, ff^, ggi, and h h^^, k k^, k k^^,l l^, 1 1^,, etc. As b is at the fourth riser, and the height is counted from the top of the first riser, make b b^ equal to three risers. (To avoid extending the drawing to inconvenient dimensions, the heights in it are made only half their actual size. As this is done uniformly throughout the drawing, this reduction will lead to no error in the desired results.) As c is on the eighth riser, therefore make c c^ equal to seven risers, and so, in like manner, make the heights dd^,ee„ and ff^ each of a height to correspond with the number of the riser at which it is placed, deduct- ing one riser. These heights fix the location of each tangent at its point of contact with the central line of the rail. But each tangent is yet free to revolve on this point of contact, up or down, as may be required to bring the ends of each pair of tangents in contact ; or, to make equal in height the edges of each pair of vertical planes, which coincide after they are revolved on their base-lines into a vertical position ; as, for example : the edges k k^ and k k^j of the planes C and D must be of equal height; so, also, the edges //, and //^, of the planes D and E must be of equal height. The method of establishing these heights will now be shown. To this end let it be observed, that of the horizontal pro- jection of any pair of intersecting tangents, their lengths, from the point of intersection to the points of contact with the circle, are equal ; for example : of the two tangents h k and Ik, the distances from k, their point of intersection, to b and c, their points of contact with the circle, are equal ; and so also r/ equals d I, dm equals e in, etc. It will be observed * A tangent is a line perpendicular to the radius, drawn from the point of contact. THE FALLING-MOULD FOR THE RAIL. 28 1 that this equality is not dependent on b^ c, d, etc., the points of contact, being disposed at equal distances ; for, in this example, they are placed at unequal distances, some being- at three treads apart and others at four ; and yet while this un- equal distribution of the points b, c, d, etc., has the effect of causing the point of contact, as b, c, or c, to divide each whole tangent into two unequal parts, it does not disturb the equality of the two adjoining parts of any two adjacent tan- gents. Now, because of this equality of the two adjoining parts of a pair of tangents, the height to be overcome in passing from one point of contact to the next must be divided equally between the two ; each tangent takes half the distance. Therefore, for stairs of this kind, the arrange- ment being symmetrical, we have this rule by which to fix the height of the ends of any two adjoining tangents, namely : To the height at the lower point of contact add half the dif- ference between the heights at the two points of contact ; the sum will be the required height of the two adjoining ends of tangents. For example: the heights at b and r, two adjacent points of contact, are respectively three and seven risers ; the difference is four risers ; half this added to three, the height of the lower rise, gives five risers as the height of k k^, kkjj, the height at the adjoining ends of the tangents h k and Ik. Again, the heights at c and d are re- spectively seven and ten risers ; their difference is three ; half of which, or one and a half risers, added to seven, the height at the lower point of contact, makes nine and a half risers as the heights //^, //^^, at the ends of the adjoining tangents k I and ;;/ /. In a similar manner are established the heights of the tangents at in, ;/, and /. The rule for finding the heights of tangents as just given is applicable to circular stairs in Avhich the treads are di- vided equally at the front-string, as in Fig. 158. Stairs of irregular plan require to have drawn an elevation of the rail, stretched out into a plane, upon which the tangents can be located. This will be shown farther on. The locations of the joints c, d, c, in this example, were disposed at unequal distances merely to show the effect on the tangents as before noticed. In practice it is proper to 282 STAIRS. locate them at equal distances, for then one face-mould in such a stairs will serve for each wreath. When the tangent at G has been drawn, the level tangent for the landing maybe obtained in this manner: As the joint /is located at the eighteenth riser, one riser below the landing, draw a horizontal line at ^, one riser above the point /, and at half a riser above this draw the level line at /^ ; then this line is the level tangent, and p its point of intersection with the raking tangent. Draw the vertical line/^/, and from/ draw the tangent/^, w^hich is the horizontal projec- tion of the tangent p^ g, on plane H (which, to avoid undue enlargement of the drawing, is reduced in height), where //// equals// . To obtain the horizontal tangent t il at the newel, pro- ceed thus : Fix the point r, in the tangent r k^, at a height above b t equal to the elevation of the centre of the newel above the height of a short baluster — for example, from 5 to 8 inches- — and draw a line through r parallel to b t ] this is a horizontal line through the middle of the height of the newel-cap, and upon which and the rake the easement to the newel is formed. Perpendicular to b t draAv r t, and join / and 7c ; then t it is the horizontal tangent. 277.— Face-Moulds for Circular Stair§. — At Fig. 159 the plan of the newel and the adjacent hand-rail are repeated, but upon an enlarged scale ; and in which b b^ is the reduced height of the point by or is equal to bb^ less t r, Fig. 158, and the angle bb^ ^equals the angle bb^r of Fig. 158. In this plan the actual heights must now be taken. Join / and iL ; then 1 7t is the level tangent, as also the line of intersection of the cutting plane C and the horizontal plane A. Perpen- dicular to 1 71, at a point t or anywhere above it, draw u^ b^^. Parallel with 1 71 draw b b^^^ ; make b^^ b^^^ equal to b b/, join bjjj and 71^ ; then the angle b b^^^ 71^ is the angle which the plank in position makes with a vertical line, or what is usually termed the phuitb-bevil. Perpendicular to b^^^ 7i^ draw 7L^ 71^^ and b^^^ b^^^^ ; make b^^, b^^^^ equal to b b^^ ; make 71^ t^ equal to 7t^ /, and 71^ 71^^ to 71^ 71 ; join b^^^^ and /, ; then b^^^^ t^ is the tangent in the cutting plane, the horizontal projection of FACE-MOULD FOR FIRST SECTION 283 which is b t. The butt-joint at b^.^^ is drawn square to the tangent b^^^^ t ^. Parallel to the intersecting line / ti, draw ordinates across the plane A from as many points as desir- able, and extend them to the rake-line 7i, b^^^ ; through the points of their intersection with this line, and perpendicular to it, draw corresponding ordinates across the plane C. Make ^.i^ui equal to<^; d, and so in like manner, for all other points, Fig. 159. obtain in the plane C for each point in the horizontal plane A its corresponding point in the plane C \ in each case taking the distance to the point in the plane A from the line Jl b^i and applying it in the plane C from the rake-line ti^ b^^^. For the curves bend a flexible strip to coincide with the several points obtained, and draw the curve by the side of the strip. The point of the mitre is at d,,^, the mitre-joint is 284 STAIRS. shown at hd^^^ and d^^^ c^^. The line f c^^ is drawn through c^j, the most projecting point of the mitre, and parallel to the rake-line tt^ b^^^. Additional wood is left attached, extending from h to /; this is an allowance to cover the mitre, which has to be cut vertically ; the butt-joint at b^^^^ and the face at f Cj, are both to be cut square through the plank. The face fCjj^ because it is parallel to the rake-line u^ b^^^, is a vertical face, as well as being perpendicular to the surface of the plank. On it, therefore, lines drawn according to the rake, or like the angle ?/^ b^^^ b^^, will be vertical and will give the direction of the mitre-faces. We now have at C the face- mould for the railing over the plan from b io d \n A, The mould thus found is that made upon a cutting plane C, passed through the plank, parallel to its face, but at the middle of its thickness. To put it in position, let the plane C be lifted by its upper edge c^^ and revolved upon the line u^ b^^^ until it stands perpendicular to the plane B. Now revolve both C and B (kept in this relative position during the revolution) upon the line zti b^^ until the plane B stands perpendicular to the plane A. Then every point upon plane C will be verti- cally over its corresponding point in the plane A. For ex- ample, the point b^^^^ will be vertically over b, t^ over t, and so of all other points. To show the application of the face-mould to the plank, make b^^^ b^. equal to half the thick- ness of the plank; parallel to ti^ b^^^ draw b^Cy a line which represents the upper surface of the plank, for the Une it^ b^^, is at the middle of the thickness. Through b^^^^, and parallel with b^^^ 21., draw the line c^ b^^^^ and extend it across the face- mould ; make b^^^^ c^ equal \.ob^c\ through c^, and parallel with b^^^^ tj, draw c^ e. Now, m n o^p is an end view of the plank, showing the face view of the butt-joint at b^^^^. Through r, the centre, draw a line parallel with the sides. Then b^.-^ rep- resents the point b^^^^ ; make b^-^ e^ equal to b^^^^ e ; through r, the centre, draw c, r across the face of the joint ; then e^ r is a vertical line (see Art. 284), parallel and perpendicular to which the four sides of the squared-up wreath are to be drawn as shown. In applying the face-mould to the plank at first, for the purpose of marking b}^ its edges the form of the face-mould, it will be observed that the face-mould is under- stood to have the position indicated by the hne ti^ b^^^, or at FACE-MOULDS FOR CIRCULAR STAIRS. 285 the middle of the thickness of the plank. By this marking the rail-piece is cut square through the plank, and this cut- ting gives the correct form of the wreath, but only at the middle of the thickness of the plank. After it is cut square through the plank, then, to obtain the form at the upper and under surfaces, the face-mould is required to be moved end- wise, but parallel with the auxihary plane B, and so far as to bring the face-mould into a position vertically over or under its true position at the middle of the thickness of the plank. For example, the point ^,,,,, if the mould were placed at the middle of the thickness of the plank, would be at the height of the point h^^^ ; but when upon the top of the plank, the point b^^^^ would have to be at the height of the point c^, therefore the mould must be so moved that the point b^^^^ shall pass from b^ to <: ; consequently b^, c is the distance the mould must be moved, or, as it is technically termed, the sliding distance; hence b^^^^ c^, which is equal to b^c, is the distance the mould is to be moved : up when on top, and down when underneath. This is more ^uUy explained in Art. 284. 278. — Face-Mowlds for Circular Stairs. — At Fig. 160 so much of the horizontal projection of the hand-railing of stairs in Fig. 158 is repeated as extends from the joint b to that at d, but at an enlarged scale. Upon the tangent c k set up the heights as given in Fig. 158 ; for example, make kk^ equal to k^^^k^^ of Fig. 158, and cc^ equal to c^^c^ of Fig. 158. Join c^ and k^ and extend the line to meet ck, extended, in a. Join a and b ; then ab \^ the line of intersection of the cutting and horizontal planes ; it is therefore a horizon- tal line, parallel to which the ordinates are to be drawn. Perpendicular \,o ab draw b^c^^^^. Parallel to <^^ draw cc^^ and kk^i ; join b^ and c^^ ; the angle c c^^ b^ is the plumb-bevil ; perpendicular to b^ c^. draw b^ b^^, k^^ k^^^, and c^^ c^^^ ; make b, b^^ equal to b^ b, and so of the other two points, k^^^ and c.^^, make them respectively equal to their horizontal projections upon the plane A. Join c^.^ and k^/, also, k^^ and b^/, then b^.k.^^ and k^^^Cj^^ are the tangents. From c^^^ draw the line c^^J)^^ parallel to b^c^^ ; this is the slide-line. In this example, this 286 STAIRS. line passes through the point b,/, the sHde-line does not always pass through the ends of the two tangents; it is not required to pass through both, but it is indispensable that it be drawn parallel with the rake- line b^c^^. The lines for the joints at each end are drawn square to the tangent lines. Points in the curves, as many as are desirable, are now to be found by ordinates as shown in the figure, and as before explained for the points in the tangents. The curves are made by drawing a line against the side of a flexible strip bent to coincide with the points. The face-mould may be put in position by revolving the planes C and B, as explained in the last article, for the rail at the newel. The face-mould for the rail over the plan from c to d is to FACE-MOULDS CONTINUED. 287 be obtained in a similar manner, taking the heights from Fig. 158. For example, make d d^ equal to di^d^ of Ftg. 158, and 11^ equal to Z.^, /^^ of Fig. 158 (taking the heights at their actual measurement now). Join d^ and /;, and extend the line to meet the line dl extended in r ; join r and c ; then re is the line of intersection, and parallel to which the ordinates are to be drawn. The points in the face-mould may now be obtained as in the previous cases, giving attention first to the tangent and slide-line ; drawing the lines for the joints perpendicular to the tangents. It may be remarked here that the chord-line be is parallel with the measuring line bjC^^^^y and that the line ^ X' bisects the chord-line; so, also, the line ^/bisects the chord-line cd. This coincidence is not accidental ; it will always occur in a regular circular stairs. Hence in cases of this kind it is not necessary to go through the preliminaries by which to obtain the intersect- ing line ab, but draw it at once parallel to the line ok, bisecting the chord be and passing through the point of intersection of the two tangents. For the distance to slide the mould in its after-application, the lines are given at e^^ and dj^j and their use is explained in the last article, and more fully in Art. 284. 279.— Face-Moulds for Circular Stairs, again. — At Fig. 161 so much of the plan of the hand-railing of the stairs of Fig. 158 is repeated as is required to show the rail from / to^, but drawn at a larger scale. To prepare for the face- moulds, perpendicular to // draw pp^, and make pp^ equal ^^ PujPu o^ ^^^' 15^ (taking this height now at its actual measurement); join p^ and /; then //^ is the tangent of the vertical plane C, and / is a point in the cutting plane at its intersection with the base-plane A. Now since r s, the tan- gent over pg, is horizontal and is in the cutting plane, therefore from / draw fa parallel with r s ov pg\ then fa is the line of intersection of the cutting and horizontal planes, and gives direction to the ordinates. Draw f,p.^, perpendicular to fa ; make Pmp,, equal to pp^ ; join p^^ and /^ ; then the angle pp.^f, is the plumb-bevil ; perpendicular 288 STAIRS. to A,/, draw X/, and /,,/,,,, ; make p^J^^^^ equal to p^^^g, Pn^ equal to /,,,/; join d and /, ; then ^/^ and dp^^^, are the tangents. Make /,, ^ equal to half the thickness of the plank ; draw /, a parallel with //^^ ; make /, a equal to ^<: ; draw aCj parallel with the tangent/,^; through /,,, per- pendicular to /,, d, draw the line for the butt-joint ; then f^^c^ is the distance required to determine the vertical line on the face of the joint at f^^, as shown at A, Through /^^^^, per- pendicular to the tangent p^^,^ d, draw the line for the butt- joint ; make /,,,, <^ equal to ^^; then p^^^^b is the distance required for determining the vertical line on the face of the joint at /,,^^, as shown at B (see Art. 284). The curved lines are obtained by drawing a line against the edges of a flexi- ble rod bent to as many points as desirable, obtained by measuring the ordinates of the plan at A and transferring them to the face-mould by the corresponding ordinates, as before explained. RAILING FOR QUARTER-CIRCLE STAIRS. 289 280. — Hand-Railings for ^Vinding^ Stair§. — The term winding is applied more particularly to a stairs having steps of parallel width compounded with those which taper in w^idth, as in Fig. 135, and as is here shown in Fig. 162, in which f abc represents the central line of the rail around the cylinder, and the quadrant dc, distant from the first quadrant 20 inches, is the tread-line, upon which from d, a point taken at pleasure, the treads are run off. Through c, perpendicu- lar to af, draw ae (the occurrence here of one of the points of division on the tread-line perpendicularly opposite a, the spring of the circle, is only an accidental coincidence) ; make a a^ equal to two risers ; join a^ and /. With the diameter <^r, on ^ as a centre, describe the arc at g, crossing ac extended; through /; draw gb^\ then ab^ is the stretch- out, or development of the quadrant a b. Through Ji draw // /, tending toward the centre of the 290 STAIRS. cylinder; make b^ i^ equal to bi\ perpendicular to fh^ draw b^ b^^ and i^ i^^. As there are four risers from e to Ji, make a^ a^^ equal to four risers, and draw a^, i^^ parallel with fa ; through / ^ draw a^ b^^ ; by intersecting lines, or in any con- venient manner, ease off to any extent the angle fa^ i^^. Through j\ a point in this curve (chosen so as to be perpen- dicularly over in, a point between a and /, nearer to a), draw k ly a tangent to the curve. Perpendicularly to this tangent, through j\ draw the line for a butt-joint ; also through bj^, and perpendicularly to a^ b^^, draw the line for the joint at the centre of the half circle. On the line aa^^ set up points of division for the riser heights, and through these points of division draw horizontal lines to the line b,Jf. From these points of contact drop perpendiculars to the line fa b^, and transfer such of them as require it to the circle ai, by drawing lines tending to^. Through these points of intersection with the central line of the rail, and through the points of division on the tread-line, draw the riser-lines uie, an, etc. At half a riser above the floor-line, on top of the upper riser draw a horizontal line, and ease off the angle as shown ; the intersection of the floor-line "with this curve gives the position of the top riser at the centre of the rail. This completes the plan of the steps and the elevation of the rail — requisite preliminaries for the face-moulds. The gradu- ation of the treads from flyers to winders obviates an abrupt angle at their junction in the rail and front-string. The objection to the graduation, that it interferes with the regularity of stepping at the tread-line, is not realized in practice. 281. — Face-M®Mlds for \¥indiiig Stairs. — At Fig. 163 so much of the plan at Fig. 162 is repeated as is required for the face-moulds, but for perspicuity at twice the size. The hori- zontal projection of the tangents for the first wreath are ad and d b drawn at right angles to each other, tangent to the circle at a and b. Let those tangents be extended beyond d\ through ;//, the lower end of the wreath, draw nt d^, mak- ing an angle with nid equal to that in Fig. 162, between the FACE-MOULDS FOR THE TWISTS. 291 line af and aj\ or let the angle dmd^ equal afa^ of Fig. 162. Make dd,^ equal to dd^. Make bh^^ equal to ^,,, <^,, of Fig. 162 ; join ^^^ and b^^ and extend the line to e^^ ; make b^, equal to ^,,<^,,,, of Fig 162, and draw <^v^^, parallel with de. From r^ draw e^^e parallel with b^^b\ through e and / draw ef tangent to the circle at /; then be and ef are the horizontal projections of the tangents for the upper wreath. Then if the plane B be revolved on ad, the plane C on de, 292 STAIRS. and the plane D on cf until they each stand vertical to the plane A, the lines ind^, d^^e^^, and e^.J^ will constitute the tangents of the two wreaths in position. This arrangement locates the upper joint of the vipper wreath at /, leaving /<:, a part of the circle, to be worked as a part of the long level rail on the landing. As the tangent over e f is level, the raking part of the rail will all be included in the wreath bf, so that at the joint / the rail terminates on the level. The portion fc, therefore, is a level rail requiring no canting, and it requires no other face-mould than that afforded by the plan from / to c. For the face-mould for the rail over vi a b, let the line c^^ d^^ be extended to m^, a point in the base-line b m^ ; then in^ is a point in the base-plane A, as well as in the cutting plane E\ therefore the line m^ 111 is the intersecting line parallel to which all the ordinates on plane A are to be drawn. Per- pendicular to this intersecting line in^. in, at any convenient place draw in, b^ ; make b^ b^^^ parallel to ;//,, in and equal to bbjj] connect b^^, with m^, a point at the intersection of the lines m^ in and b^ in^ ; then the angle b b^^^ in^ is the plumb- bevil. Through d, parallel to m^ in, dmw d d^^/, from the three points in^, d^^^, and Z'^.^ draw lines perpendicular to in. b^/j ; make in^ in^^ equal to in^ in ; make b^^^ b^^^^ equal to b^ b. Since the measuring base-line in^ b^ passes through d, the point of the angle formed by the two tangents, d^^^ is the point of this angle in the cutting plane E\ therefore join in^^ and d^^^, also d^^^ and b^^^^ ; then b^^^^ d^^^ and d^^^ m^, are the two tangents at right angles to which the joints at in^^ and b^i^i are drawn. The curves of the face-mould are now found as usual, by transferring the distances by ordinates, as shown, from the plane A to the plane E, making the distance from the rake-line w b^^^ to each point in plane E equal to the dis- tance from the corresponding point in the plane A to the measuring base-line in^ b^. Now, to obtain the sliding distance and the vertical line upon the butt-joints, make b^^^ b,. equal to half the thickness of the plank; parallel with in^ b^^^ draw b^ /'vi ; ^Iso, b^^^^ ^vii and in^^ m^^^ ; make b^^^^ <^vu and in^^ m^^^ each equal to b^ b^-^\ through /;,.h and ni^^^, and parallel to the respective tangents, draw /Vu ^x and m^.^ ^^^,^1'^ t^"'^" ^^ and FACE-MOULDS FOR WIND^iNG STAIRS. 293 m^^^t are the points from which, through the centre of the butt-joints, a line is to be drawn which will be vertical when the wreath is in position. (See Art. 284.) For the face-mould for the upper quarter, through b, Fig. 163, draw b c^ parallel with d,^ c^^ ; make e c^^^ equal to e e^\ draw ^//^// parallel with e f. Now, since c^^^ f^ is a horizon- tal line and is in the cutting plane F, therefore, parallel with e^j^ f^ and through b^, draw b n ; then b n is the required in- tersecting line. Extend c f to/^^ ; make//^^ equal to f f^ ; join/^^ and n ; then the angle //^^ ;/ is the plumb-bevil. Per- pendicular to 11 f,^ draw/^^/^/ and n n^, and make these lines respectively equal to c f and b n ; join f^^ and f^^^ ; also f^^^ and n^ ; then f^^ f^^^ and f^^^ n^ are the required tangents. The butt-joints at/^ and n^ are drawn perpendicular to their respective tangents. To get the slide distance and vertical lines on the butt-joints, make/^^/^ equal to half the thickness of the plank ; parallel with n /^^, through /,, draw/,./^^^^ ; also, through n^ draw n^ n^^ ; make n^ n.^ equal to f^f^jj/, through ;^,,, parallel with ;^^/^^^, draw ;/^^ n^^.] then ;/^^ , is the point through which a line is to be drawn to the centre of the butt-joint, and this line will be in the vertical plane contain- ing the tangent. So, also, parallel with the tangent f^^ /^^^, and through /^^^^, draw /^^^^ /,.i ; then/vj is the point through which a line is to be drawn to the centre of the butt-joint (see Art. 284). The curve is now to be obtained by the ordinates, as before explained. 282. — Faec-lIould§ for Windaiag Slair§, ag^asoi. — In the last article, in getting the face-moulds for a w^inding stairs, the two wu-eaths are found to be very dissimilar in length. This dissimilarity may be obviated by a judicious location of the butt-joint connecting the two wreaths, as shown in Fig. 164. Instead of locating the joint precisely at the middle of the half circle, as was done in Ftg. 163, place it farther down,' say at n, which is at it in Fig. 162, two risers down from the top, or at any other point at will. Then through 71 in the plan draw in^ s tangent to the circle at n ; and perpendicu- lar to this tano^ent draw n it,,, and d d,, ; make 11 n , equal to n^ It of Fig. 162 ; from d erect d d^ perpendicular to in d; 294 STAIRS. make the angle d m d^ equal to that of h^^^ j I of Fig. 162. Make d d^^ equal to d d, ; join d^^ and 71^^ and extend the line to m^^ a point of intersection with the base-line n n^ ; then 71^ is a point in the base-plane, as also in the cutting plane ; therefore 7?tj 7Ji is the intersecting line parallel to which all the ordinates of the plan are to be drawn, and perpendicular to which 771^^ n^, the measuring base-line, is drawn. Make Ttj 71^^^^ equal to 71 n^^ ; connect w^^ and n.^^^, and then transfer CARE REQUIRED IX DRAWIxXG. 295 by the ordinates to the cutting" plane ;;/ d and ;/ the three points of the plan at the ends of the tangents, as before de- scribed, as also such points in the curve as may be required to mark the curve upon the face-mould, all as shown in previ- ous examples. For the face-mould of the upper wreath, make n^i n^ij equal to nn^^ of Fig. 162. From n^^^ draw n^^^ s^^ par- allel with in^ s ; extend the line d^. /r^ to intersect n^^^ 5^ in s^^ ; parallel with n^^^ n draw s.^ s ; from s draw s r tangent to the circle at r is n equals s r) ; through 7', tending to the centre of the cylinder, draw the butt-joint ; then r s and s n are the horizontal projections of the tangents for the upper wreath- piece, the tangent s r being level and, consequently, parallel to the intersecting line drawn through ;/. Perpendicular to r s draw r^p ; parallel with n^^ s^^ draw 71 s^ ; make r^ r ^ equal to s s,; join r^^ and /. From this line and the measuring base- line r^ p^ the points for the tangents are first to be obtained and then the points in the curve, all as before described. The part of the circle from r to . But the more important test is in the length of the chord-line joining the ends of the tw^o tangents ; as, for ex- ample, the chord w^^ b^^^^ of Fig. 163, the horizontal projec- tion of which is the chord 7n b in plane A. Perpendicular to m b draw b g\ make b g equal to b b^^, and join g and m ; then m^j ^////» the chord of the face-mould, should be equal to m g. After fully testing the accuracy of the drawing for the face- mould, choose a well-seasoned thin piece of white-wood, or any other wood not Hable to split, and plane it to an even thickness throughout ; mark upon it the curves, joints, tan- gents, and slide-hne, and cut the edges true to the curve- THE FACE-MOULD APPLIED TO PLANK. 297 lines and joints square through the board ; then square over such marks as are required to draw each tangent and the sUde-hne also upon the reverse side of the board. This completes the face-mould. 284. — Application of tlie Face-Mould. — In order that a more comprehensive idea of the lines given for applying a face-mould may be had, let A, Fig. 165, represent one end of a wreath-piece as it appears when first cut from a plank, and when held up in the position it is to occupy at completion over the stairs. Also, let B represent the corresponding face-mould, laid upon the wreath-piece A in the position which it should have after sliding. And, for the purpose of a clearer illustration, let it be supposed that the two pieces, A and B, are transparent. Then let a^ a b d Cj e, represent a solid of wedge form, having a triangular level base, a b d, upon the three lines of which stand these three vertical planes, namely: on the line a b the plane a^ a bc^, upon the line a d the plane a^ a d e^, and on the line d b the plane b d c^ c^ ; the top of the solid is an inclined plane, a^ c^ e^, and the vertical line a^ a is the edge of the wedge. Now, it will be observed that the point a in the base of the solid is identical with a, the centre of the butt-joint, and the point a^ (at the intersection of two vertical planes and the inclined plane of the solid) is vertically over a, and is identical with a^, a poini in the upper surface of the plank. Also, the inclined plane c^ c^ a^, which forms the top of the solid, coincides with the upper surface of the plank A, from Avhich the Avreath-piece has been squared ; and the line c^ a^ (at the angle formed by the inclined plane e^ c^ a^ and the vertical plane a^ab c^i coin- cides with / g, the slide-line drawn upon the top of the plank ; also, the line e^ a^ (at the angle formed by the in- clined plane c^ c, a^ and the vertical plane a^ a d e), coincides with a^ k, the tangent line upon the underside of the face- mould after it has been slid to its new position, vertically over its true position at the middle of the thickness of the plank. From a the line a c is. drawn parallel with a^ c^ ; so, also, the line ^ ^ is drawn parallel with a^ c^ ; consequently the line ^ <; is parallel with e^ c^ ; and the plane e c a\^ parallel 298 STAIRS. with the plane c^ c^ a,, and coincides with a plane passing- through the middle of the thickness of the plank, and, conse- quently, is the cutting plane referred to in previous articles, upon which the lines are drawn which give shape to the Fig. i6- face-mould. When the face-mould is first laid upon the plank, the line i^j\ coincides with i^^ j\^, and when in that position, its form marked upon the plank is the form by which the plank is sawed square through ; but this gives the form of THE SLIDING OF THE FACE-MOULD. 299 the wreath, not as it is at the surface of the plank, but as it is at the middle of the thickness of the plank, or upon the plane ac c\ so that, for example, the line inj\, represents the line ij drawn through a^ the centre of the butt-joint ; and when the mould B is slid to the position shown in the figure, the line ij\ comes into a position vertically over ij\ hence the three lines i^ i, a^ a, and j j' are each vertical and in a vertical plane, i i^jj- .By these considerations it will be seen that the face-mould By located as shown in the figure, is in its true position for the second marking, by which the addi- tional cutting is now to be performed vertically. This being established, it will now be shown how to get upon the butt- joint a line in the vertical plane containing the tangent. If the top and bottom lines of the vertical plane a^a b c^ be ex- tended, they will meet in the point /, and will extend the plane into a triangle lb c^, cutting the upper edge of the butt-joint in/, the end of the tangent, and the point in which the point a^ of the underside of the face-mould was located when the mould was first applied to the plank. The \\\\^ fa on the butt-joint is perpendicular to i j ox i^^ j,i> Again, if the top and bottom lines of the plane a^ a d c^ be extended, they will meet in /, and w411 extend the plane into the tri- angle p d e^, cutting the edge of the butt-joint in h, a point from which, if a line be drawn upon the butt-joint to a, its centre, this line will be in the vertical plane / d c^, which plane contains the tangent perpendicular to which the butt- joint is drawn ; consequently lines upon the butt-joint par- allel to Ji a will each be in a vertical plane parallel to the vertical tangent plane, and lines drawn upon the butt-joint perpendicular to these lines will be horizontal lines ; hence the line Jl a is the required line by which to square the wreath at the butt-joint. Now, it will be observed that the triangle a f a^ is like that given in the various figures for ob- taining face-moulds, to regulate the sliding of the face-mould and the squaring at the butt-joint. For example, in Fig. 163, the right-angled triangle b^^^ b^ b^^ is the one referred to. This triangle is in a vertical plane pgirallel to one containing the slide-line ; its longer side is a vertical line ; one of the sides containing the right angle is equal to half the thickness 300 STAIRS. of the plank, while the other, drawn parallel to the face of the plank, is the distance the face-mould is required to slide. Precisely like this, the triangle a f a^ of Fig. 165 is in the vertical plane / b c^, containing f g, the slide-line ; its longer side, a J a, is a vertical line ; fa, one of the sides containing the right angle, is equal to half the thickness of the plank, while the other side, drawn coincident with the surface of Fig. 166. the plank, is the distance to slide the face-mould. Therefore the triangle a^f a of Fig. 165 gives the required lines by which to regulate the application of the face-moulds. The relative position of the more important of these lines is geo- metrically shown in Fig. 166, in which A and B are upon the horizontal plane of the paper, C is in a vertical plane stand- ing on the ground-line b d, and i? is a plan of the butt-joint, revolved upon the line i^^ j\^ into the horizontal plane, and BLOCKING-OUT OF THE RAIL. 30I then perpendicularly removed to the distance//^. The let- tering- corresponds with that in Fig. 165. The shaded part of D shows the end of the squared wreath. When the blocked piece has been marked by the face-mould in its second application, its edges are to be trimmed vertically as shown in Fig. 167, after which the top and bottom surfaces of the wreath are to be formed from the shape marked 6n the butt-joints. Fig. 167. 285. — Face-Mould Curves are Elliptical. — The curves of the face-mould for the hand-railing of any stairs of circular plan are elliptical, and may be drawn by a trammel, or in any other convenient manner. The trouble, however, attending the process of obtaining the axes, so as to be able to employ the trammel in describing the curves, is, in many cases, greater than it would be to obtain the curves through points found by ordinates, in the usual manner. But as 302 STAIRS. this method for certain reasons may be preferred by some, an example is here given in which the curves are drawn by a trammel, and in Avhich the method of obtaining the axes is shown. Let Fig. i68 represent the plan of a hand-rail around part Fig. i68. of a cylinder and with the heights set up, the intersection line obtained, the measuring base-Hne drawn, the rake-line estabhshed, and the tangents on the face-mould located — all in the usual manner as hereinbefore described. Then, to prepare for the trammel, from o, the centre of the cylinder, draw o b^ parallel with the intersecting line, and b^ o^ perpen- FACE-MOULDS FOR ROUND RAILS. 303 diciilar to l\f^, the rake-line ; make b^o^ equal to bo, and o^a^ equal \.o oa\ through 0^ draw o^Ji parallel with b^f^. From draw oc perpendicular to ob/, continue the central circular line of the rail around to e\ parallel with ob^ draw c f, and from /^, the point of intersection of e f with b^f^, and perpendicular to bj^, draw / r^ ; make f^L\ equal to fc\ then o^ is the centre of the ellipse, and o^ a^ the semi-conju- gate diameter and 0^ e^ the semi-transverse diameter of an ellipse drawn through the centre of the face-mould. To get the diameters for the edges of the face-mould, make a^ c^ and a^d^ each equal to half the widtii of the rail, as at cad\ par- allel to a line drawn from a^ to e^^ and through c^, draw the line c^g\ also, parallel with a line drawn from a^ to e^ draw dji (see Art. 559); then for the curve at the inner edge of the face-mould, o^g is the semi-transverse diameter, and o^c^ the semi-conjugate ; while for the curve at the outer edge 0^ h is the semi-transverse diameter, and o^ d^ the semi-conju- gate. So much of the edges of the face-mould as are straight are parallel with the tangent. Now, placing the trammel at the centre, as shown in the figure, and making the distance on the rod from the pencil to the first pin equal to the semi-conjugate diameter, and the distance to the second pin equal to the semi-transverse diameter, each curve may be drawn as shown. (See Art. 549.) 286.— Face-Moiiltls for Roaiiid Rails. — The previous ex- amples given for finding face-moulds are intended for moiddcd rails. For round rails the same process is to be followed, with this difference : that instead of finding curves on the face-mould for the sides of the rail, find one for a centre-line and describe circles upon it, as at Fig. 145 ; then trace the sides of the mould by the points so found. The thickness of stuff for the twists of a round rail is the same as for the straight part. The tw^ists are to be sawed square through. 287. — Posiiion of llie Butt-Joint. — When a block for the wreath of a hand-rail is sawed square through the plank, the joint, in all cases, is to be laid on the face-mould square to the tangent and cut square through the plank. 304 STAIRS, Managed in this way, the butt-joint is in a plane pierced perpendicularly by the tangent. But if the block be not sawed square through, but vertically from the edges of the Fig. 169, face-mould, then, especially, care is required in locating the joint. The method of sawing square through is attended with so many advantages that it is now generally followed ; yet, as it is possible that for certain reasons some may prefer, POSITION OF THE BUTT-JOINT. 305 in some cases, to saw vertically, it is proper that the method of finding the position of the joint for that purpose should be given. Therefore, let A, Fig. 169, be the plan of the rail, and B the elevation, showing its side; in which kz is the direction of the butt-joint. From /' draw kb parallel to lo, and ke at right angles to kb\ from h draw b f, tending to the centre of the plan, and from / draw f e parallel to bk\ from /, through r, draw I i, and from i draw id parallel to e f\ join ^and b, and db will be the proper direction for the joint on the plan. The direction of the joint on the other side, ac^ can be found by transferring the distances xb and d to xa and oc. Then the allowance for over wood to cover the butt-joint is shown as that which is included be- tween the lines ox and db. The face-mould must be so drawn as to cover the plan to the line b d for the wreath at the left, and to the line ac for that at the right. By some the direction of the joint is made to radiate toward the centre of the cylinder ; indeed, even Mr. Nicholson, in his Carpenter s Guide, so advised. That this is an error may be shown as follows : In Fig. 170, a rji is the plan of a part of the rail about the joint, s u is the stretch-out oi a /, and gp is the helinet, or vertical projection of the plan arji. This is found by drawing a horizontal line from the height set upon each perpendicular standing upon the stretch-out line S7i. The lines upon the plan arji are drawn radiating to the centre of the cylinder, and therefore correspond to the horizontal lines of the helinet drawn upon its upper and under surfaces. Bisect r^ on the ordinate drawn from the centre of the plan, and through the middle draw cb at right angles to gv ; from b and c draw cd and ^^ at risfht ans^les to j // ; from d and e draw lines radiating toward the centre of the plan ; then do and em will be the direction of the joint on the plan, according to Nicholson, and eb its direction on the falling-mould. It must be admitted that all the lines on the upper or the lower side of the rail w^hich radiate toward the centre of the cylinder, as do^ em, or if., are level; for instance, the level line w v, on the top of the rail in the helinet, is a true representation of the radiating line 72 on 3o6 STAIRS. the plan. The line bh, therefore, on the top of the rail in the helinet, is a true representation of cm on the plan, and kc on the bottom of the rail truly represents ^c. From k draw /^/parallel to cb, and from h draw Jif parallel \.obc\ Fig. 170. join / and b, also c and /; then cklb will be a true repre- sentation of the end of the lower piece, B, and cfhb of the end of the upper piece, A ; and fk or JlI will show how much the joint is open on the inner, or concave, side of the rail. CORRECT LINES FOR BUTT-JOINT. 307 To show that the process followed in Art. 287 is correct, let do and em {Fig. 171) be the direction of the butt-joint found as at Fig. i6g. Now, to project, on the top of the rail in the helinet, a line that does not radiate toward the centre of the cylinder, as jk, draw vertical lines from j and /c to w and /i, and join zu and /i ; then it will be evident that 7v /i is a true representation in the helinet of j7^ on the plan, it being in the same plane as j'k, and also in the same winding- surface as za2'. The line /;/, also, is a true representation on 3o8 STAIRS. the bottom of the helinet of the line j k in the plan. The line of the joint c in, therefore, is projected in the same way, and truly", by ib on the top of the helinet, and the line do by c a oxi the bottom. Join a and /, and then it will be seen that the lines c a,a i, and ib exactly coincide with c b, the line of the joint on the convex side of the rail ; thus proving the lower end of the upper piece. A, and the upper end of the lower piece, B, to be in one and the same plane, and that the direction of the joint on the plan is the true one. By refer- ence to Fig. 169 it will be seen that the line li corresponds to xi in Fig. 171 ; and that c k in that figure is a representa- tion oi fby and ik of db. 288. — Scroll§ for IIanfl-Rail§ : Ocneral Rule for Size and Position of the Regulatings Square. — The breadth which the scroll is to occupy, the number of its revolutions, and the relative size of the regulating square to the eye of the scroll being given, multiply the number of revolutions by 4, and to the product add the number of times a side of the square is contained in the diameter of the eye, and the sum will be the number of equal parts into which the breadth is to be divided. Make a side of the regulating square equal to one of these parts. To the breadth of the scroll add one of the parts thus found, and half the sum will be the length of the longest ordinate. 4 Fig. 172. 289. — Centres In Regulating Square. — Let a 2 \ b (Fig. 172) be the size of a regulating square, found according to the previous rule, the required number of revolutions being SCROLLS AT NEWEL. 309 I J. Divide two adjacent sides, as a 2 and 2 i, into as many equal parts as there are quarters in the number of revolu- tions, as seven ; from those points of division draw lines across the square at right angles to the lines divided ; then I being the first centre, 2, 3,4, 5, 6, and 7 are the centres for the other quarters, and 8 is the centre for the eye ; the heavy lines that determine these centres being each one part less in length than its preceding line. Fig. 173, 290, — Scroll for Hand-Rail Over Curtail Step. — Let a b {Fig. 173) be the given breadth, if the given number of revolutions, and let the relative size of the regulating square to the eye be \ of the diameter of the eye. Then, by the rule, if multipled by 4 gives 7, and 3, the number of times a side of the square is contained in the eye, being added, the sum is 10. Divide a b, therefore, into 10 equal parts, and set one from b \.o c \ bisect a c \n e-, then a c will be the length of the longest ordinate (i ^^ or i r). From a draw a d, from e draw e i, and from b draw b f, all at right angles to a b ; make e i equal to a, and through i draw i d parallel 3IO STAIRS. to a b\ stt b c from i to 2, and upon i 2 complete the regu- lating square; divide this square as at Fig. 172; then de- scribe the arcs that compose the scroll, as follows : upon i describe d e, upon 2 describe e f, upon 3 describe / g, upon 4 describe ^/^, etc.; make dl equal to the width of the rail, and upon i describe / ;^/, upon 2 describe mn, etc.; de- scribe the eye upon 8, and the scroll is completed. 291.— Scroll for Curtail Step.— Bisect d I {Fig. 173) in 0, and make v equal to ^ of the diameter of a baluster ; make V w equal to the projection of the nosing, and c x equal to zvl\ upon I describe w J, and upon 2 describe 7 -c ; also, upon 2 describe x i, upon 3 describe ij\ and so around to z ; and the scroll for the step will be completed. 292. — Po§iU©n of Balusters Under Scroll. — Bisect d I {Fig. 173) in 0, and upon i, with i for radius, describe the circle r ii \ set the baluster at p fair with the face of the second riser, <:", and from/, with half the tread in the divi- ders, space off as at 0, q, r, s, t, zCy etc., as far as ^- ; upon 2, 3,4, and 5 describe the centre-line of the rail around to the eye of the scroll ; from the points of division in the circle r 71 draw lines to the centre-line of the rail, tending to 8, the centre of the eye ; then the intersection of these radiating lines Avith the centre-line of the rail will determine the posi- tion of the balusters, as shown in the figure. 293. — Falling-Mould for Raking Part of Scroll. — Tangi- cal to the rail at h {Fig. 173) draw h k parallel to ^^^ ; then k a"- will be the joint between the twist and the other part of the scroll. Make d e'^ equal to the stretch-out of d c, and upon d c" find the position of the point /', as at k" \ at Fig. 174, make ^<^ equal to the angle at which the work is splayed, and b d, on the upper edge of the board, is at right angles to ^ ^ ; make the angle f g j equal \.o ab c, and from / draw f Ji parallel to e a ; from b draw ^ ^ at right angles X.O a b\ through draw ie parallel to c b^ and join c and d\ then the angle a e d will be the proper bevil for the ends from the inside, or k d e from the outside. If a mitre-joint is required, set f g, the thickness of the stuff on the level, from e to ;//, and join ;;/ and d\ then kdni will be the proper bevil for a mitre-joint. If the upper edge of the splayed work is to be bevelled, so as to be horizontal when the work is placed in its proper position, then f g j\ the same as ^ <^ r, will be the proper bevel for that purpose. Suppose, therefore, that a piece in- dicated by the lines k g, g f, and f h were taken off; then a line drawn upon the bevelled surface from d at right angles to k d would show the true position of the joint, because it would be in the direction of the board for the other side ; but a line so drawn would pass through the point o, thus proving the principle correct. So, if a line were drawn upon the bevelled surface from d at an angle of 45 degrees to k d, it would pass through the point n. / Arf-LJi/. /^:!/iru. i}c/ VIEW IN THE ALHAMBRA. SECTION IV.— DOORS AND WINDOWS. DOORS. 298- — General Requiremenls. — Among the architectural arrangements of an edifice, the door is by no means the least in importance ; and if properly constructed, it is not only an article of use, but also of ornament, adding materially to the regularity and elegance of the apartments. The dimen- sions and style of finish of a door should be in accordance with the size and style of the building, or the apartment for which it is designed. As regards the utility of doors, the principal door to a public building should be of suffi- cient width to admit of a free passage for a crowd of people ; while that of a private apartment will be wide enough if it permit one person to pass without being incommoded. Ex- perience has determined that the least width allowable for this is 2 feet 8 inches ; although doors leading to inferior and unimportant rooms may, if circumstances require it, be as narrow as 2 feet 6 inches ; and doors for closets, where an entrance is seldom required, may be but 2 feet wide. The width of the principal door to a public building may be from 6 to 12 feet, according to the size of the building ; and the width of doors for a dwelling may be from 2 feet 8 inches to 3 feet 6 inches. If the importance of an apart- ment in a dwelling be such as to require a door of greater width than 3 feet 6 inches, the opening should be closed with two doors, or a door in two folds ; generally, in such cases, where the opening is from 5 to 8 feet, folding or slid- ing doors are adopted. As to the height of a door, it should in no case be less than about 6 feet 3 inches ; and generally not less than 6 feet 8 inches. 299. — The Proportion between Width and Heig^ht: of single doors, for a dwelling, should be as 2 is to 5 ; and, for entrance-doors to public buildings, as i is to 2. If the width is given and the height required of a door for a i6 DOORS AND WINDOWS. dwelling, multiply the width by 5, and divide the product by 2 ; but if the height is given and the width required, divide by 5 and multiply by 2. Where two or more doors of different widths show in the same room, it is well to pro- portion the dimensions of the more important by the above rule, and make the narrower doors of the same height as the wider ones ; as all the doors in a suit of apartments, except the folding or sliding doors, have the best appear- ance when of one height. The proportions for folding or sliding doors should be such that the width may be equal to f of the height ; yet this rule needs some qualification ; for if the width of the opening be greater than one half the width of the room, there will not be a sufficient space left Fig. 179. for opening the doors ; also, the height should be about one tenth greater than that of the adjacent single doors. 300. — Panels. — Where doors have but two panels in width, let the stiles and muntins be each J- of the width ; or, whatever number of panels there may be, let the united widths of the stiles and the muntins, or the whole Avidth of the solid, be equal to | of the width of the door. Thus : in a door 35 inches wide, containing two panels in wndth, the stiles should be 5 inches wide ; and in a door 3 feet 6 inches wide, the stiles should be 6 inches. If a door 3 feet 6 inches wide is to have 3 panels in width, the stiles and muntins should be each 4^ inches wide, each panel being 8 inches. The bottom rail and the lock-rail ought to be each equal in width to j^ of the height of the door ; and the top TRIMMINGS FOR DOORS. 31/ rail, and all others, of the same width as the stiles. The moulding on the panel should be equal in width to ^ of the width of the stile. 301. — Trimiiiing^§. — Fig. 179 shows a method of trimming doors : a is the door-stud ; b, the lath and plaster ; r, the ground; d, the jamb ; e, the stop ; f and g, architrave casings ; and //, the door-stile. It is customary in ordinar}^ work to form the stop for the door by rebating the jamb. But when the door is thick and heavy, a better plan is to nail on a piece as at e in the figure. This piece can be fitted to the door and put on after the door is hung ; so, should the door be a trifle winding, this will correct the evil, and the door be made to shut solid. Fig. 180 is an elevation of a door and trimmings suitable for the best rooms of a dwelling. (For trimmings generally, see Sect. V.) The number of panels into which a door should be divided may be fixed at pleasure ; yet the present style of finishing requires that the number be as small as a proper regard for strength will admit. In some of our best dwellings, doors have been made having only two upright panels. A few years' experience, however, has proved that the omission of the lock-rail is at the expense of the strength and durability of the door ; a four-panel door, therefore, is the best that can be made. 302. — IIangin§^ I>oor$. — Doors should all be hung so as to open into the principal rooms ; and, in general, no door should be hung to open into the hall, or passage. As to the proper edge of the door on which to affix the hinges, no general rule can be assigned. WINDOWS. 303. — Requirements for Lig^lit. — A window should be of such dimensions, and in such a position, as to admit a sufficiency of light to that part of the apartment for which it is designed. No definite rule for the size can well be given that will answer in all cases ; yet, as an approxima- DOORS AND WINDOWS. tion, the following has been used for general purposes. Multiply together the length and the breadth in feet of the apartment to be lighted; and the product by the height in feet; then the square root of this product will show the required number of square feet of glass. 304. — Window-Frames. — For the size of window-frames, add 4^ inches to the width of the glass for their width, and WIDTH OF INSIDE SHUTTERS. 319 6J inches to the height of the glass for their height. These give the dimensions, in the clear, of ordinary frames for 12- light windows ; the height being taken at the inside edge of the sill. In a brick wall, the width of the opening is 8 inches more than the width of the glass — 4^ for the stiles of the sash, and 3^ for hanging stiles — and the height between the stone sill and lintel is about loj- inches more than the height of the glass, it being varied according to the thick- ness of the sill of the frame. 305. — Iii§i<]c SBawtters. — Inside shutters folding into doxes require to have the box-shutter about one inch wider than the flap, in order that the flap may not interfere when both are folded into the box. The usual margin shown be- tween the face of the shutter when folded into the box and the quirk of the stop-bead, or edge of the casing, is half an inch ; and, in the usual method of letting the whole of the thickness of the butt hinge into the edge of the box-shutter, it is necessary to make allowance for the throw of the hinge. This may, in general, be estimated at \ of an inch at each hinging ; which being added to the margin, the entire width of the shutters will be \\ inches more than the width of the frame in the clear. Then, to ascertain the width of the box- shutter, add i^ inches to the width of the frame in the clear, between the pulley-stiles ; divide this product by 4, and add half an inch to the quotient, and the last product will be the required width. For example, suppose the window to have 3 lights in Avidth, 11 inches each. Then, 3 times 11 is 33, and 4J added for the wood of the sash gives 37^ ; 37^ and li is 39, and 39 divided by 4 gives 9J ; to which add half an inch, and the result will be loj inches, the width required for the box-shutter. 306. — Proportion: "Widtli and 9Icight. — In disposing and locating windows in the walls of a building, the rules of architectural taste require that they be of different heights in different stories, but generally of the same width. The windows of the upper stories should all range perpendicu- larly over those of the first, or principal, story ; and they 320 DOORS AND WINDOWS. should be disposed so as to exhibit a balance of parts throughout the front of the building. To aid in this it is always proper to place the front door in the middle of the front of the building ; and, where the size of the house will admit of it, this plan should be adopted. (See the latter part of y^r/. 50.) The proportion that the height should bear to the Avidth may be, in accordance with general usage, as follows : he height of basement wind ows , li of the width. principal-Story '' -8 il il u second-story il Ij il li u third-story il It " u a fourth-story 11 I* il (t ii attic-story 11 the same as the width. But, in determining the height of the windows for the several stories, it is necessary to take into consideration the height of the story in wdiich the window is to be placed. For, in addition to the height Irom the floor, which is gen- erally required to be from 28 to 30 inches, room is wanted above the head of the window for the window-trimming and the cornice of the room, besides some respectable space which there ought to be between these. 307. — Circsaflar Meads. — Doors and windows usually ter- minate in a horizontal line at top. These require no special directions for their trimmings. But circular-headed doors and windows are more difficult of execution, and require some attention. If the jambs of a door or window be placed at right angles to the face of the wall, the edges of the so^'it, or surface of the head, would be straight, and its length be found by getting the stretch-out of the circle {Art. 524); but when the jambs are placed obliquely to the face of the wall, occasioned by the demand for light in an oblique direction, the form of the soffit Avill be obtained by the fol- lowing article ; as also when the face of the wall is circular, as shown in the succeeding figure. OBLIQUE SOFFITS OF WINDOWS. 32 308.— Form of Soffit for Circular Window-Heads. When the light is received in an oblique direction, let ad cd {Fig. 181) be the ground-plan of a given window, and efa a vertical section taken at right angles to the face of the jambs. Fig. 181. From a, through e, draw ag at right angles to ^^; obtain the sti-etch-out of efa, and make ^^ equal to it; divide eg and efa each into a like number of equal parts, and drop perpendiculars from the points of division in each ; from the points of intersection, i, 2, 3, etc., in the line ad, f ' e / ^ 7 "T^ — , — — — 1 \^ ■"^z "?^^ 1 ■ ■ ■"-"^U "^ 1 ' ^ 2 ^ / 2 ' \ 1 — 1 r ^ ^=^- - -1 - — -■rzz^V^^ ^"^ Fig. 182. draw horizontal lines to meet corresponding perpendicu- lars from eg\ then those points of intersection will give the curve line dg, which will be the one required for the edge of the sofBt. The other edge, ch, is found in the same manner. 322 DOORS AND WINDOWS. For the form of the sofht for circular window-heads, when the face of the wall is curved, let ah c d (Fig. 182) be the ground-plan of a given window, and e f a 7\. vertical sec- tion of the head taken at right angles to the face of the jambs. Proceed as in the foregoing article to obtain the line dg\ then that will be the curve required for the edge of the soffit, the other edge being found in the same manner. If the given vertical section be taken in a line with the face of the wall, instead of at right angles to the face of the jambs, place it upon the line cb {Fig. 181), and, having drawn ordinates at right angles to cb, transfer them to efa ; in this way a section at right angles to the jambs can be obtained. SECTION v.— MOULDINGS AND CORNICES. MOULDINGS. 309. — ifloiildings : are so called because they are of the same determinate shape throughout their length, as though the whole had been cast in the same mould or form. The regular mouldings, as found in remains of classic architec- ture, are eight in number, and are known by the following names : Fig. 183. Annulet, band, cincture, fillet, listel or square. Fig. 184. Astragal or bead. ) Torus or tore. Fig. 185. Pj^ jg^ Scotia, trochilus or mouth. Fig. 1S7. Ovolo, quarter-round or echinus. Cavetto, cove or hollow. Fig. 188. 324 MOULDINGS AND CORNICES. Fig. 189. Cymatium, or cyma-recta. Ogee. I Inverted cymatium, or cyma-reversa. Fig. 190. Some of the terms are derived thus : Fillet, from the French word fil, thread. Astragal, from astragalos, a bone of the heel — or the curvature of the heel. Bead, because this moulding, when properly carved, resembles a string of beads. Torus, or tore, the Greek for rope, which it resembles when on the base of a column. Scotia, from skotia, darkness, be- cause of the strong shadow which its depth produces, and which is increased by the projection of the torus above it. Ovolo, from oviuii, an ^%^^ which this member resembles, when carved, as in the Ionic capital. Cavetto, from caviis, hollow. Cymatium, from kinnaton, a wave. 3(0. — Cliai'aetcrl§tic§ of Mouldiii^§. — Neither of these mouldings is peculiar to any one of the orders of architect- ure ; and although each has its appropriate use, yet it is by no means confined to any certain position in an assemblage of mouldings. The use of the fillet is to bind the parts, as also that of the astragal and torus, which resemble ropes. The ovolo and cyma-reversa are strong at their upper ex- tremities, and are therefore used to support projecting parts above them. The cyma-recta and cavetto, being weak at their upper extremities, are not used as supporters, but are placed uppermost to cover and shelter the other parts. The scotia is introduced in the base of a column to separate the upper and lower torus, and to produce a pleasing variety and relief. The form of the bead and that of the torus is the same ; the reasons for giving distinct names to them are that the torus, in every order, is always considerabl}^ larger than the bead, and is placed among the base mouldings. GRECIAN MOULDINGS. 325 whereas the bead is never placed there, but on the capital or entablature ; the torus, also, is seldom carved, whereas the bead is ; and while the torus among the Greeks is frequently elliptical in its form, the bead retains its circular shape. While the scotia is the reverse of the torus, the cavetto is the re- verse of the ovolo, and the cyma-recta and cyma-reversa are combinations of the ovolo and cavetto. Fig. 191. The curves of mouldings, in Roman architecture, were most generally composed of parts of circles ; while those of the Greeks were almost always elliptical, or of some one of the conic sections, but rarely circular, except in the case of the bead, which Avas always, among both Greeks and Ro- mans, of the form of a semicircle. Sections of the cone af- ford a greater variety of forms than those of the sphere ; and perhaps this is one reason why the Grecian architecture so 326 MOULDINGS AND CORNICES. much excels the Roman. The quick turnings of the ovolo and cyma-reversa, in particular, when exposed to a bright sun, cause those narrow, well-defined streaks of light which give life and splendor to the whole. 311. — A Profile: is an assemblage of essential parts and mouldings. That profile produces the happiest effect which d l^i a FiGo 192. Fig. 193. is composed of but few members, varied in form and size, and arranged so that the plane and the curved surfaces suc- ceed each other alternatelv. 312. — The Grecian Torus and Scotia. — Join the extremi- ties a and b {Fig. 191), and from /, the given projection of the moulding, draw/^ at right angles to the fillets ; from b \>. 1 n ^ ' Fig. 194. Fig. 195. draw bh at right angles to ab\ bisect abvac, join /and r, and upon c, with the radius cf, describe the ^yc fh, cutting bh in // ; through c draw de parallel with the fillets ; make dc and c c each equal to b h\ then de and a b will be conju- gate diameters of the required ellipse. To describe the curve by intersection of lines, proceed as directed at Art. THE GRECIAN ECHINUS. 327 551 and note ; by a trammel, see Art. 549 ; and to find the foci, in order to describe it with a string, see Art. 548. 313. — The Grecian i:clnnu§. — Figs. 192 to 199 exhibit, va- riously modified, the Grecian ovolo, or echinus. Figs. 192 to 196 are elliptical, a b and b c being given tangents to the curve ; parallel to which the semi-conjugate diameters, ad Siud dc, fd V / Fig. 196. Fig. 197. are drawn. In Figs. 192 and 193 the lines a d Sind dc are semi- axes, the tangents, ab and be, being at right angles to each other. To draw the curve, see ^r/. 551. In Fig. 197 the curve is parabolical, and is drawn according to Art. 560. In Figs. 198 and 199 the curve is hyperbolical, being described according to Art. 561. The length of the transverse axis, a b. Fig. 198. Fig. 199. being taken at pleasure in order to flatten the curve, a b should be made short in proportion to ac, 3(4 — The Grecian Cavetto.— In order to describe this, Figs.2Qo^Vid 201, having the height and projection given, see Art. 551. 315.— The Grecian Cyma-Recta.— When the projection is more than the height, as at Fig. 202, make ^<^ equal to the 328 MOULDINGS AND CORNICES. height, and divide abed into four equal parallelograms ; then proceed as directed in note to ^r/. 551. When the projec- tion is less than the height, draw da {Fig, 203) at right angles Fig. 201. Fig. 200. to ^^; complete the rectangle, abcd\ divide this into four equal rectangles, and proceed according \.o Art. 551. 316. — The CJreciaii Cyiiia-Rever§a. — When the projection 1 \Mf' Im WW V c ■ 1 1 i ■' Fig. 202. Fig. 203. is more than the height, as at Fig. 204, proceed as directed for the last figure ; the curve being the same as that, the position only being changed. When the projection is less 1 1 ^ J5^ - a ^ #s ^^ Fig. 204. d Fig. 205. than the height, draw a d (Fig. 205) at right angles to the fillet ; make a d equal to the projection of the moulding ; then proceed as directed for Fig. 202. FORMS OF ROMAN MOULDINGS. 329 317. — Roman Moulding^s : are composed of parts of circles, and have, therefore, less beauty of form than the Grecian. The bead and torus are of the form of the semicircle, and the scotia, also, in some instances ; but the latter is often composed of two quadrants, having different radii, as at Figs. 206 and 207, which resemble the elliptical curve. The ovolo and ca- FiG. 206. Fig. 207. vetto are generally a quadrant, but often less. When they are less, as at Fig: 210, the centre is found thus : join the extrem- ities, a and d, and bisect ad in c; from c, and at right angles to a d, draw c d, cutting a level line drawn from aind; then d will be the centre. This moulding projects less than its height. When the projection is more than the height, as at Fig. 212, extend the line from c until it cuts a perpendicular Fig. 208. Fig. 209. drawn from <^, as at <^; and that will be the centre of the curve. In a similar manner, the centres are found for the mouldings at Figs. 207,211, 213, 216, 217, 218, and 219. The centres for the curves at Figs. 220 and 221 are found thus : bisect the line ^ ^ at ^ ; upon a, c and b successively, with a c or cb for radius, describe arcs intersecting at d and d\ then those intersections will be the centres. 330 MOULDINGS AND CORNICES. ^ 1 Fig. 210. Fig. 211. a \ 1 }>^ a Fig. 212. Fig. 213. ffl Fig. 214. Fig. 215 Fig. 216. Fig. 217. FORMS OF MODERN MOULDINGS. 331 318. — Modern Iflouldiiigs : are represented in Figs. 222 to 229. They have been quite extensively and successfully used in inside finishing. Fig. 222 is appropriate for a bed- moulding under a low projecting shelf, and is frequently used under mantel-shelves. The tangent i h is found thus : bisect the line ab at c, and b c 'sX d\ from d draw dc at right angles to ^ <^ ; from b draw bf parallel to ^<^; upon b. Fig. 219. with b d for radius, describe the arc d f\ divide this arc into 7 equal parts, and set one of the parts from s, the limit of the projection, to ; make h equal to e\ from //, through c, draw the tangent h i; divide b h, lie, c i, and ia each into a like number of equal parts, and draw the intersecting lines as directed at Art. ^21. If a bolder form is desired, draw the tangent, i h, nearer horizontal, and describe an elliptic Fig.^ 220. curve as shown in Figs. 191 and 224. Fig. 223 is much used on base, or skirting, of rooms, and in deep panelling. The curve is found in the same manner as that of Fig. 222. In this case, however, where the moulding has so little projec- tion in comparison with its height, the point e being found as in the last figure, h s may be made equal to s c, instead of {? ^ as in the last figure. Fig. 224 is appropriate for a crown MOULDINGS AND CORNICES. Fig. 223. Fig. -224. PLAIN MOULDINGS. 333 moulding of a cornice. In this figure the height and pro- jection are given; the direction of the diameter, ab, drawn Fig. 225. Fig. 226. through the middle of the diagonal, e f, is taken at pleasure ; and ^<; is parallel to ac. To find the length oi dc^ draw b h Fig. 227. Fig. 228. Fig. 229. at right angles \.o ab ; upon 0, with of for radius, describe the arc, f h, cutting b Ji in Ji ; then make oc and d each 334 MOULDINGS AND CORNICES. equal to bh'^ To draw the curve, see note \.oArt. 551. Figs, 225 to 229 are peculiarly distinct from ancient mouldings, being composed principally of straight lines ; the few curves they possess are quite short and quick. Figs. 230 and 231 are designs for antce caps. The di- ameter of the antae is divided into 20 equal parts, and the height and projection of the members are regulated in ac- cordance with those parts, as denoted under H and P, height and projection. The projection is measured from the mid- dle of the antas. These will be found appropriate for por- ticos, doorways, mantelpieces, door and window trimmings, H.P. 5 15 4 121 J 2 11 1 9 101 T 1 10 , Fig. 230. Fig. 231. etc. The height of the antae for mantelpieces should be from 5 to 6 diameters, having an entablature of from 2 to 2\ diameters. This is a good proportion, it being similar to the Doric order. But for a portico these proportions are * The manner of ascertaining the length of the conjugate diameter, d c, in this figure, and also in Figs. 191, 241, and 242 is new, and is important in this application. It is founded upon well-known mathematical principles, viz. : All the parallelograms that may be circumscribed about an ellipsis are equal to one another, and consequently any one is equal to the rectangle of the two axes. And again : The sum of the squares of every pair of conjugate diame- ters is equal to the sum of the squares of the two axes. EAVE CORNICES. 335 much too heavy : an antae 1 5 diameters high and an entab- lature of 3 diameters will have a better appearance. CORNICES. 319- — Designs for Cornices. — Figs. 232 to 240 are designs for eave cornices, and Figs. 241 and 242 are for stucco cor- nices for the inside finish of rooms. In some of these the projection of the uppermost member from the facia is divided into twenty equal parts, and the various members ! i \ . . 1 1' r > ^ Fig. 232. are proportioned according to those parts, as figured under 77 and P. 320 -—Eave Cornices Proportioned to Height of Build- ing. — Draw the line ac {Fig. 243), and make be and ba each equal to 36 inches ; from <^ draw bd Tit right angles to ac, and equal in length to f of ac\ bisect b dm c, and from a, through e, draw af\ upon a, Avith ac for radius, describe the arc cf, and upon e, with cfiov radius, describe the arc/<^; divide the curve dfc, into 7 equal parts, as at 10, 20, 30, etc., and from these points of division draw lines to ^^ 336 MOULDINGS AND CORNICES. Fig. 233. 7 1 ki/%}/j%/j^U^i^ JULJlEJlIUUIJUUUJU'JUUUlI Fig. 234, -EXAMPLES OF CORNICES. 337 7 riG. 235. 7 Fig. 236, 338 MOULDINGS AND CORNICES. Fig. 237. H P. n 20 5 n 11 8 ~i 25 m f J 1 16J ; ...) ^ c ■"•••-.. ) Fig. 238. VARIOUS DESIGNS OF CORNICES. 339 H. P. U20 1 17 3^31 H 2i ^3- Fig. 239. H.P. 3|20 16 nn ^ UiLXtLMJ Fig. 240. U 340 MOULDINGS AND CORNICES H. P. OODOODOO Fig. 241. H. P Fic. 242. PROPORTION OF CORNICES. 341 parallel to db\ then the distance b i is the projection of a cornice for a building 10 feet high ; b 2, the projection at 20 feet high ; b 3, the projection at 30 feet, etc. If the projec- tion of a cornice for a building 34 feet high is required, divide the arc between 30 and 40 into 10 equal parts, and from the fourth point from 30 draw a line to the base, b c, parallel with b d\ then the distance of the point at which that line cuts the base from b will be the projection re- quired. So proceed for a cornice of any height within 70 feet. The above is based on the supposition that 36 inches Fig. 244. is the proper projection for a cornice 70 feet high. This, for general purposes, will be found correct ; still, the length of the line be may be varied to suit the judgment of those who think differently. Having obtained the projection of a cornice, divide it into 20 equal parts, and apportion the several members 342 MOULDINGS AND CORNICES. according to its destination — as is shown at Figs. 238, 239, and 240. 321. — Cornice Proportioned to a g^iven Cornice. — Let the cornice at Fig. 244 be the given cornice. Upon any point in the lowest Hne of the lowest member, as at a, with the height of the required cornice for radius, describe an intersect- ing arc across the uppermost line, as at <^ ; join a and b ; then b i will be the perpendicular height of the upper fillet for the proposed cornice, i 2 the height of the crown mould- ing — and so of all the members requiring to be enlarged to the sizes indicated on this line. For the projection of the 245- proposed cornice, draw ^ <^ at right angles to a b, and c d dit right angles to ^ ^ ; parallel with c d draw lines from each projection of the given cornice to the line ad\ then ^<^ will be the required projection for the proposed cornice, and the perpendicular lines falling upon e d will indicate the proper projection for the members. To proportion a cornice according to a larger given cor- nice, let A {Fig.. 245) be the given cornice. Extend ao to b, and draw ^ <^ at right angles to ^ ^ ; extend the horizontal lines of the cornice, A, until they touch o d] place the height of the proposed cornice from to e, and join y and e\ upon Oy with the projection of the given cornice, a, for radius, TO FIND THE ANGLE BRACKET. ;43 describe the quadrant ad\ from ^ draw <^^ parallel to/^; upon 0, with o b for radius, describe the quadrant b c ; then c will be the proper projection for the proposed cornice. Join a and c ; draw lines from the projection of the different members of the given cornice to « 6* parallel Vo od\ from these divisions on the line a o draw lines to the line o c parallel \.o ac\ from the divisions on the line of draw lines to the line oe parallel to the line/r; then the divisions on the lines o e and o c will indicate the proper height and pro- jection for the different members of the proposed cornice. In this process, we have assumed the height, o e, of the pro- posed cornice to be given ; but if the projection, o c, alone Fig. 247. be given, we can obtain the same result b}' a different pro- cess. Thus: upon d?, with (? <: for radius, describe the quad- rant cb\ upon 0, with a for radius, describe the quadrant ad\ join d and b\ from /draw fc parallel to db\ then oc will be the proper height for the proposed cornice, and the height and projection of the different members can be obtained by the above directions. By this problem, a cor- nice can be proportioned according to a smaller given one as well as to a larger ; but the method described in the pre- vious article is much more simple for that purpose. 322. — Angle Bracket in a Built Cornice. 246) be the wall of the -Let A {Fi^ building, and B the given bracket, 344 OULDINGS AND CORNICES. which, for the present purpose, is turned down horizontally. The angle-bracket, C, is obtained thus : through the ex- tremity, a, and parallel with the wall,/<^, draw the line a b ; make^<: equal af, and through c draw cb parallel with ed\ join d and b^ and from the several angular points in B draw ordinates to cut db'm i, 2, and 3 ; at those points erect lines perpendicular to db\ from h draw hg parallel to fa; take Fig. 248. the ordinates, 10,20, etc., at B, and transfer them to C, and the angle-bracket, C, will be defined. In the same manner, the angle-bracket for an internal cornice, or the angle-rib of a coved ceiling, or of groins, as at Fig. 247, can be found. 323,— Raking Mouldings matched with Level Returns.— Let A {Fig. 248) be the given moulding, and A b the rake of CROWN MOULDING ON THE RAKE. 345 the roof. Divide the curve of the given moulding into any number of parts, equal or unequal, as at i, 2, and 3 ; from these points draw horizontal lines to a perpendicular erected from c ; at any convenient place on the rake, as at B, draw a c ?X right angles to A b\ also from b draw the horizontal line b a-, place the thickness, d a^ of the moulding at A from b to <^, and from <^ draw the perpendicular line ae\ from the points i, 2, 3, at ^, draw lines to C parallel X.O A b \ make a \, a2, and a 3, at B, and at C, equal to rt; i, etc., at A ; through the points, i, 2, and 3, at B, trace the curve — this will be the proper form for the raking moulding. From i, 2, and 3, at C, drop perpendiculars to the corresponding ordinates from i, 2, and 3, at y^ ; through the points of inter- section, trace the curve — this will be the proper form for the return at the top. PART II SECTION VI.— GEOMETRY. 324. — Mathematics Essential. — In this and the following Sections, which will constitute Part II., there are treated of certain matters which may be considered as elementary. They are all very necessary to be understood and acquired by the builder, and are here compactly presented in a shape which, it is believed, will aid him in his studies, and at the same time prove to be a great convenience as a matter of reference. The many geometrical forms which enter into the composition of a building suggest a knowledge of Elemen- tary Geometry as essential to an intelligent comprehension of its plan and purpose. One of the prime requisites of a building is stability, a quality which depends upon a proper distribution of the material of which the building is con- structed ; hence a knowledge of the laws of pressure and the strength of materials is essential ; and as these are based upon the laws of proportion and are expressed more con- cisely in algebraic language, a knowledge of Proportion and of Algebra are likewise indispensable to a comprehensive understanding of the subject. There will be found in this work, however, only so much of these parts of mathematics as have been deemed of the most obvious utility in the Science of Building. For a more exhaustive treatment of the subjects named, the reader is referred to the many able works, readily accessible, which make these subjects their specialties. 325.— Elementary Geometry. — In all reasoning defini- tions are necessary, in order to insure in the minds of the 348 GEOMETRY. proponent and respondent identity of ideas. A corollary is an inference deduced from a previous course of reasoning. An axiom is a proposition evident at first sight. In the fol- lowing demonstrations there are many axioms taken for granted (such as, things equal to the same thing are equal to one another, etc.) ; these it was thought not necessary to introduce in form. 326. — Definition. — If a straight line, as A B {Fig. 249), stand upon another straight line, as CD, so that the two C B D Fig. 249. angles made at the point B are equal — A B C to A B D {Art. 499, obtuse angle) — then each of the two angles is called a right angle. 327. — Definition. — The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; hence a semicircle contains 180 degrees, a quad- rant 90, etc. E 328. — Definition, — The measure of an angle is the num- ber of degrees contained between its two sides, using the angular point as a centre upon which to describe the arc. Thus the arc C E {Fig. 250) is the measure of the angle CB E, EAoi the angle E B A, and A D oi the angle A B D. 329. — Corollary. — As the two angles at B {Fig. 249) are right angles, and as the semicircle, CAD, contains 180 de- grees {Art. 327), the measure of two right angles, therefore, is RIGHT ANGLES AND OBLIQUE ANGLES. 349 180 degrees; of one right angle, 90 degrees; of half a right angle, 45 ; of one third of a right angle, 30, etc. 330. — Definition. — In measuring an angle {Art. 328), no regard is to be had to the length of its sides, but only to the degree of their inclination. Hence eqiial angles are such as have the same degree of inclination, without regard to the length of their sides. 331- — Axiom.— If two straight lines parallel to one another, a.s A B and CD {Fig. 251), stand upon another straight line, as E F, the angles A B F and CDF are equal, and the angle A B F is equal to the angle CDF. 332. — Definition. — If a straight line, as A B {Fig. 250), stand obliquely upon another straight line, as CD, then one A c / E B D P Fig. 251. of the angles, sls A B C, is called a7i obtuse angle, and the other, as A B Dy an acute angle. 333.— Axiom. — The two angles^ B DaxvdiA B C {Fig. 2^0) are together equal to two right angles {Arts. 326, 329) ; also, the three angles A B D, E B A, and CBE are together equal to two right angles. 334. — Corollary. — Hence all the angles that can be made upon one side of a line, meeting in a point in that line, are together equal to two right angles. 336, — Corollary. — Hence all the angles that can be made on both sides of a line, at a point in that line, or all the angles that can be made about a point, are together equal to four right angles. 350 GEOMETRY. 336. — Propo§ition. — If to each of two equal angles a third angle be added, their sums will be equal. Let ABC and D E F {Fig. 252) be equal angles, and the angle I J K t\\Q one to be added. Make the angles G B A and H E D each equal to the given angle I J K\ then the angle G B C will be equal to the angle H E F\ for if ABC and D E F hQ angles of 90 degrees, and IJK 30, then the angles GBC and H EF will be each equal to 90 and 30 added, viz., 120 degrees. 337- — Proposition. — Triangles that have two of their sides and the angle contained between them respectively equal, have also their third sides and the two remaining angles equal ; and consequently one triangle will every way equal the other. Let ABC {Fig. 253) and D E F he two given triangles, having the angle at A equal to the angle at D, the side A B equal to the side D E, and the side A C equal to the side D F\ then the third side of one, B C, is equal to the third side of the other, E F\ the angle at B is equal to the angle at E. and the angle at C is equal to the angle at EQUAL TRIANGLES IN PARALLELOGRAMS. 35 F. For if one triangle be applied to the other, the three points B, A, C, coinciding with the three points E, D^ F, the line ^ 6^ must coincide with the line EF] the angle at B with the angle at E\ the angle at C with the angle at F\ and the triangle B A C hQ every way equal to the triangle EDF. 338. — Propo§itioii. — The two angles at the base of an isosceles triangle are equal. Let ABC {Fig. 254) be an isosceles triangle, of which the sides, A B and A C, are equal. Bisect the angle {Art. 506) BAChy the line A D. Then, the line B A being equal to the line A C, the line A D oi the triangle E being equal to the line A D oi the triangle F (being common to each), the angle BAD being equal to the angle DA Cy — the line B D must, according to Art. 337, be C D D C Fig. 255. equal to the line D C, and the angle at B must be equal to the angle at C. 339. — Proposition. — A diagonal crossing a parallelogram divides it into two equal triangles. Let CD EF {Fig. 255) be a given parallelogram, and C F 7i line crossing it diag- onally. Then, as ^ 6" is equal to F D, and E F io C D, the angle at E to the angle at D, the triangle A must, according to Art. 337, be equal to the triangle B. 352 GEOMETRY. 340 Proposition. — Let J K LM {Fig. 256) be a given parallelogram, and K L^ diagonal. At any distance between yATand LM draw N P parallel to J K\ through the point G, the intersection of the lines K L and N P, draw HI parallel to K M. In every parallelogram thus divided, the parallelogram A is equal to the parallelogram B. For, ac- cording to Art. 339, the triangle J K L \s equal to the tri- angle K L M, the triangle C to the triangle D, and E to F; H K this being the case, take D and F from the triangle K LM, and C diud E from the triangle jf K L, and what remains in one must be equal to what remains in the other ; therefore, the parallelogram A is equal to the parallelogram B. 34(, — Proposition. — Parallelograms standing upon the same base and between the same parallels are equal. Let A BCD and EFCD (Fig. 257) be given parallelograms Fig. 257. standing upon the same base, C D^ and between the same parallels, A F and CD. Then A B and E F, being equal to CD, are equal to one another; BE being added to both A B and E F, A E equals BE] the line A C being equal to B D, and A E to B F, and the angle C A E being equal {Art. 331) to the angle Z>j5/^, the triangle ^4 ^ C must be equal {Art. 337) to the triangle BED; these two triangles being equal, take the same amount, the triangle BEG, from each, TRIANGLE EQUAL TO QUADRANGLE. 353 and what remains in one, A B G C, must be equal to what remains in the other, E F D G ; these two quadrangles being equal, add the same amount, the triangle C G D, to each, and they must still be equal ; therefore, the parallelogram A B CD is equal to the parallelogram E F C D. 342. — Corollary. — Hence, if a parallelogram and triangle stand upon the same base and between the same parallels, Fig. 258. the parallelogram will be equal to double the triangle. Thus, the parallelogram A D {Fig. 257) is double {Art, 339) the triangle C E D. 343. — Proposition. — Let F G H D {Fig. 258) be a given quadrangle with the diagonal F D. From G draw G E IE /Q- Fig. 259. parallel to FD-, extend H D to E\ join i^and E\ then the triangle F E H \n\\\ be equal in area to the quadrangle FGHD. For since the triangles FDG and FDE stand upon the same base, F D, and between the same parallels,. FD and G E, they are therefore equal {Arts. 341, 342) ; and since the triangle C is common to both, the remaining, tri- 54 GEOMETRY. angles, A and B, are therefore equal ; then, B being equal to A, the triangle F E H is equal to the quadrangle F G H D. 34-4-. — Proposition. — If two straight lines cut each other, as FG and H J {Fig. 259), the vertical, or opposite angles, A and C, are equal. Thus, FE, standing upon H J^ forms the angles B and C, which together amount {Art. 333) to two right angles ; in the same manner, the angles A and B form two right angles ; since the angles A and B are equal to B and C^ take the same amount, the angle B, from each pair, and what remains of one pair is equal to what remains of the other ; therefore, the angle A is equal to the angle C. The same can be proved of the opposite angles B and D. 345. — Proposition. — The three angles of any triangle are equal to two right angles. Let ABC {Fig. 260) be a given triangle, with its sides extended to /% E and D, and the line (7 6^ drawn parallel t'o BE. As 6* 6^ is parallel to EB,\\\q angle at H is equal {Art. 331) to the angle at Z ; as the lines FC 2indi BE cut one another at A, the opposite angles at M and A^are equal {Art. 334) ; as the angle at A^ is equal {Art. 331) to the angle at J, the angle at J is equal to the angle at M\ therefore, the three angles meeting at C are equal to the three angles of the triangle A B C \ and since the three angles at C are equal (y^r/. 333) to two right angles, the three angles of the triangle ABC must likewise be equal to two right angles. Any triangle can be subjected to the same proof. 346- — Corollary. — Hence, if one angle of a triangle be a right angle, the other two angles amount to just one right angle. RIGHT ANGLE IN SEMICIRCLE. 355 347._CoroIlary.— If one angle of a triangle be a 'right ano-le and the two remaining angles are equal to one another, these are each equal to half a right angle. 348.— Corollary — If any two angles of a triangle amount to a right angle, the remaining one is a right angle. 349. —Corollary.— If any two angles of a triangle are to- gether equal to the remaining angle, that remaining angle is a right angle. 350.— Corollary.— If any two angles of a triangle are each equal to two thirds of a right angle, the remaining angle is also equal to tw^o thirds of a right angle. 35 [, —Corollary. — Hence, the angles of an equilateral triangle are each equal to two thirds of a right angle. 352. — Proposition. — If from the extremities of the di- ameter of a semicircle two straight lines be drawn to any point in the circumference, the angle formed by them at that point will be a right angle. Let ABC {Fig. 261) be a given semicircle; and ^ ^ and i5 (7 lines drawn from the extrem- ities of the diameter A C to the given point B ; the angle formed at that point by these lines is a right angle. Join the point B and the centre D ; the lines DA, D B, and D C, being radii of the same circle, are equal; the angle at A is, therefore, equal {Art. 338) to the angle at E ; also, the angle at C is, for the same reason, equal to the angle at F; the angle ABC, being equal to the angles at A and C taken to- gether, must, therefore {Art. 349), be a right angle. 353. — Proposition. — The square on the hypothenuse of a right-angled triangle is equal to the squares on the two re- 35<^ GEOMETRY. maining sides. Let ABC {Fig. 262) be a given right-angled triangle, having a square formed on each of its sides ; then the square ^^ is equal to the squares //(7and GB taken together. This can be proved by showing that the parallelo- gram B L is equal to the square GB ; and that the parallelo- gram CL is equal to the square HC. The angle CBB is a right angle, and the angle A B Fis aright angle ; add to each of these the angle ABC; then the angle FB C will evidently be equal {Art. 336) to the angle ABB; the triangle FB C and the square G B, being both upon the same base, FB, and between the same parallels, FB and G C, the square G B is equal {Art. ^^42) to twice the triangle FBC; the triangle A B D and the parallelogram B L, being both upon the same Fig. 262. base, B D, and between the same parallels, B D and A L, the parallelogram B L is equal to twice the triangle ABD; the triangles, FB C and ABD, being equal to one another {Art. 337), the square G B is equal to the parallelogram B L, either being equal to twice the triangle F B C ox A B D. The method of proving Z/^' equal to C L is exactly similar — thus proving the square ^ ^ equal to the squares // (7 and 6^ ^, taken together. This problem, which is the 47th of the First Book of Euclid, is said to have been demonstrated first by Pythago- ras. It is stated (but the story is of doubtful authority) that as a thank-offering for its discovery he sacrificed a hun- dred oxen to the gods. From this circumstance it is some- times called the Jiecatomb problem. It is of great value in DIAGONAL OF SQUARE FORMING OCTAGON. 357 the exact sciences, more especially in Mensuration and As- tronomy, in which many otherwise intricate calculations are by it made easy of solution. 354. — Proposition. — In an equilateral octagon the semi- diagonal of a circumscribed square, having its sides coinci- dent with four of the sides of the octagon, equals the dis- tance along a side of the square from its corner to the more remote angle of the octagon occurring on that side of the square. Let Fig. 263 represent the square referred to ; in which O is the centre of each ; then A O equals A D. To prove this, it need only be shown that the triangle A O D \^ an isosceles triangle having its sides A O and A D equal. The Fig. 263. octagon being equilateral, it is also equiangular, therefore the angles B C O, E C O, A D O, etc., are all equal. Of the right-angled triangle FEC,FC and FE being equal, the two angles FE C ^nd EC E, :xrQ equal {Art. 338), and are therefore {Af't. 347) each equal to half a right angle. In like manner it may be shown that FA B and FB A are also each equal to half a right angle. And since FE C and FA B are equal angles, therefore the lines E C and A B are parallel {Art. 331,) and hence the angles ECO and A OD are equal. These being equal, and the angles ^(S' (9 ?ind A D O being equal by construction, as before shown, therefore the angles A O D and ADO are equal, and consequently the lines A and A D are equal. {Art. 338.) 358 GEOMETRY. . 355- — Proposition. — An angle at the circumference of a circle is measured by half the arc that subtends it ; that is, the angle ABC {Fig. 264) is equal to half the angle ADC. Through the centre D draw the diameter BE. The tri- angle A B D \s an isosceles triangle, A Z>and B D being ra- dii, and therefore equal ; hence, the two angles at F and G are equal {Art. 338), and the sum of these two angles is equal to the angle at H {Art. 345), and therefore one of them, G, is equal to the half of H. The angles at H and at G (or ABE) are both subtended by the arc A E. Now, since the angle Fig, 264, at H is measured by the arc A E, which subtends it, there- fore the half of the angle at H would be measured by the half of the arc A E ; and since G is equal to the half of //, therefore G or A B E is measured by the half of the arc A E. It maybe shown in like manner that the angle E B C is measured by half the arc E C, and hence it follows that the angle A B C is measured by half the arc, A C, that sub- tends it. 356. — Proposition. — In a circle all the inscribed angles, A, B, and C {Fig. 265), which stand upon the same side of the EQUAL ANGLES IX CIRCLES. 359 chord i?^ are equal. For each angle is measured by half the arc D F E {Art. 355). Hence the angles are all equal. 357- — Corollary. — Equal chords, in the same circle, sub- tend equal angles. Fig. 265. 358- — Proposition. — The angle formed by a chord and tangent is equal to any inscribed angle in the opposite seg- ment of the circle ; that is, the angle D {Fig. 266) equals the angle A. Let H F be the chord, and E G the tangent ; draw the diameter JH\ then JHG is a right angle, also J F H \s 36o GEOMETRY. a right angle. {Art. 352.) The angles A and B together equal a right angle {Art. 346) ; also the angles B and D together equal a right angle (equal to the angle JHG): therefore, the sum oi A and B equals the sum of B and D. From each of these two equals, taking the like quantity B, the remainders A and D are equal. Thus, it is proved for the angle at A ; it is also true for any other angle ; for, since all other in- scribed angles on that side of the chord line H F equal the angle A {Art. 356), therefore the angle formed by a chord and tangent equals any angle in the opposite segment of the circle. This being proved for the acute angle D, it is also true for the obtuse angle EH F\ for, from any point, K {Fig, 267) in the arc H K F, draw lines to J, F and H ; now, if it can be proved that the angle £i7i^ equals the angle FKH, the entire proposition is proved, for the angle F K H equals any of all the inscribed angles that can be drawn on that side of the chord. {Art. 356.) To prove, then, that jE"///^ equals H KF\ the angle ^///^ equals the sum of the angles A and B\ also the angle 77 A" /^equals the sum of the angles C and D. The angles B and D, being inscribed angles on the same chord, J F, are equal. The angles C and A, being right angles {Art. 352), are likewise equal. Now, since A equals 6" and B equals 7>, therefore the sum of A and B equals the sum of C and D — or the angle ^777^ equals the angle H K F. 359. — Proposition. — The areas of parallelograms of equal altitude are to each other as the bases of the parallelo- PARALLELOGRAMS PROPORTIONATE TO BASES. 361 grams. In Fig. 268 the areas of the rectangles A B C D and B E D F are to each other as the bases CD and D F. For, putting the two bases in form of a fraction and reducing this fraction to its lowest terms, then the numerator and denomi- nator of the reduced fraction will be the numbers of equal parts into which the two bases respectively may be divided. For example, let the two given bases be 12 and 9 feet respect- ively, then ^ = f , and this gives four parts for the larger base and three parts for the smaller one. So, in Fig. 268, divide the base CD into four equal parts, and the base D F into three equal parts ; then the length of any one of the parts in CD will equal the length of any one of the parts in D F. Now, parallel with A C, draw lines from each point of division to the line A F, These lines will evidently divide the whole figure into seven equal parts, four of them occupy- A B E C 1 D • Fig. 26S. ing the area A B C D, and three of them occupying the area B E D F, Now it is evident that the areas of the two rect- angles are in proportion as the number of parts respectively into which the base-lines are divided, or that — A B C D : B E D F : : C D : D F. The areas in this particular case are as 4 to 3. But in gen- eral the proportion will be as the lengths of the bases. Thus the proposition is proved in regard to rectangles, but it has been shown (Art-. 341) that all parallelograms of equal base and altitude are equal. Therefore the proposition is proved in regard to parallelograms generally, including rect- angles. 360. — Proposition. — Triangles of equal altitude are to each other as their bases. It has been shown {Arf. 359) that 362 GEOMETRY. parallelograms of equal altitude are in proportion as their bases, and it has also been shown {ArL 342) that of a triangle and parallelogram, when of equal base and altitude, the parallelogram is equal to double the triangle. Therefore triangles of equal altitude are to each other as their bases. 36(. — Propo§moii. — Homologous triangles have their corresponding sides in proportion. Let the line CD {Fig. 269) be drawn parallel with A B. Then the angles E C D and E A B are equal {Art. 331), also the angles E D C and E B A are equal. Therefore the triangles ECD and E AB are homologous, or have their corresponding angles equal. For, join C to B^ and A to D, then the triangles A C D and ^ (7 i9, standing on the same base, (T/^, and between the same parallels, CD and A B, are equal in area. To each of these equals join the common area C D E, and the sums A DE and BCE will be equal. The triangles CD E and A D E, having the same altitude, are to each other as their bases C E and A E {Art. 360), or — CDE : ADE : : CE : AE. Also the triangles CDE and BCE, having the same alti- tude, are to each other as their bases D E and BE, or — CDE : BCE : : DE : BE. CHORDS GIVING EQUAL RECTANGLES. 363 And, since the triangles A D E and B C E are equal, as before shown, therefore, substituting in the last proportion A D E for B C E, we have — CDE : ADE : : DE : BE. The first two factors here being identical with the first two in the first proportion above, we have, comparing the two proportions — CE : AE : : DE : BE; or, we have the corresponding sides of one triangle, CB E, in proportion to the corresponding sides of the other, /^ BE. Fig. 270. 362. — Proposition. — Two chords, ^/^ and CD {Eig. 270), intersecting, the parallelogram or rectangle formed by the two parts of one is equal to the rectangle formed by the two parts of the other. That is, the product oi C G multi- plied by G D is equal to the product oi E G multiplied by G F. The triangle A is similar to the triangle B, because it has corresponding angles. The angle H equals the angle G {Art. 344) ; the angle at J equals the angle at K, because they stand upon the same chord, D F {Art. 356) ; for the same 364 GEOMETRY. reason the angle M equals the angle Z, for each stands upon the same chord, E C. Therefore, the triangle A having the same angles as the triangle B^ the length of the sides of one are in like proportion as the length of the sides in the other {Art. 361). So— CG : EG : '. GF \ G D, Hence, as the product of the means equals the product of the extremes {Art, 373), E G multiplied by G F \^ equal to C G multiplied by G D, 363. — Proposition. — If the sides of a quadrangle are bisected, and lines drawn joining the points of bisection in the adjacent sides, these lines will form a parallelogram. Draw the diagonals A B and CD {Fig. 271). It will here be perceived that the two triangles A E O and ACD are homol- ogous, having like angles and proportionate sides. Two of the sides of one triangle lie coincident with the two corres- ponding sides of the other triangle, therefore the contained angles between these sides in each triangle are identical. By construction, these corresponding sides are proportion- ate ; A C being equal to twice A E, and A D being equal to twice A O ; therefore the remaining sides are proportionate, CD being equal to twice E O, hence the remaining corres- ponding angles are equal. Since, then, the angles A E O and ACD are equal, therefore the line E is parallel with PARALLELOGRAM IN QUADRANGLE. 365 the diagonal CD — so, likewise, the line^^A^is parallel to the same diagonal, CD. If, therefore, these two lines, EO and M N, are parallel to the same line, CD, they must be parallel to each other. In the same manner the lines ON and EM are proved parallel to the diagonal A B, and to each other ; therefore the inscribed figure ME ON is 3, parallelogram. It may be remarked, also, that the parallelogram so formed will contain just one half the area of the circumscribing quadrangle. SECTION VII.— RATIO, OR PROPORTION. 364-. — mercliaiKlise. — A carpenter buys 9 pounds of nails for 45 cents. He afterwards buys 87 pounds at the same rate. How much did he pay for them ? An answer to this question is readily found by multiply- ing the ^y pounds by 45 cents, the price of the 9 pounds, and dividing the product, 391 5, by 9; the quotient, 435 cents, is the answer to the question. 365. — Tlie "Rule of Three." — The process by which this problem is solved is known as the Rule of Three, or Proportion. In cases of this kind there are three quantities given, to find a fourth. Previous to working the question it is usual to make a statement, placing the three given quantities in such order that the quantity which is of like kind with the answer shall occupy the second place ; the quantity upon which this depends for its value is put in the first place, and the remaining quantity, which is of like kind with that in the first place, is assigned to the third place. When thus arranged, the second and third quantities are multiplied together and the product is divided by the first quantity ; the quotient, the answer to the question, is a fourth quantity. These four quantities are related to each other in this manner, namely : the first is in proportion to the second as the third is to the fourth ; or, taking the quantities of the given example, and putting them in a for- mal statement with the customary marks between them, we have — 9 : 45 :: 87 : 435, which is read: 9 is to 45 as 87 is to 435 ; or, 9 is in propor- tion to 45 as 87 is to 435 ; or, 9 bears the same relation to 45 as 87 does to 435. EQUALITY OF RATIOS. 367 366. — Couples: Antecedent, Consequent. — These four quantities are termed Proportionals, and may be divided into two couples ; the first and second quantities forming one couple, and the third and fourth the other couple. Of each couple the first quantity is termed the antecedent, and the last the consequent. Thus 9 is an antecedent and 45 its con- sequent ; so, also, Zj is an antecedent and 435 its consequent. 367. — Equal Couples: an Equation. — These four quan- tities may be put in form thus : 9 87 Each couple is here stated as a fraction : each has its ante- cedent beneath its consequent, and the two couples are separated by a sign, two short parallel lines, signifying equality. This is an equation, and is read thus : 45 divided by 9 is equal to 435 divided by 87 ; or, as ordinary fractions : 45 ninths are equal to 435 eighty-sevenths. 368. — Equality of Ratios. — Each couple is also termed a Ratio, and the two the Equality of Ratios. Thus the ratio — is equal to the ratio -~^-. If the division indicated in 9 87 these two ratios be actually performed, the equality between the two will at once be apparent, for the quotient in each case is 5. The resolution of each couple into its simplest form by actual division is shown thus : 435 _ , 87 -5- These are read : 45 divided by 9 equals 5 ; and 435 divided by 87 equals 5. 369. — Equals Iflultiplied by Equals Give Equals. — If two equal quantities be each multiplied by a given quantity, the 368 RATIO, OR PROPORTION. two products will be equal. For example, the fractions f and |- are each equal to i, and are therefore equal to each other. If these two equal quantities be each multiplied by any given number, say, for example, by 4, we shall have 4 times f equals |, and 4 times f equals -^2. ; these products, f and -V^- are each equal to 2, and therefore equal to each other. 370. — Multiplying^ an Equation. — The quantity on each side of the sign = is called a meinber of the equation. If each member be multiplied by the same quantity, the equality of the two members is not thereby disturbed {Art. 369) ; therefore, if the two members of the equation — = ^^ {Art. 367) be each multiplied by 87, or be modified thus : 45x87 ^ 435 X 87 9 87 ' in which x, the sign for multiplication, indicates that the quantities between which it is placed are to be multiplied together ; this ddition to each member of the equation does not destroy the equality ; the members are still equal, though considerably enlarged. The equality may be easily tested by performing the operations indicated in the equa- tion. For example : for the first member, we have 45 times 87 equals 3915, and this divided by 9 equals 435. Again, for the second member we have 435 times 87 equals 37845, and this divided by 87 equals 435, the same result as that for the first member. Thus the multiplication has not interfered with the equality of the members. 371. — Multiplying: and I>iTi«Sing: one Member of an Equation : Caneelling. — If a quantity be multiplied by a given number, and the product be divided by the same given number, the quotient will equal the original quantity. For example : if 8 be multiplied by 3, the product w^ill be 24 ; then if this product be divided by 3, the quotient w^ill be 8, the original quantity. Thus the value of a quantity is not TRANSFERRING FACTORS IN EQUATIONS. 369 changed by multiplying it by a number, provided it be also divided by the same number. From this, also, we learn that the value of a quantity which is required to be multiplied and divided by the same number will not be changed if the multiplication and divis- ion be both omitted ; one cancels the other. Therefore the number 87, appearing in the second member of the equation in the last article both as a multiplier and a divisor, may be omitted without destroying the equality of the two mem- bers. The equation thus treated will be reduced to — 45 X87 ^- = 435. This expression is read : the product of 45 times 87 divided by 9 equals 435. It will be observed that we have here the four terms of the problem in Art. 365, three of them in the first member, and the fourth, the answer to the problem, in the second member. 372. — Transferring a Factor. — Each of the four quan- tities in the aforesaid equation is termed 2i factor. Compar- ing the equation of the last article with that of Art. 43, it appears that the two are alike excepting that the factor 87 has been transferred from one member of the equation to the other, and that, Avhereas it was before a divisor, it has now become a multiplier. From this we learn that a factor may be transferred from one member of an equation to the other, provided that in the transfer its relative position to the horizontal line above or below it be also changed ; that is, if, before the transfer, it be below the line, it must be put above the line in the other member; or, if above the line, it must be put below, in the other member. For example : in the equation of the last article let the factor 9 be removed to the second member of the equation. It stands as a divi- sor in the first member ; therefore, by the rule, it must appear as a multipher after the transfer ; or — 45 X 87 = 9 X 435; 370 RATIO, OR PROPORTION. which is read, 45 times 87 equals 9 times 435. By actually performing the operations here indicated, we find that each member gives the same product, 3915; thus proving that the equality of the two members was not interfered with by the transfer. 373. — Eqwality of Products: Mcaais and Extremes. — In Art. 366, the four factors are put in the usual form of four proportionals. A comparison of these with the four factors as they appear in the equation in the last article, shows that the first member contains the second and third of the four proportionals, and the second member contains the first and the fourth ; or, the first contains what are termed the means, and the second, the extremes. From this we learn that in any set of four proportionals, the product of the means equals the product of the extremes. As for example, 3 6 -:=:ii; SO, also, - = i-J, an equality of ratios: hence the four factors, 2, 3, 4, 6, are four proportionals, and may be put thus : Extreme, mean, mean, extreme. 2:3:14:6 and, as above stated, the product of the means (3x4 — ) 12, equals the product of the extremes (2x6=) 12. 374 Ilomoloiii^ous Triangles Proportionate. — The discussion of the subject of Ratios has thus far been con- fined to its relations with the mercantile problem of Art. 364. The rules of proportion or the equality of ratios apply equally to questions other than those of a mercantile character. They apply alike to all questions in which quan- tities of any kind are comparable. For example, in geome- try, lines, surfaces, and solids bear a certain fixed relation to one another, and are, therefore, fit subjects for the rules of proportion. It is shown, in Art. 361, that the correspond- ing sides of homologous triangles are in proportion to one another. Hence, when, of two similar triangles, two corres- ponding sides and one other side are given, then by the equality of ratios the side corresponding to this other side RATIOS APPLIED TO TRIANGLES. 37 1 may be computed. For example : in two triangles, such as BCD and EAB (Fig. 269), having- their corresponding angles equal, let the side E C, in the triangle BCD, equal 12 feet, and the corresponding side B A, in the triangle B A B, equal 16 feet, and the side B D, of triangle E CD, equal 14 feet. Now, having these three sides given, how can we find the fourth ? Putting them in proportion, we have, as in Art. 361 — CE : AE : : BE : BE; and, substituting for the known sides, their dimensions, we have — 12 : 16 : : 14 : ^ii ; and, by zir/. 373 — I2x B E = 16 X 14, Dividing each member by 12, gives — 12 Performing the multiplication and division indicated, we have — BB = ^^= i8|. 12 Thus we have the fourth side equal to i8f feet. 375. — The Steelyard. — An example of tour proportion- als may also be found in the relation existing between the arms of a lever and the weights suspended at their ends. A familiar example of a lever is seen in the common steelyard used by merchants in weighing goods. This is a bar, A B, of steel, arranged as in Big. 272, with hooks and links, and a suspended platform to carry R, the article to be weighed ; and with a weight P, suspended by a link at B, from the bar A B, along which the weight P is movable. The entire load is sustained by links attached to the ful- crum, or point of suspension C. The apparatus is in equi- librium without R and P. In weighing any article, R, the 3/2 RATIO, OR PROPORTION. weight P is moved along- the bar B C until the weight just balances the load, or until the bar A B will remain in a hori- zontal position. If the weight P be too far from the fulcrum C the end of the bar B will fall, but if it be too near it will rise. 376, — The liCver Exemplified toy the Steelyard. — To exemplify the principle of the lever, let the bar A B {Fig. 272) be balanced accurately with the scale platform, but without the weights R and P. Then, placing the article R upon the platform, move the weight P along the beam until there is an equilibrium. Suppose the distances A C and B C are found to be 2 and 40 inches respectively, and suppose Fig, 272, the weight P to equal 5 pounds, what at this pomt will be the weight of i^ ? By trial we shall find that R = 100 pounds. Again, if a portion of R be removed, then the weight P would have to be moved along the bar B C to produce an equilibrium ; suppose it be moved until its distance from C be found to be 20 inches, then the weight of R would be found to be 50 pounds, or — 7? = 50 pounds. Again, suppose a part of the weight taken from R be re- stored, and the weight P, on being moved to a point re- quired for equilibrium, be found to measure 30 inches from C, then we shall find that — R— 'j^ pounds. RATIOS OF THE LEVER. 373 Thus when — ^(7 = 40, ;?= 100; or, —-=2.5; 40 BC=zo, R = 7S\ or, ^=2.5; BC^2o, R = So; or, 1^ =2.5; showing an equality of ratios ; or, in general, ^ 67 is in pro- portion to R, or — BC : R. If, instead of moving P along B C, its position be permanent, and the weight P be reduced as needed to produce equilib- rium with the various articles, R, which in turn may be put upon the scale ; then we shall find that if when the weight P equals 5 pounds the article R equals 100, and there is an equilibrium, then \vhen — p_- — ^^—.^,^^ j^ ^^jjl equal — x 100 = 90; 8 8 P = — x5 = 4, 7? will equal — x 100 = 80; 10 ^ ^10 7 7 P— — X 5 = 3 . 5, R will equal — x 100 =: 70 ; and so on for other proportions^ and in every case we shall have the ratio -p equal 20, thus— R P = 90 4-5' = 20- R P = 80 4 : 20; R P = 70 3-5" = 20. 374 RATIO, OR PROPORTION. Thus we have an equahty of ratios in comparing the weights. Again, if the weight P and the article R be permanent in weight, and the distances A Cy B C he made to vary, then if there be an equiUbrium when A C is 2 and B C is 40, we shall find that when — 8 8 AC— — X 2 = 1-6; B C will equal — x 40 = 32 ^ A C = — x2=i-2; B C will equal — x 40 = 24 ; 10 ^10 AC— ~ x2==o-8; BC will equal — x 40 =16 ; 10 ^ 10 ^ and so on for other proportions, and in every case we shall have the ratio -j-^r = 20; thus — BC _ A C 32 1.6 1= 20; BC AC 24 1-2 = 20; B C 16 = 20: AC~Q'% producing thus an equality of ratios in comparing the arms of the lever. From these experiments we have found, in comparing the article weighed with an arm of the lever, the constant ratio B C '. R, and when comparing the weights we have found the constant ratio P : R. Again, in com- paring the arms of the lever, w^e find the constant ratio A C : B C. Putting two of these couples in proportion, we have — A C : B C : : P : R. Hence {A?'t. 373) we have — ACxR = BCxP. PRINCIPLE OF THE LEVER DEMONSTRATED. Dividing both members by A C, we have — BCx P 375 R A C ' In a steelyard the short arm, A C, and the weight, or poise, P, are unvarying ; therefore we have — R BCx P AT' or, when A C is constant, we have- R : B C. Zn , — The Licver Principle Demoiistrated. — 1 he rela- tion between the weights and their arms of leverage may be demonstrated as follows : * M " G>f? — © Fig, 273. Let A B G H, Fig. 273, represent a beam of homogeneous material, of equal sectional area throughout, and suspended upon an axle or pin at Cy its centre. This beam is evidently in a state of equilibrium. Of the part of the beam A D G K, let E be the centre of gravity ; and of the remaining part, D B K H, let F be the centre of gravity. If the weight of the material in A B^ G Khe concentrated at F, its centre of gravity, and the weight of the material in * The principle upon which this demonstration is based may be found in an article written by the author and published in \}ciQ Matheinatical MontJily, Cam- bridge, U. S., for 1858, p. 77. 376 RATIO, OR PROPORTION. DBKH be concentrated in F, its centre of gravity, the state of equilibrium will not be interfered with. Therefore let the ball R be equal in weight to the part A D G K, and the ball P equal to the weight of the part D B KH \ and let these two balls be connected by the rod E F. Then these two balls and rod, supported at C^ will evidently be in a state of equilibrium (the rod E F being supposed to be with- out weight). NoAv, it is proposed to show that i^ is to P as C F is to C E. This can be proved; for, since R equals the area AD G K and P equals the area DBKH, therefore R is in proportion to A D, ?iS P is to D B {Art. 359) ; or, taking the halves of these lines, R is in proportion to A y as P is to LB. Also, y L equals half the length of the beam ; for J D is the half of A D, and D L \s the half oi DB\ thus these two parts {JD + DL) equal the half of the two parts (AD-\-D B)\ or, J L equals the half of A B ; or, we have — •^ 2 ' 2 Adding these two equations together, we have — Now, JD-r DL^ 7L, and AD + DB = AB', therefore, Thus we have A M — J L. From each of these equals take J M, common to both, then the remainders, A J and ML, will be equal ; therefore, AJ—CF. We have also MB z=z J L. From each of these equals take ML, common to both, and the remainders, JM and L B, will be equal ; therefore, LB = E C. As Avas above shown — R '. AJ '.'. P '. LB. TO FIND A FOURTH PROPORTIONAL. 377 Substituting for A y and LB their values, as just found, we have — R : CF :: P : EC; from which we have {Art. 373) — FxCF=:Rx£C Thus it is demonstrated that the product of one weight into its arm of leverage, is equal to the product of the other weight into its arm of leverage : a proposition which is known as the law of the lever. 378. — Any One of Four Proportionals may be Found. — Any three of four proportionals being given^ the fourth may be found ; for either one of the four factors may be made to stand alone ; thus, taking the equation of the last article, if we divide both members by CF {Art, 371), we have — Px CF _ RxEC CF ~ CF ' In the first member C F, in both numerator and denominator, cancel each other {Art. 371), therefore — „ RxEC CF ' so likewise we may obtain— R^ CF^ EC^ Px CF EC RxEC - P px CF - rj • R w^^mm^mm^m^mmi^m^^m^m^mm SECTION VIII.— FRACTIONS. 379. — A Fraction ©efined. — As a fracture is a break or division into parts, so a fraction is literally a piece broken off; a part of the whole. The figures which are generally used to express a frac- tion show what portion of the whole, or of an integer, the fraction is : for example, let the line A B, {Fig. 274), be divided into five equal parts, then the line A C, containing three of those parts, will be three fifths of the whole line A B, and may be expressed by the figures 3 and 5, placed thus, -, which is known as a fraction and is read, three fifths. The number 5 below the line denotes the number of parts into which an integer or unit, A B, is supposed to be divided ; it 1 I DEC B Fig. 274. is therefore called the denoviinator, and expresses th3 denom- ination or kind, whether fifths, sixths, ninths, or any number, into which a unit is supposed to be divided. The number 3 above the line, denoting the number of parts contained in the fraction, is termed the imvierator, and expresses the number of parts taken, as 2, 3, 4, or any other number. 380. — Crrapliieal llcpresciitatioii of Fractions : Effect of MultipBication. — In Fig. 275, let the line A B h^ di- vided into three equal parts ; the line CD into six equal parts; the line EF into nine equal parts; the line 6^ //into twelve equal parts, and the line J Kx^X-O fifteen equal parts. The lines AB, CD, EF, GH, and J K, being all of equal length. FRACTIONS ILLUSTRATED. 379 Then the parts of these hnes, A L, CM, EN, etc., may be expressed respectively by the fractions-,^,-, - and ---. ^ ^ •" -^ 3 6 9 12 15 In each case the figure below the line, as, 3, 6, 9, 12, or 15, expresses the number of parts into which the whole is di- vided, and the figure above the line, as 1,2, 3, 4, or 5, the I I I I I Fig. 275. number of the parts taken ; and, as the lines A L, CM, EN, etc., are all equal to each other, therefore these fractions are all equal to each other. If the numerator and denominator of the first fraction be each multiplied by 2, the products will equal the numerator and denominator of the second fraction ; thus — so, also, I X 2 nz 2 3 X -7 = 6 I X 3 = 3. 3 X 3 :=. 9 I X 4 — 4 3 X 4 — 12 I X 5 — 5 3 X 5 = 15 and and Thus it is shown that when the numerator and denomi- nator of a fraction are each multiplied by the same factor, the product forms a new fraction which is of equal value with the original. In like manner we have, -, — , --, --, etc., each equal to 8 12 16 20 one fourth ; and which may be found by multiplying the numerator and denominator of - successively by 2, 3, 4, 5, etc. 4 38o FRACTIONS. 381. — Form of Fraction Chang^cd toy Division. — By an operation the reverse of that in the last article, we may re- duce several equal fractions to one of equal value. Thus, if in each we divide the numerator and denominator by the same number, we reduce it to a fraction of equal value, but with smaller factors. For example, taking the fractions of the last article, f , -f, tV» tV> ^^t each be divided by a number which will divide both numerator and denominator without a remainder.* Thus, 2 -^2 = I 6-2 = 3' 3-f-3 = I 9-3 = 3 4^4= I 5-^5 = 1 I2-^4 = 3'' 15-^5 = 3 As these fractions are shown {Art. 380) to be equal, and as the operation of dividing each factor by a common num- ber produces quotients which in each case form the same fraction, i, we therefore conclude that the numerator and denominator of a fraction may be divided by a common number without changing the value of the fraction. 382. — Improper Fractions, — The fractions f , ^, f, etc., all fractions which have the numerator larger than the de- nominator are termed improper fractions. They are not im- proper arithmetically, but they are so named because it is an improper use of language to call that depart which is greater than the whole. As expressions of this kind, however, are subject to the same rules as those which are fractions proper, it is custom- ary to include them all under the technical X.^r\x\oi fractions. Expressions like these — all expressions in which one number is separated by a horizontal line from another number below it, or one set of numbers is thus separated from another set below it — may be called fractions, and are always to be un- derstood as . indicating division, or that the quantity above the line is to be divided by the quantity below the line. * Division is indicated by this sign -j-, which is read "divided by." IMPROPER FRACTIONS. 38 1 ^, Q 17 24 3 X 8 X 4 17X 82 ^ „ , . ^ . Thus,-. —^1 -~^> -y — ' etc., are all fractions, tech- 3 5 3 2 X 12 125 ' ' nically, although each may be greater than unity. And it is understood in each case that the operation of division is re- 1 T-i 9 24 ^3x8x4 ^,^, , ,. . quired. 1 hus, - = 3, ^ — — 8, = 4. When the divis- ^ 3 3 2x12 ^ ion cannot be made without a remainder, then the fraction, by cutting the numerator into two, may be separated into two parts, one of which may be exactly divided, and the other 1 7 will be a fraction proper. Thus, the fraction — ■ is equal to J c 2 I c h— (for 15 + 2 — 17) ; and since — equals 3, therefore, 17 15 2 '2 2 — - =-— + -= 3+-= 3-. So, likewise, the fraction 17x82 ^1394^ 1375^ 19 ^ ^^ _^ J^_^ jj J_9_ 125 125 125 125 125 125- 383. — Reduction of Mixed Niinitoers to Fractions. — By an operation the reverse oi that in the last article, a given mixed number (a whole number and fraction) can be put into the form of an improper fraction. This is done by multiplying the whole number by the de- nominator of the fraction, the product being the numerator of a fraction equal in value to the whole number ; the de- nominator of this fraction being the same as that of the given fraction. The numerator of this fraction being added to the numerator of the given fraction, the sum will be the numera- tor of the required improper fraction, the denominator of which is the same as that of the given fraction. For example, the required numerator for — 2I is 2 X 3 + I = 7. So 2I = |. 2j, is 2 X 4 + I = 9. So 2^ = f . 3f, is 3 X 5 +2 = 17. So 3-1 = 'i. 384-. — Division Indicated by the Factors put as a Frac- 2 c 1 20 lion. — Factors placed in the form of a fraction as -, -, -^^or ^ 5 3 75 382 FRACTIONS. indicate division {Art. 382) ; the denominator (the fac- tor below the line) being the divisor, and the numerator (the factor above the line) the dividend, while the value of the fraction is the quotient. Thus of the fraction, -^- = 20, 41 is the divisor, 820 the dividend, and 20 the quotient. From this we learn that division may always be indicated by placing the factors in the form of a fraction, so that the divisor shall form the denominator and the dividend the nu- merator. 385. — Aenominator§ B^iualized. — The rules just given for the addition and subtraction of fractions re- quire that the given fractions have like denominators. When the denominators are unlike it is required, before add- ing or substracting, that the fractions be modified so as to make the denominators equal. For example ; Let it be re- 2 2 quired to find the sum of - and --. Bv reference to Fio-. 3 9- 2 . 6 275, we find that - on line A B is equal to- on line E F. 3 9 6 2 These being equal, we may therefore substitute - for -. Then we have — 6 2 _ 8 9 ^ 9 "~ 9 384 FRACTIONS. Now, it will be seen that the fraction - may be had by mul- tiplying both numerator and denominator of the given frac- 2, , 2x3=6 tion - by 3, for ^ ^ ;^ _ - ; and we have seen {Art. 380) that this operation does not change the value of the fraction. From this we learn that t/ie denominators may be made equal by multiplying the smaller defwminator and its numerator by any number wJiich will effect such a result. For example 15 15"^ 15 15 27 14 7 21 and --+ — = — + — = — ; 5 35 35 35 35 ' '^'''^ 4 12 "^ 16 - 16 + 16 +T6 - 16 - ' i6- In this example the second fraction is changed by multiply- ing by i^. 388, — Reduction of Fractions to tlieir L-oivest Terms. — The process resorted to in the last article to equalize the denominators, is not always successful. What is needed for a common denominator is to find the smallest number which shall be divisible by each of the given denominators. Before seeking this number, let each given fraction be reduced to its lowest terms, by dividing each factor by a common number. For example : — may, by dividing by 5, I . . . 21 . be reduced to -, which is its equivalent. So, also, — , by di- 3 2b 3 viding by 7, is reduced to -, its lowest terms. 389. — Licast Common Denominator. — To find the least common denominator, place the several fractions in the order of their denominators, increasing toward the right. If the largest denominator be not divisible by each of the others, double it ; if the division cannot now be performed, treble LEAST COMMON DENOMINATOR. 385 it, and so proceed until it is multiplied by some number which will make it divisible by each of the other denomina- tors. This number multiplied by the largest denominator zvill be the least common denominator. To raise the denominator of each fraction to this, divide the common denominator by the de- nominator of one of the fractions, the quotient will be the number by which that fraction is to be multiplied, both numerator and denominator, and so proceed with each frac- tion. For example : What is the sum of the fractions -, -, — , - ? One of these, — , may be reduced, by divid- 2' 4' 12' 8 ' 12' -^ * -^ ing by 2, to -p. Therefore, the series is -, -, ^, -. On trial we find that 8, the largest denominator, is divisible by the first and by the second, but not by the third, therefore the largest denominator is to be doubled : 2x8=16. This is not yet divisible by the third ; therefore 3 x 8 = 24. This now is divisible by the third as well as by the first and the second ; 24 is therefore the least common denominator. Now dividing 24 by 2, the first denominator, the quotient 12 is the factor by which the terms of the first fraction are 1 X 12= 12 to be raised, or, - " -^. For the second we have 2 X 12 = 24 24 -f- 4 = 6, and - ^ — . For the third we have 24 -j- 6 = ^^ 4x6 = 24 ^ 4, and ^ _ — ; and for the fourth, 24-^8 = 3, and 0x4 — ^4 7 X ■^ =: 2 I Thus the fractions in their reduced form are : x3 =24 12 18 20 21 71 23 24 24 24 24 "" 24 ~ 24' 390. — L.ea$t Common Denominator Ag^ain. — When the denominators are not divisible by one another, then to ob- tain a common denominator, it is requisite to multiply to- gether all of the denominators which zvill not divide any of the other denominators. For example : What is the sum of the fractions -, -, -, and -? 386 FRACTIONS. In this case the first denominator will divide the last, but the others are prime to each other. Therefore, for the common denominator, multiply together all but the first ; or — 5 x7x9 = 3i5 the common denominator ; and — 315-^3 = 105, common factor for the first fraction ; 315 -^ 5 =63, common factor for the second fraction ; 315--7 = 45> common factor for the third 315-^-9 = 35, common factor for the fourth. And, then — _i X 105 = 105 ^ ^ X 63 = 1^26 _3 X 45 = 1^5 _ 4 X 35 = 240 3 X 105 = 315 ' 5 X 63 = 315 ' 7 X 45 = 315 ' 9 X 35 = 315 105 126 "31^5 "^ 3^15 i_35 315 140 3"^ 506 _ 191 391. — Fractions Multiplied Graphically. — Let A B C D {Fig. 276) be a rectangle of equal sides, or A B equal A C and each equal one foot. Then A B multiplied hy A C will Fig. 276. equal the area A B C D, or i x i = i square foot. Let the line B Fhe parallel with A B, and midway between A B and CB. Then AB x A £ equals half the area of AB CD, or J X I = J^. Again ; let G H be parallel with E C, and mid- way between E C and FD. Then EGy.EC = iy.i equals the area E G C H, which is equal to a quarter of the area MULTIPLICATION OF FRACTIONS. ,87 A B C D; or i X i = i; which is a quarter of the siiperhcial area. The product here obtained is less than either of the factors producing it. It must be remembered, however, that while the factors represent /hies, the product represents superficial area. The correctness of the result may be recognized by an inspection of the diagram. 392. — Fractions Multiplied Graphically. — In i^^> 277 let A B equal 8 feet and A C equal 5 feet ; then the rect- F A a fl Fig. 277. angle A BCD contains 5 x 8 = 40 feet. The interior lines divide the space included within A B CD into 40 equal squares of one foot each. Let A E equal 3 feet or - of ^ C. Let A G equal 7 feet or | of A B. Then the rectangle 1 1 ? T E F A G contains -- x (- = — , or twenty-one fortieths of the 5 8 40 whole area ABC D, Thus, while the factor fractions -; and - 21 represent lines, it is shown that the product fraction -— rep- ■^ T resents surface. Thus — is a fraction, E FA G, of the whole 40 surface, C D A B. 393.— Rule for Mutiplication of Fractions, and Exam- ple. — In the example given in the last article it will be ob- 388 FRACTIONS. served that the product of the denominators of the two given fractions equals the area of the whole figure {A B CD), while the product of the numerators equals the area of the rectangle {E F A G), the sides of which are equal respec- tively to the given fractions. From this we obtain for the product of fractions this — Rule. — Mtdtiply togetJier the denominators for the new de- nominator, and tJie numerators for a new numerator. For example : what is the product of —and — ? Here w^e have 20x21=420 for the new denominator, and 7 X 13 = 91 for the new numerator ; therefore the product of— 2 1 20 420 ' or, of a rectangular area divided one way into 20 parts and the other way into 21 parts, thus containing 420 rectangles, 13 7 the product of the two fractions — and — is equal to 91 of these rectano:les, or of the whole. ^ ' 420 394. — Fractions Divided Graphically. — Division is the reverse of multiplication ; or, while multiplication requires the product of two given factors, division requires one of the factors when the other and the product are given. Or (referring to Fig. 277) in division we have the area of the rectangle, E FA G, and one side, E A, given, to find the other side, A G. Now it is required to find the number of times E A \s contained in E FA G. By inspection of the figure we per- ceive the answer to be, A G times ; for E A y. A G — E FA G, 2 1 the given area. Or, when E A EG is given as — and E A as -, we have as the given problem — 40 DIVISION OF FRACTIONS. 3S9 Since division is the reverse of multiplication, instead of multiplying we divide the factors, and have — 21 -^ ^ 7 40-^ 5 « Thus, to divide one fraction by another, for the numerator of the required factor, divide the numerator of tJie product by the numerator of the given factor, and for the denominator of the required factor divide the denominator of the product by the denominator of the given factor. For example : 10 2 5 Divide 7- by -. Answer, -. 63-9 7 Divide ^^ by — . Answer, — . 27 -^ 9 3 395. — Rule for Divmoii of Fractions. — The rule just given does not work well when the factors are not commen- • • 5 2 surable. For example, if it be required to divide - by - we ^ 7-^9 have by the above rule — _5_ 5 -=- 2 _ 2 7-9 ~ y * 9 Producing fractional numerators and denominators for the resulting fraction, which require modification in order to reach those composed only of whole numbers. If the nu- merators, 5 and 7, of this compound fraction be multiplied by 9 (the denominator of the denominator fraction), or the compound fraction by 9, we shall have — 5 5x9 X 9 7 7x9 390 FRACTIONS. And, if these be again multiplied by 2 (the denominator of the numerator fraction), we shall have — 5x9 5x9x2 2 2 X 2 = 7x9 7x9x2 9 9 Like figures above and below in each fraction cancel each other {Art. 371), therefore, the result reduces to — 5 X 9 7x2' in which we find the factors of the two original fractions. In one fraction — we have the factors in position as given, 2 but in the other — they are inverted. The fraction in which the factors are inverted is the divisor. Hence, for division of fractions, we have this — Rule. — Invert the factors of the divisors and then, as in multiplication^ multiply the numerators together for the numera- tor of the required fraction, aiid the denominators for the de- nominator of the required fractiofi. f . 2 Thus, as before, if - is required to be divided by -, we have — 5_x9^45 7 X 2 14* 2"^ 7 And, to divide — by — , we have — 47 ^ 9 23 X 9 _ 207 "47 X 7 "" 329* As^ain, to divide -- by -, we have — ^ 45 -^ 9 _2 s X 9 ^ 225 ^ ^ ^ ^ 45 X 8 ~ 360 ~ 40 ~ 8* CANCELLING IN ALGEBRA. 39I This last example has two factors, 9 and 45, one of which 45 25 measures the other ; also, the first fraction — is not in its lowest terms ; when reduced it is — . The question, there- fore, may be stated thus : 5 x9 _ 5 9 X 8 ~ 8' for the two 9's cancel each other. SECTION IX.— ALGEBRA. 396. — Algebra Dcflned. — It occurs sometimes that a student familiar only with computation' by numerals is needlessly puzzled, in approaching the subject of Algebra, to comprehend how it is possible to multiply letters together, or to divide them. To remove this difficulty, it may be suf- ficient for them to learn that their perplexity arises from a misunderstanding in supposing the letters themselves are ever multiplied or divided. It is true that in treatises on the subject it is usual to speak as though these operations were actually performed upon the letters. It is always un- derstood, however, that it is not the letters, but the qtian- tities represented by the letters, which are to be multiplied or divided. For example, in Art. 361 it is shown, in comparing similar sides of homologous triangles, that the bases of the two tri- angles are to each other as the corresponding sides, or, referring to Fig. 269, Ave have C E -. A E -. -. D E \ B E. Now, let the two bases C E and A E he represented respec- tively by a and d, and the two corresponding sides E) E and B E by c and d respectively ; or, for — CE : AE : : DE : BE, put — a \ b \ \ c \ d\ and, by Art. 373, we have — b X c ^ ax d^ which may be written — be =^ ad', for X, the sign for multiplication, is not needed between let- ters, as it is between numeral factors. The operation of APPLICATION OF ALGEBRA. 393 multiplication is always understood when letters are placed side by side. Now, here we have an equation in which, as usually read, we have the product of b and c equal to the product of a and d. But the meaning is that the product of the quantities represented by b and c is equal to the product of the quan- tities represented by a and d, and that this equation is in- tended to represent the relation subsisting between the four proportionals, C E, A E, D E, and BE, o{ Eig. 269. In order to secure greater conciseness and clearness, the four small letters are substituted for the four pair of capital letters, which are used to indicate the lines of the figures referred to. 397- — Example : Application. — It was shown in the last article that the four letters a, b, c, and d represent the cor- responding sides of the two triangles of Eig. 269, and that — b c ^ a d. Now, let each member of this equation be divided by a, then {Art. 371)— a If now the dimensions of the three sides represented by a, b, and c are known, and it is required to ascertain from these the length of the side represented by d, let the three given dimensions be severally substituted for the letters repre- senting them. For example, let a = 40 feet; b = ^2 feet, and c = 4S feet ; then — be 52 X 45 2340 d = — = ^—^= -^^ = 58.5 feet. a 40 40 -^ -^ The quantities being here substituted for the letters ; we have but to perform the arithmetical processes indicated to obtain the arithmetical value of d. From this example it is seen that before any practical use can be made of an algebraical formula in computing dimensions, it is requisite to substitute numerals for the letters and actually perform arithmetically such operations as are only indicated by the letters. 394 ALGEBRA. 398, — Alg^ebra Useful in Constructing Rules. — In all problems to be solved there are certain conditions or quan- tities given, by means of which an unknown quantity is to be evolved. For example, in the problem in Art. 397, there were three certain lines given to find a fourth, based upon the condition that the four lines were four proportionals. Now, it has been found that the relation between quantities and the conditions of a question can better be stated by let- ters than by numerals ; and it is the office of algebra to present by letters a concise statement of a question, and by certain processes of comparison, substitution and elimina- tion, to condense the statement to its smallest compass, and at last to present it in a formula or rule, which exhibits the known quantities on one side as equal to the unknown on the other side. Here algebra ends, at the completion of the rule. To 2ise the rule is the office of arithmetic. For, in using the rule, each quantity in numerals must be substi- tuted for the letter representing it, and the arithmetical processes indicated performed, as was done in Art. 397. 399.— Algebraic Rules are General. — One advantage derived from algebra is that the rules made are general in their application. For example, the rule of Art. 397, — = d, is applicable to all cases of homologous triangles, however they may differ in size or shape from those given in Fig. 269 — and not only this, but it is also applicable in all cases where four quantities are in proportion so as to con- stitute four proportionals. For example, the case of the four proportionals constituting the arms of a lever and the weights attached {Arts. 375-378). For, taking the rela- tion as expressed in Art. 377 — PxCF= RxEC, we may substitute for C F the letter n, and for E C the letter in, then 7?t will represent the arm of the lever E C {Fig. 262), and n the arm of the lever F C. Then we have — Pn^Rm, SYMBOLS CHOSEN AT PLEASURE. 395 and from this, dividing by 11 {A^rt. 372), we have — P=.R-) (no.) or, dividing by m, we have — R = ^'-, (.11.) which is a rule for computing the weight of R, when P and the two arms of leverage, 7n and n, are known. For example, let the weight represented by P be 1200 pounds, the length of the arm m be 4 feet, and that of n be 8 feet, then we have — Pn 1200x8 , R — — ■ = • = 2400 pounds. m 4 t r Pn This rule, R = — , is precisely like that in ArL 397 — — = d — in which three quantities are given to find a fourth, the four constituting a set of four proportionals. 400. — Symbols Chosen at Pleasure, — The particular letter assigned to represent a particular quantity is a matter of no consequence. Any letter at will may be taken ; but w^hen taken, it must be firmly adhered to to represent that par- ticular quantity, throughout all the modifications which may be requisite in condensing the statement into which it enters into a formula for use. For example, the two rules named in Art. 399 are precisely alike — three quantities given to find a fourth — yet they are represented by different letters. In one, /? and /^ represent the two weights, and 7n and ?/ the arms of leverage at which they act ; while in the other the letters a, <^, <:, and <^ represent severally the four lines which constitute two similar sides of two homologous triangles. The two rules are alike in working, and they might have been con- stituted with the same letters. And instead of the letters chosen any others might have been taken, which con- venience or mere caprice might have dictated. In some 39^ ALGEBRA. questions it is usual to put the first letters, as a, b, c, etc., to represent known quantities, and the last letters, as x, y, z, for the quantities sought. In works on the strength of materials it is customary to represent weights by capital letters, as P, R, U, W, etc., and lines or linear dimensions by the small letters, as b, d, /, for the breadth, depth, and length, respectively, of a beam. Any other letters may be put to represent these quantities, although the initial letter of the word serves to assist the memory in recognizing the partic- ular dimensions intended. 401.^ — Arithmetical Proce§§e§ Indicated by Signs. — In algebra, the four processes of addition, subtraction, multi- plication, and division, are frequently required ; and when the required process cannot be actually performed upon the letters themselves, a certain method has been adopted by which the process is indicated. For example, in additon, when it is required to add a to by the two letters cannot be intermingled as numerals may be, and their sum presented ; but the process of addition is simply indicated by placing between the two letters this sign, +, which is called plus, meaning added to ; therefore, to add ^ to ^ we have — a^b, Avhich is read a plus b, or the sum of a and b. When the quantities represented b}^ a and b are substituted for them — and not till then — they can be condensed into one sura. For example, let a equal 4 and b equal 3, then for — a-^b we have — 4+3; and we may at once write their sum 7, instead of 4+ 3. So, likewise, in the process of subtraction, one letter can- not be taken from another letter so as to show how much of this other letter there will be left as a remainder ; but the process of subtraction can be indicated by a sign, as this, — , which is called minus, less, meaning subtracted from. For ALGEBRAICAL SIGNS. 39/ example, let it be required to subtract b from a. To do this we have — a — b\ which is read a minus b, and when the values of a and b are substituted for them, we have, when a equals 4, and b equals 3 — a — b, or — 4-3; and now, instead of 4 — 3, we may put the value of the two, which is unit}^, or i. The algebraic signs most frequently used are as follows : + , //?^^, signifies addition, and that the two quantities be- tween which it stands are to be added together ; as a-^b, read a added to b. — , minus, signifies subtraction, or that of the two quantities between which it occurs, the latter is to be subtracted from the former ; 2iS> a — b, read a minus b. X, multiplied by, or the sign of multiplication. It denotes that the two quantities between which it occurs are to be multiplied together ; as <^ x ^, read a multiplied by b, or a times b. This sign is usually omitted between symbols or letters, and is then understood, as a b. This has the same meaning as a x b. It is never omitted between arithmetical numbers; as 9x5, read nine times five. -^, divided by, or the sign of division, and denotes that of the two quantities between which it occurs, the former is to be divided by the latter ; as a-^b, read a divided by b. Division is also represented thus : -, in the form of a fraction. This signifies that a is to be divided by b. When more than one symbol occurs above or below the line, or both, as , it denotes evi that the product of the symbols above the line is to be divided by the product of those below the line. 39^ ALGEBRA. = , is equal to, or sign of equality, and denotes that the quantit}^ or quantities on its left are equal to those on its right ; as ^ — ^ = r, read a minus b is equal to c, or equals c \ or, 9 — 5 = 4, read nine minus five equals four. This sign, together with the symbols on each side of it, when spoken of as a whole, is called an equation. a" denotes a squared, or a multiplied by a, or the second power of a, and a^ denotes a cubed, or a multiplied by a and again multi- plied by a, or the third power of a. The small figure, 2, 3, or 4, etc., is termed the index or exponent of the power. It indicates how many times the symbol is to be taken. Thus, a^ — a a, a^ — a a a, a" — a a a a. \/ is the radical sign, and denotes that the square root of the quantity following it is to be extracted, and 4/ denotes that the cube root of the quantity following it is to be extracted. Thus, 1/9 = 3, and ^\/2j = 3. The extraction of roots is also denoted by a fractional in- dex or exponent, thus — rt:^ denotes the square root of a, a^ denotes the cube root of a, a^ denotes the cube root of the square of a, etc. 402a — ExampSe in Addition and ISubtraction : Cancel- ling. — Let there be some question which requires a state- ment to represent it, like this — a + d =: c — b, which indicates that if the quantity represented by a be added to the quantity represented by d, the sum will be equal to the quantity represented by c, after there has been subtracted from it the quantity represented by b ; or, as it is usually read, a plus d equals c minus b; or the sum of a and d equals the difference between c and b. For illustration, take in place of these four letters, in the order they stand, the numerals 4, 2, 9, 3, and we shall have by substitution — a -{- d ^= c — b, 4+2 = 9 — 3, or adding and subtracting — 6 — 6. TRANSFERRING SYMBOLS. 399 If it be required to add to each member of the equation the quantity represented by b, this will not interfere with the equality of the members. For a^d are equal to f — d, and if to each of these two equals a common quantity be added, the sums must be equal ; therefore— a + d+b =c — b-^by or by numerals — 4 + 2 + 3 = 9-3 + 3» or — 9 = 9- It will be observed that the right hand member contains the quantity — b and + b. This shows that the quantity b is to be subtracted and then added. Now, if 3 be subtracted from 9, the remainder v/ill be 6, and then if 3 be added, the sum will be 9, the original quantity. Thus it is seen that whejz in the same member of an equation a symbol appears as a minus quantity and also as a plus quantity^ the two cancel each other, and may be omitted. Therefore, the expression — a + d + b ^=^ c — b + b becomes — a + d+ b —c. 403. — Transferring a Symbol to the Opposite Meintoer. — In comparing, in the last article, the first equation with the last, it will be seen that the same symbols are contained in each, but differently arranged : that while in the first equa- tion b appears in the right hand member and with a minus or negative sign, in the last equation it appears in the left hand member and with a plus or positive sign. Thus it is seen that in the operation performed b has been made to pass from one member to the other, but in its passage it has been changed. A similar change may be made with another of the symbols. For example, from the last equation, let d be subtracted, or this process indicated, thus — a-\-d-\-b — d^=^c — d. 400 ALGEBRA. The plus and minus d, in the left hand member cancel each other, therefore — a-\-b =^ c — d^ or, by numerals — 4 + 3 = 9-2. Reducing — By this we learn that any quantity (connected by + or — ) may be passed from one member of the equation to the other, pro- vided the sign be changed. 404. — Sig^ns of Symtools to be Changed wlien tliey are to toe Subtracted, — As an example in subtraction, let the quantities represented hy ^-b — a — /+ c, be taken from the quantities represented by +a+b — c—f This may be written — {+a + b — c —f) — {-{-b — a — /+ c), an expression showing that the quantities enclosed within the second pair of parentheses are to be subtracted from those mcluded within the first pair. Let the quantities represent- ed in the first pair of parentheses for convenience be repre- sented by y4 , or, a + b — c — / = A . Now, by the terms of the problem, we are required to subtract from A the quantities enclosed within the second pair of parentheses. To do this take first the positive quantity, b, and subtract it or indicate the subtraction, thus — A-b', we will then subtract the positive quantity c, or indicate the subtraction, thus — A-b-c. We have yet to subtract — a and — /, two negative quanti- ties. The method by which this can be accomplished may be discovered by considering the requirements of the problem. The plus quantities b and c, before being subtracted from A, were required to have the two negative quantities <3: and /de- THE OPERATION TESTED. 4OI ducted from them. It is evident, therefore, that in subtract- ing b and c, before this deduction was made, too much has been taken from A, and that the excess taken is equal to the sum of a and /. To correct the error, therefore, it is neces- sary to add just the amount of the excess, or to add the sum of a and/", or annex them by the plus sign, thus — A — b — c + a+f. To test the correctness of the operation as here performed, let numerals be substituted for the symbols ; l^t a-=2,b— 3, c =z i^ f— ^] then the given quantities to be subtracted, — (+b — a—f+c), become — (+3-2-1+1), which reduces to — (4 - 2i) = li. Thus the quantity to be substracted equals i^. Applying the numerals to the above expression — A — b+a-^f— c becomes — A — 3 + 2 +i— I = A — 4-^2^ = A — i^. A correct result ; it is the same as before. Restoring now the symbols represented by A, we have for the whole ex- pression — + a+ b — c —f— b + a+f— c, which, by cancelling {Art. 403) and by adding like symbols with like signs, reduces to — 2 a — 2 c. To test this result, let the quantity which was represented; by A have the proper numerals substituted, thus : + a-^b — c —/, 402 ALGEBRA. The sum of the given quantity required to be subtracted was before found to amount to i^^, therefore — A-ii becomes — 3^-11=: 2. And the result by the symbols as above was — 2 a — 2 c, which becomes — 2X2 — 2X1, or — 4 — 2 = 2; a result the same as before, proving the work correct. An examination of the signs in the above expression, which de- notes the problem performed, will show that the sign of each symbol which was required to be subtracted has been changea in the operation of subtraction. Before subtract- ing they were — (+/^ — a—/+c); after subtraction they are — {— d + a-rf— c). By this result we learn, that to sttbtract a quantity we have but to change its sign and annex it to the quantity from which it was required to be subtracted. Example : Subtract a — b from c + d. Answer, c + d— a\ b. If numerals be substituted, say a— J,b— a,, c— ^, and d—<^, then — c + d becomes 5+9= 14, a-b - ;_4=3, So, also, — becomes — c ^ d — {a — b') ^= 14 ■ c + d — a-\- b 5+9-7 + 4 = FRACTIONS ADDED AND SUBTRACTED. 403 405. — Algebraic Fractions: Added and Subtracted. — When algebraic fractions of like denominators are to be added or subtracted, the same rules {Arts. 385 and 386) are to be observed as in the addition or subtraction of numeri- cal fractions — namely, add or subtract the numerators for a new numerator, and place beneath the sum or difference the common denominator. For example, what is the sum of -7,-7,-7? 000 For this we have — ~b * Subtract -j from ^. For this we have — b-c d * What is the algebraical sum of — be 1^ J 1: d^ - d^ ^"^ For these we have — b + c — 7t — r ~d • To exemplify this, let b represent 9, ^ = 8, ;/ = 2, r = 3, and d= 12. Then, for the algebraic sum, we have — 9 + 8 — 2 — 3 _ 12^ _ 12 ~ \2~ ^' Now, taking the positive and negative fractions sep- arately, we have — -9 , A. ^ iZ . 12 12 12 ' and — -2 -3^-5 12 12 12 ' r? 404 ALGEBRA Tog-ether — i; -5 ^ 12 ^ ^ 12 12 12 ' as before. 406, — The Lieast Common Denominator. — When the denominators of algebraic fractions differ it is necessary be- fore addition or subtraction can be performed to harmonize them, as in the reduction of the denominators of numerical fractions {Arts. 388-390). For example, add together the tractions v— , y, — . In these denommators we perceive that they collectively contain the letters a, b and c, and no others. It will be requisite, therefore, that each of the frac- tions be modified so that its denominator shall have these three factors. To effect this it will be seen that it is neces- sary to multiply each fraction by that one of these letters which is lacking in its denominator. Thus, in the first, a is 11- 1 r r ^ ^ a yi a =^ a a ^ . . lacking, therefore {Art. 380) j— _ -y-. In the second a (J C y^ CL ' CI' C/ € y^ Ct C — CL C € and c are lacking, therefore j _ —7-, and in the third T X b ^=^ T b b is lackinsr, therefore — r —r- Placing them now ^' ac y. b — abc ^ together we have — aa + ace + br _ a e r a be. ~ b c b ac The factor a a may be represented thus a", which means that a occurs twice, the small figure at the top indicating the number of times the letter occurs ; ^' is called a squared, a a a = a^y and is called a cubed. In order to show that the above fraction, resulting as the sum of the three given fractions, is correct, let ^ = 2, ^ = 3, c = 4, ^ = 5, and r = 6. Then the three given fractions are — 3x4 3 2x4 6 3 4 FRACTIONS SUBTRACTED. 405 In equalizing these denominators we multiply the second fraction by 2, and the third by i^^, which will give — then- Now the sum of the fractions is — a"^ + ace + br ab c 5 X 2 = ID 3 X li 4* 3 X 2 = 6 '4x I* 6 ' I . ID 4* I :^*' ^* 7 6 + 6 -^6 ■ — ~ 6 ~ : 2 6 = 2 12' or. 2^ + 2x4x5 + 3x6 2x3x4 4 + 40+18 62 14 7 or, =:^ ■ z=: 2 = 2 ! 24 24 24 12 ' the same result as before, thus showing that the reduction was rightly made. 407- — Algebraic Fractions Subtracted. — To exemplify the subtraction of fractions, let it be required to find the algebraic sum of - — -> — -p. These denominators all dif- cdf fer. The fractions, therefore, require to be modified, so that each denominator shall contain them all. To accom- plish this, the first fraction will need to be thus treated : a X df= adf the second — c X df= cdf bxcf= bcf the third— dxcf= cdf The sum of these is- e X c d = cde f X c d - cdf adf — bcf— cde cdf 406 ALGEBRA. That this is a correct answer, let the result be proved by figures ; thus, for ^ put 15 ; ^, 2 ; <:, 3 ; ^,4; ^, 5 ; /, 6. Then we shall have — a b ^ _ 15 2 5 c ^~'7~3~4~6* It will be observed that these denominators may be equal- ized by multiplying the first fraction by 2, and . the second by \\, therefore we have — JO _ 3 _ 5. 6 ^ ^' To make the required subtraction we are to deduct from 30 (the numerator of the positive fraction), first 3, then 5 ; or, the sum of the numerators of the negative fractions ; or for the numerator of the new fraction we have 30 — 8 = 22. The required result, therefore, is — 6-3-38. To apply this test to the algebraic sum we have — a d f — bcf— cde_ 15x4x6 + 2x3x6+3x4x5 c df ~ 3x4x6 ' which by multiplication reduces to — 360 — 36 — 60 264 22 1 1 72 72 6 3 = 31 a result the same as before, proving the work correct. An- other example : From take -, — and - ; 71 VI n vt n or, find the algebraic sum of— a ,b c d e n ni 71 m ti a c e a — c — e n n n ~ n b d b-d — z=. 7)1 m m DENOMINATORS HARMONIZED. 407 The fractions which have the same denominator may be grouped together thus : and- To harmonize these two denominators, m and n^ the first fraction must be multiplied by vi and the last by n, or — m(a — c — e) n {b — <:/) _ in {a — c — e) + n {b —d) 7n n ' vi n ~ m n In the polynomialfactor within the parentheses (^ — c — e)viQ have the positive quantity a, from which is to be taken the two negatives c and ^, or their sum is to be taken from a^ or {a — {c ^ e) ). With this modification we have for the alge- braic sum of the five given fractions — m {a — {c + c)) + n {b — d) mn To test the accuracy of this result, let the value of the sev- eral letters respectively be as follows: ^ = ii,/^ = 9, c = 3^ ^ = 4, ^ = 5, ?;^ = 10, and n = 8. Then the sum is — 10(11 -(3 + 5)) + 8(9-4) ^70^7 10 X 8 80 8* n n n 8 \8 8/ 8 8 ~ 8 Now, taking the fractions separately, we have — \8 ^ 87 8 b d _ g 4 5 ' m m ~ 10 ~ ~io ~ To' or, together we have, as the sum of these two results- 8 ^ 10 4o8 ALGEBRA. To harmonize these denominators we may multiply the first fraction by 5, and the second by 4, thus: 8 X 5 = 40' 10 X 4 =40' and then the sum is — il ^ _ il — Z. 40 "^ 40 ~" 40 ~ 8 ' the same result as before, thus the accuracy of the work is established. 4-08, — Graphical Representation of Multiplication. — In Fig, 278, let ABCD,2i rectangle, have its sides A B and Fig. 278. A C divided into equal parts. Then the area of the figure will be obtained by multiplying one side by the other, or putting a for the side A By and b for the side A C, then the area will be a x b, or a b. This will be the correct area of the figure, whatever the length of the sides may be. If, as shown, the area be divided into 4 x 7 == 28 ^ual rectangles, then a would equal 7, and b equal 4, and ^ ^ = 7 x 4 = 28, the area. \i A B equal 28 and A C equal 16, then will a = 28, and b = 16, and a b = 2S x 16 = 448, the area. 4-09. — Graphical Multiplication : Three Factors. — Let A B CD E FG {Fig. 279) represent a rectangular solid which may be supposed divided into numerous small cubes as shown. Now, if a be put for the edge A B^b for the edge A C, and c for the edge CD, then the cubical solidity of the MULTIPLICATION OF A BINOMIAL. 409 whole figure will be represented hy axbxc = abc. If the edge A B measures 6, the edge A C '^, and the edge (7i> 4, then ^^^==6x3x4— 72 = the cubic contents of the figure, or the number of small cubes contained in it. A B ////// /\ c / / / / / / / / / / / ////// y ' V / / li^ / V D £ Fig. 279. 4-10. — Graphic Repre§ei]tatioii : Two and Three Fac- tors. — Figs, 278 and 279 serve to illustrate the algebraic ex- pressions a b and a b c. In the former it is shown that the multiplication of two lines produces a rectangular surface, or that if a and b represent lines, then a b may represent a rectangular surface {Fig. 278) having sides respectively equal to a and b. And so if a, b^ and c represent three sev- eral lines, then ab c may represent a rectangular solid {Fig. 279) having edges respectively equal to a, b^ and c. C Fig. 280. 411. — Graphical Multiplication of a Binomial. — Let A B C D {Fig. 280) be a rectangular surface, and B E D F an- other rectangular surface, adjoining the first. The area of the whole figure is evidently equal to — {AB + BE)-xAC. ^^O Ai^GEBKA. The area is also equal to — ABxAC + BExBD', or, since A C = B D, the area equals — ABxAC + BExAC\ or, if symbols be put to represent the lines, say a for A B, b for B E, and c for A Cy then the two representatives of the area, as above shown, become : The first — {a Jr b) xc =z area ; and the last — {ax c) j^ {b xc)^= area. Hence we have — {a + b) c ^ a c + b c. This result exemplifies the algebraic multiplication of a bi- nomial, which is performed thus : Let ^ + ^^ be multiplied by c. The problem is stated thus : {a + b) c. To perform the multiplication indicated we Droceed thus : a -^ b c ac -\- b c multiplying each of the factors of the multiplicand sepa- rately and annexing them by the sign for addition. Putting the two together, or showing the problem and its answer in an equation, we have — (aA-b^c^ac-vbc, producing the same result, above shown, as derived from the graphic representation. 4(2. — Graphical ISquaring^ of a Binomial. — Let EG C J {Fig. 281) be a rectangle of equal sides, and within it draw SQUARING OF A BINOMIAL 411 the two lines, A H and F D, parallel with the Unes of the rectangle, and at such a distance from them that the sides, A B and B D, of the rectangle, A B CD, shall be of equal length. We then have in this figure the three squares, E G C y, A B CD, and FGBH, also the two equal rect- angles, EFAB and BHD J. Let F Fhe represented by a and FGhy b, then the area of A B C D will be axa — a"" \ the area of FG B H will be b xb = b^' \ the area of EFAB will hQ ax b = ab, and that • A B \ C 1 [) J Fig. 281. of BHD y will be the same. Putting these areas together thus — a"" ■\-2ab ^b"", the sum equals the area of the whole figure — equals the prod- uct of EG X E C — equals the product — {a-^b) x{a + b). So, therefore, we have — {a + b) {a + b) = a' + 2 a b + b"" ] (112.) or, in general, the square of a binomial equals the square of the first, plus twice the first by the second, plus the square of the second. This result is obtained graphically. The same re- sult may be obtained by algebraic n;ultiplication, combining 412 ALGEBRA. each factor of the multiplier with each factor of the multi- plicand and adding the products, thus — a -^ b a ■\- b ab + b' a"" ^ 2 ab + b\ The same result as above shown by graphical representa- tion. 413. — Graphical Squaring of tlie Difference of Tivo Factors. — Let the line E C {Fig. 281) be represented by c, and the line A E and A C 2ls before respectively by 3 and a, then — EC-AE = AC. c — b ^^ a. From this, squaring both sides, we have — [c-by = a\ The area of the square A B C D may be obtained thus : From the square E G CJ take the rectangle E G x E A and the rectangle F G x D J, minus the square F G B H, or from c"^ take the rectangle cb, and the rectangle c b^ minus the square, b ^ and the remainder will be the square, a ^ ; or, in proper form — c^ — c b — c b ^ b"" = a"^ In deducting from c"" the rectangle cb twice, we have taken away the small square twice ; therefore, to correct this error, we have to add the small square, or b I Then, when reduced, the expression becomes — c' -2cb-^b'' = a''^{c-b)\ This result is obtained graphically. The result by algebraic PRODUCT OF THE SUM AND DIFFERENCE. 413 process will now be sought. The square of a quantity may be obtained by multiplying the quantity by itself, or— c - b c- b > - be -bc-vb' {c-by=:c'-2bc + b\ (113.) In this process, as before, each factor of the multiplier is combined with each factor of the multipHcand and the sev- eral products annexed with their proper signs {Art. 415), and thus, by algebraic process, a result is obtained precisely like that obtained graphically. This result is the square of the difference of c and b ; and since c and b may represent any quantities whatever, we have this general — Rule. — The square of the difference of two quantities is equal to the sum of the squares of the tzvo quantities^ minus twice their product. E H 1^ Fig. 282. 4.(4, — Graphical ProdM<«t ol* the Sum and Differenc« of Two Quantities. — Let the rectangle A B C D {Fig. 282) have its sides each equal to a. Let the line E F hQ parallel with A B and at the distance b from it, also, the line F G made parallel with B D, and at the distance b from it. Then the line E F equals a -\- b, and the line E C equals a — b. Therefore the area of the rectangle E F C G equals a -f /;, 414 ALGEBRA. multiplied hy a — b. From the figure, for the area of this rectangle, we have — EFCG\ ABCD -ABEH+HFDG or, by substitution of the symbols, a"" — a b + b {a - -b). Multiply the last quantity thus — a-b b ab-b' =b{a^b). Substituting this in the above we have — a'' — ab ^ ab — b"" = { a-r b) x {a — b). Two of these like quantities, having contrary signs, cancel each other and disappear, reducing the expression to this — a'-b' = {a-{-b)x{a-b). The correctness of this result is made manifest by an inspec- tion of the figure, in which it is seen that the rectangle E FC G is equal to the square ABCD minus the square BJHF. For ABEH equals BJDG. Now, if from the square A B C D wQ take away A B E H^ and place it so as to cover BJDG, we shall have the rectangle EFCG plus the square BJHF] showing that the square ABCD is equal to the rectangle EFCG plus the square BJHF] or — a"" ={a + b) X {a — b) + b\ The last quantity may be transferred to the first member of the equation by changing its sign {Art. 403). Therefore — a-" -b'' = {a + b)x {a-b), as was before shown. MULTIPLICATION— PLUS AND MINUS. 41 The result here obtained is derived from the geometrical figure, or graphically. Precisely the same result may be obtained algebraically ; thus — a -v b a — b a -\-ao — ab {a + b) X {a — b) = a' (114.) Here the two like quantities, having unlike signs, cancel each other and disappear, leaving as the result only the dif- ference of the squares. The result here obtained is general ; hence we have this — Rule. — TJie product of the sum and differe^ue of two quan- tities equals the difference of their squares. Fig. 283. 415- — Plus and minus Sig^ns in multiplication. — In pre- vious articles the signs in multiplication have been given to products in accordance with this rule, namely: Like signs give plus ; unlike signs^ minus. This rule may be illustrated graphically, thus : In the rectangular Fig. 283, let it be re- quired to show the area of the rectangle A G C H, in terms of the several parts of the whole figure. Thus the area of AGE7 qq^utA ABEF-GB7F2Lnd the area of EJCH equals EFCD - J F H D. And the areas of AGEJ^ E JCH equals the area of ^ 6^ CH. Therefore the sum of the two former expressions equals AG CH. Thus ABEF— GBJF+EFCD-JFHD^AGCH. Let the several lines now be represented bv algebraic symbols; for example, 4i6 ALGEBRA. \^t AB=^EF=a', \QtGB = yF=b', \tt A E =^ G J = c ■ and E C ^^ J H = d, and let these symbols be substituted for the lines they represent, thus A BEE— G BJ F ^ E EC D — JFHD^AGCH. ac b c + ad — b d ^^ ij^ — ^) x {c -\- d). An inspection of the figure shows this to be a correct result. It will now be shown that an algebraical multiplica- tion of the two binomials, allotting the signs in accordance with the rule given, will produce a like result. For example — a — b c + d a c — b c -\- ad d. Fig. 284. 4-16. — Equality of Squares on Oypotlienusc and j^icle§ of Right-Angled Triangle. — The truth of this proposition has been proved geometrically in Art. 353. It will now be shown graphically and proved algebraically. Let A BCD {Fig. 284) be a rectangle of equal sides, and BED the right-angled triangle, the squares upon the sides of which, it is proposed to consider. Extend the side BE to F\ parallel with BE draw Z>6^, CK, and A L. Parallel with ED draw A J and L G. These lines produce triangles, AHB, AC J, ALC, CKD, and CGD, each equal to the given triangle BED {Art. 337). Now, if from the square SQUARES ON RIGHT-ANGLED TRIANGLE. 417 A B C D V7e take A ^// and place \t:itCDG\ and if we take BED and place it at ALC we will modify the square A BCD, so as to produce the figure LG D E HA L, which is made up of two squares, namely, the square D E EG and the square ALEH, and these two squares are evidently equal to the square A B CD. Now, the square DE EG is the square upon ED, the base of the given right-angled triangle, and the square A L EH is the square upon A H ^^ B E, the perpendicular of the given right-angled triangle, while the square A B C D is the square upon B D, the hypothenuse of the given right-angled triangle. Thus, graphically, it is shown that tJie square iipon the hypothenuse of a right-angled triangle is equal to the stun of the squares up07i the remaining two sides. To show this algebraically, let B E, the perpendicular of the given right-angled triangle, be represented hy a\ E D, the base, b}^ b, and B D, the hypothenuse, by c. Then it is required to show that — e' = a'+b\ Now, since D K ^ B E ^^^ a, therefore, E K ^= E D — D K = b — a, and the square ^ A" y 77 equals (b — «)^ which {Art. 413) equals b"" — 2ab + a^. This is the value of the square E KJ H which, with the four triangles surrounding it, make up the area of the square A B C D. Placing the triangle A B H oi this square outside ot it at CD G, and the triangle B E D "dX A L C, we have the four triangles, grouped two and two, and thus forming the two rectangles C G D K and A L CJ. Each of these rect- angles has its shorter side {A L, C G) equal to B E — a, and its longer side L C, G D, equal to E D — b ; and the sum of the two rectangles is ab ^ ab =. 2ab. This represents the area of the two rectangles, which are equal to the four tri- angles, which, together with the square EKJH, equal the square A BC D\ or — ABCD^EKyn^-CGDK+ALCJ, 4l8 ALGEBRA. or — c"" — {b — of ^ a b ^ a b, or— c''^\b-ay-\-2ab. Then, substituting for {b — of, its equivalent as above, we have — c""— b""— 2ab^a''-\-2ab, Remove the two like quantities with unlike signs {Art. 402), and we have — c'^ =^ b' -r a' \ (115.) which was to be proved. 417. — I>ivi§i©ii the Reverse of MuUiplieatiota. — As di- vision is the reverse of multiplication, so to divide one quan- tity by another is but to retrace the steps taken in multipli- cation. If we have the area ab {Fig. 278), and one of the factors a given to find the other, we have but to remove from a b the factor a, and write the answer b. If we have the cubic contents of a solid abc {Fig. 279), and one of the factors a given to find the area represented by the other two, we have but to remove a, and write the others, be, as the answer. If there be given the area represented by a {b^c) (see Art. 411), and one of the factors a to find the other, we have but to remove a and write the answer b^ c. Sometimes, how- ever, a {b + c) is written ab + ac. Then the given factor is to be removed from each monomial and the answer written b + c. If there be given the area represented hy a"^ + 2 a b + b"" to find the factors, then we know by Art. 412 that this area is that of a square the sides of which measure a + b, and that the area is the product of <^ + ^ hy a + b \ or, that a + b is the square root oi a^ + 2 ab + b"'. • If there be given the area a^ — 2 a b + b~ to find its fac- tors, then we know by Art. 413 that this area is that of a square whose sides measure a — b, or that it is the product o^ a — b by. a — b, or :the square of ^ — b. PROCESSES IN DIVISION. 419 If there be given the difference of the squares of two quantities, or the area represented hy a- — b"-, to find its fac- tors, then we know by Art. 414 that this is the area pro- duced by the multiplication oi a — bhy a + b. 418. — I>ivi§ion : Statement of Quotient. — In any case of division the requirement may be represented as a fraction ; thus : To divide c + d — fhy a — b we write the quotient thus — c + d — f For example, to illustrate by numerals, let <^ = 7, (5 = 3, <: = 4, ^T^^S^ 424 ALGEBRA. a =- = I, a a'=-= i. a a' a a a a^' 3 a~' I I ^ <^ =- =-^r- = -:, etc. a a a a Showing that a quantity with a negative exponent may have substituted for it the same quantity with a positive exponent, but used as a denominator to a fraction having unity for tJie numerator. 4-23. — Addition and Subtraction of Exponential Quan- tities. — Equal quantities raised to the same power may be added or subtracted; as, cr -\- 2 <^^ = 3^;^; but expressions in which the powers differ cannot be reduced ; thus, a^ ^^a — a" cannot be condensed. 424. — HuBtipHcation of Exponcntia3 Quantities. — It will be observed in Art. 421 that in the series of powers, the index or exponent increases by unity ; thus, d, d^, a^, a", etc. ; and that this increase is effected by multiplying by the root, or original quantity. From this we learn that to multiply two quantities having equal roots we simply add their exponents. Thus the product of a^ d\ and d^ is a' x d^ y^ a"" — a\ The product of a~', <^^ and a"" is a"^ x a^ x d" =^ a\ The exponents here, are : — 2 + 3 + 5=8 — 2 = 6. 425. — I^ivision of Exponential Quantities. — As division IS the reverse of multiplication, to divide equal quantities raised to various poivers, we need simply to subtract the expo- nent of the divisor from that of the dividend. Thus, to divide a^ by a^ we have d"^ = a^. That this is correct is manifest ; for the two factors, d^ x a^, in their product, ^^ produce the dividend. To divide ci by a", we have a^''" = ^"^ which is equal to - a" EXPLANATION OF LOGARITHMS. 4^5 (see Art. 422). The same result may be had by stating the question in the usual form. Thus, to divide a" by a" we have - , a fraction which is not in the lowest terms, for it may be a 2 2 put thus, -^-^ = — , by which it is seen that it has in both its a a a numerator and denominator the quantity a\ which cancel each other {Art. 371). Therefore, -, = - ; the same result as before. 4-26. — Extraction of Radicals. — We have seen that the square of a \s a^ x a^ ^= a^ \ oi 2 a'^ is 2 a"^ x 2 a^ = 4 a'' ; in each case the square is obtained by doubling the exponent. To obtain the square root the converse follows, namely, take half of the exponent. Thus the square root of a'^ is a\ of a^ is a, of a^ is a\ The same rule, when the exponent is an odd number, gives a fractional exponent, thus : the square root of a^ is a^ ; or, of a\ is ai. So, also, the square root of a, or a\ is ai- Therefore, we have a^ = Va, equals the square root of a, and the cube root of a^ z= ai ^= Va. 427. — E.ogaritlQm§. — We have seen in the last article the nature of fractional exponents. Thus the square root of a^ equals a^, which may be put a'^^. In this way we may have an exponent of any fraction whatever, as a^k Between the exponents 2 and 3, we may have any number of fractional exponents all less than 3 and more than 2. So, also, the same between 3 and 4, or any other two consecutive num- bers. The consideration of fractional exponents or indices has led to the making of a series of decimal numbers called logarithms, Avhich are treated in the manner in which expo- nents are treated ; namely — To multiply numbers add their logarithms. To divide numbers, subtract the logarithm of the divisor from the logarithm of the dividend. 426 ALGEBRA. To raise any number to a given pozver^ multiply its logaritJim by the exponent of that poiver. To obtain the root of any poiver^ divide the logarithm of the given number by the exponent of the given power. As an example by which to exemplify the use of loga- rithms : What is the product of 25 by 375 ? We first make this statement : Log. of 25- = I- - 375. =2. Putting at the left of the decimal point the integer char- aeteristic, or whole number of the logarithm at one less than the number of figures in the given number at the left of its decimal point. To find the decimal part of the required logarithm we seek in a book of Logarithms (such as that of Law's, in Weale's Series, London) in the column of numbers for the given number 25, .or 250 (which is the same as to the man- tissa) and opposite to this and in the next column we find 7940 and a place for two other figures, which a few lines above are seen to be 39 ; annex these and the whole number is 0-397940. These we place as below : Log. of 25. = 1.397940. Now, to find the logarithm of 375, the other factor, we turn to 375 in the column ol numbers and find the figures opposite to it, 4031, which are to be preceded by 57-, the two figures found a few lines above, making the whole, -574031, which are placed as below, and added together. Log. of 25. = 1-397940 '' 375- = 2-574 031 The sum — 3-971971 This sum is the logarithm of the product. To find the product, we seek in the column of logarithms, headed o-,, for -97 1 97 1, the decimal part. We find first 97, the first two EXAMPLES OF LOGARITHMS. 427 figures, and a little below seeking for 197 1, the remaining four figures, we find 1740, those which are the next less, and opposite these, to the left, we find 7, and above 93, or together, 937 ; these are the first three figures of the required product. For the fourth figure we seek in the horizontal column opposite 7 and 1740 for 1971, the remaining four figures of the logarithm, and find them in the column headed 5. This figure 5 is the fourth of the product and completes it, as there are only four figures required when the integer number of the logarithm is 3. The completed statement therefore is — Log. of 25. = 1.397940, " " 375- = 2.574031, " " 9375 = 3-971971. Another example in the use of logarithms. What is the product of 3957 by 94360? The preliminary statement, as explained in last article, is — Log. 3957 = 3. '' 94360 = 4. In the book of logarithms seek in the column of numbers for 3957. In the first column we find only 395, and opposite to this, in the next column, we find a blank for two figures, above which are found 59. Take these two figures as the first two of the mantissa, or decimal part of the required logarithm, thus, 0-59. Again, opposite 395 and in the col- umn headed by 7 (the fourth figure of the given number), we have the four figures 7366. These are to be annexed to (o. 59) the first two obtained. The decimal part of the loga- rithm, therefore, is 0-597366. To obtain the logarithm for 94360, the other given num- ber, we proceed in a similar manner, and, opposite 943, we find 0-97; then, opposite 943 and in column headed 6, we find 4788, or, together, the logarithm is 0-974788. The whole is now stated thus — 428 ALGEBRA. Log. of 3957 = 3-597366 " " 94360 = 4.974788 " '' 373382000 = 8-572154 = sum of logs. The two logarithms are here added together, and their sum is the logarithm of the product of the two given factors. The number corresponding to the above resultant logarithm may be found thus: Look in the column headed o for 57, the first two numbers of the mantissa, then in the same column, farther down, seek 2154, the other four figures of the mantissa; or, the four (1709) which are the next less than the four sought, and opposite these to the left, in the column of numbers, will be found 373, the first three figures of the product ; opposite these, to the right, seek the four figures next less than 2154, the other four figures of the man- tissa. These are found in the column headed 3 and are 2058. The 3 at the head of the column is the fourth figure in the product. From 2154, the last four figures of the man- tissa, deduct the above 2058, or — 2154, 2058, Remainder, 96. At the bottom of the page, opposite the next less number (3727) to that contained in 3733, the answer already found, seek the number next less to the above remainder, 96. This is 92-8, and is in the column headed 8. Then 8 is the next number in the product. From 96 deduct 92-8, and multi- ply it by 10, or — 96 92-8 3-2 X 10 =ri 32. Then, in the same horizontal column, seek for 32 or its next less number. This is 23-2, found in column 2. This 2 is the next figure in the product. Additional figures may be ob- tained by the table of proportional parts, but they cannot be THE SQUARE OF A BINOMIAL. 429 depended upon for accuracy beyond two or three figures. We therefore arrest the process here. The product requires one more figure than the integer of the logarithm indicates; as the integer is 8, there must be nine figures in the product. We have already six ; to make the requisite number nine we annex three ciphers, giving the completed product — 3957 X 94360 = 373382000. By actual multiplication we find that the true product in the last article is 373382520. In a book of logarithms, carried to seven places, the required result is found to be 373382500^ which is more nearly exact. The utility of logarithms is more apparent when there are more than two factors to be multiplied, as, in that case, the operation is performed all in one statement. Thus : What is the product of 3-75, 432-95, 1712, and 0-0327 ? The statement is as follows : Log. 3-75 == 0-574031 432-95 = 2-636438 1712- = 3-233504 •032 7 = 8-51454 8 Product = 90891. = 4-958521 16 5. Explanations of working are given more in detail in most of the books of logarithms. 4-28. — Completing the Square of a Binomial. — We have seen in Art. 412 that the square of a binomial {a + b) equals a"" + 2 ab ^ b"" — a trinomial — the first and last terms of which are each the square of one of the two quantities, Avhile the second term contains the second quantity multi- plied by twice the first quantity — In analytical investigations it frequently occurs that an expression will be obtained which may be reduced to this form : 430 ALGEBRA. a"^ + m a b =^ f, (i i8.) in which m is the coefficient of the second term, and a and b are two quantities represented by a and b or any other two symbols. A comparison of this expression with the square of a bi- nomial (ii2.) contained in Art. 412, shows that the member at the left comprises two out of the three terms of the square of a binomial ; as thus — a^ + 2 a b Jr ^"^y but with a coefficient m instead of 2. It is desirable, as will be seen, to ascertain a proper third term for the given ex- pression ; or, as it is termed, '' to complete the square." The method by which this is done will now be shown. A consideration of the above trinomial shows that the third term is equal to the square of the quotient obtained by dividing the second term by twice the square root of the first ; or — /2 abV _ \ 2 a J ~ Now a third term to the above binomial, equation (118.), may be obtained by this same rule. For example — /7n a bV / m bV The rule for the third term then is: Divide the sccoiid term by twice the square root of the first, and square the quotient. As an example, let it be required to find the third term required to complete the square in the expression — 6 n X + AfX'^ -= fy in which n and / are known quantities and x unknown. Putting it in this form — ^ x"^ ^ 6 n X = fy and dividing by 4, we have — BINOMIALS CONTINUED. 43I X +-„x = -, which reduces to — 1-2 3 / 2 4 Now applying the above rule for finding the third term, we have — (*^)=e»y 2 X which is the required third term. To complete the square we add this third term to both members of the above re- duced expression, and have — The member of this expression at the left is the completed square of a binomial, the two quantities constituting which are the square roots of the first and third terms respectively ; or X and | 7i, and we therefore have — 2 ^4 ^^ 4 and now taking the square root of both sides of the expres- sion, we have — 3 X + —n 4 1''^ e-" )' and, by transferring the second quantity to the right mem- ber, we have — ^ A \A J A ' an expression in which x, the unknown quantity, is made to stand alone and equal to known quantities. The process of completing the square is useful, as has Deen shown, in developing the value of an unknown quan- 432 ALGEBRA. tity where it enters into an expression in two forms, one as the square of the other. As an example to test the above result, let/ = 256 and n — 8. Then we have by the last expression for the value of ^ — = |/^^+(3-x8y-3-x8 ^ ZL VA / A. ;4 = 1/64+ 36 _ 6, = \/ 100 — 6, ' X := 10 — 6 = 4. Now this value of x may be tested in the original expres- sion — 6 nx ■{■ 4 ;ir ^ = y, for which we have — 6x8x4 + 4x42=/, 192 + 64 = / 256=/; the correct value as above. PROGRESSION. 429. — Ariilimetical Progre§sion. — In a series of num- bers, as 1,3, 5, 7, 9, etc., proceeding in regular order, in- creasing by a common difference, the series is called an arithmetical progression ; the quantity by which one num- ber is increased beyond the preceding one is termed the difference. If d represent the difference and a the first term, then the progression may be stated thus — Terms— i, 2, 3, 4, 5, a, a + d, a ^ 2d, a + -i^ d, a + 4.d, etc. The coefficient of d is equal to the number of terms preced- ing the one in which it occupies a place. Thus the fifth term is ^ + ^d, in which the coefficient 4 equals the number of the preceding terms. From this we learn the rule by which at once to desig- ARITHMETICAL PROGRESSION. 433 nate any term withoutfinding all the preceding terms. For the one hundredth term we should have « + 99 <^, or, if the number of terms be represented by 71^ then the last term would be represented by — I =z a + {n — i) d. (119-) For example, in a progression where a, the first term, equals I, d the difference, 2, and n, the number of terms, 90, the last term will be — I — a + {n — \) d = I + (90 — 1)2 — 179. Therefore, to find the last term : To the first term add the product of the common difference into the number of terms less one. By a transposition of the terms in the above expression, so as to give it this form — a = I — (n — \) dy (120.) we have a rule by which to find the first term, which, in words, is — Midtiply the number of terms less one by tlie common differ- ence, and deduct the product from the last term ; the remainder will be the first term. By a transposition of the terms of the former expression to this form — I — a ^ {n — i) d, and dividing both members by {n - i), we have- I — a ~~ n - I ' (121.,). which is a rule for the common difference, and which, ini words, is — Subtract the first term from the last, and divide the remain- der by the 7iumber of terms less one ; the quotient will be the com- mon difference. Multiplying both members of the equation (121.) by {n — i) and dividing by d, we obtain — 434 ALGEBRA. I — a n — \ — Transferring i to the second member, we have — I — a , . n^—^+i', (122.) which is a rule for finding the number of terms, and which, in words, is — Divide the differejtce between the first and last terms by the common difference ; to the quotient add unity, a7td the sum will be the number of terms. Thus it has been shown, in equations (119,) (120), (121), and (122), that when, of the four quantities in arithmetical progression, any three are given, the fourth may be found. The sum of the terms of an arithmetical progression may be ascertained by adding them ; but it may also be had by a shorter process. If the terms are written in order in a hori- zontal line, and then repeated in another horizontal line be- neath the first, but in reversed order, as follows : i» 3, 5, 7, 9. II. 13, 15. 15, 13, II, 9, 7, 5, 3, I, 16, 16, 16, 16, 16, 16, 16, 16, and the vertical columns added, the sums will be equal. In this case the sum of each vertical couple is 16, and there are 8 couples; hence the sum of these 8 couples is 8 x 16 = 128. And in general the sum will be the product of one of the couples into the number of couples. It w^ll be observed that the first couple contains the first and last terms, i and 15 ; therefore the sum of the double series is equal to the product of the sum of the first and last terms into the number of terms. Or if 5 be put to represent the sum of the series, we shall have — 2 S — (a \ I) n, and, dividing both sides by 2 — 5 = (^ + /)^; (123.) GEOMETRICAL TROGRESSION EXPLAINED. 435 Or, in words : The sum of an arithmetical scries equals the prod- uct of the stun of the first aud last terms j into half the number of terms. 4-30. — Geometrical Progression. — A series of numbers, such as I, 2, 4, 8, 16, 32, 64, 128, 256, etc., in whicli any one of the terms is obtained by multiplying the preceding one b}^ a constant quantity, is termed a Geometrical Progression. The constant quantity is termed the common Ratio, and is equal to any term divided by the preceding one. Thus in , 16 8 4 the above example -5- or - or - = 2, equals the common ra- 04^ tio of the above series. In the series, r, 3, 9, 27, etc., we have for the ratio — _27 ^ 9 ^ 3 ^ 931 which is the common ratio of this series. A geometrical series may be put thus : Terms: i, 2, 3, 4; Progress. : i, i x 3, i x 3 x 3, i x 3 x 3 x 3 ; or thus — Terms: i, 2, 3, 4: Progress.: i, i x 3, i x 3^ i x 3^ in which the common ratio, in this case 3, appears in each term and with an exponent which is equal to the number of terms preceding that in which it occupies a place. If the first term be represented by a and the common ra- tio by r, then the following will represent any geometrical progression — a, ar, ar-, ar^, ar'^, etc. (124.) For example, let a = 2 and r == 4 ; then the progression will be — 2, 8, 32, 128, 512, etc. 43^ ALGEBRA. li r = unity, then when a = 2 the progression becomes — 2, 2, 2, 2, 2, etc. If r be less than unity, then the progression will be a de- creasing one. For example, let a = 2 and r =z ^. Then we have for the progression — I I I I ' ' 2' 4' 8' 16' ^^^• If the number of terms be represented by ?/, and the last by /, then the last term w4ll be — / = ar^-\ (125.) For example, let 71 equal 6, then the progression will be — Terms: i, 2, 3, 4, 5, 6; Progress.: a, ar, ar^% ar^, ar"^, ar^\ in which the exponent of the last term equals n — \ — 6-1 = 5. If wS be put for the sum of a geometrical progression, we will have — Multiply each member by r, then — Sr =. ar -^ ar^ + j^ cir^~^ -^r ar^~^ Jr cif^- Subtract the upper line from the lower ; then — 5r= ar + ar- -^ ^ ar^'^ -\. ar^'^ + ar'^^ S z^ a -^ ar ^ ar"" -\- +^?'"~^ + <^r""' Sr — s — — a "^ '^ ^ + ar'\ Sr — s = — a + ar'\ S {r — i)= — a + (Z r^ = ar^ — a, ar^ — a • o — • r — \ GEOMETRICAL PROGRESSION CONTINUED. 437 The last term (equation (125.)) equals I— ar^-\ and since ar"" = r X ar"^' '' = r I, therefore — 5=^f. (.26.) Thus, to find the sum of a geometrical progression : Multi- ply the last term by the ratio ; from the product deduct the first term, and divide the remainder by the ratio less unity. For example, the sum of the geometrical progression — 5 = I + 3 + 9 + 27 + 81 + 243 + 729 = 1093 by actual addition. To obtain it by the above rule — rl — a 3 X 720 — I S — — — - — ^^ = 1003, r — I 3 — 1 the correct result. If there be a decreasing geometrical progression, as i, ^, I, ^j, etc., in which the ratio equals ^, the sum will be — ^ - I I I I ■ r ' 5=1 » T + ~ + ^ + "gY' ^tc, to mhnity. Multiply this by 3, and subtract the first from the last — 25 = 3+1+- + - + — + — - + to infinity. 3927^ 81 ^ I I I 3"^ 9 "^ 27 2 5 = 3, or 5 = 1^. S = i+- + -_|_-^ + --^ + to infinity. In a decreasing progression let r, the common ratio, be represented by - {b less than e), and the first term by a, then 'he sum will be — b b^ b' . ^ . S=-a + a-^a-^-\-a-^+, etc., to mhnity. 43^ ALGEBRA. Multiply this by -, and subtract the product from the above — ^b b b' F . , . 0- — « — -i-^— +<3:^ 4- etc., to innnity. b b^ b' . ^ . S ^ a ^ a — -f-rt^-^ + t^— + to mnnity. b b b' b' — z=i a - c c S — ^=^ a — \- a —-\- a -^ + \.o infinity. C Or— 6" (i - -) -: ^ a ■5 = r-7. (127.) I -b_ C For example, let the first term of a geometrical progression equal 2, and the ratio equal J, then the sum will be — 5 __«__,? From this, therefore, we have this rule for the sum of an in- finite geometrical progression, namely : Divide the first term by miity less the ratio. SECTION X.— POLYGONS. 431. — Relation of Sum and Bfifference of Two Liines. — Let AB and CD {Fig. 285) be two given lines; make EH B D E 1— 1—1 G H J F Fig. 285. equal to AB, and HG equal to CD\ then £ (S^ equals the sum of the two lines. Make FG equal to A B, which is equal to EH, Bisect EGixi y\ then, also, J bisects HF\ for — E7= JG, and — EH=FG, Subtract the latter from the former ; then — Ey- EH=yG-FG\ but— Ey-EH = Hy, and — yG-~FG = yF] therefore — Hy-JF. Now, E y IS half the sum of the two lines, and Hy is half the difference ; and — Ey-Hy = EH=AB. Or : Ha// the sum of two quantities, minus half their dif- ference, equals the smaller of the two quantities. 440 POLYGONS. Let the shorter line be designated by ^, and the longer by b ; then the proposition is expressed by — a — a+ b (128.) We also have EJ + JF— EF— CD\ or, half the sum of two quantities, plus half their differ ence, equals the larger quantity, 432. — Perpendicular, in Triangle of Known Sides. — Let ABC (Fig. 286) be the given triangle, and CE a perpen- dicular let fall upon A B, the base. Let the several lines of the figure be represented by the symbols a, by c, d, g, and /, as shown. Then, since A EC and B E C are right-angled triangles, we have {Art. 416) the following two equations, and, by subtracting one from the other, the third — f^ + d^=a\ f^-^'^a'-b'. Then {Art. 414), by substitution, we have — By division we obtain — (a + 6) {a — b) f-g^ f + g RULE FOR TRIGONS. 44I According to Art. 431, equation (128.), we have — ^2 2 ' In this expression let the value of / — ^, as above, be substituted, then we will have — ^^/+^ _ {a + b ){a- h) Multiply the first fraction by {f + g), then join the two fractions, when we will have — , ^ {f + gf-{a-\-b){a- b) The lines / and g, in the figure, together equal the line c ; therefore, by substitution — c" — (a + b^ (a — b) , . g= —YT "• ^'^^'^ This is the value of the line g. It may be expressed in words, thus: The shorter of the two parts into which the base of a triangle is divided by a perpendicular let fall from the apex upon the base, equals the quotient arising from a division by twice the base, of the differ- ence betiveen the square of the base and the product of the sum and difference of the two inclined lines. As an example to show the application of this rule, let ^ = 9, ^ z= 6, and c = \2\ then equation (129.) becomes — ^ i2--(9 + 6)(9- 6) ^ 2 X 12 ^^ 144- 15 X 3 24 "^ 24 ^8 442 POLYGONS. Now, to obtain the length of 287) be a given trigon or triangle with its circumscribed and inscribed circles. Draw the lines A D F, D B and D C. The three triangles, A B D, A CD, :ind B D C, have their apexes converging at D, and form there the three angles, A D B, ADC, and B D C. These three angles together form four right angles {Art. 335), and each of them, therefore, equals f of a right angle. The angles of the triangle BDC together equal two right angles {Art. 345). As above, the angle BDC equals | of a right angle, hence 2 — ^ = ^^^ = f of a right angle, equals the sum of the two remaining angles at B and C The triangle BDC is isoceles {Art. 338); for the two sides B D and D C, being radii, are equal ; therefore the two angles at the base B and C are equal, and as their sum, as above, equals f of a right angle, therefore each angle equals -J of a right angle. Draw the two lines FC and FB. Nov/, be- 444 POLYGONS. cause D C and D F are radii, they are equal, hence DFC is an isoceles triangle. It was before shown that the angle B D C equals f of a right angle ; now, since the diameter A F bisects the chord B C, the angles B D E and E D C are equal, and each equals the half of the angle B D C; or, -J of f of a right angle equals f of a right angle. Deducting this from two right angles (the sum of the three angles of the triangle), or 2 — f = ji nr |- of a right angle equals the sum of the angles at F and C\ hence each equals the half of f, or | of a right angle; therefore the triangle D FC is, equilateral. The triangles DBF and DFC are equal. The angles B D C and B FC are equal; the line B C is perpendicular to DF and bisects it, making DE and EF equal; hence DE equals half DF, or DB, radii of the circumscribing circle. Therefore, putting R to represent B D, the radius of the circumscribing circle, and I? = B C, a side of the triangle A B C, by Art. 416, we have — BD'' = BE'+ DE\ ^■=(D"-(iy Transferring and reducing — R (f)=ej 4 4 ' V 4/ 4 ^R' = -b\ 4 4 3 4 3 3 .^±. Or, The Radius of the circumscribing circle of a regular trigon or equilateral triangle, equals a side of the triangle divided by the square root of 3. AREA OF EQUILATERAL TRIANGLE. • 445 By reference to Fig. 287 it will be observed, as was above shown, that D E — E F= = ; or, D E, the ra- dius of the inscribed circle, equals half the radius of the circumscribed circle; or, again, dividing equation (131.) by 2, we have — R b 2 2 ^3 ' and, putting r for the radius of the inscribed circle, we have — (132.) 2 ^3 Or: TJie radhis oi the inscribed circle of a regular trigofi equals the half of a side of the trigon divided by the square root of 3. .To obtain the area of a trigon or equilateral triangle ; we have {Art. 408) the area of a parallelogram by multiplying its base into its height ; and {Arts. 341 and 342) the area of a triangle is equal to half that of a parallelogram of equal base and height, therefore, the area of the triangle B D 0{Fig. 287) is obtained by multiplying B C, the base, into the half of ED, its height. Or, when iV^is put for the area — N= BCx^^, 2 or- N=bx^; 4 substituting for R its value (131.) — N= bx N = 4 V3 ^2 4V3 This is the area of the triangle BBC. The triangle A B C is compounded of three equal tri- angles, one of which is the tr'mngleBDC; therefore the area of the triangle ABC equals three times the area of the triangle BBC; or, when A represents the area — 446 POLYGONS. 3^ 4 Vi (•33-) Or: The area of a regular trigon or equilateral triangle equals tJiree fourtJis of the square of a side of the triangle di- vided by the sqiiare root of 3. 434. — Tetragon ; Radius of Circumscribed and In- scribed Circles: Area. — Let AB CD {Fig. 288) be a given tetragon or square, with its circumscribed and inscribed Fig. 288. circles, of which ^ ^ is the radius of the former and E F ihdit of the latter. The point F bisects A B, the side of the square. A /^equals F F and equals half A B, a side of the square. Putting R for the radius of the circumscribed circle and b for A B, we have {Art. 416)— AE^^ = AF'^-\-EF\ -•=(4)MI)'-(I)' 2 /; b- R=: v (134.) Or: The radius of the circutuscribed c\yc\q of a regular tetra- gon equals a side of the square divided by the square root of 2. SIDE AND AREA OF HEXAGON. 447 By referring to the figure it will be seen that the radius of the inscribed circle equals half a side of the square — (•35-) The area of the square equals the square of a side — A^b\ (136.) 435. — Hexagon: Radius of Circumscribed and In- scribed Circles : Area.— Let ABODE F{Fig. 289) be an equi- lateral hexagon with its circumscribed and inscribed circles, of which EG xs the radius of the former, and 6^ //that of the latter. The three hues, AD, B E, ^nd C/^, divide the Fig. 289. hexagon into six equal triangles w^ith their apexes converg- ing at G. The six angles thus formed at G are equal, and since their sum about the point G amounts to four right angles {Art. 335), therefore each angle equals -f or | of a right angle. The sides of the six triangles radiating from G are the radii of the circle, hence they are equal ; therefore, each of the triangles is isosceles {Art. 338), having equal angles at the base. In the triangle EGD, the sum of the three angles being equal to two right angles {Art. 345), and the angle at 6^ being, as above shown, equal to f of a right angle, therefore the sum of the two angles at E and D equals 2 — f = f of a right angle ; and, since they equal each other. 448 POLYGONS. therefore each equals f of a right angle and equals the angle at G ; therefore E G D is an equilateral triangle. Hence E D^2i side of a Jiexagon, equals E G, the radius of the circum- scribing circle — R=b. (137.) As to the radius of the inscribed circle, represented by G Hy a perpendicular from the centre upon ED, the base; the point H bisects E D. Therefore, E H equals half of a side of the hexagon, equals half the radius of the circumscribing circle. Let R = this radius, and r the radius of the inscribed circle, while d = a, side of the hexagon ; then we have (Arts. 353 and 416)— GII^ = EG'-EH\ '-=R (4)' ^2 ._ Now, i? = = d, therefore- - r=/i^ ^3 (138.) Or : The radius of the inscribed circle of a regular hexagon equals the half of a side of the hexagon, multiplied by the square root of 3. As to the area of the hexagon, it will be observed that the six triangles, A B G, B G C, etc., converging at G, the centre, are together equal to the area of the hexagon. The area of E G D, one of these triangles, is equal to the product of E D, the base, into the half of G H, the perpendicular; or, when N is put to equal the area — N — EDy. , SIDE AND AREA OF OCTAGON. 449 2 and, since r, as above, equals ^ 3 — , 2 4 This is the area of one of the six equal triangles ; therefore, when A is put to represent the area of the hexagon, we have — ^ = 6x |/"3 — , ^3 3^ (I39-) Or: The area of a regular hexagon equals three halves of the square of a side multiplied by the square root of '^^ Fig. 2go. 4-36b — Octagon : Radius of Circiimscribed and In- scribed Circles: Area. — Let C E D B F {Fig. 290) represent a quarter of a regular octagon, in which i^is the centre, ED a side, and C E and D B each half a side, while C F and B F are radii of the inscribed circle, and EF and DF are radii of the circumscribed circle. 450 POLYGONS. Let R represent the latter, and r the former ; also let I? represent £ B, one of the sides, and 7i be put for A D, and for A E. Then we have — AD^DB^CF. b or — n=^r — — • 2 Since ^ Z^^is a right-angled triangle {Art. 416), we have- AE' + AD' = ED\ n'+ii' = b\ 2 7l' = b\ n — — 2 n^^\ Placing the value of n, equal to the value before found, we have — -? = i/^, r = '/^*j + ^=il*!+^, - A ^_ V o V2 2 (7; * -;)'■ This coefficient may be reduced by multiplying the first fraction bv 1/2, thus — V2 "\/~2 ~ 2 RULES FOR OCTAGON. 45 1 therefore — V~2 \ 2 2 / 2 r^{V2 + i)— . , (140.) Or: The radius of the inscribed circle of a regular octagon equals half a side of the octagon multiplied by the sum of unity plus the square root of 2. In regard to the radius of the cir- cumscribed circle, by Art. 416 we have — In this expression substituting for r\ its value as above, we have — The square of the coefficient ( -/2 + i ) by Art. 412 equals 2 + 24/2+1 = 2 -/a +3, then — R ^ = [(2 4/3 + 3) + I] (4-)'- ie^ = (2Vj+4)(Ay. i?=|/2l/2' + 4-^. (141.) Or: The radius oi the circtcmscribed c\yc\q oi a regular d?^/^^'^?;^ equals half a side of the octagon multiplied by the square root of the sum of twice the square root of 2 plus 4. In regard to the area of the octagon, the figure shows that one eighth of it is contained in the triangle D E F. 452 POLYGONS. The area oi D E F^ putting it equal to N, is- ^ ^ BF N — E D ^ , 2 ^^^.(^2+,) 2 X 2 N^ {V2+ I)--. 4 This is the area of one eighth of the octagon ; the whole area, therefore, is — 4 A =: ( 4/2"+ l) 2 ^\ (142.) Or: The area of a regular octagon equals twice the square of a side, multiplied by the sum of the square root of 2 added to unity. When a side of the enclosing square, or diameter of the inscribed circle, is given, a side of the octagon may be found ; for from equation (140.), multiplying by two, we have — 2 r — {\^2 -\- \) b. Dividing by S^ 2 ^ i, gives — T 2r , . ^ = -— -— . (143.) r 2 + I The numerator, 2 r, equals the diameter of the inscribed circle, or a side of the enclosing square ; therefore : The side of a regular octagon, equals a side of the enclosing square divided by the sum of the square root of 2 added to unity, 437,— Dodecagon : Radiu§ of Circuniseribed and In- scribed Circles: Area. — Let A B C {Fig. 291) be an equilat- SIDE AND AREA OF DODECAGON. 453 eral triangle. Bisect A B in F; draw C F D ; with radius A C describe the arc A D B. Join A and D, also D and B ; bisect A F> in E ; with the radius £ C describe the arc E G, Then A D and D B are sides of a regular dodecagon, or twelve- sided polygon ; of which A C, D C, and B C are radii of the circumscribing circle, while ^ 6" is a radius of the inscribed circle. The line A B is the side of a regular hexagon (Arl. 435). Putting R equal to A C the radius of the circumscribing cir- cle \ r, =^ E C, the radius of the inscribed circle ; i^, = A D, 3. side of the dodecagon, and ;/ — B> F. Then comparing the Fig. 291. homologous triangles, ABE and A EC (the angle ADF equals the angle E A C, and the angles BE A and A E C are right angles); therefore, the two remaining angles DAE and A C E must be equal, and the two triangles homologous {Art. 345). Thus we have — DF '.DA :: AE : A C, n \ b \\~ \ R, 2 R b^ 2 n 454 POLYGONS. In Art. 435 it was shown that FC {Fig. 291), or G H oi _ b Fig. 289, the radius of the inscribed hexagon, equals 1^ 3 ~, and in which its <^ = i? ; Fc = |/7 — . Now ( Ftg. 291) — DF= DC - FC, or — Substituting this value of n^ in the above expression, we have — ^^ ^1 2 i? (I - i V 3)* Multiplying by R and reducing, we have — R^^ = V7 R = '/— ^^ ^. (144.) 2-1/3 Or : The radius of the circumscribed circle of a regular dodec- agon, equals a side of the dodecagon multiplied by the square root of a fraction, having unity for its numerator and for its denominator 2 mimis the square root of 3. Comparing the same triangles, as above, we have — FD \ FA :: EA \ E C, or — R b n : — : : — '. r, 2 2 Rb Rb r = An 4^(1 -it' 3)* r = ~.^. (145.) 4-21/3 RULE FOR DODECAGON. 455 Or : The radius of the inscribed circle of a regular dodecagon equals a side of the dodecagon divided by the difference between 4 and the square root of 3. The area of a dodecagon is equal to twelve times the area of the triangle ADC {Fig. 291). The area of this triangle is equal to half the base by its perpendicular ; or, A E x EC\ or — b 2 ""' or, where N equals the area — N=^br. Or, for the area of the whole dodecagon— 12 N — 6 br, A =6br. Substituting for r its value as above, we have — A = --^-- b^. (146.) 4 — 24/3 Or : The area of a regular dodecagon equals the square of a side of the dodecagon, multiplied by a fraction having 6 for its numerator, and for its denominator, 4 minus twice the square root of T^. 438a — Hecadecagon : !&ad§ii§ of 4 ircumscribed and In- scribed Circles: Area. — Let A BCD {Fig. 292) be a square enclosing a quarter of a regular octagon C E F B^ E F being one of its sides, and C E and F B each half a side, while ED is the radius of the circumscribed circle, and J D the radius of the inscribed circle of the octagon. Draw the diagonal A D ; with D F ior radius, describe the circumscribed circle EGF', join G Avith F and with E; then E G and G F W\\\ each be a side of a regular hecadecagon, or polygon of six- teen sides. An expression for F D, the radius of the circumscribed 456 POLYGONS. circle, may be obtained thus: Putting FD = R; H D =r r GF = b; Gy = n\ and JF = - {Art. 416), we have— C D Fig. 292. Comparing the two homologous (ArL 361) triangles, G^F and FH D {Art. 374), we have — GJ '. GF '.: HF : FD, n \ b -.'. '- '. R, n = 2R' n" = 4R'- Putting this value of ;/' in an equation against the former value, we have — ^■"•-& In Art. 436, the value of F D, as the radius of the cir- cumscribed circle of a regular octagon, is given in equation (141.) as— R=^2V2+4-y 2 SIDE AND AREA OF HECADECAGON. 457 in which b represents a side of the octagon, or E Fy for which we have put s. Substituting s for b and putting the numerical coefficient under the radical, equal to B^ we have — Squaring each member gives — From which, by transposition, we have — Substituting in the above expression for (-- ) , this value of it, gives — -^-^b^-^\ AR' B Transposmg, we have— ^--^^'^b\ 4R' B Multiplying the first term by B, and the second by 4 7^', we have — 4BR' 4BR' Bb'-^^ R^ _ 4BR' ~ ' Bb' +^R'^^BR'b\ Transposing, we have — ^R'-^BR'b'= -Bb\ 458 POLYGONS. To complete the square {ArL 428) we proceed thus — R'-BR'y-= -\Bb\ R'-BR"'h' + {l-Bby=^ {IBby -lBb\ Taking the square root, we have — R'-^Bb'= V^B'b'-lBb R'= ViB'b'-iBb'+ iBb\ R' = bWiB'-iB + iBb\ R''=b\V-lB-'-iB + iB), R'=b"'{V\B{B - i) + ^B). Restoring B to its value, 2 4/2 + 4 as above, we have — \B = ^^\^+ I, ^—1 = 21/24-3; multiply these — 2 + 2 V2, tV 3 + t ^ 2 j^(i?- I) = 5 + i-^2, ^B= V2 + 2. Therefore— R'-^b'{^S ■\-lV~2+V2 + 2), R=b\/ ^\ + I V2 + V~2 + 2. (147.) RULES FOR HECADECAGON. 459 Or: The radms of the circumscribed circle of a regular Jiccadecagon equals a side of the Jiecadecagon multiplied by the square root of the sum of two quantities, one of which is the square root of 2 added to 2, and the other is the square root of the sum of seven halves of the square root of 2 added to 5. To obtain the radius of the inscribed circle we have {Fig. 292)— HD' ^ FD'- HF\ Substituting for R"" its value as above, we have — r^ = b^[_{ViB{B-i)+iB)-{iYl The coefficient of b is the same as in the case above, ex- cept the — \ ; therefore its numerical value will be i less, or — ^ V ^'5 + i- 1/2 + 1/2 + if. (148). Or: The radius of the inscribed circle ot a regular Jiecadeca- gon equals a side of the Jiecadecagon multiplied by tJie square root of two quantities^ one of ivJiicJi is tJie square root of 2 added to if, and tJie otJier is the square root of the sum of seven Jialves of tJie square root of 2 added to 5 . To obtain the area of the hecadecagon it will be observed that the area of the triangle G FD {Fig. 292) equals HD x H F, and 'that this is the -^-^ part of the polygon ; we there- fore have — A = i6HDxHF, A = i6r-= Srb. 460 POLYGONS. The value of r is shown in (148.) ; therefore we have — A 8^ j/ ^5 + -1- 1^2 + \'2 + if. (149.) Or: The area oi a regular Jiecadccagon equals eight times the square of its side^ multiplied by the square root of two quan- tities, one of which is the square root of 2 added to if, and the other is the square root of the stem of seven halves of the square root of 2 added to 5. 439.— Polygons ^ Radius of Circumscribed and Inscribed Circles : Area. — In Arts. 433 to 438 the relation of the radii to a side in a trigon, tetragon, hexagon, octagon, dodeca- gon and hecadecagon have been shown by methods based upon geometrical proportions. This relation in polygons of seven, nine, ten, eleven, thirteen, fourteen and fifteen sides, cannot be so readily shown by geometry, but can be easily obtained by trigonometry — as also said relation of the parts in a regular polygon of any number of sides. The na- ture of trigonometrical tables is discussed in Arts, 473 and 474. So much as is required for the present purpose will here be stated. Let ABC {Fig. 293) represent one of the triangles into which any polygon may be divided, in which B C =^ b = 71 side of the polygon; A C — R = the radius of the circum- scribed circle ; and A D — r = the radius of the inscribed circle. GENERA.L RULES FOR POLYGONS. 461 Make ^6^ equal unity ; on 6^ as a centre describe the arc E F\ draw F H and E G perpendicular to B C, or parallel to A D ; then for the uses of trigonometry E G is called the tangent of we have n — 5, and — .° = 90° -^=90-36 = 54°. For the radius of the circumscribed circle, we have (150.)- b R = 2 COS. c 464 POLYGONS. 2 COS. 54 ie = ^ ^ — ^. 2 COS. 54 Using a table of logarithmic sines and tangents {Art. 427), we have — Log. 2 = 0-3010300 Cos. 54° = 9-7692187 Their sum = 0-0702487 — subtracted from Log. I = o- 0000000 0-85065 =9-9297513 Therefore — i?= 0-85065 <^. Or : The radius of the circumscribed circle of a regular pcnta- gon equals a side of the pentagon multiplied by the decimal o - 8 5065 . For the radius of the inscribed circle, we have (151.) — b r = — tan. c, 2 ' , tan. 54° r — b 2 For this we have — Log. tan. 54° = 0-1387390 Log. 2 = 0-3010300 0-68819 = 9-8377090. Therefore — 19 b. Or: The radius oi i\\Q inscribed circle of a regular /^^^/^^<^;/ equals a side of the pentagon multiplied by the decimal 0'62)^ig. For the area we have (152.) — A =1 b'^n tan. r, ^ = i X 5 tan, 54° ^', ^=f tan. 54° b\ TABLE FOR REGULAR POLYGONS. 465 For this we have — Log. 5. = 0-6989700 Log. tan. 54° =-- 0-1387390 0-8377090 Log. 4 = 0-6020600 I -72048 11=0-2356490 A — 1. 72048 1? \ Therefore Or: The area of a regu\a.r pentagon equa/s tJie square of its side multiplied by i • 72048. /|-42- — Polygons: Table of Constant MnUipHers. — ^ To obtain expressions for the radii of the circumscribed and in- scribed circles, and for the area for polygons of 7, 9, 10, 11, 13, 14, and 15 sides, a process would be needed such pre- cisely as that just shown in the last article for a pentagon, except in the value of n and r, which are the only factors which require change for each individual case. No useful purpose, therefore, can be subserved by ex- hibiting the details of the process required for these several polygons. The values of the constants required for the radii and for the areas of these polygons have been com- puted, and the results, together with those for the polygons treated in former articles, gathered in the annexed Table of Regular Polygons. REGULAR POLYGONS. Sides. 3- 4. 5. 6. 7. 8. 9- 10. II. 12. 13. 14. 15- 16. Trigon Tetragon Pentagon Hexagon i i Heptagon Octagon Nonagon Decagon Undecagon. . . Dodecagon . . Tredecagon. . , Tetradecagon. Pentadecagon Hecadecagon. 57735 70711 85065 00000 15238 30656 46190 61803 77473 93185 2-08929 2-24698 2-40487 2-56292 .28868 • 50000 •68819 • 86603 1 03826 I -20711 1-37374 1-53884 1-70284 I • 86603 2-02858 2 - 1 9064 2-35231 2-51367 •43301 I • 00000 1-72048 2-59808 3-63391 4.82843 6-18182 7-69421 9-36564 11-19615 13-18577 15-33451 17-64236 20-10936 466 POLYGONS. In this table R represents the radius of the circumscribed circle ; r the radius of the inscribed circle ; b one of the sides, and A the area of the polygon. By the aid of the constants of this table, R, the radius of the circumscribed circle of any of the polygons named, may be found when a side of the polygon is given. For this purpose, putting m for any constant of the table, we have — R — bin. (154-) As an example : let it be required to find R, for a penta- gon having each side equal to 5 feet ; then the above expres- sion becomes — i? = 5 X 0-85065, R = 4-25325. The radius will be 4 feet 3 inches and a small fraction. In like manner the radius of the inscribed circle will be — r = bm; (155.) and for a pentagon with sides of 5 feet, we have — r = 5 X 0-68819, r =3.44095. Or, the radius of the inscribed circle will be 3 ft. -^q and a small fraction. Or, multiplying the decimal by 12, 3 ft. 5 in. j^-^Q and a small fraction. The area of any polygon of the table may be obtained by this expression — A = b'7;i; (156.) and, applying this to the pentagon as before, we have — ^ = 5^x1- 72048, A = 43-012. EXPLANATION OF THE TABLE. 467 Or, the area of a pentagon having its sides equal to 5 feet, is 43 feet and yi|-g- of a foot. By the constants of the table a side of any of its poly- gons may be found, when either of the radii, or the area, are known. When R is known, we have — " = -■ ('57.) When r is known, we have — b = --. (158.) in When the area is known, we have — m (159.) SECTION XL— THE CIRCLE. 443. — Circle§: Diameter and Perpendicular: mean Proportional. — Let ABC {Fig. 294) be a semicircle. From C, any point in the curve, draw a line to A and another to B\ then ABC will be a right-angled triangle {Art. 352). Draw the line CD perpendicular to the diameter A B ; then CD will divide the triangle ABC into two triangles. A CD and C B D, which are homologous. For, let the triangle C B D be revolved on Z> as a centre until its line CD shall come to the position i^^ /?, and the hue DB oc- cupy the position D F, each in a position at right angles to its former position, the point B describing the curve B F, and the point C the curve C E, and each forming a quadrant or angle of ninety degrees. Since these points have revolved ninety degrees, therefore the three lines of the triangle 6'^Z>have revolved into a position at right angles to that which they before occupied ; hence the line FF is at right angles to CB, and (from the fact that A CB is a right angle) parallel with A C. Since the triangle E FD equals the triangle CB D. and since the lines of E FD are parallel respectively to the corresponding lines of A CD, therefore the triangles^ CD and CB D are homologous. Comparing the lines of these triangles and putting a = A B, y = CD, and x — D B, we have — RADIUS FROM CHORD AND VERSED SINE. DB '. D C-.'. DC : AD, X \ y '. \ y \ a — X, 469 y'' — X {a — x). (160.) Or, in a semicircle, 2i perpendicular to the ^/<^;;/^/^t terminated by the diameter and the curve is a geometric mean, or mean proportional, between the two parts into zvJiicJi the perpendicular divides the diameter. 444. — Circle : Radius from Given Chord and Versed Sine.— Let A B {Fig. 295) be a given chord line and CD a versed sine. Extend CD to the opposite side of the circle ; it will pass through F, the centre. Join A and C, also E and B. The line A />, perpendicular to the diameter C E, is a mean proportional between the two parts C D ^nd DE {Art. 443) ; or, putting a r^ A D, b = C D, 2ind r equal the radius FE, we have — CD : AD :: AD : DE b : a : : a : 2 r — b, a' = b{2 r-b\ a' = 2 rb-b\ a'' ^-b'' — 2rb, a^ + b- r = ■ ^ — . 2 b (161; 470 THE CIRCLE. Or : The radius of a circle equals the sum of the squares of half the chord a7id the versed sine, divided by twice the versed sine. Another expression for the radius may be obtained ; for the two triangles C B D and C E B {Fig. 295) are homologous (Art. 443) and their corresponding lines in proportion. Put- ting/for CB, we have — CD or — or — and — CB : : CB : CE, :/: :/: 2 ^y /■- = -- 2rv, r = 2V' (162.) Or : The radius of a circle equals the square of the chord of half the arc divided by twice the versed sine. 4-45. — Circle: Segment from Ordiiiate§. — When the curve of a segment of a circle is required for which the radius can- not be used, either by reason of its extreme length, or be- D^P Fig. 296. cause the centre of the circle is inaccessible, it is desirable to obtain the curve without the use of the radius. This may be done by calculating ordinates, a rule for which will now be developed. Let DC B {Fig. 296) be a right angle, and A DB ^ cir- cular arc described from (7 as a centre, with the radius BC^CD= CP. Draw PM parallel with DC, and AG parallel with C B. Now, in the segment A D G, we have given A G,.iis chord, and D E^ its versed sine, and it is re- RULE FOR ORDINATES. 47 1 quired to find an expression by which its ordinates, as P F, may be computed. From Art. 416, we have — or, putting for these lines their usual symbols — ^2 __ ^2_ ^2^ y^ ^ r^-x\ now we have — EC^DC-DE, EC= FM, FM=DC-DE, FM^r-b. Then we have — PF^PM- FM or, putting t for P/^and substituting for PJ/and P J/ their values as above, we have — t = y-{r-b), and for y, substituting its value as above, we have — t:=^ \/r'^ -x^ -(r-b), (163.) Or : The ordinate in the segment equals the square root of the difference of the squares of the radius and the abscissa niimis the difference of the radius and the versed sine. For example : let the chord A G {Fig. 296) in a given case equal 20 feet, and the versed sine, b, or the rise D E^ equal 4 feet ; and let the ordinates be located at every 2 feet along the chord line, A G. In solving this problem Ave require first to find the radius. This is obtained by means of equation (161.) — a'^^b^^ 472 THE CIRCLE. For a, half the chord, we have lo feet ; for h, the versed sine, we have 4 feet ; and, substituting these values, we have — 10^ + 4- 116 r — ^— — = 14- 5 2x4 8 ^ ^ The radius equals — 14-5 The versed sine equals — 4-0 {r — b)-= 10.5 The square of 14-5, the radius, equals 210-25. Now we have, substituting these values in equation (163.) — i— 1/210-25 — ;ir- — 10-5. The respective values of x^ as above required, are o, 2, 4, 6, 8 and 10 Substituting successively for x one of these values, we shall have, when — X — 0; 2 ; 4; 6; 8; ^ = V 210-25 — 0-— 10-5 = 4- X — ^ = V 210-25 — 2- — IO-5 = 3-8614 X := ^ — V 210-25 -4'- IO-5 = 3-4374 X = ^ = V 210-25 — 6-— IO-5 = 2-7004 X = ^ = V 210-25 - -8^- 10-5 = 1-5934 X — lu , I — j/210-25 — IO-— 10-5=0-0 Values for t may be taken at points as numerous as desira- ble for accuracy. In ordinary cases, however, they need not be nearer than in this example. After the points are secured, let a flexible piece of wood be bent so as to coincide with at least four of the points at a time, and then draw the curve against the strip. 446. — Circle : Relation of Diameter Jo Cireiiinfereiice. — In Art. 439 it is shown that the area of a polygon equals the radius of the inscribed circle multiplied by half of a side of the polygon and by the number of the sides ; or, TO FIND THE CIRCUMFERENCE. 473 b r A = r X —n = bn ; or, the area equals half the I'adius by a 22 ^ side into the number of sides ; or, half the radius into the periphery of the polygon. Now, if a polygon have ver}^ small sides and many of them, its periphery will approxi- mate the circumference of the circle inscribed within it; in- deed when the number of sides becomes infinite, and conse- quently infinitely small, the periphery and circumference become equal. Consequently, for the area of the circle, we have — A = T-., (164.) where c represents the circumference. By computing the area of a polygon inscribed within a given circle, and that of one circumscribed about the circle, the area of one will approximate the area of the other in proportion as the number of the sides of the polygon are increased. For example : if polygons of 4 sides be inscribed within and circumscribed about a circle, the radius of which is i, the areas will be respectively 2 and 4. If the polygons have 16 sides, the areas are each 3 and a fraction, the fractions being unlike; when they have 128 sides the areas are each 3 • 14 and with unlike fractions ; when the sides are increased to 2048, the areas each equal 3-1415 and unlike fractions, and when the sides reach 32768 in number the areas are equal each to 3-1415926, having Hke decimals to seven places. The computations have been continued to 127 places (Gregory's " Math, for Practical Men "), but for all possible uses in building operations seven places will be found to be sufficient. From this result we have the diameter in proportion to the circumference as i : 3- 141 5926, or as — I : 3-i4i59i. I : 3-I4I59' I : 3-1416. Of these proportions, that one may be used which will give 474 THE CIRCLE. a result most nearly approximating the degree of accuracy required. For many purposes the last proportion will be sufficiently near the truth. For ordinary purposes the proportion 7 : 22 is very use- ful, and is correct for two places of decimals ; it fails in the third place. The proportion 113 : 355 is correct to six places of deci- mals. For the quantity 3-1415926 putting the Greek letter it (called py)y and 2r = d for the diameter, we have — c — 7t d. (165.) To apply this : in a circle of 50 feet diameter, what is the circumference ? r = 3- 1416 X 50 c = 157.08//. If the more accurate value of tt be used, we have — ^ = 3' 141 5926 X 50, c = 1 57- 07963- The difference between the two results is 0-00037, which for all ordinary purposes, would be inappreciable. By the rule of 7 : 22, we have — c == 5ox\2. =: i5;.i42857i, an excess over the more accurate result above, of 0-0632271, Avhich is about f of an inch. Bv the rule of 113 : 355, we have — ^ = 5oxm= 157-079646. This result gives an excess of only o-ooooi6; it is sufficiently near for any use required in building. From these results we have these rules, namely: To obtain the circumference of a circle, multiply its diameter by TO FIND THE AREA. 4/5 22, and divide the product by y ; or, more accurately, multiply the diameter by i^^ and divide the produet by ii^; or, by mul- tiplication only, multiply the diameter by 3-1416; or, by 3.14159^; or, by 3-1415926; according to the degree of accuracy required. And conversely : To obtain the diameter from the cir- cumference, multiply the circumference by 7 and divide the product by 22', or, multiply by 113 and divide by '^^^\ or, di- vide the circumference by 3-1416; or, by 3-i4i59:|-; or, by 3 -141 5926. 447- — Circle : LiCnglh of an Arc. — Considering the cir- cle divided into 360°, the length of an arc of one degree in a circle the diameter of which is unity may be thus found. The circumference for 360° is 3 - 141 59265 ; 3-14159265^^.^^872664625; 360 which equals an arc of one degree in a circle having unity as its diameter; or, for ordinary use the decimal 0-008727 or o-oo87i may be taken ; or putting a for the arc and g for the number of degrees, we have — a — 0-00872665 dg. (166.) Wherefore : To obtain the length of an arc of a circle, multiply the diameter of the circle by the number of degrees in the arc, and by the decimal 0-0087^, or, instead thereof, by 0-008727. 448. — Circle : Area. — The area of a circle maybe ob- tained in a manner similar to that for the area of polygons {Art. 439), in which A^^Bn\ ^5 = r — , or — A =z ^ b nr, where b equals a side of the polygon and n the number of sides ; so that b n equals the perimeter of the polygon. Now, if for the perimeter of the polygon there be sub- 476 THE CIRCLE. stitiited the circumference of the circle, we shall have, put- ting for the circumference 3- 1416 d, or, n d {Art. 446) — A =^7tdr, in which r is the radius. Since 2 r — d, the diameter, and r = -, we have- 2 And since- or — Therefore- d A —\7td — , 2 A ^\nd\ TT = 3.14159265, i^ = 0-78539816, ^ TT = Q. 7854, nearly. A = 0-7854^1 (167.) Or: The area of a circle equals the square of the diameter utiil- tiplied by 0-7854. D Fig. 297. As an example, the area of a circle 10 feet in diameter is found thus — 10 X 10 = 100. 100 X 0-7854 = 78-54 feet. 449. — Circle: Area of a Sector. — The area oi A B C D {Fig. 297), a sector of a circle, is proportionate to that of the whole circle. For, as the circumference of the whole circle is to its area, so is the arc A B C to the area of A B C D. AREA OF SECTOR. 47/ The circumference of a circle is (165.) C = 7t d. The area of a circle is (167.) A = • 7854 d^\ For the arc ABC put a, and for the area of A B C D put s. Then we have from the above-named proportion — It d : -7854 d'^ \ \ a \ s^ .7854 rf' K d The coefficient 0-7854 is — {Art. 448). 4 Therefore, multiplying the fraction by 4, we have — n d'- S — , a ; 4 n d or — S=^\da — \r a. (168.) Wherefore : To obtain the area of a sector of a circle, multiply a quarter of the diameter by the lejtgth of the arc. Thus: let ^ Z> equal 10; also let A B C = a, equal 12. Then the area oi A B C D is — 5 = ^ X 10 X 12, 5 = 60. The length of the arc may be had by the rule in Art. 44^. 450. — Circle: Area of a Segment. — In the last article, A BCD {Fig. 297) is called the sector of a circle. Of this the portion included within A E C B \s> ^ is a diameter conjugate to the diameter PJ, and KR, RQ, QL, and Ziif are tangents at the extremities of these conjugate diameters. Now it is shown by writers on Conic Sections {Fig. 302) that— ZT' + Wc'' = 'nc' + Yc^ or — Or : The sum of the squares of the two axes equals the sum of the squares of any two conjugate diameters. From this it is also shown that the area of the parallelo- gram K C equals the rectangle A C x B C; or, that a paral- lelogram formed by tangents at the extremities of any two conjugate diameters is equal to the rectangle of the axes. 458- — Ellipse ; Area. — Let E equal the area of an ellipse ; A the area of a circle, of which the radius a equals the semi- major axis of the ellipse, and let b equal the semi-minor axis. Then it has been shown that — E : A '.: b '. a, E = At a The area of a circle {Art. 448) is — A — \ n dr -^ It r'^, and when the radius equals a — A ^ 7t a"-, This value of A, substituted in the above equation, gives — E = 7ta''-, a E — n ab. ' (178.) PRACTICAL SUGGESTIONS. 489 Or: The area of an ellipse equals 3- 141 59 J times the product of the semi-axes ; or 0-7854 times the product of the axes. 459. — Ellipse : Practical §ugg[cstioiis. — In order to de- scribe the curve of an ellipse, it is essential to have the two axes ; or, the major axis and the parameter ; or, the major axis and the focal distance. If the two axes are given, then with the semi-major axis for radius, from B {Fig. 299) as centre an arc may be made at i^and F' , the foci ; and then the curve may be described by any of the various methods given at Arts. 548 to 552. If the major axis only and the parameter are given, then {Art. 454) since — b^'^ay, we have b — Vay. (179-) Or : The semi-minor axis of an ellipse equals the square root of the product of the semi-major axis into the semi-parameter. Then, having both of the axes, proceed as before. If the major axis and the focal distance are given, or the location of the foci ; then with the semi-major axis for ra- dius and from the focal points as centres, describe arcs cut- ting each other at B and O {Fig. 299). The intersection of the arcs gives the limit to B O, the minor axis. With the two axes proceed as before. Points in the curve may be found by computing the length of the ordinates, and then the curve drawn by the side of a flexible rod bent to coin- cide with the several points. For example, let it be required to find points in the curve of an ellipse, the axes of which are 12 and 20 feet ; or 490 THE ELLIPSE. the semi-axes 6 and lo feet, or 6 x 12 = 72 inches, and 10 x 12 = 120 inches. Fix the positions of the points N N', etc., along the semi- major axis C 31 {Fig. 303) at any distances apart desirable. It is better to so place them that the ordinates when drawn shall divide the curve ^Pt^ into parts approximately equal. If CM be divided into eight parts as shown, these parts measured from C will be well graded if made equal severally to the following decimals multiplied by CM. In this case CM= 120; therefore — ^A^ =z 120 X 0-3 =■ 2>^' — x' C N' — 120 X 0-475 = 57- = x' C N" :=^ 120 X 0'62i, = 75. =y Etc., = 120 X 0-75 = 90- = x' 120 X 0-85 = 102- = x' 120x0-925 = III- = .f ' 120 x 0-975 =: 117- — x' 120 X I -O = 120- = ;r'. The equation of the ellipse having the origin of co-ordi- nates at the centre {Art. 454) is — or, dividing by a a'y" = b^{a^-x'\ y' = --M'-^'% or — / 7,2 y -^ \ —Aa^ - x"'^ °^— y^-^/ a'^-x'^'-, (180.) in which a and b represent the semi-axes. Substituting for these their values in this case, we have — 72 y=-^o ^'2o^- j^ = o-6 1/14400 •/ 2 LENGTH OF ORDINATES. 491 Now, substituting in this equation the several values of x'^ successively, the values of the corresponding ordinates will be obtained. For example, taking 36, the first value of x\ as above, we have — y= 0-6 1/14400 36^ 7 = 68-684; j; — 0-6 1^14400 - 57^ y = 63-359; and so in like manner compute the others. The ordinates for this case are as follows, viz. : When .v X — o, _y = 72-0 r:= 36,7 = 68.684 = 57.7=^63.359 =. 7Sy y = 56-205 = 90, JJ/ == 47-624 = 102, J/ = 37.928 = III, y =z 27.358 = 117, J/ = 15-999 = 120, J 1= o-o. The computation of these ordinates is accomplished easi- ly by the help of a table of square roots and of logarithms. For example, the work for one ordinate is all comprisjd within the following, viz. : 7 0-6 r 14400 120- = 14400 36- = 1296 36- = 68.684. 13104 = 4. 1 174039 Half = 2-0587020 0.6 = 9.7781513 68.684 = 1-8368533. The logarithm of 13104 = 4-1 174039. The half of this is the logarithm of the square root of 13 104. To the half log- arithm add the logarithm of o-6; the sum is the logarithm of 68-684 found in the table (see Art. 427). SECTION XIII.— THE PARABOLA. 460. — Parabola : I>efinitioiis. — The parabola is one of the most interesting of the curves derived from the sections of a cone. The several curves thus produced are as fol- lows : When cut parallel with its base the outline is a circle ; when the plane passes obliquely through the cone, it is an ellipse ; when the plane is parallel with the axis, but not in the axis, it is a hyperbola ; while that which is produced by 304. a plane cutting it parallel with one side of the cone is a parabola. Let the lines L M and L N (Fig. 304) be at right angles ; draw CFB parallel with LM; make LQ = LF\ draw QB parallel with LF\ then FB — B Q. Now let the line A L move from F L^ but remain parallel with it, and as it moves let it gradually increase in length in such manner that the point A shall constantly be equally distant from the line LM and from the point F. Then A BP, the curve described by the point A, will be a semi-parabola. For example, the lines FB and B Q are equal ; the lines FPand PM^vq equal, and so of lines similarly drawn from any point in the curve A BP. Let PNhQ drawn parallel with LM; then for the EQUATION TO THE CURVE. 493 point P, AN IS the abscissa and NP its ordinate {see Ar/. 452). The double ordinate CB drawn through F, the focus, is the para^neter. A F is the focal distance. A is the vertex of the curve. The line L M is the directrix. 4-61. — Parabola : Equation to tlie Curve. — In Fig, 304 F PN is a right-angled triangle, therefore — NP"" = FP"" - FN''; but- FP^ MP= LN=A N-\-AL', and— FN= AN- AF. Therefore — NP' = A.N-hA L'-A N- A F' ; or- _^.^=.(.^^ + i/)^-Gr-l/)^ / being put for the distance LF— FB (see Art. 452). As in Arts. 412 and 413, we have — {x-v \py = ,r"' ^px-Y\p'' {x-^py^x^-px+ip-' y'' — 2px (181.) by subtraction. This is the usual equation to the parabola, in which we have the rule : The square of the ordinate equals the rectangle of the corresponding abscissa with the parani- eter. From (181.) we have — X : y : : y '. 2p, or: Th.e parameter is a third proportional \,o the abscissa and its corresponding ordinate. 462. — Parabola ; Tangent. — From AT, any point in the directrix, draw a line to F, the focus {Fig. 305) ; bisect M F in R, and through R draw U T perpendicular to M F, then the line T U will be a tangent to the curve. For, draw AI D 494 THE PARABOLA. perpendicular to L V, and from P, the point of its intersection with the line T [/, draw a line to F, the focus ; then, because RP is a. perpendicular from the middle of MF, MPF is an isosceles triangle, and therefore the lines J/Pand FP are equal, or the point P is equidistant from the focus and from the directrix, and therefore is a point in the curve. To show that the line T U touches the curve but does not pass through it, take U, any point in the line T U, other than Fig. 305. the point P\ join U to i/and to F. Then, since f/ is a point in the line T U, M U F, for reasons above given, is an isosce- les triangle ; from C/draw U F perpendicular to L V. Now, if the point ^be also in the curve, the lines C/Fand U F, by the law of the curve, must be equal ; but UF, as before shown, is equal to UM, a line evidently longer than U V; therefore, it is evident that the point ^is not in the curve. A similar absurd result will be reached if any other point than the point [/ in the line U The assigned, excepting the RULE FOR THE TANGENT. 495 point P. Therefore the line T P touches the curve in only one point, P ; hence it is a tangent. Parallel with L F, from A, draw ^4 5, the vertical tangent. Now ^ vS bisects M F or intersects it in the point i'?. For the two right-angled triangles F L J^/and FA R are homolo- gous ; and because FA ■=^ A L, by construction, therefore FR^RM. Or : The vertical tangent bisects all lines which can be drawn from the focus to the directrix. The lines P/^and FT are equal ; for the lines J/Pand N T being parallel, therefore the alternate angles MP T and N TP Sire equal (Art. 345) ; and because the line P T bi- sects MF, the base of an isosceles triangle, therefore the angles ^P 7" and T^P 7^ are equal. We thus have the two angles N TP and FPT each equal to the angle MPT; therefore the two angles N TP and FP T are equal to each other ; hence the triangle PF T is an isosceles triangle, hav- ing the points T and P equidistant from F, the focus. Also because the line MF is perpendicular to P T, there- fore the line MF bisects the tangent PT in the point R. And because TR = R P, therefore, comparing triangles TRF and TPO, TF= FO. The opposite angles MP T and U PD made by the two in- tersecting lines U T and M D {Art. 344) are equal, and since the angles MP T and FP T are equal, as before shown, therefore the angles FP T and U P D are equal. It is because these two angles are equal, that, in reflectors, rays of light and heat proceeding from F, the focus, are re- flected from the parabolic surface in lines parallel with the axis. For an equation expressing the value of the tangent, we have — TP = TN + NP\ f^ = (2xY+y\ Or: The tangent to a parabola equals the square root of tJie sum of four times the square of the abscissa added to the square of the ordinate. 496 THE PARABOLA. 463. — Parabola: Subtangent. — The line T N {Fig. 305), the portion of the axis intercepted between T, the point of intersection of the tangent, and N, the foot of a perpendicu- lar to the axis from P, the point of contact, is the subtangent. The subtangent is bisected by the vertex, or TA = A N. For, the two triangles TRA and T P N 2irQ homologous; and, as shown in the last article, the line jl/i^ bisects P Tin R', or TR = RP, Therefore, we have — TR : TA :: TP : TN, TR X TN= TA X T P, but— TR=i TP; therefore — ^TPx TN=z TA X TP, i TN= TA. Or : The subtangent of a parabola is bisected by the vertex ; or is equal to twice the abscissa. And because of the similarity of the two triangles TRA and TP N, as above shown, we have — NP=^ 2 AR, y = 2 AR. Or : The ordinate equals twice the vertical tangent. 464. — ParaboBa: Normal and Subnormal. — The line PO {Fig. 305) perpendicular to P T, is the normal and NO, the part of the axis intercepted between the normal and the ordinate, is the subnormal. For the normal, from similar triangles, we have — TN : NP :: T P : P O, i\^Px TP ^^ ~ TN ' P0 = ^-^. 2X . DIAMETER AND RECTANGLE OF BASE. 497 Or : The 7iormal equals the rectangle of the ordinate and taji- gcnty divided by tivicc the abscissa. The subnormal equals half the parameter. For (181.) — or— NP"- = 2 FB ' AN. Dividing by 2 XN gives— NP'- 2 AN In the similar, triangles {Art. 443) 6^/^iVand P T N, we have — NO : iV/' : : NP : NT, NO^EIl. NT As shown in the previous article, N T — 2 A N\ therefore — NO^Z^. (B.) 2AN Comparing equations (A.) and (B.), we have — NO=^FB. Or : The subnormal of a parabola equals Jialf the parameter, a constant quantity for the subnormal to all points of the curve. 465 — Parabola: Diameters. — In the parabola BAC* {Fig. 306), PD, a diameter (a line parallel with the axis) to the point P, is in proportion to B T> x B C, the rectangle of the two parts into which the base of the parabola is divkled by the diameter. This may be shown in the following manner:. nP=£N:=FA-NA. (A.) 498 THE PARABOLA. For^^ we have, taking the co-ordinates, for the point C, (i8i.)— or — -^ = X or- ^- ^EA. (B.) For N A we have, taking the co-ordinates to the point or — -—- — X, 2p N P'' or— -—---^NA, (C.) Using these values (B.) and (C.) in (A.), we have — D P=EA —NA, EC NP' _ EC'-NP' ~ 2p 2/ ~ 2/ If / be put for B C and n for D C, then — NP=EC-DC=^l- 72. TO FIND THE CONSTITUENT PARTS. 499 and — 2/ then {Art. 412,)- — 2/ or {Art. 415) — 2/ 2/ 2/ ' DP- DCxBD 2/ Now, since 2 /, the parameter, is constant, we have D P, the diameter, in proportion to D C x B Dy the two parts of the base. Putting d for the diameter, we have — d='l^-J^, (183.) 2/ Or : The diameter of a parabola equals the quotient obtained by dividing th.Q rectangle — formed by the two parts into which the diameter divides the base — by the parameter. It has been shown by writers on Conic Sections that a diameter, P J {Fig, 307), to any point Pm a parabola bisects all chord lines, SG,B£, etc., drawn parallel with the tan- gent to the point P; the diameter being parallel with the axis of the parabola. 466. — Parabola : Elemems. — From any given parabola, to find the axis, tangent, directrix, parameter and focus, draw any two parallel lines or chords, SG and DE (Fig. 307), and bisect them in N and J; through these points draw JP; then y/'will be a diameter of the parabola — a 500 THE PARABOLA. line parallel with the axis. Perpendicular to Py draw the double ordinate P Q and bisect it in A^; through iVand par- allel with Py draw TO, cutting the curve in A ; then TO will be the axis. Make AT^AN, join Tand P\ then TP will be the tangent to the point P; from P draw P O per- pendicular to P T \ then P O will be the normal, and NO the subnormal. With NO for radius, from N as a centre, describe the quadrant O Ps.\ draw R C parallel with A O, cutting the curve Fig. 307. in C; from C draw CB perpendicular to A O, cutting A O in F\ then Pwill be the focus and C B Xh^ parameter. Make AL — AF\ draw L M perpendicular to T0\ then Z J/ will be the directrix. Extend PJ to meet Z J/at M', join Pand F\ then, if the work has been properly performed, F P will equal MP. 467. — Parabola: Described Meeliaiiically. — With NP {Fig. 308) a given base, and N A a given height, set perpen- THE CURVE DESCRIBED MECHANICALLY. ;oi dicularly to the base, extend N A beyond A, and make A T equal to N A ; join Zand P; from /^perpendicularly to TP drawP6^; bisect ON\\\R\ make AL and y^/'each equal X.O N R\ through Z, perpendicular to L O, draw D E, the di- rectrix. Let the ruler CDBShe laid to the line DE, then with J G H, ^ set-square, the curve may be described in the fol- lowing manner : Placing the square against the ruler and with its edge Fig. 30S. y// coincident with the line MP, fasten to it a fine cord on the edge PR, and extend it from P to F, the focus, and se- cure it to a pin fixed in F. The cord FP will equal the edge MP. To describe the curve set the triangle J G H at MPE, slide it gently along the ruler towards D, keeping the edge y G in contact with the ruler, and, as the square is moved, keep the cord stretched tight, holding for this pur- pose a pencil, as at K, against the cord. Thus held, as the square is moved the pencil will describe the curve. That this operation will produce the true curve we have but to 502 THE PARABOLA. consider that at all points the line F K will equal KJ, which is the law of the curve (Art. 460). 468. — Parabola : Described from Points. — With given base, N P (Fig. 309), and given height, A N, to find the points D, F, M, etc., and describe the curve. Make A T equal to A N (Art. 462) ; join T and P\ perpendicular to TP draw Fig. 309. PO; make A B equal to twice N 0\ take G, any point in the axis A O, and bisect B G in J \ on y as a centre describe the semi-circle B CG cutting A Z, a perpendicular to B O in C ; on A C and A G complete the, rectangle A C D G. Then D is a point in the curve. Take H, another point in the axis ; bisect B H\n K] on K as a centre describe the semi-circle B E H cutting A L\xi E\ this by E F and HE, gives F, an- THE CURVE DESCRIBED FROM ARCS. 503 Other point in the curve ; in like manner procure M, and as many other points as may be desired. This simple and accurate method of obtaining points in the curve depends upon two well-established equations ; one, the equation to the parabola, and the other, the equation to the circle. The line G D, an ordinate in the parabola, is equal to A C, an ordinate in the circle BCG\ AG, the abscissa of the para- bola, is also the abscissa of the circle ; in which we have {Art. 443)- AG \ AC \\ AC : AB, X : y : : y : a — Xy j/^ =z x(a — x). For the parabola, we have (181.) — JK" = 2pX. Comparing- these two equations, we have— X {a — x) = 2px, a — ;ir = 2 /, or — BG~AG=2p. By construction AB equals 2 N O, or twice the subnor- mal ; the subnormal {Ai't. 464) equals half the parameter. Hence, twice the subnormal equals the parameter — equals 2 p. Therefore, the method shown in Fig. 309 is correct. 4-69- — Parabola: Described from Arc§. — Let NP {Fig. 310) be the given base and A iVthe given height of the par- abola. Make A T {Art. 462) equal A N. Join TtoP\ draw P O perpendicular to P T\ bisect N O \n R\ make A L and A F each equal to N R ; then L M, drawn perpendicular to TO, will be the directrix. Parallel to LM draw the lines B D, C E, etc., at discretion. Then with the distance B L for 504 THE PARABOLA. radius, and on i^ as a centre, mark the line B D with an arc ; the intersection of the arc and the line will be a point in the curve {Art. 460). Again, with C L for radius and on i^as a centre, mark the line C E with an arc ; this gives another point in the curve. In like manner, mark each horizontal Fig, 3ro, line from /^ as a centre by a radius equal to the perpendicu- lar distance between that line and L M, the directrix. Then a curve traced through the points of intersection thus ob- tained will be the required parabola. 470. — Parabola: De§cribed from Orc1iiiate§. — With a given base, N P {Fig. 311), and height, A N, ci parabola may be drawn through points J, H, G, etc., which are the extrem- ities of the ordinates B J, C H, D G, etc. ;. the lengths of the ordinates being computed from the equation to the curve, (181.)- jk' = 2px. For any given parabola, in base and height, the value of THE CURVE FORMED BY ORDINATES. 505 / may be had by dividing both members of the equation by 2 ;ir; by which we have — p = ^~.- NP' 2X 2AN' (A.) from which, TVPand A iV being known,/ may be computed. With the value of/, a constant quantity, determined, the equation is rendered practicable. For, taking the square root of each member of equation (181.), we have — y= V 2p X. (B.) which by computation will produce the value of jf, for every assigned value of ;r, 2iS> A B, A C, A D, etc. Fig. 311. As an example : let it be required to compute the ordi- nates in a parabola in which the base, N P, equals 8 feet, and the height, A N, equals 10 feet. With these values equation (A.) as above becomes — _ NP^ ^ ~ Jan 2 X 10 20 2 p — 6-4. 5o6 THE PARABOLA. Then, with this value in (B.) as above, we have, for each or- dinate — J= V6 4x. In order to assign values to x, let A N be divided into any number of parts at B, C, D, etc., say, for convenience in this example, in ten equal parts ; then each part will equal one foot, and we shall have the consecutive values oi x = i, 2, 3, 4, etc., to ID, and the corresponding values of 7 will be as follows. When — I, J^= V6 X — 2, J/ = 4/6 X 3, J = \'6 X :=^ 4, f = V6 X 5, J= V6 X =z 6, y = ^'-^ X ^ 8, J = |/6 X = 10, y = |/6 4x I = V 6.4= 2.5297 = ^7, 3-5777 = CH, 4 X 2 = |/ 12- 4x 3= v' 19-2 = 4-3^18 = Z> 6^, 4x 4= |/ 25-6 = 5-0596 = ^^, = 5-6569 = (etc.). 4 X 5=1/32 4x6=1/ 38-4 = 6.1968 = ■ 1/^4^8 = 6-6933 = 4x 7 4x8=4/ 51-2 = 7-1554 4x9= |/l7^ = 7-5895 4x10— '^Z 64 8-0 ^NP. With these values of y, respectivelv, set on the correspond- ing horizontal lines B J, CH, DG, £ S, etc., points in the curve y, //, G, 5, etc., are obtained, through which the curve may be drawn. The decimals above shown are the decimals of a foot ; they may be changed to inches and decimals of an inch by multiplying each by 12. For example : 12x0-5297 = 6-3564 equals 6 inches and the decimal 0-3564 of an inch, which equals nearly § of an inch. Near the top of the curve, owing to its rapid change in direction and to the approximation of the direction of the curve to a parallel with the direction of the ordinates, it is THE CURVE FOUND BY DIAMETERS. 507 desirable to obtain points in the curve more frequent than those obtained by dividing the axis into equal parts. Instead, therefore, of dividing the axis into equal parts, it is better to divide it into parts made gradually smaller toward the apex of the curve— or, to obtain points for this part of the curve as shown in the following article. 471. — Parabola: Dci^cribed from Diameters. — Let EC {Fig. 312) be the given base and A E the given height, placed perpendicularly to E C. Divide E C \w several parts at pleasure, and from the points of division erect perpendicu- lars to E C. The problem is to compute the length of these diameters, as DP, and thereby obtain points in the curve, as at P. For this purpose we have equation (183.), which gives D C Fig. 312. the length of the diameters, and in which n equals D C {Fig. 312), /equals twice E C\ and / equals half the parameter of the curve. The value of p is given in equation (A.), {Art, 470), in which y equals EC {Fig. 312), and x equals A E. Substituting these symbols in equation (A.), we have — / y EC 2x 2xAE 11 2 /r where h — EC, the base, and h— A E, the height. For /, substituting this, its value, in equation (183.), we have — 71 {I — 11) n {I — ri) 2p h n (2 /; — fi) 2 h (184.) 508 THE PARABOLA. As an example : let it be required in a parabola in which the base equals 12 feet and the height 8 feet, to compute the length of several diameters, and through their extremities describe the curve. Then Ji will equal 8, and b 12. If the base be divided into 6 equal parts, as in Fig. 312, each part will equal 2 feet. Then we have — and- IL - 8 _ 8 _ I b~^~ ~\2} ~ i44~ 18 d= -j-^ n {2 b- n)y n(24-n) a= — — 7: . In this equation, substituting the consecutive values of ji, we have, when — - O X 24 ;/ = o, a = = o I o 2 X 22 71 = 2, a = 18 =2.444 71= 4, d = 4x 20 18 =4.444 71 =r. 6, d = 6x 18 18 = 6. 71 =z 8, d = 8x 16 1 Q =37.111 lox 14 12 X 12 ;^=rl2, ^=— Yg ^=:8.0 The several diameters, as P D, in Fig. 312, may now be made equal respectively to these computed values of d, and the curve traced through their extremities. AREA EQUALS TWO THIRDS OF RECTANGLE. 5Q9 472. — Parabola: Area, — From (181.), the equation to the parabola, and by the aid of the calculus, it has been shown that the area of a parabola is equal to two thirds of the circumscribing rectangle. For example : if the height, AE {Fig. 312), equals 8 feet, and EC, the base, equals 12 feet, then the area of the part included within the figure A PC E A equals f of 8 x 1 2 = | x 96 = 64 feet ; or, it is equal to f of the rectangle A B C E. SECTION XIV.— TRIGONOMETRY. 473. — Riglit-Ang^led Triaiigle§ : The Sides. — In right- angled triangles, when two sides are given, the third side may be found by the relation of equality which exists of the squares of the sides (Arts. 353 and 416). For example, Fig. 313. if the sides a and b {Fig. 313) are given, c, the third side, may be computed from equation (115.) — Extracting the square root, we have — c = Vd-U-a\ When the hypothenuse and one side are given, by transposi- tion of the factors in (115.), we have — a c'- b'; or- a= Vc"- — b b' = c V'c a a (A.) (B.) TWO SIDES GIVEN TO FIND THE THIRD. 5 I I Owing to the factors being- involved to the second power in this expression, the labor of computation is greater than that in a more simple method, which will now be shown. In equation (A.) or (B.) the factors under the radical may be simplified. By equation (i 14.) we have — Therefore, equation (A.) becomes — rt = y (^ + b) {c — b), a form easy of solution. For example: let c equal 29-732 and b equal 13-216, then we have — 29-732 13-216 The sum = 42-948 The difference = 16-516 By the use of a table of logarithms {Art. 427) the problem may be easily solved ; thus — Log. 42-948 = 1-6329429 16-516 = I -2179049 To get the square root — 2)2 - 8508478 a = 26-6332 = 1-4254239 This method is applicable to the sides of a triangle, only ; for the hypothenuse it will not serve. The length of the hypothenuse as well as that of either side may, however, be obtained by proportion ; provided a triangle of known di- mensions and with like angles be also given. For example: in Fig. 314, in which the two sides a and b are known, let it be required to find c, the hypothenuse. Draw the line D E parallel with A C, then the two trian- gles B DE and B A C are homologous; consequently their 512 TRIGOXOiMETRY. corresponding sides are in proportion (Art. 361). Hence, if d equals unity, we have — d : / : : a : c, = ^/, from which, when a and / are known, c is obtained by sim- ple multiplication. 4-74. — Riglit- Angeled Triangles: Trigonometrical Ta- bles. — To render the simple method last named available, the lengths of 7 sin, B , . sin. A : sin. B : : a : b =. a -r- — -- . ( IQ^.) sin. A ^ ^^ ^ ^ • -n 7 sin. B , . sin. 6 : sin. B \ \ c \ b ^^ c —. — . {IQ4.) sin. C ^ ^-^ J . . ^ sin. C , . sin. A : sin. 6 : : a : c = a —. . iiQ^.) sm. A ^ ^^ ^ sin. B : sin. C : : b : c = b . ' . (iq6.) sin. j5 ^ ^ / These expressions give the values of the three sides respec- tively ; two expressions for each, one for each of the two remaining sides ; that is to be used which contains the ^zven side. From these expressions we derive the values of the sines; thus — (I97-) (198.) (199-) (200.) sin. A = sin. ^-■ sin. A = sin. c sin. B = sin. a'-. a sin. B = sin. ct. c sin. C = sin. A -. (201.) sin, C = sin. B -. (202.) 477. — Oblique - Angled Triangles: First Class. — The problems arising in the treatment of oblique-angled trian- gles have been divided into four classes, one of which, the TO FIND THE TWO SIDES. 52 1 first, will here be referred to. The problems of the first class are those in which a side and tzvo angles are given, to find the remaining angle and sides. As to the required angle, since the three angles of every triangle amount to just two right angles {Art. 345), or 180°, the third angle may be found simply by deducting the sum of the two given angles from 180°. For example : referring to Fig. 318, if angle ^ = 18° and angle B — 42°, then their sum is 18 + 42 = 60, and 180 — 60 = 120° = the angle A C B. To find the tw^o sides : if ^ be the given side, then to find the side b we have, equation (193.) — sm. B ^ A sm. A or, the side b equals the product of the side a into the quo- tient obtained by a division of the sine of the angle opposite b by the sine of the angle opposite a. For example: in a triangle {Fig. 318) in Avhich the angle A = 18°, the angle B — 42° (and, consequently {Art. 345) the angle C= 120°), and the given side a equals 43 feet; what are the lengths of the sides b and c? Equation (193.) gives — , sin. B b = a— — —J. sm. A Performing the problem by logarithms {Art. 427), we have — Log. a (= 43) =^-6334685 Sin. B{— 42°) = 9-8255109 £•4589794 Sin. A (= 18°) = 9.4899824 Log. ^ (= 93 • 1102) = 1 .9689970. Thus the side b equals 93.1102 feet, or 93 feet i inch and nearly one third of an inch. 522 TRIGONOMETRY. For the side c, we have, equation (195.)- sin. C c ^ a sin. A ' or- Log. «(=43) =£-6334685 Sin. C{= 120°) = 9-9375306 £•5709991 Sin. A (= 18°) ^ 9.4899824 Log. ^(== 120-508) =: 2-0810167 or, the base c equals 120 feet 6 inches and one tenth of an inch, nearly. But if instead of a the side b be given, then for a use equation (191.), and for c use equation (196.). And, lastly, if c be the given side, then for a use equation (192.), and for b use equation (194.)- 478. — Oblique-Angled Triangles: Second Class. — The problems which comprise the second class are those in which tivo sides and an angle opposite to one of them are given, to find the two remaining angles and the third side. The only requirement really needed here is to find a second angle ; for, with this second angle found, the problem is reduced to one of the first class ; and the third side may then be found under rules given in Art. 477. To find a second angle, use one of the equations (197.) to (202.). For example : in the triangle ABC {Fig. 318), let a(j=. 43) and ^(=93-11) be the two given sides, and A, the angle op- posite a, be the given angle (= 18°). Then to find the angle B, we have equation (199.) — (selecting that which in the right hand member contains the given angle and sides) — sin. B = sin. A - a = sin.^?llLi. 43 OBLIQUE-ANGLED TRLVNGLES. 523 By logarithms {Art. 427), we have — Log. sin. A (^ 18°) = 9.4899824 ^' 93.11 = 1.96899/O 1.4589794 ' 43 = 1-6334685 '' sin. ^(=42') = 9.8255109 By reference to the log. tables, the last line of figures, as above, is found to be the sine of 42° ; therefore, the required angled is 42°. Then 180° - (18^ + 42°)= 120° = the angle <;. With these angles, or with any two of them, the third side c may be found by rules given in Art. 477. M^-- «/ R \1 D N r ^ k<- 'A ^ p V / J Fig. 319. 479. — Otoliqiie-Anijled TriiingJes : Sum and Difference of T%vo Anglc§. — Preliminary to a consideration of prob- lems in the third class of triangles, it is requisite to show the relation between the siii?i and difference of two angles. In Fig. 319, let the angle A JM and the angle A JN be the two given angles ; and let A J M be called angle ^4, and A J N, angle B. Now the sum and difference of the angles may be ascertained by the use of the sum and difference of the sines of the angles, and by the sum and difference of the tangents. In the diagram, in which the radius A J equals 524 TRIGONOMETRY. unity, we have MP, the sine of angle A {— A J M), and NQ^ RP,thQ sine of angle B{= A J N). Then— IMP- RP=MR equals the difference of the sines of the angles ; and since PM' = PM— PM'+RP= RM\ equals the sum of the sines of the angles. With the radius yC' describe the arc JDE, and tangent to this arc draw FH parallel with MM', or perpendicular to^^. Then FD is the tangent of the angle M C N, and D H is the tangent of the angle N CM'. Now since an angle at the circumference is equal to half the angle at the centre standing on the same arc {Art. 355), therefore the measure of the angle M C N is the half oiMN, equals — ^{AM—AN)=^{A-B). Similarly, we have — ^{AM'+AN) = ^{A +B\ for the angle TV C: J/'. Therefore we have for the tangent of the angle M C N— FD = i?in. i{A -B), and, for the tangent of the angle N C M' — DH = tan. \{A + B). And, because FC D and M C R are homologous triangles, as, also, D CH and R CM\ therefore— M' R : MR : : DH : D F, SUM AND DIFFERENCE OF TWO ANGLES. 525 sin. ^ + sin. B : sin. A — sin. B : : tan. i{A + B) : tan. ^{A — B), from which we have — sin. A — sin. B __ tan. ^ (A — B) ,^ sin. A + sin. B ~ tan. i {A + B)' ^ '^ To obtain a proper substitute for the first member of this expression we have, equation (195.) — sin. C c — a — , sm.yi or — c sin. A = a sin. C. (M.) We also have, equation (196.) — 7 sin. C sm. B or — c sin. B — b sin. C. (N.) These two equations, (M.) and (N.), added, give — c sin. A ^ c sin. B = a sin. C -\- b sin. C. or — c (sin. A + sin. B) = sin. C(a -{- b). (P.) But, if equation (N.) be subtracted from equation (M.), we have — c sin. A — c sin. B = a sin. C — b sin. Cy or — c{?,in. A — sin. .5) =r ^ sin. C {a — b). (R.) If equation (R.) be divided by equation (P.), we have — <:(sin. A — sin. B) _ sin. C {a — b) c{s\n. A + sin. B) ~~ sin. C {a + b)* 526 TRIGONOMETRY, which reduces to — sin. A — sin. B a sin. A + sin. B a + b The first member of this equation is identical with the first member of the above equation (D.), and therefore its equal, the second member, may be substituted for it ; thus — a — b _ t an. ^{A - B) 'cT+l? ~ tan. i(yf + B)" From which we have — tan. 1 (^ - ^) z= tan. 1{A + B) ""-—^ . ' (203.) We have {Art. 431) the proposition, that if half the differ- ence of two quantities be subtracted from half their sum, the remainder will equal the smaller quantity. For example : if A represent the larger quantity and B the smaller, then — i{A+B)-i(A-B)^B; (204.) and, again, we also have {Art. 431) — ■h{A +B) + ^{A-B)= A. (205.) 480.— Oblique-Angled Triaiigle§: Third Cla§s.^The t/izrd class of problems comprises all those cases in which two sides of a triangle and their included angle are given, to find the other side and angles. In this case, as in the problems of the second class, the only requirement here is to find a second angle ; for then the problem becomes one belonging to the first class. But the finding of the second angle, in problems of the third class, is attended with more computation than it is in prob- lems of the second class. The process is as follows : Hav- ing one angle of a triangle, the sn^/i of the two remaining OBLIQUE-ANGLED TRIANGLES. 527 angles is obtained by subtracting the given angle from 180° — the sum of the three angles. Then with equation (203.) the difference of the two angles is obtained. And then, having the sum and dffcrence of the two angles, either may be found by one of the equations (204.) and (205.). For example : let Fig. 320 represent the triangle in which a {— 36 feet) and d{= 27 feet) are the given sides ; and C {= 105°) the angle included between the given sides, a and d. The sum of the two angles A and B, therefore, will be — {A+B) = 180- 105 =^75°, and the half of the sum of A and B \s ^- = 37° 30'. The sum of the given sides is 36 + 27 = 63, and their dif- ference is 36 — 27 = 9. Then from equation (203.) we have — tan. i{A-B)=^ tan. 37° 30^/. Solving this by logs. {Art. 427), we have — Log. tan. 37° 30' = 9.8849805 9 = 0-9542425 0.8392230 ■■ 63 = 1.7993405 tan. ^{A - i?)(=6° 15' 20- 5'') = 9-0398825 Thus half the difference of A and ^ is 6° 15' 20- 5'', nearly. 528 TRIGONOMETRY. By equation (204.) — 37° 30' 6° 15' 20.5^ The difference, 31° 14' 39*5'' = ^, and by equation (205.) — 37-30 6.15.20.5 The sum, 43.45.20-5 = A From above, 31.14.39. 5 = B The given angle, 105. 0.0 = C The three angles, 180. o. o Thus, by adding together the three angles, the work is tested and proved. Having the three angles, the third side may now be found by the rule for problems of the first class. 481. — Oblique-Anglcd Triangles : Fourth Class. — The fourth class comprises those problems in which the three sides of the triangle are given, to find the three angles. The method by which the problems of the fourth class are solved is to divide the triangle into two right-angled triangles; then, by the use of equation (129.), to find one side of one of these triangles, and then with this side to find one of the angles, then by rules for the second class prob- lems, obtain the second and third angles. Thus, from equation (129.), we have-^ ^ 2C By the relation of sines to sides {Art. 476), we have (Fig. 321)- b \ g '. \ sin. E : sin. F. TRIANGLES — FOURTH CLASS. 529 But the angle ^5" is a right angle, of which the sine is unity, therefore — b '. g \ '. I : sin. F —J. Substituting for g its value as above, we have — sin./-^^'-(^±i) (^-^) . (206.) 2 be To illustrate: let a, b, c {Fig. 321) be the three given sides of the triangle ABC, respectively equal to 12, 8 and 16 feet. With these, equation (206.) becomes — . ^ 16' — (12 + 8)(i2 - 8) sin. F = ^—^^ -* 2 X 8 X 16 „ 256 — (20 X 4) sin. F — — 256 sin. F — 256 Solving this by logarithms {Art, 427), we have- Log. 176 = 2.2455127 '' 256 = 2.4082400 Log. sin. 43° 26' = 9.8372727 or, the angle at F equals 43° 26', nearly. Of the triangle 530 TRIGONOMETRY. A C E {Fig. 321), ^ is a right angle, therefore the sum of F and A, the two remaining angles, equals 90° {Art. 346). Hence, for the angle at y^, we have — ^=90° -43° 26^ = 46° 34'. We now have two sides a and b and A, an angle opposite to one of them, to find B, a second angle. For this, equa- tion (199.) is appropriate. Thus — sin. B — sin. A - a This may be solved as shown in Art. 478. And, when the second angle is obtained, the third angle is found by subtracting the sum of the first and second an- gles from 180°. But to test the accuracy of the work, it is well to cojh- pute the angle C from the angle A, and the sides a and c. For this, equation (201.) will be appropriate. 482. — Trigonometric Formulae : Rig^ht-Aiig^led Trian- gles. — For facility of reference the formulce of previous articles are here presented in tabular form. The symbols referred to are those of Fig. 322. FORMULAE IN TABULAR FORM. 531 Right-Angled Triangles. Given. Required. Formula. a, b, a, c, b, c, b, a. c ^ Va'+b\ b — V{c + a) {c — a). a = \/{c -^b){c- b). A, B, B, A. B= go"" -A. ^ = 90" - B. B, a, b, b — a tan. B, a c — COS. B ' B,b, a, b b ^ ~ sin. B ' B,c, a, a — c COS. B. b — c sin. B. 483. — Trigonometrical Formulae: First Cia§§, Oblique. C • — The symbols of the formulae of the following table indi cate quantities represented in Fig. 323 by like symbols. 532 TRIGONOMETRY. Oblique-Angled Triangles: First Class. 1 Given. I Required. 1 Formula. A,B, A,C, B, C, c, A. C = i^o - A + B. B = i%o- A-\- C. A= i^o - B + C. A, B, b, A, C, c. a, J sin. A 1 a = b-. — --. 1 sin. B sin. A a = c -. --. sin. C A, B, a, B, C, c, b, b, J sin. B b — a -. . sin. A J sin. B b = C-. --. sm. 6 A, C, a, B, C, b, Cy sin. C c — a -. — - - . sin. A _ 7 sin. C sin. B 484. — Trigonometrical Formulae : Second €la§s, Oblique. — The symbols in the formulae of the following table refer to quantities represented in Fig. 323, by like symbols. J FORMUL/E FOR TRIANGLES, SECOND CLASS. $33 Oblique-Angled Triangles: Second Class. Given. Required. FORMUL.^. B, a, b, i A, A, sin. A — sin. B -. b Sin. A — sin. C -. A, a, b, C, b, c, B, B, sin. B — sin. A -. a Sin. B = sin. C -. c A, a, c, B. b, c, sin. C = sin. A -. a sin. C — sin. B-. b B, C, A,C, A, B, A = iSo - B -{- C. B = iSo-A + C. A,B, C = iSo- A +B. For — a, See Formulae, First Class. 534 TKIGONOMETRY. 4-86. — Trigonometrical Formulae : Third €las$«, Oblique. — The symbols in the formulse of the following table refer to quantities shown by like symbols in Fig, 323. Oblique- Angled Triangles: Third Class. Given, Required. Formula. A+B, ^ +^= 180- C C, a, b, A-B, tan. iM - ^) =: tan. ^{A-^ B)^~^. a -V b A, A=\{A-\-B)-r \{A -B). B, B =i{A-\-B)-^{A -B). C +B, C + B = 1^0 - A. A, b, c, C-B, tan. 1{C ^)-tan. i((f +i?)^""^. c, C = ^{C + B) +i{C-B). B, B = ^{C-^B)-i{C-B). C +A. C -{- A = iSo - B. B, a, c, C-A, tan. 1{C A) = i^n.i{C+A)'~'^. c, C=i{C + A) + h{C-A), A, A=i{C-hA) + i(C-A). For the remaining side consult formulas for the Jirst class. 486. — Trigonometrical Formulae : Fourtli Cla§s, Ob- lique. — The symbols in the formulae of the following table refer to quantities shown by like symbols in Fzg. 321. formula for triangles, fourth class. 535 Oblique-Angled Triangles: Fourth Class. Given a, b^ c, to find A,B, C. 2bc A = go - F. sin. B = sin. A -. a sin. C — sin. A -. a C = \Zo-yA + B). SECTION XV.— DRAWING. 487. — General Remarks. — A knowledge of the proper- ties and principles of lijies can best be acquired by practice. Although the various diagrams throughout this work may be understood by inspection, yet they will be impressed upon the mind with much greater force, if they are actually drawn out with pencil and paper by the student. Science is acquired by study — art by practice ; he, therefore, who would have anything more than a theoretical (which must of necessity be a superficial) knowledge of carpentry and geometry, will provide himself with the articles here speci- fied, and perform all the operations described in the fore- going and following pages. Many of the problems may appear, at the first reading, somewhat confused and intricate ; but by making one line at a time, according to the explana- tions, the student will not only succeed in copying the fig- ures correctly, but by ordinary attention will learn the principles upon which they are based, and thus be able to make them available in any unexpected case to which they may apply. 488. — Articles Required. — The following articles are necessary for drawing, viz. : a drawing-board, paper, draw- ing-pins or mouth-glue, a sponge, a T-square, a set-square, two straight-edges, or flat rulers, a lead pencil, a piece of india-rubber, a cake of india-ink, a set of drawing-instru- ments, and a scale of equal parts. 489. — The I>raxvicig-Board. — The size of the drawing- board must be regulated according to the size of the draw- ings which are to be made upon it. Yet for ordinary prac- tice, in learning to draw, a board about fifteen by twenty inches, and one inch thick, will be found large enough, and DRAWING PAPER. 53/ more convenient than a larger one. This board should be well seasoned, perfectly square at the corners, and without clamps on the ends. A board is better without clamps, because the little service they are supposed to render by preventing the board from warping is overbalanced by the consideration that the shrinking of the panel leaves the ends of the clamps projecting beyond the edge of the board, and thus interfering with the proper working of the stock of the T-square. When the stuff is well-seasoned, the warp- ing of the board will be but trifling ; and by exposing the rounding side to the fire, or to the sun, it may be brought back to its proper shape. 490. — Drawing-Paper. — For mere line drawings, it is unnecessary to use the best drawing-paper ; and since, where much is used, the expense will be considerable, it is desirable for economy to procure a paper of as low a price as will be suitable for the purpose. The best paper is made in Eng- land and water-marked " Whatman." This is a hand-made paper. There is also a machine-made paper at about half- price, and the manilla paper, of various tints of russet color, is still less in price. These papers are of the various sizes needed, and are quite sufficient for ordinary drawings. 491. — To Secure tlie Paper to tlie :Board. — A drawing- pin is a small brass button, having a steel pin projecting from the underside. By having one of these at each corner, the paper can be fixed to the board ; but this can be done in a better manner with mouth-ghie. The pins will prevent the paper from changing its position on the board ; but, more than this, the glue keeps the paper perfectly tight and smooth, thus making it so much the more pleasant to work on. To attach the paper with mouth-glue, lay it with the bottom side up, on the board ; and with a straight-edge and penknife cut off the rough and uneven edge. With a sponge moderately wet rub all the surface of the paper, except a strip around the edge about half an inch wide. As soon as the glistening of the water disappears turn the sheet 533 DRAWING. over and place it upon the board just where you wish it glued. Commence upon one of the longest sides, and pro- ceed thus : lay a flat ruler upon the paper, parallel to the edge, and within a quarter of an inch of it. With a knife, or anything similar, turn up the edge of the paper against the edge of the ruler, and put one end of the cake of mouth- glue between your lips to dampen it. Then holding it up- right, rub it against and along the entire edge of the paper that is turned up against the ruler, bearing moderately against the edge of the ruler, Avhich must be held firmly with the left hand. Moisten the glue as often as it becomes dry, until a sufficiency of it is rubbed on the edge of the paper. Take away the ruler, restore the turned-up edge to the level of the board, and lay upon it a strip of pretty stiff paper. By rubbing upon this, not very hard but pretty rapidly, with the thumb-nail of the right hand, so as to cause a gentle friction and heat to be imparted to the glue that is on the edge of the paper, you will make it adhere to the board. The other edges in succession must be treated in the same manner. Some short distances along one or more of the edges may afterward be found loose ; if so, the glue must again be applied, and the paper rubbed until it adheres. The board must then be laid away in a warm or dry place ; and in a short time the surface of the paper will be drawn out, perfectly tight and smooth, and ready for use. The paper dries best when the board is laid level. When the drawing is finished lay a straight-edge upon the paper and cut it from the board, leaving the glued strip still attached. This may afterward be taken off by wetting it freely with the sponge, which will soak the glue and loosen the paper. Do this as soon as the drawing is taken off, in order that the board may be dry when it is wanted for use again. Care must be taken that, in applying the glue, the edge of the paper does not become damper than the rest ; if it should, the paper nmst be laid aside to dry (to use at another time) and another sheet be used in its place. Sometimes, especially when the drawing-board is new, the paper will not stick very readily ; but by persevering THE T-SQUARE. 539 this difficulty may be overcome. In the place of the mouth- glue a strong solution of gum-arabic may be used, and on some accounts is to be preferred ; for the edges of the paper need not be kept dry, and it adheres more readily. Dissolve the gum in a sufficiency of warm water to make it of the consistency of linseed-oil. It must be apphed to the paper with a brush, when the edge is turned up against the ruler, as was described for the mouth-glue. If two drawing-boards are used, one may be in use while the other is laid away to dry ; and as they may be cheaply made, it is advisable to have two. The drawing-board having a frame around it, commonly called a panel board, may afford rather more facility in attaching the paper when this is of the size to Fig. 324. suit ; yet it has objections which overbalance that consid- eration. 492. — The T-Square. — A T-squarc of mahogany, at once simple in its construction and affording all necessary service, may be thus made : let the stock or handle be seven inches long, two and a quarter inches wide, and three eighths of an inch thick ; the blade, twenty inches long (exclusive of the stock), two inches wide, and one eighth of an inch thick. In joining the blade to the stock, a very firm and simple joint may be made by dovetailing it — as shown at Fig. 324. 493. — The Set-Square. — The set-square is in the form of a right-angled triangle ; and is commonly made of mahogany, 540 DRAWING. one eighth of an inch in thickness. The size that is most convenient for general use is six inches and three inches respectively for the sides which contain the right angle, although a particular length for the sides is by no means necessary. Care should be taken to have the square corner exactly true. This, as also the T-square and rulers, should have a hole bored through them, by which to hang them upon a nail when not in use. 494. — The Rulers. — One of the rulers may be about twenty inches long, and the other six inches. The pencil ought to be hard enough to retain a fine point, and yet not so hard as to leave ineffaceable marks. It should be used lightly, so that the extra marks that are not needed when the drawing is inked, may be easily rubbed off with the rubber. The best kind of india-ink is that which wall easily rub off upon the plate ; and, w^hen the cake is rubbed against the teeth, will be free from grit. 495. — The Instruments. — The draiving-instriimeiits may be purchased of mathematical instrument makers at various prices ; from one to one hundred dollars a set. In choosing a set, remember that the lowest price articles are not always the cheapest. A set, comprising a. sufficient number of instruments for ordinary use, well made and fitted in a ma- hogany box, may be purchased of the mathematical instru- ment makers in New York for four or five dollars. But for permanent use those which come at ten or twelve dollars will be found to be better. 496. — The ScaBe of Equal Parts. — The best scale of equal parts for carpenters' use, is one that has one eighth, three sixteenths, one fourth, three eighths, one half, five eighths, three fourths, and seven eighths of an inch, and one inch, severally divided into tivelftJis, instead of being divided, as they usually are, into tenths. By this, if it be required to proportion a drawing so that every foot of the object represented will upon the paper measure one fourth of an inch, use that part of the scale which is divided into one THE SET-SQUARE. 54 1 fourths of an inch, taking for every foot one of those divis- ions, and for every inch one of the subdivisions into twelfths ; and proceed in hke manner in proportioning a drawing to any of the other divisions of the scale. An instrument in the form of a semi-circle, called a protractor, and used for laying down and measuring angles, is of much service to surveyors, and occasionally to carpenters. 4-97. — The Use of tlie Set-Square. — In drawing parallel lines, when they are to be parallel to either side of the board, use the T-square ; but when it is required to draw lines parallel to a line which is drawn in a direction oblique Fig. 325. to either side of the board, the set-square must be used. Let ab {Fig. 325) be a line, parallel to which it is desired to draw one or more lines. Place any edge, as cd, of the set- square even with said line ; then place the ruler gh against one of the other sides, as ce, and hold it firmly ; slide the set-square along the edge of the ruler as far as it is desired, as at /; and a line draw^n by the edge if will be parallel to a b. To draw a line, as kl {Fig. 326), perpendicular to another, as a bj set the shortest edge of the set-square at the line a b ; place the ruler against the longest side (the hypothenuse of the right-angled triangle) ; hold the ruler firmly, and slide the set-square along until the side cd touches the point k] then the line Ik^ drawn by it, will be perpendicular to ab. 542 DRAWING. In like manner, the drawing of other problems may be facil- itated, as will be discovered in using the instruments. 498. — Directions for Drawing. — In drawing a problem, proceed, with the pencil sharpened to a point, to lay down the several lines until the Avhole figure is completed, ob- serving to let the lines cross each other at the several angles, instead of merely meeting. By this, the length of every Hne will be clearly defined. With a drop or two of water, rub one end of the cake of ink upon a plate or saucer, until a sufficiency adheres to it. Be careful to dry the cake of Fig 326. ink ; because if it is left wet it will crack and crumble in pieces. With an inferior camel's-hair pencil add a little water to the ink that w^as rubbed on the plate, and mix it well. It should be diluted sufficiently to flow freely from the pen, and yet be thick enough to make a black line. With the hair pencil place a little of the ink between the nibs of the drawing-pen, and screw the nibs together until the pen makes a fine line. Beginning Avith the curved lines, proceed to ink all the lines of the figure, being careful now to make every line of its requisite length. If they are a trifle too short or too long the drawing will have a ragged appear- ance ; and this is opposed to that neatness and accuracy which is indispensable to a good drawing. When the ink is dry efface the pencil-marks with the india-rubber. If the PUTTING THE DRAWING IN INK. 543 pencil is used lightly they will all rub off, leaving those lines only that were inked. In problems all auxiliary lines are drawn light ; while the lines given and those sought, in order to be distinguished at a glance, are made much heavier. The heavy lines are made so by passing over them a second time, having the nibs of the pen separated far enough to make the lines as heavy as desired. If the heavy lines are made before the drawing is cleaned with the rubber they will not appear so black and neat, because the india-rubber takes away part of the ink. If the drawing is a ground-plan or elevation of a house, the shade-lines, as they are termed, should not be put in until the drawing is shaded ; as there is danger of the heavy lines spreading when the brush, in shading or color- ing, passes over them. If the lines are inked with common writing-ink they will, however fine they may be made, be subject to the same evil ; for which reason india-ink is the only kind to be used. SECTION XVI.— PRACTICAL GEOMETRY. 4-99. — Definitions. — Geometry treats of the properties of magnitudes. A point has neither length, breadth, nor thickness. A line has length only. Superficies has length and breadth only. A plane is a surface, perfectly straight and even in every direction ; as the face of a panel when not warped nor winding. A solid has length, breadth, and thickness. A rights or straight, line is the shortest that can be drawn between two points. Parallel lines are equidistant throughout their length. Fig, 327. Fig. 328. Fig. 329. An angle is the inclination of two lines towards one an- other {Fig. 327). A right angle has one line perpendicular to the other {Fig 328). Kxi oblique angle is either greater or less than a right angle {Figs. 327 and 329). An acute angle is less than a right angle {Fig. 327). An obtuse angle is greater than a right angle {Fig. 329). When an angle is denoted by three letters, the middle one, in the order they stand, denotes the angular point, and the other two the sides containing the angle ; thus, let a, b, c {Fig. 327) be the angle, then b will be the angular point, and ab and be will be the two sides containing that angle. TRIANGLES AND RECTANGLES. 545 A triangle is a superficies having three sides and angles [Figs. 330, 331, 332, and 333). An equilateral triangle has its three sides equal {Fig. 330). An isosceles triangle has only two sides equal (Fig. 331). Fig. 330. Fig. 331, A scalene triangle has all its sides unequal {Fig. 332). A right-angled triangle has one right angle {Fig. 333). An acute-angled triangle has all its angles acute {Figs. 330 and 331). Fig. 332. Fig. 333. An obtuse-angled triangle has one obtuse angle {Fig. 332). A quadrangle has four sides and four angles {Figs. 334 to 339)- A parallelogram is a quadrangle having its opposite sides parallel {Figs. 334 to 337). Fig. 334. Fig. 335. being right A rectangle is a parallelogram, its angles angles {Figs. 334 and 335). A square is a rectangle having equal sides {Fig. 334). A rhombus is an equilateral parallelogram having oblique angles {Fig. 336). 546 PRACTICAL GEOMETRY. A rhomboid is a parallelogram having oblique angles {Fig. 337). A trapezoid is a quadrangle having only two of its sides parallel {Fig. 338). Fig. 336. 7 Fig. 337. A trapezium is a quadrangle which has no two of its sides parallel {Fig, 339). A polygon is a figure bounded by right lines. A regular polygo?t has its sides and angles equal. An irregular polygon has its sides and angles unequal. Fig. 338. Fig. 339. A trigon is a polygon of three sides {Figs. 330 to 333) ; a tetragon has four sides {Figs. 334 to 339) ; a pentagon has five {Fig. 340) ; a hexagon six (Z^^^. 341) ; a heptagon seven (F/^. 342) ; an octagon eight (i^^^. 343) ; a nonagon nine ; a decagon ten ; an undecagon eleven ; and a dodecagon twelve sides. Fig. 340. Fig. 341. Fig. 342. Fig. 343. A circle is a figure bounded by a curved line, called the circumference, which is everywhere equidistant from a cer- tain point within, called its centre. The circumference is also called the periphery, and some- times the circle. PARTS OF THE CIRCLE. 547 The radius of a circle is a right line drawn from the centre to any point in the circumference {ab^ Fig. 334). All the radii of a circle are equal. The diameter is a right line passing through the centre, and terminating at two opposite points in the circumference. Hence it is twice the length of the radius {cd^ Fig. 344.) Fig. 344. An arc of a circle is a part of the circumference {c b, or bed, Fig. 344). A chord is a right line joining the extremities of an arc {b d, Fig. zAA)- ^ A segment is any part of a circle bounded by an arc and its chord {A, Fig. 344). Fig. 345. A sector is any part of a circle bounded by an arc and two radii, drawn to its extremities {B, Fig. 344). A quadrant, or quarter of a circle, is a sector having a quarter of the circumference for its arc [C, Fig. 344). A tangent is a right line which, in passing a curve, touches, without cutting it {f g, Fig. 344). 548 PRACTICAL GEOMETRY. A cone is a solid figure standing upon a circular base di- minishing in straight lines to a point at the top, called its vertex {Fig. 345). The axis of a cone is a right line passing through it, from the vertex to the centre of the circle at the base. An ellipsis is described if a cone be cut by a plane, not parallel to its base, passing quite through the curved surface {a b, Fig. 346). A parabola is described if a cone be cut by a plane, par- allel to a plane touching the curved surface {cd, Fig. 346 — cd being parallel to f g). An hyperbola is described if a cone be cut by a plane. Fi( 347- parallel to any plane within the cone that passes through its vertex (eh, Fig. 346). Foci are the points at which the pins are placed in de- scribing an ellipse (see Art. 548, and /,/, Fig. 347). The transverse axis is the longest diameter of the ellipsis {a b, Fig. 347). The conjugate axis is the shortest diameter of the ellipsis ; and is, therefore, at right angles to the transverse axis {cd, J\?' 347). The parameter is a right line passing through the focus of an ellipsis, at right angles to the transverse axis, and ter- minated by the curve {gh and gt, Fig. 347). RIGHT LINES AND ANGLES. 549 A diameter of an ellipsis is any right line passing through the centre, and terminated by the curve [kl, or ;;/ ;/, Fig. 347). A diameter is conjugate to another when it is parallel to a tangent drawn at the extremity of that other — thus, the di- ameter inn {Fig. 347) being parallel to the tangent op, is therefore conjugate to the diameter kl. A double ordinate is any right line, crossing a diameter of an ellipsis, and drawn parallel to a tangent at the extremity of that diameter {i t, Fig. 347). A cylinder is a solid generated by the revolution of a right-angled parallelogram, or rectangle, about one of its Fig. 348. Fig. 349. sides ; and consequently the ends of the cylinder are equal circles {Fig. 348). The axis of a cylinder is a right line passing through it from the centres of the two circles which form the ends. A segjnent of a cylinder is comprehended under three planes, and the curved surface of the cylinder. Two of these are segments of circles ; the other plane is a parallelo- gram, called by way of distinction, the plane of the segment. The circular segments are called the ends of the cylinder {Fig. 349). PROBLEMS. RIGHT LINES AND ANGLES. 500. — To Bisect a l.iiie. — Upon the ends of the line ab {Fig. 350) as centres, with any distance for radius greater than half ab, describe arcs cutting each other in c and d\ 550 PRACTICAL GEOMETRY draw the line cd, and the point e, where it cuts ab, will be the middle of the line ab. In practice, a line is generally divided with the com- passes, or dividers ; but this problem is useful where it is desired to draw, at the middle of another line, one at right angles to it. (See Art. 514.) 601. — To Erect a Perpendicular. — From the point a {Fig. 351) set off any distance, as ab, and the same distance from ^ to ^ ; upon r, as a centre, with any distance for radius greater than ca, describe an arc at d\ upon b, with the same Fig. 351. radius, describe another at d\ join d and a, and the line da will be the perpendicular required. This, and. the three following problems, ai'e more easily performed by the use of the set-square (see Art. 493). Yet they are useful when the operation is so large that a set- square cannot be used. TO ERECT A PERPENDICULAR. 551 502. — To let Fall a Perpendicular. — Let a {Fig. 352) be the point above the line be from which the perpendicular is required to fall. Upon a^ with any radius greater than ad, describe an arc, cutting b c 2X e and /; upon the points e and /, with any radius greater than ed^ describe arcs, cutting c t h \ d r ^f g Fig. 352. each other at g", join a and g, and the line ad wiW be the perpendicular required. 603. — To Erect a Perpendicular at the End of a Line. — Let a {Fig. 353), at the end of the line c a, be the point at which the perpendicular is to be erected. Take any point, as b, above the line cay and with the radius ba- describe the arc dae\ through d and b draw the line de\ join e and «, then e a will be the perpendicular required. The principle here made use of is a very important one, and is applied in many other cases (see Art. 510, 3d, and Art. 513. For proof of its correctness, see Art. 352). A second method. Let b (Fig. 354), at the end of the hne a b, be the point at which it is required to erect a perpendic- ular. Upon b, with any radius less than b a^ describe the arc ce d\ upon c, with the same radius, describe the small arc at^. 552 PRACTICAL GEOMETRY. and upon e, another at d ; upon e and d, with the same or any other radius greater than half e d, describe arcs intersecting at /; join /and b, and the line fb will be the perpendicular required. This method of erecting a perpendicular, and that of the following article, depend for accuracy upon the Fig. 354. fact that the side of a hexagon is equal to the radius of the circumscribing circle. A third method. Let b {Fig. 355) be the given point at which it is required to erect a perpendicular. Upon b, with any radius less than ba, describe the quadrant def; upon d, with the same radius, describe an arc at e, and upon e an- other at c\ through d and e draw dc, cutting the arc in c\ join c and b, then c b will be the perpendicular required. Fig. 355. This problem can be solved by the six, eight and ten rule, as it is called, which is founded upon the same principle as the problems at Arts, 536, 537, and is applied as follows: let ad {Fig. 353) equal eight, and ae, six; then, if de equals ten, the angle ead \s> 2i right angle. Because the square of six and that of eight, added together, equal the square of EQUAL ANGLES. 553 ten, thus : 6 x 6 = 36, and 8 x 8 = 64 ; 36 + 64 = 100, and 10 X 10 = 100. Any sizes, taken in the same proportion, as six, eight and ten, will produce the same effect; as 3, 4 and 5, or 12, 16 and 20. (See Art. 536.) By the process shown at Fig. 353, the end of a board may be squared without a carpenters'-square. All that is neces- sary is a pair of compasses and a ruler. Let <;^ be the edge of the board, and a the point at which it is required to be squared. Take the point b as near as possible at an angle of forty-five degrees, or on a initre-\\nQ from a, and at about the middle of the board. This is not necessary to the work- ing of the problem, nor does it affect its accuracy, but the result is more easily obtained. Stretch the compasses from /; to Uy and then bring the leg at a around to d; draw a line from d, through b, out indefinitely ; take the distance db and place it from b to e -, join e and a ; then ea will be at right angles to ca. In squaring the foundation of a building, or laying out a garden, a rod and chalk-line may be used in- stead of compasses and ruler. 504. — To let Fall a Perpendicular near the End of a Line. — Let e {Fig. 353) be the point above the line c a, from which the perpendicular is required to fall. From e draw any line, as ed, obliquely to the fine c a; bisect ^<^at ^ ; upon by with the radius be, describe the arc ead\ join ^ and a] then ea will be the perpendicular required. 605. — To Make an Angle (a§ edf, Fig. 356) Equal to a OiTcn Angle (as b a c), — From the angular point «, with any Fig. 356. radius, describe the arc bc\ and with the same radius, on the line de, and from the point d, describe the arc/^; take the distance be, and upon g, describe the small arc at /; 554 PRACTICAL GEOMETRY. join / and d\ and the angle e df will be equal to the angle bac. If the given line upon which the angle is to be made is situated parallel to the similar line of the given angle, this may be performed more readily with the set-square. (See Art, 497.) 606. — To Bisect an Angle. — Let abc {Fig. 357) be the angle to be bisected. Upon b^ with any radius, describe the Fig. 357. arc a c ; upon a and c, with a radius greater than half a c, describe arcs cutting each other at d\ join b and d\ and bd will bisect the angle abc^ as was required. This problem is frequently made use of in solving other problems ; it should therefore be well impressed upon the memory. 507 — To Trisect a Right Angle.— Upon a {Fig. 358), with any radius, describe the arc b c ; upon b and Cy with the Fig. 358. same radius, describe arcs cutting the arc be 2it d and e; from d and e draw lines to a, and they will trisect the angle, as was required. TO DIVIDE A GIVEN LINE. 555 The truth of this is made evident by the following oper- ation : divide a circle into quadrants ; also, take the radius in the dividers, and space off the circumference. This will divide the circumference into just six parts. A semi-circum- ference, therefore, is equal to three, and a quadrant to one and a half of those parts. The radius, therefore, is equal to two thirds of a quadrant ; and this is equal to a right angle. 508. — Tliroug^li a Given Point, to Draw a Line Parallel to a Given Le§cribe a Segrmcnt of a Circle by a Set-Tri- angle. — Let a b {Fig. 369) be the chord, and c d the height Fig. 369. of the segment. Secure two straight-edges, or rulers, in the position ^^ and c f, by nailing them together at ^, and affixing a brace from ^ to /; put in pins at a and b ; move the angu- lar point c in the direction acb\ keeping the edges of the triangle hard against the pins a and b ; a pencil held at c will describe the arc acb. A curve described by this process is accurately circular, and is not a mere approximation to a circular arc, as some may suppose. This method produces a circular curve, be- cause all inscribed angles on one side of a chord-line are equal {Art. 356). To obtain the radius from a chord and its yersed sine, see Art. 444. If the angle formed by the rulers at <: be a right angle, TO FIND THE VERSED SINE, 561 the segment described will be a semi-circle. This problem is useful in describing centres for brick arches, when they are required to be rather flat. Also, for the head hang- ing-stile of a window-frame, where a brick arch, instead of a stone lintel, is to be placed over it. 517. — To Find the Radius of an Arc of a Circle when the Chord and Versed Sine are Oiven. — The radius is equal to the sum of the squares of half the chord and of the versed sine, divided by twice the versed sine. This is expressed, algebraically, thus : r = — , where r is the radius, c the chord, and v the versed sine {ArL 444.). Exainple.-^\\i a given arc of a circle a chord of 12 feet has the rise at the middle, or the versed sine, equal to 2 feet, what is the radius ? Half the chord equals 6, the square of 6 is, 6 x 6 = 36 The square of the versed sine is, 2x2=4 Their sum equals, 40 Twice the versed sine equals 4, and 40 divided b}' 4 equals ID. Therefore the radius, in this case, is 10 feet. This result is shown in less space and more neatly by using the above algebraical formula. For the letters substituting C^Y " / 1 9\2 2 their value, the formula r — - — =^-— becomes r — ^' ^ ^ , 2V 2X2 and performing the arithmetical operations here indicated equals — 6'+2'_36 + 4_4o 4 "~ 4 ~ "^ ~ ^^• 5(8. — To Find the Versed Sine of an Arc of a Circle when the Radius and Chord are Oiven. — The versed sine is equal to the radius, less the square root of the difference of the squares of the radius and half chord ; expressed alge- braically thus: v = r — Vr ' - {t)\ where r is the radius, v the versed sine, and c the chord. (Equation (161.) reduced.) 562 PRACTICAL GEOMETRY. Example. — In an arc of a circle whose radius is 75 feet, what is the versed sine to a chord of 120 feet? By the table in the Appendix it will be seen that — The square of the radius, 75, equals . . 5625 The square of half the chord, 60, equals . 3600 The difference is 2025 The square root of this is . . . -45 This deducted from the radius ... 75 The remainder is the versed sine, = 30 This is expressed by the formula, thus — 2/ = 75 - 1/75 ^_ (i|oy =: 75 - 1/5625 - 3600 ::= 75 - 45 rrr 30. 519. — To Describe the Segment of a Circle by Intersec- tion of Lines. — Let ab {Fig. 370) be the chord, and cd the ^ £r 1 1 k f height of the segment. Through c draw e f parallel \,o ab\ draw bf at right angles to cb\ make ce equal to cf\ draw ag and bh at right angles io ab\ divide ce^ cf, da, db, ag, and bh, each into a like number of equal parts, as four; draw the lines i 1,22, etc., and from the points 0, 0, and 0, draw lines to c, at the intersection of these lines trace the curve, a cb, which will be the segment required. In very large work, or in laying out ornamental gar- dens, etc., this will be found useful ; and where the centre of the proposed arc of a circle is inaccessible it will be inval- uable. (To trace the curve, see note at Art. 550.) The lines ea, cd, and fb, would, were they extended, meet in a point, and that point would be in the opposite side of the circumference of the circle of which acb is a ORDINATES TO AN ARC. 563 segment. The lines i i, 2 2, 3 3, would likewise, if extended, meet in the same point. The line cd^ if extended to the op- posite side of the circle, would become a diameter. The line fb forms, by construction, a right angle with be, and hence the extension of fb would also form a right angle with be, on the opposite side of <^^; and this right angle would be the inscribed angle in the semi-circle ; and since this is re- quired to be a right angle {Art. 352), therefore the construc- tion thus far is correct, and it will be found likewise that at each point in the curve formed by the intersection of the radiating lines, these intersecting lines are at right angles. 520. — Ordinates. — Points in the circumference of a circle may be obtained arithmetically, and positively accu- rate, by the calculation of oreimates, or the parallel lines o i, 2 / d f Fig. 371. 02, 03, 04 (i^/^. 371). These ordinates are drawn at right angles to the chord-line a b, and they may be drawn at any distance apart, either equally distant or unequally, and there may be as many of them as is desirable ; the more there are the more points in the curve will be obtained. If they are located in pairs, equally distant from the versed sine e d, calculation need be made only for those on one side of ed, as those on the opposite side will be of equal lengths, re- spectively ; for example: o i, on the left-hand side of ed, is equal to o i on the right-hand side, o 2 on the right equals o 2 on the left, and in like manner for the others. The length of any ordinate is equal to the square root of the difference of the squares of the radius and abscissa, less the difference between the radius and versed sine {Art. 445). The abscissa being the distance from the foot of the versed sine to the foot of the ordinate. Algebraically, 564 PRACTICAL GEOMETRY. / = yr"^ — x"^ — (r — d), where t is put to represent the ordi- nate ; X, the abscissa ; d, the versed sine ; and r, the radius. Example. — An arc of a circle has its chord ab {Fig. 371) 100 feet long, and its versed sine cd, ^ feet. It is required to ascertain the length of ordinates for a sufficient number of points through which to describe the curve. To this end it is requisite, first, to ascertain the radius. This is readily /«\2 2 done in accordance with Art. 517. For becomes 2,2 2 ^ ~ — 252-5 = radius. Havins: the radius, the curve 2x5 ^ :> ^ might at once be described without the ordinate points, but for the impracticability that usually occurs, in large, flat segments of the circle, of getting a location for the centre, the centre usually being inaccessible. The ordinates are, therefore, to be calculated. In Fig. 371 the ordinates are located equidistant, and are 10 feet apart. It will only be requisite, therefore, to calculate those on one side of the versed sine cd. For the first ordinate o i, the formula t — Vr"" — x^ — {r — b) becomes — /= 4/252.5'- 10'- (252.5 - 5). = 1^63756.25 — 100 — 247.5. = 252.3019-247.5. = 4 . 8019 = the first ordinate, o i . For the second — t = 4/252.5' -20' -(252.5 - 5). =z 251.7066 — 247.5. = 4. 2066 = the second ordinate, o 2. For the third— / = 4/252.5'- 30'- 247.5. = 250.7115 -247.5. = 3.21 15 = the third ordinate, 03. TO DESCRIBE A TANGED CURVE. 565 For the fourth — / = 4/252.5' — 40' — 247.5. = 249.3115 - 247.5. = 1-8115= the fourth ordinate, o 4. The results here obtained are in feet and decimals of a foot. To reduce these to feet, inches, and eighths of an inch, proceed as at Reduction of Decimals in the Appendix. If the two-feet rule, used by carpenters and others, were decimally divided, there would be no necessity of this re- duction, and it is to be hoped that the rule will yet be thus divided, as such a reform would much lessen the labor of computations, and insure more accurate measurements. Versed sine ^ <^ = ft. 5 -o = ft. 5 -o inches. Ordinates 01= 4-8019 = 4.91 inches, nearly. *' 2=r 4.2066= 4-2^ inches, nearly. " 03= 3-2115= 3 • 2|- inches, nearly. " 04= 1. 81 15 = 1 .9I inches, nearly. 521. — In a Given Angle, to Describe a Tanged Curve. — Let abc{Fig, 372) be the given angle, and i in the line a ^, Fig. 372. and 5 in the line be, the termination of the curve. Divide I b and b 5 into a like number of equal parts, as at i, 2, 3, 4, and 5 ; join i and i, 2 and 2, 3 and 3, etc. ; and a regular curve will be formed that will be tangical to the line a b, at the point i, and to ^^ at 5. This is of much use in stair-building, in easing the angles formed between the wall-string and the base of the hall, also between the front string and level facia, and in many other instances. The curve is not circular, but of the form of the parabola {Fig. 418) ; yet in large angles the difference $66 PRACTICAL GEOMETRY. is not perceptible. This problem can be applied to describ- ing the curve for door-heads, window-heads, etc., to rather better advantage th^n Art, 516. For instance, let ad {Fig. 373) be the width of the opening, and c d the height of the arc. Extend c d, and make de equal to cd\ join a and e, also e and b ; and proceed as directed above. 622, — To De§cribe a Circle within any Given Triangle, §0 that the Sides of the Triangle shall toe Tangieal. — Let abc {Fig. 374) be the given triangle. Bisect the angles Fig. 374. a and b according to Art. 506 ; upon d, the point of intersec- tion of the bisecting lines, with the radius d e, describe the required circle. 523. — About a Given Circle, to Describe an Equilateral Triangle. — Let ad be {Fig. 375) be the given circle. Draw the diameter c d] upon d, with the radius of the given circle, describe the arc aeb\ join a and b\ draw /^at right angles to dc ; make fc and eg each equal to ^^; from /, through «, draw fh, also from g, through b, draw gh\ then fgh will be the triangle required. 624-, — To Find a Right Liine nearly Equal to the Cir- cumference of a Circle. — Let abed {Fig. 376) be the given A RIGHT LINE EQUAL TO A CIRCUMFERENCE. 567 circle. Draw the diameter ac\ on this erect an equilateral triangle aec according to Art, 525 ; draw^/ parallel \.q ac\ extend ec to /, also ea to g\ then gf will be nearly the Fig. 375. length of the semi-circle adc\ and twice gf will nearly equal the circumference of the circle ab c d,2iS was required. Lines drawn from e, through any points in the circle, as bd is to be. This is usually stated thus: ab : b d : : bd : be, and since the product of the means equals the product of the extremes, therefore, abxbe^^bd^- This is shown geometrically at Art. 538. CONIC SECTIONS. 544. — Definitions. — If a cone, standing upon a base that is at right angles with its axis, be cut by a plane, per- AXIS AND BASE OF PARABOLA. 585 pendicular to its base and passing through its axis, the sec- tion will be an isosceles triangle (as a be, Fig. 404) ; and the base will be a semi-circle. If a cone be cut by a plane in the direction e f the section will be an ellipsis ; if in the direction ml, the section will be a parabola; and if in the direction TO, an hyperbola. (See Art. 499.) If the cutting planes be at right angles with the plane ab e, then — 545. — To Find the Axes of the Ellipsis: bisect e f {Fig. 404) in g\ through g draw Ji i parallel \.o ab\ bisect h i in 7; Fig. 404. upon y, withy// for radius, describe the semi-circle }iki\ from ^ draw gk 2iX. right angles to Jii\ then twice gk will be the conjugate axis and e f the transverse. 546. — To Find the Axis and Base of the Parabola. — Let m I {Fig. 404), parallel to ae, be the direction of the cut- ting plane. From m draw m d at right angles to ab; then Im will be the axis and height, and nid an ordinate and half the base, as at Figs. 417, 418. 547. — To Find the Height, Base, and Transverse Axis of an Hyperbola. — Let r {Fig. 404) be the direction of the 586 PRACTICAL GEOMETRY. cutting plane. Extend o r and a c till they meet at ;/ ; from draw op at right angles \.o ab\ then r o will be the height, ;/ r the transverse axis, and op half the base ; as at Fig. 419. 548. — Tlie Axes toeing Giieii, to Find the Foci, and to Describe an Ellipsis Vvitli a String. — Let ab {Fig. 405) and cdhQ the given axes. Upon c, with ^^ or be for radius, de- scribe the arc //; then / and /, the points at which the arc cuts the transverse axis, will be the foci. At / and / place two pins, and another at c \ tie a string about the three pins, so as to form the triangle ffc ; remove the pin from c and place a pencil in its stead ; keeping the string taut, move the pencil in the direction cga; it will then describe the required ellipsis. The lines fg and gf show the posi- tion of the string when the pencil arrives at g. This method, when performed correctly, is perfectly ac- curate ; but the string is liable to stretch, and is, therefore, not so good to use as the trammel. In making an ellipse by a string or twine, that kind should be used which has the least tendenc}^ to elasticity. For this reason, a cotton cord, such as chalk-lines are commonly made of, is not proper for the purpose ; a linen or flaxen cord is much better. 54-9, — Tlie Axes toeing CJiven, to Describe an JCllipsis with a Trammel. — Let ab and cd {Fig. 406) be the given axes. Place the trammel so that a line passing through the centre ot the grooves would coincide with the axes ; make ELLIPSE BY TRAMMEL. 587 the distance from the pencil c to the nut/ equal to half cd\ also, from the pencil e to the nut g equal to half a b ; letting the pins under the nuts slide in the grooves, move the tram- mel eg in the direction cbd\ then the pencil at c will de- scribe the required ellipse. A trammel may be constructed thus: take two straight strips of board, and make a groove on their face, in the cen- tre of their width ; join them together, in the middle of their length, at right angles to one another ; as is seen at Fig. 406. A rod is then to be prepared, having two movable nuts made of wood, with a mortise through them of the size of the rod, and pins under them large enough to fill the grooves. Make a hole at one end of the rod, in which to Fig. 406. place a pencil. In the absence of a regular trammel a tem- porar}^ one may be made, which, for any short job, will an- swer every purpose. Fasten two straight-edges at right angles to one another. Lay them so as to coincide with the axes of the proposed ellipse, having the angular point at the centre. Then, in a rod having a hole for the pencil at one end, place two brad-awls at the distances described at Art. 549. While the pencil is moved in the direction of the curve, keep the brad-awls hard against the straight-edges, as directed for using the trammel-rod, and one quarter of the ellipse will be drawn. Then, by shifting the straight- edges, the other three quarters in succession may be drawn. If the required ellipse be not too large, a carpenters'-square may be made use of, in place of the straight-edges. An improved method of constructing the trammel is as 588 PRACTICAL GEOMETRY follows : make the sides of the grooves bevelling from the face of the stuff, or dove-tailing instead of square. Prepare two slips of wood, each about two inches long, which shall be of a shape to just fill the groove when slipped in at the end. These, instead of pins, are to be attached one to each of the movable nuts with a screw, loose enough for the nut to move freely about the screw as an axis. The advantage of this contrivance is, in preventing the nuts from slipping out of their places during the operation of describing the curve. 550. — To De§cribe an Ellipsis by Ordinate^. — Let ab and cd {Fig. 407) be given axes. With c e qv c d for radius /12 3a describe the quadrant f gJi ; divide /"//, ac^ and e b, each into a like number of equal parts, as at i, 2, and 3 ; through these points draw ordinates parallel \.o cd and fg\ take the distance i i and place it at i /, transfer 27 to 2 vi, and 3 /^ to 3 ;/ ; through the points a, 11, in, I, and c, trace a curve, and the eUipsis will be completed. The greater the number of divisions on a, c, etc., in this and the following problem, the more points in the curve can be found, and the more accurate the curve can be traced. If pins are placed in the points n, in, /, etc., and a thin slip of wood bent around by them, the curve can be made quite correct. This method is mostly used in tracing face-moulds for stair hand -railing. 55 (. — To Describe an Ellipsis by Intersection of Lines. ■Let ab and cd {Fig. 408) be given axes. Through c, draw ELLIPSE BY INTERSECTION OF LINES. 589 fg parallel X.o ab\ from a and b draw af and h g at right angles to ab\ divide f a^ gb, ac\ and eb, each into a like number of equal parts, as at i, 2, 3, and o, 0, o\ from i, 2, and 3, draw lines to c ; through o, 0, and o, draw lines from d, intersecting those drawn to c ; then a curve, traced through the points i, i, i, will be that of an ellipsis. Where neither trammel nor string is at hand, this, per- haps, is the most ready method of drawing an ellipsis. The divisions should be small, where accuracy is desirable. By this method an ellipsis may be traced without the axes, pro- vided that a diameter and its conjugate be given. Thus, ab and cdiFig. 409) are conjugate diameters: fg\^ drawn par- allel to ab^ instead of being at right angles to r^; also, fa and gb are drawn parallel to c d, instead of being at right angles to a b. 590 PRACTICAL GEOxMETRY. 552- — To Describe an Ellipsis by Intersecting Arcs» — Let a b and c d {Fig. 410) be given axes. Between one of the foci, / and /, and the centre r, mark any number of points, at random, as i, 2, and 3 ; upon / and /, with b i for radius, describe arcs at g, g, g, and g] upon / and /, with a i for radius, describe arcs intersecting- the others at g,g,g, and^; then these points of intersection will be in the curve of the ellipsis. The other points, Ji and /, are found in like manner, viz.: h is found by taking b2 for one radius, and a2 for the other ; i is found by taking b 3 for one radius, and a 3 for the Fig. 411. Other, always using the foci for centres. Then by tracing a curve through the points c, g, h, i, b, etc., the ellipse will be completed. This problem is founded upon the same principle as that of the string. This is obvious, when we reflect that the length of the string is equal to the transverse axis, added to TO DESCRIBE AN OVAL. 591 t\\e distance between the foci. See Fzo-, 405, in which c/ equals ae, the half of the transverse axis. 553- — To I>e§cribc a Figure ;^early in the Shape of an Ellipsis, hy a Pair of Compasses. — Let a b and c d {Fig. 41 1) be given axes. From c draw c e parallel to ab \ from a draw ae parallel to ^<^; join e and d\ bisect ea in /; join / and c, intersecting e d in i\ bisect ic in o\ from o draw og at right angles to ic, meeting ^<^ extended to g\ join i and ^, cutting the transverse axis in r ; make Jij equal to kg, and Ji k equal to Jir\ from j, through r and /t, draw/ 7;^ RvAjn; also, from g, through k, draw gl; upon g and/, with gc for radius, describe the arcs i/ and ;;/;/; upon r and /c, with ra for Fig. 412. radius, describe the arcs ini^^nd ln\ this will complete the figure. When the axes are proportioned to one another, as at 2 to 3, the extremities, c and d^ of the shortest axis, will be the centres for describing the arcs il and ni n ; and the inter- section oi e d with the transverse axis will be the centre for describing the arc in, z, etc. x\s the elliptic curve is contin- ually changing its course from that of a circle, a true ellipsis cannot be described with a pair of compasses. The above, therefore, is only an approximation. 564. — To a^raw an Oval in the Proportion Seven by Bfine. — Let cd {Fig. 412) be the given conjugate axis. Bisect 592 PRACTICAL GEOMETRY. c d in 0, and through o draw ab "dX right angles \.o c d\ bisect CO in c ; upon o, with oc for radius, describe the circle efgh ; from e, through Ji and /, draw cj and ci\ also, from ^, through Ji and /", draw ^/^ and gl\ upon ^, with gc for radius, describe the arc kl\ upon r, with ed for radius, de- scribe the arc j i ; upon h and /, with ///^ for radius, describe the arcs j k and //; this will complete the figure. Fig. 413. This is an approximation to an elHpsis ; and perhaps no method can be found by which a well-shaped oval can be drawn with greater facility. By a little variation in the process, ovals of different proportions may be obtained. If quarter of the transverse axis is taken for the radius of the circle efgh, one will be drawn in the proportion five by seven. Fig. 414. 555. — To Draiv a Tangent to an Ellipsis. — Let abed {Fig. 413) be the given ellipsis, and d the point of contact. Find the foci {Art. 548) / and /, and from them, through d,f^ draw f e and fd\ bisect the angle {Art. 506) edo with the line sr\ then sr will be the tangent required. TO FIND THE AXES OF AN ELLIPSE. 593 656. — An ElSipsis witti u Tang^ent Oivcn, to Detect the Point of Contact. — Let ci gb f {Fig. 414) be the given ellip- sis and tangent. Through the centre c draw a b parallel to the tangent; anywhere between e and / draw cd parallel to tib\ bisect cd in o\ through and e draw fg\ then g will be, the point of contact required. 657. — A Diameter of an Ellipsis Given, to Find its Conjugate. — Let ab (Fig. 414) be the .given diameter. Find the line fg by the last problem ; then f g will be the diam- eter required. 568. — Any Diameter and its Conjugate being Given, to Ascertain tlie Two Axes, and thence to Describe the ellipsis. — Let a b and c d {Fig. 415) be the given diameters, conjugate Fig. 415. to one another. Through c draw c f parallel \.o a b -, from v draw eg at right angles to cf\ make r^ equal to aJi or lib\. join g and Ji\ upon g^ with gc for radius, describe the arc ikcj\ upon Ji, with the same radius, describe the arc In-^ through the intersections / and ;/ draw no., cutting the tan- gent e f in ; upon latur(' S^laiidiii^ bib Advance of sasd Wall. — Cast rays from a and /; (Z^?^. 447). and find the point c as in the previous examples ; from d draw the ray {^e, and from e the horizontal line e/; tangical to the curve at ^ and // draw the rays j^j and /i /, and from z and /erect the per- pendiculars // andyX'; from ;// and 71 draw the rays 7/1/ and ;//', and trace the curve between /-and/; cast a ray from oto /, a vertical line from /to s, and through jr draw the horizon- tal line St ; the shadow as required will then be completed. Fig. 448. Fio-. 448 is an example of the same kind as the last, witli all the shadows filled in, according to the lines obtained in the preceding figure. 5 85. — Shadows on a Cornice. — F/o^s. 449 and 450 arc examples of the Tuscan cornice. The manner of obtaining the shadows is evident. 586.— Reflected V^ig;lit. — In shading, the finish and life of an object depend much on reflected light. This is seen to advantage in Fzg: 446, and on the column in Fig: 448. Re- 6l2 SHADOWS. fleeted rays are thrown in a direction exactly the reverse of direct rays ; therefore, on that part of an object which is subject to reflected light, the shadows are reversed. The FITTI^'WrMWTOlllWITliraTIW^ Fig. 449. fillet of the ovolo in Fig. 446 is an example of this. On the right hand side of the column, the face of the fillet is much darker than the cove directly under it. The reason of this is, the face of the fillet is deprived both of direct and re- flected light, whereas the cove is subject to the latter. Other instances of the effect of reflected light will be seen in the other examples. CONTENTS. PART I. SECTION I. — ARCH ITECTU RE Art. I. Building defined, p. 5. — 2. Antique Buildings ; Tower of Babel, p. 5. — 3. Ancient Cities and Monuments, p. 6. — 4. Architecture in Greece, p. 6. — 5, Architecture in Rome, p. 7. — 6. Rome and Greece, p. 8. — 7, Ar- chitecture debased, p. g. — 8, The Ostrogoths, p. 9. — 9. The Lombards, p. 10. — 10, The Byzantine Architects, p. 10. — 11. The Moors, p. 10. — 12, The Architecture of England, p. 11. — 13, Architecture Progressive, p. 12. — 14, Architecture in Italy, p. 12. — 15, The Renaissance, p. 13. — 16. Styles of Ar- chitecture, p. 13. — 17. Orders, p. 14. — 18. The Stylobate, p. 14. — 19, The Column, p. 14. — 20, The Entablature, p. 14. — 21, The Base, p. 14 — 22, The Shaft, p. 15.— 23. The Capital, p. 15.— 24. The Architrave, p. is.— 25. The Frieze, p. 15. — 26. The Cornice, p. 15. — 27. The Pediment, p. 15. — 28. The Tympanum, p. 15. — 29. The Attic, p. 15. — 30. Proportions in an Order, p. 15.— 31. Grecian Styles, p. 16.— 32. The Doric Order, p. 16.— 33. The Intercolumniation, p. 17. — 34, The Doric Order, p. 19 — 35. The Ionic Order, p. 19. — 36, The Intercolumniation, p. 20. — 37. To Describe the Ionic Volute, p. 20. — 38, The Corinthian Order, p. 23. — 39, Persians and Carya- tides, p. 24. — 40, Persians, p. 24. — 41, Caryatides, p. 26. — 42, Roman Styles, p. 26. — 43, Grecian Orders modified by the Romans, p. 27. — 44, The Tuscan Order, p. 30. — 45, Egyptian Style, p. 30. — 46. Building in General, p. 33. — 47. Expression, p. 35. — 48. Durability, p. 37. — 49. Dwelling- Houses, p. 37. — 50. Arranging the Stairs and Windows, p. 42. — 51. Prin- ciples of Architecture, p. 44. — 52. Arrangement, p. 44. — 53. Ventilation, p. 45- — 54. Stability, p. 45. — 55. Decoration, p. 46.-56. Elementary Parts of a Building, p. 46. — 57, The Foundation, p. 47. — 58, The Column, or Pillar, p. 47.— 59, The Wall, p. 48.— 60, The Reticulated Walls, p. 49.— 61, The Lintel, or Beam, p. 49. — 62, The Arch, p. 50. — 63, Ilooke's Theory of an Arch, p. 50. — 64. Gothic Arches, p. 51.— 65, Arch : Definitions ; Principles, p. 52.-66. An Arcade, p. 52.-67, The Vault, p. 52.-68. The Dome, p. 53. —69, The Roof, p. 54. 6 14 CONTEXTS. SECTION II.— CONSTRUCTION. Art. 70. Construction Essential, p. 56. — 71. Laws of Pressure, p. 57. — 7*2, Parallelogram of Forces, p. 59. — 73, The Resolution of Forces, p. 59. — 74. Inclination of Supports Unequal, p. Co. — 75. The Strains Exceed the Weights, p. 61. — 76. Minimum Thrust of Rafters, p. 62. — 77, Practical Method of Determining Strains, p. 62. — 7§. Horizontal Thrust, p. 63. — 79, Position of Supports, p. 65. — 80, The Composition of Forces, p. 66. — 81. Another Example, p. 67. — 82, Ties and Struts, p. 68. — 83, To Distinguish Ties from Struts, p. 69. — 84. Another Example, p. 70. — 85, Centre of Gravity, p. 71.— 86. Effect of the Weight of Inclined Beams, p. 72.-87, Effect of Load on Beam, p. 74. — 88. Effect on Bearings, p. 75. — 89. Weight-Strength, p, 76. — 90. Quality of Materials, p. 76. — 91. Manner of Resisting, p. 77. — 92. Strength and Stiffness, p. 78. — 93. Experiments : Constants, p. 78. — 94, Resistance to Compression, p. 79. — 95. Resistance to Tension, p. 81. — 96. Resistance to Transverse Strains, p. 83. — 97, Resistance to Compression, p. 85. — 98, Compression Transversely to the Fibres, p. 86. — 99. The Limit of Weight, p. 86.— 100, Area of Post, p. 86.-101, Rupture by Sliding, p. 87. —102, The Limit of Weight, p. 87.— 103. Area of Surface, p. 88.-104. Tenons and Splices, p. 88. — Il®5. Stout Posts, p. 89. — 106. The Limit of Weight, p. 89.— 107. Area of Post, p. 90. — 108. Area of Round Posts, p. 90.— 109. Slender Posts, p. 91.— 110. The Limit of Weight, p. 91.— 111. Diameter of the Post: when Round, p. 92. — 112. Side of Post: when Square, p. 93. — 113. Thickness of a Rectangular Post, p. 95. — 114. Breadth of a Rectangular Post, p. 95. — 115. Resistance to Tension, p. 96. — 116. The Limit of Weight, p. 96. — 117. Sectional Area, p. 97. — 118. Weight of the Suspending Piece Included, p. 98. — 119. Area of Suspending Piece, p. 99. RESISTANCE TO TRANSVERSE STRAINS, Art. 120. Transverse Strains: Rupture, p. 99. — 121. Location of Mor- tises, p. 100. — 122. Transverse Strains : Relation of Weight to Dimensions, p. loi. — 123. Safe Weight : Load at Middle, p. 103.— 124. Breadth of Beam with Safe Load, p. 104. — 125. Depth of Beam with Safe Load, p. 104. — 126. Safe Load at any Point, p. 105. — 127. Breadth or Depth : Load at any Point, p. 106. — 128. Weight Uniformly Distributed, p. 107. — 129. Breadth or Depth: Load Uniformly Distributed, p. 108. — 130, Load per Foot Super- ficial, p. 109. — 131. Levers: Load at one End, p. no. — 132, Levers : Breadth or Depth, p. in. — 133, Deflection: Relation to Weight, p. 112. — 134, De- flection: Relation to Dimensions, p. 112. — 135, Deflection : Weight when at Middle, p. 114.— 136, Deflection : Breadth or Depth, Weight at Middle, p. 114. — 137, Deflection : When Weight is at Middle, p. 116. — 138. Deflection : Load Uniformly Distributed, p. 116. — 139, Deflection : Weight when Uni- formly Distributed, p. 117.— -140, Deflection : Breadth or Depth, Load Uni- formly Distributed, p. 117. — 141, Deflection: When Weight is Uniformly Distributed, p. 118. — 142. Deflection of Lever, p. 119. — 143. Deflection of a Lever : Load at End, p. 120. — 144. Deflection of a Lever : Weight when at End, p. 120. — 145. Deflection of a Lever: Breadth or Depth, Load at End, CONTENTS. 615 p. 121. — 146. Deflection of Levers : Weight Uniformly Distributed, p. 121. — 147. Deflection of Levers with Uniformly Distributed Load, p. 122. — 148, Deflection of Levers : Weight when Uniformly Distributed, p. 122. — 149. Deflection of Levers : Breadth or Depth, Load Uniformly Distributed, p. 122. CONSTRUCTION IN GENERAL. Art. 150. Construction: Object Clearly Defined, p. 123. — 151. Floors Described, p. 124. — 152. Floor-Beams, p. 125. — 153. Floor-Beams for Dwell- ings, p. 127. — 154. Floor-Beams for First-Class Stores, p. 128. — 155. Floor- Beams : Distance from Centres, p. 129. — 156. Framed Openings for Chimneys and Stairs, p. 130. — 157. Breadth of Headers, p. 130. — 15§. Breadth of Carriage-Beams, p, 132. — 159. Breadth of Carriage-Beams Carrying Two Sets of Tail-Beams, p. 134. — 160. Breadth of Carriage-Beam with Well-Holo at Middle, p. 136. — 161. Cross-Bridging, or Herring-Bone Bridging, p. 137, — 162. Bridging: Value to Resist Concentrated Loads, p. 137. — 163. Gird- ers, p. 140, — 164. Girders : Dimensions, p. 141. FIRE-PROOF TIMBER FLOORS. Art. 165. Solid Timber Floors, p. 143.— 166. Solid Timber Floors for Dwellings and Assembly-Rooms, p. 143. — 167. Solid Timber Floors for First- Class Stores, p. 144. — 168. Rolled-Iron Beams, p. 145. — 169. Rolled-Iron Beams: Dimensions; Weight at Middle, p. 146. — 170. Rolled-Iron Beams: Deflection when Weight is at Middle, p. 147. — 171. Rolled-Iron Beams: Weight when at Middle, p. 148. — 172. Rolled-Iron Beams: Weight at any Point, p. 148. — 1173. Rolled-Iron Beams : Dimensions; Weight at any Point, p. 149. — fl74. Rolled-Iron Beams : Dimensions ; Weight Uniformly Distrib- uted, p. 149. — 175. Rolled-Iron Beams : Deflection ; Weight Uniformly Dis- tributed, p. 150. — 176. Rolled-Iron Beams : Weight when Uniformly Distrib- uted, p. 151. — 177. Rolled-Iron Beams: Floors of Dwellings or Assemblv- Rooms, p. 151. — 178. Rolled-Iron Beams: Floors of First-Class Stores, p. 152. — 179. Floor-Arches: General Considerations, p. 153. — 180. Floor- Arches: Tic-Rods; Dwellings, p. 153. — 181, Floor-Arches: Tie-Rods; First-Class Stores, p. 153. TUBULAR IRON GIRDERS. Art. 182. Tubular Iron Girders: Description, p. 154. — 183. Tubular Iron Girders : Area of Flanges ; Load at Middle, p. 154. — 184. Tubular Iron Girders : Area of Flanges ; Load at an}- Point, p. 155. — 185. Tubular Iron Girders : Area of Flanges ; Load Uniformly Distributed, p. 156. — 186. Tu- bular Iron Girders: Shearing Strain, p. 157. — 187. Tubular Iron Girders: Thickness of Web, p. 158. — 188. Tubular Iron Girders for Floors of Dwell- ings, Assembly-Rooms, and Office Buildings, p. 159. — 189. Tubular Iron Girders for Floors of First-Class Stores, p. 160. CAST-IRON GIRDERS. Art. 190. Cast-Iron Girders: Inferior, p. 161. — 191. Cast-Iron Girder: Load at Middle, p. 161. — 192. Cast-Iron Girder: Load Uniformly Distributed, 6l6 CONTENTS. p. 163.— 193. Cast-iron Bowstring Girder, p. 163.— 194. Substitute for the Bowstring Girder, p. 163. FRAMED GIRDERS. Art. 195. Graphic Representation of Strains, p. 165. — 196. Framed Girders, p. 166.— 197. Framed Girder and Diagram of Forces, p. 167.— 198. Framed Girders : Load on Both Chords, p. 171. — 199. Framed Girders: Di- mensions of Parts, p. 173. PARTITIONS. Art. 200. Partitions, p. 174. — 201. Examples of Partitions, p. 175. ROOFS. Art. 202. Roofs, p. 178. — 203. Comparison of Roof-Trusses, p. 178.— 204. Force Diagram : Load upon Each Support, p. 179. — 205. Force Dia- gram for Truss in Fig. 59, p. 179. — 206. Force Diagram for Truss in Fig. 60, p. 180. — 207, Force Diagram for Truss in Fig. 61, p. 181. — 208. Force Dia- gram for Truss in Fig. 63, p. 183. — 209. Force Diagram for Truss in Fig. 64, p. 184. — 210. Force Diagram for Truss in Fig, 65, p. 1S5. — 211. Force Dia- gram for Truss in Fig. 66, p. 186. — 212. Roof-Truss : EfTect of Elevating the Tie-Beam, p. 187. — 213. Planning a Roof, p. 188. — 214. Load upon Roof- Truss, p. 189. — 215. Load on Roof per Superficial Foot, p. i8g. — 216. Load upon Tie-Beam, p. igo. — 217, Roof Weights in Detail, p. 191. — 218. Load per Foot Horizontal, p. 192. — 219. Weight of Truss, p. 192. — 220. Weight of Snow on Roofs, p. 193. — 221. Effect of Wind on Roofs, p. 193.— 222. Total Load per Foot Horizontal, p. 197. — 223. Strains in Roof Timbers Computed, p. 198. — 224. Strains in Roof Timbers Shown Geometrically, p. 199. — 225. Application of the Geometrical System of Strains, p. 202. — 226. Roof Timbers : the Tie-Beam, p. 204.— 227. The Rafter, p. 205.— 228. The Braces, p. 208. — 229, The Suspension Rod, p. 210. — 230. Roof-Beams, Jack-Rafters, and Purlins, p. 211. — 231. Five Examples of Roofs, p. 212. — 232. Roof-Truss with Elevated Tie-Beam, p. 2X4.-233. Hip-Roofs : Lines and Bevels, p. 215. — 234. The Backing of the Hip-Rafter, p. 216. DOMES. Art, 235. Domes, p. 216.— 236, Ribbed Dome, p. 217.— 237. Domes: Curve of Equilibrium, p. 218. — 238. Cubic Parabola Computed, p. 219. — 239. Small Domes over Stairways, p. 220. — 240. Covering for a Spherical Dome, p. 221. — 241. Polygonal Dome : Form of Angle-Rib, p. 223. Art. 242, Bridges, p. 223. — 243. Bridges : Built-Rib, p. 224. — 244. Bridges : Fram.ed Rib, p. 226. — 245. Bridges : Roadway, p. 227, — 246. Bridges : Abutments, p. 227. — 247. Centres for Stone Bridges, p. 229. — 248. Arch Stones : Joints, p. 223. JOINTS. Art. 249. Timber Joints, p. 234. CONTENTS. 617 SECTION III.— STAIRS. Art. 350. Stairs: General Requirements, p. 240. — 251. The Grade of Stairs, p. 241. — 252. Pitch-Board : Relation of Rise to Tread, p. 242. — 253. Dimensions of the Pitch-Board, p. 247. — 254. The String of a Stairs, p. 247. — 255. Step and Riser Connection, p. 248. PLATFORM STAIRS Art. 256. Platform Stairs: the Cylinder, p. 248.— 257. Form of Lower Edge of Cylinder, p. 249. — 25§. Position of the Balusters, p. 250. — 259. Wind- ing Stairs, p. 251. — 260. Regular Winding Stairs, p. 251.— 261. Winding Stairs : Shape and Position of Timbers, p. 252. — 262. Winding Stairs with Fl)^ers: Grade of Front String, p. 253. HAND-RAILING. Art. 26?l. Hand-Railing for Stairs, p. 256.-264. Hand-Railing: Defini- tions ; Planes and Solids, p. 257. — 265. Hand-Railing: Preliminar}- Consider- ations, p. 258. — 266. A Prism Cut by an Oblique Plane, p. 259.- 267. Form of Top of Prism, p. 259.-268. Face-Mould for Hand-Railing of Platform Stairs, p. 264. — 269. More Simple Method for Hand-Rail to Platform Stairs, p. 267.— 270. Hand-Railing for a Larger Cylinder, p. 271.— 271. Face- Mould without Canting the Plank, p. 272. — 272. Railing for Platform Stairs where the Rake meets the Level, p. 272. — 273. Application of Face-Moulds to Plank, p. 273. — 274. Face-Moulds for Moulded Rails upon Platform Stairs, p. 274. — 275. Application of Face-Moulds to Plank, p. 275.-276. Hand-Railing for Circular Stairs, p. 278. — 277. Face-Moulds for Circular Stairs, p. 282.-278. Face-Moulds, for Circular Stairs, p. 285.-279. Face- Moulds for Circular Stairs, again, p. 287. — 280. Hand-Railing for Winding Stairs, p. 289.— 281. Face-Moulds for Winding Stairs, p. 290.— 282. Face- Moulds for Winding Stairs, again, p. 293.-283. Face-Moulds : Test of Accu- racy, p. 295.-284. Application of the Face-Mould, p. 297.-285. Face-Mould Curves are Elliptical, p. 301.— 286. Face-Moulds for Round Rails, p. 303.— 287. Position of the Butt Joint, p. 303.-288. Scrolls for Hand-Rails : Gen- eral Rule for Size and Position of the Regulating Square, p. 308.— 289. Cen- tres in Regulating Square, p. 308.— 290. Scroll for Hand-Rail Over Curtail Step, p. 309.— 291. Scroll for Curtail Step, p. 310.— 292. Position of Balus- ters Under Scroll, p. 310.— 293, Falling-Mould for Raking Part of Scroll, p. 310.— 294. Face-Mould for the Scroll, p. 311.— 295. Form of Newel-Cap from a Section of the Rail, p. 312. — 296. Boring for Balusters in a Round Rail before it is Rounded, p. 313. SPLAYED WORK. Art. 297. The Bevels in Splayed Work, p. 314. SECTION IV.-DOORS AND WINDOWS. DOORS. Art. 298. General Requirements, p. 315. — 299. The Proportion between Width and Height, p. 315.-300. Panels, p. 316.— 301. Trimmings, p. 317. — 302. Hanging Doors, p. 317. mm 6l8 CONTENTS. WINDOWS. Art. 303. Requirements for Light, p. 317. — 304. Window Frames, p. 318. — 305. Inside Shutters, p. 319. — 306. Proportion: Width and Height, p. 319.— 307. Circular Heads, p. 320.— 308, Form of Soffit for Circular Win- dow Heads, p. 321. SECTION v.— MOULDINGS AND CORNICES. MOULDINGS. Art. 309. Mouldings, p. 323. -^-310. Characteristics of M®uldings, p. 324. — 311. A Profile, p. 326. — 312. The Grecian Torus and Scotia, p. 326, — 313. The Grecian Echinus, p. 327, — 314, The Grecian Cavetto, p. 327. — 3S5. The Grecian Cyma-Recta, p. 327. — 316. The Grecian Cyma-Reversa, p. 328.— 317. Roman Mouldings, p. 329. — 31§. Modern Mouldings, p. 331. CORNICES. Art. 319, Designs for Cornices, p. 335. — 320. Eave Cornices Propor- tioned to Height of Building, p. 335. — 321. Cornice Proportioned to a given Cornice, p. 342. — 322. Angle Bracket in a Built Cornice, p. 343. — 323. Rak- ing Mouldings Matched with Level Returns, p. 344. PART II. SECTION VI.— GEOMETRY. Art. 324. Mathematics Essential, p. 347.-325. Elementary Geometry, p. 347. — 326. Definition— Right Angles, p. 348. — 327. Definition— Degrees in a Circle, p. 348. — 328. Definition — Measure of an Angle, p. 348. — 329. Corollary — Degrees in a Right Angle, p. 348.— 330. Definition — Equal Angles, p. 349. — 331. Axiom — Equal Angles, p. 349. — 332. Definition — Obtuse and Acute Angles, p. 349. — 333. Axiom — Right Angles, p. 349. — 334. Corollary— Two Right Angles, p. 349.- 335, Corollary — Four Right Angles, p. 349. — 336. Proposition — Equal Angles, p. 350.-337. Propo- sition — Equal Triangles, p. 350. — 338. Proposition — Angles in Isosceles Triangle, p. 351. — 339. Proposition — Diagonal of Parallelogram, p. 351- — 340. Proposition — Equal Parallelograms, p. 352. — 341. Proposition — Paral- lelograms Standing on the Same Base, p. 352.-342. Corollary — Parallelo- gram and Triangle, p. 353. — 343. Proposition — Triangle Equal to Quadrangle, P- 353.-344. Proposition— Opposite Angles Equal, p. 354-— 345. Proposi- tion—Three Angles of Triangle Equal to Two Right Angles, p. 354.-346. Corollary— Right Angle in Triangle, p. 354.-347. Corollary— Half a Right CONTENTS. 619 Angle, p. 355.-348. Corollary — Right Angle in a Triangle, p. 355. — 349. Corollary — Two Angles Equal to Right Angle, p. 355. — 350, Corollary — Two Thirdsof a Right Angle, p. 355. — 351, Corollary — Equilateral Triangle, p. 355. — 352, Proposition — Right Angle in Semi-circle, p. 355. — 353, Proposition — The Square of the Hypothenuse Equal to the Squares of the Sides, p, 355. — 354. Proposition — Equilateral Octagon, p. 357. — 355, Proposition — Angle at the Circumference of a Circle, p. 35S. — 356, Proposition — Equal Chords give Equal Angles, p. 35S. — 357, Corollary of Equal Chords, p. 359.-358. Proposition — Angle Formed by a Chord .and Tangent, p. 359. — 359, Propo- sition — Areas of Parallelograms, p. 360. — 360, Proposition — Triangles ol Equal Altitude, p. 361. — 361, Proposition— Homologous Triangles, p, 362. — 362, Proposition — Parallelograms of Chords, p. 363. — 363, Proposition- Sides of Quadrangle, p. 364. SECTION VII.— RATIO, OR PROPORTION. Art. 364, Merchandise, p. 366.-365, The Rule of Three, p. 366.— 366. Couples: Antecedent, Consequent, p. 367. — 367. Equal Couples : an Equation, p. 367. — 368. Equality of Ratios, p. 367. — 369. Equals Multiplied by Equals Give Equals, p. 367. — 370, Multiplying an Equation, p. 368. — 371, Multiplying and Dividing one Member of an Equation : Cancelling, p. 368. — 372, Transferring a Factor, p. 369. — 373, Equality of Product: Means and Extremes, p. 369. — 374, Homologous Triangles Proportionate, p. 370. — 375, The Steelyard, p. 371. —376, The Lever Exemplified by the Steelyard, p. 372. — 377. The Lever Principle Demonstrated, p. 375. — 378. Any One ot Four Proportionals may be Found, p. 377. SECTION VIIL— FRACTIONS. Art. 379. A Fraction Defined, p. 378. — 380. Graphical Representation of Fractions : Effect of Multiplication, p. 378. — 381. Form of Fraction Changed by Division, p. 3S0. — 382. Improper Fractions, p. 380. — 383. Re- duction of Mixed Numbers to Fractions, p. 381. — 384. Division Indicated by the Factors put as a Fraction, p. 381. — 385. Addition of Fractions having Like Denominators, p. 382. — 386. Subtraction of Fractions of Like Denominators, p. 383. — 387. Dissimilar Denominators Equalized, p. 383. — 388. Reduction of Fractions to their Lowest Terms, p. 384. — 389, Least Common Denomina- tor, p. 384. — 390. Least Common Denominator Again, p. 385. — 391, Frac- tions Multiplied Graphically, p. 3S6, — 392, Fractions Multiplied Graphically Again, p. 387. — 393. Rule for Multiplication of Fractions, and Example, p. 387. — 394. Fractions Divided Graphically, p. 3S8, — 395. Rule for Division of Fractions, p. 389. SECTION IX.— ALGEBRA. Art. 396. Algebra Defined, p. 392. — 397. Example: Application, p. 393.-398. Algebra Useful in Constructing Rules, p. 394.-399. Algebraic Rules are General, p. 394. — 400. Symbols Chosen at Pleasure, p. 395. — 401, Arithmetical Processes Indicated by Signs, p. 396. — 402, Examples in Addi- 620 CONTENTS. tion and Subtraction : Cancelling, p. 398.— 403. Transferring a Symbol to the Opposite Member, p. 399.— 404. Signs of Symbols to be Changed when they j|re to be Subtracted, p. 400. — 405. Algebraic Fractions, Added and Sub- tracted, p. 403. — 406. The Least Common Denominator, p. 404. — 407. Alge- braic Fractions Subtracted, p. 405. — 408. Graphical Representation of Multi- plication, p. 408,— 409. Graphical Multiplication : Three Factors, p. 408. — 410. Graphic Representation: Two and Three Factors, p. 409. — 411. Graph- ical Multiplication of a Binomial, p. 409.— 412. Graphical Squaring of a Binomial, p. 410. — 413. Graphical Squaring of the Difference of Two Fac- tors, p. 412. — 414. Graphical Product of the Sum and Difference of Two Quantities, p. 413. — 415. Plus and Minus Signs in Multiplication, p. 415. — 416. Equality of Squares on Hypothenuse and Sides of Right-Angled Tri- angle, p. 416. — 417. Division the Reverse of Multiplication, p. 418. — 41§. Division: Statement of Quotient, p. 419. — 419. Division: Reduction, p. 419. —420. Proportionals : Analysis, p. 421. — 421, Raising a Quantity to any Power, p. 423. — 422. Quantities with Negative Exponents, p. 423. — 423. Addition and Subtraction of Exponential Quantities, p. 424. — 424. Multipli- cation of Exponential Quantities, p. 424. — 425. Division of Exponential Quantities, p. 424. — 426. Extraction of Radicals, p. 425. — 427. Logarithms, p. 425. — 42§. Completing the Square of a Binomial, p. 429. PROGRESSION. Art. 429. Arithmetical Progression, p. 432.— 430. Geometrical Progres- sion, p. 435. SECTION X.— POLYGONS. Art. 431. Relation of Sum and Difference of Two Lines, p. 439. — 432. Perpendicular, in Triangle of Known Sides, p. 440. — 433. Trigon : Radius of Circumscribed and Inscribed Circles : Area, p. 443. — 434. Tetragon : Radius of Circumscribed and Inscribed Circles: Area, p. 446. — 435. Hexagon : Ra- dius ot Circumscribed and Inscribed Circles : Area, p. 447. — 436. Octagon : Radius of Circumscribed and Inscribed Circles : Area, p. 449. — 437. Dodec- agon : Radius of Circumscribed and Inscribed Circles: Area, p. 452. — 438, Hecadecagon : Radius of Circumscribed and Inscribed Circles : Area, p. 455. — 439. Pol3^gons : Radius of Circumscribed and Inscribed Circles : Area, p. 460. — 440. Polygons: Their Angles, p. 462. — 441. Pentagon: Radius of the Circumscribed and Inscribed Circles ; Area, p. 463. — 442. Polygons: Table of Constant Multipliers, p. 465. SECTION XL— THE CIRCLE. Art. 443. Circles : Diameter and Perpendicular : Mean Proportional, p. 468. — 444. Circle : Radius from Given Chord and Versed Sine, p. 469. — 445. Circle: Segment from Ordinates, p. 470. — 446. Circle: Relation of Diameter to Circumference, p. 472. — 447. Circle : Length of an Arc, p. 475. • — 448. Circle: Area, p. 475. — 449. Circle: Area of a Sector, p. 476. — 450. Circle : Area of a Segment, p. 477. CONTENTS. 621 SECTION XIL— THE ELLIPSE. Art. 451. Ellipse: Definitions, p. 481. — 452. Ellipse: Equations to the Curve, p. 482. — 453. Ellipse: Relation of Axis to Abscissas of Axes, p. 484. — 454. Ellipse : Relation of Parameter and Axes, p. 485.-455. Ellipse : Relation of Tangent to the Axes, p. 4S5. — 456. Ellipse: Relation of Tangent with the Foci, p. 487. — 457. Ellipse : Relation of Axes to Conjugate Diam- eters, p. 487. — 45§. Ellipse : Area, p. 488.-459. Ellipse ; Practical Sugges- tions, p. 489. SECTION XIII.— THE PARABOLA. Art. 460. Parabola : Definitions, p. 492. — 461. Parabola : Equation to the Curve, p. 493. — 462. Parabola: Tangent, p. 493. — 463. Parabola: Sub- tangent, p. 496. — 464. Parabola: Normal and Subnormal, p. 496. — 465. Parabola : Diameters, p. 497. — 466. Parabola : Elements, p. 499. — 467. Parabola : Described Mechanically, p. 500, — 46§. Parabola : Described from Points, p. 502. — 469. Parabola : Described from Arcs, p. 503. — 470. Para- bola : Described from Ordinates, p. 504. — 471. Parabola: Described from Diameters, p. 507. — 472, Parabola : Area, p. 509. SECTION XIV.— TRIGONOMETRY. Art. 473. Right-Angled Triangles : The Sides, p, 510. — 474. Right- Angled Triangles : Trigonometrical Tables, p. 512. — 475, Right-Angled Triangles: Trigonometrical Value of Sides, p. 516. — 476. Oblique-Angled Triangles: Sines and Sides, p. 519. — 477. Oblique-Angled Triangles : First Class, p. 520. — 47§. Oblique-Angled Triangles: Second Class, p. 522. — 479. Oblique-Angled Triangles : Sum and Difference of Two Angles, p. 523. — 480. Oblique-Angled Triangles : Third Class, p. 526. — 481. Oblique- Angled Triangles : Fourth Class, p. 528. — 482. Trigonometrical Formulaj : Right-Angled Triangles, p. 530. — 483. Trigonometrical Formulae : First Class, Oblique, p. 531. — 484. Trigonometrical Formulae: Second Class, Oblique, p. 532. — 485. Trigonometrical Formulae : Third Class, Oblique, p^ 534. — 486. Trigonometrical Formulae : Fourth Class, Oblique, p. 534. SECTION XV.— DRAWING Art. 487. General Remarks, p. 536. — 488. Articles Required, p. 536. — 489. The Drawing-Board, p. 536. — 490. Drawing-Paper, p. 537. — 491. To Secure the Paper to the Board, p. 537. — 492, The T-Square, p. 539. — 493. The Set-Square, p. 539. — 494. The Rulers, p. 540.— 495. The Instruments, p. 540.— 496, The Scale of Equal Parts, p. 540.— 497. The Use of the Set- Square, p. 541. — 498, Directions for Drawing, p. 542. 622 CONTENTS. SECTION XVI.— PRACTICAL GEOMETRY. Art. 499. Definitions of Various Terms, p. 544. PROBLEMS. RIGHT LINES AND ANGLES. Art. 500. To Bisect a Line, p. 549. — 501. To Erect a Perpendicular, p. 550.— 502. To let Fall a Perpendicular, p. 551. —503. To Erect a Perpen- dicular at the End of a Line, p. 551. — 504. To let Fall a Perpendicular near the End of a Line, p. 553. — 505. To Make an Angle Equal to a Given Angle, P- 553.— 506. To Bisect an Angle, p. 554. — 507. To Trisect a Right Angle, P- 554- — 508. Through a Given Point to Draw a Line Parallel to a Given Line, p. 555. — 509. To Divide a Given Line into any Number of Equal Parts, P- 555. THE CIRCLE, Art. 510. To Find the Centre of a Circle, p. 556.— 511. At a Given Point in a Circle to Draw a Tangent thereto, p. 557. — 512, The Same, with- out making use of the Centre of the Circle, p. 557. — 513. A Circle and a Tangent Given, to Find the Point of Contact, p. 558. — 514. Through any Three Points not in a Straight Line to Draw a Circle, p. 559. — 515. Three Points not in a Straight Line being Given, to Find a Fourth that Shall, with the Three, Lie in the Circumference of a Circle, p. 559. — 516. To Describe a Segment of a Circle by a Set-Triangle, p. 560. — 517. To Find the Radius of an Arc of a Circle when the Chord and Versed Sine are Given, p. 561. — 518. To Find the Versed Sine of an Arc of a Circle when the Radius and Chord are Given, p. 561. — 519, To Describe the Segment of a Circle by Intersection of Lines, p. 562. — 520. Ordinates, p. 563. — 521. In a Given Angle to De- scribe a Tanged Curve, p. 565. — 522. To Describe a Circle within any Given Triangle, so that the Sides of the Triangle shall be Tangical, p. 566. — 523. About a Given Circle to Describe an Equilateral Triangle, p. 566. — 524. To Find a Right Line nearly Equal to the Circumference of a Circle, p. 566. POLYGONS, ETC. Art. 525. Upon a Given Line to Construct an Equilateral Triangle, p. 568. — 526. To Describe an Equilateral Rectangle, or Square, p. 568. — 527. Within a Given Circle to Inscribe an Equilateral Triangle, Hexagon, or Dodec- agon, p. 569. — 528. Within a Square to Inscribe an Octagon, p. 570.— 529. To Find the Side of a Buttressed Octagon, p. 571. — 530. Within a Given Circle to Inscribe any Regular Polygon, p. 572. — 531. Upon a Given Line to Describe any Regular Pol3'gon,p. 573. — 532. To Construct a Triangle whose Sides shall be severally Equal to Three Given Lines, p. 575. — 533. To Con- struct a Figure Equal to a Given Right-lined Figure, p. 575. — 534. To Make a Parallelogram Equal to a Given Triangle, p. 576. — 535. A Parallelogram being Given, to Construct Another Equal to it, and Having a Side Equal to a CONTENTS. 623 Given Line, p. 576. — 536. To Make a Square Equal to two or more Given Squares, p. 577. — 537. To Make a Circle Equal to two Given Circles, p. 5S0. — 538. To Construct a Square Equal to a Given Rectangle, p. 5S1. — 539. To Form a Square Equal to a Given Triangle, p. 582. — 540. Two Right Lines being Given, to Find a Third Proportional thereto, p. 582. — 541. Three Right Lines being Given, to Find a Fourth Proportional thereto, p. 583. — 542. A Line with Certain Divisions being Given, to Divide Another, Longer or Shorter, Given Line in the Same Proportion, p. 583. — 543. Between Two Given Right Lines to Find a Mean Proportional, p. 584. CONIC SECTIONS. Art. 544. Definitions, p. 5S4. — 545. To Find the Axes of the Ellipsis, p. 585.-546. To Find the Axis and Base of the Parabola, p. 585.-547. To Find the Height, Base, and Transverse Axis of an Hyperbola, p. 585. — 54S. The Axes being Given, to Find the Foci, and to Describe an Ellipsis with a String, p. 586. — 549, The Axes being Given, to Describe an Ellipsis with a Trammel, p. 5S6.— 550. To Describe an Ellipsis by Ordinates, p. 588. — 551. To Describe an Ellipsis by Intersection of Lines, p. 58S. — 552. To Describe an Ellipsis by Intersecting Arcs, p. 590. — 553. To Describe a Figure Nearly in the Shape of an Ellipsis by a Pair of Compasses, p. 591. — 554. To Draw an Oval in the Proportion Seven by Nine, p. 591. — 555. To Draw a Tangent to an Ellipsis, p. 592. — 556. An Ellipsis with a Tangent Given, to Detect the Point of Contact, p. 593. — 557, A Diameter of an Ellipsis Given, to Find its Conjugate, p. 593. — 558. Any Diameter and its Conjugate being Given, to Ascertain the Two Axes, and thence to Describe the Ellipsis, p. 593. — 559, To Describe an Ellipsis, whence Axes shall be Proportionate to the Axes of a Larger or Smaller Given One, p. 594. — 560, To Describe a Parabola by Intersection of Lines, p. 594. — 561, To Describe an Hyperbola by Intersec- tion of Lines, p. 595. SECTION XVII.— SHADOWS. Art. 562. The Art of Drawing, p. 596.-563. The Inclination of the Line of Shadow, p. 596. — 564. To Find the Line of Shadow on Mouldings and other Horizontally Straight Projections, p. 597. — 565. To Find the Line of Shadow Cast by a Shelf, p. 598.-566. To Find the Shadow Cast by a Shelf which is Wider at one End than at the Other, p. 599. — 567. To Find the Shadow of a Shelf having one End Acute or Obtuse Angled, p. 600. — 568. To Find the Shadow Cast by an Inclined Shelf,- p. 600.— 569. To Find the Shadow Cast by a Shelf inclined in its Vertical Section either Upward or Downward, p. 601.— 570. To Find the Shadow of a Shelf having its Front Edge or End Curved on the Plan, p. 602. — 571, To Find the Shadow of a Shelf Curved in the Elevation, p. 602.— 572. To Find the Shadow Cast upon a Cylindrical Wall by a Projection of any Kind, p. 603. — 573. To Find the Shadow Cast by a Shelf upon an Inclined Wall, p. 603. — 574. To Find the Shadow of a Projecting Horizontal Beam, p. 604. — 575, To Find the Shadow 624 CONTENTS. in a Recess, p. 604. — 576. To Find the Shadow in a Recess, when the Face of the Wall is Inclined, and the Back of the Recess is Vertical, p. 604. — 577, To Find the Shadow in a Fireplace, p. 605. — 578. To Find the Shadow of a Moulded Window-Lintel, p. 606. — 579. To Find the Shadow Cast by the Nosing of a Step, p. 606. — 580. To Find the Shadow Thrown by a Pedestal upon Steps, p. 6c6. — 581. To Find the Shadow Thrown on a Column by a Square Abacus, p. 607. — 582. To Find the Shadow Thrown on a Column by a Circular Abacus, p. 60S. — 583. To Find the Shadows on the Capital of a Column, p. 609. — 584. To Find the Shadow Thrown on a Vertical Wall by a Column and Entablature Standing in Advance of said Wall, p. 611. — 585o Shadows on a Cornice, p. 611. — 586. Reflected Light, p. 6ii. AMERICAN HOUSE CARPENTER. APPENDIX CONTENTS. PAGE. 627 Glossary Table of Squares, Cubes, and Roots ^3 Rules for the Reduction of Decimals Table of Circles ; Table showing the Capacity of Wells, Cisterns, etc 653 Table of the Weights of Materials ^54 647 649 GLOSSARY. Terms not found here can he found in the lists of definitions in othey parts of this book, or in common dictionaries. Abacus. — The uppermost member of a capital. Abattoir. — A slaughter-house. Abbey. — The residence of an abbot or abbess. Abutment. — That part of a pier from which the arch springs. Acanthus. — A plant called in English bear's-breech. Its leaves are employed for decorating the Corinthian and the Composite capitals. Acropolis. — The highest part of a city ; generail)^ the citadel, Acroteria. — The small pedestals placed on the extremities and apex of a pediment, originally intended as a base for sculpture. Aisle. — Passage to and from the pews of a church. In Gothic aicliitecture, the lean-to wings on the sides of the nave. Alcove. — Part of a chamber separated by an estradc, or partition of columns. Recess with seats, etc., in gardens. Altar. — A pedestal whereon sacrifice was cflfered. In modern churches; the area within the railing in front of the pulpit. Alto-relievo. — High relief; sculpture projectingfrom a surface so as to appear nearly isolated. Amphitheatre. — A double theatre, employed by the ancients for the exhibi- tion of gladiatorial fights and other shows. Ancones. — Trusses employed as an apparent support to a cornice upon the flanks of the architrave. Annulet. — A small square moulding used to separate others ; the fillets in the Doric capital under the ovolo, and those which separate the flutings of col- umns, are known by this term. Ant(r. — A pilaster attached to a wall. Apiary. — A place for keeping beehives. Arabesque. — A building after the Arabian style. Areostyle.—hx\ intercolumniation of from four to five diameters. Arcade. — A series of arches. Arch. — An arrangement of stones or other material in a curvilinear form, so as to perform the office of a lintel and carry superincumbent weights. Architrave.— '\\\^\ part of the entablature which rests upon the capital of a column, and is beneath the frieze. The casing and mouldings about a door or window. ^;r///7'^//.— The ceiling of a vault ; the under surface of an arch. ^;-^.^. —Superficial measurement. An open space, below the level of the ground, in front of basement windows. 62 8 APPENDIX. ~ Arsenal. — A public establishment for the deposition of arms and warlike stores. Astragal. — A small moulding consisting of a half-round with a fillet on each side. Attic. — A low story erected over an order of architecture. A low additional story immediately under the roof of a building. Aviary. — A place for keeping and breeding birds. Balcony. — An open gallery projecting from the front of a building. Baluster. — A small pillar or pilaster supporting a rail. Balustrade. — A series of balusters connected by a rail. Barge-course. — That part of the covering which projects over the gable of a building. Base. — The lowest part of a wall, column, etc. Basement-sto7y. — That which is immediately under the principal story, and included within the foundation of the building. Basso-relievo. — Low relief ; sculptured figures projecting from a surface one half their thickness or less. See Alto-relievo. Battering. — See Talus. Battlement. — Indentations on the top of a wall or parapet. Bay-window. — A window projecting in two or more planes, and not formi- ing the segment of a circle. Bazaar. — A species of mart or exchange for the sale of various articles of merchandise. Bead. — A circular moulding. Bed-mouldings. — Those mouldings which are between the corona and the frieze. Belfry. — That part of the steeple in which the bells arc hung ; anciently called campanile. Belvedere. — An ornamental turret or observatory commanding a pleasant prospect. Bozu-window. — A window projecting in curved lines. Bresstimvier. — A beam or iron tie supporting a wall over a gateway or other opening. Brick-nogging. — The brickwork between studs of partitions. Buttress. — A projection from a wall to give additional' strength. Cable. — A cylindrical moulding placed in flutes at the lower j)art of the col- umn. Camber. — To give a convexity to the upper surface of a beam. Campanile. — A tower for the reception of bells, usually, in Italy, separated from the church. Canopy. — An ornamental covering over a seat of state. Cantalivers. — The ends of rafters under a projecting roof. Pieces of wood or stone supporting the eaves. Capital. — The uppermost part of a column included between the shaft and the architrave. Caravansera. — In the East, a large public building for the reception of trav- ellers by caravans in the desert. GLOSSARY. 629 Cctrpentry.—{¥ \:om the Latin carpeiitum, carved wood.) That department of science and art which treats of the disposition, the construction, and the relative strength of timber. The first is called descriptive, the second con- structive, and the last mechanical carpentry. Caryatides. — Figures of women used instead cf columns to support an entablature. Casino. — A small country-house. Castellated. — Built with battlements and turrets in imitation of ancient castles. Castle. — A building fortified for military defence. A house with towers, usually encompassed with walls and moats, and having a donjon, or keep, in the centre. Catacombs. — Subterraneous places for burying the dead. Cathedral. — The principal church of a province or diocese, wherein the throne of the archbishop or bishop is placed. Cavetto. — A concave moulding comprising the quadrant of a circle. Cemetery. — An edifice or area where the dead are interred. Cenotaph. — A monument erected to the memory of a person buried in another place. Centring. — The temporary woodwork, or framing, whereon any vaulted work is constructed. Cesspool. — A well under a drain or pavement to receive the waste water and sediment. Chamfer. — The bevelled edge of anything originally right angled. Chancel. — That part of a Gothic church in which the altar is placed. Chajttry. — A little chapel in ancient churches, with an endowment for one or more priests to say mass for the relief of souls out of purgatory. Chapel. — A building for religious worship, erected separately from a Church, and served by a chaplain, Chaplet. — A moulding carved into beads, olives, etc. Cijicttire. — The. ring, listel, or fillet, at the top and bottom of a column, which divides the shaft of the column from its capital and base. Circus. — A straight, long, narrow building used by the Romans for the ex- hibition of public spectacles and chariot races. At the present day, a building enclosing an arena for the exhibition of feats of horsemanship. Clere-story. — The upper part of the nave of a church above the roofs of the aisles. Cloister. — The square space attached to a regular monastery or large church, having a peristyle or ambulatory around it, covered with a range of buildings. Coffer-dam. — A case of piling, water-tight, fixed in the bed of a river, for the purpose of excluding the water while any work, such as a wharf, wall, or the pier of a bridge, is carried up. Collar-beain. — A horizontal beam framed between two principal rafters above the tie-beam. Colonnade. — A range of columns. Columbarium. — A pigeon-house. Column. — A vertical cylindrical support under the entablature of an order. Common-rafters. — The same 3.s Jack-rafters, which see. 630 APPENDIX. Conduit. — A long, narrow, walled passage underground, for secret com- munication between different apartments. A canal or pipe for the conveyance of water. Conservatory. — A building for preserving curious and rare exotic plants. Consoles. — The same as ancones, which see. Contour. — The external lines which bound and terminate a figure. Convent. — A building for the reception of a society of religious persons. Coping. — Stones laid on the top of a wall to defend it from the weather, ■ Corbels. — Stones or timbers fixed in a wall to sustain the timbers of a lloor or roof. Cornice. — Any moulded projection which crowns or finishes the part to which it is affixed. Corona. — That part of a cornice which is between the crown-moulding and the bed-mouldings. Cornucopia. — The horn of plenty. Corridor. — An open gallery or communication to the dififerent apartments of a house. Cove. — A concave moulding. Cripple-rafters . — The short rafters which are spiked to the hip-rafter of a roof. Crockets. — In Gothic architecture, the ornaments placed along tho angles of pediments, pinnacles, etc. Crosettes. — The same as ancones, which see. Crypt. — The under or hidden part of a building. Culvert. — An arched channel of masonry or brickwork, built beneath the bed of a canal for the purpose of conducting water under it. Any arched channel for water underground. Cupola. — A small building on the top of a dome. Curtail-step. — A step with a spiral end, usually the first of the flight. Ctisps. — The pendants of a pointed arch. Cyina. — An ogee. There are two kinds ; the cyma-recta, having the upper part concave and the lower convex, and the cyma-reversa, with the upper part convex and the lower concave. Dado. — The die, or part between the base and cornice of a pedestal. Dairy. — An apartment or building for the preservation of milk, and the manufacture of it into butter, cheese, etc, Dead-shoar. — A piece of timber or stone stood vertically in brickwork, to support a superincumbent weight until the brickwork which is to carry it has set or become hard. Decastyle. — A building having ten columns in front. Dentils. — (From the Latin, denies, teeth.) Small rectangular blocks used in the bed-mouldings of some of the orders. Diastyle. — An intercolumniation of three, or, as some say, four diameters. Die. — That part of a pedestal included between the base and the cornice ; it is also called a dado. Dodecastyle. — A building having twelve columns in front. Donjon. — A massive tower within ancient castles, to which the garrison might letrcat in case of necessity. GLOSSARY. 631 Docks. — A Scotch name given to wooden brick'. Dormer. — A window placed on the roof of a house, the frame being placed vejlically on the rafters. Dormitory. — A sleeping- room. Dovecote. — A building for keeping tame pigeons. A cclumbarium. Echinns. — The Grecian ovolo. Elevation. — A geometrical projection drawn on a plane at right angles to the horizon. Entablatiire. — That part of an order which is supported by the columns ; consisting of the architrave, frieze, and cornice. Eiisiyle. — An intercolumniaiion of two and a quarter diameters. Exchange. — A building in which merchants and brokers meet to transact business. Extrados. — The exterior curve of an arch, Ea^ade. — The principal front of any building. Eace-mould. — The pattern for marking the plank out of which hand-railing is to be cut for stairs, etc. Facia, or Fascia. — A fiat member, like a band or broad fillet. Falliv.g-motdd. — The mould applied to the convex, vertical surface of the rail-piece, in order to form the back and under surface of the rail, and finish the squaring. Festoon. — An ornament representing a wreath of flowers and leaves. Fillet. — A narrow flat band, llstel, or annulet, used for the separation of one moulding from another, and to give breadth and firmness to the edges of mouldings. Flutes. — Upright channels on the shafts of columns. Flyers. — Steps in a flight ot stairs that are parallel to each other. Forum. — In ancient architecture a public market ; also, a place where the common courts were held and law pleadings carried on. Foundry. — A building in which various metals are cast into moulds or shapes. Fjieze. — That part of an entablature included between the architrave and the cornice. Gable. — The vertical, triangular piece of wall at the end of a roaf, from the level of the eaves to the summit. Gaijt. — A recess made to receive a tenon or tusk. Gallery. — A common passage to several rooms in an upper story. A long room for the reception of pictures. A platform raised on columns, pilasters, or piers. Girder. — The principal beam in a floor, for supporting the binding and other joists, whereby the bearing or length is lessened. Glyph. — A vertical, sunken channel. From their number, those in the Doric order are called triglyphs. Granary. — A building for storing grain, especially that intended to be kept for a considerable time. 632 APPENDIX. Groin. — The line formed by the intersection of two arches, which cross each other at any angle. GuttcE. — The small cylindrical pendent ornaments, otherwise called drops, used in the Doric order under the triglyphs, and also pendent from the mutuli of the cornice. Gymnasium. — Originally, a place measured out and covered with sand for the exercise of athletic games ; afterward, spacious buildings devoted to the mental as well as corporeal instruction of youth. Hall. — The first large apartment on entering a house. The public room of a corporate body. A manor-house. Ham.—K house or dwelling-place. A street or village : hence Nolting- ham, Bucking//^w, etc. Hamlet, the diminutive of ham, is a small street or village. Helix. — The small volute, or twist, under the abacus in the Corinthian capital. Hem. — The projecting spiral fillet of the Ionic capital. Hexastyle. — A building having six columns in front. Hip-rafter. — A piece of timber placed at the angle made by two adjacent inclined roofs. Homestall. — A mansion-house, or seat in the country. Hotel, or Hostel. — A large inn or place of public entertainment. A large house or palace. Hot-house. — A glass building used in gardening. Hovel. — An open shed. Hut. — A small cottage or hovel, generally constructed of earthy materials, as strong loamy clay, etc. Impost. — The capital of a pier or pilaster which supports an arch. Intaglio. — Sculpture in which the subject is hollowed out, so that the im- pression from it presents the appearance of a bas-relief. Intercolumniation. — The distance between two columns. Intrados. — The interior and lower curve of an arch. Jack-rafters. — Rafters that fill in between the principal rafters of a roof ; called also common-rafters. Jail. — A place of legal confinement. Jambs. — The vertical sides of an aperture. Joggle-piece. — A post to receive struts. Joists. — The timbers to which the boards of a floor or the laths of a ceiling are nailed. Keep. — The same as donjon, which see. Key-stone. — The highest central stone of an arch. Kiln. — A building for the accumulation and retention of heat, in order to dry or burn certain materials deposited within it. King-post. — The centre-post in a trussed roof. Knee. — A convex bend in the back of a hand-rail. See Ramp. GLOSSARY. 633 Lactariion. — The same as dairy, which see. Lantern. — A cupola having windows in the sides for lighting an apartnnent beneath. Larmier. — The same as corona, which see. Lattice. — A reticulated window for the admission of air, rather than light, as in dairies and cellars. Lever-boards. — Blind-slats ; a set of boards so fastened that they may be turned at any angle to admit more or less light, or to lap upon each other so as to exclude all air or light through apertures. Lintel. — A piece of timber or stone placed horizontally over a door, win- dow, or other opening. Listel. — The same diS, Jillet, which see. Lobby. — An enclosed space, or passage, communicating with the principal room or rooms of a house. Lodge. — A small house near and subordinate to the mansion. A cottage placed at the gate of the road leading to a mansion. Loop. — A small narrow window. Loophole is a term applied to the vertical series of doors in a warehouse, through which goods are delivered by means of a crane. Lnffer-boarding. — The same as lever-boards, which see. Lnthern. — The same as dormer, which see. ATansoletun. — A sepulchral building — so called from a very celebrated one erected to the memory of Mausolus, king of Caria, by his wife Artemisia. Metopa. — The square space in the frieze between the triglyphs of the Doric order. Mezzanine. — A story of small height introduced between two of greater height. Minaret. — A slender, lofty turret having projecting balconies, common in Mohammedan countries. Mimler. — A church to which an ecclesiastical fraternity has been or is attached. Moat. — An excavated reservoir of water, surrounding a house, castle, or town. Modillion. — A projection under the corona of the richer orders, resembling a bracket. Module. — The semi-diameter of a column, used by the architect as a meas- ure by which to proportion the parts of an order. Alonastcry. — A building or buildings appropriated to the reception of monks. Monopteron. — A circular colonnade supporting a dome without an enclos- ing wall. Mosaic. — A mode of representing objects by the inlaying of small cubes of glass, stone, marble, shells, etc. Mosque. — A Mohammedan temple or place of worship. Miilliojis. — The upright posts or bars which divide the lights in a Gothic window. Mnniment-Jiottse. — A strong, fire-proof apartment for the keeping and pres- ervation of evidences, charters, seals, etc., called muniments. 634 APPENDIX, Museum. — A repository of natural, scientific, and literary curiosities or of works of art. Afutule. — A projecting ornament of the Doric cornice supposed to repre- sent the ends of rafters. Nave. — The main body of a Gothic church. Neiuel. — A post at the starting or landing of a flight of stairs. Niche. — A cavity or hollow place in a wall for the reception of a statue, vase, etc. N'ogs. — Wooden bricks. Nosing. — The rounded and projecting edge of a step in stairs. Ntmnery. — A building or buildings appropriated for the reception of nuns. Obelisk. — A lofty pillar of a rectangular form. Octastyle. — A building with eight columns in front. Odeiun. — Among the Greeks, a species of theatre wherein the poets and musicians rehearsed their compositions previous to the public production of them. Ogee. — See cyma. Orangery. — A gallery or building in a garden or parterre fronting the south. Oriel-window. — A large bay or recessed window in a hall, chapel, or other apartment. Ovolo. — A convex projecting moulding whose profile is the quadrant of a circle. Pagoda. — A temple or place of worship in India. Palisade. — A fence of pales or stakes driven into the ground. Parapet. — A small wall of any material for protection on the sides of bridges, quays, or high buildings. Pavilion. — A turret or small building generally insulated and comprised under a single roof. Pedestal. — A square foundation used to elevate and sustain a column, statue, etc. Pediment. — The triangular crowning part of a portico or aperture which terminates vertically the sloping parts of the roof ; this, in Gothic architecture, is called di gable. Penitentiary. — A prison for the confinement of criminals whose crimes arc not of a very heinous nature. Piazza. — A square, open space surrounded by buildings. This term is often improperly used to denoic^ portico. Pier. — A rectangular pillar without any regular base or capital. The up- right, narrow portions of walls between doors and windows are known by this term. Pilaster. — A square pillar, sometimes insulated, but more commonly en- gaged in a wall, and projecting only a part of its thickness. Piles. — Large timbers driven into the ground to make a secure foundation in marshy places, or in the bed of a river. GLOSSARY. 635 Pillat. — A column of irregular form, always disengaged, and always deviat- ing from the proportions of the orders ; whence the distinction between a pillar and a column. Pinnacle. — A small spire used to ornament Gothic buildings. Planceer. — The same as soffit, which see. Plinth. — The lower square member of the base of a column, pedestal, or wall. Porch. — An exterior appendage to a building, forming a covered approach to one of its principal doorways. Portal. — The arch over a door or gate ; the framework of the gate ; the lesser gate, when there are two of difTerent dimensions at one entrance. Portcullis. — A strong timber gate to old castles, made to slide up and down vertically. Portico. — A colonnade supporting a shelter over a walk, or ambulatory. Priory. — A building similar in its constitution to a monastery or abbey, the head whereof was called a prior or prioress. Prism. — A solid bounded on the sides by parallelograms, and on the ends by polygonal figures in parallel planes. Prostyle. — A building with columns in front only. Pnrlines. — Those pieces of timber which lie under and at right angles to the rafters to prevent them from sinking. Pycnostyle. — An intercolumniation of one and a half diameters. Pyramid. — A solid body standing on a square, triangle, or pol)^gonal basis and terminating in a point at the top. Quarjy. — A place whence stones and slates are procured. Quay. — (Pronounced hey.) A bank formed towards the sea or on the side of a river for free passage, or for the purpose of unloading merchandise. Quoin. — An external angle. See Rustic (juoins. Rabbet, or Rebate. — A groove or channel in the edge of a board. Ramp. — A concave bend in the back of a hand-rail. Rampant arch. — One having abutments of different heights. Regula. — The band below the taenia in the Doric order. Riser. — In stairs, the vertical board forming the front of a step. Rostrum. — An elevated platform from which a speaker addresses an audi- ence. Rotunda. — A circular building. Rubble-wall. — A wall built of unhewn stone. Rudentiire. — The same as cable, which see. Rustic quoins. — The stones placed on the external angle of a building, pro- jecting beyond the face of the wall, and having their edges bevelled. Rustic-work. — A mode of building masonry wherein the faces of the stones are left rough, the sides only being wrought smooth where the union of the stones takes place. Salon, or Saloon. — A lofty and spacious apartment comprehending the height of two stories with two tiers of windows. 636 APPENDIX. Sa7'cophagiis. — A tomb or coffin made of one stone. Scantl'uig. — The measure to which a piece of timber is to be or has been cut. Scarjitig. — The joining of two pieces of limber by bolting or nailing trans- versely together, so that the two appear but one. Scotia. — The hollow moulding in the base of a column, between the fillets of the tori. Sa'oll. — A carved curvilinear ornament, somewhat resembling in profile the turnings of a ram's horn. Sepulchre. — A grave, tomb, or place of interment. Sewer. — A drain or conduit for carrying off soil or water from any place. Shaft. — The cylindrical part between the base and the capital of a column. Shoar. — A piece of timber placed in an oblique direction to support a building or wall. .5"///. — The horizontal piece of timber at the bottom of framing ; the timber or stone at the bottom of doors and windows. Soffit. — The underside of an architrave, corona, etc. The underside of the heads of doors, windows, etc. Siiinmer. — The lintel of a door or window ; a beam tenoned into a girder to support the ends of joists on both sides of it. Systyle. — An intercolumniation of two diameters. Tccnia. — The fillet which separates the Doric frieze from the architrave. Talus. — The slope or inclination of a wall, among workmen called bat- tering. Terrace. — An area raised before a building, above the level of the ground, to serve as a walk. Tesselated pavement. — A curious pavement of mosaic work, composed of small square stones. Tetrastyle. — A building having four columns in front. Thatch. — A covering of straw or reeds used on the roofs of cottages, barns, etc. Theatre. — A building appropriated to the representation of dramatic spectacles. Tile. — A thin piece or plate of baked clay or other material used for the external covering of a roof. Tojub. — A grave, or place for the interment of a human body, including also any commemorative monument raised over such a place. Tortis. — A moulding of semi-circular profile used in the bases of col- umns. To'ver. — A lofty building of several stories, round or polygonal. Transept. — The transverse portion of a cruciform church. Transom. — The b?am across a double-lighted window ; if the window have no transom,, it is called a clere-story window. Thread. — That part of a step which is included between the face of its riser and that of the riser above. Tj'ellis. — A reticulated framing made of thin bars of wood for screens, win- dows, etc. GLOSSARY. 63 TiiglypJt. — The vertical tablets in the Doric frieze, chamfered on the two vertical edges, and having two channels in the middle. Tripod. — A table or seat with three legs. Trochiliis. — The same as scotia, which see. Truss. — An arrangement of timbers for increasing the resistance to cross- strains, consisting of a tie, two struts, and a suspending-piece. Turret. — A small tower, often crowning the angle of a wall, etc. Ttisk. — A short projection under a tenon to increase its strength. Ty7npanum. — The naked face of a pediment, included between the level and the raking mouldings. Undcjpinning. — The wall under the ground-sills of a building. University. — An assemblage of colleges under the supervisionof a senate, etc. Vault. — A concave arched ceiling resting upon two opposite parallel walls. Venetian-door. — A door having side-lights. Venetian-window. — A window having three separate apertures. Veranda. — An awning. An open portico under the extended roof of a building. Vestibule. — An apartment which serves as a medium of communication to another room or series of rooms. Vestry. — An apartm.ent in a church, or attached to it, for the preservation of the sacred vestments and utensils. Villa. — A country-house for the residence of an opulent person. Vineiy.-—h. house for the cultivation of vines. Volute. — A spiral scroll, which forms the principal feature of the Ionic and the Composite capitals. Voussoirs. — Arch-stones. Wainscoting. — Wooden lining of walls, generally in panels. Water-table. — The stone covering to the projecting foundation or other walls of a building. Well. — The space occupied by a flight of stairs. The space left beyond the ends of the steps is called the well-hole. Wicket. — A small door made in a gate. Winders. — In stairs, steps not parallel to each other. Zophorus. — The same ?^s frieze, which see. Zystos. — Among the ancients, a portico of unusual length, commonly appro- priated to gymnastic exercises. 638 APPENDIX. TABLE OF SQUARES. CUBES. AND ROOTS. cFrom Iliittoi "s JMathcmali •K.) No. Square. Cube. [ Sq. Root. CiibeRoot.! No. Square. Cube. Sq. Root. CubeRwt. FT 1 1 1-0000000 roooooo! 68 46-24 314432 8-2462113 4-0816551 2 4 8 14142136 1-259921' 69 4761 323509 8 30662391 4 1015G6 3 9 27 l-73iJ()5U3 1'442250 70 4900 343000 8-36660031 4-121285 4 16 64 2-0000000 l'-537401 71 5041 357911 8-426 1498! 4-140318 5 25 125 2-2360r)80 1-709976 72 5184 373-248 8-48528141 4-160168 6 36 216 2-44918^7 1-817121 73 5329 389017 85440037 4-179339 7 49 343 2-6457513 1-912931 74 5476 405224 8-6023253 4- 193336 8 64 512 2-8234271 2000000 75 5625 421375 8-6602540 4 217163 9 81 729 300000 JO 2-080034 76 5776 433976 8-7177979 4 2353-24 10 100 1000 3-1622777 2-154435 77 5929 456533 8-7749644 4-254321 n 121 1331 3-3l6')243 2-223J30 73 6034 474552 8-8317609 4-272659 12 144 1723 3-4641016 2-2 -(9429 79 6241 493039 8-83319441 4-2-:HXS10| 13 169 2197 3 6,155513 2 351335 80 6400 512000 8-944-2719 4-303%9 14 196 2744 3-7416574i 2410142 81 6561 531441 9-tMH)0()00 4-3j6"/49 15 225 3375 3 8729333 2-466212 82 6724 551368 9-0553351 4-344481 16 256 40J6 400J0000 2-519342 83 6339 571787 9-1104336! 4-362071 17 239 4913 4- 123! 056 2-571232 84 7056 592704 9-1651514 4-379519 18 324 5332 4-2426407 2-6-2074 li 85 7225 614125 9-2195445 4396330 19 361 6359 ■4-358_-i989 2-66 -(402 86 7396 63605:5 9-2736185 4-414005 20 4L)0 8000 4-472 ISoO 2-714418; 87 7569 653503 9-3273791 4-1,31048 21 441 9261 4-5825757 2^758924: 83 7744 681472 9-3808315 4-4.:796[) 22 434 10643 4-6904153 2-802039: 89 7921 704969 9-4339311 4-164745 23 529 12167 4-7953315 2-8433671 90 8100 729000 9-436333() 4 -J 8 1405 24 576 13321 4-898.^795 2-884499 91 8281 753571 9-5393920 4-497941 25 625 15625 5-OJOOOOO 2-924018 92 8464 773633 9-5916630 4 514357] 9 64365081 4-530655| 26 676 17576 5-0990195 2-962496, 93 8649 804357 27 729 19633 5-1961524 3000000' 94 8836 8305 S4 9-6953597 4 516336 23 731 21952 5 2915:)2i') 3036539 95 9025 857375 9-7467943 4-562903 29 841 24 38 J 5-3351648 3-0723 17i 96 9216 834736 9-7979590 4-573357 30 900 27000 5-4772-25"i 3-107232' 97 9409 912673 9-8483573 4-594701 31 961 29791 5 5677644 3141331! 98 9604 941192 9 8994949 4-610435 32 1024 32763 5-6568542 3-174802 99 9301 970299 9-9493744 4-626065 33 1089 35937 5-74456-26 3-2J7534I 100 10000 lOOOOOO lO-OOOOOOOl 4-641539 34 1156 393J4 5-8309519 323J612 101 10201 103)301 10-0498755 4-657009 35 1225 42375 5-9160798 3-271066 102 10404 1061208 10-099504'): 4 '072329 36 1296 4665() 6OOO0O0O 3 3J1927 103 10609 1092727 10- 14839 iO; 4-G37548 37 1369 50653 60S27625 3-332222 104 10816 1124864 10-1 9803- io! 4-702659 • 33 1444 54372 6-1644140 3361975 105 1 1025 1157625 10-24695 J;3" 4-717694 39 1521 59319 6-2449980 3-3J1211; 106 11236 1191016 10-29553D-J 4-732623 40 1600 640J0 6-3245553 3 419952 107 11449 1 •2-25043 10-3440301] 4-747459 41 , 1681 63921 6-4031242 3448217 103 11664 1259712 10-392334.3; 4762203 \i 1764 74038 6-4307407 3-476027 109 11831 1295029 10-4403J65' 4-776856 43 1819 79507 6-5574335 3503393 110 12100 1331000 10-48308-^5 4-791420 44 1936 85181 6 6332496 3-53)313 111 12321 1367631 10-535r5:iSl 4-305895 45 20^5 91125 6-7032039 3556393 112 12514 1404923 10-5330052! 4-a20284 46 2116 9733C) 6-7323300 3-533948 113 12769 144-2397 10-6301453' 4-834588 47 2209 103 S23 6-8556546 3-60S326 114 1-2996 1431544 10-6770733' 4-848308 43 23i)4 1 10592 692S2032 3 634-241 115 13225 1520375 10 7-233i;53i 4-862944 49 2401 117r)49 70000000 3-6593J6 116 13456 1560396 10-77032.V:! 4-876999 50 2500 125000 7-0710678 3 6-!403l 117 13639 1601613 10-816:!53>^i 4-890973 51 2601 132J51 7- 14 14234 3-70343.) 118 13924 1643032 10-86-278()5! 4-90486S 52 2704 140i>03 7-2111026 3-732511 119 14161 1685159 10-9037 12 ij 4-913685 53 231)9 143377 7-2^010J9 3-756236 120 14400 1723000 10-954151"! 4-93 i 124 54 2916 157464 7-3181692 3-779763 121 14641 1771561 ll-OO0O-:>:>;; 4-946087 55 3025 166375 7-4161935 3-S02952 1-22 14384 1815348 ll-0453i>l(; 4-959676 56 3136 175616 7-4833143 3-8258.i2 123 15129 1860367 11-0905365 4-973190 57 3^19 185193 7-5193341 3-34S50II 121 15376 1906624 11- 1355-237 4-936631 58 3364 195112 7-6157/31 3 •8703771 125 15625 19531-25 11-180339-J 5-000000 59 3431 205379 7-6311457 3-892996' 126 15376 2000376 11-2249722 5-013293 60 3(J00 2 1600 J 7-7459667 3-914863] 127 16129 2048333 11-2694277 5-026526 61 3721 226931 7-8102197 39364J7| 123 16334 2097152 11-3137085 5-039684 62 3S41 23S323 7-8740079 3-957891 129 16611 2146639 11 -3573! 6? 5 052774 ( 63 |64 3969 250017 7-9372539 3-979057 130 16900 2197000 114017543 5 065797 4iiy(J 262144 8 0000000 4-000000 131 17L61 2-243091 11-4455231 5-078753 65 4225 274625 8-0622577 4-020726! 132 17421 2299963 11-4391253 5-091643 «■)(■) 435f) 2374 'j6 8-12403^4 4-041240 133 17689 235:^637 11-53256261 5 KM 469 !'■'" 4»89 300763 8-l«5352i 4061548 134 17956 24i)6|04 1I-575S369! 5 117230 TABLE OF SQUARES, CUBES, AND ROOTS. 6;g No. Sqiiiirc. Cube. S(l. Root. CubeRootJ i i\0. Square. Cube. Sq. Root. CubcRool. 135 18225 2460375 11-6189500 5-129928: 202 40804 824-2408 14-21-26704 5-867461 136 18496 2515456 11-661903S 5-142563 203 41209 83654-27 14-2478088 5-877131 137 13769 2571353 11-7046999 5-155137i 204 41616 8489664 14-2323569 5-836765 133 19044 2628072 11-7473401 5-1676491 205 42025 8615125 14-3178211 5-896368 139 19321 2685619 11-7898261 5-1801011 206 42l3f. 8741816 14.35-27001 5-905941 140 19600 2744000 11-8321598 5-192494 207 42849 8869743 14-3374946 5-915432 141 19881 2803221 11-8743422 5-204828: 203 43-264 8998912 lt-4222051 5-924902 142 20164 2S63283 11-9163753 5-217103 209 43881 9129329 14-4568323 5-934473 143 20449 2924207 11-9582607 5-2-29321: 210 44100 9261000 14-4913767 5-943022 144 20736 2985984 12-000()000 5-241483 211 44521 9393931 14-5258390 5-953342 145 21025 3048625 12-0415946 5-253533 212 44944 95-28123 14-5602198 5-96-2732 146 21316 3112136 12-0830460 5-265637 213 45369 9663597 14-59451951 5-97-2093| 147 21609 3176523 12-1243557 5-277632 214 45796 9800344 14-6-287333 5-931424 148 21904 3241792 12-1655-251 5-239572 215 46i25 9933375 14-682S783 59S0726 149 22201 3307949 12-2065558 5 301459 216 46656 10077696 14-6969335 6-OOOOOU 150 22500 3375000 12-2474487 5-313203 217 47089 10218313 14-7309199 6-000245 151 22,S01 3442951 12-2832057 5-3-25074 218 47524 10.380232 14-7648231 6-013462 152 23104 3511808 12-3288280 5-336803 219 47961 10503459 14-7986486 6-027650 153 23409 3581577 12-3693169 5-348481 220 48400 10648000 14-8323970 6-036811 154 23716 3652264 12-4096738 5-380103 221 48S41 10793381 14-8660687 6045943 155 24025 3723875 12-449899.. 5-371685 222 49234 1U941048 14-8906644 6-055049 156 24336 3796416 12-4399960 5-333213 223 49729 11030567 14-9331845 6-064127 157 24619 3869393 12-5299641 5-394691J 224 50176 11239424 149638295 6073178 158 24964 3944312 12 •5698051 5-406120 225 506-25 113906-25 15-0000000 6-082202 159 25281 4019679 12-6095202 5-417501 226 51076 11543176 15-0332964 6-091 19y 160 25600 4096000 12-6491106 5-428835: 227 51529 11897083 150865192 6-100170 161 25921 4173281 12-6385775 5-440122 223 51934 11852352 15-0996639 6109115 162 25244 4251528 12-7279221 5-451332 229 52441 12008939 151327460 6-113033 163 26569 4330747 12-7671453 5-462556 230 5290U 12167000 15- 1657509 6-126928 16i !i6896 4410944 12-8062485 5-473704! 231 53361 123-28391 15 1936342 6-135702 165 27225 4492125 12-8452326 5-434807 232 53824 12487168 15-2315 462 6-144634 166 £7558 4574296 12-8840987 5-495365 233 54-389 12649337 15-2643375 6-153449 167 27839 4657463 12 9-228480 5-506878 234 54758 1281-2904 15 2070585 6-162210' 168 2S224 4741632 12-9614814 5-517848 235 55225 12977875 15-3207097 6-171008 169 2^561 4826809 130000000 5-528775' 235 55896 13144256 15 3322015 6-179747 170 28900 4913000 13 0384048 5-539658; 237 56169 1331-2053 15-3J43043 6- 183463 171 29211 5000211 13-0766968 5-550490: 233 38644 13481272 15-4272486 6-197154 172 29584 5088448 13-1148770 5-581293 239 57121 13851919 15-4506248 6-205822 173 29929 5177717 13-1529464 5-572055 240 57600 13324000 15-4919334 6214465 174 30276 5268024 13-1909060 5 532770; 241 53081 139.*7521 15-5241747 8-223034 175 30525 5359375 13-2-287568 5-593445! 242 58564 14172433 15 5583492 8-231630 176 30976 5451776 13-2664992 5-604079 243 59049 14348907 15-5334573 6-240251 177 3132'J 5545233 13-3041347 5-614672' 244 39336 14526784 15-6-204994 6-24S3O0 178 31(:3i 5639752 13-3416341 5 •625-226 245 60023 147061-25 15-6524753 6-257325 179 32041 5735339 13-3790882 5-635741! 246 60516 14836936 15-6343S71 6-285327 180 32400 5832000 13-4164079 5-646216! 247 61009 15089223 15-7162338 6-274305 181 32751 5929741 13-4536240 5-656653! 248 61504 15252902 15-7480157 8-232761 182 33 121 6028568 13 4907376 5-667051 249 6-2001 15433249 15-7797.333 6-291195 183 33439 6128487 13-5277493 5-677411 250 62500 15825000 15-8113-!33 8-200605 181 33356 6229504 13-5648600 5-637734 251 63001 15313251 15-84-29795 6-307994 185 34225 6331625 13-6014705 5-693019 252 63504 16003008 158745079 6-316360 18^\ 34596 6434856 13-6331817 5-708267| 253 64009 18194277 15-9059737 6-324704 187 34989 6539203 13-6747943 5-718479 254 64516 18337064 15-9373775 6-333026 188 35344 6644672 13-7113092 5-7238541 255 65025 16531375 15 9687194 6-341323 189 35721 6751269 13-7477271 5-733794 256 65536 16777216 16 0000000 6-349604 190 36100 6859000 13-7840483 5-748397] 257 68049 16074593 16 0312195 6357861 191 36481 6967871 13-8202750 5-758965| 253 66564 17173512 16-0623784 6-366007 192 36384 7077838 13-8564065 5-763998 5-778996 259 67081 17373079 16-0034760 6 374311 193 372 39 7189057 13-8924 440 260 67600 17576000 16-1245155 6-382504 194 37,'.36 7301384 13-9283383 5-788960 261 63121 17779581 1615.34044 8-300676 195 38025 7414875 13-964-2400 5-798390 262 6S644 17984723 18-1864141 8-398323 196 33416 7529536 14-0000000 5-808736 283 69169 18191447 16-217-2747 8-406953 197 38809 7645373 14-0356683 5-818648 264 60696 18399744 18-2430763 6-415069 198 39204 7762392 14-0712473 5-823477 263 702-25 18609625 18-27d8206 8-423 15-^ 199 39n01 7880599 141067360 5-833272 268 70756 18821098 16-3005064 6-43122- 24000 18-4390889 6-979532 274 75076 20570824 16'55-29454 6-495065 341 116281 39651821 184661853 6-986368 275 75025 20796875 16-5831240 6-502957 342 116964 40001688 18-49324-20 6-993191 276 76176 21024576 16-6132477 6-510830 342 117649 40353607 18-5202592 7-000000 277 76729 212539.33 16-6433170 6-518634 344 1 18336 40707584 18-5472370 7-006796 278 77284 21484952 16-6733320 6-526519, 345 119025 410636-25 18-5741756 7-013579 279 77811 21717633 16-7032931 6-534335 346 119716 41421736 18-6010752 7-020349 230 78400 21952000 16-7332005 6-5121331 ^47 120409 41781923 18-6279360 7-027106 281 78961 22188041 16-7630546 6-549912 343 121104 42144192 18-6547581 7-033850 282 79524 22425763 16-7923556 6-557672 349 121801 42508549 18-6815417 7-040581 283 80089 22665187 16-82-26033 6-555414 350 122500 42875000 18-7082869 7-047299 284 80656 229063M 16-85-22995 6-573139 351 123201 43243551 18-7349940 7-054004 235 81225 23149125 16-8819430 169115345 6-580344 352 123904 43614-208 18-7616530 7-060697 286 81796 23393656 6-533532: 353 124609 43985977 18-7832942 70673771 237 82369 23639903 16-94107431 6-596202! 354 125316 44361864 18-8148877 7-074044 283 82944 23387872 16-9705627 6-603354 355 126025 44738875 18-8414437 7-080699 239 83521 24137569 17-0000000 6-611489 356 126736 45118016 18-8679623 7-087a41 290 84100 24389000 17-0293864 6-619106 357 127449 45499-293 18-8944436 7-093971 291 84681 24642171 17-0537221 6-626705; 358 128164 4538-2712 18-9208379 7-100583 292 85264 24897083 17-0880075 6-634237j 359 6 641852^1 360 128881 46268279 18-947-2953 7-107194 293 85849 25153757 17-1172428 129600 46656000 18-9736660 7-113787 294 86136 25412184 17-1464232 6-6494001 361 130321 47045331 19-0000000 7-1-20367 295 87025 25672375 17-1755640 6-656930 352 131044 47437928 19-0262976 7-1-26935 296 87616 25934336 17-2046505 6-664444 353 131769 47832147 19-05-25589 7-133492 297 83209 25198073 17-2335379 6-671940 354 132495 48223544 19-0737840 7-140037 293 83304 26463592 17-2626765 6-679420' 1 355 13.3225 436271-25 19-1049732 7-146569 299 89401 26730899 17-2916165 6-635383 • 366 133956 49027395 19-1311265! 7 1530931 300 90000 2700000-0 17-3205081 6-694329 367 134639 49430863 19-1572441 7-1595991 301 90601 27270901 17-3493516 6-701759 368 135424 49836032 19-1833-261 7-166096 302 91204 27543603 17-3731472 6-7091731 369 136161 50243409 19 20937-27 7-17-2581 303 91809 27818127 17-4068952 6-716570![ 370 136900 50653000 19-2353341 7-179054 304 92416 28094464 17-4355958 6-723951 [I 371 137641 51064811 19-2613503 7-185516 305 93025 28372625 17-4642492 6-7313l6j| 372 138334 51478848 19-23730151 7-1919661 336 93636 23652616 17-4928557 6-733664!! 373 139129 51895117 19-3132079 7-198405 307 94249 23934443 17-5214155 6-7459971 374 139876 52313624 19-3390796 7-204832 308 94864 29218112 17-5499288 6-753313, 375 140625 52734375 19-3649167 7-211248 309 95481 29503629 17-5783958 6-760614 376 141376 53157376 19-3907194 7-217652 310 96100 29791000 17-6068169 6-767899 37T 142129 53582633 19-4164878 7-224045 311 96721 30080231 17-6351921 6-775169 3:8 142834 54010152 19-442-2221 7-230427 312 97344 30371328 17-6635217 6-782423'! 379 143541 54439939 19-4679223 7-236797 313 97969 30664297 17-6918060 6-789661 ! 330 144400 54872000 19-4935337 7-243156 3H 98596 30959144 17-7200451 6-796834! 331 145161 55306341 195192213 7-249504 315 99225 31255375 17-7432393 6-804092 332 145924 5574-2953 19-5443-203: 7255341 3Iii 99856 31554496 17-7763333 6-811235' 333 146639 56181837 I9-5703353I 7-262167 317 100489 31855013 17-8044933 6-818462, 334 147456 55623104 19-5959179| 7-263432 318 101124 32157432 17-8325545 6-8-256-24 335 148-225 57066625 19-6214169, 7-274786 319 101761 32461759 17-8605711 6-83-2771 336 148996 5751-2456 19-64633-27, 7231079 320 102400 32763000 17-8835433 6-839904 337 149769 57960603 1967-23156 7-237362 321 103041 33076161 17-9164729 6-847021 333 150544 58411072 19-6977156 7-293633 322 103584 33336248 17-9443534 6854124 339 151321 53363859 19-7-230829 7-299894 323 104329 33598267 17-97-22 J08 6-861212 390 152100 59319000 19-7434177 7-306144 324 104976 34012224 18-0000000 6-858235 3J1 15-2831 59776471 19-7737199 7 312333 325 105625 34323125 18-0277564 6-875344 392 153564 60236233 19-7939399 7-318611 326 106276 34645976 18-0554701 6-832339 393 154449 60693457 19-8242276 7-3248-29 327 10C>929 34965733 180331413 6-889419 394 155-236 6116-2984 19-8494332 7 331037 323 107584 35287552 18-1107703 6-896435 395 1-.6025 61629375 19-8746069 7-337-234 329 108241 3561 12S9 181333571 6 903436 396 15'^816 , 6-2099135 19-8997487 7-3434-20 3311 108900 35937000 181659021 6-910423 397 157609 62570773 19-9243533 7-349597 331 109561 36264691 18-1934054 6-917396 308 158404 63044792 19-9499373 7-355762 332 110224 36594368 18-2208672 6-924356 399 159201 63521199 19-9749844 7361918 333 110839 36926037 18-2482376 6-931301 400 160000 64000000 20-0000000 7-363063 :^3i 111556 37259704 18-2756669 6-933232 ! 401 160801 64431201 20 0249344 7-374193 335 112225 37595375 18-3030052 6-945150 402 161604 1 64954303 20*^99377 7-330323 TABLE OF SQUARES, CUBES, AND ROOTS. 641 No. Square. S(i. Root. CubeRoot. No. I Squar 431 441 444 44.0 446 447 44S 1971361 198025; 1939161 199809; 200704' 449; 20 160 1; 450 202500 451 203401 452| 204304 453 21)5209 4541 206116 453 207025 456i 207936 4571 208849 458| 209764 4591 210681 460! 211600 461 212521 4621 213444 463! 21436i) 464| 215296 465! 2 i 6225 466| 217156 46? 218:)89 468 2190241 469 2199611 65450327 ?5939264 C643J125I 669231161 67419143 67917312 68417929 68921000 69426/31 699341 li 70444i}97 70J57944 71173375 71991296 72511713 73034632 73560059 74038000 74618461 75151448 75636967 76225024 76765625 773037761 77854483 78402752 78953589 79507000 80082991 80621568 811827371 81746504 82312875' 82881856 83453453 84027672 846015191 85184000' 8576 •) 121 1 86350388 869383071 87528384' 83121125 8871653<; 89314623 89915392 90518319 91125000 91733851 92345403; 929598771 93576664 94196375 94818816' 95443993 96071912' 96702579 97336000 97972181 98:")11123 99252847| 99897344^ 100514625 101194696 101817563 102503232 103161709 20-0748599 20091)7512 201246118 20-1494417 20-1742410 20-1990099 2J-2237484 20-2484567 20-2731349 20-2977831 20-32-24014 20-3469899 20-3715488 20-3960781 20-4205779 20-4450483 20-46;)4395 20-4939015 20-5182845 20-5426336 20-5669638 20-5912603 20-6155281 20-6397674 20-6633783 20-6831609 20-7123152 20-7364414 20-7605395 20-7846097 20-8086520 20-8326667 20-856653O 20-8806130 20-9045450 20-9284495 20-9523268 20-9761770 21-00000001 21-0237960 21-0475652' 21-0713075; 21-09502311 21-11871211 211423745 21-1660105 21-1896201 21-2132034 21-2367606 21-2802916 21-28379671 21-3072758! 21-3307290' 21-35415551 21 ■3775533 21-40093461 21-4242353 21-4476106 21-4709106 21-4941853 21-5174318! 21-54065921 21-56335871 21-5370331 21-6101823 21-6333077 , 21-6564078 7-3S6437 7-392542 7-398636 7-404721 7-410795 7-416859 7-422914 7-428959 7-434994 7-441019 7-447034 7-453040 7-459033 7-465022 7-470999 7-476966 7-4829-24 7-488872 7-494311 7-500741 7-506661 7-5 12571 7-518473 7-524365 7-530243 7-538122! 7-541987' 7-547842 7-553639 7-559526 7 585355 7-571174 7-576985 7-532786 7-583579 7-5J4363 7600133 7-()05905 7-611663 7-617412 7-6-23152:! 510i 7-6233841I 511 7-634607ii 512 7 640321! 7-646027| 7-C51725! 7-657414; 7-663094; 7-663766! 518 7-6744301' 519 470 Cube. 471 472 473 474 475 476 477 478 479 4i0 4S1 4S2 4S3 4S4 435 486 437 438 439 490 491 492 493 1 494 495 496 497 493 499 500 501 502 503 504 505 506 507 508 509 513 514 515 516 517 -680086 520 7-685733:1 521 7-691372i| 522 7-6J70021i 7-702625; I 7-7082391 j 7-713345: 7-719443'| 7-725032; 1 7-7306 14 i! 7-736183:1 530 7-7417531 531 7-747311^ 5-32 7-75-286 ill 533 7-7584021 1 531 7-763j36i 535 7-769462; I 536 523 5-24 525 526 5-27 523 5-29 2209001 221841 222784 223729 2-24676 225825 226576 227529 223484 2-29441 230400 231361 232324 233289 234256 2352-25 236196 237169 238144 239121 240100 241031 212064 243049 244036 245025 246016 247009 243004 249001 250000 251001 252004 253009 254016 255025 256036 257049 253064 259031 260100 261121 262144 263169 264196 265225 266256 267289 263324 269331 270400 271441 272434 273529 274576 275625 276676 277729 278734 279341 230900 281961 283024] 234089 •285156' 236-225' 287206 Sq. Root. CubeRooi 103323000 104487111 105154048 105823817 106496424 107^71875 107850176 108531333 109215352 109902239 110592000 111-284641 111930163 11-2678537 113379004 1140841-25 114791256 115501303 116214272 116930169 117649000 118370771 119005483 119823157 1-20553784 121287375 122023036 122763473 123505902 1-24251499 125000000 125751501 126506008 127-263527 1-28024064 128787625 129554216 130323843 131096512 131872-229 13-2651000 133432331 134217728 135005697 135796744 136500875 137383098 133183413 138991832 139798359 14060800U 141420761 14-2236648 143055667 143377824 144703125 145531576 146363183 147197952 148035889 148877000 149721291 150588763 151419437 152-273304 153130375; 153990656 21-6794334 21-70-25344 21-7-255610 21-7485632 21-7715411 21-7914947 21-8174242 21-8403207 21-8632111 21-8880836 21 -9080023 21-93171-22 21-9544934 21-9772610 22-OOOuOOO 22-0227155 22 0454077 22-0680765 22-0907220 2-2-1133444 22- 1359438 2-2-1585193 22- 181073 j! 22-2038033| 22-228110^ 22-2485055 22-2710575 22-20349631 22-3159138 22-3333079 22-3608798 22-3330-293 22-4053565 22-4276615 22-4499443 22-4722051 22-4944438 22-5166805 22-533^553 22-5810283 22-53317981 22-6053091 22-6274 170| 22-6195033 22-67158311 22-6936 1141 2271563341 22-7376340; 22-75081341 22-7815715 22-8033035' 22-8254244' 22-84731'j3 22-8891933 22-8010463 22-91-23785' 22-93 16 V09 22-95,i48;6 22-9732508 23 0000000 23-021728J1 230434372 •23-065 12 J 2' 23-0887923 231084400 23-1300870 23-1516738 7-7749 7-730490 7-785993 7-791487 7-798974 7-802454 7-8079-25 7-813389 7-813846 7 -82 4-294 7-829735 7-835169 7 840595 7-846013 7-851424 7-356823 7-832'224 7-867613 7-872994 7-878368 7-833735 7-839095 7-894447 7-899792 7-9051-29 7-910460 7-915783 7-92109i) 7-928403 7-931710 7-937005 7-94-2293 7-947574 7-952848 7-953114 7-963374 7-968627 7-973373 7-97911 7-934314 7-989570 7-994788 8 000001 8-0052(1: 8-0L>4o:« 8-015595 8-020779 8-025957 8-0311-29 803820/ 8-041451 8-016803 8-051748 8 058836 806201b 8067143 8-072262 8077374 8 08-2430 8 037579 8-09267^ 8-097759 8-l0233y 8-107913 8-11-2930 8-118041 , 8-12309; 642 APPENDIX. No. "537 Square. Cube. Sq. Root. CuheRoot. No. Square. Cube. Sq. Rcct. CuleRoou 283369 154854153 231732605 8-123145 604 3.'>4816 220348864 24-5764115 8 45302^ 538 2'!9444 155720872 231948270 8-1331371 605 3(16025 221445125 24-5967478 8-457691 539 293521 156590819 23-2163735 8-13S223! 606 3'17236 222545016 24-6170673 8-462348 540 291600 157464000 23-2379001 8-143-253i 607 368449 223643543 24-63r3700! 8-4670O0| 541 292681 158340421 23-2594067 8-148276! 608 369664 224755712 24-6576560 8-471647 542 293764 159220083 23-2.S03935 8-153294 609 370831 225865529 24-6779254 8-476289 543 294849 160103007 23-3023604 8-153305 610 372100 226981000 246981781 8-480926 544 295936 160939184 23-323S076 8163310 611 373321 223099131 24-7184142 8-485558 545 297025 161873625 23-345-2351 8-168309 612 374554 229220928 24-7385338 8-490185 546 298116 162771336 23-3665429 8-173302 613 3757fl9 230346397 24-7538358 8-494806 547 299209 163667323 23-33303111 8-1782391 611 376996 231475544 24-7790234 8-499423 548 300304 164566592 23-4093998 8-183269 615 373225 232608375 24-7991935 8-504035 549 301401 165469149 23-4307490 •8-188244 616 379456 233744895 24-81934-73 8-508642 550 302500 166375000 23-4520788 8-193213 617 330689 234885113 24-8394347 8-513243 551 3J3601 167234151 23-4733392 8198175 618 331924 235029032 24-8596053 8-517840 552 304704 163196608 23-4946302 8-203132 619 333161 237176659 24-8797106 8-522432 553 335809 16911237.7 23-5159520 8-203032 620 331400 233328060 24-8997992 8-527019 554 306916 170031464 23-5372046 8-213027 631 335641 239483061 24-9198716 8-531601 555 303025 170953375 23 5534330 8-217966 G32 386384 240641848 24-9399278 8-536178 556 309136 171879616 23-5796522 8-2-32893 623 38^129 241804367 24-959b679 8-540750 557 310249 172808693 23-6003474 8-227825 634 339376 242970624 24-9799920 8-545317 558 311.364 173741112 23 6220236 8-232746 625 390825 244140625 25-0000000 8-549880 559 312431 174676379 23-6431808 8-237661 626 391876 245314376 25-0199920 8-554437 560 313600 175616000 23-6643191 8-242571 627 393129 246491833 25-0399681 8-558990 561 314721 176558481 23-6854336 8-247474 623 394334 247673152 25-0599282 8-563538 562 315344 177504328 23-7065392 8-252371 629 395641 248853189 25-0798724 8-568081 563 316969 178453547 23-7276210 8-257263 630 396900 250047000 250993003 8-572619 564 318096 179406144 23-7486842 8-262149 631 393161 251239591 25-1197134 8-577152 565 319225 180362125 23-7697283 8-267029 632 399424 252435958 25-1395102 8-581681 566 320356 181321496 23-7907515 8-271904 633 400689 253636137 25-1594913 8-586205 567 321489 182284263 23-8117618 8-276773 634 401956 254840104 25-1793566 8-590724 558 322624 183250432 23-8327536 8-281635 635 4U3225 256047375 25-1992063 8-595233 569 323761 184220009 23-8537209 8-236493 636 404196 257259456 25-2190404 8-599748 570 324900 185193000 23-8746723 8-291344 637 405769 258474853 25 2333539 8-604252 571 326041 186169411 23-8956063 8-2.)6190 633 407044 259694072 25-2585619 8-608753 572 327184 187149248 23-9165215 8-331033 639 408331 260917119 25 2734493 8-613248 573 333329 183133517 23-9374184 8-335365 640 409600 262144000 25.2932213 8-617739 574 329476 189119224 23-9532971 8-310694 641 410381 263374721 25-3179778 8-622225 575 330625 190109375 23-9791576 8-31551? 642 412164 264609283 25-3377189 8-626706 576 331776 191102976 24-0000000 8-32'33;^5 643 413449 265847707 25-3574447 8-631183 577 332929 192100033 24-0208243 8-325147 644 414735 267089984 25-3771551 8-635655 578 334034 193100552 24-0416306 8-329954 645 416025 253336125 25-3968502 8.640123 579 335241 194104539 240524188 8-334755 646 417316 269586136 25-4165301 8-644585 580 336400 195112000 24-0831891 8-339551 ! 647 418509 270340023 25-4351947 8-649044 581 337561 196122941 241039416 8-344341 ! 648 419904 272097792 25-4553441 8-653497 582 333724 197137368 24-1246762 8-349126|^ 649 421-201 273359449 254754734 8-657946 583 339339 193155237 241453929 8-353905 i 650 422500 274625000 25-4950976 8-662391 584 a41056 199176704 24-1660919 8 -353678! i 651 423301 275894451 25-5147016 8-666331 585 343225 200201625 24-1867732 8-353447 f 652 425104 2771678081 25-5342907 8-671266 586 343396 201230055 24-2374369 8-353-309 653 425409 273445077] 25-5533647 8-675697 587 344569 202262003 24-2230829 8-3-72967 , 654 4-27716 2797262641 25-5734237 8-680124 588 345744 203297472 24 2487113 8-377719 1 655 429035 281011375! 25-5929578 8-684546 589 346921 204336469 24-2593222 8-332465 : 656 43.336 282300416 25-6124969 8-688963 590 348100 205379000 24-2899156 8-337206 i 657 431649 283593393 25-6330112 8-693376 591 349281 1 206425071 24-3104916 8-391942 I 653 432964 2348903121 25-6515107 8-697784 •^92 350464 1 207474688 24-3310501 8-396673 1 659 434281 2361911791 25-6709953 8-702188 593 351649 1 208527857 24-3515913 8-4013981 660 8-406 Uh! 661 435600 237496000! 25-6904652 8-706538 594 352836 ; 209584584 24-3721152 436921 2388047811 25-7099-303 8-710983 595 354025 210644875 24-3926218 8-410333i 662 438244 2901175-281 25-7293607 8-715373 596 355216 21170873e 24-4131112 8-415542! 663 439559 2914342471 25-7487854 8-719760 597 356409 212776173 24-4335834 8-42024fr 664 440396 2937519441 25768 1975 8724141 598 357604 ! 213347192 24-4540385 8-4-24945^ 655 442225 294079625! 25-7875939 8-728518 599 353801 214921799 24.4744765 8-4296331 666 443556 295408296' 25 8069758 8-732892 601 360000 216000000 24 4943974 8-434337: 667 444889 296740963 25-8363431 8-737260 601 361-201 217081801 24-5153J13 8-4390l0!| 663 446224 298077632 25-8455960 8741625 602 362404 218167-208 24-5356i33 8-44368311 669 447561 299418309 25-8650343 8-745985 603 363609 1 219256227 24-5560533 8 41836UI' 670 448900 300763000 25-8843582 8-750340 TABLE OF SQUARES, CUBES, AND ROOTS. 643 No, Square. 67 J 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 t>94 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721' ' 7-/31 724| 725! 726 1 727 728 729 730 731 732 733 734 735 736 737 Cube. Sq. Root. ICubeRoot. 450241 451584 452929 454276 455625 45697G 458329 459684 461041 46240U 463761 465124 466489 467856 469225 470596 471969 473344 474721 476100 477481 478864 480249 48163C 483025 484416 4858C9 487201 488601 490000 491401 492^04 494209 495616 497025 498436 499849 501264 502681 504100 5U5521 506944 508369 5097^6 511225 515524 516961 518400 519841 521284 522729 524176 525625 527076 528529 529984 5314411 532900 302111711 303464448 3U4821217 306182024 307546875 308915776 310288733 311665752 313046839 314432000 315S21241 317214568 31861198- 320013504 321419125 322828856 3242427U3 3^5661672 327082769 328509000 3299393' 331373888 332812557 3342553^S4 335702U75 337153536 338608873 34006839:^ 341532099 343000000 344472101 345948408 34742892' 348913664 35040-2625 351895816 353393243 334894912 356400829 357911000 359425131 360944128 362407097 363994344 365525875 5126561 367061696 514089 368601813 37014623:i 371694959 373248U00 374805361 37636704^ 377933067 379503424 381078125 382657176 38424058;: 385828352 387420489 389017C00 5343611 390617891 535S24I 392223168 5"^72H9' 393832837 53:^756 3y54469t>4 540225' 397065375 541 69.)! 398688256 543169; 400315553 25-9035677I 8-754691 25-9229628! 8-75903S 25-9422435 25-9615100 No. Square. 25-9807621 26 0000000 26-0192237 26-0384331 26.0576-284 26-0768096 26-0959767 26-1151-297 •26-134-2687 26-1533937 26-1725047 26-1916017 26-2106848 26--2297541 26-2488095 26-2678511 26-2868789 26-3058929 26-3248932 26-3438797 26-3628527 26-3818119 26-4007576 26-4196896 26-4386081 26-4575131 26-4764046 26-4952826 26-5141472 26 5329983 26-5518361 26-5706605 26-5894716 26-6082694 26-6-270539 26-6458-<>5-i 26-6645833 26-6833281 26-7020598 26-7207784 ?" -7394839 26-7581763 26-7768557 26-7955-220 26-8141754 26-83-2815' 26-8514432 26-8700577 26-8886593 26-9072481 26-9258240 26-9443872 26-9629375 26-9814751 27-0000000 27-0185122 27-0370117 27-0554985 27-073'j727 27-0924344 27.1108834 27 1293199 27 1477439 8-763381; 8-767719 8-772053 8-776383 8-780708 8-785030 8-7893471 8-793659! 8-7979681 8-80-2272 8-8J6,572| 8-810868! 8-815160 8-819447 8-823731 8-8-28010 8-832285 8-836556 8-840823 8 845085 8-84i)344| 8-853598 8-857849 8-86-2095 8.866337 8-870576 8-874810 8-879040 8-883266 8-887483 8-891706 8-895920 8-900130 8-904337 8-908539 8-912737 8-916931 8-921121 8-925308 8-929490 8-933669 8-937843 8-942014 8-946181 8-950344 8-954503 8-958658 8-962809 8-966957 8-971101 8-975241 8-979377 8-983509 8-987637 8-991762 8-995883 9-000000 9-004113 9-008223 9-012329 9-016431 9-020529 y 02462 4 9-028715 9-032802 738 739 740 741 742 743 744 745 746 7^17 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 7 780 781 782 783 784 785 786 78/ 788 789 790 791 792 793 791 795 796 79 798 799 800 801 802 803 804 Cube. 544644 546121 547600 549081 550564 552049 553536 555025 556516 558009 559504 561001 562500 564001 565504 567009 568516 570025 571536 573049 574564 576081 57760C 579121 580644 582169 583696 585:^25 586756 588289 589824 591361 592900 59444 595984 597529 599076 600625 6021761 603729] 605-284 606841 6084U0 60J961 611524 613089 614656 616-225 617796 619369 620944 622521 624100 625681 627264 6-28849 630436 632025 633616 635-209 636804 638401 640000 641601 643204 644809 646416 Sq. Rod. 401947272 403583419 405224000 406869021 408518488 4101724071 411830784 413493625 415160936! 416832723 418508992 420189749 421875000 423564751 425259008 426957777 428661064 430368875 432081216 433798093 435519512 437245479 438976000 440711081 4424507-28 444194947 445943744 447697125 449455096 451217663 452984832 454756609 456533000 458314011 460099648 461889917 4636848-24 465484375 467288576 469097433 470910952 47i'7-29139 47455-2000 47637y541 470211768 480048687 481890304 4837366: 485587656 487443403 489303872 491169069 4y 30 39000 494913671 496793080 49867725/ 500566184 502459875 504358336 506-261573 508169592 510082399 512000000 51392-2401 515849608 517781627 519718464 CubeRooL 9-036886 9040965 9-045042 9-049114 9-053183 27-1661554 27-1845544 27-2029110 27-2213152 27-2396769 27-2580263; 9-057248 27-27636341 9061310 27-2946881 27-3130006 27-3313007 27-3495887 27-3678644 27-3861279 27-4043792 27-42-26184 27-4408455 27-4590604 27-4772633 27-4954542 275136330 27-5317998 27-5499546 27-5680975 27-5862284 27-6043475 276224546 27-640549d ■27-6586334 27-6767050 27-6947648 27-71-28129 27-7308492 27-7488739 27-7668868 27-784:5880 27-8028775 27-8208555 27-8388218 27-8567766 27-8747197 27-8926514 27-9105715 27-9284801 27-9463772 27-9642629 27-9821372 28-0000000 28 017o515 28-0356^15 28-0535203 28-0713377 28-0891438 28-l06i)386 28-1247222 28-1424946 28-1602557 28-1780056 28-1957444 28-2134720 28-2311884 28-2488933 28-2665881 28-2842712 28-3019434 28-319tj045 28-3372546 28-3548938 9-065368 9 069422 9-073473 9077520 9 081563 9-085603 9 08963y 9093672 9-097701 9-101726 9-105748 9-109767 9-113782 9-117793 9-121801 9-125805 9-129806 9-133803 9-l377y 9-141787 9-145774 9-149750 9-15373' 9-.»577l4 9-16168 9-165656 9-16962:i 9-17358J 9-177544 9-181500 9-185453 9-18040. 9-ly33i7 9-197290 9-20r22i, 9-205i0. 9-20y09t 9-2130-25 9-2l6y5v. 9-2208/ 9-z-2-i7Ji 9-2-2:370< y-2020li. 9-2Jb52b 9-210435 9-244333 9 -248231 9-25213V. 9-250022 9-2:^9011 9-26.)7y 9-267680 9-27 155y 9-275435 9-2793UO 9-283170 9-237044 9-290.^07 9-29476- 9-298624 644 APPENDIX. No. 805 Square. 1 Cube. | Sq. Root. CubeRoot.' No. Square. Cute. 1 Sq. Root. jCubeRool] 648025 521660125 28 3725219 9-302477 872 760334 0030548481 29-5296401 9-553712 806 649636 5236U0616I 28 3901391 9-300328 873 7621291 055338617 29-5465734 9-557363 807 651249 5255579431 284077454 9-310175 874 763376 6676276-24 29-5634910 9-561011 808 652864 5275141121 28 4253408 9-3140191 875 765525 669921875 29-5803989 9-564650 809 654481 529475129 28-4429253 9-317860 876 767370 ()72221376 29-5972972 9-568298 810 656100 531441000 28-4604989 9-321097 877 709129 674526133 29-6141858 9-571938 811 657721 533411731 28-4780017 9-3255321 878 770884 670830152 29-6310648 9-575574 812 659314 535387328 28-4956137 9-32J363 879 772641 079151439 29-6479342' 9-5792081 813 C60969 537367797 28-5131549 9-3331921 880 774400 0814720C0 29-66479391 9-58-2840 814 662596 539353144 28-5300852 9-337017; 881 776161 083797841 29-6816442 9-:86468 815 664225 541343375 28-5482048 9-340839 882 777924 080128968 29.6984843 9-590094 816 665856 543338496 28-5657137 9-3446571 883 779689 688465387 29-7153159 9-593717 817 667489 545335513 28-5832119 9-3-8473' 8-Srl 781456 690807101 29-7321375 9-597337 818 669124 547313432 28-6006993 9-352-280 8-35 783225 6931541-25 29-7489496 9-600955 819 670761 549353259 28-6181760 9-356095 886 78499B 095506456 29-7657521 9-604570 820 672400 551368000 28-6356421 9-359902 887 786769 697854103 29-7825452 9-608182 821 674041 553387661 28-6530976 9-3037051 838 788544 700227072 29-7993289 9-611791 822 675684 555412248 28-0705424 9307505 889 790321 702595369 29-8161030 9-61539'' 823 677329 557441767 28-0879760 9-371302 890 792100 704969000 29-8328678 9-619002 824 078976 559476224 28-7054002 9-375090 891 793881 707347971 29-8496231 9-622003 825 680625 561515625 28-72-28132 9-378887 1 892 795664 709732288 29-8663690 9-626202 826 682276 563559976 28-7402157 9-382075 893 797449 712121957 29-8831056 9-629797 827 683J29 565609283 28-7570077 9-380460! 894 799236 714516984 29-8998328 9-633391 828 685584 567663552 28-7749S91 9-390242 895 801025 716917375 29-9165506 9-636981 829 687241 569722789 28-7923001 9-39402 li 896 802816 719323136 29.-33-2591 9-040569 830 6889)0 571787000 28-8097206 9-3..7796 897 804609 721734273 29-9499533 9-044154 831 690561 573856191 28-8270706 9-4015691 898 806404 724150792 29-9666481 9-047737 832 692224 575930368 28-8444102 9-4053391 899 808201 72657269S 29-9S33-.J87 9-051317 833 693889 578009537 28-8617394 9-4091051 900 810000 729000000 30-0000000 9-054894 834 695555 580093704 28-8790582 9-4128691 901 811801 73143-2701 30-0166620 9-658408 835 697225 582182875 28-8903666 9-416630 902 813604 733870808 30-0333148 9-00-2040 836 698896 584277056 28-9136646 9-4203871 903 815409 736314327 30-0499584 9-005010 837 700569 586376253 28-9309523 9-424142 904 817216 738763264 30-0665923 9-669176 838 702244 588480472 28-9482297 9-4278d4i 905 819025 741217625 30-0832179 9-672740 839 703921 590589719 28-9654967 9-431642' 906 820836 743677410 30-0998333 9-676302 840 705600 592704000 28-9827535 9-435388 907 822649 746142643 30-1164407 9-679860 841 707281 594823321 29-0000000 9-439131' 908 824464 748613312 30-1330383 9-r.83417 842 708964 596947688 29 0172363 9-442870 909 826-281 751089429 30-1496263 9-686970 843 710649 599077107 29-0344623 9-446607 910 828100 753571C00 30-1662063 9-690521 844 712336 601211584 29-0516781 9-450341 911 829921 756053031 30-18-27765 9-694069 845 714025 0U3351125 29-0688837 9-454072 912 831744 758550528 30-1993377 9-697615 846 715716 005495736 29-0860791 9-457800 913 833569 761048497 30-2158399 9-701158 847 717409 607645423 29-1032644 9-4615-25 9U 835396 763551944 30-232432^ 9-704699 848 719104 609800192 29-1204396 9-465247 915 837225 766060875 30-2489669 9-708237 849 720801 611960049 29-1376046 9-468966 916 839056 768575296 30-2654919 9-711772 850 722500 614125000 29-1547595 9-472682 917 840889 771095213 30-2820079 9-715305 851 724201 616295051 29-1719043 9-476390 918 842724 773620632 30-2985148 9-718835 852 725904 618470208 29-1890390 9-480106 919 844561 77615 155S 30315012a 9-722363 853 727609 620650477 29-2061637 9-483814 921 846400 77863300L 30-3315018 9-725388 854 729316 022835804 29-2232784 9-487518 921 848241 781229961 30-3479818 9-729411 855 7310^5 025026375 29-2403830 9-491220 922 850084 783777448 30-364452 30-6207857 1 9-78S909 TABLE OF SQUARES, CUBES, AND ROOTS. 645 No. 939 Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq. Root. C5beRooiJ 831721 82793G019 30-6431069 9 792386 970 940900 9126730CO 31-1448230 9-898983 910 8b3600 830584000 30-6594194 9-795361 971 942 ■i41 915498611 31-160372.* 9-902333 941 8H5481 833237621 30-6757233 9-799334 972 944784 918330048 311769145 9-905782 912 83"'364 835S9688S 30-6920185 9-802304 973 946729 921167317 31-1929479 9-909178 943 889249 838561807 30-7083051 9-806271 974 948676 924010424 31-2039731 9-912571 944 891136 841232334 30-7245330 9-809736 975 950625 926859375 31-2249900 9-915962 945 893025 843D08625 30-7408523 9-813199 976 952576 929714176 31-2409987 9-919351 946 894916 84G590536 30-7571130 9-816659 977 954529 932574833 31-2569992 9-922738 947 8'J6S09 849278123 30-7733651 9-820117 978 956484 935441352 31-2729915 9-926122 948 898704 851971392 30-7896086 9-823572 979 958141; 933313739 31-288.'757 9-929504 949 900601 854670349 30-8053436 9-827025 980 960400 91 1192000 31-3049517 9-932834 950 90:-500 857375000 30-8220700 9-830476 931 96236 l| 944076141 31-3209195 9-936261 951 904101 860085351 30 8382879 9-8339-24 982 964.3241 946966168 31-3368792 9-939636 952 9U6304 862801408 30-8544972 9-837369 983 966289! 949862087 31-3528308 9-943009 953 908209 865523177 30-8706981 9-840813 934 9682561 95-2763-)04 31-3687743 9-916380 954 910116 868250664 30-8868904 9-844254 935 970225 955671625 31-3847097 9-949748 955 912025 870983875 30-9030743 9-8476:12 986 972196 958535256 31-4006,369 9-953114 956 913936 873722816 30-9192497 9-8511^:8 987 974169! 961504803 31-4165561 9-956477 957 915849 876467493 30-9354166 9-854562 238 976144 9644302^2 31-43246731 9-959839| 958 917764 879217912 30-9515751 9-857993 989 978121 967361669 31-4483704 9-963 19S 959 919681 831974079 30-9677251 9-861422 990 980100 970299000 31-4642654 9-96655.5 960 92ia)0 884736000 30-9838668 9-864843 991 982081 973242271 31-4801525 9-969909 961 923521 887503G31 310000000 9-868272 992 984064 976191488 31-4960315 9-973262 962 925444 ^•90277128 310161248 9-871694 993 986049 979146657 31-5119025 9-976612 963 927369 893056347 310322413 9-875113 994 988036 982107784 31-5277655 9-979960 964 9292^.6 5J5341344 31-0483494 9-878530 995 990-025 985071875 31-5436206 9-933305 965 931225 898632125 31-0644491 9-881945 996 992016 983047936 31-5594677 9-986649 966 933156 901423696 310805405 9-835357 997 994009 991026973 31-5753068 9-989990 967 935089 904231063 31-0966235 9-883767: 993 996004 994011992 31-5911330 9-993329' 968 937U24 907039232 31-1126984 9-892175 999 993001 997002999 31-6069613 9-996666, 10000000 969 93S961 909853209 31-1287648 9-895530 1000 1000000 ! 1000000000 31-6227766 The following rules are for finding the squares, cubes, and roots of num- bers exceeding 1000. To find the square of any number divisible without a remainder. Rule. — Di- vide the given number by such a number from the foregoing table as will divide it without a remainder ; then the square of the quotient, multiplied by the square of the number found in the table, will give the answer. Exatnple. — What is the square of 2000 ? 2000, divided by 1000, a number found in the table, gives a quotient of 2, the square of which is 4, and the square of 1000 is 1,000,000, therefore : 4 X 1,000,000 = 4,000,000: the Ans. Another Example. — What is the square of 1230? 1230, being divided by 123, the quotient will be 10, the square of which is 100, and the square of 123 is 15,129, therefore : 100 X 15,129 = 1,512,900: the Ans. To find the square of any number not divisible without a remainder. Rule. — Add together the squares of such two adjoining numbers from the table as shall together equal the given number, and multiply the sum by 2 ; then this product, less i, will be the answer. Example. — What is the square of 1487 ? The adjoining numbers, 743 and 744, added together, equal the given number, 1487, and the square of 743 = 552,049, the square of 744 = 553,536, and these added = 1,105,585, therefore : 1,105,585 X 2 = 2,211,170 — I = 2,211,169 : the Ans. To find the cube of any number divisible without a remainder. Rule. — Divide the given number by such a number from the foregoing table as will divide 646 APPENDIX. it without a remainder ; then the cube of the quotient, multiplied by the cube of the number found in the table, will give the answer. Z'jr^w//^.— What is the cube of 2700? 2700, being divided by 900, the quo- tient is 3, the cube of which is 27 and the cube of 900 is 729,000,000, there- fore : 27 X 729,000,000 = 19,683,000,000 : the Ans, To find the square or cube root of numbers higher than is found in the table, /^uk.— Select, in the column of squares or cubes, as the case may require, that number which is nearest the given number ; then the answer, when decimals are not of importance, will be found directly opposite, in the column of num- bers. £xamJ>le.—Whsit is the square root of 87,620? In the column of squares, 87,616 is nearest to the given number ; therefore, 296, immediately opposite in the column of numbers, is the answer, nearly. Another example. — What is the cube root of 110,591? In the column of cubes, 110,592 is found to be nearest to the given number ; therefore, 48, the number opposite, is the answer, nearly. To find the cube root more accurately. Rule. — Select from the column of cubes that number which is nearest the given number, and add twice the number so selected to the given number ; also, add twice the given number to the number selected from the table. Then, as the former product is to the latter, so is the root of the number selected to the root of the number given. Example. — What is the cube root of 9200? The nearest number in the col- umn of cubes is 9261, the root of which is 21, therefore : 9261 9200 18522 18400 9200 9261 As 27,722 is to 27,661, so is 21 to 20-953 -f , the Ans. Thus, 27,661 X 21 = 580,881, and this divided by 27,722 = 20-953 +. To fijid the square or cube root of a whole number with decimals. Rule. — Sub- tract the root of the whole number from the root of the next higher number, and multiply the remainder by the given decimal ; then the product, added to the root of the given whole number, will give the answer correctly to three places of decimals in the square root, and to seven in the cube root. Example. — What is the square root of 11-14? The square root of 11 if 3-3166, and the square root of the next higher number, 12, is 3-4641 ; the for- mer from the latter, the remainder is 0-1475, and this by 0-14 equals 0-02065. This added to 3-3166, the sum, 3-33725, is the square root of 11-14. To find the roots of decimals by the use of the table. Rule. — Seek for the given decimal in the column of numbers, and opposite in the columns of roots will be found the answer, correct as to the figures, but requiring the decimal point to be shifted. The transposition of the decimal point is to be performed thus : For every place the decimal point is removed in the root, remove it in the number two places for the square root and three places for the cube root. J THE REDUCTION OF DECIMALS. 647 Examples. — By the table, the square root of 86 -o is 9-2736, consequently by the rule the square root of o-86 is 0-92736. The square root of 9* is 3-, hence the square root of 0-09 is 0-3. For the square root of 0-0657 we have 0*25632, found opposite No. 657. So, also, the square root of 0-000927 is 0-030446, found opposite No. 927. And the square root of 8-73 (whole num- ber with decimals) is 2-9546, found opposite No. 873. The cube root of o-8 is 0-928, found at No. 800 ; the cube root of 0-08 is 0-4308, found opposite No. 80, and the cube root of 0-008 is 0-2, as 2-0 is the cube root of 8-0. So also the cube root of 0-047 is 0-36088, found opposite No. 47. RULES FOR THE REDUCTION OF DECIMALS. To reduce a fraction to its equivalent deci?nal. Rule. — Divide the numerator by the denominator, annexing cyphers as required. Example. — What is the decimal of a foot equivalent to three inches ? 3 inches is jg- of a foot, therefore : ^i , . . 12)3.00 •25 Ans. Another example, — What is the equivalent decimal of | of an inch? I . . . 8)7-000 •875 Ans. To reduce a compound fraction to its equivalent decijnal. Rule. — In accordance with the preceding rule, reduce each fraction, commencing at the lowest, to the decimal of the next higher denomination, to which add the numerator of the next higher fraction, and reduce the sum to the decimal of the next higher denomination, and so proceed to the last ; and the final product will be the answer. Exajnple. — What is the decimal of a foot equivalent to five inches, f and ^ of an inch ? The fractions in this case are, \ of an eighth, f of an inch, and -/^ of a foot, therefore : \ 2) i-o •5 3- eighths. f 8) 3 • 5000 •4375 5. inches. ■h 12) 5-4 37500 -453125 Ans. The process may be condensed, thus : write the numerators of the given 648 APPENDIX. fractions, from the least to the greatest, under each other, and place each de- nominator to the left of its numerator, thus : 10 3 • 5000 5-437500 •453125 Ans. To reduce a decimal to its equivalent in terms of lower denominations. Rule. — Multiply the given decimal by the number of parts in the next less denomi- nation, and point off from the product as many figures to the right hand as there are in the given decimal ; then multiply the figures pointed off by the number of parts in the next lower denomination, and point off as before, and so proceed to the end ; then the several figures pointed off to the left will be the answer. Example, — What is the expression in inches of 0-390625 feet ? Feet 0-390625 12 inches in a foot. Inches 4-687500 8 eighths in an inch. Eighths 5-5000 2 sixteenths in an eighth. Sixteenth i-o Ans., 4 inches, f and -fg-. Another example. — What is the expression, in fractions of an inch of 0'6875 inches? Inches 0-6875 8 eignths in an inch. Eighths 5 • 5000 2 sixteenths in an eighth. Sixteenth 10 Ans., t and -j^. \ TABLE OF CIRCLES. (From Gregory's Mathematics.) From this table may be found by inspection the area or circumference of a circle of any diameter, and the side of a square equal to the area of any given circle from i to lOO inches, feet, yards, miles, etc. If the given diameter is ia inches, the area, circumference, etc., set opposite, will be inches ; if in feet, then feet, etc. Side of Side or Diam. Area. Circum. equal sq. Diam. Area. Circum. cfjual scj. '¥j •01908 •78539 •22155 -75 90-76257 3377212 9-5-2693 •5 •19635 1-57079 -44311 11- 95-03317 34-55751 9-74349 •75 •44178 1 2-35619 •66467 -25 99-40195 35-34-291 9-97005 1- •7853'J 3-14159 •886-22 •5 103-86890 36-12S31 10-19160 •25 1-22718 3-92699 1-10778 •75 108-43403 38-91371 10-41316 •5 1-7G714 4-71-238 1-32934 12- 113 09733 37-69911 10-63472 •75 2-40528 5-49773 1 --55983 -25 llr85381 38-43451 1085627 2- 314159 6-23318 1-77245 -5 1^22-71846 39-26990 11-07783 •25 3-97fi07 7-06358 1-99401 -75 127-676'28 4005530 11-29939 •5 4-90373 7-85393 2-21556 13- 132-73228 40-84070 ll-'>2iYJ5 •75 5-93957 8-63937 2-43712 -25 137-83646 41-6-2610 11-74-250 3^ 7-0685S 942477 2-65388 -5 143-13331 4-2-4 1150 11-98406 •25 8-29576 10-21017 2-88023 -75 143-48934 43-19639 12-18562 •5 9-62112 10-99557 310179 14- 153-93804 43 98-2-29 12-40717 •75 1104466 11-78097 3-3-2335 •25 159-48491 44-76769 12-82373 4- 12-56637 12-56637 3-54490 •5 165-1»2996 45-55309 12-85029 •25 14- 186-25 13-35176 3-76646 •75 170-87318 46-33349 13-07184 •5 15-90431 14-13716 3-98802; 15^ 176-71458 47-12338 13-29340 •75 17-72054 14-92256 4-20957! •25 182-65418 47-90923 13-51496 5- 19-63195 15-70796 4-431 13i -5 188-69190 48-694C)8 13-73651 •25 21-64753 16-493.36 4-65269] •75 194-8-2783 49-48003 13-95307 •5 23-75329 17-27875 4-87424' 15^ 201-06192 50-26548 14-17963 •75 25-96722 18-06415 5-095»a •25 207-394-20 51-05;)83 14-10118 6^ 28-27433 18-84955 5-31736 •5 213-82464 51 -833 27 14-62-274 •25 30-67961 19-63495 5-53891 •75 220-.35327 52 62167 14-84430 •5 33-18307 20-42035 5-760471 17^ 226-93008 53-40707 15-06535 •75 35-78470 21-20575 5-9321)31 •25 233-70504 54-19-247 15-28741 7- 33-48156 21-99114 6-20358 •5 240-52818 54-97737 15-50897 ■25 41-28249 27-77654 6^4-2514 6-64670 •75 247-44950 55 76326 15-73052 •5 41-17861 23-56194 18- 254-46900 58-54368 15-95-208 •75 47-17-297 24-34734 6-86325 •25 261-58667 57-33406 16-17364 8- 50-26548 25-13274 7-08981 •5 268-80252 53-1 19 46 1639519 25 53-45616 25-91813 7-31137 -75 276-11654 53-90486 16-61875 •5 56-74501 26-70353 7-53292 19- 233-5-2873 59-69028 16-83331 •75 6013204 27-48893 7-75448 •25 291-03910 60-47565 17-05936 9- 63-61725 28-27433 7-976J4 -5 298-64765 61-28105 17-28142 •25 67-20063 29-0597a 8-19759 -75 306-35437 62-04645 17-50298 •5 70-83218 29-84513 8-41915 20- 314-15926 62-83185 17-72153 •75 74-66191 30-63052 8-64071 •25 322-06233 63-617-25 17-91809 10^ 78-53981 31-41592 8-86226 -5 330-06357 64-40284 18-16765 •25 82-51539 32-20132 9-08382 -75 333-16-299 65-18301 18-389-20 •5 1 86-59014 32-98672 9-30538! 21- 346-36059 65-97314 18-61076 650 APPENDIX. Side of 1 Side of Diam. Area. Circam. equal sq. Diam. Area. Circum. enual sq. Tr25 354-65635 66-758841 18 83232 ~38~~ 1134-114941 119-380521 .33 67662 •5 363-05030 67-54424 19-05387 •25 1149-08660 120-16591 33-89317 •75 371-54241 68-32964 19-27543 •5 1164^15642 120-95131 34-11973 22- 38013271 69- 11503! 19-49699 •75 1179.32442 121-73671 34-34129 •25 388-82 117 69-900431 19-71854 39^ 1194-59060 122-5-2211 34-56285 •5 3J7-6078-2 70-68583! 19-94010 •25 1209-95495 1-23-3 J751 31-78440 •75 406-41)263 71-471231 20-16166: •5 1225-41748 124-09-290 35-00596 23- 415-475:'.2 72-25663 20-38321 •75 1240-97818 124-87830 35-22752 •25 424-55679 73-04202 20-60477 40^ 1255-63704 125-66370 35 44907 •5 433-73613 73-82742 20-8-2633 •25 1272-39411 126-44910 3567063 •75 443-01365 74-61282 2104788 •5 1288-24933 127-23450 35-89219 24- 452-38934 75-398-22 21-26944 •75 1304-20273 128-01999 36-11374 •25 461-8r)320 76-18362 21-49100 41- 1320-25431 123-80529 36-33530 •5 471-43524 76-96902 2171255 -25 1336-40406 129-59069 36-55686 •75 481-10546 77-75441 21-93111 •5 1352-65198 130-37609 36-77841 25- 490-87335 78-53J81 22-15567 •75 1368-99808 131-16149 ,36-99997 •25 500-74041 79-32521 22-377-22 42- 1385-44236 131-94689 37-22153 •5 510-70515 80-11061 22-59878 •25 1401-98480 132-73228 37-44308 •75 520-76806 80 89601 22-82034 •5 1418-62543 133 51768 37 66464 26- 530-92915 81-68140 23-04190 •75 1435-36423 131-30308 37-88620 •25 541-18842 82-46680 23-26345 43^ 1452-20120 135-08348 33-10775 •5 551-51586 83-25-220 23-48501 •25 1469-13635 135 87338 33-32931 •75 562-00147 84 03760 23 70657 -5 1486-16967 135-65928 38-55087' 27- 572-55526 84-82300 23-92812 •75 1503-30117 137-44467 33-77-242 •25 583-20722 85-60839 24- 14968 44- 1520-53084 133-23007 38-99398 ■5 593-1:5736 86-39379 24-37124 •25 1537-85369 139-01547 39-21554 •75 604-80567 87-17919 24-59279 •5 1556-23471 139-80087 39-43709 28- 615-75216 87-96459 24-81135: •75 1572-80890 140-58527 3965865 •25 626-79582 8874999 25-03591 45- 1590 43128 141-37166 39-88021 •5 637-931:65 89-53539 25-25745 •25 1608-15182 142-15706 40-10176 •75 649-18066 90-32078 25-47i;02 •5 1625-97054 142-94-246 40-32332 29^ 660 51985 9M0S18 25-70053 •75 164388744 143-72785 4054488 •25 671-95721 9 39153 25-92213 46- 1661-90251 144513-26 40-76643 •5 683-49275 92-67698 26-143(59 •25 1680-01575 145-29866 40-98799 •75 6:5-12646 93-4623S 26-36525 •5 1698-22717 146-08405 41-20955 30^ 706-85834 94-24777 25-58680' •75 1716-53677 146-86945 41-43110 •25 718 68810 9503317 26-80836 47- 1734-944M 147-65485 41-65266 •5 730-61664 95-81857 27-02992: •25 1753-45048 148-44025 41-87422 •75 742 64305 96-60397 27-251471 ■5 1772-05460 149-2-2565 42-09577 31- 751-76763 766-99039' 97-38937 27-47303 •75 1790-75639 150-01104 42-31733 •25 98-17477 27-694591 43- 1809 55736 150-79644 42-53889 •5 779-31132 9896016 27-91614: •25 1828-45601 151-53184 42-76044 •75 791-73043 99-74556 28-13770; •5 1847-45282 152-36724 42-98200 32^ 804-24771 100-53096 23-35926' •75 1866-54782 153-15264 43-20356 •25 816-86317 101-31636 2.S-58081i 49- 1885-74099 153-93804 43 42511 •5 829-57681 102-10176 2S-h0237' •25 1905-83233 154-72343 43-64667 •75 842-38861 102-88715 29i)23J3 -5 19-24-42184 155-50883 45-86823 33^ 855-29859 10367255 29-21548 •75 1943-90954 156-29423 44-08978 •25 868-3(;675 104-45795 29-46704* 50- 1963-49540 157-07963 44-31134 •5 881-41308 105-21335 29-68860' •25 1983-17944 157-96503 44-53290 •75 894-61759 106-02875 29-91015 •5 2G02-96I66 153-65042 44-75445 34- 907-921)27 106-81415 30-13171 •75 2022-84205 159-43582 44-97601 25 921-32113 107-59954 30-35327 51- 2042-82062 160-221-22 45-19757 •5 934-82016 108-38494 30-57482 •25 2062-89735 161-00662 45-41912 •75 943 41736 109-17034 30-79638 •5 2083-07227 161-79202 45-64068 35^ 962-11275 109-95574 31-01794 •75 2103-34536 162-57741 45-85-224 •25 975-90630 110-74114 31-23949 52^ 2123-71663 163-36-281 46-0g:!80 •5 989-79803 111-52653 31-46105 •25 2144-18607 164-14821 46-30535 •75 1003-78794 11231193 31-68-261 •5 2164-75368 164-93361 46 5-2691 36- 1017-87601 113 09733 31-90416 •75 2185-41947 165-71901 46-74847 •25 1032-06227 11388273 32-12572 53^ 2205-18344 166-50441 46-97002 •5 1046-34670 114 66813 32-34728 •25 2227-04557 167-28980 47-19158 •75 1060-72J30 115-45353 32-55883 •5 2248-00589 16807520 47-41314 37- 1075-21008 1 16-238 J2 32-7903J •75 2-269-06438 168-85060 47-63469 •25 1089-78903 117-02432 33-01195 54^ 2290-22104 169-61600 47-85625 •5 1104-46616 117-80972 33-23350 •25 • 2311-475S8 170-43140 48-0778'l •75 1119-2414" 1 18-59572 33-45506 •5 2332-82889 171-21679 48-2993(i TABLE OF CIRCLES. 651 Side of Side of DiF.m. Arcii. Circi-.];;. 17200219 equal sq. 48-52092' Diam. Area. Circum, e(iuul eq. 63-36522 54-75 "2354 -28008 ~n~ 4015-1517O 2-24 -623^7 55- 2375-82J44 172-78759 48-74248 •75 404327883 225-40J-27 63-53678 •25 2397-47698 173-57299 48-96403 72^ 4071-50407 2-26-19467 63-80833 •5 241 9^22209 174-35839 49-18559 •25 4099-8-2750 226-98006 6 1-0-2989 •75 2441-06657 175-14379 49-40715 •5 41-28-24909 227-7(:546 64-25145 56- 2463 00861 175-92918 49-62870 •75 4156-76886 2-2855(:86 64-47300 •25 2485-04887 176-71458 49-85026 73^ 4185 33681 229-33626 64-69456 •5 2507-18728 177-49998 5007182 •25 4214-10-293 230-12 16i'. 64-91612 •75 2520-4'2337 178-28538 50-29337 •5 4242-91722 230-9()7(:6 65-13767 57- 2551-75S63 179-07078 50-51493 •75 4271-82969 231-69-245 <)5-35923 •25 2574-19156 179-85617 50-73649 74^ 4300-84034 232-47785 65-58079 •5 2596-7-2'267 180-64157 50-95304 •25 4329-94916 23326325 65 80234 •75 2619-35196 181-4-2697 51-17960 •5 4359-15615 23 4-0481' 5 66-C2390 53^ 264207942 182 21237 51-40116 •75 4388-46132 234-83105 65-24546 •25 2664 90505 182-99777 51-6-2271 75^ 4417-86466 235-61944 66-46701 •5 2687-8-28S6 183-78317 51844-27 •25 4447-36618 236-40484 66-63857 •75 2710-85084 184-56856 52-06583 •5 4476-96538 237-19024 66-91043 59- 2733-97100 185-353J6 52-28738 •75 4506-66374 237-97564 67-13168 ■25 2757-18933 186-13936 52-50894 76^ 4536-45979 238-76104 67-35324 •5 2780-50584 186-92476 52-73050 •25 4566 35400 239-54613 67-57180 •75 280392053 187-71016 52-95-205 •5 459634640 240-33183 67-79635 60- 2327-43338 ia3-49555 5317364 •75 4626-43696 241-11723 63 01791 •25 2851-04442 189-28095 5339517 77^ 465562571 241-90-263 68-23947 •5 2874-75362 19;)-06635 53-61672 •25 46S691262 242-638;)3 63-46102 •75 2898-56100 190-85175 53-83328 •5 4717-29771 243-47343 6868258 61- 2922-46656 191-63715 54 05984 •75 4747-78093 244-25332 68-90414 •25 2946-47029 192-42255 54-28139 78^ 477836242 245 044-22 69-1-2570 •5 2970-572-20 193-20794 54-50295 •25 4809-04204 245-82962 69 34725 •75 2994-77223 193-99334 54-7-2451 •5 4839-81983 246-61592 69-56881 62- 301907054 194-77874 54 946C6 •75 4S7()-79579 247-40042 69-79037 •25 3043 46697 195-56414 5516762 79^ 4901-66993 248-18531 70-01192 •5 3067-96157 196-34954 55-33918 •25 4932-74225 248-97121 70-23348 •75 3092-55435 197-13493 5561073 •5 4963-91274 249-73661 70-45504 63- 3117-24531 197-9-2033 55-83229 •75 4995-18140 250-34-201 70-67659 •25 314203444 198-70573 56053^5 80- 5026-548-24 251-32741 70-89815 •5 3166-92174 199-49113 56-27540 •25 5058-01325 252-11231 71-11971 •75 3 191 •907-22 200-27653 56-496 J6i •5 5089-57644 25289820 71-34126 64- 3216-99087 201-06192 56-71852 •75 5121-23781 253-63360 71-56282 •25 3242- 17-270 201-84732 56-94007 81^ 5152-99735 254-46900 71-78433 •5 3267-45270 20263272 57-16163 •25 5184-8551)6 255-25440 72-00593 •75 3292 83088 203-41812 57-333191 •5 5216-81095 256-03J80 72-22749 65- 3318-30724 204-20352 5760475 •75 5248-86501 255-82579 72-44905 •25 3313-88176 204-98392 57-82630 82- 5281-01725 257-61059 72-67060 •5 3369-55447 205-77431 53-04786 •25 531326766 258-39599 72-89216 •75 339532534 206-55971 53-26942 •5 5345-616-24 259-18139 73 11372 66- 3421 -19439 207-34511 53-49097 •75 5378-06301 259-96679 73-33527 •25 344716162 203-13051 53-71-253 83^ 5410-60794 260-75219 73-55633 •5 347322702 208-91591 58-93409 •25 544325105 261-53753 7377839 •75 349939060 209-70130 59-15584 •5 5475-99234 262-32-298 73-99994 67- 3525-65235 21048670 59-37720 •75 5508-83180 263-10333 74-22150 25 3552^01228 211-27210 59-59876 84- 5541-76944 263-89378 74-44306 •5 3578-47033 212-05750 59-82031 •25 5574-80525 264-67918 74-66461 •75 350502665 212-84-290 6004187 •5 5607-93923 265-46457 74-88617 68- 3631-68110 213-62^30 60-26343 •75 5641-J7139 266-24997 75-10773 •25 3658-43373 214-41369 60-434981 85^ 5674-50173 267-03537 75-32928 •5 3685-28453 215-19909 60-70654 •25 5707-93023 267-82077 75-55084 •75 3712 23350 215-98449 60-92310 •5 5741-45692 268-60617 75 77240 69- 3-'39-28065 21676939 61-14965 •75 5775-08178 269-39157 75-99395 •25 3766-4-2597 217-555-29 61-37121 86^ 5808-80481 270-17696 76 21551 •5 379CS-66947 218-34068 61-59-277 •25 5342-62602 270-96236 76-43707 •75 332101115 2il'- 12608 61-81432 •5 5876-54540 271-74776 76-65362 70 3348-45100 219-91143 62-03533 •75 5;) 10-56296 272 53316 76-88/J18 •25 3875-98902 2-20-69683 62-25744 87^ 5944 67869 273-31856 77-10174 •5 3'J03-62522 221-48-228 62-47899 •25 5973-89260 274-103J5 77-32329 •75 3931-35959 222-2JV/63 62-70055 •5 6013-20468 274-88935 77-54485 71 3959-19214 223-05307 62-9-2211 •75 604761494 275-67475 777664J •2'- 3987-12286 223 83347 6314366 88^ 6082-12337 276-46015 77 987 96 652 APPENDIX. "Side of" Side ot Diam. Area. Circum. equal eq. Diam. Area. Circum. equa) sq. 83-52688 l8~25 6116-72993 277-24555 78-20952 ~94-25 6976-74097 296-09510 •5 6151-43476 278-03094 78-43103 •5 7012-80194 296-88050 83-74844 •75 6186-23772 278-81634 7865263 -75 7050-96109 297-66590 83-97000 89- 6221 -13885 279-60174 78-87419 95- 7083-21842 298-45130 84-19155 •35 6256-13315 230-33714 79 09575 •25 7125-57992 299-23670 84-41313 •5 6291-23563 231-17254 79-31730 -5 7163-02759 300-02209 84-63467 •75 6326-43129 281-95794 79-53836 •75 7200-57944 300-80749 84-856-22 90^ 6361-7-2512 282-74333 79-76042 96- 7238-22947 3U 1-59239 85-0VV78 •25 639711712 233-52873 79-98193 •25 7275-97767 302-37829 85-29934 •5 6432-60730 284-31413 80-20353 -5 7313-82404 303-16369 85-52089 •75 6463-19556 285-09953 80-42509 -75 7351-76859 303-94908 85-74-245 91^ 6503-83219 285-83493 80-646-59 97- 7389-81131 304-73448 85-96401 .25 653J-65689 286-67032 80-85820 -25 7427-95221 305-5 1983 86-18556 •5 6575-51977 237-45572 81-03976 -5 7466-191-29 306-305-23 86-40712 •75 6811-53382 288-24112 81-31132 -75 7504-52853 307-09068 86-6-2868 92^ 6347-61005 239-02652 81-53237 98- 7542-96396 307-87603 86-85023 •25 6633-73745 289-81192 81-75443 -25 7581-49755 308-66147 87-07179 •5 67-20-06303 290-59732 81-97599 -5 76-20-12933 309-44687 87-29335 •75 6756-43678 291-33271 82-19734 -75 7653-85927 310-23227 87-51490 93- 6792-90871 292-16811 82-41910 99- 7697-68739 311-01767 87-73646 •25 6829-47831 292-95351 8-2-0 1066 -25 7736-61369 311-80307 87-95802 •5 6866-14709 293-73391 82-86221 -5 7775-63816 312-58346 88-17957 •75 6902-91354 294-52431 83-08377 •75 7814-76031 313-37336 88-40113 94- 6939-77817 295-30970 83 3053311100- 7353-98163 314-15926 88-62269 The following rules are for extending the use of the above table. To find the area, circumference, or side of equal square, of a circle having a diameter of more than 100 inches, feet, etc. Rule. — Divide the given diameter by a number that will give a quotient equal to some one of the diameters in the table ; then the circumference or side of equal square, opposite that diameter, multiplied by that divisor, or the area opposite that diameter, multiplied by the square of- the aforesaid divisor, will give the answer. Example. — What is the circumference of a circle whose diameter is 228 feet ? 228, divided by 3, gives 76, a diameter of the table, the circumference of which is 238-761, therefore : 238-761 3 716-283 feet. Ans. Another example. — What is the area of a circle having a diameter of 150 inches? 150, divided by 10, gives 15, one of the diameters in the table, the area of which is 176-71458, therefore: 176.71458 10 X 10 17,671-45800 inches. Ans. To find the area, circumference, or side of equal square, of a circle having an intermediate diameter to those in the table. -/?z^/^.— Multiply the given diameter by a number that will give a product equal to some one of the diameters ia the table ; then the circumference or side of equal square opposite that diame- ter, divided by that multiplier, or the area opposite that diameter divided by the square of the aforesaid multiplier, will give the answer. CAPACITY OF WELLS, CISTERNS, ETC. 653 Example. — What is the circumference of a circle whose diameter is 6|, or 6-125 inches ? 6-125, multiplied by 2, gives 12-25, one of the diameters of the table, whose circumference is 38-484, therefore : 2 )38-484 19-242 inches. Ans. Another exarnple.—Wh^i is the area of a circle, the diameter of which is 3-2 feet? 3-2, multiplied by 5, gives 16. and the area of 16 is 201 'OGig, therefore ;- 5 X 5 =25)201-0619(8-0424+ feet. Ans. 200 106 100 61 50 119 100 Note. — The diameter of a circle, multiplied by 3-14159, will give its cir- cumference,; the square of the diameter, multiplied by -78539, will give its area; and the diameter, multiplied by -88622, will give the side of a square equal to the area of the circle. TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, ETC. The gallon of the State of New York, by an act passed April 11, 1851, is leqiiired to conform to the standard gallon of the United States government. This standard gallon contains 231 cubic inches. In conformity with this standard the following table has been computed. One foot in depth of a cistern of 3 feet diameter will contain 52-872 gallons. 3^^ 4 4i 5 5l 6 6i 7 8 9 10 - 71-965 ■ 93-995 .iiS-963 ,146-868 ,177-710 211 -490 2^8-207 287-861 375-982 .475-852 ■587-472 •845-959 Note. — To reduce cubic feet to gallons, multiply by 7-48. The weight of a gallon of water is 8-355 lbs. To find the contents of a round cistern, multi- ply the square of the diameter by the height, both in feet, and this product by 5-875- 654 APPENDIX. TABLE OF WEIGHTS. MATERIALS USED IN THE CONSTRUCTION OR LOADING OF BUILDINGS. Weights per Cubic Foot. As per Barlow, Gallier, Jlaswell, Hurst, Rankine, Tredgold, Wood and the Author. Material. WOODS. Acacia Alder Apple-tree Ash Beech Birch.... Box.. " French Brazil-wood ' .. Cedar ' 27 " Canadian 47 " Palestine ' 30 " Virginia Red I .. Cherry j 32 Chestnut, Horse . ' 29 " Sweet i 27 Cork ' .. Cypress..... j 27 '' Spanish ! . . Deal, Christiania j . . '* English I . . " (Norway Spruce). ' 21 Dogwood t .. Ebony ' C9 Elder Elm Fir (Norway Spruce) " (Red Pine) 30 " Riga Gum, Blue " Water Hackmatack . Hemlock Hickory Lance-wood Larch " Red /' White ■. Lignum-vitse 41 Locust 4) Logwood Mahogany, Honduras 35 « 1 1 <; H > < 51 46 51 38 51 60 57 49 ss 4G 49 43 6^ 62 83 (i4 35 31 57 52 ^3 34 40 46 39 41 'A^ 55 41 15 41 34 40 44 29 33 27 47 8^ 7G 43 59 46 33 27 44 37 47 63 62 37 3^ 26 58 49 6^ 62 35 33 54 43 23 83 62 51 46 57 40 38 LIaterial. Mahogany, St. Domingo. Maple Mulberry Oak, Adriatic " Black Bog " Canadian " Dantzic " English " Live " Red " White Olive Orange Pear-tree Pine, Georgia (pitch) " Mar Forest " Memel and Riga. .. " Red " Scotch " White " Yellow Plum Poplar Quince Redwood Rosewood Sassafras Satin wood Spruce Sycamore Teak Tulip-tree Vine Walnut, Black "• White Whitewood Yew METALS. Bismuth, Cast Brass, Cast '* (Gun-metal) " Plate Bronze 487 534 524 55 41 45 62 63 54 47 54 68 51 50 58 44 42 48 43 32 37 39 28 33 45 30 44 23 45 30 5 7 30 38 51 30 80 33 49 27 50 614 506 544 531 516 WEIGHT OF MATERIALS. 655 TABLE OF WEIGHTS.— (Conizmted.) MATERIALS USED IN THE CONSTRUCTION OR LOADING OF BUILDINGS. Weights per Cubic Foot. ^s per Barlow, Gallier, Ilaszucll, Ilurst, Rankiiic, TredgolJ, Wood and the Attlhor. Material. > < Copper, Cast 537 549 487 474 486 ... 492 468 449 180 103 304 187 156 ^34 iig 543 550 644 1 1206 1108 509 481 454! 4751 480 709! 7171 713 85l! 849! 837j 488 453 975 1345 1379 142 636 655 658 644 489 462 439 173 156 80i 277 171 139 129 160 102 138 107 100 105 '' Hammered " Plate Gold '^ Standard Gun-metal . Iron, Bar 475 434 "• Cast " Malleable " Wrought 474 Lead Cast *' En-Ush Cast " LliUed . . " " 60" " 212° Nickel Cast Pewter " Pure . " Rolled Silver, Parisian Standard... " " " Hammered Steel 486 4i>6 429 165 Tin, Ca<^t Zinc, Cabt STONES, EARTHS, Etc. Alabaster ... 57 250 155 122 124 '85 Parytes, Sulphate of Basalt .. Bath Stone Beton Coii:;net Blue Stone, Common Brick "- Fire- " N. R. common hard. . . " " Salmon " Philadelphia Front Material. Brick-work "■ dry " in Cement " in Mortar Caen Stone Cement, Portland " Roman, Cast " " and Sand equal parts. . Chalk Clay... " with Gravel Coal, Anthracite "• Bituminous " Cannel " Cumberland Coke Concrete, Cement Coquina Earth, Common " Loamy " with Gravel Emer}"- Feld^^par Flagging, Silver Gray Flinc. ... Glass, Crown " Flint " Green "■ Plate " _ White Granite "■ Aberdeen " Egyptian Red " Guernsey " Quincy Gravel Grindstone Gypsum Lime, Unslaked Limestone " Aubigne " Limerick Marble '• Brocatcl " Cw^rrara 96 95 104 100 112 110 130 81 100 113 145 122 160 96 83 79 85 54 130 106 110 126 126 250 160 185 163 160 183 105 163 174 165 164 166 185 166 105 134 140 52 169 146 162 170 166 170 656 APPENDIX. TABLE OF WEIGHTS.— (Confmued,) MATERIALS USED IN THE CONSTRUCTION OR LOADING OF BUILDINGS. Weights per Cubic Foot. As per Barlow, Gallier, Jlaszaeli, Hiirst, Rankine, Ti'edgold, Wood and the Author. Material. 167 Marble, Eastchester. . . Egyptian " French " Italian Marl Masonry Mica Millstone Mortar *' dry " new '* Hair, incl. Lath and Nails, per foot sup.{ 7 " Hair, dry *' new " Sand 3 and Lime paste 2 .. u 2 " » " 2 well beat together. . Peat, Hard Petrified Wood Pitch Plaster, Cast Porphyry, Green Red Portland Stone . Pumice-stone Puzzolana Quartz, Crystallized Rotten-stone Sand, Coarse *' Common " Dry " Moist " Mortar •• Pit *■ Quartz "• with Gravel Sandstone . . Amherst, O Belleville, N.J... Berea, O " Dorchester, N. S. " Little Falls, N. J. " Marietta, O " Middletown, Ct. . 92 «• 1 < H u <; 178 1-73 167 16« i6q 167 170 140 140 125 175 155 109 98 118 103 107 11 86 105 100 118 83 146 72 80 180 175 161 147 56 165 165 124 112 118 105 120 105 128 123 105 101 »7 172 126 1^8 144 133 142 134 141 ... 134 162 150 Material. Serpentine.. Chester, Pa. " Green Shingle Slate '" Common " Cornwall " Welsh. Stone, Artificial *' Pavmg Stone-work Hewn Rubble Sulphur, Melted Tiles, Common plain.. Trap Rock Tufa, Roman MISCELLANEOUS. Ashes. Wood Bark, Peruvian Butter Camphor Charcoal • Cotton, baled Fat Gunpowder Gutta-percha Hay. baled India Rubber Isinglass Ivory. Plaster of Paris Plumbago Red Lead Resin Rock Crystal Salt Saltpetre Snow Sugar Water, Rain .. '' Sea Whalebone 60 165 144 152 95 159 167 157 180 135 151 140 160 140 124 115 170 76 58 49 59 62 26 20 58 57 61 17 61 69 114 73 131 559 68 171 133 131 14 80 62^ 64 81 INDEX. PAGE Abscissas of Axes, Ellipse 484 Abutments, Bridges, Strength of. 227 Abutments, Houses, Strength of.. 53 Acute Angle Defined 349, 544 Acute-angled Triangle Defined... 545 Acute or Lancet Arch 51 Algebra, Addition 398 Algebra, Application of 393 Algebra, Binomial, Multiplica- tion of 409 Algebra, Binomial, Square of a... 429 Algebra, Binomial, Squaring a. . . 410 Algebra Defined 392 Algebra, Denominator, Least Common 404 Algebra, Division, the Quotient.. 419 Algebra, Division, Reduction. . . . 419 Algebra, Division, Reverse of Multiplication 418 Algebra, Factors, Multiplication of Two and Three 409 Algebra, Factors, P*Iultiplication of Three 408 Algebra, Factors, Squaring Differ- ence of Two 412 Algebra, Fractions Added and Subtracted 403 Algebra, Fractions, Denominators 407 Algebra, Fractions Subtracted... 405 Algebra, Hypothenuse, Equality of Squares on 416 Algebra, Letters, Customary Uses of 396 Algebra, Logarithms Explained.. 425 Algebra, Logarithms, Examples in 426 Algebra, Multiplication, Graphical 408 PAGE Algebra, Progression, Arithmeti- cal 432 Algebra, Progression, Geometrical 435 Algebra, Proportion Essential. . . . 347 Algebra, Proportionals, Lever Formula , .... 421 Algebra, Quantities, Addition and Subtraction 424 Algebra, Quantities, Division of. 424 Algebra, Quantities, Multiplica- tion of 424 Algebra, Quantities with Negative Exponents 423 Algebra, Quantity, Raising to any Power 423 Algebra, Radicals, Extraction of., 425 Algebra, Rules are General 394 Algebra, Rules, Useful Construc- ting 391 Algebra, Signs 397 Algebra, Signs, Arithmetical Pro- cess by. 396 Algebra, Signs, Changed when Subtracted 400 Algebra, Signs, Multiplication of Plus and Minus 415 Algebra, Squares on Right-Angled Triangle 417 Algebra, Subtraction 39S Algebra, Sum and Difference, Pro- duct of 413 Algebra, Symbols Chosen at Pleas- ure 395 Algebra, Symbol, Transferring a.. 399 Algebra, Triangle, Squares on Right-angled 417 558 INDEX. PAGE Alhambra, or Red House, Ancient Palace of the ii Ancient Cities, Historical Ac- counts of 6 Ancient Monuments, their Archi- tects 6 Angle at Circumference of Circle. 358 Angle Defined 544 Angle to Bracket of Cornice, To Obtain 343 Angle, To Measure a. Geometry.. 348 Angle-rib to Polygonal Dome. . . . 223 Angle-rib, Shape of Polygonal Domes 223 Amulet or Fillet, Classic Mould- ing 323 Antfe Cap, Modern Moulding. .. . 334 Antique Columns, Forms of 48 Antiquity of Building 5 Arabian and Moorish Styles, An- tiquities of II Arseostyle, Intercolumniations. . . 20 Arc of Circle Defined 547 Arc of Circle, Length, Rule for. . 475 Arc, Radius of. To Find 561 Arc, Versed Sine, To Find (Geom- etry) 561 Arcade 52 Arcade of Arches, Resistance in.. 52 Arcade in Bridges, Strength of Piers 52 Arch 50 Arch, Acute or Lancet 51 Arch, Archivolt in 52 Arch, Bridge, Pressure on 51 Arch, Building, Manner of 50 Arch, Catenary 51 Arch, Construction of 50 Arch, Definitions and Principles of 52 Arch, Extrados of 52 Arch, Form of 50 Arch, Formation in Bridges 51 Arch, Hooke's Theory of an-. 50 Arch, Horseshoe or Moorish 51 Arch, Impost in. 52 Arch, Intrados of 52 Arch, Keystone, Position of 50 Arch, Lateral Thrust in 52 PAGE Arch, Ogee 51 Arch, Rampant 51 Arch, Span of an 52 Arch, Spring in an 52 Arch, Stone Bridges 230 Arch-stones, Bridges, Jointing. . . 233 Arch, Strength of 50 Arch of Titus, Composite Order. . 28 Arch, Uses of 50 Arch, Voussoir in 52 Architect and Builder, Construc- tion Necessary to 56 Architect, Derivation of the Word 5 Architects of Italy, 14th Century. 12 Architecture, Classic Mouldings in 323 Architecture, Ecclesiastical, Origin of 14 Architecture, Egyptian, Character of 33 Architecture, Egyptian, Features of 30 Architecture, English, Ccttage Style 35 Architecture, English, Early 11 Architecture, Grecian and Roman 8 Architecture, Grecian, Historj- of. 6 Architecture, Hindoo, Character of 30 Architecture, Order, Three Princi- pal parts of 14 Architecture, Principles of 44 Architecture, Roman, Ruins of. . . 11 Architecture in Rome Defined. . . 7 Architecture, Result of Necessity. 13 Architrave Defined 15 Area of Circle, To Find 475 Area of Post, Rule for Finding. . , 90 Area of Round Post, Rule 90 Area of Surface, Sliding Rupture, Rule 88 Arithmetical Progression (Alge- bra) 432 Astragal, or Bead, Classic Mould- ing 323 Athens, Parthenon, Columns of. . 48 Attic, a Small Order, Top of Building 15 INDEX. 659 PAGE Attic Story, Upper Story 15 Axes of Ellipse (Geometry) 585 Axiom Defined (Geometry) 348 Axis Defined 548 Balusters, Handrailing, Winding Stairs 310 Baluster, Platform Stairs, Position of 250 Baluster in Round Rail, Winding Stairs 313 Base, Shaft, and Capital Defined. 14 Bathing, Necessary Arrangements for 45 Baths of Diocletian, Splendor of.. 27 Bead, or Astragal, Classic Mould- ings 323 Beams, Bearings of, Rules for Pressure 75 Beams, Breaking Weight on 74 Beams, Framed, Rules for Thick- ness 130 Beams, Framed, Position of Mor- tise 236 Beams, Headers Defined 130 Beams, Horizontal Thrust, Rules for 72 Beams, Inclined, Effect of Weight on 72 Beams, Load on. Effect of 74 Beams, Splicing 235 Beams, Tail, Defined 130 Beams, Trimmers or Carriage, De- fined 130 Beams, Weight on. Proportion of. 130 Beams, White Pine. Table of Weights 177 Beams, Wooden, Use of Limited. 154 Bearings for Girders 141 Binomials, Multiplication of (Al- gebra) 409 Binomials, Square of (Algebra). . . 429 Binomials, Squaring (Algebra)... 410 Bisect an Angle (Geometry) 554 Bisect a Line (Geometry) 549 Blocking out Rail, Winding Stairs 301 Blondel's Method, Rise and Tread in Stairs 242 rAc.ii: Bottom Rail for Doors, Rule for Width 316 Bow, Mr., On Economics and Construction 166 Bowstring Girder, Cast -Iron, Should not be Used 163 Brace, Length of, To Find (Geom- etry) 579 Braces, Rafters, etc.. To Find Length 5 So Braces in Roof, Rule for. Same as Rafter 208 Breaking Weight Defined 84 Brick or Stone Buildings 37 Brick Walls, Modern 49 Bridge Abutments, Strength of.. . 227 Bridge Arches, Formation of 51 Bridge Arch-stones, Joints of 233 Bridges, Construction of Various. 223 Bridge, London, Age of Piles under 229 Bridge Piers, Construction and Sizes 228 Bridge, Rib-built 224 Bridge, Rib, Construction of. . . . 225 Bridge, Rib, Framed, Construction and Distance 226 Bridge, Rib, Radials of 226 Bridge, Rib, Table of Least Rise in 224 Bridge, Rib, Rule for Area of 225 Bridge, Rib, Rule for Depth of. . . 226 Bridge, Roadway, Width of 227 Bridge, Stone, Arch Construction 230 Bridge, Stone, Arch-stones, Table of Pressures on 230 Bridge, Stone, Arch, Centres for. Bad Construction 229 Bridge, Arch, Spring of 247 Bridge, Stone, Strength of Truss- ing 232 Bridge, Weight, Greatest on 225 Bridge, without Tie-Beam 224 Bridging, Cross-, Additional Strength by 137 Bridging, Cross-, Defined 137 Bridging, Cross-, Resistance by Adjoining Beams. 139 66o INDEX. PAGE Building, Antiquity of 5 Building, Elementary Parts of a.. 46 Building, Expression in a 35 Building by the Greeks 35 Building, Modes of, Defined 9 Building by the Romans 26 Building, Style of. Selected 10 Suit Destination 35 Butt-joint on Handrail to Stairs. . 303 Butt-joint, Handrail, Stairs, Posi- tion of 307 Byzantine St3'le, Lombard 10 Campanile, or Leaning Tower, Twelfth Century 12 Capital, Uppermost Part of a Column 15 Carriage Beam, Well-Hole in Mid- dle, Find Breadth 136 Carriage Beam, One Header, Rule for Breadth 133 Carriage Beam or Trimmer De- fined r30 Carriage Beam, Rule for Breadth. 132 Carriage Beam, Two Sets of Tail Beams, Rule for Breadth 134 Caryatides, Description and Ori- gin of 26 Cast-iron Bowstring Girder, Should not be Used 163 Cast-Iron Girder, Load at Middle, Size of Flanges. .... 162 Cast-Iron Girder, Load Uniform, Size of Flanges 163 Cast-iron Girder, Manner of Mak- ing a 161 Cast-Iron Girder, Proper Form... 161 Cast-iron, Tensile Strength of . . . . 161 Cast-Iron Untrustworthy 161 Catenary Arch, Hooke's Theory of. 51 Cathedral of Cologne ii Cathedrals, Domes of 53 Cathedrals of Pisa, Erection in 1016 12 Cavern, The Original Place of Shelter , 13 Cavetto or Cove, Classic Moulding 323 Cavetto, Grecian Moulding 327 PAGE Cavetto, Roman Moulding 329 Ceiling, Cracking, How to Pre- vent 125 Centre of Circles, To Find (Ge- ometry) 556 Centre of Gravity, Position of ... . 71 Centre of Gravity, Rule for Find- ing, Examples 71 Chimneys, How Arranged 42 Chinese Structure, The Tent the Model of 14 Chord of Circle Defined 547 Chords Giving Equal Rectangles. 363 Circle, Arc, Rule for Length of.. . 475 Circle, Area, Circumference, etc.. Examples 652 Circle, Area, Rule for. Length of Arc Given 478 Circle, Area, To Find 475 Circle, Circumference, To Find . . 473 Circle Defined 546 Circle, Describe within Triangle.. 566 Circle, Diameter and Circumfer- ence 472 Circle, Diameter and Perpendicu- lar 468 Circle Equal Given Circles, To Make 580 Circle, Ordinates, Rule for 471 Circle, Radius from Chord and Versed Sine 469 Circle, Sector, Area of 476 Circle, Segment, Area of. 477 Circle, Segment from Ordinates. . 470 Circle, Segment, Rule for Area of. 479 Circles, Table of 649-652 Circle through Given Points 559 Circular Headed Doors 320 Circular Headed Doors, To Form Soffit 321 Circular Headed Windows 320 Circular Headed Windows, To Form Soffit 321 Circular Stairs, Face Mould for (i). 282 Circular Stairs, Face Mould for (2). 285 Circular Stairs, Face Mould for (3). 287 Circular Stairs, Face Mould, First Section ... 283 INDEX. 66 1 Circular Stairs, Falling Mould for Rail 2S1 Circular Stairs, Handrailing for . . 278 Circular Stairs, Plan of 279 Circular Stairs, Plumb Bevel De- fined 282 Circular Stairs, Timbers Put in after Erection 253 Cisterns, Wells, etc., Table of Capacity of 653 City Houses, General Idea of, . . , . 42 City Houses, Arrangements for.. . 37 Civil Architecture Defined 5 Classic Architecture, Mouldings in 323 Classic Moulding, Annulet or Fil- let 323 Classic Moulding, Astragal or Bead 323 Classic Moulding, Cavetto or Cove 323 Classic Moulding, Cyma-Recta. . . 324 Classic Moulding, Cyma-Reversa. 324 Classic Moulding, Ogee . 324 Classic Moulding, Ovolo 323 Classic Moulding, Scotia 323 Classic Moulding, Torus 323 Coffer Walls 49 Cohesive Strength of Materials. . . 76 Collar Beam in Truss 238 Cologne, Cathedral of 11 Columns, Antique, Form of 48 Column, Base, Shaft and Capital. 14 Columns, Egyptian, Dimensions, etc 33 Column, Gothic Pillar, Form of.. 48 Column, Outline of. . . 47 Columns, Parthenon at Athens, Forms of 48 Column or Pillar 47 Column, Resistance of 47 Column, Shaft, Form of. 47 Column, Shaft, Swell of, Called Entasis 48 Complex, or Ground Vault 52 Composite Arch of Titus 28 Composite, Corinthian or Roman Order 28 PAGE Compression, Resistance to 77 Compression, Resistance to Crush- ing and Bending 85 Compression, Resistance to, Pres- sures Classified 83 Compression, Resistance to, Table of 79 Compression, Resistance to, in Proportion to Depth loi Compression at Right Angles and Parallel to Length 206 Compression of Stout Posts 89 Compression and Tension, Framed Girders 174 Compression Transversely to Fi- bres SO Cone Defined 548 Conic Sections 584 Conjugate Axis Defined 548 Conjugate Diameters to Axes of Ellipse 487 Construction Essential 56 Construction of Floors, Roof, etc.. Economy Important 123 Construction, Framing,' Heav}'^ Weight 56 Construction, Joints, Effect of Many 123 Construction, Object of Defined. . 123 Construction, Simplest Form Best. 123 Construction, Superfluous Mate- rial 56 Contents, Table of, General. . .613-624 Corinthian Capital, Fanciful Ori- gin of. 24 Corinthian Order Appropriate in Buildings 24 Corinthian Order, Character of.... 16 Corinthian Order, Description of. 23 Corinthian Order, Elegance of. . . . 23 Corinthian Order, The Favorite at Rome 27 Corinthian Order, Grecian Origin of 16 Corinthian Order, Modification of. 27 Cornice, Angle Bracket, To Ob- tain the 343 Cornice, Eaves, To Find Depth of. 335 662 INDEX. PAGE Cornice, Mouldings, Depth of. . . . 342 Cornice, Projection, To Find 342 Cornice, Projecting Part of En- tablature 15 Cornice, Rake and Level Mould- ings, To Match 344 Cornice, Shading, Rule for 611 Cornice, Stucco, for Interior, De- signs 340 Corollary Defined (Geometry) 348 Corollary of Triangle and Right Angle 355 Cottage Style, English 35 Country-Seat, Style of a 37 Cross-Bridging, Additional Strength by , 137 Cross-Bridging, Furring Impor- tant 137 Cross-Bridging, Resistance of Ad- joining Beams 139 Cross- or Herring-Bone Bridging Defined 137 Cross-Furring Defined 125 Cross-Strains, Resistance to. . . .77, 99 Crushing and Bending Pressure.. 85 Crushing, Liability of Rafter to. . . 205 Crushing Strength of Stout Posts. 89 Cube Root, Examples in 645 Cubes, Squares and Roots, Table of 638-645 Cubic Feet to Gallons, To Reduce. 653 Cupola or Dome 53 Curb or Mansard Roof 54 Curve Ellipse, Equations to 482 Curve Equilibrium of Dome 218 Cylinder, Denned 549 Cylinder, Platform Stairs 24S Cylinder, Platform Stairs, Lower Edge of 249 Cylinders and Prisms, Stair-Build- ing 257 Cyma-Recta, Classic Moulding... 324 Cyma-Reversa, Classic Moulding. 324 Cyma-Recta, Grecian Moulding . . 327 Cyma-Reversa, Grecian Moulding. 328 Deafening, Weight per Foot 177 Decagon, Defined 546 PAGE Decimals, Reduction of. Examples 647 Decorated Style, 14th Century 11 Decoration, Attention to be given to 46 Decoration, Roman 27 Deflection, Defined 112 Deflection, Differs in Different Ma- terials 113 Deflection, Elasticity not Dimin- ished by 112 Deflection, Floor-beams, Dwell- ings, Dimensions 127 Deflection, Floor-beams, First- class Stores, Dimensions 128 Deflection, Floor-beams, Ordinary Stores, Dimensions 127 Deflection, Lever, Principle of. . . 119 Deflection, Lever and Beam, Rela- tion Between 119 Deflection, Lever, To Find, Load at End 120 Deflection, Lever, Breadth or Depth, Load at End 121 Deflection, Lever, Load Uniform. 121 Deflection, Lever, Breadth or Depth, Load Uniform 122 Deflection, Lever, for Certain, Load Uniform 122 Deflection, Load Uniform or at Middle, Proportion of 116 Deflection, Load Uniform, Breadth and Depth 117 Deflection, Load Uniform or at Middle, Proportion of 119 Deflection, in Proportion to Weight 112 Deflection, Resistance to. Rule for 113 Deflection, Safe Weight for Pre- vention... no Deflection, Weight at Middle, Breadth and Depth 114 Deflection, Weight at Middle, for Certain 114 Deflection, Weight at Middle, Cer- tain, for ii5 Deflection, Weight Uniform, for Certain 117 INDEX. 663 PAGIt Deflection, Weight Uniform, Cer- tain, for 118 Denominator, Least Common (Al- gebra) 404 Denominator, Least Common (Fractions) 384 Dentils, Teeth-like Mouldings in Cornice 20 Diagonal Crossing Parallelogram (Geometry) 35 1 Diagonal of Square Forming Oc- tagon 357 Diagram of Forces, Example 166 Diameter, Circle, Defined 547 Diameter, Ellipse, Defined 549 Diast)'le, Explanation of the Word 19 Diastyle, Intercolumniation 20 Diocletian, Baths, Splendor of 27 Division, Fractions. Rule for 3S9 Division, by Factors (Fractions).. 381 Division, Quotient (Algebra) 419 Division, Reduction (Algebra).,. 419 Dodecagon, Defined 546 Dodecagon, To Inscribe 569 Dodecagon, Radius of Circles (Polygons) 452 Dodecagon, Side and Area (Poly- gons) 453 Dome, Abutments, Strength of. . . 53 Domes of Cathedrals 53 Dome, Character of 53 Dome, Construction and Form of 216 Dome, Construction and Strength of 53 Dome, Cubic Parabola computed 219 Dome or Cupola, the 53 Dome, Curve of Equilibrium, rule for 218 Dome, Halle du Bled, Paris 54 Dome, Pantheon at Rome 53 Dome, Pendentives of 53 Dome, Polygonal, Shape of An- gle Rib 223 Dome, Ribbed, Form and Con- struction 217 Dome, Scantling for. Table of Thickness 218 PAGE Dome, Small, over Stairways, Form of 220 Dome, Spherical, To Form...... 221 Dome, St. Paul's, London ..... 54 Dome, Strains on. Tendencies of. 219 Domes, Wooden 54 Doors, Circular Head 320 Doors, Circular Head, to Form Soffit 321 Doors, Construction of 317 Doors, Folding and Sliding, Pro- portions 316 Doors, Front, Location of 320 Doors, Height, Rule for. Width Given 315 Door Hanging, Manner of 317 Doors, Panel, Bottom and Lock Rail, Width 316 Doors, Panel, Four Necessary... 317 Doors, Panel, Mouldings, Width. 317 Doors, Panel, Styles and Muntins, Width 316 Doors, Panel, Top Rail, Width. .. 317 Doors, Stop for. How to Form... 317 Doors, Single and Double, height of 316 Doors, Trimmings Explained. .. . 317 Doors, Uses and Requirements of 315 Doors, Width of 315 Doors, Should not be Winding. .. 317 Doors, Width and Height, Propor- tion of 315 Doors, Width, Rule for, Height Given 316 Doric Order, Character of 16 Doric Order, Grecian Origin of. . 16 Doric Order, Modified by the Ro- mans 27 Doric Order, Used by Greeks only at First 19 Doric Order, Peculiarities of 17 Doric Order, Rudeness of 30 Doric Order, Specimen Buildings in 19 Doric Temples, Fanciful Origin of 17 Doric Temples 19 Drawing, Articles Required 536 664 INDEX. PAGE Drawing-board Better without Clamps 537 Drawing-board Liable to Warp, How Remedied 537 Drawing-board, Difficulty in Stretching Paper 539 Drawing-board, Ordinary Size.... 536 Drawing, Diagrams aid Under- standing 536 Drawing, Inking in 542 Drawing, Laying Out the 541 Drawing, the Paper 537 Drawing in Pencil, To Make Lines 542 Drawing, Secure Paper to Board. 537 Drawing, Shade Lining 543 Drawing, Stretching Paper 537 Durability in a Building. 37 Dwelling, Arrangement of Rooms 38 Dwellings, Floor-beams, To Find Dimensions 127 Dwellings, Floor-beams, Safe Weight for 126 Dwelling-houses, Dimensions and Style 37 Eaves Cornice, Designs for 335 Eaves Cornice, Rule for Depth... 335 Ecclesiastical Architecture, Point- ed Style II Ecclesiastical Style, Origin of. . . . 14 Echinus, Grecian Moulding 327 Economy, Construction Floors, Roofs, Bridges 123 Eddystone and Bell Rock Light House 48 Egyptian Architecture 30 Egyptian Architecture, Appropri- ate Buildings for 33 Egyptian Architecture, Character of 33 Egyptian Architecture, Origin in Caverns , 14 Egyptian Architecture. Principal Features of 30 Egyptian Columns, Dimensions and Proportions 33 Egyptian Walls, Massiveness of. . 33 Egyptian Works of Art 30 PAGE Elasticity of Materials 84 Elasticity not Diminished by De- flection 112 Elasticity, Result of Exceeding Limit 120 Elevation, a Front View 37 Elevated Tie-beam Roof Truss Objectionable 214 Ellipse, Area 48S Ellipse, Axes, Two, To Find, Di- ameter and Conjugate Given. .. 593 Ellipse Defined 481 Ellipse, Equations to the Curve. . 482 Ellipse, Major and Minor Axes Defined 481 Ellipse, Ordinates, Length of . . . 491 Ellipse, Parameter and Axis, Re- lation of o 485 Ellipse, Practical Suggestions. ... 489 Ellipse, Semi-major, Axis Defined 486 Ellipse, Subtangent Defined 486 Ellipse, Tangent to Axes, Rela- tion of 485 Ellipse, Tangent with Foci, Rela- tion of 487 Ellipsis, Axes of, To Find (Geom- etry) 585 Ellipsis, Conjugate Diameters (Ge- ometry) 593 Ellipsis Defined 548, 585 Ellipsis, Diameter Defined 549 Ellipsis, Foci, To Find 586 Ellipsis, by Intersecting Arcs 590 Ellipsis, by Intersecting Lines... 588 Ellipsis, by Ordinates 588 Ellipsis, Point of Contact with Tangent, To Find 593 Ellipsis, Proportionate Axes, to Describe with 594 Ellipsis, Trammel, to Find, Axes Given 586 Elliptical Arch, Joints, Direction of 233 English Architecture, Early 11 English Cottage Style Extensive- ly Used 35 England and France, Fourteenth Century I2 INDEX. 66- PAGE j PAGE Entablature, above Columns and Flanges, Area of, Tubular Iron Horizontal 14 Entasis, Swell of Shaft of Column 48 Equal Angles Defined 349 Equal Angles, Example in 350 Equal Angles, in Circle 358 Girder 155 Flanges, Area of Bottom, Tubular Iron Girder 159 Flanges, Load at Middle of Cast- iron Girder, Sizes 162 Equal Angles (Geometry) 553 Flanges, Load Uniform on Tubu- Equilateral Rectangle, to De- ! lar Iron Girder, Sizes 156 scribe 56S ! Flanges, Proportion of, Tubular Equilateral Triangle Defined (Ge- ] Iron Girder. ,. ." 157 ometry) 545 Flexure, Compared with Rup- Equilateral Triangle, to Construct ture. . . 84 (Geometry) 568 Flexureof Rafter 205 Equilateral Triangle, to Describe j Flexure, Resistance to. Defined. . 145 (Geometry) 566 j Floor-arches, How Constructed. . 153 Eqilateral Triangle, to Inscribe i Floor-arches, Tie-rods, Dwellings, (Geometry) 5691 Sizes 153 Equilateral Triangle (Polygons).. 445 Eustyle Defined 20 Exponents, Quantities with Nega- tive (Algebra) 423 Extrados of an Arch 52 Face Mould, Accuracy of. Wind- ing Stairs 295 Face Mould, Curves Elliptical, Winding Stairs 301 Face Mould, Drawing of, Winding Stairs 296 Face Mould, Sliding of. Winding Stairs 299 Face Mould, Application of, Plat- form Stairs 275 Face Mould, a Simple, Kell's Method for 268 Factors, Multiplication (Algebra) 409 Factors, Two, Squaring Difference of (Algebra) 412 Fibrous Structure of Materials. . . 76 Figure Equal, Given Figure (Ge- Floor-arches, Tie-rods, First-class Stores, Sizes 153 Floor-beams, Distance from Cen- tres, Sizes Fixed 129 Floor-beams, Dwellings, Safe Weight for 126 Floor-beams, Dwellings, Deflec- tion Given, Sizes 127 Floor-beams, First-class Stores, Deflection Given, Sizes 128 Floor-beams, Ordinary Stores, De- flection Given, Sizes 127 Floor-beams, Stores, Safe Weight for 126 Floor-beams, Reference to Rules for Sizes 125 Floor-beams, Reference to Trans- verse Strains 126 Floor-beams, Proportion of Weight on All 130 Floors Constructed, Single or Double 124 Floors, Fire-proof Iron, Action of ometry) 575 j Fire on 143 Figure, Nearly Elliptical, To Make Floors, Framed, Seldom Used. . . 124 (Geometry) 591 Fillet or Amulet, Classic Mould- Floors, Framed, Openings in 130 Floors, Headers, Defined 130 ing. 323 Floors, Ordinary, Efl^ect of Fire Fire-proof Floors, Action of Fire on 143 on 143 1 Floors, Solid Timber, Dwellings Flanges, Cast-iron Girder 1631 and Assembly, Depth 143 666 INDEX. PAGE Floors, Solid Timber, First-class Stores, Depth 144 Floors, Solid Timber, to Make Fire-proof 143 Floors, Tail-beams Defined 130 Floors, Trimmers or Carriage- beams Defined 130 Floors, Wooden, More Fire-proof than Iron, Some Cases 143 Fl)'ers and Winders, Winding Stairs 251 Foci Defined 548 Foci of Ellipsis To Find 586 Foci of Ellipse, Tangent 487 Force Diagram, Load on Each Support 179 Force Diagram, Truss, Figs. 59, 68 and 69 179 Force Diagram, Truss, Figs. 60, 70 and 71 180 Force Diagram, Truss, Figs. 61, 72 and 73 181 Force Diagram Truss, Figs. 63 74 and 75 183 Force Diagram, Truss, Figs. 64, 77 and 78 184 Force Diagram, Truss, Figs. 65, 78 and 79 185 Force Diagram, Truss, Figs. 66, 80 and 81 186 Forces, Parallelogram cf 59 Forces, Composition of 66 Forces, Composition, Reverse of Resolution 67 Forces, Resolution of 59 Forces, Resolution of. Oblique Pressure 59 Foundations, Description of . . . . 47 Foundations in Marshes, Timbers Used 47 Fractions, Addition, Like Denom- inators 382 Fractions Added and Subtracted (Algebra) 403 Fractions Changed by Division.. 380 Fractions Defined 378 Fractions, Division, Rule for 389 Fractions, Division by Factors. .. 381 PAGE Fractions Divided Graphically... 388 Fractions Graphically Expressed. 378 Fractions, Improper, Defined.... 380 Fractions, Least Common Denom- inator 384 Fractions, Multiplication, Rule. . . 387 Fractions Multiplied Graphically. 386 Fractions, Numerator and Denom- inator 378 Fractions, Reduce Mixed Num- bers.- 381 Fractions, Reduction to Lowest Terms 384 Fractions Subtracted (Algebra). . . 405 Fractions, Subtraction Like De- nominators 383 Fractions, Unlike Denominators Equalized 383 Framed Beams, Thickness of, Rules 130 Framed Girder, Bays Defined.... 167 Framed Girders, Compression and Tension, Dimensions 174 Framed Girders, Construction and Uses 166 Framed Girders, Height and Depth 167 Framed Girders, Kinds of Pres- sure 173 Framed Girders, Long, Construc- tion of 174 Framed Girders, Panels on Under Chord, Table of 167 Framed Girders, Ties and Struts, Effect of 174 Framed Girders, Triangular Pres- sure, Upper Chord 168 Framed Girders, Triangular Pres- sure, Both Chords 171 Framed Openings in Floors 130 Framing Beams, Effect of Splic- ing 235 Framing Roof Truss 237 Framing Roof Truss, Iron Straps, Size of o 239 France and England, Fourteenth Century 12 Friction, Effect of 82 INDEX. 667 PAGE 1 Frieze between Architrave and Cornice Furring Defined Gable, a Pediment in Gothic Ar- chitecture Gaining a Beam Defined General Contents, Table of. . . .613- Geometrical Progression (Alge- bra) Geometry, Angles of Triangle, Three, Equal Right Angle Geometry Chords Giving Equal Rectangles Geometry Defined Geometry, Divide a Given Line. . Geometry, Divisions in Line Pro- portionate Geometry, Elementary Geometry, Equal Angles Geometry, Equal Angles, Ex- ample Geometry, Figure Equal to Given Figure, Construct Geometry, Figure Nearly Ellipti- cal by Compasses Geometry, Measure an Angle. . . . Geometry Necessary in Handrail- ing. Stairs Geometry, Opposite Angles Equal. Geometry, Parallel Lines Geometry, a Perpendicular, To Erect Geometry,Perpendicular,let Fall a. Geometry, Perpendicular, Erect at End of Line Geometry, Perpendicular, Let Fall Near End of Line. . . , , Geometr}', Plane Defined (Stairs). Geometry, Point of Contact Geometry, Points, Three Given, Find Fourth Geometry, Right Line Equal Cir- cumference Geometry, Right Lines, Propor- tion Between Geometry, Right Lines, Two Given, Find Third 15 125 15 100 ■624 435 354 363 544 583 347 553 350 575 591 34S 257 354 555 550 551 551 553 257 558 559 566 584 582 Geometry, Square Equal Rec- tangle, To Make 5S1 Geometry, Square Equal Given Squares, To make 577 Geometry, Square Equal Triangle, To Make 582 German or Romantic Style, Thir- teenth and Fourteenth Centuries. 11 Girder, Bearings, Space Allowed for 141 Girder, Bow-String, Cast-iron, Should not be Used 163 Girder, Bow-String, Substitute for. 163 Girder, Construction with Long Bearings 140 555 i Girder, Cast-Iron, Load Uniform, Flanges 163 Girder, Cast-Iron, Load at Middle, Flanges 162 Girder, Cast-Iron, Proper Form of iCi Girder Defined, Position and Use of 140 Girder, Different Supports for 140 Girder, Dwellings, Sizes for 141 Girder, Framed, Bays Defined... . 167 Girders, Framed, Compression and Tension, Dimensions 174 Girder, Framed, Construction of.. 140 Girder, Framed, Construction and Uses 166 Girder, Framed, Construction of Long 174 Girder, Framed, Kinds of Pres- sure 173 Girders, Framed, Height and Depth 167 Girders, Framed, Panels on Under Chord, Table of 1C7 Girders, Framed, Triangular Pres- sure Upper Chord lOS Girders, Framed, Triangular Pres- sure Both Chords 171 Girders, Framed, and Tubular Iron 140 Girders, First-Class Stores, Sizes for 141 Girders, Sizes, To Obtain 141 668 INDEX. PAGE Girders, Strengthening, Manner of 140 Girders, Supports, Length of. Rule. 157 Girders, Tubular Iron, Construc- tion of 154 Girders, Tubular Iron, Area of Flange, Load at Middle 154 Girders, Tubular Iron, Area of Flange, Load at any Point 155 Girders, Tubular Iron, Area of Flange, Load Uniform 156 Girders, Tubular Iron, Dwellings, Area of Bottom Flange 159 Girders, Tubular Iron, First-Class Stores, Area of Bottom Flange.. 160 Girders, Tubular Iron, Rivets, Al- lowance for 157 Girders, Tubular Iron, Flanges, Proportion of 157 Girders, Tubular Iron, Shearing Strain I57 Girders, Tubular Iron.Web, Thick- ness of 158 Girders, Weakening, Manner of. . 140 Girders, Wooden, Objectionable. 154 Girders, Wooden, Supporting, Manner of 154 Glossary of Terms 627-637 Gothic Arches 51 Gothic Buildings, Roofs of 55 Gothic and Norman Roofs, Con- struction of 178 Gothic Pillar, Form of 48 Gothic Style, Characteristics of. . . 12 Goths, Ruins Caused by 12 Granular Structure of Materials. . 76 Gravity, Centre of. Position 71 Gravity, Centre of. Examples, and Rule for 71 Grecian Architecture, History of. 6 Grecian Art, Elegance of 27 Grecian Moulding, Cyma-Recta. . 327 Grecian Moulding, Cyma-Re- versa 328 Grecian Moulding, Echinus and Cavelto 327 Grecian Moulding, Scotia 326 Grecian Moulding, Torus 326 PACE Grecian Orders' Modified by the Romans 27 Grecian Origin of the Doric Or der t6 Grecian Origin of Ionic Order ... 16 Grecian Style in America 13 Grecian Styles, their Different Orders j5 Greek Architecture, Doric Order Used iQ Greek Building 35 Greek Moulding, Form of 325 Greek, Persian, and Caryatides Orders 24 Greek Style Originally in Wood.. 14 Greek Styles Only Known by Them 16 Groined or Complex Vault 52 Halle du Bled, Paris, Dome of, . . 54 Halls of Justice, N. Y. C, Speci- men of Egyptian Architecture. . 8 Handrailing, Circular Stairs 278 Handrailing, Platform Stairs. ... 269 Handrailing, Platform Stairs, Face Mould 264 Handrailing, Platform Stairs, Large Cylinder 271 Handrailing Stairs, Geometry Necessary 257 Handrailing Stairs, " Out of Wind" Defined 257 Handrailing Stairs, Tools Used. . 257 Handrailing, Winding Stairs. .256, 289 Handrailing Winding Stairs, Bal- usters Under Scroll 310 Handrailing, Winding Stairs, Centres in Square 308 Handrailing, Winding Stairs, Face for Scroll 311 Handrailing, Winding Stairs, Fall- ing Mould 310 Handrailing, Winding Stairs, Gen- eral Considerations 258 Handrailing, Winding Stairs, Scroll for 308 Handrailing, Winding Stairs, Scroll at Newel 309 INDEX. 669 PACE Handrailing, Winding Stairs, Scroll Over Curtail Step 309 Handrailing, Winding Stairs, Scroll for Curtail Step 310 Headers, Breadth of 130 Headers Defined 130 Headers, Mortises, Allowance for Weakening by 131 Headers, Stores and Dwellings, Same for Both 132 Hecadecagon, Complete Square (Polygons) 458 Hecadecagon, Radius of Circles (Polygons) 455 Hecadecagon, Rules (Polygons). . 459 Hecadecagon, Side and Area (Polygons) 457 Height and Projection, Numbers of an Order 16 Hemlock, Weight per Foot Super- ficial 177 Heptagon Defined 546 Herring-bone Bridging Defined... 137 Hexagon Defined 546 Hexagon, To Inscribe 569 Hexagons, Radius of Circles 447 Hexastyle, Intercolumniation. . . . 20 Hindoo Architecture, Ancient, Character of 30 Hip-Rafter, Backing of 216 Hip-Roofs, Diagram and Expla- nation 215 History of Architecture. , 44 Hogged Ridge in Roof Truss. . . . 238 Homologous Triangles (Geom- etry) 362 Homologous Triangles (Ratio and Proportion) 370 Hooke's Theory of an Arch 50 Hooke's Theory, Bridge Arch, Pressure on 51 Hooke's Theory, Catenary Arch. . 51 Horizontal and Inclined Roofing, Weight igo Horizontal Pressure on Roof, To Remove 74 Horizontal Thrust in Beams 72 Horizontal Thrust, Tendency of. . 88 1>AGE Hut, Original Habitation 13 Hydraulic Method, Testing Woods 80 Hyperbola Defined 548, 585 Hyperbola, Height, To Find, Base and Axis Given 585 Hyperbola by Intersecting Lines. 595 Hypothenuse, Equality of Squares (Algebra) 416 Hypothenuse, Formula for (Trig- onometry) 516 Hypothenuse, Side, To Find (Ge- ometr)^) 579 Hypothenuse, Right Angled Tri- angle (Geometr}') 355 Hypothenuse, Triangle (Trigo- nometry) 518 Ichnographic Projection, Ground Plan 37 Improper Fractions Defined 380 India Ink in Drawing 540 Inertia, Moment of. Defined 145 Inking-in Drawing 542 Inside Shutters for Windows, Re- quirements 319 Instruments in Drawing 540 Intercolumniation Defined 17 Intercolumniation of Orders 20 Intrados of Arch 52 Ionic Order, Character of. 16 Ionic Order, Grecian Origin of. . . 16 Ionic Order Modified by the Ro- mans 27 Ionic Order, Origin of 20 Ionic Order, Suitable for What Buildings 20 Ionic Volute, To Describe an. ... 20 Iron Beams, Breaking Weight at Middle 148 Iron Beams, Deflection, To Find, Weight at Middle 147 Iron Beams, Deflection, To Find, Weight Uniform 150 Iron Beams, Dimensions, To Find, Weight any Point 149 Iron Beams, Dimensions, To Find, Weight Uniform 149 670 indp:x. PAGE Iron Beams, Dwellings, Distance from Centres 151 Iron Beams, First-Class Stores, Distance from Centres 152 Iron Beams, Rectangular Cross- Section , , 145 Iron Beams, Rolled, Sizes 145 Iron Beams, Safe Weight, Load any Point 148 Iron Beams, Safe Weight, Load Uniform. .' 151 Iron Beams, Table IV 146 Iron Beams, Weight at Middle, Deflection Given 146 Iron Fire-Proof Floors, Action of Fire On 143 Iron Straps, Framing, to Prevent Rusting 239 Irregular Polygon, Trigon (Geom- etry) 546 Isosceles Triangle Defined 545, 584 Italian Architecture, Thirteenth, Fourteenth, and Fifteenth Cen- turies 12 Italian Use of Roman Styles 13 Italy, Tuscan Order the Principal Style 30 Jack-Rafters, Location of 212 Jack-Rafters and Purlins in Roof. 211 Jack-Rafters, Weight per Superfi- cial Foot 189 Joists and Studs Defined 174 Jupiter, Temple of, at Thebes, Ex- tent of 33 Kell's Method, Simple Face Mould, Stairs 268 Keystone for Arch, Position of . . . . 50 King-Post, Bad Framing, Effect of 237 King-Post, Location of 213 King-Post in Roof. 54 Lamina in Girders Defined 174 Lancet Arch 51 Lateral Thrust in Arch 52 PAGE Laws of Pressure 57 Laws of Pressure, Inclined, Ex- amples 57 Laws of Pressure, Vertical, Exam- ples 57 Leaning Tower or Campanile, Twelfth Century 12 Length, Breadth, or Thickness, Relation to Pressure 78 Lever, Breadth or Depth, To Find Ill Lever, Deflection as Relating to Beam 119 Lever, Deflection, Load at End.. 120 Lever, Deflection, Load Uniform. 121 Lever, Deflection, Breadth or Depth, Load at End 121 Lever, Deflection, Breadth or Depth, Load Uniform 122 Lever, Deflection, Load Required. 122 Lever Formula, Proportionals in (Algebra) 421 Lever Load Uniformly Distrib- uted Ill Lever, Load at One End no Lever Principle Demonstrated (Ratio) 375 Lever, Support, Relative Strength of One no Light-Houses, Eddystone and Bell Rock 48 Line Defined (Geometry) 544 Lines, Divisions in. Proportionate (Geometry) 583 Lintel, Position of 49 Lintel, Strength of 49 Load, per foot, Horizontal 192 Load on Roof Truss, per Superfi- cial Foot c 189 Load on Tie-Beam, Ceiling, etc. . 190 Lock Rail for Doors, Width 316 Logarithms Explained (Algebra).. 425 Logarithms, Examples 426 Logarithms, Sine and Tangents (Polygons) 464 Lombard, Byzantine Style 10 Lombard Style, Seventh Century. 10 London Bridge, Piles, Age of. . . . 229 INDEX. 671 J'AGE Materials, Cohesive Strength of. . 76 Materials, Compression, Resist- ance to 77 Materials, Cross-strain, Resistance to 77 Materials, Structure of 76 Materials, Tension, Resistance to. 77 Materials Tested, General De- scription 80 Materials, Weights, Table of. . . . 654 Major and Minor Axes of Ellipse Denned 4S1 Marshes, Foundation for Timbers in 47 Mathematics Essential 347 Maxwell, Prof. I. Clerk, Diagrams of Forces, etc. , 165 Memphis, Pyramids of, Estimate of Stone in 33 Minster, Tower of Strassburg. ... 11 Minutes, Sixty Equal Parts, to Proportion an Order 15 Mixed Numbers in Fractions, To Reduce 381 Modern Architecture, First Ap- pearance of 9 Modern Tuscan, Appropriate for Buildings 30 Moment of Inertia Defined 145 Mono-triglyph, Explanation of the Word 19 Monuments, Ancient, Their Archi- tects 6 Moorish and Arabian Styles, An- tiquities of II Mortises, Proper Location of. • . . 100 Mortising, Beam, Effect on Strength of 100 Mortising Beam at Top, Injurious Effect of 100 Mortising Beam, Effect of 231 Mortising, Beam, Position of. . . . 236 Mortising Headers, Allowance for Weakening 131 Moulding, Classic, Astragal or Bead 323 Moulding, Classic, Annulet or Fillet 323 I PAGE Mouldings, Classic Architecture. 323 Moulding, Classic, Cavctto or Cove 323 Moulding, Classic, Cyma-Recta. . 324 Moulding, Classic, C)'ma-Revcrsa. 324 Moulding, Classic, Ogee 324 Moulding, Classic, Ovolo 323 oNIoulding, Classic, Scotia 323 ^loulding, Classic, Torus 323 Mouldings, Common to all Or- ders 324 Mouldings Defined 323 Mouldings, Diagrams of. 330 Mouldings, Doors, Rule for Width. 317 Moulding; Grecian, Cyma-Rccta. 327 Moulding, Grecian, Cyma-Re- versa 328 Moulding, Grecian Echinus and Cavetto 327 Mouldings, Greek, Form of. 325 Mouldings, Grecian Torus and Scotia 326 Mouldings, Modern 331 Moulding, Modern, Antac Cap... 334 Mouldings, Modern Interior, Dia- grams 332 Mouldings, Modern, Plain 333 Mouldings,Names, Derivations of. 324 Mouldings,' Profile Defined 326 Mouldings, Roman, Forms of. . . . 325 Mouldings, Roman, Comments on. 329 Mouldings, Roman, Ovolo and Cavetto 329 Mouldings,Uses and Positions of. 324 Multiplication (Algebra) 408 Multiplication, Plus and Minus (Algebra) 415 Multiplication, Three Factors (Al- gebra) 408 Multiplication, Fractions 387 Newel Cap, Form of. Winding Stairs 312 Nicholson's Method, Plane Through Cylinder (Stairs) 259 Nicholson's Method, Twists in Stairs 259 Nonagon Defined 546 6j: INDEX. PAGE Normal and Subnormal in Para- bola 496 Norman and Gothic Construction of Roofs 17S Norman Style, Peculiarities of . . . 11 Nosing and Tread, Position in Stairs 241 Oblique Angle Defined 544 Oblique Pressure, Resolution of Forces 59 Oblique Triangle, Difference Two Angles (Trigonometr}^ 523 Oblique Triangle, First Class (Trigonometry) 520 Oblique Triangles, First Class, Formulae (Trigonometry) 531 Oblique Triangles, Second Class (Trigonometr)') 522 Oblique Triangles, Second Class, Formulae (Trigonometry) 532 Oblique Triangles, Third Class (Trigonometry) 526 Oblique Triangles, Third Class, Formulae (Trigonometry) 534 Oblique Triangles, Fourth Class (Trigonometry) 528 Oblique Triangles, Fourth Class, Formulae (Trigonometry) 534 Oblique Triangles, Two Sides (Trigonometry) 521 Oblique Triangles, Sines and Sides (Trigonometry) 519 Obtuse Angle Defined 349, 544 Obtuse Angled Triangle Defined. 545 Octagon, Buttressed, Find Side (Geometry) 571 Octagon Defined 546 Octagon, Diagonal of Square Forming 357 Octagon, Inscribe a (Geometry). . 570 Octagon, Rules (Polygons) 451 Octagon, Radius of Circles (Poly- gons) 449 Octastyle, Intercolumniation 20 Ogee Mouldings, Classic 324 Opposite Angles Equal (Geome- try) 354 Order of Architecture, Three Principal Parts 14 Orders of Architecture, Persians and Caryatides 24 Ordinates to an Arc (Geometry). . 563 Ordinates, Circle, Rule for 471 Ordinates of Ellipse 491 Ostrogoths, Style of the 9 Oval, To Describe a (Geometry). . 591 Ovolo, Classic Moulding 323 Ovolo, Roman Moulding 329 Paper, The, in Drawing, Secure to Board 537 Pantheon at Rome, Dome of, and Walls. 53 Pantheon and Roman Buildings, Walls of 49 Parabola, Arcs Described from... 503 Parabola, Area, Rule for 509 Parabola, Axis and Base, to find (Geometr)') 585 Parabola, Curve, Equations to... . 493 Parabola Defined 492 Parabola Defined (Geometry).. 548, 585 Parabola, Diameters 497 Parabola Described from Ordi- nates 504 Parabola Described from Diame- ters 507 Parabola Described from Points.. 502 Parabola of Dome Computed. . . . 219 Parabola, General Rules 499 Parabola by Intersecting Lines. . . 594 Parabola Mechanically Described. 500 Parabola, Normal and Subnor- mal 496 Parabola, Ordinate Defined. .... 496 Parabola, Subtangent. 496 Parabola, Tangent 493 Parabola, Vertical Tangent De- fined 495 Parabolic Arch, Direction of Joints 234 Parallel Lines Defined 544 Parallel Lines (Geometry) 555 Parallelogram, Construct a 576 Parallelogram Defined 545 INDEX. 67. PAGE Parallelogram Equal to Triangles, To Make 576 Parallelogram of Forces, Strains by 165 Parallelograms Proportioned to Bases (Geometry) 360 Parallelogram in Quadrangle (Geometry) 364 Parallelogram, Same Base (Geom- etry) 352 Parameter Defined 54S Parameter, Axes (Ellipse) 485 Parthenon at Athens, Columns of. 48 Partitions, Bracing and Trussing. 176 Partitions, How Constructed 174 Partition, Door in Middle, Con- struction 175 Partition, Doors at End, Construc- tion of 176 Partition, Great Strength, Con- struction 176 Partitions, Location and Connec- tion 175 Partitions, Materials, Quality of. . 175 Partitions, Plastered, Proper Sup- ports for 175 Partitions, Pressure on. Rules.... 177 Partitions, Principal, of what Com- posed 175 Partitions, Trussing in. Effects of. 175 Pedestal, a Separate Substruc- ture 14 Pediment, Triangular End of Building 15 Pencil and Rulers, Drawing 540 Pentagon Defined 546 Pentagon, Circumscribed Circles (Polygons) 463 Perpendicular Height of Roof, To find 579 Perpendicular, Erect a 550 Perpendicular, Erect a, at End of Line 551 Perpendicular, Let Fall a 551 Perpendicular, Let Fall a, at End of Line 553 Perpendicular Style, Fifteenth Century 12 Perpendicular in Triangle (Poly- gons) 440 Persians, Origin and Description of 24 Persians and Caryatides, Orders Used by Greeks 24 Piers, Arrangement, in City Front of House 44 Piers, Bridges, Construction and Sizes 228 Piles, London Bridge, Age of. . . . 229 Pine, White, Beams, Table of Weights for 177 Pisa, Cathedral of. Eleventh Cen- tury 12 Pisa, Cathedral of, Erection in 1016 12 Pise Wall of France 49 Pitch Board, To Make, for Stairs. 247 Pitch Board, Winding Stairs 252 Plane Defined 257 Plane Defined (Geometry) 544 Plank,Weightof,on Roof, per foot. 189 Plastering, Defective, To what Due 174 Plastering, Strength of 174 Plastering, Weight per foot 177 Platform Stairs, Baluster, Posi- tion of 250 Platform Stairs Beneficial 240 Platform Stairs, Cylinder of. 248 Platform Stairs, Cylinder, Lower Edge 249. Platform Stairs, Face Mould, Ap- plication of Plank 273 : Platform Stairs, Face Mould, Handrailing in 264 Platform Stairs, Face Mould, Sim- ple Method 267 ■ Platform Stairs, Face Mould, Moulded Rails 274 Platform Stairs, Face Mould, Ap- plication of 275 Platform Stairs, Face Mould With- out Canting Plank 272- Platform Stairs, Handrail to 269 Platform Stairs, Handrailing Large Cylinder 271 674 INDEX. PAGE Platform Stairs, Railing Where Rake Meets Level 272 Platform Stairs, Twist-Rail, Cut- ting of 277 Platform Stairs, Wreath of Round Rail 267 Point of Contact (Geometry) 558 Point Defined (Geometry) 544 Pointed Style, Ecclesiastical Arch- itecture II Polygons, Angles of 462 Polygons, Circumscribed and In- scribed Circles, Radius of 460 Polygons Defined (Geometry) 546 Polygons, Equilateral Triangle. . . 445 Polygons, General Rules 461 Polygons, Irregular, Trigon (Ge- ometry) 546 Polygons, Perpendicular in Tri- angle 440 Polygon, Regular, Defined (Geom- etry) 546 Polygons, Regular, To Describe (Geometry) 573 Polygons, Regular, To Inscribe in Circle (Geometry) 572 Polygons, Sum and Difference, Two Lines 439 Polygons, Table Explained 466 Polygons, Table of Multipliers. . . 465 Polygons, Triangle, Altitude of.. 442 Polygonal Dome, Shape of Angle- Rib 223 Posts, Area, To Find 86 Posts, Diameter, To Find 92 Posts, Rectangular, Safe Weight.. 92 Posts, Rectangular,To Find Thick- ness 94 Posts, Rectangular, Breadth Less than Thickness 96 Posts, Rectangular, To Find Breadth 95 Posts, To Find Side 93 Posts, Slender, Safe Weight for. . 91 Posts, Stout, Crushing Strength of. 89 Pressures Classified 85 Pressure, Oblique, Resolution of Forces 59 PACK Pressure, Triangular, Framed Girders 171 Pressure, Upper Chord, Triangu- lar Girder 168 Prisms Cut by Oblique Plane 259 Prisms and Cylinders, Stair-Build- i"g 257 Prisms Defined (Stairs) 257, 259 Prism, Top, Form of, in Perspec- tive 259 Profile of Mouldings Defined 326 Progression, Arithmetical (Alge- bra) 432 Progression,Geometrical (Algebra) 435 Projection and Height, Members of Orders of Architecture 16 Protractor, Useful in Drawing... 541 Purlins and Jack-Rafters in Roof. 211 Purlins, Location of 212 Pyramids of Memphis, Amount of Stone in 33 Pycnostyle, Explanation of 20 Quadrangle Defined 545 Quadrangle Equal Triangle 353 Quadrant Defined 547 Quantities, Addition and Sub- traction (Algebra) 424 Quantities, Division of (Algebra). 424 Quantities, Multiplication of(Al. gebra) 424 Queen-Post, Location of 213 Queen-Post in Roof 54 Radials of Rib in Bridge 226 Radials of Rib for Wedges 226 Radicals, Extraction of (Algebra). 425 Radius of Arc, To Find 561 Radius of Circle Defined 547 Rafters, Braces, etc.. Length, To Find 580 Rafters, Least Thrust, Rule for. . . 62 Rafters, Length of, To Find 578 Rafters, Liability to Crush Other Materials 205 Rafters, Liability to Being Crushed 205 Rafters, Liability to Flexure 205 Rafters, Minimum Thrust of 62 INDEX. 6/5 PAGE Rafters in Roof, EfTect of Weight on 179 Rafters in Roof, Strains Subjected to 205 Rafters and Tie-Beams, Safe Weight 87 Rafters, Uses in Roof 54 Rake in Cornice Matched with Level Mouldings 344 Railing, Platform Stairs Rake Meets Level 272 Ratio or Proportion, Equals Mul- tiplied 367 Ratio or Proportion, Equality of Products 370 Ratio or Proportion, Equality of Ratios 367 Ratio or Proportion Equation, Form of 367 Ratio or Proportion, Examples.. . 366 Ratio or Proportion, Four Propor- tionals, to Find 377 Ratio or Proportion, Homologous Triangles 370 Ratio or Proportion, Lever Prin- ciple in 372 Ratio or Proportion, Lever Prin- ciple Demonstrated 375 Ratio or Proportion, Multiply an Equation 368 Ratio or Proportion, Multiply and Divide One Number 368 Ratio or Proportion, Rule of Three 366 Ratio or Proportion, Steelyard as Example in 371 Ratio or Proportion, Terms of Quantities 3C7 Ratio or Proportion, Transfer a Factor 369 Rectangle Defined 545 Rectangle, Equilateral, To De- scribe 568 Rectangular Cross-Section, Iron Beams 145 Reduction Cubic Feet to Gallons, Rule 653 Reduction Decimals, Examples.. 647 Reflected Light, Opposite of Shade 611 Regular Polygon in Circle, To In- scribe (Geometry) 572 Regular Polygon Defined (Geom- etry) 54^ Regular Polygons, To Describe (Geometry) 573 Resistance, Capability of 86 Resistance to Compression, Ap- plication of Pressure 85 Resistance to Compression, Crushing and Bending 85 Resistance to Compression, Mate- rials 77 Resistance to Compression, Pres- sure Classified 85 Resistance to Compression in Proportion to Depth lor Resistance to Compression, Stout Posts, Rule 89 Resistance to Compression, Table of Woods 79 Resistance to Cross-Strains 77 Resistance to Cross-Strains De- fined 99 Resistance to Deflection, Rule.... 113 Resistance Depending on Com- pactness and Cohesion 78 Resistance Depending on Loca- tion, Soil, etc 79 Resistance to Flexure Defined. . . 145 Resistance Inversely in Propor- tion to Length 102 Resistance to Oblique Force 206 Resistance, Power of, How Ob- tained 78 Resistance, Proportion to Area. . . 86 Resistance, Strains, To What Due. 78 Resistance to Tension Greatest in Direction of Length 8i Resistance to Tension, Proportion in Materials 81 Resistance to Tension, Table of Materials 82 Resistance to Tension, Materials. 77 Resistance to Tension, Results from Trattsverse Strains 82 676 INDEX. Resistance to Transverse Strains, Table of 83 Resistance to Transverse Strains, Description of Table 84 Resistance Variable in One Ma- terial 79 Reticulated Walls 49 Rhomboid Defined 546 Rhombus Defined 545 Ribbed Bridge, Area of Rule .... 225 Ribbed Bridge, Built 224 Ribbed Bridge, Least Rise, Table of 224 Right Angle Defined 348, 544 Right Angle in Semicircle (Ge- ometry) 355 Right Angle, To Trisect a 554 Right Angled Triangle Defined . . 545 Right Angled Triangle, Squares on (Algebra) 417 Right Angled Triangles (Trigo- nometry) 510 Right Angled Triangles, Formula for (Trigonometry) 530 j Right Lines (Geometry) 584 ; Right Line Equal Circumference. 566 Right Lines, Mean Proportionals Between 584 Right Lines, Two Given, Find Third 582 Right Lines, Three Given, Find Fourth 5S3 Right or Straight Line Defined. . . 544 Right Prism Defined (Stairs) 257 Risers, Number of. Rule to Ob- tain (Stairs) 246 Rise and Tread (Stairs) 241 Rise and Tread, Connection of (Stairs) 248 Rise and Tread, Blondel's Method of Finding (Stairs) 242 Rise and Tread, Table of, for Shops and Dwellings (Stairs). . . 245 Rise and Tread, To Obtain (Wind- ing Stairs) 251 Rolled Iron Beams, Extensive Use of 161 Roman Architecture Defined 7 Roman Architecture, Ruins of. . . 11 Roman Architecture, Excess of Eniichm.ent 46 Roman Building 26 Roman Composite and Corinthian Orders 28 Roman Decoration 27 Roman Empire, Overthrow of 13 Romans, Ionic Order Modified by 27 Roman Moulding, Cavetto 329 Roman Mouldings, Comments on. 329 Roman Moulding, Ovolo 329 Roman Mouldings, Forms of. 325 Roman Pantheon, etc.. Walls of. . 49 Roman Styles of Architecture. ... 26 Roman Styles Spread by the Ital- ians 13 Romantic or German Style, Thir- teenth and Fourteenth Centu- ries II Rome, Ancient Buildings of. 12 Rome and Greece, Architecture of 8 Roof, The 54 Roofs, Ancient Norman and Gothic, Construction 178 Roof Beams, Weight per Super- ficial Foot i8g Roof, Brace in, Rule Same as for Rafter 208 Roofs. Construction of. 55 Roof Covering, Mode of i83 Roof Covering, Weights, Table of. 191 Roof, Curb or Mansard 54 Roofs, Diagrams and Description of 212 Roof, Gothic Buildings 55 Roofs, Gothic and Norman Build- ings, Construction 178 Roofs, Hip, Diagram and Expla- nation 215 Roof, Hip 54 Roof, Horizontal Pressure, To Re- move from 74 Roof, Jack-Rafters and Purlins.. . 211 Roof, King-Post in 54 Roof, Load per Foot Horizontal, Rule iq2 INDEX. ^77 Roof, Load, Total per Foot Hori- zontal, Rule 197 Roofs, Modern, Trussing Neces- sary 178 Roofs, Norman and Gothic Build- ings 178 Roof, Pent, To Find 54 Roof, Perpendicular Height, To Find 579 Roof Plank, Weight per Super- ficial Foot 189 Roof, Planning a 188 Roof, Pressure on 55 Roof, Queen-Post in 54 Roof, Rafters in 54 Roof, Sagging, To Prevent 54 Roof, Slope Should Vary Accord- ing to Climate, 191 Roof Supports, Distance between. 189 Roof, Suspension Rods, Safe Weight for 210 Roof, Tie-Beam in 54 Roof, Tie-Beam, Tensile Strain, Rule 204 Roof Timbers, Mortising 55 Roof Timbers, Scarfing of 55 Roof Timbers, Splicing of 55 Roof Timbers, Strains by Parallel- ogram of forces *. . . 198 Roof Timbers, Strain Shown Ge- ometrically 199, 202 Roof Truss, Arched Ceiling 214 Roof Truss, Elevated Tie-Beam Objectionable 214 Roof Truss, Elevating Tie-Beam, Effect of 187 Roof Truss, Force Diagram, Figs. 59, 68, and 69 179 Roof Truss, Force Diagram, Figs. 60, 70, and 71 180 Roof Truss, Force Diagram, Figs. 61, 72, and 73 181 Roof Truss, Force Diagram, Figs. 63, 74, and 75 183 Roof Truss, Force Diagram, Figs. 64, 77, and 78 184 Roof Truss, Force Diagram, Figs. 66, 80, and 81 186 PAGE Roof Truss, Load on 189 Roof Trusses, Strains, Effect of, on Different 179 Roof Truss, Weights, Table of, per Superficial Foot 189 Roof Truss, Weight per Superfi- cial Foot 190 Roof, Trussing in 54 Roof Trussing, Designs for 178 Roof Trussing, Framing for 237 Roof Trussing, Hogged Ridge.... 238 Roof Trussing, King-Post, Effect of Bad Framing on 237 Roofs, United States 55 Roof, Vertical Pressure of Wind on, Effect of. 194 Roof, Snow, Weight per Horizon- tal Foot 193 Roof Weight on Rafter, Effect of.. 179 Roof, Wind, Horizontal and Verti- cal Pressure of 193 Roofing, Weight of Horizontal and Inclined 190 Roofing, Weight per Superficial Foot 190 Roots, Cubes, and Squares, Table of 638-645 Round Post, Area of 90 Rubble Walls 48 Rulers and Pencil in Drawing... . 540 Rupture Compared with Flexure. 84 Rupture, Crushing, Safe Weight.. 89 Rupture, Sliding, Safe Weight 87 Rupture, Transverse, Safe Weight. 86 Rusting Iron Framing Straps, To Prevent 239 Safe Load for Material 81 Safe Weight, Allowance for 84 Safe Weight at Any Point, Rule. . 106 Safe Weight, Beam at Middle 103 Safe Weight, Bending 91 Safe Weight, Beam, Breadth of, To Find 104 Safe Weight, Beam, Depth, To Find 104 Safe Weight, Breadth or Depth, To Find, Load at Middle 106 6/8 INDEX. PAGE Safe "Weight, Breadth or Depth, To Find, Load Uniform io8 Safe Weight, Crushing Rupture. . 89 Safe Weight Defined 84 Safe Weight, Deflection, To Pre- vent no Safe Weight, Floor Beams, Dwell- ings 126 Safe Weight, Floor Beams, Stores. 126 Safe Weight, Load Evenly Distri- buted 107 Safe Weight, Load per Superficial Foot of Floor 109 Safe Weight, Margin Greater than Table 104 Safe Weight, Rafter and Tie-Beam, Example 87 Safe Weight, Rectangular Posts. . 92 Safe Weight, Slender Posts 91 Safe Weight, Sliding Rupture. ... 87 Safe Weight, Strain at Middle of Beam 105 Safe Weight, Suspension Rods in Roof 210 Safe Weight, Tensile Strain 96 Safe Weight, Tensile Strain, To Find 97 Safe Weight, Transverse Rupture. 86 Scale of Equal Parts, for an Order 15 Scale, Use of, in Drawing 540 Scalene Triangle Defined 545 Scantling, Thickness for Domes. , 218 Scotia, Classic Moulding 323 Scotia, Grecian Moulding 326 Sectional Area, Explanation of. . . 97 Sector, Area in Circle 476 Sector Defined 547 Segment of Circle, Area, To Find. 479 Segment of Circle Defined 547 Segment of Circle, To Describe (Geometry) 560, 562 Segment of Circle from Ordinates. 470 Segment of Cylinder Defined 549 Semi-major Axes of Ellipse De- fined 486 Shade Lining, Drawing 543 Shadow on Capital of Column.. . . 609 PAGE Shadow of Circular Abacus on Column 608 Shadow of Column on Wall 611 Shadow on Cornice 611 Shadow in Fireplace 605 Shadow of Horizontal Beam 604 Shadow, Inclination of the Line of. 596 Shadow of Moulded Window Lin- tel 606 Shadow of Nosing of Step 606 Shadow of Pedestal on Steps 606 Shadow of Projection on Cylindri- cal Wall 603 Shadow in Recess 604 Shadow in Recess, Back Vertical. 604 Shadows, Reflected Light Opposite of 611 Shadow and Shade, Distinction Between , 597 Shadow of Shelf 598 Shadow of Shelf, Acute Angled. . Coo Shadow of Shelf, Front Edge Curved 602 Shadow of Shelf Inclined. ....... 600 Shadow of Shelf Inclined in Verti- cal Section 601 Shadow of Shelf on Inclined Wall. 603 Shadow of Square Abacus on Col- umn 607 Shadow on Straight Projections and Mouldings 587 Shadow on Shelf of Uneven Width 599 Shadows, Usefulness of 596 Shaft, Base and Capital of Column Defined 14 Shaft of Column 47 Shaft, Upright Part of Column. . . 15 Shearing Strain, Tubular Iron Girder I57 Shutters, Inside, for Windows, Requirements 319 Sines and Tangents, Logarithms (Polygons) 464 Slate, Weight per Superficial Foot. 189 Snow on Roof, Weight per Hori- zontal Foot 193 Snow on Roof, Weight per Super- ficial Foot 190 INDEX. 679 Soffit of Circular Headed Win- dows 321 Solid Defined (Geometry) 544 Solid Timber Floors, First-Class Stores, Depth 144 Solid Timber Floors, Dwellings and Assembly, Depth 143 Solid Timber Floors, Fire-Proof, To Make I43 Span of Arch 52 Spherical Dome, Shape, To Find. 221 Splicing Beams 235 Splicing, Depth of Indents, To Find 235 Splicing, Effect of 236 Splicing Roof Timbers 55 Splicing or Scarfing Tie-Beam. . . . 234 Spring in Arch. 52 Square Defined 545 Square or Cube Roots, Example. 645 Squares, Cubes and Roots, Table of 638-645 Square Equal Rectangle, To Make (Geometry) 581 Square Equal Given Squares, To Make (Geometry) 581 Square Equal Triangle, To Make (Geometry) 582 Stability, Principles of 45 Stairs, Circular, Face Mould, First Section 283 Stairs, Circular, Face Mould for 2S2, 2S5, 287 Stairs, Circular, Falling Mould for Rail 281 Stairs, Circular, Ilandrailing for. 27S Stairs, Circular, Plan of 279 Stairs, Circular, Plumb Bevel De- fined 282 Stairs, Circular, Timbers put in After Erection 253 Stairs for Dwellings, Rule 244 Stairs, Face Mould, Kell's Method for Simple 268 Stairs, Handrailing, Geometry Necessary 257 Stairs, Handrailing, "Out of Wind" Defined 257 PAGE Stairs, Handrailing, Tools Used.. 257 Stairs, Light and Ventilation 240 Stairs, for Men and 'Women, Rise and Tread 243 Stairs, Nosing and Tread, Posi- tion of 241 Stairs, Pitch Board, To Make 247 Stairs, Plan of. Defined 257 Stairs, Plane and Cylinder, Nich- olson's Method 259 Stairs, Platform, Baluster, Posi- tion of 250 Stairs, Platform, Beneficial 240 Stairs, Platform, Cutting Twist- Rail 277 Stairs, Platform, Cylinder of 24S Stairs, Platform, Face IMould Without Canting Plank 272 Stairs, Platform, Face Mould, Ap- plication to Plank 273 Stairs, Platform, Face Mould, Ap- plication to Plank 275 Stairs, Platform, Face INIould for Handrail 264 Stairs, Platform, Face Mould, Moulded Rail 274 Stairs, Platform, Face Mould, Sim- ple Method 267 Stairs, Platform, Handrail to 269 Stairs, Platform, Handrail, Large Cylinder 271 Stairs, Platform, Handrail Where Rake Meets Level 272 Stairs, Platform, Lower Edge of Cylinder 249 Stairs, Platform, Wreath for Round Rail 267 Stairs, Position and Require- ments 240 Stairs, Prisms and Cylinders 257 Stairs, Rise and Tread 241 Stairs, Rise and Tread, Blondel's Method 242 Stairs, Rise and Tread, Table for Shops and Dwellings 245 Stairs, Rises, Number of. Height Given 246 Stairs, Shops, Rise for 244 680 INDEX. PAGE Stairs, Space for Timber and Plas- ter 247 Stairs, Stone, Public Building 240 Stairs, String of, To Make 247 Stairs, Tread, To Find, Rise Given 242, 246 j Stairs, Tread and Riser Connec- tion. , 248 Stairs, Width, Rule for 241 Stairs, Winding, Balusters in Round Rail 313 Stairs, Winding, Bevels in Splayed Work 314 Stairs, Winding, Blocking Out Rail 301 I Stairs, Winding, Butt-joint on j Hand rail 303 Stairs, Winding, Butt-joint, Cor- rect Lines for 307 Stairs, Winding, Diagrams Ex- plained 263 Stairs, Winding, Face Mould, Ac- curacy of 295 Stairs, Winding, Face Mould, Application 297 Stairs, Winding, Face Mould, Care in Drawing 295 Stairs, Winding, Face Mould, Curves Elliptical 301 Stairs, Winding, Face Mould for 290, 293 Stairs, Winding, Face Mould, Round Rail 303 Stairs, Winding, Face Mould for Twist 291 Stairs, Winding, Flyers and Winders 251 Stairs, Winding, Front String, Grade of 253 Stairs, Winding, Handrailing.256, 289 Stairs, Winding, Handrailing, Bal- usters Under Scroll 310 Stairs. Winding, Handrailing, , Centres for Square 308 Stairs, Winding, Handrailing, Face Mould for Scroll 311 Stairs, Winding, Handrailing, Fall- ing Mould for Raking Scroll. . . 310 Stairs, Winding, Handrailing, Gen- eral Considerations 258 Stairs, Winding, Handrailing, Scrolls for 308 Stairs, Winding, Handrailing, Scroll Over Curtail Step 309 Stairs, Winding, Handrailing, Scroll for Curtail Step. 310 Stairs, Winding, Scroll at Newel. 309 Stairs, Winding, Illustrations by Planes 261 Stairs, Winding, Moulds for Quarter Circle 255 Stairs, Winding, Newel Cap, Form of 312 Stairs, Winding, Objectionable. . 240 Stairs, Winding, Pitch Board, To Obtain 252 Stairs, Winding, Rise and Tread, To Obtain 251 Stairs, Winding, Sliding of Face Mould 299 Stairs, Winding, String, To Ob- tain 252 Stairs, Winding, Timbers, Posi- tion of 252 Stairs and Windows, How Ar- ranged 42 Stiles of Windows, Allowance for. 319 St. Mark, Tenth or Eleventh Cen- tury 12 Stone Bridge Building, Truss Work 232 Stone Bridge, Building Arch.... 230 Stone Bridge, Centres for. Con- struction 229 Stone Bridge, Pressure on Arch Stones 230 Stop for Doors 317 Stores, Floor Beams, Safe Weight. 126 Stores, Ordinary, Floor-Beams, Sizes, To Find 127 Stores, First-Class, Floor-Beams, Sizes, To Find 128 St. Paul's, London, Dome of. . . . 54 St. Peter's, Rome, Fourteenth and Fifteenth Centuries 12 Straight or Right Line Defined. . . 544 INDEX. 68 1 PAGE Strains, Cross, Resistance to 77 Strains on Domes, Tendency of. . 219 Strains Exceed Weights 61 Strains, Graphic Representation.. 165 Strain Greatest at Middle of Beam. 105 Strains by Parallelogram of Forces 165 Strains, Practical Method of De- termining 62 Strains of Rafter in Roof 205 Strains, Resistance, To What Due. 78 Strain on Roof Timbers Shown Geometrically 190 Strains on Roof Timbers Geomet- rically Applied 202 Strains on Roof Timbers, Parallel- ogram of Forces. . 198 Strain, Shearing, Tubular Iron Girder 157 Strain Unequal, Cause of .... 83 Straps, Iron, Roof Truss 239 Strassburg, Cathedral of 12 Strassburg, Towers of the Min- ster II Strength and Stiffness of Mate- rials 7S Structure of Materials 76 Struts Defined 173 Struts and Ties t)8 Struts and Ties, Difference Be- tween 69 St. Sophia, Sixth Century 12 Stucco Cornice for Interior 340 Studs and Joists Defined 174 Styles, Grecian, Only Known by Them 16 Stylobate, Substructure for Col- umns 14 Subnormal and Normal (in Para- bola) 496 Subtangent, Parabola 496 Subtangent of Ellipse Defined. . . 486 Subtraction and Addition (Alge- bra) ; 398 Superficies Defined (Geometry)... 544 Supports, Girders, Length, Rule.. 157 Supports, Position of 65 Supports, Inclination of, Unequal. 60 Suspension Rods, Location in Roof.. 212 Suspension Rods in Roof, Safe Weight 210 Symbols Chosen at Pleasure (Al- I gebra) 395 Symbols, Transferring (Algebra). 399 1 Systyle, Explanation of 20 i Table of Circles 649-652 I Table of Contents 6x3-624 i Table of Capacity of Wells, Cis- j terns, etc 653 I Table of Squares, Cubes, and I Roots 638-645 ' Table of Woods, Description of.. 80 Tail-Beams Defined 130 Tanged Curve, To Describe (Ge- ometry) 565 Tangent to Axes, Ellipse 485 Tangent Defined 547 I Tangent with Foci, Ellipse 4S7 I Tangent to Ellipse, To Draw 592 Tangent at Given Point in Cir- I cle 557 Tangent at Given Point, Without Centre 557 Tangent of Parabola 493 Tangents and Sines, Logarithms (Polygons) 464 Temples Built in the Doric Style. 19 Temple, Doric, Origin of the 17 Temple of Jupiter at Thebes 33 Tenons and Splices, Knowledge Important 88 Tensile Strain, Area of Piece, To Find 99 Tensile Strain, Compressed Ma- terial 100 j Tensile Strain, Condition of Sus- pended Piece 98 Tensile Strain, Safe Weight 96 Tensile Strain, Safe Weight, To Compute 97 Tensile Strain, Sectional Area, To Obtain 97 Tensile Strain, Suspended, Mate- terial Extended 100 682 INDEX. PACE Tensile Strain on Tie-Beam in Roof Truss 204 Tensile Strain, Weight of Suspend- ed Piece 9S Tensile Strength of Cast Iron 161 Tension and Compression, Fran:ied Girders 174 Tension, Resistance to 77 Tension, Resistance to, Table of Materials 82 Tension, Resistance to. Results Obtained 82 Tension, Resistance to. Proportion in Materials 81 Tent, Habitation of the Shepherd. 13 Testing Machine, Description in Transverse Strains 80 Testing Materials, Hydraulic Method 80 Testing Materials, Dates of 80 Testing Materials, Manner of 80 Tetragon Defined 546 Tetragon, Radius of Circles (Polygons) 446 Tetrastyle, Intercolumniation. ... 20 Thebes, Thickness of Walls at. . . 33 Thrust, Horizontal 63 Thrust, Horizontal, Examples. ... 64 Thrust, Horizontal, Tendency of.. 88 Tie-Beam in Ceiling, Load on, . . 190 Tie-Beam and Rafter, Safe Weight. 87 Tie-Beam in Roof 54 Tie-Beam in Roof, Tensile Strain. 204 Tie-Rods, Diameter, To Find. . . . 164 Tie-Rods, Floor Arches, Dwell- ings 153 Tie-Reds, Floor Arches, First- Class Stores 153 Tie-Rods, Wrought Iron 164 Ties Defined 173 Ties and Struts, To Distinguish.. 69 Ties and Struts, Framed Girders.. 174 Ties and Struts, Principles of. . . . 63 Ties, Timbers in a State of Ten- sion 68 Titus, Composite Arch of 28 Trimmer, Breadth, To Find, Two Sets Tail-Beams 134 TAGE Top Rail, Doors, Width, Rule 317 Torus, Classic Moulding 323 Torus, Grecian Moulding 326 Tower of Babel, History of 5 Towers of the Minster, Strassburg. ii Transverse Axis Defined 548 Transverse Strains, Compressed and Extended, Material 100 Transverse Strains, Defined 99 Transverse Strains, Explanation of Table III loi Transverse Strains, Greater Strength of One Piece loi Transverse Strains, Neutral Line Defined 100 Transverse Strains, Proportion to Bread th loi Transverse Slraiiis, Hatfield's, Reference to. .80, 121, 133,138, 143, 144, 145, 146, 148 Transverse Strains, Resistance to. Table of 83 Transverse Strains, Description of Table 84 Transverse Strains, Strength Di- minished by Division loi Trapezoid Defined 546 Trapezium Defined 546 Tread, To Find, Rise Given (Stairs) 242, 246 Tread and Nosing, Position of (Stairs) 241 Tread and Rise, To Find, Winding Stairs 251 Tread and Rise, To Find, Blon- del's Method , 242 Tread and Rise, Table for Shops and Dwellings 245 Tread and Riser, Connection of (Stairs) 248 Triangle, Altitude of (Poh'gons). 442 Triangles, Base, Formula for (Trig- onometry) 516 Triangle, Construct a (Geometry). 587 Triangle, Construct Equal-Sided (Geometry) 575 Triangle Defined 545 Triangle, Examples (Geometry).. . 2>^ INDEX. 683 Triangles, Equal Altitude 361 Triangle Equal Quadrangle 353 Triangles, Equation of (Trigo- nometry) 515 Triangles, Homologous (Geom- etry) 362 Triangles, Hypothenuse, Formula for 5i(J Triangles, Hypothenuse, To Find (Trigonometry) 518 Triangles, Perpendicular, To Find (Trigonometry) 517 Triangle or Set-Square in Draw- ing 539 Triangle or Set-Square, Use of. . . 541 Triangles, Terms Defined (Trigo- nometry) 512 Triangles, Three Angles Equal Right Angle 354 Triangles, Value of Sides (Trigo- nometry) 516 Trigon, Irregular Polygons (Ge- ometry) 546 Trigon, Radius of Circle (Poly- gons) 443 Trigon, Rule (Polygons) 441 Trigonometry, Oblique Triangles, Two Angles 523 Trigonometry, Oblique Triangles, Two Sides 521 Trigonometry, Oblique Triangles, First Class 520 Trigonometry, Oblique Triangles, Second Class 522 Trigonometry, Oblique Triangles, Third Class 526 Trigonometry, Oblique Triangles, Fourth Class 528 Trigonometry, Oblique Triangles, Sines and Sides 519 Trigonometry, Oblique Triangles, Formula, First Class 531 Trigonometry, Oblique Triangles, Formula, Second Class 532 Trigonometry, Oblique Triangles, Formula, Third Class 534 Trigonometry, Oblique Triangles, j Formula, Fourth Class 534 i Trigonometry, Right Angled Tri- angles 510 Trigonometry, Right Angled Tri- angles, Third Side, To Find... 511 Trigonometry, Right Angled Tri- angle, Formula 530 Trigonometry, Tables 513 Trigonometry, Triangles, Base, Formula for 516 Trigonometry, Triangles, Equa- tions of 515 Trigonometry, Triangles, Hypoth- enuse, Formula 516 Trigonometry, Triangles, Hypoth- enuse, To Find 518 Trigonometry, Triangles, Perpen- dicular, To Find 517 Trigonometry, Triangles, Terms Defined 512 Trigonometry, Triangles, Value of Sides 516 Trimmer or Carriage Beam, Breadth, To Find 132 Trimmer or Carriage Beams De- fined 130 Trimmer, One Header, Breadth, To Find, Dwellings and Stores. 133 Trimmer, Well-Hole in Middle, Breadth, To Find 136 Trisect a Right Angle 5154 Truss, Diagram of 200 Truss, Force Diagrams, Figs. 59, 68 and 69 179 Figs. 60, 70 and 71 iBo Figs. 61, 72 and 73 181 Figs. 63, 74 and 75 183 Figs. 64, 77 and 78 184 Figs. 65, 78 and 79 185 Figs. 66, 80 and 81 186 Truss, Roof, Framing for 237 Truss, Roof, Iron Straps 239 Truss, Weight, per Horizontal Foot, To Find 192 Truss Work, Stone Bridge Build- ing 232 Trussing and Framing, Gravity and Resistance 76 Trussing Partitions, Effect of.... 175 684 INDEX. PAGE Trussing Roofs, Effect of 178 T-Square, How to Make 539 Tubular Iron Girder, Area of Bot- tom Flange, Dwellings 159 Tubular Iron Girder, Area of Bot- tom Flange, First-Class Stores. 160 Tubular Iron Girder, Arc of Flange, Load at Middle 154 Tubular Iron Girder, Area of Flange, Load Any Point 155 Tubular Iron Girder, Area of Flange, Load Uniform 156 Tubular Iron Girder, Flanges, Proportion of 157 Tubular Iron Girder, Construction of 154 Tubular Iron Girder, Rivets, Al- lowance for 157 Tubular Iron Girder, Shearing Strain 157 Tubular Iron Girders, Web of. . . 158 Tuscan, Modern, Appropriate for Buildings 30 Tuscan Order, Introduction of the. 30 Tuscan Order, Principal Style in Italy 30 Twelfth Century, Buildings in the. 11 Twist Rail, Platform Stairs 277 Twists, Stairs, Nicholson's Method for 259 Undecagon Defined 546 United States, Roofs in 55 Vault, Simple, Groined or Com- plex 52 Ventilation, Proper Arrangement for 45 Versed Sine of Arc, To Find 561 Vertical Pressure of Wind on Roof 194 Vertical Tangent of Parabola De- fined 495 Volutes, To Describe the 20 Voussoir of an Arch 52 Wall, The 48 Walls, Coffer 49 Walls, Construction and Forma- tion 48 Walls, Eddystone and Bell Rock Lighthouses 48 Walls, Egyptian, Massiveness of. 33 Walls, Modern Brick 49 Walls of Pantheon and Roman Buildings 49 Walls of Pantheon at Rome 53 Walls, Pise, of France 49 Walls, Reticulated 49 Walls, Rubble 48 Walls, Strength of 48 Walls, Various Kinds 49 Walls, Wooden 49 Weakening Girder, Manner of. . . 140 Web of Tubular Iron Girder, Thickness of 15S Weight of Materials for Building Table of 654-656 Wells, Cisteins, etc.. Table of Capacity 653 White Pine, Weights of Beams Table of 177 i Wind, Greatest Pressure, per Su- perficial Foot 90 Wind on Roof, Effect of Vertical j Pressure 194 Wind on Roof, Horizontal and ( Vertical Pressure 193 I Winders in Stairs, How to Place ' the 42 Winders and Flyers, Stairs 251 I Windows, Arrangement of 44 i Windows, Circular Headed 320 Windows, Circular Headed, To Form Soffit 321 Windows, Dimensions, To Find. 318 Window-Frame, Size of 318 Windows, Front of Building, Ef- fect of 320 Windows, Heights, Table of, Width Given 320 Windows, Height from Floor. . . . 320 Windows, Inside Shutters, Re- quirement 319 Windows, Position and Light from 317 INDEX. 685 Windows and Stairs, How Ar- ranged 42 Windows, Sliles, Allowance lor.. 3191 Windows, Width Uniform, Height I Varying 319 Winding Stairs, BaluGters in Round Rail 313 Winding Stairs, Bevels in Splayed Work 314 Winding Stairs, Blocking Out Rail 301 Winding Stairs, Butt Joint, Posi- tion of. 303 Winding Stairs, Butt Joint 307 Winding Stairs, [Diagram of. Ex- plained 263 Winding Stairs, Face Mould for 290, 293 Winding Stairs, Face Mould, Ac- curacy of. 295 Winding Stairs, Face Mould, Ap- plication of 297 Winding Stairs, Face Mould, Curves Elliptical 30T Winding Stairs, Face Mould, Drawing 296 Winding Stairs, Face Mould, Round Rail .... 303 Winding Stairs, Face Mould, Slid- ing of 299 Winding Stairs, Face Mould for Twist 291 Winding Stairs, Flyers and Wind- ers 251 Winding Stairs, Front String, Grade of. 253 Winding Stairs, Handrailing . c 256, 2S9 Winding Stairs, Handrailing, oal- usters Under Scroll 310 Winding Stairs, Handrailing, Cen- tres in Square 308 Winding Stairs, Handrailing, Face Mould for Scroll 311 Winding Stairs, Handrailing, Fall- ing Mould 310 Winding Stairs, Handrailing, General Considerations 258 Winding Stairs, Handrailing, Scrolls for 308 Winding Stairs, Handrailing, Scroll Over Curtail Step 309 Winding Stairs, Handrailing, Scroll for Curtail Step 310 Winding Stairs, Handrailing, Scrolls at Newel 309 Winding Stairs, Illustrations by Planes 261 Winding Stairs, Moulds for Quar- ter Circle 255 Winding Stairs, Newel Cap, Form of. 312 Winding Stairs Objectionable. . . . 240 Winding Stairs, Pitch Board, To Obtain 252 Winding Stairs, Rise and Tread, To Obtain 251 Winding Stairs, String, To Obtain. 252 Winding Stairs, Timbers, Posi- tion of. 252 Wood, Destruction by Fire 37 Wooden Beams, Use Limited 154 Woods, Hydraulic Method of Testing 80 Wreath for Round Rail, Platform Stairs 267 THE END. DEC 4 - 1950 c?^.^_,^".<^^ 00^.^:,:^;/^% c?\\l;i'^% cp^^.^i: ^^0^ ^ 0^ ^ ^^0^ ^ .^G^ ^f^'^^'-f^^^ cp^.^l:i:^% o°^o^z:">^ cp^v^i:^'^ ■^^o'< » H Q o.*^ -^ # ^ 'c?^^^:;:^% ^''^^:.;;^^^% c?^^:^;^^^ ^^ ^:^;i::v ^^0^ ^"^111^^^ >^o^ ^^^la^^ "%o^ ^^^la^^ '^AO^ ^^ ^^^9. ^^^ ^^ <^ ^^ .^^ ^^ %^' ^ ^ i\\<^' ^Miv\^ ;^M£z%<^ ^M£;^\<^ -^ '■ 5 o^ I .^ ^ '« % .^^ >^^ v^ %'*%rs^.- ^-^^ ^^ 'v.''^^^,* ,^^' v^ o^^ cHa ^ '^o,.<^^ : .^?^ °^ .^^ N^ : "^-^ ^^^ : {' '% .^' .*kfe °, "^M^^ - ^ 0^ . LIBRARY OF CONGRESS 019 450 831 1