https://archive.org/details/carpentersbuildeOOgoul CARPENTER'S AND BUILDER'S ASSISTANT, AND WOOD WORKER'S GUIDE. Revised and Enlarged BY LUCIUS D, GOULD, Architect and Practical Builder. NEW YORK: WILLIAM T. COMSTOCK, Architectural Designer and Publisher, 23 WARREN STREET. 1897. Copyright, 1888. LUCIUS D. GOULD, PREFACE. The experience of workmen generally will testify that books have, as yet, furnished them but small assistance in the theory and art of construction. The object of the author in publishing this work, is to furnish them with rules for finding sections of pieces placed in any position ; for cutting every description of joints ; for finding the form of the raking mould at any point divergent from the straight line ; for springing and bending mouldings ; for mitering circular mouldings, and planes oblique to the base at any angle; and an easy system of building stairs and railing for straight and platform stairs. And eight plates containing steel square problems. Together with these rules, the author presents tables of the weight, and cohesive strength, of the different materials used in the construction of buildings, as well as the weight required to crush said materials ; a treatise on the adhesion of nails, screws, iron pins and glue; and a geometrical and mathematical demonstra- tion for finding the circumference, and squaring the circle. There can be but little doubt that a work of this kind is needed by Architects and Builders, and especially by Carpenters and Wood-workers, who are inexperienced in the different kinds of labor which they are called upon to perform. It is but due to acknowledge that we have consulted the valuable works of Thomas Tredgold for the articles on the strength and weight of materials ; also Mr. Nicholson, of London, for the glossary of technical terms. CONTENTS. I S T OIF 1 PLAT TH'S J PAGE. Plate i 6. To form an actagonal prism without instruments or tools. To find the size of a piece, to form a six sided prism when one of the sides is given. Plate 2 a To find the backing of hip rafters for obtuse and acute angled buildings. To find the length of common rafters. Plate 3 io To square the circle, also the ellipse. To find the mitre line of a right angle. To find the diagonal or mitre line for an octagonal prism. Plate 4 12 To find the intersecting or mitre line for hexagon and three-sided prism. To find the degree of elevation with the square. Plate 5 14 To find the mitre and butt joint of a mill hopper when placed at 45 elevation. To find the mitre and butt joint of a piece placed at any angle of elevation. Plate 6 16 To find the mitre and butt joint of a piece oblique to the base over an acute angle. Plate 7 18 To find the mitre and butt joint of a piece placed oblique to the base over an obtuse angle. Plate 8 20 To find the he ; ght of an obelisk with the square. To find the distance to an inaccessible object. To find the distance between two inaccessible objects. Plate 9 — The carpenter's square 22 Plate 10 24 To form an ellipse. To draw a polygon. To form a false ellipse. Plate 11 26 Timber foundation for a frame building. Plate 12 — Balloon frames 28 Plate 13 — Cutting and jointing timbers 30 Plate 14 — Framing 32 Plate 15 34 Figure I. Hip and jack rafters. Figure 2, Circul r stairs. Figure 3, Roof framing span 40 to 70 feet. Plate 16 — Roof with internal angles 36 Plate 17 — Plan and elevation of obtuse and acute angled buildings, projection of rafters and braces 38 CONTENTS. 5 PAGE. Plate 18 — Mitre -box — octagonal and hexagonal roofs 40 Plate 19— Spires 42 Plate 20— Curve of sprung moulding 44 Plate 21 — Bevels : acute and obtuse angles 46 Plate 22 — Area of a circle and contents of a globe 48 Plate 23 — Brackets 50 Plate 24 — The raking mould 52 Plate 25 — Rule for finding mitre lines 54 Plate 26 — Circular and square pans 56 Plate 27 — Circular desk and seat 58 Plate 28 — Angle of rafter for French roof 60 Plate 29 — Mitreing of circular mouldings. 62 Plate 30 — Sash and Door tools 64 Plate 31 — Corinthian truss 66 Plate 32— Stairs 6S Plate 33 — Straight and Platform stairs 70 Plate 34 — Hand railing 72 Plate 35 — Groin arches 74 Plate 36 — Squaring the circle 76 TABLES AND MISCELLANEOUS MATTER. Table showing length of brace 7 Practical method of finding contents in cubic feet 9 Superficial contents II Construction of roofs 13 Roof coverings 15 Long measure 17 Square measure 19 Strength of materials 21 Posts 23 Weights of materials. 25 Adhesion of nails 27 & 29 Adhesion of screws and iron pins and length of iron nails 33 Adhesion of glue 35, 37 & 39 Metric system of weights and measures 41 & 43 Protection against rust 43 Properties of various woods 45 How to measure grain bins 45 Miscellaneous notes and rules 47 & 49 Terms used in carpentry • 5 1 to 75 Valuation of plasterers' work , 66 PROBLEMS. PLATE 1. STEEL SQUARE PROBLEMS. The Carpenter s Square is an instrument in general use,, and is as important and valuable to the workman as the clock is to the time-keeper, or the compass to the mariner. The square consists of a blade and tongue, placed at right angles to each other. The blade is two feet long; the tongue twelve to sixteen inches long, divided into inches and eighths of an inch. This and the following plates will demonstrate a few of the uses to which the square may be applied. Figure i. — Having a piece of wood three (3) inches square ; wishing to form an octagonal prism, not having any instruments or tools convenient, I bisected the sides of the piece, and drew the diagonal lines ; after which I removed a section of the piece and placed the bisected lines on the diagonal lines, and drew the lines to form the octagonal prism required. Having the side of a hexagon, or six-sided prism given ; to find the size of the piece, and the angles required. Draw A B, Fig. 2, equal in length to 2 of the given sides. Place the square on the points A and B, with the given side B C on the tongue; then A B and AC determines the size of the piece, and ABC the angle required to form the prism. The other sides are found by the same operation with the square, or by dividing the line A B into four equal parts, and from the points drawing the diagonal line C G, and the perpendicular lines C D and C F. To find the area of the prism, multiply the length of the blade C A by six, the number of sides ; the product wiU be the area required. TABLE Showing the length of brace when the run is given, also the length of run when the brace is given. RUN. JjJ\..rYv^JCi. r>p ACE RUN. 2 ft. x 2 ft. 2.8248 2 ft. 1 .4142 X I 4142 2 ft 2 in. V 2 ft z in 3.I8I9 2 ft. 3 in. X I ,CQOQ 2 ft. 6 in. 2 ft. 6 in. 3-5749 2 ft. 6 in. 1.7870 • / / y X 1.7870 2 ft. 9 in. v 2 ft. 9 in. 3- 8 9°3 2 ft. 9 in. 1 .04 c; 1 x 1.04^ I 2 ft x 2 ft 4.2426 3 ft- 2.1213 x 2. 1 2 13 2 ft 3 in. V y\ 2 ft in 4.5961 3 ft. 3 in. 2.2980 x 2. 2980 1 ft. 6 in. x 3 ft. 6 in. 4-9497 3 ft. 6 in. 2.4748 x 2.4748 2 ft. in. x 2 ft. o in. O w * y 5-3 r 4i 3 ft. 9 in. 2 600 x 2. 6^70 4 ft. x 4 ft. 5.6568 4 ft. 2.8784 x 2.8784 4. ft. 2 in. x 4 ft in 6.0103 4 ft. 3 in - 3.oo £ e Ien §£h- . o ) 4, the number of pieces. 12)15 12)50-0 i*3 Multiply 4-2 By i-3 4*2 1*2-6 5-4-6, gives 5 ft. 4 in. and 6 parts, the solidity. 10 CARPENTRY. PLATE 3 Exhibits a practical demonstration of squaring the circle j and an inscribed ellipse. Also, of finding the intersecting or mitre lines for square and octagonal figures. To determine the exact size of a square, the contents of which shall equal the contents of a circle in square measure, is practi- cally demonstrated at Fig. i, which represents a circle, and an inscribed ellipse. To find the side of square, divide the radius A B into seven (7) equal parts ; square up from the point 3, cut- ting the arc A D at S ; join C S, the side of the square, equal in area to the area of the circle. Describe the ellipse any size, cut- ting the line C S at N ; then C N equals the sides of a square equal in area to the area of the inscribed ellipse. Place the square on the diameter, with the heel at the point S, and find the exact size of the square required. To find the intersecting line or mitre, for a right angle Fig. 2, place the square at equal distances from the heel, then the blade and tongue gives the lines required. Fig. 3. To find the side of an octagonal prism, when the side of the square piece is given: Bisect the sides of the piece ; place the square on the side A B, with the length bisected on the blade and tongue ; then the tongue cuts the side at the point to gauge for the piece to be removed for the prism required. To find the size of square required for an octagonal prism, when the side is given: Let C D equal the given side; place the square on the line of the side, with one-half of the side on the blade and tongue; then the tongue cuts the line at the point B, which determines the size of the square and the piece to be removed. To find the area of an octagonal prism: Multiply the given side by eight, the number of sides ; the product by half of the altitude of the isosceles triangle formed by the side and diagonal lines. The product equals the area required. BY CALCULATION. Suppose I have a lot of ground 40 feet square. Wanting to know how far to measure from the angles to form an octagon: 40-^-2=20X20=400+400=^800=28.28 — 40=11.72 feet, the dis- tance required. Wanting to know the size of lot when the side is given: Say 20 feet-4-2=ioXio=ioo+ 100=^/200=14.14+20=34. 14+14.14=48.28. feet square, the size required. Fig. 2. Fig. 3. PROBLEMS. 11 A PRACTICAL METHOD Of finding the superficial contents of boards and timber. For boards, multiply the width, in inches, by the length, in feet, and divide by 12. Example. — Find the number of feet in a board 1 inch thick, 9 inches wide and 13 feet long. 9 12)117 9-9=9 feet 9 inches. Example 2d. — Find the number of feet in a piece of timber 3x10 inches, 21 feet long. 10 inches wide. 3 " thick. 12)30 2*6 inches in each foot in length. 21 feet long. 42 io-6 52*6 gives 52 feet 6 inches, the number of feet in the piece. 12 CARPENTRY. PLATE 4. Having the side of a six-sided prism, to find the diagonal or mitre line, with the square. Figure I. — Place the square on the side given, with one- half of the side on the tongue ; then the tongue and the side of the hexagon gives the angle to cut for the mitre. Fig. 2 represents the operation of finding the mitre for an equilateral triangle, by placing the square on the side, with one-half the distance on the blade ; then the tongue and the sides give the angle required to cut the mitre. Fig. 3 represents a quadrant of 90 , with the steel square placed equally distant from the heel, on the tongue and blade, to intersect the arc at 45°. To find the run to any degree of elevation, slide the square to and from the centre of quadrant, for the run and height required, which will be found useful to workmen in finding the elevation of roofs, etc., when specified in degrees by the architect ROOFS. Construction of Roofs. In old Gothic buildings, the roof always had a high pitch, its outline formed a striking feature, and in general had a graceful proportion with the magnitude of the building; sometimes, however, it presented a plain sur- face of too great extent, as the roof of Westminster Hall. Though a high roof is in perfect unison with the aspiring and pyramidal character of Gothic architecture, in the more chaste and classic style of the Greek, it is a less con- spicuous object. Many of the Grecian buildings were never intended to be roofed at all; but where a roof is necessary, it was not attempted to be hidden, but consti- tuted one of the most ornamental parts of the building. Of timber roofs, we have no examples in Grecian build- ings ; but the beautiful stone roof of the Octagon Tower of Andronicus Cyrrhestes, and that of the Choragic Monu- ment of Lysicrates, are sufficient to show that they were more inclined to ornament than to hide this essential part of a building. The height of roofs, at the present time, is seldom above one-third of the span, and should never be less than one- sixth. The most usual pitch is when the height is one- fourth of the span, or when the angle with the horizon is 26% degrees. The pediments of the Greek temples make an angle of from 12 to 16 degrees with the horizon ; the latter corres- ponds nearly with one-seventh of the span. The pedi- ments of the Roman buildings vary from 23 to 24 degrees ; 24 degrees is nearly two-ninths of the span. 14 CARPENTRY. PLATE 5 Exhibits the operation of finding the mitre and butt joints for a mill hopper, when the sides are placed at an angle of 45 / also to find the mitre and butt joints to a piece placed at any angle from a horizontal to a perpendicular, with the steel square. Figure i. — The elevation of a mill hopper. To find the mitre for the edge and sides, place the square on the line of the edge and sides, with A B on the blade and A C on the tongue ; then the tongue gives the lines for the face and edge required. To find the angle for the butt joint, set off from the heel of the square equal to A D on the blade and C D on the tongue ; then the tongue gives the line required. Figure 2. — Exhibits the section of a piece placed oblique to the base, to be mitred at a right angle. To find the line to cut the edge, place the square on the line A S, with A C on the blade and A B on the tongue ; then the tongue gives the line required. To find the line to cut the side of the piece, place the square on the line H J, with A C on the blade and B C on the tongue ; then the tongue gives the line required. To find the line to cut the edge for the butt joints, place the square on the line P R, with E D on the blade and B D on the tongue ; then the tongue gives the line required. ROOFING. 15 ROOF COVERINGS. The kinds of covering used for timber roofs, are cop- per, lead, iron, tinned iron, slates of different kinds, tiles, shingles, gravel, felt and cement. Taking the angle for slates to be 26y 2 degrees, the following table will show the degree of inclination that may be given for other materials. Kind of covering. Inclination to the ho- rizon, in degrees. Height of roof in parts of the span. Weight upon a square of roofing. Deg. Min. Tin 3 5° 1 18 pounds. 3 50 1 18 IOO u 3 50 1 18 700 it 22 OO 1 5 II20 it " ordinary.... 26 33 1 1 9OO tt " fine 26 33 i 500 u 29 4i 2 7 I78o 4( Gravel Felt and Cement. . . Felt and Cement or Gravel Roofing can be used at almost any inclination that other materials are used. 16 CARPENTRY. PLATE 6. Exhibits the operation of finding the lines for the edge and side of a piece placed oblique to the base, to be mitred over an acute angle. At Figure i, draw the base and perpendicular indefi- nitely ; place the side of the piece A B at the angle re- quired ; draw A C at right angles to A B ; produce B A to D ; draw D E parallel to C B ; draw the plan of the acute angle required. To find the line to cut the edge, place the square on the line, with F G on the tongue and A C on the blade ; then the tongue gives the line required. To find the line to cut the side A B, place the square on the line, with F G on the tongue and A B on the blade ; then the tongue gives the line required. The angle to cut the edge for a butt joint, is shown at E H A. TABLES. 1 1 LONG MEASURE. Long measure is used in measuring length or distance only, without regard to breadth or depth. Its denomina- tions are leagues, miles, furlongs, rods, yards, feet and inches* 12 inches - make i foot. 3 feet - "i yard. 5^ yards, or \6% feet, - " I rod. 40 rods ... - " i furlong. 8 furlongs, or 320 rods, - " 1 mile. 3 miles - " 1 league. Note. — 4 inches make 1 hand ; 9 inches 1 span ; 18 inches 1 cubit ; 6 feet 1 fathom ; 4 rods, or 100 links, 1 chain ; 25 links 1 rod ; 7S inches, 1 link. The chain is commonly used in measuring roads and land, and is called Gunter's chain, from the name of the inventor. A knot, in sea phrase, answers to a nautical or geographi- cal mile of 5,280 feet. Mariner's measure is a kind of long measure used in es- timating distances at sea. 6 feet - make 1 fathom. 120 fathoms - < 1 cable-length. 880 fathoms, or y\ cable, 4 1 mile. 18 CARPENTRY. PLATE 7 Exhibits a piece placed oblique to the base, to be mitred over an obtuse angle with the steel square. To find the line to cut the edge of the piece A B, Figure i, place the square on the line, with E D on the tongue and B C on the blade ; then the tongue gives the line required. To find the line to cut the side A B, place the square on the line with E D on the tongue, and A B on the blade; then the tongue gives the line required. To find the line to cut the edge for a butt joint, place the square on the line, with F H on the tongue, and F G on the blade; then the tongue gives the line required. Figure 2. — To find the centre when an arc is given : Draw two chord lines six inches long ; then place the blade of the square three inches from the heel on each of the chords, and at the intersection of the tongues will be found the centre required. To divide a piece into any number of equal parts, place the square on the piece, with the points on the edges ; then if 4 equal parts are required, mark the piece from the points 6, 12 and 18. If five pieces are required, place the heel of the square and the figure 20 on the edge, then mark from the points 4, 8, 12 and 16. By this rule the piece can be divided into any number from 2 to 24 equal parts without the dividers. To find the distance to gauge from the angles of a square piece to form an octagonal prism : Place the square on the side of the piece diagonally ; then gauge from the points 7 and 17, the distance required. TABLES. 19 SQUARE MEASURE. Square measure is used in measuring surfaces, or things whose length and breadth are considered, without regard to heighth or depth : as land, flooring, plastering, etc. Its denominations are Acres, Roods, Square Rods, Yards, Square feet, and Square iuchts* 144 square inches - make i square foot 9 square feet - " 1 " yard 30X square yards or ) « j 1 sc l uare roc ** 272^ square feet j ( perch or pole. 40 square rods - " 1 rood. 4 roods, or 160 square rods " 1 acre. 640 acres - - " 1 square mile. Note. — 16 square rods make 1 square chain; 10 square chains, or 100,000 square links, make an acre. Flooring, roofing, plastering, etc., are frequently estimated by the "square," which contains 100 square feet. Note. — A chain is 66 feet in length, and is divided into 100 equal parts, or links. The length of a link is, therefore, 7.92 inches. 20 CARPENTRY. PLATE 8. FIGURE i. — Wishing to know the height of an obelisk situated on a horizontal plane, I measured 70 feet in a right line from the centre of its base, and raised a perpen- dicular five feet high, and placed the square twelve inches from the heel, at right angles to the perpendicular ; then with a straight edge took the angle of elevation of the top,, which I found to be 8.7 inches to the foot. Multiplied by 70=609—12=50.75+5=55.75 feet, the height required. Figure 2. — Wanting to know the distance between two inaccessible objects, A and B, from the point C, I draw CA and CB ; at right angles to CA and CB, I measured thirty feet to the points E and S, where I placed the square ; then with the straight edge took the observation, and found that 12 inches on E C gave 13.2 inches on the line C AX30=396-J-i2=33 feet from C to A, and by the same operation on the line C S determines the length of C B 50 feet. Having found the angle and two sides of the triangle C B A, the other side can be found by drawing to a scale, or by trigonometry, where two sides and the in- cluded angle being given, to find the other angles and side. Figure 3. — Being on the side of a river, and wanting to know the distance to a tree on the other side, I measured 40 feet at right angles from the tree and station ; placed the square at the point, and found by observation that the square gave 22.7 inches to the foot, which multiplied by 40=908^-12=75.75 feet, the distance required. TABLES. 21 WEIGHT OR FORCE REQUIRED TO TEAR ASUNDER ONE SQUARE INCH OF THE DIFFERENT MATERIALS USED IN THE CONSTRUCTION OF BUILDINGS. WOODS AND METALS. Oak, American, 17,300 Swedish Iron, 78,850 Oak, English, - 19,800 English Iron, - 55,772 Beech, 17,700 French Iron, - 61,041 Ash, - 16,700 Russian Iron, - 59,472 Elm, - 13,489 Cast Iron, 42,000 Walnut, - - 8,130 Steel, Soft, - - 120,000 Norway Pine, 14,300 Ivory, 16,000 Georgia Pine, - 7,8i8 Marble, 8,700 White Pine, 8,800 Whalebone, 7,600 Iron Wire, - - n3,o77 To find the strength of Cohesion : Multiply area of section, in inches, by the weight required to tear one inch asunder, and the product is the strength in pounds. WEIGHTS 'REQUIRED TO CRUSH ONE CUBIC INCH OF SEVERAL MATERIALS USED IN THE CONSTRUCTION OF BUILDINGS. METALS. WOODS. Cast Iron, Brass, Copper, Cast, Lead, Cast, - Freestone, Limestone, Black, Granite, Blue, - 116,700 154,784 116,102 8,042 Elm, American Pine, White Deal, - White Oak, - English Oak, - STONES. 18,006 Brick, hard, - 19,450 Brick, soft, 20,890 Chalk, 1,284 1,606 1,928 3, 2 4o 3,860 i,754 1,224 1,040 ,22 CARPENTRY. . _ PLATE 9. Figure i. — Exhibits the use of the square to divide a board into any number of equal parts. For example, to divide a board into four equal parts, place the points of the blade on the edges of the piece, then 6, 12 and 18, will be the points of division. If five pieces are required, place the heel of the square and the figure 20 on the edges of the piece, then 4, 8, 12 and 16 are the points of division. - « Figure 2. — Exhibits the application of the square to find the points for eight- squaring timber. Also to cut a piece to fit any angle, by extending the line of the blade to A : place the square on the piece, transfer the distance extended, and draw the line A B, the angle required. Figure 3. — Exhibits the application of the square to find the angles of the octagonal figure. To find the cuts in the mitre-box. — At Figure 4, place the square at equal distances from the heel, on the line A B. Ta prove the truth of the lines, reverse the bevel. To find the per- pendicular and horizontal cuts of rafters with the square, take half the width of the building for the run, on the blade, and the rise on the tongue. Figure 5. — Exhibits two rules for finding the backing of hip- rafters ; one with ' the square, as follows : Place the square on the line D E, with the height H B on the tongue, and the length A B on the blade ; then the direction of the tongue gives the angle required. For an obtuse and acute angled roof ; for the obtuse angled hip, place the length of the acute angled hip rafter on the blade, and the height on the tongue, then the tongue gives the angle required. The same operation on* the obtuse angled hip rafter gives the angle to bevel the acute angled rafter. The other rule is geometrical and applies to right, obtuse, and acute angles where the pitches are the same, as follows : From the point D as centre, describe an arc from the line L K ; tangent to the arc, draw the dotted line parallel to D A, cutting the line A H at I ; draw I J parallel to AB ; then the line I. J gives the distance to gauge the rafter for the backing, as shown at section G. POSTS. 23 POSTS. According to the experiments of Rondelet, when the height of a square post is less than about seven or eight times the size of its base, it cannot be bent by any pres- sure less than that which would crush it. The internal mechanism of the resisting forces when timber yields by crushing is not exactly understood. In timber, the resist^ ance to crushing is less than the cohesive force. The resistance of timber to crushing appears to increase in a higher ratio than that ol the area of its section. The load a piece of timber will bear, when pressed in the direction of its length, without risk of being crushed, may be found by the following rule: Multiply the area of the piece of timber, in inches, by the weight that is capable of crushing a square inch of the same kind of wood, then one-fourth of the product will give the load, in pounds, that the piece would bear with safety. If the area that would support a given weight be re- quired, divide four times the weight by the number of pounds that would crush a square inch, and the quotient is the area in inches. The length should never exceed ten times the side of the section, to give the above results ; for, when the length is greater than about ten times the thickness, the piece will bend before it crushes. CARPENTRY. PLATE 10. To form an ellipse with a thread or string. At Fig. i, draw the major and minor axes, A B and C D. To find the points for the pins, to describe the ellipse : from the point C as centre, with E B as radius, describe arcs cutting the major axis at 2 and 3, the points required ; around the pins and the point C place a cord ; with the pencil placed at the point C, describe the ellipse re- quired. Care should be taken to keep the cord at an even tension. To draw a polygon of any number of sides. To form a polygon of five sides. — From the point A, Fig. 2, as centre, with the given side A B as radius, de- scribe a semi-circle, which divide into five equal parts ; through the points of division, draw A 2, A 3 and A 4, in- definitely ; parallel to A 3 and A 4, draw B C and 2 D ; join C D, which completes the polygon required. To form the false ellipse. Figure 3. Draw the major and minor axes, A B and C D ; join B C, and divide into three equal parts ; draw N E at right angles to C B : from the point E as centre, with E C as radius, describe the arc R N : from the point S as centre, with S B as radius, describe the arc N P. The opposite sides are found in the same manner. Figures 4, 5 and 6 are simple geometrical operations, aji inspection of which is sufficient for their comprehen- sion. / TABLES 25 WEIGHT IN POUNDS OF A CUBIC FOOT OF WOOD OR STONE. WOOD. STONE. Apple-tree, 49.6 Flint, 163.2 Ash, » - 5 2 -9 Blue Granite, - 164. 1 Birch, 33- 2 Limestone, 199. American Cedar, - - 35-i Grindstone, - 134. Elm, 42. Slate Stone, 167. White Pine, - 35-6 Marble, - 170. "Yellow Pine, - 4f.t Freestone, 150. Mahogany, - 66.5 African Marble, - - 169.2 Maple, - 47- Egyptian Marble, 166.8 Mulberry, . - 561 Italian Marble, - 166.1 Oak, - 58-74 Roman Marble, 172.2 Live Oak, - 70- OTHER SUBSTANCES. Cast Iron, - 450-55 Air, - .07529 Wrought Iron, - - 486.65 Steam, - .03689 Steel, - 489.8 Loose Earth or Sand, 95- Copper, - - 555- Common Soil, 124. Lead, - 708.75 Strong Soil, 127. Brass, - 537-75 Clay, - - 135. Tin, - 45 6 - Clay and Stones, 160. Salt-water, (Sea,) - 64.3 Cork, 15. Fresh-water, - 62.5 Brick, - "5- Tallow, - 59- 26 CARPENTRY. PLATE 11. Shows a timber foundation for a frame building, with two> side elevations, framed in the visual manner for good houses. — The object of this and the following Plates, is first to give the inexperienced workman the names used among car- penters and joiners, of the different pieces of timber used in framing, and where they are placed ; also to show the method of constructing what is called a balloon frame. Figure i. — Shows a timber plan of foundation support- ed by brick or stone walls. The outside timbers are called sills; and, if there are no openings, all other timbers are called beams; but when there are openings for chim- neys or stair-ways, the workman will be required to mor- tise and tenon the timbers together as shown on the plan. The first piece of timber to prepare will be the trimmer t shown at A, which is tenoned into the trimmer-beams \ shown at BE. The short beams tenoned into the trim7ner are called tail-beams.' Figs. 2 and 3 are the front and a portion of the side elevation of the frame standing on the foundation, showing the posts, beams, enter-ties, plates, rafters and braces in their proper places. The timbers shown at A A, Fig. 2, are called frame-beams; D D, corner-posts, and C C, rafters. At Fig. 3, A shows what should be called an intermediate post; the pieces of timber called enter-ties, are shown at E E ; the piece of timber supporting the rafters at C, represents the plate, and B B the sills; the oblique pieces of timber shown on the ele- vations, are called braces; the timbers shown on each side of the openings are called joists, and- termed door and window joist; those placed between doors and windows,, are called intermediate joists, or furrings ; all joists cut under or over the braces are called cripples; a piece of timber placed on piers for the purpose of supporting other timbers or partitions, are called summers; a piece of tim- ber placed on a truss-frame, for the purpose of supporting the common rafters, is called a purlin. J / NAILS. 2? ADHESION OF NAILS, Every carpenter is familiar with the use of nails, and possesses a practical knowledge, more or less accurate of the force of adhesion of different nails, and in different substances, so as to deci.de, without difficulty, what num- ber, and of what length, may be sufficient to fasten to- gether substances of various shapes, and subject to various strains. But interesting as this subject unquestionably is, it has not been till very recently that the necessary experi- • ments have been made to determine : ist, the adhesive force of different nails, when driven into wood of different^ species ; 2d, the actual weight, without impulse, necessary ' to force a nail a given depth ; and 3d, the force required to extract the nail when so driven. The obtaining of this useful knowledge was reserved for Mr. B. Bevan, a gen- tleman well known in the mechanical and scientific world : for the accuracy with which his experiments are con- : ducted. Mr. Bevan observes, that the theoretical investigation points out an equality of resistance to the entrance and ex~ traction of a nail,, supposing the thickness to be invariable ; but as the general shape of nails is tapering towards the point, the resistance of entrance necessarily becomes greater than that of extraction ; in some experiments he found the ratio to be about 6 to 5. The percussive force required to drive the common six- penny nail to the depth of one inch and a half, into dry Christiana deal, with a cast iron weight of 6.275 lt> s - was four blows, or strokes, falling freely the space of 1 2 inches; and the steady pressure to produce the same effect was 400 lbs. {Continued on page 29.) 28 CARPENTRY. PLATE 12. Shows the method of constructing what is termed a balloon frame. Fig-, i shows the timber plan ; Figs. 2 and 3, the front and side elevations. The foundation timbers should be ot white pine ; all other timbers, of spruce or Eastern pine. All the tools.the workman requires to construct a frame of this kind, are a saw, hammer and chisel. The side-sills should be 4x4 inches ; front and rear-sills, four inches thick; beams 2x8 or ten inches, according to their length and the load they are required to carry. Corner post 4x4 inches ; door and window joists, 3x4 inches ; all other in- termediate joists, 2x4 inches r plates, 4x4 inches ; rafters, 3x5 inches. The two outside beams, in second story, are spiked to the joists ; those resting on the plates are spiked to the rafters. The enter-ties require to be 1^x4 inches let into the joists to support second story beams. Each tier of beams should have one or two courses of bridging. When the frame is completed and sheathed with one inch worked boards, placed diagonally and securely nailed to -every joist, it will be quite as substantial and safe as a frame made in the usual manner. NAILS. 28 ADHESION OF NAILS. A sixpenny nail driven into dry elm, to the depth of one inch, across the grain, required a pressure of 327 pounds to extract it; and the same nail, driven endways, or longi- tudinally, into the same wood, was extracted by a force of 257 pounds. The same nail driven two inches, endways, into dry Christiana deal, was drawn by a force of 257 pounds ; and to draw out one inch, under like circumstances, took 87 pounds only. The relative adhesion, therefore, in the same wood, when driven transversely and longitudinally, is 100 to 78, or about 4 to 3, in dry elm ; and 100 to 46, or about 2 to 1, in deal; and, in like circumstances, the relative ad- hesion to elm and deal is as 2 or 3 to 1. The progressive depths of a sixpenny nail into dry Christiana deal by simple pressure were as follows : — One-quarter of an inch, a pressure of 24 lbs. Half an inch, - - - - 76 ' 4 One inch, ----- 235 " One inch and a half, - 400 " Two inches, - 610 " In the above experiments, great care was taken by Mr* Bevan to apply the weight steadily ; and towards the conclusion of each experiment, the additions did not ex- ceed 10 pounds at one time ; with a moderative interval between, generally about one minute, sometimes 10 or 20 minutes. In other species of wood, the requsite force to extract the nail was different. Thus, to extract a com- mon sixpenny nail from a depth of one inch out of Dry Oak, required - - - 507 lbs. Dry Beech, - 667 " Green Sycamore, - 313 " From these experiments, we may infer that a common sixpenny nail, driven two inches into oak, would require a force of more than one-half a ton to extract it by a steady force. 80 CARPENTRY. PLATE 13. Carpentry is the art of ' cutting and jointing timbers ZTi, the construction of buildings. ' To cut timbers and adapt them to their various situa- tions, so that one of the sides of every piece shall be ar- ranged according to a given plane or surface shown in the designs of the architect, is a department of carpentry which requires a thorough knowledge of the finding of sections of solids, their coverings, and the various methods ot connecting timbers, etc. The art of combining pieces of timber to increase their strength and firmness, is called framing. The form of a frame should be adapted to the nature cf the load which it is designed to carry. In carpentry, the load is usually distributed over the whole length of the framing, but it is generally supported from point to point, by short beams or joists. First, let us consider a case where the load is collected at one point of the frame ; and, in order that the advantage of framing may be more obvious, let us suppose all the parts of a certain piece of frame-work to be cut out of a single beam, which, in a solid mass, would be too weak for the purpose. Let Fig. i be a piece of timber, cut in the various direc- tions indicated by the lines passing through it, and let the triangular piece shown at Eand F be removed ; then raise the pieces A E and A F till they make close joints at E and F, and increase their lengths till they form a frame, or truss, as represented at Fig. 2. A small rod of iron with suitable nuts, will be required to support the centre of the tie, as seen in the drawing. If the depth of the frame at the middle be double the depth of the beam, the strength of the frame will be a little more than eight times as great as that of the beam. If the depth of the frame be three times the depth of the beam, as represented at Fig. 2, it will be about six times as strong as the beam, and about eighteen times as firm ; that is, it will bend only an eight- eenth part of the distance which the beam would bend, under the same weight. To render the strength more equal, and to obtain two points of support, there may be a level piece of timber placed between the inclining ones, as shown at Fig. 3 ; but if a greater weight be placed at G than at H, there will be a tendency to spring upwards at H, and inwards at A, which may be effectually prevented by the suspension rod Ji A, as shcvn in the same figure* Tig.6. FRAMING. 31 It now remains to show why the strength of a piece of timber is increased by forming it into a truss; and to have a clear conception of this subject is of the utmost impor- tance in the science of carpentry. Let ABC, Fig. 4, be a truss to support a weight applied at A. It is evident that the force of the weight will tend to spread the abutments, B and C, and the nearer we reduce the angle A B C to a straight line, the greater will be the pressure, or tendency to spread or increase at A. On the contrary, if the height be increased, as at Fig. 5, the tendency to spread the abutment will be less. The advantage of framing timbers together for the purpose of giving strength and firmness having been shown, let us proceed to explain how the strain on any part may be measured. To find the pressure on oblique supports or parts of trusses, frames, etc. Let A C, Fig. 6, be a heavy beam supported by two posts, A C and B D, placed at equal distances from E, the centre of the beam. The pressure on each post will obviously be equal to half the weight of the beam. But if the posts be placed obliquely, as in Fig. 7, the pressure on each post will be increased in the same proportion as its length is increased, the height A C being the same as before; that is, when A F is double AC, the pressure on the post in the direction of its length is double half the weight of the beam. Hence it is very easy to find the pressure in the direction of an inclined strut, for it is as many times half the weight supported as A C is contained in A F. Therefore, if the depth A C of a truss to support a weight of two tons be only one foot, and A F be ten feet, the pressure in the direction of A F will be ten tons. It will be observed that when the beam is supported by oblique posts, as in Fig. 7, these posts will slide out at the bottom, and together at the top, if not prevented by proper abutments. The force with which the foot F tends to slide out is to half the weight of the beam A B, as F C is to A C. Therefore, when F C is equal to A C, the tendency to slide out is equal to half the weight sup- ported ; and if F C be ten times A C, the tendency to spread out would be ten times the weight supported. Hence it is evident that a flat truss requires a tie of im- mense strength to prevent it from spreading. If a flat truss produces any degree of stretching in the tie, the truss must obviously settle, and by settling it becomes flat, and consequently exerts a greater strain. In a flat truss, therefore, too much caution cannot be used in fit- ting the joints and choosing good materials. 82 CARPENTRY. PLATE 14. In framing, all pieces placed at right angles to each other are cut square or beveled ; but when placed diag- onally and oblique to the base, require a geometrical operation to find a section of the piece whose sides shall be in the plane of those it is connected with. It is intended, therefore, to present, at this time, a new and complete system of lines for finding sections" and cuts of pieces placed in any position, from the horizontal to the perpendicular, by means of tangents and circles. Let A B C D, Fig. i, represent the plan of a right- angled hip-roof, and B F C the elevation. To find a sec- tion of the hip-rafters, draw G H at right angles to B E; from the point H as centre, with H J as radius, describe an arc ; from the point G, draw the tangent, cutting the line B E at R ; join H R, which forms the angle for the section required. To find the lengths of the hip and jack-rafters. Draw D L, Fig. i, equal to the common rafter C F, Fig. 2 ; join CL for the length of the hip-rafters. To find the lengths of the jack-rafters, divide the common rafter D L into as many parts as there are jack-rafters required. To find the bevels for the hip and jack-rafters. Draw C N, Fig. 1, equal to C E, and LN equal to P F, Fig. 2; then in the angle at L is the down bevel, and at C the face bevel, for the hip-rafters. The face and down bevels for the jack rafters are shown at E and F. Figure 3 exhibits the application of the foregoing system to an obtuse and acute-angled plan ; the operation is precisely the same, and consequently needs no further explanation. SCREWS. 33 ADHESION OF SCREWS. A common screw, of one-fifth of an inch, was found to have an adhesive force of about three times that of a six- penny nail. ADHESION OF IRON PINS. The force necessary to break or tear out a half-inch iron pin, applied in the manner of a pin to a tenon in the mortise, has likewise obtained the attention of the same celebrated experimentalist. The thickness of the board was 0.87 inch, and the distance of the center of the hole from the end of the board, 1.05 inch. The force required was 916 lbs. As the strength of a tenon from the pin-hole may be considered in proportion to the distance from the end, and also as the thickness, we may, for this species of wood, obtain the breaking force in pounds, nearly, by multiplying together one thousand times the distance of the hole from the end, by the thickness of the tenon, in inches. LENGTH OF IRON NAILS. AND NUMBER TO A POUND. SIZE. LENGTH. NO. SIZE. LENGTH. NO. 3 d 1 J in. 420 io d 3 in - 65 4 d \\ in. 270 12 d 34 in. 5* 5 mmended. He used it on air-pumps telescopes, and various other apparatus : — Take any quantity of good lard, and, to every pound or so, add of common resin ("rosin") an amount about equal to half the size of an egg, or less — a little more or less is of no consequence. Melt them slowly together, stirring as they cool. Apply this with a cloth, or otherwise, just enough to give a thin coating to the metal surface to be pro- tected. It can be wiped off nearly clean from surfaces where it will be un- desirable, as in the case of knives and forks, etc. The resin prevents rancidity, and the mixture obviates the ready access of air and mo'sture. A fresh appli- cation may be needed when the coat ng is washed off by the friction of beating storms, or otherw se. There was talk of patenting this recipe, at one time, but Prof. Olmstead dec ded to publ sh it for the general good. 44 CARPENTRY PLATE 20. Exhibits the operation of finding the curve of what are termed, among carpenters, sprung mouldings for circular cornices. The stuff from which they are obtained is thinner than if the angular piece were worked on the moulding. These mouldings require brackets, as at Fig. i, placed at proper distances, either in a straight or curved line. If they are curved, the moulding will require to be bent as in cover- ing the frustrum of a cone. Figure 2. — Represents the plan and elevation of a cir- cular moulding. To find the radius to describe the curve, produce B D to C: from the point C as centre, describe the curves required. The curve of the moulding, when in position, is shown at D H, and will require to be kerfed at proper distances, a rule for which is given in plate 7, Fig. 2. Figure 3. — Exhibits the elevation of an Ogee cornice. The centres from which the curves are described are found in the same manner as in the preceding figure. A tangent to a circle being given, to find the point of contact. From the centre A, Fig. 4, describe the circle : draw the tangent B D, indefinitely ; bisect A B ; from the point C, describe the arc A B cutting the circle at D, the point required. WOODS. 45 VARIOUS WOODS. The following are interest'ng items concerning the commercial value and properties of the better known woods, as laid down by the American Builder. Elasticity : Ash, hickory, hazel, lancewood, chestnut (smjll), yew, snakewood. Elasticity and toughness : Oak, beech, elm, lignum-vitae, walnut, hornbeam. Even Grain (for carving and engraving) : Pear, pine, box, lime-tree. Durability (in dry works) : Cedar, oak, yellow pine, chestnut. Building (ship-building) : Cedar, pine (deal), fir, larch, elm, oak, locust, teak. Wet construction (as piles, foundations, flumes, etc.): Elm, alden, beech, oak, whitewood, chestnut, ash, spruce, sycamore. Machinery and Millwork (frames): Ash, beech, birch, pine, elm, oak. Rollers, etc. : Box, lignum-vitee, mahogany. Teeth of wheels : Crab-tree, hornbeam, locust. Foundry patterns : Alden, pine, mahogany. Furniture (common): Beech, birch, cedar, cherry, pine, whitewood. Best furniture : Amboyna, walnut, oak, rosewood, satinwood, sandalwood, chestnut, cedar, tulip-wood, zebra-wood, ebony. Of these varieties, those that chiefly enter into commerce in this country are oak, hickory, ash, elm, cedar, black-walnut, maple-cherry, butternut, etc. To Measure Grain Bins. — A cubical box I2| inches each way will hold a bushel. Hence, to ascertain the contents of a bin, take a stick or rule I2| inches long, and divide it by marks into tenths and hundredths. Measure the length, breadth and depth with this rule ; multiply the three dimensions together, and the product will be bushels. This is the most convenient and easiest method known. Use the rule as though it were feet and inches. Suppose, for example, a bin measures 8.5 in length, 5.7 in width, and 4.9 in depth. The product of these is 237.405, or about 237.4 bushels. Every farmer should make such a rule, and use it in all cases where the contents of bins or boxes are required. It is a common thing, when a screw or staple becomes loose, to draw it out, plug up the hole with wood, and re-insert it. It has been found that a much better way is to fill up the holes tightly with cork. Screws and irons so secured are said to remain perfectly tight as long as when put into new wood. To find the length when the width is given, to contain a given number of square feet. For example : required the length of a piece 32 inches wide, to contain 8 square feet. 8 X 1 2 = 9^ X 12 = 1 T 5 2 -!- 3 2 — 3° inches, the length required. A weight of 36,000 pounds attached to a bar of iron, one inch square and 1,000 inches in length, will draw it out one inch ; 45,000 pounds will stretch it two inches ; 54,000 pounds, four inches ; 63,000 pounds, eight inches, and 72,000 pounds, sixteen inches, when it will finally break. 40 CARPENTRY. PLATE 21. Represents the geometrical operation of finding the lines re- quired for the sides and edges of pieces placed at a given angle oblique to the base. — To butt or mitre over obtuse and acute angles. To find the bevels required for the obtuse angle, F G H, Fig. i. Draw the base A B, indefinitely. Draw C D, the angle and height required. To find the angle, to cut the face of the piece C D : from the point C as centre, with C D as radius, describe an arc cutting the base at R ; then R G C forms the angle required. To find the angle to cut the edge of the piece : from the point H as centre, and with H J as radius, describe an arc ; tangent to the arc, and parallel to N J, draw the dotted line to intersect the base at A ; join A G and N F ; then A G C forms the angle required. The bevel required for the butt joint is given at A. Join A G; then A G C forms the angle required. The operation of finding the bevels for the acute-angled plan at Fig. 2 is nearly the same, and consequently needs no explanation. These rules will be found useful to workmen in con- structing boxes where the sides are required to be placed oblique to the base. Also for mitring or butting purlins or other timbers when placed in similar positions. 7 RULES. 4? Miscellaneous Notes & Rules, The greatest force produced by the wind on a vertical wall is equal to 40 lbs. to the square foot. When a summer or beam has settled one-fortieth of its length it is liable to break. Laths for plastering will lay 48 feet to the bundle, equal to 53 square yards. One barrel of lime to one cubic yard of sand, will plas- ter 17 square yards with two coats. It requires 14 bricks to lay 1 foot in length and 1 foot in height of an 8 inch wall; 20 bricks for a 12 inch wall, and 27 bricks for a 16 inch wall. An acre of ground is 2083 feet square, and contains 43,560 square feet. In water, sound passes 4,766 feet per second ; in air, 1,146 feet per second. A Winchester bushel is 183 inches in diameter, 8 inches deep, and contains 2,1505 cubic inches. A box 16X16 inches square, 8J inches deep, will hold a bushel. A box 12X12 inches square, j\ inches deep, wiii hold half bushel. A box 9X9 inches square, 6\ inches deep, will hold one peck. A box 7X7 inches square, si inches deep, will hold 4 qts., or half peck. A pile of wood 8 feet long, 4 feet wide and 4 feet high, contains one cord=to 128 cubic feet. A cistern 5 feet diameter, and 6 feet deep, will hold 30 barrels, of 32 gallons each. A cistern 6 feet diameter, and 6 feet deep, will hold 39 barrels. A cistern 7 feet diameter, and 6 feet deep, will hold 54 barrels. {Continued on page 49.) 4H CARPENTRY. PLATE 22- FIGURE i. — Represents a geometrical demonstration of finding the side of a square, the area of which shall be equal to the area of the circle. Also to find the side of a cube, the con- tent of which shall be equal to the content of a globe, or ball, as follows : From the point A as centre, with the radius of the cir- cle, describe an arc cutting the circle at C : from the point C as centre, with C D as radius, describe an arc cutting the circle at E. Draw E G parallel to A E, and G S at right angles to A B ; join H S, the side required. To find the side of a cube : from the point F as centre describe an arc cutting the circle at L ; join H L, the side required. Figures 2, 3. — Represent the plan and elevation of a box, the sides of which are placed at different angles. To find the face-bevel for the side 5 L, draw 2 E equal to L 5 : from the point 2 as centre, describe the arc E T ; square over from T to N ; join 2 N, the angle required. The face-bevel for the side 2 3 is given at S, The bevel re- quired to miter the edges is drawn at P. To find the angle for butt joints, draw 6 G at right angles to 2 H : from the points 6 and G as centres, describe arcs touching the lines 2 3 and 2 E ; tangent to the arcs, draw lines from 6 and G, intersecting on the line 2 H, forming the angle required. The bevel to be applied at right angles to the joint. RULES. 49 At the depth of 45 feet the temperature of the earth is uniform throughout the year. Dimensions of drawings for patents in the United States, 8.5x12 inches. The lap of slates varies from 2 to 4 inches ; the standard is assumed to be 3 inches. The pitch of a slate roof should not be less than 1 inch in height to 4 inches in length. According to the last census, there are 2,000 Architects, 350,000 Carpenters, 45,000 Cabinet makers, and 46,000 Carriage makers in the United States. The strength of a horse is equivalent to that of 5 men ; the daily allowance of water for a horse should be 4 gal- lons. Elasticity and Strength. — The component parts of a rigid body adhere to each other with a force which is termed cohesion. Elasticity is the resistance which a body opposes to a change of form. Strength is the resistance which a body opposes to a permanent separation of its parts. A horse can draw upon a plank road three times the load that he can upon an ordinary broken stone or macad- amized road. 50 CARPENTRY. PLATE 23. Exhibits the plan and elevation of the angle brackets required for internal and external angles, formed with a cord or string. To find the points for the pins, to describe the elliptic curve required for the angle bracket, square up from S to H, Fig. 1, equal to B D : from the point H as centre, with S C as radius, describe arcs cutting the major axis at 2 and 3, the points required. Figure 2. — Exhibits the plan and elevation for an inter- nal angle ; the elliptic curve of the bracket is found in the same manner as Fig. 1. FIGURE 3. — Exhibits a geometrical demonstration of finding the centre of a circle when lost. Take any points, A, C, D, equally distant from each other, as centres, from which describe arcs cutting each other ; through the points of intersection draw lines to intersect at J, the point required. To erect a perpendicular from the extremity of-a given line. Draw the line A B, Fig. 4. To find the perpendic- ular B C ; from any point D as centre, with D B as radius, describe an arc, cutting the given line at A; join AD, and extend to C ; join B C, the perpendicular required. 7 Terms Used in Carpentry. Abutment. — The junction or meeting of two pieces of timber, of which the fibres of one extend perpendicular to the joint, and those of the other, parallel to it. Arris. — The line of concourse or meeting- of two surfaces. Back of a Hand-rail. — The upper side of it. Back of a Hip. — The upper edge of a rafter, between the two sides of a hipped roof, formed to an angle, so as to range with the rafters on each side of it. Back-Shutters or Back-Flaps. — Additional breadths, hinged to the front shutters, for covering the aperture compieteiy when required to be shut. Back of a Window. — The board, or wainscoting be- tween the sash-frame and the floor, uniting with the two elbows, and forming part of the finish of a room. When framed, it has commonly a single panel, with mouldings on the framing, corresponding with the doors, shutters, etc., in the apartment in which it is fixed. Basil. — The sloping edge of a chisel, or of the iron of a plane. Batten. — A scantling of stuff from two inches to seven inches in breadth, and from half an inch to one inch and a half in thickness. Baulk. — A piece of fir or deal, from four to ten inches square, being the trunk of a tree of that species of wood, generally brought to a square for the use of building. Bead. — A round moulding commonly made upon the edge of a piece of stuff. Of beads there are two kinds; one flush with the surface, called a quirk-bead, and the other raised, called a cock-bead. Beam. — A horizontal timber, used to resist a force or weight; as a tie-beam, where it acts as a string or chain by its tension ; as a collar-beam, where it acts by compression ; Continued on page 53. 52 CARPENTRY. PLATE 24. Exhibits rules for finding a section of the raking mould to intersect the horizontal moulding, at any angle of elevation, for right-angled buildings. Also for finding a section of the raking moulding for the table, placed at any intermediate point, diverg- ing from the straight line to a right angle. To find a section of the raking moulding to intersect the horizontal moulding for right-angled buildings. At Fig. i, the plan and elevation of the gable are given. Also the horizontal and raking moulds required to intersect each other when in position. The rules for drawing and trans- ferring the distances to form the raking moulding at B Fig. i, are simple geometrical operations which the work man will find no difficulty in comprehending. To find the raking mould for the gable placed on the diverging lines i, 2, 3, etc. Produce B S to C equal to S E. Divide the quadrant F H into any number of parts. Extend the line D F, Fig. i r to G, equal to the develop- ment of the arc F H. Produce the lines E S and G H, to intersect each other: from the point of intersection draw the radiating lines 11, 22, etc.; join GE: parallel to GE, di~aw lines from the points 1, 2, 3, etc., to intersect the line EF: from the points of intersection, draw lines parallel to E D, cutting the line F D at the points 1, 2, 3, etc.; join S 1, S 2, etc. Then the line S 1 is the angle of elevation from which to draw the raking mould for the gable S D E, Fig. 1, placed on the diverging line S J, Fig. 2. The angle of elevation from which to draw the raking mould for the gable S D E, placed on any of the diverging lines, is found at the corresponding figures on the line F D,. Fig. 1. Note. — If the horizontal moulding were continued in the straight line S H, though elevated to the angle of the gable, it would not require a change of form. But if the elevated line were to diverge from the straight line, it would begin to form the right angle, and consequently commence to chnnge its form from the horizontal to the raking mould required for the right angle. Figure 3. — To find a veneer for a Gothic head-jamb splayed alike all around. Produce the splay from B to A, the radius to describe the veneer required to cover the circular jamb. TERMS USED IN CARPENTRY. 53 as a bressicmmer, where it resists a transverse insisting weight. Bearer. — Anything used by way of support to another. Bearing. — The dr-tance in which a beam or rafter is suspended in the clear ; thus, if a piece of timber rests upon two opposite walls, the span of the void is called the bearing, and not the whole length of the timber. Bench. — A platform supported on four legs, and used for planing upon, etc. Bevel. — One side is said to be bevelled with respect to another, when the angle formed by these two sides is greater or less than a right angle. Bird's Mouth. — An interior angle, formed on the end of a piece of timber, so that it may rest firmly upon the exterior angle of another piece. Blade. — Any part of a tool that is broad and thin ; as the blade of an axe, of an adze, of a chisel, etc.; but the blade of a saw is generally called a plate. Blockings. — Small pieces of wood, fitted in, or glued, •or fixed, to the interior angle of two boards or other pieces, in order to give strength to the joint. Board. — A substance of wood contained between two parallel planes ; as when the baulk is divided into several pieces by the pit saw, the pieces are called boards. The •section of boards is sometimes, however, of a triangular, or rather trapezoidal, form ; that is, with one edge very thin ; these are called feather-edged boards. Bond-Timbers. — Horizontal pieces, built in stone or brick walls, for strengthening them, and securing the bat- tening, lath, plaster, etc. Bottom Rail. — The lowest rail of a door. Boxings of a Window. — The two cases, one on each side of a window, into which the shutters are folded. Brace. — A piece of slanting timber, used in truss-par- titions, or in framed roofs, in order to form a triangle, and thereby rendering the frame immovable ; when a brace is used by way of support to a rafter, it is called a strut. [Continued on page 55.) 54 CARPENTRY. PLATE 25. Exhibits rules for finding the lines to cut the sides and edges of a piece placed at a given angle oblique to the base. — To miter over right, acute and obtuse angles. Draw the acute angle ABC, Fig. i ; join B D, the line of intersection ; draw I J, the pitch required. At right angles to I J draw I A; from the point I as centre, with I J and I A as radii, describe the arcs J K and AG; tangent to the arcs, draw lines parallel to A B, indefinitely ; from the point B draw a line at right angles to A B, cutting the tangents in L and H ; join L D and H D, the angles required. The bevel for the sides of the piece is shown at H ; for the edges of the piece, at L. Figures 2 and 3 are examples of obtuse and right-angled figures; the operation of finding the angles for the bevels is the same. Fig. 4 represents the rule for finding the lines for a butt joint. The bevel to be applied at right angles to the lines on the sides of the piece. TERMS USED IN CARPENTRY. 55 Braces, in partitions and spanroofs, are always, or should be, disposed in pairs and placed in opposite directions. Brace and Bits. — The same as stock and bits, as ex- plained hereafter. Brad. — A small nail, having no head except on one edge. The intention is to drive it within the surface of the wood by means of a hammer and punch, and to fill the cavity flush to the surface with putty. Breaking Down, in sawing, is dividing the baulk into boards or planks ; but, if planks are sawed longitudinally, through their thickness, the saw-way is called a ripping-cut and the former a breaking-cut. To Break-in. — To cut or break a hole in brick-work, with the ripping chisel, for inserting timber, etc. Breaking Joint. — Is the joint formed by the meeting of several heading joints in one continued line, which is sometimes the case in folded doors. Bressummer or Breastsummer.— A beam supporting a superincumbent part of an exterior wall, and running longitudinally below that part. — See Summer. Bridged Gutters. — Gutters made with boards sup- ported below with bearers, and covered over with lead. Bridging Floors. — Floors in which bridging joists are used. Bridging Joists. — The smallest joints in naked floor- ing, for supporting the boarding for walking upon. Butting Joint. — The junction formed by the surfaces of tw r o pieces of wood, of which one surface is perpen- dicular to the fibres, and the other in their direction, or making with them an oblique angle. Chamber. — The convexity of a beam upon the upper edge, in order to prevent its becoming straight or con- cave by its own weight, or by the burden it may have to sustain, in course of time. Chamber Beams. — Those beams used in the flats of truncated roots, and raised in the middle with an obtuse angle, for discharging the rain-water towards both sides of the roof. (Continued on page 57.) 56 CARPENTRY. PLATE 26. FIGURE i. — Represents the geometrical operation of finding the curve and length of the body or side of a circular pan. Also the side of a square pail the content of which shall be equal to the content of the circular pan. To find the side of a square pan. From the point F as centre, with the radius of the circle, describe an arc cutting the circle at H: from the point H as centre, with H S as radius, describe an arc cutting- the circle at J : draw J R parallel to D F, and R P at right angles to G F ; join G P, the side required. The angle for the joints is given at A. To find the curve required for the body of the circular pan, produce the sides C A and D B to intersect at E: from the point of intersection, describe arcs from D and B, indefinitely. To find the length of the body, join P L: then from the point D as centre, with P L as radius, describe an arc cutting the curve at N, one-fourth of the length required.* Figure 2. — Represents three circles, and three inscribed squares. The second square equals half the area of the first ; the third square equals one-fourth the area of the first square. The same rule applies to the circles. An inspection of the figure is sufficient for its comprehension. Figure 3. — Shows a practical rule for finding the bevels for mitering pieces placed oblique to the base. Draw A B, the angle required ; at right angles to A B, draw B C : from the points A and C as centres, describe the arcs B D and B E ; tangent to the arcs, draw D S and E H ; join A S and C H. The bevel for the face A B is shown at S ; the bevel for the edge is shown at H. If butt joints at the angles are required, join A H for the bevel at A. * Add all necessary material for edges and seams. TERMS USED IN CARPENTRY. 57 Cantalevers. — Horizontal rows of timber, projecting at right angles from the naked part ot a wall, for sustain- ing the eaves or other mouldings. Sometimes they are planed on the horizontal and vertical sides, and sometimes the carpentry is rough and cased with joinery. Carriage of a Stair. — The timber-work which sup- ports the steps. Carcase of a Building. — The naked walls and the rough timber-work of the flooring and quarter partitions, before the building is plastered or the floors laid. Carry-up. — A term used among builders or workmen, denoting that the walls or other parts, are intended to be built to i certain given height: thus, the carpenter will say to the brick-layer, Carry-up that wall ; carry-up that stack of chimneys ; which means, build up that wall or stack of chimneys. Casting or warping. — The bending of the surfaces of a piece of wood from their original position, either by the weight of the wood, or by an unequal exposure to the weather or by an unequal texture of the wood. Chamfering. — Cutting the edge of any thing, origi- nally right-angled, aslope or bevelled. Clamp. — A piece of wood fixed to the end of a thin board, by mortise and tenon, or by groove and tongue, so that the fibres of the one piece, thus fixed, traverse those of the board, and by this means prevent it from casting : the piece at the end is called a clamp, and the board is said to be clamped. Clear Story Windows are those that have no transom. Cross-Grained Stuff is that which has its fibres run- ning in contrary positions to the surfaces ; and, conse- quently, cannot be made perfectly smooth, when planed in one direction, without turning it or turning the plane, Crown-Post— The middle post of a trussed roof.— See King-Post. {Continued on page 59.) 58 CARPENTRY. PLATE 27. Exhibits the plan and elevation of a circular desk. Also the plan and elevation of a circular seat. Figures i, 2. — Represent the plan and elevation of the circular desk. To find the radii of the arcs required for the ribs to form the drum, to bend the circular inclining top, A B C D, Fig. 2. Draw G H, the angle shown on the elevation, Fig. 1. Square over from I to L ; also from H to M. From the points L and M as centres, describe arcs touching the line G H ; tangent to the arcs, and at right angles to GH, draw N J and R K, the radii required. To find the centres from which to describe the ribs. From the points A and B as centres, with R K as radius, describe arcs cutting each other at P, the centre required ; from which describe the arc A S B for the rib placed over the chord A B. The rib placed over the chord C D is found in the same manner. The ribs wilL require beveling at the points of contact, A, B. Figure 3. — Represents the piece required for the cir- cular inclining top. The radii to describe the outside and inside curves are taken from G I and G H, Fig. 2. The radiated lines shown on the piece are grooves for the keys required to shape the piece. Figures 4, 5. — Exhibit the plan and elevation of a circular seat with an inclining back. The rules for find- ing the radii to describe the seat and back pieces, placed parallel to each other when in position, are the same as those used for finding the veneer for a Gothic head-jamb* splayed alike all around. / TERMS USED IN CARPENTRY. 5$ Curling Stuff. — That which is occasioned by the winding or coiling- of the fibres round the boughs of the tree, when they begin to shoot from the trunk. Deal Timber. — The timber of the fir tree, as cut into boards, planks, etc., for the use of building. Discharge. — A post trimmed up under a beam, or part of a building which is weak or overcharged by weight. Door-Frame. — The surrounding case of a door, into which, and out of which, the door shuts and opens. Dormer, or Dormer Window. — A projecting window in the roof of a house ; the glass frame, or casements, be- ing set vertically, and not in the inclined sides of the roofs : thus dormers are distinguished from skylights, which have their sides inclined to the horizon. Drag. — A door is said to drag when it rubs on the floor. This arises from the loosening of the hinges, or the settling of the building. Dragon-Beam. — The piece of timber which supports the hip-rafter, and bisects the angle formed by the wall- plates. Dragon-Piece. — A beam bisecting the wall-plate, for receiving the heel or foot of the hip-rafters. Edging. — Reducing the edges of ribs or rafters, exter* nally or internally, so as to range in a plane, or in any curved surface required. Enter. — When the end of a tenon is put into a mortise, it is said to enter the mortise. Face-Mould. — A mould for drawing the proper figure of a hand-rail on both sides of the plank ; so that when cut by a saw, held at a required inclination, the two sur- faces of the rail-piece, when laid in the right position, will be everywhere perpendicular to the plan. Fang. — The narrow part of the iron of any instrument which passes into the stock. Feather-edged Boards.— Boards, thicker at one edge than the other, and commonly used in the facing of wooden walls, and for the covering of inclined roofs, etc. (Continued on page 61.) 60 CARPENTRY. PLATE 28. Exhibits the operation of finding the angle rafter for French Roofs. The plan and elevation of the common raftefs are shown at Figs, i and 2. To find the major and minor axes of the elliptic curve required for the angle rafter A B, Fig. r. Draw A D at right angles to A B, equal to S H, Fig. 2; from the point D draw a line parallel to A B, indefinitely. Through the point P draw C R parallel to D A, equal to P B ; then C D equals half of the major axis, and C R equals half of the minor axis, of the elliptic curve required. To find the points for the pins, to describe the elliptic curve: from the point R as centre, with C D as radius, describe arcs cutting the major axis at 2 and 3, the points required. To form the angle rafter by ordinates, draw any number, 11, 22, etc. ; transfer the distances, and through the points trace the elliptic curve required. Figure 3. — Represents a simple and easy rule for find- ing the section of a semi-cylinder cut at a given angle oblique to the base. From the points A, B, C, on the plan, draw lines at right angles to A C, indefinitely. Draw D E, the angle required ; also the oblique angle C F. To find the direction of the major axis, set off from 1 to 2, equal to 3,4; from the point 2 square up to 5, equal to 3 B ; join 1, 5, the minor axis; through the point 1 draw the major axis at right angles to 1, 5, indefinitely. To find the points for the pins, to describe the semi- ellipse: from the point 5 as centre, with 1 D as radius, describe arcs cutting the major axis at 6 and 7, the points required. / TERMS USED IN CARPENTRY. 61 Fence of a Plane. — A guard which obliges it to work to a certain horizontal breadth from the arris. Filling-in Pieces. — Short timbers less than the full length ; as the jack-rafters of a roof, the puncheons or short quarters, in partitions, between braces and sills, or head pieces. Fine-set. — A plane is said to be fine-set, when the sole of the plane so projects as to take a very thin broad shaving. Fir Poles. — Small trunks of fir trees, from ten to six- teen feet in length, used in rustic buildings and out- houses. Free Stuff. — That timber or stuff which is quite clean, or without knots, and works easily without tearing. Frowy Siuff. — The same as free stuff. Furrings. — Slips of timber nailed to joists or rafters, in order to bring them to a level, and to range them into a straight surface, when the timbers are sagged, either by casting, or by a set which they have obtained by their weight, in length of time. Girder. — The principal beam in a floor for supporting the binding joists. Glue. — A tenacious viscid matter, which is used as a cement, by carpenters, joiners, etc. Grind-Stone. — A cylindrical stone, by which, on its being turned round its axis, edge-tools are sharpened, by applying the basil to the convex surface. Ground-Plate or Sill. — The lowest plate of a wood- en building for supporting the principal and other posts. Grounds. — Pieces of wood concealed in a wall, to which the facings or finishings are attached, and having their surfaces flush with the plaster. Handspike. — A lever for carrying a beam, or other body, the weights being placed in the middle, and sup- ported at each end by a man. Hanging Stile. — The stile of a door or shutter to which the hinge is fastened ; also, a narrow stile fixed to the jamb on which a door or shutter is frequently hung. {Continued on page 63.) C2 carpentry PLATE 29. Mitring of Circular Mouldings. Some twenty years have elapsed since I first published the House Carpenter's Assistant, in which rules were given for the mitring- of circular mouldings. The idea, I think, originated with me. It being seldom that the work- man is required to perform the operation of mitring cir- cular mouldings, yet if he ever should be, a knowledge of the rules here given will make it an agreeable occupation rather than an unpleasant task attended with anxiety and uncertainty. Figure i. — Represents the rule for finding the centres, from which to describe the intersecting line, and is appli- cable to all cases. Figure 4. — Shows how nearly impossible it is to accu- rately perform work of this kind, without the use of the compasses for describing the intersecting lines. / TERMS USED IN CARPENTRY. 63 Hip-Roof. — A roof the ends of which rise immediately from the wall-plate, with the same inclination to the hori- zon, and its other two sides. The Backing of a Hip is the angle made on its upper edge to the range with the two sides or planes of the roof between which it is placed. Hoarding. — An enclosure of wood about a building, while erecting or repairing. Jack-Rafters. — All those short rafters which meet the hips. Jack Ribs. — Those short ribs which meet the angle ribs, as in groins, domes, etc. Jack Timber. — A timber shorter than the whole length of other pieces in the same range. Inter-tie or Enter-tie. — A horizontal piece of timber, framed between two posts, in order to tie them together. Joggle-Piece. — A truss post, with shoulders and sockets for abutting and fixing the lower ends of the struts. Joists. —Those beams in a floor which support, or are necessary in the supporting of, the boarding or ceiling ; as the binding, bridging and ceiling joists ; girders are, however, to be excepted, as not being joists. Juffers. — Stuff of about four or five inches square, and of several lengths. This term is out of use, though fre- quently found in old books. Kerf. — The way which a saw makes in dividing a piece of wood into two parts. King-Post. — The middle post of a trussed roof, for supporting the tie-beam at the middle and the lower ends of the struts. Knee. — A piece of timber cut at an angle, or having grooves to an angle. In hand-railing a knee is part of the back, with a convex curvature, and therefore the reverse of a ramp, which is hollow on the back, now called over or under easing. Knot. — That part of a piece of timber where a branch had issued out of the trunk. {Continued on page 65.) 64 CARPENTRY. PLATE 30. The designs drawn in this plate are given to show what can be done with the saw, a few chisels and the plough^ which are all the tools required to construct the sash. Although the bead and rosette, placed in the panels, will add very much to the appearance of the door, the work- man requires no sash tools to construct the sash, or mould- ing planes for the door, such as are necessary to make the common paneled doors and sash now in general use. The sash need not be over one and one-fourth inches in thick- ness. The splays can be tinted, and when done by an artistic painter, present both taste and style in the design ; to add to which, the door may be tinted and shaded in three or four colors. The designs here given can be seen in the building now occupied by the author in Newark, N. J. TERMS USED IN CARPENTRY. 65 Lining of a Wall. — A timber boarding-, of which the edges are either rebated or grooved and tongued. Lintels. — Short beams over the heads of doors and windows, for supporting the inside of an exterior wall ; and the super-incumbent part over doors, in brick or stone partitions. Lower Rail. — The rail at the foot of a door next to the floor. Lying Panel. — A panel with the fibres of the wood dis- posed horizontally. Margins or Margents. — The flat part of the stiles and rails of framed work. Middle Rail. — The rail of a door which is upon a level with the hand when hanging freely and bending the joint of the wrist. The lock of the door is generally fixed in this rail. Mitre. — If two pieces of wood be formed to equal an- gles, or if the two sides of each piece form an equal in- clination, and two sides, one of each piece, be joined to- gether at their common vertex so as to make an angle, or an inclination double that of either piece, they are said to be mitred together, and the joint is called the viitre. Mortise and Tenon. — The tenon, in general, may be taken at about one-third of the thickness of the stuff. When the mortise and tenon are to lie horizontally, as the juncture will thus be unsupported, the tenon should not be more than one-fifth of the thickness of the stuff; in order that the strain on the upper surface of the tenoned piece may not split off the under cheek of the mortise. When the piece that is tenoned is not to pass the end of the mortised piece, the tenon should be reduced one- third or one-fourth of its breadth, to prevent the necessity of opening one side of the tenon. As there is always some danger of splitting the end of the piece in which the mortise is made, the end beyond the mortise should, as often as possible, be made considerably longer than it is intended to remain ; so that the tenon may be driven tightly in, and the superfluous wood cut off afterwards. {Continued on page 67.) ORNAMENTAL WORK. PLATE 31. Exhibits the method of constructing a Corinthian truss. A represents the eye of its volute at large, with the centres numbered on which the curves are described. B and C are geometrical views showing the front and side elevation. A careful inspection of which will enable the workman to construct one of any size he may require. PLASTERER'S WORK. The measuring and valuation of plasterer's work is con- ducted by surveyors. All common plastering is measured by the yard square, of nine feet ; this includes the par- titions and ceilings of rooms, stuccoing, internally and externally, etc., etc. Cornices are measured by the foot superficial, girting their members to ascertain their widths, which multiplied by their lengths, will produce the super- ficial contents. Running measures consist of beads, quirks, arrises, and small mouldings. Ornamental cornices are frequently valued in this way ; that is, by the running foot. The labor in plasterer's work is frequently of more con- sideration than the materials ; hence it becomes requisite to note down the exact time which is consumed in effect- ing particular portions, so that an adequate and proper value may be put upon the work. ! TERMS USED IN CARPENTRY. 67 But the above regulations may be varied, according as the tenoned or mortised piece is weaker or stronger. The labor of making deep mortises, in hard wood, may be lessened, by first boring a number of holes with the auger in the part to be mortised, as the compartments be- tween may then more easily be cut away by the chisel. Before employing the saw to cut the shoulder of a tenon, in neat work, if the line of its entrance be correctly determined by nicking the place with a paring chisel, there will be no danger of the wood being torn at the edges by the saw. As the neatness and durability of a juncture depend entirely on the sides of the mortise coming exactly in con- tact with the sides of the tenon ; and, as this is not easily performed when a mortise is to pass entirely through a piece of stuff, the space allotted for it should be first of all correctly gauged on both sides. One half is then to be cut from one side, and the other half from the opposite side ; and as any irregularities, which may arise from an error in the direction of the chisel, will thus be confined to the middle of the mortise, they will be of very little hindrance to the exact fitting of the sides of the mortise and tenon. Moreover, as the tenon is expanded by wedges after it is driven in, the sides of the mortise may, in a small degree, be inclined towards each other, near the shoulders of the tenon. Mullion OR Munnion. — A large vertical bar of a win- dow frame, separating two casements, or glass-frames, from each other. Vertical mullions are called munnions ; and those which extend horizontally are transoms. Muntins OR MONTANTS. — The vertical pieces of the frame of a door between the stiles. Naked Flooring.— The timber-work of a floor for sup- porting the boarding or ceiling, or both. Newel.— The post, in a dog-legged stairs, where the winders terminate, and to which the adjacent string-boards are fixed, {Continued on page 69.) STAIRS. PLATE 32- To build stairs, the workman will first get the size ot the room and the height of the story which determines the width of the steps and risers ; the length of which and the size of the opening are a matter of taste or con- venience. The cylinder is staved up and secured with glue and screws. The string pieces are secured to the cylinder in the same manner. To find the development or stretchout of the cylinder,. Fig. i, describe the arcs 2, 3, and 4, 5 ; from the point 5,. draw the diagonal 5, 7, at an angle of 45 ; then 7, 8, equals one-half of the semi-circle that forms the cylinder ; set off from 2 to A and B equal to 7, 8 ; draw the elevation of the steps and risers, Fig. 2, below and above the plat- form. Then the front string-piece should be wide enough to receive suitable width of timber to support the stairs. Form the easing on the stretchout of the cylinder, which completes the elevation for a platform stairs. Figure 3. — Is an elevation of the cylinder and easing for a straight flight of stairs. The back string-piece should be mortised about y% of an inch deep, and large enough to receive wedges, glued, to secure the steps and risers. The workman should in all cases imagine that he sees what he wants and can do it ; now suppose we place the centre of the cylinder, Fig. 2, over the centre of the plan, then bring the lower and upper ends of the string-pieces around until the lines from A and B stand over the points 9 and 8, and the steps correspond with the steps on the plan, which will be the case if executed according to the drawings. TERMS USED IN CA R TEN TRY. 60 Ogee. — A moulding-, the transverse section of which consists of two curves of contrary flexure. Panel. — A thin board having all its edges inserted in the groove of a surrounding frame. Pitch of a Roof. — The inclination which the sloping sides make with the plane, or level of the wall-plate ; or it is the ratio which arises by dividing the span by the height. Thus, if it be asked : What is the pitch of such a roof? the answer is, one-quarter, one-third, or half. When the pitch is half, the roof is a square, which is the highest that is now in use, or that is necessary in practice. Plank. — All boards above an inch thick are called planks. Plate. — A horizontal piece of timber in a wall, gener- ally flush with the inside, for resting the ends of beams, joists or rafters, upon ; and, therefore, denominated floor or roof plates. Posts. — All upright or vertical pieces of timber what- ever ; as truss-posts, door-posts, quarters in partitions, etc. Brick Posts. — Intermediate posts in a wooden build- ing, framed between principal posts. Principal Posts. — The corner posts of a wooden building. PuDLAIES. — Pieces of timber to serve the purpose of hand-spikes. Puncheons. — Any short post of timber. The small quarterings in a stud partition, above the head of a door, are also called puncheons. Purlins. — The horizontal timbers in the sides of a roof, for supporting the spars or small rafters. Quartering. — The stud work of a partition. Quarters. — The timbers to be used in stud partitions, bond in walls, etc. Rafters. — All the inclined timbers in the sides of a roof ; as principal rafters, hip rafters, and common rafters ; the latter are called in most countries, spars. Rails. — The horizontal pieces which contain the tenons {Continued 011 page 71.) CARPENTRY. PLATE 33- Exhibits the plan and elevation for a platform stair-case. To find the point to bore for the first short baluster on. the second step. At Fig. i, place the point of the pitch- board ; at B, the centre of the newel post, set up on the rise from C to A equal to the difference in the heights of the newel post and the short baluster, say six inches. Then the riser intersects the rail at the point required. To form the face mould for the wreath. At Fig. 3, place the pitch-board, and draw the pitch line C D ; transfer the distances from the plan, Fig. 2, for the width of the mould at the joints. The elliptical curves for the outside and inside of the mould are drawn with a cord or string. The points for the pins are found in the same manner as in plate 1, Fig. 1. The application of the mould and bevel drawn at Fig. 3, are demonstrated at Fig. 4. The plank sawed square, place the bevel on the joint, and draw the perpendicular line ; set off from the centre of the plank, each way, half the width and thickness of the rail. Apply the mould, and mark the piece for the corners to be removed ; the same operation is required for the opposite side. Tack the mould on the side opposite the corners to be removed. Care should be taken to keep the saw or plane perpen- dicular to the plane of the rail when in position. Re- move the surplus wood on the upper and lower si'tes of the plank, and form the wreaths required at Fig. 5. The casing on second floor terminates half the height of the riser above the point to bore for the first baluster 24.82 area of circle. ) Proof: 28X22=616 ; the square root of which is 24.8193. Add the quotient when it consists of two or more figures. EXAMPLE. Diameter 224-^-14=16 Product 1 6Xn= 176 One-fourth (%) of the circum. 22.4 One-tenth do) of the diam. Side of square ) 16 Quotient added. equal in area to > area of circle. ) 198.56 78 MATHEMATICAL DEMONSTRATION. Proof : 224^176=39424 the square root of which is 198.5547. 4. Eleven-fourteenths (}}) of the area of the circle equals the area of a square whose sides are equal to the circum- ference. 5. Seven-elevenths (J) of the area of the circle equals the area of an inscribed square. 6. One-fourth {%) of the circumference multiplied by nine (9), the product divided by ten (10), equals the side of an inscribed squaie, nearly. 7. To fi?id the diameter when the circumference is given. Multiply by seven (7) and divide by twenty-two (22.) 8. To find the diameter when the area of the circle is given. Divide by fourteen (14), and multiply the quotient by eleven (11); the square root of the product equals one- fourth {%) of the circumference ; to find the diameter proceed as in Rule 7. These rules give the exact circumference of the circle, where the diameters are 1, 2, 3,4, etc., multiplied by seven (7), with as much certainty as you can find the root of a rational number, and will be found very useful to work- men. 4 »