A treatise on the art of building. Comprising styles of architecture, strength of materials, the theory and practice of the construction of floors, framed girders, roof trusses, rolled-iron beams, tubular-iron girders, cast-iron girders, stairs, doors, windows, mouldings, and cornices; a compend of mathematics. a manual for the practical use of architects, carpenters, stair-builders, and others.

Title | The American House Carpenter |

Author | R. G. Hatfield |

Publisher | John Wiley & Sons |

Year | 1883 |

Copyright | 1880, The Estate Of R. G. Hatfield |

Amazon | The American House Carpenter |

Eighth Edition, Rewritten And Enlarged.

By R. G. Hatfield, Architect, Late Fellow Of The American Institute Of Architects, Member Of The American Society Of Civil Engineers, Etc. Author Of "Transverse Strains."

Edited by O. P. HATFIELD, F.A.I.A., Architect.

- Preface
- Since the publication of the first edition of this work, six subsequent editions have been issued; but, although from time to time many additions . to its pages and revisions of its subject-matter hav...

- Part I. Section I. - Architecture
- 1. - Building Defined. - Building and Architecture are technical terms by some thought to be synonymous; but there is a distinction. Architecture has been defined to be - the art of building; but mo...

- 2. - Antique Buildings; Tower Of Babel
- Building is one of the most ancient of the arts: the Scriptures inform us of its existence at a very early period. Cain, the son of Adam, builded a city, and called the name of the city after the na...

- 3. - Ancient Cities And Monuments
- Historical accounts of ancient cities, such as Babylon, Palmyra, and Nineveh of the Assyrians; Sidon, Tyre, Aradus, and Serepta of the Phoenicians; and Jerusalem, with its splendid temple, of the Isra...

- 4. - Architecture In Greece
- Among the Greeks, architecture was cultivated as a fine art. Dignity and grace were added to stability and magnificence. In the Doric order, their first style of building, this is fully exemplified. P...

- 5. - Architecture In Rome
- While the Greeks illustrated their knowledge of architecture in the erection of their temples and other public buildings, the Romans gave their attention to the science in the construction of the many...

- 7. - Architecture Debased
- The Goths and Vandals overran Italy, Greece, Asia, and Africa, destroying most of their works of ancient architecture. Cultivating no art but that of war, these savage hordes could not be expected to ...

- 8. - The Ostrogoths
- Theodoric, a friend of the arts, who reigned in Italy from A.D. 493 to 525, endeavored to restore and preserve some of the ancient buildings; and erected others, the ruins of which are still seen at V...

- 11. - The Moors
- When the Arabs and Moors destroyed the kingdom of the Goths, the arts and sciences were mostly in possession of the Musselmen-conquerors; at which time there were three kinds of architecture practised...

- 13. - Architecture Progressive
- The styles erroneously termed Gothic were distinguished by peculiar characteristics as well as by different names. The first symptoms of a desire to return to a pure style in architecture, after the r...

- 14. - Architecture In Italy
- In the 14th and 15th centuries, architecture in Italy was greatly revived. The masters began to study the remains of ancient Roman edifices; and many splendid buildings were erected, which displayed a...

- 15. - The Renaissance
- The Italian masters and numerous artists who had visited Italy for the purpose, spread the Roman style over various countries of Europe; which was gradually received into favor in place of the pointed...

- 16. - Styles Of Architecture
- It is generally acknowledged that the various styles in architecture were the results of necessity, and originated in accordance with the different pursuits of the early inhabitants of the earth; and ...

- 30. - Proportion 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 ...

- 37. - To Describe The Ionie Volute
- Draw a perpendicular from a to s (Fig. 4), and make a s equal to 20 min. or to 4/7 of the whole height, a c; draw s 0 at right angles to s a, and equal to 1 1/4 min.; upon 0, with 2 1/2 min. for radiu...

- 39. - Persians And Caryatides
- In addition to the three regular orders of architecture, it was customary among the Greeks and other nations to employ representations of the human form, instead of columns, to support entablatures; t...

- 42. - Roman Styles
- Strictly speaking, Rome had no architecture of her own; all she possessed was borrowed from other nations. Before the Romans exchanged intercourse with the Greeks, they possessed some edifices of cons...

- 43. - Grecian Orders Modified By The Romans
- The orders of Greece were introduced into Rome in all their perfection. But the luxurious Romans, not satisfied with the simple elegance of their refined proportions, sought to improve upon them by la...

- 46. - Buildings In General
- In selecting a style for an edifice, its peculiar requirements must be allowed to govern. Fig. 12. - Modified Tuscan Order. Fitness Of Styles That style of architecture is to be preferred in whic...

- 47. - Expression
- Every building should manifest its destination. If it be intended for national purposes, it should be magnificent - grand; for a private residence, neat and modest; for a banqueting-house, gay and spl...

- 48. - Durability
- Europeans express surprise that we build so much with wood. And yet, in a new country, where wood is plenty, that this should be so is no cause for wonder. Still the practice should not be encouraged....

- 50. - Arranging The Slains And Windows
- There should be at least as much room in the passage at the side of the stairs as upon them; and in regard to the length of the passage in the second story, there must be room for the doors which open...

- 51. - Principles Of Architecture
- To build well requires close attention and much experience. The science of building is the result of centuries of study. Its progress towards perfection must have been exceedingly slow. In the constru...

- 52. - Arrangement
- In all designs for buildings of importance, utility, durability, and beauty, the first great principles, should be pre-eminent. In order that the edifice be useful, commodious, and comfortable, the ar...

- 53. - Ventilation
- Attention should be given to such arrangements as are calculated to promote health: among these, ventilation is by no means the least. For this purpose, the ceilings of the apartments should have a re...

- 54. - Stability
- To secure this, an edifice should be designed upon well-known geometrical principles: such as science has demonstrated to be necessary and sufficient for firmness and durability. It is well, also, tha...

- 55. - Decoration
- The elegance of a design, although chiefly depending upon a just proportion and harmony of the parts, will be promoted by the introduction of ornaments, provided this be judiciously performed; for enr...

- 56. - Elementary Parts Of A Building
- The builder should be acquainted with the principles upon which the essential, elementary parts of a building are founded. A scientific knowledge of these will insure certainty and security, and enabl...

- 63. - Hooke's Theory Of An Arch
- The simplest theory of an arch supporting itself only is that of Dr. Hooke. The arch, when it has only its own weight to bear, may be considered as the inversion of a chain, suspended at each end. The...

- 64. - Gothic Arches
- Besides these arches, various others are in use. The acute or lancet arch, much used in Gothic architecture, is described usually from two centres outside the arch. It is a strong arch for supporting ...

- Section II. - Construction
- Art. 70. - Construction Essential. - 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 o...

- 71. - Laws Of Pressure
- (1.) A heavy body always exerts a pressure equal to its own weight in a vertical direction. Example: Suppose an iron ball weighing 100 lbs. be supported upon the top of a perpendicular post (Fig. 22-A...

- 72. - Parallelogram Of Forces
- This relation between lines and pressures is applicable in ascertaining the pressures induced by known weights throughout any system of framing. The parallelogram b e df is called the Parallelogram of...

- 74. - Inclination Of Supports Unequal
- 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...

- 75. - The Strains Exceed The Weights
- Thus the united pressures upon the two inclined supports always exceed the weight. In the last case, 2 3/4 tons exert a pressure of 2 1/2 and 2 tons, equal together to 4 1/2 tons; and in the former ca...

- 76. - Minimum Thrust Of Rafters
- Ordinarily, as in roofs, the load is not concentrated, it being that of the framing itself. Here the amount of the load will be in proportion to the length of the rafter, and this will increase with t...

- 77. - Practical Method Of Determining Strains
- A comparison 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 ...

- 78. - Horizontal Thrust
- In Fig. 24, the weight W presses the struts in the direction of their length; their feet, n n, therefore, tend to move in the direction n 0, and would so move were they not opposed by a sufficient res...

- 79. - Position Of Supports
- Figs. 26 and 27 exhibit two methods of supporting the equal weights, W and W. Let it be required to measure and compare the strains produced on the pieces, A B and A C. Construct the parallelogram of ...

- 82. - Ties And Struts
- Timbers in a state of tension are called ties, while such as are in a state of compression are termed struts. This subject can be illustrated in the following manner: Let A and B (Fig. 30) represent ...

- 83. - To Distinguish Ties From Struts
- This may be done by the following rule. In Fig. 22-B, the timbers C and D are the sustaining forces, and the weight W is the straining force; and if the support be removed, the straining force would m...

- 85. - Centre Of Gravity
- The centre of gravity of a uniform prism or cylinder is in its axis, at the middle of its length; that of a triangle is in a line drawn from one angle to the middle of the opposite side, and at one th...

- 88. - Effect On Bearings
- When a uniformly loaded beam is supported at each end on level bearings (the beam itself being either horizontal or inclined), the amount of pressure caused by the load on each point of support is equ...

- 90. - Quality Of Materials
- Materials used in construction are constituted in their structure either of fibres (threads) or of grains, and are termed, the former fibrous, the latter granular. All woods and wrought metals are fib...

- 91. - Manner Of Resisting
- In the strain applied to a post supporting a weight imposed upon it (Fig. 36), we have an instance of an essay to shorten the fibres of which the timber is composed. The strength of the timber in this...

- 92. - Strength And Stiffness
- The strength of materials is their power to resist fracture, while the stiffness of materials is their capability to resist deflection or sagging. A knowledge of their strength is useful, in order to ...

- 94. - Resistance To Compression
- The following table exhibits the results of experiments made to test the resistance to compression of such woods as are in common use in this country for the purposes of construction. Table: Resistan...

- 95. - Resistance 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 resistance to tension. In fibrous materials, this force wil...

- Values Of Materials For Cross-Strains. 96. - Resistance To Transverse Strains
- In the following 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 aut...

- 97. - Resistance To Compression
- The resistance of materials 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 o...

- 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 requir...

- 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 11. - Divide the given weight in p...

- 101. - Rupture By Sliding
- In this the second class of rupture by compression, it has been ascertained by experiment that the resistance is in proportion to the area of the surface separated without regard to the form of the su...

- 102. - The Limit Of Weight
- 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 ...

- 103. - Area Of Surface
- To ascertain the area of surface that is required to sustain a given weight safely, when the weight tends to split off the part pressed against, by causing, in case of fracture, one surface to slide o...

- 105. - Stout Posts
- These comprise the third class of objects 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 t...

- 106. - The Limit Of Weight
- 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...

- 107. - Area Of Post
- To find the area of the cross-section 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, th...

- 110. - The Limit Of Weight
- To ascertain the weight that can be sustained safely by a post the height of which is at least ten times its least side if rectangular, or ten times its diameter if round, the direction of the pressur...

- 113. - Thickness Of A Rectangular Post
- This may be definitely ascertained when the proportion which the thickness shall bear to the breadth shall have been previously determined. For example, when the proportion is as 6 to 8, then 1 1/3 ti...

- 114. - Breadth Of A Rectangular Post
- When the thickness 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 coincides...

- 115. - Resistanee To Tension
- In Art. 95 are recorded the results of experiments made to test the resistance of various materials to tensile strain, showing in each case the capability to such resistance per square inch of section...

- 116. - The Limit Of Weight
- To ascertain the weight or pressure that may be safely applied to a beam or rod as a tensile strain, we have - Rule XII. - Multiply the area of the cross-section of the beam or rod in inches by the v...

- 117. - Sectional 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 - Rule XII I. - Multiply the given weight in pounds by the factor o...

- 118. - Weight Of The Suspending Piece Included
- Pieces subjected to a tensile strain are frequently suspended vertically. In this case, at the upper end, the strain is due not only to the weight attached at the lower end, but also to the weight of ...

- 119. - 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 s...

- 121. - Location 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 a...

- 124. - Breadth Of Beam With Safe Load
- By a simple transposition of the factors in equation (20.), we obtain - b=Wal/Bd2 (21.) a rule for the breadth of the beam. Therefore, to ascertain what should be the breadth of a beam of given dep...

- 125. - Depth Of Beam With Safe Load
- A transposition of the factors in equation (21.), and marking it for extraction of the square root, gives a rule for the depth of a beam. Therefore, to ascertain what should be the depth of a beam of ...

- 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 decreases gradually as it approaches one of the bearings, and w...

- 123. - Weight Uniformly Distributed
- 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. Therefor...

- 130. - Load 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 equally distributed ov...

- 133. - Deflection; Relation To Weight
- When a load is placed upon a beam supported at each end, the beam bends more or less; the distance that the beam descends under the operation of the load, measured at the middle of .its length, is ter...

- 133. - Deflection; Relation To Weight. Continued
- 137. - Deflection: when Weight is at middle. - By a transposition of the factors in (32.), we obtain - = wl3/Fbd3, (33.) a rule by which the deflection of any given beam may be ascertained, an...

- 142. - Deflection Of Levers
- The deflection of a lever is the same as that of a beam of the same breadth and depth, but of twice the length, and loaded at the middle with a load equal to twice that which is at the end of the leve...

- 151. - Floors Described
- Floors are most generally con, structed single; that is, simply a series of parallel beams, each spanning the width of the building, as seen at Fig. 39, Occasionally floors are constructed double, as ...

- 152. - Floor-Beams
- The size of floor-beams can be ascertained 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 ma...

- 153. - Floor-Beams For Dwellings
- To find the dimensions 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 ...

- 156. - Framed Openings For Chimneys 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 trimmer...

- 157. - Breadth 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 ...

- 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 a...

- 859. - Breadth Of Carriage-Beams Carrying Two Sets Of Tail-Beams
- A rule for this is the same as that for a carriage-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 becom...

- 161. - Cross-Bridging, Or Herring-Bone Bridging
- The diagonal struts set between floor-beams, as in Fig. 43, are known as cross-bridging, or herringbone bridging. By connecting the beams thus at intervals, say, of from 5 to 8 feet, the stiffness of ...

- 163. - Crirtlcrs
- 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 suppor...

- 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 inten...

- 166. - Solid Timber Floors For Dwellings And Assembly-Rooms
- From Transverse Strains, Art. 702, we have - d3=(82+yd)l3/0.576F, which may be modified so as to take this form: (64.) which is a rule for the depth or thickness of solid timber floors for dwelli...

- 167. - Solid Timber Floors For First-Class Stores
- The equation given for first-class stores, in Transverse Strains, Art. 702, is - d3=(263+yd)l3/.768 F which may be changed to this form: (65.) in which y is as before, and k for - Georgi...

- 168. - Rolled-Iron Beams
- The dimensions of iron beams, whether wrought or cast, are to be ascertained by the rules already given, when the beams are of rectangular form in their cross-section; these rules are applicable alike...

- 168. - Rolled-Iron Beams. Part 2
- W= 186000 I/lmn= 186000 x 292.0.5 x 1.5/25x10x15=21728.52; or, the required weight is, say, 21,730 pounds. 173. - Rolled-Iron Beams: Dimensions; Weight at any Point. - By transposition of fact...

- 168. - Rolled-Iron Beams. Part 3
- 180. - Floor - Arches; Tie-Rods: Dwellings. - From Transverse Strains, Art. 507, we have - (79). which is a rule for the diameter in inches of a tie-rod for an arch in the floor of a bank, office ...

- 168. - Rolled-Iron Beams. Part 4
- a' = U mn/2 dkl = 120000 mn/2x3 1/2x9000x50=0.038095 mn. Now, when m = n = 25, we have the middle point; then - a' = 0.038095 m n = 0.038095 x 25 x 25 = 23.81; or, the area of the bottom flange at ...

- 188. - Tubular Iron Girders, For Floors Of Dwellings Assembly-Rooms, And Office 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 ...

- 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 horizontal line is a wrought-iron tie-rod attac...

- 194. - Substitute For The Bowstring Girder
- As the cast-iron arch of a bowstring girder serves only to resist compression, 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...

- 195. - 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, s...

- 196. - Framed Girders
- Girders of solid timber are useful for the support of floors only where posts are admissible as supports, at intervals of from 8 to 15 feet. For unobstructed long spans it becomes requisite to constru...

- 196. - Framed Girders. Continued
- In this last polygon, a peculiarity seems to indicate an error: the line FG has no length; it begins and ends at the same point; or, rather, the polygon is complete without it. This is easily understo...

- 200. - Partitions
- Such partitions as are required for the divisions in ordinary houses are usually formed by timber of small size, termed studs or joists. These are placed upright at 12 or 16 inches from centres, and w...

- 201. - Examples Of Partitions
- Fig. 56 represents a partition having a door in the middle. Its construction is simple but effective. Fig. 57 shows the manner of constructing a partition having doors near the ends. The truss is form...

- 202. - Roofs
- In ancient Norman and Gothic buildings, the walls and buttresses were erected so massive and firm that it was customary to construct their roofs without a tie-beam, the walls being abundantly capable ...

- 203. - Comparison Of Roof-Trusses
- Designs 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 67 by having a horizo...

- 207. - Force Diagram For Truss In Fig 61
- For this truss we have, in Fig. 72, a similar design, properly prepared by weights and lettering; and in Fig. 73 the force diagram appropriate to it. In the construction of this diagram, proceed as d...

- 213. - Planning A Roof
- In designing a roof for a building, the first point requiring attention is the location of the trusses. These should be so placed as to secure solid bearings upon the walls; care being taken not to pl...

- 214. - Load Upon Roof-Truss
- In constructing the force diagram for any truss, it is requisite to determine the points of the truss which are to serve as points of support (see Figs. 70, 72, etc.), and to ascertain the amount of s...

- 216. - Load Upon Tie-Beam
- The load upon the tie-beam must of course be estimated according to the requirements of each case. If the timber is to be exposed to view, the load to be carried will be that only of the tie-beam and ...

- 217. - Roof-Weights In Detail
- The load to be sustained by a roof-truss has been referred to in the previous three articles in general terms. It will now be treated more in detail. But first a few words regarding the slope of the r...

- 218. - Load Per Foot Horizontal
- The weight of the covering as referred to in the last article is the weight per foot on the inclined surface; but it is desirable to know how much per foot, measured horizontally, this is equal to. Th...

- 219. - Weight Of Truss
- The weight of the framed truss will be in proportion to the load and to the span. This, for the weight upon a foot horizontal, will about equal - T = 0.077 Cs; which equals the weight in pounds per ...

- 220. - Weight Of Snow On Roofs
- The weight of snow will be in proportion to the depth it acquires, which will be in proportion to the rigor of the climate of the place where the building is to be erected. Upon roofs of ordinary incl...

- 221. - Effect Of Wind On Roofs
- The direction of wind is horizontal, or nearly so, when unobstructed. Precipitous mountains or tall buildings deflect the wind considerably from its usual horizontal direction. Its power usually does ...

- 222. - Total Load Per Foot Horizontal
- The various items comprising the total load upon a roof are the covering, the truss, the wind, snow, the plastering or other kind of ceiling, and the load which may be deposited upon a floor formed in...

- 223. - Strains In Roof-Timbers Computed
- The graphic method of obtaining the strains, as shown in Arts. 205 to 211, is, for its conciseness and simplicity, to be preferred to any other method; yet, on some accounts, the method of obtaining t...

- 224. - Strains 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 F B the axes of the braces, and D G, FE, and CB th...

- 225. - Application Of The 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 ascertain the loa...

- 227. - The Rafter
- A rafter, like a post, is subject to a compressive force, and is liable to fail in three ways, namely: by flexure, by being crushed, or by crushing the material against which it presses. To render it ...

- 228. - The Braces
- Each brace is subject to compression, 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...

- 229. - The Suspension-Rods
- 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 ...

- 230. - Roof-Beams, Jack-Rafters, And Purlins
- These timbers are subject to loads nearly uniformly distributed, and their dimensions may be obtained by Rule XXX., equation (35.), Art. 140. In this equation, U= cfl (Art. 152). Substituting this val...

- 232. - Roof-Truss With Elevated Tie-Beain
- Designs such as are shown in Fig. 91 have the tie elevated for the accommodation of an arch in the ceiling. This and all similar designs are seriously objectionable, and should always be avoided; as t...

- 234. - The Backing Of The Hip-Rafter
- At any convenient place in be (Fig. 92), as 0, draw m n at right angles to be; from 0, tangical to bh, describe a semicircle, cutting b e in s; join m and s and n and s; then these lines will form at ...

- 235. - Domes
- The usual form for domes is that of the sphere; the base circular. When the interior dome does not rise too high, a horizontal tie may be thrown across, by which any degree of strength required may be...

- 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 seen at Figs. 95 and 96 - the latter being a plan a...

- 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 de 26, ef 18 1/4 1/2,fg 12 1/4, 12 1/4, h i 10 3/4, ij 9 1/2, and the balance, 8 ...

- 240. - 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 a c at right angles to a b; find the stretch-out (Art. 524) of 0 b, and make dc equal to it; divide the ...

- 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 u...

- 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...

- 249. - Timber Joints
- 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 shr...

- Section III. - Stairs
- 250. - Stairs: General Requirements. - 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 ...

- 251. - The Grade 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. He...

- 251. - The Grade Of Stairs. Continued
- (108, A.) (109, A.) For the scale for dwellings, we have, for those occurring between H and D - r = 1/2(34 - t); (108, B.) t = 24 - 2 r; (109, B.) and for those between H and J. we have - (1...

- 253. - Dimensions Of The Pitch-Board
- The first thing in commencing to build a stairs is to make the pitc/i-board; this is done in the following manner: Obtain very accurately, in feet and inches, the rise, or perpendicular height, of the...

- 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, therefore, at 5 or 6 inches, as the case requires, and run it on...

- 255. - Step And Riser 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 inches long, two or three of which, in the length of the step, ...

- 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, ed; and place the centre of the other balus...

- 260. - Regular Winding Stairs
- In Fig. 133, abcd represents 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 a e ...

- 263. - Hand-Railing For Stairs
- A piece of hand-railing intended for the curved part of a stairs, when properly shaped, has a twisted form, deviating widely from plane surfaces. If laid upon a table it may easily be rocked to and fr...

- 266. - A Prism Cut By An Oblique 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, bcf, cdef, and adeg are the vertical sides; while efbg is the top,...

- 268. - Face-Mould For Hand-Railing Of Platform Stairs
- Let j k and l m, Fig. 142, represent the central or axial lines of the hand-rails of the two flights, one above, the other below the platform; and let the semicircle j d l be the central line of the r...

- 269. - More Simple Method For Hand-Rail To Platform Stairs
- In Fig. 144, j'ge represents a pitch-board of the first flight, and d and i the pitch-board of the second flight of a platform stairs, the line e f being the top of the platform; and abc is the plan o...

- 270. - Hand-Railing For A Larger Cylinder
- Fig. 147 represents a plan and a vertical section of a line passing through the centre of the rail as before. From b draw bk parallel to cd; extend the lines id and jc until they meet k b in k and f; ...

- 271. - Faee-Mould Without Canting The Plank
- Instead of placing the platform-risers at the spring of the cylinder, a . more easy and graceful appearance may be given to the rail, and the necessity of canting either of the twists entirely obviate...

- 272. - Railing For Platform Stairs Where The Rake Meets The Level
- In Fig. 150, abc is the plan of a line passing 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 eh in d, and ...

- 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 plank sawed squ...

- 274. - Face-Moulds For Moulded Rails Upon Platform Stairs
- In Fig. 152, a bc is the plan of a line passing through the centre of the rail around the cylinder, as before, and the lines above it are a vertical section of steps, risers, and platform, with the li...

- 275. - Application Of Face-Moulds To Plank
- In Fig. 152 make a drawing, from d to h, of the cross-section of the hand-rail, and tangent to the lower corner draw the line gh The distance between the lines je and g h is the thickness of the plank...

- 276. - Hand-Railing For Circular Stairs
- Let it be required 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 o...

- 277. - Face-Moulds For Circular Stairs
- At Fig. 159 the plan of the newel and the adjacent hand-rail are repeated, but upon an enlarged scale; and in which bb' is the reduced height of the point b, or is equal to bb' less t r, Fig. 158, and...

- 278. - Face-Moulds 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 ck set up th...

- 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 g, but drawn at a larger scale. To prepare for the face-moulds, perp...

- 280. - Hand-Railing For Winding Stairs
- The term winding is applied more particularly to a stairs having steps of parallel width compounded with those which taper in width, as in Fig. 135, and as is here shown in Fig. 162, in which fabc rep...

- 281. - Face-Moulds For Winding 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 horizontal projection of the tangents for the first wreath are ad...

- 282. - Face-Moulds For Winding Stairs, Again
- In the last article, in getting the face-moulds for a winding stairs, the two wreaths are found to be very dissimilar in length. This dissimilarity may be obviated by a judicious location of the butt-...

- 284. - Application Of The 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 whe...

- 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 p...

- 286. - Face-Moulds For Round Rails
- The previous examples given for finding face-moulds are intended for moulded rails. For round rails the same process is to be followed, with this difference: that instead of finding curves on the face...

- 287. - Position Of The 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. Managed ...

- 289. - Centres In Regulating Square
- Let a 2 I 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 1 3/4. Divide two adjacent sides, as a 2 an...

- 290. - Scroll For Hand-Rail Over Curtail Step
- Let a b (Fig, 173) be the given breadth, 1 3/4 the given number of revolutions, and let the relative size of the regulating square to the eye be 1/3 of the diameter of the eye. Then, by the rule, 1 3/...

- 293. - Falling-Mould For Raking Part Of Scroll
- Tangi-cal to the rail at h (Fig. 173) draw h k parallel to da; then ka2 will be the joint between the twist and the other part of the scroll. Make de2 equal to the stretch-out of de, and upon d e2 fin...

- 294. - Face-Mould For The Scroll
- At Fig. 173, from k draw kr2 at right angles to r2 d. at Fig. 172, make h r equal to h2 r2 in Fig. 173, and from r draw r s at right angles tor rh; from the intersection of r s with the level line m q...

- 295. - Form Of Newel-Cap From A Section Of The Rail
- Draw a b (Fig. 176) through the widest part of the given section, and parallel to c d; bisect a b in e, and through a, e, and b draw h i, f g, and k j at right angles to a b; at a convenfent place on ...

- 296. - Boring For Balusters In A Round Rail Before It Is Rounded
- Make the angle o c t (Fig. 177) equal to the angle o c t at Fig. 144; upon c describe a circle with a radius equal to half the thickness of the rail; draw the tangent b d parallel to t c, and complete...

- 297. - The Bevels In Splayed Work
- The principles employed in finding the lines in stairs are nearly allied to those required to find the bevels for splayed work - such as hoppers, bread-trays, etc. A method by which these may be obtai...

- Section IV. - Doors And Windows. Doors. 298. - General Requirements
- 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 materia...

- 300. - Panefls
- Where doors have but two panels in width, let the stiles and muntins be each 1/7 of the width; or, whatever number of panels there may be, let the united widths of the stiles and the muntins, or the w...

- 301. - Trimmings
- Fig. 179 shows a method of trimming doors: a is the door-stud; b, the lath and plaster; c, the ground; d, the jamb; e, the stop; fand g, architrave casings; and h, the door-stile. It is customary in o...

- Windows. 303. - Requirements For Light
- 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 give...

- 305. - Inside Shutters
- Inside shutters folding into boxes 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 show...

- 307. - Circular Heads
- Doors and windows usually terminate 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 req...

- 308. - Form Of Soffit For Cireular Window-Heads
- When the light is received in an oblique direction, let abed (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 ...

- Section V. - Mouldings And Cornices. Mouldings
- 309. - Mouldings: 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 ...

- 310. - Characteristics Of Mouldings
- Neither of these mouldings is peculiar to any one of the orders of architecture; and although each has its appropriate use, yet it is by no means confined to any certain position in an assemblage of m...

- 312. - The Grecian Torus And Scotia
- Join the extremities a and b (Fig. 191), and from f, the given projection of the moulding, draw/0 at right angles to the fillets; from b draw bh at right angles to a b; bisect a b in c; join f and c, ...

- 313. - The Grecian Echinus
- Figs. 192 to 199 exhibit, variously 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 diamet...

- Forms Of Roman Mouldings
- 317. - Roman Mouldings: 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 ...

- Cornices. 319. - Designs For Cornices
- Figs. 232 to 240 are designs for eave cornices, and Figs. 241 and 242 are for stucco cornices for the inside finish of rooms. In some of these the projection of the uppermost member from the facia is ...

- 321. - Cornice Proportioned To A Given Corniee
- Let the cornice at Fig. 244 be the given cornice. Upon any point in the lowest line of the lowest member, as at a, with the height of the required cornice for radius, describe an intersecting arc acro...

- 322. - Angle Bracket In A Built Cornice
- Let A (Fig. 246) be the wall of the building, and B the given bracket, which, for the present purpose, is turned down horizontally. The angle-bracket, C, is obtained thus: through the extremity, a, an...

- 323. - Raking Mouldings Matched With Level Returns
- Let A (Fig. 248) be the given moulding, and A b the rake of the roof. Divide the curve of the given moulding into any number of parts, equal or unequal, as at 1, 2, and 3; from these points draw horiz...

- 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 acquire...

- 325. - Elementary Geometry
- In all reasoning definitions are necessary, in order to insure in the minds of the proponent and respondent identity of ideas. A corollary is an inference deduced from a previous course of reasoning. ...

- 325. - Elementary Geometry. Part 2
- Fig. 256. 341. - Proposition Parallelograms standing upon the same base and between the same parallels are equal. Let. A BCD and EFCD (Fig. 257) be given parallelograms standing upon the same base...

- 325. - Elementary Geometry. Part 3
- Fig. 262. This problem, which is the 47th of the First Book of Euclid, is said to have been demonstrated first by Pythagoras. It is stated (but the story is of doubtful authority) that as a thank-o...

- 325. - Elementary Geometry. Part 4
- Fig. 268. A B C D: B E D F:: Cd: Df. The areas in this particular case are as 4 to 3. But in general the proportion will be as the lengths of the bases. Thus the proposition is proved in regard to...

- Section VII. - Ratio, Or Proportion. 364. - Merchandise
- 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 multiplying the 87 pounds ...

- 368. - Equality Of Ratios
- Each couple is also termed a Ratio, and the two the Equality of Ratios. Thus the ratio 45/9is equal to the ratio 435/87. If the division indicated in these two ratios be actually performed, the equal...

- 370. - Multiplying An Equation
- The quantity on each side of the sign = is called a member of the equation. If each member be multiplied by the same quantity, the equality of the two members is not thereby disturbed (Art. 369); ther...

- 372. - Transferring A Factor
- -Each of the four quantities in the aforesaid equation is termed a factor. Comparing the equation of the last article with that of Art. 43, it appears that the two are alike excepting that the factor ...

- 374. - Homologous Triangles Proportionate
- The discussion of the subject of Ratios has thus far been confined to its relations with the mercantile problem of Art. 364. The rules of proportion or the equality of ratios apply equally to question...

- 375. - The Steelyard
- An example of lour proportionals 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 ste...

- 376. - The Lever Exemplified By 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...

- 377. - The Lever Principle Demonstrated
- The relation between the weights and their arms of leverage may be demonstrated as follows: * Fig. 273. Let A B G H, Fig. 273, represent a beam of homogeneous material, of equal sectional area thr...

- Section VIII. - Fractions. 379. - A Fraction Denned
- 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 fraction show what portion of th...

- 381. - Form Of Fraction Changed By Division
- By an operation the reverse of that in the last article, we may reduce several equal fractions to one of equal value. Thus, if in each we divide the numerator and denominator by the same number, we re...

- 382. - Improper Fractions
- The fractions 9/3, 17/5, 24/3, etc., all fractions which have the numerator larger than the denominator are termed improper fractions. They are not improper arithmetically, but they are so named becau...

- 383. - Reduction Of Mixed Numbers To Fractions
- By an operation the reverse of 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 num...

- 384. - Division Indicated By The Factors Put As A Fraction
- Factors placed in the form of a fraction as3/5,5/3,120/75 or 820/41 indicate division (Art. 382); the denominator (the factor below the line) being the divisor, and the numerator (the factor above th...

- 385. - Addition Of Fractions Having Like Denominators
- Let it be required to add the fractions 1/5 and 2/5. By referring to Art. 379 we see that A D (Fig. 274), is one of the five parts into which the whole line A B is divided; it is, therefore, 1/5 We al...

- 386. - Subtraction Of Fractions Of Like Denominators
- Subtraction is the reverse of addition; therefore, to subtract fractions a reverse operation is required to that had in the process of addition; or simply to subtract instead of adding. For example, ...

- 387. - Dissimilar Denominators Equalized
- The rules just given for the addition and subtraction of fractions require that the given fractions have like denominators. When the denominators are unlike it is required, before adding or substracti...

- 388. - Reduction Of Fractions To Their Lowest 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 e...

- 389. - Least Common Denominator
- To find the hast 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, doub...

- 390. - Least Common Denominator Again
- When the denominators are not divisible by one another, then to obtain a common denominator, it is requisite to multiply to-get her all of the denominators which will not divide any of the other denom...

- 391. - Fractions Multiplied Graphically
- Let A B CD (Fig. 276) be a rectangle of equal sides, or A £ equal A C and each equal one foot. Then A B multiplied by A C will equal the area A B CD, or 1 x 1 = 1 square foot. Let the line E F be par...

- 392. - Fractions Multiplied Graphically
- In Fig. 277 let A B equal 8 feet and A C equal 5 feet; then the rectangle A BCD contains 5 x 8 = 40 feet. The interior lines divide the space included within A BCD into 40 equal squares of one foot ea...

- 393. - Rule For Mutiplication Of Fractions And Example
- In the example given in the last article it will be observed 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 nume...

- 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 (referr...

- 395. - Rule For Division Of Fractions
- The rule just given does not work well when the factors are not commensurable. For example, if it be required to divide 5/7 by 2/9 we have by the above rule - 5 ÷ 2 = 5/2 7 ÷ 9 7/9 . Producing frac...

- Section IX. - Algebra. 396. - Algebra Defined
- 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 togethe...

- 398. - Algebra Useful In Constructing Rules
- In all problems to be solved there are certain conditions or quantities given, by means of which an unknown quantity is to be evolved. For example, in the problem in Art. 397, there were three certain...

- 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, bc/a = d, is applicable to all cases of homologous triangles, however the...

- 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 when taken, it must be firmly adhered to to represent that particu...

- 401. - Arithmetical Processes Indicated By Signs
- In algebra, the four processes of addition, subtraction, multiplication, and division, are frequently required; and when the required process cannot be actually performed upon the letters themselves, ...

- Transferring Symbols. 403. - Transferring A Symbol To The Opposite Member
- 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 equation b appears i...

- 404. - Signs Of Symbols To Be Changed When They Are To Be Subtracted
- As an example in subtraction, let the quantities represented by + b - a - f+ c, be taken from the quantities represented by + a+b - c - f This may be written - (+ a + b - c - f) - (+b - a - /+ c), a...

- 406. - The Least Common Denominator
- When the denominators of algebraic fractions differ it is necessary before addition or subtraction can be performed to harmonize them, as in the reduction of the denominators of numerical fractions (A...

- 407. - Algebraic Fractions Subtracted
- To exemplify the subtraction of fractions, let it be required to find the algebraic sum of a/c-b/d-e/f. These denominators all differ. The fractions, therefore, require to be modified, so that each de...

- 408. - Graphical Representation Of Multiplication
- In Fig. 278, let A BCD, a rectangle, have its sides A B and 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...

- 411. - Graphical Multiplication Of A Binomial
- Let A B C D (Fig. 280) be a rectangular surface, and B E D F another rectangular surface, adjoining the first. The area of the whole figure is evidently equal to - (AB + BE)xAC. The area is also eq...

- 412. - Graphical Squaring Of A Binomial
- Let EGC J (Fig. 281) be a rectangle of equal sides, and within it draw the two lines, A H and F D, parallel with the lines of the rectangle, and at such a distance from them that the sides, A B and B...

- 415. - Plus And Minus Signs In Multiplication
- In previous 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, ...

- 416. - Equality Of Squares On Hypotltenuse And Sides 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 rig...

- 417. - Division The Reverse Of Multiplication
- As division is the reverse of multiplication, so to divide one quantity by another is but to retrace the steps taken in multiplication. If we have the area ab (Fig. 278), and one of the factors a give...

- 421. - Raising A Quantity To Any Power
- When a quantity is required to be multiplied by its equal, the product is called the square of the quantity. Thus a x a = a* (Art. 412). If the square be multiplied by the original quantity the result...

- 422. - Quantities With Negative Exponents
- The series of powers, by division, may be extended backward. Thus, if we divide a5/a= a 4; a4/a = a3; a3/a = a2; a2/a= a1;a1/a = a0; a0/a= a-1; a-1/a = a-2; a-2/a = a-3, etc. In this series we have a/...

- 426. - Extraction Of Radicals
- We have seen that the square of a is a1 x a1 = a2; of 2 a3 is 2 a3 x 2 a3 = 4 a6; in each case the square is obtained by doubling the exponent. To obtain the square root the converse follows, namely,...

- 427. - Logarithms
- We have seen in the last article the nature of fractional exponents. Thus the square root of a5 equals a5/2 which may be put a2 1/2. In this way we may have an exponent of any fraction whatever, as a1...

- 428. - Completing The Square Of A Binomial
- We have seen in Art. 412 that the square of a binomial (a + b) equals a2 + 2 ab + b2 - a trinomial - the first and last terms of which are each the square of one of the two quantities, while the secon...

- Progression. 429. - Arithmetical Progression
- In a series of numbers, as 1, 3, 5, 7, 9, etc., proceeding in regular order, increasing by a common difference, the series is called an arithmetical progression; the quantity by which one number is in...

- 430. - Geometrical Progrression
- A series of numbers, such as 1, 2, 4, 8, 16, 32, 64, 128, 256, etc., in which any one of the terms is obtained by multiplying the preceding one by a constant quantity, is termed a Geometrical Progress...

- Section X. - Polygons. 431. - Relation Of Sum And Difference Of Two Lines
- Let AB and CD (Fig. 285) be two given lines; make EH equal to A B, and HG equal to CD; then E G equals the sum of the two lines. Make FG equal to A B, which is equal to EH. Bisect E G in J; then, ...

- 432. - Perpendicular, In Triangle Of Known Sides
- Let ABC (Fig. 286) be the given triangle, and CE a perpendicular let fall upon A B, the base. Let the several lines of the figure be represented by the symbols a, b, c, d, g, and f as shown. Then, sin...

- 432. - Perpendicular, In Triangle Of Known Sides. Part 2
- and, putting r for the radius of the inscribed circle, we have - (132.) Or: The radius of the inscribed circle of a regular trigon equals the half of a side of the trigon divided by the square r...

- 432. - Perpendicular, In Triangle Of Known Sides. Part 3
- Fig. 290. 436.- - Octagon: Radius of Circumscribed and Inscribed Circles: Area. - Let CEDBF(Fig. 290) represent a quarter of a regular octagon, in which F is the centre, ED a side, and CE and D B eac...

- 432. - Perpendicular, In Triangle Of Known Sides. Part 4
- Now ( Fig. 291) - DF=DC-FC, or - Substituting this value of n, in the above expression, we have - Multiplying by R and reducing, we have - (144.) Or: The radius of the circumscribed circl...

- 432. - Perpendicular, In Triangle Of Known Sides. Part 5
- (148). Or: The radius of the inscribed circle of a regular hecadecagon equals a side of the hecadecagon multiplied by the square root of two quantities, one of which is the square root of 2 added to ...

- 432. - Perpendicular, In Triangle Of Known Sides. Part 6
- 10 For the radius of the circumscribed circle, we have (150.) - R = b/2cos.c, 464. POLYGONS. R = b/cos.54o R = b I/2cos.54o Using a table of logarithmic sines and tangents (Art. 427), we have - ...

- Section XI. - The Circle
- 443. - Circles: 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 ...

- - The Circle. Continued
- To Find The Circumference A = r x b/2n =r/2bn or, the area equals half the radius by a side into the number of sides; or, half the radius into the periphery of the polygon. Now, if a polygon have ver...

- Section XII. - The Ellipse
- 451. - Ellipse: Definitions. - Let two lines, PF, PF' (Fig. 299), be drawn from any point P to any two fixed points FF', and let the point P move in such a manner that the sum of the two lines, PF, ...

- 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::...

- Section XIII. - The Parabola
- 460. - Parabola: Definitions - 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 follows: When cut parallel with ...

- - The Parabola. Part 2
- Therefore, we have - Tr: Ta :: Tp: Tn, TR x TN = TA x TP, but - TR=1/2 TP; therefore - 1/2 TP x TN = TA x TP, 1/2 TN = TA. Or: The snbtangent of a parabola is bisected by the vertex; or is equal...

- - The Parabola. Part 3
- 468. - Parabola: Described from Points. - With given base, NP (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 ...

- Section XIV. - Trigonometry
- 473. - Right-Angled Triangles: 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....

- - Trigonometry. Part 2
- Fig. 317. Having, in this case, the base and perpendicular known, by referring to the above proportions we find that with these two sides we may obtain the tangent; therefore - Tan. B = b/a = 6/8 =0...

- - Trigonometry. Part 3
- To find the two sides: if a be the given side, then to find the side b we have, equation (193.) - b = a sin. B/sin. A; or, the side b equals the product of the side a into the quotient obtained by a...

- - Trigonometry. Part 4
- 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 ...

- Section XV. - Drawing
- 487. - General Remarks A knowledge of the properties and principles of lines can best be acquired by practice. Although the various diagrams throughout this work may be understood by inspection, yet ...

- 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 pri...

- 492. - The T-Square
- A T-square 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 thre...

- 498. - Directions For Drawing
- In drawing a problem, proceed, with the pencil sharpened to a point, to lay down the several lines until the whole figure is completed, observing to let the lines cross each other at the several angle...

- Section XVI. - Practical Geometry. 499. - 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, perfectl...

- Right Lines And Angles
- 500. - To Bisect A Line 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; draw the line cd, and the ...

- 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 c draw any line, as e d, obliquely to the line c a; bisect c d at b; upon b, with the radius be...

- 513. - A Circle And A Tangent Given, To Find The Point Of Contact
- From any point, as a (Fig. 366), in the tangent bc, draw a line to the centre d; bisect ad at e; upon e, with the radius ea, describe the arc afd; f is the point of contact required. Fig. 366. If ...

- 516. - To Describe A Segment Of A Circle By A Set-Triangle
- Let a b (Fig. 369) be the chord, and c d the height of the segment. Secure two straight-edges, or rulers, in the position c e and c f, by nailing them together at c, and affixing a brace from c to f; ...

- 517. - To Find The Radius Of An Arc Of A Circle When The Chord And Versed Sine Are Given
- 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 = (C/2)2+V2/2V, where r is the radiu...

- 518. - To Find The Versed Sine Of An Arc Of A Circle When The Radius And Chord Are Given
- 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 algebraically thus: where r is the radius, v the versed sine, and...

- 519. - To Describe The Segment Of A Circle By Intersc-Tion Of Lines
- Let ab (Fig. 370) be the chord, and cd the height of the segment. Through c draw e f parallel to a b; draw bf at right angles to cb; make ce equal to cf; draw ag and bh at right angles to a b; divide ...

- 521. - In A Given Angle, To Describe A Tanged Curve
- Let a b c (Fig. 372) be the given angle, and 1 in the line a b, and 5 in the line bc, the termination of the curve. Divide 1 b and b 5 into a like number of equal parts, as at 1, 2, 3, 4, and 5; join ...

- 527. - With In A Given Circle, To Insribe An Equilateral Triangle, Hexagon Or Dodecagon
- Let abed (Fig. 380) be the given circle. Draw the diameter bd; upon b, with the radius of the given circle, describe the arc aec; join a and c, also a and d, and c and d - and the triangle is complete...

- 529. - To Find The Side Of A Buttressed Octagon
- Let ABCDE (Fig. 382) represent one quarter of an octagon structure, having a buttress HFGJ at each angle. The distance MH, between the buttresses, being given, as also FG, the width of a buttress; to ...

- 531. - Upon A Given Line To Describe Any Regular Polygon
- Let a b (Figs. 386, 387, and 388) be given lines, equal to a side of the required figure. From b draw bc at right angles to a b; upon a and b, with a b for radius, describe the arcs acd and feb; divid...

- 534. - To Make A Parallelogram Equal To A Given Triangle
- Let abc (Fig. 392) be the given triangle. From a draw a d at right angles to b c; bisect a d in e; through e draw fg parallel to bc, from b and c draw b f and cg parallel to de; then bfgc will be a pa...

- 537. - To Make A Circle Equal To Two Given Circles
- Let A and B (Fig. 396) be the given circles. In the right-angled triangle a bc make a b equal to the diameter of the circle B, and cb equal to the diameter of the circle A; then the hypothenuse a c wi...

- 542. - A Line With Certain Divisions Being Given, To Divide Another, Longer Or Shorter, Given Line In The Same Proportion
- Let A (Fig. 402) be the line to be divided, and B the line with its divisions. Make ab equal to B with all its divisions, as at 1, 2, 3', etc.; from a draw ac at any angle with a b; make a c equal to ...

- 549. - The Axes Being Given, To Describe An Ellipsis With A Trammel
- Let ab and cd (Fig. 406) be the given axes. Place the trammel so that a line passing through the centre of the grooves would coincide with the axes; make the distance from the pencil e to the nut f eq...

- 550. - To Describe An Ellipsis By Ordinates
- Let ab and cd (Fig. 407) be given axes. With ce or ed for radius describe the quadrant fgh; divide fh, ae, and eb, each into a like number of equal parts, as at 1, 2, and 3; through these points draw ...

- 551. - To Describe An Ellipsis By Intersection Of Lines
- Let ab and cd (Fig. 408) be given axes. Through c, draw fg parallel to.ab; from a and b draw af and bg at right angles to ab; divide fa, gb, ae, and eb, each into a like number of equal parts, as at 1...

- 552. - To Describe An Ellipsis By Intersecting Arcs
- Let a b and c d (Fig. 410) be given axes. Between one of the foci, f and f, and the centre e, mark any number of points, at random, as 1, 2, and 3; upon f and f, with b 1 for radius, describe arcs at ...

- 553. - To Describe A Figure Nearly In The Shape Of An Ellipsis, By A Pair Of Compasses
- Let ab and c d (Fig. 411) be given axes. From c draw ce parallel to a b; from a draw ae parallel to cd; join e and d; bisect ea in f; join f and c, intersecting e d in i; bisect ic in o from o draw og...

- 554. - To Draw An Oval In The Proportion Seven By Nine
- Let cd (Fig. 412) be the given conjugate axis. Bisect c d in o, and through o draw ab at right angles to cd; bisect co in e; upon o, with oe for radius, describe the circle efgh; from e, through h and...

- 559. - To Describe An Ellipsis, Whose Axes Shall Be Proportionate To The Axes Of A Larger Or Smaller Given One
- Let a c b d (Fig. 416) be the given ellipsis and axes, and ij the transverse axis of a proposed smaller one. Join a and c; from i draw ie parallel to ac; make of equal to oe; then ef will be the conju...

- Section XVII. - Shadows
- 562. - The Art of Drawing consists in representing solids Upon a plane surface, so that a curious and nice adjustment of lines is made to present the same appearance to the eye as does the human ...

- 563. - The Inclination Of The Line Of Shadow
- This is always, in architectural drawing, 45 degrees, both on the elevation and on the plan; and the sun is supposed to be behind the spectator, and over his left shoulder. This can be illustrated by ...

- 565. - To Find The Line Of Shadow Cast By A Shelf
- In Fig. 425, A is the plan and B is the elevation of a shelf attached to a wall. From a and c draw a b and c d, according to the angle previously directed; from b erect a perpendicular intersecting c ...

- 571. - To Find The Shadow Of A Shelf Curved In The Elevation
- In Fig. 433 find the points of intersection, e, e and e, as in the last examples, and a curve traced through them will define the shadow. Fig. 434 The preceding examples show how to find shadows w...

- 580. - To Find The Shadow Thrown By A Pedestal Upon Steps
- From a (Fig. 442) in the plan, and from c in the elevation, draw the rays ab and ce; then ao will show the extent of the shadow on the first riser, as at A; fg will determine the shadow on the second ...

- 584. - To Find The Shadow Thrown On A Vertical Wall By A Column And Entablature Standing In Advance Of Said Wall
- Cast rays from a and b (Fig. 447), and find the point c as in the previous examples; from d draw the ray de, and from e the horizontal line ef; tangical to the curve at g and h draw the rays gj and h ...

- 586. - Reflected Light
- In shading, the finish and life of an object depend much on reflected light. This is seen to advantage in Fig. 446, and on the column in Fig. 448. Refleeted rays are thrown in a direction exactly the ...

- Table Of Squares, Cubes. And Roots
- (From Hutton's Mathematics.) No. Square. Cube. Sq. Root. CubeRoot. 1 1 1 1.0000000 1.000000 2 4 8 1.4142136 1.25992...

- Rules For The Reduction Of Decimals
- To reduce a fraction to its equivalent decimal. Rule. - Divide the numerator by the denominator, annexing cyphers as required. Example. - What is the decimal of a foot equivalent to three inches? 3 i...

- 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 1 t...

- Table Showing The Capacity Of Wells, Cisterns, Etc
- The gallon of the State of New York, by an act passed April 11, 1851, is required to conform to the standard gallon of the United States government. This standard gallon contains 231 cubic inches. In ...

- Table Of Weights
- Materials Used In The Construction Or Loading Of Buildings Weights per Cubic Foot. As per Barlow, Gallier, Haswell, Hurst, Rankine, Tredgold, Wood and the Author. Material. From To ...

- Carpentry Glossary
- Terms not found here can be found in the lists of definitions in other parts of this book, or in common dictionaries. Abacus The uppermost member of a capital. Abattoir A slaughter-house. Abbey ...

- Carpentry Glossary. Part 2
- 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 va...

- Carpentry Glossary. Part 3
- Festoon An ornament representing a wreath of flowers and leaves. Fillet A narrow flat band, listel, or annulet, used for the separation of one moulding from another, and to give breadth and firmnes...

- Carpentry Glossary. Part 4
- Monastery A building or buildings appropriated to the reception of monks. Monopteron A circular colonnade supporting a dome without an enclosing wall. Mosaic A mode of representing objects by the...

- Carpentry Glossary. Part 5
- The measure to which a piece of timber is to be or has been cut. Scarfing The joining of two pieces of timber by bolting or nailing transversely together, so that the two appear but one. Scotia Th...