^ 
 
THE 
 
 AMERICAN HOUSE CARPENTER. 
 
 A TREATISE 
 
 ON THE 
 
 ART OF BUILDING. 
 
 COMPRISING 
 
 STYLES OF ARCHITECTURE, STRENGTH OF MATERIALS, 
 
 B D 
 
 THE THEORY AND PRACTICE OF THE CONSTRUCTION OF FLOORS, FRAMED 
 
 GIRDERS, ROOF TRUSSES, ROLLED-IRON BEAMS, TUBULAR-IRON 
 
 GIRDKRS, 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. 
 
 EIGHTH EDITION, 
 REWRITTEN AND ENLARGED. 
 
 BY 
 
 R. G. HATFIELD, ARCHITECT, 
 
 l\ 
 
 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. 
 NINTH EDITION. 
 
 NEW YORK: 
 JOHN WILEY & SONS, 15 ASTOR PLACE 
 
 1883. 
 
^/.o 
 
 \ 
 
 COPYRIGHT 1880, 
 BY THE ESTATE OF R. G. HAITI ELD. 
 
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 have been made, still its sev- 
 eral issues have always been printed substantially from the original stereotype 
 plates. In this edition, however, the book has been extensively remodelled 
 and expanded, the greater portion of it rewritten, and the whole put in a new 
 dress by being newly set up in type uniform in style with that of the late 
 author's recent work, Transverse Strains. To this revision a labor of love to 
 him he devoted all the time he could spare from his other pressing engage- 
 ments for a year or more, and by close and arduous application brought the 
 book to a successful termination, notwithstanding the engrossing nature of 
 his customary business avocations. Although essentially an elementary work, 
 and intended originally for a class of minds not generally favored with oppor- 
 tunities for securing a very extended form of education, either in the store of 
 
 * <r * N x* N: 
 
 information acquired or in the discipline of mind which culture confers, still 
 it has been his aim to embody in its pages so complete and exhaustive a treat- 
 ment of the various subjects discussed, and so practical and useful a collection 
 of data and the rules governing their application, as to make it also not un- 
 worthy the attention of those who have been more highly favored in that 
 respect. 
 
 In all the various trades connected with building it is the intelligent 
 workman that commands the greatest respect, and who receives in all cases 
 the highest remuneration. As apprentice, journeyman, and master-builder, 
 his course is upward and onward, and success crowns his efforts in all that he 
 undertakes. There is a kind of freemasonry in the very air that surrounds 
 the skilful, intelligent man, that gives him a pass at once into the appreciation 
 and recognition of all those whose regard is valuable. We admire and respect 
 the plodding toil of the honest, patient laborer, whose humble task may tax 
 his muscles though not his mind, but we yield a deeper homage to the skilful 
 hand and tutored eye that accomplish wonders in art and science through per- 
 severance in aspiring studies. It was to excite in the minds of workmen like 
 these an ambition to excel in their calling, and to point out to them the surest 
 path to -that consummation, that the preparation of this volume was under- 
 taken ; that all its tendencies are in that direction, and that it cannot well fail 
 
11 PREFACE. 
 
 of its purpose when judiciously used, must be the conviction of all who will 
 take the trouble to examine its pages. 
 
 In the first part of the book matters more particularly relating to building 
 are treated of. The first section is in the nature of an introduction, serving 
 by its historical references to excite an interest in the general subject, while 
 in the second are presented the methods of erecting edifices in accordance 
 with the acknowledged principles of sound construction. In the remaining 
 sections of Part I. the several well-defined branches of house-building, as 
 stairs, doors and windows, etc., are illustrated and explained. In the second 
 part the more useful rules and simple problems of mathematics are reduced 
 to an easily acquired form, and adapted to the necessities of the ordinary 
 workman. By studying the latter, the young mechanic may not only improve 
 and strengthen his mind, but grow more self-reliant daily, demonstrating in 
 his own experience that scientific knowledge gives power. By carefully 
 studying this part of the book he will see how easy it is to acquire the knowl- 
 edge of solving problems by signs and symbols, commonly called Algebra 
 (although looked upon by the uninitiated as almost incomprehensible), and 
 thus find it easy to understand all the illustrations of the various subjects 
 wherein those condensed forms of expression are used. Useful problems in 
 geometry, described in simple^ language, and hints upon the subject of draw- 
 ing and shading, are also to be found in Part II. A glossary of architectural 
 terms and many useful tables are provided in the Appendix, and finally, an 
 Index is added to aid in referring to special subjects. The full-plate illustra- 
 tions are inserted to make it attractive to the general reader, and at the same 
 time to serve as explanatory of the historical portion of the volume. 
 
 It will not be denied that the class of information herein furnished is one 
 of the most instructive and useful that can.be presented to the practical mind 
 of a workingman, or to any mind engaged in mechanical pursuits. The im- 
 press stamped upon it by the author's peculiar line of study is not to be 
 effaced, but this has given it characteristics of originality and strength not 
 to be found in a mere compilation. 
 
 THE EDITOR. 
 
 NEW YORK, 31 Pine Street, 
 January 6, 1880. 
 
CONTENTS. 
 
 For a Table of Contents more in detail, see p. 613, and for Index, see p. 657. 
 
 PART I. 
 
 PAGE. 
 
 SECTION I. Architecture 5 
 
 II. Construction 57 
 
 III. Stairs 240 
 
 " IV. Doors and Windows 315 
 
 " V. Mouldings and Cornices 323 
 
 PART II. 
 
 SECTION VI. Geometry 347 
 
 VII. Ratio or Proportion 366 
 
 " VIII. Fractions 378 
 
 IX. Algebra 392 
 
 X. Polygons 439 
 
 XL The Circle 468 
 
 ' XII. The Ellipse 481 
 
 " XIII. The Parabola 49 2 
 
 " XIV. Trigonometry 5 IQ 
 
 XV. Drawing 53^ 
 
 " XVI. Practical Geometry 544 
 
 " XVII. Shadows..... S9& 
 
 Table of Contents in detail . . 6l 3 
 
iv .CONTENTS. 
 
 APPENDIX. 
 
 Glossary 627 
 
 Table of Squares, Cubes and Roots 638 
 
 Rules for the Reduction of Decimals 647 
 
 Table of Circles 649 
 
 Table showing the Capacity'of Wells, Cisterns, etc 653 
 
 Table of the Weights of Materials 654 
 
 Index 657 
 
((UNIVERSITY 
 
UNIVERSITY 
 
 PART I. 
 
 SECTION L ARCHITECTURE. 
 
 ART. I. 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 more correctly it is "the 
 art of designing and constructing buildings, in accordance 
 with such principles as constitute stability, utility, and 
 beauty." The literal signification of the Greek word arclii- 
 tecton, from which the word architect is derived, is chief- 
 carpenter ; and the architect who designs and builds well 
 may truly be considered the chief builder. Of the three 
 classes into which architecture has >een divided viz., Civil, 
 Military, and Naval the first is that which refers to the 
 construction of edifices known as dwellings, churches, and 
 other public buildings, bridges, etc., for the accommodation 
 of civilized man and is the subject of the remarks which 
 follow. . 
 
 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 name of his son, Enoch ;" but of the peculiar style 
 or manner of building we are not informed. It is presumed 
 that it was not remarkable for beauty, but that utility and 
 perhaps stability were its characteristics. Soon after the 
 deluge that memorable event, which removed from ex- 
 istence all traces of the works of man the Tower of Babel 
 
6 ARCHITECTURE. 
 
 was commenced. This was a work of such magnitude that 
 the gathering of the materials, according to some writers, 
 occupied three years; the period from its commencement 
 until the work was abandoned was twenty-two years ; and 
 the bricks were like blocks of stone, being twenty feet long, 
 fifteen broad, and seven thick. Learned men have given it 
 as their opinion that the tower in the temple of Belus at 
 Babylon was the same as that which in the Scriptures is 
 called the Tower of Babel. The tower of the temple of 
 Belus was square at its base, each side measuring one 
 furlong, and consequently half a mile in circumference. Its 
 form was that of a pyramid, and its height was 660 feet. It 
 had a winding passage on the outside from the base to the 
 summit, which was wide enough for two carriages. 
 
 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 Israelites show that architecture among them had 
 made great advances. Ancient monuments of the art are 
 found also among other nations ; the subterraneous temples 
 of the Hindoos upon the islands Elephanta and Salsetta ; 
 the ruins of Persepolis in Persia ; pyramids, obelisks, tem- 
 ples, palaces, and sepulchres in Egypt all prove that the 
 architects of those early times were possessed of skill and 
 judgment highly cultivated. The principal characteristics 
 of their works are gigantic dimensions, immovable solidity, 
 and, in some instances, harmonious splendor. The extra- 
 ordinary size of some is illustrated in the pyramids of Egypt. 
 The largest of these stands not far from the city of Cairo : 
 its base, which is square, covers about ir acres, and its 
 height is nearly 500 feet. The stones of which it is built 
 are immense the smallest being full thirty feet long. 
 
 4. Architecture in Greece. Among the Greeks, archi- 
 tecture 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. 
 Phidias, Ictinus, and Calicrates are spoken of as masters in 
 
GRECIAN AND ROMAN BUILDINGS. 7 
 
 the art at this period : the encouragement and support of 
 Pericles stimulated them to a noble emulation. The beauti- 
 ful temple of Minerva, called the Parthenon, erected upon 
 the acropolis of Athens, the Propyleum, the Odeum, and 
 others, were lasting monuments of their success. The Ionic 
 and Corinthian orders were added to the Doric, and many 
 magnificent edifices arose. These exemplified, in their 
 chaste proportions, the elegant refinement of Grecian taste. 
 Improvement in Grecian architecture continued to advance 
 until perfection seems to have been attained. The speci- 
 mens which have been partially preserved exhibit a com- 
 bination of elegant proportion, dignified simplicity, and 
 majestic grandeur. Architecture among the Greeks was at 
 the height of its glory at the period immediately preceding 
 the Peloponnesian war ; after which the art declined. An 
 excess of enrichment succeeded its former simple grandeur ; 
 yet a strict regularity was maintained amid the profusion of 
 ornament. After the death of Alexander, 323 B.C., a love 
 of gaudy splendor increased : the consequent decline of the 
 art was visible, and the Greeks afterwards paid but little 
 attention to the science. 
 
 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 
 aqueducts and sewers with which Rome abounded ; build- 
 ing no such splendid edifices as adorned Athens, Corinth, 
 and Ephesus, until about 200 years B.C., when their inter- 
 course with the Greeks became more extended. Grecian 
 architecture was introduced into Rome by Sylla ; by whom, 
 as also by Marius and Caesar, many large edifices were 
 erected in various cities of Italy. But under Csesar Augus- 
 tus, at about the beginning of the Christian era, the art 
 arose to the greatest perfection it ever attained in Italy. 
 Under his patronage Grecian artists were encouraged, and 
 many emigrated to Rome. It was at about this time that 
 Solomon's temple at Jerusalem was rebuilt by Herod a 
 Roman. This was 46 years in the erection, and was most 
 probably of ' the Grecian style of building perhaps of the 
 
CATHEDRAL OF NOTRE 
 
THE GOTHS AND VANDALS. 9 
 
 provement ; but very soon after his reign the art began 
 rapidly to decline, as particularly evidenced in the mean 
 and trifling character of the ornaments. 
 
 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 take any interest in the beautiful forms and proportions 
 of their habitations. From this time architecture assumed 
 an entirely different aspect. The celebrated styles of Greece 
 were unappreciated and forgotten ; and modern architec- 
 ture made its first appearance on the stage of existence. 
 The Goths, in their conquering invasions, gradually ex- 
 tended it over Italy, France, Spain, Portugal, and Ger- 
 many, into England. From the reign of Galienus may be 
 reckoned the total extinction of the arts among the Romans. 
 From this time until the sixth or seventh century, architec- 
 ture was almost entirely neglected. The buildings which 
 were erected during this suspension of the arts were very 
 rude. Being constructed of the fragments of the edifices 
 which had been demolished by the Visigoths in their unre- 
 strained fury, and the builders being destitute of a proper 
 knowledge of architecture, many sad blunders and exten- 
 sive patch-work might have been seen in their construction 
 entablatures inverted, columns standing on their wrong 
 ends, and other ridiculous arrangements characterized their 
 clumsy work. The vast number of columns which the ruins 
 around them afforded they used as piers in the construction 
 of arcades which by some is thought, after having passed 
 through various changes, to have been the origin of the 
 plan of the Gothic cathedral. Buildings generally, which 
 are not of the classical styles, and which were erected after 
 the fall of the Roman empire, have by some been indiscrim- 
 inately included under the term Gothic. But the changes 
 which architecture underwent during the Mediaeval age 
 show that there were then several distinct modes of building. 
 
 8. The Otrogoths. Theodoric, a friend of the arts, 
 who reigned in Italy from A.D. 493 to 525, endeavored to 
 
10 ARCHITECTURE. 
 
 restore and preserve some of the ancient buildings; and 
 erected others, the ruins of which are still seen at Verona 
 and Ravenna. Simplicity and strength are the character- 
 istics of the structures erected by him ; they are, however, 
 devoid of grandeur and elegance, or fine proportions. 
 These are properly of the GOTHIC style ; by some called 
 the old Gothic, to distinguish it from the pointed Gothic. 
 
 9. The Lombard*, who ruled in Italy from A.D. 568, 
 had no taste for architecture nor respect for antiquities. 
 Accordingly, they pulled down the splendid monuments of 
 classic architecture which they found standing, and erected 
 in their stead huge buildings of stone which were greatly 
 destitute of proportion, elegance, or utility their charac- 
 teristics being scarcely anything more than stability and 
 immensity combined with ornaments of a puerile character. 
 Their churches were decorated with rows of small columns 
 along the cornice of the pediment, small doors and win- 
 dows with circular heads, roofs supported by arches having 
 arched buttresses to resist their thrust, and a lavish display 
 of incongruous ornaments. This kind of architecture is 
 called the LOMBARD style, and was employed in the seventh 
 century in Pavia, the chief city of the Lombards ; at which 
 city, as also at many other places, a great many edifices 
 were erected in accordance with its peculiar forms. 
 
 10. Tlic Byzantine Architects, of Byzantium, Constan- 
 tinople, erected many spacious edifices; among which are 
 included the cathedrals of Bamberg, Worms, and Mentz, 
 and the most ancient part of the minster at Strassburg ; in 
 all of these they combined the classic styles with the crude 
 Lombardian. This style is called the LOMBARD-BYZANTINE. 
 To the last style there were afterwards added cupolas sim- 
 ilar to those used in the East, together with numerous slen- 
 der pillars with elaborate capitals, and the many minarets 
 which are the characteristics of the proper Byzantine, or 
 Oriental style. 
 
 H. The Moor*. When the Arabs and Moors destroyed 
 the kingdom of the Goths, the arts and sciences were mostly 
 
E LI 
 
 DIVERSITY) 
 
 >, ^ K / 
 
 MOSQUE AT CAIRO. 
 
THE MEDLEVAL STYLES. II 
 
 in possession of the Musselmen-conquerors ; at which time 
 there were three kinds of architecture practised ; viz. : the 
 Arabian, the Moorish, and the Lombardian. The ARABIAN 
 style was formed from Greek models, having circular arches 
 added, and towers which terminated with globes and mina- 
 rets. The MOORISH is very similar to the Arabian, being 
 distinguished from it by arches in the form of a horseshoe. 
 It originated in Spain in the erection of buildings with the 
 ruins of Roman architecture, and is seen in all its splendor 
 in the ancient palace of the Mohammedan monarchs at 
 Grenada, called the Alhambra, or red-house. The style which 
 was originated by the Visigoths in Spain by a combination 
 of the Arabian and Moorish styles, was introduced by Charle- 
 magne into Germany. On account of the changes and im- 
 provements it there underwent, it Was, at about the I3th or 
 I4th century, termed the German or romantic style. It is ex- 
 hibited in great perfection in the towers of the minster of 
 Strassburg, the cathedral of Cologne and other edifices. 
 The most remarkable features of this lofty and aspiring style 
 are the lancet or pointed arch, clustered pillars, lofty towers, 
 and flying buttresses. It was principally employed in eccle- 
 siastical architecture, and in this capacity introduced into 
 France, Italy, Spain, and England. 
 
 12. Ttie Architecture of England: is divided into the 
 Norman, the Early-English, the Decorated, and the Perpendic- 
 ular styles. The Norman is principally distinguished by 
 the character of its ornaments the chevron, or zigzag, being 
 the most common. Buildings in this style were erected in 
 the 1 2th century. The Early-English is celebrated for the 
 beauty of its edifices, the chaste simplicity and purity of 
 design which they display, and the peculiarly graceful char- 
 acter of its foliage. This style is of the isth century. The 
 Decorated style, as its name implies, is characterized by a 
 great profusion of enrichment, which consists principally of 
 the crocket, or feathered-ornament, and ball-flower. It was 
 mostly in use in the Hth century. The Perpendicular style, 
 which dates from the I5th century, is distinguished by its 
 high towers, and parapets surmounted with spires similar in 
 number and grouping to oriental minarets. 
 
12 ARCHITECTURE. 
 
 13. Architecture Progresiiive. The styles erroneously 
 termed Gothic were distinguished by peculiar characteris- 
 tics as well as by different names. The first symptoms of a 
 desire to return to a pure style in architecture, after the 
 ruin caused by the Goths, was manifested in the character 
 of the art as displayed in the church of St. Sophia at Con- 
 stantinople, which was erected by Justinian in the 6th 
 century. The church of St. Mark at Venice, which arose 
 in the loth or nth century, is a most remarkable building; 
 a compound of many of the forms of ancient architecture. 
 The cathedral at Pisa, a wonderful structure for the age, 
 was erected by a Grecian architect in 1016. The marble 
 with which the walls of this building were faced, and of 
 which the four rows of columns that support the roof are 
 composed, is said to be of an excellent character. The 
 Campanile, or leaning-tower as it is usually called, was 
 erected near the cathedral in the I2th century. Its inclina- 
 tion is generally supposed to have arisen from a poor foun- 
 dation ; although by some it is said to have been thus con- 
 structed originally, in order to inspire in the minds of the 
 beholder sensations of sublimity and awe. In the I3th cen- 
 tury, the science in Italy was slowly progressing ; many fine 
 churches were erected, the style of which displayed a de- 
 cided advance in the progress towards pure classical archi- 
 tecture. In other parts of Europe, the Gothic, or pointed 
 style was prevalent. The cathedral at Strassburg, designed 
 by Irwin Steinbeck, was erected in the I3th and I4th cen- 
 turies. In France and England during the I4th century, 
 many very superior edifices were erected in this style. 
 
 14-. Architecture in Italy. In the I4th and 1 5th cen- 
 turies, architecture in Italy was greatly revived. The mas- 
 ters began to study the remains of ancient Roman edifices ; 
 and many splendid buildings were erected, which displayed 
 a purer taste in the science. Among others, St. Peter's of 
 Rome, which was built about this time, is a lasting monu- 
 ment of the architectural skill of the age. Giocondo, Mi- 
 chael Angelo, Palladio, Vignola, and other celebrated archi- 
 tects, each in their turn, did much to restore the art to its 
 
INTERIOR OF ST 
 
 SOPHIA, CONSTANTINOPLE. 
 
-gjpl Lift 
 
 Y^ 
 
 UNIVERSITY] 
 
ORIGIN OF STYLES. 13 
 
 former excellence. In the edifices which were erected under 
 their direction, however, it is plainly to be seen that they 
 studied not from the pure models of Greece, but from the 
 remains of the deteriorated architecture of Rome. The 
 high pedestal, the coupled columns, the rounded pediment, 
 the many curved-and-twisted enrichments, and the convex 
 frieze, were unknown to pure Grecian architecture. Yet 
 their efforts were serviceable in correcting, to a good de- 
 gree, the very impure taste that had prevailed since the over- 
 throw of the Roman empire. 
 
 15. The Renaissance. The Italian masters and numer- 
 ous 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 Gothic. 
 This fell into disuse ; although it has of late years been 
 again cultivated. It requires a building of great magnitude 
 and complexity for a perfect display of its beauties. In 
 America, the pure Grecian style was at first more or less 
 studied ; and perhaps the simplicity of its principles would 
 be better adapted to a republican country than the more 
 intricate mediaeval styles ; yet these, during the last quarter 
 of a century, have been extensively studied, and now wholly 
 supersede the Grecian styles. 
 
 16. Style of Arehiteeture. It is generally acknowl- 
 edged 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 were 
 brought by their descendants to their present state of per- 
 fection, through the propensity for imitation and desire of 
 emulation which are found more or less among- all nations. 
 Those that followed agricultural pursuits, from being em- 
 ployed constantly upon the same piece of land, needed a 
 permanent residence, and the wooden hut was the offspring 
 of their wants ; while the shepherd, who followed his flocks 
 and was compelled to traverse large tracts of country for 
 pasture, found the tent to be the most portable habitation ; 
 again, the man devoted to hunting and fishing an idle and 
 vagabond way of living is naturally supposed to have been 
 
14 ARCHITECTURE. 
 
 content with the cavern as a place of shelter. .The latter is 
 said to have been the origin of'the Egyptian style; while 
 the curved roof of Chinese structures gives a strong indica- 
 tion of their having had the tent for their model ; and the 
 simplicity of the original style of the Greeks (the Doric) 
 shows quite conclusively, as is generally conceded, that its 
 original was of wood. The pointed, or ecclesiastical style, 
 is said to have originated in an attempt to imitate the bower, 
 or grove of trees, in which the ancients performed their idol- 
 worship. But it is more probably the result of repeated 
 scientific attempts to secure real strength with apparent 
 lightness ; thus giving a graceful, aspiring effect. 
 
 17. Order: or styles, in architecture are numerous; 
 and a knowledge of the peculiarities of each is important to 
 the student in the art. An ORDER, in architecture, is com- 
 posed of three principal parts, viz. : the Stylobate, the Col- 
 umn, and the Entablature. This appertains chiefly to the 
 classic styles. 
 
 18. The Stylobate: is the substructure, or basement, 
 upon which the columns of an order are arranged. In 
 Roman architecture especially in the interior of an edi- 
 fice it frequently occurs that each column has a separate 
 substructure ; this is called a pedestal. If possible, the ped- 
 estal should be avoided in all cases ; because it gives to the 
 column the appearance of having been originally designed 
 for a small building, and afterwards pieced out to make it 
 long enough for a larger one. 
 
 19. The Column : is composed of the base, shaft, and 
 capital. 
 
 20. Tlie Entablature: above and supported by the 
 columns, is horizontal ; and is composed of the architrave, 
 frieze, and cornice. These principal parts are again divided 
 into various members and mouldings. 
 
 21. The Base: of a column is so called from basis, a 
 foundation or footing. 
 
^t ^"4j>2\ 
 
 7A > 
 
 UNIVERSITY) 
 
 INTERIOR OF ST. STEPHENS, PARIS. 
 
PARTS OF AN ORDER. 15 
 
 22. The Shaft: the upright part of a column standing 
 upon the base and crowned with the capital, is from shafto, 
 to dig in the manner of a well, whose inside is not unlike 
 the form of a column. 
 
 23. The Capital : from kephale or caput, the head, is the 
 uppermost and crowning part of the column. 
 
 24. The Architrave : from archi, chief or principal, 
 and trabs, a beam, is that part of the entablature which lies 
 in immediate connection with the column. 
 
 25. The Frieze: from fibron, a fringe or border, is that 
 part of the entablature which is immediately above the 
 architrave and beneath the cornice. It was called by some 
 of the ancients zophorus, because it was usually enriched 
 with sculptured animals. 
 
 26. The Corniee: from corona, a crown, is the upper 
 and projecting part of the entablature being also the upper- 
 most and crowning part of the whole order. 
 
 27. The Pediment: above the entablature, is the tri- 
 angular portion which is formed by the inclined edges of 
 the roof at the end of the building. In Gothic architecture, 
 the pediment is called a gable. 
 
 28. The Tympanum: is the perpendicular triangular 
 surface which is enclosed by the cornice of the pediment. 
 
 29. The Attic : is a small order, consisting of pilasters 
 and entablature, raised above a larger order, instead of a 
 pediment. An attic story is the upper story, its windows 
 being usually square. 
 
 30. Proportions in an Order. An order has its several 
 members proportioned to one another by a scale of 60 equal 
 parts, which are called minutes. If the height of buildings 
 were always the same, the scale of equal parts would be a 
 fixed quantity an exact number of feet and inches. But as 
 buildings are erected of different heights, the column and 
 
1 6 ARCHITECTURE. 
 
 its accompaniments are required to be of different dimen- 
 sions. To ascertain the scale of equal parts, it is necessary 
 to know the height to which the whole order is to be 
 erected. This must be divided by the number of diameters 
 which is directed for the order under consideration. Then 
 the quotient obtained by such division is the length of the 
 scale of equal parts and is, also, the diameter of the column 
 next above the base. For instance, in the Grecian Doric 
 order the whole height, including column and entablature, 
 is 8 diameters. Suppose now it were desirable to construct 
 an example of this order, forty feet high. Then 40 feet 
 divided by 8 gives 5 feet for the length of the scale ; and 
 this being divided by 60, the scale is completed. The up- 
 right columns of figures, marked H and P, by the side of 
 the drawings illustrating the orders, designate the height 
 and the projection of the members. The projection of each 
 member is reckoned from a line passing through the axis of 
 the column, and extending above it to the top of the entab- 
 lature. The figures represent minutes, or 6oths, of the 
 major diameter of the shaft of the column. 
 
 31. Grecian Styles. The original method of building 
 among the Greeks was in what is called the Doric order : 
 to this were afterwards added the Ionic and the Corinthian. 
 These three were the only styles known among them. Each 
 is distinguished from the other two by not only a peculiar- 
 ity of some one or more of its principal parts, but also by a 
 particular destination. The character of the Doric is robust, 
 manly, and Herculean-like ; that of the Ionic is more deli- 
 cate, feminine, matronly; while that of the Corinthian is 
 extremely delicate, youthful, and virgin-like. However 
 they may differ in their general character, they are alike 
 famous for grace and dignity, elegance and grandeur, to a 
 high degree of perfection. 
 
 32 The Doric Order: (Fig. 2,) is so ancient that its 
 origin is unknown although some have pretended to have 
 discovered it. But the most general opinion is, that it is 
 an improvement upon the original wooden buildings of the 
 
FANCIFUL ORIGIN OF THE DORIC. I/ 
 
 Grecians. These no doubt were very rude, and perhaps 
 not unlike the following figure. 
 
 FIG i. SUPPOSED ORIGIN OF DORIC TEMPLE. 
 
 The trunks of trees, set perpendicularly to support the 
 roof, may be taken for columns ; the tree laid upon the 
 tops of the perpendicular ones, the architrave ; the ends 
 of the cross-beams which rest upon the architrave, the 
 triglyphs ; the tree laid on the cross-beams as a support for 
 the ends of the rafters, the bed-moulding of the cornice ; the 
 ends of the rafters which project beyond the bed-moulding, 
 the mutules ; and perhaps the projection of the roof in 
 front, to screen the entrance from the weather, gave origin 
 to the portico. 
 
 The peculiarities of the Doric order are the triglyphs 
 those parts of the frieze which have perpendicular channels 
 cut in their surface ; the absence of a base to the column 
 as also of fillets between the flutings of the column ; and the 
 plainness of the capital. The triglyphs should be so dis- 
 posed that the width of the metopes the space between 
 the triglyphs shall be equal to their height. 
 
 33. The Intercolumniation : or space between the col- 
 umns, is regulated by placing the centres of the columns 
 under the centres of the triglyphs except at the angle of 
 the building ; where, as may be seen in Fig. 2, one edge of 
 
18 
 
 ARCHITECTURE. 
 
 FIG. 2. GRECIAN DORIC. 
 
PECULIARITIES OF THE DORIC. 19 
 
 the triglyph must be over the centre of the column.* 
 Where the columns are so disposed that one of them stands 
 beneath every other triglyph, the arrangement is called 
 mono-triglyph and is most common. When a column is 
 placed beneath every third triglyph, the arrangement is 
 called diastyle ; and when beneath every fourth, arceostyle. 
 This last style is the worst, and is seldom adopted. 
 
 34-. The Doric Order: is suitable for buildings that 
 are destined for national purposes, for banking-houses, etc. 
 Its appearance, though massive and grand, is nevertheless 
 rich and graceful. The Patent Office at Washington, and 
 the Treasury at New York, are good specimens of this 
 order. 
 
 35. The Ionic Order. (Fig. 3.) The Doric was for 
 some time the only order in use among the Greeks. They 
 gave their attention to the cultivation of it, until perfection 
 seems to have been attained. Their temples were the prin- 
 
 * GRECIAN DORIC ORDER. When the width to be occupied by the whole front 
 is limited, to determine the diameter of the column. 
 
 The relation between the parts may be expressed thus : 
 
 _ 60 a 
 
 ~~ ~d(b '+ c) + (60 c) 
 
 Where a equals the width in feet occupied by the columns, and their inter- 
 columniations taken collectively, measured at the base ; b equals the width 
 of the metope, in minutes ; c equals the width of the triglyphs in minutes ; d 
 equals the number of metopes, and x equals the diameter in feet. 
 
 Example. A front of six columns hexastyle 61 feet wide ; the frieze 
 having one triglyph over each intercolumniation, or mono-triglyph. In this 
 case, there being five intercolumniations and two metopes over each, therefore 
 there are 5 x 2 = 10 metopes. Let the metope equal 42 minutes and the 
 triglyph equal 28. Then a = 61 ; b = 42 ; c = 28 ; and d = 10 ; and the formula 
 above becomes 
 
 60 x 61 60 x 61 3660 
 
 x . --- - - -- - - ST = - = - = 5 feet = the d lameter 
 10(42 + 28) + (60 28) 10x70 + 32 732 
 
 required. 
 
 Example. An octastyle front, 8 columns, 184 feet wide, three metopes 
 over each, intercolumniation, 21 in all, and the metope and triglyph 42 and 
 28, as before. Then 
 
 l84 - = H212 = 7.35-rigir feet = the diameter required. 
 
 21 (42 + 28) + (60 - 28) 1502 
 
20 ARCHITECTURE. 
 
 cipal objects upon which their skill in the art was displayed ; 
 and as the Doric order seems to have been well fitted, by its 
 massive proportions, to represent the character of their 
 male deities rather than the female, there seems to have 
 been a necessity for another style which should be emble- 
 matical of feminine graces, and with which they might 
 decorate such temples as were dedicated to the goddesses. 
 Hence the origin of the Ionic order. This was invented, 
 according to historians, by Hermogenes of Alabanda ; and 
 he being a native of Caria, then in the possession of the 
 lonians, the order was called the Ionic. 
 
 The distinguishing features of this order are* the volutes 
 or spirals of the capital ; and the dentils among the bed- 
 mouldings of the cornice: although in some instances 
 dentils are wanting. The volutes are said to have been 
 designed as a representation of curls of hair on the head of 
 a matron, of whom the whole column is taken as a sem- 
 blance. 
 
 The Ionic order is appropriate for churches, colleges, 
 seminaries, libraries, all edifices dedicated to literature and 
 the arts, and all places of peace and tranquillity. The front 
 of the Custom-House, New York City, is a good specimen 
 of this order. 
 
 36. The Intercolumniation : of this and the other 
 orders both Roman and Grecian, with the exception of 
 the Doric are distinguished as follows. When the interval 
 is one and a half diameters, it is called pycnostyle, or columns 
 thick-set; when two diameters, systyle ; when two and a 
 quarter diameters, eustyle ; when three diameters, diastyle ; 
 and when more than three diameters, arceostyle, or columns 
 thin-set. In all the orders, when there are four columns in 
 one row, the arrangement is called tetrastyle ; when there 
 are six in a row, hcxastyle ; and when eight, octastyle. 
 
 37. To Describe the Ionic Volute. Draw a perpen- 
 dicular from a to s (Fig. 4), and make a s equal to 20 min. 
 or to $ of the whole height, a c ; draw s o at right angles to 
 s a, and equal to I min. ; upon o, with 2^ min. for radius, 
 
PROPORTIONS OF GRECIAN IONIC. 
 
 v 
 
 FIG. 3. GRECIAN IONIC. 
 
22 
 
 ARCHITECTURE. 
 
 describe the eye of the volute ; about o, the centre of the 
 eye, draw the square, r t i 2, with sides equal to half the 
 diameter of the eye, viz. 2j min., and divide it into 144 equal 
 parts, as shown at Fig. 5. The several centres in rotation are 
 at the angles formed by the heavy lines, as figured, i, 2, 3, 
 4, 5, 6, etc. The position of these angles is determined by 
 commencing at the point, i, and making each heavy line one 
 part less in length than the preceding one. No. i is the 
 
 FIG. 4. IONIC VOLUTE. 
 
THE IONIC VOLUTE. 23 
 
 centre for the arc a b (Fig. 4 ;) 2 is the centre for the arc 
 be; and so on to the last. The inside spiral line is to be 
 described from the centres, x, x, x, etc. (Fig. 5), being the 
 centre of the first small square towards the middle of the 
 eye from the centre for the outside arc. The breadth of the 
 fillet at aj is to be made equal to 2 T 3 min. This is for a spiral 
 of three revolutions ; but one of any number of revolutions, 
 as 4 or 6, may be drawn, by dividing of (Fig. 5) into a cor- 
 responding number of equal parts. Then divide the part 
 nearest the centre, o, into two parts, as at h ; join o and i, 
 also o and 2 ; draw h 3 parallel to o i, and h 4 parallel to o 
 
 FIG. 5. EYE OF VOLUTE. 
 
 2 ; then the lines o i, o 2, // 3, h 4 will determine the length 
 of the heavy lines, and the place of the centres. (See Art. 
 288.) 
 
 38. The Corinthian Order : (Fig. /,) is in general like 
 the Ionic, though the proportions are lighter. The Corin- 
 thian displays a more airy elegance, a richer appearance ; 
 but its distinguishing feature is its beautiful capital, 
 is generally supposed to have had its origin in the capitals 
 
24 ARCHITECTURE. 
 
 of the columns of Egyptian temples, which, though not ap- 
 proaching it in elegance, have yet a similarity of form with 
 the Corinthian. The oft-repeated story of its origin which 
 
 is told by Vitruvius an architect 
 who flourished in Rome in the days 
 of Augustus Caesar though pretty 
 generally considered to be fabu- 
 lous, is nevertheless worthy of be- 
 ing again recited. It is this : A 
 young lady of Corinth was sick, and 
 finally died. Her nurse gathered 
 , into a deep basket such trinkets and 
 keepsakes as the lady had been 
 
 fond of when alive, and placed them upon her grave, cover- 
 ing the basket with a flat stone or tile, that its contents 
 might not be disturbed. The basket was placed accident- 
 ally upon the stem of an acanthus plant, which, shooting 
 forth, enclosed the basket with its foliage, some of which, 
 reaching the tile, turned gracefully over in the form of a 
 volute. 
 
 A celebrated sculptor, Calimachus, saw the basket thus 
 decorated, and from the hint which it suggested conceived 
 and constructed a capital for a column. This was called 
 Corinthian, from the fact that it was invented and first made 
 use of at Corinth. 
 
 The Corinthian being the gayest, the richest, the most 
 lovely of all the orders, it is appropriate for edifices which 
 are dedicated to amusement, banqueting, and festivity for 
 all places where delicacy, gavety, and splendor are desir- 
 able. 
 
 39. Pcrian 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 ; 
 these were called Persians and Caryatides. 
 
 40. Persian* : are statues of men, and are so called in 
 commemoration of a victory gained over the Persians by 
 Pausanias. The Persian prisoners were brought to Athens 
 
PROPORTIONS OF GRECIAN CORINTHIAN. 
 
 x 
 
 s l 
 
 - 
 
 J 
 
 FIG. "j. GRECIAN CORINTHIAN. 
 
26 ARCHITECTURE. 
 
 and condemned to abject slavery ; and in order to represent . 
 them in the lowest state of servitude and degradation, the 
 statues were loaded with the heaviest entablature, the Doric. 
 
 41. Caryatides: are statues of women dressed in long 
 robes after the Asiatic manner. Their origin is as follows : 
 In a war between the Greeks and the Caryans, the latter 
 were totally vanquished, their male population extinguished, 
 and their females carried to Athens. To perpetuate the 
 memory of this event, statues of females, having the form 
 and dress of the Caryans, were erected, and crowned with 
 the Ionic or Corinthian entablature. The caryatides were 
 generally formed of about the human size, but the persians 
 much larger, in order to produce the greater awe and 
 astonishment in the beholder. The entablatures were pro- 
 portioned to a statue in like manner as to a column of the 
 same height. 
 
 These semblances of slavery have been in frequent use 
 among moderns as well as ancients ; and, as a relief from 
 the stateliness and formality of the regular orders, are capa- 
 ble of forming a thousand varieties ; yet in a land of liberty 
 such marks of human degradation ought not to be perpetu- 
 ated. 
 
 42. Roman Styles. Strictly speaking, Rome had no 
 architecture of her own ; all she possessed was borrowed 
 from other nations. Before the Romans exchanged inter- 
 course with the Greeks, they possessed some edifices of 
 considerable extent and merit, which were erected by archi- 
 tects from Etruria ; but Rome was principally indebted to 
 Greece for what she acquired of the art. Although there is 
 no such thing as an architecture of Roman invention, yet 
 no nation, perhaps, ever was so devoted to the cultivation 
 of the art as the Roman. Whether we consider the number 
 and extent of their structures, or the lavish richness and 
 splendor with which they were adorned, we are compelled 
 to yield to them our admiration and praise. At one time, 
 under the consuls and emperors, Rome employed 400 ar- 
 chitects. The public works such as theatres, circuses, 
 baths, aqueducts, etc. were, in extent and grandeur, be- 
 
PORTICO OF THE ERECTHEUM, ATHENS. 
 
CHANGE OF STYLES BY THE ROMANS. 2? 
 
 yond anything- attempted in modern times. Aqueducts 
 were built to convey water from a distance of 60 miles or 
 more. In the prosecution of this work rocks and mountains 
 were tunnelled, and valleys bridged. Some of the latter 
 descended 200 feet below the level of the water; and. in 
 passing them the canals were supported by an arcade, or 
 succession of arches. Public baths are spoken of as large as 
 cities, being fitted up with numerous conveniences for ex- 
 ercise and amusement. Their decorations were most splen- 
 did ; indeed, the exuberance of the ornaments alone was 
 offensive to good taste. So overloaded with enrichments 
 were the baths of Diocletian that on one occasion of public 
 festivity great quantities of sculpture fell from the ceilings 
 and entablatures, killing many of the people. 
 
 43. Grecian Order 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 lavish displays of ornament. They 
 transformed in many instances the true elegance of the 
 Grecian art into a gaudy splendor, better suited to their 
 less refined taste. The Romans remodelled each of the 
 orders : the Doric (Fig. 8) was modified by increasing the 
 height of the column to 8 diameters ; by changing the 
 echinus of the capital for an ovolo, or quarter-round, and 
 adding an astragal and neck below it ; by placing the centre, 
 instead of one edge, of the first triglyph over the centre of 
 the column ; and introducing horizontal instead of inclined 
 mutules in the cornice, and in some instances dispensing 
 with them altogether. The Ionic was modified by diminish- 
 ing the size of the volutes, and, in some specimens, intro- 
 ducing a new capital in which the volutes were diagonally 
 arranged (Fig. 9). This new capital has been termed modern 
 Ionic. The favorite order at Rome and her colonies was 
 the Corinthian (Fig. 10). But this order the Roman artists, 
 in their search for novelty, subjected to many alterations 
 especially in the foliage of its capital. Into the upper part 
 of this they introduced the modified Ionic capital ; thus 
 
28 
 
 ARCHITECTURE. 
 
 combining the two in one. This change was dignified with 
 the importance of an order, and received the appellation 
 
 + 
 
 [29. 
 
 Vft 
 
 
 ' 
 
 j^'* 
 
 n ><y 
 
 x>L->xL 
 
 FIG. 8. ROMAN DORIC. 
 
 of COMPOSITE, or Roman : the best specimen of which is 
 found in the Arch of Titus (Fig. n). This style was not 
 
PROPORTIONS OF THE ROMAN IONIC. 
 
 2 9 
 
 If 4 
 
 
 .-._ 
 
 aygpg^^ 
 
 ]^>34&*^&&1&1>S<1&1&Z> 
 
 wmm: 
 
 
 FIG, 9. ROMAN IONIC. 
 
30 ARCHITECTURE. 
 
 much used among the Romans themselves, and is but 
 slightly appreciated now. 
 
 44. Tlie Tuscan Order: is said to have been intro- 
 duced to the Romans by the Etruscan architects, and to 
 have been the only style used in Italy before the introduc- 
 tion of the Grecian orders. However this may be, its simi- 
 larity to the Doric order gives strong indications of its 
 having been a rude imitation of that style : this is very prob- 
 able, since history informs us that the Etruscans held inter- 
 course with the Greeks at a remote period. The rudeness 
 of this order prevented its extensive use in Italy. All that 
 is known concerning it is from Vitruvius, no remains of 
 buildings in this style being found among ancient ruins. 
 
 For mills, factories, markets, barns, stables, etc., where 
 utility and strength are of more importance than beauty, 
 the improved modification of this order, called the modern 
 Tuscan (Fig. 12), will be useful; and its simplicity recom- 
 mends it where economy is desirable. 
 
 45. Egyptian Style. The architecture of the ancient 
 Egyptians to which that of the ancient Hindoos bears 
 some resemblance is characterized by boldness of outline, 
 solidity, and grandeur. The amazing labyrinths and exten- 
 sive artificial lakes, the splendid palaces and gloomy ceme- 
 teries, the gigantic pyramids and towering obelisks, of the 
 Egyptians were works of immensity and durability ; and 
 their extensive remains are enduring proofs of the enlight- 
 ened skill of this once-powerful but long since extinct na- 
 tion. The principal features of the Egyptian style of archi- 
 tecture are uniformity of plan, never deviating from right 
 lines and angles ; thick walls, having the outer surface 
 slightly deviating inwardly from the perpendicular ; the 
 whole building low ; roof flat, composed of stones reaching 
 in one piece from pier to pier, these being supported by 
 enormous columns, very stout in proportion to their height ; 
 the shaft sometimes polygonal, having no base but with a 
 great variety of handsome capitals, the foliage of these being 
 of the palm, lotus, and other leaves ; entablatures having 
 simply an architrave, crowned with a huge cavetto orna- 
 
PROPORTIONS OF THE ROMAN CORINTHIAN. 
 
 UNIVERSITY 
 
 FIG. io. ROMAN CORINTHIAN. 
 
ARCHITECTURE. 
 
 * WMWKM^-Q^WW&AWW^^ 
 
 Fie. ii. COMPOSITE ORDER ARCH OF TITUS. 
 
MASSIVENESS OF EGYPTIAN STRUCTURES. 33 
 
 mented with sculpture ; and the intercolumniation very nar- 
 row, usually i diameters and seldom exceeding 2^. In the 
 remains of a temple the walls were found to be 24 feet thick ; 
 and at the gates of Thebes, the walls at the foundation were 
 50 feet thick and perfectly solid. The immense stones of 
 which these, as well as Egyptian walls generally, were built, 
 had both their inside and outside surfaces faced, and the 
 oints throughout the body of the wall as perfectly close as 
 upon the outer surface. For this reason, as well as that the 
 buildings generally partake of the pyramidal form, arise 
 their great solidity and durability. The dimensions and ex- 
 tent of the buildings may be judged from the temple of 
 Jupiter at Thebes, which was 1400 feet long and 300 feet 
 wide exclusive of the porticos, of which there was a great 
 number. 
 
 It is estimated by Mr. Gliddon, U. S. Consul in Egypt, 
 that not less than 25,000,000 tons of hewn stone were em- 
 ployed in the erection of the Pyramids of Memphis alone 
 or enough to construct 3000 Bunker Hill monuments. Some 
 of the blocks are 40 feet long, and polished with emery to a 
 surprising degree. It is conjectured that the stone for these 
 pyramids was brought, by rafts and canals, from a distance 
 of six or seven hundred miles. 
 
 The general appearance of the Egyptian style of archi- 
 tecture is that of solemn grandeur amounting sometimes to 
 sepulchral gloom. For this reason it is appropriate for cem- 
 eteries, prisons, etc. ; and being adopted for these purposes, 
 it is gradually gaining favor. 
 
 A great dissimilarity exists in the proportion, form, and 
 general features of Egyptian columns. In some instances, 
 there is no uniformity even in those of the same building, 
 each differing from the others either in its shaft or capital. 
 For practical use in this country, Fig. 13 may be taken as a 
 standard of this style. The Halls of Justice in Centre 
 Street, New York City, is a building in general accordance 
 with the principles of Egyptian architecture. 
 
 46. Buildings in General. In selecting a style for an 
 edifice, its peculiar requirements must be allowed to govern. 
 
34 
 
 ARCHITECTURE. 
 
 736* 
 
 in 
 
 ir, 
 
 41 
 
 Fu;. 12. MODIFIED TUSCAN ORDER. 
 
FITNESS OF STYLES. 35 
 
 That style of architecture is to be preferred in which utility, 
 stability, and regularity are gracefully blended with gran- 
 deur and elegance. But as an arrangement designed for a 
 warm country would be inappropriate for a colder climate, 
 it would seem that the style of building ought to be modified 
 to suit the wants of the people for whom it is. designed. 
 High roofs to resist the pressure of heavy snows, and ar- 
 rangements for artificial heat, are indispensable in northern 
 climes ; while they would be regarded as entirely out of 
 place in buildings at the equator. 
 
 Among the Greeks, architecture was employed chiefly 
 upon their temples and other large buildings ; and the pro- 
 portions of the orders, as determined by them, when execu- 
 ted to such large dimensions, have the happiest effect. But 
 when used for small buildings, porticos, porches, etc., espe- 
 cially in country places, they are rather heavy and clumsy ; 
 in such cases, more slender proportions will be found to pro- 
 duce a better effect. The English cottage-style is rather 
 more appropriate, and is becoming extensively practised for 
 small buildings in the country. 
 
 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 splendid ; for 
 a monument or cemetery, gloomy melancholy ; or, if for a 
 church, majestic and graceful by some it has been said, 
 " somewhat dark and gloomy, as being favorable to a devo- 
 tional state of feeling ;" but such impressions can only re- 
 sult from a misapprehension of the nature of true devotion. 
 " Her ways are ways of pleasantness, and all her paths are 
 peace." The church should rather be a type of that 
 brighter world to which it leads. Simply for purposes of 
 contemplation, however, the glare of the noonday light 
 should be excluded, that the worshipper may, with Milton 
 
 "Love the high, embowed roof, 
 With antique pillars massy pr*f, 
 And storied windows richlyjj^ght, 
 Casting a dim, religious ligHt." 
 
ARCHITECTURE. 
 
 H.P. 
 
PREVALENCE OF WOODEN DWELLINGS. 37 
 
 However happily the several parts of an edifice may be 
 disposed, and however pleasing it may appear as a whole, 
 yet much depends upon its site, as also upon the character 
 and style of the structures in its immediate vicinity, and the 
 degree of cultivation of the adjacent country. A splendid 
 country-seat should have the out-houses and fences in the 
 same style with itself, the trees and shrubbery neatly 
 trimmed, and the grounds well cultivated. 
 
 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. Build- 
 ings erected with brick or stone are far preferable to those 
 of wood : they are more durable ; not so liable to injury by 
 fire, nor to need repairs ; and will be found in the end quite 
 as economical. A wooden house is suitable for a temporary 
 residence only ; and those who would bequeath a dwelling 
 to their children will endeavor to build with a more dura- 
 ble material. Wooden cornices and gutters, attached to 
 brick houses, are objectionable not only on account of their 
 frail nature, but also because they render the building liable 
 to destruction by fire. 
 
 4-9. Dwelling-Houses : are built of various dimensions 
 and styles, according to their destination ; and to give de- 
 signs and directions for their erection, it is necessary to know 
 their situation and object. A dwelling intended for a gar- 
 dener would require very different dimensions and arrange- 
 ments from one intended for a retired gentleman with his 
 servants, horses, etc. ; nor would a house designed for the 
 city be appropriate for the country. For city houses, ar- 
 rangements that would be convenient for one family might 
 be very inconvenient for two or more. Figs. 14, 15, 16, 17, 
 1 8, and 19 represent the icJinograpJiical projection, or ground- 
 plan, of the floors of an ordinary city house, designed to be 
 occupied by one family only. Fig. 21 is an elevation, or 
 front view, of the same house. All these plans are drawn at 
 the same scale which is that at the bottom of 
 
38 ARCHITECTURE. 
 
 Fig. 14 is a Plan of the Under-Cellar. 
 
 a, is the coal-vault, 6 by 10 feet. 
 
 b, is the furnace for heating the house. 
 <:, d, are front and rear areas. 
 
 Fig. 15 is a Plan of the Basement. 
 
 a, is the library, or ordinary dining-room, 15 by 20 feet. 
 by is the kitchen, 15 by 22 feet. 
 
 c, is the store-room, 6 by 9 feet. 
 
 d, is the pantry, 4 by 7 feet. 
 
 e, is the china closet, 4 by 7 feet. 
 fj is the servants' water-closet. 
 g, is a closet. 
 
 //, is a closet with a dumb-waiter to the first story above. 
 i, is an ash closet under the front stoop. 
 j, is the kitchen-range. 
 
 k, is the sink for washing and drawing water. 
 /, are wash-trays. 
 
 Fig. 1 6 is a Plan of the First Story. 
 
 a, is the parlor, 1 5 by 34 feet. 
 
 b, is the dining-room, 16 by 23 feet. 
 
 c, is the vestibule. 
 
 <, is the closet containing the dumb-waiter from the base- 
 ment. 
 
 /, is the closet containing butler's sink. 
 g, gy are closets. 
 
 //, is a closet for hats and cloaks. 
 iyjy are front and rear balconies. 
 
 Fig. 17 is the Second Story. 
 
 a, <?, are chambers, 15 by 13 feet. 
 
 b, is a bed-room, 7^ by 13 feet. 
 
 c, is the bath-room, 7^ by 13 feet. 
 
 d, dy are dressing-rooms, 6 by 7^ feet. 
 Cy e, are closets. 
 
 /, /, are wardrobes. 
 g, g, ve cupboards. 
 
PLANS OF A CITY HOUSE. 
 
 39 
 
 FIG. 14. 
 
 UNDBR-CELLAR. 
 
 FIG- 15- 
 
 BASEMENT. 
 
 CITY DWELLING. 
 
ARCHITECTURE. 
 
 CITY DWELLING. 
 
 FIG. 17. 
 
 SECOND STORY. 
 
UPPER STORIES OF A CITY HOUSE. 
 
 Fig. 1 8 is the Third Story. 
 
 a, a, are chambers, 15 by 19 feet. 
 
 b, b, are bed-rooms, ; by 13 feet. 
 
 c, c, are closets. 
 
 d, is a linen-closet, 5 by 7 feet. 
 
 FIG. 19. 
 FOURTH STORY. 
 CITY DWELLING. 
 
 e, e, are dressing-closets. 
 f,f, are wardrobes. 
 g, g, are cupboards. 
 
 Fig. 19 is the Fourth Story. 
 a, a, are chambers, 14 by 17 feet. 
 
42 ARCHITECTURE. 
 
 b, b, are bed-rooms, 8 by 17 feet. 
 
 c, c, c, are closets. 
 
 d, is the step-ladder to the roof. 
 
 Fig. 20 is the Section of the House showing the heights 
 of the several stories. 
 
 Fig. 2 1 is the Front Elevation. 
 
 The size of the house is 25 feet front by 55 feet deep ; this 
 is about the average depth, although some are extended to 
 60 and 65 feet in depth. 
 
 These are introduced to give some general ideas of the 
 principles to be followed in designing city houses. In plac- 
 ing the chimneys in the parlors, set the chimney-breasts 
 equidistant from the ends of the room. The basement 
 chimney-breasts may be placed nearly in the middle of the 
 side of the room, as there is but one flue to pass through 
 the chimney-breast above ; but in the second story, as there 
 are two flues, one from the basement and one from the par- 
 lor, the breast will have to be placed nearly perpendicular 
 over the parlor breast, so as to receive the flues within the 
 jambs of the fire-place. As it is desirable to have the chim- 
 ney-breast as near the middle of the room as possible, it may 
 be placed a few inches towards that point from over the 
 breast below. So in arranging those of the stories above, 
 always make provision for the flues from below. 
 
 50. Arranging the Stairs and Window. 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 pas- 
 sage in the second story, there must be room for the doors 
 which open from each of the principal rooms into the hall, 
 and more if the stairs require it. Having assigned a posi- 
 tion for the stairs of the second story, now generally placed 
 in the centre of the depth of the house, let the winders of 
 the other stories be placed perpendicularly over and under 
 them ; and be careful to provide for head-room. To ascer- 
 tain this, when it is doubtful, it is well to draw a vertical 
 section of the whole stairs ; but in ordinary cases this is not 
 
FRONT OF A CITY HOUSE. 
 
 43 
 
 CITY DWELLING. 
 
44 ARCHITECTURE. 
 
 necessary. To dispose the windows properly, the middle 
 window of each story should be exactly in the middle of the 
 front ; but the pier between the two windows which light 
 the parlor should be in the centre of that room ; because 
 when chandeliers or any similar ornaments hang from the 
 centre-pieces of the parlor ceilings, it is important, in order 
 to give the better effect, that the pier-glasses at the front 
 and rear be in a range with them. If both these objects 
 cannot be attained, an approximation to each must be at- 
 tempted. The piers should in no case be less in width than 
 the window openings, else the blinds or shutters, when 
 thrown open, will interfere with one another ; in general 
 practice, it is well to make the outside piers f of the width 
 of one of the middle piers. When this is desirable, deduct 
 the amount of the three openings from the width of the 
 front, and the remainder will be the amount of the width of 
 all the piers ; divide this by 10, and the product will be ^ of 
 a middle pier ; and then, if the parlor arrangements do not 
 interfere, give twice this amount to each corner pier, and 
 three times the same amount to each of the middle piers. 
 
 51. Principles of Architecture. To build well requires 
 close attention and much experience. The science of build- 
 ing is the result of centuries of study. Its progress towards 
 perfection must have been exceedingly slow. In the con- 
 struction of the first frail and rude habitations of men, the 
 primary object was, doubtless, utility a mere shelter from 
 sun and rain. But as successive storms shattered his poor 
 tenement, man was taught by experience the necessity of 
 building with an idea to durability. And as the symmetry, 
 proportion, and beauty of nature met his admiring gaze, 
 contrasting so strangely with the misshapen and dispropor- 
 tioned work of his own hands, he was led to make gradual 
 changes, till his abode was rendered not only commodious 
 and durable, but pleasant in its appearance ; and building 
 became a fine art, having utility for its basis. 
 
 52. Arrangement. In all designs for buildings of im- 
 portance, utility, durability, and beauty, the first great prin- 
 ciples, should be pre-eminent. In order that the edifice be 
 
Ox 
 
 UNIVERSITY 
 
ESSENTIAL REQUIREMENTS OF A BUILDING. 45 
 
 useful, commodious, and comfortable, the arrangement of the 
 apartments should be such as to fit them for their several 
 destinations ; for publio assemblies, oratory, state, visitors, 
 retiring, eating, reading, sleeping, bathing, dressing, etc. 
 these should each have its own peculiar form and situation. 
 To accomplish this, and at the same time to make their 
 relative situation agreeable and pleasant, producing regu- 
 larity and harmony, require 'in some instances much skill 
 and sound judgment. Convenience and regularity are very 
 important, and each should have due attention ; yet when 
 both cannot be obtained, the latter should in most cases 
 give place to the former. A building that is neither con- 
 venient nor regular, whatever other good qualities it may 
 possess, will be sure of disapprobation. 
 
 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 pur- 
 pose, the ceilings of the apartments should have a respect- 
 able height ; and the sky-light, or any part of the roof that 
 can be made movable, should be arranged with cord and 
 pullies, so as to be easily raised and lowered. Small open- 
 ings near the ceiling, that may be closed at pleasure, should 
 be made in the partitions that separate the rooms from the 
 passages especially for those rooms which are used for 
 sleeping apartments. All the apartments should be so ar- 
 ranged as to secure their being easily kept dry and clean. 
 In dwellings, suitable apartments should be fitted up for 
 bathing with all the necessary apparatus for conveying 
 water. 
 
 54. Stability. To secure this, an edifice should be de- 
 signed upon well-known geometrical principles : such as 
 science has demonstrated to be necessary and sufficient for 
 firmness and durability. It is well, also, that it have the 
 appearance of stability as well as the reality ; for should it 
 seem tottering and unsafe, the sensation of fear, rather than 
 those of admiration and pleasure, Avill be excited in the be- 
 holder. To secure certainty and accuracy in the applica- 
 tion of those principles, a knowledge of the strength and 
 
46 ARCHITECTURE. 
 
 other properties of the materials used is indispensable ; and 
 in order that the whole design be so made as to be capable 
 of execution, a practical knowledge of the requisite mechan- 
 ical operations is quite important. 
 
 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 orna- 
 ments, provided this be judiciously performed ; for enrich- 
 ments should not only be ot a proper character to suit the 
 style of the building, but should also have their true posi- 
 tion, and be bestowed in proper quantity. The most com- 
 mon fault, and one which is prominent in Roman architec- 
 ture, is an excess of enrichment : an error which is carefully 
 to be guarded against. But those who take the Grecian 
 models for their standard will not be liable to go to that 
 extreme. In ornamenting a cornice, or any other assem- 
 blage ol mouldings, at least every alternate member should 
 be left plain ; and those that are near the eye should be more 
 finished than those which are distant. Although the charac- 
 teristics of good architecture are utility and elegance, in 
 connection with durability, yet some buildings are designed 
 expressly for use, and others again for ornament : in the 
 former, utility, and in the latter, beauty, should be the gov- 
 erning principle. 
 
 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 secu- 
 rity, and enable the mechanic to erect the most extensive 
 and lofty edifices with confidence. The more important 
 parts are the foundation, the column, the wall, the lintel, 
 the arch, the vault, the dome, and the roof. A separate 
 description of the peculiarities of each would seem to be 
 necessary, and cannot perhaps be better expressed than in 
 the following language of a modern writer on this subject, 
 slightly modified : 
 

 UNIVERSITY 
 
STRUCTURAL FEATURES OF A BUILDING. 47 
 
 57. The Foundation: of a building should be begun 
 at a certain depth in the earth, to secure a solid basis, below 
 the reach of frost and common accidents. The most solid 
 basis is rock, or gravel which has not been moved. Next 
 to these are clay and sand, provided no other excavations 
 have been made in the immediate neighborhood. From 
 this basis a stone wall is carried up to the surface of the 
 ground, and constitutes the foundation. Where it is in- 
 tended that the superstructure shall press unequally, as at 
 its piers, chimneys, or columns, it is sometimes of use to 
 occupy the space between the points of pressure by an 
 inverted arch. This distributes the pressure equally, and 
 prevents the foundation from springing between the differ- 
 ent points. In loose or muddy situations, it is always un- 
 safe to build, unless we can reach the solid bottom below. 
 In salt marshes and flats, this is done by depositing timbers, 
 or driving wooden piles into the earth, and raising walls 
 upon them. The preservative quality of the salt will keep 
 these timbers unimpaired for a great length of time, and 
 makes the foundation equally secure with one of brick or 
 stone. 
 
 58. The Column, or Pillar: is the simplest member in 
 any building, though by no means an essential one to all. 
 This is a perpendicular part, commonly of equal breadth 
 and thickness, not intended for the purpose of enclosure, 
 but simply for the support of some part of the superstruc- 
 ture. The principal force which a column has to resist is 
 that of perpendicular pressure. In its shape, the shaft of a 
 column should not be exactly cylindrical, but, since the 
 lower part must support the weight of the superior part, in 
 addition to the weight which presses equally on the whole 
 column, the thickness should gradually decrease from bot- 
 tom to top. The outline of columns should be a little 
 curved, so as to represent a portion of a very long spheroid, 
 or paraboloid, rather than of a cone. This figure is the joint 
 result of two calculations, independent of beauty of appear- 
 ance. One of these is, that the form best adapted for sta- 
 bility of base is that of a cone ; the other is, that the figure, 
 
48 ARCHITECTURE. 
 
 which would be of equal strength throughout for support- 
 ing a superincumbent weight, would be generated by the 
 revolution of two parabolas round the axis of the column, 
 the vertices of the curves being at its extremities. The 
 swell of the shafts of columns was called the entasis by the 
 ancients. It has been lately found that the columns of the 
 Parthenon, at Athens, which have been commonly supposed 
 straight, deviate about an inch from a straight line, and that 
 their greatest swell is at about one third of their height. 
 Columns in the antique orders are usually made to diminish 
 one sixth or one seventh of their diameter, and sometimes 
 even one fourth. The Gothic pillar is commonly of equal 
 thickness throughout. 
 
 59. The Wall : another elementary part of a building, 
 may be considered as the lateral continuation of the column, 
 answering the purpose both of enclosure and support. A 
 wall must diminish as it rises, for the same reasons, and in 
 the same proportion, as the column. It must diminish still 
 more rapidly if it extends through several stories, support- 
 ing weights at different heights. A wall, to possess the 
 greatest strength, must also consist of pieces, the upper and 
 lower surfaces of which are horizontal and regular, not 
 rounded nor oblique. The walls of most of the ancient 
 structures which have stood to the present time are con- 
 structed in this manner, and frequently have their stones 
 bound together with bolts and clamps of iron. The same 
 method is adopted in such modern structures as are intended 
 to possess great strength and durability, and, in some cases, 
 the stones are even dovetailed together, as in the light- 
 houses at Eddy stone and Bell Rock. But many of our 
 modern stone walls, for the sake of cheapness, have only one 
 face of the stones squared, the inner half of the wall being 
 completed with brick ; so that they can, in reality, be con- 
 sidered only as brick walls faced with stone. Such walls are 
 said to be liable to become convex outwardly, from the dif- 
 ference in the shrinking of the cement. Rubble walls are 
 made of rough, irregular stones, laid in mortar. The stones 
 should be broken, if possible, so as to produce horizontal 
 
VARIOUS METHODS OF ERECTING WALLS. 49 
 
 surfaces. The coffer walls of the ancient Romans were made 
 by enclosing successive portions of the intended wall in a 
 box, and filling it with stones, sand, and mortar promis- 
 cuously. This kind of structure must have been extremely 
 insecure. The Pantheon and various other Roman build- 
 ings are surrounded with a double brick wall, having its 
 vacancy filled up with loose bricks and cement. The whole 
 has gradually consolidated into a mass of great firmness. 
 
 60. The Reticulated Walls : of the Romans com- 
 posed of bricks with oblique surfaces would, at the present 
 day, be thought highly unphilosophical. Indeed, they could 
 not long have stood, had it not been for the great strength 
 of their cement. Modern brick walls are laid with great 
 precision, and depend for firmness more upon their position 
 than upon the strength of their cement. The bricks being 
 laid in horizontal courses, and continually overlaying each 
 other, or breaking joints, the whole mass is strongly inter- 
 woven, and bound together. Wooden walls, composed of 
 timbers covered with boards, are a common but more per- 
 ishable kind. They require to be constantly covered with a 
 coating of a foreign substance, as paint or plaster, to pre- 
 serve them from spontaneous decomposition. In some parts 
 of France, and elsewhere, a kind of wall is made of earth, 
 rendered compact by ramming it in moulds or cases. This 
 method is called building in pise, and is much more durable 
 than the nature of the material would lead us to suppose. 
 Walls of all kinds are greatly strengthened by angles and 
 curves, also by projections, such as pilasters, chimneys, and 
 buttresses. These projections serve to increase the breadth 
 of the foundation, and are always to be made use of in large 
 buildings, and in walls of considerable length. 
 
 61. The Lintel, or Beam: extends in a right line over 
 a vacant space, from one column or wall to another. The 
 strength of the lintel will be greater in proportion as its 
 transverse vertical diameter exceeds the horizontal, the 
 strength being always as the square of the depth. The 
 floor is the lateral continuation or connection of beams by 
 means of a covering of boards. 
 
50 ARCHITECTURE. 
 
 62. The Arch : is a transverse member of a building, 
 answering the same purpose as the lintel, but vastly exceed- 
 ing it in strength. The arch, unlike the lintel, may consist 
 of any number of constituent pieces, without impairing its 
 strength. It is, however, necessary that all the pieces should 
 possess a uniform shape, the shape of a portion of a 
 wedge, and that the joints, formed by the contact of their 
 surfaces, should point towards a common centre. In this 
 case, no one portion of the arch can be displaced or forced 
 inward ; and the arch cannot be broken by any force which 
 is not sufficient to crush the materials of which it is made. 
 In arches made of common bricks, the sides of which are 
 parallel, any one of the bricks might be forced inward, were 
 it not for the adhesion of the cement. Any two of the bricks, 
 however, by the disposition of their mortar, cannot collect- 
 ively be forced inward. An arch of the proper form, when 
 complete, is rendered stronger, instead of weaker, by the 
 pressure of a considerable weight, provided this pressure be 
 uniform. While building, however, it requires to be sup- 
 ported by a centring of the shape of its internal surface, 
 until it is complete. The upper stone of an arch is called 
 the keystone, but is not more essential than any other. In 
 regard to the shape of the arch, its most simple form is that 
 of the semicircle. It is, however, very frequently a smaller 
 arc of a circle, or a portion of an ellipse. 
 
 63. Hooke' 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 chain hangs in such a form that the weight of 
 each link or portion is held in equilibrium by the result of 
 two forces acting at its extremities ; and these forces, or 
 tensions, are produced, the one by the weight of the portion 
 of the chain below the link, the other by the same weight 
 increased by that of the link itself, both of them acting ori- 
 ginally in a vertical direction. Now, supposing the chain 
 inverted, so as to constitute an arch of the same form and 
 weight, the relative situations of the forces will be the same, 
 
VIADUCT AT CHAUMONT. 
 
PECULIARITIES OF THE ARCH. 51 
 
 only they will act in contrary directions, so that they are 
 compounded in a similar manner, and balance each other on 
 the same conditions. 
 
 The arch thus formed is denominated a catenary arch. 
 In common cases, it differs but little from a circular arch of 
 the extent of about one third of a whole circle, and rising 
 from the abutments with an obliquity of about 30 degrees 
 from a perpendicular. But though the catenary arch is the 
 best form for supporting its own weight, and also all addi- 
 tional weight which presses in a vertical direction, it is not 
 the best form to resist lateral pressure, or pressure like that 
 of fluids, acting equally in all directions. Thus the arches 
 of bridges and similar structures, when covered with loose 
 stones and earth, are pressed sideways, as well as vertically, 
 in the same manner as if they supported a weight of fluid. 
 In this case, it is necessary that the arch should arise more 
 perpendicularly from the abutment, and that its general fig- 
 ure should be that of the longitudinal segment of an ellipse. 
 In small arches, in common buildings, where the disturbing 
 force is not great, it is of little consequence what is the 
 shape of the curve. The outlines may even be perfectly 
 straight, as in the tier of bricks which we frequently see 
 over a window. This is, strictly speaking, a real arch, pro- 
 vided the surfaces of the bricks tend toward a common 
 centre. It is the weakest kind of arch, and a part of it 
 is necessarily superfluous, since no greater portion can act 
 in supporting a weight above it than can be included be- 
 tween two curved or arched lines. 
 
 f 
 
 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 cen- 
 tres outside the arch. It is a strong arch for supporting 
 vertical pressure. The rampant arch is one in which the two 
 ends spring from unequal heights. The Jwrseshoe or Moorisli 
 arch is described from one or more 'centres placed above the 
 base line. In this arch, the lower parts are in danger of 
 being forced inward. The ogee arch is concavo-convex, and 
 therefore fit only for ornament. 
 
52 ARCHITECTURE. 
 
 65. Arch: Definition*; Principles. The upper sur- 
 face is called the extrados, and the inner, the intrados. 
 The spring is where the intrados meets the abutments. The 
 span is the distance between the abutments. The wedge- 
 shaped stones which form an arch are sometimes called 
 vonssoirs, the uppermost being- the keystone. The part of a 
 pier from which an arch springs is called the impost, and the 
 curve formed by the under side of the voussoirs, the archi- 
 volt. It is necessary that the walls, abutments, and piers on 
 which arches are supported should be so firm as to resist the 
 lateral thrust, as well as vertical pressure, of the arch. 
 It will at once be seen that the lateral or sideway pressure 
 of an arch is very considerable, when we recollect that every 
 stone, or portion of the arch, is a wedge, a part of whose 
 force acts to separate the abutments. For want of attention 
 to this circumstance, important mistakes have been commit- 
 ted, the strength of buildings materially impaired, and their 
 ruin accelerated. In some cases, the want of lateral firmness 
 in the walls is compensated by a bar of iron stretched across 
 the span of the arch, and connecting the abutments, like the 
 tie-beam of a roof. This is the case in the cathedral of Milan 
 and some other Gothic buildings. 
 
 66. An Arcade : or continuation of arches, needs only 
 that the outer supports of the terminal arches should be 
 strong enough to resist horizontal pressure. In the inter- 
 mediate arches, the lateral force of each arch is counter- 
 acted by the opposing lateral force of the one contiguous to 
 it. In bridges, however, where individual arches are liable 
 to be destroyed by accident, it is desirable that each of the 
 piers should possess sufficient horizontal strength to resist 
 the lateral pressure of the adjoining arches. 
 
 67. The Vault: is the lateral continuation of an arch, 
 serving to cover an area or passage, and bearing the same 
 relation to the arch that the wall does to the column. A 
 simple vault is constructed on the principles of the arch, and 
 distributes its pressure equally along the walls or abutments. 
 A complex or groined vault is made by two vaults intersect^ 
 ing each other, in which case the pressure is thrown upon 
 
VARIOUS CONSTRUCTIONS OF THE DOME. 53 
 
 springing points, and is greatly increased at those points. 
 The groined vault is common in Gothic architecture. 
 
 68. The Dome: sometimes called cupola,'^ a concave 
 covering to a building, or part of it, and may be either a 
 segment of a sphere, of a spheroid, or of any similar figure. 
 When built of stone, it is a very strong kind of structure, 
 even more so than the arch, since the tendency of each part 
 to fall is counteracted, not only by those above and below it, 
 but also by those on each side. It is only necessary that 
 the constituent pieces should have a common form, and that 
 this form should be somewhat like the frustum of a pyra- 
 mid, so that, when placed in its situation, its four angles may 
 point toward the centre, or axis, of the dome. During the 
 erection of a dome, it is not necessary that it should be sup- 
 ported by a centring, until complete, as is done in the arch. 
 Each circle of stones, when laid, is capable of supporting 
 itself without aid from those above it. It follows that the 
 dome may be left open at top, without a keystone, and 
 yet be perfectly secure in this respect, being the reverse of 
 the arch. The dome of the Pantheon, at Rome, has been 
 always open at top, and yet has stood unimpaired for nearly 
 2006 years. The upper circle of stones, though apparently 
 the weakest, is nevertheless often made to support the addi- 
 tional weight of a lantern or tower above it. In several of 
 the largest cathedrals, there are two domes, one within the 
 other, which contribute their joint support to the lantern, 
 which rests upon the top. In these buildings, the dome 
 rests upon a circular wall, which is supported, in its turn, by 
 arches upon massive pillars or piers. This construction is 
 called building upon pendentivcs, and gives open space and 
 room for passage beneath the dome.' The remarks which 
 have been made in regard to the abutments of the arch 
 apply equally to the walls immediately supporting a dome. 
 They must be of sufficient thickness and solidity to resist 
 the lateral pressure of the dome, which is very great. The 
 walls of the Roman Pantheon are of great depth and solid- 
 ity. In order that a dome in itself should be perfectly 
 secure, its lower parts must not be too nearly vertical, since, 
 
54 ARCHITECTURE. 
 
 in this case, they- partake of the nature of perpendicular 
 walls, and are acted upon by the spreading force of the 
 parts above them. The dome of St. Paul's Church, in Lon- 
 don, and some others of similar construction, are bound with 
 chains or hoops of iron, to prevent them from spreading at 
 bottom. Domes which are made of wood depend, in part* 
 for their strength on their internal carpentry. The Halle 
 du Bled, in Paris, had originally a wooden dome more than 
 200 feet in diameter, and only one foot in thickness. This 
 has since been replaced by a dome of iron. (See Art- 
 
 235.) 
 
 69. Tlie Roof: is the most common and cheap method 
 of covering buildings, to protect them from rain and other 
 effects of the weather. It is sometimes flat, but more fre- 
 quently oblique, in its shape. The flat or platform roof is 
 the least advantageous for shedding rain, and is seldom used 
 in northern countries. The pent roof, consisting of two 
 oblique sides meeting at top, is the most common form. 
 These roofs are made steepest in cold climates, where they 
 are liable to be loaded with snow. Where the four sides of 
 the roof are all oblique, it is denominated a hipped roof, and 
 where there are two portions to the roof, of different ob- 
 liquity, it is a curb, or mansard roof. In modern times, roofs 
 are made almost exclusively of wood, though frequently 
 covered with incombustible materials. The internal struc- 
 ture or carpentry of roofs is a subject of considerable me- 
 chanical contrivance. The roof is supported by rafters, 
 which abut on the walls on each side, like the extremities of 
 an arch. If no other timbers existed except the rafters, 
 they would exert a strong lateral pressure on the walls, 
 tending to separate and overthrow them. To counteract 
 this lateral force, a tic-beam, as it is called, extends across, 
 receiving the ends of the rafters, and protecting the wall 
 from their horizontal thrust. To prevent the tie-beam from 
 sagging, or bending downward with its own weight, a king- 
 post is erected from this beam, to the upper angle of the 
 rafters, serving to connect the whole, and to suspend the 
 weight of the beam. This is called trussing. Queen-posts 
 
MANNER OF CONSTRUCTING ROOFS. 55 
 
 are sometimes added, parallel to the king-post, in large roofs I 
 also various other connecting timbers. In Gothic buildings, 
 where the vaults do not admit of the use of a tie-beam, the 
 rafters are prevented from spreading, as in an arch, by the 
 strength of the buttresses. 
 
 In comparing the lateral pressure of a high roof with 
 that of a low one, the length of the tie-beam being the same, 
 it will be seen that a high roof, from its containing most 
 materials, may produce the greatest pressure, as far as 
 weight is concerned. On the other hand, if the weight of 
 both be equal, then the low roof will exert the greater press- 
 ure ; and this will increase in proportion to the distance of 
 the point at which perpendiculars, drawn from the end of 
 each rafter, would meet. In roofs, as well as in wooden 
 domes and bridges, the materials are subjected to an inter- 
 nal strain, to resist which the cohesive strength of the ma- 
 terial is relied on. On this account, beams should, when 
 possible, be of one piece. Where this cannot be effected, 
 two or more beams are connected together by splicing. 
 Spliced beams are never so strong as whole ones, yet they 
 may be made to approach the same strength, by affixing lat- 
 eral pieces, or by making the ends overlay each other, and 
 connecting them with bolts and straps of iron. The ten- 
 dency to separate is also resisted, by letting the two pieces 
 into each other by the process called scarfing. Mortices, in- 
 tended to truss or suspend one piece by another, should be 
 formed upon similar principles. 
 
 Roofs in the United States, after being boarded, receive 
 a secondary covering of shingles. When intended to be 
 incombustible, they are covered with slates or earthen tiles, 
 or with sheets of lead, copper, or tinned iron. Slates are 
 preferable to tiles, being lighter, and absorbing less moisture. 
 Metallic sheets are chiefly used for flat roofs, wooden domes, 
 and curved and angular surfaces, which require a flexible 
 material to cover them, or have not a sufficient pitch to shed 
 the rain from slates or shingles. Various artificial composi- 
 tions are occasionally used to cover roofs, the most common 
 of which are mixtures of tar with lime, and sometimes with 
 sand and gravel. Ency. Am. (See Art. 202.) 
 
SECTION II. CONSTRUCTION. 
 
 ART. 70. Contructioii EseiUiai. Construction is that 
 part of the Science of Building which treats of the Laws of 
 Pressure and the strength of materials. To the architect 
 and builder a knowledge of it is absolutely essential. It de- 
 serves a larger place in a volume of this kind than is gene- 
 rally allotted to it. Something, indeed, has been said upon 
 the styles and principles, by which the best arrangements 
 may be ascertained ; yet, besides this, there is much to be 
 learned. For however precise or workmanlike the several 
 parts may be made, what will it avail, should the system of 
 framing, from deficient material, or an erroneous position of 
 its timbers, fail to sustain even its own weight ? Hence the 
 necessity for a knowledge of the laws of pressure and the 
 strength of materials. These being once understood, we 
 can with confidence determine the best position and dimen- 
 sions for the several pieces which compose a floor or a roof, 
 a partition or a bridge. As systems of framing are more or 
 less exposed to heavy weights and strains, and, in case of 
 failure, cause not only a loss of labor and material, but fre- 
 quently that of life itself, it is vety important that the mate- 
 rials employed be of the proper quantity and quality to serve 
 their destination. And, on the other hand, any superfluous 
 material is not only useless, but a positive injury, as it is an 
 unnecessary load upon the points of support. It is neces- 
 sary, therefore, to know the least quantity of material that 
 will suffice for strength. Not the least common fault in 
 framing is that of using an excess of material. Economy, at 
 least, would seem to require that this evil be abated. < 
 
 Before proceeding to consider the principles upon which 
 a system of framing should be constructed, let us attend to 
 a few of the elementary laws in Mechanics, which will be 
 found to be of great value in determining those principles. 
 
INTERIOR OF THE CATHEDRAL, SIENNA. 
 
DIRECT AND OBLIQUE SUPPORTS. 
 
 57 
 
 71. ILaws of Preure. (i.) A heavy body always ex- 
 erts a pressure equal to its own weight in a vertical direc- 
 tion. Example : Suppose an iron ball weighing 100 Ibs. be 
 supported upon the top of a perpendicular post (Fig. 22-A) ; 
 then the pressure exerted upon that post will be equal to 
 the weight of the ball, viz., 100 Ibs. (2.) But if two inclined 
 posts (Fig. 22-B) be substituted for the perpendicular sup- 
 port, the united pressures upon these posts will be more 
 than equal to the weight, and will be in proportion to their 
 position. The farther apart their feet are spread the greater 
 will be the pressure, and vice versa. Hence tremendous 
 strains may be exerted by a comparatively small weight. 
 And it follows, therefore, that a piece of timber intended 
 for a strut or post should be so placed that its axis may 
 coincide, as nearly as possible, with the direction of the 
 pressure. The direction of the pressure of the weight W 
 (Fig. 22-B) is in the vertical line b d\ and the weight W 
 would fall in that line if the two posts were removed ; hence 
 the best position for a support for the weight would be in 
 
 A. 
 
 FlG. 22. 
 
 that line. But as it rarely occurs in systems of framing 
 that weights can be supported by any single resistance, they 
 requiring generally two or more supports (as in the case of 
 a roof supported by its rafters), it becomes important, there- 
 fore, to know the exact amount of pressure any certain 
 weight is capable of exerting upon oblique supports. Now, 
 it has been ascertained that the three lines of a triangle, 
 drawn parallel with the direction of three concurring forces 
 in equilibrium, are in proportion respectively to these 
 
58 CONSTRUCTION. 
 
 forces. For example, in Fig. 22-B, we have a representation 
 of three forces concurring in a point, which forces are in 
 equilibrium and at rest ; thus, the weight W is one force, 
 and the resistances exerted by the two pieces of timber are 
 the other two forces. The direction in which the first force 
 acts is vertical downwards ; the direction of the two other 
 forces is in the axis of each piece of timber respectively. 
 These three forces all tend towards the point b. 
 
 Draw the axes a b and b c of the two supports ; make b d 
 vertical, and from d draw d e and d f parallel with the axes 
 b c and b a repectively. Then the triangle b d e has its 
 lines parallel respectively with the direction of the three 
 forces ; thus, bd is in the direction of the weight W, d e paral- 
 lel with the axis of the timber D, and e b is in the direction of 
 the timber C. In accordance with the principle above stated, 
 the lengths of the sides of the triangle b d e are in propor- 
 tion respectively to the three forces aforesaid ; thus 
 
 As the length of the line b d 
 
 Is to the number of pounds in the weight W, 
 
 So is the length of the line e 
 
 To the number of pounds' pressure resisted by the 
 
 timber C. 
 Again 
 
 As the length of the line b d 
 
 Is to the number of pounds in the weight W, 
 
 So is the length of the line d c 
 
 To the number of pounds' pressure resisted by the 
 
 timber D. 
 And again 
 
 As the length of the line b e 
 
 Is to the pounds' pressure resisted by C, 
 
 So is the length of the line d e 
 
 To the pounds' pressure resisted by D. 
 
 These proportions are more briefly stated thus 
 \st. bd\ W\\be\P, 
 
 P being used as a symbol to represent the number of pounds' 
 pressure resisted by the timber C. 
 
PARALLELOGRAM OF FORCES. 59 
 
 2d. b d : W : : d e : Q, 
 
 Q representing the number of pounds' pressure resisted by 
 the timber D. 
 
 3d. b e \P\\de\ Q. 
 
 72. Parallelogram of Forces. This relation between 
 lines and pressures is applicable in ascertaining the pres- 
 sures induced by known weights throughout any system of 
 framing. The parallelogram b e df is called the Parallelo- 
 %ram of Forces ; the two lines If e- and bf being called the 
 components, and the line b d the resultant. Where it is re- 
 quired to find the components from a given resultant (Fig. 
 22- B\ the fourth line df/need not be drawn, for the triangle 
 b d e gives the desired result. But when the resultant is to 
 be ascertained from given components (Fig. 28), it is more 
 convenient to draw the fourth line. 
 
 73. The Resolution of Forces : Is the finding of two 
 or more forces which, acting in different directions, shall 
 exactly balance the pressure of any given single force. To 
 make a practical application of this, let it be required to 
 ascertain the oblique pressure in Fig. 22-B. In this figure the 
 line bd measures half an inch (0-5 inch), and the line be 
 three tenths of an inch (0-3 inch). Now if the weight W 
 be supposed to be 1200 pounds, then the first stated propor- 
 tion above, 
 
 b d\ W: : b e : P, becomes 0-5 : 1200 : : 0-3 : P. 
 
 And since the product of the means divided by one of the 
 extremes gives the other extreme, this proportion may be 
 put in the form of an equation^ thus 
 
 I200X 0-3 _ p 
 
 0-5 
 
 Performing the arithmetical operation here indicated -that 
 is, multiplying together the two quantities above the line, 
 and dividing the product by the quantity under the line the 
 
6o 
 
 CONSTRUCTION. 
 
 quotient will be equal to the quantity represented by P, viz., 
 the pressure resisted by the timber C. Thus 
 
 1 200 
 
 0-5)360-0 
 
 720 = P. 
 
 The strain upon the timber C is, therefore, equal to 720 
 pounds; and since, in this case (the two timbers being in- 
 clined equally from the vertical), the line e d is equal to the 
 line b e, therefore the strain upon the other timber D is 
 also 720 pounds. 
 
 FIG. 23. 
 
 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 supported ; but generally long timbers 
 are not so capable of resistance as shorter ones. For, not 
 being so stiff, they bend more readily, and, since the com- 
 pression is in proportion to the length, they therefore 
 shorten more. To ascertain the pressures in Fig. 23, let the 
 weight suspended from b d be equal to two and three quarter 
 tons (2-75 tons). The line b d measures five and a half 
 tenths of an inch (0-55 inch), and the line b e half an inch 
 (0-5 inch). Therefore, the proportion 
 
 b d\ W \\ b e : P becomes 0-55 : 2-75 : : 0-5 : P, 
 
STRAIN IN PROPORTION TO INCLINATION. r 
 
 and 2.75x0.5 / 
 
 0-55 (UNIVERSITY 
 
 0'55)i -375(2- 5 = ^. 
 
 I 10 
 
 275 
 
 275 
 
 The strain upon the timber A. is, therefore, equal to two 
 and a half tons. 
 
 Again, the line e d measures four tenths of an inch (0-4 
 inch) ; therefore, the proportion 
 
 b d \ W \ : e d \ Q becomes 0-55 : 2-75 : : 0-4 : Q, 
 
 and 2./5 x 0-4 __ ^ 
 
 o-55 
 
 2-75 
 0-4 
 
 0-55) 1-100(2 = Q. 
 
 I 10 
 
 The strain upon the timber B is, therefore, equal to two 
 tons. 
 
 '/5. The Strains Exceed tBie Weights. Thus the united 
 pressures upon the two inclined supports always exceed the 
 weight. In the last case, 2f tons exert a pressure of 2\ and 
 2 tons, equal together to 4^ tons ; and in the former case, 
 1200 pounds exert a pressure of twice 720 pounds, equal to 
 1440 pounds. The smaller the angle of inclination to the ' 
 horizontal, the greater will be the pressure upon the sup- 
 ports. So, in the frame of a roof, the strain upon the rafters 
 decreases gradually with the increase of the angle of incli- 
 nation to the horizon, the length of the rafter remaining the 
 
 same. 
 
62 CONSTRUCTION. 
 
 This is true in comparing one system of framing with 
 another ; but in a system where the concentrated weight to 
 be supported is not in the middle (see Fig. 23), and, in con- 
 sequence, the supports are not inclined equally, the strain 
 will be greatest upon that support which has the greatest 
 inclination to the horizon. 
 
 76. Minimum Thrust of Rafter. Ordinarily, as in 
 roofs, the load is not concentrated, it being that of the fram- 
 ing itself. Here the amount of the load will be in proportion 
 to the length of the rafter, and this will increase with the 
 increase of the angle of inclination, the span remaining the 
 same. So it is seen that in enlarging the angle of inclina- 
 tion to the horizon in order to lessen the oblique thrust, the 
 load is increased in consequence of the elongation of the 
 rafter, thus increasing the oblique thrust. Hence there is 
 an economical angle of inclination. A rafter will have the 
 least oblique thrust when its angle of inclination .to the 
 horizon is 35 16' nearly. This angle is attained very nearly 
 when the rafter rises 8J inches per foot, or when the height 
 B C (Fig. 32) is to the base A C as 8 is to 12, or as 0-7071 is 
 to i-o. 
 
 77. Practical Method of Determining Strain. A com- 
 parison of pressures in timbers, according to their position, 
 may be readily made by drawing various designs of framing 
 and estimating the several strains in accordance with the 
 parallelogram of forces, always drawing the triangle b d e 
 so that the three lines shall be parallel with the three forces 
 or pressures respectively. The length of the lines forming 
 this triangle is unimportant, but it will be found more con- 
 venient if the line drawn parallel with the known force is 
 made to contain as many inches as the known force contains 
 pounds, or as many tenths of an inch as pounds, or as many 
 inches as to is, or tenths of an inch as tons; or, in general, 
 as many divisions of any convenient scale as there are units 
 of weight or pressure in the known force. If drawn in this 
 manner, then the number of divisions of the same scale 
 found in the other two lines of the triangle will equal the 
 units of pressure or weight of the other two forces respect- 
 
HORIZONTAL THRUST OF RAFTERS. 63 
 
 ively, and the pressures sought will be ascertained simply by 
 applying the scale to the lines of the triangle. 
 
 For example, in Fig. 23, the vertical line b d, of the tri- 
 angle, measures fifty-five hundredths of an inch (0-55 inch) ; 
 the line b e, fifty hundredths (0-50 inch); and the line e d> 
 forty (0-40 inch). Now, if it be supposed that the vertical 
 pressure, or the weight suspended below b d, is equal to 55 
 pounds, then the pressure on A will equal 50 pounds, and 
 that on B will equal 40 pounds ; for, by the proportion above 
 stated, 
 
 b d: W-.-.b e:P, 
 55 : 55 :: 50: 50; 
 
 and so of the other pressure. 
 
 If a scale cannot be had of equal proportions with the 
 forces, the arithmetical process will be shortened somewhat 
 by making the line of the triangle that represents the known 
 weight equal to unity of a decimally divided scale, then the 
 other lines will be measured in tenths or hundredths ; and 
 in the numerical statement of the proportions between the 
 lines and forces, the first term being unity, the fourth term 
 will be ascertained simply by multiplying the second and 
 third terms together. 
 
 For example, if the three lines are i, 0-7, and 1-3, and 
 the known weight is 6 tons, then 
 
 b d : W : : b e : P becomes 
 I : 6 : : 0-7 : P 4-2, 
 
 equals four and two tenths tons. Again 
 
 bd\ W : : e d : Q becomes 
 I :6:: 1-3: Q = 7-8, 
 
 equals seven and eight tenths tons. 
 
 78. Horizontal Thrust. In Fig. 24, the weight ^presses 
 the struts in the direction of their length ; their feet, n n, 
 therefore, tend to move in the direction n o, and would so 
 move were they not opposed by a sufficient resistance from 
 the blocks, A and A. If a piece of each block be cut off at 
 
64 CONSTRUCTION. 
 
 the horizontal line, a n, the feet of the struts would slide 
 away from each other along that line, in the direction na; 
 but if, instead of these, two pieces were cut off at the verti- 
 cal line, nb, then the struts would descend vertically. To 
 estimate the horizontal and the vertical pressures exerted by 
 the struts, let n o be made equal (upon any scale of equal 
 parts) to the number of tons with which the strut is pressed ; 
 
 FIG. 24. 
 
 construct the parallelogram of forces by drawing o c parallel 
 to an, and of parallel to bn; then nf (by the same scale) 
 shows the number of tons pressure that is exerted by the 
 strut in the direction n a, and n e shows the amount exerted 
 in the direction n b. By constructing designs similar to this, 
 giving various and dissimilar positions to the struts, and 
 then estimating the pressures, it will be found in every case 
 that the horizontal pressure of one strut is exactly equal to 
 that of the other, however much one strut may be inclined 
 more than the other; and also, that the united vertical 
 pressure of the two struts is exactly equal to the weight W. 
 (In this calculation the weight of the timbers has not been 
 taken into consideration, simply to avoid complication to 
 the learner. In practice it is requisite to include the weight 
 of the framing with the load upon the framing.) 
 
 Suppose that the two 1 struts, B and B (Fig. 24), were 
 rafters of a roof, and that instead of the blocks, A and A, the 
 walls of a building were the supports: then, to prevent 
 
TIES DESIRABLE IN ROOFS. 65 
 
 the walls from being thrown over by the thrust of B and B, 
 it would be desirable to remove the horizontal pressure. 
 Tnis may be done by uniting- the feet of the rafters with a 
 
 u 
 
 FIG. 25. 
 
 rope, iron rod, or piece of timber, as in Fig. 25. This figure 
 is similar to the truss of a roof. The horizontal strains on 
 the tie-beam, tending to pull it asunder in the direction of 
 its length, may be measured at the foot of the rafter, as was 
 shown at Fig* 24 ; but, it can be more readily and as accu- 
 rately measured by drawing from f and e horizontal lines to 
 the vertical line, b d, meeting it in o and o ; then/0 will be 
 the horizontal thrust at B, and co at A ; these will be found 
 to equal one another. When the rafters of a roof are thus 
 connected, all tendency to thrust out the walls horizontally 
 is removed, the only pressure on them is in a vertical direc- 
 tion, being equal to the weight of the roof and whatever it 
 has to support. This pressure is beneficial rather than 
 otherwise, as a roof having trusses thus formed, and the 
 trusses well braced to each other, tends to steady the walls. 
 
 79. Position of Supports./^. 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 
 forces, ebfd, according to Art. 71. Then 3/will show the 
 
66 
 
 CONSTRUCTION. 
 
 strain on A B, and b e the strain on A C. By comparing the 
 figures, bd being equal in each, it will be seen that the 
 strains in Fig. 26 are about three times as great as those in 
 
 B 
 
 FIG. 27. 
 
 Fig. 27 ; the position of the pieces, A B and A C, in Fig. 27, 
 is therefore far preferable. 
 
 FIG. 28. 
 
 80. The Composition of Force : consists in ascertain- 
 ing the direction and amount of one force which shall be 
 just capable of balancing tivo or more given forces, acting in 
 different directions. This is only the reverse of the resolu- 
 
STRAINS INVOLVED IN CRANE. 
 
 tion of forces ; and the two are founded on one and the 
 same principle, and may be solved in the same manner. For 
 example, let A and B (Fig. 28) be two pieces of timber* 
 pressed in the direction of their length towards b A by a 
 force equal to 6 tons weight, and B 9 tons. To find the 
 direction and amount of pressure they would unitedly exert, 
 draw the lines b e and b f in a line with the axes of the 
 timbers, and make b e equal to the pressure exerted by B, 
 viz., 9 ; also make b f equal to the pressure on A, viz., 6, and 
 complete the parallelogram of forces ebfd; then bd, the 
 diagonal of the parallelogram, will be the direction, and its 
 length, 9-25, will be the amount, of the united pressures of 
 A and of B. The line b d is termed the resultant of the two 
 forces b f and be. If A and B are to be supported by one 
 post, C t the best position for that post will be in the direc- 
 tion of the diagonal bd\ and it will require to be sufficiently 
 strong to support the united pressures of A and of B, which 
 are equal to 9-25 or 9^ tons. 
 
 FIG. 29. 
 
 81. Another Example. Let Fig. 29 represent a piece of 
 Laming commonly called a crane, which is used for hoist- 
 ing heavy weights by means of the rope, B bf, which passes 
 over a pulley at b. This, though similar to Figs. 26 and 27, is, 
 however, still materially different. In those figures, the 
 strain is in one direction only, viz., from b to d\ but in this 
 there are two strains, from A to B and from A to W. ^ The 
 strain in the direction A B is evidently equal to that in the 
 
68 
 
 CONSTRUCTION. 
 
 direction A W. To ascertain the best position for the strut 
 A C, make b e equal to bf, and complete the parallelogram of 
 forces ebfd; then draw the diagonal bd, and it will be the 
 position required. Should the foot, C, of the strut be placed 
 either higher or lower, the strain on A C would be in- 
 creased. In constructing cranes, it is advisable, in order 
 that the piece B A may be under a gentle pressure, to place 
 the foot of the. strut a trifle lower than where the diagonal 
 bd would indicate, but never higher. 
 
 W 
 
 FIG. 30. 
 
 82. Tie and Strut. 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 follow- 
 ing manner : 
 
 Let A and B (Fig. 30) represent beams of timber sup- 
 porting the weights W, W, and W ' ; A having but one sup- 
 port, which is in the middle of its length, and B two, one at 
 each end. To show the nature of the strains, let each beam 
 be sawed in the middle from a to b. The effects are obvious: 
 the cut in the beam A will open, whereas that in B will 
 close. If the weights are heavy enough, the beam A will 
 break at b ; while the cut in B will be closed perfectly tight 
 at a, and the beam be very little injured by it. But if, on 
 the other hand, the cuts be made in the bottom edge of the 
 timbers, from c to b, B will be seriously injured, while A 
 will scarcely be affected. . By this it appears evident that, 
 in a piece of timber subject to a pressure across the direction 
 of its length, the fibres are exposed to contrary strains. If 
 the timber is supported at both ends, as at B, those from the 
 top edge down to the middle are compressed in the direction 
 
TIES AND STRUTS. 69 
 
 of their length, while those from the middle to the bottom 
 edge are in a state of tension ; but if the beam is supported 
 as at A, the contrary effect is produced ; while the fibres at 
 the middle of either beam are not at all strained. The strains 
 in a framed truss are of the same nature as those in a single 
 beam. The truss for a roof, being supported at each end, 
 has its tie-beam in a state of tension, while its rafters are 
 compressed in the direction of their length. By this, it 
 appears highly important that pieces in a state of tension 
 should be distinguished from such as are compressed, in 
 order that the former may be preserved continuous. A strut 
 may be constructed of two or more pieces ; yet, where there 
 are many joints, it will not resist compression so well. 
 
 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 Fis the straining force ; 
 and if the support be removed, the straining force would 
 move from the point of support b towards d. Let it be 
 required to ascertain whether the sustaining forces arc 
 stretched or pressed by the straining force. Rule : Upon the 
 direction of the straining force b d, as a diagonal, construct 
 a parallelogram ebfd whose sides shall be parallel with the 
 direction of the sustaining forces C and D\ through the 
 point b draw a line parallel to the diagonal ef\ this may 
 then be called the dividing line between ties and struts. 
 Because all those supports which are on that side of the 
 dividing line which the straining force would occupy if 
 unresisted are compressed, while those on the other side of 
 the dividing line are stretched. 
 
 In Fig. 22-B, the supports are both compressed, being on 
 that side of the dividing line which the straining force would 
 occupy if unresisted. In Figs. 26 and 27, in which A B and 
 A C are the sustaining forces, A C is compressed, whereas 
 A B is in a state of tension ; A C being on that side of the 
 line h i which the straining force would occupy if unresisted, 
 and A B on the opposite side. The place of the latter might 
 be supplied by a chain or rope. In Fig. 25, the foot of the 
 rafter at A is sustained by two forces, the wall and the tie- 
 
70 CONSTRUCTION. 
 
 beam, one perpendicular and the other horizontal: the 
 direction of the straining force is indicated by the line b a. 
 The dividing line h i, ascertained by the rule, shows that the 
 wall is pressed and the tie-beam stretched. 
 
 FIG. 31. 
 
 84. Another Example. Let E A B F (Fig. 31) represent 
 a gate, supported by hinges at A and E. In this case, the 
 straining force is the weight of the materials, and the direc- 
 tion of course vertical. Ascertain the dividing line at the 
 several points, G, B, /, J, H, and F. It will then appear that 
 the force at G is sustained by A G and G E, and the dividing 
 line shows that th'e former is stretched and the latter com- 
 pressed. The force at H is supported by A H and HE the 
 former stretched and the latter compressed. The force at B 
 is opposed by H B and A B, one pressed, the other stretched. 
 The force at F is sustained by G F and FE, G F being 
 stretched and F E pressed. By this it appears that A B is in 
 a state of tension, and E F of compression ; also, that A H 
 and G F are stretched, while B H and G E are compressed : 
 which shows the necessity of having A H and G F each in 
 one whole length, while B H and G E may be, as they are 
 shown, each in two pieces. The force at J is sustained by 
 GJ and J H, the former stretched and the latter compressed. 
 The piece C D is neither stretched nor pressed, and could 
 be dispensed with if the joinings at J and / could be made 
 
TO FIND THE CENTRE OF GRAVITY. 71 
 
 as effectually without it. In case A B should fail, then C D 
 would be in a state of tension. 
 
 85. Centre of Gravity. The centre of gravity of a uni- 
 form 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 third of the 
 length of the line from that side ; that of a right-angled tri- 
 angle, at a point distant from the perpendicular equal to one 
 third of the base, and distant from the base equal to one 
 third of the perpendicular ; that of a pyramid or cone, in 
 the axis and at one quarter of the height from the base. 
 
 The centre of gravity of a trapezoid (a four-sided, figure 
 having only two of its sides parallel) is in a line joining the 
 centres of the two parallel sides, and at a distance from the 
 longest of the parallel sides equal to the product of the 
 length in the sum of twice the shorter added to the longer 
 of the parallel sides, divided by three times the sum of the 
 two parallel sides. Algebraically thus : 
 
 ,_'_(2_ 
 - 3 ( + 
 
 where d equals the distance from the longest of the parallel 
 sides, / the length of the line joining the two parallel sides, 
 and a the shorter and b the longer of the parallel sides. 
 
 Example. A rafter 25 feet long has the larger end 14 
 inches wide, and the smaller end 10 inches wide: how far 
 from the larger end is the centre of gravity located ? 
 
 Here / = 25, a |f, and b = |f, 
 
 - _ . 25_x_ = 25x34, 
 
 " WP) ' 3(W + ) " 3xff 3x24 - 
 
 =11-8 = 11 feet 9! inches nearly. 
 
 In irregular bodies with plain sides, the centre of gravity 
 may be found by balancing them upon the edge of a prism 
 
 upon the edge of a table in two positions, making a line 
 each time upon the body in a line with the edge of the prism, 
 and the intersection of those lines will indicate the point re- 
 
72 CONSTRUCTION. 
 
 quired. Or suspend the article by a cord or thread attached 
 to one corner or edge ; also from the same point of suspen- 
 sion hang a plumb-line, and mark its position on the face of 
 the article; again, suspend the article from another corner 
 or side (nearly at right angles to its former position), and 
 mark the position of the plumb-line upon its face ; then the 
 intersection of the two lines will be the centre of gravity. 
 
 FIG. 32. 
 
 86. Effect of the Weight of Inclined Beam. An in- 
 clined post or strut supporting some heavy pressure applied 
 at its upper end, as at Fig. 25, exerts a pressure at its foot in 
 the direction of its length, or nearly so. But when such a 
 beam is loaded uniformly over its whole length, as the rafter 
 of a roof, the pressure at its foot varies considerably from 
 the direction of its length. For example, let A B (Fig. 32) 
 be a beam leaning against the wall B c, and supported at its 
 foot by the abutment A, in the beam A c, and let o be the 
 centre of gravity of the beam. Through o draw the verti- 
 cal line b d y and from B draw the horizontal line B b, cutting 
 b dm b\ join b and A, and b A will be the direction of the 
 thrust. To prevent the beam from loosing its footing, the 
 joint at A should be made at right angles to b A. The 
 amount of pressure will be found thus : Let b d (by any scale 
 of equal parts) equal the number of tons upon the beam 
 A B\ draw d e parallel to B b ; then /; c (by the same scale) 
 equals the pressure in the direction b A ; and e d the pres- 
 sure against the wall at B and also the horizontal thrust at 
 A^ as these are always equal in a construction of this kind. 
 
 The horizontal thrust of an inclined beam (Fig. 32) the 
 effect of its own weight may be calculated thus : 
 
 Rule. Multiply the weight of the beam in pounds by 
 
THRUST OF INCLINED BEAMS. 73 
 
 its base, A C, in feet, and by the distance in feet of its centre 
 of gravity, o (see Art. 85), from the lower end, at A, and 
 divide this product by the product of the length, A B, into 
 the height, B C, and the quotient will be the horizontal 
 
 thrust in pounds. This may be stated thus : H = , 
 
 where d equals the distance of the centre of gravity, 0, from 
 the lower end ; b equals the base, A C ; iv equals the weight 
 of the beam ; h equals the height, D C ; /equals the length 
 of the beam ; and H equals the horizontal thrust. 
 
 Example. A beam 20 feet long weighs 300 pounds; its 
 centre of gravity is at 9 feet from its lower end ; it is so in- 
 clined that its base is 16 feet and its height 12 feet : what is 
 the horizontal thrust ? 
 
 TT d b w , o x 16 x 300 Q x 4 x 25 
 
 Here becomes - 0x4x5 
 
 hi 12x20 5 
 
 = 180 =H =. the horizontal thrust. 
 
 This rule is for cases where the centre of gravity does 
 not occur at the middle of the length of the beam, although 
 it is applicable when it does occur at the middle ; yet a 
 shorter rule will suffice in this case, and it is thus: 
 
 Rule. Multiply the weight of the rafter in pounds by 
 the base, A C (Fig. 32), in feet, and divide the product by 
 twice the height, B C, in feet, and the quotient will be the 
 horizontal thrust, when the centre of gravity occurs at the 
 middle of the beam. 
 
 If the inclined beam is loaded with an equally distributed 
 load, add this load to the weight of the beam, and use this 
 total weight in the rule instead of the weight of the beam. 
 And generally, if the centre of gravity of the combined 
 weights of the beam and load does not occur at the centre 
 of the length of the beam, then the former rule is to be used. 
 
 In Fig. 33, two equal beams are supported at their feet by 
 the abutments in the tie-beam. This case is similar to the 
 last ; for it is obvious that each beam is in precisely the 
 position of the beam in Fig. 32. The horizontal pressures at 
 B, being equal and opposite, balance one another ; and their 
 horizontal thrusts at the tie-beam are also equal. (See Art. 
 
CONSTRUCTION. 
 
 78 Fig. 25.) When the height of a roof (Fig. 33) is one 
 fourth of the span, or of a shed (Fig. 32) is one halt the 
 span, the horizontal thrust of a rafter, whose centre of grav- 
 
 FIG. 33. 
 
 ity is at the middle of its length, is exactly equal to the 
 weight distributed uniformly over its surface. 
 
 In shed or lean-to roofs, as Fig. 32, the horizontal pressure 
 will be entirely removed if the bearings of the rafters, as A 
 and B (Fig. 34), are made horizontal provided, however, 
 
 FIG. 34. 
 
 that the rafters and other framing do not bend between the 
 points of support. If a beam or rafter have a natural curve, 
 the convex or rounding edge should be laid uppermost. 
 
 87. Effect of Load 011 Beam. The strain in a uniformly 
 loaded beam, supported at each end, is greatest at the 
 middle of its length. Hence mortices, large knots, and other 
 defects should be kept as far as possible from that point ; 
 and in resting a load upon a beam, as a partition upon a 
 floor-beam, the weight should be so adjusted, if possible, 
 that it will bear at or near the ends. 
 
 Twice the weight that will break a beam, acting at the 
 centre of its length, is required to break it when equally 
 
VARYING PRESSURE ON BEARINGS. 75 
 
 distributed over its length ; and precisely the same deflec- 
 tion or .rag- 'will be produced on a beam by a load equally 
 distributed that five eighths ot the load will produce if act- 
 ing at the centre of its length. 
 
 88. Effect 011 Bern-ings. 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 
 equal to one half the load ; and this is also the ase when 
 the load is concentrated at the middle of the beam, or has 
 its centre of gravity at the middle of the beam ; but when 
 the load is unequally distributed, or concentrated so that its 
 centre of gravity occurs at some other point than the middle 
 of the beam, then the amount of pressure caused by the 
 load on one of the points of support is unequal to that on 
 the other. The precise amount on each may be ascertained 
 by the following rule. 
 
 Rule. Multiply the weight W (Fig. 35) by its distance, 
 C B, from its nearest point of support, B y and divide the pro-, 
 duct by the length, A B, of the beam, and the quotient will 
 
 FIG 35. 
 
 be the amount of pressure on the remote point of support, A. 
 Again, deduct this amount from the weight W, and the re- 
 mainder will be the amount of pressure on the near point of 
 support, B ; or, multiply the weight W by its distance, A C, 
 from the remote point of support, A, and divide the pro- 
 duct by the length, A B, and the quotient will be the amount 
 of pressure on the near point of support, B. 
 
 When / equals the length between the bearings A and B, 
 n AC, m C B, and W the load ; then 
 
7 g CONSTRUCTION. 
 
 v/ I 
 
 JL_/- A = the amount of pressure at A, and 
 
 . ? = ,5 = the amount of pressure at ^?. 
 
 Example. A beam 20 feet long between the bearings 
 has a load of 100 pounds concentrated at 3 feet from one of 
 the bearings : what is the portion of this weight sustained by 
 each bearing ? 
 
 Here W 100 ; , 17 ; m, 3 ; and /, 20. 
 
 W m 
 Hence A=-j- 
 
 Load on A = 15 pounds. 
 Load on ^ = 85 pounds, 
 Total weight = 100 pounds. 
 
 RESISTANCE OF MATERIALS. 
 
 89. Weight Strength. Preliminary to designing a roof- 
 truss or other piece of framing, a knowledge of two subjects 
 is essential : one is, the effect of gravity acting upon the 
 various parts of the intended structure ; the other, the power 
 of resistance possessed by the materials of which the framing 
 is to be constructed. The former subject having been 
 treated of in the preceding pages, it remains now to call at- 
 tention to the latter. 
 
 90. Quality of Materials. Materials used in construc- 
 tion 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 
 fibrous, while cast iron, stone, glass, etc., are granular. The 
 strength of a granular material lies in the power of attrac- 
 tion acting among the grains of matter of which the mate- 
 rial is composed, by which it resists any attempt to separate 
 its grains or particles of matter. A fibre of wood or ot 
 
UNIVERSITY 
 
 
 THE THREE KINDS OF ST 
 
 wrought metal has a strength by which it resists being com- 
 pressed or shortened, and finally crushed ; also a strength 
 by which it resists being extended or made longer, and 
 finally sundered. There is another kind of strength in a 
 fibrous material : it is the adhesion of one fibre to another 
 along their sides, or the lateral adhesion of the fibres. 
 
 91. Manner of Rcisting. 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 case is termed the resistance to compression. In the strain 
 on a piece of timber like a king-post or suspending piece 
 (A, Fig. 37), we have an instance of an essay to extend or 
 lengthen the fibres of the material. The strength here ex- 
 hibited is termed the resistance to tension. When a piece of 
 timber is strained like a floor-beam or any horizontal piece 
 
 FIG 37. 
 
 FIG 38. 
 
 carrying a load (Fig. 38). we have an instance in which the 
 two strains of compression and tension are both brought 
 into action ; the fibres of the upper portion of the beam be- 
 ing compressed, and those of the under part being stretched. 
 
78 CONSTRUCTION. 
 
 This kind of strength of timber is termed resistance to cross- 
 strains. In each of these three kinds of strain to which tim- 
 ber is subjected, the power of resistance is in a measure due 
 to the lateral adhesion of the fibres, not so much perhaps in 
 the simple tensile strain, yet to a considerable degree in the 
 compressive and cross strains. But the power of timber, 
 by which it resists -a pressure acting compressively in the 
 direction of the length of the fibres, tending to separate the 
 timber by splitting off a part, as in the case of the end of a 
 tie-beam, against which the foot of the rafter presses, is 
 wholly due to the lateral adhesion of the fibres. 
 
 92. Strength and Stiffne. The strengtJi of materials 
 is their power to resist fracture, while the stiffness of mate- 
 rials is their capability to resist deflection or sagging. A 
 knowledge of their strengtJi is useful, in order to determine 
 their limits of size to sustain given weights safely ; but a 
 knowledge of their stiffness is more important, as in almost 
 all constructions it is desirable not only that the load be safely 
 sustained, but that no appearance of weakness be manifested 
 by any sensible deflection or sagging. 
 
 93. Experiments : Constants In the investigation of 
 the laws applicable to the resistance of materials, it is found 
 that the dimensions length, breadth, and thickness bear 
 certain relations to the weight or pressure to which the 
 piece is subjected. These relations are general ; they exist 
 quite independently of the peculiarities of any specific piece 
 of material. These proportions between the dimensions 
 and the load are found to exist alike in wood, metal, stone, 
 and glass, or other material. One law applies alike to all 
 materials ; but the capability of materials to resist differs in 
 accordance with the compactness and cohesion of particles, 
 and the tenacity and adhesion of fibres, those qualities upon 
 which depends the superiority of one kind of material over 
 another. The capability of each particular kind of material 
 is ascertained by experiments, made upon several specimens, 
 and an average of the results thus obtained is taken as an 
 index of the capability of that material, and is introduced 
 in the rules as a constant number, each specific kind of ma- 
 
VALUES OF WOODS FOR COMPRESSION. 
 
 79 
 
 terial having- its own special constant, obtained by ex- 
 perimenting- on specimens of that peculiar material. The 
 results of experiments made to test the resistance of various 
 materials useful in construction their capability to resist 
 the three strains before named will now be introduced. 
 
 94. Resistance to Compression. The following table 
 exhibits the results of experiments made to test the resist- 
 ance to compression of such woods as are in common use in 
 this country for the purposes of construction. 
 
 TABLE I. RESISTANCE TO COMPRESSION. 
 
 MATERIAL. 
 
 >, 
 
 "> 
 Z 
 O 
 u 
 
 te 
 
 1 
 
 To crush fibres ~ 
 longitudinally. 
 
 To separate fibres 
 
 by sliding. fS 
 
 To crush fibres trans- 
 versely 3^ inch deep. 
 
 Value of .Pin Rules. 
 Sensible Impres- ^ 
 sion. 
 
 Georgia Pine 
 
 0-611 
 
 Pounds 
 per inch. 
 95OO 
 
 Pounds 
 per inch. 
 840 
 
 Pounds 
 per inch. 
 225O 
 
 QOO 
 
 
 0-762 
 
 II7OO 
 
 1160 
 
 28OO 
 
 1 1 2O 
 
 White Oak 
 
 O- 774 
 
 8000 
 
 I25O 
 
 2650 
 
 IO6O 
 
 Spruce 
 
 0-360 
 
 7850 
 
 540 
 
 650 
 
 26O 
 
 White Pine 
 
 O-^88 
 
 6650 
 
 480 
 
 800 
 
 32O 
 
 Hemlock 
 
 O-423 
 
 57OO 
 
 370 
 
 800 
 
 32O 
 
 White Wood 
 
 o- 307 
 
 3400 
 
 
 800 
 
 320 
 
 Chestnut 
 
 O-4QI 
 
 6700 
 
 
 1250 
 
 5OO 
 
 Ash 
 
 o- 517 
 
 5850 
 
 
 3'OO 
 
 1240 
 
 Maple 
 
 O-574 
 
 8450 
 
 
 2700 
 
 I080 
 
 Hickory 
 
 O-877 
 
 13750 
 
 
 4100 
 
 1640 
 
 Cherry 
 
 O*4Q4 
 
 QO5O 
 
 
 2500 
 
 IOOO 
 
 Black Walnut 
 
 O-42I 
 
 7800 
 
 
 2IOO 
 
 840 
 
 Mahogany (St. Domingo) 
 (Bay Wood) 
 
 0-837 
 O-43Q 
 
 Il6oO 
 4900 
 
 
 
 5700 
 I7OO 
 
 2280 
 680 
 
 Live Oak 
 
 o 016 
 
 IIIOO 
 
 
 6800 
 
 2720 
 
 Lignum Vita? . . 
 
 \J yj.w 
 
 1-282 
 
 I2IOO 
 
 
 7700 
 
 3080 
 
 
 
 
 
 
 
 The resistance of timber of the same name varies much ; 
 depending as it obviously must on the soil in which it grew 
 on its age before and after cutting, on the time of year 
 when cut, and on the manner in which it has been kept since 
 it was cut. And of wood from the same tree much depends 
 upon its location, whether at the butt or towards the limbs, 
 and whether at the heart or of the sap, or at a point mid- 
 way from the centre to the circumference of the tree. The 
 
8O CONSTRUCTION. 
 
 pieces submitted to experiment were of ordinary good 
 quality, such as would be deemed proper to be used in 
 framing. The prisms crushed were generally small, about 
 2 inches long, and from I inch to i-J inches square ; some 
 were wider one way than the other, but all containing in 
 area of cross section from I to 2 inches. The weight given 
 in the table is the average weight per superficial inch. 
 
 Of the first six woods named, there were nine specimens 
 of each tested ; of the others, generally three specimens. 
 
 The results for the first six woods named are taken 
 from the author's work on Transverse Strains, published by 
 John Wiley & Sons, New York. The results for these six 
 woods, as well as those for all the others named in the table, 
 were obtained by experiments carefully made by the author. 
 The first six woods named were tested in 1874 and 1876, and 
 upon a testing machine, in which the power is transmitted 
 to the pieces tested, by levers acting upon knife-edges. 
 For a description of this machine, see Transverse Strains, 
 Art. 704. The woods named in the table, other than the 
 first six, were tested some twenty years since, and upon a 
 hydraulic press, which, owing to friction, gave results too 
 low. 
 
 The results, as thus ascertained, were given to the public 
 in the 7th edition of this work, in 1857. I n the present edi- 
 tion, the figures in Table I., for these woods, are those 
 which have resulted by adding to the results given by the 
 hydraulic press a certain quantity thought to be requisite 
 to compensate for the loss by friction. Thus corrected, the 
 figures in the table may be taken as sufficiently near approx- 
 imations for use in the rules, although not so trustworthy 
 as the results given for the first six woods named, as these 
 were obtained upon a superior testing machine, as above 
 stated. 
 
 In the preceding table, the second column contains the 
 specific gravity of the several kinds of wood, showing their 
 comparative density. The weight in pounds of a cubic foot 
 of any kind of wood or other material is equal to its 
 specific gravity multiplied by 62-5, this number being the 
 weight in pounds of a cubic foot of water. The third column 
 
EXPLANATION OF TABLE I. 8 1 
 
 contains the weight in pounds required to crush a prism 
 having a base of one inch square ; the pressure applied to 
 the fibres longitudinally. In practice, it is usual never to 
 load material exposed to compression with more than one 
 fourth of the crushing weight, and generally with from one 
 sixth to one tenth only. The fourth column contains the 
 weight in pounds w r hich, applied in line with the length of the 
 fibres, is required to force off a part of the piece, causing the 
 fibres to separate by sliding, the surface separated being one 
 inch square. The fifth column contains the weight in pounds 
 required to crush the piece when the pressure is applied to the 
 fibres transversely, the piece being one inch thick, and the 
 surface crushed being one inch square, and depressed one 
 twentieth of an inch deep. The sixth column contains the 
 value of P in the rules ; P being the weight in pounds, ap- 
 plied to the fibres transversely, which is required to make a 
 sensible impression one inch square on the side of the piece, 
 this being the greatest weight that would be proper for a 
 post to be loaded with per inch surface of bearing, resting 
 on the side of the kind of wood set opposite in the table. A 
 greater weight would, in proportion to the excess, crush the 
 side of the wood under the post, and proportionably derange 
 the framing, if not cause a total failure. It will be observed 
 that the measure of this resistance is useful in limiting the 
 load on a post according to the kind of material contained, 
 not in the post, but in the timber upon which the post presses. 
 
 95. Heitance to Tension. The resistance of materials 
 to the force of stretching, as exemplified in the case of a 
 rope from which a weight is suspended, is termed the resist- 
 ance to tension. In fibrous materials, this force will be differ- 
 ent in the same specimen, in accordance \vith the direction 
 in which the force acts, whether in the direction of the length 
 of the fibres or at right angles to the direction of their length. 
 It has been found that, in hard woods, the resistance in the 
 former direction is about eight to ten times what it is in the 
 latter; and in soft woods, straight, grained, such as white 
 pine, the resistance is from sixteen to twenty times. A 
 knowledge of the resistance in the direction of the 1 
 the most useful in practice. 
 
82 
 
 CONSTRUCTION. 
 
 In the following table are recorded the results of ex- 
 periments made to test this resistance in some of the woods 
 in common use, and also in iron, cast and wrought. Each 
 specimen of the woods was turned cylindrical, and about 2 
 inches diameter, and then the middle part reduced to about 
 f of an inch diameter, at the middle of the reduced part, 
 and thence gradually increased toward each end, where it 
 was considerably larger at its junction with the enlarged 
 end. The results, in the case of the iron and of the first six 
 woods named, are taken from the author's work, Trans- 
 verse Strains, Table XX. Experiments were made upon 
 the other three woods named by a hydraulic press, some 
 twenty years since, and the results were first published in 
 the 7th edition of this work, in 1857. These results, owing 
 to friction, were too low. Adding to them what is supposed 
 to be the loss by friction of the machine, the results thus 
 corrected are what are given for these three woods in the 
 following table, and may be taken as fair approximations, 
 but are not so trustworthy as the figures given for the other 
 six woods and for the metals. 
 
 TABLE II. RESISTANCE TO TENSION. 
 
 MATERIAL. 
 
 Specific 
 Gravity. 
 
 T. 
 
 Pounds re- 
 quired to rup- 
 ture one inch 
 square. 
 
 Georgia Pine 
 
 O-6^ 
 
 16000 
 
 Locust 
 
 O'7Q4 
 
 24800 
 
 White Oak 
 
 O 77J. 
 
 IQCQO 
 
 
 O 4^2 
 
 TOCOO 
 
 White Pine 
 
 Q. 4^8 
 
 I2OCO 
 
 Hemlock f 
 
 o 402 
 
 87OO 
 
 Hickory 
 
 O- 7^1 
 
 26OCO 
 
 Maple 
 
 O- 6o4 
 
 2OOOO 
 
 Ash 
 
 O-6o8 
 
 I5OOO 
 
 Cast Iron, American ) from 
 
 6 Q44 
 
 
 English f to 
 
 7 c84 
 
 1 7OOO 
 
 Wrought Iron, American ) from 
 
 7 6OO 
 
 
 English y to 
 
 7 7Q2 
 
 5OOOO 
 
 
 
 
 The figures in the table denote the ultimate capability of 
 a bar one inch square, or the weight in pounds required to 
 
VALUES OF MATERIALS FOR CROSS-STRAINS. 
 
 produce rupture. Just what portion of this should be taken 
 as the safe capability will depend upon the nature of the 
 strain to which the material is to be exposed. In practice it 
 is found that, through defects in workmanship, the attach- 
 ments may be so made as to cause the strain to act along one 
 side of the piece, instead of through its axis; and that in this 
 case fracture will be produced with one third of the strain 
 that can be sustained through the axis. Due to this and 
 other contingencies, it is usual to subject materials exposed 
 to tensile strain with only from one sixth to one tenth of 
 the breaking weight. 
 
 96. Resistance to Transverse Strains. In the follow- 
 ing table are recorded the results of experiments made to 
 test the capability of the various materials named to resist 
 the effects of transverse strain. The figures are taken from 
 the author's work, Transverse Strains, before referred to. 
 
 TABLE III. TRANSVERSE STRAINS. 
 
 MATERIAL. 
 
 * 
 
 Resistance 
 to 
 Rupture. 
 
 F. 
 
 Resistance 
 to 
 Flexure. 
 
 e. 
 
 Extension 
 of 
 Fibres. 
 
 a, 
 
 Margin 
 for 
 Safety. 
 
 Georgia. Pine 
 
 850 
 
 5QOO 
 
 00109 
 
 I-8 4 
 
 
 I2OO 
 
 505 
 
 OOI5 
 
 2-20 
 
 White Oak 
 
 6"iO 
 
 3IOO 
 
 00086 
 
 3'39 
 
 
 55 
 
 35OO 
 
 00098 
 
 2-23 
 
 White Pine 
 
 500 
 
 2900 
 
 0014 
 
 I-7I 
 
 Hemlock 
 
 450 
 
 2800 
 
 00095 
 
 2-35 
 
 White Wood 
 
 600 
 
 3450 
 
 00096 
 
 2-52 
 
 
 480 
 
 2550 
 
 00103 
 
 2-54 
 
 A*h 
 
 QOO 
 
 4OOO 
 
 OOIII 
 
 2 82 
 
 Maple 
 
 IIOO 
 
 5I5 
 
 0014 
 
 2-12 
 
 
 1050 
 
 3850 
 
 0013 
 
 2-QI 
 
 Cherry 
 
 650 
 
 2850 
 
 001563 
 
 2-03 
 
 Black Walnut 
 
 750 
 
 3900 
 
 00104 
 
 2-57 
 
 Mahogany (St Domingo) .... 
 
 650 
 
 3600 
 
 00116 
 
 2-16 
 
 (Bay Wood) 
 
 850 
 
 475 
 
 00109 
 
 2-28 
 
 Cast Iron, American . . 
 " English 
 
 2500 
 
 2IOO 
 
 50000 
 40000 
 
 
 
 
 26OO 
 
 62000 
 
 
 
 " English 
 
 iqoo 
 
 60000 
 
 
 
 Steel in Bars 
 
 6000 
 
 70000 
 
 1 
 
 
 
 200 
 
 
 
 
 
 59 
 
 
 
 
 
 33 
 
 
 
 
 ' ' pressed 
 
 37 
 
 
 
 
 Marble East Chester 
 
 147 
 
 
 
 
 
 
 
 
 
84 CONSTRUCTION. 
 
 The figures in the second column, headed B, denote the 
 weight in pounds required to break a unit of the material 
 named when suspended from the middle, the piece being 
 supported at each end. The unit of material is a bar one inch 
 square and one foot long between the bearings. The third 
 column, headed F, contains the values of the several mate- 
 rials named as to their resistance to flexure, as explained in 
 Arts. 302-305, Transverse Strains. These values of F, as 
 constants, are used in the rules. The fourth column, headed 
 ^, contains the values of the several materials named, denot- 
 ing the elasticity of the fibres, as explained in Art. 312, 
 Transverse Strains. These values of e, as constants, are 
 used in the rules. 
 
 The fifth column, headed a y contains for the several ma- 
 terials named the ratio of the resistance to flexure as com- 
 pared with that to rupture, and which, as constants used 
 in the rules, indicate the margin of safety to be given for 
 each kind of material. The figures given in each case show 
 the smallest possible value that may be safely given to a, the 
 factor of safety, x In practice it is generally taken higher 
 than the amount given in the table. For example, in the 
 table the value of B, the constant for rupture by transverse 
 strain for spruce, is 550. 
 
 Now, if the dimensions of a spruce beam to carry a given 
 weight be computed by the rules, using the constant B, at 
 550, the beam will be of such a size that the given weight 
 will just break it. 
 
 But if, in the computation, instead of taking the full 
 value of B, only a part of it be taken, then the beam will not 
 break immediately; and if the part taken be so small that 
 its effect upon the fibres shall not be sufficient to strain them 
 beyond their limit of elasticity, the beam will be capable of 
 sustaining the weight for an indefinite period ; in this case 
 the beam will be loaded by what is termed the safe weight. 
 Or, since the value of a for spruce is 2-23 in the table, if, in- 
 stead of taking B at 550, its full value, only the quotient 
 arising from a division of B by a be taken or 550 divided 
 by 2-23, which equals 246-6 then the beam will be of suffi- 
 cient size to carry the load safely. Therefore, while the con- 
 
THE VARIOUS CLASSES OF PRESSURES. 85 
 
 slant B is to be used for a breaking weight, for a safe load 
 
 n 
 
 the quotient of - is to be used. But, again, if a be taken at 
 
 its value as given in the table, the computed beam will be 
 loaded up to its limit of safety. So loaded that, if the load 
 be increased only in a small degree, the limit of safety will 
 be passed, and the beam liable, in time, to fail by rupture. 
 
 Therefore, as the values of a, in the table, are the smallest 
 possible, it is prudent in practice always to take a larger 
 than the table value. For example, the table value of 
 a for spruce is 2-23, but in practice let it be taken at 3 
 or 4. 
 
 97. Resistance to Compression. The resistance of ma- 
 terials to the force of compression may be considered in 
 several ways. Posts having their heights less than ten 
 times their least sides will crush before bending ; these 
 belong to one class : another class is that which com- 
 prises all posts the height of which is equal to ten times 
 their least sides, or more than ten times ; these will bend 
 before crushing. Then there remains to be considered 
 the manner in which the pressure is applied : whether in 
 line with the fibres, or transversely to them ; and, again, 
 whether the pressure tends to crush the fibres, or simply 
 to force off a part of the piece by splitting. The various 
 pressures may be comprised in the four classes following, 
 namely : 
 
 ist. When the pressure is applied to the fibres trans- 
 versely. 
 
 2d. When the pressure is applied to the fibres longi- 
 tudinally, and so as to split off the part pressed against, 
 causing the fibres to separate by sliding. 
 
 3d. When the pressure is applied to the fibres longi- 
 tudinally, and on short pieces. 
 
 4th. When the pressure is applied to the fibres longi- 
 tudinally, and on long pieces. 
 
 These four classes will now be considered in their reg- 
 ular order. 
 
86 CONSTRUCTION. 
 
 98. Compression Transversely to the Fibres. In this 
 first class of compression, experiment has shown that the 
 resistance is in proportion to the number of fibres pressed, 
 that is, in proportion to the area. For example, if 5000 
 pounds is required to crush a prism with a base i inch 
 square, it will require 20,000 pounds to crush a prism having 
 a base of 2 by 2 inches, equal to 4 inches area ; because 4 
 times 5000 equals 20000. 
 
 Therefore, if any given surface pressed be multiplied by 
 the pressure per inch which the kind of material pressed 
 may be safely trusted with, the product will be the total 
 pressure which may with safety be put upon the given sur- 
 face. Now, the capability for this kind of resistance is given 
 in column P, in Table I., for each kind of material named in 
 the table. Therefore, to find the limit of weight, proceed 
 as follows: 
 
 99. The Limit of Weight. To ascertain what weight 
 a post may be loaded with, so as not to crush the surface 
 against which it presses, we have 
 
 Rule 1. Multiply the area of the post in inches by the 
 value of P, Table I., and the product will be the weight re- 
 quired in pounds ; or 
 
 W=AP. (i.) 
 
 Example. A post, 8 by 10 inches, stands upon a white- 
 pine girder; the area equals 8 x 10 = 80 inches. This being 
 multiplied by 320, the value of P, Table I., set opposite white 
 pine, the product, 25600, is the required weight in pounds. 
 
 100. Area of Post. To ascertain what area a post must 
 have in order to prevent the post, loaded with a given 
 weight, from crushing the surface against which it presses, 
 we have 
 
 Rule II. Divide the given weight in pounds by the value 
 of P, Table I., and the quotient will be the area required in 
 inches ; or-- 
 
RESISTANCE TO RUPTURE BY SLIDING. 87 
 
 Example. A post standing on a Georgia-pine girder is 
 loaded with 100,000 pounds: what must be its area? The 
 weight, 100000, divided by 900, the value of /> Table I., set 
 opposite Georgia pine, the quotient, in- 11, is the required 
 area in inches. The post may be 10 by ii|, or 10 by 11 
 inches; or if square each side will be 10-54 inches, or II T ' 
 inches diameter if round. 
 
 101. Rupture by Sliding. In this the second class of 
 rupture by compression, it has been ascertained by ex- 
 periment that the resistance is in proportion to the area of 
 the surface separated without regard to the form of the sur- 
 face. Now, in Table I., column //, we have the ultimate 
 resistance to this strain of the several materials named. 
 But to obtain the safe load per inch, the ultimate resist- 
 ance of the table is to be divided by a factor of safety, of 
 such value as circumstances may seem to require. Gener- 
 ally this factor may be taken at 3. Then to obtain the safe 
 load for any given case, we have but to multiply the given 
 surface by the ultimate resistance, and divide by the factor 
 of safety ; therefore, proceed as follows : 
 
 102. Tle 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 to slide 
 on the other, we have 
 
 Rule III. Multiply the area of the surface by the value 
 of Hi in Table I. divide by the factor of safety, and the 
 quotient will be the weight required in pounds ; or 
 
 W = -*-Z (3-) 
 
 Example. The foot of a rafter is framed into the end of 
 its tie-beam, so that the uncut substance of the tie-beam is 
 1 5 inches long from the end of the tie-beam to the joint of 
 the rafter; the tie-beam is of white pine, and is 6 inches 
 thick: what amount of horizontal thrust will this end of 
 the tie-beam sustain, without danger of having the end oi 
 
88 CONSTRUCTION. 
 
 the tie-beam split off? Here the area of surface that sus- 
 tains the pressure is 6 by 15 inches, equal to 90 inches. 
 This multiplied by 480, the value of H, set opposite to 
 white pine, Table I., and divided by 3, as a factor of safety, 
 gives a quotient of 14400, and this is the required weight in 
 pounds. 
 
 103. Area of Surface. To ascertain the area of surface 
 that is required to sustain a given weight safely, when the 
 weight tends to split off the part pressed against, by causing, 
 in case of fracture, one 'surface to slide on the other, we 
 have 
 
 Rule IV. Divide the given weight in pounds by the 
 value of H, Table I. ; multiply the quotient by the factor of 
 safety, and the product will be the required area in inches ; 
 or 
 
 (40 
 
 Example. The load on a rafter causes a horizontal thrust 
 at its foot of 40,000 pounds, tending to split off the end of 
 the tie-beam : what must be the length of the tie-beam be- 
 yond the line where the foot of the rafter is framed into it, 
 the tie-beam being of Georgia pine, and 9 inches thick ? 
 The weight, or horizontal thrust, 40000, divided by 840, the 
 value of //, Table I., set opposite Georgia pine, gives a quo- 
 tient of 47-619, and this multiplied by 3, as a factor of safety, 
 gives a product of 142-857. This, the area of surface in 
 inches, divided by 9, the breadth of the surface strained 
 (equal to the thickness of the tie-beam), the quotient, 15.87, 
 is the length in inches from the end of the tie-beam to the 
 rafter joint, say 16 inches. 
 
 I04.-Tcnon and Splices.- A knowledge of this kind of 
 resistance of materials is useful, also, in ascertaining the 
 length of framed tenons, so as to prevent the pin, or key, 
 with which they are fastened from tearing out ; and, also, in 
 cases where tie-beams, or other timber under a tensile strain, 
 
CRUSHING STRENGTH OF STOUT POSTS. 89 
 
 are spliced, this rule gives the length of the joggle at each 
 end of the splice. 
 
 105. Stout Post. These comprise the third class of ob- 
 jects subject to compression (Art. 97), and include all posts 
 which are less than ten diameters high. The resistance to 
 compression, in this class, is ascertained to be directly in pro- 
 portion to the area of cross-section of the post. 
 
 Now in Table I., column C y the ultimate resistance to 
 crushing is given for the several kinds of materials named ; 
 from which the safe resistance per inch may be obtained by 
 dividing it by a proper factor of safety. Having the safe 
 resistance per inch, the resistance of any given post may be 
 determined by multiplying it by the area of the cross-section 
 of the post. Therefore, proceed as follows : 
 
 106. TBie 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 pressure 
 coinciding with the axis, we have 
 
 R H l e V. Multiply the area of the cross-section of the 
 post in inches by the value of C, in Table I. ; divide the pro- 
 duct by the factor of safety, and the quotient will be the re- 
 quired weight in pounds ; or 
 
 W= ^-. (5.) 
 
 Example. A Georgia-pine post is 6 feet high, and in 
 cross-section 8 x 12 inches: what weight will it safely sus- 
 tain? The height of this post, 12 x 6 = 72 inches, which is 
 less than lox 8 (the size of the narrowest side) 80 inches ; 
 it therefore belongs to the class coming under this rule. The 
 area = 8 x 12 = 96 inches ; this multiplied by 9500, the value 
 of C, in the table, set opposite Georgia pine, and divided by 
 6, as a factor of safety, the quotient, 152000, is the weight in 
 pounds required. It will be observed that the weight would 
 be the same for a Georgia-pine post of any height less than 
 
90 CONSTRUCTION. 
 
 10 times 8 inches = So inches = 6 feet 8 inches, provided its 
 breadth and thickness remain the same, 12 and 8 inches. 
 
 107. Area of Post. To find the area of the cross-sec- 
 tion of a post to sustain a given weight safely, the height of 
 the post being less than ten times the diameter if round, or 
 ten times the least side if rectangular, the pressure coinciding 
 with the axis, we have 
 
 Rule VI. Divide the given weight in pounds by the 
 value of C, in Table I. -, multiply the quotient by the factor 
 of safety, and the product will be the required area in 
 inches ; or 
 
 (6.) 
 
 Example. A weight of 40,000 pounds is to be sustained 
 by a white-pine post 4 feet high : what must be its area of 
 section to sustain the weight safely ? Here, 40000 divided 
 by 6650, the value of C, in Table I., set opposite white pine, 
 and the quotient multiplied by 6, as a factor of safety, the pro- 
 duct is 36 ; this, therefore, is the required area, and such a 
 post may be 6 x 6 inches. To find the least side, so that it 
 shall not be less than one tenth of the height, divide the 
 height, reduced to inches, by 10, and make the least side to 
 exceed this quotient. The area divided by the least side so 
 determined will give the wide side. If, however, by this 
 process, the first side found should prove to be the greatest, 
 then the size of the post is to be found by Rule IX., X., or 
 XI. 
 
 108. Area of Round Pot. In case the post is to be 
 round, its diameter may be found by reference to the Table 
 of Circles in the Appendix, in the column of diameters, op- 
 posite to the area of the post to be found in the column of 
 areas, or opposite to the next nearest area. For example, 
 suppose the required area, as just found by the example 
 under Rule VI., is 36 : by reference to the column of areas, 
 35.78 is the nearest to 36, and the diameter set opposite is 
 
CRUSHING STRENGTH OF SLENDER POSTS. 9! 
 
 6.75, which is a trifle too small. The post may therefore be, 
 say, 6| inches diameter. 
 
 109. Slender Posts. When the height of a post is less 
 than ten times its diameter, the resistance of the post to 
 crushing is approximately in proportion to its area of cross- 
 section. But when the height is equal to or more than ten 
 diameters, the resistance per square inch is diminished. The 
 resistance diminishes as the height is increased, the diameter 
 remaining the same (Transverse Strains, Art. 643). The 
 strength of a slender post consists in a combination of the 
 resistances of the material to bending and to crushing, and 
 is represented in the following rule : 
 
 110. The 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 pressure coincid- 
 ing with the axis, we have 
 
 Rule VII. Divide the height of the post in inches by the 
 diameter, or least side, in inches ; multiply the quotient by 
 itself, or take its square ; multiply the square by the value 
 of e, in Table III., set opposite the kind of material of which 
 the post is made ; to the product add the half of itself ; to 
 the sum add unity (or one) ; multiply this sum by the factor 
 of safety, and reserve the product for use, as below. Now 
 multiply the area of cross-section of the post in inches by 
 the value of C, in Table I., set opposite the material of the 
 post, and divide the product by the above reserved product; 
 the quotient will be the required weight in pounds ; or 
 
 (7.) 
 
 Example : A Round Post. What weight may be safely 
 placed upon a post of Georgia pine 10 inches diameter and 
 10 feet high, the pressure coinciding with the axis of the 
 post? The height of the post, dox 12 =) 120 inches, divided 
 by 10, its diameter, gives a quotient of 12; this multiplied 
 
92 CONSTRUCTION. 
 
 by itself gives 144, its square; and this by -00109, the value 
 of e for Georgia pine, in Table III., gives 15696 ; to which 
 adding its half, the sum is 0-23544; to which adding unity, 
 the sum is 1-23544 ; and this multiplied by 7, as a factor of 
 safety, the product is 8 -648, the reserved divisor. Now the 
 area of the post is (see Table of Areas of Circles, in the Ap- 
 pendix, opposite its diameter, 10) 78-54; this multiplied by 
 9500, the value of C for Georgia pine, in Table I., gives a 
 product of 746130; which divided by 8-648, the above re- 
 served divisor, gives a quotient of 86278, the required weight 
 in pounds. 
 
 Anotlier Example : A Rectangular Post. What weight may 
 be safely placed upon a white-pine post lox 12 inches, and 
 15 feet high, the pressure coinciding with the axis of the 
 post? Proceeding according to the rule, we find the height 
 of the post to be 180 inches, which divided by 10, the least 
 side of the post, gives 18 ; this multiplied by itself gives 324* 
 its square ; which multiplied by -0014, the value of e for 
 white pine, in Table III., gives -4536; to which adding its 
 half, the sum is -6804; to which adding unity, the sum is 
 i -6804 ; and this multiplied by 8, as a factor of safety, the pro- 
 duct is 13-4432, the reserved divisor. Now the area of the 
 post, ( 10 x 12 =) 120 inches, multiplied by 6650, the value of 
 C for white pine, in Table I., gives a product of 798,000, and 
 this divided by 13-4432, the above reserved divisor, the quo- 
 tient, 59360, is the required weight in pounds. 
 
 III. Diameter of the Post: when Round. To ascertain 
 the size of a round post to sustain safely a given weight, 
 when the height of the post is at least ten times the diameter ; 
 the direction of the pressure coinciding with the axis of the 
 post; we have 
 
 Rule VIII. Multiply the given weight by the factor of 
 safety, and divide the product by 1-5708 times the value of C 
 for the material of the post, found in Table I. ; reserve the 
 quotient, calling its value G. Now multiply 432 times the 
 value of c for the material of the post, found in Table III., 
 by the square of the height in feet, and by the above quo- 
 tient G ; to the product add the square of G : extract the 
 
SIZE OF POST FOR GIVEN WEIGHT. 93 
 
 square root of the sum, and to it add the value of G ; then 
 the square root of this sum will be the required diameter; 
 
 or 
 
 r Wa 
 
 Cr nzr 
 
 ,5708 L 
 
 d =4 / . / At? (T~f / 8 O. f^ 4- /C * (90 
 
 Example. What should be the diameter of a locust post 
 10 feet high to sustain safely 40,000 pounds, the pressure 
 coinciding with the axis ? Proceeding by the rule, the given 
 weight multiplied by 6, taken as a factor of safety, equals 
 240000. Dividing this by 1-5708 times 11700, the value of 
 C for locust, in Table I., the quotient, 13-06, is the value of 
 G, the square of which is 170-53. Now, the value of e for 
 locust, in Table III., is -0015. This multiplied by 432, by 
 100, the square of the height, and by the above value of G, 
 gives a product of 846-2 ; which added to 170-53, the above 
 square of G, gives the sum of 1016-73. To 31-89, the square 
 root of this, add the above value of G ; then 6-7, the square 
 root of this sum, is the required diameter of the post. The 
 post therefore requires to be 6-7, say 6- inches diameter. 
 
 112. Side of tlic Post: when Square. To ascertain the 
 side of a square post to sustain safely a given weight, when 
 the height of the post is at least ten times the side ; the pres- 
 sure coinciding with the axis ; we have 
 
 Rule IX. Multiply the given weight by the factor of 
 safety, and divide the product by twice the value of C for 
 the material of the post, found in Table I. ; reserve the quo- 
 tient, calling its value G.. Now multiply 432 times the value 
 of e for the material of the post, found in Table III., by the 
 square of the height in feet, and by the above quotient G\ 
 to the product add the square of G ; extract the square root 
 of the sum, and to it add the value of G ; then the square 
 root of this sum will be the required side of the post ; or 
 
 . Co-) 
 
 2 C 
 
94 CONSTRUCTION. 
 
 S =4/ 4/432 Ge I * + G' 2 + G. 
 
 Example. What should be the side of a Georgia-pine 
 square post 1 5 feet high to sustain safely 50,000 pounds, the 
 pressure coinciding with the axis of the post? Proceeding 
 by the rule, 50,000 pounds multiplied by 6, as a factor of 
 safety, gives 300000 ; this divided by 2 x 9500 (the value of 
 )= 19000, the quotient, 15-789, is the value of G. The 
 value of e for Georgia pine is -00109; tne square of the 
 height is 225 ; then, 432 times -00109 by 225 and by 15-789 
 (the above value of G) gives a product of 1672 - 86 ; the square 
 of equals 249-31 ; this added to 1672-86 gives a sum of 
 1922- 17, the square root of which is 43-843 ; which added* to 
 15-789, the value of G, gives 59-632, the square root of which 
 is 7-722, the required side of the post. The post, therefore, 
 requires to be, say, 7f inches square. 
 
 113. Thickness of a Rectangular Post. This may be 
 definitely ascertained when the proportion which the thick- 
 ness shall bear to the breadth shall have been previously 
 determined. For example, when the proportion is as 6 to 8, 
 then i J times 6 equals 8, and the proportion is as 1 to i; 
 again, when the proportion is as 8 to 10, then ij times 8 
 equals 10, and in this case the proportion is as i to ij. Let 
 the latter figure of the ratio i to ij, i to ij, etc., be called 
 n, or so that the proportion shall be as i to n y then 
 
 To ascertain the thickness of a post to sustain safely a 
 given weight, when the height is at least ten times the thick- 
 ness ; the action of the weight coinciding with the axis ; we 
 have 
 
 Rule X. Multiply the given weight by the factor of 
 safety, and divide the product by twice the value of C for 
 the material of the post, found in Table I., multiplied by ;/, 
 as above explained ; reserve the quotient, calling it G. Now 
 multiply 432 times the value of e for the material of the post, . 
 found in Table III., by the square of the height in feet, and 
 by the above quotient G ; to the product add the square of 
 G ; extract the square root of the sum, and to it add th value 
 
BREADTH OF POST FOR GIVEN THICKNESS. 
 
 95 
 
 of G ; then the square root of this sum will be the required 
 thickness of the post ; or 
 
 Wa 
 
 G-~ TT- -. (12.) 
 
 2 C n v ' 
 
 t = V 1/432 G 
 
 (130 
 
 Example. What should be the thickness of a white-pine 
 rectangular post 20 feet high to sustain safely 30,000 pounds, 
 the pressure coinciding with the axis, and the proportion 
 between the thickness and breadth to be as 10 to 12, or as I 
 to i -2 ? Proceeding according to the rule, we have the pro- 
 duct of 30000, the given weight, by 6, as a factor of safety, 
 equals 180000 ; this divided by twice Cx n, or 2 x 6650 x i -2, 
 (=15960) gives a quotient of 11-278, the value of G. Then, 
 we have c= -0014, the square of the height equals 400; 
 therefore, 432 x -0014x400x11.278 = 2728-43. Tothisadd- 
 ing 127-2, the square of , we have 2855 63, the square 
 root of which is 53-438; and this added to G gives 64-716, 
 the square root of which is 8-045, tne required thickness of 
 the post. Now, since the thickness is in proportion to the 
 breadth as i to 1-2, therefore 8-045 x 1-2 = 9-654, the re- 
 quired width. The post, therefore, may be made 8x9! 
 inches. 
 
 114-. Breadth of a Rectangular Post. When the thick- 
 ness of a post is fixed, and the breadth required ; then, to 
 ascertain the breadth of a rectangular post to sustain safely 
 a given weight, the direction of the pressure of which coin- 
 cides with the axis of the post, we have 
 
 Rule XI. Divide the height in inches by the given thick- 
 ness, and multiply the quotient by itself, or take its square ; 
 multiply this square by the value of e for the material of the 
 post, found in Table 111. ; to the product add its half, and to 
 the sum add unity ; multiply this sum by the given weight, 
 and by the factor of safety ; divide the product by the pro- 
 duct of the given thickness multiplied by the value of C for 
 
96 CONSTRUCTION. 
 
 the material of the post, found in Table I., and the quotient 
 will be the required breadth ; or 
 
 Example. What should be the breadth of a spruce post 
 1 8 feet high and 6 inches thick to sustain safely 25,000 
 pounds, the pressure coinciding with the axis of the post ? 
 According to the rule, 216 (= 12 x 18), the height in inches, 
 divided by 6, the given thickness, gives a quotient of 36, the 
 square of which is 1296; the value of e for spruce is -00098 ; 
 this multiplied by 1296, the above square, equals i -27 ; which 
 increased by -635, its half, amounts to 1-905 ; this increased 
 by unity, the sum is 2-905 ; which multiplied by the given 
 weight, and by the factor of safety, gives a product of 
 435749; and this divided by 6 (the given thickness) times 7850 
 (the value of C for spruce) = 47 1 oo, gives a quotient of 9 2 5 1 6, 
 the required breadth of the post. The post, therefore, re- 
 quires to be 6 x 9^ inches. 
 
 Observe that when the breadth obtained by the rule is 
 less than the given thickness, the result shows that the con- 
 ditions of the case are incompatible with the rule, and that 
 a new computation must be made ; taking now for the 
 breadth what was before understood to be the thickness, 
 and proceeding in this case, by Rule X., to find the thickness. 
 
 115. Resistance to Tenion. In Art. 95 are recorded the 
 results of experiments made to test the resistance of vari- 
 ous materials to tensile strain, showing in each case the ca- 
 pability to such resistance per square inch of sectional area. 
 The action of materials in resisting a tensile strain is quite 
 simple ; their resistance is found to be directly as their sec- 
 tional area. Hence 
 
 116. The rim it 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 value of 'T, Table II. ; divide 
 
AREA OF BEAM FOR TENSILE STRAIN. 97 
 
 the product by the factor of safety, and the quotient will be 
 the required weight in pounds ; or 
 
 (150 
 
 The cross-section here intended is that taken at the small- 
 est part of the beam or rod. A beam, in framing, is usually 
 cut with mortices ; the area will probably be smallest at the 
 severest cutting ; the area used in the rule must be that of the 
 uncut fibres only. 
 
 Example. The tie-beam of a roof-truss is of white pine, 
 6 x 10 inches ; the cutting for the foot of the rafter reduces 
 the uncut area to 40 inches : what amount of horizontal thrust 
 from the foot of the rafter will this tie-beam safely sustain ? 
 Here 40 times 12000, the value of T, equals 480000; this 
 divided by 6, as a factor of safety, gives 80000, the required 
 weight in pounds. 
 
 (17. 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 XIII. Multiply the given weight in pounds by the 
 factor of safety ; divide the product by the value of T, Table 
 II., and the quotient will be the area required in inches; 
 or 
 
 A=^. (16.) 
 
 This is the area of uncut fibres. If the piece is to be cut 
 for mortices, or for any other purpose, then for this an 
 adequate addition is to be made to the result found by the 
 rule. 
 
 Example. A rafter produces a thrust horizontally of 80,000 
 pounds ; the tie-beam is to be of oak : what must be the 
 area of the cross-section of the tie-beam in order to sustain 
 the rafter safely ? The given weight, 80000, multiplied by 
 10, as a factor of safety, gives 800000; this divided by 19500, 
 the value of 7", Table II., the quotient, 41, is the area of uncut 
 fibres. This should have usually one half of its amount 
 
98 CONSTRUCTION. 
 
 added to it as an allowance for cutting; therefore, 41+21 
 = 62. The tie-beam may be 6 x loj inches. 
 
 Another Example. A tie-rod of American refined 
 wrought iron is required to sustain safely 36,000 pounds : 
 what should be its area of cross-section ? Taking 7 as the 
 factor of safety, 7x36000= 252000; and this divided by 
 60000, the value of 7", Table II., gives a quotient of 4- 2 inches, 
 the required area of the rod. 
 
 118. Weight of the Suspending Piece Included. Pieces 
 
 subjected to a tensile strain are frequently suspended verti- 
 cally. In this case, at the upper end, ,the strain is due not 
 only to the weight attached at the lower end, but also to the 
 weight of the rod itself. Usually, in timber, this is small 
 in comparison with the load, and may be neglected ; 
 although in very long timbers, and where accuracy is decid- 
 edly essential, as, also, when the rod is of iron, it may form 
 a part of the rule. Taking the effect of the weight of the 
 beam into account, the relation existing between the weights 
 and the beam requires that the rule for the weight should 
 be as follows : 
 
 Rule XIV. Divide the value of T for the material of the 
 beam or rod, Table II., by the factor of safety ; from the 
 quotient subtract 0-434 times the specific gravity of the ma- 
 terial in the beam or rod multiplied by the length of the 
 beam or rod in feet ; multiply the remainder by the area of 
 cross-section in inches, and the product will be the required 
 weight in pounds ; or 
 
 W=A -0-434 
 
 N. B. This rule is based upon the condition that the sus- 
 pending piece be not cut by mortices or in any other way. 
 
 Example. What weight may be safely sustained by a 
 white-pine rod 4x6 inches, 40 feet long, suspended verti- 
 cally? For white pine the value of T is 12000; this divid- 
 ed by 8, as a factor of safety, gives 1 500 ; from which sub- 
 tracting 0-434 times 0-458 (the specific gravity of white pine, 
 Table II.) multiplied by 40, the length in feet, the remainder 
 
RESISTANCE TO TRANSVERSE STRAINS. 99 
 
 is 1492-049; which multiplied by 24 ( = 4x6, the area of 
 cross-section) equals 35,761 pounds, the required weight to be 
 carried. The weight which the rule would give, neglecting 
 the weight of the rod, would have been 36000; ordinarily, 
 so slight a difference would be quite unimportant. 
 
 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 suspending piece by the length in feet ; deduct the pro- 
 duct from the quotient arising from a division of the value 
 of T, Table II., by the factor of safety, and with the remain- 
 der divide the given weight in pounds ; the quotient will be 
 the required area in inches ; or 
 
 A = 
 
 T , ' (18.) 
 
 o-434/.y 
 
 a' 
 
 N.B. This rule is based upon the condition that the rod 
 be not injured in anywise by cutting. 
 
 Example. What should be the area of a bar of English 
 cast iron 20 feet long to sustain safely, suspended from its 
 lower end, a weight of 5000 pounds ? Taking the factor of 
 safety at 7-0, and the specific gravity also at 7, and the 
 value of T, Table II., at 17000, we have the product of 
 0-434 x 7-0 x 20 = 60-76; then 17000 divided by 7 gives 
 a quotient of 2428-57; from which deducting the above 
 60-76, there remains 2367-81 ; dividing 5000, the given 
 weight, by this remainder, we have the quotient, 2-11, which 
 is the required area in inches. 
 
 RESISTANCE TO TRANSVERSE STRAINS. 
 
 120. Tranvere Strains: Rupture. A load placed 
 
 upon a beam, laid horizontally or inclined, will bend it, and, 
 if the weight be proportionally large, will break it. The 
 power in the material that resists this bending or breaking 
 is termed the resistance to cross-strains, or transverse strains. 
 
100 CONSTRUCTION. 
 
 While in posts or struts the material is compressed or short- 
 ened, and in ties and suspending pieces it is extended or 
 lengthened, in beams subjected to cross-strains the material 
 is both compressed and extended. (See Art. 91.) When the 
 beam is bent the fibres on the concave side are compressed, 
 while those on the convex side are extended. The line 
 where these two portions of the beam meet that is, the 
 portion compressed and the portion extended the hori- 
 zontal line of juncture, is termed the neutral line or plane. 
 It is so called because at this line the fibres are neither com- 
 pressed nor extended, and hence are under no strain what- 
 ever. The location of this line or plane is not far from the 
 middle of the depth of the beam, when the strain is not suf- 
 ficient to injure the elasticity of the material ; but it re- 
 moves towards the concave or convex side of the beam as 
 the strain is increased, until, at the period of rupture, its 
 distance from the top of the beam is in proportion to its dis- 
 tance from the bottom of the beam as the tensile strength of 
 the material is to its compressive strength. 
 
 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 are aware 
 that the upper fibres of a beam are compressed when sub- 
 ject to cross-strains, that it is not injurious to cut these top 
 fibres, provided that the cutting be for the insertion of an- 
 other piece of timber as in the case of gaining the ends of 
 beams into the side of a girder. They suppose that the piece 
 filled in will as effectually resist the compression as the part 
 removed would have done, had it not been taken out. Now, 
 besides the effect of shrinkage, which of itself is quite suf- 
 ficient to prevent the proper resistance to the strain, there 
 is the mechanical difficulty of fitting the joints perfectly 
 throughout ; and, also, a great loss in the power of resist- 
 ance, as the material is so much less capable of resistance 
 when pressed at right angles to the direction of the fibres 
 than when directly with them, as the results of the experi- 
 ments in the tables show. 
 
STRENGTH OF BEAMS FOR CROSS-STRAINS. IOI 
 
 122. Transverse Strains : Relation of Weight to Di- 
 mensions. The strength of various materials, in their re- 
 sistance to cross-strains, is given in Table III., Art. 96. The 
 second column of the table contains the results of experi- 
 ments made to test their resistance to rupture. In the case 
 of each material, the figures given and represented by B 
 indicate the pounds at the middle required to break a unit 
 of the material, or a piece i inch square and i foot long 
 between the bearings upon which the piece rests. To be 
 able to use these indices of strength, in the computation of 
 the strength of large beams, it is requisite, first, to establish 
 the relation between the unit of material and the larger 
 beam. Now, it may be easily comprehended that the strength 
 of beams will be in proportion to their breadth ; that is, 
 when the length and depth remain the same, the strength 
 will be directly as the breadth ; lor it is evident that a beam 
 2 inches broad will bear twice as much as one which is only 
 i inch broad, or that one which is 6 inches broad will bear 
 three times as much as one which is 2 inches broad. This 
 establishes the relation of the weight to the breadth. With 
 the depth, however, the relation is different ; the strength is 
 greater than simply in proportion to the depth. If the 
 boards cut from a squared piece of timber be piled up in 
 the order in which they came from the timber, and be loaded 
 with a heavy weight at the middle, the boards will deflect 
 or sag much more than they would have done in the timber 
 before sawing. The greater strength of the material when 
 in a solid piece of timber is due to the cohesion of the fibres 
 at the line of separation, by which the several boards, before 
 sawing, are prevented from sliding upon each other, and 
 thus the resistance to compression and tension is made to 
 contribute to the strength. This resistance is found to be 
 in proportion to the depth. Thus the strength due to the 
 depth is, first, that which arises from the quantity of the 
 material (the greater the depth, the more the material), 
 which is in proportion to the depth ; then, that which en- 
 sues from the cohesion of the fibres in such a manner as to 
 prevent sliding ; this is also as the depth. Combining the 
 two, we have, as the total result, the resistance in proportion 
 
102 CONSTRUCTION. 
 
 to the square of the depth. The relation between the 
 weight and the length is such that the longer the beam is, 
 the less it will resist ; a beam which is 20 feet long will sus- 
 tain only half as much as one which is 10 feet long ; the 
 breadth and depth each being the same in the two beams. 
 From this it results that the resistance is inversely in pro- 
 portion to the length. To obtain, therefore, the relation 
 between the strength of the unit of material and that of a 
 larger beam, we have these facts, namely : the strength of 
 the unit is the value of B, as recorded in Table III. ; and 
 the strength of the larger beam, represented . by W, the 
 weight required to break it, is the product of the breadth 
 into the square of the depth, divided by the length ; or, 
 while for the unit we have the ratio 
 
 B\ i, 
 we have for the larger beam the ratio 
 
 Therefore, putting these ratios in an expressed proportion, 
 we have 
 
 From which (the product of the means equalling the pro- 
 duct of the extremes ; see Art. 373) we have 
 
 w = 
 
 In which W represents the pounds required to break a 
 beam, when acting at the middle between the two supports 
 upon which the beam is laid ; of which beam b represents 
 the breadth and d the depth, both in inches, and / the length 
 in feet between the supports ; and B is from Table III., and 
 represents the pounds required to break a unit of material 
 like that contained in the larger beam. 
 
fuinvERsm 
 
 LIMIT OF WEIGHT AT MIDDLE. ?&&,>' 
 
 123. Safe Weight: Load at Middle. The relation 
 established, in the last article, between the weight and the 
 dimensions is that which exists at the moment of rupture. 
 The rule (19.) derived therefrom is not, therefore, directly 
 practicable for computing the dimensions of beams for 
 buildings. From it, however, one may readily be deduced 
 which shall be practicable. In the fifth column of Table III. 
 are given the least values of a, the factor of safety, explained 
 in Art. 96. Now, if in place of B y the symbol for the break- 
 ing weight, the quotient of B divided by a be substituted, 
 then the rule at once becomes practicable ; the results now 
 being in consonance with the requirements for materials 
 used in buildings. Thus, with this modification, we have 
 
 Therefore, to ascertain the weight which a beam may be 
 safely loaded with at the centre, we have 
 
 Rule XVI. Multiply the value of B, Table HI., for the 
 kind of material in the beam by the breadth and by the 
 square of the depth of the beam in inches ; divide the pro- 
 duct by the product of the factor of safety into the length 
 of the beam between bearings in feet, and the quotient will 
 be the weight in pounds that the beam will safely sustain 
 at the middle of its length. 
 
 Example. What weight in pounds can be suspended 
 safely from the middle of a Georgia-pine beam 4x 10 inches, 
 and 20 feet long between the bearings ? For Georgia pine the 
 value of B, in Table III., is 850, and the least value of a is 
 1-84. For reasons given in Art. 96, let a be taken as high 
 as 4; then, in this case, the value of b is 4, and that of d is 
 10,' while that of/ is 20. Therefore, proceeding by the rule, 
 850 x 4 x io 2 = 340000 ; this divided by 4 x 20 ( 80) gives 
 a quotient of 4250 pounds, the required weight. 
 
 Observe that, had the value of a been taken at 3, instead 
 of 4, the result by the rule would have been a load of 5667 
 pounds, instead of 4250, and the larger amount would be 
 none too much for a safe load upon such a beam ; although, 
 
104 CONSTRUCTION. 
 
 with it, the deflection would be one third greater than with 
 the lesser load. The value of a should always be assigned 
 higher than the figures of the table, which show it at its 
 least value ; but just how much higher must depend upon 
 the firmness required and the conditions of each particular 
 case. 
 
 124. Breadth of Beam willi Safe Load. By a simple 
 transposition of the factors in equation (20.), we obtain 
 
 a rule for the breadth of the beam. 
 
 Therefore, to ascertain what should be the breadth of a 
 beam of given depth and length to safely sustain at the 
 middle a given weight, we have 
 
 Rule XVII. Multiply the given weight in pounds by 
 the factor of safety, and by the length in feet, and divide the 
 product by the square of the depth multiplied by the value 
 of B for the material in the beam, in Table III. ; the quotient 
 will be the required breadth. 
 
 Example. What should be the breadth of a white-pine 
 beam 8 inches deep and 10 feet long between bearings to 
 sustain safely 2400 pounds at the middle ? For white pine 
 the value of B, in Table III., is 500. Taking the value of a 
 at 4, and proceeding by the rule, we have 2400 x4x 10 = 
 96000 ; this divided by (8 a x 500 =) 32000 gives a quotient 
 of 3, the required breadth of the beam. 
 
 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 given breadth and length 
 to safely sustain a given weight at the middle, we have 
 
WEIGHT AT ANY POINT. 
 
 105 
 
 Rule XVIIL Multiply the given weight by the factor of 
 safety, and by the length in feet ; divide the product by the 
 product of the breadth into the value of B for the kind of 
 wood, Table III. ; then the square root of the quotient will 
 be the required depth. 
 
 Example. What should be the depth of a spruce beam 
 5 inches broad and lofeet long between bearings to sustain 
 safely, at middle, 4500 pounds ? The value of B from the table 
 is 550; taking a at 4, and proceeding by the rule, we have 
 4500 x 4 x 15 = 270000; this divided by (550 x 5 =) 2750 
 gives a quotient of 98-18, the square root of which is 9-909, 
 the required depth of the beam. The beam should be 5 x 10 
 inches. 
 
 126. Safe Load at any Point. When the load is at the 
 middle of a beam it exerts the greatest possible strain ; at 
 any other point the strain would be less. The strain de- 
 creases gradually as it approaches one of the bearings, and 
 when arrived at the bearing its effect upon the beam as a 
 cross-strain is zero. The effect of a weight upon a beam is 
 in proportion to its distance from one of the bearings, mul- 
 tiplied by the portion of the load borne by that bearing. 
 
 The load upon a beam is divided upon the two bearings, 
 as shown at Art. 88. The weight which is required to rup- 
 ture a beam is in proportion to the breadth and square of 
 the depth, b d*, as before shown, and also in proportion to 
 the length divided by 4 times the rectangle of the two parts 
 
 into which the load divides the length, or (see Fig. 35). 
 
 4 MI 11 
 
 This, when the load is at the middle, may be put as 
 
 = i a result coinciding with the relation before 
 
 4x|./xi/ / 
 
 given in Art. 122, viz. : "The resistance is inversely in pro- 
 portion to the length." The total resistance, therefore, put- 
 
 b d* I 
 ting the two statements together, is in proportion to - 
 
 A /// / 
 
 7727 
 
 There are, therefore, these two ratios, viz., W \ - - and 
 
 4 m i* 
 
 B : I, from which we have the proportion 
 
106 CONSTRUCTION. 
 
 4 m n 
 from which we have 
 
 4 ; n 
 
 (23 . } 
 
 r> 
 
 This is the relation at the point of rupture, and when is 
 
 used instead of B, the expression gives the safe weight. 
 Therefore 
 
 (24.) 
 
 4 a m n 
 
 is an expression for the safe weight. Now, to ascertain the 
 weight which may be safely borne by a beam at any point 
 in its length, we have 
 
 Rule XIX. Multiply the breadth by the square of the 
 depth, by the length in feet, and by the value of B for the 
 material of the beam, in Table III. ; divide the product by 
 the product of four times the factor of safety into the rec- 
 tangle of the two parts into which the centre of gravity of 
 the weight divides the beam, and the quotient will be the 
 required weight in pounds. 
 
 Example. What weight may be safely sustained at 3 feet 
 from one end of a Georgia-pine beam which is 4 x 10 inches, 
 and 20 feet long? The value of B for Georgia pine, in 
 Table III., is 850 ; therefore, by the rule, 4 x io a x 20 x 850 = 
 6800000. Taking the factor of safety at 4, we have 
 4x4x3x17=816. Using this as a divisor with which to 
 divide the former product, we have as a quotient 8333 
 pounds, the required weight. 
 
 127. Breadth or Depth: Load at any Point. By a 
 
 proper transposition of the factors of (24.) we obtain 
 
 , , a 4 W a in 11 
 
 an expression showing- the product of the breadth into the 
 square of the depth ; hence, to ascertain the breadth or 
 
DISTRIBUTED WEIGHT. IO/ 
 
 depth of a beam to sustain safely a given weight located at 
 any point on the beam, we have 
 
 Rule XX. Multiply four times the given weight by the 
 factor of safety, and by the rectangle of the two parts into 
 which the load divides the length ; divide the product by 
 the product of the length into the value of B for the mate- 
 rial of the beam, found in Table III., and the quotient will be 
 equal to the product of the breadth into the square of the 
 depth. Now, to obtain the breadth, divide this product by 
 the square of the depth, and the quotient will be the required 
 breadth. But if, instead of the breadth, the depth be de- 
 sired, divide the said product by the breadth ; then the 
 square root of the quotient will be the required depth. 
 
 Example. What should be the breadth (the depth being 
 8) of a white-pine beam 12 feet long to safely sustain 3500 
 pounds at 3 feet from one end ? Also, what should oe its 
 depth when the breadth is 3 inches? By the rule, taking 
 the factor of safety at 4, 4 x 3500 x 4 x 3 x 9 = 1512000. 
 The value of B for white pine, in Table III., is 500 ; there- 
 fore, 500 x 12 = 6000; with this as divisor, dividing 1512000, 
 the quotient is 252. Now, to obtain the breadth when the 
 depth is 8, 252 divided by (8 x 8 =) 64 gives a quotient of 
 3-9375, the required breadth ; or the beam may be, say, 4 x 8. 
 Again, when the breadth is 3 inches, we have for the quotient 
 of 252 divided by 3 = 84, and the square root of 84 is 9- 165, 
 or 9^ inches. For this case, therefore, the beam should be, 
 say, 3 x 9 J inches. 
 
 128. 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. 
 
 Therefore, we have W , where U represents the dis- 
 tributed load. Substituting this value of W in equation 
 (20.), we have 
 
 U__Bbd\ 
 
 2 ~~ ' a I ' 
 
 U= 2 -*** (26 .) - 
 
 a I 
 
IOS CONSTRUCTION. 
 
 Therefore, to ascertain the weight which may be safely sus- 
 tained, when uniformly distributed over the length of a 
 beam, we have 
 
 Rule XXI. Multiply twice the breadth by the square of 
 the depth, and by the value of B for the material of the 
 beam, in Table III., and divide the product by the product 
 of the length in feet by the factor of safety, and the quotient 
 will be the required weight in pounds. 
 
 Example. What weight uniformly distributed may be 
 safely sustained upon a hemlock beam 4x9 inches, and 20 
 feet long? The value of B for hemlock, in Table III., is 
 450 ; therefore, by the rule, 2 x 4 x 9' x 450 = 291600. Tak- 
 ing the factor of safety at 4, we have 4 x 20 80, the pro- 
 duct by which the former product is to be divided. This 
 division produces a quotient of 3645, the required weight. 
 
 129. Breadth or Depth : Load Uniformly Distributed. 
 
 By a proper transposition of factors in (26.), we obtain 
 
 an expression giving the value of the breadth into the square 
 of the depth. From this, therefore, to ascertain the breadth 
 or the depth of a beam to sustain safely a given weight uni- 
 formly distributed over the length of a beam, we have 
 
 Rule XXII. Multiply the given weight by the factor of 
 safety, and by the length ; divide the product by the pro- 
 duct of twice the value of B for the material of the beam, 
 in Table III., and the quotient will be equal to the breadth 
 into the square of the depth. Now, to find the breadth, 
 divide the said quotient by the square of the depth ; but if, 
 instead of the breadth, the depth be required, then divide 
 said quotient by the breadth, and the square root of this 
 quotient will be the required depth. 
 
 Example. What should be the size of a white-pine beam 20 
 feet long to sustain safely 10,000 pounds uniformly distributed 
 over its length ? The value of B for white pine, in Table III., 
 is 500. Let the factor of safety be taken at 4. Then, by the 
 rule, loooo x 4 x 20 = 800000 ; this divided by (2 x 500 =) 
 
WEIGHT PER BEAM IN FLOORS. 109 
 
 1000 gives a quotient of 800. Now, if the depth be fixed at 
 12, then the said quotient, 800, divided by (12 x 12=) 144 
 gives 5-5-, the required breadth of beam ; and the beam may 
 be, say, 5! x 12. Again, if the breadth is fixed, say, at 6, and 
 the depth is required, then the said quotient, 8co, divided by 
 6 gives 133^, the square root of which, 1 1 - 55, is the required 
 depth. The beam in this case should therefore be, say, 
 6 x I if inches. 
 
 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 over the 
 floor. 
 
 If the distance apart at which they are placed, measured 
 from the centres of the beams, be multiplied by the length 
 of the beams between bearings, the product will equal the 
 area of the floor sustained by one beam ; and if this area be 
 multiplied by the weight upon a superficial foot of the floor, 
 the product will equal the total load uniformly distributed 
 over the length of the beam ; or, if c be put to represent the 
 distance apart between the centres of the beams in feet, and 
 / represent the length in feet of the beam between bearings, 
 and/ equal the pounds per superficial foot on the floor, 
 then the product of these, or c f I, will represent the uni- 
 formly distributed load on a beam ; but this load was before 
 represented by U (Art. 128); therefore, we have cfl= U, 
 and they may be substituted for it in (26.) and (27.). Thus 
 we have 
 
 b d* = cflal 
 
 or 
 
 (28.) 
 
 Therefore, to ascertain the size of floor-beams to sustain 
 safely a given load per superficial foot, we have 
 
 Rule XXIII. Multiply the given weight per superficial 
 foot by the factor of safety, by the distance between the 
 
HO CONSTRUCTION. 
 
 centres of the beams in feet, and by the square of the length 
 in feet; divide the product by twice the value of B for the 
 material of the beams, in Table III., and the quotient will be 
 equal to the breadth into the square of the depth. Now, to 
 obtain the breadth, divide said quotient by the square of the 
 depth, and this quotient will be the required breadth. But 
 if, instead of the breadth, the depth be required, divide the 
 aforesaid quotient by the breadth ; then the square root of 
 this quotient will be the required depth. 
 
 Example. What should be the size of white-pine floor- 
 beams 20 feet long, placed i# inches from centres, to sustain 
 safely 90 pounds per superficial foot, including the weight 
 of the materials of construction the beams, flooring, plas- 
 tering, etc. ? The value of B for white pine is 500 ; the 
 factor of safety may be put at 5. Then, by the rule, we 
 have 90 x 5 x -ff x 2O 2 = 240000. This divided by (2 x 500 
 =) 1000 gives 240. Now, for the breadth, if the depth be 
 fixed at 9 inches, then 240 divided by (9* = ) 8 1 gives a 
 quotient of 2-963. The beams therefore should be, say, 
 3x9. But if the breadth be fixed, say, at 2-5 inches, then 
 240 divided by 2-5 gives a quotient of 96, the square root of 
 which is 9-8 nearly. The beams in this case would require 
 therefore to be, say, 2^ x 10 inches. 
 
 N. B. It is well to observe that the question decided 
 by Rule XXII. is simply that of strcngtli only. Floor-beams 
 computed by it will be quite safe against rupture, but they 
 will in most cases deflect much more than would be consist- 
 ent with their good appearance. Floor-beams should be 
 computed by the rules which include the effect of deflection. 
 (See Art. 152.) 
 
 131. Levers: Load at One End. The beams so far con- 
 sidered as being exposed to transverse strains have been 
 supposed to be supported at each end. When a piece is 
 held firmly at one end only, and loaded at the other, it is 
 termed a lever ; and the load which a piece so held and 
 loaded will sustain is equal to one fourth that which the 
 same piece would sustain if it were supported at each end 
 and loaded at the middle. Or, the strain in a beam sup- 
 
LEVERS TO SUSTAIN GIVEN WEIGHTS. Ill 
 
 ported at each end caused by a given weight located at the 
 middle is equal to that in a lever of the same breadth and 
 depth, when the length of the latter is equal to on6 half that 
 of the beam, and the load at its end is equal to one half of 
 that at the middle of the beam. Or, when P represents the 
 load at the end of the lever, and n its length, then W2P, 
 and l2n. Substituting these values of W and / in equa- 
 tion (20.), we have 
 
 4 
 
 2an 
 
 from which 
 
 p-Bbd* 
 
 T^P 
 
 Hence, to ascertain the weight which may be safely sus- 
 tained at the end of a lever, we have 
 
 Rule XXIV. Multiply the breadth of the lever by the 
 square of its depth, and by the value of B for the material 
 of the lever, in Table III. ; divide the product by the pro- 
 duct of four times the length in feet into the factor of safety, 
 and the quotient will be the required weight in pounds. 
 
 Example. What weight can be safely sustained at the 
 end of a maple lever of which the breadth is 2 inches, the 
 depth is 4 inches, and the length is 6 feet ? The value of B 
 for maple, in Table III., is uoo; therefore, by the rule, 
 2 x4 2 x i TOO = 35200. And, taking the factor of safety at 5, 
 4x5x6= 1 20, and 35200 divided by 120 gives a quotient of 
 2 93 '33> or 293^- pounds. 
 
 N. B. When a lever is loaded with a weight uniformly 
 distributed over its length, it will sustain just twice the load 
 which can be sustained at the end. 
 
 132. Levers: Breadth or Depth. By a proper trans- 
 position of the factors in (29.), we obtain 
 
 L (30.) 
 
 Z> 
 
 Hence, to ascertain the breadth or depth of a lever to sus- 
 tain safely a given weight, we have 
 
I I2 CONSTRUCTION. 
 
 Rule XXV. Multiply four times the given weight by 
 the length of the lever, and by the factor of safety ; divide 
 the product by the value of B for the material of the lever, 
 in Table III., and the quotient will be equal to the breadth 
 multiplied by the square of the depth. Now, if the breadth 
 be required, divide said quotient by the square of the depth, 
 and this quotient will be the required breadth ; but if, 
 instead of the breadth, the depth be required, divide the 
 said quotient by the breadth ; then the square root of this 
 quotient will be the required depth. 
 
 Example. What should be the size of a cherry lever 5 
 feet long to sustain safely 250 pounds at its end? Proceed- 
 ing by the rule, taking the factor of safety at 5, we have 
 4x250x5x5 = 25000. The value of B for cherry, in Table 
 III., is 650 ; and 25000 divided by 650 gives a quotient of 
 38-46. Now, if the depth be fixed at 4, then 38-46 divided 
 by (4x4 =) 16 gives a quotient of 2-4, the required breadth. 
 But if the breadth be fixed at 2, then 38-46 divided by 2 
 gives a quotient of 19-23, the square root of which is 4-38, 
 the required depth. Therefore, the lever maybe 2-4x4, 
 or 2 x 4-f- inches. 
 
 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 termed its deflection. In an investigation of the 
 laws of deflection it has been demonstrated, and experiments 
 have confirmed it, that while the elasticity of the material 
 remains uninjured by the pressure, or is injured in but a 
 small degree, the amount of deflection is directly in propor- 
 tion to the weight producing it ; for example, if 1000 pounds 
 laid upon a beam is found to cause it to deflect or descend at 
 the middle a quarter of an inch, then 2000 pounds will cause 
 it to deflect half an inch, 3000 pounds will deflect it three 
 fourths of an inch, and so on. 
 
 134. Deflection : Relation to Dimensions. In Table 
 III. are recorded the results of experiments made to test the 
 
THE LAW OF DEFLECTION. 113 
 
 resistance of the materials named to deflection. The fig- 
 ures in the third column designated by the letter F (for flex- 
 ure) show the number of pounds required to deflect a unit 
 of material one inch. This is an extreme state of the case, 
 for in most kinds of material this amount of depression 
 would exceed the limits of elasticity ; and hence the rule 
 would here fail to give the correct relation as between the 
 dimensions and pressure. For the law of deflection as above 
 stated (the deflections being in proportion to the weights) 
 is true only while the depressions are small in comparison 
 with the length. Nothing useful is, therefore, derived from 
 this position of the question, except to give an idea of the 
 nature of the quantity represented by the constant F\ it 
 being in reality an index of the stiffness of the kind of mate- 
 rial used in comparing one material with another. Whatever 
 be the dimensions of the beam, F will always be the same 
 quantity for the same material ; but among various materials 
 /''will vary according to the flexibility or stiffness of each 
 particular material. For example, F will be much greater 
 for iron than for wood ; and again, among the various kinds 
 of wood, it will be larger for the stiff woods than for those 
 that are flexible. The value of F, therefore, is the weight 
 which would deflect the unit of material one inch, upon the 
 supposition that the deflections, from zero to the depth of 
 one inch, continue regularly in proportion to the increments 
 of weight producing the deflections, or, for each deflection 
 
 F : I : : W : <?, 
 from which we have 
 
 in which cJ represents the deflection in inches corresponding 
 to W t the weight producing it. This is for the unit of ma- 
 terial. For beams of larger dimensions, investigations have 
 shown (Transverse Strains, Chapters XIII. and XIV.) that 
 the power of a beam to resist deflection by a weight at mid- 
 dle is in proportion to its breadth and the cube of its depth, 
 and it is inversely in proportion to the cube of the length ; 
 
1 14 CONSTRUCTION. 
 
 or, when the resistance of the unit of material is measured, 
 as above, by , we have the relation between it and a 
 larger beam of 
 
 Putting this ratio in a proportion with that of the unit of 
 material, we have 
 
 . ..; . . ; " ; i >: ' :: T ' b -r-- : 
 
 which gives 
 
 W _Fbd* 
 
 6 ~ T ' 
 from which we have 
 
 W= F -^-. (31.) 
 
 135. Deflection : Weight when at MidklQc. In equation 
 (31.) we have a rule by which to ascertain what weight is 
 required to deflect a given beam to a given depth of deflec- 
 tion ; this, in words at length, is 
 
 Rule XXVI. Multiply the breadth of the beam by the 
 cube of its depth, and by the given deflection, all in inches, 
 and by the value of .Ffor the material of the beam, in Table 
 III.; divide the product by the cube of the length in feet, 
 and the quotient will be the required weight in pounds. 
 
 Example. What weight is required at the middle of a 
 4x12 inch Georgia-pine beam 20 feet long to deflect it 
 three quarters of an inch ? The value of F for Georgia 
 pine, in Table III., is 5900; therefore, by the rule, we have 
 4 x i2 3 x 0-75 x 5900 = 30585600, which divided by (20x20 
 x 20 =) 8000 gives a quotient of 3823-2, the required weight 
 in pounds. 
 
 136. Deflection: Breadth or Depth, Weight at middle. 
 
 By a transposition of equation (31.), we obtain 
 
SIZE FOR A GIVEN DEFLECTION. 115 
 
 a rule by which may be found the breadth or depth of a 
 beam, with a given load at middle and with a given deflec- 
 tion ; this, in words at length, is 
 
 Rule XXVII. Multiply the given load by the cube of 
 the length in feet, and divide the product by the product of 
 the deflection into the value of F for the material of the 
 beam, in Table III. ; then the quotient will be equal to the 
 breadth of the beam multiplied by the cube of its depth, 
 both in inches. 
 
 Now, to obtain the breadth, divide the said quotient by 
 the cube of the depth, and this quotient will be the required 
 breadth. But if, instead of the breadth, the depth be re- 
 quired, then divide the said quotient by the breadth, and 
 the cube root of this quotient will be the required depth. 
 But if neither breadth nor depth be previously fixed, but it 
 be required that they bear a certain proportion to each 
 other ; such that d : b : : i : r, r being a decimal, then b = rd, 
 and b d* r d* ; then, to find the depth, divide the aforesaid 
 quotient by the decimal r, and the fourth root (or the square 
 root of the square root) will be the required depth, and this 
 multiplied by the decimal r will give the breadth. 
 
 Example. What should be the size of a spruce beam 20 
 feet long between bearings, sustaining 2000 pounds at the 
 middle, with a deflection of one inch ? By the rule, the 
 weight into the cube of the length is 2000 x 8000 = 16000000. 
 The value of Ffor spruce, in Table III., is 3500; this by the 
 deflection = i gives 3500, which used as a divisor in divid- 
 ing the above 16000000 gives a quotient of 4571 -43. Now, 
 if the breadth be required, the depth being fixed, say, at 10, 
 then 4571-43 divided by (lox lox 10 =) 1000 gives 4-57, the 
 required breadth. The beam should be, say, 4$ by 10 inches. 
 But if the depth be required, the breadth being fixed, say, at 
 4, then 4571-43 divided by 4 gives 1142-86, the cube root 
 of which is 10-46; so in this case, therefore, the beam is 
 required to be 4 x io inches. Again, if the breadth is to 
 bear a certain proportion to the depth, or that the ratio be- 
 tween them is to be, say, 0-6 to i, then let r = 0-6, and then 
 457i. 4 3 = o-6^ 4 , and dividing by 0-6, we have 7619-05 
 = d\ This equals d*xd*\ therefore the square root of 7619 
 
1 16 CONSTRUCTION. 
 
 is 87-29, and the square root of this is 9-343, the required 
 depth in inches. Now 9-343x0-6 equals the breadth, or 
 9.343x0-6=5-6; therefore the beam is required to be 
 5 -6 x 9- 34 inches, or, say, 5f x 9^ inches. 
 
 137. Deflection : when Weight i at Middle. By a trans- 
 position of the factors in (32.), we obtain 
 
 a rule by which the deflection of any given beam may be as- 
 certained, and which, in words at length, is 
 
 Rule XXVIIL Multiply the given weight by the cube 
 of the length in feet ; divide the product by the product of 
 the breadth into the cube of the depth in inches, multiplied 
 by the value of Fior the material of the beam, in Table III., 
 and the quotient will be the required deflection in inches! 
 
 Example. To what depth will 1000 pounds deflect a 
 3x10 inch white-pine beam 20 feet long, the weight being 
 at the middle of the beam ? By the rule, we have 1000 x 2o 3 
 = 8000000; then, since the value of F for white pine, in 
 Table III., is 2900, we have 3 x io 3 x 2900 = 8700000 ; using 
 this product as a divisor and by it dividing the former pro- 
 duct, we obtain a quotient of 0.9195, the required deflection 
 in inches. 
 
 138. Deflection: Load . Uniformly Distributed. In two 
 
 beams of equal capacity, suppose the one loaded at the 
 middle, and the other with its load uniformly distributed 
 over its length, and so loaded that the deflection in one beam 
 shall equal that in the other ; then the weight at the middle 
 of the former beam will be equal to five eighths of that on 
 the latter. This proportion between the two has been de- 
 monstrated by writers on the strength of materials. (See p. 
 484, Mechanics of Eng. and Arch., by Prof. Mosely, Am. eel. by 
 Prof. Mahan, 1856.) Hence, when /is put to represent the 
 uniformly distributed load, we have 
 
DEFLECTION FOR LOAD EQUALLY DISTRIBUTED. 1 17 
 
 or, when an equally distributed load deflects a beam to a 
 certain depth, five eighths of that load, if concentrated at 
 the middle, would cause an equal deflection. This value of 
 W may therefore be substituted for it in equation (31.), and 
 give 
 
 from which we obtain 
 
 i- 
 u = -- JT -- > (34-) 
 
 a rule for a uniformly distributed load. 
 
 139. Deflection: Weight when Uniformly Distributed. 
 
 In equation (34.) we have a rule by which we may ascertain 
 what weight is required to deflect to a given depth any 
 given beam. This, in words at length, is 
 
 Rule XXIX. Multiply 1-6 times the deflection by the 
 breadth of the beam, and by the cube of its depth, all in 
 inches, and by the value of Ffor the material of the beam, 
 in Table III. ; divide the product by the cube of the length 
 in feet, and the quotient will be the required weight in 
 pounds. 
 
 Example. What weight, uniformly distributed over the 
 length of a spruce beam, will be required to deflect it to the 
 depth ot 0-5 ot an inch, the beam being 3 x 10 inches and 10 
 feet long? The value of F ior spruce, in Table III., is 3500. 
 Therefore, by the rule, we have j -6x0-5 x 3 x io 3 x 3500 = 
 8400000, and this divided by (10x10x10=) 1000 gives 
 8400, the required weight in pounds. 
 
 (40, __ Deflection: Breadth or Depth, Load Uniformly 
 Distributed. By transposition of the factors in equation 
 (54.), we obtain 
 
 a rule for the dimensions, which, in words at length, is 
 
Il8 CONSTRUCTION. 
 
 Rule XXX. Multiply the given weight by the cube of 
 the length of the beam ; divide the product by i -6 times the 
 given deflection in inches, multiplied by the value of F for 
 the material of the beam, in Table III., and the quotient will 
 equal the breadth into the cube of the depth. Now, to ob- 
 tain the breadth, divide this quotient by the cube of the depth, 
 and the resulting quotient will be the required breadth in 
 inches. But if, instead of the breadth, the depth be required, 
 then divide the aforesaid quotient by the breadth, and the 
 cube root of the resulting quotient will be the required depth 
 in inches. Again, if neither breadth nor depth be previously 
 determined, but to be in proportion to each other at a given 
 ratio, as r to i, r being a decimal fixed at pleasure, then di- 
 vide the aforesaid quotient by the value of r, and take the 
 square root of the quotient; then the square root of this 
 square root will be the required depth in inches. The breadth 
 will equal the depth multiplied by the value of the deci- 
 mal r. 
 
 Example. What should be the size of a locust beam 10 
 feet long which is to be loaded with 6000 pounds equally 
 distributed over the length, and with which the beam is to 
 be deflected of an inch ? The value of F for locust, in 
 Table 1 1 1., is 5050. By the rule, we have 6000 x(io x 10 x 10 =) 
 1000 = 6000000, which is to be divided by (i -6xo- 75 x 5050 =) 
 6060, giving a quotient of 990-1. Now, if the depth be, say, 
 6 inches, then 990- 1 divided by (6x6x6) 216 gives a quo- 
 tient of 4-584, the required breadth in inches, say 4^. But 
 if the breadth be assumed at 4 inches, then 990- 1 divided by 
 4.gives a quotient of 247- 5 2 5, the cube root of which is 6-279, 
 the required depth in inches, or, say, 6J. And, again, if the 
 ratio between the breadth and depth be as o- 7 to i, then 990- 1 
 divided by 0-7 gives a quotient of 1414-43, the square root 
 of which is 37-609, of which the square root is 6-1326, the 
 required depth in inches, or, say, 6J- ; and then 6-1326x0-7 = 
 4-293, the required breadth in inches; or, the beam shoujd 
 be 4 T \ x 6J- inches. 
 
 14-1. Deflection : when Weight is Uniformly Distributed. 
 
 By a transposition of the factors of equation (35.), we ob- 
 tain 
 
DEFLECTION OF LEVERS AND BEAMS. 119 
 
 a result nearly the same as that in equation (33.), which is a 
 rule for deflection by a weight at middle, and which by 
 slight modifications may be used for deflection by an equally 
 distributed load. Thus by 
 
 Rule XXXI. Proceed as directed in Rule XXVIII. (Art. 
 137), using the equally distributed weight instead of a con- 
 centrated weight, and then divide the result there obtained 
 for deflection by I -6 ; then the quotient will be the required 
 deflection in inches. 
 
 Example. Taking the example given under Rule 
 XX VI 1 1., \uArt. 137, and assuming that the 1000 pounds load 
 with which the beam is loaded be equally distributed, then 
 0-9195, the result for deflection as there found, divided by i -6, 
 as by the above rule, gives 0-5747, the required deflection. 
 This result is just five eighths of 0-9195, the deflection by the 
 load at middle. 
 
 N.B. The deflection by a uniformly distributed load is 
 just five eighths of that produced by the same load when 
 concentrated at the middle of the beam; therefore, five 
 eighths of the deflection obtained by Rule XXVIII. will be 
 the deflection of the same beam when the same weight is 
 uniformly distributed. 
 
 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 lever. There- 
 fore, if P represents the weight at the end of a lever, and n 
 the length of the lever in feet, then 2 P= W smd 2 n = t, and 
 if these values of Wand /be substituted for those in equa- 
 tion (33.), we obtain 
 
 2 P x 2 n 3 
 
 which reduces to 
 
 -.,,., (37-) 
 
 Fbd" 
 
120 CONSTRUCTION. 
 
 a result 16 times that in equation (33.), which is the deflection 
 in a beam. Therefore, when a beam and a lever equal in 
 sectional area and in length be loaded by equal weights, the 
 one at the middle, the other at one end, the deflection of the 
 lever will be 16 times that of the beam. This proportion is 
 based upon the condition that neither the beam nor the lever 
 shall be deflected beyond the limits of elasticity. 
 
 14-3. Deflection of a Lever: Load at End. Equation 
 (37.), in words at length, is 
 
 Ride XXXII. Multiply 16 times the given weight by 
 the cube of the length in feet ; divide the product by the 
 product of the breadth into the cube of the depth multiplied 
 by the value of .Ffor the material of the lever, in Table III., 
 and the quotient will be the required deflection. 
 
 Example. What would be the deflection of a bar of 
 American wrought iron one inch broad, two inches deep, 
 loaded with 150 pounds at a point 5 feet distant from the 
 wall in which the bar is imbedded ? The value of F for 
 American wrought iron, in Table III., is 62000. Therefore, 
 by the rule, 16 x 150 x 5" = 300000. This divided by 
 (i x 2 3 x 62000 =) 496000 gives 0-6048, the required deflec- 
 tion nearly f of an inch. 
 
 (4-4-. Deflection of a Lever: Weight when at End. By 
 
 a transposition of the factors in equation (37.), we obtain 
 
 This result is equal to one sixteenth of that shown in equa- 
 tion (31.), a rule for the weight at the middle. Therefore, 
 for 
 
 Rule XXXIII. Proceed as directed in Rule XXVII.; 
 divide the quotient there obtained by 16, and the resulting 
 quotient will be the required weight in pounds. 
 
 Example. What weight is required at the end of a 4 x 12 
 inch Georgia-pine lever 20 feet long to deflect it three 
 quarters of an inch? Proceeding by Rule XXVII., we ob- 
 tain a quotient of 3823-2; this divided by 16 gives 
 say 239, the required weight in pounds. 
 
DEFLECTION OF LEVERS WITH UNIFORM LOAD. 121 
 
 145. Deflection of a Lever : Breadth or Depth, Load 
 at End. A transposition of the factors of equation (38.) 
 gives 
 
 , 73 i6Pn* 
 
 a rule by which to obtain the sectional area of the lever. 
 By comparison with equation (32.) it is seen that the result 
 in (39.) is 16 times that found by (32.). Therefore, the dimen- 
 sions for a lever loaded at the end may be found by 
 
 Rule XXXIV. Multiply by 16 the first quotient found 
 by Rule XXVII., and then proceed as farther directed in 
 Rule XXVII., using the product of 16 times the quotient, 
 instead of the said quotient. 
 
 Example. What should be the size of a spruce lever 20 
 feet long, between weight and wall, to sustain 2000 pounds 
 at the end with a deflection of I inch? Proceeding by Rule 
 XXVII., we obtain a first quotient of 4571-43. By Rule 
 XXXIV., 4571-43 x 16 73144-88. Now, if the depth be 
 fixed, say, at 20, then 73144-88 divided by (20 x 20 x 20 =) 
 8000 gives 9- 143, the required breadth. But to obtain the 
 depth, fixing the breadth, say, at 9, we have for 73144-88 di- 
 vided by 9 = 8127-21, the cube root of which is- 20- 1055, the 
 required depth. Again, if the breadth and depth are to be 
 in proportion, say, as 0-7 to i-o, then 73144-88 divided by 
 0-7 gives 104492-7, the square root of which is 323-254, of 
 which the square root is 17-98, the required depth in inches ; 
 and 17-98 x 0-7 = 12-586, the required breadth in inches. 
 The lever, therefore, should be, say, I2f x 18 inches. 
 
 146. Deflection of Levers: Weight Uniformly Distrib- 
 uted. A comparison of the effects of loads upon levers 
 shows (Transverse Strains, Art. 347) that the deflection by a 
 uniformly distributed load is equal to that which would be 
 produced by three eighths of that load if suspended from 
 the end of the lever. Or, P f U. Substituting this value 
 of P, in equation (37.), gives 
 
 16 x | Un* 
 
 Fb<T* ' 
 which reduces to 
 
122 CONSTRUCTION. 
 
 a rule for the deflection of levers loaded with an equally dis- 
 tributed load. 
 
 (47. _ Deflection of Severs with Uniformly Distributed 
 Load. The deflection shown in equation .(40.) is just six 
 times that shown in equation (33.). The result by (33.) mul- 
 tiplied by 6 will equal the result by (40.); therefore, we 
 have 
 
 Rule XXXV. Proceed as directed in Rule XXVIII. ; 
 the result thereby obtained multiplied by 6 will give the 
 required deflection. 
 
 Example. To what depth will 500 pounds deflect a 3 x 10 
 inch white-pine lever 10 feet long, the weight uniformly 
 distributed over the lever? Here, by Rule XXVIII. , we 
 obtain the result 0-05747 ; this multiplied by 6 gives 0-3448, 
 the required deflection. 
 
 (4-8. Deflection of Levers : Weight when Uniformly 
 Distributed. By a transposition of factors in (40.), we ob- 
 tain 
 
 This is equal to one sixth that of equation (31.) ; therefore, 
 we have 
 
 Rule XXX VI. Proceed as directed in Rule XXVI.; 
 the quotient thereby obtained divide by 6, and the quotient 
 thus obtained will be the required weight. 
 
 Example. What weight will be required to deflect a 
 4x5 inch spruce lever I inch, the weight uniformly dis- 
 tributed over its length ? Proceeding as directed in Rule 
 XXVI., the result thereby obtained is 1750; this divided by 
 6 gives 29 if, the required weight in pounds. 
 
 14-9. Deflection of Severs : ISreadth or Depth, Load 
 Uniformly Distributed. A transposition of factors in equa- 
 tion (41.) gives 
 
SIMPLICITY IN CONSTRUCTION. 123 
 
 This result is just six times that of equation (32.); we, there- 
 fore, have 
 
 Rule XXXVII. Proceed as directed in Rule XXVII. ; 
 multiply the first quotient thereby obtained by 6 ; then in 
 the subsequent directions use this multiplied quotient in- 
 stead of the said first quotient, to obtain the required breadth 
 and depth. 
 
 Example. What should be the size of a spruce lever 10 
 feet long, sustaining 2666| pounds, uniformly distributed 
 over its length, with a deflection of I inch ? Proceeding 
 by Rule XXVII., the first quotient obtained is 761-905; 
 this multiplied by 6 gives 4571-43, the multiplied quotient 
 which is to be used in place of the said first quotient. Now, 
 to obtain the breadth, the depth being fixed, say, at 10 ; 
 4571 -43 divided by (cube of 10 ) 1000, the quotient, 4-57, is 
 the required breadth. But if the breadth be fixed, say, at 
 4, then, to obtain the depth, 4571-43 divided by 4 gives 
 1142-86, the cube root of which is 10-46, the required depth. 
 Again, if the breadth and depth are to be in proportion, say, 
 as 0-6 to i -o, then 4571 -43 divided by 0-6 gives 7619-05, the 
 square root of which is 87-27, of which the square root is 
 9-343, the required depth in inches ; and 9-343 x ' 6 equals 
 5-6, the required breadth in inches; or, the lever may be, 
 say, 5f x 9- inches. 
 
 CONSTRUCTION IN GENERAL. 
 
 150. Construction: Object Clearly Defined. In the 
 
 various parts of timber construction, known as floors, par- 
 titions, roofs, bridges, etc., each has a specific object, and in 
 all designs for such constructions this object should be kept 
 clearly in view, the various parts being so disposed as to 
 serve the design with the least quantity of material. The 
 simplest form is the best, not only because it is the most 
 economical, but for many other reasons. The great number 
 of joints, in a complex design, render the construction liable 
 to derangement by multiplied compressions, shrinkage, and, 
 in consequence, highly increased oblique strains ; by which 
 its stability and durability are greatly lessened. 
 
I2 4 
 
 CONSTRUCTION. 
 FLOORS. 
 
 151. Floor Described. Floors are most generally con, 
 structed single; that is, simply a series of parallel beams, each 
 
 FIG. 39. 
 
 spanning the width of the building, as seen at Fig. 39* Oc- 
 
 FIG. 40. 
 
 casionally floors are constructed double, as at Fig. 40 ; and 
 sometimes framed, as at Fig. 41 ; but these methods are 
 
RULES APPLIED TO FLOORS. 
 
 I2 5 
 
 seldom practised, inasmuch as either of these requires more 
 timber than the single floor. Where lathing and plastering 
 is attached to the floor-beams to form a ceiling below, the 
 springing of the beams, by customary use, is liable to crack 
 the plastering. To obviate this in good dwellings, the double 
 and framed floors have been resorted to, but more in former 
 times than now, as the cross-furring (a series of narrow strips 
 of board or plank nailed transversely to the underside of 
 
 UNIVERSITY 
 
 FIG. 41. 
 
 the beams to receive the lathing for the plastering) serves a 
 like purpose very nearly as well. 
 
 152. Floor-Beams. The size of floor-beams can be as- 
 certained by the preceding rules for the stiffness of materials. 
 These rules give the required dimensions for the various 
 kinds of material in common use. The rules may be some- 
 what abridged for ordinary use, if some of the quantities 
 represented in the formula be made constant within certain 
 limits. For example, if the load per foot superficial upon 
 the floor be fixed, and the deflection, then these, together 
 with the constant 'represented by F y may be reduced to one 
 
126 CONSTRUCTION. 
 
 constant. For dwellings, the load per foot may be taken at 
 70 pounds, the weight proper to be allowed for a crowd 
 of people on their feet. (Transverse Strains, Art. 114.) To 
 this add 20 for the weight of the material of which the floor 
 is composed, and the sum, 90, is the value of f, or the weight 
 per foot superficial for dwellings. Then eft-. U (Art. 130). 
 The rate of deflection allowable for this load may be fixed 
 at 0-03 inch per foot of the length, or d =. 0-03 /. Substitut- 
 ing these values in equation (35.), we obtain 
 
 b d*= cfl * 9 c/3 = 1875 */ 3 
 
 i -6 Fx -03 / ~ ~ i -6 x -03 /^ ~ F 
 
 or 
 
 . , ' i<r=2*ZL f r. ( 43 .) 
 
 T X*7 C 
 
 Putting/ to represent , we have 
 
 (44.) 
 
 T 8*7 C 
 
 Now, by reducing -, for the six woods in common use, 
 the value of j for each is found as follows: 
 
 Georgia Pine ......................... / = 0-32 
 
 Locust ...... ......................... j = 0-37 
 
 White Oak.... ........................ j = 0-6 
 
 Spruce ............................... j 0-54 
 
 White Pine ............ ' ............... j = 0-65 
 
 Hemlock .... .......................... j ^0-67 
 
 Equation (44.") is a rule for the floor-beams of dwellings ; 
 it may be used also to obtain the dimensions of beams for 
 stores for all ordinary business- for it will require from 3 to 
 5 times the weight used in this rule, or from 200 to 400 
 (average 300) pounds to increase the deflection to the limit 
 of elasticity in beams of the usual depths and lengths. For 
 light stores, therefore, loaded, say, to 150 pounds per foot, 
 the beams would be safe, but the deflection would be in- 
 
CONSTANTS FOR USE IN THE RULES. 127 
 
 creased to 0-06 per foot. When so great a deflection as this, 
 would not be objectionable to the eye, then this rule (44.) 
 will serve for the beams of light stores. But for first-class 
 stores, taking the rate of deflection at -04 per foot, and * 
 fixing the weight per superficial foot at 275 pounds, includ- 
 ing the weight of the material of which the floor is con- 
 structed, and letting k represent the constant, then 
 
 bd*=kcl\ (45.) 
 
 and for 
 
 Georgia Pine k = 0-73 
 
 Locust k = 0-85 
 
 White Oak k = 1-38 
 
 Spruce k 1-48 
 
 White Pine k = 1-23 
 
 Hemlock. . , k 1-53 
 
 153. Floor-Beam^ for Dwellings To find the dimen- 
 sions of floor-beams for dwellings, when the rate of deflection" 
 is 0-03 inch per foot, or for ordinary stores when the load is 
 about 150 pounds per foot, and the deflection caused by this 
 weight is within the limits of the elasticity of the material, 
 we have the following rule : 
 
 Rule XXXVIIL Multiply the cube of the length by 
 the distance apart between the beams (from centres), both in 
 feet, and multiply the product by the value of/ (Art. 152) 
 for the material of the beam, and the product will equal the 
 product of the breadth into the cube of the depth. Now, 
 to find the breadth, divide this product by the cube of the 
 depth in inches, and the quotient will be the breadth in 
 inches. But if the depth is sought, divide the said product 
 by the breadth in inches, and the cube root of the quotient 
 will be the depth in inches ; or if the breadth and depth are 
 to be in proportion as r is to unity, r representing any re- 
 quired decimal, then divide the aforesaid product by the 
 value of r, and extract the square root of the quotient, and 
 the square root of this square root will be the depth re- 
 quired in inches, and the depth multiplied by the value of i 
 will be the breadth in inches. 
 
128 CONSTRUCTION. 
 
 Example. In a dwelling Or ordinary stOre, what must be 
 the breadth of the beams, when placed 15 inches from 
 centres, to support a floor covering a span of 16 feet, the 
 depth being 1 1 inches, the beams of white oak ? By the 
 rule, 4096, the cube of the length, by i|, the distance from 
 centres, and by 0-6, the value of j for white oak, equals 
 3072. This divided by 1331, the cube of the depth, equals 
 2-31 inches, or 2 T ^ inches, the required breadth. But if, in- 
 stead of the breadth, the depth be required, the breadth 
 being fixed at 3 inches, then the product, 3072, as above, di- 
 vided by 3, the breadth, equals 1024 ; the cube root of this 
 is 10-08, or, say, 10 inches nearly. But if the breadth and 
 depth are to be in proportion, say, as 0-3 to i-o, then the 
 aforesaid product, 3072, divided by 0-3, the value of r, 
 equals 10240, the square root of which is 101-2, and the 
 square root of this is 10-06, the required depth. This 
 multiplied by 0-3, the value of r, equals 3-02, the re- 
 quired breadth ; the beam is therefore to be, say, 3 x 10 
 inches. 
 
 154. Floor-Beams for First-Class Stores. To find the 
 breadth and depth of the beams for a floor of a first-class 
 store sufficient to sustain 250 pounds per foot superficial 
 (exclusive of the weight of the material in the floor), with 
 a deflection of 0-04 inch per foot of the length, we have 
 
 Rule XXXIX. The same as XXXVIIL, with the ex- 
 ception that the value of k (Art. 152) is to be used instead 
 of the value of j. 
 
 Example. The beams of the floor of a first-class store 
 are to be of Georgia pine, with a clear bearing between the 
 walls of 18 feet, and placed 14 inches from centres: what 
 must be the breadth when the depth is 1 1 inches ? By the 
 rule, 5832, the cube of the length, and i, the distance from 
 centres, and 0-73, the value of k for Georgia pine, all multi- 
 plied together equal 4966-92 ; and this product divided by 
 1331, the cube of the depth, equals 3-732, the required 
 breadth, or 3| inches. 
 
 But if, instead of the breadth, the depth be required : 
 what must be the depth when the breadth is 3 inches ? 
 
DISTANCE APART OF FLOOR-BEAMS. 129 
 
 The said product, 4966-92, divided by 3, the breadth, equals 
 1655-64, and the cube root of this, 11-83, or > sav > I2 inches, 
 is the depth required. 
 
 But if the breadth and depth are to be in a given pro- 
 portion, say 0-35 to i-o, the 4966-92 aforesaid divided by 
 0-35, the value of r, equals 14191, the square root of 
 which is 119-13, and the square root of this square root is 
 10-91, or, say, n inches, the required depth. And 10-91 
 multiplied by 0-35, the value of r, equals 3-82, the required 
 breadth, say 3^ inches. 
 
 I55 Floor -Beams: Distance from Centres. It is 
 
 sometimes desirable, when the breadth and depth of the 
 beams are fixed, or when the beams have been sawed and 
 are now ready for use, to know the distance from cen- 
 tres at which such beams should be placed in order that the 
 floor be sufficiently stiff. By a transposition of the factors 
 in equation (44.), we obtain 
 
 bd* 
 
 In like manner, equation (45.) produces 
 
 _bd* 
 
 (470 
 
 These, in words at length, are as follows : 
 
 Rule XL. Multiply the cube of the depth by the breadth, 
 both in inches, and divide the product by the cube of the 
 length in feet multiplied by the value of /, for dwellings 
 and for ordinary stores, or by k, for first-class stores, and 
 the quotient will be the distance apart from centres in feet. 
 
 Example. K span of 17 feet, in a dwelling, is to be cov- 
 ered by white-pine beams 3x12 inches: at what distance 
 apart from centres should they be placed? By the rule, 
 1728, the cube of the depth, multiplied by 3, the breadth, 
 equals 5184. The cube of 17 is 4913 ; this by 0-65, the value 
 of j for white pine, equals 3193-45- The aforesaid 5184 
 divided by this 3193-45 equals 1-6233 feet, or, say, 20 inches. 
 
CONSTRUCTION. 
 
 156. Framed Openings for Chimney* and Stairs. 
 
 Where chimneys, flues, stairs, etc., occur to interrupt the 
 bearing, the beams are framed into a piece, b (Fig. 42), called 
 a header. The beams, a a, into which the header is framed 
 are called trimmers or carriage-beams. These framed beams 
 require to be made thicker than the common beams. The 
 header must be strong enough to sustain one half of the 
 weight that is sustained upon the &w7-beams, c c (the wall at 
 the opposite end or another header there sustaining the other 
 half), and the trimmers must each sustain one half of the 
 weight sustained by the header in addition to the weight it 
 supports as a common beam. It is usual in practice to make 
 
 these framed beams one inch thicker than the common beams 
 for dwellings, and two inches thicker for heavy stores. This 
 practice in ordinary cases answers very well, but in extreme 
 cases these dimensions are not proper. Rules applicable 
 generally must be deduced from the conditions of the case 
 the load to be sustained and the strength of the material. 
 
 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 equal to 
 the length of the header multiplied by half the length of the 
 tail-beams, and by the load per superficial foot. Putting g 
 
DIMENSIONS OF HEADERS. Ijl 
 
 for the length of the header, n for the length of the tail- 
 beams, and / for the load per superficial foot ; U, the uni- 
 formly distributed load carried by the header, will equal 
 f n g. By substituting for /, in equation (35.), this value of 
 it, we obtain 
 
 The symbols g and / here both represent the same thing, 
 the length of the header ; combining these, and for # putting 
 its value gr y we obtain 
 
 . 
 
 3-2 Fr 
 
 To allow for the weakening of the header by the mor- 
 tices for the tail-beams (which should be cut as near the 
 middle of the depth of the header as practicable), the depth 
 should be taken at, say, one inch less than the actual depth. 
 With this modification, we obtain 
 
 If /be taken at 90, and r at 0-03, we have, by reducing 
 
 h _ 937- 5 "g* ( 4Q .) 
 
 -~' 
 
 which is a rule for the breadth of headers for dwellings and 
 for ordinary stores. This, in words, is as follows : 
 
 Rule XLL Multiply 937-5 times the length of the tail- 
 beams by the cube of the'length of the header, both in feet. 
 The product divided by the cube of one less than the depth 
 multiplied by the value of F, Table III., will equal the 
 breadth of the header in inches for dwellings or ordinary 
 stores. 
 
 Example. K header of white pine, for a dwelling, is 10 
 feet long, and sustains tail-beams 20 feet long ; its depth is 
 \2 inches: what must be its breadth? By the rule, 
 937. 5x20x10*= 18750000.' This divided by (12- i) 3 x 2900^ 
 
CONSTRUCTION. 
 
 3859900, equals 4-858, say 5 inches, the required breadth. 
 F or first-class stores,/ should be taken at 275, and r at 0-04. 
 With these values the constants in equation (48.) reduce to 
 2I48-4375, or, say, 2150. This gives 
 
 a rule for the breadth of a header for first-class stores. It 
 is the same as that for dwellings, except that the constant 
 2150 is to be used in place of 937-5. Taking the same ex- 
 ample, and using the constant 2150 instead of 937-5, we 
 obtain 1 1 14 as the required breadth of the header for a first- 
 class store. Modifying the question by using Georgia pine 
 instead of white pine, we obtain 5 -476 as the required thick- 
 ness, say 5^ inches. 
 
 158. Breadth of Carriage-Beams. A carriage-beam or 
 trimmer, in addition to its load as a common beam, carries 
 one half of the load on the header, which, as has been 
 seen in the last article, is equal to one half of the superficial 
 area of the floor supported by the tail-beams multiplied by 
 the weight per superficial foot of the load upon the floor ; 
 therefore, when the length of the header in feet is repre- 
 sented by g, and the length of the tail-beams by n, w equals 
 
 - x - x /, equals f g n* 
 
 For a load not at middle, we have (25.) 
 
 4 W ' amn 
 b d = BJ 
 
 *The load from the header, instead of being \fg n, is, more accurately, 
 i/(g c) : because the surface of floor carried by the header is only 
 that which occurs between the surfaces carried by the carriage-beams, each of 
 which carries so much of the floor as extends half way to the first tail-beam 
 
 from it, or the distance - ; therefore, the width of the surface carried equals 
 
 the length of the header less ( 2 x - = W, or g c. When, however, it is con- 
 sidered that the carriage-beam is liable to receive some weight from a stairs or 
 other article in the well-hole, the small additional load above referred to is 
 not only not objectionable, but is really quite necessary to be included in the 
 calculation. 
 
THICKNESS OF CARRIAGE-BEAMS. 133 
 
 This is a rule based upon resistance to rupture. By substi- 
 
 7? / 
 tuting for a, the factor of safety, -p-j-, its value in terms of 
 
 resistance to flexure {Transverse Strains, (154.)), we have 
 
 h j* 4 W B Im n __ 4 Wm n . 
 BIFdr Fdr 
 
 
 In this expression, W is a concentrated weight at the dis- 
 tances m and n from the two ends of the beam. Taking the 
 load upon a carriage-beam due to the load from the header, 
 as above found, and substituting it for W, we obtain 
 
 ,,* __ fgmn* 
 
 bd = -- 
 
 This is the expression required for the concentrated load. 
 To this is to be added the uniformly distributed load upon 
 the carriage-beam ; this is given in equation (35.). Substi- 
 tuting for U of this equation its value, fc /, gives 
 
 , ,, 
 
 -T67^" Fr 
 Combining these two equations, we have for the total load 
 
 r r 
 
 If, in this equation, /be taken at 90, and r at 0-03, these 
 reduce to 3000 ; therefore, with this value of -, we have 
 
 / N 
 
 This rule for the breadth of carriage-beams with one 
 header, for dwellings and for ordinary stores, is put in words 
 as follows : 
 
134 CONSTRUCTION. 
 
 Rule XLII. Multiply the length of the framed opening 
 by its breadth, and by the square of the length of the tail- 
 beams ; to this product add f of the cube of the length into 
 the distance of the common beams from centres all in feet ; 
 divide 3000 times the sum by the cube of the depth in inches 
 multiplied by the value of F for the material of the beam, in 
 Table III., and the quotient will be the breadth in inches. 
 
 Example. In a tier of 3 x 10 inch beams, placed 14 inches 
 from centres, what should be the breadth of a Georgia-pine 
 carriage-beam 20 feet long, carrying a header 12 feet long, 
 .having tail-beams 1 5 feet long? Here the framed opening 
 is 5-x 1 2 feet. Therefore, according to the rule, 12 x 5 x 15' = 
 13500; to which add(|x2o 3 x jf =)5833i; the sum is 19333^, 
 and this by 3000= 58000000. The value of F for Georgia 
 pine, in Table III., is 5900; the cube of the depth is 1000; 
 the product of these two is 5900000; therefore, dividing 
 the above 58000000 by 5900000 gives a quotient of 9.83, 
 the required breadth in inches. If, in equation (51.), f be 
 
 taken at 275, and r at 0-04, then - becomes 6875, and the 
 equation becomes 
 
 , 
 ~~ 
 
 Fd* 
 
 a rule for the breadth of carriage-beams for first-c lass stores ; 
 the same as that for dwellings, except that the constant is 
 6875 instead of 3000. 
 
 fl59. Breadth of Carriage-Beams Carrying Two Sets 
 of Tail-Beams. r-A rule for this is the same as that for a car- 
 riage-beam carrying one set of tail-beams, if to it there be 
 added the effect of the second set of tail-beams. Equation 
 (51.) with the additibn named becomes 
 
 , , 
 
 (54 
 
 in which n is the length of one set of tail-beams, and s the 
 length of the other set ; and m + n = I. 
 
CARRIAGE-BEAMS WITH TWO HEADERS. 135 
 
 If / be taken at 90, and r at 0-03, these two reduce to 
 3000, and we have 
 
 _ 3000 [,;/ (;;?;; -Kr 2 )-f | r/ 3 ] 
 Pd*~ 
 
 9 
 
 a rule for the breadth of a carriage-beam carrying two sets 
 of headers, for dwellings and for ordinary stores. It may 
 be stated in words as follows : 
 
 Rule XLIII. Multiply the length of the longer set of 
 tail-beams by the difference between this length and the 
 length of the carriage-beam, and to the product add the 
 square of the length of the shorter set of tail-beams ; mul- 
 tiply the sum by the length of the longer set of tail-beams, 
 and by the length of the header ; to this product add f of 
 the product of the cube of the length of the carriage- 
 beam into the distance apart from centres of the common 
 beams ; multiply this sum by 3000 ; divide this product by 
 the product of the cube of the depth in inches into the value 
 of F ior the material of the carriage-beam, in Table III., and 
 the quotient will be the required breadth. 
 
 Example. In a tier of 3 x 12 inch beams, placed 14 inches 
 from centres, what should be the breadth of a spruce car- 
 riage-beam 20 feet long in the clear of the bearings, carry- 
 ing two sets of tail-beams, one of them 9 feet long, the 
 other 5 feet ; the headers being 15 feet long ? The difference 
 between the longer set of tail-beams and the carriage-beam 
 is (20 9 =) 1 1 feet. Therefore, by the rule, 9 x 1 1 + 5* = 
 124; then (124x9x15=) 16740 + (f x 20 3 x |f ) 5833^- = 
 22 573i; then 22573^x3000 = 67720000. Now the value of 
 F for spruce, Table III., is 3500; this by I2 3 , the cube of 
 the depth, equals 6048000; by this dividing the aforesaid 
 67720000, we obtain a quotient of 11-197, the required 
 breadth of the carriage-beam. If, in equation (54.), / be 
 taken at 275, and > at 0-04, these reduce to 6875, and we 
 obtain 
 
 a rule for the breadth of carriage -beams carrying two sets 
 
CONSTRUCTION. 
 
 of tail-beams, in the floors of first-class stores. -This is like 
 the rule for dwellings, except that the constant is 6875 in- 
 stead of 3000. 
 
 160. _ Breadth o Carriage - Beam willi Well-Hole at 
 middle. When the framed opening between the two sets of 
 tail-beams occurs at the middle, or when the lengths of the 
 two sets of tail-beams are equal, then equation (54.) reduces 
 to 
 
 and if /be taken at 90, and r at 0-03, these reduce to 3000, 
 and we have 
 
 a rule for the breadth of a carriage-beam carrying two sets 
 of tail-beams of equal length, in the floor of a dwelling or of 
 an ordinary store ; and which in words is as follows: 
 
 Rule XLIV. Multiply the length of the header by the 
 square of the length of the tail-beams, and to the product 
 add | of the product of the square of the length of the car- 
 riage-beam by the distance apart from centres of the com- 
 mon beams; multiply the sum by 3000 times the length of 
 the carriage-beam ; divide the product by the product of 
 the cube of the depth into the value of F for the material of 
 the carriage-beam, in Table III., and the quotient will be the 
 required breadth. 
 
 Example. In a tier of 3x12 inch beams, placed 12 inches 
 from centres, what must be the thickness of a hemlock car- 
 riage-beam 20 feet long, carrying two sets of tail-beams, 
 each 8 feet long, with headers 10 feet long? By the rule, 
 10 x 8" + x i x 20* 890 ; 890 x 3000 x 20 = 53400000. Now, 
 the value of F, in Table III., for hemlock is 2800 ; this by the 
 cube of the depth, 1728, equals 4838400; by this dividing 
 the former product, 53400000, and the quotient, 11-0367, is 
 the required breadth of the carriage-beam. 
 
CROSS-BRIDGING. 
 
 If, in equation (57.), /be taken at 275, and rat 0.04, these 
 will reduce, to 6875, and we shall have 
 
 6875 
 
 (59-) 
 
 a result the same as in equation (58.), except that the 
 constant is 6875 instead of 3000. Equation (59.) is a rule for 
 the breadth of carriage-beams carrying two sets of tail-beams 
 of equal length, in the floor of a first-class store. In words 
 at length, it is the same as Rule XLI V., except that the con- 
 stant 6875 is to be used in place of 3000. 
 
 161. ro-Bridgiiig, or Herring-Bone Bridging. The 
 
 diagonal struts set between floor-beams, as in Fig. 43, are 
 known as cross- bridging, or herring- 
 bone bridging. By connecting the 
 beams thus at intervals, say, of from 
 5 to 8 feet, the stiffness of the floor 
 is greatly increased. The absolute 
 strength of a tier of beams to resist a 
 weight uniformly distributed over 
 the whole tier is augmented but lit- 
 tle by cross-bridging ; but the power 
 of any one beam in the tier to re- 
 sist a concentrated load upon it, as a heavy article of fur- 
 niture or an iron safe, is greatly increased by the cross- 
 bridging; for this device, by connecting the loaded beam 
 with the adjacent beams on each side, causes these beams to 
 assist in carrying the load. To secure the full benefit of the 
 diagonal struts, it is very important that the beams be well 
 secured from separating laterally, by having strips, such as 
 cross-furring, firmly nailed to the under edges of the beams. 
 The tie thus made, together with that of the floor-plank on 
 the top edges, will prevent the thrust of the struts from sep- 
 arating the beams. 
 
 162. Bridging: Value to Resit Concentrated L,oad. 
 
 A rule for determining the additional load which any one 
 beam connected by bridging will be capable of sustaining, 
 by the assistance derived from the other beams, through the 
 
 FIG. 43. 
 
138 CONSTRUCTION. 
 
 bridging, may be found in Chapter XVIII., Transverse Strains. 
 This rule may be stated thus : * 
 
 R = -(1 + 22 + 3V 4 2 + etc.) ; (60.) 
 
 in which R is the increased resistance, equal to the addi- 
 tional load which may be put upon the loaded beam ; c is the 
 distance from centres in feet at which the beams in the tier 
 are placed ; /is the load in pounds per superficial foot upon 
 the floor ; / is the length of the beams in feet ; and d is the 
 depth of the beams in inches. The squares within the 
 bracket are to be extended to as many places as there are 
 beams on each side which contribute assistance through the 
 bridging. The rule given in the work referred to, for ascer- 
 taining the number of spaces between the beams, is 
 
 d f ,- \ 
 
 = 7 r; (61.) 
 
 or, the depth of the beam in inches divided by the square of 
 the distance from centres, in leet, at which the beams are 
 placed will give the number of spaces between the beams 
 which contribute on each side in sustaining the concentrated 
 load. The nearest whole number, minus unity, will equal 
 the required number of beams. 
 
 The value of c for beams in floors of dwellings is given in 
 equation (46.), and lor those in first-class stores in equation 
 (47.). By a modification of equation (34.), putting c f I for 
 U, we have 
 
 and- 
 
 c = 
 
 = (6 3 .) 
 
 These equations give general rules for the value of c. 
 
INCREASED LOAD BY CROSS-BRIDGING. 139 
 
 Now, the rule, in words at length, for the resistance offered 
 by the adjoining beams to a weight concentrated upon one 
 of the beams sustained by cross-bridging to the others, is 
 
 Rule XLV. Divide the depth of the beam in inches by 
 the square of the distance apart from centres in feet at 
 which the floor-beams are placed ; from the quotient deduct 
 unity, and call the whole number nearest to the remainder 
 the First Result. Take the sum of the squares of the con- 
 secutive numbers from unity to as many places as shall equal 
 the above first result ; multiply this sum by 5 times the 
 length in feet, by the load per foot superficial upon the floor, 
 and by the fifth power of the distance apart from centres in 
 feet at which the beams are placed ; divide the product by 
 4 times the square of the depth in inches, and the quotient 
 will be the weight in pounds required. 
 
 Example. In a tier of 3 x 12 inch floor-beams 20 feet 
 long, placed in a dwelling 16 inches from centres and well 
 bridged, what load maybe uniformly distributed upon one of 
 the beams, additional to the load which that beam is capable 
 of sustaining safely when unassisted by bridging? Here, 
 according to the rule, 12 divided by (ij+ ij = ) i-J equals 
 6f ; 6J i 5|, the nearest whole number to which is 6, the 
 first result. The sum of the square of the first 6 numbers 
 equals (i + 2 2 + 3' + 4* + 5 + 6 2 =) I +4 + 9+ 16 + 25 + 36 = 91. 
 Therefore, 91 x 5 x 20 x 90 x (|-) r ' = 345 1266.* The square of 
 the depth (12 x 12 = ) H4x4 = 576; by this dividing the 
 above 3451266, we have the quotient 5991-78, say 5992 
 pounds, the required weight. This is the additional load 
 which may be placed upon the beam. At 90 pounds per 
 superficial foot, the common load on each beam, we have 
 
 * The value of c, 16 inches, equals feet. The fifth power of this, or () 6 , 
 is obtained by involving both numerator and denominator to the fifth power, 
 and dividing the fifth power of the former by the fifth power of the latter ; for 
 
 (i) 5 = i_. For the numerator we have 4x4x4x4x4=1024, and for the de- 
 
 3 5 
 nominator 3x3x3x3x3 = 243. The former divided by the latter gives as a 
 
 quotient 4-214, the value of (j) 5 . The process of involving a number to a high 
 power, or the reverse operation of extracting high roots, may be performed by 
 logarithms with great facility. (See Art. 427.) 
 
I4O CONSTRUCTION. 
 
 90 x 20 x | = 2400 as the common load. To this add 5992, 
 the load sustained through the bridging by the other beams, 
 and the sum, 8392 pounds, will be the total load which may 
 be safely sustained, uniformly distributed, upon one beam 
 nearly 3^ times the common load. 
 
 163. Crirclers. When the distance between the walls of 
 a building is greater than that which would be the limit for 
 the length of ordinary single beams, it becomes requisite to 
 introduce one or more additional supports. Where sup- 
 ports are needed for a floor and partitions are not desirable, 
 it is usual to use a large piece of timber called a girder, sus- 
 tained by posts set at intervals of from 8 to 1 5 feet ; or, when 
 posts are objectionable, a framed construction called a 
 framed girder (Art. 196) ; or an iron box called a tubular 
 iron girder (Art. 182). When a simple timber girder is used 
 it is advisable, if it be large, to divide it vertically from end 
 to end and reverse the two pieces, exposing the heart of the 
 timber to the aii in order that it may dry quickly, and also 
 to detect decay at the heart. When the halves are bolted 
 together, thin slips of wood should be inserted between 
 them at the several points at which they are bolted, in order 
 to leave sufficient space for the air to circulate freely in the 
 space thus formed between them. This tends to prevent 
 decay, which will be found first at such parts as are not 
 exactly tight, nor yet far enough apart to permit the escape 
 of moisture. When girders are required for a long bear- 
 ing, it is usual to truss them ; that is, to insert between 
 the halves two pieces of oak which are inclined towards each 
 other, and which meet at the centre of the length of the 
 girder like the rafters of a roof-truss, though nearly if not 
 quite concealed within the girder. This and many similar 
 methods, though extensively practised, are generally worse 
 than useless ; since it has been ascertained that, in nearly all 
 such cases, the operation has positively weakened the girder. 
 
 A girder may be strengthened by mechanical contrivance, 
 when its depth is required to be greater than any one piece 
 of timber will allow. Fig. 44 shows a very simple yet invalu- 
 able method of doing this. The two pieces of which the gir- 
 
CONSTRUCTION OF GIRDERS. 14! 
 
 der is composed are bolted or pinned together,, having keys 
 inserted between to prevent the pieces from sliding. The 
 keys should be of hard wood, well seasoned. The two 
 pieces should be about equal in depth, in order that the 
 joint between them may be in the neutral line. (See Arts. 
 120, 121.) The thickness of the keys should be about half 
 their breadth, and the amount of their united thickness 
 should be equal to a trifle over the depth and one third of 
 the depth of the girder. Instead of bolts orpins, iron hoops 
 are sometimes used ; and when they can be procured, they 
 are far preferable. In this case, the girder is diminished at 
 the ends, and the hoops driven from each end towards the 
 middle. A girder may be spliced if timber of a sufficient 
 length cannot be obtained ; though not at or near the mid- 
 
 FIG. 44- 
 
 die, if it can be avoided. (See Art. 87.) Girders should 
 rest from 9 to 12 inches on each wall, and a space should be 
 left for the air to circulate around the ends, that the damp- 
 ness may evaporate. 
 
 164. Girders : IMmeniions. The size of a girder, for 
 any special case, may be determined by equations (21.), (22.), 
 (25,), (27.), and (28.), to resist rupture ; and to resist deflection, 
 by equations (32.) and (35.). For girders in dwellings, equa- 
 tion (44.) may be used. In this case, the value of c is to be 
 taken equal to the width of floor supported by the girder, 
 which is equal to the sum of the distances half way to the 
 wall or next bearing on each side. When there is but one 
 
142 CONSTRUCTION. 
 
 girder between the two walls, the value of c is equal to half 
 the distance between the walls. The rule for girders for 
 dwellings, in words, is 
 
 Rule XLVL Multiply the cube of the length of the gir- 
 der by the sum of the distances from the girder half way to 
 the next bearing on each side, and by the value of j for the 
 material of the girder, in Art. 152; the product -will equal 
 the product of the breadth of the girder into the cube of the 
 depth. To obtain the breadth, divide this product by the 
 cube of the depth ; the quotient will be the breadth. To 
 obtain the depth, divide the said product by the breadth ; 
 the cube root of the quotient will be the depth. If the 
 breadth and depth are to be in a given proportion, say as 
 r : i-o, then divide the aforesaid quotient by the value of 
 r ; take the square root of the quotient; then the square root 
 of this square root will be the depth, and the depth multi- 
 plied by the value of r will be. the breadth. 
 
 Example. In the floor of a dwelling, what should be the 
 size of a Georgia-pine girder 14 feet long between posts, 
 placed at 10 feet from one wall and 20 feet from the other? 
 The value of c here is V~ + % = J 5- The value of j 
 for Georgia pine (Art. 152) is 0-32. By the rule, 14' x 15 x 
 0-32 = 13171-2. Now, to find the breadth when the depth 
 is 12 inches ; 13171 -2 divided by the cube of 12, or by 1728, 
 gives a quotient of 7-622, or 7$, the required breadth. 
 Again,, to find the depth, when the breadth is 8 inches : 
 13171-2 divided by 8 gives 1646-4, the cube root of which is 
 1 1 -808, or, say, uj inches, the required depth. But if 
 neither breadth nor depth have been previously determined, 
 except as to their proportion, say as 0-7 to i-o, then 
 13171-2 divided by 0-7 gives 18816, of which the square 
 root is 137-171, and of this the square root is 11-712, or, say, 
 I if inches, the required depth. For the breadth, we have 
 11-712 by 0-7 equals 8-198, or, say, 8J, the required 
 breadth. Thus the girder is required to be 7$- x 12, 8 x iij, 
 or 8J xi if inches. This example is one in a dwelling or 
 ordinary store ; (or first-class stores the rule for girders is the 
 same as the last, except that the value of k is to be taken 
 instead ofy, in Art. 152. 
 
FIRE-PROOF TIMBER FLOORS. 143 
 
 165. Solid Timber Floors. Floors constructed with 
 rolled-iron beams and brick arches are proof against fire 
 only to a limited degree ; for experience has shown that the 
 heat, in an extensive conflagration, is sufficiently intense to 
 deprive the iron of its rigidity, and consequently of its 
 strength. Singular as it may seem, it is nevertheless true 
 that wood, under certain circumstances, has a greater fire- 
 resisting quality than iron. Floors of timber constructed, 
 as is usual, with the beams set apart, have but little power 
 to resist fire, but if the spaces between the beams be filled 
 up solid with other beams, which thus close the openings 
 against the passage of the flames, and the under surface be 
 coated with plastering mortar containing a large portion of 
 plaster of Paris, and finished smooth, then this wooden 
 floor will resist the action of fire longer than a floor of iron 
 beams and brick arches. The wooden beams should be se- 
 cured to each other by dowels or spikes. 
 
 166. Solid Timber Floors for Dwellings and Assem- 
 bly-Rooms. From Transverse Strains, Art. 702, we have 
 
 which may be modified so as to take this form : 
 
 which is a rule for the depth or thickness of solid timber 
 floors for dwellings, assembly-rooms, or office buildings, and 
 in which y and h are constants depending upon the mate- 
 rial ; thus, for 
 
 Georgia Pine ................ y 4 and h = 
 
 Spruce ...................... ^ = 2i, " // = 0-365 
 
 White Pine .................. y = 2j, " // = 0-389 
 
 Hemlock ..................... y = 2, ' 7*=o-39 
 
 The rule may be stated in words thus : 
 
 Rule XLVIL Multiply the length by the value of 
 
144 CONSTRUCTION. 
 
 and by the value of /i, as above given ; to the product add 
 82 ; multiply the sum by the cube of the length ; divide this 
 product by 0-576 times the value of F, in Table III. ; then 
 the cube root of the quotient will be the required depth in 
 inches. 
 
 Example. What depth is required for a solid Georgia- 
 pine floor to cover a span of 20 feet ? For Georgia pine 
 F= 5900 ; y, as above given, equals 4, and h equals 0-314 ; 
 therefore, by the rule 
 
 d - _ = 6.318; 
 
 0-576x5900 339 8 '4 
 
 or, the depth required is, say, 6-32 or 6^ inches. 
 
 167. Solid Timber Floors for First-Clas Si ores. The 
 
 equation given for first-class stores, in Transverse Strains, 
 Art. 702, is 
 
 -(263 
 
 which may be changed to this form : 
 
 in which y is as before, and k for 
 
 Georgia Pine equals 0-4 
 
 Spruce equals o 472 
 
 White Pine equals 0-502 
 
 Hemlock equals o 506 
 
 This rule may be put in words the same as Rule XLVIL, 
 except as to the constants, which require that 263 be used in 
 place of 82, that k be used in place of /*, and that 0-768 be 
 used in place of 0-576. Table XXI. of Transverse Strains 
 contains the results of computation showing the depths of 
 solid timber floors for dwellings and assembly-rooms and 
 for first-class stores, in floors of spans varying from 8 to 30 
 feet, and for the four kinds of timber before named. 
 
IRON FLOOR-BEAMS. 145 
 
 I6ff. Rolled - Iron Beam*. The dimensions of iron 
 beams, whether Avrought 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 
 to wood and iron (Art. 93), and may be used for any mate- 
 rial, provided the constant appropriate to the given mate- 
 rial be used. But when the form of cross- 
 section is such as that which is usual for 
 rolled-iron beams (Fig. 45), the rules 
 need modifying. Without' attempting 
 to explain these modifications (referring 
 for this to Transverse Strains, Art. 457 
 and following article), it may be re- 
 marked that the elements of resistance 
 to flexure in a beam constitute what is 
 
 termed the Moment of Inertia. This, in 
 
 . FIG. 45. 
 
 a beam of rectangular cross-section, is 
 
 equal to -J^ of the breadth into the cube of the depth ; or 
 I=^bd\ (66.) 
 
 This would be appropriate to rolled-iron beams if the 
 hollow on each side were filled with metal, so as to complete 
 the form of cross-section into a rectangle. The proper ex- 
 pression for them may be obtained by taking first the 
 moment for the beam as if it were a solid rectangle, and 
 from this deducting the moment for the part which on each 
 side is wanting, or for the rectangles of the hollows. In 
 accordance with this view of the case, we have 
 
 b,d^; (67.) 
 
 in which b is the breadth of the beam or width of the 
 flanges ; b, is the breadth of the two hollows, or is equal to# 
 less the thickness of the web or stem ; d is the depth includ- 
 ing top and bottom flanges ; and d t is the depth in the clear 
 between the top and bottom flanges. 
 
 Now, if equation (32.) be divided by 12, we shall have 
 
146 CONSTRUCTION. 
 
 and since -^ b d* represents the moment of inertia, we have- 
 
 (68.) 
 
 12 Fd 
 
 This gives the value of / for a beam of any form in cross- 
 section loaded at the middle. By this equation the values 
 of / have been computed for rolled -iron beams of many 
 sizes, and the results recorded in Table XVII., Transverse 
 Strains. A few of these are included in Table IV., as follows : 
 
 TABLE IV. ROLLED- IRON BEAMS. 
 
 NAME. 
 
 Depth. 
 
 Weight 
 per yard. 
 
 / = 
 
 NAME. 
 
 Depth. 
 
 Weight 
 per yard. 
 
 / = 
 
 Trenton 
 
 
 qo 
 
 7-8d 
 
 Buffalo 
 
 
 QO 
 
 109- 1 17 
 
 Paterson . . 
 Phoenix 
 
 5 
 t 
 
 3 
 36 
 
 12-082 
 
 I 4'3 I 7 
 
 Phoenix 
 Buffalo 
 
 T?* 
 
 150 
 QO 
 
 190-63 
 151 -436 
 
 Trenton 
 
 6 
 
 -1O 
 
 v 761 
 
 Buffalo 
 
 ioi 
 
 IO^ 
 
 17^ -6-1^ 
 
 Phoenix . . 
 Trenton . . 
 Buffalo 
 
 7 
 
 7 
 
 8 
 
 55 
 60 
 65 
 
 42-43 
 46-012 
 64- 526 
 
 Trenton 
 iBuffalo 
 Paterson. . . . 
 
 1 
 loj 
 
 12$ 
 
 I2 
 
 135 
 
 125 
 125 
 
 241-478 
 286-OI9 
 292-05 
 
 Paterson. . . 
 Phoenix ... . 
 Phoenix . . . 
 
 8 
 9 
 9 
 
 So 
 ?o 
 
 84 
 
 84-735 
 92-207 
 107-793 
 
 Paterson. ... 
 iBuffalo 
 Trenton 
 
 12* 
 12* 
 
 !5fV 
 
 170 
 1 80 
 150 
 
 398-93^ 
 418-945 
 528-223 
 
 169. Rolled -Iron Beams: Diiuciitioii* ; Weight at 
 
 middle. If, in equation (68.), there be substituted for F its 
 value for wrought iron, as in Table TIL, we shall have 
 
 or 
 
 " 1 2 x 62000 <y ' 
 
 Wl* 
 
 744000 rf 
 
 (69.) 
 
 This is a rule by which to ascertain the size of a rolled-iron 
 beam to sustain a given weight at middle with a given de- 
 flection, and, in words at length, is as follows : 
 
 Rule XLVII1. Multiply the weight in pounds by the 
 cube of the length in feet ; divide the product by 744000 
 times the deflection in inches, and the quotient will be the 
 
DEFLECTION OF IRON BEAMS. 147 
 
 moment of inertia of the required beam, and may be found, 
 or the next nearest number, in Table IV. in column headed 
 7. Opposite to the number thus found, to the left, will be 
 found the name, -depth, and weight per yard of the required 
 beam. 
 
 Example. Which of the beams of Table IV. would be 
 proper to carry 10,000 pounds at the middle with a deflection 
 of one inch, the length between bearings being 20 feet ? 
 Here we have, substituting for the symbols their values 
 
 W Y 3 _ioooox2o 3 _ 80000000 
 ~ 744000 tf ~~~ 744000 x i ~~ 744000 3 ' ' 
 
 or, the moment of inertia of the required beam is 107-527, the 
 nearest to which, in the table, is 107-793, pertaining to the 
 Phoenix 9-inch, 84-pound beam. This, then, is the required 
 beam. 
 
 170. Rolled-Iron Beams: Deflection when Weight is 
 at Middle. By a transposition of symbols in equation (69.), 
 we have 
 
 W l z 
 
 ~-~ (70.) 
 
 or a rule for the deflection of rolled-iron beams when the 
 weight is at the middle. This, in words, is 
 
 Rule XLIX. Multiply the weight in pounds by the 
 cube of the length in feet ; divide the product by 744000 
 times the value of / for the given beam, and the quotient 
 will be the required deflection in inches. 
 
 Example.-JN\Mi\. will be the deflection of a Phoenix 9- 
 inch, 70-pound beam 20 feet long, loaded at the middle with 
 7500 pounds ? The value of / for this beam, in Table IV., 
 is 92-207; therefore, substituting for the symbols their val- 
 ues, and proceeding by the rule, we have 
 
 __ ___. = 
 " 744000 / " " 744000 x 92 - 207 
 
 or, the deflection will be, say, f of an inch. 
 
148 CONSTRUCTION. 
 
 171. Rolled -Iron Beams : Weight when at middle. 
 
 A transposition of factors in equation (70.) gives 
 
 IV 744QQQ / <? 
 
 / 3 (71.) 
 
 This is a rule for the weight at middle, and, in words, is 
 Rule L. Multiply 744000 times the value of 7 by the 
 
 deflection in inches ; divide the product by the cube of the 
 
 length, and the quotient will be the required weight in 
 
 pounds. 
 
 Example. What weight at the middle of a Buffalo g-inch, 
 
 go-pound beam will deflect it one inch, the length between 
 
 bearings being 20 feet ? The value of 7 for this beam, in 
 
 Table IV., 18109-117; therefore 
 
 744000 /<? 744000x109-117x1 
 
 ^ 20 3 : IO147 -WSJ 
 
 or, the required weight is, say, 10,148 pounds. 
 
 172. Rolled -Iron Beam : Weight at any Point. The 
 
 equation for a load at any point is (Transverse Strains, Art. 
 
 485)- 
 
 186000 IS 
 
 r^r,r - (72.) 
 
 in which m and n represent the two parts m feet into which 
 the point where the load rests divides the length. This, in 
 words, is as follows : 
 
 Rule LI. Multiply 186000 times the value of 7 by the 
 deflection in inches ; divide the product by the product of 
 the length into the rectangle formed by the two parts into 
 which the point where the load rests divides the length ; 
 the quotient will be the required weight in pounds. 
 
 Example. What weight is required, located at 10 feet 
 from one end, to deflect i inches a Paterson 12^-inch, 125- 
 pound beam 25 feet long between bearings ? The value of 
 7 for this beam, in Table IV., is 292-05 ; m 10, and n = I 
 - m = 25 10 ^ 15 ; therefore 
 
VARYING WEIGHTS ON IRON BEAMS. 149 
 
 ,,, I86OOO Id I86OOO X 2Q2-O5 X I -s 
 
 - - 
 
 or, the required weight is, say, 21,730 pounds. 
 
 (73. Rolled-Iron Beams: Dimension* Weight at any 
 Point. By transposition of factors in equation (72.), we ob- 
 tain 
 
 Wlmn 
 '" 186000 tf* (73-) 
 
 This may be expressed in words as follows : 
 
 Rule LI I. Multiply the weight by the length, and by the 
 rectangle of the two parts into which the point where the 
 weight rests divides the length ; divide the product by 186000 
 times the deflection, and the quotient will be the value of 7, 
 which (or its next nearest number) may be found in Table 
 IV., opposite to which will be found the required beam. 
 
 Example. What beam 10 feet long will be required to 
 carry 5000 pounds at 3 feet from one end with a deflection 
 of 0-4 inch? Here we have m equal 3, and n equal 7; 
 therefore 
 
 Wlmn 5000x10x3x7 
 
 - . U ' - - J A . J I ? 
 
 1 86000 tf " 1 86000 x o 4 
 
 The value of /is 14-113, the nearest number to which in 
 the table, is 14-317, the moment of inertia of the Phoenix 5- 
 inch, 36-pound beam ; this, therefore, is the .beam required. 
 
 174. Rolled-Iron Beams: Dimensions; Weight Uniform- 
 ly Distributed. Since f U = W(Art. 138), equation (69.) may 
 be modified by the substitution of this value of W, when we 
 obtain 
 
 which reduces to 
 
 7 = 
 
 744000 $' 
 
 / = p (74.) 
 
 1 190400 6 
 
150 CONSTRUCTION. 
 
 a rule for the dimensions of a beam for a uniformly distrib- 
 uted load, which, in words, is as follows : 
 
 Rule LIII. Multiply the uniformly distributed load by 
 the cube of the length ; divide the product by 1 190400 times 
 the deflection, and the quotient will be the value of /, corre- 
 sponding to which, or to its next nearest number will be 
 found in Table IV/the required beam. 
 
 Example. What beam 10 feet long is required to sus- 
 tain an equally distributed load of 14,000 pounds with a de- 
 flection of half an inch ? For this we have 
 
 14000 x io 8 
 
 1190400x0-5 
 
 This is the moment of inertia of the required beam ; nearly. 
 the same as 23-761, in Table IV., the value of / fora Tren- 
 ton 6-inch, Ao-pound beam, which will serve as the re- 
 quired beam. 
 
 175. Rolled-Iron Beam* : Deflection ; Weight Uniformly 
 
 Distributed. A transposition of the factors in equation (74.) 
 gives 
 
 *= UIS 
 
 11904007' (75-) 
 
 a rule for the deflection of a uniformly loaded beam, and 
 which may be put in these words, namely : 
 
 Rule LIV. Multiply the uniformly distributed load by 
 the cube of the length ; divide the product by 1190400 times 
 the value of /, Table IV., and the quotient will be the re- 
 quired deflection. 
 
 Example. To what depth will 14,000 pounds, uniformly 
 distributed, deflect a Buffalo lo^-inch, 9O-pound beam 20 
 feet long? The value of /for this beam, as per the table, is 
 151- 436 ; therefore 
 
 14000 X 20 3 
 
 ?= 0-6213 ; 
 
 1 190400 x 151 -436 
 or, the required deflection is, say, of an inch. 
 
IRON FLOOR-BEAMS FOR DWELLINGS. 151 
 
 ! 7 6. Rolled -Iron Beam: Weight when Uniformly 
 
 Distributed. Equation (75.), by a transposition of factors, 
 gives 
 
 * 9 (76 .) 
 
 a rule for the weight uniformly distributed, and which may 
 be worded thus : 
 
 Rule LV. Multiply 1190400 times the value of /, Table 
 IV., by the deflection ; divide the product by the cube of 
 the length, and the quotient will be the required weight. 
 
 Example. What weight uniformly distributed upon a 
 Buffalo loj-inch, 105 -pound beam 25 feet long between 
 bearings will deflect it f of an inch ? 
 
 The value of / for this beam, as per Table IV., is 175-645 ; 
 therefore 
 
 1190400 17 
 
 or, the required weight is, say, 10,036 pounds. 
 
 (77. Rolled-Iron Beam: Floors of Dwellings or As- 
 semfoly- Rooms. From Transverse Strains, Art. 500, we 
 have 
 
 a rule for the distance from centres of rolled-iron beams in 
 floors of dwellings, assembly-rooms, or offices, where the 
 spaces between the beams are filled in with brick arches and 
 concrete. In the equation, c is the distance apart from cen- 
 tres in feet, and y is the weight per yard of the beam. This, 
 in words, is thus expressed : 
 
 Rule LVL Divide 255 times the value of / by the cube 
 of the length ; from the quotient deduct one 42oth part of 
 the weight of the beam per yard, and the remainder will 
 the required distance apart from centres. 
 
 Example. What should be the distance apart from cen- 
 
1 5 2 CONSTRUCTION. 
 
 tres of Buffalo I2j-inch, 125 -pound beams 25 feet lo.ng 
 between bearings, in the floor of an assembly-room ? For 
 these beams, in Table IV., / equals 286-019, and y= 125; 
 therefore 
 
 _ 255 x 286-oi9_ 125 m 
 ~^J ~ 420 ' 
 
 - -298 = 4-37; 
 
 15625 420 
 
 or, the required distance from centres is, say, 4 feet 4^ 
 inches. 
 
 178. Rolled-Iron Beams : Floor of First-Class Stores. 
 
 From Transverse Strains, Art. 504, we have 
 
 a rule for the distance from centres of rolled-iron beams in 
 the floor of a first-class store ; the spaces between the beams 
 being filled with brick arches and concrete. This rule may 
 be put in words as follows: 
 
 Rule LVIL Divide 148-8 times the value of 7 by the 
 cube of the length ; from the quotient deduct one 96oth 
 part of the weight of the beam per yard, and the remainder 
 will be the distance apart of the beams from centres in 
 feet. 
 
 Example. What should be the distance apart from cen- 
 tres of Buffalo I2j-inch, i8o-pound beams 20 feet long 
 between bearings, in the floor of a first-class store? For 
 these beams the value of /, Table IV., is 418-945, and the 
 value of y is 180; therefore 
 
 c = I48 ' 8 x 418-945 _ 180 6o . 
 
 20 3 960 
 
 or, the required distance from centres is, say, 7 feet /| 
 inches. 
 
TIE-RODS FOR IRON BEAMS. 153 
 
 179. Floor- A roll es : Oeneral Considerations. In fill- 
 ing the spaces between the iron beams of a floor, the arches 
 should be constructed with hard whole brick of good shape, 
 laid upon the supporting centre in contact with each other, 
 and the joints thoroughly filled with cement grout, and 
 keyed with slate. Made in this manner, the arches need not 
 be over four inches thick at the crown for spans extending 
 to 7 or 8 feet, and 8 inches thick at the springing, where 
 they should be started upon a proper skew-back. The rise 
 of the arch should not be less than i inches for each foot 
 of the span. 
 
 180. Floor - Arches ; Tie -Rods: Dwellings. From 
 Transverse Strains ', Art. 507, we have 
 
 d = V 0-0198 c s, (79.) 
 
 which is a rule for the diameter in inches of a tie-rod for 
 an arch in the floor of a bank, office building, or assembly- 
 room ; in which d is the diameter in inches of the rod, s is 
 the span of the arch, and c is the distance apart between the 
 rods (s and c both in feet). This rule requires that the arch 
 rise i^ inches per foot of the span, and that the brick-work 
 and the superimposed load each weigh 70 pounds, or to- 
 gether 140 pounds. This rule, in words, is -as follows: 
 
 Rule LVIII. Multiply the span of the arch by the dis- 
 tance apart at which the rods are placed, and by the decimal 
 0-0198 ; the square root of the product will be the diameter 
 of the required rod. 
 
 Example. What should be the diameter of the wrought- 
 iron ties of brick arches of 5 feet span, in a bank or hall of 
 assembly, where the ties are 8 feet apart ? For this we 
 
 have 
 
 d 1/0-0198 x^8 x 5 = 1/-792 = 0-89; 
 
 or, the diameter of the required rods should be, say, -J of an 
 inch. 
 
 181. Floor-Arches ; Tie-Rods: First-Class Stores. From 
 the same source as in last article, we have 
 
 d = V o- 04527 c~s, (80.) 
 
154 
 
 CONSTRUCTION. 
 
 which is a rule for the size of tie-rods for the brick arches 
 of the floors of first-class stores, where the arches have a 
 rise of i inches for each foot of the span, and where the 
 weight of the brick arch and concrete is not over 70 pounds 
 per superficial foot of the floor, and the loading does not 
 exceed 250 pounds per superficial foot. As the rule is the 
 same as the one in the preceding article, except the deci- 
 mal, a recital of the rule, in words, is not here needed. To 
 obtain the required diameter, proceed as directed in Rule 
 LVIII., using the decimal 0-04527 instead of the one there 
 given. 
 
 TUBULAR IRON GIRDERS. 
 
 182. Tubular Iron Girders: Decripfion. The use of 
 
 wooden beams for floors is limited to spans of about 25 feet. 
 When greater spans than this are to be covered, some expe- 
 dient must be resorted to by which 
 intermediate bearings for the floor- 
 beams may be provided. Wooden 
 girders may be used, but these 
 need to be supported by posts at 
 intervals of from 10 to 15 feet, 
 unless the girders are trussed, or 
 made up of top and bottom chords, 
 struts, and ties. And even this 
 is objectionable, owing to the 
 height such a piece of framing 
 requires, and which encumbers 
 the otherwise free space of the 
 hall. A substitute for the framed 
 girder has been found in the 
 tubular iron girder, as in Fig. 46, made of rolled plate 
 iron and angle irons, riveted. They require to be stiffened 
 by an occasional upright T iron along eachside, and a 
 cross-head at least at each bearing. 
 
 183. Tubular Iron Girders: Area of Flangc ; Load 
 at Middle. In wrought-iron tubular girders it is usual to 
 make the top and bottom flanges of equal thickness. From 
 Transverse Strains, Art. 551, we have 
 
 FIG. 46. 
 
TUBULAR IRON GIRDERS. 1 55 
 
 a rule for the area of the bottom flange ; in which a' equals 
 the area of the flange in inches, W the weight in pounds at 
 the middle, / the length and d the depth of the girder, both 
 in feet, and k the saTe load in pounds per inch with which 
 the metal may be loaded, and which is usually taken at 
 9000. The rule may be stated thus : 
 
 Rule LIX. Multiply the weight by the length ; divide 
 the product by 4 times the depth into the value of k, and 
 the quotient will be the required area of the bottom flange. 
 
 Example. In a girder 40 feet long and 3 feet high, to 
 carry 75,000 pounds at the middle, what area of metal is 
 required in the bottom flange, putting k at 9000? For this 
 we have, by the rule 
 
 . W I 75000 x 40 
 f,' j__j_ ~ , ^7 77'* 
 
 4 d k 4 x 3 x 9000 
 
 or, the area required is 27$ inches. This is the amount of 
 uncut metal. An allowance is required for that which will 
 be cut by rivet-holes. This is usually an addition of one 
 sixth. 
 
 184. Tubular Iron Girders : Area of Flanges; Load at 
 
 any Point. The equation suitable for this (Transverse 
 Strains, Art. 553) is 
 
 (82 - } 
 
 in which m and n are the distances respectively from the lo- 
 cation of the load to the two ends of the girder. The other 
 symbols are the same as in the last article. This rule may 
 be thus stated : 
 
 Rule LX. Multiply the weight by the values of m and 
 of n ; divide the product by the product of the depth into 
 the length and into the value of k, and the quotient will be 
 the required area of the bottom flange. 
 
 Example. In a girder 50 feet long between bearings and 
 
156 CONSTRUCTION. 
 
 3J- feet high, what area of metal is required in the bottom 
 flange to sustain 50,000 pounds at 20 feet from one end, when 
 k equals 9000 * By the rule, we have 
 
 rr m n 50000 x 20 x 30 
 
 or, each flange requires 19 inches of solid metal uncut for 
 rivets. 
 
 185. Tubular Iron Girder : Area of Flange ; Load 
 Uniformly Distributed. The equation appropriate here is 
 (Transverse Strains, Art, 555) 
 
 This is a rule by which to obtain the area of cross-sec- 
 tion of the bottom flange at any point in the length of the 
 girder, the load uniformly distributed ; m and n being the 
 respective distances from the point measured to the two 
 ends of the girder, and U representing the uniformly dis- 
 tributed load in pounds. This, in words, is described as 
 follows : 
 
 Rule LXI. Divide the weight by the product of twice 
 the depth into the length and into the value of k ; then the 
 quotient multiplied by the values of m and of n will be the 
 required area of the bottom flange at the point measured, 
 the distance of which from the ends equals m and n. 
 
 Examplc.~\n a girder 50 feet long and 3^ feet high, to 
 carry a uniformly distributed load of 120,000 pounds, what 
 area of cross-section is required in the bottom flange, at the 
 middle and at intervals of 5 feet thence, to each support ; k 
 being taken at 9000? Here we have, first 
 
 . m n 1 20000 m n 
 
 a = U -jT-r = - = 0-038005 m n. 
 
 2dkl 2x3^x9000x50 
 
 Now, when m = n = 25, we have the middle point ; then 
 a' = 0-038095 m n = 0-038095 x 25 x 25 = 23-81 ; 
 
/Til w T i 
 
 SHEARING STRAIN. 
 
 j& "^ 
 
 \v v 
 
 SRJ^r 
 
 or, the area of the bottom flange at mid-length is 23 
 inches. 
 
 When ;;/ = 20, then n = 30, and 
 
 a' = 0-038095 X2ox 30 = 22-86; 
 
 or, the required area, at 5 feet either way from the middle, 
 is 22-J inches. 
 
 When m = 15, then n = 35, and 
 
 a = 0-038095 x 15 x 35 = 20-0 ; 
 
 or, at 10 feet either way from the middle, the required area 
 is 20 inches. 
 
 When m= 10, then n = 40, and 
 
 a' 0-038095 x 10x40 = 15-24 ; 
 
 or, at 15 feet either way from the middle, the required area 
 is 15^ inches. 
 
 When m = 5, then n 45, and 
 
 a' = 0-038095 x 5x45 =8-57; 
 
 or, at 20 feet each side of the middle, the required area is 8f 
 inches. 
 
 The area of cross-section found in every case is that of 
 the uncut fibres ; to this is to be added as much as will be 
 cut by the rivets. This is usually about one sixth of the area 
 given by the rule. The top flange is to be made equal in 
 area to the bottom flange. The flanges are unvarying in 
 width from end to end, the variation of area being obtained 
 by varying the thickness of the flanges, and this being at- 
 tained by building the flange in lamina, or plates ; but these 
 should not be less than a quarter of an inch thick. There 
 should be added to the length of the girder, in the clear, 
 about one tenth of its length for supports on the walls : thus, 
 a girder 30 feet long requires 3 feet added for supports, or 
 1 8 inches on each wall. 
 
 186. Tubular Iron Girdcr: Shearing Strain. The top 
 
 and bottom flanges are provided of sufficient size to resist 
 
158 CONSTRUCTION. 
 
 the transverse strain ; the two upright plates, technically 
 termed the web, need, therefore, to be thick enough to resist 
 only the shearing strain. This, upon a beam uniformly 
 loaded, is at the middle theoretically nothing, but from 
 thence it increases regularly towards each support, where it 
 equals half the whole weight. For example, the girder of 
 Art. 185, 50 feet long between supports, carries 120,000 
 pounds uniformly distributed over its length. In this case 
 the shearing strain at the wall at each end is the half of 
 120,000 pounds, or 60,000 pounds; at 5 feet from the wall it 
 is -fg or -J less, or 48,000 pounds ; at 10 feet from the wall it 
 is f less, or 36,000 pounds ; at 15 feet it is 24,000 ; at 20 feet 
 it is 12,000; and at 25 feet or the middle, it is nothing. 
 
 187. Tubular Iron Girder: Thickness of Wefo. The 
 
 equation appropriate for this is 
 
 in which t is the thickness of the web (equal to the sum of 
 the thicknesses of the two side plates), d is the height of the 
 plate (/ and d both in inches), G is the shearing strain, and k' 
 is the effective resistance of wrought iron to shearing per 
 inch of cross-section. This may be put in words as follows : 
 
 Rule LXII. Divide the shearing strain by the product of 
 the depth in inches into the value of /', and the quotient 
 will be the thickness of the web, or of the two side plates 
 taken together. 
 
 Example. What is the required thickness of web in a 
 girder 50 feet between bearings, side plates 38 inches high 
 between top and bottom flanges, and to carry 120,000 pounds, 
 uniformly distributed ? Here, putting the shearing resistance 
 of the plates at 7000 pounds per inch, we have 
 
 dk' ' 38x7000 266000* 
 
 The shearing strain at the supports, as in last article, is 
 60000 ; therefore, we have for this point 
 
LIGHT IRON GIRDERS. 159 
 
 60000 
 1 "266000" ^ ' 225 ' 
 
 When G = 48000, then 
 
 48000 
 
 : "^660^0"- ' I8; 
 and when G = 36000, then 
 
 36000 
 
 / -? 0-135. 
 
 266000 
 
 Those nearer the middle of the girder are still less than 
 these ; and these are all below the practicable thickness, 
 which is half an inch for the two plates. The plates ought 
 not in practice ever to be made less than a quarter of an 
 inch thick. 
 
 188. Tubular Iron Girder, for Floors of 
 Acmbly-Room, 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 of 
 the bottom flange of the girder : 
 
 in which a' is in inches, and c, c , d, /, m, and n are in feet. 
 Also, a' is the area required ; y is the weight per yard of the 
 rolled-iron beam of the floor ; c, their distances from centres ; 
 c', the distance from centres at which the tubular girders are 
 placed, or the breadth of floor carried by one girder; d, the 
 depth of the girder; k, the effective resistance of the metal 
 per inch in the flanges of the girder ; and ;// and n are the 
 distances respectively from the two ends of the girder to the 
 point at which the area of cross-section of the bottom flange 
 is required. The rule may be thus described : 
 
 Rule LXIIL Divide the weight per yard of the rolled- 
 iron beams by 3 times their distance from centres; to the 
 quotient add 140 and reserve the sum ; deduct the length 
 in feet from 700, and with the remainder as a divisor divide 
 700 ; multiply the quotient by the above reserved sum, and 
 
l6o CONSTRUCTION. 
 
 by the value of c' ; divide the product by the product of 
 twice the depth into the value of k, and the quotient multi- 
 plied by the values of m and of n will be the required area 
 of cross-section of the bottom flange at the point in the 
 length distant from the two ends equal to m and n respec- 
 tively. 
 
 Example. In a floor of 9-inch, /o-pound beams, 4 feet 
 from centres, what ought to be the area of the bottom flange 
 of a tubular girder 40 feet long between bearings, 2 feet 8 
 inches deep, and placed 17 feet from the walls or from other 
 girders ; the area of the flange to be ascertained at every 5 
 feet of the length : the value of k to be put at 9000 ? Here 
 y =. 70, c = 4, c f = 17, 7=40, and d= 2f. Therefore, by 
 the rule 
 
 / 70 \ 700 17 
 
 a = I 14.0+ ) x o x m n : 
 
 V 3 x 47 700 - 40 2 x 2f x 9000 
 
 a' 145 8^- x I -0606 x 0-0003 54 r f x mn\ 
 a' = 0-05478 m n. 
 
 The values of m and n are 
 
 At the middle m = 20 ; n = 20 
 
 5 feet from middle m 15; n = 25 
 
 10 " " " m = 10 ; n = 30 
 
 15 . " " " m= 5 ; n = 35 
 
 These give 
 
 At the middle a' 0-05478 x 20 x 20 = 21 -91 
 
 " 5 feet from middle a' = 0-05478 x 15 x 25 = 20-54 
 
 "10 " " " a' = 0-05478 x 10 x 30 16-43 
 
 "15 " " " *' = 0-05478 x 5x35= 9-59 
 
 These are the areas of uncut fibres at the points named, in 
 the lower flange ; the upper flange requires the same sizes. 
 
 (89. Tubular Iron Girders, for Floors of First-Class 
 
 Stores. The equation proper for this is (Transverse Strains, 
 Art. 570) 
 
CAST IRON COMPARED WITH WROUGHT. l6l 
 
 a rule the same in form as that of the previous article ; hence 
 it needs no particular exemplification. 
 
 Rule LXIII. of last article may be used for this case, 
 simply by using the constant 320 in place of that of 140. 
 
 CAST-IRON GIRDERS. 
 
 190. Cast-iron Girders: Inferior. Rolled- iron beams 
 have been so extensively introduced within a few years as 
 to have superseded almost entirely the formerly much used 
 cast-iron beam or girder. The tensile strength of cast iron 
 is far inferior to that of wrought iron. This inferiority and 
 the contingencies to which the metal is subject in casting 
 render it very untrustworthy ; it should not be used where 
 rolled-iron beams can be procured. A very substantial gir- 
 der to carry a brick wall is made by placing two or more 
 rolled-iron beams side by side, and securing them together 
 by bolts at mid-height of the web ; placing thimbles or sep- 
 arators at each bolt. As there may be cases, however, in 
 which cast-iron girders will be used, a few rules for them 
 will here be given. 
 
 1 9 1. _Cast-Iron Girder: Load at Middle. The form of 
 cross-section given to this girder usually is as shown in 
 Fig. 47. 
 
 In the cross-section, the bottom flange 
 is made to contain in area four times as 
 much as the top flange. The strength 
 will be in proportion to the area of the 
 bottom flange, and to the height or 
 depth of the girder at middle. Hence, 
 to obtain the greater strength from a 
 given amount of material, it is requisite 
 to make the upright part, or the web, 
 rather thin ; yet, in order to prevent 
 injurious strains in the casting while it 
 is cooling, the parts should be nearly 
 equal in thickness. The thickness of the three parts web, 
 
1 62 CONSTRUCTION. 
 
 top flange, and bottom flange may be made in proportion 
 as 5, 6, and 8. 
 
 For a weight at middle, the form of the web should be 
 that of a triangle ; the top flange forming two straight lines 
 declining from the centre each way to the bottom flange at 
 the ends, like the rafters of a roof to its tie-beam. From 
 Transverse Strains, Art. 583, we have 
 
 
 which is a rule for the area in inches of the bottom flange, 
 for a load at middle ; the area of the top flange is to be equal 
 to one fourth of that of the bottom flange. To secure this, 
 make the width of the top flange equal to one third of the 
 width of the bottom flange ; the thickness of the former, 
 as before directed, being made equal to f or f of the lat- 
 ter. The weight W is in pounds ; the length / is in feet ; 
 and the depth d is in inches. The factor of safety a should 
 be taken at not less than 3 ; better at 4 or 5. 
 
 The equation in words may be as follows : 
 
 Rule LXIV. Multiply the weight by the length, and by 
 the factor of safety ; divide the product by 4850 times the 
 depth at middle, and the quotient will be the area in inches 
 of the bottom flange ; divide this area by the width of the 
 bottom flange, and the quotient will be its thickness. Of the 
 top flange make its width equal one third that of the bot- 
 tom flange, and its thickness equal to three quarters that of 
 the latter. Make the thickness of the web equal to f- that 
 of the bottom flange. 
 
 Example. What should be the dimensions of the cross- 
 section of a cast-iron girder 20 feet long between bearings, 
 and 24 inches high at middle, where 30,000 pounds is to be 
 carried ; the factor of safety being put at 5 ? 
 
 Here we have W 30000 ; a 5 ; / 20 ; and d = 
 24 ; therefore, by the rule 
 
 30000 x 5 x 20 
 
THE BOWSTRING GIRDER. 
 
 I6 3 
 
 This is the area of the bottom flange. If the width of this 
 flange be 12 inches, then 25-773 divided by 12 gives 2 -15, or 
 2-J- full, as the thickness. One third of 12 equals 4, equals 
 the width of the top flange ; and j- of 2 1 5 equals i 6 1 , or i f 
 its thickness. The thickness of the web equals -| x 2-15 = 
 i -34 or i-J inches. 
 
 192 Cast-iron Girder: Load Uniformly Distributed. 
 
 The equation suitable to this is 
 
 Ual 
 
 
 9700 (f 
 
 (88.) 
 
 a rule of like form with that of the last article ; therefore, 
 Rule LXIV. may be used for this case, simply by substitut- 
 ing 9700 for 4850. 
 
 193. Cast -Iron Bowstring Girder. An arched girder, 
 such as that in Fig. 48, is technically termed a " bowstring 
 girder." The curved part is a cast-iron beam of T form in 
 section, and the horizon- 
 tal line is a wrought-iron 
 tie-rod attached to the 
 ends of the arch. This 
 girder has but little to 
 commend it, and is by no 
 means worthy the confi- 
 
 dence placed in it by FlG- 48 ' 
 
 builders, with many of whom it is quite popular. The brick 
 arch usually turned over it is adequate to sustain the entire 
 compressive force induced from the load (the brick wall 
 built above it), and it thereby supersedes the necessity for the 
 iron arch, which is a useless expense. The tie-rod is the 
 only useful part of the bowstring girder, but it is usually 
 made too small, and not infrequently is seriously injured by 
 the needless strain to which it is subjected when it is 
 " shrunk in" to the sockets in the ends of the arch. The bow- 
 string girder, therefore, should never be used. 
 
 (94. Substitute for tlie Bowstring Girder. As the cast- 
 iron arch of a bowstring girder serves only to resist com- 
 
164 
 
 CONSTRUCTION. 
 
 pression, its place can as well be filled by an arch of brick, 
 footed on a pair of cast-iron skew-backs ; and these held 
 in position by a pair of wrought-iron tie-rods, as shown in 
 Fig. 49. This system of construction is preferable to the 
 
 bowstring girder, in that the 
 tie-rods are not liable to injury 
 by " shrinking in," and the 
 cost is less. From Transverse 
 Strains, Art. 596, we have 
 
 D = 
 
 Ul 
 
 9425 d 
 
 (89.) 
 
 FIG. 49. 
 
 an equation in which D is the 
 diameter in inches of each of 
 the two tie-rods of the brick 
 arch ; U is the load in pounds 
 uniformly distributed over the arch ; / is the span of the 
 arch in feet ; and d, in inches, is its versed sine, or its height 
 measured from the centre of the tie-rod to the centre of the 
 thickness or height of the arch at middle. 
 
 This equation may be put in words as follows : 
 Rule LXV. Multiply the weight by the length ; divide 
 the product by 9425 times the depth, and the square root of 
 the quotient will be the diameter of each rod. 
 
 Example. What should be the diameter of each of the 
 pair of tie-rods required to sustain a brick arch 20 feet span 
 from centres, with a versed sine or height at middle of 30 
 inches, to carry a brick wall 12 inches thick and 30 feet high, 
 weighing TOO pounds per cubic foot? The load upon this 
 arch will be for so much of the wall as will occur over the 
 opening, which will be about one foot less than the span of 
 the arch, or 20 i = 19 feet. Therefore, the load will 
 equal 19 x 3ox i x 100 = 57,000. pounds ; and hence, U 
 57000, / = 20, d 30, and, by the rule 
 
 57000 x 20 
 
 __. -- = 1/4-0318 = 2. 008; 
 
 9425 x 30 
 
 or, the diameter of each rod is required to be 2 inches. 
 
STRAINS REPRESENTED GRAPHICALLY. 165 
 
 FRAMED GIRDERS. 
 
 195 Graphic Representation of Strain In the first 
 
 part of this section, commencing at Art. 71, the metfiod was 
 developed of ascertaining the strains in the various parts of 
 a frame by the parallelogram or triangle of forces. The 
 method, so far as there explained, is adequate to solve sim- 
 ple cases ; but when more than three pieces of a frame con- 
 verge in one point, the task bv that method becomes difficult. 
 This difficulty, however, disappears when recourse is had to 
 the method known as that of 4< Reciprocal Figures, Frames, 
 and Diagrams of Forces," proposed by Professor I. Clerk 
 Maxwell in 1867. This is an extension of the method by 
 the triangle of forces, and may be illustrated as follows : 
 
 FIG. 50. FIG. 51. 
 
 Let the lines in Fig. 50 represent, in direction and 
 amount, four converging forces in equilibrium in any frame, 
 as, for example, the truss of a roof; let the lines in Fig. 51 
 be drawn parallel to those in Fig. 50, in the manner fol- 
 lowing, namely : Let the line A B be drawn parallel with the 
 line of Fig. 50 which is between the corresponding letters 
 A and B, and let it be of corresponding length ; from B draw 
 the line B C parallel with the line of Fig. 50 which is be- 
 tween the letters B and C, and of corresponding length ; 
 then from C draw CD, and from A draw A D, respectively 
 parallel with the lines of Fig. 50 designated by the corre- 
 sponding letters, and extend them till they intersect at D. 
 The lengths of these two lines, the last two drawn, are de- 
 termined by the point D where they intersect; their lengths, 
 therefore, need not be previously known,. The lengths of 
 the lines in Fig. 51 are respectively in proportion to the 
 
l66 CONSTRUCTION. 
 
 several strains in Fig. 50, provided these strains are in 
 equilibrium. Fig. 5 1 is termed a closed polygon of forces. 
 A system of such polygons, one for each point, in the frame 
 where forces converge, so constructed that no line repre- 
 senting a force shall be repeated, is termed a diagram of 
 forces. This diagram of forces is a reciprocal of the frame 
 from which it is drawn, its lines and angles being the same. 
 The facility of tracing the forces in the diagram of forces 
 depends materially upon the system of lettering here shown, 
 and which was proposed by Mr. Bow, in his excellent work 
 on the Economics of Construction. In this system each 
 line of the frame is designated by the two letters which it 
 separates ; thus the line between A and B is called line A B ; 
 that between C and D is called line CD ; and so of others ; 
 and in the diagram the corresponding lines are called by the 
 same letters, but here the letters designating the line are, as 
 usual, at the ends of the line. Any point in a frame where 
 forces converge is designated by the several letters which 
 cluster around it; as, for example, in Fig. 50, the point of 
 convergence there shown is designated as point A B C D. 
 
 This invaluable method of denning graphically the 
 strains in the various pieces composing a frame, such as a 
 girder or roof-truss, is remarkably simple, and is of general 
 application. Its utility will now be exemplified in its appli- 
 cation to framed girders, and afterwards to roof-trusses. 
 
 196. Framed 4*irder. Girders of solid timber are use- 
 ful for the support of floors only where posts are admissible 
 as supports, at intervals of from 8 to 15 feet. For unob- 
 structed long spans it becomes requisite to construct a frame 
 to serve as a girder (Arts. 163, 182). A frame of this kind 
 requires two horizontal pieces, a top and a bottom chord, 
 and a system of struts and suspension-pieces by which 
 the top and bottom chords are held in position, and the 
 strains from the load are transmitted to the bearings at the 
 ends of the girders. Various methods of arranging these 
 struts and ties have been proposed. One of the most simple 
 and effective is shown in Fig. 52, forming a series of isos- 
 celes triangles. The proportion between the length and 
 height of a girder is important as an element of economy 
 
RULES FOR FRAMED, GIRDERS. 167 
 
 both of space and cost. When circumstances do not control 
 in limiting- the height, it may be determined by this equation 
 from Transverse Strains, Art. 624 
 
 , (I75+/)/ 
 
 2400 ; (90.) 
 
 in which d is the depth or height between the axes of the 
 top and bottom chords, and / is the length between the cen- 
 tres of bearings at the supports (d and / both in feet). This 
 equation in words is as follows : 
 
 Rule LXVI. To the length add 175 ; multiply the sum 
 by the length ; divide the product by 2400, and the quotient 
 will be the required height between the axes of the top and 
 bottom chords. 
 
 Example. What should be the depth of a girder which 
 is 40 feet long between the centres of action at the supports? 
 For this the rule gives 
 
 (175+40^40 s 
 
 2400 
 
 or, the proper depth for economy of material is 3 feet and 
 7 inches. 
 
 The number of bays, panels, or triangles into which the 
 bottom chord may be divided is a matter of some considera- 
 tion. Usually girders from 
 
 20 to 59 feet long should have 5 bays. 
 
 59 85 " " " " 6 " 
 
 85 " 107 " " " " 7 " 
 
 107 " 127 " " " " 8 " 
 
 127 " 146 " " " 9 
 
 (97. Framed Girder and Diagram of Force. Let Fig. 
 52 represent a framed girder of six bays of, say, n feet 
 each, or of a total length of 66 feet. 
 
 The lines shown are the axial lines, or the imaginary lines 
 passing through the axes of the several pieces composing the 
 frame. The. six arrows indicate the six pressures into which 
 the equally distributed load is supposed to be divided. Each 
 of these is at the apex of a triangle, the base of which 1 
 along the lower chord. 
 
1 68 
 
 CONSTRUCTION. 
 
 The spaces between the arrows are lettered ; so, also, the 
 space between the last arrow at either end and the point of 
 support has a letter, and so has each triangle, and there is 
 one for the space beneath the lower chord. These letters 
 are to be used in describing the diagram of forces, as was 
 explained in Art. 195. The diagram of forces (Fig. 53) for 
 this girder-frame is drawn as follows, namely : Upon a verti- 
 
 FIG. 52. 
 
 cal line A ^Vmark the points A, O, P, Q, R, S, and N, at equal 
 distances, to represent the six equal vertical pressures indi- 
 cated by the arrows in Fig. 52. The equal distances A O, 
 OP, etc., may be made of any convenient size; but it will 
 
 HGrF 
 
 FIG. 53. 
 
 serve to facilitate the measurement of the forces in the dia- 
 gram if they are made by a scale of equal parts, and the 
 number of parts given to each division be made equal to the 
 number of tons of 2000 pounds each which is contained in 
 the pressure indicated by each arrow. On this vertical line 
 the distance A represents the load at the apex of the tri- 
 angle B, or the point A OCB (Art. 195); the distance OP 
 
GRAPHICAL DIAGRAMS OF FORCES. 169 
 
 represents the weight at the second arrow, or at the point 
 O PR D C, and so of the rest. If the weights upon the 
 points in the upper chord had been unequal, then the divi- 
 sion of the vertical line A N would have had to be corre- 
 spondingly unequal, each division being laid off by the scale, 
 to accord with the weight represented by each. The line 
 of loads, A N, being adjusted, the other lines are drawn from 
 it (Art. 195), so as to make a closed polygon for the forces 
 converging at each point of the frame, Fig. 52 commenc- 
 ing with the point A B T, Fig. 52, where there are three 
 forces, namely, the force acting through the inclined strut 
 A B, the horizontal force in B 7", and the vertical reaction 
 A T at the point of support. This last is equal to half the 
 entire load, or equal to the pressure indicated by the three 
 arrows, A O, O P, and P Q, and is represented in Fig. 53 by 
 A Q or A T. From the point Q draw a horizontal line Q B ; 
 this is parallel with the force B T of Fig. 52, in the lower 
 chord. From the point A draw A B parallel with the strut 
 A B of Fig. 52. This line intersects the line B T in B and 
 closes the polygon A B TA ; the point B defines the length 
 of the lines A B and B 7", and these lines measured by the 
 scale by which the line of loads was constructed give the 
 required pressures in the corresponding lines, A B and B T, 
 of Fig. 52. 
 
 Taking next, the point ABCO, where four forces meet, 
 of which we already have two, namely, the force in the 
 strut A B and the load A O from the point O draw the hori- 
 zontal line O C ; this is parallel to the horizontal force O C 
 of Fig. 52. Now from B draw B C parallel with the suspen- 
 sion-piece B C of Fig. 52. This line intersects O C in C, and 
 the point C limits the lines O C and B C and closes the poly- 
 gon A B C O A, the four sides of which are respectively in 
 proportion to the four forces converging at the point A B CO 
 of Fig. 52, and when measured by the scale by which the 
 line of loads was constructed give the required strains re- 
 spectively in each. Taking next the point B C D T, where 
 four forces converge, of which we already have two, B C 
 and B T from B extend the horizontal line TB to D\ from 
 C draw CD parallel with CD of Fig. 52, and extend it to in- 
 tersect TD in D, and thus close the polygon T B C D T. 
 
I/O CONSTRUCTION. 
 
 The lines in a part of this polygon coincide those from 
 B to T\ this is because the two strains B T and D T, Fig. 52, 
 lie in the same horizontal line. Again, taking the point 
 OC D EP, where five forces meet, three of which, O P, O C, 
 and CD, we already have draw from D the line D E parallel 
 with D E of Fig. 52, and from P the line PE horizontally or 
 parallel with P E of Fig. 52. These two lines intersect at E 
 and close the polygon PO C D E P, the sides of which meas- 
 ure the forces converging in the point PO CD E, Fig: 52. 
 Next in order is the point D E F T, Fig. 52, where four forces 
 meet, two of which, T D and D E, are known. From E draw 
 EF parallel with EF'm Fig. 52; and from 7 1 , TF parallel 
 with TFin Fig. 52 ; these two lines meet in F and close the 
 polygon TD E F T, the sides of which measure the required 
 strains in the lines converging at the point DEFT, Fig. 52. 
 Taking next the point PEFG Q 9 Fig. 52, where five forces 
 meet, of which we already have three, QP, P,and E F 
 Irom F draw a line parallel with F G of Fig. 52, and from Q 
 a line parallel with Q G of Fig. 52. These two intersect at G 
 and complete the polygon QPEFG Q, the lines of which 
 measure the forces converging at PEFG Q in Fig. 52. 
 
 In this last polygon, a peculiarity seems to indicate an 
 error: the line FG has no length ; it begins and ends at the 
 same point ; or, rather, the polygon is complete without it. 
 This is easily understood when it is considered that the two 
 lines FG and G H do not contribute any strength towards 
 sustaining the loads P Q and QR, and in so far as these 
 weights are concerned they might be dispensed with, and 
 the space occupied by the three triangles F , G, and H left 
 free, and be designated by only one letter instead of three. 
 Thus it appears that there are only four instead of five forces 
 at the point PEFG Q, and that the four are represented by 
 the lines of the polygon QPEFQ. 
 
 The peculiarity above explained arises from considering 
 loads only on the top chord : the analysis of the case is cor- 
 rect as worked from the premises given ; but in practice 
 there is always more or less load on the bottom chord at the 
 middle, which should be considered. This will be included 
 in a case proposed in the next article. One half of the dia- 
 
LOAD ON BOTH CHORDS. 
 
 I/I 
 
 gram of forces is now complete. The other half being ex- 
 actly the same, except that it is in reversed order, need not 
 here be drawn. 
 
 198. Framed Girders: Load on Both Chords Let 
 
 Fig. 54 represent the axial lines of a girder carrying an 
 
 FIG. 54. 
 
 equally distributed load on each chord, represented by the 
 arrows and balls shown in the figure. Let each bay measure 
 IO feet, or the length ot the girder be 50 feet, and its height 
 
 /\7 
 
 XXX 
 
 M 
 
 FIG. 55- 
 
 be 4J- feet. The diagram of forces (Fig. 55) for this girder 
 is obtained thus : 
 
 The plan of the girder, Fig. 54> requires to be lettered 
 as shown ; having one letter within each panel and outside 
 the frame, and one between every two weights or strains. 
 Then, in Fig. 55, mark the vertical line K V at L, M, N, V, 
 
1 72 CONSTRUCTION. 
 
 and P t dividing it by scale into equal parts, corresponding 
 with the weights on the top chord represented by the ar- 
 rows. For example, if the load at each arrow equals 6J 
 tons, make K L, L M, M N, etc., each equal to 6^ parts of the 
 scale. Then K P will equal the total load on the top flange. 
 Make the distance P V equal to the sum of the loads on the 
 bottom chord. Then K V equals the total load on the gir- 
 der. Bisect K Fin U\ then K U or U Fequals half the total 
 load ; consequently, equals the reaction of the bearing at K 
 or P of Fig. 54. 
 
 Now, to obtain the polygon of forces converging at 
 K A U, Fig. 54, we have one of these forces, K U, or the re- 
 action of the bearing at KA U, equal to K U, Fig. 55. From 
 f/draw UA parallel with U A of Fig. 54, and from 1C draw 
 KA parallel with the strut KA, Fig. 54, and intersecting the 
 line UA at A, a point which marks the limit of K A and UA, 
 and closes the polygon K A UK, the sides of which are in 
 proportion respectively to the three strains which converge 
 at the point A UK, Fig. 54. For example, since the line K U 
 by scale measures the vertical reaction, K U, of the bearing 
 at A UK, Fig. 54, therefore the line K A of the diagram of 
 forces by the same scale measures the strain in the strut KA, 
 Fig. 54, and the line A U of the diagram by the same scale 
 measures the strain in the bottom chord at A U, Fig. 54. For 
 the strains converging at K A B L, Fig. 54, of which two, 
 KA and K L, are already known, we draw from A the line 
 A B parallel with the line A B, Fig. 54, and from L draw L B 
 parallel with L B, Fig. 54, meeting A B at B, a point which 
 limits the two lines and closes the polygon K A B L K, the 
 lines of which are in proportion respectively to the strains 
 converging at the point KA B L, Fig. 54, as before explained. 
 Of the five strains converging at U A B C T, we already have 
 three T U, UA,and AB* to obtain the other two, make 
 UQ equal to PV, equal to the total load upon the lower 
 flange ; divide U Q into four equal parts, QR, RS, S T, and 
 T U, corresponding with the four weights on the lower 
 chord, and represented by the four balls, Fig. 54. Now, 
 from T, the point marking the first of these divisions, draw 
 TC parallel with T C, Fig. 54, and from B draw B C paral- 
 
VARIOUS STRAINS IN FRAMED GIRDERS. 1/3 
 
 lei with the strut EC, Fig. 54, meeting TC in C, a point 
 which limits the lines B C and TC and closes the polygon 
 T U A B C T, the sides of which are in proportion respectively 
 to the strains converging in the point T U A B C T, Fig. 54. 
 Of the five forces converging at MLB CD, we already have 
 three ML, LB, and B C\ to obtain the other two, from M 
 draw M D parallel with M D, Fig. 54, and from C draw CD 
 parallel with CD, Fig. 54, meeting MD at D, a point limit- 
 ing the lines M D and CD and closing the polygon 
 MLBCDM,i\\Q sides of which are in proportion to the 
 strains converging at the point MLB CD, Fig. 54. Of the 
 five forces converging at the point 5 TCD E, three S T, 
 T C, and CD are known; to obtain the other two, from 5 
 draw SE parallel with SE, Fig. 54, and from D draw D E, 
 parallel with the strut D E, Fig. 54, meeting the line SE in 
 E t a point limiting the two lines S E and D E and closing the 
 polygon 5 TCD E S, the sides of which are in proportion to 
 the strains converging at 5 TCDE, Fig. 54. One half of the 
 strains in Fig. 54 are now shown in its diagram of forces, Fig. 
 55 ; and since the two halves of the girder are symmetrical, 
 the forces in one half corresponding to those in the other, 
 hence the lines of the diagram for one half of the forces 
 may be used for the corresponding forces of the other half. 
 
 199. Framed Girders : Dimensions of Parl. The 
 
 parts of a framed girder are the two horizontal chords (top 
 and bottom) and the diagonals the struts and ties. The top 
 chord is in a state of compression, while the bottom chord 
 experiences a tensile strain. Those of the diagonal pieces 
 which have a direction from the top to the bottom chord, 
 and from the middle towards one of the bearings of the 
 girder, as KA, B C, or D E, Fig. 54, are struts, and are sub- 
 jected to compression. The diagonal pieces which have a 
 direction from the bottom to the top chord, and from the 
 middle towards one of the supports, as A B or CD, Fig. 54. 
 are ties, and are subjected to extension, (Art. 83). The 
 amount of strain in each piece in a framed girder having 
 been ascertained in a diagram of forces, as shown in Arts. 
 197 and 198, the dimensions of each piece may be obtained 
 
174 CONST RUCTION. 
 
 by rules already given. The dimensions of the pieces in a 
 state of compression are to be ascertained by the rules for 
 posts in Arts. 107 to 114, and those in a state of tension by 
 A: m ts. 117 to 119 (see Arts. 226 to 229). Care is required, in 
 obtaining the size of the lower chord, to allow for the joints 
 which necessarily occur in long ties, for the reason that tim- 
 ber is not readily obtained sufficiently long without splicing. 
 Usually, in cases where the length of the girder is too great 
 to obtain a bottom chord in one piece, the chord is made up 
 of vertical lamina, and in as long lengths as practicable, and 
 secured with bolts. A chord thus made will usually require 
 about twice the material ; or, its sectional area of cross-sec- 
 tion will require to be twice the size of a chord which is in 
 one whole piece ; and in this chord it is usual to put the fac- 
 tor of safety at from 8 to 10. 
 
 The diagonal ties are usually made of wrought iron, and 
 it is well to secure the struts, especially the end ones, with 
 iron stirrups and bolts. And, to prevent the evil effects of 
 shrinkage, it is well to provide iron bearings extending 
 through the depth of each chord, so shaped that the struts 
 and rods may have their bearings upon it, instead of upon 
 the wood. 
 
 PARTITIONS. 
 
 200 Partitions Such partitions as are required for 
 the divisions in ordinary houses are usually formed by tim- 
 ber of small size, termed studs or joists. These are placed 
 upright at 12 or 16 inches from centres, and Avell nailed. 
 Upon these studs lath are nailed, and these are covered 
 with plastering. The strength of the plastering depends in 
 a great measure upon the clinch iormed by the mortar which 
 has been pressed through between the lath. That this 
 clinch may be interfered with in the least possible degree, it 
 is proper that the edges of the partition-joists which are 
 presented to receive the lath should be as narrow as prac- 
 ticable ; those which are necessarily large should be reduced 
 by chamfering the corners. The derangements in floors, 
 plastering, and doors which too frequently disfigure the 
 interior of pretentious houses with gaping cracks in the 
 
FRAMED PARTITIONS. 
 
 175 
 
 plastering and in the door-casings are due in nearly all cases 
 to defective partitions, and to the shrinkage of floor-timbers. 
 A plastered partition is too heavy to be trusted upon an ordi- 
 nary tier of beams, unless so braced as to prevent its weight 
 from pressing upon the beams. This precaution becomes es- 
 pecially important when, in addition to its own weight, the 
 partition serves as a girder to carry the weight of the floor- 
 beams next above it. In order to reduce to the smallest 
 practicable degree the derangements named, it is important 
 that the studs in a partition should be trussed or braced so 
 as to throw the weight upon firmly sustained points in the 
 construction beneath, and that the timber in both partitions 
 and floors should be well seasoned and carefully framed. 
 To avoid the settlement due to the shrinkage of a tier of 
 beams, it is important, in a partition standing over one in the 
 story below or over a girder, that the studs pass between 
 the beams to the plate of the lower partition, or to the 
 girder ; and, to be able to do this, it is also important to ar- 
 range the partitions of the several stories vertically over 
 each other. All principal partitions should be of brick, 
 especially such as are required to assist in sustaining the 
 floors of the building. 
 
 FIG. 56. 
 
 201. Examples of Partition &.Fig. 56 represents a par- 
 tition having a door in the middle. Its construction is simple 
 but effective. Fig. 57 shows the manner of constructing a 
 
I 7 6 
 
 CONSTRUCTION. 
 
 partition having- doors near the ends. The truss is formed 
 above the door-heads, and the lower parts are suspended 
 from it. The posts a and b are halved, and nailed to the 
 tie c d and the sill ef. The braces in a trussed partition 
 
 r 
 
 a b 
 
 FIG 57- 
 
 should be placed so as to form, as near as possible, an angle 
 of 40 degrees with the horizon. The braces in a partition 
 should be so placed as to discharge the weight upon the 
 
 FIG. 58. 
 
 points of support. All oblique pieces that fail to do this 
 should be omitted. 
 
 When the principal timbers of a partition require to be 
 large for the purpose of greater strength, it is a good plan 
 
WEIGHT UPON PARTITIONS. 177 
 
 to omit the upright filling-in pieces, and in their stead to 
 place a few horizontal pieces, as in Fig. 58, in order that upon 
 these and the principal timbers upright battens may be 
 nailed at the proper distances for lathing. A partition thus 
 constructed requires a little more space than others ; but it 
 has the advantage of insuring greater stability to the plas- 
 tering, and also of preventing to a good degree the conver- 
 sation of one room from being overheard in the adjoining 
 one. Ordinary partitions are constructed with 3x4, 3x5, 
 or 4x6 inch joists, for the principal pieces, and with 2x4, 
 2x5, or 2x6 filling-in studs, well strutted at intervals of 
 about 5 feet. When a partition is required to support, in 
 addition to its own weight, that of a floor or some other 
 burden resting upon it, the dimensions of the timbers should 
 be ascertained, by applying the principles which regulate 
 the laws of pressure and those of the resistance of timber, 
 as explained in the first part of this section, and in Arts. 196 
 to 199 for framed girders. The following data may assist in 
 calculating the amount of pressure upon partitions : 
 
 White-pine timber weighs from 22 to 32 pounds per cubic 
 loot, varying in accordance with the amount of seasoning it 
 has had. Assuming it to weigh 30 pounds, the weight of 
 the beams and floor-plank in every superficial foot of the 
 flooring will be 
 
 6 pounds when the beams are 3 x 8 inches, and placed 20 inches from centres 
 7\ " " " 3 x 10 " " 18 
 
 9 " " 3 x 12 " " 16 
 
 II " 3 XI2 " 12 
 
 I 3 " " " 4X12 " " 12 
 
 I 3 4x14 " 14 
 
 In addition to the beams and plank, there is generally the 
 plastering, of the ceiling of the apartments beneath, and some- 
 times the deafening. Plastering may be assumed to weigh 9 
 pounds per superficial foot, and deafening 1 1 pounds. 
 
 Hemlock weighs about the same as white pine. A parti- 
 tion of 3x4 joists of hemlock, set 12 inches from centres, 
 therefore, will weigh about 2j pounds per foot superficial 
 and when plastered on both sides, 2o pounds. 
 
CONSTRUCTION. 
 
 ROOFS. 
 
 202. Roof*. 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 of resisting the 
 lateral pressure exerted by the rafters. But in modern 
 buildings, usually the walls are so slightly built as to be in- 
 capable of resisting much if any oblique pressure ; hence 
 the necessity of care in constructing the roof so as to avoid 
 oblique and lateral strains. The roof so constructed, instead 
 of tending to separate the walls, will bind and steady them. 
 
 FIG. 59. 
 
 FIG. 60. 
 
 FIG. 61. 
 
 FIG. 62. 
 
 FIG. 63. 
 
 FIG. 64, 
 
 FIG. 65. 
 
 FIG. 66. 
 
 FIG. 67. 
 
 203. Comparison of Roof-Truses. 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 horizontal tie-beam. In the 
 latter group, and in all designs similarly destitute of the 
 horizontal tie at the foot of the rafters,- the strains are much 
 greater than in those having the tie, unless the truss be pro- 
 
VARIOUS FORMS OF ROOF-TRUSSES. 179 
 
 tected by exterior resistance, such as may be afforded by 
 competent buttresses. 
 
 To the uninitiated it may appear preferable, in Fig. 64, 
 to extend the inclined ties to the rafters, as shown by the 
 dotted lines. But this would not be beneficial ; on the con- 
 trary, it would be injurious. The point of the rafter where 
 the tie would be attached is near the middle of its length, 
 and consequently is a point the least capable of resisting 
 transverse strains. The weight of the roofing itself tends to 
 bend the rafter ; and the inclined tie, were it attached to the 
 rafter, would, by its tension, have a tendency to increase this 
 bending. As a necessary consequence, the feet of the rafters 
 would separate, and the ridge descend. 
 
 In Fig. 65 the inclined ties are extended to the rafters ; 
 but here the horizontal strut or straining beam, located at 
 the points of contact between the ties and rafters, counteracts 
 the bending tendency of the rafters and renders these points 
 stable. In this design, therefore, and only in such designs, it 
 is permissible to extend the ties through to the rafters. 
 Even here it is not advisable to do so, because of the in- 
 creased strain produced. (See Figs. 77 and 79.) The design 
 in Fig. 64, 66, or 67 is to be preferred to that in Fig. 65. 
 
 204. Force Diagram : L,oacl upon Each Support. By a 
 
 comparison of the force diagrams hereinafter given, of each 
 of the foregoing designs, we* may see that the strains in the 
 trusses without horizontal tie-beams at the feet of the rafters 
 are greatly in excess of those having the tie. In constructing 
 these diagrams, the first step is to ascertain the reaction of, 
 or load carried by, each of the supports at the ends of the 
 truss. In symmetrically loaded trusses, the weight upon 
 each support is always just one half of the whole load. 
 
 205. Force Diagram for Trus in Fig. 59. To obtain 
 the force diagram appropriate to the design in Fig. 59, first 
 letter the figure as directed in Art. 195, and as in Fig. 68. 
 Then draw a vertical line, EF (Fig. 69), equal to the weight W 
 at the apex of roof ; or (which is the same thing in effect) 
 equal to the sum of the two loads of the roof, one extending 
 on each side of W half-way to the foot of the rafter. Di- 
 
1 8o 
 
 CONSTRUCTION. 
 
 vide E F into two equal parts at G. Make G C and G D eacTi 
 equal to one half of the weight N. Now, since G is equal 
 to one half of the upper load, and G D to one half of the low- 
 er load, therefore their sum, E G + G D ED, is equal to 
 one half of the total load, or to the reaction of each support, 
 E or F. From D draw DA parallel with DA of Fig. 68, and 
 from E draw EA parallel with EA of Fig. 68. The three 
 lines of the triangle A E D represent the strains, respectively, 
 in the three lines converging at the point A D E of Fig. 68. 
 Draw the other lines of the diagram parallel with the lines of 
 
 c- 
 
 FIG. 69. 
 
 Fig. 68, and as directed in Arts. 195 and 197. The various 
 lines of Fig. 69 will represent the forces in the corresponding 
 lines of Fig. 68 ; bearing in mind (Art. 195.) that while a line 
 in the force diagram is designated in the usual manner by the 
 letters at the two ends of it, a line of the frame diagram is 
 designated by the two letters between which it passes. Thus, 
 the horizontal lines A D, the vertical lines A B, and the in- 
 clined lines A E have these letters at their ends in Fig. 69, 
 while they pass between these letters in Fig. 68. 
 
 206. Force Diagram for Truss in Fig. 60. For this truss 
 we have, in Fig. 70, a like design, repeated and lettered as 
 required. We here have one load on the tie-beam, and three 
 loads above the truss: one on each rafter and one at the 
 ridge. In the force diagram, Fig. 71, make G H, H J, and 
 J K, by any convenient scale, equal respectively to the 
 weights GH, HJ, and J K oi Fig. 70. Divide G K into two 
 equal parts at L. Make LE and ZFeach equal to one half 
 the weight E F (Fig. 70). Then G Fis equal to one half the 
 
FORCE DIAGRAMS OF TRUSSES. jgl 
 
 total load, or to the load upon the support G (Art. 205). 
 Complete the diagram by drawing its several lines parallel 
 with the lines of Fig. 70, as indicated by the letters (see Art. 
 205), commencing with G F, the load on the support G (Fig. 
 70). Draw from F and G the two lines FA and GA paral- 
 lel with these lines in Fig. 70. Their point of intersection 
 defines the point A. From this the several points B, C, and 
 D are developed, and the figure completed. Then the lines 
 m Fig. 71 will represent the forces in the corresponding lines 
 of Fig. 70, as indicated by the lettering. (See Art. 195.) 
 
 K A 
 
 FIG. 70. 
 
 FIG. 71. 
 
 207. Force Diagram for Tru 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 ap- 
 propriate to it. 
 
 In the construction of this diagram, proceed as directed 
 in the previous example, by first constructing N S, the ver- 
 tical line of weights ; in which line NO y OP,PQ,QR, and R S 
 are made respectively equal to the several weights above 
 the truss in Fig. 72. Then divide NS into two equal parts at T. 
 Make 7^ and TL each equal to the half of the weight K L. 
 Make J K and L M equal to the weights J K and L M of Fig. 
 72. Now, since M " N is equal to one half of the weights above 
 the truss plus one half of the weights below the truss, or half of 
 the whole weight, it is therefore the weight upon the support 
 N (Fig. 72), and represents the reaction of that support. A 
 horizontal line drawn from M will meet the inclined line 
 drawn from N, parallel with the rafter A N (Fig. 72), in the 
 
1-82 
 
 CONSTRUCTION. 
 
 point^, and the three sides of the triangle A MN, Fig. 73, will 
 give the strains in the three corresponding lines meeting at 
 the point A MN, Fig. 72. The sides of the triangle HJ 5, Fig. 
 
 FIG. 72. 
 
 73, give likewise the strains in the three corresponding lines 
 meeting at the point H J 5, Fig. 72. Continuing the con- 
 struction, draw all the other lines of the force diagram parallel 
 
 FIG. 73. 
 
 with the corresponding lines of Fig. 72, and as directed in 
 Art. 195. The completed diagram will measure the strains 
 in all the lines of Fig. 72. 
 
FORCE DIAGRAMS CONTINUED. 183 
 
 208, Force Diagram for Truss in Fig. 63. The roof 
 truss indicated at Fig. 63 is repeated in Fig. 74, with the ad 
 
 FIG. 74. 
 
 FIG. 75- 
 
 dition of the lettering required for the construction of 
 force diagram, Fig. 75. 
 
1 84 CONSTRUCTION. . 
 
 In this case there are seven weights, or loads, above the 
 truss, and three below. Divide the vertical line O V at W 
 into two equal parts, and place the lower loads in two equal 
 parts on each side of W. Owing to the middle one of these 
 loads not being on the tie-beam with the other two, but on 
 the upper tie-beam, the line G H, its representative in the 
 force diagram, has to be removed to the vertical BJ, and 
 the letter M is duplicated. The line NO equals half the 
 whole weight of the truss, or 3^ of the upper loads, plus one 
 of the lower loads, plus half of the load at the upper tie- 
 beam. It is, therefore, the true reaction of the support NO, 
 and A N is the horizontal strain in the beam there. It will 
 be observed also that while H ' M and G M (Fig. 75), which 
 are equal lines, show the strain in the lower tie-beam at the 
 middle of the truss, the lines C H and FG, also equal 'but 
 considerably shorter lines, show the strains in the upper 
 tie-beam. Ordinarily, in a truss of this design, the strain in 
 the upper beam would be equal to that in the lower one, 
 which becomes true when the rafters and braces above the 
 upper beam are omitted. In the present case, the thrusts of 
 the upper rafters produce tension in the upper beam equal 
 to C M or F M of Fig. 75, and thus, by counteracting the 
 compression in the beam, reduce it to C H or FG of the 
 force diagram, as shown. 
 
 209. Force Diagram for Truss in Fig. 64. The force 
 diagram for the roof-truss at Fig. 64 is given in Fig". 77, 
 while Fig. 78 is the truss reproduced, with the lettering 
 requisite for the construction of Fig. 77. 
 
 The vertical E F (Fig. 77) represents the load at the 
 ridge. Divide this equally at W, and place half the lower 
 weight each side of W, so that CD equals the lower 
 weight. Then ED is equal to half the whole load, and 
 equal to the reaction of the support E (Fig. 76). The lines 
 in the triangle A D E give the strains in the corresponding 
 lines converging at the point A D E of Fig. 76. The other 
 lines, according to the lettering, give the strains in the cor- 
 responding lines of the truss. (See Art. 195.) 
 
FORCE DIAGRAMS CONTINUED. 
 
 i8 5 
 
 210. Force Diagram for Trus in Fig. 65. This truss is 
 reproduced in Fig. 78, with the letters proper for use in the 
 force diagram, Fig. 79. 
 
 < 
 
 FIG. 79. 
 
 Here the vertical G K, containing the three upper loads 
 GH, HJ, and J K, is divided equally at W, and the lower 
 
1 86 CONSTRUCTION. 
 
 load E F is placed half on each side of W, and extends from E 
 to F. Then FG represents one half of the whole load of the 
 truss, and therefore the reaction of the support G (Fig. 78). 
 Drawing the several lines of Fig. 79 parallel with the corre- 
 sponding lines of Fig. 78, the force diagram is complete, and 
 the strains in the several lines of 78 are measured by the cor- 
 responding lines of 79. (See Art. 195.) 
 
 A comparison of the force diagram of the truss in Fig. 76 
 with that of the truss in Fig. 78 shows much greater strains 
 in the latter, and we thus see that Fig. 76 or 64 is the more 
 economical form. 
 
 FIG. 80. 
 
 211. Force Diagram for Truss in Fig. 66. This truss is 
 reproduced and prepared by proper lettering in Fig. 80, and 
 its force diagram is given in Fig. 81. 
 
 Here the vertical J M contains the three upper loads 
 JK, KL, and LM. Divide 7 J/ into two equal parts at 
 G, and make FG and G H respectively equal to the two 
 loads FG and GH of Fig.Zo. Then HJ represents one 
 half of the whole weight of the truss, and therefore the reac- 
 tion of the support J. From H and J draw lines par- 
 allel with A H and A J of Fig. 80, and the sides of the tri- 
 angle A H J will give the strains in the three lines concen- 
 trating in the point A H J (Fig. 80). The other lines of Fig, 
 
EFFECT OF ELEVATING THE TIE-BEAM. 187 
 
 8 1 are all drawn parallel with their corresponding lines in 
 Fig. 80, as indicated by the lettering. (See Art. 195.) 
 
 FIG. 81. 
 
 212 Roof-Truss: Effect of Elevating tbe Tie-Beam. 
 
 From Arts. 670, 671, Transverse Strains, it appears that the 
 
 FIG. 83 
 
 effect of substituting inclined ties for the horizontal tie at 
 feet of rafters is 
 
1 88 CONSTRUCTION. 
 
 in which P represents half the weight of the whole truss and 
 the load upon it ; a + & = height of the truss at middle above 
 a horizontal line drawn at the feet of the rafters ; a equals 
 the height from this line to the point where the two inclined 
 ties meet ; b, the height thence to the top of the truss ; and 
 F, the additional vertical strain at the middle of the truss 
 due to elevating the tie from a horizontal line. 
 
 Examples are given to show that when the elevation of 
 the tie equals i of the whole height, the vertical strain there- 
 by induced is equal to a weight which equals \ of half the 
 whole load ; and that when the elevation equals half the 
 whole height, the vertical strain is equal to half the whole 
 load. This is the strain in the vertical rod at middle. The 
 strains in the rafters and inclined ties are proportionately 
 increased. 
 
 213. Planning a Roof. In designing a roof for a build- 
 ing, the first point requiring attention is the location of the 
 trusses. These should be so placed as to secure solid bear- 
 ings upon the walls ; care being taken not to place either of 
 the trusses over an opening, such as those for windows or 
 doors, in the wall below. Ordinarily, trusses are placed so 
 as to be centrally over the piers between the windows ; the 
 number of windows consequently ruling in determining the 
 number of trusses and their distances from centres. This 
 distance should be from ten to twenty feet ; fifteen feet apart 
 being a suitable medium distance. The farther apart the 
 trusses are placed, the more they will have to carry ; not 
 only in having a larger surface to support, but also in that 
 the roof-timbers will be heavier ; for the size and weight, of 
 the roof-beams will increase with the span over which they 
 have to reach. 
 
 In the roof-covering itself, the roof-planking may be laid 
 upon jack-rafters, carried by purlins supported by the 
 trusses ; or upon roof-beams laid directly upon the back of 
 the principal rafters in the trusses. In either case, proper 
 
LOAD UPON ROOFS. 189 
 
 struts should be provided, and set at proper intervals to re- 
 sist the bending of the rafter. Jn case purlins are used, one of 
 these struts should be placed at the location of each purlin. 
 
 The number of these points of support rules largely in 
 determining the design for the truss, thus : 
 
 For a short span, where the rafter will not require sup- 
 port at an intermediate point, Fig. 59 or 64 will be proper. 
 
 For a span in which the rafter requires supporting at one 
 intermediate point, take Fig. 60, 65, or 66. 
 
 For a span with two intermediate points of support for 
 the rafter, take Fig. 61 or 67. 
 
 For a span with three intermediate points, take Fig. 63. 
 
 Generally, it is found convenient to locate these points of 
 support at nine to twelve feet apart. They should be suffi- 
 ciently close to make it certain that the rafter will not be sub- 
 ject to the possibility of bending. 
 
 214. Load upon Roof-Tru. 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 strain, or 
 loading, which will occur at every such point. 
 
 The points of support along the rafters will be required 
 to sustain the roofing timbers, the planking, the slating, the 
 snow, and the force of the wind. The points along the tie- 
 beam will have to sustain the weight of the ceiling and the 
 flooring of a loft within the roof, if there be one, together 
 with the loading upon this floor. The weight of the truss 
 itself must be added to the weight of roof and ceiling. 
 
 215,-lLoacl on Roof per Superficial Foot In any im- 
 portant work, each of the items in Art. 214 should be care- 
 fully estimated, in making up the load to be carried. For 
 ordinary roofs, the weights may be taken per foot superficial, 
 as follows : 
 
 Slate, about 7-0 pounds. 
 
 Roof-plank, 
 
 Roof-beams or jack-rafters, 2 - 3 
 
 In all, I2 pounds. 
 
CONSTRUCTION. 
 
 This is for the superficial foot of the inclined roof. For the 
 foot horizontal, the augmentation of load due to the angle of 
 the roof will be in proportion to its steepness. In ordinary 
 cases, the twelve pounds of the inclined surface will not be 
 far from fifteen pounds upon the horizontal foot. 
 For the roof-load we may take as follows : 
 
 Roofing, about 15 pounds. 
 
 Roof-truss, " 5 
 
 Snow, " 20 " 
 
 Wind, " 10 . 
 
 Total on roof, 50 pounds 
 
 per square foot horizontal. 
 
 This estimate is for a roof of moderate inclination, say 
 one in which the height does not exceed J of the span. 
 Upon a steeper roof the snow would not gather so heavily, 
 but the wind, on the contrary, would exert a greater force. 
 Again, the wind acting on one side of a roof may drift the 
 snow from that side, and perhaps add it to that already 
 lodged upon the opposite side. These two, the wind and 
 the snow, are compensating forces. The action of the snow 
 is vertical : that of the wind is horizontal, or nearly so. The 
 power of the wind in this latitude is not more than thirty 
 pounds upon a superficial foot of a vertical surface ; except, 
 perhaps, on elevated places, as mountain-tops for example, 
 where it should be taken as high as fifty pounds per foot of 
 vertical surface. 
 
 216. Load upon Tic-Beam. The load upon the tie- 
 beam must of course be estimated according to the require- 
 ments 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 
 the timber struts resting upon it. If there is to be a ceiling 
 attached to the tie-beam, the weight to be added will be in 
 accordance with the material composing the ceiling. If of 
 wood, it need not weigh more than two or three pounds per 
 foot. If of lath and plaster, it will weigh about nine pounds ; 
 and if of iron, from ten to fifteen pounds, according to the 
 
WEIGHT UPON ROOFS, IN DETAIL. 191 
 
 thickness of the metal. Again, if there is to be a loft in the 
 roof, the requisite flooring may be taken at five pounds, and 
 the load upon the floor at from twenty-five to seventy 
 pounds, according to the purpose for which it is to be used. 
 
 217 Roof- Weights in Beta51. 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 fehe slope of the 
 roof. In a severe climate, roofs ought to be constructed 
 steeper than in a milder one, in order that snow may have a 
 tendency to slide off before it becomes of sufficient weight 
 to endanger the safety of the roof. In selecting the material 
 with which -the roof is to be covered, regard should be had 
 to the requirements of the inclination : slate and shingles 
 cannot be used safely on roofs of small rise. The smallest 
 inclination of the various kinds of covering is here given, 
 together with the weight per superficial foot of each. 
 
 MATERIAL. 
 
 Least Inclination. 
 
 Weight upon a 
 square foot. 
 
 Tin ; 
 
 F 
 
 se i inch to 
 
 a f c 
 
 or 
 
 1 to ii 1 
 
 3S. 
 
 CopDcr 
 
 
 i " 
 
 
 
 i to i^ 
 
 
 v^uppci 
 
 Lead .... . . 
 
 
 2 inches 
 
 
 
 4 to 7 
 
 < 
 
 Zinc 
 
 
 3 " 
 
 
 
 i to 2 
 
 M 
 
 Short pine shingles 
 
 
 5 " 
 
 
 
 l tO 2 
 
 w 
 
 Long cypress shingles . . 
 
 
 ' 6 " 
 
 
 
 2 to 3 
 
 
 
 Slate 
 
 
 6 " 
 
 
 
 5 to 9 
 
 < 
 
 
 
 
 
 
 
 
 The weight of the covering as here estimated includes 
 the weight of whatever is used to fix it in place, such as 
 nails, etc. The weight of that which the covering is laid 
 upon, such as plank, boards, or lath, is not included. The 
 weight of plank is about 3 pounds per foot superficial ; of 
 boards, 2 pounds ; and lath, about half a pound. 
 
 Generally, for a slate roof, the weight of the covering, in- 
 cluding plank and jack-rafters, amounts to about 12 pounds, 
 as stated in Art. 215 ; but in every case, the weight of each 
 article of the covering should be estimated, and the full load 
 ascertained by summing up these weights. 
 
1 92 CONSTRUCTION. 
 
 218. Load per Foot Horizontal. The weight ot 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. The 
 horizontal measure of one foot of the inclined surface is 
 equal to the cosine of the angle of inclination. Then, to ob- 
 tain the inclined measure corresponding to one foot horizon- 
 tal, we have 
 
 cos. : I : : p : C = 
 
 cos. 
 
 where / represents the pressure on a foot of the inclined 
 surface, and C the weight of so much of the inclined cover- 
 ing as corresponds to one foot horizontal. The cosine of an 
 angle is equal to the base of the right-angled triangle divided 
 by the hypothenuse (see Trigonometrical Terms, Art. 474), 
 which in this case is half the span divided by the length of 
 
 the rafter, or -. , where s is the span, and / the length of the 
 rafter. Hence, the load per foot horizontal equals 
 
 p p _2 Ip 
 
 ^c^sT^T 1 ~T~ (92.) 
 
 2/ 
 
 or, twice the pressure per foot of inclined surface multiplied 
 by the length of the rafter and divided by the span, both in 
 feet, will give the weight per foot measured horizontally. 
 
 219. WeigBit of Tru. 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 foot horizontal to be 
 allowed for a wooden truss with iron suspension-rods and a 
 horizontal tie-beam, near enough for the requirements of our 
 present purpose ; where s- equals the length or span -of the 
 
EFFECT OF WIND ON ROOFS. 193 
 
 truss, and C the weight per foot horizontal of the roof cover- 
 ing, as in equation (92.). Substituting for Cits value, as in 
 (92.), we have 
 
 T= 0-0077^; 
 or 
 
 T = 0-0154 lp\ (93.) 
 
 which equals the weight in pounds per foot horizontal to 
 be allowed for the truss. 
 
 220. Weight of Snow on Rooffc. 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 incli- 
 nation, snow, if deposited in the absence, of wind, will not 
 slide off ; at least until after it has acquired some depth, and 
 then the tendency to slide will be in proportion to the angle 
 of inclination. The weight of snow may be taken, therefore, 
 at its weight per cubic foot (8 pounds) multiplied by the 
 depth it is usual for it to acquire. This, in the latitude of 
 New York, may be taken at about 2-J- feet. Its. weight 
 would, therefore, be 20 pounds per foot superficial, meas- 
 ured horizontally. 
 
 221. Effect of Wind on Roof*. 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 
 not exceed 30 pounds per superficial foot except on ele- 
 vated places, where it sometimes reaches 50 pounds or more. 
 This is the pressure upon a vertical surface ; roofs, however, 
 generally present to the wind an inclined surface. The ef- 
 fect of a horizontal force on an inclined surface is in pro- 
 portion to the sine of the angle of inclination ; the direction 
 of this effect being at right angles to the inclined surface. 
 The force thus acting may be resolved into forces acting in 
 two directions the one horizontal, the other vertical ; the 
 former tending, in the case of a roof, to thrust aside the walls 
 
194 
 
 CONSTRUCTION. 
 
 on which the roof rests, and the latter acting directly on the 
 materials of which the roof is constructed this latter force 
 being in proportion to the sine of the angle of inclination 
 multiplied by the cosine. This will be made clear by the 
 
 following explanation. Re- 
 ferring to Fig. 83, let D KE 
 be the angle of inclination 
 of the roof, D E being equal 
 to one foot. Bisect DK at 
 A ; draw A L parallel with 
 FIG. 83. EK\ make A L equal to the 
 
 horizontal pressure of the wind upon one foot superficial of 
 a vertical plane. Draw A C perpendicular to D K, and LF 
 parallel with A C from F draw FC parallel with EK\ draw 
 A B parallel with D E. The sides of the triangle LA F rep- 
 resent the three several forces in equilibrium : LA is the 
 force of the wind ; L F is the pressure upon the roof ; and 
 A F is the force with which the wind moves on up the roof 
 towards D. Now, to find the relation of the force of the 
 wind to the strain produced by it in the direction A C, we 
 have 
 
 rad. 
 
 rad. : sin. \\FC\AC\ 
 F C = LA; therefore 
 : sin. : : L A : A C L A sin. ; 
 AC = Fsin.- t 
 
 or, the strain perpendicular to the surface of the roof equals 
 the force of the wind multiplied by the sine of the angle of 
 inclination. When A C represents this strain, then, of the 
 two forces referred to above, B C represents the horizontal 
 force, and A B the vertical force. To obtain this last force, 
 we have 
 
 rad. : cos. \\AC\AB. 
 Putting for A C its value as above, we have 
 
 rad. : cos. : : /^sin. : A B = F sin. cos.; 
 V = F sin. cos. ; 
 
FORCE OF THE WIND. 195 
 
 or, the vertical effect is equal to the product of the force of 
 the wind upon a superficial foot into the sine and the cosine 
 of the angle of inclination. This result is that which is due 
 to the pressure of the wind upon so much of the inclined 
 surface as is covered by one square foot of a vertical sur- 
 face. The wind, acting horizontally through one foot super- 
 ficial of vertical section, acts on an area of inclined surface 
 equal to the reciprocal of the sine of inclination, and the 
 horizontal measurement of this inclined surface is equal to 
 the cosine of the angle of inclination divided by the sine. 
 This may be illustrated from Fig. 83, thus 
 
 sin. : rad. \\DE\DK. 
 
 D E equals I foot ; therefore .+. 
 
 sin. : rad. : : I : D K = ! ; 
 
 sin. 
 
 or, the surface acted upon by one square foot of sectional 
 area equals the reciprocal of the sine of the angle of incli- 
 nation. Again, the horizontal measure of this inclined sur- 
 face may be obtained thus 
 
 cos. 
 
 sin. : cos. : : D E : K E = ; 
 
 sin. 
 
 or, KE, the horizontal measurement, equals the cosine of the 
 angle of inclination divided by the sine. 
 
 In tile figure, make K G equal to one foot ; then we 
 have 
 
 K E : KG : : V '. W\ 
 
 in which V, as above, represents the vertical pressure due to 
 the wind acting upon the surface KD, and W the vertical 
 pressure due to the wind acting upon the surface KH, or 
 so much as covers KG, one foot horizontal. 
 
 Now we have, as above, K E equal to - , K. G = i, and 
 
196 CONSTRUCTION. 
 
 V F sin. cos. Substituting these values, we have, instead 
 of the above proportion- 
 
 cos. _ . 
 
 : i : : F sin. cos. : W\ 
 
 sin. 
 
 from which 
 
 'f? (94.) 
 
 sn. 
 
 or, the vertical effect of the wind upon so much of the roof 
 as covers each square foot horizontal, is equal to the pro- 
 duct of the force of the wind per square foot into the square 
 of the sine of the angle of inclination. 
 
 Example. When the force of the wind upon a square 
 foot of vertical surface is 30 pounds, what will be the verti- 
 cal effect per square foot horizontal upon a roof the inclina- 
 tion of which is 26 33', or 6 inches to the foot? 
 
 Here we have F = 30, and the sine of 26 33' is 0-44698 ; 
 therefore 
 
 W 30 x 0-44698 2 = 5-9937- 
 This is conveniently solved by logarithms ; thus 
 
 log. 30 = 1^-4771213 
 0-44698 = 9-6502868 
 0-44698 = 9^-6502868 
 
 5-9937 = 0-7776949 
 
 or, the vertical effect is (5 -9937, or) 6 pounds. 
 
 The form of equation (94.) may be changed ; for, in a right- 
 angled triangle, the sine of the angle at the base is equal to 
 the perpendicular divided by the hypothenuse ; which, in 
 the case of a roof, is the height divided by the length of the 
 rafter; or 
 
 height h 
 Sme = TaTteF = 2 
 
LOAD UPON ROOFS. 197 
 
 Therefore, equation (94.) may be changed to 
 
 
 (950 
 
 or, the vertical effect upon each square foot of a roof is equal 
 to the product of the force of the wind per foot into the 
 square of the height of the roof at the ridge, divided by the 
 square of the length of the rafter (the height and length both 
 in feet.) 
 
 Example. When the force of the wind is 30 pounds, the 
 height of the roof 10 feet, and the length of the rafter 22-36 
 feet, what will be the vertical effect of the wind ? Here we 
 have F ~ 30, h = 10, and / = 22-36 ; and 
 
 222. Total Load per Foot Horizontal. The various 
 items comprising the total load upon a roof are the cover- 
 ing, the truss, the wind, snow, the plastering or other kind 
 of ceiling, and the load which may be deposited upon a floor 
 formed in the roof ; or, the total load 
 
 M= C+T + W+S + P+L. 
 
 The value per foot horizontal for these has been found as 
 follows: C=^; T= 0-0154 //; W=F^. For 5 the 
 
 value must be taken according to circumstances, as in Art. 
 220. So, also, the value of P and L are to be assigned as 
 required for each particular case, as in Art. 216. The total 
 load, therefore, with these substitutions, will be 
 
 M = 
 
 which reduces to 
 
 M = lp (- + 0-0154) + F^ t \-S + P+L; (96.) 
 
 * S / l> 
 
198 CONSTRUCTION. 
 
 in which / is the length of the rafter ; / is the weight of the 
 covering per foot superficial, including the roof boards or 
 slats, the jack-rafters, etc. ; s is the span of the roof ; ~h is the 
 vertical height above a horizontal line passing through the 
 feet of the rafters ; F is the force of the wind per square foot 
 against a vertical surface ; 5 is the weight of snow per 
 square foot horizontal ; P is the weight per superficial foot 
 of the ceiling at the tie-beam ; and L, the load per superficial 
 foot in the roof, including weight of flooring and floor- 
 timbers. The dimensions, s, /, and /i, are each in feet ; the 
 weight of /, F, 5, P, and L are each in pounds. The value 
 of/ is for a square foot of the inclined surface. 
 
 223. Strain 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 obtain- 
 ing the strains by the parallelogram of forces and by arith- 
 metical computations will be found useful, and will now be 
 referred to. 
 
 By the parallelogram of forces, the weight of the roof is 
 in proportion to the oblique thrust or pressure in the axis of 
 the rafter as twice the height of the roof is to the length of 
 the rafter ; or 
 
 R: F:: 2 A: /; 
 or, transposing 
 
 2&:i::R:Y=j; (97-) 
 
 where F equals the pressure in the axis of the rafter, and R 
 the weight of one truss and its load. Again, the weight of 
 the roof is in proportion to the horizontal thrust in the tie- 
 beam as twice the height of the roof is to half the span ; or 
 
 2 
 
 or, transposing 
 
 2/i: S -::R:H=~', (98.) 
 
THE STRAINS SHOWN GEOMETRICALLY. 199 
 
 where H equals the horizontal thrust in the tie-beam. To 
 obtain R, the weight of the roof, multiply M, the load per 
 foot, as in equation (96.), by s, the span, and by c, the dis- 
 tance from centres at which the trusses are placed ; or 
 
 R = M c s. 
 With this value of R substituted for it, we have 
 
 K= ' 
 
 and 
 
 TT M c s* , . 
 
 H = T ; (loo.) 
 
 4 It 
 
 in which F equals the strain in the axis of the rafter, and H 
 the strain in the tie-beam. These are the greatest strains 
 in the rafter and tie-beam. At certain parts of these pieces 
 the strains are less, as will be shown in the next article. 
 
 224. Strains in Roof-Timber 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 DG, FE, and C B the axes of the suspension-rods. In 
 this design for a truss, the distance A B is divided into three 
 equal parts, and the rods located at the two points of division, 
 G and E. By this arrangement the rafter A 7 is supported at 
 equidistant points, D and F. The point D supports the rafter 
 for a distance extending half-way to A and half-way to F, and 
 the point F sustains half-way to D and half-way to C. Also, 
 the point C sustains half-way to F, and, on the other rafter, 
 half-way to the corresponding point to F. And because these 
 points of support are located at equal distances apart, there- 
 fore the load on each is the same in amount. On D G make 
 Da equal by any decimally divided scale to the number of 
 hundreds of pounds in the load on D, and draw the parallel- 
 ogram abDc. Then, by the same scale, Db represents 
 (Art. 71) the pressure in the axis of the rafter by the load at 
 
2OO 
 
 CONSTRUCTION, 
 
 FIG. 84. 
 
STRAINS IN A. TRUSS. 2OI 
 
 D\ also, DC the pressure in the brace D E. Draw cd hori- 
 zontal ; then D d is the vertical pressure exerted by the brace 
 D E at E. The point F sustains, besides the common load 
 represented by D a, also the vertical pressure exerted by the 
 brace D E ; therefore, make Fe equal to the sum of D a and 
 Dd, and draw the parallelogram F gef. Then Fg, meas- 
 ured by the scale, is the pressure in the axis of the rafter 
 caused by the load at F, and F f is the load in the axis of the 
 brace FB. Draw fh horizontal ; then Fh is the vertical 
 pressure exerted by the brace Ffiat B. The point C, besides 
 the common load represented by D a, sustains the vertical 
 pressure Fh caused by, the brace FB, and a like amount 
 from the corresponding brace on the opposite side. There- 
 fore, make Cj equal to the sum of Da and twice Fit, and 
 draw jk parallel to the opposite rafter. Then Ck is the 
 pressure in the axis of the rafter at C. This is not the only 
 pressure in the rafter, although it is the total pressure at its 
 head C. At the point F, besides the pressure C k, there is 
 F g. At the point D, besides these two pressures, there is 
 the pressure D b. At the foot, at A, there is still an addi- 
 tional pressure ; for while the point D sustains the load half- 
 way to F and half-way to A, the point A sustains the load 
 half-way to D. This load is, in this case, just half the load 
 at D. Therefore draw A m vertical, and equal, by the scale, 
 to half of Da. Extend CA to/; draw ml horizontal. 
 Then A I is the pressure in the rafter at A caused by the 
 weight of the roof from A half-way to D. Now the total of 
 the pressures in the rafter is equal to the sum of A 1+ D b + 
 Fg added to C k. Therefore make kn equal to the sum of 
 A l+Db + F g, and draw no parallel with the opposite raf- 
 ter, and nj horizontal. Then Co, measured by the same 
 scale, will be found equal to the total weight of the roof on 
 both sides of C B. Since Da represents s, the portion of the 
 weight borne by the point D, therefore Co, representing the 
 whole weight of the roof, should equal six times Da, as it 
 does, because D supports just one sixth of the whole load. 
 Since C n is the total oblique thrust in the axis of the rafter 
 at its foot, therefore nj is the horizontal thrust in the tie- 
 beam at A. 
 
2O2 CONSTRUCTION. 
 
 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 as- 
 certain the load at the head of each brace, as represented by 
 the line D a, Fig. 84. The load corresponding to any part 
 of the roof is equal to the product of the superficial area of 
 that particular part (measured horizontally) multiplied by 
 the weight per square foot of the roof. Or, when M equals 
 the weight per square foot, c the distance from centres at 
 which the trusses are placed, and n the horizontal distance 
 between the heads of the braces, then the total load at the 
 head of a brace is represented by 
 
 NMcn. (101.) 
 
 The value of M is given in general terms in equation (96.). 
 To show its actual value, let it be required to find the weight 
 per square foot upon a root 52 feet span and 13 feet high at 
 middle ; or (Fig. 84), where A B equals half the space, or 26 
 feet, and C B 13 feet, then ^4 C, the length of the rafter, will 
 be 26-069, nearly. And where the weight of covering per 
 square foot, on the inclination, is 12 pounds, the force of the 
 wind against a vertical plane is 30 pounds ; the weight of 
 snow per foot horizontal is 20 pounds ; the weight of the 
 plastering forming the ceiling at the tie-beam is 9 pounds ; 
 and the load in the roof is nothing ; with these quantities 
 substituted, equation (96.) becomes 
 
 M 2Q'o6gx 12 (-- 4- 0-0154) 4- 30 1^ + 20 49 4-0 ; 
 \52 / 29-069' 
 
 M = (29-069 X 12 X 0-05386) 4 (30 x 0-2) + 20 4- 9; 
 
 M = 18-788 4-6 + 29 = 53-788; 
 
 or, say, 53-8 pounds. Then if <r, the distance from centres 
 between trusses, is 10 feet, and ;/, the distance between 
 braces, is one third of A B, Fig. 84, or 2 / = 8f , the total load 
 at the head of a brace will be, as per equation (101.) 
 
 N 53-8 x 10 x 8f = 4663; 
 
TABLE OF STRAINS. 203 
 
 or, say, 4650 pounds. Now, by any decimally divided scale, 
 make D a, Fig. 84, equal to 46^ parts of the scale ; this being 
 the number of hundreds of pounds contained in the weight 
 at D, as above. Then, by the same scale, the several lines 
 in the figure drawn as before shown will be found to repre- 
 sent respectively the weights here set opposite to them, as 
 follows : 
 
 D d da he 23^, and represents 2325 pounds; 
 
 " 4650 
 
 " 5200 " 
 
 F e = D a + D d 69$ " 6975 
 
 Ff = 6 5 | 6575 
 
 Cj 3 D a 139-1- " 13950 
 
 CK = $ D b = \$6 " 15600 
 
 C n = C k +- Fg+ D b + A 1= 312 " 31200 " 
 Cn^Ck + $Db=6 Db=2 Ck 
 
 = 312 " 31200 
 
 Nj C o 6 D a 6^ 4.6% 279 " 27900 *' 
 
 It should be observed here that the equality of the lines nj 
 and Co is a coincidence dependent upon the relation which 
 in this particular case the line CB happens to bear to the 
 line A B ; A B being equal to twice C B. And so of some 
 other lines in the figure. If the inclination of the roof were 
 made greater or less, the equality of the lines referred to 
 would disappear. It should also fye observed that the strains 
 above found are not quite exact ; they are, however, correct 
 to within a fraction of a hundred pounds, which is a suffi- 
 ciently near approximation for the purpose intended. From 
 the results obtained above, we ascertain that the strain in 
 the rafter, from F to C, is represented by C K, and is equal 
 to 15,600 pounds ; while the strain at the foot of the rafter, 
 from A to D, is represented by C n, and equals 3 1,200 pounds, 
 or double that which is at the head of the rafter. We ascer- 
 tain, also, that the maximum strain in the tie-beam, repre- 
 sented by 11 j, is 27,900 pounds; that that in the brace D E, 
 represented by DC, is 5200 pounds; and that that in the 
 brace F B, represented by Ff, is 6575 pounds. The strain 
 
204 CONSTRUCTION. 
 
 in the vertical rod D G is theoretically nothing. There is, 
 however, a small strain in it, for it has to carry a part of the 
 tie-beam and so much of the ceiling as depends for support 
 upon that part. But the manner of locating the weights, 
 adopted in this article, does not recognize any load located 
 at the point G. This is an objection to this system, but it 
 is not material. 
 
 For a recognition of weights at the tie-beam, see Arts. 
 
 205 to 211. The load at G may be found by obtaining the 
 product of the surface carried into the weight per foot of 
 the ceiling; or, say, 10 c n = 10 x lox 8f = 867 pounds. 
 The load to be carried by the rod F E is shown at D d= he, 
 which above is found to be 2325 pounds. To this is to be 
 added 867 pounds for the ceiling at E, as before found for the 
 ceiling at G\ or, together, 3192 pounds. The central rod 
 C B has to carrv the two loads brought to B by the two 
 braces footed there ; and also the weight of the ceiling sup- 
 ported by B. The vertical strain from the brace F B is rep- 
 resented at Fh, and equals 4650 pounds ; therefore, the 
 total load on CB is 4650 + 4650 -f 867 = 10,167 pounds. 
 
 226. Roof-Timber* : the Tie-Beam. The roof-timbers 
 comprised in the truss shown in Fig. 84 are the rafters, 
 tie-beam, two braces, and three rods. Of these, taking first 
 the tie-beam, we have a piece subject to tension and some- 
 times to cross-strain (see Art. 682, Transverse Strains}. In 
 this case the tensile strain "only need be considered. For 
 this a rule is given in Art. 117. In this rule, if the factor of 
 safety be taken at 20, the result will be sufficiently large to 
 allow for necessary cuttings at the joints. Therefore, if the 
 beam be of Georgia pine, equation (16.), Art. 117, becomes 
 
 __ 27900x20 _ 
 ~~ ~ 
 
 or, say, 35 inches. This is ample to resist the tensile strain ; 
 but, to resist the transverse strains to which such a long 
 piece of timber is subjected in the hands of the workman, 
 it would be proper to make it, say, 6x9. 
 
STRAIN UPON THE RAFTER. 205 
 
 227. The Rafter. A rafter, like a post, is subject to a 
 compressive force, and is liable to fail in three ways, name- 
 ly : by flexure, by being crushed, or by crushing the material 
 against which it presses. To render it entirely safe, there- 
 fore, it is requisite to ascertain the requirements for resisting 
 failure in each of these three ways. 
 
 Of these it will be convenient to consider, first, that of 
 the liability to being crushed. The rule for this is found in 
 Art. 107. Let the rafter be of Georgia pine, then the vajue 
 of C, Table I., will be 9500. The strain in the rafter (Art. 
 22$) is 31,200 pounds. Now, taking the value of a, the fac- 
 tor of safety, at 10, we have, by Rule VI. (Art. 107.) 
 
 31200 x 10 
 A = 9500 = 32 ' 737: 
 
 or, 33 inches area of cross-section. This is the size of the 
 rafter at its smallest section ; for example, at any one of the 
 joints where it is customary to reduce the area by cutting 
 for the struts and rods. 
 
 Again : Let the liability of the rafter to flexure be now 
 considered. For this we have a rule in Art. 114. The 
 length of the rafter between unsupported points is nearly 9$ 
 feet, or 9! x 12 = 1 16 inches. Let the thickness of the rafter 
 be taken at 6 inches. Then, by Rule XI. (Art. 114), we 
 have 
 
 + fyer 3 ) _ 31200 x 10(14- f x -00109 x r*). 
 
 c r~ 9500 x 6 
 
 / 116 
 r= 7 --g-=i9i; 19* * : 373-8- 
 
 Then, f x -00109 x 373-8 = 0-611127 
 adding unity = I 
 
 1-611127 
 
 Substituting this, we have 
 
 31200 x IPX 1-611127 _. 50267 1 '624 _ 
 " 9500 x~6~ "57000" 
 
206 CONSTRUCTION. 
 
 or, to resist flexure the breadth is required to be 8-82, or, 
 say, 9 inches ; or, the rafter is to be 6 xg inches at the foot. 
 The strain in the rafter at the upper end is only half that at 
 the foot ; the area of cross-section, therefore, at the head 
 need not be more than half that which is required at the 
 foot ; but it is usual to make it there about f of the size at the 
 foot. In this case it would be, therefore, 6x6 inches at the 
 upper end. 
 
 Lastly, the requirement to resist crushing the surfaces 
 against which the rafter presses is to be considered. 
 
 The fibres of timber yield much more readily when 
 pressed together by a force acting at right angles to the di- 
 rection of their length than when it acts in a line with their 
 length. 
 
 The value of timber subjected to pressure in these two 
 ways is shown in Arts. 94, 98. In Table I., the value per 
 square inch of the first stated resistance is expressed by P, 
 
 and the ultimate resistance of the other by . The value 
 of timber per square inch to safely resist crushing may be ex- 
 pressed by , in which a is the factor of safety. Timber 
 pressed in an oblique direction will resist a force exceeding 
 
 
 
 that expressed by P, and less than that expressed by - . 
 
 ct 
 
 When the angle of inclination at which the force acts is just 
 45, then the force will be an average between P and . 
 
 And for any angle of inclination, the force will vary inverse- 
 ly as the angle ; approaching P as the angle is enlarged, 
 
 but approaching - - as the angle is diminished. It will be 
 
 equal to -- when the angle becomes zero, and equal P 
 
 when the angle becomes 90. The resistance of timber per 
 square inch to an oblique force is therefore expressed by 
 
 M = 
 
RESISTANCE OF SURFACES. 2O? 
 
 where A equals the complement of the angle of inclination. 
 In a roof, A is the acute angle formed by the rafter with 
 a vertical line. If no convenient instrument be at hand to 
 measure the angle, describe an arc upon the plan of the 
 truss thus : with C B (Fig. 84) for radius, describe the arc 
 B g, and get the length of this arc in feet by stepping it off 
 with a pair of dividers. Then 
 
 where k equals the length of the arc, and h equals B C, the 
 height of the roof. Therefore 
 
 M = 
 
 equals the value of timber per square inch in a tie-beam, C 
 and P being obtained from Table I., Art. 94. When C for 
 the kind of wood in the tie-beam exceeds C set opposite the 
 kind of wood in the rafter, then the latter is to be used in 
 the rules instead of the former. 
 
 The value of M, equation (103.),. is the resistance per 
 square inch of the surface pressed at the foot of the rafter. 
 The resistance of the entire surface will therefore be MA, 
 where A equals the area of the joint. Then, when the re- 
 sistance equals the strain, we will have 
 
 from which we have 
 
 in which 5 is the strain to be resisted. 
 
 Now, the end of the rafter must be of sufficient size to 
 afford a joint the area of which will not be less than that 
 expressed by A in equation (104.). 
 
 For example, the strain to which the rafter, Fig. 84, is 
 subject at its foot is ascertained to be (Art. 225) 31,200 pounds. 
 For Georgia pine, the material of the tie-beam, P = 900 
 (Art. 94, Table /.), and ^ = 9500. 
 
208 CONSTRUCTION. 
 
 The length of the arc Bg is about 14-4 feet; the height 
 B Cis 13 feet. Let a, the factor of safety, be taken at 10, 
 then we have (104.) 
 
 31200 
 
 900 + (o-6 3 | 
 
 = 
 
 x 50) - 
 
 or, the superficial area of the bearing at the joint required 
 to prevent crushing the tie-beam is 33^- inches. 
 
 The results of the computations show that the rafter is 
 required to be 6 inches thick, 9 inches wide at the foot, and 
 6 inches wide at the top. It is also ascertained that, in cut- 
 ting for the bearing for the struts and boring for the sus- 
 pension-rods, it is required that there shall be at least 33 
 inches area of cross-section left intact ; and, farther, that the 
 area of the surface of the joint against the tie-beam should 
 not be less than 33^ inches. 
 
 228. The Braces. Each brace is subject to compres- 
 sion, and is liable to fail if too small, in the same manner as the 
 rafter. Its size is to be ascertained, therefore, in the manner 
 described for the rafter ; which need not be here repeated, 
 except, perhaps, as to the liability to fail by flexure ; for in this 
 case we have the breadth given, and need to find the thick- 
 ness. The breadth of the brace is fixed by the thickness of 
 the rafter, for it is usual to have the two pieces flush with 
 each other. Rule XI. (Art. 114) is to be used, but with this 
 difference, namely : instead of the thickness, use the breadth 
 as one of the factors in the divisor. Thus 
 
 (105.) 
 
 In working this rule, it is required, in order to get the 
 value of r, the ratio between the height and thickness, to 
 assume the thickness before it is ascertained ; and after com- 
 putation, if the result shows that the assumed value was not 
 a near approximation, a second trial will have to be made. 
 Usually the first trial will be sufficient. 
 
STRAIN UPON BRACES. 209 
 
 For example, the brace D E is about 9$ feet or 1 16 inches 
 long. As the strain in it is only 5200 pounds, the thickness 
 will probably be not over 3 inches. Assuming it at this, we 
 
 have r = -= -*-J-& = 38$ ; the square of whichxis about 
 Therefore, we have- 4 N I VE R S I T Y 
 
 |-XO-OOI09XI495=2 
 
 add unity = i. 
 
 ^444S 
 The equation reduces, therefore, to this 
 
 , = 5200 x 10XV4445 
 
 9500 x 6 
 
 or, the required thickness of the brace is 3^- inches, or the 
 brace should be, say, 3 J x 6 inches. In this case the result is 
 so near the assumed value, a second trial. is not needed. 
 
 For the second brace, we have the length equal to about 
 \2\ feet or 147 inches; and the strain equal to 6575 pounds 
 (Art. 22$). The ratio, therefore, may be obtained by assum- 
 ing the thickness, say, at 4. With this, we have 
 
 i^ 36-75 ; the square of which is 1350^' 
 With this value of r 2 
 
 | x 00109 x 1350^ = 2- 2081 
 add unity = I. 
 
 3-2081 
 Then 
 
 6575 x lox 3-2081 . 
 
 t= .^L2 _^ - = 3.7006. 
 
 9500x6 
 
 Comparing this result with the assumed value of / = 4 
 we find the difference so great as to require a second trial. 
 As the value of r was taken too low, the result obtained is 
 correspondingly low. The true value is somewhere between 
 3 - 7 and 4. Assume it now, say, at 3 - 9. With this value, we 
 have 
 
 r = -== = 37-692 ; the square of which is 1420-7. 
 
210 CONSTRUCTION. 
 
 With this value of r* 
 
 | x -00109 x 1420-7 = 2-32282 
 add unity = i 
 
 3-32282 
 Then 
 
 , _ 6575x10x3-32282 _ 
 9500x6 
 
 This result is a trifle less than the assumed value, 3-9. The 
 true value is between these, and probably is about 3-86. 
 This is quite near enough for use. This brace, therefore, is 
 required to be 3 - 86 x 6 inches, or, say, 4x6 inches. 
 
 229. The Supension-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 only 
 9000 pounds per inch, and, as ordinarily made, these rods 
 ought not to be loaded with more than 7000 pounds. The 
 strain divided by this value per inch of the metal will give 
 the sectional area of cross-section. For example, the strain in 
 the rod D G, Fig. 84, is 867 pounds (Art. 225); therefore 
 
 867 
 
 or, the sectional area required is only an eighth of an inch. 
 By reference to the table of areas of circles in the Appen- 
 dix, the diameter of a rod containing the required area, as 
 above, will be found to be a little less than half an inch. A 
 rod half an inch in diameter will therefore be of ample 
 strength. For appearance's sake, however, no rod in a truss 
 should be less than f of an inch in diameter. 
 
 The rod FE has to resist a strain of 3192 pounds. For 
 this, then, we have 
 
 A reference to the table of areas shows that a rod contain- 
 
ROOF-BEAMS. 211 
 
 ing this area would be a little more than J of an inch in di- 
 ameter ; it would be of ample strength, say, at of an inch 
 in diameter. 
 
 The rod C B, at the centre, has to carry a strain of 10,167 
 pounds. For this, then,- we have 
 
 10167 
 
 A reference to the table of areas shows that this rod should 
 be i inches in diameter. 
 
 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., equa- 
 tion (35.), Art. 140. In this equation, U=cfl(Art. 152). 
 Substituting this value for U, and r I for tf, equation (35.) be- 
 comes 
 
 and putting for r the rate of deflection, .04, we have 
 
 a formula convenient for roof-timbers. 
 
 Example. In a roof where the roofing is to be supported 
 on white-pine roof-beams 10 feet long, placed 2\ feet from 
 centres, and where the load per foot superficial is to be 40 
 pounds, including wind and snow : what should be the di- 
 mensions of the roof-beams? By equation (106.) 
 
 bd = 
 
 Now if b, the breadth, be fixed, say, at 3, then 
 
 d = 5-64 nearly. 
 
212 
 
 CONSTRUCTION. 
 
 The roof-beams, therefore, require to be 3 x 5$, or, say, 3x6. 
 All pieces of timber subject to cross-strains will sustain 
 safely much greater strains when extended in one piece over 
 two, three, or more distances between bearings ; therefore, 
 roof-beams, jack-rafters, and purlins should, if possible, be 
 made in as long lengths as practicable ; the roof-beams and 
 purlins laid on, not framed into, the principal rafters, and 
 extended over at least two spaces, the joints alternating on 
 the trusses ; and likewise the jack-rafters laid on the purlins 
 in long lengths. 
 
 231. Five Examples of Rooft: are shown at Figs. 85, 86, 
 87, 88, and 89. In Fig. 85, a is an iron suspension-rod, b y b are 
 
 braces. In Fig. 86, a, a, and 
 b are iron rods, and d, d, c, c 
 are braces. In Fig. 87, a, b 
 are iron rods, d, d braces, and 
 c the straining beam. In 
 FIG. 85. Fig. 88, a, a, b, b are iron rods, 
 
 , e> d, d are braces, and c is a straining beam. In Fig. 89, pur- 
 
 out 
 
 S<rft. I 
 
 lins are located at PP, etc. ; the inclined beam that lies upon 
 them is the jack-rafter; the post at the ridge is the king- 
 
TRUSS WITH BUILT-RIB. 
 
 2I 3 
 
 rTi 
 
 ?k 
 
 o a 
 
 post, the others are queen-posts. In this design the tie-beam 
 is increased in height along the middle by a strengthening 
 piece (Art. 163), for the purpose of sustaining additional 
 weight placed in the room form- 
 ed in the truss (Art. 216). 
 
 Fig. 90 shows a method of 
 constructing a truss having a 
 built-rib in the place of prin- 
 cipal rafters. The proper form 
 for the curve is that of the par- 
 abola (Art. 560). This curve, 
 when as flat as is described in 
 the figure, approximates so close- 
 ly to that of the circle that the 
 latter may be used in its stead. 
 The height, a b, is just half of 
 a c, the curve to pass through 
 the middle of the rib. The rib 
 is composed of two series of 
 abutting pieces, bolted together, oo ^ 
 These pieces should be as long 
 as the dimensions of the timber 
 will admit, in order that there 
 may be but few joints. The sus- 
 pending pieces are in halves, 
 notched and bolted to the tie- 
 beam and rib, and a purlin is 
 framed upon the upper end of 
 each. A truss of this construc- 
 tion needs, for ordinary roofs, 
 no diagonal braces between the 
 suspending pieces, but if extra 
 strength is required the braces 
 may be added. The best place \~^~.\ 
 for the suspending pieces is at 
 
 the joints of the rib. A rib of this kind will be sufficiently 
 strong if the area of its section contain about one fourth 
 more timber than is required for that of a rafter for a roof 
 of the same size. The proportion of the depth to the thick- 
 ness should be about as 10 to 7. 
 
214 
 
 CONSTRUCTION. 
 
 232. Roof-Trus with Elevated Tic-Beam. Designs 
 such as are shown in Fig. 91 have the tie elevated for the ac- 
 commodation of an arch in the ceiling. This and all similar 
 designs are seriously objectionable, and should always be 
 
 avoided ; as the smail height gained by the omission of the 
 tie-beam can never compensate for the powerful lateral 
 strains which are exerted by the oblique position of the 
 
 supports, tending to separate the walls. Where an arch is 
 required in the ceiling, the best plan is to carry up the 
 walls as hi^h as the top of the arch. Then, by using a 
 horizontal tie-beam, the oblique strains will be entirely re- 
 
HIP-ROOFS. 
 
 215 
 
 moved. It is well known that many a public building has 
 been all but ruined by the settling of the roof, consequent 
 upon a defective plan in the formation of the truss in this 
 respect. It is very necessary, therefore, that the horizontal 
 
 FIG. 91. 
 
 tie-beam be used, except where the walls are made so strong 
 and firm by buttresses, or other support, as to prevent a 
 possibility of their separating. (See Art. 212.) 
 
 233. Hip-Roofs: Lines and Bevils. The lines a b and 
 be in Fig. 92, represent the walls at the angle of a building; 
 b e is the seat of the hip-rafter, and gfoi a jack or cnppl( 
 rafter. Draw e h at right angles to be, and make it equal 
 
2l6 
 
 CONSTRUCTION. 
 
 to the rise of the roof ; join b and //, and h b will be the 
 length of the hip-rafter. Through e draw di at right angles 
 to b c\ upon b, with the radius b h, describe the arc hi, 
 cutting di in z ; join b and i, and extend gfto meet b i in/; 
 then gj will be the length of the jack-rafter. The length 
 of each jack-rafter is found in the same manner by extend- 
 ing its seat to cut the line b i. From / draw fk at right 
 angles tofg, also// at right angles to be\ make/ equal 
 to// by the arc Ik, or make g k equal to gj by the arc jk ; 
 then the angle at / will be the top-bevil of the jack-rafters, 
 and the one at k will be the down-bevil* 
 
 234. The Backing of the Hip-Rafter. At any con- 
 venient place in be (Fig. 92), as o, draw m n at right angles to 
 b e ; from o, tangical to b h, describe a semicircle, cutting b e 
 in s ; join m and s and n and s\ then these lines will form at 
 s the proper angle for bevilling the top of the hip-rafter. 
 
 DOMES.f 
 
 235. Domes. The usual form for domes is that of the 
 sphere ; the base circular. When the interior dome does not 
 
 FIG. 93. 
 
 rise too high, a horizontal tie may be thrown across, by 
 which any degree of strength required may be obtained. 
 
 * The lengths and bevils of rafters for root-valleys can also be found by the 
 above process. 
 
 f See also Art. 68. 
 
CONSTRUCTION OF DOMES. 
 
 217 
 
 - 93 shows a section, and Fig. 94 the plan, of a dome of 
 this kind, a b being the tie-beam in both. Two trusses of 
 this kind (Fig. 93), parallel to each other, are to be placed 
 one on each side of the opening in the top of the dome. 
 Upon these the whole framework is to depend for support, 
 
 FIG. 94. 
 
 u u 
 
 and their strength must be calculated accordingly. (See 
 Arts. 70 to 80 and 214 to 222.) If the dome is large and of 
 importance, two other trusses may be introduced at right 
 angles to the foregoing, the tie-beams being preserved in 
 
 ; i*/ FIG. 95. 
 
 one continuous length by framing them high enough to pass 
 over the others. 
 
 236. Ribbed Dome. When the interior must be kept 
 free, then the framing may be composed of a succession of 
 ribs standing upon a continuous circular curb of timber, as 
 
218 
 
 CONSTRUCTION. 
 
 seen at Figs. 95 and 96 the latter being a plan and the former 
 a section. This curb must be well secured, as it serves hi 
 the place of a tie-beam to resist the lateral thrust of the ribs. 
 In small domes these ribs may be easily cut from wide 
 plank ; but where an extensive structure is required, they 
 must be built in two thicknesses so as to break joints, in the 
 same manner as is described for a roof at Art. 231. They 
 should be placed at about two feet apart at the base, and 
 strutted as at a in Fig. 95. 
 
 FIG. .96. 
 
 The scantling of each thickness of the rib may be as fol- 
 lows : 
 
 For domes of 24 feet diameter, i x 8 inches. 
 36 " " i x 10 " 
 
 2 x 13 " 
 
 a 
 
 " 60 
 " 90 
 
 "108 
 
 3 x 13 
 
 237. Dome: Curve of Equilibrium. The surfaces of a 
 dome may be finished to any curve that may be desired, but 
 the framing should be constructed of such form that the 
 curve of equilibrium shall be sure to pass through the middle 
 of the depth of the framing. The nature of this curve is 
 such that, if an arch or dome be constructed in accordance 
 with it, no one part of the structure will be less capable than 
 another of resisting the strains and pressures to which the 
 whole fabric may be exposed. The curve of equilibrium for 
 an arched vault or a roof, where the load is equally diffused 
 
CURVE OF EQUILIBRIUM. 
 
 2I 9 
 
 over the whole surface, is that of a. parabola (Art. 460}-, for 
 a dome having no lantern, tower, or cupola above it, a cubic 
 parabola (Fig. 97) ; and for one having a tower, etc., above it, 
 a curve approaching that of an hyperbola must be adopted, 
 as the greatest strength is required at its upper parts. If 
 the curve of a dome be circular (as in the vertical section, 
 ^95)> the pressure will have a tendency to burst the dome 
 outwards at about one third of its height. Therefore, when 
 this form is used in the construction of an extensive dome, 
 an iron band should be placed around the framework at 
 that height ; and whatever may be the form of the curve, a 
 band or tie of some kind is necessary around or across the 
 base. 
 
 
 
 o/ 
 
 
 
 / 
 
 /I 
 
 
 1 
 
 
 
 
 
 of \ 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 s f 
 
 FIG. 97. 
 
 If the framing be of a form less convex than the curve of 
 equilibrium, the weight will have a tendency to crush the 
 ribs inwards, but this pressure may be effectually overcome 
 by strutting between the ribs ; and hence it is important 
 that the struts be so placed as to form continuous horizontal 
 circles. 
 
 238. Cubic Parabola Computed. Let a b (Fig. 97) be 
 the base, and b c the height. Bisect a b at d, and divide a d 
 into 100 equal parts ; of these give d e 26, ef \%\,fg i&,gh 
 \2\, h i lof, ij 9^, and the balance, 8j, to/# ; divide be into 
 8 equal parts, and from the points of division draw lines 
 parallel to a b, to meet perpendiculars from the several points 
 
22O 
 
 CONSTRUCTION. 
 
 of division in a b, at the points o, o, o y etc. Then a curve 
 traced through these points will be the one required. 
 
 239. Small Domes over Stairways : are frequently made 
 elliptical in both plan and section ; and as no two of the ribs 
 
 in one quarter of the dome are alike in form, a method for 
 obtaining the curves may be useful. 
 
 FIG. 99, 
 
 To find the curves for the ribs of an elliptical dome, let 
 abed (Fig. 98) be the plan of a dome, and ef the seat of 
 one of the ribs, Then take c /for the transverse axis and 
 twice the rise, og> of the dome for the conjugate, and de- 
 
COVERING OF DOMES. 
 
 221 
 
 scribe (according to Arts. 548, 549, etc.) the semi-ellipse 
 e gf, which will be the curve required for the rib e gf. The 
 other ribs are found in the same manner. 
 
 240. Covering for a Spherical Dome. To find the 
 shape, let^4 C^T-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 o b, and make dc equal to it; divide 
 the arc o b and the line d c each into a like number of equal 
 parts, as 5 (a large number will insure greater accuracy than 
 a small one) ; upon c, through the several points of division 
 in cd, describe the arcs o do, i e I, 2/2, etc. ; make do equal 
 to half the width of one of the boards, and draw o s parallel 
 
 to a c ; join s and #, and from the points of division in the arc 
 <?drop perpendiculars, meeting a sinij kl\ from these 
 points draw z 4,/3, etc., parallel to ac ; make do t ei, etc., on 
 the lower side of a c, equal to do,e\, etc., on the upper side ; 
 trace a curve through the points o, 1,2, 3, 4, c, on each side 
 of dc ; then o c o will be the proper shape for the board. By 
 dividing the circumference of the base A into equal parts, 
 and making the bottom, o d o, of the board of a size equal to 
 one of those parts, every board may be made of the same 
 size. In the same manner as the above, the shape of the 
 covering for sections of another form may be found, such as 
 an ogee, cove, etc. 
 
 To find the curve of the boards when laid in horizontal 
 courses, let A B C (Fig. 100) be the section of a given dome, 
 
222 
 
 CONSTRUCTION. 
 
 and DB its axis. Divide B C into as many parts as there 
 are to be courses of boards, in the points i, 2, 3, etc. ; through 
 i and 2 draw a line to meet the axis extended at a\ then a 
 will be the centre for describing the edges of the board F. 
 Through 3 and 2 draw 3 b; then b will be the centre for de- 
 
 FlG. IOI. 
 
 scribing F. Through 4 and 3 draw 4^; then d will be the 
 centre for G. B is the centre for the arc i o. If this method 
 is taken to find the centres for the boards at the base of the 
 dome, they would occur so distant as to make it impracti- 
 cable ; the following method is preferable for this purpose : 
 G being the last board obtained by the above method, ex- 
 tend the curve of its inner edge until it meets the axis, D B, 
 
 in e ; from 3, through e, draw 3 f, meeting the arc A B in f ; 
 join f and 4, f and 5 , and /and 6, cutting the axis, D B, in s, n, 
 and m ; from 4, 5, and 6 draw lines parallel to A C and cutting 
 the axis in c, p, and r ; make c 4 (Fig. 101) equal to c 4 in the pre- 
 vious figure, and c s equal to c s also in the previous figure ; 
 then describe the inner edge of the board //, according to 
 Art. 516; the outer edge can be obtained by gauging from 
 the inner edge. Tn like manner proceed to obtain the next 
 
DESIGNS OF BRIDGES. 
 
 22 3 
 
 board taking/ 5 for half the chord, and p n for the height 
 of the segment. Should the segment be too large to be de- 
 scribed easily, reduce it by finding intermediate points in the 
 curve, as at Art. 515. 
 
 241. Polygonal Dome: Form of Angle-Rib. To ob- 
 tain the shape of this rib, let A G H (Fig. 102) be the plan of 
 a given dome, and C D a vertical section taken at the line 
 ef. From i, 2, 3, etc., in the arc CD draw ordinates, paral- 
 lel to A D, to meet f-G ; from the points of intersection on 
 fG draw ordinates at right angles to/ G\ make s i equal 
 to o i, s 2 equal to o 2, etc. ; then GfB, obtained in this way, 
 will be the angle-rib required. The best position for the 
 sheathing-boards for a dome of this kind is horizontal, but if 
 they are required to be bent from the base to the vertex, 
 their shape may be found in a similar manner to that shown 
 at Fig. 99. 
 
 BRIDGES. 
 
 FIG. 103 
 
 242. Bridges. Of plans for the construction of bridges, 
 perhaps the following are the most useful. Fig. 103 shows a 
 method of constructing wooden bridges where the banks 
 of the river are high enough to permit the use of the tie- 
 beam, a b. The upright pieces, c d, are notched and bolted 
 on in pairs, for the support of the tie-beam. A bridge ot 
 this construction exerts no lateral pressure upon the abut- 
 ments. This method may be employed even where the banks 
 of the river are low, by letting the timbers for the roadway 
 rest immediately upon the tie-beam. In this case the irame- 
 work above will serve the purpose of a railing. 
 
22 4 
 
 CONSTRUCTION. 
 
 Fig. 104 exhibits a wooden bridge without a tie-beam. 
 Where staunch buttresses can be obtained this method may 
 be recommended ; but if there is any doubt ot their stability, 
 it should not be attempted, as it is evident that such a sys- 
 tem of framing is capable of a tremendous lateral thrust. 
 
 FIG. 104. 
 
 243, Bridge: Built-Rib. Fig. 105 represents a bridge 
 with a built-rib (see Art. 231) as a chief support. The curve 
 of equilibrium will not differ much from that of a parabola ; 
 this, therefore, may be used especially if the rib is made 
 
 FIG. 105. 
 
 gradually a little stronger as it approaches the buttresses. 
 As it is desirable that a bridge be kept low, the following 
 table is given to show the least rise that may be given to the 
 rib. 
 
 Span in Feet. 
 
 Least Rise in Feet 
 
 Span in Feet. 
 
 Least Rise in Feet 
 
 Span in Feet. ! Least Rise in Feet 
 
 ( 
 
 30 
 
 o-5 
 
 1 2O 
 
 7 
 
 280 
 
 24 
 
 40 
 
 0-8 
 
 I4O 
 
 8 
 
 3OO 
 
 28 
 
 50 
 
 1-4 
 
 1 60 
 
 10 
 
 320 
 
 32 
 
 60 
 
 2 
 
 1 80 
 
 ii 
 
 350 
 
 39 
 
 70 
 
 21 
 
 200 
 
 12 
 
 380 
 
 47 
 
 80 
 
 3 
 
 2 2O 
 
 14 
 
 400 
 
 53 
 
 9 
 
 4 
 
 240 
 
 17 
 
 
 
 100 
 
 5 
 
 260 
 
 20 
 
 i 
 
W 
 ((UNIVERSITY 
 
 DIMENSIONS OF THE BUILT-Rl] 
 
 The rise should never be made less than this, but in all 
 cases greater if practicable ; as a small rise requires a greater 
 quantity of timber to make the bridge equally strong. The 
 greatest uniform weight with which a bridge is likely to be 
 loaded is, probably, that of a dense crowd of people. This 
 may be estimated at 70 pounds per square foot, and the fram- 
 ing and gravelled roadway at 230 pounds more ; which 
 amounts to 300 pounds on a square foot. The following 
 rule, based upon this estimate, may be useful in determining 
 the area of the ribs. 
 
 Rule LXVIL Multiply the width of the bridge by the 
 square of half the span, both in feet, and divide this pro- 
 duct by the rise in feet multiplied by the number of ribs ; 
 the quotient multiplied by the decimal o-oon will give the 
 area of each rib in feet. When the roadway is only planked, 
 use the decimal 0-0007 instead of o-ooii, 
 
 Example. What should be the area of the ribs for a 
 bridge of 200 feet span, to rise 15 feet and be 30 feet wide, 
 with three curved ribs ? The half of the span is 100, and 
 its square is 10000 ; this multiplied by 30 gives 300000, and 
 15 multiplied by 3 gives 45; then 300000 divided by 45 
 gives 6666|, which multiplied by o-ooii gives 7-333 feet or 
 1056 inches for the area of each rib. Such a rib may be 24 
 inches thick by 44 inches deep, and composed of 6 pieces, 
 2 in width and 3 in depth. 
 
 The above rule gives the area of a rib that would be 
 requisite to support the greatest possible tiniform load. 
 But in large bridges, a variable load, such as a heavy wagon, 
 is capable of exerting much greater strains ; in such cases, 
 therefore, the rib should be made larger.* 
 
 In constructing these ribs, if the span be not over 50 feet, 
 each rib may be made in two or three thicknesses of timber 
 (three thicknesses is preferable), of convenient lengths bolted 
 together ; but in larger spans, where the rib will be such as 
 to render it difficult to procure timber of sufficient breadth, 
 they may be constructed by bending the pieces to the proper 
 curve and bolting them together. In this case, where tim- 
 
 * See Tredgold's Carpentry by Hurst, Arts. 174 to 177. 
 
226 
 
 CONSTRUCTION. 
 
 her of sufficient length to span the opening- cannot be ob- 
 tained, and scarfing is necessary, such joints must be made 
 as will resist both tension and compression (see Fig. 1 14). 
 To ascertain the greatest depth for the pieces which compose 
 the rib, so that the process of bending may not injure their 
 elasticity, multiply the radius of curvature in feet by the 
 decimal 0-05, and the product will be the depth in inches. 
 Example. Suppose the curve of the rib to be described 
 with a radius of 100 feet, then what should be the depth ? The 
 radius in feet, 100, multiplied by 0-05 gives a product of 5 
 inches. White pine or oak timber 5 inches thick would 
 freely bend to the above curve ; and if the required depth 
 of such a rib be 20 inches, it would have to be composed of at 
 least 4 pieces. Pitch pine is not quite so elastic as white 
 pine or oak its thickness may be found by using the deci- 
 mal 0-046 instead of 0-05. 
 
 FIG. 106. 
 
 244. Bridges: Framed Rib. In spans of over 250 
 feet, SL framed r\b y as in Fig. 106, would be preferable to the 
 foregoing. Of this, the upper and the lower edges are 
 formed as just described, by bending the timber to the proper 
 curve. The pieces that tend to the centre of the curve, 
 called radials, are notched and bolted on in pairs, and the 
 cross-braces are halved together in the middle, and abut end 
 to end between the radials. The distance between the ribs 
 of a bridge should not exceed about 8 feet. The roadway 
 should be supported by vertical standards bolted to the ribs 
 
THE ROADWAY AND ABUTMENTS. 22/ 
 
 at about every 10 to 15 feet. At the place where they rest 
 on the ribs, a double, horizontal tie should be notched and 
 bolted on the back of the ribs, and also another on the un- 
 derside ; and diagonal braces should be framed between the 
 standards, over the space between the ribs, to prevent lat- 
 eral motion. The timbers for the roadway may be as light 
 as their situation will admit, as all useless timber is only an 
 unnecessary load upon the arch. 
 
 245. Bridges: Roadway. If a roadway be 18 feet 
 wide, two carriages can pass without inconvenience. Its 
 width, therefore, should be either 9, 18, 27, or 36 feet, ac- 
 cording to the amount of travel. The width of the foot- 
 path should be two feet for every person. When a stream 
 of water has a rapid current, as few piers as practicable 
 should be allowed to obstruct its course ; otherwise the 
 bridge will be liable to be swept away by freshets. When 
 the span is not over 300 feet, and the banks of the river are 
 of sufficient height to admit of it, only one arch should be 
 employed. The rise of the arch is limited by the form of 
 the roadway, and by the height of the banks of the river 
 (see Art. 243). The rise of the roadway should not exceed 
 one in 24 feet, but as the framing settles about one in 72, the 
 roadway should be framed to rise one in 18, that it may be 
 one in 24 after settling. The commencement of the arch at 
 the abutments the spring, as it is termed should not be 
 below high-water mark ; and the bridge should be placed at 
 right angles with the course of the current. 
 
 246. Bridges: Abutments. The best material for the 
 abutments and piers of a bridge is stone ; and no other 
 should be used. The following rule is to determine the ex- 
 tent of the abutments, they being rectangular, and built with 
 stone weighing 120 pounds to a cubic foot. 
 
 Rule LXVIII Multiply the square of the height of the 
 abutment by 160, and divide this product by the weight of a 
 square foot of the arch, and by the rise of the arch ; add 
 unity to the quotient, and extract the square root. Dimin- 
 ish the square root by unity, and multiply the root so dimin- 
 
228 CONSTRUCTION. 
 
 ished by half the span o-f the arch, and by the weight of a 
 square foot of the arch. Divide the last product by 120 
 times the height of the abutment, and the quotient will be 
 the thickness of the abutment. 
 
 Example. Let the height of the abutment from the base 
 to the springing of the arch be 20 feet, hall the span 100 feet, 
 the weight of a square foot of the arch, including the great- 
 est possible load upon it, 300 pounds, and the rise of the arch 
 18 feet: what should be its thickness? The square of the 
 height of the abutment, 400, multiplied by 160 gives 64000, 
 and 300 by 18 gives 5400; 64000 divided by 5400 gives a 
 quotient of 11-852; one added to this makes 12-852, the 
 square root of which is 3-6 ; this, less one is 2-6 ; this mul- 
 tiplied by 100 gives 260, and this again by 300 gives 78000; 
 this divided by 120 times the height of the abutment, 2400, 
 gives 32 feet 6 inches, the thickness required. 
 
 The dimensions of a pier will be found by the same rule ; 
 for, although the thrust of an arch may be balanced by an 
 adjoining arch when the bridge is finished, and while it re- 
 mains uninjured, yet, during the erection, and in the event 
 of one arch being destroyed, the pier should be capable of 
 sustaining the entire thrust of the other. 
 
 Piers are sometimes constructed of timber their princi- 
 pal strength depending on piles driven into the earth ; but 
 such piers should never be adopted where it is possible to 
 avoid them ; for, being alternately wet and dry, they decay 
 much sooner than the upper parts of the bridge. Spruce 
 and elm are considered good for piles. Where the height 
 from the bottom of the river to the roadway is great, it is a 
 good plan to cut them off at a little below low-water mark, 
 cap them with a horizontal tie, and upon this erect the posts 
 for the support of the roadway. This method cuts off the 
 pan that is continually wet from that which is only occa- 
 sionally so, and thus affords an opportunity for replacing the 
 upper part. The pieces which are immersed will last a 
 great length of time, especially when of elm ; for it is a 
 well-established fact that timber is less durable when subject 
 to alternate dryness and moisture than when it is either con- 
 tinually wet or continually dry. It has been ascertained that 
 
CENTRING FOR BRIDGES. 
 
 22 9 
 
 the piles under London Bridge, after having been driven 
 about 600 years, were not materially decayed. These piles 
 are chiefly of elm, and wholly immersed. 
 
 247. Centre for Stone Bridges. Fig. 107 is a design 
 for a centre for a stone bridge where intermediate supports, 
 as piles driven into the bed of the river, are practicable. Its 
 timbers are so distributed as to sustain the weight of the 
 arch-stones as they are being laid, without destroying the 
 original form of the centre ; and also to prevent its destruc- 
 tion or settlement, should any of the piles be swept away. 
 The most usual error in badly-constructed centres is that 
 the timbers are disposed so as to cause the framing to 
 rise at the crown during the laying of the arch-stones up 
 
 FIG. 107. 
 
 the sides. To remedy this evil, some have loaded the crown 
 with heavy stones ; but a centre properly constructed will 
 need no such precaution. 
 
 Experiments have shown that an arch-stone does not press 
 upon the centring until its bed is inclined to the horizon at 
 an angle of from 30 to 45 degrees, according to the hardness 
 of the stone, and whether it is laid in mortar or not. For 
 general purposes, the point at which the pressure com- 
 mences may be considered to be at that joint which forms 
 an angle of 32 degrees with the horizon. At this point the 
 pressure is inconsiderable, but gradually increases towards 
 the crown. The following table gives the portion of the 
 weight of the arch-stones that presses upon the framing at 
 the various angles of inclination formed by the bed of the 
 
230 CONSTRUCTION. 
 
 stone with the horizon. The pressure perpendicular to the 
 curve is equal to the weight of the arch-stone multiplied by 
 the decimal 
 
 o, when the angle of inclination is 32 degrees. 
 .04 " " " 34 
 
 08 " " " 36 " 
 
 12 " " " 38 " 
 17 " " " 40 " 
 21 " " " 42 " 
 
 -25 " - 44 
 
 29 " " " 46 " 
 
 33 " ;" " 48 " 
 
 37 " " " 50 " 
 
 4 " " " 52 " 
 
 44 " 4t " 54 " 
 
 48 " " " 56 
 
 52 " " u 58 " 
 -54 " " " 60 " 
 
 From this it is seen that at the inclination of 44 degrees the 
 pressure equals one quarter the weight of the stone ; at 57 
 degrees, half the weight ; and when a vertical line, as a b 
 (Fig. 1 08), passing through the centre of 
 gravity of the arch-stone, does not fall 
 within its bed, c d, the pressure may be con- 
 sidered equal to the whole weight of the 
 stone. This will be the case at about 60 
 degrees, when the depth of the stone is 
 double its breadth. The direction of these 
 pressures is considered in a line with the radius of the curve. 
 The weight upon a centre being known, the pressure may be 
 estimated and the timber calculated accordingly. But it 
 must be remembered that the whole weight is never placed 
 upon the framing at once as seems to have been the idea 
 had in view by the designers of some centres. In building 
 the arch, it should be commenced at each buttress at the 
 same time (as is generally the case), and each side should 
 progress equally towards the crown. In designing the fram- 
 
CENTRE FOR A STONE BRIDGE. 
 
 231 
 
 ing, the effect produced by each successive layer of stone 
 should be considered. The pressure of the stones upon one 
 side should, by the arrangement of the struts, be counter- 
 poised by that of the stones upon the other side. 
 
 Over a river whose stream is rapid, or where it is neces- 
 sary to preserve an uninterrupted passage for the purposes 
 of navigation, the centre must be constructed without in- 
 termediate supports, and without a continued horizontal tie 
 at the base ; such a centre is shown at Fig. 109. In laying 
 the stones from the base up to a and c, the pieces bd and 
 bd act as ties to prevent any rising at b. After this, while 
 the stones are being laid from a and from c to b, they act as 
 struts; the piece f g is added for additional security. 
 Upon this plan, with some variation to suit circumstances, 
 
 FIG. 109. 
 
 centres may be constructed for any span usual in stone- 
 bridge building. 
 
 In bridge centres, the principal timbers should abut, and 
 not be intercepted by a suspension or radial piece between. 
 These should be in halves, notched on each side and bolted. 
 The timbers should intersect as little as possible, for the 
 more joints the greater is the settling ; and halving them 
 together is a bad practice, as it destroys nearly one half the 
 strength of the timber. Ties should be introduced across, 
 especially where many timbers 'meet ; and as the centre is 
 to serve but a temporary purpose, the whole should be de- 
 signed with a view to employ the timber afterwards for 
 other uses. For this reason, all unnecessary cutting should 
 be avoided. 
 
232 CONSTRUCTION. 
 
 Centres should be sufficiently strong- to preserve a 
 staunch and steady form during the whole process of build- 
 ing ; for any shaking or trembling will have a tendency to 
 prevent the mortar or cement from setting. For this pur- 
 pose, also, the centre should be lowered a trifle immedi- 
 ately after the key-stone is laid, in order that the stones may 
 take their bearing before the mortar is set ; otherwise the 
 joints will open on the underside. The trusses, in centring, 
 are placed at the distance of from 4 to 6 feet apart, accord- 
 ing to their strength and the weight of the arch. Between 
 every two trusses diagonal braces should be introduced to 
 prevent lateral motion. 
 
 In order that the centre may be easily lowered, the 
 frames, or trusses, should be placed upon wedge-formed 
 sills, as is shown at d (Fig. 109). These are contrived so as 
 to admit of the settling of the frame by driving the wedge 
 d with a maul, or, in large centres, with a piece of timber 
 mounted as a battering-ram. The operation of lowering a 
 centre should be very slowly performed, in order that the 
 parts of the arch may take their bearing uniformly. The 
 wedge pieces, instead of being placed parallel with the 
 truss, are sometimes made sufficiently long and laid through 
 the arch, in a direction at right angles to that shown at Fig. 
 109. This method obviates the necessity of stationing men 
 beneath the arch during the process of lowering ; and was 
 originally adopted with success soon after the occurrence of 
 an accident, in lowering a centre, by which nine men were 
 killed. 
 
 To give some idea of the manner of estimating the pres- 
 sures, in order to select timber of the proper scantling, cal- 
 culate the pressure (Art. 247) of the arch-stones from i to b 
 (Fig. 109), and suppose half this pressure concentrated at a, 
 and acting in the direction a f. Then, by the parallelogram 
 of forces (Art. 71), the strain in the several pieces compos- 
 ing the frame bda may be computed. Again, calculate 
 the pressure of that portion of the arch included between a 
 and c, and consider half of it collected at , and acting in a 
 vertical direction ; then, by the parallelogram of forces, the 
 pressure on the beams bd and </may be found. Add the 
 
JOINTS OF THE ARCH-STONES. 
 
 233 
 
 pressure of that portion of the arch which is included be- 
 tween i and b to half the weight of the centre, and consider 
 this amount concentrated at d, and acting in a vertical direc- 
 tion ; then, by constructing the parallelogram of forces, the 
 pressure upon dj may be ascertained. 
 
 The strains having been obtained, the dimensions of the 
 several pieces in the frames #</and bed may be found by 
 computation, as directed in the case of roof-trusses, from 
 Arts. 226 to 229. The tie-beams b d, b d, if made of suffi- 
 cient size to resist the compressive strain acting upon them 
 from the load at b, will be more than large enough to resist 
 the tensile strain upon them during the laying of the first 
 part of the arch-stones below a and c. 
 
 248. Arcli-Stones : JToint. In an arch, the arch-stones 
 are so shaped that the joints between them are perpendicu- 
 lar to the curve of the arch, or to its tangent at the point at 
 which the joint intersects the curve. In a circular arch, the 
 
 FIG. no. 
 
 joints tend toward the centre of the circle ; in an elliptical 
 arch, the joints may be found by the following process : 
 To find the direction of the joints for an elliptical arch ; 
 
 FIG. in. 
 
 a joint being wanted at a (Fig. no), draw lines from that 
 point to the foci, /and/; bisect the angle faf with the 
 line a b ; then a b will be the direction of the joint. 
 
234 CONSTRUCTION. 
 
 To find the direction of the joints for a parabolic arch ; 
 a joint being- wanted at a (Fig. 1 1 1), draw a e at right angles 
 to the axis eg; make eg equal to c e, and join a and g\ 
 draw a h at right angles to ag\ then a h will be the 
 direction of the joint. The direction of the joint from b is 
 found in the same manner. The lines a gaud b f are tan- 
 gents to the curve at those points respectively ; and any 
 number of joints in the curve may be obtained by first 
 ascertaining the tangents, and then drawing lines at right 
 angles to them. (See Art. 462.) 
 
 JOINTS. 
 
 24-9. 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 shrink to 
 
 FIG. ii2. 
 
 even a small degree, the strength would depend altogether 
 on the bolts. It would be made much stronger by indent- 
 ing the pieces together, as at the upper edge of the tie-beam 
 
 FIG. 113. 
 
 in Fig. 113, or by placing keys in the joints, as at the lower 
 edge in the same figure. This process, however, weakens 
 the beam in proportion to the depth of the indents. 
 
 FIG. 114 
 
 Fig. 114 shows a method of scarfing, or splicing, a tie- 
 beam without bolts. The keys are to be of well-seasoned, 
 hard wood, and, if possible, very cross-grained. The addi- 
 
THE SPLICING OF TIMBER. 235. 
 
 tion of bolts would make this a very strong splice, or even 
 white-oak pins would add materially to its strength. 
 
 Fig. 1 1 5 shows about as strong a splice, perhaps, as can 
 well be made. It is to be recommended for its simplicity ; 
 as, on account of there being no oblique joints in it, it can 
 be readily and accurately executed. A complicated joint is 
 the worst that can be adopted ; still, some have proposed 
 joints that seem to have little else besides complication to 
 recommend them. 
 
 In proportioning the parts of these scarfs, the depths of 
 
 FIG. 115. 
 
 all the indents taken together should be equal to one third 
 of the depth of the beam. In oak, ash or elm, the whole 
 length of the scarf should be six times the depth, or thick- 
 ness, of the beam, when there are no bolts ; but, if bolts in- 
 stead of indents are used, then three times the breadth ; and 
 when both methods are combined, twice the depth of the 
 beam. The length of the scarf in pine and similar soft 
 woods, depending wholly on indents, should be about 12 
 times the thickness, or depth, of the beam ; when depend- 
 ing wholly on bolts, 6 times the breadth ; and when both 
 methods are combined, 4 times the depth. 
 
 FIG. 116. 
 
 Sometimes beams have to be pieced that are required 
 to resist cross-strains such as a girder, or the tie-beam of a 
 roof when supporting the ceiling. In such beams, the 
 fibres of the wood in the upper part are compressed ; and 
 therefore a simple butt joint at that place (as in Fig. 116) 
 is far preferable to any other. In such case, an oblique 
 joint is the very worst. The under side of the beam being 
 in a state of tension, it must be indented or bolted, or both ; 
 and an iron plate under the heads of the bolts gives a great 
 addition of strength. 
 
236 CONSTRUCTION. 
 
 Scarfing requires accuracy and care, as all the indents 
 should bear equally ; otherwise, one being strained more 
 than another, there would be a tendency to splinter off the 
 parts. Hence the simplest form that will attain the object 
 is by far the best. In all beams that are compressed end- 
 wise, abutting joints, formed at right angles to the direction 
 of their length, are at once the simplest and the best. For a 
 temporary purpose, Fig. 112 would do very well ; it would 
 be improved, however, by having a piece bolted on all four 
 sides. Fig. 113, and indeed each of the others, since they 
 have no oblique joints, would resist compression well. 
 
 In framing one beam into another for bearing purposes, 
 such as a floor-beam into a trimmer, the best place to make 
 the mortise in the trimmer is in the neutral line (Arts. 120, 
 121), which is in the middle of its depth. Some have 
 thought that, as the fibres of the upper edge are compressed, 
 
 FIG. 117. 
 
 a mortise might be made there, and the tenon driven in 
 tight enough to make the parts as capable of resisting the 
 compression as they would be without it ; and they have 
 therefore concluded that plan to be the best. This could 
 not be the case, even if the tenon would not shrink ; for a 
 joint between two pieces cannot possibly be made to resist 
 compression so well as a solid piece without joints. The 
 proper place, therefore, for the mortise is at the middle of 
 the depth of the beam ; but the best place for the tenon, in 
 the floor-beam, is at its bottom edge. For the nearer this 
 is placed to the upper edge, the greater is the liability for it 
 to splinter off; if the joint is formed, therefore, as at Fig. \ 17, 
 it will combine all the advantages that can be obtained. Dou- 
 ble tenons are objectionable, because the piece framed into 
 is needlessly weakened, and the tenons are seldom so accu- 
 rately made as to bear equally. For this reason, unless the tusk 
 
THE FRAMING IN A ROOF-TRUSS. 
 
 237 
 
 at a in the figure fits exactly, so as to bear equally with the 
 tenon, it had better be omitted. And in sawing the shoulders 
 care should be taken not to saw into the tenon in the least, 
 as it would wound the beam in the place least able to bear it. 
 
 Thus it will be seen that framing weakens both pieces, 
 more or less. It should, therefore, be avoided as much as 
 possible , and where it is practicable one piece should rest 
 upon the other, rather than be framed into it. This re- 
 mark applies to the bearing of floor-beams on a girder, to 
 the purlins and jack-rafters of a roof, etc. 
 
 In a framed truss for a roof, bridge, partition, etc., the 
 joints should be so constructed as to direct the pressures 
 through the axes of the several pieces, and also to avoid 
 every tendency of the parts to slide. To attain this object, 
 
 FIG. 118. 
 
 FIG. 119. 
 
 FIG. 120. 
 
 the abutting surface on the end of a strut should be at 
 right angles to the direction of the pressure ; as at the joint 
 shown in Fig. 1 18 for the foot of a rafter (see Art. 86), in Fig. 
 1 19 for the head of a rafter, and in Fig. 120 for the foot of a 
 strut or brace. The joint at Fig. 118 is not cut completely 
 across the tie-beam, but a narrow lip is left standing in the 
 middle, and a corresponding indent is made in the rafter, to 
 prevent the parts from separating sideways. The abutting 
 surface should be made as large as the attainment of other 
 necessary objects will admit. The iron strap is added to 
 prevent the rafter sliding out, should the end of the tie- 
 beam, by decay or otherwise, splinter off. In making- the 
 joint shown at Fig. 119, it should be left a little open at a, 
 so as to bring the parts to a fair bearing at the settling of 
 the truss, which must necessarily take place from the shrink- 
 ing of the king-post and other parts. If the joint is made 
 fair at first, when the truss settles it will cause it to open at 
 
CONSTRUCTION. 
 
 the under side of the rafter, thus throwing the whole pres- 
 sure upon the sharp edge at a. This will cause an indenta- 
 tion in the king-post, by which the truss will be made to 
 settle further ; and this pressure not being in the axis of the 
 rafter, it will be greatly increased, thereby rendering the 
 rafter liable to split and break. 
 
 FIG. 121. 
 
 FIG. 122. 
 
 FIG. 123. 
 
 If the rafters and struts were made to abut end to end, 
 as in Figs. 121, 122 and 123, and' the king or queen post 
 notched on in halves -and bolted, the ill effects of shrinking 
 would be avoided. This method has been practised with 
 success in some of the most celebrated bridges and roofs in 
 Europe ; and, were its use adopted in this country, the un- 
 seemly sight of a hogged ridge would seldom be met with. 
 
 FIG. 124. 
 
 FIG. 125. 
 
 A plate of cast-iron between the abutting surfaces will 
 equalize the pressure. 
 
 Fig. 124 is a proper joint for a collar-beam in a small 
 roof : the principle shown here should characterize all tie- 
 joints. The dovetail joint, although extensively practised 
 in the above and similar cases, is the very worst that can be 
 employed. The shrinking of the timber, if only to a small 
 
WHITE-OAK PINS AND IRON STRAPS. 239 
 
 degree, permits the tie to withdraw as is shown at Fig. 
 12$. The dotted line shows the position of the tie after it 
 has shrunk. 
 
 Locust and white-oak pins are great additions to the 
 strength of a joint. In many cases they would supply the 
 place of iron bolts ; and, on account of their small cost, they 
 should be used in preference wherever the strength ef iron 
 is not requisite. In small framing, good cut nails are of 
 great service at the joints ; but they should not be trusted 
 to bear any considerable pressure, as they are apt to be 
 brittle. Iron straps are seldom necessary, as all the joinings 
 in carpentry may be made without them. They can be 
 used to advantage, however, at the foot of suspending-pieces, 
 and for the rafter at the end of the tie-beam. In roofs for 
 ordinary purposes, the iron straps for suspending-pieces 
 may be as follows : When the longest unsupported part of 
 the tie-beam is 
 
 10 feet, the strap may' be i inch wide by T 3 T thick. 
 
 15 " " " i " " i " 
 
 2Q 2 u JL " 
 
 In fastening a strap, its hold on the suspending-piece will be 
 much increased by turning its ends into the wood. Iron 
 straps should be protected from rust ; for thin plates of iron 
 decay very soon, especially when exposed to dampness. 
 For this purpose, as soon as the strap is made let it be 
 heated to about a blue heat, and, while it is hot, pour over 
 its entire surface raw linseed oil, or rub it with beeswax. 
 Either of these will give it a coating which dampness will 
 not penetrate. 
 
SECTION III. STAIRS. 
 
 250. tair : 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 determined with discrimi- 
 nating taste, add greatly to the comfort and elegance of a 
 structure. As regards their position, the first object should 
 be to have them near the middle of the building, in order 
 that they may afford an equally easy access to all the rooms 
 and passages. Next in importance is light ; to obtain which 
 they would seem to be best situated near an outer wall, in 
 which windows might be constructed for the purpose ; yet 
 a skylight, or opening in the roof, would not only provide 
 light, and so secure a central position for the stairs, but may 
 be made, also, to assist materially as an ornament to the 
 building, and, what is of more importance, afford an oppor- 
 tunity for better ventilation. 
 
 All stairs, especially those of the most important build- 
 ings, should be erected of stone or some equally durable and 
 fire-resisting material, that the means of egress from a burn- 
 ing building may not be too rapidly destroyed. 
 
 Winding stairs, or those in which the direction is gradu- 
 ally changed by means of winders, or steps which taper in 
 width, are interesting by reason of the greater skill required 
 in their construction ; but are objectionable, for the reason 
 that children are exposed to accident by their liability to fall 
 when passing over the narrow ends of the steps. Stairs of 
 this kind should be tolerated only where there is not suffi- 
 cient space for those with flyers, or steps of parallel width. 
 
 Stairs in one long continuous flight are also objection- 
 able. Platforms or landings should be introduced at inter- 
 vals, so that any one flight may not contain more than about 
 twelve or fifteen steps. 
 
 The width of stairs should be in accordance with the im- 
 
KHORSABAD. ASSYRIAN TEMPLE, RESTORED. 
 
THE GRADE OF STAIRS. 241 
 
 portance of the building in which they are placed, varying 
 from 3 to 12 feet. Where two persons are expected to pass 
 each other conveniently the least width admissible is 3 feet. 
 Still, in crowded cities, where land is valuable, the space 
 allowed for passages is correspondingly small, and in these 
 stairs are sometimes made as narrow as 2\ feet. 
 
 From 3 to 4 feet is a suitable width for a good dwelling ; 
 while 5 feet will be found ample for stairs in buildings occu- 
 pied by many people ; and from 8 to 12 feet is sufficient for 
 the width of stairs in halls of assembly. 
 
 To avoid tripping or stumbling, care should be exer- 
 cised, in the planning of a stairs, to secure an even grade. 
 To this end, the nosing, or outer edge, of each step should be 
 exactly in line with all the other nosings. In stairs com- 
 posed of both flyers and winders, precaution in this regard 
 is especially needed. In such stairs, the steps flyers and 
 winders alike should be of one width on the line along 
 which a person would naturally walk when having his hand 
 upon the rail. This tread-line, consequently, would be paral- 
 lel with the hand-rail, and is usually taken at a distance of 
 from 1 8 to 20 inches from the centre of it. In the plan of 
 the stairs this tread-line should be drawn and divided into 
 equal parts, each part being the tread, or width of a flyer 
 from the face of one riser to the face of the next. 
 
 251. The 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. Hence 
 the difficulty of ascent will be in proportion to the height of 
 each step, or to the rise, as it is termed. To facilitate the 
 operation of going up stairs, therefore, the -risers should be 
 low. The grade of a stairs, or its angle of ascent, depends 
 not only upon the height of the riser, but also upon the 
 width of the step ; and this has a certain relation to the 
 riser ; for the width of a step should be in proportion to the 
 smallness of the angle of ascent. 
 
 The distance from the top of one riser to the top of the 
 next is the distance travelled at each step taken, and this dis- 
 
242 STAIRS. 
 
 tance should vary as the grade of the stairs ; for a person 
 who in climbing a ladder, or a nearly vertical stairs, can 
 travel only 12 inches, or less, at a step, will be able with 
 equal or greater facility to travel at least twice this distance 
 on level ground. The distance travelled, therefore, should 
 be in proportion inversely to the angle of ascent ; or, the di- 
 mensions of riser and step should be reciprocal : a low rise 
 should have a wide step, and a high rise a narrow step. 
 
 252. Pitch-Board : Relation of Rise to Tread. Among 
 the various devices for determining the relation of the rise 
 to the tread, or net width of step, one is to make the sum of 
 the two equal to 18 inches. 
 
 For example, for a rise of 6 inches the tread should be 
 12, for 7 inches the tread should be n ; or 
 
 6 + 12 = 18 8 + 10 = 18 
 6J + iii= 18 8i + 9i = 18 
 
 7 + ii = 18 9 + 9 = 18 
 ;+ ioi= 18 Qj + 8i= 18 
 
 This rule is simple, but the results in extreme cases are not 
 satisfactory. If the ascent of a stairs be gradual and easy, 
 the length from the top of one rise to that of another, or the 
 hypothenuse of the pitch-board, may be proportionally long ; 
 but if the stairs be steep, the length must be shorter. 
 
 There is a French method, introduced by Blondel in his 
 Cours d' Architecture. It is referred to in Gwilt's Encyclo- 
 pedia, Art. 2813. 
 
 This method is based upon the assumed distance of 24 
 inches as being a convenient step upon level ground, and 
 upon 12 inches as the most convenient height to rise when 
 the ascent is vertical. These are French inches, old system. 
 The 24 inches French equals about 25^ inches English. 
 
 With these distances as base and perpendicular, a right- 
 angled triangle is formed, which is used as a scale upon 
 which the proportions of a pitch-board are found. For 
 example, let a line be drawn from any point in the hypothe- 
 nuse of this triangle to the right angle of the triangle ; then 
 this line will equal the length of the pitch-board, along the 
 
PROPORTIONS OF THE PITCH-BOARD. 243 
 
 rake, for a stairs having a grade equal to the angle formed 
 by this line and the base-line of the scale. 
 
 In the absence of the triangular scale, the lengths of the 
 pitch-boards, as found by this rule, may be computed by this 
 expression 
 
 W=2$3 r -2A; (107.) 
 
 in which W equals the tread, or base of the pitch-board, and 
 h the riser, or its perpendicular height. 
 For example, let // = 6 ; then 
 
 ^=25^-2x6=13^. 
 
 This result is greater than would be proper in some cases. 
 
 The length of the hypothenuse of the pitch-board should 
 be proportional not only to the angle of ascent (Art. 251), but 
 also to the strength and height of the class of people who 
 are to use the stairs. Tall and strong persons will take 
 longer steps than short and feeble people. The hypothe- 
 nuse of the pitch-board should be made in proportion to 
 the distance taken at a step on level ground by the persons 
 who are to use the stairs. 
 
 If people are divided into two classes, one composed of 
 robust workmen and the other of delicate women and in- 
 firm men, then there may be two scales formed for the pitch- 
 boards of stairs one to be used for shops and factories, and 
 the other for dwellings. The distance on level ground trav- 
 elled per step, by men, varies from about 26 to 32 inches, or 
 on an average 28 inches. The height to which men are 
 accustomed to rise on ladders is from 12 to 16 inches at each 
 step, or on the average 14 inches. 
 
 With these dimensions, therefore, of 14 and 28 inches, a 
 scale may be formed for pitch-boards for stairs, in buildings 
 to be used exclusively by robust workmen. And with 12 
 and 24 inches another scale may be formed for pitch-boards 
 for stairs, in buildings to be used by women and feeble 
 people. These two scales are both shown in Fig. 126. 
 They are made thus : Let C A B be a right angle. Make A 
 B equal to 28 inches, and A C equal to 14 ; then join B and 
 
244 
 
 STAIRS. 
 
 C. At right angles to C B, from A, draw A F\ then with 
 A F for radius describe the arc F G. Then a' line, as A K 
 or A L, drawn from A at any angle with A B and limited by 
 the line G FB will give the length of the hypothenuse of 
 the pitch-board, for shop stairs of a grade equal to the angle 
 which said line makes with A B. From K, perpendicular to 
 A B, draw K N "; then K N will be the proper riser for a 
 pitch-board of which A N is the tread. So, likewise, L M 
 will be the appropriate riser for the tread A M. The arc F G 
 is introduced to limit the rake-line of pitch-boards occur- 
 ring between F and C, in order to avoid making them longer 
 than the one at F. The scale for the stairs for dwellings is 
 made in the same manner ; A D 24 inches being the base, 
 A E = 12 inches the rise, and J H D the line limiting the 
 rake-lines of pitch-boards. 
 
 M 
 
 N 
 
 FIG. 126. 
 
 To compute the length of risers and treads, we have for 
 the scale for shops, for those occurring between F and B 
 
 r = 4(28-/): (108.) 
 
 fi=28-2rj (109.) 
 
 and for those between F and G, we have 
 
 (108, A.) 
 (109, A.) 
 
 r= 
 
 -/ 2 ; 
 
 1/756^8- 
 
 For the scale for dwellings, we have, for those occurring 
 between H and D 
 
 r =i(24-/); (108, B.) 
 
 t = 24 2 r ; (109, B.) 
 
STAIRS FOR SHOPS AND FOR DWELLINGS. 
 
 and for those between H and J, we have 
 
 t = 
 
 245 
 
 , c.) 
 
 (109, C.) 
 
 where, in each equation, r represents the riser, and t the 
 tread, or net step. 
 
 By these formulae, the following tables have been com- 
 puted : 
 
 STAIRS FOR SHOPS. 
 
 Rise. 
 
 Tread. 
 
 Ratio Rise to Tread. 
 
 Rise. 
 
 Tread. 
 
 Ratio Rise to Tread. 
 
 2- 
 
 24- 
 
 to 12- 
 
 7-40 
 
 13-20 
 
 to -78 
 
 3' 
 
 22- 
 
 " 7-33 
 
 7-60 
 
 12-80 
 
 " -68 
 
 3-50 
 
 21- 
 
 " 6- 
 
 7-80 
 
 12-40 
 
 " -59 
 
 4- 
 
 20- 
 
 " 5- 
 
 8- 
 
 12- 
 
 50 
 
 4-50 
 
 19. 
 
 " 4-22 
 
 8-20 
 
 II-6 
 
 " -41 
 
 5' 
 
 18- 
 
 " 3'6o 
 
 8-50 
 
 II- 
 
 29 
 
 5-4 
 
 17-20 
 
 3-19 
 
 8-80 
 
 10-40 
 
 " -18 
 
 5-7 
 
 16-60 
 
 ' 2-91 
 
 9' 
 
 10- 
 
 " -ii 
 
 6- 
 
 16- 
 
 ' 2-67 
 
 9-30 
 
 9-40 
 
 " -01 
 
 6-25 
 
 I5-50 
 
 : 2-48 
 
 9-60 
 
 8-80 
 
 " 0-92 
 
 6-50 
 
 15- 
 
 ' 2-31 
 
 10- 
 
 8- 
 
 . ' 0-80 
 
 6-70 
 
 14-60 
 
 ' 2-18 
 
 10-50 
 
 7' 
 
 4 0-67 
 
 6-90 
 
 14-20 
 
 ' 2-06 
 
 ii- 
 
 6- 
 
 1 '55 
 
 7- 
 
 14- 
 
 ' 2- 
 
 11-50 
 
 4-95 
 
 ' 0-43 
 
 7-20 
 
 13-60 
 
 ' I-8 9 
 
 12- 
 
 3-58 
 
 i ' 0-30 
 
 STAIRS FOR DWELLINGS. 
 
 Rise. 
 
 Tread. 
 
 Ratio Rise to Tread. 
 
 Rise. 
 
 Tread. 
 
 Ratio Rise to Tread. 
 
 2- 
 
 2O- 
 
 I to 10- 
 
 7-40 
 
 9-20 
 
 i .to -24 
 
 3" 
 
 18- 
 
 I " 6- 
 
 7-50 
 
 9' 
 
 I " -2O 
 
 3-50 
 
 17- 
 
 i ' 4-86 
 
 7'60 
 
 8-80 
 
 " .16 
 
 4' 
 
 16- 
 
 I ' 4- 
 
 7-70 
 
 8-60 
 
 " -12 
 
 4-50 
 
 15- 
 
 ' 3-33 
 
 7-80 
 
 8-40 
 
 " -08 
 
 5' 
 
 14- 
 
 2-80 
 
 7-90 
 
 8-20 
 
 " -04 
 
 5-40 
 
 13-20 
 
 ' 2-44 
 
 8- 
 
 8- 
 
 
 
 5-70 
 
 12-60 
 
 " 2-21 
 
 8-10 
 
 7-80 
 
 " 0-96 
 
 6- 
 
 12- 
 
 " 2- 
 
 8-30 
 
 7-40 
 
 " 0-89 
 
 6-25 
 
 II-5O 
 
 " -8 4 
 
 8-50 
 
 7- 
 
 " 0-82 
 
 6-50 
 
 II- 
 
 " -6 9 
 
 8-75 
 
 6-50 
 
 " o-74 
 
 6-75 
 
 IO-5O 
 
 ' -56 
 
 9' 
 
 6- 
 
 " 0-67 
 
 7- 
 
 10- 
 
 43 
 
 9-30 
 
 5-40 
 
 " 0-58 
 
 7-10 
 
 9-80 
 
 ' -38 
 
 9-60 
 
 4-80 
 
 " 0-50 
 
 7-20 
 
 9-60 
 
 33 
 
 10- 
 
 3.90 
 
 " 0-39 
 
 7-30 
 
 9.40 
 
 -29 
 
 10-50 
 
 2 -2O 
 
 ' 0-21 
 
 These tables will be useful in determining questions in- 
 
246 STAIRS. 
 
 volving the proportion between the rise and tread of a 
 pitch-board. 
 
 For stairs in which the run is limited, to determine the 
 number of risers which would give an easy ascent : Divide 
 the run by the height, and find in the proper table, above, 
 the ratio nearest to the quotient, and in a line with this ratio, 
 in the second column to the left, will be found the corre- 
 sponding riser. With this divide the rise in inches ; the quo- 
 tient, or the nearest whole number thereto, will be the required 
 number of risers in the stairs. 
 
 Example. For the stairs in a dwelling, let the rise be 12' 
 8", or I524nches. Let the run between the extreme risers 
 be 17' 2". To this, for the purpose of obtaining the correct 
 angle of ascent, by having an equal number of risers and 
 treads, add, for one more tread, say 10 inches, its probable 
 width; thus making the total run 1 8 feet, or 216 inches. 
 Thus we have for the run 216, and for the rise 152. Divid- 
 ing the former by the latter gives i -42 nearly. In the table 
 of stairs for dwellings, the ratio nearest to this is I -43, and in 
 the line to the left, in the second column, is 7, the approxi- 
 mate size of riser appropriate to this case. Dividing the 
 rise, 152 inches, by this 7, we have 2 if as the quotient. 
 
 This is nearer to 22 than to 21 ; therefore, the number of 
 risers required is 22. 
 
 When the number of risers is determined, then the rise 
 divided by this number will give the height of each riser ; 
 thus, in the above case, the rise is 152 inches. This divided 
 by 22 gives 6-909 inches for the height of the riser. 
 
 When the height of the riser is known, then, if the run is 
 unlimited, the width of tread will be found in the proper table 
 above. For example, if the riser is 7 inches or nearly that, 
 then in the table of stairs for dwellings, in the next column 
 to the right, and opposite 7 in the column of risers, is found 
 10, the approximate width of tread. By the use of equation 
 (109, B.), the width may be had exactly according to the 
 scale. For example, equation (109, B.) with 6-91 for the 
 riser, becomes 
 
 t 24 2 x 6-91 = io- 18, 
 or about io T 3 ^ inches. 
 
TO CONSTRUCT THE PITCH-BOARD. 247 
 
 When the run is limited and the number of risers is 
 known, then the width of tread is obtained by dividing the 
 run by the number of treads. There are always of treads 
 one less than there are of risers, in each flight. 
 
 253. Dimensions of the Pitch-Board. The first thing 
 in commencing to build a stairs is to make the //fc/j-board ; 
 this is done in the following manner : Obtain very accurate- 
 ly, in feet and inches, the rise, or perpendicular height, of the 
 story in which the stairs are to be placed. This must be 
 taken from the top of the lower floor to the top of the upper 
 floor. Then, to obtain the number of rises and treads and 
 their size, proceed as directed in Art. 252. Having obtained 
 these, the pitch-board may be made in the following man- 
 ner: Upon a piece of well-seasoned board about -| of an 
 inch thick, having one edge jointed 
 straight and square, lay the corner 
 of a steel square, as shown at Fig. 127. 
 Make a b equal to the riser, and b c 
 equal to the tread ; mark along the 
 edges with a knife, and cut by the FlG< I27 ' 
 
 marks, making the edges of the pitch - board perfectly 
 square. The grain of the wood should run in the direction 
 indicated in the figure, because, in case of shrinkage, the 
 rise and the tread will be equally affected by it. When a 
 pitch-board is first made, the dimensions of the riser and 
 tread should be preserved in figures, in order that, in case 
 of shrinkage or damage otherwise, a second may be 
 made. 
 
 254. The String of a Stairs. The space required for 
 timber and plastering under the steps is about 5 inches for 
 ordinary stairs, or 6 inches if furred ; set a gauge, there- 
 fore, at 5 or 6 inches, as the case requires, and run it on the 
 lower edge of the plank, as ab (Fig. 128). Commencing at 
 one end, lay the longest side of the pitch-board against the 
 gauge-mark, a b, as at c, and draw by the edges the lines for 
 the first rise and tread ; then place it successively as at d, e, 
 
248 
 
 STAIRS. 
 
 and f, until the required number of risers shall be laid down. 
 To insure accuracy, it is well to ascertain the theoretical 
 raking length of the pitch-board by computation, as in note 
 to Art. 536, by getting the square root of the sum of the 
 squares of the rise and run, and using this by which to 
 divide the line ab into equal parts. 
 
 f 
 
 FIG. 128. 
 
 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 
 
 FIG. 129, 
 
 inches long, two or three of which, in the length of the 
 step, are glued in the corner. The cove at b is planed up 
 square, glued in, and stuck or moulded after the glue is 
 set. 
 
 PLATFORM STAIRS. 
 
 256. Platform Stair* : the Cylinder. A platform stairs 
 ascends from one story to another in two or more flights, 
 having platforms or landings between for resting and to 
 change their direction. This kind of stairs, being simple, is 
 
CYLINDER OF PLATFORM STAIRS. 
 
 249 
 
 easily constructed, and at the same time is to be preferred 
 
 to those with winders, for the convenience it affords in use 
 
 (Art. 250). The cylinder may be of 
 
 any diameter desirable, from a few 
 
 inches to 3 or more feet, but it is 
 
 generally small, about 6 inches. It 
 
 may be worked out of one solid 
 
 piece, but a better way is to glue 
 
 together 3 pieces, as in Fig. 130 ; in 
 
 which the pieces a, b, and c compose 
 
 the cylinder, and d and e represent 
 
 parts of the strings. The strings, 
 
 after being glued to the cylinder, are secured with screws. 
 
 The joining at o and o is the most proper for that kind of 
 
 joint. 
 
 FIG. 130. 
 
 FIG. 131. 
 
 257. Form of Lower Edge of Cylinder. Find the 
 
 stretch-out, de (Fig. 131), of the face of the cylinder, a be, 
 
2?O STAIRS. 
 
 according to Art. 524; from d and e draw df and eg at 
 right angles to d^; draw kg parallel to de, and make hf 
 and -2 each equal to one riser; from i and f draw ij and 
 / parallel to hg\ place the tread of the pitch-board at these 
 last lines, and draw by the lower edge the lines kh and il ; 
 parallel to these draw m n and op, at the requisite distance 
 for the dimensions of the string ; from s, the centre of the 
 plan, draw sq parallel to df\ divide // q and qg each into two 
 equal parts, as at v and w ; from v and w draw v n and w o 
 parallel to fd\ join n and o, cutting qs in r ; then the angles 
 unr and rot, being eased off according to Art. 521, will give 
 the proper curve for the bottom edge of the cylinder. A 
 centre may be found upon which to describe these curves, 
 thus : from u draw u x at right angles to m n ; from r draw 
 r x at right angles to no ; then x will be the centre for the 
 curve ur. The centre for the curve rt may be found in a 
 similar manner. Centres from which to strike these curves 
 are usually quite unnecessary ; an experienced workman 
 will readily form the curves guided alone by his practised 
 eye. 
 
 FIG. 132. 
 
 258. Position of the Baluters. 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, e d\ and place the centre of the other baluster, a, 
 half the tread from the centre of the first. A line through 
 the centre of the rail will occur vertically over the centres 
 of the balusters. The usual length of the balusters is 2 feet 
 5 inches and 2 feet 9 inches respectively, for the short and 
 long balusters. Their length may be greater than is here 
 indicated, but, for safety, should never be less. The differ- 
 ence in length between the short and long balusters is 
 equal to one half the height of a riser. 
 
CONSTRUCTION OF WINDING STAIRS. 
 
 251 
 
 259. Winding stairs: have the steps narrower atone 
 end than at the other. In some stairs there are steps of 
 parallel width incorporated with the tapering steps ; in this 
 case the former are called flyers, and the latter winder:. 
 
 260. Regular Winding Stairs In Fig. 133, abed rep- 
 resents the inner surface of the wall enclosing the space 
 allotted to the stairs, a e the length of the steps, and efgk 
 the cylinder, or face of the front-string. The line a e is given 
 as the face of the first riser, and the point / for the limit of 
 
 FIG. 133 
 
 the last. Make e i equal to 18 inches, and upon o, with o i 
 for radius, describe the arc i j \ obtain the number of risers 
 and of treads required to ascend to the floor at j, according 
 to Art. 252, and divide the arc ij into the same number of 
 equal parts as there are to be treads : through the points of 
 division, 1,2, 3, etc., and from the wall-string to the front- 
 string, draw lines tending to the centre, o : then these lines 
 will represent the face of each riser, and determine the form 
 and width of the steps. Allow the necessary projection for 
 the nosing beyond a e, which should be equal to the thick- 
 
STAIRS. 
 
 ness of the step, and then a e I k will be the dimensions for 
 each step. Make a pitch-board for the wall-string having a k 
 for the tread, and the rise as previously ascertained : with 
 this lay out on a thicknessed plank the several risers and 
 treads, as at Fig. 128, gauging from the upper edge of the 
 string for the line at which to set the pitch-board. 
 
 Upon the back of the string, with a ij-inch dado plane, 
 make a succession of grooves ij inches apart, and parallel 
 with the lines for the risers on the face. These grooves 
 must be cut along the whole length of the plank, and deep 
 enough to admit of the plank's bending around the curve 
 abed. Then construct a drum, or cylinder, of any com- 
 mon kind of stuff, made to fit a curve with a radius the 
 thickness of the string less than oa ; upon this the string must 
 be bent, and the grooves filled with strips of wood, called 
 keys, which must be very nicely fitted and glued in. After 
 it has dried, a board thin enough to bend around on the out- 
 side of the string must be glued on from one end to the 
 other, and nailed with clout-nails. In doing this, be careful 
 not to nail into any place opposite to where a riser or step is 
 to enter on the face. 
 
 After the string has been on the drum a sufficient time 
 for the glue to set, take it off, and cut the mortices for the 
 steps and risers on the face at the lines previously made ; 
 which may be done by boring with a centre-bit half through 
 the string, and nicely chiselling to the line. The drum need 
 not be made to extend over the whole space occupied by the 
 stairs, but merely so far as requisite to receive one piece of 
 the wall-string at a time ; for it is evident that more than 
 One will be required. The front-string may be constructed 
 in the same manner ; taking e I instead of a k for the tread of 
 the pitch-board, dadoing it with a smaller dado plane, and 
 bending it on a drum of the proper size. 
 
 261. Winding Stairs : Shape and Position of Timbers. 
 
 The dotted lines in Fig. 133 show the position of the timbers 
 as regards the plan ; the shape of each is obtained as follows: 
 In Fig. 134, the line i a is equal to a riser, less the thickness 
 ot the floor, and the lines 2 m, 3 n, 4 <?, 5 /, and 6 q are each 
 
TIMBERS FOR WINDING STAIRS. 253 
 
 equal to one riser. The line a 2 is equal to a m in Fig. 133, 
 the line m 3 to m n in that figure, etc. In drawing this 
 figure, commence at a, and make the lines a i and a 2 of the 
 length above s*pecified, and draw them at right angles to 
 each other ; draw 2 m at right angles to a 2, and m 3 at 
 right angles to ;;/ 2, and make 2 m and m 3 of the lengths 
 as above specified ; and so proceed to the end. Then 
 through the points i, 2, 3, 4, 5, and 6 trace the line \b\ upon 
 the points i, 2, 3, 4, etc., with the size of the timber for 
 radius, describe arcs as shown in the figure, and by these 
 the lower line may be traced parallel to the upper. This 
 will give the proper shape for the timber, a b, in Fig. 133 ; 
 and that of the others may be found in a similar manner. In 
 ordinary cases, the shape of one face of the timber will be 
 sufficient, for a good workman can easily hew it to its 
 proper level by that ; but where great accuracy is desirable, 
 a pattern for the other side may be found in the same man- 
 
 FIG. 134. 
 
 ner as for the first. In many cases, the timbers beneath cir- 
 cular stairs are put up after the stairs are erected, and with- 
 out previously giving them the required form ; the work- 
 man in shaping them being guided by the form marked out 
 by the lower edge of the risers. 
 
 262. Winding Stairs with Flyers : Grade of Front- 
 String. In stairs of this kind, if the winders are confined to 
 the quarter circle, the transition from the winders to the 
 flyers is too abrupt for convenience, as well as in appear- 
 ance. To remove this unsightly bend in the rail and string, 
 it is usual to take in among the winders one or more of the 
 flyers, and thus graduate the width of the winders to that of 
 the flyers. But this is not always done so as to secure the 
 best results. By the method now to be shown, both rail and 
 strings will be gracefully graded. In Fig. 135, a b repre- 
 sents the line of the facia along the floor of the upper story, 
 
254 STAIRS. 
 
 bee the face of the cylinder, and c d the face of the front- 
 string. Make gb equal to of the diameter of the baluster, 
 and parallel to a b, b e c, and c d draw the centre-line of the 
 rail, fg> g k z, and ij\ make gk and gl each equal to half the 
 width of the rail, and through k and /, parallel to the centre- 
 line, draw lines for the convex and the concave sides of the 
 rail ; tangical to the convex side of the rail, and parallel to 
 k m, draw ;/ o ; obtain the stretch-out, q r, of the semicircle, 
 k p m, according to Art. 524; extend a b to /, and k m to s; 
 make c s equal to the length of the steps, and i u equal to 18 
 inches, and parallel to m p describe the arcs s t and u 6 ; 
 from / draw / w, tending to the centre of the cylinder ; from 
 6, and on the line 6 u x, run off the regular tread, as at 5, 4, 
 3, 2, i, and v\ make ' u x equal to half the arc u 6, and make 
 the point of division nearest to x, as v, the limit of the par- 
 allel steps, or flyers ; make r o equal to m z; from o draw o 
 a** at right angles to ;/ o, and equal to one riser; from a 2 
 draw a* s parallel to n o, and equal to one tread; from s, 
 through o, draw s b*. 
 
 Then from w draw w c 2 at right angles to ;/ o, and set up 
 on the line w c^ the same number of risers that the floor, A, 
 is above the first winder, B, as at i, 2, 3, 4, 5, and 6 ; through 
 5 (on the arc 6 u) draw d* e*, tending to the centre of the 
 cylinder; from e*~ draw erf* at right angles to 110, and 
 through 5 (on the line w c*} draw g* f 2 parallel to n o\ 
 through 6 (on the line zu c 2 ) and/ 2 draw the line h* b*\ 
 make 6 c~ equal to half a riser, and from c* and 6 draw c' 2 i' 2 
 and 6/ 2 parallel to n o\ make h* i* equal to 7/ 2 / 2 ; from i- 
 draw i* k* at right angles to i* h-, and from f~ draw / 2 k* 
 at right angles to / 2 /* 2 ; upon k~, with 2 / 2 for radius, de- 
 scribe the arc / 2 * 2 ; make b~ / 2 equal to 2 / 2 , and ease oft 
 the angle at b* by the curve/ 2 / 2 . In the figure, the curve 
 is described from a centre, but as this might be imprac- 
 ticable in a full-size plan, the curve may be obtained accord- 
 
 * In the references a 2 , />', etc., a new form is introduced for the first time. 
 During the time taken to refer to the figure, the memory of the form of these 
 may pass from the mind, while that of the sound alone remains; they may 
 then be mistaken for a 2, b 2, etc. This can be avoided in reading by giving 
 them a sound corresponding to their meaning, which is a second, b second, etc. 
 
MOULDS FOR QUARTER-CIRCLE STAIRS. 
 
 2 55 
 
 ing to Art. 521. Then from i, 2, 3, and 4 (on the line w c~) 
 draw lines parallel to n o, meeting the curve in w 2 , n~, o 2 , 
 and / 2 ; from these points draw lines at right angles to ;/ o, 
 and meeting it in jr 2 , r 2 , s z , and / 2 ; from x~ and r 2 draw 
 
 FIG. 135. 
 
 lines tending to z/ 2 , and meeting the convex side of the rail 
 in j/ 2 and 2 ; make m ?> 2 equal to r s~, and m w* equal to 
 rt*-, fromj/ W 2 , and w 2 , through 4, 3, 2, and i, draw lines 
 meeting the line of the wall-string in a 3 , 3 , c\ and rt 78 ; from 
 
256 STAIRS. 
 
 e 3 , where the centre-line of the rail crosses the line of the 
 floor, draw e 3 / 3 at right angles to n o, and from/ 3 , through 
 6, draw f*g~\ then the heavy lines f*g\ e* d\ y* a*, Z* b\ 
 v*c 3 , w 2 d 3 , and z y will be the lines for the risers, which, 
 being extended to the line of the front-string, b e c d, will 
 give the dimensions of the winders and the grading of the 
 front-string, as was required. 
 
 HAND-RAILING. 
 
 263. Hand-Railing for Stairs. A piece of hand-rail- 
 ' ing intended for the curved part of a stairs, when properly 
 shaped, has a twisted form, deviating widely from plane sur- 
 faces. If laid upon a table it may easily be rocked to and 
 fro, and can be made to coincide with the surface of the 
 table in only three points. And yet it is usual to cut such 
 twisted pieces from ordinary parallel-faced plank ; and to 
 cut the plank in form according to a face-mould, previously 
 formed from given dimensions obtained from the plan of the 
 stairs. The shape of the finished wreath differs so widely 
 from the piece when first cut from the plank as to make it 
 appear to a novice a matter of exceeding difficulty, if not an 
 impossibility, to design a face-mould which shall cover accu- 
 rately the form of the completed wreath. But he will find, 
 as he progresses in a study of the subject, that it is not only 
 a possibility, but that the science has been reduced to such 
 a system that all necessary moulds may be obtained with 
 great facility. To attain to this proficiency, however, re- 
 quires close attention and continued persistent study, yet no 
 more than this important science deserves. The young car- 
 penter may entertain a less worthy ambition than that of 
 desiring to be able to form from planks of black-walnut or 
 mahogany those pieces of hand-railing which, when secured 
 together with rail-screws, shall, on applying them over the 
 stairs for which they are intended, be found to fit their 
 places exactly, and to form graceful curves at the cylinders. 
 That railing which requires to be placed upon the stairs 
 before cutting the joints, or which requires the curves or 
 butt-joints to be refitted after leaving the shop, is discredit- 
 
PRINCIPLES OF HAND-RAILING. 257 
 
 able to the workman who makes it. No true mechanic will 
 be content until he shall be proved able to form the curves 
 and cut the joints in the shop, and so accurately that no altera- 
 tion shall be needed when the railing is brought to its place 
 on the stairs. The science of hand-railing requires some 
 knowledge of descriptive geometry that branch of geometry 
 which has for its object the solution of problems involving 
 three dimensions by means of intersecting planes. The 
 method of obtaining the lengths and bevils of hip and valley 
 rafters, etc., as in Art. 233, is a practical example of descrip- 
 tive geometry. The lines and angles to be developed in 
 problems of hand-railing are to be obtained by methods 
 dependent upon like principles. 
 
 264. Hand-Railing: Definitions; Planes and Solids. 
 
 Preliminary to an exposition of the method for drawing 
 the face-moulds of a hand-rail wreath, certain terms used in 
 descriptive geometry need to be denned. Among the tools 
 used by a carpenter are those well-known implements called 
 planes, such as the jack-plane, fore-plane, smoothing-plane, 
 etc. These enable the workman to straighten and smooth 
 the faces of boards and plank, and to dress them out of 
 wind, or so that their surfaces shall be true and unwinding. 
 The term plane, as used in descriptive geometry, however, 
 refers not to the implement aforesaid, but to the unwinding 
 surface formed by these implements. A plane in geometry 
 is defined to be such a surface that if any two points in it be 
 joined by a straight line, this line will be in contact with the 
 surface at every point in its length. With like results lines 
 may be drawn in all possible directions upon such a sur- 
 face. This can be done only upon an unwinding surface ; 
 therefore, a plane is an unwinding surface. Planes are 
 understood to be unlimited in their extent, and to pass freely 
 through other planes encountered. 
 
 The science of stair-building has to do with prisms and 
 cylinders, examples of which are shown in Figs. 136, 137, and 
 138. A right prism (Figs. 136 and 137) is a solid standing 
 upon a horizontal plane, and with faces each of which is a 
 plane. Two of these faces top and bottom are horizontal 
 
2 5 8 
 
 STAIRS. 
 
 and are equal polygons, having their corresponding sides 
 parallel. 
 
 The other faces of the prism are parallelograms, each of 
 which -is a vertical plane. When the vertical sides of a 
 prism are of equal width, and in number increased indefi- 
 nitely, the two polygonal faces of the prism do not differ 
 essentially from circles, and thence the prism becomes a 
 cylinder. Thus a right cylinder may be defined to be a 
 prism, with circles for the horizontal faces (Fig. 138). 
 
 FIG. 136. 
 
 FIG. 137- 
 
 FIG. 138. 
 
 265. Hand - Railing : Preliminary Considerations. 
 
 If within the well-hole, or stair-opening, of a circular stairs a 
 solid cylinder be constructed of such diameter as shall fill the 
 well-hole completely, touching the hand-railing at all points, 
 and then if the top of this cylinder be cut off on a line with 
 the top of the hand-railing, the upper end of the cylinder 
 would present a winding surface. But if, instead of cutting 
 the cylinder as suggested, it be cut by several planes, each 
 of which shall extend so as to cover only one of the wreaths 
 of the railing, and be so inclined as to touch its top in three 
 points, then the form of each of these planes, at its intersec- 
 tion with the vertical sides of the cylinder, would present 
 the shape of the concave edge of the face-mould for that 
 particular piece of hand - railing covered by the plane. 
 Again, if a hollow cylinder be constructed so as to be in 
 contact with the outer edge of the hand-railing throughout 
 its length, and this cylinder be also cut by the aforesaid 
 
FACE-MOULDS FOR HAND-RAILS. 259 
 
 planes, then each of said planes at its intersection with this 
 latter cylinder would present the form of the convex edge 
 of the said face-mould. A plank of proper thickness may 
 now have marked upon it the shape of this face-mould, and 
 the piece covered by the face-mould, when cut from the 
 plank, will evidently contain a wreath like that over which 
 the face-mould was formed, and which, by cutting away the 
 surplus material above and below, may be gradually wrought 
 into the graceful form of the required wreath. 
 
 By the considerations here presented some general idea 
 may be had of the method pursued, by which the form of a 
 face-mould for hand-railing is obtained. A little reflection 
 upon what has been advanced will show that the problem 
 to be solved is to pass a plane obliquely through a cylinder 
 at certain given points, and find its shape at its intersection 
 with the vertical surface of \he cylinder. Peter Nicholson 
 was the first to show how this might be done, and for the 
 invention was rewarded, by a scientific society of London, 
 with a gold medal. Other writers have suggested some 
 slight improvements on Nicholson's methods. The method 
 to which preference is now given, for its simplicity ot work- 
 ing and certainty of results, is that which deals with the 
 tangents to the curves, instead of with the curves themselves; 
 so we do not pass a plane through a cylinder, but through a 
 prism the vertical sides of which are tangent to the cylinder, 
 and contain the controlling tangents of the face-moulds. The 
 task, therefore, is confined principally to finding the tangents 
 upon the face-mould. This accomplished, the rest is easy, as 
 will be seen. 
 
 The method by which is found the form of the top of a 
 prism cut by an oblique plane will now be shown. 
 
 266. A Prim Cut toy 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, cdcf, and adeg are the vertical sides; while 
 efbg is the top, the form of which is to be shown. 
 
 267. Form of Top of Prim In Fig. 139 the form of 
 the top of the prism is shown as it appears in perspective.. 
 
26o 
 
 STAIRS. 
 
 not in its real shape ; this is now to be developed. In Fig. 
 140, let the sauare a b c d represent by scale the actual form 
 
 FIG. 139. 
 
 and size of the base, a b cd, of the prism shown in Fig. 1 39. 
 Make c c, and dd t respectively equal to the actual heights at 
 
 FIG. 140. 
 
 cf and de, Fig. 139 ; the lines dd t and c c, being set up per- 
 pendicular to the line dc. Extend the lines dc and d t c t until 
 
ILLUSTRATION BY PLANES. 26 1 
 
 they meet in h ; join b and h. Now this line b h is the inter- 
 section of two planes : one, the base, or horizontal plane upon 
 which the prism stands ; the other, the cutting plane, or the 
 plane which, passing- obliquely through the prism, cuts it so 
 as to produce, by intersecting the vertical sides of the prism, 
 the form b fe g, Fig. 139. 
 
 To show that b k is the line of intersection of these two 
 planes, let the paper on which the triangle dhd t is drawn 
 (designated by the letter B) be lifted by the point d t and 
 revolved on the line dk until d t stands vertically over d, and 
 c t over c\ then B will be a plane standing on the line dh, 
 vertical to the base-plane A. The point h being in the line 
 cd extended, and the line cd being in the base-plane A, there- 
 fore h is in the base-plane A. Now the line d t c t represents 
 the line cf of Fig. 139, and is therefore in the cutting plane ; 
 consequently the point //, being also in the line d t c, ex- 
 tended, is also in the cutting plane. By reference to Fig. 
 139 it will be seen that the point b is in both the cutting and 
 base planes ; we must therefore conclude that, since the two 
 points b and h are in both the cutting and base planes, a line 
 joining these two points must be the intersection of these two 
 planes. The determination of the line of intersection of the 
 base and cutting planes is very important, as it is a control- 
 ling line ; as will be seen in denning the lines upon which 
 the form of the face-mould depends. Care should therefore 
 be taken that the method of obtaining it be clearly under- 
 stood. 
 
 It will be observed that the intersecting line bh, being in 
 the horizontal plane A, is therefore a horizontal line. Also, 
 that this horizontal line b h being a line in the cutting plane, 
 therefore all lines upon the cutting plane which are drawn 
 parallel to b h must also be horizontal lines. The import- 
 ance of this will shortly be seen. Through a, perpendicular 
 to bh, draw the line b n d^ and parallel with this line draw 
 dd ini ; on d as centre describe the arc d t d ltil ; draw d ltll d v 
 parallel with dd tlJ and extend the latter to d tll ; on d {l as 
 centre describe the arc d v d nl ; join b n and d ul . We now 
 have three vertical planes which are to be brought into 
 position around the base-plane A, 'as follows: Revolve B 
 
262 STAIRS. 
 
 upon dh, E upon dd it , and C upon b n d tn each until it stands 
 perpendicular to the plane -A. Then the points d t and d illt 
 will coincide and be vertically over d\ the points d llt and d v 
 will coincide and stand vertically over d n ; and c t will cover c. 
 These vertical planes will enclose a wedge-shaped figure, 
 lying with one face, b^d^dh, horizontal and coincident with 
 the base-plane A, and three vertical faces, b lt d u d ti/y dd n d^ d jiit , 
 and hdd t . By drawing the figure upon a piece of stout 
 paper, cutting it out at the outer edges, making creases in 
 the lines hd, dd tl , d u b t/J then folding the three planes B, E, 
 and C at right angles to A, the relation of the lines will be 
 readily seen. Now, to obtain the form of the top or cover to 
 the wedge-shaped figure, perpendicular to b n d iti draw b,,h, 
 and d tll e\ on b tl as centre describe the arc hh i ; make d llt e 
 equal to d lt d\ join e and h t . Now the form of the top of the 
 wedge-shaped figure is shown within the bounds d in b i{ h t c. 
 By revolving this plane D on the line b n d llt until it is at 
 a right angle to the plane C, and this while the latter is 
 supposed to be vertical to the plane A, it will be perceived 
 that this movement will place the plane D on top of the 
 wedge-shaped figure, and in such a manner as that the point 
 e will coincide with d lllt d { , and the point h t will fall upon and 
 be coincident with the point h, and the lines of the cover 
 will coincide with the corresponding lines of the top edges 
 of the sides of the figure; for example, the line b li d lll is 
 common to the top and the side C; the line d ltl e equals d tl d, 
 which equals d v d ttil \ therefore, the line d itt e will coincide 
 with d v d i/u of the side E\ the line eh t will coincide with d,h 
 of the side B\ and the line b l ,h l will coincide with the line 
 b tl h. Thus the figure D bounded by b ll d lll ch l will exactly 
 fit as a cover to the wedge-shaped figure. Upon this cover 
 we may now develop the form of the top of the prism. 
 
 Preliminary thereto, however, it will be observed, as was 
 before remarked, that lines upon the cutting plane which 
 are parallel to the intersecting line b tl h t are horizontal ; 
 and each, therefore, must be of the same length as the line 
 in the base-plane A vertically beneath it. For example, the 
 line d llt e { is a line in the cutting plane D, parallel with the 
 line b lt h t in the same plane, and this line b ll li l will (when the 
 
EXPLANATION OF THE DIAGRAMS. 263 
 
 cutting plane D is revolved into its proper position) be co- 
 incident with the intersecting line b lt h ; therefore, the line 
 d til e is a line in the cutting plane D, drawn parallel with the 
 intersecting line b u h. Now this line d llt e, when in position, 
 will be coincident with the line d ltll d^ which lies vertically 
 over the line d t ,d-ol the base-plane A ; its length, therefore, 
 is equal to that of the latter. In like manner it may be 
 shown that the length of any line on the plane D parallel 
 to b tl h n is equal in length to the corresponding line upon 
 the plane A vertically beneath it. 
 
 Therefore, to obtain the form of the top of the prism, we 
 proceed as follows : Perpendicular to b tl d v draw c c lu and 
 aa ttl \ perpendicular to b tl d ni draw c tll f and equal toc,,c; 
 on b lt as centre describe the arc b b t ; join b t a, n , b t f, and a l4l e. 
 Now we have here in plane D the form of the top of the 
 prism, as shown in the figure bounded by the lines a^'fije. 
 This will be readily seen when the plane D is revolved into 
 position. Then the point a tll will be vertically over a ; the 
 point e coincident with d t d ltll and vertically over d; the point 
 / coincident with c / and vertically over c ; while b t will coin- 
 cide with b of the base-plane A. 
 
 The figure a ni b t fe, therefore, represents correctly both 
 in form and size the top of the prism as it is shown in per- 
 spective at bfeg, Fig. 139. The line ef, Fig. 140, is equal to 
 the line d t c t , and so of the other lines bounding the edges of 
 the figure. 
 
 The cutting plane b f e g, Fig. 139, may be taken to repre- 
 sent the surface of the plank from which the wreath of hand- 
 railing is to be cut ; the wreath curving around from b to c t 
 as shown in Fig. 141, the lines b g and ge being tangent to 
 the curve in the cutting plane; while ab and ad are tan- 
 gents to the curve on the base plane, or plane of the cylin- 
 der. The location of the cutting plane, however, is usually 
 not at the upper surface of the plank, but midway between 
 the upper and under surfaces. The tangents in the plane 
 are found to be more conveniently located here for deter- 
 mining the position of the butt-joints. For a moulded rail 
 two curved lines, each with a pair of tangents, are required 
 upon the cutting plane, one for the outer edge of the rail, 
 
264 STAIRS. 
 
 and the other for the inner edge ; but for a round rail only 
 one curve with its tangents is required, as that from b to e 
 in Fig. 141, which is taken to represent the curved line run- 
 ning through the centre of the cross-section of the rail. As 
 an easy application of the principles regarding the prism, 
 just developed, an example will now be given. 
 
 268. Face-Mould for Hand-Railing of Platform Stairs. 
 
 Let/ and / ;;/, Fig. 142, represent the central or axial lines 
 of the hand-rails of the two flights, one above, the other be- 
 low the platform ; and let the semicircle/df/ be the central 
 line of the rail around the cylinder at the platform, the risers 
 at the platform being located at j and /. Vertically over the 
 platform risers draw gg t ; make gr t equal to a riser of the 
 lower flight, and r t g t and ss t each equal to a riser of the 
 upper flight. Draw g t s and gk t horizontal and equal 
 each to a tread of each flight respectively. Through r, draw 
 k, a u , and through g y draw s t t t . Vertically over d draw a t t r 
 Horizontally draw a tl a nll and t t t tl . 
 
 It is usual to extend the wreath of the cylinder so as to 
 include a part of the straight rail such a part as convenience 
 may require. Let the straight part here to be included ex- 
 tend from / to b on the plan. Vertically over b draw b t c tlli , 
 and horizontally draw b / w- tl ; at any point on b t w tt locate w n , 
 and make w tl w f equal to j I, and bisect it in w; erect the 
 perpendiculars iv t a ltil , w d v/J , and w /7 v ; join t lt and a tlll ; from 
 d vil horizontally draw d vil d v/ ; parallel with r t k t draw 
 dv, c tlll ' We now have the plan and elevations of the prism, 
 
FACE-MOULD FOR PLATFORM STAIRS. 
 
 265 
 
 containing at its angles the tangents required for the wreath 
 extending from b to d on the plan. The elevation F is a view 
 of the cylinder looking in the direction dc. 
 
 FIG. 142. 
 
 Comparing Fig. 142 with Fig. 141* the line b, w /t is the 
 trace, upon a vertical plane, of the horizontal plane abed 
 
266 STAIRS. 
 
 of Fig. 141, or is the ground-line from which the heights of 
 the prism are to be taken. 
 
 The triangle^ b t a u is represented in Fig. 141 at ab g, and 
 the inclined line b, a tl is the tangent of the rail of the lower 
 flight, and is represented in Fig. 141 at bg ; while a nil t n is 
 the tangent of the railing around the cylinder, and the half 
 of it is represented in Fig. 141 at ge. The height b t c tilJ is 
 shown in Fig. 141 at c f, while the height iv d v , t , or a/d Vl , is 
 shown in Fig. 141 at de. 
 
 The vertical planes EEC may now be constructed about 
 the prism as in Fig. 140, proceeding thus :. Make c c f equal to 
 b t c iul , and dd t equal to a t d v/ ; through c t draw d i h\ through 
 b draw h b tl ; perpendicular to h b lt through a draw b n d v ; 
 from ^parallel with b n */ v draw d d jiti ; on d as centre describe 
 the arc d t d nil \ draw d itll d v , also d d ltl , parallel with hb n \ 
 on d ti as centre describe the arc d v d tj ; join d w to b jt . Par- 
 allel with b n h draw from each important point of the plan, 
 as shown, an ordinate extending to the line b tl d tji , and thence 
 across plane D draw ordinates perpendicular to b lt d tll , and 
 make them respectively equal to the corresponding ordinates 
 of the plane A, measured from the line b n d v ; join e to/, a ll{ 
 to b^ a ltl to e, and b, to/; also join /, to r t . Then a in b, is the 
 tangent standing over a b, and a lit e is Jthe tangent standing 
 over ad. The line b l l i is the part of the tangent which 
 stands over bl t , the portion of the wreath which is straight. 
 The curve en l p l l l is the trace upon the cutting plane of the 
 quarter circle dnpl, traced through the points /*,/,, and as 
 many more as desirable, found by ordinates as any other 
 point in the plane A. Thus we have complete the line 
 b t I, n t e, the central line of the wreath extending from b to d 
 in the plan. This is the essential part of the face-mould, which 
 is now to be drawn as follows: At Fig. 143 repeat the par- 
 allelogram a tll b t fe of Fig. 142, and, with a radius equal to 
 half the diameter of the rail, describe, from centres taken on 
 the central line, the several circles shown ; and tangent to 
 these circles draw the outer and inner edges of the rail. 
 The joint at b t is to be drawn perpendicular to the tangent 
 b. a in , while that at e is to be perpendicular to the tangent 
 ^a tll . This completes the face-mould for the wreath over 
 
WREATHS FOR A ROUND RAIL. 267 
 
 bind of the plan. If the pitch-board of the upper flight be 
 the same as that of the lower flight, the face-mould at Fig. 
 143 will, reversed, serve also for the wreath over the other 
 half of the cylinder. 
 
 In using this face-mould, place it upon a plank equal in 
 thickness to the diameter of the rail, mark its form upon the 
 plank, and saw square through ; then chamfer the wreath to 
 an octagonal form, after which carefully remove the angles 
 so as to produce the required round form. The joints, as well 
 as the curved edges, are to be cut square through the plank. 
 
 Many more lines have been used in obtaining this face- 
 mould than were really necessary for so simple a case, but no 
 more than was deemed advisable in order properly to eluci- 
 date the general principles involved. A very simple method 
 
 FIG. 143. 
 
 for face-moulds of platform stairs with small cylinders will 
 now be shown. 
 
 269. More Simple method for Hand-Rail to Platform 
 Stairs. In Fig. 144, jge represents a pitch-board of the first 
 flight, and d and i the pitch-board of the second flight of a plat- 
 form stairs, the line e f being the top of the platform ; and 
 abc is the plan of a line passing through the centre of 
 the rail around the cylinder. Through i and d draw i k, 
 and through y and e drawy k ; from k draw k I parallel to/e; 
 from b draw bm parallel to gd ; from / draw Ir parallel to 
 kj ; from n draw nt at right Angles toy/6; on the line ob 
 make ot equal to nt ; join c and t ; on the line jc, Fig. 145, 
 make ec equal to en at Fig. 144; from c draw c t at right 
 angles to j c, and make ct equal to c t at Fig. 144; through / 
 draw p I parallel to j c, and make // equal to / / at Fig. 144 ; 
 join /and c, and complete the parallelogram eels ; find the 
 points o, o, o, according to Art. 551 ; upon e, o, o, o, and /, 
 

 268 
 
 STAIRS. 
 
 successively, with a radius equal to half the width of the 
 rail, describe the circles shown in the figure ; then a curve 
 traced on both sides of these circles, and just touching them, 
 
 FIG, 144. 
 
 Avill give the proper form for the mould, 
 drawn at right angles to c /. 
 
 The joint at / is 
 
 FIG. 145. 
 
 This simple method for obtaining the face-moulds for the 
 hand-rail of a platform stairs appeared first in the early edi- 
 tions of this work. It was invented by a Mr. Kells, an 
 
HAND-RAIL TO PLATFORM STAIRS. 
 
 269 
 
 eminent stair-builder of this city. A comparison with Fig. 
 142 will explain the use of the few lines introduced. For a 
 full comprehension of it reference is made to Fig. 146, in 
 which the cylinder, for this purpose, is made rectangular 
 
 FIG. 146. 
 
 instead of circular. ,The figure gives a perspective view of 
 a part of the upper and of the lower flights, and a part of 
 the platform about the cylinder. The heavy lines, ////, me, 
 and cj, show the direction of the rail, and are supposed to 
 pass through the centre of it. Assuming that the rake of 
 
270 
 
 STAIRS. 
 
 the second flight is the same as that of the first, as is gener- 
 ally the case, the- face-mould for the lower twist will, when 
 reversed, do for the upper flight ; that part of the rail, there- 
 fore, which passes from e to c, and from c to /, is all that will 
 need explanation. 
 
 Suppose, then, that the parallelogram eaoc represent a 
 plane lying perpendicularly over 'eabf, being inclined in 
 the direction ec, and level in the direction co ; suppose this 
 
 FIG. 147. 
 
 plane eaoc be revolved on ec as an axis, in the manner indi- 
 cated by the arcs o n and a x, until it coincides with the 
 plane ertc\ the line ao will then be represented by the line 
 x n ; then add the parallelogram xrtn, and the triangle ctl, 
 deducting the triangle ers\ then the edges of the plane cslc, 
 inclined in the direction ec, and also in the direction c I, will 
 lie perpendicularly over the plane eabf. From this we 
 gather that the line co, being at right angles to ec, must, -in 
 
HAND-RAIL FOR LARGE CYLINDER. 2JI 
 
 order to reach the point /, be lengthened the distance nt, 
 and the right angle ect be made obtuse by the addition to 
 it of the angle tc /. By reference to Fig. 144, it will be seen 
 that this lengthening is performed by forming the right- 
 angled triangle cot, corresponding to the triangle cot in 
 Fig. 146. The line ct is then transferred to Fig. 145, and 
 placed at right angles tov^r; this angle ect is then increased 
 by adding the angle tcl, corresponding to tcl, Fig. 146. 
 Thus the point / is reached, and the proper position and 
 length of the lines ec and ^/obtained. To obtain the face- 
 mould for a rail over a cylindrical well-hole, the same process 
 is necessary to be followed until the length and position of 
 these lines are found ; then, by forming the parallelogram 
 eels, and describing a quarter of an ellipse therein, the 
 proper form will be given. 
 
 FIG. 148. 
 
 270. Hand-Railling 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 zWandyV until they meet kb 
 in k and /; from ;/ draw nl parallel to ob; through / draw 
 // parallel to j k; from k draw kt at right angles to/; on 
 the line ob make ot equal to kt. Make ec (Fig. 148) equal 
 to ek at Fig. 147 ; from c draw ct at right angles to ec, and 
 equal to ct at Fig. 147 ; from / draw // parallel to ce, and 
 make tl equal to //at Fig. 147 ; complete the parallelogram 
 eels, and find the points o, 0, o, as before ; then describe the 
 circles and complete the mould as in Fig. 145. The difference 
 between this and Case I is that the line ct, instead of being 
 raised and thrown out, is lowered and drawn in. A method 
 of planning a cylinder so as to avoid the necessity of cant- 
 ing the plank, either up or down, will now be shown. 
 
2/2 
 
 STAIRS. 
 
 271. Faee-moMld 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 
 obviated, by fixing the place of the above risers at a certain 
 distance within the cylinder, as shown in Fig. 149 the lines 
 indicating the face of the risers cutting the cylinder at k and 
 /, instead of at / and q, the spring of the cylinder. To 
 ascertain the position of the risers, let abc be the pitch- 
 board of the lower flight, and cde that of the upper flight, 
 
 these being placed so that b c and 
 cd shall form a right line. Extend 
 a c to cut de in f; draw fg parallel 
 *to db, and of indefinite length ; 
 draw go at right angles to fg, and 
 equal in length to the radius of the 
 circle formed by the centre of the 
 rail in passing around the cylinder ; 
 on o as centre describe the semi- 
 circle /^z'/ through o draw is par- 
 allel to db; make oh equal to the 
 radius of the cylinder, and describe 
 on o the face of the cylinder phq; 
 then extend db across the cylinder, 
 cutting it in / and k giving the 
 position of the face of the risers, 
 as required. To find the face- 
 mould for the twists is simple and 
 obvious : it being merely a quarter 
 of an ellipse, having oj for semi- 
 minor axis, and sf for the semi-major axis; or, at Fig. 151, 
 let dci"be<\. right angle ; make c i equal to oj, Fig. 149, and dc 
 equal to sf, Fig. 149; then draw do parallel to ci, and com- 
 plete the curve as before. 
 
 272. Railing for Platform Stair* where the Rake 
 meets the Level. In Fig. 150, abc is the plan of a line pass- 
 ing through the centre of the rail around the cylinder as 
 before, and je is a vertical section of two steps starting 
 from the floor, kg. Bisect eh in d, and through d draw df 
 
 FIG. 149. 
 
HAND-RAIL AT RAKE AND LEVEL. 
 
 273 
 
 parallel to hg\ bisect /# in /, and from / draw It parallel 
 to nj\ from n draw nt at right angles to jn ; on the line ob 
 make ot equal to nt. Then, to obtain a mould for the twist 
 going up the flight, proceed as at Fig. 145 ; making ec in 
 that figure equal to en in Fig. 150, and the other lines of a 
 length and position such as is indicated by the letters of 
 reference in each figure. To obtain the mould for the level 
 
 FIG. 150. 
 
 rail, extend bo (Fig. 150) to i ; make oi equal to //, and join 
 z'andc; vcw&e c i (Fig. 151) equal to civ&Fig. 150; thiough 
 
 FIG. 151. 
 
 c draw cd at right angles to ci\ make dc equal to df at 
 Fig. 150, and complete the parallelogram odd; then pro- 
 ceed as in the previous cases to find the mould. 
 
 273. Application of Face-lWoiilds 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 
 
274 
 
 STAIRS. 
 
 plank sawed square through, the joints being cut square 
 from the face of the plank. A twist thus cut and truly 
 rounded will hang in a proper position over the plan, and 
 present a perfect and graceful wreath. 
 
 274. Face-Moulds for Moulded Rails upon Platform 
 Stairs. In Fig. i$2,abcis the plan of a line passing through 
 
 FIG. 152. 
 
 the centre of the rail around the cylinder, as before, and the 
 lines above it are a vertical section of steps, risers, and plat- 
 form, with the lines for the rail obtained as in Fig. 144. Set 
 half the width of the rail from b to / and from b to r, and 
 from / and r draw .fc and rd parallel to ca. At Fig. 153 
 the centre-lines of the rail jc and cl are obtained as in the 
 previous examples, making jc equal jn of Fig. 152, ct 
 
FACE-MOULD APPLIED TO PLANK. 2/5 
 
 equal ct of Fig. 152, and tl equal si of Fig. 152. Make ci 
 and ck each equal to <:z at /^. 152, and draw the lines im 
 and /& parallel to cj ; make /<? and /r equal to ne and ?z</ at 
 7%. 152, and draw dn and eq parallel to lc\ also, through j 
 draw <? parallel to lc; then, in the parallelograms mnro 
 and goe q, find the elliptic curves, d?? and *, according to 
 Art. 551, and they will define the curves. The line dp, 
 being drawn through / perpendicular to Ic, defines the 
 joint which is to be cut square through the plank. 
 
 275. Application of Face-Moulds to Plank. In Fig. 
 152 make a drawing, from d to /*, of the cross-section of the 
 hand-rail, and tangent to the lower corner draw the line gh. 
 The distance between the lines/* and^/z is the thickness of 
 the plank from which the rail is to be cut. Lay the face 1 
 mould upon the plank, mark its shape upon the plank, and 
 
 FIG. 153. 
 
 saw it square through. To proceed strictly in accordance 
 with the requirements of the principles upon which the face- 
 mould is formed, the cutting ought to be made vertically 
 through the plank, the latter being in the position which it 
 would occupy when upon the stairs. Formerly it was the 
 custom to cut it thus, with its long raking lines. But, owing 
 to the great labor and inconvenience of this method, efforts 
 were made to secure an easier process. By investigation it 
 was found that it was possible, without change in the face- 
 mould, to cut the plank square through and still obtain the 
 correct figure for the railing, and this method is the one now 
 usually pursued. Not only is the labor of sawing much re- 
 duced by this change ; but to the workman it is an entire re- 
 lief, as he now, after marking the form of the wreath upon the 
 plank, sends it to a steam saw-mill, and, at a small cost, has it 
 
STAIRS. 
 
 cut out with an upright scroll-saw. When thus cut out in 
 the square, the upper surface of the plank is to be faced up 
 true and unwinding, and the outer edge jointed straight 
 and square from the face. Then a figure of the cross-section 
 of the hand-railing is to be carefully drawn on the ends of the 
 squared block as shown in Figs. 154 and 155, and which 
 are regulated so as to be correctly in position, as follows. 
 First, as to the end h of the straight part hj\ In Fig. 154, 
 let a b c d be an end view of the squared block, of which a e fd 
 is the shape of the end of the straight part. Let the point g 
 be the centre of this end of the straight part ; through g 
 draw upon the end a efd the line jk, so that the angle bjk 
 shall be equal to the angle kt c, Fig. 152. This is the angle 
 at which the plank is required to be canted, revolving it on 
 
 FIG. 154. 
 
 FIG. 155. 
 
 the axis of the straight part of the rail. Through g draw 
 the line ;/ h parallel with a b. Upon a thin sheet of metal 
 (zinc is preferable) mark carefully the exact figure of the 
 cross-section of the rail, drawing a vertical line through its 
 centre, cut away the surplus metal, then, with this template 
 as a pattern, mark upon the end a efd, Fig. \ 54, the figure of 
 the rail as shown, the vertical line upon the template being 
 made to coincide with the line//. From n and h draw the 
 vertical lines // in and /;/ parallel with j k. 
 
 Now, as to the other end of the square block : Let b c f e, 
 Fig. 155, represent the block, of which bcvn is the form of 
 the end at the curved part, and o its centre. Through o 
 draiv/^, so that the angle epq shall be equal to the angle 
 j n /7, Fig. 152. Also, through o draw d h parallel with e b\ 
 
CUTTING THE TWIST-RAIL. 277 
 
 from d and h draw the vertical lines h r and ds parallel with 
 pq. Place the template on bcvn, the end of the block, so 
 that the vertical line through its centre shall coincide with 
 pq\ mark its form, then from y, at mid-thickness, draw wy 
 parallel with p q. 
 
 In applying- the mould, let Fig. 156 represent the upper 
 face of the squared block, 
 with the face-mould lying 
 upon it. With the distance 
 a /, Fig. 154, and by the 
 edge a x, mark a gauge-line 
 upon the upper face of the FIG. 156. 
 
 squared block. Set the outer edge of the face-mould to coin- 
 cide with this gauge-line. Let the end of the face-mould be 
 set at w, e w being equal to e w, Fig. 155; then mark the 
 block by the edge of the face-mould. 
 
 Now turn the block over and apply the face-mould to the 
 underside, as in Fig. 157. With the distance d r/t t Fig. 154, 
 and by the outer edge of 
 the block, mark a gauge- 
 line from m, Fig. 157. Set 
 the inner edge of the face- 
 mould to this gauge-line, 
 and slide it endwise till the FlG - J 57. 
 
 distance em shall equal ew, Fig. 155, then mark the block by 
 the edges of the face-mould. The over wood may now be re- 
 moved as indicated by the vertical lines at the sides of the 
 cross-section marked on each end of the block (see also Fig. 
 167) : the direction of the cutting at the curves must be verti- 
 cal ; the inner curve will require a round-faced plane. A com- 
 parison of the several figures referred to, with the directions 
 given, together with a little reflection, will manifest the 
 reasons for the method here given for applying the face- 
 mould. Especially so when it is remembered that the face- 
 mould was obtained not for the top of the rail, but for the rail 
 at the mid-thickness of the block. So, therefore, in the 
 application to the upper surface of the block, the face-mould 
 is slid up the rake far enough to put the mould in position 
 vertically over its true positional mid-thickness ; and on the 
 
2/8 STAIRS. 
 
 contrary, in applying the face-mould to the underside of the 
 plank, it is slid down until it is vertically beneath its true 
 position at the mid-thickness of the block. 
 
 When the vertical faces are completed, the over wood 
 above and below the wreath is to be removed. In doing 
 this, the form at the ends, as given by the template, is a suf- 
 ficient guide there. Between these the upper and under 
 surfaces are to be warped from one end to the other, so as 
 to form a graceful' curve. With a little practice an intelli- 
 gent mechanic will be able to work these surfaces with 
 facility. The form of cross-section produced by this opera- 
 tion is that of a parallelogram, tangent to the top, bottom, 
 and two sides of the rail ; and which at and near the ends 
 of the block is not quite full. The next operation is that of 
 working the moulding at the sides and on top, first by re- 
 bates at the sides, then chamfering, and finally moulding the 
 curves. Templates to fit the rail, one at the sides, another 
 on top, are useful as checks against cutting away too much 
 6f the wood. f 
 
 The joints are all to be worked square through the plank 
 in the line drawn perpendicular to the tangent, as shown in 
 153- 
 
 276. Hand-Railing for Circular Stairs. Let it be re- 
 
 quired to furnish the face-moulds for a circular stairs similar 
 to that shown in Fig. 133. 
 
 Preliminary to making the face-moulds it is requisite to 
 make a plan, or horizontal projection of the stairs, and on 
 this to locate the projections of the tangents and develop 
 their vertical projections. For this purpose let b c d e f g, 
 Fig. 158, be the horizontal projection of the centre of the 
 rail, and the lines numbered from i to 19 be the risers. At 
 any point, a, on an extension of the line of the first riser 
 locate the centre of the newel. On a as a centre describe the 
 two circles; the larger one equal in diameter to the diame- 
 ter of the newel-cap, the inner one distant from the outer 
 one equal to half the width of the rail. Let the first joint in 
 the hand-rail be located at b, at the fourth riser ; through b 
 draw h k tangent to the circle. Select a point, h, on this 
 
PLAN OF CIRCULAR STAIRS. 
 
 279 
 
 tangent which shall be equally distant from b and from the 
 inner circle of the newel-cap, measured on a line tending to 
 a ; join h and a, and from a point, <?, on the line b o describe 
 
 ft, 
 
 10 
 
 
 FIG. 158. 
 
 the curve from b to the point of the mitre of the newel-cap, 
 the curve being tangent, at this point, to the line a h. Select 
 positions for the other joints in the hand-rail as at c, d, c, and /. 
 
280 STAIRS. 
 
 Through these draw lines tangent to the circle.* Then the 
 horizontal projection of the tangents will be the lines hk, kl, 
 /;//, m n, and np. Now, if a vertical plane stand upon each 
 of these lines, these planes would form a prism not quite 
 complete standing upon the base-plane, A. Upon these ver- 
 tical planes, C, D, E, F, G, and H, lines may be drawn which 
 at each joint shall be tangent to the central line of the rail. 
 These are the tangents now to be sought. Perpendicular to 
 the tangents at , c, d, etc., draw the lines b b t , c c^ dd jy e e t , ff /y 
 gg n and h h tl , kk t , k k lt ,l l t ,l l in etc. As b is at the fourth riser, 
 and the height is counted from the top of the first riser, 
 make b b t equal to three risers. (To avoid extending the 
 drawing to inconvenient dimensions, the heights in it are 
 made only half their actual size. As this is done uniformly 
 throughout the drawing, this reduction will lead to no error 
 in the desired results.) As c is on the eighth riser, therefore 
 make c c t equal to seven risers, and so, in like manner, make 
 the heights dd lt ee^ and ff t each of a height to correspond 
 with the number of the riser at which it is placed, deduct- 
 ing one riser. These heights fix the location of each tangent 
 at its point of contact with the central line of the rail. But 
 each tangent is yet free to revolve on this point of contact, 
 up or down, as may be required to bring the ends of each 
 pair of tangents in contact; or, to make equal in height the 
 edges of each pair of vertical planes, which coincide after 
 they are revolved on their base-lines into a vertical position ; 
 as, for example : the edges k k t and k k it of the planes C and 
 D must be of equal height; so, also, the edges ll f and ll tl of 
 the planes D and E must be of equal height. The method 
 of establishing these heights will now be shown. 
 
 To this end let it be observed, that of the horizontal pro- 
 jection of any pair of intersecting tangents, their lengths, 
 from the point of intersection to the points of contact with 
 the circle, are equal ; for example : of the two tangents // k 
 and Ik, the distances from k, their point of intersection, to b 
 and c, their points of contact with the circle, are equal ; and 
 so also cl equals <//, dm equals e m, etc. It will be observed 
 
 * A tangent is a line perpendicular to the radius, drawn from the point of 
 contact. 
 
THE FALLING-MOULD FOR THE RAIL. 28 1 
 
 that this equality is not dependent on b,c, d, etc., the points 
 of contact, being disposed at equal distances ; for, in this 
 example, they are placed at unequal distances, some being at 
 three treads apart and others at four; and yet while this un- 
 equal distribution of the points b, c, d, etc., has the effect of 
 causing the point of contact, as b, c, or e y to divide each whole 
 tangent into two unequal parts, it does not disturb the 
 equality of the two adjoining parts of any two adjacent tan- 
 gents. Now, because of this equality of the two adjoining 
 parts of a pair of tangents, the height to be overcome in 
 passing from one point of contact to the next must be 
 divided equally between the two ; each tangent takes half 
 the distance. Therefore, for stairs of this kind, the arrange- 
 ment being symmetrical, we have this rule by which to fix 
 the height of the ends of any two adjoining tangents, namely : 
 To the height at the lower point of contact add half the dif- 
 ference between the heights at the two points of contact ; 
 the sum will be the required height of the two adjoining 
 ends of tangents. For example: the heights at b and c, 
 two adjacent points of contact, are respectively three and 
 seven risers; the difference is four risers; half this added to 
 three, the height of the lower rise, gives five risers as the 
 height of k k r kk u , the height at the adjoining ends of the 
 tangents h k and / k. Again, the heights at c and d are re- 
 spectively seven and ten risers ; their difference is three ; 
 half of which, or one and a half 'risers, added to seven, the 
 height at the lower point of contact, makes nine and a half 
 risers as the heights //,, //, at the ends of the adjoining 
 tangents k I and m I. In a similar manner are established 
 the heights of the tangents at ;;/, ;/, and /. 
 
 The rule for finding the heights of tangents as just given 
 is applicable to circular stairs in which the treads are di- 
 vided equally at the front-string, as in Fig. 158. Stairs of 
 irregular plan require to have drawn an elevation of the 
 rail, stretched out into a plane, upon which the tangents can 
 be located. This will be shown farther on. 
 
 The locations of the joints c, d, c, in this example, were 
 disposed at unequal distances merely to show the effect on 
 the tangents as before noticed. In practice it is proper to 
 
282 STAIRS. 
 
 locate them at equal distances, for then one face-mould in 
 such a stairs will serve for each wreath. 
 
 When the tangent at G has been drawn, the level tangent 
 for the landing maybe obtained in this manner: As the 
 joint f is located at the eighteenth riser, one riser below the 
 landing, draw a horizontal line at s, one riser above the point 
 f^ and at half a riser above this draw the level line at p t \ then 
 this line is the level tangent, and p its point of intersection 
 with the raking tangent. Draw the vertical line /,/, and 
 from/ draw the tangent /-, which is the horizontal projec- 
 tion of the tangent p t g, on plane H (which, to avoid undue 
 enlargement of the drawing, is reduced in height), where 
 ////equals//,,. 
 
 To obtain the horizontal tangent / u at the newel, pro- 
 ceed thus : Fix the point r, in the tangent r k t , at a height 
 above b t equal to the elevation of the centre of the newel 
 above the height of a short baluster for example, from 5 
 to 8 inches and draw a line through r parallel to b t ; this 
 is a horizontal line through the middle of the height of the 
 newel-cap, and upon which and the rake the easement to 
 the newel is formed. Perpendicular to b t draw r t, and join 
 / and u ; then / u is the horizontal tangent. 
 
 277. Face-Moulds for Circular tair. At Fig. 159 the 
 plan of the newel and the adjacent hand-rail are repeated, 
 but upon an enlarged scale ; and in which b b t is the reduced 
 height of the point &, or is equal to b b t less tr, Fig. 158, 
 and the angle b b t /equals the angle bb t r of Fig. 158. In 
 this plan the actual heights must now be taken. Join t and 
 u ; then / u is the level tangent, as also the line of intersection 
 of the cutting plane C and the horizontal plane A. Perpen- 
 dicular to / u, at a point / or anywhere above it, draw u, b /r 
 Parallel with / u draw b b ltl ; make b n b ltl equal to b b t ; join 
 b ltl and u^ ; then the angle b b ul u t is the angle which the 
 plank in position makes with a vertical line, or what is 
 usually termed the plumb-beviL Perpendicular to b ltl u j 
 draw , and b ltl b ltll ; make b tll b tlll equal to bb tl ; make u, 
 t t equal to u t t, and u j u n to u / u ; join b lltl and t t ; then b ini t t is 
 the tangent in the cutting plane, the horizontal projection of 
 
FACE-MOULD FOR FIRST SECTION. 
 
 28 3 
 
 which is bt. The butt-joint at b ltil is drawn square to the 
 tangent b ilu t r Parallel to the intersecting line / u, draw 
 ordinates across the plane A from as many points as desir- 
 able, and extend them to the rake-line u t b ni ; through the 
 points of their intersection with this line, and perpendicular 
 to it, draw corresponding ordinates across the plane C. Make 
 d u d tll equal to d, d, and so in like manner, for all other points, 
 
 FIG. 159. 
 
 obtain in the plane C for each point in the horizontal plane 
 A its corresponding point in the plane C: in each case 
 taking the distance to the point in the plane A from the line 
 h b tl and applying it in the plane C from the rake-line u, b nl . 
 For the curves bend a flexible strip to coincide with the 
 several points obtained, and draw the curve by the side of 
 the strip. The point of the mitre is at */,, the mitre-joint is 
 
284 STAIRS. 
 
 shown at hd tll and d tll c lt . The line f c lt is drawn through 
 c //t the most projecting point of the mitre, and parallel to the 
 rake-line u t b lir Additional wood is left attached, extending 
 from h to f\ this is an allowance to cover the mitre, which 
 has to be cut vertically ; the butt-joint at b ltll and the face at 
 fc tl are both to be cut square through the plank. The face 
 /<:, because it is parallel to the rake-line u f b ltl , is a vertical 
 face, as well as being perpendicular to the surface of the 
 plank. On it, therefore, lines drawn according to the rake, 
 or like the angle u t b tll b it , will be vertical and will give the 
 direction of the mitre-faces. We now have at C the face- 
 mould for the railing over the plan from b to d in A. The 
 mould thus found is that made upon a cutting plane C, passed 
 through the plank, parallel to its face, but at the middle of 
 its thickness. To put it in position, let the plane C be lifted 
 by its upper edge c tl and revolved upon the line n t b tll until 
 it stands perpendicular to the plane B. Now revolve both 
 C and B (kept in this relative position during the revolution) 
 upon the line u t b t . until the plane B stands perpendicular to 
 the plane A. Then every point upon plane C will be verti- 
 cally over its corresponding point in the plane A. For ex- 
 ample, the point b lllt will be vertically over b, t, over /, 
 and so of all other points. To show the application of the 
 face-mould to the plank, make b ltl b v equal to half the thick- 
 ness of the plank; parallel to u t b tll draw b v c, a line which 
 represents the upper surface of the plank, for the line u t b ul 
 is at the middle of the thickness. Through b itll , and parallel 
 with b ltl u t , draw the line c t b ltll and extend it across the face- 
 mould ; make b litl c t equal to b v c ; through c^ and parallel with 
 b ull t /t draw c t e. Now, ;// n o t p is an end view of the plank, 
 showing the face view of the butt-joint at b i{ll . Through r, 
 the centre, draw a line parallel with the sides. Then # vl rep- 
 resents the point b ltll \ make vi e t equal to b tlil c\ through r, 
 the centre, draw c, r across the face of the joint ; then e t r is 
 a vertical line (see Art. 284), parallel and perpendicular to 
 which the four sides of the squared-up wreath are to be 
 drawn as shown. In applying the face-mould to the plank at 
 first, for the purpose of marking by its edges the form of the 
 face-mould, it will be observed that the face-mould is under- 
 stood to have the position indicated by the line u t b lu , or at 
 
FACE-MOULDS FOR CIRCULAR STAIRS. 285 
 
 the middle of the thickness of the plank. By this marking 
 the rail-piece is cut square through the plank, and this cut- 
 ting gives the correct form of the wreath, but only at the 
 middle of the thickness of the plank. After it is cut square 
 through the plank, then, to obtain the form at the upper and 
 under surfaces, the face-mould is required to be moved end- 
 wise, but parallel with the auxiliary plane ^,-and so far as to 
 bring the face-mould into a position vertically over or under 
 its true position at the middle of the thickness of the plank. 
 For example, the point b tlll ^ if the mould were placed at the 
 middle of the thickness of the plank, would be at the height 
 of the point b nl ; but when upon the top of the plank, the 
 point b tlll would have to be at the height of the point c t , 
 therefore the mould must be so moved that the point b lltl 
 shall pass from b v to c ; consequently b v c is the distance 
 the mould must be moved, or, as it is technically termed, 
 the sliding distance; hence b lltl c,, which is equal to b v c, is 
 the distance the mould is to be moved: up when on top, 
 and down when underneath. This is more fully explained 
 in Art. 284. 
 
 278. Face-Moulds for Circular Stairs. At Fig. 1 60 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 the heights as given in Fig. 158; for example, make 
 kk t equal to k ni k tl of Fig. 158, and cc, equal to c ll c l of Fig. 
 158. Join c t and k t and extend the line to meet ck, extended, 
 in a. Join a and b ; then ab is the line of intersection of 
 the cutting and horizontal planes ; it is therefore a horizon- 
 tal line, parallel to which the ordinates are to be drawn. 
 Perpendicular to ab draw b,c, til . Parallel to ab draw cc u 
 and kk lt ; join b, and c lt ; the angle cc tl b f is the plumb-bevil ; 
 perpendicular to b, c f . draw b, b lt , k tl k in , and c tl c llt ; make b t b u 
 equal to b t b, and so of the other two points, k ltl and r ///} make 
 them respectively equal to their horizontal projections upon 
 the plane A. Join c iu and k fl ; also, k it and b tl ; then b tl k ltl 
 and k tll c ltl are the tangents. From t //f draw the line c ltl b lt 
 parallel to b t c tl ; this is the slide-line. In this example, this 
 
286 
 
 STAIRS. 
 
 line passes through the point b lt ; the slide-line does not 
 always pass through the ends of the two tangents ; it is not 
 required to pass through both, but it is indispensable that it 
 be drawn parallel with the rake-line b t c tl . The lines for the 
 joints at each end are drawn square to the tangent lines. 
 Points in the curves, as many as are desirable, are now to 
 be found by ordinates as shown in the figure, and as before 
 
 explained for the points in the tangents. The curves are 
 made by drawing a line against the side of a flexible strip 
 bent to coincide with the points. 
 
 The face-mould may be put in position by revolving the 
 planes C and B, as explained in the last article, for the rail 
 at the newel. 
 
 The face-mould for the rail over the plan from c to d is to 
 
FACE-MOULDS CONTINUED. 287 
 
 be obtained in a similar manner, taking the heights from Fig. 
 158. For example, make d d / equal to d tl d t of Fig. 15.8, and 
 //, equal to /,/ of Fig. 158 (taking the heights at their 
 actual measurement now). Join d t and / and extend the 
 line to meet the line dl extended in r ; join r and c\ then re 
 is the line of intersection, and parallel to which the ordinates 
 are to be drawn. The points in the face-mould may now be 
 obtained as in the previous cases, giving attention first to 
 the tangent and slide-line ; drawing the lines for the joints 
 perpendicular to the tangents. 
 
 It may be remarked here that the chord-line <:is parallel 
 with the measuring line b t c ilin and that the line o k bisects 
 the chord-line; so, also, the line ol bisects the chord-line cd. 
 This coincidence is not accidental ; it will always occur in a 
 regular circular stairs. 
 
 Hence in cases of this kind it is not necessary to go 
 through the preliminaries by which to obtain the intersect- 
 ing line ab, but draw it at once parallel to the line ok, 
 bisecting the chord be and passing through the point of 
 intersection of the two tangents! For the distance to slide 
 the mould in its after-application, the lines are given at c tl 
 and d lfj and their use is explained in the last article, and 
 more fully in Art. 284. 
 
 279. Face-Mould 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 f 
 to g y but drawn at a larger scale. To prepare for the face- 
 moulds, perpendicular to // draw //., and make pp t equal 
 t PuiPii f F*8- ! 5^ (taking this height now at its actual 
 measurement) ; join p t and /; then fp t is the tangent of the 
 vertical plane C, and / is a point in the cutting plane at its 
 intersection with the bass-plane A. Now since rs, the tan- 
 gent over pg, is horizontal and is in the cutting plane, 
 therefore from / draw fa parallel with r s or pg\ then fa 
 is the line of intersection of the cutting and horizontal 
 planes, and gives direction to the ordinates. Draw //, 
 perpendicular to fa ; make p lll p jl equal to pp t ; join / and 
 /,; then the angle //// is the plumb-bevil ; perpendicular 
 
288 
 
 STAIRS. 
 
 to //, draw // and // ; make // equal to p tu g, 
 p it d equal to p llt p\ join d and / ; then *// and dp lin are 
 the tangents. Make p tl e equal to half the thickness of the 
 plank ; draw f tl a parallel Avith f t p lt ; make f tl a equal to e c ; 
 draw ac t parallel with the tangent f n d\ through /, per- 
 pendicular to f ti d, draw the line for the butt-joint ; then f u c, 
 is the distance required to determine the vertical line on the 
 face of the joint at /, as shown at A. Through p ltli , per- 
 
 pendicular to the tangent p ull d, draw the line for the butt- 
 joint ; make p nil b equal to ec\ then p lltl b is the distance 
 required for determining the vertical line on the face of the 
 joint at / //7/ , as shown at B (see Art. 284). The curved lines 
 are obtained by drawing a line against, the edges of a flexi- 
 ble rod bent to as many points as desirable, obtained by 
 measuring the ordinates of the plan at A and transferring 
 them to the face-mould by the corresponding ordinates, as 
 before explained. 
 
RAILING FOR QUARTER-CIRCLE STAIRS. 
 
 28 9 
 
 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 fa be represents the central line of the rail around the 
 cylinder, and the quadrant de, distant from the first quadrant 
 20 inches, is the tread-line, upon which from d, a point taken 
 at pleasure, the treads are run off. Through e, perpendicu- 
 
 FIG. 162. 
 
 lar to af, draw ae (the occurrence here of one of the 
 points of division on the tread-line perpendicularly opposite 
 a, the spring of the circle, is only an accidental coincidence) ; 
 make a a, equal to two risers ; join a, and /. With the 
 diameter a c, on b as a centre, describe the arc at g, crossing 
 ac extended ; through b draw gb, ; then ab, is the stretch- 
 out, or development of the quadrant a b. 
 
 Through // draw h i, tending toward the centre of the 
 
STAIRS. 
 
 cylinder; make b t i t equal to bi\ perpendicular to fb t draw 
 b t b it and i t i tl . As there are four risers from e to //, make 
 a t a ti equal to four risers, and draw a tl i u parallel with fa ; 
 through i it draw a t b i( ; by intersecting lines, or in any con- 
 venient manner, ease off to any extent the angle fa t i u . 
 Through j, a point in this curve (chosen so as to be perpen- 
 dicularly over m, a point between a and /, nearer to a), 
 draw kly a tangent to the curve. Perpendicularly to this 
 tangent, through /, draw the line for a butt-joint ; also 
 through //t and perpendicularly to a, b lt , draw the line for 
 the joint at the centre of the half circle. On the line aa n 
 set up points of division for the riser heights, and through 
 these points of division draw horizontal lines to the line 
 
 From these points of contact drop perpendiculars to the 
 line fa b t , and transfer such of them as require it to the circle 
 at, by drawing lines tending to g. Through these points of 
 intersection with the central line of the rail, and through the 
 points of division on the tread-line, draw the riser-lines me, 
 a n, etc. At half a riser above the floor-line, on top of the 
 upper riser draw a horizontal line, and ease off the angle as 
 shown ; the intersection of the floor-line with this curve 
 gives the position of the top riser at the centre of the rail. 
 This completes the plan of the steps and the elevation of the 
 rail requisite preliminaries for the face-moulds. The gradu- 
 ation of the treads from flyers to winders obviates an abrupt 
 angle at their junction in the rail and front-string. The 
 objection to the graduation, that it interferes with the 
 regularity of stepping at the tread-line, is not realized in 
 practice. 
 
 281. Face-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 hori- 
 zontal projection of the tangents for the first wreath are ad 
 and db drawn at right angles to each other, tangent to the 
 circle at a and b. Let those tangents be extended beyond 
 d; through ;;/, the lower end of the wreath, draw md jy mak- 
 ing an angle with md equal to that in Fig. 162, between the 
 
FACE-MOULDS FOR THE TWISTS. 
 
 2 9 I 
 
 line af and #,./; or let the angle dmd t equal a fa, of Fig. 
 162. Make dd lt equal to dd r Make bb tl equal to b tll b tl of 
 Fig. 162 ; join d lt and b tl and extend the line to e 4l ; make 
 b lt b v equal to b u b llu of Fig. 162, and draw b^e tl parallel with 
 
 FIG. 163. 
 
 </*. From ^ /y draw e tl e parallel with b lt b\ through e and / 
 draw ef tangent to the circle at /; then b e and ef are the 
 horizontal projections of the tangents for the upper wreath. 
 Then if the plane B be revolved on ad, the plane C on de, 
 
2Q2 STAIRS. 
 
 and the plane D on cf until they each stand vertical to the 
 plane A, the lines md n d tl e in and e ni f t will constitute the 
 tangents of the two wreaths in position. This arrangement 
 locates the upper joint of the upper wreath at /, leaving fc, 
 a part of the circle, to be worked as a part of the long level 
 rail on the landing. As the tangent over ef is level, the 
 raking part of the rail will all be included in the wreath bf, 
 so that at the joint / the rail terminates on the level. 
 
 The portion fc, therefore, is a level rail requiring no 
 canting, and it requires no other face-mould than that afforded 
 by the plan from / to c. 
 
 For the face-mould for the rail over ;;/ a b, let the line e ff d tl 
 be extended to m v , a point in the base-line b m v ; then ;// v is a 
 point in the base-plane A, as well as in the cutting plane E; 
 therefore the line m v m is the intersecting line parallel to 
 which all the ordinates on plane A are to be drawn. Per- 
 pendicular to this intersecting line m v m,at any convenient 
 place draw m, b, ; make b t b ni parallel to m v m and equal to 
 b b tl ; connect b ilt with ;//,, a point at the intersection of the 
 lines m v m and b, m t ; then the angle b b nl m / is the plumb- 
 bevil. Through d, parallel to m v m, draw^^; from the 
 three points /#,, d iti , and b til draw lines perpendicular to 
 m f b llt ; make m t m tl equal to m t m ; make b itl b llu equal to b t b. 
 Since the measuring base-line m, b, passes through d, the 
 point of the angle formed by the two tangents, d til is the 
 point of this angle in the cutting plane E\ therefore join m tl 
 and </,, also d ltl and b lltl ; then b ltll d ni and d ltl m n are .the 
 two tangents at right angles to which the joints at m n and 
 b lin are drawn. The curves of the face-mould are now found 
 as usual, by transferring the distances by ordinates, as shown, 
 from the plane A to the plane E, making the distance from 
 the rake-line m / b ltl to each point in plane E equal to the dis- 
 tance from the corresponding point in the plane A to the 
 measuring base-line m, b t . Now, to obtain the sliding distance 
 and the vertical line upon the butt-joints, make b tll b v equal 
 to half the thickness of the plank ; parallel with m t b tll draw 
 b y b^ ; also, b llu vii and m 4l m tll ; make b tlll b vYi and * m llt 
 each equal to b v vi ; through ^ vii and m n/ , and parallel to the 
 respective tangents, draw vil b^ and m llt m liu ; then b* and 
 
FACE-MOULDS FOR WINDING STAIRS. 293 
 
 m, ni are the points from which, through the centre of the 
 butt-joints, a line is to be drawn which will be vertical when 
 the wreath is in position. (See Art. 284.) 
 
 For the face-mould for the upper quarter, through b, Fig. 
 
 163, draw b e t parallel with d u e tl ; make e e lit equal to e e t ; 
 draw e lil f l parallel with e f. Now, since e tll f t is a horizon- 
 tal line and is in the cutting plane F, therefore, parallel with 
 e til f t and through b tJ draw b n ; then b n is the required in- 
 tersecting line. Extend e f to / ; make // equal to //, ; 
 join/,, and n ; then the angle ff l{ n is the plumb-bevil. Per- 
 pendicular to nf t/ draw /"/", and n n,, and make these lines 
 respectively equal to e f and b n ; join f tl and f ltl ; also /, 
 and n t ; then f n f til and f nl n { are the required tangents. 
 The butt-joints at/) and n t are drawn perpendicular to their 
 respective tangents. To get the slide distance and vertical 
 lines on the butt-joints, make/), f v equal to half the thickness 
 of the plank ; parallel with n /),, through / v draw/ v /) yy/ ; also, 
 through n t draw n t n ti ; make n, ii' u equal to / v f ntl ; through 
 /; , parallel with n t f ltlJ draw n u n lt .\ then n ln is the point 
 through which a line is to be drawn to the centre of the 
 butt-joint, and this line will be in the vertical plane contain- 
 ing the tangent. So, also, parallel with the tangent f tl f ltl , 
 and through / ///y , draw f lllt / vi ; then/ vi is the point through 
 which a line is to be drawn to the centre of the butt-joint 
 (see Art. 284). The curve is now to be obtained by the 
 ordinates, as before explained. 
 
 282. Face-]Woiilci 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-joint connecting the two wreaths, as shown in Fig. 
 
 164. Instead of locating the joint precisely at the middle of 
 the half circle, as was done in Fig. 163, place it farther down, 
 say at ;/, which is at n in Fig. 162, two risers down from the 
 top, or at any other point at will. Then through n in the 
 plan draw m t s tangent to the circle at n ; and perpendicu- 
 lar to this tangent draw ;/ n in and dd n ; make n n tl equal to 
 n t n of Fig. 162; from d erect d d t perpendicular to m d\ 
 
294 
 
 STAIRS. 
 
 make the angle d m d t equal to that of b in j I of Fig. 162. 
 Make d d n equal to d d t ; join d n and n n and extend the line 
 to ;;/,, a point of intersection with the base-line n n t ; then n t 
 is a point in the base-plane, as also in the cutting plane ; 
 
 FIG. 164. 
 
 therefore m t m is the intersecting line parallel to which all 
 the ordinates of the plan are to be drawn, and perpendicular 
 to which m /{ n t , the measuring base-line, is drawn. Make 
 n t n tlil equal to n n n ; connect m tl and n /(/i , and then transfer 
 
CARE REQUIRED IN DRAWING. 295 
 
 by the ordinates to the cutting plane m d and n the three 
 points of the plan at the ends of the tangents, as before de- 
 scribed, as also such points in the curve as may be required 
 to mark the curve upon the face-mould, all as shown in previ- 
 ous examples. For the face-mould of the upper wreath, make 
 n tl n nl equal to n n n of Fig. 162. From n tll draw n nl s tl par- 
 allel with m t s ; extend the line d lt n lt to intersect n tll s u in s lt ; 
 parallel with n iu n draw s /f s ; from s draw s r tangent to the 
 circle at r (s n equals s r) ; through r, tending to the centre of 
 the cylinder, draw the butt-joint ; then r s and s n are the 
 horizontal projections of the tangents for the upper wreath- 
 piece, the tangent s r being level and, consequently, parallel 
 to the intersecting line drawn through n. Perpendicular to 
 r s draw r t p ; parallel with n u s it draw ns t ; make r t r lt equal 
 toss,; join r n and /. From this line and the measuring base- 
 line r, p y the points for the tangents are first to be obtained 
 and then the points in the curve, all as before described. 
 The part of the circle from r to c is on the level, as before 
 shown, and may be worked upon the end of the long level 
 rail, its form being just what is shown in the plan from c to r. 
 
 283. Face-Moiild : Tet of Accuracy. The methods 
 which have been advanced for obtaining face-moulds are 
 based upon principles of such undoubted correctness that 
 there can be no question as to the results, when the methods 
 given are thoroughly followed. And yet, notwithstanding 
 the correctness of the system and its thorough comprehen- 
 sion by the stair-builder, he will fail of success unless he 
 exercises the greatest care in getting his dimensions, his per- 
 pendiculars, and his angles. The slightest deviation in a 
 perpendicular terminated by an oblique line will result in a 
 magnified error at the oblique line. To secure the greatest 
 possible degree of accuracy, care must be exercised in the 
 choice of the instruments by which the drawings are to be 
 made : care to know that a straight-edge is what it purports 
 to be ; that a square, or right-angle, is truly a right-angle ; 
 that the compasses or dividers be well made, the joint per- 
 fect, and the ends neatly ground to a point. Then let the 
 drawing-board be carefully planed to a true surface ; and, 
 
296 STAIRS. 
 
 if possible, let the drawing, full size, be made upon large, 
 stout roll-paper rather than upon the drawing-board itself, 
 as then the points for the face-mould may be pricked through 
 upon the board out of which the face-mould is to be cut, and 
 thus a correct transfer be made. For long straight lines it 
 is better to use a fine chalk-line than the edge of a wooden 
 straight-edge. The line is more trustworthy. Perpendicu- 
 lars, especially when long, are better obtained by measure- 
 ment or by calculation (Art. 503) than by a square. The 
 pencil used should be of fine quality rather hard, in order 
 that its point may be kept fine. With these precautions in 
 regard to the instruments used, and with due care in the 
 manipulations, the face-moulds may be correctly drawn, 
 accurate in size and form. As a test of the accuracy of the 
 work, it will be well to observe in regard to the tangents, 
 that the length of a tangent, as found upon the face-mould, 
 should always equal its length as shown upon the vertical 
 plane. For example, in Fig. 160, the tangent k lt c til on the 
 face-mould should be equal to k t c t , the tangent on the vertical 
 plane B ; and in cases like this, where the stairs are quite 
 regular, with equal treads at the front-string, the two tan- 
 gents of a face-mould are-equal to each other, or k lf c in equals 
 k tl b tl ; and in this case, the line b tl c llt should equal the rake- 
 line b, *. 
 
 Again, as another example, in Fig. 161, d f in the tangent 
 upon the face-mould, should be equal to//,, the tangent of 
 the vertical plane C\ while d / //;/ , the other tangent on the 
 face-mould, should be equal to r s, the tangent of the vertical 
 plane D. But the more important test is in the length of the 
 chord-line joining the ends of the two tangents ; as, for ex- 
 ample, the chord m tl b inl of Fig. 163, the horizontal projec- 
 tion of which is the chord ;;/ b in plane A. Perpendicular to 
 m b draw b g\ make b g equal to b b lt , and join g and m ; then 
 m^ b lilt , the chord of the face-mould, should be equal to ;// g. 
 After fully testing the accuracy of the drawing for the face- 
 mould, choose a well-seasoned thin piece of white-wood, or 
 any other wood not liable to split, and plane it to an even 
 thickness throughout ; mark upon it the curves, joints, tan- 
 gents, and slide-line, and cut the edges true to the curve- 
 
THE FACE-MOULD APPLIED TO PLANK. 297 
 
 lines and joints square through the board ; then square over 
 such marks as are required to draw each tangent and the 
 slide-line also upon the reverse side of the board. This 
 completes the face-mould. 
 
 284. Application of the Faee-UIoulcl. In order that a 
 more comprehensive idea of the lines given for applying a 
 face-mould may be had, let A, Fig. 165, represent one end of 
 a wreath-piece as it appears when first cut from a plank, and 
 when held up in the position it is to occupy at completion 
 over the stairs. Also, let B represent the corresponding 
 face-mould, laid upon the wreath-piece A in the position 
 which it should have after sliding. And, for the purpose of 
 a clearer illustration, let it be supposed that the two pieces, 
 A and B, are transparent. Then let a t a b d c^ e t represent a 
 solid of wedge form, having a triangular level base, a b d, 
 upon the three lines of which stand these three vertical 
 planes, namely : on the line a b the plane a t a b c^ upon the 
 line a d the plane a, a d e t , and on the line d b the plane b d 
 e t c, ; the top of the solid is an inclined plane, a i c t e n and the 
 vertical line a t a is the edge of the wedge. Now, it will be 
 observed that the point a in the base of the solid is identical 
 with a, the centre of the butt-joint, and the point a, (at the 
 intersection of two vertical planes and the inclined plane of 
 the solid) is vertically over a, and is identical with a,, a poim 
 in the upper surface of the plank. Also, the inclined plane 
 e / c / a t , which forms the top of the solid, coincides with the 
 upper surface of the plank A, from which the wreath-piece 
 has been squared ; and the line c, a t (at the angle formed by 
 the inclined plane e, c, a, and the vertical plane a f a b c,) coin- 
 cides with / g, the slide-line drawn upon the top of the 
 plank ; also, the line e, a, (at the angle formed by the in- 
 clined plane e, c, a t and the vertical plane a t a d e^ coincides 
 with a t k, the tangent line upon the underside of the face- 
 mould after it has been slid to its new position, vertically 
 over its true position at the middle of the thickness of the 
 plank. From a the line a c is drawn parallel with a, c t ; so, 
 also, the line a e is drawn parallel with a, e, ; consequently 
 the line e c is parallel with e, c t ; and the plane e c a is parallel 
 
298 
 
 STAIRS. 
 
 with the plane e { c f a t , and coincides with a plane passing 
 through the middle of the thickness of the plank, and, conse- 
 quently, is the cutting plane referred to in previous articles, 
 upon which the lines are drawn which give shape to the 
 
 FIG. 165. 
 
 face-mould. When the face-mould is first laid upon the plank, 
 the line i,j, coincides with i lt j /t , and Hvhen in that position, 
 its form marked upon the plank is the form by which the 
 plank is sawed square through ; but this gives the form of 
 
THE SLIDING OF THE FACE-MOULD. 299 
 
 the wreath, not as it is at the surface of the plank, but as it 
 is at the middle of the thickness of the plank, or upon the 
 plane ace-, so that, for example, the line i it j tl represents the 
 line ij drawn through a, the centre of the butt-joint ; and 
 when the mould B is slid to the position shown in the figure, 
 the line i l j l comes into a position vertically over ij\ hence 
 the three lines i f i, a t a, and j t j are each vertical and in a 
 vertical plane, Hjj^j- By these considerations it will be seen 
 that the face-mould B, located as shown in the figure, is in its 
 true position for the second marking, by which the addi- 
 tional cutting is now to be performed vertically. This being 
 established, it will now be shown how to get upon the butt- 
 joint a line in the vertical plane containing the tangent. If 
 the top and bottom lines of the vertical plane a t a b c t be ex- 
 tended, they will meet in the point /, and will extend the 
 plane into a triangle Ib c t , cutting the upper edge of the 
 butt-joint in/, the end of the tangent, and the point in which 
 the point a t of the underside of the face-mould was located 
 when the mould was first applied to the plank. The line/Vz 
 on the butt-joint is perpendicular to i j or i lt j u . Again, if 
 the top and bottom lines of the plane a t a d e, be extended, 
 they will meet in /, and will extend the plane into the tri- 
 angle p d e t , cutting the edge of the butt-joint in h, a point 
 from which, if a line be drawn upon the butt-joint to a, its 
 centre, this line will be in the vertical plane pde t , which 
 plane contains the tangent perpendicular to which the butt- 
 joint is drawn ; consequently lines upon the butt-joint par- 
 allel to h a will each be in a vertical plane parallel to the 
 vertical tangent plane, and lines drawn upon the butt-joint 
 perpendicular to these lines will be horizontal lines ; hence 
 the line h a is the required line by which to square the 
 wreath at the butt-joint. Now, it will be observed that the 
 triangle af a, is like that given in the various figures for obr 
 taining face-moulds, to regulate the sliding of the face-mould 
 and the squaring at the butt-joint. For example, in Fig. 
 163, the right-angled triangle b ul v vi is the one referred to. 
 This triangle is in a vertical plane parallel to one containing 
 the slide-line; its longer side is a vertical line ; one of the 
 sides containing the right angle is equal to half the thickness 
 
300 
 
 STAIRS. 
 
 of the plank, while the other, drawn parallel to the face of 
 the plank, is the distance the face-mould is required to slide. 
 Precisely like this, the triangle a f a t of Fig. 165 is in the 
 vertical plane / b c n containing fg, the slide-line ; its longer 
 side, a, a, is a vertical line ; fa, one of the sides containing 
 the right angle, is equal to half the thickness of the plank, 
 while the other side, drawn coincident with the surface of 
 
 FIG. 166, 
 
 the plank, is the distance to slide the face-mould. Therefore 
 the triangle a t f a of Fig. 165 gives the required lines by 
 which to regulate the application of the face-moulds. The 
 relative position of the more important of these lines is geo- 
 metrically shown in Fig. 166, in which A and B are upon the 
 horizontal plane of the paper, C is in a vertical plane stand- 
 ing on the ground-line b d, and D is a plan of the butt-joint, 
 revolved upon the line i tt j n into the horizontal plane, and 
 
BLOCKING-OUT OF THE RAIL. 3<DI 
 
 then perpendicularly removed to the distance //,. The let- 
 tering corresponds with that in Fig. 165. The shaded part 
 of D shows the end of the squared wreath. When the 
 blocked piece has been marked by the face-mould in its 
 second application, its edges are to be trimmed vertically as 
 shown in Fig. 167, after which the top and bottom surfaces 
 of the wreath are to be formed from the shape marked on 
 the butt-joints. 
 
 FIG. 167. 
 
 285. Face-Mould Curves are Elliptical. The curves of 
 the face-mould for the hand-railing of any stairs of circular 
 plan are elliptical, and may be drawn by a trammel, or 
 in anv other convenient manner. The trouble, however, 
 attending the process of obtaining the axes, so as to be able 
 to employ the trammel in describing the curves, is, in many 
 cases, greater than it would be to obtain the curves through 
 points found by ordinates, in the usual manner. But as 
 
302 
 
 STAIRS. 
 
 this method for certain reasons may be preferred by some, 
 an example is here given in which the curves are drawn by 
 a trammel, and in which the method of obtaining the axes is 
 shown. 
 
 Let Fig. 168 represent the plan of a hand-rail around part 
 
 FIG. 168. 
 
 of a cylinder and with the heights set up, the intersection 
 line obtained, the measuring base-line drawn, the rake-line 
 established, and the tangents on the face-mould located all 
 in the usual manner as hereinbefore described. Then, to 
 prepare for the trammel, from o, the centre of the cylinder, 
 draw o b t parallel with the intersecting line, and b t o t perpen- 
 
FACE-MOULDS FOR ROUND RAILS. 303 
 
 dicular to ,/,, the rake-line ; make b t o t equal to bo, and o l a l 
 equal to oa; through o t draw o t h parallel with b t f r From 
 o draw oe perpendicular toob,; continue the central circular 
 line of the rail around to e; parallel with ob t draw e f y 
 and from / the point of intersection of ef with b t f t , and 
 perpendicular to #,/,, draw f t e t \ make f l e l equal to fe\ 
 then 0, is the centre of the ellipse, and o t a, the semi-conju- 
 gate diameter and o t e, the semi-transverse diameter of an 
 ellipse drawn through the centre of the face-mould. To get 
 the diameters for the edges of the face-mould, make a / c, and 
 a / d i each equal to half the width of the rail, as at cad\ par- 
 allel to a line drawn from a t to e t , and through c r draw the 
 line c t g\ also, parallel with a line drawn from a t to e t draw 
 d t h (see Art. 559); then for the curve at the inner edge of 
 the face-mould, o t g is the semi-transverse diameter, and o l c l 
 the semi-conjugate ; while for the curve at the outer edge 
 o.h is the semi-transverse diameter, and o t d t the semi-conju- 
 gate. So much of the edges of the face-mould as are straight 
 are parallel with the tangent. Now, placing the trammel at 
 the centre, as shown in the figure, and making the distance 
 on the rod from the pencil to the first pin equal to the 
 semi-conjugate diameter, and the distance to the second pin 
 equal to the semi-transverse diameter, each curve may be 
 drawn as shown. (See Art. 549.) 
 
 286. Face-Moulds for Round Hails. The previous ex- 
 amples 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-mould for the sides of the rail, find one for a centre-line 
 and describe circles upon it, as at Fig. 145 ; then trace the 
 sides of the mould by the' points so found. The thickness of 
 stuff for the twists of a round rail is the same as for the 
 straight part. The twists are to be sawed square through. 
 
 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. 
 
304 
 
 STAIRS. 
 
 Managed in this way, the butt-joint is in a plane pierced 
 perpendicularly by the tangent. But if the block be not 
 sawed square through, but vertically from the edges of the 
 
 FIG. 169. 
 
 face-mould, then, especially, care is required in locating the 
 joint. The method of sawing square through is attended 
 with so many advantages that it is now generally followed ; 
 yet, as it is possible that for certain reasons some may prefer, 
 
rv 
 
 [(UNI VET? si 
 
 POSITION OF THE BUTT-JOINT." 305 
 
 in some cases, to saw vertically, it is proper that the method 
 of finding the position of the joint for that purpose should 
 be given. Therefore, let A, Fig. 169, be the plan of the rail, 
 and B the elevation, showing its side; in which kz is the 
 direction of the butt-joint. From k draw kb parallel to /<?, 
 and ke at right angles to kb\ from b draw b f, tending to 
 the centre of the plan, and from / draw fe parallel to bk; 
 from /, through e, draw I i, and from i draw id parallel to 
 ef-, join dfand b, and db will be the proper direction for the 
 joint on the plan. The direction of the joint on the other 
 side, a c, can be found by transferring the distances x b and 
 od to xa and oc. Then the allowance for over wood to 
 cover the butt-joint is shown as that which is included be- 
 tween the lines ox and db. The face-mould must be so 
 drawn as to cover the plan to the line b d for the wreath at 
 the left, and to the line a c for that at the right. By some 
 the direction of the joint is made to radiate toward the 
 centre of the cylinder ; indeed, even Mr. Nicholson, in his 
 Carpenter s Guide, so advised. That this is an error may be 
 shown as follows : In Fig. 170, arji is the plan of a part of the 
 rail about the joint, s u is the stretch-out of a i, and gp is the 
 helinet, or vertical projection of the plan arji. This is 
 found by drawing a horizontal line from the height set upon 
 each perpendicular standing upon the stretch-out line su. 
 The lines upon the plan arji are drawn radiating to the 
 centre of the cylinder, and therefore correspond to the 
 horizontal lines of the helinet drawn upon its upper and 
 under surfaces. 
 
 Bisect rt on the ordinate drawn from the centre of the 
 plan, and through the middle draw cb at right angles to gv ; 
 from b and c draw cd and be at right angles to su ; from d 
 and e draw lines radiating toward the centre of the plan ; 
 then do and em will be the direction of the joint on the 
 plan, according to Nicholson, and cb its direction on the 
 falling-mould. It must be admitted that all the lines on the 
 upper or the lower side of the rail which radiate toward 
 the centre of the cylinder, as do, cm, or ij\ are level; for 
 instance, the level line wv, on the top of the rail in the 
 helinet, is a true representation of the radiating line// on 
 
306 
 
 STAIRS. 
 
 the plan. The line bh, therefore, on the top of the rail in 
 the helinet, is a true representation of e m on the plan, and 
 kc on the bottom of the rail truly represents do. From k 
 draw / parallel to cb, and from h draw hf parallel to bc\ 
 
 FIG. 170. 
 
 join / and b, also c and /; then cklb will be a true repre- 
 sentation of the end of the lower piece, B, and cfh b of the 
 end of the upper piece, A ; and fk or hi will show how 
 much the joint is open on the inner, or concave, side of the 
 rail. 
 
CORRECT LINES FOR BUTT-JOINT. 
 
 307 
 
 To show that the process followed in Art. 287 is correct, 
 let do and em (Fig. 171) be the direction of the butt-joint 
 found as at Fig. 169. Now, to project, on the top of the rail 
 in the helinet, a line that does not radiate toward the centre 
 
 of the cylinder, as jk, draw vertical lines from j and k to w 
 and A, and join w and h ; then it will be evident that wh is 
 a true representation in the helinet of jk on the plan, it 
 being in the same plane as j k, and also in the same winding- 
 surface as wv. The line /, also, is a true representation on 
 
308 
 
 STAIRS. 
 
 the bottom of the helinet of the line jk in the plan. The 
 line of the joint e m, therefore, is projected in the same way, 
 and truly, by ib on the top of the helmet, and the line do 
 by ca on the bottom. Join a and i, and then it will be seen 
 that the lines c a, a t, and ib exactly coincide with cb, the line 
 of the joint on the convex side of the rail ; thus proving the 
 lower end of the upper piece, A, and the upper end of the 
 lower piece, B, to be in one and the same plane, and that the 
 direction of the joint on the plan is the true one. By refer- 
 ence to Fig. 169 it will be seen that the line li corresponds 
 to xi in Fig. 171 ; and that e k in that figure is a representa- 
 tion of fb, and ik of db. 
 
 288. Scrolls for Hand-Rail : General Rule for Size 
 and Position of the Regulating Square. The breadth 
 which the scroll is to occupy, the number of its revolutions, 
 and the relative size of the regulating square to the eye of 
 the scroll being given, multiply the number of revolutions 
 by 4, and to the product add the number of times a side of 
 the square is contained in the diameter of the eye, and the 
 sum will be the number of equal parts into which the breadth 
 is to be divided. Make a side of the regulating square 
 equal to one of these parts. To the breadth of the scroll 
 add one of the parts thus found, and half the sum will be the 
 length of the longest ordinate. 
 
 FIG. 172. 
 
 289. Centres in Regulating Square. Let a 2 I b (Fig. 
 T72) be the size of a regulating square, found according to 
 the previous rule, the required number of revolutions being 
 
SCROLLS AT NEWEL. 
 
 309 
 
 if. Divide two adjacent sides, as a 2 and 2 i, into as many 
 equal parts as there are quarters in the number of revolu- 
 tions, as seven ; from those points of division draw lines 
 across the square at right angles to the lines divided ; then 
 i being the first centre, 2, 3,4, 5, 6, and 7 are the centres for 
 the other quarters, and 8 is the centre for the eye ; the heavy 
 lines that determine these centres being each one part less 
 in length than its preceding line. 
 
 FIG. 173- 
 
 290. Scroll for Hand-Rail Over Curtail Step. Let 
 a b (Fig. 173) be the given breadth, if the given number of 
 revolutions, and let the relative size of the regulating square 
 to the eye be \ of the diameter of the eye. Then, by the 
 rule, if multipled by 4 gives 7, and 3, the number of times a 
 side of the square is contained in the eye, being added, the 
 sum is 10. Divide a b, therefore, into 10 equal parts, and set 
 one from b to c ; bisect a c in e ; then a e will be the length 
 of the longest ordinate (i d or i e). From a draw a d, 
 from e draw e i, and from b draw b f, all at right angles to 
 a b ; make e \ equal to a, and through i draw i d parallel 
 
310 STAIRS. 
 
 to a b ; set b c from i to 2, and upon i 2 complete the regu 
 lating square; divide this square as at Fig. 172; then de- 
 scribe the arcs that compose the scroll, as follows : upon i 
 describe d e, upon 2 describe e /, upon 3 describe / g, 
 upon 4 describe -//, etc.; make dl equal to the width of the 
 rail, and upon i describe / ;//, upon 2 describe mn, etc.; de- 
 scribe the eye upon 8, and the scroll is completed. 
 
 291. Scroll for Curtail Step. Bisect d I (Fig. 173) in <?, 
 and make o v equal to of the diameter of a baluster ; make 
 v w equal to the projection of the nosing, and e x equal to 
 wl\ upon i describe ivy, and upon 2 describe y z ; also, upon 
 2 describe x i, upon 3 describe ij\ and so around to z ; and 
 the scroll for the step will be completed. 
 
 292. Position of Baluster* Under Scroll. Bisect dl 
 (Fig. 173) in 0, and upon i, with i o for radius, describe the 
 circle o r u ; set the baluster at p fair with the face of the 
 second riser, r 2 , and from/, with half the tread in the divi- 
 ders, space off as at o, q, r, s, /, u, etc., as far as f ; upon 2, 3,4, 
 and 5 describe the centre-line of the rail around to the eye 
 of the scroll ; from the points of division in the circle o r u 
 draw lines to the centre-line of the rail, tending to 8, the 
 centre of the eye ; then the intersection of these radiating 
 lines with the centre-line of the rail will determine the posi- 
 tion of the balusters, as shown in the figure. 
 
 293. Fal ling-Mould for Raking Part of Scroll. Tangi- 
 cal to the rail at h (Fig. 173) draw h k parallel to da ; then 
 ha* will be the joint between the twist and the other part 
 of the scroll. Make de* equal to the stretch-out of d e, and 
 upon d e* find the position of the point k, as at k* ; at Fig. 
 174, make e d equal to ^ 2 d in Fig. 173, and d c equal to d c* 
 in that figure ; from c draw c a at right angles to e c, and 
 equal to one rise ; make c b equal to one tread, and from b, 
 through a, draw bj\ bisect a c in /, and through / draw m q 
 parallel to eh', m q is the height of the level part of a 
 scroll, which should always be about 3^ feet from the floor ; 
 ease off the angle mfj, according to Art. 521, and draw 
 
FACE-MOULD FOR THE SCROLL. 
 
 gw n parallel to' m x j, and. at a distance equal to the thick- 
 ness of the rail ; at a convenient place for the joint, as *, 
 draw in at right angles to bj\ through n draw/// at right 
 angles to e h ; make dk equal to d k~ in Fig. 173, and from k 
 draw ko at right angles toe A; at Fig. 173, make <//; 2 equal 
 to d h in ^. 174, and draw // 2 2 at right angles to 
 
 FIG. 174. 
 
 then k a* and /i 2 b* will be the position of the joints on the 
 plan, and, at Fig. 174, op and in their position on the falling- 
 mould ; and po in (Fig. 174) will be the required falling- 
 mould which is to be bent upon the vertical surface from h 2 
 to k (Fig. 173). 
 
 FIG. 175. 
 
 294. Face-Mould for the Scroll. At Fig. 173, from k 
 draw kr* at right angles to r 2 d- at Fig. 172, make h r 
 equal to h* r* in Fig. 173, and from r draw r s at right 
 angles to rh ; from the intersection of r s with the level line 
 m q, through /, draw s t ; at Fig. 173, make // 2 b* equal to ^ / 
 in Fig. 172, and join b* and r 2 ; from rt 2 , and from as many 
 
3 I2 
 
 STAIRS. 
 
 other points in the arcs, a * I and k d, as is thought neces- 
 sary, draw ordinates to r 2 d at right angles to the latter; 
 make r b (Fig. 175) equal in its length and in its divisions to 
 the line r 2 d 2 in Fig. 173 ; from r, n, o, p, q, and / draw the 
 lines r k, n d, o a, p e, q f, and / c at right angles to r b, and 
 equal to r 2 k, d* s-,f* a", etc., in Fig. 173 ; through the points 
 thus found trace the curves k I and a c, and complete the 
 face-mould, as shown in the figure. This mould is to be ap- 
 plied to a square-edged plank, with the edge, / b parallel to 
 the edge of the plank. The rake-lines upon the edge of the 
 plank are to be made to correspond to the angle s t h in 
 Fig. 174. The thickness of stuff required for this mould is 
 shown at Fig. 174, between the lines s t and u v u v being 
 drawn parallel to s t. 
 
 f 7' 
 
 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 con- 
 
BORING FOR BALUSTERS. 
 
 313 
 
 venient place on the line fg, as o, with a radius equal to 
 half the width of the cap, describe the circle ijg\ make r I 
 equal to e b or e a ; join / and /, also / and i\ from the curve 
 / b to the line / j draw as many ordinates as is thought 
 necessary parallel to fg' from the points at which these 
 ordinates meet the line I j, and upon the centre, 0, describe 
 arcs in continuation to meet op ; from n t x, etc., draw ns,tu, 
 etc., parallel tofg; make n s, t u, etc., equal to ef, wv, etc.; 
 make x y, etc., equal to z d, etc.; make o 2, o 3, etc., equal to 
 o n, o t, etc.; make 2 4 equal to n s, and in this way find the 
 length of the lines crossing o m ; through the points thus 
 found describe the section of the newel-cap as shown in 
 the figure. 
 
 FIG. 177. 
 
 296. Boring for Balusters in a Round Rail before it 
 is Rounded. Make the angle oct (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 tan- 
 gent b d parallel to / c, and complete the rectangle e b d f, 
 having sides tangical to the circle ; from c draw c a at right 
 angles to o c; then, b d being the bottom of the rail, set a 
 gauge from b to a, and run it the whole length of the stuff ; 
 in boring, place the centre of the bit in the gauge-mark at a, 
 and bore in the direction a c. To do this easily, make chucks 
 as represented in the figure, the bottom edge,^/*, being par- 
 allel to o c, and having a place sawed out, as e f, to receive 
 the rail. These being nailed to the bench, the rail will be 
 held steadily in its proper place for boring vertically. The 
 distance apart that the balusters require to be, on the under- 
 side of the rail, is one half the length of the rake-side of the 
 pitch-board. 
 
STAIRS. 
 SPLAYED WORK. 
 
 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 
 
 FIG. 178. 
 
 obtained will, therefore, here be shown. In Fig. 178, a b c is 
 the angle at which the work is splayed, and b d, on the 
 upper edge of the board, is at right angles to a b ; make the 
 angle f g.j equal to a b c, and from f draw f h parallel to 
 e a ; from b draw b o at right angles to a b ; through o draw 
 ie parallel to c b, and join e and d\ then the angle a e d will 
 be the proper bevil for the ends from the inside, or k d e 
 from the outside. If a mitre-joint is required, set f g, the 
 thickness of the stuff on the level, from e to ;, and join m 
 and d-, then kdm will be the proper bevil for a mitre-joint. 
 If the upper edge of the splayed work is to be bevelled, 
 so as to be horizontal when the work is placed in its proper 
 position, then f gj, the same as a b c, will be the proper 
 bevel for that purpose. Suppose, therefore, that a piece in- 
 dicated by the lines kg, g f, and f h were taken off ; then a 
 line drawn upon the bevelled surface from d at right angles 
 to k d would show the true position of the joint, because it 
 would be in the direction of the board for the other side ; 
 but a line so drawn would pass through the point o, thus 
 proving the principle correct. So, if a line were drawn upon 
 the bevelled surface from d at an angle of 45 degrees to k d, 
 it would pass through the point n. 
 
VIEW IN THE ALHAMBRA. 
 
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 materially to 
 the regularity and elegance of the apartments. The dimen- 
 sions and style of finish of a door should be in accordance 
 with the size and style of the building, or the apartment 
 for which it is designed. As regards the utility of doors, 
 the principal door to a public building should be of suffi- 
 cient width to admit of a free passage for a crowd of people ; 
 while that of a private apartment will be wide enough if it 
 permit one person to pass without being incommoded. Ex- 
 perience has determined that the least width allowable for 
 this is 2 feet 8 inches ; although doors leading to inferior 
 and unimportant rooms may, if circumstances require it, be 
 as narrow as 2 feet 6 inches ; and doors for closets, where an 
 entrance is seldom required, may be but 2 feet wide. The 
 width of the principal door to a public building may be 
 from 6 to 12 feet, according to the size of the building; and 
 the width of doors for a dwelling may be from 2 feet 8 
 inches to 3 feet 6 inches. If the importance of an apart- 
 ment in a dwelling be such as to require a door of greater 
 width than 3 feet 6 inches, the opening should be closed 
 with two doors, or a door in two folds ; generally, in such 
 cases, where the opening is from 5 to 8 feet, folding or slid- 
 ing doors are adopted. As to*he height of a door, it should 
 in no case be less than about 6 feet 3 inches ; and generally 
 not less than 6 feet 8 inches. 
 
 299. The Proportion between Width and Height: of 
 
 single doors, for a dwelling, should be as 2 is to 5 ; and, for 
 entrance-doors to public buildings, as I is to 2. If the 
 width is given and the height required of a door for a 
 
3 i6 
 
 DOORS AND WINDOWS. 
 
 dwelling, multiply the width by 5, and divide the product 
 by 2 ; but if the height is given and the width required, 
 divide by 5 and multiply by 2. Where two or more doors 
 of different widths show in the same room, it is well to pro- 
 portion the dimensions of the more important by the above 
 rule, and make the narrower doors of the same height as 
 the wider ones; as all the doors in a suit of apartments, 
 except the folding or sliding doors, have the best appear- 
 ance when of one height. The proportions for folding or 
 sliding doors should be such that the width may be equal 
 to |- of the height ; yet this rule needs some qualification ; 
 for if the width of the opening be greater than one half the 
 width of the room, there will not be a sufficient space left 
 
 FIG. 179. 
 
 for opening the doors ; also, the height should be about one 
 tenth greater than that of the adjacent single doors. 
 
 300. Panefls. Where doors have but two panels in 
 width, let the stiles and muntins be each \- of the width ; or, 
 whatever number of panels there may be, let the united 
 widths of the stiles and the muntins, or the whole width of 
 the solid, be equal to -| of tt^e width of the door. Thus : in 
 a door 35 inches wide, containing two panels in width, the 
 stiles should be 5 inches wide ; and in a door 3 feet 6 inches 
 wide, the stiles should be 6 inches. If a door 3 feet 6 
 inches wide is to have 3 panels in width, the stiles and 
 muntins should be each 4^- inches wide, each panel being 8 
 inches. The bottom rail and the lock-rail ought to be each 
 equal in width to T V of the height of the door ; and the top 
 
TRIMMINGS FOR DOORS. 317 
 
 rail, and all others, of the same width as the stiles. The 
 moulding on the panel should be equal in width to j of the 
 width of the stile. 
 
 301. TTrimmings. Fig. 179 shows a method of trimming 
 doors : a is the door-stud ; b, the lath and plaster ; <:, the 
 ground ; d, the jamb ; ^, the stop ; /and g, architrave casings ; 
 and h, the door -stile. It is customary in ordinary work to 
 form the stop for the door by rebating the jamb. But when 
 the door is thick and heavy, a better plan is to nail on a 
 piece as at e in the figure. This piece can be fitted to the 
 door and put on after the door is hung ; so, should the door 
 be a trifle winding, this will correct the evil, and the door be 
 made to shut solid. 
 
 Fig. 1 80 is an elevation of a door and trimmings suitable 
 for the best rooms of a dwelling. (For trimmings generally, 
 see Sect. V.) The number of panels into which a door 
 should be divided may be fixed at pleasure ; yet the present 
 style of finishing requires that the number be as small as a 
 proper regard for strength will admit. In some of our best 
 dwellings, doors have been made having only two upright 
 panels. A few years' experience, however, has proved that 
 the omission of the lock-rail is at the expense of the strength 
 and durability of the door ; a four-panel door, therefore, is 
 the best that can be made. 
 
 302. Hanging l>oor*. Doors should all be hung so as 
 to open into the principal rooms ; and, in general, no door 
 should be hung to open into the hall, or passage. As to the 
 proper edge of the door on which to affix the hinges, no 
 general rule can be assigned. 
 
 WINDOWS. 
 
 303. Requirement* 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 
 given that will answer in all cases ; yet, as an apprpxima- 
 
DOORS AND WINDOWS. 
 
 tion, the following has been used for general purposes. 
 Multiply together the length and the breadth in feet of the 
 apartment to be lighted, and the product by the height in 
 
 FIG. 1 80. 
 
 feet; then the square root of this product will show the 
 required number of square feet of glass. 
 
 304. Winclow-Frame. For the size of window-frames, 
 add 4-J inches to the width of the glass for their width, and 
 
WIDTH OF .INSIDE SHUTTERS. 319 
 
 6 inches to the height of the glass for their height. These 
 give the dimensions, in the clear, of ordinary frames for 12- 
 light windows ; the height being taken at the inside edge of 
 the sill. In a brick wall, the width of the opening is 8 
 inches more than the width of the glass 4^ for the stiles of 
 the sash, and 3^ for hanging stiles and the height between 
 the stone sill and lintel is about io| inches more than the 
 height of the glass, it being varied according to the thick- 
 ness of the sill of the frame. 
 
 305. Inside Shuiter. 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 shown be- 
 tween the face of the shutter when folded into the box and 
 the quirk of the stop-bead, or edge of the casing, is half an 
 inch ; and, in the usual method of letting the whole of the 
 thickness of the butt hinge into the edge of the box-shutter, 
 it is necessary to make allowance for the tlirow of the hinge. 
 This may, in general, be estimated at \ of an inch at each 
 hinging ; which being added to the margin, the entire width 
 of the shutters will be i J inches more than the width of the 
 frame in the clear. Then, to ascertain the width of the box- 
 shutter, add i-J inches to the width of the frame in the clear, 
 between the pulley-stiles ; divide this product by 4, and add 
 half an inch to the quotient, and the last product will be 
 the required width. For example, suppose the window to 
 have 3 lights in width, 1 1 inches each. Then, 3 times 1 1 is 
 33, and 4^ added for the wood of the sash gives 37^ ; 37^ 
 and 1^ is 39, and 39 divided by 4 gives 9! ; to which add 
 half an inch, and the result will be loj inches, the width 
 required for the box-shutter. 
 
 306. Proportion: Width and Height. In disposing 
 and locating windows in the walls of a building, the rules of 
 architectural taste require that they be of different heights 
 in different stories, but generally of the same width. The 
 windows of the upper stories should all range perpendicu- 
 larly over those of the first, or principal, story ; and they 
 
320 DOORS AND WINDOWS. 
 
 should be disposed so as to exhibit a balance of parts 
 throughout the front of the building. To aid in this it is 
 always proper to place the front door in the middle of the 
 front of the building ; and, where the size of the house will 
 admit of it, this plan should be adopted. (See the latter 
 part of Art. 50.) The proportion that the height should 
 bear to the width may be, in accordance with general usage, 
 as follows : 
 
 of the width. 
 
 The height 
 
 of basement windows, i -J 
 
 i< 
 
 principal-story " 
 
 2* 
 
 n 
 
 second -story " 
 
 '1 
 
 u 
 
 third-story 
 
 If 
 
 
 
 fourth-story " 
 
 I| 
 
 
 
 attic-story " 
 
 th( 
 
 the same as the width. 
 
 But, in determining the height of the windows for the 
 several stories, it is necessary to take into consideration the 
 .height of the story in which the window is to be placed. 
 For, in addition to the height from the floor, which is gen- 
 erally required to be from 28 to 30 inches, room is wanted 
 above the head of the window for the window-trimming 
 and the cornice of the room, besides some respectable space 
 which there ought to be between these. 
 
 307. Circular Heads. Doors and windows usually ter- 
 minate in a horizontal line at top. These require no special 
 directions for their trimmings. But circular-headed doors 
 and windows are more difficult of execution, and require 
 some attention. If the jambs of a door or window be placed 
 at right angles to the face of the wall, the edges of the soffit, 
 or surface of the head, would be straight, and its length be 
 found by getting the stretch-out of the circle (Art. 524) ; 
 but when the jambs are placed obliquely to the face of the 
 Avail, occasioned by the demand for light in an oblique 
 direction, the form of the soffit will be obtained by the fol- 
 lowing article ; as also when the face of the wall is circular, 
 as shown in the succeeding figure. 
 
OBLIQUE SOFFITS OF WINDOWS. 
 
 321 
 
 308. Form of Soffit for Circular 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. 
 
 From a, through e, draw ag at righ't angles to ab\ obtain 
 the stretch-out of efa, and make eg equal to it; divide eg- 
 and efa each into a like number of equal parts, and drop 
 perpendiculars from the points of division in each ; from 
 the points of intersection, i, 2, 3, etc., in the line ad, 
 
 FIG. 182. 
 
 draw horizontal lines to meet corresponding perpendicu- 
 lars from eg\ then those points of intersection will give the 
 curve line dg, which will be the on* 3 required for the edge 
 of the soffit. The other edge, ch t is found in the same 
 manner. 
 
322 DOORS AND WINDOWS. 
 
 For the form of the soffit for circular window-heads, 
 when the face of the wall is curved, let abed (Fig. 182) be 
 the ground-plan of a given window, and e f a a vertical sec- 
 tion of the head taken at right angles to the face of the 
 jambs. Proceed as in the foregoing article to obtain the 
 line dg\ then that will be the curve required for the edge of 
 the soffit, the other edge being found in the same manner. 
 
 If the given vertical section be taken in a line with the 
 face of the wall, instead of at right angles to the face of the 
 jambs, place it upon the line cb (Fig. 181), and, having drawn 
 ordinates at right angles to cb, transfer them to efa ; in this 
 way a section at right angles to the jambs can be obtained. 
 
SECTION V. MOULDINGS AND CORNICES. 
 
 MOULDINGS. 
 
 3O9, 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 remains of classic architec- 
 ture, are eight in number, and are known by the following 
 names : 
 
 FIG 183. Annulet, band, cincture, fillet, listel or square. 
 
 FIG. 184. ^ Astragal or bead. 
 
 Torus or tore. 
 
 FIG. 185. 
 
 FIG 186 Scotia, trochilus or mouth. 
 
 Ovolo, quarter-round or echinus. 
 
 FIG. 187. 
 
 Cavetto, cove or hollow. 
 
 FIG. 188. 
 
324 MOULDINGS AND CORNICES. 
 
 Cymatium, or cyma-recta. 
 
 FIG. 189. 
 
 Ogee. 
 
 Inverted cymatium, or cyma-reversa. 
 FIG. 190. 
 
 Some of the terms are derived thus : Fillet, from the French 
 word fil, thread. Astragal, from astragalos, a bone of the 
 heel or the curvature of the heel. Bead, because this 
 moulding, when properly carved, resembles a string of beads. 
 Torus, or tore, the Greek for rope, which it resembles when 
 on the base of a column. Scotia, from skotia, darkness, be- 
 cause of the strong shadow which its depth produces, and 
 which is increased by the projection of the torus above it. 
 Ovolo, from ovum, an egg, which this member resembles, 
 when carved, as in the Ionic capital. Cavetto, from cavus, 
 hollow. Cymatium, from kumaton, a wave. 
 
 310. Characteristics of Mouldings. Neither of these 
 mouldings is peculiar to any one of the orders of architect- 
 ure ; and although each has its appropriate use, yet it is by 
 no means confined to any certain position in an assemblage 
 of mouldings. The use of the fillet is to bind the parts, as 
 also that of- the astragal and torus, which resemble ropes. 
 The ovolo and cyma-reversa are strong at their upper ex- 
 tremities, and are therefore used to support projecting parts 
 above them. The cyma-recta and cavetto, being weak at 
 their upper extremities, are not used as supporters, but are 
 placed uppermost to cover and shelter the other parts. The 
 scotia is introduced in the base of a column to separate the 
 upper and lower torus, and to produce a pleasing variety 
 and relief. The form of the bead and that of the torus is the 
 same ; the reasons for giving distinct names to them are 
 that the torus, in every order, is always considerably larger 
 than the bead, and is placed among the base mouldings, 
 
GRECIAN MOULDINGS. 
 
 325 
 
 whereas the bead is never placed there, but on the capital or 
 entablature ; the torus, also, is seldom carved, whereas the 
 bead is ; and while the torus among the Greeks is frequently 
 elliptical in its form, the bead retains its circular shape. While 
 the scotia is the reverse of the torus, the cavetto is the re- 
 verse of the ovolo, and the cyma-recta and cyma-reversa are 
 combinations of the ovolo and cavetto. 
 
 FIG. 191. 
 
 The curves of mouldings, in Roman architecture, were 
 most generally composed of parts of circles ; while those of 
 the Greeks were almost always elliptical, or of some one of 
 the conic sections, but rarely circular, except in the case of 
 the bead, which was always, among both Greeks and Ro- 
 mans, of the form of a semicircle. Sections of the cone af- 
 ford a greater variety of forms than those of the sphere ; and 
 perhaps this is one reason why the Grecian architecture so 
 
326 
 
 MOULDINGS AND CORNICES. 
 
 much excels the Roman. The quick turnings of the ovolo 
 and cyma-reversa, in particular, when exposed to a bright 
 sun, cause those narrow, well-defined streaks of light which 
 give life and splendor to the whole. 
 
 311. A Profile: is an assemblage of essential parts and 
 mouldings. That profile produces the happiest effect which 
 
 FIG, 192. 
 
 is composed of but few members, varied in form and size, 
 and arranged so that the plane and the curved surfaces suc- 
 ceed each other alternately. 
 
 312. The Grecian Torus and Scotia. Join the extremi- 
 ties a and b (Fig. 191), and from /, the given projection of 
 the moulding, draw/0 at right angles to the fillets ; from b 
 
 FJG. 194. 
 
 FIG. 195. 
 
 draw bh at right angles to a b ; bisect a b in c ; join / and c, 
 and upon c, with the radius cf, describe the arc/^, cutting 
 bh'mh', through c draw de parallel with the fillets; make 
 dc and ce each equal to b //; then de and a b will be conju- 
 gate diameters of the required ellipse. To describe the 
 curve by intersection of lines, proceed as directed at Art. 
 
THE GRECIAN ECHINUS. 
 
 551 and note ; by a trammel, see Art. 549; and to find the 
 foci, in order to describe it with a string, see Art. 548. 
 
 313. The Grecian Echinus. Figs. 192 to 199 exhibit, va- 
 riously modified, the Grecian ovolo, or echinus. Figs. 192 to 
 196 are elliptical, a b and b c being given tangents to the curve ; 
 parallel to which the semi-conjugate diameters, ad and dc, 
 
 IMG. 196. 
 
 FIG. 197. 
 
 are drawn. In Figs. 192 and 193 the lines a d and dc are semi- 
 axes, the tangents, ab and be, being at right angles to each 
 other. To draw the curve, see Art. 551. In Fig. 197 the 
 curve is parabolical, and is drawn according to Art. 560. In 
 Figs. 198 and 199 the curve is hyperbolical, being described 
 according to Art. 561. The length of the transverse ax's, a b y 
 
 FIG. i 
 
 FIG. 199. 
 
 being taken at pleasure in order to flatten the curve, a b 
 should be made short in proportion to ac. 
 
 314. The Grecian Cavetto. In order to describe this, 
 Figs. 200 and 201, having the height and projection given, 
 see Art. 551. 
 
 315. The Grecian Cynia-Rccta. When the projection 
 is more than the height, as at Fig. 202, make a b equal to the 
 
328 
 
 MOULDINGS AND CORNICES. 
 
 height, and divide abed into four equal parallelograms ; then 
 proceed as directed in note to Art. 551. When the projec- 
 tion is less than the height, draw da (Fig. 203) at right angles 
 
 FIG. 201. 
 
 FIG. 200. 
 
 to ab\ complete the rectangle, abed; divide this into four 
 equal rectangles, and proceed according to Art. 551. 
 
 316. The Grecian Cyma-Reversa. When the projection 
 
 FIG. 203. 
 
 is more than the height, as at Fig. 204, proceed as directed 
 for the last figure ; the curve being the same as that, the 
 position only being changed. When the projection is less 
 
 FIG. 204. 
 
 FIG. 205. 
 
 than the height, draw a d (Fig. 205) at right angles to the 
 fillet ; make a d equal to the projection of the moulding ; then 
 proceed as directed for Fig. 202. 
 
FORMS OF ROMAN MOULDINGS. 
 
 329 
 
 317. Roman Mouldings : are composed pf parts of circles, 
 and have, therefore, less beauty of form than the Grecian. 
 The bead and torus are of the form of the semicircle, and the 
 scotia, also, in some instances ; but the latter is often composed 
 of two quadrants, having different radii, as at Figs. 206 and 
 207, which resemble the elliptical curve. The ovolo and ca- 
 
 FIG. 206. 
 
 FIG. 207. 
 
 vetto are generally a quadrant, but often less. When they are 
 less, as at Fig. 210, the centre is found thus : join the extrem- 
 ities, a and b, and bisect a b in c ; from c, and at right angles 
 to a b, draw c d, cutting a level line drawn from a in d ; then d 
 will be the centre. This moulding projects less than its 
 height. When the projection is more than the height, as at 
 Fig. 212, extend the line from c until it cuts a perpendicular 
 
 FIG. 208. 
 
 FIG. 209. 
 
 drawn from a, as at d\ and that will be the centre of the 
 curve. In a similar manner, the centres are found for the 
 mouldings at Figs. 2QJ, 211, 213, 216, 217, 218, and 219. The 
 centres for the curves at Figs. 220 and 221 are found thus: 
 bisect the line a b at c ; upon a, c and b successively, with a c 
 or cb for radius, describe arcs intersecting at d and d\ then 
 those intersections will be the centres. 
 
330 
 
 MOULDINGS AND CORNICES. 
 
 FIG. 210. 
 
 FIG. 211. 
 
 FIG. 212. 
 
 FIG. 213. 
 
 FIG. 214. 
 
 FIG. 215 
 
 FIG. 216. 
 
 FIG. 217. 
 
FORMS OF MODERN MOULDINGS. 
 
 331 
 
 3(8. Modern Mouldings: are represented in Figs. 222 
 to 229. They have been quite extensively and successfully 
 used in inside finishing. Fig. 222 is appropriate for a bed- 
 moulding under a low projecting shelf, and is frequently 
 used under mantel-shelves. The tangent i h is found thus : 
 bisect the line ab at c, and be at d-, from d draw de at 
 right angles to eb\ from b draw bf parallel to ed\ upon b, 
 
 FIG. 218. 
 
 FIG. 219. 
 
 with b d for radius, describe the arc d f\ divide this arc 
 into 7 equal parts, and set one of the parts from s, the limit 
 of the projection, to o ; make o h equal to o e ; from h, through 
 c, draw the tangent ki\ divide b h, hc,ci, and ia each into 
 a like number of equal parts, and draw the intersecting lines 
 as directed at Art. 521. If a bolder form is desired, draw 
 the tangent, i h, nearer horizontal, and describe an elliptic 
 
 FIG. 220. 
 
 FIG. 221. 
 
 curve as shown in Figs. 191 and 224. Fig. 223 is much used 
 on base, or skirting, of rooms, and in deep panelling. The 
 curve is found in the same manner as that of Fig. 222. In 
 this case, however, where the moulding has so little projec- 
 tion in comparison with its height, the point e being found 
 as in the last figure, h s may be made equal to s e, instead of 
 o e as in the last figure. Fig. 224 is appropriate for a crown 
 
332 
 
 MOULDINGS AND CORNICES. 
 
 FIG. 223. 
 
 FIG. 224. 
 
PLAIN MOULDINGS. 
 
 333 
 
 moulding of a cornice. In this figure the height and pro- 
 jection are given; the direction of the diameter, ab, drawn 
 
 FIG. 225. 
 
 FIG. 226. 
 
 through the middle of the diagonal, ef, is taken at pleasure ; 
 and dc is parallel to ae. To find the length of dc, draw b h 
 
 FIG. 227. 
 
 FIG. 228. 
 
 FIG. 229. 
 
 at right angles toab; upon 0, with of for radius, describe 
 the arc, ///, cutting bh in h; then make o c and od each 
 
334 
 
 MOULDINGS AND CORNICES. 
 
 equal to bh.* To draw the curve, see note to Art. 551. Figs. 
 22$ to 229 are peculiarly distinct from ancient mouldings, 
 being composed principally of straight lines ; the few curves 
 they possess are quite short and quick. 
 
 Figs. 230 and 231 are designs for antae caps. The di- 
 ameter of the antas is divided into 20 equal parts, and the 
 height and projection of the members are regulated in ac- 
 cordance with those parts, as denoted under H and P, height 
 and projection. The projection is measured from the mid- 
 dle of the antse. These will be found appropriate for por- 
 ticos, doorways, mantelpieces, door and window trimmings, 
 
 H.P. 
 
 n. 
 
 15 
 
 8l'l4f! 
 
 *"*i 
 
 910J 
 
 10 
 
 Hi 15 
 
 . 
 
 
 ^|- H 
 HH~ 
 
 910J 
 
 FIG. 230. 
 
 FIG. 231. 
 
 etc. The height of the antae for mantelpieces should be 
 from 5 to 6 diameters, having an entablature of from 2 to 
 2J- diameters. This is a good proportion, it being similar to 
 the Doric order. But for a portico these proportions are 
 
 * The manner of ascertaining the length of the conjugate diameter, dc, in 
 this figure, and also in Figs. 191, 241, and 242 is' new, and is important in this 
 application. It is founded upon well-known mathematical principles, viz.: All 
 the parallelograms that may be circumscribed about an ellipsis are equal to 
 one another, and consequently any one is equal to the rectangle of the two 
 axes. And again : The sum of the squares of every pair of conjugate diame- 
 ters is equal to the sum of the squares of the two axes. 
 
EAVE CORNICES. 
 
 335 
 
 much too heavy : ah antse 1 5 diameters high and an entab- 
 lature of 3 diameters will have a better appearance. 
 
 CORNICES. 
 
 319. Idesigns for Cornice. Figs. 232 to 240 are designs 
 for eave cornices, and Figs. 241 and 242 are for stucco cor- 
 nices for the inside finish of rooms. In some of these the 
 projection of the uppermost member from the facia is 
 divided into twenty equal parts, and the various members 
 
 FIG. 232. 
 
 are proportioned according to those parts, as figured under 
 //and P. 
 
 320. Eave Cornices Proportioned to Height of Build- 
 
 ing. Draw the line ac (Fig. 243), and make be and ba each 
 equal to 36 inches ; from b draw bdvk right angles to ac, 
 and equal in length to f of a c ; bisect b d in ^, and from a, 
 through >, draw af\ upon a, with ac for radius, describe the 
 arc cf, and upon e, with ef-iwc radius, describe the arc/W; 
 divide the curve dfc, into 7 equal parts, as at IO, 20, 30, 
 etc., and from these points of division draw lines to be 
 
336 
 
 MOULDINGS AND CORNICES. 
 
 J J 
 
 FIG. 233. 
 
 DDOIfi 
 
 JLJLILJLILIULILJLILJU 
 
 JUUUUULI 
 
 FIG. 234. 
 
EXAMPLES OF CORNICES. 
 
 337 
 
 FIG. 235. 
 
 FIG. 236. 
 
338 
 
 MOULDINGS AND CORNICES. 
 
 FIG. 237. 
 
 H. P. 
 
 10J 
 
 FIG. 238. 
 
VARIOUS DESIGNS OF CORNICES. 
 
 339 
 
 H. P. 
 
 17 
 
 716 
 
 3*31 
 
 FIG. 239. 
 
 H.P. 
 
 FIG. 240. 
 
340 
 
 MOULDINGS AND CORNICES 
 
 H. P. 
 
 FIG. 241. 
 
 H. P. 
 
 FIG. 242. 
 
PROPORTION OF CORNICES. 
 
 341 
 
 parallel to db; then the distance b i is the projection of a 
 cornice for a building 10 feet high ; b 2, the projection at 20 
 feet high ; b 3, the projection at 30 feet, etc. If the projec- 
 tion of a cornice for a building 34 feet high is required, 
 divide the arc between 30 and 40 into 10 equal parts, and 
 
 V 
 
 a b i 2 3 4 c 
 
 FIG. 243. 
 
 from the fourth point from 30 draw a line to the base, b c, 
 parallel with bd\ then the distance o/ the point at which 
 that line cuts the base from b will be the projection re- 
 quired. So proceed for a cornice of any height within 70 
 feet. The above is based on the supposition that 36 inches 
 
 FIG. 244. 
 
 is the proper projection for a cornice 70 feet high. This, 
 for general purposes, will be found correct ; still, the length 
 of the line be may be varied to suit the judgment of those 
 who think differently. 
 
 Having obtained the projection of a cornice, divide it 
 into 20 equal parts, and apportion the several members 
 
342 
 
 MOULDINGS AND CORNICES. 
 
 according to its destination as is shown at Figs. 238, 239, 
 and 240. 
 
 32 L Cornice Proportioned to a given Cornice. Let 
 
 the cornice at Fig. 244 be the given cornice. Upon any 
 point in the lowest line of the lowest member, as at #, with the 
 height of the required cornice for radius, describe an intersect- 
 ing arc across the uppermost line, as at b ; join a and b ; 
 then b \ will be the perpendicular height of the upper fillet 
 for the proposed cornice, I 2 the height of the crown mould- 
 ing and so of all the members requiring to be enlarged to 
 the sizes indicated on this line. For the projection of the 
 
 \ 
 
 FIG. 245. 
 
 proposed cornice, draw a d at right angles to a b, and c d at 
 right angles to b c ; parallel with c d draw lines from each 
 projection of the given cornice to the line ad-, then e d \v\\\ 
 be the required projection for the proposed cornice, and the 
 perpendicular lines falling upon c d will indicate the proper 
 projection for the members. 
 
 To proportion a cornice according to a larger given cor- 
 nice, let A (Fig. 245) be the given cornice. Extend a o to b, 
 and draw c d at right angles to a b ; extend the horizontal 
 lines of the cornice, A, until they touch o d\ place the height 
 of the proposed cornice from o to e, and join /"and e\ upon 
 o, with the projection of the given cornice, o a, for radius, 
 
TO FIND THE ANGLE BRACKET. 
 
 describe the quadrant ad\ from d draw db parallel tofe; 
 upon o t with o b for radius, describe the quadrant b c ; then 
 o c will be the proper projection for the proposed cornice. 
 Join a and c ; draw lines from the projection of the different 
 members of the given cornice to ao parallel to od\ from 
 these divisions on the line ao draw lines to the line oc 
 parallel to a c ; from the divisions on the line of draw lines 
 to the line o e parallel to the line fe\ then the divisions on 
 the lines o e and o c will indicate the proper height and pro- 
 jection for the different members of the proposed cornice. 
 In this process, we have assumed the height, o e, of the pro- 
 posed cornice to be given; but if the projection, o c, alone 
 
 FIG. 247. 
 
 be given, we can obtain the same result by a different pro- 
 cess. Thus: upon o, with oc for radius, describe the quad- 
 rant c b ; upon <?, with o a for radius, describe the quadrant 
 ad\ join d and b ; from /draw fe parallel to.db\ then oe 
 will be the proper height for the proposed cornice, and the 
 height and projection of the different members can be 
 obtained by the above directions. By this problem, a cor- 
 nice can be proportioned according to a smaller given one 
 as well as to a larger ; but the method described in the pre- 
 vious article is much more simple for that purpose. 
 
 322. Angle Bracket in a Built Cornice. Let A (Fig. 
 246) be the wall of the building, and B the given bracket, 
 
344 
 
 OULDINGS AND CORNICES. 
 
 which, for the present purpose, is turned down horizontally. 
 The angle-bracket, C, is obtained thus : through the ex- 
 tremity, #, and parallel with the wall, fd, draw the line a b ; 
 make^r equal af, and through c draw cb parallel with ed\ 
 join d and , and from the several angular points in B draw 
 ordinates to cut db in i, 2, and 3 ; at those points erect lines 
 perpendicular to db', from h draw kg parallel to fa-, take 
 
 FIG. 248. 
 
 the ordinates, i 0, 2 0, etc., at B, and transfer them to C, and 
 the angle-bracket, C, will be denned. In the same manner, 
 the angle-bracket for an internal cornice, or the angle-rib of 
 a coved ceiling, or of groins, as at Fig. 247, can be found. 
 
 323. Raking mouldings matched with Level Returns 
 
 Let A (Fig. 248) be the given moulding, and A b the rake of 
 
CROWN MOULDING ON THE RAKE. 345 
 
 the roof. Divide the curve of the given moulding into any 
 number of parts, equal or unequal, as at i, 2, and 3 ; from 
 these points draw horizontal lines to a perpendicular erected 
 from c ; at any convenient place on the rake, as at B, draw 
 a c at right angles to A b ; also from b draw the horizontal 
 line b a ; place the thickness, da, of the moulding at A from 
 b to a, and from a draw the perpendicular line a e ; from 
 the points i, 2, 3, at A, draw lines to C parallel to A b ; 
 make a i, a 2, and a 3, at B, and at C, equal to ^ui, etc., at A ; 
 through the points, i, 2, and 3, at B, trace the curve this 
 will be the proper form for the raking moulding, From i, 
 2, and 3, at C, drop perpendiculars to the corresponding 
 ordinates from i, 2, and 3, at A ; through the points of inter- 
 section, trace the curve this will be the proper form for the 
 return at the top. 
 
' 
 
 PART II. 
 
 SECTION VI. GEOMETRY. 
 
 324. Mathematics Essential. In this and the following 
 Sections, which will constitute Part II., there are treated of 
 certain matters which may be considered as elementary. 
 They are all very necessary to be understood and acquired 
 by the builder, and are here compactly presented in a shape 
 which, it is believed, will aid him in his studies, and at the 
 same .time prove to be a great convenience as a matter of 
 reference. 
 
 The many geometrical forms which enter into the 
 composition of a building suggest a knowledge of .Elemen- 
 tary Geometry as essential to an intelligent comprehension 
 of its plan and purpose. One of the prime requisites of a 
 building is stability, a quality which depends upon a proper 
 distribution of the material of which the building is con- 
 structed ; hence a knowledge of the laws of pressure and 
 the strength of materials is essential ; and as these are based 
 upon the laws of proportion and are expressed more con- 
 cisely in algebraic language, a knowledge of Proportion and 
 of Algebra are likewise indispensable to a comprehensive 
 understanding of the subject. There will be found in this 
 work, however, only so much of these parts of mathematics 
 as have been deemed of the most obvious utility in the 
 Science of Building. For a more exhaustive treatment of 
 the subjects named, the reader is referred to the many able 
 works, readily accessible, which make these subjects their 
 specialties. 
 
 325. Elementary Geometry. In all reasoning defini- 
 tions are necessary, in order to insure in the minds of the 
 
348 GEOMETRY. 
 
 proponent and respondent identity of ideas. A corollary is 
 an inference deduced from a previous course of reasoning. 
 An axiom is a proposition evident at first sight. In the fol- 
 lowing demonstrations there are many axioms taken for 
 granted (such as, things equal to the same thing are equal to 
 one another, etc.) ; these it was thought not necessary to 
 introduce in form. 
 
 326. Definition. If a straight line, as A B (Fig. 249), 
 stand upon another straight line, as CD, so that the two 
 
 angles made at the point B are equal A B C to A B D (Art. 
 499, obtuse angle] then each of the two angles is called a 
 right angle. 
 
 327. Definition. The circumference of every circle 
 is supposed to be divided into 360 equal parts, called 
 degrees ; hence a semicircle contains 180 degrees, a quad- 
 rant 90, etc. 
 
 C B D 
 
 FIG. 250. 
 
 328. Definition. The measure of an angle is the num- 
 ber of degrees contained between its two sides, using the 
 angular point as a centre upon which to describe the arc. 
 Thus the arc C E (Fig. 250) is the measure of the angle 
 C B E, E A of the angle E B A, and A D of the angle A B D. 
 
 329. Corollary. As the two angles at B (Fig. 249) are 
 right angles, and as the semicircle, CAD, contains 180 de- 
 grees (Art. 327), the measure of two right angles, therefore, is 
 
HE 
 
 'UNIVERSITY 
 
 RI6HT ANGLES AND OBLIQUE ANG^^ 
 
 ^4, .._. 
 
 1 80 degrees ; of one right angle, 90 degrees ; of half a right 
 angle, 4$ ; of one third of a right angle, 30, etc. 
 
 330. Definition. In measuring an angle (Art. 328), no 
 regard is to be had to the length of its sides, but only to the 
 degree of their inclination. Hence equal angles are such as 
 have the same degree of inclination, without regard to the 
 length of their sides. 
 
 331. Axiom. If two straight lines parallel to one 
 another, as A B and CD (Fig. 251), stand upon another 
 straight line, as E F, the angles A B F and CDF are equal, 
 and the angle A B E is equal to the angle CD E. 
 
 332. Definition. If a straight line, as A B (Fig. 250), 
 stand obliquely upon another straight line, as CD, then one 
 
 A C 
 
 / 
 
 B ~ D 
 
 FIG. 251. 
 
 of the angles, as A B C, is called an obtuse angle, and the 
 other, as A B D, an acute angle. 
 
 333. Axiom. The two angles ABDzndABC (Fig. 250) 
 are together equal to two right angles (Arts. 326, 329) ; also, 
 the three angles A B D, E B A, and CBE are together 
 equal to two right angles. 
 
 334.. corollary. Hence all the angles that can be 
 
 made upon one side of a line, meeting in a point in that 
 line, are together equal to two right angles. 
 
 335, corollary. Hence all the angles that can be made 
 
 on both sides of a line, at a point in that line, or all the 
 angles that can be made about a point, are together equal to 
 four right angles. 
 
350 
 
 GEOMETRY. 
 
 336. Proposition. If to each of two equal angles a 
 third angle be added, their sums will be equal. Let ABC 
 and D E F (Fig. 252) be equal angles, and the angle I J K the 
 one to be added. Make the angles G B A and H E D each 
 equal to the given angle IJ K\ then the angle G B C will be 
 equal to the angle HE F; for if ABC and D E F be angles 
 
 A~ 
 
 C 
 
 FIG. 252. 
 
 of 90 degrees, and IJK 30, then the angles GBC and 
 HEF will be each equal to 90 and 30 added, viz., 120 
 degrees. 
 
 337. Proportion. Triangles that have two of their 
 sides and the angle contained between them respectively 
 equal, have also their third sides and the two remaining 
 
 FIG. 253. 
 
 angles equal ; and consequently one triangle will every way 
 equal the other. Let ABC (Fig. 253) and DEF be two 
 given triangles, having the angle at A equal to the angle at 
 D, the side A B equal to the side D E, and the side A C 
 equal to the side D F\ then the third side of one, B C, is equal 
 to the third side of the other, E F\ the angle at B is equal to 
 the angle at * and the angle at C is equal to the angle at 
 
EQUAL TRIANGLES IN PARALLELOGRAMS. 351 
 
 F. For if one triangle be applied to the other, the three 
 points B, A, C, coinciding with the three points E, D, F, the 
 line Bf must coincide with the line EF\ the angle at B 
 with the angle at E; the angle at C with the angle at F\ 
 and the triangle B A C be every way equal to the triangle 
 EDF. 
 
 338. Proposition. The two angles at the base of an 
 isosceles triangle are equal. Let ABC (Fig. 254) be an 
 
 B D C 
 
 FIG. 254. 
 
 isosceles triangle, of which the sides, A B and A C, are equal. 
 Bisect the angle (Art. 506) BA C by the line A D. Then, the 
 \ineJ5A being equal to the line A C, the line A D of the 
 triangle E being equal to the line A D of the triangle F 
 (being common to each), the angle BAD being equal to the 
 angle DA C, the line B D must, according to Art. 337, be 
 
 D D c 
 
 FIG. 255. 
 
 equal to the line D C, and the angle at B must be equal to 
 the angle at C. 
 
 339. Proportion. A diagonal crossing a parallelogram 
 divides it into two equal triangles. Let C D E F (Fig. 255) 
 be a given parallelogram, and C F a line crossing it diag- 
 onally. Then, as E C is equal to F D, and EF to CD, the 
 angle at E to the angle at D, the triangle A must, according 
 to Art. 337, be equal to the triangle B. 
 
352 
 
 GEOMETRY. 
 
 340 Proposition. Let J KL M (Fig. 256) be a given 
 parallelogram, and K L a diagonal. At any distance between 
 J K and L M draw N P parallel to J K\ through the point 
 G, the intersection of the lines KL and N P, draw HI 
 parallel to K M. In every parallelogram thus divided, the 
 parallelogram A is equal to the parallelogram B. For, ac- 
 cording to Art. 339, the triangle JKL is equal to the tri- 
 angle K L M, the triangle C to the triangle D, and E to F\ 
 
 M 
 
 this being the case, take D and F from the triangle K LM, 
 and C and from the triangle JKL, and what remains in 
 one must be equal to what remains in the other ; therefore, 
 the parallelogram A is equal to the parallelogram B. 
 
 34-1. Proposition. Parallelograms standing upon the 
 same base and between the same parallels are equal. Let. 
 A BCD and EFCD (Fig. 257) be given parallelograms 
 
 FIG. 257. 
 
 standing upon the same base, CD, and between the same 
 parallels, A F and CD. Then A B and E F, being equal to 
 CD, are equal to one another; BE being added to both 
 A B and E F, A E equals B F; the line A C being equal to 
 B D, and A E to B F, and the angle C A E being equal (Art. 
 331) to the angle D B F, the triangle A E C must be equal 
 (Art. 337) to the triangle B F D\ these two triangles being 
 equal, take the same amount, the triangle B E G, from each, 
 
TRIANGLE EQUAL TO QUADRANGLE. 
 
 353 
 
 and what remains in one, A B G C, must be equal to what 
 remains in the other, E F D G\ these two quadrangles being 
 equal, add the same amount, the triangle C G D, to each, and 
 they must still be equal ; therefore, the parallelogram 
 A B CD is equal to the parallelogram E F C D. 
 
 342. Corollary. Hence, if a parallelogram and triangle 
 stand upon the same base and between the same parallels, 
 
 H 
 
 D 
 
 FIG. 258. 
 
 the parallelogram will be equal to double the triangle. 
 Thus, the parallelogram A D (Fig. 257) is double (Art. 339) 
 the triangle C E D. 
 
 343. proposition. Let FG H D (Fig. 258) be a given 
 
 quadrangle with the diagonal F D. From G draw G E 
 
 FIG. 259. 
 
 parallel toFD- extend HD to E\ join F and E ; then the 
 triangle .F^^will be equal in area to the quadrangle 
 FGHD. For since the triangles FDG and FDE stand 
 upon the same base, F D, and between the same parallels, 
 FD and G E, they are therefore equal (Arts. 341, 342); and 
 since the triangle C is common to both, the remaining tri- 
 
354 GEOMETRY. 
 
 angles, A and B, are therefore equal ; then, B being equal to 
 A, the triangle F E H is equal to the quadrangle F G H D. 
 
 344. Proposition. If two straight lines cut each other, 
 as FG and H J (Fig. 259), the vertical, or opposite angles, 
 A and C, are equal. Thus, FE, standing upon H y, forms 
 the angles B and C, which together amount (Art. 333) to two 
 right angles ; in the same manner, the angles A and B form 
 two right angles ; since the angles A and B are equal to B 
 and C, take the same amount, the angle B, from each pair, 
 and what remains of one pair is equal to what remains of 
 the other ; therefore, the angle A is equal to the angle C. 
 The same can be proved of the opposite angles B and D. 
 
 345. Propo8ition. The three angles of any triangle are 
 equal to two right angles. Let ABC (Fig. 260) be a given 
 triangle, with its sides extended to F, E and D, and the line 
 C G drawn parallel to BE. As G C is parallel, to EB, the 
 angle at //is equal (Art. 331) to the angle at L ; as the lines 
 FC and BE cut one another at A, the opposite angles at M 
 and TV" are equal (Art. 334) ; as the angle at N is equal (Art. 
 331) to the angle at J y the angle at J is equal to the angle at 
 M] therefore, the three angles meeting at 7 are equal to the 
 three.angles of the triangle A B C ; and since the three angles 
 at C are equal (Art. 333) to two right angles, the three angles 
 of the triangle ABC must likewise be equal to two right 
 angles. Any triangle can be subjected to the same proof. 
 
 346. Corollary. Hence, if one angle of a triangle be a 
 right angle, the other two angles amount to just one right 
 angle. 
 
RIGHT ANGLE IN SEMICIRCLE. 355 
 
 34-7. Corollary. If one angle of a triangle be a 'right 
 angle and the two remaining angles are equal to one another, 
 these are each equal to half a right angle. 
 
 348. Corollary. If any two angles of a triangle amount 
 to a right angle, the remaining one is a right angle. 
 
 349. Corollary. If any two angles of a triangle are to- 
 gether equal to the remaining angle, that remaining angle is 
 a right angle. 
 
 350. Corollary. If any two angles of a triangle are each 
 equal to two thirds of a right angle, the remaining angle is 
 also equal to two thirds of a right angle. 
 
 351. Corollary. Hence, the angles of an equilateral 
 triangle are each equal to two thirds of a right angle. 
 
 FIG. 261. 
 
 352. Proposition. If from the extremities of the di- 
 ameter of a semicircle two straight lines be drawn to any 
 point in the circumference, the angle formed by them at that 
 point will be a right angle. Let ABC (Fig. 261) be a given 
 semicircle ; and A B and B C lines drawn from the extrem- 
 ities of the diameter A C to the given point B; the angle 
 formed at that point by these lines is a right angle. Join 
 the point B and the centre D ; the lines DA, D B, and D C, 
 being radii of the same circle, are equal; the angle at A is, 
 therefore, equal (Art. 338) to the angle at E\ also, the angle 
 at C is, for the same reason, equal to the angle at F\ the 
 angle A B C, being equal to the angles at A and C taken to- 
 gether, must, therefore (Art. 349), be a right angle. 
 
 353. Proportion. The square on the hypothenuse of a 
 right-angled triangle is equal to the squares on the two re- 
 
356' 
 
 GEOMETRY. 
 
 maining sides. Let ABC (Fig. 262) be a given right-angled 
 triangle, having a square formed on each of its sides ; then 
 the square BE is equal to the squares NCand GB taken 
 together. This can be proved by showing- that the parallelo- 
 gram B L is equal to the square G B ; and that the parallelo- 
 gram C L is equal to the square H C. The angle C B D is a 
 right angle, and the angle A B F\s a right angle ; add to each 
 of these the angle ABC] then the angle F B C will evidently 
 be equal (Art. 336) to the angle ABD\ the triangle FB C 
 and the square G B, being both upon the same base, F B, and 
 between the same parallels, F B and G C, the square G B is 
 equal (A rt. 342) to twice the triangle F B C '; the triangle 
 A B D and the parallelogram B L, being both upon the same 
 
 FIG. 262. 
 
 base, B D, and between the same parallels, B D and A L, the 
 parallelogram BL is equal to twice the triangle A B D ; 
 the triangles, FB C and AB D, being equal to one another 
 (Art. 337), the square G B is equal to the parallelogram B L, 
 either being equal to twice the triangle F B C or A B D. The 
 method of proving H C equal to C L is exactly similar thus 
 proving the square B E equal to the squares H C and G B, 
 taken together. 
 
 This problem, which is the 4/th of the First Book of 
 Euclid, is said to have been demonstrated first by Pythago- 
 ras. It is stated (but the story is of doubtful authority) 
 that as a thank-offering for its discovery he sacrificed a hun- 
 dred oxen to the gods. From this circumstance it is some- 
 times called the Jiecatomb problem. It is of great value in 
 
DIAGONAL OF SQUARE FORMING OCTAGON. 
 
 357 
 
 the exact sciences, more especially in Mensuration and As- 
 tronomy, in which many otherwise intricate calculations are 
 by it made easy of solution. 
 
 354. Proposition. In an equilateral octagon the semi- 
 diagonal of a circumscribed square, having its sides coinci- 
 dent with four of the sides of the octagon, equals the dis- 
 tance along a side of the square from its corner to the more 
 remote angle of the octagon occurring on that side of the 
 square. Let Fig. 263 represent the square referred to ; in 
 which is the centre of each ; then A O equals A D. To 
 prove this, it need only be shown that the triangle A O D is 
 an isosceles triangle having its sides A O and A D equal. The 
 
 FIG. 263. 
 
 octagon being equilateral, it is also equiangular, therefore 
 the angles BCO,ECO,ADO, etc., are all equal. Of the 
 right-angled triangle FEC,FC and FE being equal, the 
 two angles FECand FCE, are equal (Art. 338), and are 
 therefore (Art. 347) each equal to half a right angle. In like 
 manner it may be shown that FA B and FR A are also each 
 equal to half a right angle. And since FE C and FA B are 
 equal angles, therefore the lines E C and A B are parallel 
 (Art. 331,) and hence the angles E CO and A OD are equal. 
 These being equal, and the angles ECO and A D O being 
 equal by construction, as before shown, therefore the angles 
 A OD and A D are equal, and consequently the lines A O 
 and A D are equal. (Art. 338.) 
 
35$ GEOMETRY. 
 
 355. Proposition. An angle at the circumference of a 
 circle is measured by half the arc that subtends it ; that is, 
 the angle A B C (Fig. 264) is equal to half the angle A D C. 
 Through the centre D draw the diameter BE. The tri- 
 angle A B D is an isosceles triangle, A Z?and B D being ra- 
 dii, and therefore equal ; hence, the two angles at F and G 
 are equal (Art. 338), and the sum of these two angles is equal 
 to the angle at H (Art. 345), and therefore one of them, G t is 
 equal to the half of H. The angles at H and at G (or A BE) 
 are both subtended by the arc A E. Now, since the angle 
 
 FIG. 264. 
 
 at H is measured by the arc A E, which subtends it, there- 
 fore the half of the angle at H would be measured by the 
 half of the arc A E ; and since G is equal to the half of H, 
 therefore G or A BE is measured by the half of the arc A E. 
 It maybe shown in like manner that the angle E B C is 
 measured by half the arc E C, and hence it follows that the 
 angle A B C is measured by half the arc, A C, that sub- 
 tends it. 
 
 356. Proposition. In a circle all the inscribed angles, 
 A, B, and C (Fig. 265), which stand upon the same side of the 
 
EQUAL ANGLES IN CIRCLES. 
 
 359 
 
 chord DE are equal. For each angle is measured by half 
 the arc D F E (Art. 355). Hence the angles are all equal. 
 
 357. Corollary. Equal chords, in the same circle, sub- 
 tend equal angles. 
 
 FIG. 265. 
 
 358. Proposition. The angle formed by a chord and 
 tangent is equal to any inscribed angle in the opposite seg- 
 
 ment of the circle ; that is, the angle D (Fig. 266) equals the 
 angle A. Let H F be the chord, and E 6 the tangent ; draw 
 the diameter y H '; then JH G is a right angle, also J F H \$ 
 
3 6o 
 
 GEOMETRY. 
 
 a right angle. (Art. 352.) The angles A and B together equal 
 a right angle (Art. 346) ; also the angles B and D together 
 equal a right angle (equal to the angle J HG) ; therefore, the 
 sum of A and B equals the sum of B and D. From each of 
 these two equals, taking the like quantity B, the remainders 
 A and D are equal. Thus, it is proved for the angle at A ; 
 it is also true for any other angle ; for, since all other in- 
 scribed angles on that side of the chord line H F equal the 
 angle A (Art. 356), therefore the angle formed by a chord 
 and tangent equals any angle in the opposite segment of the 
 circle. This being proved for the acute angle D, it is also 
 true for the obtuse angle EHF-, for, from any point, K (Fig. 
 267) in the arcHKF, draw lines to 7, F and H ; now, if it can 
 
 be proved that the angle EH F equals the angle FK H, the 
 entire proposition is proved, for the angle FKH equals any 
 of all the inscribed angles that can be drawn on that side of 
 the chord. (Art. 356.) To prove, then, that EH F equals 
 H KF\ the angle EH F equals the sum of the angles A and 
 B ; also the angle H K F equals the sum of the angles C and 
 D. The angles B and D, being inscribed angles on the same 
 chord, J F, are equal. The angles C and A, being right angles 
 (Art. 352), are likewise equal. Now, since A equals C and B 
 equals D, therefore the sum of A and B equals the sum of C 
 and DOY the angle E H F equals the angle H K F. 
 
 359. Propokion. The areas of parallelograms of 
 equal altitude are to each other as the bases of the parallelo- 
 
PARALLELOGRAMS PROPORTIONATE TO BASES. 
 
 361 
 
 grams. In Fig. 268 the areas of the rectangles A B CD and 
 B E D F are to each other as the bases CD and D F. For, 
 putting the two bases in form of a fraction and reducing this 
 fraction to its lowest terms, then the numerator and denomi- 
 nator of the reduced fraction will be the numbers of equal 
 parts into which the two bases respectively may be divided. 
 For example, let the two given bases be 12 and 9 feet respect- 
 ively, then -ijp- = f , and this gives four parts for the larger 
 base and three parts for the smaller one. So, in Fig. 268, 
 divide the base CD into four equal parts, and the base D F 
 into three equal parts ; then the length of any one of the 
 parts in CD will equal the length of any one of the parts in 
 D F. Now, parallel with A C, draw lines from each point of 
 division to the line A E. These lines will evidently divide 
 the whole figure into seven equal parts, four of them occupy- 
 
 FIG. 268. 
 
 ing the area A B C D, and three of them occupying the area 
 B E D F. Now it is evident that the areas of the two rect- 
 angles are in proportion as the number of parts respectively 
 into which the base-lines are divided, or that . 
 
 A B C D \ B E D F \ : C D \ D F. 
 
 The areas in this particular case are as 4 to 3. But in gen- 
 eral the proportion will be as the lengths of the bases. 
 Thus the proposition is proved in regard to rectangles, but 
 it has been shown (Art. 341) that all parallelograms of equal 
 base and altitude are equal. Therefore the proposition is 
 proved in regard to parallelograms generally, including rect- 
 angles. 
 
 360. Proposition. Triangles of equal altitude are to 
 each other as their bases. It has been shown (Art. 359) that 
 
362 GEOMETRY. 
 
 parallelograms of equal altitude are in proportion as their 
 bases, and it has also been shown (Art. 342) that of a triangle 
 and parallelogram, when of equal base and altitude, the 
 parallelogram is equal to double the triangle. Therefore 
 triangles of equal altitude are to each other as their bases. 
 
 361. Proposition. Homologous triangles have their 
 corresponding sides in proportion. Let the line CD (Fig. 
 269) be drawn parallel with. A B. Then the angles E CD 
 and E A B are equal (Art. 331), also the angles E D C and 
 E B A are equal. Therefore the triangles E CD and E A B 
 are homologous, or have their corresponding angles equal. 
 
 FIG. 269. 
 
 For, join C to- B^ and A to A then the triangles A C D and 
 B C D) standing on the same base, C D, and between the 
 same parallels, CD and A B, are equal in area. To each 
 of these equals join the common area C D E, and the sums 
 A DE and B CE will be equal. The triangles CD E and 
 A D E, having the same altitude, are to each other as 
 their bases C E and A E (Art. 360), or 
 
 CDE : ADE : : CE : A E. 
 
 Also the triangles CDE and B C E, having the same alti- 
 tude, are to each other as their bases D E and BE, or 
 
 CDE ; BCE : : DE : BE. 
 
CHORDS GIVING EQUAL RECTANGLES. 
 
 363 
 
 And, since the triangles A D E and B C E are equal, as before 
 shown, therefore, substituting in the last proportion A D E 
 for B CE, we have 
 
 CD E : AD E : : D E : BE. 
 
 The first two factors here being identical with the first two 
 in the first proportion above, we have, comparing the two 
 proportions 
 
 CE : AE : : DE : B E\ 
 
 or, we have the corresponding sides of one triangle, CD E, 
 in proportion to the corresponding sides of the other, A BE. 
 
 FIG. 270. 
 
 362. Proposition. Two chords, E F and CD (Fig. 270), 
 intersecting, the parallelogram or rectangle formed by the 
 two parts of one is equal to the rectangle formed by the 
 two parts of the other. That is, the product of C G multi- 
 plied by G D is equal to the product of E G multiplied by 
 G F. The triangle A is similar to the triangle B, because it 
 has corresponding angles. The angle H equals the angle G 
 (Art. 344); the angle at J equals the angle at K, because 
 they stand upon the same chord, D F (Art. 356) ; for the same 
 
364 GEOMETRY. 
 
 reason the angle M equals the angle L, for each stands upon 
 the same chord, E C. Therefore, the triangle A having the 
 same angles as the triangle B, the length of the sides of one 
 are in like proportion as the length of the sides in the other 
 (Art. 361). So 
 
 CG : EG : : GF : G D. 
 
 Hence, as the product of the means equals the product of 
 the extremes (Art. 373), E G multiplied by GF is equal to 
 C G multiplied by G D. 
 
 363. Proposition. If the sides of a quadrangle are 
 bisected, and lines drawn joining the points of bisection in 
 the adjacent sides, these lines will form a parallelogram. 
 
 FIG. 271. 
 
 Draw the diagonals A B and CD (Fig. 271). It will here be 
 perceived that the two triangles A E O and A C D are homol- 
 ogous, having like angles and proportionate sides. Two of 
 the sides of one triangle lie coincident with the two corres- 
 ponding sides of the other triangle, therefore the contained 
 angles between these sides in each triangle are identical. 
 By construction, these corresponding sides are proportion- 
 ate ; AC being equal to twice A E, and A D being equal to 
 twice A O ; therefore the remaining sides are proportionate, 
 CD being equal to twice E O, hence the remaining corres- 
 ponding angles are equal. Since, then, the angles A E O 
 and A C D are equal, therefore the line E is parallel with 
 
PARALLELOGRAM IN QUADRANGLE. 365 
 
 the diagonal CD so, likewise, the line MNis parallel to the 
 same diagonal, CD. If, therefore, these two lines, EO and 
 MN, are parallel to the same line, CD, they must be parallel 
 to each other. In the same manner the lines ON and EM 
 are proved parallel to the diagonal A B, and to each other ; 
 therefore the inscribed figure ME ON is a parallelogram. 
 It may be remarked, also, that the parallelogram so formed 
 will contain just one half the area of the circumscribing 
 quadrangle. 
 
SECTION VII. RATIO, OR PROPORTION. 
 
 364. 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 multiply- 
 ing the 87 pounds by 45 cents, the price of the 9 pounds, 
 and dividing the product, 3915, by 9 ; the quotient, 435 cents, 
 is the answer to the question. 
 
 365. The "Rule of Three." The process by which 
 this problem is solved is known as the Rule of Three, or 
 Proportion. 
 
 In cases of this kind there are three quantities given,, to 
 find a fourth. Previous to working the question it is usual 
 to make a statement, placing the three given quantities in 
 such order that the quantity which is of like kind with the 
 answer shall occupy the second place; the quantity upon 
 which this depends for its value is put m the first place, and 
 the remaining quantity, which is of like kind with that in the. 
 first place, is assigned to the third place. 
 
 When thus arranged, the second and third quantities are 
 multiplied together and the product is divided by the first 
 quantity ; the quotient, the answer to the question, is a 
 fourth quantity. These four quantities are related to each 
 other in this manner, namely : the first is in proportion to 
 the second as the third is to the fourth ; or, taking the 
 quantities of the given example, and putting them in a for- 
 mal statement with the customary marks between them, we 
 have 
 
 9 : 45 : : 87 : 435, 
 
 which is read: 9 is to 45 as 87 is to 435 ; or, 9 is in propor- 
 tion to 45 as 87 is to 435 ; or, 9 bears the same relation to 45 
 as 87 does to 435. 
 
EQUALITY OF RATIOS. 367 
 
 366. Couple*: Antecedent, Consequent. These four 
 quantities are termed Proportionals, and may be divided into 
 two couples ; the first and second quantities forming one 
 couple, and the third and fourth the other couple. Of each 
 couple the first quantity is termed the antecedent, and the 
 last the consequent. Thus 9 is an antecedent and 45 its con- 
 sequent ; so, also, 87 is an antecedent and 435 its consequent. 
 
 367. Equal Couples : an Equation. These four quan- 
 tities may be put in form thus : 
 
 45. = 435 
 9 - 87 
 
 Each couple is here stated as a fraction : each has its ante- 
 cedent beneath its consequent, and the two couples are 
 separated by a sign, two short parallel lines, signifying 
 equality. This is an equation, and is read thus : 45 divided 
 by 9 is equal to 435 divided by 87 ; or, as ordinary fractions : 
 45 ninths are equal to 435 eighty-sevenths. 
 
 368. Equality of Ratios. Each couple is also termed a 
 Ratio, and the two the Equality of Ratios. Thus the ratio 
 
 is equal to the ratio ^-. If the division indicated in 
 9 87 
 
 these two ratios be actually performed, the equality between 
 the two will at once be apparent, for the quotient in each 
 case is 5. The resolution of each couple into its simplest 
 form by actual division is shown thus : 
 
 f- 
 
 These are read : 45 divided by 9 equals 5 ; and 435 divided 
 by 87 equals 5. 
 
 369. Equali multiplied by Equal Give Equals. If two 
 
 equal quantities be each multiplied by a given quantity, the 
 
368 RATIO, OR PROPORTION. 
 
 two products will be equal. For example, the fractions f 
 and f are each equal to ^, and are therefore equal to each 
 other. If these two equal quantities be each multiplied by 
 any given number, say, for example, by 4, we shall have 4 
 times f equals f , and 4 times f equals - 1 /- ; these products, f 
 and \ 2 - are each equal to 2, and therefore equal to each 
 other. 
 
 370. Multiplying an Equation. The quantity on each 
 side of the sign = is called a 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); therefore, if the two members of the equation 
 
 45 -415. (Art. 367) be each multiplied by 87, or be modified 
 9 7 
 thus: 
 
 45x87^435x87 
 9 , 87 
 
 in which x, the sign for multiplication, indicates that the 
 quantities between which it is placed are to be multiplied 
 together ; this ddition to each member of the equation does 
 not destroy the equality ; the members are still equal, 
 though considerably enlarged. The equality may be easily 
 tested by performing the operations indicated in the equa- 
 tion. For example : for the first member, we have 45 times 
 87 equals 3915, and this divided by 9 equals 435. Again, for 
 the second member we have 435 times 87 equals 37845, and 
 this divided by 87 equals 435, the same result as that for the 
 first member. Thus the multiplication has not interfered 
 with the equality of the members. 
 
 371. Multiplying and Dividing one Member of an 
 Equation: Cancelling. If a quantity be multiplied by a 
 given number, and the product be divided by the same 
 given number, the quotient will equal the original quantity. 
 For example : if 8 be multiplied by 3, the product will be 24 ; 
 then if this product be divided by 3, the quotient will be 8, 
 the original quantity. Thus the value of a quantity is not 
 
TRANSFERRING FACTORS IN EQUATIONS. 369 
 
 changed by multiplying it by a number, provided it be also 
 divided by the same number. 
 
 From this, also, we learn that the value of a quantity 
 which is required to be multiplied and divided by the same 
 number will not be changed if the multiplication and divis- 
 ion be both omitted ; one cancels the other. Therefore the 
 number 87, appearing in the second member of the equation 
 in the last article both as a multiplier and a divisor, may be 
 omitted without destroying the equality of the two mem- 
 bers. The equation thus treated will be reduced to 
 
 45 x8 7 
 
 This expression is read : the product of 45 times 87 divided 
 by 9 equals 435. It will be observed that we have here the 
 four terms of the problem in Art. 365, three of them in the 
 first member, and the fourth, the answer to the problem, in 
 the second member. 
 
 372. Transferring a Factor .Each of the four quan- 
 tities in the aforesaid equation is termed a factor. Compar- 
 ing the equation of the last article with that of Art. 43, it 
 appears that the two are alike excepting that the factor 87 
 has been transferred from one member of the equation to 
 the other, and that, whereas it was before a divisor, it has 
 now become a multiplier. From this we learn that a factor 
 may be transferred from one member of an equation to the 
 other, provided that in the transfer its relative position to 
 the horizontal line above or below it be also changed ; that 
 is, if, before the transfer, it be below the line, it must be put 
 above the line in the other member; or, if above the line, it 
 must be put below, in the other member. For example : in 
 the equation of the last article let the factor 9 be removed 
 to the second member of the equation. It stands as a divi- 
 sor in the first member ; therefore, by the rule, it must appear 
 as a multiplier after the transfer ; or 
 
 45 x 87 = 9 x 435; 
 
3/O RATIO, OR PROPORTION. 
 
 which is read, 45 times 87 equals 9 times 435. By actually 
 performing the operations here indicated, we find that each 
 member gives the same product, 3915; thus proving that 
 the equality of the two members was not interfered with 
 by the transfer. 
 
 373. Equality of Products: Means and Extremes. In 
 
 Art. 366, the four factors are put in the usual form of four 
 proportionals. A comparison of these with the four factors 
 as they appear in the equation in the last article, shows that 
 the first member contains the second and third of the four 
 proportionals, and the second member contains the first and 
 the fourth ; or, the first contains what are termed the 
 means, and the second, the extremes. From this we learn 
 that in any set of four proportionals, the product of the 
 means equals the product of the extremes. As for example, 
 
 - = i^ ; so, also, - = i, an equality of ratios : hence the 
 
 four factors, 2, 3, 4, 6, are four proportionals, and may be 
 put thus : 
 
 Extreme, /nean, mean, extreme. 
 2:31:4:6 
 
 and, as above stated, the product of the means (3x4=) 12, 
 equals the product of the extremes (2x6=) 12. 
 
 374-. Homologous Triangles Proportionate. The 
 
 discussion of the subject of Ratios has thus far been con- 
 fined to its relations with the mercantile problem of Art. 
 364. The rules of proportion or the equality of ratios 
 apply equally to questions other than those of a mercantile 
 character. They apply alike to all questions in which quan- 
 tities of any kind are comparable. For example, in geome- 
 try, lines, surfaces, and solids bear a certain fixed relation to 
 one another, and are, therefore, fit subjects for the rules of 
 proportion. It is shown, in Art. 361, that the correspond- 
 ing sides of homologous triangles are in proportion to one 
 another. Hence, when, of two similar triangles, two corres- 
 ponding sides and one other side are given, then by the 
 equality of ratios the side corresponding to this other side 
 
RATIOS APPLIED TO TRIANGLES. 3/1 
 
 may be computed. For example : in two triangles, such as 
 ECD and EAB (Fig. 269), having their corresponding 
 angles equal, let the side E C, in the triangle ECD, equal 12 
 feet, and the corresponding side E A, in the triangle E A B, 
 equal 16 feet, and the side ED, of triangle ECD, equal 14 
 feet. Now, having these three sides given, how can we find 
 the fourth ?, Putting them in proportion, we have, as in 
 Art. 361 
 
 CE : AE : : DE : BE', 
 
 and, substituting for the known sides, their dimensions, we 
 have 
 
 12 : 16 : : 14 : B E ; 
 
 and, by Art. 373 
 
 12 x BE = 16 x 14. 
 
 Dividing each member by 12, gives 
 
 Performing the multiplication and division indicated, we 
 have 
 
 Thus we have the fourth side.equal to i8| feet. 
 
 375. _ The Steelyard. An example of lour proportion- 
 als may also be found in the relation existing between the 
 arms of a lever and the weights suspended at their ends. A 
 familiar example of a lever is seen in the common steelyard 
 used by merchants in weighing goods. This is a bar, A B, 
 of steel, arranged as in Fig. 272, with hooks and links, and a 
 suspended platform to carry R, the article to be weighed ; 
 and with a weight P, suspended by a link at B, from the bar 
 A B, along which the weight P is movable. 
 
 The entire load is sustained by links attached to the ful- 
 crum, or point of suspension C. The apparatus is in equi- 
 librium without R and P. In weighing any article, R, the 
 
3/2 RATIO, OR PROPORTION. 
 
 weight P is moved along the bar B C until the weight just 
 balances the load, or until the bar A B will remain in a hori- 
 zontal position. If the weight P be too far from the fulcrum 
 C the end of the bar B will fall, but if it be too near it will 
 rise. 
 
 376. The L<ever Exemplified l>y the Steelyard. To 
 
 exemplify the principle of the lever, let the bar A B (Fig. 
 272) be balanced accurately with the scale platform, but 
 without the weights R and P. Then, placing the article R 
 upon the platform, move the weight P along the beam until 
 there is an equilibrium. Suppose the distances A C and B C 
 are found to be 2 and 40 inches respectively, and suppose 
 
 FIG. 272. 
 
 the weight P to equal 5 pounds, what at this point will be 
 the weight of R? By trial we shall find that R = 100 pounds. 
 Again, if a portion of R be removed, then the weight P 
 would have to be moved along the bar B C to produce an 
 equilibrium ; suppose it be moved until its distance from C 
 be found to be 20 inches, then the weight of R would be 
 found to be 50 pounds, or 
 
 R = 50 pounds. 
 
 Again, suppose a part of the weight taken from R be re- 
 stored, and the weight P, on being moved to a point re- 
 quired for equilibrium, be found to measure 30 inches from 
 C, then we shall find that 
 
 R = 75 pounds. 
 
RATIOS OF THE LEVER. 373 
 
 Thus when 
 
 B C = 40, R = loo ; or, -- = 2 5 ; 
 
 40 
 
 BC=2o, R = $o> J or, -=2-5; 
 
 showing an equality of ratios ; or, in general, B C is in pro- 
 portion to R) or 
 
 BC : R. 
 
 If, instead of moving P along B C, its position be permanent, 
 and the weight P be reduced as needed to produce equilib- 
 rium with the various articles, R, which in turn may be 
 put upon the scale ; then we shall find that if when the 
 weight P equals 5 pounds the article R equals 100, and there 
 is an equilibrium, then when 
 
 P- x 5 =4-5, R will equal -^*x 100 =-90; 
 
 8 8 
 
 P = x 5 = 4, R will equal x 100 = 80; 
 
 P= -^ x 5 r= 3 5, R will equal x 100 = 70 ; 
 and so on for other proportions; and in every case we shall 
 
 r> 
 
 have the ratio - equal 20, thus 
 
 R 90 
 
 -75- = '^ = 20 ' 
 
 P 4-5 
 
 R 80 
 P=T = 2 ; 
 
 ^? 70 
 
 7 = 3^ = 2 - 
 
374 RATIO, OR PROPORTION. 
 
 Thus we have an equality of ratios in comparing the 
 weights. 
 
 Again, if the weight P and the article R be permanent in 
 weight, and the distances A C, B C be made to vary, then if 
 there be an equilibrium when A C is 2 and B C is 40, we 
 shall find that when 
 
 o O 
 
 A C = - x 2 = I -6 ; B C will equal x 40 = 32 , 
 A C ' = x 2 = i 2 ; /? C" will equal x 40 = 24 ; 
 
 AC= ~ x2 = ' 8; BC wil1 ec l ual x 40 = 16 ; 
 
 and so on for other proportions, and in every case we shall 
 
 BC 
 
 have the ratio -jyr = 20 ; thus 
 
 B C 32 
 
 _ - ^ - *}f\ . 
 
 - _,-, ~ -- v/ 
 
 ^4 T 1-6 
 
 ^C~ 0-8 
 
 producing thus an equality of ratios in comparing the arms 
 of the lever. From these experiments we have found, in 
 comparing the article weighed with an arm of the lever, the 
 constant ratio B C : R, and when comparing the weights 
 we have found the constant ratio P : R. Again, in com- 
 paring the arms of the lever, we find the constant ratio 
 A C : B C. Putting two of these couples in proportion, we 
 have 
 
 A C : B C : : P : R. 
 
 Hence (Art. 373) we have 
 
PRINCIPLE OF THE LEVER DEMONSTRATED. 375 
 
 Dividing both members by A C, we have 
 
 BC*P 
 ~AC 
 
 In a steelyard the short arm, A C, and the weight, or poise, 
 P, are unvarying ; therefore we have 
 
 or, when ^ is constant, we have 
 
 R : B C. 
 
 377. The L,cvcr Principle Bcmontrated. The rela- 
 tion 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 throughout, and suspended 
 upon an axle or pin at C, its centre. This beam is evidently 
 in a state of equilibrium. Of the part of the beam A D G K, 
 let E be the centre of gravity ; and of the remaining part, 
 D D K H, let F be the centre of gravity. 
 
 If the weight of tne material in A D G K\>t concentrated 
 at E, its centre of gravity, and the weight of the material in 
 
 * The principle upon which this demonstration is based may be found in an 
 article written by the author and published in the Mathematical Monthly, Cam- 
 bridge, U. S., fori8s8, p. 77. 
 
376 RATIO, OR PROPORTION. 
 
 DBKH be concentrated in F, its centre of gravity, the 
 state of equilibrium will not be interfered with. Therefore 
 let the ball R be equal in weight to the part A D G K, and 
 the ball P equal to the weight of the part D BKH\ and let 
 these two balls be connected by the rod E F. Then these 
 two balls and rod, supported at C, will evidently be in a 
 state of equilibrium (the rod EF being supposed to be with- 
 out weight). 
 
 Now, it is proposed to show that R is to P as C F is to 
 C E. This can be proved; for, since R equals the area 
 ADGK and P equals the area DBKH, therefore R is in 
 proportion to A D, as P is to D B (Art. 359) ; or, taking the 
 halves of these lines, R is in proportion to A J as P is to 
 LB. 
 
 Also, J L equals half the length of the beam ; for J D is 
 the half of A D, and D L is the half of DB; thus these two 
 parts (JD + DL) equal the half of the two parts (AD + DB)\ 
 or, y L equals the half of A B\ or, we have 
 
 Adding these two equations together, we have 
 
 Now, JD + DL = JL, and AD + DB = AB\ therefore, 
 
 Thus we have A M = J L. From each of these equals 
 take J M, common to both, then the remainders, A J and 
 ML, will be equal ; therefore, A J = C F. 
 
 We have also MB = J L. From each of these equals 
 take ML, common to both, and the remainders, J M and 
 L B, will be equal ; therefore, L B = E C. As was above 
 shown 
 
 RiAy'iiP-iL.9. 
 
TO FIND A FOURTH PROPORTIONAL. 377 
 
 Substituting for A J and LB their values, as just found, 
 we have 
 
 R : CF : : P : EC', 
 
 from which we have (Art. 373) 
 
 Px CF= R x E C. 
 
 Thus it is demonstrated that the product of one weight into 
 its arm of leverage, is equal to the product of the other 
 weight into its arm of leverage : a proposition which is 
 known <is the law of the lever. 
 
 378. Any One of Four Proportionals may be Found. 
 
 Any three of four proportionals being given, the fourth 
 may be found ; for either one of the four factors may be 
 made to stand alone ; thus, taking the equation of the last 
 article, if we divide both members by CF (Art. 371), we 
 have 
 
 Px CF'_RxEC 
 C.F CF ' 
 
 In the first member C F, in both numerator and denominator, 
 cancel each other (Art. 371), therefore 
 
 so likewise we may obtain 
 
 Px CF 
 
SECTION VIII. FRACTIONS. 
 
 379. A Fraction Defined. As a fracture is a break or 
 division into parts, so a fraction is literally a piece broken off; 
 a part of the whole. 
 
 The figures which are generally used to express a frac- 
 tion show what portion of the whole, or of an integer, the 
 fraction is : for example, let the line A B, (Fig. 274), be divided 
 into five equal parts, then the line A C, containing three of 
 those parts, will be three fifths of the whole line A B, and 
 
 3 
 
 may be expressed by the figures 3 and 5, placed thus, , 
 
 which is known as a fraction and is read, three fifths. The 
 number 5 below the line deno'tes the number of parts into 
 which an integer or unit, A B, is supposed to be divided ; it 
 
 riii 1 1 
 
 A D E C B 
 
 FIG. 274. 
 
 is therefore called the denominator, and expresses th3 denom- 
 ination or kind, whether fifths, sixths, ninths, or any number, 
 into which a unit is supposed to be divided. The number 
 3 above the line, denoting the number of parts contained in 
 the fraction, is termed the numerator, and expresses the 
 number of parts taken, as 2, 3, 4, or any other number. 
 
 380. Graphical Repreentation of Fractions : Effect 
 of Multiplication. In Fig. 275, let the line A B be di- 
 vided into three equal parts ; the line CD into six equal 
 parts; the line EF into nine equal parts; the line G H into 
 twelve equal parts, and the line y A' into fifteen equal parts. 
 The lines AB, CD, EF,GH, and J K, being all of equal 
 length. 
 
FRACTIONS ILLUSTRATED. 
 
 379 
 
 Then the parts of these lines, A L, CM, EN, etc., may 
 
 be expressed respectively by the fractions-,^,-, - and --. 
 
 369 12 15 
 
 In each case the figure below the line, as, 3, 6, 9, 12, or 15, 
 expresses the number of parts into which the whole is di- 
 vided, and the figure above the line, as 1,2, 3, 4, or 5, the 
 
 L 
 
 
 M 
 
 
 
 N 
 
 
 
 D 
 
 
 
 P 
 
 
 FIG. 275. 
 
 number of the parts taken ; and, as the lines A Z, CM, EN, 
 etc., are all equal to each other, therefore these fractions are 
 all equal to each other. If the numerator and denominator 
 of the first fraction be each multiplied by 2, the products 
 will equal the numerator and denominator of the second 
 fraction ; thus 
 
 1X2 = 2 
 3X2 = 6' 
 
 so, also, 
 
 and 
 
 and 
 
 I>< 3 ~3. 
 3x3=9' 
 
 i_x 4= 4^ 
 3 x 4 = 12 
 
 i x 5 =_i_ 
 
 3 x 5 =-15 
 
 Thus it is shown that when the numerator and denomi- 
 nator of a fraction are each multiplied by the same factor, 
 the product forms a new fraction which is of equal value 
 with the original. 
 
 In like manner we have, |, , A --, etc., each equal to 
 
 o 12 It) 2O 
 
 one fourth; and which may be found by multiplying' the 
 
 numerator and denominator of - successively by 2, 3, 4, 5, etc. 
 
 4 
 
380 FRACTIONS. 
 
 381. Form of Fraction Changed by Division. By an 
 
 operation the reverse of that in the last article, we may re- 
 duce several equal fractions to one of equal value. Thus, if 
 in each we divide the numerator and denominator by the 
 same number, we reduce it to a fraction of equal value, but 
 with smaller factors. 
 
 For example, taking the fractions of the last article, f , f, 
 iV xV> l e t eacn De divided by a number which will divide 
 both numerator and denominator without a remainder.* 
 
 Thus, ^"^ 2== I 1~"~3 = l 
 
 6 + 2*3' 9-^3 = 3" 
 
 _4/r-4= l J_^-5 =1 
 12-4=3' 15-^5 = 3* 
 
 As these fractions are shown (Art. 380) to be equal, and 
 as the operation of dividing each factor by a common num- 
 ber produces quotients which in each case form the same 
 fraction, -J-, we therefore conclude that the numerator and 
 denominator of a fraction may be divided by a common 
 number without changing the value of the fraction. 
 
 382. Improper Fractions. The fractions f , ^, ~, etc., 
 all fractions which have the numerator larger than the de- 
 nominator are termed improper fractions. They are not im- 
 proper arithmetically, but they are so named because it is an 
 improper use of language to call that &part which is greater 
 than the whole. 
 
 As expressions of this kind, however, are sabject to the 
 same rules as those which are fractions proper, it is custom- 
 ary to include them all under the technical term of fractions. 
 Expressions like these all expressions in which one number 
 is separated by a 'horizontal line from another number below 
 it, or one set of numbers is thus separated from another set 
 below it may be called fractions, and are always to be un- 
 derstood as indicating division, or that the quantity above 
 the line is to be divided by the quantity below the line. 
 
 Division is indicated by this sign -r-, which is read "divided by." 
 
IMPROPER FRACTIONS. 381 
 
 Q 17 2A 3x8x4 17x82 
 
 Thus, z > > * ^ - > etc., are all fractions, tech- 
 
 nically, although each may be greater than unity. And it is 
 understood in each case that the operation of division is re- 
 quired. Thus, - = 3, -- 8, = 4. When the divis- 
 
 33 % , 
 
 ion cannot be made without a remainder, then the fraction, 
 by cutting the numerator into two, may be separated into two 
 parts, one of which may be exactly divided, and the other 
 
 will be a fraction proper. Thus, the fraction -~ is equal to 
 
 1 (for 15 + 2 17); and since equals 3, therefore, 
 
 17 15 2 22 
 
 = + - = 3 + - = 3- So, likewise, the fraction 
 
 17x82 __ 1394 := i375 + J9_. .J_9_. J_9_ 
 
 125 125 125 125 125 125' 
 
 383. Reduction of Mixed Numbers to Fractions By 
 
 an operation the reverse ot that in the last article, a given 
 mixed number (a whole number and fraction) can be put 
 into the form of an improper fraction. 
 
 This is done by multiplying the whole number by the de- 
 nominator of the fraction, the product being the numerator 
 of a fraction equal in value to the whole number ; the de- 
 nominator of this fraction being the same as that of the given 
 fraction. The numerator of this fraction being added to the 
 numerator of the given fraction, the sum will be the numera- 
 tor of the required improper fraction, the denominator of 
 which is the same as that of the given fraction. For example, 
 the required numerator for 
 
 2 J, is 2 x 3 + I 7. So 2-3- = -J. 
 
 2j, is 2 X 4 + I = 9. So 2\ = f. 
 
 3i is 3 x 5 +2 = 17. So3f = $. 
 
 384. Division Indicated by the Factors put as a Frac- 
 tion. Factors placed in the form of a fraction as , -, - or 
 
382 FRACTIONS. 
 
 - indicate division (Art. 382) ; the denominator (the fac- 
 tor below the line) being the divisor, and the numerator 
 (the factor above the line) the dividend, while the value of 
 
 the fraction is the quotient. Thus of the fraction, = 20, 
 
 9 41 
 
 41 is the divisor, 820 the dividend, and 20 the quotient. 
 From this we learn that division may always be indicated 
 by placing the factors in the form of a fraction, so that the 
 divisor shall form the denominator and the dividend the nu- 
 merator. 
 
 385. Addition of Fractions having Like Denomina- 
 tors Let it be required to add the fractions - and -. By 
 
 referring to Art. 379 we see that ^4 D (Fig. 274), is one of the 
 five parts into which the whole line A B is divided ; it is, 
 
 therefore, . We also see that D C contains two of the five 
 
 2 
 
 parts ; it is, therefore, -. We also see that AD + D C ' A C, 
 which contains three of the five parts, or A C = of A B. 
 
 12 3 
 
 We therefore conclude that + = . In this operation it 
 
 is seen that the denominator is not changed, and that the 
 resultant fraction has for a numerator a number equal to the 
 sum of the numerators of the fractions which were required 
 to be added. 
 
 By this it is shown that to add fractions we simply take 
 the sum of the numerators for the new numerator, making the 
 denominator of the resultant fraction the same as that of the 
 fractions to be added. For example : What is the sum of the 
 
 fractions , and - ? Here we have 14-3+4 8 for the 
 numerator, therefore 
 
 999 
 
SUBTRACTION AND ADDITION OF FRACTIONS. 383 
 
 386. Subtraction of Fractions O f Like Denominators. 
 
 Subtraction is the reverse of addition ; therefore, to sub- 
 tract fractions a reverse operation is required to that had in 
 the process of addition ; or simply to subtract instead of 
 adding. 
 
 2 ^ 
 
 For example, if - be required to be su>tracted from 
 we have 
 
 UNIVERSITY 
 
 5 5 " 
 
 By reference to Fig. 274 an exemplification of^tkis-wiit' 
 
 ? 2 T 
 
 seen where we have A C = , A E = , and E C = , and 
 we have 
 
 3 _2 = 
 
 5 5 5' 
 
 We therefore have this rule for the substraction of frac- 
 tions : Subtract the less from the greater numerator ; the remain- 
 der will be the numerator of the required fraction. The denom- 
 inator to be the same as that of the given fractions. 
 
 387. Dissimilar Denominators Equalized. The rules 
 just given for the addition and subtraction of fractions re- 
 quire that the given fractions have like denominators. 
 When the denominators are unlike it is required, before add- 
 ing or substracting, that the fractions be modified so as to 
 make the denominators equal. For example : Let it be re- 
 quired to find the sum of - and -. By reference to Fig. 
 
 2 6 
 
 275, we find that on line A B is equal to - on line E F. 
 
 These being equal, we may therefore substitute for -. 
 Then we have 
 
 6 2_ _ 8 
 9 + 9 " 9 
 
384 FRACTIONS. 
 
 Now, it will be seen that the fraction - may be had by mul- 
 tiplying both numerator and denominator of the given frac- 
 
 2 , 2X^ = 6 
 
 tion- by 3, for 3 x - = -; 
 
 and we have seen (Art. 380) that this operation does not 
 change the value of the fraction. From this we learn that 
 
 the denominators may be made equal by multiplying the smaller 
 denominator and Its numerator by any number which will effect 
 such a result. 
 
 For example : ^-+- = + = -^- ; 
 27 14 7 21 
 
 3 ,3 7 12 4 7 23 7 
 
 4 + I7 + T6 =: T6 + 7^ + 7^ = ^ z '76' 
 
 In this example the second fraction is changed by multiply- 
 ing by i j. 
 
 388. Reduction of Fraction to tlieir 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 each of the given denominators. 
 Before seeking this number, let each given fraction be 
 reduced to its lowest terms, by dividing each factor by a 
 
 common number. For example: may, by dividing by 5, 
 
 be reduced to , which is its equivalent. So, also, , by di- 
 3 2o 
 
 viding by 7, is reduced to , its lowest terms. 
 
 389. Leat Common Denominator. To find the least 
 common denominator ^V&w the several fractions in the order 
 of their denominators, increasing toward the right. If the 
 largest denominator be not divisible by each of the others, 
 double it ; if the division cannot now be performed, treble 
 
LEAST COMMON DENOMINATOR. 385 
 
 it, and so proceed until it is multiplied by some number 
 which will make it divisible by each of the other denomina- 
 tors. This number multiplied by the largest denominator will be 
 the least common denominator. To raise the denominator of 
 each fraction to this, divide the common denominator by the de- 
 nominator of one of the fractions, the quotient will be the 
 number by which that fraction is to be multiplied, both 
 numerator and denominator, and so proceed with each frac- 
 tion. For example : What is the sum of the fractions 
 
 -, -, , -g? One of these, , may be reduced, by divid- 
 ing by 2, to ^. Therefore, the series is -, -, -| ~. On trial 
 o 2 A. \) o 
 
 we find that 8, the largest denominator, is divisible by the 
 first and by the second, but not by the third, therefore the 
 largest denominator is to be doubled: 2x8= 16. This is 
 not yet divisible by the third ; therefore 3 x 8 = 24. This 
 now is divisible by the third as well as by the first and the 
 second ; 24 is therefore the least common denominator. 
 
 Now dividing 24 by 2, the first denominator, the quotient 
 12 is the factor by which the terms of the first fraction are 
 
 to be raised, or, - ~ -. For the second we have 
 
 24-5-4 6, and - ^ . For the third we have 24 -*- 6 = 
 4x0 = 24 
 
 4, and ~ X 7 J an< ^ f r tne fourth, 24-^-8 = 3, and 
 o x 4 24 
 
 7 X 3 -21 
 
 ^ ~_ . Thus the fractions in their reduced form are : 
 12 18 20 21 7i 23 
 
 I I I . ... n ^ 
 
 24 24 24 24 ~~ 24 ~~ 24* 
 
 390. Leat Common Denominator Again. When the 
 denominators are not divisible by one another, then to ob- 
 tain a common denominator, it is requisite to multiply to- 
 gether all of the denominators which will not divide any of the 
 other denominators. For example : What is the sum of the 
 
 fractions -, -, -, and -? 
 
3 86 
 
 FRACTIONS. 
 
 In this case the first denominator will divide the last, but 
 the others are prime to each other. Therefore, for the 
 common denominator, multiply, together all but the first ; 
 or 
 
 5x7x9 = 315 the common denominator ; 
 and 
 
 315 _:_ 3 105, common factor for the first fraction ; 
 315 -=- 5 = 63, common factor for the second fraction ; 
 315 _i_ 7 =. 45, common factor for the third; . 
 315-5-9 = 35, common factor for the fourth. 
 
 And, then 
 
 
 
 i x 105 = 105 t 2 x 63 = 126 3 x 45 = 135 ^ 4 x 35 = 140 
 
 1 x 105 = 315 ' "5 x 63 = 3"i~5 ' 7 x 45 - 3^5 ' 9 x 35 = " 
 
 105 126 135 140 _ 506 
 + 315 " h 3i5 * 3i5 ~~ 3^5 
 
 191 
 
 39(. Fraction multiplied Graphically. Let A B CD 
 
 (Fig. 276) be a rectangle of equal sides, or A B equal A C 
 and each equal one foot. Then A B multiplied by A C will 
 
 G 
 
 
 
 C H ( 
 FIG. 276. 
 
 equal the area A B CD, or i x i = i square foot. Let the 
 line E F be parallel with A B, and midway between -A B and 
 CD. Then A B x A equals half the area of A B CD, or 
 i x J = -J. Again ; let G H be parallel with E C, and mid- 
 way between E C and FD. Then E G x E C = i x equals 
 the area E G C H, which is equal to a quarter of the area 
 
MULTIPLICATION OF FRACTIONS. 
 
 387 
 
 A B C D; or % x % = J; which is a quarter of the superficial 
 area. 
 
 The product here obtained is less than either of the 
 factors producing it. It must be remembered, however, 
 that while the factors represent lines, the product represents 
 superficial area. The correctness of the result may be 
 recognized by an inspection of the diagram. 
 
 392. Fraction multiplied Graphically. In Fig. 277 
 let A B equal 8 feet and A C equal 5 feet ; then the rect- 
 
 G 
 
 FIG. 277. 
 
 angle A B CD contains 5 x 8 = 40 feet. The interior lines 
 divide the space included within A B CD into 40 equal 
 
 squares of one foot each. Let A E equal 3 feet or - of A C. 
 Let A G equal 7 feet or ~ of A B. Then the rectangle 
 
 3 7 21 
 
 E F A G contains x ' , or twenty-one fortieths of the 
 5 04 
 
 <+ *j 
 
 whole area A B CD. Thus, while the factor fractions -- and -^ 
 
 5 o 
 
 represent lines, it is shown that the product fraction rep- 
 
 40 
 
 21 . 
 
 resents surface. Thus is a fraction, E FA G, of the whole 
 
 40 
 
 surface, CDAB. 
 
 393. Rule for Miitiplication of Fraction*, and Exam- 
 ple. In the example given in the last article it will be ob- 
 
388 FRACTIONS. 
 
 served that the product of the denominators of the two 
 given fractions equals the area of the whole figure (A B C D\ 
 while the product of the numerators equals the area of the 
 rectangle (E 'FA G), the sides of which are equal respec- 
 tively to the given fractions. From this we obtain for the 
 product of fractions this 
 
 RULE. Multiply together the denominators for the new de- 
 nominator, and the numerators for a new numerator. 
 
 j ? j 
 
 For example: what is the product of and ? Here 
 
 we have 20x21420 for the new denominator, and 
 7 x 13 = 91 for the new numerator; therefore the product 
 of 
 
 il x _7 = _L. 
 
 21 2O 420 ' 
 
 or, of a rectangular area divided one way into 20 parts and 
 the other way into 21 parts, thus containing 420 rectangles, 
 
 1 3 7 
 
 the product of the two fractions and - is equal to 91 of 
 
 these rectangles, or - of the whole. 
 
 394. Fraction Divided Graphically. Division is the 
 reverse of multiplication ; or, while multiplication requires 
 the product of two given factors, division requires one of 
 the factors when the other and the product are given. Or 
 (referring to Fig. 277) in division we have the area of the 
 rectangle, E FA G, and one side, E A; given, to find the 
 other side, A G. 
 
 Now it is required to find the number of times E A is 
 contained in E FA G. By inspection of the figure we per- 
 ceive the answer to be, A G times ; for E A xAG EFA G, 
 
 2 I 
 
 the given area. Or, when E A F G is given as and E A 
 as -, we have as the given problem 
 
 ILL.! 
 
 40 ' 5* 
 
DIVISION OF FRACTIONS. 389 
 
 Since division is the reverse of multiplication, instead of 
 multiplying we divide the factors, and have 
 
 21 -i- 3 = 7 
 40 -r- 5 " 8* 
 
 Thus, to divide one fraction by another, for the numerator of 
 the required factor, divide the numerator of t/ie product by the 
 numerator of the given factor, and for the denominator of the 
 required factor divide the denominator of the product by the 
 denominator of the given factor. For example : 
 
 10 . 2 5 
 
 Divide ^~ by -. Answer, . 
 
 o Q >| / 
 
 Divide - - by . Answer, . 
 
 395. Rule for Division of Fractions. The rule just 
 given does not work well when the factors are not commen- 
 
 5 2 
 surable. For example, if it be required to divide by we 
 
 have by the above rule 
 
 7-9 " 7 ' 
 9. 
 
 Producing fractional numerators and denominators for the 
 resulting fraction, which require modification in order to 
 reach those composed only of whole numbers. If the nu- 
 merators, 5 and 7, of this compound fraction be multiplied 
 by 9 (the denominator of the denominator fraction), or the 
 compound fraction by 9, we shall have 
 
 5 ><9 
 
390 FRACTIONS. 
 
 And, if these be again multiplied by 2 (the denominator of 
 the numerator fraction), we shall have 
 
 5 X 9 
 
 2 
 
 X 2 = 
 
 7x9 7x9x2 
 
 ~9~ ~9~ 
 
 Like figures above and below in each fraction cancel each 
 other (Art. 371), therefore, the result reduces to 
 
 5 x 9 
 
 7x2' 
 
 in which we find the factors of the two original fractions. 
 In one fraction we have the factors in position as given, 
 
 but in the other they are inverted. The fraction in which. 
 
 the factors are inverted is the divisor. Hence, for division 
 of fractions, we have this 
 
 RULE. Invert the factors of the divisor, and then, as in 
 multiplication, multiply the numerators together for the numera- 
 tor of the required fraction, and the denominators for the de- 
 nominator of the required fraction. 
 
 c 2 
 
 Thus, as before, if - is required to be divided by -, we 
 
 have 
 
 i x 9 45 
 7x2 14' 
 
 And, to divide by , we have 
 
 23 x 9 _ 207 
 47 x 7 329 
 
 2 8 
 
 Again, to divide by , we have 
 
 2$ x 9 _ 225 = _25 = _s 
 45 x 8 "" 360 "40 "8" 
 
CANCELLING IN ALGEBRA. 39! 
 
 This last example has two factors, 9 and 45, one of which 
 
 measures the other ; also, the first fraction - - is not in its 
 
 45 
 
 lowest terms; when reduced it is . The question, there- 
 i fore, may be stated thus : 
 
 5 x 9 1 . 
 9 x 8 "" 8 ' 
 
 for the two 9*8 cancel each other. 
 
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 together, 
 or to divide them. To remove this difficulty, it may be suf- 
 ficient for them to learn that their perplexity arises from a 
 misunderstanding in supposing the letters themselves are 
 ever multiplied or divided. It is true that in treatises on 
 the subject it is usual to speak as though these operations 
 were actually performed upon the letters. It is always un- 
 derstood, however, that it is not the letters, but the quan- 
 tities represented by the letters, which are to be multiplied 
 or divided. 
 
 For example, in Art. 361 it is shown, in comparing similar 
 sides of homologous triangles, that the bases of the two tri- 
 angles are to each other as the corresponding sides, or, 
 referring to Fig. 269, we have C E : A E : : D E : B E. 
 Now, let the two bases C E and A E be represented respec- 
 tively by a and b, and the two corresponding sides D E and 
 B E by c and d respectively ; or, for 
 
 CE : AE : : DE : BE, 
 put 
 
 a : b : : c : d\ 
 
 and, by Art. 373, we have 
 
 b x c = a x d, 
 which may be written 
 
 be = ad\ 
 
 for x, the sign for multiplication, is not needed between let- 
 ters, as it is between numeral factors. The operation of 
 
APPLICATION OF ALGEBRA. 393 
 
 multiplication is always understood when letters are placed 
 side by side. 
 
 Now, here we have an equation in which, as usually read, 
 we have the product of b and c equal to the product of a 
 and d. But the meaning is that the product of the quantities 
 represented by b and c is equal to the product of the quan- 
 tities represented by a and d, and that this equation is in- 
 tended to represent the relation subsisting between the four 
 proportionals, C , A E, D E, and BE, of Fig. 269. In order 
 to secure greater conciseness and clearness, the four small 
 letters are substituted for the four pair of capital letters, 
 which are used to indicate the lines of the figures referred to. 
 
 397. Example : Application. It was shown in the last 
 article that the four letters a, , c, and d represent the cor- 
 responding sides of the two triangles of Fig..2g, and that 
 
 b c = a d. 
 
 Now, let each member of this equation be divided by a, then 
 (Art. 371)- 
 
 If now the dimensions of the three sides represented by a, 
 b, and c are known, and it is required to ascertain from these 
 the length of the side represented by d, let the three given 
 dimensions be severally substituted for the letters repre- 
 senting them. For example, let a = 40 feet ; b = 52 feet, 
 and c = 45 feet ; then 
 
 be 52 x 45 
 
 d = = - = -~ = 58 
 a 40 40 
 
 The quantities being here substituted for the letters ; we have 
 but to perform the arithmetical processes indicated to obtain 
 the arithmetical value of d. From this example it is seen 
 that before any practical use can be made of an algebraical 
 formula in computing dimensions, it is requisite to substitute 
 numerals for the letters and actually perform arithmetically 
 such operations as are only indicated by the letters. 
 
394 ALGEBRA. 
 
 398. Algebra Useful in Constructing Rules. In all 
 
 problems to be solved there are certain conditions or quan- 
 tities given, by means of which an unknown quantity is to 
 be evolved. For example, in the problem in Art. 397, there 
 were three certain lines given to find a fourth, based upon 
 the condition that the four lines were four proportionals. 
 Now, it has been found that the relation between quantities 
 and the conditions of a question can better be stated by let- 
 ters than by numerals ; and it is the office of algebra to 
 present by letters a concise statement of a question, and by 
 certain processes of comparison, substitution and elimina- 
 tion, to condense the statement to its smallest compass, and 
 at last to present it in a formula or rule, which exhibits the 
 known quantities on one side as equal to the unknown on 
 the other side. Here algebra ends, at the completion of the 
 rule. To use the rule is the office of arithmetic. For, in 
 using the rule, each quantity in numerals must be substi- 
 tuted for the letter representing it, and the arithmetical 
 processes indicated performed, as was done in Art. 397. 
 
 399. Algebraic Rule are General. One advantage 
 derived from algebra is that the rules made are general 
 in their application, For example, the rule of Art. 397, 
 
 = d, is applicable to all cases of homologous triangles, 
 
 however they may differ in size or shape from those given in 
 Fig. 269 and not only this, but it is also applicable in all 
 cases where four quantities are in proportion so as to con- 
 stitute four proportionals. For example, the case of the 
 four proportionals constituting the arms of a lever and the 
 weights attached (Arts. 375-378). For, taking the rela- 
 tion as expressed in Art. 377 
 
 PxCF= RxEC, 
 
 we may substitute for C F the letter n, and for E C the letter 
 m, then m will represent the arm of the lever E C (Fig. 262), 
 aid n *he arm of the lever F C. Then we have 
 
SYMBOLS CHOSEN AT PLEASURE. 395 
 
 and from this, dividing by n (/Irt. 372), we have 
 
 or, dividing by m, we have 
 
 (in.) - 
 
 v ' 
 
 m 
 
 which is a rule for computing the weight of R, when P and 
 the two arms of leverage, m and n, are known. For example, 
 let the weight represented by P be 1200 pounds, the length 
 of the arm m be 4 feet, and that of n be 8 feet, then we have 
 
 Pn 1 200 x 8 
 
 R = - - = - = 2400 pounds. 
 m 4 
 
 Pn 
 This rule, R =. -, is precisely like that in Art. 397 
 
 d in which three quantities are given to find a fourth, 
 the four constituting a set of four proportionals. 
 
 400. Symbols hoen 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 par- 
 ticular quantity, throughout all the modifications which may 
 be requisite in condensing the statement into which it enters 
 into a formula for use. For example, the two rules named in 
 Art. 399 are precisely alike three quantities given to find a 
 fourth yet they are represented by different letters. In one, 
 R and P represent the two weights, and m and ft the arms of 
 leverage at which they act ; while in the other the letters 
 a, b,c, and ^represent severally the four lines which constitute 
 two similar sides of two homologous triangles. The two 
 rules are alike in working, and they might have been con- 
 stituted with the same letters. And instead of the letters 
 chosen any others might have been taken, which con- 
 venience or mere caprice might have dictated. In some 
 
ALGEBRA. 
 
 questions it is usual to put the first letters, as a, b, c, etc., to 
 represent known quantities, and the last letters, as x, y, z, 
 for the quantities sought. In works on the strength of 
 materials it is customary to represent weights by capital 
 letters, as P, R, U, W, etc., and lines or linear dimensions by 
 the small letters, as b, d, /, for the breadth, depth, and length, 
 respectively, of a beam. Any other letters may be put to 
 represent these quantities, although the initial letter of the 
 word serves to assist the memory in recognizing the partic- 
 ular dimensions intended. 
 
 40 1. Arithmetical Processe Indicated by Sign. In 
 
 algebra, the four processes of addition, subtraction, multi- 
 plication, and division, are frequently required ; and when 
 the required process cannot be actually performed upon the 
 letters themselves, a certain method has been adopted by 
 which the process is indicated. For example, in additon, 
 when it is required to add a to b, the two letters cannot be 
 intermingled as numerals may be, and their sum presented ; 
 but the process of addition is simply indicated by placing 
 between the two letters this sign, +, which is called plus, 
 meaning added to ; therefore, to add a to b we have 
 
 which is read a plus b, or the sum of a and b. When the 
 quantities represented b}^ a and b are substituted for them 
 and not till tHen they can be condensed into one sum. 
 For example, let a equal 4 and b equal 3, then for 
 
 a-\-b 
 we have 
 
 4+3; 
 
 and we may at once write their sum 7, instead of 4 + 3. 
 
 So, likewise, in the process of subtraction, one letter can- 
 not be taken from another letter so as to show how much of 
 this other letter there will be left as a remainder ; but the 
 process of subtraction can be indicated by a sign, as this, , 
 which is called minus, less, meaning subtracted from. For 
 
ALGEBRAICAL SIGNS. 397 
 
 example, let it be required to subtract b. from a. To do 
 this we have 
 
 which is read a minus b, and when the values of a and b are 
 substituted for them, we have, when a equals 4, and b 
 equals 3 
 
 a-b, 
 or 
 
 4-3; 
 
 and now, instead of 4 3, we may put the value of the 
 which is unity, or i. 
 
 The algebraic signs most frequently used are as follows : 
 
 + ,//#.$, signifies addition, and that the two quantities be- 
 tween which it stands are to be added together; as 
 a + b, read a added to b. 
 
 , minus, signifies subtraction, or that of the two quantities 
 between which it occurs> the latter is to be subtracted 
 from the former ; as a b, read a minus b. 
 
 X, multiplied by, or the sign of multiplication. It denotes 
 that the two quantities between which it occurs are to 
 be multiplied together ; as a x b, read a multiplied by b, 
 ' or a times b. This sign is usually omitted between 
 symbols or letters, and is then understood, as a b. This 
 has the same meaning as a x b. It is never omitted 
 between arithmetical numbers; as 9x5, read nine 
 times five. 
 
 -^, divided by, or the sign of division, and denotes that of the 
 two quantities between which it occurs, the former is 
 to be divided by the latter; as a~b, read a divided by 
 b. Division is also represented thus : 
 
 -, in the form of a fraction. This signifies that a is to be 
 divided by b. When more than one symbol occurs 
 
 above or below the line, or both, as -- , it denotes 
 
 that the product of the symbols above the line is to be 
 divided by the product of those below the line. 
 
39^ ALGEBRA. 
 
 = , is equal to, or sign of equality, and denotes that the 
 quantity or quantities on its left are equal to those on 
 its right ; as a b = c, read a minus b is equal to c y or 
 equals c ; or, 9 5 = 4, read nine minus five equals 
 four. This sign, together with the symbols on each 
 side of it, when spoken of as a whole, is called an 
 equation. 
 
 a 1 denotes a squared, or a multiplied by a, or the second 
 power of a, and 
 
 a* denotes a cubed, or a multiplied by a and again multi- 
 plied by a, or the third power of a. The small figure, 
 2, 3, or 4, etc., is termed the index or exponent of the 
 power. It indicates how many times the symbol is to 
 be taken. Thus, d 1 = a a, a 3 = a a a, a" a a a a. 
 
 \/ is the radical sign, and denotes that the square root of the 
 quantity following it is to be extracted, and 
 
 I/ denotes that the cube root of the quantity following it is 
 to be extracted. Thus, 4/9 = 3, and V 2 7 3- The 
 extraction of roots is also denoted by a fractional in- 
 dex or exponent, thus 
 
 a 1 /* denotes the square root "of a, 
 
 a* denotes the cube root of a, 
 
 a* denotes the cube root of the square of a, etc. 
 
 402. Example in Addition and Subtraction : Cancel- 
 ling. Let there be some question which requires a state- 
 ment to represent it, like this 
 
 a-i-d = c b, 
 
 which indicates that if the quantity represented by a be 
 added to the quantity represented by d, the sum will be 
 equal to the quantity represented by c, after there has been 
 subtracted from it the quantity represented by b ; or, as it is 
 usually read, a plus d equals c minus b ; or the sum of a and 
 d equals the difference between c and b. For illustration, 
 take in place of these four letters, in the order they stand, 
 the numerals 4, 2, 9, 3, and we shall havq by substitution - 
 
 a + d c b, 
 
 4+29 3, or adding 
 
 and subtracting 6 = 6. 
 
TRANSFERRING SYMBOLS. 399 
 
 If it be required to add to each member of the equation 
 the quantity represented by b, this will not interfere with 
 the equality of the members. For a + d are equal to c d, 
 and if to each of these two equals a com-mon quantity be 
 added, the sums must be equal ; therefore 
 
 a + d+ b = c b-^b, 
 or by numerals 
 
 4 + 2 + 3 = 9-3 + 3, 
 or 
 
 9 = 9- 
 
 It will be observed that the right hand member contains 
 the quantity b and + b. This shows that the quantity b is 
 to be subtracted and then added. Now, if 3 be subtracted 
 from 9, the remainder will be 6, and then if 3 be added, the 
 sum will be 9, the original quantity. Thus it is seen that 
 when in the same member of an equation a symbol appears as a 
 minus quantity and also as a plus quantity, the two cancel each 
 other, and may be omitted. Therefore, the expression 
 
 b = c b + b 
 becomes 
 
 a + d+ b = c. 
 
 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 equa- 
 tion b appears in the right hand member and with a minus 
 or negative sign, in the last equation it appears in the left 
 hand member and with a plus or positive sign. Thus it is 
 seen that in the operation performed b has been made to 
 pass from one member to the other, but in its passage it has 
 been changed. A similar change may be made with another 
 of the symbols. For example, from the last equation, let d 
 be subtracted, or this process indicated, thus 
 
 a + d+b d = c d. 
 
400 ALGEBRA. 
 
 The plus and minus d, in the left hand member cancel each 
 other, therefore 
 
 a + b = c d, 
 or, by numerals 
 
 4+3=9 2 - 
 Reducing 
 
 7 = 7. 
 
 By this we learn that any quantity (connected by + or ) 
 may be passed from one member of the equation to the other, pro- 
 vided the sign be changed. 
 
 404. Signs of Symbols to be Changed when they are 
 to e 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 af+c), 
 
 an expression showing that the quantities enclosed within the 
 second pair of parentheses are to be .subtracted from those 
 included within the first pair. Let the quantities represent- 
 ed in the first pair of parentheses for convenience be repre- 
 sented by A , or, a + b c f A . Now, by the terms of the 
 problem, we are required to subtract from A. the quantities 
 enclosed within the second pair of parentheses. To do this 
 take first the positive quantity, b, and subtract it or indicate 
 the subtraction, thus 
 
 A-b; 
 
 we will then subtract the positive quantity c, or indicate the 
 subtraction, thus 
 
 A-b-c. 
 
 We have yet to subtract a and /, two negative quanti- 
 ties. 
 
 The method by which this can be accomplished may be 
 discovered by considering the requirements of the problem. 
 The plus quantities b and , before being subtracted from A, 
 were required to have the two negative quantities # and /de- 
 
THE OPERATION TESTED. 40 1 
 
 ducted from them. It is evident, therefore, that in subtract- 
 ing b and c, before this deduction was made, too much has 
 been taken from A, and that the excess taken is equal to the 
 sum of a and /. To correct the error, therefore, it is neces- 
 sary to add just the amount of the excess, or to add the sum 
 of a and/, or annex them by the plus sign, thus 
 
 A b c + a+f. 
 
 To test the correctness of the operation as here performed, 
 let numerals be substituted for the symbols ; let a = 2, b = 3, 
 c =r i,/= ; then the given quantities to be subtracted, 
 
 become 
 
 (+3 -2-i+i), 
 which reduces to 
 
 (4 - 24) = ij. 
 
 Thus the quantity to be substracted equals ij. Applying 
 the numerals to the above expression 
 
 A b+ a +f c 
 becomes 
 
 A 3 + 2 + -J i =A 4+2% = A i-J. 
 
 A correct result ; it is the same as before. Restoring now 
 the symbols represented by A> we have for the whole ex- 
 pression 
 
 which, by cancelling (Art. 403) and by adding like symbols 
 with like signs, reduces to 
 
 2 a 2 c. 
 
 To test this result, let the quantity which was represented 
 by A have the proper numerals substituted, thus : 
 
 + + b c /, 
 
 + 2 + 3 ~ i -4=5- i* = 3i- 
 
4O2 ALGEBRA. . 
 
 The sum of the given quantity required to be subtracted 
 was before found to amount to i^, therefore 
 
 A -1$ 
 becomes 
 
 And the result by the symbols as above was 
 
 2 a 2 c, 
 which becomes 
 
 2X2 2X1, 
 
 or 
 
 4 2 2; 
 
 a result the same as before, proving the work correct. An 
 examination of the signs in the above expression, which de- 
 notes the problem performed, will show that the sign of each 
 symbol which was required to be subtracted has been 
 changed in the operation of subtraction. Before subtract- 
 ing they were 
 
 after subtraction they are 
 
 (-b + a+f-c). 
 
 By this result we learn, that to subtract a quantity we have 
 but to change its sign and annex it to the quantity from 
 which it was required to be subtracted. 
 
 Example : Subtract a b from c + d. Answer, c + d a + b. 
 
 If numerals be substituted, say a = 7, b = 4, c = 5, and 
 d=g, then 
 
 c+d becomes 5+9=14, 
 a-b " ;_ 4= 3, 
 
 c + d (a b) 14 3 ii, 
 So, also, 
 
 c + d a + b 
 becomes 
 
 + + = ii. 
 
FRACTIONS ADDED AND SUBTRACTED. 403 
 
 405. Algebraic Fraction*: Added and Subtracted. 
 
 When algebraic fractions of like denominators are to be 
 added or subtracted, the same rules (Arts. 385 and 386) are 
 to be observed as in the addition or subtraction of numeri- 
 cal fractions namely, add or subtract the numerators for a 
 new numerator, and place beneath the sum or difference the 
 common denominator. 
 
 For example, what is the sum of T> T T? 
 
 bob 
 
 For this we have 
 
 Subtract -> from -;. For this we have 
 a a 
 
 b-c 
 
 d ' 
 
 What is the algebraical sum of 
 
 ben r^ 
 
 - - - - and - 5 ? 
 
 For these we have 
 
 b + c n r 
 
 To exemplify this, let b represent 9, c = 8, n = 2, r = 3, 
 and d 12. 
 
 Then, for the algebraic sum, we have 
 
 + 8 2 3 12 
 
 = = i. 
 
 12 12 
 
 Now, taking the positive and negative fractions sep- 
 arately, we have 
 
 .. *- = !?. 
 
 12 " h 12 12 ' 
 
 and 
 
 n2 nJ^-I 
 
 12 12 12 " 
 
404 ALGEBRA 
 
 Together 
 
 12. -5 _ _^ = 
 
 12 12 ~~ 12 ~~ ' 
 
 as before. 
 
 406. The Least Common Denominator. When the 
 denominators of algebraic fractions differ it is necessary be- 
 fore addition or subtraction can be performed to harmonize 
 them, as in the reduction of the denominators of numerical 
 fractions (Arts. 388-390). For example, add together the 
 
 /7i @ y 
 
 fractions 7, -7, . In these denominators we perceive 
 oc o ac 
 
 that they collectively contain the letters a, b and c, and no 
 others. It will be requisite, therefore, that each of the frac- 
 tions be modified so that its denominator shall have these 
 three factors. To effect this it will be seen that it is neces- 
 sary to multiply each fraction by that one of these letters 
 which is lacking in its denominator. Thus, in the first, a is 
 
 lacking, therefore (Art. 380) T 7. In the second a 
 
 and c are lacking, therefore T r-, and in the third 
 
 by,ac-=-abc 
 
 r x b rb . 
 
 b is lacking, therefore , _ ^-^> Placing them now 
 
 together we have 
 
 aa + acebr a e r 
 
 t | 
 
 a b c. be b a c 
 
 The factor a a may be represented thus # 2 , which means 
 that a occurs twice, the small figure at the top indicating 
 the number of times the letter occurs ; a 2 is called a squared, 
 a a a = a*, and is called a cubed. 
 
 In order to show that the above fraction, resulting as the 
 sum of the three given fractions, is correct, let a = 2, b 3, 
 c = 4, e = 5, and r = 6. Then the three given fractions 
 are 
 
 2 JU 6 =: 1:+ 1 + 1 
 
 3x4 3 2x4 6 3 4 
 
FRACTIONS SUBTRACTED. 405 
 
 In equalizing these denominators we multiply the second 
 fraction by 2, and the third by i, which will give 
 
 5_ x 2 = 10^ 3 x ^i_4i. 
 3x2= 6 ' 4 x i ~ ' 6 ' 
 then 
 
 1 1_ ,44 ij_4 3i _7_ 
 6 + 6 " 6 ' 6 6 " 12* 
 
 Now the sum of the fractions i 
 
 s 
 
 2 2 +2 X 4 X 5 + 3 X6 
 tJi , 
 
 2X3X4 
 
 4 + 40+ 1 8 __ 62 _ 14 _ 7 
 24 24 " 2 24 ~ 2 12 ' 
 
 the same result as before, thus showing that the reduction 
 was rightly made. 
 
 407. Algebraic Fractions Subtracted. To exemplify 
 the subtraction of fractions, let it be required to find the 
 
 algebraic sum of - - -% j. These denominators all dif- 
 fer. The fractions, therefore, require to be modified, so 
 that each denominator shall contain them all. To accom- 
 plish this, the first fraction will need to be thus treated : 
 
 ax df= adf 
 7x 
 the second 
 
 _ b_xcf= _ bcf 
 
 ~ dXCf= :,. Cdf'' 
 
 the third 
 
 e x c d = c d e 
 
 ~ fxcd= ~ 7d~f 
 The sum of these is 
 
 adf bcf c de 
 
 ~Tdf~ ' 
 
406 ALGEBRA. 
 
 That this is a correct answer, let the result be proved by 
 figures ; thus, for a put 15 ; b, 2 ; c, 3 ; d, 4; e, 5 ; /, 6. Then' 
 we shall have 
 
 a b e 15 25 
 ~c~~d~J~- T " 4" 6* 
 
 It will be observed that these denominators may be equal- 
 ized by multiplying the first fraction by 2, and the second 
 by ij, therefore we have 
 
 j$o _3 _ 5. 
 6 " 6 6' 
 
 To make the required subtraction we are to deduct from 30 
 (the numerator of the positive fraction), first 3, then 5 ; or, 
 the sum of the numerators of the negative fractions ; or for 
 the numerator of the new fraction we have 30 8 22. 
 The required result, therefore, is 
 
 -.~, 
 63 
 
 To apply this test to the algebraic sum we have 
 
 a d f b c f c d e I5x4x6 + 2x 3x64.3x4x5 
 " ~ cdf 3x4x6 
 
 which by multiplication reduces to 
 
 360 36 60 _ 264 _ 22_ ri. a 
 ~W '- 72" : : 6 : : 3 : 
 
 a result the same as before, proving the work correct. An 
 other example : 
 
 a b c d . e 
 
 From ----- take -, and - ; 
 n m n m n 
 
 or. find the algebraic sum of 
 
 a . b c d_ e_ 
 n m n , m n 
 
DENOMINATORS HARMONIZED. 407 
 
 The fractions which have the same denominator may be 
 grouped together thus : 
 
 a c e a c c 
 
 n n n n 
 
 and 
 
 * A ^L b ~ d 
 
 m m m 
 
 To harmonize these two denominators, m and n, the first 
 fraction must be multiplied by m and the last by ;/, or 
 
 m (a c e) n (b d} m (a c e) + n (b d) 
 m n m n mn 
 
 In the polynomial factor within the parentheses (a c e) we 
 have the positive quantity a, from which is to be taken the 
 two negatives c and e, or their sum is to be taken from #, or 
 (a (c + e) ). With this modification we have for the alge- 
 braic sum of the five given fractions 
 
 m(a (c +c)) + n (b d) 
 mn 
 
 To test the accuracy of this result, let the value of the sev- 
 eral letters respectively be as follows : a = n, b =g, c = 3, 
 d = 4, e = 5, m = 10, and n 8. Then the sum is 
 
 10 (11 -(3 + 5)) + 8 (9-4) __ TO __ 7 
 10 x 8 "80 8* 
 
 Now, taking the fractions separately, we have 
 
 <*__. _Ii (A,l\ ii_ 8 1 
 n n n ~ 8 V8 + 8/ " 8 8 " 8' 
 
 ^^945 
 again- - - = = - == ; 
 
 or, together we have, as the sum of these two results 
 
 8 + lo' 
 
408 
 
 ALGEBRA. 
 
 To harmonize these denominators we may multiply the first 
 fraction by 5, and the second by 4, thus : 
 
 8x5=4' io x 4 = 4' 
 and then the sum is 
 
 Jl - 35. = 1_. 
 40 + 40 "" 40 8 ' 
 
 the same result as before, thus the accuracy of the work is 
 established. 
 
 408. Graphical Representation of multiplication. 
 
 In Fig. 278, let A BC D, a rectangle, have its sides A B and 
 
 A B 
 
 FIG. 278. 
 
 A C divided into equal parts. Then the area of the figure 
 will be obtained by multiplying one side by the other, or 
 putting a for the side A B, and b for the side A C, then the 
 area will be a x b, or ab. This will be the correct area of 
 the figure, whatever the length of the sides may be. If, as 
 shown, the area be divided into 4 x 7 = 28 equal rectangles, 
 then a would equal 7, and b equal 4, and a b = 7 x 4 = 28, the 
 area. If A B equal 28 and A C equal 16, then will a 28, 
 and b = 16, and a b = 28 x 16 = 448, the area. 
 
 409. Graphical Multiplication : Three Factor. Let 
 
 A B C D E FG (Fig. 279) represent a rectangular solid which 
 may be supposed divided into numerous small cubes as 
 shown. Now, if a be put for the edge A B, b for the edge 
 A C, and c for the edge CD, then the cubical solidity of the 
 
MULTIPLICATION OF A BINOMIAL. 
 
 409 
 
 whole figure will be represented by a x b x c = a b c. If the 
 edge A B measures 6, the edge A C 3, and the edge CD 4, 
 then abc = 6x3x4 = 72 = the cubic contents of the 
 figure, or the number of small cubes contained in it. 
 
 D L 
 
 /G 
 
 FIG. 279. 
 
 410. Graphic Representation: Two and Three Fac- 
 tors Figs. 278 and 279 serve to illustrate the algebraic ex- 
 pressions a b and a b c. In the former it is shown that the 
 multiplication of two lines produces a rectangular surface, 
 or that if a and b represent lines, then a b may represent a 
 rectangular surface (Fig. 278) having sides respectively 
 equal to a and b. And so if a, b, and c represent three sev- 
 eral lines, then a b c may represent a rectangular solid 
 279) having edges respectively equal to #, , and c. 
 
 A BE 
 
 FIG. 280. 
 
 4f|. Graphical Multiplication of a Binomial. Let 
 
 A B CD (Fig. 280) be a rectangular surface, and BED F an- 
 other rectangular surface, adjoining the first. The area of 
 the whole figure is evidently equal to 
 
 (A B + B E) x A C. 
 
410 ALGEBRA. 
 
 The area is also equal to 
 
 ABxA C + BExBD: 
 or, since A C B D, the area equals 
 
 ABxAC + BExAC; 
 
 or, if symbols be put to represent the lines; say a for A B, 
 b for B E, and c for A C, then the two representatives of the 
 area, as above shown, become : The first 
 
 (a + b} x c = area ; 
 and the last 
 
 (a x c) + (b x c) = area. 
 Hence we have 
 
 (a + b} c = a c -\- b c. 
 
 This result exemplifies the algebraic multiplication of a bi- 
 nomial, which is performed thus : Let a + b be multiplied 
 by c. 
 
 The problem is stated thus : 
 
 (a + b) c. 
 To perform the multiplication indicated we Droceed thus : 
 
 a + b 
 c 
 
 ac + be 
 
 multiplying each of the factors of the multiplicand sepa- 
 rately and annexing them by the sign for addition. Putting 
 the two together, or showing the problem and its answer in 
 an equation, we have 
 
 (a + b) c == a c + b c, 
 
 producing the same result, above shown, as derived from 
 the graphic representation. 
 
 412. Graphical Squaring of a Binomial. Let EGCJ 
 (Fig: 281) be a rectangle of equal sides, and within it draw 
 
SQUARING OF A BINOMIAL. 411 
 
 the two lines, A H and F D, parallel with the lines of the 
 rectangle, and at such a distance from them that the sides, 
 A B and B D, of the rectangle, A B C D, shall be of equal 
 length. We then have in this figure the three squares, 
 E GCJ, AB CD, and FGBH, also the two equal rect- 
 angles, EFA B and BHD J. 
 
 Let E F be represented by a and F G by b, then the area 
 of ABC D will be axa = a* \ the area of FGBH will be 
 b x b b 2 ; the area of E F A B will be a x b = a b, and that 
 
 
 
 A B 
 
 
 
 C 1 
 
 5 
 
 FIG. 281. 
 
 of B H D y will be the same. Putting these areas together 
 thus 
 
 the sum equals the area of the whole figure equals the prod- 
 uct of EG x E * -equals the product 
 
 (a + b) x(a + b). 
 So, therefore, we have 
 
 (a + b) (a + b} = a* + 2a& + &'; (112.) 
 
 or, in general, the square of a binomial equals the square of 
 the first, plus twice the first by the second, plus the square of the 
 second. This result is obtained graphically. The same re- 
 sult may be obtained by algebraic multiplication, combining 
 
412 ALGEBRA. 
 
 each factor of the multiplier with each factor of the multi- 
 plicand and adding the products, thus 
 
 a + b 
 a + b 
 
 a b 
 
 The same result as above shown by graphical representa- 
 tion. 
 
 413. Graphical Squaring of the Difference of Two 
 Factors. Let the line E C (Fig. 281) be represented by c, 
 and the line A E and A C as before respectively by b and 
 a, then 
 
 c b a. 
 From this, squaring both sides, we have 
 
 The area of the square A B C D may be obtained thus : 
 From the square E G C J take the rectangle E G x E A and 
 the rectangle F G x D y, minus the square F G B H, or 
 from c* take the rectangle cb, and the rectangle c b, minus 
 the square, b a , and the remainder will be the square, a 8 ; or, 
 in proper form 
 
 In deducting from c* the rectangle cb twice, we have taken 
 away the small square twice ; therefore, to correct this 
 error, we have to add the small square, or b*. Then, when 
 reduced, the expression becomes 
 
 This result is obtained graphically. The result by algebraic 
 
PRODUCT OF THE SUM AND DIFFERENCE. 
 
 413 
 
 process will now be sought. The square of a quantity may 
 be obtained by multiplying the quantity by itself, or 
 
 (c - ) 2 = - 
 
 (US-) 
 
 In this process, as before, each factor of the multiplier is 
 combined with each factor of the multiplicand and the sev- 
 eral products annexed with their proper signs (Art. 415), 
 and thus, by algebraic process, a result is obtained precisely 
 like that obtained graphically. This result is the square of 
 the difference of c and b ; and since c and b may represent 
 any quantities whatever, we have this general 
 
 RULE. The square of the difference of two quantities is 
 equal to the sum of the squares of the two quantities, minus twice 
 their product. 
 
 FIG. 282. 
 
 4(4. Graphical Product of the Sum and Difference 
 
 of Two Quantities. Let the rectangle A B C D (Fig. 282) 
 have its sides each equal to a. Let the line E F be parallel 
 with A B and at the distance b from it, also, the line F G 
 made parallel with B D, and at the distance b from it. Then 
 the line E F equals a -f /;, and the line E C equals a b. 
 Therefore the area of the rectangle E F C G equals n + b, 
 
4H ALGEBRA. 
 
 multiplied by a b. From the figure, for the area of this 
 rectangle, we have 
 
 ABCD-ABEH+HFDG = RFC G\ 
 or, by substitution of the symbols, 
 
 a 2 a b + b (a B). 
 Multiply the last quantity thus 
 
 a-b 
 b 
 
 ab-b* = b(a-b}. 
 Substituting this in the above we have 
 
 a* a b + a b b* = ( a T &) x (a b). 
 
 Two of these like quantities, having contrary signs, cancel 
 each other and disappear, reducing the expression to this 
 
 The correctness of this result is made manifest by an inspec- 
 tion of the figure, in which it is seen that the rectangle E FC G 
 is equal to the square A BCD minus the square BJHF. 
 For ABEH equals BJDG. Now, if from the square 
 A B CD we take away A B E H, and place it so as to cover 
 BJDG, we shall have the rectangle E FC G plus the square 
 BJHF-, showing that the square A BCD is equal to the 
 rectangle EFC ' G plus the square B J H F '; or 
 
 a*=(a + b) x (a-b) + b\ 
 
 The last quantity may be transferred to the first member of 
 the equation by changing its sign (Art. 403). Therefore - 
 
 + b} x (a - b\ 
 as was before shown. 
 
MULTIPLICATION PLUS AND MINUS. 
 
 415 
 
 The result here obtained is derived from the geometrical 
 figure, or graphically. * Precisely the same result may be 
 obtained algebraically ; thus 
 
 a + b 
 a- b 
 
 a* +ab 
 -ab-b* 
 
 (.114.) 
 
 Here the two like quantities, having unlike signs, cancel 
 each other and disappear, leaving as the result only the dif- 
 ference of the squares. 
 
 The result here obtained is general ; hence we have this 
 
 RuL.E. The product of the sum and difference of two quan- 
 tities equals the difference of their squares. 
 
 ^ G 1 
 
 
 
 E J 
 
 F 
 
 FIG. 283. 
 
 415. PIu and Minus Sign in Multiplication. In pre- 
 vious articles the signs in multiplication have been given to 
 products in accordance with this rule, namely : Like signs 
 give plus ; unlike signs, minus. This rule may be illustrated 
 graphically, thus : In the rectangular Fig. 283, let it be re- 
 quired to show the area of the rectangle A G C H, in terms 
 of the several parts of the whole figure. Thus the area of 
 A GE J equal ABEF-GBJF*n& the area of EJCH 
 equals E FCD - J F H D. And the areas of A G E J -r 
 EJCH equals the area of A G C H. Therefore the sum of 
 the two former expressions equals A G C H. Thus A BF 
 GBJF+ EFCD JFHD = AGCH. Let the several 
 lines now be represented by algebraic symbols ; for example, 
 
416 ALGEBRA. 
 
 let AB = EF=a; let GB = ?F= l>; let A E = G J = c ; 
 and EC J H = d, and let these symbols be substituted for 
 the lines they represent, thus ABEF- GBJF+EFCD - 
 JFHD = AGCH. 
 
 ac be + ad b d = (a b) x (c + d). 
 
 An inspection of the figure shows this to be a correct 
 result. It will now be shown that an algebraical multiplica- 
 tion of. the two binomials, allotting the signs in accordance 
 with the rule given, will produce a like result. For example 
 
 a-b 
 
 c + d 
 
 ac be + ad b d. 
 
 416. Equality of Squares on Hj potliemise 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 right-angled triangle, the squares upon the sides 
 of which, it is proposed to consider. Extend the side BE 
 to F; parallel with BF draw DG, C K, and A L. Parallel 
 with ED draw A J and L G. These lines produce triangles, 
 AHB, AC?, ALC, CKD, and C G D, each equal to the 
 given triangle BED (Art. 337). Now, if from the square 
 
SQUARES ON RIGHT-ANGLED TRIANGLE. 417 
 
 A B CD we take Afiffand place it at CD G ; and if we take 
 BED and place it at A L C we will modify the square 
 A BCD, so as to produce the figure LGDEHAL, which 
 is made up of two squares, namely, the square D E FG and 
 the square ALFH, and these two squares are evidently 
 equal to the square A B CD. Now, the square D E FG is the 
 square upon ED, the base of the given right-angled triangle, 
 and the square A L F H is the square upon A H = BE, the 
 perpendicular of the given right-angled triangle, while the 
 square A B C D is the square upon B D, the hypothenuse of 
 the given right-angled triangle. Thus, graphically, it is shown 
 that the square upon the hypothenuse of a right-angled triangle 
 is equal to the sum of the squares upon the remaining two sides. 
 To show this algebraically, let B E, the perpendicular of 
 the given right-angled triangle, be represented by a ; E D, 
 the base, by b, and B D, the hypothenuse, by c. Then it is 
 required to show that 
 
 Now, since D K == B E = a, therefore, E K = E D - 
 D K = b a, and the square E K J H equals (b #) 2 , which 
 (Art. 413) equals 
 
 This is the value of the square EKJ H which, with the four 
 triangles surrounding it, make up the area of the square 
 A B C D. Placing the triangle A B H of this square outside 
 of it at CD G, and the triangle B E D at A L C, we have the 
 four triangles, grouped two and two, and thus forming the 
 two rectangles C G D K and A L C J. Each of these rect- 
 angles has its shorter side (A L, C G) equal to BE a, and 
 its longer side L C, G D, equal to E D = b ; and the sum of 
 the two rectangles is ab + ab=2ab. This represents the 
 area of the two rectangles, which are equal to the four tri- 
 angles, which, together with the square EKJH, equal the 
 square ABCD\ or 
 
 ABCD^EKJH+CGDK+ALCJ, 
 
ALGEBRA. 
 
 or c* (b a)* + a b + a b, or 
 
 c*= (b - tf) 2 + 2ab. 
 
 Then, substituting for (b #) 2 , its equivalent as above, we 
 have 
 
 c* = b* 2ab + a*+ 2ab. 
 
 Remove the two like quantities with unlike signs (Art. 402), 
 and we have 
 
 c*=b*+a*-, (115.) 
 
 which was to be proved. 
 
 417. Division the Reverse of Multiplication. As di- 
 vision is the reverse of multiplication, so to divide one quan- 
 tity by another is but to retrace the steps taken in multipli- 
 cation. If we have the area ab (Fig. 278), and one of the 
 factors a given to find the other, we have but to remove 
 from a b the factor a, and write the answer b. 
 
 If we have the cubic contents of a solid abc (Fig. 279), 
 and one of the factors a given to find the area represented 
 by the other two, we have but to remove a, and write the 
 others, b c, as the answer. 
 
 If there be given the area represented by a (b + c) (see 
 Art. 41 1), and one of the factors a to find the other, we have 
 but to remove a and write the answer b + c. Sometimes, how- 
 ever, a (b + c) is written ab + ac. Then the given factor is 
 to be removed from each monomial and the answer written 
 b + c. 
 
 If there be given the area represented by a* + 2 ab + b* 
 to find the factors, then we know by Art. 412 that this area 
 is that of a square the sides of which measure a+ b, and that 
 the area is the product of a + b by a + b ; or, that a + b is 
 the square root of a 1 + 2 a b + b*. 
 
 If there be given the area a~ - 2 ab + b~ to find its fac- 
 tors, then we know by Art. 413 that this area is that of a 
 square whose sides measure a b, or that it is the product 
 of a b bv a b, or the square of a b' 
 
PROCESSES IN DIVISION. 419 
 
 If there be given the difference, of the squares of two 
 quantities, or the area represented by a* b* y to find its fac- 
 tors, then we know by Art. 414 that this is the area pro- 
 duced by the multiplication of a b by a + b. 
 
 4-18. BH vi si on : Statement of Quotient. In any case of 
 division the requirement may be represented as a fraction ; 
 thus : To divide c + d /by a ^ we write the quotient 
 thus 
 
 c + d-f 
 a- b 
 
 For example, to illustrate by numerals, let a = 7, b = 3, 
 c = 4, d = 5, and/ = 6. Then the above becomes 
 
 7-3 "4* 
 
 4 1 9. Division ; Reduction. When each monomial in 
 either the numerator or denominator contains a common 
 quantity, that quantity may be removed and placed outside 
 of parentheses containing the monomials from which it was 
 taken ; thus, in 
 
 2 ab -\. ^ ac 8 ad 
 
 ~T~ 
 
 we have 2 and a factors common to each monomial of the 
 numerator. Therefore the expression may be reduced to 
 
 2 a (b + 2 c 4^) 
 
 To test this arithmetically we willl put a = 9, b = 7, c 5, 
 d = 4, and / = 6. Then for the first expression we have 
 
 2x9x7 + 4x9x5 8x9x4 
 ~6~~ 
 
 which equals 
 
 126 + 1 80 288 
 
 
42O ALGEBRA. 
 
 And for the second expression 
 
 2 x 9 (7 + 2 x 5 4 x 4) 
 
 6 
 which equals 
 
 18 (17 & 1 6) 18 
 
 the same result as before. It will be observed that in this 
 process of removing all common factors algebra furnishes 
 the means of performing the work arithmetically with many 
 less figures. The reduction is greater when the common 
 factors are found in both numerator and denominator. For 
 example, in the expression 
 
 $ an + 9 fin 15 en 
 
 12 dn 
 
 we have 3 n a factor common to each monomial in the nu- 
 merator and denominator ; therefore the expression reduces 
 to 
 
 And now, since 3 n is a factor common to both numerator 
 and denominator, these cancel each other ; therefore (Art. 
 371) the expression reduces to 
 
 5 
 
 To test these reductions arithmetically, let a = 9, b = 8, 
 c 4, d 6, /= 3, and n = .5. Then the first expression 
 becomes 
 
 3x9x5 _+_9_ x 8 x 5 - 15x4x5 
 12x6x5 18x3x5 
 
 which equals 
 
 135 + 360-300^ *95 _ i_. 
 360 - 270 90 6 ' 
 
FORMULA OF THE LEVER. 421 
 
 and the second expression becomes 
 
 9 + 3x8 5x4 
 4x6 6x3 ' 
 
 which equals 
 
 9 + 24 20 __ ^S __ 1 
 2418 6 6* 
 
 The same result, but with many less figures. 
 
 420. Proportionals : Analysis. In the formula of the 
 lever (Art. 377), P x CF = R x E C. Let n be put for the 
 arm of leverage 6^and m for E C. Then we have 
 
 Pn = Rm, 
 from which by division (Art. 372) we have (Art. 399) 
 
 and ~ 
 
 (in.) 
 
 Suppose there be a case in which neither R nor P severally 
 are known, but that their sum is known ; and it is required 
 from this and the m and n to find R and P. Let 
 
 W = R + P, 
 
 then W- R = P. (See Art. 403.) 
 
 The value of P was above found to be 
 
 Since P = R and also equals W - - R, therefore- 
 
422 ALGEBRA. 
 
 Transferring R to the opposite member (Art. 403) we hav< 
 
 Here R appears as a common factor and may be separated 
 by division (Art. 419) ; thus 
 
 W= I 
 
 By division the factor ( i + J may be transferred to the 
 opposite member (Art. 371). Thus we have 
 
 W 
 
 by which we find the value of R developed. As an example, 
 let W = looo pounds, m = 3 feet and n 7 feet ; then 
 
 1000 _ looo 
 ~- i+l = "ToT ' 
 
 Multiplying the numerator and denominator by 7, we get 
 
 7 x 1000 
 
 
 =700 . 
 
 Since R + P 1000, 
 
 and R = 700, 
 
 then P 300. 
 
 But a process similar to the above develops an expression 
 for the value of P, which is 
 
 i + n 
 
 ~Z ("70 
 
 Putting this to the test of figures, we have 
 looo looo 3000 
 
 r> _ _ __ if _ _ inn 
 
 i + i - y 10 - 3 - 
 
NEGATIVE EXPONENTS. 423 
 
 4-21. Raising a Quantity to any Power. When a 
 quantity is required to be multiplied by its equal, the prod- 
 uct 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 is a cube ; or, a 9 x a = a 3 ; or, generally, 
 for- 
 
 a, a a, a a a, aaaa, aaaaa, 
 we put 
 
 in which the small number at the upper right-hand cor- 
 ner indicates the number of times the quantity occurs in 
 the expression. Thus, if a 2, then # a = 2 x 2 = 4, 
 a 3 = 4 x 2 = 8, a* = 8 x 2 = 16, a" 16 x 2 = 32 ; any term 
 in the series of powers may be found by multiplying the 
 preceding one by a, or by dividing the succeeding one by a. 
 
 Thus a* x a = a*, and - a\ 
 a 
 
 4-22. Quantities with Negative Exponents. The series 
 of powers, by division, may be extended backward. Thus, 
 
 a 6 a* 3 a* * a* , a 1 a 
 
 if we divide = # ; = a ; - = a ; = a ', - = a ; - -a ; 
 a # # a a a 
 
 f?lW; "-=>, etc. 
 a a 
 
 In this series we have - = a*. But a quantity divided by 
 
 its equal gives unity for quotient, or - = i. Therefore, = i, 
 
 and a i. This result is remarkable, and holds good re- 
 gardless of the value of a. 
 
 From this and the preceding negative exponents we de- 
 rive the following : 
 
4 2 4 ALGEBRA. 
 
 . 
 
 a = = -, 
 
 a a 
 
 3 a~" I I 
 
 a - 3 =- = -, etc. 
 
 Showing that # quantity with a negative exponent may have 
 substituted for it the same quantity with a positive exponent, but 
 used as a denominator to a fraction having unity for the 
 numerator. 
 
 423. Addition and Subtraction of Exponential Quan- 
 tities. Equal quantities raised to the same power may be 
 added or subtracted; as, 2 + 2 2 = 3# 2 ; but expressions in 
 which the powers differ cannot be reduced ; thus, a* + a a* 
 cannot be condensed. 
 
 4-24-. multiplication of Exponential Quantities. It 
 
 will be observed in Art. 421 that in the series of powers, the 
 index or exponent increases by unity ; thus, a 1 , a\ a\ a\ etc. ; 
 and that this increase is effected by multiplying by the root, 
 or original quantity. From this we learn that to multiply 
 two quantities having equal roots we simply add their exponents. 
 Thus the product of a, a'\ and a 3 is a' x a* x a* a\ 
 The product of a~ 2 , # 3 , and a b is a~* x a* x a* = a*. 
 The exponents here, are : 2 + 3 + 5 8 2 = 6. 
 
 425. Division of Exponential Quantities. As division 
 is the reverse of multiplication, to divide equal quantities 
 raised to various powers, we need simply to subtract the expo- 
 nent of the divisor from that of the dividend. Thus, to divide 
 a" by a 9 we have a*"* = a\ That this is correct is manifest ; 
 for the two factors, a* x a\ in their product, a\ produce the 
 dividend. 
 
 To divide a by a*, we have a^ = a~\ which is equal to - 3 
 
EXPLANATION OF LOGARITHMS. 425 
 
 (see Art. 422). The same result may be had by stating the 
 question in the usual form. Thus, to divide a 1 by a" we have 
 
 6 , a fraction which is not in the lowest terms, for it may be 
 
 put thus, - T 5 = , by which it is seen that it has in both its 
 a a a 
 
 numerator and denominator the quantity a\ which cancel 
 each other (Art. 371). Therefore, - 6 = L ; the same result 
 as before. 
 
 426. Extraction of Radicals. We have seen that the 
 square of a is a 1 x a 1 = a* ; of 2 a 3 is 2 a* x 2 a 3 = 4 a 6 ; in each 
 case the square is obtained by doubling the exponent. 
 
 To obtain the square root the converse follows, namely, 
 take half of the exponent. 
 
 Thus the square root of a* is a 1 , of a? is a, of a 6 is a 3 . 
 
 The same rule, when the exponent is an odd number, 
 gives a fractional exponent, thus : the square root of a 3 is cfr ; 
 or, of a b , is a*. So, also, the square root of a, or a 1 , is a*. 
 Therefore, we have <?* = Va, equals the square root of a, and 
 the cube root of a 1 = cfr = Va. 
 
 427. Logarithms. We have seen in the last article the 
 nature of fractional exponents. Thus the square root of a" 
 equals eft, which may be put a**. In this way we may have 
 an exponent of any fraction whatever, as #**. Between the 
 exponents 2 and 3, we may have any number of fractional 
 exponents all less than 3 and more than 2. So, also, the 
 same between 3 and 4, or any other two consecutive num- 
 bers. 
 
 The consideration of fractional exponents or indices has 
 led to the making of a series of decimal numbers called 
 logarithms, which are treated in the manner in which expo- 
 nents are treated ; namely 
 
 To multiply numbers add their logarithms. 
 
 To divide numbers, subtract the logarithm of the divisor from 
 the logarithm of the dividend. 
 
426 ALGEBRA. 
 
 To raise any number to a given power, multiply its logarithm 
 by the exponent of that po^ver. 
 
 To obtain the root of any power, divide the logarithm of the 
 given number by the exponent of the given power. 
 
 As an example by which to exemplify the use of loga- 
 rithms : What is the product of 25 by 375 ? 
 
 We first make this statement : 
 
 Log. of 25- = i- 
 " 375* =2. 
 
 Putting at the left of the decimal point the integer char- 
 acteristic, or whole number of the logarithm at one less than 
 the number of figures in the given number at the left of its 
 decimal point. 
 
 To find the decimal part of the required logarithm we 
 seek in a book of Logarithms (such as that of Law's, in 
 Weale's Series, London) in the column of numbers for the 
 given number 25, or 250 (which is the same as to the man- 
 tissa) and opposite to this and in the next column we find 
 7940 and a .place for two other figures, which a few lines 
 above are seen to be 39 ; annex these and the whole number 
 is 0-397940. These we place as below : 
 
 Log. of 25 = i 397940. 
 
 Now, to find the logarithm of 375, the other factor, we 
 turn to 375 in the column ol numbers and find the figures 
 opposite to it, 4031, which are to be preceded by 57, the two 
 figures found a few lines above, making the whole, -574031, 
 which are placed as below, and added together. 
 
 Log. of 25- = 1-397940 
 
 11 375- = 2-574031 
 
 The sum = 3-971971 
 
 This sum is the logarithm of the product. To find the 
 product, we seek in the column of logarithms, headed o-, 
 for -971971, the decimal part. We find first 97, the first two 
 
EXAMPLES OF LOGARITHMS. 427 
 
 figures, and a little below seeking for 1971, the remaining 
 four figures, we find 1740, those which are the next less, 
 and opposite these, to the left, we find 7, and above 93, or 
 together, 937 ; these are the first three figures of the required 
 product. 
 
 For the fourth figure we seek in the horizontal column 
 opposite 7 and 1740 for 1971, the remaining four figures of 
 the logarithm, and find them in the column headed 5. 
 
 This figure 5 is the fourth of the product and completes 
 it, as there are only four figures required when the integer 
 number of the logarithm is 3. The completed statement 
 therefore is 
 
 Log. of 25-' = 1-397940, 
 " " 375- = 2-574031, 
 
 9375 = 3-97I97I- 
 
 Another example in the use of logarithms. What is the 
 product of 3957 by 94360? 
 
 The preliminary statement, as explained in last article, is 
 
 Log- 3957 = 3- 
 " 94360 = 4- 
 
 In the book of logarithms seek in the column of numbers 
 for 3957. In the first column we find only 395, and opposite 
 to this, in the next column, we find a blank for two figures, 
 above which are found 59. Take these two figures as the 
 first two of the mantissa, or decimal part of the required 
 logarithm, thus, 0-59. Again, opposite 395 and in the col- 
 umn headed by 7 (the fourth figure of the given number), 
 we have the four figures 7366. These are to be annexed to 
 (o- 59) the first two obtained. The decimal part of the loga- 
 rithm, therefore, is 0-597366. 
 
 To obtain the logarithm for 94360, the other given num- 
 ber, we proceed in a similar manner, and, opposite 943, we 
 find 0-97; then, opposite 943 and in column headed 6, we 
 find 4788, or, together, the logarithm is 0-974788. The 
 whole is now stated thus 
 
428 ALGEBRA. 
 
 Log. of 3957 = 3-597366 
 
 94360= 4-974788 
 
 " " 373382000 = 8-572154 = sum of logs. 
 
 The two logarithms are here added together, and their sum 
 is the logarithm of the product of the two given factors. 
 The number corresponding to the above resultant logarithm 
 may be found thus: Look in the column headed o for 57, 
 the first two numbers of the mantissa, then in the same 
 column, farther down, seek 2154, the other four figures of 
 the mantissa; or, the four (1709) which are the next less 
 than the four sought, and opposite these to the left, in the 
 column of numbers, will be found 373, the first three figures 
 of the product ; opposite these, to the right, seek the four 
 figures next less than 2154, the other four figures of the man- 
 tissa. These are found in the column headed 3 and are 
 2058. The 3 at the head of the column is the fourth figure 
 in the product. From 2154, the last four figures of the man- 
 tissa, deduct the above 2058, or 
 
 2154, 
 2058, 
 
 Remainder, 96. 
 
 At the bottom of the page, opposite the next less number 
 (3727) to that contained in 3733, the answer already found, 
 seek the number next less to the above remainder, 96. This 
 is 92-8, and is in the column headed 8. Then 8 is the next 
 number in the product. From 96 deduct 92-8, and multi- 
 ply it by 10, or 
 
 96 
 
 92-8 
 
 3-2 x 10= 32. 
 
 Then, in the same horizontal column, seek for 32 or its next 
 less number. This is 23-2, found in column 2. This 2 is the 
 next figure in the product. Additional figures may be ob- 
 tained by the table of proportional parts, but they cannot be 
 
THE SQUARE OF A BINOMIAL. 429 
 
 depended upon for accuracy beyond two or three figures. 
 We therefore arrest the process here. 
 
 The product requires one more figure than the integer of 
 the logarithm indicates; as the integer is 8, there must be 
 nine figures in the product. We have already six ; to make 
 the requisite number nine we annex three ciphers, giving 
 the completed product 
 
 3957 x 9436o = 373382000. 
 
 By actual multiplication we find that the true product in the 
 last article is 373382520. In a book of logarithms, carried to 
 seven places, the required result is found to be 373382500^ 
 which is more nearly exact. 
 
 The utility of logarithms is more apparent when there 
 are more than two factors to be multiplied, as, in that case, 
 the operation is performed all in one statement. Thus: 
 What is the product of 3-75, 432-95, 1712, and 0-0327 ? 
 
 The statement is as follows : 
 
 Log. 3-75 = 0-574031 
 
 432-95 = 2-636438 
 
 1712- = 3-233504 
 
 0327 = 8- 
 
 Product = 90891. = 4-958521 
 
 16 
 
 Explanations of working are given more in detail in most of 
 the books of logarithms. 
 
 428. Completing the Square of a Binomial. We 
 
 have seen in Art. 412 that the square of a binomial (a + &) 
 equals a" + 2 ab + b* a trinomial the first and last terms 
 of which are each the square of one of the two quantities, 
 while the second term contains the second quantity multi- 
 plied by twice the first quantity 
 
 In analytical investigations it frequently occurs that an 
 expression will be obtained which may be reduced to this 
 form : 
 
43 ALGEBRA. 
 
 a* + mab=f, (nS.) 
 
 in which m is the coefficient of the second term, and a and b 
 are two quantities represented by a and b or any other two 
 symbols. 
 
 A comparison of this expression with the square of a bi- 
 nomial (112.) contained in Art. 412, shows that the member 
 at the left comprises two out of the three terms of the square 
 of a binomial ; as thus 
 
 a* + 2 a b + b*, 
 
 but with a coefficient m instead of 2. It is desirable, as will 
 be seen, to ascertain a proper third term for the given ex- 
 pression ; or, as it is termed, " to complete the square." The 
 method by which this is done will now be shown. 
 
 A consideration of the above trinomial shows that the 
 third term is equal to the square of the quotient obtained by 
 dividing the second term by twice the square root of the 
 first ; or 
 
 Now a third term to the above binomial, equation (i 18.), may 
 be obtained by this same rule. For. example 
 
 The rule for the third term then is: Divide the second term 
 by twice the square root of the first, and square the quotient. 
 
 As an example, let it; be required to find the third term 
 required to complete the square in the expression 
 
 6 n x -f- 4** =/, 
 
 in which n and / are known quantities and x unknown. 
 Putting it in this form 
 
 4_r 2 + 6 nx = /, 
 and dividing by 4, we have 
 
BINOMIALS CONTINUED. 431 
 
 4 4 
 
 which reduces to 
 
 Now applying the above rule for finding the third term, we 
 have 
 
 which is the required third term. To complete the square 
 we add this third term to both members of the above re- 
 duced expression, and have 
 
 The member of this expression at the left is the completed 
 square of a binomial, the two quantities constituting which 
 are the square roots of the first and third terms respectively ; 
 or x and f n, and we therefore have 
 
 - 2 n*+\-H,- 
 
 and now taking the square root of both sides of the expres- 
 sion, we have 
 
 and, by transferring the second quantity to the right mem- 
 ber, we have 
 
 an expression in which x, the unknown quantity, is made to 
 stand alone and equal to known quantities. 
 
 The process of completing the square is useful, as has 
 been shown, in developing the value of an unknown quan- 
 
432 ALGEBRA. 
 
 tity where it enters into an expression in two forms, one as 
 the square of the other. 
 
 As an example to test the above result, let/ 256 and 
 n = 8. Then we have by the last expression for the value 
 of* 
 
 __ 
 
 4 
 
 = 1/64+36 - 6, 
 
 = y ioo 6, 
 
 x = 10 6 = 4. 
 
 Now this value of x may be tested in the original expres- 
 sion 
 
 6 nx + 4** = /, 
 
 for which we have 
 
 6x8x4 + 4x42 = /, 
 192 + 64 = /, 
 
 the correct value as above. 
 
 PROGRESSION. 
 
 429. Arithmetical Progreion. In a series of num- 
 bers, as i, 3, 5, 7, 9, etc., proceeding in regular order, in- 
 creasing by a common difference, the series is called an 
 arithmetical progression ; the quantity by which one num- 
 ber is increased beyond the preceding one is termed the 
 difference. If d represent the difference and a the first term, 
 then the progression may be stated thus 
 
 Terms i, 2, 3, 4, 5, 
 
 a, a + d, a + 2 d, a + 3 d, a + 4 d, etc. 
 
 The coefficient of d is equal to the number of terms preced- 
 ing the one in which it occupies a place. Thus the fifth 
 term is a + ^d, in which the coefficient 4 equals the number 
 of the preceding terms. 
 
 From this we learn the rule by which at once to desig- 
 
ARITHMETICAL PROGRESSION. 433 
 
 nate any term without finding all the preceding terms. For 
 the one hundredth term we should have a + 99 d, or, if the 
 number of terms be represented by n> then the last term 
 would be represented by 
 
 / = a + (n i) d. (H9-) 
 
 For example, in a progression where a, the first term, equals 
 i, d the difference, 2, and n, the number of terms, 90, the last 
 term will be 
 
 / = a + (n -- i) d = i + (.90 i) 2 = 179. 
 
 Therefore, to find the last term : 
 
 To the first term add the product of the common difference 
 into the number of terms less one. 
 
 By a transposition of the terms in the above expression, 
 so as to give it this form 
 
 a = / - (n - i)</, (120.) 
 
 we have a rule by which to find the first term, which, in 
 words, is 
 
 Multiply the number of terms less one by the common differ- 
 ence, and deduct the product from the last term ; the remainder 
 will be the first term. 
 
 By a transposition of the terms of the former expression 
 to this form 
 
 / - a = ( n - i) d, 
 
 and dividing both members by (n - i), we have 
 
 rf=^-=^. (121.) 
 
 n - i ' 
 
 which is a rule for the common difference, and which, in 
 words, is 
 
 Subtract the first term from the last, and divide the remain- 
 der by the number of terms less one ; the quotient will be the com- 
 mon difference. 
 
 Multiplying both members of the equation (121.) by 
 (n i) and dividing by d, we obtain 
 
434 ALGEBRA. 
 
 l-a 
 n - 1 = -J-' 
 
 Transferring i to the second member, we have 
 
 l-a 
 n = , + i ; (122.) 
 
 which is a rule for finding the number of terms, and which, 
 in words, is 
 
 Divide the difference between the first and last terms by the 
 common difference ; to the quotient add unity, and the sum will 
 be the number of terms. 
 
 Thus it has been shown, in equations (119,) (120), (121), 
 and (122), that when, of the four quantities in arithmetical 
 progression, any three are given, the fourth may be found. 
 
 The sum of the terms of an arithmetical progression may 
 be ascertained by adding them ; but it may also be had by a 
 shorter process. If the terms are written in order in a hori- 
 zontal line, and then repeated in another horizontal line be- 
 neath the first, but in reversed order, as follows : 
 
 i, 3, 5, 7, 9, 11, 13, 15, 
 
 15, 13, 11, 9, 7, 5, 3, i, 
 
 16, 16, 16, 16, 16, 16, 16, 16, 
 
 and the vertical columns added, the sums will be equal. In 
 this case the sum of each vertical couple is 16, and there are 
 8 couples ; hence the sum of these 8 couples is 8 x 16 = 128. 
 And in general the sum will be the product of one of the 
 couples into the number of couples. It will be observed 
 that the first couple contains the first and last terms, I and 
 15 ; therefore the sum of the double series is equal to the 
 product of the sum of the first and last terms into the 
 number of terms. Or if 5 be put to represent the sum of 
 the series, we shall have 
 
 2 S = (a + I) n, 
 and, dividing both sides by 2 
 
 -S = ( + /); (I2 3 .) 
 
GEOMETRICAL PROGRESSION EXPLAINED. 435 
 
 Or, in words : The sum of an arithmetical series equals the prod- 
 uct of the sum of the first and last terms, into half the number 
 of terms. 
 
 430. Geometrical Progression. A series of numbers, 
 such as i, 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 Progression. 
 The constant quantity is termed the common Ratio, and is 
 equal to any term divided by the preceding one. Thus in 
 
 the above example -- or Or 2, equals the common ra- 
 
 o 42 
 
 tio of the above series. In the series, i, 3, 9, 27, etc., we 
 have for the ratio 
 
 27 _ 9 _ 3 
 ~9 "3^1 : 
 
 which is the common ratio of this series. 
 A geometrical series may be put thus : 
 
 Terms: I, 2, 3, 4; 
 
 Progress. : i, i x 3, i x 3 x 3, i x 3 x 3 x 3 ; 
 
 or thus 
 
 Terms: I, 2, 3, 4: 
 
 Progress. : i, i x 3, i x 3 2 , i x 3 8 ; 
 
 in which the common ratio, in this case 3, appears in each 
 term and with an exponent which is equal to the number of 
 terms preceding that in which it occupies a place. 
 
 If the first term be represented by a and the common ra- 
 tio by r, then the following will represent any geometrical 
 progression 
 
 a, ar, ar z , ar 3 , #r 4 , etc. (124.) 
 
 For example, let a = 2 and r = 4 ; then the progression 
 will be 
 
 2, 8, 32, 128, 512, etc. 
 
43 6 ALGEBRA. 
 
 If r = unity, then when a = 2 the progression becomes 
 2, 2, 2, 2, 2, etc. 
 
 If r be less than unity, then the progression will be a de- 
 creasing one. 
 
 For example, let a =. 2 and r = -J. Then we have for 
 the progression 
 
 1 1 L JL 
 
 2j Iy 2' 4' 8' 16' e 
 
 If the number of terms be represented by n, and the last by 
 /, then the last term will be 
 
 For example, let n equal 6, then the progression will be 
 
 Terms: i, 2, 3, 4, 5, 6; 
 
 Progress.: a, ar, ar* y ar 3 , ar\ ar*\ 
 
 in which the .exponent of the last term equals n i = 
 6 _ i = 5. 
 
 If 5 be put for the sum of a geometrical progression, we 
 will have 
 
 S = a + ar + ar z + ..... + a r n ~* + a r ll ~ l . 
 Multiply each member by r y then 
 
 Sr = ar + ar 2 + ..... + a r n ~ 2 + a r*~ l + ar n . 
 Subtract the upper line from the lower; then 
 
 S r = ar + ar 2 + ..... + ar n ~ 2 + a r 11 ' 1 + a r*, 
 
 S =g + ar+ar 2 + ..... + ar n ~ 2 + ar*~ l _ 
 Sr - s = - a *"* * ~* + ar", 
 
 Sr s = a + ar n , 
 
 S (r i) = a + a r n = ar n a, 
 
 ar " ~~ a 
 
GEOMETRICAL PROGRESSION CONTINUED. 437 
 
 The last term (equation (125.)) equals I ar n ~ l , and since 
 ar n = r x ar n ~ * = r /, therefore 
 
 Thus, to find the sum of a geometrical progression : Multi- 
 ply the last term by the ratio ; from tJie product deduct the first 
 term, and divide the remainder by the ratio less unity. 
 
 For example, the sum of the geometrical progression 
 
 S = i + 3 + 9 + 27 + 81 + 243 +729 = 1093 
 
 by actual addition. 
 
 To obtain it by the above rule 
 
 rl a 3 x 729 I 
 
 s== : TfST II093 ' 
 
 the correct result. 
 
 If there be a decreasing geometrical progression, as i, -J, 
 J, ^ T , etc., in which the ratio equals ^, the sum will be 
 
 S=i T + " + ^ + -g7> etc -> to infinity. 
 Multiply this by 3, and subtract the first from the last 
 
 35=3+1+- + - + + + to infinity. 
 
 S= i + i + i + ^ + g I f + to infinity. 
 
 2 5 = 3 or 5 = if 
 
 In a decreasing progression let r, the common ratio, be 
 represented by (b less than c), and the first term by a, then 
 ihe sum will be 
 
 b b 2 b 3 
 S=a + a- + a- + a- 3 +, etc., to infinity. 
 
438 ALGEBRA. 
 
 Multiply this by -, and subtract the product from the 
 
 above 
 
 ^b b d* b* 
 S- = a- + a- + a-^ + etc., to infinity. 
 c c c c 
 
 b b* l> 3 
 =z a + a + a 2 + a-j + to infinity. 
 
 b b* b 3 
 
 S a \- a + <2 -j + to infinity. 
 cere * 
 
 Or s(i --)*, 
 
 For example, let the first term of a geometrical progression 
 equal 2, and the ratio equal , then the sum will be 
 
 From this, therefore, we have this rule for the sum of an in- 
 finite geometrical progression, namely : Divide the first term 
 by unity less the ratio. 
 
SECTION X. POLYGONS. 
 
 431. Relation of Sum and Difference of Two Lines. 
 
 Let AB and CD (Fig. 285) be two given lines; make EH 
 
 -B 
 D 
 
 E - ] |- 
 H J 
 
 FIG. 285. 
 
 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, also, J bisects HF\ for 
 
 and 
 
 EH=FG. 
 
 Subtract the latter from the former ; then 
 
 EJ- 
 but 
 
 E 
 
 and 
 therefore 
 
 Now, E J is half the sum of the two lines, and HJ is half 
 the difference ; and 
 
 Ey-Hy=EH=AB. 
 
 Or : Half the sum of two quantities, minus half their dif- 
 ference, equals the smaller of the two quantities. 
 
440 
 
 POLYGONS. 
 
 Let the shorter line be designated by a, and the longer 
 by b ; then the proposition is expressed by 
 
 _a+b b a 
 
 2 2 
 
 (128.) 
 
 We also have EJ+JFEF CD\ or, half the mm 
 of two quantities, plus half their difference, equals the larger 
 quantity. 
 
 432. Perpendicular, in Triangle of Known Side. 
 
 Let ABC (Fig. 286) be the given triangle, and CE a perpen- 
 dicular let fall upon A B, the base. Let the several lines of 
 
 the figure be represented by the symbols a, b, c, d, g, and f, 
 as shown. Then, since A EC and BEC are right-angled 
 triangles, we have (Art. 416) the following two equations, 
 and, by subtracting one fr.om the other, the third 
 
 Then (Art. 414), by substitution, we have 
 
 (f + f)(f-e) = ( + *)(-*) 
 
 By division we obtain 
 
 _ 
 
 a- 6) 
 
 f+g 
 
RULE FOR TRIGONS. 441 
 
 According to Art. 431, equation (128.), we have 
 
 In this expression let the value of / g, as above, be 
 substituted, then we will have 
 
 ~ = f+* (<* + *) (a - b) 
 
 Multiply the first fraction by (f + g), then join the two 
 fractions, when we will have 
 
 The lines f and g, in the figure, together equal the line 
 c ; therefore, by substitution 
 
 f - (a + b) (a - b] 
 g = -^- -i. (129.) 
 
 This is the value of the line g. 
 
 It may be expressed in words, thus: The shorter of the 
 two parts into which the base of a triangle is divided by a 
 perpendicular let fall from the apex upon the base, equals the 
 quotient arising from a division by twice the base, of the differ- 
 ence between the square of the base and the product of the sum 
 and difference of the two inclined lines. 
 
 As an example to show the application of this rule, let 
 a 9, b = 6, and c = 12 ; then equation (129.) becomes 
 
 12' - (9 + 6) (9 - 6) 
 
 2 X 12 
 
 . .144 - iT><"3 
 
 - 
 
 99- 
 
44 2 POLYGONS. 
 
 Now, to obtain the length of d, the perpendicular, by the 
 figure, we have 
 
 -and, extracting the square root 
 
 or, in words : The altitude of a triangle equals the square root 
 of the difference of the squares of one of the inclined sides and 
 its base. 
 
 As an example, take the same dimensions as before, then 
 equation (130.) becomes 
 
 The square of 6 = 36- 
 
 " 4i = 17-015625 
 
 6 2 -4i 2 = 18^984375, 
 the square root of which is 4-44234; therefore 
 
 d= t-- 4^ = 4. 44234. 
 
 This may be tested by applying the rule to the other in 
 clined side and its base 
 
 c = 12 
 * = 4* 
 /= 71- 
 
 Then, ^- 
 
 9' = 81- 
 
 ?%*= 62-015625 
 9' - 7F = ^8-984375. 
 
TRIGON RADIUS OF CIRCLES. 
 
 443 
 
 The same result as before, producing for its square root 
 the same, 4-44234, the value of d\ therefore 
 
 433. Trigon : Radiu of ircumcribed and Incribed 
 Circles: Area. Let A B C (Fig. 287) be a given trigon or 
 triangle with its circumscribed and inscribed circles. Draw 
 the lines A D F, DB and D C. 
 
 The three triangles, A B D, A CD, and B D C, have their 
 apexes converging at D, and form there the three angles, 
 A DB, ADC, and B D C. These three angles together form 
 
 four right angles (Art. 335), and each of them, therefore, 
 equals f of a right angle. 
 
 The angles of the triangle BDC together equal two 
 right angles (Art. 345). As above, the angle BDC equals | 
 of a right angle, hence 2 -J = ^^ | of a right angle, 
 equals the sum of the two remaining angles at B and C. 
 The triangle BDC is isoceles (Art. 338); for the two sides 
 B D and D C, being radii, are equal ; therefore the two angles 
 at the base B and C are equal, and as their sum, as above, 
 equals f of a right angle, therefore each angle equals -J of a 
 right angle. Draw the two lines FC and F B. Now, be- 
 
444 POLYGONS. 
 
 cause Z> C and DF are radii, they are equal, hence DFC is 
 an isoceles triangle. 
 
 It was before shown that the angle B D C equals | of a 
 right angle; now, since the diameter A F bisects the chord 
 B C, the angles B D E ^nd E D C are equal, and each equals 
 the half of the angle B D C\ or, \ of f of a right angle equals 
 of a right angle. Deducting this from two right angles 
 (the sum of the three angles of the triangle), or 2 f = 
 \\ | of a right angle equals the sum of the angles at F and 
 C', hence each equals the half of f, or f of a right angle; 
 therefore the triangle DFC is equilateral. The triangles 
 DBF and DFC are equal. The angles B D C and B F C are 
 equal; the line BC is perpendicular to D F and bisects it, 
 making DE and EF equal; hence DE equals half D F, or 
 DB, radii of the circumscribing circle. Therefore, putting 
 R to represent B D, the radius of the circumscribing circle, 
 and b = B C, a side of the triangle A B C, by Art. 416, we 
 have 
 
 + DE 
 
 Transferring and reducing 
 
 4 " 4' 
 
 4/ 4 
 Ijpt^i^ 
 
 4 ~4 
 
 4 x I^ == i3 == ^ 
 34 ~3 ~3 ' 
 
 Or, The Radius of the circumscribing circle of a regular trigon 
 or equilateral triangle, equals a side of the triangle divided by 
 the square root of 3. 
 
AREA OF EQUILATERAL TRIANGLE. 
 
 445 
 
 By reference to Fig. 287 it will be observed, as was 
 above shown, that D E = E F= - = - ; or, D E, the ra- 
 
 dius of the inscribed circle, equals half the radius of the 
 circumscribed circle; or, again, dividing equation (131.) by 
 2, we have 
 
 R _ b 
 
 2 " 2 t/y- 
 
 and, putting r for the radius of the inscribed circle, we 
 have 
 
 Or: The radius of the inscribed circle of a regular trigon equals 
 the half of a side of the trigon divided by the square root of 3. 
 To obtain the area of a trigon or equilateral triangle ; we 
 have (Art. 408) the area of a parallelogram by multiplying 
 its base into its height ; and (Arts. 341 and 342) the area of a 
 triangle is equal to half that of a parallelogram of equal base 
 and height, therefore, the area of the triangle BD &(Fig. 287) 
 is obtained by multiplying B C, the base, into the half of 
 ED, its height. Or, when A^ is put for the area 
 
 or _ ,:;/;' *=>*., 
 
 substituting for R its value (131.)" 
 
 jr = *x-4= 
 
 4 1/3 
 
 4^3 
 
 This is the area of the triangle BD C. 
 
 The triangle A B C is compounded of three equal tri- 
 angles, one of which is the triangle B D C ; therefore the 
 area of the triangle ABC equals three times the area of the 
 triangle B DC\ or, when A represents the area 
 
446 
 
 POLYGONS. 
 
 4 1/3 
 
 (1330 
 
 Or: The area of a regular /rz^wz or equilateral triangle 
 equals three fourths of the square of a side of the triangle di- 
 vided by the square root of 3. 
 
 Tetragon ; Radius of Circumscribed and In- 
 scribed Circles: Area. Let A B CD (Fig. 288) be a given 
 tetragon or square, with its circumscribed and inscribed 
 
 FIG. 288. 
 
 circles, of which A E is the radius of the former and EF that 
 of the latter. The point F bisects A B, the side of the 
 square. A F equals EF and equals half A B, a side of the 
 square. Putting R for the radius of the circumscribed 
 circle and b for A B, we have (Art. 416) 
 
 : 7T: <'34-) 
 
 Or: The radius of the circumscribed circle of a regular tetra- 
 gon equals a side of the square divided by the square root of 2. 
 
SIDE AND AREA OF HEXAGON. 
 
 447 
 
 By referring to the figure it will be seen that the radius 
 of the inscribed circle equals half a side of the square 
 
 b 
 
 r 
 
 (135.) 
 
 The. area of the square equals the square of a side 
 
 A = b*. (136.) 
 
 435. Hexagon: Radius of Circumscribed and In- 
 scribed Circles: Area. Let A B C D EF(Fig. 289) be an equi- 
 lateral hexagon with its circumscribed and inscribed circles, 
 of which EG is the radius of the former, and G H that of 
 the latter. The three lines, A D, BE, and C F, divide the 
 
 FIG. 289. 
 
 hexagon into six equal triangles with their apexes converg- 
 ing at G. The six angles thus formed at G are equal, and 
 since their sum about the point G amounts to four right 
 angles (Art. 335), therefore each angle equals or f of a 
 right angle. The sides of the six triangles radiating from G 
 are the radii of the circle, hence they are equal ; therefore, 
 each of the triangles is isosceles (Art. 338), having equal angles 
 at the base. In the triangle EGD, the sum of the three 
 angles being equal to two right angles (Art. 345), and the 
 angle at G being, as above shown, equal to f of a right angle, 
 therefore the sum of the two angles at E and D equals 
 2 = J of a right angle ; and, since they equal each other, 
 
44^ POLYGONS. 
 
 therefore each equals f of a right angle and equals the angle 
 at G ; therefore E G D is an equilateral triangle. Hence 
 ED, a side of a hexagon, equals E G, the radius of the circum- 
 scribing circle 
 
 R=b. (137.) 
 
 As to the radius of the inscribed circle, represented by G H, 
 a perpendicular from the centre upon ED, the base; the 
 point H bisects E D. Therefore, E H equals half of a side 
 of the hexagon, equals half the radius of the circumscribing 
 circle. Let R = this radius, and r the radius of the inscribed 
 circle, while b = a side of the hexagon ; then we have (Arts. 
 353 and 416) 
 
 i 
 
 r 
 
 Now, R =1 b, therefore 
 
 r =^-- ' (138.) 
 
 Or : The radius of the inscribed circle of a regular hexagon 
 equals the half of a side of the hexagon, multiplied by the 
 square root of 3. 
 
 As to the area of the hexagon, it will be observed that the 
 six triangles, A B G, B G C, etc., converging at G, the centre, 
 are together equal to the area of the hexagon. The area of 
 E G D, one of these triangles, is equal to the product of D, 
 the base, into the half of G H, the perpendicular ; or, when 
 N is put to equal the area 
 
 G H 
 
SIDE AND AREA OF OCTAGON. 
 
 449 
 
 and, since r t as above, equals * 3 , 
 
 2 
 
 This is the area of one of the six equal triangles ; therefore, 
 when A is put to represent the area of the hexagon, we have 
 
 A = 
 
 (139.) 
 
 Or : The area of a regular hexagon equals three lialves of the 
 square of a side multiplied by the square root of $. 
 
 FIG. 290. 
 
 4-36- Octagon: Radius of Circumscribed and In- 
 icribed Circles: Area. Let C E D B F (Fig. 290) represent a 
 quarter of a regular octagon, in which ^is the centre, ED 
 a side, and CE and DB each half a side, while CF and 
 Fare radii of the inscribed circle, and BF and DF are 
 radii of the circumscribed circle. 
 
450 POLYGONS. 
 
 Let R represent the latter, and r the former ; also let b 
 represent ED, one of the sides, and n be put for A D, and 
 for A E. Then we have 
 
 b_ 
 
 2~ T * 
 b 
 
 or- n = r--> 
 
 Since A D Eis a right-angled triangle (Art. 416), we have 
 
 T = ED\ 
 
 n* = b\ 
 b* 
 
 Placing the value of n, equal to the value before found, 
 we have 
 
 b b 
 
 f i i\ 
 -=f - + - ) 
 
 2 V 1/2 2 / 
 
 This coefficient may be reduced by multiplying the first 
 fraction by ^2, thus 
 
 JL x t5 = .^i 
 
 V2 X 2 X> 
 
RULES FOR OCTAGON. 451 
 
 therefore 
 
 r = 
 
 Or : The radius of the inscribed circle of a regular octagon 
 equals half a side of the octagon multiplied by the sum of unity 
 plus the square root of 2. In regard to the radius of the cir- 
 cumscribed circle, by Art. 416 we have 
 
 In this expression substituting for r a , its value as above, we 
 have 
 
 The square of the coefficient ( t/2 + i ) by Art. 412 equals 
 21/2+1 =21/2 + 3, then 
 
 Or : The radius of the circumscribed circle of a regular octagon 
 equals half a side of the octagon multiplied by the square root of 
 the sum of twice the square root of 2 plus 4. 
 
 In regard to the area of the octagon, the figure shows 
 that one eighth of it is contained in the triangle D E F. 
 
45 2 POLYGONS. 
 
 The area of D E F, putting it equal to N, is 
 
 B F 
 N = EDx -, 
 
 N = b x , 
 
 AT = ( 1/2" + i) . 
 4 
 
 This is the area of one eighth of the octagon ; the whole 
 area, therefore, is 
 
 . 
 
 4 
 A = (V~2+i)2b\ (142.) 
 
 Or : The area of a regular octagon equals twice the square of a 
 side, multiplied by the sum of the square root of 2 added to unity. 
 When a side of the enclosing square, or diameter of the 
 inscribed circle, is given, a side of the octagon may be found ; 
 for from equation (140.), multiplying by two, we have 
 
 2 r - ( V~2 + i) b. 
 Dividing by V.2 + i, gives 
 
 The numerator, 2 r, equals the diameter of the inscribed 
 circle, or a side of the enclosing square ; therefore : 
 
 The side of a regular octagon, equals a side of the enclosing 
 square divided by the sum of the square root of 2 added to unity. 
 
 437. Dodecagon : Radius of Circumscribed and In- 
 scribed Circles: Area. Let A B C (Fig. 291) be an equilat- 
 
SIDE AND AREA OF DODECAGON. 453. 
 
 era! triangle. Bisect A B in F\ draw C FD ; with radius A C 
 describe the arc A D B. Join A and D, also D and B ; bisect 
 A D in E ; with the radius E C describe the arc E G. Then 
 A D and D B are sides of a regular dodecagon, or twelve- 
 sided polygon ; of which A C, D C, and B C are radii of the 
 circumscribing circle, while E C is a radius of the inscribed 
 circle. 
 
 The line A B is the side of a regular hexagon (Art. 435). 
 Putting R equal to A C the radius of the circumscribing cir- 
 cle ; r, = E C, the radius of the inscribed circle ; , = A D, a 
 side of the dodecagon, and n D F. Then comparing the 
 
 FIG. 291. 
 
 homologous triangles, ADF and A EC (the angle ADF 
 equals the angle EA C, and the angles DFA and A E C are 
 right angles); therefore, the two remaining angles DAF 
 and A CE must be equal, and the two triangles homologous 
 (Art. 345). Thus we have 
 
 DF : DA : : AE : A C, 
 n : b : : : R, 
 
 **& 
 
454 POLYGONS. 
 
 In Art. 435 it was shown that FC (Fig. 291), or G H oi 
 Fig. 289, the radius of the inscribed hexagon, equals V~l ~, 
 
 n 
 
 and in which its b = R ; Fc VJ . 
 
 Now ( Fig. 291) 
 
 = DC - FC, 
 
 or 
 
 n = R-^~ 
 
 Substituting this value of n, in the above expression, we 
 have 
 
 R- 
 ~ 
 
 Multiplying by R and reducing, we have 
 
 R = |_ ^_ b. (144.) 
 
 Or : The radius of the circumscribed circle of a regular dodec- 
 agon, equals . a side of the dodecagon multiplied by the square 
 root of a fraction, having unity for its numerator and for its 
 denominator 2 minus the square root of 3. 
 
 Comparing the same triangles, as above, we have 
 
 FD \ FA : : EA ; EC, 
 or 
 
 R 
 
 . b 
 
 n : 
 
 2 
 
 : : - : r, 
 
 Rb 
 
 Rb 
 
 ~ 4 n ~ * 
 
 \R(\ -i|/^ 
 
 
 b 
 
 ('450 
 
RULE FOR DODECAGON. 455 
 
 Or : The radius of the inscribed circle of a regular dodecagon 
 equals a side of the dodecagon divided by the difference between 
 4 and the square root of 3. 
 
 The area of a dodecagon is equal to twelve times the area 
 of the triangle ADC (Fig. 291). The area of this triangle is 
 equal to half the base by its perpendicular ; or, A E x E C ; 
 or 
 
 b 
 
 or, where N equals the area 
 
 Or, for the area of the whole dodecagon 
 
 12 N 6 br, 
 A =6br. 
 
 Substituting for r its value as above, we have 
 
 Or : The area of a regular dodecagon equals the square of a 
 side of the dodecagon, multiplied by a fraction having 6 for its 
 numerator, and for its denominator, 4 minus twice the square 
 root of 3. 
 
 438. Hecadecagon : Radius of Circumscribed and Iii- 
 cribed Circle : Area. Let A B CD (Fig. 292) be a square 
 enclosing a quarter of a regular octagon C EFB, E F being 
 one of its sides, and C E and FB each half a side, while F D 
 is the radius of the circumscribed circle, and J D the radius 
 of the inscribed circle of the octagon. Draw the diagonal 
 A D ; with DFior radius, describe the circumscribed circle 
 EGF\ join G with F and with E ; then EG and GFvfill 
 each be a side of a regular hecadecagon, or polygon of six- 
 teen sides. 
 
 An expression for F D, the radius of the circumscribed 
 
456 
 
 POLYGONS. 
 
 circle, may be obtained thus: Putting FD = R] H D = r; 
 
 G F = b\ GJ n\ and J F - (Art. 416), we have 
 
 GT = GF* JF\ 
 
 = - (9- 
 
 C D 
 
 FIG. 292. 
 
 Comparing the two homologous (Art. 361) triangles, GJF 
 and F H D (Art. 374), we have 
 
 Gy : GF : : HF : FD, 
 n, b :: I : *, 
 
 Putting this value of n' in an equation against the former 
 value, we have 
 
 In Art. 436, the value of F D, as the radius of the cir- 
 cumscribed circle of a regular octagon, is given in equation 
 (141.) as 
 
 b 
 
 R 
 
 V2 
 
SIDE AND AREA OF HECADECAGON. 457 
 
 in which b represents a side of the octagon, or E F t for 
 which we have put s. Substituting s for b and putting the 
 numerical coefficient under the radical, equal to B, we 
 have 
 
 Squaring each member gives 
 
 From which, by transposition, we have 
 
 2 
 
 Substituting in the above expression for ( ) , this value 
 
 \2 / 
 
 of it, gives 
 
 ^- = t>>-*\ 
 4^ 2 ~ B . 
 
 Transposing, we have 
 
 -*1 + *'=*-. 
 
 4R* B 
 
 Multiplying the first term by B, and the second by 
 we have 
 
 * 
 
 Bb* + 4^ 4 _ ^ 
 
 Transposing, we have 
 
 2 = -Bb\ 
 
458 POLYGONS. 
 
 To complete the square (Art. 428) we proceed thus 
 
 Taking the square root, we have 
 
 Restoring B to its value, 2 I/I + 4 as above, we have 
 
 B I = 2^2 
 
 multiply these 
 
 2 + 2 i/J, 
 
 3 + 
 
 = 4/2"+ 2. 
 Therefore 
 
 2. (147.) 
 
RULES FOR HECADECAGON. 459 
 
 Or: The radius of the circumscribed circle of a regular 
 hecadecagon equals a side of the hecadecagon multiplied by the 
 square root of the sum of two quantities, one of which is the 
 square root of 2 added to 2, and the other is the square root of 
 the sum of seven halves of the square root of 2 added to 5. 
 
 To obtain the radius of the inscribed circle we have (Fig. 
 292) 
 
 H D* = FD* H F\ 
 
 Substituting for R a its value as above, we have 
 
 r a = &* ( B(B} + * B) - , 
 
 The coefficient of b is the same as in the case above, ex- 
 cept the i; therefore its numericaK value will be i less, 
 or 
 
 r = b \/ ^S + I V2 + V2 + if. (148). 
 
 Or: The radius of the inscribed circle of a regular hecadeca- 
 gon 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 
 if, and the other is the square root of the sum of seven halves of 
 the square root of 2 added to 5. 
 
 To obtain the area of the hecadecagon it will be observed 
 that the area of the triangle G FD (Fig. 292) equals HD x 
 H F, and that this is the T V part of the polygon ; we there- 
 fore have 
 
 A = \6HDxHF, 
 A = i6r- = Sr&. 
 
 2 
 
460 POLYGONS. 
 
 The value of r is shown in (148.); therefore we have 
 
 A = 8 b 
 
 + 1 . 
 
 + if. 
 
 (H9-) 
 
 Or : The area of a regular hecadecagon equals eight times 
 the square of its side, multiplied by the square root of two qtian- 
 tities, one of which is the square root of 2 added to if, and the 
 other is the square root of the sum of seven halves of the square 
 root of 2 added to 5. 
 
 439. Polygon* : Radius of Circumscribed and I u scribed 
 Circles : Area. In Arts. 433 to 438 the relation of the radii 
 to a side in a trigon, tetragon, hexagon, octagon, dodeca- 
 gon and hecadecagon have been shown by methods based 
 
 upon geometrical proportions. This relation in polygons 
 of seven, nine, ten, eleven, thirteen, fourteen and fifteen 
 sides, cannot be so readily shown by geometry, but can be 
 easily obtained by trigonometry as also said relation of the 
 parts in a regular polygon of any number of sides. The na- 
 ture of trigonometrical tables is discussed in Arts. 473 and 
 474. So much as is required for the present purpose will 
 here be stated. 
 
 Let ABC (Fig. 293) represent one of the triangles into 
 which any polygon may be divided, in which B C = b = a 
 side of the polygon ; A C R = the radius of the circum- 
 scribed circle ; and A D = r = the radius of the inscribed 
 circle. 
 
GENERAL RULES FOR POLYGONS. 461 
 
 Make E C equal unity ; on C as a centre describe the arc 
 E F\ draw FH and E G perpendicular to B C, or parallel to 
 A D ; then for the uses of trigonometry E G is called the 
 tangent of c, or of the angle A C B, and FHis the sine, and 
 H C the cosine of the same angle. 
 
 These trigonometrical quantities for angles varying from 
 zero up to ninety degrees have been computed and are to be 
 found in trigonometrical tables. 
 
 Referring now to Fig. 293 we have 
 
 HC : FC : : DC : AC, 
 
 b 
 cos. c : I : : : R, 
 
 (150.) 
 
 Again 
 
 E C : E G : : D C : A D, 
 
 b 
 I : tan. c : : - : r, 
 
 r = - tan. c. (151.) 
 
 These two equations give the required radii of the cir- 
 cumscribed and inscribed circles. They may be stated thus : 
 
 The radius of the circumscribed circle of any regular poly- 
 gon equals a side of the polygon divided by twice the cosine of 
 the angle formed by a side of the polygon and a radius from one 
 end of the side. 
 
 The radius of the inscribed circle of any regular polygon 
 equals half of a side of the polygon imiltiplied by the tangent of 
 the angle formed by a side of the polygon and a radius from one 
 end of the side. 
 
 The area of a polygon equals the area of the triangle 
 ABC (Fig. 293), (of which B C is one side of the polygon 
 and A is the centre), multiplied by the number of sides in 
 the polygon ; or, if n be put to represent the number of the 
 sides and A the area, then we have 
 
POLYGONS. 
 
 A = Bn, 
 
 in which B equals the area of the triangle. The area of A 
 B C (Fig. 293) is equal to AD x B D, or 
 
 For r substituting its value, as in equation (151.), we have 
 
 b b i 7 
 
 B = - tan. c- = b" tan. c. 
 2 24 
 
 Therefore, by substitution 
 
 A =-b*n tan.*. (152.) 
 
 Or : The area of a regular polygon equals the square of a 
 side of the polygon, multiplied by one fourth of the number of 
 its sides, and by the tangent of the angle formed by a side of the 
 polygon, and a radius from one end of the sides. 
 
 440. Polygons : Their Angles. Let a line be drawn 
 from each angle of a regular polygon to its centre, then 
 these lines form with each other angles at the centre, which 
 taken together amount to four right angles, or to 360 de- 
 grees (Arts. 327, 335). 
 
 If this 360 degrees be divided by the number of the sides 
 of the polygon, the quotient will equal the angle at the cen- 
 tre of the polygon, of each triangle formed by a side and two 
 radii drawn from the ends of the side. For example: if 
 ABC (Fig. 293) be one of the triangles referred to, having 
 B C one of the sides of the polygon and the point A the cen- 
 tre of the polygon, then the angle B A (Twill be equal to 360 
 degrees divided by the number of the sides of the polygon. 
 If the polygon has six sides, then the angle B A C will contain 
 
 = 60 degrees ; or if there be 10 sides, then the angle at 
 A, the centre, will contain - ---- = 36 degrees. The angle 
 
SIDE AND AREA OF PENTAGON. 463 
 
 BAD equals half the angle B A C, or, when n equals the 
 number of sides, the angle BAG equals 
 
 360 
 
 
 
 n 
 
 B A C 
 and the triangle B A D = , equals 
 
 360 
 
 2 n 
 
 Now the angles B A D + D B A equal one right angle 
 (Art. 346), or 90 degrees. Hence the angle DBA =90 - 
 BAD,or the angle c equals 
 
 (1530 
 
 2 n 
 
 For example, if n equal 6, or the polygon have six sides, 
 then 
 
 Therefore, the angle c, contained in equations (150.), (151.)* 
 and (152.), equals 90 degrees, less the quotient derived from a di- 
 vision of 360 by twice the number of sides to the polygon. 
 
 441. Pentagon: Radius of the Circumscribed and In- 
 scribed Circles: Area. The rules for polygons developed 
 in the two former articles will here be exemplified in their 
 application to the case of a regular pentagon, or polygon of 
 five sides. 
 
 To obtain the angle c (153.), we have n = 5, and 
 
 ,>= 90- 3g = go- 3 6 = 54. 
 
 For the radius of the circumscribed circle, we have 
 
 (150.)- 
 
 2 COS. C 
 
464 POLYGONS. 
 
 b 
 
 2 cos. 54 C 
 
 i 
 2 cos. 54 
 
 Using a table of logarithmic sines and tangents (Art. 427), 
 we have 
 
 Log. 2 =0-3010300 
 Cos. 54 = 9-7692187 
 
 Their sum = 0-0702487 subtracted from 
 Log. i = o-ooooooo 
 
 0-85065 =9-9297513 
 
 Therefore 
 
 , = 0-85065 . 
 
 Or : The radius of the circumscribed circle of a regular/^ta- 
 gon- equals a side of the pentagon multiplied by the decimal o 8 5065 . 
 
 For the radius of the inscribed circle, we have (151.) 
 
 = - tan. c t 
 
 .tan. 54 
 r = o 
 
 For this we have 
 
 Log. tan. 54 = 0-1387390 
 Log. 2 = 0-3010300 
 
 0-68819 = 9-8377090. 
 Therefore 
 
 r = 0-68819 b. 
 
 Or: The ra&usofthe inscribed circle of a regular pentagon 
 equals a side of the pentagon multiplied by the decimal 0-68819. 
 For the area we have (152.) 
 
 A =%fr*n tan. c, 
 A = J x 5 tan. 54 b\ 
 A =ftan. 54 b\ 
 
TABLE FOR REGULAR POLYGONS. 465 
 
 For this we have 
 
 Log. 5. = 0-6989700 
 Log. tan. 54 =_- o- 1387390 
 
 0-8377090 
 Log. 4 = 0-6020600 
 
 1-72048 0-2356490 
 Therefore 
 
 A = i- 72048 b \ 
 
 Or: The area of a regular pentagon equals the square of its 
 side multiplied by I 72048. 
 
 1-42. Polygons Table of ontant multipliers. To 
 
 obtain expressions for the radii of the circumscribed and in- 
 scribed circles, and for the area for polygons of 7, 9, 10, n, 
 13, 14, and 15 sides/a process would be needed such pre- 
 cisely as that just shown in the last article for a pentagon, 
 except in the value of n and c, which are the only factors 
 which require change for each individual case. 
 
 No useful purpose, therefore, can be subserved by ex- 
 hibiting the details of the process required for these several 
 polygons. The values of the constants required for the 
 radii and for the areas of these polygons have been com- 
 puted, and the results, together with those for the polygons 
 treated in former articles, gathered in the annexed Table of 
 Regular Polygons. 
 
 REGULAR POLYGONS. 
 
 SIDES. 
 
 R 
 
 b . 
 
 r 
 1> = 
 
 A 
 b~*~ 
 
 o Trie-on 
 
 ^77^ 
 
 28868 
 
 4TJOI 
 
 A Tetragon 
 
 7O7II 
 
 5OOOO 
 
 I -OOOOO 
 
 5 Pentagon 
 
 .8co6t; 
 
 68819 
 
 I 72O48 
 
 
 ooooo 
 
 . 86603 
 
 2. cr\So8 
 
 
 152-18 
 
 I 03826 
 
 3-6T3QI 
 
 8 Octagon 
 
 * 30656 
 
 I 2O7 I I 
 
 4- 8284^ 
 
 
 4.6 1 QO 
 
 I 37'374 
 
 6- 18182 
 
 10 Decagon ... . 
 
 61803 
 
 i -^884 
 
 
 II. Undecagon 
 
 7747^ 
 
 i 70284 
 
 <J 94^ J - 
 
 9. a6c6d. 
 
 12 Dodecagon 
 
 . Q^lS^ 
 
 i 86603 
 
 
 13. Tredecagon 
 
 2 08020 
 
 2-02858 
 
 11 ' I 8^77 
 
 14 Xetradecagon ... . 
 
 2 24608 
 
 2 I 0064 
 
 
 15. Pentadecagon 
 
 2 4OJ.8 7 
 
 2 ^^2^1 
 
 X D JJ45 1 
 17 6j.2^6 
 
 16 Hecadecagon 
 
 2 56202 
 
 2. e T'l67 
 
 
 
 
 
 
466 POLYGONS. 
 
 In this table R represents the radius of the circumscribed 
 circle ; r the radius of the inscribed circle ; b one of the 
 sides, and A the area of the polygon. By the aid of the 
 constants of this table, R, the radius of the circumscribed 
 circle of any of the polygons named, may be found when a 
 side of the polygon is given. For this purpose, putting m 
 for any constant of the table, we have 
 
 R = bm. ( 1 S4-) 
 
 As an example : let it be required to find R, for a penta- 
 gon having each side equal to 5 feet ; then the above expres- 
 sion becomes 
 
 R 5 x 0-85065, 
 R = 4-25325- 
 
 The radius will be 4 feet 3 inches and a small fraction. 
 In like manner the radius of the inscribed circle will be 
 
 r = bm; (1550 
 
 and for a pentagon with sides of 5 feet, we have 
 
 r = 5 x 0-68819, 
 r 3-44095- 
 
 Or, the radius of the inscribed circle will be 3 ft. -j^ and a 
 small fraction. Or, multiplying the decimal by 12, 3 ft. 5 in. 
 -f^Q and a small fraction. 
 
 The area of any polygon of the table may be obtained 
 by this expression 
 
 A^b*m; (156.) 
 
 and, applying this to the pentagon as before, we have 
 
 A 5 2 x i 72048, 
 A 43-012. 
 
EXPLANATION OF THE TABLE. 467 
 
 Or, the area of a pentagon having its sides equal to 5 feet, 
 is 43 feet and T -^f 7 of a foot. 
 
 By the constants of the table a side of any of its poly- 
 gons may be found, when either of the radii, or the area, 
 are known. 
 
 When R is known, we have 
 
 When r is known, we have- 
 
 When the area is known, we have 
 
SECTION XL 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 triangle (Art. 352). 
 Draw the line CD perpendicular to the diameter AB\ 
 then C D w\\\ divide the triangle A B C into two triangles, 
 A CD and C B D, which are homologous. For, let the 
 triangle C B D be revolved on D as a centre until its line 
 CD shall come to the position E D, and the line DB oc- 
 cupy the position D F, each in a position at right angles 
 
 to its former position, the point B describing the curve 
 B F, and the point C the curve C E, and each forming a 
 quadrant or angle of ninety degrees. Since these points 
 have revolved ninety degrees, therefore the three lines of 
 the triangle CBDhave revolved into a position at right 
 angles to that which they before occupied ; hence the line 
 EFis at right angles to CB } and (from the fact that A C B is 
 a right angle) parallel with A C, Since the triangle EFD 
 equals the triangle C B D. and since the lines of E FD are 
 parallel respectively to the corresponding lines of A CD, 
 therefore the triangles^ CD and C B D are homologous. 
 
 Comparing the lines of these triangles and putting a = 
 A B, y = C D, and x D B, we have 
 
RADIUS FROM CHORD AND VERSED SINE. 469 
 
 DB : D C : : D C : A D, 
 
 x : y : : y : a x, 
 y* = x (a x], (160.) 
 
 Or, in a semicircle, a perpendicular to the diameter terminated 
 by the diameter and the curve is a geometric mean, or mean 
 proportional, between the two parts into which the perpendicular 
 divides the diameter. 
 
 444. Circle : Radius from Given Chord and Versed 
 Sine. Let A B (Fig. 295) be a given chord line and CD a 
 versed sine. Extend CD to the opposite side of the circle ; 
 it will pass through F, the centre. Join A and C, also E and 
 
 B 
 
 B. The line A D, perpendicular to the diameter C E, is 
 a mean proportional between the two parts CD and DE 
 (Art. 443) ; or, putting a = A D, b = C D, and r equal the 
 radius FE, we have 
 
 C D \ AD \\ AD \ DE\ 
 
 b : a : : a : 2 r b, 
 
 -2 rb-b*, 
 + l>* = 2rb, 
 
 r - 
 
 (161.) 
 
470 
 
 THE CIRCLE. 
 
 Or : The radius of a circle equals the sum of the squares of half 
 the chord and the versed sine, divided by twice the 'versed sine. 
 
 Another expression for the radius may be obtained ; for 
 the two triangles C B D and C E B (Fig. 295) are homologous 
 (Art. 443) and their corresponding lines in proportion. Put- 
 ting/for CB, we have 
 
 or 
 or 
 and 
 
 CD : CB : : CB : C E, 
 
 v :/::/: 2 r, 
 
 f ' 2 rv, 
 
 r = 
 
 2V 
 
 (162.) 
 
 Or : The radius of a circle equals the square of the chord of half 
 the arc divided by twice the versed sine. 
 
 4-45. Circle: Segment from Ordinate. When the curve 
 of a segment of a circle is required for which the radius can- 
 not be used, either by reason of its extreme length, or be- 
 
 FIG. 296. 
 
 cause the centre of the circle is inaccessible, it is desirable 
 to obtain the curve without the use of the radius. This may 
 be done by calculating ordinates, a rule for which will now 
 be developed. 
 
 Let DC B (Fig. 296) be a right angle, and A DB a cir- 
 cular arc described from C as a centre, with the radius 
 B C= CD = CP. Draw PM parallel with DC, and A G 
 parallel with C B. Now, in the segment A D G, we have 
 given A G, its chord, and D E, its versed sine, and it is re- 
 
RULE FOR ORDINATES. 471 
 
 quired to find an expression by which its ordinates, as P F, 
 may be computed. From Art. 416, we have 
 
 PM*=CP*-CJf*<, 
 or, putting for these lines their usual symbols 
 
 now we have 
 
 EC= FM, 
 FM=DCDE, 
 
 FM =r-b. 
 Then we have 
 
 or, putting t for P F and substituting for PM and FM their 
 values as above, we have 
 
 t = y-(r-b\ 
 and for y, substituting its value as above, we have 
 
 /r* -x* -(r-b). (163.) 
 
 Or: The ordinate in the segment equals the square root of the 
 difference of the squares of the radius and the abscissa minus 
 the difference of the radius and the versed sine. 
 
 For example : let the chord A G (Fig. 296) in a given case 
 equal 20 feet, and the versed sine, b, or the rise D E, equal 4 
 feet ; and let the ordinates be located at every 2 feet along 
 the chord line, A G. 
 
 In solving this problem we require 'first to find the radius. 
 This is obtained by means of equation 
 
 2b 
 
4/2 THE CIRCLE. 
 
 For a, half the chord, we have 10 feet ; for b, the versed 
 sine, we have 4 feet ; and, substituting these values, we 
 have 
 
 The radius equals H'5 
 
 The versed sine equals 4-0 
 
 (r-b}= 10-5 
 
 The square of 14-5, the radius, equals 210-25. Now we 
 have, substituting these values in equation (163.) 
 
 -~- 1210-25 x^ 10-5. 
 
 The respective values of x, as above required, are o, 2, 
 4, 6, 8 and 10 Substituting successively for x one of these 
 values, we shall have, when 
 
 x o; t -- y 210-25 o 2 10-5 = 4. 
 
 x - 2 ; /-= |/ 210-25 2 2 10- 5 = 3-8614 
 
 4; '=- V 210-2$ 4*-- 10-5 = 3-4374 
 
 # = 6; / = |/ 210-25 6 2 10-5 =: 2-7004 
 
 * = 8 ; / = 4/ 210-25 8 2 10- 5 = i 5934 
 
 r I0 ' * - : 1/210-25 io 2 -- 10-5 = o-o 
 
 Values for / may be taken at points as numerous as desira- 
 ble for accuracy. 
 
 In ordinary cases, however, they need not be nearer than 
 in this example. 
 
 After the points are secured, let a flexible piece of wood 
 be bent so as to coincide with at least four of the points at a 
 time, and then draw the curve against the strip. 
 
 446. Circle : Relation of Diameter to Circumference. 
 
 In Art. 439 it is* shown that the area of a polygon equals 
 the radius of the inscribed circle multiplied by half of a 
 side of the polygon and by the number of the sides ; or, 
 
TO FIND THE CIRCUMFERENCE. 473 
 
 A = r x n = b n ; or, the area equals half the radius by a 
 2 2 
 
 side into the number of sides ; or, half the radius into the 
 periphery of the polygon. Now, if a polygon have very 
 small sides and many of them, its periphery will approxi- 
 mate the circumference of the circle inscribed within it ; in- 
 deed when the number of sides becomes infinite, and conse- 
 quently infinitely small, the periphery and circumference 
 become equal. Consequently, for the area of the circle, we 
 have 
 
 A = r --c, (164.) 
 
 where c represents the circumference. 
 
 By computing the area of a polygon inscribed within a 
 given circle, and that of one circumscribed about the circle, 
 the area of one will approximate the area of the other in 
 proportion as the number of the sides of the polygon are 
 increased. 
 
 For example : if polygons of 4 sides be inscribed within 
 and circumscribed about a circle, the radius of which is I, 
 the areas will be respectively 2 and 4. If the polygons have 
 1 6 sides, the areas are each 3 and a fraction, the fractions 
 being unlike; when they have 128 sides the areas are each 
 3 14 and with unlike fractions ; when the sides are increased 
 to 2048, the areas each equal 3-1415 and unlike fractions, 
 and when the sides reach 32768 in number the areas are 
 equal each to 3-1415926, having like decimals to seven 
 places. The computations have been continued to 127 
 places (Gregory's " Math, for Practical Men "), but for all 
 possible uses in building operations seven places will be found 
 to be sufficient. From this result we have the diameter in 
 proportion to the circumference as i : 3- 1415926, or as 
 
 I : 3 
 i : 3 
 1:3- 1416. 
 
 Of these proportions, that one may be used which will give 
 
474 THE CIRCLE. 
 
 a result most nearly approximating the degree of accuracy 
 required. For many purposes the last proportion will be 
 sufficiently near the truth. 
 
 For ordinary purposes the proportion 7 : 22 is very use- 
 ful, and is correct for two places of decimals; it fails in the 
 third place. 
 
 The proportion 113 : 355 is correct to six places of deci- 
 mals. 
 
 For the quantity 3-1415926 putting the Greek letter n 
 (called py\ and 2 r = d for the diameter, we have 
 
 c n d. (165.) 
 
 To apply this : in a circle of 50 feet diameter, what is 
 the circumference ? 
 
 c = 3-1416 x 50 
 c = 1 57-08 ft. 
 
 If the more accurate value of n be used, we have 
 
 c = 3-1415926 x 50, 
 c = i 57- 07963. 
 
 The difference between the two results is 0-00037, which 
 for all ordinary purposes, would be inappreciable. 
 By the rule of 7 : 22, we have 
 
 c = 5ox- 3 T 2 - _ 157.1428571, 
 
 an excess over the more accurate result above, of 0-0632271, 
 which is about of an inch. 
 
 Bv the rule of 113 : 355, we have 
 
 c = 50 x fff = 157-079646. 
 
 This result gives an excess of only 0-000016; it is sufficiently 
 near for any use required in building. 
 
 From these results we have these rules, namely : To 
 obtain the circumference of a circle, multiply its diameter by 
 
TO FIND THE AREA. 475 
 
 22, and divide the product by 7 ; or, more accurately, multiply 
 the diameter ^355 and divide the product by 113; or, by mul- 
 tiplication only, m ultiply the diameter by 3-1416; or, by 
 3-14159^; or, by 3-1415926; according to the degree of 
 accuracy required. 
 
 And conversely: To obtain the diameter from the cir- 
 cumference, multiply the circumference by 7 and divide the 
 product by 22 ; or, multiply by 113 and divide by 355 ; or, di- 
 vide the circumference by 3-1416; or, by 3-14159^; or, by 
 3-1415926. 
 
 4-47. Circle : Length of an Arc. Considering the cir- 
 cle divided into 360, the length of an arc of one degree in 
 a circle the diameter of which is unity may be thus found. 
 
 The circumference for 360 is 3- 14159265 ; 
 
 3. 14159265 = . oo8726fi4625; . 
 
 which equals an arc of one degree in a circle having unity 
 as its diameter; or, for ordinary use the decimal 0-008727 
 or 0-0087^ may be taken ; or putting a for the arc and g for 
 the number of degrees, we have 
 
 a = 0-00872665 dg. (166.) 
 
 Wherefore : To obtain the length of an arc of a circle, 
 
 multiply the diameter of the circle by the number of degrees in 
 the arc, and by the decimal 0-0087^, or, instead thereof, by 
 0-008727. 
 
 4.43. Circle: Area. The area of a circle may be ob- 
 tained in a manner similar to that for the area of polygons 
 
 (Art. 439), in which ABn\ B r , or 
 
 A = % b n r, 
 
 where b equals a side of the polygon and n the number of 
 sides ; so that b n equals the perimeter of the polygon. 
 
 Now, if for the perimeter of the polygon there be sub- 
 
476 THE CIRCLE. 
 
 stituted the circumference of the circle, we shall have, put- 
 ting for the circumference 3- 1416 d y or, n d (Art. 446) 
 
 A = \n dr, 
 
 in which r is the radius. Since 2 r d, the diameter, and 
 r = -, we have 
 
 d 
 
 And since 
 
 ^ = 3.14159265, 
 
 \7t = 0-78539816, 
 or 
 
 \ n = 0-7854, nearly. 
 Therefore 
 
 (167.) 
 
 Or: The area of a circle equals the square of the diameter mul- 
 tiplied by 0-7854. 
 
 B 
 
 As an example, the area of a circle 10 feet in diameter is 
 found thus 
 
 IOX IO = IOO. 
 
 100x0-7854 = 78 -54 feet. 
 
 449. Circle: Area of a Sector. The area of A B CD 
 (Fig. 297), a sector of a circle, is proportionate to that of the 
 whole circle. For, as the circumference of the whole circle 
 is to its area, so is the arc A B C to the area of A B C D. 
 
AREA OF SECTOR. 477 
 
 The circumference of a circle is (165.) C= ir d. The area 
 of a circle is (167.) A = -7854 d*. For the arc ABC put a, 
 and for the area of A B CD put s. Then we have from the 
 above-named proportion 
 
 7t d : 
 
 _ 
 
 J - 
 
 * 
 
 Tt d 
 
 The coefficient 0-7854 is J - (^4rA 448). 
 
 4 
 
 Therefore, multiplying the fraction by 4, we have 
 5- * d \ 
 
 O - 7 ft 
 
 4 TTdf 
 
 or S = \da = \ra. (168.) 
 
 Wherefore : To obtain the #raz of a sector of a circle, 
 multiply a quarter of the diameter by the length of the arc. 
 
 Thus: let A D equal 10; also let A B C = a, equal 12. 
 Then the area of A CD is 
 
 S =%x lox 12, 
 
 S = 6o. 
 The length of the arc may be had by the rule in Art. 447. 
 
 450. Circle: Area of a Segment. In the last article, 
 A BCD (Fig. 297) is called the sector of a circle. Of this 
 the portion included within A E C B is a segment of a circle. 
 The area of this equals the area of the sector minus the area 
 of the triangle A D C ; or, putting M for the area of the seg- 
 ment, S for the area of the sector, and T for the area of the 
 triangle, then 
 
 M=S- T. 
 
 Putting c for A C (Fig. 297) and h for D E, then T = ~ h. 
 In the last article, s ra, in which a = the length of the 
 
478 
 
 THE CIRCLE. 
 
 arc ABC. 
 have 
 
 Substituting this value of s in the above, we 
 
 ar 
 
 Or : When the length of the arc is known, also that of the 
 chord and the perpendicular from the centre of the circle, 
 then the area of the segment equals the difference between the 
 product of half the arc into the radius, aud half the chord into 
 its perpendicular to the centre of the circle. 
 
 But ordinarily the length of the arc and of the chord are 
 unknown. If in this case the number of degrees contained 
 between the two radii, DA,DC>wz known, then the area of 
 
 the segment may be found by a rule which will now be de- 
 veloped. 
 
 In Fig. 298 (a repetition of Fig. 297) upon D as a centre, 
 and with D F = unity for a radius, describe the arc H F. 
 Then GFis the sine of the angle C D B, and D G is the co- 
 sine ; and we have 
 
 or 
 
 Again 
 
 or 
 
 DF : GF : : DC : EC, 
 
 I : sin : : r : - = r sin. 
 
 DF : DG : : DC : D E y 
 i : cos : : r : // = r cos. 
 
RULE FOR AREA OF SEGMENT. 479 
 
 By equation (166.) we have 
 
 a = 0-00872665 dg, 
 
 in which a is the length of the arc ; g the number of degrees 
 contained in the arc ; and d is the diameter of the circle. 
 Since d = 2 r, therefore 
 
 a = 0-0174533 rg. 
 
 Putting B for the decimal coefficient, we have 
 
 a = Br g. 
 
 The expression (169.), by substitution of values as above, 
 becomes 
 
 a c 
 M = -r h. 
 
 2 2 
 
 B rg 
 M = r r sin. x r cos. 
 
 M \ B gr* sin. cos. r 2 
 
 M = r* (%*Bg sin. cos.) 
 
 M r * (o 00872665 g sin. cos.) ( 1 70.) 
 
 Or : The area of a segment of a circle equals the square of the 
 radius into the difference between 0-00872665 times the number 
 of degrees contained in the arc of the circle, and the product of 
 the sine and cosine of half the arc. 
 
 When the number of degrees subtended by the arc is 
 unknown, or tables of sines and cosines are not accessible, 
 then the area may be obtained by equation (169.), provided 
 the chord and versed sine are known ; but before this equa- 
 tion can be used, for this purpose, expressions giving their 
 values in terms of the chord and versed sine must be ob- 
 tained, for a, the arc, r, the radius, and h, the perpendicular 
 to the chord from the centre of the circle. 
 
 For the value of the arc we have (from " Penny Cycl.," 
 Art. Segment] as a close approximation 
 
480 THE CIRCLE. 
 
 By equation (162.) we have 
 
 = 2^' 
 
 Then 
 
 h = r v, 
 or 
 
 h = f--v. 
 
 2 V 
 
 Substituting these values in equation (169.) we have 
 
 This rule is the rule (169.) expanded. 
 
 The written rule for equation (169.) may be used, substi- 
 tuting for " half the arc" one sixth of the difference between 
 eight times the chord of half the arc and the chord (or \ of 8 
 times A >, Fig. 298, minus A C, the chord). Also substitute 
 for " the radius" the square of the chord of half the arc divided 
 by twice the versed sine. Also, tor. "its perpendicular to the 
 centre of the circle" substitute, the quotient of the square of the 
 chord of half the arc divided by twice the versed sine, minus 
 the versed sine. 
 
 When the arc is small the curve approximates that of a 
 parabola. In this case the equation for the area of the par- 
 abola, which is quite simple, may be used. It is this 
 
 Or, in segments of circles where the versed sine is small in 
 comparison with the chord, the area equals approximately two 
 thirds of the chord into the versed sine. 
 
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'y and let the point P move in such a manner that the sum 
 of the two lines, PF, PF', shall remain a constant quantity ; 
 then the curve P M KO G A D B P, traced by P, will be an 
 Ellipse ; the two fixed points F, F' , the Foci ; the point C at 
 
 FIG. 299. 
 
 the middle of FF', the centre ; the line A M drawn through 
 F F' and terminated by the curve, the Major or Transverse 
 Axis ; the line B O, drawn through C and at right angles to 
 A M, the Minor or Conjugate Axis; the line G P, drawn 
 through Pand C and terminated by the curve, the Diameter 
 to the point P; the line D K drawn through C, parallel with 
 the tangent P T, and terminated by the curve, the diameter 
 Conjugate to P G\ the line EH R drawn parallel with D K 
 is a double ordinate to the abscissas G H "and H Poi the di- 
 ameter GP(EH = HR) ; the line JL drawn through Fat a 
 
482 
 
 THE ELLIPSE. 
 
 right angle to A M and terminated by the curve, the Param- 
 eter, or Latus Rectum. 
 
 When the point P reaches and coincides with B, the two 
 lines PFand PF' become equal. 
 
 The proportion between the major and minor axes de- 
 pends upon the relative position of F,F f , the foci ; the nearer 
 these are placed to the extremities of the major axis the 
 smaller will the minor axis be in comparison with the major 
 axis. The nearer F, F' approach C, the centre, the nearer 
 will the minor axis approach the length of the major axis. 
 When F, F' reach and coincide with the centre, the minor 
 axis will equal the major axis, and the ellipse will become a 
 
 circle. Then we have PF = PF' = B C= A C. From this 
 we \earnPF+PF'=2A C=AM- t also, when PF= PF', 
 thenPF=F=AC. 
 
 From this we may, with given major and minor axes, 
 find the position of F and F f . To do this, on B, as a centre, 
 with A C for radius, mark the major axis at F and F' . 
 
 452. Ellipe : Equations to the Curve. An equation to 
 a curve is an expression containing factors two of which, 
 called co-ordinates, measure the distance to any point in the 
 curve. For example : in a circle it has been shown (Art. 
 443) that P N is a mean proportional to A A^and N B. Or, 
 putting x A N, y = PN, and a A B, we have 
 
 AN : PN : : PN : NB y 
 
 or 
 or 
 
 x : y : : y : a x, 
 
 y a X (a X}. 
 
EQUATION TO THE ELLIPSE. 483 
 
 This is the equation to the circle having the origin of x 
 and y, the co-ordinates at A, the vertex of the curve. It will 
 be observed that the factors are of such nature in this equa- 
 tion, that it may be employed to measure the distance, rect- 
 angularly, to (*, wherever in the curve the point P may be 
 located. By this equation the rectangular distance to any 
 and every point in the curve may be measured ; or, having 
 the curve and one of the lines ;ror y, the other may be com- 
 puted. 
 
 From this example, the nature and utility of an equation 
 to any curve may be understood. The equation to the 
 ellipse having the origin of co-ordinates at the vertex, is 
 similar to that for the circle. In the form usually given by 
 writers on Conic Sections, it is 
 
 in which a 'A C (Fig. 299) ; b = B C\ x equals A N, and y = 
 PN. 
 
 If, as before suggested, the loci be drawn towards the cen- 
 tre and finally made to coincide with it, the minor axis would 
 then become equal to the major axis, changing the ellipse into 
 a circle. In this case, the factors a and b in the equation would 
 
 become equal; and the fraction 5- would equal , = i,and 
 
 a a 
 
 hence the equation would become 
 
 or 
 
 y a = x (2 a x) ; 
 
 precisely the same as in the equation to the circle above 
 shown. The 2 a of this equation is equivalent to a of the 
 circle ; for a in the ellipse represents only half the major 
 axis ; while in the equation to the circle a represents the 
 diameter. The relation between the ellipse and the circle 
 is thus shown ; indeed, the circle has been said to be an 
 ellipse in its extreme conditions. 
 
484 
 
 THE ELLIPSE. 
 
 453. Ellipse : Relation of Axi to Abscissas of Axe 
 
 Multiplying equation (173.) by a* we have 
 
 a* y* = b* (2 ax-x*\ 
 or # 2 j/ 2 = b*x(2 a x). ^ 
 
 These four factors may be put in a proportion, thus 
 
 rf a : b* : : x (2 a x) : y\ 
 
 representing 
 
 A~C* 
 
 7TC 
 
 NX NM : P N\ 
 
 Or : The rectangle of the two parts into which the ordinate 
 divides the axis major is in proportion to the square of the 
 ordtnate, as the square of the semi-axis major is to the square 
 of the semi-axis minor. 
 
 It is shown by writers on Conic Sections that this rela- 
 tion is found to subsist, not only with the axes and ordinate, 
 but also between an ordinate to any diameter and the ab- 
 scissas of that diameter ; for example, referring to Fig. 299 
 
 If A B' P'M (Fig. 301) be a semi-circle, then (Art. 443) 
 * = A 
 
 Substituting this value of A NX N Min 
 
 TC* : ~B~C* - v^W a : ~PN*> 
 we have 
 
 A C i BC : : P'N : PN\ 
 
RELATION OF TANGENT TO AXIS. 485 
 
 Or : The ordinate in the circle is in proportion to its correspond- 
 ing ordinate in the ellipse, as the semi-axis major is to the semi- 
 axis minor, or as the axis major is to the axis minor. 
 
 454. Ellipse : Relation of Parameter and Axe. The 
 
 equation to the ellipse when the origin of the co-ordinates 
 is at the centre is, as shown by writers on Conic Sections, 
 thus 
 
 a* y* = a *b*-b* x'\ (174.) 
 
 or a* y" 1 b* (a* x' 2 ). 
 
 If x' equal C-F (Fig. 299) then the ordinate will be located 
 
 ^- '^^ 
 
 01 THE ^f 
 
 .'UNIVERSITY 
 
 Then- 
 
 This is shown also by the figure. 
 
 Substituting in the above this value of a* x'*, we have 
 
 a*y* = &*&* = b\ 
 From which, taking the square root 
 
 ay = b\ 
 
 or a : b : : b : y. 
 
 Now y, located at FJ, is the semi-parameter; hence we 
 have the semi-minor axis a third proportional to the semi- 
 major axis and the semi-parameter. Or : The parameter is a 
 third proportional to the two axes of an ellipse. 
 
 455. Cllipe: Relation of Tangent to the Axes. Let 
 
 T T' (Fig. 301) be a tangent to P, a point in the ellipse ; then, 
 as has been shown by writers on Conic Sections 
 
 or 
 
 CM : CT :: CN : CM. 
 
486 THE ELLIPSE. 
 
 Or : The semi-major axis is a mean proportional between the ab- 
 scissa C N and C T, the part of the axis intercepted between tJie 
 centre and the tangent. 
 
 This relation is found also to subsist between the similar 
 parts of the minor axis ; for 
 
 This relation affords an easy rule for finding the point T, 
 or T' ; for from the above we have 
 
 - CN' 
 or, putting / for C T, we have 
 
 : ? y ;y / = 75.) 
 
 or 
 
 t' = --. (176.) 
 
 y 
 
 Since the value of t is not dependent upon y nor upon b, 
 therefore / is constant for all ellipses which may be de- 
 scribed upon the same major axis A M\ and since the circle 
 is an ellipse (Art. 452) with equal major and minor axes, 
 therefore rule (175.) is applicable also to a circle, as shown 
 in Fig. 301. 
 
 The equation (175.) gives the value of / = C T. From 
 this deducting CN = x' , we have N T, the sub tangent, or 
 
 CT- CN = NT, 
 
 t - X > = S ; 
 
 or, substituting for t its value in (175.), we have 
 
 Or: The subtangent to an ellipse equals the difference between 
 the quotient of the square of the semi-major axis divided by tlie 
 abscissa, and the abscissa ; the origin of the co-ordinates being 
 at C, the centre. 
 
AXES TO CONJUGATE DIAMETER. 
 
 487 
 
 456. Ellipse : Relation of Tangent witli the Foci. Let 
 
 the two lines from the foci to P (Fig. 302), any point in the 
 ellipse, be extended beyond P. With the radius P F' de- 
 
 FIG. 302. 
 
 scribe from P the arc F' G, and bisect it in H. Then the 
 line P T, drawn through H, will be a tangent to the ellipse 
 
 Sit P. 
 
 This has been shown by writers on Conic Sections. The 
 construction here shown affords a ready method of drawing 
 a tangent. And from the principle here given we learn 
 that a tangent makes equal angles with the lines from the 
 tangential point to the two foci. 
 
 For, because GH= HF', we have the angle F' PH = 
 HPG. The angles H PG and KPF are opposite, and 
 hence (Art. 344) are equal ; and since the two triangles 
 F'PffandKPFare each equal to HPG, therefore F' PH 
 and KPF are equal to each other. Or: A tangent to an 
 ellipse makes equal angles with the tivo lines drawn from the 
 point of tangency to the two foci. 
 
 Experience shows that light shining from one focus is 
 reflected from the ellipse into the other focus. It is for this 
 reason that the two points F and F' are called foci, the plu- 
 ral oifoczts, a fireplace. 
 
 457. Ellipse : Relation of Axes to Conjugate Diame- 
 terParallel with K T (Fig. 302) let D E be drawn through 
 
488 THE ELLIPSE. 
 
 C, the centre, and L Q through y, one end of the diameter 
 from the point P. Parallel with this diameter PJ draw L K 
 and QR through the extremities of the diameter D E. Then 
 D E is a diameter conjugate to the diameter PJ, and K R, 
 R Q, QL, and L K are tangents at the extremities of these 
 conjugate diameters. 
 
 Now it is shown by writers on Conic Sections (Fig. 302) 
 that 
 
 PC*, 
 
 or 
 
 Or : The sum of the squares of the two axes equals the sum 
 of the squares of any two conjugate diameters. 
 
 From this it is also shown that the area of the parallelo- 
 gram K C equals the rectangle A C x B C', or, that a paral- 
 lelogram formed by tangents at the extremities of any two 
 conjugate diameters is equal to the rectangle of the axes. 
 
 458. Ellipse ; Area. Let E equal the area of an ellipse ; 
 A the area of a circle, of which the radius a equals the semi- 
 major axis of the ellipse, and let b equal the semi-minor axis. 
 Then it has been shown that 
 
 E : A : : b : a, 
 
 E=A b -. 
 a 
 
 The area of a circle (Art. 448) is 
 
 A \ n dr = TT r*, 
 and when the radius equals a 
 
 A = n a 2 , 
 This value of A, substituted in the above equation, gives 
 
 E = TTtf 2 -, 
 
 a 
 
 E = n ab. (178.) 
 
PRACTICAL SUGGESTIONS. 
 
 489 
 
 Or: The area of an ellipse equals 3- 141 59^ times the product 
 of the semi-axes ; or 0-7854 times the product of the axes. 
 
 459. Ellipse : Practical Suggestion*. In order to de- 
 scribe the curve of an ellipse, it is essential to have the two 
 axes ; or, the major axis and the parameter ; or, the major 
 axis and the focal distance. 
 
 If the two axes are given, then with the semi-major axis 
 for radius, from B (Fig. 299) as centre an arc may be made 
 at F and F' t the foci ; and then the curve may be described 
 by any of the various methods given at Arts. 548 to 552. 
 
 If the major axis only and the parameter are given, then 
 (Art. 454) since 
 
 * = ay, 
 we have 
 
 = Vay. 
 
 (I79-) 
 
 Or : The semi-minor axis of an ellipse equals the square root 
 of the product of the semi-major axis into the semi-parameter. 
 Then, having both of the axes, proceed as before. 
 
 If the major axis and the focal distance are given, or the 
 location of the foci ; then with the semi-major axis for ra- 
 
 N N 
 
 FIG. 303. 
 
 dius and from the focal points as centres, describe arcs cut- 
 ting each other at B and O (Fig. 299). The intersection of 
 the arcs gives the limit to B O, the minor axis. With the 
 two axes proceed as before. Points in the curve may be 
 found by computing the length of the ordinates, and then 
 the curve drawn by the side of a flexible rod bent to coin- 
 cide with the several points.. 
 
 For example, let it be required to find points in the 
 curve of an ellipse, the axes of which are 12 and 20 feet ; or 
 
490 THE ELLIPSE. 
 
 the semi-axes 6 and 10 feet, or 6 x 12 = 72 inches, and 10 x 
 12 = 1 20 inches. 
 
 Fix the positions of the points N N', etc., along the semi- 
 major axis C ' M (Fig. 303) at any distances apart desirable. 
 It is better to so place them that the ordinates when drawn 
 shall divide the curve B PM'mto parts approximately equal. 
 If CM be divided into eight parts as shown, these parts 
 measured from C will be well graded if made equal severally 
 to the following decimals multiplied by CM. In this case 
 CM= 120; therefore 
 
 C N 1 20 x 0-3 = 36- = x' 
 
 C N' 120 x 0-475 = 57- = x' 
 
 C N ff = 120 x 0-625 = 75 - = x' 
 
 Etc., = 120x0-75 = 90- x f 
 
 120 X 0-85 = 102- = X f 
 
 120 x 0-925 = in - = x' 
 120x0-975 = 117- = _x' 
 
 1 2O X I-O = 1 2O- X' . 
 
 The equation of the ellipse having the origin of co-ordi- 
 nates at the centre (Art. 454) is 
 
 or, dividing by a*- 
 
 a 
 
 or 
 
 or - y=-\/ a*~x r *; (i go.) 
 
 in which a and b represent the semi-axes. Substituting for 
 these their values in this case, we have 
 
LENGTH OF ORDINATES. 491 
 
 Now, substituting in this equation the several values of 
 x* successively, the values of the corresponding ordinates 
 will be obtained. For example, taking 36, the first value of 
 x', as above, we have 
 
 y = 0-6 ^74400 36* 
 y = 68-684; 
 
 y =. 0-6 V 14400 57 2 
 
 y = 63-359; 
 
 and so in like manner compute the others. 
 
 The ordinates for this case are as follows, viz. : 
 
 When x' o, y 72-0 
 " x' 36, y 68-684 
 " n= 57,^ = 63.359 
 
 " x ' = 75, y = 56-205 
 " * = 90, y = 47-624 
 
 " X' = 102, J/ = 37-928 
 
 " x' = ill, 7 = 27-358 
 
 " /=; 117, y = 15-999 
 
 " *' = 120, J/ 0-0. 
 
 The computation of these ordinates is accomplished easv 
 ly by the help of a table of square roots and of logarithms. 
 
 For example, the work for one ordinate is all comprised 
 within the following, viz. : 
 
 y 0-6 1/14400 36 2 = 68-684. 
 
 I2O 2 = I44OO 
 36 2 = I2Q6 
 
 I3I04 = 4-JI74Q39 
 
 Half = 2-0587020 
 
 0-6 = 9-7781513 
 
 68-684 1-8368533. 
 
 The logarithm of 13104 = 4-1 174039. The half of this is 
 the logarithm of the square root of 13104. To the half log- 
 arithm add the logarithm of c-6; the sum is the logarithm 
 of 68-684 found in the table (see Art. 427). 
 
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 fol- 
 lows : When cut parallel with its base the outline is a circle ; 
 when the plane passes obliquely through the cone, it is an 
 ellipse ; when the plane is parallel with the axis, but not in 
 the axis, it is a hyperbola ; while that which is produced by 
 
 FIG. 304. 
 
 a plane cutting it parallel with one side of the cone is a 
 parabola. 
 
 Let the lines L M and L N (Fig. 304) be at right angles ; 
 draw CFB parallel with L M\ make LQ LF\ draw QB 
 parallel with LF\ then FB = B Q. Now let the line A L 
 move from F L, but remain parallel with it, and as it moves 
 let it gradually increase in length in such manner that the 
 point A shall constantly be equally distant from the line LM 
 and from the point F. Then A BP, the curve described by 
 the point A, will be a semi-parabola. For example, the 
 lines FB and B Q are equal ; the lines PPand PJ/are equal, 
 and so of lines similarly drawn from any point in the curve 
 A B P. Let PNbe drawn parallel with LM ; then for the 
 
EQUATION TO THE CURVE. 493 
 
 point P, A Nis the abscissa and N P its ordinate (see Art. 
 452). 
 
 The double ordinate C B drawn through F, the focus, is 
 the parameter. A F is the focal distance. A is the vertex of 
 the curve. The line L M is the directrix. 
 
 4-6 1. Parabola : Equation to the Curve. In Fig. 304 
 FPN is a right-angled triangle, therefore 
 
 = FP* - 
 
 but- FP = .MP=LN=AN+AL\ 
 
 and FN= A N A F. 
 
 Therefore 
 
 NP* = A N+AL*- A N- A F* ; 
 or- ' = * 
 
 / being put for the distance LF= FB (see Art. 452). As 
 in Arts. 412 and 413, we have 
 
 y* = 2px (181.) 
 
 by subtraction. This is the usual equation to the parabola, 
 in which we have the rule : The square of the ordinate equals 
 the rectangle of the corresponding abscissa with the param- 
 eter. 
 
 From (181.) we have 
 
 x : y : : y : 2p, 
 
 or: 1\\e parameter is a third proportional \.o the abscissa and its 
 corresponding ordinate. 
 
 462. Parabola : Tangent. From M, any point in the 
 directrix, draw a line to F t the focus (Fig. 305) ; bisect M F 
 in R, and through R draw U T perpendicular to MF, then 
 the line T U will be a tangent to the curve. For, draw M D 
 
494 
 
 THE PARABOLA. 
 
 perpendicular to L V, and from P, the point of its intersection 
 with the line TU, draw a line to F, the focus ; then, because 
 fiPis a perpendicular from the middle of MF, MPFis an 
 isosceles triangle, and therefore the lines MPand FP are 
 equal, or the point P is equidistant from the focus and from 
 the directrix, and therefore is a point in the curve. 
 
 To show that the line TU touches the curve but does not 
 pass through it, take 7, any point in the line T U, other than 
 
 FIG. 305. 
 
 the point P; join [7 to Mand to F. Then, since U is a point 
 in the line T U, M U F, for reasons above given, is an isosce- 
 les triangle ; from [/draw U F perpendicular to L V. Now, 
 if the point /be also in the curve, the lines Wand U F, 
 by the law of the curve, must be equal ; but U F, as before 
 shown, is equal to U M, a line evidently longer than UV\ 
 therefore, it is evident that the point U is riot in the curve. 
 A similar absurd result will be reached if any other point 
 than the point U in the line U T be assigned, excepting the 
 
RULE FOR THE TANGENT. 495 
 
 point P. Therefore the line T P touches the curve in only 
 one point, P ; hence it is a tangent. 
 
 Parallel with L V, from A , draw A S, the vertical tangent. 
 Now A S bisects M F or intersects it in the point R. For 
 the two right-angled triangles FL J/and FA R are homolo- 
 gous ; and because FA = A Z, by construction, therefore 
 FR = RM. 
 
 Or : The vertical tangent bisects all lines which can be drawn 
 from the focus to the directrix. 
 
 The lines PFand FT are equal ; for the lines MPand 
 N T being parallel, therefore the alternate angles MPT 
 and N TPare equal (Art. 345) ; and because the line P T bi- 
 sects M F, the base of an isosceles triangle, therefore the 
 angles M P T and FPTafe equal. We thus have the two 
 angles N TP and FP T each equal to the angle MPT-, 
 therefore the two angles N TP and FP T are equal to each 
 other ; hence the triangle P F T is an isosceles triangle, hav- 
 ing the points T and P equidistant from F, the focus. 
 
 Also because the line MFis perpendicular to P T, there- 
 fore the line M F bisects the tangent PT in the point R. 
 And because TR = R P, therefore, comparing triangles 
 TRFund TPO, TF= F O. 
 
 The opposite angles MPT and U PD made by the two in- 
 tersecting lines U T and M D (Art. 344) are equal, and since 
 the angles M P T and F P T are equal, as before shown, 
 therefore the angles FP T and U PD are equal. 
 
 It is because these two angles are equal, that, in reflectors, 
 rays of light and heat proceeding from /% the focus, are re- 
 flected from the parabolic surface in lines parallel with the 
 axis. 
 
 For an equation expressing the value of the tangent, we 
 have 
 
 Or : The tangent to a parabola equals the square root of tJie 
 sum of four times the square of the abscissa added to the square 
 of the ordinate. 
 
496 THE PARABOLA. 
 
 463. Parabola: Subtangeiit. The line T N (Fig. 305), 
 the portion of the axis intercepted between T, the point of 
 intersection of the tangent, and N, the foot of a perpendicu- 
 lar to the axis from P, the point of contact, is the subtangent. 
 The subtangent is bisected by the vertex, or TA = A N. 
 For, the two triangles TRA and TPNare homologous; 
 and, as shown in the last article, the line M F bisects PTin 
 R-, or TR = R P. 
 
 Therefore, we have 
 
 TR : TA : : TP 
 
 TR x TN = TA x TP, 
 but TR=TP; 
 
 therefore- J- TP x TN= TA x TP, 
 
 | TN = TA. 
 
 Or : The subtangent of z. parabola is bisected by the vertex ; or 
 is equal to twice the abscissa. 
 
 And because of the similarity of the two triangles TRA 
 and TP N, as above shown, we have 
 
 NP= 2AR, 
 y = 2 A R. 
 Or : The ordinate equals twice the vertical tangent. 
 
 464. Parabola: Normal and Subnormal. The line 
 PO (Fig. 305) perpendicular to P T, is the normal and NO, 
 the part of the axis intercepted between the normal and the 
 ordinate, is the subnormal. For the normal, from similar 
 triangles, we have 
 
 TN : NP : :. TP : P O, 
 TP 
 
 2X 
 
DIAMETER AND RECTANGLE OF BASE. 497 
 
 Or : The normal equals the rectangle of the ordinate and tan- 
 gent, divided by twice the abscissa. 
 
 The subnormal equals half the parameter. For (181.) 
 
 or NP* = 2 FB A N. 
 
 Dividing by 2 A N gives 
 
 I FB = ttf (A ' } 
 
 In the similar triangles (Art. 443) OPNand PT N, we 
 have 
 
 NO : NP : : NP : NT, 
 
 NO = N~P\ 
 NT 
 
 As shown in the previous article, N T 2 A N; therefore 
 
 (B.) 
 
 2AN 
 
 Comparing equations (A.) and (B.), we have 
 
 NO = FB. 
 
 Or : The subnormal of a parabola equals half the parameter, 
 a constant quantity for the subnormal to all points of the 
 curve. 
 
 465. Parabola: Diameters. In the parabola BAC 
 (Fig. 306), P D, a diameter (a line parallel with the axis) to 
 the point P, is in proportion to B D x D C, the rectangle of 
 the two parts into which the base of the parabola is divided 
 by the diameter. 
 
 This may be shown in the following manner : 
 
 DP=EN=EA-NA. (A.) 
 
498 
 
 THE PARABOLA. 
 
 For EA we have, taking the co-ordinates, for the point 
 C, (i8i.)~ 
 
 or- 
 
 = X 
 
 or 
 
 EC 
 
 = EA. 
 
 C 
 
 For N A we have, taking the co-ordinates to the point 
 />, (181.)- 
 
 or 
 
 = X, 
 
 NP'- 
 
 or- 
 
 (C.) 
 
 Using these values (B.) and (C.j in (A.), we have 
 
 =A -NA, 
 
 ' NP* EC*-NP* 
 
 If / be put for B C and n for D C, then 
 
TO FIND THE CONSTITUENT PARTS. 499 
 
 and- 
 then (Art. 413) 
 
 ft 
 
 or (Art. 415) 
 
 2 P 
 
 = n(l- n) 
 2 P 
 
 Dp _DCxBD 
 
 ~ 
 
 Now, since 2 /, the parameter, is constant, we have D P, 
 the diameter, in proportion to D C x B D, the two parts of 
 the base. 
 
 Putting d for the diameter, we have 
 
 (183.) 
 
 2p 
 
 Or : The diameter of a parabola equals the quotient obtained 
 by dividing \\\Q rectangle formed by the two parts into which 
 the diameter divides the base by the parameter. 
 
 It has been shown by writers on Conic Sections that a 
 diameter, P J (Fig. 307), to any point Pin a parabola bisects 
 all chord lines, SG,D, etc., drawn parallel with the tan- 
 gent to the point/*; the diameter being parallel with the axis 
 of the parabola. 
 
 466. Parabola : Elements. From any given parabola, 
 to find the axis, tangent, directrix, parameter and focus, 
 draw any two parallel lines or chords, SG and D E (Fig. 
 307), and bisect them in H and J\ through these points 
 draw JP\ then JP will be a diameter of the parabola a 
 
5oo 
 
 THE PARABOLA. 
 
 line parallel with the axis. Perpendicular to P J draw the 
 double ordinate PQ and bisect it in N\ through N and par- 
 allel with PJ draw TO, cutting the curve in A\ then TO 
 will be the axis. Make AT- AN, join T and P\ then TP 
 will be the tangent to the point P\ from P draw PO per- 
 pendicular to P T\ then P O will be the normal, and NO the 
 subnormal. 
 
 With NO for radius, from N as a centre, describe the 
 quadrant O R ; draw R C parallel with A O, cutting the curve 
 
 FIG. 307. 
 
 in C\ from C draw C B perpendicular to A O, cutting A O in 
 F; then ^will be the focus and C B the parameter. Make 
 A L A F; draw L M perpendicular to TO\ then LM will 
 be the directrix. Extend PJ to meet LM at M\ join P and , 
 F\ then, if the work has been properly performed, F P will 
 equal M P. 
 
 4-67. Parabola : Described mechanically. With N P 
 (Fig. 308) a given base, and N A a given height, set perpen- 
 
THE CURVE DESCRIBED MECHANICALLY. 
 
 501 
 
 dicularly to the base, extend N A beyond A, and make A T 
 equal to NA ; join T and/*; from P perpendicularly to TP 
 drawP(9; bisect ON in R; make AL and A F each equal 
 to N R ; through L, perpendicular to L O, draw D E, the di- 
 rectrix. 
 
 Let the ruler CDESbe laid to the line DE, then with 
 J G H, a set-square, the curve may be described in the fol- 
 lowing manner : 
 
 Placing the square against the ruler and with its edge 
 
 FIG. 308. 
 
 J H coincident with the line MP y fasten to it a fine cord on 
 the edge PE, and extend it from P to F y the focus, and se- 
 cure it to a pin fixed in F. The cord FP will equal the 
 edge M P. To describe the curve set the triangle J G H at 
 M P E, slide it gently along the ruler towards D, keeping the 
 edge J G in contact with the ruler, and, as the square is 
 moved, keep the cord stretched tight, holding for this pur- 
 pose a pencil, as at K, against the cord. Thus held, as the 
 square is moved the pencil will describe the curve. That 
 this operation will produce the true curve we have but to 
 
502 
 
 THE PARABOLA. 
 
 consider that at all points the line FK will equal KJ t which 
 is the law of the curve (Art. 460). 
 
 468. Parabola : Decribed from Point. With given 
 base, N P (Fig. 309), and given height, A N, to find the points 
 D, F, M, etc., and describe the curve. Make A T equal to 
 A N (Art. 462); join T and P', perpendicular to TP draw 
 
 FIG. 309. 
 
 PO ; make A B equal to twice NO ; take G, any point in the 
 axis A O, and bisect B G in J ; on y as a centre describe the 
 semi-circle B C G cutting A L, a perpendicular to BO in C ; 
 on A C and A G complete the rectangle A C DG. Then D is 
 a point in the curve. Take H, another point in the axis ; 
 bisect B H in. K\ on A" as a centre describe the semi-circle 
 B E H cutting A L in E ; this by E F and H F, gives F, an- 
 
THE CURVE DESCRIBED FROM ARCS. 503 
 
 other point in the curve ; in like manner procure M, and as 
 many other points as may be desired. This simple and 
 accurate method of obtaining points in the curve depends 
 upon two well-established equations ; one, the equation to 
 the parabola, and the other, the equation to the circle. The 
 line G D y an ordinate in the parabola, is equal to A C, an 
 ordinate in the circle B CG; A G, the abscissa of the para- 
 bola, is also the abscissa of the circle ; in which we have 
 (Art. 443)- 
 
 AG : AC :: AC : A B, 
 
 x : y : : y : a x, 
 
 For the parabola, we have (181.) 
 
 y*=2px. 
 Comparing" these two equations, we have 
 
 x(a x] = 2px, 
 
 a x = 2p, 
 or 
 
 EG A G= 2 p. 
 
 By construction A B equals 2 NO, or twice the subnor- 
 mal ; the subnormal (Art. 464) equals half the parameter. 
 Hence, twice the subnormal equals the parameter equals 
 2 p. Therefore, the method shown in Fig. 309 is correct. 
 
 469. Parabola: Described from Arcg. Let N P (Fig. 
 310) be the given base and A TV the given height of the par- 
 abola. Make A T(Art. 462) equal A N. Join TtoP; draw 
 PO perpendicular to PT\ bisect N O in R ; make A L and 
 A F each equal to N R ; then L M, drawn perpendicular to 
 TO, will be the directrix. Parallel to L M draw the lines 
 B D, C E, etc., at discretion. Then with the distance B L for 
 
504 
 
 THE PARABOLA. 
 
 radius, and on F as a centre, mark the line B D with an arc ; 
 the intersection of the arc and the line will be a point in the 
 curve (Art. 460). Again, with C L for radius and on .Fas a 
 centre, mark the line CE with an arc ; this gives another 
 point in the curve. In like manner, mark each horizontal 
 
 FIG. 310. 
 
 line from F as a centre by a radius equal to the perpendicu- 
 lar distance between that line and L M, the directrix. Then 
 a curve traced through the points of intersection thus ob- 
 tained will be the required parabola. 
 
 4-70. Parabola : Dccribed from Ordinates. With a 
 given base, NP(Fig. 311), and height, A N, a parabola may 
 be drawn through points J, H, G, etc., which are the extrem- 
 ities of the ordinates B J, C H, D G, etc. ; the lengths of the 
 ordinates being computed from the equation to the curve, 
 (181.)- 
 
 For any given parabola, in base and height, the value of 
 
THE CURVE FORMED BY ORDINATES. 
 
 505 
 
 / may be had by dividing both members of the equation by 
 2 4;; by which we have 
 
 y 
 
 NP* 
 
 2AN' 
 
 (A.) 
 
 from which, NPand A N being known,/ may be computed. 
 With the value of/, a constant quantity, determined, the 
 equation is rendered practicable. For, taking the square 
 root of each member of equation (181.), we have 
 
 y = V 2p x. ( B -) 
 
 which by computation will produce the value of y, for every 
 assigned value of x, as A JS, A C, A D, etc. 
 
 FIG. 311. 
 
 As an example : let it be required to compute the ordi- 
 nates in a parabola in which the base, N ' P, equals 8 feet, and 
 the height, A N, equals 10 feet. With these values equation 
 (A.) as above becomes 
 
 NP* 
 
 2AN 
 
 8" _64 
 
 2 X 10 ~ 2O " ^' * 
 
506 THE PARABOLA. 
 
 Then, with this value in (B.) as above, we have, for each or- 
 dinate 
 
 y ' z ' V 6.4*. 
 
 In order to assign values to x, let A N be divided into 
 any number of parts at B, C, D, etc., say, for convenience in 
 this example, in ten equal parts ; then each part will equal 
 one foot, and we shall have the consecutive values of x = i , 
 2, 3, 4, etc., to 10, and the corresponding values of y will be 
 as follows. When 
 
 x = I, y= 4/6. 4 x i = 
 
 3= 
 x = 4, y = V6-4X "4 = 1/ 25-6 = 5 -059 6 = 
 
 ~32~ = 5 -6569 = (etc.), 
 38-4 = 6- J 9 68 = 
 "^8 = 6>6 933 = 
 
 = 7-5^95 = 
 
 With these values of y, respectively, set on the correspond- 
 ing horizontal lines By, C H, DG, E S, etc., points in the 
 curve y, //, G, S, etc., are obtained, through which the curve 
 may be drawn. The decimals above shown are the decimals 
 of a foot ; they may be changed to inches and decimals of an 
 inch by multiplying each by 12. For example: 12x0-5297 
 = 6-3564 equals 6 inches and the decimal 0-3564 of an inch, 
 which equals nearly | of an inch. 
 
 Near the top of the curve, owing to its rapid change in 
 direction and to the approximation of the direction of the 
 curve to a parallel with the direction of the ordinates, it is 
 
THE CURVE FOUND BY DIAMETERS. 
 
 SO/ 
 
 desirable to obtain points in the curve more frequent than 
 those obtained by dividing the axis into equal parts. 
 
 Instead, therefore, of dividing the axis into equal parts, 
 it is better to divide it into parts made gradually smaller 
 toward the apex of the curve or, to obtain points for this 
 part of the curve as shown in the following article. 
 
 471. Parabola: Described from Diameters. Let EC 
 
 (Fig. 312) be the given base and A E the given height, placed 
 perpendicularly to E C. Divide E C in several parts at 
 pleasure, and from the points of division erect perpendicu- 
 lars to E C. The problem is to compute the length of these 
 diameters, as DP, and thereby obtain points in the curve, as 
 at P. For this purpose we have equation (183.), which gives 
 
 B 
 
 \ 
 
 FIG. 312. 
 
 the length of the diameters, and in which n equals D C (Fig. 
 312), / equals twice E C, and / equals half the parameter of 
 the curve. The value of p is given in equation (A.), (Art, 
 470), in which y equals EC (Fig. 312), and ^equals A E. 
 Substituting these symbols in equation (A.), we have 
 
 y> EC' _^ 
 
 p ~~'' 2x ~ 2xAE~ 2 /T' 
 
 where b EC, the base, and hAE, the height. For 
 substituting this, its value, in equation (183.), we have 
 
 hn.(2b n) 
 
 (184.) 
 
508 THE PARABOLA. 
 
 As an example : let it be required in a parabola in which 
 the base equals 12 feet and the height 8 feet, to compute the 
 length of several diameters, and through their extremities 
 describe the curve. Then h will equal 8, and b 12. 
 
 If the base be divided into 6 equal parts, as in Fig. 312, 
 each part will equal 2 feet. Then we have 
 
 h 8 8 i 
 
 b*~ I2*~~ i44~ 18 ' 
 and 
 
 _ h . N 
 d -i n (2b-n), 
 
 In this equation, substituting the consecutive values of #, 
 we have, when 
 
 0x24 
 n= O, d-- -jg- =0 
 
 2 X 22 
 
 ti z, 
 
 18 
 
 
 4X 20 
 
 M A 
 
 /y _...- - . A . A A A 
 
 , , 6x18 
 
 *-; 6, ^-^g- 
 
 8x16 
 n= 8, ^=-73 =7-ni 
 
 12 X 12 
 
 n= 12, d= -- = 8-0 
 
 The several diameters, as P ' D, in Fig. 312, may now be 
 made equal respectively to these computed values of d, and 
 the curve traced through their extremities. 
 
AREA EQUALS TWO THIRDS OF RECTANGLE. 509 
 
 472. Parabola: Area. From (181.), the equation to 
 the parabola, and by the aid of the calculus, it has been 
 shown that the area of a parabola is equal to two thirds of 
 the circumscribing rectangle. For example : if the height, 
 A E (Fig. 312), equals 8 feet, and EC, the base, equals 12 
 feet, then the area of the part included within the figure 
 APC E A equals f of 8x 12 = 1x96 = 64 feet; or, it is equal 
 to f of the rectangle A B C E. 
 
SECTION XIV. TRIGONOMETRY. 
 
 473. Right- Angled Triangle: The Sides. In right- 
 angled triangles, when two sides are given, the third side 
 may be found by the relation of equality which exists of 
 the squares of the sides (Arts. 353 and 416). For example, 
 
 if the sides a and b (Fig. 313) are given, c y the third side, 
 may be computed from equation (115.) 
 
 Extracting the square root, we have 
 
 When the hypothenuse and one side are given, by transposi 
 tion of the factors in (115.), we have 
 
 (A.) 
 
 or 
 
TWO SIDES GIVEN TO FIND THE THIRD. 511 
 
 Owing to the factors being involved to the second power in 
 this expression, the labor of computation is greater than 
 that in a more simple method, which will now be shown. 
 
 In equation (A.) or (B.) the factors under the radical may 
 be simplified. By equation (i 14.) we have 
 
 Therefore, equation (A.) becomes 
 
 a = \( c + ) (V _ fy 
 
 a form easy of solution. 
 
 For example : let c equal 29-732 and b equal 13-216, then 
 we have 
 
 29-732 
 
 13-216 
 
 The sum = 42-948 
 The difference = 16-516 
 
 By the use of a table of logarithms (Art. 427) the problem 
 may be easily solved ; thus 
 
 Log. 42-948 = 1-6329429 
 16-516 = i -2179049 
 
 To get the square root 2)2- 8508478 
 a 26-6332 = i -4254239 
 
 This method is applicable to the sides of a triangle, only ; 
 for the hypothenuse it will not serve. The length of the 
 hypothenuse as well as that of either side may, however, be 
 obtained by proportion ; provided a triangle of known di- 
 mensions and with like angles be also given. 
 
 For example: in Fig. 314, in which the two sides a and 
 b are known, let it be required to find c, the hypothenuse. 
 
 Draw the line D E parallel with A C, then the two trian- 
 gles B DE and BAG are homologous; consequently their 
 
512 
 
 TRIGONOMETRY. 
 
 corresponding sides are in proportion (Art. 361). Hence, if 
 d equals unity, we have 
 
 d : f : : a : c, 
 = */,' 
 
 from which, when a and / are known, c is obtained by sim- 
 ple multiplication. 
 
 474. Right- Angled Triangle: Trigonometrical Ta- 
 bles To render the simple method last named available, 
 the lengths of d, e and f (Fig. 314) have been computed for 
 triangles of all possible angles, and the results arranged in 
 
 FIG. 314. 
 
 tables, termed Trigonometrical Tables. The lines d, e, and 
 /, are known as sines, cosine, tangents, cotangents, etc., as 
 shown in Fig. 315 wheVe A B is the radius of the circle 
 B C H. Draw a line A F, from A, through any point, C, of the 
 arc B G. From C draw CD perpendicular to A B ; from B 
 draw BE perpendicular to A B ; and from G draw G F per- 
 pendicular to A G. 
 
 Then, lor the angle FA B, when the radius A C equals 
 unity, CD is the sine; AD the cosine; DB the versed sine ; 
 BE the tangent; GF the cotangent ; AE the secant; and 
 A Fihe cosecant. 
 
 But if the angle be larger than one right angle, yet less 
 than two right angles, as BAH, extend HA to K and E B 
 to K, and from H draw H J perpendicular to A J. 
 
TRIGONOMETRICAL TABLES. 
 
 513 
 
 Then, for the angle BAH, when the radius A H equals 
 unity, HJ is the sine ; A J the cosine; BJ the versed sine ; 
 B K the tangent ; and A K the secant. 
 
 When the number of degrees contained in a given angle 
 is known, the value of the sine, cosine, etc., corresponding to 
 that angle, may be found in a table of Natural Sines, CO- 
 
 FIG. 315. 
 
 sines, etc. Or, the logarithms of the sines, cosines, etc., may 
 be found in logarithmic tables. 
 
 In the absence of such a table, and when the degrees 
 contained in the given angle are unknown, the values of 
 the sine, cosine, etc., may be found by computation, as fol- 
 lows: Let ABC (Fig. 316) be the given angle. At any 
 distance from B draw b perpendicular to B C. By any scale 
 of equal parts obtain the length of each of the three lines a, 
 b, c. Then for the angle at B we have, by proportion 
 
5M- 
 
 TRIGONOMETRY. 
 
 c : b : : i -o : sin. B = . 
 
 c 
 
 c : a : : I o : cos. B = . 
 
 c 
 
 a : b : : i -o : tan. B = . 
 
 a 
 
 b : a : : I -o : cot. B -. 
 
 a i c : : i -o : sec. B = -. 
 
 a 
 
 b : c : : i o : cosec. B --. 
 
 Or, in any right-angled triangle, for the angle contained 
 between the base and hypothenuse 
 
 When perp. divided by hyp., the quotient equals the sine. 
 
 base " " hyp., " " " cosine. 
 
 " perp. " " base, " " tangent. 
 
 " base " " perp., " " cotangent. 
 
 " hyp. " " base, " " " secant. 
 
 " hyp. " " perp., " " cosecant. 
 
 To designate the angle to which a trigonometrical term 
 applies, the letter at the intended angle is annexed to the 
 
 c 
 FIG. 316. 
 
 name of the trigonometrical term ; thus, in the above exam- 
 ple, for the sine of A B C we write sin. B ; for the cosine, 
 cos. B, etc. 
 
EQUATIONS TO RIGHT-ANGLED TRIANGLES. 
 
 515 
 
 By these proportions the two acute angles of a right- 
 angled triangle may be computed, provided two of the 
 sides are known. For when the perpendicular and hypoth- 
 enuse are known, the sine and cosecant may be obtained. 
 When the base and hypothenuse are known, the cosine and 
 secant may be computed. And when the base and perpen- 
 dicular are known, the tangent and cotangent may be com- 
 puted. 
 
 Either one of these, thus obtained, shows by the trigo- 
 nometrical tables the number of degrees in the angle ; and, 
 deducting the angle thus found from 90, the remainder will 
 be the angle of the other acute angle of the triangle. For 
 
 M D 
 
 FIG. 317. 
 
 example : in a right-angled triangle, of which the base is 8 
 feet and the perpendicular 6 feet, how many degrees are 
 contained in each of the acute angles ? 
 
 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 
 
 Referring to the trigonometrical tables, we find that 0-75 is 
 the tangent of 36 52' 12", nearly ; therefore 
 
 The quadrant equals 90- o- o 
 The angle B equals 36-52-12 
 
 The angle A equals 53-07-48 
 
TRIGONOMETRY. 
 
 475. Right-Angled Triangle : Trigonometrical Value 
 of Side. In the triangle A B C (Fig. 317), with BP = i for 
 radius, and on B as a centre, describe the arc P D, and from 
 its intersection with the lines A B and B C, draw PM and 
 T D perpendicular to the line B C. Then from homologous 
 triangles we have these proportions for the perpendicular 
 
 BD : DT : : BC : CA, 
 r : tan. B : : base : perp., 
 
 I : tan. B : : a : b = a tan. B. ( 1 %S-) 
 
 Also * 
 
 - BP : PM \\BA\AC, 
 
 r : sin. B : : hyp. : perp., 
 I : sin. B : : c : b = c sin. B. (186.) 
 
 For the base, we have 
 
 BP : BM : : BA : B C, 
 r : cos. B : : hyp. : base, 
 
 I : cos. B : : c : a = c cos. B. ( l %7-) 
 
 Again 
 
 TD : BD :: AC : B C, 
 
 tan. B : r : : perp. : base, 
 tan.*:!::*:,^ L_. (.88.) 
 
 For the hypothenuse, we have 
 
 PM : PB : : A C : : A B, 
 sin. B : r : : perp. : hyp., 
 
RULE FOR THE PERPENDICULAR. 517 
 
 sin. B : i : : b : c = -, . (180.) 
 
 sm. B 
 
 Again 
 
 BD \BT\\BC\ BA, 
 
 r : sec. B : : base : hyp., 
 
 I : sec. B : : a : c = a sec. B - . (IQO.) 
 
 cos. B 
 
 This substitution of the cos. for the sec. is needed because 
 tables of secants are not always accessible. That it is an 
 equivalent is clear ; for we have 
 
 BM \ BP :-. BD : BT, 
 
 cos. \r\\r\ sec. = 
 
 cos. 
 
 By these equations either side of a right-angled triangle 
 may be computed, provided there are certain parts of the 
 triangle given. As, for example : of the six parts of a tri- 
 angle (the three sides and the three angles), three must be 
 given, and at least one of these must be a side. 
 
 As an example : let it be required to find two sides of a 
 right-angled triangle of which the base is 100 feet, and the 
 acute angle at the base is 35 degrees. Here we have given 
 one side and two angles (the base, acute angle, and the right 
 angle) to find the other two sides, the perpendicular and the 
 hypothenuse. 
 
 Among the above rules we have, in equation (185.), for 
 the perpendicular 
 
 b a tan. B. 
 
 Or : The perpendicular equals the product of the base into the 
 tangent of the acute angle at the base. 
 
5 1 8 TRIGONOMETRY. 
 
 Then (Art. 427) 
 
 The logarithmic tangent of B ( 35) is 9-8452268 
 Log. of a ( 100) is 2-0000000 
 
 Perpendicular, b (= 70-02075) = I -8452268 
 
 And for the hypothenuse, taking equation (190.), we 
 have 
 
 cos. B 
 
 Or : The hypothenuse equals the quotient of the base divided 
 by the cosine of the acute angle at the base. 
 For this we have- 
 Log, of a (= 100) is 2-0000000 
 " cos. B (= 35) is 9-9133645 
 Hypothenuse c(~ 122-0775) = 2-0866355 
 
 We thus find that a right-angled triangle, having an angle 
 of 35 degrees at the base, has its three sides, the perpendic- 
 
 ular, baseband hypothenuse, respectively equal to 70-02075, 
 loo, and 122-0775. 
 
 N.B. The angle at A (Fig^ij) is obtained by deducting 
 the angle at B from 90 (Art. 346). Thus, 90 35 = 55 ; 
 this is the angle at A, in the above case. 
 
 If the perpendicular be given, then for the base use 
 equation (188.), and for the hypothenuse use equation (189.). 
 If the hypothenuse be given, then for the base use equation 
 (187.) and for the perpendicular use equation (186.). 
 
USEFUL RULE FOR THE SIDES. 519 
 
 476. Oblique-Angled Triangle* : Sine* and Side*. In 
 
 the oblique-angled triangle A BC (Fig. 318) from C and per- 
 pendicular to A B draw CD. This line divides the oblique- 
 angled triangle into two right-angled triangles, the lines and 
 angles of which may be treated by the rules already given ; 
 but there is a still more simple method, as will now be 
 shown. 
 
 As shown in Art. ^4: " When the perpendicular is di- 
 vided by the hypothenuse the quotient equals the sine." 
 Applying this to Fig. 318, we have 
 
 jNIVERSITY 
 
 Let the former be divided by the latter ; then 
 
 d 
 
 sin. A __ b_ 
 ~sm7 ~ d ' 
 a 
 
 or, reducing, we have 
 
 sin. A 
 
 or, putting the equation in the form of a proportion- 
 sin. B : sin. A : : b : a ; 
 
 or ; the sines are in proportion as the sides, respectively op- 
 posite. Or, as commonly stated, the sines are in proportion 
 as the sides which subtend them. 
 
 This is a rule of great utility ; by it we obtain the follow- 
 
 ing : 
 
 Referring to Fig. 318, we have- 
 
 sin. B : sin. A : : b : a = b *-. - . (191.) 
 
520 TRIGONOMETRY. 
 
 ~ sin. A 
 
 sin. C : sin. A : : c : a = c- ~ . (192.) 
 
 sin. C 
 
 , sin. B 
 
 sin. A : sin. B : : a : b a . . (ICH.) 
 
 sin. A 
 
 ~ . , sin. B 
 
 sin. C : sin. B : : c : b = - -. (iQ4.) 
 
 sin. (7 
 
 , . ^ sin. 6" , N 
 
 sin. A : sin. c : : a : c = a-^ -~ . (iQ5.) 
 
 sin. A 
 
 n ' r L 7 Sm - C 
 
 sin. B : sin. C \ \ b \ c b . 
 
 sin. B 
 
 These expressions give the values of the three sides respec- 
 tively ; two expressions for each, one for each of the two 
 remaining sides; that is to be used which contains the given 
 side. 
 
 From these expressions we derive the values of the 
 sines \ thus 
 
 sin. A = sin. B a -. ( 1 97-) 
 
 b 
 
 sin. A = sin. C . ( 1 9%-) 
 
 sin. B = sin. A . ( I 99) 
 
 ci 
 
 sin. B = sin. C '-. (200.) 
 
 sin. C sin. A -. (201.) 
 
 sin. C sin. B C -. (202.) 
 
 477. Oblique - Angled Triangles : First Class. The 
 
 problems arising in the treatment of oblique-angled trian- 
 gles have been divided into four classes, one of which, the 
 
TO FIND THE TWO SIDES. 521 
 
 first, will here be referred to. The problems of the first 
 class are those in which a side and two angles are given, to 
 find the remaining angle and sides. 
 
 As to the required angle, since the three angles of every 
 triangle amount to just two right angles (Art. 345), or 180, 
 the third angle may be found simply by deducting the sum 
 of the two given angles from 180. 
 
 For example : referring to Fig. 318, if angle A = 18 and 
 angle B = 42, then their sum is 18 + 42 = 60, and 180 - 
 60 = 120 = the angle AC B. 
 
 To find the two sides : if a be the given side, then to find 
 the side b we have, equation (193.) 
 
 sin. B 
 
 b a ~ ; 
 
 sin. A 
 
 or, the side b equals the product of the side a into the quo- 
 tient obtained by a division of the sine of the angle opposite 
 b by the sine of the angle opposite a. 
 
 For example: in a triangle (Fig. 318) in which the angle 
 A = 1 8, the angle B 42 (and, consequently (Art. 345) the 
 angle C= 120), and the given side a equals 43 feet; what 
 are the lengths of the sides b and cl Equation (193.) gives- 
 sin.' B 
 
 b~a- --. 
 sin. A 
 
 Performing the problem by logarithms (Art. 427), we 
 have 
 
 Log. a(= 43)= I -6334685 
 Sin. B (- 42) = 9-8255109 
 
 -4589794 
 Sin. A (= 1 8) = 9-4899824 
 
 Log. b O 93 1 102) = i -9689970. 
 
 Thus the side b equals 93-1102 feet, or 93 feet I inch and 
 nearly one third of ah inch. 
 
522 TRIGONOMETRY. 
 
 For the side c, we have, equation (195.) 
 
 sin. C 
 
 c = a . ; 
 sin. A 
 
 or 
 
 Log. *(= 43) =1-6334685 
 Sin. C(= 120) = 9-9375306 
 
 1' 5 70999 1 
 Sin. A ( 1 8) = 9-4899824 
 
 Log. c(= 120-508) 2-0810167 
 
 or, the base c equals 120 feet 6 inches and one tenth of an 
 inch, nearly. But if instead of a the side b be given, then 
 for a use equation (191.), and for c use equation (196.). 
 
 And, lastly, if c be the given side, then for a use equation 
 (192.), and for b use equation (194.). t 
 
 478. Oblique-Angled Triangles: Second Clas. The 
 problems which comprise the second class are those in which 
 .two sides ^ax\& an angle opposite to one of them are given, to 
 find the two remaining angles and the third side. 
 
 The only requirement really needed here is to find a 
 second angle ; for, with this second angle found, the problem 
 is reduced to one of the firt class ; and the third side may 
 then be found under rules given in Art. 477. 
 
 To find a second angle, use one of the equations (197.) to 
 
 (202.). 
 
 For example : in the triangle ABC (Fig. 318), let a (= 43) 
 and b (= 93 1 1) be the two given sides, and A, the angle op- 
 posite a, be the given angle (= 18). Then to find the angle 
 B, we have equation (199.) (selecting that which in the 
 right hand member contains the given angle and sides) 
 
 sin. B = sin. A 
 
 43 
 
OBLIQUE-ANGLED TRIANGLES. 523 
 
 By logarithms (Art. 427), we have 
 
 Log. sin. A (= 1 8) = 9-4899824 
 
 " 93-n = 1.9689970 
 
 1-4589794 
 
 43 = 1-6334685 
 
 " sin. 5 (=42) = 9-8255109 
 
 By reference to the log. tables, the last line of figures, as 
 above, is found to be the sine of 42 ; therefore, the required 
 angle B is 42. Then 180 - (18 + 42) = 120 = the angle C. 
 
 With these angles, or with any two of them, the third 
 side c may be found by rules given in Art. 477. 
 
 FIG. 319. 
 
 479. Oblique-Angled Triangles : Sum and Difference 
 of Two Angles. Preliminary to a consideration of prob- 
 lems in the third class of triangles, it is requisite to show the 
 relation between the sum and difference of two angles. 
 
 In Fig. 319, let the angle A JM and the angle A JN be 
 the two given angles ; and let A J M be called angle A, and 
 AJN, angle B. Now the sum and difference of the angles 
 may be ascertained by the use of the sum and difference of 
 the sines of the angles, and by the sum and difference of the 
 tangents. In the diagram, in which the radius A J equals 
 
524 TRIGONOMETRY. 
 
 unity, we have MP, the sine of angle A (= A y M), and 
 NQ = RP, the sine of angle B(= A y N). Then 
 
 MP- RP= MR 
 
 equals the difference of the sines of the angles ; and since 
 PM' = PM 
 
 PM'+RP = RM', 
 
 equals the sum of the sines of the angles. 
 
 With the radius JC describe the arc J D E, and tangent 
 to this arc draw FH parallel with MM', or perpendicular 
 to AB. 
 
 Then FD is the tangent of the angle M C N, and D H is 
 the tangent of the angle NCM'. 
 
 Now since an angle at the circumference is equal to half 
 the angle at the centre standing on the same arc (Art. 355), 
 therefore the measure of the angle M C N is the half of M N, 
 equals 
 
 -B). 
 Similarly, we have 
 
 for the angle NCM'. 
 
 Therefore we have for the tangent of the angle M C N 
 
 -B\ 
 
 and, for the tangent of the angle N C M'- 
 DH = tan. \(A + B). 
 
 And, because FC D and M C R are homologous triangles, as, 
 also, DCH and R'CM', therefore 
 
 M' R : MR : : DH : D F, 
 
SUM AND DIFFERENCE OF TWO ANGLES. 525 
 
 sin. A 4- sin. B : sin. A sin. B : : tan. %(A + B) : tan. (A B), 
 
 from which we have 
 
 sin. A sin. B __ tan. % (A B) ,~ . 
 
 sin. ^4 + sin. ./? . tan. % (A + B)' 
 
 To obtain a proper substitute for the first member of this 
 expression we have, equation (195.) 
 
 sin. C 
 
 / 7 > 
 
 C- * LI ; ~ . 
 
 sin. A 
 or 
 
 c sin. A a sin. C. (M.) 
 
 We also have, equation (196.) 
 
 ,sin. C 
 
 c = b -. , 
 
 sin. B 
 
 or 
 
 c sin. B b sin. C. (N.) 
 
 These two equations, (M.) and (N.), added, give 
 
 c sin. A -t- c sin. B a sin. (7 + b sin. C 
 or 
 
 c (sin. ^ + sin. B) = sin. C( + b). (P.) 
 
 But, if equation (N.) be subtracted from equation (M.), we 
 have 
 
 c sin. A c sin. B a sin. C b sin. C, 
 or 
 
 f(sin. A sin. B) = A sin. C (a ).' (R.) 
 
 If equation (R.) be divided by equation (P.), we have 
 
 <r(sin. A sin. B) __ sin. C(a b) 
 <r(sin. A -f sin. B) ~ sin. C (a + b) ' 
 
526 TRIGONOMETRY. 
 
 which reduces to 
 
 sin. A sin. B _ a b 
 sin. A + sin. B ~ a + b' 
 
 The first member of this equation is identical with the first 
 member of the above equation (D.), and therefore its equal, 
 the second member, may be substituted for it ; thus 
 
 a b _ tan, j- (A - B) 
 a +~b~ tan. ^(A~+^)" 
 
 From which we have t 
 
 tan. $(A) = tan. J (A + B) ~ . (203.) 
 
 We have (Art. 431) the proposition, that if half the differ- 
 ence of two quantities be subtracted from half their sum, the 
 remainder will equal the smaller quantity. For example : 
 if A represent the larger quantity and B the smaller, then 
 
 )> Jfj (204.) 
 
 and, again, we also have (Art. 431) 
 
 B)=A. (205.) 
 
 480. Oblique-Angled Triangles; : Third < In**. The 
 
 third class of problems comprises all those cases in which two 
 sides of a triangle and their included angle are given, to 
 find the other side and angles. 
 
 In this case, as in the problems of the second class, the 
 only requirement here is to find a second angle ; for then 
 the problem becomes one belonging to the first class. But 
 the finding of the second angle, in problems of the third 
 class, is attended with more computation than it is in prob- 
 lems of the second class. The process is as follows : Hav- 
 ing one angle of a triangle, the sum of the two remaining 
 
OBLIQUE-ANGLED TRIANGLES. 527 
 
 angles is obtained by subtracting the given angle from 
 1 80 the sum of the three angles. 
 
 Then with equation (203.) the difference of the two angles 
 is obtained. And then, having the sum and difference of the 
 two angles, either may be found by one of the equations 
 (204.) and (205.). 
 
 For example : let Fig. 320 represent the triangle in 
 which a (= 36 feet) and b(= 27 feet) are the given sides ; and 
 
 C (= 105) the angle included between the given sides, a and 
 b. The sum of the two angles A and B, therefore, will be 
 
 (A + ) = 180- 105 ^75, 
 
 and the half of the sum of A and B is -V - 37 30'. 
 
 The sum of the given sides is 36 + 27 = 63, and their dif- 
 ference is 36 27 = 9. 
 
 Then from equation (203.) we have 
 
 . tan. (A-B) = tan. 37 
 Solving this by logs. (Art. 427), we have- 
 
 Log. tan. 37 30' = 9-8849805 
 9 =0-9542425 
 
 0-8392230 
 
 63 = I-79934Q5 
 
 tan. $(A- B)(= 6 15' 20-5") = 9-0398825 
 
 Thus half the difference of A and B is 6 15' 20- 5", nearly, 
 
528 TRIGONOMETRY. 
 
 By equation (204.) 
 
 37 30' 
 6 15' 20- 5" 
 
 The difference, 31 14' 39'$" 
 and by equation (205.) 
 
 37-30 
 6.15.20-5 
 
 The sum, 43.45.20-5 A 
 
 From above, 31.14.39-5 ^ 
 
 The given angle, 105. o. o C 
 
 The three angles, 180. o. o 
 
 Thus, by adding together the three angles, the work is 
 tested and proved. 
 
 Having the three angles, the third side may now be 
 found by the rule for problems of the first class. 
 
 Oblique- Angled Triangles: Fourth Class. The 
 
 fourth class comprises those problems in which the three 
 sides of the triangle are given, to find the three angles. 
 
 The method by which the problems of the fourth class 
 are solved is to divide the triangle into two right-angled 
 triangles; then, by the use of equation (129.), to find one 
 side of one of these triangles, and then with this side to find 
 one of the angles, then by rules for the second class prob- 
 lems, obtain the second and third angles. 
 
 Thus, from equation (129.), we have 
 
 By the relation of sines to sides (Art. 476), we have (Fig. 
 b : g : : sin. E : sin. F. 
 
TRIANGLES FOURTH CLASS. 
 
 529 
 
 But the angle E is a right angle, of which the sine is unity, 
 therefore^ 
 
 b : g : : I : sin. F = -. 
 
 Substituting for g its value as above, we have 
 
 *sin. F = 
 
 q- b) 
 
 2bc 
 
 (206.) 
 
 To illustrate: let a, b, c (Fig. 321) be the three given sides 
 
 of the triangle ABC, respectively equal to 12,' 8 and 16 feet. 
 With these, equation (206.) becomes 
 
 I 6 2 -(i2 + 8)(i2-8) 
 
 sin. F = *-*-< ^ , 
 
 2 x 8 x 16 
 
 sin. /* = 
 
 sin. F 
 
 176 
 
 Solving this by logarithms (Art. 427), we have- 
 Log. 176 = 2-2455127 
 " 256 = 2-4082400 
 
 Log. sin. 43 26' = 9*8372727 
 or, the angle at F equals 43 26', nearly. Of the triangle 
 
530 
 
 TRIGONOMETRY. 
 
 A C E (Fig. 321), E is a right angle, therefore the sum of F 
 and A, the two remaining angles, equals 90 (Art. 346). 
 Hence, for the angle at A, we have 
 
 =90 -43 26' = 46 34'- 
 
 We now have two sides a and b and A, an angle opposite 
 to one of them, to find B, a second angle. For this, equa- 
 tion (199.) is appropriate. Thus 
 
 sin. B = sin. A . 
 a 
 
 This may be solved as shown in Art. 478. 
 
 And, when the second angle is obtained, the third angle 
 is found by subtracting the sum of the first and second an- 
 gles from 1 80. 
 
 But to test the accuracy of the work, it is well to com- 
 pute the angle 'C from the angle A, and the sides a and c. 
 For this, equation (201.) will be appropriate. 
 
 482. Trigonometric Formulae: Right- Angled Trian- 
 gles. For facility of reference the formulas of previous 
 
 articles are here presented in tabular form. The symbols 
 referred to are those of Fig. 322. 
 
FORMULA IN TABULAR FORM. 
 
 531 
 
 RIGHT-ANGLED TRIANGLES. 
 
 GIVEN. 
 
 REQUIRED. 
 
 FORMULA. 
 
 a, b, 
 a, c, 
 b, c, 
 
 ; 
 
 
 
 c = fV+J*. 
 
 = V(c + *) (c 
 
 -a). 
 
 a = V(c + ^) (^ 
 
 -b). 
 
 A, 
 
 A, 
 
 = 90 -A. 
 A = 90 ^. 
 
 B, a, 
 
 *' 
 
 ^ = tan. ^. 
 
 rt: 
 
 
 " cos. B ' 
 
 B,b, 
 
 *, 
 
 b 
 
 
 tan. ^ * 
 b 
 
 B.,. 
 
 a, 
 b, 
 
 a = c cos. B, 
 b c sin. B. 
 
 483. Trigonometrical Formulae: First Class, Oblique. 
 
 C 
 
 The symbols of the formulae of the following- table indi- 
 cate quantities represented in Fig. 323 by like symbols. 
 
532 
 
 TRIGONOMETRY. 
 
 OBLIQUE-ANGLED TRIANGLES: FIRST CLASS. 
 
 GIVEN, i REQUIRED. 
 
 FORMULA. 
 
 ; 
 
 
 A "D /~> 
 
 fJ. y Jj) O. 
 
 ^=180 ^+^. 
 
 A, C, B, 
 
 B = iSo-A + C. 
 
 B, C, A, 
 
 A= 180 - B + C. 
 
 A B b \ a 
 
 ..sin. A 
 
 n. t jj f t/y \ u-, 
 
 [ 
 
 A C c \ a 
 
 sin. B' 
 sin. A 
 
 JJ.) W', C-, | C*, 
 
 ' . W- <s ^-, 
 
 sm. (7 
 
 1 
 A B // h 
 
 , sin. ^ 
 
 A /7 
 
 f* 9 JJj l*j \ I/, 
 
 1 
 
 B C c b 
 
 sm. ^4 
 , sin. B 
 
 h r 
 
 U) C-, t, C/j 
 
 sm. C 
 
 A C /r r 
 
 sin. 7 
 
 si, L,, a, c y 
 
 \ 
 
 sin. A 
 ,sin. 7 
 
 '*''. ' 
 
 t u . Ty . 
 
 sm. j5 
 
 484. Trigonometrical Formulae : Second Cla, Oblique. 
 
 The symbols in the formulae of the following table refer 
 to quantities represented in Fig. 323, by like symbols. 
 
FORMULAE FOR TRIANGLES, SECOND CLASS. 533 
 
 OBLIQUE-ANGLED TRIANGLES: SECOND CLASS. 
 
 1 
 
 GIVEN. 
 
 REQUIRED 
 
 FORMULAE. 
 
 B, a, b, 
 
 A, 
 
 sin. A = sin. B . 
 b 
 
 C, a, c, 
 
 A, 
 
 sin. A sin. C . 
 c 
 
 A, a, b, 
 
 By 
 
 sin. B = sin. A . 
 a 
 
 C, b, c, 
 
 By 
 
 sin. B = sin. C . 
 c 
 
 A, a, c, 
 
 c, 
 
 sin. C = sin. A . 
 a 
 
 B, b, c, 
 
 Cy 
 
 sin. C sin. B-. 
 
 B, C, 
 
 A, 
 
 
 A 180 B + C. 
 
 A, C, 
 Ay By 
 
 By 
 
 B = 1 80 A + C. 
 
 C = 1 80 A + B. 
 
 For 
 
 
 
 ay 
 
 See Formulae, First Class. 
 
534 
 
 TRIGONOMETRY. 
 
 485. Trigonometrical Formulae : Third Class, Oblique. 
 
 The symbols in the formulae of the following table refer 
 to quantities shown by like symbols in Fig. 323. 
 
 OBLIQUE-ANGLED TRIANGLES: THIRD CLASS. 
 
 GIVEN. 
 
 REQUIRED. 
 
 FORMULA. 
 
 C, a, b, 
 
 A +B, 
 
 A- B, 
 
 A, 
 
 tan. $(A-B) = tan. $(A+ B) ~-^. 
 
 
 B, 
 
 B =$(A+B) i(A B). 
 
 A, b, c, 
 
 C +B, 
 C-B, 
 
 C + B = 1 80 A. 
 
 c b 
 tan. 4- (C B) = tan. (C + B) -. . 
 c + b 
 
 r 1 ( f JL. 7?'\ _J_ 1 /" f 7?\ 
 
 D 1 ( i^ i /?\ 1 /'/^ J2\ 
 -D "2" Ivx T *'l 2 \^ **1* 
 
 B, a, c, 
 
 C + A. 
 r A 
 
 C + A 180 B. 
 
 ft 
 
 U SI, 
 
 c, 
 
 c + a 
 
 
 A, 
 
 A=t(C + A) + i(C-A). 
 
 For the remaining side consult formulas for the first 
 class. 
 
 486. Trigonometrical Formulae: Fourtli Class, Ob- 
 lique. The symbols in the formulae of the following table 
 refer to quantities shown by like symbols in Fig. 321. 
 
FORMULA FOR TRIANGLES, FOURTH CLASS. 535 
 
 OBLIQUE-ANGLED TRIANGLES: FOURTH CLASS. 
 
 Given , #, c, to find A, B, C. 
 
 A = 90 - F. 
 sin. ^ = sin. A . 
 
 ^ /i c 
 
 sin. c = sin. ^4 -. 
 
 = 180 (A + B}. 
 
SECTION XV. DRAWING. 
 
 487. General Remark. A knowledge of the proper- 
 ties and principles of lines can best be acquired by practice. 
 Although the various diagrams throughout this work may 
 be understood by inspection, yet they will be impressed 
 upon the mind with much greater force, if they are actually 
 drawn out with pencil and paper by the student. Science 
 is acquired by study art by practice ; he, therefore, who 
 would have anything more than a theoretical (which must 
 of necessity be a superficial) knowledge of carpentry and 
 geometry, will provide himself with the articles here speci- 
 fied, and perform all the operations described in the fore- 
 going and following pages. Many of the problems may 
 appear, at the first reading, somewhat confused and intricate ; 
 but by making one line at a time, according to the explana- 
 tions, the student will not only succeed in copying the fig- 
 ures correctly, but by ordinary attention will learn the 
 principles upon which they are based, and thus be able to 
 make them available in any unexpected case to which they 
 may apply. 
 
 488. Articles Required. The following articles are 
 necessary for drawing, viz. : a drawing-board, paper, draw- 
 ing-pins or mouth-glue, a sponge, a T-square, a set-square, 
 two straight-edges, or flat rulers, a lead pencil, a piece of 
 india-rubber, a cake of india-ink, a set of drawing-instru- 
 ments, and a scale of equal parts. 
 
 489. The Drawing-Board. The size of the drawing- 
 board must be regulated according to the size of the draw- 
 ings which are to be made upon it. Yet for ordinary prac- 
 tice, in learning to draw, a board about fifteen by twenty 
 inches, and one inch thick, will be found large enough, and 
 
DRAWING PAPER. 537 
 
 more convenient than a larger one. This board should be 
 well seasoned, perfectly square at the corners, and without 
 clamps on the ends. A board is better without clamps, 
 because the little service they are supposed to render by 
 preventing the board from warping is overbalanced by the 
 consideration that the shrinking of the panel leaves the 
 ends of the clamps projecting beyond the edge of the board, 
 and thus interfering with the proper working of the stock 
 of the T-square. When the stuff is well-seasoned, the warp- 
 ing of the board will be but trifling ; and by exposing the 
 rounding side to the fire, or to the sun, it may be brought 
 back to its proper shape. 
 
 490. Drawing-Paper. For mere line drawings, it is 
 unnecessary to use the best drawing-paper ; and since, where 
 much is used, the expense will be considerable, it is desirable 
 for economy to procure a paper of as low a price as will be 
 suitable for the purpose. The best paper is made in Eng- 
 land and water-marked " Whatman." This is a hand-made 
 paper. There is also a machine-made paper at about half- 
 price, and the manilla paper, of various tints of russet color, 
 is still less in price. These papers are of the various sAzes 
 needed, and are quite sufficient for ordinary drawings. 
 
 49 1. To Secure the Paper to the Board. A drawing- 
 pin is a small brass button, having a steel pin projecting from 
 the underside. By having one of these at each corner, the 
 paper can be fixed to the board ; but this can be done in a 
 better manner with moutJi-glue. The pins will prevent the 
 paper from changing its position on the board ; but, more 
 than this,, the glue keeps the paper perfectly tight and 
 smooth, thus making it so much the more pleasant to work 
 on. 
 
 To attach the paper with mouth-glue, lay it with the 
 bottom side up, on the board ; and with a straight-edge and 
 penknife cut off the rough and uneven edge. With a 
 sponge moderately wet rub all the surface* of the paper, 
 except a strip around the edge about half an inch wide. As 
 soon as the glistening of the water disappears turn the sheet 
 
DRAWING. 
 
 over and place it upon the board just where you wish it 
 glued. Commence upon one of the longest sides, and pro- 
 ceed thus : lay a flat ruler upon the paper, parallel to the 
 edge, and within a quarter of an inch of it. With a knife, 
 or anything similar, turn up. the edge of the paper against 
 the edge of the ruler, and put one end of the cake of mouth- 
 glue between your lips to dampen it. Then holding it up- 
 right, rub it against and along the entire edge of the paper 
 that is turned up against the ruler, bearing moderately 
 against the edge of the ruler, which must be held firmly 
 with the left hand. Moisten the glue as often as it becomes 
 dry, until a sufficiency of it is rubbed on the edge of the 
 paper. -Take away the ruler, restore the turned-up edge to 
 the level of the board, and lay upon it a strip of pretty stiff 
 paper. By rubbing upon this, not very hard but pretty 
 rapidly, with the thumb-nail of the right hand, so as to cause 
 a gentle friction and heat to be imparted to the glue that is 
 on the edge of the paper, you will make it adhere to the 
 board. The other edges in succession must be treated in 
 the same manner. 
 
 Some short distances along one or more of the edges 
 may afterward be found loose ; if so, the glue must again 
 be applied, and the paper rubbed until it adheres. The 
 board must then be laid away in a warm or dry place ; and 
 in a short, time the surface of the paper will be drawn out, 
 perfectly tight and smooth, and ready for use. The paper 
 dries best when the board is laid level. When the drawing 
 is finished lay a straight-edge upon the paper and cut it 
 from the board, leaving the glued strip still attached. This 
 may afterward be taken off by wetting it freely with the 
 sponge, which will soak the glue and loosen the paper. Do 
 this as soon as the drawing is taken off, in order that the 
 board may be dry when it is wanted for use again. Care 
 must be taken that, in applying the glue, the edge of the 
 paper does not become damper than the rest ; if it should, 
 the paper must be laid aside to dry (to use at another time) 
 and another sheet be used in its place. 
 
 Sometimes, especially when the draAving-board is new, 
 the paper will not stick very readily ; but by persevering 
 
THE T-SQUARE. 539 
 
 this difficulty may be overcome. In the place of the mouth- 
 glue a strong solution of gum-arabic may be used, and on 
 some accounts is to be preferred ; for the edges of the paper 
 need not be kept dry, and it adheres more readily. Dissolve 
 the gum in a sufficiency of warm water to make it of the 
 consistency of linseed-oil. It must be applied to the paper 
 with a brush, when the edge is turned up against the ruler, 
 as was described for the mouth-glue. If two drawing-boards 
 are used, one may be in use while the other is laid away to 
 dry; and as' they may be cheaply made, it is advisable to 
 have two. The drawing-board having a frame around it, 
 commonly called a panel board, may afford rather more 
 facility in attaching the paper when this is of the size to 
 
 FIG. 324. 
 
 suit ; yet it has objections which overbalance that consid- 
 eration. 
 
 492. The T-Square. A T-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 three eighths of an 
 inch thick ; the blade, twenty inches long (exclusive of the 
 stock), two inches wide, and one eighth of an inch thick. In 
 joining the blade to the stock, a very firm and simple joint 
 may be made by dovetailing it as shown at Fig. 324. 
 
 493. Tlie Set-Square. The set-square is in the form of 
 a right-angled triangle ; and is commonly made of mahogany, 
 
540 DRAWING. 
 
 one eighth of an inch in thickness. The size that is most 
 convenient for general use is six inches and three inches 
 respectively for the sides which contain the right angle, 
 although a particular length for the sides is by no means 
 necessary. Care should be taken to have the square corner 
 exactly true. This, as also the T-square and rulers, should 
 have a hole bored through them, by which to hang them 
 upon a nail when not in use. 
 
 494. The Rulers. One of the rulers may be about 
 twenty inches long, and the other six inches. The pencil 
 ought to be hard enough to retain a fine point, and yet not 
 so hard as to leave ineffaceable marks. It should be used 
 lightly, so that the extra marks that are not needed when 
 the drawing is inked, may be easily rubbed off with the 
 rubber. The best kind of india-ink is that which will easily 
 rub off upon the plate ; and, when the cake is rubbed against 
 the teeth, will be free from grit. 
 
 495. The Instruments. The drawing-instruments may 
 be purchased of mathematical instrument makers at various 
 prices ; from one to one hundred dollars a set. In choosing 
 a set, remember that the lowest price articles are not always 
 the cheapest. A set, comprising a. sufficient number of 
 instruments for ordinary use, well made and fitted in a ma- 
 hogany box, may be purchased of the mathematical instru- 
 ment makers in New York for four or five dollars. But for 
 permanent use those which come at ten or twelve dollars 
 will be found to be better. 
 
 496. The Scale of Equal Parts. The best scale of 
 equal parts for carpenters' use, is one that has one eighth, 
 three sixteenths, one fourth, three eighths, one half, five 
 eighths, three fourths, and seven eighths of an inch, arid one 
 inch, severally divided into twelfths, instead of being divided, 
 as they usually are, into tenths. By this, if it be required 
 to proportion a drawing so that every foot of the object 
 represented will upon the paper measure one fourth of an 
 inch*, use that part of the scale which is divided into one 
 
THE SET-SQUARE. 541 
 
 fourths of an inch, taking for every foot one of those divis- 
 ions, and for every inch one of the subdivisions into twelfths ; 
 and proceed in like manner in proportioning a drawing to 
 any of the other divisions of the scale. An instrument in 
 the form of a semi-circle, called a protractor, and used for 
 laying down and measuring angles, is of much service to 
 surveyors, and occasionally to carpenters. 
 
 497. The Ue of the Set-Square. In drawing parallel 
 lines, when they are to be parallel to either side of the 
 board, use the T-square ; but when it is required to draw 
 lines parallel to a line which is drawn in a direction oblique 
 
 FIG. 325. 
 
 to either side of the board, the set-square must be used. 
 Let ab (Fig. 325) be a line, parallel to which it is desired to 
 draw one or more lines. Place any edge, as c d, of the set- 
 square even with said line ; then place the ruler gh against 
 one of the other sides, as ce, and hold it firmly ; slide the 
 set-square along the edge of the ruler as far as it is desired, 
 as at /; and a line drawn by the edge *'/ will be parallel 
 to a b. 
 
 To draw a line, as kl (Fig. 326), perpendicular to another, 
 as a b, set the shortest edge of the set-square at the line a b ; 
 place the ruler against the longest side (the hypothenuse of 
 the right-angled triangle); hold the ruler firmly, and slide 
 the set-square along until the side ed touches the point k\ 
 then the line Ik, drawn by it, will be perpendicular to ab. 
 
542 DRAWING. 
 
 In like manner, the drawing of other problems may be facil- 
 itated, as will be discovered in using the instruments. 
 
 498. Directions for Drawing. In drawing a problem, 
 proceed, with the pencil sharpened to a point, to lay down 
 the several lines until the whole figure is completed, ob- 
 serving to let the lines cross each other at the several angles, 
 instead of merely meeting. By this, the length of every 
 line will be clearly defined. With a drop or two of water, 
 rub one end of the cake of ink upon a plate or saucer, until 
 a sufficiency adheres to it. Be careful to dry the cake of 
 
 FIG 326. 
 
 ink ; because if it is left wet it will crack and crumble in 
 pieces. With an inferior camel's-hair pencil add a little 
 water to the ink that was rubbed on the plate, and mix it 
 well. It should be diluted sufficiently to flow freely from 
 the pen, and yet be thick enough to make a black line. With 
 the hair pencil place a little of the ink between the nibs of 
 the drawing-pen, and screw the nibs together until the pen 
 makes a fine line. Beginning with the curved lines, proceed 
 to ink all the lines of the figure, being careful now to make 
 every line of its requisite length. If tl.ey are a trifle too 
 short or too long the drawing will have a ragged appear- 
 ance ; and this is opposed to that neatness and accuracy 
 which is indispensable -to a good drawing. When the ink 
 is dry efface the pencil-marks with the india-rubber. If the 
 
PUTTING THE DRAWING IN INK. 543 
 
 pencil is used lightly they will all rub off, leaving those lines 
 only that were inked. 
 
 In problems all auxiliary lines are drawn light ; while the 
 lines given and those sought, in order to be distinguished at 
 a glance, are made much heavier. The heavy lines are 
 made so by passing over them a second time, having the 
 nibs of the pen separated far enough to make the lines as 
 heavy as desired. If the heavy lines are made before the 
 drawing is cleaned with the rubber they will not appear so 
 black and neat, because the india-rubber takes away part 
 of the ink. If the drawing is a ground-plan or elevation of 
 a house, the shade-lines, as they are termed, should not be 
 put in until the drawing is shaded ; as there is danger of the 
 heavy lines spreading when the brush, in shading or color- 
 ing, passes over them. If the lines are inked with common 
 writing-ink they will, however fine they may be made, be 
 subject to the same evil ; for which reason india-ink is the 
 only kind to be used. 
 
SECTION XVI. PRACTICAL GEOMETRY. 
 
 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, perfectly straight and even in every 
 direction ; as the face of a panel when not warped nor 
 winding. 
 
 A solid has length, breadth, and thickness. 
 
 A right, or straight, line is the shortest that can be drawn 
 between two points. 
 
 Parallel lines are equidistant throughout their length. 
 
 FIG. 327. FIG. 328. FIG. 329. 
 
 An angle is the inclination of two lines towards one an- 
 other (Fig. 327). 
 
 A right angle has one line perpendicular to the other 
 (Fig. 328). 
 
 An oblique angle is either greater or less than a right 
 angle (Figs. 327 and 329). 
 
 An acute angle is less than a right angle (Fig. 327). 
 
 An obtuse angle is greater than a right angle (Fig. 329). 
 
 When an angle is denoted by three letters, the middle 
 one, in the order they stand, denotes the angular point, and 
 the other two the sides containing the angle ; thus, let a, b, c 
 (Fig. 327) be the angle, then b will be the angular point, and 
 ab and be will be the two sides containing that angle. 
 
TRIANGLES AND RECTANGLES. 
 
 545 
 
 A triangle -is a superficies having three sides and angles 
 (Figs. 330, 331, 332, and 333). 
 
 An equilateral triangle has its three sides equal (Fig. 330). 
 An isosceles triangle has only two sides equal (Fig. 331). 
 
 FIG. 330. 
 
 FIG 331. 
 
 A scalene triangle has all its sides unequal (Fig. 332). 
 A right-angled triangle has one right angle (Fig. 333). 
 An acute-angled triangle has all its angles acute (Figs. 330 
 and 331). 
 
 FIG. 332. 
 
 FIG. 333. 
 
 An obtuse-angled triangle has one obtuse angle (Fig. 332). 
 A quadrangle has four sides and four angles (Figs. 334 to 
 
 339). 
 
 A parallelogram is a quadrangle having its opposite sides 
 
 parallel (Figs. 334 to 337). 
 
 FIG. 334- 
 
 FIG. 335. 
 
 A rectangle is a parallelogram, its angles being right 
 angles (Figs. 334 and 335). 
 
 A square is a rectangle having equal sides (Fig. 334). 
 
 A rhombus is an equilateral parallelogram having oblique 
 angles (Fig. 336). 
 
546 PRACTICAL GEOMETRY. 
 
 A rhomboid is a parallelogram having oblique angles 
 
 (Fig- 337). 
 
 A trapezoid is a quadrangle having only two of its sides 
 parallel (Fig. 338). 
 
 FIG. 336. FIG. 337. 
 
 A trapezium is a quadrangle which has no two of its sides 
 parallel (Fig. 339). 
 
 A polygon is a figure bounded by right lines. 
 
 A regular polygon has its sides and angles equal. 
 
 An irregular polygon has its sides and angles unequal. 
 
 FIG. 338. FIG. 339. 
 
 A trigon is a polygon of three sides (Figs. 330 to 333) ; a 
 tetragon has four sides (Figs. 334 to 339) ; a pentagon has five 
 (Fig. 340) ; a hexagon six (Fig. 341) ; a heptagon seven (/%-. 
 342) ; an octagon eight (/ r /~. 343) ; a nonagon nine ; a decagon 
 ten ; an undecagon eleven ; and a dodecagon twelve sides. 
 
 FIG. 340. FIG. 341. FIG. 342. FIG. 343. 
 
 A circle is a figure bounded by a curved line, called the 
 circumference, which is everywhere equidistant from a cer- 
 tain point within, called its centre. 
 
 The circumference is also called the periphery, and some- 
 times the circle. 
 
PARTS OF THE CIRCLE. 
 
 547 
 
 The radius of a circle is a right line drawn from the 
 centre to any point in the circumference (ab, Fig. 334). 
 
 All the radii of a circle are equal. 
 
 The diameter is a right line passing through the centre, 
 and terminating at two opposite points in the circumference. 
 Hence it is twice the length of the radius (cd. Fig. 344.) 
 
 FIG. 344. 
 
 An arc of a circle is a part of the circumference (cb, or 
 bed, Fig. 344). 
 
 A chord is a right line joining the extremities of an arc 
 (b d, Fig. 344). 
 
 A segment is any part of a circle bounded by an arc and 
 its chord (A, Fig. 344). 
 
 FIG. 345. 
 
 A sector is any part of a circle bounded by an arc and 
 two radii, drawn to its extremities (B, Fig. 344). 
 
 A quadrant, or quarter of a circle, is a sector having a 
 quarter of the circumference for its arc (C, Fig. 344). 
 
 A tangent is a right line which, in passing a curve, 
 touches, withont cutting it (fg, Fig. 344). 
 
PRACTICAL GEOMETRY. 
 
 A cone is a solid figure standing upon a circular base di- 
 minishing in straight lines to a point at the top, called its 
 vertex (Fig: 345). 
 
 The axis of a cone is a right line passing through It, 
 from the vertex to the centre of the circle at the base. 
 
 An ellipsis is described if a cone be cut by a plane, not 
 parallel to its base, passing quite through the curved surface 
 (a b, Fig. 346). 
 
 A parabola is described if a cone be cut by a plane, par- 
 allel to a plane touching the curved surface (c d, Fig. 346 
 cd being parallel to f g\ 
 
 An hyperbola is described if a cone be cut by a plane, 
 
 FIG. 347. t 
 
 parallel to any plane within the cone that passes through its 
 vertex (e/t, Fig. 346). 
 
 Foci are the points at which the pins are placed in de- 
 scribing an ellipse (see Art. 548, and /,/, Fig. 347). 
 
 The transverse axis is the longest diameter of the ellipsis 
 (a b, Fig. 347). 
 
 The conjugate axis is the shortest diameter of the ellipsis ; 
 and is, therefore, at right angles to the transverse axis (cd, 
 
 Fig. 347). 
 
 The parameter is a right line passing through the focus 
 of an ellipsis, at right angles to the transverse axis, and ter- 
 minated by the curve (gk and gt, Fig. 347). 
 
RIGHT LINES AND ANGLES. 
 
 549 
 
 A diameter of an ellipsis is any right line passing through 
 the centre, and terminated by the curve (kl, or ;;/ n, Fig. 347). 
 
 A diameter is conjugate to another when it is parallel to a 
 tangent drawn at the extremity of that other thus, the di- 
 ameter mn (Fig. 347) being parallel to the tangent op, is 
 therefore conjugate to the diameter kl. 
 
 A double ordinate is any right line, crossing a diameter of 
 an ellipsis, and drawn parallel to a tangent at the extremity 
 of that diameter (it, Fig. 347). 
 
 A cylinder is a solid generated by the % revolution of a 
 right-angled parallelogram, or rectangle, about one of its 
 
 FIG. 348. 
 
 FIG. 349. 
 
 sides ; and consequently the ends of the cylinder are equal 
 circles (Fig. 348). 
 
 The axis of a cylinder is a right line passing through it 
 from the centres of the two circles which form the ends. 
 
 A segment of a cylinder is comprehended under three 
 planes, and the curved surface of the cylinder. Two of 
 these are segments of circles ; the other plane is a parallelo- 
 gram, called by way of distinction, the plane of tJie segment. 
 The circular segments are called the ends of the cylinder 
 
 (Fig. 349)- 
 
 PROBLEMS. 
 
 RIGHT LINES AND ANGLES. 
 
 500. To Biect 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 
 
550 
 
 PRACTICAL GEOMETRY. 
 
 draw the line cd, and the point e, where it cuts ab } will be 
 the middle of the line ab. 
 
 In practice, a line is generally divided with the com- 
 passes, or dividers ; but this problem is useful where it is 
 
 desired to draw, at the middle of another line, one at right 
 angles to it. (See Art. 514.) 
 
 501. To Erect a Perpendicular. From the point a 
 (Fig. 351) set off any distance, as ab, and the same distance 
 from a to c ; upon c, as a centre, with any distance for radius 
 greater than ca, describe an arc at d\ upon b, with the same 
 
 FIG. 351. 
 
 radius, describe another at d\ join d and a, and the line da 
 will be the perpendicular required. 
 
 This, and the three following problems, are more easily 
 performed by the use of the set-square (see Art. 493). Yet 
 they are useful when the operation is so large that a set- 
 square cannot be used. 
 
TO ERECT A PERPENDICULAR. 
 
 551 
 
 502. To let Fall a Perpendicular. Let a (Fig. 352) be 
 the point above the line be from which the perpendicular is 
 required to fall. Upon a, with any radius greater than ad, 
 describe an arc, cutting be at ^and /; upon the points e and 
 /, with any radius greater than ed, describe arcs, cutting 
 
 FIG. 352. 
 
 each other at g\ join a and g, and the line ad will be the 
 perpendicular required. 
 
 503. To Erect a Perpendicular at the End of a Line. 
 
 Let a (Fig. 353), at the end of the line c a, be the point at 
 which the perpendicular is to be erected. Take any point, 
 as by above the line ca, and with the radius ba describe the 
 arc dae; through d and b draw the line de\ join e and a, 
 then e a will be the perpendicular required. 
 
 FIG. 353- 
 
 The principle here made use of is a very important one, 
 and is applied in many other cases (see Art. 510, 3d, and Art. 
 513. For proof of its correctness, see Art. 352). 
 
 A second method. Let b (Fig. 354), at the end of the line 
 a b, be the point at which it is required to erect a perpendic- 
 ular. Upon b, with any radius less than b a, describe the arc 
 ced\ upon c t with the same radius, describe the small arc at*/ 
 
552 
 
 PRACTICAL GEOMETRY. 
 
 and upon c, another at d ; upon e and d, with the same or any 
 other radius greater than half e d, describe arcs intersecting 
 at f' join /and b, and the line fb will be the perpendicular 
 required. This method of erecting a perpendicular, and 
 that of the following article, depend for accuracy upon the 
 
 c b 
 
 FIG. 354. 
 
 fact that the side of a hexagon is equal to the radius of the 
 circumscribing circle. 
 
 A third method. Let b (Fig. 355) be the given point at 
 which it is required to erect a perpendicular. Upon b, with 
 any radius less than ba, describe the quadrant def\ upon d, 
 with the same radius, describe an arc at e, and upon e an- 
 other at c ; through d and e draw dc, cutting the arc in c ; 
 join c and 3, then cb will be the perpendicular required. 
 
 d b 
 
 FIG. 355. 
 
 This problem can be solved by the six, eight and ten rule, 
 as it is called, which is founded upon the same principle as 
 the problems at Arts. 536, 537, and is applied as follows: 
 let ad (Fig. 353) equal eight, and ae, six ; then, if de equals 
 ten, the angle cad is a right angle. Because the square of 
 six and that of eight, added together, equal the square of 
 
EQUAL ANGLES. 553 
 
 ten, thus : 6 x 6 = 36, and S x 8 64; 36 + 64 = 100, and 
 10 x 10 = 100. Any sizes, taken in the same proportion, as 
 six, eight and ten, will produce the same effect ; as 3, 4 and 
 5, or 12, 1 6 and 20. (See Art. 536.) 
 
 By the process shown at Fig. 353, the end of a board may 
 be squared without a carpenters'-square. All that is neces- 
 sary is a pair of compasses and a ruler. Let ca be the edge 
 of the board, and a the point at which it is required to be 
 squared. Take the point b as near as possible at an angle 
 of forty-five degrees, or on a mitre-line from a, and at about 
 the middle of the board. This is not necessary to the work- 
 ing of the problem, nor does it affect its accuracy, but the 
 result is more easily obtained. Stretch the compasses from 
 b to a, and then bring the leg at a around to d\ draw a line 
 from d, through b, out indefinitely ; take the distance db and 
 place it from b to c ; join e and a ; then ca will be at right 
 angles to c a. In squaring the foundation of a building, or 
 laying out a garden, a rod and chalk-line may be used in- 
 stead of compasses and ruler. 
 
 504. To let Fall a Perpendicular near the End of a 
 Line. Let e (Fig. 353) be the point above the line c a, from 
 which the perpendicular is required to fall. From e draw 
 any line, as e d, obliquely to the line ca; bisect edai b\ upon 
 b, with the radius be, describe the arc ead\ join e and #; 
 then ea will be the perpendicular required. 
 
 505. To Make an Angle (a edf, Fig. 356*) Equal to a 
 Given Angle (as b a c). From the angular point a, with any 
 
 FIG. 356. 
 
 radius, describe the arc b c ; and with the same radius, on 
 the line dc, and from the point d, describe the arc/^-; take 
 the distance be, and upon g t describe the small arc at/; 
 
554 
 
 PRACTICAL GEOMETRY. 
 
 join f and d\ and the angle edf will be equal to the angle 
 bac. 
 
 If the given line upon which the angle is to be made is 
 situated parallel to the. similar line of the given angle, this 
 may be performed more readily with the set-square. (See 
 Art. 497.) 
 
 506. To Bisect an Angle. Let a be (Fig. 357) be the 
 angle to be bisected. Upon b, with any radius, describe the 
 
 FIG. 357- 
 
 arc a c ; upon a and c, with a radius greater than half a c, 
 describe arcs cutting each other at d; join b and d\ and bd 
 will bisect the angle a be, as was required. 
 
 This problem is frequently made use of in solving other 
 problems ; it should therefore be well impressed upon the 
 memory. 
 
 507 To Trisect a Right Angle. Upon a (Fig. 358), 
 with any radius, describe the arc b c ; upon b and c, with the 
 
 FIG. 358. 
 
 same radius, describe arcs cutting the arc be at d and e\ 
 from d and e draw lines to a, and they will trisect the angle, 
 as was required. 
 
TO DIVIDE A GIVEN LINE 
 
 The truth of this is made evident by the following oper- 
 ation : divide a circle into quadrants ; also, take the radius 
 in the dividers, and space off the circumference. This will 
 divide the circumference into just six parts. A semi-circum- 
 ference, therefore, is equal to three, and a quadrant to one 
 and a half of those parts. The radius, therefore, is equal to 
 two thirds of a quadrant ; and this is equal to a right angle. 
 
 508. Through a Given Point, to Draw a Line Parallel 
 to a Given Line. Let a (Fig. 359) be the given point, and 
 
 FIG. 359. 
 
 be the given line. Upon any point, as d, in the line be, with 
 the radius da, describe the arc ac\ upon a, with the same 
 radius, describe the arc de\ make de equal to ac\ through 
 e and a draw the line ea, which will be the line required. 
 This is upon the same principle as Art. 505. 
 
 509. To Divide a Given Line into any Number of 
 Equal Part. Let a b (Fig. 360) be the given line, and 5 the 
 number of parts. Draw ac at any angle to a b ; on ac, from 
 
 J A' 
 
 FIG. 360. 
 
 a, set off five equal parts of any length, as at i, 2, 3, 4 and c ; 
 join c and b\ through the points I, 2, 3, and 4, draw I e , 2/, 
 3^ and 4//, parallel to cb\ which will divide the line ab, as 
 was required. 
 
556 
 
 PRACTICAL GEOMETRY. 
 
 The lines ab and ac are divided in the same proportion. 
 (See Art. 542.) 
 
 THE CIRCLE. 
 
 510. To Find the Centre of a Circle. Draw any chord, 
 as ab (Fig. 361), and bisect it with the perpendicular cd\ bi- 
 
 sect cd with the line e f, as at g\ then g is the centre, as was 
 required. 
 
 A second method. Upon any two points in the circumfer- 
 ence nearly opposite, as a and b (Fig. 362), describe arcs cut- 
 
 ting each other at c and d; take aay other two points, as e 
 and f y and describe arcs intersecting, as at g and h ; join g 
 and h and c and d\ the intersection o is the centre. 
 
 This is upon the same principle as Art. 514. 
 
 A third method. Draw any chord, as ab (Fig. 363), and 
 from the point a draw ac at right angles to ab ; join c and 
 b\ bisect c b at d which will be the centre of the circle. 
 
A TANGENT AT A GIVEN POINT. 
 
 557 
 
 If a circle be not too large for the purpose, its centre 
 may very readily be ascertained by the help of a carpenters'- 
 square, thus : apply the corner of the square to any point in 
 the circumference, as at a ; by the edges of the square 
 (which the lines ab and ac represent) draw lines cutting the 
 
 FIG. 363. 
 
 circle, as at b and c ; join b and c ; then if be is bisected, as at 
 d, the point d will be the centre. (See Art. 352.) 
 
 5 II At a Given Point in a Circle to Draw a Tangent 
 thereto. Let a (Fig. 364) be the given point, and b the cen- 
 
 FIG. 364. 
 
 tre of the circle. Join a and b ; through the point a, and at 
 right angles to a b, draw cd ; then c d is the tangent required. 
 
 512. The Same, without making use of the Centre of 
 the Circle. Let a (Fig. 365) be the given point. From a set 
 off any distance to b, and the same from b to c ; join a and 
 c ; upon a, with ab for radius, describe the arc dbc-, make 
 db equal to bc\ through a and d draw a line; this will be 
 the tangent required. 
 
558 PRACTICAL GEOMETRY. 
 
 The correctness of this method depends upon the fact 
 that the angle formed by a chord and tangent is equal to any 
 inscribed angle in the opposite segment of the circle (Art. 
 358); ab being the chord, and bca the angle in the opposite 
 segment of the circle. Now, the angles dab and bca are 
 equal, because the angles dab and bac are, by construction, 
 
 FIG. 365. 
 
 equal; and the angles bac and bca are equal, because the 
 triangle abc is an isosceles triangle, having its two sides, ab 
 and be, by construction equal ; therefore the angles dab and 
 bca are equal. 
 
 513. A Circle and a Tangent Given, to Find Hie Point 
 of Contact. From any point, as a (Fig. 366), in the tangent 
 
 FIG. 366. 
 
 be, draw a line to the centre d\ bisect ad at r; upon e, with 
 the radius ea, describe the arc afd\ f is the point of con- 
 tact required. 
 
 If/ and// were joined, the line would form right angles 
 with the tangent be. (See Art. 352.) 
 
A CIRCLE THROUGH GIVEN POINTS. 
 
 559 
 
 514. Through any Three Points not in a Straight Line, 
 to Draw a Circle. Let a, b and c (Fig. 367) be the three 
 given points. Upon a and b, with any radius greater than 
 half a b, describe arcs intersecting at d and e ; upon b and c, 
 with any radius greater than half be, describe arcs intersect- 
 ing at / and g\ through d and e draw a right line, also 
 
 .. FIG. 367. 
 
 another through / and g\ upon the intersection //, with the 
 radius ha, describe the circle a be, and it will be the one re- 
 quired. 
 
 515. Three Points not in a Straight Line being Given, 
 to Find a Fourth that shall, with the Three, Lie in the 
 Circumference of a Circle. Let a b c (Fig. 368) be the given 
 points. Connect them with right lines, forming the triangle 
 
 FIG. 368. 
 
 acb', bisect the angle cba (Art. 506) with the line bd\ also 
 bisect c a in e, and erect ed perpendicular to ac, cutting bd 
 in d; then d is the fourth point required. 
 
 A fifth point may be found, as at /, by assuming a, d and 
 b, as the three given points, and proceeding as before. So, 
 
560 PRACTICAL GEOMETRY. 
 
 also, any number of points may be found simply by using 
 any three already found. This problem will be serviceable 
 in obtaining short pieces of very flat sweeps. (See Art. 240.) 
 The proof of the correctness of this method is found in 
 the fact that equal chords subtend equal angles (Art. 357). 
 Join d and c; then since ae and ec are, by construction, 
 equal, therefore the chords a d and dc are equal ; hence the 
 angles they subtend, dba and d b c, are equal. So, like- 
 wise, chords drawn from a to /, and from / to d, are equal, 
 and subtend the equal angles dbf and fba. Additional 
 points beyond a or b may be obtained on the same principle. 
 To obtain a point beyond a, on b, as a centre, describe with 
 any radius the arc ion ; make on equal to o i ; through b and 
 n draw b g\ on a as centre and with af for radius, describe 
 the arc, cutting gb at g t then g. is the point sought. 
 
 516. To Describe a Segment of a Circle toy a Set-Tri- 
 angle. Let a b (Fig. 369) be the chord, and c d the height 
 
 FIG. 369. 
 
 of the segment. Secure two straight-edges, or rulers, in the 
 position ce and c f, by nailing them together at c, and affixing 
 a brace from c to /; put in pins at a and b ; move the angu- 
 lar point c in the direction acb\ keeping the edges of the 
 triangle hard against the pins a and b ; a pencil held at c 
 will describe the arc acb. 
 
 A curve described by this process is accurately circular, 
 and is not a mere approximation to a circular arc, as some 
 may suppose. This method produces a circular curve, be- 
 cause all inscribed angles on one side of a chord-line are 
 equal (Art. 356). To obtain the radius from a chord and its 
 versed sine, see Art. 444. 
 
 If the angle formed by the rulers at c be a right angle, 
 
TO FIND THE VERSED SINE. 561 
 
 the segment described will be a semi-circle. This problem 
 is useful in describing centres for brick arches, when they 
 are required to be rather flat. Also, for the head hang- 
 ing-stile of a window-frame, where a brick arch, instead of a 
 stone lintel, is to be placed over it. 
 
 517. To Find the Radiu of an Arc of a Circle when 
 the Chord and Vered 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, 
 
 (-}* a 
 algebraically, thus : r = , where r is the radius, c the 
 
 chord, and v the versed sine (Art. 444). 
 
 Example. In a given arc of a circle a chord of 12 feet 
 has the rise at the middle, or the versed sine, equal to 2 feet, 
 what is the radius ? 
 
 Half the chord equals 6, the square of 6 is, 6 x 6 = 36 
 The square of the versed sine is, 2x2=4 
 
 Their sum equals, 40 
 
 Twice the versed sine equals 4, and 40 divided by 4 equals 
 10. Therefore the radius, in this case, is 10 feet. This 
 result is shown in less space and more neatly by using the 
 above algebraical formula. For the letters substituting 
 
 their value, the formula r = i- becomes r = sj 
 
 2V 2X2 
 
 and performing the arithmetical operations here indicated 
 equals 
 
 6 a -f 2 2 _ 36 -t- 4 _ 40 _ 
 
 4 44" 
 
 518. To Find the Vered 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 alge- 
 braically thus : v = r Vr 3 - (l) a , where r is the radius, v 
 the versed sine, and c the chord. (Equation (161.) reduced.) 
 
562 PRACTICAL GEOMETRY. 
 
 Example. In an arc of a circle whose radius is 75 feet, 
 what is the versed sine to a chord of 120 feet? By the table 
 in the Appendix it will be seen that 
 
 The square of the radius, 75, equals . . 5625 
 The square of half the chord, 60, equals . 5600 
 
 The .difference is ..... 2025 
 
 The square root of this is . . . -45 
 
 This deducted from the radius ... 75 
 
 
 The remainder is the versed sine, = 30 
 
 This is expressed by the formula, thus 
 
 v = 75 - ^75 " - FF? = 75 - ^5625-3600 = 75 - 45 = 30. 
 
 519. To Describe the Segment of a Circle by Intersec- 
 tion of Lines. Let ab (Fig. 370) be the chord, and cd the 
 
 height of the segment. Through c draw ef 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 ce, cf t da, db, a g, 
 and bh, each into a like number of equal parts, as four; 
 draw the lines i 1,22, etc., and from the points o, o, and o, 
 draw lines to c\ at the intersection of these lines trace the 
 curve, acb, which will be the segment required. 
 
 In very large work, or in laying out ornamental gar- 
 dens, etc., this will be found useful ; and where the centre 
 of the proposed arc of a circle is inaccessible it will be inval- 
 uable. (To trace the curve, see note at Art. 550.) 
 
 The lines e a, c d, and fb, would, were they extended, 
 meet in a point, and that point would be in the opposite 
 side of the circumference of the circle of which acb is a 
 
ORDINATES TO AN ARC. 
 
 563 
 
 segment. The lines i i, 2 2, 3 3, would likewise, if extended, 
 meet in the same point. The line cd, if extended to the op- 
 posite side of the circle, would become a diameter. The line 
 fb forms, by construction, a right angle with be, and hence 
 the extension of fb would also form a right angle with be, 
 on the opposite side of bc\ and this right angle would be 
 the inscribed angle in the semi-circle ; and since this is re- 
 quired to be a right angle (Art. 352), therefore the construc- 
 tion thus far is correct, and it will be found likewise that at 
 each point in the curve formed by the intersection of the 
 radiating lines, these intersecting lines are at right angles. 
 
 520. Ordinate. Points in the circumference of a 
 circle may be obtained arithmetically, and positively accu- 
 rate, by the calculation of ordinates, or the parallel lines o i, 
 
 -f J 2 / (1725 
 
 FIG. 371. 
 
 02, 03, 04 (Fig. 3/i). These ordinates are drawn at right 
 angles to the chord-line a b, and they may be drawn at any 
 distance apart, either equally distant or unequally, and there 
 may be as many of them as is desirable ; the more there are 
 the more points in the curve will be obtained. If they are 
 located in pairs, equally distant from the versed sine c d, 
 calculation need be made only for those on one side of cd, 
 as those on the opposite side will be of equal lengths, re- 
 spectively ; for example: o i, on the left-hand side of cd, is 
 equal to o i on the right-hand side, o 2 on the right equals 
 o 2 on the left, and in like manner for the others. 
 
 The length of any ordinate is equal to the square root 
 of the difference of the squares of the radius and abscissa, 
 less the difference between the radius and versed sine (Art. 
 445). The abscissa being the distance from the foot of 
 the versed sine to the foot of the ordinate. Algebraically, 
 
564 PRACTICAL GEOMETRY. 
 
 t = Vr * - x* (r b\ where t is put to represent the ordi- 
 nate ; x, the abscissa ; b, the versed sine ; and r, the radius. 
 
 Example. An arc of a circle has its chord ab (Fig. 371) 
 100 feet long, and its versed sine cd, 5 feet. It is required 
 to ascertain the length of ordinates for a sufficient number 
 of points through which to describe the curve. To this end 
 it is requisite, first, to ascertain the radius. This is readily 
 
 /\2 2 
 
 done in accordance with Art. 517. For - becomes 
 
 - = 252-5 = radius. Having the radius, the curve 
 
 2x5 
 
 might at once be described without the ordinate points, but 
 for the impracticability that usually occurs, in large, flat 
 segments of the circle, of getting a location for the centre, 
 the centre usually being inaccessible. The ordinates are, 
 therefore, to be calculated. In Fig. 371 the ordinates are 
 located equidistant, and are 10 feet apart. It will only 
 be requisite, therefore, to calculate those on one side of 
 the versed sine cd. For the first ordinate 01, the formula 
 / = Vr^ x* (r &) becomes 
 
 2 - io 2 -(252-5 -5). 
 
 = 1/63756-25 100 247.5. 
 
 252.3019-247.5. 
 
 4.8019 = the first ordinate, o i. 
 For the second 
 
 t ^252- 5 2 20 2 (252-5 5). 
 = 251-7066 247.5. 
 
 4-2066 = the second ordinate, 02. 
 For the third 
 
 / = 1/^52-5 a - 3 o 2 - 247. 5. 
 = 250-7115-247.5. 
 
 3-2115 = the third ordinate, 03. 
 
TO DESCRIBE A TANGED CURVE. 565 
 
 For the fourth 
 
 / = 1/252. 5 2 -40 2 - 247- 5. 
 = 249-3115 - 247.5. 
 
 1-8115 = the fourth ordinate, o 4. 
 
 The results here obtained are in feet and decimals of a 
 foot. To reduce these to feet, inches, and eighths of an 
 inch, proceed as at Reduction of Decimals in the Appendix. 
 If the two-feet rule, used by carpenters and others, were 
 decimally divided, there would be no necessity of this re- 
 duction, and it is to be hoped that the rule will yet be thus 
 divided, as such a reform would much lessen the labor of 
 computations, and insure more accurate measurements. 
 
 Versed sine c d = ft. 5 o = ft. 5 -o inches. 
 Ordinates o I = 4-8019= 4-9! inches, nearly. 
 
 " 02= 4-2066 = 4. 2\ inches, nearly. 
 
 " 03= 3-2115 3 -2j inches, nearly. 
 
 " 04= 1-8115 = I- 9f inches, nearly. 
 
 521. In a Given Angle, to Describe a Tanged Curve. 
 
 Let a b c (Fig. 372) be the given angle, and I in the line a b, 
 
 FIG. 372. 
 
 and 5 in the line be, the termination of the curve. Divide 
 i b and b 5 into a like number of equal parts, as at i, 2, 3, 4, 
 and 5 ; join i and i, 2 and 2, 3 and 3, etc. ; and a regular 
 curve wiU be formed that will be tangical to the line ab, at 
 the point i, and to be at 5. 
 
 This is of much use in stair-building, in easing the angles 
 formed between the wall-string and the base of the hall, 
 also between the front string and level facia, and in many 
 other instances. The curve is not circular, but of the form 
 of the parabola (Fig. 418) ; yet in large angles the difference 
 
PRACTICAL GEOMETRY. 
 
 is not perceptible. This problem can be applied to describ- 
 ing- the curve for door-heads, window-heads, etc., to rather 
 better advantage .than Art. 516. For instance, let ab (Fig. 
 373) be the width of the opening, and c d the height of the 
 
 c b 
 
 FIG. 373. 
 
 arc. Extend c d, and make de equal to cd\ join a and e, 
 also e and b ; and proceed as directed above. 
 
 522. To Deeribe a Circle within any Given Triangle, 
 so that the Sides of the Triangle shall be Tangical. 
 
 Let a be (Fig. 374) be the given triangle. Bisect the angles 
 
 FIG. 374. 
 
 a and b according to Art. 506; upon d, the point of intersec- 
 tion of the bisecting lines, with the radius d e, describe the 
 required circle. 
 
 523. About a Given Circle, to Describe an Equilateral 
 Triangle. Let adb c (Fig. 375) be the given circle. Draw 
 the diameter c d; upon'af, with the radius of the given circle, 
 describe the arc aeb\ join a and b ; draw fg at right angles 
 to dc ; make fc and eg each equal to ab ; from f t through 
 a, draw //*, also from g, through b, draw gh\ then fgh 
 will be the triangle required. 
 
 524. To Find a Right L.ine nearly Equal to the Cir- 
 cumference of a Circle. Let abed (Fig. 376) be the given 
 
A RIGHT LINE EQUAL TO A CIRCUMFERENCE. 567 
 
 circle. Draw the diameter ac\ on this erect an equilateral 
 triangle aec according to Art. 525 ; draw gf parallel to ac\ 
 extend ec to /, also ea to g\ then gf will be nearly the 
 
 * ^ FIG. 375. 
 
 length of the semi-circle adc\ and twice gf will nearly 
 equal the circumference of the circle a b c d, as was required. 
 Lines drawn from *, through any points in the circle, as 
 o, o and o, to /,/ and /, will divide gf in the same way as 
 the semi-circle adc is divided. So, any portion of a circle 
 may be transferred to a straight line. This is a very useful 
 
 ff P P d p f 
 
 FIG. 376. 
 
 problem, and should be well studied, as it is frequently 
 used to solve problems on stairs, domes, etc. 
 
 Another method. Let a bfc (Fig. 377) be the given circle. 
 Draw the diameter ac\ from d, the centre, and at right an- 
 
568 PRACTICAL GEOMETRY. 
 
 gles to ac, draw db\ join b and c\ bisect be at e\ from d, 
 through e, draw df\ then ef added to three times the di- 
 ameter, will equal the circumference of the circle sufficiently 
 near for many uses. The result is a trifle too large. If the 
 
 FIG. 377. 
 
 circumference found by this rule be divided by 648-22 the 
 quotient will be the excess. Deduct this excess, and the 
 remainder will be the true circumference. This problem is 
 rather more curious than useful, as it is less labor to perform 
 the operation arithmetically, simply multiplying the given 
 diameter by 3- 1416, or, where a greater degree of accuracy 
 is needed, by 3-1415926. (See Art. 446.) 
 
 POLYGONS, ETC. 
 
 525. Upon a Given Line to Construct an Equilateral 
 Triangle. Let a b (Fig. 378) be the given line. Upon a and 
 
 
 
 FIG. 378. 
 
 b, with a b for radius, describe arcs, intersecting at c ; join a 
 and c, also c and b ; then acb will be the triangle required. 
 
 526. To Decribe an Equilateral Rectangle, or Square. 
 
 Let a b (Fig. 379) be the length of a side of the proposed 
 
POLYGONS IN CIRCUMSCRIBING CIRCLES. 
 
 square. Upon a and b, with a b for radius, describe the arcs 
 a d and b c ; bisect the arc ae in f.\ upon e, with ef for ra- 
 dius, describe the arc cfd\ join a and <:, c and </, </and ; 
 then acdb will be the square required. 
 
 FIG. 379. 
 
 527. Within a Given Circle, to Inscribe 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 ae c ; join a and c, 
 also a and </, and c and </ and the triangle is completed. 
 For the hexagon: from a, also from c, through e, draw the 
 lines af and cg\ join # and b, b and <:, c and /, etc., and the 
 
 hexagon is completed. The dodecagon may be formed by 
 bisecting the sides of the hexagon. 
 
 Each side of a regular hexagon is exactly equal to the 
 radius of the circle that circumscribes the figure. For the 
 radius is equal to a chord of an arc of 60 degrees ; and, as 
 every circle is supposed to be divided into 360 degrees, there 
 is just 6 times 60, or 6 arcs of 60 degrees, in the whole cir- 
 cumference. A line drawn from each angle of the hexagon 
 
570 
 
 PRACTICAL GEOMETRY. 
 
 to the centre (as in the figure) divides it into six equal, equi- 
 lateral triangles. 
 
 528. Within a Square to Inscribe an Octagon. Let 
 
 abed (Fig. 381) be the given square. Draw the diagonals 
 
 a d and b c ; upon a, b, c, and d, with a e for radius, describe 
 arcs cutting the sides of the square at 1,2, 3, 4, 5, 6, 7, and 8 ; 
 join i and 2, 3 and 4, 5 and 6, etc., and the figure is com- 
 pleted. 
 
 In order to eight-square a hand-rail, or any piece that is 
 
 FIG. 382. 
 
 to be afterwards rounded, draw the diagonals a d and b c 
 upon the end of it, after it has been squared-up. Set a 
 gauge to the distance ae and run it upon the whole length 
 of the stuff, from each corner both ways. This will show 
 
BUTTRESSED OCTAGON. 5/1 
 
 how much is to be chamfered off, in order to make the piece 
 octagonal (Art. 354). 
 
 529. To Find the Side of a Buttressed Octagon. Let 
 
 ABCDE (Fig. 382) represent one quarter of an octagon 
 structure, having a buttress H FGJ at each angle. The 
 distance M H, between the buttresses, being given, as also 
 F G, the width of a buttress ; to find H C or C J, in order to 
 obtain B C, the side of the octagon. Let B C, a side of the 
 octagon, be represented by b ; or D C by b. Let M H = a ; 
 or J D = a\ and 
 Then we have 
 
 JD+ JC = CD, 
 
 . i a + x = i b, 
 
 a + 2 x = b. 
 
 For FG put p\ ^r LG K J \p. 
 
 Now D is the radius of an inscribed circle and, as per 
 
 equation (140.), equals r ( 1/2 + i) -. 
 
 Also, (7 is the radius of a circumscribed circle, and, as 
 per equation (141.)* equals R ^2^2 + 4-. 
 
 The two triangles, CJK and C ED, are homologous; 
 for the angles at C are common and the angles at AT and D 
 are right angles. Having thus two angles of one equal 
 respectively to the two angles of the other, therefore (Art. 
 345) the remaining angles must be equal. Hence, the sides 
 of the triangles are proportionate, or 
 
 ED : EC :: JK : CJ 
 r R '.'. \} 
 
 The value of the side, as above, is 
 
 R 
 
 I = a + 2 x = a + p - - , 
 
572 pkACTICAL GEOMETRY. 
 
 And taking the value of R and r, as above, we have 
 
 (1/2 
 
 +1 
 
 r> 
 
 Substituting this for , we have 
 
 V2 + I 
 
 The numerical coefficient of / reduces to 1-0823923 or 
 i -0824, nearly. 
 
 Therefore we have 
 
 b = a + i -o824/. (207.) 
 
 Or: The side of a buttressed octagon equals the distance be- 
 tween the buttresses plus \ -0824 times the width of the faced 
 the buttress. 
 
 For example : let there be an octagon building, which 
 measures between the buttresses, as at M H, 18 feet, and the 
 face of the buttresses, as FG, equals 3 feet ; what, in such a 
 building, is the length of a side B Cl For this, using equa- 
 tion (207.), we have 
 
 b = 1 8 + I -0824 x 3 
 = 18 + 3-2472 
 = 21-2472. 
 
 Or : The side of the octagon B C equals 21 feet and nearly 3 
 inches. 
 
 530. Within a Given Circle to Incribe any Regular 
 Polygon. Let abc 2 (Figs. 383* 384, and 385) be given circles. 
 Draw the diameter a c ; upon this erect an equilateral trian- 
 gle aec, according to A rt. 525 ; divide ac into as many equal 
 parts as the polygon is to have sides, as at i, 2, 3, 4, etc.; 
 from e, through each even number, as 2, 4, 6, etc., draw lines 
 
TO DESCRIBE ANY REGULAR POLYGON. 
 
 573 
 
 cutting the circle in the points 2, 4, etc.; from these points 
 and at right angles to a c draw lines to the opposite part 
 of the circle ; this will give the remaining points for the 
 polygon, as b, /, etc. 
 
 In forming a hexagon, the sides of the triangle erected 
 
 FIG. 383. 
 
 upon ac (as at Fig. 384) mark the points b and f. This 
 method of locating the angles of a polygon is an approxima- 
 tion sufficiently near for many purposes ; it is based upon 
 the like principle with the method of obtaining a right line 
 nearly equal to a circle (Art. 524). The method shown at 
 Art. 531 is accurate. 
 
 FIG. 386. 
 
 FIG. 387.- 
 
 FIG. 388 
 
 531. Upon a Given Line to Decrifoe any Regular 
 Polygon. Let a b (Figs. 386, 387, and 388) be (riven lines, 
 equal to a side of the required figure. From b draw be at 
 right angles to a b ; upon a and &, with a b for radius, describe 
 the arcs acd and feb\ divide ac into as many equal parts 
 
5/4 PRACTICAL GEOMETRY. 
 
 as the polygon is to have sides, and extend those divisions 
 from c towards d\ from the second point of division, count- 
 ing from c towards a, as 3 (Fig. 386), 4 (Fig. 387), and 5 (Fig. 
 388), draw a line to b ; take the distance from said point of 
 division to a, and set it from b to e ; join e and a ; upon the 
 intersection o with the 'radius oa, describe the circle afdb; 
 then radiating lines, drawn from b through the even numbers 
 on the arc a d, will cut the circle at the several angles of the 
 required figure. 
 
 In the hexagon (Fig. 387), the divisions on the arc ad are 
 not necessary ; for the point o is at the intersection of the 
 arcs ad and fb, the points f and d are determined by the 
 intersection of those arcs with the circle, and the points 
 above g and h can be found by drawing lines from a and b 
 through the centre o. In polygons of a greater number of 
 sides than the hexagon the intersection o comes above the 
 arcs ; in such case, therefore, the lines a e and b 5 (Fig. 388) 
 have to be extended before they will intersect. This method 
 of describing polygons is founded on correct principles, and 
 is therefore accurate. In the circle equal arcs subtend 
 equal angles (Arts. 357 and 515). Although this method is 
 accurate, yet polygons may be described as accurately and 
 more simply in the following manner. It will be observed 
 that much of the process in this method is for the purpose 
 of ascertaining the centre of a circle that will circumscribe 
 the proposed polygon. By reference to the Table of Poly- 
 gons in Art. 442 it will be seen ho-w this centre may be ob- 
 tained arithmetically. This is the rule : multiply the given 
 side by the tabular radius for polygons of a like number of 
 sides with the proposed figure, and the product will be the 
 radius of the required circumscribing circle. Divide this 
 circle into as many equal parts as the polygon is to have 
 sides, connect the points of division by straight lines, and 
 the figure is complete. For example : It is desired to de- 
 scribe a polygon of 7 sides, and 20 inches a side. The tabu- 
 lar radius is 1-15238. This multiplied by 20, the product, 
 23-0476 is Ihe required radius in inches. The Rules for 
 the Reduction of Decimals, in the Appendix, show how to 
 change decimals to the fractions of a foot or an inch. From 
 
EQUAL FIGURES. 
 
 575 
 
 this, 23 -0476 is equal to 23 T V inches, nearly. It is not needed 
 to take all the decimals in the table, three or four of them 
 will give a result sufficiently near for all ordinary practice. 
 
 532. To ontruct a Triangle whose Side shall be 
 everally Equal to Three Given Lines. Let a, b and c (Fig. 
 389) be the given lines. Draw the line de and make it equal 
 
 FIG. 389. 
 
 c\ upon e, with b for radius, describe an arc at /; upon d, 
 with a for radius, describe an arc intersecting the other at/; 
 join d and /, also f and e ; then dfe will be the triangle 
 required. 
 
 533 To Construct a Figure Equal to a Given, Right- 
 lined Figure. Let abed (Fig. 390) be the given figure. 
 Make ef (Fig. 391) equal to cd-, upon /, with da for radius, 
 
 FIG. 390. 
 
 FIG. 391. 
 
 describe an arc at^; upon r, with ca for radius, describe an 
 arc intersecting the other at^; join ^and e,\ upon f and g, 
 with db and ab for radius, describe arcs intersecting at // ; 
 join g and /*, also h and /; then Fig. 391 will every way 
 equal Fig. 390. 
 
 So, right-lined figures of any number of sides may be 
 copied, by first dividing them into triangles, and then pro- 
 
57 6 
 
 PRACTICAL GEOMETRY. 
 
 ceeding as above. The shape of the floor of any room, or 
 of any piece of land, etc., may be accurately laid out by this 
 problem, at a scale upon paper ; and the contents in square 
 feet be ascertained by the next. 
 
 534. To Make a Parallelogram equal to a Given 
 Triangle. Let a be (Fig. 392) be the given triangle. From 
 a draw a d at right angles to b c ; bisect a d in e ; through e 
 
 f 
 
 FIG. 392. 
 
 draw fg parallel to be-, from b and c draw b f and eg ^par- 
 allel to de\ then bfgc will be a parallelogram containing a 
 surface exactly equal to that of the triangle a be. 
 
 Unless the parallelogram is required to be a rectangle, 
 the lines bf and eg need not be drawn parallel to d e. If a 
 rhomboid is desired they may be drawn at an oblique angle, 
 provided they be parallel to one another. To ascertain the 
 area of a triangle, multiply the base be by half the perpen- 
 
 d 
 FIG. 393. 
 
 dicular height da. 
 is taken for base. 
 
 In doing this it matters not which side 
 
 535. A Parallelogram being Given, to Construct An- 
 other Equal to it, and Having a Side Equal to a Given Line. 
 
 Let A (Fig. 393) be the given parallelogram, and B the 
 given line. Produce the sides of the parallelogram, as at 
 
SQUARE EQUAL TO TWO OR MORE SQUARES. 577 
 
 a, b, c, and d'\ make ed equal to B ; through d draw cf par- 
 allel to gb-, through e draw the diagonal ca\ from a draw 
 af parallel to ed; then C will be equal to A. (See Art. 340.) 
 
 536. To Make a Square Equal to two or more Given 
 Squares. Let A and B (Fig. 394) be two given squares. 
 
 FIG. 394. 
 
 Place them so as to form a right angle, as at a ; Join b ai<d c ; 
 then the square C, formed upon the line be, wi'J 1 be equal in 
 extent to the squares A and B added together. Again : if 
 a b (Fig. 395) be equal to the side of a given square, c a, placed 
 at right angles to a b, be the side of another given square, 
 
 and cd, placed at right angles to cb, be the side of a third 
 given square, then the square A, formed upon the line db> 
 will be equal to the three given squares. (See Art. 353.) 
 
 The usefulness and importance of this problem are pro- 
 verbial. To ascertain the length of braces and of rafters in 
 
578 PRACTICAL GEOMETRY. 
 
 framing, the length of stair-strings, etc., are some of the pur- 
 poses to which it may be applied in carpentry. (See note 
 to Art. 503.) If the lengths of any two sides of a right- 
 angled triangle are known, that of the third can be ascer- 
 tained. Because the square of the hypothenuse is equal to 
 the united squares of the two sides that contain the right 
 angle. 
 
 (i.) The two sides containing the right angle being 
 known, to find the hypothenuse. 
 
 Rule. Square each given side, add the squares together, 
 and from the product extract the square root ; this will be 
 the answer. 
 
 For instance, suppose it were required to find the length 
 of a rafter for a house, 34 feet wide the ridge of the roof 
 to be 9 feet high, above the level of the wall-plates. Then 
 17 feet, half of the span, is one, and 9 feet, the height, is the 
 other of the sides that contain the right angle. Proceed as 
 directed by the rule : 
 
 17 9 
 
 17 _9 
 
 119 8 1 = square of 9. 
 
 17 289 = square of 17. 
 
 289 square of 17. 370 Product. 
 
 i ) 370 ( 19-235 + = square root of 370 ; equal 19 feet 2-J in., 
 i i nearly ; which would be the required 
 
 20 )~270 length of the rafter. 
 
 9 261 
 382). -900 
 
 _ 2 ^1 
 3843) 13600 
 
 38465)- 207 100 
 192325 
 
TO FIND THE LENGTH OF A RAFTER. 579 
 
 (By reference to the table of square roots in the Appen- 
 dix, the root of almost any number may be found ready 
 calculated ; also, to change the decimals of a foot to inches 
 and parts, see Rules for the Reduction of Decimals in the 
 Appendix.) 
 
 Again : suppose it be required, in a frame building, to 
 find the length of a brace having a run of three feet each 
 way from the point of the right angle. The length of the 
 sides containing the right angle will be each 3 feet ; then, as 
 before 
 
 3 
 _3 
 
 9 = square of one side. 
 3 times 3 = 9 = square of the other side. 
 
 1 8 Product : the square root of which is 4 2426-}- ft., 
 or 4 feet 2 inches and -J full. 
 
 (2.) The hypothenuse and one side being known, to find 
 the other side. 
 
 Rule. Subtract the square of the given side from the 
 square of the hypothenuse, and the square root of the prod- 
 uct will be the answer. 
 
 Suppose it were required to ascertain the greatest per- 
 pendicular height a roof of a given span may have, when 
 pieces of timber of a given length are to be used as rafters. 
 Let the span be 20 feet, and the rafters of 3 x 4 hemlock 
 joist. These come about 13 feet long. The known hy- 
 pothenuse, then, is 13 feet, and the known side, 10 feet 
 that being half the span of the building. 
 
 13 
 13 
 
 39 
 13 
 
 169 = square of hypothenuse. 
 10 times 10 = 100 square of the given side. 
 
 69 Product : the square root of which is 
 
580 PRACTICAL GEOMETRY. 
 
 8 3066+ feet, or 8 feet 3 inches and | full. This will be the 
 greatest perpendicular height, as required. Again : suppose 
 that in a story of 8 feet, from floor to floor, a step-ladder is 
 required, the strings of which are to be of plank 12 feet 
 long, and it is desirable to know the greatest run such a 
 length of string will afford. In this case, the two given 
 sides are hypothenuse 12, perpendicular 8 feet. 
 
 12 times 12 144 square of hypothenuse, 
 8 times 8 = 64 = square of perpendicular. 
 
 80 Product : the square root of which is 
 8-9442+ feet, or 8 feet n inches and ^ the answer, as re- 
 quired. 
 
 Many other cases might be adduced to show the utility 
 of this problem. A practical and ready method of ascer- 
 taining the length of braces, rafters, etc., when not of a great 
 length, is to apply a rule across the carpenters' -square. 
 Suppose, for the length of a rafter, the base be 12 feet and 
 the height 7. Apply the rule diagonally on the square, so 
 that it touches 12 inches from the corner on one side, and 7 
 inches from the corner on the other. The number of inches 
 on the rule which are intercepted by the sides of the square, 
 13!-, nearly, wilt be the length of the rafter in feet ; viz., 13 
 feet and. -J- of a foot. If the dimensions are large, as 30 feet 
 and 20, take the half of each on the sides of the square, viz., 
 15 and 10 inches; then the length in inches across will be 
 one half the number of feet the rafter is long. This method 
 is just as accurate as the preceding ; but when the length of 
 a very long rafter is sought, it requires great care and pre- 
 cision to ascertain the fractions. For the least variation on 
 the square, or in the length taken on the rule, would make 
 perhaps several inches difference in the length of the rafter. 
 For shorter dimensions, however, the result will be true 
 enough. 
 
 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 abc make ab equal to the diameter of the 
 
SIMILAR FIGURES. 
 
 circle B, and cb equal to the diameter of the cin 
 
 the hypothenuse a c will be the diameter of a circle C, which 
 
 will be equal in area to the two circles A and B, added 
 
 together. 
 
 FIG. 396. 
 
 Any polygonal figure, as A (Fig. 397), formed on the hy- 
 pothenuse of a right-angled triangle, will be equal to two 
 similar figures,* as B and C, formed on the two legs of the 
 triangle. 
 
 FIG. 397. 
 
 538. To ontruct a Square Equal to a Given Rect- 
 angle. Let A (Fig. 398) be the given rectangle. Extend 
 the side ab and make be equal to be\ bisect a c in /, and 
 upon /, with the radius fa, describe the semi-circle agc\ 
 extend eb till it cuts the curve in g\ then a square bghd, 
 formed on the line bg, will be equal in area to the rectan- 
 glcA. 
 
 * Similar figures are such as have their several angles respectively equal, 
 and their sides respectively proportionate. 
 
582 
 
 PRACTICAL GEOMETRY. 
 
 Another method. Let A (Fig. 399) be the given rectangle. 
 Extend the side a b and make a d equal to a c ; bisect a d in 
 e\ upon e, with the radius ea, describe the semi-circle afd\ 
 extend gb till it cuts the curve in /; join a and f\ then 
 
 FIG. 398. 
 
 the square B, formed on the line af, will be equal in area to 
 the rectangle A. (See Arts. 352 and 353.) 
 
 539. To Form a Square Equal to a Given Triangle- 
 Let ab (Fig. 398) equal the base of the given triangle, and be 
 
 equal half its perpendicular height (see Fig. 392) ; then pro- 
 ceed as directed at Art. 538. 
 
 540. Two Right Lines being Given, to Find a Third 
 Proportional Thereto. Let A and B (Fig. 400) be the given 
 lines. Make a b equal to A ; from a draw a c at any angle 
 
PROPORTIONATE DIVISIONS IN LINES. 583 
 
 with ab-, make ac and ad each equal to B; join c and ; 
 from d draw de parallel to c b ; then # e will be the third 
 proportional required. That is, ae bears the same propor- 
 tion to B as B does to A. 
 
 FIG. 400. 
 
 541. Three Right Lines being Given, to Find a Fourth 
 Proportional Thereto. Let A, B, and C (Fig. 401) be the 
 given lines. Make ab equal to A ; from a draw ac at any 
 
 angle with a b ; make # c equal to j9 and a e equal to (7 ; join 
 c and ; from e draw */ parallel to cb\ then 0/ will be the 
 fourth proportional required. That is, af bears the same 
 proportion to C as B does to A. 
 
 To apply this problem, suppose the two axes of a given 
 ellipsis and the longer axis of a proposed ellipsis are given. 
 Then, by this problem, the length of the shorter axis to the 
 proposed ellipsis can be found ; so that it will bear the same 
 proportion to the longer axis as the shorter of the given 
 ellipsis does to its longer. (See also Art. 559.) 
 
 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 di- 
 vided, and B the line with its divisions. Make a b equal to 
 B with all its divisions, as at i, 2, 3', etc.; from a draw ac at 
 any angle with a b ; make a c equal to A ; join c and b ; from 
 
584 
 
 PRACTICAL GEOMETRY. 
 
 the points i, 2, 3, etc., draw lines parallel to cb', then these 
 will divide the line ac in the same proportion as B is divided 
 as was required. 
 
 This problem will be found useful in proportioning the 
 
 members of a proposed cornice, in the same proportion as 
 those of a given cornice of another size. (See Art. 321.) So 
 of a pilaster, architrave, etc. 
 
 543. Between Two Given Right Lines, to Find a 
 Mean Proportional. Let A and B (Fig. 403) be the given 
 lines. On the line ac make ab equal to A and be equal to 
 B ; bisect ac in e\ upon e, with ea for radius, describe the 
 semi-circle adc\ at b erect b d at right angles to a c ; then 
 
 bd will be the mean proportional between A and B. That 
 is, ab is to bd as bd is to be. This is usually stated thus: 
 ab : bd : : bd : be, and since the product of the means 
 equals the product of the extremes, therefore, abxbe = bd*- 
 This is shown geometrically at Art. 538. 
 
 CONIC SECTIONS. 
 
 544. Definitions. If a cone, standing upon a base 
 that is at right angles with its axis, be cut by a plane, per- 
 
AXIS AND BASE OF PARABOLA. 
 
 585 
 
 pendicular to its base and passing through its axis, the sec- 
 tion will be an isosceles triangle (as a be, Fig. 404) ; and the 
 base will be a semi-circle. If a cone be cut by a plane in the 
 direction ef the section will be an ellipsis ; if in the direction 
 ml, the section will be a parabola; and if in the direction 
 ro, an hyperbola. (See Art. 499.) If the cutting planes be 
 at right angles with the plane a be, then 
 
 545. To Find the Axe of the Ellipsi: bisect ef (Fig. 
 404) in g\ through g draw h i parallel to ab\ bisect h i in/; 
 
 FIG. 404. 
 
 upon j, with jh for radius, describe the semi-circle hki\ 
 from ^-draw gk.at right angles to hi\ then twice gk will 
 be the conjugate axis and ef the transverse. 
 
 546. To Find the Axis and Bae of the Parabola. 
 
 Let ;;/ / (Fig. 404), parallel to ac, be the direction of the cut- 
 ting plane. From ;;/ draw m d at right angles to a b ; then 
 l-m will be the axis and height, and md an ordinate and half 
 the base, as at Figs. 417, 418. 
 
 547. To Find the Height, Bae, and Transverse Axis 
 of an Hyperbola. Let o r (Fig. 404) be the direction of the 
 
586 
 
 PRACTICAL GEOMETRY. 
 
 cutting plane. Extend or and ac till they meet at w; from 
 o draw op at right angles to a b ; then r o will be the height, 
 n r the transverse axis, and op half the base ; as at Fig. 419. 
 
 54-8. The Axes being Given, to Find the Foci, and to 
 Describe an Ellipsis with a String. Let ab (Fig. 405) and 
 cd be the given axes. Upon c, with a e or be for radius, de- 
 scribe the arc //; then / and /, the points at which the 
 arc cuts the transverse axis, will be the foci. At f and f 
 place two pins, and another at c\ tie a string about the three 
 pins, so as to form the triangle ffc ; remove the pin from c 
 and place a pencil in its stead ; keeping the string taut, 
 
 move the pencil in the direction cga\ it will then describe 
 the required ellipsis. The lines fg and gf show the posi- 
 tion of the string when the pencil arrives at g. 
 
 This method, when performed correctly, is perfectly ac- 
 curate ; but the string is liable to stretch, and is, therefore, 
 not so good to use as the trammel. In making an ellipse by 
 a string or twine, that kind should be used which has the 
 least tendency to elasticity. For this reason, a cotton cord, 
 such as chalk-lines are commonly made of, is not proper for 
 the purpose ; a linen or flaxen cord is much better. 
 
 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 ol the grooves would coincide with the axes ; make 
 
ELLIPSE BY TRAMMEL. 
 
 587 
 
 the distance from the pencil e to the nut/ equal to half c d\ 
 also, from the pencil e to the nut g equal to half a b ; letting 
 the pins under the nuts slide in the grooves, move the tram- 
 mel eg in the direction cbd\ then the pencil at e will de- 
 scribe the required ellipse. 
 
 A trammel may be constructed thus : take two straight 
 strips of board, and make a groove on their face, in the cen- 
 tre of their width ; join them together, in the middle of their 
 length, at right angles to one another ; as is seen at Fig. 406. 
 A rod is then to be prepared, having two movable nuts 
 made of wood, with a mortise through them of the size of 
 the rod, and pins under them large enough to fill the 
 grooves. Make a hole at one end of the rod, in which to 
 
 FIG. 406. 
 
 place a pencil. In the absence of a regular trammel a tem- 
 porary one may be made, which, for any short job, will an- 
 swer every purpose. Fasten two straight-edges at right 
 angles to one another. Lay them so as to coincide with the 
 axes of the proposed ellipse, having the angular point at the 
 centre. Then, in a rod having a hole for the pencil at one 
 end, place two brad-awls at the distances described at Art, 
 549. While the pencil is moved in the direction of the 
 curve, keep the brad-awls hard against the straight-edges, 
 as directed for using the trammel-rod, and one quarter of 
 the ellipse will be drawn. Then, by shifting the straight- 
 edges, the other three quarters in succession may be drawn. 
 If the required ellipse be not too large, a carpenters'-square 
 may be made use of, in place of the straight-edges. 
 
 An improved method of constructing the trammel is as 
 
588 
 
 PRACTICAL GEOMETRY. 
 
 follows: make the sides of the grooves bevelling from the 
 face of the stuff, or dove-tailing instead of square. Prepare 
 two slips of wood, each about two inches long, which shall 
 be of a shape to just fill the groove when slipped in at the 
 end. These, instead of pins, are to be attached one to each 
 of the movable nuts with a screw, loose enough for the nut 
 to move freely about the screw as an axis. The advantage 
 of this contrivance is, in preventing the nuts from slipping 
 out oftheir places during the operation of describing the 
 curve. 
 
 550. To Describe an Ellipsis by Ordiiiute*. Let ab 
 
 and cd (Fig. 407) be given axes. With c e or ' e d for radius 
 
 describe the quadrant fgh ; divide f/i, ac, and eb, each into 
 a like number of equal parts, as at I, 2, and 3 ; through 
 these points draw ordinates parallel to cd and -fg\ take the 
 distance I i and place it at i /, transfer 27 to 2 /#, and 3 k to 
 3 n ; through the points # , n, m, /, and c, trace a curve, and 
 the ellipsis will be completed. 
 
 The greater the number of divisions on a, e, etc., in this 
 and the following problem, the more points in the curve can 
 be found, and the more accurate the curve can be traced. 
 If pins are placed in the points n, m, /, etc., and a thin slip 
 of wood bent around by them, the curve can be made quite 
 correct. This method is mostly used in tracing face-moulds 
 for stair hand -railing. 
 
 551. To Describe an Ellipsis by Intersection of Lines. 
 
 Let ab and cd (Fig. 408) be given axes. Through c, draw 
 
ELLIPSE BY INTERSECTION OF LINES. 
 
 589 
 
 fg parallel to.ab', from a and b draw af and b g at right 
 angles to a b ; divide fa, gb, ae, and eb, each into a like 
 number of equal parts, as at i, 2, 3, and 0, <?,<?; from I, 2, 
 and 3, draw lines to c; through 0, 0, and 0, draw lines from d, 
 
 intersecting those drawn to c ; then a curve, traced through 
 the points i, i, i, will be that of an ellipsis. 
 
 Where neither trammel nor string is at hand, this, per- 
 haps, is the most ready method of drawing an ellipsis. The 
 divisions should be small, where accuracy is desirable. By 
 this method an ellipsis may be traced without the axes, pro- 
 vided that a diameter and its conjugate be given. Thus, ab 
 
 and cd (Fig. 409) are conjugate diameters: fg\s drawn par- 
 allel to ab, instead of being at right angles to cd\ also, fa 
 and g b are drawn parallel to c d, instead of being at right 
 angles to a b. 
 
590 
 
 PRACTICAL GEOMETRY. 
 
 552. To Describe an Ellipsis by Intersecting Arcs. 
 
 Let a b and cd (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 I, 2, and 3 ; upon f and /, with b i for radius, 
 describe arcs at g t g, g, and g ; upon f and /, with a I for 
 
 radius, describe arcs intersecting the others at g>g,g, and g; 
 then these points of intersection will be in the curve of the 
 ellipsis. The other points, h and i, are found in like manner, 
 viz.: h is found by taking b2 for one radius, and 0,2 for the 
 other ; i is found by taking b 3 for one radius, and a 3 for the 
 
 other, always using the foci for centres. Then by tracing a 
 curve through the points c, g, //, i, b, etc., the ellipse will be 
 completed. 
 
 This problem is founded upon the same principle as that 
 of the string. This is obvious, when we reflect that the 
 length of the string is equal to the transverse axis, added to 
 
TO DESCRIBE AN OVAL. 
 
 591 
 
 the distance between the foci. See Fig. 405, in which cf 
 equals ae, the half of the transverse axis. 
 
 553. To Describe a Figure Nearly in the Shape of an 
 Ellipsis, by a Pair of Compasses. Let ab and c d (Fig. 41 1) 
 be given axes. From c draw c e parallel to a b ; from a draw 
 ae parallel to cd\ join e and d\ bisect ea in /; join / and c, 
 intersecting edvn. i\ bisect ic in,*?; from o draw og at right, 
 angles to ic, meeting cd extended to g\ join i and g, cutting 
 the transverse axis in r ; make hj equal to Jig, and h k equal 
 to//r; from j, through r and k, draw//;z and/#; also, from 
 g, through /, draw gl; upon g and/, with gc for radius, 
 describe the arcs il and mn\ upon r and , with ?-# for 
 
 radius, describe the arcs ;;/*and /;/ ; this will complete the 
 figure. 
 
 When the axes are proportioned to one another, as at 2 
 to 3, the extremities, c and d, of the shortest axis, will be 
 the centres for describing the arcs il and m n ; and the inter- 
 section of ed with the transverse axis will be the centre for 
 describing the arc m, i, etc. As the elliptic curve is contin- 
 ually changing its course from that of a circle, a true ellipsis 
 cannot be described with a pair of compasses. The above, 
 therefore, is only an approximation. 
 
 554. To Draw an Oval in the Proportion Seven by 
 Nine. Let cd (Fig. 412) be the given conjugate axis. Bisect 
 
592 
 
 PRACTICAL GEOMETRY. 
 
 cdin o, and through o draw ab at right angles to cd\ bisect 
 co in e ; upon o, with 0^ for radius, describe the circle efgh ; 
 from e, through h and f t draw <?/ and ei\ also, from ^, 
 through h and /, draw ^ and gl\ upon , with gc for 
 radius, describe the arc kl] upon *, with e d for radius, de- 
 scribe the arc ji ; upon h and /, with h k for radius, describe 
 the arcs jk and /*'; this will complete the figure. 
 
 This is an approximation to an ellipsis ; and perhaps no 
 method can be found by which* a well-shaped oval can be 
 drawn with greater facility. By a little variation in the 
 process, ovals of different proportions may be obtained. If 
 quarter of the transverse axis is taken for the radius of the 
 circle efgh, one will be drawn in the proportion five by 
 seven. 
 
 FIG. 414. 
 
 555. To Draw a Tangent to an Ellipsis. Let abed 
 (Fig. 413) be the given ellipsis, and d the point of contact. 
 Find the foci (Art. 548) / and f t and from them, through d, 
 draw fe and fd\ bisect the angle (Art. 506) edo with the 
 line sr ; then sr will be the tangent required. 
 
TO FIND THE AXES OF AN ELLIPSE. 
 
 593 
 
 556. An Ellipsis with a Tangent Given, to Detect the 
 Point of Contact. Let a gb f (Fig. 414) be the given ellip- 
 sis and tangent. Through the centre e draw a b parallel to 
 the tangent; anywhere between e and / draw cd parallel to 
 a b ; bisect cd in o ; through o and e draw fg] then g will 
 be the point of contact required. 
 
 557. A Diameter of an Ellipsi Given, to Find its 
 Conjugate. Let a b (Fig. 414) be the given diameter. Find 
 the line fg by the last problem ; then fg will be the diam- 
 eter required. 
 
 558. Any Diameter and its Conjugate being Given, to 
 Ascertain the Two Axes, and thence to Describe the Ellipsis. 
 
 Let a b and cd(Fig. 415) be the given diameters, conjugate 
 
 FIG. 415. 
 
 to one another. Through c draw cf parallel to a b ; from c 
 draw eg at right angles to ef\ make eg equal to ah or Jib\ 
 join g and h ; upon g, with gc for radius, describe the arc 
 ikcj\ upon h, with the same radius, describe the arc In 
 through the intersections / and n draw n o, cutting the tan- 
 gent ef in o ; upon o, with ogfor radius, describe the semi- 
 circle e igf\ join e and g, also g and f, cutting the arc icj 
 in k and / ; from r, through //, draw e m, also from /, through 
 h, draw // ; from k and t draw kr and ts parallel to gk 
 
594 
 
 PRACTICAL GEOMETRY. 
 
 cutting em in r, and // in s ; make h m equal to hr, and hp 
 equal to hs\ then r;;/ and s p will be the axes required, by 
 which the ellipsis may be drawn in the usual way. 
 
 559. To Describe an Ellipsis, whose Axes shall toe 
 Proportionate to the Axes of a Larger or SmalDer Given 
 One. Let a cbd(Fig. 416) be the given ellipsis and axes, and 
 
 FIG. 416. 
 
 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 conjugate axis required, and will bear the 
 same proportion to ij as cd does to ab. (See Art. 541.) 
 
 560. To Describe a Parabola by Intersection of Lines. 
 
 Let ml (Fig. 417) be the axis and height (see Fig. 404) and 
 
 i 2 
 
 / 3 2 1 
 
 1 2 3 in 3 2 
 
 FIG. 417. 
 
 dd a double ordinate and base of the proposed parabola. 
 Through / draw a a parallel to dd\ through d and d draw 
 da and da parallel to ml\ divide ad and dm, each into a 
 like number of equal parts ; from each point of division in 
 
TO DESCRIBE AN HYPERBOLA. 
 
 595 
 
 dm draw the lines i i, 22, etc., parallel to ;;//; from each 
 point of division in da draw lines to/; then a curve traced 
 through the points of intersection o, o, and o, will be that of 
 a parabola. 
 
 Another method. Let m I (Fig. 418) be the axis and height, 
 and dd the base. Extend m I and make la equal to m I ; 
 join a and d, and a and d\ divide ad and ad, each into a 
 like number of equal parts, as at i, 2, 3, etc. ; join i and i, 2 
 and 2, etc., and the parabola will be completed. (See Arts. 
 460 to 472.) 
 
 561. To Describe an Hyperbola by Intergection of 
 Lines. Let ro (Fig. 419) be the height,// the base, and nr 
 the transverse axis. (See Fig. 404.) Through r draw a a 
 
 m 
 
 FIG. 418. 
 
 1 2 3 o 3 2 \ p 
 
 FIG. 419. 
 
 parallel to pp\ from / draw ap parallel to ro\ divide ap 
 and po, each into a like number of equal parts ; from each 
 of the points of division in the base, draw lines to ;/ ; from 
 each of the points of division in ap, draw lines to r\ then 
 a curve traced through the points of intersection o, o, etc., 
 will be that of an hyperbola. 
 
 The parabola and hyperbola afford handsome curves for 
 various mouldings. (See Figs. 191 to 205 ; 222 to 224; 241 
 and 242 ; also note to Art. 318.) 
 
SECTION XVII. SHADOWS. 
 
 562. The Art of Drawing consists in representing 
 solids Upon a plane surface, so that a curious and nice ad- 
 justment of lines is made to present the same appearance to 
 the eye as does the human figure, a tree, or a house. It is 
 by the effects of light, in its reflection, shade, and shadow, 
 that the presence of an object is made known to us; so 
 upon paper it is necessary, in order that the delineation 
 may appear real, to represent fully all the shades and shad- 
 ows that would be seen upon the object itself. In this sec- 
 tion I propose to illustrate, by a few plain examples, the 
 simple elementary principles upon which shading, in archi- 
 tectural subjects, is based. The necessary knowledge of 
 drawing, preliminary to this subject, is treated of in Section 
 XV., from Arts. 487 to 498. 
 
 563. The Inclination of the JLlne 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 reference to Fig. 420, in which A repre- 
 sents a horizontal plane, and B and C two vertical planes 
 placed at right angles to each other. A represents the plan, 
 C the elevation, and B a vertical projection from the eleva- 
 tion. In finding the shadow of the plane B, the line a b is 
 drawn at an angle of 45 degrees with the horizon, and the 
 liner^ at the same angle with the vertical planed. The 
 plane B being a rectangle, this makes the true direction of 
 the sun's rays to be in a course parallel to db, which direc- 
 tion has been proved to be at an angle of 35 degrees and 
 16 minutes with the horizon. It is convenient, in shading, 
 to have a set-square with the two sides that contain the 
 
CONVENTIONAL PLANES OF SHADOW. 
 
 597 
 
 right angle of equal length; this will make the two acute 
 angles each 45 degrees, and will give the requisite bevel 
 when worked upon the edge of the T-square. One reason 
 why this angle is chosen in preference to another is that 
 when shadows are properly made upon the drawing by it, 
 the depth of every recess is more readily known, since the 
 breadth of shadow and the depth of the recess will be equal. 
 To distinguish between the terms shade and shadow, it will 
 be understood that all such parts of a body as are not e5c- 
 posed to the direct action of the sun's rays are in shade ; 
 
 while those parts which are deprived of light by the inter- 
 position of other bodies are in shadow. 
 
 564. To Find the Line of Shadow 011 Mouldings and 
 other Horizontally Straight Projections. Figs. 421, 422, 
 423, and 424 represent various mouldings in elevation, re- 
 turned at the left, in the usual manner of mitering around a 
 projection. A mere inspection of the figures is sufficient to 
 see how the line of shadow is obtained, bearing in mind that 
 the ray a b is drawn from the projections at an angle of 45 
 
59 8 
 
 SHADOWS. 
 
 degrees. When there is no return at the end, it is neces- 
 sary to draw a section, at any place in the length of the 
 mouldings, and find the line of shadow from that. 
 
 565. To Find tlie 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, accord- 
 ing to the angle previously directed ; from b erect a per- 
 pendicular intersecting c d <& d\ from d draw de parallel to 
 
 FIG. 421, 
 
 FIG. 422. 
 
 FIG. 423. 
 
 FIG. 424. 
 
 the shelf; then the lines cd and de will define the shadow 
 cast by the shelf. There is another method of finding the 
 shadow, without the plan A. Extend the lower line of the 
 shelf to //and make cf equal to the projection of the shelf 
 from the wall ; from/ draw fg at the customary angle, and 
 from c drop the vertical line eg intersecting fg^g\ from 
 g draw ge parallel to the shelf, and from c draw c d at the 
 usual angle; then the lines cd and de will determine the 
 extent of the shadow as before. 
 
SHADOWS OF STRAIGHT AND OBLIQUE SHELVES. 599 
 
 .566. To Find the Shadow Cast by a Shelf which i* 
 Wider at one End than at the Other. In Fig. 426, A is the 
 plan, and B the elevation. Find the point d, as in the pre- 
 
 B 
 
 FIG. 425. 
 
 vious example, and from any other point in the front of the 
 shelf, as a, erect the perpendicular a e ; from a and e draw a b 
 and e c, at the proper angle, and from b erect the perpendicu- 
 
 FIG. 426. 
 
 lar be, intersecting ec in c\ from d, through c, dra\v do\ 
 then the lines id and do will give the limit of the shadow 
 cast by the shelf. 
 
6oo 
 
 SHADOWS. 
 
 567. To Find tlie Shadow of a Shelf having one End 
 Aeute or Obtuse Angled. Fig. 427 shows the plan and ele- 
 vation of an acute-angled shelf. Find the line eg as before ; 
 
 FIG. 427. 
 
 from a erect the perpendicular ab\ join b and e\ then be 
 and eg will define the boundary of shadow. 
 
 568. To Find the shadow Cast by an Inclined Shelf. 
 
 In Fig. 428 the plan and elevation of such a shelf are shown, 
 having also one end wider than the other. Proceed as di- 
 
 FIG 428. 
 
 rected for finding the shadows of Fig. 426, and find the points 
 ^/and c ; then ad and dc will be the shadow required. If 
 the shelf had been parallel in width on the plan, then the 
 line dc would have been parallel with the shelf a b. 
 
SHADOWS OF INCLINED AND CURVED SHELVES. 6oi 
 
 569. To Find the Shadow Cast by a Shelf Inclined in 
 its Vertical Section either Upward or Downward. From 
 a (Figs. 429 and 430) draw a b at the usual angle, and from b 
 draw be parallel with the shelf; obtain the point e by draw- 
 
 FIG. 429. 
 
 FIG. 430. 
 
 ing -a line from d at the usual angle. In Fig. 429 join e and 
 i ;. then ic and cc will define the shadow. In Fig. 430, from 
 o draw oi parallel with the shelf ; join i and e ; then ie and 
 cc will be the shadow required. 
 
 The projections in these several examples are bounded 
 
 FIG. 431- 
 
 FIG. 432. 
 
 by straight lines ; but the shadows of curved lines may be 
 found in the same manner, by projecting shadows from sev- 
 eral points in the curved line, and tracing the curve .of 
 shadow through these points. (Figs. 431 and 432.) 
 
602 
 
 SHADOWS. 
 
 570. To Find the Shallow of a Shelf having its Front 
 Edge, or End, Curved on the Plan, In Figs. 431 and 432 
 A and A show an example of each kind. From several 
 points, as a, a, in the plan, and from the corresponding- points 
 0, o in the elevation, draw rays and perpendiculars intersect- 
 
 FIG. 433. 
 
 ing at e, <:, etc. ; through these points of intersection trace 
 the curve, and it will define the shadow. 
 
 57 L To Find the Shadow of a Shelf Curved in the Ele- 
 vation. In Fig. 433 find the points of intersection, e, c and 
 
 FIG. 434 
 
 e, as in the last examples, and a curve traced through them 
 will define the shadow. 
 
 The preceding examples show how to find shadows when 
 cast upon a vertical plane ; shadows thrown upon curved sur- 
 faces are ascertained in a similar manner. (Fig. 434.) 
 
SHADOW UPON AN INCLINED WALL. 
 
 603 
 
 572. To Find the Shadow Cast upon a Cylindrical 
 Wall by a Projection of any Kind. By an inspection of 
 Fig. 434, it will be seen that the only difference between this 
 and the last examples is that the rays in the plan die against 
 the circle ab, instead of a straight line. 
 
 573. To Find the Shadow Cat by a Shelf upon an In- 
 clined Wall. Cast the ray ab (Fig. 435) from the end of the 
 shelf to the face of the wall, and from b draw be parallel to 
 the shelf; cast the ray de from the end of the shelf; then 
 the lines de and ec will define the shadow. 
 
 FIG. 436. 
 
 These examples might be multiplied, but enough has 
 been given to illustrate the general principle by which shad- 
 ows in all instances are found. Let us attend now to the 
 application of this principle to such familiar objects as are 
 likely to occur in practice. 
 
004 SHADOWS. 
 
 574. To Find the Shadow of a Projecting Horizontal 
 Ream. From the points a, a, etc. (Fig. 436), cast rays upon 
 the wall ; the intersections e, e, e of those rays with the per- 
 pendiculars drawn from the plan will define the shadow. If 
 the beam be inclined, either on the plan or elevation, at any 
 angle other than a right angle, the difference in the manner 
 
 FIG. 437. 
 
 of proceeding can be seen by reference to the preceding 
 examples of inclined shelves, etc. 
 
 575. To Find the Shadow in a Recess. From the point 
 a (Fig. 437) in the plan, and b in the elevation, draw the rays 
 acand be; from c erect the perpendicular ce, and from e 
 
 FIG. 438. 
 
 draw the horizontal line ed; then the lines r^.and ed will 
 show the extent of the shadow. This applies only where 
 the back of the recess is parallel with the face of the wall. 
 
 576. To Find the Shadow in a Rece, when the Face 
 of the Wall Is Inclined, and the Baek of the Recess i 
 Vertical. In Fig. 438, A shows the section and B the eleva- 
 
SHADOW IN A FIREPLACE. 
 
 60 5 
 
 tion of a recess of this kind. From b, and from any other 
 point in the line ba, as a, draw the rays be and ae ; from c, 
 a, and e draw the horizontal lines eg, af, and eh; from d 
 
 FIG. 439. 
 
 and /cast the rays di and ///; from i, through h, draw is; 
 then s i and ig will define the shadow. 
 
 577. TO Find the Shadow in a Fireplace. From a and 
 
 b (Fig. 439) cast the rays a c and b e, and from c erect the 
 
 FIG. 440. 
 
 perpendicular cc\ from c draw the horizontal line eo, and 
 join o and </; then c c, eo, and <?</ will give the extent of the 
 shadow. 
 
6o6 
 
 SHADOWS. 
 
 578. To Find I lie Shadow of a moulded Window-Lin- 
 tel. Cast rays from the projections a, o, etc., in the plan 
 (Fig. 440), and d, e, etc., in the elevation, and draw the usual 
 perpendiculars intersecting the rays at z, i, and i ; these in- 
 tersections connected, and horizontal lines drawn from them, 
 will define the shadow. The shadow on the face of the lin- 
 tel is found by casting a ray back from i to s, and drawing 
 the horizontal line s n. 
 
 579. To Find the Shadow Cast by the Noing of a Step. 
 
 From a (Fig. 441) and its corresponding point c, cast the 
 
 FIG. 441. 
 
 rays a b and cd, and from b erect the perpendicular b d; tan- 
 gical to the curve at e cast the ray ef, and from c drop the 
 perpendicular e o, meeting the mitre-line ag in o ; cast a ray 
 from o to /', and from /erect the perpendicular if\ from /t 
 draw the ray // k ; from /to d and from d to k trace the 
 curve as shown in the figure ; from k and h draw the hori- 
 zontal lines kn and hs\ then the limit of the shadow will be 
 completed. 
 
 580. To Find the Shadow Thrown by a Pedetal upon 
 Step. From a (Fig. 442) in the plan, and from c in the ele- 
 vation, draw the rays ab and c e ; then ao will show the ex- 
 
SHADOWS ON STEPS AND COLUMNS. 
 
 60 7 
 
 tent of the shadow on the first riser, as at A ; fg will deter- 
 mine the shadow on the second riser, as at B\ cd gives the 
 amount of shadow on the first tread, as at 7, and // i that on 
 the second tread, as at D ; which completes the shadow of 
 
 FIG. 442. 
 
 the left-hand pedestal, both on the plan and elevation. A 
 mere inspection of the figure will be sufficient to show how 
 the shadow of the right-hand pedestal is obtained. 
 
 FIG. 443. 
 
 FIG. 444. 
 
 681. To Find the Shadow Thrown on a Column by a 
 Square Abacus. From a and b (Fig. 443) draw the rays ac 
 and b e, and from c erect the perpendicular c e ; tangical to 
 the curve at d draw the ray df, and from //, corresponding 
 to /in the plan, draw the ray ho; take any point between a 
 and f y as i, and from this, as also from a corresponding point 
 
6o8 
 
 SHADOWS. 
 
 n, draw the rays ir and ns ; from r and from d erect the 
 perpendiculars rs and do-, through the points e, s, and o 
 trace the curve as shown in the figure ; then the extent of 
 the shadow will be defined. 
 
 m 
 X. 
 
 
 N/^ 
 
 A. /" 
 
 
 
 
 
 
 
 J 
 
 ^^^^^ ^TlWk 
 
 FIG. 445. 
 
 582. To Find the Shadow Tin-own on a Column by a 
 Circular Abacus. This is so nearly like the last example 
 that no explanation will be necessary, farther than a refer- 
 ence to the preceding article. 
 
SHADOWS ON THE CAPITAL OF A COLUMN". 
 
 609 
 
 583. To Find the Shadows on the Capital of a Columii. 
 
 This may be done according- to the principles explained 
 in the examples already given ; a quicker way of doing it, 
 however, is as follows : if we take into consideration one 
 ray of light in connection with all those perpendicularly 
 under and over it, it is evident that these several rays would 
 form a vertical plane, standing at an angle of 45 degrees 
 with the face of the elevation. Now we may suppose the 
 column to be sliced, so to speak, with planes of this nature 
 cutting it in the lines a b, c d, etc. (Fig. 445), and, in the ele- 
 
 FIG. 446. 
 
 vation, find by squaring up from the plan, the lines of section 
 which these planes would make thereupon. For instance : 
 in finding upon the elevation the line-of section a I), the plane 
 cuts the ovolo at e, and therefore / will be the correspond- 
 ing point upon the elevation ; h corresponds with g, i withy, 
 o with s, and / with b. Now, to find the shadows upon tfcis 
 line of section, cast from m the ray m , from // the ray h o, 
 etc. ; then that part of the section indicated by the letters 
 m f i n, and that part also between // and o will be under 
 
6io 
 
 SHADOWS. 
 
 shadow. By an inspection of the figure, it will be seen that 
 the same process is applied to each line of section, and in 
 that way the points /, r, t, u, v, ?v, x, as also i, 2, 3, etc., are 
 
 FIG. 447. 
 
 successively found, and the lines of shadow traced through 
 them. 
 
 Fig. 446 is an example of the same capital with all the 
 shadows finished in accordance with the lines obtained on 
 Fig. 445. 
 
SHADOW OF A COLUMN ON A WALL. 
 
 584. To Find the Shadow Thrown on a Vertical Wall 
 l>y a Column and Entablature Standing in Advance of aid 
 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 i, and from i and j erect the per- 
 pendiculars il and/; from m and n draw the rays mf and 
 nk, and trace the curve between and/; cast a ray from o to 
 /, a vertical line from/ to s, and through s draw the horizon- 
 tal line s t ; the shadow as required will then be completed. 
 
 FIG. 448. 
 
 Fig. >| /| K is an example of the same kind as the last, with 
 all the shadows filled in, according to the lines obtained in 
 the preceding figure. 
 
 585. Shadows on a Cornice. Figs. 449 and 450 are 
 
 examples of the Tuscan cornice. The manner of obtaining 
 the shadows is evident. 
 
 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. Re- 
 
6l2 
 
 SHADOWS. 
 
 fleeted rays are thrown in a direction exactly the reverse 
 of direct rays ; therefore, on that part of an object which is 
 subject to reflected light, the shadows are reversed. The 
 
 FIG. 449. 
 
 fillet of the ovolo in Fig. 446 is an example of this. On the 
 right hand side of the column, the^ace of the fillet is much 
 darker than the cove directly under it. The reason of this 
 
 FIG. 450, 
 
 is, the face of the fillet is deprived both of direct and re- 
 flected light, whereas the cove is subject to the latter. Other 
 instances of the effect of reflected light will be seen in the 
 other examples. 
 
CONTENTS. 
 
 PART I. 
 
 SECTION I. ARC H ITECTU RE. 
 
 Art. 1. Building defined, p. 5. 2. Antique Builcings ; Tower of Babel, 
 p. 5. 3. Ancient Cities and Monuments, p. 6. 4. Architecture in Greece, 
 p. 6. 5. Architecture in Rome, p. 7. 6. Rome and Greece, p. 8. 7. Ar- 
 chitecture debased, p. 9. . The Ostrogoths, p. 9. 9. The Lombards, p. 10. 
 1O. The Byzantine Architects, p. 10. 11. The Moors, p. 10. 12. The 
 Architecture of England, p. n. 13. Architecture Progressive, p. 12. 14. 
 Architecture in Italy, p. 12. 15. The Renaissance, p. 13. 16. Styles of Ar- 
 chitecture, p. 13. 17. Orders, p. 14. 18. The Stylobatc, p. 14. 19. The 
 Column, p. 14. 2O. The Entablature, p. 14. 21. The Base, p. 1422. 
 The Shaft, p. 15. 23. The Capital, p. 15. 24. The Architrave, p. 15. 25. 
 The Frieze, p. 15. 26. The Cornice, p. 15. 27. The Pediment, p. 15. 28. 
 The Tympanum, p. 15. 29. The Attic, p. 15. 3O. Proportions in an Order, 
 p. 15. 31. Grecian Styles, p. 16. 32. The Doric Order, p. 16. 33. The 
 Intercolumniation, p. 17. 34. The Doric Order, p. 19 35. The Ionic 
 Order, p. 19. 36. The Intercolumniation, p. 20. 37. To Describe the Ionic 
 Volute, p. 20. 38. The Corinthian Order, p. 23. 39. Persians and Carya- 
 tides, p. 24. 4O. Persians, p. 24. 41, Caryatides, p. 26. 42. Roman 
 Styles, p. 26. 43. Grecian Orders modified by the Romans, p. 27. 44. The 
 Tuscan Order, p. 30. 45. Egyptian Style, p. 30. 46. Building in General, 
 p. 33. 47. Expression, p. 35. 48. Durability, p. 37. 49. Dwelling- 
 Houses, p. 37. 5O. Arranging the Stairs and Windows, p. 42. 51. Prin- 
 ciples of Architecture, p. 44. 52. Arrangement, p. 44. 53. Ventilation, p. 
 45. 54. Stability, p. 45. 55. Decoration, p. 46. 56. Elementary Parts of 
 a Building, p. 46. 57. The Foundation, p. 47. 58. The Column, or Pillar, 
 p. 47. 59. The Wall, p. 48. 6O. The Reticulated Walls, p. 49. 61. The 
 Lintel, or Beam, p. 49. 62. The Arch, p. 50. 63. Hookc's Theory of an 
 Arch, p. 50. 64. Gothic Arches, p. 51. 65. Arch : Definitions ; Principles, 
 p. 52. 66. An Arcade, p. 52. 67. The Vault, p. 52. 68. The Dome, p. 53. 
 69. The Roof, p. 54. 
 
614 CONTENTS. 
 
 SECTION II. CONSTRUCTION. 
 
 Art. TO. Construction Essential, p. 56. 71. Laws of Pressure, p. 57. 
 72. Parallelogram of Forces, p. 59. 73. The Resolution of Forces, p. 59. 
 74. Inclination of Supports Unequal, p. 60. 75. The Strains Exceed the 
 Weights, p. 61. 76. Minimum Thrust of Rafters, p. 62. 77. Practical 
 Method of Determining Strains, p. 62. 78. Horizontal Thrust, p. 63. 79. 
 Position of Supports, p. 65. 8O. The Composition of Forces, p. 66. 81. 
 Another Example, p. 67. 82. Ties and Struts, p. 68. 83. To Distinguish 
 Ties from Struts, p. 69. 84. Another Example, p. 70. 85. Centre of Gravity, 
 p. 7I ._86. Effect of the Weight of Inclined Beams, p. 72. 87. Effect of 
 Load on Beam, p. 74. 88. Effect on Bearings, p. 75. 89. Weight-Strength, 
 p. 76. 9O. Quality of Materials, p. 76. 91. Manner of Resisting, p. 77. 
 92. Strength and Stiffness, p. 78. 93. Experiments : Constants, p. 78. 94. 
 Resistance to Compression, p. 79. 95. Resistance to Tension, p. 81. 96. 
 Resistance to Transverse Strains, p. 83. 97. Resistance to Compression, p. 
 85. 98. Compression Transversely to the Fibres, p. 86. 99. The Limit of 
 Weight, p. 86. 1OO. Area of Post, p. 86. 1O1. Rupture by Sliding, p. 87. 
 1O2. The Limit of Weight, p. 87. 1O3. Area of Surface, p. 88. 1O4. 
 Tenons and Splices, p. 88. HO5. Stout Posts, p. 89. IO6. The Limit of 
 Weight, p. 89. 17. Area of Post, p. 90. 1O8. Area of Round Posts, p. 
 90. 1O9. Slender Posts, p. 91. 11O. The Limit of Weight, p. 91. 111. 
 Diameter of the Post; when Round, p. 92. 112. Side of Post: when 
 Square, p. 93. 113. Thickness of a Rectangular Post, p. 95. 114. Breadth 
 of a Rectangular Post, p. 95. 115. Resistance to Tension, p. 96. 116. 
 .The Limit of Weight, p. 96. 117. Sectional Area, p. 97. 118. Weight of 
 the Suspending Piece Included, p. 98. 119. Area of Suspending Piece, 
 p. 99. 
 
 RESISTANCE TO TRANSVERSE STRAINS. 
 
 Art. H2O. Transverse Strains: Rupture, p. 99. H21L. Location of Mor- 
 tises, p. 100. 122. Transverse Strains : Relation of Weight to Dimensions, 
 p. loi 123. Safe Weight : Load at Middle, p. 103. 124. Breadth of Beam 
 with Safe Load, p. 104. fl25. Depth of Beam with Safe Load, p. 104. 126. 
 Safe Load at any Point, p. 105. 127. Breadth or Depth : Load at any Point, 
 p. 106. 128. Weight Uniformly Distributed, p. 107. 129. Breadth or 
 Depth: Load Uniformly Distributed, p. 108. 13O. Load per Foot Super- 
 ficial, p. 109. 131. Levers : Load at one End, p. no. 132. Levers : Breadth 
 or Depth, p. in. 133. Deflection: Relation to Weight, p. 112. 134. De- 
 flection: Relation t<) Dimensions, p. 112. 135. Deflection : Weight when at 
 Middle, p. 114. 136. Deflection : Breadth or Depth, Weight at Middle, p. 
 114. 137. Deflection : When Weight is at Middle, p. 116. 138. Deflection : 
 Load Uniformly Distributed, p. 116. 139. Deflection : Weight when Uni- 
 formly Distributed, p. 117. I4O. Deflection: Breadth or Depth, Load Uni- 
 formly Distributed, p. 117. 14i. Deflection: When Weight is Uniformly 
 Distributed, p. 118. 142. Deflection of Lever, p. 119. 143. Deflection of 
 a Lever: Load at End, p. 120. 144. Deflection of a Lever : Weight when at 
 End, p. 120. 145. Deflection of a Lever : Breadth or Depth, Load at End, 
 
CONTENTS. 615 
 
 p. i2i. 146. Deflection of Levers: Weight Uniformly Distributed, p. 121. 
 147. Deflection of Levers with Uniformly Distributed Load, p. 122.. 148. 
 Deflection of Levers: Weight when Uniformly Distributed, p. 122. 149. 
 Deflection of Levers : Breadth or Depth, Load Uniformly Distributed, p. 122. 
 
 CONSTRUCTION IN GENERAL. 
 
 Art. 15O. Construction : Object Clearly Denned, p. 123. 151. Floors 
 Described, p. 124. 152. Floor-Beams, p. 125. 153. Floor-Beams for Dwell- 
 ings, p. 127. 154. Floor-Beams for First-Class Stores, p. 128. 155. Floor- 
 Beams : Distance from Centres, p. 129. 156. Framed Openings for Chimneys 
 and Stairs, p. 130. 157. Breadth of Headers, p. 130. 15. Breadth of 
 Carriage-Beams, p. 132. 159. Breadth of Carriage-Beams Carrying Two 
 Sets of Tail-Beams, p. 134. 16O. Breadth of Carriage-Beam with Well-Hole 
 at Middle, p. 136. 161. Cross- Bridging, or Herring-Bone Bridging, p. 137. 
 162. Bridging : Value to Resist Concentrated Loads, p. 137. 163. Gird- 
 ers, p. 140. 164. Girders : Dimensions, p. 141. 
 
 FIRE-PROOF TIMBER FLOORS. 
 
 Art. 165. Solid Timber Floors, p. 143. 166. Solid Timber Floors for 
 Dwellings and Assembly-Rooms, p. 143. 167. Solid Timber Floors for First- 
 Class Stores, p. 144. 168. Rolled-Iron Beams, p. 145. 169. Rolled-Iron 
 Beams: Dimensions; Weight at Middle, p. 146. 17O. Rolled-Iron Beams: 
 Deflection when Weight is at Middle, p. 147. 171. Rolled-Iron Beams: 
 Weight when at Middle, p. 148. 172. Rolled-Iron Beams: Weight at any 
 Point, p. 148. 173. Rolled-Iron Beams: Dimensions ; Weight at any Point, 
 p ^g. B.74. Rolled-Iron Beams : Dimensions ; Weight Uniformly Distrib- 
 uted, p. 149. 175. Rolled-Iron Beams : Deflection ; Weight Uniformly Dis- 
 tributed, p. 150. 176. Rolled-Iron Beams : Weight when Uniformly Distrib- 
 uted, p. 151. 177. Rolled-Iron Beams: Floors of Dwellings or Assembly- 
 Rooms, p. 151. 178. Rolled-Iron Beams : Floors of First-Class Stores, p. 
 152. 179. Floor-Arches: General Considerations, p. 153. 18O. Floor- 
 Arches: Tie-Rods; Dwellings, p. 153. 181. Floor-Arches: Tie-Rods; 
 First Class Stores, p. 153. 
 
 TUBULAR IRON GIRDERS. 
 
 Art. 182. Tubular Iron Girders: Description, p. 154. 183. Tubular 
 Iron Girders : Area of Flanges ; Load at Middle, p. 154. 184. Tubular Iron 
 Girders : Area of Flanges ; Load at any Point, p. 155. 185. Tubular Iron 
 Girders : Area of Flanges ; Load Uniformly Distributed, p. 156. 186. Tu- 
 bular Iron Girders: Shearing Strain, p. 157. 187. Tubular Iron Girders: 
 Thickness of Web, p. 158. 188. Tubular Iron Girders for Floors of Dwell- 
 ings, Assembly-Rooms, and Office Buildings, p. 159. 189. Tubular Iron 
 Girders for Floors of First-Class Stores, p. 160. 
 
 CAST-IRON GIRDERS. 
 
 Art. 19O. Cast-Iron Girders: Inferior, p. 161. 191. Cast-Iron Girder: 
 Load at Middle, p. 161. 192. Cast-Iron Girder: Load Uniformly Distributed, 
 
6l6 CONTENTS. 
 
 p. 163. 193. Cast-Iron Bowstring Girder, p. 163. 194. Substitute for the 
 Bowstring Girder, p. 163. 
 
 FRAMED GIRDERS. 
 
 Art. 195. Graphic Representation of Strains, p. 165. 196. Framed 
 Girders, p. 166. 197. Framed Girder and Diagram of Forces, p. 167. 198. 
 Framed Girders : Load on Both Chords, p. 171. 199. Framed Girders: Di- 
 mensions of Parts, p. 173. 
 
 PARTITIONS. 
 
 Art. 2OO. Partitions, p. 174. 2O1. Examples of Partitions, p. 175. 
 
 ROOFS. 
 
 Art. 2O2. Roofs, p. 178. 2O3. Comparison of Roof-Trusses, p. 178. 
 2O4. Force Diagram : Load upon Each Support, p. 179. 2O5. Force Dia- 
 gram for Truss in Fig. 59, p. 179. 2O6. Force Diagram for Truss in Fig. 60, 
 p. 180. 2O7. Force Diagram for Truss in JFtg.Gi, p. 181. 2O8. Force Dia- 
 gram for Truss in Fig. 63, p. 183. 299. Force Diagram for Truss in Fig. 64, 
 p. 184. 21O. Forc Diagram for Truss in Fig. 65, p. 185. 211. Force Dia- 
 gram for Truss in Fig. 66, p. 186. 212. Roof-Truss : Effect of Elevating the 
 Tie-Beam, p. 187. 213. Planning a Roof, p. 188. 214. Load upon Roof- 
 Truss, p. 189. 215. Load on Roof per Superficial Foot, p. 189. 216. Load 
 upon Tie-Beam, p. 190. 217. Roof Weights in Detail, p. 191. 21. Load 
 per Foot Horizontal, p. 192. 219. Weight of Truss, p. 192. 22O. Weight 
 of Snow on Roofs, p. 193. 221. Effect of Wind on Roofs, p. 193. 222. 
 Total Load per Foot Horizontal, p. 197. 223. Strains in Roof Timbers 
 Computed, p. 198. 224. Strains in Roof Timbers Shown Geometrically, p. 
 199. 225. Application of the Geometrical System of Strains, p. 202. 226. 
 Roof Timbers : the Tie-Beam, p. 204. 227. The Rafter, p. 205. 228. The 
 Braces, p. 208. 229. The Suspension Rod, p. 210. 23O. Roof-Beams, 
 Jack-Rafters, and Purlins, p. 211. 231. Five Examples of Roofs, p. 212. 
 232. Roof-Truss with Elevated Tie-Beam, p. 214. 233. Hip-Roofs : Lines 
 and Bevels, p. 215. 234. The Backing of the Hip-Rafter, p. 216. 
 
 DOMES. 
 
 Art. 235. Domes, p. 216. 236. Ribbed Dome, p. 217. 237. Domes: 
 Curve of Equilibrium, p. 218. 238. Cubic Parabola Computed, p. 219. 
 239. Small Domes over Stairways, p. 220. 24O. Covering for a Spherical 
 Dome, p. 221. 241. Polygonal Dome : Form of Angle-Rib, p. 223. 
 
 BRIDGES. 
 
 Art. 242. Bridges, p. 223. 243. Bridges : Built-Rib, p. 224. 244. 
 Bridges : Framed Rib, p. 226. 245. Bridges : Roadway, p. 227. 246. 
 Bridges : Abutments, p. 227. 247. Centres for Stone Bridges, p. 229. 248. 
 Arch Stones : Joints, p. 223. 
 
 JOINTS. 
 
 Art. 249. Timber Joints, p. 234. 
 
CONTENTS. 6i; 
 
 SECTION III. STAIRS. 
 
 Art. 250. Stairs : General Requirements, p. 240. 251. The Grade of 
 Stairs, p. 241. 252. Pitch-Board : Relation of Rise to Tread, p. 242. 253. 
 Dimensions of the Pitch-Board, p. 247. 254. The String of a Stairs, p. 247. 
 255. Step and Riser Connection, p. 248. 
 
 PLATFORM STAIRS 
 
 Art. 256. Platform Stairs : the Cylinder, p. 248. 257. Form of Lower 
 Edge of Cylinder, p. 249. 25. Position of the Balusters, p. 250. 259. Wind- 
 ing Stairs, p. 251. 260. Regular Winding Stairs, p. 251. 261. Winding 
 Stairs : Shape and Position of Timbers, p. 252. 262. Winding Stairs with 
 Flyers: Grade of Front String, p. 253. 
 
 HAND-RAILING. 
 
 Art. 263. Hand-Railing for Stairs, p. 256. 264. Hand-Railing : Defini- 
 tions ; Planes and Solids, p. 257. 265. Hand-Railing: Preliminary Consider- 
 ations, p. 258. 266. A Prism Cut by an Oblique Plane, p. 259. 267. Form 
 of Top of Prism, p. 259. 26. Face-Mould for Hand-Railing of Platform 
 Stairs, p. 264. 269. More Simple Method for Hand-Rail to Platform Stairs, 
 p. 267. 27O. Hand-Railing for a Larger Cylinder, p. 271. 271. Face- 
 Mould without Canting the Plank, p. 272. 272. Railing for Platform Stairs 
 where the Rake meets the Level, p. 272. 273. Application of Face-Moulds 
 to Plank, p. 273. 274. Face-Moulds for Moulded Rails upon Platform 
 Stairs, p. 274. 275. Application of Face-Moulds to Plank, p. 275. 276. 
 Hand-Railing for Circular Stairs, p. 278. 277. Face-Moulds for Circular 
 Stairs, p. 282. 27. Face-Moulds, for Circular Stairs, p. 285279. Face- 
 Moulds for Circular Stairs, again, p. 287. 280. Hand-Railing for Winding 
 Stairs, p. 289. 21. Face-Moulds for Windjng Stairs, p. 290. 22. Face- 
 Moulds for Winding Stairs, again, p. 293. 283. Face-Moulds : Test of Accu- 
 racy, p. 295. 284. Application of the Face-Mould, p. 297. 285. Face-Mould 
 Curves are Elliptical, p. 301. 286. Face-Moulds for Round Rails, p. 303. 
 287. Position of the Butt Joint, p. 303. 288. Scrolls for Hand-Rails: Gen- 
 eral Rule for Size and Position of the Regulating Square, p. 308. 289. Cen- 
 tres in Regulating Square, p. 308. 29O. Scroll for Hand-Rail Over Curtail 
 Step, p. 309. 291. Scroll for Curtail Step, p. 310. 292. Position of Balus- 
 ters Under Scroll, p. 310 293. Falling-Mould for Raking Part of Scroll, p. 
 310. 294. Face-Mould for the Scroll, p. 311. 295. Form of Newel-Cap 
 from a Section of the Rail, p. 312. 296. Boring for Balusters in a Round Rail 
 before it is Rounded, p. 313. 
 
 SPLAYED WORK. 
 
 Art. 297. The Bevels in Splayed Work, p. 314. 
 
 SECTION IV. DOORS AND WINDOWS. 
 
 DOORS. 
 
 Art. 298. General Requirements, p. 315. 299. The Proportion between 
 Width and Height, p. 315. 3OO. Panels, p. 316. 3O1. Trimmings, p. 317. 
 302. Hanging Doors, p. 317. 
 
6l8 CONTENTS. 
 
 WINDOWS. 
 
 Art. 3O3. Requirements for Light, p. 317. 3O4. Window Frames, p. 
 318. 3O5. Inside Shutters, p. 319. 3O6. Proportion: Width and Height, 
 p. 319. 3O7. Circular Heads, p. 320. 3O8. Form of Soffit for Circular Win- 
 dow Heads, p. 321. 
 
 SECTION V. MOULDINGS AND CORNICES. 
 
 MOULDINGS. 
 
 Art. 309. Mouldings, p. 323. 31 0. Characteristics of Mouldings, p. 
 324. 311. A Profile, p. 326. 312. The Grecian Torus and Scotia, p. 326. 
 313. The Grecian Echinus, p. 327. 314. The Grecian Cavetto, p. 327. 
 315. The Grecian Cyma- Recta, p. 327. 316. The Grecian Cyma-Reversa, 
 p. 328. 3S.7. Roman Mouldings, p. 329. 318. Modern Mouldings, p. 331. 
 
 CORNICES. 
 
 Art. 3H9. Designs for Cornices, p. 335. 32O. Eave Cornices Propor- 
 tioned to Height of Building, p. 335. 321. Cornice Proportioned to a given 
 Cornice, p. 342. 322. Angle Bracket in a Built Cornice, p. 343. 323. Rak- 
 ing Mouldings Matched with Level Returns, p. 344. 
 
 PART II. 
 
 SECTION VI.-GEOMETRY. 
 
 Art. 324. Mathematics Essential, p. 347. 325. Elementary Geometry, 
 p. 347. 326. Definition Right Angles, p. 348. 327. Definition Degrees 
 in a Circle, p. 348. 328. Definition Measure of an Angle, p. 348. 329. 
 Corollary Degrees in a Right Angle, p. 348. 33O. Definition Equal 
 Angles, p. 349. 331. Axiom Equal Angles, p. 349. 332. Definition 
 Obtuse and Acute Angles, p. 349. 333. Axiom Right Angles, p. 349. 
 334. Corollary Two Right Angles, p. 349. 335. Corollary Four Right 
 Angles, p. 349. 336. Proposition Equal Angles, p. 350. 337. Propo- 
 sition Equal Triangles, p. 350. 338. Proposition Angles in Isosceles 
 Triangle, p. 351. 339. Proposition Diagonal of Parallelogram, p. 351- 
 34O. Proposition Equal Parallelograms, p. 352. 341. Proposition Paral- 
 lelograms Standing on the Same Base, p. 352. 342. Corollary Parallelo- 
 gram and Triangle, p. 353. 343. Proposition Triangle Equal to Quadrangle, 
 p. 353. 344. Proposition Opposite Angles Equal, p. 354. 345. Proposi- 
 tion Three Angles of Triangle Equal to Two Right Angles, p. 354. 346. 
 Corollary Right Angle in Triangle, p. 354. 347. Corollary Half a Right 
 
CONTENTS. 619 
 
 Angle, p. 355. 348. Corollary Right Angle in a Triangle, p. 355. 349. 
 Corollary Two Angles Equal to Right Angle, p. 355. 35O. Corollary Two 
 Thirds of a Right Angle, p. 355. 351. Corollary Equilateral Triangle, p. 355. 
 352. Proposition Right Angle in Semi-circle, p. 355. 353. Proposition- 
 The Square of the Hypothenuse Equal to the Squares of the Sides, p. 355. 
 354. Proposition Equilateral Octagon, p. 357. 355. Proposition Angle 
 at the Circumference of a Circle, p. 358. 356. Proposition Equal Chords 
 give Equal Angles, p. 358. 357. Corollary of Equal Chords, p. 359. 35. 
 Proposition Angle Formed by a Chord and Tangent, p. 359. 359. Propo- 
 sition Areas of Parallelograms, p. 360. 360. Proposition Triangles ot 
 Equal Altitude, p. 361. 361. Proposition Homologous Triangles, p. 362. 
 362. Proposition Parallelograms of Chords, p. 363. 363. Proposition 
 Sides of Quadrangle, p. 364. 
 
 SECTION VII. RATIO, OR PROPORTION. 
 
 Art. 364. Merchandise, p. 366. 365. The Rule of Three, p. 366. 
 366. Couples: Antecedent, Consequent, p. 367. 367. Equal Couples : an 
 Equation, p. 367. 36. Equality of Ratios, p. 367. 369. Equals Multiplied 
 by Equals Give Equals, p. 367. 37O. Multiplying an Equation, p. 368. 371. 
 Multiplying and Dividing one Member of an Equation : Cancelling, p. 368. 
 372. Transferring a Factor, p. 369. 373. Equality of Product : Means and 
 Extremes, p. 369. 374. Homologous Triangles Proportionate, p. 370. 
 375. The Steelyard, p. 371. 376. The Lever Exemplified by the Steelyard, 
 p. 372. 377. The Lever Principle Demonstrated, p. 375. 378. Any One or 
 Four Proportionals may be Found, p. 377. 
 
 SECTION VIII. FRACTIONS. 
 
 Art. 379. A Fraction Defined, p. 378. 38O. Graphical Representation 
 of Fractions : Effect of Multiplication, p. 378. 381. Form of Fraction 
 Changed by Division, p. 380. 382. Improper Fractions, p. 380. 383. Re- 
 duction of Mixed Numbers to Fractions, p. 381. 384. Division Indicated by 
 the Factors put as a Fraction, p. 381. 385. Addition of Fractions having Like 
 Denominators, p. 382. 386. Subtraction of Fractions of Like Denominators, 
 p> 383. 387. Dissimilar Denominators Equalized, p. 383. 388. Reduction 
 of Fractions to their Lowest Terms, p. 384. 389. Least Common Denomina- 
 tor, p. 384. 390. Least Common Denominator Again, p. 385. 391. Frac- 
 tions Multiplied Graphically, p. 386. 392. Fractions Multiplied Graphically 
 Again, p. 387. 393. Rule for Multiplication of Fractions, and Example, p. 
 387. 394. Fractions Divided Graphically, p. 388. 395. Rule for Division 
 of Fractions, p. 389. 
 
 SECTION IX. ALGEBRA. 
 
 Art. 396. Algebra Defined, p. 392. 397. Example: Application, p. 
 393> 398. Algebra Useful in Constructing Rules, p. 394. 399. Algebraic 
 Rules are General, p. 394. 4OO. Symbols Chosen at Pleasure, p. 395. 4O1. 
 Arithmetical Processes Indicated by Signs, p. 396. 4O2. Examples in Addi- 
 
C2O CONTENTS. 
 
 tion and Subtraction : Cancelling, p. 398. 403. Transferring a Symbol to the 
 Opposite Member, p. 399. 4O4. Signs of Symbols to be Changed when they 
 are to be Subtracted, p. 400. 4O5. Algebraic Fractions, Added and Sub- 
 tracted, p. 403. 4O6. The Least Common Denominator., p. 404. 4O7. Alge- 
 braic Fractions Subtracted, p. 405. 4O. Graphical Representation of Multi- 
 plication, p. 408. 4O9. Graphical Multiplication : Three Factors, p. 408. 
 41O. Graphic Representation : Two and Three Factors, p. 409. 411. Graph- 
 ical Multiplication of a Binomial, p. 409. 412. Graphical Squaring of a 
 Binomial, p. 410. 413. Graphical Squaring of the Difference of Two Fac- 
 tors, p. 412. 414. Graphical Product of the Sum and Difference of Two 
 Quantities, p. 413. 415. Plus and Minus Signs in Multiplication, p. 415, 
 416. Equality of Squares on Hypothenuse and Sides of Right-Angled Tri- 
 angle, p. 416. 417*. Division the Reverse of Multiplication, p. 418. 418. 
 Division: Statement of Quotient, p. 419. 419. Division: Reduction, p. 419. 
 420. Proportionals : Analysis, p. 421. 421. Raising a Quantity to any 
 Power, p. 423. 422. Quantities with Negative Exponents, p. 423. 423. 
 Addition and Subtraction of Exponential Quantities, p. 424. 424. Multipli- 
 cation of Exponential Quantities, p. 424. 425. Division of Exponential 
 Quantities, p. 424. 426. Extraction of Radicals, p. 425. 427. Logarithms, 
 p. 425. 428. Completing the Square of a Binomial, p. 429. 
 
 PROGRESSION. 
 
 Art. 429. Arithmetical Progression, p. 432. 43O. Geometrical ProgreS' 
 sion, p. 435. 
 
 SECTION X.--POLYGONS. 
 
 Art. 431. Relation of Sum and Difference of Two Lines, p. 439. 432. 
 
 Perpendicular, in Triangle of Known Sides, p. 440. 433. Trigon : Radius of 
 Circumscribed and Inscribed Circles : Area, p. 443. 434. Tetragon : Radius 
 of Circumscribed and Inscribed Circles: Area, p. 446. 435. Hexagon : Ra- 
 dius ot Circumscribed and Inscribed Circles : Area, p. 447. 436. Octagon : 
 Radius of Circumscribed and Inscribed Circles : Area, p. 449. 437. Dodec- 
 agon: Radius of Circumscribed and Inscribed Circles: Area, p. 452. 438. 
 Hecadecagon : Radius of Circumscribed and Inscribed Circles : Area, p. 455. 
 439. Polygons : Radius of Circumscribed and Inscribed Circles : Area, p. 
 460. 440. Polygons : Their Angles, p. 462. 441. Pentagon: Radius of the 
 Circumscribed and Inscribed Circles: Area, p. 463. 442. Polygons: Table 
 of Constant Multipliers, p. 465. 
 
 SECTION XL THE CIRCLE. 
 
 Art. 443. Circles : Diameter and Perpendicular : Mean Proportional, p. 
 468. 444. Circle : Radius from Given Chord and Versed Sine, p. 469. 
 445. Circle : Segment from Ordinates, p. 470. 446. Circle : Relation of 
 Diameter to Circumference, p. 472. 447. Circle : Length of an Arc, p. 475. 
 448. Circle : Area, p. 475. 449. Circle: Area of a Sector, p. 476. 45O. 
 Circle : Area of a Segment, p. 477. 
 
CONTENTS. 621 
 
 SECTION XII. THE ELLIPSE. 
 
 Art. 451. Ellipse: Definitions, p. 481. 452. Ellipse: Equations to the 
 Curve, p. 482. 453. Ellipse : Relation of Axis to Abscissas of Axes, p. 484. 
 454. Ellipse : Relation of Parameter and Axes, p. 485. 455. Ellipse : 
 Relation of Tangent to the Axes, p. 485. 456. Ellipse: Relation of Tangent 
 with the Foci, p. 487. 457. Ellipse : Relation of Axes to Conjugate Diam- 
 eters, p. 487. 458. Ellipse : Area, p. 488. 459. Ellipse : Practical Sugges- 
 tions, p. 489. 
 
 SECTION XIII. THE PARABOLA. 
 
 Art. 46O. Parabola : Definitions, p. 492. 461. Parabola : Equation to 
 the Curve, p. 493. 462. Parabola : Tangent, p. 493. 463. Parabola : Sub- 
 tangent, p. 496. 464. Parabola : Normal and Subnormal, p. 496. 465. 
 Parabola : Diameters, p. 497. 466. Parabola : Elements, p. 499. 467. 
 Parabola : Described Mechanically, p. 500. 468. Parabola : Described from 
 Points, p. 502. 469. Parabola : Described from Arcs, p. 503. 47O. Para- 
 bola : Described from Ordinates, p. 504. 471. Parabola : Described from 
 Diameters, p. 507. 472. Parabola : Area, p. 509. 
 
 SECTION XIV. TRIGONOMETRY. 
 
 Art. 473. Right-Angled Triangles: The Sides, p. 510. 474. Right- 
 Angled Triangles : Trigonometrical Tables, p. 512. 475* Right-Angled 
 Triangles: Trigonometrical Value of Sides, p. 516. 476. Oblique- Angled 
 Triangles: Sines and Sides, p. 519. 477. Oblique-Angled Triangles : First 
 Class, p. 520. 478. Oblique-Angled Triangles: Second Class, p. 522. 
 479. Oblique-Angled Triangles : Sum and Difference of Two Angles, p. 523. 
 48O. Oblique-Angled Triangles : Third Class, p. 526. 481. Oblique- 
 Angled Triangles : Fourth Class, p. 528. 482* Trigonometrical Formulae : 
 Right-Angled Triangles, p. 530. 483. Trigonometrical Formula; : First 
 Class, Oblique, p. 531. 484. Trigonometrical Formulae: Second Class, 
 Oblique, p. 532. 485. Trigonometrical Formulae : Third Class, Oblique, p. 
 534. 486. Trigonometrical Formulas : Fourth Class, Oblique, p. 534. 
 
 SECTION XV. DRAWING 
 
 Art. 487. General Remarks, p. 536. 488. Articles Required, p. 536. 
 489. The Drawing-Board, p. 536. 49O. Drawing- Paper, p. 537. 491. To 
 
 Secure the Paper to the Board, p. 537. 492. The T-Square, p. 539. 493. 
 The Set-Square, p. 539. 494. The Rulers, p. 540. 495. The Instruments, 
 p. 540. 496. The Scale of Equal Parts, p. 540. 497. The Use of the Set- 
 Square, p.. 541. 498. Directions for Drawing, p. 542. 
 
622 CONTENTS. 
 
 SECTION XVI. PRACTICAL GEOMETRY. 
 Art. 499. Definitions of Various Terms, p. 544. 
 
 PROBLEMS. 
 
 RIGHT LINES AND ANGLES. 
 
 Art. 5OO. To Bisect a Line, p. 549. 5O1. To Erect a Perpendicular, p. 
 550. 5O2. To let Fall a Perpendicular, p. 551. 5O3. To Erect a Perpen- 
 dicular at the End of a Line, p. 551. 5O4. To let Fall a Perpendicular near 
 the End of a Line, p. 553. 5O5. To Make an Angle Equal to a Given Angle, 
 p. 553. 5O6. To Bisect an Angle, p. 554. 5O7. To Trisect a Right Angle, 
 p. 554. 5O8. Through a Given Point to Draw a Line Parallel to a Given 
 Line, p. 555. 509. To Divide a Given Line into any Number of Equal Parts, 
 P- 555- 
 
 THE CIRCLE. 
 
 Art. 5IO. To Find the Centre of a Circle, p. 556. 511. At a Given 
 Point in a Circle to Draw a Tangent thereto, p. 557. 512. The Same, with- 
 out making use of the Centre of the Circle, p. 557. 513. A Circle and a 
 Tangent Given, to Find the Point of Contact, p. 558. 514. Through any 
 Three Points not in a Straight Line to Draw a Circle, p. 559. 515. Three 
 Points not in a Straight Line being Given, to Find a Fourth that Shall, with 
 the Three, Lie in the Circumference of a Circle, p. 559. 516. To Describe 
 a Segment of a Circle by a Set-Triangle, p. 560. 517. To Find the Radius of 
 an Ate of a Circle when the Chord and Versed Sine are Given, p. 561. 518. 
 To Find the Versed Sine of an Arc of a Circle when the Radius and Chord 
 are Given, p. 561. 519. To Describe the Segment of a Circle by Intersection 
 of Lines, p. 562. 52O. Ordinates, p. 563. 521. In a Given Angle to De- 
 scribe a Tanged Curve, p. 565. 522. To Describe a Circle within any Given 
 Triangle, so that the Sides of the Triangle shall be Tangical, p. 566. 523. 
 About a Given Circle to Describe an Equilateral Triangle, p. 566. 524. To 
 Find a Right Line nearly Equal to the Circumference of a Circle, p. 566. 
 
 POLYGONS, ETC. 
 
 Art. 525. Upon a Given Line to Construct an Equilateral Triangle, p. 
 568. 526. To Describe an Equi'lateral Rectangle, or Square, p. 568. 527. 
 Within a Given Circle to Inscribe an Equilateral Triangle, Hexagon, or Dodec- 
 agon, p. 569. 52. Within a Square to Inscribe an Octagon, p. 570. 529. 
 To Find the Side of a Buttressed Octagon, p. 571. 5ilO. Within a Given 
 Circle to Inscribe any Regular Polygon, p. 572. 531. Upon a Given Line to 
 Describe any Regular Polygon, p. 573. 532. To Construct a Triangle whose 
 Sides shall be severally Equal to Three Given Lines, p. 575. 533. To Con- 
 struct a Figure Equal to a Given Right-lined Figure, p. 575. 534. To Make 
 a Parallelogram Equal to a Given Triangle, p. 576. 535. A Parallelogram 
 being Given, to Construct Another Equal to it, and Having a Side Ecjual to a 
 
CONTENTS. 623 
 
 Given Line, p. 576. 536. To Make a Square Equal to two or more Given 
 Squares, p. 577. 537. To Make a Circle Equal to two Given Circles, p. 580. 
 53. To Construct a Square Equal to a Given Rectangle, p. 581. 539. To 
 Form a Square Equal to a Given Triangle, p. 582. 54O. Two Right Lines 
 being Given, to Find a Third Proportional thereto, p. 582. 541. Three Right 
 Lines being Given, to Find a Fourth Proportional thereto, p. 583. 542. A 
 Line with Certain Divisions being Given, to Divide Another, Longer or 
 Shorter, Given Line in the Same Proportion, p. 583. 543. Between Two 
 Given Right Lines to Find a Mean Proportional, p. 584. 
 
 CONIC SECTIONS. 
 
 Art. 544. Definitions, p. 584. 545. To Find the Axes of the Ellipsis, 
 p. 585. 546. To Find the Axis and Base of the Parabola, p. 585. 547. To 
 Find the Height, Base, and Transverse Axis of an Hyperbola, p. 585. 54. 
 The Axes being Given, to Find the Foci, and to Describe an Ellipsis with a 
 String, p. 586. 549. The Axes being Given, to Describe an Ellipsis with a 
 Trammel, p. 586. 55O. To Describe an Ellipsis by Ordinates, p. 588. 551. 
 To Describe an Ellipsis by Intersection of Lines, p. 588. 552. To Describe 
 an Ellipsis by Intersecting Arcs, p. 590. 553. To Describe; a Figure Nearly 
 in the Shape of an Ellipsis by a Pair of Compasses, p. 591. 554. To Draw 
 an Oval in the Proportion Seven by Nine, p. 591. 555. To Draw a Tangent 
 to an Ellipsis, p. 592. 556. An Ellipsis with a Tangent Given, to Detect the 
 Point of Contact, p. 593. 557. A Diameter of an Ellipsis Given, to Find its 
 Conjugate, p. 593. 558. Any Diameter and its Conjugate being Given, to 
 Ascertain the Two Axes, and thence to Describe the Ellipsis, p. 593. 559. 
 To Describe an Ellipsis, whence Axes shall be Proportionate to the Axes of 
 a Larger or Smaller Given One, p. 594. 56O. To Describe a Parabola by 
 Intersection of Lines, p. 594. 561. To Describe an Hyperbola by Intersec- 
 tion of Lines, p. 595. 
 
 SECTION XVII. SHADOWS. 
 
 Art. 562. The Art of Drawing, p. 596. 563. The Inclination of the Line 
 of Shadow, p. 596. 564. To Find the Line of Shadow on Mouldings and 
 other Horizontally Straight Projections, p. 597. 565. To Find the Line of 
 Shadow Cast by a Shelf, p. 598. 566. To Find the Shadow Cast by a Shelf 
 which is Wider at one End than at the Other, p. 599. 567. To Find the 
 Shadow of a Shelf having one End Acute or Obtuse Angled, p. 600. 56. 
 To Find the Shadow Cast by an Inclined Shelf, p. 600. 569. To Find the 
 Shadow Cast by a Shelf inclined in its Vertical Section either Upward or 
 Downward, p. 601. 57O. To Find the Shadow of a Shelf having its Front 
 Edge or End Curved on the Plan, p. 602. 571. To Find the Shadow of a 
 Shelf Curved in the Elevation, p. 602. 572. To Find the Shadow Cast upon 
 a Cylindrical Wall by a Projection of any Kind, p. 603. 573. To Find the 
 Shadow Cast by a Shelf upon an Inclined Wall. p. 603. 574. To Find the 
 Shadow of a Projecting Horizontal Beam, p 604. 575. To Find the Shadow 
 
624 CONTENTS. 
 
 in a Recess, p. 604. 576. To Find the Shadow in a Recess, when the Face of 
 the Wall is Inclined, and the Back of the Recess is Vertical, p. 604. 577. 
 To Find the Shadow in a Fireplace, p. 605. 578. To Find the Shadow of a 
 Moulded Window-Lintel, p. 606. 579. To Find the Shadow Cast by the 
 Nosing of a Step, p. 606. 58O. To Find the Shadow Thrown by a Pedestal 
 upon Steps, p. 6c6. 51. To Find the Shadow Thrown on a Column by a 
 Square Abacus, p. 607. 582. To Find the Shadow Thrown on a Column by 
 a Circular Abacus, p. 608. 583. To Find the Shadows on the Capital of a 
 Column, p. 609. 584. To Find the Shadow Thrown on a Vertical Wall by a 
 Column and Entablature Standing in Advance of said Wall, p. 611. 585* 
 Shadows on a Cornice, p. 611. 596. Reflected Light, p. 611. 
 
AMERICAN HOUSE CARPENTER. 
 
 APPENDIX. 
 
 UNIVERSITY 
 
CONTENTS. 
 
 PAGE. 
 
 GLOSSARY 627 
 
 TABLE OF SQUARES, CUBES, AND ROOTS 638 
 
 RULES FOR THE REDUCTION OF DECIMALS 647 
 
 TABLE OF CIRCLES 649 
 
 TABLE SHOWING THE CAPAC T TY OF WELLS, CISTERNS, ETC 653 
 
 TABLE OF THE WEIGHTS OF MATERIALS 654 
 
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. The residence of an abbot or abbess. 
 
 Abutment. That part of a pier from which the arch springs. 
 
 Acanthus. A plant called in English bear' s-breech. Its leaves are employed 
 for decorating the Corinthian and the Composite capitals. 
 
 Acropolis. The highest part of a city ; generally the citadel. 
 
 Acroteria. The small pedestals placed on the extremities and apex of a 
 pediment, originally intended as a base for sculpture. 
 
 Aisle. Passage to and from the pews of a church. In Gothic architecture, 
 the lean-to wings on the sides of the nave. 
 
 Alcove. Part of a chamber separated by an estrade, or partition of columns. 
 Recess with seats, etc., in gardens. 
 
 Altar. A pedestal whereon sacrifice was offered. In modern churches, the 
 area within the railing in front of the pulpit. 
 
 Alto-relievo. High relief ; sculpture projecting from a surface so as to appear 
 nearly isolated. 
 
 Amphitheatre. A double theatre, employed by the ancients for the exhibi- 
 tion of gladiatorial fights and other shows. 
 
 Ancones. Trusses employed as an apparent support to a cornice upon the 
 flanks of the architrave. 
 
 Annulet. A small square moulding used to separate others ; the fillets in 
 the Doric capital under the ovolo, and those which separate the flutings of col- 
 umns, are known by this term. 
 
 Antce. A pilaster attached to a wall. 
 
 Apiary. A place for keeping beehives. 
 
 Arabesque. A building after the Arabian style. 
 
 Areostyle. An intercolumniation of from four to five diameters. 
 
 Arcade. A series of arches. 
 
 Arch. An arrangement of stones or other material in a curvilinear form, so 
 
 as to perform the office of a lintel and carry superincumbent weights. 
 
 Architrave. That part of the entablature which rests upon the capital of a 
 column, and is beneath the frieze. The casing and mouldings about a door or 
 window. 
 
 Archivolt.The ceiling of a vault ; the under surface of an arch. 
 
 Area. Superficial measurement. An open space, below the level of the 
 ground, in front of basement windows. 
 
628 APPENDIX. 
 
 Arsenal. A public establishment for the deposition of arms and warlike 
 stores. 
 
 Astragal. A small moulding consisting of a half-round with a fillet on each 
 side. 
 
 Attic. A. low story erected over an order of architecture. A low additional 
 story immediately under the roof of a building. 
 
 Aviary. A place for keeping and breeding birds. 
 
 Balcony. An open gallery projecting from the front of a building. 
 
 Baluster. A small pillar or pilaster supporting a rail. 
 
 Balustrade. A series of balusters connected by a rail. 
 
 Barge-course. That part of the covering which projects over the gable of a 
 building. 
 
 Base. The lowest part of a wall, column, etc. 
 
 Basement-story. That which is immediately under the principal story, and 
 included within the foundation of the building. 
 
 Basso-relievo. Low relief ; sculptured figures projecting from a surface one 
 half their thickness or less. See Alto-relievo. 
 
 Battering. See Talus. 
 
 Battlement. Indentations on the top of a wall or parapet. 
 
 Bay-window. A window projecting in two or more planes, and not form- 
 ing the segment of a circle. 
 
 Bazaar. A species of mart or exchange for the sale of various articles of 
 merchandise. 
 
 Bead. A circular moulding. 
 
 Bed-mouldings. Those mouldings which are between the corona and the 
 frieze. 
 
 Belfry. That part of the steeple in which the bells are hung ; anciently 
 called campanile. 
 
 Belvedere. An ornamental turret or observatory commanding a pleasant 
 prospect. 
 
 Bow-window. A window projecting in curved lines. 
 
 Bressummer. A beam or iron tie supporting a wall over a gateway or other 
 opening. 
 
 Brick-nogging. The brickwork between studs of partitions. 
 
 Buttress. A projection from a wall to give additional strength. 
 
 Cable. A cylindrical moulding placed in flutes at the lower part of the col- 
 umn. 
 
 Camber. To give a convexity to the upper surface of a beam. 
 
 Campanile. A tower for the reception of bells, usually, in Italy, separated 
 from the church. 
 
 Canopy. An ornamental covering over a seat of state. 
 
 Cantalivers. The ends of rafters under a projecting roof. Pieces of wood 
 or stone supporting the eaves. 
 
 Capital. The uppermost part of a column included between the shaft and 
 the architrave. 
 
 Caravansera. In the East, a large public building for the reception of trav- 
 ellers by caravans in the desert. 
 
GLOSSARY. 629 
 
 Carpentry. (From the Latin carpentum, carved wood.) That department 
 of science and art which treats of the disposition, the construction, and the 
 relative strength of timber. The first is called descriptive, the second con- 
 structive, and the last mechanical carpentry. 
 
 Caryatides. Figures of women used instead of columns to support an 
 entablature. 
 
 Casino. A small country-house. 
 
 Castellated. Built with battlements and turrets in imitation of ancient 
 castles. 
 
 Castle. A building fortified for military defence. A house with towers, 
 usually encompassed with walls and moats, and having a donjon, or keep, in 
 the centre. 
 
 Catacombs. Subterraneous places for burying the dead. 
 
 Cathedral. The principal church of a province or diocese, wherein* the 
 throne of the archbishop or bishop is placed. 
 
 Cavetto. A concave moulding comprising the quadrant of a circle. 
 
 Cemetery. An edifice or area where the dead are interred. 
 
 Cenotaph. A monument erected to the memory of a person buried in 
 another place. 
 
 Centring. The temporary woodwork, or framing, whereon any vaulted 
 work is constructed. 
 
 Cesspool. A well under a drain or pavement to receive the waste water and 
 sediment. 
 
 Chamfer. The bevelled edge of anything originally right angled. 
 
 Chancel. That part of a Gothic church in which the altar is placed. 
 
 Chantry. A little chapel in ancient churches, with an endowment for one 
 or more priests to say mass for the relief of souls out of purgatory. 
 
 Chapel. A building for religious worship, erected separately from a church, 
 and served by a chaplain. 
 
 Chaplet. A moulding carved into beads, olives, etc. 
 
 Cincture. The ring, listel, or fillet, at the top and bottom of a column, 
 which divides the shaft of the column from its capital and base. 
 
 Circus. A straight, long, narrow building used by the Romans for the ex- 
 hibition of public spectacles and chariot races. At the present day, a building 
 enclosing an arena for the exhibition of feats of horsemanship. 
 
 Clere-story. The upper part of the nave of a church above the roofs of the 
 aisles. 
 
 Cloister. The square space attached to a regular monastery or large church, 
 having a peristyle or ambulatory around it, covered with a range of buildings. 
 
 Coffer-dam. A case of piling, water-tight, fixed in the bed of a river, for the 
 purpose of excluding the water while any work, such as a wharf, wall, or the 
 pier of a bridge, is carried up. 
 
 Collar-beam. A horizontal beam framed between two principal rafters above 
 the tie-beam. 
 
 Colonnade. A range of columns. 
 
 Columbarium. A pigeon-house. 
 
 Column. A vertical cylindrical support under the entablature of an order. 
 
 Common-rafters. The same as jack-rafters, which see. 
 
630 APPENDIX. 
 
 Conduit. A long, narrow, walled passage underground, for secret com- 
 munication between different apartments. A canal or pipe for the conveyance 
 of water. 
 
 Conservatory. A building for preserving curious and rare exotic plants. 
 
 Consoles. The same as ancones, which see. 
 
 Contour. The external lines which bound and terminate a figure. 
 
 Convent. A building for the reception of a society of religious persons. 
 
 Coping. Stones laid on the top of a wall to defend it from the weather. 
 
 Corbels. Stones or timbers fixed in a wall to sustain the timbers of a floor 
 or roof. 
 
 Cornice. Any moulded projection which crowns or finishes the part to 
 which it is affixed. 
 
 Corona. That part of a cornice which is between the crown-moulding and 
 the bed-mouldings. 
 
 Cornucopia. The horn of plenty. 
 
 Corridor. An open gallery or communication to the different apartments of 
 a house. 
 
 Cove. A concave moulding. 
 
 Cripple-rafters. The short rafters which are spiked to the hip-rafter of a 
 roof. 
 
 Crockets. In Gothic architecture, the ornaments placed along the angles of 
 pediments, pinnacles, etc. 
 
 Crosettes. The same as ancones, which see. 
 
 Crypt. The under or hidden part of a building. 
 
 Culvert. An arched channel of masonry or brickwork, built beneath the 
 bed of a canal for the purpose of conducting water under it. Any arched 
 channel for water underground. 
 
 Cupola. A small building on the top of a dome. 
 
 Curtail-step. A step with a spiral end, usually the first of the flight. 
 
 Cusps. The pendants of a pointed arch. 
 
 Cyma. An ogee. There are two kinds ; the cyma-recta, having the upper 
 part concave and the lower convex, and the cyma-reversa, with the upper part 
 convex and the lower concave. 
 
 Dado. The die, or part between the base and cornice of a pedestal. 
 
 Dairy. An apartment or building for the preservation of milk, and the 
 manufacture of it into butter, cheese, etc. 
 
 Dead-shoar. A piece of timber or stone stood vertically in brickwork, to 
 support a superincumbent weight until the brickwork which is to carry it 
 has set or become hard. 
 
 Decastyle. A building having ten columns in front. 
 
 Dentils. (From the Latin, dentes, teeth.) Small rectangular blocks used in 
 the bed-mouldings of some of the orders. 
 
 Diastyle. An intercolumniation of three, or, as some say, four diameters. 
 
 Die. That part of a pedestal included between the base and the cornice ; it 
 is also called a dado. 
 
 Dodecastyle. A building having twelve columns in front. 
 
 Donjon. A massive tower within ancient castles, to which the garrison 
 might retreat in case of necessity. 
 
GLOSSARY. 631 
 
 Dcoks. A Scotch name given to wooden brick*. 
 
 Dormer. A window placed on the roof of a house, the frame being placed 
 vertically on the rafters. 
 
 Dormitory. A sleeping-room. 
 
 Dovecote. A building for keeping tarrje pigeons. A columbarium. 
 
 Echinus. The Grecian ovolo. 
 
 Elevation. A geometrical projection drawn on a plane at right angles to 
 the horizon. 
 
 Entablature. That part of an order which is supported by the columns ; 
 consisting of the architrave,lrieze, and cornice. 
 
 Etistyle.An intercolumniation of two and a quarter diameters. 
 
 Exchange. A building in which merchants and brokers meet to transact 
 business. 
 
 Extrados. The exterior curve of an arch. 
 
 Facade. The principal front of any building. 
 
 Face-mould. The pattern for marking the plank out of which hand-railing 
 is to be cut for stairs, etc. 
 
 Facia, or Fascia. A flat member, like a band or broad fillet. 
 
 Falling-mould. The mould applied to the convex, vertical surface of the 
 rail-piece, in order to form the back and under surface of the rail, and finish 
 the squaring. 
 
 Festoon. An ornament representing a wreath of flowers and leaves. 
 
 Fillet. A narrow flat band, listel, or annulet, used for the separation of 
 one moulding from another, and to give breadth and firmness to the edges of 
 mouldings. 
 
 Fl u t es . Upright channels on the shafts of columns. 
 
 Flyers. Steps in a flight ot stairs that are parallel to each other. 
 
 Forum. In ancient architecture a public market ; also, a place where the 
 common courts were held and law pleadings carried on. 
 
 Foundry. A building in which various metals are cast into moulds or 
 shapes. 
 
 Fneze. That part of an entablature included between the architrave and 
 the corn.ice. 
 
 
 
 Gable. The vertical, triangular piece of wall at the end of a roof, from the 
 level of the eaves to the summit. 
 
 Gain. A recess made to receive a tenon or tusk. 
 
 Gallery. A common passage to several rooms in an upper story. A long 
 room for the reception of pictures. A platform raised on columns, pilasters, 
 or piers. 
 
 Girder. -rite principal beam in a floor, for supporting the binding and 
 other joists, whereby the bearing or length is lessened. 
 
 Glyph A vertical, sunken channel. From their number, those in the 
 Doric order are called triglyphs. 
 
 Granary. A building for storing grain, especially that intended to be 
 kept for a considerable time. 
 
632 APPENDIX. 
 
 Groin. The line formed by the intersection of two arches, which cross each 
 other at any angle. 
 
 Gutta. The small cylindrical pendent ornaments, otherwise called drops, 
 used in the Doric order under the triglyphs, and also pendent from the mutult 
 of the cornice. 
 
 Gymnasium. Originally, a place measured out and covered with sand for 
 the exercise of athletic games ; afterward, spacious buildings devoted to the 
 mental as well as corporeal instruction of youth. 
 
 Hall. The first large apartment on entering a house. The public room of 
 a corporate body. A manor-house. 
 
 Ham. A. house or dwelling-place. A street or village : hence Notting- 
 ham, Buckingham, etc. Hamlet, the diminutive of ham, is a small street or 
 village. 
 
 Helix. The small volute, or twist, under the abacus in the Corinthian 
 capital. 
 
 Hem. The projecting spiral fillet of the Ionic capital. 
 
 Hexastyle. A building having six columns in front. 
 
 Hip-rafter. A piece of timber placed at the angle made by two adjacent 
 inclined roofs. 
 
 Homestall. A mansion-house, or seat in the country. 
 
 Hotel, or Hostel. A large inn or place of public entertainment. A large 
 house or palace. 
 
 Hot-house. A glass building used in gardening. 
 
 Hovel. An open shed. 
 
 Hut. A small cottage or hovel, generally constructed of earthy materials, 
 as strong loamy clay, etc. 
 
 Impost. The capital of a pier or pilaster which supports an arch. 
 Intaglio. Sculpture in which the subject is hollowed out, so that the im- 
 pression from it presents the appearance of a bas-relief. 
 Intercolumniation. The distance between two columns. 
 Intrados. The interior and lower curve of an arch. 
 
 Jack-rafters. Rafters that fill in between the principal rafters of a roof; 
 called also common-rafters. 
 
 Jail. A place of legal confinement. 
 
 Jambs. The vertical sides of an aperture. 
 
 Joggle-piece. A post to receive struts. 
 
 Joists. The timbers to which the boards of a floor or the laths of a ceiling 
 are nailed. 
 
 Keep. The same as donjon, which see. 
 Key-stone. The highest central stone of an arch. 
 
 Kiln. A building for the accumulation and retention of heat, in order to 
 dry or burn certain materials deposited within it. 
 King-post. The centre-post in a trussed roof. 
 Knee. A convex bend in the back of a hand-rail. See Ramp. 
 
GLOSSARY. 633 
 
 Lactarium. The same as dairy, which see. 
 
 Lantern. A cupola having windows in the sides for lighting an apartment 
 beneath. 
 
 Larmier. The same as corona, which see. 
 
 Lattice. A reticulated window for the admission of air, rather than light, 
 as in dairies and cellars. 
 
 Lever-boards. Blind-slats; a set of boards so fastened that they maybe 
 turned at any angle to admit more or less light, or to lap upon each other so 
 as to exclude all air or light through apertures. 
 
 Lintel. A piece of timber or stone placed horizontally over a door, win- 
 dow, or other opening. 
 
 Listel. The same asyf//<?/, which see. 
 
 Lobby. An enclosed space, or passage, communicating with the principal 
 room or rooms of a house. 
 
 Lodge. A small house near and subordinate to the mansion. A cottage 
 placed at the gate of the road leading to a mansion. 
 
 Loof. A. small narrow window. Loophole is a term applied to the vertical 
 series of doors in a warehouse, through which goods are delivered by means 
 of a crane. 
 
 Luffer-boarding. The same as lever-boards, which see. 
 
 Luthetn. The same as dormer, which see. 
 
 Mausoleum. A sepulchral building so called from a very celebrated one 
 erected to the memory of Mausolus, king of Caria, by his wife Artemisia. 
 
 Melopa. The square space in the frieze between the triglyphs of the Doric 
 order. 
 
 Mezzanine. A story of small height introduced between two of greater 
 height. 
 
 Minaret. A slender, lofty turret having projecting balconies, common in 
 Mohammedan countries. 
 
 Minster. A church to which an ecclesiastical fraternity has been or is 
 attached. 
 
 Moat. An excavated reservoir of water, surrounding a house, castle, or 
 
 town. 
 
 Modillion. A projection under the corona of the richer orders, resembling 
 a bracket. 
 
 Module. The semi-diameter of a column, used by the architect as a meas- 
 ure by which to proportion the parts of an order. 
 
 Monastery. A building or buildings appropriated to the reception of 
 monks. 
 
 Monopteron. A circular colonnade supporting a dome without an enclos- 
 ing wall. 
 
 Mosaic. A mode of representing objects by the inlaying of small cubes of 
 glass, stone, marble, shells, etc. 
 
 Mosque. A Mohammedan temple or place of worship. 
 
 Midlions.Thz upright posts or bars which divide the lights in a Gothic 
 window. 
 
 Mtiniment-house.k strong, fire-proof apartment for the keeping and pres- 
 ervation of evidences, charters, seals, etc., called muniments. 
 
634 APPENDIX. 
 
 Museum. A repository of natural, scientific, and literary curiosities or of 
 works of art. 
 
 Mutule. A projecting ornament of the Doric cornice supposed to repre- 
 sent the ends of rafters. 
 
 Nave. The main body of a Gothic church. 
 Newel. A post at the starting or landing of a flight of stairs. 
 Niche. A cavity or hollow place in a wall for the reception of a statue, 
 vase, etc. 
 
 Nogs. Wooden bricks. 
 
 Nosing. The rounded and projecting edge of a step in stairs. 
 
 Nunnery. A building or buildings appropriated for the reception of nuns. 
 
 Obelisk. A lofty pillar of a rectangular form. 
 
 Octastyle. A building with eight columns in front. 
 
 Odeum. Among the Greeks, a species of theatre wherein the poets and 
 musicians rehearsed their compositions previous to the public production of 
 them. 
 
 Ogee. See cyma. 
 
 Orangery. A gallery or building in a garden or parterre fronting the 
 south. 
 
 Oriel-window. A large bay or recessed window in a hall, chapel, or other 
 apartment. 
 
 Ovolo. A convex projecting moulding whose profile is the quadrant of a 
 circle. 
 
 Pagoda. A temple or place of worship in India. 
 
 Palisade. A fence of pales or stakes driven into the ground. 
 
 Parapet. A small wall of any material for protection on the sides of 
 bridges, quays, or high buildings. 
 
 Pavilion. A turret or small building generally insulated and comprised 
 under a single roof. 
 
 Pedestal. A square foundation used to elevate and sustain a column, 
 statue, etc. 
 
 Pediment. The triangular crowning part of a portico or aperture which 
 terminates vertically the sloping parts of the roof; this, in Gothic architecture, 
 is called a gable. 
 
 Penitentiary. A prison for the confinement of criminals whose crimes are 
 not of a very heinous nature. 
 
 Piazza. A square, open space surrounded by buildings. This term is 
 often improperly used to denote a portico. 
 
 Pier. A rectangular pillar without any regular base or capital. The up- 
 right, narrow portions of walls between doors and windows are known by this 
 term. 
 
 Pilaster. A square pillar, sometimes insulated, but more commonly en- 
 gaged in a wall, and projecting only a part of its thickness. 
 
 Piles. Large timbers driven into the ground to make a secure foundation 
 in marshy places, or in the bed of a river. 
 
GLOSSARY. 635 
 
 Pitta*. A column of irregular form, always disengaged, and always deviat- 
 ing from the proportions of the orders; whence the distinction between a 
 pillar and a column. 
 
 Pinnacle. A small spire used to ornament Gothic buildings. 
 
 Planceer. The same as soffit, which see. 
 
 Plinth. The lower square member of the base of a column, pedestal or 
 wall. 
 
 Porch. An exterior appendage to a building, forming a covered approach 
 to one of its principal doorways. 
 
 Portal. The arch over a door or gate ; the framework of the gate ; the 
 lesser gate, when there are two of different dimensions at one entrance. 
 
 Portcullis. A strong timber gate to old castles, made to slide up and 
 down vertically. 
 
 Portico. A colonnade supporting a shelter over a walk, or ambulatory. 
 
 Priory. A building similar in its constitution to a monastery or abbey, 
 the head whereof was called a prior or prioress. 
 
 Prism. A solid bounded on the sides by parallelograms, and on the ends 
 by polygonal figures in parallel planes. 
 
 Prostyle. A building with columns in front only. 
 
 Purlines. Those pieces of timber which lie under and at right angles to 
 the rafters to prevent them from sinking. 
 
 Pycnostyle. An intercolumniation of one and a half diameters. 
 
 Pyramid. A solid body standing on a square, triangle, or potygonal basis 
 and terminating in a point at the top. 
 
 Quarry. A place whence stones and slates are procured. 
 Quay. (Pronounced key.) A bank formed towards the sea or on the side 
 of a river for free passage, or for the purpose of unloading merchandise. 
 Quoin. An external angle. See Rustic quoins. 
 
 Rabbet, or Rebate. A groove or channel in the edge of a board. 
 
 Ramp. A concave bend in the back of a hand-rail. 
 
 Rampant arch. One having abutments of different heights. . 
 
 Regula. The band below the taenia in the Doric order. 
 
 Riser. In stairs, the vertical board forming the front of a step. 
 
 Rostrum. An elevated platform from which a speaker addresses an audi- 
 ence. 
 
 Rotunda. A circular building. 
 
 Rubble-wall. A wall built of unhewn stone. 
 
 Rudenture. The same as cable, which see. 
 
 Rustic quoins. The stones placed on the external angle of a building, pro- 
 jecting beyond the face of the wall, and having their edges bevelled. 
 
 Rustic-work. A mode of building masonry wherein the faces of the stones 
 are left rough, the sides only being wrought smooth where the union of the 
 stones takes place. 
 
 Salon, or Saloon. A lofty and spacious apartment comprehending the 
 height of two stories with two tiers of windows. 
 
636 APPENDIX. 
 
 Sarcophagus. A tomb or coffin made of one stone. 
 
 Scantling. 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 trans- 
 versely together, so that the two appear but one. 
 
 Scotia. The hollow moulding in the base of a column, between the fillets 
 of the tori. 
 
 Scroll. A carved curvilinear ornament, somewhat resembling in profile 
 the turnings of a ram's horn. 
 
 Sepulchre. A grave, tomb, or place of interment. 
 
 Sewer. A drain or conduit for carrying off soil or water from any place. 
 
 Shaft. The cylindrical part between the base and the capital of a column. 
 
 Shoar. A piece of timber placed in an oblique direction to support a 
 building or wall. 
 
 Sill. The horizontal piece of timber at the bottom of framing ; the timber 
 or stone at the bottom of doors and windows. 
 
 Soffit. The underside of an architrave, corona, etc. The underside of 
 the heads of doors, windows, etc. 
 
 Summer. The lintel of a door of window ; a beam tenoned into a girder 
 to support the ends of joists on both sides of it. 
 
 Systyle.- An intercolumniation of two diameters. 
 
 Tania. The fillet which separates the Doric frieze from the architrave. 
 
 Talus. The slope or inclination of a wall, among workmen called bat- 
 tering. 
 
 Terrace. An area raised before a building, above the level of the ground, 
 to serve as a walk. 
 
 Tesselated pavement. A curious pavement of mosaic work, composed of 
 small square stones. 
 
 Tetrastyle. A building having four columns in front. 
 
 Thatch. A covering of straw or reeds used on the roofs of cottages, 
 barns, etc. 
 
 Theatre. A building appropriated to the representation of dramatic 
 spectacles. 
 
 Tile. A thin piece or plate of baked clay or other material used for the 
 external covering of a roof. 
 
 Tomb. A grave, or place for the interment of a human body, including 
 also any commemorative monument raised over such a place. 
 
 Torus. A moulding of semi-circular profile used in the bases of col- 
 umns. 
 
 Tower. A lofty building of several stories, round or polygonal. 
 
 Transept. The transverse portion of a cruciform church. 
 
 Transom. The b^am across a double-lighted window ; if the window 
 have no transom, it is called a clere-story window. 
 
 Thread. That part of a step which is included between the face of its riser 
 and that of the riser above. 
 
 Trellis. A reticulated framing made of thin bars of wood for screens, win- 
 dows, etc. 
 
GLOSSARY. 637 
 
 \ 
 
 Tiiglyph.i:\\e. vertical tablets in the Doric frieze, chamfered on the two 
 vertical edges, and having two channels in the middle. 
 
 Tripod. A table or seat with three legs. 
 
 Trochilus. The same as scotia, which see. 
 
 Truss. An arrangement of timbers for increasing the resistance to cross- 
 strains, consisting of a tie, two struts, and a suspending-piece. 
 
 Turret. A small tower, often crowning the angle of a wall, etc. 
 
 Tusk. A short projection under a tenon to increase its strength. 
 
 Tympanum. The naked face of a pediment, included between the level and 
 the raking mouldings. 
 
 Underpinning. The wall under the ground-sills of a building. 
 
 University. An assemblage of colleges under the supervision of a senate, etc. 
 
 Vault. A concave arched ceiling resting upon two opposite parallel walls. 
 
 Venetian-door. A door having side-lights. 
 
 Venetian-window. A window having three separate apertures. 
 
 Veranda. An awning. An open portico under the extended roof of a 
 building. 
 
 Vestibule. An apartment which serves as a medium of communication to 
 another room or series of rooms. 
 
 Vestry. An apartment in a church, or attached to it, for the preservation 
 of the sacred vestments and utensils. 
 
 Villa. A country-house for the residence of an opulent person. 
 
 Vinery. A house for the cultivation of vines. 
 
 Volute. A spiral scroll, which forms the principal feature of the Ionic and 
 the Composite capitals. 
 
 Vottssoirs. Arch-stones. 
 
 Wainscoting. Wooden lining of walls, generally in panels. 
 
 Water-table. The stone covering to the projecting foundation or other walls 
 of a building. 
 
 Well. The space occupied by a flight of stairs. The space left beyond the 
 ends of the steps is called the well-hole. 
 
 Wicket. A small door made in a gate. 
 
 Winders. In stairs, steps not parallel to each other. 
 
 Zophorus. The same as frieze, which see. 
 
 Zystos. Among the ancients, a portico of unusual length, commonly appro- 
 priated to gymnastic exercises. 
 
638 APPENDIX. 
 
 TABLE OF SQUARES, CUBES. AND ROOTS. 
 
 (From Button's Mathematics.) 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. jCubeRooJ| No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeR-wt. 
 
 rr 
 
 1 
 
 1 
 
 1-0000000 1-000000 
 
 68 
 
 4624 
 
 314432 
 
 8-2462113 
 
 4-081655 
 
 2 
 
 4 
 
 8 
 
 1 4142136 
 
 1-259921 
 
 69 
 
 4761 
 
 328509 
 
 8-3066239 
 
 4 1015G6 
 
 3 
 
 9 
 
 27 
 
 1-7320508 
 
 1-442250 
 
 70 
 
 4900 
 
 343000 
 
 8-3666003 4-121285 
 
 4 
 
 16 
 
 64 
 
 2-0000000 
 
 1-537401 
 
 71 
 
 5041 
 
 357911 
 
 8-4261498! 4-140818 
 
 5 
 
 25 
 
 125 
 
 2-2360680 
 
 1-709976 
 
 72 
 
 5184 
 
 373248 
 
 8-4852814J 4-160168 
 
 6 
 
 36 
 
 216 
 
 2-4494897 
 
 1-817121 
 
 73 
 
 5329 
 
 389017 
 
 8-54400371 4-179339 
 
 7 
 
 49 
 
 343 
 
 2-6457513 
 
 1-912931 
 
 74 
 
 5476 
 
 405224 
 
 8-6023253] 4-198336 
 
 8 
 
 64 
 
 512 
 
 2-8284271 
 
 2-000000 
 
 75 
 
 5625 
 
 421875 
 
 8-6602540 4 217163 
 
 9 
 
 81 
 
 729 
 
 3-0000000 
 
 2-080034 
 
 76 
 
 5776 
 
 433976 
 
 8-7177979 4235324 
 
 10 
 
 100 
 
 1000 
 
 3-1622777 
 
 2-154435 
 
 77 
 
 5929 
 
 456533 
 
 8-7749644 
 
 4-254321 
 
 11 
 
 121 
 
 1331 
 
 3-3166243 
 
 2-223330 
 
 78 
 
 6084 
 
 474552 
 
 8-8317609! 4-272659 
 
 12 
 
 144 
 
 1728 
 
 3-4641016 
 
 2-239429 
 
 79 
 
 6241 
 
 493039 
 
 8-8881944 4-290840 
 
 13 
 
 169 
 
 2197 
 
 36055513 2351335 
 
 80 
 
 6400 
 
 512000 
 
 8-9442719! 4-30SS69 
 
 14 
 
 196 
 
 2744 
 
 3-74165741 2-410142 
 
 81 
 
 6561 
 
 531441 
 
 9-(M)0()000| 4-3*6749 
 
 15 
 
 225 
 
 3375 
 
 38729833! 2-466212 
 
 82 
 
 6724 
 
 551358 
 
 9-05538511 4-344431 
 
 16 
 
 256 
 
 4096 
 
 4'OOOOOOOi 2-519842; 
 
 83 
 
 6339 
 
 571787 
 
 9-1104336' 4-362071 
 
 17 
 
 239 
 
 4913 
 
 4-1231056 1 2-571232 
 
 84 
 
 7056 
 
 592704 
 
 9-1651514! 4-379519 
 
 18 
 
 324 
 
 5332 
 
 4-2426407 
 
 2-62074 1 ' 
 
 85 
 
 7225 
 
 614125 
 
 9-2195445 4-396830 
 
 19 
 
 361 
 
 635J 
 
 4-3533989 2-663402! 
 
 86 
 
 7396 
 
 636055 
 
 9-2736185 4-414005 
 
 20 
 
 400 
 
 8000 
 
 4-4721350 2-714118 
 
 87 
 
 7569 
 
 653503 
 
 9-3273791 4-431048 
 
 21 
 
 441 
 
 9261 
 
 4-5825757 
 
 2-758924 
 
 88 
 
 7744 
 
 681472 
 
 9-38J8315J 4-447960 
 
 22 
 
 484 
 
 10643 
 
 4-6904158 2-802033 
 
 89 
 
 7921 
 
 704969 
 
 9-4339311! 4 "464745 
 
 23 
 
 529 
 
 12167 
 
 4-79533151 2-843367 
 
 90 
 
 8100 
 
 729000 
 
 9-4363330 4 -58 1405 
 
 24 
 
 576 
 
 13324 
 
 4-8989795] 2-881499 
 
 91 
 
 8281 
 
 753571 
 
 9-5393920] 4-497941 
 
 25 
 
 625 
 
 15625 
 
 5-0000000! 2-924018 
 
 92 
 
 8464 
 
 778688 
 
 9-5916630 4514357 
 
 26 
 
 676 
 
 17576 
 
 5 -09901 95 j 2-962496 
 
 93 
 
 8649 
 
 804357 
 
 9-64365081 4-530055 
 
 27 
 
 729 
 
 19633 
 
 5-1961524' 3-000000 
 
 94 
 
 8836 
 
 830534 
 
 9-6953597 
 
 4 516336 
 
 28 
 
 784 
 
 21952 
 
 529151)26 3-036539 
 
 95 
 
 9025 
 
 857375 
 
 9-7467943 
 
 4-5o2903 
 
 29 
 
 841 
 
 24389 
 
 5-3851648 3-072317 
 
 96 
 
 9216 
 
 884736 
 
 9-7979590 
 
 4-573357 
 
 30 
 
 900 
 
 27000 
 
 5-4772256 3-107232 
 
 97 
 
 9409 
 
 912673 
 
 9-8483578 
 
 4-591701 
 
 31 
 
 961 
 
 29791 
 
 556776441 3-14133l| 
 
 98 
 
 9604 
 
 941192 
 
 9-8994949 
 
 4-610436 
 
 32 
 
 1024 
 
 32763 
 
 5-6568542 3-174802 1 
 
 99 
 
 9801 
 
 970299 
 
 9-9493744 
 
 4-626065 
 
 33 
 
 1089 
 
 35937 
 
 57445626 3-20753i! 100 
 
 10000 
 
 1000000 
 
 10-0000000 
 
 4641589 
 
 34 
 
 1156 
 
 39304 
 
 5-83J9519 3233612 101 
 
 10201 
 
 1030301 
 
 10-0498756 
 
 4-657009 
 
 35 
 
 1225 
 
 42875 
 
 5-9160798 3-271066 
 
 102 
 
 10404 
 
 1061208 
 
 10-0995049 41372329 
 
 36 
 
 1296 
 
 46656 
 
 6-0000000 
 
 3-331927 
 
 103 
 
 10609 
 
 1092727 
 
 10- 14839 1GJ 4-687548 
 
 37 
 
 1369 
 
 50653 
 
 6-0327625 
 
 3-332222 
 
 104 
 
 10816 
 
 1124864 
 
 10-1980330! 4-702659 
 
 1 38 
 
 1444 
 
 54872 
 
 6-1644140 
 
 3-361975 
 
 105 
 
 11025 
 
 1157625 
 
 10-2469503 4-717694 
 
 9 
 
 1521 
 
 59319 
 
 6-2449980 
 
 3-331211 
 
 106 
 
 11236 
 
 1191016 
 
 10-2956301! 4-732623 
 
 40 
 
 1600 
 
 6401)0 
 
 6-3245553 
 
 3419952 
 
 107 
 
 11449 
 
 1225043 
 
 10-3440801] 4-747459 
 
 41 
 
 1681 
 
 68921 
 
 6-4031242 3448217 
 
 108 
 
 11664 
 
 1259712 
 
 10-3923343' 4-762203 
 
 \J 
 
 1764 
 
 74088 
 
 6-4307407 3-476027 
 
 109 
 
 11831 
 
 1295029 
 
 10-4403J65' 4-776356 
 
 ^3 
 
 1849 
 
 79507 
 
 6-5574335 
 
 3503398 
 
 110 
 
 12100 
 
 1331000 
 
 10-4880335 4-791420 
 
 44 
 
 1936 
 
 85184 
 
 6-6332496 
 
 3530348 
 
 111 
 
 12321 
 
 1367631 
 
 10-535G53S! 4-805895 
 
 45 
 
 2025 
 
 91125 
 
 6-7082039 
 
 3-556893 
 
 112 
 
 12544 
 
 1404928 
 
 10-58300521 4-820284 
 
 46 
 
 2116 
 
 97336 
 
 6-7823300 
 
 3-533J48 
 
 113 
 
 12769 
 
 1442897 
 
 10-6301453 4-834533 
 
 47 
 
 2209 
 
 103423 
 
 6-8556546 3-608325 
 
 114 
 
 12996 
 
 1481544 
 
 10-0770783 
 
 4-848808 
 
 48 
 
 2304 
 
 1 10592 
 
 69232032 3634241 
 
 115 
 
 13225 
 
 1520875 
 
 107238053 
 
 4-862944 
 
 49 
 
 2401 
 
 117649 
 
 7-0000000; 3-6593J6 
 
 116 
 
 13456 
 
 1560896 
 
 10-7703296 
 
 4-87G999 
 
 50 
 
 2500 
 
 125000 
 
 7-0710678 3634031 
 
 117 
 
 13689 
 
 1601613 
 
 10-816553$ 
 
 4-890973 
 
 51 
 
 2601 
 
 132651 
 
 7-1414284 3-70843J 
 
 118 
 
 13924 
 
 1643032 
 
 10-8627805 
 
 4-904863 
 
 52 
 
 2704 
 
 140608 
 
 7-2111026; 3-732511 
 
 119 
 
 14161 
 
 1685159 
 
 10-9087121, 
 
 4-913685 
 
 53 
 
 2809 
 
 148877 
 
 7-2301099 3-756236 
 
 120 
 
 14400 
 
 1728000 
 
 10-9544512 
 
 4-93-3424 
 
 54 
 
 2916 
 
 157464 
 
 7-3181692 3-779763 121 
 
 14641 
 
 1771561 
 
 11-0000000 
 
 4-946087 
 
 55 
 
 3025 
 
 166375 
 
 7-4161985! 3-502952 ! 122 
 
 14884 
 
 1815848 
 
 11-0453610 
 
 4-959676 
 
 56 
 
 .3136 
 
 175616 
 
 7-4833148 
 
 3-8258521! 123 
 
 15129 
 
 1860867 
 
 11*0905365 
 
 4-973190 
 
 57 
 
 3219 
 
 185193 
 
 7-549334 1 
 
 3-343501 1! 124 
 
 15376 
 
 1906624 
 
 11-1355287 
 
 4-986631 
 
 58 
 
 3364 
 
 195112 
 
 7-6157731 
 
 3-870877 125 
 
 15625 
 
 1953125 
 
 11-1803399 
 
 5-000000 
 
 59 
 
 3481 
 
 205379 
 
 7-6311457 
 
 3-892996 j 126 
 
 15876 
 
 2000376 
 
 11-2219722 
 
 5-013293 
 
 60 
 
 3600 
 
 216001) 
 
 7-7459667 
 
 3-914868 127 
 
 16129 
 
 2048383 
 
 11-2694277 
 
 5-026526 
 
 61 
 
 3721 
 
 226981 
 
 7-8102497 
 
 3-936497 
 
 128 
 
 16334 
 
 2097152 
 
 11-3137085 
 
 5-039684 
 
 62 
 
 3344 
 
 233328 
 
 7-8740079 
 
 3-957891 
 
 129 
 
 16641 
 
 2146689 
 
 1 1-35 Vd 167 
 
 5-052774 
 
 63 
 
 3969 
 
 250017 
 
 7-9372539 
 
 3-979057 
 
 130 
 
 16900 
 
 2197000 114017543 
 
 5-065797 
 
 64 
 
 4096 
 
 262144 
 
 8-0000000 
 
 4-000000 
 
 131 
 
 17161 
 
 2248091 11*4455331 
 
 5-078753 
 
 65 
 
 4225 
 
 274625 
 
 8-0622577 
 
 4-020726 132 
 
 17424 
 
 2299968 11-4891253 
 
 5-091643 
 
 66 
 
 4356 
 
 2374% 
 
 8-1240334 
 
 4-041240 133 
 
 17689 
 
 2252637 11-5325626 1 5104469 
 
 67 
 
 448'J 
 
 300763 
 
 8-1853523 4-061548|! 134 
 
 17956 2406I04 1 ]i-575369l 5' 11 7230 
 
TABLE OF SQUARES, CUBES, AND ROOTS. 
 
 639 
 
 tfo. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. CubeRoot. 
 
 No. 
 
 Square. 
 
 40804 
 41209 
 
 Cube. Sq. Root. CubcRooL 
 
 135 
 136 
 
 18225 
 18496 
 
 2460375 
 2515156 
 
 11-6189500 5-129928 
 11-6619033! 5-142563 
 
 202 
 203 
 
 ! 8242406 
 8365427 
 
 14-2126704 5-867464 
 14-24780681 5-877131 
 
 137 
 
 18769 
 
 2571353 
 
 11-7016999J 5-155137 
 
 204 
 
 4I616J 8489664 
 
 14-23285691 5-836765 
 
 133 
 
 19044 
 
 2628072 
 
 11-7473401 
 
 5- 167649 
 
 205 
 
 42023 
 
 8615125! 14-3178211 
 
 5-896368 
 
 139 
 
 19321 
 
 2635619 
 
 11-7898261 
 
 5-180101 
 
 206 42 1ST 
 
 8741816! 14-3527001 
 
 5-905941 
 
 140 
 
 19600 
 
 27-14000 
 
 11-83215% 
 
 5-192494 
 
 207 
 
 42849 
 
 8369743 14'337494fi 
 
 5-915432 
 
 141 
 
 19881 
 
 2803221 
 
 11-8743422 
 
 5-204328 
 
 203 
 
 43264 
 
 8998912 11-4222051 
 
 5-924992 
 
 142 
 
 20164 
 
 2S63283 
 
 11-9163753 
 
 5-217103 
 
 209 
 
 43681 
 
 9129329 14-4568322 
 
 5-934473 
 
 143 
 
 20449 
 
 2924207 
 
 11-9582607 
 
 5-229321 
 
 210 
 
 44100 
 
 9261000 14-4913767 
 
 5-943J22 
 
 144 
 
 20736 
 
 2985934 
 
 12-0000000 
 
 5-241483 
 
 211 
 
 44521 
 
 9393931] 14-5253390 
 
 5-953342 
 
 145 
 
 21025 
 
 3048625 
 
 12-0415946 
 
 5-253533 
 
 212 
 
 44944 
 
 9528123; 14-5602193! 5-962732 
 
 146 
 
 21316 
 
 3112136 
 
 12-0830460J 5-265637 
 
 213 
 
 45369 
 
 9663597J 14-5945195! 5-972093 
 
 147 
 
 21609 
 
 317C>523 
 
 12-1243557 5-277632 
 
 214 
 
 45796 
 
 9800344! 146287338! 5-931424 
 
 148 
 
 21904 
 
 3241792 
 
 12-1655251 
 
 5-289572! 
 
 215 
 
 46225 
 
 99333751 14-6623783! S'%0726 
 
 149 
 
 22201 
 
 3307949 
 
 12-2065556 
 
 5301459! 
 
 216 
 
 46656 
 
 10077696 
 
 14-69693851 6-000000 
 
 150 
 
 22500 
 
 3375000 
 
 12-2174487 
 
 5-3132931 
 
 217 
 
 47089 
 
 10218313 
 
 14-73091991 6-009245 
 
 1511 22301 
 
 3142951 
 
 12-2332057 
 
 5-325074 
 
 218 
 
 47524 
 
 10.360232! 14-7648231 
 
 6-018462 
 
 152i 23104 
 
 3511808 
 
 12-3238280 
 
 5-336803^ 
 
 219 
 
 47961 
 
 10503459! 14-7986486 
 
 6-027650 
 
 153 23409 
 
 3531577 
 
 12-3693169 
 
 5-348481! 
 
 220 
 
 48400 10648000 14-8323970 
 
 6-036811 
 
 154 23710 
 
 3652264 
 
 12-4096736 
 
 5-360108! 
 
 221 
 
 48341, 10793861 14-8660687 
 
 6-045943 
 
 155 24025 3723375 
 
 12-449899. 
 
 5-371685 
 
 222 
 
 49234 10941048! 14-89 J6044 
 
 6-055049 
 
 156 24336 
 
 37% 416i 12-4399960 
 
 5-383213 
 
 223 
 
 49729 
 
 11039567 14-9331845 
 
 6-064127 
 
 157 24649 3869393 
 
 12-5299641 
 
 5-394691 
 
 224 
 
 50176 
 
 11239424 149666295 6-073178 
 
 153 24964 
 
 3944312 
 
 12-5698051 
 
 5-406120J 
 
 225 
 
 50625 
 
 11390625 15-000000ol 6-082202 
 
 159 25281 
 
 4019679 
 
 12-61)95202 
 
 5-417501J 
 
 226 
 
 51076 
 
 11543176 15-0332964 6-091199 
 
 160J 25600 
 
 4096000 
 
 12-6191106 
 
 5-428335! 
 
 227 
 
 51529 
 
 11697083 15-05651921 6-100170 
 
 161 
 
 25921 
 
 4173281 
 
 12-6385775 
 
 5-440122 
 
 223 
 
 51984 
 
 11852352 15-0996639J 6109115 
 
 162 
 
 26344 
 
 4251523 
 
 12-7279221 
 
 5-451362 
 
 229 
 
 52441 
 
 12008939 15-1327460' 6-118033 
 
 163 26569 
 
 4330747 
 
 12-7671453 
 
 5-462556 
 
 230 
 
 52900 
 
 12167000 15-1657509 
 
 6-126926 
 
 164 
 
 26896 
 
 4410944 
 
 12-8062485 
 
 5-473704 231 
 
 53361 
 
 12326391 
 
 151936342 
 
 6-135792 
 
 165 
 
 27225 
 
 4492125 
 
 12-8452326 
 
 5-484807 232 
 
 53824 
 
 12487168 15-2315462 
 
 6-144634 
 
 166 
 
 87556 
 
 4574296! 12-8340987 
 
 5-495365 233 
 
 54289 
 
 12649337 152643375 
 
 6-153449 
 
 167 27839 
 
 4657463 129228480 
 
 5-506878! 
 
 234 
 
 54756 
 
 12812904 
 
 15-2970585 
 
 6-162240 
 
 168 23^4 
 
 4741632 12-9614814 
 
 5-51781H -235 
 
 55225 
 
 12977875 
 
 15-32J70J7 
 
 6-171006 
 
 169 28561 
 
 4826309 13-0000000 
 
 5-528775! 236 
 
 55696 
 
 13144256 153622915 6-179747 
 
 170 28'JOO 
 
 4913000 13-0331048 
 
 5-539658 237 
 
 56169 
 
 13312053! 15-3J43043I 6-183463 
 
 171 
 
 2;>2 1 1 
 
 5000211 13-0766968 
 
 5-550499 233 
 
 56644 
 
 134812721 154272486 6-197154 
 
 172 
 
 29581 
 
 5083148 13-1143770 
 
 5-561293! 239 
 
 57121 
 
 13651919! 15-4596248 6-205822 
 
 173 89929] 5177717 
 
 1741 30276 52681)24 
 
 13-1529464 
 
 13-1909060 
 
 5-572055 
 5 532770 
 
 240 
 241 
 
 57600 
 53031 
 
 13324000 15-4919334! 6-214465 
 139/7521: 155241747 6223084 
 
 175 
 
 30625 
 
 5359375 
 
 13-2287565 
 
 5-593445' 
 
 242 
 
 58564 
 
 11172433! 15-5563492 6231630 
 
 176 
 
 30976 
 
 5151776 13-2664992 5-604079! 
 
 243 
 
 59049 
 
 14348907 
 
 l.V.'.ssif,:;} J6-210J51 
 
 177 
 
 31329 5515233 13-3041347 
 
 31C81 5639752 13-3416541 
 
 5-614672! 
 5-625226 
 
 244 
 245 
 
 59536 
 
 60025 
 
 14526784 
 14706125 
 
 15-6204994 6243300 
 15-6524753 6-25 7325 
 
 179 
 
 32041 5735339! 13-3790332 
 
 5-635741 
 
 246 
 
 60516 
 
 14836936 
 
 15-6343371 6-265327 
 
 130 
 
 32103 5832000 
 
 13-4161071) 
 
 5-6462 16 
 
 247 
 
 61009 
 
 15069223 
 
 15-7162335; 6-274305 
 
 181! 32761] 5921)711 
 
 13-4536240 5-656653|| 248 
 
 61504 
 
 152529J2 
 
 15-74801571 6--N2701 
 
 182i 33124 6023568 
 
 134907376 5-6670511 249 
 
 62001 
 
 15433249 
 
 15-7797333 6-291195 
 
 183; 33 kd 6128487 
 
 13-5277493! 5-67741 1| 250 
 
 62500 
 
 15525000 
 
 15-8113333! 6-299605 
 
 181 33356 
 
 6229504 
 
 13-5646600! 5-6S7734 251 
 
 63001 
 
 15313251 
 
 15-8429795 6-307994 
 
 185 3 1225 
 
 6331625 
 
 13-6014705J 5-693019 
 
 252 
 
 63504 
 
 16003008 
 
 158745079 6316360 
 
 18* 
 
 34596 
 
 613H56 
 
 13-63318171 5-70821'.; 253 
 
 64009 
 
 16194277 
 
 -737 
 
 6-321704 
 
 187 
 
 3iy;>'.i 
 
 653921)3 
 
 13-07 17J 13 5-718 17'J -J.M 
 
 64516 
 
 16337064 
 
 15-9373775 
 
 6-333026 
 
 IMS 35314 
 
 6644672 
 
 13-7113092 5-728.-,:,l 
 
 255 65025 
 
 16581375 
 
 159687194 
 
 6341326 
 
 isj 35721 
 
 6751269 
 
 13-7477271 5-7337J4 
 
 256 65536 
 
 16777216 160000000 
 
 6-349604 
 
 190J 36100 
 
 6859000 
 
 13-78404831 5-1 18391 
 
 257 65049 
 
 16.)74593 
 
 16-0312195 
 
 6-357861 
 
 191 36181 
 192 36864 
 
 6967871 
 7077838 
 
 13-S202750 5-758966 
 13-85640651 5-76-. 
 
 253 66564 
 259 67031 
 
 17173312 16-06237S1 
 17373979 16-0934769 
 
 6-366097 
 
 6374311 
 
 193i 372)9 
 
 7189057 
 
 13-8924140! 5 -77S.96 200' 67600 
 
 l?f>76000 
 
 16-1215155 
 
 6-382504 
 
 194i 37,136 
 
 73.H331 
 
 13-9283383; 5-783960 
 
 261 63121 
 
 17779.-.S1 1C,- 1. V.I.I 11 
 
 6-390676 
 
 195 3302.", 
 
 7414875 
 
 13-9642400J 5-798390 
 
 2621 6S644 
 
 17984723 16-1884141 
 
 6-398889 
 
 196J 38416 
 
 7529536 
 
 14-0000000! 5-808786 
 
 263 
 
 69169 
 
 18191447 16-2172717 
 
 6-406953 
 
 197j 33909 
 
 7615373 
 
 14-0356683 5-818648 
 
 264 
 
 69696 
 
 1SW9711 l:-.2H07t-,< 
 
 6-415069 
 
 198 392u4 
 
 77IWW2 
 
 14-0712473 5-S2-U77 
 
 265 
 
 70225 
 
 18609625 16-2738206 
 
 (,12315- 
 
 199) 39H01 
 
 7880599 
 
 14- 10673(50 
 
 5-833272 
 
 265 
 
 70756 188210961 16-30.).",. >C, I 6-1312;!- 
 
 2XJ' 10000 
 
 8000000 
 
 14-1421356 
 
 5-848035 
 
 267 
 
 7I-2S9 190341631 16-3401346 6-439277 
 
 201 40101 
 
 8120601 
 
 14-1774469 
 
 5-857766 
 
 268 71824 
 
 19248832, 16 -37070."):) 6-447306 
 
640 
 
 APPENDIX. 
 
 No. Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 No. Square. 
 
 Cube. 
 
 Sq Root. IcubeRnot- 
 
 269' 72361 
 
 19465109 
 
 16-40121951 6-455315 
 
 336 
 
 1 12896 
 
 37933056 
 
 18-3303028 
 
 6952053 
 
 270 
 
 72900 
 
 19633000 
 
 16-43167671 6-463304 
 
 337 
 
 113569 
 
 38272753 
 
 183575598 
 
 6-958943 
 
 271 
 
 73441 
 
 19902511 
 
 16-4620776 6-471274 
 
 333 
 
 114244 
 
 33614472 
 
 18-3347763 
 
 6-965820 
 
 272 
 
 73384 
 
 20123648 
 
 16-4924225! 6-479224 
 
 339 
 
 114921 
 
 38908219 
 
 18-4119526 
 
 6-972683 
 
 273 
 
 74529 
 
 20346417 
 
 16-5227116 
 
 6-487154 
 
 340 
 
 115600 
 
 393(UOOO 
 
 18-4390889 
 
 6-979532 
 
 274 
 
 75076 
 
 20570824 
 
 16-5529454 
 
 6-495065 
 
 341 
 
 116281 
 
 39651821 
 
 184661853 
 
 6-986368 
 
 275 
 
 75625 
 
 20796375 
 
 16-5331240 
 
 6-502357 
 
 342 
 
 116964 
 
 40001638 
 
 18-4932420 
 
 6-993191 
 
 276 
 
 76176 
 
 21024576 
 
 16-6132477 
 
 6-510833 
 
 343 
 
 117649 
 
 40353607 
 
 18-5202592 
 
 7-000000 
 
 277 
 
 7^729 
 
 21253933 
 
 16-6433170 
 
 6-518634 
 
 344 
 
 1 18336 
 
 40707584 
 
 18-5472370 
 
 7-006796 
 
 278 
 
 77234 
 
 21484952 
 
 16-6733320 
 
 6-526519 
 
 345 
 
 119025 
 
 41063625 
 
 18-5741756 
 
 7-013579 
 
 279 
 
 77811 
 
 21717639 
 
 16-7032931 
 
 6534335 
 
 346 
 
 119716 
 
 41421736 
 
 18-6010752 
 
 7-020349 
 
 230 
 
 78400 
 
 21952000 
 
 16-7332005 
 
 6-542133 
 
 347 
 
 120409 
 
 41781923 
 
 18-6279360 
 
 7027106 
 
 281 
 
 78961 
 
 22188041 
 
 16-7630546 
 
 6-549912 
 
 343 
 
 121104 
 
 42144192 
 
 18-6547581 
 
 7-033850 
 
 282 
 
 79524 
 
 22425763 
 
 16-7923556 
 
 6-557672 
 
 349 
 
 121801 
 
 42508549 
 
 18-8815417 
 
 7-040581 
 
 233 
 
 80089 
 
 22665187 
 
 16-8226033 
 
 6565414 
 
 350 
 
 122500 
 
 42875000 
 
 18-7032869 
 
 7-047299 
 
 284 
 
 80656 
 
 22906334 
 
 16-8522995 
 
 6-573139 
 
 351 
 
 123201 
 
 43243551 
 
 18-7349940 
 
 7-054004 
 
 235 
 
 81225 
 
 23149125 
 
 16-8819430 
 
 6-530344 
 
 352 
 
 123904 
 
 43614208 
 
 18-7616630 
 
 7-060697 
 
 286 
 
 81796 
 
 23393656 
 
 169115345 
 
 6538532 
 
 353 
 
 124609 
 
 43985977 
 
 18-7882942 
 
 7-067377 
 
 287 
 
 82369 
 
 23639903 
 
 16-9410743 
 
 6-596202 
 
 354 
 
 125316 
 
 44361864 
 
 18-8148877 
 
 7-074044 
 
 283 
 
 82944 
 
 23387872 
 
 16-9705627 
 
 6-603354 
 
 355 
 
 126025 
 
 44738875 
 
 18-8414437 
 
 7-080699 
 
 239 
 
 83^.21 
 
 24137569 
 
 17-0000000 
 
 6-611489 
 
 356 
 
 126736 
 
 45118016 
 
 18-8679623 
 
 7-087341 
 
 290 
 
 84100, 24389000 
 
 17-0293864 6-619106 
 
 357 
 
 127449 
 
 45499293 
 
 18-8944436 
 
 7-093971 
 
 291 
 
 84681 24642171 
 
 17-0537221 6-626705 
 
 358 
 
 128164 
 
 45882712 
 
 18-9208379 
 
 7-100588 
 
 292 
 
 85264 24897083 
 
 17-0380075 6634237 
 
 359 
 
 128881 
 
 46268279 
 
 18-9472953 
 
 7-107194 
 
 293 85849 251^3757 17-1172428 
 
 6641852 
 
 360 
 
 129600 
 
 46656000 
 
 18-9736660 
 
 7-113787 
 
 294 86 136 25*12184! 17-1464232 
 
 6-649400 
 
 361 
 
 130321 
 
 47045381 
 
 19-0000000 
 
 7-120367 
 
 295 87025 
 
 25672375 
 
 17-1755640 
 
 6-656930 
 
 352 
 
 131044 
 
 47437928 
 
 19-0262976 
 
 7-126936 
 
 296 87616 
 
 25934336 
 
 17-2046505 
 
 6-664444 
 
 363 
 
 131769 
 
 47832147 
 
 19-0525589 
 
 7-133492 
 
 297. 83209 
 
 26198073 
 
 17-2336879 
 
 6-671940 
 
 354 
 
 132496 
 
 48228544 
 
 19-0787840 
 
 7-140037 
 
 293 83804 
 
 26463592 
 
 17-2626765 
 
 6-679420 
 
 365 
 
 133225 
 
 48627125 
 
 19-1049732 
 
 7-146569 
 
 299! 89401 
 
 26730899 
 
 17-2916165 
 
 6-686833 
 
 366 
 
 133956 
 
 49027396 
 
 19-1311265 
 
 7153090 
 
 300 90000 
 
 27000000 
 
 17-3205081 
 
 6-694329 
 
 367 
 
 134689 
 
 49430863 
 
 19-1572441 
 
 7-159599 
 
 301 
 
 90601 
 
 27270901 
 
 17-3493516 
 
 6-701759 
 
 368 
 
 135424 
 
 49835032 
 
 1'.)- 1833261 
 
 7-166096 
 
 302 
 
 91204 
 
 27543603 
 
 17-3781472 
 
 6-709173 
 
 369 
 
 136161 
 
 53243409 
 
 192093727 
 
 7-172531 
 
 303 
 
 91809 27818127 
 
 17-4068952 
 
 6-716570 
 
 370 
 
 136900 
 
 50653000 
 
 19-2353341 
 
 7-179054 
 
 304 
 
 92416 
 
 28094464J 17-4355953 
 
 6-723951 
 
 371 
 
 137641 
 
 51064811 
 
 19-2613603 
 
 7-185516 
 
 305 
 
 93025 
 
 28372625 17-4642492 
 
 6-731316 
 
 372 
 
 138384 
 
 51478848 
 
 19-2373015 
 
 7-191966 
 
 306 
 
 93636 
 
 23652616 17-4928557! 6-733664 
 
 373 
 
 139129 
 
 51895117 
 
 19-3132079 
 
 7-198405 
 
 307 
 
 94249 
 
 28934443. 17-5214155 
 
 6-745997 
 
 374 
 
 139876 
 
 52313624 
 
 19-3390796 
 
 7-204832 
 
 308 
 
 94864 
 
 29218112 17-5499288 
 
 6-753313 
 
 375 
 
 140625 
 
 52734375 
 
 19-3649167 
 
 7-211248 
 
 309 
 
 95481 
 
 2951)3629 17-5783953 
 
 6-760614 
 
 376 
 
 141376 
 
 53157376 
 
 19-3307194 
 
 7-217652 
 
 310 
 
 96100 
 
 297910001 17-6068169 
 
 6-767899 
 
 377 
 
 142129 
 
 53582633 
 
 19-4164878 
 
 7-224045 
 
 311 
 
 96721 
 
 30080231 
 
 17-6351921 
 
 6-775169 
 
 378 
 
 142884 
 
 54010152 
 
 19-4422221 
 
 7-233427 
 
 312 
 
 97344 
 
 33371328 
 
 17-6635217 
 
 6-782423 
 
 379 
 
 143541 
 
 54439939 
 
 19-4679223 
 
 7-236797 
 
 313 
 
 97969; 30664297 
 
 17-6918060 
 
 6-789661 
 
 380 
 
 144400 
 
 54872000 
 
 19-4935837 
 
 7-243156 
 
 311 
 
 985961 30959144 
 
 17-7200451 
 
 6-796834 
 
 331 
 
 145161 
 
 55306341 
 
 19-5192213 
 
 7-249504 
 
 315 
 
 99225 
 
 31255375 
 
 17-7482393 
 
 6-804092 
 
 332 
 
 145924 
 
 55742968 
 
 19-5448203 
 
 7-255841 
 
 31li 
 
 99856 
 
 31554496 17-7763383 
 
 6-811235 
 
 333 
 
 146639 
 
 56181837 
 
 19-5703353 
 
 7-262167 
 
 317 
 
 100489 
 
 31855013 17-8044933 
 
 6-818462 
 
 334 
 
 147456 
 
 56623104 
 
 19-5959179 
 
 7-268482 
 
 318 101124 
 
 32157432 17-8325545 
 
 6-825624 
 
 335 
 
 148225 
 
 57066625 
 
 19-6214169 
 
 7-274786 
 
 3191 101761 
 
 32461759 17-8605711 
 
 6-832771 
 
 336 
 
 148996 
 
 57512456 
 
 19-6468327 
 
 7-231079 
 
 320! 102400 
 
 32763000 17-8835438 
 
 6-839904 
 
 3S7 
 
 149769 
 
 57960603 
 
 19-6723156 
 
 7287362 
 
 321| 103041 
 
 33076161 17-9164729 
 
 6847021 
 
 338 
 
 150544 
 
 58411072 
 
 19-6977 15f 
 
 7-293633 
 
 322! 103684 
 
 33336248i 17-9443534 
 
 6854124 
 
 339 
 
 151321 
 
 53863869 19-7230829 
 
 7-299894 
 
 323 104329 
 
 33598267 17-9722308 
 
 6-861212 
 
 390 
 
 152100 
 
 59319000 19-7434177 
 
 7-306144 
 
 324 104976 
 
 34012224 18-0000000 
 
 6-868235 
 
 391 
 
 152831 
 
 59776471 19-7737199 
 
 7312333 
 
 325 105625 
 
 34323125 18-0277564 
 
 6-875344 
 
 392 
 
 153664 
 
 60236238 19-7989899 
 
 7-318611 
 
 326 
 
 106276 
 
 34645976 
 
 18-0554701 
 
 6-882339 
 
 393 
 
 154449 
 
 60693457 19-8242276 
 
 7-324829 
 
 327 
 
 106929 
 
 34965783 18-0831413 
 
 6-889419 
 
 394 
 
 155236 61162984J 19-8494332 
 
 7331037 
 
 328 
 
 107584 
 
 35287552 
 
 18-1107703 
 
 6-896435 
 
 395 
 
 156025 61623875 
 
 19-8746069 
 
 7-337234 
 
 329 
 
 108241 
 
 35611239 
 
 18-1333571 
 
 6903436 
 
 396 
 
 155816 
 
 62099136 
 
 19-8997487 
 
 7-343420 
 
 33C 
 
 108900 
 
 35937000 
 
 18-1659021 
 
 6-910423 
 
 1 397 
 
 157609 
 
 62570773 
 
 19-9243538 
 
 7-349597 
 
 331 
 
 109561 
 
 36264691 18-1934054 
 
 6-917396 
 
 ! 398 
 
 158404 
 
 63044792 
 
 19-9499373 
 
 7-355762 
 
 332 110224 
 
 365943681 18-2208672 
 
 6-924356 
 
 399 
 
 159201 
 
 63521199 
 
 19-9749844 
 
 7361918 
 
 333 110832 
 
 36926037 18-248237G 
 
 6-93130 
 
 400 
 
 160000 
 
 64000000 
 
 20-0030001 
 
 7-363063 
 
 331 
 
 11155fi 
 
 37259704 18-2756669 
 
 6-933232 
 
 401 
 
 160801 
 
 644H1201 
 
 200249844 
 
 7-374198 
 
 335 
 
 112225 
 
 37595375 18-3030052 
 
 6-945150 
 
 ! 402 
 
 161604! 64964308 
 
 20 049937* 
 
 7-330323 
 
TABLE OF SQUARES, CUBES, AND ROOTS. 
 
 641 
 
 0. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 No. Square. 
 
 Cube. Sq. Root. 
 
 CubeRoot. 
 
 403 
 
 1024091 65450327 
 
 20-0748599 
 
 7336437 
 
 470! 22J900! 10332300o| 21-6794334 
 
 7-774930 
 
 404 
 405 
 
 1632 If 
 164025 
 
 F5939264 
 
 C643J125 
 
 20-099751-2 
 20-1246118 
 
 7-392542 
 7-398636 
 
 471 221841 1044871111 21-7025344 
 472 222784! 105154048! 21-7255510 
 
 7-780490 
 7-785993 
 
 406 
 
 164836 
 
 66923416) 20-1494417 
 
 7 "104721 
 
 473| 2237291 105823817 21 -74850*2 
 
 7-79148? 
 
 0? 
 
 165 49 
 
 67419143 
 
 20-1742410 
 
 7-410795 
 
 474J 224676 106496424 
 
 21-7715411 
 
 7-790974 
 
 08 
 
 1564641 67917312 
 
 20-1990099 
 
 7-416859 
 
 475! 225625 
 
 107171875 
 
 21-7944947 7-80245- 
 
 409 
 
 167281 68417929 
 
 20-2237484 
 
 7-422914 
 
 476 226576 
 
 107850176 
 
 21-8174242) 7-807925 
 
 410 
 
 168100! 68921000 
 
 20-2484567 
 
 7-428959 
 
 477 
 
 227529 
 
 108531333 
 
 21 84032 J7 7-313389 
 
 411 
 
 168921 69426i 31 
 
 20-2731349 
 
 7-434994 
 
 47fc 
 
 223484 
 
 109215352 
 
 21-8632111 7-818846 
 
 412 
 
 169744 
 
 69934J 23 
 
 20-2977831 
 
 7-441019 
 
 47S 
 
 229441 
 
 109902239 
 
 21-8850686 7-82429- 
 
 413 
 411 
 
 170569 
 
 171396 
 
 70444997 
 70957944 
 
 20-3224014 
 20-3469899 
 
 7-447034 
 7-453040 
 
 430 
 431 
 
 230400 
 231361 
 
 1 105920UO 
 111284641 
 
 21-908J023 7-829735 
 21-9317122! 7-H:16j 
 
 415 
 
 172225 
 
 71178375 
 
 20-3715488 
 
 7-459035 
 
 432 
 
 232324 
 
 111930168 
 
 21-9544934 
 
 7-840595 
 
 41- 
 
 173055 
 
 71991296 
 
 20-3960781 
 
 7-465022 
 
 433 
 
 233289 
 
 112678537 
 
 21-9772610 
 
 7846013 
 
 41? 
 
 173389 
 
 72511713 
 
 20-4205779 
 
 7-470999 
 
 484 
 
 234256 
 
 113379J04 
 
 22-OOOuOOO 
 
 7851424 
 
 418 174724 
 
 73034632 
 
 20-4450483 
 
 7-476966 
 
 4S5 235225 
 
 114084125 y-2'0227155 
 
 7-356823 
 
 419! 175561 
 
 73560059 
 
 20-4694895 
 
 7-482924 
 
 486 236196 114791256 
 
 220454077 
 
 7-862224 
 
 420i 176400 
 
 74083000 
 
 20-4939015 
 
 7-488872 
 
 487 
 
 237169 
 
 115501303) ^-0680765 
 
 7-867613 
 
 421 
 
 177241 
 
 74618461 
 
 20-5182845 
 
 7-494311 
 
 488 
 
 238144 
 
 116214272 22-OJ07220 
 
 7-H72994 
 
 422 
 
 178084 
 
 75151448 
 
 20-5426336 
 
 7-500741 
 
 439 
 
 239121 
 
 116930169 22-1133444 
 
 7-878368 
 
 423 178929 
 
 75636967 
 
 20-5669638 
 
 7-506661 
 
 490 
 
 240100 
 
 117649000 22 1359435 
 
 7-883735 
 
 424! 179776 
 
 76225024 
 
 20-5912603 
 
 7-512571 
 
 491 
 
 241031 
 
 118370771! 22-1585193 
 
 7-839095 
 
 425 180625 
 
 76765625 
 
 20-6155281 
 
 7-518473 
 
 492 
 
 242064 
 
 119095438' 22-1811)730 
 
 7-894447 
 
 426 181476 
 
 77303776 
 
 20-6397674 
 
 7-524365 
 
 493 
 
 243049 
 
 119823157: 22^035033 
 
 7-899792 
 
 427t 182329 
 
 77854483 
 
 20-6639783 7-530248 
 
 494 
 
 244036 
 
 120553784 222251KH 
 
 7-905129 
 
 428 
 
 183184 
 
 78402752 
 
 20-6881609 7-535122 
 
 495 
 
 245025! 121287375J 22^35955 
 
 7-910460 
 
 429 
 
 1840 11 
 
 78953539 
 
 20-7123152 7-541987 
 
 4% 
 
 246010! 122023936 222710575 
 
 7-915733 
 
 43li 
 
 184900 
 
 79507000 
 
 20-7364414 
 
 7-547842 
 
 497 
 
 247009 
 
 122703473 222934963 
 
 7-921099 
 
 431 
 
 185761 
 
 80062991 
 
 20-7605395 
 
 7-553639 
 
 493 
 
 243004 
 
 1235059 M 22-3159135 
 
 7-926403 
 
 432 1850*24 
 
 80621568! 
 
 20-7846097 
 
 7-559526 
 
 499 
 
 249001 124251499) 22^333^79 
 
 7-931710 
 
 433, 187489 
 
 81182737 
 
 20-8086520 
 
 7565355 500 
 
 250000 125000000 22 360o798 
 
 7-937005 
 
 434 183356 
 
 81746504 
 
 20-8326667 7571174 
 
 501 
 
 251001 
 
 125751501 22333J293 
 
 7-942293 
 
 435; 189225 
 
 82312875 
 
 20-8566536 7'576985 
 
 502 
 
 252004 
 
 126506008 224053505; 
 
 7-947574 
 
 436 190096 82881856 
 
 20-8806130| 7-582786 
 
 503 
 
 253009 127263527) 22-4270615) 
 
 7-952848 
 
 437 190969 
 
 83453453 
 
 20-9045450 
 
 7-583579 
 
 504 
 
 254016 1280^4004 224499443 
 
 7953114 
 
 433 191844 
 
 84027672 
 
 20-9284495 
 
 7-594363 505 255025 12878702 >i 22-472205 ll 
 
 7-963374 
 
 439 192721 84604519 
 
 20-9523268! 7600133; 506 
 
 250036; 129554216 22-4944433 
 
 7-968627 
 
 440 193600 85184000 
 
 20-9761770 
 
 7-005905 507 
 
 257049 1303233431 22 516651)5 
 
 7-973373 
 
 441! 194481 8576)121 
 442; 195364 86350388 
 
 21-0000000 7-611663 508 
 21-0237960) 7-617412 509 
 
 253064 131090512! 22 5333553' 
 259031 131872229) 225510243 
 
 7-9791 U 
 7-934344 
 
 443! 196249 
 
 86938307 
 
 21-0475652! 7-623152 510! 2601001 132651000) 22'5331795| 
 
 7-989570 
 
 444 197136 87528334 
 
 21-0713075! 7-023384 511 
 
 261121 133432:131! 22-6053091! 
 
 7-994788 
 
 445 198025 88121 125 ! 
 
 21-0950231 
 
 7-63460? 512 
 
 202144 134217728 22-0274 17o! 
 
 8000000 
 
 446! 1939161 88716536 
 
 21-1187121 
 
 7010321 513 
 
 263169 13500569?) 22-6195033 
 
 8-00521 * 
 
 447 
 
 199809; 89314623 
 
 21-1423745 7010027)! 514 
 
 264190 1357:Mi74l 
 
 220; 15531) 
 
 
 448 
 
 200704 1 89915392 
 
 21-1660105 7-051725! 515 
 
 265225 136590875 
 
 22-6936 1 1 4 
 
 8-0155'JJ 
 
 449 2016011 90518349 
 
 21-1896201 
 
 7657414 510 
 
 266256 137333096 
 
 227156334 
 
 8-020779 
 
 450 202500! 91125000 
 
 212132034 7-663094 517 
 
 2672891 13318*413 
 
 22-7376340! 02595? 
 
 i:-l 203401 91733851 
 
 21-2367606 7-663760 5 Is 
 
 263324 138991832 
 
 22-75J6134) 
 
 8-031129 
 
 452 204304 92345403 
 
 21-2502916 7-071430- 519 
 
 269351 S 139?'.K!.-.' 
 
 22-7815715 
 
 8-03529/ 
 
 453 305209) 92959677 
 
 21-2837967; 7'080036 520 
 
 270400 1406U8000 
 
 22-8u350a5 
 
 M-041451 
 
 454 206116 93576664 
 
 21-3072758' 7'685733 521 
 
 271441 141420761 228254244 
 
 8-040003 
 
 4531 207025 94196375 
 
 21-3307290 7'69137'2 52'2 
 
 272 4^1 142236648 
 
 22-847311*3 
 
 8-051748 
 
 456 21)7936 94818816 
 
 21-3541565; 7 6J7002 523 
 
 273529 1 UUV.OO; 
 
 22-3691933 
 
 8 056336 
 
 457! 208349 95443993 
 
 21-3775533 7'702025 524 
 
 274576 143377824 
 
 22-89104 63 1 
 
 8062018 
 
 458; 209764 
 
 96071912 
 
 21-4009340 7-708239 5J5 
 
 2750-25 1417.131,5 
 
 2-2-9128785 
 
 8-067143 
 
 459i 210581 
 
 96702579! 
 
 21-4242353 7713S15 526 
 
 276676 145531570 
 
 22-93 10 ^99 
 
 8-07-2262 
 
 460 211600 
 
 97336000 
 
 2 1-4 176 UK; 7-719413 5-27 
 
 277729 145363183 
 
 22-95.i4S.j6 
 
 8077374 
 
 461 212521 
 
 97972181 
 
 21-4709 UK") ?-725:>32 528 
 
 278784 147197952 
 
 22-973^500 
 
 8-082430 
 
 462 213444 
 
 ys.il 1123 
 
 21-4941853 7 730014 529 
 
 279341 148035339 
 
 2300JOJOO, 
 
 
 463 214369 
 
 99252347 
 
 21-51743J8 77361HH -,30 
 
 280900 148877000 
 
 23-0217.^9 
 
 809267* 
 
 464 215296 
 
 99897344 
 
 21-5406592 7711753 531 
 
 281961J 149721291 
 
 23(1134372 
 
 8-097759 
 
 465! 216225 1005! 1625 
 
 21-56335871 7'7473lir 532 
 
 283024! 150558763 
 
 23-065 1 -2 J 2 
 
 8-102339 
 
 466; 217156) 101194696 
 
 21-5370331 
 
 7-752861 533 
 
 284089 15 14 1'9 i:i7 
 
 23080792.3 
 
 8-107913 
 
 46? 21&J39J 101847563 
 
 21-6101828 
 
 7-758402 531 
 
 285156 152273304 
 
 23 1084400 
 
 B'ltsfctiO 
 
 46S 2190241 102503232 
 
 21-0333077 
 
 7-703.-30 5!55 
 
 286225 153130375 
 
 23 13, "0070 
 
 8-118041 
 
 469 219961 10)161709 
 
 21-6564078 
 
 7-769462 536 
 
 287296 153990656 23 1516738 
 
 
6 4 2 
 
 APPENDIX. 
 
 No. 
 
 Square. 
 
 Cub*. 
 
 Sq. Root. 
 
 CuheRoot-' No. Squar,;. 
 
 Cube. 1 S.]. Rcu. 
 
 CutijKoot. 
 
 537 
 
 2333691 154854153 23-1732605' 8-123145J 604 
 
 3-4816 
 
 220348864 
 
 245764115 
 
 8 453026 
 
 33 
 
 239444| 155720872; 23-1948^70 8-133187| 605 
 
 3;. 6025 
 
 221445125 
 
 24-5967478 8-457691 
 
 539 
 
 2935211 156590819; 23-2163735 8-1332231 
 
 606 
 
 3<>7236 
 
 222545016 
 
 24-6170673! 8-462318 
 
 540 2916001 157464030; 23-23790011 8-143253 
 
 607 
 
 368449 
 
 223648543 
 
 24-63737001 8-467001) 
 
 541 292631* 158340421; 23-2594067. 8-148276 
 
 608 
 
 369664 
 
 224755712 
 
 24-6576560 
 
 8-471647 
 
 542 293764', 1592201)881 23-2H03935! 8-153294 
 
 609 
 
 370881! 225866529 
 
 24-6779254 
 
 8-476289 
 
 543| 294349' 160103007 23-3023604! 8-158305 
 
 610 
 
 372100] 226981000 
 
 246981781 
 
 8-480926 
 
 544 295936 
 
 160939184 23-3233076 8-163310! 611 
 
 373321 
 
 228099131 
 
 24-7184142 
 
 8-485558 
 
 545 297025 
 
 161873625 23-3452351 
 
 8-168309 612 
 
 374554 
 
 229220928 
 
 24-7386338 
 
 8490185 
 
 546 298116 
 
 162771336J 23-3665429 
 
 8-173302 613 
 
 375769 
 
 230346397 
 
 24-7538358 
 
 8-494806 
 
 547J 299209 
 
 163667323; 23-3330311 
 
 8-178239 
 
 614 
 
 376996 
 
 231475544 
 
 24-7790234 
 
 8-499423 
 
 548, 300304 
 
 1645665921 23-4093998 
 
 8-183269 
 
 615 
 
 373225 
 
 232608375 
 
 24-7991935 
 
 8-504035 
 
 5 19| 301401 
 .550 302500 
 
 165469149 
 166375000 
 
 23-4307490 
 23-4520788 
 
 8-188244 
 8-193213 
 
 616 
 617 
 
 379456 
 330689 
 
 2337448961 24-8193473 
 234885113 24-8394347 
 
 8-508642 
 8-513243 
 
 551 
 
 3J3601 
 
 1672841511 23-4733392 
 
 8-198175) 618 
 
 331924 
 
 236029032! 24-8596058 
 
 8-517840 
 
 552 
 
 304704! 163196608 
 
 23-4946802 
 
 8-2031321 619 
 
 333161 
 
 237176659 24-8797106 
 
 8-522432 
 
 553 3J5809 
 
 16911237.7 
 
 23-5159520 
 
 8-203032 620 
 
 334400 
 
 238328UOO] 24-8997992! 8-527019 
 
 554 306916 
 
 170031464 
 
 235372046 
 
 8-213027 621 
 
 385641 
 
 239433061 24-91987161 8-531601 
 
 555 303025 
 
 170953375 
 
 235534333 
 
 8-217966 622 
 
 386884 
 
 240641848 
 
 24-9399278 8-535178 
 
 556 309136 
 
 171879616 
 
 23-5/96522 
 
 8-222893J 623 
 
 38^129 
 
 241804367 
 
 24-9591679 8-540750 
 
 557 
 
 310249 
 
 172308693 
 
 23-6003474 
 
 8-227825 
 
 i 624 
 
 339376 
 
 242970624 
 
 24-9799920 8'5453l7 
 
 558 
 
 311364 
 
 173741112 
 
 23 6220236 
 
 8-232746 
 
 625 ! 390625 
 
 244140625 
 
 25-0000000 8-549880 
 
 559 312431 
 
 174676879! 236431808] 8-237661 
 
 626 1 391876 
 
 245314376 
 
 25-01-99920 8-554437 
 
 560 313600 
 
 175516000 23-6543191 
 
 8-242571 
 
 627 393129 
 
 246491833 
 
 25-0399681 
 
 8-553990 
 
 561 314721 
 
 176553481 
 
 23-6854336 
 
 8-247474 
 
 623 394334 
 
 247673152 
 
 25-0599282 
 
 8-563538 
 
 562! 315344 
 
 177504328 
 
 23-7065392 
 
 8-252371 
 
 629 395641 
 
 248853189 
 
 25-0793724 
 
 8-568081 
 
 563 316969 
 
 178453547 23-7276210 
 
 8-257263 
 
 630 
 
 396900 
 
 250047000 
 
 25-0993003 
 
 8-572619 
 
 564 318096 
 
 17941)6144 23-7481x312 8^62149 
 
 631 
 
 393161 
 
 251239591 
 
 25-1197134 
 
 8577152 
 
 565! 319225 
 
 180362125 23769728.1 
 
 8-267029! 632 
 
 3J9424 
 
 252435968 
 
 25-1398102 
 
 8-581681 
 
 566 320356 
 
 1813214% 23-7J07515 8-2719041! 633 
 
 40068^ 
 
 253638137 
 
 25-1594913 
 
 8-536205 
 
 567 321489 
 
 182284263 
 
 23-81176 18| 8-2767731' 634 
 
 401956 
 
 254810104 25-1793566 8-530724 
 
 568 
 
 322624 
 
 183250432 
 
 23-83275:;6: 8-281635! 635 
 
 403225 
 
 256047375 25-1992C63J 8'595233 
 
 569 
 
 323761 184220009 
 
 33*537409 8-236493 636 
 
 404 96 
 
 257259456 
 
 25-2190404 8'599748 
 
 570 
 
 324900 185193000 
 
 23-8746728! 8-2 9 1344 
 
 637 
 
 405769 
 
 258474853 
 
 25-2333539 8'f>042V2 
 
 571 
 
 326011! 186169411 
 
 238.156063 8-2J6190 
 
 638 
 
 407044 
 
 239694072 
 
 25-25366191 8-608753 
 
 572 
 
 327184 
 
 187149248 
 
 23-9165215 8-3J1030 
 
 639! 408321 
 
 260917119 
 
 25 2731493 8-613248 
 
 573 
 
 328329 
 
 183132517 
 
 23-9374184 8-3J5865 640 409630 
 
 262144000 
 
 25.2932213! 8-617739 
 
 574 
 
 3294761 189119224 
 
 23-9532 J71 8-310694 
 
 641 410381 
 
 263374721 
 
 25-3179778J 8'622225 
 
 575 
 
 330625 190109375 
 
 239791576 8-315517 
 
 642J 412164 
 
 264609283 
 
 25-3377189 8-626706 
 
 576 331776 191102976 
 
 24-0000030 8-3203M31 643 
 
 413419 265847707 
 
 25-35741471 8-631183 
 
 577! 332929 
 
 192103033 
 
 24-0208213 
 
 8-325147! 64-1 
 
 414736 
 
 267089984 
 
 25-3771551 
 
 8-635655 
 
 578 334034 
 579! 335241 
 
 193100552 
 194104539 
 
 21-0416306 
 24-0524183 
 
 8-329954 645 416025 
 8-33475J;; 616: 417316 
 
 268336125 
 269586136 
 
 25-3968502 8.640123 
 25-41653.il 8-6445S5 
 
 580| 336400 195112000 
 
 24-0331891 8-339551 ! 647 418509 
 
 270340023 25 -4 3,5 1947 
 
 8-649044 
 
 5811 337581, 196122941 
 
 24-1039416 8-344341 648; 419904 
 
 272097792 25 '45581411 8-653497 
 
 582) 333724! 197137368 
 
 24-1246762! 8-349126! 649 421201 
 
 2733594491 254751784: 8-657946 
 
 533 1 339839 
 
 193155237 
 
 24-1453929 8-353905i 650i 422500 
 
 274625000 25'4950976i 8-662391 
 
 534J 341056 199176704 
 
 24-1660919 
 
 8-353678 65li 423301 
 
 275894451 25-5147016: 8-666331 
 
 535 342225! 200201625 
 
 24-18(57732 
 
 83534471 652! 425104 
 
 2771(57808 
 
 25-53429071 8-671266 
 
 586 343395 201230055 
 
 242)74359 
 
 8-333209! 653 425409! 278445077 
 
 25-5533647! 8 675697 
 
 587 3445691 202262003 
 
 24-2230829 
 
 8-372967: 654 427716! 2*79726264! 25'5734237l 8-680124 
 
 538 3457441 203297472 
 
 242487113 8-377719: 655 4290251 281011375! 25'5929678; 8-68454f 
 
 539J 346921 204335469 
 
 24-2593222 8-332465 
 
 1 656 43,336 282300416 25*6124969 8-683963 
 
 590 343100, 20537'JOOOJ 242399156 8-337206 657i 431619! 2835933931 25-6320112 8693376 
 591 349281 206425071 24-3104916 8-3919421' 658' 432964 234890312 25'6515107 8-697781 
 
 *92' 350464: 207474683 213310501 8-3J6673! 659< 4342811 236191179! 25'6709953 8-702188 
 
 593! 3516491 208527357 
 
 24-3515913 8-4013981 660i 435600! 287496000! 25-6904652; 8-706538 
 
 594i 352836 209584584 
 
 24-3721152; 840611$,! 661| 436921 
 
 288804781 25*7099203 8-710988 
 
 595 
 
 354025 210644875 
 
 24-3926218! 8-410333J 6621 438244 
 
 290117528 25-7293607! 8-715373 
 
 596 
 
 355216! 211703731 
 
 24-4131112 8-415542! 663 439569 
 
 291434247 25'7487854 8-719760 
 
 59? 
 
 356409 212776172 
 
 24-4335834 8-420246! 664 440896 
 
 292754944: 25'7681975 8724141 
 
 59$ 
 
 357604 213347192 
 
 24-45403851 8-424945 665 442225 
 
 294079625! 25'7875939 ! 8-728518 
 
 59 
 
 353801! 21492179ii 
 
 21.4744765 8-429633! 666 443556 
 
 295408296; 25-8069758' 8-732392 
 
 601 
 
 360001 
 
 216000001 
 
 244948974 
 
 8-4343271; 667 444889 
 
 296740963! 258263431; 8-737260 
 
 GO] 
 
 361201 
 
 217081801 
 
 24-5153012 
 
 8-439010! i 668 446224 298077632 ; 25 8456 1 J6J! 8741625 
 
 60S 
 
 362404 21816720? 
 
 2 4 -5356 -HI 
 
 8-443688!' 669, 447561 299418309 25-8(5503431 8-745985 
 
 605 
 
 363609; 21925622? 
 
 24-5560532 
 
 8-448360J 670 448900 300763000! 25 "884 35 821 8-753340 
 
TABLE OF SQUARES, CUBES, AND ROOTS. 
 
 643 
 
 No, 
 
 Square. 
 
 Cube. | Sq. Root. CubeRoot. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Ro<t. CubeRool. 
 
 671 
 
 450241 
 
 302111711 
 
 25-90366771 8-754691 
 
 738 
 
 544644 401947*72 27-1661554 1 9-036880 
 
 672 
 
 451534 
 
 303464448 
 
 25-9229623! 8-759033 
 
 739 
 
 546121 
 
 403533419 27-1845544' 9 040965 
 
 673 
 
 674 
 
 452929 
 454276 
 
 304821217 
 306182024 
 
 25-y422435| 8-763331 
 25-9615100 8-767719 
 
 740 
 741 
 
 547600 
 549031 
 
 405224COU 27-2029410! 9'04504* 
 4U6369021 27-2213152 9-049114 
 
 675 
 
 455625 
 
 307546375 
 
 25-98076211 8-7720531 742 
 
 550564 
 
 408518488 27-23J6769 9'053l83 
 
 676 
 
 45b'97<i 
 
 308915776 
 
 26-0000000' 8-776333' 743! 552J49 
 
 410172407 27-2530263' 9-057245 
 
 677 
 
 453329 
 
 310288733 
 
 26-0192237 878J708 744 553536 
 
 411830784 27-2763634 
 
 9061310 
 
 678 
 
 459684 311665752 
 
 26-03813311 8-785030 1 745! 555025 
 
 413493625; 27-2946881 
 
 
 679 
 630 
 
 461041 
 462401. 
 
 313046839 
 314432000 
 
 26.0576284J 8-789347 
 26-07630%! 8-793659 
 
 746 5565 lb 
 747| 5J80(,9 
 
 415160936! 27-31300L6 
 416332723! 27-33130i)7 
 
 9-073473 
 
 681 
 
 463761 
 
 315821241 
 
 26-0959767 8-797968 
 
 74UJ 5J9504 
 
 4185U8992 
 
 27-34y5337 
 
 y0775*0 
 
 632 
 
 46-H24 
 
 bl7214568 
 
 26-1151297 8-802272 
 
 749 561001 
 
 42Jl8974y 
 
 273670644 
 
 
 683 
 
 466489 
 
 313611987 
 
 26-1342687 1 8-8U6572 
 
 1 750 5625 JO 
 
 42187.XXX) 
 
 27-336127J 
 
 y-085bOb 
 
 684 
 
 4678561 320013504 
 
 26-1533937 
 
 8-810868!) 7511 564o01 
 
 4*3564751 
 
 27-40437y2 
 
 9-08^63j 
 
 685 
 
 469225 
 
 321419125 
 
 26-1725047 
 
 8-815160 
 
 752 565504 
 
 4*5^59008 
 
 27-4226134 
 
 y-0j36<2 
 
 686! 470596 
 
 322828356 
 
 26-1916017 
 
 8-819447 
 
 753 
 
 o.j/uoy 
 
 426957777 
 
 27-4403455 
 
 9-Oi)770l 
 
 <>87i 471961) 
 688J 473344 
 
 3*4242703 
 32566(;672 
 
 262106848 
 26-2297541 
 
 8-823731 
 8-823010 
 
 754 
 755 
 
 568516 
 570025 
 
 4*^661064 
 430J68375 
 
 2745906o4 
 27-47/2633 
 
 9-l0172b 
 9-105740 
 
 609 
 
 474721 
 
 327082769 
 
 26-2483095 
 
 8-832285 
 
 756 
 
 571536 
 
 432084215 
 
 27-4^54542 9-109767 
 
 690 
 
 476100 
 
 323509UGO 
 
 26-*673511 
 
 8-836556 
 
 757 
 
 573U49 
 
 433798093 
 
 
 9-113702 
 
 691 
 
 477431 
 
 329939371 
 
 26-2868789 
 
 8-840823 
 
 758 
 
 574564 
 
 43551951* 
 
 2/-5317yy8 
 
 y-l!773 
 
 692 
 
 478361 
 
 331373888 
 
 26-3053929 8-845085 
 
 759 
 
 57608 1 
 
 437*45479 
 
 275499546 
 
 9-121801 
 
 693 
 
 480249 
 
 33*2812557 
 
 26-3248932! b'Ws>344ll 760 
 
 577600 
 
 438976000 
 
 2 T -5680975 
 
 y- 125805 
 
 <94 
 
 481636 
 
 334255384 
 
 26-34387^7 
 
 8-853598 761 
 
 57^121 
 
 440711081 
 
 27'53b2234 
 
 9-12y8b6 
 
 695 
 
 483J25 
 
 335702U75 
 
 26-3628527 
 
 8-857849 762 
 
 530644 
 
 442450728 
 
 27-6043475 
 
 y-13380b 
 
 696 
 
 484416 
 
 337153536 
 
 26-3318119 
 
 8-862095 763 
 
 532469 
 
 444194947 
 
 276224546 
 
 y- 1377^7 
 
 697 
 
 485809 
 
 338608373 
 
 26-4007576 
 
 8.866337 764 
 
 533696 
 
 4459437^4 
 
 
 9-441787 
 
 698 
 
 48720-i 
 
 3400GS392 
 
 26-4196896 
 
 8-8705761 765 
 
 585225 
 
 447697425 
 
 ^7 'Ojt^o33-i 
 
 9- 44577 1 
 
 699 
 
 488601 
 
 341532099 
 
 26-4386081 
 
 8-874810 766 
 
 586756 
 
 44*4550b6 
 
 27 b7o705b 
 
 9-l4975o 
 
 700 
 
 49UL'00 
 
 343000000 
 
 26-4575131 
 
 8-879040 767 
 
 533^09 
 
 45121.663 
 
 *7'694764o 
 
 y-45o737 
 
 701 
 
 491401 
 
 344472101 
 
 *6-4764046 
 
 8-883266 
 
 768 
 
 589824 
 
 452984332 
 
 *7-7l23i*y 
 
 y-.'5/Vi^ 
 
 702 
 
 492i04 
 
 345948408 
 
 26-4952826 
 
 8-887483 
 
 769 
 
 591361 
 
 454756609 
 
 27-730849* 
 
 y-lblbo/ 
 
 703 491209 
 
 347428927 
 
 26-5141472 
 
 8-891706 
 
 770 
 
 592900 
 
 456533000 
 
 27-7480739 
 
 9-165656 
 
 704 
 
 495616 
 
 348913664 
 
 26 5329983 
 
 8-895920 
 
 771 
 
 594441 
 
 45334401 1 
 
 27-7663860 
 
 9'169b2.i 
 
 705 
 
 497025 
 
 b5u40~625 
 
 26-5518361 
 
 8-900130 
 
 772 
 
 5J5984 
 
 4b009b648 
 
 
 9-17353J 
 
 706 
 
 498436 
 
 351895316 
 
 26-57C6605 
 
 8-904337 
 
 773 
 
 597529 
 
 46183i)9i7 
 
 278023775 
 
 9-17/014 
 
 707 
 
 499849 
 
 353393243 
 
 26-5394716 
 
 8-908539 
 
 774 
 
 599076 
 
 463684824 
 
 27 "8200555 
 
 y 401500 
 
 708 
 
 501264 
 
 354894912 
 
 26-6082694 
 
 8-912737 
 
 775 
 
 600625 
 
 465481375 
 
 27 -0 33021 8 
 
 y 105453 
 
 709 
 
 502681 
 
 356400829 
 
 266270539 
 
 8-916931 
 
 776 
 
 602176 
 
 4b7*8e5/6 
 
 27 o5b776o 
 
 9 18940^ 
 
 710 
 
 504100 
 
 357911000 
 
 26-6453*52 
 
 8-921121 
 
 .777 
 
 603729 
 
 46 1 J097433 
 
 27-8747197 
 
 y I9b347 
 
 711 
 
 5U5521 
 
 35942543! 
 
 266645333 
 
 8-925308 
 
 778 
 
 605284 
 
 47U9l09;)2 
 
 27'0b26ol4 
 
 'j-197~9c. 
 
 712 
 
 5t6944! 360944128 
 
 266833231! 8-929490 
 
 779 
 
 606841 
 
 47V729139 
 
 27-9ib57J5 
 
 V*0l22 l j 
 
 713 
 
 500369 362467097 
 
 26-7020598 
 
 8-9b3669 
 
 780 
 
 6034 b 
 
 4/4552UOO 
 
 27-9284801 
 
 > j-2Ujlt> 
 
 714 
 
 5097^6: 353994344 
 
 26-7207784 
 
 8-937843 
 
 781 
 
 60^961 
 
 47t 37^541 
 
 27-y4b3772 
 
 y^o^o-Jb 
 
 715 
 
 511225! 365525375 
 
 5T -739483 J 
 
 8-942014 
 
 782 
 
 6115^4 
 
 47o^ll768 
 
 27-9642629 
 
 y-*i3o~o 
 
 716 
 
 512656 367061696 
 
 26-7581763 
 
 8-946181 
 
 783 
 
 613089 
 
 40X04060V 
 
 27'i821b7*! y-*ibi)Jv, 
 
 717 
 
 514089 368601813 
 
 26-7763557! 8-950344 
 
 ; 784 
 
 614656 
 
 481890^04 
 
 23-OCOOObO 
 
 y-*2O8/o 
 
 718 
 
 515524 370146232 
 
 26-7955220 8-9545o3|: 785 
 
 616225 
 
 40o/b66*5 
 
 28 Ol7o5l5 
 
 > j-2l7v-l 
 
 719 
 
 516%l| 37169495U 
 
 26-81447.-)! 
 
 8-958653 706 
 
 617796 
 
 4855o'/b5o 
 
 28 "0356 yl 5 
 
 y-*2o7^/ 
 
 720 
 
 518400; 373248UOO 
 
 26-8328157 
 
 8-962809! 787 
 
 61936'J 
 
 407443-lob 
 
 28-0535*00 
 
 .'_. 'Jo i'.. 
 
 721 
 
 519341 37181)5361 
 
 26-8514432 
 
 8-966957 
 
 i 788 
 
 620944 
 
 48X)3o3i72 
 
 28-071 33 // 
 
 y-2ot5^6 
 
 722 521234 37t.3,'.70v- 
 
 26-8700577 8-971101 
 
 789 
 
 62*521 4yil6906'j 
 
 28-089 143^ 
 
 y-24o4b;> 
 
 723 522729 377933i>r>7 
 
 26-8886593 8-975241 
 
 i 790 
 
 6*4100 4y3b3yOoO 
 
 23'106y38b 
 
 9-*14b3o 
 
 724! 524176 379503424 
 
 26-9072481 
 
 8-979377 
 
 i 7<J1 
 
 625631 
 
 494940671 
 
 if8'12472*2 
 
 y-*48*b4 
 
 725 525625 381078125 
 
 26-9258240 
 
 8-983509 
 
 i 792 
 
 627264 
 
 49b'/'yi)oM^ 
 
 23'l42494b 
 
 y-2o2i3t 
 
 726 527076 382657176 
 
 *6-9 143872 8'9876:>7 793 
 
 628849 4906/7*57 
 
 23-1602557 
 
 y-^obo2* 
 
 727 528529 38424053 
 
 2-3-9629375 8-99176^ 7J< 
 
 63Ji:-.t} o005b6404 
 
 28'170bOJb 
 
 9-*j99ll 
 
 728! 529984' 38582835:. 
 
 .6-98 14751 8-995883| : ?y5 
 
 6320*5 50*459375 
 
 28- 495 i 444 
 
 y-*6.uy< 
 
 729 531441 3^7420489 
 
 27-000(K)00 y-COOOOb 
 
 ?y6 
 
 6336 1<) 
 
 5D4353336 
 
 20-2134720 
 
 y-2b7bOv 
 
 730 532900 38^0171-00 
 
 27-0185122 9-004113 
 
 79. 
 
 635*0'J 
 
 50626l57> 
 
 28-2311004 
 
 '.I -. l-'-'v 
 
 731 534361; 39C6l7s.ll 
 
 270370117: 9-008223 
 
 I 798 
 
 636oOi 
 
 5084bu5* 
 
 28'248893o 
 
 y-^ij4oo 
 
 7321 535324 392223168 
 
 :.7-c<:>.V;985 9-012329 
 
 1 7yy 
 
 638401 
 
 5iot8*3yy 
 
 28*005801 
 
 9-i7y3oc 
 
 733! 5272*9 39383^837 
 
 27-G73V727 9-016431 
 
 > 800 
 
 64000C 
 
 512000000 
 
 28-204271* 
 
 y-2o3i7o 
 
 734! 53/7;6 3.i54469i! 
 
 27-0924341 'J -02052'. 
 
 801 
 
 641604J 513922401 
 
 28-3019434 
 
 y-2a 7o-*- 
 
 735 : 540^25 3J7065375 
 
 27.1108834 9-021621 
 
 802 
 
 643204 51534l6bO 
 
 20-3 1 1 .* oi.) 
 
 9-290.0* 
 
 736 541*i> 39'0,s ( su;>(i 271293199 9-0*a7l5 
 
 i 803 
 
 644809 51778162/ 
 
 28-o3/*54b 
 
 9-*'j4<b< 
 
 737, f>431t.9 40031555:> 27 147743.* 9-032802 
 
 804] 64641b 
 
 5197181b4 
 
 *tt-354oy30 
 
 y-2y8ti*i 
 
644 
 
 APPENDIX. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRootJ 
 
 No. 
 
 Square.) Cube. 
 
 1 
 
 Sq. Root CubeKoot. 
 
 805 
 
 6430251 52166J125 
 
 2837*5219 9-302477 
 
 872| 761)334 
 
 663.)M348| 29-5296461 
 
 9-553712 
 
 806 
 
 649636 52360G616: 
 
 283901391 
 
 9-306323 
 
 873 762129 
 
 665338617! 29-5465734 
 
 9-557363 
 
 807 
 
 651249 525557943; 
 
 28-4077454 
 
 9-310175 
 
 874 
 
 763376 
 
 667627624 29-5634910 
 
 9-561011 
 
 808 
 
 652864 
 
 527514112 
 
 28 4253408 
 
 9-3140191 
 
 875 765625 
 
 669921875 
 
 29-5303989 
 
 9-564656 
 
 809 
 
 654481 
 
 529475129 
 
 28-4429253 
 
 9-317860; 
 
 876 
 
 767376 
 
 <>?2221376 
 
 29-59729721 9-568298 
 
 810 
 
 656100 
 
 531441000 
 
 28-4604989 
 
 9-321697 877 
 
 769129 
 
 674526133 
 
 29-6141853 9-571938 
 
 811 
 
 657721 
 
 533411731 
 
 28-4780617 
 
 9-325532 
 
 878 770884 
 
 676836152 
 
 29-6310648 1 9-575574 
 
 812 
 
 659344 
 
 535387328 
 
 23-4956137 
 
 9-32J363 
 
 879 772641 
 
 679151439 
 
 29-6479342, 9-579208 
 
 813 
 
 660969 
 
 537367797 
 
 23-5131549 
 
 9-333192 
 
 880 774400 
 
 681472000 
 
 29-6647939! 9-582840 
 
 814 
 
 662596 
 
 539353144 
 
 23-5306852 
 
 9-347017 
 
 b81 776 Hi 1 
 
 683797341 
 
 29-6816442 9-:86468 
 
 815 664225 
 
 541343375 
 
 23-5482048 
 
 93408391 
 
 832 777924 
 
 686128968 
 
 29.6984843 
 
 9-590094 
 
 816 
 
 665356 
 
 543338496 
 
 23-5657137 
 
 9-344657! 
 
 883 779689 
 
 688465387 
 
 29-7153159 9-593717 
 
 81? 
 
 6l>7489 
 
 545333513 
 
 28-5832119 
 
 9-3:8473 
 
 834 781456 
 
 690807104 
 
 29-7321375' 9-597337 
 
 818 
 
 669124 
 
 547343432 
 
 23-6006993 
 
 9-352286 
 
 8b5j 783225 
 
 693154125 
 
 29-7439496! 9-600955 
 
 819 
 
 670761 
 
 549353259 
 
 28-6181760 
 
 9-356095 
 
 886 784996 
 
 695506456 
 
 29-7657521 9-604570 
 
 820 
 
 672400 
 
 551368000 
 
 28-6356421 
 
 9-359902! 
 
 88? 
 
 786769 
 
 697864103 
 
 29-7825452 
 
 9-603182 
 
 821 
 
 674041 
 
 553387661 
 
 28-6530976 
 
 9363705 
 
 838 
 
 788544 
 
 700227072] 29-7993239 
 
 9-61*791 
 
 822 
 
 675684 
 
 555412248 
 
 28-6705424 
 
 9367505: 
 
 889 
 
 790321 
 
 702595369 
 
 29-816103:) 
 
 9-6153'J'r 
 
 823 
 
 677329 
 
 557441767 
 
 23-63797G6 
 
 9-371302 
 
 890 
 
 7 1 J2100 
 
 704969000 29-8323678 
 
 9-619002 
 
 824 
 
 678976 
 
 559476224 
 
 23-7054002 
 
 9375096 1 
 
 891 
 
 793881 
 
 707347971 29-8496231 
 
 9-622603 
 
 825 
 
 680625 
 
 561515625 
 
 28-7228132 
 
 9-373387 
 
 892 
 
 795664 
 
 709732288 29-8663690 
 
 9-626202 
 
 826 
 
 632276 
 
 563559976 
 
 28-740215? 
 
 9-382675 
 
 893 
 
 797449 
 
 712121957 29-8831050 
 
 9-629797 
 
 827 
 
 633429 
 
 565609283 
 
 28-7576077 
 
 9-336460 
 
 894 
 
 799236 
 
 714516934 
 
 29-8998328 
 
 9-633391 
 
 823 
 
 685584 
 
 567663552 
 
 28-7749891 
 
 9-390242 
 
 895 
 
 801025 
 
 716917375 29-9165506 
 
 9-636981 
 
 829 
 
 687241 
 
 569722789 
 
 28-7923601 
 
 9-394021! 
 
 896 
 
 802816 
 
 719323136 2l ^332591 
 
 9-640563 
 
 830 
 
 6839 >0 
 
 571787000 
 
 28-8097206 
 
 9-3. 7796! 
 
 897 
 
 804609 
 
 721734273 
 
 29-9499583 
 
 9-644154 
 
 831 
 
 690561 
 
 5?385619l| 28-8270706 9-401569 
 
 898 
 
 806404 
 
 724150792 
 
 29-9666481 
 
 9-647737 
 
 832 
 833 
 
 69i224 
 
 693889 
 
 5?5930368| 28-8444102; 9-405339 
 5780095371 28-86173941 9-409105 
 
 899 
 900 
 
 808201 726572699 
 810000| 729000000 
 
 29-98332-37 9-65131? 
 30-0000000! 9-654894 
 
 834 
 
 695556 
 
 530093704 
 
 28-8790582! 9-412869 
 
 901 
 
 811801 731432701 
 
 30-0166620 9-658468 
 
 835 697225 
 
 582182875 
 
 28-8963666! 9-416630 
 
 902 
 
 813604! 733870808 
 
 30-0333148 9-662040 
 
 836: 698896 
 
 584277056 
 
 28-9136646 9-420337 
 
 903 
 
 815409 
 
 73631432? 
 
 30-0499584 9-665610 
 
 837 700569 
 
 586376253 
 
 28-9309523 
 
 9-424142 
 
 904 
 
 817216 
 
 738763264 
 
 30-0665923 9-669176 
 
 838 i 702244 
 
 5^8480472 
 
 28-948229? 
 
 9-4278J4 
 
 905 
 
 819025 
 
 741217625 
 
 30-0332179 
 
 9-672740 
 
 839 703921 
 
 590589719; 23-965496? 
 
 9-431642 
 
 906 
 
 820836 
 
 74367741<i 
 
 30-0998333 
 
 9-676302 
 
 840! 705600 
 
 592704000 28-9827535 9-435338 
 
 90? 
 
 822649 
 
 746142643 
 
 30-1164407 
 
 9-679860 
 
 841 707281 
 
 594823321 
 
 29-OOOOOOC 
 
 9-439131 
 
 908 
 
 824464| 748613312 
 
 30-1330383 
 
 9-f>83417 
 
 842J 708^64 
 
 596947688 
 
 29-0172362 
 
 9-442370 
 
 909 
 
 826281 
 
 751089429 
 
 30-149626'J 
 
 9-036970 
 
 843 710649 
 
 599077107 
 
 29-0344622 
 
 9-446607 
 
 910 
 
 828100 
 
 753571000 
 
 30-1662063 
 
 9-690521 
 
 844j 712336! 6012 1 1584 
 
 29-0516781 
 
 9-450341 
 
 911 
 
 829921 
 
 756058031 
 
 30-1827765 
 
 9-694069 
 
 8451 7140251 603351125 
 
 29-0688837 
 
 9-454072 
 
 912 
 
 831744 
 
 758550528 
 
 30-1993377 
 
 9-697615 
 
 846 1 715716J 605495736 
 
 29-0860791 
 
 9-457800 
 
 913 
 
 833S6S 
 
 761048497 
 
 30-2158393 
 
 9-701158 
 
 847 717409 607645123 29-1032644 
 
 9-461525 
 
 914 
 
 835396] 763551944 
 
 30-232432b 
 
 9-704699 
 
 84fc 
 
 719104 
 
 609300192) 29-1204396 
 
 9-465247 
 
 915 
 
 8372251 766060875 
 
 30-248966S 
 
 9-708237 
 
 84 
 
 720801 
 
 611960049 29-137604f 
 
 9-463966 
 
 916 
 
 839056 768575-296 
 
 30-26549 It 
 
 9-711772 
 
 85C 
 
 > 72250C 
 
 614125000 29-154759= 
 
 9-47*682 
 
 917 
 
 840889 771095213 
 
 30-282007L 
 
 9-715305 
 
 851! 724201 
 
 6162950511 29- 17 19042 
 
 t 9-476396 
 
 918 
 
 842724 
 
 773620632 
 
 3J-2 ( J8514fc 
 
 9-718835 
 
 852! 725904 
 
 618470208! 29 1890390 9-480106 
 
 919 
 
 8445611 7761515511 
 
 30-3150 12b 
 
 9-722363 
 
 853 72760'j 
 
 6206504771 29-2U61637J 9-483814 
 
 92u 
 
 84640C 
 
 1 773683000 
 
 30-3315016 
 
 9-725388 
 
 854 
 
 7293 If 
 
 622835364 29-2232784 9'48?518 
 
 921 
 
 848241 
 
 781229961 
 
 30-347981^ 
 
 9-729411 
 
 85f 
 
 7310^' 
 
 625026375 29-2403830 9-491220 
 
 922J 850084 78377744ti 
 
 3J-364452L 
 
 9732931 
 
 856 
 
 , 73273f 
 
 627222016 29-2574777 9-494919 
 
 923 851929 786330467 
 
 303809151 
 
 9-736448 
 
 85' 
 
 ' 73444i 
 
 629422793J 29-2745623! 9-498615 
 
 924 
 
 853776: 78388902-4 
 
 30-3J7363? 
 
 9-73J963 
 
 85 
 
 J 736164 
 
 631623712! 29'2916370| 9-5U23J8 
 
 925 
 
 855625! 791453125 
 
 30-4138127 
 
 9-743476 
 
 85i 
 
 > 737881 
 
 633339779 29-3087018! 9-505998 
 
 926 
 
 857476 79402277f 
 
 3J-43J2481) 9-746986 
 
 860 73960C 
 
 6360560(X)| 29 3257566 9-509685 927 
 
 859329 796597982 
 
 3J-4466747 9-750493 
 
 861 74132] 
 
 6382773811 29-3423015J 9-513370 
 
 928 
 
 861184 
 
 79917875. 
 
 3U-4630924I 9-753J98 
 
 862 743044 
 
 640503928 29-3598365 9517051 
 
 92'J 
 
 fc63041 
 
 801765031 
 
 30-47y5013| 9-757500 
 
 863 74476i 
 
 6427356471 29-3768616 9-520730 
 
 931) 
 
 864900 804357000 30-4i/59014| 9-761000 
 
 864 1 746496! 644972544! 29-3933769 9-524406 
 
 931 
 
 866761' 806954491 30-5122926J 9-76449? 
 
 865 748225 647214625 1 29-4103823 9-528079 
 
 932 
 
 868624. 8095575681 30-5285750 9-767992 
 
 866 749956 649461896 29-4278779 9-531750 
 
 932 
 
 87J4*-j 812i6G237| 30-545048? 9-771484 
 
 867 75168i 
 
 > 651714363 29-4448637 9-535417 
 
 934 
 
 872356; 814780504 30-5:5 14 136! 9-?74i>74 
 
 868; 75342^ 
 
 [ 653972032' 29-4618397 9-539082 
 
 933 
 
 874225! 817400375 
 
 30-67776911 9-778462 
 
 869! 75516] 
 
 656234909 29-4788059 9-542744 
 
 93fc 
 
 876096: 820025y5f 
 
 30-5;;4Ii71 978194? 
 
 87C 1 75690( 
 
 ) 658503000 29-4957624 9 "5 46403 
 
 937 
 
 877969: 82265615; 
 
 3D-610455?; 9 785429 
 
 87 1\ 75864 
 
 660776311! 29-51270'Jli 9-55UU5!) 
 
 933 H79H44 825293675! 
 
 30-6267857 9-736908 
 
TABLE OF SQUARES, CUBES, AND ROOTS. 
 
 645 
 
 No. 
 939 
 
 Square. 
 
 881721 
 
 Cube. | Sq. Root. 
 
 CubeRoot. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CabeRooiJ 
 
 827936019: 30-6431069 
 
 9-792386 
 
 970 
 
 940900 
 
 912673000 
 
 3H448230J 9-838983 1 
 
 940 
 
 883600 
 
 830584000! 33-6594194 
 
 9-795861 
 
 971 
 
 942341 
 
 915498611 
 
 31-160872'J 
 
 9-902333 
 
 941 
 
 885481 
 
 8332376211 30-6757233 
 
 9-799334 
 
 972 
 
 944784 
 
 918330048 
 
 31-1769145 
 
 9-905782 
 
 942 
 
 837364 
 
 835896888 30-6920185 
 
 9-802804 
 
 973 
 
 946729 
 
 921167317| 31-1929479 
 
 9-909173 
 
 943 
 
 889249 
 
 83856 1807J 30-7083051 
 
 9-806271 
 
 974 
 
 948676 
 
 924010424! 31-2039731 
 
 9-912571 
 
 944 
 
 8U1136 
 
 8412323341 30-7245830 
 
 9-809736 
 
 975 
 
 950625 
 
 926859375J 31-2249900 
 
 9-915962 
 
 945 
 
 893025 
 
 843908625 30-7408523 
 
 9-813199 
 
 976 
 
 952576 
 
 929714176 
 
 31-2409987 
 
 9-919351 
 
 946 
 
 894916 
 
 846590536 30-7571130 
 
 9-816659 
 
 977 
 
 954529 
 
 932574833 
 
 31-2569992! 9-922733 
 
 947 
 
 896809 
 
 849278123 30-7733651 
 
 9-820117 
 
 978 
 
 956484 
 
 935441352 
 
 31-27299151 9-926122 
 
 948 
 
 898704 
 
 851971392! 30-7896086 
 
 9-823572 
 
 979 
 
 958141 938313739 
 
 31-288.^757 
 
 9-929504 
 
 949 
 
 900601 
 
 8546703491 30-8058436 
 
 9-827025 
 
 980 
 
 960400 
 
 941192000 
 
 31-3049517 
 
 9-9328S4 
 
 950 
 
 90:2500 
 
 8573750001 30-82207001 9-830476 
 
 981 
 
 962361 944076141 
 
 31-3209195 9-936261 
 
 951 
 
 904401 
 
 860085351 308382879 9'833924 
 
 982 
 
 964324! 946966168 
 
 31-3368792 9-939636 
 
 952 
 
 906304 
 
 862801408: 30-85449721 9-837369 
 
 983 
 
 966289 949862087 
 
 31-3528308 
 
 9943009 
 
 953 
 
 908209 
 
 865523177 30-870698l' 9-840813 
 
 984 
 
 968256! 952763904 
 
 31-36877431 9-946380 
 
 954 
 
 910116 
 
 868250664 30-8868904 1 9'844254 
 
 985 
 
 970225 955671625; 31-3847097! 9-949748 
 
 K55 
 
 912025 
 
 870983875 30-9030743, 9-847692 
 
 986! 972196 
 
 958535256 
 
 31-400H369) 9-953114 
 
 956 
 
 913936 
 
 873722816 30-9192497J 9-851128 
 
 987; 974169 
 
 961504803 
 
 31-4165561 9-956477 
 
 957 
 
 915849 
 
 876467493; 30-9354166! 9-854562 
 
 2881 9761441 9644302"2 
 
 31-4324673! 9-959839 
 
 958 
 
 917764 
 
 879217912 30-95 15751J 9-857993 
 
 989 
 
 9781211 967361669 
 
 31-4483704 9-963198 
 
 959 
 
 919681 
 
 881974079 30-9f>77251| 9-861422 
 
 990 
 
 980100 
 
 970299000 
 
 31-4642654 9-96655. l > 
 
 960 
 
 921SOO 
 
 884736000^ 30-9^36668 9-864848 
 
 991 982081 
 
 973242271 
 
 31-4801525 9-969909 
 
 961 
 
 923521 
 
 887503681 31-OOOOOOOi 9-868272 
 
 992 984064 
 
 976191488 
 
 31-4960315! 9-973262 
 
 962 
 
 925444 
 
 NW277128 31-01612481 9-871694 
 
 993i 986049 
 
 979146657 
 
 31-5119025 
 
 9-976612 
 
 963 
 
 927369 
 
 ;< < .:J056347 3l-0322413i 9875113 
 
 994 988036 
 
 982107784 
 
 31-5277655 
 
 9-979960 
 
 964 
 
 929296 
 
 S;). r )841344 31-0483494J 9-878530 
 
 995 990025 
 
 985074875 
 
 31-5436206 
 
 9-983305 
 
 965 
 
 931225 
 
 898632125 31-0644491 9-881945 
 
 9% 
 
 992016 
 
 988047936 
 
 31-5594677 
 
 9-986649 
 
 966 
 
 933156 
 
 90M28696 21-08054051 9-885357 
 
 9971 994009 
 
 991026973J 31-5753068 
 
 9-989990 
 
 967 
 
 935089 
 
 104231063 31-0966236 9-888767 
 
 998 996004 
 
 994011992 
 
 31-5911330 
 
 9-993329 1 
 
 968 
 
 937024 
 
 907039232 31-1126984 9-892175 
 
 999 
 
 998001 
 
 997002999 
 
 31-6069613 
 
 9-996666, 
 
 969 938961 
 
 909853209 31-1287648! 9-895580 1000 
 
 1000000 1000000000 
 
 31 -6227766 lO-OOOOOOJ 
 
 The following rules are for finding the squares, cubes, and roots of num- 
 bers exceeding 1000. 
 
 To find the square of any number divisible without a remainder. Rule. Di- 
 vide the given number by such a number from the foregoing table as will 
 divide it without a remainder ; then the square of the quotient, multiplied by 
 the square of the number found in the table, will give the answer. 
 
 Examfle.Whzt is the square of 2000 ? 2000, divided by 1000. a number 
 found in the table, gives a quotient of 2, the square of which is 4, and the 
 square of 1000 is 1,000,000, therefore : 
 
 4 x 1,000,000 = 4,000,000: the Ans. 
 
 Another Example. What is the square of 1230? 1230, being divided by 
 123, the quotient will be 10, the square of which is 100, and the square of 123 
 
 is 15,129, therefore : 
 
 100 x 15,129= 1,512,900: the Ans. 
 
 To find the square of any number not divisible without a remainder. Rule. 
 Add together the squares of such two adjoining numbers from the table as 
 shall together equal the given number, and multiply the sum by 2 ; then this 
 product, less i, will be the answer. 
 
 Example. What is the square of 1487 ? The adjoining numbers, 743 and 
 744, added together, equal the given number, 1487, and the square of 743 = 
 552,049, the square of 744 = 553,536, and these added = 1,105,585, therefore : 
 
 1,105,585 x 2 2,211,170 i = 2,211,169 : the Ans. 
 
 To find the ciibc of any number divisible without a remainder. Ritle. Divide 
 the given number by such a number from the foregoing table as will divide 
 
646 APPENDIX. 
 
 it without a remainder ; then the cube of the quotient, multiplied by the cube 
 of the number found in the table, will give the answer. 
 
 Example. What is the cube of 2700? 2700, being divided by 900, the quo- 
 tient is 3, the cube of which is 27 and the cube of 900 is 729,000,000, there- 
 fore : 
 
 27 x 729,000,000 = 19,683,000,000 : the Ans. 
 
 To find the square or cztbe root of numbers higher than is found in the table. 
 Rule. Select, in the column of squares or cubes, as the case may require, that 
 number which is nearest the given number ; then the answer, when decimals 
 are not of importance, will be found directly opposite, in the column of num- 
 bers. 
 
 Example. What is the square root of 87,620? In the column of squares, 
 87,616 is nearest to the given number ; therefore, 296, immediately opposite 
 in the column of numbers, is the answer, nearly. 
 
 Another example. What is the cube root of 110,591? In the column of 
 cubes, 110,592 is found to be nearest to the given number ; therefore, 48, the 
 number opposite, is the answer, nearly. 
 
 To find the cube root more accurately. Rule. Select from the column of 
 cubes that number which is nearest the given number, and add twice the 
 number so selected to the given number ; also, add twice the given number 
 to the number selected from the table. Then, as the former product is to the 
 latter, so is the root of the number selected to the root of the number given. 
 
 Example. What is the cube root of 9200? The nearest number in the col- 
 umn of cubes is 9261, the root of which is 21, therefore : 
 
 9261 9200 
 
 2 2 
 
 18522 18400 
 9200 9261 
 
 As 27,722 is to '27,661, so is 21 to 20-953 + , the Ans. 
 
 Thus, 27,661 x 21 = 580,881, and this divided by 27,722 = 20-953 -f . 
 
 To find the square or cube root of a whole number with decimals. Rule. Sub- 
 tract the root of the whole number from the root of the next higher number, 
 and multiply the remainder by the given decimal ; then the product, added to 
 the root of the given whole number, will give the answer correctly to three 
 places of decimals in the square root, and to seven in the cube root. 
 
 Example. What is the square root of 11-14? The square root of n is 
 3-3166, and the square root of the next higher number, 12, is 3-4641 ; the for- 
 mer from the latter, the remainder is 0-1475, and this by 0-14 equals 0-02065. 
 This added to 3-3166, the sum, 3-33725, is the square root of 11-14. 
 
 To find the roots of decimals by the use of the table. Rule. Seek for the 
 given decimal in the column of numbers, and opposite in the columns of roots 
 will be found the answer, correct as to the figures, but requiring the decimal 
 point to be shifted. The transposition of the decimal point is to be performed 
 thus : For every place the decimal point is removed in the root, remove it in 
 the number t^vo places for the square root and three places for the cube root. 
 
THE REDUCTION OF DECIMALS. 647 
 
 Examples. By the table, the square root of 86-0 is 9-2736, consequently by 
 the rule the square root of 0-86 is 0-92736. The square root of 9- is 3-, hence 
 the square root of 0-09 is 0-3. For the square root of 0-0657 we have 
 0*25632, found opposite No. 657. So, also, the square root of 0-000927 is 
 0-030446, found opposite No. 927. And the square root of 8-73 (whole num- 
 ber with decimals) is 2-9546, found opposite No 873. The cube root of 0-8 
 is 0-928, found at No. 800 ; the cube root of 0-08 is 0-4308, found opposite 
 No. 80, and the cube root of 0-008 is 0-2, as 2-0 is the cube root of 8-0. So 
 also the cube root of 0-047 ' s 0-36088, found opposite No. 47. 
 
 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 inches is j\ of a foot, therefore : 
 
 & . . . 12)3-00 
 
 25 Ans. 
 Another example. What is the equivalent decimal of $ of an inch? 
 
 1 ... 8)7-000 
 
 875 Ans. 
 
 To reduce a compound fraction to its equivalent decimal. Rule. In accordance 
 with the preceding rule, reduce each fraction, commencing at the lowest, to 
 the decimal of the next higher denomination, to which add the numerator of 
 the next higher fraction, and reduce the sum to the decimal of the next higher 
 denomination, and so proceed to the last ; and the final product will be the 
 answer. 
 
 Example. What is the decimal of a foot equivalent to five inches, f and T V 
 of an inch ? 
 
 The fractions in this case are, i of an eighth, | of an inch, and T r v of a foot, 
 therefore : 
 
 eighths. 
 
 inches. 
 
 rV 12)5-437500 
 
 -453125 Ans. 
 
 The process may be condensed, thus : write the numerators of the given 
 
648 APPENDIX. 
 
 fractions, from the least to the greatest, under each other, and place each de 
 nominator to the left of its numerator, thus : 
 
 8 
 
 3. CQOO 
 
 
 
 12 
 
 5-437500 
 
 
 453125 Ans. 
 
 To reduce a decimal to its equivalent in terms of lower denominations. Rule. 
 Multiply the given decimal by the number of parts in the next less denomi- 
 nation, and point off from the product as many figures to the right hand as 
 there are in the given decimal ; then multiply the figures pointed off by the 
 number of parts in the next lower denomination, and point off as before, and 
 so proceed to the end ; then the several figures pointed off to the left will be 
 the answer. 
 
 Example, What is the expression in inches of 0-390625 feet? 
 
 Feet 0-390625 
 
 12 inches in a foot. 
 
 Inches 4-687500 
 
 8 eighths in an inch. 
 
 Eighths 5 5000 
 
 2 sixteenths in an eighth. 
 
 Sixteenth i-o 
 
 Ans., 4 inches, and $. 
 
 Another example. What is the expression, in fractions of an inch of 
 0-6875 inches? 
 
 Inches 0-6875 
 
 8 eignths in an inch. 
 
 Eighths 5-5000 
 
 2 sixteenths in an eighth. 
 
 Sixteenth i-o 
 
 Ans., f and ^ 
 
TABLE OF CIRCLES. 
 
 (From Gregory's Mathematics.) 
 
 FROM this table may be found by inspection the area or circumference of a 
 circle of any diameter, and the side of a square equal to the area of any given 
 circle from i to 100 inches, feet, yards, miles, etc. If the given diameter is in 
 inches, the area, circumference, etc., set opposite, will be inches ; if in feet, 
 then feet, etc. 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 Side of 
 equal sq. 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 Side of 
 equal sq. 
 
 25 
 
 04908 
 
 78539 
 
 22155 
 
 75 
 
 90-7625? 
 
 33-7721-2 
 
 9-52693 
 
 5 
 
 19635 
 
 1-57079 
 
 44311 
 
 11- 
 
 95-03317 
 
 34-55751 
 
 9-74S1'.* 
 
 75| '44178 
 
 235619 
 
 66467 
 
 25 
 
 99-40195 
 
 35-34291 
 
 9-97005 
 
 ] 
 
 78539 
 
 3-14159 
 
 88622 
 
 5 
 
 103-86890 
 
 36-12331 
 
 10-19160 
 
 25 
 
 1-22718 
 
 3-92699 
 
 1-10778 
 
 75 
 
 108-43403 
 
 36-91371 
 
 10-41316 
 
 5 
 
 1-76714 
 
 4-71238 
 
 1-32934 
 
 12- 
 
 11309733 
 
 37-69911 
 
 10-63472 
 
 75 
 
 2-40528 
 
 5-49778 
 
 1-55089 
 
 25 
 
 117-85831 
 
 3848451 
 
 10-85627 
 
 2- 
 
 3-14159 
 
 6-28318 
 
 1-77245 
 
 5 
 
 122-71846 
 
 3926990 
 
 11-07733 
 
 25 
 
 3-9760? 
 
 7-06858 
 
 1-99401 
 
 75 
 
 127-676281 40-05530 
 
 11-29939 
 
 5 
 
 4-90873 
 
 7-85393 
 
 2-21556 
 
 13- 
 
 132-73228 
 
 40-84070 
 
 11-5-2095 
 
 75 
 
 5-93957 
 
 8-63937 
 
 2-43712 
 
 25 
 
 137-88646 
 
 41-62610 
 
 11-7425;) 
 
 3- 
 
 7-068581 942477 
 
 2-65868 
 
 -.-) 
 
 14313881 
 
 42-41150 
 
 11-96406 
 
 25 
 
 8-29576 
 
 10-21017 
 
 2-88023 
 
 75 
 
 148-48934 
 
 43-19689 
 
 12-1856-2 
 
 5 
 
 9-62112 
 
 10-99557 
 
 3-10179 
 
 14- 
 
 153-93804 
 
 43-98229 
 
 12-40717 
 
 75 
 
 11-044661 11-78097 
 
 3-32335 
 
 25 
 
 159-48491 
 
 44-76769 
 
 12-62S73 
 
 4- 
 
 12-56637 12-56637 3-54490 
 
 5 
 
 165-12996 
 
 45-55309 
 
 12-85029 
 
 25 
 
 14-186251 13-35176J 3-76646! 
 
 75 
 
 170-87318 
 
 46-33849J 13-07184 
 
 5 
 
 15-90431 
 
 14-13716| 3-98802 
 
 15- 
 
 176-71458 
 
 47-12338! 13-21)340 
 
 75 
 
 17-72054 
 
 14-922561 4-20957! 
 
 25 
 
 182-65416 
 
 47-909281 13-51496 
 
 5- 
 
 19-63195 
 
 15-70796 4-431 13j 
 
 -. r > 
 
 188-69190 
 
 48-69468 13-73651 
 
 25 
 
 21-64753 
 
 16-49336 4-65269J 
 
 75 
 
 194-82783 
 
 49-48003 13-95307 
 
 5 
 
 23-75829 
 
 17-27875 
 
 4-87424 
 
 IS- 
 
 201-06192 
 
 50-26548 14-17963 
 
 75 
 
 25-96722 
 
 18-06415 
 
 5 -095 WO, 
 
 25 
 
 207-:W4'20 
 
 51-05088: 14-101 IS 
 
 6- 
 
 28-27433 
 
 18-84955 5-31736 
 
 5 
 
 213-82464 
 
 51-83027 14-62274 
 
 25 
 
 30-67961 
 
 19-63495! 5-53891 
 
 75 
 
 220-35327 
 
 5262167 
 
 14-84430 
 
 5 
 
 33-18307 
 
 20-42035 5-76047 17- 
 
 226-98006 
 
 53-40707 
 
 15-065-5 
 
 75 
 
 35-78470 
 
 21-20575. 5-9S2()3i' -25 
 
 233-70504 
 
 54-19247 
 
 15-28741 
 
 7- 
 
 33-48456 
 
 21-99114 
 
 6-20358! ! -5 
 
 240-52818 
 
 54-97787 15-50897 
 
 25 
 
 41-28249 
 
 27-77654 
 
 6-42514 j -75 
 
 217-44950 
 
 55763261 15-73052 
 
 .K 
 
 44-17H61 
 
 23-56194 
 
 6-646701: 18- 
 
 254-46900 
 
 56-54866! 15-9.V20S 
 
 75 
 
 47-17297 
 
 24-34734 
 
 6-86825 
 
 25 
 
 261-58667 
 
 57-33406 
 
 16-17361 
 
 8- 
 
 50-26518 
 
 25-13274 
 
 7-08981 
 
 5 
 
 26880252 
 
 53-11946 
 
 1639519 
 
 25 
 
 53-45616 
 
 25-91813 
 
 7-31137 
 
 75 27()-llC,54 
 
 58-90486 
 
 16-61675 
 
 5 
 
 5S-74501 
 
 26-70353 
 
 7-53292 
 
 19- 2-<3-52873 
 
 59-69026 
 
 16-8:KH 
 
 75 
 
 60-13204 
 
 27-48893 7754 18 
 
 25 291-03910 
 
 6047565 
 
 17-05986 
 
 9- 
 
 63617-2") 
 
 28-27433 7-976J4 
 
 5 i 298-64765 
 
 61-26105 
 
 17-28 U2 
 
 25 
 
 07-20063 
 
 29-0.'>97* 8-197.')9 
 
 75 
 
 300-:{513? 
 
 62-04645 
 
 17-5;-2'J8 
 
 5 
 
 70-83218 
 
 29-84513 8-41915 
 
 20- 
 
 314-15926 
 
 62-83185 
 
 17-72453 
 
 75 
 
 74-66191 
 
 30-63052 8-64071 
 
 "J.-> 322-00-233 
 
 6361725 
 
 17-94609 
 
 10- 
 
 78-53981 
 
 31-41592 886226 
 
 5 
 
 330-06357 
 
 64-40264 
 
 18-16765 
 
 25 
 
 82-51589 
 
 38*01321 9-08382 
 
 75 
 
 338-16-299 
 
 65-18804 
 
 18-38920 
 
 5 86-59014 
 
 32-986721 9-3i):>33 21- I 346-36059 
 
 6 j 97341 
 
 18-61076 
 
650 
 
 APPENDIX. 
 
 )iam. 
 
 Area. 
 
 Circam. 
 
 Side of 
 equal sq. 
 
 Diam. 
 
 Area. 
 
 Side of 
 Circum. enual sq. 
 
 21-25 
 
 354-65635 
 
 66-75834 
 
 1883232 
 
 38- 
 
 1134-114941 119-38052! 3367662 
 
 5 
 
 363-05030 
 
 67-54424 
 
 19-05337 
 
 25 
 
 1149-086601 120- 16591 i 33 893 17 
 
 75 
 
 371-54241 
 
 68-32964 
 
 19-27543 
 
 5 
 
 1164-15642 
 
 120-95131 34- IS 973 
 
 22- 380-13271 
 
 69-11503 
 
 19-49699 
 
 75 
 
 1179-32442 121-73671 3434129 
 
 25 1 388-82117 
 
 69-90043 
 
 19-71854 
 
 39- 
 
 1194-59060 122-52211 
 
 34-56285 
 
 5 
 
 337-60782! 70-68583 
 
 19-94010 
 
 25 
 
 1209-95495 
 
 123-30751 
 
 34-78440 
 
 75 
 
 406-49263 
 
 71-47123 
 
 20-16166 
 
 5 
 
 1225-41748 124-09290 
 
 35-00596 
 
 23- 
 
 415-475^2 
 
 72-25663 
 
 20-38321 
 
 75 
 
 1210-97818 
 
 124-878301 35-22752 
 
 25 
 
 424-55679 
 
 73-04202 
 
 20-60477 
 
 40- 
 
 1255-63704! 125-663701 35 44907 
 
 5 
 
 433-736131 73-82742 
 
 20-82633 
 
 25 
 
 1272-39411 
 
 126-449101 35-67063T 
 
 75 
 
 443-01365| 74-61282 
 
 21-04788 
 
 5 
 
 1288-24933 
 
 127-23450 35-89219 
 
 24- 
 
 452-38934J 75-39822 
 
 21-26944 
 
 75 
 
 : 1304-20273 
 
 128-01990 36-11374 
 
 25 
 
 461-86320 
 
 76-18362 
 
 21-49100 
 
 41- 
 
 1320-25431 123-80529J 36-33530 
 
 5 
 
 471-43524 
 
 76-96902 
 
 21-71205 
 
 25 
 
 1336-40406 
 
 129-59069 36-55686 
 
 75 
 
 481-105461 77-75441 
 
 2193411 
 
 5 
 
 1352-65198 
 
 1 30-3760') 36-77841 
 
 25- 
 
 490-87385! 78-53981 
 
 22-15567 
 
 75 
 
 1368-99808 
 
 131-16149 36-99997 
 
 25 
 
 500-740411 79-32521 
 
 22-37722 
 
 42- 
 
 1385-44236 
 
 131-94689 37-2*2153 
 
 5 
 
 . 510-70515! 80-11061 
 
 22-59878 
 
 25 
 
 1401-98480 
 
 13273228 37-44308 
 
 75 
 
 520-76306! 80-89601 
 
 22-82034 
 
 5 
 
 1418-625431 133-51768! 3766464 
 
 26- 
 
 530-92915 81-68140 
 
 23-04190 
 
 75 
 
 1435-36423 
 
 134-303081 37-88620 
 
 25 
 
 541-18842 
 
 82-46680 
 
 23-26345 
 
 4o- 
 
 1452-20120 
 
 135-08348 38-10775 
 
 5 
 
 551-54586 
 
 83-25220 
 
 23-48501: 
 
 25 
 
 1469-13635 
 
 135-87338 38-32931 
 
 75 
 
 562-00147 
 
 84-03760 
 
 2370657 
 
 5 
 
 1486-16967 
 
 136-65928! 38-55087' 
 
 27- 
 
 572-55526 
 
 84-82300 
 
 23-92812; 
 
 75 
 
 1503-30117 
 
 137-44467J 33-77242 
 
 25 
 
 583-20722 
 
 85-60839 
 
 24-14968 
 
 44- 
 
 1520-53084 
 
 138-83007 38-993^8 
 
 5 
 
 593-95736 
 
 86-39379 
 
 24-37124 
 
 25 
 
 1537-85869 
 
 1P9-01547J 39-21554 
 
 75 
 
 604-80567 
 
 87-17919 
 
 24-59279; 
 
 5 
 
 1556-23471 139-80087! 39-43709 
 
 28- 
 
 615-75216 
 
 87-96459 
 
 24-81435! 
 
 75 
 
 1572-80890 
 
 140-58627 
 
 3965865 
 
 25 
 
 626-79682 
 
 88-74999 
 
 25-03591 
 
 45- 
 
 152043128 
 
 141-37166 
 
 39-88021 
 
 5 637-93965 
 
 89-53539 
 
 25-25746 
 
 25 
 
 1608-15182 
 
 142-15706 40-10176 
 
 75 
 
 649-18066 
 
 90-32078 
 
 25-47'J02 
 
 5 
 
 1625-97054 
 
 142-94246! 40-32332 
 
 29- 
 
 66051985 
 
 9M0613 
 
 25-70058 
 
 75 
 
 164388744 
 
 143-727861 4054488 
 
 25 
 
 671-95721 
 
 9 39153 
 
 25-92213 
 
 46- 
 
 1661-90251 14451326! 40-76643 
 
 5 
 
 683-49275 
 
 92-67698 
 
 26-14359 
 
 25 1680-01575 145'29866l 40-98799 
 
 75 
 
 695-12646 
 
 93-46233 
 
 26-36525 
 
 5 1698-22717 
 
 146-08405 41-20955 
 
 30- 
 
 706-85834 
 
 94-24777 
 
 26-58680 
 
 75 1716-53677 
 
 146-86945 41-43110 
 
 25 
 
 71868840 
 
 9503317 
 
 26-80836 
 
 47- 1 1734-94454 
 
 147-65485 41-65266 
 
 5 
 
 730-61664 
 
 95-81857 
 
 27-02992 
 
 251 1753-45048 
 
 148-44025! 4187422 
 
 75 
 
 74264305 
 
 96-603971 27-25147; 
 
 5 
 
 1772-05460 
 
 149-22565 
 
 42-09577 
 
 31- 
 
 751-76763 97-38937; 27-47303 i -75 1790-75639 
 
 150-01104 
 
 4231733 
 
 25 
 
 766-99039" 98-17477 
 
 27-69459! 48- i 180955736 
 
 150-79644 
 
 42-53889 
 
 5 
 
 779-31132 
 
 9896016 
 
 27-91614 ; -25 '' 1828-45601 
 
 151-58184 
 
 42-76044 
 
 75 
 
 791-73043 
 
 99-74556 
 
 28-13770 
 
 5 1847-45282 
 
 152-36724 
 
 42-98200 
 
 32- 
 
 804-24771 
 
 100-53096 
 
 2S-M3926; -75 1866-54782 153-15264 
 
 43-20356 
 
 25 
 
 816-86317 
 
 101-31636 
 
 2M/59081I 49-' 1885-74099 153'93804 
 
 4342511 
 
 5 
 
 829-57631 
 
 102-10176 
 
 28-h0237 ; -25 ! 1905-83233 154-72343 
 
 43-64667 
 
 75 
 
 842-38861 
 
 102-887151 2902393 
 
 5 1924-42184 
 
 155-50883 43-86823 
 
 *33- 
 
 855-29859 
 
 10367255 
 
 29-24548 
 
 75 
 
 1943-90954 
 
 156-29423 44-08978 
 
 25 
 
 868-30675 
 
 104-45795 
 
 29-46704! 
 
 50- 
 
 1963-49540 
 
 157-07963] 44-31134 
 
 5 
 
 881-41308 
 
 105-24335 
 
 29-68860 
 
 25 
 
 1983-17944 
 
 157-96503] 44-53290 
 
 75 
 
 89461759 
 
 106-028751 29-91015 
 
 5 
 
 2002-96166 
 
 153-65042 44-75445 
 
 34- 
 
 907-92027 106-81415 
 
 30-13171 
 
 75! 2022-84205 
 
 159-43582! 44-97601 
 
 25 
 
 921-321131 107-59954 
 
 30-35327 
 
 51- 
 
 2042-82062 
 
 160-22122 
 
 45-19757 
 
 5 
 
 934-82016 
 
 308-33494 
 
 30-57482 
 
 25 
 
 2062-89736 
 
 161-00662 
 
 45-41912 
 
 75 
 
 94841736 
 
 109-17034 
 
 30-796'*8 
 
 5 ! 2083-07227 
 
 161-79202 
 
 45-64068 
 
 35- 
 
 962-11275 
 
 109-95574 
 
 31-01794J ! -75 1 210334536 
 
 162-57741 
 
 43-85224 
 
 25 
 
 975-90630 
 
 110-74114 
 
 31-23949 52- 1 2123-71663 
 
 163-36281 
 
 46-08380 
 
 5 
 
 989-79803 
 
 111-52653! 31-46105 
 
 25| 2144-18607 
 
 164-14821 
 
 46-30535 
 
 75 
 
 1003-78794 
 
 112311931 31-68261 
 
 5 1 2164-75368 
 
 184-93361 
 
 46 52691 
 
 36 
 
 1017-87601 
 
 11309733 
 
 31-90416 
 
 75! 2185-41947 
 
 165-71901 
 
 46-74847 
 
 25 
 
 1032-06227 
 
 113-88-273 
 
 32-12572 
 
 53- 2206-18344 
 
 166-50441 
 
 46-97002 
 
 5 
 
 75 
 
 1046-34670 11466813 
 1060-729301 115-45353 
 
 32-34728 
 
 32-56383 
 
 25 2227-04557 
 5 2248-00589 
 
 167-2-8980 
 168-07520 
 
 47-19158 
 47-41314 
 
 37- 
 
 1075-21008 
 
 116-23892 
 
 32-7903J 
 
 75 
 
 2269-06433 
 
 168-86060 
 
 47-63469 
 
 25 
 
 1089-78903 
 
 117-024 3-2 
 
 33-01195 
 
 54- 
 
 2290-22104 
 
 169-64600 
 
 47-85625 
 
 5 
 
 1104-46616 
 
 117-80972 
 
 33-23350 
 
 25: 2311-475^8 
 
 170-43140 
 
 48-07781 
 
 75 
 
 1119-2414 1 - 118-59572 
 
 33-45506 
 
 5 ! 2332-82889 171-21679 48-29936 
 
TABLE OF CIRCLES. 
 
 6 5 I 
 
 
 Side of 
 
 
 &<leo/ ' 
 
 Du.ni. 
 
 Arciu 
 
 Circuir,, 
 
 equal sq. Diam. Area. 
 
 Circum. 
 
 equal !. 
 
 "sflb 
 
 2354-2800S 172-00219 
 
 "~4~8l>2092 
 
 71-5 
 
 4015-1517f> 
 
 ~1^^23^! ""63-36522 
 
 55- 
 
 2375-82344 172-78759 
 
 48-74248 
 
 75 
 
 404327833 
 
 225-40927 i 63-58678 
 
 25 
 
 2397-47698! 173-57299 
 
 48-96403 
 
 72- 
 
 4071-50407 
 
 226-19467 
 
 63-80333 
 
 5 2419-22269 174 -3583 J 
 
 49-18559 
 
 25 
 
 4099-82750 
 
 226-98006 
 
 64-02989 
 
 75 2141-0665?! 175-14379 
 
 49-40715 
 
 5 
 
 4128-24909 
 
 227-76546 
 
 64-25145 
 
 56- 
 
 246300864! 175-92918 
 
 49-62870 
 
 75 
 
 4156-76886 
 
 22855086 
 
 64-47300 
 
 25 
 
 2185-04887' 17<r7145Sl 49-85026 
 
 73- 
 
 4185-38681 
 
 229-33626 
 
 64-69456 
 
 5 
 
 2507-18728 177-49998 5007182 
 
 25 
 
 4214-10293 
 
 230-12166 
 
 64-91612 
 
 75 2520-452337 i 178-28538! 50-29337 
 
 5 
 
 4242-91722 
 
 230-90706 
 
 65-13767 
 
 57- 2551-75363I 179-07078 50-51493 
 
 75 
 
 4271-82969 
 
 231-69245 
 
 65-35923 
 
 25| 2574-19156 179-85617 
 
 50-73649 
 
 74- 
 
 4300-84034 
 
 232-47785 
 
 65-58079 
 
 5 
 
 2596-72267 180-64157 
 
 . 50-95804 
 
 25 
 
 4329-94916 
 
 23326325 
 
 65-80234 
 
 75 
 
 2619-35196 181-426b7| 51-17960 
 
 5 
 
 4359-15615 
 
 234-04865 
 
 66-02390 
 
 58- 
 
 264207942! 18221237 51-40116 
 
 75 
 
 4388-46132 
 
 23483405 
 
 66-24510 
 
 25| 26(54 90505; 182-99777 
 
 51-62271 
 
 75- 
 
 4417-86466 
 
 235-61944 
 
 66-46701 
 
 5 
 
 2687-82836 183-78317 
 
 51 84427 
 
 25 
 
 4447-36618 
 
 236-40484 
 
 66-6885? 
 
 75 
 
 2710-85084 - 184-56856 
 
 52-06583 
 
 5 
 
 4476-965S8 
 
 237-19024 
 
 66-91043 
 
 59- 
 
 2733-97100 185-353 J6 
 
 52-28738 
 
 75 
 
 4506-66374 
 
 237-97f)64 
 
 67-13168 
 
 25 
 
 275718933 18613936 
 
 52-50894 
 
 76- 
 
 4536-45979 
 
 238-76104 
 
 67-3532 1 
 
 5 
 
 2780-50584 186-92476 
 
 52-73D50 
 
 25 
 
 456635400 
 
 239-54643 
 
 67-57480 
 
 75 
 
 280392053 187-71016 
 
 52-95205 
 
 5 
 
 45D6 34640 
 
 240-33183 
 
 67-79635 
 
 60- 
 
 2327-43338 188-49555 53-17364 
 
 75 
 
 462643696 
 
 241-11723 
 
 6301791 
 
 25 
 
 2851-04442 189-28095 
 
 5339517 
 
 77- 
 
 465562571 
 
 241-90263 
 
 68-23J47 
 
 5 
 
 2874-75362 193-06635 
 
 5361672! 
 
 25 
 
 468691262 
 
 242-6H8U3 
 
 68-46102 
 
 75 
 
 2898-56101) 193-85175 
 
 53-83328 ! 
 
 5 
 
 471729771 
 
 243-47343 
 
 68-68258 
 
 61 
 
 2922-46656 
 
 191-63715 
 
 54-05H84! 
 
 "75 
 
 4747-78098 
 
 24425382 
 
 68-90414 
 
 25 
 
 2946-47029 
 
 192-42255 
 
 5428139 
 
 78- 
 
 477836242 
 
 245 04422 
 
 63-12570 
 
 5 
 
 2970-57220 
 
 193-20794 
 
 5451)295 
 
 25 
 
 4809-04204 
 
 24582962 
 
 6931725 
 
 75 
 
 2994-77228 
 
 193-99334 
 
 54-724511 
 
 5 
 
 4839-81983 
 
 24661502 69-56881 
 
 62- 
 
 3019-07054 
 
 194-77874 
 
 54<MG','6 
 
 75 
 
 4870-79579 
 
 247-40042 69-79037 
 
 25 
 
 304346697 
 
 195-56414 
 
 55-167621 
 
 79- 
 
 4901-66993 
 
 248-18591 
 
 70-01192 
 
 5 
 
 3J67- ( J6I57 
 
 196-34954 
 
 55-33918 
 
 25 
 
 4;>32-74225 
 
 248-97121 
 
 70-23318 
 
 75 
 
 309255135 
 
 197-13493 
 
 5561073 
 
 5 
 
 4963-91274 
 
 24975661 
 
 70-45504 
 
 63 
 
 3117-21531 
 
 197-92033 55-83229 
 
 75 
 
 4995-18140 
 
 250-34201 
 
 70-67659 
 
 25 
 
 314203444 
 
 198-70573 55-053-5 
 
 80- 
 
 5026-54824 
 
 251-32741 
 
 70-39815 
 
 5 
 
 3166-92174 
 
 199-49113 56-27510 
 
 25 
 
 505801325 
 
 252-11281 
 
 71-11971 
 
 75 
 
 3191-90722 
 
 200-27653 
 
 56-49698 
 
 5 
 
 503957644: 
 
 252-89820 
 
 71-34126 
 
 64- 
 
 3*16-99087 
 
 201-061^2 
 
 5671852 
 
 75 
 
 5 121 23781 
 
 25368360 
 
 71-56282 
 
 25 
 
 3242-17270 
 
 201-84732 56-9100? 
 
 81- 
 
 5152-99735 
 
 254*46900 
 
 71-78433 
 
 5 
 
 326745270 
 
 20263272 57-16163 
 
 25 
 
 5184-85506 
 
 255-25440 
 
 72-00593 
 
 ~5 
 
 3292 83088 
 
 203-41812 57-3 S3 19 
 
 5 
 
 5216-SI<t'.l.~> 
 
 !2^6-()3JSO 
 
 72 -22749 
 
 ; 35- 
 
 3318-30784 
 
 204-20352 57-60475 
 
 75 
 
 5248-86501; 
 
 25682579 
 
 72-14905 
 
 25 
 
 334388176 
 
 204-98892 57-8263<)| 
 
 82- 
 
 5281-01725 
 
 257-61059 
 
 2-67060 
 
 5 
 
 3369-55447 
 
 20577431 
 
 5804786' 
 
 25 
 
 531326766 
 
 258-3i)599 
 
 72-89216 
 
 75 
 
 339532534 
 
 206-55971 
 
 53-26942 
 
 5 
 
 5345-r.ir.-ji 
 
 259-18139 
 
 7311372 
 
 66- 
 
 3421 19439 
 
 207-34511 
 
 5349097 
 
 75 
 
 5378-06301 
 
 259-96679 
 
 ?:<:52? 
 
 25 
 
 344716162 
 
 203-130511 53-712:-):! 
 
 83- 
 
 541060794 
 
 260-75219 
 
 ? 3 55633 
 
 5 
 
 3473 22702 
 
 208-915911 58-93 lO'.i 
 
 25 
 
 :m-2f>K!:i 
 
 261-53753 
 
 73 77839 
 
 75 
 
 349939060 
 
 209-7013!) 5.H:V>!il 
 
 5 
 
 r.l?.Vii9234 
 
 262-32298 
 
 73*99994 
 
 67- 
 
 3525-65235 
 
 210 48670 
 
 59-37720 
 
 75 
 
 5508-83180 
 
 263-10338 
 
 74-22150 
 
 25 
 
 3552-01228 
 
 211-27210 
 
 595;S76 
 
 84- 
 
 5.') 1 1 ?tV.t 1 1 
 
 263-89378 
 
 71:14306 
 
 5 
 
 3573-47033 
 
 212-05750 
 
 59-821)31 
 
 25 
 
 55?480f>2:> 
 
 264-67918 
 
 74-66461 
 
 *T5 
 
 3505 026i55 
 
 212-84290 60-04 I*? 
 
 5 
 
 5607-US923 
 
 265-46457 
 
 74-8861? 
 
 68- 
 
 3631-68110 
 
 2i3-r,->>3o r,o-2i;:m 
 
 75 
 
 5641-17I3J 
 
 266-24997 
 
 75-10773 
 
 25 
 
 9658-43373 
 
 214-4 Kiy CO-l-il'.tS 
 
 85- 
 
 ."67 1-50 173 
 
 267-03537 
 
 75'32:'-> 
 
 5 
 
 S68528453 
 
 21.")- 19.109 60-70C,;:! 
 
 25 
 
 5707-y;{o-j:j 
 
 2G7-82077 
 
 75-55: iM 
 
 75 
 
 371!} 23350 
 
 2J598449 60-92SIU 
 
 5 
 
 5741 1 :,r,. -j 
 
 268-60617 
 
 75-772 li 
 
 69- 
 
 3^39-28065 
 
 216761IS9 C.l-ll.G.") 
 
 75 
 
 5775-08178 
 
 269-3915? 
 
 75-99395 
 
 25 
 
 '576 C -42597 
 
 217-f)5;-.29 61-37121 
 
 86- 
 
 :.s(is-H0481. 
 
 270-176% 7621551 
 
 5 
 
 379JJ-66947 
 
 21834068 
 
 61-59277 
 
 25 
 
 5Sl262(i<)2 
 
 270-96236 764370? 
 
 75 
 
 332 '01 115 
 
 21. 
 
 61-81132 
 
 5 
 
 5876-54540 
 
 271-74770 
 
 76-65362 
 
 70 
 
 :MS-|.-,KM) vMH-'jiiis 1 62-03588 
 
 75 
 
 5.nor,r,29r. 
 
 272-53316 
 
 76-88018 
 
 25 
 
 3S7:>-<J3902j 2-20-tVjr.8S 62"J:>71I 
 
 87- 
 
 :.'J 1 1 67869 
 
 27331856 
 
 77- 10174 
 
 5 
 
 3i)03-62522 2 i 1-43228 62-478'J9 
 
 25 
 
 5978-892C.O 
 
 274-10395 
 
 77-::*2'. 
 
 75 
 
 3931-35959 MB-JJCTW 62-70055 
 
 5 
 
 6013-20468 
 
 274-88935 
 
 77*54485 
 
 71 
 
 3959-19214' J-.'^or.WT r>2'J2211 
 
 75 
 
 604761 J91 
 
 27567475 
 
 7776641 
 
 2? 
 
 3987- 12286 223 &i i47 03 1 136l> 
 
 88- 
 
 6082-12337, 
 
 276-16015 
 
 77-98796 
 
6 5 2 
 
 APPENDIX. 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 Side of 
 equal sq. 
 
 Dinm. 
 
 Area. 
 
 Circum. 
 
 Side ot 
 equal sq. 
 
 88-25 
 
 6116-729931 277-24555 
 
 78-20952 
 
 94-25 
 
 6976-74097 
 
 2-6-09510 
 
 83-52688 
 
 5 
 
 6151-43476 27803094 
 
 78-43103 
 
 5 
 
 7012-80194 
 
 296-88050 
 
 83-74844 
 
 75 
 
 6186-23772 278-81634 
 
 7865263 
 
 75 
 
 7050-96109 
 
 297-66590 
 
 83-97000 
 
 89- 
 
 6-22M3385 279-60174 
 
 78-87419 
 
 95- 
 
 7083-21842 
 
 298-45130 
 
 84-19155 
 
 35! 6256-13S15I 23033714 
 
 79-09575 
 
 25 
 
 7125-57992 
 
 299-23070 
 
 84-4131 1 
 
 5 
 
 6291-23563 
 
 231-17254 
 
 79-31730 
 
 5 
 
 7163-02759 
 
 300-02209 
 
 84-63467 
 
 75 
 
 63-26-43129 
 
 281-95794 
 
 79-53386 
 
 75 
 
 7200-57944 
 
 300-80749 
 
 84-85622 
 
 90- 
 
 6361-7-2512 
 
 28-2-74333 
 
 79-76042 
 
 96- 
 
 7233-22947 
 
 3U 1-59239 
 
 85-07778 
 
 25 
 
 6397-11712 
 
 233-52873 
 
 79-98198 
 
 25 
 
 7275-97767 
 
 302-37829 
 
 85-291)34 
 
 5 
 
 6432-60730 
 
 284-31413 
 
 80-20353 
 
 5 
 
 7313-82404 
 
 303-16369 85-52039 
 
 75 
 
 6463-19586 
 
 285-09953 
 
 80-42509 
 
 75) 7351-76859 
 
 303-94908 
 
 85-74245 
 
 91- 
 
 6503-83219 
 
 285-83493 
 
 80-64669 
 
 97- 
 
 7389-81131 
 
 304-73448 
 
 85-96401 
 
 .25 
 
 653J-66639 
 
 286-67032 
 
 80-85820 
 
 25 
 
 7427-95221 
 
 305-51988 
 
 86-18556 
 
 5 
 
 6575-54977 
 
 237-45572 
 
 81-08976 
 
 5 
 
 7466-19129 
 
 306-30523 
 
 86-40712 
 
 75 
 
 6611-53382 
 
 288-24112 
 
 81-31132 
 
 75 
 
 7504-52853 
 
 307-09068 
 
 86-62868 
 
 92- 
 
 6647-61005 
 
 239-02652 
 
 81-53287 
 
 98- 
 
 7542-96396 
 
 307-87608 
 
 86-85023 
 
 25 
 
 6683-78745 
 
 289-31192 
 
 81-75443 
 
 25 
 
 7581-49755 
 
 - 308-66147 
 
 87-0717t' 
 
 5 
 
 6720-06303 
 
 2:)0-5 1 J7:)'2 
 
 81-975^'J 
 
 5 
 
 7620-12933 
 
 309-44637 
 
 87-29335 
 
 75 
 
 6756-43678 
 
 291-33271 
 
 82-VJ754 
 
 75 
 
 7653-85927 
 
 310-23227 
 
 87-51490 
 
 93- 
 
 6792-90871 
 
 292-16811 
 
 82-41910 
 
 99- 
 
 7697-68739 
 
 311-01767 
 
 87-73646 
 
 25 
 
 6829-47831 
 
 292-95351 
 
 82-04066 
 
 25 
 
 7736-61369 
 
 311-80307 
 
 87-95802 
 
 5 
 
 6866-14709 
 
 293-7339} 
 
 82-86-221 
 
 5 
 
 7775-63816 
 
 312-58346 
 
 88-17957 
 
 75 
 
 6902-91354 
 
 294-52431 
 
 83-08377 
 
 75 
 
 7814-760311 313-37336 
 
 88-40113 
 
 94- 
 
 6939-77317 295-30970, 83 30533 jj 100- 
 
 7653-98163| 314-15926 
 
 88-62269 
 
 The following rules are for extending the use of the above table. 
 
 To find the area, circumference, or side of equal square, of a circle having a 
 diameter of more than 100 inches, feet, etc. Rule. Divide the given diameter by 
 a number that will give a quotient equal to some one of the diameters in the 
 table ; then the circumference or side of equal square, opposite that diameter, 
 multiplied by that divisor, or the area opposite that diameter, multiplied by 
 the square of the aforesaid divisor, will give the answer. 
 
 Example. What is the circumference of a circle whose diameter is 228 
 feet ? 228, divided by 3, gives 76, a diameter of the table, the circumference 
 of which is 238-761, therefore : 
 
 238-761 
 3 
 
 716 -283 feet. Ans. 
 
 Another example. What is the area of a circle having a diameter of 150 
 inches? 150, divided by 10, gives 15, one of the diameters in the table, the 
 area of which is 176-71458, therefore: 
 
 176-71458 
 
 100 = TO X IO 
 
 17,671-45800 inches. Ans. 
 
 To find the area, circumference, or side of equal square, of a circle having an. 
 intermediate diameter to those in the table. Rule. Multiply the given diameter 
 by a number that will give a product equal to some one of the diameters in 
 the table ; then the circumference or side of equal square opposite that diame- 
 ter, divided by that multiplier, or the area opposite that diameter divided by 
 the square of the aforesaid multiplier, will give the answer. 
 
CAPACITY OF WELLS, CISTERNS, ETC. 653 
 
 Example. What is the circumference of a circle whose diameter is 6J, or 
 6-125 inches? 6-125, multiplied by 2, gives 12-25, one of the diameters of the 
 table, whose circumference is 38-484, therefore : 
 
 2)38-484 
 19-242 inches. Ans. 
 
 Another example. Wh?t is the area of a circle, the diameter of which is 3-2 
 feet? 3-2, multiplied by 5, gives 16, and the area of 16 is 201-0619, therefore : 
 
 5 x 5 = 25)201-0619(8-0424 + feet. Ans. 
 200 
 
 106 
 
 100 
 
 61 
 50 
 
 Note. The diameter of a circle, multiplied by 3-14159, will give its cir- 
 cumference ; the square of the diameter, multiplied by -78539, will give its 
 area; and the diameter, multiplied by -88622, will give the side of a square 
 equal to the area of the circle. 
 
 TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, ETC. 
 
 The gallon of the State of New York, by an act passed April n, 1851, is required to conform to 
 the standard gallon of the United States government. This standard gallon contains 231 cubic 
 inches. In conformity with this standard the following table has been computed. 
 
 One foot in depth of a cistern of 
 
 3 feet diameter will contain ........................ 52 872 gallons. 
 
 3* " " ....................... 7I-965 
 
 4 " ........................ 93-995 " 
 
 4i " ........................ 118-963 
 
 5 " " ........................ 146-868 
 
 5i " " ........................ 177-710 
 
 6 " " ..................... 211-490 " 
 
 6 " " ....................... 2*8-207 
 
 7 " " ....................... 287-861 " 
 
 8 " " ....................... 375-982 
 
 9 ....................... 475-852 
 
 10 " " ............ : ............ 587-472 
 
 12 " " ....................... 845-959 
 
 Note. To reduce cubic feet to gallons, multiply by 7-48. The weight of a 
 gallon of water is 8-355 Ibs. To find the contents of a round cistern, multi- 
 ply the square of the diameter by the height, both in feet, and this product by 
 
 5-875- 
 
654 
 
 APPENDIX. 
 
 TABLE OF WEIGHTS. 
 
 MATERIALS USED IN THE CONSTRUCTION OR LOADING OF 
 
 BUILDINGS. 
 
 WEIGHTS PER CUBIC FOOT. 
 
 As per Barloiv, Gallier, Jlaswell, Jfurst, Rankine, Tredgold, Wood 
 and tJie AutJwr. 
 
 MATERIAL. 
 
 | 
 In 
 
 
 
 AVERAGE. 
 
 MATERIAL. 
 
 S 
 
 
 K 
 
 to 
 
 o 
 H 
 
 AVERAGE. 
 
 WOODS. 
 
 
 
 
 Mahogany, St. Domingo. . . . 
 
 45 
 
 65 
 49 
 
 55 
 41 
 
 
 41 
 
 51 
 
 46 
 
 Mulberry 
 
 35 
 
 55 
 
 45 
 
 Alder 
 
 35 
 
 51 
 
 38 
 
 Oak, Adriatic 
 
 
 
 62 
 
 
 49 
 
 51 
 
 5O 
 
 " Black Bog 
 
 60 
 
 66 
 
 63 
 
 Ash 
 
 41 
 
 57 
 
 49 
 
 " Canadian 
 
 
 
 54 
 
 Beech 
 
 
 
 46 
 
 
 
 
 47 
 
 Birch. 
 
 35 
 
 49 
 
 42 
 
 " English .. 
 
 S 
 
 7 
 
 54 
 
 Box . 
 
 59 
 
 65 
 
 62 
 
 Live 
 
 57 
 
 79 
 
 68 
 
 
 
 
 83 
 
 " Red 
 
 47 
 
 
 51 
 
 
 
 
 64 
 
 " White. 
 
 
 
 50 
 
 Cedar 
 
 27 
 
 35 
 
 31 
 
 Olive.. 
 
 
 
 58 
 
 
 47 
 
 
 52 
 
 
 
 
 44 
 
 " Palestine 
 u Virginia Red ...... 
 Cherry 
 
 30 
 32 
 
 . 38 
 46 
 
 34 
 40 
 39 
 
 Pear-tree . .' 
 Pine, Georgia (pitch) 
 '* Mar Forest 
 
 40 
 
 38 
 
 \ 
 
 42 
 
 48 
 43 
 
 Chestnut, Horse 
 
 29 
 
 
 35 
 
 " Memel and Riga 
 
 29 
 
 ** 
 
 32 
 
 Sweet 
 Cork . ... 
 
 27 
 
 55 
 
 41 
 15 
 
 " Red 
 " Scotch 
 
 27 
 
 
 37 
 39 
 
 Cypress . 
 
 27 
 
 
 34 
 
 " White 
 
 
 
 28 
 
 *' Spanish 
 Deal, Christiania. . 
 
 
 
 40 
 44 
 
 " Yellow 
 
 27 
 4 1 
 
 39 
 
 33 
 45 
 
 '* English 
 41 (Norway Spruce). 
 
 21 
 
 
 t9 
 7 
 
 Poplar 
 
 23 
 
 37 
 
 30 
 44 
 
 Dogwood 
 
 
 
 47 
 
 
 
 
 23 
 
 Ebony 
 
 69 
 
 8^ 
 
 76 
 
 
 
 
 45 
 
 Elder 
 
 
 
 43 
 
 
 
 
 30 
 
 Elm 
 
 31 
 
 59 
 
 46 
 
 Satinwood 
 
 e;5 
 
 
 57 
 
 
 
 
 27 
 
 
 
 oA 
 
 30 
 
 " (Red Pine) 
 
 3 
 
 
 37 
 
 
 36 
 
 
 38 
 
 " Riga 
 
 
 
 47 
 
 Teak 
 
 
 61 
 
 51 
 
 
 
 
 53 
 
 Tulip-tree 
 
 
 
 30 
 
 " ' Water 
 
 
 
 62 
 
 Vine. 
 
 
 81 
 
 so 
 
 
 
 
 37 
 
 Walnut Black 
 
 26 
 
 
 33 
 
 Hemlock 
 
 21 
 
 
 26 
 
 " White'. 
 
 
 eg 
 
 49 
 
 
 40 
 
 la 
 
 49 
 
 Whitewood 
 
 
 
 27 
 
 
 
 te 
 
 52 
 
 Yew 
 
 
 
 50 
 
 Larch . ... 
 
 31 
 
 
 33 
 
 
 
 
 
 " Red 
 
 31 
 
 
 43 
 
 METALS. 
 
 
 
 
 " White 
 
 
 
 23 
 
 Bismuth, Cast 
 
 
 
 614 
 
 
 41 
 
 8 3 
 
 62 
 
 
 487 
 
 
 5O6 
 
 
 41 
 
 
 -?6 
 
 
 
 
 544 
 
 
 
 
 57 
 
 k< Plate 
 
 q2 8 
 
 
 531 
 
 
 35 
 
 
 38 
 
 Bronze . 
 
 508 
 
 
 516 
 
 
 
 
 
 
 
 
 
WEIGHT OF MATERIALS. 
 
 655 
 
 TABLE OF WEIGHTS. (Continued.) 
 
 MATERIALS USED IN THE CONSTRUCTION OR LOADING OF 
 
 BUILDINGS. 
 
 WEIGHTS PER CUBIC FOOT. 
 
 As per Barlow, Gallier, Jfaswell, Hurst, Rankine, Tredgold, Wood 
 and the Author. 
 
 MATERIAL. 
 
 Z 
 
 1 
 
 o 
 H 
 
 AVERAGE. 
 
 MATERIAL. 
 
 I 
 
 t* 
 
 O 
 
 H 
 
 AVERAGE. 
 
 Copper Cast 
 
 537 
 
 549 
 
 543 
 
 
 06 
 
 u_ 
 
 1O4 
 
 
 
 
 556 
 
 dry 
 
 
 
 1OO 
 
 " Plate 
 
 
 
 644 
 
 44 in Cenfent . 
 
 
 
 112 
 
 Gold 
 
 
 
 1206 
 
 " in Mortar 
 
 
 
 1 1O 
 
 
 
 
 110S 
 
 Caen Stone 
 
 
 
 13O 
 
 
 
 
 509 
 
 
 
 
 81 
 
 
 475 
 
 487 
 
 481 
 
 " Roman, Cast. 
 
 
 
 100 1 
 
 Cast 
 
 434 
 
 474 
 
 454 
 
 
 
 
 
 " Malleable 
 " Wrought 
 
 
 486 
 
 475 
 
 480 
 
 equal parts.. 
 Chalk 
 
 116 
 
 
 113 
 
 14 * 
 
 1 Lead Cast 
 
 
 
 709 
 
 Clay 
 
 
 
 122 
 
 Er^lish Cast 
 
 
 
 717 
 
 " with Gravel 
 
 
 
 16O 
 
 " M ficd 
 
 
 
 713 
 
 
 
 
 90 
 
 
 
 
 851 
 
 
 ?6 
 
 102 
 
 83 
 
 " " 60 
 
 
 
 849 
 
 
 
 8i 
 
 79 
 
 " " 212 
 
 
 
 837 
 
 
 
 
 85 
 
 Nickel Cast 
 
 
 
 488 
 
 Coke 
 
 ^.6 
 
 62 
 
 54 I 
 
 
 
 
 453 
 
 Concrete, Cement 
 
 
 
 13O 
 
 
 
 
 975 
 
 
 
 
 1O6 
 
 
 
 
 1345 
 
 
 
 
 U o 
 
 n ' Rolled 
 
 
 
 1379 
 
 
 
 
 126 
 
 
 
 
 142 
 
 " with Gravel 
 
 
 
 126 
 
 
 
 
 636 
 
 
 
 
 25O 
 
 *' ' Pure Cast 
 
 
 
 655 
 
 Feldspar 
 
 
 
 16O 
 
 " " " Hammered 
 " Standard 
 
 
 ... 
 
 658 
 644 
 
 Flagging, Silver Gray. . . . 
 Flint 
 
 
 - - 
 
 185 
 163 
 
 Steel 
 
 <86 
 
 402 
 
 489 
 
 
 
 l6e 
 
 160 
 
 Tin Cast 
 
 !,6 
 
 1 < 
 
 462 
 
 " Flint 
 
 
 
 183 
 
 
 429 
 
 449 
 
 439 
 
 " Green 
 
 
 
 li; 
 
 Zinc, Cast 
 
 
 
 
 ' Plate 
 
 
 
 163 
 
 
 
 
 
 " White 
 
 167 
 
 181 
 
 174 
 
 STONES, EARTHS, ETC. 
 
 
 
 
 Granite 
 
 nR 
 
 
 Hi 5 
 
 
 
 
 
 
 
 
 161 
 
 
 x6c 
 
 180 
 
 173 
 
 
 
 
 166 
 
 
 
 
 156 
 
 " Guernsey 
 
 
 
 185 
 
 
 
 
 8O 
 
 
 
 
 166 
 
 
 
 
 277 
 
 Gnvel 
 
 9 
 
 12O 
 
 105 
 
 Basalt 
 
 
 187 
 
 
 
 
 
 134 
 
 
 
 J!-5 
 
 i { M 
 
 
 135 
 
 1^5 
 
 14O 
 
 t)atn i one..^.. 
 
 
 
 129 
 
 Lime Unslaked 
 
 
 
 52 
 
 r>f c r' 
 
 
 
 16O 
 
 
 11Q 
 
 
 169 
 
 Brick ' 
 
 8 5 
 
 
 102 
 
 Aubigne 
 
 
 
 146 
 
 
 
 
 138 
 
 
 
 
 162 
 
 ,i -T TJ h ' d 
 
 
 
 107 
 
 Marble 
 
 161 
 
 178 
 
 17O 
 
 " " Salmon 
 
 
 ' 
 
 1OO 
 
 
 
 
 166 
 
 
 
 
 105 
 
 
 
 
 17O 
 
 
 
 
 
 
 
 
 
656 
 
 APPENDIX. 
 
 TABLE OF WEIGHTS. (Continued^ 
 
 MATERIALS USED IN THE CONSTRUCTION OR LOADING OF 
 
 BUILDINGS. 
 
 WEIGHTS PER CUBIC FOOT. 
 
 As per BarloTV, Gallier, Haswell, Hurst, Rankine> Tredgold, Wood 
 and the Author. 
 
 MATERIAL. 
 
 1 
 
 S 
 
 AVERAGE. 
 
 MATERIAL. 
 
 6 
 
 N 
 
 CK 
 
 o 
 H 
 
 AVERAGE. 
 
 Marble Enstchester 
 
 167 
 i6 3 
 
 IOO 
 110 
 
 178 
 169 
 
 179 
 140 
 
 173 
 167 
 166 
 167 
 140 
 125 
 175 
 155 
 98 
 103 
 107 
 
 9 
 86 
 105 
 1OO 
 
 118 
 83 
 146 
 72 
 8O 
 180 
 175 
 147 
 56 
 165 
 165 
 124 
 112 
 1O5 
 105 
 123 
 1O5 
 97 
 172 
 126 
 144 
 133 
 142 
 134 
 141 
 134 
 162 
 150 
 
 Serpentine 
 " Chester Pa 
 
 
 ... 
 
 165 
 144 
 152 
 95 
 159 
 167 
 157 
 18O 
 135 
 151 
 14O 
 16O 
 14O 
 124 
 115 
 170 
 76 
 
 SS 
 49 
 59 
 62 
 26 
 2O 
 58 
 57 
 61 
 17 
 61 
 69 
 114 
 73 
 131 
 559 
 68 
 171 
 133 
 131 
 14 
 8O 
 62} 
 64 
 81 
 
 Egyptian 
 " French 
 
 
 
 
 Shingle 
 
 137 
 
 181 
 
 Marl 
 
 Slate 
 
 
 
 Mica 
 
 " Cornwall 
 
 
 
 
 109 
 
 118 
 
 " Welsh 
 
 
 
 
 87 
 
 83 
 
 Stone, Artificial 
 
 120 
 
 150 
 
 
 " Paving . ,. 
 
 
 Stone-work . 
 
 120 
 
 io 
 
 " Hair, incl. Lath and 
 Nails, per foot sup. 
 
 7 
 
 ii 
 
 " Rubble 
 Sulphur, Melted 
 
 
 
 
 
 
 
 
 " Sand 3 and Lime paste 2 
 " " 3 " " " 2 
 well beat together. . 
 Peat Hard 
 
 
 
 Trap Rock 
 
 
 
 
 
 
 MISCELLANEOUS. 
 
 
 
 Petrified Wood 
 
 
 
 Pitch 
 
 
 
 Plaster Cast 
 
 
 
 Bark, Peruvian 
 
 
 
 
 
 
 
 
 
 " ' Red 
 
 1 3 2 
 
 161 
 
 
 17 
 
 4 
 
 34 
 25 
 
 62 
 
 24 
 66 
 
 
 Charcoal 
 
 Pumice-stone 
 
 Cotton, baled. . . 
 Fat 
 
 
 
 
 
 
 
 
 52 
 
 JO 
 
 56 
 
 Rotten-stone 
 Sand, Coarse 
 
 92 
 90 
 118 
 
 ii8 
 
 120 
 128 
 
 Gutta-percha 
 Hay baled 
 
 India Rubber 
 
 ' Dry 
 
 
 Moist 
 
 
 
 
 
 Pit 
 
 92 
 
 101 
 
 
 
 
 Quartz 
 
 Red Lead 
 
 
 
 
 
 158 
 
 
 
 
 130 
 
 Rock Crystal 
 
 
 " Amherst, O. . . . 
 " Belleville, N. J... 
 Berea, O 
 " Dorchester, N. S 
 " Little Falls, N. J. 
 u Marietta, O. . . . 
 " Middletown, Ct.. 
 
 Salt 
 
 
 20 
 100 
 
 Saltpetre 
 Snow 
 
 "a 
 
 60 
 
 Water, Ram 
 " Sea 
 
 Whalebone 
 
 
 
 
 
 
INDEX. 
 
 Abscissas of Axes, Ellipse 484 
 
 Abutments, Bridges,. Strength of. 227 
 Abutments, Houses, Strength of.. 53 
 
 Acute Angle Denned 349, 544 
 
 Acute-angled Triangle Denned... 545 
 
 Acute or Lancet Arch 51 
 
 Algebra, Addition 398 
 
 Algebra, Application of 393 
 
 Algebra, Binomial, Multiplica- 
 tion of , 409 
 
 Algebra, Binomial, Square of a... 429 
 Algebra, Binomial, Squaring a. . . 410 
 
 Algebra Defined 392 
 
 Algebra, Denominator, Least 
 
 Common 404 
 
 Algebra, Division, the Quotient.. 419 
 Algebra, Division, Reduction.... 419 
 Algebra, Division, Reverse of 
 
 Multiplication 418 
 
 Algebra, Factors, Multiplication 
 
 of Two and Three 409 
 
 Algebra, Factors, Multiplication 
 
 of Three . 408 
 
 Algebra, Factors, Squaring Differ- 
 ence of Two 412 
 
 Algebra, Fractions Added and 
 
 Subtracted 403 
 
 Algebra, Fractions, Denominators 407 
 Algebra, Fractions Subtracted... 405 
 Algebra, Hypothenuse, Equality 
 
 of Squares on 416 
 
 Algebra, Letters, Customary Uses 
 
 of 396 
 
 Algebra, Logarithms Explained.. 425 
 Algebra, Logarithms, Examples in 426 
 Algebra, Multiplication, Graphical 408 
 
 Algebra, Progression, Arithmeti- 
 cal... 432 
 
 Algebra, Progression, Geometrical 435 
 
 Algebra, Proportion Essential 347 
 
 Algebra, Proportionals, Lever 
 
 Formula 421 
 
 Algebra, Quantities, Addition and 
 
 Subtraction 424 
 
 Algebra, Quantities, Division of. 424 
 Algebra, Quantities, Multiplica- 
 tion of 424 
 
 Algebra, Quantities with Negative 
 
 Exponents 423 
 
 Algebra, Quantity, Raising to any 
 
 Power 423 
 
 Algebra, Radicals, Extraction of.. 425 
 
 Algebra, Rules are General 394 
 
 Algebra, Rules, Useful Construc- 
 ting 394 
 
 Algebra, Signs 397 
 
 Algebra, Signs, Arithmetical Pro- 
 cess by 396 
 
 Algebra, 'Signs, Changed when 
 
 Subtracted .... 400 
 
 Algebra, Signs, Multiplication of 
 
 Plus and Minus. .. 415 
 
 Algebra, Squares on Right- Angled 
 
 Triangle 417 
 
 Algebra, Subtraction 398 
 
 Algebra, Sum and Difference, Pro- 
 duct of 413 
 
 Algebra, Symbols Chosen at Pleas- 
 ure 395 
 
 Algebra, Symbol, Transferring a.. 399 
 Algebra, Triangle, Squares on 
 Right-angled 417 
 
658 
 
 INDEX. 
 
 Alhambra, or Red House, Ancient 
 
 Palace of the n 
 
 Ancient Cities, Historical Ac- 
 counts of 6 
 
 Ancient Monuments, their Archi- 
 tects 6 
 
 Angle at Circumference of Circle. 358 
 
 Angle Defined 544 
 
 Angle to Bracket of Cornice, To 
 
 Obtain 343 
 
 Angle, To Measure a, Geometry.. 348 
 Angle rib to Polygonal Dome. . . . 223 
 Angle-rib, Shape of Polygonal 
 
 Domes 223 
 
 Amulet or Fillet, Classic Mould- 
 ing 323 
 
 Antae Cap, Modern Moulding. ... 334 
 
 Antique Columns, Forms of 48 
 
 Antiquity of Building 5 
 
 Arabian and Moorish Styles, An- 
 tiquities of ii 
 
 Araeostyle, Intercolumniations. . . 20 
 
 Arc of Circle Defined 547 
 
 Arc of Circle, Length, Rule for. . 475 
 
 Arc, Radius of, To Find 561 
 
 Arc, Versed Sine, To Find (Geom- 
 etry) 561 
 
 Arcade 52 
 
 Arcade of Arches, Resistance in. . 52 
 Arcade in Bridges, Strength of 
 
 Piers 52 
 
 Arch 50 
 
 Arch, Acute or Lancet 51 
 
 Arch, Archivolt in 52 
 
 Arch, Bridge, Pressure on 51 
 
 Arch, Building, Manner of 50 
 
 Arch, Catenary 51 
 
 Arch, Construction of 50 
 
 Arch, Definitions and Principles of 52 
 
 Arch, Extrados of 52 
 
 Arch, Form of 50 
 
 Arch, Formation in Bridges 51 
 
 Arch, Hooke's Theory of an 50 
 
 Arch, Horseshoe or Moorish 51 
 
 Arch, Impost in 52 
 
 Arch, Intrados of 52 
 
 Arch, Keystone, Position of 50 
 
 Arch, Lateral Thrust in . 52 
 
 Arch, Ogee 51 
 
 Arch, Rampant 51 
 
 Arch, Span of an 52 
 
 Arch, Spring in an 52 
 
 Arch, Stone Bridges 230 
 
 Arch-stones, Bridges, Jointing. .. 233 
 
 Arch, Strength of 50 
 
 Arch of Titus, Composite Order. . 28 
 
 Arch, Uses of 50 
 
 Arch, Voussoir in 52 
 
 Architect and Builder, Construc- 
 tion Necessary to 56 
 
 Architect, Derivation of the Word 5 
 
 Architects of Italy, I4th Century. 12 
 Architecture, Classic Mouldings 
 
 in 323 
 
 Architecture, Ecclesiastical, Origin 
 
 of 14 
 
 Architecture, Egyptian, Character 
 
 of 33 
 
 Architecture, Egyptian, Features 
 
 of 30 
 
 Architecture, English, Ccttage 
 
 Style 35 
 
 Architecture, English, Early n 
 
 Architecture, Grecian and Roman 8 
 
 Architecture, Grecian, History of. 6 
 Architecture, Hindoo, Character 
 
 of . 30 
 
 Architecture, Order, Three Princi- 
 pal parts of 14 
 
 Architecture, Principles of 44 
 
 Architecture, Roman, Ruins of... II 
 
 Architecture in Rome Defined. .. 7 
 
 Architecture, Result of Necessity. 13 
 
 Architrave Defined 15 
 
 Area of Circle, To Find 475 
 
 Area of Post, Rule for Finding. . . 90 
 
 Area of Round Post, Rule 90 
 
 Area of Surface, Sliding Rupture, 
 
 Rule 88 
 
 Arithmetical Progression (Alge- 
 bra) 432 
 
 Astragal, or Bead, Classic Mould- 
 ing 323 
 
 Athens, Parthenon, Columns of. . 48 
 Attic, a Small Order, Top of 
 
 Building 15 
 
INDEX. 
 
 659 
 
 Attic Story, Upper Story 15 
 
 Axes of Ellipse (Geometry) . . 585 
 
 Axiom Defined (Geometry) 348 
 
 Axis Defined 548 
 
 Balusters, Handrailing, Winding 
 Stairs , 310 
 
 Baluster, Platform Stairs, Position 
 of 250 
 
 Baluster in Round Rail, Winding 
 Stairs 313 
 
 Base, Shaft, and Capital Defined . 14 
 
 Bathing, Necessary Arrangements 
 
 for 45 
 
 Baths of Diocletian, Splendor of.. 27 
 Bead, or Astragal, Classic Mould- 
 ings 323 
 
 Beams, Bearings of, Rules for 
 
 Pressure 75 
 
 Beams, Breaking Weight on 74 
 
 Beams, Framed, Rules for Thick- 
 ness 130 
 
 Beams, Framed, Position of Mor- 
 tise 236 
 
 Beams, Headers Defined 130 
 
 Beams, Horizontal Thrust, Rule.s 
 
 for 72 
 
 Beams, Inclined, Effect of Weight 
 
 on 72 
 
 Beams, Load on, Effect of 74 
 
 Beams, Splicing 235 
 
 Beams, Tail, Defined 130 
 
 Beams, Trimmers or Carriage, De- 
 fined 130 
 
 Beams, Weight on, Proportion of. 130 
 Beams, White Pine, Table of 
 
 Weights 177 
 
 Beams, Wooden, Use of Limited. 154 
 
 Bearings for Girders 141 
 
 Binomials, Multiplication of (Al- 
 gebra) 409 
 
 Binomials, Square of (Algebra). . . 429 
 Binomials, Squaring (Algebra)... 410 
 
 Bisect an Angle (Geometry) 554 
 
 Bis'ect a Line (Geometry) 549 
 
 Blocking out Rail, Winding Stairs 301 
 Blondel's Method, Rise and Tread 
 in Stairs 242 
 
 Bottom Rail for Doors, Rule for 
 
 Width 316 
 
 Bow, Mr., On Economics and 
 
 Construction 166 
 
 Bowstring Girder, Cast -Iron, 
 
 Should not be Used 163 
 
 Brace, Length of, To Find (Geom- 
 etry) 579 
 
 Braces, Rafters, etc., To Find 
 
 Length 580 
 
 Braces in Roof, Rule for, Same as 
 
 Rafter 208 
 
 Breaking Weight Defined 84 
 
 Brick or Stone Buildings 37 
 
 Brick Walls, Modern 49 
 
 Bridge Abutments, Strength of.. . 227 
 
 Bridge Arches, Formation of 51 
 
 Bridge Arch-stones, Joints of .... 233 
 Bridges, Construction of Various. 223 
 Bridge, London, Age of Piles 
 
 under 229 
 
 Bridge Piers, Construction and 
 
 Sizes 228* 
 
 Bridge, Rib-built 224 
 
 Bridge, Rib, Construction of. . . . 225 
 Bridge, Rib, Framed, Construction 
 
 and Distance 226 
 
 Bridge, Rib, Radials of 226 
 
 Bridge, Rib, Table of Least Rise 
 
 in 224 
 
 Bridge, Rib, Rule for Area of 225 
 
 Bridge, Rib, Rule for Depth of. . . 226 
 
 Bridge, Roadway, Width of 227 
 
 Bridge, Stone, Arch Construction 230 
 Bridge, Stone, Arch-stones, Table 
 
 of Pressures on 230 
 
 Bridge, Stone, Arch, Centres for, 
 
 Bad Construction 229 
 
 Bridge, Arch, Spring of 247 
 
 Bridge, Stone, Strength of Truss- 
 ing 232 
 
 Bridge, Weight, Greatest on! 225 
 
 Bridge, without Tie-Beam 224 
 
 Bridging, Cross-, Additional 
 
 Strength by 137 
 
 Bridging, Cross-, Defined 137 
 
 Bridging, Cross-, Resistance by 
 Adjoining Beams 139 
 
66o 
 
 INDEX. 
 
 Building, Antiquity of 5 
 
 Building, Elementary Parts of a.. 46 
 
 Building, Expression in a 35 
 
 Building by the Greeks 35 
 
 Building, Modes of, Defined 9 
 
 Building by the Romans 26 
 
 Building, Style of, Selected to Suit 
 
 Destination 35 
 
 Butt-joint on Handrail to Stairs. . 303 
 Butt-joint, Handrail, Stairs, Posi- 
 tion of 307 
 
 Byzantine Style, Lombard 10 
 
 Campanile, or Leaning Tower, 
 
 Twelfth Century 12 
 
 Capital, Uppermost Part of a 
 
 Column 15 
 
 Carriage Beam, Well-Hole in Mid- 
 dle, Find Breadth 136 
 
 Carriage Beam, One Header, Rule 
 
 for Breadth 133 
 
 Carriage Beam or Trimmer De- 
 fined rso 
 
 Carriage Beam, Rule for Breadth. 132 
 Carriage Beam, Two Sets of Tail 
 
 Beams, Rule for Breadth 134 
 
 Caryatides, Description and Ori- 
 
 . gin of 26 
 
 Cast-Iron Bowstring Girder, 
 
 Should not be Used 163 
 
 Cast-Iron Girder, Load at Middle, 
 
 Size of Flanges. 162 
 
 Cast-Iron Girder, Load Uniform, 
 
 Size of Flanges 163 
 
 Cast-iron Girder, Manner of Mak- 
 ing a 161 
 
 Cast-Iron Girder, Proper Form... 161 
 
 Cast-iron, Tensile Strength of. ... 161 
 
 Cast-Iron Untrustworthy 161 
 
 Catenary Arch, Hooke's Theory of. 51 
 
 Cathedral of Cologne n 
 
 Cathedrals, Domes of 53 
 
 Cathedrals of Pisa, Erection in 
 
 1016 12 
 
 Cavern, The Original Place of 
 
 Shelter 13 
 
 Cavetto or Cove, Classic Moulding 323 
 
 Cavetto, Grecian Moulding 327 
 
 Cavetto, Roman Moulding 329 
 
 Ceiling, Cracking, How to Pre- 
 vent 125 
 
 Centre of Circles, To Find (Ge- 
 ometry).... 556 
 
 Centre of Gravity, Position of 71 
 
 Centre of Gravity, Rule for Find- 
 ing, Examp^es 71 
 
 Chimneys, How Arranged, 42 
 
 Chinese Structure, The Tent the 
 
 Model of . . *. 14 
 
 Chord of Circle Defined 547 
 
 Chords Giving Equal Rectangles. 363 
 Circle, Arc, Rule for Length of. . . 475 
 Circle, Area, Circumference, etc., 
 
 Examples 652 
 
 Circle, Area, Rule for, Length of 
 
 Arc Given 478 
 
 Circle, Area, To Find 475 
 
 Circle > Circumference, To Find . . 473 
 
 Circle Defined 546 
 
 Circle, Describe within Triangle.. 566 
 Circle, Diameter and Circumfer- 
 ence 472 
 
 Circle, Diameter and Perpendicu- 
 lar 468 
 
 Circle Equal Given Circles, To 
 
 Make 580 
 
 Circle, Ordinates, Rule for 471 
 
 Circle, Radius from Chord and 
 
 Versed Sine 469 
 
 Circle, Sector, Area of 476 
 
 Circle, Segment, Area of 477 
 
 Circle, Segment from Ordinates. . 470 
 Circle, Segment, Rule for Area of. 479 
 
 Circles, Table of 649-652 
 
 Circle through Given Points 559 
 
 Circular Headed Doors 320 
 
 Circular Headed Doors, To Form 
 
 Soffit 321 
 
 Circular Headed Windows 320 
 
 Circular Headed Windows, To 
 
 Form Soffit 321 
 
 Circular Stairs, Face Mould for (i). 282 
 Circular Stairs, Face Mould for (2). 285 
 Circular Stairs, Face Mould for (3). -287 
 Circular Stairs, Face Mould, First 
 Section 283 
 
INDEX. 
 
 66l 
 
 Circular Stairs, Falling Mould for 
 
 Rail 281 
 
 Circular Stairs, Handrailing for . . 278 
 
 Circular Stairs, Plan of 279 
 
 Circular Stairs, Plumb Bevel De- 
 fined 282 
 
 Circular Stairs, Timbers Put in 
 
 after Erection *. . . 253 
 
 Cisterns, Wells, etc., Table of 
 
 Capacity of 653 
 
 City Houses, General Idea of.. ... 42 
 City Houses, Arrangements for.. . 37 
 
 Civil Architecture Defined 5 
 
 Classic Architecture, Mouldings 
 
 in 323 
 
 Classic Moulding, Annulet or Fil- 
 let 323 
 
 Classic Moulding, Astragal or 
 
 Bead 323 
 
 Classic Moulding, Cavetto or 
 
 Cove 323 
 
 Classic Moulding, Cyma-Recta. . . 324 
 
 Classic Moulding, Cyma-Reversa. 324 
 
 Classic Moulding, Ogee 324 
 
 Classic Moulding, Ovolo 323 
 
 Classic Moulding, Scotia 323 
 
 Classic Moulding, Torus 323 
 
 Coffer Walls 49 
 
 Cohesive Strength of Materials. . . 76 
 
 Collar Beam in Truss 238 
 
 Cologne, Cathedral of u 
 
 Columns, Antique, Form of 48 
 
 Column, Base, Shaft and Capital. 14 
 Columns, Egyptian, Dimensions, 
 
 etc 33 
 
 Column, Gothic Pillar, Form of.. 48 
 
 Column, Outline of. 47 
 
 Columns, Parthenon at Athens, 
 
 Forms of 48 
 
 Column or Pillar 47 
 
 Column, Resistance of 47 
 
 Column, Shaft, Form of. 47 
 
 Column, Shaft, Swell of, Called 
 
 Entasis 48 
 
 Complex, or Ground Vault 52 
 
 Composite Arch of Titus 28 
 
 Composite, Corinthian or Roman 
 
 Order.. 28 
 
 Compression, Resistance to 77 
 
 Compression, Resistance to Crush- 
 ing and Bending.*. 85 
 
 Compression, Resistance to, Pres- 
 sures Classified 83 
 
 Compression, Resistance to, Table 
 
 of 79 
 
 Compression, Resistance to, in 
 
 Proportion to Depth 101 
 
 Compression at Right Angles and 
 
 Parallel to Length 206 
 
 Compression of Stout Posts 89 
 
 Compression and Tension, 
 
 Framed Girders 174 
 
 Compression Transversely to Fi- 
 bres 86 
 
 Cone Defined 548 
 
 Conic Sections 584 
 
 Conjugate Axis Defined 548 
 
 Conjugate Diameters to Axes of 
 
 Ellipse 487 
 
 Construction Essential 56 
 
 Construction of Floors, 'Roof, 
 
 etc., Economy Important 123 
 
 Construction, Framing, Heavy 
 
 Weight 56 
 
 Construction, Joints, Effect of 
 
 Many 123 
 
 Construction, Object of Defined. . 123 
 Construction, Simplest Form Best. 123 
 Construction, Superfluous Mate- 
 rial 56 
 
 ontents, Table of, General. . .613-624 
 orinthian Capital, Fanciful Ori- 
 gin of. 24 
 
 orinthian Order Appropriate in 
 
 Buildings.. . . . 24 
 
 orinthian Order, Character of.... 16 
 orinthian Order, Description of. 23 
 orinthian Order, Elegance of. ... 23 
 orinthian Order, The Favorite 
 
 at Rome 27 
 
 orinthian Order, Grecian Origin 
 
 of 16 
 
 orinthian Order, Modification of. 27 
 Cornice, Angle Bracket, To Ob- 
 tain the 343 
 
 Cornice, Eaves, To Find Depth of. 335 
 
662 
 
 INDEX. 
 
 Cornice, Mouldings, Depth of. . . . 342 
 Cornice, Projection, To Find .... 342 
 Cornice, Projecting Part of En- 
 tablature 15 
 
 Cornice, Rake and Level Mould- 
 ings, To Match 344 
 
 Cornice, Shading, Rule for . 6n 
 
 Cornice, Stucco, for Interior, De- 
 signs 340 
 
 Corollary Defined (Geometry) 348 
 
 Corollary of Triangle and Right 
 
 Angle 355 
 
 Cottage Style, English 35 
 
 Country-Seat, Style of a 37 
 
 Cross-Bridging, Additional 
 
 Strength by 137 
 
 Cross-Bridging, Furring Impor- 
 tant. .;.. 137 
 
 Cross-Bridging, Resistance of Ad- 
 joining Beams ... 139 
 
 Cross- or Herring-Bone Bridging 
 
 Defined 137 
 
 Cross-Furring Denned 125 
 
 Cross-Strains, Resistance to. . . .77, 99 
 Crushing and Bending Pressure.. 85 
 Crushing, Liability of Rafter to. . . 205 
 Crushing Strength of Stout Posts. 89 
 
 Cube Root, Examples in 645 
 
 Cubes, Squares and Roots, Table 
 
 of 638-645 
 
 Cubic Feet to Gallons, To Reduce. 653 
 
 Cupola or Dome 53 
 
 Curb or Mansard Roof 54 
 
 Curve Ellipse, Equations to 482 
 
 Curve Equilibrium of Dome 218 
 
 Cylinder, Defined 549 
 
 Cylinder, Platform Stairs 248 
 
 Cylinder, Platform Stairs, Lower 
 
 Edge of 249 
 
 Cylinders and Prisms, Stair-Build- 
 ing 257 
 
 Cyma-Recta, Classic Moulding... 324 
 Cyma-Reversa, Classic Moulding. 324 
 Cyma-Recta, Grecian Moulding . . 327 
 Cyma-Reversa, Grecian Moulding. 328 
 
 Deafening, Weight per Foot 177 
 
 Decagon, Defined 546 
 
 Decimals, Reduction of, Examples 647 
 
 Decorated Style, i4th Century. ... u 
 
 Decoration, Attention to be given 
 to.. .: 46 
 
 Decoration, Roman 27 
 
 Deflection, Defined 112 
 
 Deflection, Differs in Different Ma- 
 terials 113 
 
 Deflection, Elasticity not Dimin- 
 ished by 112 
 
 Deflection, Floor-beams, Dwell- 
 ings, Dimensions 127 
 
 Deflection, Floor-beams, First- 
 class Stores, Dimensions 128 
 
 Deflection, Floor-beams, Ordinary 
 Stores, Dimensions 127 
 
 Deflection, Lever, Principle of. . . 119 
 
 Deflection, Lever and Beam, Rela- 
 tion Between 119 
 
 Deflection, Lever, To Find, Load 
 at End 120 
 
 Deflection, Lever, Breadth or 
 Depth, Load at End 121 
 
 Deflection, Lever, Load Uniform. 121 
 
 Deflection, Lever, Breadth or 
 Depth, Load Uniform 122 
 
 Deflection, Lever, for Certain, 
 Load Uniform 122 
 
 Deflection, Load Uniform or at 
 Middle, Proportion of 116 
 
 Deflection, Load Uniform, Breadth 
 and Depth 117 
 
 Deflection, Load Uniform or at 
 Middle, Proportion of 119 
 
 Deflection, in Proportion to 
 Weight '. . 112 
 
 Deflection, Resistance to, Rule 
 for 113 
 
 Deflection, Safe Weight for Pre- 
 vention.. no 
 
 Deflection, Weight at Middle, 
 Breadth and Depth 114 
 
 Deflection, Weight at Middle, for 
 Certain 114 
 
 Deflection, Weight at Middle, Cer- 
 tain, for 116 
 
 Deflection, Weight Uniform, for 
 Certain , 117 
 
INDEX. 
 
 Deflection, Weight Uniform, Cer- 
 tain, for 
 
 Denominator, Least Common (Al- 
 gebra) '. 
 
 Denominator, Least Common 
 (Fractions) 
 
 Dentils, Teeth-like Mouldings in 
 Cornice 
 
 Diagonal Crossing Parallelogram 
 (Geometry) 
 
 Diagonal of Square Forming Oc- 
 tagon 
 
 Diagram of Forces, Example.... 
 
 Diameter, Circle, Denned 
 
 Diameter, Ellipse, Denned 
 
 Diastyle, Explanation of the Word 
 
 Diastyle, Intercolumniation 
 
 Diocletian, Baths, Splendor of. ... 
 
 Division, Fractions. Rule for 
 
 Division, by Factors (Fractions).. 
 
 Division, Quotient (Algebra) 
 
 Division, Reduction (Algebra)... 
 
 Dodecagon, Denned 
 
 Dodecagon, To Inscribe 
 
 Dodecagon, Radius of Circles 
 (Polygons) 
 
 Dodecagon, Side and Area (Poly- 
 gons) 
 
 Dome, Abutments, Strength of. . . 
 
 Domes of Cathedrals 
 
 Dome, Character of '. 
 
 Dome, Construction and Form 
 
 118 
 
 404 
 
 384 
 
 357 
 1 66 
 
 547 
 
 549 
 
 19 
 
 20 
 
 27 
 
 389 
 
 381 
 
 419 
 
 Dome, Small, over Stairways, Form 
 
 of 220 
 
 Dome, Spherical, To Form 221 
 
 Dome, St. Paul's, London 54 
 
 Dome, Strains on, Tendencies of. 219 
 
 Domes, Wooden 54 
 
 Doors, Circular Head 320 
 
 Doors, Circular Head, to Form 
 
 Soffit 321 
 
 Doors, Construction of 317 
 
 Doors, Folding and Sliding, Pro- 
 portions 316 
 
 Doors, Front, Location of 320 
 
 Doors, Height, Rule for, Width 
 
 Given 315 
 
 Door Hanging, Manner of 317 
 
 Doors, Panel, Bottom and Lock 
 
 Rail, Width 316 
 
 Doors, Panel, Four Necessary. . . 317 
 Doors, Panel, Mouldings, Width. 317 
 Doors, Panel, Styles and Muntins, 
 
 419 Width 316 
 
 546 Doors, Panel, Top Rail, Width. .. 317 
 569 Doors, Stop for. How to Form... 317 
 Doors, Single and Double, height 
 
 452 of 316 
 
 Doors, Trimmings Explained. ... 317 
 
 453 I Doors, Uses and Requirements of 315 
 
 Doors, Width of 315 
 
 of. 
 
 216 
 
 Dome, Construction and Strength 
 of 
 
 Dome, Cubic Parabola computed 
 
 Dome or Cupola, the 
 
 Dome, Curve of Equilibrium, rule 
 for 
 
 Dome, Halle du Bled, Paris 
 
 Dome, Pantheon at Rome 
 
 Dome, Pendentives of 
 
 Dome, Polygonal, Shape of An- 
 gle Rib 
 
 Dome, Ribbed, Form and Con- 
 struction 
 
 Dome, Scantling for, Table of 
 Thickness 
 
 53 
 219 
 
 53 
 
 218 
 
 54 
 53 
 53 
 
 223 
 
 217 
 
 218 
 
 Doors, Should not be Winding. .. 317 
 Doors, Width and Height, Propor- 
 tion of 315 
 
 Doors, Width, Rule for, Height 
 
 Given 316 
 
 Doric Order, Character of 16 
 
 Doric Order, Grecian Origin of. . 16 
 Doric Order, Modified by the Ro- 
 mans 27 
 
 Doric Order, Used by Greeks only 
 
 at First 19 
 
 Doric Order. Peculiarities of 17 
 
 Doric Order, Rudeness of 30 
 
 Doric Order, Specimen Buildings 
 
 in 19 
 
 Doric Temples, Fanciful Origin 
 
 of 17 
 
 Doric Temples 19 
 
 Drawing, Articles Required 536 
 
664 
 
 INDEX. 
 
 Drawing-board Better without 
 
 Clamps 537 
 
 Drawing-board Liable to Warp, 
 
 How Remedied 537 
 
 Drawing-board, Difficulty in 
 
 Stretching Paper 539 
 
 Drawing-board, Ordinary Size 536 
 
 Drawing, Diagrams aid Under- 
 standing 536 
 
 Drawing, Inking in 542 
 
 Drawing, Laying Out the 541 
 
 Drawing, the Paper 537 
 
 Drawing in Pencil, To Make Lines 542 
 
 Drawing, Secure Paper to Board. 537 
 
 Drawing, Shade Lining 543 
 
 Drawing, Stretching Paper 537 
 
 Durability in a Building 37 
 
 Dwelling, Arrangement of Rooms 38 
 Dwellings, Floor-beams, To Find 
 
 Dimensions 127 
 
 Dwellings, Floor-beams, Safe 
 
 Weight for 126 
 
 Dwelling-houses, Dimensions and 
 
 Style . 37 
 
 Eaves Cornice, Designs for 335 
 
 Eaves Cornice, Rule for Depth... 335 
 Ecclesiastical Architecture, Point- 
 ed Style ii 
 
 Ecclesiastical Style, Origin of. ... 14 
 
 Echinus, Grecian Moulding 327 
 
 Economy, Construction Floors, 
 
 Roofs, Bridges 123 
 
 Eddystone and Bell Rock Light 
 
 House 48 
 
 Egyptian Architecture 30 
 
 Egyptian Architecture, Appropri- 
 ate Buildings for 33 
 
 Egyptian Architecture, Character 
 
 of. 
 
 33 
 
 Egyptian Architecture, Origin in 
 
 Caverns 14 
 
 Egyptian Architecture. Principal 
 
 Features of 30 
 
 Egyptian Columns, Dimensions 
 
 and Proportions 33 
 
 Egyptian Walls, Massiveness of. . 33 
 
 Egyptian Works of Art 30 
 
 Elasticity of Materials 84 
 
 Elasticity not Diminished by De- 
 flection 112 
 
 Elasticity, Result of Exceeding 
 
 Limit 120 
 
 Elevation, a Front View 37 
 
 Elevated Tie-beam Roof Truss 
 
 Objectionable 214 
 
 Ellipse, Area 488 
 
 Ellipse, Axes, Two, To Find, Di- 
 ameter and Conjugate Given. .. 593 
 
 Ellipse Defined 481 
 
 Ellipse, Equations to the Curve. . 482 
 Ellipse, Major and Minor Axes 
 
 Defined : .. 481 
 
 Ellipse, Ordinates, Length of ... 491 
 Ellipse, Parameter and Axis, Re- 
 lation of 485 
 
 Ellipse, Practical Suggestions.... 489 
 Ellipse, Semi-major, Axis Defined 486 
 
 Ellipse, Subtangent Defined 486 
 
 "Ellipse, Tangent to Axes, Rela- 
 tion of 485 
 
 Ellipse, Tangent with Foci, Rela- 
 tion of 487 
 
 Ellipsis, Axes of, To Find (Geom- 
 etry) 585 
 
 Ellipsis, Conjugate Diameters (Ge- 
 ometry) 593 
 
 Ellipsis Defined 548, 585 
 
 Ellipsis, Diameter Defined 549 
 
 Ellipsis, Foci, To Find 586 
 
 Ellipsis, by Intersecting Arcs. . . . 590 
 Ellipsis, by Intersecting Lines... 588 
 
 Ellipsis, by Ordinates 588 
 
 Ellipsis, Point of Contact with 
 
 Tangent, To Find 593 
 
 Ellipsis, Proportionate Axes, to 
 
 Describe with 594 
 
 Ellipsis, Trammel, to Find, Axes 
 
 Given 586 
 
 Elliptical Arch, Joints, Direction 
 
 of 233 
 
 English Architecture, Early n 
 
 English Cottage Style Extensive- 
 ly Used 35 
 
 j England and France, Fourteenth 
 Century 12 
 
INDEX. 
 
 66 5 
 
 Entablature, above Columns and 
 
 Horizontal 14 
 
 Entasis, Swell of Shaft of Column 48 
 
 Equal Angles Defined 349 
 
 Equal Angles, Example in 350 
 
 Equal Angles, in Circle 358 
 
 Equal Angles (Geometry) 553 
 
 Equilateral Rectangle, to De- 
 scribe 568 
 
 Equilateral Triangle Defined (Ge 
 
 ometry) 545 
 
 Equilateral Triangle, to Construct 
 
 (Geometry) 568 
 
 Equilateral Triangle, to Describe 
 
 (Geometry) 566 
 
 Eqilateral Triangle, to Inscribe 
 
 (Geometry) 569 
 
 Equilateral Triangle (Polygons).. 445 
 
 Eustyle Defined 20 
 
 Exponents, Quantities with Nega- 
 tive (Algebra) 423 
 
 Extrados of an Arch 52 
 
 Face Mould, Accuracy of, Wind- 
 ing Stairs 295 
 
 Face Mould, Curves Elliptical, 
 Winding Stairs 301 
 
 Face Mould, Drawing of, Winding 
 Stairs 296 
 
 Face Mould, Sliding of, Winding 
 Stairs 299 
 
 Face Mould, Application of, Plat- 
 form Stairs 275 
 
 Face Mould, a Simple, Kell's 
 Method for 268 
 
 Factors, Multiplication (Algebra) 409 
 
 Factors, Two, Squaring Difference 
 of (Algebra) 412 
 
 Fibrous Structure of Materials.. . 76 
 
 Figure Equal, Given Figure (Ge- 
 ometry) 575 
 
 Figure, Nearly Elliptical, To Make 
 (Geometry) 59 1 
 
 Fillet or Amulet, Classic Mould- 
 ing 323 
 
 Fire-proof Floors, Action of Fire 
 on JJ3 
 
 Flanges, Cast-iron Girder 163 
 
 Flanges, Area of, Tubular Iron 
 Girder 155 
 
 Flanges, Area of Bottom, Tubular 
 Iron Girder 159 
 
 Flanges, Load at Middle of Cast- 
 iron Girder, Sizes 162 
 
 Flanges, Load Uniform on Tubu- 
 lar Iron Girder, Sizes 156 
 
 Flanges, Proportion of, Tubular 
 Iron Girder 157 
 
 Flexure, Compared with Rup- 
 ture 84 
 
 Flexureof Rafter 205 
 
 Flexure, Resistance to, Defined. . 145 
 
 Floor-arches, How Constructed. . 153 
 
 Floor-arches, Tie-rods, Dwellings, 
 Sizes 153 
 
 Floor-arches, Tie-rods, First-class 
 Stores, Sizes 153 
 
 Floor-beams, Distance from Cen- 
 tres, Sizes Fixed 129 
 
 Floor-beams, Dwellings, Safe 
 Weight for 126 
 
 Floor-beams, Dwellings, Deflec- 
 tion Given, Sizes 127 
 
 Floor-beams, First-class Stores, 
 Deflection Given, Sizes 128 
 
 Floor-beams, Ordinary Stores, De- 
 flection Given, Sizes 127 
 
 Floor-beams, Stores, Safe Weight 
 for 126 
 
 Floor-beams, Reference to Rules 
 for Sizes 125 
 
 Floor-beams, Reference to Trans- 
 verse Strains 126 
 
 Floor-beams, Proportion of 
 Weight on All 130 
 
 Floors Constructed, Single or 
 Double 124. 
 
 Floors, Fire-proof Iron, Action of 
 Fire on 143 
 
 Floors, Framed, Seldom Used. .. 124 
 
 Floors, Framed, Openings in 130 
 
 Floors, Headers, Defined 130 
 
 Floofs, Ordinary, Effect of Fire 
 on 143 
 
 Floors, Solid Timber, Dwellings 
 and Assembly, Depth 143 
 
666 
 
 INDEX. 
 
 Floors, Solid Timber, First-class 
 
 Stores, Depth 144 
 
 Floors, Solid Timber, to Make 
 
 Fire-proof 143 
 
 Floors, Tail-beams Defined 130 
 
 Floors, Trimmers or Carriage- 
 beams Defined 130 
 
 Floors, Wooden, More Fire-proof 
 
 than Iron, Some* Cases 143 
 
 Flyers and Winders, Winding 
 
 Stairs 251 
 
 Foci Defined 548 
 
 Foci of Ellipsis To Find 586 
 
 Foci of Ellipse, Tangent 487 
 
 Force Diagram, Load on Each 
 
 Support 179 
 
 Force Diagram, Truss, Figs. 59, 
 
 68 and 69 179 
 
 Force Diagram, Truss, Figs. 60, 70 
 
 and 71 180 
 
 Force Diagram, Truss, Figs. 61, 
 
 72 and 73 181 
 
 Force Diagram Truss, Figs. 63 
 
 74 and 75 183 
 
 Force Diagram, Truss, Figs. 64, 
 
 77 and 78 184 
 
 Force Diagram, Truss, Figs. 65, 
 
 78 and 79 185 
 
 Force Diagram, Truss, Figs. 66, 
 
 8oand8i 186 
 
 Forces, Parallelogram of 59 
 
 Forces, Composition of 66 
 
 Forces, Composition, Reverse of 
 
 Resolution 67 
 
 Forces, Resolution of 59 
 
 Forces, Resolution of, Oblique 
 
 Pressure 59 
 
 Foundations, Description of 47 
 
 . Foundations in Marshes, Timbers 
 
 Used 47 
 
 Fractions, Addition, Like Denom- 
 inators 382 
 
 Fractions Added and Subtracted 
 
 (Algebra) 403 
 
 Fractions Changed by Division!. 380 
 
 Fractions Defined 378 
 
 Fractions, Division, Rule for 389 
 
 Fractions, Division by Factors... 381 
 
 Fractions Divided Graphically. . . 388 
 Fractions Graphically Expressed. 378 
 Fractions, Improper, Defined.... 380 
 Fractions, Least Common Denom- 
 inator , 384 
 
 Fractions, Multiplication, Rule. . . 387 
 
 Fractions Multiplied Graphically. 386 
 
 Fractions, Numeratorand Denom- 
 inator 378 
 
 Fractions, Reduce Mixed Num- 
 bers 381 
 
 Fractions, Reduction to Lowest 
 Terms 384 
 
 Fractions Subtracted (Algebra). . . 405 
 
 Fractions, Subtraction Like De- 
 nominators 383 
 
 Fractions, Unlike Denominators 
 Equalized 383 
 
 Framed Beams, Thickness of, 
 Rules 130 
 
 Framed Girder, Bays Defined 167 
 
 Framed Girders, Compression and 
 Tension, Dimensions 174 
 
 Framed Girders, Construction 
 and Uses 166 
 
 Framed Girders, Height and 
 Depth 167 
 
 Framed Girders, Kinds of Pres- 
 sure 173 
 
 Framed Girders, Long, Construc- 
 tion of '. 174 
 
 Framed Girders, Panels on Under 
 Chord, Table of 167 
 
 Framed Girders, Ties and Struts, 
 Effect of 174 
 
 Framed Girders, Triangular Pres- 
 sure, Upper Chord 168 
 
 Framed Girders, Triangular Pres- 
 sure, Both Chords 171 
 
 Framed Openings in Floors 130 
 
 Framing Beams, Effect of Splic- 
 ing 235 
 
 Framing Roof Truss 237 
 
 Framing Roof Truss, Iron Straps, 
 Size of 239 
 
 France and England, Fourteenth 
 Century 12 
 
 Friction, Effect of 82 
 
INDEX. 
 
 667 
 
 Frieze between Architrave and 
 
 Cornice 15 
 
 Furring Denned 125 
 
 Gable, a Pediment in Gothic Ar- 
 chitecture 15 
 
 Gaining a Beam Denned 100 
 
 General Contents, Table of 613-624 
 
 Geometrical Progression (Alge- 
 bra)- 435 
 
 Geometry, Angles of Triangle, 
 
 Three, Equal Right Angle 354 
 
 Geometry Chords Giving Equal 
 
 Rectangles 363 
 
 Geometry Denned 544 
 
 Geometry, Divide a Given Line. . 555 
 Geometry, Divisions in Line Pro- 
 portionate 583 
 
 Geometry, Elementary 347 
 
 Geometry, Equal Angles 553 
 
 Geometry, Equal Angles, Ex- 
 ample 350 
 
 Geometry, Figure Equal to Given 
 
 Figure, Construct 575 
 
 Geometry, Figure Nearly Ellipti- 
 cal by Compasses 591 
 
 Geometry, Measure an Angle. . . . 348 
 Geometry Necessary in Handrail- 
 ing, Stairs 257 
 
 Geometry, Opposite Angles Equal. 354 
 
 Geometry, Parallel Lines 555 
 
 Geometry, a Perpendicular, To 
 
 Erect 550 
 
 Geometry.Perpendicular.let Fall a. 551 
 Geometry, Perpendicular, Erect at 
 
 End of Line 551 
 
 Geometry, Perpendicular, Let Fall 
 
 Near End of Line 553 
 
 Geometry, Plane Denned (Stairs). 257 
 
 Geometry, Point of Contact 558 
 
 Geometry, Points, Three Given, 
 
 Find Fourth 559 
 
 Geometry, Right Line Equal Cir- 
 cumference 566 
 
 Geometry, Right Lines, Propor- 
 tion Between 584 
 
 Geometry, Right Lines, Two 
 Given, Find Third 582 
 
 Geometry, Square Equal Rec- 
 tangle, To Make 581 
 
 Geometry, Square Equal Given 
 
 Squares, To make 577 
 
 Geometry, Square Equal Triangle, 
 
 To Make .. 582 
 
 German or Romantic Style, Thir- 
 teenth and Fourteenth Centuries, n 
 Girder, Bearings, Space Allowed 
 
 for 141 
 
 Girder, Bow-String, Cast-Iron, 
 
 Should not be Used 163 
 
 Girder, Bow-String, Substitute for. 163 
 Girder, Construction with Long 
 
 Bearings 140 
 
 Girder, Cast-Iron, Load Uniform, 
 
 Flanges 163 
 
 Girder, Cast-Iron, Load at Middle, 
 
 Flanges 162 
 
 Girder, Cast-Iron, Proper Form 
 
 of 161 
 
 Girder Denned, Position and Use 
 
 of 140 
 
 Girder, Different Supports for. . . . 140 
 
 Girder, Dwellings, Sizes for 141 
 
 Girder, Framed, Bays Denned 167 
 
 Girders, Framed, Compression 
 
 and Tension, Dimensions 174 
 
 Girder, Framed, Construction of.. 140 
 Girder, Framed, Construction and 
 
 Uses 166 
 
 Girder, Framed, Construction of 
 
 Long 174 
 
 Girder, Framed, Kinds of Pres- 
 sure 173 
 
 Girders, Framed, Height and 
 
 Depth 167 
 
 Girders, Framed, Panels on Under 
 
 Chord, Table of. 167 
 
 Girders, Framed, Triangular Pres- 
 sure Upper Chord 108 
 
 Girders, Framed, Triangular Pres- 
 sure Both Chords 171 
 
 Girders, Framed, and Tubular 
 
 Iron 140 
 
 Girders, First-Class Stores, Sizes 
 
 for 141 
 
 Girders, Sizes, To Obtain 141 
 
668 
 
 INDEX. 
 
 Girders, Strengthening, Manner 
 
 of. 140 
 
 Girders, Supports, Length of, Rule. 157 
 Girders, Tubular Iron, Construc- 
 tion of 154 
 
 Girders, Tubular Iron, Area of 
 
 Flange. Load at Middle 154 
 
 Girders, Tubular Iron, Area of 
 
 Flange, Load at any Point 155 
 
 Girders, Tubular Iron, Area of 
 
 Flange, Load Uniform 156 
 
 Girders, Tubular Iron, Dwellings, 
 
 Area of Bottom Flange 159 
 
 Girders, Tubular Iron, First-Class 
 
 Stores, Area of Bottom Flange.. 160 
 Girders, Tubular Iron, Rivets, Al- 
 lowance for 157 
 
 Girders, Tubular Iron, Flanges, 
 
 Proportion of 157 
 
 Girders, Tubular Iron, Shearing 
 
 Strain 15? 
 
 Girders, Tubular Iron.Web, Thick- 
 ness of i5 8 
 
 Girders, Weakening, Manner of. . 140 
 Girders, Wooden, Objectionable. 154 
 Girders, Wooden, Supporting, 
 
 Manner of 154 
 
 Glossary of Terms 627-637 
 
 Gothic Arches 51 
 
 Gothic Buildings, Roofs of 55 
 
 Gothic and Norman Roofs, Con- 
 struction of 178 
 
 Gothic Pillar, Form of 48 
 
 Gothic Style, Characteristics of. . . 12 
 
 Goths, Ruins Caused by 12 
 
 Granular Structure of Materials. . 76 
 
 Gravity, Centre of, Position 71 
 
 Gravity, Centre of, Examples, and 
 
 Rule for 71 
 
 Grecian Architecture, History of. 6 
 
 Grecian Art, Elegance of 27 
 
 Grecian Moulding, Cyma-Recta.. 327 
 Grecian Moulding, Cyma-Re- 
 
 versa. . . -. 328 
 
 Grecian Moulding, Echinus and 
 
 Cavetto 327 
 
 Grecian Moulding, Scotia 326 
 
 Grecian Moulding, Torus 326 
 
 Grecian Orders Modified by the 
 
 Romans 27 
 
 Grecian Origin of the Doric Or 
 
 der 16 
 
 Grecian Origin of Ionic Order ... 16 
 
 Grecian Style in America 13 
 
 Grecian Styles, their Different 
 
 Orders 16 
 
 Greek Architecture, Doric Order 
 
 Used 19 
 
 Greek Building 35 
 
 Greek Moulding, Form of 325 
 
 Greek, Persian, and Caryatides 
 
 Orders 24 
 
 Greek Style Originally in Wood.. 14 
 Greek Styles Only Known by 
 
 Them 16 
 
 Groined or Complex Vault 52 
 
 Halle du Bled, Paris, Dome of. . . 54 
 
 Halls of Justice, N. Y. C., Speci- 
 men of Egyptian Architecture. . 8 
 
 Handrailing, Circular Stairs 278 
 
 Handrailing, Platform Stairs. ... 269 
 
 Handrailing, Platform Stairs, Face 
 Mould 264 
 
 Handrailing, Platform Stairs, 
 Large Cylinder 271 
 
 Handrailing Stairs, Geometry 
 Necessary 257 
 
 Handrailing Stairs," Out of Wind" 
 Defined. 257 
 
 Handrailing Stairs, Tools Used. . 257 
 
 Handrailing, Winding Stairs. .256, 289 
 
 Handrailing Winding Stairs, Bal- 
 usters Under Scroll 310 
 
 Handrailing, Winding Stairs, 
 Centres in Square 308 
 
 Handrailing, Winding Stairs, Face 
 for Scroll 311 
 
 Handrailing, Winding Stairs, Fall- 
 ing Mould 310 
 
 Handrailing, Winding Stairs, Gen- 
 eral Considerations 258 
 
 Handrailing, Winding Stairs, 
 Scroll for 308 
 
 Handrailing, Winding Stairs, 
 Scroll at Newel 309 
 
INDEX. 
 
 669 
 
 Handrailing, Winding Stairs, 
 
 Scroll Over Curtail Step 309 
 
 Handrailing, Winding Stairs, 
 
 Scroll for Curtail Step 310 
 
 Headers, Breadth of. 130 
 
 Headers Defined 130 
 
 Headers, Mortises, Allowance for 
 
 Weakening by 131 
 
 Headers, Stores and Dwellings, 
 
 Same for Both 132 
 
 Hecadecagon, Complete Square 
 
 (Polygons) 458 
 
 Hecadecagon, Radius of Circles 
 
 (Polygons) 455 
 
 Hecadecagon, Rules (Polygons). . 459 
 Hecadecagon, Side and Area 
 
 (Polygons) 457 
 
 Height and Projection, Numbers 
 
 of an Order 16 
 
 Hemlock, Weight per Foot Super- 
 ficial 177 
 
 Heptagon Defined 546 
 
 Herring-bone Bridging Defined... 137 
 
 Hexagon Defined 546 
 
 Hexagon, To Inscribe 569 
 
 Hexagons, Radius of Circles 447 
 
 Hexastyle, Intercolumniation. . . . 20 
 Hindoo Architecture, Ancient, 
 
 Character of 30 
 
 Hip-Rafter, Backing of 216 
 
 Hip-Roofs, Diagram and Expla- 
 nation 215 
 
 History of Architecture 44 
 
 Hogged Ridge in Roof Truss 238 
 
 Homologous Triangles (Geom- 
 etry) 362 
 
 Homologous Triangles (Ratio 
 
 and Proportion) 370 
 
 Hooke's Theory of an Arch 50 
 
 Hooke's Theory, Bridge Arch, 
 
 Pressure on. ... 51 
 
 Hooke's Theory, Catenary Arch. . 51 
 Horizontal and Inclined Roofing, 
 
 Weight 190 
 
 Horizontal Pressure on Roof, To 
 
 Remove 74 
 
 Horizontal Thrust in Beams 72 i 
 
 Horizontal Thrust, Tendency of. . 88 i 
 
 Hut, Original Habitation 13 
 
 Hydraulic Method, Testing 
 Woods 80 
 
 Hyperbola Defined 548, 585 
 
 Hyperbola, Height, To Find, Base 
 and Axis Given 585 
 
 Hyperbola by Intersecting Lines. 595 
 
 Hypothenuse, Equality of Squares 
 (Algebra) 416 
 
 Hypothenuse, Formula for (Trig- 
 onometry) 516 
 
 Hypothenuse, Side, To Find (Ge- 
 ometry) 579 
 
 Hypothenuse, Right Angled Tri- 
 angle (Geometry) 355 
 
 Hypothenuse, Triangle (Trigo- 
 nometry) 518 
 
 Ichnographic Projection, Ground 
 
 Plan 37 
 
 Improper Fractions Defined 380 
 
 India Ink in Drawing 540 
 
 Inertia, Moment of, Defined 145 
 
 Inking-in Drawing . . 542 
 
 Inside Shutters for Windows, Re- 
 quirements 319 
 
 Instruments in Drawing 540 
 
 Intercolumniation Defined 17 
 
 Intercolumniation of Orders 20 
 
 Intrados of Arch 52 
 
 Ionic Order, Character of. 16 
 
 Ionic Order, Grecian Origin of. . . 16 
 Ionic Order Modified by the Ro- 
 mans. ...i 27 
 
 Ionic Order, Origin of 20 
 
 Ionic Order, Suitable for What 
 
 Buildings 20 
 
 Ionic Volute, To Describe an. ... 20 
 Iron Beams, Breaking Weight at 
 
 Middle 148 
 
 Irpn Beams, Deflection, To Find, 
 
 Weight at Middle 147 
 
 Iron Beams, Deflection, To Find, 
 
 Weight Uniform 150 
 
 Iron Beams, Dimensions, To Find, 
 
 Weight any Point 149 
 
 Iron Beams, Dimensions, To Find, 
 Weight Uniform 149 
 
6 ;o 
 
 INDEX. 
 
 Iron Beams, Dwellings, Distance 
 from Centres 151 
 
 Iron Beams, First-Class Stores, 
 Distance from Centres 152 
 
 Iron Beams, Rectangular Cross- 
 Section 145 
 
 Iron Beams, Rolled, Sizes 145 
 
 Iron Beams, Safe Weight, Load 
 any Point 148 
 
 Iron Beams, Safe Weight, Load 
 Uniform 151 
 
 Iron Beams, Table IV 146 
 
 Iron Beams, Weight at Middle, 
 Deflection Given 146 
 
 Iron Fire-Proof Floors, Action of 
 Fire On 143 
 
 Iron Straps, Framing, to Prevent 
 Rusting 239 
 
 Irregular Polygon, Trigon (Geom- 
 etry) 546 
 
 Isosceles Triangle Denned 545, 584 
 
 Italian Architecture, Thirteenth, 
 Fourteenth, and Fifteenth Cen- 
 turies 12 
 
 Italian Use of Roman Styles 13 
 
 Italy, Tuscan Order the Principal 
 Style 30 
 
 Jack-Rafters, Location of 212 
 
 jack-Rafters and Purlins in Roof. 211 
 Jack-Rafters, Weight per Superfi- 
 cial Foot 189 
 
 Joists and Studs Defined 174 
 
 Jupiter, Temple of, at Thebes, Ex- 
 tent of 33 
 
 Kell's Method, Simple Face 
 
 Mould, Stairs 268 
 
 Keystone for Arch, Position of. ... 50 
 King-Post, Bad Framing, Effect 
 
 of ... 237 
 
 King-Post, Location of 213 
 
 King-Post in Roof 54 
 
 Lamina in Girders Defined 174 
 
 Lancet Arch 51 
 
 Lateral Thrust in Arch 52 
 
 Laws of Pressure 57 
 
 Laws of Pressure, Inclined, Ex- 
 amples 57 
 
 Laws of Pressure, Vertical, Exam- 
 ples 57 
 
 Leaning Tower or Campanile, 
 
 Twelfth Century 12 
 
 Length, Breadth, or Thickness, 
 
 Relation to Pressure 78 
 
 Lever, Breadth or Depth, To 
 
 Find in 
 
 Lever, Deflection as Relating to 
 
 Beam 1 19 
 
 Lever, Deflection, Load at End.. 120 
 Lever, Deflection, Load Uniform. 121 
 Lever, Deflection, Breadth or 
 
 Depth, Load at End 121 
 
 Lever, Deflection, Breadth or 
 
 Depth, Load Uniform 122 
 
 Lever, Deflection, Load Required. 122 
 Lever Formula, Proportionals in 
 
 (Algebra) 421 
 
 Lever Load Uniformly Distrib- 
 uted in 
 
 Lever, Load at One End no 
 
 Lever Principle Demonstrated 
 
 (Ratio) 375 
 
 Lever, Support, Relative Strength 
 
 of One no 
 
 Light-Houses, Eddystone and Bell 
 
 Rock 48 
 
 Line Defined (Geometry) 544 
 
 Lines, Divisions in, Proportionate 
 
 (Geometry) 583 
 
 Lintel, Position of 49 
 
 Lintel, Strength of 49 
 
 Load, per foot, Horizontal 192 
 
 Load on Roof Truss, per Superfi- 
 cial Foot 189 
 
 Load on Tie-Beam, Ceiling, etc. . 190 
 
 Lock Rail for Doors, Width 316 
 
 Logarithms Explained (Algebra).. 425 
 
 Logarithms, Examples 426 
 
 Logarithms, Sine and Tangents 
 
 (Polygons) 464 
 
 Lombard, Byzantine Style 10 
 
 Lombard Style, Seventh Century. 10 
 London Bridge, Piles, Age of 229 
 
INDEX. 
 
 6 7 I 
 
 Materials, Cohesive Strength of. . 76 
 Materials, Compression, Resist- 
 ance to. 77 
 
 Materials, Cross-strain, Resistance 
 
 to 77 
 
 Materials, Structure of 76 
 
 Materials, Tension, Resistance to. 77 
 Materials Tested, General De- 
 scription 80 
 
 Materials, Weights, Table of. . . . 654 
 Major and Minor Axes of Ellipse 
 
 Denned 481 
 
 Marshes, Foundation for Timbers 
 
 in 47 
 
 Mathematics Essential 347 
 
 Maxwell, Prof. I. Clerk, Diagrams 
 
 of Forces, etc 165 
 
 Memphis, Pyramids of, Estimate 
 
 of Stone in 33 
 
 Minster, Tower of Strassburg n 
 
 Minutes, Sixty Equal Parts, to 
 
 Proportion an Order 15 
 
 Mixed Numbers in Fractions, To 
 
 Reduce 381 
 
 Modern Architecture, First Ap- 
 pearance of 9 
 
 Modern Tuscan, Appropriate for 
 
 Buildings 30 
 
 Moment of Inertia Defined 145 
 
 Mono-triglyph, Explanation of the 
 
 Word 19 
 
 Monuments, Ancient, Their Archi- 
 tects 6 
 
 Moorish and Arabian Styles, An- 
 tiquities of ii 
 
 Mortises, Proper Location of . 100 
 Mortising, Beam, Effect on 
 
 Strength of 100 
 
 Mortising Beam at Top, Injurious 
 
 Effect of 100 
 
 Mortising Beam, Effect of 231 
 
 Mortising, Beam, Position of 236 
 
 Mortising Headers, Allowance for 
 
 Weakening 131 
 
 Moulding, Classic, Astragal or 
 
 Bead 323 
 
 Moulding, Classic, Annulet or 
 Fillet 323 
 
 Mouldings, Classic Architecture. 323 
 Moulding, Classic, Cavctto or 
 
 Cove 323 
 
 Moulding, Classic, Cyma-Recta. . 324 
 Moulding, Classic, Cyma-Reversa. 324 
 
 Moulding, Classic, Ogee 324 
 
 Moulding, Classic, Ovolo 323 
 
 Moulding, Classic, Scotia 323 
 
 Moulding, Classic, Torus 323 
 
 Mouldings, Common to all Or- 
 ders 324 
 
 Mouldings Defined 323 
 
 Mouldings, Diagrams of 330 
 
 Mouldings, Doors, Rule for Width. 317 
 Moulding, Grecian, Cyma-Recta. 327 
 Moulding, Grecian, Cyma-Rc- 
 
 versa 328 
 
 Moulding, Grecian Echinus and 
 
 Cavetto 327 
 
 Mouldings, Greek, Form of. 325 
 
 Mouldings, Grecian Torus and 
 
 Scotia 326 
 
 Mouldings, Modern 331 
 
 Moulding, Modern, Antae Cap... 334 
 Mouldings, Mbdern Interior, Dia- 
 grams 332 
 
 Mouldings, Modern, Plain 333 
 
 Mouldings,Names, Derivations of. 324 
 
 Mouldings, Profile Defined 326 
 
 Mouldings, Roman, Forms of. .. . 325 
 Mouldings, Roman, Comments on. 329 
 Mouldings, Roman, Ovolo and 
 
 Cavetto 329 
 
 Mouldings,Uses and Positions of. 324 
 
 Multiplication (Algebra) 408 
 
 Multiplication, Plus and Minus 
 
 (Algebra) 415 
 
 Multiplication, Three Factors (Al- 
 gebra) 408 
 
 Multiplication, Fractions 387 
 
 Newel Cap, Form of, Winding 
 Stairs 312 
 
 Nicholson's Method, Plane 
 Through Cylinder (Stairs) 259 
 
 Nicholson's Method, Twists in 
 Stairs 259 
 
 Nonagon Defined 546 
 
672 
 
 INDEX". 
 
 Normal and Subnormal in Para- 
 bola 496 
 
 Norman and Gothic Construction 
 of Roofs 178 
 
 Norman Style, Peculiarities of. . . n 
 
 Nosing and Tread, Position in 
 Stairs 241 
 
 Oblique Angle Defined 544 
 
 Oblique Pressure, Resolution of 
 
 Forces . . . . 59 
 
 Oblique Triangle, Difference Two 
 
 Angles (Trigonometry) 523 
 
 Oblique Triangle, First Class 
 
 (Trigonometry) , 520 
 
 Oblique Triangles, First Class, 
 
 Formulae (Trigonometry) 531 
 
 Oblique Triangles, Second Class 
 
 (Trigonometry) 522 
 
 Oblique Triangles, Second Class, 
 
 Formulae (Trigonometry) 532 
 
 Oblique Triangles, Third Class 
 
 (Trigonometry) 526 
 
 Oblique Triangles, Third Class, 
 
 Formula; (Trigonometry) 534 
 
 Oblique Triangles, Fourth Class 
 
 (Trigonometry) 528 
 
 Oblique Triangles, Fourth Class, 
 
 Formulae (Trigonometry) 534 
 
 Oblique Triangles, Two Sides 
 
 (Trigonometry) 521 
 
 Oblique Triangles, Sines and 
 
 Sides (Trigonometry) 519 
 
 Obtuse Angle Denned 349, 544 
 
 Obtuse Angled Triangle Denned. 545 
 Octagon, Buttressed, Find Side 
 
 (Geometry) 571 
 
 Octagon Defined 546 
 
 Octagon, Diagonal of Square 
 
 Forming 357 
 
 Octagon, Inscribe a (Geometry). . 570 
 
 Octagon, Rules (Polygons) 451 
 
 Octagon, Radius of Circles (Poly- 
 gons) 449 
 
 Octastyle, Intercolumniation 20 
 
 Ogee Mouldings, Classic 324 
 
 Opposite Angles Equal (Geome- 
 try) 354 
 
 Order of Architecture, Three 
 
 Principal Parts 14 
 
 Orders of Architecture, Persians 
 
 and Caryatides 24 
 
 Ordinates to an Arc (Geometry). . 563 
 
 Ordinates, Circle, Rule for 471 
 
 Ordinates of Ellipse 491 
 
 Ostrogoths, Style of the 9 
 
 Oval, To Describe a (Geometry). . 591 
 
 Ovolo, Classic Moulding 323 
 
 Ovolo, Roman Moulding 329 
 
 Paper, The, in Drawing, Secure to 
 
 Board , . . 537 
 
 Pantheon at Rome, Dome of, and 
 
 Walls 53 
 
 Pantheon and Roman Buildings, 
 
 Walls of 49 
 
 Parabola, Arcs Described from... 503 
 
 Parabola, Area, Rule for 509 
 
 Parabola, Axis and Base, to find 
 
 (Geometry) 585 
 
 Parabola, Curve, Equations to... . 493 
 
 Parabola Defined 492 
 
 Parabola Defined (Geometry).. 548, 585 
 
 Parabola, Diameters 497 
 
 Parabola Described from Ordi- 
 nates 504 
 
 Parabola Described from Diame- 
 ters 507 
 
 Parabola Described from Points.. 502 
 
 Parabola of Dome Computed 219 
 
 Parabola, General Rules 499 
 
 Parabola by Intersecting Lines. . . 594 
 Parabola Mechanically Described. 500 
 Parabola, Normal and Subnor- 
 mal 496 
 
 Parabola, Ordinate Defined 496 
 
 Parabola, Subtangent 496 
 
 Parabola, Tangent 493 
 
 Parabola, Vertical Tangent De- 
 fined 495 
 
 Parabolic Arch, Direction of 
 
 Joints 234 
 
 Parallel Lines Defined 544 
 
 Parallel Lines (Geometry) 555 
 
 Parallelogram, Construct a 576 
 
 Parallelogram Defined 545 
 
INDEX. 
 
 673 
 
 Parallelogram Equal to Triangles, 
 
 To Make.. 576 
 
 Parallelogram of Forces, Strains 
 
 by 165 
 
 Parallelograms Proportioned to 
 
 Bases (Geometry) 360 
 
 Parallelogram in Quadrangle 
 
 (Geometry) 364 
 
 Parallelogram, Same Base (Geom- 
 etry) 352 
 
 Parameter Defined 548 
 
 Parameter, Axes (Ellipse) 485 
 
 Parthenon at Athens, Columns of. 48 
 Partitions, Bracing and Trussing. 176 
 
 Partitions, How Constructed 174 
 
 Partition, Door in Middle, Con- 
 struction 175 
 
 Partition, Doors at End, Construc- 
 tion of 176 
 
 Partition, Great Strength, Con- 
 struction 176 
 
 Partitions, Location and Connec- 
 tion . 175 
 
 Partitions, Materials, Quality of. . 175 
 Partitions, Plastered, Proper Sup- 
 ports for. 175 
 
 Partitions, Pressure on. Rules 177 
 
 Partitions, Principal, of what Com- 
 posed 175 
 
 Partitions, Trussing in, Effects of. 175 
 Pedestal, a Separate Substruc- 
 ture 14 
 
 Pediment, Triangular End of 
 
 Building 15 
 
 Pencil and Rulers, Drawing 540 
 
 Pentagon Defined 546 
 
 Pentagon, Circumscribed Circles 
 
 (Polygons) 463 
 
 Perpendicular Height of Roof, To 
 
 find 579 
 
 Perpendicular, Erect a 550 
 
 Perpendicular, Erect a, at End of 
 
 Line 551 
 
 Perpendicular, Let Fall a 551 
 
 Perpendicular, Let Fall a, at End 
 
 of Line 553 
 
 Perpendicular Style, Fifteenth 
 Century 12 
 
 Perpendicular in Triangle (Poly- 
 gons) 440 
 
 Persians, Origin and Description 
 
 of 24 
 
 Persians and Caryatides, Orders 
 
 Used by Greeks 24 
 
 Piers, Arrangement, in City Front 
 
 of House 44 
 
 Piers, Bridges, Construction and 
 
 Sizes 228 
 
 Piles, London Bridge, Age of. ... 229 
 Pine, White, Beams, Table of 
 
 Weights for. 177 
 
 Pisa, Cathedral of, Eleventh Cen- 
 tury 12 
 
 Pisa, Cathedral of, Erection in 
 
 1016 12 
 
 Pise Wall of France 49 
 
 Pitch Board, To Make, for Stairs. 247 
 
 Pitch Board, Winding Stairs 252 
 
 Plane Defined 257 
 
 Plane Defined (Geometry) 544 
 
 Plank.Weightof.on Roof, per foot. 189 
 Plastering, Defective, To what 
 
 Due 174 
 
 Plastering, Strength of 174 
 
 Plastering, Weight per foot 177 
 
 Platform Stairs, Baluster, Posi- 
 tion of 250 
 
 Platform Stairs Beneficial 240 
 
 Platform Stairs, Cylinder of. 248 
 
 Platform Stairs, Cylinder, Lower 
 
 Edge 249 
 
 Platform Stairs, Face Mould, Ap- 
 plication of Plank 273 
 
 Platform Stairs, Face Mould, 
 
 Handrailing in 264 
 
 Platform Stairs, Face Mould, Sim- 
 ple Method 267 
 
 Platform Stairs, Face Mould, 
 
 Moulded Rails 274 
 
 Platform Stairs, Face Mould, Ap- 
 plication of 275 
 
 Platform Stairs, Face Mould With- 
 out Canting Plank 272 
 
 Platform Stairs, Handrail to 269 
 
 Platform Stairs, Handrailing Large 
 Cylinder 271 
 
674 
 
 INDEX. 
 
 PAGE I 
 
 Platform Stairs, Railing Where 
 
 Rake Meets Level 272 
 
 Platform Stairs, Twist-Rail, Cut- 
 ting of 277 
 
 Platform Stairs, Wreath of Round 
 
 Rail 267 
 
 Point of Contact (Geometry) 558 
 
 Point Denned (Geometry) 544 
 
 Pointed Style, Ecclesiastical Arch- 
 itecture ii 
 
 Polygons, Angles of 462 
 
 Polygons, Circumscribed and In- 
 scribed Circles, Radius of 460 
 
 Polygons Defined (Geometry). . . . 546 
 Polygons, Equilateral Triangle. .. 445 
 
 Polygons, General Rules 461 
 
 Polygons, Irregular, Trigon (Ge- 
 ometry) 546 
 
 Polygons, Perpendicular in Tri- 
 angle 440 
 
 Polygon, Regular, Defined (Geom- 
 etry) 546 
 
 Polygons, Regular, To Describe 
 
 (Geometry) 573 
 
 Polygons, Regular, To Inscribe in 
 
 Circle (Geometry) 572 
 
 Polygons, Sum and Difference, 
 
 Two Lines 439 
 
 Polygons, Table Explained 466 
 
 Polygons, Table of Multipliers. .. 465 
 Polygons, Triangle, Altitude of.. 442 
 Polygonal Dome, Shape of Angle- 
 Rib 223 
 
 Posts, Area, To Find 86 
 
 Posts, Diameter, To Find 92 
 
 Posts, Rectangular, Safe Weight.. 92 
 Posts, Rectangular.To Find Thick- 
 ness 94 
 
 Posts, Rectangular, Breadth Less 
 
 than Thickness 96 
 
 Posts, Rectangular, To Find 
 
 Breadth 95 
 
 Posts, To Find Side 93 
 
 Posts, Slender, Safe Weight for. . 91 
 Posts, Stout. Crushing Strength of. 89 
 
 Pressures Classified 85 
 
 Pressure, Oblique, Resolution of 
 Forces 59 
 
 Pressure, Triangular, Framed 
 
 Girders 171 
 
 Pressure, Upper Chord, Triangu- 
 lar Girder 168 
 
 Prisms Cut by Oblique Plane 259 
 
 Prisms and Cylinders, Stair-Build- 
 ing 257 
 
 Prisms Defined (Stairs) 257, 259 
 
 Prism, Top, Form of, in Perspec- 
 tive 259 
 
 Profile of Mouldings Defined. . . . 326 
 Progression, Arithmetical (Alge- 
 bra) 432 
 
 Progression.Geometrical (Algebra) 435 
 Projection and Height, Members 
 
 of Orders of Architecture 16 
 
 Protractor, Useful in Drawing. .. 541 
 Purlins and Jack-Rafters in Roof. 211 
 
 Purlins, Location of 212 
 
 Pyramids of Memphis, Amount of 
 
 Stone in 33 
 
 Pycnostyle, Explanation of 20 
 
 Quadrangle Defined 545 
 
 Quadrangle Equal Triangle 353 
 
 Quadrant Defined 547 
 
 Quantities, Addition and Sub- 
 traction (Algebra) 424 
 
 Quantities, Division of (Algebra). 424 
 Quantities, Multiplication of (Al- 
 gebra) 424 
 
 Queen-Post, Location of 213 
 
 Queen-Post in Roof 54 
 
 Radials of Rib in Bridge 226 
 
 Radials of Rib for Wedges 226 
 
 Radicals, Extraction of (Algebra). 425 
 
 Radius of Arc, To Find 561 
 
 Radius of Circle Defined 547 
 
 Rafters, Braces, etc., Length, To 
 
 Find 580 
 
 Rafters, Least Thrust, Rule for. . . 62 
 
 Rafters, Length of, To Find 578 
 
 Rafters, Liability to Crush Other 
 
 Materials 205 
 
 Rafters, Liability to Being Crushed 205 
 
 Rafters, Liability to Flexure 205 
 
 Rafters, Minimum Thrust of 62 
 
INDEX. 
 
 6/5 
 
 PAGE 
 
 Rafters in Roof, Effect of Weight 
 on 179 
 
 Rafters in Roof, Strains Subjected 
 to , 205 
 
 Rafters and Tie-Beams, Safe 
 Weight 87 
 
 Rafters, Uses in Roof 54 
 
 Rake in Cornice Matched with 
 Level Mouldings 344 
 
 Railing, Platform Stairs Rake 
 Meets Level 272 
 
 Ratio or Proportion, Equals Mul- 
 tiplied 367 
 
 Ratio or Proportion, Equality of 
 Products 370 
 
 Ratio or Proportion, Equality of 
 Ratios 367 
 
 Ratio or Proportion Equation, 
 Form of 367 
 
 Ratio or Proportion, Examples.. . 366 
 
 Ratio or Proportion, Four Propor- 
 tionals, to Find 377 
 
 Ratio or Proportion, Homologous 
 Triangles 370 
 
 Ratio or Proportion, Lever Prin- 
 ciple in 372 
 
 Ratio or Proportion, Lever Prin- 
 ciple Demonstrated 375 
 
 Ratio or Proportion, Multiply an 
 Equation 368 
 
 Ratio or Proportion, Multiply and 
 Divide One Number 368 
 
 Ratio or Proportion, Rule of 
 Three 366 
 
 Ratio or Proportion, Steelyard as 
 Example in 371 
 
 Ratio or Proportion, Terms of 
 Quantities 367 
 
 Ratio or Proportion, Transfer a 
 Factor 369 
 
 Rectangle Defined 545 
 
 Rectangle, Equilateral, To De- 
 scribe 58 
 
 Rectangular Cross-Section, Iron 
 Beams 145 
 
 Reduction Cubic Feet to Gallons, 
 Rule 653 
 
 Reduction Decimals, Examples.. 647 
 
 Reflected Light, Opposite of 
 Shade 611 
 
 Regular Polygon in Circle, To In- 
 scribe (Geometry) 572 
 
 Regular Polygon Defined (Geom- 
 etry) 546 
 
 Regular Polygons, To Describe 
 (Geometry) 573 
 
 Resistance, Capability of 86 
 
 Resistance to Compression, Ap- 
 plication of Pressure 85 
 
 Resistance to Compression, 
 Crushing and Bending 85 
 
 Resistance to Compression, Mate- 
 rials 77 
 
 Resistance to Compression, Pres- 
 sure Classified 85 
 
 Resistance to Compression in 
 Proportion to Depth 101 
 
 Resistance to Compression, Stout 
 Posts, Rule 89 
 
 Resistance to Compression, Table 
 of Woods 79 
 
 Resistance to Cross-Strains 77 
 
 Resistance to Cross-Strains De- 
 fined 99 
 
 Resistance to Deflection, Rule... . 113 
 
 Resistance Depending on Com- 
 pactness and Cohesion 78 
 
 Resistance Depending on Loca- 
 tion, Soil, etc 79 
 
 Resistance to Flexure Defined. . . 145 
 
 Resistance Inversely in Propor- 
 tion to Length 102 
 
 Resistance to Oblique Force 206 
 
 Resistance, Power of, Hew Ob- 
 tained 78 
 
 Resistance, Proportion to Area. . . 86 
 
 Resistance, Strains, To What Due. 78 
 
 Resistance to Tension Greatest 
 in Direction of Length 81 
 
 Resistance to Tension, Proportion 
 in Materials 81 
 
 Resistance to Tension, Table of 
 Materials 82 
 
 Resistance to Tension, Materials. 77 
 
 Resistance to Tension, Results 
 from Transverse Strains . . 82 
 
6;6 
 
 INDEX. 
 
 PAGE 
 
 Resistance to Transverse Strains, Roman Architecture, Ruins of. . . n 
 
 Table of. 83 Roman Architecture, Excess of 
 
 Resistance to Transverse Strains, I Emichment 46 
 
 Description of Table 84 Roman Building 26 
 
 Resistance Variable in One Ma- Roman Composite and Corinthian 
 
 terial.... 79 Orders.. . 28 
 
 Reticulated Walls . < 49 j Roman Decoration 27 
 
 Rhomboid Defined 546 Roman Empire, Overthrow of. ... 13 
 
 Romans, Ionic Order Modified by 27 
 
 Roman Moulding, Cavetto 329 
 
 Roman Mouldings, Comments on. 329 
 Roman Moulding, Ovolo 329 
 
 Rhombus Defined ' . . . . 545 
 
 Ribbed Bridge, Area of Rule 225 
 
 Ribbed Bridge, Built 224 
 
 Ribbed Bridge, Least Rise, Table 
 
 of 224 
 
 Right Angle Defined 348, 544 
 
 Right Angle in Semicircle (Ge- 
 ometry) 355 
 
 Right Angle, To Trisect a 554 
 
 Right Angled Triangle Defined . . 545 
 Right Angled Triangle, Squares 
 
 on (Algebra) 417 
 
 Right Angled Triangles (Trigo- 
 nometry) 510 
 
 Right Angled Triangles, Formula 
 
 for (Trigonometry) 530 
 
 Right Lines (Geometry) 584 
 
 Right Line Equal Circumference. 566 
 Right Lines, Mean Proportionals 
 
 Between 584 
 
 Right Lines, Two Given, Find 
 
 Third 582 
 
 Right Lines, Three Given, Find 
 
 Fourth 583 
 
 Right or Straight Line Defined. . . 544 
 
 Right Prism Defined (Stairs) 257 
 
 Risers, Number of, Rule to Ob- 
 tain (Stairs) 246 
 
 Rise and Tread (Stairs) 241 
 
 Rise and Tread, Connection of 
 
 (Stairs) 248 
 
 Rise and Tread, Blondel's Method 
 
 of Finding (Stairs) 242 
 
 Rise and Tread, Table of, for 
 
 Shops and Dwellings (Stairs). . . 245 
 Rise and Tread, To Obtain (Wind- 
 ing Stairs) 251 
 
 Rolled Iron Beams, Extensive Use 
 
 Roman Mouldings, Forms of.. ... 325 
 
 49 
 26 
 
 13 
 
 of. 
 
 161 
 
 Roman Architecture Defined. .... 7 
 
 Roman Pantheon, etc., Walls of. . 
 Roman Styles of Architecture. . . . 
 Roman Styles Spread by the Ital- 
 ians 
 
 Romantic or German Style, Thir- 
 teenth and Fourteenth Centu- 
 ries ii 
 
 Rome, Ancient Buildings of. 12 
 
 Rome and Greece, Architecture 
 
 of 8 
 
 Roof, The 54 
 
 Roofs, Ancient Norman and 
 
 Gothic, Construction 178 
 
 Roof Beams, Weight per Super- 
 ficial Foot 189 
 
 Roof, Brace in, Rule Same as for 
 
 Rafter 208 
 
 Roofs, Construction of. 55 
 
 Roof Covering, Mode of 188 
 
 Roof Covering, Weights, Table of. 191 
 
 Roof, Curb or Mansard 54 
 
 Roofs, Diagrams and Description 
 
 of 212 
 
 Roof, Gothic Buildings 55 
 
 Roofs, Gothic and Norman Puild- 
 
 ings, Construction 178 
 
 Roofs, Hip, Diagram and Exj 
 
 nation 215 
 
 Roof, Hip 54 
 
 Roof, Horizontal Pressure, To Re- 
 move from 74 
 
 Roof, Jack-Rafters and Purlins.. . 211 
 Roof, King-Post in 54 
 
 Roof, Load per Foot Horizontal, 
 
 Rule, 
 
 IQ2 
 
INDEX. 
 
 6 77 
 
 Roof, Load, Total per Foot Hori- 
 zontal, Rule 197 
 
 Roofs, Modern, Trussing Neces- 
 sary 178 
 
 Roofs, Norman and Gothic Build- 
 ings, 178 
 
 Roof, Pent, To Find 54 
 
 Roof, Perpendicular Height, To 
 
 Find 579 
 
 Roof Plank, Weight per Super- 
 ficial Foot 189 
 
 Roof, Planning a 188 
 
 Roof, Pressure on 55 
 
 Roof, Queen-Post in 54 
 
 Roof, Rafters in 54 
 
 Roof, Sagging, To Prevent. ...... 54 
 
 Roof, Slope Should Vary Accord- 
 ing to Climate 191 
 
 Roof Supports. Distance between. 189 
 Roof, Suspension Rods, Safe 
 
 Weight for 210 
 
 Roof, Tie-Beam in 54 
 
 Roof, Tie-Beam, Tensile Strain, 
 
 Rule ,. 204 
 
 Roof Timbers, Mortising 55 
 
 Roof Timbers, Scarfing of 55 
 
 Roof Timbers, Splicing of 55 
 
 Roof Timbers, Strains by Parallel- 
 ogram of forces 198 
 
 Roof Timbers, Strain Shown Ge- 
 ometrically 199, 202 
 
 Roof Truss, Arched Ceiling 214 
 
 Roof Truss, Elevated Tie-Beam 
 
 Objectionable 214 
 
 Roof Truss, Elevating Tie-Beam, 
 
 Effect of 187 
 
 Roof Truss, Force Diagram, Figs. 
 
 59, 68, and 69 179 
 
 Roof Truss, Force Diagram, Figs. 
 
 60, 70, and 7 r 180 
 
 Roof Truss, Force Diagram, Figs. 
 
 61, 72, and 73 181 
 
 Roof Truss, Force Diagram, Figs. 
 
 63,74, and 75 183 
 
 Roof Truss, Force Diagram, Figs. 
 
 64, 77, and 78 184 
 
 Roof Truss, Force Diagram, Figs. 
 
 66, 80, and 81 186 
 
 Roof Truss, Load on 189 
 
 Roof Trusses, Strains, Effect of, on 
 
 Different 179 
 
 Roof Truss, Weights, Table of, per 
 
 Superficial Foot 189 
 
 Roof Truss, Weight per Superfi- 
 cial Foot 190 
 
 Roof, Trussing in 54 
 
 Roof Trussing, Designs for 178 
 
 Roof Trussing, Framing for 237 
 
 Roof Trussing, Hogged Ridge.... 238 
 Roof Trussing, King-Post, Effect 
 
 of Bad Framing on 237 
 
 Roofs, United States 55 
 
 Roof, Vertical Pressure of Wind 
 
 on, Effect of. 194 
 
 Roof, Snow, Weight per Horizon- 
 tal Foot 193 
 
 Roof Weight on Rafter, Effect of.. 179 
 Roof, Wind, Horizontal and Verti- 
 cal Pressure of 193 
 
 Roofing, Weight of Horizontal and 
 
 Inclined 190 
 
 Roofing, Weight per Superficial 
 
 Foot 190 
 
 Roots, Cubes, and Squares, Table 
 
 of 638-645 
 
 Round Post, Area of 90 
 
 Rubble Walls 48 
 
 Rulers and Pencil in Drawing.... 540 
 Rupture Compared with Flexure. 84 
 Rupture, Crushing, Safe Weight.. 89 
 Rupture, Sliding, Safe Weight. ... 87 
 Rupture, Transverse, Safe Weight. 86 
 Rusting Iron Framing Straps, To 
 Prevent 239 
 
 Safe Load for Material 81 
 
 Safe Weight, Allowance for 84 
 
 Safe Weight at Any Point, Rule. . 106 
 
 Safe Weight, Beam at Middle 103 
 
 Safe Weight, Bending 91 
 
 Safe Weight, Beam, Breadth of, 
 
 To Find 104 
 
 Safe Weight, Beam, Depth, To 
 
 Find 104 
 
 Safe Weight, Breadth or Depth, To 
 
 Find, Load at Middle 106 
 
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68o 
 
 INDEX. 
 
 PAGE 
 
 247 
 240 
 
 247 
 
 Stairs, Space for Timber and Plas- 
 ter 
 
 Stairs, Stone, Public Building 
 
 Stairs, String of, To Make 
 
 Stairs, Tread, To Find, Rise Given 
 242, 246 
 
 Stairs, Tread and Riser Connec- 
 tion 248 
 
 Stairs, Width, Rule for 241 
 
 Stairs, Winding, Balusters in 
 Round Rail 313 
 
 Stairs, Winding, Bevels in Splayed 
 Work ... 314 
 
 Stairs, Winding, Blocking Out 
 Rail.. 301 
 
 Stairs, Winding, Butt-joint on 
 Handrail 303 
 
 Stairs, Winding, Butt-joint, Cor- 
 rect Lines for 307 
 
 Stairs, Winding, Diagrams Ex- 
 plained .. 263 
 
 Stairs, Winding, Face Mould, Ac- 
 curacy of 295 
 
 Stairs, Winding, Face Mould, 
 Application 297 
 
 Stairs, Winding, Face Mould, 
 Care in Drawing 295 
 
 Stairs, Winding, Face Mould, 
 Curves Elliptical 301 
 
 Stairs, Winding, Face Mould 
 for 290, 293 
 
 Stairs, Winding, Face Mould, 
 Round Rail 303 
 
 Stairs, Winding, Face Mould for 
 Twist \ . . 291 
 
 Stairs, Winding, Flyers and 
 Winders 251 
 
 Stairs, Winding, Front String, 
 Grade of 253 
 
 Stairs, Winding, Handrailing. 256, 289 
 
 Stairs, Winding, Handrailing, Bal- 
 usters Under Scroll 310 
 
 Stairs, Winding, Handrailing, 
 Centres for Square 368 
 
 Stairs, Winding, Handrailing, 
 Face Mould for Scroll 311 
 
 Stairs, Winding, Handrailing, Fall- 
 ing Mould for Raking Scroll. . . 310 
 
 Stairs, Winding, Handrailing, Gen- 
 eral Considerations 258 
 
 Stairs, Winding, Handrailing, 
 Scrolls for 308 
 
 Stairs, Winding, Handrailing, 
 Scroll Over Curtail Step 309 
 
 Stairs, Winding,, Handrailing, 
 Scroll for Curtail Step 310 
 
 Stairs, Winding, Scroll at Newel. 309 
 
 Stairs, Winding, Illustrations by 
 Planes 261 
 
 Stairs, Winding, Moulds for 
 Quarter Circle 255 
 
 Stairs, Winding, Newel Cap, Form 
 of 312 
 
 Stairs, Winding, Objectionable. . 240 
 
 Stairs, Winding, Pitch Board, To 
 Obtain 252 
 
 Stairs, Winding, Rise and Tread, 
 To Obtain 251 
 
 Stairs, Winding, Sliding of Face 
 Mould. 299 
 
 Stairs, Winding, String, To Ob- 
 tain 252 
 
 Stairs, Winding, Timbers, Posi- 
 tion of 252 
 
 Stairs and Windows, How. Ar- 
 ranged 42 
 
 Stiles of Windows, Allowance for. 319 
 
 St. Mark, Tenth or Eleventh Cen- 
 tury 12 
 
 Stone Bridge Building, Truss 
 Work 232 
 
 Stone Bridge, Building Arch.... 230 
 
 Stone Bridge, Centres for, Con- 
 struction 229 
 
 Stone Bridge, Pressure on Arch 
 Stones 230 
 
 Stop for Doors 317 
 
 Stores, Floor Beams, Safe Weight. 126 
 
 Stores, Ordinary, Floor-Beams, 
 Sizes, To Find 127 
 
 Stores, First-Class, Floor-Beams, 
 Sizes, To Find 128 
 
 St. Paul's, London, Dome of 54 
 
 St. Peter's, Rome, Fourteenth and 
 Fifteenth Centuries 12 
 
 Straight or Right Line Defined. . . 544 
 
INDEX. 
 
 681 
 
 Strains, Cross, Resistance to 77 
 
 Strains on Domes, Tendency of. . 219 
 
 Strains Exceed Weights 61 
 
 Strains, Graphic Representation.. 165 
 Strain Greatest at Middle of Beam. 105 
 Strains by Parallelogram of 
 
 Forces 165 
 
 Strains, Practical Method cf De- 
 termining 62 
 
 Strains of Rafter in Roof 205 
 
 Strains, Resistance, To What Due. 78 | 
 Strain on Roof Timbers Shown 
 
 Geometrically 190 
 
 Strains on Roof Timbers Geomet- 
 rically Applied 202 j 
 
 Strains on Roof Timbers, Parallel- 
 ogram of Forces 198 ; 
 
 Strain, Shearing, Tubular Iron 
 
 Girder 157 
 
 Strain Unequal, Cause of 83 
 
 Straps, Iron, Roof Truss 239 j 
 
 Strassburg, Cathedral of 12 j 
 
 Strassburg, Towers of the Min- 
 ster II I 
 
 Strength and Stiffness of Mate- 
 rials..- ."... 78 j 
 
 Structure of Materials 76 | 
 
 Struts Denned 173 j 
 
 Struts and Ties 68 i 
 
 Struts and Ties, Difference Be- 
 tween 69 : 
 
 St. Sophia, Sixth Century 12 i 
 
 Stucco Cornice for Interior 340 ; 
 
 Studs and Joists Defined 174 ' 
 
 Styles, Grecian, Only Known by 
 
 Them 16 ! 
 
 Stylobate, Substructure 'for Col- 
 umns 14 
 
 Subnormal and Normal (in Para- 
 bola) 496 I 
 
 Subtangent, Parabola 496 
 
 Subtangent of Ellipse Defined... 486 
 Subtraction and Addition (Alge- 
 bra) 398 
 
 Superficies Defined (Geometry)... 544 
 Supports, Girders, Length, Rule.. 157 
 
 Supports, Position of 65 
 
 Supports, Inclination of, Unequal. 60 
 
 Suspension Rods, Location in 
 Roof 212 
 
 Suspension Rods in Roof, Safe 
 Weight 210 
 
 Symbols Chosen at Pleasure (Al- 
 gebra) 395 
 
 Symbols, Transferring (Algebra). 399 
 
 Systyle, Explanation of 20 
 
 Table of Circles 649-652 
 
 Table of Contents 6^3-624 
 
 Table of Capacity of Wells, Cis- 
 terns, etc 653 
 
 Table of Squares, Cubes, and 
 
 Roots 638-645 
 
 Table of Woods, Description of. . 80 
 
 Tail-Beams Defined 130 
 
 Tanged Curve, To Describe (Ge- 
 ometry) 565 
 
 Tangent to Axes, Ellipse . . . . 485 
 
 Tangent Defined 547 
 
 Tangent with Foci, Ellipse 487 
 
 Tangent to Ellipse. To Draw 592 
 
 Tangent at Given Point in Cir- 
 cle 557 
 
 Tangent at Given Point, Without 
 
 Centre 557 
 
 Tangent of Parabola 493 
 
 Tangents and Sines, Logarithms 
 
 (Polygons) 464 
 
 Temples Built in the Doric Style. 19 
 
 Temple, Doric, Origin of the 17 
 
 Temple of Jupiter at Thebes 33 
 
 Tenons and Splices, Knowledge 
 
 Important 88 
 
 Tensile Strain, Area of Piece, To 
 
 Find 99 
 
 Tensile Strain, Compressed Ma- 
 terial ioo 
 
 Tensile Strain, Condition of Sus- 
 pended Piece 98 
 
 Tensile Strain, Safe Weight 96 
 
 Tensile Strain, Safe Weight, To 
 
 Compute 97 
 
 Tensile Strain, Sectional Area, To 
 
 Obtain 97 
 
 Tensile Strain, Suspended, Mte- 
 terial Extended ioo 
 
682 
 
 INDEX. 
 
 PAGE 
 
 Tensile Strain on Tie-Beam in Roof 
 
 Truss 204 
 
 Tensile Strain, Weight of Suspend- 
 ed Piece 98 
 
 Tensile Strength of Cast Iron 161 
 
 Tension and Compression, Framed 
 
 Girders 174 
 
 Tension, Resistance to 77 
 
 Tension, Resistance to, Table of 
 
 Materials 82 
 
 Tension, Resistance to, Results 
 
 Obtained 82 
 
 Tension, Resistance to, Proportion 
 
 in Materials 81 
 
 Tent, Habitation of the Shepherd. 13 
 Testing Machine, Description in 
 
 Transverse Strains 80 
 
 Testing Materials, Hydraulic 
 
 Method 80 
 
 Testing Materials, Dates of 80 
 
 Testing Materials, Manner of. ... 80 
 
 Tetragon Defined 546 
 
 Tetragon, Radius of Circles 
 
 (Polygons) 446 
 
 Tetrastyle, Intercolumniation 20 
 
 Thebes, Thickness of Walls at. . . 33 
 
 Thrust, Horizontal 63 
 
 Thrust, Horizontal, Examples. ... 64 
 Thrust, Horizontal, Tendency of.. 88 
 Tie-Beam in Ceiling, Load on... 190 
 Tie-Beam and Rafter, Safe Weight. 87 
 
 Tie-Beam in Roof 54 
 
 Tie-Beam in Roof, Tensile Strain. 204 
 
 Tie-Rods, Diameter, To Find 164 
 
 Tie-Rods, Floor Arches, Dwell- 
 ings 153 
 
 Tie-Reds, Floor Arches, First- 
 
 Class Stores 153 
 
 Tie-Rods, Wrought Iron 164 
 
 Ties Defined 173 
 
 Ties and Struts, To Distinguish.. 69 
 Ties and Struts, Framed Girders.. 174 
 Ties and Struts, Principles of. ... 68 
 Ties, Timbers in a State of Ten- 
 sion 68 
 
 Titus, Composite Arch of 28 
 
 Trimme/, Breadth, To Find, Two 
 Sets Tail-Beams 134 
 
 Top Rail, Doors, Width, Rule 317 
 
 Torus, Classic Moulding 323 
 
 Torus, Grecian Moulding. 326 
 
 Tower of Babel, History of 5 
 
 Towers of the Minster, Strassburg. n 
 
 Transverse Axis Defined 548 
 
 Transverse Strains, Compressed 
 
 and Extended, Material 100 
 
 Transverse Strains, Defined 99 
 
 Transverse Strains, Explanation 
 
 of Table III 101 
 
 Transverse Strains, Greater 
 
 Strength of One Piece 101 
 
 Transverse Strains, Neutral Line 
 
 Defined 100 
 
 Transverse Strains, Proportion to 
 
 Breadth 101 
 
 Transverse Strains, Hatfield's, 
 
 Reference to. .80, 121, 133, 138, 
 
 143, 144, 145, 146, 148 
 Transverse Strains, Resistance to, 
 
 Table of 83 
 
 Transverse Strains, Description of 
 
 Table 84 
 
 Transverse Strains, Strength Di- 
 minished by Division 101 
 
 Trapezoid Defined 546 
 
 Trapezium Defined 546 
 
 Tread, To Find, Rise Given 
 
 (Stairs) 242,246 
 
 Tread and Nosing, Position of 
 
 (Stairs) 241 
 
 Tread and Rise, To Find, Winding 
 
 Stairs 251 
 
 Tread and Rise, To Find, Blon- 
 
 del's Method 242 
 
 Tread and Rfcse, Table for Shops \ 
 
 and Dwellings 245 
 
 Tread and Riser, Connection of 
 
 (Stairs) 248 
 
 Triangle, Altitude of (Polygons). 442 
 Triangles, Base, Formula for (Trig- 
 onometry) 516 
 
 Triangle, Construct a (Geometry). 587 
 Triangle, Construct Equal-Sided 
 
 (Geometry) 575 
 
 Triangle Defined 545 
 
 Triangle, Examples (Geometry).. . 350 
 
INDEX. 
 
 68 3 
 
 PAGE 
 
 Triangles, Equal Altitude 361 
 
 Triangle Equal Quadrangle 353 
 
 Triangles, Equation of (Trigo- 
 nometry) 515 
 
 Triangles, Homologous (Geom- 
 etry) 362 
 
 Triangles, Hypothenuse, Formula 
 for 516 
 
 Triangles, Hypothenuse, To Find 
 (Trigonometry) 518 
 
 Triangles, Perpendicular, To Find 
 (Trigonometry) 517 
 
 Triangle or Set-Square in Draw- 
 ing 539 
 
 Triangle or Set-Square, Use of. . . 541 
 
 Triangles, Terms Denned (Trigo- 
 nometry) 512 
 
 Triangles, Three Angles Equal 
 Right Angle 354 
 
 Triangles, Value of Sides (Trigo- 
 nometry) 516 
 
 Trigon, Irregular Polygons (Ge- 
 ometry) 546 
 
 Trigon, Radius of Circle (Poly- 
 gons) 443 
 
 Trigon, Rule (Polygons) 441 
 
 Trigonometry, Oblique Triangles, 
 Two Angles 523 
 
 Trigonometry, Oblique Triangles, 
 Two Sides 521 
 
 Trigonometry, Oblique Triangles, 
 First Class 520 
 
 Trigonometry, Oblique Triangles, 
 Second Class 522 
 
 Trigonometry, Oblique Triangles, 
 Third Class 526 
 
 Trigonometry, Oblique Triangles, 
 Fourth Class 
 
 Trigonometry, Oblique Triangles, 
 Sines and Sides 
 
 Trigonometry, Oblique Triangles, 
 Formula, First Class 531 
 
 Trigonometry, Oblique Triangles, 
 Formula, Second Class 532 
 
 Trigonometry, Oblique Triangles, 
 Formula, Third Class 534 
 
 Trigonometry, Oblique Triangles, 
 Formula, Fourth Class 534 
 
 S*\ 
 
 519 
 
 Trigonometry, Right Angled Tri- 
 angles 510 
 
 Trigonometry, Right Angled Tri- 
 angles, Third Side, To Find... 511 
 
 Trigonometry, Right Angled Tri- 
 angle, Formula 530 
 
 Trigonometry, Tables 513 
 
 Trigonometry, Triangles, Base, 
 Formula for 516 
 
 Trigonometry, Triangles, Equa- 
 tions of 515 
 
 Trigonometry, Triangles, Hypoth- 
 enuse, Formula 516 
 
 Trigonometry, Triangles, Hypoth- 
 enuse, To Find 518 
 
 Trigonometry, Triangles, Perpen- 
 dicular, To Find 517 
 
 Trigonometry, Triangles, Terms 
 Denned 512 
 
 Trigonometry, Triangles, Value of 
 Sides 516 
 
 Trimmer or Carriage Beam, 
 Breadth, To Find 132 
 
 Trimmer or Carriage Beams De- 
 nned 130 
 
 Trimmer, One Header, Breadth, 
 To Find, Dwellings and Stores. 133 
 
 Trimmer, Well-Hole in Middle, 
 Breadth, To Find 136 
 
 Trisect a Right Angle 5^4 
 
 Truss, Diagram of. 200 
 
 Truss, Force Diagrams, Figs. 59, 
 
 68 and 69 179 
 
 Figs. 60, 70 and 71 iSo 
 
 Figs. 61, 72 and 73 181 
 
 Figs. 63, 74 and 75 183 
 
 Figs. 64, 77 and 78 184 
 
 Figs. 65, 78 and 79 185 
 
 Figs. 66, 80 and 81 186 
 
 Truss, Roof, Framing for 237 
 
 Truss, Roof, Iron Straps 239 
 
 Truss, Weight, per Horizontal 
 Foot, To Find 192 
 
 Truss Work, Stone Bridge Build- 
 ing 232 
 
 Trussing and Framing, Gravity 
 and Resistance 76 
 
 Trussing Partitions, Effect of 175 
 
684 
 
 INDEX. 
 
 Trussing Roofs, Effect of 178 
 
 T-Square, How to Make 539 
 
 Tubular Iron Girder, Area of Bot- 
 tom Flange, Dwellings 159 
 
 Tubular Iron Girder, Area of Bot- 
 tom Flange, First-Class Stores. 160 
 
 Tubular Iron Girder, Arc of 
 Flange, Load at Middle 154 
 
 Tubular Iron Girder, Area of 
 Flange, Load Any Point 155 
 
 Tubular Iron Girder, Area of 
 Flange, Load Uniform 156 
 
 Tubular Iron Girder, Flanges, 
 Proportion of. 157 
 
 Tubular Iron Girder, Construction 
 
 of 154 
 
 Tubular Iron Girder, Rivets, Al- 
 lowance for 157 
 
 Tubular Iron Girder, Shearing 
 
 Strain 157 
 
 Tubular Iron Girders, Web of. . . 158 
 Tuscan, Modern, Appropriate for 
 
 Buildings 30 
 
 Tuscan Order, Introduction of the. 30 
 Tuscan Order, Principal Style in 
 
 Italy 30 
 
 Twelfth Century, Buildings in the. n 
 
 Twist Rail, Platform Stairs 277 
 
 Twists, Stairs, Nicholson's Method 
 for 259 
 
 Undecagon Defined 
 
 United States, Roofs in. 
 
 Vault, Simple, Groined or Com- 
 plex 
 
 Ventilation, Proper Arrangement 
 for...!. 
 
 Versed Sine of Arc, To Find 
 
 Vertical Pressure of Wind on 
 Roof. 
 
 Vertical Tangent of Parabola De- 
 fined 
 
 Volutes, To Describe the 
 
 Voussoir of an Arch. . , 
 
 Wall, The... 
 Walls, Coffer. 
 
 546 
 
 55 
 
 52 
 
 45 
 56i 
 
 194 
 
 495 ! 
 
 20 * 
 
 52 I 
 
 | 
 
 48 I 
 49! 
 
 PAGE 
 
 Walls, Construction and Forma- 
 tion 48 
 
 Walls, Eddystone and Bell Rock 
 
 Lighthouses 48 
 
 Walls, Egyptian, Massiveness of.. 33 
 
 Walls, Modern Brick 49 
 
 Walls of Pantheon and Roman 
 
 Buildings 49 
 
 Walls of Pantheon at Rome 53 
 
 Walls, Pise, of France 49 
 
 Walls, Reticulated 49 
 
 Walls, Rubble -. ... 48 
 
 Walls, Strength of. 48 
 
 Walls, Various Kinds 49 
 
 Walls, Wooden 49 
 
 Weakening Girder, Manner of. . . 140 
 Web of Tubular Iron Girder, 
 
 Thickness of 158 
 
 Weight of Materials for Building 
 
 Table of 654-656 
 
 Wells, Cisteins, etc., Table of 
 
 Capacity 653 
 
 White Pine, Weights of Beams 
 
 Table of 177 
 
 Wind, Greatest Pressure, per Su- 
 perficial Foot 90 
 
 Wind on Roof, Effect of Vertical 
 
 Pressure 194 
 
 Wind on Roof, Horizontal and 
 
 Vertical Pressure 193 
 
 Winders in Stairs, How to Place 
 
 the 42 
 
 Winders and Flyers, Stairs 251 
 
 Windows, Arrangement of 44 
 
 Windows, Circular Headed 320 
 
 Windows, Circular Headed, To 
 
 Form Soffit 321 
 
 Windows, Dimensions, To Find. 318 
 
 Window-Frame, Size of 318 
 
 Windows, Front of Building, Ef- 
 fect of 320 
 
 Windows, Heights, Table of, 
 
 Width Given 320 
 
 Windows, Height from Floor. . . . 320 
 Windows, Inside Shutters, Re- 
 quirement 319 
 
 Windows, Position and Light 
 from.. 317 
 
INDEX. 
 
 685 
 
 Windows and Stairs, How Ar- 
 ranged 42 
 
 Windows, Stiles, Allowance lor.. 319 
 
 Windows, Width Uniform, Height 
 Varying . . . : 319 
 
 Winding Stairs, Balusters in 
 Round Rail 313 
 
 Winding Stairs, Bevels in Splayed 
 Work 314 
 
 Winding Stairs, Blocking Out 
 Rail.. 301 
 
 Winding Stairs, Butt Joint, Posi- 
 tion of. 303 
 
 Winding Stairs, Butt Joint 307 
 
 Winding Stairs, |Diagram of, Ex- 
 plained 263 
 
 Winding Stairs, Face Mould for 
 290,293 
 
 Winding Stairs, Face Mould, Ac- 
 curacy of. 295 
 
 Winding Stairs, Face Mould, Ap- 
 plication of 297 
 
 Winding Stairs, Face Mould, 
 Curves Elliptical 301 
 
 Winding Stairs, Face Mould, 
 Drawing 296 
 
 Winding Stairs, Face Mould, 
 Round Rail 303 
 
 Winding Stairs, Face Mould, Slid- 
 ing of. 299 
 
 Winding Stairs, Face Mould for 
 Twist 291 
 
 Winding Stairs, Flyers and Wind- 
 ers 251 
 
 Winding Stairs, Front String, 
 Grade of. 253 
 
 Winding Stairs, Handrailing 
 . ; 256, 289 
 
 Winding Stairs, Handrailing, nal- 
 
 usters Under Scroll 310 
 
 Winding Stairs, Handrailing, Cen- 
 tres in Square 308 
 
 Winding Stairs, Handrailing, Face 
 
 Mould for Scroll. .., 311 
 
 Winding Stairs, Handrailing, Fall- 
 ing Mould. . . 310 
 
 Winding Stairs, Handrailing, 
 
 General Considerations 258 
 
 Winding Stairs, Handrailing, 
 
 Scrolls for 308 
 
 Winding Stairs, Handrailing, 
 
 Scroll Over Curtail Step 309 
 
 Winding Stairs, Handrailing, 
 
 Scroll for Curtail Step 310 
 
 Winding Stairs, Handrailing, 
 
 Scrolls at Newel 309 
 
 Winding Stairs, Illustrations by 
 
 Planes 261 
 
 Winding Stairs, Moulds for Quar- 
 ter Circle 255 
 
 Winding Stairs, Newel Cap, Form 
 
 of 312 
 
 Winding Stairs Objectionable. . . . 240 
 Winding Stairs, Pitch Board, To 
 
 Obtain 252 
 
 Winding Stairs, Rise and Tread, 
 
 To Obtain 251 
 
 Winding Stairs, String, To Obtain. 252 
 Winding Stairs, Timbers, Posi- 
 tion of. 252 
 
 Wood, Destruction by Fire 37 
 
 Wooden Beams, Use Limited 154 
 
 Woods, Hydraulic Method of 
 
 Testing 80 
 
 Wreath for Round Rail, Platform 
 Stairs 267 
 
 THE END. 
 
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