OF 4 NEW THEOREMS, TABLES, AND DIAGRAMS, FOR THE COMPUTATION OF EARTH-WORIK. DESIGNED FOR THIE USE OF ENGINEERS IN PRELIMINARY AND FINAL ESTIMATES, OF STUDENTS IN ENGINEERING, AND OF CONTRACTORS AND OTHER NON-PROFESSIONAL COMPUTERS. IN TWO PARTS, WITH AN APPENDIX. PART I.-A PRACTICAL TREATISE; PART II. —A THEORETICAL TREATISE; AND THE APPENDIX. CONTAINING NOTES TO THE RULES AND EXAMPLES OF PART I,; EXPLANATIONS OF THE CONSTRUCTION OF SCALES, TABLES, AND DIAGRAMIS, AND A TREATISE UPON EQUIVALENT SQUARE BASES AND EQUIVALENT LEVEL HEIGHTS. THE WHOLE ILLUSTRATED BY NUMEROUS ORIGINAL ENGRAVINGS, COMPRISING EXPLANATORY CUTS FOR DEFINITIONS AND PROBLEMS, STEREOMETRIC SCALES AND DIAGRAMS, AND A SERIES OF LITHOGRAPHIC DRAWINGS FROM MODELS; SHOWING ALL THE COMBINATIONS OF SOLID FORMS WHIICI OCCUR IN RAILROAD EXCAVATIONS AND EMBANKIMENTS. BY JOHN WARNER, A.M. MINING AND MECHANICAL ENGINEER; AUTHOR OF STUDIES IN ORGANIC MORPHOLOGY. PHILADELPHIA: J. B. LIPPINCOTT & CO. 1861. Entered according to Act of Congress. in the year 1861, by JOHN WARNER, in the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. PREFACE. THE main object of this work is to present both a theoretical and practical solution of some of the most important problems of earth-work. The rules given are, however, applicable to other computations of solidity for solids having the same form as those of earth-work; but cubic yards must be converted into other measure when required. It will doubtless be conceded that there is yet room for improvement in the theory of our subject; and that previous treatises do not satisfy the wants of computers, is evident from the demand so constantly heard for "' something practical." This indicates that a system is yet desired which, though sufficient for the ordinary practice of engineering, does not require of the computer either unusual mathematical knowledge or extraordinary patience. It has appeared to the writer that such a system followed by a theoretical treatise, would constitute a work suitable both as a text-book for students and as a manual for computers. Such a work we believe to be needed, and to produce it has been our principal aim. In attempting this, the author has endeavored, both by study and by oral consultation, to ascertain the wants of the practical man, and to reconcile them with the theoretical exigencies of the subject. No method of earth-work computations is likely to be generally adopted which does not reduce them to an operation of routine. To this end, uniformity is necessary in the processes of calculation. This the author has sought to attain by founding those processes, as far as possible, on a few ideas easily understood and remembered. One of them is the use of the sum and difference of end dimensions. The manner and the object of the arrangement of the work will be obvious without a minute explanation here. In general, we have endeavored to make the several parts available either separately or collectively, according to the wants of the reader. A portion of the First Part is put in small type,-most of which may be passed over if desired; and we have also avoided, as far as practicable, the use of algebraic signs and the phraseology of mathematical enunciation. Some further remarks upon the work and its subject will be found in the Introduction. It is believed that the rules and explanations of the First Part will also enable non-professional computers to perform the most necessary 3 4 PREFACE. calculations, and that the work will thus meet the wants of a large class of persons to whom its subject has hitherto appeared difficult or inaccessible. We have endeavored, in the prosecution of the work, to remember that most readers will judge it by the standard of utility rather than of novelty. We desire, however, to say that it is considered to be in no proper sense a compilation, or a mere amplification of known methods, but as fairly original as any performance of the kind which does not lay claim to absolute novelty. The mathematical investigations are original, and, it is believed, are mostly new in method, results, and practical application. The tables as a body, and, for the most part, individually, are also thought to be new. They are adapted to the author's formulae; they were all computed by him,* and faithfully revised with the assistance of an experienced proof-reader, and, it is believed, they may be relied on. The Subdivision of Sections has been but slightly noticed by former writers, yet the want of explicit rules for it is daily felt. Our treatment of this subject is original, and we have much increased the size of the work by devoting the necessary space to it, believing that a brief disposition of it would be unsatisfactory. Several gentlemen are especially entitled to the author's thanks for assistance and encouragement. Professors Vethake and Franck, of the Polytechnic College of Pennsylvania, and Professor Kirkwood, of Indiana University, have examined considerable portions of the work in proof or in manuscript; and Messrs. Ellwood Morris and N. F. Jones, civil engineers, have also rendered assistance. Whilst the author was employed under the direction of Mr. Jones, that gentleman kindly assumed a task from which his position as chief engineer might have exempted him: he tested the author's system by actual computations, and devoted much time to assist in perfecting it. The author is assured that in profiting, to the best of his ability, by the suggestions of these able gentlemen, he has acquired an additional claim to public favor. THE AUTHOR. PHILADELPHIA, May, 1861. *With two exceptions. Messrs. Uriah Hunt & Son have kindly granted the use of Table XXVI. from their last edition (1860) of Gummere's Surveying; and Table A is a table of Natural Tangents. CONTENTS. INTRODUCTION. PAGE Remarks upon the origin and prosecution of this work.....9................................................... 9 Notices of British and American writers, including remarks upon the Prismoidal Formula......... 9 Remarks upon the methods of this work............................................................................ 11 On the mathematical definition of the surface in Earthwork computations.............................. 13 Table of dimensions for forty-two varieties of Excavation and Embankment.......................... 14 PART I.-PRACTICAL TREATISE. CHAPTER I. PRELIMINARY DEFINITION AND DESCRIPTION-SUBDIVISION OF SECTIONS. PRELIMINARY DEFINITION AND DESCRIPTION. ARTICLE 1. Data................................................................................................................. 5 2. Elements defined..............,,,,,............................................................................,,........ 15 3. " preparation of, by construction and calculation............................................ 15 4. Sections defined, different methods of computation..................................................... 16 5. " bounding surfaces of.............................................................. 16 6. Ground-Surface, usual conception of...................................................................... 16 78. " "c plane...........................................6................................. 16 8. " " warped.............................................................. 16 9. " " warped, particular kind of..17 i. Mein ln d edi P.......a. dined o................................................................. 17 10. Mi\edian Plane defined........17..........,,,,,,,,,,,,,,...,...............,......... 17 11. Marginal Planes and Heights defined.........7................................................ 17 12. Median Surface-Line defined.................................................................................... 17 13. Stations defined...................................................... 17 14. Profile defined.................................................................................................. 17 15. Directrices defined............................................................................... 17 16. Ground-Trace defined.............................................................1............................ 17 17. Grade-Level and Grade defined............................................................................... 17 18. Grade-Line defined....................................................................... 18 19. Earthwork Solids, names and description of.................................................. 18 20. Extension of illustration by imagining the inversion of diagrams..........................,....... 18 21. Redundant Prism, nature and use of....................................................................... 18 22. Centre-Height defined.. 19 23. Plus and Minus Signs defined.......................................................................... 19 24. Distance Out defined.......................... 19 25. Side-Slope defined................................................................................................... 19 26. Transverse Ground-Slope defined............................................................................. 19 27. Cross-Sections defined............................................................................................ 20 28. Full, Partial, and Neutral Cross-Sections defined....................................................... 20 SUBDIVISION OF SECTIONS. 29, Subdivision of Sections, for what purpose, and when necessary.................................... 20 30. Three Cases of Subdivision, how distinguished..................................... 20 31. Twelve Varieties of Subdivision, how formed............................................................. 21 32. Sub-Varieties not described........................................................................... 21 33. General Scheme for determining the Cases and Varieties of Subdivision......................... 21 34. Application of the General Scheme.......................................................................... 22 35. Determination of the Variety.................................................................................. 22 36. Union of several Component Solids and Reduction to a Plane Surface. 22 37. Wedges and Wedge-Shaped Masses formed by Subdivision of Sections................... 23 5 6 CONTENTS. CHAPTER II. AUXILIARY CONSTRUCTIONS-DETERMINATION OF VARIETIES AND PREPARATION OF ELEMENTS BY CONSTRUCTION AND BY TABULATION. ARTICLE AUXILIARY CONSTRUCTIONS. PAGE 38. Drawing Profiles................................................................ 24 39. Reduction of Oblique Length of Sections to Direct Length.................................. 24 40. Finding the Direct Length by Construction............................................ 24 41. To draw the Side-Slope when the rate of slope is given.............................................. 25 42. To draw a Cros-Section.....................................................................25 43. To find the Augmented Centre-Height and the Surface-Slope of the Mid Cross-Section...... 26 44. To find the Distances Out of the Mid-Section......................................................,, 26 45. Cross-Sections drawn on the Plane of the Roadbed................................... 26 46. To find the Marginal Heights at the ends of a Section................................................. 27 47. To find the Centre-Height of the Neutral Cross-Section................................................ 27 48. To find the Distance Out of the Grade-Point.............................................................. 27 49. To find the Base of a Sub-Section............................................................................ 27 50. To find the Abscissa of the Neutral Cross-Section....................................................... 27 51. To find the Abscissa of the Median Grade-Point............................................ 28 52. To find Abscissas when the Ground-Surface is a Plane................................................. 28 53. To find the Total Variation of Surface-Slope and the Mean Slope....................... 28 DETERMINATION OF VARIETIES AND PREPARATION OF ELEMENTS BY CONSTRUCTION. 54. Determination of Varieties by drawing the Cross-Sections............................................ 29 55. Whole Sections, preparation of Elements for........................................................ 30 56. Sub-Sections,.31 56. Sub-Sections, 4" ".............................................................I 31 57. " length of, found by drawing Profiles......................................... 31 58. Cross-Sections and Profiles upon a prepared Diagram.................................................. 31 59. Conjoint use of diagrams and Tables............................................................... 31 DETERMI1NATION OF VARIETIES AND PREPARATION OF ELEMENTS BY TABULATION. 60. The Tabular Method, wherein it consists............................................................................ 32 61. Determination of Variety by tabulating the Cross-Sections........................................... 32 62. Preparation of Elements; finding the Abscissas........................................................ 32 63. Finding the Marginal Heights and their Differences.................................................... 32 64. Difference of Centre-Heights equivalent to their Numerical Sum in finding Median Abscissas 32 65. Finding the Abscissas by Table IV........................................................................... 32 66. Succeeding Consecutive Sections treated similarly to the first section of the series........... 32 67. Arbitrary Assumption of the Direction of Slope in Tabulation....................... 33 68. Double Tabulation necessary with two widths of Roadbed............................................ 34 69. Marginal Heights and Bases under Plane Ground. I......................................I..... 34 70. To find the Sum and Difference of Bases for Sub-Sections under Plane Ground..... 34 71. To find the Sum and Difference of End-Heights for Sub-Sections under Plane Ground...... 35 72. To find the Distance Out, either Horizontal or Sloping................................................. 36 73. To find the Distances Out of the Mid Cross-Section for a Plane Surface.......................... 37 74. To find the Distances Out for Triangular Cross-Sections............................................... 37 75. To find the Augmented Height of the Mid Cross-Section.............................................. 37 76. To find the Augmented Centre-Height of a Cross-Section whose Distances Out are given... 38 77. To find the Augmented Side-Height when the Distance Out is given.. 38 78. To find twice the Difference of the Side-Heights..................39...................................... 79. To find the Surface-Slope of a Cross-Section............................................................... 39 CHAPTER III. COMPUTATION OF EXCAVATION AND EMBANKMENT BY TRANSVERSE GROUND-SLOPES. STRAIGHT WORK AND PLANE GROUND-SURFACE. 80. Three Different Methods of Computation,-by Tables, by Scale, by Logarithms............... 40 81. Necessary References to previous Chapters and to Chapter V....................................... 40 82. Cubical Content proportional to the Length of the Work.............................................. 40 83. Transverse Ground-Slope in Whole Degrees..................4......0..................... 40 84. Preliminaries necessary to Computation.......................................................... 40 Computation by Tables of Cubical C'ontent. 85. General Rule for Computation by Transverse Ground-Slopes................4........................ 86. Examples in Whole and Sub Sections under the General Rule................................... 41 87. The same Examples for Sub-Sections by Heights....................................................... 43 88. Further Examples in Whole and Sub Sections............................................. 44 89. Special Tables in Whole Sections for a particular width of Roadbed...............4............... 44 90. Rule for Computation by Special Tables.................................................................. 45 Unusutal Cases. 91. Unequal End-Widths...........................................................,,,,, 45 92. Inclined Surface of the Redundant Prismoid.................................................... 46 93. Reduction of an Irregular Surface to a Sloping Plane Surface............................... 4.... 46 94. Applicability of the Methods by Scale and by Logarithms........................................... 47 CONTENTS. -; ARTICLE Computation by Scale. PAcE 95. Method of use and description of Scales.................................................................... 47 96. General Scale for Whole and Sub Sections, Rule for use of........................................... 47 97. Special Scales for particular widths of Roadbed, Rule for use of........................... 48 98. Computation by Equivalent Level-Heights, description of the method......................... 49 Computation by Logarithms. 99. Elements required for Computation by Logarithms..................................................... 49 100. Rule for Computation by Logarithms..................4...................................9....... 49 101. Omission of the Correction for the Difference of the End-Heights................................. 50 102. The place of Special Tables supplied by Logarithms..........................................., 51 Straight Work and Curved Ground-Surface. 103. Reduction to a Plane Surface-Sub-Sections not considered...................................... 51 104. Rule for Computation under a Curved Ground-Surface.............................., 52 Curves. 105. Reduction to Straight Work.....................................................,..,........ 53 106. Preparation for finding the Correction for Whole Sections............................................ 53 107. To find the Correction for Whole and Sub Sections.............................................., to 55 CHAPTER IV. COMPUTATION OF EXCAVATION AND EMBANKMENT BY CENTRE AND SIDE HEIGHTS. STRAIGHT WORK. 108. Data................................................................................................... 56 109. Enumeration and Preparation of Elements for Whole and Sub Sections.............. 56 to 58 110. To find the Sum and Difference of Bases for Sub-Sections............................................ 58 111. Preparation of Elements, recapitulation of, for Whole-Section Work.................. 59 112. Rule for the Arithmetical Computation of Whole Sections........................................ 59 113. Rule for the Tabular Computation of Whole Sections................................................... 60 114. Sub-Sections-Preparatory Recapitulation and Description.......................................... 60 115. Rule for the Arithmetical Computation of Sub-Sections............................................... 61 116. Elements for Sub-Sections when the Cross-Section is Irregular............................. 62 117. Arithmetical Rule for Sub-Sections when the Cross-Section is Irregular......................... 63 118. Rule for the Tabular Computation of Sub-Sections.................................................... 64 119. Method for the same with Heights not taken both on the Median Line........................... 65 Curves. 120. Reduction to Straight Work.................................................................................... 66 121. Preparation for finding the Correction for Whole Sections........................ 6 122. To find the Correction for Whole and Sub Sections............................................... 66 to 69 Unusual Cases, and Computation by Logarithms. 123. Description of Unusual Cases............................................................................... 69 124. Two Intermediate Level-Heights at each End in Whole-Section Work......................... 69 125. Work having the Cross-Section everywhere Rectangular............................................, 71 126. When the Rectangular Cross-Section is everywhere the same....................................... 71 127. When the Cross-Section is everywhere Square........................................................... 72 128. Rule for Computation by Logarithms, for Whole and Sub Sections................................ 72 CHAPTER V. THE TABLES AND THEIR USE. 129. General Description of the Tables....................................74................................. 74 130. Technical Terms, definition of........................................................................... 75. 5. Interpolation by Proportional Parts explained..................................................... 75 131. Finding the Tabular Quantity explained for each Table.............................................. 76 Arrangement of the Tables, and Index........................................................................83, 274 PART II.-THEORETICAL TREATISE. AUXILIARY CONSTRUCTIONS AND CALCULATIONS. 1. Profiles.....................................................................................t.................... 275 2. Tabulation of Cross-Sections and of Marginal and Median Profiles.................................. 276 3, Trace of the Ground-Surface upon the Plane of the Roadbed......................................... 277 COMPUTATION OF SOLIDITY. Straight Work. 4. Whole Sections by Transverse Ground-Slopes....................................................... 278 5. Sub-Section or Partial Work by Transverse Ground-Slopes............................................. 281 6. Computation of Solidity by Centre and Side Heights..................................................... 283 8 CONTENTS. ARTICLE Curves, and Straight Solids of Revolution. PAGE 7. Curves..................................................................................................... 284 8. Straight Work considered as a Solid of Revolution........................................... 288 APPENDIX. NOTES TO THE RULES AND EXAMPLES OF PART I. 1. Examples in Whole and Sub Sections.................,,,,.,,,,,....,.,.........,,,. 291 2, Unusual Cases by Transverse Ground-Slopes................................................., 291 3, Unusual Cases by Equidistant Level-Heights:-Preparation of Elements......................... 292 4. Rule for Computation of Solidity by Equidistant Level-Heights..................................... 292 5. Curvature of the Median Profile of Whole Sections................................................. 292 6. Correction for Curved Surface-Dimensions of the Secondary Prism found by Construction... 293 7. Secondary Dimensions found by Calculation....... 294 8. Comparison of the Methods by Transverse Slopes and by Centre and Side Heights...... 294 9. Application of Tables to various Formulwe....................296...................................... 296 CATALOGUE OF FORMULAE AND CONSTRUCTION OF THE TABLES. 10. Catalogue of Formule............................................................. 296 11. Construction of the Tables....................................................................................... 297, 1. Tables proceeding in the Direct Ratio of the Argument............................... 297 Q 2. " " as the Square of the Argument..298, 3. Correct Interpolation for Height in Tables proceeding as the Square of the Argument 299 Q 4. Logarithmic Tables-Explanation of Formulse for Tables XXTII. and XXIV............ 300 DISCUSSION OF GRAPHICAL PROCESSES. CONSTRUCTION OF SCALES. 12. Construction of General and Special Scales......................................................... 800. 1. Sub-Section Diagram of Plate VII. and Mid-Section Diagram of Plate VIII............. 300 Q 2. General Stereometric Scale for Whole and Sub Sections......................................... 300 3. Special Scales, Plate IX............................................................. 301 13. Given Dimensions-Application of the Diagrams in the following Discussions... 302 BASES OF EQUIVALENT SQUIARE PRISMS. 14. To find the Base of an Equivalent Square Prism by Mean Proportionals.......................... 302 3 1. For Whole Sections.................... 302, 2. For Sub-Sections............................................................................................ 303 15. To find the Content by Diagram................................................................................ 303 3 1. For Whole Sections...................................... 303 Q 2. For Sub-Sections..........................................................................303 16. Bases of Equivalent Square Prisms and Equivalent Heights found by Geometric Loci 304 Q 1. Investigation of Formulse for Whole Sections...................................................... 304 Q 2. To construct a Diagram for finding Bases of Equivalent Square Prisms for Whole Sections 304 3 3. To find by Construction Equivalent Heights for New Surface-Slopes....................... 305 Q 4. To find Equivalent Level-Heights by Construction..305 Q 5. Geometrical Basis of these Constructions a property of the Hyperbola..................... 305 17. Examples in finding the Content of Whole Sections by the Diagram for Equivalent Square Bases 305 18. Bases of Equivalent Square Prisms found by Geometric Loci 306 Q 1. Investigation of Formulse for Sub-Sections 30'6 Q 2. To construct Diagram for finding Bases of Equivalent Square Prisms for Sub-Sections 306 19. Examples in finding the Content of Sub-Sections by the Diagram for Equivalent Square Bases 307 CONVERSION OF SLOPES. 20. Conversion of Slopes defined..........................................3......... 308 21. Nature of the Problem of Conversion of Slopes.........................................30.................. 308 22. General Construction for Conversion of Slopes by Mean Proportionals........................ 308 23. Investigation of Formulse for Conversion of Slopes by Auxiliary Angles........... 308 Q 1. For Construction............................................................................................ 309 2. Logarithmic Formule..................................................................................... 309 24. Construction by Auxiliary Angles...............................3..............................10 Q 1. General Construction for Double Conversion.................. 310 Q 2. Single Conversion-Arbitrary Change of Surface-Slope......................................... 310 } 3. Double Conversion, with Surface-Slope made Level 310 4. SinglI Conversion, with Surface-Slope made Level.............................................. 310 Q 5. To construct a Diagram for finding Equivalent Level-Heights 311 25. To find the Solidity by a Diagram of Equivalent Level-Heights.. 311 1. With Tables adapted to particular Side-Slopes...... 311. 2. With Table XV., adapted to all Side-Slopes......................................... 311 Q 4. Remarks on the Method of Equivalent Level-Heights.......................................... 312 Q 5. Bases of Equivalent Square Prisms found by Auxiliary Angles............................ 312 26. Examples in Conversion of Slopes by Logarithmic Formulse.................................. 313 Notes................................................................................................................. 314 INTRODUCTION. THE author's employment in the practical duties of computation engaged him in the special studies which have resulted in the production of this work, -commenced several years since. The completion of it has been delayed by business and by the prosecution and publication of other researches. In May, 1858, it was submitted to the Pottsville Scientific Association. Since then additions have been made, and the text has been in a good measure re-written. The first printed notice of the work is contained in the Annual of Scientific Discovery, for 1859, page 196. In the summer of 1859, a portion of the work, containing rules and tables for computation by centre and side heights, and also scales and the sub-section diagram, was distributed, in pamphlet form, among the author's friends. The time employed upon the original labors of the work left inadequate leisure for the collection of historical materials; and of those now presented, a considerable portion were not known, or if known, but slightly regarded, until the work had been nearly completed. The writer therefore hopes to be excused if interesting historical facts or meritorious works have been overlooked. As far as the writer is informed by British authorities, the study of rules for earthwork computations began to engage special attention about the commencement of railway-enterprises in England, some thirty-five years ago. Among British authors who have furnished tables, or who have otherwise labored on the subject in question, may be mentioned Macneil, Bidder, Huntingdon, Hughes, Bashforth, Sibley, Rutherford, Law, Lowe, and Baker. This list is from Baker's works.* He appears to have labored with zeal and success. Macneil's works are frequently cited. Of the writings of our own countrymen, those of Mr. Morris deserve especial attention, as well for their valuable information and suggestions as for their early historical place in our engineering literature. This writer, by original research, revived the discussion of the prismoidal formula, showed its importance in earthwork computations, and noticed its applicability to solids included under curved surfaces.t Professor Gillespie has shown its application to the hyperbolic paraboloid;~ and Mr. Chauncey Wright has recently much pronooted our knowledge of the subject by his papers on the Prismoidal Formula.~ * Rudimentary Treatise on Land Engineering, by T. Baker, C.E. London, 1850, p. 188. Also Baker's Railway Engineering. The critical student should not omit to consult Baker's works. t Journal of Franklin Institute, vol. xxv., 2d series, pp. 25, 387, and vol. xxiii., 3d series, p. 240. I Journal of Franklin Institute, vol. xxxiv., third series, p. 372. Mathematical Monthly, vol. i., No. 1, p. 21, and No. 2, p. 53. 9 10 INTRODUCTION. It was, however, previously known, but not generally taught in text-books of Mensuration or of Engineering, that the formula holds for various solids contained under ruled or other curved surfaces. UIp to the time of his paper, two methods (says Mr. Morris) had come into general use. They are thus defined by him as adapted to dimensions given in feet:* 1. Arithmetical Average. —Multiply the sum of the end areas by their distance apart, and divide the product by 6 and by 9; the result will give, approximately, the number of cubic yards in the given length of excavation or embankment. 2. Geometrical Average.-Multiply the sum of the end areas and the square root of their product, by the distance apart, and divide the product by 9 and by 9. The result will be, nearly, the number of cubic yards in the given length of excavation or embankment. We may here digress for a moment from our notice of this paper, to record the prismoidal rule, and to give some historical references pertaining to it. 8. The Prismoidal Rule. Add together the end areas and four times the area of the mid cross-section. Multiply the sum thus found by the distance between the end areas, and divide by 6, by 9, and by 3. The result will be the number of cubic yards in the given length of excavation or embankment as nearly, in our opinion, as by any general rule known. Cardan finds the frustum of a pyramid, or of a cone, by computing the whole solid; and also the redundant part, and then subtracting the latter.t Hutton gives for this problem the rule of geometrical average before cited,t but subsequently applies the prismoidal formula to the mensuration of prismoids and cylindroids.~ He afterwards extends it to the hyperbolic conoid,ll and finally to the frusta of solids generated by the revolution of conic sections about their axes.4 Bonnycastle follows Hutton in several of these applications,** and refers to Simpson for the demonstration in regard to the prismloid,tt-the earliest notice we have found. The methods for the quadrature of curves, which began with Newton's theorem of interpolation, seem to have led to more comprehensive formule for the mensuration of solids.St Montucla gives an interesting review of the labors of Newton, Cotes, and Simpson, in relation to the quadrature of curves by equidistant ordinates;~~ but at present I am unable to refer to other authorities than those already cited, for the extension of this method to solids. The *Journal of Franklin Institute, vol. xxv., second series, p. 28. t " In pyramide autem curta sciemus magnitudinem totius pyramidis, et partis defficientis, unde detracta parte defficiente a tota pyramide, remanebit pyramis curta."-Cardani Opera, vol. iv. p. 133. 1 Treatise of Mensuration, 1770, page 159. g Ibid. p. 163. II Ibid. p. 389. iT Ibid. p. 456. **An Introduction to Mensuration and Practical Geometry; 3d edition, London,1791, pp. 135, 141, 178, and note, p. 180. tt Doctrine and Application of Fluxions. London, 1750, p. 178. 4$ Hutton, pp. 458 to 468, and Gregory's Mathematics for Practical Men. London, 1848, p. 133. The later editions of Hutton and Simpson do not, as far as we have examined, contain much new matter on these subjects. ~ Histoire des Recherches sur la Quadrature du Cercle. Paris, 1831, p. 176. When only three ordinates are taken, the series of Cotes and of Simpson reduce to the prismoidal formula. The notes of the author and of the editor refer to numerous original sources; among others, to an extensiye series of memoirs, by MM. Kramp and B~rard: their formulas appear to us similar to those of Cotes and Stirling.-(Harmonia Mensurarum, 1722, 2d part, p. 33; Phil. Trans., vol. xxx., 1720, p. 1063.) A report to the French Academy upon the method of M. Berard (1817) may be found in vol. VIII. INTRODUCTION. 11 incorrectness of No. 1 and No. 2, tried by No. 3 as a standard, is shown by Mr. Morris in numerous examples. Both Baker and Macneil, we believe, have also examined the errors of these methods.* Mr. Morris has suggested the general application of the prismoidal rule, and has given instructions for its employment in practice. To find the area of the mid-section, which is here required, is, however, not always an easy operation; and this may be a principal reason why his method has not prevailed. Whatever be the objections to it, it is, if I do not err, the only general one yet proposed applicable to all ground, and with any number of end level-heights and their corresponding distances out. Subsequent to Mr. Morris, among American engineers, lMr. Trautwine, and, after him, Mr. Lyons, are well known for their various publications upon Earthwork.t Mr. Henck is also prominent among recent American authors who have published methods of earthwork cornputation.4 His rules appear to us in general original, ingenious, and accurate. We have not, however, found them, as a system, gaining the favor which they seem to deserve. The works of Messrs. Trautwine and Lyons are devoted to the method of computation by transverse ground-slopes. To the former we are indebted for the introduction of this method. As far as whole-section work is concerned, both writers proceed upon the principle of finding centre-heights which, under level ground, would contain the same area as the heights of the given cross-sections under sloping ground. Mr. Trautwine obtains these heights by the aid of diagrams. Mr. Lyons finds them from a table. The heights, being found by either method, are used to enter a table adapted to level ground. The Transverse-Slope method has acquired considerable popularity, especially for preliminary estimates. This appears to depend on the fact that it possesses several of the requisites of a practical system, such as, in the Preface, we have said is still desired by computers. As regards the labor of observation upon the field, the method employs only a few data, which can be collected with facility and despatch. It also permits simple modes of computation. Hence an approximate result is obtained with comparatively small labor, both on the field and in the office. These remarks apply to what we have termed whole-section work. The subdivision of sections is part of the work of computation; for it must be remembered that to subdivide sections upon the field would scarcely consist with the expedition required in preliminary surveys. A large portion of the present work is devoted to the method of transverse slopes. The writer has endeavored to supply a code of rules for the subdivision of sections in the office, and to increase by original methods the comprehensiveness and the practical facilities of the whole system. of Gergonne's Annales. The method is considered useful, and its publication is recommended, by the examiners Ampere and Poinsot. In some of Mr. Wright's formulae, by adding in pairs the areas of parallel sections equidistant from the mid-area, the coefficients of Cotes and Stirling are reproduced. Mr. Wright shows the degree of approximation attainable by his formulae. * See also a paper by Professor Gillespie, Journal of Franklin Institute, vol. xxxvii., third series, p. 11. t A New Method of Calculating the Cubic Contents of Excavations and Embankments. By John C. Trautwine. Also, for the same, Journal of Franklin Institute, vol. xxii., third series, pp. 1 and 80. Earthwork Tables, by M. E. Lyons, C.E. Printed by J. Knabb, Reading, Pa., 1855. 1 Field-Book for Railroad Engineers, by John B. Henck, A.M., Civil Engineer. New York, 1854. 12 INTRODUCTION. It is doubtful whether sub-section work can be treated in such a manner as to reconcile entirely theoretical and practical requirements. The rules given in Art. 36, and the modes of computation adopted in these cases, are intended as a practical disposition of the subject by giving a running estimate. It may not, however, be out of place to offler here some remarks on the detailed computation of sub-sections. Our methods of finding abscissas are accurate and consistent with our hypothesis concerning the surface, provided the true width of roadbed be taken. The surface-slope of neutral cross-sections, or of other sections where the slope may be required, may be approximately*found by assuming the variation of slope to be proportional to the distance passed over in the direction of the roadbed, (Art. 53.) The means would therefore be provided for finding the dimensions of each solid of a subdivided section. Those solids which take the form of prismoids might then be treated by the rules for whole-section work. The other solids might be treated as pyramids and truncated pyramids. The use, in computation, of solids not representing real excavation or embankment may doubtless be of advantage. The writer has, in the main text, confined himself to a single one, —viz., the Redundant Prism. This being always subtractive, and easily imagined, the employment of it can rarely lead -to error. The observance of algebraic signs requires care and thought, even with trained computers; but the use of these signs cannot be entirely dispensed with. Where several quantities, each of which may be positive or negative, are to be combined,-either quantitatively or characteristically, or in both ways at once, —the attainment of correct results may necessarily be somewhat difficult and tedious. This applies especially to the purely tabular treatment of sub-sections; but neither these difficulties nor the necessary labor of research, it seems to us, would have justified the omission of the tabular processes given in small type in the latter part of Chapter II. We have intimated that the direct use of the mid-section has so far complicated the rules given by Mr. Morris as to cause them to be neglected. The direct use of the mid-area may be avoided by confining the problem to the case where the same number of level-heights are taken at each end of the work, and equidistant on their respective cross-sections. Our rule for proceeding by this method is given in Art. 4 of the Appendix. It has not been presented as the principal rule for irregular cross-sections. The method given in Art. 124, Part I., employs but one redundant solid; more than five heights are seldom required on a cross-section; the mode of computation consists in repeating a process previously understood, does not require equidistant heights, and may be extended to various polygonal cross-sections. We have therefore preferred it as a rule of routine. The method of equidistant heights is, however, easily applicable when both of the surfaces which terminate the heights, or ordinates, are irregular. Besides this, auxiliary solids may be employed, (as explained in the Appendix,) and by this means many diverse and irregular forms of cross-sections may be treated.t The rule will frequently be useful in the estimation of excavations for borrowing. Another useful e Or accurately by construction or by calculation. (Arts. 52, p. 28, and 3, p. 277.) t The use of auxiliary solids may, indeed, be avoided by putting the external heights -0; but every angle of the cross-section must be upon an ordinate. When the process is not strictly practicable, an empirical modification of cross-sections (Art. 93) may permit this spacing of ordinates. INTRODUCTION. 13 application may be mentioned, not connected with the special object of this work, —viz., the computation of the cubic contents of coal-veins. The width of the cross-section will, according to general usage in such measurements, correspond to the distance across the strike, (or height of breast, as it is termed in our coal-regions;) the heights will be the thiness of thee vein taken at regular intervals, and the length will be measured in the direction of the strike. A few new technical terms have been introduced. This will be sufficiently justified, if it shall be found to serve a useful purpose. The scales and diagrams designed to assist calculation, although carefully executed, are intended rather for illustration than for actual use. The author is prepared to issue them in form and variety adapted to the wants of computers, should they be required separate from the volume. The lithographic plates, showing the subdivision of sections, were taken fromn models made expressly for this purpose.* These plates, or stereoscopic views of similar models, will also be furnished separately if necessary. We shall now offer some remarks upon the necessity of adopting a standard ground-surface. The writer is of opinion that in most estimates a considerable sacrifice of accuracy has been, and is likely to be, made for the sake of expedition. It is evident that there must be some limit to the collection of data and the labor of computation. Therefore the surface cannot be treated as a network of indefinitely small surfaces, discontinuous or heterogeneous, belonging to a polyhedral solid. Hence there must be some sacrifice of theoretical accuracy: the question is, how much? This question, as far as I know, has never been either definitely proposed or answered. To reply to it strictly would require the mathematical definition of theoretical accuracy; that is, the determination of a surface or of an assemblage of surfaces which, with a given number of heights, distances out, and lengths, (co-ordinates,) would include a solidity most nearly approximating that which it is desired to compute. Some of the usages of engineering would restrict the inquiries suggested by the general question,-as, for instance, that of taking levels only on two cross-sections forming the end-planes of a piece of work. A desideratum yet to be attained seems, therefore, to be the following. That there should be established and adopted a surface whose contained solid sufficiently approximates the actual content, and at the same time permits a ready computation of solidity from a moderate number of data. The writer has not undertaken to decide what this standard surface should be: he has adopted that of an hyperbolic paraboloid generated by the surfaceline of the cross-section, modified upon curves, by assuming curvilinear directrices, and, having made this choice, has merely shown what data his rules require, and has there left this subject.t *The working drawings for these models were executed, after the author's designs, by E. T. Quilitch, Esq., Engineer to the Forest Improvement Company. To Professor J. P. Lesley, of the University of Penna., I am indebted for the suggestion of showing the contour-lines by horizontal strata. My thanks are due to both gentlemen for these and other friendly offices requiring time and attention. At the time of making these models I was acquainted with the cono-cuneus of Wallis only by a brief citation, and was not aware of Mr. Pett's treatment of that solid. The passages cited in the note on page 288 also escaped notice until too late for other recognition. t By diversifying the mode of generation of ruled surfaces, a satisfactory imitation of most of the 14 INTRODUCTION. For the purpose of exhibiting the most usual characteristic dimensions of railroad excavation and embankment, and also of showing that these are all embraced in the author's tables for transverse slopes, it may be proper to give, in tabular form, a list furnished him by an engineer of experience and ability.* "Widths of Base and Ratio of Side-Slopes desirable to be included in a series of Tables for computing Excavation and Embankment. Kind of Work. Material. Single-Track Bases. Double-Track Bases.! Side-Slopes. 16 feet. 26 feet. 2 to 1, 1~ to 1, 1 to 1, 1 to 1. 18 feet. 28 feet. 2 to 1, 11 to 1, 11 to 1, 1 to 1. 20 feet.. 30 feet. 2 to 1, 1] to 1, 14 to 1, 1 to 1. 14 feet. 24 feet. i to 1, I to 1. 16 feet. 26 feet. i to 1, ~ to 1. 18 feet. 28 feet. A to 1, i to 1. 12 feet. 22 feet. 11 to 1. 14 feet. 24 feet. 1~ to 1. i a~ i ~ r < f i 16 feet. 26 feet. 1~ to 1. "Making in all forty-two varieties." These forty-two varieties may all be computed by aid of Tables XV., XVI., and GXVII. Several of these varieties, however, are not of very frequent occurrence. If, therefore, it were proposed to compute a tolerably complete set of tables similar to VII. and VIII., or, for whole sections alone, similar to XIV., the work would be altogether practicable. In the first case, the number of tables would be double that of the selected side-slopes; in the second, the same as the number of selected varieties. solids of earthwork could probably be obtained. (Note, p. 288.) The following notices and citations may perhaps be interesting and suggestive in this connection. The cono-cuneus is a solid shaped like a cone at the base and like a wedge at. the vertex. It was invented by Mr. Pett, a Naval Architect, who submitted it to Wallis in the belief that the curves resulting from its section might be useful in ship-building. To exhibit these curves, the inventor cut the cono-cuneus from a block made of slabs glued together. Afterwards he dissolved the glue and obtained the slabs cut in the curved forms desired. Wallis devotes considerable attention to the geometrical delineation of these curves. His memoir appears to contain one of the earlier steps in the generalization of ruled surfaces,-a subject, it would seem, not familiar to mathematicians in his time; for, in giving his reasons for undertaking the treatment of his subject, he says, " Quod eo potius suscepi, quoniam hoc solidum est de novo excogitatum, quod nescio an quisquam prior considerandum proposuerit. Atque exemplo sit, aliis ejusmodi solidis considerandis, siquando postulaverit occasio." The delineations given by Wallis appear interesting in relation to the progress of Descriltive Geometry. Ruled surfaces continue to be much discussed, and are important in the arts. The solids devised by Gregory St. Vincent in his attempted quadrature of the circle, appear to me suggestive of a general investigation of ruled surfaces. These are the principal references of this kind which I have been able to collect. — allisii Opera, vol. ii. p. 681; 1662. AI. Tinseau,.Alemoires presentds d l'Academie par divers savans, Tome ix. p. 625; 1780. P. Gregorii A Sto Vincentio Opus, 1647; also, p. 82 of Montucla's Histoire, &c., before cited. *Ellwood Morris. September 9, 1856. NE W THEOREMS, TABLES, AND DIAGRAMS FOR THE COMPUTATION OF EARTHWORK. PART I.-PRACTICAL TREATISE. COMPUTATION BY TRANSVERSE GROUND-SLOPES, AND BY CENTRE AND SIDE HEIGHTS. CHAPTER I. PRELIMINARY DEFINITION AND DESCRIPTION-SUBDIVISION OF SECTIONS. 1. DATA are the given dimensions upon which computations are founded. (1.) Arbitrary Data are determined by expediency alone. Such are the width of roadbed and rate of side-slope. (See the Table, page 14.) (2.) Observed Data are obtained by measurement or observation upon the field. 2. ELEMENTS. Earthwork computations may be facilitated by deriving from the data other quantities proper for the purposes of calculation. Quantities thus prepared are called elements. In some cases the quantities necessary for computation are not directly given among the data, and must be found by auxiliary processes. For example, when a section is to be subdivided, (Art. 29,) the lengths of the different parts must be found, in order to compute the solidity. 3. PREPARATION OF ELEMENTS BY CONSTRUCTION* AND CALCULATION. The derivation of elements from the original data may be performed either by construction or calculation, or by both conjointly. In Chapter II. we shall treat at length of the preparation of elements. * Construction, or the drawing of mathematical figures, requires a knowledge of the use of the scale and graduated arc, and the method of drawing angles, parallel lines, and perpendiculars. The reader who is unprepared in this respect should consult some suitable work upon Mensuration, or on Practical Geometry. 15 16 PRELIMINARY DEFINITION AND DESCRIPTION. 4. SECTIONS. Regular lengths of ground staked out for excavation or embankment, or the quantities of work enclosed, are called sections. We shall understand by a section, any given length of ground marked out for excavation or embankment. The length of a section is supposed to be taken horizontally. It is generally one hundred feet. To facilitate the computation of the solid contents of sections is the principal object of this work. We shall treat of two methods of computation,-viz., by Transverse Slopes, in Chapter III., and by Centre and Side Heights, in Chapter IV. The meaning of these terms will duly appear. The present and the following chapter, as their titles import, will be devoted to preliminary subjects. 5. BOUNDING SURFACES OF SECTIONS. The solid to be excavated or embanked in a section is contained by six surfaces,-viz.: By the natural, or ground surface; by the plane of the roadbed;* by the two sloping side-planes and by two vertical and parallel end-planes. On curves, the side-walls are curved, and the end-planes converge. Unless otherwise understood, straight work is meant. 6. THE GROUND-SURFACE. It is not known what kind of mathematical surface most nearly resembles the actual surface of the ground. It is usual to assume the surface to be either a plane or a warped surface. The groundsurface is frequently called simply the surface. 7. PLANE GROUND-SURFACE. The ground-surface is seldom a true plane, but it may frequently be as well represented by a plane as by any other surface of easy definition. 8. WARPED GROUND-SURFACE. When the ground is not supposed to be a plane, it is assumed to be a warped surface. Fig. 1. Suppose ABCDEH I K (Fig. 1) to represent a section P / / \ of excavation. B C K I is the roadbed. BAHI and CDEK _ to_"\ M are the side-planes. ABCD, HIKE are the end-planes. AD E H is the surface of the Ii7 ground. A warped surface is supposed to be generated B 1 4 0 by the motion of a straight line LM parallel to the end-plane A B CD, and guided by the straight sidelines A H, D E. Such a surface is called, in Geometry, an hyperbolic paraboloid. Surfaces generated by a straight line moving parallel to a fixed plane, as in this case, are also called ruled surfaces. Since a ruler applied to a flat table may be slid over its surface in the manner described, a plane may be considered as a variety of the paraboloid. Ruled surfaces, therefore, have this advantage,-that they include, under a simple definition, both plane and curved ground. * We shall speak only of railroad-work, leaving the reader to make the proper application of our rules to other computations. PRELIMINARY DEFINITION AND DESCRIPTION. 17 9. PARTICULAR KIND OF WARPED SURFACE. WVhen the work of a section is all excavation or all embankment, we suppose the lines A HII, DE to join the corners of the section, as in Fig. Fig. 2. 1. But when a section contains R E both excavation and embankment., it may simplify the calculation to sup- It pose A H, DE to be parallel to a ver- \ tical plane NM I L (Fig. 2) passing D through the centre of the roadbed. It will also be convenient to suppose AH and DE to lie respectively B N in the vertical planes B A H I, CDE K which pass through the sides B I, C K of the roadbed. (For illustrations, see Plates and Description, page 316.) 10. MEDIAN PLANE. The vertical plane NM R L is called the median plane. 11. MARGINAL PLANES AND HEIGHTS. B A H I, C D E K are called marginal planes. B A, C D, K E, I H are called marginal heights. 12. MEDIAN SURFACE-LINE. The line P Q, in which the median plane intersects the surface, is called the median surface-line. (See now Fig. 1.) 13. STATIONS. Upon the field, the ends P, Q of a section are marked by stakes which are numbered. In a continuous line of railroad, the first stake is numbered 0, the second 1, and so on as far as required. No. 1 will then designate the first section, No. 2 the second, and thus continuously. 14, PROFILE. A profile contains such a figure as N 0 P Q, for a single section, or for a continuous series of sections. Profiles drawn on the median plane are called median profiles. Profiles drawn on the marginal planes are called marginal profiles. N 0 P Q (Fig. 1) is a median profile, A H I B (Fig. 2) a marginal profile. 15. DIRECTRICES. Each of the lines D E, A H is called a directrix, because it directs the motion of the line supposed to generate the surface. In Fig. 1 these lines may properly also be called external surface-lines, because they there form the external boundaries of the surface. 16. GROUND-TRACE. A trace is the line of intersection of two surfaces. The ground-trace is the line which the surface of the ground would mark upon the plane of the roadbed if both were continued so as to meet. In the case of warped surfaces, the ground-trace is a curved line called an hyperbola. When the ground-surface is a plane, the trace is a straight line. 17. GRADE-LEVEL. The roadbed is considered to be level in a direction transverse to the length of the road; but in the direction of the length the roadbed frequently has a vertical rise or fall, which is called grade. Grade is measured by-the number of feet, rise or fall, per mile. Grade-level, at any point, is the height which the roadbed-plane there has compared with a B 18 PRELIMINARY DEFINITION AND DESCRIPTION. standard level-plane called the datum-plane. This is a plane which is imagined to pass through some one known fixed point. 18. GRADE-LINE is generally understood to mean a line of considerable extent, marked at every one hundred feet, or other regular interval, and following the grade of a proposed roadbed. Such lines assist preliminary exploration. 19. NAMES OF EARTHWORK SOLIDS. When all the bounding surfaces are Fig. 3. planes, these solids are either prisms, prismoids, pyramids, truncated pyramids, or wedges. AB C D E I (Fig. 3) is a triangular prism. In a prism, the ends ABC, IDEK are of equal size and simiKxS lar shape. BLIMCDENO is also a prism. If the ends were unequal in size or dissimilar in shape, the solid would properly be called a 2prismoid. A AB CD (Fig. 4) is a triangular pyramicd. Fig. 4. D If the point of the pyramid be cut off by the plane KL M, the part A B C M L K is a truncated pyramid. Fig. 5 represents a wedge-shaped mass. It is a / K prismoid, in which one of the endL / \ surfaces dwindles nearly or quite to a line or sharp edge. /When the ground-surface or the side/ walls are curved, earthwork solids will still resemble the geometrical figures A o mentioned; but the same names cannot Fig. 5. be strictly applied. They may, however, by a license of language, be emG-~'~ is "`~ —---— ~. // ployed, provided they clearly indicate what is meant. 20. EXTENSION OF ILLUSTRATION. The general form is the same for embankment as for excavation; but if the eye be accustomed to consider only excavation, embankment will appear as inverted excavation,-and vice versa. The reader must, when necessary, endeavor to extend the illustrations afforded by the diagrams, by imagining the inversion. 21. REDUNDANT PRISM is the prism AL M 0 N K (Fig. 3) lying below the roadbed in excavation and above it in embankment. If this be subtracted from the solid A B C D E K, there is left the prismoid B L M C D E N O, which is the solid to be excavated or embanked. Hence, as the content of the redun. dant prism is easily found, it frequently assists computation to calculate the content of the solid A B C D E K, and then subtract the redundant prism. PRELIMINARY DEFINITION AND DESCRIPTION. 19 22. CENTRE-ITEIGHT is the vertical distance P S, (Fig. 3,) measured on the median plane, from the roadbed to the surface. S A is called the Augmented Centre-Height. It is equal to P S augmented by PA, the height of the redundant prism. Centre-heights may be taken anywhere between the ends of a section, but are understood to be taken at the ends, unless otherwise stated. They may be called end-heights, also, when taken at the ends. Centre-heights are said to be plus when they are above the roadbed, or in excavation, and minus when they are below, or in embankment. 23. PLUS AND MINUS SIGNS. These signs are borrowed from Algebra. +(plus) indicates addition, or an additive quantity. - (minus) indicates subtraction, or a subtractive quantity. WVhen applied in Geometry, or in Algebra combined with Geometry, plus and minus are also used as relative terms, indicating opposite directions. When direction is taken into account, distances are measured from a common point of beginning, called the Origin. Plus is called the positive sign, and minus, the negative sign.* 24. DISTANCE OUT is the distance from the median plane to the outside corners of an excavation or embankment. Fig. 6. AD or E C (Fig. 6) is the distance out on the E C right. A M or K B is the distance out on the left. Distances out, when necessary, may be taken to points intermediate between B B. _-.. or C and the centre-line AE. The point where B C meets the roadbed P Q is called the grade-point of the cross-section, and its distance out the distance out of the gradepoint.' poin............................. M A D 25. SIDE-SLOPE is the inclination of the side A C or AB (Fig. 6) to the roadbed P Q, or to the horizontal line M D. It is generally the same on both sides. Side-slope is measured by comparing horizontal with vertical distance. If a point be supposed to move along AC from A toward C, so that it departs horizontally from AE one and a half feet in the same time that it rises or falls vertically one foot from A D, the side-slope is said to be one and one-half to one. 26. TRANSVERSE GROUND-SLOPE is the inclination of the line B C (Fig. 6) to the horizontal line M D. It is for brevity called ground-slope, surface-slope, cross-slope, top or bottom slope, or simply slope, when this term is not ambiguous. This slope is measured, by a slope-instrument, in degrees of the quadrant. Similar and Opposite Slopes.-The slopes may or may not incline toward the same side of the roadbed. If they incline toward the same side, they are called similar; if they incline toward opposite sides, they are called opposite. * A reference to treatises on Algebra and Algebraic Geometry will afford the student a fuller explanation of the meaning of positive and negative signs. -20 PRELIMINARY DEFINITION AND DESCRIPTION. 27. CROSS-SECTIONS are the plane figures formed by cutting earthwork solids transversely. B C Q P (Fig. 6) is a cross-section. When the surface-line B C is a single straight line, the form and area of a cross-section depend upon four quantities,-viz., the width of roadbed, the centre-height, the side-slope, and the Fig. 7. surface-slope. When the surface-line is not D straight, instead of using the surface-slope K\H F > j ~C to draw the cross-section, the heights and L,,., the distances out of certain points in the surface-line are employed. In the crosssection AB CDEHIKLA, (Fig. 7,) besides the centre-height HII M, the heights C N,'1' A S.M r DO, E Q, KS, L T, and also the distances out of the points C, D, E, K, L, would be used to draw the figure. C N and L T are called side-heights. 28. FULL, PARTIAL, AND NEUTRAL CROSS-SECTIONS. When the cross-section Fig. 8. is wholly in excavation or wholly in embankment, it is said to be full. Fig. 8. When the cross-section is partially in excavation and partially in embankment, it is said to be partial. Fig. 9. Fig. 9. Excavation and embankment may be called full or partial, according as the cross-section of the work is full or partial. When the surface-line just touches the edge of the roadbed, the crosssection is said to be neutral. Fig. 10. Fig. 10. Neutral cross-sections may with propriety also be regarded as full, because they really fill out the whole figure; but occasionally it may be convenient to have a special designation for them. Work having a neutral cross-section in all parts might without impropriety be called neutral; but such work requires no distinctive appellation. The solid in such work is a prism. It may be computed either as full work or as partial work. Subdivision of Sections. 29. SUBDIVISION OF SECTIONS is so dividing theni as to show the formi and dimensions of the component excavation and embankment solids. Subdivision is necessary whenever a partial cross-section can be made at any part of the work. Otherwise, it is not necessary. If there be a partial cross-section, it will be shown by the methods indicated in Articles ^34, 47, and 54. 30. THREE CASES include all the varieties of work, whether full or partial, which can occur in any section. These cases are distinguished by the nature of the end cross-sections. The end cross-sections of any piece of work must be either alike or different. Like cross-sections may be either fall or partial. Hence, there are three combinations of cross-sections,-viz.: Case I., both full. Case II., one full, and the other partial. Case III., both partial. SUBDIVISION OF SECTIONS. 21 31. TWELVE VARIETIES, four to each case, belong to these cases. This results from combining~ for each case, the sign (plus or minus) of the centre-height, with similar or opposite surface-slopes. Four combinations of signs and slopes are possible. It may facilitate the study of the subject to view the neutral section as a full one, with a small bal visible side. By this means the prismoids (F, f, on the Plates) may be imagined to exist, if wanting, but to be evanescent. In this way the twelve varieties will include all the combinations. End-heights equal to 0 may have either sign prefixed. 32. SUB-VARIETIES. If the neutral cross-section be kept distinct, we must, in order to a strict classification, either make use of this section in characterizing the cases, or introduce it among the characteristics of sub-varieties. The complete discussion would probably exceed the limits of practical expediency. 33, GENERAL SCHEME. The cases and their varieties are exhibited in the following general scheme. (See Plates I., II., III.) G ]E NE ]RA L S C H IE IE, FOR ALL VARIETIES OF FULL AND PARTIAL WORK IN A SECTION. CASE I. Cross-Section _Full at Both Ends. Signs of Directlon of Nunmber of Signs of Direction of Npmber of Enumeration and Description of Component Solids. Centre-Heights. Slopes. Diagram. Like. f Similar. No. 1. One prismoid F. Opposite. No. 2. One prismoid F. Similar. No. 3. One prismoid F, and one pyramid P. Unlike. {. tOne prismoid f, and one pyramid p. Opposite. No. 4. One prismoid F, and one pyramid P. One prismoid f, and one pyramid p. CASE II. Cross-Section _Full at One End, Partial at the Other. Signs of Direction of Number of CeSnIigns of Direction of Number of Enumeration and Description of Component Solids. Centre-Heights. Slopes. Diagram. Like. Similar. No. 5. One prismoid F, one truncated pyramid T, and one pyramid P. Opposite. No. 6. One prismoid F, one truncated pyramid T, and one pyramid P. U lik f Similar. No. 7. One prismoid F, one truncated pyramid T, and one pyramid P. Unlke. X Opposite. No. 8. One prismoid F, one truncated pyramid T, and one pyramid P. CASE III. Cross-Section Partial at Both Ends. Signs of Direction of Number of Enumeration and Description of Component Solids. Centre-Heights. Slopes. Diagram. Like. { Similar. No. 9. Two truncated pyramids T and t. Opposite. No. 10. One prismoid F, and two truncated pyramids T and t. Two pyramids P and p. Unlike. f Similar. No. 11. Two truncated pyramids T and t. Opposite. No. 12. One prismoid F, and two truncated pyramids T and t. Two pyramids P and p. _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 22 SUBDIVISION OF SECTIONS. This scheme is adapted to a mean roadbed. It may be studied by first observing, on all the diagrams belonging to each case, the characteristic end cross-sections of that case. Then examine, in the scheme, the repetition of the four combinations of centre heights and slopes, and follow it on the diagrams. Finally, apply to its proper diagram the description of each variety given in the scheme, and observe the component solids. Solids are separated by neutral cross-sections, because a solid continues of the same kind as long as its cross-section does, and the cross-section becomes neutral at the point where it changes its character. 34. APPLICATION OF TIlE GENERAL SCHEME. The scheme shows that unlike signs of the end-heights always require subdivision of the section. When the signs are alike, the necessity for subdivision must be examined, by observing whether one of the end cross-sections is partial.* If there be a partial crosssection, it will be shown by construction, (Art. 47;) also, by observing (Table III.) whether either end-height be less than the height of the neutral crosssection. 35. DETERMINATION OF THE VARIETY. If subdivision be necessary, assume a mean roadbed, and find the abscissas of the neutral cross-section. (Art. 50 or 52.) The abscissas being mar]ked on the diagram, (Fig. 16,) together with the surface-slopes, the computer may sketch the ground-trace through the ends of the abscissas and through the grade-points of the end cross-sections. If desirable, the side-slopes may be added, in order to give relief to the diagram. Then consult the drawings illustrative of the Case to which the section belongs, and determine its variety. 36. UNION OF SEVERAL COMPONENT SOLIDS AND REDUCTION TO A PLANE SURFACE. Having found the variety, we endeavor to find a single solid with a plane surface, which shall be approximately equal, in the case of excavation and embankment respectively, to the sum of the component solids. This is done as followsfor the several cases of the General Scheme:CASE I. Solids reduced to Prismoids. —Nos. 1 and 2 are whole sections, requiring no reduction: they are prismoids. Nos. 3 and 4. The median grade-point is on the division-line between the pyramids P and p. Assume that if these pyramids were cut off at this point, the apex of each would contain material enough to fill out the basal portion to a neutral cross-section. Thus a single prismoid would be formed with a full cross-section at one end, neutral at the other, and consisting of F or f and the basal part of P or p. The length of it is the abscissa of the median grade-point. (Arts. 9 and 51.) The greater end-height is given; * A little exercise of memory will assist us in judging whether it is worth while to draw cross-sections or to, refer to Table III. in order to find whether there can be a partial cross-section. The widths for double track are: excavation, 28 feet; embankment, 24 feet. 28 feet roadbed and 350 surface-slope give, by Table III., say 10 feet for height of neutral cross-section; and 24 feet and 180, say 4 feet. Hence, in general, no height in excavation over 10 feet, and none in embankment over 4 feet, require examination. This method of judging can be applied according to circumstances. SUBDIVISION OF' SECTIONS. 23 the lesser is the height of the neutral cross-section. For the surface-slope, assume a mean slope. CASE II. Solids reduced to Prismoids and Pyramids. —Nos. 5, 6, 7, 8. Assume, in a manner similar to the previous, that if T be cut in half, the prismoid F would be filled out to a neutral cross-section. The dimensions then are, greater end-height given; lesser, the height of neutral cross-section; length, the length of whole section less half the abscissa of PT.- Assume a mean surface-slope. If T should predominate over F, then F and T together may be taken as a truncated pyramid having its end-heights given. The length, the whole length of the section. Assume a mean slope, as before. The pyramid P may take the surface-slope of its base. Its length. is shown by the abscissa. It needs no further remark here. CASE III. Solids reduced to Truncated Pyramids or to Prisms and to Pyramids. Nos. 9 and 11. Assume a mean slope, and these become truncated pyramids. They retain their original heights and length. Nos. 10 and 12. The parts T, F, and t may be taken together as a prism. It is assumed equal to a prism erected upon a triangle having a mean base between the widths shown at the ends of the section. Its length, that of the section; the slope, the half-sum of the given slopes. The pyramids P and p may be treated in a similar manner. Assume them equal to a pyramid of mean base and a slope equal to the half-sum of the given slopes; the length equal to the united length of P and p. In No. 10, the end-heights, being of the same sign, show to what kind of work-whether excavation or embankment-the parts T, F, and t belong. In No. 12, from whichever end the section be regarded, the longer marginal abscissa, counting from that end, shows the character of the work. The work will be of that kind in which the longer abscissa is contained.,curved ASfections.-(See Arts. 5, 105, 120, 128.)-The rules for curved solids require the length to be measured on the circle of alignment; but it is nearly correct to find the length as if for straight work, which gives the length somewhat too short.* It is usual and generally sufficient-at least for preliminary estimates-to disregard curvature' and compute as if for straight work. For pyramids and truncated pyramids, it is a better approximation to modify the process for straight work, thus: —Find the mnarginal radius, by increasing or diminishing the radius of alignment by the half-width of roadbed; multiply the length for straight work by the marginal radius belonging to the solid, and divide by the radius of alignment to find a corrected length. (Approximately, the curved marginal length.) As these solids are not greatly distorted, the plates illustrative of straight solids will suffice. 37. WEDGES.AND WEDGE-SHAPED MASsEs.-True wedges rarely occur. Solids which may be so termed have one end-height practically inconsiderable, and a nearly level surface-slope. We may sometimes assume the surface-slope to be level, or 0~, and one end-height nothing. The sum and difference of the endheights are then the same, —viz., equal to the greater end-height. *By employing the marginal radius, and the distances out counted from the margin, we might proceed as in Arts. 107, 122. (See ~ 2, Art. 7, p. 286.) 24 AUXILIARY CONSTRUCTIONS. CHAPTER II. AUXILIARY CONSTRUCTIONS-DETERMINATION OF VARIETIES AND PREPARATION OF ELEMENTS BY CONSTRUCTION AND BY TABULATION. Auxiliary Constructions. 38. DRAWING PROFILES. The method of doing this is sufficiently indicated by the figures NOPQ (Art. 8) and A H I B, (Art. 9.) The lines PQ, AH, D E (Fig. 1) are all drawn straight. In reality, PQ would be curved in such surfaces as are described in Art. 8; but this is not regarded in drawing profiles. It is usual, when a continuous profile is constructed for numerous sections, to draw the heights O P, N Q upon a larger scale than the length N O, in order to economize space. For computing solidities, the scale for the heights should be sufficiently large to permit the estimation of tenths of feet. The scale for the length should permit the estimation of single feet. 39. REDUCTION OF OBLIQUE TO DIRECT LENGTH. The length of a section is supposed to be measured in a direction perpendicular to the parallel planes of the end cross-sections. Generally, it is taken on the median surface-line. This makes the direction perpendicular to the end planes, as far as any deviation to the right or left is concerned. But in most cases there is a vertical deviation on account of the rising or falling of the surface lengthwise of the section. This vertical deviation is not generally regarded; the length is supposed to have been measured horizontally upon the field, by taking care to hold the measuring-chain in a horizontal position. The chain cannot be held, with ordinary care, strictly horizontal, but sufficiently so for practical purposes. Cases may arise, however, where the length is known to have been measured so obliquely as to require correction. The direct length for a measured length of 100 feet and a horizontal or vertical deviation of 10 feet will be 991 feet; and for other lengths in proportion, with the same rate of deviation. 40. FINDING THE DIRECT LENGTH BY CONSTRUCTION. Let AB (Fig. 11) be the measured length. Take C in the middle of AB, and from C describe a portion BD of a circle. Then, taking in the dividers a length equal to the vertical Fig. 11. rise from A to B, place a point of the dividers in B B, and describe an are E D meeting the first are -~, If there be no deviation to the right or left, this is the direct length required. If there be any' x' — I MA deviation right or left, repeat the process. Take k"h \,,,@' *-0"~- H in the middle of AD, and from H describe a portion DK of a circle. Then, taking in the dividers a length equal to the horizontal deviation in measuring from A to B, place a point of the dividers in D, and describe AUXILIARY CONSTRUCTIONS. 25 an arc L K meeting D K in K. Join AK. AX: is the direct length, reduced both for vertical and horizontal deviation. -Explanation.-By a known property of the circle, if B and D be joined, B DA is a right angle: hence B D is the vertical height in a vertical position, and A D is the horizontal length corresponding to the sloping length AB. In the same manner DK, if joined, would represent the end crosssection, perpendicular to the median line. AK would be the direct length corresponding to the oblique length AD. 41. To DRAW THE SIDE-SLOPE WHEN THE RATE OF SLOPE IS GIVEN. Suppose the rate of slope to be 1~- to 1. Let AB repre- Fig. 12. sent part of the roadbed upon the cross-section. D B is the side of the roadbed. Prolong AB and make B C, by any convenient scale of equal parts, equal 11. Draw CD perpendicular to AC and A B a equal 1. Join B D. B D is the side-slope required. 42. To DRAW A CROSS-SECTION, the necessary data are the width of roadbed, the rate of side-slope, the centre-height, and either the side-heights or the surface-slope. (1.) When. the side-heights are givee*. Make AB equal to the width of roadbed. Draw the centre line CD perpendicular Fig. 13. to A B, and take C E and CK equal to the respective side-heights. Through E and K, -----------—; —--. parallel to A B, drasw E N, K M to intersect the Nside-slopes, taking care to transport thus the heights CE, C K each to its proper side. Make CD equal to the centre-height, and join DM, A c B DN. AB MDN is the cross-section required. (2.) When the surface-slope is given. We shall suppose the roadbed, the sideslopes, and the centre-line drawn in the same manner as when the side-heights are given; and that these lines, so drawn, are ready for use upon a diagram or an engraving as in Fig. 14. Upon Fig. 14. the centre-line CE is a scale of equal E parts, counting from C toward E. Through HI, the intersection of the side-slopes, passes the diameter I L parallel to AB. From this diameter are reckoned the graduated arcs IK, LM. To construct the cross-section, draw, or suppose to be drawn, the radius HIN, making the angle NHL A L of the same magnitude as the given surface-slope, and the direction from -,,. —N to IH to correspond with that of the surface. Then, through the point D, indicating upon the scale the given * When the surface-slope is not given, the known points on the surface must be found by giving to each its proper height and distance out. Art. 27. 26 AUXILIARY CONSTRUCTIONS. centre-height, draw PQ parallel to HN. A B Q P is the cross-section required. The distance H C is called the augment for the centre-height. Twice this distance is the augment for the sum of the end-heights. It is given in Table II. The augmented height H D may be measured upon the scale C E. 43. To FIND THE AUGMENTED CENTRE-HEIGHT AND THE SURFACE-SLOPE OF THE Fig... MID CROSS-SECTION. Draw the surface-slopes P Q, RS (Fig. 15) *at their respective end-heights, as v N? 7..4..............-U in Art. 42. Take T half-way be\v H tween P and R, and U half-way between Q and S. ABUT is the mid cross-section. The augmented centre-height H V and the surface-slope TU can be measured upon a diagram, as in Art. 42. 44. To FIND THE DISTANCES OUT OF THE MID-SECTION. Having drawn TU, (Fig. 15,) as in Art. 43, draw TK and UN parallel to AB. TK and UN are the distances out of the mid-section, and can be measured on a scale. If the cross-section be not full, one of the distances, as UN, will be found in the manner described. The other will become the distance out of the gradepoint. (Art. 48.) It is evident that the distance out of the mid cross-section is equal to the half-sum of the end-distances out on the same side. 45. CROSS-SECTIONS DRAWN ON THE PLANE OF THE ROADBED. Let AB CD (Fig. 16) represent the roadbed, the sides of which may be divided into feet. Fig. 16. At each end are graduated arcs whose centres are H and K. The cross-section at each end may be v drawn in the manner described in, Ft E =: _K! ---— /~ Art. 42, observing that plus centre- _ D heights count in the direction KE, and minus heights in the direction H L; also observing the proper directions of the surface-slopes. They are here supposed to be opposite. The side-slopes, if desired, may be drawn through the corners A, B, C, D of the roadbed. Crosssections will thus show their dimeni-i... R 1sions as accurately as if drawn separately from the roadbed, and may be perfectly understood. If this mode of representation appeal strange, the reader need only imagine the plane of the cross-section turned upon AR or C S, AUXILIARY CONSTRUCTIONS. 27 upward or downward, as may be required, at right angles to the plane of the roadbed. The distances out of the grade-points R and S will remain unchanged. 46. To FIND THE MARGINAL tHEIGHTS AT THE ENDS OF A SECTION. Draw, as above, (Fig. 16,) the surface-slopes T R, VS. A T, B U, CV, DW represent the marginal heights, and may be measured on one of the scales of the diagram. 47. To FIND THE CENTRE-HEIGHT OF THE NEUTRAL CROSS-SECTION. Draw the surface-slope, in its proper direction, through the corner of the road-bed. The height may be read off from the central scale. (Fig. 16.) This dimension is the tabular quantity of Table III. 48. TO FIND TIHE DISTANCE OUT OF THE GRADE-POINT. Apply HR or KS to the scale. This distance is the tabular quantity of Table V. Added to or subtracted from the half-width of roadbed, as may be required, it gives the base. 49. To FIND THE BASE OF A SUB-SECTION. When R and S (Fig. 16) fall within the roadbed, the bases AR and CS require to be measured. The sum and the difference of the bases at the ends of a section are elements for computing solidity. They are easily found after measuring the bases. The sum and difference of marginal heights (Art. 46) may be used instead, with suitable rules. 50. TO FIND THE ABSCISSA OF THE NEUTRAL CROSS-SECTION. To simplify this subject, we shall adopt a mean roadbed. There can Fig. 17. then never be more than two marginal grade- V points belonging to the same section. Let AB CD W (Fig. 17) represent the roadbed, and AT, B U, c CV, DW the marginal heights found, as in Fig. 16. By considering the mode of generation of the ground-surface, as described in Art. 9, it is plain that the abscissas required are those of the grade-point of the directrix. In the present example, because the marginal heights B U, DW are both of the same sign, there can be no such grade-point within the section on the marginal plane through B U and D W, Take, therefore, the plane through A B AT and CV. Let AT CAV (Fig. 18) represent the side view of the side AC, Fig.17. Care must be taken to observe the plus and minus signs in drawing A T, CV. Join TV, and P will be the grade-point of the directrix, because AC represents the plane of the roadbed, and TV is the direc- Fig. 18. trix. The abscissa AP or CP may be measured on A the side-scale of Fig. 16. Instead of drawing a sepa- c A rate diagram, as here shown, the operation may be performed at once, on the side-scale A C of Fig. 16. T It will only be necessary there to transfer AT in the direction AlH, and CVin 28 AUXILIARY CONSTRUCTIONS. the opposite direction. The point P will then fall on the scale, and the abscissa can be at once read. The inner-side lines of Fig. 16 may, if desired, be drawn to represent embankment-width. The abscissa for each width of roadbed may be thus determined by separate operations. The abscissa is required in the subdivision of sections. (Art. 29.) 51. To FIND THE ABSCISSA OF THE MEDIAN GRADE-POINT. If, in Fig. 18, AT, C V represent the end centre-heights, AT VC is the median plane, and TV the median surface-line. AP and CP are the abscissas required. It is plain that the finding of the abscissas in this manner is equivalent to drawing the marginal and median profiles. In Fig. 2, Art. 9, R is the marginal gradepoint, and BR31 is the abscissa of the neutral cross-section. 52. A METHOD FOR FINDING ABSCISSAS WHEN THE GROUND-SURFACE IS A PLANE. We have seen (Art. 8) that a plane surface may be generated by Fig. 19. means of directrices, in the same imanner as a curved surface. The above methods for finding E abscissas therefore apply to plane as well as to curved surfaces. But we know (Art. 16) that s r.) Call this whole hyperbolic area Ah, and let A' and A" be the end-areas: we shall have, by formula 39, V= L (Ah+ A/ + Al/)... 45. Prismoidal Formula.-Let the area of the mid cross-section be A,, and its radius-vector p"'. If U be now supposed to trace the surface-line, H U will always be a mean between H Q and H S; that is, p' + hence, the area between the limits o', y" will be A0 P+P ) d (= f (p22pP ) d. 4, Or, AH1 = I pp' d +' x (p2+p'2) d = A (A+A") 47, whence Ah = 2 A - (A' + A"); and, therefore, by formula 45, V=L (Ah + A' + A") (4 A + A' +A").... 48, The last member of which is the prismoidal formula. (See p. 279.) Equations 39, and 45 to 48, are not founded on any relation existing between p, p', and (5: hence, in order that these shall hold, it is not necessary that the directrices B C, ED, (Fig. 3,) be of any particular form. This tends to confirm the belief in the practical sufficiency of the prismoidal formula; but the choice of a ground-surface from among those surfaces to which the formula applies, is left to individual judgment. (Art. 8, Appendix.) The point U (Fig. 15) describes an hyperbolic arc with its convexity towards HI.* *Every cross-section gives an hyperbola, and the projections P Q, R S, of the end-slopes upon the cross-section are parallel to the asymptotes. This is because the parts Q U, U S, of the intercept Q S, are always in a constant ratio; that is, in the ratio of the lengths of the two parts into which the solid is divided by the cross-section. We omit the demonstration, intending to publish it elsewhere; but will here remark upon a simple and, we believe, new method of describing an hyperbola'when the axes are given. Suppose HK to be the transverse axis; through one extremity, as K,,draw straight lines through the ends of the conjugate axis; these will, therefore, be parallel to the'asymptotes of the required hyperbola, and they will correspond to the projections of equal and -opposite end-slopes having equal centre-heights. Bisect the intercept Q S constantly in U, and the locus of U will be an hyperbola with the given axes. APPENDIX. Notes to the Rules and Examples of Part I. 1. FURTHER EXAMPLES IN WHOLE AND SUB SECTIONS. (See Art. 88, Part I., and the Table, page 33.) Sections 1, 2, 4, and 9 exhibit all the varieties of Case II. In Nos. 1 and 4, the length of the pyramid is the same as that of the marginal abscissa, —viz.: for No. 1, 23 feet; for No. 4, 46 feet. The length of the reduced full-work solid, ~ T + F,* will be, for these sections respectively, 23 + 100- 23- 100 23 = 88.5; and 46 + 100-46 100 6 _ 77. InNos.2and9, 2 2 2 the length of the abscissa is, for No. 2, 49 feet; for No. 9, 23 feet. The length-of the reduced 100 -49 100 - 23 full-work solid F ~ T will be respectively, 49 + 2 = 74.5, and 23 + 2 61.5. The / 2 2 length of the pyramid will be, for No. 2,1 100- 49 —51; for No. 9, 100 -23 - 77. The lesser end-height of the full work, in all these sections, is the height of the neutral cross-section belonging to the work and to the mean slope. All these are not unusual forms. Sections 3 and 11 are varieties of Case III. They represent forms of frequent occurrence,viz., truncated pyramids running from station to station. The length is given by the length of the section. The bases are easily found by constructing the. cross-sections, or by calculation. Sections 5 and 8.-Two varieties of Case I. These forms will be met with on ground which rises or falls rapidly in the direction of the length. The work reduces to two full-work portions. The length of one part is the median abscissa; of the other, the length of the section less the abscissa. The lesser end-height, in each part, is the height of the neutral cross-section of that part, under the mean slope. We have, for No. 5, the median abscissa 43 feet; the length of the other part is therefore 57 feet. For No. 8, the median abscissa is 67 feet; the length of the remainder, 33 feet. Sections 6 and 7.-The two other varieties of Case I. These are frequent forms. They are whole sections not requiring subdivision. Sections 10 and! 12.-The two remaining varieties of Case III. The work which runs entirely through is treated as a truncated pyramid similar to those of Sections 3 and 11, which also belong to this Case. The correction for the difference of the bases, however, is here omitted, as it is supposed the cubic content found from the sum of the bases will sufficiently well represent the whole solidity. To find the sum of the bases by calculation, proceed as directed in Art. 70, Part I., using a slope equal to the half-sum of the given slopes. The sum of the bases of the two pyramids will be found in the same way; but it is the half-sum, or mean base, which is required. In like manner, the rule (Art. 71, Part I.) may be applied to finding the augmented sum of the heights of the thorough work, and thence the mean height of the pyramids. In finding by construction the mean base or mean height of the pyramids, it will be sufficient to take the dimensions as they appear on the cross-sections drawn with the given slopes. The rule for these figures is only intended to give a running estimate. 2. UNUSUAL CASES. Calculation of the Elements for Unequal End-Widths. (See Example, Art. 91, Part I.)-The elements may be thus found by calculation. The height of the:redundant prismoid at each end, is evidently half the augment from Table II. for a roadbed equal to the given Width A B or E F. The heights L V, R U are -respectively the augments from Table III. for a roadbed equal to twice QV, and twice N U; that is, for a roadbed 2 (I A B - K B) - A B *,This is merely a designation intended to point out, by reference to the diagrams, the mode of combination of the reduced solids; and having no strict quantitative application to solidity. 291 292 APPENDIX. 2 K B 33.4 28 = 5.4; and for a roadbed - 2(~ EF- EM) EF- 2 EM-40 - 31.8 = 8.2. Hence 33.4 = A B gives half the augment from Table II. =... 16.7' 40.0 E F gives half the augment from Table II. = P T.. 20.0 3 3 difference. Sum of the end-heights of the redundant prismoid 36.7 2 Q V 5.4, under 100, in Table III., gives LV. — 0.5 K L, the given end-height.. 17.6 SQ........ 17.1 2 NU = 8.2, under 100, in Table III., gives R U..... 0.7 M N, the given end-height.. 9.2 TR..9.9 Q-OS 0 + SQ = 16.7 + 17.1 = 33.8: PR = PT + TR = 20 + 9.9 - 29.9 O Q + PR = Augmented sum of heights for the whole solid = 33.8 + 29.9 = 63.7 0 Q - PR = Difference of heights for the whole solid = 33.8 - 29.9-= 3.9 3. UJNsuAL CASES BY EQUIDISTANT LEVEL-HEIGHTS. Each end cross-section is divided into the same number of spaces by equidistant level-heights,-as many as desirable. If the heights have not been taken on the field, they may be found after constructing the cross-sections. Preparation of Elements. Form the sum and difference of total bases, as L M, (Fig. 30, p. 56, and Art. 111, Part I.) Form two level-totals,-one for each end,-by adding together the two external heights B M, D L, with twice the sum of all the intermediate heights. Take the sum and difference of the level-totals. 4. RULE FOR COMPUTATION OF SOLIDITY BY EQUIDISTANT LEVEL-HEIGHTS. Proceed as in Art. 112 or 113, Part I., retaining the sum and difference of total bases, and substituting the sum and difference of level-totals for the sum and difference of heights there employed. Observe that the second term is to be added if the greater level-total is at the same end of the work with the greater total base, and subtracted if otherwise. Divide the result by the number of spaces on the cross-section; that is, by the number of level-heights at either end, less one. The result is the whole solidity between the vertical side-planes B M, D L. If the content upon a figure like B A E D K is required, the redundant solids upon B A M and DEL must be found, (by Art. 115 or 118,) and deducted. If BA or DE-one or both-were sloping opposite to the present direction, one or both of the external solids must then be added. EXAMPLE. First End. Outside heights, 16, 17; intermediate heights, 13, 15, 4, 5, 18, 14; interval of heights, 6. We find, Total Base = 42; Level-Total = (16 + 17) + 2 (13 + 15 + 4 + 5 +- 18 + 14) =171. Second End. Outside heights, 10, 13; intermediate heights, 11, 7, 8, 15, 14, 12; interval of heights, 10. We find, Total Base = 70; Level-Total -(10 + 13) + 2 (11 +- 7 + 8 + 15 + 14 i12) = 157. Then, Sum of Total Bases — 112; Difference = 28$; Sum of Level-Totals 328; Difference =14. Table XXI., opposite 112.0 under 3 (hundreds) 2 (tens) 8 (units)... 17007.4." " " 28.0 take one-third for 1 (ten) 4 (units).. -60.5 Divide by 7, the number of spaces.. 7)16946.9 Content between the vertical side-planes. 2421.0 5. CURVATURE OF THE MEDIAN PROFILE OF WHOLE SECTIONS. (See Articles 103, Part I., and 1, Part II.) When the external surface-lines incline in different directions, they will, if produced, meet the intersection of the side-slopes in two points at opposite ends of the work. The line which passes through these points and through the ends of the centre-heights will obviously be APPENDIX. 293 curved, with the concave side towards the roadbed. When the points where the median surfaceline meets the intersection of the side-slopes are both at the same end of the work, consider the point which is nearest to the work, and imagine a curve passing through this and through the ends of the centre-heights. 6. CORRECTION FOR CURVED SURFACE. (See Note, Art. 104, Part I.) The correction for all straight work included under surfaces generated according to the method of Art. 8, Part I., may'be made in the same general manner. A plane surface is only a particular variety of the surface there described; but, when the surface is plane, the method of correction takes a simpler form than for curved surfaces, although the same in principle. Let ABC (Fig. 45) represent one entire end cross-section of any piece of whole-section work, and ADE the other. Through E, one of the points where AC, Fig. 45. a side-slope of one of the end cross-sections, 49 55 37 01 A 25.59 107.39 ABC, is cut by the surface-slope of the other, ADE, draw a line EM parallel to the othelr side-slope; and make E M equal to B D, the.. —--;.; distance cut off by the surface-slope of ABC D' —'' - M from the side-slope of ADE. Join M with N K the external angle C of ABC. The correction is equal to one-twelfth part of a prism erected upon the triangle C M E as a base, and of the same length as the work. c Draw E K parallel to the centre-height AN. E K is the augmented centre-height of this prissm. If we should enter the tables with double the height E K, under the surface-slope of C M, (which can be measured as in Art. 43, Part I.,) the tabular quantity would be equal to the content of a prism 100 feet long, standing on the base C IE. But it is one-twelfth of this tabular quantity which is required. Now, the tabular quantities are as the squares of the arguments in the side column; hence, if we enter the tables with the height E K, the tabular quantity is one-fourth the solidity of the prism upon C M E. One-third of this is the twelfth part required for the correction. The correction is to be added if both external surface-lines rise or fall together towards the same end of the section, and subtracted if one of them rises whilst the other falls towards the same end. When the ground becomes a plane, as in Fig. 46, Fig. 46. the surface-slopes B C, DE are parallel; hence A E K becomes equal to 0 N, which is the difference of the end-heights: the slope of C M is the original D surface-slope. These things being known, it is un- B 0 necessary to find E K and the surface-slope, by A' I construction. Explanation.-By formula 9, page 279, it appears that the term which is neglected, in the Rule, Art. 104, Part I., is air sin 2 a p q L3. Put G the lengths of the side-slopes (Fig. 45) respectively, A D = M, A B = M', A C = N, A E = N/: M - M N-N' I 12 then- and q=. Hence the neglected term -X 1 sin 2 r LL2 pq 1X L L 12 2 12 2 sin 2 a (M — M/) (N - N') L. This represents the twelfth part of a prism whose length is L, and whose triangular base has an angle =- — r - 2 a, with the sides about this angle respectively equal to M - M' and N - N'. But C E M is, by construction, such a triangular base. The correction is additive if p and q are of the same sign,-that is, if the directrices are inclined in the same direction,-and subtractive otherwise. If A B = A D, M E and the base C E MI both vanish. Fig. 46, where the slope is the half-sum of the slopes of Fig. 45, may represent the end crosssections of Example 1, Art. 86, Part I., and Fig. 45 those of Examples 1 and 2, Articles 72 and 104, Part I. These references for Fig. 45 will be useful in studying the following examples:Example 1.-Take Example 1, Art. 104, Part I. E K is found - 12.7, and the surface-slope CM - 24~. 294 APPENDIX. Opposite 12.7, under 240~, in: Table VII.. 3)404.9Correction additive...... 134.9 Opposite 59.68, under 160, in Table VII.. 6070.1 Whole content corrected. 6205.0 Redundant prism.. 355.6 Corrected residual prismoid... 5849.4 The error in working according to the rule (Art. 104, Part I.) is 134.9, or about 2 per cent., in defect on the whole content, 6205. Example 2.-Take Example 2, Art. 104, Part I. E K is found =14.5, and the surface-slope CM.- 26G. Opposite 14.5, under 26~, in Table VII... 3)629.1 Correction subtractive... 209.7 Opposite 69.9, under 80, in Table VII. 7101.8 Whole content corrected.. 6892.1 Redundant prism..355.6 Corrected residual prismoid.., 6536.5 The error in working according to the rule (Art. 104, Part I.) is 209.7, or about 3 per cent., in excess on the whole content, 6892.1. 7. SECONDARY DIMENSIONS FOUND BY CALCULATION.:The dimensions of the base C M E (Fig. 45) of- the secondary prism may be found by calculation. Call the side-heights and distances out of the whole work primary, and those of the base C M E secondary, dimensions; also, call those heights and distances which lie on the same side of the work homogeneous.: The side-heights of the base of the secondary prism are equal respectively to the difference of the homogeneous primary side-heights: the same for the distances out. Secondary Distances Out. —In the above examples, we find, for No. 1, the distances out 57.84 and 11.42; for No. 2, 81.80 and 12.54. (See Art. 72, Part I., for the primary distances out.) Secondary Centre-Height.-Then (see Examples 1 and 2, Art. 76, Part I.) the heights are found, for No. 1, =12.7; and for No. 2, =14.5. Difference of Secondary Side-ileights. —This is required in order to find the secondary surfaceslope. The mode of proceeding is evident from Art. 78, Part I. The division by the rate of sideslope directed -in Art. 78 may be deferred until after the division by the sum of distances out directed in Art. 79, Part I. Secondary Suq:face-Slope. (See Art. 79, Part I.)-The difference of secondary distances out is, for No. 1, _ 46.42, and the sum of the same, 69.26. Then 46.42 divided by 69.26, and again by the rate of side-slope, 1I, gives.447 = tan 240, the secondary surface-slope. For No. 2, difference, = 69.26; sum, - 94.34. Then, 69.26 divided by 94.34, and again by the rate of side-slope, 14, gives.489 =-tan 260, the secondary surface-slope. C8. COMPARISON OF THE METHODS BY TRANSVERSE SLOPES AND BY CENTRE AND SIDE HEIGHTS. The rule of Art. 112, Part I. is founded on a very usual supposition concerning the nature of. the surface,-viz., that the median and external surface-lines are straight; also, that the surface-line of the cross-section is straight on each side, from the centre out. Computation upon this hypothesis has generally been considered more accurate than by transverse ground-slopes, supposing each method to be correctly carried out. Having, then, by the method of transverse slopes, determined the points B, C, D, E on the end cross-sections, (Fig. 45,) these points ought to be adopted, as if they had been observed upon the field; but their adoption is not a sufficient reason for giving up the supposition of a straight median surface-line. This supposition is, APPENDIX. 295 however, violated by assuming the surface to b6 generated as in Art. 8, Part I. (See Art. 5.) In Example 1, Art. 104, Part I., the centre-height of the mid cross-section was found = 29.84; but, on the supposition-of a straight median line, that height would have been 29.6. In Example 2 of the same article, the mid-height was found - 34.96. With a straight median line, the midheight would have been 29.6, the same as before; for the difference between these examples is only in the direction of the surface-slope of the ends. To show in what degree this increase of centre-height affects the result, if the method by centre and:side-heights be taken as a standard, we shall compute the above examples by that method. One computation will suffice, because none of the elements for this computation are changed by reversing the direction of the slopes. The distances out are given in Art. 72, Part I. We have, for the elements, augmented sum of heights, 59.2; difference of heights, 14.2; sum of total bases, 219.6; difference of total bases, 69.3. Opposite 59.2, under 200, in Table XXI.. 5481.5..... 10, " "....... 274.1. "..- 9,'" "... 246.7..... 6, " "..... v.. 16.4 " 69.3, " 10, (add one-third of tabular quantity,)... 106.9 " " " 4,........... 42.8 (L (( L( ~.2)' (( cc (~ e e ~ 2.1 Content of whole ground by centre and side heights.. 6170.5 Redundant prism...... 355.6 Residual prismoid, by centre and side heights..5814.9 Assuming this result (6170.5 yards) to be the true content of the whole ground, we have the following comparison with the results obtained in Article 6. Example 1. True whole content by centre and side heights..6170.5 Corrected whole content by transverse slopes..6205.0 Error in excess (about A6 per cent. on 6170.5).34.5 Example 2. True whole content by centre and side heights..6170.5 Corrected whole content by transverse slopes..6892.1 Error in excess (about 12 per cent. on 6170.5). 721.6 The error committed in working according to the approximate rule is thus shown. Example 1. True whole content by centre and side heights..6170.5 Approximate whole content by Art. 104, Part I... 6070.1 Error in defect (about 19 per cent. on 6170.5). 100.4 Example 2. True whole content by centre and side heights..6170.5 Approximate whole content by Art. 104, Part I.. 7101.8 Error in excess (about 15 per cent. on 6170.5).931.3 The above per centages of error on the whole ground become more important as the width of roadbed is increased; that is, as the amount of work to be done is lessened, whilst the absolute error of computation remains the same. It will be observed that the error in the corrected result by transverse slopes, for the second 296 APPENDIX. example, is 721.6, whilst the content of the secondary prism, (see Art. 6,) which is the correction for the approximate rule, is 209.7; that is, the error of approximate calculation under the rule, for the paraboloidal surface adopted, (209.7,) is here considerably less than the absolute error (721.6) caused by the adoption of that surface. This surface has, we believe, been recommended by high professional authority. Whether or not it is the proper one, its adoption appears to us to be in contradiction to the hypothesis made in the method by centre and side heights,-viz., that the median surface-line between observed points should be considered straight. This difficulty is avoided by adopting the surface proposed in & 2, Art. 8, Part II., whilst straightness of outline and longitudinal straightness are preserved. The solidity is less than by either of the above methods. Formula 45 gives for the whole ground,-say, Example 1, 6150 yds., Example 2, 5952 yds. The straight cross-surface of the paraboloid may be entitled to some preference, because of the tendency to straightness of hill-side surfaces from natural and artificial causes. Q7onclusions.-The use of the method by transverse slopes is generally for preliminary estimates. We are not prepared to decide what amount of error is permissible in computation by this method, nor what standard of comparison should be adopted in estimating it. It is well to run the risk of making preliminary estimates too high, rather than too low. On the whole, we are inclined to think that the rule of Article 104, Part I., may be adopted in practice. Taking the mid-section slope in whole degrees, and the allowance of a tenth or two in the sum of augmented centreheights, present means either of empirically approximating the paraboloid more nearly, or of finding an intermediate solidity between that and the solid of revolution. 9. APPLICATION OF TABLES TO VARIOUs FoRMULA.-Several formulae in use for earthwork computations embrace terms containing a multiple or submultiple of the square of a given linear dimension, or of the product of two such dimensions. It may be useful to indicate how our tables can sometimes be employed in such cases. Let a and b represent heights, widths, or other linear dimensions in feet. Let I be the length of work =-100; and let q = 27 Q be the number of cubic feet in the term in question. Q is the tabular quantity of cubic yards. For Table XV.-For the First Part, q -= l a2; for the Second Part, q -= 1 a2. Make a = 2 a'; then, for the First Part, q -= 14 a'2 -- Y a/2; for the Second Part, q = — 2 1 4 i a2 = i 1 a/2 Table XXV. contains logarithms proper for the same computations..For Table XXI.-Here, q - lab. If a = 2 a' and b-2 b', q = 4 a' b/- la' b'. Table XXIII. contains logarithms proper for the same computations. For Table XXII.-Here, q = I a b; whence any multiple or submultiple of this term may be found. Table XXIV. contains logarithms proper for the same computations. Catalogue of.Formulce and Construction of the Tables. 10. CATALOGUE OF FORMULE. This contains formulae for the computation of the several tables. By attending to the references of the catalogue, and by consulting Chapter V., Part I., the rules for the application of the tables will be understood. The following notation has been employed: Q = tabular quantity, B - width of roadbed, L = length of the section, Sh = sum of heights, Sb = sum of bases, Dh = difference of heights, a = angle of side-slope, y = angle of surfaceslope. The tabular length of the section is 100 feet; the tabular solidities are in cubic yards. Table I. 27 Q L B 2 tan a. See formula 14, page 280, where 4 B2 tan a represents the area of the cross-section of' the redundant prism. Table II. Q= B tan a = 2 A G. See Fig. 41, page 278. Table III. Q = Q B tan y = C D. See Fig. 40, page 276. Table IV. Q = L. See formula 1, page 275, wherein 7 =- either the marginal or the h-h' centre height at the origin, and h' = the corresponding. height at the other end of the section. Table V. Q = h cot y C P. See Fig. 40, page 276, wherein 7h = centre-height, C E. Table VI. Q = h cot a D= H, A M, E Q, or A N. See Fig. 41, page 278, wherein h = the side-height, augmented or unaugmented; that is, B H, B 1M, C Q, or C N. APPENDIX. 297 Table VII. 27Q=- L Sb2 X Sin( 2sia-. See formula 13, page 280. sin (a + =) sin (ay -.y) Table VIII. 27 Q= L Sb2 X e sii( s- y.. See formula 21, page 282. sin (' -- y) TableIX. 27 Q= L Sh2 X sin.2 Cs2 Y See formula 13, page 280. sill (a- + y) sin ( — -y) Table X. 27 Q L S.2 X sn....... See formula 21, page 282. sin (a - y) Table XI. 27 Q = L S2 Xi sin 2 a-c S. See formula 13, page 280. Table XIV. 27 Q - L Sh2 sin (a- +- y) sin (a- - y) Table XII. 27 Q L Sh2 X F -si ac2y.... See formula 23, page 282. sin -y sin (a - y) Table XIII. 27 Q = _ L S2 X i in (..... See formula 23, page 282. sin y sin (a- -. Table XIV. 27 Q L (Sh 2-B2 tan2 a + B2 tan2 + D2 ) x sin 2 a- cos2y - See formulae 16 and 18, page 281, and 13, page 280. Table XV. 27 Q L Sh2. For the Second Part, 27Q L See formula 13, page 280. Tab l e X sinVIC _ Accos. F =sin 2 s- Co=2. Table XVI. Q = log. n 2 COS2... See formula 13, page 280.* sin (a- + y) sin (a - y) Table XVII. Q log. _ i COsY. See formula 23, page 282.* sin o sin (a- -?) Table XVIII. Q =h cos (- or += o0); wherein h = AF. (Fig. 41, page 278.) In the triangles A F B, A F C, we have, by Trigonometry, FC=AF X~i +o l cos a -; F B = AFX sin FAB h c -1 fn C A / sin ( — y) sn sinFBA sin (a' - y) Table XIX. Q log. 4 CO ( COS', orQ= log. a COS - COS Employing the same notation, and sin ( -y) sill ( + y) referring to the same figure as before, we have h cos a cos y; h cos a-coo y: CK FCIX cos7 hco.- c OB. B =FB cos y — _ whence sin (a - -y) sin (a- +y) log. CI K = log. 2 h + log. I Cos C0os, and log. 0 B - log. 2 h ~ log. cos a cossin (a'- y) sin (a- +y) Table XX. Q = log. CO or Q = log. By Table XVIII., we have s ino(a-g.' sin o (a- + y) log. FC=log. 2h + log. - sin'; log. FB = log. 2h - log. s +. sin (O- - -Y) Y sin (ar - ~y) Table XXI. 27 Q - I L Sh S...... See formula 27, page 283. Table XXII. 27 Q = L Sh Sb........... See Art. 126, Part I., and Art. 9. Table XXIII. Q = log. (Sn or S) +- log. 10 - 4 log. 216. See formula 27, page 283.t Table XXIV. Q= log. (Sh or Sb) + log. 10 - 4 log. 27. See Art. 126, Part I., and Art. 9.tTable XXV. Q = log. 1 L Sh..... See formula 13, page 280. 216 Tables XXVI., A and B. Tables XXV1. and A are well known. For Table B, see NAte, Art. 25. 11. CONSTRUCTION OF THE TABLES. (1.) Tables proceedinag in the direct ratio of the argument.The method of continued addition was employed in the construction of Tables II., III., IV., V., VI., XVIII., XXI., XXII. These proceed regularly, either downwards or across, and, in some tables, in both ways, with the tabular number in direct proportion to the argument. Tables thus constructed admit of proof at the end of every interval of ten lines or of ten columns, beginning with the first line or column. At the end of every such interval, divide both the argument and *See Note 1, page 314. t See, also, page 300,. 4, Art. 11. 298 APPEN-DIX. tabular number by ten. The resulting argument and number should, respectively, equal an argument and its number found above. (2.) Tables proceeding as the square of the argumnent.-Tables I., VII., VIII., IX., X., XI., XII., XIII., XV. proceed with the tabular number proportional to the square of the argument in the side column. They admit of proof at intervals of ten lines, nearly as the tables previously mentioned. When the argument is divided by ten, the tabular number must be divided by one hundred. This mode of proof applies only partially in Table XV. Table XIV. is properly classed among the tables here enumerated. (See the formula, Art. 10. Also, Art. 89, Part I.) Such tables can be accurately, and probably most readily, computed by a method of continued addition in two columns simultaneously formed. In the equation of Table IX., for example, put F - 100 sin 2r cos --; then Q = Sh2 F. Let Q, Q2, Q2, Q.Qn-1, be successive 216 sin (ao + y) sin ( - y)h tabular numbers of the same column, corresponding to the successive values of the argument Sh, Sh + 1, Sh + 2, Sh + 3, down to Sh + n — 1 and Sh + n. We have, by the formula, Sh2 F........... Q (Sh + 1) 2 F Sh2 F + (2 Sh + 1) F =. Q1 Q + (2 Sh — 1 ) F. { (Sh + 1)+ 1} F= (Sh + 1) F+ {2 (Sh + 1)+ 1}F. Q2 Q1 + (2Sh + 3) F. (Sh + 1) + 1 + F= (Sh + 1) -+ 1 2 F + 2 (Sh +l) + l + 1 F Q3 =Q2 - (2 Sh + 5) F. The law of this series being such that 81, + (n- 1) 2F = Sh2+ 2 Sh (n-:1) + (n-1) 2 F = (Sh + 2 Sh n + b2 - 2 Sh - 2 n + 1) F Q-1, and Sh + (n- 1) + 1 2 F= (S +n) F — (Sh2+ 2Shn n) F=Q, Q,_ + (2 Sh + 2n-1) F. Sums. 90 Sh 0.00.95 (2 Sh + 1) F I F.95 1.90 2F Sh + I= 1.95 2.85 (2 Sh + 3) F 3 F 2.85 1.90 2 F Sh-+ 2= 2 3.80 4. 75 (2 Sh-+ 5) F= 5 F 4.75 1.90 2 F Sh-+ 3 3 8.55 6.65 (2Sh + 7)F= 7F 6.65 1.90 2 F 8h+~ 4- 4 15.20 8.55 (2 Sh-+ 9) F 9F 8.55 1.90 2F Sh + 5- 5 23.75 10.45 (2 Sh-+ 11) F - 11 F 10.45 1.90 2 F Sh +- 6 6 83 4.20 5 _ (2 S +- 13) F _ 13 F 12.35 1.90 2 F Sh + 7- 7 46.55 14.25 (2 Sh + 15) F = 15 F 14.25 1.90 2 F Sh-+ 8= 1 60.80 16.15 -(2Sh +17) F 17F 16.15 1.90 2 F Sh + 9= 9 76.95 18.05 (2 Sh + 19) F = 19 F 18.o5 1.90 2 F Sh-~10- 10 95.00 19.95 (25 h+21) F=21F APPENDIX. 299 The increment to be added to any tabular quantity Q.-1, in order to produce the next Q,, is, therefore, Qn-Q,_- -(2 Sh + 2 n-l) F. The increments are formed on the right-hand side of the annexed scheme by the continued addition of 2 F to the first increment. Take the 9~ column of Table IX. Let Sh=0, and suppose F=.95; then Q S2h F=-0. Take n successively = 1, 2, 3, &c.: the first increment is (2 Sh -+ 2 —1) F — (2 Sh+ 1) F =F. The differences for one-tenth of a foot are formed by dividing the increments by 10. In actual computation we have taken F to five or six decimals. The following examples show the methods of proof: —The argument 10 gives the tabular quantity 95.00; one-tenth of the argument is 1, opposite to which is.95, the one-hundredth of 95.00. For the increments, we have 21 F = 20 F + F = 2 F X 10 + F = 19.00 +.95 = 19.95. The increments should be proved when an: error occurs in the tabular quantities.* (3.) Correct Interpolation for Heights in Tables proceeding as the Square of the Argument. As before, let Sh, Sh + 1, be two consecutive side numbers:: Q. Qtheir respective tabular quantities. Further, let Sh+ - be a side number, for whose number t of odd tenths interpolation is required; and let the tabular quantity Qt correspond to the argument Sh + 1. Also, let x be the true quantity to be interpolated for the odd tenths, so, so that Qt = Q + x. Make p = the quantity which would be interpolated for t by proportional parts; and assume e =p -x. It is required to find the correction e, which must be applied after proportional interpolation. We have Qt Q -=(Sha -) Sa} F-(- lo +)Ft; and 10 X (Sh)+ 1) 10 10} P - t- {t 2sh-1) S —s I Sh+ F t Therefore, substituting these values of x and p in e p - x, we find e X -X== Sh +l F t_ (2 Sh + t F t ( F 1 t O — t F. 10 10 0l 100 The maximum value of e is when t = 5, in which case t t-, -.25. Hence the error by proportional interpolation cannot exceed one-fourth the first tabular number in the given column. e is positive; that is, p is greater than x, because t is always less than 10. Hence the following RULE. Interpolate by proportional parts, and reserve the tabular quantity thus found for subsequent correction. Multiply the tabular quantity for the side number one foot, by ten less the number of odd tenths, and again by the number of odd tenths itself. Divide the product by one hundred, and subtract this result from the tabular number reserved for correction. Example.-What is the tabular quantity for 45.6, under 150, in Table VII.? *Another Demonstration. —(See Art. 130, Part I.)-Let x be the argument, lk its increment, and x =. Q n-l,: the tabular quantity for x; also, let ab (x + k) = Q. be the tabular quantity for the argument x -+k. By Taylor's theorem, (x:+k) -- p x-k d z -+ 2 d2 + &c. We havehere x= x2 F; whence d ~ x - 2 x = 2 F. Hence, Qn- Q-1 = 2 k x F + k2 F, which, if x = Sh qn - 1, and k — =, becomes — = (2 S8,- 2 n -1) F. And this is the increment of the above scheme: also, if x be any argument, and k-= 1, the increment = (2 x + 1) F; which is evident from the scheme. If I x=x F, and k= 1, then d =- F., d — 0, and the increment is Q- Qn-i-F; which gives the rule of 21 for computation by continued addition. The Table of Pythagoras, or common Multiplication Table, may be thus formed. The Tabuls Arithmeticrc of Herwart (1610) are an extension of the Pythagorean Table for factors from 1 to 1000. By proceeding in a manner somewhat similar to that explained for our Tables XXI. and XXII. (page 81), Herwart converts the multiplication of larger factors than 1000, into addition. He also performs division by inspection of the tables, or by the aid of subtraction. Montucla has reviewed this work, (Histoire des Math., tom. ii. p. 13,) and gives as reasons for its falling into disuse, the invention of logarithms, and the unwieldiness of the work in plan and dimensions, being a large folio of 1000 pages. It may be seen in the Astor Library. 300 APPENDIX. Opposite 45, under 150.. 3354.4 Opposite 45, under 150, take six times the difference.90.4 Quantity by proportional interpolation.. 3444.8 Ten less six, the number of odd tenths, is 4. The tabular number for one foot, under 150, is.1.7 Multiply by 4. 4 6.8 And again by 6, the number of odd tenths, and divide by 100... 6 Correction..408 Quantity by proportional interpolation, as above. 3444.8 Corrected tabular quantity.. 3444.4 (4.) Logarithmic Tables.-The inspection of the proper formula will generally explain sufficiently the construction of these tables. Table XXIII. contains the logarithm of one of the factors of Table XXI. If, in this latter, 100 10 Sh 10 Sb L- 100, Q = 2 SSb = - 2 X V/216. The logarithm of one of these factors may be thus expressed:10 Sh log. -216 log. Sh + log. 10 - ~ log. 216 log. Sl - 1.83277. Table XXIV. contains the logarithm of one of the factors of Table XXII. If in this latter 1 10 Sh 10 Sb L _ 100, Q= - Sh Sb = /27 X v27 The logarithm of one of these factors may be thus expressed:10 Sh log. - log. Sh ~ log. 10- ~ log. 27 = log. Sh + 0.28432. By adding log. 8 = 0.45155 to the logarithm of Table XXIII., the logarithm of Table XXIV. is produced. In this way the correctness of both tables may be tested. Discussion of Graphical Processes. CONSTRUCTION OF SCALES. 12. CONSTRUCTION OF GENERAL AND SPECIAL SCALES. (1.) Sub-Section Diagram, Plate VII., and Mid-Section I)agram, Plate VIII.-The explanation of these figures is sufficiently obvious from Articles 43, 45, 50, 52, and others, Part I. (2.) General Stereometric Scale for Whole and Sub Sections. First Form.-This scale may be exhibited in two forms. We shall first take the most simple, which is not given on the plate. The object of the scale is to find an augmented sum of heights, or a sum of bases, which, in the table, under the given surface-slope, shall contain the same quantity as would be derived by adding together the quantities corresponding to the sum and to the difference, according to the General Rule, Art. 85, Part I. By formula 13, page 280, and 21 and 23, page 282, the solidity of the whole ground in whole sections, and of the work in sub-sections, may be put under the general form V = (S + D') F:.... 1, wherein S is the augmented sum of heights or sum of bases, and D the difference of heights or bases. F is a factor depending upon the side-slope and surface-slope. We have only to find a new sum of heights or bases, S1; such that S12 = S2 + DI or S= -S2I+ D: that is, to extract the square root of S2 + -2 by construction. The extraction of the square root is performed, in APPENDIX. 301 Geometry, by means of a right-angled triangle. Let SAD (Fig. 47) be a Fig. 47. right-angle. Take any desirable length A S, and divide it into equal parts, D C A each of which represents one foot. Take A D to A S in the same proportion as -- to 1, and divide A D into a like number of equal parts with A S. Take A B = S and A C = D, according to the respective scales. Then, taking the unit of A S as a standard, AC -_ 3. By the property of right-angled tri- B angles, B C2 = A B2 + A C2, and BC-=- AB 2 + AC2 =\ S2 + = SpS.... 1, the new sum of heights or bases which was required. s Second Form, Plate VIII.-Equation 1 may be modified so as to allow the use of two rightangled triangles. By this means, the operation upon the scale is made similar to the operation upon the special scales, Plate IX.; and the divisions upon the scales for differences may be enlarged. We have JS2+ D2 JS2 + (,+2 + 1)D2 - 12 J=S2 +Qz2 +1)2D2 )2D2; wherein n is a number taken at pleasure. Take, therefore, (Fig. 48,) A D: A S:: +: 1; and divide each scale into the same number of equal parts. Take AB = S and AC = D, according to the respective Fig. 48. 2 1) 2 D C A MF scales: then B C2 S2 + (n2 + 1) D Make now AF: A S::: 1; and divide A F into a like number of equal parts with A S. Take A M = D, according to the scale A F. Then AM =- -. Make M K = B C. We have A K2 = MK2 - A M2; that is, K A K = B C2 - A M2 = S2 + (122 + 1) 2 _ -2 D2; and 3 3 AK= S2 + (n2 + 1) D2 _ = D = S2+ D2 = S1, the new sum of heights or bases which was required. On the plate, n = /3; so that the divisions on the shorter side of the scale for differences are equal to those of the scale of sums. (3.) Special Scales, Plate IX.-These scales are constructed upon the formula v= 1 L (S2 - B2 tan2 C + B2 tan2 Y + D2) t2tan a -. (See formula 16, page 281.) tan2 a - tabn2 y We have to find a new sum of heights which shall contain the same solidity under the same slope; that is, we have to extract the square root of the quantity within the parenthesis. For this purpose, we may transform that quantity into S2 + D2 - (B2 tan2a - B2 tan2 y). We shall first construct /B2 tan2 a - B2 tan2 y. Upon the diameter AB= B tan a (Fig. 49) describe a Fig. 49. semicircle. Make A C = B, and draw C D, making DCA = y: then AD = B tan y. From A as a centre D describe the arc D K: join A K and B K. A K B, being in K - a semicircle, is, by Geometry, a right-angled triangle, and BK2- AB2 —AK2A B2-AD2 =B2tan2a-Bg2tan2y; / hence, B K B2 tan2 a - B2 tan2 y. A B C 302 APPENDIX. Draw and divide A S and AD (Fig. 50) as in Fig. 47, and take, as before, A B = S, A C D: Fig. 50. then B C= S2 + D2 Make T H F a right angle, and let D C AH M F H T be a scale like A S. On H F take H M = B K, (of Fig. 49, which is supposed of the same scale as A S, Fig. 50,) and make M K = B C (of the present figure.) Then, H2 = MK2 IM2 = S2 + 2 (B2 tan2 -B2 tan2 y); B3 K and HK S2- B2 tan2 a + B2 tan2 y + S the new sum of heights which was required. On the scales, Plate IX., this construction is only modified s T by leaving, between the zero-point of the scale A S and the side A D, a blank space equal to the tabular augment, from Table II., for the sum of the heights. The distance A B is not altered by this arrangement. 13. GIVEN DIMENSIONS.-APPLICATION OF THE DIAGRAMS. In the following discussions, the given sum of heights for whole-section work may be either the augmented sum, which. gives an approximate content for the whole ground, (see Art. 85, Part I., and Formula 13, page 280;) or the sum found by General Scale, (see Art. 96, Part I.,) which gives the true content of the whole ground; or the sum found by Special Scale, (see Art. 97, Part I., and Formula 16, page 281,) which gives the content of the residual prismoid. For sub-sections, the given sum may be either the sum of the bases, or the augmented sum of the end-heights. These may be employed to find the approximate content. For the true content, the sum must be found by General Scale. (See Art. 96, Part I.) The rules and diagrams apply to straight work and plane ground, or to all work after reduction to this form. The length of work in the examples is understood to be 100 feet. BASES OF EQUIVALENT SQUARE PRISMS. 14. —To FIND THE BASE OF AN EQUIVALENT SQUARE PRISM BY MEAN PROPORTIONALS. W~Te shall not, at present, actually exhibit this base, but will find the value of the base multiplied by /8. This quantity is proper for entering Table XV. (1.) Whole Sections. —On the plane of the cross-section, draw the horizontal line R O L, (Plate X.,) the vertical centre-line OH, and the side-slopes O A, OB. Also draw 0 P, of indefinite length, perpendicular to OA; and, parallel to O P, draw MN through any point T on the sideslope OB. Divide OA, O H, OB, OR according to the same scale of equal parts. Make N K = N T, and N I = O T. On I I describe a semicircle cutting NT in M. Then N T = O T X sin A O T. Also, by Geometry, N M is a mean proportional between N K and N I, that is, between N T and OT; andNM -= OTX NT - OT2X sinAOT OTX l/sin AOT. Make OQ=OT, and 0 G N M. Join G Q, and draw T P parallel, to G Q. Then, by similar triangles, OP: OQ:: OT: OG; whence P QXOT T OT2 OT Hence, ifOP 0o NM OT X 1/sinAOT 1/ sin A T be divided into the same number of equal parts as OT, the length of the unit on 0 P will be 2/ i AOT~; the unit on 0 T being the standard. And the reading of any length 1, transferred from the scale OT to the scale O P, will be l 1/sin AO T. Let OD = 2 H be the sum of the augmented end-heights. Then the triangle A OB, having a given surface-slope A B, is four times as great in area as the similar triangle, having the height H, which latter triangle is that of the mid cross-section. Therefore, putting the area of A 0 B = A',and that of the mid cross-section = A, A'- 4 sin AOT X OA X OB = 4A; whence A= 4 -sin AOT X OA X OB. Put / sin A p T X 0 A X O B = S; then A = 1 S2. From what precedes, we know that if l/O AX OB, measured on 0 A or 0 B, be transferred to O P, the reading will be 1/0 A X 0 B v/sin A O T; that APPENDIX. 303 is, the reading will give the value of S. It remains to find 1/O A X 0 B. This is a mean proportional between 0 A and 0 B, and may be found by taking O U = O B; then describing a semicircle on A U, cutting O P in S. 0 S is the mean proportional required, and is transferred to the scale 0 P. Hence, upon the scale 0 P, 0 S - S, and A = -G O S2. It will not be necessary actually to lay off 0 U, or to describe the whole semicircle. We may add the reading of 0 B to the reading of 0 A, and set off the half-sum of the readings from A to Y. Then from the centre Y, with radius YA find the point S. (2.) Sub-Sections.-A similar construction applies. It may either be made separately, or may be exhibited on the same diagram in the following manner:Having the sum of heights or of bases and the surface-slope given, draw FV, observing to take the sum of heights, or of bases, for the marginal height or the base of the triangle; and let fall the perpendicular F C. The area of F O V is equal to four times the area of the mid cross-section; hence, calling the area of F OV A, and the area of the mid-section A, 4 A = A' = G OV X F C, and A - OV X F C. A mean proportional is now to be found between OV and F C. Graduate OL so that the reading may give 0C = OF X sin AOL = FPC as read upon the scale OR or OH. Add the reading of 0 C to the reading of OV. The half-sum of these readings is the radius, which set off from V to X. With the centre X and radius XV find the point Z -on the scale OH. OZ = /OV X FC; whence A =- O Z2. Since Table XV. contains the eighth part of a square prism, the side of whose base is the argument, it is obvious the solidity will be found by entering this table with 0 S or 0 Z. If B be the base of an equivalent square prism, then A - (O S or O Z)2 = B2, and O S or O Z = / 8 B. Because the distance 0 S (not the reading) = / OA X 0B; if 0 S be laid off from 0 on OA and OB, an isosceles triangle may be formed, the product of whose two equal sides is equal to OA X OB; and, consequently, the area of the triangle equal to the area of AOB. Hence, the height of the isosceles triangle will be the equivalent sum of heights under level ground. 15. To FIND THE CONTENT BY DIAGRAM. (Plates X. and XI.) RULE. (1.) For Whole Sections.-On the scale 0 H, take 0D, equal to the given sum of centre-heights. By aid of the graduated arc, whose centre is 0, draw ADB with the given surface-slope. Read OA and AB on their scales, and add the readings together. Lay off the half-sum of the readings from A to Y. From Y as a centre, with radius YA describe an arc cutting 0 P in S. With the number shown by OS on its scale, enter Table XV. and take out the content, which will correspond to the given sum of heights. NOTE.-In Plate XI., the lines O P and O T fall together. (2.) For Sub-Sections.-Take O V equal to the sum of bases, or take the augmented sum of the centre-heights on the scale O H; and draw the surface-slope F V. Draw F C parallel to O H. Add together the readings of OV and OC. Measure the half-sum of the readings by the scale OR, and lay it off from V to X. From the centre X, with radius XV describe an arc cutting OGH in Z. WVith the number shown by O Z on its scale, enter Table XV. and take out the required content. Example 1. —The augmented sum of heights for a piece of whole-section work is 30, the surfaceslope 150, the side-slope 1' to 1. WThat is the content of the whole ground? On Plate X., make OD equal to 30. Draw ADB with the surface-slope of 15~. OA reads 90.4; OB reads 39.0. The half-sum of these readings is 64.7. Make AY equal to 64.7, and with the centre Y and radius 64.7 find O S equal to 56.8. Then, in Table XV., opposite 56 and under 8, find 1493.6, which is the whole content nearly.'Example 2. The sum of the bases of a piece of sub-section work is 50, the surface-slope 10%, the side-slope 1~ to 1. What is the content? The example is drawn and lettered on Plate X. OV is 50; the reading of 00 is 12.0. The radius V X is therefore: 31. From the centre X, with this radius find 0 Z, equal to 24.5.,Then, in Table XV., with 24.5 take out 277.9, which is the content nearly. Two other examples are drawn on Plate XI.' In regard to this diagram, it is only necessary to remark that 0 S is to be taken upon the side-slope. 304 APPENDIX. Example 3. —The augmented sum of heights for a piece of whole-section work is 40, the surfaceslope 253, the side-slope 1 to 1. What is the whole content between the surface and the intersection of the side-slopes? O S is found to be 63.9, which, in Table XV., gives 1890.4. Example 4.-The sum of the bases of a piece of sub-section work is 30, the surface-slope 20~, the side-slope 1 to 1. What is the content? OZ is found to be 22.7, which, in Table XV., gives 238.6. 16. BASES OF EQUIVALENT SQUARE PRISMS AND EQUIVALENT HEIGHTS FOUND BY GEOMETRIC LOCI. (1.) Investigation of Formulce fQr Whole Sections.-Let the solidity be V -= L S2 X tan ta tan2 -- - tan2 y (See formulae 16 and 19, page 281.) As we propose to find the square cross-section of a prism which is equal to the solid represented by this equation, and whose length is L, we need only seek the side tan - wherein A/ is the area of this square base, and may treat the equation A' S2 X tan2 - wherein Ais the area of the square. In order to transform this expression, we have tan aor tan - tan or tan (r tan a tan a tan2 a -- tan2 y sec2 a-(1 - tan2 y se2c2 a - (1c tan y) sec a a sec 2 a cosa y —1 cos2 y cos2 y tan a X os2 7 cot a X tan2 C X> cos2 cot a X sec2 a -1 X cos2 Y see2 a cost y-1 sec aC cyos- -- 1 seca a Cos2 Y -1 Whence A' = - S2 cot a X eet2 - 1 X cos2 y sees a cos2 - 1 N sec acos y —1 If A be the semi-transverse axis of an hyperbola whose eccentricity is e and polar angle y, the polar equation of the curve referred to the centre as a pole is ea - 1 p A e2 cos2 7 — 1 Therefore, making A -= 1/ S2 cot a and e = sec a, equation 1 becomes A' - (p cos y)2. When y = a, p = o; whence the asymptotes make an angle = a with the polar axis. When a = 45~, or when the side-slope is 1 to 1, the hyperbola is equilateral. It now remains to find p cos y, which is the required side of the square. (2.) To construct a Diagram for finding the bases of Equivalent Square Prisms for Whole Sections.-Let AO B (Plate XII.) be a horizontal line upon the mid cross-section, and passing through the intersection 0 of the side-slopes. OY is the vertical centre-line; O S and OT the sideslopes. Upon OY take any convenient length OH, and draw H D parallel to O B. HD = OH X cot SOB = O H X cot a. Upon 0 H produced make 0 C = I D, and describe the semicircle HBC. By Geometry, OB = /OH X OC = /OH X I-ID = 1/OH2 X cot a. Draw BK parallel to OH, and from the centre 0 describe through K an arc K F' cutting OB in F'. Make OA- OB and OF = OF'. - Put OK = OF'= c, OB = A; and make C OK A e: then e B Osee SOB = sec a: BK = / 2 A2. By the property of the hyperbola, if e = sec a be the eccentricity, A = OB = 0 A is the semi-transverse axis, B = B K, the semi-conjugate; F and F' are the foci, and 0 S and 0 T the asymptotes. Construct the hyperbola; draw any radius vector 0 y= p, making the polar angle P 0 y = y; and draw the ordinate P y: then 0 P = p cos y. If OHI= S, O P is the line required whose square = A'. For we have found above, OB = A = VO H 2 X cot a; whence, if O II = ~ S, A = 1 S2 cot a; also, we have made, by construction, e = see a: but A = V/ i S2 cot a and e = see a are the conditions required by the previous investigation, in order that A' = (p cos y)2 = O P2. (See also ] 5, Art. 25.) The areas of similar plane figures are proportional to the squares of their homologous lilies. Hence we infer that for given values of a and y, the area of the whole cross-section will be, for APPENDIX. 305 different values of S, proportional to S2, land hence, also, to 4 S2. The side of the equivalent square base will then be proportional to 4 S: this is-also evident from equation 1, ~ 1. It is, therefore, not necessary to make OH -- the half-sum of heights in order to find the side of the base. For, let O H= S - the augmented. centre-height of a cross-section similar to the one proposed, of which latter let the augmented centre-height be ~ S'. It is now required to find O N, the side of the square base equal in area to the proposed cross-section. For any sum S and surface-slope y, find O P, as before;we shall then evidently have OP: ON:: S S S: S::S: S'. The same:conclusion may be reached by considering that, whilst SOB is constant, the hyperbolas corresponding to different values of 0 H will all be similar; and, therefore, with P Oy = y constant, the abscissas O P will be to each other as the respective values of O H. But these values of O H are to each other as the respective sums of heights. The above proportion will then be obtained, by regarding O N as the abscissa corresponding to the surface-slope y, for an hyperbola in which O H has been made equal to the half-sum of augmented heights. To satisfy this proportion, take, upon the scale 0 T, OH' equal or in a given ratio to O H- 5; also take OM in the'same ratio to 4 S'. Draw II' P, and parallel to it draw M N. The similar triangles O MN, OH' P give the required proportion. To avoid the necessity of finding 4 S' when S' is given, we may make OR 2 OH = S: then make O V - S'; join R P, and draw VN parallel to R P. The First Part of Table XV. contains the eighth parts of prisms 100 feet long, upon a square base. If -the scale OP be so graduated that its divisions are to those of OT as 1 to V/8, we may enter Table XV. with O N as read upon its own scale, and may take out the required content for 100 feet. The necessity of this graduation of O P might be avoided by preparing a Table similar to XV., but containing the whole solidities of square prisms. (3.)' Equivalent Heights for New Surface-Slopes.-If it be required to find a new sum of heights which, with a new surface-slope, shall contain the original solidity, let POL be the new surfaceslope. Find OP, and thence ON, as before, for the original sum of heights and slope; then find OU for the new slope POL, as OP was found for the original slope. Draw UR, and NW parallel to it, cutting off from the scale 0 T the distance OW, equal to the required new sum of heights. It is evident, from the construction shown in section 2, that POL may be regarded as the surface-slope of a cross-section for which ON is the side of the equivalent base, and in which OW -- the sum of heights. By joining U Hi' and drawing through N a line parallel to U H', we should find the half-sum of the new heights. (4.) Equivalent Level-Heights.-If the new surface-slope be level, or = 00, L and U fall upon A. Therefore, after finding O P, and thence ON, for the original cross-section, join AR, and through N draw a line parallel to AR, cutting off from 0 T the required new sum of heights. To find the half-sum, or augmented height of the new cross-section, join A H, and draw a parallel to it * through N. (5.) Geometrical Basis of these Constructions.-Since cot a -= -B A, Anwe have, ( 2,) K'vB A OH 2"-X cota= 0H 2 X A; whence 0 H 2 = A X- B; that is, O H is a mean proportional between the semi-axes. Hence these constructions depend on the following property of the hyperbola. Take from the centre 0, upon the line of the conjugate axis, 0 H a mean proportional between the semi-axes. The area of any triangle included between the asymptotes and a third side drawn through H, is equal to the square of the abscissa O P subtended by the radius O y drawn parallel to this third side. 17, EXAMPLES IN FINDING THE CONTENT OF WHOLE SECTIONS BY THE DIAGRAM. (See Plate XII.) The diagram is adapted to the side-slope 11 to 1. Example 1.'Equivalent Square Bases.-The augmented sum of heights is 45; surface-slope, 250; side-slope, 14 to 1. Required the side of the equivalent square base, and the content of the section. U 306 APPENDIX. Draw 0 y, making the angle POy equal to 250, the given surface-slope.* Mark the point of intersection y with the curved line A L y. Draw Py parallel to OI, and join PR. Make OV equal to 45, the given sum of heights, and through V draw V N parallel to P R. Read the number (say 109.1) indicated by N upon the scale ON. Then, Opposite 109, under 1, in Table XV., First Part, find the solidity.. 5510.6 Example 2. Equivalent Heights for New ~Su'face-Slopes. —If, in Example 1, the surface-slope were changed to 150, what must be the augmented sum of heights to contain the same solidity? Find the point N, as before, for the given sum 45 and surface-slope 250. Make POL equal to 150, the new surface-slope, and draw LU parallel to OH. Draw UR, and parallel to it draw NW. Read the number (say 55.7) equal to the required sum, indicated byW upon the scale OT. Then, Opposite 57.7, under 150, in Table VII., find the solidity.... 5515.3 Examjple 3. Equivalent Level-leights.-If, in Example 1, the surface-slope were made level, what must be the augmented sum of heights to contain the same solidity? Proceed, as in Examples 1 and 2, to find P and N for the given sum 45 and surface-slope 25~; the point U will now fall upon A. Join A R, and through N draw a parallel to AR, cutting the scale O T in a point indicating, say 63.0, the new sum required. Then, Opposite 63.0, under 00, in Table VII., find the solidity. 5512.5 18. BASES OF EQUIVALENT SQUARE PRISMS FOUND BY GEOMETRIC LOCI. (1.) Investigation of Formulce for Sub-Sections.-The solidity for sub-sections (see formula 24, page 282) is V- I LSb2 X 1 cot y - cot a wherein Sb is put for the sum of the bases. Leaving out of consideration the length, and proceeding as before to find the side of a square equal in area to A, the area of the triangle, we have A=- Sb2X.1 Sb tan 7 X b ot S -.cot 1- cot a cot y Because y is always less than a, we may put -cot cot a tan y cos, which gives, by substitution, A = i Sb tan y X l-b.... 1. In which sb is the radius vector of a parabola whose semi-parameter is SW. 1-cos If this equation be first multiplied and then divided by tan2 y, the term Sb2 tan2 y applies to computation by the augmented sum of centre-heights. (See equations 21, 22, 23, page 282.) Therefore, designating the sum of the augmented centre-heights by Sh, we have A = Sh cot y h 1-....2. If p' be the radius vector of the parabola in this last equation, and p in the previous, and if the side of the square which is equal to A be put, respectively, b — S, Sh we shall have SbI/ X V/P Sb tan y; and Sh= — / X /p/p Sh cot y. As we propose to enter Table XV., we shall proceed to construct, not Sb/ and Sh/, but Sb/ 1/8 and Sh' y/8, equal, respectively, to V/p Sb tan y and /p/' S cot y. (2.) To construct a Diagram for finding the Bases of Equivalent Square Prisms for SubSections.-About the focus 0 and axis O X, (Fig. 1, Plate XIII.,) construct a parabola P 0, having the semi-parameter 0 P of any convenient length. Upon the line of the parameter, make OM - unity, and describe a circular quadrant MQ. Make the angle YO0y = y, and take OY = cot.a. Draw Y y parallel to O X, and y N parallel to 0 Y, cutting the circular quadrant in N; also, * If desirable, a graduated arc may be added to the diagram, in order to facilitate the drawing of slopes; or, the requisite number of slopes may be permanently indicated by radii. APPENDIX. 307 through N draw O f N, cutting the parabola in ~: then XO 0- =, because cos XO = Yy - cot a tan y. The radius vector O p oP * whence o 1= 1 —cos O~P 1-coso Upon OX construct a scale of equal parts, and, according to this scale, take 0 B = Sb. Draw P and B K parallel to it, cutting 0 q in K: then, by similar triangles, OK 0 0 or OK=2- O; whence OK = Sb X 0.- sb OB OP Sb OP 0]2 1 —cos~ Draw B T parallel to 0 X, and T H parallel to 0 Y: then B T = OH = O B tan y = Sb tan y. Make OL =OH and OX -OK sb On LX describe a semicircle cutting OY in F: 1- cos q then OF = OL X OX = 1/Sb tan y X S, which is the quantity sought = V8 Sb' = VA. (See equation 1, { 1.) If Sh be given instead of Sb, let OH = Sh, instead of taking OB - Sb. Draw iHT parallel to OB, and TB parallel to OX; then H T = — OB — Sh cot y. Make OS = OH = Sh, and draw SR parallel to P l, which is found as before: then, by similar triangles, OR= SX O - S. X )<0 - Sh OP OP 1 —cos Make OU = OR, and describe the semicircle B V U, cutting O X in V: then OV =V VOB X OU = V/Sa cot y X 1 s 1 - Cos 0 which is the quantity sought, = r/8 S h'- V/A. (See equation 2, ] 1.) By making permanent the parabola and various radii P 0 for the different surface-slopes required, there would remain the other parts of the construction to be performed in particular cases. 19. EXAMPLES IN FINDING THE CONTENT OF SuB-SECTIONS BY THE DIAGRAM. (See Fig. 1, Plate XIII.) If radii be drawn similarly to OK for the requisite number of surface-slopes, the diagram will apply to a particular rate of side-slope. But if no such radii have been drawn, and the diagram contains only the curved line, the curved line will apply to all side-slopes. -For the method of drawing such radii, see ] 2, Art. 18. In the following examples the radius OK is supposed to correspond to a surface-slope of 200, and to be one of a series adapted to the sideslope 1~ to 1. Example 1.-The sum of the end-bases is 40, the surface-slope 200, and the side-slope 1~ to 1. Required the argument for Table XV., and the corresponding content. Make 0 B equal to 40, the sum of the bases, according to the scale upon 0 X; and draw 0 T, making the angle TOB equal to 20~, the given surface-slope. Draw, also, B T parallel to OX. Draw B K parallel to P 0, and make O X equal to O K; also make 0 L equal to B T. On L X describe a semicircle cutting 0 B in F. Measure 0 F on the scale 0 X, and find the required argument, say 35.8: then Opposite 35, under 8, in Table XV., find the solidity. 593.4 Example 2.-The augmented sum of heights is 14.6, the surface-slope 200, and the side-slope 1j to 1. Required the argument for Table XV. and the corresponding content. Make O i equal to the sum of the heights, upon the scale OX; and draw 0 T, making the angle TO B equal to 200, the given surface-slope. Draw, also, H T parallel to OY. Make O S equal-to OH, and draw SR parallel to P. Make OU equal to OR, and O B equal to tIT. On B U describe a semicircle cutting O X in V. OV read upon the scale O X is the required augment,, say 35.8: then, Opposite 35, under 8, in Table XV., find the solidity. 593.4 .308 APPENDIX. CONVERSION OF SLOPES. 20. CONVERSION OF SLOPES is changing one or both slopes of a cross-section, whilst the area either remains unchanged, or is mad-e equal to another given area. There are two cases, —viz., single conversion, when only one slope is changed, and double conversion, when both slopes are changed. This problem requires the finding a new sum of heights or bases, by which the solidity may be obtained from the tables under new slopes. The practical application of this method has generally been confined to changing the surface to a level one. The conversion of slopes has already been referred to. (See Art. 98, Part I., Art. 14, ] 2, Art. 16, ~ 3, and Art. 17.) We propose, in the following articles, to consider further the conversion of slopes. 21. NATURE OF THE PROBLEM OF CONVERSION OF SLOPES. The problem is as follows:Having given a triangle A OB, (Fig. 50,) to find another triangle LO M, (Fig. 51,) equal in area to A O B, and similar to a third triangle H O K. A O B is the given base of a prism containing the solidity of the work, and L 0 M is the required base of another prism containing the same solidity. 22. GENERAL CONSTRUCTION BY MEAN PROPORTIONALS. This applies both to whole and sub sections; the use of mean proportionals has also been shown in Art. 14. Fig. 50. Fig. 51. F -—......"" -....... p.............. A As M O'%, 0 From one of the angles B of the given cross-section A 0 B draw B S perpendicular to O B, one of the sides adjacent to B. Make B F equal to A C, the altitude of the triangle. On O F describe the semicircle F S O, cutting B S in S. B S is — a mean proportional between B O and AC; that is, the square of B S is equal to the rectangle of BO and A C, or equal to twice the area of AG B. In a similar manner find (Fig. 51) ON, the side of a square equal to twice the area of 1O K, the triangle which exhibits the new slope or slopes. Upon the line of ON, take O P, equal to B S. - Join N H. and draw P L parallel to N H; then,. by the similar triangles O L1 P, Oi H N, OL OP 0L2 OP2 BS2 OH = ON whence_- - - OH2 ON2 ON2 But L2 Area LO M,B S2 AreaAOB.2 Area HO- 0 - Area HOK' whence, Area L O-M = Area AO B, which was required. The length of the new centre-height O D can now be measured. If it be required to find the area L-O M = Area A O B X m, the construction may be varied by making OP = B S X I/m. 23. INVESTIGATION OF FORMULAE BY AUXILIARY ANGLES. The following articles on this subject APPENDIX. 309 apply to whole sections only: the method of auxiliary angles has been partially discussed in Art. 18. (1.) For Construction.-We have for the solidity of whole sections, (see formula 19, page 281,) yV = L S2 X cot v 1-taiii y tan2 a Because a is always greater than y, we may replace tan ~ by the sine of an auxiliary angle. Puttan a ting tan _- sin a, the above equation becomes tan a V-4 L S2 X cot L S2 X COt cot -I L S2cot ff sec2. 1I-sin2 4 coe82 Let-a' and y' be new slopes; and let S1 be the new sum of heights which, with these slopes, contains the original volume V. Put tan =' sin i'. We have, by the previous reasoning and by tan at equating the two expressions for V, -LS12 X cot a' sec2 /-'LS2 X cot sec2. Whence S1= S eot a tan a'X se. 1. sec. This formula is applicable either to single or double conversion. If the side-slope is unchanged, the equation takes the form S1=S' X ~' 2. see d) If the new surface-slope be level, that is, if y' - O, then t' = 0, and see'-= 1: equation 1 becomes S =- S 1/cot a tan a' X sec..... 3. And equation 2 becomes S1= S see 9................4. We have, then, the following cases of conversion, of which the number designates the formula:1. General Form: change of one or both slopes. 2. Side-slope unchanged; surface-slope changed arbitrarily. 3. Side-slope changed arbitrarily; surface-slope made level. 4. Side-slope unchanged; surface-slope made level. The finding of the new sum S1 is then reduced to the construction or the numerical solution of these equations. For numerical solution, it will be convenient to have formulta adapted to logarithmic computation. (2.) Logarithmic Formulce. —Since sin 9 - tt and sin..""Y.', wehave sin tan y cota tan a- tan a'ycot and sin p' = tan y' cot a'; whence log. sin p = log. tan y + log. cot a; and log. sin' =_ log. tan y' + log. cot a'. By these last formulae we may find 0 and 0'; when log. cos p and log. cos 0' can be readily found. Now, sec 0= cos ft; and, therefore, log. sec = log. cos' - log. cos b. sec' cos ~ see O' Hence, by substituting successively in formule -1, 2, 3, and 4, ~ 1 the required logarithms, we have log. S1 = log. S + ~ log. cot a +- log. tan a' -+ log. cos If- log. cos 0. 1. log. S1 = log. S + log. cos p'- log. cos 0.............. 2. log. SI = log. S + y log. cot a + - log. tan at- log. cos ~..... 3. log. S1 log. S- log. cos t............ 4. (3.) Interpolation for Parts of a Degree in tables similar to Table VII. may be made by auxiliary tables constructed from the formula for conversion. The side-slope remains unchanged. Let y indicate the angle in whole degrees for a given surface-slope y which contains a fraction. By means of Formula 2, an auxiliary table might be constructed containing the values of S,, or of 310 APPENDIX. an increment for S, for a limited series of values of S and for consecutive values of y with intervals of one degree. It would, however, probably be better at once to extend the principal table. (4.) Auxiliary Tables for Equivalent Level-Heights.*-By means of Formula 4, an auxiliary table might be formed containing the equivalent sum of level-heights S1 for a series of values of S. The heights thus obtained might be employed with a table adapted to particular side-slopes. (See ~ 1, Art. 25.) These latter might be made to contain tenths of feet in the argument, in a manner similar to that of Table XV. Or, Table XV., or one constructed on a similar plan suitable for all side-slopes, might be adopted. (See ~ 2, Art. 25.) Instead of adapting the table of cubical content to odd tenths of height, we might employ its several columns for graduated lengths of work. 24. CONSTRUCTION BY AUXILIARY ANGLES. (1.) General Construction for Double Conversion. (See Formula 1, & 1, Art. 23.)-From any convenient scale take CA = 1, (Fig. 2, Plate XIII.,) and with centre C describe an arc A B. Make AC y = y, the original surface-slope, and produce a y. On the same scale take C E — cot a, and perpendicular to C A draw E H = cot a tan y sin b, cutting C y in H. Draw H I parallel to C A, cutting the arc A B in I; and through I draw CIO. I C A =?, because the sine of this angle to radius C I = C A = 1 is equal to E H = sin p. In the same manner, make A C y- = y' the new surface-slope. Make C E- = cot a'. Draw E' H', H I', and C I' ~', making I' C A = ~'. Draw the tangent A M M'; and draw I' P parallel to A M M'. Then, from the similar triangles C M M', C P I'z;2L CM; whence, because CI= C A= 1, CP= sec_ sec I' Draw N CO 0 perpendicular to C A. On any scale, make CO S cot a, and C N = S tan ea. Upon N 0 describe a semicircle cutting C A in R. C R is a mean proportional between C N and CO; that is, C R -1/= C O X C N = VS cot a S tan a' = S /cot a tan a'. Draw RI, and PT parallel to it. Because of the similar triangles C R I, C T P; C -_ R; but CI= 1, CP= sec and CR= S cot atan a': hence, sec 4t'~ C T = S ~cot a tan a'X ec= Si. In the example represented upon the diagram, the augmented end-heights are 95 and 64; the surface-slope 350; side-slope 4 to 1. The sum of end-heights containing the same solidity is found by Art. 96, Part I.; say 160. It is required to find a sum of heights which will contain the same solidity under a surface-slope of 18~ and side-slope 1~ to 1. The construction gives the equivalent sum, say 57.9. For the same by computation, see Example 1, Art. 26. (2.) Single Conversion of Suzface-Slope. (See Formula 2, ~ 1, Art. 23.)-Find C P _- see as sec 4' before, observing that C E/ C E when the side-slope is unchanged. No construction is necessary to find C R, which may be at once taken = S. We then have CT CRX see S. sec 0! (3.) Double Conversion with Surface-Slope made Level. (See Formula 3, ~ 1, Art. 23.)-Omit that part of the construction pertaining to the finding of 0', which is now = 0. The point I' will fall upon A, and P will coincide with M. Through M draw a parallel to IR cutting C A in a new point T', (not marked on the figure:) then CT'=CR X cM = CR X see = S —cotatan a' X sec = S,. (4.) Single Conversion with Surface-Slope made Level. (See Formula 4, ~ 1, Art. 23.)-Omit that part of the construction pertaining to the finding of 0', which is now = 0. The point I' * The writer has not had the advantage of consulting the work of Macneil, cited on this subject by Mr. Morris, to whose paper and to the work of Mr. Lyons we have referred. (See Introduction.) APPENDIX. 311 will fall upon A, and P will coincide with M. Take C R = S, and through M draw a parallel to I R cutting C A in T': then CT -'= CRX M = CR X see = S sec = S1. This construction may be simplified and applied to drawing a diagram for finding equivalent level-heights. (5.) To construct a Diagram for finding Equivalent Level-Heights.-Upon any convenient scale take C A - 1. (Plate XIV.) With centre C and radius C A describe an arc A B. Make the angle A C y y, the original surface-slope, and prolong C y as far as required. Take C E = the number denoting the rate of side-slope; that is, = 1~ for 1 to 1; and for other side-slopes in the same manner. Draw EH perpendicular and H I parallel to CE. Through I draw CI: then AC I = p. On any suitable scale make C K = S; draw K 0 perpendicular to C K: then C - CK X sec -= S sec - = S1. To complete the diagram, find the radius C I ~ for each degree of surface-slope, from 00 upwards as far as required; and, in some convenient part, mark each of these radii with the degree of original surface-slope to which it corresponds. Also, graduate the scale C E, and through the points of division, at suitable regular intervals, draw perpendiculars to CE. The radius for a given sum of heights and given surface-slope may be measured by applying a scale from C. Or the scale may be hinged at C as a movable radius, upon a tablet; and the radii omitted. 25. To FIND THE SOLIDITY BY A DIAGRAM OF EQUIVALENT LEVEL-HEIGHTS. (1.) With Tables adapted to particular Side-Slopes. —Tables VII., IX., and XI. are of this kind. Upon the diagram (Plate XIV.) find the radius marked with the given degree of surface-slope. Apply a scale, graduated like the scale C E, along the radius; and measure the distance intercepted between C and a line, perpendicular to C E, which passes through that division of the scale C E indicating the given sum of augmented heights. The distance measured upon the radius is the equivalent sum of level heights. If the distance upon C E be taken equal to the augmented height of a crosssection, the length measured upon the radius will be the equivalent level-height of the crosssection. If there is a fraction in the number of degrees indicating the surflce-slope, the space between the two adjacent radii containing the slope may be divided, by estimation, by the edge of the scale applied from C in the direction of the radii. The spaces between the perpendiculars to the scale C E may also be divided by estimation. After finding the equivalent sum of level-heights, enter the proper table, under 0~, with this sum, and take out the tabular quantity. Example 1. —The augmented end-heights are 38.5 and 20.6; surface-slope, 230; side-slope, 1} to 1. Required the equivalent sum of level-heights and the content of the whole ground. The sum of heights containing the same solidity is found by Art. 96, Part I.; say 60. With 60 on the scale CE and the radius of 230, find C p upon the diagram equal to, say 77.8 +; then Opposite 77.8 +, under 00, in Table VII., find the solidity v.. 8406.9 + Example 2.-The first end-height, from the roadbed, is 28.7; surface-slope, 180 to the right. Second end-height, 14.5; surface-slope, 12~ to the right. The side-slope is 14 to 1; roadbed, 24. Required the equivalent sum of level-heights and the content. The surface is here curved, because the surface-slopes at the ends are different. The solidity of the whole ground may be approximately found (see Example 1, Art. 104, Part I.) by an augmented sum of heights 59.68 and a surface-slope of 16~. (2.) With Table XV., adapted to all Side-Slopes.-Find the equivalent sum of level-heights as directed in section 1. With this equivalent sum take the tabular quantity from the First Part of Table XV., and multiply the quantity thus found by twice the rate of the given side-slope. NoTE.-Twice the rate of side-slope may be taken from the auxiliary table B, page 179. The radial scale may, however, be so divided that no multiplier is required. (See ~ 5, page 312.) 312 APPENDIX. Example 1.-Take Example 1, ~ 1. The equivalent sum of level-heights is found as before, 77.8 +; then Opposite 77, under 8 +, in Table XV., say... 2802.96 Multiplier equal to twice the rate of side-slope. 3 Content of the whole ground..8408.88 Example 2.-Take Example 2, ~ 1. For the operation, see Example 1, Art. 104, Part I. Explanation. —The tabular quantity of the First Part of Table XV. is - L S2. If this be multiplied; by twice'the rate of side-slope- 2 cot a, there results I L S2 cot a, which is the solidity when the end-heights are equal and the surface-slope level. (See Formula 19, page 281.) The Redundant Prism may obviously be found by the above method, as it is contained under a level surface-slope. The sum of the end-heights is the tabular number from Table II. (3.) Depths of Equivalent Prisms found by Construction. Let SI be the equivalent sum of level-heights, and S the sum of heights for the redundant prism. It is required to find the depth of a prism equal to the solidity contained between the surface and roadbed. The solidity of the whole ground is I L S12 cot a, and of the redundant prism 4 L S2 cot a. The difference is 4 L (S12 _ S2) cot a. Put S2 - S12- S2; then the solidity of the residual prismoid is 4 L S22 X cot a. It is obvious that S2 may be found by the construction of ~ 3, Art. 12, after obtaining SI and S. Having found S2, we may proceed as above directed to find the content. We may, however, proceed at once (Art. 97, Part I.) to find an equivalent depth under the original surfaceslope, and then work as above directed. (See Art. 13.) (4.) Remarks on the Method of Equivalent Level-Heights.-The facility of constructing a diagram for any required side-slope, together with the use of Table XV. as above shown, make this a method of general application. As a purely tabular method, such as indicated in ~ 4, Art. 23, it has advantages in the compendiousness of the tables of cubical content. To make it satisfactory, however, the auxiliary tables would require to be as extensive as a series constructed on the plan of Table VII. and others similar, embracing the same number of side-slopes. (5.) Bases of Equivalent Square Prisms found by Auxiliary Angles.-Fromn the equation VT 4 LS2 cot asee2, (21, Art. 23,) we have for the side of the square base A', V/A'=-/ 4-S2cot a X sec 6. If V/ 4 Sc2 ot a be found as in Art. 16, and then put for S in the formula of section 4, Art. 24, the construction according to section 5 will give the side of the Equivalent Square Base instead of the sum of Equivalent Level Heights. The argument required for Table XV. is l/ A' X 1/S — S see p X /4- cot a X V18 - S see 9 X I/ 2 cot a. Hence, to find the content, enter Table XV. with S sec I, and multiply the result,by 2 cot a, which is the rule above given in Sect. 2. (See also the Explanation, page 312.) Or if (as indicated in the Note to ] 2) the radial scale (~ 5, Art. 24) be so graduated that its 1 unit is to the unit of C K (Plate XIV.) as _/o to 1, the reading of the radial scale will be V2 cot a:S sec X / 2 cot a, and the quantity taken from Table XV. with this argument will be the required content; and the method of equivalent square bases is substituted for that of equivalent level heights. We have already employed a similar method of graduation. (See the Scale 0 P, Arts. 14 and 16.) The Second Part of Table XV. might be used in a similar manner to find the second term or correction, (Art. 85, Part I.,) when the sum employed for the First Part is the augmented sum of heights. Observe that the Redundant Prism is to be subtracted when the content obtained is that of the whole ground, (Art. 13.) APPENDIX. 313 Example.-Take Example 1, -- 1. The radial scale must here be graduated so.,asto read S sec > X' 2 X 1~= S sec 1/1. With this scale measure the radius corresponding to 60 on the scale C K and to 23~, and find, say 134.8-. Then opposite 134, under 8- in Table XV., find the whole content, say......8...... 8408.88 If, as suggested in Art. 16, a table similar to XV. were made to contain the term L S2, the multiplier (~ 2, Art. 25) would become 4 cot a; which would not, in general, be more convenient than 2 cot a which we have employed. Were it desired to dispense with the multiplier by the method here explained, the unit of the radial scale should, for the proposed table, bear to that of 1 2 C K the proportion to 1 = to 1; which would increase the size of the divisions of I/ cota J/cot a the radial scale. A table of the kind mentioned would frequently be convenient, (Art. 9; and Arts. 126, 127, Part I.) On a future occasion we may perhaps supply it. By bisecting the angle of surface-slope on a partial cross-section, a median line may be found homologous with the augmented centre-height of a full cross-section. This line might be obtained by construction, or by the aid of auxiliary tables. Thus both whole and sub sections may, theoretically, be included under one method; but the practical application of this idea would perhaps be too complex. 26. EXAMPLES IN CONVERSION OF SLOPES BY LOGARITHMIC FORMIULE. The formulae for the following examples will be found in ~ 2, Art. 23. Example 1. Do2%ble Conversion. (See Formula 1.)-The augmented end-heights are 95 and 64; surface-slope, 350; side-slope, 4 to 1. The sum of end-heights containing the same solidity is found by Art. 96, Part I.; say 160. It is required to find a sum of heights which will contain the same solidity under a surface-slope of 18~ and side-slope 14 to 1. The construction for this example is shown in Fig. 2, Plate XIII. (See ~ 1, Art. 26.) We have first to find log. cos ~ and log. cos I'. Log. tan y, 35..9.8452268 Log. cot a log... — 1.3979400 Log. sin 4, (say 100 4/ 54")..9.2431668 Whence log. cos 0 = 9.9932419. Log. tan y7, 180..9.5117760 Log. cot a' log. 1.0.1760913 Log. sin q', (say 29 10' 7").. 9.6878673 Whence log. cos p' = 9.9411084. Having now log. cos 0 and log. cos Id, we proceed to find those logarithms which are yet wanting, and to subtract log. cos 0 from the sum of all the others. Log. S- log. 160...... 2.2041200 4 log. cot a log. 1. -1.6989700 1log. tan a' - log..-. 1.9119544 Log. cos 0'..9.9411084 11.7561528 Log. cos.. 9.9932419 Log. S1, the required sum, 57.93.. 1.7629109 160, under 35~, in Table XI., gives.... 6113.3 57.93, under 18~, in Table VII., gives.6113.1 :314 APPENDIX. Example 2. Single Conversion. (See Formula 4.)-The augmented end-heights are 38.5 and 20.6; surface-slope, 230; side-slope, 1-4 to 1. Required the equivalent sum of level-heights and the content. The sum of heights containing the same solidity is found by Art. 96, Part I., or by computation; say 60. The construction for this example is shown on the diagram, Plate XIV. (See Example 1, ~ 1, Art. 25.) We have to find log. cos I, and to add to it log. S. Log. tan y, 230.....9.6278519 Log. cot a = log. 14. 0.1760913 Log. sin p 390 32' 491".4.. ~ 9.8039432 Whence log. cos 0 = 9.8871118. Log. S - log. 60..... 1.7781513 Log. cos 9 (= tabular log. - 10)...... — 1.8871118 Log. S1, the required sum, 77.81.......1.8910395 60, under 230, in Table VII., gives.. 8409.1 77.81, under 00, in Table VII., gives... 8409.1 NOTES. We here record some formulae which have been omitted in the text. Those of Note 1, although early known to us, we had not actually employed until, when too late for convenient insertion, they were used to verify some of the tabular computations. Those of Note 2 were at one time embodied, at greater length, in the text to which they refer; but were afterwards rejected as not entirely appropriate. In closing the work we have finally decided to preserve them all in the following form:cot a (1.) Article 10, page 296.-From V= — L S2X cost (page 309,) we find for the logarithm of 2 cot a Table XVI., log. co - log. 2 cot a-2 log. cos 0. The formulae of the catalogue, for the computation of Tables XVI., XVII., XIX., and XX., are frequently less expeditious than the following, in which tan a and cot a are derivable from the rate of side-slope, and where tan y and cot y may be taken from a table of natural tangents and cotangents. For Table XVI.-The logarithm required is log. tan a log. 2tan _a tan2 o - tan2 y (tan'a + tan y (tan - tally) log. 2 tan a- {log. (tan a + tan y) + log. (tan a — tan y) } cot2 y For Table XVIL —The logarithm required is (by the formulae of Art. 18) log. cot -cot 2 log. cot y - log. (cot y - cot a). A similar form for computation by bases, derived from the same 1 formule, would give log. -t =log. 1 -log. (cot y-cot a). See note, page 49. cot c — cot oFor Tables XIX. and XX —By aid of the formula tan a ~_ tan b sin (ab) the expression cos a cos b log. cs 4 s) may be transformed to log. - log. (tan a _~ tan y); which serves for Table XIX. In a similar manner log. cosy, becomes log. 4 —log. (tan a _ tan y) -log. cos y; which serves sin (a. + for Table XX. If the logarithm for Table XIX. be denoted by log. XIX., and that for Table XX. by log. XX., we have log. XX. = log. XIX. -log. cos. y. APPENDIX. 315 (2.) Surface-Slope not given in Degrees, (pp. 281, 282.) —If the surface-slope were given by its rate similarly to the side-slope, we might determine the tangent, and find the degree of surfaceslope by Table A; and then proceed, by any suitable mode of construction or computation, to the solution of any problems embraced by our rules. The following substitutions, which directly employ the rate of surface-slope, may, however, be noted. Let r denote the rate of side-slope, and r' the rate of surface-slope; that is, let r =cot a, tan a rr, 2 1 and r=cot y: we shall find ta2 tan2 rrt2 (Formula 16.) Also 1 co (Fortan a-tan~y 2cot2 a r — mulae 21, 24;) and this divided by tan2 y gives cot - v- _:=r, (Formula 22.) cot -y - ct cr r P-rFrul 2. DESCRIPTION OF THE PLATES. In order to view Plates I., II., and III., turn the top of the volume towards the left, (or direct the sight across the page,) so as to bring the right-hand margin of the plate upwards. Plates I., II., III., and IV.-These represent the warped surfaces of Art. 9, page 17, and also the solid forms to be considered in excavation and embankment upon such surfaces. The drawings are from models made of rectangular blocks, the blocks being formed of veneers piled together. The veneers represent horizontal strata, and the surface of the ground is shown by curved bands which represent the outcroppings of the strata. The curved lines of division between the strata are called by topographers Contour Lines. That contour line which extends on to the roadbed is the Grouznd-Trace, (Art. 16, page 17.) The ends of the models are presented towards the spectator, and are numbered. Plates I., II., and III. represent excavation and embankment upon the same (or mean) width of roadbed. The numbers designate the Varieties of the General Scheme, page 21. Nos. 1 and 2 give a satisfactory (though not strict) illustration of the surfaces described in Art. 8, page 16. The lines drawn across the roadbed and continued on the side of the excavation or embankment denote the place of neutral cross-sections. Attention to these lines will aid the study of Art. 32, page 21, by supposing, when necessary, the neutral cross-section to be at the end of the work. Plate IV. represents earthwork as it is really made; viz. with extra width in excavation, to permit drainage. No. 13 belongs to Variety 3 of the General Scheme, and shows a plane surface with straight ground-trace. No. 14 belongs to Variety 4 of the General Scheme. The remarks of Art. 20, page 18, should be remembered in viewing these plates. Plate V. See Art. 54, page 29; and Art. 88, page 44. Plate VI. See Art. 57, page 31; and Art. 88, page 44. Plate VII. See Art. 58, page 31; and Art. 12, page 300. Plate VIII. See Art. 55, page 30, for the Mid-Section Diagram; and Arts. 96, page 47, and 12, page 301, for the General Scale. Plate IX. See Art. 97, page 48; and Art. 12, page 301. Plate X. See Art. 14, page 302; and Art. 15, page 303. Plate XI. See Art. 14, page 302; and Art. 15, page 303. Plate'XII. See Art. 16, page 304; and Art. 17, page 305. Plate XIII. See Art. 18, ~ 2, page 306, for Fig. 1; and Art. 24, page 310, for Fig. 2. Plate XIV. 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