iw^^^M^uVsi^.^W^WA.kr^i 
 
JVo. 
 
 Division 
 Range 
 
 Shelf. 
 
 Received 
 
 mtoig 
 
 OF THE 
 
 
 
EAST EULES 
 
 FOR THE 
 
 MEASUREMENT OF EARTHWORKS, 
 
 BY MEANS OF TIIE 
 
 PRISMOIDAL FORMULA. 
 
 ILLUSTRATED WITH NUMEROUS WOODCUTS, PROBLEMS, AND EX- 
 AMPLES, AND CONCLUDED BY AN EXTENSIVE TABLE 
 FOR FINDING THE SOLIDITY IN CUBIC 
 YARDS FROM MEAN AREAS. 
 
 THE WHOLH 
 
 BEING ADAPTED FOR CONVENIENT USE BY ENGINEERS, SURVEYORS. 
 
 CONTRACTORS, AND OTHERS NEEDING CORRECT 
 
 MEASUREMENTS OF EARTHWORK. 
 
 L I B R A II V 
 
 j UNIVERSITY OF 
 
 CALIFORNIA. 
 
 ELLWOOD MORRIS, CIVIL ENGINEER. 
 
 PHILADELPHIA: 
 
 T. R. CALLENDER & CO., THIRD AND WALNUT STS. 
 
 LONDON: TRUBNER A CO., 60 PATERNOSTER ROW. 
 1872. 
 

 Entered according to Act of Congress, in the year 1871, by 
 
 ELLWOOD MORRIS, CIVIL ENGINEER, OF CAMDEN, N. J., 
 
 In the Office of the Librarian of Congress, at Washington, D. C. 
 
RESPECTFULLY DEDICATED 
 
 TO THE 
 
 ENGINEERS, SURVEYORS, AND CONTRACTORS 
 
 OF 
 
 THE UNITED STATES, 
 BY ONE WHO IS WELL ACQUAINTED 
 
 THEIR ABILITIES AND WORTH. 
 
TABLE OF CONTENTS, 
 
 BY CHAPTER, AETICLE, PAGE, AND REFERENCE TO ILLUSTRATIONS. 
 
 CHAPTER T. 
 
 Art. 1. 
 
 Art. 2. 
 Art. 3. 
 
 Art 4. 
 
 Art. 5. 
 
 Art. 6. 
 
 Art. 7. 
 
 Art. 8. 
 
 Art. 9. 
 Art. 10. 
 
 Art. 11. 
 Art. 12. 
 
 Art. 13. 
 
 PRELIMINAKY PROBLEMS. 
 
 PACE 
 
 Of the Prismoid. Origin probably ancient. Prismoidal Rule, first devised and 7 
 demonstrated by THOMAS SIMPSON (1750). Generalized and improved by CHARLES 
 HUITON (1770). Now known as the Prismoidal Formula,. A Prismoid denned, 
 described, and illustrated by Fig. I. The Prismoid, a frustum of a wedge, which 
 latter is itself a truncated triangular Prism. Example of rectangular Prismoids, 
 by Simpson's Rule and other processes. 
 
 Simpson's I*rismoidal RuU. Applicable chiefly to rectangular and triangular 10 
 Prismoids, with examples of each. Illustrated by Fig. 2. 11 
 
 Huttoris Prismoidal Rules. His definition of a Prismoid. General Rule, in which 12 
 the hypothetical mid-section, deduced from the ends, is first introduced by him, 
 thus generalizing the rule, and materially extending its usefulness. Particular 
 Rule, deduced from the wedge. Rectangular Prismoid, composed of two wedges. 
 Initial Prismoids, to determine generality of rule. Applicability of Button's Rule 13 
 to various solids, named by him. Example of computation by General and Par- 15 
 ticular rules; also, by Initial Prismoids. The T Prismoid, example of its compu- 16 
 tation. Figs. 3. 4, and 5 illustrate this article. 
 
 The Prismoid adapted to Earthwork. By Sir John Macneill, C. E. (Fig.6), his figure 18 
 of a prismoid composed of a prism and superposed wedge. Demonstrates the 
 Prismoidal Rule, and uses it in an extensive series of Tables (1833). 
 
 The Prismoid in its Simplest Form. Computed by several rules. Illustrated by 
 Fig. 7. 
 
 Further Illustration of MacneilUs Prismoid. Double cross-sectioned in Fig. 8. Con- 21 
 cise rule for earthwork wedges, confirmed by Chauvenet's Geometry. Brevity of 
 computation of earthwork solids on level ground, by combining the rules for 
 Wedge and Prism. Wedges applicable also to irregular ground. 
 
 Trapezoidal Prismoid of Earthwork, considered as two Wedges. Button's Particular 24 
 Rule, modified for earthwork. Illustrated by Fig. 9, and by examples. 
 
 26 
 
 Areas of Railroad Cross-sections (within Diedral Angles), whether Triangular, 26 
 Quadrangular, or Irregular. Triangles and Trapeziums computable by their 28 
 hights and widths, taken rectangularly, however they may be placed (see Figs. 30 
 10, 11, 12. and 13). Exact method of equalizing irregular ground surface, by a 32 
 single right line (see Figs. 14 and 15). Calculation of areas of crows-sections 33 
 (Figs. 16 and 17). Very irregular cross-sections require division into elementary 34 
 figures, these to be separately computed and totalized. 35 
 
 Further Illustration of the Modification of Simpson's Rule, etc. Example illustrated 
 by Fig. 18. Diagram of Simpson's mid-section, Fig. 19, showing its curious com- 37 
 position. Diagram of the formulas of Simpson and Hutton, Fig. 19%. Formula 
 deduced from it. 
 
 Adaptation of the Prismoidal Formula to Quadrature, Oubature, etc., by Simpson 40 
 and Hutton. Explanation of it by Hutton, under the head of Equidistant Ordi- 42 
 nates, Fig. 20. Example of two stations of earthwork. Figs. 21 to 26. Coincidence 44 
 with the Prismoidal Formula. Earthwork calculations by Roots and Squares, 46 
 explained and illustrated by examples, and by Figs, from 27 to 38 ; also, by com- 47 
 parison with other methods. Computation of the passages at grade points, from 53 
 excavation to embankment. Figs. 39 to 42. 
 
 Cross-sections in Diedral Angles, to find Mid-section, etc. Illustrated by Figs. 43 55 
 and 44. 
 
 To find Prismoidal Mean Area, from Arithmetical or Geometrical Means, from 
 the Mid-section, or by Corrective Fractions. Examples, illustrated by Figs. 45 
 and 46. 
 
 Applicability of the Prismoidal Formula to other Solids. The three round bodies 
 computable by it. Examples given and illustrated by Fig. 47. This formula also 
 applies to the three square, or angular bodies. 
 
TABLE OF CONTENTS. 
 
 Transformation of Solids into Equivalents more easily Computable Prismoidally. 
 Equivalency explained. With examples illustrated by Figs. 48, 49, 50, and 51. 
 
 Equivalence of some important Formulas, with that for the I*rismoid. Coincidence 
 of the Pyramidal and Prisnioidal Formulas to a certain extent, and similitude of 
 their results within certain limits. Illustrated by Fig. 52 (in projection), and 
 followed by examples calculated both Prismoidally and Pyramidally, and by other 
 rules; all equivalent in solidity upon level ground. 
 
 Summary of Rules and Formulas in Chapter I. Indicated by numbers from I. to 
 XII., and by reference to the various articles of the chapter in which they may 
 be found. 
 
 CHAPTER. II. 
 
 FIRST METHOD OF COMPUTATION, BY MID-SECTIONS DRAWN AND CALCULATED FOB AREA 
 ON THE BASIS OF HUTTON'8 GENERAL RULE. 
 
 encrality and accuracy of the Prismoidal Formula. Manner of collecting the field 
 data. Kules for cross-sectioning. Correction for curvature. Vital importance of 
 judicious cross-sectioning. Ground to be sectioned so as to reduce it practically 
 to plane surfaces. Warped surfaces and opposite slopes deemed inadmissible in 
 general. Expression for the Prismoidal Formula, as generalized by HUTTON. 
 Classification of the ground surface. Mensuration by cross-sections, long used by 
 mathematicians, and early adopted by engineers. 
 
 Examples in computation by our First Method. Illustrated by Figs. 53 to 64. 
 
 Connected calculation of contiguous portions of Excavation and Embankment, with 
 the passage from one to the other. Illustrated by Figs. 65 to 71. End sections to 
 have, or receive, the same number of sides before calculation. Observations on 
 mid sections. Importance of verifying all calculations. The Prisnioidal Formula, 
 the standard test for solidity. Correction for centres of gravity referred to aa a 
 refinement promoting accuracy, but not employed by engineers. This First 
 Method easily applicable to masonry calculations. Borden's Problem, a striking 
 example of bud cross-sectioning. 
 
 CHAPTER III. 
 
 SECOND METHOD OF COMPUTATION, BY HIOHT8 AND WIDTHS, AFTER SIMPSON'S 
 ORIGINAL RULE. 
 
 Prismoidal Rule, originated and demonstrated by Simpson (1750) for rectangular 
 prismoids. Its transformation for triangular areas. Modification by direct and 
 cross multiplication. 
 
 Examples of computation by our Second Method. Illustrated by Figs. 72 to 75. 
 
 Observations upon Simpson's Rule Ground surfaces to be equalized to a single 
 line. Peculiar solid where Simpson's Rule fails. See Figs. 81 and 82. Examples 
 of rough ground equalized by a single line. Illustrated by Figs. 43 and 44. 
 Example of a heavy embankment, from Warner's Earthwork, on uniform ground, 
 sloping 15 transversely. 
 
 CHAPTER IV. 
 
 THIRD METHOD OF COMPUTATION, BY MEANS OF ROOTS AND SQUARES J A PECULIAR MODI- 
 FICATION OF THE PRISMOIDAL FORMULA, WHICH WILL BE FOUND IN PRACTICE TO U 
 BOTH EXPEDITIOUS AND CORRECT, IN ORDINARY CASES. 
 
 Enunciation of thfi formnla used. Comparison with the Prismoidal Formula. Use 
 of Simpson's rule for cubatnre. as adopted by Hutton. Skeleton tabular arrange- 
 ment of data. Tabulations for solidity. Numbered places for end-sections and 
 mid-sections. Example of a heavy embankment and rock-cut (profiled in Fig. 
 76), and each computed in a body by this method. Relations of the squares of 
 lines, or parts of lines. Re-computation by Roots and Squares of the examples of 
 Chapter II. (illustrated by Figs. 53 to 64), showing their close agreement with 
 this method. 
 
 CHAPTER V. 
 
 FOURTH METHOD OF COMPUTATION, BY REGARDING THE PRISMOID AS BEING COMPOSED 
 OF A PRISM, WITH A WEDGE SUPERPOSED, OB OF A WEDGE AND PYRAMID COMBINED. 
 
 Macneill's wedge superposed upon a prism to form a prismoid. Formula for wedge 
 
 and prism to find solidity of prismoid. Discussion of the inclined wedge (Fig. 79), 
 
 and calculation of-it. 
 
 Rules for computation by Wedge and Priam. 
 Discussion of the wedge in connection with Chauvenet's Theorem. Illustrated by 
 
 9 small ciits. Showing the mode of computing various wedges. 
 Examples of Wedge and Prism computation. Illustrated by Figs. 80, 14, 43, and 44. 
 Peculiar solid cross-sectioned (Fig. 81), and in projection (Fig. 82). Examples of 
 
 modes of computing it. 
 
TABLE OF CONTENTS. 
 
 Art. 30. 
 
 Aw. 31. 
 
 Art. 32. 
 
 Ait. 33. 
 Art. 34. 
 
 Art. 35. 
 
 Art. 36. 
 
 Art. 37. 
 Art. 38. 
 
 ?he Rhomboidal Wedge and Pyramid introduced. Examples of calculation by using 
 them to find the solidity of a prismoid. 
 
 e method of using the Rhomboidal Wedge and Pyramid combined. (Figs. 81 
 and 82.) 
 
 Any irregular earthwork, within certain limits, computable by Wedge and Pyramid. 
 Process of computation. Notation of rule. The Rule itself. Its limits of range. 
 
 Examples of computation by Wedge and Pyramid ; direct and reverse. Re-com- 
 puting by this method the examples of Chapter II., and showing their near coin- 
 cidence. Illustrated by Figs, from 53 to 64, in Chapter II. 
 
 Ixample of heavy embankment (from Warner's Earthwork), computed by Wedge 
 and Pyramid. 
 
 eculiar solid again ; illustrating Case 2, of the rule for Wedge and Pyramid, which 
 correctly gives its volume. 
 
 CHAPTER VI. 
 
 PROFESSOE GILLESPIE'S FOUR USUAL RULES, WITH THEIR CORRECTIONS, AND A COMPARI- 
 SON OF HIS CHIEF EXAMPLE WITH OUR THIRD METHOD OF COMPUTATION BY ROOTS 
 AND SQUARES. 
 
 Che late Professor Gillespie's ability and labors. The four usual rules, which he 
 found in use. 1. Arithmetical Average, a rule very much used, but very incor- 
 rect. The correction necessary for true results. Tabulation of an example by the 
 Arithmetical Average, with Gillespie's correction. 2. Middle Areas, rule explained 
 and formula for correction to give true solidity. Tabulation of example by Middle 
 Areas, with correction. 3. The Prismoidal Formula, as generalized by Hutton. 
 The test rule for all. Tabulation of example by it. 4. Mean Proportionals, or 
 Geometrical Average. Explanation of the rule. Gillespie's condemnation of Geo- 
 metrical Average, and of Equivalent Level Eights, evidently hasty. 
 
 Ele-computation of example by Geometrical Average, using the grade triangle, and 
 tabulating to intersection of side-slopes. Solidity the same as given by the Pris- 
 moidal Formula, as calculated by Gillespie himself. Generalization from which 
 this rule flows. Equivalent Level Hights. Tabulation of an example by them to 
 intersection of side-slopes. Resulting in solidity the same as by the Prismoidal 
 Formula, calculated by Professor Gillespie. 
 
 Five reliable rules for earthworks, deduced from this Chapter. Brief reference to 
 the Core and Slope Method, one of the earliest proposed, and often re-produced, 
 but not in general use. 
 
 Tabulated comparison of Gillespie's example by the method of Roots and Squares, 
 agreeing with the Prismoidal Formula, and may be used with Simpson's Multi- 
 pliers. Gillespie's examples usually refer to level ground, and hence are easily 
 computed. 
 
 .CHAPTER VII. 
 
 PRELIMINARY OR HASTY ESTIMATES, COMPUTED BY SIMPSON'S RULE FOR CUBATURE. 
 
 Nature of Preliminary Estimates. Rough and expeditious, but giving quantities 
 proximately correct. Quantities to be always full, but never excessive. Simp- 
 son's rule for cubature. Explanation and notation. Mode of numbering the 
 cross-sections, even and odd. Cuts and Banks to be computed separately, each 
 in a body. General Mean Area to be found, and multiplied by length for volume. 
 
 Simpson's Multipliers, their convenience and application. Diagrams indicating a 
 probable construction of the rule for cubature, and its intimate connection with 
 the Prismoidal Formula (Figs. 77 and 78.) 
 
 Rough profile to be sketched, involving- bights and ground slopes. Computation of 
 a Bank by Simpson's Multipliers. Rock-cut calculated in the same way. Both 
 illustrated by Fig. 76. 
 
 Reference to the grade prism. Earth cutting, Fig. 29, computed by both Roots and 
 Squares and the Hasty Process, differing only one and a half per cent. 
 
 Tables (four in number) for use in Preliminary or Hasty Estimates. 
 
 PAGH 
 
 133 
 
 Following this Chapter, and closing the Book, will be found an extensive TABLE OF 
 CUBIC YARDS to mean areas for 100 feet stations (entirely clear of error, it is 
 believed), giving the Cubic Yards for every foot and tenth of mean area from to 
 1000, by direct inspection. And being computed accurately to three decimal 
 places, ranges correctly up to 100,000 square feet of mean area, or to a cut 1000 
 feet wide, and 100 feet deep. Table preceded by explanations, and some examples 
 of its use. This Table also operates as a general one for the conversion of any sum 
 of cubic feet into Cubic Yards, by simply dividing by 100 and using the quotient 
 as a mean area,. 
 
UNIVERSITY OF 
 
 CALIFORNIA. 
 
 EASY RULES 
 
 FOR THE 
 
 MEASUREMENT OF EARTHWORKS, 
 
 BY MEANS OF THE PRISMOIDAL FORMULA. 
 
 CHAPTER L 
 
 PRELIMINARY PROBLEMS. 
 
 1. Of the Prismoid. Although this solid probably originated \vith 
 the ancient geometers THOMAS SIMPSON (1750), an eminent mathe- 
 matician of the last century, appears to have been the first, in later 
 days, to demonstrate the rule for its solidity,* now accepted by 
 modern mensurators ; and he was soon followed by Hutton, in his 
 quarto treatise on Mensuration,f who by another process again 
 demonstrated the Prismoidal Rule, and at the same time laid the 
 foundations of modern mensuration, in a manner so solid, that it has 
 come down to our time, through various editors and commentators, 
 substantially (in many cases literally) the same as established by Hut- 
 ton in his famous work of 1770. 
 
 Simpson's rule for the prismoid has been variously transformed, 
 and written, and is now generally known by the name of the prismoi- 
 dal formula, of which we will give hereafter the usual expressions, as 
 well as some useful modifications, the same in substance, but often 
 more convenient for practical purposes. 
 
 The solid called a Prismoid (from its general resemblance to a 
 prism, and in like manner named from its base, triangular, rectangu- 
 lar, trapezoidal, etc.) is a body contained between two parallel planes t 
 
 * Simpson's Doctrine of Fluxions. (1750), 8vo, London. 
 f Button's Mensuration. (1770), 4to, Newcastle upon Tyne. 
 
 7 
 
8 
 
 MEASUREMENT OF EARTHWORKS. 
 
 its hight being their perpendicular distance apart, its ends rectangles* 
 and its faces plane trapezoids ; and this seems to be a sufficient defini- 
 tion. As to such form, all prismoids may be reduced or made equiva- 
 lent; but although this simple definition answers our purpose of intro- 
 ducing the rectangular prismoid, HUTTON'S, Art. 3, is the authorita- 
 tive one. 
 
 This solid is usually the frustum of a wedge ; but as the proportions 
 of the ends are changed, it may become a frustum of a pyramid, a 
 complete pyramid, a wedge, or a prism ; and hence it is indispensably 
 necessary that the rule for its solidity should also hold for all these 
 solids, which, in fact, it does. 
 
 The ends may be, and often are, irregular polygons, but they must 
 always coincide with the limiting parallel planes ; and though the 
 solid may be quite oblique, its hight must be taken normal to the 
 end planes. The faces are usually straight longitudinally, but this 
 condition is not absolute, since the remarkable formula, deduced from 
 the prismoid for its solidity, applies as well to the volume of many 
 curved solids in an extraordinary manner, of which the limits are not 
 yet known, though more than a century has elapsed since Simpson 
 developed it. _____ 
 
 The mid-section, inclu- 
 ded by the usual prismoi- 
 dal formula, must be in 
 a plane parallel to, and 
 equally distant from, those 
 containing the ends, and 
 is deduced from the arith- 
 metical average of like 
 parts in them. It is en- 
 tirely hypothetical, or as- 
 sumed for the purposes of 
 computation, and has no 
 actual existence in the 
 body itself. 
 
 The rectangular pris- 
 moid (usually regarded as 
 the elementary figure of 
 this solid) is a frustum 
 of the wedge. 
 
 (a.) Thus the prismoid AB (Fig. 1) is a frustum of the 
 
 wedge AEC. 
 
CHAP. I. PRELIM. PROBS.-ART. 1. 9 
 
 The wedge AEG itself being a triangular prism, truncated twice, 
 the rectangular prismoid then is a triangular prism, trebly truncated : 
 1st, by two cutting planes, reduced to a wedge; and 2nd, by another 
 plane, to a prismoid (AB), the latter being parallel to the base, and 
 by its section forming the top of the solid at B. 
 
 The prismoid, therefore, may be computed as a truncated triangu- 
 lar prism or wedge, and the part cut off deducted, in like manner as 
 the frustum of a pyramid may be calculated as though the pyramid 
 was complete, and then the truncated part computed separately and 
 subtracted, leaving only the solidity of the frustum, subject, like the 
 prismoid, to calculation, by more concise rules, if expedient. 
 
 Referring now to Fig. 1. 
 
 Let Abode/be the original triangular prism, truncated right and 
 left by planes passing through A b and ef, reducing it first to the 
 wedge AE ; and secondly, by passing the plane B 2, parallel to the 
 base eb, leaving as the residual solid, after three truncations, the 
 Prismoid AB. 
 
 Then, in the wedge AEC, the right section has a base of 4, a hight 
 of 12, and area of 24, which, multiplied by the sum of the lateral 
 edges * (or 6), gives a solidity of 160 ; while the wedge BCE, cut 
 off, has a base of 2, and hjght of 6, in its right section, or area of 6, 
 which, multiplied by i the sum of its lateral edges (or 5i), gives a 
 volume of 32. 
 
 Now, 160 32 = 128, the solidity of the Prismoid AB, as is shown 
 (more concisely) 05 follows : 
 
 By Simpson's Rule 
 
 lit*. Widths. 
 
 Base, 8 X 4 = 32 
 
 Top, 6 X 2 = 12 
 
 ums, equivalent to ) + , ^ R ft . 
 
 -J f -I* X O O-4 
 
 id. sec., . . . . J 
 
 Product of sums, equivalent to 
 
 A j.' -J 
 
 4 times mid. 
 
 128 
 
 Multiplied by i h. . .i.T ;.**..= 1 
 
 Solidity, . . . : . >/} . .'J- . *;*;. . = 128 
 
 (The same as above.) 
 
 Precisely the same result is also reached by means of the centre of 
 gravity of the right section, flowing with that section along a line 
 
 * Chauvenet's Geom. (1871), vii. 22. A wedge, whether trapezoidal or rectangular, 
 being merely a truncated triangular prism, this rule of Chauvenet's is probably the 
 most concise,, and fast for ordinary uw. 
 
10 
 
 MEASUREMENT OF EARTHWORKS. 
 
 curved with an infinite radius, according to Button's Problem.* The 
 right section of the prismoid AB (Fig. 1) is a plane trapezoid (18 in 
 area), of which (from the dimensions given in the figure) the centre 
 of gravity is found in a perpendicular line, drawn from the middle 
 of A 6, and at the distance of 21 feet vertically from it. Now, the 
 length of a straight line, drawn from face to face of the prismoid, 
 parallel to the plane of the base also to its edges and at a vertical 
 distance of 2f feet, will be 7J feet, by which the right section (18) 
 being multiplied, we have for the solidity = 128, as before. 
 
 2. THOMAS SIMPSON'S Prismoidal Rule. In his work on Fluxions 
 and their Applications 
 (1750), Simpson demon- 
 strates the following rule 
 for the solidity of a pris- 
 moid, referring to Fig. 2. 
 
 This rule for the pris- 
 moid, as demonstrated 
 by Simpson, renders the 
 formation of the hypo- 
 thetical mid-section un- 
 necessary, though con- 
 taining it, in effect, as 
 marked upon the figure, 
 for illustration. 
 
 wA. 
 
 Simpson's Rule is as 
 follows: Fig. 2. 
 (AB X AD) + (EH X EF) + (AB+EH X AD + EF) X 
 
 i h = Solidity, (I.) 
 
 Or, 
 
 /hight X width \ , /hight X width \ , 
 \ of one end, j ' ( -*- Al - J 
 
 of one end, / 
 
 / sum of bights X sum of widths \ 
 \ of both ends, / 
 
 of other end, ) 
 
 / T \ 
 
 (I.) 
 
 Here AB X AD = area of base. EH X EF = area of top. While 
 the product of their sums = (AB + EH) X (AD + EF) = four 
 times the area of the mid-section. 
 
 * Button's Mens. (1770), part iv. sec. 3. 
 
CHAP. I. PRELIM. PROBS. ART. 2. 11 
 
 EXAMPLE 1. 
 
 Let AB and EH be called the widths, AD and EF the highte, 
 and take the dimensions marked upon Fig. 2. Then, by Simpson's 
 rule, we have for the solidity of this rectangular prismoid the fol- 
 lowing : 
 
 Widths. Ht8. 
 
 20 X 16 = 320 = area of base. 
 18 X 12 = 216 = do. top. 
 
 Sums of hts. and widths = 38 X 28 = 1064 = four times mid-sec. 
 
 1600 = sum of areas. 
 Multiplied by i h = 2 e 4 , . . . . = 4 = i h. 
 
 Solidity, = 6400 = volume. 
 
 (a.) The above is a rectangular prismoid, or one in which all the 
 
 parallel sections are rectangles. Now, suppose this prismoid to be 
 cut diagonally by a plane, FHBD, dividing it into two triangular 
 prismoids, each equal to the other, and to one-half of the rectangular 
 prismoid. 
 
 Then (AB X AD) = double the base; (EH X EF) =. double 
 the top; and (AB -f EH) X (AD + EF) = eight times the mid- 
 section. 
 
 Hence, Simpson's rule, thojugh applicable to any prismoid, by 
 reducing the ends to equivalent rectangles, seems especially suitable to 
 triangular prismoids, since the double area of every triangle is equal 
 to the product of its bight and width, taken rectangularly; while 
 the product of the sums of those bights and widths, multiplied to- 
 gether, gives eight times the area of the mid-section, without the ne- 
 cessity of forming it by arithmetical averages. 
 
 Accordingly, with triangular sections, a slight transformation of 
 this rule will often be more convenient for use with given areas. 
 
 Thus, 
 
 Let double the area of the base = 2 b. 
 
 top . .-'.; v . . . = 2 t. 
 
 Eight times the area of the mid-sec, v ^V *; n' . = 8 m. 
 And the final divisor (12), or if used as above, . = ^ h. 
 
 Then, to find, in the first instance, the mean area of the prismoid. 
 
 We have the formula, - - = mean area . . (II.) 
 
 And this mean area, being multiplied by the bight or length (h), 
 of the whole prismoid between the end planes, gives the solidity. 
 
12 MEASUREMENT OF EARTHWORKS. 
 
 Thus, in the case of the two triangular prismoids, into which the 
 diagonal plane FB (Fig. 2) divides Simpson's rectangular prismoid, 
 we have, by taking the dimensions marked upon the figure, the fol- 
 lowing : 
 
 EXAMPLE 2. 
 
 Calculation of the triangular prismoid ABDFHE, or of its equal 
 GD = 3200, Solidity. 
 
 Hts. Widths. 
 
 16 X 20 = 320 = 2 b. 
 
 12 X IS = 216 = 2 t. 
 
 Sums, . . 28 X 38 = 1064 = 8 m. 
 
 12)1600 
 Mean area, . . = 1331 X h = 24 = 3200, Solidity. 
 
 And 3200 X 2 = 6400 = the solidity of the whole rectangular 
 prismoid, as above. 
 
 3. CHARLES HUTTON'S Prismoidal Rules. In his famous quarto 
 Mensuration (Newcastle-upon-Tyne, 1770), Hutton gives the follow- 
 ing definition : 
 
 "A prismoid is a solid having for its two ends any dissimilar par- 
 allel plane figures of the same number of sides, and all the sides of 
 the solid, plane figures also." 
 
 He adds : " It is evident that the sides of this solid are all trape- 
 zoids ;" and : " If the ends of the prismoid be bounded by curves, as 
 ellipses, etc., the number of its sides, or trapezoids, will be infinite, 
 and it is then called, sometimes, a cylindroid." 
 
 Hutton gives two rules for the solidity of the body (so defined), 
 one general, and the other he calls the particular rule he also indi- 
 cates a third, by means of initial prismoids, which, by a little develop- 
 ment, can be made quite useful. 
 
 Button's General Mule. 
 
 " To the sum of the areas of the two ends add four times the area 
 of a section parallel to, and equally distant from, both ends, mul- 
 tiply the last sum by the hight, and i of the product will be the 
 solidity, (III.) 
 
 In this shape, and nearly in the same words, through Bonnycastle, 
 and other writers on Mensuration, the Prismoidal Formula has come 
 down to our time. 
 
 In the work above cited, Hutton also (part iv. prop. 3) shows that 
 
CHAP. I. PRELIM. PROBS.-ART. 3. 
 
 13 
 
 Fig. 3 
 
 
 t of the sum of the end areas, and four times the mid-section, gives 
 the mean area of any prismoidal solid, which, multiplied by its length, 
 will equal the solidity. 
 
 The particular rule, referred to above, is directly deduced from that 
 given by him for the solidity of a wedge. 
 
 Thus, referring to Fig. 3 (copied by us from the original work 
 of 1770). 
 
 Hutton says, where L and I represent two corresponding dimen- 
 sions of the end rectangles, B and b the others, and h. the bight or 
 length of the prismoid, 
 
 Then, 
 
 (217+1 X B + 27TL X b) X t h = Solidity, 
 
 which is the particular rule, . . . . . .- . (IV.) 
 
 A note, on page 163, referring to this, 
 says: 
 
 "It is evident that the rectangular 
 prismoid is composed of two wedges, 
 whose bases are the two ends of the 
 prismoid, and whose hights are each equal 
 to that of the prismoid." 
 
 It might be added, that the edges of 
 these two wedges are formed by two 
 diagonally opposite sides of the rectangu- 
 lar ends. 
 
 Hutton notes also, 
 
 That - - = M, and - - = m, the sides of the mid-section, so 
 
 _ L 
 
 that the correspondence of the General and Particular Kules becomes 
 evident. 
 
 (a.) At page 164 of the quarto Mensuration, cited above, 
 
 reference is made to the General Rule as follows : 
 
 " This rule will serve for any prismoid or cylindroid, of whatever 
 figure the ends may be, inasmuch as they may be conceived to be com- 
 posed of an infinite number of rectangular prismoids. Which is the 
 General Rule." 
 
 This method of considering any prismoid to be composed of a great 
 number of rectangular prismoids, of the same common length, has pre- 
 vailed from Hutton's time down to the present day. 
 
 Thus, we find in Davies Legendre,* chapter on the Mensuration 
 
 * Davies Legendre. (1853), 8vo : New York. 
 
14 
 
 MEASUREMENT OF EARTHWORKS. 
 
 of Solids, in treating of prismoids, where he copies Hutton's figure, 
 and both Particular and General Rules, the following : 
 
 " This rule (the general one} may be applied to any prismoid what- 
 ever. For whatever the form of the bases, there may be inscribed in 
 each the same number of rectangles, and the number of these 
 rectangles may be made so great that their sum in each base will 
 differ from that base by less than any assignable quantity. Now, if 
 on these .rectangles rectangular prismoids be constructed, their sum 
 will differ from the given prismoid by less than any assignable quan- 
 tity. Hence, the rule is general." 
 
 In his remarkable chapter on the cubature of curves (Mens., part 
 iv. page 457), Hutton shows that the prismoidal formula is applica- 
 ble to the frusta of all solids 
 generated by the revolution 
 of a conic section (as well 
 as to the complete solids); 
 also, to all pyramids and 
 cones, and in short to all 
 solids (right or oblique), of 
 which the parallel sections 
 are similar figures. 
 
 We will now illustrate 
 Hutton's Rules, by means 
 of a figure and examples, to 
 find the solidity of a pris- 
 moid, with very dissimilar 
 (See Fig. 4.) * 
 
 Kg. 4. 
 
 1. By General Rule* 
 
 40 X 30 = 1200 = b. 
 80 X 4 320 = t. 
 60 X 17 X 4 = 4080 = 4 m. 
 
 6)5600 
 
 Multiplied by ll 
 Solidity 
 
 60 
 
 56000 
 
 2. By Particular Rule. 
 
 As -two Wedges. 
 
 40 80 
 
 2 2 
 
 160 
 40 
 
 200 
 4 
 
 800 
 10 
 
 48000 
 8000 
 
 8000 
 
 Solidity 56000 of whole pris- 
 moid. 
 
CHAP. I. PRELIM. PROBS. ART. 3. 
 
 15 
 
 3. By means of Initial Prismoids ...... (V.) (To be further explained.) 
 
 (1) Areas of ends, b = 1200, and t = 320. 
 , ON f Rights = 30 ) , = 4 K 
 
 (3) Assumed squares in larger end, 1200 of 1 X 
 
 t 320 
 
 (4) Ratio of ends, - = =-2667. 
 
 UNIT E R S I T 
 
 80 
 
 (5) Proportional rectangles in small end (1200 in number), 
 
 40 
 
 2, 
 
 = -13333, 2 X '13333 
 
 oO 
 
 '26667 = area of these, being equiva- 
 lent to the ratio of the ends 1 to '2667. [See (4).] 
 
 1-1-2 
 
 (6) Mid-section, dimensions of proportional rectangle, -- = 1 
 
 1 + -13333 
 
 5667, and 1 '5 X '5667 = '85 = rectangular area of 
 
 b' =1 X 1 
 
 mid-section of initial prismoid. 
 
 Then for the solidity of the initial prismoid, by General 
 Rule. 
 
 Call these areas 
 b', m', and t', to 
 distinguish them 
 
 4m' = -85 X 4 . = 3-4 
 
 f = -13333 X 2 = -26667 
 6) 4-66667 
 
 Mean area, . . . . = -77778 
 Multiplied by h . . = 60 
 
 Volume of one, *= 46-66680 
 
 Mult, by No. initial prismoids, assumed = 1200 
 
 (7) 
 
 from those of the 
 main solid. 
 
 Solidity of the whole prismoid, as above = 56000-16000 
 In computing initial prismoids it is necessary to em- 
 ploy sufficient decimals, but 4 or 5 places are usually 
 enough. 
 
 (b.) These initial prismoids are supposed to be constructed upon 
 
 small rectangles in the two ends, equal in number in each, and of pro- 
 portional areas. 
 
 In the base, or larger end (though either end may be used), it will 
 be most convenient to assume these to be squares formed upon the 
 unit of measure, while at the top they must be rectangles proportional 
 both in dimensions and area, by the view we have herein taken (as 
 indicated at (5) above). 
 
16 
 
 MEASUREMENT OF EARTHWORKS. 
 
 The end areas of the main prismoid being always given, or com- 
 putable, they must be proximately reduced to rectangles before we 
 can properly apply the principle of initial prismoids to calculate, or 
 verify, their solidity ; and the solid will then become, in effect, a 
 rectangular prismoid like those of Simpson and Hutton. 
 
 In doing this, it will be sufficient to dermine a width and hight, 
 apparently proportional to the shape of the cross section (which in 
 some species of earthwork is extremely irregular), but this hight 
 and width must be such that, used as factors, they reproduce the 
 given area, even though of themselves they may not be exactly geo- 
 metrical equivalents, for the dimensions of the section. 
 
 Having thus (as it were) rectified the solid proximately, we may 
 proceed with it as a rectangular prismoid, by the method of initial 
 prismoids, briefly as follows : Determine the rectangular hights and 
 widths, such as will proximate the figure, and by multiplication reproduce 
 the areas. Assume one end as base, to be divided into squares of super- 
 ficial units, and the others into proportional rectangles; upon these con- 
 struct (or imagine) ini- 
 tial prismoids, and having 
 ascertained the volume 
 of one, multiply by num- 
 ber, for solidity of main 
 prismoid, as shown in de- 
 tail above. . . . (V.) 
 
 (C.) We will 
 
 further illustrate tiiis 
 subject by presenting an 
 outline of a T-shaped 
 prismoid ; a solid (Fig. 
 5), with a figure so pecu- 
 liar that none of the 
 usual methods of averag- 
 ing could even proximate 
 
 its solidity, which 
 
 can only be dealt with by the Prismoidal Formula, or some cog- 
 nate rules. 
 
 This we will calculate as a prismoid by Simpson's General Rule, 
 by Hutton's Particular Rule, and by the Method of Initial Prismoids. 
 
 nu&seo 
 
CHAP. I. PRELIM. PROBS. ART. 3. 
 
 17 
 
 By Hutton's Particular Rule. 
 
 100 
 2 
 
 200 
 
 208 
 6 
 
 1248 
 100 
 
 As two Wedges. 
 
 8 
 
 2 
 
 16 
 
 100 
 
 116 
 50 
 
 5800 
 100 
 
 6) 124800 6 ) 580000 
 
 20800 96666J 
 20800 
 
 Solidity = 117466S 
 
 By Simpson's General Rule. 
 
 As a Rectangular Prismoid. 
 Hts. Wds. 
 6 X 100 . . = 
 
 50 X 8 . . = 
 
 Sums, 56 X 108 = 
 4 times mid-sec. 
 
 600 
 400 
 
 6048 
 7048 
 
 Solidity,. . . = 1174661 
 
 By the Method of Initial Prismoids. Let their number be 400, the 
 same as the superficies of A. Suppose them constructed upon 
 squares at A. (on a side equal to the unit of measure), and upon pro- 
 portional rectangles at BC. 
 
 Then, 600 -*- 400 = 1'5, the ratio of A. to BC. and of initial squares 
 at one end to rectangles at the other. 
 
 And in the 3 main sections of the prismoidal solid, Fig. 5, 
 We have for similar sections of the initial prismoids = 
 
 Representative. Dimensions of initial sections. Initial areas. No. Alain areas. 
 
 End A . . . = squares of 1 X 1 . . . . = 1' X 400 = 400. 
 
 " BC . . = propor. rectans. 12'5 X '12 = \o X 400 = 600. 
 
 Mid-section . = " " 6'75 X '56 = 3'78 X 400 = 1512. 
 
 It will be seen that the main areas result as above calculated ; and 
 having these and the common length h., it is easy to compute the pris- 
 raoid by Simpson's General Rule, as shown before. 
 
 We may add here, as being indicative of the difficulty of comput- 
 ing such a solid, by ordinary average rules (which answer tolerably 
 well), in common cases. 
 
 That the Arithmetical Mean of the end areas = 500, the Geomet- 
 rical Mean = 490 ; while the Prismoidal Mid-section = 1512, and 
 the Prismoidal Mean Area = 1174s; which, multiplied by the length, 
 or hight, h. = 100 : makes the solidity, above = 1174662, or more 
 than twice as much as would result from multiplying the arithmetical 
 mean by the length. 
 
18 
 
 MEASUREMENT OF EARTHWORKS. 
 
 4. The Prismoid adapted to Earthwork. Sir John Macneill, a dis- 
 tinguished English engineer, as early as 1833, soon after the intro- 
 duction of railroads, when the necessity became apparent of having 
 ready and correct methods at hand for computing the volume of the 
 vast quantities of earth, removed or supplied, in grading them, pre- 
 pared and published three series of Tables (in 8vo), computed by 
 rn^ans of the Prismoidal Formula. These Tables were systematically 
 arranged, and have been extensively used abroad. 
 
 He considered the Earthwork Prismoid as being composed of a 
 Prism, with a wedge superposed : since the lower portion of the cross 
 section of a railroad, canal, or road is generally symmetrical and 
 regular, the ground surface alone being relatively variable. 
 
 In this diagram (Fig. 6) the reduced surface of the ground (taken 
 as level, crosswise, or made so) is shown by the plane AFGE, and 
 the cross section of the road by ABCG, these are supposed to be 
 transparent, in order to show the road-bed and mid-section, as well as 
 the far end of the trapezoidal prismoid. 
 
 Sir John Macneill commences his work, by referring to a represen- 
 tation of the Earthwork Prismoid (copied above), as follows : 
 
 " Let ABCGFKDE represent a prismoid or solid figure, similar 
 to that which is formed in excavations or embankments, in which 
 BCDK represents the roadway, and ABCG, FKDE, parallel cross 
 sections at each end. The cubic content of this solid is equal to 
 
CHAP. I. PRELIM. PROBS. ART. 4. 19 
 
 The area ABCG -j- area FKDE -f 4 times area a beg, 
 Mutipliedby^R: 
 
 " If, then, we suppose a plane, HIEF, to be drawn through the 
 lines HI, and EF, it will be parallel to the base BCKD, and will 
 divide the solid, ABCGFKDE, into two others, one of which will be 
 the regular prism, HBCIFKDE, and the other will be a wedge, the 
 base of which will be the trapezium, AHIG, the length IE or CD, 
 the length of the prismoid, and the edge FE, the breadth of the cut- 
 ting at the lower end of the section." 
 
 The prismoid, then, being assumed as composed of a regular prism, 
 with a wedge superposed, he demonstrates in the usual manner the 
 formula for the volume of these two solids, and shows that by addi- 
 tion they result in the Prismoidal Formula, which he uses in the com- 
 putation of the three series of Tables .which form the bulk of his neat 
 octavo volume (London, 1833). 
 
 It will be observed that all Macneill's prismoids refer to ground 
 sloping longitudinally, but level transversely: to apply them, there- 
 fore, to an irregular surface, it must be first reduced to a level cross- 
 wise, or assumed to be so, practically. 
 
 The above extract from Sir John Macneill's work of 1833 is made, 
 not only for its intrinsic value, but on account of its being the first 
 regular and successful attempt to adapt the Prwmoidal Formula to 
 the computation of modern earthworks: which is followed out through 
 a series of practical Tables, comprising 239 pages, and extending to 
 50 feet of hight or depth : an embankment being considered as an 
 excavation inverted. 
 
 This meritorious work of Sir John Macneill was speedily followed 
 by other writers in England, and later by several in this country.* 
 All, or most of these productions being based upon the Prwmoidal 
 Formula (or some modification of it), which is now universally 
 acknowledged to be the only consistent and exact method for com- 
 puting the volume of solids employed in modern earthworks, and 
 even those authors who employ pyramidal rules are but using a par- 
 ticular case of the former. 
 
 * Bidder, Baker^ Bashforth, Henderson, Sibley, Rutherford, Hughes, Huntington, 
 Law, Dempsey, Haskoll, Morrison, Rankine, Graham, Macgregor, and others, in England. 
 While in this country, Long, Johnson, Borden, Trautwine, Gillespie, Henck, Davies, P. 
 Lyon, Cross, M. E. Lyons, Byrne, Warner, Rice, and others (besides the present writer), 
 have dealt with this subject. Amongst these, however, the most comprehensive, and the 
 best in many particulars, is the work of John Warner, A. M., a well printed and hand- 
 somely illustrated 8vo, Philadelphia, 1861, containing 28 valuable and useful Tables, and 
 14 plates of great importance to every student of engineering. 
 
20 
 
 MEASUREMENT OF EARTHWORKS. 
 
 5. The Prismoid in its Simplest Form. The unexpected manner in 
 which the Prismoidal Formula applies to the cubature of other solids, 
 totally 1 dissimilar in form and appearance (as to the sphere, taking 
 the poles as end sections at zero, and the mid-section as a great circle), 
 justifies its consideration under various aspects, which would be 
 superfluous in any other body, and hence we give below a figure 
 illustrating the Prismoid, in what may be deemed its simplest form 
 (when not contained within a diedral angle). See Fig. 7, where the 
 solid is level transversely, but sloping longitudinally, and may be 
 supposed to represent (proximately~) one of Button's Initial Prismoids, 
 square at one end, and with a proportional rectangle at the other. 
 
 1 
 
 I 
 
 
 B 
 
 3 
 
 4 
 
 Fi&7. 
 
 """*"" -~^__m. 
 
 B 
 
 A. 
 
 - rrrr^fl 
 
 *i A. ^ i 
 
 a. 
 
 6 1 
 
 Here the prismoid is composed of a prism on a square base, with a 
 side of 1, and length of 6, and of a wedge, superposed, with a square 
 back, on a side of 1, its edge also 1, and hight 6, the common length 
 of the two combined as a prismoid. 
 
 )AA Represent the prism. 
 BB The wedge. 
 m The mid-section of the prismoid. 
 
 Then we have for the volume of this solid, by several of the rules 
 already given. 
 
 Formulas. 
 
 Cubic ft 
 
 (I.) (1 X 2) + (1 X 1) + [(1 + 1) X (2 + 1)] X -g- =9 
 (II.) (2 X 2) + (2 X 1) + (1-5 X 1 X 8) -4- 12 X 6 . . = 9 
 (III.) 2 + 1 + (1-5 X1X4)X-|- -9 
 
 (IV.) (2X1 + 1 X 2) + (2X1 + 1 X 1) X |- . = 9 
 
 Divided ( Prism =1X1X6 =6"J 
 
 .=3^= 9 
 
 All, of course, resulting in the same solidity for this simple pris- 
 moid = 9 cubic feet. 
 
CHAP. I. PRELIM. PROBS. ART. 6. 
 
 21 
 
 6. Further Illustration of MacneilUs Prismoid. In computing the 
 quantities of earthwork for railroads, etc., it is often useful (and gen- 
 erally desirable) to consider the side slopes, continued to their 
 intersection, above or below the road-bed (as has been done by T. 
 Baker, C. E.,* and other writers), thus forming a constant triangle 
 at the intersection, which is deductive from the general triangular 
 figure formed by the slopes, and ground, in order to obtain the regu- 
 lar cross section of excavation or embankment, from ground to grade ; 
 and this triangle also forms the right section of the grade prism, ter- 
 minating the earthwork solid at edge of diedral angle, formed by the 
 side slope planes containing it. 
 
 To explain this more clearly, we give a figure in which both end 
 areas are drawn upon the same plane (Fig. 8). 
 
 Double cross section of a railroad cut (in fact, Macneill's pris- 
 moid on level ground) with road-bed of 20, and slopes of 1 to 1. 
 
 Prism. 
 
 G^Prism. 
 
 lot o/ slopes. -10 
 
 References. 
 
 A = Altitude of grade triangle. 
 
 B = Level top, sloping forward in 100 feet to b. 
 
 b = Level top of forward cross section. 
 
 G = Grade, or road-bed, 20 feet wide. 
 
 c = Grade triangle, or constant end, of grade prism. 
 
 H h = Breadth of back of trapezoidal wedge. 
 
 r = Slope ratio, or in this case 1. 
 
 * Railway Engineering and Earthwork, by T. Baker, C. E. London, 1840. Wherein 
 he develops a very compendious and excellent system of computing the earthwork of 
 railways, which has been extensively copied. 
 
22 MEASUREMENT OF EARTHWORKS. 
 
 CC = Centre line of road. 
 
 I = Intersection of side slopes, or edge of diedral angle formed 
 by them. 
 
 To find the equivalent level hight no matter how irregu- 
 lar the ground may be. 
 Let 
 
 a = Whole area, to the intersection of slopes. 
 
 r = Slope ratio. 
 
 h = Equivalent level hight. 
 
 Then, \/ = h. 
 r 
 
 Let B and b represent the level tops of two cross sections of a rail- 
 road cut, 100 feet apart sections, and lying within the same diedral 
 angle of 90, formed by side slopes of 1 to 1, continued to their inter- 
 section, or edge at I. 
 
 Now, supposing B and b, to have been originally a very irregular 
 surface, reduced, by any exact method, to the level tops represented. 
 
 Then, below b we have a regular prism, .on a triangular base, 
 extending down to I ; and above b, a regular wedge (back and edge 
 parallel), upon a trapezoidal back, of which the base b is equal to the 
 edge b, representing the top of the forward cross section, 100 feet 
 distant. 
 
 Then, in the wedge above b, by the properties of that solid, consid- 
 ered as * a truncated triangular prism, and applicable either to rectan- 
 gular or trapezoidal wedges, 
 We have, 
 
 Mean Area. 
 
 (B + 6-f 6)X(H 7<) (44 '+ 32 +32) X (22 16) 
 
 -g- -g- . 108. 
 
 And in the prism beloiv b, down to I (including the grade 
 triangle) 
 
 We have, 
 
 (/j, 2 r) = 256. 1 
 
 Deduct the grade triangle = 100. f = 15 - 
 
 Leaves area of prism (above grade) from G to b = 156. 
 
 Finally, then, we have the mean area of the trapezoidal 
 earthwork solid, above grade, or road-bed = 264. 
 
 Cubic Ft. 
 
 Then, 264 X 100 = 26400. The solidity of this Prismoid. 
 
 * Chauvenet's Gcom., vii. 22 (1871), easily reducible to the text. 
 
CHAP. I. PRELIM. PROBS. ART. 6. 23 
 
 If more convenient, we might exclude entirely the grade triangle, 
 and stop the calculation at G (the road-bed), but as a system of com- 
 putation, and in view of the simplicity of the geometrical relations of 
 triangles, it will usually be found best to include the grade triangle 
 as above, and ultimately to deduct it, in some form. 
 
 The employment of the method of this article enables us to find a 
 mean area to the prismoid without using a mid-section and this 
 mean area, when multiplied by the length, gives the volume of the 
 whole solid. 
 
 Thus we may assume any level trapezoidal prismoid of unequal 
 parallel ends (as Macneill does), to be composed of two solids a 
 prism, with a wedge superposed. 
 
 1. A Triangular Prism, with a cross section, equivalent to the 
 
 lesser end, supposing the slopes to intersect, and embracing 
 the grade triangle. 
 
 2. A Trapezoidal Wedge, superposed upon the prism, having an 
 
 area of back equivalent to the difference of the ends, its 
 edge being the level top of the smaller, and equal to the 
 base of the back. 
 
 The length being common to both partial solids, and to the whole 
 prismoid. 
 
 Then, for the mean area of the wedge, we have, 
 (B + & + ft) X (H A)* 
 
 6 
 
 and for that of the prism to intersection of slopes (A 2 r grade 
 triangle), and by addition ,f 
 
 the common length = The Solidity of the Prismoid .... (VI.) 
 
 Or, in words, The sum of the mean areas of the prism, and super- 
 posed wedge, multiplied by the common length, equals the solidity of this 
 prismoid. 
 
 * Chauvenet's Geom., vii. 22 (1871). 
 
 f B and b are always the widths between top slopes at the ends. 
 
 And H h (however irregular the ground line of the ends may be) is obtained by 
 
 dividing the difference of end areas by half the sum of their top widths, or ( + V 
 See note at foot of this Article 6. 
 
24 MEASUREMENT OF EARTHWORKS. 
 
 Note. When the ground surface, or upper side of the superposed 
 wedge, is very irregular (as in Figs. 43 and 44) ascertain the hori- 
 zontal widths of -each end at top slope. Then the difference between 
 the areas of the two ends is the surface of the back of the superposed 
 wedge, and this, divided by the average of the two horizontal widths 
 above, gives the vertical hight of the back, or altitude of the trian- 
 gular section, of which the length of the prismoid is the base, giving 
 at once the means of computing its area, and this, multiplied by one- 
 third of the sum of the lateral edges, gives the solidity of the superposed 
 wedge. (Chauvenct, Geom.,vii. 22.) 
 
 7. Trapezoidal Prismoid of Earthwork, considered as two Wedges. 
 On ground, either level crosswise, or reduced to an equivalent level 
 by any correct process, an Earthwork Prismoid, within the limits of 
 its slopes, road-bed, and ground surface, may readily be computed aa 
 two wedges (Hutton's Particular Rule), without an assumed mid-sec- 
 tion, or even the end areas. 
 
 And in this there is some advantage, as the width of road-bed at 
 the end sections may be unequal to any extent, provided the widening 
 is gradual. 
 
 Thus, let Fig. 9 represent a regular station of a railroad cut, 100 
 feet in length, with slopes of 1 to 1, and in the near end section a 
 depth of 40 feet, and road-bed of 20, while in the far one it has a 
 depth of 30, and road-bed of 40 feet wide. 
 
 Hutton's Particular Rule, modified for application to earthwork, 
 may be expressed in words at length as follows : 
 
 Rule. 
 
 Add road-bed -f- top width + road- 
 
 r ., . ,. I bed of 2d section; multiply the sum 
 
 In 1st cross section -< v i i i ^ 
 
 of these three by level hight of sec- 
 tion, and reserve the product. 
 
 Add road-bed -j- top width + top 
 
 T m I width of 1st section ; multiply the sum 
 
 In Id cross section - xl - , , , , \ *_ 
 
 of these three by level hight of sec- 
 tion, and reserve the product. 
 
 Finally, add the two products reserved, and i of their sum is 
 the mean area of the Prismoid, which, multiplied by length = 
 Solidity (VII.) 
 
CHAP. I. PRELIM. PROBS. ART. 7. 
 
 25 
 
 Eeferring to Fig. 9, the line CC is the centre line traced upon the 
 ground, and below it the road-bed gradually widened from 20 to 40 
 feet, in the length of 100 ; the figures marked show the dimensions 
 assumed for illustration, and the dotted lines the edges of a plane 
 supposed to be passed, so as to convert this solid into two wedges. 
 
 The nearest having a trapezoidal back, standing on a road-bed of 
 20, with a hight of 40, and its edge being the road-bed of 40 feet 
 wide, belonging to the far cross section. 
 
 The farthest wedge, above the dotted lines, having for its- back the 
 
 far section, standing on a road-bed of 40, with hight of 30, and its 
 edge being the top-width of the near cross section, 100 feet wide, at 
 ground line. 
 
 [In Chapter 5 we shall consider further, and more in detail, the 
 subject of Wedges ; and their application to the computation of earth- 
 work solids, and illustrate it by several examples. Comparing also 
 the results obtained with those derived from the use of BUTTON'S 
 General Rule: which is the accepted standard for accuracy in such 
 work.] 
 
26 
 
 MEASUREMENT OF EARTHWORKS. 
 
 EXAMPLE. 
 
 By Our Modification of Hutton's 
 Rule (VII.) 
 
 In 1st cross section 
 
 In 2d cross section 
 
 20 
 100 
 40 
 
 160 
 40 
 
 6400 
 
 40 
 100 
 100 
 
 240 
 30 
 
 7200 
 
 6400 
 7200 
 
 6)13600" 
 
 Mean Area = 
 
 2266-67 
 100 
 
 By Button's Particular Rule. (IV.) 
 
 Reducing Trapezoids to Rectangles. 
 
 Mean breadths = 
 
 60 
 
 2 
 
 120 
 40 
 
 160 
 40 
 
 70 
 
 2 
 
 140 
 100 
 
 240 
 30 
 
 6400 7200 
 
 6400 
 
 7200 
 
 Solidity . . 
 
 13600 
 100 
 
 6)1360000 
 226667 
 
 Solidity . . = 226667'00 
 
 8. Areas of Railroad Cross-sections (within Diedral Angles] 
 whether Triangular, Quadrangular, or Irregular. 
 
 All railroad sections are contained within diedral angles, formed by 
 side slope planes, of a given divergency determined by the slope 
 ratio (r). The edge of this diedral angle is a right line, parallel to 
 the grade, and prolonged forward indefinitely from I, the intersection 
 of the side slopes (in a right section), until the end of the cut or fill 
 is attained. Here, at the grade point, it changes its position to a 
 corresponding parallel above, or below, as the case may be. Consid- 
 ering, with Sir John Macneill, an embankment to be, in effect, an 
 excavation inverted, the situation of the edge of the diedral angle, or 
 intersection of the slopes, will generally (in our examples) be found 
 below the road-bed, but always parallel to the grade line, and at the 
 same distance from it, as long as the side slopes continue uniform. 
 
 (a.) From the geometrical relations of triangles and rect- 
 angles, it is obvious that in a triangle situated as in Fig. 10 con- 
 
CHAP. I. PRELIM. PROBS. ART. 8. 
 
 27 
 
 tained within rectangular axes and their parallels, and divided into 
 
 two by the central axis h, the area of the whole is equivalent to 
 
 2i. 
 
 the parallels a and b, to the centre line h, limiting the triangle 
 laterally. 
 
 The same rule, precisely, applies to quadrangles, which may always 
 be cut by a diagonal into two triangles. 
 
 This rule (in fact), equally applicable both to triangles and trape- 
 ziums, is that laid down by Hutton (1770) for trapeziums. 
 
 In Fig. 10, h X w = double area of the whole triangle, whose ver- 
 tex is at I, the intersection of the slopes, and its sides, the side-slopes, 
 and the ground line. Thus, let h = 20, w = 45, then 20 X 45 = 
 900 -f- 2 = 450, area of whole triangle ; but it is often more conve- 
 
 nient, in calculations, to use double areas alone, until the close of the 
 operation, as in many problems of land surveying. 
 
 In a triangle, the direct axes h or h' may take any position, pro- 
 vided the parallels through the lateral vertices are made to follow, 
 and the tranverse axes, w and w', remain rectangular. 
 
 But in a quadrangle, the position of the direct axis is fixed by that 
 of the opposite vertices, through which it passes, and with it the axis 
 of width, and its limiting parallels, are also fixed. 
 
 In Fig. 10, suppose the direct axis and its parallels to revolve upon 
 I, into the position h', and that h' becomes 22*1 then it will be found 
 
 thatuf has become40'73, 
 
 will be 22>1 X 40 ' 73 450j 
 
 area of whole triangle, as before. 
 
28 
 
 MEASUREMENT OF EARTHWORKS. 
 
 In both these cases, Figs. 10 and 11, each figure is divided by the 
 centre line, or direct axis, into two triangles, having a common base, 
 and contained between parallels to it, drawn through the opposite 
 vertices. 
 
 In both Figs. 10 and 11, h X w = double area of the figure to 
 which they relate, as these are rectangular factors, for determining 
 the content of the wholly or partially circumscribing rectangles 
 (between the same parallels), of which the triangle or trapezium 
 represented, is each equivalent to one-half. 
 
 This rule is, in fact, the simplest possible, being, substantially, the 
 definition of a plane surface, length X breadth (which indicates 
 superficial extension), and from its extreme simplicity, there seems to 
 
 be no adequate reason why it should not be more generally employed, 
 for although its application to ^triangular surfaces necessarily gives 
 double areas, a division by two is the briefest imaginable. 
 
 Right and left of centre each triangle is obviously equal to half the 
 rectangle of the hight and width on that side (the triangle and rect- 
 angle having a common base, and lying between the same parallels, 
 a and b), and by addition, the double area of the whole trapezium = 
 hight X width. 
 
 (b.) In view of the rule just recited, for finding the areas 
 
 of triangles and trapeziums, by hights and widths, it becomes of some 
 importance to have a concise rule* for determining the distances out 
 of the vertices from the axis, when the hight and slopes alone are 
 
 * GHlespie, Roads and Railroads (1847), gives rules analogous to ours, but they had 
 long before been kiwion. 
 
CHAP. I. PRELIM. PROBS. ART. 8. 
 
 29 
 
 given : in this there is little difficulty, as engineers have long been 
 possessed of formulas for the purpose, similar to those which will be 
 seen below, referring to Figs. 12 and 13, and these distances out, when 
 added together, form the width w, of the rule above. 
 
 In Fig. 12. 
 
 Ht. TTid. 
 
 40 X 60-8 
 
 2432 
 2 
 
 Area. 
 
 1216. 
 
 Both in trapeziums and triangles the diagonal X the sum of per- 
 pendiculars from the opposite angles = double area. 
 Or, centre Light X the total width = double area. 
 
 Suppose, in both these figures, the side-slopes, ground-slopes, and 
 centre hight, or axis, given, and the side-slopes intersected at I, then 
 to find the distances out, right and left of cejitre, take each side sepa- 
 rately. Consider the centre line, or axis, to be a meridian (as in a 
 map), imagine also an east or west line, drawn through the origin of 
 each slope (side or ground). 
 
 Then, 
 
 If the slopes incline towards the same compass quarter : 
 
 Hight 
 
 ^5 jr& j p-j = distance out = cL 
 
 By difference of nat. tans, of slopes 
 
 If the slopes incline towards adjacent compass quarters: 
 
 Hight 
 
 ^ -7 F~l = distance out d. 
 
 Uy sum oi nat. tans, of slopes 
 
 These results on both sides of centre, added together, give the total 
 width of the whole trapezium. 
 
30 
 
 MEASUREMENT OF EARTHWORKS. 
 
 In Fig. 13. 
 
 Ht. Wdt. Area. 
 
 30 X 88-2 2646 
 2 " 2 * 
 
 1323. 
 
 These rules also furnish a concise and easy method of finding the 
 half breadths, a matter deemed quite important by foreign engineers. 
 
 (C.) The side slopes (bounding the diedral angle) remain- 
 ing plane surfaces as usual in the cross-sections of earthwork, we 
 sometimes find the ground surface very irregular, but even these 
 cases, upon the principle of equivalency, may be correctly dealt with, 
 so as to reduce them easily to the plane figures of the elements of 
 geometry. 
 
 Thus, although, as far as we have shown, the rule of , applies 
 
 only to a line once broken, so as to change the figure considered, from 
 an oblique triangle into a trapezium ; nevertheless, it is not difficult 
 to reduce or equalize a surface line, very much broken, by a single one 
 properly drawn, which shall contain within it an area exactly equal to 
 that bounded by the irregular outline, and thus bring it within the 
 rule. 
 
 In Fig. 14, let ABCDEFGH be the cross-section of a rail- 
 road cut, base 20, slopes 1 to 1, intersecting at I, the centre line being 
 marked CC (this area looks irregular enough, but had it been ten 
 times more so, the process below would have equalized it exactly.') 
 
 Then, from the top of the shortest side hight at H (adopted for 
 convenience), draw a line HK parallel to the road-bed, or base AB, 
 
CHAP. I. PRELIM. PROBS. ART. 8. 
 
 31 
 
 making a level trapezoid 10 feet high upon the section, or ABKH = 
 300 in area. 
 
 Now, we will find, by a common calculation, the area of the whole 
 cross-section between base AB, side slopes, and broken ground line 
 to contain = 654 area. Neglecting in this case the grade triangle at 
 I, as being a common quantity, not affecting the result : (but adding 
 the grade triangle (100), the area, from the ground line down to the 
 edge of the diedrul angle at I = 754). 
 
 Then, 654 300 = 354, the area of the partial cross-section above 
 UK, extending to the irregular outline, which is to be correctly equal- 
 ized, by a single sloping line drawn from H. 
 
 40 
 
 7- 
 
 Clrcum:Gr: 
 n-S 
 
 Area I 
 
 *Tr* xT!al:Ia.Cir. 
 
 Now, = 17-7 
 
 LM, the altitude of a triangle HKM, on 
 
 the base HK, which is exactly equivalent in area to the partial cross- 
 section above HK. 
 
 So that HM is a single equalizing line, drawn from H, equivalent 
 to the broken line of ground, and including the same area exactly. 
 Another way of finding the point M the terminus of the equaliz- 
 
 f Double area = 1508 IM 
 ing line is the following : \ = 53'3 ] and 
 
 ( IHXsin.ofI 
 this is a very concise method, as IH is easily found.* 
 
 VI) 
 
 4,, 
 
 * This rule will be found useful as a verification of the process of Fig. 14. 
 
32 MEASUREMENT OF EARTHWORKS. 
 
 If the degree of equivalent surface slope be desired (as it usually is), 
 
 Then, ^- = cot. 17 (nearly) = 3'26. 
 
 The slope of the equalizing line HAI being 17 ascending from H, 
 we easily find FN =6'135, and adding FI = 20, we have IN or h = 
 
 26-135, and w = 57'7. Then, 
 
 h X iv = 26-135 X 57-7 
 
 754, and 
 
 deducting the grade triangle (ABI = 100), we have, finally, the area 
 of the whole cross-section above the road-bed = 654, thus verifying 
 
 the original calculation as before given, and, by using the radii of 
 inscribed and circumscribed circles, we can prove it, if necessary : 
 
 (Fig. 14). 
 
 (d.) It is sometimes desirable, by means of an equalizing 
 
 line, to deal with the boundary alone, without the rest of the cross- 
 section, and this is not difficult, for we may consider the broken line 
 HKM (Fig. 14), or aeg (Fig. 15), as a base of ordinates, preserving, 
 however, their parallelism, and taking all the distances horizontally 
 as though the base were straight (see Fig. 15) ; but the process of Fig. 
 14 is generally preferable. 
 
CHAP. I. PRELIM. PROBS. 
 
 , 
 
 Q& U ^ 
 
 It is often useful to equalize a section by a level top line, or slope 
 o/0. This can be done as shown in Art. 6. 
 
 Whole area = a. 
 
 Slope ratio = r. 
 
 Level hight = h. 
 
 Then h 
 
 The ordinates marked upon Fig. 15 are deduced from those of 
 Fig. 14, and the calculations of the irregular area, a eg, are made by 
 successive trapezoids, and double areas, as follows : 
 
 Ordinates in f a _}_ J b -\- C C + d d + e 6 +/ / +^ 
 
 Ease 3 Hne7ai!l + 5 5 + 2 2 + 6 6 + 16 16+16 16 + 
 broken at e. . . . ( 5 7 8 22 32 16 
 
 10 10 10 10 4 10 
 
 50+70 + 80+220+128 + 160 
 
 Then,* 
 
 Sum of double areas = 708 
 
 T5 1- r^ = \ -77. = 1 r7 = ft A:, as before. 
 
 .base or equalizing triangle, a e = 40 
 
 And ak is the equalizing line, ascending from a, with a slope of 
 17, which is equivalent to HM, of Fig. 14. 
 
 (6.) We may now briefly refer to the computation of cross- 
 
 Horizontal distances 
 part 
 
 Double areas (total 
 70S) 
 
 sections. These are usually taken in the field with the rod, level, and 
 tape; they designate by levels, and distances out, the prominent 
 
 * With equal abscisao, Simpson's well-known rule, or that of Davies Legendre, would 
 conveniently apply. 
 
 3 
 
34 MEASUREMENT OF EARTHWORKS. 
 
 points, or features of the ground, and fix the intersection of the side 
 slopes, or place of the slope stake, which bounds the limits of excava- 
 tion or embankment ; and on regular ground, the clinometer may be 
 used, but is less correct and satisfactory. 
 
 On plain ground, but three levels are taken, the centre and side 
 hights, and this has been called three-level ground. It is the prac- 
 tice of many engineers (and it is a good one) to take angle levels and 
 distances over the edges of the road-bed, this, then becomes five-level 
 ground; and where more than five levels are necessarily taken, the 
 cross-section is usually deemed irregular, though the point where 
 sections become irregular is not well defined, and may be safely left 
 to the judgment of the engineer. 
 
 In this case (Fig. 16), the centre and side hights, and the right and 
 left distances out to the slope stakes, are always given, and the calcu- 
 lation becomes simple and rapid. 
 
 The following is the method long ago used by engineers, and pub- 
 lished by Trautwine * and others, twenty years since. 
 
 KULE for area of cross-section, with uniform road-bed and centre 
 and side hights given. 
 
 Half the centre cutting X by right and left distance, plus right 
 and left cuttings X one-fourth of road-bed. 
 
 Thus, in Fig. 16, 
 
 We have, by this rule, 
 
 5 X 64 = 320. 
 44 X 5 = 220. 
 
 Area. . = 540. 
 
 And by using the grade triangle and 
 hights and widths, as in Figs. 10 and 11, 
 
 We have, 
 
 20 X 64 
 
 .=640. 
 
 w 64. J Less grade triangle . = 100. 
 Area. . = 54a 
 
 (f.) To find the area of cross-sections, where angle levels 
 
 have been taken,f or Jive-level ground (which angle levels have long 
 been used by engineers, and are recommended by Prof. Davies in his 
 new surveying), we will give an example for illustration, from which 
 the rule of this method will be evident. (See Cross, Eng. Field 
 Book, N. Y., 1855.) 
 
 * Trautwine's New Method of Ex. and Em. (1851). 
 
 f Davies' New Surveying (1870), cross-section levelling. 
 
CHAP. I. PRELIM. PROBS. ART. 8. 
 
 35 
 
 Now, to calculate the area of this cross-section, Fig. 17, by double 
 areas, 
 
 Equivalent to, 
 Triangle, 15 X 10 
 
 We have, 
 
 By divid- 
 
 20 X 15 = 300. 
 
 ing the figure 
 
 20 X 12 = 240. 
 
 into six trian- 
 
 34 X 16 = 544. 
 
 gles, or three 
 
 2)1084. 
 
 trapeziums. 
 
 Area. = 542. 
 
 Trapezoid, 27 X 10 
 
 Triangle, 
 
 28 X 10 
 16 X 24 
 
 = 150. 
 
 = 270. 
 
 = 280. 
 
 = 384. 
 
 2)1084. 
 
 Area. 
 
 = 542. 
 
 To compute this area in the usual method by successive trapezoida 
 and deductive triangles, is much longer and less satisfactory. 
 
 -i<* 
 
 (g.) For very irregular cross-sections, no definite rule can 
 
 be given, they are usually reduced to elementary forms, which, being 
 separately computed, and finally totalized, give the whole area in 
 the end. 
 
 This reduction is usually made to trapezoids and triangles (additive 
 or deductive], while the calculations are the simplest possible, though, 
 from the multitude of figures, necessarily tedious. 
 
 In the most irregular sections, involving heavy rock-work on side- 
 hill, the several cuttings (or level hights), transversely, are fre- 
 quently taken at ten feet only, or some such uniform distance apart, 
 and in these cases the mean hights of a number of contiguous trape- 
 zoids may be ascertained, and multiplied by the uniform distance 
 (agreeably to the rules of mensuration for irregular areas), and thus 
 abbreviate somewhat the labor of such computations ; which, how- 
 ever, in their origin, and indispensable verifications, are often laborious 
 enough, though, fortunately, so simple and elementary as to be within 
 the comprehension of all the members of an engineer party, which 
 enables us to bring many hands to the work. 
 
36 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Not nnfrequently, too, in rock -work (proximating a cost of a dollar 
 per cubic yard), it has been deemed necessary to take independent 
 cross-sections, at only ten feet apart forward, over the roughest por- 
 tions of the work. 
 
 In that event, although the calculations become voluminous, we 
 have the satisfaction of knowing that the solidity is correctly obtained ; 
 since, in such short spaces, no ordinary rules would produce any 
 important variation in the final result ; supposing, of course, the 
 cross-sections to be correctly laid out, and measured with accuracy, 
 both horizontally and vertically a matter of no small difficulty on 
 steep, rocky hill-sides, when cleaned for ivork. 
 
 9. Further Illustration of the Modification of Simpson's Rule (II.)> 
 with a Diagram Representing it, and also one of the Regular Formula, 
 and another Modification. 
 
 Here let us take the triangular prismoid, cross-sectioned, in Fig. 8 
 (and shown below), and suppose its length 100 feet (A) the end 
 
 Tig 18. 
 
 e.---* 
 
 cross-sections being dimensioned as before. With road-bed of 20, and 
 slopes of 1 to 1. The whole, shown in projection, to give a better 
 idea of the nature of the solid. 
 
CHAP. I. PRELIM. PROBS. ART. 9. 37 
 
 References. 
 
 CC = Centre line and edge diedral angle. 
 
 ACCB = Grade prism. 
 
 AB = Road-bed, 20. 
 
 AE = Side-slope plane, 1 to 1. 
 
 EF = Ground plane, assumed as level. 
 
 ea&E = Wedge of Fig. 8. 
 
 Then, for the volume of this solid, we have, by the modification of 
 Simpson's Rule (II.), 
 
 Ilightg. Widths. 
 
 Near end (double area), 22 X 44 . . . = 968 = 2 b. 
 Far end, " 16 X 32 '. . . = 512 = 2*. 
 
 8 times mid-section, . . 38 X 76 1 
 
 . , > = zooo = o in. 
 = sum hts. X sum wids. j 
 
 12)4368 
 
 Mean area. . . =* 364 
 Length h. . . = 100 
 Whole triangular solid to intersection ) 
 
 of slopes. .' / 
 
 Deduct grade prism under road-bed. . . = 10000 
 
 Leaves volume above road-bed, or Trape- ) 
 
 -j i r> -j f 77- ti 7 r 20400 = The same 
 
 zoidal Prismoid of Earthwork. . . j 
 
 solidity, as before computed, Art. 6. 
 
 (a.) The transformation or modification of Simpson's Rulo 
 
 (II.) may, in its mid-section term, be conveniently represented by \\ 
 diagram (perhaps more curious than useful). Thus, continuing the 
 side-slopes through the intersection, so as to form the end cross-sec- 
 tions, one above the other. 
 
 So, in Fig. 19, dimensioned as in Fig. 8, we have, 
 
 The triangle IEF = The larger end section, or area. 
 
 " " ICD = The smaller one. 
 
 ; . " rectangle KLMN = 8 times the area of the mid-section, 
 
 or the circumscribing rectangle 
 formed by sunn of hights X sum 
 of widths. 
 
 The road-beds . . . = The dotted lines, and may be 
 
 assumed (parallel) anywhere. 
 
38 
 
 MEASUREMENT OF EARTHWORKS. 
 
 The parallelogram IFEP = Higlit X width of larger 
 
 end, or double area of . A. 
 IDCO = Hight X width of smaller, 
 
 or double area of. . . B. 
 
 " rectangle KLMN = HG X OP, or sum Lights X 
 
 sum widths, = 8 times the 
 mid-section. 
 
 Here it is evident that IH X FE = Double area of larger end 
 section, or = IFEP ...... and IG X CD = same of smaller = 
 
 IDCO. 
 
 While (CD + FE) X (GI + IH) = the circumscribing rect- 
 angle KLMN = HG X OP, or the rectangle of sum of hights and 
 sum of widths. 
 
 Also, 
 
 /HI -f- IG\ "/FE + CD^ 
 
 19 
 
 = 861, the mid-sec. 
 
 < \ 2 / /N V 2 
 ( HG X OP, or 38 X 76 = 2888, or 8 times mid-sec. : 
 
 The triangles Q and R taken together = the Arithmetical Mean of A 
 and B, the end areas = (16 X 8) -f (22 X 11) = 128 + 242 = 370, or 
 
 484+256 740 , .,. 4 . , lf 
 
 j- = = 370, the Arithmetical Mean. 
 
CHAP. I. PRELIM. PROBS. ART. 9. 39 
 
 The triangles T and T are each equal to the Geometrical Mean of 
 the end sections A and B = \^484 X 256 = 352. 
 
 While U and V added together proximately equal the Harmonic 
 Mean between A and B, or = 334. 
 
 So that the circumscribing rectangle, KLMN, representing the 
 mid-section term, of Simpson's Transformed Rule (II.)* contains, or is 
 composed of, the following areas. 
 
 Double area of A. 
 " " B. 
 
 C 484 
 ' \ 484 
 f 256 
 ( 256 
 
 (The two end sections.) 
 Arithmetical Mean ...... 370 
 
 Geometrical Mean X 2. . . -{352 
 Harmonic Mean ....... 334 
 
 Total 8 times the mid-sec., 
 or 361 X 8. = 2888 
 
 In this case : 
 = Double areas 
 of both ends + 
 4 times the Geo- 
 metrical Mean 
 = 2888. 
 
 Some curious inferences may be drawn from this diagram, but their 
 practical results can be more concisely obtained in other forms. 
 
 Diagram of the regular Prismoidal Formula of Simpson and Hutton. 
 
 As applied to a triangular prismoid, formed by a diagonal cutting 
 plane, from the rectangular prismoid, Fig. 2, and shown again in Figs, 
 22, 24, and 52, with side-slopes of li to 1. 
 
40 MEASUREMENT OF EARTHWORKS. 
 
 Let 1 (Fig. 19*) Be the larger end section (Fig. 22), transformed 
 
 into an equivalent right triangle. 
 
 3 The smaller end (Fig. 24), also transformed : 4 and 5, 
 additive triangles, making up the trapezium ABCD (Fig. 
 19 J), equivalent in area to four times the prismoidal mid- 
 section (Fig. 23). 
 
 From this diagram we readily deduce a simple modification of the 
 prismoidal formula, equivalent in remit, for triangular prismoids. 
 
 Higlits. Widths. 
 
 -P.. f h = 30 X 90 = w 
 
 Dimensions of <% 
 
 Figs. 22 and 24. J . /\ 
 
 j K = 20 X 60 == w' 
 
 (^ Length = 100, usually. 
 
 Then, - - x length = Solidity. . VIII, 
 
 This operates very simply in figures, by direct and cross multiplica- 
 tion of hights and widths. 
 
 Substituting the numbers, Solidity 95000, as hereafter computed, 
 Art. 10 (a). 
 
 10. Adaptation of the Prismoidal Formula to the Quadrature and 
 Cubature of Curves, and also Solids, where the Ordinates are equivalent 
 to Sections by the Method of Simpson, as explained by Hutton. 
 
 The eminent mathematician, THOMAS SIMPSON, to whom we are 
 indebted for the Prismoidal Formula, also devised a method for the 
 quadrature of irregular curves by 'means of equidistant ordinates, or 
 for their cubature, by using equivalent sections of irregular'solids, at 
 equal distances, instead of ordinates ; such solids being bounded oppo- 
 site the base by a general curved outline. 
 
 This method, although a century old, is still the simplest and best 
 yet known for proximating the area of irregular curves, or the volume 
 of unusual solids, it has attained great celebrity, and been of much 
 service to philosophers and calculators, ever since its origin in 1750. 
 
 It has long been used by military engineers for ascertaining the 
 volume of warlike earthworks, and is regularly quoted in the leading 
 text books of that important profession.* 
 
 Also by naval architects in determining the nice problem of the 
 displacement of ships ; by mechanical philosophers, like Morin and 
 
 * LaisnS, Aide Memoire, du G6nie. Eds., 1831-61. 
 
CHAP. I. PRELIM. PROBS. ART. 10. 41 
 
 Poncelet, etc. by these it has been deemed of much importance, not 
 only for the quadrature uf irregular areas, but also for the ''Cubature 
 of solids of irregular excavations, embankments, etc." * 
 
 It forms a leading feature in Button's remarkable chapter on the 
 cubature of curves (who seems to have fully adopted it), under the 
 name of the method of equidistant ordinates. (See 4to Mens., 1770, 
 sec. 2, part iv. page 458.) We are much indebted to Hutton for the 
 practical development of this important problem, and he gives several 
 examples of its utility. Amongst others, computing the area of a 
 quadrant of a circle, with radius = 1, which, by Simpson's method, 
 using 11 ordinates, gives *7817 area, instead of '7854 "pretty near 
 the truth " (says Hutton). 
 
 "We will describe this method from the (4to Mens., 1770, p. 458). 
 
 "If any right line, AN, be divided into any even number of 
 
 equal parts, AC, CE, EG, etc., and at the points of division be 
 
 erected perpendicular ordinates, AB, CD, EF, etc., terminated 
 
 by any curve, BDF, etc." 
 
 Then, the sum of the first and last ordinates, plus 4 times sum of 
 even ordinates, plus 2 times sum of odd ones, -f. by 3, and X by AC, 
 one of the equal parts ; the resulting product will equal the area, 
 ABON, "very nearly." 
 
 That is to say, if 
 
 The sum of the two extreme ordinates . . = A. | ,-, 
 
 " of all the even numbered " . . =* B. 
 
 f 11 *u jj u j />. r tne " rst anc > 
 
 " of all the odd numbered " . . = C. I . 
 
 mi T- A / T T-X last irom (j. ) 
 
 The common distance apart of ordinates . . = D. I 
 
 Then the rule is, 
 A + 4B + 2C 
 
 X D (or AC) = Area, ABOK . . . (IX.) 
 
 And if more convenient (as it may be\ we transform this into its 
 equivalent, 
 
 A + 4B + 2C 
 
 D (or AE) _ AreRj ABOK m m (X j 
 
 n applying this formula, it is desirable to draw a figure, and num- 
 ber all the ordinates (as below), commencing with 1. 
 
 * Morin's Mechanics (Bennett's Trans., I860). See also Gregory, Math. Prac. Men. 
 (1825). 
 
42 
 
 MEASUREMENT OF EARTHWORKS. 
 
 " The same theorem will also obtain, for the contents of all solids, 
 by using the sections perpendicular to the axe, instead of the ordi- 
 nates." 
 
 In this form it becomes applicable to excavations and embankments, 
 or any similar solids relating to a guiding line, centre, or base line, 
 to which the cross-sections representing ordinates are perpendicular. 
 
 See Fig. 20, copied below from 
 Hutton, page 458. 
 
 Button's Example 3, p. 462. 
 
 " Given the length of five 
 equidistant ordinates of an area, 
 or sections of a solid, 10, 11, 14, 
 16, 16, and the length of the 
 whole base, 20." 
 
 Then, 
 
 26 + 108 -f 28 ^ 
 
 X 5 = 2/0. 
 
 " The area or solidity required" 
 
 This formula of Simpson (adopted by Hutton) is evidently derived 
 from ike Prismoidal Formula, or it may be, originated it, both having 
 the same author, and their precedence unknown. 
 
 (a.) AYe will now give an example of Hutton's Method of 
 
 Equidistant Ordinates (adopted from Simpson), giving two stations 
 of a railroad cut (each 100 feet long, with a road-bed of 18, and side- 
 
 + 24) 
 
 Fig. a 
 
 Hor: sea: Vcr. 
 
CHAP. I. PRELIM. PROBS. ART. 10. 
 
 43 
 
 slopes U to 1), .shown both in profile and cross-sections. (See Figs. 
 21 to 26, inclusive.) 
 
 The above figure is a profile, or vertical section (of two stations), 
 upon the centre line of a railroad cut, with a road-bed of 18, and side- 
 slopes of 1J to 1. The horizontal scale (/or convenience) being made 
 t of the vertical. 
 
 Firstly : Computing each station separately, by Simpson's Rule (II.) 
 
 Stations 1 to 3 = 100 = h. 
 
 Ills. WIds. 
 
 30 X 90 = 2700 = 26. 
 20 X 60 = 1200 = 2 t. 
 
 Stations 3 to 5 = 100 = h. 
 
 II ts. Wids. 
 
 20 X 60= 1200 = 26. 
 10 X 30 = 300 = 2 t. 
 
 50 X 150 = 7500 =* 8 m. 
 
 30 X 90 = 2700 = 8 m. 
 
 -5- by 12)11400 
 
 -T- by 12)4200 
 
 Mean Area . . = 950 
 
 X by h . = 100 
 
 Mean Area . . = 350 
 X by h . = 100 
 
 Solidity in c. ft. = 95000 
 
 Solidity in c. ft. = 35000 
 
 -4- 27 . . = 3519 
 Deduct Grade 
 Prism for 100 
 
 feet . . . = 200 
 
 -^ 27 . . = 1296 
 Deduct Grade 
 Prism for 100 
 feet . . . . = 200 
 
 Solidity inc. yds. = 3319 
 
 Solidity in c. yds. = 1096 
 
 Then, 3319 + 1096 = 4415 cubic yards, whole solidity of cut from 
 1 to 5 inclusive. 
 
 Secondly: Now computing the same, in a body, by Hutton's 
 Rule (X.). 
 
 Data. 
 (1350 
 L= JjL50 
 
 (1500 
 
 B 
 
 5 
 337-5 
 
 ( 937- 
 J . 337- 
 (.1275 
 
 X 4 = 5100 
 
 600 X 2 = 1200 
 
 We have, 
 
 1500 + 5100 + 1200 
 
 X 100 
 
 Now, -4- by 27 
 
 Deduct Grade Prism, 200 X 2 stations. 
 
 Solidity in cubic yards 
 
 (The same as above.) 
 
 C. feet. 
 
 130,000 
 
 4,815 
 
 400 
 
 4,415 
 
44 
 
 MEASUREMENT OF EARTHWORKS. 
 
 (CROSS SECTIONS.) 
 
 (b.) The preceding example clearly shows that Hutton's 
 
 method of equidistant ordinates is merely the Prismoidal Formula 
 extended to several stations, instead of confining it to one. 
 
 There is another mode of considering this question where the cross- 
 sections are triangular, and the ground level transversely. 
 
 Thus, in any station, let h and h' be the end hights from the inter- 
 section of the side-slopes to the ground, then, 7i 2 r and h f2 r = the cor- 
 responding areas (r being the slope ratio, which, in the preceding 
 example = 1), then omitting r, a common factor, we have in A 2 and 
 h n vertical lines, or ordinates, representative of the end areas, and in 
 
 ( - J of the mid-section. 
 
CHAP. I. PRELIM. PROBS. ART. 1O. 45 
 
 The square roots, then, of the areas (however computed, and what- 
 ever be the ratio (r) of the side slopes), correctly represent them; 
 since these roots form the side of an equivalent square (or half base 
 of an equivalent triangle, with 1 to 1 side-slopes) squaring which, 
 obviously re-produces the areas they are the roots of. 
 
 Hence, the end areas being given in any station, or number of 
 stations, their square roots may represent them in Hutton's rule of 
 cubature, and any pair of roots added together, and their sum squared, 
 gives 4 times the mid-section between them ; which is precisely what we 
 need in the Prismoidal Formula. 
 
 This is evident, from Fig. 
 27, where we suppose h and k' 
 placed in a continuous line, then, 
 
 _v_ 
 
 /h -4- h'\ 2 . . ,, *+~ 50 
 
 ( - j = \ the square of (h -^^ 
 
 -f h f ), or equivalent to the pro- 
 position of geometry that the square of a whole line equals 4 times the 
 square of half. 
 
 f Let h = 30, and h' = 20, then h -f h' = 50, h + h ' = 25 
 
 4-^)'= (25) 2 = the mid-sec. = 625, and X 4 = 2500 \ 
 
 (A + h')* = (50)' = 2500 j 
 
 VWhile /i 2 = 900 = one end area, and h n = 400, the other. 
 
 Also, 
 
 ( h 2 + h' 2 -f 2 (h X h') *) 
 
 J = 900 -f 400 -f- 1200 = 2500 I 
 (= (h + hj ,, . ; .. . = 2500 j 
 
 From all which, we readily draw the following: 
 
 Rule. Compute the end areas at each regular station (numbered 
 upon a diagram on Hutton's plan, by the odd numbers, 1, 
 3, 5, 7, etc., marking also the even numbers intermediately, 
 which are, in fact, half stations, or the places of mid-sec- 
 tions), find the square roots of these end areas: add any 
 two adjacent roots, and their sum squared equals 4 times the 
 area of the mid-section, between the regular stations. 
 
MEASUREMENT OF EARTHWORKS. 
 
 Let Fig. 28 be the profile of one station of cutting, from intersection 
 
 of slope to ground. 
 h and h' = The end hights, or representative square roots of 
 
 the areas, at regular stations, numbered odd. 
 m The place of the mid-section, numbered even, and repre- 
 sented by its ordinate. 
 Length = usually, 100, between principal stations. 
 
 . 28. 
 
 Whence, 
 h* + h' 2 + 4m? 
 
 Length. 
 
 6 
 
 X 100 = Solidity, by the Prismoidal Formula. 
 
 
 
 
 XI. 
 
 Which, for one station, is equivalent to Hutton's Ride. 
 
 (C.) ...... So that having the end areas given, we deduce at once 
 
 the mid-section, by a table of roots and squares,* and can proceed 
 station by station, prismoidally , to find the solidity. Or combining 
 them as in Hutton's Rule for cubature, we may calculate in a body the 
 whole of a cut or bank. 
 
 Thus, taking the preceding example, and tabulating it (see Figs. 
 21 to 26). 
 
 Stations. 
 
 Areas. 
 
 
 
 Even Nos. 
 
 Odd. 
 
 Even. 
 
 Extreme. | Odd Nos. 
 
 
 
 Squares, or 
 Mid-see. 
 
 1 
 
 
 1350 
 
 
 36-7423 
 
 
 Areas. 
 
 
 2 
 
 
 
 
 61-24 
 
 3750 
 
 3 
 
 
 
 600 
 
 24-4949 
 
 
 
 
 4 
 
 
 
 
 36-74 
 
 1350 
 
 5 
 
 
 150 
 
 
 12-2475 
 
 
 
 
 
 1500 
 
 600 
 
 
 
 5100 
 
 
 
 
 2 
 
 
 
 
 
 
 
 1200 
 
 
 
 
 
 
 A. 
 
 20. 
 
 
 
 4 B. 
 
 This tabulation may be made in any more convenient form, or the 
 data may be written upon the working profile of the line with advantage. 
 
 * Such as Barlow's (Prof. De Morgan's Ed., London, 1860), which is the most con- 
 venient and extensive, or any like tables. 
 
. CHAP. I. PRELIM. PROBS. ART. 10. 47 
 
 Then, 
 
 A -f- 4B -f 20 Mean Area. Length of Sta. Cub. Ft. 
 
 1500 + 5100 + 1200 __ 1300 X 100 = 130000 = by Hutton's 
 6 Rule X. 
 
 'Now, dividing by 27, = 4815 
 
 Deduct grade prism for two stations . . = 400 
 
 Leaves solidity in cubic yards (as before) = 4415. From 1 to 5 
 = 200 feet. 
 
 The division by 6 in the first term results in a mean area, which X 
 by length, gives the solidity and enables us to use a table of cubic 
 yards to mean areas, as soon as we have found the latter, in order to 
 obtain the cubic yards more readily by inspection. 
 
 (d.) In further illustration of this important method of 
 
 computation in earthworks, we will submit another example, repre- 
 senting an entire railroad cut, with 20 feet road-bed, and side-slopes 
 of 1 to 1, laid off in regular stations of 100 feet, and truncated at 
 both ends in light cutting (at selected stations), so as to secure full 
 cross-sections throughout; and also an even number of equal distances 
 (apart sections), each 100 feet, or regular and uniform stations, what- 
 ever their length. 
 
 These truncations are made before proceeding to the calculation, 
 so that all the cross-sections shall be complete (or have some side 
 slope however small at both edges of the road-bed), which simplifies 
 the main calculation, while in the end the truncated volumes may be 
 computed independently, and added in with the rest. 
 
 Again, if the ground should have required the insertion of interme- 
 diates in any one or more of the regular stations, it will be best to 
 draw a pencil line around all such whole stations upon the diagram, 
 and compute them separately from the main body the places of such 
 stations being considered vacant for the time (omitting distance, mid- 
 section, and end areas, so far as they apply to the assumed vacancy), 
 and thus the cut will be computable under our rule, in one or more 
 masses (as though a single mass originally), according to the number 
 of vacant spaces. A little practice will familiarize this matter better 
 than further explanation, as the object to be attained is evident. 
 
48 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Generally, we may compute the cut, or bank, in one principal 
 mass, and then calculate separately, and add. 
 
 1. The solidity in the special stations containing intermediates. 
 
 2. The quantities of work of the same kind, at the passages from 
 
 excavation to embankment, at both ends of the cut (as will 
 be further explained). 
 
 In all such cases (indeed, in all cases of heavy work), it is necessary 
 to draw diagrams, as below, and these (in cross-sections) will usually 
 have a scale of 20 feet to the inch, which long practice has shown to 
 be entirely suitable ; but any preferred scale may be employed, or the 
 cross-section paper in common use amongst engineers which carries 
 its own scale and which will be found convenient in many respects, 
 either bound up for the purpose, or in loose sheets, to be ultimately 
 tacked together, including a mile forward, or thereabouts. 
 
 Profile of 8 stations of railroad cut; base 20, side-slopes 1 to 1. 
 
 a b = Intersection of side-slopes, or edge of diedral angle, formed 
 
 by their planes meeting. 
 
 c d Grade, or formation line of the road-bed = + O'O. 
 ef = Surface line of ground, as cut by centre plane. 
 gp = Grade prism deductive for solidity 
 
 Tig. 29 / 
 
 V? 
 
 
 ^^^- * 5O 
 
 Hor: b c a: tb VBT. 415 ^f^ 
 
 
 
 ^s 
 
 k 
 
 * 
 
 LS,^^ 
 
 ^^ 
 
 
 
 
 
 X Erojile 
 
 41OXX* 
 
 ^ 
 
 
 
 
 
 
 
 
 X+1D 
 
 41 
 
 J \ 
 
 r 
 
 
 
 S 
 
 
 a 
 
 
 8 
 
 
 > 
 
 X^ 
 
 45^^-^**n 
 
 
 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 C3 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 If 
 
 n * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 sT 
 
 C 
 50 
 
 so 
 
 S 
 
 
 
 
 
 
 
 
 
 
 R 
 
 
 50 
 
 so p 
 
 a 
 
 ,-10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _m 
 
 h 
 
 1. S) 3 (4) 5 (6) 
 
 7 43) 00) U 02) 13 Hi) 15 OC) 17 
 
 Regular stations designated by odd numbers (1, 3, 5, etc.). 
 Mid-section places by even numbers (2, 4, 6, etc.) 
 
f 
 
 CHAP. I. PRELIM. PROBS. ART. 10. 
 
 49 
 
 The ordinates show the level hlghts from grade to ground, to which 
 add always the common hight of grade triangle. 
 
 Transverse slopes are shown on cross-sections. 
 
 f Regular Stations = ! 
 Cross-sectirm Areas = 232*5 
 I Square Boott = 15 25 1 
 
 1 Sums of Roots = 33-94 
 
 3- 5- 7' 9' 11' 13- 15- 17- 
 
 349-2 412-7 720-5 844-8 1085- 901-5 516- 259-5 
 18-69 20-31 26-84 29-06 32-94 30-02 22-72 16-09 
 39-00 47-15 55-90 62-00 6296 52'74 38-81 
 
 Squares of Sums = 1151-9 1521-0 2223-1 3124-8 3844-0 3964*0 2781-5 1506-2 
 ^ These squares are each equal to 4 times the mid-section, between regular stations. 
 
 All hights and areas taken to intersection of slopes. 
 
 Mean areas computed separately 
 
 General Mean Area computed 
 
 for each regular station, by Simp- 
 
 by Button's Rule, 
 
 son's Rule. 
 
 
 
 A -f 4B -f 2C 
 
 232-5 ^ 
 
 
 6 
 
 (1 to 3) 349-2 
 
 
 
 1151-9 
 
 
 
 Tubulated fbr the numerator by 
 
 6)1733-6 
 
 
 successive additions equivalent to 
 
 
 
 multiplication. 
 
 Mean Area = 288'9 , 
 
 349-2 > 
 
 
 1 . . 232-5 
 
 
 
 
 2 1151-9 
 
 
 (3 to 5) 412-7 
 
 
 ^ 1 1 - ' I 
 
 o f 349-2 
 
 
 1521-0 
 
 
 3 ' '{ 349-2 
 
 
 6)2282-9 
 
 
 4 . . 1521-0 
 
 
 Mean Area = 380*5 , 
 
 
 r f 412-7 
 
 O . . -\ t -4 n fv 
 
 1 to 9 
 
 
 1 412-7 
 
 
 412-7 1 
 
 
 6 . . 2223-1 
 
 
 (5 to 7) 720-5 
 2223-1 
 
 
 7 f 720-5 
 ' ' '( 720-5 
 8 . . 3124-8 
 
 
 6)3356-3 
 
 
 9 . . 844-8 
 
 
 Mean Area = 559'4 
 
 
 6) 12062-9 
 
 720-5 ' 
 
 
 1 to 9 = 2010-5 Gen. Mean 
 
 (7 to 9) 844-8 
 
 
 Area. 
 
 3124-8 
 
 . 
 
 Separate Mean Areas. 
 
 6)4690-1 
 
 
 ( 288-9 
 
 Mean Area = 781*7 
 
 
 lto'9 J 380 ' 5 
 
 . 
 
 ' ' * 1 559-4 
 
 
 I 781-7 
 
 
 Same as above = 2010*5 
 
50 
 
 MEASUREMENT OF EARTHWORKS. 
 
CHAP. I. PRELIM. PROBS. ART. 10. 
 
 51 
 
 Mean areas computed separately 
 for each regular station, by Simp- 
 son's Kule. 
 
 (9 to 11) 
 
 Mean Area = 
 (11 to 13) 
 
 Mean Area = 
 (13 to 15) 
 
 Mean Area = 
 (15 to 17) 
 
 Mean Area = 
 
 844-8 ' 
 1085-0 
 3844-0 
 
 
 
 6)5773-8 
 
 962-3 J 
 
 1085-0 ' 
 901-5 
 3964-0 
 
 6)5950-5 
 
 991-8 j 
 
 901-5 1 
 516-0 
 2781-5 
 
 6)4199-0 
 
 699-8 , 
 
 516-0 > 
 259-5 
 1506-2 
 
 6)2281-7 
 
 380-3 , 
 
 General Mean Area computed 
 by Button's Kule. 
 
 A + 4B + 2C 
 6 
 
 Tabulated for the numerator by 
 successive additions equivalent to 
 multiplication. 
 
 Bro't over 1 tO 9 = 12062-9 
 
 9 . . 844-8 
 3844-0 
 1085-0 
 1085-0 
 3964-0 
 901-5 
 901-5 
 2781-5 
 516-0 
 516-0 
 1506-2 
 259-5 
 
 1 to 17 
 
 11 
 
 13 
 
 15 
 
 17 
 
 Gen. Mean Area 
 
 Separate Mean Areas. 
 
 6) 30267-9 
 5044-7 
 
 1 to 17 
 
 Brought over = 2010'5 
 962-3 
 991-8 
 699-8 
 380-3 
 
 \ Total . . = 
 (Same as above.) 
 
 Then, Mean Area. 
 
 5044-7 X 100 _ 
 
 27 "" 
 
 5044-7 
 
 C. yards. 
 
 18684-1 
 
 = 2963-2 
 
 Deduct prade Prism 
 for 8 stations = 
 370-4 X 8 ... 
 
 Solidity = 15721- 
 
 in cubic yards from 
 1 to 17. 
 
 So that the final solidity of this cut (as shown) from grade to 
 ground, vertically, and from 1 to 17 (8 stations), horizontally = 15721 
 cubic yards (excluding for the present the grade passages). A com- 
 
52 
 
 MEASUREMENT OF EARTHWORKS. 
 
 F5|.35 
 
 cot 
 
 00' 
 
 Fig. 38 
 
 Cross -sections 
 
/^n> l/l> 
 
 CHAP. I. PRELIM. PROBS. ART. 1O. ^ ^^63/ > 
 
 ^XA 
 
 parison of the calculated work, by Separate Mean Areas, and by 
 General Mean Area, while resulting alike, evinces the superiority of 
 the latter, in point of brevity. 
 
 In the tabulation for General Mean Area, it will be observed that 
 the extreme end areas are written but once (equivalent to addition) 
 the odd numbered areas twice (equivalent to X by 2), while the 
 even numbered areas are written, in effect, 4 times, as squares of 
 sums of adjacent representative hights, because in that shape they each 
 equal 4 times the area of the prismoidal mid-section. 
 
 (6.) We must now consider the passages from excavation 
 
 to embankment at both extremities of the cut, near the regular sta- 
 tions, 1 and 17, where it was assumed to be truncated, in order to sim- 
 plify its computation. 
 
 Figs. 39 to 42 show these passages so clearly, in the assumed case, 
 as to need little explanation. 
 
 On plain ground the line of passage a c will often be so nearly 
 normal to the centre that, having set the grade peg in the centre line 
 at e (the entrance of the cut), we may place those for the edges ofthe 
 road-bed (as a and c), at right angles in many cases, where the ground 
 differs in level only a few tenths of a foot; the error being merely a 
 change of some yards from excavation to embankment, which is quite 
 immaterial, since their values differ little per cubic yard. 
 
 But where the ground is much inclined, in either direction, the 
 grade pegs aec must be set on an oblique line, broken at e, if neces- 
 sary. 
 
 Precise rules can scarcely be furnished for such cases, but the 
 quantities being usually small, and the distances short, any of the 
 ordinary methods may be safely employed. 
 
 In the case before us, we have made the computation from 17 to a, 
 and from 1 to a, by the Arithmetical Mean, and for the parts from 
 a to c as pyramids. 
 
 In this manner we have found the volume of excavation, at the 
 passage at Fig. 39, to be = 321 cubic yards. 
 
 And at Fig. 41 = 622 " 
 
 Total, in the whole length of the passages 
 
 (230 feet) = 943 cubic yards. 
 
54 
 
 MEASUREMENT OF EARTHWORKS. 
 
 So that, finally, we have for the solidity of the entire railroad cut, 
 under consideration, the following result : 
 
 From 1 to 17 (as before computed) = 15721 cubic yards. 
 In the passages from excavation to 
 
 embankment, at both ends (230 
 
 feet long in all) = 943 " " 
 
 Whole solidity of the cut from grade 
 to grade, on both sides . . . = 16664 cubic yards. 
 
 We will now illustrate the passages from excavation to embank- 
 ment, at both ends of the cut (shown in profile at Fig. 29.) 
 
 In Figs. 39 to 42 all letters refer to similar parts. 
 1 and 17 = Places of cross-sections, at the selected regular stations, 
 
 where the cut was truncated, to obtain full work. 
 a a = Cross-section, where one edge of road-bed runs to grade. 
 
 c = Grade point at the other edge, or opposite side. 
 a c = Line of junction of cut and bank, at grade level. 
 bb Slopes of cut. 
 d d = Slopes of bank. 
 e = Grade point at centre. 
 
 Total length of cut between the extreme grade points forming the 
 vertices of the small pyramids at c and c = 1030 feet. 
 
CHAP. I. PRELIM. PROBS. ART. 11. 
 
 55 
 
 Other modes may be used for treating the question of passages 
 between excavation and embankment, but the above is as simple as 
 any, and may be easily modified for particular cases. 
 
 11. With Railroad Cross-sections in Diedral Angles to find the mid- 
 section of the Prismoidal Formula, by a brief calculation from the End 
 Areas, without a Special Diagram. 
 
 In all railroad cross-sections, instrumental data of adequate extent 
 are first obtained in the field by well-known processes, and these data 
 enable us in the office, subsequently, to draw them as diagrams, by a 
 suitable scale, and to compute their superficies. 
 
 The length of each separate solid of earthwork, and its position 
 upon the centre or guiding line, is also known. 
 
 With these given data, the Prismoidal Formula requires the deduc- 
 tion of a hypothetical mid-section, in some form, for use under the 
 general rule, or its modifications. 
 
 As mentioned previously, this mid-section is usually derived from 
 the Arithmetical Average of like parts in the end sections, and even 
 in extremely irregular ground, to find this leading section of an Earth- 
 work Prismoid, is not very difficult when the diagrams of the end 
 cross-sections are correctly drawn (as in heavy work they always 
 should be), or even from the field notes of the engineer, since the posi- 
 tion of every leading point of ground, transversely, is always fixed 
 and recorded by level bights, and distances out from centre, and their 
 average position is always reproduced, proportionally, -in the mid- 
 section. 
 
 Nevertheless, some judgment is required in deducing the mid-sec- 
 tions from the end ones, by Arithmetical Means, since the points to 
 
56 MEASUREMENT OF EARTHWORKS. 
 
 average upon are often in doubt, the process, too, including finding 
 its area, is like most others connected with earthwork computations, 
 very often tedious, so that some shrewd mathematicians, while con- 
 ceding the accuracy of this method, when properly carried out, have, 
 nevertheless, deemed it unsatisfactory in some respects.* 
 
 It is well, therefore, to have the means of operating with given end 
 areas, to find the mid-section, without the necessity of arithmetically 
 deducing, or even of sketching it. 
 
 We, therefore, now submit some rules and examples by which the 
 area of the mid-section may be computed from the ends, without 
 deriving it in the usual way, or drawing for it a special diagram. 
 
 These rules are intended only for Earthwork Prismoids, within die- 
 dral angles ; and though their range is clearly more extensive, the 
 variety of prismoidal solids is so great that it is probably best to limit 
 our rules and examples to the object before us. 
 
 The broken ground line of very irregular cross-sections should 
 always be reduced to a uniform slope, by a single equalizing line (or 
 at most by two), containing exactly the same superficies, by the method 
 of Art. 8, and the bights and widths ascertained for each section 
 (by the equalizing line), and verified by multiplication to re-produce 
 the area equalized, see 8 (a), these bights and widths enable us at 
 once to compute the volume of the prismoid by Simpson's Rule (their 
 product giving end areas) (Art. 2 (a) ) and the sums of these 
 bights and widths, when multiplied together, producing always 8 
 times the mid-section (without directly deducing it). 
 
 Having given then the end areas, or the bights and widths which 
 produce them, we readily find the Prismoidal Mid-section by the 
 following : 
 
 x Arithmetical Mean -f Geometrical Mean 
 
 (1.) = . = Mid-sec. 
 
 2 
 
 (Sum of square roots of end areas) 2 _ . 
 
 tSum end bights X sum end widths ,_, , 
 
 (3.) . . Mia-sec. 
 
 (4.) By the method of Initial Prismoids Art. 3 (a). 
 
 I * Warner's Earthwork (1861). Davies' New Surveying (1870). 
 
 f These bights and widths (used in 3) are those connected with the equalizing line 
 of the equivalent triangular section the product of which, at each cross-section, re-pro- 
 duces exactly the double area of the whole surface, from the side-slopes to the broken 
 ground line; and the product of their sums always equals eight times the mid-section. 
 
 Rules. 
 
CHAP. L PRELIM. PROBS. ART. 11. 
 
 57 
 
 Other rules might be given, but these Jour appear to be the simplest 
 and best for use in earthwork, under the view we have herein taken. 
 
 Having then found the mid-section, and having the end areas and 
 length previously given, we can easily compute the volume of any 
 earthwork solid, by the Prismoidal Formula, or its numerous modifi- 
 cations. 
 
 1. A Prism . = Base. 
 
 By Geometry, we have 
 for the mid-sections of 
 
 f A Wedge, with back ^ 
 
 2. < and edge equal and > = i Base, 
 v. parallel . . . . J 
 
 3. A Pyramid = 1 Base. 
 
 Fig. 43 shows the end cross-sections of one station of a railroad cut, 
 upon irregular ground, both upon one diagram*, road-bed 20, side- 
 slopes 1 to 1. Length of station, 100 feet. 
 
 g g'g" cent: of grav: 
 
 Centre higbts to intersection of slopes. 
 
 -f 37-5 
 
 4- 31-6 
 
 + 25-7 
 
 from equalizing line. 
 
 Total widths from side to side. 
 
 80 
 68 
 56 
 
58 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Note: 
 
 Both in Figa. 43 and 44 the same letters refer to like parts. 
 CC = Centre line of railroad, or guiding line of earthwork, 
 a b sm Equalizing line of broken ground surface of larger end . 
 / = " " " " " of smaller end . 
 
 d = " " " " " of mid-section . 
 
 14 2' slope. 
 , 15 57' " 
 14 50' " 
 
 Fig. 44, like the preceding, shows both end sections of a railroad cut, 
 upon one diagram. Road-bed = 20, side-slopes 1 i to 1. Length = 100. 
 
 Centre bights to intersection of slopes. 
 
 + 22*02 
 + 26-07 
 + 29-81 
 from equalizing line. 
 
 Total widths from side to side. 
 
 66- 
 
 78-7 
 
 90-7 
 
 In this figure (44) the line ef has a minus slope, which is always 
 the case when the area assumed up to .the equalizing point is greater 
 than that to be equalized. 
 
 In both of the above figures, I is the intersection of the side-slopes, 
 or edge of the diedral angle, containing the earthwork prismoids. 
 
 The constant area of the grade triangle, with side-slopes of 1 to 1 
 (Fig. 43) = 100. While, with side-slopes of 11 to 1 (Fig. 44) = 
 66f. The road-bed, or graded width, in both cases being 20 feet. 
 The altitude of this triangle for 1 to 1 = 10, and for IHo 1 = 6f. 
 
CHAP. I. PRELIM. PROBS. ART. 12. 59 
 
 The rules (numbered) above, for the figures shown, give the follow- 
 ing results : 
 
 ( Fig. 43 gives Mid-sections (1) = 1074-5; (2) = 1074-5 ; (3) = 1074-4 ; (4) = 1074-6 
 \ Fig. 44 gives Mid-sections (1) = 1015' ; (2) = 1014-74 j (3) = 1015-22; (4) = 1015- 
 
 The small variations arise from the decimals not being sufficiently 
 extended. 
 
 12. To find the Prismoidal Mean Area from the Arithmetical or 
 Geometrical Means, or the Mid-section, by Corrective Fractions of the 
 Square of the Difference of End Hights. 
 
 In all cases we suppose the end areas of the Prismoid to be given, 
 and that the Prismoid itself is contained within a diedral angle, the 
 plane angle measuring it being supplemental to double the angle of 
 side-slope, as in the Figs. 43 and 44. 
 
 The simplest, and probably by far the most generally employed 
 method of finding a mean area between two others, is by the Arith 
 metical Mean which is itself half the sum of any two magnitudes. 
 
 Adopting the Arithmetical Mean as being the simplest known 
 base, and forming all sections of earthwork by prolonging the planes 
 of the side-slopes to their intersection (or supposing them to be), so 
 as to bring the computed prismoids within diedral angles of given 
 divergency. 
 
 We have, from the relations between the sums or differences 
 of the squares, or rectangles of lines producing areas, some rules, 
 which may often be useful in the calculation of earthwork, for cor 
 recting mean areas to be used in finding the solidity. 
 
 This correction being always equivalent to some fraction of the square 
 of the difference of the end hights. 
 
 While these end hights are always to be deemed and taken as the squart 
 roots of the end areas, and are, in fact (as before mentioned), a side of 
 an equivalent square, or half base of an equivalent triangle, having 
 side-slopes of 1 to 1 (or a diedral angle of 90), for (we repeat), no 
 matter what may be the ratio of actual side-slope, nor how irregular 
 the ground surface, the square root of the area is invariably the true 
 representative hight whichx rectifies the section, and which, when 
 squared, reproduces the area. 
 
 See Art 10 (a) (b) etc., where much use is made of these square 
 roots, or representative hights. 
 
60 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Having, then, the end areas given, and their square roots or hights 
 ascertained, 
 
 D = Difference of hights. 
 
 D 2 = The square of the difference of hights. 
 
 Rules: 
 
 (1) Arithmetical Mean 
 
 Sum end areas 
 
 (2) 
 (3) 
 (4) 
 
 (5) 
 
 (6) 
 
 Then the Prismoidal Mean Area. 
 . . = Arithmetical Mean I D 2 . 
 . . = Mid-section . . . -f ^ D 2 . 
 . . s=s Geometrical Mean + J D 2 . 
 
 Prismoidal Mid-section. 
 . . = Arithmetical Mean J D 2 . 
 
 Geometrical Mean. 
 . . = Arithmetical Mean \ D 2 . 
 
 ^ For Fig. 43 these rules give, For Fig. 44 these rules give, 
 
 (1)=1039- =Arith. Mean. 
 
 (2) = 1022-9 ) 
 
 (3) = 1023- V= Pris. Mean. 
 
 (4) = 1023-2 ) 
 
 (5) = 1014-8 = Pris. Mid-sec. 
 (6)= 991- =Geom. Mean. 
 
 In these numerical illustrations (as in others) slight variations 
 arise from insufficient decimals. 
 
 Baker* gives yet another rule for the Prismoidal Mean Areas, as 
 follows : 
 
 (1) = 1110- 
 
 = Arith. Mean. 
 
 (2) = 1086-4 
 
 i 
 
 (3) = 1086-3 
 
 = Pris. Mean. 
 
 .(4) = 1086-4 
 
 \ 
 
 (5) = 1074-6 
 
 = Pris. Mid-sec. 
 
 (6) = 1039-2 
 
 ==Geom. Mean. 
 
 Sum end areas -f Rectangle hights 
 
 Prismoidal Mean. 
 
 And we may repeat, as another modification of the Prismoidal 
 Formula, arising from this discussion, the following (same as XI., 
 before given) : 
 
 XII Solidity 
 
 _ (Sum of squares of hights) -f (Square of sum of hights) 
 _ . y^ fa 
 
 * Baker's Railway Engineering and Earthwork (London, 1848). Other writers have 
 given the same, and it is deducible from Button's Mens., Prob. 7, as most of these For- 
 mulcts are. 
 
CHAP. I. PRELIM. PROBS. ART. 12. 
 
 61 
 
 2 (Sum sqs.) + 2 (Rect. hights) 
 
 , or -f- 2 = 
 
 This is equivalent to 
 
 (Sum of sqs.) -f- (Rect. hights) 
 
 , which is Baker s rule above, or Bid- 
 der's, as quoted by Dempsey (Practical Railway Engineering (4th 
 edition) 1855). 
 
 We may illustrate this matter further by two simple figures. 
 
 Here Fig. 45 represents a 1 to 1 side-slope diedral angle 90 ; and 
 Fig. 46 a side-slope of H to 1 diedral angle 112 38'. 
 
 In both these diagrams the same letters refer to like parts. 
 
 
 / ..LJ 
 
 References. 
 CO = Centre line. 
 
 I = Intersection of planes of side-slope. 
 a b = Ground line of one end section. 
 c d = " " % of the other. 
 m s " " of the mid-section. 
 Hights and areas both extend to the intersection at I. 
 
62 MEASUREMENT OF EARTHWORKS. 
 
 In Fig. 45, The end areas are 1600 and 400 the higlits 40 and 
 20 and by the rules herein, Arithmetical Mean = 
 1000, Geometrical Mean = 800, Mid-section = 900, 
 Prismoidal Mean Area = 933, by all the rules. 
 
 In Fig. 46, The end areas are 2400 and -600 the hights = 48'99 
 and 24'99, being the square roots of the respective 
 end areas and by the rules herein, Arithmetical 
 Mean = 1500, Geometrical Mean = 1200, Mid-sec- 
 tion = 1350, Prismoidal Mean Area 1400, by all thg 
 rules. 
 
 The areas and hights, in both examples, are contained between the 
 ground lines, and the intersection of the planes of side-slope, or edge 
 of diedral angle, including the Prismoid of Earthwork. 
 
 13. Applicability of the Prismoidal Formula to find the Solidity of 
 Various Solids other than Prismoids. 
 
 The Prismoidal Formula appears to be the fundamental rule for the 
 mensuration of all right-lined solids, and the special rules given, in 
 works on mensuration, for ascertaining the volume of solids in general 
 use, seem like mere cases of the former ; though their relation has never 
 been demonstrated in plain terms by mathematicians so as to con- 
 nect them directly further than prisms, pyramids, and wedges, which 
 has already been done by the present writer in Jour. Frank. Inst., 
 1840. 
 
 Nevertheless, Hutton (1770) has indicated numerous applications, 
 and various writers have since shown the applicability of the Pris- 
 moidal Formula to ordinary solids, and also its coincidence with many 
 special rules of the books, when proper algebraic substitutions are 
 made; and it has been further shown to hold for certain warped 
 solids, to which its application was not expected.* 
 
 As an evidence of its remarkable flexibility, we may show, briefly, 
 its application to the three round bodies, illustrated by a diagram. 
 
 (1) The volume of a cone equals the product of its base X i its hight.'f 
 The prismoidal mid-section of a cone = \ the area of the base. The 
 section at the top, or vertex = 0. Then, the sum of these areas used 
 prismoidally = 2 base, which, X i h = base X i hight, which is 
 the geometrical rule. 
 
 * Gillespie, Frank. Inst. Jour. (1857 and 1859). Warner's Earthwork (1861). 
 f Chauvenet, ix. 3, 7, 14, Geom. (1871). Borden's Useful Formulas (1851). Henck'a 
 Field Book (1854), Art. 112. 
 
CHAP. I. PRELIM. PROBS. ART. 13. 
 
 63 
 
 (2) The volume of a sphere equals 4 great circles X i fa radius.* 
 Now, the prismoidal sections at the poles are both = 0. While four 
 times the mid-section = 4 great circles. Then, the prismoidal sum 
 of areas = 4 great circles, which X i hight, or diameter, or radius, 
 is the geometrical rule. 
 
 (3) The volume of a cylinder equals the product of its base by its 
 hight.* Now, by the Prismoidal Formula, base -f- top + 4 times 
 mid-section = 6 base (for all the sections are alike), and 6 base X ^ 
 h = base X hight, which is the geometrical rule. 
 
 So that there can be no doubt of the applicability of the Prismoidal 
 Formula to the three round bodies; and in a similar manner it is easy 
 to show its coincidence with many special rules for solids, but a direct 
 mathematical demonstration connecting all these together, and exhib- 
 iting their geometrical relations, has never come under the writer's 
 notice ; though indirectly, and perhaps quite as satisfactorily, this con- 
 nection has been clearly established for all the leading solids in prac" 
 tical use. 
 
 Numerical calculation of the three round bodies, supposing each to 
 have a diameter of 1, and an altitude of 1. 
 
 CONE. 
 
 SPHERE. 
 
 CYLINDER. 
 
 Prisnioidfilly. 
 
 Geom. Rule. 
 
 Prismoidally. 
 
 Geom. Rule. 
 
 Prismoiditlly. 
 
 Geom. Rule. 
 
 Top . . = -0 . 
 Mid.X 4 = '7854 
 Base. .= 7854 
 6)1-6708 
 
 Base = -7854 
 
 ix^ 
 
 Solidity = "2618 
 
 Top . ..^.-O- 
 Mid.X 4 =3-1416 
 Base. . =0- 
 
 4 great circles 
 = 3-1416 
 
 X l At*A 
 
 Top. .= -7854 
 
 Mid.X 4 "= 3 ' 1416 
 Base. .= -7854 
 
 Base . = -7854 
 1 
 
 6)3-i41G 
 
 Solidity** -6-236 
 
 6)4-7121 
 
 Shdity= -7854 
 
 2618 
 1 
 
 Solidity = -26T8 
 
 523i 
 .' : ' > 
 Solidity = -5236 
 
 XI 
 Solidity = -7854 
 
 
 
 
 
 
 
 
 
 c 
 
 
 
 
 a b = 
 
 The Base. 
 
 Flg.W. 
 
 /w\ 
 
 
 
 c d = 
 
 " Top. 
 
 IU 
 
 V/ V 
 
 s* 
 
 
 ' m s = 
 
 " Mid-section. 
 
 
 
 h 
 
 
 
 
 
 1. 
 
 
 
 The common rules of mensuration are drawn from geometry but 
 geometry also teaches that a cone, a sphere, and a cylinder, dimen- 
 sioned and situated as shown by their right sections, in Fig. 47, have 
 
 . Chauvenet, ix. 3, 7, 14, Geom. (1871). Borden's Useful Formulas (1851). Henok'a 
 Field Book (1854), art. 112. 
 
64 MEASUREMENT OF EARTHWORKS. 
 
 their volumes in the ratio of the numbers 1, 2, and 3. Now, the 
 above calculations show the same result numerically, which, with the 
 preceding observations, furnish an adequate demonstration. 
 
 In like manner we might show that the Prismoidal Formula applies 
 to all the separate geometrical solids, which, when aggregated, form 
 the irregular prismoid known as an Earthivork Solid. 
 
 Now, considering this species of solid as a prismoid, within the 
 limits of Button's definition (1770), we find that all such admit of 
 decomposition into Prisms, Prismoids,* Pyramids, or Wedges (complete 
 or truncated), or some combination of them, having a common length, 
 or hight, equal to the distance between the end areas or cross-sections, 
 and either separately or together computable by the Prismoidal Formula 
 as a general rule for all. 
 
 By a similar analogy (to the three round bodies), we find somewhat 
 like relations to obtain between what we may call the three square or 
 angular bodies; which geometry-shows to exist alike amongst them 
 all, the round bodies being referred to the cylinder; the square or 
 angular ones to the cube. But the wedge requires this special defini- 
 tion, that the edge be double the back. 
 
 1. A Pyramid, with a square base, on a side of 1, and 
 
 having also an altitude of 1, has a volume . . . .. = |. 
 
 2. A Wedge, doubled on the edge, with a square back, on a 
 
 side of 1, the edge parallel = 2 (or double the back), 
 and an altitude of 1, has a volume = I. 
 
 3. A Cube, or Hexaedron, with its six square faces, each 
 
 formed upon a side of 1, has a volume = 1. 
 
 So that, finally, we have, both in the three round, and in the three 
 square bodies (as defined) where unity is the controlling dimension, 
 like ratios of volume. 
 
 Thus, these six bodies, 
 
 ( Cone and 
 \ Pyramid. 
 
 Sphere and 
 
 Wedge 
 (doubled on the edge). 
 
 Cylinder "] Solids of 
 and Cube. | Circular 
 and 
 
 ra"oTof th voiuml = 1- 2 - 3 J Square Bases. 
 
 And of each and all of these alike, the Prismoidal Formula gives the 
 Solidity, 
 
 * The Rectangular Prismoid being always divisible into two \vedges. 
 
CHAP. L PRELIM. PROBS. ART. 14. 65 
 
 14. Transformation of Areas into Equivalent ones, Simpler in Form, 
 and of Solids into Equivalents, more readily Computable by the Pris- 
 moidal Formula, or its Modifications. 
 
 Hutton hath defined a Prismoid as follows: 
 
 " A Prismoid is a solid having for its two ends any dissimilar 
 plane figures of the same number of sides, and all the sides of 
 the solid plane figures also." (Quarto Mens., 1770.) 
 
 This is the oldest and best definition of the Prismoid which we are 
 able to find on record.* 
 
 Under this definition, for which the General Rule (coinciding with 
 Simpson's) was framed by Hutton, it is clear that we ought not to 
 expect of the Prismoidal Formula the cubature of curvilinear solids, 
 though, by a happy coincidence, it applies to many such, which are 
 not prismoids at all, nor in the least resemble them, geometrically. 
 
 But though often true of this remarkable formula, where a correct 
 mid-section can be first obtained, it by no means follows that its 
 numerous modifications (all framed for right-lined solids) will, like 
 their principal, also hold, as it does in many singular cases exactly, and 
 in most others approximately. 
 
 It was early discovered that it would materially simplify the com- 
 putation of irregular prismoids, to transform them into equivalent 
 right-lined bodies, of which the nature was better known, and the 
 forms more regular and simple. 
 
 As the calculations for level ground were obviously the most easy, 
 Sir John Macneill, in his Tables of 1833, adopted for the end sections 
 the principle of transformation into level hights, to contain equivalent 
 level areas and was, in fact, the originator of what has since been 
 known as the Method of Equivalent Level Hights by means of which, 
 the end sections of irregular prismoids of earthwork are transformed 
 into level trapezoids, which are then employed to compute an equiva- 
 lent solid of the same length, and transversely level, at top or bottom, 
 according as it may be excavation or embankment each, however, 
 representing the other, when inverted. 
 
 Sir John Macneill has been followed, more or less closely, by most 
 of the authors of Earthwork tables, the bulk of which are applicable 
 to level ground alone, or ground reduced to such ; though Watner's 
 System of Earthwork Computation (1861) deals with ground how- 
 ever sloping, or even warped, within certain limits. 
 
 * See also Henck's Field Book (1854). Davies Legendre (1853). Haswoll's Mens. 
 (1863). Bonnycastle's Mens. (1807). Hawnev's Mens. (1798). All define the Pris- 
 moid ni a right-lined solid. 
 
 5 
 
66 MEASUREMENT OF EARTHWORKS. 
 
 The method of using Equivalent Level Hights (when the cross- 
 section of the ground is not level) has been concisely explained, by a 
 recent writer, to consist in finding* 
 
 1. " The area of a cross-section at each end of the mass." 
 
 2. " The hight of a section, level at the top, equivalent in area to 
 
 each of these end sections." 
 
 3. " From the average of these two hights, the middle area of 
 
 the mass." 
 
 " And, lastly, in applying the Prismoidal Formula to find the 
 contents." 
 
 It is obviously necessary then to understand what is meant by 
 equivalency and this we find from Geometry .f 
 
 1 . " Equivalent (plane) figures are those which have the same 
 
 surface measured by the area." 
 
 2. "Equivalent solids are those which have the same bulk or 
 
 magnitude." 
 
 " Theorem: If two solids have equal bases and hights, and if 
 their sections made by any plane parallel to the 
 common plane of their bases are equal, they are 
 equivalent." 
 
 Now, the transformation of triangular prismoids of earthwork, by 
 means of Equivalent Level Hights, meets every point of Professor 
 Peirce's definitions of equivalency, and hence the solid they produce 
 may be regarded as equivalent to the original defined by Hutton : in 
 the above theorem, equality of sections evidently means equality in 
 area, and not geometrical equality, which is somewhat different. 
 
 Some writers have doubted the accuracy of the transformation or 
 equivalency produced by Equivalent Level Hights,J but it is because 
 the solids, which they found in error, were either not prismoids at all, 
 or else the data used were inadequate to the solution of the problem. 
 
 An error in this direction is not surprising ; for when we know that 
 the Prismoidal Formula applies correctly to a solid, we are apt to 
 infer that its modifications also do, and here the error lies. 
 
 For.instance, we know this formula does apply correctly to a sphere, 
 but if we test that solid, by the method of Equivalent Level Hights, 
 we should find that the end sections being 0, have a hight of 0, and 
 that the mid-section being constructed on a mean of like parts in the 
 
 * Henck's Field Book (1854). f Peirce's Plane and Solid Geom. (1837). 
 
 J Gillespie, Frank. Inst. Jour. (1859). 
 
CHAP. I. PRELIM. PROBS. ART. 14. 
 
 67 
 
 ends must also equal 0, and hence we might in this way legitimately 
 come to the conclusion that the globe itself had a solidity of ! This 
 shows that Equivalent Level Hights are limited in range. 
 
 The error obviously is that all, or most of the transformations and 
 modifications of the Prisrnoidal Formula, are intended for right-lined 
 solids, " varying uniformly " from end to end, like a stick of timber 
 dressed off tapering, and to all such rectilinear solids they do apply 
 correctly ; but not to those which bulge out, or curve in, by laws 
 unknoivn to Huttoris definition of the Prismoid. 
 
 It would be easy to illustrate this by examples, and to show that, 
 confined within proper limits, the usual modifications of the Prismoidal 
 Formula are correct enough for practical use ; but they have not the 
 wide range of their principal; nor must they be expected to apply 
 either to the three round bodies, or to warped solids, but only to right- 
 lined ones, varying uniformly, or nearly so, from end to end. 
 
 One important point, however, must not be overlooked in applying 
 the Prismoidal Formula (or its modifications) to cases of earthwork: 
 that is, the ground must be properly cross-sectioned; or, have its sections 
 judiciously located, while the hights and distances of its controlling 
 points are correctly measured and recorded, prior to undertaking the 
 calculations of solidity. 
 
 It is in this point that Borden's ridge and holloiv problem fails* 
 Had one or more intermediate cross-sections been adopted there, no 
 difficulty would have existed in its calculation, either by Borden him- 
 self, or by subsequent students. 
 
 To illustrate this subject, we will 
 give an example, drawn from Simp- 
 son's original Prismoid of 1750, on 
 which he founded the Prismoidal 
 Formula, or used to explain it. 
 Art. 2, Fig. 2. (And see Figs. 48, 
 49, 50, 51.) 
 
 2.1. 82 
 
 Here we will take the Prismoid as 
 being cut in two, by the diagonal 
 plane, through DB, so as to divide it 
 into triangular prismoids, and then 
 calculate one of these halves in three 
 ways. 
 
 Fig. 49 
 
 * Borden's Useful Formulas, etc. (1851). Henck's Field Book (1854). 
 
68 
 
 MEASUREMENT OF EARTHWORKS. 
 
 1. By Simpson's Rule, as the 
 
 half of a rectangular pris- 
 moid, dimensioned as in 
 Fig. 2. 
 
 2. By Hights and Widths, as a 
 
 triangular earthwork solid, 
 with unequal side-slopes. 
 (See Figs. 48, 49.) 
 
 3. By Equivalent Level Hights 
 
 purely as an equivalent tri- 
 angular prismoid, or earth- 
 work solid, within a diedral 
 angle of 90, and having 
 equal side-slopes of 1 to 1. 
 
 In all these figures the angle A = 90. 
 B and B, Figs. 48 and 49 = 38 40', and 33 41'. 
 ' 48 and 50 = 320. 
 
 The common hight of the prismoids being h = 24. All the calcu- 
 lations being carried out in detail ; all having the same end areas, 
 320 and 216; and all dimensioned as marked upon the figures. 
 
 We find, then, by all these calculations, the Solidity to be the same 
 = 3200, varying but a few small decimals, and agreeing with the 
 results already ascertained in Art. 2. 
 
 This exhibits the equivalency we have been discussing (the figures 
 being quite unlike), and might readily be extended to more compli- 
 cated examples, with a like result. 
 
 15. Equivalence of some important Formulas, for .computing the 
 Solidity of Triangular Prismoids of Earthwork, contained within 
 Diedral Angles, formed by Prolonging the Side-slope Planes to an Edge. 
 
 Equivalent Formulas are those which reach the same results by 
 unlike steps and in mathematical processes it is often found that a 
 general formula will hold in many cases, usually governed by concise 
 special rules, and yet produce identical results. 
 
 This is equivalency, and relates^ in mensuration especially to the 
 Prismoidal Formula, which appears to have a sort of concurrent juris- 
 diction over the domain of solid geometry, along with the special 
 rules for the volume of each separate solid, producing exactly the 
 same results, though by different steps. 
 
CHAP. I. PRELIM. PROBS. ART. 15. 69 
 
 Such is particularly the case in earthwork solids, contained (as 
 they mostly are) in diedral angles formed by uniform planes, called 
 side-slopes, and having a general triangular section two sides being 
 the inclined lateral planes, known as side-slopes (continued to inter- 
 sect for computation), and these slopes being usually alike in inclina- 
 tion, while the contained angle is equal ; the third side, or ground 
 line, alone being variable, and often irregular. 
 
 By geometry, triangles having an angle common or equal, and the 
 containing sides proportional, are similar ; and the areas of similar 
 triangles are always proportional to the squares of any similar or 
 homologous lines, or to the rectangles of such as have like positions 
 and relations to each other : as the squares of perpendiculars from 
 the equal angles, or their bisectors, the rectangles of containing sides, 
 the product of hights and widths, etc. 
 
 Now, these triangular sections of an earthwork solid, extending 
 (for computation) from the ground surface to the intersection of the 
 side-slopes prolonged to an edge, are sections of triangular pyramids, as 
 well as of prismoids ; and to such solids the rules for Pyramids, and 
 their frusta, as well as the Prismoidal Formula, and its modifica- 
 tions, apply concurrently, and either may be used at will, with correct 
 results. 
 
 These considerations regarding the equivalency of Pyramidal and 
 Prismoidal Formulas in such cases are important, and require to be 
 well considered by computers of earthwork. 
 
 Hutton's definition of the Prismoid is based on three conditions: 
 
 1. The two ends must be dissimilar parallel plane figures. 
 
 2. They must have an equal number of sides. 
 
 3. The faces, or sides of the solid, must be plane figures also. 
 Usually, says Hutton, the faces are plane trapezoids. 
 
 Considering, now, a regular prismoid as being composed of known 
 elementary solids. 
 
 Macneill regards it as formed of a prism, with a wedge superposed. 
 Art. 4 (and this is also the case with a frustum of a pyramid, turned 
 upon its edge). 
 
 Hutton, of two wedges, formed by a single cutting plane passed in 
 a diagonal direction, Art. 3. 
 
 The writer, as a triangular prism trebly truncated, Art. 1. 
 
 Simpson (the father of the prismoid) gives no special definition, 
 but figures in his work of 1750 a rectangular prismoid (the same or 
 
70 MEASUREMENT OF EARTHWORKS. 
 
 similar to that adopted and figured by Hutton, 1770); and by a 
 single diagonal plane, convertible into two triangular prismoids. 
 (See Fig. 2.) 
 
 Now, as a triangle is the simplest of all polygons, so a prismoid 
 within a diedral angle (triangular in section) may be considered as 
 the simplest of all prismoids, though the rectangular prismoid is 
 nearly so. 
 
 The simplest case of the ordinary trapezoidal prismoid of earthwork 
 is in, or upon, ground level transversely. 
 
 In that case, the cross-sections are level trapezoids, and the solid is 
 obviously composed of a prism and superposed wedge, as in Macneill's 
 solid, Art. 4. 
 
 Its volume may be computed by Simpson's, or by Button's general 
 rules, because this solid then is strictly a prismoid within the scope 
 of Hutton's definition, and as a whole computable only by prismoidal 
 rules. 
 
 But suppose the assumed road-bed was taken less and less, until we 
 reached the edge of the diedral angle, and it became zero. 
 
 Then, the cross-section from a trapezoid becomes a triangle, and the 
 prismoid changes at once into a fmstum of a pyramid a solid known 
 since the days of Euclid. 
 
 This solid becomes then computable by Euclid's geometry, as the 
 frustum of a pyramid or by Equivalent Level Hights by roots and 
 squares by geometrical average all of which are equivalent, as are 
 the similar rules of Bidder, Baker, Bash forth, and others ; or, by 
 wedge and prism, by hights and widths (Simpson), by Hutton's par- 
 ticular rule, by the method of initial prismoids, or, finally, by the 
 Prismoidal Formula itself, which always holds alike for prismoids, 
 pyramids, or pyramidal frusta. 
 
 Hutton (4to Mens., 1770, p. 155) shows that in similar sections of 
 a pyramidal frustum (say triangular) the squares of similar lines, as 
 the bisector of an equal angle (which the centre line of a railroad 
 generally is), are as the areas of the cross-sections, or, conversely, the 
 areas are as the squares of similar lines (Chauvenet's Geom. iv. 7). 
 
 Then, from Hutton's prob. 7, cor. 2, we have a formula (for pyra- 
 midal frusta) in which, substituting Bidder's and Baker's notation, 
 we have, by a slight reduction, the identical rules given by those 
 authors for the computation of earthwork.* 
 
 * Bidder, quoted in Dempsey's Prnc. Rail. Eng., London, 1855. Baker, in his Rail- 
 way Eng. and Earthwork, London, 1848. 
 
CHAP. I. PRELIM. PROBS. ART. 15. 
 
 71 
 
 We will now give a diagram to illustrate the equivalency of prismoi- 
 dal and pyramidal formulas. 
 
 Fig. 52. 
 
 Road-Tied-lO. 
 
 Fig. 52 represents the full station of earthwork, already shown in 
 Figs. 22 and 24, having a road-bed of 18 feet, and side-slopes of li to 
 1, with other dimensions as marked upon the figures. 
 
 Suppose, in all cases (as in Fig. 52), the trapezoidal sections of the 
 ends above the road-bed to be carried down by prolonging the side- 
 slopes to their intersection at I I, the edge of the diedral angle. 
 
 ( c c = Top of larger end, and h = its hight = 30 feet. 
 Let\bb = Top of smaller end, and h' = its hight = 20 feet. 
 ( I = The intersection of side-slopes, of 1J to 1. 
 
 Then, suppose a horizontal plane to be passed parallel to 1 1, through 
 bbb b, then ccbbb b, the part cut off, is a wedge, its edge being b b, 
 the top of the forward cross-section ; while h h' = the hight of the 
 back c c b b, and as a wedge it may easily be calculated. 
 
 Now, suppose the plane b bb b moves downward, parallel always to 
 its first position at the distance h f from I, then the solid immediately 
 becomes a prismoid being then a prism with a wedge superposed, as 
 in Art. 4 (or analogous to it). 
 
 Continue this parallel movement of the plane downward until we 
 reach the position a a a, assumed for the road-bed, and then we have 
 the precise case of A rt. 4 Sir John Macneill's figure of 1833. To 
 this of course the Prismoidal Formula applies, but the Pyramidal For- 
 mulas do not. 
 
 Continue on again, with the movement of our supposed horizontal 
 plane downwards, until it comes to I, I, (the junction of the side-slopes), 
 then the solid becomes the frustum of a pyramid, triangular in sec- 
 tion, and the wedge is absorbed ; nevertheless, a frustum of a pyramid 
 
72 MEASUREMENT OF EARTHWORKS. 
 
 is also in tins respect like unto a prismoid, and may, if we choose, be 
 regarded as a prism with a wedge superposed, and forming the top 
 of the solid. 
 
 Taking the horizontal plane, supposed to move parallel downwards, 
 at three particular points of its progress, at b, a, and I, the calcula- 
 tions for volume would be, 
 
 1. For the wedge alone = ccb bbb 
 
 2. " wedge and prism, or prismoid = ccaaabb. 
 
 3. " frustum of a pyramid alone, both wedge and prism being 
 
 merged in it and in such case this is the simplest and 
 best form of calculation, for volume. 
 
 We may here remark that so long as the end cross-sections contain 
 a road-bed of definite width, the solid is a real prismoid, and must be 
 computed as such by prismoidal rules alone; but the moment the 
 angle at I becomes common to both, then the solid becomes a regular 
 frustum of a pyramid, and all the pyramidal rules apply, as well as 
 the prismoidal ones, to which they are strictly equivalent, whenever I, 
 the diedral edge, is common to both. 
 
 Now, suppose the case reversed, and that the horizontal plane was 
 originally passed through I, I, (edge of diedral angle), and moves 
 gradually upwards, parallel. 
 
 At every step of its progress, the solid, cut off above I, is always a 
 prism, until its limit has been reached, at b b b b, the top of the 
 smaller end here the moving horizontal plane ceases to be longer 
 useful in illustration; and becoming fixed at one end, on the top of 
 the far end section as an axis, opens wider and wider at the near end, 
 until it attains the line cc (the top of the main solid), and completes 
 the wedge we have referred to, and the pyramidal frustum with it. 
 
 In this position the whole solid is undeniably a prismoid (if we 
 allow to it an infinitesimal road-bed). So, also, it is a frustum of a trian- 
 gular pyramid, both being strictly equivalent, and both computable by 
 the regular rules for either* 
 
 We will now illustrate this equivalence of the Prismoidal and Pyra- 
 midal Formulas, in their application to earthwork solids, within 
 diedral angles, by a few examples. 
 
 Taking the dimensions of Figs. 22 and 24, with 1 to 1 side-slopes, 
 and road-bed of 18, for thfe numbers to be employed the diedral 
 angle being common to both. 
 
 * As might be inferred from Button's remarkable chapter on the Cubature of Curves 
 (4to Hens., 1770). 
 
CHAP. I. PRELIM. PROBS. ART. 15. 73 
 
 1. Priwnoidally. By the direct and cross multiplication of Hights 
 and Widths. Formula at the end of Art. 9 ...... VIII. 
 
 TT . ( h = 30 \s w = 90 ) w . ,,, 
 
 Hights | /t , = 20 X u'' = 60 j Wldths - 
 
 30 20 30 90 2700 
 
 90 60 60 20 1200 
 
 2700 1200 2)1800 + 1800 
 
 6)5700 
 
 950 X 100 =- 95000 = 
 Solidity, as before computed. 
 
 2. Pyramidally. By the rules of Baker's Earthwork. 
 
 30 
 30 
 
 900 
 
 r 
 I 
 
 20 30 
 20 20 
 
 400 600 
 
 = H 
 = 100 
 
 900 
 400 
 600 
 
 1900 
 50 
 
 95000 
 
 3)150 
 ~~50 
 
 Solidity, as before computed 
 
 3. Prismoidally. By Simpson's rule, modified for triangular solids. 
 
 Ilights. Widths. 
 
 30 X 90 = 2700 
 20 X 60 = 1200 
 
 Sums, 50 X 150 = 7500 
 12)11400 
 
 950 X 100 = 95000 = Solidity, as 
 before computed. 
 
 4. Pyramidally. By Roots and Squares, Art. 10 (c). 
 
 End Areas . . = 1350 600 
 Roots . . = 36-74 24-50 
 
 Sum ....== 61-24 
 Square of Sum = 3750 
 End Areas. . = { ^ 
 
 6)5700 
 
 950 X 100 = 95000 = Solidity, as 
 before computed. 
 
74 MEASUREMENT OF EARTHWORKS. 
 
 5. Finally, by Warner's Earthwork, Art. 112. 
 
 Hts. Wds. 
 
 Difference = 10 j ^ x 60 } Difference = 30 - 
 
 Sums . 50 X 150 . . = 7500 
 
 937-5 = 1st term. 
 
 X 100 = 95000 = Solidity. 
 
 So, we may safely assume that the Pyramidal Formulas of Bidder, 
 Baker, and others, the Geometrical Average, Equivalent Level Rights, 
 Euclid's rule for the frustum of a pyramid, etc., are all strictly equiva- 
 lent to the Prismoidal Formula, and its modifications, when applied to 
 earthwork solids, within diedral angles, on ground transversely level. 
 
 16. Summary of Rules and Formulas from the Preliminary Problems. 
 
 It will be found convenient to use, substantially, the same notation 
 for the Prismoidal Formula, and its numerous modifications, wher- 
 ever practicable. 
 
 b = Base, or area of end assumed for such. 
 t = Top, or area at the other end. 
 
 Thus let^ m = Hypothetical Mid-section, used in computation. 
 h = Length or hight of the Prismoid. 
 S = Solidity or volume. 
 
 Then, the Prismoidal Formula can always be in substance expressed 
 by ^ - X h S, when a mean area is desired, or by 
 
 (b -f 4 m -\- f) X i h S, for rectangular prismoids, or equivalent 
 solids; or, when triangular prismoids are under computation, 
 
 2 b + 2 t f 8 ?TI 
 
 ----- X h = fe, equivalent in using triangular sections 
 
 and double areas, to this rule in words : The separate products of hights 
 by widths at each end, plus product of sums of hights and widths at both 
 ends, and the sum of these three products, multiplied by ^ h = Solidity. 
 The following modification of this rule may be sometimes useful in 
 computing the volume of triangular earthwork solids : The products 
 of the direct multiplication of hight by width at each end, plus sum of 
 half products of the cross multiplications of alternate hights and widths a*, 
 
CHAP. I. PRELIM. PROBS. ART. 16. 
 
 75 
 
 both ends, multiplied by h = solidity from ground to intersection of 
 slopes, and mimis the grade prism = solidity from road-bed to ground. 
 
 Many other expressions are assumed for special purposes by the 
 Prismoidal Formula ; but no matter into what shape it be transformed, 
 the essential idea must always be borne in mind that this formula, in 
 words, concisely is, 
 
 " The sum of the areas of the two ends, and four times the sec- 
 tion in the middle, multiplied into h = S." (Hutton, 1770.) 
 
 Such is the simple expression of this celebrated formula given a 
 century ago which applies not only to all prismoids, but to all right- 
 lined solids, and many curved ones too.* 
 
 
 SUMMARY. 
 
 For rectangular prismoids, or any prismoid, reduced 
 to an equivalent rectangular section, we have Simp- 
 son's original rule expressed by sides of the end rect- 
 angles, referring to Fig. 2, Art. 2. But it is more 
 convenient, perhaps, for our purpose, to designate 
 these sides relatively, as hights and widths, and in this 
 form we ma^ write Simpson's rule as follows : 
 
 (Hight X Width of one end) + (Right X Width 
 of other end) -f (Sum of Hights X Sum of Widths 
 of both ends) X h == S. 
 
 And the transformation of this formula, for use in 
 the computation of triangular prismoids (like earth- 
 work), placing it in Button's form. 
 
 TViq TVTpflll A*OQ art A N/ ^ . Ks\7frJ*fai 
 
 Article. I Formula. 
 
 2. 
 
 2. 
 3. 
 3. 
 
 I. 
 
 II. 
 III. 
 IV. 
 
 2 
 
 For rectangular prismoids, considered as two wedges. 
 We have Hutton's General Rule for any prismoid, 
 (6 + < + 4m)X/> 
 
 6 
 We have also Hutton's Particular Rule. 
 
 (2L + 1 X B + 21 + L X 6) X $h = S. 
 
 * The English engineers have for many years unhesitatingly applied this formula to 
 the warped solids of earthwork. See Dempsey's Practical Railway Engineer, 4th edition, 
 4to, London (1855), pp. 71 to 74. And in this country, Prof. Gillespie (1857), and John 
 Warner, A. M. (1861), have also discussed the subject of Warped Solids of Earthwork. 
 
76 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Article. 
 
 Formula. 
 
 SUMMARY Continued. 
 
 3. 
 
 V. 
 
 For unusual and irregular prismoids we have the 
 
 
 
 method of " Initial 
 
 Prismoids" deduced from Hutton. 
 
 6. 
 
 VI. 
 
 For a prismoid, 
 
 composed of a prism and wedge, 
 
 
 
 superposed. 
 
 
 
 
 (B + b -f b) X (H 
 
 1 ( n w fipvnfir* f'vinnn'lo^ \S 
 
 6 
 
 
 7. 
 
 VII. 
 
 For a trapezoidal prismoid of earthwork, taken as 
 
 
 
 two wedges. 
 
 
 
 
 We have the following Rule : 
 
 
 
 
 Add road-bed + top- width -f- 
 
 
 
 In 1st cross-section 
 
 road-bed of 2d section ; multiply the 
 sum of these three by level hight 
 
 
 
 
 of section, and reserve the product. 
 
 
 
 
 Add road-bed -f top-width -f top- 
 
 
 
 In 2d cross-section 
 
 width *of 1st section; multiply the 
 sum of these three by level hight 
 
 
 
 
 of section, and reserve the product. 
 
 
 
 Finally, add the 
 
 two products reserved, and of 
 
 
 
 their sum is the mean area of the Prismoid, which, 
 
 
 
 multiplied by length = Solidity. 
 
 
 
 For a triangular prismoid of earthwork, we have 
 
 
 
 the following modification of the Prismoidal Formula, 
 
 
 
 operating by direct and cross-multiplication of hights 
 
 ' 
 
 
 and widths. All 
 
 hights being taken at centre from 
 
 
 
 ground to intersection of slopes, and all widths from 
 
 
 
 top to top of slopes on both sides of centre. 
 
 
 
 Let h and h r = 
 
 the hights. w and w f = the widths. 
 
 
 
 Then, 
 
 
 
 
 Hights. Widths. \ 
 
 
 
 
 h X iv 
 
 h w r -4- Ti r w 
 
 9. 
 
 VIII. 
 
 X 
 
 h' X w' 
 
 /> -)/, _i_ // /!// _L_ 
 
 
 ,and 6 
 
 
 
 Length = 100, 
 
 length = S. 
 
 
 
 usually. / 
 
 
Article 
 
 Formula. 
 
 CHAP. I. PRELIM. PROBS. ART. 16. - 
 SUMMARY Continued 
 
 10. 
 
 10. 
 
 IX. 
 
 10. 
 
 XI. 
 
 Simpson's Rule, for the Quadrature and Cubature 
 of Curves (adopted by Hutton), and copied from the 
 4to Mens. (1770). 
 
 Sum extreme ordinates = A. "| 
 " all even " = B. [A-f 4B + 2C 
 " all odd " = C. [ ~3~~ 
 Common distance = D. J D = area or solidity. 
 
 For convenience we may transform this into, 
 
 X 2 D = area or solidity. 
 
 To find the solidity of a triangular prisraoid by 
 roots and squares. 
 
 ' h and h' = The end hights or representative 
 square roots of the areas of the ends (between 
 ground and intersection of slopes), at regular 
 stations, numbered even. 
 
 m Place of mid-section, represented by its ordi- 
 nate, and numbered odd. 
 
 Length = Usually, 100, between principal sta- 
 tions. 
 
 &' -f fr' 2 + (h + hj vx , 
 
 |^ - X length = S. 
 
 "Which, for one station, is equivalent to Hutton's 
 rule above. This is a very important transformation 
 of the Prismoidal Formula, and should be well con- 
 sidered, with the examples in Art. 1O. 
 
 One of the earliest followers, in the path projected 
 by Sir John Macneill, of using the Prismoidal For- 
 mula, with auxiliary tables, for correctly computing 
 the volume of earthwork solids, was G. P. Bidder, 
 C. E., who adopted the obvious plan of imagining the 
 side-slopes to be moved parallel inward, to intersect at 
 grade, and then computing the triangular solid thus 
 formed as a prismoid, or the frustum of a pyramid 
 (both being equivalent in these circumstances) ; finally, 
 calculating the centre part (or core) as a prism sepa- 
 rately, and adding the two for the volume of the whole. 
 The core being computed for one foot wide only, 
 
MEASUREMENT OF EARTHWORKS. 
 
 Article. 
 
 Formula. 
 
 SUMMARY Continued. 
 
 and then multiplied by the width of road-bed intended 
 to be given.* (This is the plan of Macneill's second 
 series of Tables, for various side-slopes, and base of 
 one foot.) 
 
 Bidder's formula for the slopes united is, [ (a -f &) 2 
 a 5] || = S, in cubic yards for a 66 foot chain, a 
 and b being the hights or depths at the ends. 
 
 This is identical with the formulas of Baker, Bash- 
 forth, and others, of subsequent writers : = (a 2 -f a b 
 + 6' 2 ) II = S, in cubic yards, and is in fact the alge- 
 braic expression for the volume of the frustum of a tri- 
 angular pyramid, demonstrated in all the elements of 
 geometry supposed to have been originated by Euclid 
 (about 300 B. c.), and known in this country as the 
 method of Geometrical Average. 
 
 These formulas are equivalent to the following, men- 
 tioned in Art. 12. 
 
 (Sum of sqs. of hts.) -f (Sq. of sum of hts.) ^ a 
 
 12. 
 
 XII. 
 
 6 
 
 _ 2 (Sum sqs.) -f 2 (Rect. of hights) 
 
 (Sum sqs. of hights) + (Rect. of hights) w ^ Q 
 
 X "- k, 
 
 which, for a four pole chain, and cubic yards, becomes 
 equivalent to the formulas above, by introducing the 
 proper fractional multipliers the hights are the square 
 roots of the areas. 
 
 * A similar plan of computing and tabulating the slopes and core 
 separately : the latter on a base of unity, to be subsequently multi- 
 plied, by any road-bed, is also that of E. F. Johnson, C. E. the 
 pioneer of Earthwork Tables in this country (New York, 1840) and 
 has been followed by several other writers; indeed, it is a method so 
 obvious as to be likely to occur to any student. This core and slope 
 method originated by Bidder and Johnson (some 30 years ago), and 
 since repeated by numerous 1 writers, is now again reiterated by the 
 latest compiler of Earthwork Tables, E. C. Rice, C. E. (St. Louis, 
 Mo., 1870). 
 
CHAPTER II. 
 
 FIRST METHOD OF COMPUTATION BY MID-SECTIONS, DRAWN AND 
 CALCULATED FOR AREA, ON THE BASIS OF BUTTON'S GENERAL 
 RULE. 
 
 17 Since 1833 the date of publication of Sir John Macneill's 
 
 meritorious volume on the mensuration of earthworks, for canals, 
 roads, and railroads the investigations of numerous able writers in 
 various countries have shown, conclusively, that the Prismoidal For- 
 mula (adopted by Macneill) furnishes the most convenient, if not the 
 only correct rule for the measurement of the immense bodies of mate- 
 rial employed in earthworks, and removed from, or supplied to, the 
 irregularities of the ground encountered by the location of lines, 
 under the general name of excavation or embankment. 
 
 The writer, as long ago as 1840, in the Journal of the Franklin 
 Institute of Pennsylvania, repeated the demonstration of the formula 
 referred to, by means of a simple figure, and established its connection 
 with the ordinary rules for the volume of the three principal right- 
 lined bodies, known to solid mensuration the Prism, Wedge, and 
 Pyramid (to all of which, whether complete or truncated, the Pris- 
 moidal Formula correctly applies) ; these are the elementary solids 
 which enter into the composition of a station of earthwork, and sepa- 
 rately, or together, are all computable by the same rule. 
 
 He also showed, by numerous examples (worked out in detail) of 
 the leading forms assumed by railroad earthworks, that by means of 
 hypothetical mid-sections, deduced from the usual cross-sections taken 
 in the field (and diagrammed between them if necessary), the volumes 
 of excavation and embankment solids could be computed correctly 
 without unusual labor, and with more than usual accuracy. This 
 method was made to depend essentially upon two points : * 
 
 * Journal of the Franklin Institute (Philadelphia, 1840). 
 
 79 
 
gO MEASUREMENT OF EARTHWORKS. 
 
 1. "That the formula expressing the capacity of a prismoid is 
 the fundamental rule for the mensuration of all right-lined solids, 
 \vhose terminations lie in parallel planes, and is equally applica- 
 ble to each." 
 
 2. " That any solid whatever, bounded by planes, and parallel 
 ends, may be regarded as composed of some combination of 
 prisms, prism oids, pyramids, and wedges, or their frusta, having 
 a common altitude, and hence capable of computation by the gen- 
 eral rule for prismoids." 
 
 All excavation and embankment solids come within the scope of 
 these definitions, and all are computable with ease and accuracy by 
 means of the Prismoidal Formula. 
 
 These views have met with general acceptance from most practical 
 writers, but many useful transformations and modifications have 
 naturally been indicated ; all grounded upon the same formula which 
 appears to have originated with THOMAS SIMPSON, an eminent mathe- 
 matician, and was demonstrated and published by him (for rectangular 
 prismoids) in London, 1750 (Arts. 1 and 2), but generalized and 
 made more useful by HUTTON, in 1770 (Art. 3). 
 
 This extraordinary formula is not only the fundamental rule for 
 all right-lined solids, but reaches also to many curved bodies and 
 warped surfaces (as before mentioned), so that it may safely be 
 assumed as correct for all the earthwork solids in common use, which, 
 indeed, are invariably laid out with the view of reducing the ground, 
 however irregular, to equivalent planes (as near as may be), by means 
 of leyels and sections, taken at short distances ; and though this effort 
 may not be entirely successful in practice, it must be so nearly so that 
 the warped surfaces, remaining involved in the solid, can only differ 
 slightly (if at all) from those for which the Prismoidal Formula is 
 known to hold. 
 
 As a general rule, it may therefore be considered as close an 
 approximation to existing facts as is admitted by any convenient 
 method within the present range of human knowledge, and far more 
 accurate than any of the proximate rules, which have been extensively 
 employed for the solution of the complicated problems of earthwork. 
 
 As a preliminary matter, it is necessary now to make some remarks 
 on the manner of collecting data in the field, for subsequent use in 
 calculating the quantities of earthwork solids. 
 
 The centre or guiding line of the road or work having been care- 
 fully located upon the ground, and marked off in regular stations 
 
CHAP. II. FIRST METH. COMP. ART. 17. 81 
 
 usually of one hundred feet each the next operation is to cross-section 
 the work, with level, rod, and tape; most engineers also using the 
 clinometer, or slope level, as an auxiliary, in some stages of the pro- 
 cess. The centre line is assumed in all cases to be straight, from point 
 to point, and generally to be a tangent line, to which the cross-sec- 
 tions are perpendicular, but owing to the convergence of the radii 
 upon curves, this is not strictly correct though within the limits of 
 the work staked out, that convergence is but slight ; nevertheless, the 
 cross-sections (before proceeding to level them) should be set out 
 approximately, normal to the tangents, and radial to the curves ; and 
 upon all curves, or at least on all of small radius, intermediates at half 
 distance should be placed, or, if the curves are unusually sharp, even 
 at the quarter of a regular station. 
 
 Some engineer manuals furnish formula for the correction of quan- 
 tities upon curved lines,* but they are rarely used ; a simple reduction 
 of distance between the cross-sections, or a closer assemblage of them, 
 being usually deemed sufficient. 
 
 The surface of the ground f is regarded by the engineer as being 
 composed of planes variously disposed, with relation to each other, so 
 
 * The simplest and most convenient rule for this purpose, is that of Warner's Earth- 
 work (1861). This rule has been adopted, and somewhat simplified, by Prof. Rankine, 
 in Useful Rules, etc. (London, 1866). 
 
 The process is : First, to calculate the solidity of the earthwork to the intersection of 
 the slopes (as though the line were straight), and then to multiply it by a factor, which 
 corrects for curvature. 
 
 Difference slope distances , 
 
 This factor is found thus : - - - - 'jbl The corrective quotient 
 
 6 Radius of curve. 
 
 being added to unity, when the greater slope distance lies outward from the curve, or 
 subtracted, if otherwise. 
 
 For example, take a curve of 700 feet radius, lying upon a heavy embankment, along 
 a ground surface sloping uniformly inwards, towards the centre of the curve, at the rate 
 of 15. The road-bed being 24 feet wide, and side-slopes 1 J to 1. 
 
 Let the difference of slope distances be 42 feet, the greater being inwards, and suppose 
 the whole volume, for straight work = 5917 cubic yards to intersection of slope. Then, 
 
 = -02, and 1 -02 = -98, the factor required. Then, 5917 X *9 8 
 
 6 X < "" 
 
 cubic yards, and 5799 grade prism (356) = 5443 cubic yards, the volume, corrected for 
 curvature. The difference in this case, produced by the curvature of the line, being 118 
 cubic yards, for the station computed. 
 
 The correction for other curves would be inversely as their radii, and for a 1 curve, 
 similarly situated, about 15 cubic yards, per station. 
 
 The difference of the distances out from the centre are the same thing as Prof. Ran- 
 kine's difference of slope distances since the former involve an equivalent quantity on 
 both sides of centre, equal to half the road-bed. 
 
 f Journal Franklin Institute (1840). 
 6 
 
82 MEASUREMENT OF EARTHWORKS. 
 
 that any vertical section will exhibit a rectilineal figure, more or less 
 regular. This supposition, though not strictly correct, is sufficiently 
 accurate for practical purposes. 
 
 Upon the cross-sections (taken near enough together to define posi- 
 tively the general figure of the surface), sufficient level points are 
 obtained transversely, by level and rod, their distances out from 
 centre being simultaneously measured, with a tape line ; in this man- 
 ner, both vertically and horizontally, in relation to established planes, 
 the position of all the points necessary to determine the configuration 
 of the ground is well ascertained. 
 
 These points of elevation, or depression, are commonly called plus 
 or minus cuttings (or simply cuttings), and the horizontal distances 
 which fix their relation to the centre are shortly called distances out. 
 
 The details of the operation Staking the cuttings, or cross-sectioning 
 the work (a matter of vital importance in correct measurement), 
 require good judgment and accuracy ; but are so well known to prac- 
 tical engineers as to render unnecessary a description at length. This 
 operation, however, is the absolute foundation upon which the whole 
 fabric of computation rests, and if it be not judiciously executed, all 
 rules are vain. 
 
 We may here mention a general maxim, which should never be 
 neglected, if accurate results are desired, viz. : At every change of sur- 
 face slope, transversely, single cuttings and distances out must be taken ; 
 and at every longitudinal change, sections of cuttings, or cross-sections. 
 
 Upon very rough ground it is customary to make the lateral dis- 
 tances apart of the cuttings, uniformly 10 feet, which materially 
 facilitates the subsequent calculations ; so much so, indeed, that on a 
 rock side hill it is often advisable to use this distance, even though 
 the ground seems not actually to need it; the cuttings and distances 
 out are commonly taken in feet and tenths, and the regular stations 
 of one hundred feet are subdivided by cross-sections into shorter 
 lengths, if the ground requires it, as is frequently the case. One foot 
 being usually the unit of linear measure, one hundred feet a regular 
 station, and the cubic yard the unit of solidity, in earthwork. 
 
 Though not indispensably necessary, it will be found convenient 
 in using the prismoidal method of calculation, as well as conducive 
 both to expedition and accuracy, to observe the following rules in 
 "taking the cuttings," as far as the character of the surface will 
 admit, viz. : 
 
CHAP. II. FIRST METH. COMP. ART. 17. 83 
 
 1. On side-hill, at each cross-section, where the work runs 
 partly in filling and partly in cutting, ascertain the point where 
 grade, or bottom, strikes ground surface. 
 
 2. On every cross-section, take a cutting at both edges of the 
 road, or at the distance out right and left of one-half the base. 
 
 3. Always take a cross-section, whenever either edge of the road- 
 bed strikes ground surface, and set a grade peg there to guide 
 the workmen. 
 
 4. On rough side-hill, or wherever the ground appears to 
 require it, take the cuttings (not otherwise provided for) at ten 
 feet apart. 
 
 5. Wherever the ground admits, place the cross-sections at 
 some decimal division of 100 feet apart, as 10, 20, 30, etc. 
 
 6. Endeavor to take the same number of cuttings, in each 
 * adjacent cross-section, to facilitate the computation. 
 
 7. On plain and regular ground, take three cuttings only at 
 centre and both slopes. 
 
 If these simple directions are observed by the field engineer, and 
 the work carefully done, much labor will be saved, both to him, and 
 to the computer in the office. 
 
 In all cases of side-long ground, we suppose it to slope in the same 
 general direction, between the end sections, and do not admit of oppo- 
 site surface slopes, because, under the general rule, the field engineer 
 would place a cross-section at the point of change slope, and render 
 the consideration of opposite slopes, and the warped surfaces they 
 always produce, entirely unnecessary ; indeed, by more closely assem- 
 bling the cross-sections together, we can practically reduce even the 
 most irregular surface to a series of planes coincident with it. 
 
 Nevertheless, an able writer * has shown that warped solids of a 
 certain kind are computable by his rules ; and the late Professor 
 Gillespie, in several valuable essays, has demonstrated that hyper- 
 bolic paraboloids at least could be correctly calculated by the Pris- 
 raoidal Formula ; while English engineers have long used this rule 
 for computing the volume of earthwork solids, ivith warped surfaces;^ 
 it appears, however, to be more certain and satisfactory if we confine 
 the operations of this formula to solids bounded by plane surfaces as 
 nearly as circumstances admit; but it is fortunate that our rule is 
 
 * John Warner, A. M., Computation of Earthwork (1861). Prof. Gillespie, Manual 
 of Roads and Railroad?, 10th edition (1871). 
 
 f Dempsey, Practical Railway Engineer (London, 1855). 
 
84 MEASUREMENT OF EARTHWORKS. 
 
 known to hold for some descriptions of warped ground, and hence can 
 hardly fail to proximate results, near unto the truth, however much 
 the surface may be warped, between the cross-sections, if they have 
 been judiciously placed by the field engineer. 
 
 a ....... The modification of the Prismoidal Formula, which we 
 
 shall employ in this first method of computation, will be that designed 
 to find a mean area, to be subsequently employed by the aid of our 
 Table, at the end, to ascertain the cubic yards of volume. 
 
 This formula comes from that generalized by Hutton (1770) through 
 the special mid-section, and is expressed in the beginning of Art. 16 
 as follows : * 
 
 Summarily expressed in words as follows; One-sixth the sum of 
 end areas, and quadruple mid-section, multiplied by length, gives the 
 Solidity. 
 
 This general formula (identical with one of Hutton's) requires three 
 areas (one, the mid-section, deduced from the others), and also the 
 hight or length of the Prismoid to be given; and by its aid we pro- 
 pose in illustration to furnish five examples of calculation. 
 
 1. Of a regular station, of three-level ground. 
 
 2. Of the same length, of five-level ground. 
 
 3. Of seven-level ground. 
 
 4. Of nine-level ground. 
 
 5. Of a portion of excavation and of embankment adjacent, 
 with an oblique passage between them, from one to the other. 
 
 We here follow a classification of ground nearly resembling that 
 adopted by the late Prof. Gillespie (one of our ablest writers upon 
 earthwork), who enumerates four classes only, under the simple 
 nomenclature of, 1, one-level; 2, two-level; 3, three-level; 4, irregular 
 ground; and under these four classes, he dealt with the problems of 
 earthwork in his excellent lectures " to the Civil Engineering Classes 
 in Union College." f 
 
 * " This rule," says Prof. Rankine, in Useful Rules and Tables, 2d edition, London, 
 1867, p. 74, " applies generally to any solid bounded endwise by a pair of parallel planes, 
 and sideways by a conical, spherical, or ellipsoidal surface, or by any number of 
 planes." 
 
 j- Manual of Roads and Railroads, 10th edition (1871). 
 
CHAP. II. FIRST METH. COMP. ART. 18. 85 
 
 We think, however, that few engineers would be willing to class 
 ordinary five-level ground as irregular ; for such ground would in fact 
 be produced simply by the angle levels commonly taken, which at 
 once convert the plainest three-level into five-level ground. 
 
 But ground requiring more than five cuttings on one cross-section, 
 all would probably agree in classifying as irregular y aiid such is the 
 view taken by the present writer. 
 
 This would bring all ground whatever within the scope of five 
 classes, and make but a slight variation in Gillespie's nomenclature. 
 1. Level ground, where the centre cutting alone is sufficient for vol- 
 ume. 2. Ground slightly inclined, where side-high ts only may have 
 been taken. 3. Ordinary ground, requiring centre and side-hights. 
 4. Same as 3, with the addition of angle levels, or one cutting right 
 and left of centre, besides those at the slope stakes. 5. Irregular 
 ground, such, or any similar classification would somewhat simplify 
 the matter of earthwork, but it is not indispensable. Centre cuttings, 
 or level bights at the centre, are, however, invariably taken in the 
 field, and recorded at the time, whether they be subsequently used or 
 not, so that class 2 would seldom occur on original ground. 
 
 The method of measuring the capacity of long irregular solids, by 
 means of normal sections, at short distances, has long been used by 
 mathematicians ; of which numerous examples may be found in Hut- 
 ton (1770), as well as in the demonstration and use of Simpson's rule 
 for quadrature and cubature, referred to in many works, both civil 
 and military. 
 
 This method then was naturally adopted by the earlier engineers 
 for the mensuration of earthwork, and has been continued down to 
 the present day with little chance of being superseded ; as the areas 
 of the sections, commonly known to the engineer as cross-sections, are 
 not only useful in the computation of solidity, but also in many other 
 ways, during the progress of earthworks ; and consequently those rules 
 which disregard the areas of cross-sections, and aim directly at the 
 volume alone of excavation and embankment, are less useful (even if 
 more concise} than those which require the sectional areas to be first com- 
 puted. 
 
 18. Examples in Computation by the First Method. 
 
 In computing by this method, the Grade Prism is not required, and 
 is not used, but it may be employed in verification. 
 
 Example 1. We will now give three figures (Figs. 53, 54, and 55), 
 representing three cross-sections, upon one regular station of 100 feet 
 
86 
 
 MEASUREMENT OF EARTHWORKS. 
 
 in length, of a railroad cut with side-slopes of 1 to 1, and road-bed of 
 20 feet the other dimensions being as marked upon the figures. 
 
 In these, the first and last represent the end cross-sections of the 
 100 feet station, supposed to have been regularly taken in the field. 
 
 The other (Fig. 54) being the hypothetical mid-section, deduced from 
 the end ones, as required by HUTTON'S General Rule. 
 
 Bg.53 
 
 ~ ...._ 3 -- "/ 
 
 t 
 
 These cross-sections are marked as follows: 
 
 b = 890 Area. 
 m = 625 " 
 t = 400 " 
 Length, 100 feet 
 
 1 
 
 Example 1. 
 
CHAP. IL FIRST METH. COMP. ART. 18. 
 
 87 
 
 And the calculations for solidity are as below: 
 
 890 = b. 
 
 400 = t. 
 
 2500 = 4 in. 
 Calculations, 
 
 631'7 = Prismoidal Mean Area. 
 
 2339-6 = Cubic Yards (by Table) for 100 feet. 
 
 The above example. is for plain ground of "three levels" as classed 
 by Professor Gillespie. 
 
 Example 2. We will now give an example of a railroad cut, with 
 the same road-bed (20) and ratio of side-slopes (1 to 1), in five-level 
 ground. 
 
 The three cross-sections, upon the regular station of 100 feet, are 
 numbered, Figs. 56, 57, and 58, and marked b, m, and t, the middle 
 
88 
 
 MEASUREMENT OF EARTHWORKS. 
 
 one being Hutton's hypothetical mid-section, deduced by Arithmetica\ 
 Averages from b and t, the cross-sections, assumed to have been taken 
 in the field, with rod, level, and tape, in the usual manner. 
 
 Cross-sections. 
 b = 244 Area. 
 Example 2 m = 286 " 
 t = 331 " 
 Length 100 feet = h. 
 
 And the calculations for solidity are as follows : 
 
 244 = b. 
 
 1144 = 47M. 
 
 331 = t. 
 
 6)1719 
 
 286*5 = Prismoidal Mean Area. 
 And for Cubic Yards, in 100 feet long, per Table = 1061-1. 
 
 Example 3. We will now give an example of a railroad cut, simi- 
 lar to the preceding, base 20, slope ratio r = 1, in seven-level 
 ground. 
 
 Cross-sections and areas. 
 
 b = 524 
 
 m = 537 
 
 * = 551 
 
 Length, 100 feet = h. 
 
 Example 3 
 
 Calculations for solidity : 
 
 524 = b. 
 2148 = 4 m. 
 551 = t. 
 
 6)3223 
 
 537-2 = Prismoidal Mean Area. 
 And for Cubic Yards, in 100 feet long, per Table = 1989'6. 
 
 Example 4. Although embankment is merely excavation inverted, 
 and governed in its computation by precisely the same principles, we 
 will now give an example of embankment on irregular or nine-level 
 ground, road-bed 16, side-slopes 1J to 1, and ground surface supposed 
 to be jagged masses of rock. CC represents as usual the centre or 
 guiding line of the road, the cross-sections being dimensioned * 
 
CHAP. II. FIRST METII. COUP. ART. 18. 
 
 89 
 
 Fig. 59 
 
 marked upon the figures (62, 63, 64), the distance between the end 
 sections being a regular station of 100 feet, and m (Fig. 63) being the 
 hypothetical raid-section, deduced from the two others, supposed to 
 have been regularly measured by the field engineer, and furnished to 
 the computer by him from his note book. 
 
 The areas of the sections being given, having been previously cal 
 culated in the customary manner. 
 
 Example 4 
 
 Cross-sections and. areas. 
 b =602 
 
 m = 691 
 * = 786 
 Length, 100 feet 
 
 h. 
 
90 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Calculations for solidity ; 
 
 602 = b. 
 2764 = 4 m. 
 
 786 = t. 
 6)4152 
 
 692 = Prismoidal Mean Area. 
 And for Cubic Yards, in 100 feet long, per Table = 2562*9. 
 
 Fig. 62. 
 
 . 63. 
 
 17. 
 
 10 
 
 1 
 
 10 
 
 As has been observed ^before, b and t are correlative, and either 
 might be taken as base ; the calculations of quantity are 'usually 
 
CHAP. II. FIRST METH. COMP. ART. 19. 91 
 
 made in the direction in which the numbers run, or the one nearest 
 to us of any pair may be assumed as b, and the other as t it is quite 
 immaterial which but during the pendency of the computation, to 
 which they are subject, the special designation must remain for the 
 time unchanged. 
 
 The surface of ground, assumed in this example, appears to be suf- 
 ficiently irregular to test any rule (though rougher ones will occur to 
 the memory of most engineers), and we might proceed to give illus- 
 trations of such, but enough has been done in this way to indicate the 
 principles on which we work, and which can readily be applied to 
 any case which may occur in practice. Nor does it seem necessary 
 here to define and classify the numerous distinct cases of earthwork 
 the Prismoidal Formula holds for all, and it is left to the judgment 
 of the engineer to make the application. 
 
 19. Connected Calculation of Contiguous Portions of Excavation and 
 Embankment, with the Passage from one to the other. 
 
 Example 5. See Figs. 65 to 71. 
 
 In Fig. 65, ABC, a portion of a railroad cut, road-bed = 20, side- 
 slopes 1 to 1. BCD, a portion of a railroad^, road-bed = 14, slopes 
 1 to 1. Grade points four in number, besides the centre. 
 
 In Figs. 66 to 71, six cross-sections, 3 of excavation and 3 of 
 embankment, are shown, and all dimensioned as marked. Fig. 68 is 
 the base of the closing pyramid of excavation in the passage from 
 excavation and embankment, the vertex of which is at the grade 
 point B. Fig. 69 is the base of the closing pyramid of embankment, 
 in the passage from embankment to excavation, the vertex of which 
 is at the grade point C. 
 
 The other cross-sections are those necessary to compute the portions 
 of excavation and embankment shown upon the plan, Fig. 65. One 
 of them only is at a regular station, called station (10), Fig. 68, the 
 others are all intermediates, supposed to have been required by the 
 configuration of the ground. 
 
 The scale is 20 feet to the inch. 
 
 On the centre line, the excavation shown is 61 feet in length but 
 the closirlg pyramid of cutting runs 11 feet further to its vertex at the 
 grade point B. While in like manner the embankment is 48 feet 
 long on the centre, and the closing pyramid of filling extends 7 feet 
 further to its vertex at the grade point C. 
 
 This over-lapping of the closing pyramids is an inconvenience, but 
 it is sometimes unavoidable. 
 
92 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Plan 
 
 Cross jSecs. 
 
 Fig.es. 
 
 Sta: 
 
 9+ SO 
 
 9+75 
 
CHAF. 11. FIRST METH. COMP. ART. 19. 93 
 
 Calculations for Solidity. 
 
 Position of Cross-sec- Distances Cross-section 
 
 tions upon the centre. apart. Areas, etc. 
 
 9 + 50 ... ... 342 = b. "] 
 9 + 75 ... 25 ... 907 = 4 w. [ 
 10 Reg. Sta. . . 25 . . . 106 = t. }~ Excavation. 
 
 Length = 50 f>)1355 
 
 225'8 = Prism. Mean Area. 
 
 418-1 = Cubic Yards, by 
 
 Table for /& feet = 418'1 
 10+11 Grade at centre. 
 
 (Paf9age t etc., from Excavation to Embankment.) 
 Closing Pyramid of Excavation, vertex at B r Fig. 6& 
 Area of base at 10 = 106. Then y 
 
 1 C\f _L 1 Oft i f\ Mean Area. 
 
 tJp - = 35-3 Xlength y 22=by Table 1307 Xr 2 2 o= 28'8 
 Total Solidity of Excavation = 446*9 
 
 Now, commence the embankment with the closing pyra- 
 mid in the passage, altitude or length 15 feet, and vertex at 
 C, Fig. 65. Area of base at 10 + 19 = 46. Then, 
 
 AP _|_ Afi _l_ ^ ean A reR - 
 
 LL Z = 15-3 X length, 15 = by Table 56'7 X T \ft = 8*5 
 
 10 + 19 . . '. ... 46 = b. 
 
 10 + 39 . .-. 20 ... 504 = 4m. 
 
 10 + 59 ... 20 ... 215-5 = t. } Embankment. 
 
 Length =40 6)765'5 
 
 127 '6 = Pris. Mean Area. 
 
 189-0 = Cubic Yards, by 
 
 . Table for ffo = 189'0 
 
 Total Solidity of Embankment = 197'5 
 
 And this closes the computation of Cubic Yards in the portion of 
 Excavation and Embankment, from A to D (Fig. 65), including the 
 passage between them, and comprising in all two prismoids and two 
 closing pyramids. 
 
 In concluding this branch of the subject, we may mention that as 
 HUTTON defines " a pri&moid " to have in its end sections " an equal 
 number of sides" (Arts. 3 and 14), a like number of level hights, or 
 
1)4 MEASUREMENT OF EARTHWORKS. 
 
 cuttings, ought always to be taken in adjacent cross-sections, but 
 should that have been omitted in the field, additional cuttings may 
 be computed or drawn upon the sections obtained, so that previous 
 to calculating their areas, there shall be the same number of cuttings in 
 all the adjacent cross-sections, and we shall then, have for solidity a correct 
 prismoid. 
 
 a In verifying the work given in the first four examples 
 
 preceding illustrated by Figs. 53 to 63 inclusive the end areas and 
 length being correctly given in all, it is only necessary to prove the 
 mid-section ; as an agreement there necessitates a like result when 
 used with the given d&ta,prismoidally, to find the solidity. 
 
 This proof may be made either by our 2d method of computation 
 (Rights and Widths), or 3d method (Roots and Squares) the latter 
 being generally the most convenient, though the former may often 
 be used with advantage. 
 
 No single calculation, truly says Prof. Gillespie, ought ever to 
 be relied on by the engineer, and proof of the correctness of every 
 computation should always be obtained before employing it in work. 
 
 It is often the case when railroads follow the rugged margins of 
 rivers that many miles of side-hill work present themselves, where 
 the road-bed, located above the flood line, lays in rock excavation on 
 one side, and heavy embankment upon the other to such cases the 
 preceding method of computation will be found peculiarly applicable ; 
 both cutting and filling showing themselves. upon the end cross-sec- 
 tions of every station and intermediate, while the mid-section may be 
 diagrammed between them with great facility. 
 
 In continuing this chapter we may state That in any right-lined 
 solid whatever, lying between two parallel planes (according to the 
 definition of a prismoid), whenever a mid-section can be correctly 
 deduced between two given end sections, situated in the limiting 
 planes (and by taking pains it always can be), there,' our First Method 
 of Computation will be found to apply strictly for solidity. 
 
 So that this method is a standard test for all other rules, and has been 
 accepted as such by Prof. Gillespie, and other able writers. 
 
 Hence, we may repeat that the formula employed in this chapter 
 i? the fundamental rule for the mensuration of all right-lined solids, 
 within parallel planes, and applicable also to many warped figures, 
 and other curvilinear bodies, in a manner so unexpected as to have 
 excited the surprise of some able geometers, whose attention had not 
 been specially directed to that subject before. 
 
CHAP. II. FIRST METH. COMP. ART. 19. 95 
 
 Cases often occur in heavy work, where it is evident from the cross- 
 sections, that the bulk of the solid under consideration lays consider- 
 ably on one side of the centre line (or where, in common phrase, the 
 sections are lop-sided), and it would seem in such cases as if some 
 correction ought to be made for the position of the centres of gravity 
 (as indicated upon Figs. 43 and 44, Chapter I.) ; for it is most obvious 
 that in a long line of heavy work the path of gravity centres would 
 frequently Gross and re-cross the guiding line of the work, and hence 
 would necessarily be longer. 
 
 So that if the line of magnitude should be assumed as the true 
 line of calculation, the centres of gravity ought to be assembled 
 upon the centre line, in effect, at every station, and this correction 
 would probably be found by multiplying the projections of the points 
 of gravity upon the centre, by their distances from it (-f when oil 
 the same side when opposite) ; but this is a refinement which has 
 never been employed by engineers, in dealing with the huge masses 
 in question. 
 
 What the engineer most needs in earthworks appears* to be not 
 astronomical accuracy, but the systematic use of some rule for solidity, 
 which shall always be consistent with itself, and closely proximate 
 the truth, without involving those stupendous discrepancies (men- 
 tioned by many writers), as flowing from the employment of the 
 average methods, which have been so much (and as it always appeared 
 to the writer) so unnecessarily, used in the ordinary computations of 
 earthwork. 
 
 The method of computation developed in this chapter finds appro- 
 priate application also in masonry calculations. In this manner the 
 writer once computed the contents of a heavy stone aqueduct, con- 
 taining over 4000 perches, with numerous projections and off-sets, and 
 walls battered, both inside and outside. 
 
 The process taken was by drawing to a scale accurate horizontal 
 plans, at all the off-set levels, at the skewbacks, and other breaks in 
 the contour deducing mid-sections between these, and multiplying 
 together each set of three, in accordance with the Prismoidal For- 
 mula, etc. 
 
 This gave a very satisfactory exhibit of the work, and a correct 
 result in volume, with less labor, and greater accuracy, than any other 
 modes he found in use at the time. 
 
 In calculating stone culverts, and bridge abutments also, this 
 method will be found quite useful. 
 
96 MEASUREMENT OF EARTHWORKS. 
 
 In fact, in computing the volume of solid bodies of any kind, the 
 engineer will find the Prismoidal Formula to be either strictly correct, 
 or a very close approximation. 
 
 b We now conclude this chapter by some remarks upon 
 
 Borden's Problem. 
 
 Some examples acquire celebrity from being apposite in themselves, 
 for the illustration of important processes, and are consequently 
 copied by others ; besides, there is an evident advantage to the reader 
 in re-producing examples, which, having been before discussed, are 
 more generally known ; amongst such is Borderis Problem, first pub- 
 lished by Simeon Borden, C. E. (Boston, 1851), in his " System of 
 Useful Formulae" (Art. 63). 
 
 He treats this example at great length (14 pages), and commits 
 some errors, which were subsequently pointed out and corrected in 
 Henck's Field Book (Boston, 1854). 
 
 This example was also adopted by John Warner, A. M., in hia 
 Earthwork (Philadelphia, 1861, Art. 112), without comment. 
 
 The problem appears to have given Mr. Borden some trouble, 
 involving a number of his " blind pyramids," and also some errors, ag 
 Mr. Henck hath shown. 
 
 Nevertheless, it is simply a case of injudicious cross-sectioning for 
 had Borden, instead of attempting to compute its full length of 100 
 feet, imagined an intermediate at 50 feet (for which he gave all the 
 data necessary), all difficulty would have vanished, and he would 
 neither have stumbled over his own blind pyramids, nor been shortly 
 corrected by a subsequent author. 
 
 Indeed, Mr. Borden admits, page 186, of his work of 1851, that 
 " the engineer would be likely to divide the section into two or three " 
 and this the present writer deems to be not only likely, but absolutely 
 certain. 
 
 Now, taking the end areas alone (100 feet apart), and disregarding 
 (for the moment) the irregularities of the ground, which ought to 
 have been intercepted a*nd brought out, by an intermediate at 50 
 feet we find : 
 
 Warner, in Art. 112, of his Earthwork, gives for 
 
 the volume . . . = 1155'9 C. Yards. 
 
 By Hutton's General Rule (as in this chapter) = 1155'9 " 
 Difference . . . . = 
 
 But Henck, in his Engineer's Field Book (after noting Borden's 
 mistake of 360 cubic feet), finds by his own process the solidity = 
 
CHAP II FIRST METH. COMP. ART. 19. 97 
 
 32,820 cubic feet = 1215*5 cubic yards ; or, the former are in a 
 deficiency of 59*6 cubic yards, an error inadmissible in the quan- 
 tity before us. 
 
 In this problem Borden makes two theoretical suppositions, and 
 two summations of results, based upon his hypothetical view of the 
 effect upon solidity of the irregularities of the ground surface, between 
 the end sections, but he gives no opinion on either. 
 
 The Prismoidal Formula of Hutton (computed on the whole sta- 
 tion of 100 feet) gives precisely an Arithmetical Mean between the two 
 suppositions of Borden, but is considerably in defect of the true vol- 
 ume as given by Henck's Formula. 
 
 And here we come to the point of the importance of properly cross 
 sectioning a solid, before we begin to calculate it; for if we sketch 
 from Borden's data an intermediate at 50 feet, of which we find the 
 area to be 335*6 then all difficulties are at once resolved, and we pro- 
 ceed prismoidally in a few lines to reach a correct result, which Mr. 
 Borden failed to attain in fourteen pages. 
 
 Considered in connection with an intermediate at 50 feet, Borden's 
 Problem stands as follows : Two end areas = 387 and 240. One 
 intermediate area = 335*6. Now, deducing between these (by Bor- 
 den's data) the hypothetical mid-sections, required by Button's Gen- 
 eral Rule, we find they have areas of 293*5 and 366*5, and working 
 prismoidally with them we quickly find the solidity of the entire body 
 to be 32,820 cubic feet, or 1215*5 cubic yards precisely the same as 
 Henck makes it by his own formula, and as Borden would have made 
 it had he been aware of the errors into which his own "blind pyra- 
 mids" led him. 
 
 As this problem is a well-known one, and has not a very irregular 
 appearance in Borden's diagram, we think this a suitable place to 
 urge upon all engineers the great importance of judicious cross-sectioning. 
 
 In terminating this chapter, we may safely state that Button's 
 General Rule, as applied to earthworks by the methods detailed 
 herein, is ONE WHICH NEVER FAILS WHEN THE DATA is CORRECT. 
 7 
 
CHAPTER III. 
 
 SECOND METHOD OF COMPUTATION, BY HIGHTS AND WIDTHS, AFTER 
 SIMPSON'S ORIGINAL RULE. 
 
 20 ....... The Prismoidal Formula, as originally demonstrated 
 
 by Simpson (1750) see Art. 2 was evidently designed for the rect- 
 angular prismoid (Fig. 2) its end areas were obtained by multiply- 
 ing together the Sights and Widths; and four times its mid-section 
 by multiplying the sum of the Hights by the sum of the Widths. 
 
 To adapt it more conveniently to the triangular prismoids of Earth- 
 works, with side-slopes drawn to intersect each other, the original 
 formula of Simpson (1750), reduced to the form subsequently enun- 
 ciated by Hutton, as a general rule (1770), is multiplied by 2, on the 
 left side only, changing its divisor at the same time. 
 
 Thus, 
 
 2 _ 
 
 ~~ 
 
 This is the same thing, in effect, as the original formula of Simp- 
 son (when arranged for a mean area) ; for if we suppose the rectan- 
 gular prismoid (Fig. 2) cut in half by a plane through the diagonals 
 of its end areas, FB, etc., so as to convert it into two triangular pris- 
 moids (each with one right angle), the Hights X Widths from the 
 right angle would give double the triangular area of each end, while 
 their sums, multiplied together, would equal 8 times the triangular 
 mid-section, the divisor becoming 6 X 2 = 12. 
 
 * It would evidently be a much better notation for earthwork to adopt I inttead of h, 
 because the greatest extent of an earthwork solid usually lays along the ground (length- 
 wise) ; but Simpson and Hutton, the fathers of these formulas, have both used h they 
 dealing generally with prismoids of small dimensions, supposed to stand erect upon a 
 base (as in Figs. 1 and 3), and have been followed by most writers, and necessarily for 
 the most part also here ; though we have occasionally used I (to avoid confusion), and 
 this must be taken as correllative with the h of Simpson and Hutton, in the cases in 
 which it has been employed; but some care will be needed to avoid confounding the h 
 indicating the length of the prismoid, with the same letter often used as a symbol for 
 {light In cross sections. 
 
 98 
 
CHAP. III. SECOND METH. COMP.-ART. 20. 99 
 
 Now, as shown in Art. 8, a, it is an equivalent process to imagine 
 the triangular section, partially revolved, so as to bring the edge of 
 the diedral angle downwards, and to cause its bisector (the centre line) 
 to become the perpendicular hight (h) of the cross-section, while the 
 extreme breadth to ground edges of side-slopes, horizontally, becomes 
 the width (w) then, by Art. 8,* we have h X w = double area of 
 triangular section to intersection of side-slopes. 
 
 This is the position occupied by the triangular areas of the cross- 
 sections of the solids forming the earthworks of railroads, the centre 
 line being the bisector, or hight (Ji), and the sum of the distances out, 
 to the ground edges of the side-slopes of an equivalent triangle, being 
 the width (w). 
 
 The equivalent triangle is often formed by means of an equalizing 
 line, drawn (for convenience) through the lowest side-bight of the 
 cross-section, so as to form a figure of only three sides, exactly equiva- 
 lent in area to the cross-section of earthwork, which is nearly always 
 more or less irregular on the top, and frequently has numerous sides 
 for its ground line ; the side-slopes, however, remaining generally 
 uniform and even, from station to station (see Fig. 14). 
 
 The equation for Hights and Widths may often take another form 
 (already mentioned in Art. 9), which, at times, will be found convenient. 
 
 h = Hight at one end. 
 h' = " " other end. 
 w = Width at one end. 
 w' = " other end. 
 I = Length of mass, usually 
 i, HO denoted by (//) = 
 100, generally. 
 
 7 ,, t . hu! + h' w 
 
 h w -f h' w' -f - ' 
 
 Then, X I = S. 
 
 o 
 
 Let 
 
 * In any A, however situated : If one angle coincides with the intersection (or 
 origin,) of two rectangular axes (such as a Meridian, and an East and West line, or centre 
 line, and base of levels), and the co-ordinates of the other angles are known (as by their 
 Lat. and Dep., or level bights and distances out) ; then, the area of any such A is easily 
 found. 
 
 Thus, calling the first angle 0, and the others in succession 1 and 2. 
 
 (Lat. of IX Dep. of 2) -(Lat. of2X Dep. of 1) 
 We have, J L_ J = Area of A required. 
 
 But, in the single case of either rectangular axis cutting the 6, then, instead of 
 between the products (forming the numerator above) put -f . With this exception, the 
 
100 
 
 MEASUREMENT OF EARTHWORKS. 
 
 This formula may be briefly called (from a leading feature in the 
 process), the direct and cross multiplication of Hights and WidtJis, which 
 
 may be represented as below ; and then, (x ~), or one-sixth the whole 
 being taken = Solidity. 
 
 Thus, 
 
 For example, take Figs. 72 and 73 (dimensioned as marked). 
 1. By Direct and Cross Multiplication of Hights and Widths. 
 
 ( h w = 23-4 X 47 . . . . = 1100 Double area. 
 ' \h'w'= 27-6 X 55-5 . . . . 1532 " 
 and 
 
 Cross 
 
 oss Multi- f W = 23-4 X 55'5 = 1299 \ 
 plication. \ h' w = 27'6 X 47 = 1297 J 
 
 Let 
 
 h = + 23-4 
 w = 47 
 h' -B + 27-6 
 vf = 55-5 
 
 2)2596 f Kepresenta- 
 
 1298 = 1298 I tive product 
 6)3930 I for mid-sec. 
 
 ( Including the 
 
 Prism. Mean Area = 655 < grade trian. 
 
 of 100 area. 
 
 i 
 
 2. Proof by Simpson's Formula (modified for triangles). 
 
 Eights. Widths. 
 
 23-4 X 47 = 1100 
 27-6 X 55-5 = 1532 
 51 X 102-5 = 5228 
 
 12)7~860 
 
 Prism. Mean Area 
 
 655 05 above, including 
 grade triangle. 
 
 Then, the mean area X length = 100 feet between sections 
 Solidity = 65,500 cubic feet. 
 
 rule is general, and finds ready application in computing the areas of irregular cross-sec- 
 tions, and the contents of LAND SURVEYS. 
 
 (Prob. V., Young's Analyt. Geom., London, 1833. Prof. Johnson's ed. of Weisbach, 
 Philada., 1848, article 107.) 
 
CHAP. III. SECOND METH. COMP. ART. 21. 1Q1 
 
 21 Examples of the Application of Simpson's Rule to Earthworks. 
 
 In further illustration of this subject, suppose Figs. 72, 73, 74, and 
 75, to be cross-sections upon a railroad line, in stations of 100 feet, 
 apart sections, with road-bed of 20, side-slopes 1 to 1, and other data 
 as dimensioned upon the figures given ; with equalizing lines properly 
 drawn, reducing them to equivalent triangles, and with centre hights 
 correctly ascertained. 
 
 Then, to find the End Areas to Intersection of Slopes. 
 
 Hights. Widths. Sq. Ft. 
 
 Fig. 72 = 23-4 X 47 = 1100 
 73 = 27-6 X 55-5 = 1532 
 74 = 28-8 X 59-9 = 1725 
 75 = 27-25 X 54-6 = 1488 
 
 Double Areas 
 in 
 Whole numbers. 
 
 Or, they may be computed, as is usual with engineers, by means 
 of trapezoids and triangles, as they have been, indeed, in this case for 
 the purpose of verification, and found to agree in whole numbers: 
 there being, as usual, small differences in the decimal places. 
 
 When the ground surface is irregular, as shown in these cross-sec- 
 tions, the successive processes are as follows: 
 
 1. Find the equalizing line by Art. 8. 
 
 2. Ascertain the centre hight from intersection of slopes to 
 equalizing line. 
 
 3. Find the extreme width, or sum of distances out, to the 
 edges of tops of slopes, where they cut the equalizing line. 
 
 4. Find the double areas of the cross-sections, by multiplying 
 together the hights and widths, or h X w. 
 
 5. Find 8 times the mid-section, by means of sum of Hights X 
 sum of Widths. 
 
 6. Then, for Solidity, proceed prismoidally, by Simpson's For- 
 mula as modified, for triangular solids. 
 
 The areas of the cross-sections haying been duly verified, we may 
 proceed to the calculation of some examples, as follows: 
 
102 
 
 MEASUREMENT OF EARTHWORKS. 
 
CHAP. III. SECOND METH. COMP. ART. 21. 1Q3 
 
 EXAMPLES. 
 Figs. 72 and 73. 
 
 (Hights. Widths. 
 
 23-4 X 47 = 1100 = Double Area of top. 
 27-6 X 55-5 1532 = " " base. 
 
 51 X 102-5 = 5228 = 8 times mid-section. 
 12)7860 
 
 655 = Prismoidal Mean Area. 
 100 Distance apart sections. 
 65500 = Solidity in Cubic Feet. 
 
 Figs. 73 and 74. 
 
 Rights. Widths. 
 
 27-6 X 55-5 = 1532 = 2 t. 
 28-8 X 59-9 = 1725 =26. 
 56-4 X 115-4 = 6509 = 8 m. 
 12)9766 
 
 814 = Prismoidal Mean. 
 100 
 
 SHOO = Solidity. 
 Figs. 74 and 75. 
 
 Hights. 
 
 28-8 
 27-25 
 
 X 
 X 
 
 Widths. 
 
 59-9 = 
 54-6 = 
 
 1125 
 1488 
 6418 
 
 = 26. 
 = 8m. 
 
 56-05 
 
 X 
 
 114-5 
 
 = 
 
 12)9631 
 
 
 
 
 
 803 
 100 
 
 = Prismoidal Mean. 
 
 80300 = Solidity. 
 
 (Cub. Ft. ~\ Grade Prism to be deducted, 
 SU nd * voiume^ftomnmd. 
 ftHQnn I bed to ground. 
 oUoUU J 
 227200 = Sum of quantities. 
 
 Grade Prism. 
 
 (Then, 227,200 30,000 =. ~ = 7304 Cubic Yards. 
 
 Tabulated by our 3d Method of Computation (Roots and Squares), 
 the sum of the quantities, from Fig. 72 to Fig. 75 = 227,170 Cubic 
 Feet (including Grade Prism) ; the slight difference of 30 Cubic Feet 
 
104 MEASUREMENT OF EARTHWORKS. 
 
 arising from neglect of decimals on both sides ; had these been car- 
 ried further, the results would probably have been identical, or very 
 nearly so. 
 
 We may also verify this calculation by means of multipliers, 
 modelled after Simpson's, and applied to the areas, as given in the 
 examples, as follows: 
 
 Cross-sections figured in Nos. 72, 73, 74, and 75, stations 100 feet. 
 
 Double 
 Sta. Areas, etc. Mults. Sq. Ft. 
 
 72 1100 X 0-5 = 550 
 8 times mid-sec. 5228 X 0'5 = 2615 
 
 73 1532 X 1 = 1532 
 8 times mid-sec. 6509 X 0'5 = 3255 
 
 74 1725 X 1 = 1725 
 8 times mid-sec. 6418 X 0'5 = 3209 
 
 75 1488 X 0-5 = 744 
 
 6)13630 
 2272 
 
 100 Double Interval. 
 Solidity, in Cubic Feet = 227,200, same as before. 
 
 The intervals are subdivided by the mid-sections into 50 feet 
 epaces, or single interval. The regular stations of 100 feet forming a 
 double interval in this case. 
 
 The Grade Prism being deducted (30,000 Cubic Feet), and the 
 remainder divided by 27, we have as before, a volume of 7304 Cubic 
 Yards. 
 
 22. Observations upon Simpson's Rule. SIMPSON appears to have 
 framed his rule for application to rectangular prismoids, and as such 
 he demonstrated it in reference to a diagram like Fig. 2, Art. 2 
 including of course those right triangles which are the halves of 
 rectangles. 
 
 He could have had no conception of the vast masses of earthwork 
 needed upon the public works of later days ; nor of providing a rule 
 for the mensuration of such ; nor, indeed, of the immense range the 
 Prismoidal Formula has since taken. 
 
 His rule (see Art. 2), though wonderfully flexible when applied to 
 rectangular or triangular figures, has no leading lines, common with 
 
CHAP. III. SECOND METH. COMR ART. 22. 1Q5 
 
 irregular ground; such surfaces then require to be equalized, by a 
 single line on the principle of Fig. 14* converting the sections 
 bounded by them into equivalent triangles before they can be com- 
 puted by the Rights and Widths of Simpson's Rule, though we find 
 occasionally that trapezium sections also, when not very much dis- 
 torted, are often computable by the rule mentioned. 
 
 But, in applying such a rule to the rude masses of earthwork, so 
 common at the present day, failing cases were to be expected, and the 
 peculiar solid shown in Figs. 81 and 82 furnishes an example in point. 
 
 Figs. 81 and 82, Chap. V., computed by Simpson's Rule. 
 
 Eights. Widths. 
 
 60 X 40 = 2400 
 
 30 X 60 = 1800 
 
 90 X 100 = 9000 
 12)13200" 
 
 Prism. Mean Area = 11QO 
 Common length . = 100 
 
 Solidity . . . . = 110,000 Cubic Feet. 
 
 But, by various 
 examples, in Arts 
 29 and 30, Chap. 
 V., the Solidity = 
 130,000 Cubic Feet. 
 
 So that, in the case of this peculiar solid, Figs. 81 and 82, Simp- 
 eon's Rule falls short = 20,000 Cubic Feet. 
 
 As the solid referred to has one end section a Rhomboid the mid- 
 section a Pentagon and the other end a Triangle. 
 
 We could hardly expect Simpson's Rule, framed for rectangular and 
 triangular sections, to answer in a case like this, and hence we men- 
 tion it especially. 
 
 For all the solids which present sections, such as Simpson con- 
 templated, his rule is unquestionably correct, while it is remarkably 
 plain and simple in its application. 
 
 Further to illustrate what may be expected from Simpson's Rule, 
 when applied by equalizing lines to rough and heavy sections, we will 
 now compute the cases shown by Figs. 43 and 44, Chapter I. 
 
 Example, Illustrated by Fig. 43, Chapter L 
 
 Side-slopes 1 to 1. No road-bed designated. Proximate Computa- 
 tion, by Simpson's Rule, to intersection of slopes ; other dimensions as 
 in Fig. 43. 
 
 Equalizing line of base = b = 14 2' asc. 
 top = t = 15 57' asc. 
 
 * In substance, this method is found in Button's Land Surveying (1770), quarto Mens. 
 
106 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Both these lines being drawn from the lowest side-hight, so as to 
 equalize the areas, as per Fig. 14, Chapter I. 
 
 Hights. Widths. 
 
 f 1500 = 6. i = 37.5 x 80 = 3000 
 
 Areas < 720 = t. t = 25-7 X 56 = 1440 
 
 (Length, 100 feet. 63'2 X 136 = 8595*2 
 
 Prism Mean Area = 1086*3 
 
 Length ....== 100 
 
 Solidity . . . . = 108630 
 Same, by BUTTON = 108667 
 Difference . . . = 37 
 
 Example, Illustrated by Fig. 44, Chapter I. 
 
 Side-slope Is to 1. No road-bed designated. Proximate Computa- 
 tion, by Simpson's Rule, to intersection of slopes, other dimensions as 
 in Fig. 44. 
 
 Equalizing line of the base b = 4 30' asc. 
 top t = 1 5' des. 
 
 Both these lines being drawn from the 
 lowest side-hight, so as to equalize the areas, 
 as per Fig. 14, Chapter I. 
 
 Highta. Widths. 
 
 22-02 X 66 = 1453 
 29-81 X 90-7 = 2704 
 8122 
 
 f 1352 = 6. 
 Areas < 726= t. % 
 
 (Length, 100ft. 
 
 51-83 X 156-7 = 
 
 12) 12279 
 
 Prismoidal Mean Area = 1023'25 
 
 Length = 100 
 
 Solidity = 102325 
 
 By Wedge and Pyramid = 102363 
 Difference . . . . . = ~ 38 
 With several other methods, this proximate calculation agrees within 
 a few cubic yards. 
 
 Example from Warner's Earthwork, Art. 86. 
 A heavy embankment. For details, see Chapter V., near the close. 
 
 (2411 = b. 
 907 = t. 
 Length, 100 feet 
 Surface slope, 15. 
 
CHAP III. SECOND METH. COMP. ART. 22. 107 
 
 Hignts. Widths. 
 
 36-7 X 131-4 = 4822 
 
 22-5 X 80-6 = 1814 
 
 59-2 X 212-0 = 12550 
 12)19186 
 Prismoidal Mean "Area . . = 1599 
 
 Length = 100 
 
 Solidity = 159900 Cubic Feet. 
 
 For Cubic Yards -5- 27 . . = 5922 
 Deduct vol. of Grade Prism = 356 
 
 Solidity = 5566 Cubic Yards. 
 
 By Hutton's Rule . . . . = 5566 
 Difference = 
 
 In calculating by Simpson's Rule, the example figured by Figs. 74 
 and 75 which agrees very nearly with BUTTON we observe, by 
 reference to the figures, that the ground slope at the end sections 
 differs about 9. So that we may safely assume that where the 
 equalizing lines (representing the ground) have a nearly similar 
 slope, and in the same direction, which do not differ more than 10 in 
 their inclination, SIMPSON'S Rule may be safely used this appears to 
 be a sure limit, and we might perhaps go higher. 
 
 When the work happens to be upon uniform ground, or the equal- 
 izing lines have the same slope, as in the case cited from Warner's 
 Earthwork, where the ground slope itself is uniform at 15, the 
 results obtained by Simpson's Rule ought to be exact^andf they appear 
 to be so. 
 
LIBKAU 5f 
 
 VERSITY OF 
 
 CALIFORNIA. 
 
 CHAPTER IV. 
 
 THIRD METHOD OF COMPUTATION, BY MEANS OF ROOTS AND SQUARES J 
 A PECULIAR MODIFICATION OF THE PRISMOIDAL FORMULA, WHICH 
 WILL BE FOUND IN PRACTICE TO BE BOTH EXPEDITIOUS A!N D 
 CORRECT, IN ORDINARY CASES. 
 
 23 This method of computation, by Roots and Squares,* 
 
 appears to be the most rapid and compendious one treated by us, 
 while it requires less data and preliminary work, and agrees in its 
 results (for usual field work) with computations made direct by the 
 Prismoidal Formula, of which, indeed, it is only a special modification, 
 more concise and rapid in use, but at the same time less accurate. 
 The formula for the Rule of Roots and Squares has been already 
 described in the Preliminary Problems, Art. 10, where it is num- 
 bered XI., and is as follows: 
 
 -g- - X / = S. 
 
 Where, 
 
 h* = Representative square of area of top, 
 
 from ground to intersection of slopes = (f). 
 h- 2 = Representative square of area of base, 
 
 from ground to intersection of slopes = (b). 
 (h -f h'y = Representative square of 4 times mid-sec. = (4m). 
 I = Distance apart sections usually desig- 
 nated as (h) by the earlier writers, 
 and hence continued by us to some 
 extent ; though I is clearly a more 
 suitable symbol for earthwork, which, 
 with a comparatively small cross-sec- 
 tion, extends its length along the 
 ground. 
 
 * This method is materially aided in its use by a good Table of Squares and Roots.- 
 Prof. De Morgan's stereotyped edition of Barlow's Tables (8vo, London, 1860) is 
 believed to be the best: a very large edition was published, and this valuable work can 
 be obtained from any of our importing booksellers at quite a low price. 
 
 When the numbers are large, the well known method of Logarithms gives the simplest 
 process for Involution or Evolution. 
 
 108 
 
CHAP. IV. THIRD METH. COMP. ART. 23. 109 
 
 Note. That the bights of the end sections in this chapter are 
 always to be considered as extending from the ground to intersection 
 of slopes, or be representative of such. 
 
 The most important item in this notation is (h -f /i') 2 , which, by 
 geometry, we know to be equivalent to 4 ^ - j , while - 
 is the representative in the mid-section of a line similar to h and h'. 
 
 So that this formula (for a single station) is, in fact, equivalent to 
 the Prismoidal Formula, as heretofore expressed, viz. : 
 
 but for exact work (our formula above) requires the end sections to 
 be triangles, with a uniform ground slope. 
 
 Let us now apply the above formula to an entire cut or bank, to 
 be computed by Mutton's Kule (adopted from Simpson) see Art. 10, 
 Formula IX. 
 
 Where A +4B-f 20 ^ I)vM6 interval = S. 
 
 D 
 
 Here, for a case of 6 single or 3 double intervals, as shown in the 
 skeleton table below. 
 
 We have, for 3 double intervals or even spaces between stations of 
 equal length : 
 
 A 2 -f 7t' 2 . . . = A. The sum of extreme sections, each desig- 
 
 nating one end. 
 
 3 (h -f h') 2 . . = 4 B. Mid-sections, standing on even numbers. 
 2 (h'y + 2 (7i) 2 = 2 C. Kegular Cross-sections, standing on odd 
 
 numbers. 
 
 Double Interval Any one of the uniform spaces, from 1 to 3, or 
 3 to 5, etc., being the odd numbers where the regular cross-sec- 
 tions stand. 
 S = Solidity of entire cut of 3 equal stations in length. 
 
 Example 1 ...... Being a simple case (on irregular ground) of 
 
 three uniform stations, or double intervals, of 100 feet each, the mid- 
 sections falling in between, and dividing the length of 300 feet into 
 single intervals of 50 feet each ; for which we will tabulate the exam- 
 ple represented by Figs. 72, 73, 74, and 75, of Chapter III. in a 
 skeleton table as follows : 
 
110 
 
 MEASUREMENT OF EARTHWORKS. 
 
 
 / 
 
 (/>+><')' 
 
 1,'* 
 
 (h+h')* 
 
 /i a 
 
 (h+h 1 )* 
 
 h 
 
 
 1 
 
 
 3 
 
 
 5 
 
 
 7 
 
 Regular stations designated by 
 the numbers of the figures. 
 
 72 
 
 
 73 
 
 
 74 
 
 
 75 
 
 Places of mid-sections, on even 
 numbers. 
 
 
 2 
 
 
 4 
 
 
 6 
 
 
 Regular cross-section areas, upon 
 the odd numbers. 
 
 550- 
 
 
 766- 
 
 
 862-5 
 
 
 744- 
 
 Square roots of areas ot regular 
 cross-sections. 
 
 23-45 
 
 
 27-68 
 
 
 29-37 
 
 
 27-28 
 
 Sums of square roots. 
 
 
 51-13 
 
 
 57-05 
 
 
 5665 
 
 
 * Squares of sums, or 4 times the 
 proper mid-section. 
 
 
 2615- 
 
 
 3255- 
 
 
 3209- 
 
 
 
 
 Extra 
 decimals 
 thrown 
 together 
 here. 
 
 
 
 
 
 
 Having given the skeleton table of data, we will now tabulate for 
 solidity on three different plans, any one of which may be adopted, 01 
 in fact any other which truly represents the formula given. 
 
 Tabulation for Solidify. 
 
 On the plan of Art. 10, iu Chap- 
 ter I. 
 
 By Simpson's Rule (as given by 
 Hntton). 
 
 By Multipliers, modelled after 
 Simpson's. 
 
 Sta. 72. Areas . . = 550 
 4micJ-8ec. . . 2615 
 
 73. . // ;j 
 
 A. 4 B. 2 C. 
 
 550 2615 766 
 744 3255 766 
 
 End areas, 
 and 4 times 
 mid-section. Mults. Results 
 
 4 mid-sec. . . 3255 
 7 f 862-5 
 
 1294 3209 862-5 
 4 B = 9079 862-5 
 
 2615 X 1 = 2615 
 766 X 2 = 1532 
 
 4 mid-sec. . . 3209 
 75 744 
 
 2C =- 3257 3257 
 A = 1294 
 
 3255 X 1 = 3255 
 862-5 X 2 = 1725 
 3209 X 1 3209 
 
 6)13630 
 
 6)13630 
 
 744 X 1 = 744 
 
 General Mean Area = 2271-7 
 Double Interval. . = 100 
 
 2271-7 
 100 Double Int. 
 
 6)13630 
 2271-7 
 
 Solidity in C. Feet = 227.170 
 
 Solidity = 227,170 in C. Feet. 
 
 Double Interval . . = 100 
 
 Whole length of cut 300 feet. 
 
 Whole length of cut 300 feet. 
 
 Solidity in C. Feet = 227,170 
 Whole length of cut 300 feet. 
 
 24. Now, for further illustration : Take any cut or bank say of 
 6 (or any even number of) equal stations their termini being tem- 
 
 * HtiTTON and other geometers have shown that the square of any line equals 4 times 
 that of half the line ; and that similar triangles are to each other not only as the squares 
 of their like sides, but also as the squares of any similar lines; and these principles of 
 Geometry lay at the foundation of the method of computation, developed in this Chap- 
 ter IV. (as already indicated in the Preliminary Problems). 
 
CHAP. IV. THIRD METH. COMP. ART. 24. 
 
 Ill 
 
 porarily numbered in the series of odd numbers, while the interme- 
 diate spaces (or places of mid-sections) are also temporarily numbered 
 in the series of even numbers, and the places of cross-sections and mid- 
 sections, as well as those of the symbols used in the formula, all 
 regularly marked, as follows : 
 
 Regular stations. 
 Places of cross-sees. 
 " mid-sees. 
 Symbols of formula 
 
 10 6 1 
 
 (h+h'f\ h* (/i+A') 3 | 
 
 10 
 
 19 
 
 This little skeleton table shows the positions of the representative 
 squares equivalent to the areas of the several regular cross-sections 
 computed, and also of 4 times the proper mid-sections, which belong 
 between them, and it will indicate the manner in which they are 
 combined relatively to the odd numbers, which represent the regular 
 stations ; so that having computed the regular cross-sections, we can 
 readily assemble them in a skeleton table, compute from them by 
 Roots and Squares the other data demanded by the formula, and 
 proceed to tabulate for Solidity, as has been already shown, and will 
 be more conspicuously exhibited hereafter. 
 
 JJpon the foregoing principles we will now proceed with an entire 
 piece of heavy embankment, succeeded by a rock cut, as shown in 
 the annexed, Fig. 76. 
 
 Example 2. ... BANK = 1000 feet long. . . . Fig. 76. 
 
 Skeleton Table of Data, Given or Computed. 
 
 Length of regular ftationi 100 feet intervals produced by Mid-sections 50 feet. 
 
 Regular stations of 100 feet = 1 2 3 45 6 7 8 9 10 11 
 
 Temporary numbers . . . = 1 3 5 7 9 11 13 15 17 19 21 
 
 Regular Cross-section Areas = 24 185 495 1467 3123 3123 3123 1978 1197 391 24 
 Places of mid-sees., inter- \ 
 mediates at 50 ft. (really). J 
 
 the Cross-sec-) 
 
 
 11' 
 
 || 
 
 4-90 13-60 22-25 38-30 55'88 55'88 
 
 44-47 34-60 19-77 4-90 
 
 tion Areas / " 
 
 gums of Roots = 18-50 35-85 60-55 94-18 111-76 111-76 100-35 79-07 64-37 24-67 
 
 Squares of Sums, or 4 time 
 the Mid-section Areas. 
 
 342-25 1285-22 3666-30 8869-87 12490-30 12490-30 10070-12 6252-06 2956-10 608-61 
 
 * For Figs. 77 and 78, illustrating a supposed basis of the Prismoidal Formula, and 
 its connexion with Simpson's Rule for Cubature (see Chap. VII.). 
 
112 
 
 MEASUREMENT OF EARTHWORKS. 
 
CHAP. IV THIRD METH. COMP. ART. 24. 
 
 Tabulations for Solidity ; 
 
 113 
 
 1. 
 
 Regular stations 
 of 100 feet. 
 1 ... 
 
 Cross-sect iou 
 Areas. 
 . . 24 
 
 2. By M! 
 
 Mnl 
 
 1 
 
 2 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 2 
 1 
 1 
 Proof 
 Gen.mea 
 
 Solidity 
 
 4 times mid-section 
 2 ... 
 
 . . = 342-25 
 _ f 185 
 
 3 ... 
 
 ' ' (185 
 . .= 1285-82 
 _ f 495 
 
 it a 
 4 
 
 I 495 
 . . = 3666-30 
 f 1467 
 
 t 
 
 5 . . 
 
 ( 1467 
 . . = 8869-87 
 _f 3123 
 
 6 ... 
 
 * ' { 3123 
 . . = 12490 
 _f 3123 
 
 7 ... 
 
 ' " { 3123 
 . . = 12490 
 _ f 3123 
 
 U 
 
 8 ... 
 
 . . = 10070-12 
 _ { 1978 
 
 9 . . 
 
 . . = 6252-06 
 
 10 . . 
 
 ' ' { 1197 
 . . = 2956-10 
 f 391 
 
 
 ' * ( 391 
 
 . . = 608-61 
 24 
 
 
 
 6^89243-13 
 
 Gen.mean area to int.of slopes = 14874 
 100 
 
 Solidity in c.ft.to int.of slopet 1487400 of 
 BANK. 
 
 By 100 feet stations, or bOfeet intervals. 
 
 2. By Multipliers, modelled after Simpson's. 
 Results. 
 
 = 2t 
 
 = 342 
 
 = 370 
 
 = 1285 
 
 = 990 
 
 = 3667 
 
 = 2934 
 
 = 8870 
 
 = 6246 
 
 = 12490 
 
 = 6246 
 
 . = 12490 
 
 = 10070 
 
 = 3956 
 
 . . . . .== 6252 
 
 = 2394 
 
 = 2956 
 
 = 782 
 
 = 609 
 
 = 24 
 
 6)89243 
 Gen.mean area to int.of slopes = 14874 
 
 100 
 
 Solidity in c.ft. to int.of elope* = 1487400 of 
 
 BANK. 
 
 . Fig. 76. 
 
 Example 2 Continued. ROCK Cur = 1000 feet long. . 
 Skeleton Table of Data, Given or Computed. 
 
 Length of regular stations 100 feet ; which, by means of the Hypothetical M id- sections, cover the ground with 50 
 
 feet intervals. 
 
 Regular stations of 100 feet = 11 12 13 14 15 16 17 18 19 20 21 
 Temporary numbers . . .= 1 3 5 7 9 11 13 15 17 19 21 
 
 Regular Cross-section Areas =, 192 386 646 801 975 768 589 706 771 433 192 
 Places of mid-sees., Inter-) 
 
 mediates at 50 ft.(really). j = 
 V Roots of the Cross-section ) 
 
 Areas ...... -J : 
 
 Sums of Roots ..... = 
 
 Squares of Sums, or 4 times 1 
 
 the Mid-section Areas. } 
 8 
 
 10 
 
 H 
 
 13-86 19-65 25-42 28-31 31-23 27-71 24-27 
 
 33-51 45-07 53-73 59-54 58-96 51-98 
 1122-92 2031-30 2886-91 3545'01 3476'28 2701-92 2584-70 2952-83 2360-01 1202-01 
 
 26-57 
 50-84 
 
 27'77 
 54-34 
 
 20-81 13-86 
 48-58 34-87 
 
114 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Tabulations for Solidity : 
 
 By 100 feet stations, or 50 feet intervals. 
 
 1. 
 
 Regular stations -~- 
 
 of 100 feet. 
 
 n . . . . . 
 
 4 times mid-section . . . 
 
 Gen.mean area to int.of slopes 
 Solidity in c.ft.to int. of slopes 
 
 Cross-section 
 
 2. By Mi 
 
 Mul 
 
 Areas. 
 
 1 
 
 : 192 
 
 1 
 
 1122-92 
 
 2 
 
 f 386 
 
 1 
 
 1 386 
 
 2 
 
 2031-30 
 
 1 
 
 f 646 
 
 { 646 
 
 2 
 
 2886-91 
 
 1 
 
 f 801 
 { 801 
 
 2 
 1 
 
 3545-01 
 
 2 
 
 f 975 
 
 1 
 
 i 975 
 
 2 
 
 : 3476-28 
 
 1 
 
 f 768 
 
 2 
 
 1 768 
 
 1 
 
 : 2701-92 
 
 2 
 
 f 589 
 1 589 
 
 1 
 
 2 
 
 2584-70 
 
 1 
 
 f 706 
 
 
 I 706 
 
 1 
 
 2952-83 
 
 Proof 
 
 f 771 
 
 Gen.mea 
 
 \ 771 
 
 
 2360-01 
 
 Solidity 
 
 f 433 
 
 
 '{ 433 
 
 
 1202-01 
 
 
 192 
 
 
 6)37397-89 
 
 
 = 6233 
 
 
 100 
 
 
 = 623300 of 
 
 
 ROCK CUT. 
 
 
 2. By Multipliers, modelled after Simpson's 
 Results. 
 
 = 192 
 
 . . . . . = 1123 
 
 .... . . = 772 
 
 1 = 2031 
 
 . . . . . = 1292 
 
 1 = 2887 
 
 2 = 1602 
 
 . ... .= 3545 
 
 ... . .= 1950 
 
 . . .. . .= 3476 
 
 = 1536 
 
 1 .......= 2702 
 
 = 1178 
 
 1 = 2585 
 
 = 1412 
 
 . . . . ' . = 2953 
 = 1542 
 
 1 = 2360 
 
 .....= 866 
 
 . . . . '.== 1202 
 
 . = 192 
 
 6)37398 
 Gen.mean area to int.of slopes = 6233 
 
 100 
 
 Solidity in c.ft. to int.of slopes = 623300 of 
 KOCK CUT. 
 
 25. In the preceding example, the side-slopes of the BANK are 1 J 
 to 1 road-bed = 12 ; while in the ROCK CUT, the side-slopes are i 
 to 1 road-bed = 16 ; and in all these calculations (we repeat), the 
 sectional areas, in every case, are taken from ground line to intersection 
 of side-slopes ; and the hights, from the vertex of the common angle 
 thus formed to the line, or lines, representing the surface of the ground. 
 
CHAP. IV. THIRD METH. COMR ART. 25. H5 
 
 So that in all such computations if the contents above or below 
 a given road-bed be desired in the results, then the volume of the 
 grade prism (being included in the summation) must in every case 
 be duly deducted. 
 
 The volume of the grade prism depends upon its sectional area, 
 and the length of the bank or cut these calculations are very simple, 
 and once made, remain unchanged as long as the road-bed and side- 
 slopes continue uniform. 
 
 Geometers having shown that the areas of similar triangles are to 
 each other, not only as the squares of like sides, but also as the 
 squares of any similar lines in each, and these often occurring in 
 earthwork solids, when their cross-sections are converted into trian- 
 gular areas, by the prolongation (to a junction) of the side-slopes, it 
 becomes of importance to classify the relations existing among lines 
 and their squares, as well as the squares and rectangles of their sums 
 and differences ; this has been well done in J. R. Young's Geometry 
 (London, 1827), in several successive propositions : Book II., 4, 5, 
 6, 7, and 8. 
 
 Now, suppose any line to be divided into two parts, h and h' then, 
 by these propositions, we have : 
 
 1. (h + hj = 2 (h + h') X 
 
 2. (h + lij = A 2 -{- h* + 2h h'. 
 
 3. (h hj = h 2 -f h 2/i h'. 
 
 4. 7i 2 h" 2 = (h -f h') X (/* h'). 
 
 5. 7i 2 -f h'* = l(h + h'y -f l(h /O 2 - 
 
 6. 2 (/i 2 -f 7i' 2 ) = (h + hj + (h h')\ 
 
 As these lines, or parts of lines, may, and often do, occupy in simi- 
 lar triangles the relation of like lines, they become of some conse- 
 quence in earthwork calculations, and in various forms can be 
 traced through many of the formulas now before the public. 
 
 We will now give an example from Warner's Earthwork (Art. 
 124), to show that small variances may be expected in employing the 
 Rule of this Chapter upon irregular ground : indeed, it is only in 
 uniform sections that an exact agreement of Rules can be antici- 
 pated, but the variations (always small) are not unlikely to balance 
 themselves in computing considerable lengths of line. 
 
116 MEASUREMENT OF EARTHWORKS. 
 
 C End areas to grade . . . . = 846'5 . . = 915.5 
 J Grade Triangle to add . . . = 196 . . = 196 
 
 ( End areas to int. of slopes . . = 1042'5 . . = IIH'5 
 
 H&r6) Square Roots = 32'29 . . = 33'34 
 
 Sums of Roots. . ,>,V^H^ . . = 65'63 
 Square of sum, or 
 
 quadruple mid-section = 4308 
 
 \ Length, 100 feet. 
 
 Then, Prismoidally, 
 
 Sum end areas ...'.... = 2154 
 Quadruple Mid-section . . . . = 4308 
 
 6)6462 
 1077 
 Length = 100 
 
 107700 
 
 Off Grade Prism = 19600 
 
 27)88100" 
 Solidity in Cubic Yards . . . . = 3263 
 
 As computed by Warner (3274, C. Y.) ; and also by Button's 
 General Rule (3274, C. Y.), the difference made by our Rule of this 
 Chapter is, 11 Cubic Yards, or about $ of one per cent. 
 
 Comparison of the method of this Chapter with the test examples 
 of Chapter II., as computed by Button's General Rule (each for 100 
 feet in length). 
 
 1. Three-level Ground. 
 
 (See Figs. 53, 54, and 55.) c . Yards. 
 
 Computed by Roots and Squares (method of this Chapter) = 2337'6 
 " Button's General Rule (Chapter II.) . . . = 2339.6 
 
 Difference = 2 
 
 2. Five-level Ground. 
 (See Figs. 56, 57, and 58.) 
 
 Computed by Roots and Sqirares (this Chapter) . . . . = 
 " " Button's General Rule (Chapter II.) . . . = 
 
 Difference = 
 
CHAP. IV. THIRD METH. COMP ART. 25. 117 
 
 3. Seven-level Ground. c. Yards 
 Computed by Roots and Squares (this Chapter) . . . . = 1990' 
 
 " " Button's General Rule (Chapter II.) . . . = 1989'6 
 
 Difference = -f- 0'4 
 
 4. Nine-level Ground. c. Yards. 
 Computed by Roots and Squares (this Chapter) . . . . = 2562*9 
 
 " Button's General Rule (Chapter II.) . . . = 2562'9 
 
 Difference = 
 
 We will now give another example from Warner's Earthwork, 
 computed by the method of this chapter. 
 
 Heavy Embankment (Art. 86). 
 
 Areas = 2411 907 
 
 \/Roots~ . . . . = 49-10 30-12 
 
 Sums of Roots = 79'22 
 
 Square of sum, ^ 
 
 or quadruple >....= 6276 
 
 mid-section. ) 
 
 Then, Prismoidally, 
 
 Sum of ends . . . = 3318 
 Quadruple Mid-sec. = 6276 
 6)9594 
 
 X length . . . . = 159900 
 
 -T- 27 for C. Yards = 5566 = Same as Hutton s Gen. Rule. 
 
 From the above it will be observed that, with a Table of Powers 
 and Roots at hand, the method of this chapter affords a very convenient 
 and speedy test for volumes, found by other processes, and it is a proxi- 
 mately correct one. 
 
 
CHAPTER V. 
 
 FOURTH METHOD OP COMPUTATION, BY REGARDING THE PRISMOID AS 
 BEING COMPOSED OP A PRISM WITH A WEDGE SUPERPOSED, OR OF A 
 WEDGE AND PYRAMID COMBINED. 
 
 26 ....... Sir John Macneill (1833) hath shown that a Prismoid 
 
 of Earthwork is really a prism with a wedge superposed (as we have 
 already mentioned in Art. 4) that the wedge is also divisible into 
 two pyramids and that the formulas for volume, in these three 
 chief bodies of solid geometry, form, by addition, the Prismoidal 
 Formula. 
 
 Regarding the Prismoid in this way, and assuming it to have been 
 diagrammed as shown in Fig. 8, Art. 6 (both end sections upon one 
 drawing), it is easily computable when reduced to a level on the top, 
 and the back of the wedge is a trapezoid, by means of Formula VI., 
 Art. 6. 
 
 This Formula is : 
 
 - + (Vr _ Grade . Triang]e) x t _ Solidityi 
 
 to road-bed, and omitting G. T. to intersection of slopes. 
 
 Where, 
 B = Top-width of back, or larger parallel side of trapezoid, 
 
 measured horizontally. 
 
 b = Bottom- width of back, or lesser side of trapezoid, equal 
 
 also to the edge, which is the horizontal top-width 
 of smaller end section, at a distance forward = to 
 the common length of wedge and prism. 
 H and h = Vertical hights of the end sections to intersection of 
 
 slopes. 
 
 H h = Hight of back of wedge. 
 
 r = Ratio of side-slopes to unity, or cot. of slope angle. 
 
 h 2 r = Area of prism to intersection of slopes, and less Grade 
 Triangle = area of section from ground to road-bed. 
 118 
 
CHAP. V FOURTH METH. COMP. ART. 26. 119 
 
 In calculating by this Formula we may omit the Grade Triangle 
 if we choose (though we should have to supply a more complicated 
 expression for A 2 r), and might, perhaps, somewhat simplify the com- 
 putation thereby; but if used in area, we must be careful to account 
 tor it in volume; while the bights need only be extended from ground 
 to road-bed ; though as their difference only is used here, that is not 
 material and altogether we would gain so little by the change as to 
 make it unadvisable. 
 
 In words, this Formula ^ 
 
 may be expressed as jot- V (Mean Area Wtdge -f Mean Area of 
 lows : ) Prism) X Common Length = Solidity, 
 
 of the Prismoid, to intersection of slopes, 
 and minus G. T. to Road-bed. 
 
 Inasmuch, however, as a trapezoid is always reducible to an equiva- 
 lent rectangle, we may consider this matter of the superposed wedge 
 in a more general manner, without the necessity of first reducing the 
 trapezoidal, or triangular, cross-section to a level on the top, or 
 slope of 0. 
 
 Before entering upon this branch of the subject we may, however, 
 state that the reason why, in a wedge with a trapezoidal back, we 
 sum up all the three parallel sides of back and edge X by hight of 
 back -i- by 6, and finally multiply by length for volume is drawn 
 from the common rule for a wedge (Twice width of back -j- edge 
 X by hight of back -r- by 6, and X by length = Volume.} But in a 
 wedge with a trapezoidal back the sum of top and bottom parallel 
 sides X 2 = simply the sum of those parallel sides ; and, as in an 
 earthwork solid, the lesser parallel side also (generally) equals the 
 edge, that being the top line of the smaller end section, situated at a 
 distance of the length forward. Hence, B + b + b is usually equiva- 
 lent to X 2 -f (b the length of the edge) which will be found 
 2i 
 
 in substance as a term in Button's Rule for wedges (4to Mens., 1770) ; 
 but more concisely expressed in Chauvenet's Theorem. 
 
 References to Fig. 79.* 
 
 a d = End view of the back of a rectangular wedge. 
 af = Equivalent parallelogram, of which a g is the base, 
 and a D the altitude. 
 
 * For Figs. 7 and 78, see Chapter VII. 
 
120 
 
 MEASUREMENT OF EARTHWORKS. 
 
 a D = Horizontal projection (7O71), or width of a b (the back). 
 a I = Horizontal projection (35'36), or width of a h (the edge) 
 a e g k = The initial square of 50 square feet area, which is con- 
 
 707 
 
 tained in the back = 
 
 A B f Vertical and horizontal 
 C D \ rectangular axes. 
 
 50 
 
 = 14*14 times. 
 
 Fi 79. 
 
 aedb *, Back of Wedge _ area =701. 
 
 agfb EqqiV:FaTall: _ do. 707. 
 
 aegk c=: Initial Square _ do. 50. 
 
 aeg^ 
 
 S' = Equal ^Xa 
 b/d J 
 
 ac ..^a JHor.'projrofTjaclc 
 
 al * do: ede. 
 
 as. 
 
 :o 
 
 The triangles, a eg and 6 df, are identical, and the one cut off, and 
 the other added, make the two parallelograms, a d and a/, precisely 
 equivalent = 707 area, for each. 
 
CHAP. V. FOURTH METH. COMP. ART. 26. 121 
 
 a b = Width of back of rectangular wedge, inclined at an angle 
 
 of 45 = 100. 
 a h = Width of edge, or top of forward, or smaller, section = 50. 
 
 Now (as above mentioned), a trapezoid being always reducible to an 
 equivalent rectangle, we may consider in this place the superposed 
 wedge (with reference to Fig. 79), without the necessity of first equal- 
 izing the end cross-sections, by level lines on the top, as will be more 
 clearly seen further on. 
 
 However much the back or edge of a rectangular wedge may be 
 inclined from a level plane, the resulting volume is still the same by 
 using their projections upon the horizontal one of two rectangular 
 axes (as C D), instead of the actual widths of back or edge, whilst the 
 hight of the back becomes the base of an equivalent parallelogram, 
 of which the projection is the altitude ; this will become evident by 
 reference to Fig. 79. 
 
 For example, let us now compute the wedge shown in the figure: 
 1st, As though it were upon a level, and the back a rectangle. 2d, 
 As an oblique parallelogram on the back, and inclined at 45 from a 
 level line. 
 
 1. Rectangular back supposed to be level. Length of wedge = 
 100. Breadth of back = 100. Edge = 50. Hight of back = 
 7-071. 
 
 Here we have : Sum of the 3 parallel sides of edge and back -r- 3. 
 
 100 )_.-, f 7-071 = Altitude. 
 
 100 } - 100 = Length. 
 
 50 = Edge. Right Section 1 2)707-100 
 
 83i = Average multiplier * . . = 83 
 
 Volume = 29,463 = C. Feet 
 
 Computed after Chauvenet's Theorem (Geom., VII. 22). 
 
122 MEASUREMENT OF EARTHWORKS. 
 
 2. Oblique-angled Parallelogram for Back, and inclined 45. Length 
 of wedge .= 100. Hight of back = 10. Horizontal projection of 
 back = 70*71. Horizontal projection of edge = 35'36. 
 
 -Sum of the 3 parallel sides or edges 
 ~1T~ 
 
 70 ' 71 1 -R v f 10 = Altitude. 
 
 70-71 I " 100 = Length. 
 
 35-36 = Edge. Eight Section 1 2^1000 
 3)1778 
 
 58-927 = Average multiplier . = 58-927 
 
 Volume = 29,463 
 
 It is evident, from a consideration of the above case of a rectan- 
 gular wedge, whether level or inclined, that the same process would 
 apply to the trapezoidal wedge (usual in earthworks), either by its 
 reduction to an equivalent rectangular one, or (when diagrammed 
 together) by projecting both sides of the back, and also the edge, 
 upon the horizontal axis, and ascertaining the respective lengths of 
 these three projections, to be used in the computation of volume, by 
 Chauvenet's Theorem,* instead of their actual measured lengths, this is 
 in fact the method of the engineer, who usually disregards the incli- 
 nation of the ground, and takes all his measures horizontally and 
 vertically. 
 
 The hight of the back of the inclined wedge being in the case 
 above, ascertained by dividing the known area of the back of the 
 rectangular wedge, by the Arithmetical Mean of the horizontal pro- 
 jections of its top and bottom breadths ; both equal in the above 
 rectangular back, but always unequal in a trapezoidal one. 
 
 With these preliminary observations, we will now give the rule 
 for finding the volume of the superposed wqdge in ordinary earth- 
 works, with examples to show how, by the simple addition of the 
 under-prism, the solidity of the entire earthwork, between any two 
 cross-sections of given area, and distance apart, is easily ascertained, in 
 all cases, within a limit hereafter discussed (Art. 29). 
 
 27 Eulesfor Computation by Wedge and Prism. The data 
 
 required to be given will be as follows : 
 
 * Chauvenet's Geoin., VII. 22 (Philada., 1871). 
 
CHAP. V. FOURTH METH. COMP. ART. 27. 123 
 
 1. Areas of end cross-sections. 
 
 2. Distance apart, or common length of wedge and prism. 
 
 3. Sum of distances out, to ground edges of side-slopes, which 
 are, in fact, the projections or horizontal widths of back and 
 edge, as well as the right and left distances of the field engineer. 
 
 The first is obtained by well-known processes, and the two latter 
 are always supplied by the Field Book of the engineer. 
 
 Then, as preliminary steps: (1) Find the difference of the areas 
 of the end cross-sections, which difference is the area of the back of the 
 superposed wedge. (2) Divide this difference of area by half the 
 sum of the widths of the back (or horizontal projections), which gives 
 the vertical mean hight of the back. Now, the lower side of the 
 back (when both sections are diagrammed together) equals the edge 
 (or top-width of the smaller end section) supposed to be forward, at 
 a distance equal to the common length. So that if B = top-width of 
 larger end section, b will equal its bottom width (and also that of 
 the edge} so that B + b -p- b, for the wedge-shaped part, would give 
 the sum of the three parallel edges (or, in reality, their horizontal 
 projections) to be divided by 3, for use in ChauveneVs Theorem. 
 
 RULE. When the width of the large end is equal to or greater than 
 that of the small one. 
 
 1. Vertical mean hight X distance apart sections 
 
 ~2~ 
 
 Sum of the three parallel edges T . . 
 
 5 a - = Volume of Superposed Wedge. 
 
 o 
 
 2. Smaller end area X length (or distance apart sections) = Vol- 
 ume of Prism. 
 
 These two results, added together = Solidity of the whole Prismoid. 
 
 a Prior to giving examples in illustration of our rule, it 
 
 appears necessary in this place to make some explanations to show 
 the generality of the application of the rule drawn from Chauvenet's 
 Theorem (Geom., VII. 22) for the volume of wedges. 
 
 Wedges are always formed by the truncation of triangular prisms, 
 which may be termed their elementary body ; and are usually desig- 
 nated by the outlines of their backs as Rectangular, Triangular, 
 Trapezoidal, etc. The Initial Wedge may be assumed to have a square 
 back; by successive transformations of which, several varieties are 
 easily formed. 
 
124 
 
 MEASUREMENT OF EARTHWORKS. 
 
 (1) Let the back of u rectan- 
 gular wedge (or the initial wedge) 
 be a square, on a side of 6, edge 
 12, length 20. Then, the right 
 section = (6 X 20) -T- 2 = 60. 
 One-third of the sum of the lat- 
 eral edges = (6 + 6 + 1 2) -H 
 3 = 8 ; and 60 X 8 = 480 == 
 Volume of the Square Wedge. 
 
 (2) Now, suppose the edge of 
 (1) to be contracted to a point; 
 then, the wedge becomes a pyra- 
 mid, for which case the rule also 
 holds; thus, right section = 
 
 3 = 4; and 60 
 
 60 i sum of edges = (6 -f 6 + 0) 
 
 X 4 = 240 = Volume. 
 
 Proof: By the common rule for pyramids, we have, base (6 
 X 6) -* 3 = 12 ; and X by altitude 20 = 240 = Volume, the 
 same as before. 
 
 (3) Suppose the back of the 
 square wedge (1) to be con- 
 verted into an isosceles triangle, 
 on a base of 6, and hight of 6 
 other dimensions as in (1) 
 
 then right section = 60 
 
 i sum of edges = (6 -f -f 12) 
 -r- 3 = 6 ; and 60 X 6 = 360 
 = Volume. 
 
 Proof: Now, the inscription of the isosceles triangle, within the 
 square back, evidently cuts off two pyramids, of which the volume 
 of each = (3X6)-f-2 = 9-v-3X20 length X 2 in number 
 = 120 Volume, of pyramids cut away from the square wedge (1) ; 
 then, 480 120 = 360 = Volume, the same as before. 
 
 (4) Now, suppose (1) and (2) 
 to be placed in contact sidewise, 
 then they form together a rect- 
 angular wedge, back, 12 by 6; 
 edge, 12 ; length, 20 : right sec- 
 tion = 60 i sum of edges 
 
 = (12 +12 + 12) -3 = 12; 
 and 60 X 12 = 720 = Volume. 
 
CHAP. V. FOURTH METH. COMP. ART. 27. 
 
 125 
 
 Proof: By two Pyramids = (72 +- 3 X 20 = 480) -f (60 -* 
 3 X 12 = 240) = 720, the same Volume; or, by addition of (1) 
 and (2) = 480 -f 240 = 720, Volume as before. 
 
 (5) Suppose now the vertical 
 sides of the square back of (1) to 
 close in gradually until they 
 meet and coincide in a single 
 vertical line ; then the back has 
 vanished, and become a vertical 
 edge, while the original one 
 remains horizontal, dimensioned 
 along with the other parts as in (1) and we have right-section 
 
 60 J sum of edges = (12 -f + 0) -*- 3 = 4 ; and 60 
 
 X 4 = 240 = Volume of this peculiar double-edged wedge; 
 which is composed of, or may be decomposed into, two pyramids, 
 based on the right-section, as common to both, and each having 
 an altitude of half the edge, or 6 (though such equal division of 
 edge is not essential) ; hence, we may assume the edge 12 to be 
 
 a double altitude; and (~ X 12) = 240 Volume of both 
 the same as before. 
 
 (6) Now, suppose the vertical 
 sides of the square (1) to become 
 inclined (at any angle that will 
 not extinguish the base of the 
 back), say at an angle of i to 1 
 side-slope, thus reducing the base 
 from 6 to 2, then we have the right- 
 section as before = 60 ...... I 
 
 sum of edges = (6 + 2 -f 12) 
 
 6t ; and 60 X 6 = 
 
 400 = Volume of Trapezoidal Wedge. 
 
 Proof: In this case two triangular pyramids are cut away from 
 the original solid, by the sloping sides, having together a base of 
 4, and altitude of 6 ; then, (6 X 4) -4- 2 = 12, which -f- 3 and 
 X 20 common length = 80 Volume cut away but Volume of 
 (1) = 480 80 = residual Volume = 400, as before. 
 
 (7) Now, suppose two sides of the square back of (1) to gradu- 
 ally reduce their contained angle, and finally to vanish upon the 
 
126 
 
 MEASUREMENT OF EARTHWORKS. 
 
 diagonal then the back be- 
 comes a right-angled triangle 
 (the side joining the right-angle, 
 say perpendicular to the edge), 
 and this wedge has two edges (one 
 original, and the other now 
 formed at the side connecting 
 with the acute angle, both being horizontal edges). Then, the 
 
 right-section = 60 i sum of edges (6 -f -f- 12) -t- 3 
 
 = 6 ; and 60 X 6 = 360 = Volume. 
 
 Proof: Divided by a plane diagonally through the vertex of 
 the triangular back, and opposite corner of the edge, we may 
 decompose this wedge into two pyramids the one with a base 
 = the right-section = 60, and altitude = the original edge = 
 12 ; then, 60 X 12 -4- 3 = Volume = 240 
 
 The other, with a base equal to the triangular back, or 
 (6 X 6) -i- 2 = 18, and an altitude = the length = 20 ; 
 then, 18 -T- 3 = 6, and X length 20 = Volume . . . = 120 
 
 Total Volume of both Pyramids =360 
 
 the same as before. 
 
 (8) A Rhomboid Wedge is 
 computed in a similar manner : 
 thus, let the rhomboidal back 
 have a vertical diagonal = 12; 
 the other = 4 ; an edge of 12; 
 length = 20 ; and the side-slopes 
 being $ to 1. 
 
 Then, the right-section = 
 12 X 20 
 
 ~2~~ 
 120 X fr 
 
 =120 ...... i sum of edges, 
 
 = 640 = Volume. 
 
 44-12 
 
 ; and 
 
 Now, by cutting off from the rhomboid, near the lower angle, 
 any given triangle, we have remaining a Pentagonal Wedge. 
 
 Thus, suppose we cut off a triangular wedge having the base 
 of its back uppermost = 2 ; altitude = 3 ; common length and 
 edge = 20 and 12. 
 
 Then its right-section = ?->< 2 - 2 + 12 
 
 X 
 
 = 140 Vol- 
 
 ume, cut off. And 640 140 
 Pentagonal Wedge. 
 
 500 = the Volume of the residual 
 
CHAP. V. FOURTH METII. COMP. ART. 28. 
 
 127 
 
 (9) Let us now consider a 
 Trapezoidal Wedge dimensioned 
 like (8), with side-slopes of i to 
 1, forming the top of the back, 
 while its base = 2. 
 
 Let one side-high t = 12 above 
 intersection of slopes; the other 
 = 6 ; the edge = 12 ; and the 
 length = 20. 
 
 Now, we may compute this wedge in two parts as follows: 
 
 1. As a triangular wedge, above the level of the lowest 
 side-hight. 
 
 x 
 
 12 
 
 320 
 
 2. As a trapezoidal wedge, between the level 
 mentioned and the base of the back. 
 
 2+12 
 
 Total Volume 
 
 180 
 500 
 
 Or, as in (8), we may compute the body as a Ehomboidal 
 Wedge, and deduct the triangular wedge cut away below the 
 base of 2, as in fact we did in (8), the resulting volume being 
 500, the same as herein found. 
 
 Finally, we perceive that from (1) the square or initial wedge we may 
 easily deduce several varieties of wedges, and might go further. 
 
 After this necessary digression, indicative of the simplicity, gen- 
 erality, and value of Chauvenet's Theorem, we will now proceed to 
 illustrate our own rule (deduced from this theorem), as applied to 
 Earthworks, by several examples. 
 
 28. 
 
 Here follows the calculation of some examples. 
 
 Example 1. Computation by Wedge and Prism, tested by Hights 
 and Widths, under Simpson's Rule 
 
128 
 
 MEASUREMENT OF EARTHWORKS. 
 References to Fig. 80. 
 
 In this case equal slopes of 1 in 4 form a ridge in the larger end 
 section, and a hollow in the lesser one. 
 
 Dimensioned as shown in the figure annexed. 
 
 2000 area. 
 JacliI 12OO area. 
 Tadbl = mid.8ec.16OO ar; 
 
 -sr 
 
 I.=ini: of sip: 
 
 Data. 
 
 Sq. Ft. 
 
 ( Differences of areas of end sections ......... = 800 
 
 < Widths, or horizontal projections, equal for both sections . = 80 
 I Distance apart sections ............ . = 100 
 
 To find the vertical mean hight of back of wedge. 
 
 C 2000 ) 
 End Areas = j -^oo f Difference of Areas. 
 
 Half sum of widths - 80) 800 
 
 10 = Vertical Mean Hight of Back. 
 
CHAP. V. FOURTH METH. COMP. ART. 28. 129 
 
 Then, by the Rule above, and Chauvenefs Theorem. 
 Sum of 3 parallel sides of edge and back -r- 3. 
 
 80 ) -p , f f Vertical Mean 
 
 }. = Back. 10 = < TJ. , , /, -r, i 
 
 80 j ( Right of Back. 
 
 80 = Edge. Right Section i 100 = Common length. 
 
 3)240 2)1000 
 
 80 = Average breadth 500 = Area of right sec. 
 
 Right section X Mean breadth = 500 X 80 . . . = 40,000 = Volume of Wedge* 
 Smaller end area = 1200 X 100, length . . . . = 120,000 = " " Prism. 
 
 Solidify of entire prismoid = 160,000 Cubic Feet. 
 
 Proof, by Hights and Widths (SIMPSON). 
 
 Hights. Widths. 
 
 Larger cross-section . = 50 X 80 = 4000 =26. 
 Smaller " " . = 30 X 80 = 2400 = 2 1. 
 
 Sums of his. and wids. = 80 X 160 = 12800 = 8m. 
 Divisor 12)19200 
 
 "T600 = Prism. Mean Area. 
 100 = Common length. 
 Solidity of entire Prismoid (as above) = 160,000 Cubic Feet. 
 
 Note. By BUTTON'S General Rule we have the same Solidity -=* 
 160,000 Cubic Feet. 
 
 Example 2. Let us now take the case figured for another purpose, 
 by Fig. 14, Art. 8. 
 
 Areas. 
 
 Large end section -= 654 to road-bed only. 
 
 Small " " = 300 " 
 
 Difference, or area of back 1 __ 
 of superposed wedge . ) 
 
 Supposing the smaller end, at a distance of 100 feet forward, to be 
 ABKH = 300 in area. While the larger end ABCDEFGHA = 
 654 area. Common length = 100 feet. 
 
 Widths. 
 fU _L 4ft 
 
 Then, -~-- = 47, Mean width of back. 
 
 , 7-532 X 100 length Ri ^ on - 
 and = oTo'b 
 
 = 7-532, Vertical Mean Hight of Back. 
 
 47 
 
130 MEASUREMENT OF EARTHWORKS. 
 
 54 -4- 40 -f 40 = Sum of the three parallel sides 
 
 ^r- . = 44f feet 
 
 o 
 
 p . ( 376-6 X 44f . . . = 16822 = Volume of Wedge. 
 a y> \ 300 X 100 length == 30000 = " " Prism. 
 
 Solidity of the whole Prismoid, \ ~~ 
 
 f j i j 4 j T C = 46822 = Cubic feet to road-bed. 
 
 jrom road-bed to ground line J 
 
 or 56,822 to inter- 
 section of slopes. 
 
 Now, roughly computing this example, both by Hights and Widths, 
 and by Roots and Squares, we find for the Solidity about the same 
 result, the difference being small in the whole body of earthwork con- 
 sidered. 
 
 In like manner, roughly calculating Figs. 43 and 44, which have 
 very irregular ground lines, with both end sections in each case dia- 
 grammed upon one figure. We find that computed by Wedge and 
 Prism, and some other methods, as a proximate test, they all coincide 
 within a few cubic yards. 
 
 So that this rule for calculating Prismoids of Earthwork by means 
 of a Prism and Wedge, superposed, may be accepted as proximately 
 correct in all ordinary* cases, and it is in practice a very simple one, 
 as may be noticed in the examples. 
 
 Requiring for data give,n merely the areas of the end cross-sections, 
 their distance apart, and their total widths across, horizontally, to 
 ground edges of slopes : no matter how irregular the surface may be. 
 
 In all the computations above (as well as in the methods of pre- 
 ceding chapters), so soon as the mean area of an earthwork solid is 
 ascertained, it will be found conducive, both to expedition and to 
 accuracy, to resort with it to the table of cubic yards for mean areas 
 (at the end of the book), to obtain cubic yards, if they should be 
 required in the resulting volume. 
 
 In this connection it may be observed that the transverse area of 
 the under-prism being always given in the data (and usually given as 
 that of the smaller cross-section), whilst the distance apart sections 
 is also known, it is better, where cubic yards are desired in the ulti- 
 mate solidity, always to find them from the table in the manner shown 
 by the directions for its use; and the superposed wedge may be also 
 treated in a similar way by computing its mean area. 
 
 * Where the cross-sections appear to be unusually distorted, so as to render doubtful, 
 the application of any ordinary rules, then we must endeavor to sketch an accurate mid- 
 section, and use our First Method of Computation (Chapter II.) which never fails when 
 the data is correct. 
 
CHAP. V FOURTH METH. COMP. ART. 29. 
 
 131 
 
 29 Although the foregoing rule for the computation of a 
 
 Prismoid, by "Wedge and Prism, is proximately correct in all ordinary 
 cases, it has limits which must be observed, when exact results are 
 sought. These limits are: That the extreme horizontal width of the 
 smaller end section shall always be equal to, or less than, that of the larger 
 end, and never greater, where our rule is used as written above. 
 
 Thus, in all the cases computed in the above examples, the width 
 of smaller end is less, except in the figure next preceding, where it is 
 equal but in none of the examples is it greater, and hence they are 
 all clearly within the limits of the rule. 
 
 In the following figure (Fig. 81), however, the horizontal width 
 of the smaller end is, in this unusual case, greater than that of the 
 
 Pig. 81. 
 
 larger one to such cases then our rule above stated does not apply 
 directly in the form as written. 
 
132 MEASUREMENT OF EARTHWORKS. 
 
 A consideration of the figure annexed, where both end sections and 
 the mid-section are diagrammed together, will make the reason 
 evident. 
 
 It is simply this, that whenever the horizontal top line of the 
 smaller end exceeds in width that of the larger one, or lays above it 
 (in a cut), when diagrammed together in one figure, with the diedral 
 angle common to both, then the smaller end ceases to be the section of a 
 prism, and becomes that of a prismoid. 
 
 But as a prismoid is formed of an under prism, with a wedge super- 
 posed, we have then in this solid (such as is sectioned in Fig. 81) a 
 prism with two wedges superposed the upper one carrying the ground 
 surface of the earthwork solid. 
 
 The prism in this case has for its cross-section the portion of the 
 solid below the line c 6, marking the extreme breadth of the larger 
 end section, while the two superposed wedges are reversed in position 
 that in contact with the under prism having its edge in the line c b, 
 the width of the larger, while that carrying the ground surface has its 
 edge in e d, the width of the smaller end section ; and therefore the 
 wedges are reversed in position, though having the same length in com- 
 mon with the prism, which underlies both. 
 
 Example 3, Fig. 81. 
 
 Cross-section of prism below c b = 400. 
 " " smaller end = 900. 
 
 Data { " " larger end = 1200. 
 
 Common length of all = 100 feet; other dimensions as in 
 Fig. 81. 
 
 (1) By Prwnoidal Formula First Method Computation, Chapter 
 II. (Button's General Rule) which is an accepted standard for 
 accuracy. 
 
 Smaller end section . . . = 900 = t. 
 Larger " " . . . = 1200 = b. 
 Mid-section deduced, being 
 a mansard figure flat on 
 the top = 1425 X 4 . . = 5700 = 4 m. 
 
 6)7800 
 T300 == Prism. Mean Area. 
 
 100 = Common length. 
 Solidity ....... = 130,000 Cubic Feet. 
 
CHAP. V. FOURTH METH. COMP. ART. 30. 133 
 
 (2) By Chauvenet's Theorem, and our rule drawn from it. 
 
 I (1) = The top wedge (at ground) = Right 
 section (40 X 100 -H 2 = 2000) 
 X i sum of edges = (60 + 40 
 
 + -H 3 = 33i) = 66,667 C. Feet- 
 
 (2) = The intermediate wedge, adjoining the 
 prism (as in our rule). Difference 
 of areas -f- siim of widths = 500 
 
 I 
 
 I 
 
 S, Then, by the rule (from Chauve- 
 
 net), (10 X 100 -s- 2 = 500) X 
 
 50 = 10, Mean Hight of wedge. 
 Tl 
 
 a 
 
 i sum of edges = (60 + 40 + 40 
 
 -f- 3 = 463) = 23,333 
 
 (3) = The prism, which underlies both = 
 
 400 area X 100 length . . . . = 40,000 " " 
 Totality of this solid, containing two 
 wedges and one prism = Solidity = 130,000 C. Feet. 
 
 In examining the solid body terminated by the cross-sections figured 
 (in Fig. 81), it will be found to be bounded upon every side by planes, 
 passed through three common points, so connected that the faces con- 
 tain no warped surfaces whatever. 
 
 30. It would appear that in peculiar solids, like that in Fig. 81. 
 we might omit the prism entirely, and decompose the body into a 
 species of double triangular or rhomboidal wedge (with base of back, 
 and also the edge, common to two triangular wedges superposed, and 
 inverted with their bases in contact, one on the other), and this 
 double triangular wedge, with a single pyramid based upon the 
 smaller end (or in fact on either end), all having a common length, 
 would form the whole earthwork solid, and simplify the calculation 
 in such special cases if not in all cases of irregular ground. 
 
 Thus, examining the large end I b a c, we find it to consist of the backs 
 of two triangular wedges, joined together at their bases c b, and hav- 
 ing a common edge at 100 feet forward, equal to d e, the top of the 
 smaller end. 
 
 Below this double wedge we find a pyramid whose base is I e d I, and 
 vertex at I, with the common length of 100 the calculation of 
 solidity is as follows: 
 
134 MEASUREMENT OF EARTHWORKS. 
 
 Example 4 (Fig. 81). 
 
 (1) The Double (Triangular or Rhomboidal) Wedge. 
 The mean breadth being common both to the upper and lower tri- 
 angular part of the larger cross-section, then we have, 
 
 o 
 
 -= 33*. 
 
 And the whole hight of the double triangular wedge is composed 
 of the hights of the two separate parts = 40 -f~ 20 = 60, forming a 
 Rhomboid. 
 
 Then, - -^ - = 3000 = Right Section. 
 
 C. Feet. 
 
 And right section = 3000 X * sum edges = 33* . . . = 100,000 
 (2) The Pyramid, based on smaller end = X 100 . = 30,000 
 
 Solidity of the whole Prismoid = 130,000 
 
 (Being the same as in Example 3.) 
 
 "We might also divide this solid into two wedges and a pyramid by 
 other cutting planes, with the same result. Thus : 
 
 Example 5 (Fig. 81). 
 
 Rt. Sec. % sum edges. C. Feet. 
 
 (1) lfe*r l*dfc*L>^?- 2000 x( 4 + f + ) = 66,667 
 
 Rt. Sec. % sum edges. 
 
 (2) termed. W^e, 5*1? = 1500 x( 60 + 4 3 + ) = 50,000 
 
 (3) Pyramid underlying both = ---== 133* X 100 length = 13,333 
 
 Solidity of the whole Prismoid = 130,000- 
 
 (Being the same as in Examples 3 and 4.) 
 
 Suppose now upon the smaller end section (Fig. 81) we place a 
 triangle of 60 feet base, and 10 feet altitude, the vertex representing the 
 termination of the crest of the ridge coming from the apex of the taller 
 section, and thus augment the area of the lesser end to an equality with 
 the other, or make each = 1200 in area the addition in Solidity 
 being a Pyramid. 
 
 Then, although the end areas are now equal, the horizontal widths 
 between the ground edges of the side-slopes remain unequal, as before ; 
 the big end having least width. 
 
CHAP. V. FOURTH METH. COMR ART. 30. 
 
 And the computation of this solid is as follows: 
 Example 6 (Fig. 81). 
 
 135 
 
 x 4 =6000 = 4 m. 
 6)8400 
 1400 Pris. Mean. 
 
 100 Length. 
 Sol.= 140,000 C. Feet. 
 
 By known Geometrical Solids, gov- 
 erned by Familiar Rules. 
 
 Pyramid (super-added) base 300. 
 Then, 
 
 300 Length. 
 
 X 100 = 10,000 
 
 (1) Top Wedge = 66,667 
 
 (2) Intermediate Wedge . . . = 23,333 
 
 (3) Prism . . = 40,000 
 
 Solidity in C. Feet . . = 140,000 
 
 By Hutton's General Rut 
 
 , = 1200 = t. 
 Find Areas . . 
 
 in, The mid-sec- 
 tion deduced, 
 being a man- 
 sard figure, 
 peaked upon 
 the top = 
 1500 in area. 
 
 50 + 30 _ 
 
 40 X 20 = 800 
 
 ?!2L_ 5 = 75 
 
 A of 25' =625 
 
 ~1500 
 
 In all the above examples (except Example 2), the computation 
 for solidity extends from ground surface to intersection of slopes, 
 without regard to the road-bed. But any width of road-bed may 
 be assumed, the volume of the grade prism ascertained, and being 
 deducted, will leave the solidity from road-bed to ground all the same, 
 as if it had been specially calculated in that way. 
 
 a Of the Rhomboidal Wedge and Pyramid. 
 
 A close examination of the solid, cross-sectioned in Fig. 81, and 
 shown in isometrical projection by Fig. 82, will make it evident that 
 beginning with the larger end section, the three cross-sections required 
 by HUTTON'S General Prismoidal Rule will be a Rhomboid, a Penta- 
 gon, and a Triangle, dimensioned as shown in the figures. 
 
 And the solidity of this body by BUTTON'S Rule, as shown in 
 Example 3, Art. 29 = 130,000 Cubic Feet. 
 
 It is also evident, from Example 4, of this article, that this compu- 
 tation can be made for solidity with the same result (130,000 Cubic 
 Feet), by decomposing the body into a Rhomboidal Wedge and two 
 Pyramids, which may be aggregated and calculated as one, so that, as 
 in Example 4, this solid can be computed as though it were composed 
 of a single Rhomboidal Wedge, having its edge in the width line of 
 the smaller end section; and of a single Pyramid upon a base equiva- 
 lent to the latter in area, and its vertex at the foot of the rhomboidal 
 
136 
 
 MEASUREMENT OF EARTHWORKS. 
 
 back which forms the area of the larger cross-section, or one equiva- 
 lent thereto, and standing (as both end sections do) with the vertices 
 of one of their vertical angles coincident with the line of intersection 
 of the side-slopes prolonged. 
 
 Kg. 82 
 
 Sea. TOT Inch. 
 
 By means of Wedge and Prism, or Wedge and Pyramid (especially 
 the latter), we have already indicated the process of reaching the vol- 
 ume of an earthwork solid, and we will now continue our examples 
 until the simple combination of Wedge and Pyramid, in computing 
 solidity upon the usual earthworks, is fully illustrated. 
 
 Although solids resembling Fig. 81 in their cross-sections admit of 
 being easily computed by their own dimensions, either by Wedge, 
 Prism, and Pyramid, or by HUTTON'S General Rule, which is a stan- 
 dard for volume; nevertheless, as earthwork sections generally pre- 
 sent themselves in a somewhat different form, it becomes desirable to 
 devise a rule which, within a long range, will apply to all earthwork 
 with uniform slopes, and shall include within its limits the great 
 majority of cases which come under the notice of the engineer. 
 
CHAP. V FOURTH METH. COMP. ART. 30. 137 
 
 Extremely irregular and distorted solids, however, have sometimes 
 to be subjected^ to calculation, which seem almost incommensurable 
 by any fixed rule, and such exceptional cases must be left to inde- 
 pendent methods adopted at the time ; though it is obvious that any 
 solid may be so sectioned, and divided into limited portions, as to 
 admit of computation by many processes, without material error. 
 
 b Statement. In any earthwork solid contained within a 
 
 diedral angle (formed by the intersection of uniform side-slopes), 
 however irregular the ground may be, if the side-slopes continue uni- 
 form and we have given, the length I, the areas of the cross-sections 
 at the ends A and A', and the slope ratio r. We may compute the 
 volume of such solid as a double Triangular, or single Rhomboidal 
 Wedge in combination with a single Pyramid (the latter also usually 
 Rhomboidal but sometimes Triangular). 
 
 Process. Take any pair of irregular cross-sections, judiciously 
 located and measured by the field engineer, so as correctly to define 
 the ground, and of which all the necessary dimensions are known, as 
 well as the distance apart sections. 
 
 1. Ascertain the areas of the cross-sections to intersection of 
 side-slopes. 
 
 2. Find the proper hight from intersection of slopes, to include 
 one-half the area, also the proper width, and assume this as the 
 base of the back of a double Triangular, or Rhomboidal Wedge 
 in the larger end, and as the edge of the same in the smaller one. 
 
 3. Compute from the larger, or from either end section, a 
 Rhomboidal Wedge, by Chauvenet's Theorem. (See Example, 
 Art. 27, a, paragraph 8.) 
 
 4. Then, to the solidity of this Rhomboidal Wedge, add that 
 of a Pyramid, based upon the other end section, and having for 
 its altitude the common length, or distance apart sections. (See 
 rule following.) . 
 
 The sum of the altitudes of the double triangles (joined at their 
 bases) forms the vertical diagonals, or hights of back, of the rhomboi- 
 dal wedges, while their horizontal "diagonals form the width of back 
 at one end, and of the edge at the other, the angular points of the 
 Rhomboid, vertically, being zero. Either end may be calculated from, 
 while the other area is the base of a pyramid (Rhomboidal, Triangu- 
 lar, or Irregular), having for altitude the common length I. For 
 proof of the work we should always make both direct and reverse calcu- 
 
138 MEASUREMENT OF EARTHWORKS. 
 
 lations, taking either end alternately as the base, and though they 
 will seldom agree exactly, owing to the decimals coining in a different 
 order (unless we use a cumbrous number of places) ; nevertheless, 
 the agreement will be found close enough for a verification of such 
 work. 
 
 To compute the Rhomboidal Wedge and Pyramid in an Earthwork. 
 Adopt either end for Base, and call the other the Top = b and t, of 
 former notations. 
 
 Present notation : 
 
 A. Area of cross-section assumed for the Base. 
 A! = " " " " " Top. 
 
 I = Common length, or distance apart sections. 
 
 These are all the data required to be given, the remainder needed 
 
 are easily computable. 
 
 h \ Vertical diagonals of the equivalent Rhomboids, into which 
 h' J the end areas are transformed. 
 
 , > Horizontal diagonals of the same. 
 Then, by computation : 
 
 From the foregoing it is evident that w = h r, and w' = h' r. 
 Also, when the slopes are 1 to 1, then h = \/2 A; if 1 to 1, 
 h = V-fAT; and if 2 to 1, h = VAT The use of these will often be 
 convenient. 
 
 RULE. Case 1. Where width of big end is equal to, or greater 
 than, that of small end. 
 
 1 (Half product of vertical diagonal of base, by distance apart 
 sections) X (One-third the sum of horizontal diagonals of 
 both ends) = Solidity of Rhomboidal Wedge ; 
 
CHAP. V. FOURTH METH. COMP. ART. 30. 139 
 
 2 (One- third of area of top) X (Distance apart sections) = 
 Solidity of Pyramid ; 
 
 3. Add together the two solidities above (1 and 2) for the solidity 
 of the entire Prismoid : from ground to intersection of slopes, 
 and minus the volume of the grade prism, gives solidity from 
 road-bed to ground. 
 
 RULE. Case 2. Where width of big end is equal to, or less than, 
 that of small end. 
 
 In this case the multiplier for edges (No. 1, Case 1) is to be 
 
 (w + w') + (w w) . (w + ,, .. 
 
 -, instead of simply - - . . While to 
 6 6 
 
 the volume produced by the Rule of Case 1 modified in the 
 multiplier as just mentioned we must add a final correction, 
 as follows : (Difference of actual horizontal widths X Difference 
 of their hights from intersection of slopes) X length this 
 final product, added to the volume resulting from the rule above, 
 gives the solidity for Case 2. 
 
 The application of these corrections will be shown hereafter by 
 an example, drawn from the peculiar solid, figured in Figs. 81 
 and 82. 
 
 The results produced by these corrections, when added to those 
 obtained by the Rule of Case l,will give the solidity, whenever 
 the actual width of the smaller end section does not exceed three 
 times that of the greater one. 
 
 Within these limits the rules and corrections above will apply, and 
 they will be found to cover the great majority of practical cases; but 
 where thl end sections are even more distorted, we must then com- 
 pute by Mutton's General Rule, or by the actual dimensions of the 
 solid, decomposing it into elementary bodies. 
 
 As the Prism, Wedge, and Pyramid, are the solid elements from 
 which every great-lined body is composed, and into which it may be 
 again resolved, it follows by parity of reasoning (as in the case of the 
 Prismoidal Formula) that for all earthwork solids, bounded by planes, 
 the rules of this chapter hold. 
 
 C ...... We will now illustrate our method of Wedge and Pyra- 
 
 mid, by computing the cases of Chapter II., figured from 53 to 64 
 inclusive, and all originally computed by HUTTON'S General Rule- 
 the standard for accuracy. 
 
140 MEASUREMENT OF EARTHWORKS. 
 
 All of these examples (as indeed is the fact with most others in 
 practice) come under our Rule and Case 1 the width of the larger 
 end section being in every instance greater than that of the smaller 
 one. (See Figs. 53 to 64, Art. 18. 
 
 Art. 18. Example, illustrated by Figs. 53 to 55. 
 
 Given areas f b = 990 = A ^ Vertical diago- ( h = 44-50 1 Horizontal dia- ( w = 44'50 j 
 to intersection It = 500 = A' [ na ls computed. \ h' = 31-62 J gonals computed. \w f = 31-62 J 
 of slopes, etc. ( I =100 feet. J 
 
 The road-bed being 20 feet ; the side-slopes 1 to 1 in this case, as 
 in all where r = 1 ; the Rhomboid becomes a square, and the diago- 
 nals equal. 
 
 Direct calculations. 
 
 h X I w + *? o 
 
 ~ X = S ' 
 
 44-50 X 100 44-50 + 31-62 
 
 -g - X g - . . = 56,471 
 
 A 7 
 
 - x I = & of Pyramid. 
 o 
 
 500 
 
 -^ X 100 ........... = 16,667 = Pyramid. 
 
 Total . ...... . .. > . . = 73,138 C. Feet. 
 
 Deduct Grade Prism ........ = 10,000 
 
 Leaves Solidity of Earthwork ..... = 63,138 
 As computed in Art. 18, Chapter II. . . = 63,170 
 
 Difference ......... = 32 
 
 Reverse calculations. 
 
 31-62 X 100 - 31-62 + 44-50 
 
 ^ - X - g - . . . . = 40,126 == Wedge. 
 
 QQA 
 
 ~ X 100 ........... = 33,000 = Pyramid. 
 
 Total ..... . ...... = 73,126 C. Feet. 
 
 Deduct Grade Prism ........ = 10,000 
 
 Leaves Solidity of Earthwork ..... = 63,126 ' 
 As computed in Art. 18, Chapter 11... = 63,170 
 
 Difference ......... = 44 
 
 The above example represents an earth-cut upon three-level ground. 
 
CHAP. V. FOURTH METH. COMP. ART. 30. 
 Art. 18. Example, illustrated by Figs. 56 to 58. 
 
 141 
 
 This example represents an earth-cut on -five-level ground, having a 
 road-bed of 20 ; slopes of 1 to 1 ; length 100 feet. 
 
 Computed by our Rule, Case 1, we have. 
 
 Direct calculations. 
 
 Wedge . . = 24,306 
 
 Pyramid . . = 14,367 
 
 38,673 
 
 Deduct G. P. = 10,000 
 
 Solidity . = 28,673 
 By Art. 18 . = 28,650 
 
 Difference. = + 23 C. Feet. 
 
 Reverse calculations. 
 
 (Wedge . . = 27,254 
 Pyramid. . = 11,467 
 38,721 
 , Deduct G. P. = 10,000 
 j Solidity. . = 28,721 
 [ EyArt. 18 . = 28,650 
 \ Difference. = -f 71 C. Feet. 
 
 Art. 18. Example, illustrated by Figs. 59 to 61. 
 
 This example represents an earth-cut on seven-level ground, dimen- 
 sioned as above. . 
 
 Computed by our Rule, Case 1, we have: 
 
 Direct calculations. 
 
 Reverse calculations. 
 
 Wedge . . = 42,048 
 Pyramid . . =* 21,700 
 
 
 Wedge 
 Pyramid 
 
 Deduct C 
 Solidity 
 By Art. ] 
 
 Differe 
 
 63,748 
 Deduct G. P. = 10,000 
 Solidity. . = 53,748 
 By Art. 18 . = 53,733 
 
 Difference = + 15 C. Feet. 
 
 42,935 
 20,800 
 63,735 
 10,000 
 53,735 
 53,733 
 
 -f 2 C. Feet. 
 
 Art. IS. Example, illustrated by Figs. 62 to 64. 
 
 This example represents an embankment upon nine-level ground, 
 very rough. Road-bed 16 feet; side-slopes 1 to 1; length 100 feet. 
 
 Areas given f t = 
 to intersection \ b = 
 of slopes, etc. 
 
 iven f t = 828% = A ") Vertical diag 
 tion \ b = 644% = A' I nals computed, 
 tc. (I =100 feet. J 
 
 Vertical diago- f h = 33-24 1 Horizontal dia- f w = 49-86 1 
 h' = 29-<W J gonals computed. | / = 43-98 
 
142 MEASUREMENT OF EARTHWORKS. 
 
 Direct calculations. 
 
 33 ' 24 >< 10 X = 51,987 Wedge. 
 
 2 o 
 
 644 ' 67 X 100 . ;' v v "' -* ki v V * . . = 21,489 Pyramid. 
 
 73^476 
 Deduct Grade Prism. . . .'..'. . . = 4,267 
 
 Solidity . . . = 69,209 C. Feet. 
 
 As computed in Art. 18, Chapter II. . . . = 69,200 
 
 Difference . ... C "!'. = ~+~9 C. Feet. 
 
 Reverse calculations. 
 
 29-32 X 100 49-86 + 43-98 
 
 2 3 = 45,856 Wedge. 
 
 828-67 
 
 3 
 
 X 100 .....= 27,622 Pyramid. 
 
 73,478 
 
 Deduct Grade Prism. - ." = 4,267 
 
 Solidity = 69,211 C. Feet. 
 
 As computed in Art. 18, Chapter II. ... = 69,200 
 
 Difference = -f 11 C. Feet, 
 
 t 
 
 d We have thus compared the whole four of the examples 
 
 illustrated in Chapter II., and all computed by HUTTON'S General 
 Rule. These we find to agree with the calculations by Wedge and 
 Pyramid, in every instance within a few cubic feet, and had the deci- 
 mals (into which all these computations run) been carried further, 
 the agreement would probably have been closer. 
 
 We will now compute by Wedge and Pyramid the example of a 
 heavy embankment, taken from Warner's Earthwork, Art. 86. 
 
 " Prismoid. First end-hight 28'7 ; second end-hight 14'5 ; 
 surface-slope 15 ; side-slope H to 1; road-bed 24 feet." 
 
 Data computed f b 2411 = A ~\ Vertical diago- f h = 56-70 j Horizontal dia- f w = 85-05 J 
 to intersection of -I t 907 = A' > nalg computed. { h' 34'78 j gonals computed. ( w' = 5217 } 
 slooes.etc. U= 100 feet. ) 
 
CHAP. V. FOURTH METH. COMP. ART. 30. 143 
 
 Direct calculations. 
 
 56 ' 7 9 X 10 X **+* . . . = 12% Wedge. 
 
 L o 
 
 - X 100 = 30,233 Pyramid. 
 
 159,906 
 
 For Cubic Yards -=-27 = 5,923 
 
 Deduct volume of Grade Prism = 356 
 
 Solidity = ~5,567C. Yards. 
 
 By Hutton's General Rule = 5,566 
 
 \ Difference = + 1 C. Yard. 
 
 Reverse calculations. 
 
 34-78 X 100 52-17 + 85-05 ** 
 
 X . . . = 7y,o4J 
 
 < 100 . = 80,367 Pyramid. 
 
 Jf 9,909. 
 
 For Cubic Yards -r- 27 = 5,923 
 
 Deduct volume of Grade Prism = 356 
 
 Solidity = 5^67 
 
 By Hutton's General Rule = 5,566 . 
 
 Difference = -f 1 C. Yard. 
 
 Mr. "Warner (in Art. 86 quoted) makes the volume here computed 
 : 5562 Cubic Yards. 
 
 6 All of the above examples come under Case 1, of our 
 
 Rule, as ordinary earthwork sections usually do. But we will now 
 compute a single example by Case 2 where the width of the greater 
 end is less than that of the smaller one. This condition will be found 
 in the solid figured in Figs. 81 and 82. 
 
 In this example, illustrative of the rule in Case 2, the corrections 
 therein named have been duly embodied. 
 
144 MEASUREMENT OF EARTHWORKS. 
 
 Example of Case 2 (Fig. 81).* 
 4W8_XJOO x 4M>8+ 4*42 + 6-66 _ _ = 
 
 h X I (tfl + tip + (M i*Q 
 
 QAA 
 
 irr X 100 ............ = 30,000 Pyramid. 
 
 o - 
 
 = ^ x i. 110,000 
 
 Final correction, 10 X 10 X 20 X 100 . . = 20,000 
 
 Solidity . . . . . . . "... . = 130,000 C. Feet. 
 
 The same as computed before ...... = 130,000 
 
 It would appear, then, from the discussion in this chapter, the 
 examples given, and the simplicity and conciseness of the rules for 
 computing earthworks, by means of the Prism, Wedge, and Pyramid, 
 that they deserve to rank amongst the best employed for the purpose. 
 
 * Although this solid (Figs. 81 and 82) is bounded on all sides by plane surfaces, and 
 is composed simply of a Rhomboidal Wedge, superposed upon a Pyramid very few of 
 the E,ules or Tables, of the numerous writers on Earthwork, furnish means for comput- 
 ing its solidity which can only be readily ascertained by BUTTON'S General Rule, or by 
 decomposition into elementary solids, of which the rules for volume have been long 
 established. 
 
. 
 
 =r sp* 
 
 CHAPTER VI. 
 
 PROFESSOR GILLESPIE'S FOUR USUAL RULES, WITH THEIR CORREC- 
 TIONS, AND A COMPARISON OF HIS CHIEF EXAMPLE WITH OUR 
 THIRD METHOD OF COMPUTATION OR ROOTS AND SQUARES (CHAP- 
 TER IV.). 
 
 31 The late Professor IV. M. Gillespie, of Union College, 
 
 Schenectady, N. Y., was an able teacher of Civil Engineering, and a 
 sound practical writer on that and cognate subjects, as may witness 
 his Roads and Railroads (1847), 10 editions; Land Surveying 
 (1855), 8 editions; Higher Surveying, etc. (1870), posthumous, 1 
 edition ; and numerous valuable papers, read before the American 
 Scientific Association, or printed in scientific journals. 
 
 In 1847 he published his first edition of Roads and Railroads, and, 
 as an appendix to it, in about 25 pages, he gave a practical summary 
 of various methods of computing Excavation and Embankment, 
 accompanied by valuable corrections and suggestions, which were 
 together so explicit and so well grounded that this Appendix has 
 become the basis of several works upon the subject, whose authors, 
 without much acknowledgment (often without any), have freely 
 availed themselves of Professor Gillespie's labors. 
 
 His work on Roads and Railroads, well printed and cheaply pub- 
 lished, has had a great circulation; it has already filled 10 editions, 
 and is probably better known in the offices of engineers, all over this 
 country, than any other similar book. In the Appendix, on Excava- 
 tion and Embankment, Professor Gillespie recognizes "four usual 
 methods of calculation" 
 
 1. Calculation by Averaging End Areas (or Arithmetical Average). 
 
 2. " " Middle Areas. 
 
 3. " " Prismoidal Formula. 
 
 4. " Mean Proportionals (or Geometrical Average). 
 
 And we will now proceed to give his views substantially, but not 
 literally, upon these four rules, which he found in use when he took 
 up this subject in 1847, and which, indeed, had long before been known, 
 as follows: 
 
 10 145 
 
146 MEASUREMENT OF EARTHWORKS. 
 
 1st. Arithmetical Average. This consists simply in adding together 
 the areas of any two adjacent cross-sections, taking half their sum 
 for a mean area, and multiplying it by the length of the station, or 
 distance apart sections, to find the Solidity. 
 
 As generally used by engineers, instead of adding the end areas, 
 halving their sum, etc., they employ the sum of the two, or double 
 areas, and merely double one of the divisors in working for Cubic 
 Yards, as follows : 
 
 Engineers' Rule. 
 
 (Take the sum of the areas of any two adjacent cross-sections, 
 multiply these double areas by the length (which, if a full station 
 \ of 100 feet, is done mentally, or by removing the decimal point 
 / two places to the right). Divide by 6 and by 9, and the last quo- 
 \ tient gives the volume in Cubic Yards. <A 
 
 This Rule has been by far the most used of any other in our coun- 
 try ; with tables of Cubic Yards, for double areas, it is very expedi- 
 tious, and has found numerous advocates amongst engineers on 
 account of its simplicity and convenience ; it usually gives a result 
 in excess of the truth, and where the disparity of areas is great, very 
 much in excess; even this well-known error has found commendatory 
 advocates, on the ground that it is like the merchant giving good 
 measure to the customer, and that this excess in quantity being well 
 understood, would be compensated for by a reduced price, whenever 
 the work was executed by contract but these arguments are clearly 
 unsound. 
 
 Professor Gillespie has, however, indicated a simple correction, by 
 means of which the result of a computation, by Arithmetical Aver- 
 age can be reduced to the truth. 
 
 Thus, let 
 
 d = Difference of centre hights, supposing all the cross-sections to 
 
 be reduced to an equivalent level top. 
 3* -= Ratio of the side-slopes (or cot. of angle) s to 1. 
 I = Length of the cut or fill between sections. 
 
 * Engineers and writers have pretty generally, of late years, agreed to designate the 
 ratio of side-slopes as r (and this we have usually employed), while the symbol is con- 
 fined to slopes of ground, or surface slopes, but in the present case Professor Gillespie's 
 notation is adhered to. 
 
CHAP. VI. GILLESPIE'S RULES, ETC. ART. 31. 147 
 
 s d? I 
 Then, ^ is the proper correction for the results of Arithmetical 
 
 Average, which correction, if computed for each mass so calculated, 
 and then deducted therefrom, will give the true solidity the same pre- 
 cisely as if calculated direct by the Prismoidal Formula itself. 
 
 The chief example computed by Professor Gillespie under the sev- 
 eral heads of his subject, has the same data in all, as shown by the 
 first four columns of the following Tables the cross-sections in all 
 cases being assumed to be equivalent level trapezoids by him. 
 
 1. Arithmetical Average. 
 
 Table 1, computed in illustration of the corrections proposed, 
 including an entire section of a supposed railroad, 4219 feet in length. 
 
 1.' Road-bed 50; side-slopes of excavation 1J to 1; of embank- 
 ment 2 to 1. 
 
 Sta. 
 
 ~r 
 
 2 
 3 
 4 
 5 
 
 6 
 7 
 
 t 
 
 Dis- 
 tance 
 in 
 feet. 
 
 Cut. 
 + 
 in 
 feet. 
 
 
 
 20 
 
 
 -sT 
 
 Fill. 
 
 in 
 ft. 
 
 
 
 19 
 
 8 
 
 _o 
 
 27 
 
 End 
 Areas, 
 or 
 QAM* 
 
 sees. 
 
 Excava- 
 tion. 
 C. Feet. 
 
 Em- 
 bank- 
 ment. 
 C. Feet. 
 
 CORRECTIONS. 
 
 Corrected quanti- 
 ties, agreeing 
 with the 
 Prismoidal 
 Formula. 
 
 Sq. Ft. 
 
 Computed 
 by 
 Arith. Average. 
 
 By Formula 
 scP I 
 ~6~ 
 
 Amounts in 
 Cubic Feet, 
 deductive. 
 
 Excava- 
 tion. 
 C. Feet. 
 
 Embkt. 
 Cubic 
 Feet. 
 
 561 
 
 858 
 825 
 820 
 825 
 330 
 
 o 
 
 1386 
 1600 
 
 1672 
 528 
 
 
 388,773 
 1,280,994 
 660,000 
 
 685,520 
 907,500 
 87,120 
 
 HXIS'XSGl 
 
 45,441 
 
 858 
 82,500 
 
 128,79t) 
 
 98,673 
 33,275 
 
 7,040 
 13pS8 
 
 343,332 
 1,280,136 
 577,500 
 
 586,847 
 874,225 
 
 80,080 
 1,541,152 
 
 9 
 
 UX 2'X858 
 
 6 
 
 li X 20 3 X 825 
 
 6 
 
 2 X19 9 X820 
 
 6 
 
 2 X H s X 825 
 
 6 
 2 X 8"X33 
 
 4219 
 
 + 2986 
 2200 
 
 2,329,767 
 
 1,680,140 
 
 6 
 
 2,200,9ti8 
 
 From this Table it will be perceived that the error of the process 
 of Arithmetical Average, in this example, amounts in Excavation to 
 6 per cent., and in Embankment to 9 per cent., above the true solidity. 
 
 2d. Calculation by the Middle Areas. The second method of calcu- 
 lation is to deduce the middle areas (commonly called mid-sections) 
 of each Prismoidal mass, from the middle hight, or Arithmetical 
 Mean of the extreme hights of the solid, and multiply the middle 
 area thus found by length for volume. The results thus obtained are 
 too small; their deficiency being equal to just half the excess of the 
 first method. 
 
148 
 
 MEASUREMENT OF EARTHWORKS. 
 
 s d 2 1 
 
 Here the corrective formula is,; and 
 
 3.22 
 
 corrections thus calcu- 
 
 lated being added to the results obtained, by the process of middle 
 areas, would make them coincide with the true volume given by the 
 Prismoidal Formula. 
 
 2. Middle Areas. 
 
 Table 2, computed and corrected in illustration of the above, including- 
 an entire section of a supposed railroad = 4219 feet in length. 
 
 2. Road-bed 50 ; side-slopes of excavation 1 J to 1 ; of embank- 
 ment 2 to 1. 
 
 Sta. 
 
 Dis- 
 tance 
 in 
 
 feet. 
 
 Cut. 
 
 i 
 
 feet. 
 
 Fill. 
 
 in 
 ft. 
 
 Middle 
 Areas. 
 
 Computed 
 by 
 Middle Areas. 
 
 CORRECTIONS. 
 
 Corrected quanti- 
 ties, agreeing 
 with the 
 Prismoidal 
 Formula; 
 
 Exca- 
 va- 
 tion. 
 
 Em- 
 bank- 
 ment. 
 
 By Formula 
 sd*l 
 
 "12" 
 
 Amounts in 
 Cubic Feet, 
 additive. 
 
 Ex- 
 cava- 
 tion. 
 
 Em- 
 bank- 
 mont. 
 
 Sq. Ft. 
 
 Cubic Feet. 
 
 Ex. 
 
 Em. 
 
 C. Feet. 
 
 C. Feet, 
 
 1 
 2 
 
 3 
 
 4 
 5 
 
 6 
 
 7 
 
 561 
 858 
 825 
 820 
 825 
 330 
 
 8 
 
 20 
 
 
 
 o 
 
 19 
 8 
 
 
 571-5 
 1491-5 
 650 
 655-5 
 1039-5 
 232 
 
 320,611 
 1,279,707 
 536,250 
 
 537,510 
 857,587 
 76,560 
 
 liX!S a X561 
 
 22,721 
 429 
 41,250 
 
 49,337 
 16,638 
 3,520 
 
 343,332 
 1,280,136 
 577.500 
 
 586,847 
 874,225 
 80,080 
 
 12 
 
 1JX 2 s X 858 
 
 12 
 
 liX29 3 X825 
 
 12 
 2 X 19* X 820 
 
 12 
 
 2 X 11 s X 825 
 
 12 
 2 X 89 X 330 
 
 4219 
 
 38 
 
 27 
 
 +2713-0 
 1927-0 
 
 2,136,568 
 
 1,471,657 
 
 12 
 
 64,400 
 
 69,495 
 
 2,200,968 
 
 1,541,152 
 
 From the above Table it will be perceived that this process of 
 Middle Areas is a closer one than that of Arithmetical Average ; but 
 being in deficiency, while the former was in excess, the difference in 
 this case, from the true solidity, being about 3 per cent, less in Exca- 
 vation, and about 4 per cent, less in Embankment, 
 
 3d. Calculation by the Prismoidal Formula. The mass of which the 
 volume is demanded is a true Prismoid, and its contents will there- 
 fore be given by the well-known Prismoidal Formula. 
 b + 4m -f t 
 
 6 
 
 Where, 
 
 X length = Volume. 
 
 b = Area of Base. 
 m = Mid-section. 
 t = Area of top. 
 
CHAP. VI. GILLESPIE'S RULES, ETC. ART. 31. 149 
 
 Retaining the same data for the example as has been used in the 
 preceding tabulations, and will be continued throughout this discus- 
 sion, we refer to the following Table (3), where the results obtained 
 from the data given, by means of the Prism oidal Formula, are pro- 
 perly tabulated. 
 
 3. Prismoidal Formula. 
 
 Table 3, in illustration of the computation by it. Including an 
 entire section of a supposed railroad = 4219 feet in length. 
 
 3. Road-bed 50 ; side-slopes of excavation 1 to 1 ; of embank- 
 ment 2 to 1. 
 
 
 Dis- 
 
 
 
 
 
 
 
 
 
 
 
 Mid- 
 
 QUANTITIES. 
 
 Sta. 
 
 tance 
 
 in 
 
 Cut. 
 
 + 
 
 Fill. 
 
 End 
 Areas. 
 
 dle 
 Areas. 
 
 
 Excava- 
 
 Embank- 
 
 
 
 
 
 
 
 tion. 
 
 ment. 
 
 
 feet. 
 
 
 
 Sq.Ft. 
 
 Sq. Ft. 
 
 C. Feet. 
 
 C. Feet. 
 
 1 
 
 2 
 
 561 
 
 i 
 
 
 +1386 
 
 + 571-5 
 
 343,332 
 
 
 3 
 
 858 
 
 20 
 
 
 +1600 
 
 + 1491-5 
 
 1,280,136 
 
 
 4 
 
 825 
 
 o 
 
 O 
 
 O - 
 
 + 650 
 
 677,500 
 
 
 5 
 
 820 
 
 
 19 
 
 1672 
 
 655-5 
 
 
 586,847 
 
 6 
 
 825 
 
 
 8 
 
 528 
 
 1039-5 
 
 
 874,225 
 
 7 
 
 330 
 
 
 
 
 
 232 
 
 
 80,080 
 
 
 4219 
 
 4-38 
 
 27 
 
 +2986 
 
 + 2714 
 
 2,200,968 
 
 1,541,452 
 
 
 
 
 
 2100 
 
 1927 
 
 
 
 This Table 3, computed by the Prismoidal Formula itself, is the 
 standard for all the others, and gives the true solidities in the section 
 of railroad under consideration. 
 
 4th. Calculation by Mean Proportionals (or Geometrical Average). 
 Professor Gillespie says a fourth method, called that of "Mean 
 Proportionals" is sometimes, though very improperly, employed. 
 He gives the following rule for Mean Proportionals. 
 
 Rule. Add together the areas of the two ends, and a Mean 
 Proportional between them (found by extracting the Square 
 Root of their product) ; multiply the sum of these three areas by 
 the length of the Frustum, and divide the product by three.* 
 As used by engineers, in working for Cubic Yards as the result, 
 tnis rule takes a somewhat different shape, as follows: 
 
 Rule. Multiply the sum of the end areas, and the Square 
 Root of their product, by the distance apart, and divide this final 
 product by 9 and by 9. 
 
 * This is, substantially, Euclid's Rule for the Frustum of a Pyramid; Davies' 
 Legendre, VII. 18. 
 
150 MEASUREMENT OF EARTHWORKS. 
 
 The result is always much less than the truth (supposing the areas 
 taken between ground line and road-bed), for it treats as Pyramids, 
 or thirds of Prisms, the wedge-shaped pieces which are really halves 
 of Prisms, and is farthest from the truth when one of the areas = 0.* 
 So far. the Professor. 
 
 And this is all correct when the cross-sections are limited between 
 road-bed and ground surface ; but if they are extended to the inter- 
 section of the side-slopes, or edge of the diedral angle containing the 
 earthwork solid, an entirely different state of affairs takes place, for if 
 the road-bed be imagined to be gradually narrowed, so that eventu- 
 ally it vanishes at the intersection of the side-slopes; then, at that 
 point, both Pyramid and Prismoid coincide, or become equivalent, 
 whilst their rules become correlative (or mutually interchangeable), 
 and either may be used with the same results in point of solidity ; and 
 this is also the case with the "Equivalent Level Hights," much used by 
 engineers since the publication of Sir John Macneill's work (London, 
 1833), but likewise condemned by Professor Gillespie, rather hastily 
 as it seems to the writer, and hardly upon sufficient grounds. 
 
 It seems singular that this able Professor should have overlooked 
 the facts mentioned above, as he was well acquainted with the method 
 of continuing calculations to junction of side-slopes, including the 
 Grade Prism in the earlier stages of the computation, but rejecting it 
 at the close (as may be seen in his paper on Warped Solids (1859) ). 
 
 Now, so long as the cross-section of the earthwork remains trape- 
 zoidal in figure, the strictures of Professor Gillespie upon this rule 
 (commonly called the Geometrical Average) are undoubtedly correct; 
 but whenever the cross-section becomes triangular they fail entirely, as 
 also does his similar censure on " Equivalent Level Hights." 
 
 In evidence of this, we have tabulated (for ourselves) the same 
 general example as heretofore given both for the Geometrical 
 
 * Now, taking a case of precisely this kind (only continued to intersection of slopes) 
 hight at one end 34-5, at the other 0, with road-bed of 30 feet, slopes of 2 to 1, a length 
 of 66 feet, and level on the top. 
 
 If we compute this solid, either prismoidally, or by the usual rule for wedges, we have 
 for its volume 3205 Cubic Yards in round numbers. 
 
 And if we compute it by Baker's Rule (who treats such cases as Frusta of Pyramids, 
 but with the important addition of the Grade Prism), we find the resulting volume to be 
 the same to the nearest Cubic Yard. 
 
 For this pyramidal rule see Baker's Earthwork, London, 1848, whose rule is similar 
 to that of Bidder and others, which have always been accepted as correct by English 
 engineers, and most certainly they are so. 
 
CHAP. VI. GILLESPIE'S 'RULES, ETC. ART. 32. 151 
 
 Average, and for the Equivalent Level Higlits, merely carrying the 
 areas to the intersection of the side-slopes, in both cases, including at 
 first the Grade Prism, but excluding it after as a common quantity. 
 
 32 By these Tables we find the solidity of Gillespie's exam- 
 ple to be precisely the same as computed by him with the Prismoidal 
 Formula (Table 3 above), and which he has very properly adopted as 
 the correct standard for all. 
 
 4. Mean Proportionals (or Geometrical Average). 
 
 Table 4, in illustration of computation by them, including an entire 
 section of a supposed railroad = 4219 feet in length. 
 
 4. Road-bed 50 ; side-slopes of excavation 1 to 1 ; of embank- 
 ment 2 to 1. 
 
 Sta. 
 
 Dis- 
 tance 
 in 
 feet. 
 
 To the 
 Road-bed. 
 
 To intersec- 
 tion 
 of 
 slopes. 
 
 End Areas to 
 intersection 
 of 
 slopes. 
 
 Geomet- 
 rical 
 Mean 
 Area. 
 
 Quantities agreeing 
 with those of the 
 Prismoidal Formula. 
 
 Cut. 
 
 + 
 
 Fill. 
 
 Cut. 
 
 + 
 
 Fill. 
 
 Sq. Feet. 
 
 Sq. Feet. 
 
 Sq.Feet. 
 
 Kxcava. 
 Cub. Feet. 
 
 Embank. 
 Cub. Feet. 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 
 561 
 858 
 825 
 820 
 825 
 330 
 
 8 
 
 20 
 
 
 
 +38 
 
 oso 
 
 1K 
 
 34%: 
 
 31 y, 
 
 20V| 
 1'42 
 
 416666 
 1802-666 
 2(116-666 
 416666 
 
 3125 
 
 19845 
 840-5 
 312-5 
 
 + 816-666 
 + 1906-666 
 + 916-333 
 787 5 
 12915 
 512-5 
 
 343332 
 1,280,136 
 577,500 
 
 586847 
 874,225 
 80,080 
 
 4219 
 
 27 
 
 + 104% 
 
 77 
 
 + 4652-654 
 
 3450-0 
 
 + 3639-665 
 2591-5 
 
 2,200,968 
 
 1,541,152 
 
 In this Table the Grade Prism is included at first, and excluded 
 afterwards. Its sectional area is as follows : 
 
 Grade Prism of Cut = 416*666 Square Feet. 
 " " Bank = 312'5 " " 
 
 To be multiplied for volume by length of mass to which it belongs. 
 
 Altitudes of the Grade Prism in the Cut = 161 feet; on Bank = 
 mfeet. 
 
 In computing quantities by Geometrical Average, the following 
 generalization has occurred to the writer, which indeed may possibly 
 be a germ from which the Prismoidal Formula might have sprung 
 since both the Arithmetical and Geometrical Means were known in 
 the days of Euclid ( 200 B. c.), while the original Prismoidal For- 
 mula (so far as we know) was devised by Simpson, as late as A. D. 
 1750. 
 
152 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Thus, 
 Double the sum of End Areas -f- Double Geom. Mean 
 
 6 
 
 X h = Solidity. 
 
 Let 
 
 ( A = Sum of End Areas. "I Then the above f 2 A + 2 B 
 \ B = Geometrical Mean. / becomes . .-\ 6 
 
 Or, in its lowest terms, - - X h = S, which is the Geometrical 
 
 o 
 
 Average; or, in substance, Euclid's Rule for the Frustum of a Pyra- 
 mid ; and by the aid of the Grade Prism strictly applicable to earth- 
 works of a general triangular section in ordinary cases. 
 
 5 Equivalent Level Higlits. 
 Table 5, in illustration of computation by them. 
 
 5. Road-bed 50; side-slopes of excavation 1J to 1; of embank- 
 ment 2 to 1. 
 
 r 
 
 Sta. 
 
 Dis- 
 
 t'noe 
 in 
 ft. 
 
 To the 
 Road- 
 bed. 
 
 To intersec- 
 tion 
 of 
 
 slopes. 
 
 End Areas to inter- 
 section of 
 slopes. 
 
 Mid. hta. 
 to inter- 
 section 
 of sl'pes. 
 
 Mid-sections, or 
 areas to the inter- 
 section of slopes. 
 
 Quaiititi 
 ing wit 
 of the P 
 Fom 
 
 js agree- 
 h those 
 ismoidal 
 ntila. 
 
 Embkt. 
 
 0. Ft^T. 
 
 Cut. 
 
 + 
 
 Fill. 
 
 Cut. 
 
 Fill. 
 
 Cut. 
 
 + 
 
 Fill. 
 
 Feet. 
 
 Sq. Feet. 
 
 Sq. Feet. 
 
 Excava. 
 C. Feet". 
 
 1 
 2 
 
 3 
 4 
 5 
 6 
 
 7 
 
 561 
 
 858 
 825 
 820 
 825 
 330 
 
 O 
 
 18 
 20 
 O 
 
 +38 
 
 O 
 19 
 8 
 
 O 
 
 34% 
 
 i<y 
 
 12> 
 31^ 
 
 20^ 
 
 12% 
 
 416666 
 1802-666 
 2016-666 
 416-666 
 
 3125 
 
 1984-5 
 8405 
 312-5 
 
 + 25-666 
 -r 35-666 
 + 26 666 
 22000 
 26-000 
 16 500 
 
 988166 
 1908-166 
 1066 666 
 
 968-0 
 1352 
 544-5 
 
 343,332 
 1.280,136 
 
 577,500 
 
 586,847 
 874,225 
 80,080 
 
 4219 
 
 27 
 
 + 104% 
 
 77 
 
 + 4652-654 
 
 3450-000 
 
 + 87-998 
 64-500 
 
 + 3962 998 
 
 2864-5 
 
 2,200 ; 968 1,541,152 
 
 In this Table the Grade Prism is included in the earlier operations, 
 and excluded in the later ones. Its sectional area is as follows: 
 
 Grade Prism of Cut = 416'66 Square Feet. 
 Fill = 312-50 " 
 
 To be multiplied for volume by the length of mass to which it 
 belongs. 
 
 Altitudes of the Grade Prism in the Cut = 16f feet; on Bank = 
 12* feet. 
 
 33. From the preceding discussion in the present chapter we are 
 justified in declaring that all the following rules and formulas 
 (detailed above) are equivalent in their results for volume when pro- 
 
CHAP. VI.-GILLESPIE'S RULES, ETC. ART. 33. 153 
 
 perly corrected and appropriately used ; and that they all give the 
 same solidity in the end as No. 3 does, which is the standard for ALL. 
 
 1. Arithmetical Average to Road-bed (with correction). 
 
 2. Middle Areas to Road-bed (with correction). 
 
 3. Prismoidal Formula (the standard for all) to Road-bed, or to 
 
 the intersection of slopes either. 
 
 4. Geometrical Average to intersection of slopes. 
 
 5. Equivalent Level Hights to intersection of slopes. 
 
 All these are fully described above, and the tabular statements 
 bearing the same number show in each case the results of the calcu- 
 lations for volume, agreeing uniformly with the computations for 
 solidity, made by means of the Prismoidal formula. 
 
 In concluding his notices of the method of computing the contents 
 of earthworks, by means of the Prismoidal Formula, Professor Gilles- 
 pie gives some special rules, transformed from it, which are doubtless 
 valuable in certain cases, but do not appear to be of general applica- 
 tion; he also gives formulas for a series of equal distances apart sta- 
 tions, such as are usually found in the location of railroads. 
 
 These are intended to be applied to a central core, or body of the 
 work, based upon the road-bed, to be calculated by itself, and then 
 the slopes, to be computed separately or together, and added in with 
 the core, so as to form finally the volume of the ivhole prismoidal mass. 
 
 This idea of separating the core or body from the slopes, calculating 
 them independently, and adding them together, seems to have occurred 
 to a great many .engineers,* and forms the theme of nearly a dozen 
 books on the subject of Earthwork Measurements here or abroad. 
 
 Indeed, the very first special work on the mensuration of earth- 
 works, which was published in this country that of E. F. Johnson, 
 C. E. (New York, 1840), adopted this system, and furnished a series of 
 Tables to facilitate its operation ; it was, however, briefly explained 
 before, in Lieut.-Col. Long's valuable Railroad Manual (Balti- 
 more, 1828), which was the first to treat the subject in this country, 
 and was, in fact, the pioneer of technical railroad literature in the 
 UNITED STATES. 
 
 Nevertheless, the method of Core and Slopes has never come into 
 general use, though often revived from time to time by new writers, 
 apparently unacquainted with the literature of this subject. 
 
 * Amongst others, it is the method of Bidder, who followed Macneill in the earlier 
 days of English railroads. 
 
154 
 
 MEASUREMENT OF EARTHWORKS. 
 
 34. . . 1 . . Comparison of Gillespie's Main Example and the Method^ 
 of Roots and Squares. 
 
 Professor Gillespie's chief example, of a heavy Cut and Fill, form- 
 ing an entire section of railroad, 4219 feet long, must by this time 
 be so familiar to engineers, and others, in consequence of the exten- 
 sive circulation of his Manual of Roads and Railroads, since its origi- 
 nal publication in 1847, that we have selected it as the most suitable, 
 or at least the best known* for the purpose of comparison with our 
 Third Method of Computation that by Roots and Squares. 
 
 We therefore give a Table No. 6 (below), which contains in the 
 first 5 columns the data given by Professor Gillespie, and in the last 
 6 the results of the computation by Roots and Squares, which will be 
 found to agree exactly with those obtained above, by means of the 
 Prismoidal Formula accepted as being a correct standard for com- 
 parison. 
 
 6. Comparison of Example, with Hoots and Squares. 
 
 Including (as before) an entire section of a supposed railroad 
 4219 feet in length. 
 
 6. Road-bed 50; side-slopes of excavation II to 1 ; of embank- 
 ment 2 to 1. 
 
 Sta. 
 
 Dis- 
 tance 
 in 
 feet. 
 
 End 
 Areas 
 in 
 Sq. Tt. 
 
 Centre 
 Rights 
 in feet. 
 
 End 
 Areas 
 increased 
 by Grade 
 Triangle. 
 
 Square 
 Roots 
 of 
 End 
 Areas. 
 
 Suma 
 of 
 Square 
 
 Roots. 
 
 Squares of 
 sums, or 4 
 times the 
 mid- 
 section. 
 
 Quantities agree- 
 ing with those 
 given by the 
 Prismoidal 
 Formula. 
 
 Cut + 
 Fill 
 
 Cut 
 
 + 
 
 Fill 
 
 O 
 
 19 
 
 8 
 O 
 
 Sq. Feet. 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 C. Feet. 
 
 0. Feet. 
 
 1 
 2 
 3 
 
 4 
 
 5 
 6 
 
 7 
 
 561 
 
 858 
 
 825 
 
 820 
 825 
 330 
 
 o 
 
 +1386 
 -j-1600 
 
 
 1672 
 528 
 
 
 
 g 
 
 20 
 
 
 +38 
 
 + 416% 
 +1802% 
 +2016% 
 < + 416% 
 ) 312J> 
 1984U 
 840^ 
 312} 
 
 + 20-42 
 + 42-46 
 + 44-91 
 + 20-42 
 17-68 
 44-55 
 28-99 
 17-68 
 
 62-88 
 87-37 
 65-33 
 
 62-23 
 73-54 
 46-67 
 
 3954 
 7634 
 
 4268 
 
 3872 
 
 5408 
 2178 
 
 343,332 
 
 1,280,136 
 
 577,500 
 
 586,847 
 874,225 
 80,080 
 
 4219 
 
 +2986 
 2200 
 
 27 
 
 +4652% 
 3450 
 
 + 128-21 
 108-90 
 
 215-58 
 182-44 
 
 15856 
 11458 
 
 2,200,968 
 
 1,541,152 
 
 In the above Table (as in the others), the cross-sections in the 
 data given being level trapezoids from ground to road-bed, we neces- 
 
 * Besides, this example, originated by F. W. Simms, C. E. (London, 1836), has been 
 before the public for many years, having been first published in our country in Alexan- 
 der's edition of Simms on Levelling (Baltimore, 1837) ; from which, or the original, it was 
 copied by Professor Gillespie. In the work above mentioned, Mr. Alexander gives every 
 detail of the computation of this example, by the Prismoidal Formula, at great length, 
 and so indeed does Simms. 
 
CHAP. VI. GILLESPIE'S RULES, ETC. ART. 34. 155 
 
 sarily add in this mode of computation (to intersection of slopes) the 
 Grade Triangle, and deduct it again near the close of the operation. 
 
 Road-bed 50 ; side-slopes of excavation = 1 J to 1 ; of embank- 
 ment = 2 to 1. 
 
 Grade Triangle of Cut, area = 416f Sq. Ft. altitude = 16$ Feet. 
 " Fill, " = 312 " " " = 12* " 
 
 Where the distances apart stations are uniform in length and even 
 in number, the method of Roots and Squares enables us to employ a 
 very simple modification of Simpson's Multipliers, as has been already 
 shown in Chapter IV., so as to compute with ease and expedition an 
 entire cut or fill, at a single operation, or one station only, at pleasure. 
 
CHAPTER VII. 
 
 PRELIMINARY OR HASTY ESTIMATES, COMPUTED BY SIMPSON'S RULE 
 
 FOR CUBATURE. 
 
 35 Preliminary, and often hasty estimates of earthworks, 
 
 are constantly required by engineers prior to deciding upon railroad 
 routes, or their modifications, and indeed are generally necessary in 
 determining the relative merits of engineering lines (amongst which 
 there are always alternatives} since few can undertake to settle pro- 
 perly any important questions relating to their comparative value, 
 without some serious consideration, for which the Preliminary Esti- 
 mates, on various lines surveyed, supply a proximate foundation, by 
 aiding without controlling the judgment of the engineer. 
 
 Exploring Lines, preparatory to the final location of a railway, are 
 indispensable, and in a difficult country may extend to tenfold the 
 length of the final line, while the time allowed to engineers being usually 
 extremely short, the estimates of quantities on these Preliminary Sur- 
 veys are necessarily hasty, and consequently imperfect but neverthe- 
 less demand rapidity in execution, however made. 
 
 For this there seems to be no remedy ; all we can do is to endeavor 
 to point out a method for hasty estimates, more correct and more expe- 
 ditious than those usually employed, and to this we shall confine our- 
 selves in the present chapter. 
 
 Exploring lines are usually traced with stations at double distance, 
 or 200 feet apart and, indeed, sometimes on plain ground the dis- 
 tance apart stations has been stretched (to save time) as far as 400 
 or 600 feet ; and as this last distance is about the longest range 
 which gives distinct vision for the Engineer Levels in use in this 
 country, it ought rarely to be exceeded, as a general rule; while at 
 least, the distance of 200 feet apart stations, or double distance of loca- 
 156 
 
CHAP. VII. HASTY ESTIMATES. ART. 35. 157 
 
 tion, furnishes good information of the ground, and also enables the 
 exploring party to proceed rapidly enough to gain an adequate know- 
 ledge of the country, without much loss of time. 
 
 Nevertheless, the rules we suggest will apply to any uniform dis- 
 tance apart stations of exploring line, which may be deemed advisable 
 by the engineer in charge: but the longer the distance between sta- 
 tions, the less accurate will be the estimate in general. 
 
 We propose to apply Simpson's celebrated rule for cubature (the 
 accuracy of which is well known) to Preliminary or Hasty Estimates, 
 taking as data the centre hights and surface slopes alone; the former 
 to the nearest foot of hight or depth, from ground to intersection of 
 side-slopes, and the latter to the nearest 5 of average ground slope 
 across the line, leaving special cases to be dealt with by the engineer, 
 according to rules of his own. 
 
 We have provided proximate tables (very nearly correct) to facili- 
 tate these hasty operations, and would also suggest that, in all cases 
 of Preliminary Estimates, the resulting quantities of earthwork should 
 be augmented ten per cent.: this addition will give full quantities, 
 and has been shown by long experience to be ample to meet the usual 
 contingencies which always arise in the construction, and cannot be 
 foreseen, and of which, in fact, it must be confessed, the engineer in 
 charge (often unknown to himself) almost invariably takes the most 
 favorable view', and hence the greater necessity exists for some appro- 
 priate allowance beyond the net result of the calculations. 
 
 Simpson's Rule for Cubature, using cross-sections instead of ordi- 
 nates (as we have before shown), is as follows: 
 
 i --Jt X D = Solidity. 
 
 o 
 
 ( Sometimes 2 D, and 6 for divisor, are used, and are equivalent.) 
 
 A = Sum of extreme end ordinates, or sections. 
 B = Sum of cross-sections standing on even numbers. 
 C = Sum of " " " " odd numbers. 
 D = The common interval, or distance apart sections. 
 
 Simpson's rule above is limited to an even number of equal spaces. 
 
158 MEASUREMENT OF EARTHWORKS. 
 
 And it must be observed that in its application it is always best to 
 prepare a rough profile of the line run, and under the regular num- 
 bers to pencil forward, from the beginning of the cut or fill to be 
 computed, the series of numbers 1, 2, 3, 4, etc. No. 1 always stand- 
 ing at the place of beginning ; it is this series of numbers, so arranged, 
 which are referred to in the rule above as even and odd. 
 
 By this rule it is best to compute entire and separately each cut and 
 each fill encountered by the line ; and if the whole number of equal 
 intervals or stations, in any cut or fill, should be an odd number, then 
 one station of the common length, at beginning or end (or indeed any 
 where deemed most suitable), should be struck off temporarily, and 
 reserved for separate calculation ; while the body of the work thus 
 reduced, to an even number of common intervals, comes directly within 
 the rule, and can be calculated as a whole, while the detached sta- 
 tion, computed by itself, may be added in near the close of the ope- 
 ration. 
 
 It will always be found briefer and better in using this and similar 
 rules, to aim first at finding a General Mean Area, which, multiplied 
 by the proper length or distance, will give the solidity ; but it is still 
 better, having the General Mean Area in square feet, to use our Table 
 at the end when the result is desired in Cubic Yards. 
 
 36 Instead of employing Simpson's Formula, as it stands 
 
 above, it will be often more convenient to use the multipliers which 
 represent it these are known as Simpson's Multipliers* and are as 
 follows : 
 
 For two equal int6rvals, apart sections, Mults. 
 
 ( Divisors 6; qiiot?ent,Mean 
 I, 4, l.< Areas ; factors for length 
 ( = double interval. 
 
 " four" " " " " = 1, 4, 2, 4, 1. f Divisors 3; quotient, 
 
 s ix ' " " " " = 1. 4, 2, 4, 2, 4, 1. J Mean Areas ; factors for 
 
 eiaM " " " " " = 1, 4, 2, 4, 2, 4, 2, 4, 1. 1 length = single inter- 
 
 ten " " " " " = 1, 4, 2, 4, 2, 4, 2, 4, 2, 4, 1. [veil. 
 
 The first set of multipliers, their divisors, and factors for length, are 
 clearly those of the Prismoidal Formula, which evidently forms the 
 basis of this famous rule. 
 
 Indeed, it is easy to show by diagrams how this rule may probably 
 have been formed, by the eminent mathematician, with whom it 
 originated, about the year 1750 ; and also how intimately it appears 
 to be connected with the Prismoidal Formula. 
 
 * Rankine's Useful Rules and Tables, 2d edition, London, 1867, page 64. 
 
CHAP. VII. HASTY ESTIMATES. ART. 36. 
 
 159 
 
 See Figs. 11 and 78, following. 
 
 Suppose Figs. 11 and 78 to represent front views of four planes, A, 
 B, C, D, or of four solids with a thickness of unity, all standing on 
 the level base line EF, and that their respective ordinates, or cross- 
 sections (correllative in Simpson's Rule for Cubature), are dimen- 
 sioned as marked upon the figures. 
 
 Tig. 77; 
 
 10, 10. I 10. 10. jtO. I 1O. 1O. 10. 30 
 
 31O 
 
 Kg. 73. 
 
 1. Suppose the solids to be separated from each other by the dis- 
 tance of 10 feet (or any other), and let each be computed 
 independently by means of Simpson's Multipliers, or as they 
 are all exactly alike, let one be computed and multiplied by 
 4, as follows : 
 
 This is clearly 
 
 a 
 Prismoidal Compulation. 
 
 Cross-sees. Simpson's Results in 
 in Sq. Ft. Mults. Sq. Ft. 
 
 X 1 
 X 4 
 X 1 
 
 = 1 
 
 = 16 
 
 6)18 
 Mean Area = 3 X 20 
 
 60 A. 
 
 60 X 4 = 240 Cubic Feet = 
 
160 MEASUREMENT OF EARTHWORKS. 
 
 2. Now, suppose the solids to be slid along the base line EF, 
 until they come in actual contact with each other, as shown 
 in Fig. 78. Then it becomes evident that the intermediate 
 sections at odd numbers (1, 3, etc.), which, in the detached 
 solids, Fig. 77, were used but once, are here, when combined, 
 to be used twice; while the mid-sections, or those at even 
 numbers, are to be used four times, and the extreme end sec- 
 tions only once each ; so that they become, in effect, when 
 treated thus, the Multipliers of Simpson ; while the divisor 
 is changed to 3, because the common interval is reduced 
 one-half; and the volume of the four solids, when aggre- 
 gated together, so as to form a single body, would be com- 
 puted by Simpson's Rule, or by his Multipliers, as follows: 
 
 By Sim 
 
 Common 
 9 4- 64 -4- 6 Interval. 
 ,/, Rut* V 10 
 
 240, o.s above. 
 
 s. Sq. Ft. 
 -j 
 
 = 16 
 
 = 2 
 
 Sees. Multi 
 
 /I X 1 
 |4 X 4 
 1X2 
 
 
 \4 X 
 
 4 
 
 = 16 
 
 
 By 
 
 Simpson's Multipliers, 1 1 X 
 
 2 
 
 = 2 
 
 
 with 8 equal intervals. \ 4 X 
 
 4 
 
 = 16 , 
 
 
 
 (1 X 
 
 
 = 2 
 
 
 
 4 X 
 
 4 
 
 = 16 
 
 
 
 1 X 
 
 1 
 
 = 1 
 
 
 
 
 
 3)72 
 
 
 
 General Mean Area . 
 
 t 
 
 = 24 
 
 
 
 Common Interval . . 
 
 . 
 
 = 10 
 
 
 
 Result same as before . 
 
 . 
 
 = 240 C. 
 
 Feet. 
 
 As Simpson's Rule is an important one, we hope the above digres- 
 sion to explain it fully, and the foundation on which it rests, will be 
 excused by the reader. 
 
 37. Having then taken off from a rough profile of the line run the 
 centre bights to the nearest foot, and from the field notes ascertained 
 the average surface slope at each station to the nearest 5, we enter 
 Tables 2, 3, and 4, and obtain the triangular areas to the intersection 
 of the side-slopes (supposed to be prolonged to meet), to the nearest 
 foot of area, for rock cutting, earth cutting, or embankment each of 
 
CHAP. VIL HASTY ESTIMATES. ART. 37. 161 
 
 these, that we may require, we set down separately in a column, and 
 where a case occurs of a hight exceeding the limits of the Tables 
 named, then we resort to the initial triangles of Table 1, by means of 
 which the area due to any hight whatever may easily be ascertained ; 
 then, if we find we have an even number of equal stations, we apply 
 Simpson's Multipliers to the column of areas, and speedily compute 
 the solidity. 
 
 But if the equal intervals or stations are found to be uneven in 
 number, strike off one station temporarily for independent calcula- 
 tion, and then the number of intervals becoming eren t we are ready 
 to apply Simpson's Multipliers, in a column parallel to that of areas, 
 and beginning at 1, as 1, 4, 2, 4, 2, 4, etc., multiplying each cross-sec- 
 tion by its proper factor, and placing the results in a third parallel 
 column, which we sum up and divide the total by 3 (giving a Mean 
 Area as the quotient), add to this the mean area of the station 
 reserved (if any), which gives a General Mean Area, to be multiplied 
 by the equal interval, or length of station say 200 feet, or whatever 
 distance has been adopted and used as a common interval or station 
 the result will be cubic feet, from which cubic yards (if desired) 
 can easily be found. 
 
 But, inasmuch as the quotient of 3 (with the mean area of the 
 reserved station (if any) added in) is a General Mean Area usually 
 in square feet it will be found more convenient, and usually more 
 accurate, to use it in connection with our Table 5, at the end of the 
 Book, to find the cubic yards which may be desired, according to the 
 directions preceding the Table. 
 
 We will now proceed to give examples of the process above 
 explained, and for this purpose we will take the adjacent bank and 
 rock cut, profiled on Fiy. 76, Art. 24, as being an appropriate exam- 
 ple of this expeditious method of computing an embankment, or an 
 excavation in a single body, with sufficient accuracy for the purpose 
 contemplated, and without unusual delay. 
 
 Fig. 76. BANK. 
 
 Here we find the Bank to be 1000 feet in length between the grade 
 points, or 5 intervals of 200 feet each ; the number of intervals being 
 uneven, we must temporarily omit one station to bring this case within 
 the rule ; let the station omitted, and to be calculated independently, 
 be from 5 to 7 = 200 feet. 
 11 
 
162 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Tabulation. 
 
 Sta. Areas. 
 1 
 
 3 
 
 Mults. 
 
 24 X 1 = 
 
 495 X 4 
 
 5 and 7 3123 X 2 = 
 united. 
 
 9 1197 X 4 = 
 
 11 24 X 1 = 
 
 Sq. Feet. 
 
 24 
 
 1980 
 6246 
 
 4788 
 24 
 
 3)13062 
 4354 
 
 Partial Mean Area. 
 
 Add area of reserved station. 
 
 The hight of the embank- 
 ment and the surface-slope at 
 5 and 7 being the same, this 
 reserved station is a Prism, of 
 which the base, or sectional 
 area, is 3123 square feet, and 
 length = 200 feet .... 
 
 General Mean Area. . . 
 
 Solidity .-.... ; >'--.-; 
 
 Or, ; ; .: 
 
 Tabulated, by Roots and 
 Squares, in 100 feet stations . 
 
 Difference about the half of 
 one per cent, more 
 
 = 3123 = Mean Area, reserved 
 
 station. 
 = 7477 Square Feet. 
 
 200 Common Interval. 
 
 = 1495400 Cubic Feet. 
 = 55385 Cubic Yards. 
 
 = 55088 
 
 = -f297 " " 
 
 Tabulated by Roots and Squares in 100 feet stations, as though for 
 a final estimate, the Bank in our example contains 55,088 Cubic 
 Yards, while by our hasty process the result is 55,385 Cubic Yards, 
 or 297 Cubic Yards more. As this difference is but little more than the 
 half of one per cent, upon the true amount, it can hardly be consid- 
 ered as excessive for a method as brief and simple as that under con- 
 sideration here. 
 
 Fig. 76. ROCK-CUT. 
 
 The Rock-Cut, like the Bank connected with it, and tabulated 
 above, is 1000 feet in length between the grade points, or 5 intervals 
 of 200 feet each, which, being an uneven number, we must tempora- 
 
CHAP. VII.-HASTY ESTIMATES. ART. 38. 
 
 163 
 
 rily omit one station, and calculate it separately, to make the number 
 of intervals even, and bring it within the scope of Simpson's Rule. 
 Let the station reserved be from 19 to 21 = 200 feet. 
 
 Tabulation. 
 
 Sta. Areas. Mults. 
 
 11 192 X 1 
 13 646 X 4 
 
 15 975 
 
 17 589 X 4 
 19 771 X 1 
 
 Station reserved from 19 to 
 21, to make the number of in- 
 tervals even, as required by 
 the Rule of Simpson. 
 
 19 = 771 X 1 = 771 
 
 20 = 433 X 4 = 1732 
 
 21 = 192 X 1 = 192 
 
 6)2695 
 
 Mean Area = 449 
 General Mean Area 
 
 Solidity . . . . 
 
 Tabulated by Roots and 
 
 Squares, in stations of 100 feet 
 
 Diff. about 1 per cent, less 
 
 Sq. Feet. 
 
 = 192 
 
 = 2584 
 
 = 1950 
 
 = 2356 
 
 = 771 
 
 3)7853 
 
 2618 
 
 Partial Mean Area, 
 
 449 
 
 3067 
 200 
 
 Mean Area, reserved 
 station. 
 
 Square Feet. 
 Common Interval. 
 
 = 613400 = 22718 Cubic Yards. 
 
 = 623298 = 23085 " 
 = 9898 = 367 " 
 
 38 It will be observed that in the preceding computations 
 
 the Grade Prism is not taken into the account, as it is deductive on 
 both sides, and the only object in hand is a comparison. 
 
 The triangular section, or area of the Grade Prism, is the minimum 
 area found, in the methods of computation which go down to the 
 junction of the side-slopes, and always occurs when the road-bed 
 comes to grade, or the level hight on the centre line is 0. 
 
 And we repeat, it is necessary to be careful that the volume of the 
 Grade Prism (always included in the earlier steps of such calcula- 
 tions) is duly deducted before the close of the operation, in order to 
 determine the solidity above the road-bed in cutting, or below it in 
 filling. 
 
164 MEASUREMENT OF EARTHWORKS. 
 
 We may here add that the earth cutting profiled ante, and 
 there correctly computed by Roots and Squares, if calculated 
 with Simpson's Multipliers by the hasty process above given, in sta- 
 tions of 200 feet, as though it were part of an exploring line, would 
 give as follows : 
 
 Volume of Grade Prism omitted in both. 
 
 C. Yards. 
 
 /Tabulated ante, in 100 feet stations == 18684 
 " by our Hasty Process, 200 feet stations . . . = 18378 
 
 i 
 
 Difference about H per cent, less = 306 
 
 So that this brief and hasty process, being very expeditious and 
 proximately correct (usually varying only 1 or 2 per cent, from the 
 truth), may be safely accepted as adequate for the determination of 
 the quantities of earthwork, which may be needed in rough estimates, 
 or for the comparison of exploring lines. 
 
 For the purpose of furnishing additional aid in expediting Prelimi- 
 nary Estimates, we annex four small Tables, which will be found 
 quite convenient. 
 
TABLES. 165 
 
 "LIBRA u v 
 
 UNIVERSITY OJ 
 
 TABLES 
 
 CALIFORNIA. 
 
 1, 2, 3, and 4. 
 
 For use in Hasty or Preliminary Estimates. 
 
 Viz: 1. Initial Triangles to a hight of unity, and various side and 
 surface slopes. 
 
 Triangular Areas to Intersection of Slopes. 
 
 Side-slopes. Surface-slopes. 
 
 2. Kock Cut i to 1, and 0, 5, 10, 15, 20. 
 
 3. Earth Cut 1 to 1, and " " " " 
 
 4. Embankment and " 
 
 In using Tables 2, 3, and 4, the centre hight is generally to be 
 taken to the nearest foot (though tenths might be used), and the 
 ground surface slope to the nearest 5 these being thought sufficient 
 for rough estimates and if the centre hight should exceed the limits 
 of the Tables, then, by using the Initial Triangles of Table 1, the area 
 of the cross-section for any hight whatever can be easily ascertained. 
 If the centre hights necessarily contain tenths of feet, they may be 
 proportioned for by the columns in the Tables for that purpose. 
 
 Note. All the triangular areas in Tables 2, 3, and 4, extend from 
 ground line to junction of side-slopes prolonged, or edge of the diedral 
 angle, which, with ground surface, bounds on every side the earth work 
 solid. The road-bed, or grade line, may be assumed to cross the tri- 
 angle at any given distance from the angle of intersection ; but the 
 volume of the Grade Prism must always be ascertained and deducted 
 at the close of the operation, in every calculation involving the trian- 
 gular areas of the Tables. The altitude of the Grade Triangle is 
 invariably = road-bed -f- 2 r, and its area will be found opposite to 
 this hierht in the column of the Tables. 
 
166 
 
 MEASUREMENT OF EARTHWORKS. 
 
 TABLE 1. 
 
 Initial Triangles, to a hight of unity, with side-slopes of i to 1 for 
 Rock; 1 to 1 for earth; 1& to 1 for embankment; and ground sur- 
 face slopes of 0, 5, 10, 15, 20. All computed to six places of 
 decimals, and all extending from ground line to intersection of side- 
 slopes. 
 
 Side-slopes. 
 
 Ground Surface-elopes. 
 
 Ratio. 
 
 Angle. 
 
 Cot. 
 
 Tan. 
 
 
 
 5 
 
 10 
 
 15 
 
 20 
 
 of Trian. Tables. 
 
 Tan. 
 = () 
 
 Tan. 
 
 = -0875 
 
 Tan. 
 = -1763 
 
 Tan. 
 
 = -2679 
 
 Tan. 
 
 = 3640 
 
 Y 3 to 1 
 1 to 1 
 1^ to 1 
 
 71 34' 
 45 
 33 41' 
 
 0-3333 
 1 
 1-5 
 
 3 
 1 
 6666 
 
 0'333333 
 1-5 
 
 0-333586 
 1-007713 
 1-526688 
 
 0-334457 
 1-032088 
 1-613298 
 
 0-335682 
 1-077350 
 1-790002 
 
 0-338280 
 1-152663 
 2-137798 
 
 Note. A similar Table may easily be extended to any other side, 
 or surface-slope, and such extension would often be found useful to 
 the engineer. 
 
 Application of the above Table. 
 
 Hide. For any given hight, to find the triangular area, when con- 
 ditioned as above. 
 
 Multiply the Square of the Given Hight by the Tabular Area of the 
 Initial Triangle. 
 
 Example. 
 
 Let the given hight be 26'4 feet, the side-slope 1 to 1, and the 
 ground surface-slope 20. 
 
 Then, (26'4) 2 X 1*152663 = 803*36 square feet = area of triangle 
 required. 
 
TABLES. 
 
 167 
 
 Triangular Areas, in square feet, for side-slopes of i to 1, to intersec- 
 tion of slopes, (r = i) Slope angle = 71 34'. 
 
 TABLE 2Ri>ck-cirt. 
 
 Hight 
 
 iii 
 
 Surf.-slope O. 
 
 Surf.-elope 5. 
 
 Surf.-elope 10. 
 
 Surf.-slope 15. 
 
 Surf.-slope 0. 
 
 Hight 
 in 
 
 feet. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 feet. 
 
 
 Areas, 
 
 for-1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 
 1 
 
 3333 
 
 03 
 
 3336 
 
 03 
 
 3346 
 
 03 
 
 3357 
 
 03 
 
 3383 
 
 03 1 
 
 2 
 
 1-3333 
 
 10 
 
 1-3 
 
 10 
 
 1-3 
 
 10 
 
 13 
 
 10 
 
 14 
 
 10 
 
 2 
 
 3 
 
 3 
 
 17 
 
 3 
 
 17 
 
 3 
 
 17 
 
 3 
 
 17 
 
 3 
 
 17 
 
 3 
 
 4 
 
 5-3333 
 
 23 
 
 5 
 
 23 
 
 5 
 
 23 
 
 5 
 
 23 
 
 5 
 
 24 
 
 4 
 
 5 
 
 83333 
 
 30 
 
 8 
 
 30 
 
 8 
 
 30 
 
 8 
 
 30 
 
 8 
 
 30 
 
 5 
 
 6 
 
 12 
 
 37 
 
 12 
 
 37 
 
 12 
 
 37 
 
 12 
 
 37 
 
 12 
 
 37 
 
 6 
 
 7 
 
 163333 
 
 43 
 
 16 
 
 43 
 
 16 
 
 43 
 
 16 
 
 44 
 
 17 
 
 44 
 
 7 
 
 8 
 
 21-3333 
 
 50 
 
 21 
 
 50 
 
 21 
 
 to 
 
 22 
 
 50 
 
 22 
 
 51 
 
 8 
 
 9 
 
 27 
 
 57 
 
 27 
 
 57 
 
 27 
 
 57 
 
 27 
 
 57 
 
 28 
 
 58 
 
 9 
 
 10 
 
 33-3333 
 
 63 
 
 33 
 
 63 
 
 33 
 
 64 
 
 34 
 
 64 
 
 34 
 
 64 
 
 10 
 
 11 
 
 4O3333 
 
 70 
 
 40 
 
 70 
 
 41 
 
 70 
 
 41 
 
 71 
 
 41 
 
 71 
 
 11 
 
 12 
 
 48 
 
 77 
 
 48 
 
 77 
 
 48 
 
 77 
 
 48 
 
 77 
 
 49 
 
 78 
 
 12 
 
 13 
 
 56-3333 
 
 83 
 
 56 
 
 83 
 
 57 
 
 84 
 
 57 
 
 84 
 
 67 
 
 85 
 
 13 
 
 14 
 
 65-3333 
 
 90 
 
 65 
 
 90 
 
 66 
 
 90 
 
 66 
 
 91 
 
 66 
 
 91 
 
 14 
 
 15 
 
 75 
 
 97 
 
 75 
 
 97 
 
 75 
 
 97 
 
 76 
 
 98 
 
 76 
 
 98 
 
 15 
 
 16 
 
 85-3333 
 
 1-03 
 
 85 
 
 1-03 
 
 86 
 
 1-04 
 
 86 
 
 1-04 
 
 87 
 
 1-05 
 
 16 
 
 17 
 
 96-3333 
 
 1-10 
 
 96 
 
 1-10 
 
 97 
 
 1-10 
 
 97 
 
 1-11 
 
 98 
 
 I'll 
 
 17 
 
 18 
 
 K>8 
 
 1-17 
 
 108 
 
 1-17 
 
 108 
 
 1-17 
 
 109 
 
 1-18 
 
 110 
 
 1-18 
 
 18 
 
 19 
 
 120-333.'} 
 
 1-23 
 
 121 
 
 1-23 
 
 121 
 
 1-24 
 
 121 
 
 124 
 
 122 
 
 1-25 
 
 19 
 
 20 
 
 133-3333 
 
 1-30 
 
 133 
 
 1-30 
 
 134 
 
 1-30 
 
 l; 
 
 1-30 
 
 135 
 
 131 
 
 20 
 
 21 
 
 147 
 
 1-37 
 
 147 
 
 1-37 
 
 148 
 
 1-37 
 
 148 
 
 1-37 
 
 149 
 
 1-38 
 
 21 
 
 
 161-3333 
 
 1-43 
 
 161 
 
 1-43 
 
 162 
 
 1-44 
 
 163 
 
 1-44 
 
 164 
 
 1-45 
 
 22 
 
 23 
 
 176-3333 
 
 1-50 
 
 176 
 
 1-50 
 
 177 
 
 1-50 
 
 178 
 
 1-51 
 
 179 
 
 1-52 
 
 23 
 
 24 
 
 192 
 
 1-57 
 
 192 
 
 1-57 
 
 193 
 
 157 
 
 194 
 
 1-68 
 
 195 
 
 1-59 
 
 24 
 
 25 
 
 2 >8-3333 
 
 1-63 
 
 209 
 
 1-63 
 
 209 
 
 164 
 
 210 
 
 1-64 
 
 212 
 
 1-66 
 
 25 
 
 26 
 
 225-3333 
 
 1-70 
 
 226 
 
 1-70 
 
 226 
 
 V70 
 
 227 
 
 1-71 
 
 229 
 
 1-72 
 
 26 
 
 27 
 
 243 
 
 177 
 
 243 
 
 1-77 
 
 244 
 
 1-77 
 
 245 
 
 1-78 
 
 247 
 
 1-79 
 
 27 
 
 23 
 
 261-3333 
 
 1-83 
 
 262 
 
 1-84 
 
 262 
 
 1-84 
 
 263 
 
 1-85 
 
 265 
 
 1-86 
 
 28 
 
 29 
 
 280-3333 
 
 1-9!) 
 
 281 
 
 1-90 
 
 281 
 
 1-91 
 
 282 
 
 1-91 
 
 285 
 
 1-93 
 
 29 
 
 31) 
 
 300 
 
 1-97 
 
 300 
 
 1-97 
 
 301 
 
 1-97 
 
 302 
 
 1-98 
 
 305 
 
 2-00 
 
 30 
 
 31 
 
 320-3333 
 
 2-03 
 
 321 
 
 2-04 
 
 322 
 
 2-04 
 
 323 
 
 2-05 
 
 325 
 
 2-06 
 
 31 
 
 32 
 
 341-3333 
 
 2-10 
 
 342 
 
 2-10 
 
 343 
 
 2-11 
 
 344 
 
 2-12 
 
 346 
 
 2-13 
 
 32 
 
 33 
 
 363 
 
 2-17 
 
 363 
 
 2-17 
 
 36i 
 
 2-17 
 
 366 
 
 2-18 
 
 308 
 
 2-20 
 
 33 
 
 34 
 
 385-3333 
 
 2-23 
 
 386 
 
 2-24 
 
 387 
 
 224 
 
 388 
 
 225 
 
 391 
 
 2-27 
 
 34 
 
 35 
 
 408-3333 
 
 2-30 
 
 409 
 
 2-30 
 
 410 
 
 2-31 
 
 412 
 
 2-32 
 
 415 
 
 2-34 
 
 35 
 
 36 
 
 432 
 
 2-37 
 
 433 
 
 2-37 
 
 434 
 
 2-38 
 
 436 
 
 2-39 
 
 439 
 
 2-40 
 
 36 
 
 37 
 
 456-3333 
 
 2-43 
 
 457 
 
 2-44 
 
 458 
 
 2-44 
 
 460 
 
 2-45 
 
 463 
 
 2-47 
 
 37 
 
 38 
 
 481-3333 
 
 2-50 
 
 482 
 
 2-50 
 
 483 
 
 2-51 
 
 485 
 
 2-52 
 
 489 
 
 2-54 
 
 38 
 
 39 
 
 507 
 
 2-57 
 
 508 
 
 2-57 
 
 51 >9 
 
 2-58 
 
 611 
 
 2-59 
 
 515 
 
 2-61 
 
 39 
 
 40 
 
 533-3333 
 
 2-63 
 
 534 
 
 2-64 
 
 535 
 
 2-64 
 
 538 
 
 2-66 
 
 641 
 
 2-67 
 
 40' 
 
 41 
 
 560-3333 
 
 2-70 
 
 561 
 
 2-70 
 
 562 
 
 2-71 
 
 565 
 
 2-72 
 
 569 
 
 2-74 
 
 41 
 
 42 
 
 588 
 
 2-77 
 
 589 
 
 2-77 
 
 590 
 
 2-77 
 
 593 
 
 2-79 
 
 597 
 
 2-81 
 
 42 
 
 43 
 
 616-3333 
 
 2-83 
 
 017 
 
 2-84 
 
 618 
 
 '2-84 
 
 621 
 
 2-86 
 
 625 
 
 2-88 
 
 43 
 
 44 
 
 645-3333 
 
 2-90 
 
 646 
 
 2-90 
 
 648 
 
 2-91 
 
 6.H 
 
 2-92 
 
 655 
 
 2-94 
 
 44 
 
 45 
 
 675 
 
 2-97 
 
 676 
 
 2-97 
 
 677 
 
 2-98 
 
 680 
 
 2-99 
 
 685 
 
 3-01 
 
 45 
 
 46 
 
 7U53333 
 
 3-03 
 
 706 
 
 3-04 
 
 708 
 
 3-04 
 
 711 
 
 3-06 
 
 716 
 
 3-08 
 
 46 
 
 47 
 
 736-3333 
 
 3-10 
 
 737 
 
 3-10 
 
 739 
 
 3-11 
 
 742 
 
 3-13 
 
 747 
 
 3-15 
 
 47 
 
 48 
 
 768 
 
 317 
 
 769 
 
 3-17 
 
 771 
 
 3-18 
 
 774 
 
 3-19 
 
 780 
 
 3-21 
 
 48 
 
 49 
 
 800-3333 
 
 3-23 
 
 801 
 
 3-24 
 
 803 
 
 324 
 
 807 
 
 3-26 
 
 812 
 
 3-28 
 
 49 
 
 50 
 
 833-3333 
 
 330 
 
 834 
 
 331 
 
 836 
 
 331 
 
 840 
 
 333 
 
 846 
 
 335 
 
 50 
 
 Hight 
 
 
 
 
 
 
 Hight 
 
 in 
 feet. 
 
 Surf.-elope 0. 
 
 Surf.-elope 5. 
 
 Surf.-elopelO . 
 
 Surf.-slope 15. 
 
 Surf.-slope 20. 
 
 in 
 
 feet. 
 
168 
 
 MEASUREMENT OF EARTHWORKS. 
 
 Triangular Areas, in square feet, for side-slopes of 1 to 1, to inter- 
 section of slopes, (r = 1.) Slope angle = *45. 
 
 TABLE 3 Earth-cut. 
 
 Hight 
 
 Surf.-slope 0. 
 
 Surf.-slope 5. 
 
 Surf.-slope 10. 
 
 Surf.-slope 15. 
 
 Surf.-slope 20. 
 
 Hight 
 
 in 
 
 
 
 
 
 
 in 
 
 feet. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 i'lo. 
 
 feet 
 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 
 1 
 
 1-UOOO 
 
 10 
 
 1-0077 
 
 .10 
 
 1-0321 
 
 10 
 
 1-0773 
 
 11 
 
 1.1527 
 
 12 
 
 1 
 
 2 
 
 4 
 
 30 
 
 4 
 
 30 
 
 4 
 
 31 
 
 4 
 
 32 
 
 5 
 
 35 
 
 2 
 
 3 
 
 9 
 
 50 
 
 9 
 
 50 
 
 9 
 
 52 
 
 10 
 
 54 
 
 11 
 
 58 
 
 3 
 
 4 
 
 16 
 
 70 
 
 16 
 
 70 
 
 17 
 
 72 
 
 17 
 
 75 
 
 18 
 
 81 
 
 4 
 
 5 
 
 25 
 
 90 
 
 25 
 
 90 
 
 26 
 
 93 
 
 27 
 
 97 
 
 29 
 
 1-04 
 
 5 
 
 6 
 
 36 
 
 1-10 
 
 36 
 
 1-11 
 
 37 
 
 1-14 
 
 39 
 
 1-19 
 
 42 
 
 1-27 
 
 6 
 
 7 
 
 49 
 
 1-30 
 
 49 
 
 1-31 
 
 51 
 
 1-34 
 
 53 
 
 1-40 
 
 56 
 
 1-50 
 
 7 
 
 8 
 
 64 
 
 1-50 
 
 64 
 
 1-51 
 
 66 
 
 1-55 
 
 69 
 
 1-62 
 
 74 
 
 1-73 
 
 8 
 
 9 
 
 81 
 
 1-70 
 
 82 
 
 1-71 
 
 84 
 
 1-75 
 
 87 
 
 1-83 
 
 93 
 
 1-96 
 
 9 
 
 10 
 
 100 
 
 1-90 
 
 101 
 
 1-91 
 
 103 
 
 1-96 
 
 108 
 
 2-05 
 
 115 
 
 2-19 
 
 10 
 
 11 
 
 121 
 
 2-10 
 
 122 
 
 2-12 
 
 125 
 
 2-17 
 
 130 
 
 2-26 
 
 139 
 
 2-42 
 
 11 
 
 12 
 
 144 
 
 2-30 
 
 145 
 
 232 
 
 149 
 
 2-37 
 
 155 
 
 2-48 
 
 166 
 
 2-65 
 
 12 
 
 13 
 
 169 
 
 2-50 
 
 170 
 
 2-52 
 
 174 
 
 2-58 
 
 182 
 
 269 
 
 195 
 
 2-88 
 
 13 
 
 14 
 
 196 
 
 2-70 
 
 198 
 
 2-72 
 
 202 
 
 2-79 
 
 211 
 
 2-91 
 
 226 
 
 3-11 
 
 14 
 
 15 
 
 225 
 
 2-90 
 
 227 
 
 2-92 
 
 232 
 
 2-99 
 
 242 
 
 3-12 
 
 259 
 
 3-34 
 
 15 
 
 16 
 
 256 
 
 3-10 
 
 258 
 
 3-12 
 
 264 
 
 3-20 
 
 276 
 
 3-34 
 
 295 
 
 357 
 
 16 
 
 17 
 
 289 
 
 3-30 
 
 291 
 
 3-33 
 
 298 
 
 3-41 
 
 311 
 
 3-56 
 
 333 
 
 3-80 
 
 17 
 
 18 
 
 324 
 
 3-50 
 
 327 
 
 3-53 
 
 334 
 
 3-61 
 
 34d 
 
 3-77 
 
 373 
 
 4-03 
 
 18 
 
 19 
 
 361 
 
 3-70 
 
 364 
 
 3-73 
 
 373 
 
 3-82 
 
 389 
 
 3-99 
 
 416 
 
 4-27 
 
 19 
 
 20 
 
 400 
 
 3-90 
 
 403 
 
 3-93 
 
 413 
 
 4-02 
 
 431 
 
 420 
 
 461 
 
 4-50 
 
 20 
 
 21 
 
 441 
 
 4-10 
 
 444 
 
 4-13 
 
 455 
 
 4-23 
 
 475 
 
 4-42 
 
 508 
 
 4-73 
 
 21 
 
 22 
 
 484 
 
 4-30 
 
 488 
 
 4-33 
 
 499 
 
 4-44 
 
 521 
 
 4-63 
 
 558 
 
 4-96 
 
 22 
 
 23 
 
 529 
 
 4-50 
 
 533 
 
 4-53 
 
 546 
 
 4-64 
 
 570 
 
 4-85 
 
 610 
 
 5-19 
 
 23 
 
 24 
 
 576 
 
 4-70 
 
 580 
 
 4-74 
 
 594 
 
 4-85 
 
 621 
 
 5-06 
 
 664 
 
 5-42 
 
 24 
 
 25 
 
 625 
 
 4-90 
 
 630 
 
 4-94 
 
 645 
 
 5-06 
 
 673 
 
 5-28 
 
 720 
 
 5-G5 
 
 25 
 
 26 
 
 676 
 
 5-10 
 
 681 
 
 5-14 
 
 698 
 
 5-26 
 
 728 
 
 5-49 
 
 779 
 
 5-88 
 
 26 
 
 27 
 
 729 
 
 5-30 
 
 735 
 
 5-34 
 
 752 
 
 5-47 
 
 785 
 
 5-71 
 
 840 
 
 6-11 
 
 27 
 
 28 
 
 784 
 
 5-50 
 
 790 
 
 5-54 
 
 809 
 
 5-68 
 
 845 
 
 5-92 
 
 904 
 
 6-34 
 
 28 
 
 29 
 
 841 
 
 5-70 
 
 848 
 
 5-74 
 
 868 
 
 5-88 
 
 906 
 
 6-14 
 
 9C9 
 
 6-57 
 
 29 
 
 30 
 
 900 
 
 5-90 
 
 907 
 
 5-95 
 
 929 
 
 6-09 
 
 970 
 
 6-36 
 
 1037 
 
 6-80 
 
 30 
 
 31 
 
 961 
 
 6-10 
 
 968 
 
 6-15 
 
 992 
 
 6-30 
 
 1035 
 
 6-57 
 
 1108 
 
 7-03 
 
 31 
 
 32 
 
 1024 
 
 6-30 
 
 1032 
 
 6-35 
 
 1057 
 
 6-50 
 
 1103 
 
 6-79 
 
 1180 
 
 7-26 
 
 32 
 
 33 
 
 1089 
 
 6-50 
 
 log- 
 
 6-55 
 
 1124 
 
 6-71 
 
 1173 
 
 7-00 
 
 1255 
 
 7-49 
 
 33 
 
 34 
 
 1156 
 
 6-70 
 
 lies 
 
 6-75 
 
 1193 
 
 6-91 
 
 1245 
 
 722 
 
 1333 
 
 7-72 
 
 34 
 
 35 
 
 1225 
 
 6-90 
 
 1234 
 
 6-95 
 
 1264 
 
 7-12 
 
 1320 
 
 7 ; 43 
 
 1412 
 
 7-95 
 
 35 
 
 36 
 
 1296 
 
 7-10 
 
 1306 
 
 7-15 
 
 1338 
 
 7-33 
 
 1396 
 
 7-65 
 
 1494 
 
 8-18 
 
 36 
 
 37 
 
 1369 
 
 7-30 
 
 1380 
 
 7-36 
 
 1413 
 
 7-53 
 
 1475 
 
 7-86 
 
 1578 
 
 8-41 
 
 37 
 
 38 
 
 1444 
 
 7-50 
 
 1455 
 
 7-56 
 
 1490 
 
 7-74 
 
 1556 
 
 8-08 
 
 1665 
 
 8-64 
 
 38 
 
 39 
 
 1521 
 
 7-70 
 
 1533 
 
 7-76 
 
 1570 
 
 7-95 
 
 11)39 
 
 8-29 
 
 1753 
 
 8-88 
 
 39 
 
 40 
 
 1600 
 
 7-90 
 
 1612 
 
 7-96 
 
 1651. 
 
 8-15 
 
 1724 
 
 8-51 
 
 1844 
 
 9-11 
 
 40 
 
 41 
 
 1681 
 
 8-10 
 
 1694 
 
 8-16 
 
 1735 
 
 8-36 
 
 1811 
 
 8-73 
 
 1938 
 
 934 
 
 41 
 
 42 
 
 1764 
 
 8-30 
 
 1778 
 
 8-36 
 
 1820: 
 
 8-57 
 
 1900 
 
 894 
 
 2033 
 
 9-57 
 
 42 
 
 43 
 
 1S49 
 
 8-50 
 
 T863 
 
 8-56 
 
 1908 
 
 8-77 
 
 1992 
 
 9-10 
 
 2131 
 
 9-80 
 
 43 
 
 44 
 
 19o6 
 
 8-70 
 
 1951 
 
 8-77 
 
 1998 
 
 8-98 
 
 2086 
 
 9-37 
 
 2232 
 
 10-03 
 
 44 
 
 45 
 
 2025 
 
 8-90 
 
 2041 
 
 8-97 
 
 2090 
 
 9-18 
 
 2182 
 
 9-59 
 
 2334 
 
 10-26 
 
 45 
 
 46 
 
 2116 
 
 9-10 
 
 2132 
 
 9-17 
 
 2184 
 
 9-39 
 
 2280 
 
 980 
 
 2439 
 
 10-49 
 
 46 
 
 47 
 
 2209 
 
 930 
 
 2226 
 
 9-37 
 
 2280 
 
 9-60 
 
 2380 
 
 10-02 
 
 2546 
 
 10-72 
 
 47 
 
 48 
 
 2304 
 
 9-50 
 
 2322 
 
 9-57 
 
 2378 
 
 980 
 
 2482 
 
 10-23 
 
 2656 
 
 10-95 
 
 48 
 
 49 
 
 2401 
 
 9-70 
 
 2420 
 
 9-77 
 
 2478 
 
 10-01 
 
 2587 
 
 10-45 
 
 2768 
 
 11-18 
 
 49 
 
 50 
 
 2500 
 
 9-90 
 
 2519 
 
 9-97 
 
 2580 
 
 10-22 
 
 2693 
 
 10-67 
 
 2882 
 
 11-41 
 
 50 
 
 Hight 
 
 
 
 
 
 
 liight 
 
 in 
 feet. 
 
 Surf.-slope 0. 
 
 Surf.-slope 5. 
 
 Surf.-slope 10. 
 
 Surf.-slope 15. 
 
 Surf-slope 20. 
 
 in 
 (Vet. 
 
TABLES. 
 
 169 
 
 Triangular areas, in square feet, for side-slopes of 
 section of slopes, (r = H.) Slope angle = 33 41'. 
 
 to 1, to inter- 
 
 TAJ1LE 4 Bank. 
 
 Right 
 
 Surf-slope 0. 
 
 Surf.-slope 5. 
 
 Surf.-fllope 10. 
 
 Surf.-fllope 15. 
 
 Surf.-slope 30. 
 
 Hight 
 
 in 
 
 
 
 
 
 
 in 
 
 feet. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 
 Pro. 
 
 feet. 
 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 Areas. 
 
 for -1. 
 
 
 1 
 
 1-5000 
 
 16 
 
 1-52U7 
 
 15 
 
 1-6133 
 
 16 
 
 1-7900 
 
 18 
 
 2-1378 
 
 21 
 
 1 
 
 2 
 
 6 
 
 45 
 
 6 
 
 46 
 
 6 
 
 48 
 
 7 
 
 54 
 
 9 
 
 6i 
 
 2 
 
 3 
 
 13-5 
 
 75 
 
 14 
 
 76 
 
 15 
 
 81 
 
 16 
 
 89 
 
 19 
 
 1-07 
 
 3 
 
 4 
 
 24 
 
 1-05 
 
 25 
 
 1-07 
 
 26 
 
 1-13 
 
 29 
 
 1-25 
 
 34 
 
 1-50 
 
 4 
 
 5 
 
 37-5 
 
 1-35 
 
 38 
 
 1-37 
 
 40 
 
 145 
 
 45 
 
 1-61 
 
 54 
 
 1-92 
 
 5 
 
 6 
 
 54 
 
 1-65 
 
 55 
 
 1-68 
 
 58 
 
 1-78 
 
 64 
 
 1-97 
 
 77 
 
 2-35 
 
 6 
 
 7 
 
 73-5 
 
 1-95 
 
 75 
 
 1-98 
 
 79 
 
 2-10 
 
 88 
 
 233 
 
 105 
 
 2-78 
 
 7 
 
 8 
 
 96 
 
 2-25 
 
 98 
 
 2-29 
 
 103 
 
 2-42 
 
 115 
 
 2-68 
 
 137 
 
 321 
 
 8 
 
 9 
 
 121-5 
 
 2-55 
 
 124 
 
 2-59 
 
 131 
 
 2-74 
 
 145 
 
 304 
 
 173 
 
 363 
 
 9 
 
 10 
 
 150 
 
 2-85 
 
 153 
 
 2-90 
 
 161 
 
 3-06 
 
 179 
 
 3-39 
 
 214 
 
 4-06 
 
 10 
 
 11 
 
 181-5 
 
 315 
 
 185 
 
 3-20 
 
 195 
 
 3-39 
 
 217 
 
 3-76 
 
 259 
 
 4-49 
 
 11 
 
 12 
 
 216 
 
 3-45 
 
 220 
 
 351 
 
 232 
 
 3-71 
 
 258 
 
 4-12 
 
 308 
 
 4-92 
 
 12 
 
 13 
 
 253-5 
 
 3-75 
 
 258 
 
 3-82 
 
 273 
 
 403 
 
 302 
 
 447 
 
 361 
 
 5-34 
 
 13 
 
 14 
 
 294 
 
 4-05 
 
 299 
 
 4-12 
 
 316 
 
 4-36 
 
 351 
 
 4-83 
 
 419 
 
 5-77 
 
 14 
 
 15 
 
 337-5 
 
 4-35 
 
 344 
 
 4-43 
 
 363 
 
 4-68 
 
 403 
 
 5-19 
 
 481 
 
 620 
 
 15 
 
 16 
 
 384 
 
 465 
 
 391 
 
 4-73 
 
 413 
 
 5-00 
 
 458 
 
 5-55 
 
 647 
 
 6-63 
 
 16 
 
 17 
 
 433-5 
 
 4-95 
 
 441 
 
 5-04 
 
 466 
 
 - 5-32 
 
 617 
 
 5-92 
 
 618 
 
 7-05 
 
 17 
 
 18 
 
 486 
 
 5-25 
 
 495 
 
 .5-34 
 
 523 
 
 5-65 
 
 580 
 
 626 
 
 693 
 
 7-48 
 
 18 
 
 19 
 
 5415 
 
 5-55 
 
 551 
 
 5-65 
 
 582 
 
 5-97 
 
 646 
 
 6-62 
 
 772 
 
 7-91 
 
 19 
 
 20 
 
 600 
 
 5-85 
 
 611 
 
 5-95 
 
 645 
 
 6-29 
 
 716 
 
 6-98 
 
 855 
 
 8-34 
 
 20 
 
 21 
 
 661-5 
 
 6-15 
 
 673 
 
 6-26 
 
 711 
 
 6-61 
 
 789 
 
 7-34 
 
 943 
 
 8-76 
 
 21 
 
 22 
 
 726 
 
 6-45 
 
 739 
 
 6-56 
 
 781 
 
 6-94 
 
 866 
 
 7-69 
 
 1035 
 
 9-19 
 
 22 
 
 23 
 
 793-5 
 
 6-75 
 
 808 
 
 6-87 
 
 853 
 
 7-26 
 
 947 
 
 8-05 
 
 1131 
 
 962 
 
 23 
 
 24 
 
 864 
 
 7-05 
 
 879 
 
 7-17 
 
 929 
 
 7-58 
 
 1031 
 
 8-41 
 
 1231 
 
 10-05 
 
 24 
 
 25 
 
 937-5 
 
 7-35 
 
 954 
 
 7-48 
 
 1008 
 
 7-90 
 
 1118 
 
 8-77 
 
 1336 
 
 10-47 
 
 25 
 
 26 
 
 1014 
 
 7-6.5 
 
 1032 
 
 7-79 
 
 1090 
 
 8-23 
 
 1210 
 
 9-13 
 
 1445] 
 
 10-90 
 
 26 
 
 27 
 
 1093-5 
 
 7-95 
 
 1113 
 
 8-09 
 
 1176 
 
 8-55 
 
 1304 
 
 9-48 
 
 1558 
 
 1133 
 
 27 
 
 28 
 
 1176 
 
 8-25 
 
 1197 
 
 8-40 
 
 1265 
 
 8-87 
 
 1403 
 
 9-84 
 
 1676 
 
 11-76 
 
 28 
 
 29 
 
 1261-5 
 
 8-55 
 
 1284 
 
 8-70 
 
 1357 
 
 9-19 
 
 1505 
 
 10-20 
 
 1798 
 
 1218 
 
 29 
 
 30 
 
 1350 
 
 8-85 
 
 1374 
 
 9-00 
 
 1452 
 
 9-52 
 
 1610 
 
 10-55 
 
 1924 
 
 12-61 
 
 30 
 
 31 
 
 1441-5 
 
 9-15 
 
 1467 
 
 9-31 
 
 1550 
 
 9-84 
 
 1719 
 
 10-91 
 
 2054 
 
 1304 
 
 31 
 
 32 
 
 1536 
 
 9-45 
 
 1563 
 
 9-62 
 
 1652 
 
 10-16 
 
 1832 
 
 11-27 
 
 2189 
 
 13-47 
 
 32 
 
 33 
 
 1633-5 
 
 975 
 
 1662 
 
 9-92 
 
 1757 
 
 10-48 
 
 1948 
 
 11-63 
 
 2328 
 
 13-89 
 
 33 
 
 34 
 
 1734 
 
 10-05 
 
 1765 
 
 10-23 
 
 1865 
 
 10-81 
 
 2068 
 
 1199 
 
 2471 
 
 1432 
 
 34 
 
 35 
 
 l37-5 
 
 10-35 
 
 1870 
 
 10-53 
 
 1976 
 
 11-13 
 
 2192 
 
 12-35 
 
 2619- 
 
 14-75 
 
 35 
 
 36 
 
 1944 
 
 10-65 
 
 1978 
 
 10-84 
 
 2090 
 
 11-45 
 
 2319 
 
 12-70 
 
 2770 
 
 15-18 
 
 36 
 
 37 
 
 I10&9 
 
 10-95 
 
 2090 
 
 11-14 
 
 2208 
 
 11-77 
 
 2449 
 
 13-06 
 
 2926 
 
 15-60 
 
 37 
 
 38 
 
 2166 
 
 11-23 
 
 2204 
 
 11-45 
 
 2329 
 
 12-10 
 
 2584 
 
 1342 
 
 3087 
 
 1603 
 
 38 
 
 39 
 
 2281-5 
 
 11-55 
 
 2322 
 
 11-76 
 
 2453 
 
 12-42 
 
 2721 
 
 13-78 
 
 3251 
 
 1646 
 
 39 
 
 40 
 
 2400 
 
 11-85 
 
 2442 
 
 12-06 
 
 2581 
 
 12-74 
 
 2863 
 
 14-14 
 
 3420 
 
 16-89 
 
 40 
 
 41 
 
 2521-5 
 
 12-15 
 
 2566 
 
 12-36 
 
 2711 
 
 13-06 
 
 3008 
 
 14-50 
 
 3593 
 
 17-31 
 
 41 
 
 42 
 
 2646 
 
 12-45 
 
 2693 
 
 1267 
 
 2845 
 
 13-39 
 
 3156 
 
 14-85 
 
 3771 
 
 17-74 
 
 42 
 
 43 
 
 2773-5 
 
 12-75 
 
 2823 
 
 1298 
 
 2982 
 
 1371 
 
 3308 
 
 15-21 
 
 3952 
 
 18-17 
 
 43 
 
 44 
 
 2904 
 
 13-05 
 
 2955 
 
 13-28 
 
 3123 
 
 14-03 
 
 3464 
 
 15-57 
 
 4138 
 
 18-60 
 
 44 
 
 45 
 
 3037-5 
 
 13-35 
 
 3091 
 
 13-59 
 
 3266 
 
 14-35 
 
 3623 
 
 15-92 
 
 4329 
 
 1902 
 
 45 
 
 46 
 
 3174 
 
 13-65 
 
 3230 
 
 1389 
 
 3413 
 
 14-68 
 
 3786 
 
 16-28 
 
 4523 
 
 1945 
 
 46 
 
 47 
 
 3313-5 
 
 13-95 
 
 3372 
 
 14-20 
 
 3563 
 
 15-00 
 
 3952 
 
 16-64 
 
 4722 
 
 19-88 
 
 47 
 
 48 
 
 3456 
 
 14-25 
 
 3517 
 
 1450 
 
 3716 
 
 15-32 
 
 4122 
 
 16-99 
 
 4925 
 
 20-31 
 
 48 
 
 49 
 
 3601-5 
 
 ;u-55 
 
 3665 
 
 14-81 
 
 3873 
 
 15-64 
 
 4296 
 
 17-35 
 
 5132 
 
 2074 
 
 49 
 
 50 
 
 3750 
 
 14-85 
 
 3816 
 
 15-12 
 
 4032 
 
 15-97 
 
 4473 
 
 17-71 
 
 5344 
 
 2116 
 
 50 
 
 Hight 
 
 
 
 
 
 
 Right 
 
 in 
 
 feet. 
 
 Surf.-slope 0. 
 
 Snrf.-Blope 5. 
 
 Surf.-slope 10. 
 
 Surf.-elope 15. 
 
 Surf.-slope 0. 
 
 in 
 
 feet. 
 
TABLE OF CUBIC YARDS 
 
 IN FULL STATIONS, OR LENGTHS OF WO FEET. 
 
 CALCULATED FOR EVERY FOOT AND TENTH OF MEAN AREA, 
 FROM 0- TO 1000' SUPERFICIAL FEET. 
 
 Note. On every page of the Table, the columns on both sides headed M.A. contain the 
 Mean Areas, in square, or superficial feet. 
 
 The horizontal lines at top and bottom show the tenths of square feet of 
 Mean Area. 
 
 And the figures in the body of the Table, computed to three places of decimals, 
 are the Cubic Yards (for 100- feet), corresponding to the feet and tenths of Mean 
 Area, indicated in the side columns, and lines of tenths at top and bottom. 
 
 EXPLANATION OF THE TABLE OF CUBIC YARDS, 
 To Mean Areas, in lengtJis of 100* feet, and of its Applications. 
 
 This Table is computed to facilitate the conversion into Cubic Yards 
 of the content of any solid 100 feet in length, of which the Mean Area, 
 in superficial feet has been ascertained. It applies directly to all 
 Mean Areas from 0' to 1000' square feet (including tenths of feet), 
 and being calculated to three decimal places, it extends indirectly to 
 100,000* superficial feet of Mean Area, as will be shown hereafter. 
 
 EXAMPLE 1. 
 
 Cubic yards for 
 full stations 
 
 To find the Cubic Yards, belonging to 579' 8 
 sup. ft. of Mean Area, for a full station, or length 
 of 100- feet : 
 
 Opposite 579' and under *8 we find the con- 
 tent, or solidity =2147 '407 cubic yards. 
 
 Which is equal to 
 
 579- 8 sq. ft. of Mean Area X 100' feet long, 
 
 and divided by 27. 
 170 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 171 
 
 EXAMPLE 2. 
 
 Cubic yards for 
 
 short stations 
 
 (-100-) 
 
 EXAMPLE 3. 
 
 Cubic yards for 
 
 long stations 
 
 (+ 100-) 
 
 / Let the Mean Area of any solid, be 98* T sq. ft. 
 
 and its length 84 ft. lineal : (being a short station). 
 
 Then at 98' 7 we find 365*556 cubic yards, 
 
 which being multiplied by '84 taken decimally, 
 
 gives 365-556 X '84 =307'067 cubic yards. 
 
 Equal to... *7X* 
 
 Again, let the Mean Area be 88* 6 and the 
 length 259- feet (or a long station) ; then for 88' 6 
 sq. ft. of Mean Area, we have 328*148 cubic 
 yards, which multiplied by 2*59 (decimal) 
 gives =849-903 cubic yards. 
 
 Equal to... 
 
 88-6 X 259- 
 27- 
 
 Th is Table is especially useful in the computation of the Earth- 
 work of Railroads, and other Public Works, where cross-sections 
 have been taken normal to a guide line, at distances (generally) of 
 100- lineal feet (or full stations), and the Mean Area calculated in 
 superficial feet and parts: but it is also applicable to any solid of 
 which the mean section is known in square feet, and the length 100* 
 feet, or any decimal part thereof. 
 
 For, if the distances apart of cross-sections, or lengths of stations, 
 be more, or less, than 100' feet, we have only to take them decimally, 
 as in the above examples, and by a simple multiplication, of the 
 tabular quantity, belonging to the known area, the correct number 
 of cubic yards will be ascertained. 
 
 The Table being calculated to three places of decimals, readily ad- 
 mits of being used for Mean Areas, much exceeding its direct range 
 of 1000- superficial feet (as follows) : 
 
 EXAMPLE 4. Suppose the Mean Area to be 98,967* * sq. ft. (repre- 
 senting a cut 98' 9 feet deep, and 1000" feet wide). 
 
 Then for 98,900' (by moving the decimal point 
 of the tabular quantity of cubic yards for 989' 
 two figures to the right) 
 
 We have, area 98,900' = 366,296- 3 cubic yds. 
 Add 67- 4 = 249- 8 " 
 
 Total, for sq. ft... 98,967' <= 366,545- 
 Equal to '- 
 
172 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 Again, take a Mean Area, of 100,048' 9 sq. ft. (representing a cut 
 100- feet deep, and 1000' feet wide). 
 
 Then for 100,000 sq. ft. (by moving the deci- 
 mal point of the tabular quantity of cubic yards 
 for 1000* two figures to the right), 
 We have, 100,000 Area == 370,370' 4 cub. yds. 
 
 Add 48; 9 " = 181' l " " 
 
 Total for 100,048' 9 " = 370,551' 5 " " 
 
 Equal to... 100^X100, 
 
 Example 4, shows the easy application of the Table, to Mean 
 Areas, which may be called immense, by merely moving the decimal 
 point, and a simple addition, as shown above. 
 
 Other methods of using the Table will occur to the reader, but the 
 examples given seem sufficient for illustration. 
 
 Much pains have been taken to make this Table correct, to the 
 nearest decimal, and we believe it may be safely depended on. 
 
 Note. Besides its special application to Earthworks, the extensive 
 Table following is also a general Table for the conversion of any sum 
 of Cub.ic Feet into Cubic Yards. Thus, in the example at page 103, 
 the reduced quantities of Cubic Feet sum up 227,200 30,000 = 
 197,200 Cubic Feet. 
 
 In such cases we have only to cut off two figures from the right 
 (or H- by 100), and we have 1972, the mean area, which, in 100 feet 
 length, would have produced the quantity given. 
 
 With 197*2 we enter the Table following, and find 730'370 Cubic 
 Yards ; now, moving the decimal point one place to the right, we 
 have 7303-70 Cubic Yards, or in round numbers, 7304 Cubic Yards, 
 as already given on page 103. 
 
 In like manner the Cubic Yards for any sum whatever of Cubic 
 Feet can readily be obtained, and the Table being in itself strictly 
 correct, the result will be reliable. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 173 
 
 TABT.E OF CUIilC YARDS, in full Station*, or length ft of 1OO fret: for every 
 foot ami ti-nth of Menu Arm, from O to 1OOO Snjurftcinl Fret. 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 | -6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 
 
 o-ooo 
 
 0-370 
 
 0-741 
 
 1-111 
 
 1-481 
 
 1-852 
 
 2-222 
 
 2-593 
 
 2-963 
 
 3-333 
 
 
 
 1 
 
 3-704 
 
 4-074 
 
 4-444 
 
 4-815 
 
 5-185 
 
 5-556 
 
 5-926 
 
 6-296 
 
 6-667 
 
 7-037 
 
 1 
 
 2 
 
 7-407 
 
 7-778 
 
 8-148 
 
 8-519 
 
 8-889 
 
 9-259 
 
 9630 
 
 10- 
 
 10.370 
 
 10741 
 
 2 
 
 3 
 
 11-111 
 
 11481 
 
 11-852 
 
 12-222 
 
 12-593 
 
 12-963 
 
 13-333 
 
 13-704 
 
 14-074 
 
 14-444 
 
 3 
 
 4 
 
 14-815 
 
 15-185 
 
 15-556 
 
 15-920 
 
 16-296 
 
 16-667 
 
 17-037 
 
 17-407 
 
 17-778 
 
 18148 
 
 4 
 
 5 
 
 18-519 
 
 18-889 
 
 19-259 
 
 19-630 
 
 20- 
 
 20370 
 
 20-741 
 
 21-111 
 
 21481 
 
 21-852 
 
 5 
 
 6 
 
 22-222 
 
 22-593 
 
 22-963 
 
 23-333 
 
 23-704 
 
 24-074 
 
 24-444 
 
 24-815 
 
 25-185 
 
 25-556 
 
 6 
 
 7 
 
 25920 
 
 26-296 
 
 26-667 
 
 27037 
 
 27-407 
 
 27-778 
 
 28-148 
 
 28519 
 
 28-889 
 
 29-259 
 
 7 
 
 8 
 
 29-630 
 
 30- 
 
 30-370 
 
 30-741 
 
 31-111 
 
 31-481 
 
 31-852 
 
 32-222 
 
 32-593 
 
 32963 
 
 8 
 
 9 
 
 33-333 
 
 33-704 
 
 34-074 
 
 34-444 
 
 34-815 
 
 35-185 
 
 35-556 
 
 35-92b 
 
 36-296 
 
 36-667 
 
 9 
 
 10 
 
 37037 
 
 37-407 
 
 37-778 
 
 38-148 
 
 38-519 
 
 38-889 
 
 39-259 
 
 39-630 
 
 40- 
 
 40-370 
 
 10 
 
 11 
 
 40-741 
 
 41-111 
 
 41-481 
 
 41-852 
 
 42-222 
 
 42-593 
 
 42-963 
 
 43-333 
 
 43704 
 
 44-074 
 
 11 
 
 12 
 
 44444 
 
 44-815 
 
 45-185 
 
 45-556 
 
 45-926 
 
 46-296 
 
 46-667 
 
 47-037 
 
 47-407 
 
 47-778 
 
 12 
 
 13 
 
 48-148 
 
 48-519 
 
 48-889 
 
 49-259 
 
 49630 
 
 50 
 
 60-370 
 
 60-741 
 
 51-111 
 
 51-481 
 
 13 
 
 14 
 
 51-852 
 
 52 222 
 
 52-593 
 
 52-963 
 
 53333 
 
 53-704 
 
 64-074 
 
 54-444 
 
 54*815 
 
 55-185 
 
 14 
 
 15 
 
 55-556 
 
 55-926 
 
 56-296 
 
 56-667 
 
 57-037 
 
 57-407 
 
 57-778 
 
 68-148 
 
 58-519 
 
 68-889 
 
 15 
 
 16 
 
 59-259 
 
 59-630 
 
 60- 
 
 60-370 
 
 60-741 
 
 61-111 
 
 61-481 
 
 61852 
 
 02-222 
 
 62-593 
 
 16 
 
 17 
 
 62-903 
 
 63-3:53 
 
 63-704 
 
 64-074 
 
 64-444 
 
 64815 
 
 65185 
 
 65-556 
 
 65-926 
 
 66-296 
 
 17 
 
 18 
 
 66-667 
 
 67.037 
 
 67-407 
 
 67-778 
 
 68-148 
 
 (8-519 
 
 68-889 
 
 69-259 
 
 69-630 
 
 70- 
 
 18 
 
 19 
 
 70-370 
 
 70-741 
 
 71-111 
 
 71-481 
 
 71-852 
 
 72-222 
 
 72-593 
 
 72-963 
 
 73-333 
 
 73-704 
 
 19 
 
 20 
 
 74-074 
 
 74-444 
 
 74-815 
 
 75-185 
 
 75-556 
 
 75-926 
 
 76-29G 
 
 76-667 
 
 77-037 
 
 77-407 
 
 20 
 
 21 
 
 77-778 
 
 78-148 
 
 78-519 
 
 78-889 
 
 79-259 
 
 79-630 
 
 80- 
 
 80-370 
 
 80-741 
 
 81111 
 
 21 
 
 22 
 
 81-481 
 
 81-852 
 
 82-222 
 
 82593 
 
 82-963 
 
 83-3: J3 
 
 83-704 
 
 84-074 
 
 84-444 
 
 84-815 
 
 22 
 
 23 
 
 85-185 
 
 85-556 
 
 85-92f. 
 
 86-296 
 
 86-667 
 
 87-037 
 
 87-407 
 
 87-778 
 
 88-148 
 
 88-519 
 
 23 
 
 24 
 
 88-889 
 
 89-259 
 
 89-630 
 
 9J- 
 
 90-370 
 
 90-741 
 
 91-111 
 
 91-481 
 
 91-852 
 
 92-222 
 
 24 
 
 25 
 
 92-593 
 
 92-963 
 
 93-333 
 
 93.704 
 
 94-074 
 
 94-444 
 
 94-815 
 
 95-185 
 
 96-556 
 
 95926 
 
 25 
 
 26 
 
 96-296 
 
 96-667 
 
 97-037 
 
 97-407 
 
 97-778 
 
 98-148 
 
 985i 9 
 
 98-889 
 
 99-259 
 
 99-630 
 
 26 
 
 27 
 
 100- 
 
 100.370 
 
 100741 
 
 101-111 
 
 101-481 
 
 101-852 
 
 102-222 
 
 102.593 
 
 102-963 
 
 103-333 
 
 27 
 
 28 
 
 103-704 
 
 104-074 
 
 104-444 
 
 104-815 
 
 105-186 
 
 105-556 
 
 105-920 
 
 106-296 
 
 106-667 
 
 107-037 
 
 28 
 
 29 
 
 107-407 
 
 107-778 
 
 108-148 
 
 108-519 
 
 108-889 
 
 109-259 
 
 109-630 
 
 110- 
 
 110-370 
 
 110-741 
 
 29 
 
 30 
 
 Ill-Ill 
 
 111-481 
 
 111-852 
 
 112-222 
 
 112-593 
 
 112-963 
 
 113-333 
 
 113-704 
 
 114-074 
 
 114-444 
 
 30 
 
 31 
 
 114-815 
 
 115-185 
 
 115-556 
 
 115-926 
 
 116-296 
 
 116-667 
 
 117-037 
 
 117-407 
 
 117-778 
 
 118-148 
 
 31 
 
 32 
 
 118-519 
 
 118-889 
 
 119259 
 
 119-630 
 
 120- 
 
 120-370 
 
 120-741 
 
 121-111 
 
 121-481 
 
 121-852 
 
 32 
 
 33 
 
 122-222 
 
 122-593 
 
 122-963 
 
 123-333 
 
 123-704 
 
 124-074 
 
 124-444 
 
 124-815 
 
 125-lSo 
 
 125*-556 
 
 33 
 
 34 
 
 125-92b 
 
 126-296 
 
 126-667 
 
 127-037 
 
 127-407 
 
 127-778 
 
 128-148 
 
 128-519 
 
 128-889 
 
 129-259 
 
 34 
 
 35 
 
 129-630 
 
 130- 
 
 130-370 
 
 130-741 
 
 131-111 
 
 131-481 
 
 131-852 
 
 132-222 
 
 132-593 
 
 132.963 
 
 35 
 
 36 
 
 133-333 
 
 133-704 
 
 134-074 
 
 134-444 
 
 134-815 
 
 135-185 
 
 135-556 
 
 135-926 
 
 136-296 
 
 136-667 
 
 36 
 
 37 
 
 137-037 
 
 137-40" 
 
 137-778 
 
 138-148 
 
 138-519 
 
 138-889 
 
 139-259 
 
 139-630 
 
 140- 
 
 140-370 
 
 37 
 
 38 
 
 140-741 
 
 141-111 
 
 141-481 
 
 141-852 
 
 142-222 
 
 142-593 
 
 142-963 
 
 143-33o 
 
 143-704 
 
 144-074 
 
 38 
 
 39 
 
 144-444 
 
 144 815 
 
 145-185 
 
 145-556 
 
 145-926 
 
 146-296 
 
 146-667 
 
 147-03" 
 
 147-407 
 
 147-778 
 
 39 
 
 40 
 
 148-148 
 
 148-519 
 
 148-889 
 
 149-259 
 
 149-630 
 
 150- 
 
 150-37 
 
 160-74 
 
 161-11 
 
 151-481 
 
 40 
 
 41 
 
 151-8&* 
 
 152-22 
 
 152-593 
 
 152-963 
 
 153-333 
 
 153-704 
 
 154-07 
 
 154-444 
 
 154-81 
 
 155-185 
 
 41 
 
 42 
 
 155-55 
 
 15592 
 
 156296 
 
 156-66' 
 
 157-037 
 
 157-407 
 
 157-77 
 
 158-14 
 
 168-51 
 
 168-889 
 
 42 
 
 4;i 
 
 159-25 
 
 159-63 
 
 160- 
 
 160-370 
 
 160-741 
 
 161-11 
 
 161-48 
 
 161-85 
 
 162-222 
 
 162-593 
 
 43 
 
 44 
 
 162-%. 
 
 163-333 
 
 163-704 
 
 164-074 
 
 164-444 
 
 164-81 
 
 165-18 
 
 165-55 
 
 165-92 
 
 166-296 
 
 44 
 
 45 
 
 16666 
 
 107-03 
 
 167-40" 
 
 167-77S 
 
 168-148 
 
 168-61 
 
 168-88 
 
 169-25 
 
 169-630 
 
 170- 
 
 46 
 
 46 
 
 170-37 
 
 170-74 
 
 171-111 
 
 171-481 
 
 171-852 
 
 17222 
 
 172-59 
 
 172-96. 
 
 173-33 
 
 173-704 
 
 46 
 
 47 
 
 174-07 
 
 174-444 
 
 174-815 
 
 175-18o 
 
 175-556 
 
 175-92 
 
 176-29 
 
 176-66 
 
 177-03 
 
 177-40 
 
 47 
 
 43 
 
 177-77 
 
 178-14 
 
 178-519 
 
 178-889 
 
 179-259 
 
 179-630 
 
 180- 
 
 18037 
 
 180-74 
 
 181-111 
 
 48 
 
 49 
 
 181-48 
 
 181-85 
 
 182-222 
 
 182-593 
 
 182-963 
 
 183-333 
 
 183-70 
 
 184-07 
 
 184-444 
 
 184-815 
 
 49 
 
 50 
 
 185-lSa 
 
 185-55 
 
 185-926 
 
 186-29b 
 
 186-667 
 
 187-03 
 
 187-40 
 
 187-77 
 
 188-14 
 
 188-519 
 
 60 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 51 
 
 188-88 
 
 189-259 
 
 189-630 
 
 190- 
 
 190-37 
 
 190-74 
 
 191-11 
 
 191-48 
 
 191-85 
 
 192-222 
 
 51 
 
 52 
 
 192-59 
 
 192-96 
 
 193-333 
 
 193-704 
 
 194-074 
 
 194-444 
 
 194-81 
 
 195-18 
 
 195-55 
 
 195-926 
 
 62 
 
 
 196-29 
 
 196-66 
 
 197-037 
 
 197-407 
 
 197-77 
 
 198-14 
 
 198-51 
 
 198-88 
 
 199-25 
 
 199-630 
 
 53 
 
 54 
 
 200- 
 
 200-37 
 
 200-74 
 
 201-11 
 
 201-48 
 
 201-85 
 
 202- 22* 
 
 202-59 
 
 202-96 
 
 203-333 
 
 54 
 
 55 
 
 203-70 
 
 204-07 
 
 204-444 
 
 204-81 
 
 205-18 
 
 205-55 
 
 205-92 
 
 206-29 
 
 206-66 
 
 207 03" 
 
 65 
 
 56 
 
 207-40 
 
 207-77 
 
 208-14 
 
 208-51 
 
 208-88 
 
 209-25 
 
 209-630 
 
 210- 
 
 210-37 
 
 210-74 
 
 56 
 
 57 
 
 211-11 
 
 211-48 
 
 211-85 
 
 212-222 
 
 212-59 
 
 212-96, 
 
 213-333i 213-704 
 
 214-07 
 
 214-444 
 
 67 
 
 58 
 
 214-81 
 
 215-18 
 
 215-55 
 
 215-92 
 
 216-29 
 
 216-66 
 
 217-037 
 
 217-40 
 
 217-77 
 
 218-14 
 
 58 
 
 59 
 
 218-51 
 
 218-88 
 
 219-25 
 
 219-63 
 
 220- 
 
 220-37 
 
 220-741 
 
 221-11 
 
 221-48 
 
 221-85 
 
 59 
 
 CO 
 
 222-22 
 
 222-59 
 
 222-96- 
 
 223-333 
 
 223-70 
 
 224-07 
 
 224-444 
 
 224-81 
 
 225-18 
 
 225-55 
 
 60 
 
 M.A 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS O to 6O. 
 
174 
 
 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR WO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 3 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 61 
 
 225-926 
 
 226-296 
 
 226-667 
 
 227-037 
 
 227-407 
 
 227-778 
 
 228-148 
 
 228-519 
 
 228-889 
 
 229-259 
 
 61 
 
 62 
 
 229-630 
 
 230- 
 
 230-370 
 
 230-741 
 
 231-111 
 
 231-481 
 
 231-852 
 
 232-222 
 
 232-593 
 
 232-963 
 
 62 
 
 63 
 
 233-333 
 
 233-704 
 
 234-074 
 
 234-444 
 
 234-815 
 
 235-185 
 
 235-556 
 
 235-926 
 
 236-296 
 
 236-667 
 
 63 
 
 64 
 
 237-037 
 
 237-407 
 
 237-778 
 
 238-148 
 
 238-519 
 
 238-889 
 
 239-259 
 
 239-630 
 
 240- 
 
 240-370 
 
 64 
 
 65 
 
 240741 
 
 241-111 
 
 241-481 
 
 241-852 
 
 242-222 
 
 242-593 
 
 242-963 
 
 243-333 
 
 243704 
 
 244-074 
 
 65 
 
 66 
 
 244-444 
 
 244-815 
 
 245-185 
 
 245-556 
 
 245-926 
 
 246-296 
 
 246-667 
 
 247-037 
 
 247-407 
 
 247-778 
 
 66 
 
 67 
 
 248-148 
 
 248519 
 
 248-889 
 
 249-259 
 
 249-630 
 
 250- 
 
 250-370 
 
 250-741 
 
 251-111 
 
 251-481 
 
 67 
 
 6S 
 
 251-852 
 
 252-222 
 
 252-593 
 
 252-963 
 
 253-33o 
 
 253-704 
 
 254-074 
 
 254-444 
 
 254-815 
 
 255-185 
 
 68 
 
 69 
 
 255-556 
 
 255-926 
 
 256-296 
 
 256-667 
 
 257-037 
 
 257-407 
 
 257-778 
 
 258-148 
 
 258-519 
 
 25K-889 
 
 C9 
 
 70 
 
 259-259 
 
 259-630 
 
 260- 
 
 260-370 
 
 260-741 
 
 261-111 
 
 261-481 
 
 261-852 
 
 262-222 
 
 262-593 
 
 70 
 
 71 
 
 262-963 
 
 263-333 
 
 263-704 
 
 264-074 
 
 264-444 
 
 264-815 
 
 265-185 
 
 265-556 
 
 265-926 
 
 266-296 
 
 71 
 
 72 
 
 266-667 
 
 267-037 
 
 267-407 
 
 267-778 
 
 268-148 
 
 268-519 
 
 268-889 
 
 269-259 
 
 269-630 
 
 270- 
 
 72 
 
 73 
 
 270-370 
 
 270741 
 
 271-111 
 
 271-481 
 
 271-852 
 
 272-222 
 
 272-593 
 
 272-963 
 
 273-333 
 
 273-704 
 
 73 
 
 74 
 
 274-074 
 
 274-444 
 
 274-815 
 
 275-185 
 
 275-556 
 
 275-926 
 
 276-296 
 
 276-667 
 
 277-037 
 
 277-407 
 
 74 
 
 75 
 
 277-778 
 
 278-148 
 
 278-519 
 
 278-889 
 
 279-259 
 
 279-630 
 
 280- 
 
 280-370 
 
 280-741 
 
 281-111 
 
 75 
 
 76 
 
 281-4S1 
 
 281-852 
 
 282-222 
 
 282-593 
 
 282-963 
 
 283 333 
 
 283-704 
 
 284-074 
 
 284-444 
 
 284-815 
 
 76 
 
 77 
 
 285-185 
 
 285-556 
 
 285-926 
 
 286-296 
 
 286-667 
 
 287-037 
 
 287-407 
 
 287-778 
 
 288-148 
 
 288-519 
 
 77 
 
 78 
 
 288-889 
 
 289-259 
 
 289-630 
 
 290- 
 
 290-370 
 
 290-741 
 
 291-111 
 
 291-481 
 
 291-852 
 
 292-222 
 
 78 
 
 79 
 
 292-593 
 
 292-963 
 
 293-333 
 
 293-704 
 
 294-074 
 
 294-444 
 
 294-815 
 
 '295-185 
 
 295-556 
 
 295-926 
 
 79 
 
 80 
 
 296-296 
 
 296-667 
 
 297-037 
 
 297-407 
 
 297-778 
 
 298-148 
 
 298-519 
 
 298-889 
 
 299-259 
 
 299-630 
 
 80 
 
 81 
 
 300- 
 
 300-370 
 
 300-741 
 
 301-111 
 
 301-481 
 
 301-852 
 
 302-222 
 
 302-593 
 
 302-963 
 
 303-333 
 
 81 
 
 82 
 
 303-704 
 
 304-074 
 
 304444 
 
 304-815 
 
 305-185 
 
 305-556 
 
 305-926 
 
 306-296 
 
 306-667 
 
 307 037 
 
 82 
 
 83 
 
 307-407 
 
 307-778 
 
 308-148 
 
 308-519 
 
 308-889 
 
 309-259 
 
 309-630 
 
 310- 
 
 310-370 
 
 310-741 
 
 83 
 
 84 
 
 311-111 
 
 311-481 
 
 311-852 
 
 312-222 
 
 312-593 
 
 312-963 
 
 313-333 
 
 313-704 
 
 314-074 
 
 314-444 
 
 84 
 
 85 
 
 314-815 
 
 315-185 
 
 315-556 
 
 315-926 
 
 316-296 
 
 316-667 
 
 317-037 
 
 317-407 
 
 317-778 
 
 318-148 
 
 85 
 
 86 
 
 318-519 
 
 318-889 
 
 319-259 
 
 319-630 
 
 320- 
 
 320-370 
 
 320-741 
 
 321-111 
 
 321-481 
 
 321-852 
 
 86 
 
 87 
 
 322-222 
 
 322-593 
 
 322-963 
 
 323-333 
 
 323-704 
 
 324-074 
 
 324-444 
 
 324-815 
 
 325-185 
 
 325-556 
 
 87 
 
 88 
 
 325' 926 
 
 320-296 
 
 326-667 
 
 327-037 
 
 327-407 
 
 327-778 
 
 328-148 
 
 328-519 
 
 328-889 
 
 329-259 
 
 88 
 
 89 
 
 329-630 
 
 330: 
 
 330-370 
 
 330-741 
 
 331-111 
 
 331-481 
 
 331-852 
 
 332-222 
 
 332-593 
 
 332-963 
 
 89 
 
 90 
 
 333-333 
 
 333-704 
 
 334-074 
 
 334-444 
 
 334-815 
 
 335-185 
 
 335-556 
 
 335-926 
 
 336-296 
 
 336-667 
 
 90 
 
 91 
 
 337-037 
 
 337-407 
 
 337-778 
 
 338-148 
 
 338-519 
 
 338-889 
 
 339-259 
 
 339-630 
 
 340- 
 
 340-370 
 
 91 
 
 92 
 
 340-741 
 
 341-111 
 
 341-481 
 
 341-852 
 
 342-222 
 
 342-593 
 
 342-963 
 
 343-333 
 
 343-704 
 
 344-074 
 
 92 
 
 93 
 
 344-444 
 
 344-815 
 
 345-185 
 
 345556 
 
 345-926 
 
 346-296 
 
 346-667 
 
 347-037 
 
 347-407 
 
 347-778 
 
 93 
 
 94 
 
 348-148 
 
 348-519 
 
 348-889 
 
 349-259 
 
 349-630 
 
 350- 
 
 350'370 
 
 350-741 
 
 351-111 
 
 351-481 
 
 94 
 
 95 
 
 351-852 
 
 352-222 
 
 352-593 
 
 352-963 
 
 353-333 
 
 353-704 
 
 354-074 
 
 354-444 
 
 354-815 
 
 355-185 
 
 95 
 
 96 
 
 355-556 
 
 355-926 
 
 356-296 
 
 356-667 
 
 357-037 
 
 357-407 
 
 357-778 
 
 358-148 
 
 358-519 
 
 358-889 
 
 96 
 
 97 
 
 359-259 
 
 359-630 
 
 360- 
 
 360-37C 
 
 360-741 
 
 361-111 
 
 361-481 
 
 361-852 
 
 362-222 
 
 362-593 
 
 97 
 
 98 
 
 362-963 
 
 363-333 
 
 363-704 
 
 364-074 
 
 364-444 
 
 364-815 
 
 365-185 
 
 365-556 
 
 365-926 
 
 366-296 
 
 98 
 
 99 
 
 366-667 
 
 367-037 
 
 367-407 
 
 367-778 
 
 368-148 
 
 368-519 
 
 368-889 
 
 369-259 
 
 369-630 
 
 370- 
 
 99 
 
 100 
 
 370-370 
 
 370-741 
 
 371.111 
 
 371-481 
 
 371-852 
 
 372-222 
 
 372-593 
 
 372-963 
 
 373-333 
 
 373-704 
 
 100 
 
 101 
 
 374-074 
 
 374-444 
 
 374-815 
 
 375-185 
 
 375-556 
 
 375-926 
 
 376-296 
 
 376-667 
 
 377-037 
 
 377-407 
 
 101 
 
 102 
 
 377-778 
 
 378-148 
 
 378-519 
 
 378-889 
 
 379-259 
 
 379-630 
 
 380- 
 
 380-370 
 
 380-741 
 
 381-111 
 
 102 
 
 103 
 
 381-481 
 
 381-852 
 
 382-222 
 
 382593 
 
 382-963 
 
 383-333 
 
 383-704 
 
 384-074 
 
 384-444 
 
 384-815 
 
 103 
 
 104 
 
 385185 
 
 385-55b 
 
 385-926 
 
 386-296 
 
 386-667 
 
 387-037 
 
 387-407 
 
 387-778 
 
 388-148 
 
 388-519 
 
 104 
 
 105 
 
 388-889 
 
 389-259 
 
 389-630 
 
 390- 
 
 390-370 
 
 390-741 
 
 391-111 
 
 391-481 
 
 391-852 
 
 392-222 
 
 105 
 
 106 
 
 392-593 
 
 392-963 
 
 393-333 
 
 393-704 
 
 394-074 
 
 394-444 
 
 394-815 
 
 395-185 
 
 395-556 
 
 395-926 
 
 106 
 
 107 
 
 390-296 
 
 396-667 
 
 397-037 
 
 397-407 
 
 397-778 
 
 398-148 
 
 398-519 
 
 398-889 
 
 399-259 
 
 399630 
 
 107 
 
 108 
 
 400- 
 
 400-370 
 
 400-741 
 
 401-111 
 
 401-481 
 
 401-852 
 
 402-222 
 
 402-593 
 
 402-963 
 
 403-333| 108 
 
 109 
 
 403-704 
 
 404-074 
 
 404-444 
 
 404-815 
 
 405-185 
 
 405-556 
 
 405-926 
 
 406-296 
 
 406-667 
 
 407-037 
 
 109 
 
 110 
 
 407-407 
 
 407-778 
 
 408-148 
 
 408-519 
 
 408-889 
 
 409-259 
 
 409-630 
 
 410- 
 
 410-370 
 
 410-741 
 
 110 
 
 111 
 
 411-111 
 
 411-481 
 
 411-852 
 
 412-222 
 
 412-593 
 
 412963 
 
 413-333 
 
 413-704 
 
 414-074 
 
 414-444 
 
 111 
 
 112 
 
 414-815 
 
 415-185 
 
 415-55C 
 
 415-926 
 
 416-296 
 
 416-667 
 
 417-037 
 
 417-407 
 
 417-778 
 
 418-148 
 
 112 
 
 113 
 
 418-519 
 
 418-889 
 
 419-259 
 
 419-600 
 
 420- 
 
 420-370 
 
 420-741 
 
 421-111 
 
 421-481 
 
 421-852 
 
 113 
 
 114 
 
 422-222 
 
 422-593 
 
 422-963 
 
 423-333 
 
 423-704 
 
 424-074 
 
 424-444 
 
 424-815 
 
 425-185 
 
 425-556 
 
 114 
 
 115 
 
 425-926 
 
 426-296 
 
 426-667 
 
 427-037 
 
 427-40- 
 
 427-778 
 
 428-148 
 
 428-519 
 
 428-889 
 
 429-259 
 
 115 
 
 116 
 
 429-630 
 
 430- 
 
 430-370 
 
 430-741 
 
 431-11 
 
 431-481 
 
 431-852 
 
 432-222 
 
 432-593 
 
 432-963 
 
 116 
 
 117 
 
 433-333 
 
 433-704 
 
 434-074 
 
 434-444 
 
 434-815 
 
 435-185 
 
 435-556 
 
 435-926 
 
 436-296 
 
 436-667 
 
 117 
 
 118 
 
 437-03" 
 
 437-407 
 
 437-778 
 
 458-148 
 
 438-519 
 
 438-889 
 
 439-259 
 
 439-630 
 
 440- 
 
 440-370 
 
 118 
 
 119 
 
 440-741 
 
 441-111 
 
 441-481 
 
 441-852 
 
 442-222 
 
 442-593 
 
 442-963 
 
 443-333 
 
 443-704 
 
 444-074 
 
 119 
 
 120 
 
 444-444 
 
 444-815 
 
 445-185 
 
 445-556 
 
 445-926 
 
 446-296 
 
 446-667 
 
 447-037 
 
 447-407 
 
 447-778 
 
 120 
 
 M.A 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 61 to 12O. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 175 
 
 CUBIC YARDS TO MEAN AREAS FOR 1OO FEET Iff LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 121 
 
 448-148 
 
 448-519 
 
 448-889 
 
 449259 
 
 449.630 
 
 450- 
 
 450-370 
 
 450-741 
 
 451-111 
 
 451-481 
 
 121 
 
 122 
 
 451-852 
 
 452-222 
 
 452-593 
 
 452-963 
 
 453-333 
 
 453-704 
 
 454-074 
 
 454-444 
 
 454-815 
 
 455-185 
 
 122 
 
 123 
 
 455-556 
 
 455-926 
 
 456-296 
 
 456-667 
 
 457-037 
 
 457-407 
 
 457-778 
 
 458-148 
 
 458-519 
 
 458-889 
 
 123 
 
 124 
 
 459-259 
 
 459-630 
 
 460- 
 
 400-370 
 
 460-741 
 
 461-111 
 
 461-481 
 
 461-852 
 
 462-222 
 
 462593 
 
 124 
 
 125 
 
 462-963 
 
 463-333 
 
 463-704 
 
 464-074 
 
 464-444 
 
 464-815 
 
 465-185 
 
 465-556 
 
 465-926 
 
 466-296 
 
 125 
 
 126 
 
 466-667 
 
 467-037 
 
 467-407 
 
 4H7-778 
 
 468-148 
 
 468-519 
 
 468-889 
 
 469-259 
 
 469-630 
 
 470- 
 
 12(1 
 
 127 
 
 470 370 
 
 470-741 
 
 471-111 
 
 471-481 
 
 471-852 
 
 472--222 
 
 472-593 
 
 472-963 
 
 473-333 
 
 473-704 
 
 127 
 
 128 
 
 474-074 
 
 474-444 
 
 474-815 
 
 475-185 
 
 475-556 
 
 475-926 
 
 476-296 
 
 476-667 
 
 477-037 
 
 477-407 
 
 128 
 
 129 
 
 477-778 
 
 478-148 
 
 478-519 
 
 478-889 
 
 479-259 
 
 479-630 
 
 480- 
 
 480-370 
 
 480-741 
 
 481-111 
 
 129 
 
 130 
 
 481-481 
 
 481-852 
 
 482-222 
 
 482593 
 
 482-963 
 
 483-333 
 
 483-704 
 
 484-074 
 
 484-444 
 
 484-815 
 
 130 
 
 131 
 
 485-185 
 
 485556 
 
 485-926 
 
 486-296 
 
 486-667 
 
 487-037 
 
 487-407 
 
 487-778 
 
 488-148 
 
 488-51 9 
 
 131 
 
 132 
 
 488-889 1 489-259 
 
 489-630 
 
 490- 
 
 490-370 
 
 490-741 
 
 491-111 
 
 491-481 
 
 491-852 
 
 492-222 
 
 132 
 
 133 
 
 492-593 
 
 492-963 
 
 493-333 
 
 493-704 
 
 494-074 
 
 494-444 
 
 494-815 
 
 495-185 
 
 495-656 
 
 495926 
 
 133 
 
 134 
 
 496-296 
 
 495-667 
 
 497-037 
 
 497-407 
 
 497-778 
 
 498-148 
 
 498-519 
 
 498-8S9 
 
 499-269 
 
 499-630 
 
 134 
 
 135 
 
 500- 
 
 500-370 
 
 500-741 
 
 501-111 
 
 501-481 
 
 601-852 
 
 502222 
 
 602-593 
 
 602-9B3 
 
 503-333 
 
 135 
 
 136 
 
 503-704 
 
 504-074 
 
 504444 
 
 504-815 
 
 605-185 
 
 605-556 
 
 505926 
 
 506-296 
 
 506-667 
 
 507-037 
 
 136 
 
 137 
 
 5D7-407 
 
 507-778 
 
 508-148 
 
 608-519 
 
 508-889 
 
 609-259 
 
 609630 
 
 610- 
 
 610-370 
 
 510-741 
 
 137 
 
 138 
 
 511-111 
 
 511-481 
 
 511-852 
 
 512-J-22 
 
 512-593 
 
 512-963 
 
 613-333 
 
 513-704 
 
 514-074 
 
 514-444 
 
 138 
 
 139 
 
 514-815 
 
 515-185 
 
 615-556 
 
 515-926 
 
 516-296 
 
 516-667 
 
 517-037 
 
 617-407 
 
 517778 
 
 518-148 
 
 139 
 
 140 
 
 518-519 
 
 518-889 
 
 519-259 
 
 519-630 
 
 520- 
 
 520-370 
 
 520-741 
 
 521-111 
 
 521-481 
 
 521-852 
 
 140 
 
 Ul 
 
 522-222 
 
 522-593 
 
 522-963 
 
 523-333 
 
 523-704 
 
 524-074 
 
 624-444 
 
 524-815 
 
 625-185 
 
 625-556 
 
 141 
 
 142 
 
 525-926 
 
 526-296 
 
 526-667 
 
 527-037 
 
 527-407 
 
 627-778 
 
 628-148 
 
 628-519 
 
 528-889 
 
 529-259 
 
 142 
 
 143 
 
 529-630 
 
 530- 
 
 530-370 
 
 630-741 
 
 531111 
 
 631-481 
 
 631-852 
 
 632-222 
 
 632-593 
 
 632-9C3 
 
 143 
 
 144 
 
 633-333 
 
 533-704 
 
 534-074 
 
 534-444 
 
 634-815 
 
 635-185 
 
 635-556 
 
 635926 
 
 636296 
 
 536-667 
 
 144 
 
 145 
 
 537-037 
 
 537-407 
 
 537-778 
 
 538148 
 
 538-519 
 
 538-889 
 
 539.259 
 
 639-030 
 
 640- 
 
 540-370 
 
 145 
 
 146 
 
 540-741 
 
 541-111 
 
 541-481 
 
 541-852 
 
 542-222 
 
 642-593 
 
 642-963 
 
 543333 
 
 643-704 
 
 544-074 
 
 146 
 
 147 
 
 544-444 
 
 544-815 
 
 545-185 
 
 545-556 
 
 545-926 
 
 646-296 
 
 546-667 
 
 547-037 
 
 547-407 
 
 547-77S 
 
 147 
 
 148 
 
 548-148 
 
 548-519 
 
 648-889 
 
 549-259 
 
 549-630 
 
 650- 
 
 550-370 
 
 650-741 
 
 651-111 
 
 551-481 
 
 148 
 
 149 
 
 551-852 
 
 552-222 
 
 552-593 
 
 652-963 
 
 553-333 
 
 553-704 
 
 554-074 
 
 654-444 
 
 554-815 
 
 655-185 
 
 149 
 
 150 
 
 555-556 
 
 555-926 
 
 656-296 
 
 556-667 
 
 557-037 
 
 657-407 
 
 657-778 
 
 658-148 
 
 658-519 
 
 658-889 
 
 150 
 
 151 
 
 559-259 
 
 559-630 
 
 560- 
 
 560-370 
 
 560-741 
 
 661-111 
 
 661-481 
 
 561-862 
 
 662-222 
 
 662-693 
 
 151 
 
 152 
 
 562-963 
 
 563-333 
 
 663-704 
 
 564-074 
 
 664-444 
 
 664-815 
 
 665-185 
 
 565556 
 
 665-926 
 
 566-296 
 
 152 
 
 153 
 
 566-667 
 
 567-037 
 
 567-407 
 
 567-778 
 
 568-148 
 
 668-519 
 
 568-889 
 
 569-259 
 
 669-630 
 
 670- 
 
 153 
 
 154 
 
 570-370 
 
 570-741 
 
 571-111 
 
 571-481 
 
 671-852 
 
 672-222 
 
 672-593 
 
 672-963 
 
 573-333 
 
 573704 
 
 154 
 
 155 
 
 574-074 
 
 574-444 
 
 574-815 
 
 575-185 
 
 675-556 
 
 675-926 
 
 676-296 
 
 676-607 
 
 677 037 
 
 577-407 
 
 165 
 
 156 
 
 577-778 
 
 578-148 
 
 578-519 
 
 678-889 
 
 579-259 
 
 679-630 
 
 MO 
 
 580-370 
 
 580-741 
 
 581-111 
 
 156 
 
 157 
 
 681-481 
 
 581-852 
 
 582-222 
 
 582-593 
 
 582-963 
 
 583333 
 
 683-704 
 
 684-074 
 
 684-444 
 
 684-815 
 
 157 
 
 158 
 
 585-185 
 
 585-556 
 
 585-926 
 
 586-296 
 
 586-667 
 
 5S7-037 
 
 587-407 
 
 587-778 
 
 588-148 
 
 588-519 
 
 158 
 
 159 
 
 588-889 
 
 589-259 
 
 589-630 
 
 590- 
 
 590-370 
 
 690-741 
 
 691-111 
 
 691-481 
 
 691-852 
 
 592-222 
 
 159 
 
 160 
 
 592-593 
 
 592-963 
 
 593*333 
 
 593-704 
 
 594 074 
 
 594-444 
 
 594815 
 
 595-185 
 
 595-556 
 
 595-926 
 
 160 
 
 161 
 
 596-296 
 
 596-667 
 
 597-037 
 
 597-407 
 
 597-778 
 
 598-148 
 
 698-519 
 
 698-889 
 
 599-259 
 
 699630 
 
 161 
 
 162 
 
 600- 
 
 600-370 
 
 600-741 
 
 601-111 
 
 601-4S1 
 
 601-852 
 
 602-22'J 
 
 602-593 
 
 6U2-963 
 
 603-333 
 
 162 
 
 163 
 
 603-704 
 
 604-074 
 
 604-444 
 
 604-815 
 
 605-185 
 
 605-556 
 
 C05-926 
 
 606-296 
 
 606-667 
 
 607-037 
 
 163 
 
 164 
 
 607-407 
 
 607-778 
 
 608-148 
 
 608-519 
 
 608-889 
 
 609-259 
 
 609-630 
 
 610- 
 
 610-370 
 
 610-741 
 
 164 
 
 165 
 
 611-111 
 
 611-481 
 
 611-852 
 
 612-222 
 
 612-593 
 
 612-963 
 
 613-3*33 
 
 613-704 
 
 614-074 
 
 614-444 
 
 165 
 
 166 
 
 614-815 
 
 615-185 
 
 615556 
 
 615-926 
 
 616-296 
 
 616-667 
 
 617-037 
 
 617-407 
 
 617-778 
 
 618-148 
 
 166 
 
 167 
 
 618519 
 
 618-889 
 
 619 259 
 
 619-630 
 
 620- 
 
 620-370 
 
 620-741 
 
 621-111 
 
 621-481 
 
 621-852 
 
 167 
 
 168 
 
 622 222 
 
 622-593 
 
 622-963 
 
 623-333 
 
 62:1-704 
 
 624-074 
 
 624444 
 
 624-815 
 
 625-185 
 
 625556 
 
 168 
 
 169 
 
 625-926 
 
 626-296 
 
 626-667 
 
 627 -037 
 
 627-407 
 
 627-778 
 
 628-148 
 
 628-519 
 
 628-889 
 
 629-259 
 
 169 
 
 170 
 
 629-630 
 
 630- 
 
 630-370 
 
 630741 
 
 631-111 
 
 631-481 
 
 631-852 
 
 632-222 
 
 632-593 
 
 632-963 
 
 170 
 
 171 
 
 633-333 
 
 633-704 
 
 634-074 
 
 634-444 
 
 634-815 
 
 635-185 
 
 635-556 
 
 635-926 
 
 636-296 
 
 636-667 
 
 171 
 
 172 
 
 637-037 
 
 637-407 
 
 637-778 
 
 638-148 
 
 638-519 
 
 638-889 
 
 639-259 
 
 639-630 
 
 640- 
 
 640-370 
 
 172 
 
 173 
 
 640-741 
 
 641-111 
 
 641-481 
 
 641-852 
 
 642-222 
 
 642-593 
 
 642-963 
 
 643 333 
 
 643-704 
 
 644-074 
 
 173 
 
 174 
 
 644-444 
 
 644-815 
 
 645-185 
 
 645-556 
 
 645-926 
 
 646-296 
 
 646-667 
 
 647-037 
 
 647-407 
 
 647-778 
 
 174 
 
 175 
 
 648-148 
 
 648519 
 
 648-889 
 
 649-259 
 
 649-630 
 
 650- 
 
 650-370 
 
 650-741 
 
 651-111 
 
 651-481 
 
 175 
 
 176 
 
 651-852 
 
 652-222 
 
 652-593 
 
 652963 
 
 653-333 
 
 653-704 
 
 654-074 
 
 654-444 
 
 654-815 
 
 655-185 
 
 176 
 
 177 
 
 655-556 
 
 655-926 
 
 656-296 
 
 656-667 
 
 657-037 
 
 657-407 
 
 657-778 
 
 658-148 
 
 658-519 
 
 658-889 
 
 177 
 
 178 
 
 659-259 
 
 659-630 
 
 660- 
 
 660-370 
 
 660-741 
 
 661-111 
 
 661-481 
 
 661-852 
 
 662-222 
 
 662-593 
 
 178 
 
 179 
 
 662-963 
 
 663-S33 
 
 663-704 
 
 664-074 
 
 664-444 
 
 664-815 
 
 665-185 
 
 665-556 
 
 665-926 
 
 666-296 
 
 179 
 
 180 
 
 666-667 
 
 667-037 
 
 667-407 
 
 667-778 
 
 668-148 
 
 668 519 
 
 668-889 
 
 669-259 
 
 669-630 
 
 670- 
 
 180 
 
 M.A 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 121 to ISO. 
 
176 
 
 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR WO FEET IN LENGTH. 
 
 M.A. 
 
 O 
 
 1 
 
 til 
 
 a 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 181 
 
 670-370 
 
 670-741 
 
 671-111 
 
 671-481 
 
 71-852 
 
 672-222 
 
 672-593 
 
 672-963 
 
 673-333 
 
 673-704 
 
 181 
 
 182 
 
 674-074 
 
 674-444 
 
 674-815 
 
 675-185 
 
 675-556 
 
 675-926 
 
 676-296 
 
 676-667 
 
 677-037 
 
 677-407 
 
 182 
 
 183 
 
 677-778 
 
 678-148 
 
 678-519 
 
 678889 
 
 679259 
 
 679-630 
 
 680- 
 
 680-370 
 
 680-741 
 
 681-111 
 
 183 
 
 184 
 
 681-481 
 
 681-852 
 
 682-222 
 
 68-2-593 
 
 682-963 
 
 6-3-333 
 
 683-704 
 
 684-074 
 
 684-444 
 
 684-815 
 
 184 
 
 185 
 
 685-185 
 
 685-556 
 
 685-926 
 
 686-296 
 
 686-667 
 
 687-037 
 
 687-407 
 
 687-778 
 
 688-148 
 
 688-519 
 
 185 
 
 186 
 
 688-889 
 
 689-259 
 
 689 630 
 
 690- 
 
 690-370 
 
 690-741 
 
 691-111 
 
 91-481 
 
 691-852 
 
 692-222 
 
 186 
 
 187 
 
 692-593 
 
 692-963 
 
 693-333 
 
 693704 
 
 694 074 
 
 694444 
 
 694-815 
 
 695-185 
 
 695-556 
 
 695-926 
 
 187 
 
 188 
 
 696-296 
 
 696-667 
 
 697-037 
 
 697-407 
 
 697-778 
 
 698-148 
 
 69S-519 
 
 698-889 
 
 699-259 
 
 699-630 
 
 188 
 
 189 
 
 700- 
 
 700-370 
 
 700-741 
 
 701-111 
 
 701-481 
 
 701-852 
 
 702-22-2 
 
 702-593 
 
 702-963 
 
 703-333 
 
 189 
 
 190 
 
 703-704 
 
 704-074 
 
 704-444 
 
 704-815 
 
 705-185 
 
 705-556 
 
 705-926 
 
 706-296 
 
 706-667 
 
 707-037 
 
 190 
 
 191 
 
 707-407 
 
 707-778 
 
 708-148 
 
 708-519 
 
 708-889 
 
 709-259 
 
 709-630 
 
 710- 
 
 710370 
 
 710-741 
 
 191 
 
 192 
 
 711-111 
 
 711-481 
 
 711-852 
 
 712-222 
 
 712-593 
 
 712-96:^ 
 
 713-333 
 
 713-704 
 
 714-074 
 
 714-444 
 
 192 
 
 193 
 
 714-815 
 
 715-185 
 
 715-556 
 
 715-926 
 
 716-296 
 
 716-667 
 
 717-037 
 
 717-407 
 
 717-778 
 
 718-148 
 
 193 
 
 194 
 
 718-519 718-889 
 
 719 ; 259 
 
 719-630 
 
 720- 
 
 720-370 
 
 7'20-741 
 
 721-111 
 
 7'21-481 
 
 721-852 
 
 194 
 
 195 
 
 722-222 
 
 722-593 
 
 722-963 
 
 723-333 
 
 723-704 
 
 724-074 
 
 724-444 
 
 724-815 
 
 725-185 
 
 725-55fi 
 
 195 
 
 196 
 
 725-926 
 
 726-296 
 
 726-667 
 
 727037 
 
 727-407 
 
 727-778 
 
 728-148 
 
 728-519 
 
 728-F89 
 
 729-259 
 
 196 
 
 197 
 
 729-630 
 
 730- 
 
 730-370 
 
 730-741 
 
 731-111 
 
 731-481 
 
 731-852 
 
 732-222 
 
 732-593 
 
 732963 
 
 197 
 
 198 
 
 733-333 
 
 733-704 
 
 734-074 
 
 734-444 
 
 734815 
 
 735-185 
 
 735-556 
 
 735-926 
 
 736-296 
 
 736-667 
 
 198 
 
 199 
 
 737-037 
 
 737-407 
 
 737-778 
 
 738-148 
 
 738-519 
 
 738-889 
 
 739-259 
 
 739-630 
 
 740- 
 
 740-370 
 
 199 
 
 200 
 
 740-741 
 
 741-111 
 
 741-481 
 
 741-852 
 
 742-222 
 
 742-593 
 
 742-963 
 
 743333 
 
 743-704 
 
 744-074 
 
 200 
 
 201 
 
 744-444 
 
 744-815 
 
 745-185 
 
 745-556 
 
 745-926 
 
 746-296 
 
 746667 
 
 747-037 
 
 747-407 
 
 747-778 
 
 201 
 
 202 
 
 748-148 
 
 748-519 
 
 748-889 
 
 749-259 
 
 749-631 
 
 750- 
 
 750-370 
 
 750-741 
 
 751-111 
 
 751-481 
 
 202 
 
 203 
 
 751-852 
 
 752-222 
 
 752-593 
 
 752-963 
 
 753-333 
 
 753-704 
 
 754-074 
 
 754-444 
 
 754-815 
 
 755-185 
 
 203 
 
 204 
 
 755-556 
 
 755-926 
 
 756-296 
 
 756-667 
 
 757-037 
 
 757-407 
 
 757-778 
 
 758-148 
 
 758-519 
 
 758 889 
 
 204 
 
 205 
 
 759-259 
 
 759-630 
 
 760 
 
 760-370 
 
 760-741 
 
 761-111 
 
 761-481 
 
 761-852 
 
 762-222 
 
 76-2-593 
 
 205 
 
 206 
 
 762-963 
 
 763-333 
 
 763-704 
 
 764-074 
 
 764-444 
 
 764-815 
 
 765-185 
 
 765556 
 
 765-92P 
 
 766-296 
 
 206 
 
 207 
 
 766-667 
 
 767-037 
 
 767-407 
 
 767-778 
 
 768-148 
 
 768-519 
 
 768-889 
 
 769-259 
 
 769-63( 
 
 770- 
 
 207 
 
 208 
 
 770-370 
 
 770741 
 
 771-111 
 
 771-481 
 
 771-85-2 
 
 772-222 
 
 772-593 
 
 772-963 
 
 773-333 
 
 773-704 
 
 208 
 
 209 
 
 774 074 
 
 774-444 
 
 774-815 
 
 775-185 
 
 775-556 
 
 775-920 
 
 776-296 
 
 776-667 
 
 777-037 
 
 777-407 
 
 209 
 
 210 
 
 777-778 
 
 778-148 
 
 778-519 
 
 778-889 
 
 779-259 
 
 779-630 
 
 780- 
 
 780-370 
 
 780-741 
 
 781-111 
 
 210 
 
 211 
 
 781-481 
 
 781-852 
 
 782-222 
 
 782-593 
 
 782-963 
 
 783-333 
 
 783-704 
 
 784-074 
 
 784-444 
 
 784-815 
 
 '211 
 
 212 
 
 785-185 
 
 785-556 
 
 785-926 
 
 786-296 
 
 786-667 
 
 787-03' 
 
 787-40' 
 
 787-778 
 
 788-148 
 
 788-519 
 
 212 
 
 213 
 
 788-889 
 
 789-259 
 
 789-630 
 
 790- 
 
 790-370 
 
 790-74 
 
 791-11 
 
 791-481 
 
 791-85^ 
 
 792-22'- 
 
 213 
 
 214 
 
 792-593 
 
 792-963 
 
 793333 
 
 793-704 
 
 794-074 
 
 794-444 
 
 794-81o 
 
 795-185 
 
 795-556 
 
 795-9-20 
 
 214 
 
 215 
 
 796-296 
 
 796667 
 
 797-037 
 
 797-407 
 
 797-778 
 
 798-14 
 
 79S-519 
 
 798-889 
 
 799 259 
 
 799-630 
 
 215 
 
 216 
 
 800- 
 
 800-37( 
 
 800-741 
 
 801-111 
 
 801-481 
 
 801-85 
 
 802-22 
 
 802-593 
 
 802-963 
 
 8U3-333 
 
 216 
 
 217 
 
 803-704 
 
 804-074 
 
 804-444 
 
 804-815 
 
 805-185 
 
 80555 
 
 805-92 
 
 806-296 
 
 80606" 
 
 807-037 
 
 217 
 
 218 
 
 807-407 
 
 807-778 
 
 808-148 
 
 808-519 
 
 808-889 
 
 809-25 
 
 809-63 
 
 810- 
 
 810-370 
 
 810-741 
 
 218 
 
 219 
 
 811-111 
 
 811-481 
 
 811-852 
 
 812-222 
 
 812-593 
 
 812-96 
 
 813-33 
 
 813-704 
 
 814-074 
 
 814-444 
 
 219 
 
 220 
 
 814-815 
 
 815-185 
 
 815-556 
 
 815-926 
 
 816-296 
 
 816-66 
 
 817-03 
 
 817-407 
 
 817-77 
 
 818-148 
 
 220 
 
 221 
 
 818-519 
 
 818-889 
 
 819-259 
 
 819-630 
 
 820- 
 
 820-37 
 
 820-74 
 
 821-111 
 
 821-48 
 
 821-85'' 
 
 221 
 
 222 
 
 822-22- 
 
 822-593 
 
 822-963 
 
 823-333 
 
 823-704 
 
 824-07 
 
 824-44 
 
 824-815 
 
 825-18 
 
 825-55 
 
 222 
 
 223 
 
 825-926 
 
 826-29 
 
 826-667 
 
 827-037 
 
 827-407 
 
 827-77 
 
 828-14 
 
 828-519 
 
 828-88 
 
 829-259 
 
 223 
 
 224 
 
 829-630 
 
 830- 
 
 830-370 
 
 830-741 
 
 831-111 
 
 831-48 
 
 831-85 
 
 832-222 
 
 832-59 
 
 832-96 
 
 224 
 
 225 
 
 833-333 
 
 833-70 
 
 834-074 
 
 834-444 
 
 834-815 
 
 835-18 
 
 835-55 
 
 835-926 
 
 836-29 
 
 836-66 
 
 225 
 
 226 
 
 837-037 
 
 837-40 
 
 837-778 
 
 838-148 
 
 838-519 
 
 838-88 
 
 839-25 
 
 839-630 
 
 840- 
 
 840 37( 
 
 226 
 
 227 
 
 840-741 
 
 841-11 
 
 841-481 
 
 841-852 
 
 842-222 
 
 842-59 
 
 842-963 
 
 843-H33 
 
 843-70 
 
 844-07 
 
 227 
 
 228 
 
 844-444 
 
 844-81 
 
 845-185 
 
 845-556 
 
 845926 
 
 846-29 
 
 846-66 
 
 847-037 
 
 847 -JO 
 
 847-77 
 
 228 
 
 229 
 
 848-148 
 
 848-51 
 
 848-889 
 
 849-259 
 
 849-630 
 
 850- 
 
 850-37 
 
 850-741 
 
 851-11 
 
 851-48 
 
 229 
 
 230 
 
 851-852 
 
 852-22 
 
 852-593 
 
 852-963 
 
 853-333 
 
 85370 
 
 854-07 
 
 854-444 
 
 854-81 
 
 855-18 
 
 230 
 
 231 
 
 855-556 
 
 855-92 
 
 856-296 
 
 856-667 
 
 857-037 
 
 857-40 
 
 857-77 
 
 858-148 
 
 858-51 
 
 858-889 
 
 231 
 
 232 
 
 859259 
 
 859-63 
 
 860- 
 
 860-370 
 
 860-741 
 
 861-11 
 
 861-48 
 
 861-852 
 
 862-22 
 
 862-593 
 
 232 
 
 233 
 
 862-963 
 
 86333 
 
 863-704 
 
 864-074 
 
 864-444 
 
 864-81 
 
 865 18. 
 
 865-55G 
 
 865-92 
 
 866-296 
 
 233 
 
 234 
 
 866-667 
 
 867-03 
 
 867-407 
 
 867-778 
 
 868-148 
 
 868-51 
 
 868-8 
 
 869-259 
 
 869-63 
 
 870- 
 
 234 
 
 235 
 
 870-370 
 
 870-74 
 
 871-111 
 
 871-481 
 
 871-85 
 
 872-22 
 
 872-59 
 
 872-963 
 
 873-33 
 
 873-704 
 
 235 
 
 2:56 
 
 874-074 
 
 874-44 
 
 874-815 
 
 875-185 
 
 875-556 
 
 875-92 
 
 876-29 
 
 876-667 
 
 877-03 
 
 877-407 
 
 236 
 
 237 
 
 877-778 
 
 878-14 
 
 878-519 
 
 878-889 
 
 879-250 
 
 879-63 
 
 880- 
 
 880-370 
 
 880-74 
 
 881-111 
 
 237 
 
 238 
 
 881-481 
 
 881-85 
 
 882-222 
 
 882-593 
 
 882-96: 
 
 883-33 
 
 883-70 
 
 SS4-074 
 
 884-444 
 
 884-815 
 
 238 
 
 239 
 
 885-185 
 
 885-55 
 
 885-926 
 
 886-29e 
 
 886-667 
 
 887-03 
 
 887-40 
 
 887-778 
 
 888-14 
 
 888-51S 
 
 239 
 
 240 
 
 888-889 
 
 889-25 
 
 8S9-63C 
 
 890- 
 
 890-37C 
 
 890-74 
 
 891-11 
 
 891-481 
 
 891-85 
 
 892-222 
 
 240 
 
 M.A 
 
 O 
 
 1 
 
 ft 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 181 to 24O. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 m 
 
 CUBIC TARDS TO MEAN AREAS FOJl 1OO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 8 
 
 9 
 
 M.A. 
 
 241 
 
 892-593 892-963 
 
 893-333 
 
 893-704 
 
 894074 
 
 894-444 
 
 894-815 
 
 895-185 
 
 895-556 
 
 895-926 241 
 
 242 
 
 896-296 896-667 
 
 897-037 
 
 897-407 
 
 897-778 
 
 898-148 
 
 898-519 
 
 898-889 
 
 899-259 
 
 899-630 242 
 
 243 
 
 900- 900-370 
 
 900-741 
 
 901-111 
 
 901481 
 
 901-852 
 
 902-222 
 
 902-593 
 
 902-963 
 
 903-333 243 
 
 214 
 
 903-704 904-074 
 
 904-444 
 
 904-815 
 
 905-188 
 
 905-556 
 
 905-926 
 
 906-296 
 
 900-667 
 
 907-037 
 
 244 
 
 245 
 
 907-407 907-778 
 
 908-148 
 
 908-519 
 
 908-889 
 
 909-259 
 
 909-630 
 
 910- 
 
 910-370 
 
 910-741 
 
 245 
 
 246 
 
 911-111 911-4S1 
 
 911852 
 
 912-222 
 
 912-593 
 
 912-963 
 
 913-333 
 
 913-704 
 
 914-074 
 
 914-444 
 
 246 
 
 247 
 
 914-815 915-185 
 
 915-556 
 
 915-926 
 
 916-296 
 
 916-667 
 
 917-037 
 
 917-407 
 
 917-778 
 
 918-148 
 
 247 
 
 24* 
 
 918-519! 918-889 
 
 919-259 
 
 .919-630 
 
 920- 
 
 920-370 
 
 920741 
 
 921-111 
 
 921-481 
 
 921-852 
 
 248 
 
 249 
 
 922-222 
 
 922-593 
 
 922-963 
 
 923-3:33 
 
 923704 
 
 924-074 
 
 924-444 
 
 924-815 
 
 925-185 
 
 925-556 
 
 249 
 
 250 
 
 925-926 
 
 926-296 
 
 926-667 
 
 927-037 
 
 927-407 
 
 927-778 
 
 928-148 
 
 928519 
 
 928-889 
 
 929-259 
 
 250 
 
 251 
 
 929-630 
 
 930- 
 
 930-370 
 
 930-741 
 
 931-111 
 
 931-481 
 
 931-852 
 
 932-222 
 
 932-593 
 
 932-963 
 
 251 
 
 252 
 
 933-333 
 
 933-70-1 
 
 9:34-1(74 
 
 934-444 
 
 9:34-815 
 
 935-185 
 
 935-556 
 
 935-926 
 
 936-296 
 
 936-667 
 
 252 
 
 253 
 
 937-037 
 
 937-407 
 
 937-778 
 
 9.J8-148 
 
 938 519 
 
 938-889 
 
 939-259 
 
 939630 
 
 940- 
 
 940-370 
 
 2f3 
 
 254 
 
 940-741 
 
 911-111 
 
 941-481 
 
 941-852 
 
 942-222 
 
 942593 
 
 942-963 
 
 943-333 
 
 943-704 
 
 944074 
 
 254 
 
 255 
 
 944-444 944-815 
 
 945-185 
 
 945-556 
 
 945-926 
 
 946-296 
 
 946-667 
 
 947-037 
 
 947-407 
 
 947-778 
 
 255 
 
 256 
 
 948-148 j 948-519 
 
 948-889 
 
 949259 
 
 949-630 
 
 950- 
 
 950-370 
 
 950-741 
 
 951-111 
 
 951-481 
 
 256 
 
 257 
 
 951-852 i 952222 
 
 952-693 
 
 952-963 
 
 953-333 
 
 953704 
 
 954 074 
 
 954-444 
 
 954815 
 
 955-185 
 
 257 
 
 258 
 
 955-556 
 
 955-926 
 
 956-296 
 
 956 667 
 
 957-037 
 
 957-407 
 
 957-778 958-148 
 
 958-519 
 
 958-889 
 
 i>e ,Q 
 
 259 
 
 959-259 
 
 959-630 
 
 960- 
 
 960-370 
 
 960-741 
 
 96M11 
 
 961-481 
 
 961-852 
 
 962-222 
 
 962-593 
 
 259 
 
 260 
 
 962-963 
 
 963-333 
 
 963-704 
 
 964-074 
 
 964-444 
 
 964-815 
 
 965-185 
 
 965-556 
 
 965926 
 
 966-296 
 
 260 
 
 261 
 
 966-667 
 
 967-037 
 
 967-407 
 
 967-778 
 
 968-148 
 
 968519 
 
 968-889 
 
 969-259 
 
 9R9-630 
 
 970- 
 
 261 
 
 262 
 
 970-370 
 
 970-741 
 
 971-111 
 
 971-481 
 
 971-852 
 
 972-222 
 
 972-593 
 
 972963 
 
 973333 
 
 973-704 
 
 262 
 
 263 
 
 974-074 974 111 
 
 974-815 
 
 975-185 
 
 975-556 
 
 975-926 
 
 976-296 
 
 676-667 
 
 977-037 
 
 977-407 
 
 263 
 
 264 
 
 977-778 978148 
 
 978-519 
 
 978-889 
 
 979-259 
 
 979-630 
 
 980- 
 
 980-370 
 
 980-741 
 
 981-111 
 
 264 
 
 265 
 
 981-481 981-852 
 
 982-222 
 
 982-593 
 
 982-963 
 
 933-333 
 
 983-704 
 
 9K4-074 
 
 984-444 
 
 984-815 
 
 265 
 
 266 
 
 985-185 985-556 
 
 98.V926 
 
 986-296 
 
 986-667 
 
 9H7-037 
 
 9S7407 
 
 987778 
 
 988-148 
 
 988-519 
 
 266 
 
 267 
 
 988-889 ! 989-259 
 
 989630 
 
 990- 
 
 990-370 
 
 990-741 
 
 991-111 
 
 991-481 
 
 991-852 
 
 992-222 
 
 267 
 
 263 
 
 992-593 992-963 
 
 993-3:33 
 
 993-704 
 
 994-074 
 
 994-444 
 
 994-815 
 
 995-185 
 
 995-556 
 
 995-926 
 
 268 
 
 269 
 
 996-296' 996667 
 
 997-037 
 
 997-407 
 
 997-778 
 
 998-148 
 
 998-519 
 
 998889 
 
 999-259 
 
 999-630 
 
 269 
 
 270 
 
 1000- 
 
 1000-370 
 
 1000-741 
 
 001-111 
 
 1001-481 
 
 1001-852 
 
 1002222 
 
 1002-593 
 
 1002-963 
 
 1003-333 
 
 270 
 
 271 
 
 1003-704 
 
 1004-074 
 
 1004-444 
 
 1004-815 
 
 1005-185 
 
 1005-556 
 
 1005-926 
 
 1006-296 
 
 1006-667 
 
 1007-037 
 
 271 
 
 272 
 
 1007-407 
 
 1007-778 
 
 1008-148 
 
 1008-519 
 
 1008-889 
 
 1009-259 
 
 1009-630 
 
 1010- 
 
 1010-370 
 
 1010-741 
 
 272 
 
 273 
 
 1011-111 
 
 1011-481 
 
 1011-852 
 
 1012-222 
 
 1012-593 
 
 1012-963 
 
 1013*333 
 
 1013704 
 
 1014-074 
 
 1014-444 
 
 273 
 
 274 
 
 1014815 1015-1N5 
 
 1015556 
 
 1015-926 
 
 1016-296 
 
 1016-667 
 
 1017-037 
 
 1017-407 
 
 1017-778 
 
 101S-148 
 
 274 
 
 275 
 
 1018-51911018-889 
 
 1019-259 
 
 1019-630 
 
 1020- 
 
 102<)-37l 
 
 1020-741 
 
 1021-111 
 
 1021-481 
 
 1021-852 
 
 275 
 
 276 
 
 1022-222 
 
 1022-593 
 
 1022-963 
 
 1023-333 
 
 1023-704 
 
 1024-074 
 
 1024-444 
 
 1024-815 
 
 102;') -is: 
 
 1025-556 276 
 
 277 
 
 1025-926 
 
 1026-296 
 
 1026-667 
 
 1027-037 
 
 1027-407 
 
 1027-778 
 
 1028-148 
 
 1028-519 
 
 1028-8f ; 9 
 
 1029-259 
 
 277 
 
 278 
 
 1029-630 
 
 1030- 
 
 1030-370 
 
 1030-741 
 
 1031-111 
 
 1031 481 1031-852 
 
 1032-222 
 
 1032-593 
 
 1032-963 
 
 278 
 
 279 
 
 1033-333 
 
 10:53-704 
 
 1034-074 
 
 1034-444 
 
 1034-815 
 
 1035-185 
 
 1035-556 
 
 1035-926 
 
 1036-296 
 
 1036-667 
 
 279 
 
 2SO 
 
 1037-037 
 
 1037-407 
 
 1037-778 
 
 1038-148 
 
 1038-519 
 
 1038-889 
 
 1039-259 
 
 1039-630 
 
 1040- 
 
 1040-370 
 
 2SO 
 
 281 
 
 1040-741 
 
 1041-111 
 
 1041-481 
 
 1041-852 
 
 1042-222 
 
 1042-593 
 
 1042963 
 
 1043-333 
 
 1043-704 
 
 1044-074 
 
 281 
 
 282 
 
 1014444 
 
 1044-815 
 
 1045-185 
 
 1045-556 
 
 1045-926 1046-296 
 
 1046-667 
 
 1047-037 
 
 1047-407 
 
 1047-778 
 
 282 
 
 283 
 
 1048-148 
 
 1048-519 
 
 1048-889 
 
 1049-259 
 
 1049-630 1050- 
 
 ior,o-370 
 
 1050-741 
 
 1051-111 
 
 1051-481 
 
 283 
 
 284 
 
 1051-852 j 1052-222 
 
 1052-593 
 
 1052-963 
 
 1053-3:33 
 
 1053-704 
 
 1054-074 
 
 1054-444 
 
 1054-815 
 
 1055-1S5 
 
 284 
 
 285 
 
 1055-556 1055-926 
 
 I056-29d 
 
 1056-667 
 
 1057-037 
 
 1057-407 
 
 1057-778 
 
 1058-148 
 
 1058-519 
 
 1058-889 
 
 285 
 
 286 
 
 1059259 
 
 1059-630 
 
 1060- 
 
 1060-370 
 
 1060-741 
 
 1061-111 
 
 1061-481 
 
 1061-852 1062-222 
 
 1062-593 
 
 286 
 
 287 
 
 1062-963 
 
 1063-333 
 
 1063704 
 
 1064-074 
 
 1064-444 
 
 1064-815 
 
 1065-185 
 
 1005-556 
 
 1065-926 
 
 1066-296 
 
 287 
 
 288 
 
 1066-667 
 
 1067-037 
 
 10.J7407 
 
 1067-778 
 
 106S-148 
 
 1068-519 1068-889 
 
 1069259 
 
 1009-630 1070- 
 
 288 
 
 289 
 
 1070-370 
 
 1070-741 
 
 1071-111 
 
 1071-481 
 
 1071-852 
 
 1072-222 
 
 1072-593 1072-963 
 
 1073-333 1073-704 
 
 289 
 
 290 
 
 1074-074 
 
 1074-444 
 
 1074-815 
 
 1075-185 
 
 1075-556 
 
 1075-926 
 
 1076-296 
 
 1076667 
 
 1077037 
 
 1077-407 
 
 290 
 
 291 
 
 1077-778 
 
 1078-148 
 
 1078-519 
 
 1078-889 
 
 1079-259 
 
 1079-630 
 
 1080- 
 
 10*0-370 
 
 1080-741 
 
 108M11 
 
 291 
 
 292 
 
 1081-481 
 
 1081-852 
 
 1082-222 
 
 1082-593 
 
 1082-963 
 
 1083-333 
 
 1083-704 
 
 1084-074 
 
 1084-444 
 
 1084-815 
 
 292 
 
 293 
 
 1085-185 
 
 10H5-55b 
 
 1085-926 
 
 1086-296 
 
 1086-667 
 
 1087-037 
 
 1087-407 
 
 1087-778 
 
 1088-148 
 
 1088-519 
 
 293 
 
 294 
 
 1088-889 
 
 1089-259 
 
 1089-630 
 
 1090- 
 
 1090-370 
 
 1090-741 
 
 1091-111 
 
 1091-481 
 
 1091-852 
 
 1092-222 
 
 294 
 
 295 
 
 1092-593 
 
 1092-963 1093-333 
 
 1093-704 
 
 1094-074 
 
 1094-444 
 
 1094-815 
 
 1095-185 
 
 1095-550 
 
 1095-926 
 
 295 
 
 296 
 
 1096-296 
 
 1096-667 1097-037 
 
 1097-407 
 
 1097-778 
 
 1098-148 
 
 1098-519 
 
 1098-889 
 
 1099-259 
 
 1099-630 
 
 296 
 
 297 
 
 1100- 
 
 1100370 1100-741 
 
 1101111 
 
 1101-481 
 
 1101-852 
 
 1102-222 
 
 1102-593 
 
 1102-963 
 
 1103-333 
 
 297 
 
 298 
 
 1103-704 
 
 1104 074 '1104-444 
 
 1104-815 
 
 1105-185 
 
 1105-550 
 
 1105-924 
 
 1 106-296 
 
 1100-667 
 
 1107-037 
 
 298 
 
 299 
 
 1107-407 
 
 1107-77H 1 1108-148 
 
 110S-519 
 
 1108-889 
 
 1109-259 
 
 1109-63 
 
 1110- 
 
 1110-37( 
 
 1110-741 
 
 299 
 
 300 
 
 1111-111 
 
 1111-48111111-852 
 
 1112222 
 
 1112-593 
 
 1112-963 
 
 1113-333 
 
 1113-704 
 
 1114-074 
 
 1114-444 
 
 300 
 
 M.A.| '0 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 8 
 
 
 
 M.A. 
 
 MEAN AREAS 241 to 3OO. 
 
178 
 
 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR 1OO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 2 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 301 
 
 1114-815 
 
 1115-185 
 
 1115-556 
 
 1115-926 
 
 1116-296 
 
 1116 667 
 
 1117-037 
 
 1117-407 1117-778 
 
 1118-148 
 
 301 
 
 302 
 
 1118519 
 
 1118-889 
 
 1119-259 
 
 1119630 
 
 1120- 
 
 1120-370 
 
 1120-741 
 
 1121-111 1121-481 
 
 1121-852 
 
 302 
 
 303 
 
 1122-222 
 
 1122593 
 
 1122-963 
 
 1123-333 
 
 1123704 
 
 1124-074 
 
 1124-444 
 
 1124-815 
 
 1125-lbS 
 
 1125-556 
 
 303 
 
 304 
 
 1125-926 
 
 1126296 
 
 1126-667 
 
 1127-037 
 
 1127 407 
 
 1127-778 
 
 1128-148 
 
 1128-519 
 
 1128-889 
 
 1129-259 
 
 304 
 
 305 
 
 1129-630 
 
 1130- 
 
 1130370 
 
 1130-741 
 
 1131-111 
 
 1131-481 
 
 1131-852 
 
 1132-222 
 
 11 32-593 
 
 1132-9113 
 
 SOS 
 
 306 
 
 1133-33:5 
 
 1133-704 
 
 1134-074 
 
 1134-444 
 
 1134815 
 
 1135-185 
 
 1135-556 
 
 1135-926 
 
 1136-296 
 
 1136-667 
 
 306 
 
 307 
 
 1137-037 
 
 1137407 
 
 1137-778 
 
 1138-148 
 
 1138-519 
 
 1138-889 
 
 1139-259 
 
 1139-630 
 
 1140- 
 
 1140-370 
 
 307 
 
 308 
 
 1140 741| 1141-111 
 
 1141-481 
 
 1141-852 
 
 1142-222 
 
 1142-593 
 
 1142-963 
 
 1143-333 
 
 1143-704 
 
 1144-074 
 
 308 
 
 309 
 
 1144-444 1144-815 
 
 1145-1S5 
 
 1145-556 
 
 1145-926 
 
 1146-296 
 
 1146-667 
 
 1147-037 
 
 1147-407 
 
 1147 778 
 
 309 
 
 310 
 
 1148-148 
 
 1148-519 
 
 1148-889 
 
 1149-259 
 
 1149-630 
 
 1150- 
 
 1150-370 
 
 1150-741 
 
 1151-111 
 
 1151-481 
 
 310 
 
 311 
 
 1151-852 
 
 1152-222 
 
 1152-593 
 
 1152-963 
 
 1153-333 
 
 1153-704 
 
 1154-074 
 
 1154-444 
 
 1154-815 
 
 1155-185 
 
 311 
 
 312 
 
 1155-556 
 
 1155-926 
 
 1156-296 
 
 1156-667 
 
 1157-037 
 
 1157-407 
 
 1157-778 
 
 1158-148 
 
 1158-519 
 
 115S-8S9 
 
 312 
 
 313 
 
 1159259 
 
 1159-630 
 
 1160- 
 
 11 60-370 
 
 1160-741 
 
 1161-111 
 
 1161481 
 
 1161-852 
 
 1162-222 
 
 1162-593 
 
 313 
 
 314 
 
 1162-963 1163-333 
 
 1163-704 
 
 1164074 
 
 1164-444 
 
 1164-815 
 
 1165-185 
 
 1165-556 
 
 1165-926 
 
 1166-296 
 
 314 
 
 315 
 
 1166-66711167-037 
 
 1167-407 
 
 1167-778 
 
 1168-148 
 
 1168-519 
 
 1168-889 
 
 1169-259 
 
 1169-630 
 
 1170- 
 
 315 
 
 316 
 
 1170-370 
 
 1170-741 
 
 1171-111 
 
 1171-481 
 
 Ii7 1-852 
 
 1172-222 
 
 1172-593 
 
 1172-963 
 
 1173-333 
 
 1173-704 
 
 316 
 
 317 
 
 1174-074 
 
 1174-444 
 
 1174-815 
 
 1175-185 
 
 1175-556 
 
 1175-926 
 
 1176-296 
 
 1176-667 
 
 1177-037 
 
 1177-407 
 
 317 
 
 318 
 
 1177-778 
 
 1178-148 
 
 1178519 
 
 178-889 
 
 1179-259 
 
 1179-630 
 
 1180- 
 
 11SO-370 
 
 1180-741 
 
 1181-111 
 
 318 
 
 319 
 
 1181-481 
 
 1181-852 
 
 1182-222 
 
 1182-593 
 
 11829rt3 
 
 1183-333 
 
 1183-704 
 
 1184074 
 
 1184-444 
 
 1184-815 
 
 319 
 
 320 
 
 1185-185 
 
 1185-55(3 
 
 1185 926 
 
 1186-296 
 
 1186-667 
 
 1187-037 
 
 1187-407 
 
 1187-778 
 
 1188-148 
 
 1188-519 
 
 320 
 
 321 
 
 1188-889 
 
 1189-259 
 
 1189-630 
 
 1190- 
 
 1190-370 
 
 1190-741 
 
 1191-111 
 
 1191-581 
 
 1191-852 
 
 1192-222 
 
 321 
 
 322 
 323 
 
 1192593 
 1196-296 
 
 1192-903 
 1196-667 
 
 1193-333 
 1197-037 
 
 1193-704 
 1197-407 
 
 1194-074 
 1197-778 
 
 1194-444 
 1198-148 
 
 1194-815 
 1198-519 
 
 1195-185 1195-556 
 1198-889 1199-259 
 
 1195-926 
 1199-630 
 
 322 
 323 
 
 324 
 
 1200- 
 
 1200-370 
 
 1200-741 
 
 1201-111 
 
 1201-481 
 
 1201-852 
 
 1202-222 
 
 1202-593) 1202-963 
 
 1203-333 
 
 324 
 
 325 
 
 1203-704 
 
 1204-074 
 
 1204-444 
 
 1204-815 
 
 1205-185 
 
 1205-556 
 
 1205-926 
 
 1206-296 
 
 1206-667 
 
 1207-037 
 
 325 
 
 326 
 
 1207-407 
 
 1207-778 
 
 1208-148 
 
 1208-519 
 
 1208-889 
 
 1209-259 
 
 1209-630 
 
 1210- 
 
 1210-370 
 
 1210-741 
 
 3-26 
 
 327 
 
 1211-111 
 
 1211-481 
 
 1211-852 
 
 212222 
 
 1212-593 
 
 1212-963 
 
 1213-333 
 
 1213-704 
 
 1214-074 
 
 1214-444 
 
 327 
 
 328 
 
 1214-815 
 
 1215-185 
 
 1215-556 
 
 1215-926 
 
 1216-296 
 
 1216-667 
 
 1217-037 
 
 1217-407 
 
 1217-778 
 
 1218-148 
 
 328 
 
 329 
 
 1218-519 
 
 1218-889 
 
 1219-259 
 
 1219-630 
 
 1220- 
 
 1220-370 
 
 1220-741 
 
 1221-111 
 
 1221-481 
 
 1221-852 
 
 329 
 
 330 
 
 1222-222 
 
 1222-593 
 
 1222-963 
 
 1223-333 
 
 1223-704 
 
 1224-074 
 
 1224-444 
 
 1224-815 
 
 1225-185 
 
 1225-556 
 
 330 
 
 331 
 
 1225-926 
 
 1226-296 
 
 1226-667 
 
 1227-037 
 
 1227-407 
 
 1227-778 
 
 1228-148 
 
 1228-519 
 
 1228-889 
 
 1229-259 
 
 331 
 
 332 
 
 1229-630 
 
 1230- 
 
 1230-370 
 
 1230-741 
 
 1231-111 
 
 1231-481 
 
 1231-852 
 
 1232-222 
 
 1232-593 
 
 1232-963 
 
 332 
 
 333 
 
 1233-333 
 
 1233-704 
 
 1234-074 
 
 1234-444 
 
 1234-815 
 
 1235-185 
 
 1235-556 
 
 1235-926 
 
 1236-296 
 
 1236-667 
 
 333 
 
 334 
 
 1237-037 
 
 1237-407 
 
 1237*778 
 
 1238-148 
 
 1238-519 
 
 1238-889 
 
 1239-259 
 
 1239-630 
 
 1240- 
 
 1240-370 
 
 334 
 
 335 
 
 1240-741 
 
 1241-111 
 
 1241-481 
 
 1241-852 
 
 1242-222 
 
 1242-593 
 
 1242-963 
 
 124S-333 
 
 1243-704 
 
 1244074 
 
 335 
 
 336 
 
 1244-444 
 
 1244-815 
 
 1245-185 
 
 1245-556 
 
 1245-926 
 
 1246-296 
 
 1246-667 
 
 1247-037 
 
 1247-407 
 
 1247-778 
 
 336 
 
 337 
 
 1248-148 
 
 1248-519 
 
 1248-889 
 
 1249-259 
 
 1249-630 
 
 1250- 
 
 1250-370 
 
 1250-741 
 
 1251-111 
 
 1251-481 
 
 337 
 
 338 
 
 1251-852 
 
 1252222 
 
 1252-593 
 
 1252-963 
 
 1253-333 
 
 1253-704 
 
 1254-074 
 
 1254-444 
 
 1254-815 
 
 1255-185 
 
 338 
 
 339 
 
 1255-556 
 
 1255926 
 
 1256-296 
 
 1256-667 
 
 1257-037 
 
 1257-407 
 
 1257-778 
 
 1258-148 
 
 1258-519 
 
 1258-889 
 
 339 
 
 340 
 
 1259-259 
 
 1259-630 
 
 1260- 
 
 1260-370 
 
 1260-741 
 
 1261-111 
 
 1261-481 
 
 1261-852 
 
 1262-222 
 
 1262-593 
 
 340 
 
 341 
 
 1262-963 
 
 1263-333 
 
 1263-704 
 
 1264-074 
 
 1264-444 
 
 1264-815 
 
 1265-185 
 
 1265-556 
 
 1265-926 
 
 1266-296 
 
 341 
 
 342 
 
 1266-667 
 
 1267-037 
 
 1267-407 
 
 1267-778 
 
 1268-148 
 
 1268-519 
 
 1268-889 
 
 1269-259 
 
 1269-630 
 
 1270- 
 
 342 
 
 343 
 
 1270-370 
 
 1270-741 
 
 1271-111 
 
 1271-481 
 
 1271-852 
 
 1272-222 
 
 1272-593 
 
 1272-963 
 
 1273-333 
 
 1273-704 
 
 343 
 
 314 
 
 1274-074 
 
 1274-444 
 
 1274-815 
 
 1275-185 
 
 1275-556 
 
 1275-926 
 
 1276-296 
 
 1276-667 
 
 1277-037 
 
 1277-407 
 
 344 
 
 345 
 
 1277-778 
 
 1278-148 
 
 1278-519 
 
 1278-889 
 
 1279-259 
 
 1279630 
 
 1280- 
 
 1280-370 
 
 1280-741 
 
 1281-111 
 
 345 
 
 346 
 
 1281-481 
 
 1281-852 
 
 1282-222 
 
 1282-593 
 
 1282-963 
 
 1283-333 
 
 1283-704 
 
 1284-074 
 
 1284-444 
 
 1284-815 
 
 346 
 
 347 
 
 1285-185 
 
 1285-556 
 
 1285-926 
 
 1286-296 
 
 1286-667 
 
 1287-037 
 
 1287-407 
 
 1287-778 
 
 1288-148 
 
 1288-519 
 
 347 
 
 348 1288-889 
 
 1289-259 
 
 1289-630 
 
 1290- 
 
 1290-370 
 
 1290-741 
 
 1291-111 
 
 1291-481 
 
 1291-852 
 
 1292-222 
 
 348 
 
 349 
 
 1292-593 
 
 1292-963 
 
 1293-333 
 
 1293-704 
 
 1294-074 
 
 1294-444 
 
 1294-815 
 
 1295-185 
 
 1295-556 
 
 1295-926 
 
 349 
 
 350 
 
 1296-296 
 
 1296-667 
 
 1297-037 
 
 1297-407 
 
 1297-778 
 
 1298-148 
 
 1298-519 
 
 1298-889 
 
 1299-259 
 
 1299-630 
 
 350 
 
 351 
 
 1300- 
 
 1300370 
 
 1300-741 
 
 1301-111 
 
 1301-481 
 
 1301-852 
 
 1302-222 
 
 1302-593 
 
 1302-963 
 
 1303-333 
 
 351 
 
 352 
 
 1303-704 
 
 1304-074 
 
 1304-444 
 
 1304-815 
 
 1305-185 
 
 1305-556 
 
 1305-926 
 
 1306-296 
 
 1306-667 
 
 1307-037 
 
 352 
 
 353 
 
 1307-407 
 
 1307-778 
 
 1308-148 
 
 1308-519 
 
 1308-889 
 
 1309-259 
 
 1309-630 
 
 1310- 
 
 1310-370 
 
 1310-741 
 
 353 
 
 354 
 
 1311-111 
 
 1311-481 
 
 1311-852 
 
 1312-222 
 
 1312-593 
 
 1312-963 
 
 1313-333 
 
 1313-704 
 
 1314-074 
 
 1314-444 
 
 354 
 
 355 
 
 1314-815 
 
 1315-185 
 
 1315-556 
 
 1315-926 
 
 1316-296 
 
 1316-667 
 
 1317-037 
 
 1317-407 
 
 1317-778 
 
 1318-148 
 
 355 
 
 356 
 
 1318-519 
 
 1318-889 
 
 1319-259 
 
 1319-630 
 
 1320- 
 
 1320-370 
 
 1320-741 
 
 1321-111 
 
 1321-481 
 
 1321-852 
 
 356 
 
 357 
 
 1322-222 
 
 1322-593 
 
 1322-963 
 
 1323-333 
 
 1323-704 
 
 1324-074 
 
 1324-444 
 
 1324-815 
 
 1325-185 
 
 1325-556 
 
 357 
 
 358 
 
 1325-926 
 
 1326-296 
 
 1326-667 
 
 1327-037 
 
 1327-407 
 
 1327-778 
 
 1328-148 
 
 1328-519 
 
 1328-889 
 
 1329-259 
 
 358 
 
 359 
 
 1329-630 
 
 1330- 
 
 1330-370 
 
 1330-741 
 
 13X1-111 
 
 1331-481 
 
 1331-852 
 
 1332-222 
 
 1332-593 
 
 1332-963 
 
 359 
 
 360 
 
 1333-333 
 
 1333-704 
 
 1334-074 
 
 1334-444 
 
 1334-815 
 
 1335-185 
 
 1335-556 
 
 1335-926 
 
 1336-296 
 
 1336-667 
 
 360 
 
 M.A. 
 
 O 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 3O1 to 36O. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 179 
 
 CUBIC YARDS TO MEAN AREAS FOR 1OO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A 
 
 361 
 
 1337-037 1337-407 
 
 1337-778 
 
 1338-148 
 
 1338-519 
 
 1338-889 
 
 1339259 
 
 1339-630 
 
 1340- 
 
 1340-370 
 
 361 
 
 362 
 
 1340-741 11341-111 
 
 1341481 
 
 1341-852 
 
 1342-222 
 
 1342-593 
 
 1342-WJ:! 
 
 1343-333 
 
 1343-704 
 
 1344-074 
 
 362 
 
 363 
 
 1344-444 1344-815 
 
 1345-185 
 
 1345-556 
 
 1345-926 
 
 1346-296 
 
 1346-667 
 
 1347-037 
 
 1347-407 
 
 1347-778 
 
 363 
 
 304 
 
 1348-148 1348-519 
 
 1348-889 
 
 1349-259 
 
 1349-630 
 
 1350- 
 
 1350-370 
 
 1350-741 
 
 1331-111 
 
 1351-481 
 
 364 
 
 365 
 
 1351-852 1 352-2-22 
 
 1352-593 
 
 1352-963 
 
 1353-333 
 
 1353-704 
 
 1354-074 
 
 1354-444 
 
 1354-815 
 
 1355-185 
 
 365 
 
 366 
 
 1355-556 1355-926 
 
 1356-296 
 
 1356667 
 
 1357-037 
 
 1357-407 
 
 1357778 
 
 1358-148 
 
 1358 519 
 
 1358-889 
 
 366 
 
 367 
 
 1359-259 1359-630 
 
 1360- 
 
 1360-370 
 
 1360741 
 
 1361-111 
 
 1361481 
 
 1361-852 
 
 1362-222 
 
 1362593 
 
 367 
 
 368 
 
 1362-963 
 
 1303-333 
 
 1363-704 
 
 1364-074 
 
 1364-444 
 
 1364-815 
 
 1365185 
 
 1365-556 
 
 1365926 
 
 1366-296 
 
 368 
 
 369 
 
 1366-667 
 
 1367-037 
 
 1367-407 
 
 1367-778 
 
 I3n8-148 
 
 1368519 
 
 1368-889 
 
 1369259 
 
 1369-630 
 
 1370- 
 
 369 
 
 370 
 
 1370-370 
 
 1370-741 
 
 1371-111 
 
 1371-481 
 
 1371-852 
 
 1372-222 
 
 1372-593 
 
 1372963 
 
 1373-333 
 
 1373-704 
 
 370 
 
 371 
 
 1374-074 
 
 1374-444 
 
 1374-815 
 
 1375-185 
 
 1375-556 
 
 1375-926 
 
 1376-296 
 
 1376-667 
 
 1377-037 
 
 1377-407 
 
 371 
 
 372 
 
 1377778 
 
 1378-148 
 
 1378-519 
 
 1378.8S9 
 
 1379-259 
 
 1379630 
 
 1380- 
 
 1380-370 
 
 1380-741 
 
 1381-111 
 
 372 
 
 373 
 
 1381-481 
 
 1381-852 
 
 13V2-222 
 
 1382-593 
 
 1382-963 
 
 1383-333 
 
 1383-704 
 
 1384-074 
 
 1384-444 
 
 1384-815 
 
 373 
 
 374 
 
 1385-185 
 
 1385-556 
 
 1385-926 
 
 1386-296 
 
 1386-667 
 
 1387-037 
 
 387-407 
 
 1387-778 
 
 1388148 
 
 1388-519 
 
 374 
 
 375 
 
 138S-889 1389-259 
 
 1389-630 
 
 1390- 
 
 1390-370 
 
 1390-741 
 
 1391-111 
 
 1391-481 
 
 1391-852 
 
 1392-222 
 
 375 
 
 376 
 
 1392-593 
 
 L3921KI8 
 
 1393-333 
 
 1393-70* 
 
 1394-074 
 
 1394-444 
 
 1394-815 
 
 1395185 
 
 1395-556 
 
 1395-926 
 
 376 
 
 377 
 
 1396-296 
 
 1396-667 
 
 1397-037 
 
 1397-407 
 
 1397-778 
 
 1398-148 
 
 139S-519 
 
 1398-889 
 
 1399-259 
 
 1399-630 
 
 377 
 
 378 
 
 1400- 
 
 1400-370 
 
 1400-741 
 
 1401-111 
 
 1401-481 
 
 1401-852 
 
 1402-222 
 
 1402-593 
 
 1402-963 
 
 1403-333 
 
 378 
 
 379 
 
 1403-704 
 
 1404-074 
 
 1404-444 
 
 1404-815 
 
 1 405- IK.", 
 
 1405-556 
 
 1405-926 
 
 1406-296 
 
 1406-667 
 
 1407-037 
 
 379 
 
 380 
 
 1407-407 
 
 1407-778 
 
 1408-148 
 
 1408-519 
 
 1408-889 
 
 1409-259 
 
 1409-630 
 
 1410- 
 
 1410-370 
 
 1410-741 
 
 380 
 
 381 
 
 1411-111 
 
 1411-481 
 
 1411-852 
 
 1412-222 
 
 1412-593 
 
 1412-963 
 
 1413-333 
 
 1413-704 
 
 1414-074 
 
 1414-444 
 
 381 
 
 382 
 
 1414-815 
 
 1415-185 
 
 1415-556 
 
 1415-926 
 
 1416-296 
 
 1416-667 
 
 1417-037 
 
 1417-407 
 
 1417-778 
 
 1418-148 
 
 382 
 
 3X3 
 
 1418-519 
 
 1418-889 
 
 1419-259 
 
 1419-630 
 
 1420- 
 
 1420-370 
 
 1420-741 
 
 1421-111 
 
 1421-481 
 
 1421-852 
 
 383 
 
 384 
 
 1422-222 
 
 1422-593 
 
 1422-963 
 
 1423-333 
 
 1423-704 
 
 1424074 
 
 1424-444 
 
 1424-815 
 
 1425-185 
 
 1425-556 
 
 384 
 
 ;>>*;> 
 
 1425-926 
 
 1426-296 
 
 1426-667 
 
 1427-037 
 
 1427-407 
 
 1427-778 
 
 1428-148 
 
 1428-519 
 
 1428-889 
 
 1429-259 
 
 385 
 
 386 
 
 1429-630 
 
 1430- 
 
 1430370 
 
 1430-741 
 
 1431-111 
 
 1431-481 
 
 1431 852 
 
 1432-222 
 
 1432-593 
 
 1432963 
 
 386 
 
 387 
 
 14313-333 
 
 1433704 
 
 1434074 
 
 1434-444 
 
 1434-815 
 
 1435-185 
 
 1435-556 
 
 1435-926 
 
 1436-296 
 
 1436-667 
 
 387 
 
 388 
 
 1437-037 
 
 1437407 
 
 1437-778 
 
 1438-148 
 
 1438-519 
 
 1438-889 
 
 1439-259 
 
 1439-630 
 
 1440- 
 
 1440-370 
 
 388 
 
 3S9 
 
 1440-741 
 
 1441-111 
 
 1441-481 
 
 1441-852 
 
 1442-222 
 
 1442-593 
 
 1442-963 
 
 1443-333 
 
 1443-704 
 
 1444-074 
 
 389 
 
 390 
 
 1444-444 
 
 1444-815 
 
 1445-185 
 
 1445-556 
 
 1445926 
 
 1446296 
 
 1446667 
 
 1447-037 
 
 1447-407 
 
 1447-778 
 
 390 
 
 301 
 
 1448-148 
 
 1448-519 
 
 1448:889 
 
 1449-259 
 
 1449-630 
 
 1450- 
 
 1450-370 
 
 1450-741 
 
 1451-111 
 
 1451-481 
 
 391 
 
 392 
 
 1451-852 
 
 1468*828 
 
 1452-593 
 
 1452-963 
 
 1453-333 
 
 1453-704 
 
 1454-074 
 
 1454-444 
 
 1454-815 
 
 1455-185 
 
 592 
 
 393 
 
 1455-556 
 
 1455-926 
 
 1456296 
 
 1456-667 
 
 1457*039 
 
 1457-407 
 
 1457-77S 
 
 1458-148 
 
 1458-519 
 
 1458889 
 
 393 
 
 394 
 
 1459-259 
 
 1459-630 
 
 1400- 
 
 1460-370 
 
 1460-741 
 
 1461-111 
 
 1461-481 
 
 1461-852 
 
 1462-222 
 
 1462-593 
 
 394 
 
 899 
 
 1462-963 
 
 1463-333 
 
 1463-704 
 
 1464-074 
 
 1464-444 
 
 1464-815 
 
 1465-185 
 
 1465-556 
 
 1465-926 
 
 1466-296 
 
 395 
 
 396 
 
 1466-667 
 
 1467-037 
 
 1467-407 
 
 1467-778 
 
 1468148 
 
 1468-519 
 
 1468-889 
 
 1469-259 
 
 1469-630 
 
 1470- 
 
 396 
 
 397 
 
 1470-370 
 
 1470-741 
 
 1471-111 
 
 1471-481 
 
 1471-852 
 
 1472-222 
 
 1 472-59:1 
 
 1472-903 
 
 1473-333 
 
 1473-704 
 
 397 
 
 398 
 
 1474-074 
 
 1474-444 
 
 1474-815 
 
 1475-185 
 
 1475-556 
 
 1475-926 1476-296 
 
 1476-6G7 
 
 1477-037 
 
 1477-407 
 
 398 
 
 399 
 
 1477-778 
 
 1478-148 
 
 1478-519 
 
 1478-889 
 
 1479-259 
 
 1479-630 11480- 
 
 1480-370 
 
 1480-741 
 
 1481-111 
 
 399 
 
 400 
 
 1481-481 
 
 1481-852 
 
 1482-222 
 
 1482-593 
 
 1482-963 
 
 1483-333 
 
 1483-704 
 
 1484-074 
 
 1484-444 
 
 1484-815 
 
 400 
 
 401 
 
 1485-185 
 
 1485-556 
 
 1485-926 
 
 1486-296 
 
 1486-667 
 
 1487-037 
 
 1487-407 
 
 1487-778 
 
 1488-148 
 
 1488-519 
 
 401 
 
 402 
 
 1488-889 
 
 1489-259 
 
 1489-630 
 
 1490- 
 
 1490-370 
 
 1490-741 
 
 1491-111 
 
 1491-481 
 
 1491-862 
 
 1492-222 
 
 402 
 
 403 
 
 1492-593 
 
 1492-903 
 
 1493-333 
 
 1493-704 
 
 1494-074 
 
 1494-444 
 
 1494-815 
 
 1495-185 
 
 1495-556 
 
 1495-926 
 
 403 
 
 404 
 
 1496-296 
 
 1496-667 
 
 1497-037 
 
 1497-407 
 
 1497-778 
 
 1498-148 
 
 U 98-51 9 
 
 1498-889 
 
 1499-259 
 
 1499-630 
 
 404 
 
 405 
 
 1500- 
 
 1500-370 
 
 1500-741 
 
 1501-111 
 
 1501-481 
 
 1501-852 
 
 1502-222 
 
 1502-593 
 
 1502-963 
 
 1503-333 
 
 405 
 
 406 
 
 1503-704 
 
 1504-074 
 
 1504-444 
 
 1504-815 
 
 1505-185 
 
 1505-556 
 
 1505-926 
 
 1506-296 
 
 1506-667 
 
 1507-037 
 
 406 
 
 407 
 
 1507-407 
 
 1507778 
 
 1508- 14S 
 
 1508-519 
 
 1508-889 
 
 1509-259 
 
 1509-630 
 
 1510- 
 
 1610-370 
 
 1510-741 
 
 407 
 
 408 
 
 1511-111 
 
 1511-481 
 
 1511-852 
 
 1512-222 
 
 1512-593 
 
 1512-963 
 
 1513-333" 
 
 1513-704 
 
 1614-074 
 
 1514444 
 
 408 
 
 409 
 
 1514-815 
 
 1515-185 
 
 1515-556 
 
 1515 92C 
 
 1516-296 
 
 1516-667 
 
 1517-037 
 
 1517-407 
 
 1517-778 
 
 1518-148 
 
 409 
 
 410 
 
 1518-519 
 
 1518-889 
 
 1519-259 
 
 1519-630 
 
 1520- 
 
 1520-370 
 
 1520-741 
 
 1521-111 
 
 1521-481 
 
 1621-852 
 
 410 
 
 411 
 
 1522-222 
 
 1522-593 
 
 1522-963 
 
 1523-333 
 
 1523-704 
 
 1524-074 
 
 1524-444 
 
 1524-815 
 
 1525-185 
 
 1525-656 
 
 411 
 
 412 
 
 1525-926 
 
 1526-296 
 
 1526-667 
 
 1527-037 
 
 1527-407 
 
 1527-778 
 
 1528-148 
 
 1528-519 
 
 1528-889 
 
 1529-259 
 
 412 
 
 413 
 
 1529-630 
 
 1530- 
 
 1530-370 
 
 1530-741 
 
 1531-111 
 
 1531-481 
 
 15:;l-852 
 
 1532-222 
 
 1532-593 
 
 1532-963 
 
 413 
 
 414 
 
 1533-333 
 
 1533-704 
 
 1534074 
 
 1534-444 
 
 1534-815 
 
 1535-185 
 
 1535-556 
 
 1535-926 
 
 1536-296 
 
 1536-667 
 
 414 
 
 415 
 
 1537-037 
 
 1537-407 
 
 1537-778 
 
 1538-148 
 
 1538-519 
 
 1538-889 
 
 1539-259 
 
 1539-630 
 
 1540- 
 
 1540-370 
 
 415 
 
 416 
 
 1540-741 
 
 1541-111 
 
 1541-481 
 
 1541-852 
 
 1542-222 
 
 1542-593 
 
 1542-963 
 
 1543-333 
 
 1543-704 
 
 1544-074 
 
 416 
 
 417 
 
 1544-444 
 
 1544-815 
 
 1545-185 
 
 1545-556 
 
 1545-926 
 
 1546-296 
 
 1546667 
 
 1547-037 
 
 1547-407 
 
 1547-778 
 
 417 
 
 418 
 
 1548-14* 
 
 1548-619 
 
 1548-889 
 
 1549-259 
 
 1549-630 
 
 1550- 
 
 1550-370 
 
 1550-741 
 
 1.55M11 
 
 1551-481 
 
 418 
 
 419 
 
 1551-852 
 
 1 552- 222 1552-593 
 
 1fM2-%3 
 
 1553-333 
 
 1553-704 
 
 1554-074| 1554-444 
 
 1554-815 
 
 1555-185 
 
 419 
 
 420 
 
 155o-a56 
 
 ir,55-02rt l. r ).->6-2'J6 
 
 1556667 
 
 1557-037 
 
 1557-407 
 
 1557-778 
 
 1558-148 
 
 1558-519 
 
 1558-889 
 
 420 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 361 to 42O. 
 
180 
 
 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS J OR 1OO FEET IN LENGTH. 
 
 M.A. 
 
 | .i 
 
 a 
 
 3 1 * 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 l.A. 
 
 421 
 
 559-259 1559 630 
 
 1560- 
 
 1560-37011560-741 
 
 1561-111 
 
 561-481 
 
 1561-852 ! 1562-222 
 
 562-593 
 
 421 
 
 422 
 
 562-963 1563-333 
 
 1563-704 
 
 1564-074 11564-444 
 
 1564-815 
 
 1565-185 
 
 565-55611565-926 
 
 566-296 
 
 422 
 
 423 
 
 566-667 11567-037 
 
 1567-407 
 
 1507-778 1568-148 
 
 1568-519 
 
 568-889 
 
 569-259 '1569-630 
 
 570- 
 
 423 
 
 424 
 
 570 370 1 1570-741 
 
 571-111 
 
 1571-481 
 
 1571-852 
 
 1572-222 
 
 1572-593 
 
 1572-963 
 
 1573-333 
 
 573-704 
 
 424 
 
 425 
 
 574-07411574-444 
 
 574-815 
 
 1575-185 
 
 1575556 
 
 1575-926 
 
 576-296 
 
 576-667 
 
 1577-037 
 
 577-407 
 
 425 
 
 426 
 
 577-778 1578-148 
 
 1578-519 
 
 1578-889! 1579-259 
 
 1579-630 
 
 1580- 
 
 1580-370 1580-741 
 
 581-111 
 
 426 
 
 427 
 
 581-481 1581-852 
 
 5S2-222 
 
 1582-593 1582-963 
 
 1583-333 
 
 1583-704 
 
 584-074 
 
 1584-444 
 
 584-815 
 
 427 
 
 428 
 
 585-185 1585-556 
 
 585-926 
 
 1586-296 1586-667 
 
 1587-037 
 
 1587-407 
 
 1587-778 
 
 1588-148 
 
 1588-519 
 
 428 
 
 429 
 
 588-889 1589-259 
 
 589-630 
 
 1590- 
 
 1590-370 
 
 1590-741 
 
 1591-111 
 
 1591-481 
 
 1591-852 
 
 1592-222 
 
 429 
 
 430 
 
 592-593 
 
 1592-963 
 
 1593-333 
 
 1593-704 
 
 1594-074 
 
 1594-444 
 
 1594-815 
 
 1595-185 
 
 1595-556 
 
 1595-926 
 
 430 
 
 431 
 
 596296 
 
 1596-667 
 
 597-037 
 
 597-407 
 
 1597-778 
 
 1 598-1 4S 
 
 1598-519 
 
 1598-889 
 
 1599-259 
 
 1599-630 
 
 431 
 
 432 
 433 
 
 600- 1600-370 
 (303-704 1604-074 
 
 600-741 
 604-444 
 
 601-111 
 604-815 
 
 1001-481 
 1605-185 
 
 1601-852 
 1605-556 
 
 1602-222 
 1605-926 
 
 1602-593 1602-963 
 1606-296 1606-667 
 
 1603-333 
 607-037 
 
 432 
 433 
 
 434 
 
 607-407 j 1607-778 
 
 1608-148 
 
 608-519 
 
 1608-889 
 
 1609-259 
 
 1609630 
 
 1610- 
 
 1610-370 
 
 1610-741 
 
 434 
 
 435 
 
 611111 
 
 1611-481 
 
 1611-852 
 
 612-222 
 
 1612-593 
 
 1612-963 
 
 1613-333 
 
 1613-704 
 
 1614-074 
 
 1614-444 
 
 435 
 
 436 
 
 614-815 
 
 1615-185 
 
 615-556 
 
 615-926 
 
 1616-296 
 
 1616-667 
 
 K17-037 
 
 1617-407 
 
 1617-778 
 
 1618-148 
 
 436 
 
 437 
 
 618-519 
 
 1618-889 
 
 619-259 
 
 619 630 
 
 1620- 
 
 1620-370 
 
 1620-741 
 
 1621-111 
 
 1621-481 
 
 1621-852 
 
 437 
 
 438 
 
 622-222 
 
 1622-593 
 
 1622-963 
 
 623-333 
 
 1623-704 
 
 1624-074 
 
 1C24-444 
 
 1624-815 
 
 1625-185 
 
 1625-556 
 
 438 
 
 439 
 
 625-926 
 
 1626-296 
 
 1626-667 
 
 627-037 
 
 1627-407 
 
 1627-778 
 
 1628-148 
 
 1628-519 
 
 1628-889 
 
 1629-259 
 
 439 
 
 440 
 
 1629-630 
 
 1630- 
 
 1630-370 
 
 630-741 
 
 1631-111 
 
 1631-481 
 
 1631-852 
 
 1632-222 
 
 1632-593 
 
 1632-963 
 
 440 
 
 441 
 
 1633-333 
 
 1633-704 
 
 1634-074 
 
 634-444 
 
 1634-815 
 
 1635-185 
 
 1635-556 
 
 1635-926 
 
 1636-296 
 
 1636-667 
 
 441 
 
 442 
 
 1637-007 
 
 1637-407 
 
 637-778 
 
 638-148 
 
 1638-519 
 
 1638-889 
 
 1639-259 
 
 1639-630 
 
 1640- 
 
 1640-370 
 
 442 
 
 443 
 
 1640-741 
 
 1641-111 
 
 1641-481 
 
 641-852 
 
 1642-222 
 
 1642-593 
 
 1642-963 
 
 1643-333 
 
 1643-704 
 
 1644-074 
 
 443 
 
 444 
 
 1644-444 
 
 1644-815 
 
 .645-185 
 
 645-556 
 
 1645-926 
 
 1646-296 
 
 1646-667 
 
 1647-037 
 
 1647-407 
 
 1647-778 
 
 444 
 
 445 
 
 1648-148 
 
 1648-519 
 
 1648-889 
 
 649259 
 
 1649-630 
 
 1650- 
 
 1650-370 
 
 1650-741 
 
 1651-111 
 
 1651-481 
 
 445 
 
 446 
 
 1651-852 
 
 1652-222 
 
 1652-593 
 
 652-963 
 
 1653-333 
 
 1653-704 
 
 1654-074 
 
 1654-444 
 
 1654-815 
 
 1655-185 
 
 446 
 
 447 
 
 1655-556 
 
 1655-926 
 
 1656-296 
 
 656-667 1657-037 
 
 1657-407 
 
 1657-778 
 
 1658-148 
 
 1658-519 
 
 1658-889 
 
 447 
 
 448 
 
 1659-259 
 
 1659-630 
 
 1660- 
 
 1660-370' 1660-741 
 
 1661-111 
 
 1661-481 
 
 1661-852 
 
 1662-222 
 
 1662-593 
 
 448 
 
 449 
 
 1662-963 
 
 1663-333 
 
 1663-704 
 
 1664-074 
 
 1664-444 
 
 1664-815 
 
 1665-185 
 
 1665-556 
 
 1665-926 
 
 1666-296 
 
 449 
 
 450 
 
 1666-667 
 
 1667-037 
 
 1667-407 
 
 1667-778 
 
 1668-148 
 
 1668-519 
 
 1668-889 
 
 1669-259 
 
 1669-630 
 
 1670- 
 
 450 
 
 4-31 
 
 1670-370 
 
 1670-741 
 
 1671-111 
 
 1671-481 
 
 1671-852 
 
 1672-222 
 
 1672-593 
 
 1672-963 
 
 1673-333 
 
 1673-704 
 
 451 
 
 452 
 
 1674-074 
 
 1674-444 
 
 1674-815 
 
 1675-185 
 
 1675-550 
 
 1675-926 
 
 1676-296 
 
 1676-667 
 
 1677-037 
 
 1677-407 
 
 452 
 
 453 
 
 1677-778 
 
 1678-148 
 
 '1678-519 
 
 1678-889 
 
 1679-259 
 
 1679-630 
 
 1680- 
 
 1680-370 
 
 1680-741 
 
 1681-111 
 
 453 
 
 454 
 
 1681-481 
 
 1681-852 
 
 1682-222 
 
 1682-593 
 
 1682-963 
 
 1683-333 
 
 1683-704 
 
 1684-074 
 
 1 684-444 
 
 1684-815 
 
 454 
 
 455 
 
 1685-185 
 
 1685-556 
 
 1685-926 
 
 1686-296 
 
 1686-667 
 
 1687-037 
 
 1687-407 
 
 1687-778 
 
 1688-148 
 
 1688-519 
 
 455 
 
 456 
 
 1688-889 
 
 1689-259 
 
 1689-630 
 
 L690- 
 
 1690-370 
 
 1690-741 
 
 1691-111 
 
 1691-481 
 
 1691-852 
 
 1692-222 
 
 456 
 
 457 
 
 1692-593 
 
 1692-963 
 
 1693-333 
 
 1693-704 
 
 1694-074 
 
 1694-444 
 
 1694-815 
 
 1695-185 
 
 1695-556 
 
 1695-926 
 
 457 
 
 458 
 
 1696-296 
 
 1696-667 
 
 1697-037 
 
 1697-407 
 
 1697-778 
 
 1698-148 
 
 1698-519 
 
 1698-889 
 
 1699-259 
 
 1699-630 
 
 458 
 
 459 
 
 1700- 
 
 1700-370 
 
 1700-741 
 
 1701-111 
 
 1701-481 
 
 1701-852 
 
 1702-222 
 
 1702-593 
 
 1702-963 
 
 1703-333 
 
 459 
 
 460 
 
 1703-704 
 
 1704-074 
 
 1704-444 
 
 1704-815 
 
 1705-185 
 
 1705-556 
 
 1705-920 
 
 1706-296 
 
 1706-667 
 
 1707-037 
 
 460 
 
 461 
 
 1707-407 
 
 1707-778 
 
 1708-148 
 
 1708-519 
 
 1708-889 
 
 1709-259 
 
 1709-630 
 
 1710- 
 
 1710-370 
 
 1710-741 
 
 461 
 
 462 
 
 1711-111 
 
 1711-481 
 
 1711-852 
 
 1712-222 
 
 1712-593 
 
 1712-96b 
 
 1713-333 
 
 1713-704 
 
 1714-074 
 
 1714-444 
 
 462 
 
 463 
 
 1714-815 
 
 1715-185 
 
 1715-556 
 
 1715-926 
 
 1716-296 
 
 1716-667 
 
 1717-037 
 
 1717-407 
 
 1717-778 
 
 1718-148 
 
 463 
 
 464 
 
 465 
 
 1718-519 
 1722-222 
 
 1718-889 
 1722-593 
 
 1719-259 
 1722-963 
 
 1719-630 
 1723333 
 
 1720- 
 1723-704 
 
 1720-370 1720-741 
 1724-074; 1724-444 
 
 1721-111 
 1724-815 
 
 1721-481 
 1725-185 
 
 1721-852 
 1725-556 
 
 464 
 465 
 
 466 
 
 1725-926 
 
 1726-296 
 
 1726-667 
 
 1727-037 
 
 1727-407 
 
 1727-778 
 
 1728-148 
 
 1728-519 
 
 1728-889 
 
 1729-259 
 
 406 
 
 467 
 
 1729-630 
 
 1730- 
 
 1730-370 
 
 1730-741 
 
 1731-111 
 
 1731-481 
 
 1731-852 
 
 1732-222 
 
 1732-593 
 
 1732-963 
 
 467 
 
 468 
 
 1733-333 
 
 1733704 
 
 1734-074 
 
 1734-444 
 
 1734-815 
 
 1735-185 
 
 1735-556 
 
 1735-926 
 
 1736-296 
 
 1736-667 
 
 468 
 
 469 
 
 1737-037 
 
 1737-407 
 
 1737-778 
 
 1738-148 
 
 1738-519 
 
 1738-889 
 
 1739-259 
 
 1739-630 
 
 1740- 
 
 1740-370 
 
 469 
 
 470 
 
 1740-741 
 
 1741-111 
 
 1741-481 
 
 1741-852 
 
 1742-222 
 
 1742-593 
 
 1742-963 
 
 1743-333 
 
 1743-704 
 
 1744-074 
 
 470 
 
 471 
 
 1744-444 
 
 1744-815 
 
 1745-185 
 
 1745-556 
 
 1745-926 
 
 1746-296 
 
 1746-667 
 
 1747-037 
 
 1747-407 
 
 1747-778 
 
 471 
 
 472 
 
 1748-148 
 
 1748-519 
 
 1748-889 
 
 1749-259 
 
 1749-630 
 
 1750- 
 
 1750-370 
 
 1750-741 
 
 1751-111 
 
 1751-481 
 
 472 
 
 473 
 
 1751-852 
 
 1752-222 
 
 1752-593 
 
 1752-963 
 
 1753-333 
 
 1753-704 
 
 1754-074 
 
 1754-444 
 
 1754-815 
 
 1755-185 
 
 473 
 
 474 
 
 1755-556 
 
 1755-926 
 
 1756-296 
 
 1756-667 
 
 1757-037 
 
 1757-407 
 
 1757-778 
 
 1758-148 
 
 1758-519 
 
 1758-889 
 
 474 
 
 475 
 
 1759-259 
 
 1759-630 
 
 1760- 
 
 1760-370 
 
 1760-741 
 
 1761-111 
 
 1761-481 
 
 1761-852 
 
 1762-222 
 
 1762-593 
 
 475 
 
 476 
 
 1762-963 
 
 1763-333 
 
 1763-704 
 
 1764-074 
 
 1764-444 
 
 1764-815 
 
 1765-185 
 
 1765-556 
 
 1765-926 
 
 1766-296 
 
 476 
 
 477 
 
 1766-667 
 
 1767-037 
 
 1767-407 
 
 1767-778 
 
 1768-148 
 
 1768-519 
 
 1768-889 
 
 1769-259 11769-630 
 
 1770- 
 
 477 
 
 478 
 
 1770-370 
 
 1770-741 
 
 1771-111 
 
 1771-481 
 
 1771-852 
 
 1772-222 
 
 1772-593 
 
 1772-963 
 
 1773-333 
 
 1773-704 
 
 478 
 
 479 
 
 1774-074 
 
 1774-444 
 
 1774-815 
 
 1775-185 
 
 1775-556 
 
 1775-926 
 
 1776-296 
 
 1776-667 
 
 1777-037 
 
 1777-40" 
 
 479 
 
 480 
 
 1777-778 
 
 1778-148 
 
 1778-519 
 
 1778-889 
 
 1779-259 
 
 1779-630 
 
 1780- 
 
 1780-370 
 
 1780-741 
 
 1781-111 
 
 480 
 
 M.A 
 
 -O 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 421 to 48O. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 181 
 
 CUBIC YARDS TO MEAN AREAS FOR WO FEET IN LEXGTIT. 
 
 M.A. 
 
 
 
 1 
 
 2 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 .9 IM.A. 
 
 481 
 
 1781-481 
 
 1781-852 
 
 1782-222 
 
 1782-593 
 
 1782-963 
 
 1783-333 
 
 1783-704 
 
 1784-074 
 
 1784-444 
 
 1784-815 
 
 481 
 
 482 
 
 1785-185 
 
 1785-556 
 
 1785-926 
 
 1786296 
 
 1786-667 
 
 1787-037 
 
 1787-407 
 
 1787-778 
 
 1788-148 
 
 1788-519 
 
 482 
 
 483 
 
 1788-889 
 
 1789-259 
 
 1789630 
 
 1790- 
 
 1790-370 
 
 1790-741 
 
 1791-111 
 
 1791-481 
 
 1791-852 
 
 1792-222 
 
 483 
 
 484 
 
 1792-593 
 
 1792-963 
 
 1793-333 
 
 1793-704 
 
 1794-074 
 
 1794444 
 
 1794-815 
 
 1795-185 
 
 1795-556 
 
 1795-926 
 
 4^4 
 
 485 
 
 1796-296 
 
 1796-667 
 
 1797-037 
 
 1797-407 
 
 1797-778 
 
 1798148 
 
 1798-51911798-889 
 
 1799-259 
 
 1799-630 
 
 485 
 
 486 
 
 1800- 
 
 1800-370 
 
 1800-741 
 
 1801-111 
 
 1801-481 
 
 1801-852 
 
 1802-222 1802-593 
 
 1802 903 
 
 1803-333 
 
 486 
 
 487 
 
 1803-704 
 
 1804-074 
 
 1804-444 
 
 1804-815 
 
 1805-185 
 
 1805-556 
 
 1805926 
 
 1806-296 
 
 1806-607 
 
 1807-037 
 
 487 
 
 488 
 
 1807-407 
 
 1807-778 
 
 1808-148 
 
 1808-519 
 
 1808-889 
 
 1809-259 
 
 1809-630 
 
 1810- 
 
 1810-370 
 
 1810-741 
 
 488 
 
 489 
 
 1811-111 
 
 1811-481 
 
 1811-852 
 
 1812-222 
 
 1812-593 
 
 1812-903 
 
 1813-333 
 
 1813-704 
 
 1814-074 
 
 1814444 
 
 489 
 
 490 
 
 1814-815 
 
 1815-185 
 
 1815-556 
 
 1815926 
 
 1816-296 
 
 1816-667 
 
 1817-037 
 
 1817-407 
 
 1817-778 
 
 1818-148 
 
 490 
 
 491 
 
 1818-519 
 
 1818-889 
 
 1819-259 
 
 181963ft 
 
 1820- 
 
 1820-370 
 
 1820-741 
 
 1821-111 
 
 1821-481 
 
 1821-852 
 
 491 
 
 492 
 
 1822-222| 1822-593 
 
 18-22-963 
 
 1823333 
 
 1823-704 
 
 1824-074 
 
 1824-444 
 
 1824-815 
 
 1825-185 
 
 1825-556 
 
 492 
 
 493 
 
 1825-926 11826-296 
 
 1826-667 
 
 1827-037 
 
 1827-407 
 
 1827 778 
 
 1828-148 
 
 1828-619 
 
 1828-889 
 
 1829-259 
 
 493 
 
 494 
 
 1829-6301 1830- 
 
 1830-370 
 
 1830-741 
 
 1831-11111831-481 
 
 1831-852 
 
 1832 222 
 
 1832-593 
 
 1832963 
 
 494 
 
 495 
 
 1833-333 1833-704 
 
 1834074 
 
 1834-444 
 
 1834-815 l-:;.vis;, 
 
 1835-556 
 
 1835-926 
 
 1836-296 
 
 1836 667 
 
 495 
 
 496 
 
 1837-037 1837-407 
 
 1837-778 
 
 1838-148 
 
 1838-519 
 
 1838-88U 
 
 1839-259 
 
 1839-630 
 
 1840- 
 
 1840-370 
 
 496 
 
 497 
 
 1840-741 '1841-111 
 
 1841-481 
 
 1841-852 
 
 1842-222 
 
 1842-593 
 
 1842-963 
 
 1843-333 
 
 1843-704 
 
 1844074 
 
 4<7 
 
 498 
 
 1844-444;1844-815 
 
 1845-185 
 
 1845-556 
 
 1845-926 
 
 1846296 
 
 1846-667 
 
 1847-037 
 
 1847-407 
 
 1847-778 
 
 498 
 
 499 
 
 1848-148il848-519 
 
 1848-889 
 
 1849-259 
 
 1849-630 
 
 1850- 
 
 1850-370 
 
 1850-741 
 
 1851111 
 
 1851-481 
 
 499 
 
 500 
 
 1851-852 
 
 1852-222 
 
 1852-593 
 
 1852-963 
 
 1853-333 
 
 1853-704 
 
 1854-074 
 
 1854-444 
 
 1864-815 
 
 1855-185 
 
 500 
 
 501 
 
 1856-660 
 
 1855-926 
 
 1856-296 
 
 1856-667 
 
 1857-037 
 
 1857-407 
 
 1857-778 
 
 1868-148 
 
 1858-519 
 
 1858-889 
 
 501 
 
 502 
 
 1859-259! 1859-630 
 
 I860- 
 
 1860-370 
 
 1860-741 
 
 1861-111 
 
 1861-481 
 
 1861-852 
 
 1862-222 
 
 1862-593 
 
 502 
 
 503 
 
 1862-963 ,1863 333 
 
 1863-704 
 
 1864-074 
 
 li- 64-444 
 
 1864-815 
 
 186*186 
 
 1865-556 
 
 1865-926 
 
 1866-296 
 
 503 
 
 504 
 
 1866-667; 1867 '037 
 
 1867-407 
 
 1867-778 
 
 1868-148 
 
 1868-519 
 
 1868-889 
 
 1869-259 
 
 1869-630 
 
 1870- 
 
 504 
 
 505 
 
 187o-;j?i 1870741 
 
 1871-111 
 
 1871-481 
 
 1871-852 
 
 1872-222 
 
 1872-593 
 
 1872-9W 
 
 1873-333 
 
 1873-704 
 
 505 
 
 506 
 
 1874-074 1874-444 
 
 1874-815 
 
 1876-185 
 
 1875-556 
 
 1875-926 
 
 1876-296 
 
 1876-667 
 
 1877-037 
 
 1877-407 
 
 506 
 
 507 
 
 1877-778 1878-148 
 
 187S-519 
 
 1878-889 
 
 1879259 
 
 1879-630 
 
 1880- 
 
 1880-370 
 
 1880-741 
 
 1881-111 
 
 507 
 
 508 
 
 1881-481 1881-852 
 
 lss-2-j-j-j 
 
 18S2-593 
 
 ISBfrMfl 
 
 is*:;-:;:;:' 
 
 1883-704 
 
 L-sH-74 
 
 1 884-444 
 
 1884815 
 
 508 
 
 509 
 
 1885-185 1885-556 
 
 1-85-926 
 
 1886-296 
 
 18*6-667 
 
 1887-081 
 
 1887-407 
 
 1887-778 
 
 1888-148 
 
 1888-519 
 
 509 
 
 510 
 
 1888-8S9 
 
 1889-259 
 
 1889-630 
 
 1890- 
 
 1890 370 
 
 1890-741 
 
 1891-111 
 
 1891-481 
 
 1891-852 
 
 1892-222 
 
 510 
 
 511 
 
 1892-593 
 
 1892-963 
 
 1893-333 
 
 1803-704 
 
 1894-074 
 
 1894-444 
 
 1894-815 
 
 1895-185 
 
 1895-556 
 
 1895-926 
 
 511 
 
 612 
 
 1896-296 
 
 189*601 
 
 1897-037 
 
 1897-407 
 
 1897-77-S 
 
 1898-148 
 
 1898-519 
 
 1898-889 
 
 1899-259 
 
 1899-630 
 
 512 
 
 513 
 
 1900- 
 
 1900-370 
 
 1900-741 
 
 1901-111 
 
 1901-481 
 
 1901-852 
 
 1902-222 
 
 1902-593 
 
 1902-963 
 
 1903-333 
 
 513 
 
 514 
 
 1903-704 
 
 1904-074 
 
 1 904-444 
 
 1904-815 
 
 1905-185 
 
 1905-556 
 
 1905-926 
 
 1906-290 
 
 1906-667 
 
 1907-037 
 
 514 
 
 515 
 
 1907-407 
 
 1907-778 
 
 1908-148 
 
 1908-519 
 
 1908-889 
 
 1909-259 
 
 1909-630 
 
 1910- 
 
 1910-370 
 
 1910-741 
 
 515 
 
 510 
 
 1911-111 
 
 1911-481 
 
 1911-852 
 
 1912-222 
 
 1912-593 
 
 191'2-yrw 
 
 1913-333 
 
 1913-704 
 
 1914-074 
 
 1914-444 
 
 516 
 
 517 
 
 1914-815 
 
 1915-185 
 
 1915-556 
 
 1915-926 
 
 1916296 
 
 1916-667 
 
 1917-037 
 
 1917-407 
 
 1917-778 
 
 1918-148 
 
 517 
 
 518 
 
 1918-519 
 
 1918-889 
 
 1919-259 
 
 1919-630 
 
 1920- 
 
 1920-370 
 
 1920-741 
 
 1921-111 
 
 1921-481 
 
 1921-852 
 
 518 
 
 519 
 
 1922-223 
 
 1922-593 
 
 1922963 
 
 1923-333 
 
 19-23704 
 
 1924-074 
 
 1924-444 
 
 1924815 
 
 1925-185 
 
 1925-556 
 
 519 
 
 520 
 
 1925-926 
 
 1926-296 
 
 1926-667 
 
 1927-037 
 
 1927-407 
 
 1927-778 
 
 1928-148 
 
 1928-519 
 
 1928-889 
 
 1929-259 
 
 520 
 
 521 
 
 1929-630 
 
 1930- 
 
 1930-370 
 
 1930-741 
 
 1931-111 
 
 1931-481 
 
 1931-852 
 
 1932-222 
 
 1932-593 
 
 1932963 
 
 521 
 
 522 
 
 1933333 
 
 1933-704 
 
 1934-074 
 
 1934-441 
 
 1934-815 
 
 1935-185 
 
 1935-556 
 
 1935-926 
 
 1936-296 
 
 1936-667 
 
 522 
 
 523 
 
 1937-037 
 
 1937-407 
 
 1937-778 
 
 1938-148 
 
 1938-519 
 
 1938-889 
 
 1939-259 
 
 1939-630 
 
 1940- 
 
 1940370 
 
 523 
 
 524 
 
 1940-741 
 
 1941-111 
 
 1941-481 
 
 1941-852 
 
 1942-222 
 
 1942-593 
 
 1942-963 
 
 1943-33:' 
 
 1943-704 
 
 1944-074 
 
 524 
 
 526 
 
 1944-444 
 
 1944-815 
 
 1945185 
 
 1945-556 
 
 1945-926 
 
 1946-296 
 
 1946-667 
 
 1947-037 
 
 1947-407 
 
 1947-778 
 
 tor. 
 
 526 
 
 1948-148 
 
 1948-519 
 
 1948-889 
 
 1949-259 
 
 1949-G30 
 
 1950- 
 
 1950-370 
 
 1950-741 
 
 1951-111 
 
 1951 481 
 
 526 
 
 527 
 
 1951-852 
 
 1952-222 
 
 1952-593 
 
 1952963 
 
 1953-333 
 
 1953-704 
 
 1954-074 
 
 1954-444 
 
 1954-815 
 
 1955185 
 
 5-27. 
 
 528 
 
 1955-556 
 
 1955-926 
 
 1956-296 
 
 1956-667 
 
 1957-037 
 
 1057-407 
 
 1957-778 
 
 1958-148 
 
 1958-519 
 
 1958-889 
 
 528 
 
 529 
 
 1959-259 
 
 1959-630 
 
 I960- 
 
 1960-370 
 
 1960-741 
 
 1961-111 
 
 1961-481 
 
 1961-852 
 
 1962-222 
 
 1962-593 
 
 529 
 
 530 
 
 1962963 
 
 1963333 
 
 1963-704 
 
 1964-074 
 
 1964-444 
 
 1964-815 
 
 1965-185 
 
 1965-556 
 
 1965-920 
 
 1966-296 
 
 530 
 
 531 
 
 1966-667 
 
 1967-037 
 
 1967-407 
 
 1967-778 
 
 1968-148 
 
 1968-519 
 
 1968-889 
 
 1969-259 
 
 1969-630 
 
 1970- 
 
 531 
 
 532 
 
 1970-370 
 
 1970-741 
 
 1971-111 
 
 1971481 
 
 1971-852 
 
 1972-222 
 
 1972-593 
 
 1972963 
 
 1973333 
 
 1973-704 
 
 532 
 
 533 
 
 1974-074 
 
 1974-444 
 
 1974-815 
 
 1975-185 
 
 1975-556 
 
 1975-920 
 
 1976-296 
 
 1976-667 
 
 1977-037 
 
 1977-407 
 
 533 
 
 :>34 
 
 1977-778 
 
 1978-148 
 
 1978-519 
 
 1978-889 
 
 1979-259 
 
 1979-630 
 
 1980- 
 
 1980-370 
 
 1980-741 
 
 1981-111 
 
 534 
 
 535 
 
 1981-481 
 
 1981-852 
 
 1982-222 
 
 1982-593 
 
 1982-963 
 
 1983-333 
 
 1983-704 
 
 1984-074 
 
 1984-444 
 
 1984-815 
 
 535 
 
 536 
 
 1985-185 
 
 1985-556 
 
 1 985-926 
 
 1986-296 
 
 1986667 
 
 1987-037 
 
 1987-407 
 
 1987-778 
 
 19*8-148 
 
 1988-519 
 
 536 
 
 537 
 
 1988-889 
 
 1989-259 
 
 1989-630 
 
 1990- 
 
 1990370 
 
 1990-741 
 
 1991-111 
 
 1991-481 
 
 1991-852 
 
 1992-222 
 
 537 
 
 538 
 
 1992-593 
 
 1992-963 
 
 1993-333 
 
 1993-704 
 
 1994-074 
 
 1994-444 
 
 1994-815 
 
 1995-185 
 
 1995-556 
 
 1995-926 
 
 538 , 
 
 :>39 
 
 1996-296 
 
 1996-667 
 
 1997-037 
 
 1997-407 
 
 1997*778 
 
 1998-148 
 
 199S-519 
 
 1998-889 
 
 1999-259 
 
 1999630 
 
 539 ' 
 
 540 
 
 2000- 
 
 2000-370 
 
 2000-741 
 
 2001-111 
 
 2001-481 
 
 2001-852 
 
 2002-222 
 
 2002-593 
 
 2002-963 
 
 2003-333 
 
 540 ; 
 
 M.A. 
 
 O 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 -5 
 
 -0 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 ME A If AREAS 48 1 to 54Q. 
 
182 
 
 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR WO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 % 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 541 
 
 J003-704 2004-074 
 
 2004-444 
 
 004-815 
 
 2005-185 
 
 2005-556 
 
 2005-926 
 
 006-296 
 
 2006-607 
 
 2007-037 
 
 541 
 
 542 
 
 2007-407, 2007-778 
 
 2008-148 
 
 008-519 
 
 008-889 
 
 2U09-259 
 
 2009-630 
 
 010- 
 
 2010-370 
 
 2010-741 
 
 542 
 
 543 
 
 2011-111 2011-481 
 
 2011-852 
 
 012-222 
 
 2012-593 
 
 2012-963 
 
 2013-333 
 
 013-704 
 
 2014-074 
 
 2014444 
 
 543 
 
 544 
 
 2014-815; 2015- 185 
 
 2015-556 
 
 015-926 
 
 2016-296 
 
 2016-667 
 
 2017-037 
 
 017-407 
 
 2017-778 
 
 2018-148 
 
 544 
 
 545 
 
 2018-519 2018-889 
 
 2019-259 
 
 019-630 
 
 2020- 
 
 2020-370 
 
 2020-741 
 
 021-111 
 
 2021-481 
 
 2021-852 
 
 545 
 
 546 
 
 2022-222 2022-593 
 
 2022-963 
 
 023-333 
 
 2023-704 
 
 2024-074 
 
 2024-444 
 
 024-815 
 
 2025-185 
 
 2025-550 
 
 546 
 
 547 
 
 2025-926 2026-296 
 
 2026-667 
 
 027-037 
 
 2027-407 
 
 2027-778 
 
 2028-148 
 
 028-519 2028-889 
 
 2029-259 
 
 547 
 
 518 
 
 2029-030,2030- 
 
 2030-370 
 
 030-741 
 
 2031-111 
 
 2031-481 
 
 2031-852 
 
 2032-222 12032-593 
 
 2032-963 
 
 548 
 
 549 
 
 2033 333 2033-704 
 
 2034-074 
 
 034-444 
 
 2034-815 
 
 2035-185 
 
 2035-55S 
 
 2035-926 2036-290 
 
 2036-667 
 
 549 
 
 550 
 
 2037-037 
 
 2037-407 
 
 2037-778 
 
 038-148 
 
 2038-519 
 
 2038-889 
 
 2039-259 
 
 2039-630 
 
 2040- 
 
 2040-370 
 
 550 
 
 551 
 
 2040-741 
 
 2041-111 
 
 2041-481 
 
 041-852 
 
 2042-222 
 
 2042-593 
 
 2042-963 
 
 2043-333 
 
 2043704 
 
 044-074 
 
 551 
 
 5.V2 
 
 2044-444 2044-815 
 
 2045 185 
 
 045-506 
 
 2045-926 
 
 2046-296 
 
 2046-667 
 
 2047-037 
 
 2047-407 
 
 047-778 
 
 552 
 
 553 
 
 2048-148 2048-519 
 
 2048-889 
 
 049-259 
 
 2049-630 
 
 2050- 
 
 2050-370 
 
 2050-741 
 
 2051-111 
 
 051-481 
 
 553 
 
 554 
 
 2051-852 2052-222 
 
 2052-593 
 
 052-963 
 
 2053-333 
 
 2053-704 
 
 2054-074 
 
 2054-444 
 
 2054-815 
 
 055-185 
 
 554 
 
 555 
 556 
 
 2055-556 12055-926 
 2059-259 2059-630 
 
 2056-296 
 2060- 
 
 056-667 
 060-370 
 
 2057-037 
 2060-741 
 
 057-407 
 2061-111 
 
 2057-778 
 2061-481 
 
 2058-148 
 2061-852 
 
 2058-519 
 2062-222 
 
 2058-889 
 2062-593 
 
 555 
 556 
 
 557 
 
 2062-963 
 
 2063-333 
 
 2063-704 
 
 064'07-i 
 
 2064-444 
 
 2064-815 
 
 2065-1F5 
 
 2065-550 
 
 2065-926 
 
 2000-290 
 
 557 
 
 558 
 
 206(5-667 
 
 2067-037 
 
 2067-407 
 
 2067-778 
 
 2068-148 
 
 2068-519 
 
 2068-889 
 
 2069-259 
 
 2069-630 
 
 2070- 
 
 558 
 
 559 
 
 2070-370 
 
 2070-741 
 
 2071-111 
 
 2071-481 
 
 2071-852 
 
 2072-222 
 
 2072-593 
 
 2072-96:5 
 
 2073-333 
 
 2073-704 
 
 559 
 
 500 
 
 2074-074 
 
 2074-444 
 
 2074-815 
 
 2075-185 
 
 2075-556 
 
 2075-926 
 
 2076-296 
 
 2076-667 
 
 2077-037 
 
 2077-407 
 
 560 
 
 561 
 
 2077-778 
 
 2078-148 
 
 2078-519 
 
 2078-889 
 
 2079-259 
 
 2079-630 
 
 2080- 
 
 2080-370 
 
 2080-741 
 
 2081-111 
 
 501 
 
 502 
 
 2081-481 
 
 2081-852 
 
 2082-222 
 
 2082-593 
 
 2082 963 
 
 2083-333 
 
 2083-704 
 
 2084-074 
 
 2084-444 
 
 2084-815 
 
 562 
 
 563 
 
 2085-185 
 
 2085-556 
 
 2085-926 
 
 2086-296 
 
 2086-667 
 
 2087-037 
 
 2087-407 
 
 20S7-778 
 
 2088-148 
 
 2088-519 
 
 503 
 
 564 
 
 2088-889 
 
 2089-259 
 
 2089-630 
 
 2090- 
 
 2090-370 
 
 2090 741 
 
 2091-111 
 
 2091-481 
 
 2091-852 
 
 2092-222 
 
 504 
 
 565 
 
 2092-593 
 
 2092-963 
 
 2093-333 
 
 2093-704 
 
 2094-074 
 
 2094-444 
 
 2094-815 
 
 2095-185 
 
 2095-556 
 
 2095-920 
 
 565 
 
 5fiO 
 
 2096-296 
 
 2096-667 
 
 2097-037 
 
 2097-407 
 
 2097-778 
 
 2098-148 
 
 2098-519 
 
 2098-889 
 
 2099-259 
 
 2099-6:50 
 
 566 
 
 567 
 
 2100- 
 
 2100-370 
 
 2100-741 
 
 2101-111 
 
 2101-481 
 
 2101-852 
 
 2102-222 
 
 2102-593 
 
 2102-963 
 
 210:333:5 
 
 567 
 
 568 
 
 2103-704 
 
 2104-074 
 
 2104-441 
 
 2104-815 
 
 2105-185 
 
 2105-556 
 
 2105-920 
 
 2106-296 
 
 2106-667 
 
 2107-037 
 
 508 
 
 69 
 
 2107-407 
 
 2107-778 
 
 2108-148 
 
 2108-519 
 
 2108-889 
 
 2109-259 
 
 2109-6-H 
 
 2110- 
 
 2110-370 
 
 2110-741 
 
 509 
 
 570 
 
 2111-111 
 
 2111-481 
 
 2111-852 
 
 2112-222 
 
 2112-593 
 
 2112963 
 
 2113-333 
 
 2113-704 
 
 2114-074 
 
 2114-444 
 
 570 
 
 571 
 
 2114-815 
 
 2115-185 
 
 2115-556 
 
 2115-926 
 
 2116-296 
 
 2116-C67 
 
 2117-03" 
 
 2117-407 
 
 2117-778 
 
 2118-148 
 
 571 
 
 572 
 
 2118-519 
 
 2118-869 
 
 2119-259 
 
 2119-630 
 
 2120- 
 
 2120-370 
 
 2120-741 
 
 2121-111 
 
 2121-481 
 
 2121-852 
 
 572 
 
 573 
 
 2122-222 
 
 2122-593 
 
 2122-963 
 
 2123333 
 
 2123-704 
 
 2124-074 
 
 2124-444 
 
 2124-815 
 
 2125-185 
 
 2125-556 
 
 573 
 
 574 
 
 2125926 
 
 2126-296 
 
 2126-607 
 
 2127-037 
 
 2127-407 
 
 2127-77* 
 
 2128-148 
 
 2128-519 
 
 21 28-889 
 
 212925< 
 
 574 
 
 575 
 
 21-29-630 
 
 2130- 
 
 2130-370 
 
 2130-741 
 
 2131-111 
 
 2131-481 
 
 2131-852 
 
 2132-222 
 
 2132-593 
 
 2132-963 
 
 575 
 
 576 
 
 2133-333 
 
 2133-704 
 
 2134-074 
 
 2134-444 
 
 2134-815 
 
 2135-18' 
 
 2135-556 
 
 2135-926 
 
 2136-296 
 
 2130-667 
 
 576 
 
 577 
 
 2137-037 
 
 2137-407 
 
 2137-778 
 
 2138-148 
 
 2138-519 
 
 2138-889 
 
 21o9-259 
 
 2139-630 
 
 2140- 
 
 2140-370 
 
 577 
 
 578 
 
 2140-741 
 
 2141-111 
 
 2141-481 
 
 2141-852 
 
 2142-222 
 
 2142-593 
 
 2142-96! 
 
 2143-333 
 
 2143-704 
 
 2144-074 
 
 578 
 
 579 
 
 2144-444 
 
 2144-815 
 
 2145-185 
 
 2145-556 
 
 2145 926 
 
 2146-290 
 
 2i4(/667 
 
 2147-037 
 
 2147 407 
 
 2147-778 
 
 579 
 
 580 
 
 2148-148 
 
 2148-519 
 
 2148-889 
 
 2149-259 
 
 2149-630 
 
 2150- 
 
 2150-370 
 
 2150-741 
 
 2151-111 
 
 2151-481 
 
 580 
 
 581 
 
 2151-852 
 
 2152-222 
 
 2152-593 
 
 2152-96. 
 
 2153-333 
 
 2153-704 
 
 2154-074 
 
 2154-444 
 
 2154-815 
 
 2155-185 
 
 581 
 
 582 
 
 2155-556 
 
 2155-926 
 
 2156-296 
 
 2156-667 
 
 2157-037 
 
 2157-407 
 
 2157-77 
 
 2158-14S 
 
 2158-519 
 
 2158 889 
 
 582 
 
 583 
 
 2159-259 
 
 2159-630 
 
 2160- 
 
 2160-370 
 
 2160-741 
 
 2161-111 
 
 2101-48 
 
 2161-852 
 
 2162-222 
 
 2162-593 
 
 583 
 
 584 
 
 2162-963 
 
 2163-333 
 
 2163-704 
 
 2164-074 
 
 2164-444 
 
 2164-815 
 
 2165-18 
 
 2165-556 
 
 2165-926 
 
 2166-290 
 
 584 
 
 585 
 
 21t)6-667 
 
 2167-037 
 
 2167-407 
 
 2167-778 
 
 2168-148 
 
 2168-519 
 
 2168-88 
 
 2169-259 
 
 2169-630 
 
 2170- 
 
 585 
 
 586 
 
 2170-370 
 
 2170-741 
 
 2171-111 
 
 2171-481 
 
 2171-852 
 
 2172-222 
 
 2172-59 
 
 2172-963 
 
 2173-333 
 
 2173-704 
 
 586 
 
 587 
 
 2174-074 
 
 2174-444 
 
 2174-815 
 
 2175-185 
 
 2175-556 
 
 2175-926 
 
 2176-29 
 
 2176-667 
 
 2177-037 
 
 2177-40- 
 
 587 
 
 588 2177-778 
 
 2178-148 
 
 2178-519 
 
 2178-889 
 
 2179-259 
 
 2179-630 
 
 2180- 
 
 2180-370 
 
 2180-741 
 
 2181-111 
 
 588 
 
 589 
 
 2181-481 
 
 2181-852 
 
 21S2-222 
 
 2182-593 
 
 2l82-96o 
 
 2183-333 
 
 2183-70 
 
 2184-074 
 
 2184-444 
 
 2184-815 
 
 589 
 
 590 
 
 2185-185 
 
 2185-556 
 
 2185-926 
 
 2186296 
 
 2186-667 
 
 2187-037 
 
 2187-40 
 
 2187-778 
 
 2188-148 
 
 2188-519 
 
 590 
 
 591 
 
 2188-889 
 
 2189-259 
 
 2189-630 
 
 2190- 
 
 2190-370 
 
 2190-741 
 
 2191-11 
 
 2191-481 
 
 2191-852 
 
 2192-222 
 
 591 
 
 592 
 
 2192-593 
 
 2192-963 
 
 2193-33: 
 
 2193704 
 
 2194-074 
 
 2194-444 
 
 2194-81 
 
 2195-185 
 
 2195-55t) 
 
 2195-926 
 
 592 
 
 593 
 
 2196-296 
 
 2196-667 
 
 2 197 -03" 
 
 2197-4(7 
 
 2197778 
 
 2198-148 
 
 2198-61 
 
 2198-889 
 
 2199-259 
 
 2199-631 
 
 593 
 
 594 
 
 2200- 
 
 2200-37C 
 
 2200-741 
 
 2201-111 
 
 2201-481 
 
 2201852 
 
 2202-22 
 
 2202-593 
 
 2202-963 
 
 2202-3:33 
 
 594 
 
 595 
 
 2203-704 
 
 2204-074 
 
 2204-444 
 
 2204-815 
 
 2205-185 
 
 2205-556 
 
 2205-92 
 
 2206-296 
 
 2206-667 
 
 2207-037 
 
 595 
 
 596 
 
 2207-407 
 
 2207-77* 
 
 2208-148 
 
 2208-519 
 
 -'208-889 
 
 2209-259 
 
 2209-63 
 
 2210- 
 
 2210-370 
 
 2210-741 
 
 596 
 
 597 
 
 2211-111 
 
 2211-481 
 
 2211-852 
 
 2212-222 
 
 2212-593 
 
 2212 9^3 
 
 2213-33 
 
 2213-704 
 
 2214-074 
 
 2214-444 
 
 597 
 
 598 
 
 2214815 
 
 2215-185 
 
 2215-556 
 
 2215-92C 
 
 2216-296 
 
 2216-667 
 
 2217-03 
 
 2217-407! 2217-778 
 
 2218-148 
 
 598 
 
 599 
 
 2218-519 
 
 2218-88P 
 
 2219-259 
 
 2219-630 
 
 2220- 
 
 2-J20-370 
 
 2220-74 
 
 2221-111 
 
 2221-481 
 
 2221-852 
 
 599 
 
 600 
 
 2222-222 
 
 2222-593 
 
 2222-96: 
 
 2223-333 
 
 2223-704 
 
 2224-074 
 
 2224-44 
 
 2224-815 
 
 2225-185 
 
 2225-556 
 
 6UO 
 
 M.A 
 
 O 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 54 1 to 6OO. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 183 
 
 CVTtIC YARDS TO MEAN AREAS FOR 1OO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 2 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 001 
 
 2225- 92 
 
 2220-296 
 
 2220-667 
 
 22-27-037 
 
 2227-407 
 
 2227778 
 
 2228-148 
 
 2228-519 
 
 2228-889 
 
 2229-259 
 
 601 
 
 602 
 
 2229-630 
 
 2230- 
 
 22::o-370 
 
 2230-741 
 
 2231 111 
 
 2231-481 
 
 223 1 852 
 
 2232*222 
 
 2232-593 
 
 2232-903 
 
 602 
 
 003 
 
 2233-333 
 
 2233-704 
 
 2234-074 
 
 224-444 
 
 2234-815 
 
 2235-185 
 
 2235-550 
 
 2235-92G 
 
 2236-296 
 
 2230-067 
 
 603 
 
 604 
 
 22-37-037 
 
 22.57-407 
 
 2237-778 
 
 2238-148 
 
 2238-519 
 
 2238-889 
 
 2239-259 
 
 22i>9-t30 
 
 2240- 
 
 2240370 
 
 604 
 
 HI )5 
 
 2240-741 
 
 2241-111 
 
 2241-481 
 
 2241-852 
 
 2242222 
 
 2242-593 
 
 ^242-903 
 
 2243-333 
 
 2243-704 
 
 2244-074 
 
 605 
 
 666 
 
 2J44-444 
 
 2244-815 
 
 2245- 1S5 
 
 2245-556 
 
 2245 920 
 
 2246-296 
 
 2240-667 
 
 2247-C37 
 
 2247-407 
 
 2247-778 
 
 606 
 
 f7 
 
 2248-14S 
 
 2243-519 
 
 224S-889 
 
 2249-259 
 
 2249-030 
 
 2250- 
 
 2250-370 
 
 2250-741 
 
 2251-111 
 
 2251-481 
 
 607 
 
 ('08 
 
 2''5l-85" > 
 
 2252-222 
 
 J252-593 
 
 2252-963 
 
 2253-333 
 
 2253-704 
 
 2254-074 
 
 2254-444 
 
 2254-815 
 
 2255-185 
 
 608 
 
 0)9 
 
 225.) r,5G 
 
 2255-920 
 
 2250-296 
 
 2-256-607 
 
 2257-037 
 
 2257-407 
 
 2257-778 
 
 2258-148 
 
 2258-519 
 
 2258-889 
 
 609 
 
 tilO 
 
 2259-253 
 
 2259-630 
 
 2260- 
 
 2260-370 
 
 2260-741 
 
 2261111 
 
 2261-481 
 
 2261-852 
 
 2262-222 
 
 2262-593 
 
 610 
 
 611 
 
 22^2-963 
 
 2263-333 
 
 2263-704 
 
 22R4-074 
 
 2264-444 
 
 2-264-815 
 
 2265-185 
 
 2265-556 
 
 22P5-920 
 
 2266-296 
 
 611 
 
 012 
 
 2 JGtt 667 
 
 2267037 
 
 2207-407 
 
 2207-778 
 
 22G8-14S 
 
 226S-519 
 
 2268-8*9 
 
 22C9-259 
 
 2269-630 
 
 2270- 
 
 612 
 
 B13 
 
 2270-370 
 
 2270-741 
 
 2271-11! 
 
 2271-481 
 
 2271-852 
 
 2272-222 
 
 2272-593 
 
 2272-963 
 
 2273-33o 
 
 2273704 
 
 613 
 
 014 
 
 2274-074 
 
 2274-441 
 
 2274-815 
 
 2275-185 
 
 2275-556 
 
 2275-920 
 
 2-276-296 
 
 227G-C67 
 
 2277-037 
 
 2277-407 
 
 614 
 
 015 
 
 2277-778 
 
 2278-148 
 
 2278519 
 
 227S-8S9 
 
 2279-259 
 
 2279-630 
 
 2280- 
 
 2-280-37C 
 
 2280-741 
 
 2281-111 
 
 615 
 
 616 
 
 2281-481 
 
 22S1-852 
 
 2282-'222 
 
 2282-593 
 
 2282-9H3 
 
 2283-333 
 
 2283704 
 
 2284-074 
 
 2284-444 
 
 22*4-815 
 
 616 
 
 617 
 
 2285-185 
 
 22S5-55G 
 
 22S5-92G 
 
 2286-296 
 
 22SG-C67 
 
 2287 037 
 
 2287-407 
 
 2287-778 
 
 2288-148 
 
 2288-519 
 
 617 
 
 618 
 
 228H-889 
 
 '2283-259 
 
 2289-630 
 
 2230- 
 
 2290-370 
 
 2290-741 
 
 2291-111 
 
 2291-481 
 
 2291-852 
 
 2292-222 
 
 618 
 
 019 
 
 2292 593 
 
 2292-903 
 
 2293-333 
 
 2233-704 
 
 2294-074 
 
 2294-444 
 
 2294-815 
 
 2-J9.V1X5 
 
 2295-556 
 
 2295-S-26 
 
 619 
 
 620 
 
 229o-296 
 
 2290-607 
 
 2297-037 
 
 2297-407 
 
 2297-778 
 
 2298-148 
 
 2298-519 
 
 2298 889 
 
 2299-259 
 
 2299630 
 
 620 
 
 G21 
 
 23^0- 
 
 2300-370 
 
 2300-741 
 
 2301-1 n 
 
 2301-481 
 
 2301-852 
 
 2302-222 
 
 2302-693 
 
 2302-963 
 
 2303-833 
 
 621 
 
 623 
 
 23 (3-704 
 
 2304-074 
 
 2304-444 
 
 2304-815 
 
 2305-185 
 
 2305-556 
 
 2305-926 
 
 2306296 
 
 2306-007 
 
 2307-037 
 
 622 
 
 623 
 
 23)7 -407 12.307 -77 S 
 
 23)8-1 48 
 
 2308-519 
 
 230S-8S9 
 
 2309-259 
 
 2309 G3( 
 
 2310- 
 
 2310-370 
 
 2310-741 
 
 V23 
 
 624 
 
 2311 111 
 
 2311-481 
 
 2311-852 
 
 2312-222 
 
 2312-593 
 
 2312-90.3 
 
 2313333 
 
 2313-704 
 
 2314-074 
 
 2314-444 
 
 624 
 
 625 
 
 2314815 
 
 2315-185 
 
 23 1.Y .-,:,, 
 
 2315-926 
 
 2316-296 
 
 2316-667 
 
 2317-037 
 
 2317-407 
 
 2317-778 
 
 K318-148 
 
 6-.>5 
 
 G2G 
 
 2318519 2318-8S9 
 
 211 9-253 
 
 2319-630 
 
 2320- 
 
 2320-370 
 
 2320-741 
 
 2321-111 
 
 2321-481 
 
 2321-852 
 
 626 
 
 c>n 
 
 J322-222 
 
 2322*593 
 
 2322-903 
 
 2323-333 
 
 2323704 
 
 2324-074 
 
 2324-444 
 
 2324-815 
 
 2325-185 
 
 2325-556 
 
 627 
 
 62 
 
 2325926 
 
 232>;"2'JC 
 
 2320-067 
 
 2327-037 
 
 2327 -4')7 
 
 2327-778 
 
 2328-148 
 
 2328-519 
 
 l'o28-889 
 
 2329-259 
 
 628 
 
 02J 
 
 2329-630 
 
 2330- 
 
 2330-370 
 
 2330741 
 
 2331-111 
 
 2331481 
 
 2331-852 
 
 2:'32-222 
 
 23i2-593 
 
 2332-963 
 
 629 
 
 030 
 
 2333-333 
 
 2333-70 4 
 
 2334-074 
 
 2334-444 
 
 2334-815 
 
 2335-185 
 
 2335-556 
 
 2335-926 
 
 233629t 
 
 2336-667 
 
 C30 
 
 (531 
 
 2337037 
 
 2337-407 
 
 2337-778 
 
 2338-148 
 
 2338-519 
 
 2338-889 
 
 2339-259 
 
 2339-f30 
 
 2340- 
 
 2340-370 
 
 P31 
 
 (532 
 
 2340741 
 
 2341-111 
 
 2341-4S1 
 
 3341-852 
 
 2ii 12-22- 
 
 2342-5 f j:; 
 
 2342-9GG 
 
 2343-G33 
 
 2343-704 
 
 2344-074 
 
 632 
 
 033 
 
 -1314-414 
 
 2344-815 
 
 2345-185 
 
 2o45-55G 
 
 23LV920 
 
 2346-290 
 
 234G-GG7 
 
 2347 037 
 
 2347-407 
 
 2347-778 
 
 G33 
 
 634 
 
 234vl48 
 
 2348519 
 
 2318-889 
 
 2343-25) 
 
 2319 630 
 
 2350- 
 
 2350-370 
 
 2350741 
 
 2351-111 
 
 2351-481 
 
 634 
 
 035 
 
 2351-852 
 
 2352-222 
 
 2352-593 
 
 J352-9G3 
 
 2353-33:i 
 
 23' 3 704 
 
 2354-074 
 
 2,?54-444 
 
 2G54815 
 
 2355-185 
 
 635 
 
 G3l> 
 
 2.355-556 
 
 2:555926 
 
 235G-29G 
 
 2356-667 
 
 2:j.-.7-o.:7 
 
 23;.7-407 
 
 2357-778 
 
 23;58-148 
 
 2358519 
 
 2358889 
 
 636 
 
 037 
 
 2359-259 
 
 2359-630 
 
 2360- 
 
 2360-370 
 
 2G60-741 
 
 2361-111 
 
 2361-481 
 
 ^361-852 
 
 2362-222 
 
 23C2-593 
 
 637 
 
 G!8 
 
 :;i-.2-. -3 
 
 2363-333 
 
 2363-704 
 
 2364-074 
 
 2304-444 
 
 2.364-815 
 
 23G5-185 
 
 236555P. 
 
 23C5-92* 
 
 23fO-296 
 
 638 
 
 G:59 
 
 2366-6OT 
 
 2367-037 
 
 2307-4 V? 
 
 J3G7-778 
 
 2368-148 
 
 2368-519 
 
 2368-8S9 
 
 23C9-2/9 
 
 2c69-(30 
 
 2370- 
 
 639 
 
 010 
 
 2370-370 
 
 2370-741 
 
 2371-111 
 
 2371-481 
 
 2371-852 
 
 2372-222 
 
 2u7 2-693 
 
 2372-963 
 
 2373-333 
 
 2373704 
 
 C40 
 
 641 
 
 2374-074 
 
 2374-444 
 
 2374-815 
 
 2375-185 
 
 2375-556 
 
 2375926 
 
 2376-29G 
 
 2376-f67 
 
 2377-OS7 
 
 2377-407 
 
 641 
 
 G4i 
 
 2377-778 
 
 2378-148 
 
 2378-519 
 
 2378-.S89 
 
 2379-2C9 
 
 2379-030 
 
 380- 
 
 2380-370 
 
 2350741 
 
 2381-111 
 
 (42 
 
 013 
 
 23SI-481 
 
 2381-852 
 
 23S2-222 
 
 238259:) 
 
 J382-9G3 
 
 2.J83-333 
 
 23*3-704 
 
 2384-074 
 
 2384-444 
 
 2384-815 
 
 643 
 
 014 
 
 2385- 1 85 
 
 2385-556 
 
 23S5-92G 
 
 2386-296 
 
 2386-067 
 
 1387 037 
 
 2387-407 
 
 2387-778 
 
 2388-148 
 
 2C88 51 
 
 644 
 
 Gl.i 
 
 '38S-8S9 
 
 23S92J9 
 
 2389-03:) 
 
 2393- 
 
 2390-370 
 
 2390-741 
 
 2391-111 
 
 2C91-481 
 
 2391-852 
 
 2392-222 
 
 645 
 
 (UG 
 
 2392-693 
 
 2332-963 
 
 2333-333 
 
 2393-704 
 
 2.394-074 
 
 2394-444 
 
 2394-81;' 
 
 2S95-1! S 5 
 
 2395-550 
 
 295-9-.'Q 
 
 646 
 
 oir 
 
 J39I5-29G 
 
 239:5-007 
 
 2397-037 
 
 2337-407 
 
 2397*778 
 
 2398-148 
 
 2398-519 
 
 2398-889 
 
 23t9-259 
 
 2l ! -99 630 
 
 647 
 
 (513 
 
 2400- 
 
 2400-370 
 
 2403-741 
 
 2401-111 
 
 2401-481 
 
 2401-852 
 
 2402-222 
 
 2402-593 
 
 2402-963 
 
 2403-333 
 
 648 
 
 o;j 
 
 2403-704 
 
 2404-074 
 
 2404-444 
 
 2404-815 
 
 2405-185 
 
 2405-550 
 
 2405-920 
 
 2406-296 
 
 2406-607 
 
 2407-037 
 
 649 
 
 050 
 
 24U7-407 
 
 2407-778 
 
 2408-148 
 
 2408-519 
 
 2408-889 
 
 2409-259 
 
 2409-630 
 
 2410- 
 
 2410-370 
 
 2410-741 
 
 050 
 
 G51 
 
 2111-111 
 
 2411-481 
 
 2411-852 
 
 2412-222 
 
 2412-593 
 
 2412-903 
 
 2413-333 
 
 2413-704 
 
 2414-074 
 
 2414-444 
 
 651 
 
 152 
 
 2,14815 
 
 2415-185 2115-550 
 
 2415".;2 i 
 
 2410-290 
 
 2416-667 
 
 2417-037 
 
 2417-407 
 
 2H7-778 
 
 2418-148 
 
 6C2 
 
 GC.3 
 
 211S-519 
 
 2418-883 2419 259 
 
 2419-630 
 
 2420- 
 
 2120-370 
 
 2420-741 
 
 2121-111 
 
 2121481 
 
 2J2l-8i2 
 
 653 
 
 (.51 
 
 2122222 
 
 2:22-59:J 2'22-9i'3 
 
 2123-333 
 
 2123704 
 
 2124-074 
 
 2124-444 
 
 2424-815 
 
 2 '2,5-1 85 
 
 2J2555G 
 
 054 
 
 G55 
 
 242V923 
 
 2420-230 
 
 242J-6G7 
 
 2527-037 
 
 2 '27-407 
 
 2427-778 
 
 2428-148 
 
 2428-519 
 
 2428-889 
 
 2429 2-"9 
 
 055 
 
 G'0 
 
 2121-63) 
 
 2130- 
 
 2430-370 
 
 2 430-741 
 
 2431-111 
 
 2431-481 
 
 2131-852 
 
 24.?2-222 
 
 2432-593 
 
 2432-903 
 
 656 
 
 0'7 
 
 2.133-333 
 
 2433-704 
 
 2LU-H74 
 
 2434 444 
 
 2!34-S15 
 
 2 135-185 
 
 2435-556 
 
 2435-J126 
 
 2430-296 
 
 2436-667 
 
 657 
 
 058 
 
 2137-037 
 
 2437-407 2437-778 
 
 2438-1 4 
 
 2138-519 
 
 2438-889 
 
 2439-2-9 
 
 2439-630 
 
 2!40- 
 
 2440-370 
 
 658 
 
 659 
 
 2140-741 
 
 2441-m 2441-481 
 
 2441-8^2 24-12 222 
 
 2442593 
 
 2442963 
 
 2443-33.3 
 
 2443-704 
 
 2144-074 
 
 659 
 
 COO 
 
 2444-444 
 
 2444-815 ^445-185 
 
 2445-550 
 
 2445-926 
 
 2446-296 
 
 2446-C67 
 
 2447037 
 
 2447-407 
 
 2447-778 
 
 GGO 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 i 
 
 5 
 
 G 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 6OI to 660. 
 
184 
 
 RULES FOR THE MEASUREMENT OP EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR 1OO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 
 
 3 
 
 .* 
 
 5 
 
 6 
 
 I 
 
 8 
 
 9 
 
 ALA. 
 
 fiGl 
 
 2448-148 
 
 2448-519 
 
 244S-889 
 
 2449-259 
 
 2449-630 
 
 2450- 
 
 2450-370 
 
 450-741 
 
 2451-111 
 
 2451-481 
 
 601 
 
 662 
 
 2451-852 2452-222 
 
 2452-593 
 
 2452-963 
 
 2453-333 
 
 2453-704 
 
 2454074 
 
 434-444 2454-815 
 
 2455-185 
 
 602 
 
 063 
 
 2455-556 2455-926 
 
 2456-2% 
 
 2456-667 
 
 2457-037 
 
 2457-407 
 
 2457-778 
 
 2458-148 
 
 2458-519 
 
 2458-889 
 
 663 
 
 664 
 
 2459-259 2459-630 
 
 2460- 
 
 2460-370 
 
 2460-741 
 
 2461-111 
 
 2461-481 
 
 461-852 
 
 2462-222 
 
 2462-593 
 
 664 
 
 665 
 
 2462-963 2463-333 
 
 2463-704 
 
 2464-074 
 
 2464-444 
 
 2464-815 
 
 2465-185 
 
 2465-556 
 
 2465-926 
 
 2466-296 
 
 665 
 
 666 
 
 2466 667 2467-037 
 
 2467-407 
 
 2467778 
 
 2468-148 
 
 2468-519 
 
 2468 889 
 
 2469-259 
 
 2469-630 
 
 2470- 
 
 666 
 
 667 
 
 2470-370 2470-741 
 
 471-111 
 
 2471-481 
 
 2471-852 
 
 2472-222 
 
 2472-593 
 
 2472-963 
 
 2473-333 
 
 2473704 
 
 6(17 
 
 668 
 
 2474-074 2474-444 
 
 474-815 
 
 2475-185 
 
 2475-556 
 
 2475-926 
 
 2476-296 
 
 2476-667 
 
 2477037 
 
 2477-407 
 
 668 
 
 669 
 
 2477-778 
 
 2478148 
 
 478-519 
 
 2478-889 
 
 2479-259 
 
 2479-630 
 
 2480- 
 
 2480-370 2480-741 
 
 2481-111 
 
 669 
 
 670 
 
 2481-481 
 
 2481-852 
 
 482-222 
 
 2482593 
 
 2482-963 
 
 2483-333 
 
 2483-704 
 
 2484-074 
 
 2484-444 
 
 2484-815 
 
 670 
 
 671 
 
 2485-185 
 
 2485-556 
 
 485-926 
 
 2486-296 
 
 2486-667 
 
 2487-037 
 
 2487-407 
 
 2487-778 
 
 2488-148 
 
 2488-519 
 
 671 
 
 672 
 
 2488-889 
 
 2489-259 
 
 489-630 
 
 2490- 
 
 2490-370 
 
 2490-741 
 
 2491-111 
 
 2491-481 
 
 2491-852 
 
 2492-222 
 
 672 
 
 673 
 
 2492-593 j 2492-963 
 
 493-333 
 
 2493 704 
 
 249HT74 
 
 2494-444 
 
 2494815 
 
 2495-185 
 
 2495-556 
 
 2495-9-26 
 
 673 
 
 674 
 
 496-296; 2496-667 
 
 497-037 
 
 2497-407 
 
 2497-778 
 
 2498-148 
 
 2498-519 
 
 2498-889 
 
 2499-259 
 
 2499630 
 
 674 
 
 675 
 
 500- 12500-370 
 
 500-741 
 
 2501-111 
 
 2501-481 
 
 2501-852 
 
 2502-222 
 
 2502-593 
 
 2502-963 
 
 2503-333 
 
 675 
 
 676 
 
 503-704 
 
 .504-074 
 
 504-444 
 
 2504-815 
 
 2505-185 
 
 2505-556 
 
 2505-926 
 
 2506-296 
 
 2506-667 
 
 2507-037 
 
 676 
 
 677 
 
 507-407 
 
 507-778 
 
 508-148 
 
 2508-519 
 
 2508-889 
 
 2509-259 
 
 2509-630 
 
 2510- 
 
 2510-370 
 
 2510-741 
 
 677 
 
 678 
 
 511-111 
 
 511-481 
 
 511-852 
 
 2:M2-222 
 
 2512-593 
 
 2512-963 
 
 2513-333 
 
 2513-704 
 
 2514-074 
 
 2514-444 
 
 678 
 
 679 
 
 514-815 
 
 515-185 
 
 515-556 
 
 '515-926 
 
 2516-296 
 
 2516-667 
 
 2517-037 
 
 2517-407 
 
 2517-778 
 
 2518-14* 
 
 679 
 
 630 
 
 518-519 
 
 518-889 
 
 519-259 
 
 2519-030 
 
 2520- 
 
 2520-370 
 
 2520-741 
 
 25-21-111 
 
 2521-481 
 
 2521-852 
 
 680 
 
 6S1 
 
 522-222 
 
 522-593 
 
 522-963 
 
 2523-333 
 
 2523' 704 
 
 2524-074 
 
 2524-444 
 
 2524-815 
 
 2525-185 
 
 2525-556 
 
 681 
 
 632 
 
 2525-926 
 
 526-296 
 
 526-667 
 
 2527-037 
 
 2527-407 
 
 2527-778 
 
 2528-148 
 
 2528-519 
 
 2528-889 
 
 2529-259 
 
 682 
 
 683 
 
 2529-630 
 
 530- 
 
 530-370 
 
 2530-741 
 
 2531-111 
 
 2531-481 
 
 2531-852 
 
 2532-222 
 
 2532-593 
 
 2532-963 
 
 683 
 
 68 1 
 
 533-333 
 
 533-704 
 
 534-074 
 
 2534-444 
 
 2534-815 
 
 2535-185 
 
 2535-556 
 
 2535-926 
 
 2536-296 
 
 2536-667 
 
 684 
 
 685 
 
 2537-037 
 
 537-407 
 
 537-778 
 
 2538-148 
 
 2538-519 
 
 2538-889 
 
 2539-259 
 
 2539-630 
 
 2540- 
 
 2540-370 
 
 685 
 
 686 
 
 2540-741 
 
 541-111 
 
 541-481 
 
 2541-852 
 
 2542-222 
 
 2542-593 
 
 2542-963 
 
 2543-333 
 
 2543-704 
 
 2544-074 
 
 686 
 
 687 
 
 2544-444 
 
 544-815 
 
 545-185 
 
 2545-556 
 
 2545-926 
 
 2546-296 
 
 2546-667 
 
 2547-037 
 
 2547-407 
 
 2547-77* 
 
 687 
 
 688 
 
 2548-148 
 
 548-519 
 
 2548-8S9 
 
 2549-259 
 
 2549630 
 
 2550- 
 
 2550-370 
 
 2550-741 
 
 2551-111 
 
 2551-4S1 
 
 688 
 
 689 
 
 2551-85-2 
 
 552-222 
 
 2552-593 
 
 2552-963 
 
 2553-333 
 
 2553-704 
 
 2554-074 
 
 2664*444 
 
 2554-815 
 
 2555-1 85 
 
 689 
 
 690 
 
 2555-556 
 
 555-926 
 
 2556-296 
 
 2556-667 
 
 2557-037 
 
 2557-407 
 
 2557-778 
 
 2558-148 
 
 2558-519 
 
 2558-889 
 
 690 
 
 601 
 
 2559-259 
 
 2559-630 
 
 2560' 
 
 2560-370 
 
 2560-741 
 
 2561-111 
 
 2561-481 
 
 2561-852 
 
 2562-222 
 
 2562-593 
 
 691 
 
 692 
 
 2562963 
 
 2563-333 
 
 2563-704 
 
 2564-074 
 
 2564-444 
 
 2564-815 
 
 2565-185 
 
 2565'556 
 
 2565-926 
 
 2566-296 
 
 692 
 
 6 r J3 
 
 2566-667 
 
 2567-037 
 
 2567-407 
 
 2567-778 
 
 2568-148 
 
 2568-519 
 
 2568-889 
 
 2569-259 
 
 2569-630 
 
 2570- 
 
 693 
 
 694 
 
 2570-370 
 
 2570-741 
 
 2571-111 
 
 2571-481 
 
 2571-852 
 
 2572-222 
 
 2572-593 
 
 2572-963 
 
 2573-333 
 
 2573-704 
 
 694 
 
 695 
 
 2574-074 
 
 2574-444 
 
 2574-815 
 
 2575-185 
 
 2575-556 
 
 2575-926 
 
 2576-296 
 
 2576-667 
 
 2577-037 
 
 2577-407 
 
 695 
 
 696 
 
 2577-778 
 
 2578-148 
 
 2578-519 
 
 2578-889 
 
 2579-259 
 
 2579-630 
 
 2580- 
 
 2580-370 
 
 '2580-741 
 
 2581-111 
 
 696 
 
 697 
 
 2581-481 
 
 2581-852 
 
 2582-222 
 
 2582-593 
 
 2582-963 
 
 2583-333 
 
 2583-704 
 
 2584-074 
 
 2584-444 
 
 2584-815 
 
 697 
 
 698 
 
 2585-185 
 
 2585-556 
 
 2585-926 
 
 2586-296 
 
 2586-667 
 
 2587-037 
 
 2587-407 
 
 2587-778 
 
 2588-148 
 
 2588-519 
 
 698 
 
 699 
 
 2588-889 
 
 2589-259 
 
 2589-630 
 
 2590- 
 
 2590-370 
 
 2590-741 
 
 2591-111 
 
 2591-481 
 
 2591-852 
 
 2592-22L 
 
 699 
 
 700 
 
 2592-593 
 
 2592-963 
 
 2593333 
 
 2593-704 
 
 2591-074 
 
 2594-444 
 
 2594-815 
 
 259^-185 
 
 2595-556 
 
 2595-926 
 
 700 
 
 701 
 
 2596-296 
 
 2596-667 
 
 2597-037 
 
 2597-407 
 
 2597-778 
 
 2598-148 
 
 2598-519 
 
 2598-889 
 
 2599-259 
 
 2599-630 
 
 701 
 
 702 
 
 2600- 
 
 2600-370 
 
 2600741 
 
 2001-111 
 
 2601-481 
 
 2601-852 
 
 2602-222 
 
 2602-593 
 
 2602-963 
 
 2603-333 
 
 702 
 
 703 
 
 2603-704 
 
 2604-074 
 
 2604-444 
 
 2604-815 
 
 2605-185 
 
 2605-556 
 
 2605-926 
 
 2606-296 
 
 2606-667 
 
 2607-037 
 
 703 
 
 704 
 
 2607-407 
 
 2607-778 
 
 2608-148 
 
 2608-519 
 
 2608-889 
 
 2609-259 
 
 2609-630 
 
 2610- 
 
 2610-370 
 
 2610-741 
 
 704 
 
 705 
 
 2611-111 
 
 2611-481 
 
 2611-852 
 
 2612-222 
 
 2612-593 
 
 2612-963 
 
 2613-333 
 
 2613-704 
 
 2614-074 
 
 2614-444 
 
 705 
 
 7 .16 
 
 2614-815 
 
 2615-185 
 
 2615-556 
 
 2615-926 
 
 2616-296 
 
 2616-667 
 
 2617-037 
 
 2617-407 
 
 2617-778 
 
 2618-148 
 
 706 
 
 707 
 
 2618-519 
 
 2618-889 
 
 2619-259 
 
 2619-630 
 
 2620- 
 
 2620-370 
 
 2620-741 
 
 2621-111 
 
 2621-481 
 
 2621-852 
 
 707 
 
 708 
 
 2622-222 
 
 2622-591, 
 
 2622-963 
 
 2623-33? 
 
 26-23-704 
 
 2624-074 
 
 2624-444 
 
 2624-815 
 
 2625-185 
 
 2625-556 
 
 708 
 
 709 
 
 2625-920 
 
 2626-296 
 
 2626-667 
 
 26-27-037 
 
 2627-407 
 
 2627-778 
 
 2628-148 
 
 2628-519 
 
 2628-889 
 
 2629-259 
 
 709 
 
 710 
 
 2629-630 
 
 2630- 
 
 2630-370 
 
 2630-741 
 
 2631-111 
 
 2631-481 
 
 2631-852 
 
 2632-222 
 
 2632-593 
 
 2632-963 
 
 710 
 
 711 
 
 2633-33 
 
 2633-70 
 
 2634-07-1 
 
 2634-444 
 
 2634-815 
 
 2635-185 
 
 2635-556 
 
 2635-926 
 
 2636-296 
 
 2636-667 
 
 711 
 
 712 
 
 _:637-G3 
 
 2637-40 
 
 2637-77S 
 
 2638-148 
 
 2638-519 
 
 2038-889 
 
 2639-259 
 
 2639-630 
 
 2640- 
 
 2640-370 
 
 712 
 
 713 
 
 2640-74 
 
 2641-111 
 
 2641-481 
 
 2641-852 
 
 2642-222 
 
 2642-593 
 
 2642-963 
 
 2643-333 
 
 2643-704 
 
 2644-074 
 
 713 
 
 714 
 
 2614-444 
 
 2644-Slo 
 
 2645-185 
 
 2645-556 
 
 2645-926 
 
 2646-296 
 
 2646-667 
 
 2647-037 
 
 2647-407 
 
 2647-778 
 
 714 
 
 715 
 
 2648-14 
 
 2648-519 
 
 2648-880 
 
 2649-259 
 
 2649-630 
 
 2650- 
 
 2650-370 
 
 2650-741 
 
 2651-111 
 
 2651-481 
 
 715 
 
 716 
 
 2651-85 
 
 2652-22' 
 
 2652-593 
 
 2652-963 
 
 2653-333 
 
 2653-704 
 
 2654-074 
 
 2654-444 
 
 2654-815 
 
 2655-185 
 
 716 
 
 717 
 
 2655-55 
 
 2655-926 
 
 2656-29(3 
 
 2656-667 
 
 2657-037 
 
 2657-407 
 
 2657-778 
 
 2658-148 
 
 2658-519 
 
 2658-889 
 
 717 
 
 718 
 
 2659-25 
 
 2659-630 
 
 2660- 
 
 2660-370 
 
 2660-741 
 
 2661-111 
 
 2661-481 
 
 2661-852 
 
 2662-222 
 
 2662-593 
 
 718 
 
 719 
 
 2662-96 
 
 2363-333 
 
 2663-704 
 
 2664-074 
 
 2664-444 
 
 2664-815 
 
 2665-185 
 
 2665-556 2665-926 
 
 2666-296 
 
 719 
 
 720 
 
 2C66-66 
 
 2667-03" 
 
 2667-407 
 
 2607-778 
 
 2068-148 
 
 2668-519 
 
 2668-889 
 
 2669-259 
 
 2669-630 
 
 2670- 
 
 720 
 
 M.A. 
 
 
 
 1 
 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 661 to 7XO. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 J85 
 
 CUBIC YARDS TO MEAN AREAS FOR 1OO FEET IN LENGTH. 
 
 .M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4: 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 7-21 
 
 2070-370 2670-741 
 
 2671-111 
 
 2671-481 
 
 2671-852 
 
 2tf7 2-222 
 
 2672-593 
 
 2672-963 
 
 2673-333 
 
 2673-704 
 
 72I 
 
 72-2 
 
 2074-074 2674-444 
 
 2674-815 
 
 2675-185 
 
 2675-556 12675-926 
 
 2676-296 
 
 2670-607 
 
 2677-037 
 
 2677-407 
 
 722 
 
 723 
 
 2G77-778 2678-148 
 
 2678-519 
 
 2678889 
 
 2679-259 
 
 2079-630 
 
 2680- 
 
 2680-370 
 
 2680-741 
 
 2681-111 
 
 723 
 
 724 
 
 26S1-481 2681-852 
 
 2682-222 
 
 2082-59:5 
 
 2682-963 
 
 2683-333 
 
 2683-704 
 
 2684-074 
 
 2684-444 
 
 2684-815 
 
 724 
 
 725 
 
 2685-185 2685-556 
 
 2685-926 
 
 2680-296 
 
 2686-667 
 
 2687-037 
 
 2687-407 
 
 2687-778 
 
 26-8-148 
 
 2688-519 
 
 725 
 
 720 
 
 2088-889 2689-259 
 
 2689-630 
 
 2690- 
 
 2690-370 
 
 2690-741 
 
 2691-111 
 
 2691-481 
 
 2691-852 
 
 2092-222 
 
 726 
 
 727 
 
 2(192-593 2692 963 
 
 2693-333 
 
 2693-704 
 
 2694-074 
 
 2694-444 
 
 2694-815 
 
 2695-185 
 
 2695-556 
 
 2695-926 
 
 727 
 
 728 
 
 2096-296 2696 007 
 
 2697-037 
 
 2697-407 
 
 2697-778 
 
 2698-148 
 
 2698-519 
 
 2698-889 
 
 2699-259 
 
 2699-630 
 
 728 
 
 729 
 
 2700- 
 
 2700-370 
 
 2700-741 
 
 2701-111 
 
 2701-481 
 
 2701-852 
 
 2702-222 
 
 2702-593 
 
 2702-963 
 
 2703333 
 
 7-29 
 
 730 
 
 2703-704 
 
 2704-074 
 
 2704-444 
 
 2704-815 
 
 2705-185 
 
 2705-556 
 
 2705-926 
 
 2706-296 
 
 2706-667 
 
 2707-037 
 
 730 
 
 731 
 
 2707-407 
 
 2707-778 
 
 2708-148 
 
 2708-519 
 
 2708-889 
 
 2709-259 
 
 2709-630 
 
 2710- 
 
 2710-370 
 
 2710-741 
 
 731 
 
 732 
 
 2711-111 J2711-481 
 
 2711-852 
 
 2712-2-22 
 
 2712-593 
 
 2712-963 
 
 2713-333 
 
 2713-704 
 
 2714-074 
 
 2714-444 
 
 732 
 
 733 
 
 2714-815 '27 15- 185 
 
 2715-550 
 
 2715-926 
 
 2716-296 
 
 2716-667 
 
 2717-037 
 
 2717-407 
 
 2717-778 
 
 2718-148 
 
 733 
 
 7*4 
 
 2718-519 2718-889 
 
 2719-259 
 
 2719-630 
 
 2720- 
 
 2720-370 
 
 2720-741 
 
 2721-111 
 
 2721-481 
 
 2721-852 
 
 734 
 
 735 
 
 2722 222 2722-593 
 
 2722903 
 
 272:5-:!:;;; 
 
 2723-704 127 24-H74 
 
 2724-444 
 
 2724815 
 
 2725-185 
 
 2725-556 
 
 735 
 
 736 
 
 2725-926 2726-296 
 
 2726-607 
 
 2727-037 
 
 2727-407 
 
 2727-778 
 
 2728-148 
 
 2728-519 
 
 2728-889 
 
 2729-259 
 
 736 
 
 737 
 
 2729030; 27 30- 
 
 2730-370 
 
 2730-741 
 
 2731-111 
 
 2731-481 
 
 2731-852 
 
 27:',2-2-22 
 
 2732-593 
 
 2732-963 
 
 737 
 
 738 
 
 27:33-333 2733-704 
 
 2734-1 '74 
 
 2734-444 
 
 2734-815 
 
 2735-185 
 
 2735-556 
 
 2735-926 
 
 2736-296 
 
 2736-667 
 
 738 
 
 739 
 
 2737-037 27: 17-407 
 
 2737-778 
 
 2738-148 
 
 2738-519 
 
 2738-889 
 
 2739-259 
 
 2739-630 
 
 2740- 
 
 2740-370 
 
 739 
 
 740 
 
 2740-741 
 
 2741-111 
 
 2741-481 
 
 2741-852 
 
 2742-222 
 
 2742-593 
 
 2742-963 
 
 2743-333 
 
 2743-704 
 
 2744-074 
 
 740 
 
 741 
 
 2744-444 
 
 2744-815 
 
 2745-185 
 
 2745-556 
 
 2745-926 
 
 2746-296 
 
 2746-667 
 
 2747-037 
 
 2747-407 
 
 2747-778 
 
 741 
 
 742 
 
 2748-148 2748-519 
 
 2748-889 
 
 2749-259 
 
 2749-630 
 
 2750- 
 
 2750370 
 
 2750741 
 
 2751-111 
 
 2751-481 
 
 742 
 
 743 
 
 2751-852 2752-222 
 
 27.V2-f>!KJ 
 
 2752-963 
 
 2753-333 
 
 2753-704 
 
 2754-074 
 
 2754-444 
 
 2754-815 
 
 2755-185 
 
 743 
 
 744 
 
 2755-556 2755-926 
 
 2756-296 
 
 2756-667 
 
 2757-037 
 
 2757-407 
 
 2757-778 
 
 2758 148 
 
 2758-519 
 
 2768-888 
 
 744 
 
 745 
 
 2759-259 2759-630 
 
 2760- 
 
 2760-370 
 
 2760741 
 
 2761-111 
 
 2761-481 
 
 2761-852 
 
 2762-222 
 
 2762-593 
 
 745 
 
 746 
 
 2762-963 2763-333 
 
 27K3-704 
 
 2764-074 
 
 27 64-444 12764-815 
 
 2765-185 
 
 2765 550 
 
 2765-926 
 
 2766-296 
 
 746 
 
 747 
 
 2766-667 2767'037 
 
 2767-407 
 
 2767-778 
 
 2768-148|2768-51l) 
 
 2768-889 
 
 2769-259 
 
 2769-630 
 
 2770- 
 
 747 
 
 748 
 
 2770-370 2770-741 
 
 2771-111 
 
 2771-481 
 
 2771-852 
 
 2772-222 
 
 2772-593 
 
 2772-963 
 
 2773-333 
 
 2773-704 
 
 748 
 
 749 
 
 2774-074 2774 444 
 
 2774-815 
 
 2775-185 
 
 2775-556 
 
 2775-926 
 
 2776-296 
 
 2776-667 
 
 2777037 
 
 2777-407 
 
 749 
 
 750 
 
 2777-778 
 
 2778-148 
 
 2778-519 
 
 2778-889 
 
 2779-259 
 
 2779-630 
 
 2780- 
 
 2780-370 
 
 2780-741 
 
 2781-111 
 
 750 
 
 751 
 
 2781-481 
 
 2781-852 
 
 2782-222 
 
 2782-593 
 
 2782-963 
 
 2783-333 
 
 2783-704 
 
 2784-074 
 
 2784-444 
 
 2784-815 
 
 751 
 
 752 
 
 2785-185 
 
 2785-556 
 
 2785-926 
 
 2786-296 
 
 2786-667 
 
 2787-037 
 
 2787-407 
 
 2787-778 
 
 2788-148 
 
 2788-519 
 
 752 
 
 753 
 
 2788-889 
 
 2789-259 
 
 2789-030 
 
 2790- 
 
 2790-370 
 
 2790-741 
 
 2791-111 
 
 2791-481 
 
 2791-852 
 
 2792-222 
 
 753 
 
 754 
 
 2792-593 
 
 2792-963 
 
 2793-333 
 
 2793-704 
 
 2794-074 
 
 2794-444 
 
 2794-815 
 
 2795-185 
 
 2795-556 
 
 2795-926 
 
 754 
 
 755 
 
 2796-296 
 
 2796-667 
 
 2797-037 
 
 2797-407 
 
 2797-778 
 
 2798-148 
 
 2798-519 
 
 2798 889 
 
 2799-259 
 
 2799-630 
 
 755 
 
 756 
 
 2800- 
 
 2800-370 
 
 2800-741 
 
 2801-111 
 
 2801-481 
 
 2801-852 
 
 2802-222 
 
 2802-593 
 
 2802-963 
 
 2803-333 
 
 756 
 
 757 
 
 2803-704 
 
 2804-074 
 
 2S04-444 
 
 2804-815 
 
 2805-185 
 
 2805-556 
 
 2805-926 
 
 2806-296 
 
 2806-667 
 
 2807037 
 
 757 
 
 758 
 
 2807-407 
 
 2807-778 
 
 2808-148 
 
 2808-519 
 
 2808-889 
 
 2809-259 
 
 2809-630 
 
 2810- 
 
 2810-370 
 
 2810-741 
 
 758 
 
 759 
 
 2811-111 
 
 2811-481 
 
 2811-852 
 
 2812-222 
 
 2812-593 
 
 2812-963 
 
 2813-333 
 
 2813-704 
 
 2814-074 
 
 2814-444 
 
 759 
 
 760 
 
 2814-815 
 
 2815-185 
 
 2815-556 
 
 2815-926 
 
 2816-296 
 
 2816-667 
 
 2817-037 
 
 2817-407 
 
 2817-778 
 
 2818-148 
 
 760 
 
 761 
 
 2818-519 
 
 2818-889 
 
 2819-259 
 
 2819-630 
 
 2820- 
 
 2820-370 
 
 2820-741 
 
 282M11 
 
 2821-481 
 
 2821-852 
 
 761 
 
 762 
 
 2822-222 2822-593 
 
 2822-963 
 
 2823-333 
 
 2823-704 
 
 2824-074 
 
 2824-444 
 
 2824-815 
 
 2825-185 
 
 2825-556 
 
 762 
 
 763 
 
 2825-926 2826-296 
 
 2826-667 
 
 2827-037 
 
 2827-407 
 
 2827-778 
 
 2828-148 
 
 2828-519 
 
 2828-889 
 
 2829-259 
 
 763 
 
 764 
 
 2829-630 2830- 
 
 2830-370 
 
 2830-741 
 
 2831-111 
 
 2831-481 
 
 2831-852 
 
 2832-222 
 
 2832-593 
 
 2832-963 
 
 764 
 
 765 
 
 2833-33312833-704 
 
 2834-074 
 
 2834-444 
 
 2834-815 
 
 2835-185 
 
 2835-556 
 
 2835-926 
 
 2836 296 
 
 2836 667 
 
 765 
 
 766 
 
 2837-037 
 
 2837-407 
 
 2837-778 
 
 2838-148 
 
 2838-519 
 
 2838-889 
 
 2839-259 
 
 2839-630 
 
 2840- 
 
 2840-370 
 
 766 
 
 767 
 
 2840-741 
 
 2841-111 
 
 2841-481 
 
 2841-852 
 
 2842-222 
 
 2842-593 
 
 2842-963 
 
 2843-333 
 
 2843-704 
 
 2844-074 
 
 767 
 
 7G8 
 
 2844-444 
 
 2844-815 
 
 2845-185 
 
 2845-. r >56 
 
 2845-920 
 
 2846-296 
 
 2846667 
 
 2847-037 
 
 2847-407 
 
 2847-778 
 
 768 
 
 769 
 
 2848-148 
 
 2848-51912848-889 
 
 2849-259 
 
 2849-630 
 
 2850- 
 
 2850-370 
 
 2850-741 
 
 2851111 
 
 2851-481 
 
 769 
 
 770 
 
 2851-852 
 
 2852-222 
 
 2852-593 
 
 2852-963 
 
 2853-333 
 
 2863-704 
 
 2854-074 
 
 2854-444 
 
 2854-815 
 
 2855-185 
 
 770 
 
 771 
 
 2855-556 
 
 2855-926 
 
 2856-296 
 
 2856-667 
 
 2857-037 
 
 2857-407 
 
 2857-778 
 
 2858-148 
 
 2858-519 
 
 2858-889 
 
 771 
 
 772 
 
 2859-259 
 
 2859-630 
 
 2860- 
 
 2860-370 
 
 2800-741 
 
 2861-111 
 
 2861-481 
 
 2861-852 
 
 2862-222 
 
 2802-593 
 
 772 
 
 773 
 
 2862-963 
 
 2863-333 
 
 2863-704 
 
 2864-074 
 
 2864-444 
 
 2864-815 
 
 2865-185 
 
 2865-556 
 
 2865-926 
 
 2866-296 
 
 773 
 
 774 
 
 2866-667 
 
 2867-037 
 
 2867-407 
 
 2807-778 
 
 2868-148 
 
 2868-519 
 
 >t;s-s,v, 
 
 2869-259 
 
 2869-630 
 
 2870- ' 
 
 774 
 
 775 
 
 2870-370 
 
 2870-741 
 
 2871-111 
 
 2871-481 
 
 2871-852 
 
 2872-222 
 
 2872-593 
 
 2872 963 
 
 2873-333 
 
 2873-704 
 
 775 
 
 776 
 
 2874-074 
 
 2874-444 
 
 2874-815 
 
 2875-185 
 
 2875-556 
 
 2875-926 
 
 2876-296 
 
 2876-667 
 
 2877-037 
 
 287 7 -407 
 
 776 
 
 777 
 
 2877-778 
 
 2878-148 
 
 2878-519 
 
 2878-889 
 
 2879-259 
 
 2879-630 
 
 2880- 
 
 2880-370 
 
 2880-741 
 
 2881-111 
 
 777 
 
 778 
 
 2881-481 
 
 2881-852 
 
 2882-222 
 
 28S2-593 
 
 2882-963 
 
 2883-33312883-704 
 
 2884-074 
 
 2884-444 
 
 2884-815 
 
 778 
 
 779 
 
 2885-185 
 
 2885-556|2885-926 
 
 2886-296 
 
 2886-667 
 
 js7-n:;7 287-407 
 
 2887-778 
 
 2888-148 
 
 2888-519 
 
 779 
 
 780 
 
 2888-889 
 
 2889-259 2889-630 
 
 2890- 
 
 2890-370 
 
 2890-741 
 
 2891-111 
 
 2891-481 
 
 2891-852 
 
 2892-222 
 
 780 
 
 M.A 
 
 
 
 1 
 
 
 
 3 
 
 *. 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 721 to 78O. 
 
18G 
 
 RULES FOR THE MEASUREMENT OP EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR WO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 781 
 
 2892-593 
 
 2892-963 
 
 893-333 
 
 2893-704 
 
 2894-074 
 
 2894-444 
 
 .'8S4-815 
 
 895-185 
 
 2895-556 
 
 ^895-92*', 
 
 781 
 
 782 
 
 2S96-296 2S96 667 
 
 2897-037 
 
 2897-407 
 
 2897-778 
 
 2898148 
 
 2898-519 
 
 2898-889 
 
 2899-259 
 
 2899-630 
 
 782 
 
 783 
 
 2900- 2900-370 
 
 2900-741 
 
 2901-111 
 
 2901-481 
 
 2901 852 
 
 2902-222 
 
 2902-593 
 
 2902-963 
 
 2903-3:>3 
 
 res 
 
 784 
 
 2903-704 -2904-074 
 
 904-444 
 
 2904-815 
 
 2905-185 
 
 2905-556 
 
 -905-926 
 
 2906-296 
 
 2906-667 
 
 2907-037 
 
 784 
 
 785 
 
 2907-407 2907-778 
 
 908-148 
 
 2908'5I9 
 
 2908-889 
 
 2909-259 
 
 2909-630 
 
 2910- 
 
 2910-370 
 
 2910-741 
 
 785 
 
 786 
 
 2911-111 2911-481 
 
 911-852 
 
 2912-222 
 
 2912-593 
 
 2912-963 
 
 2913-333 
 
 2913-704 
 
 2914-074 
 
 2914-444 
 
 7^6 
 
 787 
 
 2914-815 2915 185 
 
 915-556 
 
 2915-926 
 
 2916-296 
 
 2916-667 
 
 2917-037 
 
 2917-407 2917-778 
 
 2918-148 
 
 787 
 
 788 
 
 2918-519 -2918-889 
 
 919-259 
 
 2919-630 
 
 2920- 
 
 2920-370 
 
 2920741 
 
 2921-111 1 2921-481 
 
 2921 -852 
 
 788 
 
 789 
 
 2922-222 12922-593 
 
 >922-963 
 
 29-'3-333 
 
 2923-704 
 
 2924-074 
 
 -1924-444 
 
 2924-815 
 
 2925 185 
 
 2925-55G 
 
 789 
 
 790 
 
 2925-926 
 
 2926-296 
 
 2926-667 
 
 2927-037 
 
 2927-407 
 
 2927-778 
 
 2928-148 
 
 2928-519 
 
 2928-889 
 
 2929-259 
 
 790 
 
 791 
 
 2929-630 
 
 2930- 
 
 2930-370 
 
 2930-741 
 
 2931-111 
 
 2931-481 
 
 2931-852 
 
 2932-222 
 
 2932-593 
 
 2932-963 
 
 791 
 
 792 
 
 2933-333 2933-704 
 
 2934-074 
 
 2934-444 
 
 2934-815 
 
 2935-185 
 
 2935-556 
 
 2935-926 
 
 2936-296 
 
 2936-667 
 
 792 
 
 793 
 
 2937-0:37 2937-407 
 
 2937-778 
 
 2938-148 
 
 2938-519 
 
 2938-889 
 
 2939-259 
 
 2939 630 
 
 2940- 
 
 2940-370 
 
 793 
 
 704 
 
 2940-741 2941-111 
 
 2941-481 
 
 2941-852 
 
 2942-222 
 
 2942-593 
 
 2942-963 
 
 2943-333 
 
 2943-704 
 
 2944-074 
 
 7^4 
 
 795 
 
 2944-444 1'29 44-815 
 
 2945-185 
 
 _'945-556 
 
 2945-926 
 
 J946-296 
 
 2946-667 
 
 2947-037 
 
 2947-407 
 
 2947-778 
 
 795 
 
 796 
 
 2948-148 
 
 2948-519 
 
 2948-889 
 
 2949-259 
 
 2949-630 
 
 2950- 
 
 2950-M70 
 
 2950-741 
 
 2951-111 
 
 2951-481 
 
 79fi 
 
 797 
 
 2951-852 
 
 29.V2-222 
 
 2952-593 
 
 2952-963 
 
 2953-333 
 
 2953704 
 
 2954-074 
 
 ^954-444 
 
 2954-815 
 
 2955185 
 
 797 
 
 798 
 
 2955-556 
 
 2955-926 
 
 2956-296 
 
 2956-667 
 
 2957-037 
 
 2957-407 
 
 2957-778 
 
 2958-148 
 
 2958-519 
 
 2958-889 
 
 798 
 
 799 
 
 2959259 
 
 2959-630 
 
 2960- 
 
 2960-370 
 
 2960-741 
 
 2961-111 
 
 2961-481 
 
 2961-852 
 
 2962-222 
 
 2962-593 
 
 799 
 
 800 
 
 2962-963 
 
 2963-333 
 
 2963-704 
 
 2964-074 
 
 2964-444 
 
 2964-815 
 
 2965-185 
 
 2965-556 
 
 2965-926 
 
 ^966-296 
 
 800 
 
 801 
 
 2966667 
 
 2967-037 
 
 2967-407 
 
 2967-778 
 
 2968-148 
 
 2968-519 
 
 2968-889 
 
 2969-259 
 
 2969-630 
 
 2970- 
 
 801 
 
 802 
 
 2970-370 
 
 2970-741 
 
 2971-111 
 
 2971-4S1 
 
 2971-852 
 
 2972-222 
 
 2972-593 
 
 2972963 
 
 2973-033 
 
 2973-704 
 
 802 
 
 803 
 
 2974-074 
 
 2974441 
 
 2974-815 
 
 2975-185 
 
 2975-556 
 
 2975-926 
 
 2976-296 
 
 2976667 
 
 2977-037 
 
 2977-407 
 
 803 
 
 801 
 
 2977-778 
 
 2978-148 
 
 2978-519 
 
 2978-889 
 
 2979-259 
 
 2979T30 
 
 2980- 
 
 2980-370 
 
 2980-741 
 
 2981-111 
 
 804 
 
 805 
 
 2981-481 
 
 2981-852 
 
 2982 222 
 
 2982-593 
 
 2982-963 
 
 2983-333 
 
 2983-704 
 
 2984-074 1 2984-444 
 
 2984-815 
 
 805 
 
 806 
 
 2985-185 
 
 2985556 
 
 2985-926 
 
 2986-296 
 
 29S6-667 
 
 2987-037 
 
 2987-407 
 
 2987-778 
 
 2988-148 
 
 2988-519 
 
 806 
 
 807 
 
 2988-889 
 
 29H9-259 
 
 2989-630 
 
 2990- 
 
 2990 370 
 
 2990-741 
 
 2991-111 
 
 2991 -4S1 
 
 2991-85? 
 
 2992-222 
 
 807 
 
 808 
 
 2992593 
 
 2992-963 
 
 2993-333 
 
 2993-704 
 
 2994-074 
 
 J994-444 
 
 2994-815 
 
 2995-185 
 
 2995-556 
 
 2995-921 
 
 808 
 
 809 
 
 2996-296 
 
 2996-667 
 
 2997-037 
 
 2997-407 
 
 2997-778 
 
 2998-148 
 
 2998-519 
 
 2998-889 
 
 2999-259 
 
 2999-031 
 
 809 
 
 810 
 
 3000- 
 
 3000-370 
 
 3000-741 
 
 3001-111 
 
 3001-481 
 
 3001-862 
 
 3002-222 
 
 3002-593 
 
 3002-963 
 
 3003-333 
 
 810 
 
 811 
 
 3003-704 
 
 3004-074 
 
 3004-444 
 
 3004-815 
 
 3005-185 
 
 3005-5.56 
 
 3005-926 
 
 3006-296 
 
 3006-667 
 
 3007-037 
 
 811 
 
 812 
 
 3007-407 
 
 3007-778 
 
 3008-148 
 
 3008519 
 
 3008-889 
 
 3009-259 
 
 3009'63( 
 
 3010- 
 
 3010-370 
 
 3010-741 
 
 812 
 
 813 |3011-111 
 
 3011-481 
 
 3011-852 
 
 3012-222 
 
 3012-593 
 
 301 2-963 
 
 301333: 
 
 3013-704 
 
 3014-074 
 
 3014-444 
 
 813 
 
 814 
 
 3014-815 
 
 .5015-185 
 
 3015556 
 
 3015-926 
 
 3016-296 
 
 3016-667 
 
 3017-037 
 
 3017407 
 
 3017-778 
 
 3018-148 
 
 814 
 
 815 
 
 3018-519 
 
 3018-889 
 
 3019-259 
 
 3019-630 
 
 3020- 
 
 3020-370 
 
 3020-741 
 
 3021-111 
 
 3021-481 
 
 3021-852 
 
 815 
 
 816 
 
 302-2-222 
 
 3022-59} 
 
 3022-963 
 
 3023-333 
 
 3023-704 
 
 3024-074 
 
 3024444 
 
 3024-815 
 
 3025-185 
 
 3025-556 
 
 816 
 
 817 
 
 3025-926 
 
 3026-296 
 
 3026-667 
 
 3027-0.57 
 
 3027-407 
 
 3027-778 
 
 3028148 
 
 3028-519 
 
 3028-889 
 
 3029-259 
 
 817 
 
 818 
 
 3029-630 
 
 3030- 
 
 3030-370 
 
 3030-741 
 
 3031-111 
 
 3031-481 
 
 3031-852 
 
 3032-222 
 
 3032-593 
 
 3032-963 
 
 818 
 
 819 
 
 3033-333 
 
 3i 133-7 04 
 
 3034-074 
 
 3034-444 
 
 3034-815 
 
 3035-185 
 
 3035-556 
 
 3035-926 
 
 3036-296 
 
 3036-667 
 
 819 
 
 820 
 
 3037-037 
 
 3037-407 
 
 3037-778 
 
 3038-148 
 
 3038-519 
 
 3038-8SP 
 
 3039-259 
 
 3039 630 
 
 3040- 
 
 3040-370 
 
 820 
 
 821 
 
 3040-741 
 
 3041-111 
 
 3041-481 
 
 3041-852 
 
 3042-222 
 
 .1042-593 
 
 3042-963 
 
 3043-333 
 
 3043-704 
 
 3044-074 
 
 821 
 
 822 
 
 3044-444 
 
 3044-815 
 
 3045-1 85 
 
 3045-551 
 
 3045-926 
 
 3046-296 
 
 3046-667 
 
 3047-037 
 
 3047-407 
 
 3047- 77 S 
 
 822 
 
 823 
 
 3048-148 
 
 3048-519 
 
 3048-889 
 
 3049259 
 
 3049630 
 
 3050- 
 
 3050370 
 
 3050-741 
 
 3051-111 
 
 3051-481 
 
 823 
 
 824 
 
 3051-852 
 
 305222- 
 
 3052-693 
 
 3052-963 
 
 3053-333 
 
 3053-704 
 
 3054-074 
 
 3054-444 
 
 3054-815 
 
 3055-185 
 
 824 
 
 825 
 
 3055-556 
 
 3055-926 
 
 3056296 
 
 3056-067 
 
 3057-037 
 
 3057-407 
 
 3057-778 
 
 3058-148 
 
 3058-519 
 
 3058-889 
 
 825 
 
 826 
 
 3059259 
 
 3059-630 
 
 30(30- 
 
 3060-370 
 
 3060-741 
 
 3061 -111 
 
 3061-481 
 
 3061-852 
 
 3062-222 
 
 3062-593 
 
 826 
 
 827 
 
 3062-963 
 
 3063-33; 
 
 3063-704 
 
 3064-074 
 
 3064-444 
 
 3064-815 
 
 3065-185 
 
 3065-556 
 
 3065-926 
 
 3066296 
 
 827 
 
 828 
 
 3066-667 
 
 3067-037 
 
 3067-407 
 
 3067-778 
 
 3068-148 
 
 3068-519 
 
 3068-889 
 
 3069259 
 
 3069-630 
 
 3070- 
 
 828 
 
 829 
 
 3070-371 
 
 3070-74 
 
 3071-111 
 
 3071-481 
 
 3071-852 
 
 3072-222 
 
 3072-593 
 
 3072-963 
 
 3073-333 
 
 3073-704 
 
 829 
 
 830 
 
 3U74-074 
 
 3074-444 
 
 3074-815 
 
 3075-185 
 
 3075-556 
 
 3075-926 
 
 3076-296 
 
 3076-667 
 
 3077-037 
 
 3077-407 
 
 sao 
 
 R31 
 
 ?077-778 
 
 3078-148 
 
 3078-519 
 
 3078-889 
 
 3079-259 
 
 3079-630 
 
 3080- 
 
 3080-370 
 
 3080-741 
 
 308M11 
 
 831 
 
 832 
 
 3081-481 
 
 3081-85- 
 
 3082-222 
 
 3082-593 
 
 3082-9*3 
 
 3083-333 
 
 3083-704 
 
 3084-074 
 
 3084-444 
 
 3084-815 
 
 832 
 
 833 
 
 3085-1 ?5 
 
 3085-556 
 
 3085-926 
 
 W6-296 
 
 3086-667 
 
 3087-037 
 
 3087-407 
 
 3087-778 
 
 3088148 
 
 3088-519 
 
 833 
 
 834 
 
 3088-889 
 
 3089-259 
 
 3089-630 
 
 3090- 
 
 3090-370 
 
 3090-741 
 
 3091-111 
 
 3091-481 
 
 3091-85-2 
 
 3092-222 
 
 834 
 
 8i!5 
 
 3092-593 
 
 3092-96: 
 
 3093-333 
 
 3093-704 
 
 3094-074 
 
 3094-444 
 
 3<i94-81f 
 
 3095-185 
 
 3095-556 
 
 3095-926 
 
 835 
 
 836 
 
 3096-296 
 
 3096-66- 
 
 30,97-037 
 
 3097-407 
 
 3097-778 
 
 3098-148 
 
 3098-519 
 
 3098 889 
 
 3099-259 
 
 3099-630 
 
 836 
 
 837 
 
 3100- 
 
 3100-370 
 
 3100-741 
 
 3101-111 
 
 3101-481 
 
 3101-852 
 
 3102-222 
 
 31 02-593 13102-963 
 
 :-,103333 
 
 837 
 
 838 
 
 3103-704 
 
 3104-074 
 
 3104-444 
 
 3104-815 
 
 3105-185 
 
 3105-556 
 
 3105-926 
 
 3106-296 
 
 3106-667 
 
 3107-037 
 
 838 
 
 839 
 
 3107-407 
 
 3107-778 
 
 310S-14S 
 
 3108-519 
 
 3108-889 
 
 3109-259 
 
 3109-630 
 
 3110- 
 
 3110-370 
 
 3110-741 
 
 839 
 
 840 
 
 311 Mil 
 
 3111-481 
 
 3111-852 
 
 3112-222 
 
 3112-593 
 
 3112-963 
 
 3113-333 
 
 3113-704 
 
 3114-074 
 
 3114-444 
 
 840 
 
 M.A. 
 
 O 
 
 1 
 
 a 
 
 3 
 
 * 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 781 to 84O. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 187 
 
 CTTJilC YARDS TO MEAN AREAS FOR WO FEET IN LENGTH. 
 
 M.A- 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 841 
 
 3114-815 
 
 3115-185 
 
 3115-556 
 
 3115-926 
 
 3116-296 
 
 3116-667 
 
 3117037 
 
 3117-407 
 
 3117-778 
 
 3118-148 
 
 841 
 
 812 
 
 3118 519 3118-889 
 
 3119-259 
 
 3119-630 
 
 3120- 
 
 3120-370 
 
 3120-741 
 
 3121-111 
 
 3121-481 
 
 3121-852 
 
 842 
 
 843 
 
 3122-222 3122 593 
 
 3122-9U3 
 
 3123-333 
 
 3123-704 
 
 3124-H74 
 
 3124-444 
 
 3124-815 
 
 3125-185 
 
 3125-556 
 
 843 
 
 814 
 
 3125-926 
 
 3126-296 
 
 3126-667 
 
 3127-037 
 
 3127-407 
 
 3127-778 
 
 3128148 
 
 3128-519 
 
 3128-889 
 
 3129-259 
 
 844 
 
 845 
 
 3129-630 
 
 3130- 
 
 3130-370 
 
 3130-741 
 
 3131-111 
 
 3131-481 
 
 3131-852 
 
 3132-222 
 
 3132-593 
 
 3132963 
 
 845 
 
 846 
 
 3133-333 
 
 3133-704 
 
 5134-074 
 
 3134-444 
 
 3134-815 
 
 3135-185 
 
 3135-556 
 
 3135-926 
 
 3136296 
 
 3136-667 
 
 846 
 
 847 
 
 3137-037 
 
 3137-407 
 
 3137-778 
 
 3138-148 
 
 3138-519 
 
 3138-889 
 
 3139-259 
 
 3139-630 
 
 3140- 
 
 3140-370 
 
 847 
 
 848 
 
 3140741 
 
 3141-111 
 
 3141-481 
 
 3141-852 
 
 3142-222 
 
 3142-593 
 
 3142-963 
 
 3143-333 
 
 3143-704 
 
 3144074 
 
 848 
 
 849 
 
 5144-444 
 
 3144-815 
 
 3145-185 
 
 3145-556 
 
 3145-926 
 
 3146-296 
 
 3146-667 
 
 3147-037 
 
 3147-407 
 
 3147-778 
 
 849 
 
 850 
 
 3148- 148 
 
 3148-519 
 
 3148-889 
 
 3149-259 
 
 3149630 
 
 3150- 
 
 3150-370 
 
 3150-741 
 
 3151-111 
 
 3151-481 
 
 850 
 
 851 
 
 3151-852 
 
 3152-222 
 
 3152-593 
 
 3152-963 
 
 3153-333 
 
 3153-704 
 
 3154-074 
 
 3154-444 
 
 3154-815 
 
 3155-185 
 
 851 
 
 852 
 853 
 
 3 155-556 j 3155-926 
 3159-259 3159-630 
 
 3156-296 
 3160- 
 
 3156-667 
 3160-370 
 
 3157-037 
 3160-741 
 
 3157-407 
 3161-111 
 
 3157-778 
 3161-481 
 
 3158-148 
 3161-852 
 
 3158-519 
 3162-222 
 
 3158-889, 
 3162-593 
 
 852 
 853 
 
 854 
 
 J 102-963 ; 31 63-333 
 
 3163-704 
 
 3164-074 
 
 3164-444 
 
 31(34-815 
 
 3165185 
 
 3165-556 
 
 3165-926 
 
 3166-296 
 
 854 
 
 855 
 
 5166667 
 
 3167-037 
 
 3167-407 
 
 3167-778 
 
 3168-148 
 
 3168-519 
 
 3168-889 
 
 3169-259 
 
 3169-630 
 
 3170- 
 
 855 
 
 856 
 
 3170-370 
 
 3170-741 
 
 3171-111 
 
 3171-481 
 
 3171-852 
 
 3172-222 
 
 3172-593 
 
 3172-963 
 
 3173-233 
 
 3173-704 
 
 1-56 
 
 857 
 
 3174-074 
 
 3174444 
 
 3174-815 
 
 3175-185 
 
 3175-556 
 
 3175-926 
 
 3176-296 
 
 3176-667 
 
 3177-037 13177-407 
 
 857 
 
 858 
 
 3177-778 
 
 3178-148 
 
 3178-519 
 
 3178-889 
 
 Jl 79-259 
 
 3179630 
 
 3180- 
 
 3180-370 
 
 3180741 
 
 3181111 
 
 858 
 
 859 
 
 5181481 
 
 3181-852 
 
 3182-222 
 
 3182-593 
 
 3182-963 
 
 3183-333 
 
 3183-704 
 
 3184-074 
 
 3184-444 
 
 31K4-815 
 
 859 
 
 860 
 
 3185-185 
 
 3185-556 
 
 3185-926 
 
 3186-296 
 
 3186-667 
 
 3187-037 
 
 3187-407 
 
 3187-778 
 
 3188-148 
 
 3188-519 
 
 860 
 
 861 
 
 1188-889 
 
 3189-259 
 
 3189-630 
 
 3190- 
 
 3190-370 
 
 3190-741 
 
 3191-111 
 
 3191-481 
 
 3191-852 
 
 3192-222 
 
 861 
 
 862 
 
 5192-593 
 
 3192-963 
 
 3193333 
 
 3193-704 
 
 3194-074 
 
 3194-444 
 
 3194815 
 
 3195-185 
 
 3195-556 
 
 3195-926 
 
 862 
 
 863 
 
 3196-296 
 
 3196-667 
 
 3197-037 
 
 3197-407 
 
 3197-778 
 
 3198-148 
 
 3198-519 
 
 3198-889 
 
 3199-259 
 
 3199-630 
 
 863 
 
 864 
 
 3200- 
 
 3200-370 
 
 3200-741 
 
 3201-111 
 
 3201-481 
 
 3201-852 
 
 3202-222 
 
 3202-593 
 
 32U2-963 
 
 3203-333 
 
 864 
 
 865 
 
 3203-704 
 
 3204-074 
 
 3204-444 
 
 3204-815 
 
 3205185 
 
 3205-556 
 
 3205-926 
 
 3206-296 
 
 3206-667 
 
 3207-037 
 
 865 
 
 8G6 
 
 '.207-407 
 
 3207-778 
 
 3208-148 
 
 3208-519 
 
 3208-889 
 
 3209-259 
 
 3209-630 
 
 3210- 
 
 3210-370 
 
 3210741 
 
 866 
 
 867 
 
 3211-111 
 
 3211-481 
 
 3211852 
 
 :;2i2"2-2-2 
 
 3212-593 
 
 3212-963 
 
 3213-333 
 
 3213-704 
 
 3214-074 
 
 3214-444 
 
 867 
 
 868 
 
 3214-815 
 
 3215-185 
 
 3215-556 
 
 3215-926 
 
 3216-296 
 
 3216-667 
 
 3217-037 
 
 3217-407 
 
 3217-778 
 
 3218-148 
 
 868 
 
 869 
 
 3218-519 
 
 3218889 
 
 3219-259 
 
 3219-630 
 
 3220- 
 
 3220-370 
 
 3220-741 
 
 3221-111 
 
 3221-481 
 
 3221-852 
 
 869 
 
 870 
 
 3222-222 
 
 3222593 
 
 3222-963 
 
 3223-333 
 
 3223-704 
 
 3224-074 
 
 3224-444 
 
 3224-815 
 
 3225-185 
 
 3225-656 
 
 870 
 
 871 
 
 3225-926 
 
 3226-296 
 
 3226-667 
 
 3227-037 
 
 3227-407 
 
 3227-778 
 
 3228-148 
 
 3228-519 
 
 3228-889 
 
 3229-259 
 
 871 
 
 872 
 
 '3229-630 
 
 3230- 
 
 J230-370 
 
 3230-741 
 
 3-2IU-111 
 
 3231-481 
 
 3231-852 
 
 3232-222 
 
 3232-593 
 
 3232-963 
 
 872 
 
 873 
 
 3233-333 
 
 3233-704 
 
 3234-074 
 
 3234-444 
 
 3234-815 
 
 3235-185 
 
 32&V556 
 
 3235-926 
 
 3236-296 
 
 3236-667 
 
 873 
 
 874 
 
 3237-037 
 
 3237-407 
 
 3237-778 
 
 3238-148 
 
 3238-519 
 
 3238-889 
 
 3239-259 
 
 3239-630 
 
 3240- 
 
 3240 370 
 
 874 
 
 875 
 
 3240 741 
 
 3241111 
 
 3241-481 
 
 3241-852 
 
 3242-222 
 
 3242-593 
 
 3242-963 
 
 3243-333 
 
 3243-704 
 
 3244-074 
 
 875 
 
 876 
 
 52U-444 
 
 3244-K15 
 
 3245-18.") 
 
 3245-550 
 
 3245-926 
 
 3246-296 
 
 3246-667 
 
 3247-037 
 
 3247-407 
 
 3247-778 
 
 876 
 
 877 
 
 3248-148 
 
 3248-519 
 
 3248-889 
 
 3249-259 
 
 3249-630 
 
 3250- 
 
 3250-370 
 
 3250-741 
 
 3251-111 
 
 3251-481 
 
 877 
 
 878 
 
 3251-852 
 
 3-252-222 
 
 3252-593 
 
 3252-963 
 
 3253-333 
 
 3253704 
 
 3254-074 
 
 3254-444 
 
 3254-815 
 
 3255-185 
 
 878 
 
 879 
 
 3255-556 
 
 3255-926 
 
 3256-296 
 
 3256667 
 
 3257-037 
 
 3257-407 
 
 3257-778 
 
 3258-148 
 
 3258-519 
 
 3258-889 
 
 879 
 
 MO 
 
 3259-259 
 
 3259-630 
 
 3260- 
 
 3260-370 
 
 3260741 
 
 3261-111 
 
 3261-481 
 
 3261-852 
 
 3262-222 
 
 3262-593 
 
 880 
 
 881 
 
 3262-963 
 
 3263-333 
 
 3263-704 
 
 3264-074 
 
 3264-444 
 
 3264-815 
 
 3265-1 80 
 
 3265-556 
 
 3265926 
 
 3260-296 
 
 881 
 
 882 
 883 
 
 3266-667 
 3270-370 
 
 3267-037 
 3270-741 
 
 3267-407 
 3271-111 
 
 3267-778 
 3271-481 
 
 328-148 
 3271-852 
 
 3268-519 
 3272 22-2 
 
 3268-889 3269'259 
 3272-59313272-963 
 
 32*9-630 
 3273-333 
 
 3270- 
 3273-704 
 
 882 
 8S3 
 
 884 
 
 3274-074 
 
 3274-444 
 
 3274-815 
 
 3275-185 
 
 3275-556 
 
 3275-9|6 
 
 3276-296 
 
 3276-667 
 
 3277H37 
 
 3277-407 
 
 884 
 
 885 
 
 3277-778 13278-148 
 
 3278-519 
 
 3278-889 
 
 3279-259 
 
 3279-flO 
 
 3280- 
 
 3280-370 
 
 3280-741 
 
 3281-111 
 
 885 
 
 886 
 
 3281-481 3281-852 
 
 3282-222 
 
 3282-593 
 
 3282-963 
 
 3283-333 
 
 3283-704 
 
 3284-074 
 
 3284-444 
 
 3284815 
 
 886 
 
 8S7 
 
 3285-185 32^5-556 
 
 3285-926 
 
 3286-296 
 
 32866R7 
 
 3287-037 
 
 3287-407 
 
 32S7-778 
 
 3288-148 
 
 3288-519 
 
 887 
 
 888 
 
 3288-88913289-259 
 
 32^9-630 
 
 3290- 
 
 3-290-370 
 
 3290-741 
 
 3291-111 
 
 3291-481 3291-852 
 
 3292222 
 
 888 
 
 889 
 
 32<)2-593 3292-9<'3 3293 333 
 
 3293-704 
 
 3294-074 
 
 3294 444 
 
 3294-815 
 
 3295-1 85 
 
 3295-556 
 
 3^95-926 
 
 889 
 
 890 
 
 3296-29t> 
 
 3296-667 
 
 3297-037 
 
 3297-407 
 
 3297-778 
 
 3298-148 
 
 3298-519 
 
 3298-889 
 
 3299-259 
 
 3299-630 
 
 890 
 
 891 
 
 3300- 
 
 3300-370 
 
 3300-741 
 
 3301-111 
 
 3301-481 
 
 3301-852 
 
 3302-222 
 
 3302-593 
 
 3302-963 
 
 3303333 
 
 891 
 
 892 
 
 3303704 
 
 3304-074 
 
 315 14 444 
 
 3304-815 
 
 3315-185 
 
 3305-556 
 
 3305-926 
 
 .'3306-296 
 
 3306-667 
 
 3307-037 
 
 892 
 
 893 
 
 3307-407 
 
 3307-778 
 
 3308-148 
 
 3308-519 
 
 3308-889 
 
 3309-259 
 
 3309 630 
 
 3310- 
 
 3310370 
 
 13310741 
 
 893 
 
 894 
 
 3311*111 
 
 153 11-481 
 
 3311-852 
 
 3312-222 
 
 3312593 
 
 3312-963 
 
 3313333 
 
 3313-704 
 
 13314-074 
 
 3314-444 
 
 894 
 
 895 
 
 31314815 
 
 3315-185 
 
 3315-556 
 
 3315926 
 
 3316-296 
 
 331 6-667 
 
 3317 037 
 
 3317-407 
 
 3317-778 
 
 3318-148 
 
 895 
 
 896 
 
 3318-519 
 
 3318-889 
 
 3319-259 
 
 3319-630 
 
 3320- 
 
 3320-370 
 
 3320-741 
 
 3321-111 
 
 3321-481 
 
 3321-852 
 
 896 
 
 897 
 
 3322-222 
 
 3322-593 
 
 33-22-9(i3 
 
 33-23333 
 
 3323 704 
 
 3324-074 
 
 33-24-444 
 
 3324-S15 
 
 3325-185 
 
 3325-556 
 
 897 
 
 893 
 899 
 
 3325-9-2C, 
 3329-630 
 
 3326-296 
 3330- 
 
 3326-6R7 
 3330370 
 
 3327-037 
 333V741 
 
 3327-407 
 3331-111 
 
 3327-778 
 3331 481 
 
 3328-148 
 3331-852 
 
 3328-519 
 3332222 
 
 3328-889 
 3332-693 
 
 3329-259 
 3332-963 
 
 898 
 899 
 
 900 
 
 3333-333 
 
 3333-704 
 
 =3334074 
 
 3334-444 
 
 3334-815 
 
 3335-185 
 
 3335-556 
 
 3335-926 
 
 3336-296 
 
 3336-667 
 
 900 
 
 M.A 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 .6 
 
 7 
 
 8 
 
 
 
 M.A. 
 
 MEAN AREAS 841 to 000. 
 
188 
 
 RULES FOR THE MEASUREMENT OF EARTHWORKS. 
 
 CUBIC YARDS TO MEAN AREAS FOR 10O FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 3 
 
 3 
 
 4t 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 901 
 
 3337-037 
 
 3337-407 
 
 3337-778 
 
 3338-148 
 
 3338-519 
 
 3338-889 
 
 3339-259 
 
 3339-630 
 
 3340- 
 
 3340-370 
 
 901 
 
 902 
 
 3340-741 
 
 3341- }\\ 
 
 3341-481 
 
 3341-852 
 
 3342-22-2 
 
 3342-593 
 
 3342-963 
 
 3343-333 3343'704 
 
 3344-074 
 
 902 
 
 903 
 
 3344-414 
 
 3344-815 
 
 3345-185 
 
 3345-556 
 
 3345-926 
 
 3346-296 
 
 3346-667 
 
 3347-037 3347-407 
 
 3347-778 
 
 903 
 
 904 
 
 1348-148 
 
 3348-519 
 
 o348-889 
 
 3349-259 
 
 3349-630 
 
 3350- 
 
 3350-370 
 
 3350-741 
 
 3251-111 
 
 3351-481 
 
 904 
 
 905 
 
 3351-852 
 
 3352-222 
 
 3352-593 
 
 3352-963 
 
 3353-333 
 
 "3353-704 
 
 3354-074 
 
 3354-444 
 
 3354-815 
 
 rf355-185 
 
 905 
 
 906 
 
 3355-556 
 
 3355-926 
 
 3356-296 
 
 3356-667 
 
 3357-037 
 
 3357-407 
 
 3357-778 
 
 3358-148 13358-519 
 
 3358-889 
 
 906, 
 
 907 
 
 3359-259 
 
 3359-630 
 
 3360- 
 
 3360-370 
 
 3360-741 
 
 3361-111 
 
 3361-481 
 
 3361-852 ; 3362-222 
 
 3362-593 
 
 907 
 
 908 
 
 3362-963 
 
 3363-333 
 
 3363-704 
 
 3364 074 
 
 3364-444 
 
 3364-815 
 
 3365-185 
 
 33 65-556 '3:365-926 
 
 3366-296 
 
 908 
 
 909 
 
 3366-667 
 
 3367 037 
 
 3367*407 
 
 3367-778 
 
 3368-148 
 
 3368-519 
 
 3368-889 
 
 3369-259; 3369-630 
 
 3370- 
 
 909 
 
 910 
 
 3370-370 
 
 3370-741 
 
 3371-111 
 
 3371-481 
 
 3371-852 
 
 3372-222 
 
 3372-593 
 
 3372963 
 
 3373-333 
 
 3373-704 
 
 910 
 
 911 
 
 3374-074 
 
 3374-444 
 
 3374-815 
 
 3375-185 
 
 3375-556 
 
 3375-926 
 
 3376-296 
 
 3376-667 
 
 3377-037 
 
 3377-407 
 
 911 
 
 912 
 
 3377-778 
 
 3378-148 
 
 3378-519 
 
 3378-889 
 
 3379-259 
 
 3379-630 
 
 3380- 
 
 3380-370 3380-741 
 
 3381-111 
 
 912 
 
 913 
 
 3381-481 
 
 3381-852 
 
 3382-222 
 
 3382-593 
 
 3382-963 
 
 3383-333 
 
 3383-704 
 
 3384-074 3384-444 
 
 3384-815 
 
 913 
 
 914 
 
 .'33S5-185 
 
 3385-556 
 
 3385-926 
 
 3386-296 
 
 3386667 
 
 3387-037 
 
 3387-407 
 
 3387-778 3388-148 
 
 3388-519 
 
 914 
 
 915 
 
 3388-889 
 
 3389-259 
 
 3389-630 
 
 3390- 
 
 3390-370 
 
 3390-741 
 
 3391-111 
 
 3391-481 3391-852 
 
 3392-222 
 
 915 
 
 916 
 
 3392-593 
 
 3392-963 
 
 3393-333 
 
 3393-704 
 
 3394-074 
 
 3394-444 
 
 3394-815 
 
 3395-185; 3395-556 
 
 3395-926 
 
 916 
 
 917 
 
 3396-296 
 
 3390-667 
 
 3397-037 
 
 3397-407 
 
 3397-778 
 
 3398-148 
 
 3398519 
 
 3398-889! 3399-259 
 
 3399-630 
 
 917 
 
 918 
 
 3400- 
 
 3400-370 
 
 3400741 
 
 3401-111 
 
 5401-481 
 
 3401-852 
 
 3402-222 
 
 3402-593 
 
 3402-963 
 
 3403-333 
 
 918 
 
 919 
 
 3403-704 
 
 3404-074 
 
 3404-444 
 
 3404-815 
 
 3405-185 
 
 3405-556 
 
 3405-926 
 
 3406-296 
 
 3406-667 
 
 3407-037 
 
 919 
 
 920 
 
 3407-407 
 
 3407-778 
 
 3408-148 
 
 3408-519 
 
 3408-889 
 
 3409-259 
 
 3409-630 
 
 3410- 
 
 3410-370 
 
 3410-741 
 
 920 
 
 921 
 
 3411111 
 
 3411-481 
 
 3411-852 
 
 3412-222 
 
 3412-593 
 
 3412-963 
 
 3413-333 
 
 3413-704 
 
 3414-074 
 
 3414-444 
 
 921 
 
 922 
 
 3414-815 
 
 3415-185 
 
 3415-556 
 
 3415-926 
 
 3416-296 
 
 3416-667 
 
 3417037 
 
 3417-407 
 
 3417-778 
 
 3418-148 
 
 922 
 
 923 
 
 3418-519 
 
 3418-889 
 
 3419-259 
 
 3419-630 
 
 3420- 
 
 3420-370 
 
 3420-741 
 
 3421-111 
 
 3421-481 
 
 3421-852 
 
 923 
 
 924 
 
 3422-222 
 
 3422-593 
 
 3422-963 
 
 3423-333 
 
 3423-704 
 
 3424-074 
 
 3424-444 
 
 3424-815 
 
 3425-185 
 
 3425-556 
 
 924 
 
 925 
 
 3425-926 
 
 3426-296 
 
 3426-667 
 
 3427-037 
 
 3427-407 
 
 3427-778 
 
 3428-148 
 
 3428-519 
 
 3428-889 
 
 3429259 
 
 925 
 
 920 
 
 3429-630 
 
 3430- 
 
 3430-370 
 
 3430-741 
 
 3431-111 
 
 3431-481 
 
 3431-852 
 
 3432-222 
 
 3432-593 
 
 3432-963 
 
 926 
 
 927 
 
 3433-333 
 
 3433-704 
 
 3434-074 
 
 3434-444 
 
 3434-815 
 
 3435-185 
 
 3435-556 
 
 3435-92t> 
 
 3436-296 
 
 3436-667 
 
 927 
 
 928 
 
 3437-037 
 
 3437-407 
 
 3437-778 
 
 3438-148 
 
 3438-519 
 
 3438 889 
 
 3439-259 
 
 3439-630 
 
 3440- 
 
 3440-370 
 
 928 
 
 929 
 
 3440741 
 
 3441-111 
 
 3441-481 
 
 3441-852 
 
 3442-222 
 
 3442-593 
 
 3442-90:3 
 
 3443-333 
 
 3443-704 
 
 3444-074 
 
 929 
 
 930 
 
 3444-444 
 
 3444-816 
 
 3445-185 
 
 3445-556 
 
 3445-920 
 
 3446-296 
 
 3446-667 
 
 3447-037 
 
 3447-407 
 
 3447-778 
 
 930 
 
 931 
 
 3448-148 
 
 3448-519 
 
 3448-889 
 
 3449-259 
 
 3449-630 
 
 3450- 
 
 3450-370 
 
 3450-741 
 
 3451-111 
 
 3451-481 
 
 931 
 
 932 
 
 3451-852 
 
 3452-222 
 
 3452593 
 
 3452-963 
 
 3453-333 
 
 3453-704 
 
 3454-074 
 
 3454-444 
 
 3454-815 
 
 3455-185 
 
 932 
 
 933 
 
 3455-556 
 
 3455926 
 
 3456296 
 
 3456-667 
 
 3457-037 
 
 3457-407 
 
 3457-778 
 
 3458-148 
 
 3458-519 
 
 3458-889 
 
 933 
 
 934 
 
 3459-259 
 
 3459-630 
 
 3460- 
 
 3460-370 
 
 3460-741 
 
 3461-111 
 
 3461-481 
 
 3461-852 
 
 3462-222 
 
 3402-593 
 
 934 
 
 935 
 
 3462-963 
 
 3463-333 
 
 3463-704 
 
 3464-074 
 
 3464-444 
 
 3464-815 
 
 3465-185 
 
 3465-556 
 
 3465-92*. 
 
 3466-290 
 
 935 
 
 936 
 
 3466667 
 
 3467-037 
 
 3467-407 
 
 3467-778 
 
 3468-148 
 
 3468-519 
 
 3468-889 
 
 3469-259 
 
 3469-630 
 
 3470- 
 
 936 
 
 937 
 
 3470-370 
 
 3470-741 
 
 3471-111 
 
 3471-481 
 
 3471-852 
 
 3472-222 
 
 3472-593 
 
 3472-963 
 
 3473-333 
 
 3473-704 
 
 937 
 
 938 
 
 3474-074 
 
 3474-444 
 
 3474-815 
 
 3475-185 
 
 3475-556 
 
 3475-926 
 
 3476-296 
 
 3476-667 
 
 3477-037 
 
 3477-407 
 
 938 
 
 939 
 
 3477-778 
 
 3478-148 
 
 3478-519 
 
 3478-889 
 
 3479-259 
 
 3479-630 
 
 3480- 
 
 3480-370 
 
 3480-741 
 
 :348l-lll 
 
 939 
 
 940 
 
 3481-481 
 
 3481-852 
 
 3482-222 
 
 3482-593 
 
 3482-963 
 
 3483-333 
 
 3483-704 
 
 3484-074 
 
 3484-444 
 
 3484-815 
 
 940 
 
 941 
 
 3485-185 
 
 3485-556 
 
 3485-926 
 
 3486-296 
 
 3486-667 
 
 3487-037 
 
 3487-407 
 
 34S7-77S 
 
 3488-148 
 
 3488-519 
 
 941 
 
 942 
 
 3488-889 
 
 3489-259 
 
 3489-630 
 
 3490- 
 
 3490-370 
 
 3490-741 
 
 3491-111 
 
 3491-481 
 
 3491-852 
 
 3492-222 
 
 942 
 
 943 
 
 3492-593 
 
 3492-963 
 
 3493-333 
 
 3493-704 
 
 3494-074 
 
 3494-444 
 
 3494-815 
 
 3495-185 
 
 3495-556 
 
 3495-926 
 
 943 
 
 944 
 
 3496-296 
 
 3496 667 
 
 3497-037 
 
 3497-407 
 
 3497-778 
 
 3498-148 
 
 3498-519 
 
 3498-889 
 
 3499-259 
 
 3499-630 
 
 944 
 
 945 
 
 3500- 
 
 3500-370 
 
 3500-741 
 
 3501-111 
 
 1501-481 
 
 3501-852 
 
 3502-222 
 
 3502-593 
 
 3502-5)63 
 
 3503-333 
 
 945 
 
 946 
 
 3503-704 
 
 3504-074 
 
 3504-444 
 
 3504-815 
 
 3505185 
 
 3505-55b 
 
 3505-926 
 
 3506-296 
 
 3506-667 
 
 3507-037 
 
 946 
 
 947 
 
 3507-407 
 
 3507-77S 
 
 3508-148 
 
 3508-519 
 
 3508-889 
 
 3509-259 
 
 3509-630 
 
 3510- 
 
 3510-370 
 
 3510-741 
 
 947 
 
 948 
 
 3511-111 
 
 3511-481 
 
 3511-852 
 
 3512-222 
 
 3512-593 
 
 3512-963 
 
 3513-333 
 
 3513-704 
 
 3514-074 
 
 3514-444 
 
 948 
 
 949 
 
 3514-815 
 
 3515-185 
 
 3515-556 
 
 3515-926 
 
 3516-296 
 
 3516-667 
 
 3517-037 
 
 3517-407 
 
 3517-778 
 
 3518-148 
 
 949 
 
 950 
 
 3518-519 
 
 3518-889 
 
 3519-259 
 
 3519630 
 
 3520- 
 
 3520-370 
 
 3520-741 
 
 3521-111 
 
 3521-481 
 
 3521-852 
 
 950 
 
 951 
 
 3522-222 
 
 3522-593 
 
 3522-963 
 
 3523-333 
 
 3523-704 
 
 3524-074 
 
 3524-444 
 
 3524-815 
 
 3525-185 
 
 3525-556 
 
 951 
 
 952 
 
 3525-926 
 
 3526-296 
 
 3526-667 
 
 3527-037 
 
 3527-407 
 
 3527-778 
 
 3528-148 
 
 3528-519 
 
 3528-889 
 
 3529-259 
 
 952 
 
 953 
 
 3529-630 
 
 3530- 
 
 3530-370 
 
 3530-741 
 
 3531-111 
 
 3531-481 
 
 3531-852 
 
 3532-222 
 
 3532-593 
 
 3532-963 
 
 953 
 
 954 
 
 3533-333 
 
 3533-704 
 
 3534-074 
 
 3534-444 
 
 3534-815 
 
 3535-185 
 
 3535-556 
 
 3535926 
 
 3536-296 
 
 3536-667 
 
 954 
 
 955 
 
 3537-037 
 
 3537-407 
 
 3537-778 
 
 3538-148 
 
 3538-519 
 
 3538-889 
 
 3539-259 
 
 3539-630 
 
 3540- 
 
 3540-370 
 
 955 
 
 956 
 
 3540-741 
 
 3541-111 
 
 3541-481 
 
 3541-852 
 
 3542-222 
 
 3542-593 
 
 3542-963 
 
 3543-333 
 
 3543-704 
 
 3544-074 
 
 956 
 
 957 
 
 3544-444 
 
 3544-815 
 
 3545-185 
 
 3545-556 
 
 3545-926 
 
 3546-296 
 
 3546-667 
 
 3547-037 
 
 3547-407 
 
 3547-778 
 
 957 
 
 958 
 
 3548-148 
 
 3548-519 
 
 3548-889 
 
 3549-259 
 
 3549-630 
 
 3550- 
 
 3550-370 
 
 3550-741 3551-111 
 
 3551-481 
 
 958 
 
 959 
 
 3551-852 
 
 3552-222 
 
 3552-593 
 
 3552-963 
 
 3553-333 
 
 3553-704 
 
 3554-074 
 
 3554-444 3554-815 
 
 3555-185 
 
 959 
 
 960 
 
 3555-556 
 
 3555-926 
 
 3556-296 
 
 3556-667 
 
 3557-037 
 
 3557-407 
 
 3557-778 
 
 3558-148 3558-519 
 
 3558-889 
 
 9tiO 
 
 M.A 
 
 O 
 
 1 
 
 3 
 
 3 
 
 * 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 9O1 to 9GO. 
 
RULES FOR THE MEASUREMENT OF EARTHWORKS. 189 
 
 CVJSIC YARDS TO ME AX AREAS FOR 1OO FEET IN LENGTH. 
 
 M.A. 
 
 
 
 1 
 
 3 
 
 3 
 
 4: 
 
 5 
 
 G 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 9G1 
 
 3559-259 
 
 3559-630 
 
 3560- 
 
 3560-370 
 
 3560-741 
 
 3561-111 
 
 3561 481 
 
 3561-852 
 
 3562-222 
 
 3562-593 
 
 961 
 
 9f>2 
 
 3562-963 
 
 3563-333 
 
 3563-704 
 
 3564-074 
 
 3564-444 
 
 3564-815 3565-185 
 
 3565-556 
 
 3565-926 
 
 ,3566-266 
 
 962 
 
 963 
 
 3566-667 
 
 3567-037 
 
 3567-407 
 
 3567-778 
 
 3568-148 
 
 3568-519 
 
 3568-889 
 
 3569-259 
 
 3569630 
 
 3570- 
 
 9613 
 
 964 
 
 3570-370 
 
 :3570-741 
 
 3571-111 
 
 3571-481 
 
 3571-852 
 
 3572-222 
 
 3572-593 
 
 3572-963 
 
 3573-333 
 
 3573-704 
 
 964 
 
 9(55 
 906 
 
 3574-074 
 3577-778 
 
 3574-444 
 
 3578-148 
 
 3574-815 
 3578-519 
 
 3575-185 
 
 3578-889 
 
 3675-556 
 3579-259 
 
 3575-926 
 3579-630 
 
 3576-296 
 3580- 
 
 3576-667 
 3580-370 
 
 3577-037 
 3680-741 
 
 3577407 
 3581-111 
 
 965 
 966 
 
 9137 
 
 3581-481 
 
 3581 -852 
 
 3582-222 3582-593 
 
 J582-963 
 
 3583333 
 
 3583-704 
 
 3584-074 
 
 3584444 
 
 3584-815 
 
 967 
 
 968 
 969 
 
 3585-185 3585-556 
 3588-889 35*9"259 
 
 3585926 
 3589-630 
 
 3586-296 
 3590 
 
 3586-667 
 3590-370 
 
 3587-037 
 3590-741 
 
 3587-407 
 3591-111 
 
 3587-778 
 3591-481 
 
 ".588-148 
 3591-852 
 
 3588-519 
 3592-222 
 
 968 
 969 
 
 970 
 
 3592-593 
 
 3592-903 
 
 3593-333 
 
 3593-704 
 
 3594-074 
 
 3594-444 
 
 o594-815 
 
 3595-185 
 
 3595-556 
 
 3595-926 
 
 970 
 
 971 
 
 3596-296 
 
 3596-667 
 
 3597-037 
 
 3597-407 
 
 3597-778 
 
 3598-1 48 
 
 3598-519 
 
 598-889 
 
 3599-259 
 
 3599-630 
 
 971 
 
 972 
 
 3600- 
 
 3600-370 
 
 3600-741 
 
 3601-111 
 
 3601-481 
 
 3601-852 
 
 3602-222 
 
 3602-593 
 
 3602963 
 
 3603333 
 
 972 
 
 973 
 
 3603-704 
 
 3604-074 
 
 3604-444 
 
 3604-815 
 
 3605-185 
 
 3605-556 
 
 3605926 
 
 3606-296 
 
 3606-667 
 
 3607-037 
 
 973 
 
 974 
 
 1607-407 
 
 3607778 
 
 360S-148J 3608-519 
 
 3808*888 
 
 3609259 
 
 3609-630 
 
 3610- 
 
 3610-370 
 
 %10-741 
 
 974 
 
 975 
 
 3611-111 
 
 3611-481 
 
 3611-852 3612-222 
 
 3612-593 
 
 3612963 
 
 3613 333 
 
 3613-704 
 
 3614074 
 
 '3614-444 
 
 975 
 
 976 
 
 3614-815 
 
 3615 185 
 
 3615-556 3615-926 
 
 36143-296 
 
 3616-667 
 
 3617-037 
 
 3617-407 
 
 3617-778 
 
 3618-148 
 
 976 
 
 977 
 
 3618-519 
 
 3618-8S9 
 
 3619-259 3619-630 
 
 3620- 
 
 3620-370 
 
 3620-741 
 
 3621-111 
 
 3621-481 
 
 3621-852 
 
 977 
 
 978 
 
 3622-222 
 
 3622-593 
 
 3622-963 3623 :;:;:! 
 
 3623-704 
 
 3024-074 
 
 3624-444 
 
 3624-815 
 
 3625-185 
 
 3625-556 
 
 978 
 
 979 
 
 3625-926 
 
 3<526"296 
 
 3626-667 
 
 3627-037 
 
 3627-407 
 
 3627-778 
 
 3628-148 
 
 3628-519 
 
 3628-889 
 
 3629-259 
 
 979 
 
 980 
 
 3629-030 
 
 3630- 
 
 3630-370 
 
 3630-741 
 
 3631-111 
 
 3631-481 
 
 3631-852 
 
 3632-222 
 
 3632-593 
 
 3632-963 
 
 980 
 
 981 
 
 3633-333 
 
 3633-704 
 
 3634-074 
 
 3634-444 
 
 3634-815 
 
 3635-185 
 
 36:35-556 
 
 3635-926 
 
 3636-296 
 
 3636-667 
 
 9R1 
 
 982 
 
 3637037 
 
 3637-407 
 
 3637778 
 
 3638148 
 
 3638-519 
 
 3638-889 
 
 3639-259 
 
 3039-630 
 
 3640- 
 
 3640370 
 
 982 
 
 933 
 
 3640741 
 
 3641-111 
 
 3641-481 
 
 3641-852 
 
 3642-222 
 
 3642-593 
 
 3642-903 
 
 3643-333 
 
 3643704 
 
 3644074 
 
 983 
 
 984 
 
 3644-444 
 
 3644-815 
 
 3645-185 
 
 3645-556 
 
 3645-926 
 
 3646-296 
 
 3646-667 
 
 3647-037 
 
 3647-407 
 
 3647-778 
 
 984 
 
 985 
 
 3648-148 
 
 3648-519 
 
 3648-889 
 
 3649-259 
 
 3649-630 
 
 3650- 
 
 3650-370 
 
 3650-741 
 
 3651-111 
 
 3651-481 
 
 985 
 
 96 
 
 3651-852 
 
 3652-222 
 
 3652-593 
 
 3652-963 
 
 3653-333 
 
 3663-704 
 
 3654-074 
 
 3654-444 
 
 3654-815 
 
 3655-185 
 
 986 
 
 987 
 
 3655-556 
 
 3655-926 
 
 3656-296 
 
 3656-667 
 
 3657037 
 
 3657-407 
 
 3657-778 
 
 3658-148 
 
 3658-519 
 
 3a r )8 889 
 
 987 
 
 988 
 
 3659-259 
 
 3659-630 
 
 3600- 
 
 3660-370 
 
 3660741 
 
 3661-111 
 
 3661-481 
 
 3661-852 
 
 3662-222 
 
 3662593 
 
 988 
 
 989 
 
 3662-963 
 
 3663-333 
 
 3663-704 
 
 3664-074 
 
 3664-444 
 
 3664-815 
 
 3665-1 8f 
 
 3665-556 
 
 3665-926 
 
 3666-296 
 
 989 
 
 9110 
 
 3666-667 
 
 3667-037 
 
 3067-407 
 
 3667-778 
 
 3668' 148 
 
 3668-519 
 
 3668-889 
 
 3669-259 
 
 3669-630 
 
 3670- 
 
 990 
 
 991 
 
 3670-370 
 
 3670741 
 
 3671-111 
 
 3671-481 
 
 3671852 
 
 3672-222 
 
 3672-593 
 
 3672-963 
 
 3673-333 
 
 3673-704 
 
 991 
 
 992 
 
 3674-074 
 
 3674-444 
 
 3674-81) 
 
 3675-185 
 
 3675-55* 
 
 3675-926 
 
 3676-296 
 
 3676-667 
 
 3677-037 
 
 3677-407 
 
 992 
 
 993 
 
 3677-778 
 
 3678-148 3678-519 
 
 3678 889 3679 259 
 
 3679-630 
 
 3680- 
 
 3680-370 
 
 3680 741 
 
 3681-111 
 
 993 
 
 994 
 
 3681-481 
 
 3681-852 1 3682-222 
 
 3682-593 3682-963 
 
 3683-33H 
 
 3683-704 
 
 5684074 
 
 3684-444 
 
 3684815 
 
 994 
 
 995 
 
 3685-185 
 
 3685-556 
 
 3685-926 
 
 3686-296,3686-667 
 
 3687-037 
 
 3687407 
 
 3687*778 
 
 3688-148 
 
 3688-519 
 
 995 
 
 996 
 
 3688-889 
 
 3689-259 
 
 3889-630 
 
 3690- 13690-370 
 
 3690-741 
 
 3691-111 
 
 3691-481 
 
 3691-852 
 
 3692-222 
 
 996 
 
 907 
 998 
 
 3692-593 
 3696-296 
 
 3692-963 3693-333 
 3696-667 3697-037 
 
 3693704 3' 94 074 
 3697(407 3697-77^ 
 
 3694-444 
 
 3698-148 
 
 3694-81 5 13695-1 85 
 3698-519 3698-889 
 
 3695-556 
 3699-259 
 
 3695926 
 3699630 
 
 997 
 998 
 
 999 
 
 3700- 
 
 3700-370 
 
 3700-741 
 
 3701-111 
 
 3701-481 
 
 3701-852 
 
 3702-222 
 
 3702-593 
 
 3702-963 
 
 3703-333 
 
 999 
 
 1000 
 
 3703-704 
 
 3704-074 
 
 3704-444 
 
 3704-815 
 
 3705-185 
 
 3705-556 
 
 3705-920 
 
 3706-296 
 
 3706-667 
 
 3707-037 
 
 1000 
 
 M.A 
 
 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 M.A. 
 
 MEAN AREAS 961 to 1OOO. 
 
 NOTE. This Table having been carefully computed by the Author, through the 
 usual method of successive additions, and verified in the manuscript, was set up by a 
 skilful printer, and the proofs examined, and re-examined, until they were thought to 
 be free from error; finally, the plates were cast, and the revises taken from them sub- 
 mitted, page by page, to the scrutiny of a competent Civil Engineer, who examined the 
 whole, figure by figure, and ultimately reported but few slight mistakes, which were 
 immediately corrected in the plates themselves; so that every precaution having been 
 taken to secure accuracy: the Author feels justified in declaring his belief, that the 
 Table above t entirely clear of any material error. 
 
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