Identifier: manualofprincipl00gill (find matches)
Title: A manual of the principles and practice of road-making: comprising the location, consruction, and improvement of roads (common, macadam, paved, plank, etc.) and rail-roads
Year: 1853 (1850s)
Authors: Gillespie, W. M. (William Mitchell), 1816-1868
Subjects: Roads Railroads
Publisher: New York, A. S. Barnes & Co.
Contributing Library: Northeastern University, Snell Library
Digitizing Sponsor: Northeastern University, Snell Library

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igure presents a perspective view of a tapering prismoidalmass, such as is an excavation of unequal size at its two ex-tremities ; ABCD being the area of its largest end, and EFGII ofits smallest. Conceive a plane, parallel to the base of the cuttingCDHG, to be passed through EF. It would cut the larger end inthe line IJ, leaving below it a quadrangular prism, with equal basesEFGH and CDIJ. Subdivide the remaining figure, by raising thevertical lines IL and JK, and passing a plane through IL and E,and another through JK and F. The interior body thus formedappears wedge-shaped, but is a triangular prism, equal to half thequadrangular prism, which has IJKL for base, and IE or JF forheight. There remain two triangular pyramids,—one with baseALI and vertex E, and the other with base BJK and vertex F. The prismoid being thus dissected, the contents of the quadran-gular and of the triangular prisms would be correctly obtained bymultiplying the sum of the bases or end-areas by one-half the
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APPENDIX. 353 length; but to find the contents of the pyramids, their bases shouldbe multiplied by one-third of their length. The method of calcula-tion which we have employed multiplies the sum of the end-areasof the original figure, (which is composed of the prisms and pyra-mids which we are discussing) by one-half the length ; and there-fore gives a result too large by the difference between a half and athird—i. e., by a sixth—of the product of the bases of the pyra- mids by their length JK X KB + IL X LA JF 2 ^T- Representing by d the difference of the depths of the end cut-tings, the ratio of the side-slopes by a- to 1, and the length of thecutting or filling by I, the error in excess will bed X sd-\- d X sd I _sd^l2 ^6 ~ ~6~ If this be calculated for each mass, and subtracted from the resultspreviously obtained by averaging end-areas, the remainder willequal the result obtained by the correct prismoidal formula, to behereafter examined. Thus, for the mass between stations 1

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