HoGBEN. — Calculating Excavations. 603 



OE represents the product ah in the formula (Euchd, 



vi., 8). 



Mark off EF = CD. 



Then OF represents a^ -\- ah -{■ Jf. 



(1.) We can read this off and multiply by - ; or 



(2.) Use tables giving the content of pyramid with base 

 a^ + ab -\- h^ and height x ; or 



(3.) If the distance between our sections is always the 

 same— say, 162 ft. — we can graduate OH to represent cubic 

 yards — 1 in. = 40 cubic yards — and read off the cubic content 

 at once without calculation. 



If the distance between the faces is not always the same, 

 we may place the scale at the right-hand side, and have a 

 straight edge or rigid steel wire radius movable about a pivot 

 at S. Move this edge or radius about S until its edge is over 

 F. Run the eye along the edge or radius until you reach, 

 say at F, the vertical line corresponding to the perpendicular 

 distance between the faces, given at the top MHN. The 

 reading on the same horizontal line as this gives the cubic 

 content. (This is easily seen to follow by sii:nilar triangles.) 



No calculation whatever is needed, and the whole opera- 

 tion may be as readily performed as any ordinary consultation 

 of tables for rectangular areas and volumes. 



To avoid the use of a rigid wire or edge for radius — 

 (a) A silk thread may be attached at S, and stretched 

 so as to pass over F, and the reading taken as before; or 

 (6), if only one or two distances between the faces are 

 commonly used, scales of cubic yards may be constructed to 

 correspond, and the proper scale may be placed along OH, 

 and the cubic content read off thereon. 



The scales I have chosen are rather large for practical 

 needs ; but it would be easy to put various scales along AOB,- 

 MHN, and at the hne of volumes at the right, to suit different 

 kinds of work. 



Example. — The figure shows the method of calculation for 

 faces of 50 square feet and 70 square feet. With horizon- 

 tal distance 162ft. the content is 358 cubic yards; 216ft,, 

 478 cubic yards, &c. 



The method, it will be noted, is an exact one, not a mere 

 approximation ; the accuracy depends chiefly on the accuracy 

 of the original measurements, and on the accuracy with which 

 the scale can be read.* 



* The scale referred to above can be read much more nearly than 

 probably the areas of the faces can be estimated, for an error of ^ a 

 square foot in each face would give an error of about 3 cubic yards in 

 the result. 



