1816.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



335 



To facilitate the calculation of the corrections to widths, and therefore 

 of the increments to quality, I have arranged a table, available as well 

 when the ground surface is examined in the approximate manner just 

 described, as when the more detailed levels are taken for setting out the 

 earthworks of the line. This table is annexed, and its use will become 

 familiar from following the steps of an example : — 



Taking the measurements from the form of level book attached to the 

 paper on " Selling out Railways," already referred lo (see the Journal 

 for September, p. 277), we lind at the beginning of the first chain's length, 



B +rC = 30-75 ; H ^ 1-9 ; — H = 22 ; i- = 1-5. 

 Then I'5 X 1-9 = 2-85 ; and 1-5 X 2-5 = 33. 



Entering the table wiih 2-85, I find opposite 285 the tabular number 

 •903509; from which, as there are two decimal places iu the number 285, 



I cut oft' two decimals and have for )• H 2'85 "3509. 



And in the same way, entering the table with B + r C =: 30 8, I obtain 



SO-8 0324 ; and taking the difference of these two tabular numbers, 



•3509 — -3324 =-3086, which, found in the table, corresponds to 3-14, 

 the value of .t, additive. 



On the other side, for the value of — .r'. 

 r H' =: 3-3 tabular number 3030 



X 30 8 -0324 



•3354, which corresponds lo 2'98. 

 At the other end of the first chain, B + r C'= 24'45 ; H = 3 ; H' = 1 'O. 

 SxU = 4'5 •• Tab.No. .. -2222 1 9x I J Tab.No.for 2'85 .. -3509 

 deduct for, 24-5 do -0408 add ..-0409 



*, additive = 5-51 •1814 ar' snbtractive = 255 ....'Sgn 



These examples show Ihe use of the table in setting out the earthwork. 



For values of .r, .t', &€., as elements of calculation for correcting the 

 quantity, when the rate of inclination of the ground surface simply is taken 

 in the field, this rate of inclination should be expressed so as to indicate a 

 vertical height iu terms of 1000 parts horizontal : — Thus, if in a chain's 

 length of 50 feet, the inclination of the ground amounts to 1'25 feet, Ihe 

 ratio expressing the vertical height in terms of 1000 parts horizontal is 25. 

 Also, if in the length of a Gunfer's chain = 60 feet, the rise or fall mea- 

 sured 2 2 feet, the ratio in 1000 parts would be 333. This is a common 

 Rule of Three question, thus : — 



£0 : 1-25 :: looo : 25 



which, when using a Gunter's chain, and a level-staff divided into feet 

 and decimals, is abbreviated by converting the height or difference of level? 

 read off on the latter, into links, and multiplying that equivalent by 10' 

 Let us call this ratio of the inclination of the ground b ; — r the ratio of 

 slopes of cutting and embankment, as before, as likewise the other charac- 

 ters which follow. 

 The table should be entered with a number, the result of the formula 



— — , from the tabular number of which we must subtract 



the tabular number corresponding to the valuel^B -f- r C), to obtain the 



value of X ; and, on the other hand, we must add to get the value of x' 



Thus, suppose 6 = 33 3 ; and B + r C = 30.8; r = l-5; 



r6(B +rC) 33-3 X 1-5 X 30-8 o^ u xt 



1000 1^0 =■ l-5«"-Tab.No -6494 



30-8 deduct -0335 



X =• 1-63 '6169 



x'= 1-47 -6819 



Again, if i=7'64 ; r=2 ; B + r C=35-2 ; 



7-64 X 2 X 35-2 

 Yoiio ^ -538.... Tab. No -1859 



352 deduct -0284 



» additive 6'35 .... ■1575 



;r'subtractive 4'67 .... '2143 

 It is not altogether foreign to the subject of this paper to present some 

 applications of the prismoidal formula, which, as cases of frequent occur- 

 rence, are deserving of being noted for their simplicity, and for the brevity 

 of their solution in numbers; especially as they are seldom, if ever, met 

 with in works on mensuration. 



1. In a triangular prismoid, of which the area of the base is called B, 



and Ihe three heights perpendicular to the plane of that base h, k', and A", 

 the solidity = B . — ■'. 



2. In a prismoid with a trapezoidal base ; divide this base into two 

 triangles, and call one of the parallel sides of the trapezoid as the base of 

 each triangle; then add together twice the heights above the base of one 

 triangle, lo once the height above Ihe base of Ihe other triangle, and multi- 

 ply tills sum by one-sixth the area of Ihe base of the triangle ; this done 

 for each triangle, and the results added together will give the solidity of 

 Ihe prismoid. Thus, if A, /i", are the heights above the base at Ihe ends of 

 one of the parallel sides of Ihe Irapeznid ; and h'', h'", the heights at the 

 ends of the opposite parallel side; B, and B', being the areas of the tri- 

 angles, having those sides for their bases respectively ; then 



the solidity 



(.2h + 2h ' + h" + h'') (2h" + 2h- + l, + h') 



6 + ^ "i • 



This solid may have one, two, three, or four heights, and the formula 

 admits of further simplication according lo the relation of the heights to 

 each other. 



3. When the four heights, being unequal, Ihe base of the prismoid be- 

 comes a parallelogram of which the area = B, 



the solidity = B 



(h+ h- + h" + h'") 



This last formula furnishes the means of determining the solidity of a body 

 of any extent whatever, terminated by an irregular surface, but of which 

 the base is a parallelogram. With this view, we must suppose it divided 

 by two systems of equidistant vertical planes, parallel respectively to each 

 other and to the sides of Ihe parallelogram ; and if I = one of Ihe equal 

 parts, into which one of the sides of the parallelogram is divided, 



A,Aj, Aj, A3, A4, &c.,the series of vertical dimensions taken through points 

 at the distances I, in the first vertical plane ; 



A', A'l, A'2, A3', A'4, &c., the corresponding vertical dimensions in the 

 second vertical plane; and soon. 

 Then the area of each plane is, 



A = ^{^h + 2h^+2h^+2h,+...A\ 



A'= i A' + 2A\-f 2A',-f-2A'3+.... A'\ . 



and soon. Then if L represent the perpendicular distance between the 

 vertical parallel planes, 



the solidity = -^ (A -j- 2 A' -f 2 A" + 2 A'" -f A"). 



These theorems may readily be deduced as corollaries from the prismoidai 

 theorem, but they are also capable of an independent elementary proof. 

 — See Puissant Traile'de Topographie. 



4. If a line of slope (of a single slope) turns at right angles to its ori- 

 ginal direction, the solidity of the angular portion, when A = height at the 

 angle, and r the ratio of the slope, is 



r A» 

 rAx ^ 



5. If the direction of a double slope and bottom, as a ditch or trench, 

 turns off at right angles, then b being the width of the bottom, the solidity 



of the angular portion is (3 i -f 6 r A) — 2^ — . 



Example. 



In erecting some Iron Works, it became necessary to provide a proper 

 foundation for a quantity of ponderous machinery, by removing a consi- 

 derable depth of cinders, rubbish, and loose ground over the whole site of 

 the proposed buildings, which covered a rectangular space of 150 feet x80 

 feet. 



To calculate the quantity of earthwork to be executed :— The parallelo- 

 gram being set out, each of its long sides was divided into six equal parti 

 of 25 feet, and each of the short sides into four equal parts of 20 feet. 

 Note, — the calculation is much shortened by ditiding each side into an eveh 

 number of equal parts. This is shown on Ihe figure. 



The necessary depth was then ascertained by boring, and a series of 

 levels over the surface of the ground were taken, as shown by the dotted 

 lines, which being reduced to the plane at the intended depth, as a datum, 



43» 



