Tables for Determining the Cubical Content of Earthwork, in the Construction of Railways and ...
15-49 21-78 33-19 5-09 6-08 7-41 9-21 11-86 15-81 22-24 33-88 5-ifl 6-20 7-56 9-43 12-10 16-13 22-69 34-57 5-30 6-33 7-72 9-62 12-35 lG.4tt 23-15 35-27 5-40 6-46 7-87 9-81 12-59 16-79 23-61 35-98 5-51 6-59 8-03 1001 12-84 17-13 24-08 36-70 3-6; 6-72 8-18 10-21 13-10 17-46 24-56 37-4-2 5-72 0-85 8-34 10-41 13-36 17-80 25-04 38-16 5-83 6-98 8-51 10-61 13-61 18-15 25-52 38-89 5-95 7-12 8-67 10-81 13-87 18-50 26-01 39 63 6-00 7-25 e-83 11-01 14-13 18-85 26-50 40 38 6-18 7-39 9-O0 11-22 14-40 19-21 27-00 41-14 6-29 7-52 917 11-43 14-67 19-56 27-50 41-91 6'41 7-6(i a -34 11-64 14-94 19-92 2801 42-li9 6-52 7-80 fl-5l 11-85 15-21 20-28 28-52 43-46 6-64 7-94 9-68 12 07 15-49 20-65 29-04 44-25 6-76 8-OS <l-85 12-26 15-76 21-03 29-56 45-04 6-88 8-23 10-03 12-51 1604 21-40:30-08 45-84 7-00 8-37 10.20 12-72 16-33 21-77 30-61 46-65 36 Table V. For Calculating Sidecutting. breadth V BRBAnrH OF B\SB OF CUTTING IM FRKT. of base in feef • 1 2 002 3 006 4 010 5 015 6 0-22 7 0-30 8 0''40 9 10 11 000 001 0*50 0*62 0-75 1 001 002 0-04 008 013 0-19 0-27 0-35 0-45 0-56 0-69 0-82 2 002 004 0-07 012 017 0-24 0-32 0-41 .0-52 0*64 0*77 0-91 3 0-06 0-08 012 0-17 0-23 0-30 0-39 0-49 0-60 0-72 0-86 101 4 010 013 0-17 0-23 0-30 0-38 0-47 0-57 0-6m-82 0*96 1*12 5 0-15 0-19 0-24 0-30 0-38 0-46 0-56 0-67 0-80 0*93 1-08 1-24 6 0-22 0-27 0-32 0-39 0-47 0-56 0-66 0-78 0-91 1-05 1*21 1-38 7 0-30 0-35 0-41 0-49 0-57 0-67 0-78 0-91 104 1*19 1-35 1-52 8 0-40 0-45 0-52 0-60 0-69 0-80 0-91 104 1*18 1*34 1*51 1*69 9 0-50 0-56 0-64 0-72 0-82 0-93 105 1-19 1-34 1*50 1-67 1-86 10 0-62 0-69 0-77 0-86 0-96 1-08 1-21 1-35 1-51 1-69 1-85 204 11 0-75 0-82 0-91 101 112 1-24 1-38 1-52 1-69 1-86 204 2-24 12 0-89 0-97 106 1-17 1-28 1-41 1-56 1-71 1*88 206 2*25 2-45 13 104 113 1-23 1-34 1-46 1-60 1-75 1-91 2*08 2-27 2*46 2*67 14 1-21 1-30 1-41 1-52 1-65 1-80 1-95 212 2-30 2-49 2*69 2*91 15 1-39 1-49 1-60 1-72 1-86 2-01 2-17 2-34 2-52 2-72 2-93 3-15 16 1-58 1-69 1-80 1-93 207 2-23 2-39 2-57 2-77 2-97 318 3*41 17 1-78 1-90 202 215 2-30 2-46 2-63 2-82 302 3-23 3-45 3-69 18 2-00 212 2-25 2-39 a-54 2-71 2-89 3-08 3-28 3-50 3-73 3-97 19 2-23 2-35 2*49 2-64 2-80 2-97 315 3-35 3-56 3-78 4-02 4-27 20 2-47 2-60 2.74 2-90 306 3-24 3-43 3-64 3-85 4-08 4-ft 4*57 21 2-72 2-86 301 317 3-34 3-52 3-72 3-93 415 4*39 4*63 4*89 22 2-99 313 3-28 3-45 3-63 3-82 402 4-24 4-47 4-71 4*96 5-22 23 3-27 3-41 3-57 3-75 3-93 413 4-34 4.56 4*80 504 5*30 5:57 24 3-56 3-71 3-88 4-06 4-24 4-45 4-67 4-89 514 5-38 5*65 5-93 25 3-86 402 4-19 4-38 4-57 4-78 5-01 5-24 5-49 5-75 602 6-30 26 417 4-34 4-52 4-71 4-91 513 5-36 5-60 5-85 6-12 6-40 6-68 2f 4-50 4-67 4 86 5-05 5-26 5-49 5*72 5-97 6-23 6*50 6-78 7-08 28 4-84 5-02 5-21 5-41 5-63 5-86 6-10 6*35 6-62 6-90 7-19 7-49 29 5-19 5-38 5-57 5-78 6-01 6-24 6-49 6-75 7-02 7-30 7-60 7-91 30 5-56 5-75 5-95 6-17 6-40 6-64 6-89 7-15 7-43 7-72 8-02 8-34 31 5-93 6-13 6-34 6-56 6-80 7-04 7-30 7-57 7-86 8-15 8-46 8-78 32 6-32 6-52 6-74 6-97 7-21 7-46 7-73 8-00 8-30 8-59 8-90 9-23 33 6-72 6-93 7-15 7-39 7-63 7-89 8-17 8-45 8-75 9-05 9-37 9-70 34 7-14 7-35 7-58 7-82 8-07 8-34 8-62 8-91 9-21 9-52 9-85 10*19 35 7-56 7-78 8-02 8-27 8-52 8-80 9-08 9-38 9*68 1001 10-34 10-69 36 8-00 8-23 8-47 8-72 8-99 9-27 9-56 9*86 1017 10-50 10-84 11-19 37 8-45 8-69 8-93 9-19 9-46 9-75 10-04 10*35 10*67 11-01 11-35 11-71 38 8-91 9-15 9-41 9-67 9-95 10*24 10-54 10-86 11-18 11-52 11-87 12-24 39 9-39 9-64 9-90 10-17 10-45 10-75 11-05 11-38 11-71 12-06 12*41 12-78 40 9-88 10-13 10-40 10-67 10-96 11*27 11*58 11-91 12-25 12*60 12*96 13-34 41 10-38 10-64 10-91 11-19 11-49 11-80 12-12 12-45 12-80 13*15 13*53 13-91 42 10-89 1115 11-43 11-72 12-02 12*34 12-67 13-01 13-36 13*72 14*10 14-49 43 11-41 11-69 11-97 12-27 12-57 12-89 13-23 13-57 13*93 14*30 14*69 15-08 44 11-95 12-23 12-52 12-82 13*14 13*46 13-80 14*15 14*52 14*90 15*29 15-69 45 12-50 12-78 13-08 13-39 13-71 14*04 f4-39 14-75 15*12 15*50 15*90 16-30 46 13-06 13-35 13-65 13-97 14-30 14*64 14*99 15-36 15*73 16*12 16*53 16-93 47 13-64 13-93 14-24 14-56 14-90 15-24 15-60 15-97 16*35 16*75 17*16 17*58 48 14-22 14-52 14-84 15-17 15-51 15-86 16-22 16-60 16*99 17*39 17*80 18*23 49 14-82 15-13 15-45 15-78 16-13 16-49 16-86 17-24 17*64 18*04 18*46 18-89 50 15-43 15-75 16 07 16-41 16-77 17-13 17-51 17-90 18*30 18*71 19*14 19-57 Tablb V. ^^ 7^ ASK OFCUl TTno" ^^ kT- -- ~ in fwt. b^^ aittet 36 37 38 39 40 41 42 43 44 45 46 47 8-00 8-45 8-91 9-39 9-S8 10-38 10-89 11-41 11-95 12-50 13-06 13-64 1 8-23 8-69 915 9-64 10-13 10-64 11-15 II -60 12-23 12-78 13-35 13-93 a 8-47 8-93 9-41 9-90 10-40 10-91 H-43 11-97 12-52 13-08 13-65 14-2* 3 8-72 919 9-67 10-17 10-67 11-19 11-72 12-27 12-82 13-39 13-07 14-56 4 8-99 9-46 9-95 10-45 10-96 11-49 12-02 12-57 13-14 13-71 14-30 14-90 5 9-27 9-75 10-24 10-75 11-27 11-80 12-34 12-89 13-46 14-04 14-64 15-24 6 9-56 10-04 10-54 11-05 11-58 12-12 12-67 13-23 13-80 14-89 14-99 15-60 r 9-86 10-35 10-86 11-38 11-91 12-45 13-0! 13-57 14-15 14-75 15-36 15-97 8 10-17 10-67 11-18 11-71 12-25 12-80 13-36 13-03(14-52 15-12 15-73 16-35 9 10-50 11-01 11-52 13-06 12-60 1315 13-72 14-30^14-00 15-50 16-12 16-76 10 io-a4 11-35 11-87 12-41 12-96 13-53 14-10 14-69 15-29 15-90 16-53 17-16 11-19 11-71 12-24 12-78 13-34 13-91 14-49 15-08 15-69 16-30 16-03 17-58 12 11-55 12-08 12-61 13-16 13-73 14-30 15-49 16-10 16-72 17-36 18-00 13 1 1-93 [2-46 13-OJ 13-56 14-13 14-71 15-30 15-9! 16-52 17-15 17-80 18-45 14 12-32 12-86 13-41 13-97 14-54 15-13 15-73 16-31 16-96 17-60 18-25 18-91 15 12-72 13-27 13-82 14-39 14-97 15-56 16-17 16-78 17-41 18-06 18-71 10-38 la 13-14 13-69 14-25 14-82 15-41 l6-0l{l6-62 17-24 17-88118-53 19-19 19-86 ir 13-56 14-12 14-69 15-27 15-86 16-46 17-08 17-71 18-35 19-01 10-67 20-35 18 U-00 14-56 15-14 15-72 16-32 16-93 17-56 18-19 18-84 19-50 20-17 20-86 19 14-45 15 02 15-60 16-19 16-80 17-41 1804 18-69 19-34 20-01 20-60 21-38 20 14-91 15-49 16-07 16-67 17-28 17-91 18-54 19-19 19-85 20-53 21-21 21-91 21 15-39 15-97 16-56 17-17 17-78 18-41 10-06 19-7! 20-38 21-06 21-75 22-45 22 15-88 16-46 17 06 17-67 18-30 18-93 19-58 20-24120-91 21-60 22-30 23-01 33 16-38 16-97 17-57 18-19 18-82 19-46 20-12 20-78 21-46 22-15 22-86 23-57 24 16-89 17-49 18-10 18-72 19-36 20-01 20-67 21-34 22-02 22-72 28-43 2415 25 17-41 18-02 18-64 19-27 J9-91 20-56 l'l-23 21-91 22-60 23 30 24-02 24-75 26 17 95 18-56 19-19 19-82 20-47 21-13 21-80 22-49 23-19 23-00 24-62 25-35 27 ie-30 19-12 19-75 20-39 21-04 21-71 22-30 23-08 23-78 24-50 25-23 25-97 28 ig-06 19-68 20-32 20-97 21-63 22-30 22-99 23-69 24-40 25-12 25-85 26-60 29 10-63 20-26 20-91 21-56 22-23 22-91 23-60 24-30 25-02 25-75 26-49 27-24 30 20-22 20-86 21-51 22-17 22-84 23-53 24-22 24-93 25 66 26-39 27-14 27-90 31 20-82 21-46 22-12 22-78 23-46 2416 24-86 25-57 26-30 27-04 27-80 28-57 32 21-43 22-08 22-74 23-41 24-10 24-80 25-51 26-23 26-96 27-70 28-47 29-24 33 ■2205 22-71 23-38 24-06 24-75 25-46 26-17 26-90 27-64 28-39 20-15 20-03 34 22-69 23-35 24-03 24-71 25-41 26-12 26-84 27-57 28-32 29-08 20-85 30-64 35 23-34 24-01 24-69 25-38 26-08 26-80 27-52 28-27 29-02 29-78 30-56 31-35 36 24-00 24-67 25-36 26-06 26-77 27-40 28-22 28-97129-72 80-50 31-28 32-08 37 24-67 25-35 26-04 20 -751 27 -46 28-19 28-93 29-69 30-45 31-23 32-02 32-82 38 25-36 26-04 26-74 27 -45128- 1 7 28-91 29-65 30-4ll31-19 31-07 32-77 33-57 39 2606 26-75 27-45 28-17|28-90 29-64 30-39 31-15 31-93 32-72 33-52 34-33 40 26-77 27-46 28-17 28-9o!29-63 30-38 31-14 31-91 32-69 33-40 34-30 35-12 41 27-49 28-19 28-91 29-64130-38 31-13 31-90 82-67 33-46 34-27 35-08 35-01 42 28-22 28-93 29-65 30-39 3114 31-90 32-67 33-45 34-25 35-06 35-88 43 28-97 29-69 30-41 31-15 31-91 J2-67 33-45 34-24 35-04 35-86 36-69 37-53 44 29-72 30-45 3119 31-93 32-69 33-46 34-25 3504 35-85 36-67 37-51 38-35 45 30-50 31-23 31-97 32-72 33-49 84-27 35-06 35-86 36-67 37 50 88-34 3919 46 31-28 32-02 32-77 33-52 34-30 35-08 35-88 36-69 37-51 38-34 39-10 40-04 47 32-08 32-82 33-67 34-33 35-12 'la-91 36-71 17-53;38-35J39-l9 38-38139-21 40-06 40-04 40-91 4fl 32-89 33-64 34-40 35-16 35-95 36-75 37-56 40-91 41-78 49 33-71 34-46i35'23 36-00 36-80 37-60 38-41 39-24 4O-0e[40-93 41-80 42-67 50 »4-54 35-30l36-07 36-86 37-6S 38-46 39-28 40-12 40-96 41-82 43-60 43-57 40 Table VI. u. In. 3 0-0006 13 0-0298 25 6 6 0-001 1 3 0-0304 9 9 00017 6 0-0310 ^^ ^h ^% 9 0-0315 26 1 0-0023 3 3 0-0029 14 0-0321 6 6 0-0034 3 0-0327 9 0-0040 6 0333 9 0-0338 27 2 0-0046 3 3 0-0052 15 0-0344 6 6 0-0058 3 0-0350 9 0-0068 6 0-0356 ^\^\ ^\ 9 0-0361 28 3 0-0069 3 3 0-0075 16 0-0367 6 6 0-0080 3 0-0373 9 9 0-0086 6 0-0379 9 0-0384 29 4 0-0092 3 3 0.-0098 17 0-0390 6 6 0-0103 3 0-0396 9 9 0-0109 6 0-0402 9 0-0407 30 5 0-0115 3 3 0-0120 18 0-0413 6 6 0-0126 3 0-0419 9 9 0-0132 6 0-0425 tf% ■ ^\ 9 0-0430 31 6 0-0138 3 3 0*0143 19 0-0436 6 6 0-0149 3 0-0442 9 9 0-0155 6 0-0448 ^% f^ ^\. 7 0-0161 9 0-0453 32 3 3 0-0166 20 0-0459 6 6 00172 3 0-0465 9 9 0-0178 6 0-0471 8 0-0184 9 0-0476 33 3 6 0-0189 0-0194 21 0-0482 3 6 9 0-0200 3 6 0-0488 0-0494 9 9 0-0207 9 0-0499 34 3 0-0212 3 6 0-0217 22 0-0505 6 9 00223 3 6 0051 1 0-0517 9 10 0-0230 9 0-0522 35 3 0-0235 3 6 0-0241 23 0-0528 6 9 0-0246 3 0-0534 9 11 0-0252 6 0-0540 36 3 0-0258 9 0-0545 3 6 0-0263 24 0-0551 6 9 0269 3 0-0557 9 12 0-0275 6 0-0562 37 3 0-0281 9 0.0568 3 6 0-0287 25 0-0574 6 9 0-0292 3 0-0579 9 Tabular Number 0-0585 0-0591 0-0597 0-0602 0-0608 0-0614 0-0620 0-0625 0-0631 0-0637 0-0643 0-0648 0-0654 0-0659 0-0665 0-0671 0-0677 0-0682 0-0688 0-0694 0-0700 0-0705 0-071 1 0-0717 0723 0-0728 00734 0-0740 0746 0-0751 0-0757 0-0763 0-0769 0-0774 0-0780 0-0786 0-0792 0-0797 0-0803 0-0809 0-0815 0-0820 0-0826 0-0832 0-0838 0-0843 0-0849 0-0855 0-0861 0-0866 Base or Sum of Hpights ft in 38 3 6 9 39 3 6 9 40 3 6 9 41 3 6 9 42 3 6 9 43 3 6 9 44 3 6 9 45 3 6 9 46 3 6 9 47 3 6 9 48 3 6 9 49 3 6 9 50 Tabular N urn her 0-0872 0-0878 0-0884 0-0889 0-0895 0-0901 0-0907 0-0912 0-0918 0-0924 00930 0935 0-0941 0-0947 0-0953 0-0958 0-0964 0-0970 0-0976 0-0981 0-0987 0-0993 0-0999 0-1004 0-1010 0-1016 0-1022 0-1027 01 033 0-1039 0-1045 0-1050 0-1056 0-1062 0-1068 0-1073 0-1079 0-1085 0-1091 0-1096 0-1102 0-1108 0-1114 0-1119 01125 01130 0-1136 0-1141 01 147 Table VL IRISH ACRES. 41 Baaeor Base or Base or Sum of Tabular Sum of Tabular Sura of Heights Number Heights Number Heights ft. in. ft. in. ft. in. In the Cattings or Embankments for Railways, Canals, or Common Roads, the Prismoid is that solid which is applicable to the form out out in hills, or filled into hollows, in the line where the road is made. The Rale for finding the solidity of a Prismoid will be found in the description of Table h Where there is a long -line of road to be estimated, the oalcnlation of it in so many Prismoids as must be made in the most favourable ground, is tedious : hence several forms of tables have been invented and used, to render the calculation in extensive works shorter and easier. It is not thought desirable here to enter upon a critical examination of the tables already published on this subject, so as to point out the individual defects or merits of each ; neither is the author inclined to eonjure up false, or common methods, as they are called, which were never found in any mathematical work, and exhibit them, to show the superiority of a true method— the tables he has put forward it is hoped will be found satisfactory, and that is his great object. In selecting a Form for the principal Table, so as to embody the greatest oonvenienoe in the least space, regard was had chiefly to keeping the required parts on one page, so far as is practicable, so that the operator should not be obliged to turn over to other pages for the numbers he requires. Thus, every ordinary slope, in rook or earthwork, is contained in one double page as the book lies open, for 50 feet depth of cutting at intervals of 3 inches, for 20 and 30 feet bases, and with a few more figures of calculation, for every possible breadth of base. The rule that will be g^ven for working out by this table, will require a few figures of calculation in making out the slopes more than might be required by other forms of table : — thus, the content of a foot or a yard in length of the section might be found in a table with two dimensions of heights, one forming a column down at the left hand margin of the page, the other across at the top, as headings ; but to do this as fully as can be done by the present Table II, limiting the bases to a range from 10 to 50 feet, and the heights and slopes as given therein, would require so enormous an extension of the table in this form, that 43,200 pages would be necessary to comprise what is done here in 24 pages ; otherwise some breadth of base, some slope, or some interval of height must be omitted, which is included in the form adopted here. The intervals of height selected are 3 inches ; although one foot is the smallest interval of height required by the standing orders of the House of Commons on parliamentary sections ; yet this is considered too wide an interval for accuracy in practical work, especially where rock occurs, or • 44 where disputes might arise between companies and their contractors. Snppose the piece of catting marked No 2, Plate 1, Fig 1, where the heights given are 35 feet 6 inches, and 45 feet 6 inches, length 100 feet, and side slopes ]^ to 1 : — In those tables which have intervals of whole feet, a number must be taken out either 6 inches above or below the true height — ^if taken at the foot above, the content is 13941 cube yards ; if taken at the foot below, the content is 13380 cube yards, whereas the true content is 13660 cube yards, giving an error of 280 cube yards in 13660 ; or about one forty- eighth part of the whole. The surveyor in taking his levels, enters thenlhs they occur to him, in feet and hundredths, or in feet and inches ; the notes are thus taken to a great degree of accuracy, which is quite necessary for carrying on the section. This degree of accuracy is rendered useless, so far as the calculation of quantity is concerned, by adopting intervals of whole feet in the heights. On the other hand, in some tables the intervals of height are one- twentieth of a foot, and this is thought to be an affectation of accuracy which can never be attained in practice. Every surveyor knows that in the most even natural surface of ground, when he has his staff placed on the surface, it requires a nice eye to tell, in a cross section, whether he be within an inch of the true point or not ; and the case becomes worse when the intervals between the heights are taken, as in some tables, at whole chains, or as in others, at 100 feet asunder ; and yet this inch is nearly double the intervals of height given in the tables alluded to. Here in one case there is too little accuracy, in the other, too much ; and to avoid both, intervals of 3 inches, or ^ of a foot, are adopted in this table, as before stated. With regard to the intervals for the length — some have chosen in their tables 100 feet ; others have adopted whole chains or 66 feet ; either of these may be used advantageously, where the natural surface of the ground is very even ; in surveying over rock or uneven ground, however, it is found that no attention can be paid to regular intervals of this kind, and table? ealoiUated for such intervals become troublesome to manage, when the intervals on the section are different from those in the tables, which is very often the case. For these reasons, a length on the section of one foot is the interval chosen in this table, and this answers for all lengths of interval between the given transverse sections, as will be shown in the rule for using tbe tables, and the examples attached. There is also this advantage in following this form, that m every interval between the sections, the quantity of that interval is found in itself, and this i» useful in arranging the Leads ; so that, as far as is practicable, the quantity of cuttsig may be enough to fill the embankments and not more ; or, if on trial they are fovnd otherwise, that a correction of the gradients may be made to effect that object. This will be foand especially useful in cutting through rock, and could not be done by taUes adapted to equal intervals of one cham, of 100 feet, or any other. Where, however, the ground is favourable, the adoption of equal hitenrals, etpedally that of 100 feet, is very desirable ; it makes a great short cut in tlie caloolationt of the earthworic ; but for tills purpose the surveyor should use a chain of either 50 or 100 feet long, and made wilh links each a foot in kpgtb ..gmoh a chain is perhaps the best for road or canal sarTeyiag^ DESCRIPTION AND USE OF THE TABLES. TABLE I. This is a very general Table, and can be used for all varieties and dimensions of the Prismoid. It consists of two pages only, and the heights are carried np to 100 feet in intervals of 3 inches. There is bat one tabular number, but this can be used either for Cores or Slopes, according to the rule given. There is a Uttle more trouble in calculating with it than with Table II., which being considered the principal Table will be described be- fore it. TABLE IL This is the chief Table for the calculation of Earthwork : It is the same in principle as Table I., but extended and rendered more useful, by having the tabular numbers calculated more near the i^esult required. It is made out for all the usual slopes in Earthwork, from | to 1, to 3 to 1, being 12 different slopes increasing by quarters, having a double page for each. It consists of 5 columns ; the first is for the heights, in intervals of 3 indies ; the second, third, and fourth, are for the Cores or those parts that stand vertically over or under the bases, of 1, 20, and 30 feet respectively ; the fifth column is for the side slopes. The 30 feet base column is considered the principal of the core columns, that being the breadth usually adopted in roads for a double line of rails, and which was selected especially by the Irish Railway Commissioners, as the breadth for their riulroads. The colunn for a 20 feet base is added, this being sometimes the breadth 46 chosen for a single line of railroad, or a country road. But besides this, this column is useful in checking the calculation of other work, and for tMs purpose chiefly it has got a place here. Thus, if the cutting in a line of 30 feet base be all calculated by one person and in one way directly from the tables, the same may be done another way, by using the 20 feet column and adding the number corresponding to 10 feet more of breadth, which may be found by adding half the number in the 20 feet column to itself. The column for 1 foot breadth of base is put in the same page, for the purpose of suiting the table to any breadth of road, so that the calculator having the page for his side slopes open before hl^nind not turn oyer to any other table to complete his work, as will be shewn presently. The calculation of the Core is independent of the slopes ; and that of the slopes is the same whatever be the breadth of base. RULE FOR USING TABLE H. Having found the page headed by the given slope. 1 . In the column answering, to the breadth of Base, find the Tabular Number which is opposite the average Height — 2. Find the Tabular Number in the Slope column corresponding to either of the Heights, and multiply this by the other heights 3. Find the Tabular Number corresponding to the difference of the Heights, and Multiply it by one-third of the same difference — 4. Add those three results so obtained, and multiply their sum by the length in feet — the Product is the content of the Cutting or Em- bankment in cube yards. Example. Required the quantity of cutting in an excavation, the base being 30 feet, and the side Slopes 1^ to 1; the other dimensions as shown in the Section, Plate 1, Fig. 1. FORM FOB ENTERING THE DIMENSIONS FOR CALCULATION. 3 Inter- val of length. 60 100 200 HeighU 0:0 35:6 Tabular Number! 35:6 45:6 f 19-722-^ 35J 45:6^25-2^ «^0 I 2.55 '45-000 2 -528X35 J 0*556 X— ^3 25-278 2-528 X 45^ RenilU. \ 19-722 23-335 43-057 X 60 '45-000 89-744 1-853 Cube Yards 136-597x100 2583 25-278 38-341 13660 63-619x200 12724 Total of Cutting_Gube Yards, 28967 47 la the foregoing Example, having entered the distances and heights in their respective columns, open the page for Slope If to 1, and the first tabular number to be found is that answering to 17 feet 9 inches, the mean height, in the column headed Base 30 Feat. This number 19*722 is entered in the 4th column or that of Tabular Numbers, as also in the 5th, or Results. Since one of the heights is 0, the second part of the rule gives a result of 0, and therefore a dotted line is put in the space. The Tabular Number cor- responding to the difference of heights 35 ^feet 6 inches in the column for slopes is 1-972 ; this is entered next, to be multiplied by one-third of the same difference ; the product|p3*335 is entered in the fifth column. The sum of these two, 43*057 is then multiplied by the distance between the sections — 60 feet, and the product 2583, which is the content in cube yards of the first division, is entered in the sixth column. The next Tabular Number is taken from the 30 Feet Base column oppo- site the average height 40 feet 6 inches, and is 45*000 : it is entered in the fourth and fifth columns. Opposite 45 feet 6 inches, one of the heights in the column for slopes, is 2*528 ; this is to be multiplied by 35^, the height at the other end of the catting : the product 89*744 is placed in the fifth column. Opposite 10 feet the difference of the heights, in the same column is 0*556; this is entered, and multiplied by one-third of the same difference, and the pro- duct 1*853 is placed in the fifth column. The three results now in the fifth column are added together, making 136*597 ; and multiplied by 100 the length, the product is the content of the second division of the cutting — 13660 cube yards. The Tabular Number for the average height of the third division 22 feet 9 inches, as also taken from the 30 feet column, is 25*278 ; this as before is entered in the fourth and fifth oolunms. The result of the second part of the rule will be 0, in this as well as in the first division of the cuttiug. The tabular number for 45 ft. 6 in., the difference of the heights, is found in the column of slopes to be 2*528 ; it is multiplied by one-third of the same difference, and the product 38*341 is entered in the fifth column to be added to 25*278 ; the sum 63*619 is multiplied by 200 the length, and^tbe product 12724 cube yards entered in the content column — ^being the quantity con- tained in the third division of the section. By adding this to the two former contents, the total quantity of cutting in the section is found to be 28967 cube yards. This has been calculated at length, by the formula On page 49), and found to be 28966*5 cube yards, or half a cube yard less ; an error accounted for by carrying the decimals in the table to only three places. It may be here remarked, that the first result in the calculation of each division is the number of cube yards for a foot in length of the core ; and the sum of the other two results is the content of the slopes for a foot in length. When the required dimensions are not found in the Table, or when the base is neither 20 nor dOfeet wide — 1. Suppose the ni^ber corresponding to a base of 26 feet be required, it may be done three ways : — 1. As above — Multiply the number opposite the given height in the 1 foot column by 26 ; or 2. — Multiply the same number by 6, and add the product fo the number for 20 feet ; or 3 — Multiply the same number 48 by 4, and take the product from the nomber in the 30 feet column— any of these three will give the same result. This is for the Core only. 2. When the average height of a cutting or embankment exceeds 50 feet, its tabular number for the core will be equal to twice the tabular number for half that height ; or in this case, instead of entering the table with the height, it may be entered with the base, and the corresponding number multiplied by the average height. 3. For the Slopes — When both heights exceed 50 feet, half one height and double the other will give the same result and bring the dimensions withiu the limits of the Table. Or, take a third of one h«%ht and three times the other, &o. This does not apply to the core, which is based upon the average heights. 4. When the given ratio of the slopes exceeds 3tol, the calculations may be made with a slope of 1 to 1, taking care to multiply the results of the second and third parts of the Rule by the given ratio, before entering them in the result column : or, one-half or one-third the given ratio may be substituted, and the result doubled or trebled before being placed in its proper column. 5. When the slopes on both sides of a cutting or embankment are difPerent, take the mean or average of the two ratios of the slopes, and proceed with this as if both sides were so sloped alike. This and the preceding remark apply also to Table I. The following Example illustrates the first, fourth, and fifth of these observations : — Required the quantity of earthwork in a sea Embankment, 100 feet in length and 10 feet broad at top ; the height, 12 feet at each end, and the sides sloping 1 in 8 next the sea, and 1 in 2 next the land, agreeably to the Section shewn in Plate 1., Fig. 3. • Length. Heights. Tabular Numbers. Results. X 100=3 11 0-8 cube yards • 100 12 12 4-444 ^ 1111X12X2 • *• 4-444 26-664 ••* 31108 The length and heights being entered — opposite the average height 12 feet, in the 20 feet column is 8*889, the half of which is 4-444, which is placed in the third and also in the result column. The average of the two side Slopes, 8 to 1 and 2 to 1, is 5 to 1 ; but as this is not in the Table, take from Slope 2^ to 1 the number 1*111 opposite 12 feet one of the heights, and enter it in the third column, to be multiplied by 12 feet the other height, and by 2 in order to bring the result to an average Slope of 5 to 1. Had the Tabular number been taken from the 1 to 1 Slope column, a multiplier of 5 should be used instead of 2. The result of the third part of the rule will be because there is no difference of heights, as both ends are alike. Then the sum of the two results is 31*108, which multiplied by the length 100 feet, gives a content of 3110*8 cube yards. The true content of this is 3111 cube yards, the small error of one-fifth of a cube yard, arismg from the Table having been carrSd to only 3 places of decimals. ^• ^ 49 TABLE I. This Table is very general, and is applicable to Prismoids of any dimen- sions. It answers for every Slope, for every Breadth of Road or Base, for every Height, and for every Length of Interval. The usual rule for obtaining the content of a Prismoid as stated by Bonnycastle, is — *' To the sum of the areai of the two ends, add four times the area of a section parallel to the two ends and equally distant from them, and this sum being multiplied by one- sixth of the length, will gire the solid content of the prismoid.** This rule though concise in the enunciation, is tedious in its application, and is not stated in terms of the given dimensions* The following Rule, though seemingly longer, requires fewer figures, and is more explicit — Rule fob the Content op a Prismoid. 1. Multiply the breadth of the base by half the sum of the end heights — 2. Multiply together the two end heights, and the product by the ratio of the side slopes-^ 3. Square the difference of the end heights, take one*third of that square and multiply it by the ratio of the side slopes — 4. Add into one sum the three Results so found, and multiply it by the length : The product is the cubic content in the same denomination as the given dimensions — This should not perhaps be the place to introduce a rule in mensuration, even though a new one, but this rule accords with the tables, all being de- " rived from the same formula, and depending on the same subdivision of the prismoid. An operation has been performed by both rules from the following dimensions : — Suppose a piece of earthwork 200 feet long, 36 feet breadth of base, side slope 2^ to 1, heights at the ends, 15 feet 6 inches and 19 feet jr , respectively. The calculation of this by the first rule requires 191 figures, by the second rule 131 ; thus shewing that the working out by the new rule saves one-third of the Figures. ' This rule is derived from the following general expression for the content k of a prismoid : — where L and B represent its length and breadth of base respectively ; H and A, the end heights ; d, the difference of those heights ; and S, the ratio of the side slopes. The whole of the dimensions being given in feet, this expression when divided by 27 to reduce the result to cube yards, becomes I 27 ^ 27 ^ 3 • 27J * The above formula is thus demonstrated :— (Plate 1. Fig 2.) Let D A C O and FACE represent the sections at the opposite end9 of a cutting — draw A N, C I, FP, and E R, at right angles to A C. Four vertical planes passing longitudinally through these hnes, will divide the solid into five parts, yis— 1. The core, or part that stands upon the base, bounded by vertical planes at the sides and ends, as A C I N— -this is a Prism. 2 & 3. The adjoining portions of the slopes :~one, included between the two vertical planes passing through C I and R E, and bounded at the ends by the planes C I R B and C K B, which also are vertical— the other, a similar figure at the opposite side of the core : Each of these is composed of a prism C K B, and a wedge K I R B. 4 & 5. The remainder of the slopes, forming two equal pyramids, having for bases tho 50 To constniot a table from this fonnnla requires only to divide a series of numbers by 27 ; the quotients so obtained will represent the last, or indeed any factor, of each term in the above expression ; and we may denominate the first series of figures as Heights or Depths, Average Heights, Difference •f Heights, Bases, or Slopes ; the annexed tabular numbers being treated accordingly in each case. The term Heights in the table has been adopted as being most general. The form of table selected haying but one tabular number for each height, the whole has been comprbed in one page, including a range of heights up to 100 feet, in intervals of three inches, which is considered quite sufficient for every practical purpose. The first column shews the heights of cutting or embankment, and the second the corresponding tabular number. The first column is continued in the third, fifth, and seventh, &c ; and the second, im the fourth, sixth, eighth, &o. The quantity of cutting or filling in any piece of earthwork of prismoidal form, may be obtained by the table, by attending to the following rule, which consists of four parts, referring to the three members of the aboye formula, and to the lengtiiy which is a common multiplier for each. — RULE FOR USING TABLE I. 1. Multiply the Tabular Number opposite the ayerage height, by the breadth of the Base. 2. Multiply the Tabular Number opposite dither of the heights, by the other height, and the product by the ratio of the Side Slopes. 3. Multiply the Tabular Number opposite the difference of the heights by one-third of that difference, and by the ratio of the Side Slopes* 4. Add the three results so obtained, and multiply their sum by the length in feet ; the product is the content of the Cutting or Embank- ment in cube yards. triangle! F P D and E R O, and terminaliiig at the opposite end of the cutting (Hrem- bankmcnl in F and E, tlie points of Intersection of the ground and side slopes. The content of the first division or core iJ A C^-I±^xLength=B5±*x L (I) The content of the 3d and 8d division! c=2 {^5+cke}xl But 2 |L£+CKE|=K R+K 0=1 Q=K ExC 1=8 h H The content therefore is —S A H X I< (2) The content of the remaining pyramid! -2. EE Ox^=R VxJ=ll E. R O-^^^d. d SXj=S fxL.(3) The sum of these three expressions is then equal to the whole content of the piismold -L-fBEir^+sAH+sl} The formnla and rule apply equally to cuttiogi and to emhankmenti. 51 ExAHPLB. Required the content of the Catting and Embankment, the longitudinal action and dimensions of which are represented in Plate 1, Fig. 1 ; the base in the Cutting being 30 feet wide, and the Slopes 1^ to 1 ; the top of the embankment to be 36 feet wide, and the Side Slopes 2 to 1. FORM FOR THE CALCULATION. Cutting, Base 30 feet, Slopes 1^ to 1. No of Diriiion 1 Length Heights 60:0 100 200 0:0 35:6 35:6 45:6 45:6 0:0 Tabular Numben 0-657 X 30 == ••• 1-315X^XH = 1 -500 X 30 l-315X45iXU 10 0-370 X -5- X If 0-843 X 30 1-685 X-~XU Reiults 19-710 ••• 23-341 43-051 X 60 = 45-000 89-748 1-850 136-598X100 = 25-290 38-333 63-623 X 200 = Content in Cube Yards = 2583 = 13660 = 12724 Total Cutting 28967 Embankment, top 38 feet wide, Slopes 2 to 1. 6 85 115 35 145 0:0 30:6 30:6 60:6 60:6 55:0 55:0 0:0 0-565X36 = ••• 1-130X^-^X2 - 1-685X36 l-130X60iX2 30 l-lllX:s-X2 2-139X36 2-241X55 X2 0-204X 1^X2 1-019X36 ••• 2037X3-X2 20-340 22-976 43-316 X 85 = 60-660 136-730 22-220 219-610 X 115 = 77-004 246-510 0-748 324-262 X 35= 36-684 74-690 3682 25255 11349 111-374 X 145 16149 Total FilUng 56435 52 In proceeding with tlie above form, the number of the Division of Cutting is put in ^he first colunm, the length of the interval between the Sections in the second, and in the third are put the Heights at the ends of the di- vision. The average of the two first heights is 17 feet 9 inches, and its corres- ponding tabular number is 0'657 ; which enter in the fourth column, to be multiplied by 30, the breadth of the base. In this case, since one of the heights is : 0, the result of the second part of the rule would be 0, and it' is in the proper place represented by a dotted line* The difference of the heights is 35 feet & inches, and the corresponding tabular number to this is 1*315 ; which is entered in its place to be multiplied by 35^, and by the ratio of the slope 1 ^, and by one-third. The sum of these two results is 43051, which multiplied by the length in feet, gives 2583 cube yards, which is entered in the sixth column — being the content of the first division of the section. For the Second Division. — The tabular number opposite the average height 40 feet 6 inches is entered, to be multiplied by 30, the breadth of the base. The number corresponding with 35 feet 6 inches is 1'315, and is also to be entered in its place and 'multiplied by 45^ and by 1^. The difference of these heights is 10 feet, and its tabular number is 0*370, which is entered, multiplied by 10, by 1^, and by ^. These three results added make 136*598, which multiplied by 100, gives 13660 cubic yards for the content of this division. For the Thibd Division. — 22 feet 9 inches is the mean height, its tabular number is 0*843, which is entered in the fourth column, and multiplied by 30, giving a result of 25*290. Applying the second part of the rule here, the result is 0, which is represented by a dotted line, as in the first division. The difference of the heights is 45 feet 6 inches, its tabular number is 1*685 ; this is multiplied by one-third of 45^, and that by 1^, the ratio of the slopes — the result is 38*333. This added to the former 25*290, makes 63*623, which multiplied by the length 200 fiset, gives 127^ cube yards as the content of the third division. The three divisions added together make 28967 cube yards, the total content of the Cutting. The true content by the formula is 28966 cube yards, t|p|re being an error of 1 cube yard in this quantity, which is caused by carrying the decimals to only three places. For the Embankment ; — ^in order to vary the breadth of base, or rather in this case the width at top, this is taken at 36 feet, and the Slopes at 2 to 1. The mode of proceeding is precisely the same as that just described, and it is considered unnecessary to repeat the instructions. The Example for the Embankment is worked out at full length, in strict accordance with the rule ; and an inspection of the tabular numbers, showing what they are multiplied by, &c, &o, it is hoped, will be found sufficient for understanding this part of the calculation. The preceding example may be made out from the table in a variety of ways. Another form of calculation is added, in which tabular numbers for different members of the dimensions are used. The three first divisions of the section, Plate 1. Fig !•) will be sufficient to illustrate this. 03 SECOND FORM. No of Division. Length. Heights Tabular Numbers. Results. Content in cubic Yards 1 2 3 60-0 100:0 200:0 0: 35 : 6 35 : 6 45: 6 45 : 6 0:0 (1111x17} o CMllX40i n-686X35jXlJ (0-370X^^ Xli 3 (1111X22} U-685x*f*Xli 19-720 • •• 23-341 43-061 X 60= 44-995 89-725 1-850 258 4 13657 12722 136-570 X 100= 25-275 ••• 38-333 63-608 X200= Total Content 28963 The first colamn contains the number of the diyision, as before; the second contains the lengths between the sections, also as in the former example ; in the third column are entered the heights corresponding to the several lengthai against which they are placed. 