10 BARCLAY STREET, N. Y. CITY..I ~~~~ALL VARIETIES OF GLASS APPARATUS, BOTTLES, FLASKS, ETC. RARE R:E]AGENTS AND FLUXES, MINERALS, FOSSILS, Foil, Wire and Vessels, FURNACES, MORTARSAND RETORTS,' Blowpipe Tools and Reagents in Great Variety. ALSO L VERY LARGE STOCK OF CHEICAL ANDP PHYSICAL APPARATUS. Large Catalogue, Cloth BouSICALnd, $1.50 Each.......... FORMULaI - to. EXCAVATION AND EMBANKMENT BY JOHN WOODBRIDGE DAVIS, CIVIL ENGINEER. USED AS A TEXT-BOOK IN THE SCHOOL OF MINES, COLUMBIA COLLEGE. NEW YORK: J. DICKSON & BRO., BOOK AND JOB PRLNTERS, 13 NORTH WNILMIA STREET. 1876. Entered, according to Act of Congress, in the year' 876, BY JOHN WOODBRIDGE DAVIS, In the office of the Librarian of Congress, at Washington. PREFACE. TH[E METHOD Of calculating earthwork discussed in this volume originated with the necessities for expedition and accuracy, while the author was engaged in making extensive computations of this kind, and from actual study of the ground and the manner in which irregularities extend from one cross-section to another, made with the express purpose of obtaining general and accurate formulae for all shapes that occur in the engineer's practice. Thus the several portions forming the entire plan, which the author now considers complete, were gradually developed in about the order in which they appear. Although it was necessary to consult many works on mensuration and civil-engineering in order to be assured of the novelty of some of the devices, and to curtail the description of others commonly known, direct use has been made of two only. The formula of Henck for correcting the calculation of earthwork on curves has been adopted; and in the note at the end of this treatise the hint has been taken from Professor Gillespie of applying integral calculus to railroad volumes, though this would seem a natural use of that department of mathematics. In both instances, the source and value of the aid is thankfully acknowledged. In the formula for the area of the mere ordinary crosssection there could not be much originality. The rate of slope has been included, and the -values that vary from section to section reduced to two. Also the plan of combining consecutive volumes of equal length in series is one of the first that present themselves to calculators. Beyond this each topic is presented as new. These are, first, the inclusion of ~volumes of minor length in the general series, whereby the inequality in length of conse~etive volumes forms no interruption to a single and speedy operation. Next, the attainment of a single formula for all varieties of irregular and defective volumes, which not only abbreviates the computation of each, but allows them all to be included in the same series with ordinary volumes, so that any entire cut or bank can be estimated accurately in one operation. Approximate formul1e, on account of their simplicity, are used in the combinations, and the results corrected by an exceedingly easy formula, thus obtaining the contents as by the prismoidal rule. Formulh of error are often met with in discussions on mensuration. Prof. Gillespie has investigated formuTl for the errors of all common approximate methods, using thfe prismoid 1'ormula as a standard; but the shape he treats is not very frequent in earthwork, and he does not propose to use the formule as corrections. [In SONNET'S DICTIONNAIRE DES MATHE]MA-'TIQIES APPLIQUIJES ocCurs a formula of difference for earthwork volumnes, identical with that used in this treatise; but it isi exh-ibited as a formula o-f error, not of correction. For' the similar formula to be applied to irregular volumes the author has seen no counterpart. The mode of calculating level-section volumes differs slightly from the best methods in use, one column in the operation being constructed by a simpler rule than the ordinary. Complete formule for the finished work, exactly similar to t1he formule for the original computation, and capable of being joined in series, without interruption, through a whole cutting, whatever its irregularity, is believed to be entirely new. Ordinarily, the original calculations are disregarded as useless for the final estimation, unless the work has been finished exactly to the p-rescribed lines, and the advantage of exact rules is lost to the most important calculation of all. A method is here devised for using the former calculation in conjunction with the final, thus removing a large'bulk of work from the latter, and giving to it the same advantage of accuracy as to the other. An easy correcting fozrmula is'given for the rude method of assuming the centre cuttings as those of level-sections in preliminary 5 estimates, when the work is not yet staked out. The result is the same as that obtained by approximating, with end areas of volumes as actually staked out. Also a method for borrow-pits is described, founded on the principles used for combining road-bed volumes in series. Although the original idea of the author was to secure the maximum amount of brevity by combining everything without exception in series, and then eliminating every constant factor and term from the calculation of each, inventing for this purpose several devices not hitherto used, yet he has not allowed himself at any point to be tempted away -from exact work, deeming this to be of the first importance. The clain for the method is absolute accutracy joined with what brevity is shown to arise from the use of the resulting formule. Special attention is cralled to the treatment of irregular volumes, the discussion of which is so provokingly limited and obscure in most books. These are as methodically and exactly calculated as the more regular volumes. A general discussion of this subject will be found in thle note at the end of this treatise, where the errors of neglect and false assumption are shown and estimated. In inviting the attention of engineers to this methllod the author considers it. fair to state in its behalf the fact that upon carrying, as a perfect, stranger, the manuscript of the former part of this work to Gen. Francis L. Tinton, head of the Department of Civil and Mining EEngiDeering, School of lines, Columbia College, that official immediately adopted it, using it in his lectures of the winter of'75-6, and recommending it, in the shape of a text-book, for use in that Institution.' To Gen. Vinton the author must acknowledge his great obligations, for vclduable advice in the constrtuction of the work and the manner of presenting the discussions, for important research through our own and foreign methods of cornputing earthwork, and for that more material aid, without which this volume could not at present have been published by the efforts alone of its AJUTTHOB FORMULAE FOR THE CALCULATION OF RAILROAD EXCAVATION AND EMBANKMENT. TREATMENT OF INTERMEDIATE STATIONS. THE manner of calculating regular cross-sections of excavation and embankment, contained by uniform slopes, has been reduced to formnula by many authors, representing the operation in concise form; and these formula, modified by the third dimension, length, have been moulded to express the content of a volume between two such sections, and even the bulk of a series, indefinitely extended, of such volumes, lying consecutive between cross-sections equi-distant, the width of roadway and rate of slope, of course, remainining the same throughout the whole length. We now propose to unfold a method of comnputing by formula the contents of a series uninterrupted by the presence of vols. however unequal in length, and show the advantage attending this plan, after revieving as briefly as possible the method now in use. The accompanying diagram represents the crosssection of a railroad cut, b being half the width of It -._ 8 road-bed, c centre-height, r elevation of right slopestake above grade, r' its horizontal distance from nearest side of road-bed, 1 elev. left slope-stakle, l' its horizontal dist. from nearest side of road-bed, and w the entire top-width or horizontal dist. between slopestakes. The area of this section is evidently rb + Mb + c(r' + b) +1-lc(l' ~b). Let S denote the ratio of slope: then S=-,, and r=Sr', 1= Si'. Substituting and reducing, _Area Section S'b(r' + I') + Ic(r' 1' + 2b). Adding and subtracting S62 do not change its value; A. nreca Section - Sb("'+'+ 2b) - Sb2+ +c(r'+'+ 2b), or 4- Arnea Section- w(c + Sb) - 2. Supposing w', c' to represent width and centre at next station, the area of its cross-section may be expressed by a formula similar to the above: half the sum of these, multiplied by dist., D, between, and divided by 27, gives a near approximate of the volume bet. in cu. yds. Vol.- (wc + w'c' + Sb(w + wt')-4Sb2)y * Add two consecutive volumes of equal length by means of the general formula, w",c", representing the width and centre of third cross-section: Vol. (tvc + 2w'c' + w"c" + Sb(w + 2w' + w") - Sb2)TT. By continual addition we may get a formula for the sum of any number of consecutive volumes; but, letting n1 denote the number of volumes, we may at once indite a general formula for the calculation of any number of volumes consecutive. Thus we have T { w-ec + 2vC+ rc' +&c. + 2ivCnn + I.nlCi+l, D ~' *- Jr- Sb(w + 2w' +- &c. + 2wn + wn+l)-4Sb2n n 108 Divide and multiply by 2 to convert formula into more convenient shape, which now may be expressed ~mid-prods. + - end-prods. D W VoL - + Sb)(mid-widtks -V A eind-wtidt]s) - 25b' x no. o' vols. Let us illustrate this formlula by applying it to the following extract from a field book, containing colunmns of stas., centre cuts, left and right heights and dists. of slope-stakes, the road-bed being 18 ft., slope, 1 to 1, dist. apart of stas., 100 ft.: STA. CENTPBE. LEFT. RIGHT. 1 3.0 Tu. 2 5.1 i. 6 - 9 6 -(. 7.5 4 7'2 T 8.7 T- () 5 9. 0 9 o r6 (; @ 7 7.3 64t 6 6.7 T~.3 T.. 4 OPEBATION. STA. WIDTHS. CENTRE. PRODS. 1 1.1.55 x 3.0 - 34.65 2 27.8 x 5.1 = 141.78 3 31.5 x 6.4 = 201.60 4 34.1 x 7.2 _ 245.52 5 38.0 x 9.0 342.00 6 15.85 x 6.7 - 106.195 9 or Sb x 158.80 1071.745 1429.2 -810. 9)169094-.5 6)18788.3 3131.38 cu. yds. As stated in ina'or1luetory paragrapl, the method of combining consecutive vols. of equal leigtli in series is common; but the formulrel used for this purpose are various. We have been able to find none iore concise and fit for combiLnation tllat those of this paper. For iustanc e, as an example fi'omn a popular book, Mr. Hfenck, to calculate above series, would use a table of ten columnis. [See example on page 105 of his FIELD BooK.] Three of these are cols. of prods.; five cols. must be summed, the first one three separate times, omitting certailn values in each addition, the second also three times, and the third twice, raking in effect ten cols. to be added. Tlle work under the cols. is proportionally long. In the formulae of this paper the variables are reduced to two, the width and centre, and all the other values, being constant, are eliminated from the calculation of each cross-section and used each once only in a series. Evidently by this method an entire cutting or bank cannot be considered in a single series, if it contain one or more vols. of minor length, which not only interrupt the series, but must themselves be each calculated singly. From this it would seem at the outset a great convenience to be able to compute the whole cut or fill in one operation, regardless of the no. of minor vols. or the variety in their lengths; but the nature and extent of this advantage will become more clearly apparent as we discuss the manner of obtaining it. Let a represent the area of one end of a vol. whose length is ID; let b represent the area of the other end, and c the area of an intermediate cross-section, distant D' from a. Then yol. x +- X (D- D') —D (b +c)+AD'(a- b). C. Representing another intermediate by d, we obtain in the same manner for the vol. ID nb _L + ILD1 nd - b ) Waa N c_. tAILI Wt third —..,.h With third interm., e, we have ID(b4c) +-XD'(d- b) +!D"(e-c) + OD1"(a-d). It is noticed that the first terms of these formulm are identical, being in every case the prod. of half the entire length of vol. by sum of areas of last interm. and last end. The remaining terms we call the corrections for interms., each term being the correction for one interm., having for its coefficient the dist. of that interm. from first end, and for a quantity within the parenthesis the difference of sectional: areas next one on each side. This arrangement of values remains unaltered, considering any no. whatever of interms. Substituting in place of symbols a, b, c, &c., in the expression for three intermns. the formula for areas of railroad cross-sections, the positions of v, w', &C., marked in diagram, the centres of same sections accompanied by corresponding strokes, and reducing, we obtain for content of vol. with three interms this expression, (wIC + w'c'+ Sb(w" + w')- 4 Sb)10 + W"'c"' -w'c'+ Sb(w -w) 1-) + w c - w' +Sb(w-w) 1\8 The expression representing ordinary process is iD r~ (r ""c+VIC//""//+tq + sb +(IV ) -4Sb 2 108;""c""+V "'c"' + Sb(V"" I+U"') -4S!' ) iD' — i-'!10 +i ("'C"' + wl"C"l + Sb(w"' + ")- 4S2 ) _lO L + Iw"c" + V'c' +Sb(~" I+C') 48%2) ) Ds These expressions contain each four principll terms: of the latter, each term includes the subtraction, -4sb2, while only one term of the former contains this subtraction; and this term, having for coefficient the length of a full vol., D, we intend to combine in series with the other full vols. of cut or fill, where this subtraction will be eliminated fromn the termn and supplied by the common subtraction of the series. Therefore, by this new method of calculating minor vols., the subtraction, -4sb2, otheriwise unavoidably present in tlhe formula of each, is virtually eliminated from all. Again, three terms of the former contain the differences of sectional prods. and widths, instead of the sums, a very important fetature of brevity in favor of that expression. Furthllerlore, for coefficients the first, expression has D, the length of full vol., always 100 ft. or sone other easy factor, and D', D", D"', tlhe act-ual (lists. of inltermso as recorded. in field-book; whlereas the lhatter has only one coefficient as noted, the ottler three being found by subtraction. These points of brevity belonlg to this method when tl'eating a vol. with single intternl., but increase in value with the number of JnlteAms8. As lately remarked, thefirst term, whose coefficient is D, we shcall include with the general series, leaving the renmaining terms to serve as a correction. This term, as before shown, is the same whatever be the inumber of interms., always being similatr to the formula for the whole vol. except tllat the prod. and width of last interm. are used instead of those of first end. The formulr of correction may be reduced to the following RULE OF COR. for any no. of interms., being the ant. in cu. yds. to be adcled to the content found by areas of last 13 interm. and last end, multiplied by hal:f thie length of entire vol., to find the true content of a, vol. which has any no. of intermediate cross-sections. fui.ltipiy t/he dist. of' each inlterm. from first end by, the Sultm of these two diffe'rences, viz: the difference of the prod,. of the section next before and tlhe prod. of the section, next followinlg, ad the difference qof the sectional width next be/bore cqd widthv of' the next section./foblowing, the latter' cfference mnultiplied by S'). Divide all by 108. Now, it is clear that to calculate the con-tents of a cut or fill we f.may regard it as having full stas. only, and use the original rule for solution, with the exception that vols. containing interms. must be calculated by means of last interm. and last end cross-sectionls instead of by end areas; and afterward apply the rule of cor. for intermys. just enunciated. It only remains, then, to reconstruct the original rule so as to provide for the above-menitioned exceptibn to the former plan, when we shall hbrave a guide to a very simple and uniform method of calculating the contents of a cut or fill. If the first vol. of cut or fill contain interms., by the foregoing exception we disregard the width and prod. of first cross-section and use instead those of last interm.: but, if internms. occur in a middle vol., half whose first end-section belongs to t-he vol. next before, we omit half only of its prod. and width, and use instead those of last interm. This being the only alteration in the original formula for series of equal-lengthed vols., we shall now restate it, including tle exception for interm. sections. RULE for a near approx. calculation of the entire contents of a cut or fill bet. end full sias. Acld clt the nidcwidths, except those followed by interms., /f' which add the hacf-wic-thl increased by the ha.f-wvidths of last inierms. cAdd to this stum the haclf-width of last endsection; and thie hal/f-width cf first end-section, except it be.followed by interms., in which case omit it and add instead the haclf-width qf last interm. litiply this sum by Sb, Add to the above the full prods. of sections where full,widths have been used, ha!f-prods. where haf-widths have been used, and no prods. where no widths have been used. 14 From this entire sum subtract 2 Sb2 multiplied by the no. of vols. bet. full stas. Multiply remainder by dist. bet, consecutive full stas., and divide by 54. Apply the rule of coro for interms. Let us apply this rule to the following extract froom field-book, where the full stas. are 100 ft. apart, the slope 1 to 1, and roadbed 18 ft. wide. STA. CENTRE. LEFT. RIGHT. 1 4.1. +40 5.2 r-. 2 5.4. 3 5.9:35 2 4 6.3 74.6, 27 4.5 T1s +- 27 Xt8 <3T7 rF 2- + 74 7.26 7 r + 08 7.6 _-. 7 s 7.0 -6:7.- 7 T 5'.'7 TF. — 6 8.0- 9 s:1 7 8.6 -8 7.- +19 7.7 T- 1 - T + 50 7.4 7I-: 51 8 6.9.. +25 6.8. ~r 5 6 ~1.4 +75 6.0 73 20 9 5.1 T 4+ 10 4.8 - o0fQ TY.+ V STA. WIDTHS. CEN. PRODS. CORRECTIoN. 1 24.8 4.1 101.68 +,io 13.75 5.2 71.5 2902 2 27.6 27.6 5.4 149.04 149.04 3 28.7 5.9 169.33 4 15.0 30.0 6.3 94.5 189.0 — 27 27.2 4.5 122.4 - 1516 + 74 31.8 7.2 228.96 -12996 + 8 16.35 32.7 7.6 124.26 248.52 + 700 5 31.7 31.7 7.0 221.9 221.9 6 34.0 8.0 272.0 7 17.7 35.4 8.6 152.22 304.44 +19 33.2 7.7 255.64 + 2209 +5o 15.45 30.9 7.4 114.33 228.66 + 2521 8 15.85 31.7 6.9 109.365 218.73 25 31.9 6.8 216.92 + 2363 + 75 13.65 27.3 6.0 81.9 163.8 +11358 9 25.1 25.1 5.1 128.01 128.01 10 11.55 4.8 55.44 266.4 1743.795 -17414 2397.6 + 19051 --- 1458. + 1637 + 8.185 9)269158.0 6)29906.4 4984.4 cu. ydso 16 MODE OF OPERATION. Make col. of stas. and interms. as in field-book. Opposite under word WIDTHS make two cols., the second of full widths to be used for cor., the first of widths and half widths according to rule. The rule of cor. for interms. considers full widths only; therefore, since the first col. may contain either full or half widths, it is convenient to fill a second col. solely with full widths, where required, especially for the cor., the places where these full widths are necessary being the sections next one on each side of each internm. The first col. has, according to rule, entire widths at mid stas. and halfwidths at end stas., with the exception that, if the filst sta. be followed by interms., the half width of last interm. is used instead, and, if a mid sta. be followed by interms., half only of its width is used, aind in place of the other half is used the half width of last interni. section. These cols. are formed together by adding mentally fronm field-book, setting down the widths and half widths in each col. as required. But where so many interms. occur as in this exaimple it is more convenient, perhaps, to fill the second col. completely, since it must be nearly filled as it is, and afterward transfer the widths and half widths to first col. Next under word PRODS. ConstruCt two co1s., each containing the prods. of centres by the values in corresponding col. of widths. Both these cols. are also formed at once by multiplying each centre by most convenient number opposite in cols. of widths, setting prod. in corresponding col. of prods., and multiplying or dividing by 2, merely transferring or entirely omitting, as the case may be, to find the prod. for the other col. Add the first col. of widths and the first col. of prods., as before. For the cor., multiply the dist. of each interm. from next full sta. before by the dif. of prods. in second col., one on each side of interm., added to 9 times [=Sb] the dif. of widths, one on each side of internm. in second col. of widths, always subtracting the lower from the upper, using the consequent plus or minus sign, and setting result in col. of corrections. Since this col. must be divided by 108, it is unnecessary to consider decimals: 17 for, assuming the decimals to average half a unit each, 216 interms. must be present in a cut to make the sum of these fractions amount to 108 units, and this is merely equivalent to 1 cu. yd. Moreover, as these corrections have opposite signs, a far greater no. of interms. would be required to make a dif. of a cu. yd. by their decimals. For first interm., add (101.68-149.04), or -47.36, to (24.8 —27.6) x9, or -2.8 x9, or -25.2. Tile sum is -72.56: this multiplied by 40 is -2902. For internm. at sta. 4+88, we have 228.96-221.9+9 x (31.8-31.7) -7.9. 7.96 x 88-700. Instead of dividing total coy., 1637, by 108, transfer htalf to results under col. of prods., which have for a divisor 54; but, since these results must be multiplied by 100, move the decimal point of correcting quantity two places to the left. Concerning the brevity of this method, we have already demonstrated its advantages when applied to single vols. witlh interrn. cross-sections; and of course all the advantages of the parts are collectively the advantage of the whole system, -vwhen used to calculate in one series the entire cut or fill, over the old method of dividing into several series and many single vols. But there are, besides those already discussed, certain merits peculiar to the system as a wvllole thatv were not noticeable while considering the single vols. alone. These merits consist rather in avoiding several disadvantages, which in the old method are unavoidable. Thus, by the old method, the work being computed in separate series and single vols., the end sections of these series and vols., except the first and last crosssections of cut, are common each to two series or vols., and must each accordingly be included in the calculations of both; so that to estimate a cut or bank containing numerous interms. necessarily the prods. and widths of many cross-sections must be treated twice. Again, it is evident by the mere mention that a system of numerous separate series and vols., the quantities of each by itself to be added and modified by the common factors of the series or vol., the final results of all again to be combined, necessitates the employment of more time and greater space than does 2 18 this method of compacting all the quantities in one set of cols., once to be added, and once only to be modified by the factors common to all. It is likewise as evident that this method of calculating an entire cut or bank in one concise table is more neat, scientific and convenient than the ordinary manner of separate calculation; and that it is therefore also more fit in shape to preserve these calculations. For these reasons, illustrating the advantages of brevity and convenience, we advocate thile practice of this system of calculation, wherein all the full Vols., whether pure or containing intermediate cross-sections, are computed together by on.e rule, and all the interraediates'by one correction. The value of these points of brevity, stated here in the abstract, will be fully appreciated if the above example be examined with this regard. Without using the plan of including intermrs. hardly any of the advantage of combining vols. can be gained, because scarcely two together of the example have equal length. The first vol., 40 feet long, must be considered alone, its widths and prods. separately added, the former sum mutiplied by Sb, then added to tlie latter, a subtraction made, and finally the result mnust be multiplied by 40. In fact, nearly all the work ndler the cols. of the example as solved by corrections must be done for that single vol. The same is true of the next vol, 60 ft. long. The next two vols. may be calculated together, the foliowing four singly, the next two together, and all the rest singly, making fifteen separate operations to be performed, each containing all the elements of the whole example. Again, the cross-section at sta. 1 +40 must be, used to find the content of first vol.; it must also be, used in another calculation for the second vol. The: section of sta. 2 must be used with second vol., and also, with the following series. Sta. 3 need be used but; once. With the exception of the first and last stas.. always, and in this case of stas. 3 and 6, every seetion must be used twice. This alone requires 14, extra lines. And, since the calcalatioQn thus con-, 19 sists of 15 distinct operations, each requiring for the sum of its procls. one line, a second for sum of widths, a third for the subtraction, a fourth for the sum of these three, and a fifth and sixth to find the prod. of this sum by lengtlh of vol., tlere must be 15 x 6, or 90, lines to accommodate the operations under the cols., instead of the 7 of our illustration. This is an addition of 83 lines, which with the other 14 make, besides the pr'obable intervals between the different operations, the last a consumption of space not time, 97 extra lines for the calculation. After all this the results of the sinlgle vols. nmust be added and the sum divided by 108 to find the cu. yds., and thle sum of thle results of the different series must be divided by 54, the last two suims finally combined to mak:e the answer, this summitng of results being equal in space to a col. of our illustration. The relative amounts of actual figuring is thus perceived. Of the method by correction the extra col. of wid;ths and thlit of prods., although incrieasing the apparent bulk of the ope~ration, contain no real extra labor except tlhe mere writing of the numbers; for, whether to find the full or hatlf widths of sections for first col., the full widths mnst of course be first found, and we have only to set these in the second col. where required. In cols. of prods. the full proCls., half plods. and blank spaces occupy positions corresponding with the full and half widths and blank spaces of the cols. of widths, requiring a mere transfer from one col. of prods. to the other, or at most simply a doubling or halving of values. By the old method as many operations must be performed, wnith a dist. less than 100 ft. as a factor, as there are minor vols., 12 in this example: by the correcting method a dist. less than 100 ft. is used as many times only as there are interm. stas., 8. Regardingg the differences of widths and prods., used in the new method, opposed to the sums of these, as:used in the old, let us examine the example. Subtracting one width in the second col. of widths from the second above, in every instance cancels the figure of the tens place, leaving only a figure in the units place, with generally the decimal; while some g20 times, as for the interm. 4+88, where 31.7 is taken from 31.8, both the figures of tens and units places are removed. But to add two widths gives a far more considerable number to use as a factor. Differences in the col. of prods. in almost every case lose the figure of hundreds place, and occasionally of the tens place also, as for interm. 4+88. The sum of two prods. is a much more troublesome number to use. 21 CORRECTION OF APPROXIMATION. THE foregoing method. of calculating earthwork is approximate. To find the true contents we use a formula of correction, which is obtained in the following manner. It is well known that the exact content of a vol. of earthwork is obtained by use of the prismoidal formula, whether the ground be a right plane, or the more general hyperbolic paraboloid or warped surface. That is, to find true content add to the sum of end areas four times the area of a cross-section midway bet. ends, andi multiply sum by - the length of vol. Let: w(c + S) -- Sb2, w'(c' + Sb)-Sb2 be the end areas of a vol. of earthwork. Then -- (w+w'), ~(c-+c') are evidently the width and centre of mid-section, and its area is ( +,)( + Sb Sb2. Multiply this by 4, add thereto the end areas, multiply all by - D, and divide by 27, to obtain in cu. yds. ol.2(e + w'c') + (we' + w'c) D True vol.= +3Sb(wv+ w,') -12Sb2 324' This formula would prove very unwieldy to carry through the calculations for series of vols., especially in the consideration of interm. stas. The same results may be obtained in a simpler manner by using the dif. bet. true and approx. contents as a correction. Subtracting the approx. vol., (wc + w'c' + Sb(w + w') - 4Sb) 2 from the true, we have for the error or correction 22 (1VCr + W)C - C- W'C/) o (W- r ive (Ct o 321 324 The advantages of using a separate correcting formula are the following: If it be desired to make only a hasty approx. estimate, we have a short method by the approx. formula; whereas, were the table of operations constructed upon the prismoiclal formula, a great amount of unnecessary work would be unavoidable. Even if the true contents be required, it is found by trial to be much easier to approximate first and afterward correct than to use the exact formula at once. Thus, to find true content of single vol., we may use the approx. formula and afterward the correction in its simplified form on the right; but to combine these the cor. must be augmented to the difficult shape on the left in order to be taken into the parenthesis. The dif. of labor is still greater in the combinations of vols. Another very excellent advantage is the facility with which, by means of the correcting formula, it may be at once determined whether a cor. is required at all or not, thus frequently saving much unnecessary labor. Supposing the widths of end-sections to be equal, we see immediately by the correcting formula, (w -') (C - C) that the cor. is nothing, whvlile the pris. formula requires as much labor for this case as for any other. Tile cor. would also be nothing if tile centres were equal. Again, supposing the centres to vary by one-tenth [0.1], the length of vol. being 100 ft., the widths must vary by 32.4 ft., scarcely a possibility, to make an error of 1 cu. yd.: of a vol. 50 ft. long they must vary 64.8 ft. Hence it is seen that, where the widths or centres of a vol. vary by a few tenths only, the cor. is immaterial. Now, since in any series generally from ~ to I of the vols. need no cor., we are able to achieve a correct result witlhin a very small fraction with comparatively little work by the aid of this methlod of selection, as will be 23 shown in correcting the last example of approximate work. Another inference to be drawn from the correcting formula is that, where the wid-th and centre of one section are both greater than those of the next, the cor. is a minus quantity; where one measurement is greater and the other less, the cor. is plus: and, since, when one measurement is greater, the other is likely to be so too, the approximation of earthwork by end areas is generally an over-estimate. We see by the formula that to correct a vol. the dif. of widths, found by a subtraction in one direction, must be multiplied by the dif. of centres, resulting from a subtraction in the opposite direction, this prod. multiplied by the length of vol. and divided by 324. ApplyiIlg this rule to the second vol. of first exmmple, we have iwidth at sta. 2 [27.8]-width at sta. 3 [31.5] —3.7: centre sta. 3 [6.4] -centre sta. 2 [5.1] -1.3. -3.7 x +1.3 xl00=-481. Set this in an extra col. Treat each vol. in like manner, remembering first and last nuumbers in col. of widths are half widthsll. We here representf the col. of corrections of first example. The PRIS. coR. sum, —3827, divided by324 is -11.81 cu.yds. - 987 This added to the approx. result yields for 481 thle true answer 3119.56 cu. yds. In the sec- 208 ond example the dif. of widths of first vol., 702i -- 14491 by subtraction downwards, is -2.7, of cen12)3-8L7 tres, by subtraction upwards, is +-1.1. -2.7 9)319 x +1.1 x40 —119. The cor. of second vol.3 we instantly discover to be inconsiderable, 3)35.4 -11.81 Cor. because the dif. of widths 3131.37 Approx. con'ts. is only one tenth. This 3119.56 True con'ts. cor., if calculated, is found to be no more than ~ —a of a cu. yd. Bet. sta. 4+74 and sta. 5 the corrections are too small to be considered, because its vols. are short and the centres differ by a few tenths only. In this example 8 vols., of the total 17, need no cor., a fact discoverable by the formula without actual labor. The error of the 8 vols. is altogether only & of a cu. yd. 24 At the expense of ~ of a cu. yd. more, CORRECTIONS. the cor. of 4 other vols. may be dispensed IN'ERis I P1SMOJAL. with. So, using the correcting formula, -119 we may take as little trouble as we - please, or, on the other hnand, attain as' - 52 perfect a result as desired. I- 136 Instead of dividinlg the sum of this -584 0 col., -1504, by 324, move thle decimcll O point two pli:ces to the left, and, di- 230 viding by 6, place the result witlh those - 84 under col. of prods. as was done with 0 the sum of corrections for internm. stas. 0 The cor. of first example may be like- 0 wise tretated. - Decimals need not appear in the col. 0 of pris. cor, for tkhe same reason that -;0 affects the decimals in the col. of cor- 12)1504 rections for intermns. 9)125a 3)1.4 Correction= - 4.7 Approx. con'ts- 4984.4 True con'ts=4979. 7 25 LEVEL SECTIONS. THE surface line of a cross-section may be unbroken at centre stake, as shown in first diagram; and for this case some authors construct especial formulra, using the symbols of side-heights instead of centre. But such formulm require greatly more labor than those which use the centre height; and since the last is always known, and the section itself is susceptible of division into the four triangles upon which the formulae of this paper are based, there is no need to consider this an especial case. But if the surface be level, as frequently occurs in long stretches of river bottom and prairie lands, all the formulm may be appreciably reduced. Thius for the area, substituting in the formule w(c+ Sb)-Sb2 2o the value of w in terms of c, [w = 2b +,] and denoting for convenience the entire road-width by B, we obtain c2 Level area = Bc + Approx. vo. = (B(c + C) + Xc2 + 5c4) =4 26 Nvols.( ~- B(2c' &c.) 2e 2 + +. + +.) -, or ( B(icld-cen hl'es +- j- endc-cen'1tes) D (!nid-celtr'e.v2 +' e?+d-ceIntcs)) s27' Pris. coy. -—' c')2. Let us illulstrate by the following example, found on page 99 of Henck's FIELD BOOK, where B=28, —, and the centres as recorded in col. headed c, except first and last, which in agreement with the formula a,re halved: Sta. c C2 (c- C)2 0 1 2 1 4 16 4 2 7 49 9 3 6 36 1 4 10 100 16 5 7 49 9 6 6;, 36 1 7 2 8 4 43 296 41 28 - 71 1204 2882 x 433 1204 91163700. 3118188.9 6063. cu. yds. This table of calculations for level sections is identical withi Henck's and othiers' in cols. headed c and c2, but differs from all others in last col., which Henck fills with prods. of each pair of consecutive centres. The difference arises from the use here of the correcting formula instead of the exact expression for the whole contents at once: and the advantages attending the use 27 oi the separate cor. are the same for level sections as for the others. Thus, although the example above is too simple to show a marked difference, it may readily be conceived how it is far easier to get the difference between two consecutive centres, frequently having a very small remainder and with fewer digits, and square this, than to find the prod. of thle same. In proof of this we may point out the fact that, while the sum of the last, col. as above is only 44, the sum of the last col. by Henck's method for the same example is 274, results indicating directly the comparative bulks of operations necessary to fill the respective cols. It is noticed that the cor. embraced by this col. is always a minus quantity for level sections. The divisor of the sum is 6 times as great as the divisor of the next col.; wherefore divide the cor. by 6, setting quotient as obtained under next col., subtract, divide by S [=I], add in the prod. of B [=28] by sum of 2d col., and for cu. yds. divide by 27. When S=1, we need simply add 296,-71 and 1204. Henck used the value. for { in this example because it makes his formula easier. Intermediate stas. rarely occur on level ground, and more rarely would they happen to be level also, so the cor. for internms. need not be reconsidered, though if required it can easily be constructed and applied. Of course, where but several level-sections occur in a cut it is not advantageous to alter the mode of calculation or remove them from the general tables, where they are as correctly treated as the rest; but where many such sections follow each other consecutively, the method last discussed is a sensible improvement upon the former. 28 IRREGULAR CROSS-SECTIONS. OvEn very uneven ground cross-sections of earthwork must necessarily be exceedingly irreg.; and such, it is seen, must interrupt the series of vols. in which they occur, so that by the formula already established no entire cut or fill, containing one or more irreg. sections, can be calculated in one operation. As far as we have seen, no formulae have been constructed for this class of sections, mere hints being given for their convenient computation, as, regarding the diagram, to find areas of trapezoids, r u u' / r', it u' t' t, &c., and subtract from their sum the outside triangles, r ", " o, dividing into triangles, to solve r. r' u/ 1' p singly, and in pairs all the other trilunles, as r, u' t / - U u' xt',;', &c., subtracting outside tri-ngles from the Cl I ___ _,,,.! portion of an irreg. section of one " break," the broken surface line being above the straight line drawn from the surface at centre to top of slope, the latter showing the surface line of an imaginary reg. section of same 29 base, centre and slope. The area of a whole reg. section is 1w(c + Sb) -Sb2; of right portion, letting x represent width,.x(c + Sb)-,S Sb2. Adding to the latter area, the triangle above the reg. portion, we have the area of the irreg. section. Let m be the hleight of vertex of break above grade, rn' its dist. from centre. Draw a line from top of slope parallel to base, making a triangle with line to top of centre and with the excess of centre height over slope elev.; also a smaller similar triangle, having for a base part of m denoted by a. From these triangles we derive X: Xerx:: c-?: a... C(X The part of mn below the horizontal line is equal to r; therefore, subtracting r +a from in we have the portion of m included in the upper triangle, which multiplied by Ax is the area of that triangle. Area tr'iangle -Ix(n - ) + mn'(c -r). Adding to the reg. portion of section, we have Area riyht sectioz - x(m t+ Sb) + - mn'(c - r) — Sb2. If the break be below instead of above the surface line of reg. section, the Ariea triac.gle -- ~x(c - in) + ~m'(r- c); but in this case its area must be subtracted from the area of reg. section. Changing the signs, therefore, we have the same formula to add to the formula for reg. section as before. Hence, in all cases, the formula given above for the area of right-section remains true, whether the break be above or below surface line of reg. section, and, as may also be proven, whether the slope be higher or lower than centre. Consider the diagram representing a right-section 9, 9- 30 with two breaks. c m n r is the surface line, c r the top line of reg. section. Draw c n. As with last diagram, Area triangle c n r = x(n - c) +n'(c- r), Area triangle c m n= n'(c - n) + -'(n - c). If to the formula for reg. part of section the first triangle be added and the second subtracted, Area rigla section= + }'S(m -r)-Sb This formula represents any form of right-section containing two breaks. The formula for three breaks, p being the height of third from centre, and p' its dist. therefrom, is similarly found to be x(9p ~ St) ~+.( c- n) ~+ ~n'(m - p) + ~p'( - r) - Sb2; and we see that to find the area of a side-section with any no. of breaks we have only to use the height of last break from centre instead of centre height in the forrmala for a reg. section, anid afterward add one-half the prod. of the dist. of each break from centre by the dif. in height of the breaks next one on each side, the one farther from centre, on suratce line, always being subtracted from the one nearer. The same being true of the other side section, we have this RULE to find the area of an irreg. section: Multiply one-lhatf each side-width by the heighlt of last break on thlat side added to Sb. To the sum add one-hal~ the prod. Of the dist. of each break from centre by the dif. in elev. of the breaks ntext one on each side, always subtracting the one fcirther from centre from the one near~er. Subtract Si,2. This method of calculating irreg. sections is similar in piinciple to tlle manner of treating interm. stas., before described, the dist. of each break front centre being multiplied by the dif. of heights of breaks next one on each side; and some of the merits of the plan are still perceivable in its latter application. For an irreg. section of five breaks, as illustrated in first diagram, by the rule just given seven prods. must be 81 found, not so difficult as thle seven trapezoids of the old method, and instead of finding the areas of the outside triangles and subtracting them, we have only to deduct the constant quantity Sb2. When combined in series with reg. vols. this subtraction is eliminated, as also the factor Sb, making the process still simpler. But it is not to the simplification of the treatment of a single section, which must be confessed is comparatively not great in case of the irreg. section, that we now tend, but to the construction of a formula for such sections thhat may be joined with the formula for the reg., in order to avoid the inconvenience of dividing a bank or cut, having one or more irreg. sections, into separated portions, each to be considered alone, and afterward treating singly the vol. on each side of every irreg. section. To obtain this formula, let v represent the left side-width of an irreg. section, m the height of a single break on that, side, m' its dist. from centre; on the right let n and p represent the breaks. Area Left =- lv(n + Sb) A- _m'(c-)- Sb2) -rreg. 4( - Sb) ~n(c- o) Section Bight - A) - ((V)-4Sb' 2 Altering the shape of this expression, remembering that W —+X, we have PRODS. Area irreg. section = Sow +- vm +xp - Sb. n'(c-p) L p'(n-r) Area reg. section - - Sbw + we ] Sb2. It is noticed that the first and last terms of the respective formulm are identical, and that the second occur in the same col., which may be headed PRODS. Therefore, Area any section ='(Sbw + Prods.) - Sb2, which formula may be carried with perfect facility tllrougfh all the combinations discussed in former part of this paper, the only distinction bet. reg. and irreg. sections, as represented by their formula, beinug in the definition of the word Prod., which for the former means the prod. of centre by width, and for the latter the sum of 32 prods., each found by multiplying the dist. of a break by dif. of -height of two adcjacent breaks. [In case of prods. vm, xp the same rule applies, since the top of slope may be considered a break dclistant by v orx, the next break nearer m or p, and', no break being beyond, this may be considered zero, whence we have, agreeing with rule, v (m —O), x (p-O).] Therefore, the rule given for series of vols., as well as the rule for interm. stas., is perfectly applicable to irreg. vols.; and in the table of operations the widths of both fill col. of WIDTHS to be multiplied; together, by common factor Sb, the prods. of both fill col. of PRODS. to be added together, and finally the common subtrmaction, — Sb2, is made once for both, as will shortly be illustrated by an example. There is alnother class of cross-sections, which should perhaps be distinguished by the name defective or impefect, wherein the entire base of section does not appear, as in the laccompanying diagram, where the surface line dips lbelow grade in right-section. Evidently L __J p, the portion of section above grade is excavation, and that below, filling; therefore, if the surf.ace line should be considered as it is, we should obtain as a result not the amt. of excavation nor of embankment, but the dif. of their quantities. As this dif. is rarely required alone, excavation and embankment must be considered separately. This is done by regarding the points where surface line crosses grade as breaks, and the surface line to be I m c n p r, of which the values of n and p are 33 zero. The area of the section is now correctly represented by the formula for an irreg. section of corresponding breaks, viz., j 1v( + SI)))+ x(p +Sb)+ n'(c- ) t This formula may be obtained from the section directly, as well as from all tlhat follow, by means similar to those used. in connection with ir:eg. secetions; and it may be verified by substitutinng therein the actu;ll values of the symbols. Talke another instance wlleire (a portion of right-base does not ap)ear in the section of excavation. Since in,, ~ 24' the same series we must always have sections of equal bases, the base of this section must be prolonged on the right to the proper dist.; ancld now the surface line is considered to be 1 c m r, of which m is the only break, and mt and r' zero. Its formula is accordingly gv(c+Sb) + Ax(n +S,) +.m-(c-.)-Xb2. The formula for the next section represented is v(rm + Sb) + rx(nt + Sb) + m(c -1) +,n'(c r)- Sba 3 34 The next section may be regarded as reg. since its surface line is broken only at centre. Its formula is -Lw(+ Sb) _ Sb2 t: is cert, a;l. ly not preten'ndted that the formula, as applied to such simple shlapes as the last two or the next two following, simplify the same; but it is sho\wn that by means of this formula, as an expression of area,, any possible shape of cros.s-section may be, includdcl in the series and thus cause no interruption, which is thle great advantage we wish to obtain. Neitller does this formula augment to any sensible degree the calcultation of simple shapes, it'being a mere foIri serving to direct the values to their proper places in thel cols. Thus, for second section a,bove, when iti combination, the subtraction,- Sb, is elimnllate(d, as also Sb; so we lave merely to set in colo of widths the entire. width, equal to the base of section, and for col. of prods. it is instantly seen that the first two prods. are zero, and the last two, qn'c, n'c. [['he wvidth, of an impe'/ect section is the full width of roadwacy added to the width of wzhatever sloping the section may have. The formulae for the next two are in order V( ~+Sb) + XJ _ 1- + 1(3 ) - - &)2, + I n'( - p) 1+ -21, n 7- \ 0a 35 The formula for each of the following two sections is 1,'x' —-. /1 is-s~ I- m 7,r v(+ S lt e- (c+ i l) m( — is ) ~ e'(y -o ) se Srbi(z 2, which is the general formula for irreg. sections, mr being the first breal, n tlhe second, measured on szuface line.'The founlam.] for tlhe next is rather long, but just as easily constructed by the general lawv. Its surface line is 1 t p, mn c g h k: r. The last section shown is merely of side trimming, ~~~~11 36 the road-bed not being formed by excavation at all, yet the same rule applies and the same formula represents its area. The surfaee line is considered to be r c rn n p t g h l~ 1, of which the breaks are all in the left, section. The first is evidently n,, an imaginary pt., whose dist. is known to be the width of half base, and elev. zero; the second, D', the pt. where slope enters material to be removed. The formula need not be re-stated. This case is rare, but is here included to support the assertion that no possible shape of section, within limits of reg. slope and base, lie without the limits of the rule for the calculation of irreg. sections. We muay now calculate approximately in one operation anly possible cut or fill. The reg. vols. may be corrected by the formula already given; while the irreg.:may also be corrected by a somewhat similar formula, whlich Twe shall now proceed to find. First the general rule mafy be stated, which is easily suscepy'ible of proof, but for which we shall not yield space here, that the pr'ismoidc fbzor.ZncIa ay be corr''ectly at2pplied to every vol. bouizczed b/y parallel bases, however cldissi[minzar inz sh(upe anil arcea, alctl aterally coltdaled by a,szicface yel'eracted by a stracight lie movingy a7ong tVhe perineterssof the bases cus directr ices. This includes all vols. bounded. by warped surfaces and surfaces of revolution, which are generated by straight; lines, and applies to the majority of shapes occurring in earthwork and mason ry. Irreg. vols. are generally very carelessly treated, on acCount of th!e grea t labor it would incur to calcul;lte the data of mied-sections, then their areas, and afterward apply the prim.oidal formulal. YMuch of this labor may be removed by using a general formula for such sections, altlhough tljis plan does not seem to have been hitherto employed. The method, often used, of converting end areas into equal level-section areas, and applying to the resulting vol. the pris. formula, is extremely faullty. In fact, irreg. vols. should be very cautiously calculated, as the ratio of error incurred by using approx. methods is immensely greater than when the same are applied to reg. vols. In illastration of this, considec the right side of an irreg. vol., shown in diagram) with numerical values, m 9' 4-S -L 9 6 and MI being thle breaks of the two sections, supposed to represent the features of the same hollow, extending from onle section to the otlher. rn and 3l are therefore considered to be connected by a straight line dividing the two surfaces nz c c' 1O, rn r r' 31, one warping from Mn c to Jf c', the other from m r to Al r'. The half-base is 9 ft.., slope 1 to 1. Area triCigle c m r -x( m)( + -'(n-c 2 12 M, ( )! or.l5(5-4) + -.11(6-5)=13. c'r'I -- -.14(10-7) + 2.6(5-10) =6. The vol. having these triangles for bases is bounded. laterally by a surface generated by a straight line moving along perimeters of bases; hence its content should be calculated by pris. formula. Of mid seetion the centre-height is +', width elev of break't7, dist. f + 2 and slope height +- Substituting these in thee xx'( c+c an' _orn em) +rn' +m' ) r+r' _e+c or numerically, A. mid. tri.=6, 38 no greater than the area of small-end triangle. This serves as an illustration of a twisted vol., wherein the mid-section and consequently the content is much diminished by the twist,. Here we have True vo. =(13+6+4- x 6)-1 O=716 cu.ft., Approx. vo.- = (13 + 6)-LQQ = 950 cu. ft., where it is seen the approx. is a vast excess over the true content: and, since this twisted prisml must be subtracted from the vol. bet. reg. sections, the approximation in this instance for the whole irreg. vol. falls short of the true content. Let us see. A. ear irrey. sect..15(4+9)+.17 1(5-6)-40.5 _ 51.5. Afar irrey. sect. - 14(7 + 9) + 1.6(10 -5)- 40.5 86.5. Approx. irrey. vol. =(51.5+ 86.5)LQ-~-=6900 cu. ft. A. netar rey. sectio-l =-.15(5 + 9)-40.5 = 64.5. A. f:ar req. section- =.14(i0 + 9) -40.5 =92.5. Alprox. re. evol. -=(64.5 - 92.5)I%-= 7850 cu. ft. Pris. co. for b ey. vol. - (10-5)(15-14) -1o-= +413 cu. ft. T'rue reg. vo1. - 7891 " " True conltent twisted pr'ism-= -- 716-" " Tr"zte irr)ey. vo.=-265.74 cuo. yds. -7175'" Approx. irre:. vgol. =253.55 - 6 =6900 We see thlat the approx. result for the irreg. vol. falls slort 10 yds. in a true vol. of 266 cu. yds., only 11- of wlicih is recovered by application of pris. cor. to the whole reg. vol., while the 1remaincing 8-1 yds. is due to the cor. of a mere stripn of vol., whose true content is no more thil-n 261 cu. yds. _By thlis it becomes evident that it is vastly morie important to apply thle pris. cor. to vols. bet. irreg. sections tllan bet. recg. tlhe cor. in t1his case for reg. vol. beiUg but 11- ycds in 292, wvlereas the cot. for the irreg. vol. is 10 cu. yds. in a true vol. of To find the sliape of pris. formnula as applied to irreg. Vols., consider a vol. bounded by irreg. sections, of whlich c,', 1, v, x, are the centre, right and left slope elevs., left and right side-widths of first, the same 39 rid' I ~ O1 ), measurements of second correspondingly c', r', 1', v', x. In the first are one break on left, elev. mn. dist. m', two breaks on right, elevs. n, p, dists. n', p'; in the other corresponding breaks of alt. K, N, P, dists. M', N','. The areas of these by rule are respectively v(~+ Sb) + ~x(p + Sb) + m'(c - +) + I-~'(c- p) + p'(n -r) -Sb2, Iv'(M+ Sb) + Ix'(P + Sb) + Mi'(c'-l'),(lc r,- P) + IP'(NAT-r') -Sb2. The corresponding values of mid-section are found by dividing the sum of Similar measurements of endsections by 2. Its area is, therefore, WV + V ( —+ t Sb) + (x+x')( -+ Sb) + i (,n,' + K') (a' — ) + X(n,' + N') (a-~ z) + w~p/ + p' l (n+1V - 2. Multiply mid-section by 4, add thereto the sum of end areas, and multiply by - length, to find true vol. From this subtract the approx. amt., found by multiplying sum of end areas by one half length. The remainder is the pris. cor., being in cu. yds. (V -v')(fI- m) + (x -x')(P -p) 4 + (im - M')((c' -') - (c- )) + (i' -- N')((c'-P)-(c —p)) + (p'-P')((N —r') —('n-r)) This is more conveniently expressed in words. The first term is the dif. of corresponding side-widths multiplied by the dif., taken inversely, of corresponding last breaks on that side. The second term is similar. In 40 the third term one factor is the dif. of dists. of corresponding breaks from respective centres, and the other the dif., taken inversely, of the quantities, which are multiplied by those dists. to find areas of respective sections. In the formulhe for the areas of first and second sections we find the terms, 1 m' (c-l), - (c' (c'-1'), which are used in the approx. calculation, m' and l' being placed in their proper col., and c-1, c'-1', in another; so that to apply pris. cor. we have only to find a dif. by subtraction downwards in one col. and multiply it by a dif. found by subtraction- upward in the other col., a method precisely similar to that employed for reg. sections. The third term is a type of all that follow. Therefore we may construct this RULE OF PRIS. coR. for vols. bet. irreg sections. Multiply the dif. bet. each two corresponding side-widths by the dif., trken inversely, of the elevs. of last breaks on that side. To these prods. add the prod. of the dif. of dists..from respective centres of each two corr'esponding breaks by the dif., taken inversely, qf the respective factors used with these dists. to find the areas of respective sections. Multiply the sum by I- dist. bet. sections and divide by 27. This rule requires that the same no. of breaks be present in each end s( ction of a vol., and that the breaks of one end be connected with the corresponding breaks of the other by straight lines. This arrangement is true of every vol., although the breaks may not appear in one section. In the example just consideredit is perfectly obvious, wherte the three breaks of one connect with the three of the other; but sometimes one end section has more apparent brenks than' the other. ThuR, in proceeding from reg. to irreg. ground the last reg. section is succeeded. by an irreTg. section of one or more breaks, which do not, however, originate in that section itself, but gradually develop thlemselves from certain points on the profile of reg. section. These points must be ascertained in order to make a correct computation, and they represent the breaks of the reg. section corresponding to the actual breaks of the irreg.. That these ridges and hollows do extend from one section to the other is self 41 evident. The irreg. section should not represent mere local features, since its area and shape affect the vol. all the way to next section on each side. The ridges and hollows may be faint, but, if visible at all in one crosssection, their general course may be traced, and the pt. of intersection with reg. section nearly determined. The neglect of these fading ends is the source of great error, although at first thought, it would be considered perfectly immaterial to the result where the ridges and hollows might happen to fade on the surface line of reg. section. We shall give space to one illustration. r- ~~~~~~I4 L_ a_' _ i 1 A i,' ~ -',,/. i' I' A,.. I/f of irreg. section are supposed to extend in one instance; another instance to extend. Tihe approx. formula for content is Approx. vol. i= (prods. + Sb x aidtais-4rb 2. tio - whs esr e m saemrke itedaram M-,s 2,,ae;hps.ore.sio n towhc he brek o "f — irreg seio are supsd~ xedin n lne 7n, a, p, [h p1. owih h rasa upsdi 42 applying which to the example we have W C PRODS. 32-1 x 8 260 15 x 9 = 139.5 6 17 x 10 = 175. 65.5 5 x (4 -7) -15. 24 6 x (4-10) -- 36.'2620 11 x (5- 11) = -66. 1310 457.5 1572.0 1572. -1)152. 9)877.5 12)9750. Approx. ro7. in cu.?ids. =81.2.5 Now if the ridges and hollows irepresentec in irtleg. section vanish at the pts. Li, N, P, of the reg., these pts. maiy be considered correspondling brealks, and all the condlitions for the application of pris. cor. are present. Thle distances of thlese pts. are noted in field, the heights being readily deduced, as by the formula h=-a 4d(b-a), where h is the height of iniermediate pt., a th:e elev. of first end of line, 1b the elev. of secondcl end, d the dist of pt. from first end, and 1 the horizontt-dl length of line, Thus, if I. N', P', are 3, 2, 4, respectively, 3f:8.-:- 61 89, = 8+ 2 1 —8) -= 8.63, P=8+ 4( L- 89.26. Appplying the pris. cor., (15,-13)(16. - -9) 4.22 (19-171)(10-9.26) + I1.1 (5-3)((8-3)- (4-7 ) =) -- + i.00 (6 —2)((8 —9.26) -(4-10)) - + 18.96 (li-4)((8.63 — 14) - (- 511)) - 4.4-1 324)U326. pris& co')'.= + 11.2 cut. yds. When the breaks representecl in irreg. section extend to the pts. MiJ N, P, of the reg., the true content be-. is 812.5 + 11.12 -823.7 cc. yds. Let it now be supposed that these breaks extend to 43 thle pts. in, n, p, distant 11, 13 and 15, and of elev. 3.9, 12.1, 13.4. Apply cor. 5 —l13)(3.3- 9) — = 10.14 (19- 17T) (10- 13.36) = -15.04 (11-5)((4- 7)-(8 —3)) -48.00 (13- 6)((4- -10)- (8- 13.36)) = -- 4.48 (17 -11) ((5-11) -(12.1 —14))-= -24.6 324)- 10226. 2pris. cor. in cu. yds. = - 31.56 TWhen the breaks of the vol. extend to the pts. m, n, p, of the reg. section, the true content is 812.5 - 31.56- = 780.94 c.?/ds. Therefore, if,JJ -N, P, be the vanishing extremities of these breaks and remairln un oted, the approx. result falls short of the true content by 11.2 cu. yds. If, otherwise, f, ii, n, be the fadling ends, tflhe apprLox. calculation is an over-estimate by 31.56 cu. yds. The total effect on the vol. occasioned bNy movingP the exteremities of breaks from the pts. M, mA, P, to thle pts. q1, n, p, is a diminution of 43 cu. yds.; and by other arrangenmen.t of the pts. this may be made more than t50 cun. yds. It may be easily d(etermined at a glance whetl-ler the effect of shifting the vanishing extremities be great or little upon the vol. In lel't irreg. section conceive a line to be drawn from top of slope to top of centre, and a warped surface to extendl from this to tfhe left surface line of reg. section. This divides the triangular warpedf,aced wedge AIB CDE froml the main: vol.; andc the latter is evidently not affected by the shiftinlg of 1 along DZE. But the tria ngular wedge is affected to the following ext;ent. If B be moved aloong a line 3BY parallel to DE a certain dist., and 4the consequent increment or decrement to the -weclge denoted by /r, i1 remaining fixed, then the vanishing end M11 be moved along DE the same dist. in the satme direction, B remaining fixed, the second consequent increment or decremnent is equal exactly to h. Ther efore, to determine whether the neglect of noting;: the vanishing elind, Mi, of a ridge or hollow be or be not serious, it is only necessary to notice the inclination of the lines AC, DE. If they are 44 parallel, ch ange of position of J11 does not alter the content of wedge; but, if they are much inclined, thle position of i11 i's of great importance as already shown. Thus, to apply the principle, it is seen at once that the effect of shifting li in left section is far greater than can be produced by altering the positions of N and P in the right. It is thus apparent that to obt'ain all the dlata for a correct computation of an irreg. vol., both ends of every ridge and hollow must be noted, whether it fade out or not; that is, each end-section must have the same no. of corresponding pts. connected by straight lines. To take these notes in the field is simple. If we find in taking cross-sectio-ns two adjacent sections have three breaks eatch, corresponding, no remarks may be made; and in applying pris. cor. it will be talken for granted that these are respectively connected by straight lines, the nearest to the cenhte of one section wvith the nearest of the othller, the farthest with the farthest, and middle with middle, since the.y do not intersect bet. cross-sections. If one have four while the other has thlree, we have only to find the position of fading end of vanishing break in section representing but tllhree: its dist. froml centre of this section is placed in col. of remarks opposite thllis section. When the pris. cor. is applied, the four breaks of one section are connected with the three real blre:aks and the fading pt. of the other, according to their dists. from respective centres, as before. In all cases, after noticing whit breaks of one connect with what breaks of the next, (only tthe vanishiing extremities of the other breaks need be inoted, obtaining thus always an equal no. of pts. in. eacl, whliclh to connect accordiing to the dists. from their respective cent;lres. This will lhe fully illustrated in the following exmnple. The first col. of this extract1 from field book contains the nos. of stis. and interms, tlhe second the centre cutting or filling, precededl if the latter, by minius sign. The thirdl col. hns tlhe elevs. andl dists. of left slopes, with the hteights Ind dists. of infterm. breaks, aanged in their natural olrder. Thle fourthll contains the right dists.. and elevs. similarly arranged. The elevs. in all are 45 marked above the dists. of same pts. The full stas. are 100 ft. apart, the width of roadway 30 ft,, the rate of slope for the cutting is 1 hor. to 1 vert., represented hlere by L'=2 S. The slope of the filling is 12 to 1; but we propose to calculate the excavation only, wherefore the left-widths of such sections as dip below grade on that side are considered to be 15 ft. In the col. headed REMRIKs are placed the data concerning the ridges and hollows. For instance, sta. 1 being irreg. cand sta. 1+50 reg., the fading extremities of the ridges and hollows profiled in the former must be determined in the latter. The left break is found to extend to centre of sta. 1+50, and it is so marlked on a little line extendinlg to the left from a vertical line representing cenitre line. The right break extends to the top of slope, and is therefore recorded as 16.2 on thle right of vert. line (these vert. lines appear ratiher contracted in the print.) Both these records are placed at the top of vert. line, leaving the lower end for possible notes connected witih the inext following section. But here none isneeded as the sections fully correspond as they are. Sta. 3+20 is irreg., therefore the corresponding pts. of sta. 3 must be noted. The left break extends to left slope, being a grade-line bet. thde sections; the right break extends to centre, whvose dist. iS zero. Stas. 3+20 and 4 do not correspond. A pt. onl sta. 4 corrosponding with left break of sta. 3+20 must be found, and a pt. on right of sta. 3+ 20 corresponcling with second break on right of sta. 4. Thle two right breaks of sta. 4 are found to merge into thls one of sta. 3+20; the dist. 10 is therefore noted in lthle remarks. One of the two breaks of stca. 4 disappears att centre of sta. 5, and is so noted. The other runs through sta. 5 and disappears at sta. 6 at a dist. of 9 ft. from centre. Bet. stas. 6 and 7 a grade line runs fromu left slope of former to centre of latter; these pts. are noted. This grade line continues to stas. 8 and 9, appeatring in these as breaks; it must be noted in sta. 7 as a break corresponding with that on right of sta. 8. No:natural example would be likely to have so many breaks vanishing and originating in so small a dist.; it is only made so here to include all the cases. 46 FIELD Boo0: STA. CENTRE. LEFT. RIGHT. REMARKS. a I 124 20 1.0 40 30 ~-50 2 n. TL WjT l-~~ T~f 4.0 3 5 T W5. VT9. 0 1 5.10 3 - 5'100 1'6 i - 8.5 0 0 6 0 1 4~0 +20 5 AO. TVTh I 10. -0.0 70 9 30 1.0 WG5l 7 0 - 0~-~ LTV -2 - 8~40 5 s o 5o 6 -. o AO 6.-0" -30. 0 30.0 I -38~~~~~.0 85.3 8.,,~.o2 -4 p.0 I -36,0 0 2 -4'~'T. 0 PT~', 90.0 TABLE OF OPERATIONS. STA. WIDTHS. CEN. PRODUCTS. CORRECTIONS. BREAKS. INTERMS. PRISMOIDAL. 1 16. 1 16. 1 15.8 x 2.0 1 c 16.5 4 66. 0 16.2 x 2.41 -gu L 11. 0. +4.4 Ox 0.4 F >+. 9. - 1 - 9. -4.32 J 0 16.2x 0.4 J + 50 16. 32 2 32. 64. -1060 15.0 x 0.01 2 32.4 32.4 3 97.2 97.2 019.0x 5.0 I 3 17. 34. 5 85. 170. 320 155.0x 5.0 + + 20 7.5151. 0 0. 0. -1580 _3 1140 0.0 x- 3.0 J 11. 22. 6 66. 132. g 22.0 x 6.0 ) L 8. 5 20. 40. 10.0 x - 1.0. 6CQ. ~ 9 - ax5. -9 -7' 10.0X- 8.0 II 10. -9-45. - 90. 10.x - 8.0) 15. 15. 6 90. 90. &) -240 15.0 x 0.0 +el 23. 23. 9 207. 207. 15.0 x 6.0 — 15-2 +) 11. - 3- 33.. - -2 +3 24.0 x 8.0 I 15. - 9 -135. — 135. 22 0.0 x - 1.0 5 17. 17. 7 119. 119. -6 1500 12.0x — 11.0 24a. 24. 8 1992. 192. 15.0 x 6.0), IR 12. -11 -132. — 132. -227.0x 12.0 F. 6 42. 42. 6 252. 252. -21 3500 9.0 x — 18.0)' 7 45. 45. 0 0. 0. -~0, -7200 15.0 x 0.0) 8 15. 15. 0 0. 0. +98 27.0x 6.0 P 8P " 29. 29. 0 0. 0. o 150x 6.0 T 1? 5. -28 -140. -140. -10 -100015.0 x 0.0 ) 9 7.5 15. 0 0. 0. ) 30.0x 0.0:. 112.5 25. 0 0. 0O. I o. >0.0 x 0.0) o C6 B 11. -20 -110. - 220. -48 - 4800 15.0 x 0.0) l 313. 565.2 I -2640 1 44 20 30.x o0.0 i Sb x widths....... 9417. 0.0 x -30.0 -2Sb2n2.,...... -7200. Cor.for I 2t's-.200. - 13.2 EPrs. Cor.+ 600... -- 24.033 9)274496.67 6)30499.63 5083.27 cu. yd& 48 For the approx. calculation of this piece of excavation the notes in col. of remarks are of course not needed, since they do not affect the areas of the sections. According to the approx. rule on page 13, construct a table of cols., the 1st for stas., the 2d and 3d for widths, the. next for centres, two more for prods., and another for the cor. for interms. Finally add one col. for pris. cor. and one for extra notes. In 2d col. of widths set the full widths of reg. sections; and place the centres of these opposite in col. of centres. For irreg. sections place in col. of widths the side-widtihs separately, and in col. of centres tlie factors used with these side-widths to oblain prods., namely, the heights of last breaks; also in 2d col. of widths place the dist. of each break, and in col. of centres the factor used with this, namrely, the height of break next nearer centre iminus the height of break next farther. Tlhus for first section we hlave directly from field-book, 16 x 1, 16.5 x 4, il x (2 -2), 9 x (2 - 3); for sta. 3 + 20, 15x0, 22x6, 8x(5 —0), l 0x(5 —14); for the next, 15x6, 23 x9, llx(6-9), 15x(7-16); for the last, 15x0, 25 x 0, 11x (0 —20). These prods. form the 2d col. of prods. For the first cols. of widths and prods. merely transfer values in 2d cols. according to rule, that is, full widths and prods. for mid-stas., half-widths and prods. for two end stas. of work, with this exception, that half the widths and prods. of last interm. of a vol. must be used in place of half the widths and prods. of the full sta. next before. But this principle has been already discussed. Since the first sta. is one of the two end stas. and, therefore, only half its value could be used, and since this half must be omitted in favor of the half prods. and widths of the interm. following, no transfer is made for sta. 1. For sta. 1+50 transfer half the width and half the prod. For sta. 2 transfer entire values. Transfer the half values of sta. 3, since it is followed by an interm. For the interm. transfer half the width and half the prods. For all the rest except the last transfer entire values, for the last half values. In first col. of widths are placed the letters L and n opposite break dists. of sections. These letters serve two purposes. 49 First, they occupy all: space of -first col. except what should be filled with transferred quantities, thus preventing the possible mistake of transferring the break 4ists. also to first col. of widtis.' Their special use is to mark corresponding breaks while applying pris. cor. To apply cor. for interms. to first interm. we use the formula (Sb(w- w') + p -P) I where w and P are the width and prods. of next section before, or sta. 1, w', P', the same of next section following, or sta. 2, Sb =2 x 15 =30, D =50; the factor X- will be used at foot of col. So we have 0.1 x 30 + (- 24.2) -21.2; -21.2 x 50 -1060: Ofors Sta. 3 + 20, -4 x 30 + 41 -79; -79 x 20 = — 1580 To apply pris. cor. to vol. bet. stas. 2 and 3, which is reg., multiply dif. of widths -of end-sections by inverse dif. of centres, and multiply by length of vol. — 1.6 x 2 x 100 - 320. The widths of vol. above differ by a few tenths only, wherefore its cor. may be sflfely neglected. For the cor. of inreg. vol. bet. stas. 8 and 9, we have arranged before us in perfect order the values considered in formula of pris. cor. The rule requires that the dif. of corresponding widths be multiplied by inverse dif. of last breaks, or of centres, if no break be on that side; also that the dif. of corresponding brelak dists. be multiplied by inverse dif. of respective factors, viz., those in col. hecaded oENTnE, used with these dists. to find areas of respective sections. In applying this rule advantage may be taken of the fact, already mentioned in connection with reg. vols., that, if correspond-:ing values of two sections differ by a few tenths only, the cor. for the vol. bet., so far as affected by those values, is immaterial. Thus, in the last vol. the left side-widths are equal, wvhence their dif. is zero, and the cor., as affected by them, is zero. Again, although the right-widths differ, their respective factors are equal, and the cor., as affected by them, nothi ng., But for the third cbrresponding values the cor. is considerable, being A(11-5)( -` 28 ( —20)) — -48,; giving for the vo.* 4 50 -4800 to set in last col. Now, in a natural example of irreg. earthwork, the majority of vols, would have their corresponding values thus conveniently arranged in the table, making the cor. a simple matter. But here we have purposely considered numerous short breaks, making vanishing extremities at nearly every sta.; so the last is the only irreg. vol., to which the cor. can be directly applied. For the others the notes taken in col. of remarks must be consulted. For instance, the reg. sta. 1+ 50 must be treated as if irreg. to correspond with sta. 1, the dists. of its brealks being given acnd their heights founld. So its factors would be, to correspond with those in cols. of widths and centres, belonging to sta. 1, the following; 15.8 x 2, 16.2 x 2.4, 0 x 0.4, 16.2 x -0.4; that is, the sectional notes must be considered as, Left Centre Right 1.16 214 224 1i.7,, 2, T. -2 T." But it is unnecessary to place the sectional notes in this form, as the proper corresponding factors may be taken directly from field-book. These factors being obtained, we proceed as before, using them in connection with those belonging to sta. 1 and presented in the table. Thus, since 16 an.d 15.8 vary by so little, zero may be accepted as the cor., so far as these values affect it. The cor. for next values is also zero. Then we have (11-0)(0.4-0)- =+4.4, (16.2-9)(-1-(-0.4)) - -4.32. Sta. 3 must be arranged for sta. 3+20, as shown in supplementary col. headed BREAKx. Although stas. 3+20 and 4 are both irreg., their breaks do not correspond, wherefore the left of sta. 4 must be arranged as if containing a break, and right of sta. 3+20 as if containing two. The left of sta. 4 then becomes as shown in col. of breaks, where the right of sta. 3+20 is also seen. For the differences, from the left-width of sta. 3+20 as in table subtract the left width of sta. 4 as in col. of breaks; the dif., and consequently the cor., is zero; then from left break dist., 8, of sta. 3 +20 subtract left break dist., 15, of sta. 4, and multiply dif. by inverse dif. of their factQrs. For 51I the right-side measurements, use those: in the'table for sta. 4, and those in col. of breaks for sta. 3+20. The right of sta. 5 must be re-arranged for sta. 4, and the right of sta. 6 for sta. 5. -The fading end at sta. 6 is 9 ft. from centre; therefore h-.6+ 9(4-z =12, and the notes of the right sectionl are, 6,-,; -4. Tile sections at stas. 6 and 7, although r'eg.; include an irreg. vol., containinig a.hollow b6b.:its centre and left slope lines. Both sections must be arranged as if irreg. that this hollow -may not be neglected. These arrangements are both shown in col. of breaks, and must be considered together while applying the cor. Sta. 7 must also be arranged to correspond with sta.- 8. Stas. 8 and 9 correspond. By the approx. rule [see pages 32 and 13] multiply sum of widths by Sb[=30], add prod. to sum of prods., and subtract 2S,'2n, n being no. of full vols., found always, by subtracting the no. of first full sta. from the no. of: lst full sta. Next the rule directs to multiply by 100 and divide by 54; but, since the cor. for interms. and the pris. coL'. must be respectively divided by 2 x 54 and 6 x 54, place ~ the first and - the second under col. of prods., moving the decimal pt. of each two places to the left, because the quantities with which they combine will be multiplied by 100 and thus carly the decimal pt. of these corrections back to the proper place. It has been stated that a natural example would not be likely to have so many vanishing ends of breaks, and would, therefore, have its values arranged at once in convenient form for the application of pris. cor. When these vanishing ends do occur, the work of rearranging the sections containing fading ends is necessary to the true calculation by any method, as the demonstrations on this point were independent of method. But, having found them, there can be no more convenient plan of applying the pris. formula than this method of cor. by using differences of corresponding measurements, these being ranged before us in proper order. By this general method of calculating irreg. sections and including irreg. vols. in the series we have succeeded in calculating in one operation ten consecutive vols., of which four 52 are of minor length and eight are irreg. Without the method of including irreg. vols., every vol. of the example above must be calculated singly, every cross-section but first and last be used twice, and the mid-section of every irreg. vol. measured and calculated. In fact, the inclusion of irreg. vols. in series serves, in the same manner as the inclusion of minor vols., to abbreviate the computation of earthwork, and chiefly to remove interruption to the continuous calculation of a long series of vols. in one operation. By including both minor and irreg. vols. all interruption is removed from the thorough and exact calculation in one operation of any possible cut or fill; and the entire process is governed by but four rules. These are, the rule for the approx. amount with its auxiliary rule for intermediates, here remembering the distinction between Prod. as referred to reg. and to irreg. sections, the formula of pris. cor. for reg. vols., and the rule of pris. cor. for irreg. vols. 53 END VOLUMES. THE method preceding embraces all of a cutting op bank bet. end full stas., leaving to be calculated' he vols. tapering from last full stas. to grade lines. If there be a cross-section bet. last full sta. andc grade line, the approx. content bet. this section and last full sta. may be found by the formula (wc +w' + Sb(w w')-4Sb2) 2 ), if reg., or in any case by the general formula (prods. + Sb x widths —4Sb 2)T,, and the pris. cor. by the rule. If the grade line happen to be unbroken and perpendicular to centre line, it may be regarded as a reg. section, whose centre and slope elevs. are all zero, and width the width of road-way, and the rules of approxi: miation and correction accordingly applied. But, if the grade line, represented in the diagram bet. pts. E, F, G, be broken or be not perpendicular to cen2" ~4 tre line, for an approx. method suppose a plane passed: through each side slope line of last section and centre grade pt., as rF, IF in diagram. These divide end vol. into three approx. pyramids, from which is obtained this RULE for approx. calculation of vol. bet. end-section, tljis being reg., and grade line. Multiply area last section by dist. to centre grade pt.: add the prod. of, b by suzm of prods. of each slope height by dist. on that side to grade. Divide by 81. Applying this rule to the example, we gain as a result 62.59 cu. yds. This is probably the simplest possible rule. Otherwise, by passing a vertical -plane through each side grade pt. and cenitre of last section, the end vol. is again divided into three approx. pyramids,; the side sections being bases of outer ones and side dists. to grade their heigllts, and of middle pyramid the centre elev. of last section being' alt. and all portion of roadbed, unoccupied by side vols., the base. Then Conten't leftt pyramid - 1= E(c+ Si)) -ISb 2 E. Content'igy7 t pyr amid - = Gx(c t Sb) - ISb 2 G. Cortenit midldle pyrgtni(l = Fbc. Enld vot=((c+Sb)(SEv+ Gx)-Sb2(E+ G)+FBc) h, B representing full iwidth of roadway. By this formula we obtain for content of exarmple (22 x IS +17 x 10)(8 -+12) 62.81. - 144(18 + 10) + 8 x 15 x24 2 ~x=x'The average of results of these two methods, 62.7, is always the exact content; but, as in thlis case, where the surface is not much w[arped, either result is sufficiently near the trutll. Another method in use is to divide end vol. into four parts by three vertical planes, one through centre line and one thlrough each edge of base, making two outside pyramidal vols., andl two inner vols. having quadilateral bases and grade lines for edges. This requires much more work than either above approx. method. The result of this method for the present example is 62.53 cu. yds. If the surface line of last section be slightly irreg., any of the foregoing rules of approximation may be applied, the latter preferred silnce longitudinal planes are apt to cross fewer breaks than the diagonals. Bat, if the breaks be important, making the grade line irreg. as well as the surface line of section, the vol. should be divided by a diagonal plane through each warped surface. ESTIMATION OF FINISHED WORK. THE completed work seldom coincides with the prescribed lines, except in ordinary homogeneous earthcuts, and in' embankments generally. When this is so, of' course, the new cross-sections of actual excavation must be calculated; and, since these have no measurements in common but are irregular all around their perimeters, they cannot advantageously be computed in series, so that not only all the methods of brevity of the preliminary calculation are lost to the more important estimation of the final quantities but also the prismoidal correction. For instance, in the accompanying diagram 17, 1 7 the section of excavation, limited by the full lines, must be estimated in the ordinary rude way by computing the trapezoids n 1', n m', c m', c p', t p', t z', r z', and subtracting from their sum the trapezoids k r', k zA", g 1', and the triangles g g" e, h h" e'. We propose to avoid this labor and inaccuracy by considering the lower outline, 1 g e e' h k r, of the section of excavation as the surface line of a section having a regular base, e e', and slopes, e l, e' r; then calculating in one series by the method already explained, usinlg the cor. for interms. and the pris. cor., the amount of material through the whole cut remaining to be removed in order to expose the true slope and base lines; and finally subtracting this from the preliminary result to find the true content of the mass removed. The advantages of this plan are, briefly, that 57 the necessity of noticing the measurements of the upper part of perimeter is escaped, these being considered in the preliminary calculation, and also that the remaining measurements may be arranged in a series similar to the original, yielding a result with all the correctness of the prismoidal formiula. The surface line of the section of remaining excavation, represented in diagram, is 1 g e c' e h k r, of which the breaks onl left are in order e and g, on right e', h and kc. By the rule its area is 1v(g + Sb) + x(7 + Sb) + l(c'-g) + g,'(e-) This. formula is very simple, especially in combination, e, c' and e' being each zero. Sometimes the lower part of perimeter transgresses the proposed lines of slope, as in the diagram. The rule still applies to fintingo the area of such a section, its value being minus in this instance, as it should be, because a minuss mt. of material must be removed to bring the work to the proper slope lines; and this minus amt., representing the work remaining to be done, when subtracted from the original amt. prescribed, leaves an amt. greater than that originally proposed, as the case requires. Occasionally the road-bed is excavated deeper than it should be; then c, m and g become minus (in case it be determined to consider the unnecessary work.) But to every possible shape the same rule applies, yielding positive or negative quantities as it should be: so no allowance need be made for certain positions of perimeter, since the rule takes care of itself entirely, without other superintendence than to follow it exactly, as may be verified by other methods of calculating the area. Of the section last sketched, the first break on right is at g, the second at h, the third at k, where the perimeter reaches the original surface nearer centre than is the top of prescribed slope. On the left the first break is at m, where the road-bed has been cut too wide, the second at n, the third at p, where the perimeter reaches the surface beyond the slope stake at 1. Accordingly, the surface line of section to be calculated is 1 p n m c g h k r, the side-widths of new sections being always the same as those of the old, and its area is,v(p+ Sb) + x(1c + Sb) + -m'(c -n) + ln'(m-p) + Ap'(n-l) + g',(c-h)+ + (g-7c) + cu'(h-r). Of course, these sections can be arranged in a series like the original. The notes for the pris. cor. are more easily taken on the finished work than on the original surface, because the outlines are sharper, and the same features are likely to extend long distances, being occasioned by the dividing lines between strata of various hardness. The finished work may be divided into three general cases': 1~0. When the work after completion coincides with the prescribed lines, the material remaining to be removed is nothing, and the preliminary calculation serves for the final. 20. When the greater portion of the work coincides with the proper slope and base, but a few sections are found to be defective or redundant, the vols., representing the work remaining to be done, bet. the latter or affected by them should be calculated, and the result subtracted from the result of the whole original calculation. To illustrate this, snppose the finished work of the example of excavation, last given in connection with preliminary estimates, to coincide with the prescribed lines at every cross-section except the first two. Let the final notes of these, together with those of the third cross-section, used to calculate second vol., be the following: STA. CENTRE LEFT RIGHT | REMARKS ] 1 _ 1 TZlF t@ICYB T-'TW.T 02 2 0O 4-, O 0 3|iT +50 -1 T r -,N o.. o 4. 2 0 r1<.'YAt sta. 1, as shown in diagram, the base has been cut too wide on the right, and the left slope flattened too 514 /,, /' much. At sta. 1+50 the work has been cut too deep. The 2d break on left of sta. 1 runs to left slope stake of sta. 1+50 and is so recorded in REMARKzS. The sections being equi-distant, both vols. may be calculated together. Half the widths and prods. of first and last sections are used, according to rule. There being no interms., the 2d col. of prods. need not be constructed. The pris. cor. is Inconsiderable. The result is -59.7. Subtracting this from the result of original calculation,, the true content excavated is left. 60 TABLE OF OPERATIONS. S"A WI1THS. CE-N PRODS. CORECTIONS BREAIIS. 81. 1 6;. 2 16. i 15.5 x 1' i 8.25 11.5 0 _ 0_ IL k l). -2 -- 15. 15.0 -2 - L 117. -2 -17. 15.5 x -2 * IR 116. -3 - 24. ~50 15.5 15.5-1 I -15.5 16.5 16.5 -1 /-16.5 L 15. -2 11-30. i tR 15. - 4 -60. 2 7.7 15.4 0 0 " 8.5 17. 0 0 " L 15. -0.8 - 6. R 15. - 4. -30. 64.45 -198. 0 0 Sbw = -4+ 1933.5 -2Sb2n= -1800. 2)-64.5 9)-3225. 6)-358.3 -(-59.7) 5083.27 5143.' =no. of cu. yds excavated. 3~, When a large number of the sections do not coincide with base and slope lines, or when such irreg. sections are scattered and isolated, making many separate calculations necessary, it is better to make one operation of the final estimate, considering every section in the work, whether its area be zero or not. There is one important advantage available in this case. Thus, the formula for original approx. content is mid prods. + end prods D { - Sb('mid-widths + g end-widths) —2Sb2n f 4' and the formula for approx. final content is the same; whetefore, since the widths are identical in both, the terms S(nmid-widths + end-twidths) - 2S/,2n vanish by subtraciion. Hence, in the final calculation, the 1st col. of widths need not be constructed, summed 61 and multipled by Sb, and the quantity -2Sb2n needlnot be added, but the col. of prods. only considered, to this added the cor. for, interms. and the pris. cor., and the sum subtracted from the sum of prods., the cor. for interms. and the pris. cor. of the original calculltion. Between the cross-sections, which do not' coincide witll slope, there are in this case to be' considered sections which do coincide, and; whose areas are therefore zero. These are easily disposed of. In common with the other sections their widths x Sb need not be considered. Their prods. are for each V ve + X7+ 1t)(c l ) (C X r) The first two terms are always zero, and the last two are always — b(r — 1). Since the factor b is common to all these sections, the prods. of all are found at once by adding all the side-heights, prefixing the minus sign, and multiplying by b, following the rule, however, of using the half-prods. of sections at full stations preceding interms., and the half prods. of last interms. So, add to the side-heights of all not followed by interms. the half-heights of those followed by interms. and the halfheights of last interms., before multiplying by b. This: is all the work required for these sections, since the cor. for interms. and the pris. cor. always reduce to zero. The former cor., (P- P' + Sb(W- W'))TWv where P, P', are the respective prods., reduces to (-b(r-+-r'-?') +Sb(w-w'))T. iBut wV=L + +2b, w'- + -+2b. S 5 8 s Sb(w-w') =b(r +1-r'-l1'), 62 and the whole cor. reduces to zero. The pris. cor. is (V (v-v')(m'-m) + (x-x')(n'- n) D + (b-b)((c'-l') —(c-1)) +(b-b)((c'-r')-(c-r)) 4' and each term reduces to zero under the conditions. Let us suppose that the first two sections of finished work of the excavation, last treated as an example of preliminary calculation, are as noted in second case, and that the last four sections also of the same excavation do not coincide with the true slope. It now becomes more profitable to estimate the material, remaining to be excavated, in one operation. The final notes of the cut are the following. FIELD NOTES. STA. CENTRE. LEFT. RIGHT. REMARKS. 1 0?TT, -,0 i-,T-g3. +50 1 - - - ~TT ~i 1, 2 1 0 08 T,8 + i - L 3 0 -i'8 +20 0 -o - 4 0 I',5 O *T,", 6 0 A, A, 7 - O fi' f — j,2,4 8 - 0,', 9 _I 0 O 2,1 [ 63 TABLE OF OPERATIONS. STA. WIDTHS. CEN. PRODUCTS. BREAKS. -I1INTERMS. FRISMOIDAL. 1 16. 2 32. 15.5 x 1 16.5 0 0 15 x 2 L 15. - 2 -30. 15.5 x- 2 L 17. -2 -34. 16. - 3 1-48. L 15. -2 -15. 15 X.- 0 R 15.5 —4 -630. 15x 4J 2....5 15. -53.8 -807. 0 6 15. 0 0 0 17 x 0'~27. 23 +621. - 24 x 18 +I R 15. 23 - 345. 1 15 x 4 R 25 5 -_24 -612. 9 J 600 15 x -18 7 15. 0 0 01 24 x-18J 30. 22 ~ 660. +3 R 15. -22 -330. 0 1 25. -30 -750. -3 00 8 15. 0 0 0 29. 20 + 580. -2 25 x 20 ) for In -200= -0 11.95 9 15. 0 0 25. x 108 R 16. -20 -16 - Soo Pris. Cor. +600 =- 21.7 9)270755. 6)30083.9 5014. cu. yds5 64 The calculation of this remaining work is precisely similar to the original calculation, except thile omission of the 1st col. of widths. The sections of stas. 2, 3, 3+20, 4, 5, coinciding with slope lines and lying adjacent, are noted together, and their prods. joined in one. At sta. 2 the sum of side-heights is 4. 8. The sum of side-heights at sta. 3 is 8: half should be taken since it precedes an interm. Half of 14 should be taken, because it is the sum of heights of last interm. The whole heights at stas. 4 and 5 should be taken. Thle sum is 53.8, to which prefix minus sign. The common multiplier is 15. Since there is but one intelrm. to cause a cor., the 2d col. of prods. has not been constructed further than the first section. For this interm., subtract widltl of sta. 2 f~Lon that of sta. i, and multiply dif. by 30 [-Sb]. The result is 3. Subtract prods. of sta. 2 from those of sta. 1. The dif is-8..(3-8)50= —250. The pris. cor. for the first two vols. is inconsiderable, as discovered in 2d case. The cor. bet. sta. 2 and sta. 5 is nothing, because the work bet. them everywhere coincides with the true slope and base. The right of sta. 5 is arranged in col. of remarks to correspond with right of sta. 6. The right of sta. 9 is arranged to corresponld with that of sta. 8. Subtract the results of the three cols. from the results of the three corresponding cols. of the original calculation. Use the same factors with the diffierences that were used with results of same columns in original calculations. The result is the exact amount of the mass of material removed. This can be conveniently divided into the classes of material, as rock, slate, common earth, etc., from the monthly estimates or from the final notes, exactly as it is now done after obtaining the whole amt. by: the ordi-. nary method. A fair plan is to use the proportion: as the amt. of one class of work, as rock, in the sum of monthly estimates, is to the whole amt. of material in the same, so is the true amt. of rock to the true total amt. of material, as just found, unless extra trinmming of earth or other material has been done between the two measurements. But these details are well understood, and form no part of the plan of this book. 15 CHANGE OF SLOPE. It is often found necessary after the preliminary calculation to make the slope flatter than at first intended, or advisable to make it steeper, according to the matterial struck. If the surface of ground be regular, as shown in the first diagram, it is better to calculate the whole work anew witlh the new ratio of slope and the,'-7 new widths. But, if the surface of ground be very irreg., as shown in next di8granl, it becomes preferable to avoid the recalculaltion of the irreg. part, by considering the new slopes as the under part of perimeter of section of excavation, and calculating the work remaining to be done, to reach the original slopes, as in the example last given of finished work, and finally subtracting the latter result from the result of original calculation. The formula for the cross-section of remaining work in last diagram is av(p+Sb)+ Xx(z~+Sb)+ ~lb(c-n) + 27'(m-p) + 2p/(n-i- + -b(c-z) + ~z'(t-r). In this instance it represents a positive quantity, because thle new slopes are steeper than tihe old: if the new ones were flatter, the section remaininig would be negative. 5 66 RUDE PRELIMINAtRY ESTIMATES. Fon a rough, early estimate, when the widths of the sections are known, the first rule for calculating a series of equi-lengthed vols. is recommended, disregarding interms., breaks and the pris. cor., and taking the crosssections as far apart as possible. This computation can be made very rapidly. When the widths have not been calculated or staked out, and the preliminary notes only are av~ailable, these consisting of a record of centre elevation and the elevation of a point on each side of centre 50 ft. distant, at each sta., a rough but hasty estimate can be made, by placing centre heights above grade in one column, and the squares of these in another, conveniently extracted from a table of squares such a.s Henck has furnished in his FIELD BOOK, and treating their sums as in the treatment of level sections, alreadly discussed, disregarding altogether the third column, conlstrlcted for the pris. eor. When the surface line at right angles to centre line is unbroken, as sometimes occurs at a great many consecutive stas., the areas of the sections, under supposition that they are level and of thle same height as centre, are always too small, as seen in the diagram, by the triangle a, _'.-' i \ J/I' 1" I,|, 67 m r r', and, consequently, the resulting mass of material between is also smaller than the true contents. This fact has often been pointed out, but we lhave seen no formula of cor. for the error. Such an one, that serves very well both for accuracy and brevity, can be obtained in the following way. m r r' is similalr to 1 e r. eg — -S:.~. eg= Sb. LetA'represent areaof 1 er, and let the abbreviated word Cor. denote the area of m r r'. Then Co: A':: 2: (c+ Sb)2. Let A denote the area of 1' e r', equal to A'- Cor. Now by the principles of proportion Cor: A'- Cor; 2: (c -Sb)2 —z2, Or Cor-(+) 7c~+ Sb)2- 2A. If h be the elevation at 50 ft. from centre on higher side, then z: h-c:: b+-s:50, or z=7u(l;-c)(c+ S&). (5) ()( c)2(c 4 S)2(_)2 ((n) (c+ S6)2_ - ()2(c + _ Sb)2(_~)2A (S)2 (h- - 2 It is here seen that the ratio of the correcting area to (1h-)2 A is as "S_ )2 is to unity; and, if thlese areas be moved through any dist. D, the vols. generated are in the same ratio. Therefore, if the average area of m r r' in all tile sections through the work be ascertained and multiplied by D, just as the average of 1' e r' is multiplied by D, the resulting vols. are related by tile ratio 7"1-. \ S2(,)2 -1, where h is the average higher side-height, and c the average centre-height. But in calculating the vol. generated by moving the average sectional area 1' p k r' through the dist. D, a column of centres has been constructed and summed. Let C be this sum. Then c, where n is the no. of vols., is the average centreheight. Similarly, if H be the sum of side-heights, the higher one only being taken at each sta., or either if the section be level, then - is the average side-height. So the vols. hold the ratio 68 (H-C)2 1- )2 S2t_ 5)2o S2(2_(-jC)2) If, therefore, after finding the volume generated by moving the average area l'plkr' through D, and adding thereto the volume beneath road-bed, Sb2D, the sum be multiplied by -2nT ( _ —7, the prod. is the cor. to be added to the original volume. This shall be illustrated from the following notes. S is considered -, b, 10 ft. STA. LEFT. C3 NTRE. RIGHT. i 1 6 11 2 2 8 14 3 — 1 9 19 4 -5 15 35 5 15 20 25 6 18 14 10 7 12 10 8 8 11 8 5 9 12 7 2 STA. c c2 h 1 3 18 0.5 2 8 64 2. 3 9 81 -1. 4 15 225 — 5. 5 20 400 15 6 14 196 10 7 10 100 8 8 8 6i 5 9 3.5 24.5 1 90.5 1172.5 35.5 1810. 50)55.0 2982.5 1.1 800. 62.79)1.21(.0193 3782.5 6279.0193 5821 1.135 56511 34.042 1699 37.825 9)305550.2 3)33950.02 11316.67 cu. yds. 69 The col]. headed c, c2, are constructed as for levelsections, the sum of former multiplied by 2 b, the sum of latter by -, and the prods. added together. Next a col. of heights is nmade, using half the first and last as with centres. Instead of taking the higher side elevations, the lower ones have here been used, and their sumn subtracted from sum of centres: the dif. is the same andC the numbers to be dealt with smaller. This dif. divide by 50: square the quotient for a dividend: for a divisor subtract dividend from SU2J2, 64 in this examiple. The quotient multiplied by the whole volume down to where the slopes meet, is the cor. to be added to original volume. Therefore, to 2982.5 add S/,2n [-800], and multiply sum by.0193. The three partial prods. formi the cor., which, added to the 2982.5, yields a sum to be multiplied by 100 and divided by 27 to produce a close approximation to the number of cu. yds. The peculiar meirit of this plan, as applied to preliminary estimates, is the facility with which changes in the result may be made to correspond with alteration in slope, width of base, altitude of grade, etc. For instance, to alter the result for a chfange of slope, tile cols. need not be touched, but simply the values underneath modified by the new value of S instead of the old, wllere S enters the calculation. SimilTarly a change may be effected for a change of width in base. If it be desired to ascertain the effect of sinking the grade all the way through, say 10 ft., merely add to sumn of centres 10 n [=80], to the suni of squares 1.0)2t [-800] plus 2 x 1.0 x sum of centres [-1810]. The cor. ratio,.0193:1. need not be altered. Therefdre, the increase of voluLme equals 4210 x 1.0193 x 100. 27. The general formula for this example is, letting x represent the proposed increased cutting and 321 the resulting increase in the content to be excavated, [nx.2b + (nx + 2xQC)] 1 o 1 9 3 - C being the sum of col. headed c. Solving with respect to x, we have =.I / [ sLLT+ (C+ Sbn)2-(C+ Sbn)]o Sunppose we want 1000 cu. yds. more from the cutting. Let 31=1000, when x= + — 0.7-ft. To make the excavation 1000 cu. yds. less, let 31I=- -1000, when x= — 0.79 ft. For any other example simply use instead of 101.93 the proper factor, or, if the cor. is not used, take D alone, generally 100. If the centre line be moved to a position on the right or left, parallel to its present position, the new centres must be tabulated and squared, but the ratio,.0193:1, is still correct. It is seen that the cor. is not quite 3 per cent. of the true contents in this example. If such close work be not required, the cor. need not be noticed. But, if it be retained, the result is the exact approx. contents of the vols. bet. the actual cross-sections, afterward to be staked out, when the pris. cor. may be made and attacthed to the preserved result. For a hasty estimate, when the surface line is broken at centre, allowance must be made, in the centre height used, for the direction of the angle's concavity. If this be downward, the centre used for equal level-section must be diminished, and, if upward, increased. Repeated trial, tested by computation of the true areas, gives skill in this. The allowance varies with the rate of slope, width of base, centre-height, and the size of the angle's concavity. But, since the base and slope remain constant through the work, the centre-height and concavity only exert influence in a regular cutting. The cor. need not be applied in this case. Without this latter rather rough method recourse must be had to much longer rules. 71 CORRECTION FOR EXCAVATION ON CURVES. THE error of calculating earthwork on a curve, disregarding the curvature, is not great. The manner of calculating exactly the content of a vol. is the following. The diagram here accompanying represents a vol. on a curve of exaggerated degree of curvature. c JH is the radius, R, H the angle subtended by the distance. c c', -j, t i i 1f B Bli~ri~1' % I 4 72 =D, between two stations. h is tile angle of deflexion. e e', r nri' are the edges of road-bed. Conceive a straight line joining c and r'. Estimate the vol. c c r' by multiplying the area of its base, 1cc'xgr',-1-Dx'cosh, by tile average height, ~(c+c'+r'). Add to this the vol. c r r' whose base is x x 7kr', —x(R-x')sinR, and average altiLude (c+r +r'). If now the diagonal vertical plabne r c' be considered, and the vols. c c' r, c' r' r be similarly calculated, and their sum averaged with the sum of the two other vols., that average is the exact content bet. the warped surface c r r' c', the plane of grade and vertical planes through the perimeter of warped surface. This principle was also used on page 54, in connection with end-volumes. It is easy of demonstrtltion. A proof of it is given on page 379 of Gillespie's ROADS AND RAILnOADS. It is also found in SoNNET'S DICTIONNAIRE DES MATHE'MATIQUES APPLIQUEES, and in many other works. Subtract from this vol. the triatrgular prism e r r' e", whose end-section' e." is exactly equal and similar to r' e' and whose altitude is lc r'; also the pyramid e" r' e' of altitude r' and basAl alrea 1(x' —)resiaH. In the same manner compute the vol. 1 e c' 1', bounded by vertical planes, subtract the prism m n m " 1', and add thle pyramid 1' m' n". This process is entirely too laborious for any practical application except to test the accuracy of available methods. John B. Henek has instituted a formula of cor., which is very accepttable on account of its simplicity and the accuracy of its results in all ordinary cases. Expressed in his own symbols, d, d', being side-widths, hI, h', sideheigths, this formula is [lc(d —d') + IB('-/)] D(d+d') Instead of -D may be used 2sini. This, translated, becomes in cu. yds., w(v - x)(c -- Sb),.: The dif., v -x, is found by subtracting the inner sideiwidth at a sta. from the outer. The cor. is thus positive when the outer side is greater, and negative when it is the lesser. The cor. is applied at each sta., half the results being accepted at the end sta. of a curve. The following table shows the amts., resulting from the use of Henck's formula, compared with the true 73 contents, ascertained by the lengthy method explained at hlead of this article, of four sepatrate vols. on a 4~ curve. The 1st col. contains the true amts., obtained by prismoidal formula, without regard to curvature. Half of Henck's cor. is used at each stat., as this is supposed to be the end of the curve. Prefixed to the table are the field-notes of the cross-sections used. S-=, b = 10, D = 100, R=1432 69, sil-h=.0349, cosh =.99985, sizH=.06976. CENTRE. LEFT. RIGHT. 1st TVol. 10 20 30 30 0 4 2n1d ol. 10 A0 -230 4 3rd Vol. 10 2 0 10., 4th Vol. 10 ~. 10.o 10 0 TRUE AMT WHENCKIS CORRECT'D EWITHOUT TRUE AMT. ERROR. CURYATU'E colt. MT. 1st Vol 128333.331 + 174.5 12S507.83 128239 60 +268.23 2d Vol 128333.33, + 337.361128670.69 128623.27 +-47.42 3d VoTI 35000. + 3157.75 3517.75 351.9 +5.85 4th Vol{ 30000. 0 30000. 29995.5 +4.5 The surface of 1st vol. is very much warped, the outside of one section being the lower, and tlhe outer of the other the hliglier, side; and it is seen that Henck's cor. is here mulch at fault, ti.e result of the calculation without noticing tihe curvature being much nearer the true content than is the corrected amount. The next vol. is the same as last, except that its first section is reversed so that the higher sides are both outer. The surface of ground is, ther3efore, very much less warped, and, as seen, tile col. of Henck is very much nearer the truth. The next vol. is bounded by cross-sections of same size:and sllape. The surface is, therefore, two planes, and Hebck's cor. errs but little. The 4th vol. is level. The inference is': Where the suz.'cice of gyround is a plane or nearl1y.so, HIeic7c's formula is very correct. A muchwarped surface is rare; and, when it does occur, it is, perhaps, better not to attempt a correction, 74 BORRO W PITS. FoR the calculation of any class of cross-sections, whether there be a regular slope and base or not, we recommend the metlhod, employed here for intermedia, te sections and for breaks, of using the distance of each point from the base line as a factor with the dif. of elevation of the two adjacent points. The object is to secure one col. of very small factors, the differences, although the other col. receives larger factors than by the ordinary method; also to facilitate the application of the pris. cor. where required. Let the following be the field notes of a borrow-pit, whose datum plane is fixed at 20 ft. below the elevation of sta. 0 on base-line. STA. 2- - %.i _ 1. J,2 o 0 2 5 2 0 3-i. i4 7 P-6 -6-v TV 0 20\~3O o he sum of last to elevtios. om thee clclat the ol. of prods., using halthe mass of earthe pbetods. ofthis surface lastd stas. Multiply the sum by the distance bet. stas., 25 ft., and divide by 54. It is scarcely necessary to menltion the points of brevity in favor of tlis plan, as against the ordinary method of calculating such work, silce this topic has been fully discussed alreacly; but we might recall the fact that TABLE OF OPERATIONS..TA.. _ s.,CORRECTTONS. STA. DIST. h-h'| PRODS. INTERMS. PRISEOIDAL. 0 10 10 50. 30 0 0 50'_-15 - 375. 75 - 5 -187.5 100 45 2250. 25 1 15 5 75. 0 30 - 5 - 150. 25 45 -10 - 450. 0 70-5 - 350. 0 100 40 4000. 0 2 15 -2 5- 30. 50 40 -10 - 400. - 30 100 51 5100. 0 4 20-7-8 70. E 0 30-80 - 8120. 0 100 57 2850. 0 15157.5 — 50 Pris. Cot.~+6 z - 8.33 4)15149.17 9)378729.25 6)42081.03 7013.5 cu. vds. cornmmonly the areas of the several parts of each crosssection are combined to find the area of that section, thle latter value being used, while here all these partial areas are added at once. Also the factors used are more readily handled, since those of the col. of differences are nearly all composed of a single digit only, making unnecessary tlhe extra work of a separat;e multiplication, while the larger ones opposite the last distances need for the operation merely a shifting of decimal point. Besides this, the measurements are now arranged in perfect order for the easy application of the pris. cor, Simply multiply the dif. of corresponding distances by 76 the inverse dif. of corresponding factors in col. of differences, and multiply by dist. bet. sections for a value to set in col. of cor. The last paiL of factors at each sta. thus need no cor. in this example, but would require it, were the far side of the pit irregular. Sta. 2 has one point less recorded than sta. 1. The last break of latter is noticed to merge into last of former; therefore, in applying cor. the last break of sta. 2 is considered to be two breaks, and the new factors are recorded at the side of the old ones in the table, to be used only with the section before. Since the vols. are all of one length, divide cor. by 6 and add to sum of prods. Multiply sum by 25, -e~, and divide by 54. If there were intern. cross-sections, use the half prods. of stas. next before and of last interms., as in road-way calculation, and for the cor. multiply the dist. of each interme. from sta. next before by the dif. of prods. of sections adjacent. Divide sum of col. by 108. By this plan, as illustrated, it is seen that the pris. cor. is very easy of applictation, and that it is also readily discerned where the cor. is and where it is not required. For instance, where a knoll or the spur of a hill has been removed, the approximation by end areas is extremely faulty, and the cor. assumes great importance. In concluding, we must again say that the principles, upon which this plan of computing earthwork is founded, are few and simple, and the results, excepting the rough estimates, perfectly accurate; that the calculator soon becomes familiar with the processes, since the values naturally find their proper places in the tables of operation; and that these tables themselves are neat and regular, recording all the work as the ever-present authority for the results they produce, and always ready for a rapid review, while by ordinary methods the whole work is thrown away, and nothing preserved but the areas of the cross-sections or the contents of the vols. between. -- ---— "- l: e 77 GENERAL NOTE. VOLUMES BOUNDED BY WARPED SURFACES. IN staking out earthwork, the sections may be taken as far apart as the ground continues to change its slope uniformly, no matter how much the surface may warp, nor how great the distance between cross sections; that A is, while the lines AA' and BB' remain straight, and the surface from AA' slopes to BB' in straight lines. The most rational conception of the surface bet. four straight bounding lines is that it is the hyperbolic-paraboloid or warpedt suzfcace, which may be generated by moving AB, parallel to the planes of end sections, along AA', BB', till it assumes the position A'B', or by moving AA' along AB, A'B' till it becomes BB', each end traveling with speed proportional to the length of its path. To the vol. beneath such a surface the prismoiclal formula applies exactly. The assumption by Henek of one straight diagonal from A' to B or A to B', has received severe criticism, and it, moreover, has not the advantage of marling the formulea less inticate. Little fault, however, can be found with the field work of engineers, since the rule of placing cross-sections so that they shall be connected by straight lines is carefully regarded in preparing notes for any mode of calcu 78 lation, the lack of straightness in a longitudinal element showing the presence of an intermediate irreg. section. The principle that the prismoidal formula applies to vols. beneath warped surfaces was assumed on page 21, since-it has been so often used and shown before. A general proof of this application to vols. bet. three-level sections, whether covered by plane or warped surfaces, can be deduced by calculus, as Prof. Gillespie has done for a vol. bet. two trapezoids in his MANUAL OF ROADIMAKING. Let Iw(c + Sb) -S) 2, 2wV'(c' + Sb) -S) 2 be the areas of end sections. At a distance x. from the first the centre of a cross-section is c — (c' —c)x, width w +(Iw' — w). Hence its area is (I + (w' -V) ) (C ~ (C- C) + Sb) -Sb 2. Multiplying this by dx, after performing the multiplication indicated, we obtain the differential of the vol., w w(c + Sb)dx + (w'c' -wc - vc' + wc)-D 2 -- S 2Lx - (woc' — 2wzv- -'c + Sb(' — w))'' -SV D Integrating bet. the limits 0 and D, we obtain Vol= (' c+ vc' +2(v'c' -t-e) -- 3Sb(w iv u) —12Sb2 )T`,5 which is identical with the formula for the true vol. on page 21. Irreg. vols. may be divided by vertical planes through corresponding breaks into portions, which may be shown in a manner similar to the above to be subject to the prismoidal formula: therefore the composite vols. are subject to the prismoidal formula. On page 40 it is advised to note the vanishing extremities of breaks, and subsequently the importance of this is illustrated by examples. The general discussion of this topic can be readily effected by the aid of calcu].us in the manner following. The diagram represents a triangular, warped-faced wedge. Its base is ARBC, edge DE: the back of the wedge is the face ADEC, warped or plane according to the relative positions of the lines AC and DE. The remaining faces, divided by the edge BH, warp from AB to IDH and BC to HE. It will be noticed that this wedge is similar to 79 l Y the wedge A BCDE shown in the diagram of vol. on pige 41, and that all the irregularities of rail-road vols., which vtanish at one end of the vol., can be decomposed into such wedges as the one here shown. In the discussion it is convenient to make the edge DE horizontal. Using the symbols of the diagram for measurements, we have for a section distant from the base by x'a — D,w" —w+(w- )W, d" =d+(d' —d) ",h7 D i and for the area of this section Ai'B'' =A'B' G' F' + B'G' C'- A''F' = }d"(a' + IA') + Uh'(w" d")- aglw"V (a'd" + 1'w" - a'w") ad(D-x) o(d'-d)(Dx ox2) D. D2 D D 2 Multiplying by dx we obtain the differential of the vol. The indefinite integral is 80 f ad(Dx X2)'a(d't d)(1Dx 2X3) iD D2 w(Dx-_ x2) (w-_w)(,DxD2_~ )3~ Limiting this by 0 and D, we have Vol. = ( adD + a(d'- d)D + (hi - a) ( w D + (w'- v)D)) ((2 + d') + (h- a) (2w + w')). By this formula it is instantly seen that to give d' any increment m increases the vol. by amD 12' while to give d the same increment increases the vol. by 2amD 12 Therefore, to move the vanishing end of a ridge or hollow any distance alters the content qf a vol. ex actly haTf as much as to move the other end of the same ridge or hollow (ian ecual distanzce in the same direction, as stated on page 43. It is also to be remarked that when a=O, d and dc' vanish from the formula. Therefore, When DE is parallel to A C, or when the back qf the wedye is a plane surfcace, to move the vanishing end qf a ridye or hollow acry distance does not alter the content of the vol.: but the farther A C, DE are from being parallel, Or the more warped is the back qf the wedge, the greater is the effect, of moving the vanishing end of a ridge or hollow, upon. the content of a vol., as stated on page 44. We consider this an important matter since we have found no book which demonstrates the consequence of fixing these fading ends. That it has a consequence is shown by the example on page 41. The irregularity of one section in this example is made very marked in order to malke visible, if possible, the increase of content occasioned by moving m, n, p to M, N, P. But, if the irregularity were scarcely perceptible, as in tlhe most ordinary example, the increment aj would be the same, 81 since it depends entirely and only upon the difference of inclination of AC, DE, as represented in the formula by the symbol a. Prof. Gillespie proposes in an off-hand way [RoADS AND RAILROADS, page 373], considering such a vol., to conceive vertic(al planes passed through the breaks of irreg. section and cutting the surface line of tile other section proportionately, preparatory to the application of prismoidal formula. But this assumption that the ridges and hollows cut both sections proportionally has the same fault that Prof. Gillespie finds in Henck's diagonals, that tlhey would not always happen so. For, supposing them to be so now, if hereafter, proceeding from one section, the direction of the centre line be clfhanged, both the ridges of Hench and of Gillespie must change also or break the law. There is no more reason in assuming that the ridges vanish at these points than at points fixed by any other arbitrary rule. One thing is remarkable about Prof. Gillespie's method. If the approx. content of thle wedge be subtracted from the rutle, the difference is (ct (d'-d) + (h - C)(w' -t)).1 If w=w', the difference is a(d'-d). But on Prof. Gillespie's assumption d and d' are now equal also. Therefore, when the widths are equal, Prof. Gillespie's result, although obtained by much greater labor, is no better than tile approx. result, while the content may still vary by a-., cu. ft. for every difference of mn feet in the position of the fading end. Thus, the nearer the widths approach equality, the nearer is Prof. Gillespie's result to the mere approx. result, while the true content depends upon the place where the ridge or hollow terminates, not where it may be supposed to end. Taking the example of a rigllt vol., mentioned in this connection by Prof. Gillespie, which is drawn to scale in diagram, and disregarding for the present the lines BG, GC, GI, the value of a is easily found to be 5.1. Substituting this in the formnla, and making mn 1 ft., D being 100, we findL for the increment to the content -i~-=- 42.5 cu, ft., corxesponding to an increment of 1-ft. 6 82 Cf to d', measured along L)EA or 43.4 cu ft. correspondling to 1 fit. measured horizontally. Therefore, if BII happen to divide A C, DLEV proportionally, Prof. Gillespie's result is correct; but, if not, he errs by 43.4 cu. ft. for every foot of distance from the true vnlishinag point to the assiumed one. This is surely a difference worthy of being noted, especially since it; must be as easy to note the true vanishing points in the field as to calculate the false points by the rule of three. It was simply apparent to Prof. Gillespie that some point must be taken for the end H in order to obtain data for mid-section, necessary to the prismoidal formula;'but it seems that he was not aware of the importance of finding the true position of this end. This, however, might eLasily escape observation since it is a peculiarity of vols. bounded by warped surfaces. Prof. Gillespie's theory of the surface of all irreg. vols., whether bet. two irreg. cross-sections or one irreg. and one reg., is summed up in the following sentence. "Conceive a series of vertical planes to pass through all the points on each cross-section, at which the transverse slope of tthe ground changes, and at which, therefore, levels have been taken, and to cut the other cross-section so as to edivide the widths of the two proportionczay." By this treatment the surface of Prof. Gillespie's vol. is scored by a number of ridges and1 hollows equal to the 83 whole number of brenks in both end sections, against the great probability that the breaks at one end. belong to the same ridges and hollows shown at the other: all the ridges and hollows of each cross-section vanish at tile other, contrary in the highest degree to the n.atuural disposition of the ground surface: a break representintg a ridge at one end mnay be joined by proportional measurement with a break representing a hollow at the other, a perfect absurdity. His lines, fixed by an utterly unfounded assunmption, are imaginary, as he calls Henck's diagonals, and lead to error nearly as gross. Since he assumes that each ridge or hollow vanishes in the same vol., he must calculate for each vol. the heights of as many vanishing ends as there are breaks in both end-sections, instead of calculating the heights of such only as do really vanish. After his laborious computatious, first, to find the proportional distances, second, to find the height of one end of each ridge or hollow, thllird, from thlese to calculate the measui ements of mid-section, the number of whose breaks is tlhe nunmber of breaks in both end-sections, generally double the number in the true mid-section, fourth, to obtain the areas of end and mid-sections, and, fifth, to apply the prismoidal formula, the result is identical with that obtained by approximating end areas, when the widths are equal, while the true vol. may vary far from, this; and the nearer the widths are to equality the nearer is Prof. Gillespie's result to the result of mere approximnation. The ordinary methods of calcula~ting irreg. vols., by approximating with end areas, tand by finduing h.eights of level crosssections of equal areas and using tables, are both erroneeus; and assumption of any kind is entirely out of place for measurements which bear so important a relation to tlhe content. The formula P-9- may be conveniently used as a correctitlg formula in the following case. Suppose a vanishing end lhas been neglected in the field, and it is afterward discovered by thle relative dip of the lines A C, DE that this is an important point. Any point may now be assumed to satisfy the formula, for instance, the point D, already known, whose height need not be calb culated, and afterward at a convenient time the distance m may be measured and the quanutity cu. ft. or ""r cu. yds. added to the former content. a is easily calculated, or can be found graphically and measured with sufficient accuracy. To determine whether mD-. is positive or negative, conceive a line BF through. B parallel to DE; then, if moving B to the right on this line increases the area of ABC, by lengthening the perpendicular, motion of H to the right also increases thle content of the wedge. So, if D were assumed in order to avoid delaying calculation, and H were found to be the true point, a-9 is positive. For a second break G, forming the second wedge BCGEIH, the formula of correction is X-1, a' being FC or the distance of B from CK. If I were not known, it might be assumed at any point on DE, as E, H, or even D beyond H, and the correction.made by measuring the distance m from the assumed to the true point. Since motion of G to the righlt on a line parallel to DE increases the end area and consequently the content of the railroad vol., so motion of I to the righllt increases the content. Whatever the position of breaks, motion of any in the samne direction affects the content in thle same way. If both H -and I were for the present unknown, D might be assumed for both: thlen the cor. would be xD H+l x DI, DI being the increment m for the first and DI for the secoid. If D and E were assumed, the cor. would be,, x DH+ Vx -IE. To make these corrections m srhould properly be measured atlong the edge DE; but in the regular staking out of work the vanishing ends must, like real breaks, be fixed by horizoxtal measurement. TABLES' OF QUANTITIES. As for lTables of Qiutntities, it is just to say that they are little used on mailny accounts. Tile common ones, for vols. bet. level sections, are reliable and expeditious; but tlhese are the very vols. most easily calculated in series. For other sections, the areas of these must be indepe ndlently calculated, and the centre-heights of equal level-sections computed or approximately taken from cdiagrams, as Trantwine's, before the tables can' be used. This preparatoryv work is much greater than the labor of finding the whole content exactly by the method of this paper; and the level-section tables for all such cases are inaccurate, as illustrated by many authors and systematically discussed by Prof. Gillespie, the errors being almost constantly in defect and, therefore, seldom b:dlancing. This error exists in all the tables constructed by followers of the methods of Telford and Sir John MIacneill. The bulk of a complete set of tables for all slopes and bases would be enormous, the number of such tables being equal to the product of the number of different bases by the number of different slopes in use. This bulk is sometimes reduced one-hundred fold by omitting the tenths, these to be interpolated by proportion, or by diagrams wherein thle tenths are estimated by the eye. Such interpolation is not accurate. The diagrams are tiresome to the eve, and without the nicest care the producing values can not be laid off in the diagrams to the tenth part of a foot. An error of 0.1 in c, when b= 10, S-=], c_5, and D-100, produces anl error in the content of vol. 0.1 x 30 x 100 300 cu. ft., in case an equal error were madle in the other end section. If no error were made at the other end, tile error in content would be 150 cu. ft. If the error at the other end were equally far in the opposite direction, no error would exist in the result for the content. So, although there is a great cliance that the errors may balance, it is not safe to use a method whose errors are so large. The same reftsoniong may be applied to the method of finding height of equivalent level-section. The measurements of irreg. section are in feet and tenths: it would rarely happen that the height of equivalent level-section would be exactly in feet and tentlis. Tihe neglect of the extlra ftaction, if only a fourth of a tenth, would make a difference in above example of 75 cu. ft., 37.5 cu. ft. or nothing in its several cases. Some calculators have constructed auxiliary formulae, by use of which vols. of certain base and slope can be, treated by tables for other slopes and bases. These formulra represent extra work. They may increase or; 36 diminish the error of the table, according to the respective slopes and bases. Tilhe advantage of using tables of quantities is very much lessened by combining vols. in series, because thus all the constant; factors are removed from the calculation of each. Suppose, for the sake of argument, we could have a table by which the approx. content, (wC+ wUc' + Sb(w + w') - 4S 2 )T, of a vol., for a certain value of S and of b, could be obtained by mere reference, using some two mzeasurements, as in the tables for level-sections. To find the con-tents of 20 such vols. consecutive, the table must be consulted 20 times and the results added. But, since the factor T need be used once only for the 20 vols., the advantage of including it in the table is very slight. Therefore, if the table were constructed on the value within the parenthesis, the extra labor would be simply one little division. But the term Sb2 need be but once used. Therefore, if the table were constructed on the value wc+ w'c'+Sb(w+w'), the extra labor would be only one subtraction and one division. Since the prod. w'c' enters the formula for second vol., and every other mid-plod. is likewise common to two vols., and since the prod. Sb(w+w') becomes for the series a single prod. of Sb by sum of widths, the formula wc t+'c' + Sb (w + w') represents for the series 22 multiplications and the summing of 2 columns, the prods. and widths. Therefore, if we had a MULTIPLICATION TABLE extended enough for the prods. we etc., to solve the above example we must consult the multiplication table 22 times, sum 2 columns, make one subtraction and one division. To accomplish the same by a table for the whole amount of each vol., we must consult the table 20 times and sum one column. The disadvantage, then, of using a table of prods. instead of using a table of contents for n vols. is composed of the following only. The table of prods. must be consulted n + 2 times instead qof n times, an excess of 2. 2 columns mZtst be suZndmed instead of 1, an excess of 1. One subtraction and one division must be made. 87 These four extras represent very little work. The advantages of using a table of prods. instead of a table of contents are tlle following. 1. Irrey. vols. can be treated by a table of prods. as well as reg. vo7s., because the formula for irreg. vols. consists likewvise of a column, of widths multiplied by Sb and a column Qf prods. 2. Since the irreg. and minor vols. can be included, n can be mwade greater, when the disadvantages of using the table of prods. becomes comparatively less. 3. Since the values S and b are not used in the construction of the table, it applies equally well to vols. of all slopes and bases, avoiding the gryeat bulkiness of/ a complete set o' tables. The above scheme of comparative merits of a table of prods. and a table of quantities is drawn up on the supposition that a table of quantities can be constructed for such vols. to be used directly. Since this cannot be done, as is evident by inspection of the formula which contains four variables, to the advantages of a table of prods. may be added, that it obviates not only bulkiness but the rules of equating areas, the diagrams, auxiliary rules andL general inaccuracy. A table of areas might be constructed on the formula IwV(C Sb) -Sb2, since it contains only two variables. This table must be consnlted nI+l times, so the comparattive disadvantage of the table of prods. would be a little less than named above, and tihe advantages would remain the same, since the table of areas must be reconstructed for ever cllange of base or slope, etc. But the auxiliary rules, rules of equivalency, etc., would be avoided by either method. To correct the approx. results a table might be constructed on tlle formula (,V -'V (C - C); but the table of prods. would still possess all the advantages, while the disadvantage would merely be the use of the factor - once. The pris. cor. for irreg. vols., as well as tire other, can be accomplished by a table of prods., and the factor 4 used once for all. 88 We, therefore, strongly recommend the introduction and use of a table of prods. in connection with the calculation of railroad vols. in series. The multiplication table is seldom, if ever, used by calculators, for the reason that their work is desultory, changing every few seconds from one process to the other of addition, subtraction, multiplication and division; but in the method of computation recommended here, after transferring from Field-Book the widths and centres, we have in direct and uninterrupted succession a long line of multiplications to perform where the value of a table of prods. in easing the only laborious portion of the work will readily be recognized. Such a table can be accommodated on 15 pages 8vo., ranging along tops of pages from 1 to 300, and down the pages from 1 to 80, thus containing all the ordinary prods. occurring in earthwork, of factors with three digits by factors with two, the decimal point to be inserted as required. For more digits thaln three the table can be as easily used. Suppose w=147.2, c=26.3. Look at top for 263; then down column to number opposite 14,=3682; then down further in same column to number opposite 72,=18936. Set in column of prods. 36 82 This is much shorter work than to transfer 147.2 and 26.3 to a separate piece of paper, find three partial prods., add these, and bring sum back to column of prods. By use of table no addition need be made till the column has been filled. All prods. made by factors near alike can be sought out at the same time. The table would be of use in any class of calculations, where the multiplications can be congregated. FMiNs. FORMULAE FOR THE CALCULATIO I OF RAILROAD EXCAVATION AND EMBANKMENT, BY J. WOODBRIDGE DAVIS, C. E. Price (postage prepaid), in Stout Paper Bindincg, $I.oo.....'. Fine Cloth I. 5o. Colleges and Dealers supplied at the regular reduction. To be had of G. S. ROBERTS, E. M., School of Mines, Columbia College, N. Y. City, and of the principal publishers in this city. BLANK SHEETS, Ruled in columns, upon the plan of the TABLES OF CALCULATIONS in this book, have been prepared for the use of calculators, who may adopt this method. These sheets are I3xI7 ins., of stout, heavy paper, like cross-section sheets, and are neatly ruled in columns of proper widths for the values they are meant to contain. The headings of the columns are handsomely printed, as also the title of the sheets, viz.: TABLE OF EARTH-WORK CALCULATIONS. These sheets are intended to be kept on file or in port folio; and they entirely obviate the use of expensive cross-section sheets, together with the labor of plotting, since the notes of Field Book only are used. One pattern, like that used on pages 47, 60, 63 and 75, for work containing intermediate stations, or irregular cross-sections, or both, in fact, for any possible example, contains 40 lines, and serves, according to the number of intermediates, for Io to 30 stations, or I,ooo to 3,000 ft. The other pattern, like that on page 9, with the addition of the column of prismoidal corrections, for work without irregularity or intermediate stations, on same size paper is ruled and printed doubletwo similar sets of columns side by side. It, therefore, contains 80 lines, and would serve for a single cutting nearly I} miles long. 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