UNIVERSITY OF CALIFORNIA PUBLICATIONS 
 
 IN 
 
 AGRICULTURAL SCIENCES 
 
 Vol. 4, No. 9, pp. 233-245, 3 figures in text December 20, 1921 
 
 THE ALINEMENT CHART METHOD OF 
 PREPARING TREE VOLUME TABLES 1 
 
 BY 
 
 DONALD BRUCE 
 
 The chief use of the alinement chart 2 is to express with the simplest 
 possible system of lines a law the equation of which is known. The 
 underlying principle is so flexible that almost any formula can be 
 expressed thereby, although the most striking advantages over a system 
 of rectangular coordinates do not appear unless three or more variables 
 are involved. The axes may be parallel or converging, straight or 
 curved, and graduated either uniformly or with intervals which vary 
 in accordance with some given law, the form of the graph depending 
 on the form of the equation. It follows from this very flexibility that 
 such charts are, in general, unsuited for use with empirical data. The 
 following pages, however, describe an exception to this general rule in 
 which one type of alinement chart may be advantageously used in the 
 preparation of tree volume tables, although the form of the equation 
 of such a table is yet unknown. 
 
 The most suitable type of chart can be determined by working 
 out an approximate algebraic expression for the volume of a tree in 
 terms of its diameter and height. This expression is complicated by 
 the fact that, for almost all American tables, volumes must be com- 
 puted in board feet as scaled by some log rule, instead of in cubic feet. 
 The starting point must therefore be the equation of the volume of a 
 
 1 Acknowledgment is made to Professor Frank Irwin, of the Department of 
 Mathematics of this University, for assistance in connection with the analytic 
 features of this study. 
 
 2 A complete discussion of the theory of alinement charts may be found in 
 such works as: J. Lipka, Graphical and Mechanical Computation; J. B. Peddle, 
 The Construction of Graphic Charts; and M. d'Ocagne, Traite de Nomographic. 
 For a discussion of the application of certain simple types to some formulae of 
 forest mensuration see D. Bruce, "Alinement Charts in Forest Mensuration," 
 Journal of Forestry, XVII, 7, 773 (November, 1919). 
 
\ 
 
 234 University of California Publications in Agricultural Sciences [Vol. 4 
 
 log in board feet first formulated by Professor Daniels, 3 i.e., v = ad' 2 
 + bd + c (I). For simplicity, let us apply this formula to a tree of 
 uniform taper up to its merchantable top limit, that is, one which is a 
 frustum of a cone. 
 
 Let V = volume of a tree in feet b. m. 
 
 Let v = volume of a log in feet b. m. 
 
 Let D = d. i. b. stump (assumed equivalent to d. b. h.). 
 
 Let d = top diameter of a log. 
 
 Let H = height in logs of tree. 
 
 Let t = top d. i. b. of tree. 
 
 It is evident that the taper of the tree = D — t, and that the taper 
 
 per log = — z- — Therefore the top diameters of the several logs 
 
 . D—t 
 
 of the trees are the terms of the following series : t, t + — fj~ } 
 
 t _j_ 2 (^~0 f t + 3(D—t) } to jj terms, and the volumes of the 
 
 H H 
 
 same are the terms of the following series, each top diameter being 
 
 successively substituted in I : 
 at 2 +bt+c; 
 
 at' +j± (D - t) + jp(D - ty + bt + | (D - t) + c; 
 at' + *£(D-t)+ J (D -ty + U + 2 ^(D-t)+ c; 
 at 2 + to* {D _ t) + g {D _ t y + bt + g {D _ t) + c; 
 
 at 2 + 8J {D _ t) + go (Z) _ , + w + « (jD _ + c . 
 
 . . . . to H terms. 
 
 V = the sum of this series to H terms. This may be obtained by the 
 differential method in which a new series of first differences is derived 
 by subtracting each term from that which follows it, and this process 
 is repeated, successively obtaining a series of second differences, third 
 differences, etc., until all terms become zero and the series vanishes. 
 
 -Sec A. L. Daniels, Measurement of Sawlogs, Vermont A.gr. Exp. Sta., Bulletin 
 L02, L903. 
 

 1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 235 
 
 m, ,, , . 11 (n — 1) , . (n — 1) (n — 2) , , 
 Ine sum then equals na -\ ^r d x -\-n ~ - a\ + 
 
 LA LA 
 
 .... where n = number of terms, a = the first term of the original 
 series, d 1 = the first term of series of first differences, d 2 = the first 
 term of series of second differences, etc. 
 
 Series of first differences is : 
 
 ^ . (D - t) + jp (D - ty +A (Z) - 0; 
 
 2 f(D-t)+^(D-ty+±(D-t); 
 |-< (D - t) +g(Z> - ty +jj (D - t); 
 
 ~ (D - I) + J (Z) - ty + ~ (D - t); etc. 
 
 Series of second differences is : 
 
 2a (D - t)\ 2a (D - t)\ 2a (D - t) 2 
 H 2 ; H 2 ; H 2 
 
 Series of the third differences is : 
 o; o; o; etc. 
 And V = sum of this series = H (at 2 + bt + c) + — — t> 
 
 ^(D - t) +%(D - ty- + ± ( d - t) }+ H ( H ~ ^ H -V 
 
 \%(D-ty. 
 
 ; etc. 
 
 Expanding and rearranging in terms of D, this becomes : 
 
 . f (2afi + Zbt + &c)H . { Zaf- + 3bt) at 2 . 
 + 6 + 6~~ " + 6/7 ' 
 
\ 
 
 236 University of California Publications in Agricultural Sciences [Vol. 4 
 
 Rearranging in terms of H, this may also be written : 
 
 V=^{ 2aD 2 + (2at + 35) D + 2a/ 2 + Sbt + 6c } — Y 2 j aD 2 + 
 
 ID 
 
 — t(at+b) \ H (D 2 — 2tD + t 2 ) (III) 
 
 j 6# 
 
 Let us now apply this general formula to a specific case, for example, 
 that of trees scaled by the Scribner log rule to a six-inch top cutting 
 limit. A close approximation formula for this log rule (for 16-foot 
 lengths) is: 
 
 V = .765d 2 — .55d — 21. 
 
 We therefore may assume 
 
 a =.765 
 b=— .55 
 
 c = — 21 
 
 Substituting these values in III, we have : 
 
 V = H (.255Z> 2 + 1.2550 — 13.47) — (.3825D 2 — .275/) — 
 
 1 
 12.12) + — (.1275Z) 2 — 1.532) + 4.59) . (IV) 
 
 H 
 
 Typical equations for the height class curves of a volume table in 
 graphic form can now be found by substituting in IV given values 
 of H; for example : 
 
 For H = 2, F = .19125Z) 2 + 2.02Z) — 12.52 (V) 
 
 H=6, V = 1.16875/) 2 -}-7.55/) — 67.93 (VI) 
 
 H = 10, V= 2.18025/) 2 + 12.672/) — 122.121 (VII) 
 
 Similarly, typical diameter class curves are : 
 
 9 04 
 For D = 10, V = 24.58/7 - 23.38 + n^ (VIII) 
 
 ti 
 
 D = 20, V = 113.63// - 135.4 + ^° (IX) 
 
 73 4-4- 
 D = 30, V = 253.68// - 323.88 + -^P (X) 
 
 94 fi S4 
 /) = 50, 7 = 686.78// - 930.38 + =^£^ (XI) 
 
 ti 
 
1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 237 
 
 It will readibly be seen that V, VI, and VII are equations of parabolas, 
 while VIII and IX and X and XI are hyperbolas. These deductions 
 agree so well with the actual results obtained in volume tables con- 
 structed by the conventional method on a similar basis that it seems 
 probable that the general form of the equation should apply at least 
 approximately to actual trees as well as to the cone frusta on which 
 it is based. Furthermore, it has been tentatively established, and with- 
 out any conflicting evidence coming to light, that, in the case at least 
 where a fixed top cutting limit is used, frustum form factors are func- 
 tions of diameter and not of height. If this is true, such equations as 
 VIII can be corrected to apply accurately to any given species by 
 multiplying into them the proper form factors, which would merely 
 change the values of the constants without affecting the form. Finally, 
 the ease with which the alinement chart devised to apply to cone 
 frusta works out for actual trees is the best proof of the adequacy of 
 the equation. 
 
 Next, it is necessary to determine this alinement form. Unfortu- 
 nately a difficulty at once presents itself. The equation appears to be 
 one of those rare instances which cannot be thus expressed. 4 It has 
 been found by experiment, however, that if two parallel axes be as- 
 signed to V and H, the former graduated uniformly upward and the 
 latter uniformly downward, all lines expressing a single value of D 
 (taken from a table of values of volumes of cone frusta in board feet 
 or calculated by the above formulae) will intersect nearly (but not 
 quite) in a common point, and that these common points for a series 
 of values of D lie almost (but not quite) in a straight line, which if 
 produced will pass through the zero point on the V axis. Figure 1 
 illustrates this fact, although a larger scale is needed to bring out the 
 failure of the lines to intersect perfectly. 
 
 The reason for this becomes evident upon analysis. Let the lower 
 left-hand corner of figure 1 serve also as the origin of a system of 
 rectangular coordinates with one unit equaling ten of the small squares. 
 Also let b equal the width of the paper. Then any straight line 
 used in solving values by the alinement chart can be expressed 
 
 4 Only those equations can be expressed by an alinement chart that can be 
 put in the determinant form 
 
 A Or h 
 
 f 2 g 2 Ju = 
 U 9z \ 
 
 where fi, gi, and h- t are functions of x, (i = 1, 2, 3). 
 
238 University of California Publications in Agricultural Sciences [Vol. 4 
 
 f v) 
 
 as connecting the points (0, 22 — 2H) and-< b, —^ for by the equation 
 
 100 
 
 I V 1 X 
 
 Y= \~q - 22 + 2H y + 22 - 2#. 
 
 r 
 
 (XI) 
 
 10 
 
 .fcf 
 
 'S3 
 
 Figure 1 
 
 Alinement Form giving approximately correct results for volumes 
 of cone frusta in feet b.m. 
 
 O c3 
 > 
 
 Now from equation IV, V = AII -\- B -\ (where A, B, C are func- 
 
 II 
 lions of D) and the equations of two such lines corresponding to any 
 values oi* //, such as //, and II 2 , and having a common value of D, will 
 then be (from equation XI) : 
 
1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 239 
 
 Y = 
 
 AH 1 + B + 
 
 J7 
 
 and 
 
 100 
 
 AH 2 + B + 
 
 (I 
 
 100 
 
 - 22 + 2H 2 
 
 X 
 
 - 22 + 2#! I 6 
 
 + 22 - 2#i 
 
 X 
 
 + 22 - 2# 2 
 
 The point of intersection of these two lines can now be found by 
 solving these two equations simultaneously, and when this is done the 
 following values of X and Y are obtained : 
 
 X 
 
 2006 
 
 A - 
 
 C 
 
 Hi Hi 
 
 + 200 
 
 C 
 
 - =^=. !> + 25 + ^ + ^ 
 ill £li tli tl<i 
 
 Y = 
 
 A - 
 
 C 
 
 Hi Hi 
 
 + 200 
 
 (XIII) 
 
 (XIV) 
 
 It will be seen that, for such a range of values for D and H as are 
 actually encountered, these two equations approximate quite closely to 
 
 X = 
 
 2006 
 
 A + 200 
 
 22A + 2ff 
 
 A+ 200 
 
 (XV) 
 
 (XVI) 
 
 Since both X and Y in this last pair of equations are independent of 
 H, this approximation explains the approximate common intersection 
 of all lines having a given value of D. They can, moreover, be com- 
 bined to give an equation of the curve upon which all these common 
 intersections fall, but the result is in a form too complicated to be of 
 much value, although it can be readily identified as an equation of a 
 conic section (obviously a very straight portion of a hyperbola). The 
 curve can be plotted more simply by means of equations XV and XVI, 
 and will be found to be very nearly a straight line passing through 
 the zero point on the V axis. 
 
240 University of California Publications in Agricultural Sciences [Vol. 4 
 
 It seems as if a regraduation of the H axis might be made to result 
 in perfect instead of approximate intersections. It will be found that 
 this can be readily accomplished for any given value of D on the as- 
 sumption that the H graduating distance from a fixed point = KH 
 
 M 
 
 -\- L -| where K, L, and M are constants. But unfortunately these 
 
 H 
 three constants prove to be themselves functions of D. In other words, 
 
 different sets of graduations would have to be used for each value of 
 the diameter, which is obviously impracticable. 
 
 Each value of H, however, is in practice associated with a rather 
 narrow range of D values. It is therefore possible to modify the 
 positions of the H graduations empirically so as to result in a decided 
 improvement in the intersections, and at the same time a slight re- 
 adjustment of the D axis can be made with advantage. The best 
 results appear to be obtained by the following plan : 
 
 1. Graduate the V axis as already described. 
 
 2. Graduate the H axis as already described but omit all values 
 under that of 5 logs. 
 
 3. Select a few definite values of D well distributed over the de- 
 sired range, and for each calculate the volumes for three or more values 
 of H (H = 5 or over). If a table of cone frusta is available, these 
 volumes may, of course, be taken therefrom. 
 
 4. Draw straight lines from each value of H to the corresponding 
 value of V, and select points which appear to be the averages of the 
 intersections of lines relating to each common value of D. 
 
 5. Draw a curve through the points thus selected. For most work 
 a sufficiently close approximation will be found to be a straight line 
 passing through the V origin. 
 
 6. Enter on this curve or straight line the D graduations indicated 
 by the straight lines of step 4. 
 
 7. Obtain two or three values for V corresponding to the lower 
 values of H (under 5 logs) and to the values of D already graduated. 
 By drawing the appropriate straight lines their intersections with the 
 II axis will indicate the best average positions of the smaller H gradu- 
 ations. These should then be entered on that axis. 
 
 8. Complete the graduation of the D axis by intersection, using 
 for each value of D appropriate values of .IT. 
 
 The following table, worked out by the above process, may also be 
 used directly to save time and labor. 
 
1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 241 
 
 
 
 TABLE I 
 
 
 
 
 Graduation of H Axis 
 
 
 logs 
 with th< 
 
 Distance from fixed point on axis 
 
 A 
 
 Height in 
 tersection 
 
 r 
 
 Values to be used 
 
 where the heights 
 
 are in logs and 
 
 tenths of logs 
 
 3 
 
 Values to be used 
 
 where the heights 
 
 are in feet 
 
 diagonal 
 
 axis 
 
 1.26 
 
 20.16 
 
 2 
 
 
 •2.08 
 
 33.28 
 
 3 
 
 
 3.03 
 
 48.48 
 
 4 
 
 
 4.01 
 
 64.16 
 
 5 
 
 
 5 
 
 80 
 
 6 
 
 
 6 
 
 96 
 
 7 
 
 
 7 
 
 112 
 
 8 
 
 
 8 
 
 128 
 
 9 
 
 
 9 
 
 144 
 
 10 
 
 
 10 
 
 160 
 
 11 
 
 
 11 
 
 176 
 
 In the application of this theory to the making of a volume table 
 the successive steps may be as follows : 
 
 1. Prepare an incomplete alinement graph (fig. 2) such as has 
 been illustrated in Figure 1, but with the H axis graduated in accord- 
 ance with the table just given. 
 
 2. Draw a diagonal straight line representing the D axis between 
 the point representing its intersection with the H axis and the zero 
 point on the V axis. 
 
 3. Graduate the D axis as follows: Each tree measurement (a 
 sufficient number of which are supposed to be at hand) is used to draw 
 a straight line between the point on the H axis corresponding to the 
 height of the tree and the point on the V axis corresponding to the 
 volume of the tree. The intersection of this with the D axis is an 
 indication of the position of the D graduation corresponding to its 
 diameter. The indications of a number of trees will naturally be 
 more or less conflicting and the results must therefore be evened off 
 by a graduating curve such as indicated in figure 3. In this curve 
 the distance of each intersection above the base of the graph of figure 
 2 is plotted over its corresponding diameter. When all the points 
 are thus plotted a smooth curve is drawn through them, and from this 
 curve the D axis is finally graduated. 
 
242 
 
 University of California Publications in Agricultural Sciences [Vol. 4 
 
 4 £ 
 
 10 
 
 '53 
 
 7 -— 
 
 Figure 2 
 An alinement volume table. 
 
 O °3 
 
 Table II gives the basic tree data used in this illustrative instance. 
 The broken line in figure 2 shows how the first values of this table 
 are plotted. The left-hand point on figure 3 is plotted from the re- 
 sulting intersection with the D axis. 
 
 It is immaterial whether each tree is thus used to determine a point 
 on the graduating curve or whether the average volume, height, and 
 diameter of the height-diameter classes are used. In the latter case, 
 each point should, of course, be weighted in accordance with the num- 
 ber of trees which are thus averaged together. Figure 3 was drawn 
 in accordance with the latter method. 
 
 4. The volume table can now be read from the completed alinement 
 chart, the result being given in Table III. 
 
1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 243 
 
 TABLE II 
 
 Basic Tree Data, White Fir, Stanislaus National Forest 
 
 (Measurements taken by U. S. Forest Service) 
 
 o. of trees 
 in class 
 
 Average 
 D. B. H. inches 
 
 Average 
 
 merchantable 
 
 height in 16-ft. 
 
 logs 
 
 Average 
 Vol. ft. B. M. 
 
 7 
 
 18.1 
 
 4.0 
 
 280 
 
 3 
 
 19.5 
 
 3.9 
 
 330 
 
 8 
 
 19.9 
 
 4.0 
 
 340 
 
 5 
 
 19.8 
 
 5.0 
 
 500 
 
 6 
 
 21.9 
 
 4.4 
 
 540 
 
 7 
 
 22.2 
 
 5.1 
 
 640 
 
 6 
 
 21.8 
 
 6.0 
 
 670 
 
 1 
 
 22.8 
 
 6.5 
 
 810 
 
 2 
 
 23.9 
 
 4.1 
 
 510 
 
 1 
 
 23.4 
 
 4.2 
 
 490 
 
 9 
 
 23.7 
 
 5.2 
 
 640 
 
 6 
 
 24.1 
 
 6.2 
 
 970 
 
 2 
 
 24.6 
 
 7.3 
 
 1230 
 
 4 
 
 25.5 
 
 5.0 
 
 660 
 
 9 
 
 25.9 
 
 5.6 
 
 950 
 
 12 
 
 26.3 
 
 6.0 
 
 970 
 
 9 
 
 26.2 
 
 7.0 
 
 1270 
 
 1 
 
 25.8 
 
 7.6 
 
 1510 
 
 4 
 
 27.7 
 
 5.6 
 
 1050 
 
 10 
 
 28.3 
 
 6.3 
 
 1250 
 
 7 
 
 28.1 
 
 7.5 
 
 1450 
 
 1 
 
 27.6 
 
 8.3 
 
 1760 
 
 1 
 
 29.2 
 
 4.6 
 
 1090 
 
 7 
 
 29.9 
 
 6.0 
 
 1300 
 
 7 
 
 29.8 
 
 6.4 
 
 1280 
 
 12 
 
 29.7 
 
 7.3 
 
 1640 
 
 3 
 
 30.2 
 
 7.9 
 
 1720 
 
 16 31.6 6.6 1600 
 
 TABLE III 
 
 Volume Table Eead From Figure 2 
 
 Height in 16-ft. logs 
 
 . B. H. 
 
 r 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 18 
 
 280 
 
 380 
 
 
 
 
 20 
 
 350 
 
 480 
 
 
 
 
 22 
 
 430 
 
 590 
 
 740 
 
 900 
 
 
 24 
 
 510 
 
 690 
 
 880 
 
 1060 
 
 
 26 
 
 
 790 
 
 1000 
 
 1220 
 
 1430 
 
 28 
 
 
 850 
 
 1130 
 
 1370 
 
 1600 
 
 30 
 
 
 1010 
 
 1280 
 
 1550 
 
 1820 
 
 32 
 
 
 1140 
 
 1450 
 
 1750 
 
 2060 
 
244 
 
 University of California Publications in Agricultural Sciences [Vol. 4 
 
 In most cases it will be found that the graduating curve of figure 
 3 can be entered on the same sheet as the alinement chart without 
 confusion and with a saving in time and convenience. In step 3, 
 moreover, it is not necessary actually to draw the various straight lines, 
 which are apt to become confusing if many tree measurements are 
 available. Instead, a straight edge may be laid across the proper values 
 and its intersections' with the intermediate axis noted. 
 
 1300 
 1200 
 
 1100 
 
 1000 
 
 900 
 
 "1 800 
 I 700 
 5 600 
 
 ■■■ " :t'-il •■■:!"' I j !■--:' •;! i--|:"l I ■■■ -,■ ■ I ■; "I : :: i J V- 
 
 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 
 D.B.H. inches 
 
 Figure 3 
 
 The graduating curve. 
 
 There are several advantages in this method of preparing a volume 
 table. In the first place, the curve drawing is simplified. In place 
 of the system of curves which have to be harmonized in the usual plan 
 only a single graduating curve need be drawn, and since this is based 
 on all of the tree measurements available it is much better denned 
 and more easily and accurately located. As compared with the frus- 
 tum form factor method, which also uses a single curve, the use of the 
 alinement method saves considerable time, since the calculation of the 
 form factors is avoided; the result is, however, practically identical, 
 for if the frustum form factors of such a table as Table III be calcu- 
 lated they will be found to vary with diameter but not materially with 
 height. 
 
 Secondly, exterpolations are easily (perhaps almost too easily) 
 made, especially in height, and with far more certainty than is possible 
 by the normal system of curve extension. 
 
 Lastly, the resulting alinement chart can be read with great accu- 
 racy for fractions of inches in diameter and fractions of logs in height. 
 
1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 245 
 
 This is not true of the conventional method, where graphic interpo- 
 lations between the harmonized curves are both slow and inaccurate 
 and where arithmetical interpolations in the final table are exceed- 
 ingly laborious. For certain problems of forest mensuration this 
 advantage is highly important, although it is of little weight in 
 connection with ordinary timber cruising. The method appears su- 
 perior in accuracy to the ordinary plan, especially where the amount 
 of data available is limited. Volume tables have been made by both 
 methods from tree data for three species, including the species used in 
 illustrating this paper. The results appear as follows: 
 
 Aggregate difference between Average deviation between 
 
 all trees as actually scaled individual tree volumes as 
 
 and as read by table scaled and as read by table 
 
 . A . . A . 
 
 /■ ^ /■ ~s 
 
 Basic data Conventional Alinement Conventional Alinement 
 
 145 trees, western larch 1.5% 0.2% 5.8% 5.1% 
 
 166 trees, western white pine 2.1% 0.3% 3.8% 3.9% 
 
 166 trees, white fir 0.5% 0.2% 7.1% 6.3% 
 
 It will be observed that the result by the alinement method is much 
 superior as a whole for each species, and is better, on the average, in 
 detail as well.