O~CASJONAL PAPER 134 

NEW TREE-MEASUREMENT CONCEPTS : 

HEIGHT ACCUMULATION, GIANT TREE, TAPER AND SHAPE 

SOUTHERN FOREST EXPERIMENT STATION 

Philip A- Briegleb, Director 

Forest Service, U. S. Department of Agriculture 
HEIGWT AC-TION, GIAHT TREE, TAPER AMD SHAPE 

L. R. Grosenbaugh 

Southern Forest Experiment Station 

An entirely new concept of tree measurement was announced by the 

author in 1948 (11) . Since the original theory and applications have 

subsequently beerbroadened considerably, it seems advisable to publish the entire development in readily usable form, along with other 

material helpful in tree measurement. 

In its essence, the new concept consists of selecting tree 

diameters above d .b .h . in diminishing arithmetic progression (equivalent 

to saying that outside-bark diameter is treated as the independent 

variable); then tree height to each such diameter is estimated, recorded, 

and accumulated. This reverses the classical tree measurement concept 

which regarded height as the evenly-spaced independent variable, and 

which estimated and recorded diameter at regular intervals of height. 

The greatest advantage of the new concept lies in the ease with 

which individual trees can be broken down into various classes of 

material and recombined with similar portions of other trees, using 

punched cards, business machines, or both. Sections varying in length 

and grade (from different trees) can be sorted by grade-diameter groups 

within which lengths will be additive. The ability of modern business 

machines to accumulate has been largely wasted in conventional procedures 

where height is treated as the independent variable. Their capability 

can be much more fully exploited in the new procedure using diameter as 

the independent variable. 

Another advantage lies in the convenience of being able to use 

height, the sum of heights, and the sum of height-accumulations to 

calculate volume and surface of individual trees (or of classes containing portions of several trees). Final tabulations can be readily converted into surface, cubic volume, or volume by a nwnber of log rules; 

no volume tables are required. Little extra work is involved in 

tabulating volumes to several different merchantable tops, or in segregating volume by grades. 

There are already in existence dendrometers which are well 

adapted to locating diameters in diminishing progression along the bole 

of a standing tree --the Biltmore pachymeter ( 4) , the Clark dendrometer 

( 6), the Bitterlich relascope (1) and spiegel~relascope (2), and the 

~ruce slope-rotated ~ed~e-~rism-(3), tomention a few . However, anyone 

currently content to use arbitrary volume tables not based on valid 
sampling of the tree population, or to estimate butt-log formclass and accept arbitrary upper-log taper assumptions can use the new 

technique with the crudest instruments and still not exceed the error 

currently ignored or tolerated in cornonly used volume-table techniques. 

This paper not only describes the theory and application of the 

new concept but also makes available for the first time a giant-tree 

table, and discusses tree taper and shape. 
HEIGHT ACCUMSJLATION--A NEW CONCEPT 

If a peeled 16-foot log with a small-end d .i .b. (diameter inside 

bark) of 8 inches, and a large-end d . i .b . of 10 inches is stood upright 

on its large end, and if heights (H) are measured as the number of 4-foot 

sections occurring beneath diameters in the complete progressive series 

2 inches, 4 inches, 6 inches, 8 inches, and 10 inches 

the record will appear as follows: 

d.i.b. 

( inches) 

Number 

1% -4.11-1 be noticed that all diameters in the series which are 

smaller than the small end of the log have the same height as the small 

end. 

Now if these heights are differenced (upper minus lower) and if 

each difference in height is expressed as (L), the calculation will 

appear as follows: 

d.i .b. 

(inches) L H 

- - - Number - - - 

Finally, if heights are accumulated so that each entry indicates 

the progressive total (H' ) of all heights recorded beneath it, and L, H, 

and H' are then summed, the calculation will appear as follows: 

d.i .b. 

(inches) 
Now these three sums can be converted to units of volume (or 

surface) by referring to Appendix A. Where tsper-step is 2 inches, 

where height is expressed in 4-foot units, and where d*i *b ' = L d.o*b. 

(i.e., where it is not necessary to correct for bark included in 

diameter measurements), the board-f 00% volume by the Doyle log rule 

can be calculated as: 2EH1 - ~EH + EL = 80 - 80 + 16 = 16 bd. ft. 

!This is identical with the volume obtained uy the conventional Doyle 

Me for 16-foot logs : (d. i .b . - 4)* = 16. The theory behind all this 

is discussed in Appendix E. 

Of course, the height-accumulation technique offers no advantage 

over the conventional method in the simple case above, but the new 

technique has real utility in more complicated timber-cruising or treemeasurement situations. 

This technique can be .applied when d . o .be (diameter outside bark) 

is measured but volume inside bark is desired. For purposes of illustration, it will be postulated that bark constitutes 10 percent - of 

4.o.b. (i-e-, d*i*b ratio is .go), that all heights are that de0.b. 

measured above a 1/2-foot stump whose d.o .be is assumed tD be 2 inches 

greater than d .b .h., that it has been decided to adopf-a* taper-step of 

2 inches, and that lengths, heights, and height-accumlations will be 

expressed as number of 4-f oot sections. 

The heights to regularly dimini~~ing diamters of a 20-inch tree 

with five 16-foot logs (or twenty 4-foot sections) to the limit of merchantability might be estimated as in the boxed portion of the fallawing 

tabulation: 

(inches) L H Hf . 

Number of 4-foot sections 

Note that there - must be an B and an Hf for every 2-inch diameter 

step in a complete progression between (but excluding) stump and zero, 

even though no merchantable material is found outside the box beyond 

the merchantable top d.0.b. of 10-inches. Merchantable height must be 

repeated for each progressive diameter less than this merchantable top, 

2 0 20 

4 0 

143 

20 

6 

123 

0 20 

8 

103 

0 20 

merch. top 10 4 I I 83 

12 2 

5 

Id 6 

18 2 

d.b.h. 20 1 

20 = EL: 143 = 2 H: 603 = ZH': 

20 =TL 

16 

14 

9 

3 

1 

63 

43 

27 

13 

4 

1 
however. Actually, if the complete H column (commencing with the H cor- 

~zsponding to d .b .h. and including repeated heights outside the box) is 

entered in a 2-register bookkeeping machine which can transfer subtotals 

from first to second register without clearing, then ZH and zHf can be 

automatically and simultaneously computed, and EL will be the last 

entry in the R column. Some people may prefer to invert the column of 

diameters so that the corresponding H and HT columns progress downward 

as on the bookkeeping machine tape, but this is merely a matter of personal preference, as long as the arithmetic accumulation proceeds in a 

direction which is UJ $he -isree. 

Thus, H' = 603, H = 143, and L = 20 for a 20-inch f ive-log 

tree described in terms of H as 20: 1: 3: 9: 14: 16: 20, with 4 more 20's 

needed to complete the diminishing diameter progression. Appendix A 

gives coefficients A, B, C appropriate for these accumulations where 

unit-hei&t is 4 feet, where taper-step is 2 inches, and where mean 

d*i .b. 

d.0.b. ratio is .go. The appropriate A, B, C coefficients convert the 

accumulations into surface or cubic volume (inside or outside bark) and 

into board-foot volume according to International l/binch log rule, 

Scribner formula log rule, or Doyle log ae. This is demonstrated belm 

for surface, cubic volume, and International 1/4-inch volume : 

Surface inside bark = 1.88(143) + .942(20) = 288 sq. ft . i .be 

Cubic vol. inside bark = .141(603) + .0236(20) = 85.5 c'u. ft. i .b. 

Int. 1/4-inch vol. = i .29(603=) - 1.34(143) - .284(20) = 580 bd. ft. 

Although the sinqle tally and bookkeeping-machine technique just 

outlined is well adapted to many sitllations, a more useful method involves punch-card breakdm of the tree into sections classified by grade 

( or utility) and d .o .by Lengths of such sections can then be machine - 

sorted and accumulated by grade-diameter classes, and need not be converted to volume till the last step. If mrk-sensing is employed, more 

than one mark-sensed card may be needed per tree, dependinb on haw much 

information is desired. It is assumed that plot information will be 

recorded on a separate mark-sensed card. In addition to plot number, 

species, defect class, and priority class, only d.b.h. and the graded 

length of each taper-step need be recorded in the field on the marksensed tree-card. For the tree used in the preceding example, the latter 

information would be tallked as deb .he and successive L' s with their 

grades A, B, C, D, etc,, thus: 

In actual practice, of course, grade.s would be coded numerically. 

This would require only 14 of the 27 columns on a mark-sensed 

card. Oceasiomlly it might be more convenient for the estimator to 

record H instead of L, but ordinarily this would waste mark-sensed 

columns and would require differencing later, if a breakdawn by grades 
is desired. The 27-column mark-sensed cards will later be automatically 

punched, and in some cases my suffice for volume calculations, in conjunction with a 3-register bookkeeping machine . Ge nerally, however, they 

will be used to reproduce a card for each tree section, so that these can 

easily be sorted and accumulated, after which L, Z H, and zH1 can be 

directly obtained by the progressive totalling process on some accounting 

machine such as the IBM 402 or 403. 

To illustrate quite simply how the mechanics of the sorting and 

recombining procedure might work, suppose that (in addition to the 20- 

inch tree described above) there were also an 18-inch tree, with d.b.h. 

and graded lengths (L) recorded as. 18: 1A: 3: 4~: 4C: 3D: Separate 

cards would be punched for each length in each tree, with a zero-length 

master card in each summarized d.0.b.-grade group to facilitate progressive totalling. A series of machine sorts and tabulations (by grade in 

each d .o .b . class) would break dam the trees into their elements and 

would reconibine and summarize them thus: 

d.0.b. of sections 

: Sums of 

:lst and 2nd 

:progressive 

:sums: sums 

-- 

The individual tree swns and progressive sums are not needed except 

to illustrate that they agree with the later partition by grades. 

The EH: XH, and EL for each grade can be converted to volume by 

using the A, B, C coefficients in Appendix A, if an appropriate d0i.b. 

d.0.b. 

ratio is chosen. Although for most purposes this ratio is adequately 

estimated as the srzmple-based ratio \IYolumem with linear interpola- volume o . b , 

tion between tabled values of A, B, C, occasionally it may be found desirable to sample both d.i .b . and d.0.b . at regular intermls of length and 

to derive two ratios: Z(d.i.b.)2 E(d.i.b.) 

and Z(d.0.b.)' These should be 

ZCd.o.b.)2 
applied to outside-bark volume and surface f onrmlae respectively, and 

the resultant formulae should be multiplied by the log rule formulae 

given in Appendix E to derive A, B, C coefficients more appropriate 

than those obtained by linear interpolation from Appendix A. Where 

volume or surface including bark is desired, the coefficients given for 

d*i*b* ratio = 1.00 are appropriate without any adjustment. d.0.b. 

If it is desired to break dm volume or surface summaries by 

d .o.b. class, first number each d.0.b. class in the complete series consecutively from the slnallest to the largest (beginning with number l), 

then accurmilate the ordinals progressively from the beginning, then 

multiply each length (L) by its corresponding ordinal (I), or its 

ordinal accumulation ( 11) , thus : 

(1) ( 11) 

d.0.b. Numbered Acv--nulated 

( inches ) consecutively progressively L H H' 

Each L, when multiplied by its corresponding entry in column I, will give 

the appropriate H for that d.o.b., and when multip+ied by its corresponding entry in column I1 will give the appropriate H . Of course, cumulative volumes of material larger than any specified d.0.b. could be 

obtained easily by merely assigning zero? in the L column for all smaller 

d.o.bWts, and then getting ZL, ZH, tH in the usual fashion. Volumes 

or surfaces in any desired diameter class or group of diameter classes 

are then obtainea by multiplying HI, H, L or their desired sums by the 

A, B, C coefficients from Appendix A in the usual fashion. 

One shortcut might be mentioned which frequently rmy furnish an 

iz.coe7 :able approximation. Where the average d*ieb* o, ratio is exactly ,932 

inside bark ratio is .869), if taper-step of 2 inches ( Or where volume out side bark 

and unit-height of 4 feet have been adopted, the board-foot volume by 

the International 1/4-inch log rule can be easily calculated, thus : 

Board feet (1nt. 1/411) =(1.38)(zHf - ZA - - 
For the tree described on page 4, this would be: 

(lS38)(6a3 - 143 - - = (1.38)(456) = 629 bd. ft. 

There are three general situations in which slightly different 

variations of the height-accumulation technique my be found useful: 

(1) Where it is desired to grade different portions of the 

sample tree, the entire job should be done on punched cards, preferably 

with the field tally made directly on mark-sensed cards, Trees should 

be graded and described in terms of' d.b,h. and a series of graded L1s 

( rather than H s ) , thus : 

Separate cards should be reproduced for each d.0.b. starting with d.b.h, 

= 18 inches, and after desired sorts are mde, an IBM 402 or 403 accounting nachine will accumulate L, )I H, t Ht for each class yhose volume 

is desired, provided that L = 0 is supplied for any d.0.b. missing from 

the complete progression. 

(2) Where it is not desired to grade portions of a sample tree, 

an entire tree can be punched on an individual card, with each H (or L) 

recorded under its appropriate d.0.b. field starting with d.b .h. (e .g., 

starting with 18 inches for tree @ in the tabulation, on page 6), thus : 

1: 4: 8: 12: 15: (with 15: 15: 15: 15 completing the H series) 

or 1: 3: 4: 4: 3: (with 0: 0: 0: 0 completing the L series) 

Each class for which a separate volume is desired will have H (or L) 

summarized for each d.0.b. field occurring in it. The H (or L) sequence 

(including repeat heights or zeros needed to complete the progression) 

for each class can then be entered in a 2- (or 3-) register bookkeeping 

machine able to transfer subtotals from register to register without 

clearing. mles of suitable 2-register bookkeeping =chines in which 

the H's can be entered are the Burroughs Sensimatic F-50, or the Ntktional 

Cash Register 30210 (1-6). Examples of suitable multiple register machines in which the L's can be entered are Burroughs Sensimatic F-200, or 

National Cash Register 30412 (17) or 3100, any of which have more than 

the 3 registers needed. 

(3) Where it is not desired to use punched cards at all, tally 

for each sample tree will consist of its d.b .he followed by a series of 

heights, thus: 

With a tapes-step (T) = 2 inches and d .b .h. of 18 inches, there should be 

dab*h* = 9 entries, so 1, 4, 8, 12, and 15, 15, 15, 15, 15 should be T 
entered in a 2-register bookkeeping machine to obtain f. H and ZH' for 

the individual tree; L would, of course, be equal to the last entry 

(15) 

In all three situations, it will usually be found desirable to 

employ the volume sampling techniques described above merely to get 

volume/basal area ratios which are applicable to a larger sample of 

basal area which has been stratified by species and height. A brief 

discussion of this system may be found on page 7 of Shortcuts for 

Cruisers and Scalers (12). 

All the examples above have employed a taper-step of 2 inches 

and unit-height of 4 feet because such interval and unit are most convenient for United States measye, with stq height occurring 1 unit 

below breast height. However, Appendix A also gives A, By C coefficients for taper-step of 1 inch and unit-height of 1 foot. Such interval and unit are recommended only with very accurate dendrometers or 

tree diagrams where unusually precise measures are desired; 4 rather 

than 1 unit-heights will occur between d .b .h. and stump-height . Also, 

Appendix A gives A, B, C coefficients for the most convenient interval 

and unit in the metric system: taper-step of 5 centimeters and unitheight of 1 meter.. Assuming 1 unit-height below Ewropean breast-height 

of 1.3 meters would imply a stump height of about 30 centimeters. Of 

course, the retention of decimal fractions of a meter in height measurements would not change the coefficients. No board-foot volume coef'ficients were provided for metric intervals, since such units are not 

popular in countries using the metric system. 

A, By C coefficients for intervals and Wts other than those 

tabulated may be derived from the basic fommilae given in Appendix E. 
GIANT-- TAB= FOR S'URFACE AND VOLUME 

Existing tables of volume (cubic -f oot or board-f oot ) are unsatisfactory in that none of them gives volumes for every combination of 

length and diameter (measured in fractional inches) apt to be encountered 

in practice. To overcone this drawback without inflating table size 

excessively, an accumulative table is needed. The author calculated 

values in Appendix B ab initio, retaining 6 to 9 significant digits until 

the final rounding to5 digits. All columns have been checked by summation f ormlae involving the sums of numbers in arithmetic progression, 

the sums of their squares, and the sums of their cubes. Values for the 

Interrmt$.onal log rule with l/4-inch kerf are based on the formula first 

published by II. He Chapmn (I), i.e., board feet = .904762 ( .22# - .7l D) 

for a &-foot section, with taper allmnce of 1/2-inch for each 4 feet. 

To minimize accumulative and rounding error, however, the author expanded 

this to a form in which taper was an implicit joint function of length 

and diameter (12 - ) . 

In effect, Appendix B is a huge upright peeled log or tree, standon a 50-inch base and tapering upward for 400 feet at the rate of 

inch per foot. To use it, enter at the desired snall-end d .i .b . , 

mark the surface or volume found opposite this diameter, move down the 

column a distance equal to the desired log length, mrk the surface or 

volume found there, and subtract the marked wlues. 

As an example, the board-foot volume of a log 38-114 inches in 

diameter and 81 feet long would be: 

8396 

- 1447 

-board feet (Int. 1/4-inch) 

Cubic volume of this log would be 

- l;;;:: 

32.5 cubic feet 

Surface of this l.og would be : 

1085 .g 

- 167.4 

918-5 square feet 

Of course it is desirable to measure more than just one diameter 

in long logs or trees, but occasionally that is all that it is feasible 

to obtain, and a taper-assumption of 1/8-inch per foot my not be far 

amiss. 

A particularly handy use of the giant-tree table is when tree 

d.i.b. at the top of the first 16-foot log (or other reference point) is 
measured or estimated. The extra column of tabular length 0 - 80 feet 

at the end of Appendix B can be cut out and used as a sliding scale with 

the table. If it is laid over the regular length crlumn (positioned 

between the d. i .b . column and athe surface column) with 16 feet on the 

sliding scale opposite the measured d .i .b . , the volume between any specif ied heights above stump on the tree may be computed by subtracting the 

volumes which appear opposite these heights. As an example, if d.i.b. 

at the top of the first 16-foot log is 17-1/8 inches, the diameter, 

height, and cubic volume readings will look as follows after the sliding 

length (or height) scale is positioned: 

Sliding height 

d.i.b. scale Cubic volume - 

Inche s Feet - Cubic feet 

etc . etc. 

Set 16 feet at 17-118 inches 4 C- 16 

15 

The volume in the 0 - 8 foot portion of the tree would be : 

- 1 716.3 

15.1 cubic feet 

while the volume in the 9 - 18 foot portion of the tree would be: 

- 1 731.4 

16.8 cubic feet 

Higher portions not sham could be handled similarly. Grades or useclasses could be assigned in the field to sections between various tree 

heights, and the actual differencing could be done later on bookkeeping 

machines or punched-card mchines. 
When used as above, the giant-tree table has most of' the advantages of form-claee volume tablea with arbitrary upper-log tapers, yet 

only one table le qeeded, 8nd that table is in a form facilitating 

breakdawn of the tree into portions of varying lengths for each of which 

lengths both cubic and baard foot volumes can be easily computed. If 

cubic velum, eurface, and length of portions of a tree sre knm, they 

can be copmrted to Internstiom1 1/4-inch, Scribner, or Doyle volume0 

by coefficients in Appendix E. 

Ilt; is obviowr that re~ulte will, not be as accurate as those 

determined earlier by the new height-accmtion mcthod, because the 

giant-tree table aeeumee taper of 1/8 inch per foot between dimmter 

meseuremente, whereae the height-accumulation method utillzer actual 

taper. AppendiX B ie very hndy, however, in tihat it ie equally simple 

to get surface or volume for a log or tree of length, by entering 

with a single diameter (to the nesrleet 1.18 at any epecified 

point on the log or tree (em11 end, large end, middle, or any place 

else). A 118-inch (rather than ~l/l~-inch) dianrfer interval is 

Implicit in International l/k-inch. log rule aseumptions where lengths 

are desired In whole feet. 
When viewed from the butt, the taper of a tree can be defined 

as its loss in diameter divided by the 1engt.h affec22d. !Phe shape of 

different portions of a tree can be visualized as a function of taper, 

Where taper tends to increase, tree shape resembles a paraboloid; where 

it tends to remain constant, tree shape resembles a conoid; where it 

tends to decrease, tree shape resembles a neiloid. The corresponding 

profiles of the surfaces of these tree shapes viewed from outside the 

tree would be convex, linear, and concave. 

In general, trees tend to be neiloidal till butt swell disappears, then conoidal, and finally paraboloidal in the upper portions. 

This fact is helpful where deboho, merchantable height, and merchantable top (d.0.b.) have been measured, and where it is desired to distribute taper between d.beh. and top d.0.b. by eye for use in the heightaccumiLation technique. Suppose a tree 18 inches in d.b .h. has 16 fourfoot sections from a 112-foot stump to an 8-inch merchantable top d.0.b. 

It can usually be arbitrarily assumed that there is 4 two-inch diameter 

increase and one four-foot section below d.b .h. This would leeve 15 

sections above d.b .h. to be distributed among l8 - = 5 two-inch 

taper-steps . An absolutely conoidal taper (except for the neiloidal 

butt swell) would be recorded as : 

5 taper-steps above d.b.h. 

containing 15 four-foot sections 

f----- -p--- 

1:3:3:3:3:3 

However, lacking any intermediate measures, the eye may still 

detect a convex (or paraboloidal) tendency near the top, and a better 

estimate of composite tree shape would be: 

5 taper-steps above d.b,h. 

containing 15 four-foot sections 

When only d . b .he and merchantable height are estimated, and no 

information is available on top d,o.b. or shape above deboh., a crude 

conoidal assumption has of ten been used in volume -table work (especially 

with hardwoods ) with reasonably satisfactory results. The International 

log rule and the giant-tree table discussed earlier are examples of such 

an assumption (2 inches of taper per 16 feet). If a taper assumption 

of slightly less than 2 inches per 16 feet were deemed adequate for the 

hypothetical tree above, the estimate of tree shape for use in the 

height-accumulation technique would be : 

use four units of length per taper-step, 

until total 15 units is exhausted. 
The recent popularity of butt-log form-class (i.e., the ratio 

d .i .be at top of first 16 foot log as a VBriable in volume is dab .h' 

based on the obvious fact that most volume and value in trees of moderate size is contained in the butt log. However, this butt-log formclass is usually either estimted or assumed, as is shape of the first 

log and taper above the first log. If d.0.b. of the butt log is 

measured in several places, and if d*i*b* ratios for a species group do d.0.b. 

not vary excessively, the height-accumulation technique will permit 

more accurate description of both the lower and upper portions of trees 

than will the use of f orm-class volume tables which assume arbitrary 

butt-log shape and upper-log tapers, In addition, the height-accumulation 

technique is much better adapted to grading portions of the tree and to 

business-machine compilation, 

If' the effort involved in getting objective height-accurm;ilation 

measurements is deemed excessive, the height-acc~ation technique 

can be adapted to the same sort of eye-estimates and assumptions as 

are involved in the usual use of butt-log form-class volume tables. 

Below is an illustrtxtion shawing haw height-accumulation information 

about tree butts implicitly includes eye -e stimated butt -log f om-class 

information, plus additional shape and taper information. For convenience, a taper-step of 2 inches, a unit-hekght of 4 feet, and a 

d0i.b. d.i .b-. 

d.0.b. ratio of -90 have been employed; a different .o relationship wwuld, of course, lead to different form-class values. 
Lengths 

(number of 4-ft. 

sections per 2 

- inches of taper) .---..... .- ----I--- 

Taper from d.b.h. 

to d.0.b. at top 

of first 1.6-ft. log 

----*..-.---.--..." .--- . - 

- - Inches - - 

) Znpliedbutt-log form-class t 

I 

Any forester who prides himself on being able to estimate buttlog form-class can readily adapt his talent to the height-accumulation 

techniques; he can also check his eye-estimates readily with one of the 

dendrometers discussed earlier. Pole calipers or tapes of various kinds (g)(e) are already in existence, and are well adapted to checking tapers 

on the most inportant portion of a tree--its butt log. Probably the 

most convenient hand-held 'dendrome ter for use on upger as well as lower 

portions of the tree would be a slope-rotated wedge-prism (3) on which 

height or slope can be read. A horizontal target of adjusgble width 

attached to d.b .he will allow the observer to position himself or adjust 

his instrument so that the split-image of the target just fails to overlap. Re can then slawly raise his line of sight up the tree trunk, and 

the height where split-image of the trunk just fails to overlap will be 

where d .o.b. equals the desired width for which the target has been 

adjusted . 

It will be found convenient to have the hypsometer scale graduated 

to read height in number of four-f oot sections with a 40-foot base-line 

assumed, and to have the rotating wedge-grism manufactured or adjusted 

to effect a maximum deviation of 143.25 minutes (equivalent to about 

4.167 prism-diopters or enough to exactly juxtapose the direct and the 

split image of a 20-inch horizontal target at a distance of 40 feet). 
In using such an instrument with a tree 20 inches in d.b.h. on 

flat terrain, the observer would successively occupy points 40, 36, 32, 

28, 24, 20 feet, etc . , distance from tree center and would ascertain 

at each point the height at which the split image of the upper bole 

pulled apart, or separated. In each case, this figure will be height 

read frcrm the hypsomter scale multiplied by an appropriate factor 

such as 1.0, .9, .8, .7, .6, -5, etc. (computed as 

actual distance to d.b.h. 

base for which hypsometer was graduated 

!The reader can easily infer for himself modifications of this 

technique necessary where trees are more or less than 20 inches in 

d.b .ha, where the *rrain slopes, or where trees lean. 

Technicians studying the effect of erroneous assumptions as to 

taper or shape may be interested in Appendix Dy which helps in visualizlng how scaled volume estimates employing erroneous log-rule asrmmptionls may be improved by shortening the interval between measured 

diameters. 
Literature Cited 

(1) Bitterlich, W. 

1949. Das relaskop. Allgemeine Forst- und Hc:l.zwirtschaf tliche 

Zeitung 60 (516): 41-42. 

(2) 1952. Das spiegel-relaskop. Oesterreichs Forst- und 

Holzwirtschaft 7(1) : 3-7. 

( 3) Bruce, David 

1.9550 q new way to look at trees. Jour. Forestry 53: in press). 

(4) Burton, R. G. 

1906. The Biltmore pachymeter . Forestry Quarterly 4 : 8-9. 

(5) Clark, J. F. 

1906. The measurement of sawlogs . Forestry Quarterly 4 : 79- 

91. 

(6) 1913. A new dendrometer or timber scale. Forestry Quarterly 

1: 467-469. 

(7) Chapman, H. C. 

1921. Forest mensuratioii (page 493 ) . John Wiley and Sons, 

Inc., New York City. 553 pages. 

(8) Ferree, M. J. 

1946. The pole caliper. Jour . Forestry 44 : 594-595. 

(9) Godman, R. M. 

1949. The pole diameter tape. Jour . Forestry 47: 568-569. 

(10) Grosenbaugh, L. R. 

1948. RS-SS, MENSURATION, Tree studies, volume calculations. 

Mimeographed memorandum report dated August 25, 1948, 

with Appendices A through F. U. S. Forest Service, 

South. Forest Expt. ' Sta. 24 pages. 

(11) 

1948. Forest parameters and their statistical estimation. 

Proceedings of the Auburn Conference on Statistics 

Applied to Research in the Social Sciences, Plant 

Sciences, and Animal Sciences. Sept. 7-9, 1948. 

Statistical Laboratory, Alabaxm Polytechnic Institute, 

Auburn, Alabama. Abstract of paper on page k6. 

(12) 

1952. Shortcuts for cruisers aqd scalers. U. S. Forest Service, 

South. Forest Expt . Sta . Occas . Paper 126. 24 pages. 

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Giant-tree table: Disnater insidt bark (tawr: 1/8 inch per foot) via csmuhtive length, eurt.ce, and volunr 
ApPEarna a (coati-&) 

Giant-tree table: ~iamrtkr inside bark ftapcr: 118 inch per fmt) with cumulative length, surface, and volwne (eont'dl 

Giant-tree table: Dteamter %nsi& bark (wr: l/B inch mr foot1 

via cumulative length, smi"ace and volun, Icoat'dJ 

Relative NLd-diameters and Volumes of Various Solids of Revolution 

For Given Rstios of alleet Diaraeter (d) to Largest Diaraeter (D) 

*A subneiloid is a fictitious solid of revolution whose volume equals thst of a 

cylinder with diameter equal to d + D, and whose mid-diameter can be postulated 9 

on the assumption that Newton's prismatoidal fornnala holds true. Its volume is 

an easily calculated close approximation to thst of a neiloid. 

Ratio 

d 

8 

(percent) 

Ratio Frnstwn Volumt? 

Cylinder Volume 

Parsboloid Conoid 

Ratio of Middle Maaeter 

D 

o 

2.5 

5 

7.5 

10 

12.5 

15 

17-5 

20 

22.5 

25 

27.5 

30 

32.5 

3 5 

37.5 

40 

42.5 

45 

47.5 

50 

52.5 

55 

57.5 

60 

62 5 

65 

67.5 

70 

72.5 

7 5 

77.5 

80 

02.5 

85 

87.5 

90 

92.5 

95 

97.5 

100 

Heiloid 

Para - 

boloid 

% k 

7t.71 $00 35.36 35.36 

70.73 51.25 39.98 37-92 

70.80 52.50 42.79 40.35 

70.91 53.75 45.19 42.66 

71.06 55.00 47.38 44.86 

71.26 56.25 49.40 46.98 

71-50 57.50 51.33 49.02 

71.79 58.75 53.18 51.00 

72.11 60.00 9.96 52.92 

72.48 61.25 56.69 54.78 

72.80 e.50 58.37 56.60 

73.34 63.75 60.01 58.37 

73.82 65.00 61.61 60.10 

74.35 66.25 63.19 61.81 

14-92 67.50 64.73 63.47 

75.52 68.75 66.25 65.10 

76.16 70.00 67.75 66.71 

76.83 71.25 69.24 68.29 

77.54 72.50 70.70 69.84 

78.28 73.75 72.15 71.37 

79.06 75-00 73.58 72.89 

79.86 76.25 74.99 74-38 

80.70 n.50 76.39 75.85 

81.57 78.75 77.7'7 77.30 

02-46 80.00 79.15 78.74 

83.39 81.25 80.52 80.16 

84.34 &.yo 81.88 81.57 

85.31 83.75 83.22 82.96 

86.3 85.00 84.55 64.33 

87.34 86.25 85.88 85.70 

88.39 87.50 87.20 87.05 

89.46 88.75 88.50 88.39 

90.55 9.00 89.81 89.72 

91.67 91.25 91.11 91-04 

92.80 92.50 92.40 92.35 

93-96 93-75 93-68 93-65 

95.13 95.00 94-94 94-93 

96.32 96.25 96.23 96.21 

97.53 97.50 97.50 97-48 

98-76 98.75 98.75 98-75 

100.00 100.00 100.00 100.00 

% 46 k % 

50.00 33.33 25.00 25-00 

50.03 34.19 27.34 26.27 

50.12 35.08 28.92 27.56 

50.28 36.02 30.37 28.89 

50.50 37.00 31.79 30.25 

50.78 9-02 33.20 31.a 

51.12 39.08 34.62 33.06 

51.53 40.19 36.dc 34.52 

52-00 41.33 37.47 36.00 

52.53 42.52 38.93 37.52 

53.12 43.75 40.k 39.06 

53.78 45.02 41.93 40.64 

54.50 46.33 43.47 42.25 

55.28 47.69 45.04 43.89 

56.~2 49.08 46.64 45.56 

57.03 50.9 W3.27 47.27 

58.00 !j2.00 49.91 49.00 

59.03 53-52 51.64 50.77 

60.12 55.08 53.36 52.56 

61-28 55-13 54.39 

62.50 56.92 56.25 

63.78 60.02 58.74 58.14 

65.12 61.75 60'61 60.06 

66.53 63.52 62.50 62.02 

68.00 65.33 64.43 64.00 

69.53 67.19 66.40 66.02 

71.12 69.08 68-40 68.06 

72.78 71.02 70.43 70.14 

74.50 73.00 72.50 72.25 

76.28 75.02 74.60 74.39 

78.32 77.08 76.73 76.56 

80.03 79.19 78.90 78.77 

82.00 81.33 81.11 81.00 

84.03 83.52 83.35 83.27 

86.12 85.75 85.62 85.56 

68.28 88-02 87.93 87-69 

90.50 90.33 90.27 90.25 

92.78 92.69 92.65 92.64. 

95.E 95-08 95.07 95-06 

97.53 97.52 97.9 97.52 

100.00 100.00 100.00 100.00 

Sub - 

Conoid neiloid*