THE HATCHET PLANIMETER.

BY

ROBERT SPARKS, E.E.,

Instructor in Electrical Engineering, Dunham Labora tory of Electrical Engineering, Yale University.

It frequently happens tha t some one is doing a little work at home or in the field where draught ing tools are not at hand but where a knife is almost always available. I t is not in design work alone tha t planimeters are used. Whenever an indicator card is taken to determine the power ou tpu t of an engine its average height can be found only after the area of the figure is known. Again the core loss of a magnet ic circuit is obtained from the area of the hysteresis loop. There are a number of well known ways in which the work can be done, such for instance as replott ing the curve on graph paper and counting the squares inclosed. This, as are the other ways of get t ing the result, is very laborious and unnecessary trouble.

Wi th the ins t rument at hand a planimeter would usually be preferable, or possibly an integraph. Lacking these one can use his penknife and get the result as accurately by exer- cising care as with the more elaborate instruments. First it is necessary tha t the knife have one blade opened fully at one end and at the other something which will serve as a pointer to be opened half way. The latter may be a blade, a nail file, or a pipe cleaner as on many knives. This half opened blade we will call the pointer. I t is supposed tha t the figure whose area is desired is drawn on a sheet of paper. Look the drawing over and est imate the center of the figure. It is not necessary tha t this be anyth ing more than a rough estimate. From this center draw a line to the edge of the figure. (A light pencil line tha t can be erased is sufficient.) Tak ing the knife in the right hand hold it with the end of the pointer directly over the center of the figure and with the sharp edge of the other blade resting lightly on any convenient paper, ei ther the same piece tha t the drawing is on or another piece pu t in a suitable place for the purpose. With the left

6 6 I

662 ROBERT SPARKS. [J. F. I.

hand press the blade into the paper enough to leave a legible mark. I t is essential tha t the blade be held lightly as shown in the photograph. With the t h u m b and index finger of the right hand hold the lower end of the pointer very tenderly so as not to tend in any way to skid the sharp edge of the blade across the paper. The upper end of the pointer and possibly par t of the handle may rest gently against the upper par t of

the index finger. This mat te r of holding the knife correctly is the only difficult part of the operation and may be mastered in a few careful trials so tha t it becomes natural and easy. The pointer is then moved along the line to the edge of the figure, thence along the boundary of the area to be measured and back to the center, the start ing place, along the guide llne. The blade of the knife is again firmly pressed into the paper

June, I932.] THE HATCHET I:'LAN1METER. 663

to make a second mark. The motion of the knife, pointer and blade, is somewhat like tha t of a cat being chased by a dog which always heads straight for the cat but never succeeds in get t ing any closer.

I t is then necessary to measure the distance between the two imprints of the knife and also the distance from the center of the figure, tha t is from the start ing place of the pointer, to either of the two imprints. I t now remains only to multi- ply these two distances together to get the required area. If the distances are measured in inches then the area will be expressed in square inches, if the cent imeter scale was used then the area will be given in square centimeters, and so for any system of units.

There are certain precautions to be taken and limits to be considered when using the knife in this way tha t it is well to point out. I t is best tha t the greatest distance from the center of the figure to the most dis tant part of the perimeter be less than one eighth of the length of the knife and one blade. If it is otherwise, then the area may be approximated twice by this method, first encircling the area in one direction with the pointer and the second t ime encircling it in the other direc- tion. These two results should be averaged for the most prob- able value of the area. In any case it is best not to have the greatest length of the figure more than the length of the knife and one blade. Following these directions anyone can do with a knife and scale all the work tha t can be done with the most expensive planimeter and in very little more time. In the case of a very large figure the area may be divided into smaller sections or a larger knife be used. In the appendix which follows the derivation of the equations from elementry principles is given so tha t the reason for these instructions is easily seen.

APPENDIX.

This use of a penknife is known as the hatchet planimeter, and though it was devised long ago it has not become generally known because of the complicated theory and the many com- mercial types of planimeter which are simpler. For the bene- fit of those who are interested in knowing how and why it works the following mathemat ica l proof is given.

664 R O B E R T S P A R K S . [J. I7. I.

Because it will be needed in the proof we will begin by deriving the equation of the curve of pursuit. Consider in Fig. I the heavy line as a knife moved a slight distance along the line r so that the blade moves along the dotted line of the figure. Let c be the length of the knife and let dr be the dis- tance moved by the pointer. Let the angles/3 and /3' be as shown.

FIG. I .

J j s

d r

Then since dr is small the angle opposite is d/3. By the law of sines

d/3 dr

sin/3 c ' integrating,

o r

log tan 13 = r + log K, 2 6

t a n - = e ~t¢ K; 2

determining the constant

/3' tan ~ = e ./~ t a n - - , (I) 2 2

which is the required equation. In Fig. 2 let o be the point from which the knife pointer is

to be started. Let $1 be the first position of the knife and $4 the last. Suppose that the knife is moved from the first

June, I932.1 THE HATCIIET PLANIMETFR. 66 5

posit ion to a point P on the curve along a s t ra ight line making an angle 0 wi th the axis. Then let it be moved a slight a m o u n t along the curve to a point Q. The knife will t ake up posit ions as shown in the figure. We let

X O P = O, XOQ = o + do,

OPQ = ¢,

Y O S 1 = 4),

inclination S2P to O X = 4)',

" SaQ to O X = 4)' + de' ,

" $40 to O X = 4) + d4).

F I G . 2.

Y

Then we have, making use of the curve of pursuit ,

0 - - 4 ) ' O- -4 ) tan - - - e -(r/c) t a n - 2 2

and

O + d O - - ¢ ' - - d4) ' O + dO - - Ca-- d4) tan = e - (r+dr) l le tan 2 2

Different iat ing this expression

d O - d~' dr d O - d4) sin ( 0 - ¢ ' ) - c + s in (0 - - 4))"

(2)

(3)

666 ROBERT SPARKS. [J. F. 1.

T h e n the angle which the arc m a k e s wi th the rod is ¢ - 0 ~ O' a t P and it is ~ -- 0 -t- 4 ' -I- d¢ ' a t Q. T h e r e f o r e

d o ' ds

sin (¢ -- 0 -Jr- 0 ' ) c

or, w h a t is the s ame thing,

cd¢ ' = {sin ¢ cos (0 -- ¢ ' ) -- cos ¢ sin (0 -- ¢ ' ) } d s ,

a f t e r f u r t he r c o m b i n i n g

cd¢ ' = rdO cos (0 - - 0 ' ) - dr sin (0 -- O')

T h e n we p u t e q u a t i o n 4 in e q u a t i o n 3 and o b t a i n

cdO - rdO cos (0 - - O') _ dr = - - dr + c(dO - do)

( 0 - O') sln(0 -- O)

m a k i n g use of e q u a t i o n 2

d O - - d O =dO(- - , - - rc) er/C co tO - - ¢ sin (0 - - O) 2 2

(4)

- - I -Jr- e -(r/c) t a n - ; 2 2

clear ing f rac t ions

~0_ ~ - [ ( i _ :) er,~ cos2°- ~2 ÷(i ÷ ~)e ~r,c, sin20 : ~]~0

and, m a k i n g s u b s t i t u t i o n s for the d o u b l e angle,

dO-- dO = I[(I -- f-c) er/c-]- (I --Jl-~-c) e-(r/c)]dO

; [(I-]-~-c)e--(r lc)-- (I -- f-c) erlc]dO cos (0-- O) T h e t w o pa r t s are each able to be expressed in series t h u s

( : ,~ ,, ,0 ) d O - dO = 2c 2 8c 4 I44c 6 dO

- 3 f i + 3 o c + + c o s ( O - C ) do.

June, I932.] THE HATCHET PLAN1METER. 667

Integrat ing termwise

I ~ 2~" i ~ 2~" c'-(~ = A + ~c2 j U r4dO + i - ~ 4 Jo r6dO + + +

fo (r2 r ) + + 3 o c 3 + + + cos (0-- ¢) d0.

This last expression is the required result. We notice tha t the left hand side of the equat ion is the square of the length of the knife multiplied by the angle it moved through in tracing the figure. If the angle is small as it will be when r is one quar ter or less, then the chord is nearly equal to the are and the length of the knife can be multiplied by the distance from first position to last position of the blade. In some of the commercial forms of this ins t rument there is a scale en- graved on one side to read areas directly and in this case the edge is curved to measure the are. The first term of the right hand side of the equation is the value of the area sought. I t is increased by the many terms which follow. If the pointer is s tar ted from the center of the figure as was suggested above, then the value of all these remaining terms will be less than for any other start ing point. Then if the greatest length of the figure is not over about one fourth of the length of the knife all these terms will be negligible. In case of a larger figure note tha t the second series is composed of odd terms only. Therefore in tracing the figure in one direction they add to the sum but going in the other direction will subtract . So tha t averaging two readings obtained by running the pointer in alternate directions will remove the effect of these terms entirely. The remaining terms would be certainly less than

2 ~ g 4 27rg 6 + +

which, if the greatest length of the figure equals the length of the knife, an extreme case, means an error of about six per cent.

Measuring a few areas with a penknife (such as indicator cards) and then with a planimeter will show tha t the ha tchet planimeter is as accurate as the graduat ions of the best scales will permit of reading.