Manual / Guide · 1957
K&E's 32-page manual for its fixed-arm and adjustable compensating polar planimeters, including the 4242½: care, setting up, reading the dial, wheel and vernier, tracing, scale factors, indicator diagrams, pole-inside measurement with the neutral circle, accuracy, the testing rule, tracer-arm settings tables and graph, and the calculus theory.
COMPENSATING POLAR PLANIMETERS 4236, 4236M, 4237, 4242 & 42421 KEUFFEL & ESSER Co. Copyright 1949, 1957 by Keuffel & Esser Co, CONTENTS Care and Operation . Accuracy . Features Special to No, 4242 . General Theory . Index . 16 20 29 32 COMPENSATING POLAR PLANIMETERS Garriage Ball and Pole Weight. ocket Joint Tracer Grip Fig. 1 No. 4236 THE COMPENSATING POLAR PLANIMETER These instructions are written on the assumption that the reader has little or no familiarity with the planimeter. The indulgence of the engineer is asked on this account, but he may find information of value, particularly in the latter part of the text. 1. HOW IT WORKS The Polar Planimeter is a simple instrument for the accurate measurement of plane areas of any form. To measure an area it is only necessary to run a tracer point around the periphery of the figure and read the distance which a measuring wheel has revolved during the proce The planimeter consists of two units, the pole arm unit and the tracer . The pole arm unit is simply a bar, at one end of which is the pole, consisting of a weight with a needle point at th At the other end is a ball which fits into a socket in the car examine the tracer arm and carriage unit. Note that it rests on three points:—the measuring wheel rim, the guide wheel and the tracer point stop. Holding the tracer grip with thumb and second finger, push the tracer point down with the index finger. as shown in Fig. 2. The stop arm and carriage un > bottom. should be adjusted so the tracer point will just clear the paper. Swing the carriage around the tracer point. Note that the measuring wheel turns rapidly, and the dial turns slowly. Now move it parallel to the tracer arm and note that the measuring wheel slides without turning. This character- istic sliding or turning of the measuring wheel is the key to the planimeter performance. Il. CARE K&E Planimeters are built for accuracy and long service, but, like all precision instruments, they must be taken care of. [f dropped, they will almost certainly be damaged. Do not therefore use a planimeter on a sloping board. Do not leave it lying around when drawings are being moved about. When not in actual use, it is best left in its case. ‘The most critical parts are the rim of the measuring wheel and the axle bearings. The wheel rim has a matte surface of hardened steel, which should he guarded against rust or corrosion. Do not touch it with the fingers, If it has been exposed to moisture, wipe it with a soft clean cloth. If the planimeter is not to be used for some time, particularly in a humid climate, apply a coating of watch oil to the wheel rim, and wipe it off again before using the instrument. Lubricate the axle bearings occasionally with a small drop of wateh oil on a toothpick. Wipe off excess oil. The wheel should spin freely, but with- out perceptible end-play. Even after long use the bearings will not wear perceptibly. Do not attempt to adjust them. The accuracy of the instru- ment can always be checked with the testing rule as explained on page 16. [f any adjustment is necessary, return the instrument to Keuffel & Esser Co., Hoboken, N. J., or if there is not time for this, an instrument specialist or a good watchmaker should attend to it. Il. OPERATION Place the carriage a little to your right with the tracer arm perpendicular to the front edge of the board, the tracer point toward you. Insert the pole arm ball in its socket in the carriage with the pole to the right, parallel to the front edge of the board. (See Vig. 1) Press the pole needle firmly into the paper. This is the regular set-up of the planimeter. Holding the grip, move it about. Note that twisting the grip will lift the guide wheel off the paper. This should be avoided, and if it happens during a measurement, the measurement must be cancelled. Test the limits of the motion of the tracer from side to side and note that it can be moved freely whenever the tracer arm and pole arm make an angle within the wide arc between about 15° and 165°. (See Fig. 3) The shaded area in Fig. 3 represents the limit of measurable areas (“pole- outside-figure”). As an example the following table give: measurable areas of certain regular figures. the limiting No. 4237 No. 4236 No. 4236M Greate: WAxlAin. MHxTMhin, 27x27 cm. Gre 11 x16 in. 14%, x5i4 in. 48x 22cm. Greatest circle -_— 7% in. diam. 29 em. diam. How to Make Readings Move the tracer in a series of clockwis cles and, watching the dial and the wheel, stop the motion when ae pointer on the dial reads 0, and the 0 on the wheel is exactly opposite the 0 on the vernier. This is the zero position. (Fig. 4) Note that the 10 on the vernier is opposite the 9th graduation on the wheel, so that ten vernier graduations are opposite nine on the wheel. Now move the tracer very slightly to the left, stopping before the vernier 0 has reached the first wheel graduation. You will see that one and only one graduation on the vernier coincides with a wheel graduation. If the 3rd vernier graduation thu incides, you know that the wheel has turned 3/10 the distance from its 0 to its first graduation. If it is the 4th, it has turned 4/10 the distance, etc. (Fig. Thus the vernier is used just to tell you more accurately than you can judge by eye how to divide one division on the wheel into 10 equal parts. ms Sa a Tp Fig 4 Now move the tracer a little further to the left until the 0 on the vernier is opposite the first graduation on the wheel. The wheel has now moved 10 “vernier units” or 1 “wheel unit.” (Fig. 6) When the wheel has turned all the way until the graduation numbe the wheel is opposite the nier 0, it has moved LO “wheel units” or 100 “vernier units.” When the wheel has made a complete revolution past 2, 3, 4, ete. and on to the point where the two 0’s again coincide, it has moved 100 wheel units or 1000 vernier units. Notice that the dial has now moved so that the pointer is at “1” instead of at 0 (Fig. 7) 1 tion of the wheel. When the dial makes one complete revolution, the wheel makes 10 revolutions, or 10,000 vernier units. Thus in terms of vernier units we read thousands on the dial, hundreds and tens on the wheel, and units on the vernier. If the reading in wheel units is in the 80s or 90s, the dial marker will be close to the following number and it is important not to read =P? on ch of the 10 numbers on the dial corresponds to one complete revolu- 6 the dial one number too high. Thus if the wheel and vernier read 927 and the dial pointer is near 6, the correct reading is 5927, not 6927. As an example, in the illustration (Fig. 8) dial is between 3 and 4, we know the reading in vern since the vernier 0 is between 4 and 5 on the wheel we know the reading is 3400-++ vernier units; since the vernier 0 is between the 7th and 8th graduation (of the graduations between 4 and 5 on the wheel) we know the reading is 3470-+ vernier units; finally, since the 4th graduation on the vernier coincides with a wheel graduation, the complete reading 3474, This is the procedure by means of which any planimeter setting is read to four places. With a little practice the reading quickly and without error. since the pointer on the units is 3000+; an be made very “3 The Value of a Vernier Unit* K&E No. 4236 Polar Planimeter has a tracer arm of fixed length and it reads directly in square inches to hundredths. That is, 1 vernier unit = 0.01 square inches. K&E No. 4236M Polar Planimeter has a fixed tracer arm and it reads directly in square centimeters to tenths. That is, 1 vernier unit = 0.1 square centimeters. No. 4236 No. 4236M One unit on vernier = 0.01sq.in. 0.1 sq. em. Single division on wheel are OSG") TOs: Major divisions on wheel (between nos.) = 1.00 * og et One revolution of measuring wheel = 10.00% “ 1000“ “ One division on the dial = 1000 2 L000" 9 One complete revolution of the dial = 10000 “10000 > © Inadvertent eo} Intentional Compensations ——_——— Line being traced -----=Path of tracer point Fig. 9 Preparing to Trace the Area Assuming that the area to be meas ured is within the limits di: ed above, the pole can be placed in any position which will permit the tracer point to reach the entire periphery of the figure. For large areas see page 12. The following precautions should be observed: 1. Set the pole in such a position that the angle between the tracer arm and the pole arm will always he between 15° and 165°. It is better where possible to keep the angle between 30° and 150°. It is not necessary to pay any special attention to the centering of the area on the pole socket, nor to take any special precau- tions to avoid positions in which the measuring wheel is sliding. even though such precautions have been recommended in former planimeter instructions. 2. The surface of the paper with which the wheel rim comes in con- tact during measurements must be smooth, clean and uniform. If it is impossible to keep the wheel from turning off the paper, a smooth butt-joint must be made with paper of equal thickness, and the instrument positioned so the measuring wheel will roll over the joint rather than slide over it. *See page 11 for No, 4237. 3. S Draw a fine line across the periphery of the area to be measured to mark the starting and finishing point. When the tracer is nearest the pole, so that the tracer arm and pole arm make a sharp angle, it will be seen that the wheel re- volves very rapidly. The tracer should be moved very slowly over this part of the figure. Tracing the Area Having made a rough preliminary cireuit of the area, and established a starting (and finishing) point, you are ready to make a measurement. le With the tracer point precisely at the beginning line, take a read- ing and set it down, Assume the reading is 3476 vernier units. Holding the tracer grip as shown in Fig. 2, follow the outline of the figure carefully in a clockwise direction. As far as possible the eye should be in line with the direction of the motion, and the tracer point should be close to the paper for precise work. If the tracer point inadvertently leaves the line it is not necessary to go back, but instead the tracer may be intentionally moved the other way to compensate for the error. (Fig 9) Do not attempt to use a guide for straight lines, as this is apt to produce a cumulative error. Straight lines can be traced with great accuracy by holding the head so the tracer moves directly away from or toward the eye. When the figure has been circumscribed so that the tracer has returned precisely to the starting point, make a second reading. Assume it is now 7491 vernier units. . Place this number above the first reading and subtract the first from the second: Second reading 7491 First reading 3476 Difference 4015 ce 4015 indicates the area. If the instrument used was 236, in which 1 vernier unit = 0.01 sq. in., then the indicated area is 4015 x 0.01 sq. in. = 40.15 sq. in. If the instru- ment was K&E No. 4236M, in which 1 vernier unit = 0.1 sq. em., then the indicated area is 4015 x 0.1 = 401.5 sq. cm. If No, 4242 was used, the indicated area depends on the tracer arm set- ting. See page 20. It is best to repeat the tracing at least once to serve as a check. Additional tracings may be made and averaged to increase the accuracy of the results. See page 17 for a discussion of this subject. 2) 9. If the 0 of the dial passes the pointer during the tracing, add 10,000 vernier units to the second reading. Thus: Second reading 2765 12765 First reading — 74: 7452 Area 2 5313 Areas to Scale Although No, 4236 and No. 4236M always measure areas in sq. in. and sq. em. respectively, they may he used very easily to give areas to any scale, by simply multiplying the measured area by a factor. The slide rule is ideal for such work as it can be left set at the factor, and the conversion read immediately for each separate measure- ment. The factor for any map or drawing to scale is easily determined. It is the value represented on the map or drawing of 1 sq. in. for No. 1236 or 1 sq. cm. for No. 4236M. For example, suppose the map has a scale of 50 ft. = 1 in. The value of 1 sq. in. on the map is 50 X 50 — 2500 sq. ft. The factor is 2500. If the area measured was 13.76 sq. in. then the area on the map would be 13.76 2500 = 34.400 sq. ft. In the same manner, if No. 4236M were used on a map scaled 1:5000 and the result is required in sq. meters, the scale may be written 50 m = 1 cm. and 1 sq. cm. would represent 50 X 50 = 2500 sq. m. or a factor of 2500. Then if the measured area was 137.6 sq. cm. the area repre- sented would be 137.6 2500 = 344,000 sq. m. Taking several readings makes for greater y. The factor must be changed accordingly. If repeated readings give a power of 10, the computation is simplified. For example, on a scale of 200 ft. = 1 in,, if the area is traced 4 times, the factor becomes 200 < 200 + 4 = 10,000. Thus if tracing the area four times gives a total reading of 7.62 sq in., the indicated area is 7.62 x 10,000 = 76,200 sq. ft. It is also a simple matter to convert the planimeter reading directly into special units of area, such as acres. Mor example, if the scale is 100 ft. = 1 in. and there are 43,560 sq. ft. in 1 acre, the area of 1 sq. in. on the map in sis 100 & 100 + 43,560 = 0.2296 ac Thus for a reading of 21.05 sq. in. the acreage is computed as follows: 21.05 sq. in. X 0.2296 acres per sq. in. = 4.83 acres. The same method can be used to correct errors due to distortion of the paper or cloth. If the nominal scale is 200 ft. = 1 in., but the paper has stretched 0.6% in the east-west direction and shrunk 0.2% in the north-south direction, 1 sq. in. on the map represents (200 —- 1,006) (200 + 0.998) 198.8 < 200.4 = 39,840 sq. ft. = 0,915 acres. Using either of these factors with planimeter readings will automatically correct errors due to distortion of the map. For the convenience of the user, factors for commonly used architec- tural and map scales are given below. 10 Table I—FACTORS FOR SCALE DRAWINGS OR MAPS No. 4236, No. 4242 or No. 424214,—English System. FACTORS Nominal Actual Values by which sq. in. readings must be multiplied Scale of Scale to convert to units shown at column headings. Dug. or Map Ratio sq. in. ‘sq.ft. acres sq. miles Double size 5 = 3 _ 0.444 = —_ = Full size 1.0 _ _— _ Centimeters 6.45* = — — Half size 4.0 =_ - _ Quarter size 16. ee = ae 3in lf 16. 0.111 =_ _ 144 in. 64. 0.444. = = lin 144, 1.0 — — 34 in 256, 1,778 _ — ¥ in. 576. 40 = = 3 in 1024. Tl a = 7 2304. 16.0 — = Ue in, 9216. 64.0 _ =S 100. —_ = as 400. _ — =~ 1600. 0.0367 — = 2500, 0.0574 = - - 4356. 0.1 = Sa 6400. 0.147 _ ad 10,000. 0.2296 - sh 17,424. 0.4 — — 40,000. 0.918 — a 160,000. 3.673 _ —~ ee 10.0 0.0156 1:15,840 = = 40.0 0.0625 _ _- 160.0 0.25 eat — 640.0 10 == ad a 99.6 0.1557 aad = = 622.7 0.973 “his {actor is square centimeters for reading areas in the metric system. No. 4237—English Sy No, 4237 is a fixed arm planimeter for use on maps with It gives direct reading in acres. Also for reading areas directly in square inches to 0.04 sq. in. (Reading x 0.04 area in square inches.) Table II— FACTORS FOR SCALE DRAWINGS OR MAPS No. 4236M, 4242 or No. 424214—Metric System. FACTORS Values by which sq. cm. readings must be multiplied Scale Ratio to convert to units shown at column headings. of Dwg. or Map sq.mm. sq.cm sqm. hectares. sq. km. 10:1 6 0.01 = = = 4. 0.04 = — — 25. 0.25, = = = 44.4 0.444 = = 1: 100. 1.0 = : = la = 0.155t = = = 1:2 400. 4.0 - = =e 1:5 2500. 25.0 = S aS 1:10 10,000. 100.0 0.01 3 a 1:100 = 10,000. 1.0 = = 1;1000 os 100.0 Col imate a 1:10,000 = 10,000. 1.0 0.01 1:100,000 = — = 100.0 1.0 {This factor is in sq. in. for reading areas in the English system. Indicator Diagrams The mean effective pressure (MEP) indicated by the height of indicator diagrams can be easily obtained with No, 4236 or No. 4236M. (See page 25 for No. 4242. 1. Establish a factor by dividing the pressure scale by the length of the stroke on the card. Measure the area of the diagram. 3. Mul- tiply the area by the factor to obtain the MEP. Fig, 10 For example, if the pressure scale is 1 inch — 80 Ibs./in.2 and the stroke on the diagram 4 inches. the factor would h 4 = 20, If the diagram has an area of 2.31 sq. in., the MEP is 2 x 20 = 46.2 tbs./in.? The planimeters may also be used with cir mean radius, unless the diagram has great ecc area A and find the radius r from the formula A 05642 VA © + diagrams to find the Measure the On the circular diagram at a distance r inches from the center is found the mean value of the pressure, temperature, or whatever unit measured, Measuring Large Areas When the area to be me ured is larger than that described on page 5, two optional methods are available. The first is the simple expedient of dividing the area into measurable segments, measuring each separately and adding the results, This method is recommended when no more than two or three segments are re- quired, For larger areas the second method, known as the “pole-within- figure,” is described helow. It is possible to measure a large area ac rately by placing the pole near the center of the area and making a complete circuit around it with the tracer. Obviously, the area is limited to the periphery that can be reached with the tracer in this process. Thus the entire periphery of is being “The K&E 4215 Radial Planimeter is much better jor this work, as it is faster and more accurate. It can be used irrespective of the contour. 12 the figure must be of such size and shape that it can be made to lie within concentric minimum and maximum circles as indicated in the shaded areas of Fig. 11. Thus if the figure is no more than 23 in, maxi- mum length and is at least 714 in. across near the middle, it is measur- able with No. 4236 in a single setting. Fig. I The area of such a figure is measured by setting the pole m center, and running the tracer in a clockwise direction around the periphery to the starting point. However, the reading Uius obtained does not indicate the area directly, but instead ‘t indicates the difference between that of the figure and that of the neutral circle. The latter varies with e individual instrument and chown inside the lid of the case. The neutral circle is the area swept out by the tracer point n it makes a revolution around the pole, held at a uniform distance from it s that the wheel motion is all sliding. The theory is explained on page 29. A simple computation gives the desired area as explained below. On all “pole- ‘i rements it is desirable to start with the dial and wheel at 0. This is done by holding the tracer on the starting point with the right hand, lifting the carriage with the left fore- finger and turning the wheel with the thumb, as shown in Fig. 12. Do not turn the wheel while it is in contact with the paper, or touch the wheel rim at any time. Before making the measurement it is necessary to know whether the area is greater or less than that of the neutral circle. This is determined by noting whether the net motion of the wheel is plus or minus. Set the dial and wheel at 0 and make a quick approximate circuit of the area, watching the dial and noting whether its total motion has been posi- tive or negative. It may make even more than one complete revolution, and if this occurs, the fact should be duly noted. Thus, one of four possible conditions will be found. From the largest area to the smallest these conditions are: The net dial motion may be in area 13 (1) positive and more than one complete revolution; (2) positive and less than one revolution; (3) negative and less than one revolution; (4) negative and more than one revolution. The formulas to use for each of these four cases are given below. If A is the area of the figure in vernier units, C the area of the neutral in v.u., and W the final reading of dial and wheel in v.u. after ng the area beginning at 0000, Case 1. 4 = € + 10,000 v.u. +- ¥ oe +W 3. A (6% 10,000 v.u. -- W* Case 4, A — 20.000 + W* Fig. 12 The formulas are given in vernier units so they will apply to both No. 4236 and No. 4236M. For No. 4236, 1 sq. in. = 100 v.u. and for No. 4236M, 1 sq. em. = 10 v.u. For scale drawings the factors in Tables I. and IL. may be applied after A has been found. Example of Case 1. Using a No. 4236 planimeter having a neutral le area 216.91 sq. in., on a full » drawing, the dial was seen to move in a positive direction one complete revolution and about 4 units over, from a preliminary circuit of the area. When the area was care- fully traced, the reading was 43.6 in. A= C + 10,000 v.u. + W. € + 100.00 s 16.91 65, A = 360.56 sq. in ci *The negative motion of the dial and wheel must be subtracted from C. The negative motion when read positively is (10,000 — 7). Hence A = € — (10,000 — W) — 10,000 + W or A = C — 10,000 — (10,000 — W) = € — 20.000 + W. 14 Example of Case 2. Using No. 4236M planimeter having a neutral circle area 1845.0 sq. em. on a drawing to a scale of 1:50, a prelimi- nary circuit showed a positive motion of about 7 dial units. The final traced reading was 685.2 sq.cm. 4 = C + W. Cc = 1845.0 W = 685.2 om. The factor is 50 X 50 = 2500 sq. cm. = 0.25 sq. m. Area to scale is 2530.2 X 0.25 = 632.55 sq. m. Example of Case 3. With the instrument and drawing ef Example 1, the preliminary circuit showed the dial moving backward about six units. The reading after tracing was 41.64 sq. in. A = C — 10,000 v.u. +. (a 100.00 sq. in. = 116,91 W= 41.64 A = 158.55 sq. in 7 Index Line Fig. (3 Example of Case 4. With the same set-up on a smaller area the dial moved a little over 1 revolution backward. The reading was 92. A = C — 20,000 v.u. + W. C€ — 200.00 sq. in. 16.91 Ww 92.35, = 109.26 sq. in. 15 'V. ACCURACY As is the case with all scientific instruments, the accuracy of planimeter observations depends upon several factors. Four of these are discussed below: 1. The precision of the instrument. 2. The conditions affecting the measurement. 3. The personal equation. 4. The size of the figure measured. 1, The Instrument Condition The precision and accuracy of the instrument are easily determined with the aid of the Testing Rule sup- plied for the purpose. The function of the testing rule is to permit the tracer accurately to circumscribe a circle of known area. To make a test with the testing rule, proceed as follows: (a) Remove the tracer stop. (Sce Fig. 2) (b) Draw a fine straight vertical line on the paper about 4 inches long, and set up the instrument as shown in Fig. 13, with the acer pin in the prick-punched depression in the rule. (c) Bring the index line on the rule precisely to the line on the paper and set the wheel and dial exactly to 0 (page 13). (d) Holding the tracer point down firmly as shown in Fig. 2, move the testing rule at a moderate uniform speed around a complete cirele to the starting point. The wheel should make one complete revolation, and return to 0 while the dial has moved from 0 to 1. (e) Make nine more circuits and record the reading after each. Two characteristic sets of readings are given below. A B 10.02 10.00 20.02 20.00 30.02 29,99 40.02 39.99 50.01 49.99 60.01 59.98 70.02 69.97 80.02 197 90.03 89.96 100.03 99.96 Test A indicates that there was a small error in the initial setting, and that the readings show the planimeter to be accurate to the limit of the operator’s ability to read the vernier. Test B shows that there is a slight but definite tendency for the instrument to read low, the apparent error amounting to about 1 part in 2500. Both tests show that the instrument has a precision of better than 1:1000 in measuring a 10 sq. in. area. 16 (f) Further tests may be made by (a) decreasing or increasing the distance between the pole and the testing rule, (b) reversing the pole so that it is at the left instead of at the right and (c) tracing counter-clockwise instead of clockwise. Such tests will show slight variations in results, varying in different instruments. Their prin- cipal value is in determining whether the instrument is in good general condition and capable of producing consistent results. If it is found that the instrument reads high when the pole is to the right and low when it is to the left, or vice versa, an equal number of readings may be taken in each position and averaged. This is the “compensating” feature which gives these instruments their name. 2. Conditions of Measurement As important as the inherent preci- sion are the conditions external to the instrument. Perhaps the most vital is the surface upon which the measuring wheel rides. If the paper has been folded or torn or has pin-pricks or is wavy or is uneven in any way, accurate results cannot be expected. The texture of the paper makes little difference, so long as it is uniform. Working on a sloping board decreases accuracy somewhat and increases the danger of drop: ping the instrument on the floor—a serious hazard, Temperature has some effect. Instruments are calibrated to read cor ctly at 70°. 3. The Human Equation Some operators can obtain much better results than others. Good eyesight, a steady hand and patience in care- fully following the outline are important assets. Good light and com- fortable working position are also important for accurate work. Repeated measurements of the same area increase the accuracy of the result, and if tests are made by different operators, their relative efficiency can be judged. Two sets of readings are given below, both made on the same figure with the same No. 4236 planimeter but by different operators. MADE BY OPERATOR A Readings Areas Deviations 3582 4874 12.92 +.042 6162 12.88 +-.002 7454. 12.92 8737 12.83 0023 12.86 1307 12.84 2597 12.90 3883 12.86 5170 12.87 6460 12.90 Total 3582 16460 128.78 260 (Regardless of sign) Aver. 12.878 sq.in. .0260 7 MADE BY OPERATOR B Readings Areas Deviations 5784 7070 12.86 —.026 8361 12.91 ++.024 9650 12.89 -+.004 0939 12.89 +.004 2298 12.89 +-.004 3517 12.89 +-.004 4804 12.87 —.016 6090 12.86 —.026 7380 12.90 -++.014 8670 12.90 +.014 Total 5784 18670 128.86 136 (Regardless of sign) Aver. 12.886 .0136 It will be noted that the average deviation in the first set is almost double that of the second, indicating that B is the better operator. Also note that the final results of the two sets agreed within 1 part in 1500, although individual maximum and minimum readings in the first set varied as much as 1:150 and in the second set 1:400. These tests prob- ably represent a fair measure of the accuracy which can be achieved by different operators under favorable conditions on an area of the size indicated. 4. Accuracy and Size of Figure Disregarding instrumental errors, if a 1 in, square is traced with the tracer 0.01 in. outside the outline, the area would measure 1.00 + 4 < 1.01 X 0.01 = 1.04 sq. in. Tf a square 3.16 on a side were similarly measured, the result would be 10.127 instead of 10.00; for a 10.00 in, square, 100.40 instead of 100.00. The percent error drops from 4% to 1.3% to 0.4% succes- sively, and for square figures varies very nearly inversely as the square root of the area.* Thus if the average accuracy for a square figure of 10 sq. in. is 1:1000, that of a similar figure 1 sq. in. would be only 1:316, while that of a figure of 100 sq. in. would be 1:3160. *If At is the width of the tracing error, S the side of the square, 4 the true area and A; the area as measured; then S = VA and A, = (S+2A0)? = S+ 4 AES approx, Ay S+4aisS 4At AAt = et — =1 —— approx. am aS S “te com PF The general relationship given below holds for any figure but At increases with the irregularity of the figure. The error in area expressed in percent is computed as follows: PAY Ep = 100 —— approx. A When Ep = Error in percent of area, P = Perimeter, A = Area. 18 ed in the measurement of large ar s “pole within-figure” s indicated in Table ILI showing freehand measurements of large circles. Note that the average deviation is very low and is inde- pendent of size. The measured areas are in excellent agreement with the computed areas, the error apparently being due as much to the delinea- tion, diameter measurements and tracing as it is to inherent instru- mental erro ematic departure from the mean. It is also important to note that there is no impairment of at racy near the neutral circ! as has been assumed in all previous planimeter instruc- tions. We are, therefore, justified in the statement that with repeated careful measurements under favorable conditions an accuracy of 1 in 1000 can be secured with K&E No. 4236, except where the area is less than 10 sq. in. Table Ill ACTUAL FREEHAND MEASUREMENT OF CIRCLES POLE-WITHIN-FIGURE — K&E No. 4236 Nominal diameter 8” _10"__-'12"_—14”_16”__—18” | 2. Planimeter 2 ameoae readings, v.u. 3296 9595, 8455 3903-9871 «6419 3. od 319 9600, 8472 384509855415 4, om 4, 9630 8494 3870 9860-6410 bs “ 3327 9626 8468 3853 9858 6403 6. Average 3319 6139 9613 8472 3868 9861 6412 7. Neutral Circle* 1691 ‘1691-1691 11691 21691 21691 31691 8. Indicated area — — cael sq.in. 50.10 78.30 113.04 154.84 201.63 255.59 315.52 381.03 9. feasured diameter in. 8.00 9.98 11.98 14.03 16.00 18.04 20.03 22.00 10. Computed mp, area 4 5 78.23 112,72 201,06 255.60 315.10 380.13 11. Error in Jo —0.34 +-0.01 +0,28 +0.28 0.00 +013 - Average deviation dings v.u.’s i 7 15 10 11 19 5 5 20,000), (C — 10,000), C, or (C + 10,000) see page 14. 19 V. FEATURES SPECIAL TO No. 4242* The No. 4242 K&E Compensating Polar Planimeter has an adjustable tracer arm and an adjustable pole arm. In other respects it is like No. 4236 or No. 4236M. When the tracer arm is set to about 20.30 setting is shown inside the ca: (the ex. ), the planimeter will read areas “pole-outside-figure” in square inches (1 v.u. = .O1 sq. in.), and all the foregoing instructions regarding No. 4: will apply to the operation. Likewise when the tracer arm is set to about 31,50 (exact selling shown inside the case) the planimeter measures in square centimeters (1 v.u. = 0.1 sq. cm.) as No. 4236M. Included in the instrument > is a testing rule which has punch marks for 10 sq. in. 100 sq. em., and 5 sq. in. cireles. For all * oe gure” measure: ments the length of the pole arm is immaterial. Fig. 14.) Since the reader will find the general nanan of all thes ments thoroughly explained in the for sary to add here an explanation of the method of adjusting the arms and the results obtained thereby. egoing text, it is only n Adjustment of Tracer Arm When both clamp screws (A and C Fig. 15) are loosened, the carriage D is free to slide on the tracer arm E. A “setting” refers to the position of the vernier V on the tracer arm To make a setting, loosen both screws A and € and slide the carriage so that the vernier zero is near the setting desired, and tighten screw A, but do not tighten screw C. By revolving the slow motion crew B, the vernier can be moved to the e: tighten screw C. For example, to set the carriage to 28.24 loosen both clamp screws, slide the carriage so that the vernier zero is about at the setting required; then *Also applies to No. 4242 second division to the left of 28, and tighten screw A. Turn B until the vernier zero is between the second and third division to the left of 28 and the fourth division to the left of 0 on the vernier is opposite a scale division. Tighten C. For example, the setting in Fig. 15 is 20.28. The method of using the vernier is as explained on page 6, but note that in this case the scale and vernier read from right to left rather than from left to right. The tracer arm scale is in half-centimeters, so that one yernier unit represents 5 mm X 0.01 = 0.05 mm. = 0.002 inches approx. The seale is generally used in a purely arbitrary manner, because each instrument not only has its own particular character but these may vary slightly with the passage of time, due to wear and minor accidents. All variations are automatically compensated for by the proper use of the testing rule, as explained helow. The length of the pole arm when making “pole-outside-figure” measurements has no effect on the measurement. However, to assure smooth operation and maximum area coverage, the pole arm should be lengthened, if nec y, so that it is at least three inches longer than the tracer arm, Measurements to Scale (““Pole-outside-figure”) As already ex- plained, if readings are made in square inches or square centimeters, the setting of the tracer arm is as given inside the With one of these settings, are ured to any scale by using an easily computed factor, as on the fixed arm planimeters. See page 10. If it is desired to eliminate this factor, it can be done by adjusting the length of the tracer arm. n be me Table IV gives the settings for all commonly used scales in the English system. Note that the tracer arm setting is approximate, but the number of vernier units for use with the testing rule is exact. A column is provided in which the exact tracer arm setting may be put down for future reference after it has heen found by test. The procedure is to set the tracer arm to the approximate setting given, then note the exact number of vernier units for one revolution of tl sting rule. Make one revolution with the 10 sq. in. punch mark. Vary the tracer arm setting with screw B until the v.u. corresponds exactly to the num- ber given in the table. Lengthening the arm decreases the v.u. per revo- lution and vice versa. The planimeter will then give readings on the drawing or map, each v.u. having the area value given in the table, Table V_gives the settings to use when the metric system is employed. Here the unit of measurement is 1 sq. em. and the 100 sq. em. punch mark on the testing rule is used as a standard. For very short tracer arm settings (below 14) the 5 sq. in. punch mark must be used, and the reading must be multiplied by the factor 3.1, to give the reading that would result if 100 sq. cm. were used. 21 If the drawing or map seale is not found in the tables, the first step is to find the scale factor as described on page 10. In the English sys- tem the factor is the area of 1 sq. in. on the map, in the units to be used. If this factor, after multiplying it by an integral power of 10 (that is, by moving its decimal point) is in the range 565 to 2032 a setting of the tracer arm can be made in which areas can be read directly. Table IV Tracer arm settings for scale drawings or maps. No. 4242 & No. 4242/,—English System \Pracer Arm Setting] Exact Nominal Actual : ae I ; Dwg. or Map Ratio | Approx. | Exact |on Test Rule Double size 1250. 0.002 sq. in. I~ Wt 0.004 * Full size 1000 OO ess 1000 fap Ute a aes 1600 Oil SHES: Td 0.001 sq. ft. 889 U.0050) 8 1000 AY Use 889 ea ee 1000, Ode % in. ¥% in. VY in. 1600 (hile! Yin. 640 LO setecs lin. 1000 Te Ct lin. 1000 d0cueoe Lin. 1600 |100 “ “ Lin. 1250/2000 Lin. 1089 | 400 “ “ lin. 1000 0.001 acre Lin, 640 |100. sq. ft Lin. 1000 100, ae 1742 100. bia toad | 1000 0.004 acre 918 Une etiaes 918 0.04“ 1000, 0.1 ig 1000 0.4 ie 800 20a 640 | 100 a 996 10-4 1557 0.001 sq. mi. = 623 | 10.0 acre a 973 0.01 sq. mi. = 1000 Ol0 dee arene Table V Tracer arm settings for scale drawings or maps. No. 4242 & No. 424214—Metric System Scale Ratio Tracer Arm Setting Exact vt’s Value of Lv. of Drawing for 100 sq..cm. per 100 sq. cm. to scale or Map Approx. Exact on Test Rule 31.5 1000 0.1 sq. cm. 1562 CEe 2250 On 1000 Oats + 900 Soe 1600 dist 1250 Shae 1800 BOF 1600 eoke 1000 10.0 For ratios which are powers of 10 of the above, use the same setting, but change the value of 1 v.u. accordingly. For 1:200, use setting for 1:2, and 1 v.u. = 0.4 X 100 X 100 = 4,000 sq. em. = 0.4 sq. m. The factor (between 565 and 2032) represents the number of vernier units that the measuring wheel will turn for one revolution of the testing in inches: the factor is cified range, follow the vertical line numbered 900 to its intersection with the diagonal marked “square inches” and across to read 22.6 approx. at left or right. Set the tracer arm scale to 22.6 and, using the testing rule (10 sq. in.), adjust screw B until 1 revolution = 900 v.u. precisely. The instrument is now set to read areas directly to the desired scale. Thus with this setting, if a measured area reads 2756 v.u., the area represented on the drawing is 275.6 sq. in. The decimal point is set by making a rough calculation. In this case 1 . actual represents about 10 sq. in. to scale. When the metric system is used, the same principles apply, but in this case the range of factors from 875 to 3 The curve marked “square centimeters” on Fig. 16 indicates the tracer arm setting, and the 100 sq. cm. punch mark on the testing rule is used to establish the exact setting. In the cases where the factor cannot be brought within the range for direct reading it is possible to divide the factor by a convenient number such as 2, 3 or 4, and then multiply the planimeter reading by the same number. For example, if the scale is 1:2.2 the factor is 2.2 x 2.2 = 4.84. Divide 4840 by 4 to obtain 1210, and use this as a factor, multiplying all readings by 4. 23 24 This method of manipulating the factor as desired is often useful in obtaining a tracer arm setting which is most convenient; that is. near the center or near either end of the arm. Settings near the center are the best for average use, while short settings give the most accurate results, but with a limited area coverage. On the other hand, long settings cover large areas, but somewhat less accurately. Thus, if the scale of the drawing is 1:14, the factor would be 1960 which indicates a tracer arm setting of about 10.4. This setting could be used but the measurable areas would be small. To obtain a good average setting, divide 1960 by 2 to obtain a factor of 980, which gives a tra 20.7, and multiply the planime cover large areas and sacrific divided by 3 to obtain 65: r arm setting of about dings by 2. If it were desired to somewhat, the factor could be and multiply readings by er Tr Indicator Diagrams The instrument is so constructed that 7) = 2.5 in. in the formula A =IrDN (page 30). The: A L If the tracer arm is set to the length of the indicator diagram, the average height (h) of the diagram will then be ore = 2.5N h or OA h As each square inch is 1000 v.u., the instrument will then read the average height on the basis of 400 v.u. per inch. Square off the diagram with fine pencil lines. (See Fig. 10.) Set the carriage over one corner. Then slide the tracer arm to the other corner, so that the tracer arm length is that of the diagram. Tighten screws A and C. With this setting, trace the diagram and read the wheel. The M.E.P. is then equal to area in v. u. X pr ure scale. 400 Example. ‘The pressure scale is 80 Ib./in.2 per inch. ‘The traced area ead 205 Yb Te MiPais 2 x BO = 710 Iban 400 Adjustment of Pole Arm The length of the pole arm is immaterial on “pole-outside-figure” measurements. Therefore, where sp: ited, the pole arm may be shortened; or, where pole arm will be found convenient. In “pole-within-figure” measurements, the pole arm length is a vital factor in the result, as explained in the section on the theory (page 29). The pole arm setting given on the lid of the case is the one which produces a neutral circle area of 200.00 sq. in. when the tracer arm is lim- are large, a long 25 is set for reading in square inches (about 20.3). With this setting, the four formulas on page 14 become: (1) 4 = FP + 300.00 sq. in. (2) 4 = W + 200.00 sq. in. (3) A = W + 100.00 sq. in. (4) A>V where A is the area in square inches and IV is the planimeter reading in hundreds of v.u.’s. When the metric system is used and the tracer arm is set for metric measure (about 31.5) as shown on t