'1['1HIIE UJNil.WlE R~II'II'V NUMBER 5701 JANUARY 1, 1957 The Measurement of Small Angles Using Optical-Reading Theodolites and American-Design Transits BY LELAND BARCLAY Associate Professor of Civil Engineering The University of Texas Engineering Research Series No. 47 BUREAU OF ENGINEERING RESEARCH THE UNIVERSITY OF TEXAS AUSTIN Publications of The University of Texas COMMITIEE ON PUBLICATIONS L. u. HANKE H. Y. McCowN R. F. DAWSON A. MOFFIT J. R. D. Enny C. P. OLIVER J. T. LONSDALE w. P. STEWART S. A. MAcCoRKLE J. R. STOCKTON C. T. McCORMICK F. H. WARDLAW ADMINISTRATIVE PUBLICATIONS AND GENERAL RULES w. B. SHIPP c. H. EADS J. G. AsHBURNE F. H. GINASCOL c. E. LANKFORD The University publishes bulletins twice a month, so numbered that the first two digits of the number show the year of issue and the last two the position in the yearly series. (For example, No. 5701 is the first publication of the year 195 7.) 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Additional copies of this publication may be secured from the Bureau of Engineering Research, The University of Texas, Austin 12, Texas The Measurement of Small Angles Using Optical-Reading Theodolites and American-Design Transits BY LELAND BARCLAY Associate Professor of Civil Engineering The University of Texas Engineering Research Series No. 47 BUREAU OF ENGINEERING RESEARCH THE UNIVERSITY OF TEXAS AUSTIN The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preserva­tion of a free government. SAM HOUSTON Cultivated mind is the guardian genius of Democracy, and while guided and controlled by virtue, the noblest attribute of man. It is the only dictator that freemen acknowledge, and the only security which freemen desire. MIRABEAU B. LAMAR PUBLISHED BY THE UNIVERSITY TWICE A MONTH. ENTERED AS SECOND-CLASS MATTER ON MARCH 12, 1913, AT THE POST OFFICE AT AUSTIN, TEXAS, UNDER THE ACT OF AUGUST 24, 1912 Contents TOPIC PAGE I. PREFACE 5 II. CONCLUSIONS 5 III. PROCEDURE 5 IV. DATA AND RESULTS 5 Figure 1: Measured Subtended Angles 8 Figure 2: Measured Subtended Angles 9 Figure 3: Error Due to Direction Turned 9 Figure 4: Theodolite E1 Micrometer Error . 10 Figure 5: Theodolite E1 Error in Measured Angle 11 Figure 6: Theodolite E2 Micrometer Error . 12 Figure 7: Theodolite E2 Error in Measured Angle 13 Figure 8: Theodolite E3 Micrometer Error 14 Figure 9: Theodolite E3 Error in Measured Angle 15 Figure 10: Subtended Angles 16 Figure 11: Measured Subtended Angles 16 v. DISCUSSION OF CONCLUSIONS 17 Illustrations FIGURE TITLE PAGE Diagram of Optical Micrometer 7 1. Measured Subtended Angles 8 2. Measured Subtended Angles 9 3. Error Due to Direction Turned 10 4. Theodolite E1 Micrometer Error 11 5. Theodolite E1 Error in Measured Angle 12 6. Theodolite E2 Micrometer Error 13 7. Theodolite E2 Error in Measured Angle 14 8. Theodolite E3 Micrometer Error 15 9. Theodolite E3 Error in Measured Angle 15 10. Subtended Angles 16 11. Measured Subtended Angles 17 I. Preface The study was conducted under a research project titled "Investigation of the Use of American-Design Repeating Transits (ordinary transits) for Measuring Distance by the Subtense Bar Method," and was sponsored by the Bureau of Engineering Research, The University of Texas. The purpose was to develop a technique for making subtense bar measurements, using the ordinary American-design transits, to the same accuracy as may be obtained with the optical-reading theodolites such as the Wild T2 and the Kern DKM2. Also it was desired to determine if such measurements could be made as rapidly with the American-design transits as with the optical-reading the­odolites. It was proposed that the study be made in the field since it was assumed that performance of instruments under laboratory conditions was already well es­tablished and that errors would be due largely to field conditions. The study did not develop as intended. Due to obvious systematic errors that were apparent after some of the first field comparisons of the two types of instru­ments, it was decided to move into the laboratory and conduct a study to determine the sources of these systematic errors. The evaluation of the optical-reading the­odolites and the American-design repeating transits for measuring small angles finally became the purpose of the study. II. Conclusions Small angles may be measured with the American-design repeating transits with a probable error of less than one second. The instrument when operated according to the manufacturer's instructions shows no systematic errors. Small angles when measured with the optical-reading theodolites may have a systematic error of as much as eight seconds. The systematic error results from inaccuracies in the reading micrometer and will appear in the same way when measuring angles of any size. These conclusions are based on a study of three optical-reading theodolites. III. Procedure All readings reported in this study were made by the author of this report, who will be referred to as the observer. The sets of readings taken as a basis for the curves were used without discarding or adjusting any one of the readings. This procedure without doubt produced erratic points for the curves but it is thought The University of Texas Bulletin that the observer should not attempt to distinguish between blunders and errors as they appear in this study. An examination of literature discloses no evaluation of the American-design transits for measuring small angles such as those encountered in distance measure­ments by the subtense bar method. The manufacturers of optical-reading theodolites such as the Wild T2 and the Kem DKM2 state in general that when an experienced operator uses these instruments angles may be easily measured with an error of one second or less. Also articles written by practicing surveyors lead the observer to assume the surveyor is reporting measurements made with an error of one second or less when using these instruments. The observer originally assumed that angles could be measured with an error of one second or less with optical-reading theodolites such as the Wild T2 and the Kem DKM2, and he believes that most users of the optical-reading theodolites assume that the instruments are precise and that the largest contributing source of error in measuring angles is the inability of the in­strument man to point the telescope and set the reading micrometer. The data and results are presented in graphic form. This is considered desirable since a reasonable scale allows us to read the data and results to a higher precision than the precision of the measurement. Only the data and results for the final steps in the study are presented, as these are the only steps considered necessary as a basis for the essential results of this study. Several American-design repeating transits were used. There were no signifi­cant differences between these transits. Data and results from only two of the American-design transits will be included in this report. IV. Data and Results Transit Ai is an American-design repeating transit owned by The University of Texas. It is about seven years old and has been in continuous use by students. The least count of the vernier is one minute. The vernier was read by using a pocket magnifier. Transit A2 is an American-design repeating transit which was loaned by the manufacturer for the purpose of this study. The least count of the vernier is 20 seconds. The vernier was read by using an attached magnifier. Theodolite E1 is an optical-reading theodolite owned by The University of Texas. It is about ten years old and has been used by students. The smallest di­vision on the circle is 10 minutes. The total run of the optical micrometer is 10 minutes with the smallest division one second. Readings may be estimated to 1/10 second. The instrument is a Kern DKM2. Theodolite E2 is an optical-reading theodolite which was loaned by the distribu­tor for the purpose of this study. The total run of the optical micrometer is 10 minutes with the smallest division one second. Readings may be estimated to 1/ 10 second. The instrument is a Wild T2. Theodolite E3 is, as theodolite E1 , a Kern DKM2 loaned by the distributor for the purpose of this study. The optical micrometer may, for the purpose of this report, be described as follows. (See the diagram on this page.) The line of sight of a reading telescope (A) , focused on the circle to be read, is the index for reading. An optical glass plate (B) is in the line of sight of the reading telescope between the objective and the circle. Direction of 1ncrease of circle reading Diagram of Optical Micrometer The index, with the glass plate in the zero position, will usually fall between two of the marks on the circle. (The zero position is shown by the solid-line outline in the sketch.) The glass plate may be rotai:ed, using a thumb screw, to the dotted­line outline. Then the index is at coincidence with a mark on the circle. The amount of rotation of the glass plate from the zero position is the measure of the angle from the zero position of index to the mark on the circle. The amount of rotation is transcribed into angle movement of the index and read directly from the micrometer scale. The observer practiced setting and reading the optical micrometer, and then measured his ability to set and read the micrometer. The observer's probable error in setting and reading was slightly less than one second. Thus it was obvious that, when using the optical-reading theodolite, it should be easy to measure angles with an error of less than one second if the average of several measurements is taken as the correct angle. The data presented in Figure 1, Figure 2, and in Figure 11 were taken in the field, using a two-meter subtense bar to subtend the angle. The data for all of the The University of Texas Bulletin other figures were taken in the laboratory, using a metal scale to subtend the angle. The line of sight was perpendicular to the scale at approximately the mid­point of the section of scale used for the set of readings. The instrument was ap­proximately i3 feet from the scale. Figure 1: Measured Subtended Angles The subtended angles were measured, using transit AL Then theodolite E1 was placed on the same point as had been occupied by transit Ai and the sub­tended angles were measured. The subtense bar was not moved. The transit Ai was turned clockwise when turning through the angle and counterclockwise when returning to the initial point, and both tangent screws were turned clockwise. The theodolite E1 was turned clockwise and angles were read to the ends of the subtense bar, then turned beyond the end of the bar and brought back counterclockwise to read on the ends of the bar, thus giving two angles turned through. This procedure was repeated to give the total turns as indicated. Figure i appears to show the following. a. The number of repetitions above about eight does not increase the precision of measurement when using transit Ai. b. The average of four angles measured, using the theodolite Ei, gives a pre­ cision of about± 2 seconds. c. The theodolite has a systematic instrumental error. d. Systematic errors occur in one or both of the instruments. (/) QI 3011 (by repetition) oi 1° 221 c: 1° 221 I Cf) 0011 202°071 I I I I I 4 12 20 28 36 44 Number of turns through angle MEASURED SUBTENDED ANGLE FIGURE Figure 2: Measured Subtended Angles The data were collected during two different working days. The instrument and subtense bar were carefully placed the second day to the same position as on the first day. A change in subtended angle due to incorrect placing on the second day was not apparent. The theodolite EZ was turned clockwise and angles were read to the ends of the subtense bar. The line of sight was turned beyond the end of the bar and brought back counterclockwise to read at the ends of the bar, thus giving two angles turned through. Figure Z appears to show the following. a. The average of four measurements of the angle has a maximum accidental error of about ± 1% seconds. The maximum discrepancy for a group of measurements with the same initial horizontal circle reading is about 3 seconds. b. The position of the initial reading on the horizontal circle (degree or mul­tiple of 10 minutes) does not influence the size of the measured angle. c. The initial micrometer reading does influence the size of the measured angle. These data show discrepancies as large as about 9 seconds due to starting with different micrometer readings. lni tio I Horizon to I Ci re le Reading 206° 60 30 180 250 250 120 240 4 311 00 180 180 203 203 203 oo' oo oo oo 10 10 oo oo 12 35 06 18 18 00 09 09 MEASURED SUBTENDED ANGLES FIGURE 2 Figure 3: Error Due to Direction Turned All measurements were made using transit A1 . A complete circle was measured by repetition, taking readings on the fifth, tenth, fifteenth, and twentieth turn through the angle. A collimator was used to mark the point on the circle at which to start and end the 360-degree angle. The error is the measured value of the circle minus 360 degrees. a. The direction of turning does affect the accuracy of the angle measure­ ment; the instrument can be operated to produce zero error. b. This source of error should be studied to determine the exact cause and to produce a remedy. The observer, based on additional work done during this study, believes that the cause is not in the tangent screw threads as sug­ gested by some. The transit A1 produced the largest error of any transit studied. Most of the American-design transits produced maximum errors from this source of only about two seconds. The observer believes the error is of the same magnitude for angles of any size as for a complete circle. The observer did not make a quantitative study of the theodolites but be­ lieves, from superficial tests, they will show similar errors. c. This source of error can explain some of the discrepancy between angles measured with transit A1 and the same angles measured with theodolite E1 as shown in Figure 1. D1rect1on turned 3011 ~ c> c: r<> (!) (!) (J) -Initial reading 313° 01' 36.7" Error 1s plotted at m1dpo1nt of x-ln1t1al reading 313°01' 26.9 11 section of scale subtending the angle THEODOLITE EI ERROR IN MEASURED ANGLE FIGURE 5 Figure 6: Theodolite E2 Micrometer Error The errors as plotted were determined by measuring angles between successive points on a scale. Readings were made at each point on the scale as the theodolite turned clockwise and then as returned in a counterclockwise direction. The average of four readings on each point was used to determine errors. The correct angle subtended by one division of the scale was assumed to be the angle computed by dividing the measured angle subtended between the outside points used by the number of divisions between the points. The error is the measured value of one division minus the correct value of one division. The error is plotted at the mi­crometer reading midway between readings at ends of the division. The value of one division is 64 seconds and the points as plotted are actually the error in 64 seconds. Figure 6 appears to show the following. The results shown in Figure 2 can be explained in the light of this curve. For example, Figure 2 shows an angle of about 1°15'15" measured with an initial horizontal circle reading of 250°10'40" to be about 7 seconds larger than the same angle if the initial reading is 180°18'20". According to Figure 6, the error in an angle with an initial reading of 250°10'40" is the area under the curve from 00'40" to 05'55" which is about +4 seconds. The error in the angle with an initial reading of 180°18'20" is the area under the curve from 8'20" to 3'35" which is about -3 seconds. Then the discrepancy in measured angles should be, according to Figure 6, 4 seconds plus 3 seconds, equal­ling 7 seconds. o' 1' 21 31 41 5' 61 7' 8' 9' 101 Micrometer Reading THEODOLITE E2 MICROMETER ERROR FIGURE 6 Figure 7: Theodolite E2 Error in Measured Angle The theodolite E2 was pointed on the scale divisions and the horizontal circle was read as turned clockwise and then as returned in a counterclockwise direc­tion. The theodolite readings given are each the average of four readings. The correct angle subtended by one division was assumed to be the measured angle between the outside points used divided by the number of divisions of scale be­tween these points. The scale points used were selected to demonstrate that the curve, Figure 6, is approximately correct and to produce the maximum systematic error possible for this instrument. The error in measured angle should be equal to the area under the curve, Figure 6, between terminal readings of the microme­ter. The error in angles should be alternately plus and minus and should be the same size as we progress along the scale. Figure 7 appears to show the following. a. The theodolite E2 makes systematic errors about as large as 8 seconds. b. The micrometer error curve, Figure 6, is approximately correct. 201° Theodolite E2 }-3, 3' 200° 7 13 17 23 27 33 37 43 47 53 57 Reading 1" 25 I 27 3 II 31 6 31 7 33 10 35 12 · on ~ lD 0 co ~ N co lD N 0 lD Scale } ~ N <;!' lO 0 <.O I'-a:i cri cri N Reading --I I 1 I +10'I +5" <1l O' c <( o" = ..... e ..... w -5'I THEODOLITE E2 ERROR IN MEASURED ANGLE FIGURE 7 Figure 8: Theodolite E3 Micrometer Error The errors were determined by measuring angles between successive points on a scale with divisions of 1/50 inch. The angle subtended by one division was about 26 seconds. Readings were taken at each point on the scale as the theodolite turned clockwise and then as returned to the points in a counterclockwise direction. The average of eight readings on each point was used to obtain errors. Two sets of readings with different initial circle readings were taken. The correct angle sub­tended by two divisions was assumed to be the value computed for two divisions based on the angle read when enough divisions (23) were measured to give a subtended angle of about 10 minutes. The error in the micrometer per minute 60 equals " (b-a), where a is the correct angle subtended by two divisions a and b is the measured angle subtended by two divisions. The error was plotted at the micrometer reading when pointing at the midpoint of the two divisions. Figure 8 appears to show the following. An error of approximately 1 second, 2 seconds, or 3 seconds will result from several different combinations of initial and final readings of the micrometer: for example, 0'45" to 3'25", a plus error; 3'25" to 5'25", a minus error; 5'25" to 9'35", a plus error; and 9'35" to 10'45", a minus error. See Figure 9 for a demonstration of these errors. -Q) +­:::J c: .E Micrometer Reading Figure 9: Theodolite E3 Error in Measured Angle The theodolite E3 was pointed on the consecutive points of a scale as indi­cated and the horizontal circle was read as turned in a clockwise direction and then the circle was read as pointed on the scale while returning in a counter­clockwise direction. The correct angle between points was assumed to be the angle computed by using the readings on the 40.0 and 49.2 scale points as giving the correct angle for 9.2 units of the scale. For example, the correct angle sub­tended between 40.0 and 40.7 scale points equals 0.70 divided by 9.2 times the measured angle between 40.0 and 49.2 scale points. Each theodolite reading given on the figure is the average of eight readings. The total angle subtended by 9.2 divisions of the scale is 0°40'13". Figure 9 appears to show the following. a. The micrometer error curve, Figure 8, is approximately correct. See Figure 8 for the suggested readings for this demonstration. b. The maximum systematic error that the theodolite E3 micrometer will make is about 3 seconds. 00 Theodolite Reading E3} ,-o 3' 51 91 101 131 151 191 201 231 251 291 ?JJ 33' 35' 39 4d on Scale }Reading g--<::t" I'­0 ¢ ....: <;!" 0 t<) C\.iC\.i <;!" <;!" O<::t" r<) r<) <;!" <;!" t0 c.o