100 686 THS. eee torres) TRIANGULATION OF SUS Oe ezay AGRICULTURAL COLLEGE CAMPUS E. S, Anderson W. E. Frazier 1920 H Oe aw lO \420° SHESIS | - ko hn doo are “SUPPLEMENTARY MATERIAL IN BACK OF BOOK TRIANGULATION OF THE MICHIGAN AGRICULTURAL COLLEGE CAMPUS A Thesis Submitted to The Faculty of MICHIGAN AGRICULTURAL COLLEGE By E. S. Anderson W. E. Prazier Candidates for the Degree of Bachelor of Science June, 1920 THESIS TABLE OF CONTENTS. Introduction. cccsscccvccseses se Page L Apparatus Used. cc cece cccvcvsccsccesd Instruments...esccceccccccccccsccceced Principles of the Triangulation......5 Base Line Measurements... .cececcsceoe’d Measurement Of Angles... ccccccccecs eee ed ROSULEH ee ccc c ccc ccccceccesrvccsvvcell APPONdGiIX. cc. ccccerccvcvsvesecsecccesle Computation of Quad No. l...eeceeele Computation of Quad No. 2....-++ee13 Computation of Quad Noo 3.-...-e-+e14 Tabulation of ReultS..c.ccccceceeld Description of StationS.-cccccceelG - 17 Rigorous Adjustment of Quad No.1.18 Rigorous Adjustment of Quad No.2.19 Rigorous Adjustment of Quad No.3.20 Map NOe Lecce cevccccveccecccseccccad Map NOs Qeeccccccccccccccccscescete xy Hs | of) We qh peerss ‘ ‘ ‘e . : yy oe ‘ a t re io 4 “ Ld 7 ; I\ ‘ i a9 4 bE YR cy b€ owe | 93771 at. : * . g tg f Y, i £ & ta Yer E 5 % f fof 7y ro! , eo. OL Re a ow. VmeE & ww ‘ 4 h bs ‘S 3 6 ‘ad ae oe ra A’ I” ses wm \ 3h t ; ‘}! 1 ow tev f INTRODUCTION. Geodesy is that branch of science which treats of making extended measurements on the surface of the earth. Primarily the object of this work is to locate controlling points fr ex- tensive surveys. This thesis deals with the extension of the triangulation of the M. A. C. Campus and the connection of this triangula- tion system on the campus to the system which covers the M. A. C. Farm (as shown in map*). All measurements are subject to more or less unknown and unavoidable sources of error. Repeated measurements of the same quantity can not be made to agree precisely. Some method of adjustment was therefore necessary which assigned the most probable values to the unknown quantities in view of all the measurements that had been taken and the conditions which were to be satisfied. Such adjustments were made by the Rigorous Method, which is explained under the head of Triangulation Adjustments and Computation" on a later page. “See blue prints in Appendix. 1. Inlarged triangulation system. Scale 1" = 200'. 2. Sketch showing location of triangulation system. APPARATUS USED. A. Instruments: a- Transit (repeating instrument reading to 30"). b. Theodolite (repeating transit). ec. Direction instrument. de Dumpy level; (used in preliminary survey of base line). B. Flags. C. Stakes, etc. D. Stations. E. Tapes: 200' and standard. (See next page for pictures of Instruments. ) whe l. Flag. 2. Theodolite Repeating Instrument. Lease ae 3. Theodolite Direction 4. Repeating Transit. Instrument. PRINCIPLES OF THE TRIANGULATION. The word "Triangulation" as used in Geodeic surveying includes all those operations required to determine either the relative or the absolute positions of different points on the surface of the earth, and such operations are based on the properties of plane and spherical triangles. (Correction for spherical excess was not necessary in this case.) For the triangulation suitable points, called stations, were selected and marked throughout the area to be covered, the selection of these stations depended upon the character of the landscape and also the object of the survey. The stations thus selected were regarded as forming the vertices of a set of mutually connected triangles, the complete figure being called a triangulation system. At least one side and all the angles in the system were measured, using utmost care. All the remaining sides were obtained by computations of the suc- cessive triangles. The stations forming the triangulation system are called triangulation stations, and are namely: 4, 8, 24, 25, Ag. Bldg., Engin. Bldg., N. Base, and For. A map of the system is shown on Plate II. The system is made up of the quadrilateral 4, 8, 24, and 25, quadrilateral 24, 25, Ag. Bldg., and Engin. Bldg., and quadrilateral Engin. Bldg-, Ag- Bldg-., N. Base and For., making three quadrilater- als in all. The preliminary work of examining the campus for the pro- posed survey was the reconnaissance. As much information as -6- possible was obtained from existing map, and then an examin- ation of the campus helped to locate the relative positions of the probable station points. This part of the work called for the greatest care and judgement, as it practically controlled the accuracy of the survey. Every effort was therefore made to secure the best errangement of the stations. The base line is usually much smaller than the principal lines of the triangulation system, and therefore required a specially favorable location in order that its length might be accurately determined. A gently slop- ing ground, not over four per cent, is suitable for a base line. The line from stations 4 to 8 was used as a base line. In this system of triangulation four of the old stations were used, namely, For., North Base, 4 and 8. To connect up the two systems, stations 24, 25, Engin., and Ag. Bldg-, were used as intermediate stations. For a matter of formality we will call the system on the campus System Noe 1, and the system of triangulation on the farm System No. 2. Two of the stations, namely, Engin. Bldg., and Ag. Bldg. could be seen ffom System No. 2, but not from System No. l, so stations 24 and 25 were placed as intermediates, so that stations Ag. Bldg., and Engin. Bldg., and also 4 and 8 could be seen from stations 24 and 25. Stations 4, 8, 24, and 25 form Quadrilateral No. 1; 24, 25, Engin. Bldg., and Ag. Bldg. form Quadrilateral No. 2; and Ag- Bldg-, Engin. Bldg-, For., and N. Base form Quadrilateral No. 3. The intervisibility of any two stations was finally determined by observations from = Ta each station. Also each station angle was measured with a small compass mounted on a tripod to see if angles were of a reasonable magnitude. The triangulation system required permanent stations so at stations 24 and 25 concrete posts 6" x 6" x 36" with an iron center were placed six inches under the sod. A brass plate was screwed in the roof of the Agricultural Building and was used as station Ag. Bldg. A brass plug was cemented in the roof of the Engineering Building and was used as station Engin. Building. BASE LINE MEASUREMENT. The line between stations 4 and 8 was used as the base linee A preliminary survey was made of the line and it was found that the ground was quite level. Starting at station 8 we computed the elevation so that no part of the tape when used would touch the ground. Four foot 2" x 2" stations were driven firmly in the ground at every fifty feet and a nail placed at right elevations. Every 200' a 2" x 4" was driven down flush with the right elevation and a brass strip was naile ed on top, one of its edges on line between 4 and 8. These stations were made in this way to record the length while meas- uring the base line. Over station 8, it was necessary to build a station which would be directly over the original station but also at the same elevation as that of the rest of the line. This was accomplished by driving a couple of 2" x 4" pieces -8- in on each side of the station, and placing another across the tops of the two vertical ones. A hole was bored in the cross-piece over the station, a strip of brass nailed over one-half of the hole, and then by means of a plumb the sta- tion was plumbed up and a notch made on the edge of the brase directly over the original station. The tape used was a 200' wire tape. Corrections were made for absolute length, sag and temperature. The amount of sag was taken care of by standardizing the tape at the time of measuring and on the same ground. In order to do this a comparator was built, which consisted of a stretch of boards ona level piece of ground and two points, at a permanent and well determined distance of 100' apart, were fixed. Each of these points were raised as was sation 8, and a stake similar to the stakes in the course was set at the midway point. Thus the tape could be compared with the 100' length on the ground (which was-laid off with a stan- dard tape) in the same supported way in which the base lim was measured. The difference was noted so that the correction could afterwards be applied. For measuring the base line the same tape was used. The tape was stretched along the course, the rear end was fastened to a straining stake a few feet back of the rear station. The front end was connected with a spring balance for giving the desired pull of 15 lbs. The strain at this end was also re- sisted by a suitable stake beyond the forward stake. In this way no strain was allowed to come on any of the stakes. The two-by-fours at the 200' marks being set with sufficient care, -~Je the end of the tape came on the strip of brass. The strip of brass had been placed on top of the two-by-four. A scratch had been made on the strip of brass and the distance between this scratch and the end of the tape was measured with a pdir of dividers, note being made of the distance and whether plus or minus. Four measurements were made of the base line and the average taken. For each reading of the tape the temperature was also read. In measuring the base line the temperature varied more or less during the progress of the work but as a rule it has been found satisfactory to apply a correction due to the aver~ age temperature. The length was corrected to a temperature of 62° F. The accuracy possible in the determination of the length of the base line depends upon the precision with which the various constants of the measuring apparatus have been obtain- ed and the precision with which the field work is done. The precision of the measurement was judged by making four measure- ments and then the arithmetic mean was used. MEASUREMENT OF ANGLES. The measurement of angles was done mostly on afternoons and in doudy weather. Due to some disadvantages, and to the location of two of our stations it was impossible to use the larger instruments; however, a high grade transit was used and great care was used in placing flags and setting up of the -10~ instrument and in reading of anglese The angles where possi- ble were checked with the larger instrument. Two theodolites were used in checking the angles; the readings of the direction instrument were not used. These types are used for fine angle work and are considered the best. The readings which were used in tRe computations were made with a surveyor's transit reading to 30". Each angle was measured six times direct and six times inverted. The angles were also measired with B. and L. Theodolite S. T. T. 3. (See next page for Sample of N.tes.) After the angles had been measured a horizontal correction was necessary which was the dividing up of the error, and add- ing or subtracting to each angle the allotted amount, the addi- tion being made when the angles failed to make 360° and the subtraction when over 360° (see sample of notes). The three angles of a triangle, in like manner, should add up to 180°, and the interior angles of a quad should add up to 360. It was necessary, therefore, to satisfy these geometrical conditions. This was done by means of the "Rigorous Method", which is a system worked out and which satisfies all of these conditions. (See any text on Geodeic Surveying.) After the system was satisfactorily adjusted, the distance between the various stations were computed. This consisted in solving each triangle in order, as a plane triangle, by the ordinary sine ratio. In case of the quadrilaterals the two diagonals and the sides adjacent to the known side were computed from the triangles involving the base; the side opposite the base was then computed from both the triangles in which it -l0Oa~ 0°O 00 O09¢ 2°09 6S 69¢_ ¥°SO SE 942 ‘BUT 0} O8bQ°N f*00 O02 99ESBA°N 03 °Zy Ca¥G, FO LT “*3¥ 09 *3uq *I0j 78 w°S §=Se OL2 2° GS 942 O°O0T SE 942 0°O Te 6TS 0O°St BS BSE Of 62 0 o¢ 613 a 9 *Suq 03 uS°LS, 52,942 O497,62,612 O'S 62 6TZ O72 948 «= T esegen 0400 ,00,0 400 400 qd 0 €°0 02 99 8°89 61 99 Of 6S 6S¢ 0 0 0 u 9 0°6@ 0299 0°0 0 g8¢ O°S? 6S BG¢e uO, BSo4E 0 oO 8¢ qd 9 oseq -N GAbS,6T.999 OnS, 6SoLE O°S? 6S LE yO, 6T y0% 61,99 qd T 0% °3y yo 0%00,00,0 O00 400 qd 0 S°es & LT ~3°2S_¥ 4T 00 00 0000 0 Y 9 S°2S 6 AT O°ST 62 ZOT 0°00 00 0 00 62 o¢c6e got a 9 *BY 04 SESS, FoLT O4ST,6S920T O°ST 62 ZOT 400, 6a u0S, FoAT qd T “pug 0400,00,0 : 400 400 qd O *emsoto . 3 . 3. ede . owe “yu aO°d -ON 2 “40g -y aoy |) CtPUY PAdey- = eTZuY UsEH: dq *deA V *t0A *TeL eT, °° 8g WT20eTI0D $ -eSeg pusy wSTty ‘exseg pusy Je7 *“SHLON dO WIdNVS ~lle* occurred. These two values were exactly the same showing ahat no mistex® was evident. Given A, B, a. c a SINE RATIO used in c =a sin (A + B) A sin A solving of triangles. b = a sin B sin A RESULTS . In checking up our results with that of a previous survey we found that we were three-tenths of a foot off on line For. and N. Base. Our result was 1511. and the previous survey result was 1511.871- In order to come as close as possible to the average result of the two surveys we added one-fourth of the difference for Quad. No. 1; the average of the two results was taken for Quad. No. 2, and three-fourths of the aifference was added for Quad. No. 3, using lines 4 to 8 the same as measured and N. Base to For. the same as previous re- sult. ~12- APPENDIX. Computations. QUAD. NO. i, 2.7443681 log sin 555.096 (4 to 8) 9.7161248 4604 9.9996921 "6-4608008 * 208 (1/4 diff.) = 2.4608216 9.7949653 (line 4 to 24 = 288'.949) 9 .6437957 “S-611970Z log of 24 to 25. 2.74435681 9.5778567 | 9.9709621 ° + 208 = log of 8 to 25 = 2.3512835 9.9243176 8 to 25 = 224!534 9.6656105 26119698 log of 24 to 25. 26119704 one way 2-6119698 other way 2-6119701 average log 417 average difference 2.61c0115 log 24 to 25 24 to 25 = 409.279 ft. -13- QUAD. NO. 2. 2-6119701 9 .6823680 I2.294338i © 9.3950854 : 28992527 + 417 = log 25 to Ag. = 2.8992944 9.8475815 _ _— dist.25 to Ag. = 793.039 27468540 9.9744135 log Ag. to Eng. 417 = 2.7724623 + 2-6119701 9.9997075 °6 7 9.6687791 ~B-9428985 + 417 = log 24 to Eng. = 2.9429402 9.7159534 dist.24 to Eng. = 876.880 12.6588519 9.8864315 “eeW724204 checks log Ag. to Eng. + 417 = 2.7724622 Dist. = 592.192 Ag. to Eng. -14- QUAD. No. 3. 27724205 9.9729875 «7454080 9.5028001 5-2426079 + 625 (3/4 of Diff.) = 3.2426704 9 .8986549 (Ag. to N. Base Dist.) = 1748.559 9.9618499 ° 4 2°7724203 9.7996196 15.5720401 9.4679719 31050682 + 625 = log Eng. to For. = 2.1050707 9.9070570 Distance = 1273.715 T3.O11125e 9.8317125 ~S-L764IZ7 Use average =3.1794128 854 Whole diff. Dist. = 1511.871 The same as in previous survey. a —— II 948Td UO UMOUS BB T U0TIBYS 38 UsHBy ST WUTOd 0°0 BI9*PSLS- CST°SOBT+ *es¥arN SIBPELT°S TL8* TTST ss°6s ch cee) 460 oneaeH ZIS*s6ez= LES*66S+ %40g POLISHS°S 6SS*SPAT ZT°LP TO GPS = OSBATN =TER*SG9OT= GOT°ATHT+ °3V LOLOGOT*S STL* CL2T 60°6S BT OZ * 104 SSIVSLLES 261° 26S ZB°ST BS 9Se “BY yB6°COST= TIS*Trs+ cuysug 7PBZ668 °S 620° C6L eo*oy TT Tae “BY “get*ose= 30s°0s6+ °S2 SOVESHB*S 088°9L8 LL°Se BO *Bee eupsug STTOSTI*S 6L2°60P OF*SS ZO BLe SZ PSP°LEP@ CGO°STS+ °F2 GeeStse*s POS ° FSS Os°9T Le ST GZ ZS0°62S" OOS*SLE+ °S 9TZB09F°S 6b6°8ez O8°SS Sb Swe ve TS9SHPL°S 960°SSS 0°0r 9S TEZ 8 SIT°HTI= STH°OSh+ °F PIGSBHI"S GCo°StP uu SS, 690982 y Q 0 “Tl 895 BOT e0US48sTC UPUTZY 04 WZYy A a x QUutTOd SIINSaY dO NOLLTInavs -~16- DESCRIPTION OF STATIONS. Sra. 25 St7 2F Center of iron rod. 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