BOARD OF SURVEYS AND MAPS SPECIFICATIONS FOR HORIZONTAL AND VERTICAL CONTROL
The following “Specifications for Horizontal and Vertical Control” were agreed upon by the various Federal surveying and mapping agencies through the approval of the Board of Surveys and Maps of the Federal Government on May 9, 1933. Although this Board was abolished in 1942 by Executive Order (No. 9094), the basic classification of triangulation is still applicable. However, the entire specifications are reproduced here for general information only and are not to be considered as modifications of the instructions contained in the preceding chapters of this manual.
Control surveys to determine the latitude, longitude, and elevation of marked points distributed throughout an area to be mapped are necessary as a basis for all topographic, hydrographic, and cadastral surveys that cover regions of considerable extent. The control surveys should also be so designed as to give the maximum benefit to those conducting general engineering operations on a large scale.
TRIANGULATION
First-order Triangulation.-Triangulation of the first order should be executed in belts about 120 miles apart over the region to be surveyed, except under such conditions as may require the substitution of first-order traverse. Triangle closures should rarely exceed 3 seconds, and the average closing error should be not greatly in excess of 1 second. The discrepancy between the measured length of a base line and its length as computed through the triangulation from the next preceding base, after the side and angle equations have been satisfied, should not exceed one part in 25,000. Laplace stations should be selected at intervals of from 6 to 10 figures along the scheme. The accuracy of azimuths to be observed at such stations should be limited to a probable error rarely exceeding 0.3 second.
Second-order Triangulation.-Triangulation of the second order (or traverse of corresponding accuracy) should be used to subdivide the areas between belts of first-order control so that no point in such an area will be farther than about 10 to 15 miles from some station of either first-order or second-order accuracy. Triangles of second-order triangulation should close with an average error not greater than 3 seconds and a maximum error seldom exceeding 5 seconds, provided that, where single triangles are necessary, the closures shall not exceed 4 seconds. Closures in length on lines of the first-order net, on second-order lines previously adjusted, or on base lines, should not exceed one part in 10,000, after the side and angle equations have been satisfied. Arcs of second-order triangulation will, in general, not exceed 150 miles in length. One Laplace station should be selected near the center of a second-order arc of triangulation which is between 120 and 150 miles in length. Should an arc be longer than that, a Laplace station should be located at intervals of from 8 to 15 figures. The accuracy of the azimuth observations at such stations should be limited to a probable error rarely exceeding 0.5 second.
Third-order Triangulation.-Triangulation of the third order is ordinarily used for the immediate control of topographic and hydrographic surveys. Extensions of third-order triangulation or of third-order traverse, from triangulation or traverse of a higher order, should be made so as to control the entire area to be surveyed. Triangles of third-order triangulation should close with a maximum error of 10 seconds and an average error seldom exceeding 5 seconds. Closures in length on lines of first-order or second-order triangulation, on lines of third-order triangulation previously adjusted, or on base lines, should not exceed one part in 5,000, after the side and angle equations have been satisfied.
Fourth-order Triangulation.-Triangulation (or traverse) of the fourth order is used to connect the control of higher grades with the detailed surveying and mapping operations in a region. It should start from control of a higher grade and should never be carried for more than a few figures without being again connected to a control station of a higher accuracy than fourth. It may be performed with a plane table, transit, or sextant. The sole requirement of accuracy in fourth-order triangulation is that positions of points must be located with error too small to be appreciable on the resulting map.
Note: The specifications listed above were in effect in 1933. They have been modified since then so that now the recommended practice is to establish at least one control station in every 7-1/2-minute quadrangle.
BASES
Accuracy.-Bases for the control of the lengths of first-order triangulation should be measured with an accuracy represented by a probable error of not more than one part in 1,000,000 and an estimated actual error of not more than one part in 250,000 or 300,000. On second-order bases the probable error should not be greater than one part in 500,000 and the estimated actual error not greater than one part in 150,000. On third-order bases the probable error and estimated actual error should not be greater than one part in 200,000 and one part in 75,000 respectively. Any methods and instruments may be used which will secure the prescribed results. (For a statement describing what is meant by “probable error” and “estimated actual error” see appendix at the end of these specifications.)
Frequency.-The strength of figures in the triangulation between bases should be considered in determining the distance between bases, according to the method prescribed in U. S. Coast and Geodetic Survey Special Publications Nos.93 and 120.The allowable limit for the summation of R1 of the individual figures between bases for first-order triangulation should not exceed 80; for the second-order 100; and for the third-order 125, though, where a base site cannot readily be found these limits may be exceeded by 25 percent. The limit prescribed for the agreement between the measured length of a base and its length as computed from the preceding base is the controlling factor, and it may necessitate at times measuring an additional base or strengthening the triangulation between bases.
TRAVERSE
Traverse will be used, in general, only where the cost of triangulation would be excessive because of low relief and heavy timber. Traverse has the advantage of leaving marked points in locations where they are easily accessible, but it is inferior to triangulation in its checks against blunders, and it does not give control over so large an area as a belt of triangulation. Ordinarily, therefore, triangulation should be preferred to traverse on control of the two higher grades if its estimated cost does not exceed the estimated cost of traverse by more than 50 percent. For the third-order control the choice between triangulation and traverse will be determined by the local conditions, and the comparison of cost should be made on an equal basis.
Traverse of all grades of accuracy should be run in loops or connected at each end to triangulation or to traverse of the same grade as that which is being executed or of a higher grade. No traverse lines that are not thus connected or checked should be used for map control. On traverse of the two higher orders marked stations should be established at intervals of not more than 5 miles and with an average of 3 miles or less. Where a marked station occurs on a traverse, one of the adjacent stations should also be marked, in order that the distance and azimuth may be made permanently available, as well as the geographic position.
Accuracy.-First-, second-, third, and fourth-order traverse should be executed with an accuracy comparable with that of triangulation of the corresponding grades. Ordinarily, the traverse methods will give a greater accuracy for the distance between stations than the triangulation method, but it is relatively weaker in the azimuth obtained. The error of closure in position of a first-order traverse line when run in a closed loop or when starting from adjusted triangulation and ending upon adjusted triangulation should not exceed one part in 25,000 of the length of the traverse line after a preliminary adjustment has been made of the azimuth discrepancies between the stations where astronomical azimuths have been observed. This error in position may be somewhat exceeded when the length of the traverse line is small compared to the length of the arc of triangulation between the points where the traverse starts and where it ends, as there may be an appreciable error remaining in the triangulation after adjustment. On first-order traverse an astronomical azimuth should be observed at intervals of 10 to 15 stations, on second-order traverse at intervals of 25 to 40 stations, and on third-order traverse at intervals of 50 to l00 stations. The astronomical azimuth should be determined with an accuracy represented by a probable error of 0.5 second for first-order traverse, 5.0 seconds for second-order traverse, and 30.0 seconds for third-order traverse.
MARKING OF TRIANGULATION AND TRAVERSE STATIONS
Stations of either first-, second-, or third-order accuracy on triangulation and traverse should be marked by tablets of some noncorrodible metal set firmly in posts of concrete, or in large boulders or outcropping bedrock, but where a station is on a building suitable marks of a different character may be used. The tablets may be set in place by means of cement, sulphur, or lead. The concrete posts should be not less than 8 inches in diameter and should extend 30 inches below the surface of the ground. They should have the shape of a truncated prism or cone, in order that the lower end may be larger than the upper and thus better able to resist the lifting effect of frost action. For the same reason the post should be smoothly molded for the upper 10 or 12 inches of the part beneath the surface. Particular care should be taken to insure that the materials used in making the concrete are clean and well mixed. The top 12 or 15 inches of the post should be at least equivalent in strength to a 1-2-3 mixture of cement, sand, and stone; the base may be made of a somewhat leaner mixture. Where a boulder is used it should be at least as large as the concrete post prescribed and should extend to a similar depth beneath the surface. Where the tablet is to be set in bedrock care should be exercised in selecting rock that is of suitable durability, and also to make sure that what is apparently bedrock is not a small detached mass of rock.
Special Marks.-Under certain conditions special marks may be used. Where no large boulder or bedrock is available at a station and where by its location it would be unduly expensive to construct a concrete mark, a metal pipe of suitable size and of noncorrodible material may be used. The base of this pipe should be so shaped as to resist extraction of the pipe and should preferably be set in concrete. In swamps a long metal pipe, set inside a drain tile filled with hydraulic cement, may be used. Where a station mark must be set on land subject to cultivation it is better to have the top of the post entirely below the depth which can be reached by a plow-that is, about 12 inches below the surface. Where a mark of this type is set it is necessary that measurements to the center of the roadways, section lines, etc., be made in sufficient number to enable one seeking to recover the mark in the future to determine its location within a few feet. The mark itself can then be found by digging or by prodding with an iron rod.
Subsurface Marks.-Where a concrete post is used a subsurface mark should be set if possible. This mark should preferably be made of concrete, not less than 6 inches thick and 10 inches in diameter, with the station point marked by a metal tablet, copper bolt, or other durable substance. The subsurface mark should be 4 or 5 inches below the base of the concrete post, and extreme care must be taken that the subsurface mark is directly underneath the surface mark.
Reference Marks.-At least one and preferably two reference marks should be set at each station. They should be metal tablets set in concrete posts, boulders, or bedrock. The metal tablets used for the reference marks should have a different inscription upon them than the station tablets and preferably should bear an arrow pointing toward the station. Where more than one reference mark is used at a station, they should be stamped and numbered serially, clockwise as viewed from the station. Particular care should be taken in selecting sites for these reference marks where they will not be subject to disturbance. Fence lines or section lines are suitable sites.
Azimuth Marks.-At each first- or second-order triangulation station at least one azimuth mark should be established in addition to the reference marks. The mark should be, if practicable, at least 1/4 mile distant and visible from the ground at the triangulation station. It will not be required to measure the distance between the azimuth and station marks. However, the description of the azimuth mark should be sufficiently definite to enable one to recover it with ease. The directions to azimuth marks should be obtained from at least two positions of the circle. Where possible, additional azimuths of objects, such as church spires, cupolas, etc., visible from the ground at the station should be determined.
Third- and fourth-order triangulation usually consists of short lines and is extended into areas so that many of the stations are intervisible from the ground, thus making available azimuths directly between main-scheme stations and obviating the necessity for azimuth marks. Where main-scheme stations are not intervisible azimuth marks should be established. The accuracy of the directions to azimuth marks should be such that azimuths may be obtained with a probable error not in excess of 15 seconds.
Preservation of Station Marks.-The destruction of station marks that are in the way of public improvements is frequently unavoidable. By far the greater number of marks removed, however, are destroyed by malicious or thoughtless persons and this destruction represents a large economic loss to the Government. The restoration of the marks in even a small region will sometimes cost several thousand dollars, not counting the inconvenience caused to local engineers and surveyors by the lack of stations upon which to base their surveys.
There is a Federal law imposing a fine or imprisonment for the destruction of a Federal station mark, but it has never been enforced. Many States have laws directed to the same end, but even under State laws it is very difficult to secure a conviction. Such laws, however, have a notable restraining effect. It is very desirable that all the States should have laws imposing penalties for destruction of marks established by Federal surveys, and also prescribing the terms upon which Federal surveyors shall have the right of entry upon private property, together with methods for reimbursing the owner of the land for any damage sustained by such entry.
The best way of protecting station and bench marks seems to be by educating the people to understand the value of these marks. To that end as great publicity as possible should be obtained when work is being conducted in a region, to make known to the inhabitants the purpose and value of the surveys and the function of the marks in perpetuating the surveys. This publicity can be obtained by the insertion of articles in local papers, by the distribution of pamphlets illustrative of the purpose of the work, and by conversation with influential people or with owners of the land upon which the marks are located.
Selection of Names.-The name of a triangulation or traverse station should be stamped upon the metal tablet, preferably before it is set into the concrete or stone. The name should not be duplicated within the confines of a county. Names for triangulation and traverse stations should have a geographic significance wherever possible. Care should be taken by the chief of party to ascertain the name which is most prevalent for a particular geographic feature, for frequently a mountain or stream will have different names in the same region. In particular, officials of the Forest Service should be consulted regarding the names of topographic features within national or State forests or in regions adjacent thereto.
Program for Control Surveys.-First- and second-order triangulation and traverse should be completed in a region before the third-order control is developed. It is also an advantage, though not always practicable, to have the third-order control completed and computed before topographic surveys are begun in a region, in order that a better distribution of the topographic working parties may be made.
Description of Station.-A clear, concise, and complete description of each triangulation and traverse station established should be made out and filed in convenient form in the district or central office of the organization making the survey. Where search is made for a triangulation or traverse station established in a previous year, a note to that effect must be entered on an appropriate form. If the station is found, the recovery note should state the condition in which the mark was found and should give any modifications or additions to the description which would make the station more easily found in the future. If the station is not found, the note should indicate the thoroughness of the search made and give recommendations as to whether or not the station should be marked “lost” in the records.
Report of Recovery of any Federal Survey Mark.-Member organizations of the Board of Surveys and Maps should instruct their field officers to report upon the condition of station and bench marks visited by them which were established by another member organization. If a properly equipped field party of a member organization finds in poor condition a Federal triangulation or traverse station mark of a third or higher order of accuracy and if its proper location can be determined with certainty and accuracy, either by a recovered underground mark or by measurements from two or more reference marks, the party should re-mark the station if practicable. If the tablet marrking the original station is recovered, it should be reset. If an underground mark exists due care must be exercised to insure that the new surface mark is exactly centered over the subsurface mark.
A leveling bench mark should not be replaced or repaired. When a bench mark is found in poor condition by a field party from some organization represented on the Board of Surveys and Maps, a new bench mark may be established nearby, and its elevation determined by leveling from the original mark. A copy of the field notes for obtaining the elevation of the new mark should accompany the description of the new mark.
The report of the recovery of the station or bench mark, or the revised description, should be sent through official channels to the organization that established the original mark.
Elevation of Station.-The elevation above mean sea level datum should be determined for each triangulation and traverse station and for permanently marked intersection stations. The elevation of triangulation stations need be determined with only such accuracy as will enable the reduction of the directions and distances to mean sea level. It is very desirable that the elevation of first- and second-order traverse stations be determined by spirit leveling.
High Towers.-The use of high towers for instrument supports should be avoided wherever possible, both because of their expense and because usually the station cannot be used for future work without rebuilding the tower. Where local conditions make it necessary to use a tower, an azimuth mark should invariably be established in order that traverse lines may be joined to the station without the delay and expense of rebuilding the tower. If a permanent natural object is not available, an azimuth mark should be established not less than 500 feet distant from the station, in a location where it will be visible from the ground at the station. Whether a natural or artificial mark is used for the azimuth mark, it should be fully described, and either its measured or estimated distance from the station mark should be given.
Publication of Results.-Resulting positions, descriptions, and elevations of stations, together with their azimuths and distances to other stations, should be published as soon as possible and in compact form. All positions for horizonal-control stations must be given by latitude and longitude, referred to the North American datum of 1927, if possible.
Each publication containing leveling or triangulation data should include two small index maps, one of a section of the country showing the areas covered by the data in the publication, and another of the whole country showing the areas covered by each previously published report of a similar nature with reference to the numbers or names of the reports.
Density of Distribution.-The distance between belts of first- and second-order triangulation and traverse has already been specified. Third-order triangulation and traverse, which have for their function the control of detailed surveys in a region, should provide a density of distribution of stations sufficient to enable the topographer or the hydrographer to obtain the required accuracy without undue delay or expense. The nature of the terrain will determine the density of distribution of stations. No 15-minute quadrangle or area of similar extent should, however, have less than three horizontal-control stations of the third or higher order of accuracy. As many more should be established as will fulfill the requirements of the topographer. If traverse is being carried on instead of triangulation, marked traverse stations should be left at intervals averaging not more than 3 miles. The requirements of the topographer should be borne in mind on traverse as well as on triangulation.
In executing traverse or triangulation of the three higher orders particular effort should be made to locate additional stations by the intersection method. These additional stations may be either existing objects, such as water towers, church spires, or specially marked points.
The stations that are to be available for future recovery and use should be selected with two things in view-first, their use by the topographer or engineer working from the ground, and, second, their use on revision surveys by the aerial photo-topographic method. The second purpose requires as many stations as possible that could be identified from the air without special marking.
LEVELING
CLASSIFICATION:
First-order Leveling.-First-order leveling should be used in developing the main level net of the United States. The lines should be so placed that eventually no point in the country will be more than about 50 miles from a bench mark established by leveling of this order. All the lines should be divided into sections 1 to 2 kilometers in length, and each section should be run forward and backward, the two runnings of a section not to differ more than 4 mm.√K or 0.017 foot √M, where K is the length of the section in kilometers and M its length in miles.
Second-order Leveling.-Second-order leveling should be used in subdividing loops of first-order leveling until no point within the area is much more than 12-1/2 miles from a first- or second-order bench mark. Second-order leveling will include lines run by first-order methods, but in only one direction, between bench marks previously established by first-order leveling and all double lines of leveling whose sections, run in a backward and forward direction, check within the limits of 8.4 mm.√K or 0.035 foot √M where K is the length of the section in kilometers and M its length in miles.
Third-order Leveling.-Third-order leveling may be used in subdividing loops of first- or second-order leveling, where additional control may be required. Third-order lines should not be extended more than 30 miles from lines of the first or second order; they may be single-run lines but must always be loops or circuits closed upon lines of equal or higher order. Closing checks are not to exceed 12 mm. √(kilometers in circuit), or 0.05 foot √(miles in circuit).
Leveling of Lower Order.-Leveling that allows closure checks greater than the limit stated for third-order work, such as trigonometric leveling, barometric leveling, or “flying” levels, shall be considered as belonging to the lower order of work. No bench marks established by leveling that is less accurate than that of the third order, as above described, shall be marked by standard bench-mark tablets, except that in mountainous regions inaccessible to ordinary spirit-level lines standard marks may be used on mountain summits to mark elevations determined by trigonometric leveling; such marks should be stamped in a distinguishing manner. Elevations inferior to the third order in accuracy shall not be published in such a way as to be confused with standard work of the third or higher orders.
Bench Marks.-All first-, second-, and third-order level lines should be adequately marked by monuments, at average intervals of not more than 2 miles, preferably by metal tablets set into concrete posts, substantial buildings, outcropping rock, or large boulders. At places from which it is likely that future leveling lines will be extended, at least three bench marks should be established within a radius of about a half a mile, but far enough apart not to be affected by the same disturbing causes. In addition, supplementary marks on trees, bridge seats, and similar places should be left for each mile of line. In each city or town through which a line of levels passes, at least two permanent bench marks should be established, the number of bench marks for cities of large population being somewhat in proportion to the size of the city.
New leveling should be tied to bench marks of previous leveling by Government and other organizations wherever practicable, to furnish checks and provide means of correlating results.
An effort should be made to have level lines run over the routes where traverse stations have been established, in order that the elevations of those stations may be determined.
All metal bench marks should, at the time they are established, have stamped upon them some mark which will positively identify them.
A complete description should be made of each old bench mark visited.
Elevations.-All elevations should be based on mean sea level datum.
Elevations of all bench marks required by existing law (20 Stat. L., 435) in areas to be topographically mapped should be determined by leveling of the first, second, or third order or, in mountainous country, by trigonometric leveling.
Elevations above mean sea level should be stamped on each bench mark, but not until the office computations and adjustments have been made.
Publication of results.-The adjusted elevations and the descriptions of bench marks of first-, second-, and third-order leveling should be published as promptly as possible. The elevation in feet and, if desired, in meters also and the description of each bench mark should be printed on the same page.
Each publication containing leveling data should include two small index maps, one of a section of the country showing showing the area covered by the data in the publication and another of the whole country showing the areas covered by each previously published report of a similar nature, with references to the numbers or names of the reports.
The publication of the results of long spur lines of first-order leveling should await the extension of the spur line to previously established bench marks, in order to insure the results against any gross errors in the work.
CLASSIFICATION OF CONTROL
The foregoing data concerning control classification may be tabulated as follows:
| First Order | Second Order | Third Order | Fourth Order | |
|---|---|---|---|---|
| Triangulation | Average triangle closure 1″, check on base 1/25,000. | Average triangle closure 3″, check on base 1/10,000. | Average triangle closure 5″, check on base 1/5,000. | Plane table or transit. |
| Traverse | Position check 1/25,000. | Position check 1/10,000. | Position check 1/5,000. | Stadia, tape, or wheel. |
| Leveling | Error of closure of section 0.017 ft. √miles or 4 mm.√kilometers. | Error of closure of section 0.035 ft. √miles or 8.4 mm.√kilometers. | Error of closure of circuit 0.05 ft. √miles or 12 mm.√kilometers. | Flying wye levels, vertical angles. |
FIELD COMPUTATIONS
In order to insure the requisite degree of accuracy for each grade of work and to avoid the necessity of going over lines a second time to correct excessive errors, certain field computations will be necessary, but these are invariably to be considered preliminary only. On triangulation, eccentric stations should be reduced to center, spherical excesses computed, and triangle closures tested as soon as possible after the angles are measured; and distances should be computed to see that the lengths check properly through the triangulation figures.
Sufficient computations should be made for base-line measures to show whether the forward and backward measures check within proper limits, and the length of the base should be reduced to sea level from elevations determined in the field. In this way it can be learned whether the length as carried by the triangulation checks with the measured lengths within the limits required for the particular grade of work.
A progress sketch should invariably be made by the field party, whether engaged on triangulation, traverse, or leveling.
Least-square adjustments should never be made in the field.
In first-order and second-order traverse it is necessary to carry the computation far enough to check the azimuths carried by the traverse with the observed azimuths. If this is not done in the field, the office computation should closely follow the field work.
The field computations for leveling should be sufficiently complete to insure the requisite degree of accuracy for each grade of work. For each bench mark established or touched upon, a complete description should be written and the elevation computed on mean sea level datum.
RECOMMENDATIONS
Each organization to which may be delegated the task of carrying forward control of any of the classes designated should prepare detailed instructions for its own work, which, if approved by the Board of Surveys and Maps, should be published for general use. Each set of instructions should be complete in itself and should not refer to other sources for additional instructions. Fxamples of records and computations should be given for all operations.
Books of instruction for all classes of work should be furnished gratis to every technical school, college, and university that teaches engineering and should he supplied on request without charge to any interested engineer, in order to make the methods of work known to as many as possible and thus to aid in their adoption by States, counties, cities, corporations, and individuals. This will be of benefit to all concerned.
Arrangements should be made whereby any organization or individual who may wish to do control work of the first, second, or third order will be supplied by the Superintendent of Documents with field notebooks and record and computation forms at cost.
So far as practicable, the forms for records and computations used by Government surveying organizations in control surveys should be standardized.
It is just as essential to have the work of outside organizations and individuals standardized as it is to have control work by the Government bureaus standardized. The outside organizations and individuals have given very little attention to control surveys, and whatever standard specifications for control are adopted by the Government will undoubtedly be adopted to a large extent by others.
APPENDIX
To Specifications for Horizontal and Vertical Control Defining Probable and Actual Errors
Approved April 13, 1926.
The “probable error” of the measurement of a physical quantity is obtained by mathematical formulas applied to the differences between the two or more measured values of a quantity and their mean value. The probable error is a measure of the accidental errors only – that is, of those small errors which have no marked tendency to be predominantly either plus or minus. The probable error is simply a measure of the closeness of agreement among the several values of a quantity obtained by successive measurements. It will give no indication of the presence of systematic errors – for example, if a steel tape graduated at 30° C. is used at O° C. and no temperature correction is applied, the measurements may agree within a very small limit and give a very small probable error, yet the result would be in error about 0.4 inch in every 100 feet or about one part in 3,000. Neither will the probable error give any indication of blunders, for a tape length dropped from each measure will not affect adversely the probable error of the measurement of a base.
The probable error of a measured base is found in the following manner. The base is measured in sections of about 1,000 meters each, and the probable error of each section is first obtained from the formula 0.6745√(Σv2/(n(n-1))) where v is a residual-that is, it is the difference between each measured length and the mean of all the measured lengths of a section-and n is the number of measures of the section. The Greek letter Σ indicates that the sum of the squares of the residuals (v2) is to be taken. If, as is usually the case, there are only two measures of a section, then the probable error of the section is 0.6745 times one-half the difference of the measured lengths of the section. The probable error of the entire base is expressed by:
p.e. = ±√(h12 + h22 + … + hn2), where h1, h2, …hn are the probable errors of the separate sections.
This and other formula relating to the several classes of errors may be found in many textbooks on geodesy and least squares.
The “actual error” is the difference between the true value and the measured value of a physical quantity. It is the sum of all the systematic and accidental errors which have not been eliminated from the final adopted measured value. As the absolute value cannot ever be known, the actual error cannot be exactly determined, but its maximum value can always be estimated. The accuracy of the estimation depends directly and entirely upon knowledge of the maximum uncorrected effect of each source of error.
To illustrate again by the measurement of a base, suppose that the error in marking and the error in correcting for temperature of the tape are the only ones affecting the measurement. The error in marking the ends of a tape is partly systematic and partly accidental; the systematic error will be eliminated by taking the mean of an equal number of forward or backward measures if the person marking remains always on the same side of the base line, and experiments show that with proper methods the accidental error in marking a single tape end is about 0.1 millimeter, or 1 part in 500,000. For a kilometer section of 20 tape lengths, the probable error from marking errors alone would be 1 part in √20 X 500,000 or 1 part in 2,235,000.
In correcting for temperature there are three principal component sources of error-namely, the error in the calibration of the thermometer, error in reading the thermometer, and the undetermined difference between the true thermometric reading and the mean temperature of the tape. Suppose standardizations and tests show that the probable error of calibration is half a degree centigrade and that the probable error of reading is of the same magnitude. Also, that the average difference between the true thermometric readings and the mean temperature of the tape under the conditions of measurement would not exceed 2°C. but that this difference is always of one sign. The probable divergence in temperature between the tape and the thermometer readings would therefore be 2°±√((0.5)2 + (0.5)2) and would certainly not exceed 3°. If the tape to be used were made of invar, with a coefficient of expansion of 1 part in 1,000,000 per degree centigrade, the maximum error to be expected would be 3 parts in 1,000,000, or 1 part in 333,000.
After the error to be expected from each source is evaluated an estimate can be made of the “total actual error,” which is one of the criteria for base measures.