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observation after the triangulation has been extended a distance of 150 to 250 miles. Therefore a system of primary triangulation should be checked by astronomic observation at intervals not greater than this.

Primary traverse is far more liable to errors than is primary triangulation, because of the greater number of courses sighted and the consequent opportunity for the accumulation of error both in angular and in distance measPrimary traverse (Chap. XXIII) must therefore be more frequently checked by astronomic determination, and such checks should not exceed 100 miles apart.

urement.

Whereas primary triangulation and primary traverse may be executed with various degrees of accuracy, according to the distance to which a system of such control is to be propagated and according to its objects, astronomic determinations should be of the highest order of accuracy. Only the most refined instruments and methods known to science for use in the field and in permanent observatory work give results of sufficiently high quality to fulfill their purposes.

202. Base Measurement. The selection of a site for a base line is the first step towards the making of a trigonometric survey, and on its proper selection depends much of the quality of the subsequent work of triangulation.

1. The site should be reasonably level;

2. It should afford room for a base from 4 to 8 miles in length;

3. Its ends must be intervisible and so situated as to permit of the expansion of a system of primary triangulation which will form the best-conditioned figures.

The degree of accuracy with which the base measurement is to be made depends upon the uses to which the resulting triangulation is to be put.

1. If intended for geodetic purposes, the measurement must be made with the greatest attainable precision.

2. If intended only as a base for the expansion of triangu

lation over a comparatively limited area and for the making of a topographic map, this measurement should be made only with such care as will attain an accuracy such that its errors will not affect the map, although multiplied in the resulting triangulation as many times as there are stations.

3. If intended only as a base for a large-scale topographic map of but a few square miles, it will be unnecessary to determine its geodetic coordinates, as the resulting map may depend upon a plane survey.

The early method of measuring base lines consisted in the employment of wooden rods, varnished and tipped with metal, which were supported upon trestles and between the ends of which contacts were made with great care. The advantage of wooden rods consisted in the fact that their length is but slightly affected by temperature, and as they were thoroughly varnished they were only slightly affected by moisture. Later a more accurate method of base measuring was adopted, consisting in the employment of various forms of compensated rods, as the Contact-Slide Apparatus (Art. 210) of the U. S. Coast Survey and the Repsold primary base bars of the U. S. Lake Survey (Art. 212). More recently the use of steel tapes (Art. 204) has become popular, as the accuracy attainable with these has become better appreciated. The latest approved base bar apparatus is the Eimbeck duplex-bars of the U. S. Coast Survey. Finally the iced bar (Art. 211) is the highest development of base-measuring apparatus adopted by the same Survey.

203. Accuracy of Base Measurement.-The chief sources of error in base measurement, by whatsoever means made, are due to

1. Changes of temperature;

2. Difficulties of making contact; and

3. Variations of the bars or tape from the standards. The refinements of measurement consist especially in

a. Standardizing the measuring apparatus or its comparison with a standard of length.

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b. Determination of temperature or its neutralization by the use of compensating bars; and

c. Means adopted for reducing the number of contacts to the fewest possible, and of making these with the greatest degree of precision.

The inherent difficulties of measurement with bars of any kind are:

1. Necessity of measuring short bases because of the number of times which the bars must be moved.

2. Their use is expensive, requiring a considerable number of men; and

3. The measurement proceeds slowly, often occupying from a month to six weeks.

The advantages of measurements made by a steel tape are: I. Great reduction in the number of contacts, as the tapes are about three hundred feet long as compared with bars of about twelve feet;

2. Comparatively small cost because of the few persons required;

3. Shortness of the time employed, an hour to a mile being an ordinary record in actual measurement; and

4. Errors in trigonometric expansion may be reduced by increasing the length of the base from 5 miles, the average length of a bar-measured base, to 8 miles, not an uncommon length for tape-measured bases.

Prof. T. C. Mendenhall, in reviewing the qualities of the various base apparatus, stated that "The use of an iced bar applied to the measurement of considerable distances is unquestionably the method of highest precision, and its cost is not believed to be greater than that of other primary methods in use in Europe, but it will not be found necessary to resort to it in ordinary practice except for purposes of standardization.' He then goes on to state that "the metallic tape is capable of giving a result of great accuracy in the hands of experts,

and that this is evidently the best device for rapid base measurement when no great precision is aimed at."

It seems that the steel tape is capable of giving a precision indicated by a probable error of 2000 part of a measured line, while Tooooo appears to be easily and cheaply attainable with long tapes after they are standardized. This is amply sufficient for the present purposes of geodesy, and the sole obstacle in the way of much higher precision, should it be deemed essential, appears to be only the difficulty of measuring the temperature of the tape.

Bases are not measured solely for the accuracy attainable within themselves, but to attain the greatest accuracy which, when expanded through a scheme of triangulation, will not introduce into it errors of appreciable amount. Therefore it

is scarcely economic to strive at an accuracy which will be greatly in excess of that attainable in the succeeding triangulation (Art. 240). Precision of measurement represented by probable errors of 3000 to 500000 is sufficient for all practical requirements of good primary triangulation not required in the solution of geodetic problems.

204. Base Measurement with Steel Tapes. - Steel tapes offer a means of measuring base lines which is superior to that obtained by measuring bars because (Art. 203) it combines the advantages of great length and simplicity of manipulation, with the precision of the shorter laboratory standards, providing only that means be perfected for eliminating the errors of temperature and of sag in the tape. Base lines can be so conveniently and rapidly measured with long steel tapes as to permit of their being made of greater length than has been the practice with lines measured by bars, and as a result still greater errors may be introduced in tape-measured bases and yet not affect the ultimate expansion any more than will the errors in the latter, because of the greater length of the base. Primary base lines have been measured by means of long steel tapes within recent years by the U. S.

Geological Survey and the U. S. Coast and Geodetic Survey, as well as by the Missouri and the Mississippi River Commissions, and in each case with satisfactory results.

As showing the quality of this work, some measures made by Prof. R. S. Woodward of the St. Albans base, previously measured with base bars by the U. S. Coast and Geodetic Survey, showed a range of several measurements of the whole base of 24.1 millimeters or 10000 of the whole, the greatest divergence from the adopted mean being 13 millimeters or 2000. The probable error of one measure of a kilometer made at this time was 1.88 mm. It is believed that the probable error of a single tape length did not exceed ± 0.05 mm., while the probable error of the measurement of the whole base was 000000

205. Steel Tapes.-The tapes used for this work are of steel, either 300 feet or 100 meters in length. The meter tapes used by the Coast Survey are 101. 01 meters in length, 6.34 millimeters by 0.47 millimeters in cross-section, and weigh 22.3 grams per meter of length. They are subdivided into 20-meter spaces by graduations ruled on the surface of the tape, and their ends terminate in loops obtained either by turning back and annealing the tape on itself or by fastening them into brass handles. When not in use the tapes are rolled on reels for easy transportation.

The steel tapes used by the Geological Survey are similar to those used by the Coast Survey, excepting in their length, which is a little over 300 feet. They are graduated for 300 feet and are subdivided every 10 feet, the last 5 of which at either end is subdivided to feet and tenths. The various instrument-makers now carry such tapes in stock, wound on hand-reels. All tapes must be standardized before and after use by comparison with laboratory standards, and, if possible, thereafter frequently in the field by means of an iced-bar apparatus. (Art. 211.)

206. Tape-stretchers.-In measuring with steel tapes a