E, D. Preston — Measurement of th< Peruvian Arc 13 



sufficient data are not available. Take one-eighth of a foot, 

 which i> one-half the difference bet wren the results, as the 



probable error of one measure of the base line. This is com- 

 posed of errors in the Lengths of the rods and errors of meas- 

 ures properly so called. The error iii the entire base, as 

 depending on the former, varies as the Length, and as depend- 

 ing on the latter, as the square root of the Length. Assume 

 these to be equal. This would give for the uncertainty of one 

 of the rods (twenty feet) 0*0004 inches or less than l/500000th 

 part, and for the uncertainty of making contact about 1/500 

 of an inch. 



Either of these errors is not only much smaller than we can 

 expect from work done under the circumstances, but they are 

 actually less than are generally realized in modern measures. 

 Therefore when we consider the means of comparison with the 

 standard and the method of placing the bars on the ground, 

 the close agreement must be considered entirely accidental, 

 and in no wise to be taken as a criterion of the accuracy of the 

 work. 



Any error in the linear measure is transmitted through the 

 triangulation and the probable error in the last side will 

 depend on the average correction to a direction as determined 

 from the shape of the triangles and their number. To this is 

 to be added the error in the base, which transmits itself inde- 

 pendently, and its effect depends on the relation between the 

 base and the last side. The average direction error, resulting 

 from joining points in a triangulation, is about tw T ice as much 

 as the average direction error arising from closing the horizon 

 at any one point. Regarding the probable errors of the base 

 and angles as differentials of those quantities, the uncertainty 

 of any side may be computed by a formula involving these 

 differentials and known functions of the angles. 



Taking eight seconds as the probable error of an angle which 

 is less than that estimated by the observer, we calculate the 

 uncertainty of the last side, as depending on the angle equa- 

 tions alone to be slightly more than ten feet. This result is 

 based on the formula, 



r o = or sin \" <v/cot a A + cot 2 JB, 



which assumes that one of the angles in each triangle is a con- 

 cluded one. The true probable error, where all the angles are 

 measured, would be somewhat less than that given. Never- 

 theless, all the circumstances being considered we may assume 

 the uncertainty of the last line to be not far from twelve feet. 

 The chances are that this is an under-estimate. This error, as 

 we have seen, is about that discovered near the middle of the 

 chain, and which influenced all subsequent work by one-half its 



