OF ARTS AND SCIENCES. 177 



The probable error of a single measure with the comparator for 

 short lengths is about two millionths of an inch. If the probable 

 error can be taken as a measure of precision, it ought not to be diffi- 

 cult to measure one millionth of an inch with entire certainty by 

 repeating the measures a sufficient number of times. 



Let us see if this theoretical accuracy is attainable. Before pro- 

 ceeding to the discussion, it may be worth while to say that a sharp 

 distinction must be drawn between absolute accuracy and a superficial 

 appearance of accuracy. If I determine the value of a centimeter 

 within one ten-thousandth of its whole length, I can use the equivalent 

 expression, one millionth of a meter ; but it does not follow that I can 

 measure a meter within this limit. I say that a given space, cor- 

 responding to one thousandth of an inch, requires a correction of one 

 millionth of an inch ; but it makes a wide difference whether I 

 ascertain this fact by direct measurement, or whether I get it by divid- 

 ing the correction for an entire inch by one thousand. Extending the 

 number of figures in the quotient does not give a corresponding in- 

 crease of accuracy. The index of the screw of my dividing engine can 

 be set to correspond to a motion of one billionth of an inch with 

 entire certainty as far as the mechanical indication of this degree of 

 accuracy is concerned; yet previous to May, 1877, the actual errors of 

 a given ruled plate amounted, under certain conditions, to as much as 

 one seven-thousandth of an inch. Even now, after four epochs of 

 improvement, I can hardly say of a given space that it is certainly true 

 within one eighty-thousandth of an inch until a careful investigation 

 has been made with the comparator. Again, it does not follow that, 

 because the spaces of a closely ruled band of lines, like Nobert's bands, 

 appear to be equal under an objective of high power, they are there- 

 fore to be taken as the measure of the real accuracy of the gradua- 

 tions. It is far more difficult to subdivide an inch into one hundred 

 equal parts, than to make a further subdivision of one of these parts. 

 As I shall presently show, almost all of the errors of a given gradua- 

 tion are periodic in their character, but the increments proceed by 

 such minute variations in the case of closely ruled bands that they can 

 only be detected when their sum amounts to an appreciable quantity 

 Thus, if the accumulated error of a screw having a pitch of one in 

 twenty amounts to one two-thousandth of an inch for half a revolution 

 of the index, the average periodic error for each two-thousandth of 

 an inch will be one hundred-thousandth of an inch. It will thus be 

 seen that, for even the first of Nobert's bands, which are about ten 

 thousand to the inch, the systematic error for any single space is inap- 



vol. xiv. (n. s. vi.) 12 



