Comparators for Measures of Length. 209 



measure and repeat this constant with sufficient accuracy to 

 obtain the whole unit with the required precision. Suppose, 

 for example, we require the meter — even granting that the true 

 value of the millimeter could be found, it would still be im- 

 possible to get an exact value of the meter by 1000 repetitions. 

 An error of only one ten-thousandth of a millimeter in the 

 assumed value will, in the whole meter, amount to one tenth of 

 a millimeter .Through unavoidable accidental errors, the final 

 deviation from an exact meter would doubtless exceed this 

 amount. A fundamental objection to this method is found in 

 the fact that so much time would be required to complete the 

 measurements, that the changes introduced through a variation 

 of the temperature could not be neglected. 



2. We may assume the entire unit, and then obtain the sub- 

 divisions according to the following scheme: 



{a) Subdivision into 2 equal parts. 

 (/') Subdivision into 4 equal parts. 

 (c) Subdivision into 5 equal parts. 

 {d) Subdivision into 10 equal parts. 

 The fundamental principle which must govern the construc- 

 tion of a comparator, is the requirement that these large sub- 

 divisions shall be easily made with great precision and within 

 so short a time that the effect of a slight change of tempera- 

 ture can be neglected. As a check we have: 



(^) = 2 {a). 

 {^) = 2y, {a). 

 {d) = 2{c). 

 {d)-^5{a). 



{d)^2y, {b). 



When the relations between the subdivisions into 10 equal 

 parts have been found, each one of these tenths may be again 

 subdivided as before, without danger of accumulating either 

 accidental or systematic errors, or even errors which would be 

 introduced through a change of temperature, if not more than 

 ten or fifteen minutes is required for the entire operation. The 

 writer has found that for air contact with a long bar of metal, 

 whether of iron, brass, or steel, a change of several degrees in 

 the temperature, as indicated by a thermometer in air, requires 

 over thirty minutes to produce a perceptible change in the 

 length. 



When we reach those subdivisions which fall within one field 



