176 PROCEEDINGS OP THE AMERICAN ACADEMY 



been made in the absolute precision of observations with the telescope 

 for the last forty years, if we except the tentative investigations of the 

 last five or six years. Argelander's Abo Catalogue of 1830, and 

 the Pulkowa Catalogue of 1845, are as yet pre-eminent for that kind 

 of accuracy which answers to the crucial test of agreement with 

 future observations. After a lapse of nearly fifty years, Argelander's 

 positions of the thirty-six stars known as the " Maskelyne funda- 

 mental stars " are at least as near the truth as the mean of the obser- 

 vations of these stars made during the last ten years. 



The great need of instrumental astronomy is a rigid investigation 

 of all the classes of error to which observations are now subject, not 

 simply for any one observer, but for all the principal observers of the 

 world, and upon a common plan. If some competent and recognized 

 authority, like the Astronomischen Gesellschaft, would arrange a 

 scheme of observations having this object in view, and take measures 

 to secure the co-operation of all the principal observatories in this 

 work, it would seem that the foundation for a real advance might be 

 made in the precision with which observations can be made. 



In the investigation which follows I have endeavored to ascertain 

 the limits of accuracy in measurements with the microscope by a 

 process similar to that by which observers with the telescope are now 

 seeking to reach the ultimate limit of precision. The remarks already 

 made with regard to the degree of reliability to be attached to con- 

 clusions drawn from the maguitude of the probable errors of observa- 

 tion apply with equal force to measures made under the microscope. 

 Neither the probable error of a single observation nor the probable 

 error of the mean of a given number of observations furnishes a safe 

 criterion by which the real measure of accuracy may be estimated. 

 For example, with the comparator for short lengths described in the 

 April number of the American Quarterly Microscopical Journal, it 

 is the experience of the writer that, in an unlimited number of 

 repetitions of measures of the same space, the pointer will in every 

 case fall upon the same tenth of a division of the index of the screw. 

 Hence, if the readings are taken to tenths only, the resulting probable 

 error will always be zero, without regard to the value of one division. 

 In this particular instrument the value of one tenth of one division is 

 one eighty-thousandth of an inch, but the probable error would still 

 remain zero if the readings were carried to tenths of one division only 

 for any change whatever in the pitch of the screw, and consequently 

 for any reduction in the value of one division, provided the pointer 

 always falls within this tenth. 



