186 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1916. 



many observatories in upward of 20 years. It hardly requires proof 

 that with such resources as astronomy has ever commanded, or is 

 likely to command, a complete enumeration upon these lines will 

 never be attained. 



If we are to attain a conspectus of the whole, now or ever, we must 

 make a radical reduction in the demands of our problem. Now, in 

 all these catalogues the places of the stars are recorded in their two 

 coordinates, and the calculations made in each individual case which 

 are necessary to allow for precessional change in the axes of refer- 

 ence. We can not dispense with knowing where the stars are, but 

 if our interest is in their numbers and regional distribution, we can 

 dispense with recording it precisely. And if we can take an elevated 

 standpoint and eliminate the earth, like the Blessed Damozel, leaning 

 on the gold bar of heaven, and see far below 



this earth 

 Spin like a fretful midge — 



why, then, we may dispense with the troublesome calculation of pre- 

 cession. There is almost nothing left then except to count. 



But let nobody think lightly of the importance or the difficulty of 

 mere counting. When the White Queen put to Alice the question : 



How many are one and one and one and one and one and one and one and 

 one and one and one? 



Alice does not appear to have been able to answer. Counting 

 correctly is very difficult, because, so to put it, it requires from the 

 mind a simultaneous hold upon the past, present, and future. Count- 

 ing, on the other hand, done carefully is the only region of knowl- 

 edge, even of mathematics, in which we can be perfectly sure we are 

 not talking nonsense. Much that was formerly classed as geometry 

 is now classed as nonsense. A circle has no properties until we 

 say how it is generated, and we can not say how it is generated 

 until we make up our minds about continuity; and continuity, to 

 make it intelligible, is now explained in terms of discontinuity — 

 that is, of counting. By counting infinity is made comprehensible, 

 like an infinite perspective collected upon the narrow space of the 

 retina, as a sequence of converging increments — countless in their 

 number but countable in their sum or effect. 



Counting by samples is another name for the theory of statistics, 

 of averages, with their ramifications of probability, without which 

 matters so disparate as life insurance and the kinetic theory of gases 

 would be equally unmanageable. 



I need not labor my point. In counting the stars you have to 

 count a sum of which you can not tell in advance whether it will 

 prove infinite or finite; you have to count by samples; you have 

 to count by receding steps or grades as far as ,you can and then 



