i6o 



SCIENCE PROGRESS 



Considering any one column, the terms comprising it are in 

 the continued proportion of the first term to the last term in 

 the Second Table ; that is, in the proportion of 10,000 to 9,995. 

 The first term of the first column is radius " with four cyphers 

 added." The succeeding terms of the first column being in 

 the above ratio, are easily computed by methods strictly ana- 

 logous to those discussed in the formation of the First and 

 Second tables and the twenty-first term is found to be 

 9,900,473-5780. 



The first numbers in each of the 69 columns are approxi- 

 mately in the proportion of the first term to the twenty-first 

 term of the first column, i.e. in the proportion of 100 to 99. 

 The numbers which head each of the columns are therefore 

 readily calculated and the remaining twenty terms in each 

 column are. then easily filled in, as they form a descending 

 geometrical progression, of which the first term is given, con- 

 secutive terms being, as stated above, in the ratio of 10,000 

 to 9,995. Thus the table, when completed, has the form shown 

 below : 



Third Table 



The net result of all these computations is that in the Third 

 Table we have, between radius and (approximately) half-radius, 

 interposed 68 numbers in the continued proportion of 100 to 

 99; and between each pair of these numbers we have inter- 

 posed twenty other terms in the continued proportion of 

 10,000 to 9,995. Also, between the first two numbers of the 

 Third Table, which are also the first and last of the Second 

 Table, we have interposed 48 numbers in the continued pro- 

 portion of 100,000 to 99,999. And, finally, between the first 

 two numbers of the Second Table, which are also the*, first 

 and the last of the First Table, are interposed 98 nunnbers 



