July 25, 1902.] 



SCIENCE. 



135 



To find the cube root of a number the 

 rule is: 



For the iith subtrahend, multiply n by 6 

 and add the preceding subtrahend ; the last 

 remainder will be the cube root sought. 



Thus the first subtrahend is 6; the next 

 6X2 + 6=18; the third, 6X3 + 18=36; 

 and so on. 



For the fourth root, the rule is : 



To find the nth subtrahend, multiply n^ 

 by 12 and add 2 plus the preceding subtra- 

 hend; the last remainder will be the root 

 sought. 



In the paper rules are given for finding 

 the fifth, sixth, seventh, eighth, ninth and 

 tenth roots, with examples. 



General formulas for the wth subtrahend 

 of any root (the mth) are: 



s„=(»4-l)»- (»'» + !), 

 or 



Sn = s„-i+ {n+ 1)»'+ (n — l)'" — 2n"'. 



It is shown in the paper that in all cases 

 for aU roots the number of subtractions 

 to be performed is one less than the num- 

 ber of units in the root sought, and conse- 

 quently the root equals number of subtrac- 

 tions plus 1. 



A table of subtrahends containing the 

 first ten subtrahends for the first eleven 

 roots is appended to the paper. This table 

 can be extended to any desired extent by 

 the rules and formulas given. 



The paper will be published in the Math- 

 ematical Magazine. 



On the Determination of the Places of the 



Circumpolar Stars: Professor Milton 



Updegraff, U. S. Naval Observatory. 



The contents of this paper are : A sketch 



of previous work done on circumpolar stars, 



(2) a statement of the kind of work needed, 



and (3) some suggestions as to the best 



methods to be used in redeterminations of 



the coordinates of the circumpolar stars. 



The paper will be published in one of the 

 astronomical journals. 



Report on Quaternions: Professor Alex- 

 ander Macfarlane, Lehigh University. 

 This paper will be printed in full in the 



Proceedings of the Association. 



The Definite Determination of the Causes 

 of Variation in Level and Azimuth of 

 Large Meridian Instruments: Professor 

 G. W. Hough, Dearborn Observatory, 

 Evanston, 111. 



This Avas an elaborate discussion of the 

 various styles of mounting for meridian in- 

 strmnents, and of the eiiects of changes of 

 temperature in causing variation. The re- 

 sults of several long series of observations 

 upon this effect were exhibited. Professor 

 Hough's conclusion was that stone piers 

 give the best results. The paper gave rise 

 to some spirited discussion. 



A New Founding of Spherics: Professor 

 G. B. Halsted, University of Texas. 

 Professor Halsted presented under this 

 title some abstracts from a book which he 

 is about to publish. The author made a 

 simple set of assumptions: (1) of associa- 

 tion, (2) of betweenness, (3) of congru- 

 ence, and he then showed how, without the 

 assumption that the straight line is the 

 shortest distance between two points, or 

 that the shortest path between two points 

 on a sphere is on the great circle through 

 them, or even that two sides of a triangle 

 are together greater than the third, all the 

 projective and metric properties of spherics 

 are established. 



Report on the Theory of C ollineations : 

 Professor H. B. Newson, University of 

 Kansas. 



Owing to Professor Newson 's absence 

 from the meeting, this paper could be pre- 

 sented only by title. It will be printed in 

 full in the Proceedings. 



