514 



SCIENCE. 



[N. S. Vol. XVII. No. 430. 



of preserving phytogeographical data for con- 

 venient reference, and also for various edu- 

 cational purposes. In this Dr. Clements is 

 the pioneer, and deserves our congratulations 

 upon the success of this first attempt. 



It is understood that twenty-four sets (the 

 price of which was very moderate) were pre- 

 pared, of which all or nearly all have been 

 taken, about a third of them by institutions 

 abroad. W. F. Ganong. 



NOTE ON NEGATIVE DIGITS. 



In the common scale of notation 2873 

 stands for 2000 -f 800 + 70 + 3. The same 

 number might be represented by 3133 which 

 is intended to mean 3000 -- 100 — 30 + 3. It 

 might also be written 3127 or 2933, and, in- 

 deed, a great variety of_ways might easily 

 be found, but the form 3133 is most advan- 

 tageous in that the absolute values of the 

 digits are the smallest possible. It is clear 

 that any number may be written so that all 

 its digits shall be less than six in absolute 

 value. In fact, we may replace 9 by 1, 8 by 

 2, 7 by 3 and 6 by 4, leaving the others 

 unchanged. This amounts to replacing the 

 digit jE' by 10 - — K, so that we must add one 

 unit to the adjacent digit on the left. We 

 have then the following rules for changing 

 any digit from plus to minus and from minus 

 to plus: 



1. To change a digit from plus to minus, 

 subtract it from 10 and add 1 to the digit 

 on the left. 



2. To change a digit from minus to plus, 

 suhtract it from 10 and subtract 1 from the 

 digit on the left. 



In practice one begins on the right and 

 changes successively those digits which are 

 greater than 5. Thus to change 82755637 

 the 7 on the right goes into 3 and the 3 

 becomes a 4, the 6 changes to 4 and the 5 

 adjacent to it becomes 6, which goes into 4 

 and makes the second 5 a 6. This goes in 

 turn into 4 and changes 7 to 8 or 2 and the 

 2 becomes 3. The last digit on the left be- 

 comes 2, which changes the digit next to it 

 on the left (namely 0) to 1. The whole 

 process then gives 



12324444 ¥. 



The reverse process is carried out similarly, 

 and half an hour's practice will enable one 

 to make the change from one notation to the 

 other with little effort of the mind. 



The new notation is of little value in addi- 

 tion or subtraction and is entirely useless in 

 division. In multiplication its value, how- 

 ever, can hardly be overestimated. The ad- 

 vantage in using it is twofold. The digits 

 are all less than 6 and there is twice the 

 chance of repeated digits in the multiplier. 

 Thus, in the ordinary method of multiplica- 

 tion, if one has obtained the partial product 

 corresponding to a digit 3 in the multiplier, 

 one obtains the partial product corresponding 

 to a digit 3 by changing the signs of all the 

 digits in the first partial product. In the 

 short method of multiplication given in 

 Science, July 11, 1902, it is difficult to deal 

 with large digits. Thus, to find the product of 

 987593X86759 by that method would be a 

 difficult and fatiguing task. Changing to 

 negative digits, hoWever, one finds the product 

 can be written out with perfect ease, thus: 



1 oTWiTs 

 113 ¥4 T 



1143324112 



10 17 11 "3 



85682581087 



D. N. Lehmer. 

 University of Caiitornia, 

 October, 1902. 



MUSEUM NOTES. 



Part X., Volimie II., of the Annals of the 

 South African Museum is devoted to a con- 

 tinuation of ' The Moths of South Africa,' 

 by G. F. Hampson. The present instalment, 

 comprising nearly two hundred pages, deals 

 entirely with the large family Noetuidse, and 

 gives keys to the subfamilies, genera and spe- 

 cies. The descriptions are very full and in- 

 clude a great number of new species; the 

 greater number of types are in the British 

 Museum, but the location of all others is 

 noted. 



Part II. of the Memoirs of the Carnegie 

 Museum contains a detailed description of 

 the osteology of ' Oligocene Canidse,' by J. 

 B. Hatcher, including Daphnwus felinus. Pro- 



