41-2 Wisconsin Academy of Sciences, Arts, and Letters. 



of the first twenty incommensurable logarithms from a tabic of 

 five decimal places, in which we find, by comparison with the 

 sum of the same seven place logarithms, an error — 1.66; in 

 other words the sum of the five place logarithms is 1.66 less in 

 terms of the last decimal place tlian the sum of the seven place 

 logarithms of the same numbers. 1x4 us find in the same man- 

 ner the error in the sum of each succeeding twenty logarithms 

 (the rational logarithms of the numbers 1, 10, 100, being 

 omitted). 



The errors are found to be :* 



Sum of logarithms. 2 — 22 error — 1,66 



23— 42 " — 2,1(3 1 



43— 62 '♦ —0,81 



63—82 " — 2,D2 



83 — 103 *' +0,42 



101 — 123 ♦• H-0,S3 



124 — 143 " —0,07 



114 — 16:^ " 4- 0,87 



104 — 183 *« + 0,23 



181-201 " +0,01 



201 — 22] " — l,75t 



221 — 213 " + 3.23 



214 — 2o3 " — 0,12t 



2'ol — 283 »« —1,54 



2S4 — .'}i){ " +0,43t 



301 — 32 5 '* —0,12 



321-31] " +0,19 



311 — 303 *' +0,75 



3G1 - 333 •* — 1 67 



3S4 — 403 '* +0.80 



* By performing the addition from two different sets of seven place 

 tables (one of them Vega, Bremiker's 40th edition; Berlin 1887), I find con- 

 sistently these results: 



Sum of logarithms 121-113 error — 1.92 



224-213 +3.13 



I also find several mistakes in signs which I have corrected, judging them 

 to be misprints in the Latin edition. I did not correct the two given above, 

 although it seems they should be as here given. If corrected, then the 

 summary just below should read 



12 errors between and 1 



5 " " 1 " 2 



2 *' •* 2 " 3 



1 " " 3 " 4 



which agrees with the !heoretical results given a few lines below better 



than tho30 given by Bremiker. — E. F. 0. 



1 1 have changed the signs of these numbers. — E. P. C. 



