526 MR sang's new table of logarithms to 200 000. 



the last two figures only of the second differences were copied into their places ; 

 the first line of the first page, that is, the logarithm of 100 000, with the first 

 and second difference complete, was also filled up by copying. The whole table 

 was then re computed by continued summation, the results being compared, at 

 each fifth step, with the previous work ; but as this comparison was not made 

 until the result was actually written clown, the possibility of the transference of 

 an error was avoided. In this way the new computation may be held as, in all 

 probability, quite free from error, except, indeed, the minute errors inseparable 

 from interpolation, and not exceeding one or two units in the fifteenth place. 



This new computation was to serve at the compositor's desk ; wherefore, for 

 the purpose of keeping it clean and free from injury, the pages were transferred 

 by the copying press, and the copy was made to serve both for the type-setting 

 and for farther calculation. The original sheets were bound up for preservation, 

 in volumes containing each 10 000 numbers. 



The logarithms for the latter half of the new table were obtained from those 

 of the first half by adding to each alternate logarithm the logarithm of 1*5. In 

 this way each third logarithm was found ; the intermediate ones being obtained 

 by interpolation. To have constructed the table directly in this way would 

 have left us liable to unchecked individual errors. In order to avoid these, or 

 rather to convert them into running errors, which cannot fail to be detected, 

 the following plan was followed : — 



The last two figures only of the alternate logarithms were considered, and 

 the last two figures of the second differences for interpolation were adjusted by 

 trial on the slate, and, after being tested, were written in their places on the 

 prepared paper. These trials are easily made, because, at this part of the table 

 the third difference only amounts to '25 of the fifteenth place, and, towards the 

 end of the work, comes down to 10 of the same. From these terminal figures 

 the table was constructed by continuous summation as before, and each third 

 result was checked by addition after having been written. This check also 

 afforded a test of the accuracy of the preceding part, and, in point of fact, one 

 solitary error was detected by it. 



Thus, the whole manuscript table of fifteen-place logarithms, with their first 

 and second differences, was constructed by continuous summation from 100 000 

 to 200 000, and may almost be held as free from any but last-place errors. It 

 is contained in ten quarto volumes, which form the first ninth part of the manu- 

 script table of all numbers up to one million. 



The importance of having the printed table absolutely free from error, 

 naturally brought up the question of the use of calculating machinery ; and 

 that question had to be very seriously considered. All the calculating machines 

 hitherto contrived are capable merely of addition or subtraction. These opera- 



