138 On anew Method of arranging Numerical Tables. 
tive) mantissa of : which equals 1—d. When this scale had 
been drawn, the integral numbers from 100 to 1000 were marked 
on the other two columns by drawing horizontal lines opposite 
to the corresponding logarithms; the numbers up to num. 
(mant. =*5000) were placed in column I. beyond this number 
in column III, Lastly, the interval between every two such 
marks was divided into ten equal parts. Since, now, within the 
intervals « = 100 to e= 101, e=101 to x=102, &c. the 
A (log x) 
Aw 
quotients may in our case safely be taken as a con- 
stant, any Alog@ within such an interval may, together with its 
Az, be graphically represented by one and the same straight 
line, as is in fact done in the Table. It is now clear that any 
horizontal line will cut the verticals. in points which (whether 
they coincide with marks or not) are symbols for respectively a 
certain number z, log #, and a number having the same succes- 
sion of figures as = The above will afford complete informa- 
tion for the AOA use of the Table*. The division of the 
spaces between the marks on the scales 1s to be made by the eye. 
With some practice an error greater than one-tenth of an interval 
in the logarithm-scale will rarely be made; the position of the 
marks themselves may be by far more accurately determinedt ; 
the error accompanying any logarithm taken out of the Table 
will therefore scarcely ever exceed 0: 0001, and. any number 
found by its help will be correct within about S000 Of its value. 
The reliability of the results is not so dependent upon the exacti- 
tude of the drawing as one might at first sight be disposed to 
think, as only that 4 portion of every number is really graphically 
determined which in an ordinary 4-place table is found by compu- 
tative interpolation. ‘The Table is particularly handy for finding 
the values of reciprocals, as these may be obtained directly with- 
* Let us suppose the logarithm and the reciprocal of 1°0653 were to be 
found. Divide the interval between the marks 106°5 and 106°6 on the 
a-scale (in your mind) into ten equal parts, and through the third point 
from 106°5 draw a horizontal line, which is best done by bringing a straight 
line etched on one side of a piece of plate-glass into the right position. This 
line will cut the log-scale in the point ‘0274 = mant. 10653, and the 
] 1 1 
77scale in the point 938°7=1000 x T9G53- At the same time log 7-pgRy 
may be read off directly, the point of intersection in the log-scale standing 
also for mant. 10653 = 9/26. 
+ When great exactitude is required, the figure may be drawn on a cop- 
per plate by help of a dividing engine, and copies printed from the copper. 
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