ptise:. 4 
XXI. On a new Method of arranging Numerical Tables. By 
W. Dirrmar, Assistant in the Laboratory of Owens College, 
Manchester*. 
[ With a Plate. | 
HE use of numerical tables constructed in the ordinary 
manner is, for obvious reasons, inseparably connected with 
interpolation calculations. Such calculations, although by no 
means difficult, involve, as every practical mathematician knows, 
much loss of time, and often give rise to mistakes. This is 
especially the case when, from a given value of a dependent vari- 
able, the corresponding value of the independent, or of another 
dependent variable has to be found. It is obvious that all inter- 
polations could be avoided by giving all the values which can 
possibly be required; this, however, is in most cases practically 
impossible, as the tables w ould thus become inconyeniently volu- 
minous, and the chance of typographical error would be greatly 
increased. I believe that the completeness thus attainable can 
be arrived at by the use of the following graphical method, 
- while at the same time the size of the table will not extend 
beyond the ordinary limits. Let it be required to construct 
a table giving the values of several functions y, z,w...of a 
variable x. Draw a system of vertical parallels, and call them 
respectively the zw, y, z,w... line. On each of the verticals 
construct a scale, and let every point on each of the scales be the 
symbol for a number equal either to the number of divisions (and 
in general cne fraction of a division) contained between the 
origin and that point, or to a simple multiple of this number ; 
so that the marks on each scale represent the terms of an arith- 
metical series. These several scales must be so constructed that 
the corresponding values of all the variables are found in one 
and the same horizcutal line. It is true that, strictly speaking, 
this can only be the case when all the functions are linear ones ; it 
is, however, easy to show that, practically speaking, the problem 
can always be solved with any degree of exactitude required, 
provided that the functions are continuous. The mode of con- 
struction and use of such tables will perhaps be best explained 
by an example. 
Tig. 1, Plate III. represents the commencement of a Table of 
logarithms and. reciprocals, which is intended to afford about the 
same degree of exactitude as a common 4-place logarithmic table. 
Calumn UH. contains. the logarithm-scale; each of the divisions 
is 4 millims. in length, and represents a logarit thmic increment 
of 0-001; each point in this scale has a twofold meaning ; it 
stands, namely, both for the mantissa v=), and for the (posi- 
* Communicated by the Author. 
