628 
NATURE 
[Oct. 29, 1885 
This is not the place to give instructions for using the 
rule, but an outline of the method is necessary to make 
it possible to compare the different makes, many of which 
are shown at the Inventions Exhibition. 
With two similar scales of equal parts, as inches divided 
into tenths or centimetres divided into millimetres, it is 
possible to add numbers, or, conversely, to subtract num- 
bers ; thus, if the zero of one scale is placed opposite, 
say, 6°5 of the other, opposite every number 7 on the first 
will be found z+ 6°5 on the second, and so addition or 
subtraction could be performed, but there would be no 
advantage in so adding or subtracting. In the same way 
the slide of the ordinary slide rule is employed to add 
distances, but these distances do not correspond to the 
figures attached, but to the logarithms of those figures, and 
so the sum which is found by such an addition is not the 
sum of the figures apparently added, but their product. 
If the slide is placed at random, all the pairs of figures 
which are opposite to one another are in the same pro- 
portion, and the multipliers which will change either series 
into the other will be found on each scale opposite the 
divisions marked 1 on the other. It requires no great 
amount of memory to bear this in mind: however the 
slide may be set, those numbers which are opposite 
to one another are in the same proportion, ze. have 
a common quotient, which may be found opposite 
any of the divisions marked 1; and yet this is all that 
has to be remembered in multiplication, division, and 
simple proportion. The two top lines of a slide rule are 
generally identical, and they are used for these simple 
operations ; they are generally distinguished by the letters 
A and B. In general the bottom line of the slide, that is, 
the third altogether, is identical with the first two, and is 
labelled C. This arrangement is convenient, for it is 
possible to insert the slide upside down, in which case all 
numbers which are opposite one another on A and C 
have a common product, which may be found opposite 
any of the divisions marked 1. This furnishes the most 
ready mode of finding actual or approximate factors of 
numbers, and is of great use to those who have to calculate 
wheelwork ; further, by the use of the inverted C line under 
the A line any harmonical progression can at once be 
read, and any number of harmonic means can be inserted 
between two quantities. The fourth line is generally made 
different from the others in that it is on double the scale, 
and it is then distinguished by the letter D. If the units 
of the C and D line are placed opposite one another, a 
table of squares and roots is formed, or if in any other 
position the squares of the numbers on D vary in the 
same proportion as do the numbers that are opposite to 
them on C. It is in calculations made on the C and D 
lines that so much time is saved, for proportions in which 
some of the terms are squares or square roots can be 
worked out as quickly and as accurately as those in which 
simple numbers only are employed. If the slide is inverted 
so as to bring the B line opposite to the D line, then the 
square of any number on D X the number opposite to 
it on B is constant. This product may of course be found 
in B opposite 1 in D. Cube roots, among other things, 
may be found in this way. 
These four lines are all that are generally found in a 
slide rule ; occasionally others are added: thus a line on 
one third of the scale of the D line (sometimes called an 
E line) will, with the D line, enable one to directly work 
proportions in which some of the terms are cubes or cube 
roots, but this is not often required. With the usual four 
lines all arithmetical processes, except addition and 
subtraction, can be performed. There are, however, rules 
in which on the back of the slide are scales in which the 
distances are log. sines or log. tangents of the angles 
marked, then these lines being placed against an ordinary 
A line so that go° on the line of sines or 45° on the line of 
tangents is opposite 1 on the A line, a table of sines or 
tangents will be formed ; and if the slide is placed in any 
other position, the sines or tangents of the angles denoted 
by any divisions on either of these special lines will vary 
in the same proportion as do the numbers which are 
opposite them on the A line. In those rules in which 
lines of sines and tangents are given there is generally a 
scale of equal parts in which the length of the D line is 
divided into 500 or 1000 parts. If this is placed opposite 
the D line, with the ends of the two scales opposite one 
another, a table of logarithms will be seen; thus the 
logarithm of any number on the D line will be found 
opposite to it on the scale of equal parts. 
Having pointed out the chief uses of a slide rule, it 
will be possible to describe the differences in construction 
in the several varieties. The most simple possible form 
is the original Gunter’s scale to be found on any sector. 
With this and a pair of dividers calculations may be 
made, for if the dividors are set to the distance between 
any two numbers, any other pair of numbers which are 
found by the dividors to be the same distance apart will 
be in the same proportion, or have a common quotient, 
just as a common difference would be found if a scale 
of equal parts were used. This, however, is trouble- 
some ; but if the same principle is applied to a scale in 
the circular form the result is much more convenient. In 
this case angular distance takes the place of linear dist- 
ance, and a pair of arms which can be opened to any 
angle can be moved round, and every pair of numbers 
covered will bear to one another a constant proportion 
depending on the extent of the angle. This is the prin- 
ciple of some of Dixon’s rules shown at the Inventions 
Exhibition, near the arithmometers. In the well-known 
pocket instrument, the calculating circle of Boucher, an 
instrument like a watch, one hand is fixed and one is 
movable, and the face is also movable. There is another 
instrument of the same kind, in which the scale is drawn 
on a helical line. Here the scale and one hand are mov- 
able, and there is one fixed hand. This, which is Prof. 
Fuller’s spiral rule, is made and exhibited by Stanley. 
Circular instruments are also made, in which scales slide 
over one another, which are in this respect like the straight 
rules. There is more advantage in the circular form than 
appears at first. In the straight rules the A and B lines 
are each double, the first and second halves are identical ; 
this repetition of the scale is required in order that, how- 
ever the slide may be placed, the part of each opposite to 
the other may contain at least a complete scale of numbers. 
In the circular form, however, the beginning and end of a 
single logarithmic scale meet, and so the scale itself is its 
own repetition both above and below. For this reason 
the openness of the divisors in a circular instrument is 
the same as in a straight rule, of which the length is 
six times, instead of three times, the diameter of the 
circular line. 
Of the two types of instrument—one in which one slide 
works against another, generally straight, sometimes cir- 
cular, and the other in which there is no slide but only a 
line divided logarithmically with a pair of hands, which 
type is always circular—which may be called respectively 
the slide and the index types, each has certain advant- 
ages. The slide form is preferable, in that each setting 
ot the slide furnishes a complete table of pairs of related 
numbers, as, for instance, of any English and foreign 
measure, of squares and roots on any scale, such as 
diameters and areas of circles, or of sines or tangents 
on any scale, so that, without moving the slide, any 
number of results may be read off, whereas with in- 
struments of the index type the scale must be moved 
under the hands, or the hands over the scale, for each 
result. On the other hand, index instruments are more 
convenient than the usual slide rules in working out long 
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terms may be squares, cubes, sines, or tangents, for the 
terms are taken alternately from the numerator and de- 
expressions of the form , in which any of the 
