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Oct. 29, 1885 ] 
NATURE 
629 
nominator and set in order with the fixed and movable 
hand until all are worked off, when the answer is found 
under the fixed hand. There is no necessity to observe 
any result till the process is complete ; on the other hand, 
with slide instruments, each result of the form esd) 
é 
axbxXe 
exf 
operated upon by the next pair of factors. In Gravet’s 
rules, however, this disadvantage of the straight form is 
removed by the addition of a cursor or sliding index, 
which in other ways is a great comfort. 
All instruments of the index type suffer terribly from 
parallax, owing to the hands being above the face, so that 
they do not in practice give the accuracy that from the 
length of scale upon them might be expected. 
This is especially the case in small instruments : for 
instance, Boucher’s calculating circle, made in the form 
of a watch, is probably divided so accurately that on that 
score an error of one part in a thousand does not exist ; 
yet, owing to parallax, the practical limit is about 1-300. 
This instrument has, besides the ordinary line, one on a 
double and one on a treble scale for squares and 
cubes, a line of sines, and another of equal parts for 
logarithms. 
The possible accuracy of any instrument depends upon 
the length of the scale included between 1 and Io, called 
the radius, and also upon the linear accuracy with which 
a setting or reading can be made; this is at least twice 
as great in slide as in index instruments. In order to 
obtain great accuracy various means have been adopted 
whereby a great length of scale is brought within 
a small compass. Among slide instruments are Prof. 
Everett’s “Universal Proportion Table,” published by 
Longmans, Green, and Co., and General Hannyngton’s 
slide rule, made and exhibited at the Inventions Ex- 
hibition by Aston and Mauder. In these the slide is 
made in the gridiron form. In Everett’s instrument 
there are twenty bars, the total length of which is 
about 13 feet; a scale of equal parts is also printed, so 
that logarithms can be read with it. In both of these 
instruments only simple proportions can be effected, 
unless special grids, divided on a double scale or trigono- 
metrically, are provided. Far the most ingenious of all 
devices for obtaining a great length of radius in a com- 
paratively short space is due to Mr. Beauchamp Tower, 
whose name is well known in connection with the spherical 
engine. His instrument is a slide instrument consisting 
of two tapes running side by side over equal and inde- 
pendent rollers, but the tapes have a half twist in them, 
so that they have each only one surface and one edge. 
In this instrument, made privately for his own use, each 
tape is about 124 feet long, and as both sides of the tape 
are used the radius is about 25 feet, and therefore, as far 
as openness of scale is concerned, it is equivalent to a 
straight rule 50 feet long, while the instrument itself is 
only just over 6 feet in length. 
Slide rules of the index class can have a great length 
of scale more readily employed than others. Thus Prof. 
Fuller’s helical instrument has its radius equal to 424 feet, 
and is in openness of scale equivalent to a straight rule 
85 feet long, while the box which contains it is only 
17 X33 X32 inches inside measure. Dixon exhibits a 
special rule with the scale extending over 10 concentric 
circles, but with this form a less degree of accuracy is 
attainable when using the inner than when using the 
outer circle. Thus the inner circle is equivalent to a 
straight rule 30 feet long and the outer to one 60 feet 
long. There is an outer circle equally and logarithmically 
divided to find logarithms. In another of Dixon’s instru- 
ments, similar in size and form, there is the same outer 
circle for proportions and logarithms, and a series of 
inner circles divided so as to give sines, cosines, tangents, 
&c., must be read and set before it can be 
cotangents, secants, and cosecants. Each of these is on 
a board 14 inches square. Rules with very extended 
scales do not in practice give results with an accuracy 
which is proportional to their length, though the working 
accuracy is very much increased. They have this ad- 
vantage, that they can be worked to their limit with ease, 
while with a well-divided pocket rule the errors of con- 
struction are beyond the limits of vision, and so the cal- 
culator is apt to strain his eyes to get results as accurate 
as possible. For instance, results obtained by a good 
pocket-rule one foot long can be trusted to a thousandth 
part ; at the same rate Prof. Everett’s should be accurate 
toa thirteen-thousandth part, and Prof. Fuller’s to an 
eighty-five thousandth part. In practice a four and a 
ten-thousandth part are their limits. Again, instruments 
with very extended scales have only room for one line, so 
that simple proportions only and logarithms are all that 
can be directly obtained from them. For general use in 
the laboratory or elsewhere where calculations of every 
kind have to be made, the straight form, on the whole, 
seems most convenient, because of its portability, the 
quickness with which it can be worked, the diversity of 
operations that it will directly accomplish, and the extra- 
ordinary accuracy in comparison with other forms of the 
results to be obtained. Far the best instruments of this 
type that the writer has yet seen are those made by 
Tavernier-Gravet, of Paris, already alluded to. They are 
different to those generally used in England in that the 
line in the slide which works against the D line is itself a 
D line, so that squared proportions have to be performed 
by the aid of the cursor. This form has the further dis- 
advantage that the inverted slide cannot be used for 
finding factors, which is a great loss ; on the other hand, 
the two lower lines may be used for simple proportions, 
and they will give a double accuracy. On the whole, the 
original pattern with an A, B, C and D line seems pre- 
ferable. Of the straight rules shown at the Inventions 
Exhibition those made by Stanley exceed all the others 
in workmanship and they are equal in this respect to the 
Gravet rule. Among them are rules for special purposes, 
as Hudson’s scales and Ganga Ram’s rules. Hudson’s 
scales, which are made in card, each having two slides, 
are a marvel of constructive skill. Dixon shows his 
“triple radius double slide rule,” with which very complex 
operations may be readily performed. Heath shows a 
slide rule for converting sidereal to mean solar time, or 
the reverse, correct to about ‘o2 of a second, but 
this is not a slide rule proper, as the scales are not 
logarithmic. 
There is entirely a different class of slide rule shown 
by Lieut. Thomson. In this there is, as usual, an A, B, 
and C line, but instead of the D line there is a “ P” line, 
in which the distances, instead of being logarithmic, are 
logarithms of logarithms. By this instrument fractional 
powers may be found as readily as simple products or 
quotients. It has, however, this defect, that the scale 
converges so rapidly as the numbers ascend that high 
numbers can only be obtained with a proportionate 
accuracy far less than is possible with low numbers. It 
is one feature in the slide rule of ordinary construction 
that an error of reading of, say, I-10oth of an inch will pro- 
duce the same proportionate error in any part of the 
scale. This rule for involution is shown in the straight 
and circular form. It is right to mention that the same 
thing exactly was invented by the late Dr. Roget and 
published by him in the P#z/. Trans. of 1815. 
No attempt has been made to give an account of every 
special form of rule that is made; those shown at the 
Exhibition and some other well-known forms, which well 
illustrate the different kinds of development, have been 
imperfectly described and the general principles on which 
all depend sufficiently explained to make evident the 
advantages of each type of instrument. 
C. V. Boys 
