Note that "D" is used in place of a capital letter "delta" in this transcription. Also, the layout of some equations and table borders differ.
Encyclopædia Britannica With New American Supplement, 9th ed. (1898). S.v. "Calculating Machines." by P. R. S. Lang, M. A., Professor of Mathematics, University of St. Andrews.
CALCULATING MACHINES. Mathematicians and astronomers have felt in all ages the irksomeness of the labour of making necessary calculations, and this has led to the invention of various devices for shortening it. Some of these, such as the Abacus, Napier's Bones (invented by the father of logarithms), and the modern Sliding Rule, are rather aids to calcuation than calculating machines. Pascal is believed to have been the original inventor of a calculating machine; its use was limited to addition, multiplication, &c., of sums of money, and as it required the constant intervention of a human operator the results were subject to the ordinary errors of manipulation. After him came the celebrated Leibnitz, Dr Saunderson, who, blind from his childhood, became professor of mathematics in Cambridge, and others. But all their machines were completely cast into the shade by the wonderful inventions of the late Charles Babbage. He knew well the immense value that absolutely correct tables possess for the astronomer and the navigator, and that a machine which could produce them with speed was a very great desideratum. The first calculating machine he invented he called a difference engine, because it was to calculate tables of numbers by the method of differences. By setting at the outset a few figures the attendant would obtain by a mechanical operation a long series of numbers absolutely correct. The difference engine was not intended to answer special questions, but to calculate and then print numerical tables, such as logarithm tables, tables for the Nautical Almanac, &c. An interesting account of some of the errors which are found in what are considered reliable tables is given in a paper by Babbage in the Memoirs of the Astronomical Society, 1827.
Every numerical table consists of a series of numbers which
continuously increase or diminish. As an example take the squares of the
natural numbers, 1, 4, 9, 16, 25, 36, &c. Designate this series by
N.
Table of Square Numbers, N  First Differences, D^{1}  Second Differences, D^{2} 
1  3  2 
4  5  2 
9  7  2 
16  9  
25 
The main characteristics of the difference engine, designed and
partially constructed by Babbage, are these:— It consisted of several
vertical columnds of figurewheels like large "draught men"
one above another, to the number of six in each column. The natural
numbers from 0 to 9 were cut on the rims of the figure wheels; hence
each figurewheel in a column could represent a digit. Thus the lowest
wheel gave the units digit, the second wheel the tens digit. The number
5706 would be represented on the wheels of a column as in the
margin.
5 
7 
0 
6 
"The mechanism was so contrived that whatever might be the
numbers placed respectively on the figure wheels of each of the
different columns, the following succession of operations took place as
long as the handle was moved. Whatever number was found upon the column
of first differences, would be added to the number found upon the table
column. The same first difference remaining upon its own column, the
number found upon the column of second differences would be added to
that first difference." Similarly for all the other columns. For
example, suppose we are calculating the cubes of the natural numbers. At
a certain stage of the work we would find 125 shown by the wheels of the
table column, 91 by those of the first difference column, 36 by those
of the second difference column, and 6 on the lowest wheel of the third
difference column. On making a turn of the handle the 91 would be added
to the 125, which would then show 216; at the same time 36 would be
added to the 91, so that the first difference column would then show
127; moreover 6 would be simultaneously added to the 36, which would
thus become 42, and the 6 would remain unaltered. Another turn and we
would get 343, 169, 48, 6 on the different columns.
Table Column.  First Difference Column.  Second Difference Column.  Third Difference Column.  
1 2 5 
9 1 
3 6 
6 

After one Turn...  2 1 6 
1 2 7 
4 2 
6 
After two Turns...  3 4 3 
1 6 9 
4 8 
6 
It was about 1822 that Babbage having constructed a small model of his engine sent an account of it to Sir Humphrey Davy, then president of the Royal Society of London. Government heard of the invention, and, having received from the Royal Society a favourable report on the merits and utility of the engine, advanced money towards its construction. Sums of money were at irregular intervals voted for this purpose; but so great were the difficulties to be overcome, so entirely new even were many of the tools necessary, so much time was occupied in testing the value of each proposed contrivance, that in 1834 only a portion was completed. The construction of the machine here stopped, although the Royal Society had again, in 1829, reported most favourably on the engine as regards its practicability, immense utility, and the progress it had made. The Governmnet had already advanced £17,000 (over and above what Babbage had spent, besides giving his personal superintendence without any remuneration), and they saw no definite limit to the amount it would cost; and Babbage had a delicacy in pressing for the completion of the difference engine, as he had recently designed a new machine, the analytical engine, which, if completed, would entirely supersede it. The portion completed is in King's College, London.
It will be noticed that the use of the difference engine was
limited to the working of such problems as can be solved by successive
additions or subtractions. The analytical engine, on the other hand,
was designed to work out any problem that the superintendent
knew how to solve. It consits of two parts, each of a number
of vertical columns of figure wheels, similar to those of the
difference engine; on the one set called the "variables,"
which we shall designate by V_{1}, V_{2}, &c., the
numbers of the special problem or formula are placed; the other set is
called the "mill," and performs the required operations of
multiplication, division, addition, or subtraction. Its working was
directed by means of two sets of cards—"operation"
cards , which instructed the mill whether ot multiply, divide, add, or
subtract, and "variable" cards, which indicated to the mill
the particular columns, i.e., the numbers on which it was to
perform this operation. An example will make this clear. Suppose we
wish to solve the equations
ax + by = c , 
dx + fy = g . 
In the Edinburgh Review for July 1834 appeared an account
of the principles of Babbage's difference engine. Herr George
Scheutz, a printer at Stockholm, read it, and shortly afterwards he
and his son Edward set about constructing a calculating machine. By
1843 they produced one capable of calculating series with terms of
five figures, with three orders of differences, also of five figures
each, and of printing its results. Provided with a certificate to
thsi effect from the Royal Swedish Avademy of Sciences, they
endeavoured unsuccessfully to get orders for their machine. In 1853,
with the aid of grants from the Swedish Government, the
Messsrs. Scheutz finished a second machine which was exhibited in
England, and at the Paris exhibition of 1855. It eventually went to
America. It was about the size of a small pianoforte. It could
calculate series with with four order of differences each of fifteen
figures. It printed the results to eight figures, the last of which
was capable of an automatic correction where necessary for those
ommitted. "It could calculate and stereotype without a chance of
error two and a half pages of figures in the same time that a skilful
compositor would take to set up the types for one single
page."
A new machine by the Messrs Scheutz was constructed about 1860 by
Messrs Donkin for the RegistrarGeneral for the sum of £1200.
It has been used in the calculation of some of the tables in the
English Life Table, published in 1864. Dr Farr says of it,
"The machine has been extensively tried, and it has upon the
whole answered every expectation. But it is a delicate instrument,
and requires considerable skill in the manipulation. It approaches
infallibility in certain respects; but it is not infallible, except in
very skilful hands. The weakest point is the printing aparatus, and
that admits of evident improvement."
M. Staffel and M. Thomas (de Colmar) have invented machines which can perform addition, subtraction, multiplication, division, and extraction of the square root. M. Thomas's machine is extensively used.
Sir William Thomson has recently invented an instrument (no description of which has yet been printed) which is able to solve any linear differential equation with variable coefficients.
Professor Tait has also invented the principle of a machine, which, if constructed, will integrate any linear differential equation of the second order with variable coefficients.
See the article "Calculating Machines" in Walford's Insurance Cyclopædia, where many references will be found, and a translation of General Menambrea's article on Babbage's Analytical Engine in Taylor's Scientific Memoirs, vol. iii.
(P.S.L.) 
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