On the Economy of Machinery and Manufactures - Charles Babbage (interesting books to read in english .TXT) 📗

to simple numerical calculation by many individuals

employed at the same time. This section had little or nothing to

do with the actual numerical work. When its labours were

concluded, the formulae on the use of which it had decided, were

delivered to the second section.

 

Second section. This section consisted of seven or eight

persons of considerable acquaintance with mathematics: and their

duty was to convert into numbers the formulae put into their

hands by the first section an operation of great labour; and then

to deliver out these formulae to the members of the third

section, and receive from them the finished calculations. The

members of this second section had certain means of verifying the

calculations without the necessity of repeating, or even of

examining, the whole of the work done by the third section.

 

Third section. The members of this section, whose number

varied from sixty to eighty, received certain numbers from the

second section, and, using nothing more than simple addition and

subtraction, they returned to that section the tables in a

finished state. It is remarkable that nine-tenths of this class

had no knowledge of arithmetic beyond the two first rules which

they were thus called upon to exercise, and that these persons

were usually found more correct in their calculations, than those

who possessed a more extensive knowledge of the subject.

 

245. When it is stated that the tables thus computed occupy

seventeen large folio volumes, some idea may perhaps be formed of

the labour. From that part executed by the third class, which may

almost be termed mechanical, requiring the least knowledge and by

far the greatest exertions, the first class were entirely exempt.

Such labour can always be purchased at an easy rate. The duties

of the second class, although requiring considerable skill in

arithmetical operations, were yet in some measure relieved by the

higher interest naturally felt in those more difficult

operations. The exertions of the first class are not likely to

require, upon another occasion, so much skill and labour as they

did upon the first attempt to introduce such a method; but when

the completion of a calculating engine shall have produced a

substitute for the whole of the third section of computers, the

attention of analysts will naturally be directed to simplifying

its application, by a new discussion of the methods of converting

analytical formulae into numbers.

 

246. The proceeding of M. Prony, in this celebrated system of

calculation, much resembles that of a skilful person about to

construct a cotton or silk mill, or any similar establishment.

Having, by his own genius, or through the aid of his friends,

found that some improved machinery may be successfully applied to

his pursuit, he makes drawings of his plans of the machinery, and

may himself be considered as constituting the first section. He

next requires the assistance of operative engineers capable of

executing the machinery he has designed, some of whom should

understand the nature of the processes to be carried on; and

these constitute his second section. When a sufficient number of

machines have been made, a multitude of other persons, possessed

of a lower degree of skill, must be employed in using them; these

form the third section: but their work, and the just performance

of the machines, must be still superintended by the second class.

 

247. As the possibility of performing arithmetical

calculations by machinery may appear to non-mathematical readers

to be rather too large a postulate, and as it is connected with

the subject of the division of labour, I shall here endeavour, in

a few lines, to give some slight perception of the manner in

which this can be done—and thus to remove a small portion of

the veil which covers that apparent mystery.

 

248. That nearly all tables of numbers which follow any law,

however complicated, may be formed, to a greater or less extent,

solely by the proper arrangement of the successive addition and

subtraction of numbers befitting each table, is a general

principle which can be demonstrated to those only who are well

acquainted with mathematics; but the mind, even of the reader who

is but very slightly acquainted with that science, will readily

conceive that it is not impossible, by attending to the following

example.

 

The subjoined table is the beginning of one in very extensive

use, which has been printed and reprinted very frequently in many

countries, and is called a table of square numbers.

 

Terms of Table A Table B first Difference C second Difference

 

1 1

3

2 4 2

5

3 9 2

7

4 16 2

9

5 25 2

11

6 36 2

13

7 49

 

Any number in the table, column A, may be obtained, by

multiplying the number which expresses the distance of that term

from the commencement of the table by itself; thus, 25 is the

fifth term from the beginning of the table, and 5 multiplied by

itself, or by 5, is equal to 25. Let us now subtract each term of

this table from the next succeeding term, and place the results

in another column (B), which may be called first difference

column. If we again subtract each term of this first difference

from the succeeding term, we find the result is always the number

2, (column C); and that the same number will always recur in that

column, which may be called the second difference, will appear to

any person who takes the trouble to carry on the table a few

terms further. Now when once this is admitted, it is quite clear

that, provided the first term (1) of the table, the first term

(3) of the first differences, and the first term (2) of the

second or constant difference, are originally given, we can

continue the table of square numbers to any extent, merely by

addition: for the series of first differences may be formed by

repeatedly adding the constant difference (2) to (3) the first

number in column B, and we then have the series of numbers, 3, 5,

6, etc.: and again, by successively adding each of these to the

first number (1) of the table, we produce the square numbers.

 

249. Having thus, I hope, thrown some light upon the

theoretical part of the question, I shall endeavour to shew that

the mechanical execution of such an engine, as would produce this

series of numbers, is not so far removed from that of ordinary

machinery as might be conceived.(3*) Let the reader imagine three

clocks, placed on a table side by side, each having only one

hand, and each having a thousand divisions instead of twelve

hours marked on the face; and every time a string is pulled, let

them strike on a bell the numbers of the divisions to which their

hands point. Let him further suppose that two of the clocks, for

the sake of distinction called B and C, have some mechanism by

which the clock C advances the hand of the clock B one division,

for each stroke it makes upon its own bell: and let the clock B

by a similar contrivance advance the hand of the clock A one

division, for each stroke it makes on its own bell. With such an

arrangement, having set the hand of the clock A to the division

I, that of B to III, and that of C to II, let the reader imagine

the repeating parts of the clocks to be set in motion continually

in the following order: viz.—pull the string of clock A; pull

the string of clock B; pull the string of clock C.

 

The table on the following page will then express the series

of movements and their results.

 

If now only those divisions struck or pointed at by the clock

A be attended to and written down, it will be found that they

produce the series of the squares of the natural numbers. Such a

series could, of course, be carried by this mechanism only so far

as the numbers which can be expressed by three figures; but this

may be sufficient to give some idea of the construction—and

was, in fact, the point to which the first model of the

calculating engine, now in progress, extended.

 

250. We have seen, then, that the effect of the division of

labour, both in mechanical and in mental operations, is, that it

enables us to purchase and apply to each process precisely that

quantity of skill and knowledge which is required for it: we

avoid employing any part of the time of a man who can get eight

or ten shillings a day by his skill in tempering needles, in

turning a wheel, which can be done for sixpence a day; and we

equally avoid the loss arising from the employment of an

accomplished mathematician in performing the lowest processes of

arithmetic.

 

251. The division of labour cannot be successfully practised

unless there exists a great demand for its produce; and it

requires a large capital to be employed in those arts in which it

is used. In watchmaking it has been carried, perhaps, to the

greatest extent. It was stated in evidence before a committee of

the House of Commons, that there are a hundred and two distinct

branches of this art, to each of which a boy may be put

apprentice: and that he only learns his master’s department, and

is unable, after his apprenticeship has expired, without

subsequent instruction, to work at any other branch. The

watch-finisher, whose business is to put together the scattered

parts, is the only one, out of the hundred and two persons, who

can work in any other department than his own.

 

252. In one of the most difficult arts, that of mining, great

improvements have resulted from the judicious distribution of the

duties; and under the arrangments which have gradually been

introduced, the whole system of the mine and its government is

now placed under the control of the following officers.

 

1. A manager, who has the general knowledge of all that is to

be done, and who may be assisted by one or more skilful persons.

 

2. Underground captains direct the proper mining operations,

and govern the working miners.

 

3. The purser and bookkeeper manage the accounts.

 

4. The engineer erects the engines, and superintends the men

who work them.

 

5. A chief pitman has charge of the pumps and the apparatus

of the shafts.

 

6. A surface-captain, with assistants, receives the ores

raised, and directs the dressing department, the object of which

is to render them marketable.

 

7. The head carpenter superintends many constructions.

 

8. The foreman of the smiths regulates the ironwork and

tools.

 

9. A materials man selects, purchases, receives and delivers

all articles required.

 

10. The roper has charge of ropes and cordage of all sorts.

 

Notes:

 

1. An Enquiry into the Nature and Causes of the Wealth of

Nations, by Adam Smith.

 

2. Note sur la publication, proposee par le gouvernement Anglais

des grandes tables logarithmiques et trigonometriques de M de

Prony De l’imprimerie de F. Didot, December 1, 1829, p. 7

 

3. Since the publication of the second edition of this work, one

portion of the engine which I have been constructing for some

years past has been put together. It calculates, in three

columns, a table with its first and second differences. Each

column can be expressed as far as five figures, so that these

fifteen figures constitute about one ninth part of the larger

engine. The ease and precision with which it works leave no room

to doubt its success in the more extended form. Besides tables of

squares, cubes, and portions of logarithmic tables, it possesses

the power of calculating certain series whose differences are not

constant; and