Passages from the Life of a Philosopher - Charles Babbage

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1834 Dr. Lardner published, in the ‘Edinburgh Review,’12 a very elaborate description of this portion of the machine, in which he explained clearly the method of Differences.

      It is very singular that two persons, one resident in London, the other in Sweden, should both have been struck, on reading this review, with the simplicity of the math­e­mat­i­cal principle {48} of differences as applied to the calculation of Tables, and should have been so fascinated with it as to have undertaken to construct a machine of the kind.

      〈MR. DEACON—MR. SCHEUTZ.〉

      Mr. Deacon, of Beaufort House, Strand, whose mechanical skill is well known, made, for his own sat­is­fac­tion, a small model of the calculating part of such a machine, which was shown only to a few friends, and of the existence of which I was not aware until after the Swedish machine was brought to London.

      Mr. Scheutz, an eminent printer at Stockholm, had far greater difficulties to encounter. The construction of mechanism, as well as the math­e­mat­i­cal part of the question, was entirely new to him. He, however, undertook to make a machine having four differences, and fourteen places of figures, and capable of printing its own Tables.

      After many years’ indefatigable labour, and an almost ruinous expense, aided by grants from his Government, by the constant assistance of his son, and by the support of many enlightened members of the Swedish Academy, he completed his Difference Engine. It was brought to London, and some time afterwards exhibited at the great Exhibition at Paris. It was then purchased for the Dudley Observatory at Albany by an enlightened and public-spirited merchant of that city, John F. Rathbone, Esq.

      An exact copy of this machine was made by Messrs. Donkin and Co., for the English Government, and is now in use in the Registrar-General’s Department at Somerset House. It is very much to be regretted that this specimen of English workmanship was not exhibited in the International Exhibition.

      {49}

       Table of Contents

      Those who are only familiar with ordinary arithmetic may, by following out with the pen some of the examples which will be given, easily make themselves acquainted with the simple principles on which the Difference Engine acts.

      〈ARITHMETICAL TABLES.〉

      It is necessary to state distinctly at the outset, that the Difference Engine is not intended to answer special questions. Its object is to calculate and print a series of results formed according to given laws. These are called Tables—many such are in use in various trades. For example—there are collections of Tables of the amount of any number of pounds from 1 to 100 lbs. of butchers’ meat at various prices per lb. Let us examine one of these Tables: viz.—the price of meat 5 d. per lb., we find

Number. Lbs. Table. Price.
s. d.
1 0 5
2 0 10
3 1 3
4 1 8
5 2 1

      There are two ways of computing this Table:—

       1st. We might have multiplied the number of lbs. in each line by 5, the price per lb., and have put down the result in l. s. d., as in the 2nd column: or,

       2nd. We might have put down the price of 1 lb., which is 5 d., and have added five pence for each succeeding lb.

      Let us now examine the relative advantages of each plan. We shall find that if we had multiplied each number of lbs. in {50} the Table by 5, and put down the resulting amount, then every number in the Table would have been computed independently. If, therefore, an error had been committed, it would not have affected any but the single tabular number at which it had been made. On the other hand, if a single error had occurred in the system of computing by adding five at each step, any such error would have rendered the whole of the rest of the Table untrue.

      〈DIFFERENCES.〉

      Thus the system of calculating by differences, which is the easiest, is much more liable to error. It has, on the other hand, this great advantage: viz., that when the Table has been so computed, if we calculate its last term directly, and if it agree with the last term found by the continual addition of 5, we shall then be quite certain that every term throughout is correct. In the system of computing each term directly, we possess no such check upon our accuracy.

      Now the Table we have been considering is, in fact, merely a Table whose first difference is constant and equal to five. If we express it in pence it becomes—

Table. 1st Dif- ference.
1 5 5
2 10 5
3 15 5
4 20 5
5 25

      Any machine, therefore, which could add one number to another, and at the same time retain the original number called the first difference for the next operation, would be able to compute all such Tables.

      〈GROUPS OF MARBLES.〉

      Let us now consider another form of Table which might readily occur to a boy playing with his marbles, or to a young lady with the balls of her solitaire board. {51}

      The boy may place a row of his marbles on the sand, at equal distances from each other, thus—

      He might then, beginning with the second, place two other marbles under each, thus—

      He might then, beginning with the third, place three other marbles under each group, and so on; commencing always one group later, and making the addition one marble more each time. The several groups would stand thus arranged—

      He will not fail to observe that he has thus formed a series of triangular groups, every group having an equal number of marbles in each of its three sides. Also that the side of each successive group contains one more marble than that of its preceding group.

      Now an inquisitive boy would naturally count the numbers in each group and he would find them thus—

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