be required to subtract 468 from 879. The common method would be as follows: From 879 Take 468 411 The arithmetical complement of 468, or 468 deducted from 1000, is 532. To 532 1,411 and if the first figure on the left hand be struck out of the sum the result will be 411. By the introduction of logarithms and their complements, multiplication and division are also reduced to the operation of addition. It has been stated that the principle upon which the machine computes is by the method of differences, but perhaps the principle will be rendered more clearly intelligible when illustrated by some examples. If the price of wheat per quarter in England is 44s. 3d. = 44:25. Then, 1 quarter 81 82 (Constant). = 44.25 44.25 2 quarters 8 9 10 The above table has been calculated by the addition of 44.25 successively to each term, as it is by this method of differences that the machine would have arrived at the result of the price of the 13 quarters of wheat, 575.25. The results can be checked at any stage of the work by simple arithmetic. 44.25 13 13275 4425 575.25 The machine is capable of receiving four sets of differences, besides the function, as it has five tiers of figure-wheels; but for the sake of illustrating the above example, let us suppose the machine to have only two tiers of figure-wheels, viz., the top, or function row, where the results appear, and a second, or bottom row of figurewheels, which remain constant; then setting 82 44′25 on the function-wheels, and 81 44 25 on the bottom tier of figure-wheels, and turning the handle of the machine, the effect of the first half stroke will be to add the numbers of d1 to those of d2, and the result 88'050 will appear on the function-wheels, the bottom row of figure-wheels retaining their original setting for the next operation. By turning the handle of the machine continuously, the remaining terms will be successively calculated. Now, if the printing-table be released from the rack, we shall find that the results, as they appear in the above table, have been stereomoulded on a slip of papier-maché, inserted when the machine was set, the argument or number of quarters of wheat having been impressed on the paper by the numerator. Having endeavoured to explain, by a very simple illustration, the principle upon which the machine computes, let us proceed to consider its adaptation for the construction of another form of table, embracing all the tiers of figure-wheels. The machine has been principally engaged, at the General Register Office, in calculating the fundamental columns of the new English Life-table to which reference has already been made. The preliminary steps of the work, such as calculating the mortality, at decennial periods of age, and deducing the differences, are performed without the aid of the machine; but when the differences are once obtained, the machine is able to take up the work and continue the calculation by the differential method. There are numerous tables where the last order of differences would not afford a series of rigorously constant differences; but we can almost always obtain a certain number of differences which, within certain limits, remain constant for a long succession of terms; and by this principle of differences, all tables in which the series continually increase or decrease may be produced by the operation of addition, provided the first terms of the table, and of each series of differences, be given to start with. The logarithms of the probability of males living a year at the ages 20, 30, 40, and 50, having been derived directly from observation, the machine was required to calculate, by the mere addition of the differences, either the intervening logarithms of the probability of living a year, or the logarithms of the numbers living, at each of the 31 ages from 20 to 50. By adding the differences successively to each other, to the logarithm of the probability at age 20, and then to the logarithm of the numbers living at age 20, the logarithms of the numbers living at the successive ages 21, 22, 23, 24, up to 50, were calculated by the machine. Example, from the English Life-table, of the method of setting the differences in the machine, in order to interpolate the logarithms of the number of males living at each of the ages 20 to 50: But the differences for insertion in the machine have to be taken = = U20 5.5232361,5000 Setting for function. = AP198119 9·9964712,3645 Ux = 81x-1= 82x-1 83x-2 84x-2 18 9.9999161,6355 0-0000043, 1270 8418 9.9999991,6530 Taken from age 19. Taken from age 18. Example, showing the effect of each half stroke of the machine : ..... 5.5232361,5000 9.9963874,0000 5.5196235,5000=u21 9.9963874,0000=8120 9.9999196,4155=8220 0.0000034,7800=8319 9*9999991,6530=8419 After the machine has been once set as above it will go on calculating the series algo, λ,21 22, Also by turning the handle continuously. At the first half stroke, in the above example, the numbers on the 3rd and 5th tiers are added to the tiers immediately above them, and appear as the 2nd and 4th tiers in column headed "After forward motion," the second half stroke adds these new numbers to the numbers on the 1st and 3rd rows, and thus gives u11 = logarithm of the numbers living at age 21=5.5196235,5000. Besides calculating and stereoglyphing the fundamental columns of the New English Life-table, without interest, the machine has also been used to introduce the element of interest in the various tables (v), the logarithms of the values of Life Annuities depending on two lives, two males, a male and female, or two females, have also been worked out by the machine. It is able to compute tables of logarithms of the natural numbers, tables of mensuration, and of the powers of sines, natural sines, and astronomical and nautical tables, &c. &c. It is found that the machine will calculate and stereomould results in 20 minutes which would take an ordinary computer at least of an hour to calculate alone. Mr. Gravatt states that "at the average rate of working the machine 120 lines per hour of arguments and results are calculated, and actually stereotyped ready for the press. It is found on trial that the machine calculates and stereotypes, without chance of error, 24 pages of figures in the same time that a skilful compositor would take merely to set up the types for one single page." By the machine, chances of error which would undoubtedly arise in the ordinary way of computation, are avoided both in the calculation and setting up the figures for printing. In One of the most prominent features of the machine is the method by which it performs the operation of carrying the remainders. order to effect this, the figure-wheels are brought into action by arms affixed to each of two travelling-bars, which are driven by an endless chain. The machine, by an ingenious arrangement, prepares the mechanism for carrying the tens whilst making the addition, the carriages of the remainders being an after process. When any wheel in the process of addition passes from 9 to 0, a small projection on the right hand figure-wheel starts out a small lever (by touching its tail) in front of the next left hand figure-wheel, and when the arm of the travelling-bar, which glides along the front of the first and third rows of figure-wheels, finds the levers out, it is compelled to replace them, and in so doing touches one of the teeth underneath the figure-wheel, thus causing the carriage of the figure which its position indicated to be required. The same process is effected at the back of the figure-wheels for the second and fourth rows, the fifth row of figure-wheels requiring no carriage, as it remains constant. When several carriages have to be made, they are effected by the machine thus :To 5654 Add 9878 4422, after forward motion; 1111 15532, after backward motion. The remainders, or tens, are carried, one after the other, successively by the arms of the travelling-bars, after the 4422 have been computed by that part of the machine which performs the process of addition without carriage. Thus have we attempted to convey to our readers a general notion of the mathematical principle on which the machine is founded, and of the manner in which that principle is brought into practical operation. Those who are desirous of obtaining clear and precise notions of the details by which mechanism is employed as the substitute for a mental operation, are referred to the "Specification of the Patent,' No. 2216, 1854, and to the observations of Mr. Babbage on the "Mechanical Notation" of the Swedish Machine. F. J. WILLIAMS. NOTE.-The columns le for males and females, in the Table on page 14, are taken from the machine logarithms (Al) they form the basis of the English Life-table, and show, out of a given number of males and females born, the numbers attaining each age, or birthday, from the Ist to the 108th, according to the rate and laws of mortality at different ages in England and Wales, deduced from the enumerated population 1841 and 1851, and from the deaths registered in the 17 years 1838-1854. The persons were obtained by adding the males and females together. The Table may be read thus:-Of 1,000,000 children born alive, 511,745 are boys and 488,255 are girls; 428,026 boys and 422,481 girls are living at the age 1; therefore 83,719 boys and 65,774 girls die in the first year; or of 1,000,000 children born, 850,507 are alive at age 1, 149,493 having died in the first year. At age 20, only 662,750 are aliye, 337,250 having died before attaining that age, and so on. |