Much of Mr. Babbage's merit lay in the discovery that the operations of adding together any two of the ten Indian ciphers, and carrying a unit to a higher column of figures, whenever the sum of the added ciphers reached or exceeded ten, was the essential requirement in the most complex calculations-and this operation his machine can unfailingly execute. It would exceed the assigned limits of the present sketch to enter into the details of Mr. Babbage's invention. Many persons have now studied it, and two ingenious Swedish Engineers, Messrs. Scheutz, father and son, have been able to simplify the mechanical arrangement which performs the work. The object of the present sketch is to explain, that through the possession of the ten ciphers or arithmetical symbols now common, with the aid of pen and paper and simple directions for using the symbols, a computer has to attend closely to only two or three ciphers at a time, chiefly adding and subtracting and carrying tens; so that his mind is converted into a useful calculating machine, doing easy and correct work almost like that devised by Babbage. § 12. Besides the higher objects of making various scientific computations, the utility to mankind generally of having a fair amount of practical familiarity with counting and measuring, can hardly be overrated. With such knowledge, a carpenter can form more accurately his square boxes; a mason can erect better his upright walls; a mariner can steer his course more safely across the trackless waves. For want of such knowledge, persons, generally of the labouring classes, dependent for the support of their families on wages received at short intervals, are suffering multiplied difficulties and distress. These, and many others, would escape much of the unhappiness now common in the world, if they were early trained to the habit of balancing accounts of income and expenditure, as the basis of prudence and economy. It has been remarked that persons of the labouring classes, who once begin to keep accounts of expenditure, and to lay by a portion, however small, of their earnings, rarely come to want. § 13. There are but four fundamental operations in arithmetic; of these, separately, or in combination, the higher operations of computing and measuring consist, namely: Addition, Multiplication, Division. § 14. ADDITION Is the joining together, or fusing into one sum, two or more separate numbers. A ferry-boat may carry across a river in succession, the numbers of passengers here noted. A reckoner, having written. the figures in a vertical row, might begin counting from the bottom, and say-6 and 6 make 12, and 5 make 17, and 9 make 26, and 4 make 30, and lastly, 8 make 38, or three tens and eight ones or units. BA 3 8 4 9 5 6 8 After moderate practice in such work, the process is carried on almost as readily as when a person in reading connects the letters of printed lines into words. § 15. When the numbers dealt with are large, the ciphers are placed in distinct columns, as shown below, to be added and carried separately. For easy reference, lines are placed between the columns, with a letter at the top of every column. The reckoner begins at the bottom on the right hand, and finds in the column A, 18 units, or one ten and eight units over. The excess of units above ten is written at the bottom of the column, and the one ten is carried forward to the next column, B. This one, when added to the 16 tens already there, makes 17, of which the excess over ten is EDC BA 25380 23047 69661 1232 1037 8 written at the bottom, and the ten of tens is carried a step higher to the column C, which is of hundreds; and so the computer counts onward to whatever extent the numbering, in this way, is carried. The sum total in the example here given is ten thousand three hundred and seventy-eight. It is so useful to the student to be able to distinguish clearly the different columns, that this mode of illustration is continued in subsequent lessons. § 16. It is to be observed here that the cipher or zero (0) has no numerical meaning in itself, but serves to keep the other figures in their proper columns towards the left side. § 17. For general arithmetical counting, the carrying or standard number in summing up is ten. It would be an important simplification of the whole business of calculation, whether in science or for purposes of commerce, as in relation to money, weights, measures, &c., if the decimal system of subdividing large amounts and grouping smaller were adopted universally. A change towards this result is steadily progressing. Hitherto different countries have from accident adopted different systems. In England, for instance, the subdivision of money has long been into pounds, shillings, and pence, of which 12 pence form a shilling, and 20 shillings a pound. In adding amounts of money, therefore, the carrying number from pence to shillings is 12, and from shillings to pounds is 20, as exemplified in the adjoining example of four sums united into one. This is an operation the reverse of addition, telling how much is left when one number is taken from another. EDC B A 3 71 16 2 6 8 9 4 3 20 7 4 In working the two horizontal lines of figures are placed one above the other, as in addition, the larger uppermost. Then, instead of joining the two figures of the same column into one sum, as in addition, the difference of the two is noted and written at the bottom of the column. Here, in the column A of units, 4 is taken from 8, leaving the remainder or difference, 4, which is then written below. In column B of tens, 9 below has to be taken from 6 above, but as this cannot be done, a unit from the higher column, C, is borrowed, which makes 16 in the column B, from which the 9 being taken, 7 remain to be noted below. The reckoner then passing to the column C, has to deduct one (1) from the 2 above it, but this 2 having been now reduced to 1 by the borrowing for the B column, when another 1 is taken away, nothing is left. A zero (0) is therefore written below. The reckoner then proceeds to the column D, and taking 7 from 9, has 2 left to be written at the bottom of the D column. In the column E, there being no figure below to be subtracted, the figure 3 is carried down. BA 8 § 19. MULTIPLICATION. This is the operation, which, taking any number to be called the multiplicand (Latin for "what is to be multiplied"), and any other number to be called the multiplier, finds what sum or product arises when the multiplicand is added to itself as many times as there are units in the multiplier. Thus, if the multiplicand be 8 and the multiplier 6, the product is 48, for 6 times 8 makes 48. This operation may be considered as merely an abridged form of addition, and may be thus exhibited. 4 6 8 § 20. The process of multiplication is singularly abridged and facilitated when the student has committed to memory the socalled multiplication-table here exhibited. This may be regarded as a portion of the crossed-square described in page 703, having its divisions numbered at the boundary lines, commencing from one corner as A. At the meeting of the rows of small squares, proceeding inwards at right angles from the border lines, there is placed the product of the two border numbers. Thus, at the meeting of the rows, from the numbers 6 at the top and 6 on the side, the small square has the number 36 marked on it, which is the product of 6 times 6. Again, where the border numbers are 8 above, and 9 on the side, the lines meeting within show the number 72, which is the product of 8 and 9; and so for all other combinations. In this table proof is readily found of the following important facts. § 21. If two numbers (to be called factors) are multiplied together, it makes no difference which of them is taken first, that is to say, which is deemed multiplicand and which multi plier. For instance, the student's eye is led towards the same product, 63, whether he begins at the boundary from 7 or from 9, making 7 times 9 or 9 times 7. § 22. The space or area enclosed by any two lengths of the boundary lines, and corresponding lengths of parallel lines within, which space is always a rectangle, or that figure which has all its corners right angles, is exactly proportioned in 4 8 12 16 20 24 28 32 36 40 44 48 15 20 25 30 35 40 45 50 55 60 24 30 36 | 42 | 48 35 42 49 56 63 48 54 60 66 72 70 77 84 12 24 36 36 48 60 72 84 96 108 120 132 144 extent to the product of the two numbers at the extremities of the measuring lines. For instance, within the space bounded by the lines which meet within from the numbers 7 and 9, on the outside, there are just 63 small squares or equal portions of surface. Within the whole square, bounded by the lines A 12 on the top and A 12 on the side, there are just 144 of such small surfaces, and 12 times 12 make 144 in numbers. § 23. The exact relation thus discovered between the lengths of straight lines and the amount of surface bounded by them is a matter of high importance, and it further leads to the |