WHEN I was a little boy, I sometimes went for the bread to a short distance from the house. The baker would take my tallystick, put it alongside of his, and cut a notch in both. Then I would go away with my bread and the baker's account on the tallystick. At the end of a fortnight or a month the tallynotches were reckoned up and the account was settled. The number of notches represented the number of loaves of bread bought, and this number, multiplied by the price per loaf, gave the amount of money I had to take to the baker.
Although in our present article we shall make use of systems of numeration, and particularly of the decimal system, it is proper to observe that the most important properties of numbers are independent of such systems, and that they are used by the arithmetician in his calculations only for aids, as the chemist uses bottles and retorts. We give two specimens of properties of numbers, which we see illustrated in the problems called the flight of cranes and the square of the cabbages. Cranes in their flight dispose themselves regularly in triangles. We wish to get a rule for finding the number of the birds when we know the number of files; or, supposing that we have arranged the files with increasing numbers from unity to a determined limit, we seek to find the total of the unities contained in the collection. To make the matter plainer, let us seek the sum of the first six numbers, or the number of units represented to the left of the broken line in Fig. 1, by the black pawns. We will represent the same numbers, in an inverse order, by white pawns, to the right of the same line. We shall see at once that each horizontal line contains six units plus one; and, since there are six lines, the number of units in the whole square is six times seven. The number we are seeking, then, or the number in the halfsquare, is half of fortytwo, or twentyone. The same reasoning may be applied
to any number; so the sum of the first hundred numbers is half of a hundred times a hundred and one, or 5,050. Therefore, to obtain the sum of all the numbers of any series beginning with unity, all that is necessary is to take half the product of the last number of the series by the next one. An analogue of this mode of reasoning appears in the geometrical demonstration of the proposition that the area of the triangle ABC (Fig. 2) is half that of the parallelogram A B C D, of the same base and height; and reflection will show us that the arithmetical and the geometrical theorems are really one. The numbers we have just learned to calculate, which represent all collections of objects regularly disposed in triangles, are called triangular numbers. The theory of these numbers was born on the Nile at a remote epoch, and was developed by Diophantes, the father of arithmetic, at the school of Alexandria. In his treatise occurs the proposition, giving, as an Fig. 3.
Fig. 4. essential condition of such a number, that the octuple of a triangular number, augmented by unity, is a perfect square. This fact is made evident by the examination of the diagram (Fig. 3). An arithmetical progression is a series of numbers in which each member is equal to the preceding member, plus a constant number which is denominated the common difference. Thus the odd numbers one, three, five, seven, nine, eleven, form an arithmetical progression, the common difference of which is two. We can demonstrate, as was done in the preceding case, that the sum of the terms in such a procession is equal to the product of the number of terms by the halfsum of the extremes; and, in the same way, the area of a trapeze is half the area of a parallelogram of the same height, the base Fig. 5 The Square of Cabbages. of which is equal to the sum of the bases of the trapeze (Fig. 4).
Our second example is borrowed from Plato. Fig. 5 represents a square of cabbages. To get the number of cabbages contained in the square, we multiply by itself the number on one of the sides.
We have marked lines bounding the successive squares that contain one, two, three, four, five, or six cabbages to the side. Now observe the difference between the numbers of cabbages in one square and the next one. We find that the numbers included in the successive inclosures bordered by our lines are one, three, five, seven, nine, eleven; and we perceive, by reference to the short dotted lines, that the number from one inclosure to another increases by two. We come at once to the proposition that the sum of the odd numbers, beginning with unity, is the square of their number. Thus, the sum of those numbers between 1 and 199 is a hundred times 100, or 10,000.
Suppose we desire to make a table of the squares of all the numbers up to 1,000, for example, the way that first suggests itself is to make a thousand multiplications, 2 X 2, 3 X 3, to 999 X 999. But this method is of little value; it is exceedingly long, and there is no way of verifying it. We have a surer and more expeditious method. Fig. 6 represents the table for calculating the squares of the first ten numbers. The column D_{2}, which need not be written, contains 2's; column D_{1} represents the series of odd numbers, and may be written offhand; column Q may be formed after the following law, which applies to all the numbers in the table: Each number is equal to the one above it in the same column, augmented by the one that follows it in the same line. Thus, 81 ${\ce {=}}$ 64 ${\ce {+}}$ 17; and 19 ${\ce {=}}$ 17 ${\ce {+}}$ 2. A thousand additions of two numbers will then be sufficient to construct our table up to the square of 1,000. But here, it may be said, the results all depend one upon another; any error will carry itself to the next computation, and grow, like a snowball that at last becomes an avalanche, and overthrow the whole calculation. The remedy for this inconvenience is easy. When we have got the squares of the first ten numbers, we have only to add two ciphers to have the squares also of the
N. 
D. 
D_{1}. 
D_{2}. 

N. 
T. 
D_{1}. 
D. 
1 
1 
3 
2 

1 
1 
2 
1 
2 
4 
5 
2 

2 
3 
3 
1 
3 
9 
7 
2 

3 
6 
4 
1 
4 
16 
9 
2 

4 
10 
5 
1 
5 
25 
11 
2 

5 
15 
6 
1 
6 
36 
13 
2 

6 
21 
7 
1 
7 
49 
15 
2 

7 
28 
8 
1 
8 
64 
17 
2 

8 
36 
9 
1 
9 
81 
19 


9 
45 
10 

10 
100 



10 
55 


Fig. 6.—Squares. 

Fig. 7.—Triangular Numbers. 
numbers 20, 30, 40, etc., to 90; we write them immediately in the place they should occupy; and then we must get the same numbers again at the proper places in the course of our operations.
An arithmetical progression of the second order is one in which, if we form a series of the excesses of each number over the preceding one, we obtain numbers in arithmetical progression. Of this order are the series of the squares, and of the triangular numbers (Fig. 7).
There may be also arithmetical progressions of the third and fourth orders, and so on to infinity. They are all calculated in the same manner; and we take for a single example the series of the cubes of whole numbers, which form an arithmetical progression of the third order (Fig. 8). We get by direct calculation the first four terms, 1, 8, 27, 64; then, by subtraction, the first three terms of the column D^{1}, the first two of the column D^{2}, and the first term, 6, of D^{3}, The table is then completed by the law given above.
N. 
C. 
D_{1}. 
D_{2}. 
D_{3}. 
1 
1 
7 
12 
6 
2 
8 
19 
18 
6 
3 
27 
37 
24 
6 
4 
64 
61 
30 
6 
5 
125 
91 
36 
6 
6 
216 
127 
42 
6 
7 
343 
169 
48 
6 
8 
512 
217 
54 

9 
729 
271 


10 
1000 



These explanations are necessary to enable the reader to comprehend the function and classification of calculatingmachines. The method which is expounded in them appertains to the calculation of differences, and is applicable to all kinds of computations, whether of days' works, tables of interest and annuities, sinkingfunds and insurances, tables of logarithms, astronomical tables, or the resolution of numerical equations.
Numeration is based on the theory of geometrical progressions, by which name we call a series of numbers in which each member is equal to the preceding member, multiplied by a fixed number that is called the ratio. Thus the numbers 1, 10, 100, 1,000, 10,000 form a progression of which the ratio is ten, or a decimal progression. The numbers 1, 2, 4, 8, 16, 32, 64, the ratio of which is 2, form a binary progression.
The ancient Tartar hordes communicated with each other by means of sticks notched in an understood manner, so as to indicate the number of men and horses which each camp was expected to furnish. The IncaPeruvians had knotted cords of various colors, that could be tied in a thousand ways; and the number of knots, their arrangement, the tying of them with sticks, or around a central ring, permitted the expression of a variety of ideas, and of considerable series of numbers. The art of calculation is taught to children in some schools by means of apparatus consisting of a frame with ten rods, on each of which are strung ten counters; and the same kind of an apparatus is managed by the Chinese with much dexterity.
We propose, as an important aid to be used in teaching arithmetical calculation, a vertical checkerboard, the squares of which are furnished with pegs on which pawns may be slipped. No distinction is to be made between the white and black squares. "We begin by placing ten black pawns in the squares of the lower horizontal row. Lift the pawns in succession from the right, counting one, two, three, up to nine. This brings us to the top of the column. Take the pawn back to its zeroplace, and lift the pawn in the next column, up one place, calling it ten. Then we begin with the righthand pawn again, two, and count eleven, twelve, etc., to nineteen. Then bring the first pawn back to
Fig. 9.—The New Universal Abacus.
zero, and, lifting the second pawn to another square, call it twenty, to which we may add the units formed by raising the first pawn, as before. So we may go on with all the pawns, giving each successive piece, as we go to the left, ten times the value of the preceding one.
Our new abacus has the advantages that the value of its places increase in the same direction as the written numbers they represent, while the counters increase in arithmetical value as they are raised higher. As arranged in the cut the board represents the number 0,369,258,147. The capacity of this table, which is now equal to the expression of a thousand millions, may be indefinitely increased by adding columns to the left. The capacity of the board may also be changed by adding to it or subtracting from it in a vertical direction, whereby, instead of counting by tens, hundreds, and thousands, we may count by dozens, grosses, and so on, or by multiples of eight, six, four, two, or any other number. Every system of numeration is thus founded on the employment of units of different orders, each of which contains the preceding one a certain number of times, or, in other words, upon a geometrical progression, the ratio of which is called the base of the system, Aristotle observed that the number four might take the place of ten, and Weigel, in 1687, published a plan of a quaternary arithmetic. Simon Stevin, of Bruges, had previously devised a system of duodecimal numeration like the one we use in computing time and the degrees of the circle. The almost unanimous choice of the number ten as the basis of numeration was probably suggested by the ten fingers.
Instead of increasing the height of our abacus by two squares to explain the duodecimal system, let us put in its place a rectangle two squares high and of any desired width. We shall then have the system of binary numeration, and be able to write all the numbers with only two figures, and 1. The numbers one, two, three, four, five,
Fig. 10.—The Binary Abacus.
and six, may be formed on this system as in Fig. 10. This system furnishes the explanation of the Chinese symbol "JeKim, or Book of Mutations," which is attributed to the venerable Emperor FoHi. It is composed of sixtyfour figures, each formed of six horizontal lines written one over the other, some of them whole, others broken in the middle. The whole lines represent units of different degrees, rising from the lowest, and the broken lines zeros.
Fig. 11.—The Nine First Characters of the JeKim.
To return to our binary abacus: suppose the first pawn on the right weighs a gramme, the second two, the third four, and so on, doubling to the twelfth, which will weigh 512 grammes; with these twelve weights we can weigh all the whole numbers of grammes to 1023, the sum of all the preceding numbers. The principle is the same, fundamentally, as that of the wellknown ringpuzzle. A game was published toward the end of last year, professedly of IndoChinese origin, which was called the Tower of Hanoi. Fig. 12.—The Tower of Hanoi. I. The tower at the beginning of the game; II. Illustrating the process of the removal of the stories in the rebuilding of the tower; III. The tower rebuilt. The tower was composed of successive stones, decreasing in size as they rose, and represented by pawns, or buttons slipped upon a peg. The game consists in taking the pawns off from the peg and arranging them upon one of two other pegs in such a manner that only one of them shall be removed at a time, and a larger one shall never be put upon a smaller one (Fig. 12). The game is always possible, but demands twice as many removals and twice as much time for each story that is added to the tower. Having learned to rebuild the tower for eight stories by removing it from the first peg to the second one, the problem is made easy enough for one of nine stories; we first remove the eight upper stories to the second peg, then put the ninth story upon the third peg, and rebuild the eight stories upon it. To perform the operation, we must make, for a tower of two stories, at least three removals; for one of three, 7; for one of four, 15; for one of five, 31; for one of six, 63; for one of seven, 127; for one of eight stories, 255 removals. If it takes a second to make one removal, the rebuilding of a tower of eight stories will require four minutes. If the tower consists of sixtyfour stories, the readjustment will be a matter of 18,446,744,073,709,551,615 removals, and will occupy five million centuries. This prodigious number comes up again when we calculate the theory of the ringpuzzle of sixtyfour rings. According to an ancient Indian legend, the Brahmans took their turns day and night on the steps of the altar in the temple at Benares, to execute the readjustment of the sacred tower of Brahma, of sixtyfour stories, of fine gold set with diamonds. When they had done, the tower and the Brahmans would fall together, and then would be the end of the world. The principle of this game corresponds with that which is the basis of the binary system. By increasing the number of pins, and slightly modifying the rules, we can make it represent other systems.
The first machine for executing calculations by mechanical movements was invented by Pascal in 1642. It is illustrated in Diderot's "Encyclopædia," and in some editions of Pascal's works.
Every arithmetical machine is composed of four organs: the generator, the reproducer, the reverser, and the effacer. In Pascal's apparatus and in Roth's most recent modification of it the generator is very rudimentary, being nothing but a rod held in the hand. The reproducer is composed of wheels with ten or twelve cogs, mounted on parallel axes, the first wheel on the right representing units, the second tens, the third hundreds, and so on. Each of the wheels bears one or more sets of figures from 0 to 9, and has in front of it a sheet of metal pierced with an opening through which a single figure can be seen at a time. The mechanism is so adjusted that each wheel after the first one advances by one division or tooth as the wheel to the right of it advances from 0 to 9. Over the circumference of each wheel a notch in the coveringplate allows the generatorrod to be applied to the teeth of the wheel to move it as many numbers as may be desired. We can thus, by successive pushings and readings, perform any additions we wish. Multiplication is performed by successive additions, but the process is slow and tedious, on account of the inefficiency of the generator. The object of the third organ, the reverser, is to change addition into subtraction, and multiplication into division. In Pascal's machine, each of the figurebearing cylinders of the counter carries two scales, the reverse of each other, on parallel circles, the sum of the corresponding figures on which is always 9; so that the addition of four units of any order on one of the scales effects a subtraction of four units on the other scale. The object of the fourth organ, the effacer, is to bring all the numbers back to zero. In Roth's machine, 9 is made, by turning a button, to appear in the addition scale at all the openings; then the wheel is pushed around by the generator so as to add one, and appears in the place of the 9.
The Thomas arithmometer is a much more perfect and practicable machine. The generating apparatus is composed of a horizontal metallic plate, having parallel grooves, along which are written the figures from 0 to 9. Each groove has corresponding to it a button with an index, which may be slid in the direction of its length, and by means of which we can enter any number we choose. Each button is connected by a pendent wire, with a tentoothed pinion, below the tablet. By the side of each of the pinions is a cylinder with a horizontal axis, the length of which is the same with that of the grooving above it. Each of the cylinders bears projecting, upon half the circumference, nine nerves of successively increasing length, from ^{1}⁄_{10} to ^{9}⁄_{10} the length, and the motion of each cylinder is commanded by a horizontal shaft, which is turned by a crank. The cylinders make a revolution with each turn of the crank, but the pinions advance, each only according to the number of teeth marked by the corresponding index. Pinions mounted on the same axis with the index transmit the motion to the figurebearing wheels of the reproducer, which is placed under a metallic plate, prolonging the first one. Each turn of the crank produces the successive terms of an arithmetical progression. Suppose we wish to multiply 37,456 by 435. We bring down to zero the figures opposite the openings in the reproducer by means of an effacer, which is to be described. We write on the tablet the number 37,456; then turn the crank five times, when we will be able to read through the opening the product of 37,456 by 5; to get the product by 35, we would have to turn 35 times, but for an ingenious disposition by which we push the whole apparatus a notch to the right, and turn three times, which gives the product we are seeking; then pushing another notch to the right, and turning four times, we have the product by 435. The function of the reverser may be best explained by a comparison. Suppose a carriage of two wheels and an axletree, and a person is riding upon it holding an opened umbrella. As long as the umbrella is held straight over the middle of the axle, it does not move; incline it over one of the wheels, it begins to revolve; incline it over the other wheel, it will turn in the contrary direction, while the carriage will be all the time going straight ahead. In the arithmometer, the wheels of our carriage are replaced by twinpinions and the umbrella by the figurebearing wheel of the reproducer. By pushing on a little lever, we bring whichever pinion we desire into gear with the figurebearing wheel, so that each turn of the crank—it always turning in the same direction—brings successively before the openings numbers increasing or decreasing in an arithmetical progression, of which the common difference is marked on the abacus of the tablet.
Lastly, the effacer illustrates the advantage that may be drawn from a broken tooth. Below each figurebearing wheel is found, solid with it, another smaller toothed wheel in which the tooth corresponding with the below the peephole is suppressed. A roweled button pushes along a rack, that keeps the wheel turning, till the moment when the is to appear before the opening. The operation is performed with extreme rapidity, and is one among many admirable details in the performance of the machine. With this instrument, which my children learned to use when they were seven years old, the product of two numbers of ten digits can be obtained in half a minute; and it is used in numerous offices and institutions in France, the average sale of it being a hundred a year.
About a half a century ago, Charles Babbage undertook the construction of a universal calculatingmachine, which should give the successive terms of arithmetical progressions of different orders; but, having devoted the latter part of his life, and all of his fortune and income to it, died without finishing it. George Scheutz, of Stockholm, and his son, Edward Scheutz, exhibited a machine at the Paris Exposition of 1855, which was bought by a citizen of the United States and presented to the Dudley Observatory in Albany. It is shaped like a small piano, and by simply turning the handle gives out the successive terms of arithmetical progressions, not of the first order only, but of the second, third, and fourth orders.
Fig. 13.—The Table of Pythagoras on Slats.
John Napier, the inventor of logarithms, suggested an ingenious method of performing the operations of multiplication and division. The table in Fig. 13 represents the table of Pythagoras dissected into ten slats, of which the one on the left is fixed, while the others are movable and can be changed about at pleasure. Each of the squares of the table is divided by diagonals into two triangles, in the lower one of which is found the figure of the units of each of the products, and in the upper and left one the figure of the tens. If we place by the side of the fixed bar the slats bearing at the top the numbers 7, 5, and 8, we obtain almost immediately the products of 758 by all the numbers from 1 to 9. Thus, before the 6 of the fixed bar, we find, looking horizontally, 6  4/2  3/0  4/8 ; and by making the addition parallel to the diagonals, we have 4548 as the product of 758 by 6. The other products are got in the same manner. These slats then permit us to find rapidly—without having to know the table of Pythagoras, but by the simple addition of two figures—all the partial products by a number of ten figures. Thus, multiplication is again brought back to addition. This invention, however, has not become practical, because of the difficulty of finding the products when the multiplicand has two or more similar figures. An invention of our own gives it a more practical form. We have replaced the slats by square rules, containing four different numbers on each of the four faces, by which four tables of Pythagoras are included in the same space. But a little addition is still required for finding each of the partial products. M. Henri Genaille, an engineer of the state railroads, has devised a plan for substituting these additions by simple designs, that will permit all the partial products to be read instantly. The management of the rules is very simple, and may be learned at once. As perfected by us, this apparatus replaces the operations of multiplication and division by a simple addition, or a subtraction. With the boxes of the Genaille rules, each eighteen centimetres long, twelve wide, and one thick, we can obtain the partial products of all the numbers to twenty figures. With another disposition of the rules, on a larger scale, it will be possible to give all the products of numbers of ten figures by other similar numbers.—Translated for the Popular Science Monthly from the Revue Scientifique.