understanding, they merit our most scrupulous attention, even from equally important considerations; inasmuch as they lie at the very basis of our finest commercial speculations; are intimately connected with the various branches of Mechanical Philosophy; with almost all the great sources of national wealth and power; and with many of those arts which contribute to the elegance and comfort of social life.

PART I.

ARITHMETIC.

If

1. ANY single object whatever is called a unit or one. to this unit another be added, these together form the number called two; which increased in like manner by another unit, gives the next greater number denominated three. By continuing to add unity in this way to the number already obtained, the successive numbers four, five, six, &c. are formed. From this method of conceiving the formation of number, it is obvious that there is no limit to its magnitude; for, however great any number may be, it may still be augmented by a unit, or by any number of units.

2. In the early stages of society, when words, the signs of ideas, were but few, mankind would naturally represent any small number by some palpable symbols, such as nuts, pebbles, or shells arranged in a row. It would be found, however, when the objects to be represented were numerous, that this method of arranging them in a single row would only convey a very confused idea of multitude. In order, therefore, to greater precision, they would subsequently form them into two rows, or count them off by pairs. Hence the origin of the dual number, and of the terms brace, couple, &c. Thirty-one, for example, might be represented by as many small shells or pebbles, which, when ranged into successive pairs, would amount to fifteen pairs

and one over, thus:

If a pebble or shell of double the size were now taken to represent each pair, then the number, by the same process of arrangement, would be represented by seven pairs of these double-sized counters and one over, to

gether with one counter of the first size, thus:

Again, if counters of double the size of these last be taken, the number will be represented by three pairs of this third size of counters and one over; one of the se

cond size, together with one of the first,

thus:

In order to farther condensation, if each pair were represented, as before, by a counter still larger, then the whole number would be represented by three counters

of the fourth size, one of the third, one of the

second, and one of the first, thus:

Lastly, If a counter be taken equal to two of this fourth size, then the number, thirty-one, will be represented by one counter of the fifth size, one of the fourth, one of the third, one of the second, and one of the first, and

would stand thus:

Such are the steps by which we may conceive a savage, prompted by curiosity, to have proceeded in forming a condensed representation of any number of similar objects.

This mode of classifying numbers, by distributing them into successive pairs, is called the Binary scale, from the circumstance of its root or index being two.

3. The inconvenience or utter impossibility of procuring natural objects of the necessary gradation of size, would naturally suggest the idea of affixing a certain gradation of va lue to rank, independent of magnitude. This important link in the chain of improvement was made by the Greeks at a very early period, and by them communicated to the Romans, who continued, during their whole career of empire, to practise a kind of tangible Arithmetic. In this, it is evident, that the augmenting value of rank would be quite arbitrary, depending in every case on a key, to be fixed upon by convention. If it be required, for example, to represent the number thirty-one, it being agreed that the value of each successive counter, reckoning from right to left, is dou

ble of that which precedes it, it will stand thus: edc ba Where the counter a will represent one unit,

b two units, c four units, d eight units, and e sixteen units, in all thirty-one units.

4. The general method of representing any number on the Binary scale will be readily understood from a few words of explanation. Let there be a series of parallel bars, and suppose that it is required to represent the number fifty-seven. Instead of placing the fifty-seven counters on the first bar, it will amount to the same thing to place twenty-eight on the second, and one on the first, since each of the counters on the

second bar represents two units. In the same manner twenty-eight on the second bar will be represented in effect by fourteen on the third, or by seven on the fourth. Again, seven on the fourth bar will be the same as three on the fifth and one on the fourth. But three on the fifth will be equivalent to one on the sixth and one on the fifth. Hence the representation of fifty-seven on the Binary scale will be effected by one counter on the first

bar, one on the fourth, one on the fifth, and one on the sixth, thus:

By a similar process of decomposition seventy-nine would be represented thus:

One hundred and eighty-two thus :

To represent any number on

this scale, it is obvious that one

counter for each bar is sufficient; because, instead of having two counters on any one bar, we may place one counter on the next higher.

5. Similar to the Binary is the Ternary scale of numeration, which reckons by successive threes or triads. On this scale a counter on any bar is equivalent to three counters on the bar immediately below it.

Let it be required to represent the number fifty-seven on this scale. In place of having fifty-seven counters on the first bar, we may have nineteen on the second, since nineteen threes or triads are exactly fifty-seven units. Again, nineteen counters on the second bar are equivalent to six on the third, or two on the fourth bar, and one on the second; fiftyseven will therefore be represented on the Ternary scale, by one counter on the second bar, and two on the fourth, thus:

Seventy-nine would be represented thus:

One hundred and twenty-eight thus: In this scale it is evident that two sets of counters are necessary; or, in figurative

Arithmetic, two artificial characters. Let the characters 1 and 2 respectively denote one counter and two counters, and let the place of a bar on which no counter is placed be mark

ed by the cipher (0); then any number may be easily expressed on this scale.

Fifty-seven would be expressed thus:
Seventy-nine thus:

2010

2221

One hundred and twenty-eight thus: 11202

6. From what has been stated, it will be easy to represent any number, on any scale whatever. The Quaternary scale proceeds by fours-the Quinary by fives-the Senary by sixes-the Septenary by sevens-the Octary by eights— and the Nonary by nines. The Quaternary scale obviously requires three sets of counters for each bar, and, of course, three artificial characters-the Quinary five, or in general any scale requires one less than its index or root. Let it be required to represent the number one hundred and seventy-five by the counters, and also to express it by the artificial characters on these various scales. follows:

By Counters.

It will stand as

By the Artificial
Characters.

7. The Denary scale, which advances by tens, seems to have prevailed more generally, obviously from the familiar practice of counting by the fingers on both hands. This scale will evidently require nine sets of counters, and nine artificial characters.

Seventy-eight on the Denary scale would stand thus:

And by the artificial characters, thus: 78.

As the increased number of counters necessary for this scale might be inconvenient, an advantage will be gained by employing a counter of a larger size, to denote half the index of the scale, or five. Seventy-eight would then be represented in the abridged form thus:

8. The Denary scale was employed by the ancient Greeks and Romans, who performed their calculations by counters, on a board called an Abax or Abacus. A series of parallel grooves represented the bars, and the counters were originally pebbles, or small white stones, called Calculi; from which arose the verb calculare, and hence also our English term to calculate.

The Chinese use a board called a Swanpan, on which they perform calculations with astonishing rapidity. The swanpan, although a little dissimilar to the abacus in construction, is the same in principle. In it, the parallel grooves stretched transversely or across the board, from right to left, whereas in the abacus they run longitudinally, or up and

down.

This method of reckoning by palpable symbols seems to have been almost universally prevalent throughout Europe until the fifteenth century, when it began to give place to the more convenient and expeditious mode, by written characters. There are, however, several places in Russia where this tangible arithmetic still prevails, as the only means of calculation. In Shakespeare's comedy of the Winter's Tale, written at the commencement of the seventeenth century, the