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19

20

40

60

100

300

1000

2000

10,000

100,000, &c.

twenty-three.
twenty-four.
forty-eight.

seventy-two.

.. a hundred and forty-four.

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four hundred and thirty-two.

..one thousand seven hundred and twenty-eight.
three thousand four hundred and fifty-six.

twenty thousand seven hundred and thirty-six.

two hundred and forty-eight thousand eight hundred and thirty-two.

Thus the number denoted by the figures 4943 would be eighteen thousand six hundred and twenty-seven; for 4000 is eighteen thousand two hundred and eighty, 900 is one thousand two hundred and ninety-six, 40 is forty-eight, and 3 is three numbers, which if added will form the above sum.

It would be easy to form a set of rules for this new arithmetic, similar to those of common arithmetic; but as it does not seem likely that this mode of calculation will ever be brought into general use, we shall confine ourselves to what has been already said on the subject, and only add, that a book was printed in Germany, in which the common rules of arithmetic were explained in all the systems, the binary, ternary, quaternary, and so on, to the duodenary inclusively.

ON THE ARITHMETIC OF THE GREEKS.

The Greeks divided all their numbers, as we do ours, into periods of tens; but for the want of the happy idea of giving a local value to their numerical symbols, they were obliged to employ thirty-six characters, most of which were derived from their alphabet.

.....

Thus our digits .1, 2, 3, 4, 5, 6, 7, 8, 9,
Were represented by..a, 8, 7, 8, ε, s, %, n, 0,

And 10, 20, 30, 40, 50, 60, 70, 80, 90

by, x, My fly », §, 0, π, G

For hundreds they had p, 6, T, u, P, x, y, w, 71

And for the thousands they had recourse again to the characters of the simple units, with the addition of a little dash below. Thus,a= 1000,,62000, &c.

With these characters they could express any number under 10,000, or a myriad.

Thus 991 was 7 ça; 7382, τ 8; and 4001,,dx.

α M

Στοβ
M

for

In order to express myriads, they placed the letter M below the character representing the number of myriads they intended to indicate, as for 10000, 43720000. This is the notation employed by Eutocius, in his commentaries on Archimedes.

Diophantus and Pappus represented their myriads by the letters Mv, or more simply still, by a point, placed after the number, thus 43728097 is expressed by Στοβ. Μυ ης ζ, or δτοβ. η ς ζ.

The number 100,000,000 was the greatest extent of the Greek arithmetic; but Archimedes indefinitely increased it, when he invented his system of Octates, or periods of eight. He assumed 100,000,000 as a new unit, and called the numbers which he formed with it numbers of the second order; then assuming the square, cube, &c. of 100,000,000 successively as a new unit, he obtained numbers of the third, fourth, and higher orders.

This idea of Archimedes, we are informed by Pappus, Apollonius greatly improved;

6

by reducing the octates to periods of fours, and dividing all numbers into orders of myriads; thus the number...

written according to the notation of Appollonius, is

......

........

793 2.3846. 2643; ζηλβ. γωμ;. βχμγ;

the first period of four to the right being units, the next myriads, the next double myriads, or numbers of the second order. The next would be numbers of the third order, and so on indefinitely.

Having given a local value to his periods of fours, it was only necessary to have done the same for single digits, to have arrived at the system in present use; and it is astonishing that he did not perceive the advantages of doing so; and the more singular, as the use of the cipher was not unknown to the Greeks, being always employed in their sexagesimal operations, when it was necessary, as in the division of the circle, of which ours is still a representative, as is evident from the following example:

06 n" """ """" 0° 59′ 8′′ 17′′ 13""

Having given an idea of the Grecian notation for integer numbers, we next proceed to their method of representing fractions; which was by placing the denominator above, and to the right of the numerator; thus represented. But when the numerator was unity, a small dash was placed to the right of the denominator, as y for, for. And the fraction had a particular character, as c or ▲, c' or K'.

It now remains for us to explain the method employed by the ancients in performing the common rules of arithmetic with this complicated system of notation.

The examples below, of addition and subtraction, require no explanation, being performed exactly as we do ours, proceeding from right to left; but to this method, though so clearly the most simple, the Greeks did not constantly adhere, as there are many instances which make it evident that they did both addition and subtraction from left to right.

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In multiplication they usually proceeded from left to right, as we do in multiplication of algebra, and placed their successive products without much apparent order; but as each of their characters retained its own proper value wherever it was placed, this want of order only rendered the addition a little more troublesome.

In the examples which follow we will mark the myriads by an m, the thousands by "", the hundreds by ", &c., and so make the partial products in the Greek, and the translation identical. With this arrangement the reader will find it extremely easy to follow the work.

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The division of the Greeks was still more intricate than their multiplication; for which reason, it seems, they generally preferred the sexagesimal division; and no example is left at length by any of their writers, except in the latter form; but these are sufficient to throw some light on the process they followed in the division of common numbers; and Delambre, in an essay subjoined to the French translation of Archimedes, has accordingly supposed the following example:

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The example will be found, on a slight inspection, to resemble that sort of division in which we divide feet, inches, and parts by similar denominations, which, together with the number of characters they employed, must have rendered this rule extremely laborious; and that for the extraction of the square root must have been equally difficult; the principle of which was the same as ours, except in the difference of the notation; though it appears that they frequently, instead of making use of the rule, found the root by successive trials, and then squared it to prove the truth of their assumption.

CHAPTER II.

OF SOME SHORT METHODS OF PERFORMING ARITHMETICAL OPERATIONS.

SECTION I.

Method of Subtracting several Numbers from several other given Numbers, without making partial Additions.

56243

84564

A

3252

26848

2942 3654 B 2308

To give the reader an idea of this operation, one example will be sufficient. Let it be proposed to subtract all the sums below the line at B, from all those above it at A. Add, in the usual manner, all the lower figures of the first column on the right, which will make 14, and subtract their sum from the next highest number of tens, or 20. Add the remainder 6 to the corresponding column above at A, and the sum total will be 23. Write down 3 at the bottom, and because there were here two tens, as before, there is nothing to be reserved or carried. Add, in like manner, the figures of the second lower column, which will amount to 9, and this sum taken from 10 will leave I; add 1 therefore to the second column of the upper numbers, the sum of which will be 20; write down 0 at the bottom, and because there were here two tens, while in the lower column there was only one, reserve the difference, and substract it from the next column of the numbers marked B before you begin to add. In the contrary case, that is to say when there are more tens in any one of the columns marked B than in the corresponding column above it, the difference must be added. In the last place, when it happens that this difference cannot be taken from the next column below, for want of more significant figures, as is the case here in the fifth column, we must add it to the upper one, and write down the whole sum below the line. By proceeding in this manner, we shall have, in the present instance, 162003 for the remainder of the subtraction required.

SECTION II.

162003

Some Short Methods of performing Multiplication and Division.

I. Every one, in the least acquainted with arithmetic, knows, that to multiply and

number by 10, nothing is necessary but to add to it a cipher; that to multiply by 100, two must be added, and so on.

Hence it follows, that to multiply by 5, we have only to suppose a cipher added to the number, and then to divide it by 2. Thus, if it were required to multiply 127 by 5; suppose a cipher added to the former, which will give 1270, and then divide by 2: the quotient 635 will be the product required.

In like manner, to multiply any number by 25, we must suppose it multiplied by 100, or increased by two ciphers, and then divide by 4. Thus 127 multiplied by 25, will give 3175. For 127 when increased by two ciphers makes 12700, which being divided by 4, produces 3175.

According to the same principle, to multiply by 125, it will be sufficient to add three ciphers to the multiplicand, or to suppose them added, and then to divide by 8. The reason of these operations may be so readily conceived, that it is not necessary to explain it.

II. The multiplication of any number by 11 may be reduced to simple addition. For it is evident that to multiply a number by 11, is nothing else than to add the number to its decuple, that is to say, to itself followed by a cipher.

Let the proposed number, for ex., be

67583 743413

To multiply this number by eleven, say 3 and 0 make 3; write down 3 in the units place; then add 8 and 3, which makes 11; write down 1 in the place of tens, and carry 1; then 5 and and 1 carried make 14; write down 4 in the third place, or that of hundreds, and carry 1. Continue in this manner, adding every figure to its next following one, till the operation is finished, and the product will be 743413, as above.

The same number may be multiplied in like manner, by 111, if we first write down the 3, then the sum of 8 and 3, then that of 5, 8, and 3, then that of 7, 5, and 8, and so on, adding always three contiguous figures together

III. We shall only farther observe, that to multiply any number by 9, simple subtraction may be employed. Let us take, for example, the same number as before,

67583

608247

To multiply this number by 9, nothing is necessary but to suppose a cipher added to the end of it, and then to subtract each figure from that which precedes it, beginning at the right. Thus 3 from 0 or 10, leaves 7; 8 from 2 or 12, leaves 4; and if we continue in this manner, taking care to borrow 10 when the right-hand figure is too small to admit of the preceding one being subtracted from it, we shall find the product to be 608247.

The reason of these operations may be readily perceived. For it is evident, that in the first we only add the number itself to its decuple; and in the latter, we subtract it from its decuple. But in order to form a clearer idea of the matter, it may perhaps be worth while to perform the operation at full length.

Concise operations of a similar kind may be employed in certain cases of division; as in dividing, for example, a given number by any power whatever of 5. Thus, if it were required to divide 128 by 5; we must double it, which will give 256; if we then cut off the last figure, which will be a decimal, the quotient will be 25-6 or 25%. To divide the same number by 25, we must quadruple it, which will give 512; and if we then cut off the two last figures as decimals, we shall have for the quotient 5 and. To divide by 125, we must multiply the dividend by 8, and cut off three figures. In like manner we may divide a given number by any other power of 5; but it must be confessed that such short methods of calculation are attended with no great advantage.

SECTION III.

Short Method of performing Multiplication and Division by Napier's Rods or Bones.

When large numbers are to be multiplied, it is evident that the operation might be performed much more readily, by having a table previously formed of each number of the multiplicand, when doubled, tripled, quadrupled, and so on. Such a table indeed might be procured by simple addition, since nothing would be necessary but to add any number to itself, and we should have the double; then to add it to the double, and we should have the triple, &c. But unless the same figure should frequently recur in the multiplicand, this method would be more tedious than that which we wished to avoid.

The celebrated Napier, the sole object of whose researches seems to have been to shorten the operations of arithmetic and trigonometry, and to whom we are indebted for the ingenious and ever-memorable invention of logarithms, devised a method of forming a table of this kind in a moment, by means of certain rods, which he has described in his work entitled Rabdologia; printed at Edinburgh in 1617. The construction of them is as follows:

8

Fig. 1. Provide several slips of card or ivory or metal rods, about nine times as long as they are broad, and divide each of them into 9 equal squares. (Fig. 1.) Inscribe at the top, that is to say in the first square of each slip or rod, one of the numbers of the natural series 1, 2, 3, 4, &c., as far as 9 inclusively. Then divide each of the lower squares into two parts by a diagonal, drawn from the upper angle on the right hand to the lower one on the left, and inscribe in each of these triangular divisions, proceeding downwards, the double, triple, quadruple, &c. of the number inscribed at the top; taking care, when the multiple consists of only one figure, to place it in the lower triangle, and when it consists of two to place the units in the lower triangle, and the tens 64 in the upper one, as seen in the figure. It will be necessary to have one of these slips or rods, the squares of which are not divided by a diagonal, but inscribed with the natural numbers from 1 to 9. This one is called the index rod. It will be proper also to have several of these slips or rods for each figure.

Fig. 2.

1678

T

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3.

399

1/ 161818

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3/2 2 1/2 36

5 3 L 2 031 075

3

4

6

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The rods being prepared as above, let us suppose that it is required to multiply the number 6785399. Arrange the seven rods inscribed at the top with the figures 6785, &c., close to each other, and apply to them on the left hand the index rod, or that inscribed with the single figures (Fig. 2); by which means we shall have a table of all the multiples of each figure in the multiplicand; and scarcely any thing more will be necessary but to transcribe them. Thus, for example, to multiply the above number by 6; looking for 6 on the index rod, and opposite to it in the first square, on the right hand, we find 54; writing down the 4 found in the lower triangle, and adding the 5 in the upper one to the 4 in the lower triangle of the next square on the left, which makes 9; write down the 9, and then add the 5 in the upper triangle of the same square to the 8 in the lower triangle of the next one; and proceed in this manner, taking care to carry as in common addition, and we shall find the result to be 40712394, or the product of 6785399 multiplied by 6.

18 18. 7

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