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Here any figure in the first place, reckoning from right to left, denotes only its own simple value ; but that in the second place, denotes ten times its simple value ; and that in the third place, a hundred times its simple value; and so on : the value of any figure, in each successive place, being always ten times its former value.

Thus, in the number 1796, the 6 in the first place denotes only six units, or simply six ; 9 in the second place signifies nine tens, or ninety ; 7 in the third place, seven hundred ; and the I in the fourth place, one thousand : so that the whole number is read thus, one thousand seven hundred and ninety-six.

As to the cipher, O, though it signify nothing of itself, yet being joined on the right-hand side to other figures, it increases their value in the same ten-fold proportion : thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five hundred ; and so on.

For the more easily reading of large numbers, they are divided into periods and half-periods, each half-period consisting of three figures; the name of the first period being units; of the second, millions ; of the third, millions of millions, or bi-millions, contracted to billions : of the fourth, millions of millions of millions, or tri-millions, contracted to trillions, and so on. Also the first part of any period is so many units of it, and the latter part so many thousands.

The

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The following Table contains a summary of the whole doctrine.

Periods. Quadrill; Trillions;Billions ; Millions; Units.

Half-per. th. un.

th. un.

th. un.

th, un.

th. un.

Figures. | 123,456 ; 789,098 ; 765,432 ; 101,234 ; 567,890.

NUMERATION is the reading of any number in words that is proposed or set down in figures; which will be easily done by help of the following rule, deduced from the foregoing tablets and observations-viz.

Divide the figures in the proposed number, as in the summary above, into periods and half periods ; then begin at the left-hand side, and read the figures with the names set to them in the two foregoing tables.

EXAMPLES

Express in words the following numbers; viz.
34
15080

13405670
96
72003

47050023
180
109026

309025600
304
483500

4723507689
6134
2500639

274856390000
9028
7523000

6578600307024

Notation is the setting down in figures any number proposed in words ; which is done by setting down the figures instead of the words or names belonging to them in the summary above; supplying the vacant places with ciphers where any words do not occur,

EXAMPLES.

Set down in figures the following numbers : Fifty-seven. Two hundred eighty six. Nine chousand two hundred and ten. Twenty-seven thousand five hundred and ninety-four. Six hundred and forty thousand, four hundred and eighty one. Three millions, two hundred sixty thousand, one hundred and six.

Four

Four hundred and eight millions, two hundred and fifty-five

thousar.d, one hundred and ninety-two. Twenty-seven thousand and eight millions, ninety-six thou

sand two hundred and four. Two hundred thousand and five hundred and fifty millions,

one hundred and ten thousand, and sixteen. Twenty-one billions, eight hundred and ten millions, sixty

four thousand, one hundred and fifty.

OF THE ROMAN NOTATION. The Romans, like several other nations, expressed their numbers by certain letters of the alphabet. The Romans used only seven numeral letters, being the seven following capitals : viz. I for one ; V for five ; X for ten ; L for fifty ; C for an hundred ; D for five hundred : M for a thousand, The other numbers they expressed by various repetitions and combinations of these, after the following manner :

I=I
2 = II

As often as any character is re3 = III

peated, so many times is its

value repeated. 4 = IIII or IV. A less character before a great5 = V

er diminishes its value. 6 = VI

A less character after a greater 7 VII

increases its value,
8 VIII
9

IX
10 X
50 L
100 =C
500 = D or I” For every ɔ annexed, this be-

comes 10 times as many. 1000 = M or CI” For

every C and ɔ, placed one 2000 = MM

at each end, it becomes 10

times as much. 5000 =

V or 155 A bar over any number in6000 = VI

creases it 1000 fold. 10000 =

ҳ or CCIээ 50000 = 1 or 1ɔɔɔ 60000 =

-LX 100000 = Tor Ccciɔɔɔ 1000000 = M or CCCCIɔɔɔɔ 2000000 &c. &c.

= MM

EXPLANATION OF CERTAIN CHARACTERS.

There are various characters or marks used in Arithmetic, and Algebra, to denote several of the operations and proposi. tions; the chief of which are as follows :

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+ signifies plus, or addition.

minus, or subtraction. X or multiplication.

division.
proportion.
equality.
square root.
cube root, &c.
diff. between two numbers when it is not known

which is the greater.

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Thus, 5 + 3, denotes that 3 is to be added to 5. 6 2, denotes that 2 is to be taken from 6. 7 X 3, or 7.3, denotes that 7 is to be multiplied by 3. 8 • 4, denotes that 8 is to be divided by 4. 2:3 ::4:6, shows that 2 is to 3 as 4 is to 6. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. ✓ 3, or 3, denotes the square root of the number 3. Ñ 5, or 5}, denotes the cube root of the number 5. 72, denotes that the number 7 is to be squared. 89, denotes that the number 8 is to be cubed.

&c.

OF ADDITION.

ADDITION is the collecting or putting of several numbers together, in order to find their sum, or the total amount of the whole. This is done as follows :

Set or place the numbers under each other, so that each figure may stand exactly under the figures of the same value,

that

that is, units under units, tens under tens, hundreds under hundreds, &c. and draw a line under the lowest number, to separate the given numbers from their sum, when it is found. Then add up the figures in the column or row of units, and find how many tens are contained in that sum.-Set down exactly below what remains more than those tens, or if nothing remains, a cipher, and carry as many ones to the next row as there are tens-Next add up the second row, together with the number carried, in the same manner as the first. And thus proceed till the whole is finished, setting down the total amount of the last row.

TO PROVE ADDITION.

First Method.-Begin at the top, and add together all the rows of numbers downwards ; in the same manner as they were before added upwards; then if the two sums agree, it may be presumed the work is right. This method of proof is only doing the same work twice over, a little varied.

Second Method.-Draw a line below the uppermost number, and suppose it cut off. Then add all the rest of the numbers together in the usual way, and set their sum under the number to be proved.-Lastly, add this last found number and the uppermost line together; then if their sum be the same as that found by the first addition, it may be presumed the work is right. This method of proof is founded on the plain axiom, that“ The whole is equal to all its parts taken together."

Third Method.-Add the figures in the uppermost line together, and find

EXAMPLE 1, how many nines are contained in their sum.-Reject those nines, and 3497 set down the remainder towards the 6512 right-hand directly even with the 8295 figures in the line, as in the annexed example. Do the same with each 18304 of the proposed lines of numbers, setting all these excesses of nines in a co. lumn on the right-hand, as here 5, 5, 6. Then, if the excess of 9's in this sum, found as before, be equal to the excess of 9's in the total sum 18304, the work is probably right.Thus, the sum of the right-hand column, 5, 5, 6, is 16, the excess of which above 9 is 7. Also the sum of the figures in VOL. I.

the

Excess of nines.

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