AMERICAN ARITHMETIC,

AND

PRACTICAL MENSURATOR.

ARITHMETIC is (1) the art of computing by numbers; these numbers are called (2) Figures, viz: (3) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, (4)—0, of itself has no value, but when joined to the right of other numbers, it increases their value in a tenfold proportion. () Thus, 1 is one, 10 is ten, 2 is two, 20 is twenty.

There are (6) Five FRINCIPAL or FUNDAMENTAL RULES in Arithmetic, viz:-(7) NOTATION, NUMERATION, ADDITION, SUBTRACTION, MULTIPLICATION and DIVISION, and are so called () because all mathematical Questions are dependent on them for their operation.

Questions.-1. What is Arithmetic?-2. What are these numbers called?-3. Name them.-4. Describe Zero or Cypher.-5. Give an Example.-6. How many Primary or Fundamental Rules are there in Arithmetic?-7. Name them.-8. Why are they so called?

ON entering upon this most useful study, the first thing which the scholar has to regard, is

(1) NOTATION.

NOTATION is (2) the art of expressing numbers by certain characters or figures: of which there are two methods. 1. (4) The Roman method, by Letters. 2. (5) The Arabic method, by Figures. (6) The latter is that of general

use.

In the Arabic method all numbers are expressed (7) by these ten characters or figures, viz.

1

2

3

4

5

6

8

9

0

Unit; two; three; four; five; six; seven; eight; nine; cyor one S [pher or nothing. The nine first are called () significant figures or digits, each of which standing by itself or alone, invariably expresses (9) a particular or certain number; (10) thus, í sig

nifies one, 2 signifies two, 3 signifies three, and so of the rest, until you come to nine.

This will be more particularly taught by

NUMERATION.

Numeration teaches how (11) to read or write any sum or number by figures.

observations,

The value of figures in conjunction, and how to read any sum or number agreeably to the foregoing may be fully understood by the following

TABLE.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ Hundreds of Thousands.

9999999Tens of Thousands.

PPPAAAA A A A Thousands.

co co co co co co co co co co co Hundreds,
19 19 19 19 19 19 19 19 19 19 10 10(13) Tens.
co co co co co co co co co co co co co (12) Units

Those words at the head of the Table are applicable to

any sum or number, and must be committed perfectly to memory so as to be readily applied on any occasion.

The first six figures from the right hand are called the unit period, the next six the million period, after which the

trillion, quadrillion, quintillion periods, &c. may follow in their order.

Thus by the use of ten figures may be reckoned every thing which can be numbered; things, the multitude of which far exceeds the comprehension of man.

Questions. 1. What is the first thing to be regarded in the study of Arithmetic ?-2. What is Notation ?-3. How many methods of expressing numbers?-4. What is the first?-5. What is the second?-6. Which is generally used?-7. By what are all numbers expressed?-8. What are the nine first figures called?-9. What does each express?-10. Give an Example.-(11) What does Numeration teach?-(12) What does the first figure on the right hand of the Table denote?--(13) What the second? Repeat those words at the head of the Table upwards, beginning with Units.

=

Of other Characters used in Arithmetic.

Is the sign of Addition; and shews, that the number which follows the sign, must be added to the number before it. Thus 9+10 signifies that 9 and 10 are to be added.

Is the sign of Subtraction; and denotes the number following it, must be subtracted from the one before it. Thus, 16-4 signifies that 4 must be taken from 16.

Is the sign of Multiplication; and denotes that all the numbers, between which it is placed, are to be multiplied together. Thus 9X9 signifies that 9 is to be multiplied by 9, or 9×9×9 must be multiplied.

Is the sign of Division; and denotes the number standing before it, is to be divided by the number following it. Thus, 9-3 signifies that 9 is to be divided by 3.

This is the sign of equality; and signifies the sum or product of the numbers before it, is equal to the number after it, 2+4+5=11 mean that 2, 4 and 5 added, their sum would be 11; and 2X4 X5=40 mean, that 2 and 4 multiplied, and that product multiplied again by 5, the product is equal to 40; and also 9X8÷12=6, means that 9 and 8 multiplied, and the product divided by 12, the quotient will be equal to 6.

:: ::

Are the signs of proportion, as 7: 14:: 8: 16, and are read as 7 to 14, so is 8 to 16.

2 This shews the Square Root of the number is required. This shews the Cube Root of the number is required.

Required to take 9 from 15, look for 9 on the left or right hand, then follow that till you find 15, and on the remainder required.

ARITHMETIC.

SIMPLE ADDITION.

Simple addition teaches (1) to add together several sums of the same denomination.

RULE.-Set down (2) units under units, tens under tens, and hundreds under hundreds, &c. Begin to add (3) in the place of units, (4) add all in that column, carry one for every ten that is in the sum of units, to the place of tens, and set down all that is over ten, or tens; (5) add up the column of tens, carry one for every ten to the next column, and set down all that is over, thus proceed through all the columns to the last; and in the last column observe (6) to set down the whole sum.

PROOF.-() Begin at the top, add downwards, carry as before, and if the work is right, the sum will be the same as it was when added upwards.

Questions.-1. What does Simple Addition teach?-2. How do you set down sums to be added?-3. Where do you begin to add?-4. How do you add that column?-5. How the columns of tens?-6. What do you observe in the last column?-7. What is the method of Proof?

1. Add together 345674, 98786, 456752 and 7654.

Ans. 908816.

2. Add 94321, 675431, 1234 and 76432 together.

Ans. 847418.

3. Add 6421, 94742, 6752 and 875432 together.

Ans. 983347.

4. Add 67432, 98891, 65641 and 77421 together.

Ans. 309385.

5. Add 9944 2343 78764 and 94929 together.

Ans. 185980.