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


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+ signifies plus, or addition.

minus, or sobstraction.
хог multiplication.

square root.
cube root, &c.
diff. between two numbers when it is not known
which is the greater.

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. V 5, or 51, denotes the cube root of the number 5. 73, denotes that the number 7 is to be squared. 83, denotes that the number 8 is to be cubed.



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 is, units únder 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.


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 num. ber 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.'

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

EXAMPLE I. how many

nines are contained in their sum.-Reject those nines, and 3497

5 set down the remainder towards the 6512

5 right-hand directly even with the 8295 figures in the line, as in the annexed example.- Do the same with each 18304

7 of the proposed lines of numbers, setting all these excesses of nines in a column on the right-hand, as here 5, 5, 6. Then, if the excess of 9's in this eum, fuund as before, be equal to the excess of 9's in the total sum 18304, the work is probably right. Thus, the som 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. 3


Excess of nines.

the sum total 18304, is 16, the excess of which above 9 is also 7, the same as the former*.

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* This method of proof depends on a property of the number 9, which except the nunber 3, belongs to no other digit whatever; namely, that "

any number divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9;” which may be demonstrated in this manner.

Demonstration. Let there be any number proposed as 4658. This, separaled into its several parts, becomes 4000 + 600 + 50 +8. But 4000 - 4 x 1000

4* (999+1) =4 X 999 + 4. In like manner 600=6 X 99 + 6; and 50 = 5 x 9 +5. Therefore the given number 4658 =4 X 999 +4 + 6 x 99 + 6 + 5 x 9 + 5 + 8 = 4 X 999 + 6 x 99 + 5* 9 + 4 +6+ 5 + 8; and 4658 =9= (4 * 999 + 6 X 99 + 5 X9+ 4 +6 +5+8)

• 9. But 4 X 999 + 6 X 99 + 5 X 9 is evidently divisible by 9, without a remainder; therefore if the given number 4658 be divided by nine, it will leave the same reinainder as 4 + 6 + 5 + 8 divided by 9. And the same, it is evident, will bold for any other number whatever.

In like manner, the same property may be shown to belong to the number 3; but the preference is usually given to the number 9, on account of its being more convenient in practice.

Now from the demonstration above given, the reason of the rule itself is evident; for the excess of 9's in two or more numbers being taken separately, and the excess of 9's taken also out of the sum of the former excesses, it is plain that this last excess must be equal to the excess of 9's contained in the total sum of all these numbers; all the parts taken together being equal to the whole.—This rule was first given by Doctor Wallis in his Arithmetic, published in the year 1657.

Ex. 5. Add 3426 ; 9024 ; 5106 ; 8890 ; 1204, together.

Ans. 27650. 6. Add 509267; 235809 ; 72920 ; 8392 ; 420; 21; and 9, together.

Ans. 826838. 7. Add 2; 19; 817; 4298 ; 50916 ; 730205; 9180634, together.

Ans. 9966891. 8. How many days are in the twelve calendar months ?

Ans. 365. 9. How many days are there from the 15th day of April to the 24th day of November, both days included? Ans. 224.

10. An army consisting of 52714 infantry*, or foot, 5110 horse, 6250 dragoons, 3927 light horse, 928 artillery, or gunners, 1410 pioneers, 250 sappers, and 406 miners : what is the whole number of men;

Ans. 70995.



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SUBTRACTION teaches to find how much one number exceeds another called their difference, or the remainder, by taking the less from the greater. The method of doing which is as follows :

Place the less number under the greater, in the same manner as in addition, that is, units under units, tens under tens, and so on; and draw a line below them.—Begin at the righthand, and take each figure in the lower line, or number, from the figure above it, setting down the remainder below it.But if the figure in the lower line be greater than that above it, first borrow, or add, 10 to the opper one, and then take the lower figure from that sum, setting down the remainder, and carrying 1, for what was borrowed, to the next lower figure, with which proceed as before, and so on till the whole is finished.

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* The whole body of foot soldiers is denoted by the word Infantry; and all those that charge on horseback by the word Cavalry.--Some authors conjecture that the term infantry is derived from a certain Infanta of Spain, who finding that the army commanded by the king her father had been defeated by the Moors, assembled a body of the people together on foot, with which she engaged and totally routed the enemy. In honour of this event, and to distinguish the foot soldiers, who were not before held in much estimation, they received the name of Infantry, from her own title of Infanta.



App the remainder to the less number, or that which is just above it; and if the sum be equal to the greater or uppermost number, the work is right*.

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7. Sir Isaac Newton was born in the year 1642, and he died in 1727 : how old was he at the time of his decease?

Ans. 85 years. 8. Homer was born 2543 years ago, and Christ 1810 years ago : then how long before Christ was the birth of Homer?

Ans. 733 years. 9. Noah's flood happened about the year of the world 1656, and the birth of Christ about the year 4000 : then how long was the flood before Christ?

Ans. 2344 years. 10. The Arabian or Indian method of notation was first known in England about the year 1150 ; then how long is it since to this present year 1810 ?

Ans. 660 years. 11. Gunpowder was invented in the year 1330 : then how long was this before the invention of printing, which was in 1441 ?

Aps. 111 years. 12. The mariner's compass was invented in Europe in the year 1302: then how long was that before the discovery of America by Columbus, which happened in 1492 ?

Ans. 190 years.

* The reason of this method of proof is evident; for if the difference of two numbers be added to the less, it must manifestly make up a sum equal to the greater.


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