Signifies arithmetical proportion; thus 2. 4 :: 6 - 8 ;

here the meaning is, that 4-2=8—6—2.

::: Signifies geometrical proportion; thus 2:4::3:6, which is to be read, as 2 to 4, so is 3 to 6,

Signifies continual arithmetical proportion, or arithmetiçal progression; thus 2468signifies, that 2, 4, 6, and 8 are in arithmetical progression.

Signifies continual geometrical proportion, or geometrical progression; thus, 2:4:8: 16 signifies, that 2, 4, 8, 16 are in geometrical progression. .. Signifies therefore.

Signifies the second power, or square; thus, x2 sig

nifies the square of x.

73 Signifies the third power, or cube.

Signifies any power.

יך

✔, or, Signifies the square root; thus ✔x, or x signifies the square root of x,

✓, or, Signifies the cube root.

"V, or, Signifies any root.

Signifies any root of any power.

The number, or letter, belonging to the above signs of powers and roots, is called the index, or exponent.

A line, called a vinculum, drawn over several numbers, signifies, that the numbers under it are to be considered jointly; thus, 20-7+8=5; but without the vinculum, 20—7+8 21. The same thing is also sometimes expressed by a parenthesis, inclosing two or more numbers or quantities; thus, 20—(7+8)=5.

Two or more letters, joined together like those of a word, signify, that the numbers, which they represent, are to be multiplied together; thus, a baxb; and abcmaxbxc,

SIMPLE ADDITION.

Simple Addition teaches to collect several numbers of the same denomination into one number, called the sum.

RULE.*

1. Place the numbers under each other, so that units may stand under units, tens under tens, &c. and draw a line under them.

* This rule, as well as the method of proof, is founded on the known axiom," the whole is equal to the sum of all its parts." All, that requires explaining, is the method of placing the numbers, and carrying for the tens, both which are evident from the nature of notation. For any other disposition of the num bers would entirely alter their value; and carrying one for every ten, from an inferior row or column to a superior, is evidently right, since an unit in the latter case is of the same value as ten in the former.

Beside the method here given, there is another very ingenious one of proving addition by casting out the nines.

1. Add the figures in the first line, and find how many nines are contained in their sum.

2. Reject the nines and set the remainder in the same line, on the right.

3. Do the same in each of the other lines, and find the sum of the row of excesses. Then the nines of this sum, and of the sum of the given numbers being rejected, if the two excesses be equal, the addition is proved to be rightly performed.

2. Add the figures in the row of units, and find how many tens are contained in their sum.

3. Set the remainder under the line, and carry as many units to the next row, as there are tens, with which proceed as before; and so on till the whole is finished.

This method depends on a property of the number 9, which belongs to no other digit whatever, except 3, namely, that any number divided by 9 leaves the same remainder, as the sum of its figures or digits divided by 9; which may be thus demonstrated.

DEMON. Let there be any number, as 3467; this separated into its several parts becomes 3000+400+60+7; but 3000=3 X 1000=3×999+1=3×999+3. In like manner 400 4X 994, and 60=6×9+6. Therefore 3467=3×999+3+4×

99+4+6×9+6+7=3×999+4×99+6×9+3+4+6+7. And

3467 3x999+4x99+6x9, 3+4+6+7

+

19

But 3x 999+4×

99+6x9 is evidently divisible by 9; therefore 3467 divided by 9 will leave the same remainder, as 3+4+6+7 divided by 9; and the same will hold for any other number whatever. Q. E. D. The same may be demonstrated universally thus.

DEMON. Let N= any number whatever, a, b, c, &c. the digits, of which it is composed, and n= as many cyphers as a, the highest digit, is places from unity. Then Na with n Os+b with n-1 Os+c with n-2 Os, &c. by the nature of notation; =aXn 9s+a+bX 1 9s+b+cXn2 9s+c, &c. =aXn 9s+ Xn-1 9s+cxn−2 9s, &c. +a+b+ċ, &c.“but a×n 9s+bxn−1 9s +cxn−2 9s, &c. is plainly divisible by 9; therefore N divided by 9 will leave the same remainder, as a+b+c, &c. divided by 9. Q. E. D.

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

Now from the demonstration here given, the reason of the rule itself is evident; for the excess of nines in each of two or

METHOD OF PROOF.

1. Draw a line between the first and second lines of figures to cut off the first number.

2. Add all the other numbers, and set their sum under the sum of all the numbers.

3. Add the number last found and the number cut off; and if their sum be the same, as that found by the first addition, the sum is right.

more numbers being taken, and the excess of nines also in the sum of these excesses, it is plain, the last excess must be equal to the excess of nines, contained in the sum of all the numbers; the parts being equal to the whole.

This rule was first given by Dr. WALLIS in his Arithmetic, published A. D. 1657, and is a very simple, easy method; though it is liable to this inconvenience, that a wrong operation may sometimes appear to be right. For if we change the plac es of any two figures in the sum, it will still be the same. A true sum will however always appear to be true by this proof; and to make a false one appear true, there must be at least two errors, and these opposite to each other. And if there be more than two errors, they must balance among themselves; but the chance against this particular circumstance is so great, that we may pretty safely trust to this proof.

4. Add 8635, 2194, 7421, 5063, 2196, and 1245 together. Answer 26754.

5. Add 246034, 298765, 47321, 58653, 64218, 5376, 9821, and 340 together. Ans. 730528. 6. Add 562163, 21964, 56321, 18536, 4340, 279, and 83 together. Ans. 663686. 7. How many days are there in the twelve calendar months? 8. How many days are there from the 19th day of April, 1774, to the 27th day of November, 1775, both days exclusive ? Ans. 586.

Ans. 365.

SIMPLE SUBTRACTION.

Simple Subtraction teaches to take a less number from a greater of the same denomination, and thereby shows the difference or remainder. The less number, or that which is to be subtracted, is called the subtrahend; the other, the minuend; and the number, that is found by the operation, the remainder or difference.

RULE.*

1. Place the less number under the greater, so that units may stand under units, tens under tens, &c. and draw line under them.

* DEMON. 1. When all the figures of the less number are less than their correspondent figures in the greater, the differences of the figures in the several like places must, taken together, make the true difference sought; because, as the sum of the parts is equal to the whole, so must the sum of the dif rences of all the similar parts be equal to the difference of the wholes, or given numbers.

2. When any figure of the greater number is less than its correspondent figure in the less, the ten, which is added by the rule, is the value of an unit in the next higher place, by the nature of notation; and the one, that is added to the next place of the less number, is to diminish the correspondent place of the greater accordingly; and therefore the operation in this case is

VOL. I.

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