then accordingly is c twice, thrice, or four times, &c. as great as d.

To what has been thus far laid down on the signification of the signs and characters used in the algebraic notation, we may add what follows; which is equally necessary to be understood.

When any quantity is to be taken more than once, the number is to be prefixed, which shows how many times it is to be taken: thus 5a denotes that the quantity a is to be taken five times; and 3bc stands for three times bc, or the quantity which arises by multiplying bc by 3: also 7 ✔ a2 + b2 signifies that ✔a2 + b2 is to be taken seven times; and so of others.

The numbers thus prefixed are called coefficients; and that quantity which stands without a coefficient is always understood to have an unit prefixed, or to be taken once, and no more.

Those quantities are said to be like that are expressed by the same letters under the same powers, or which differ only in their coefficients: thus 3bc, 5bc, and 8bc are like quantities; and the same is to be understood of b+ the radicals 2

b+c

and 7

a

But unlike quanti

ties are those which are expressed by different letters, or by the same letters under different powers: thus 2ab, 2abc, 5ab2, and 3ba2, are all unlike.

When a quantity is expressed by a single letter, or by several single letters joined together in multiplication (without any sign between them) as a, or 2ab, it is called a simple quantity.

But that quantity which consists of two or more such simple quantities, connected by the signs + or -, is called a compound quantity; thus a-2ab+5abc is a compound quantity; whereof the simple quantities a, 2ab, and 5abc, are called the terms or members.

The letters by which any simple quantity is expressed may be ranged according to any order at pleasure, and yet the signification continue the same: thus ab may be wrote ba; for ab denotes the product of a by b, and ba the product of b by a; but it is well known, that,

when two numbers are to be multiplied together, it matters not which of them is made the multiplicand, nor which the multiplier, the product, either way, coming out the same. In like manner it will appear that abc, acb, bac, bca, cab, and cba, all express the same thing, and may be used indifferently for each other (as will be demonstrated further on ;) but it will be sometimes found convenient, in long operations, to place the several letters according to the order which they obtain in the alphabet.

Likewise the several members, or terms of which any quantity is composed, may be disposed according to any order at pleasure, and yet the signification be noways affected thereby. Thus a-2ab+5a2b may be wrote a+5a2b—2ab, or -2ab+ a + 5a2b, &c. for all these represent the same thing, viz. the quantity which remains, when, from the sum of a and 5a2b, the quantity 2ab is deducted.

Here follow some examples wherein the several forms of notation hitherto explained are promiscuously concerned, and where the signification of each is expressed in numbers.

Suppose a 6, b 5 and c = 4; then will a2 + 3ab-c2= 36 + 90 — 16

= 110,

✔2ac+c2 (or 2ac+c2] 1⁄2)=√64=8 (for 8.x 8=64)

2bc

✔2ac + c2

40

8

=2+ =7,

a2-✓b2-ac 36-1 35

2a—✓b2+ac 12-7

b2

=

7,

5

ac + ✔2ac+c2=1+8=9,

-ac + √2ac+c2=√25—24+8=3.

This method of explaining the signification of quantities I have found to be of good use to young begin

ners and would recommend it to such, who are desirous of making a proficiency in the subject, to get a clear idea of what has been thus far delivered, before they proceed farther.

SECTION II.

of addition.

ADDITION, in algebra, is performed by connecting the quantities by their proper signs, and joining into one sum such as can be united: for the more ready effecting of which, observe the following rules.

1°. If, in the quantities to be added, there are terms that are like and have all the same sign, add the coefficients of those terms together, and to their sum adjoin the letters common to each term, prefixing the common sign.

And 5a+7b Also 5a-7b added to 7a+3b added to 7a-3b

Thus
added to sa

5a

26

And the

C

7d

The reasons on which the preceding operations are grounded, will readily appear by reflecting a little on the nature and signification of the quantities to be added: for, with regard to the first example (where 3a is to be added to 5a) it is plain, that three times any quantity whatever, added to five times the same quantity, must make eight times that quantity: therefore sa, or three times the quantity denoted by a, being added to 5a, or five times the same quantity, the sum must consequently

2° When in the quantities to be added, there are like terms, whereof some are affirmative, and others negative, add together the affirmative terms (if there be more than one) and do the same by the negative ones; then take the difference of the two sums (not regarding the signs) by subtracting the coefficient of the lesser from that of the greater, and adjoining the letters common to each; to which difference prefix the sign of the greater.

1.

3.

3b

6ab + 12bc 8cd 7ab 9bc + 3cd

5bc+12cd

2.

Sum

3ab + 50c +7ab9ac

4ab -4bc

4. 5√✓ab-7✓bc + 8å

3√ab + 8✅bc 12d

7ab + 3 bc + 9d

2bc7cd. Sum 15 ab + 4√bc + 5d.

12abc 16abd25acd- 72bcd

16abc12abd + 20acd 18bcd

-13abc 26abd 15acd12bcd

make 8a, or eight times that quantity. From whence, as the sum of any two quantities is equal to the sum of all their parts, the reason of the second case, or example, is likewise obvious. But as to the third (where the given quantities are 5a-7b and 7a-3b) we are to consider, that, if the two quantities to be added together had been exactly 5a and 7a (which are the two leading terms) the sum would, then, have been just 12a; but, since the former quantity wants 76 of 5a, and the latter sb of 7a, their sum must, it is evident, want both 7b and 3b of 12a; and therefore be equal to 12a-10b, that is, equal to what remains, when the sum of the defects is deducted. And by the very same way of arguing, it is easy to conceive that the sum, which arises by adding any number

C

In the last example, and all others, where fractional and radical quantities are concerned, every such quantity, exclusive of its coefficient, is to be treated in all respects like a simple quantity expressed by a single letter.

3°. When in the quantities to be added, there are terms without others like to them, write them down with their proper signs.

Here follow a few examples for the learner's exercise, wherein all the three foregoing rules take place promis

of quantities together, will be equal to the sum of all the affirmative terms diminished by the sum of all the negative ones (considered independent of their signs ;) from whence the reason of the second general rule is apparent. As to the case where the quantities are unlike, it is plain that such quantities cannot be united into one, or otherwise added, than by their signs: thus, for example, let a be supposed to represent a crown, and b a shilling; then the sum of a and b can be neither 2a nor 2b, that is, neither two crowns nor two shillings, but one crown plus one shilling, or a + b.