number or letter, or by several letters standing together like a word, or by a number and letters so placed.

Thus, 7... ab . . . 2b, and -4xyz, are simple quantities. 16. A compound quantity is that which consists of two or more simple quantities, called its terms, connected together by the signor—.

Thus a + b is a compound quantity, having a and b for its terms; and 2x + 3y — 4 is a compound quantity, the terms of which are 2x, 3y, and

4.

It is indifferent in what order the terms of a compound quantity stand, provided every term has its proper sign préfixed; thus, a + b may be written b + a, and 2x + 3y — 4 may be written 3y+2x-4, or - 4 + 2x + 3y, &c. 17. A binomial is a compound quantity consisting of two terms, as a + z... 2b + 4c, &c. A trinomial of three terms, as x- 3y+2z, &c. A quadrinomial of four terms, as - 2a +b 3c + x. And a multinomial, or polynomial, of many. terms, as a −z + 26· c+ 4, &c.

18. A residual quantity is a binomial having one of its terms negative, as a-b, or 2x-5z, &c.

19. The powers of a quantity are the products which arise by multiplying the quantity continually into itself; they are produced and named as follows.

Every quantity is considered as being the first power of itself; thus, a is the first power of a.

If a quantity be multiplied once into itself, the product is called the second power, or square of that quantity; thus a × a, or aa, is the second power or square of a.

If a quantity be multiplied continually twice into itself, the product is called its third power, or cube; thus a xa xa, or aaa, is the third power, or cube of a.

In like manner any quantity, as a, multiplied three times into. itself, produces aaaa, the biquadrate, or fourth power of a, and so on for higher powers.

20. But the powers of quantities are frequently and more conveniently represented by small figures, called indices or exponents, placed over, and a little to the right of, the quantities. Thus a2 is the same as aa, and denotes the second power or 3 is the same as xxx, denoting the third power; yyyy, or y*, equally express the fourth power or

square
of a .
or cube of x...

biquadrate of y, and so on, where the small figures 2, 3, and 4, are the indices or exponents of the powers, each shewing how often the quantity under it is repeated.

21. The root of a quantity is that which being multiplied once or oftener into itself, produces the given quantity.

Thus, a is the square root of a2, because a × a = a2.

x is the cube root of x3, because x x x x x = x3.

y

is the fourth root of y+, because y x yxy x y = ya.

22. The root of a quantity is denoted by the character, called a radical sign, with a small figure over it, expressing what root is designed or else, by a fractional index or exponent placed over the quantity.

:

Thus a, or a, denotes the square root of a.

3 √x, or x, denotes the cube root of x.

4

√y, or y, denotes the fourth root of y, &c.

23. Quantities under the radical sign, or having a fractional index, are called radical quantities; if the root denoted by the radical sign, or index, can be found exactly, the quantity under it is called a rational quantity; but if the exact root cannot be found, it is then called an irrational quantity, or surd.

24. Like quantities are such as differ only in their numeral coefficients; thus 3a,... 5a, and 17a, are like quantities; so

are 4axy,... 3axy, and — 7axy.

25. Unlike quantities are such as differ either in their letters, or indices, or in both; thus 2a, and 3b, are unlike quantities ; so are a2x, ar2, and abr.

...

26. A vinculum is a straight line drawn over the top of two, or more quantities, to connect them together, as a + x, or 3y+4z, and signifies that the quantities under it are to be taken collectively, or considered as one quantity, with respect to the sign standing before or after the vinculum.

Thus a + bx c signifies that both a and b are to be multiplied into c; b shews that the sum of x, y,

and z, is to be multiplied into the difference of a and b.

a The square root being the first root, the small figure 2, denoting the root, is always omitted, and the root is designated simply by the sign placed before the power; but the figure denoting any higher root is never omitted.

27. The parenthesis is frequently used instead of the vinculum; thus (a + b) x c is the same as a + bx c, also (x + y + z) × (a

b) is the same as x + y + 2 x a

b.

28. Two dots are sometimes used instead of the vinculum, or parenthesis; they are sometimes placed at each end of the compound quantity, as : x + y z: X 4; and sometimes at the end only, next the sign, as x + y -z: × 4, which expressions are the same as x + y z X 4.

29. When two quantities are included in a vinculum, or parenthesis, it is usual in reading them to pronounce the word both, and the word all when there are more than two: thus a + bx c is read a plus b both into c; the expression x is 'read r minus y both, by a plus z both; a

-

y

y + z.

x + a is read a minus y plus z all, into x plus a both; and z) ÷ (α + b + c) is read 5 plus y minus z all, by a plus b plus c all.

(5+ y

30. When two quantities are compared together, the quan→ tity compared is called the antecedent, and the quantity to which it is compared is called the consequent : also the relation the two quantities bear to each other with respect to magnitude, is called their ratio; and the quotient of the antecedent divided by the consequent, being the number expressing that relation, is called the index of the ratio: the ratio itself is expressed by placing the antecedent before the consequent, with two dots placed .vertically between; thus 12: 3 expresses the ratio of 12 to 3, where 12 is the antecedent, 3 the consequent, and = 4, the index of the raiio.

12

3

31. Proportion is the equality of two ratios: thus when two quantities have the same ratio that two other quantities have, this equality or identity of ratios is called proportion, and is denoted by four dots, placed in the interval between the first two and the latter two quantities; thus 3: 6: : 1 : 2, is read 3 is to 6, as 1 is to 2, and denotes that 3 has the same ratio to 6, that 1 has to 2.

32. The sign is called greater than, and the sign less than; thus ab denotes that a is greater than b, and b【 a denotes that b is less than a.

33. The sign denotes the difference in general of two quantities between which it is placed, when it is not known

which is the greater; thus ab signifies a — b, when a is the greater; and b― a, when b is the greater.

34. The reciprocal of an integral quantity is unity divided by it, and the reciprocal of a fraction is that fraction inverted; thus

35. EXAMPLES IN NOTATION,

Wherein the signification of each letter, and of the several combinations and results, are required to be expressed in numbers.

1. To find the value of a + b + c + d in numbers?

Here, instead of a put 9, instead of b put 7, instead of c put 4, and instead of d put 3, with the proper signs between ; then proceed with these numbers (viz. add them) as the signs import.

Thus a + b + c + d = 9 + 7 + 4 + 3 = 23, the answer.

Thus a

b+c-d=9 7 +4-3= 13 10=3, ans. Here I add the numbers with the sign + (viz. 9 and 4) together into one sum, and those with the sign - (viz. 7 and 3) into another, then subtract the latter from the former. 3. Required the value of ab+bc- cd?

5. What is the value of a +b.c- e?

Thus a+b.c-e= +7.4 1 = 16 x 3 = 48. Ans.

= 15 Ans.

7. What is the value of a

c+b-5? Ans. 7.

8. To find the value of 2a + ab d? Ans. 78.

-

12a? Ans. 33.

CXC - e? Ans. 90.

bcd+cde? Ans. 15.

ab ac

cd 2e

12. Required +

2d? Ans. 174.

13. Required 4a − √3d +d2? Ans. 19.

14. Required the value of d+e2+ c? Ans. 4.

15. Required a2 — 2ab+ b2? Ans. 4.

16. Find a+b+3d); × a − b + Ge. Ans. 10.

ADDITION.

36. To add quantities which are like, and have like signs. RULE. Add all the coefficients together, and place the common sign before their sum, and the common letter or letters after it.

EXAMPLES.

1. Add together the following algebraic quantities, viz.

+ 2.a

+ 3 a

+ 5 a + 4 a + 14 a

Explanation.

I add the coefficients 2, 3, 5, and 4, together, and the sum is 14; before this I place the common sign+, and after it the common letter a, making the sum + 14 a. But, because when a quantity has no sign, + is always understood, the signs might in this case have been omitted without detriment.

e The reason of this rule is so exceedingly obvious, that he who can add numbers together will readily understand it; in the first example it is evident that 2a, 3a, 5a, and 4a, added together, will make 14a, let a represent whatever it may; 2, 3, 5, and 4 times any thing, will (being added together) evidently amount to 14 times that thing: if a represent a pound, then the sum will be 14 pounds; if a represent a yard, then 14a will imply 14 yards. With respect to the signs, it is evident that the sum will be of the same nature with the particulars which constitute it; several additive quantities being collected together, the sum will clearly be an additive quantity equal to all of them together; but if several subductive quantities are collected, the sum will as evidently be a subductive quantity equal to all the latter taken together. Therefore the sum of affirmative quantities will be +, and the sum of negative ones --.