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Next, let the segment AEB {fig. 2,) be less than a semi-circle; draw the diameter DCF, and join EF; and, because the segment ADEF is greater than a semi-circle, by the first case, the angle ADF is equal to AEF. In like manner, because the segment BEDF is greater than a semi-circle, the angle BDF is equal to the angle BEF; therefore the whole angle ADB is equal to the whole angle AEB.

Theorem 36.

93. The sum of the opposite angles of any quadrilateral, ABCD, inscribed in a circle, is equal to two right angles.

Draw the diagonals AC, BD. In the segment ABCD, the angle ABD is equal to ACD; and, in the segment CBAD, the angle CBD is equal to CAD (theorem 35); therefore the whole angle ABC is equal to the sum of the two angles ACD, CAD; and, adding ADC, the sum of the two angles, ABC, ADC, is equal to the sum of the three angles. Now these three angles are the angles of the triangle ADC, and therefore equal to two right angles (theorem 27): therefore the sum of the two angles ABC, ADC, is equal to two right angles. In the same manner it may be demonstrated that the sum of the two angles BAD, BCD, is equal to two right angles.

Theorem M.

94. An angle ABD, in a semi-circle, is a right angle; an angle BAD, in a segment greater than a semi-circle, is less than a right angle; and an angle, BED, in a segment less than a semi-circle, is greater than a right angle.

Produce AB to F, draw BC to the centre, and, because CA is equal to CB, the angle CBA is equal to CAB (Theorem 11). In like manner, because CD is equal to CB, the angle CBD is equal to CDB; therefore the sum of the two angles CBA, CBD, is equal to the sum of the two angles CAB, CDB; that is, the angle ABD is equal to the sum of the two angles CAB, CDB: but this last sum is equal to the angle DBF {theorem 27); therefore the angle ABD is equal to the angle DBF: but, when the angles are equal on each side of a

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It frequently happens that a quantity consists of several quantities of the same kind, as of two or more distances to make one distance; these must, therefore, be joined by addition, or by addition and subtraction. In order to indicate this junction, two distinct signs will be necessary.

The sign + (plus) implies that the quantity which follows it is to be added to that which goes before, and that all the quantities, when more than one, are to be added together into one sum. Thus,a + 6 shows that b is to be added to a, or that a and b are to be added together.

Again, a+b+c+d implies that b is to be added to a; c, to the sum of a and b; d, to the sum of a, b, c.

The sign — {minus) placed between two quantities, denotes that the quantity which follows it is to be subtracted from that which precedes it.

Thus, m—n denotes that the quantity represented by n is to be subtracted from that represented by m. Suppose, for instance, that m is 7, and n 3; then 7-3 will be 4. Therefore, m-n denotes the remainder arising by subtracting n from m.

Hence Subtraction is an opposite operation to Addition; and, therefore, if any quantity be both added to and subtracted from the same quantity, the quantity thus added and subtracted may be taken away entirely, by which the expression will be in its most simple form: thus, m + a—a is equivalent to m.

When two quantities are equal to each other, this equality is implied by the interposition of the double bar = between each of the quantities. Thus, w + a—a=m ; as,also, 4 + 3+6—2=11. Equal quantities, thus connected, are called Equations.

Terms are all those parts of an expression that are separated by the signs of Addition and Subtraction.

Thus, a + b + c—d, is a quantity consisting of four terms.

Quantities which contain two terms are called binomials: thus, a + b, or, a—b, are binomials.

The Multiplication of two or more factors is indicated by connecting the letters representing the factors ; as, ab denotes a product of two factors; abx a product of the three factors, a, b, and x. Thus, let a = 2, b = 3, and x=5; then abx = 30: Again, mmmm signify a product of four factors, of which all are equal: suppose m = 2, then vimmm= 16.

When any number of factors are equal to each other, instead of repeating them to that number in the representation of the product, the product will be indicated with less trouble by writing only one of the equal factors and a digit; the latter containing as many units as the factors are in number, over the right-hand side of the factor so written. Thus, instead of aa, bbb, xx, xxx, the same idea will be more conveniently expressed thus, a2, b3, x2, a?.

The continued product of equal quantities is called a Power; the quantity itself is called the Base of that power; and the digit, which indicates the number of factors, is called the Index or Exponent of that power.

When factors of a product consist of compound terms, each compound factor is enclosed within brackets, or parentheses, and the factors thus included are joined to each other by bringing the bracket on the left hand of the one nearly close to that on the right-hand of the other.

Thus, the product ofa + i + e, x + a + c + d, and a + x, is represented by {a + b + c) (x + a + c + d) (a + x). Let a=1, b 2, c = 3, d=4, and x=5, then will (a + b + c) {x + a + c+d) {a + x) = (1+2+ 3) (5 + 1 + 3+4) (l + 5) = 468.

Any Power of a compound quantity is represented in a similar manner to that of representing a simple quantity, by inclosing the compound within brackets, and writing the number which indicates the power over the righthand bracket, and on the right-hand side of that bracket: thus, (a + bf denotes the cube of a + b, and (x + y + z)* denotes the fourth power of x + y + z.

Division is represented by placing the dividend above the divisor, with a short line between them, as -b; which expression shows how often the quantity a contains the quantity b; or how often the dividend contains the divisor. Let a be 12, and b be 3, then j will be 4.

Fractions are represented in the same manner as Division, by placing the numerator above and the denominator below a short line. Thus, TM indicates a fraction, whose numerator is m, and denominator n. Let in be equal to 2, and n equal to 3; then - is equivalent to f or two-thirds: or, if we suppose m to be equal to 17, and n equal to 5, then " would be 17-fifths of unity, or by dividing the numerator m, which is equivalent to 17, by the denominator n, which is equivalent to 5: the quotient will be 3 and §.

All expressions of quantity are said to be Simple when the operations are indicated by one or more letters, either in Multiplication or Division, without the intervention of the signs + or —, as in the following: a, ab, |, ^-; which are all Simple expressions.

Known quantities are generally represented by the initial letters, a, b, c, &c. of the alphabet, or by numbers; and the unknown quantities by the final letters, v, w, x, y, z.

A Co-efficient is the number prefixed to any quantity. Thus, in the expression 5x, the number 5 is the co-efficient of x; or, if x represent a quantity to be discovered by an operation, and a a quantity already known, then, in the expression ax, the quantity a is called the co-efficient of x.

Having explained the forms which indicate the operations of Simple Quantities, we shall now explain the rules for those performed upon Compounds.

ADDITION OF ALGEBRA.

97. To add any number of simple affirmative quantities, which are of the same kind, together, or any number of quantities that have a common factor:

Prefix the sum of the co-efficients to the quantity, and the product will represent the sum; observing that, when no co-efficient is written, the coefficient is understood to be unity: and, when the co-efficients are expressed by letters, these letters are to be joined with the sign + within brackets, and the common quantity adjoined or subjoined.

In the following examples let the sum be put equal to S.

Example 1.—Add a, a, a, a, a, together; then 5a = S.

Example 2.—Add ax, 2 ax, 3 ax, together; then 6 ax = S.

Ex.3.—Add ax, bx, ex, ex, ex,fx, together; then (a+b+c+d+e+f)x=S.

98. To add any number of simple affirmative quantities of different kinds together:

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