22+8-2010: for 22 a 8 are 30, and subtracting 20, we have 10 for the remainder.

(x) The St. Andrew's cross, being two lines crossing each other at right angles, but both inclined to the horizon, stands for multiplication: thus 4x5=20, for 4 multiplied by 5 give 20.

The product of two quantities multiplied together is also expressed by writing them close together, as AM standing for the product of A× M. A quantity multiplied into itself, or squared, that is, raised to the 2d power, is thus represented A; and the cube of A, or the 3d power, would be represented by A3.

(√) This sign means the root of any number or quantity: thus A is the square root of A, and ata2 means the cube root of the sum of the quantities a and m.

All bodies, and the spaces they occupy, consist of three dimensions, viz. length, breadth, and depth or thickness. Although these three sorts of measure are absolutely inseparable from the idea of a body, yet each of them may be considered by us as existing independent of the others; thus, for instance, if we say the Thames is deep at Woolwich, we are not considering the breadth of the river; and when we say, that it would take 27 yards of carpeting to cover a room, we think only of the length and breadth,, but not of the thickness of the carpet.

From these varieties of dimension we have a line in which we consider the length alone, although it would be impos-sible to make or to conceive any line without some breadth, however small.

2. From the consideration of two dimensions, length and breadth, or height and breadth, we have the idea of a superficies or surface: thus, by considering the length and breadth of the carpet, we have a notion of its extent or surface; but the thickness of the cloth is not regarded, although no cloth can be made so thin, as to have no sensible thickness.

3. From

3. From the consideration of length, breadth, and depth or thickness, we form a conception of solid contents; thus, by observing the length, breadth, and thickness of a log of mahogany, we form an idea of the number of solid feet of timber it contains.

These three sorts of dimension all arise from what is called a mathematical point, (see A, Plate 1, figure 1), which is itself not susceptible of measurement: for although no = point can be made, by the finest instrument, which shall not be of some magnitude, and therefore must have a certain length and breadth, however minufe; yet these dimensions are not considered in our reasoning respecting the point; and by supposing this point to be moved along for a certain space, we produce a line; by moving this line through a certain space sideways, or in a direction parallel to itself, we produce a surface, and lastly, by moving this surface sideways, or in a direction any how inclined to its own place, we produce a solid body.

The extremities of lines are termed points, as are also those parts of lines which intersect each other.

The mark made by moving a point constantly in the same direction from the beginning to the end of its motion, is termed a straight, or, as in the language of mathematicians, a right line, (see AB, Plate 1, figure 2); and on the contrary, the mark made by the motion of a point which deviates, in even the smallest quantity, from the direction with which the motion began, is called a curve line, or a curve, (see BC, Plate 1, fig. 2). Hence, only one right line can be drawn between two extreme points; but the variety of curve lines which may be drawn between the same extremities is infinite. Lines consisting partly of right and partly of curve lines, are termed mixed lines, (see ABC, Pl. 1, fig. 2).

An Angle is the point when two lines meet; as (Pl. 1, fig. 3) the point B, where the line AB meets the line BC, is

called

called the angle at B, or rather the angle ABC; observing always, that in naming, or, as it is called, reading an angle, by means of three letters, that standing at the angular point is to be placed in the middle of the three; thus, the angle at B may be either read the angle ABC, or CBA. This angular point is also called the vertex of the angle. When there are at a point more angles than one, as at the point C, in fig. 4, it is necessary to mention all the three letters denoting the angle intended; for the angles ACF, FCD, DCE, and ECB, are all formed at the point C.

When one right line meets another in such a manner, that the openings formed on each side are equal, or when one line stands upon another placed horizontally, in such a way that, if it were moveable, it would have no tendency to fall over on either side: in these cases, the first line is said to be perpendicular to the second line, and the equal angles on each side of the meeting of the two lines are called right angles. Thus, in fig. 4, Pl. 1, the line DC meets the line AB in the point C in such a way, that the opening formed by the line DC, with the part AC, is equal to the opening formed by the same DC with the part CB; or if the line DC were moveable on the horizontal line AB, it would have no tendency to incline or fall towards either the point A or the point B: in such cases the line DC is said to be perpendicular to AB at the point C; and the two angles ACD and DCB are both right. But should the line FC (same figure) meet the line AB at C in such a manner, that the angle FCA should be less than the other angle FCB, the angle FCA would be less than a right angle, and termed an acute or sharp angle; while FCB being greater than a right angle, would be termed an obtuse or blunt angle. In the same way the angle ECB, which is less than DCB a right angle, is acute, and ECA, which is greater than DCA, is obtuse.

Angles, like other quantities or magnitudes, are susceptible

of

of augmentation and diminution, by addition, subtraction, multiplication, and division: thus, in the same figure 4, the angles ACF and FCD, added together, form the angle ACD; and ACF, FCD, and DCE, are together equal to the angle ACE. Again, if from the angle DCA we subtract the angle DCF, we have remaining the angle FCA. In the same manner, supposing the three angles ACF, FCD, and DCE, to be equal to each other, the whole angle ACE would be three times the angle ACF, and in the same supposition the great angle ACE, divided by 3, would give us the angle ACF.

N.B. It is always to be remembered, that when we speak of an angle, we mean only the opening formed by the meeting of two right lines, without in the least taking into consideration the relative lengths of these lines; for the above angle ACF, would be of precisely the same magnitude whether the forming lines were in length an inch, a yard, a mile, or a thousand leagues.

When two or more right lines lie in the same plane or surface in such a way that, if produced to any imaginable length either to the right or the left, they would never meet, or have a tendency to approach one another, such lines are said to be parallel: thus in fig. 5, Pl. 1, the lines AB and CD are so situated with regard to one another, that if produced indefinitely either through A and C, or through B and D, they never would approach one another; that is, that the distance from A to C would be precisely equal to the distance from B to D, or the intermediate distantes from a to b, and from c to d; in such circumstances the lines AB and CD are said to be parallel to each other.

When a line is parallel to another line, it will also be parallel to any other line which is parallel to the former; thus, in the same figure, the line EF being drawn parallel to the line CD, it will also be parallel to the line AB; for EF being parallel to CD, the distance E a will be equal to

By observing the various results in this process, the following rules have been formed for the resolution of adfected quadratic equations: viz, 1st. when the terms involving the unknown quantity are all transposed to the one side of the equation, and the known quantities to the other, and the term involving the square of the unknown quantity is positive; then if this square be multiplied by a co-efficient, all the other terms must be divided by it, that the co-efficient of this square may be 1.

2d. To both sides of the equation add the square of half the co-efficient of the unknown quantity; when that side which involves the unknown quantity will become a perfect square.

3rd. Lastly, extract the square root of both sides of the equation, thus rendering it simple with respect to the unknown quantity, and transposing, agreeably to the rules formerly given, the unknown quantity will come to stand alone on the one side of the equation, while only known quantities stand on the other side, and so the value of the. unknown quantity will be discovered.

It is to be observed, that the square root of the first side of the equation will always be equal to the sum or the difference of the unknown quantity, and half the co-efficient of the second terms; namely, that if the sign of this term be +, the root will be equal to the sum; but if the sign be, the root will be equal to the difference.

Example 1. Required the value of z from the quadratic equation + 2z = 63.

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