It follows, that each extremity of a line is a point. A straight line, or, as it is often called, a right line, is the direct, that is, the shortest, line connecting the two extremities, or extreme points, of it. A crooked line is not the direct line joining the two points which are its extremities. It may consist of two or more straight lines joined together thus, or in some other way. Or it may be what is called a curved line, no part being straight, such as or such as may be represented by a fine thread drawn tight round the trunk of a tree to measure its girth. In speaking of points we distinguish one from another by using the letters of the alphabet to mark their position; and so also with regard to lines to mark either their position or extent, or both. A single letter will fix or express a point, but two are mostly used to express a straight line. Thus, if we put A at one end of a straight line, B at the other end, the points, which are the extremities of that line, would be simply called the points A and B ; and the line would be called the line AB. Sometimes, however, a single letter may be used to denote a line, but not often. all 6. SUPERFICIES, SURFACE, or AREA. These words express the same thing, which is a subject for measurement; as, for instance, the acre-age of a field. It is obvious that this will depend upon the length and breadth of the field, but not at all upon the depth of the soil, or the thickness of the sod. And so we have the DEFINITION. A SUPERFICIES, SURFACE, or AREA, is that which hath only length and breadth. It is not meant that the body whose superficies, surface, or area, we are considering has only length and breadth, but that the dimensions of a superficies, surface, or area, are entirely dependent upon length and breadth, to the exclusion of thickness, height, or depth. Thus in speaking of the quantity of carpet which will cover a floor, the thickness of the carpet never enters into our consideration, but only the length and breadth. Hence the expression 'superficial measure' is always understood to exclude thickness. Thus, for instance, the area or surface of this page, that is, the space upon it capable of receiving the impression of type, is manifestly independent of the thickness of the paper. 7. SURFACES are of two kinds, plane and curved. A plane surface is one on which a straight line may be drawn in any part of it, wholly coincident with the surface. Or, in other words, if any two points are taken in the surface, and a straight line be drawn joining the two points, that line shall be wholly in the surface. A curved surface is one, on which if points be taken and joined by lines lying wholly on the surface, those lines are found to be curved lines. Thus the top of a table is a plane surface'; but the boundary of a globe is a 'curved surface'. Observe, it is not necessary to a curved surface that all lines drawn on it should be curved lines; there may be straight lines in particular cases. For example, the surface of a round pillar is curved, but yet the lines drawn on it in the particular direction of the length of the pillar will be straight lines, whilst all others will be curved. 8. ANGLES. A plane rectilineal angle is formed by two straight lines, which meet together, but are not in the same straight line. The angle is the measure of the inclination of the one line to the other; but how that measure is taken does not concern us at present to know. All that is here required is to know how to compare one angle with another, viz.: (1) That the angle formed by, or between, the lines AB, and AC, which meet at the point A, is equal to the angle between the lines DE, and DF, which meet at the point D, if, when the point A is 'applied to', or placed 22 upon, the point D, and the line AC upon the line DF, then also the line AB coincides with DE. (2) That the angle between AB and AC is greater or less than the angle between DE and DF, according as, when AC is applied to DF as before, AB falls farther from, or nearer to, DF, than DE does. 9. An angle is generally denoted, or expressed, by three letters of the alphabet, in the following manner: The middle letter invariably marks the point where the lines which form the angle meet together, and of the other two letters one is upon one of the lines and the other upon the other line. Thus, if the lines BA, BC, BD, meet together at the same point B, the angle between BA, and BD, is called the angle ABD, or DBA, whichever we please, only taking care that B is the middle letter; the angle be tween BA and BC is called the angle ABC, or CBA; and the angle between BD and BC is called the angle DBC or CBD. Sometimes, however, when only two lines meet together, forming only one angle, so that no mistake can arise as to the angle meant, that angle may be described by a single letter placed at the point where the lines meet. Thus, the angle formed by two lines which meet at the point A would be called the angle at A’. The point where the lines which form an angle meet together is called the angular point, or vertex of the angle; and ought to be carefully distinguished from the angle itself. Observe, the magnitude of an angle does not at all depend upon the length of the lines by which it is formed, but only upon their position. Yet the lines must be some length to be lines at all. 10. If one of the lines which form an angle be extended in the same straight line from the angular point, so as to form a second angle on the same side of it adjacent to the former, and these angles are found to be equal (8)* to each other, then each of the angles is called This will be the mode of referring to a previous paragraph, or article, as it is usually called. In this case, it is meant that the reader look back to the paragraph numbered 8, and see that a method has been there explained of comparing one angle with another. a right angle. Thus, if CA, one A perpendicular to the line CD; and again, AB is said to be at right angles with CD. An obtuse angle means an angle greater than a right angle, as EAC. (8). An acute angle means an angle less than a right angle, as EAD. (8). 11. TRIANGLES. A plane surface bounded by three straight lines meeting together at their extremities, so as entirely to enclose a space, is called a triangle; and the three straight lines are called the sides of the triangle. Thus each of the following figures is called the triangle ABC, whose sides are AB, AC, BC, the letters A, B, C, being at the three angular points. AAA B 2 Ᏼ 4 When the three sides are equal to each other, the triangle is called equilateral, or equal-sided, as in fig. 1, where 4B = AC = BC*. When two sides only are equal, as in fig. 2, where AB AC, and BC is unequal, the triangle is called isosceles', which signifies' equal-legged', as if the triangle = The following abbreviations will be used throughout the book :- + for 'added to', or to be added'. .. fortherefore'. were supposed to stand upon BC, as a base, with two legs AB, AC. A triangle, as the name implies, has also three angles within it, as the 'angle at A', the 'angle at B', and the 'angle at C', or BAC, ABC, and BCA: and triangles have received other distinctive names, besides those mentioned above, after the names of one or more of these angles. Thus, A triangle, which has one of its angles a right angle, is called a right-angled triangle, as ABC fig. 3, where the 'angle at C' is a right angle. A triangle, which has one of its angles an obtuse angle, is called an obtuse-angled triangle, as ABC fig. 4, where the angle at C' is an obtuse angle. A triangle, which has each of its angles acute, is called an acute-angled triangle, as ABC figs. 1 and 2. 12. PARALLEL straight lines are such as, being in the same plane, never meet though produced ever so far both ways. Thus the straight lines AB, CD, are parallel to each other, if, being both on the plane of this paper, they never meet however far produced either towards the right hand or the left. 13. PARALLELOGRAMS. A parallelogram is a plane surface bounded by four straight lines, called its sides, of which each opposite two are parallel. There are several kinds of parallelograms :-viz. (1) A square is a parallelogram, which has all its sides equal and all its angles right angles, as fig. 1. 2 (2) An oblong, or rectangle, is a parallelogram, which *It will be seen hereafter that no triangle can have more than one right angle. |