882. Two rays from the same origin are said to have the same direction if they coincide; otherwise, they are said to have different directions. 883. Two rays which have no point but the origin in common, and fall into the same line, are said to have opposite directions. 884. Two rays lying on parallel lines have parallel-same directions if they are on the same side of the line joining their origins. 885. Two rays lying on parallel lines have parallel-opposite directions if they are on opposite sides of the line joining their origins. 886. A sect is said to have the same direction as the ray which it is a part. of 887. A sect is definitely fixed if we know its initial point, its parallel-same direction, and its length. 888. The operation by which a sect could be traced if we knew its initial point, that is, the operation of carrying a tracing-point in a certain parallel-same direction until it passes over a given number of units for length, is called a Vector. 889. The position of B relative to A is indicated by the length and parallel-same direction of the sect AB drawn from A to B. If you start from A, and travel, in the direction indicated by the ray from A through B, and traverse the given number of units, you get to B. This parallel-same direction and length may be indicated equally well by any other sect, such as A'B', which is equal to AB and in parallel-same direction. 890. As indicating an operation, the vector AB is completely defined by the parallel-same direction and length of the transferrence. All vectors which are of the same magnitude and parallel-same direction, and only those, are regarded as equal. Thus AB is not equal to BA. PRINCIPLE OF DUALITY. JOIN OF POINTS AND OF LINES. 891. The line joining two points is called the Join of the Two Points. The point common to two intersecting lines is called the Join of the Two Lines. 892. PENCIL OF LINES. A fixed point, A, may be joined to all other points in space. We get thus all the lines which can be drawn through the point A. The aggregate of all these lines is called a Pencil of Lines. The fixed point is called the Base of the pencil. Any one of these lines is said to be a line in the pencil, and also to be a line in the fixed point. In this sense, we say not only that a point may lie in a line, but also that a line may lie in a point, meaning that the line passes through the point. 893. In most cases, we can, when one figure is given, construct another such that lines take the place of points in the first, and points the place of lines. Any theorem concerning the first thus gives rise to a corresponding theorem concerning the second figure. Figures or theorems related in this manner are called Reciprocal Figures or Reciprocal Theorems. 894. Small letters denote lines, and the join of two elements is denoted by writing the letters indicating the elements, together. Thus the join of the points A and B is the line AB, while ab denotes the join, or point of intersection, of the lines a and b. ROW OF POINTS, PENCIL OF LINES. 895. A line contains an infinite number of points, called a Row of Points, of which the line is the Base. A row is all points in a line. The reciprocal figure is all lines in a point, or all lines passing through the point. A Flat Pencil is the aggregate of all lines in a plane which pass through a given point. In plane geometry, by a pencil we mean a flat pencil. jointed, and two equal links movably pivoted at a fixed point, and at two opposite extremities of the rhombus. TO DRAW A STRAIGHT LINE, 899. Take an extra link, and, while one extremity is on the fixed point of the cell, pivot the other extremity to a fixed point. Then pivot the first end to one of the free angles of the rhombus. The opposite vertex of the rhombus will now describe a straight line, however the linkage be pushed or moved. PROOF. By the bar FD, the point D is constrained to move on the circle ADR; therefore ADR, being the angle in a semicircle, is always right. M Σ R If, now, E moves on EM & AM, Δ ADR ~ Δ ΑΜΕ, :. DA: AR :: AM : AE, :. DA. AE = RA. AM. Therefore, if AE. AD is constant, E moves on the straight line EM. But because BDCE is a rhombus, and AB AC, ..D and N are always on the variable sect AE. 900. COROLLARY. The efficacy of our cell depends on its power to keep, however it be deformed, the product of two varying sects a constant. B Therefore a Hart four-bar cell, constructed as in the accompanying figure, may be substituted for the Peancellier six-bar cell, since AE. AD equals a constant. Also, for the Hart cell may be substituted the quadruplane, four pivoted planes. CROSS-RATIO. 901. If four points are collinear, two may be taken as the extremities of a sect, which each of the others divides internally or externally in some ratio. The ratio of these two ratios is called the Cross-Ratio of the four points. The cross-ratio AC AD is written (ABCD). Distinguishing the "step" AB from BA, as of opposite "sense," and taking the points in the two groups of two in a definite order, to write out a cross-ratio, make first the two |