Parallel lines

“ The student of modern geometry is already accustomed to think of parallel lines as lines which intersect at infinity, or as lines whose inclination is zero. In speaking of the generalized geometry in a plane, we may define that two straight lines which intersect upon the fundamental conic are parallel. It thus follows that through any point two distinct parallels can be drawn to a given straight line. The only exception will arise in the case where the given line touches the fundamental conic. ”

“ Then, at the points at which the parallel lines cut the line called the horizon, the letter S is to be on the right and the letter V on the left, and from the extremity of the semicircle, at the point G, draw a line parallel to the axis, extending to the left-hand semicircle at the point H. This parallel line is called the Logotomus. ”

“ If you stand on the railway, where the lines run straight they will seem to meet in the distance, whereas we know that they always remain the same distance apart. The apparent meeting is an effect of "perspective." Some of you have no doubt learnt perspective drawing and you know that parallel lines have to be drawn to meet in a "vanishing point." So Mr. Boss's group of stars may be moving along parallel lines, without closing up: and indeed we can find out that this is the fact ”

“ By tracing parallel lines and finding vanishing points of planes and surfaces, much that bewildered you in the past will become clear and reasonable.
Planes, horizontal planes and perpendicular planes, are terms constantly used with regard to perspective.
A horizontal plane is a plane parallel with the earth; a perpendicular plane is one perpendicular to the earth. The top of a table and the ceiling of a room are horizontal planes; the walls of the room are perpendicular planes. ”

“ The theory which affirms that parallel lines converge only to one and the same point upon the eye-line is an error. It is true only of lines equidistant from the eye-line. It is true that parallel lines converge to one and the same eye-line, but meet it at different distances when more or less apart from each other. This is the true law of perspective as shown by Nature herself; any other idea is fallacious and will deceive whoever may hold and apply it to practice. ”

“ To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect.
To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line. ”

“ A perpendicular or vertical line is one perfectly upright, as a string, with a weight at the end of it, will hang from a nail, or from the hand.
In making horizontal lines, the pupil should make them parallel to the top or bottom of his slate or paper, and in making perpendiculars they should be parallel to the sides of the slate or paper. Parallel lines are lines running in the same direction equally distant from each other in every part ”

“ The paths do not appear to us to be parallel, because of perspective: they seem to radiate and spread in all directions from a fixed centre like spokes, but all these diverging streaks are really parallel lines optically foreshortened by different amounts so as to produce the radiant impression. ”

“ For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio. ”

“ If two straight lines which meet one another, be parallel to two other straight lines which meet one another, but are not in the same plane with the first two, the plane pass-ing through these is parallel to the plane passing through the others. ”

“ It may sometimes be noticed that the lines running in one direction form a system by themselves, varying very little from strict parallelism with each other, but crossing another system, more or less strongly marked, of other lines equally parallel with each other. At other times, a system of lines, strongly marked and diverging very slightly, seem to pass over another system, in which the lines form various angles with each other. ”

“ If we now connect the four points , , , by right lines we shall form a quadrilateral figure, the area of which will measure the quantity of action developed during the operation described. Now it is easy to see that and are two elements infinitely near, described upon two curves infinitely near, the equations of which will be , and They ought therefore to be considered as parallel; the two ordinates which terminate the quadrilateral figure in the other direction being also parallel, the figure is parallelogrammical, and measures . ”

“ Again, is the supposition that parallel lines can meet in one sense geometrical and from another point of view ungeometrical? ‘Ungeometrical’ is in fact an ambiguous expression, as is ‘unrhythmical.’ One thing may be ungeometrical or unrhythmical from not possessing the quality in question at all, another from having it defectively. ”

“ It is always possible when one of the functions or is known, to determine the other so that may be equal to unity. For instance, let us take the plane of and draw upon it a series of equidistant lines parallel to to represent the sections of the family by this plane. ”

“ Have you Euclid's Axiom 'all right angles are equal'?
Nie. We deduce it from 'all straight angles are equal': and that we prove by applying one straight angle to another.
Min. That is all very well, though I cannot think 'straight angles' a valuable contribution to the subject. I will now ask you to state your method of treating Pairs of Lines, as far as your proof of Euc. I. 32.
Nie. To do that we shall of course require parallel Lines: and, as our definition of them is 'Lines having the same direction,' we must begin by discussing direction. ”

“ In endeavouring, for example, to grasp the relation of the exterior angle to the two interior and opposite angles, the pupil may fail because he has not a clear knowledge of the equality of angles in connection with parallel lines. ”

“ This, however, has rendered it necessary for them to assume, as an additional axiom, some other property of parallel lines; and the unsatisfactory manner in which properties for that purpose have been selected by Euclid and others has always been deemed the opprobrium of elementary geometry. Even as a verbal definition, equidistance is a fitter property to characterize parallels by, since it is the attribute really involved in the signification of the name. ”

“ Section 15 All corners that can exist on one side of a line, as in this figure,
always add up to 180 degrees.
Section 16 The corner with the greatest angle has opposite it the longest side, and the reverse is true as well.
Section 17 The complement of an angle is the amount needed for an acute angle to reach 90 degrees.
Section 18 The remainder [שארית] of an angle is amount needed for an angle, whether acute or obtuse, to reach 180 degrees.
Section 19 Parallel lines are two lines that run next to each other separated by a constant distance. These lines will never intersect with one another. ”

“ To accomplish this he should be taught to draw, however rudely. He can begin with straight lines, of which he soon learns the properties; then, when he has drawn several lines side by side, he will learn that they are parallels and will never meet. He will learn, too, after he has drawn three intersecting lines, that the figure within them is called a triangle, that the figure formed by two parallel lines meeting two other parallels is a parallelogram, and he can go on to make and learn about polygons, etc (Fig. 3) . ”

“ Is any line parallel to u v?
Can a single line be properly called perpendicular? Parallel?
If two lines are perpendicular to each other, what angle do they form?
If parallel, what angle? If oblique?
Diagram 13.
LESSON THIRTEENTH. RELATIONS OF ANGLES. INTERIOR ANGLES. Is the angle a m n between the parallels, or outside of them? (Diagram 13.)
It is called an interior angle.
31Read three other interior angles between the same parallels.
Why is b m n an interior angle?
An interior angle is one that lies between parallel lines. ”

“ If a straight lino which joins the extremities of two equal straight lines, not parallel, make the angles on the same side of it equal to each other, the straight line which joins the other extremities will be parallel to the first. ”

“ If two pairs of opposite sides of a cyclic hexagon be respectively parallel to each other, the remaining pair of sides are also parallel.
6. If two circles intersect in the points A, B, and any two lines ACD, BFE, be drawn through A and B, cutting one of the circles in the points C, E, and the other in the points D, F, the line CE is parallel to DF.
7. ”

“ A line in motion in any direction except along itself produces a surface, which has length and breadth, but not thickness. A plane surface or plane is a surface without curvature. A straight line connecting any two of its points lies wholly within the plane. A plane surface in motion in any direction except within its plane produces a solid, which has length, breadth, and thickness. Parallel lines or surfaces are those which are everywhere equidistant. Perpendicular lines or surfaces are those which meet at right or 90° angles. ”

“ Now from the points i, f, h draw the straight lines io, fg, hl parallel to be; these lines hl, fg, io are equal to eb, db and cb, respectively; so also are the lines bo, bg, bl respectively equal to ci, df, and eh. ”

“ Why is m n g an acute angle?
What is an acute angle?
Here the attention is directed first to this particular angle; then this is taken as an example of its kind, and the idea generalized by describing the class. See also page 29.
Why are the lines e f and g h said to be parallel?
When are lines said to be parallel?
Many of the questions are intended to test the vividness of the pupil’s conception. See page 29.
Also page 78. If the circumference were divided into 360 equal parts, would each arc be large or small?
Many of the questions are intended to test the attention of the pupil. ”

“ If, in a certain diagram, whose geometrical history we know, we want to test whether two Lines, passing through a common point, have, or have not, 'the same direction,' we have simply to take any other point on one of the Lines, and observe whether the other Line does, or does not, pass through it. ”