Straightedge

“ In drawing these curves great exactitude is required to properly find their centres; nothing looks worse in a drawing than an unfair or uneven junction between curves and straight lines. To find the location for these centres, set the compasses to the required radius for the curve, and from the point or corner A draw the arcs b and c, from c mark the arc e, and from b the arc d, and where d and e cross is the centre for the curve f. ”

“ Just in the same way straightness mingled with curvature, that is to say, the close approximation of part of any curve to a straight line, gives to such curve all its spring, power, and nobleness: and even perfect straightness, limiting curves, or opposing them, is often pleasurable: yet, in the abstract, straightness is always ugly, and curvature always beautiful. ”

“ For testing the straightness, in one direction only, of a surface the straight-edge is employed. It consists in the small sizes of a piece of steel whose edges are made straight and parallel one to the other. When used to test the straightness of a surface without reference to its alignment with another one, it is simply laid upon the work and sighted by the eye, or it may have its edge coated with red marking, and be moved upon the work so that its marking will be transferred to the high spots upon the work. ”

“ If for example, when my Father, the Triangle, approaches me, he happens to present his side to me instead of his angle, then, until I have asked him to rotate, or until I have edged my eye round him, I am for the moment doubtful whether he may not be a Straight Line, or, in other words, a Woman. ”

“ But if the piece be tight in the middle and we test with the fingers and straight-edge in the same way, the middle instead of bending downwards, appears to rise up, the straight-edge showing it to be rounded. ”

“ If the exterior angle of a triangle, made by producing one of its sides, be bisected by a straight line which also cuts the base produced, the segments between the dividing straight line and the extremities of the base shall have the same ratio which the other sides of the triangle have to one another; and if the segments of the base produced have the same ratio which the other sides of the triangle have to one another, the straight line drawn from the vertex to the point of section shall bisect the exterior angle of the triangle. ”

“ But this existence of a universal representing absolute straightness is precisely one of the metaphysical cobwebs of which the discovery of non-Euclidean geometry has purged science. To illustrate this point more fully, let us assume that we think we know what is implied by a straight line. Whether we merely imagine a straight line or endeavour to realise one concretely, we are always faced with the same difficulty. ”

“ The marking will look of the darkest colour in the places where the straight-edge bears the hardest. The most refined use of the straight-edge is that of testing the alignment of one surface to the other, and as this class of work often requires straight-edges of great length, as six or ten feet, which if made of metal would bend of its own weight, therefore they are made of wood. ”

“ Euclid’s definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids. A few more notions have to be added to these, principally those of the simplest line—the straight line, and of the simplest surface—the flat surface or plane. These notions we possess, but to define them accurately is difficult. Euclid’s Definition 4, I., “A straight line is that which lies evenly between its extreme points,” must be meaningless to any one who has not the notion of straightness in his mind. ”

“ In all tests made with straight edges, the straight edge should be turned end for end and each measurement repeated, because, if the straight edge is true, turning it end for end will make no difference to the measurement, while if the straight edge is not true the measurement will vary when the straight edge is reversed. ”

“ By the position, that a straight line will bear nothing, is meant, that it receives no strength from straightness; for that many bodies, laid in straight lines, will support weight by the cohesion of their parts, every one has found, who has seen dishes on a shelf, or a thief upon the gallows. It is not denied, that stones may be so crushed together by enormous pressure on each side, that a heavy mass may safely be laid upon them ”

“ Any line intersecting the legs of a right angle is cut harmonically by any two lines through its vertex which make equal angles with either of its sides.
6. If the base of a triangle be given in magnitude and position, and the ratio of the sides, the locus of the vertex is a circle which divides the base harmonically in the ratio of the sides.
7. If a, b, c denote the sides of a triangle ABC, and D, D′ the points where the internal and external bisectors of A meet BC; prove
8. ”

“ If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another. ”

“ Appearances.We have to deal with appearances rather than realities. The edges we know to be horizontal, no longer appear so; we compare them with the pencil held horizontally until we realise the angle they make with it. The width has apparently grown much less and must be compared with the front edge. ”

“ Our Women are Straight Lines.
Our Soldiers and Lowest Classes of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle. Indeed when their bases are of the most degraded type (not more than the eighth part of an inch in size) , they can hardly be distinguished from Straight Lines or Women; so extremely pointed are their vertices. With us, as with you, these Triangles are distinguished from others by being called Isosceles ”

“ The due mixture of the curved line with the straight, the growth of one from the other, the repetition of straight lines in suitable positions, all seem to have come to them by intuition which seldom erred. ”

“ Those which declare that two straight lines cannot inclose a space, and that two straight lines which cut one another cannot both be parallel to a third, are in reality the only ones which express characteristic properties of space, and these it will be worth while to consider more nearly. Now the only clear notion we can form of straightness is uniformity of direction, for space in its ultimate analysis is nothing but an assemblage of distances and directions. ”

“ If a straight line is a thing, and a crooked line another thing, and if these two things are held to be opposed to one another, then the above logic is the most justified in the world. But who claims that there are not many straight lines which are crooked at one end, which run straight on for a certain distance and then turn? ”

“ It is evident, that all that can be meant by the three angles of a triangle being equal to two right ones, or the parabolic area to of the circumscribing parallelogram, or deduced from these positions, must always hold in future fact; and this, as observed above, is all the truth that any thing can have. In fluxional conclusions it is demonstratively evident, that the quantity under consideration cannot be greater or less by any thing assignable, than according to the fluxional conclusion ”

“ Requirements for Appendage Design
Freehand Curves
It is the unexpected curve that is welcome in all designing, provided it supports the structure and conforms to established rules. After completing a design involving appendages it should be checked from three points of view; (1) service, (2) unity between the primary mass and the appendages, and (3) variety of curvature. On this last point it is needless to say that compass curves are not desirable except in rounding small corners or in using fillets. ”

“ I. Your Lordship would treat me as if I were one of the vulgar who, being ignorant of Mathematics, suppose that a Woman is really a Straight Line, and only of One Dimension. No, no, my Lord; we Squares are better advised, and are as well aware as your Lordship that a Woman, though popularly called a Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness) . ”

“ Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of an individual ”

“ Two straight lines meeting in opposing directions give an impression of abruptness, severity, or even violence; the difference of movement being emphasized. If a third line is added, as in the sketches below, the opposition is softened and an effect of unity and completeness produced.
This combination typifies beauty itself which has been defined as consisting of elements of difference harmonized by elements of unity.
A very common example of Transition is the bracket, No. 15. The straight line is modified into curves and may be elaborated with great complexity of modelling. ”

“ This division is taken out of the 4 d j. of Euclide, where rectitude or straightnes is attributed to a line, as if from it both surfaces and bodies were to have it. And even so the rectitude of a solid figure, here-after shall be understood by a right line perpendicular from the toppe unto the center of the base. Wherefore rectitude is propper unto a line: And therefore also obliquity or crookednesse, from whence a surface is judged to be right or oblique, and a body right or oblique.
5. A right line is that which lyeth equally betweene his owne bounds: A crooked line lieth contrariwise. 4. ”

“ In the human figure there underlies the curved outline a certain element of rigidity and straightness, indicating strength in the supporting limbs and spine. Whenever firmness is required, there must be a solid structure, and straightness of form is a frequent accompaniment of solidity. The straight nose and the flat brow are subsidiary to the movement and the stability of the face. ”

“ If a right line making an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned. ”

“ If the frame is straight, this edge may be planed straight. It is not wise to take for granted the squareness or straightness of a frame. A test or series of tests may first be made with square and straight-edge. A mechanic, however, usually planes an edge until it fits the frame, testing by holding the door against the frame as near to its position as its size will allow. (3) Plane the bottom or top edge of the door until it fits the frame properly when the first planed edge is in position. (4) Measure the width of the frame at its top and bottom, Fig. ”

“ The pupil will move his pencil two or three times from the left dot to the right, before he draws the line. This precaution is more necessary when the operation is performed on paper than when on a slate, where it may be erased if wrong.
Figs 7-8 20. Make an acute angle. (fig. 7.)
Care must be taken to distinguish an angle, from what is called its point or apex. The angle is the opening between two lines that meet, and the point or apex is the point where the lines meet. A pair of dividers forms a number of different angles, by being opened more or less. ”

“ The straight-edge must not be pressed to the plate, but merely touched laterally to make it move laterally, for if we take a true straight-edge and press it vertically to a true surface while moving it, it will show the marks of contact the most plainly immediately beneath the parts where it is pressed. ”

“ We have classed the bracket as a link connecting a vertical and horizontal structure. Mouldings may likewise be considered as links connecting similar horizontal or vertical surfaces by bands of graded forms. Inasmuch as they effect the outline they are considered in this chapter. As the mouldings are to assist the eye to make the jump from one surface to another by easy steps, the position from which the mouldings are to be seen determines to some extent their design. ”