raised, and allowed to recoil by its elasticity, thus leaving upon the surface a chalked line. (7.) Another definition of a straight line is as follows: When a line is such, that the eye being placed near one extremity so as to cause it to conceal the other extremity, it shall, at the same time, hide from view all other portions of the line; then such line is called a straight line. This definition is due to PLATO. A practical application of this definition is used by artisans, in bringing the eye to range along the direction of the line under consideration, technically called sighting. If a smooth piece of pliable paper be folded, the edge formed at the fold will form a very good approximation to a straight line. (8.) A straight line is sometimes defined as one which, throughout its whole extent, does not change its direction. And, in accordance with this definition, a curved line may be regarded as one whose direction is constantly changing. (9.) If an artisan wishes to make a straight-edge, he forms two distinct rules, bringing their edges as near straight as he can by the use of the common plane, testing his work by sighting along the edge with the eye. Then, as a more accurate test, he applies the two edges together, moving the one lengthwise upon the other; then reversing the ends, he again applies them, and observing under all these cases whether they touch uniformly throughout their entire length, if so, they are straight. If they do not touch uniformly, then by observing the more prominent points, he knows just which parts must be planed off so as to bring the edge more nearly straight. Thus, by continued trials, he is enabled to produce an edge sufficiently near that of a straight line to serve all practical purposes. VI. Every line which is not a straight line is called a curved line. When we hereafter speak of a line, unless otherwise expressed, we shall mean a straight line. Thus, AB and CD are straight lines; GH and KL are curved lines. The extremities of these lines, as well as their intersections F and M, are points. + & F B M A H (10.) To the eye, many curved lines appear far more graceful and pleasing than the undeviating straight line. This accounts, in part, for the superior pleasure enjoyed in viewing the ever-varying and beautiful outlines of landscapes, as contrasted with the regular and straight outlines of architectural works. VII. A plane surface, or simply a plane, is a surface, in which, if two points be taken at pleasure, and connected by a straight line, that line will be wholly in the surface. (11.) A practical test, in accordance with this definition, is employed by artisans to determine whether a surface is plane. They take a rod or rule whose edge is straight, and apply it in various directions upon the surface under consideration, observing whether the edge of the rule, or, as it is technically called, the straight-edge, coincides in all positions with the surface; if so, the surface is a plane. (12.) The practical miller, when he wishes to dress his millstones to a plane surface, rubs the straight-edge with paint, and then applies it in various directions upon the face of the stones; thus showing, by the transfer of the paint, which are the highest portions of the stone. These portions are dressed down, and the process again repeated, until the face of the stone has been brought as near a plane surface as may be deemed necessary. The action of the carpenter's plane is founded upon the same principle. VIII. Every surface which is not plane, is called a curved surface. (13.) The plane surface may be regarded as one particular kind of surface out of the infinite varieties which can be imagined. To the eye, many of the curved surfaces are far more graceful and pleasing than the plane. IX. When two straight lines AB, AC, meet each other, the space included between the lines is called an angle. The point of intersection A, is the vertex of the angle; and the lines AB, AC are the sides of the angle. Perhaps it would be better to define an angle as the opening between two lines which meet. An angle is sometimes referred to by simply naming the letter at its vertex, as the angle A; but usually by naming the three letters, as the angle BAC, or CAB, observing to place the letter, at the vertex, in the middle. X. When a straight line AB is met by another straight line DC, so as to make the adjacent angles ACD, BCD equal to each other, each is called a right-angle. The line DC, thus meeting the line AB, is said to be perpendicular to AB. (14.) An angle is sometimes defined as follows: when two lines meet, having different directions, an angle is formed, which angle is the difference of direction of the two lines. ہے A A XI. Every angle BAC which is less than a right-angle, is called an acute angle; and every angle DFG which is greater than a right-angle, is called an obtuse angle. B D B A6 (15.) If we suppose the extremity B of the line A,B to be fixed, while the line revolves in the same plane about B, so as to take the successive positions A,B, A,B, A,B, A4B, &c., until it has made a complete revolution and returned to its first position, then will this complete revolution have caused the line A,B to pass over an angular magnitude equal to four right-angles. It is obvious that this angular magnitude has no dependence upon the length of the revolving line. (16.) Angular magnitude is expressed numerically by supposing the whole space to be divided into 360 equal portions called degrees, so that 90 degrees will be the measure of a right-angle. The degree is divided into 60 equal portions called minutes, and the minute into 60 seconds, and so on in sexagesimal divisions. The French mathematicians have thought it more convenient to divide the whole angular space into 400 degrees, and each degree into 100 minutes, each minute into 100 seconds, and so on in the centesimal division. In this division the right-angle would consist of 100 degrees. No doubt the French division is more simple than the sexagesimal division; but in many geometrical as well as physical inquiries, it is desirable to express certain aliquot parts of four right-angles in integral degrees; and, in such cases, the sexagesimal division has the preference, since 360 has more divisors than 400. A3 A right-angle, Two thirds of a right-angle, AT (17.) Hereafter, unless the contrary is expressed, we shall, when speaking of degrees, wish to be understood as referring to the usual division of the whole angular space into 360 degrees. Degrees are commonly expressed by placing over the number a small circle; thus 360° signifies 360 degrees; 90°, in the same way, denotes 90 degrees. That the student may become familiar with some of the numerical denominations of angles frequently used, we have given at one point of view the following: A2 66 66 A1 Ao A7 (18.) If two diameters of a circle be drawn at right-angles with each other, as in the adjoining figure, it is obvious that the entire space will be divided into four equal portions, each being a right-angle. Therefore the entire circumference may, with great propriety, be taken as the measure of 3600, or four rightangles. Any fractional part of the circumference will be the measure of a like fractional part of 360°; thus, one fourth of the circumference is the measure of 90°, or one rightangle. The magnitude of the circumference has nothing to do with the magnitude of the angles, since their magnitudes depend wholly upon the fractional parts of the whole angular space about the centre C. D B (19.) In the useful arts, all cutting tools have their edges formed into angles of various magnitudes, according to the materials to be cut. As a general rule, the softer the material to be divided, the more acute is the angle of the cutting edge. Chisels for cutting wood are formed with an angle of about 30°; those for cutting iron, in the lathe, at from 500 to 600; and those for brass are 800 or more. (20.) The angle which is by far the most extensively used in the arts, is the right-angle. This is the angle of mechanical equilibrium, between the direction of any impact or pressure, and the resisting surface. A force cannot be wholly counteracted by a surface, unless the surface is exactly perpendicular to the direction of the force. It is this principle which determines the erect position of natural structures of animals and plants; and it is by following out the architecture of nature, that artificial structures, raised by the hand of man, acquire stability and beauty. Buildings are erect, because the direction of their weight must be perpendicular to their support. A steeple or tower, which, by the yielding of the foundation, or any other cause, is out of the perpendicular, cannot be viewed without some sense of danger, and consequently some feelings of pain. XII. Two straight lines are said to be parallel, when, being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced. They are obviously everywhere equally distant. |