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

А C line DC, thus meeting the line AB,

B is said to be perpendicular to AB.

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.

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(11.) If we suppose the extremity B of


A2 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, 44

B AzB, A4B, &c., until it has made a com

CA plete revolution and returned to its first position, then will this complete revolution have caused the line A, B to pass over an


As angular magnitude equal to four right angles. It is obvious that this angular magnitude has no dependence upon the length of the revolving line.

Angular magnitude is expressed numerically by supposing the whole space to be divided into 360 equal portions called degrees, 80 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.

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(12.) 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: A right angle

is 90° Two right angles

• 180° Three right angles

66 270° Half a right angle

45° One third of a right angle

300 Two thirds of a right angle


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(13.) 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 A

E B equal portions, each being a right angle. Therefore the entire circumference may, with great propriety, be taken as the mea

D sure of 360°, or four right angles. 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 right angle.

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.

(14.) 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, at from 50° to 60°; and those for brass are 80° or more.

(15.) 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 every where equally distant.

(16.) The ordinary frames of windows and doors, and nearly all architectural frame work, consist of systems of parallel lines at right angles with each other. All fabrics produced in the loom, consist of two systems of parallel threads, crossing each other at right angles, so interlaced as to give strength and firmness to the cloth.

The railway consists of two or more parallel lines of iron bars, called rails.

XIII. A plane figure is a limited portion of a plane. When it is limited by straight lines, the figure is called a rectilineal figure, or a polygon; and the limiting lines, taken together, form the contour or perimeter of the polygon.

(17.) The surfaces of level fields, bounded by straight fences, are polygonal figures. Floors of buildings are” polygons, usually having four sides.

XIV. The simplest kind of polygon is one having only three sides, and is called a triangle. A polygon of four sides is called a quadrilateral ; that of five sides is called a pentagon ; that of six sides is called a heptagon ; and so on for figures of a greater number of sides.

XV. A triangle having the three sides equal, is called an equilateral triangle ; one having two sides equal, is called an isosceles triangle ; and one having no two sides equal, is called a scalene triangle.


XVI. A triangle having a right angle, is called a right-angled triangle. The side opposite the right angle is called the hypothenuse, Thus BAC is a right-angled triangle, right-angled at A ; the side BC is the hypothenuse.




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