GEOMETRY. 329 



scalene triangle has no two sides equal. A square, has four equal 

 sides, and four right angles, as ABCD, Fig. 3 ; and a rectangle, 

 has four right angles, but two of the sides longer than the other two, 

 as ABDC, Fig. 9. A rhombus, has its four sides equal, but its 

 angles oblique : a rhomboid, or parallelogram, has its opposite sides 

 equal, but its angles oblique, as ABEF, Fig. 3 ; and a trapezoid, 

 has only two of its four sides parallel, as ABCF, Fig. 3. A diago- 

 nal, is a line joining the vertices of two angles not adjacent to each 

 other. 



A circle, is* a plane surface, terminated by a curved line, called its 

 circumference, all the points of which are equidistant from a point 

 within, called the centre. Fig 4. A straight line passing through the 

 centre, and terminated by the circumference on both sides, is called a 

 diameter, (AB) ; and a straight line from the centre to the circum- 

 ference, is called a radius. (OB). An arc, is any portion of the 

 circumference, (AC) ; and the chord of an arc is a straight line join- 

 ing its extremities. A segment, is the surface between an arc and 

 its chord ; and a sector is the surface between two radii. (BOB'). A 

 tangent to a circle, is a straight line which merely touches it, (AT) ; 

 and a secant, is one which cuts the circle, as a chord prolonged. An 

 angle formed by two radii, is measured by the intercepted arc ; a 

 right angle being divided into ninety degrees. (90.) Angles of a 

 given magnitude, are usually constructed by means of a scale of 

 chords, AF, Fig. 12 ; the chord of 60, or distance A, 60, being 

 used as the radius. 



2. Of Plane Rectilinear Figures, we have only room to give 

 some of the more important measures. In every triangle, the greater 

 side is opposite to the greater angle : and either side is always less than 

 the sum of the other two. In every triangle, the sum of the three 

 angles is equal to two right angles, or 180. When two triangles 

 have the angles of the one equal to the corresponding angles of the 

 other, each to each, they are said to be similar ; and the homologous 

 or corresponding sides are proportional. The measure of a triangle, 

 is the product of its base by half its altitude ; this latter being mea- 

 sured on a line perpendicular to the base. The square on the hy- 

 pothenuse, of a right angled triangle, is equal to the sum of the squares 

 on the other two sides. The measure of a square, is the square of 

 one of its sides ; and the measure of a rectangle, is the product of 

 two of its contiguous sides, one being considered as the base, and the 

 other as the altitude. The measure of a rhombus, as well as that of 

 a parallelogram, is the product of its base by its altitude ; this latter 

 being measured on a line perpendicular to the base. Any polygon, 

 may be measured, by subdividing it into triangles, and finding the 

 sum of their areas, or measures, separately taken. In all these mea- 

 surements, the unit of surface, is a square, each side of which is the 

 adopted unit of length ; as a square foot, each side of which is one 

 foot in length. 



3. We come next to the Properties of the Circle. In the same 



circle, or in equal circles, if we take equal arcs, their chords will also 



be equal, and at equal distances from the centre ; that is, equal chords 



subtend equal arcs. The radius which is perpendicular to a chord, 



42 2m 2 



