328 MATHEMATICS. 



noticed the logarithmic spiral; and Galileo, the cycloid; which 

 was afterwards more fully investigated by Huyghens. 



In more recent times, numerous discoveries and improvements have 

 been made in Geometry, by the application of algebraic formulas to 

 geometrical figures ; the study of which belongs to the succeeding 

 branches of Mathematics. We have only room to add that the sub- 

 branch of Descriptive Geometry, was chiefly invented by Gaspar 

 Monge ; who published his treatise on this subject, about the year 

 1794. Ptolemy drew maps according to the stereographic projec- 

 tion ; but the other spherical projections are of later origin ; and the 

 globular was invented by De La Hire. The subject of Perspective, is 

 here deferred, until we come to the branch of Painting, among the Fine 

 Arts ; and Surveying, is reserved for Civil Engineering, or Viatecture. 



Elementary Geometry is sometimes divided into Longimetry, or 

 the measure of lengths, and the properties of lines ; Planimetry, 

 relating to surfaces; and Stereometry, relating to solids. We shall 

 here treat of it under the heads of 1. Preliminary Elements ; 2. Plane 

 Rectilinear Figures ; 3. The Circle and its Measure ; 4. Solid Angles 

 and Polyedrons ; and 5. The Three Round Bodies. To this division, 

 Descriptive Geometry will be regarded as an appendix. 



1. The first Elements of Geometry, are the definitions of magni- 

 tudes. A point, has no magnitude, but serves to designate a position 

 in space. A line, has length, but no breadth or thickness ; and it 

 may be considered as formed by a series of points, or generated by 

 the flowing, that is the motion, of a point. A surface, has length, 

 and breadth, but no depth or thickness ; and it may be generated by 

 the motion of a line. A solid, has length, breadth, and thickness ; 

 which are called the three dimensions of extension. A straight line, 

 is one which follows or measures the shortest distance between any 

 two of its points ; or which lies in the same direction throughout : 

 but a curved line, is one which continually changes its direction. 

 An angle, is the inclination of one line to another ; and is measured 

 by the divergence at their point of meeting, which is called the vertex 

 of the angle. In naming an angle by means of three letters, the one 

 placed at the vertex is always named in the middle place ; as ABE, 

 Plate VII. Fig. 1. When the adjacent angles, formed by the meet- 

 ing of two straight lines, are equal, they are called right angles, 

 as ABD, Fig. 1 ; and each line is said to be perpendicular to the 

 other. Oblique angles, are either obtuse, that is greater, or acute, 

 that is less, than a right angle. Lines which are not inclined to 

 each other, but have the same direction, are said to be parallel; as 

 J1D and BC, in Fig. 3. 



A plane, is a surface, with which a straight line, applied to it in 

 any direction, will entirely coincide. A plane Jigure, is a plane 

 limited on all sides by lines ; which, if straight, enclose a rectilinear 

 figure, or polygon. A polygon of three sides, is called a triangle, as 

 Fig. 2 ; one of four sides, a quadrilateral, or tetragon ; one of five 

 sides, a pentagon ; and so on. A right angled triangle, has one right 

 angle ; the side opposite to which is called the hypothenuse; as BC, 

 in the triangle BDC, Fig. 2. An equilateral triangle has its three 

 sides equal ; an isosceles triangle has only two of them equal : and a 



