28 INTRODUCTION. 



ABC : DEF=An.BC : DE.EF ; but AB : DErrBC : EF (121)^ 

 tlicrcforc ABC : DEF=AB.AB : DE.DE, or ABq : DEq. Anrt 



the same may be proved ot similar polygons, by composition (32). 



126. Definition. An indefinite right line, meeting a 

 circle and not cutting it, is called a tangent, 



127. Theorem. A right line, passing through any 

 point of a circle, and perpendicular to the radius at that 

 point, touches the circle. 



Since the perpendicular AB is shorter than any 

 other line AC, that can be drawn from A to BC 

 (103), it is evident that AC is greater than the ra- 

 dius AD, and that C, as well as every otlier point 

 of BC, besides B, is without the circle ; therefore 

 BC does not cut the circle, but touches it. 



128. Definition. BC is called the tangent of the 

 arc BD, or the angle BAD. 



129. Definition. AC is the secant of BD, or 

 BAD. 



130. Definition. DE perpendicular to AB, is the 

 sineofBDorBAD. 



131. Definition. AE is the cosine of BD or BAD. 



132. Definition. EB is the verse sine of BD or 

 BAD. 



Scholium. The circle is practically supposed to be divided into 

 360 equal parts, called degrees ; each of these into 60 minutes ; a mi- 

 nute into 60 seconds; and the division may be continued without 

 limit; thus 60"tzl\ 60'2=1°, and 90° make a right angle. Some mo- 

 dem calculators divide the quadrant into 100 equal parts, and sub- 

 divide these decimally, or rather centesimally. 



133. Theorem. The angle subtended at the centre 

 of a circle, by a given arc, is double the angle subtended 

 at the circumference. 



