OF SPACE. 



15 



A IS 



IZ\ 



B D 



124. Theorem- Equiangular parallelo- 

 "■ranis are tr each other in the ratio com- 

 pounded of the ratios of their side's. 



p. Or in the ratio of the rectangles or 



numeral products of their sides. For 



'q since AB ; BC=AD : DC (l 20;, and 



DC : dE=:DB : BE, multiplying the 



former equation by the ferms of the 



latter, AB.DB : BC.BE=AD.CE. 



125. Theorem, similar triangles, and 

 figures composed of similar triangles, are in 

 the ratio of the squares of their homologous 

 sides. 



/-< Since similar triangles are the 



F halves of equiangular parallelograms, 

 which are in the ratio compounded 

 -£ of the ratios of their sides (124), the 

 triangles are in the same ratio, or 

 ABC : DEF=AB.BC : DE.EF ; but AB : DE=BC : EF 

 (121), therefore ABC : DEF=AB.AB: DE.DE, or ABq -. 

 DEq. And the same may be proved of similar polygons, 

 by composition (32J. 



12G. 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 perpen- 

 difcular 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 (los), it 



is evident that AC is greater than the 



radius AD, and that C, as well as 



B C every other 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 tan- 

 gent of the arc BD, or the angle BAD. 



129. Definition. AC is the secant of 

 BD, or BAD. 



180. Definition. DE perpendicular to 

 A B, is the sine of BD or BAD. 



131. Definition. AE is the cosine of 

 BD or BAD. 



132. Definition. 

 of BD or BAD. 



EB is the versed sine 



Scholium, The circle is practically supposed to be di- 

 vided into 360 equal parts, called degrees, each of these 

 into 60 minutes, a minute into 60 seconds ; and the divi- 

 sion may be continued without limit ; thus 6o"=:i', 60'=: 

 1°, 90° make a right angle. Some modern calculators 

 divide the quadrant into 100 equal parts,and subdivide tliese 

 decimally. 



133. Theoreji. The angle subtended at 

 the centre of a circle by a given arc, is double 

 the angle subtended at the circumference. 



Let ABC and ADC be subtended by AC. 

 Draw the diameter DBE, then ^ ABE= 

 ADB-t-BAD(los}=:2ADB (s;). Also 

 Z. CBEzriCDB, therefore ABE— CBE= 

 2ADB— 2CDB, or ABC=2ADC. In a 

 similar manner it may be proved in other 

 positions. 



134. Theorem. The angle contained by 

 the tangent and any chord at the point of 

 contact, is equal to the angle contained in 

 the segment on the opposite side of the 

 chord. 



Draw the diaimeter AB, and join BC ; 

 then /, BCA is equal to half the angle 

 subtended at the centre by the semicircle 

 AB, or to a right angle, and AHC and 

 BAG make together another right angle 

 . (93), therefore deducting BAC, ABC= 

 CAD. And it ajipears also from the last 

 proposition that the angle contained in the lesser segment 

 CA is equal to the complement of ABC to t«o right 

 angles, or to CAE. 



135. Problem. To draw a tangent to a 

 circle from a given point without it. 



Join AB, bisect it in C, and on 

 C draw a circle, with the radius 

 CB, intersecting the former circle 

 in D, then AD shall touch the 

 circle. For the angle ADB, in a 

 semicircle, is a right angle (134, 12?), and BD is the radius 

 of the given circle. 



13f3. Theorem. In equal circles, equal 

 angles stand on equal arcs. 



