16 



or SPACE. 



/ 



Let AB and BC be the sines ot ...^ j^g 

 angles, ACB, BAC, then AC will be ti,. 

 sine of their sum CBD, qr of ABC. Now- 

 making BE perpendicular to AC, AC" 



For the chords of equal angles 



are equal (86), and the segments 



cut offby them contain equal angles 



(133) i and if a segment equal to 



AB be supposed to be described on 

 the chord CD, and on the same side with CED, it must 

 coincide with CED, for since, at each point of each arc, CD 

 subtends the same angle, the points of one arc can never be 

 within those of the other (99) ; the arcs are therefore equal. 

 Scholium. Hence it may easily be shown, that mul- 

 tiple and proportionate angles are subtended by multiple 

 and proportionate arcs. 



137. Theorem. If two chords of a given ,^^, , 



" and EF be always parallel to DG' 



circle intersect each other, the rectangles the radius of ad, and cm the cen- 

 contained by the segments of each are equal. treF, draw the circle ah; join 



Join AB and CD. Then ^AEB^: AH, then since Z-EADziiAGD 



DEC (90}, and /.BAE::=DCE (i33), =:1AFM, the chord AH will coin- 



1^ both standing on BD, therefore the cide with the chord AD (133> 



AE+EC, and rad. : cos. BAC : : AB: AE, and rad. t cos. 

 ACB: :BC:CE (139). 



141. Theorem. The ratio of the Eva- 

 nescent tangent, arc, chord, and sine, is that 

 of equality. 



' Let AB be the tangent, and CD 

 the sine of the arc AD. Let AE 

 betakenat pleasure in the tangent. 



triangles AEB, CED, are similar, and 

 AE:CE::EB:ED (li2l), therefore 

 AE.EDrzCE.EB (123). 



138. Theorem The rectangle contained 

 by the segments of a right line intercepted 

 by a circle and a given point without it, is equal 

 to the square of the tangent drawn from that 

 point. 



Join AB, AC; then ^lABCrrCAD 

 (134), and the angle at D is common, 

 therefore the triangles ABD, CAD, are 

 similar, and BD : AD : : AD : CD (121), 

 whence BD.DC=:ADq (123). 



IS9. Theorem. In every triangle the 

 sides are as the sines of their opposite angles, 

 the radius being given. 



C Take ABzzCD, and draw BE and 



134). And when DA vanishes, 

 DG coinciding with AG, EF will 

 be parallel to AF, and the angle " . 



EAH will vanish, therefore AH will coincide with AE and 

 with IH parallel to the sine CD ; and by similar triangles 

 the ratio of AB, AD, and CD, is the same as that of AE, 

 AH, and IH, and is ultimately that of equality. But the 

 arc AD is nearer to the chord AD than the figure ABD, and 

 it has no contrary flexure, therefore it is longer than the 

 line AD (79), and shorter than ABD (80>,- until their dif- 

 ference vanishes, and it coincides with both. 



Scholium. The same is obviously true of any curve 

 coinciding at a given point with any circle ; and all the 

 elements agree as well in position as in length. 



\\1. Theorem. The fluxion of the arc 

 being constant, the fluxion of the sine varies 

 as the cosine. 



B 



B 



iD 



CF perpendicular to AD, then they 

 are the sines of the angles A and D, 



The fluxion of the arc is equal to that of 

 the tangent, since their evanescent incre- 

 ments coincide (141). Let AB be the sine, 

 AC the cosine, BDthe increment of the tan- 

 gent, DE that of the sine: then /.ABC::! 

 EBD (16), and the triangles ABC, EBD, are 

 similar, and BD is to DE as BC to AC ; but the ultimate 

 ratio of the increments is that of the fluxions, therefore the 

 fluxion of the tangent, or of the arc, is to that of the sine 

 as the radius to the cosine. The same may easily be in- 



E F to tlie radius AB or CD (130), and 

 by similar triangles, AC : CF : : AB ; BE (121), or CD : BE. 

 And the same maybe shown of the other sides and angles. 



140. Theorem. The sine of the sum of 



any two arcs, is equal to the sum of the sines ferred from the theorem for finding the sine of the sum of 



of .the separate arcs, each being reduced in two arcs (i4o). 



the ratio of the radius to the cosine of the 143. Theorem. The area of a circle is 



other arc. equal to half the rectangle contained by 



