KESPECTING THE THEORY OF PARALLELS. 



443 



dicular to both parallels meets the parallels, the equality of lines at equal dis- 

 tances on either side the common perpendicular, the constant increase of these 

 lines as they recede from that perpendicular ; all these properties agree exactly 

 with the notions attached to circles. But on the other hand, the fact proved in 

 Prop. XIV., that a straight line parallel to each of the parallels bisects all these 

 lines, dissipates the idea of convexity for that line. And when it is remembered 

 that the properties of this bisecting line with respect to either the upper or the 

 lower of the given parallels are precisely the same properties as those proved to 

 exist for the given parallels themselves, we appear to have reduced parallelism 

 very exactly to square with rectilinearity, as defined by Euclid. 

 We shall now approach the subject in another form. 



Prop. XV. If a triangle he in any way divided into a number of triangles, the 

 angular defect of the whole triangle is equal to the sum of the angular defects 

 of its parts. 



Let ABC be a triangle, and, — 1 . Let it be divided into triangles by straight lines 

 drawn from one of its angles A to the opposite side : MBC — 

 ^ABE + ^AEF + MFC. For A + B + C+ all the angles 

 at E and F = all the angles of ABE, AEF and AFC. There- 

 fore, by subtracting from six right angles ^ABC = ^ ABE + 

 5AEF + 5AFC ; and in the same manner the proposition 

 may be proved, whatever be the number of triangles into which ABC is divided. 



2. Let the triangle be divided into three triangles by lines drawn from any 

 point within it: then A + B + C+ four right angles = the 

 angles of the triangles ABD, BDC, CDA. Take from six 

 right angles, and ^ABC = MBD + ^BCD + ^CAD 



3. If the triangle ABC be divided into any number of 

 tiiangles by lines drawn from the same point D, we shall 

 have the same result, by combining Case 1 with Case 2. 



4. The same demonstration applies, if one of the triangles into which ABC is 

 divided be formed by joining points in AB, AC; and a 

 similar argument would extend the proposition to other 

 cases. 



For the sake of demonstration (not of construction), 

 we are at liberty to introduce the two following postu_ 

 lates. 



Postulate 1. From the greater of two given triangles, a triangle can be cut oflP 

 ' equal to the less. 



Postulate 2. A triangle can be divided into any number of equal triangles. 



