THE MATHEMATICAL SCIENCES 143 



For example, to divide the angle ABC into three 

 equal parts (Fig. 24). First AC is drawn perpendicular 

 to BC and to AE which is parallel to BC ; then between 

 AC and AE is intercalated DE = 2AB in such a way 

 that its prolongation passes through B. F being the 

 middle of DE and the triangle ADE being a right- 

 angled triangle capable of being inscribed in a semi- 

 circle of radius FE, we have radius AF = radius 

 FE = AB by construction. The triangle ABF is 



Fig. 24. 



isoceles ; the angle ABF = AFB = twice the angle 

 AEF = twice CBD. Hence 



angle CBD = J angle CBA. 1 



By intercalation must therefore be understood " the 

 construction of a segment of a straight line of which 

 the extremities are situated on given lines and which, 

 when produced, passes through a given point. This 

 segment can easily be obtained by means of a ruler 

 (or piece of folded paper) in the following manner. 

 On the ruler two marks are first made, the space 

 between them being equal to the length of the given 

 segment, then the ruler is turned round a fixed point 

 and moved at the same time in such a way that one of 

 the marks follows exactly one of the given lines. This 



1 Pappus, Hultsch Edition, Book IV, Prop. 32. 



