OF SPACE. 17 



Scholium. With respect to all straight lines, this axiom is a dc- 

 monstrable proposition ; but since the demonstration does not extend 

 to curve Hues, it becomes necessary to assume it as an axiom. 



80. Axiom. Of any two figures meeting in the ends of 

 a straight fine, that which is nearer the Une has the shorter 

 circumference, provided there he no contrary flexure. 



81. Axiom. Two straight lines, coinciding in two 

 points, coincide in all points. 



Scholium. If tliey did not coincide in all points, the two points of 

 coincidence being at rest, and one of the lines being made the axis of 

 motion, the other must revolve round it, contrarily to the definition 

 of a straight line. Although this is sufficiently obvious, it can scarcely 

 bo called a formal demonstration. 



82. Axiom. All right angles are equal. 



83. Axiom. A straight line, cutting one of two parallel 

 lines, may be produced till it cut the other. 



84. Problem. From the greater of two right lines, 

 AB, to cut off a part equal to the less, CD. 



On the centre A describe a circle with a radius \ 



equal to CD (78), and it will cut off AEzzCD {66). ^^ -gj g 



85. Problem. On a given right line, AB, to describe 

 an equilateral triangle. 



On the centres A and B draw two circles, with 

 radii equal to AB, and to their intersection C, draw 

 AC and BC ; then ABziACizBC (66% and the tri- 

 angle ABC is equilateral. 



A B 



86. Theorem. Two triangles, having two sides and 

 the angle included, respectively equal, have also the base 

 and the other angles equal. 



