10 



OF SPACE. 



invariable, on account of changes of temperature, and of 

 other circumstances. 



77. Postulate. That a straight hne of 

 indefinite length may he diavvu through 

 any two given points. 



78. Postulate. That a circle may be 

 described on any given centre with a radius 

 equal to any given straight line. 



79. Axiom. A straight line joining two 

 poiiits is the shortest distance between them. 



Scholium. With respect to all straight lines, this 

 axiom is a demonstrable proposition; but as the demon- 

 stration does not extend to curve lines, it becomes neces- 

 sary to assume it as an axiom. 



80. Axiom. Of any two figures meeting 

 in the ends of a straight line, that which is 

 nearer the line has the shorter circumference, 

 provided there be no contrary flexure. 



81. Axiom. Two straight lines coinciding 

 in two points, coincide in all points. 



Scholium. If they 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 ling. Al- 

 though this is sufficiently obvious, it can scarcely be 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. Puoui.em. From the greater of two 

 right lines, AB, to cut oft' a part equal to 

 tlielcss, CD. 



\ On the centre A describe a circle with 



A E] B a radius equal to CD (78), and it v>iU 



C~__~_~D cutoffAE=:CD(;66). 



85. Problem. On a given right hne, AB, 

 to describe an equilateral triangle. 



C 



^ ^' On the centres A and B draw two circles, 



with radii equal to AB, and to their intersec- 

 tion C, draw AC and BC; thenAB:i:AC=:BC 

 (fio), and the triangle ABC is e<iuilateral. 



86. TiiEOKEM. Two triangles, having two 

 sides and the angle included, respectively 

 equal, have also the base and the other angles 

 equal. 



In the triangles ARC, DEF, let 

 AC=DF, BC=EF,and z. ACB= 

 DFE. Now supposing a triangle 

 equal to DEF to be constructed 

 on AC, the side equal to FE must -^ B D k, 

 coincide in position with CB, because z.ACB;=DFE, and 

 also in magnitude, for they are equal, therefore the point 

 B will be an angular point of the supposed triangle ; and 

 since the base of both triangles must be a right line, it must 

 be the same line AB (81), and the supposed triangle will 

 coincide every where with ABC ; therefore ABC^nDEF, 

 and the angles at A and B are equal to the angles at D 

 and E. 



87. Theorem. If two sides of a triangle 

 are equal, the angles opposite to them are 

 equal. 



In the sides AB and AC produced, take 

 at pleasure AD~AE, and join BE, CD ; 

 then since ADnAE, and AC~AB, and 

 the angle at A is common to the triangles 

 ADC, AEB, those triangles are equal 

 (88),and^ACD=ABE, z.ADCr:AEB, ^/ 

 and CD=:BE ; but BD=CE (16], therefore z.BCD=rBE 

 (gfl), and /cACD— BCD=ABE— CBE (10), or z.ACB= 

 ABC. 



88. Tueokem. If two angles of a triangle 

 are equal, the sides oppoivite to them are 

 equal. 



Let /.ABCn:ACD; then AC=AB. 

 If it be denied, take in the greater AC, 

 CD equal to the less AB ; then, since ^ 

 ABC:=DCB, AB=:DC, and BC is com- 

 mon, the triangle ABCzzDCB (se), the whole to a part, 

 whirh is impossible. 



89. Theorem. If two triangles have their 

 bases equal, and their sides respectively 

 equal, their angles are also respectively ■ 

 equal. 



If a triangle be supposed to 

 be constructed on AB, ilie base 

 of ABC, equal to DEF, the ver- 

 tex of the triangle must coincide 

 with C, and the whole triangle 



A 



D 



B 



