Legendre's Geometry. 295 



>oould, for a moment have doubted, whether the conclusions 

 were established with complete certainty by the evidence 

 adduced. Those who have objected most to the theory of 

 parallel lines, as usually laid down, belong to that class of 

 mathematicians who insist upon a rigour of demonstrrttion 

 not accommodated to the imperfections attending all hu- 

 man things, and which aiming at an imaginary perfection, 

 is very unreasonably dissatisfied with evidence which es- 

 tablishes its results with perfect certainty. The theorems 

 that we possess respecting the properties of parallel lines, 

 we regard as undeniably certain; any difficulties, therefore, 

 relating to the manner in which they are demonstrated, we 

 cannot but consider essentially imaginary. 



The second section, comprises the elementary proper- 

 ties of the circle, together with those of chords, of tangents 

 and the measure of angles by arcs of a circle. This sec- 

 tion contains all the principles which are of importance in 

 B.lll. of Euclid, and some others both of great use in ordin- 

 ary practice, and in the succeeding parts of the science. 

 These two sections are followed by the resolution of a 

 number of problems relating to the construction of figures. 



The third section contains the measure of surfaces, tlicir 

 comparison, the properties of a right angled triangle, those 

 of equiangular triangles, of similar figures, &:c. In this sec- 

 tion, he has blended the- properties of lines with those of 

 surfaces, but in this arrangement, he has followed the ex- 

 ample of Euclid, and the propositions in this way, admit of 

 being so well connected, that we doubt whether a better 

 arrangement can be obtained. In giving the definitions 

 ■which relate to this section, he says, "I shall call those fig- 

 ures equivalent, whose surfaces are equal. Two figures 

 may be equivalent, however dissimilar ; thus a circle may 

 be equivalent to a square, a triangle to a rectangle, <Sz;c, 

 The denomination of equal figures will be restricted to 

 those which being applied, the one to the other, coincide 

 entirely ; thus two circles having the same radius are equal, 

 and two triangles having the three sides of the one equal to 

 the three sides of the other, each to each, are also equal." 

 In the use of these definitions, he is followed by Lacroix. 

 We are persuaded that a distinction between equality by 

 equivalence, and equality by coincidence, is expedient as a 

 matter of convenience, and as a means of enlarging our 



