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4 
Legendre’s Geometry. 295 
could, 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 demonstration 
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. 
he 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.111. 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, their 
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, &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 
