THE ELEMENTS. BOOK 1 
377 
point on it. Construction as a means of proving existence is 
in evidence in the Book, not only in 1 (the equilateral triangle) 
but in 11, 12 (perpendiculars erected and let fall), and in 
22 (construction of a triangle in the general case where the 
lengths of the sides are given); 23 constructs, by means of 22, 
an angle equal to a given rectilineal angle. The propositions 
about triangles include the congruence-theorems (4, 8, 26)— 
omitting the ‘ ambiguous case ’ which is only taken into 
account in the analogous proposition (7) of Book VI—and the 
theorems (allied to 4) about two triangles in which two sides 
of the one are respectively equal to two sides of the other, but 
of the included angles (24) or of the bases (25) one is greater 
than the other, and it is proved that the triangle in which the 
included angle is greater has the greater base and vice versa. 
Proposition 7, used to prove Proposition 8, is also important as 
being the Book I equivalent of III. 10 (that two circles cannot 
intersect in more points than two). Then we have theorems 
about single triangles in 5, 6 (isosceles triangles have the 
angles opposite to the equal sides equal—Thales’s theorem— 
and the converse), the important propositions 16 (the exterior 
angle of a triangle is greater than either of the interior and 
opposite angles) and its derivative 17 (any two angles of 
a triangle are together less than two right angles), 18, 19 
(greater angle subtended by greater side and vice versa), 
20 (any two sides together greater than the third). This last 
furnishes the necessary Siopia/xoy, or criterion of possibility, of 
the problem in 22 of constructing a triangle out of three 
straight lines of given length, which problem had therefore 
to come after and not before 20. 21 (proving that the two 
sides of a triangle other than the base are together greater, 
but include a lesser angle, than the two sides of any other 
triangle on the same base but with vertex within the original 
triangle) is useful for the proof of the proposition (not stated 
in Euclid) that of all straight lines drawn from an external 
point to a given straight line the perpendicular is the 
shortest, and the nearer to the perpendicular is less than the 
more remote. 
The second group (27-32) includes the theory of parallels 
(27-31, ending with the construction through a given point 
of a parallel to a given straight line); and then, in 32, Euclid