378 
EUCLID 
proves that the sum ot‘ the three angles of a triangle is equal 
to two right angles by means of a parallel to one side drawn 
from the opposite vertex (cf. the slightly different Pytha 
gorean proof, p. 143). 
The third group of propositions (33-48) deals generally 
with parallelograms, triangles and squares with reference to 
their areas. 33, 34 amount to the proof of the existence and 
the property of a parallelogram, and then we are introduced 
to a new conception, that of equivalent figures, or figures 
equal in area though not equal in the sense of congruent : 
parallelograms on the same base or on equal bases and between 
the same parallels are equal in area (35, 36) ; the same is true 
of triangles (37, 38), and a parallelogram on the same (or an 
equal) base with a triangle and between the same parallels is 
double of the triangle (41). 39 and the interpolated 40 are 
partial converses of 37 and 38. The theorem 41 enables us 
‘ to construct in a given rectilineal angle a parallelogram 
equal to a given triangle’ (42). Propositions 44, 45 are of 
the greatest importance, being the first cases of the Pytha 
gorean method of ‘ application of areas ’, £ to apply to a given 
straight line, in a given rectilineal angle, a parallelogram 
equal to a given triangle (or rectilineal figure) The con 
struction in 44 is remarkably ingenious, being based'on that 
of 42 combined with the proposition (43) proving that the 
‘ complements of the parallelograms about the diameter ’ in any 
parallelogram are equal. We are thus enabled to transform 
a parallelogram of any shape into another with the same 
angle and of equal area but with one side of any given length, 
say a unit length ; this is the geometrical equivalent of the 
algebraic operation of dividing the product of two quantities 
by a third. Proposition 46 constructs a square on any given 
straight line as side, and is followed by the great Pythagorean 
theorem of the square on the hypotenuse of a right-angled 
triangle (47) and its converse (48). The remarkably clever 
proof of 47 by means of the well-known ‘windmill’ figure 
and the application to it of I. 41 combined with I. 4 seems to 
be due to Euclid himself ; it is really equivalent to a proof by 
the methods of Book VI (Propositions 8, 17), and Euclid’s 
achievement was that of avoiding the use of proportions and 
making the proof dependent upon Book I only.