THE ELEMENTS. BOOKS III-IV 
383 
33, 34 are problems of constructing or cutting off’ a segment 
containing a given angle, and 25 constructs the complete circle 
when a segment of it is given. 
The Book ends with three important propositions. Given 
a circle and any point 0, internal (35) or external (36), then, 
if any straight line through 0 meets the circle in P, Q, the 
rectangle PO . OQ is constant and, in the case where 0 is 
an external point, is equal to the square on the tangent from 
0 to the circle. Proposition 37 is the converse of 36. 
Book IV, consisting entirely of problems, again deals with 
circles, but in relation to rectilineal figures inscribed or circum 
scribed to them. After definitions of these terms, Euclid 
shows, in the preliminary Proposition 1, how to fit into a circle 
a chord of given length, being less than the diameter. The 
remaining problems are problems of inscribing or circum 
scribing rectilineal figures. The case of the triangle comes 
first, and we learn how to inscribe in or circumscribe about 
a circle a triangle equiangular with a given triangle (2, 3) and 
to inscribe a circle in or circumscribe a circle about a given 
triangle (4, 5), 6-9 are the same problems for a square, 11- 
14 for a regular pentagon, and 15 (with porism) for a regular 
hexagon. The porism to 15 also states that the side of the 
inscribed regular hexagon is manifestly equal to the radius 
of the circle. 16 shows how to inscribe in a circle a regular 
polygon with fifteen angles, a problem suggested by astronomy, 
since the obliquity of the ecliptic was taken to be about 24°, 
or one-fifteenth of 360°. IV. 10 is the important proposition, 
required for the construction of a regular pentagon, 1 to 
construct an isosceles triangle such that each of the base 
angles is double of the vertical angle ’, which is effected by 
dividing one of the equal sides in extreme and mean ratio 
(II. 11) and fitting into the circle with this side as radius 
a chord equal to the greater segment; the proof of the con 
struction depends on III. 32 and 37. 
We are not surprised to learn from a scholiast that the 
whole Book is c the discovery of the Pythagoreans A The 
same scholium says that ‘ it is proved in this Book that 
the perimeter of a circle is not triple of its diameter, as many 
Euclid, ed. Heib., vol. v, pp. 272-3.