412 
EUCLID 
Euclid himself, in Book XIII, makes considerable use of the 
second part of Book X dealing with apotomes; he regards a 
straight line as sufficiently defined in character if he can say 
that it is, e.g., an apotome (XIII. 17), a first apotome (XIII. 6), 
a minor straight line (XIII. 11). So does Pappus. 1 
Our description of Books XI-XIII can be shorter. They 
deal with geometry in three dimensions. The definitions, 
belonging to all three Books, come at the beginning of Book XI, 
They include those of a straight line, or a plane, at right angles 
to a plane, the inclination of a plane to a plane (dihedral angle), 
parallel planes, equal and similar solid figures, solid angle, 
pyramid, prism, sphere, cone, cylinder and parts of them, cube, 
octahedron, icosahedron and dodecahedron. Only the defini 
tion of the sphere needs special mention. Whereas it had 
previously been defined as the figure which has all points of 
its surface equidistant from its centre, Euclid, with an eye to 
his use of it in Book XIII to ‘ comprehend ’ the regular solids 
in a sphere, defines it as the figure comprehended by the revo 
lution of a semicircle about its diameter. 
The propositions of Book XI are in their order fairly 
parallel to those of Books I and VI on plane geometry. First 
we have propositions that a straight line is wholly in a plane 
if a portion of it is in the plane (1), and that two intersecting 
straight lines, and a triangle, are in one plane (2). Two 
intersecting planes cut in a straight line (3). Straight lines 
perpendicular to planes are next dealt with (4-6, 8, 11-14), 
then parallel straight lines not all in the same plane (9,10, 15), 
parallel planes (14, 16), planes at right angles to one another 
(18, 19), solid angles contained by three angles (20, 22, 23, 26) 
or by more angles (21). The rest of the Book deals mainly 
with parallelepipedal solids. It is only necessary to mention 
the more important propositions. Parallelepipedal solids on the 
same base or equal bases and between the same parallel planes 
(i.e. having the same height) are equal (29-31). Parallele 
pipedal solids of the same height are to one another as their 
bases (32). Similar parallelepipedal solids are in the tripli 
cate ratio of corresponding sides (33). In equal parallele 
pipedal solids the bases are reciprocally proportional to their 
heights and conversely (34). If four straight lines be propor- 
1 Cf. Pappus, iv, pp. 178, 182.