Of Solids 
bounded by 
Planes. 
232 
the planes BCD, bcd, the pyramid A-BCD is equal to 
the pyramid a-be d. “S 
For if they are not equal, let Z represent the solid 
which is equal to the excess of one of them, a-b ¢ d, above 
the other A-BCD ; and let a series of prisms CE, FG, 
HK, LM, of the same altitude be circumscribed about 
the pyramid A-BCD, soas to exceed it by a solid less 
than Z, which is always possible (15.); also let a series 
of prisms ce, fg, hk, 1m, equal in number to the other, 
and of the same altitude, be circumscribed about the 
pyramid a-b cd. And because the pyramids have equal 
altitudes, and the number of prisms described about 
each is the same, the altitudes of the prisms will be’ all 
equal, and the bases of the corresponding prisms in the 
two pyramids, as EF, ef, will be sections of the pyra- 
mids at-equal distances from their bases, therefore they 
are équal (14.), and the prisms themselves are equal 
(1.), and the sum of all the prisms described about the 
one pyramid is equal to the sum of all the i ne pe de- 
scribed about the other pyramid. To abridge, put P 
and p to denote the pyramids A-BCD, and a-b cd re- 
spectively, and Q and qg to express the sums of the 
‘prisms described about them. Then, because by hy- 
Fig. 164, 
Fig. 161. 
pothesis Z=p — P, and by construction Z = Q— P, 
therefore p— P=Q-—P, hence p must be greater than 
Q; but Q is equal to q, therefore p must be greater 
than gq, that is the pyramid p is greater than g, the 
sum of the prisms described about it, which is impos- 
sible ; therefore the pyramids P, pare not unequal, that 
is they are equal. 
Prop. XVII. Tueor. 
’ Every prism having a triangular base, may be di- 
vided into three pyramids that have triangular bases, 
and that are equal to one another. 
Let ABC, DEF be the opposite bases of a triangular 
prism ; join AE, EC, CD; and because AE is the di- 
agonal of a parallelogram, the triangles ABE, ADE 
are equal; therefore the pyramids C-ABE and C-ADE, 
which have a common vertex C, and the triangles ABE, 
ADE for their bases, will be equal (16.) When these 
are taken from the whole prism, there remains the py- 
ramid C-DEF, which is equal to the pyramid C-ABE, 
or E-ABC, for they have equal bases DEF, ABC, and 
the same altitude, viz. the altitude of the prism ABC- 
DEF. Therefore the three pyramids C-ABE, C-ADE, 
and C-DEF, are equal. 
Cor. 1. From this it appears, that every pyramid is 
the third part of a prism, which has the same base and 
the same altitude with it. For if the base of the prism 
be any other figure than a triangle, it may be divided 
into prisms having triangular bases. 
Cor. 2. Pyramids of equal altitudes are to one ano- 
ther as their bases ; because the prisms upon the same 
bases, and of the same altitude, are to one another as 
their bases. 
Prop. XVIII. 
Similar pyramids are to one another as the cubes of 
their homologous sides. , 
THeEor. 
If two pyramids be similar, it is evident from Def. 
13, that the lesser may be placed in the greater, so that 
they shall have a common solid angle A; and then their 
bases BCD, FGH will be parallel; for since the homo- 
logous faces are similar, the angle AFG = ABC, and 
GROMETRY. 
the angle AGH=ACD, and 80 on ; therefore the plane Of th 
FGH is parallel to the plane BCD: Hence, again, it 
will follow, that a straight line AKE perpendicular to —? 
the base of the one, will also be licular to the ~ 
base of the other, and AE, AK, the altitudes of the 
two pyramids, will have te each other the ratio of AB 
to AF, or of BC to FG, &e. Now, let P preraent a 
right prism, having the same base BCD as t pyra-- 
mid A-BCD, and the same altitude AE, ‘and similarly 
let p represent another right prism, having the same 
base 'GH as the pyramid A-FGH, and the same alti- 
tude AK: Then these prisms will manifestly be con- 
tained by the same number of similar planes, similarly 
situated, and having a like inclination to each other, 
therefore they will be similar (Def. 13.) and consequent- 
ly P is to _p as the cube of BC to the cube of FG (12.), 
but the pyramids A-BCD, A-FGH are like parts of the 
prisms (1 Cor. 17.) ; therefore the ‘pyramids are also 
to one another as the cubes of their homologous sides 
BC, FG. 
\ SECT. IIT. 
Or tne Taree Rovunp Sotins. 
- Definitions. 
1. A cylinder is a solid figure, generated by the re- Defi 
volution of a right-angled parallelogram, which re- 
Fie a one of its sides, that side remaining fixed, 
ig. 165. uy ih 
2 The es of a cylinder is the straight line about 
which the perilous revolves. 
3. The bases of a cylinder are the circles described 
by the two revolving 
Fig, : 
opposite sides of the rectangle. 
4, A cone is a solid figeire enerated by che, evans 
tion of a right-angled hats about one of the sides 
containing the right angle, which remains fixed. __ 
5. The axis of the cone is the straight line about 
which the triangle revolves. 
6. The dase of the cone is the straight line generated 
by that side containing the right angle which revolves. 
7. A-sphere is the solid figure generated by the re- 
volution of a semicircle about a diameter, which re- 
mains fixed. " 
8. The avis of a sphere is the straight line about 
which the semicircle revolves. — " 
9. The centre of the sphere is the same with that of 
the semicircle. wae 
10. Similar cones and cylinders are those which have 
the diameters of their bases and their axes propor- 
tionals. : 
Prop. I. Tuzorem. ; 
If from any point E in the circumference of AEB, 
the base of a cylinder, a straight line EF be drawn 
perpendicular to the plane of the base, it will be whol- ~ _ 
ly in the cylindric superfices. 
Let AGHD be the generating rectangle, and GH 
the axis. Because in every position of the revolving 
rectangle, the angle AGH is a right angle, GH is per- 
pendicular to the plane of the base AEB; therefore, 
AD, the line which generates the cylindric superfices, 
is in every position perpendicular to the plane of the 
base (5. 1.), and consequently, when the revolving ra- 
dius GA comes to the position GE, AD will coincide 
with EF ; therefore EF is in the cylindric superfices. 
