ds eer a poreencheniery ah, ah, would coin- 
_ _ Now, because of the similar triangles ABH, a 6h, 
and the similar figures AC, ac, we have 
rent iy AH: ah::AB:ab:: BC: bc; 
and because of the similar bases, 
_ base BCD : base bed: : BC? ; bc? (25. 4, Part 1.) 
Broan these agp peupowsians fi i « aaguepurens oy Aa 
quantities as represen y numbers, we get, (by Prop. 
1. Sect. 3. Part 1.) se 
AH x hase BCD : ah x base BCD :: BC}: bc x BC, 
ah base BCD: ahx base bed::b ex BC: be; 
therefore, ex aquali, 
AH x base BCD : ah» base be d:: BC}: bc}. 
But AH x base BCD expresses the solidity of the prism 
P;and ah x base bcd expresses the solidity of the other 
prism p, therefore, 
prism P : prism p::BC3:bc}. 
‘Cor. Similar prisms are to one another in the tri- 
plicate ratio of their homologous sides. For let Y and 
' Z be two lines, such, that BC: 6c:: bc: Y,andbec: 
Y::Y:Z; then the ratio of BC to Z is triplicate of 
the tatio of BC to bc (12: Def. Sect. 3. Part 1.) But 
__ “since BC: bc :; bc: Y, therefore BC? : bc?:: bc?: Y%, 
ae Ge 4, Part 1.) and, multiplying the antecedents by 
_ BC, and the consequents by bc, BC}: bc}::BC x 
6c? :bc x Y::: BC xbc: Y*; but Y2=bexZ (22. 4 
_., Part. 1.) ; therefore BC}:6¢3:: BCxbe:bcxZ:: 
~ BC:Z. But BC} :dc};: 
ee ' 
F prism P: prism p, therefore 
i Ye, prisms have to each other the ratio of BC to Z, that 
: triplicate ratio of BC to dc. 
- 
Wp Is, 
. Tony 
PA 7 
3 
_ Prop. XIII. Tugor. 
If a triangular pyramid A-BCD be cut by a plane 
og to its base, the section FGH is similar to the 
61. . For because the parallel planes BCD, FGH are cut 
by a third plane ABC, the sections FG, BC are paral- 
eu. 1.) In like manner it appears that FH is paral- 
to BD; e HFG is equal to the 
angle DBC (10..1.) And because the triangle ABC is 
~ similar to the triangle AFG, and the triangle ABD is 
similar to the triangle AFH, we have 
BC: BA:: FG: FA, 
and BA: BD:; FA: FH. 
Therefore, ex wquali; BC: BD :: FG: FH; now the 
* angle DBC has been shewn to be equal to the angle 
G; therefore the triangles DBC, HFG are eqnian- 
gular (20. 4, Part 1.) 
Prop. XIV. Turor. 
If two triangular pyramids A-BCD, a-bed, which 
ol - have Se ralent bases, and equal altitudes, be cut by 
" planes that are parallel to the bases, and at equal dis- 
ces from them; the sections FGH, fg h will be 
Draw AKE, ake perpendicular to the, BED, 
es, we have AE: AK :: AB: 
12. 1.) ; but, by hypothe- 
ore, AB: AF;: ab: 
tri 
Sy is 
,BC:FG:: 
33 best ; the 
bcd, meeting the cutting planes in K and & ; then, b 
HN hoy La yeas 
GEOMETRY. 
231 
FG: : : trian. BDC : trian. FAG, and in like manner 
bc? : fg? :: trian. bed: trian, fg h (25, 4. Part, 1,) 
ore 
trian. BCD : trian. FGH ; : trian. bed : trian. fg h, 
Now trian. BCD =/trian. bc d (by hypothesis) therefore 
the triangle FHG is equal to the triangle /'/ g. 
Scuorium. It is easy to see, that what is proved in 
this and the preceding Proposition is also true of poly- 
gonal pyramids. 
Pnor. XV. Turor. 
A series of prisms of the same altitude may be in- 
scribed in a pyramid, and another series may be cir- 
cumscribed al it, which shall exceed the other by 
less than any given solid, 
Let A-BCD be a pyramid, and let AC, one of its 
lateral edges, be divided into some number of equal 
parts, at po points F, G, H; through these, let planes 
pass parallel to the base BCD, making with the sides 
of the pyramid the sections QPF, SRG, UTH; which 
will be similar to one another and to the base a) 
From B, in the plane of the triangle ABC, draw B. 
parallel to CF, meeting FP prod in K; in like man- 
ner, from D draw DL parallel to CF, meeting FQ pro- 
duced in L; join KL, and the solid CBD-FKL will 
evidently be a prism. By the same construction, let 
the prisms PM, RO, TV be described: Also let the 
straight line IP, which is in the plane of the triangle 
ABC, be produced till it meet BC in h, and let MQ be 
produced till it meet DC in g; join hg, then Chg- 
FPQ will be a prism, and be equal to the prism PM. 
Th the same manner is described the prism mS equal to 
the prism RO, and the prism gU equal to the prism 
TV. Therefore the sum of all the inscribed prisms 
hQ, mS, and q U is equal to the sum of the prisms 
PM, RO, and ty ; that is, to the sum of all the circum- 
scribed prisms, except the prism BL; wherefore BL 
is the excess of the prisms circumscribed about the py- 
ramid aboye the prisms inscribed within it. 1 
Let us now suppose that Z denotes some given solid 
equal to a prism, which has the same base CBD as the 
ramid, and its altitude equal to a ndicular from 
E (a point in AC) upon the base. Then, however near 
E may be to C, it will evidently be possible to divide 
AC into such a number of equal , that one of them, 
CF, shall be less than CE ; and this being the case, the 
rism BL will evidently be less than the prism whose 
toes is the triangle CBD, and altitude a perpendicular 
from E on the base BCD; that is, less than the given 
solid Z: Therefore the excess of the circumscribed above 
the inseribed prisms may be less than the solid Z. _ 
Cor. Since the difference between .the circumscribed 
and inseribed prisms may be less than any given miag- 
nitude, and the pyramid is greater than the » and 
less than the former, it follows that a series of prisms may 
be circumscribed about the pyramid, and also a series 
of prisms may be inscribed in it, which shall differ from 
the pyramid itself by less then any given solid. 
Prop. XVI. Tutor. 
Pyramids that have equal bases and altitudes, are equal 
to one another, 
Let A-BCD, a-b¢d be two airee that have 
equal bases BCD, bcd, and equal altitudes ; viz. the 
perpendiculars drawn from the vertices A and a upon 
Of Solids 
bounded by 
Planes. 
= = 
Fig, 162. 
Fig. 163, 
