Of Solids 
bounded 
by Planes. 
—_—— 
Fig. 139. 
Fig. 159. 
280 
which will be equal, because they have equal bases 
6.):and equal altitudes, and the solid AL will contain 
three of these parallelopipeds: thus the parallelopiped 
AG will contain a part of the parallelopiped AL exact- 
ly as often as the altitude AE of the former contains a 
like part of the altitude AI of the latter, therefore the 
solids AG, AL have to each other the same ratio as 
their altitudes AE, AI. 
When the altitudes are incommensurable, it may still 
be inferred that the ratio of the solids is the same as 
that of their altitudes, for the reasons assigned in the 
conclusion of Secr. III. Parr I. 
‘Phot. TX. Teor. 
Two rectangular parallelopipeds AG, AK, which 
have the same altitude, are to one anothers their bases. 
Suppose the solids placed side by side, as in the Fi- 
: Produce the plane ONKE, until it meet the plane 
SCGH in the line PQ; thus there will be formed a third 
parallelopiped AQ, which may be compared with the 
other two. The two solids AG, AQ, having the same 
base AEHD, are to each other as their altitudes AB, 
AO (8.) In like manner the two solids AQ, AK, ha- 
ving the same base AOLE, are to otie another as their 
altitudes AD, AM: But the rectangles AC, AP, ha- 
ving the same breadth, a¥e also to each other as AB to 
AO, (3. 4. Part 1.) and similatly, the rectangles AP, 
AN are to each other as AD to AM; therefore 
sol. AG: sol. AQ: : base AC: base AP, 
sol: AQ: sol. AK: ; base AP: base AN; 
therefore, ex wquo, 
sol. AG: sol. AK: : base AC: base AN. 
Prop. X. Tueor. 
- Any two récfangular parallelopipeds are to each other 
as the products of numbers proportional to their bases 
and altitudes ; or as the products of the numbers which 
express theit three dimensions, 
Let the two rectangular parallelopipeds AG, AZ be 
so placed, that their Sl ete pe he angle 
BAE ; and let their bounding planes be produced, so 
as to form a third parallelopiped AK, having the same 
altitude as the solid AG. By the last proposition, 
sol. AG: sol. AK: : base AC : base AN; 
atid by Prop. 8, . 
sol. AK; sol. AZ :: AE: AX. 
Now, if we consider the bases AC, A'N as meastréd 
by numbers, and also their altitudés AE, AX, we shall 
have by Prop. 1. Sect. 8. Part 1, 
base AC: base AN: : AE x'base AC: AE & base AN, 
and AE: AX:: AE & baseAN: AX % base AN; 
therefore, 
sol. AG : sol. AK:: AE Xx base AC: AE & base AN, 
sol. AK: sol. AZ: 2 AE x Ddse AN: AX x base AN. 
From these two proportions, we have, ex @quo, 
sol. AG: sol. AZ :: AE & base AC PAX %& Biise AN. 
By substituting in this proportion instead of the ba- 
ses AC, AN, their numerical values AD x AB and 
AM x AO, we have also : 
gol. AG : sol. AZ:: AD xABxXAE:AMRAOX AX. 
Scnovium. Hence it appears that the product of the 
numbers which express the base of a rectangular paral- 
lelopiped, and its altitude, or the product of the nui 
bers which express its three dimensions, may be ‘takén 
GEOMETRY. 
’ 
as its numerical measure: For, if the length of the solid | 
be equal to five times a certain line, which is consider. |! 
ed as an unit, its breadth three times that anit, and its _ 
height seven times the same unit ; then the parallelopi- 
will be to a cube, whose side or edge is that unit, 
a5 x3x7tol X 1X 1, thatisass X3 % 7tol: 
Hence the parallelopiped will be equivalent to 5x 3.x 7 
= 105 timies a cube whose side is unity. 
The magnitude of a solid, its bulk, or its extension, 
constitutes what is called its solidity or its. content. Thus 
we say that the solidity or content of a igular pa- 
rallelopiped, is equal to the product of its base by its 
altitude ; or to the product of its three dimensions, 
Prop. XI. Tanda. F 
The solidity of a parallelopiped, and, in general, the 
solidity of any prism, is pater phot of its base 
by its altitude. By nid. é 
1. For any parallelopiped whatever is equivalent to 
a rectangular parallelopiped of the same alti and an 
equivalent base (7.) a the solidity of Jast has 
been proyed to be equal to the product of its base 
its altitude ; therefore the solidity of the other is 
the product of its base by its altitude. y veo 
2, Every triangular prism is half a 
which has the same altitude, atid a base twicé that 
the prism; but the solidity of this last is equal to 
product of its base by its altitude; therefore the solid 
¥ of the prism is the product of its base (half that of 
é Liang 6 A its altitude. 
8. Any p' rhatever may be divided into ‘as many 
triangular prisms of the same altitude, as there can be 
triangles in the polygon which forms its base: now the 
solidity of each prism is the product of its base by ifs 
altitude, which is common to them all; therefore the 
sum of their solidities is equal to thé sum of their bases 
I by the common altitude ; that is, the solidity 
} Bat vl prism is equal to the product of its base by 
altitude. a aca 
~ Cor. Two prisms, which have the same base, are to 
each other as their altitudes; and two prisms, which 
have the same altitude, are to each other as their bas‘ 
Let B and A be the base and altitude of a prism 
| 
g. 
& 
i] 
n 
eo] 
x 
* 
ES 
S 
x 
= 
4 
: 
NT rae etd cae : 
Note. The cube o ae ine onto kai 
séd thus, ABX AB x AB, but more commonly 
(AB, or thus, AB, ; 
Pros, XII. Turor. ; 
Similar prisms are to one another as the cube of their 
ceeds ones er 
Let P and p be two prisms, of which BC, bc are the 
homologous sides; the prism P is to the prism p as the 
cube tat to So cube va PT Si Richy 
sangles of the two prisins, draw AH, ak pe 
fete dor basés, Bor bed. Jom BH; “het 
and in the plane BHA draw ah perpendicular to BH ; 
then a% shall be dicular to the plane CBD (13. 
and 14. 1.) and equal toh, the altitude of the other 
prism ; for if the solid Band b were 
one to the other, ved which contain 
~ ge 
the 
i, ea 
i 
