GEOMETRY, 
It is sufficiently evident, that the sum of the greatest 
of the three, and either of the other two, is greater than 
the remaining angle, and it is necessary to prove 
that the sum of the angles AVC, BVC, neither of which 
is the exceeds the grt angle AVB. 
“Inthe plane AVB, make the angle BVD=BVC ; 
take any two points A, B, in the lines VA, VB, and draw 
ADB; take VC=VD, and join AC, BC. And because 
BV is common to the triangles, VBD, VBC, and VC= 
VD, and the angle BVD=BVC, therefore BD=BC; 
now AB, or AD+BD, is less than AC4-BC, therefore 
taking away the common side DB, there remains AD 
* AC. The two triangles AVC, AVD, have AV com- 
mon, VC=VD andthe base ACAD, therefore (con- 
verse of 10. 1. Part I.) the angle AVC—>AVD, and 
AVC+CVB>AVD+DVB, that is =AVB. 
Prop. XVII. Tueor. 
The sum of all the plane angles which form any solid 
angle is less than four right angles. 
Let the solid angle V be cut by any plane ABCDE; 
from. a point O taken in this p draw to all its angles 
the lines OA, OB, OC, OD, OF. The sum of the an- 
gles of the triangles AVB, BVC, &c. formed about the 
vertex V, is equivalent tothe sum of the angles ofa like 
number of triangles AOB, BOC, &c. formed about the 
point O; but at the point B, the angles OBA, OBC 
taken to , make the ABC less than the sum 
of the angles VBA, VBC (16.); in like manner.at the 
point C, we have OCB4+-OCD.<.VCB4-VCD, and so 
on with all the angles of the polygon ABCDE. Hence 
it follows, that in the triangles of which the vertex is O, 
the sum of the angles at the bases is less than the sum 
of the angles at the bases of the triangles, which have 
’ their vertex at.V; therefore, by compensation, the sum 
the angles about the point O, isgreater than the sum of 
angles about the point V ; but the sum of the angles 
about O is equal to four right angles ; therefore the sum 
_ of the plane anaes which form. the solid angle about 
the point V, is less than four right angles. 
Scuotium. This demonstration supposes, that the so- 
7a sage conan cr that the solid angle lies all on one 
sid the plane of any one of its faces.; if it, were other- 
wise, the sum of the plane angles would not be limited, 
Prop. XVIIT. Taror. 
~ ‘Tf'two solid angles be composed of three plane angles 
Which are equal, each to each, the planes in which these 
‘angles are, have thesame inclittation ‘to one another. 
Let the noe ae a ad, the angle CAB=c ab, 
and the angle BAD 6 a d ; the two planes CAB, DAB 
5 wae A each other the same inclination as the planes 
cad, dad. 
Take B any ‘in AB, and in the planes BAC, 
BAD, draw B Deke to AB, and join 
‘CD; then the : ‘is the inclination of the 
phines BAC, BAD, (Def. 4.) «Again, take aJ=AB, 
and in 'the planes bac, bad draw bc and bd perpen- 
diculars to ab, and join ed; then the angle cdd is the 
inclination of the bac, bad. ° 
The'triangles BAC, bac, have the mgle BAC=3 a c, 
Sannus oe also the side AB=ab; therefore 
the es are-equal, (7. 1. Part I.) and BC=d c, also 
AC=a‘c, Inthe same way it may be proved, that the tri- 
_‘one another, and the others are 
227 
angles BAD, bad are equal, and therefore that BD=bd, 
also AD=a d, The triangles CAD, ¢ ad, have therefore 
CA=sca, AD=ead, and the le CAD=c ad; hence 
CD=cd. Now the triangles CBD, cbd having CB= 
c 6, DB=d 4, and the base CD =e d, the angle CBD will 
be equal to the angle c6d; that is, the inclination of 
the plane BAC to the plane BAD, is equal to the in- 
clination of the plane i ac tothe plane bad. In the 
samé way it may be proved, that the other planes are 
equally inclined to one another. 
Scnorrum. If the three plane angles which contain 
the solid angles, besides being equal each to each, are 
also disposed in the same order as in Fig. 149, the solid 
will coincide when applied the one to the other, 
and they will be equal. But if the plane angles are 
dis in @ contrary order, as in Fig. 150, the so« 
lid angles will not coincide, although the theorem is 
alike true in both cases. However, in the latter case 
as well as in the former, the solid angles must be ac- 
counted equal, seeing that they are equal in every thing 
that determines their magnitude. This kind of equa- 
lity, which does not admit of superposition, and on that 
account is not absolute, may be distinguished from the 
Of Solids 
Planes. 
_—~ 
other, by calling it equality by reason 9, jand : 
two solid angles, which are contai by three plane 
angles, having the same magnitude im each, ‘but p 
in a contrary order, may be called symmetrical angles. 
What is here said, will apply to solid angles eéaitaned 
by any number of plane angles, 
SECTION IL 
Or SoLips BOUNDED By Panes. 
" Definitions. 
1. A solid is that which has length, breadth, and 
thickness. 
2.A prism is a solid contained by plane figures, of which 
two that aré opposite are equal, similar, and parallel to 
arallelograms. 
To construct this solid, let ABCDE be any rectiline- 
‘al figure, (Fig. Leah Ina plane parallel to ABC draw 
the lines ‘Gu. , &c. parallel to the sides AB, BC, 
CD, &c, thus there will be formed a figure FGHIK, si- 
tiilar to ABCDE.. Now let the vertices of the corre- 
sponding angles be joined by the lmes AF, BG, CH, 
&e, the tuces ABGF, BCHG, &c. will evidently be pa- 
ane , and the solid thus formed will be a prism. 
5. 
ze equal and parallel plane figures ABUDE. 
FGHIK, are called the bases or the prism. The other 
a or parallelograms, taken together, conetitute the 
~ lateral or convex surface of the prism. 
4. The altitude of a prism is the distance between its 
bases ; and its length is a line equal-to any one of its la- 
teral edges, as AF, or BG, &c. 
5. A prism is right, when the lateral edges AF, BG, 
&c. are perpendicular to the planes of its bases ; then 
each of them is equal to the altitude of the prism ; in 
every other case the prism is oblique. 
6. A prism is triangular, quadrangular, pentagonal, 
&c. according as the'base is a triangle, a quadrilateral, 
a pentagon, &c. 
7. A prism which has a -for its base, 
has all its faces parallelograms, and. is called a parallelo- 
Definitions. 
Fig. 151. 
piped, (Fig. 152.) A parallelopiped is rectangular, Fig. 152. 
when all its faces are rectangles. 
8, When the faces of a rectangular parallelopiped are 
squares, it is called a cube, 
5 
