S T E

415

STE

hut

*hich

are not in the same plane with the first two-jj the plane

se.8

passes through the latter is parallel to the plane whichthrough the former two lines.deposition XVI.—If two parallel planes be cut by another,l ? e > their common sections with it are parallels.

Proposition XVII.—If two straight lines be cut by parallella'ntis '

I'n

Phi

they will be cut in the same ratio.

t (1 r °position XVIII.—If a straight line be at right anglesP'une, every plane which passes through it will be at'*£* angles to that plane.

e ad ro P os iti° n XIX.—If two planes cutting each other bethem perpendicular to a third plane, their common^Jon will also be perpendicular to it.

Pla r °P os ^ on XX.—If a solid angle be contained by threeangles, any two of them are greater than the third.

Pla r °P os iti° n XXI.—Every solid angle is contained byangles, which together are less than four right angles,gi-t.. r °P oslt i°n XXII.—If every two of three plane angles bel ' lan firing and if the straight lines which con-st,. . ! l,em * Je equal, a triangle may be made of theliaes t *‘ nes t * lat J°^ n ^he extremities of those equal straight

I Properties of Solids.

8 ‘tuil a P r ' sm > parallel sections which cut the sides, are

Vo uld

and equal figures ; or, all parallel sections which

tq u .• ««t the plane of the base, if produced, are similar andj • figures.

t> ilr ,. a pyramid, all the parallel sections which are notXl to plane of the base, are unequal similar figures.If 0 ,‘ e Properties of a cone are numerous and interesting.

Out

■' iti P ara Hel to the plane of the base, the section is a circle ;H,jf lt aa y direction through the apex, the section is a planeto ti," tne< P triangle ; if the cone be cut by a plane inclinede Uipg- Plaue of the base, at any given angle, the section is an'if cut by a plane parallel to any straight line within

th,

in. j t ”1 passing through the apex, the section is denominatedM^yperbola. ; and if cut by a plane parallel to anothert>y t j ,' v fi'ch touches the curved surface, the section formedt> 0 * Position of the cutting plane is called a parabola.»ti ght tUe Purposes of stereotomy, we shall suppose the coneCor >e ; and consequently the abscissa of the curves, or4?/? S ’-' vi U l J ' sect all the double ordinates at right angles,^'•lv* an y semi-conic section be supposed to

% Sl)i e p u P on its abscissa, so as to perform an entire revolution,generated by the curve-line is called a conoid,

s Ui-f.

till

ace

Ij.f _ a bscissa the aids.

‘ion 2. —If the semi-conic section be a semi-ellipsis,

%^f £ nii,

generated is called an ellipsoid.fiie Sc J l ^°« 3.—If the generating figure be a semi-parabola,. ? palled a paraboloid.

sol^f 0 ” —If the generating figure be a semi-hyperbola,, . ' s called an hyberboloid.

P ini tion K _ah „„i:,i„ ,„i,„f,

ion 5—AH solids whatever, generated by revolving/Jy-jres upon an axis, are called solids of revolution.^' u -r es ltlon 6.—All parallel sections of conoids are similar

■0 eji n . . General Principles of Construction.

i —Solid angles, which consist of three plane

i 5 Ip tlf 1 " 6 Ca H e( I trihedrals.

c . 0r, struction of trihedrals, besides the three plane• tl lfB _ . °b form the boundaries of the solid, there are

call efi

tfioa ’'Jclinations.

These inclinations are, by way of dis-the angles ; the three boundaries are calledJts. a ’ and the sides and angles are indifferently calledI i ( ni ri( j y three of which, excepting the three angles, may| J y the following constructions:

VJ* 1 - In a right-angled trihedral are given the

' 5 x

two sides containing the right angle ; to find the acute angles ,and the side or hypothenuse which subtends the right angle.

Figure 1.—Make the angle jbh equal to one of the givensides, and the angle e b F equal to the other; draw e h perpen-dicular to e b, and e i perpendicular to b f, cutting b f at f ;from b, with the radius b h, describe an arc, cutting e i at i,and join b i ; then b i is the hypothenuse ; from e, with theradius E f, describe an arc, cutting eb in g, and join G h ;then E G H is the angle contained by the hypothenuse andthe side e b f, or the angle opposite the side e b ii : or, makef i equal to g h, and join b i.

In the same manner the angle opposite the side eb; maybe found.

The reason will appear thus : raise the plane of the triangleb e f upon b e, so as to be perpendicular to the plane beh;raise the triangle E G H upon e h, until e g fall upon E f ; thenthe plane egh will become perpendicular to b f ; revolvethe plane bif upon b f, and f i will describe a circle, whoseplane is also perpendicular to bf, from the point f ; there-fore, the plane of the circle and the plane egii will be bothin the same plane : therefore, since the point f coincides withG, the straight line f i may be made to coincide with g h ;let this coincidence take place ; and because f i is equal toG H, and the point G falls upon f, the point I will fall upon h ;therefore, the straight line b i will fall upon b h ; and theangle f b i, joining f b and b h, is the hypothenuse.

Again, it is evident from the planes thus raised, that theangle, egh, contained by the planes fee and fbi, and per-pendicular to f b, their common intersection, is the measureof the angle contained by the planes fbe and f b i.

Problem II.— Given one of the sides containing the rightangle and the angle opposite; to find the remaining sidewhich contains the right angle.

Figure 2.—Let the given side be h be : in be take anypoint, f, and make efh equal to the angle required ; drawh e perpendicular to e b; from e, with the radius e f, describean arc, f g ; draw b g a tangent at G ; and e b g is the siderequired. The demonstration is evident from the last.

Problem III.— Given one of the sides containing the rightangle and the inclination or angle adjacent; to find theremaining side which contains the right angle.

Figure 2.—Make abd equal to the given side; in a btake any point, e ; draw e g perpendicular to B d, cutting itin g, and e ii perpendicular to a b ; make ef equal to e g,and efh equal to the given angle; draw bhc and a b c willbe the measure of the plane angle opposite.

The following propositions show the construction of all thecases of trihedrals or spherical triangles, which are repre-sented by right-lined angles. In each of the cases it will befound, that two of the sides of the spherical triangle arerepresented by the tangents of the arcs drawn from the sameangle ; and the angle included by these tangents is themeasure of the spherical angle. The representation of thethird side is a line joining the extremities of the tangents ;the other two angles are measured by this proposition. Ifeach of the three plane angles be denominated a side, andeach of the three inclinations an angle, the geometrical con-struction will be the same as that of a spherical triangle, andthe manner of expressing the data of the one is the same asexpressing those of the other. The sides are always measuredby the three plane angles of the solid angle.

Problem IV.— Given two sides and the contained angle ;to find the other parts.

Figure 3.—Make the angle ABC equal to the containedangle ; draw b d perpendicular to a b, and be to b c ;make b d and b e equal to each other, the angle b d a equalto one of the containing sides, and bec equal to the other ■