PROP. XI.

If a right cone BED be cut through one side BE by a plane RAP which being produced backwards cuts the other side DB produced, the section is an hyperbola.

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Let DGEH be any circular section, BGH a triangular section through the vertex B of the cone parallel to the plane RAP.

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Then, AN EN :: BF: EF

NM: ND :: BF: FD

AN X NM: EN × ND (NP) :: BF" : EF × FD (FH') which is the property of an hyperbola, whose axis major is AM and whose conjugate axis is to AM as FH to BF.

Cor. If GT, HT, be tangents to the circle at G, H; and planes passing through GT, HT, respectively, touch the cone along the lines BG, BH; also if TB, the common section of the planes, meet AM in C: then the common sections CO, CQ, of the plane RAP extended to meet the tangent planes are the asymptotes of the hyperbola.

Draw BL parallel to DE, meeting AM in L. Then the axes of the hyperbola being in the proportion of BF to FH, the angle GBH or the equal angle OCQ is the angle between the asymptotes.

Now, by similar triangles ALB, BFE, and CLB, BFT; AL: CL:: TF : FE, and therefore AC: CL :: TE: FE, In like manner, by similar triangles MLB, BFD, and CLB, BFT; ML: CL:: TF: DF, and therefore CM: CL :: TD: DF. But by the property of the circle, TE : FE :: TD: DF. Therefore, CA CM. Hence C is the centre of the hyperbola, and CO, CQ, are the asymptotes. (Conic Sections, infra, Hyperbola, Prop. XII.)

PROP. XIT.

All pyramids and right cones of equal bases and altitudes are equal to one another

Then, by Props. VIL. and VIII.,

DO': DH' :: LM: EF,

and AN AG :: IK : BC.

But, since AN, AG3, are equal to DO3, DH'; therefore, IK: BC:: LM: EF But BC is equal to EF, by hypothesis; therefore, IK is also equal to LM.

In the same manner, it is shown that any other sections, at equal distance from the vertex, are equal to each other.

Since, then, every section in the cone, is equal to the corresponding section in the pyramids, and the heights are equal, the solids ABC, DEF, composed of those sections, must be equal also. Q. E. D.

PROP. XIII.

Every pyramid of a triangular base, is the third part of a prism of the same base and altitude.

Let ABCDEF be a prism, and BDEF a pyramid, upon the same triangular base DEF; then will the pyramid BDEF be a third part of the prism ABCDEF.

For, in the planes of the three sides of the prism, draw the diagonals BF, BD, CD. Then the two planes BDF, BCD, divide the whole prism into the three pyramids BDEF, DABC, DBCF; which are proved to be all equal to one another as follows:

Since the opposite ends of the prism are equal to each other, the pyramid whose base is ABC and vertex D, is equal to the pyramid whose base is DEF and vertex B (Prop. XII.), being pyramids of equal base and altitude.

But the latter pyramid, whose base is DEF and vertex B, is the same solid as the pyramid whose base is BEF and vertex D, and this is equal to the third pyramid, whose base is BCF and vertex D, being pyramids of the same altitude and equal bases BEF, BCF.

Consequently, all the three pyramids which compose the prism, are equal to each other, and each pyramid is the third part of the prism, or the prism is triple of the pyramid. Q. E. D.

Corol. 1. Every pyramid, whatever its figure may be, is the third part of a prism of the same base and altitude; since the base of the prism, whatever be its figure, may be divided into triangles, and the whole solid into triangular prisms and pyramids.

Cor. 2. Any right cone is the third part of a cylinder, or of a prism, of equal base and altitude; since it has been proved that a cylinder is equal to a prism, and a cone equal to a pyramid, of equal base and altitude.

SCHOLIUM. Whatever has been demonstrated of the proportionality of prisons, or cylinders, holds equally true of pyramids or cones, the former being always triple the latter; viz. that similar pyramids or cones, are as the cubes of their like linear sides, or diameters, or altitudes, &c.

PROP. XIV.

If a sphere be cut by a plane, the section will be a circle.

Because the radii of the sphere are all equal, each of them being equal to the radius of the describing semicircle, it is evident that if the section pass through the centre of the sphere, then the distance from the centre to every point in the periphery of that section will be equal to the radius of the sphere, and the section will therefore be a circle of the same radius as the sphere. But if the plane do not pass through the centre, araw a perpendicular to it from the centre, and draw any number of radii of the sphere to the intersection of its surface with the plane; then these radii are evidently the hypothenuses of a corresponding number of right-angled triangles, which have the perpendicular from the centre on the plane of the section, as a common side; consequently their other sides are all equal, and therefore the section of the sphere by the plane is a circle, whose centre is the point in which the perpendicular cuts the plane.

Cor. If two spheres intersect one another, the common section is a circle.

SCHOLIUM.

All the sections through the centre are equal to one another, and are greater than any other section which does not pass through the centre. Sections through the centre are called great circles, and the other sections small or less circles. Also, a straight line drawn through the centre of a circle of the sphere perpendicular to the plane of the circle is a diameter of the sphere, and the extremities of this diameter are called the poles of the circle. Hence it is evident that the arcs of great circles between the pole and circumference are equal, for the chords drawn in the sphere from either pole of a circle to the circumference are all equal.

PROP. XV.

Every sphere is two-thirds of its circumscribing cylinder. Let ABCD be a cylinder circumscribing the sphere EFGH; then will the sphere EFGH be two-thirds of the cylinder ABCD. For let the plane AC be a section of the sphere and cylinder through the centre I, and join AI, BI. Let FIH be parallel to AD or BC, and EIG and KL parallel to AB or DC, the base of the cylinder; the latter line KL meeting BI in M, and the circular section of the sphere in N.

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Then, if the whole plane HFBC be conceived to revolve about the line HF as an axis, the square FG will describe a cylinder AG, and the quadrant IFG will describe a hemisphere EFG, and the triangle IFB will describe a cone LAB. Also, in the rotation, the three lines, or parts, KL, KN, KM, as radii, will describe corresponding circular sections of these solids, viz. KL a section of the cylinder, KN a section of the sphere, and KM a section of the

cone.

Now, FB being equal to FI, or IG, and KL parallel to FB, then by similar triangles IK KM (Geom. Theor. 82), and IKN is a right-angled triangle; heuce IN2 is equal to IK2+KN2 (Theor. 34). But KL is equal to the radius IG or IN, and KM=IK; therefore KL is equal to KM2+KN2, or the square of the longest radius of the said circular sections, is equal to the sum of the squares of the two others. Now circles are to each other as the squares of their diameters, or of their radii, therefore the circle described by KL is equal to both the circles described by KM and KN; or the section of the cylinder is equal to both the corresponding sections of the sphere and cone. And as this is always the case in every parallel position of KL, it follows that the cylinder EB, which is composed of all the former sections, is equal to the hemisphere EFG and cone IAB, which are composed of all the latter sections.

But the cone IAB is a third part of the cylinder EB (Prop. XIII. Cor. 2) - consequently the hemisphere EFG is equal to the remaining two-thirds, or the whole sphere EFGH is equal to two-thirds of the whole cylinder ABCD.

Corol. I. A cone, hemisphere, and cylinder of the same base and altitude are to each other as the numbers 1, 2, 3.

Corol. 2. All spheres are to each other as the cubes of their diameters; all these being like parts of their circumscribing cylinders.

Corol. 3. From the foregoing demonstration it appears that the spherica zone or frustum EGNP is equal to the difference between the cylinder EGLO and the cone IMQ, all of the same common height IK. And that the spherical segment PFN is equal to the difference between the cylinder ABLO and the conic frustum AQMB, all of the same common altitude FK.

SCHOLIUM.

By the scholium to Prop. XIII. we have

cone AIB: cone QIM:: IF3: 1K3 :: FH3: (FH−2FK)3 . cone AIB: frustum ABMQ:: FH3: FH3 (FH—2FK)3 :: FH3:6FH FK-12FH.FK2+8FK3;

but cone AIB = one-third of the cylinder ABGE; hence

cylinder AG: frustum ABMQ:: 3FH3:6FH2.FK-12FH.FK+8FK3

Now cylinder AL : cylinder AG:: FK : FI

.. cylinder AL : frustum ABMQ::6FH2: 6FH2-12FH. FK+8FK2

.. cylinder AL : segment

PFN::6FH2 : 12FH.FK-8FK2, dividendo

::FH2: FK(3FH-2FK).

But cylinder AL = circular base whose diameter is AB or FH multiplied by

SPHERICAL GEOMETRY.

DEFINITIONS.

1. A SPIERE is a solid terminated by a curve surface, and is such that all the points of the surface are equally distant from an interior point, which is called the centre of the sphere.

We may conceive a sphere to be generated by the revolution of a semicircle APB about its diameter AB; for the surface described by the motion of the curve ABP will have all its points equally distant from the centre O

2. The radius of a sphere is a straight line drawn from the centre to any point on the surface.

The diameter or axis of a sphere is a straight line drawn through the centre, and terminated both ways by the surface.

It appears from Def. 1, that all the radii of the same sphere are equal, and that all the diameters are equal, and each double of the radius.

3. It will be demonstrated, (Prop. 1.), that every section of a sphere, made by a plane, is a circle; this being assumed,

A great circle of a sphere is the section made by a plane passing through the centre of the sphere.

A small circle of a sphere is the section made by a plane which does not pass through the centre of the sphere.

4. The pole of a circle of a sphere is a point on the surface of the sphere equally distant from all the points in the circumference of that circle.

It will be seen, (Prop. 11.), that all circles, whether great or small, have two poles.

5. A spherical triangle is the portion of the surface of a sphere included by the arcs of three great circles.

6. These arcs are called the sides of the triangle, and each is supposed to be less than half of the circumference.

7. The angles of a spherical triangle are the angles contained between the planes in which the sides lie.

8. A plane is said to be a tangent to a sphere, when it contains only one point in common with the surface of the sphere.

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