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lines will be a spherical pyramid, whose solidity is equal to its spherical base multiplied by the radius of the sphere; from which if we take away those portions included between the planes ABE, ABD and the centre C we shall have the ungula. But the pyramidal ABEC, is equal to the surface ABE X CF, and the pyramidal ABDC is equal the surface ABD x CL. Hence as enunciated above.

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Scholium. If an ungula is cut by two planes which pass the centre before their intersection, so as to include the centre of the sphere within the ungula, then will its solidity be equal to the spherical surface multiplied by the radius, plus the two pyramidals erected on those plane sections, and whose vertices are in the centre of the sphere, which is agreeable to our proposition; for the sign becomes changed from minus to plus according to the conditions. 2. Also, if the two intersecting planes pass the centre on one side before their intersection, so as to cut out an oblique ungula ABDG, then since the pyramidal erected on the plane AFBP, and whose A vertice is the centre, C, is considered negative, and the pyramid erected on the

B

plane GIDL is considered positive, by the proposition, then will the ungula be equal to its (spherical surface

the ra

the plane APBF x (CN) + the plane GLDF

dius) -
X CW.

3. And, generally, if the planes do not intersect each other within the sphere, the same proposition will still hold true, even though the planes may be parallel; in which case, the portion cut out will be a segment or zone of the sphere.

PROPOSITION XVI. THEOREM.

The solidity of the second segment of a sphere is equal to its spherical surface, multiplied by of the radius of its sphere minus each of the plane surfaces whose planes pass between the segment and centre of the sphere multiplied by of their respective perpendicular distances from the centre, and plus each of the plane surfaces which include the centre of the sphere on the same side with the segment, multiplied by the perpendicular distances of such planes to the centre. Let ABD be a segment of a sphere cut off by the plane ABCE, and if this segment is cut by the plane CEe perpendicular to the former plane, the two portions into which the segment is divided will be second segments. (Def. 6.)

I

K O

H

Draw AO, EO, CO, and eO, and the spherical pyramidal AE CeO will be equal to its spherical base x AO or EO. Now, from this pyramidal may be taken the pyramidal whose base is the section CeE, and whose altitude is KO, and also the pyramidal whose base is the section ACe and altitude HO, and there will remain the second segment AEeC.

Also the second segment may be shown to consist of the sectoral pyramid, whose base is the spherical surface, plus a pyramidal CEO, and minus the pyramid CeBO. Hence the proposition is true, as enunciated."

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The solidity of a parabolic prism is equal to two-thirds of its circumscribing quadrangular prism.

For the solidity of any prism or solid, all of whose sections parallel to the base are equal to the base, is equal to the base multiplied by the altitude and hence, prisms of the same altitude are as their bases; but the base of the parabolic prism, which is a parabola, is equal (Prop. VI, B. I) its circumscribing rectangle, also the rectangle circumscribing the base of the parabolic prism, is the base of the rectangular prism.

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PROPOSITION XVIII. THEOREM.

If an ungula be cut from a parabolic prism by a plane passing through the vertical line of the parabolic surface, and whose intersection forms an ordinate to the axis of the parabola of the base; this ungula is equal to two-thirds of the ungulicul complement of the prism of the same base and altitude.

Let DECF be an ungula cut from the parabolic prism DEFCAB, by the plane CDE, from the vertical line CF of the pa- Af rabolic surface, to the base DEF forming the ordinate ED to the axis FQ of the base, by its intersection with the plane of the base, and the ungula so cut will be equal to of the ungulical complement DECAB.

B

E

M

For, let a rectangle DELM be describ- D ed about the base of the ungula, and let any number of planes

NOf parallel to the base be passed through the ungula; and each of those sections made by those planes are parabolas; and if about each of those parallel sections, rectangles NRPO are described, those parabolic sections will each be equal to two-thirds of their respective circumscribing rectangles, (Prop. VI. B. I.) Now, if these parallel sections through the ungula are infinite in number and equidistant from each other, they will represent the whole ungula, and their sum may be taken as a function of the solidity of the ungula, and the sum of their several circumscribing rectangles may be taken as a similar function of the solid DECLMC; hence the ungula DEFC is equal to two-thirds of the solid CDELMC. Now, if we take the altitude FC of the ungula=QF, the axis of the parabola of the base, the side CLMC of the new solid will be a parabola equal and similar to the base DEF of the ungula, or ABC of the complement, since perpendiculars from every point in the perimeter of the base DEF, trace out the parabolic section DEC; and since perpendiculars from each point in the perimeter DNCOE to the plane CLM, includes and traces out the parabola CLM. And because QF or EL would be equal to EB, the rectangle DELM would be equal to the rectangle DEBA; hence, the two solids CABDEC, CLMDEC being similar figures on opposite sides of the same base, CDE are symmetrical, (Def. 19, B. II. El. Sol. Geom.); and hence they are equivalent. But the ungula CDEF bas been shown to be equal to two-thirds of the solid CLMDEC, it is therefore equal to two-thirds of the ungulical complement CEDABC.

Cor. 1. The above properties are true in whatever part of the base the plane CDE may cut, or whatever be the altitude of the ungula, provided the ungula and its complement have equal bases and altitudes; and because ungulas on the same base are as their altitudes, an ungula CGED is equal to the ungula GEDF, if D

C

F

the altitude CG=the altitude GF, since the two ungulas insist on or above the same base EDF.

Cor. 2. Hence the solidity of the parabolic ungula CDEF is equal to of its circumscrib- L ing prism, and the complement of a parabolic ungula is equal to of its circumscribing prism. For the parabolic prism (Prop. XVII.) is equal to of its circumscribing rectangular prism; and since the rectangular prism circumscribing the parabolic prism, may be divided into B two equal triangular prisms by a diagonal

P

G

D.

plane, identical with the plane which divides the ungula from its complement, it follows that the prism circumscribing the ungula DECF is equal to that circumscribing the complement DECI, equal half that circumscribing the parabolic prism. Let U equal the ungula, and C equal the complement, and let P equal the triangular prism circumscribing the ungula, and 2P will equal the quadrangular prism circumscribing the parabolic prism. Then will U+C=4P and U=3C, hence C=14U. Substituting the value of C in the first equation, we have 24U=4P or 15U=8P. UP,

Hence

or the ungula is equal to of its circumscribing prism. And if we substitute the value of U in terms of C in the first equation, we shall have

14C-4P or 5C=4P.

Hence C=P, or the complement of the ungula is equal to of its circumscribing triangular prism. Therefore, the portion CLBFE is equal to CFNEBL.

And the exterior portion CLEI is equal to CANELI.

of the prism

of the prism

PROPOSITION XIX. THEOREM.

A parabolic pyramid or cone is equal to two-thirds of its circumscribing rectangular pyramid.

Let a pyramid be erected on a base whose figure is a parabola, and if this base is circumscribed by a rectangle, it will =the rectangle; and because every section of each of the solids erected on those figures as bases, and whose sections are in a common point are similar figures and similar to the base, the solids erected on those bases must be in the relation of their bases; hence the parabolic pyramid is = its circumscribing quadrangular pyramid.

BOOK III.

ON REVOLOIDS AND SOLIDS FORMED BY THE REVOLUTIONS OF THE CONIC SECTIONS.

DEFINITIONS.

1. A REVOLOID is a solid generated by the continued semirevolution of a polygon on axes parallel to the sides of the polygon respectively, and passing through its centre, which is fixed, and it includes all the space that is not cut off by either side of the plane, in their several semi-revolutions.

2. Every revoloid has as many axes of rotation, as the polygon from which it is conceived to be generated, has inde pendent sides; but the axes of any two parallel sides coincide with each other and are identical.

Thus, if the quadrilaterial ABDC be made to revolve on the two axes, EF and GH, parallel to its sides respectively, the solid generated by the revolutions of those sides will be a revoloid, as represented by BCED.

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3. A revoloid is designated by the number of sides contained in the figure from whence it is conceived to be generated.

Thus a triangular revoloid is one formed by the continued semi-revolution of the sides of a triangle about their respective

axes.

A quadrangular revoloid, by the revolutions of the sides of a square about their corresponding axes.

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