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PROPOSITION IX.

THEOREM.
The Bafes and Altitudes of equal Pyramids, bav-

ing triangular Bases, are reciprocally propor-
rional; and tbose Pyramids having triangular
Bases, whose Bases and Altitudes are recipro-

cally proportional, are equal. LET there be equal Pyramids, having the trian

gular Bales A B C D E F, and Vertices the Points G, H I say, the Bases and Altitudes of the Pyramids A BCG, DEF,H, are reciprocally proportional ; that is, as the Base A B C is to the Base DEF, fo is the Altitude of the Pyramid DEFH to thie Altitude of the Pyramid A B C G.

For, compleat the folid Parallelopipedon B GML, EHPO; chen, because the Pyramid A B C G is equal to the Pyramid DEFH; and the Solid BGML is sextuple the Pyramid A BCG; and the Solid EHPO sextuple the Pyramid D EFH; the Solid B GML shall be * equal to the Solid E HPO. But the Bases and Altitudes if equal folid Parallelopipedons are reciprocally proportional; therefore, as the Base B Misto the Bale EP, rois + the Altitude of the Solid EHPO to the Alitude of the Solid BGML. But as the Base B M is to the Base E P, fo is * the Triangle A BC to the Triangle DEF; therefore as the Triangle A BC is to the Triangle DEF, so is the Altitude of the SoJid E H P O to the Altitude of the Solid B G ML. But the Altitude of the Solid EHPO is the same as the Altitude of the Pyramid D.EFH; and the Altitude of the Solid BGML, the same as the Altitude of the Pyramid. ABCG; therefore, as the Base ABC is to the Base DEF, so is the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG: Wherefore the Bases and Altitudes of the equal Pyramids ABCG, DEFH, are reciprocally proportional. And if the Bases and Altitudes of the Pyramids ABCG, DEFH, are reciprocally propor. tional; that is, if the Base A B C to sbe Base DEF,

+ 34. 11.

15. 5.

be as the Altitude of the Pyramid DEFH to the Al-
titude of the Pyramid A B CG; I say, the Pyramid
A BCG is equal to the Pyramid DEFH: For, the
fame Conftruction remaining, because the Base ABC
to the Base DEF, is as the Altitude of the Pyramid
DEFH to the Altitude of the Pyramid ABCG; and
as the Base ABC is to the Base DEF, so is the Paral-
lelogram BM to the Parallelogram EP; therefore the
Parallelogram B M to the Parallelogram EP, shall be
also as the Altitude of the Pyramid D E F H is to the
Altitude of the Pyramid ABCG. But as the Altitude
of the Pyramid DEFH is the fame as the Altitude of
the folid Parallelopipedon EPHO, and the Altitude of
the Pyramid ABCG, the same as the Altitude of the
folid Parallelopipedon BGML; therefore the Base BM
to che Base EP, will be as the Altitude of the solid Pa-
rallelopipedon EHPO to the Altitude of the folid Pa-
rallelopipedon BGML. But those folid Parallelopipe-
dons, whose Bales and Altitudes are reciprocally pro-
portional, are + equal to each other; therefore the solid + 34.11.
Parallelopipedon B G ML, is equal to the solid Paral-
lelopipedon EHPO: Now the Pyramid A B C G is a
fixth Part of the Solid BGML; and, in like manner,
the Pyramid DEFH is a fixth Part of the Solid EHPO,
therefore the Pyramid ABCG is equal to the Pyramid
DEFH. Wherefore, the Bases and Altitudes of equal
Pyramids, having triangular Bases, are reciprocally pro-
portional; and those Pyramids having triangular Bafes,
whose Bases and Altitudes are reciprocally proportional,
are equal; which was to be demonstrated.

PROPOSITION X.

THE O'R E M.
Every Cone is a third part of a Cylinder, baving.

the Jame Bose, and an equal Altitude.
LI

ET a Cone have the same Base as the Cylinder ;

viz, the Circle A B C D, and an Altitude equal to it. I say, the Cone is a third Part of the Cylinder ; that is, the Cylinder-is triple the Cone.

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For,

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For, if the Cylinder be not triple to the Cone, it shall be greater or less than triple thereof. First, let it be greater than triple to the Cone, and let the Square ABCD be described in the Circle ABCD; then the Square ABCD is greater than one half of the Circle ABCD. Now let a Prism be erected upon a Square A B C D having the same Altitude as a Cylinder, and this Prism will be greater than one half of the Cylinder; because, if a Square be circumscribed about the Circle ABCD, the inscribed Square will be one half of the circumscribed Square; and if a Prism be erected upon

the circumscribed Square of the fame Altitude as the 2. Cor. 7. Cylinder, fince Prisms are to one another as their of tbis.

Bases, the Prism erected upon the Square A B C D, is one half of the Prism erected upon the Square described about the Circle, ABCD. But the Cylinder is less than the Prism erected on the Square described about the Circle ABCD; therefore rhe Prism erected on the Square ABCD, having the same Height as the Cym linder, is greater than one half of the Cylinder. Let the Circumferences A B, BT,CD,DA, be bifected in the Points E, F, G, H; and join AE, EB, BF,

FC, CG, GD, DH, HA: Then each of the Tri+ This fol- angles A E B, BFC,CGD,DHA, is t greater than lotus from 2 the half of each of the Segments in which they stand.

Let Prisions be erected from each of the Triangles AEB B F C, CGD, DHA, of the fame Altitude as the Cylinder ; then every one of these Prisms erected is greater than half its correspondent Segment of the Cy. linder. For, because, if Parallels be drawn thro' the Points E, F, G, H, to AB, BC,CD,DA, and Pa. rallelograms be compleated on the faid AB, BC, CD, DA, on which are erected folid Parallelopipedons of the same Altitude as the Cylinder; then each of those Prisms

that are on the Triangles A E B, B FC, CGD, 32. 11. DHA, are Halves I of each of the solid Parallelepipe

dons; and the Segments of the Cylinder are less than the erected folid Parallelopipedons; and consequently, the Prisms that are on the Triangles A E B, B FC, CGD, DHA, are greater than the Halves of the Segments of the Cylinder: And so, bisecting the other Circumferences, joining Right Lines, and on every one of the Triangles erecting Prisms of the_fame Heighe as

fbe

of Ibis,

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the Cylinder, and doing this continually, we shall at last havë certain Portions of the Cylinder lest, that are less than the Excess by which the Cylinder exceeds triple the Cone.

Now, let these Portions remaining be A E, E B, BF, FC, CG, GD, DH, H A; then the Prism remaining, whose Base is the Polygon A EBFCG DH, änd Altitude equal to that of the Cylinder's is greater than the Triple of the Cone. But the Prism, whose Base is the Polygon A EBFCGDH, and Altitude the fame as that of the Cylinder's, is * triple of the Cor. 7. of

tbis, Pyramid, whose Base is the Polygon A EBFCG DH, and Vertex the same as that of the Cone; and therefore the Pyramid, whose Base is the Polygon A EBFCGDH, and Vertex the same as that of the Cope, is greater than the Cone, whose Base is the Circle ABCD: But it is less also (for it is comprehended by it) which is abfurd; therefore the Cylinder is not greater than triple the Cone. I say, it is neither less than triple the Cone: For, if it be poffible, let the Cylinder be less than triple the Cone; then (by Inverfion) the Cone shall be greater than a third Parc of the Cylinder: Let the Square A B C D be described in the Circle A BCD; then the Square A B C D is greater than half of the Circle A B CD: And let a Pyramid be erected on the Square A B C D, having the same Vertex as the Cone; then the Pyramld erected is greater than one half of the Cone ; because, as has been already demonstrated, if a Square be described about the Circle, the Square A B C D shall be half thereof: And if solid Parallelopipedons be erected upon the Squares of the fame Altitude as the Cone, which are allo Prisms; then the Prism erected on the Square ABCD is one half of that erected on the Square described about the Circle ; for they are to each other as their Bases, and so likewise are their third Parts : Therefore the Pyramid, whose Base is the Square A B C D, is one half of that Pyramid erected upon the Square described about the Circle. But the Pyramid erected upon the Square described about the Circle is greater than the Cone, for it comprehends it ; therefore the Pyramid, whose Base is the Square ABCD, and Vertex the same as that of the Cone,

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is greater than one half of the Cone. Bilect the Circumferences AB, BC, CD,D A, in the Points E, F, G, H; and join AE, EB, BF, FC, CG, GD, DH, HA ; and then each of the Triangles A E B, BFC, CGD, DHA, is greater than one half of each of the Segments they are in. Let Pyramids be erected upon, each of the Triangles A EB, BFC, CGD, DHA, having the fame Vertex as the Cone; then each of these Pyramids, thus erected, is greater than one half of the Segment of the Cone in which it is; and so, bifecting the remaining Circunferences, joining the Right Lines, and erecting Pyramids upon every of the Triangles having the same Altitude as the Cone, and doing this continually, we shall at last have Segments of the Cone left, that will be less than the Excess by which the Cone exceeds the one third Part of the Cyjinder : Ler these Segmenis be those that are on A E, EB, BF, FC, CG, GD, DH, HA; and then the remaining Pyramid, whose Base is the Polygon A EBFCGDH, and Vertex the same as that of the Cone, is greater than a third Part of the Cylinder: But the Pyramid, whose Base is the Polygon AEBFCGDH and Vertex the same as that of the Cone, is one third Part of the Prism whose Base is the Polygon A E B FCGDH, and Altitude the same as that of the Cylinder: Therefore the Prism, whose Base is the Polygon A EBFCGDH, and Altitude

the same as that of the Cylinder, is greater than the
- Cylinder, whose Base is the Circle A B C D; but it

is less allo (as being comprehended thereby); which is
absurd; therefore the Cylinder is not less than triple
of the Cone: but it has been proved also not to be
greater than triple of the Cone; therefore the Cylinder
is necetiarily triple of the Cone. Wherefore, every
Cone is a third Part of a Cylinder, having the fame Base,
and an equal Altitude; which was to be demonftrated.

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