THE

PROPOSITION IX.

THEOREM IX.

HE bafes (A B C & E F G), and altitudes (BD & F H), of equal pyramids, (A B C D & E F G H), having triangular bases, are reciprocally proportional, (that is, the base A BC: bafe E F G altitude F H : altitude B D), and triangular pyramids (A B C D & E F G H), of which the bafes (A B C & E FG), and altitudes (B D & F H), are reciprocally proportional: are equal to one another.

Hypothefis.

I. The pyrams. ABCD & EFGH are triangular.
II. The pyram. ABCD is = to the pyram. EFGH.

Preparation.

Thefis.

Bafe ABC: bafe EFG altitude
FH: altitude B D.

Complete the EBO & FK having the fame altitude with
the pyramids ABCD & EFGH; as alfo the prifms
BAPNC & FELIG.

1. DEMONSTRATION.

BECAUSE the prifms PNB & LIF, have the fame base &

altitude with the given pyramids A B C D & E F G H. (Prep).

1. Each prifm will be triple of its pyramid, (that is, the prifm P N B triple of the pyramid A B C D, & the prifin L IF triple of the SP. pyramid E F G H)..

2. Confequently, the prifm PNB is to the prifm LIF. But the BO is double of the prifm P N B, & the double of the prifm LIF.

3. Therefore, the BO is to the FK.

FK

But the equal (BO & FK) have their bafes and altitudes re-
ciprocally proportional (that is, bafe B Q: bafe F M altitude
FH altitude B D).

And thofe are each fextuple of their pyramids, (that is, the
BO is fix pyramids A B CD, & the KF = fix pyramids
EFGH. Arg 1. & 3).

7. B.12, Cor. 1. Ax.6. B. 1,

P.28. B.11.
Ax.6. B. 1.

Moreover, the base of the pyramid ABCD is the half of the base of the B O.

And the bafe of the pyramid EFGH is the half of the bafe P.41. B. 1.

of the F K.

Confequently, base A B C : base EFG alt. F H : alt. B D.

Hypothefis.

SP.15. B. 5.
P.11. B. 5.

Which was to be demonftrated.

Thefis.

1. The pyramids ABCD & EFGH are triangular. The triangular pyramid ABCD is = 11. Bafe ABC bafe EFG alt. FH: alt. BD. to the triangular pyramid EFGH.

BECAUS

II. DEMONSTRATION.

ECAUSE the AABC: AEFG=FH: BD. (Hyp. 2). And the pgr. BQ is double of the AABC, the pgr. F M double of the AEF G.

But
And

P.41. B. 1.
P.15. B. 5.

(Prep.).

P.34. B.11.

P.18. B.11.

i. It follows, that the pgr. BQ: pgr. F MFH: BD.
BO has for bafe the pgr. B Q, & for alt. B D.
FK has for bafe the pgr. FM, & for alt. F. H.
2. Confequently, the BO is to the FK.
But the BO & FK are each double of the prifms PNB &
LIF.

And thofe prifms PNB & LIF are each triple of their pyramids SP.
ABCD & E F G H.

3. Therefore, the triangular pyramid A B C D is to the triangular
pyramid E F G H.

7. B.12.

Cor. 1.

Ax.7. B. 1.

Which was to be demonftrated.

COROLLAR Y

EQUAL polygon pyramids have their bases and altitudes reciprocally propor

tional; polygon pyramids whofe bafes & altitudes are reciprocally proportional: are equal.

PROPOSITION X.

THEOREM X.

EVERY cone (BRC) is the third part of the cylinder (HG FE

ABDC) which has the fame base, (BD CA) and the fame altitude (BH) with it.

Hypothefis.

The cone BRC, & the cylinder HFADC, bave the fame base B DCA, & the fame altitude BH.

I.

If not,

Thefis.

The cone BRC is equal to the third part of the cylinder HFCABD.

DEMONSTRATION,

The cone will be <or > the third part of the cylinder, by
a part = 2.

I. Suppofition.

Let the third part of the cylinder HC be cone BRC
+ 2.

1. Preparation.

IN the bafe A BDC of the cone & cylinder, defcribe the

A B D C.

2. About the fame base describe the □POQS.
3. Upon thofe fquares erect two, the first FHB C, upon the
infcribed, & the fecond, on the circumfcribed, which will
touch the fuperior bafe with its plle. planes, in the points H, G, F,
& E, * having the fame altitude with the cylinder, & the cone.

Sf.

P. 6. B. 4

P. 7. B. 4.

* We have omitted a part of the preparation in the figure to avoid confufion.

4. Bifect the arches A TC, Cd D, D b B, & B a A, in T,d,b, & a.
5. Draw A T, & T C, &c.
6. Thro' the point T, draw the tangent ITK, which will cut BA &
DC produced, in the points I & K & complete the pgr. A K.
7. Upon the pgr. AK, erect the ALFK, & upon the ▲ AIT,
TAC, & TCK the prifins E TI, ETF, & TFK, having all
the fame altitude with the cylinder & cone.

8. Do the fame with respect to the other fegments A a B, B b B, &c.

BECAUSE the PO QS is defcribed about, & the □

POQS

BDCA defcribed in the O. (Frep. 1. & 2).

1. The PO QS is double of the BDC A.

P.30. B. 3. Pol.1. B. 1. P.17. B. 3.

the upon POQS is > the given cylinder.

And the

defcribed upon thofe fquares having the fame altitude, 2. Therefore, the upon POQS is double of the

P.47. B. 1. (Prep. 3).

upon BDCA.

P.32. B.11.

But

Ax.8. B. 1.

P.19. B. 5. P.41. B. i. (P 28. B.11. P.34. B.11. Rem.1.Cor.3. Ax.8. B. 1.

3. Therefore, the upon BDCA is > the half of the fame cylinder.
And fince the ATAC is the half of the pgr. A K.
4. The prifm ETF, defcribed upon this TAC, will be the
half of the upon the pgr. A K.

The Ed defcribed upon the pgt. A K is > the element of the
cylinder, which has for bafe the segment ATC.
5. Confequently, prifm E T F defcribed upon A T A C is > half of
the element of the cylinder which has for bafe fegment A T C.
6. Likewife, all the other prifms defcribed after the fame manner, will
be > the half of the correfponding parts or elements of the cylinder.
Therefore, there may be taken from the whole cylinder more than
the half, (viz. the upon the BDCA), & from those remain-
ing elements (viz. CFEAT, &c.) more than the half; (viz. the
prifms TF, &c.), & so on.

P.19. B. 5.

7. Until there remains feveral elements of the cylinder which together will be <Z..

Lem. B.12.

But the cylinder is to three times the cone B R C + Z. (Sup.). Therefore, if from the whole cylinder be taken thofe elements (Arg. 7.). And from three times the cone B RC+Z, the magnitude Z. 8. The remaining prifm (viz. that which has for base the polygon Aa Bb Dd CT) will be > the triple of the cone.

But this prifm is the triple of the pyramid of the fame base & altitude (viz. of the pyramid TA a B b D dC TR).

9. Confequently, the pyramid A BDCR is the given cone.
But the bafe of the cone is the in which this polygon A B DC
is infcribed, (& which is confequently> this polygon), & this cone
has the fame altitude with the pyramid.
10. Therefore, the part is the whole.
11.Which is impoffible.
12.Confequently, the cone is not

IF

13.

Let the cone be > the
Z, that is, the cone

the third part of the cylinder. II. Suppofition.

third part of the cylinder by the mgn. the third part of the cylinder + Z. II. Preparation.

Ax.4. B. 1. P. 7. B.12. {Cor. 2.

Divide the given cone into pyramids, in the fame manner.
that the cylinder was divided in the first fuppofition.

F from the given cone be taken the pyramid which has for base the
ABD C, (which is greater than the half of the whole base of
the given cone, being the half of the circumfcribed, Arg. 1. &
this being the base of the cone, Ax. 8. B. 1.), & from the
remaining fegments, the pyramids correfponding to those segments,
(as has been done in the cylinder Arg. 7.).

There will remain feveral elements of the cone which together
will be < Z.

Therefore, if from the cone thofe elements be taken which are <
Z, & from the cylinder+Z, the magnitude Z.

to the third part of the prifm,
Bb DdCT, & the fame alt.
to this prifm.

14. The remainder, viz. the pyramid Aa Bb DdCTR is to the
third part of the cylinder.
But the pyr. A a BbDdCTR is
which has for bafe the fame polyg. Aa
15.Therefore, the given cylinder, is
But the base of the given cylinder is
this fecond is infcribed in the first. (I. Prep. 4.
16. Therefore, the part is to the whole.
17.Which is impoffible.

the bafe of the prifm fince
ઇ ૬).

18. Therefore, the third part of the cylinder is not the cone.
And it has been demonftrated (Arg. 12.), that the third part of the
cylinder is not > the cone.

19. Therefore, the cone is the third part of the cylinder of the fame

bafe & altitude.

Ax.7. B. 1.

Ax.8, B. 1.

Lem. B.12

Ax.5. B. 1. SP. 7. B.12. Cor. 2. Ax.6. B. 1.

Ax.8. B. 1.

Which was to be demonftrated.