PPROPOSITION XXXI. THEOREM XXVI.

(K

ARALLELEPIPEDS (KI & N Z) which are upon equal bases

H & N q), and of the fame altitude, are equal to one another.

Thefis.

The KI is to the NZ.

DEMONSTRATION.

CASE I.

If the infifting lines AG, &c. of the

C M, &c. of the NZ, are

KI; & the infifting lines

to their bafes; or if the

inclinations of the infifting straight lines A G & M C are the

fame.

3.

B. 1.

2. At the point F in F Q, make the plane V QF R = plane VHAK. P.23. B. 1.

3. Make FRA K.

4. Complete the pgr. F QSR.

P.31. B. 1.

5. Complete likewife with the lines F Q & FD; FR & F D, the pgrs. QTD F & D FR.

P.31. B. 1.

6. Complete the DS.

7. Produce the ftraight lines F q & R S until they meet in V.

Pof.2. B. 1.

P.31. B. 1.

8. Thro' the point Q, draw X QY, plle. to V

q.

9. Produce C'q, until it meets X Y, in the point Y.

10.Complete the ZQ & VDTX.

BECAUSE the lines F Q & FR are = to AH & AK.

2. It may be demonftrated after the fame manner that the pgrs. FT & DR are & to the pgrs. A I, & A L.

SP.36. B. 1.
Į D. 1. B. 6.

Therefore, fince the three pgrs. FS, FT, & DR, of the are & to the three pgrs. A E, A I, & A L, of the (Arg. 1. & 2).

And the remaining pgrs. of the DS, likewife thofe of the KI are & to thofe already mentioned; each to each. 3. The DS, will be & to the KI.

DS

KI,

The DX & DS, have the fame bafe DQ, & their infifting lines
FV & FR, &c. are in the fame plle. directions VS, &c.
4. Confequently, DS is to the
DS is

P.24. B.11. D.10. B.II.

DX.

P.29. B.11.

But the

to the

5. Therefore the DX is alfo

to the

KI (Arg. 3).
KI.

Ax.1. B.

The MQ is cut by the plane F Z, plle. to the plane M N. 6. Confequently, the bafe N q: bafe qQMF:

P.25. B.11.

The 2X is cut by the plane DQ plle. to the plane Z Y. 7. Confequently, the base F X: base qQ=ADX:

ZQ

P.25. B.11.

But the pgr. F X is

P.35. B. 1.

And the pgr. F S is

But the 12.Therefore, the

8. Confequently, the pgr. F X is

But the bafe H K is

to the bafe q N (Hyp. 1).

9. Hence the bafe qN to the base F X,
But the bafe qN: bafe q QMP:
And the bafe qQ: bafe F X=QZQ:ED X. (Conv. Arg.

10. Hence the bafe qN: bafe F X=MF:

But the bafe q N is to the base F X (Arg. 9). 11.Confequently, the MF is to the

DX & KI are equal (Arg. 5).

to the pgr. H K. (Arg. 1).
to the pgr. HK.

Ax.1. B. 1,

If the angles of inclination of the infifting ftraight lines,
AG &c. of KI are not equal to the angles of inclination
of the infisting straight lines CM, &c. of the

UPON

MF.

PON the base K I, make a having its infifting ftraight lines, either or equally inclined with the infifting ftraight lines of the MF, & in the fame direction as thofe of KI.

:

And confequently, which will be equal to it (P. 30. B. 11).

The remainder of the conftruction, & of the demonstration, are the fame as in the foregoing cafe.

EQUAL

COROLLAR Y.

QUAL parallelepipeds which have the fame altitude, have equal bafes.

PROPOSITION XXXII. THEOREM XXVII.

PARALLELEPIPEDS (BD & EP) which have equal altitudes

(BC & F O), are to one another as their bafes (AK & EG).

1. Produce EF to M.

Pof.2. B. 1.

FL=

2. Upon F G with F M, make the
pgr. КА,
which will be in the fame direction with the EG.
pgr.

So that the pgrs. E G & FL together, form the pgr. EL. P.44. B. 1. 3. Complete the

BECAUS

FI.

DEMONSTRATION.

ECAUSE the bafe F L of the FI, is to the base A K

SIMILA

IMILAR parallelepipeds (EB & FH) are to one another in the triplicate ratio of their homologous fides (A B & G H).

S Pof.z. B. 1.
P. 3. B. 1.

1. Produce AB & make ARGH.
2. From A R defcribe the RL & to the FH,
fo that the lines AC & AI; DA & AK be in the fame
ftraight line.

3. Complete the
OK.

A O, fo as to form with

4. Complete likewife the

P.27. B.11.

RL the

AP, fo as to form with OA,
PB.

the OC, & with the EEB the

to one another & equal to the ratio of A B to G H.

6. The three ratios A B AR, AC: AI, & AD: A K, are equal

But the PB is cut by the plane A E (Prep. 4).

7. Confequently, the base CB: base QABE:AP. And the bafe CB: base QA AB: AR.

9. Therefore AB: ARBE:A P.

Which was to be demonftrated.

* See Cor. 2. of this propofition.

P.16. B. 5.

P.25. B.11.
P. i. B. 6.
P.11. B. 5.

The OC is cut by the plane RD (Prep. 4). 9. Confequently, the bafe R C: bafe A MAP:OA. And, the bafe R C: bafe AM AC: A I.

10. Therefore,

Infine, the

OK being cut by the plane A M (Prep. 4).

1.It may be demonstrated after the fame manner.

But the three ratios A B : AR, AC: A I, & AD: A K are= to the ratio AB GH (Arg. 6).

12.Confequently, the four BE, A P, A O, & A N form a feries.

of magnitudes in the fame ratio (AB: G H). 13.Therefore, they are proportionals.

14.Confequently, the BE is to the of AB to GH.

AN in the triplicate ratio

But the BE is to the FH in the triplicate ratio of A B to GH, (or as A B to G H3). *

FROM

P.25. B.11.
P. I. B. 6.
P.11. B. 5.

P.1r. B. 5. D. 6. B. 5.

D.11. B. 5.

ROM this it is manifeft, that if four flraight lines be continual proportionak, as the firft is to the fourth, fo is the parallelepiped defcribed from the first to the fimilar fimilarly defcribed parallelepiped from the fecond; because the first straight line has to the fourth, the triplicate ratio of that which it has to the fecond.

*COROLLARY II.

ALL cubes being fimilar parallelepipeds (D. IX & XXX. B. 11), fimilar pa

rallelepipeds (AB & FH) are to one another as the cubes of their homologous fides (AB GH) (expressed thus A B GH); because they are in the triplicate ratio of those fame fides.

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