Book XII.circumference ABCD to half the circumference EFGH, so is

BK 0 FL; and taking their doubles, as the circumference ABCD 94. 5. to the circumference EFGH, fois BD to FH.

PROP. IV. THEOR.

"RIANGULAR pyramids upon equal bases, and of

the same altitude, are equal to one another. Let ABC-G, DEF-H be two pyramids of the fame altitude, and having the equal triangles ABC, DEF for their bases; the pyramid ABC-G is equal to the pyramid DEF-H. If noi, one of them is greater than the other: let the

pyramid ABC-G be greater than the other pyramid by the folid Z: and complete the prism, of which ABC is the base, and AG one

11.

of its insisting lines : Then, if AG be bisected by a plane parallel a Cor. 32. to ABC, that plane will bisect a the prism; for AG and the per

pendicular from G to the plane ABC, are cut in the same ratio b 17. 11. by that parallel plane b; therefore the parts are prisms on the

fanie base, and having equal altitudes, and are therefore equal . In the same manner, may the half of the remaining prism be

cut off; and so on : Therefore, if this be done continually, there C 1. 12. fhall at length remain a prism less than the folid Z: let this

be the prison ABC-LNO, and let LN, LO meet GB, GC in P, d 31. 1. Q; and draw a PR, QS parallel to AG, and join PQ, RS ; € 13. Def. therefore ARS-LPQ is a prisme in the pyramid ABC-G: Let

the

k 4. 5.

n Cor. 326

II.

the other parts of AG equal to AL, be LK, KM, MG, and com-Book XII. plete prilms in the pyramid between the planes through L, K, M: and let the same construction be made in the pyramid DEF-H: and because LP is parallel ° to AB, AG is to GL, as 0 16.11. * AB to LP or AR: For the same reason, AG is to GL, as AC f 4. 6. to AS; therefore BA is to AR, as & CA AS; and the tri. g 11. 5. angle ABC is similar h to ARS: In the same manner, it may be h 6.6. proved, that the triangle DEF is similar to DXY: and because AG, DH are equimultiples of AL, DT; as are allo GL, HT; therefore AG is to GL, as k DH to HT; wherefore BA is to AR, as ED to DX, and consequently the triangle ABC is to ARS,

I 22. 6. as ' DEF to DXY: and the triangle ABC is equal to DEF; therefore the triangle ARS is equal n to DXY; and therefore m 14. 5. the prism LRS is equal n to the prism TXY, for they are upon equal bases, and have the fame part of the altitude of the pyramids for their altitude. In the same manner, may the other prisms in the two pyramids be proved to be equal, each to each ; therefore all the prisms in the pyramid ABC-G together are equal to all the prisms in the pyramid DEF-H. Produce RP, SQ_to a, b, and join ab; therefore LPQ-Kab is a prism; and it is equal to the prism ASR-LPQ, because they are upon the same base, and have equal altitudes : In the same manner, it

may be proved, that the other inscribed prisms are equal to the circumscribed prisms standing upon them: therefore all the inscribed prisms, together with the prism ABC-LNO, are equal to all the circumscribed prisms ; that is, they are greater than the pyramid ABC-G: But the prisms in the pyramid ABC-G are equal to those in the pyramid DEF-H, and the solid Z is greater than the prism ABC-LNO; therefore the prisms in the pyramid DEF-H, together with the solid Z, are greater thanı the pyramid ABC-G ; that is, than the pyramid DEF-H, together with the solid Z: take away the common solid Z, and the prisms in the pyramid DEF-H are greater than the pyramid itself, and they are also less; which is impossible: Therefore the pyramid ABC-G is not unequal to the pyramid DEF-H, that is, it is equal to it. C. E. D.

PROP. V. THEOR *.

E

VERY triangular prism may be divided into three

equal triangular pyramids. Let ABC-DEF be a triangular prism, of which the base is the triangle ABC, and DEF the triangle opposite to it: It may be divided into three equal triangular pyramids.

Join

Ee 2

* This is Prop. VII. Book XII. of Euclid.

D

Book XII. Join AF, FB, BD: and because BD is the diameter of the

parallelogram AE, the triangle ABD is equal - to EBD; and a 34. 1. therefore the pyramid ABD-F is equal to the b 4. 12.

pyramid EBD-F, for they are upon equal
bases, and of the same altitude: For the same
reason, the pyramid ADF-B is equal to the
pyramid ACF-B: But the pyramid ABD-F
is the same with the pyramid ADF-B: Where-
fore the pyramid ACF-B is equal to the pyra-
mid EBD-F; therefore the prism ABC-DEF
is divided into three equal pyramids ABD-F, B

A
EBD-F, ACF-B, Wherefore, &c. Q. E. D.

Cor. 1. From this it is manifeft, that every pyramid is the third

part of a prism which has the same base, and is of an equal altitude with it; for if the base of the prism be any other figure than a triangle, it may be divided into prisms having triangular bases.

Cor. 2. * Pyramids of equal altitudes are to one another as their bases; because the prisms upon the same bases, and of the same altitude, are o to one another as their bases.

* This is the 5th and 6th Proposition of Book XII. of Euclid.

C 2. Cor. 32. II.

PROP. VI. THEOR.
TF a cylinder and a parallelopiped be upon equal

bases, and of the fame altitude, they are equal to one another.

Let ABCD be a cylinder, and EF a parallelopiped of the same altitude; and let the circle ABG, which is the base of the cylinder, be equal to the base EH of the parallelopiped ; the cylinder ABCD is equal to the parallelopiped EF.

If not, it is either greater or less than the parallelopiped : First, let it be less, and let it be equal to the parallelopiped EK of the same altitude with FF, but having the base EL less than EH: and in the circle AGB let rectangles be made as in the second propofition, which together shall be greater than EL; and let the same be done in the opposite bafe CTD; and let NOPQ, RSTV be corresponding rectangles in them; therefore, because the circles are equal, the rectangle PN is equal to RT, as was shown in Propofirion III ; join NR, OS, PT, QV: and because

NQ_ is equal and parallel to RV, NR is equal and parallel to a 33. 1. QV: For the same reason, PT, OS are equal and parallel to 5-24, 11. NR or QV; therefore NT is a parallelopiped b: conftrud, in the same manner, parallelopipeds upon the other rectangles in

the

the circle AGB: and because the parallelopipeds NT, EK are of Book XII.
the fame altitude, NT is to EK, as the base NP to the base
EL: For the same reason, the parallelopipeds upon the other C 32. 11.
rectangles in the circle AGB are each of them to EK as the rect-
angle upon which it stands to the base EL ; therefore all the pa-
rallelopipeds together are to EK as all the rectangles in the d 2. Cor.
circle AGB to the base EL: But the rectangles in the circle 24. 5.
AGB are greater than EL; therefore the parallelopipeds upon
them are together greater than EK 4: and EK is equal to the e A. s.
cylinder ABCD; therefore the parallelopipeds upon the rect-
angles are greater than the cylinder; and they are also less, be.
cause they are contained in it; which is impossible : There-
fore the cylinder ABCD is not less than the parallelopiped EF.

8 1. Cor.

2. 12.

Neither is it greater ; for, if possible, let the cylinder ABCD
be equal to the parallelopiped EW of the same altitude with EF,
but upon a base EX greater than EH; that is, than the circle
AGB: and because EX is greater than the circle AGB, rect-
angles can be made about the circle, which together are lefs than
EX®: and, if upon these rectangles be made parallelopipeds in
the way that NT was made, it may be proved, as before, that
they are together less than the parallelopiped EW; that is, than
the cylinder ABCD: and they are also greater, because they
contain the cylinder; which is impossible; therefore the cylinder
is not greater than the parallelopiped EF: and it has been shown,
that it is not less than it; therefore the cylinder is equal to the
parallelopiped. Wherefore, &c. Q. E. Ď.

Cor. 1. * Hence cylinders of the saine altitude are to one ano.
ther as their bases, for the parallelopipeds upon bases equal to
them, and of the same altitude, are to one another as their
bases .

Cor. 2. Cylinders have to one another the ratio which is compounded of the ratios of their bases, and of their altitudes. For this is the ratio of the parallelopipeds, which are upon equal bases with them, and have the same altitudes.

f 2. Cor.

D. 11. * This is the uth Proposition of Book XII, of Euclid,

PROP.

BOOKXII.

PROP. VII. THEOR.

IF

fa cone and a pyramid be upon equal bases, and

of the fame altitude ; they are equal to one another.

Let the cone of which the base is the circle ABC, and the vertex D, be of the same altitude with the pyramid of which the base is the figure EFG, and the vertex H; and let the circle ABC be equal to the figure EFG: the cone ABC-D is equal to the pyramid EFG-H.

If not, let them be unequal; and, first, let the cone be less than the pyramid ; that is, let it be equal to fome pyramid FGK-H of which the base FGK is less than FGE, or than the

circle ABC; therefore, in the circle there can rectangles be a 1. Cor. made, which together are greater than FGK a: let LMNO be

one of them, and join DL, DM, DN, DO: and because the

pyramids LMNO-D and FGK-H are of the same altitude, the b 2. Cor. pyramid LMNO-D is to the pyramid FGK-H, as the base

LMNO to the base FGK: In the same manner, if pyramids be erected on all the other rectangles in the circle ABC, it may be proved, that each of them is to the pyramid FGK-H, as the

2. 12.

5. I2.

rectangle on which it stands to the base FGK: wherefore all the C 2. Cor. pyramids upon the rectangles are to the pyramid FGK-H, as Call 24. 5.

the rectangles to the base FGK: But the rectangles together are greater than the base FGK; therefore the pyramids upon them are together greater than the pyramid FGK-H; that is, than the cone ABC-D: and they are also less, for they are contained in it; which is impossible; therefore the cone ABC-D is not less

than