Then the parallelopipedons AR, LS, MC, EV, PG are all equal, having equal bases and heights. Therefore, the solid AC is to the solid EG as the number of parts in AC to the number of equal parts in EG, or as the number of parts in AB to the number of equal parts in EF; that is, as the base AB to the base EF. Q. E. D.

Note. If the bases be incommensurable, the divisions must be infinitely small.

Corol. From this proposition, and the corollary to the last, it appears that all prisms and cylinders of equal altitudes are to each other as their bases; every prism and cylinder being equal to a rectangular parallelopipedon of an equal base and height.

PROP. V.

Rectangular parallelopipedons of equal bases are in proportion to each other as their altitudes.

Let AB, CD be two rectangular parallelopipedons standing on the equal bases AE, CF; then will AB be to CD as the altitude EB is to the altitude DF.

H

B

12

D

E

C

For, let AG be a rectangu- A lar parallopipedon on the base AE, and its altitude. EG equal to the altitude FD of the solid CD.

Then AG and CD are equal, being prisms of equal bases and altitudes. But if HB, HG be considered as bases, the solids AB, AG of equal altitude AH, will be to each other as those bases HB, HG. But these bases HB, HG being parallelograms of equal altitude HE, are to each other as their bases EB, EG; and, therefore, the two prisms AB, AG are to each other as the lines EB, EG. But AG is equal CD, and EG equal FD; consequently, the prisms AB, CD are to each other as their altitudes EB, FD; that is, AB: CD:: EB: FD. Q. E. D.

Corol. From this proposition and the corollary to

Prop. 3, it appears that all prisms and cylinders of equal bases are to one another as their altitudes.

PROP. VI.

Rectangular parallelopipedons are to each other as

the products of their bases by their altitudes.

The parallelopipedon AF

is to the parallelopipedon

CE as the base AG × the altitude GF, is to the base CD X altitude DE.

For the parallelopipedons AB and CE, having the same altitude, are to each other as their bases AG and CD; and the parallelopipedons AF and AB, having the same base, are to each A

other as their altitudes GF, GB; or,

AB:CE::AG: CD,

AF:AB::GF: GB.

B

F

E

D'

Multiplying_these two proportions together and striking out AB from the two terms of the first resulting ratio, we have

AF:CE:: AGX GF: CDX GB.*

* As rectangular parallelopipeds are always to each other as the products of their bases by their altitudes, this may be taken as the measure of parallelopipeds; and as every parallelopiped is equal to a prism or cylinder of the same base and altitude (Prop. 3), it follows that the measure of any prism or cylinder is the product of its base by its altitude.

If the convex surface of a cylinder be developed, it opens out into a parallelogram, of which the circumference of the cylinder's base is the base, and the altitude of which is that of the cylinder; and as this parallelogram is measured by the product of its base by its altitude, we have for the measure of the convex surface of a cylinder the product of the circumference of its base by its altitude.

A plane determined by an element of a cylinder, and the tangent line to the base at the point where the element meets it, is a tangent plane to the cylinder.

The contact is along the whole length of the element, which is called the element of contact.

PROP. VII.

Similar prisms and cylinders are to each other as the cubes of their altitudes, or of any other like linear dimensions.

Let ABCD, EFGH be two similar prisms; then will the D. prism CD be to the prism GH as AB3 to EF3, or as AD3 to EH'.

For the solids are to each other as the products of their bases and altitudes (by the note

A

B

H

E

F

to the last Prop.), that is, as AC. AD to EG. EH. But the bases being similar planes, are to each other as the squares of their like sides, that is, AC to EG as AB to EF; therefore, substituting the ratio or fraction AB: EF2 for AC: EG, we have the solid CD to the solid GH as AB'. AD to EF. EH. But BD and FH being similar planes, have their like sides proportional, that is, AB: EF:: AD: EH, or AB': EF2:: AD2: EH'; multiply this by the identical proportion AD: EH:: AD: EH, and we have AB'. AD EF.EH::AD': EH'; and, consequently, the solid CD: solid GH:: AD3: EH3, or AB3: ÉF3. Q. E. D.

PROP. VIII.

In a pyramid, a section parallel to the base is similar to the base, and these two planes will be to each other as the squares of their distances from the vertex.

Let ABCD be a pyramid, and EFG a section parallel to the base BCD, also AIH a line perpendicular to the two planes at H and I; then will BD, EG be two simi- E lar planes, and the plane BD will be to the plane EG as AH' to AI.

For, join CH, FI. Then, because a plane cutting two parallel planes makes

B

A

G

D

H

parallel sections, therefore the plane ABC, meeting the two parallel planes BD, EG, makes the sections BC, EF parallel; in like manner, the plane ACD makes the sections CD, FG parallel. Again, because two pair of parallel lines make equal angles, the two EF, FG, which are parallel to BC, CD, make the angle EFG equal the angle BCD. And, in like manner, it is shown that each angle in the plane EG is equal to each angle in the plane BD, and, consequently, those two planes are equiangular.

Again, the three lines AB, AC, AD, making with the parallels BC, EF, and CD, FG, equal angles; and the angles at A being common, the two triangles ABC, AEF are equiangular, as also the two triangles ACD, AFG, and have, therefore, their like sides. proportional, namely, AC: AF::BC:EF::CD: FG. And, in like manner, it may be shown that all the lines in the plane EG are proportional to all the corresponding ones in the base BD. Hence these two planes, having their angles equal and their sides proportional, are similar.

But similar planes being to each other as the squares of their like sides, the plane BD: EG :: BC2: EF2: or :: AC2: AF2, by what is shown above. But the two triangles AHC, AIF, having the angles H and I right ones, and the angle A common, are equiangular, and have, therefore, their like sides proportional, namely, AC:AF::AH: AI, or AC2: AF2 :: AH2: AI2. Consequently, the two planes BD, EG, which are as the former squares AC2, AF2, will be also as the latter squares AH', AI, that is, BD: EG:: AH' : AI2.

PROP. IX.

In a right cone a section parallel to the base is a circle, and this section is to the base as the squares of their distances from the vertex.

Let ABCD be a right cone, and GHI a section parallel to the base BCD; then will GHI be a circle,

and BCD, GHI will be to each other as the squares of their distances from the vertex.

For, let the planes ACE, ADE pass through the axis of the cone AKE, meeting the section in the three points H, I, K.

B

GI

H

D

E

Then, since the section GHI is parallel to the base BCD, and the planes CK, DK meet them, HK is parallel to CE, and IK to DE. And from similar triangles, shown to be such (as in the last Prop.), KH: EC::AK: AE:: KI: ED. But EC is equal to ED, being radii of the same circle; therefore, KI is also equal to KH. And the same may be shown of any other lines drawn from the point K to the circumference of the section GHI, which is, therefore, a circle.

Again, since AK: AE:: KI: ED, hence AK': AE: KI: ED; but KI: ED':: circle GHI: circle BCD (th. 72); therefore, AK: AE:: circle GHI: circle BCD. Q. E. D.

PROP. X.

All pyramids and right cones of equal bases and altitudes are equal to one another.

equal.

For, parallel to the bases, and at equal distances, AN, DO, from the vertices, suppose the planes IK, LM to be drawn.

Then, by Prop. 8 and 9,

and

DO: DH':: LM: EF,

AN: AG::IK :BC.

But, since AN', AG' are equal to DO', DH'; there