PROPOSITION VIII. THEOREM.

Right parallelopipeds, having the same base, are to each other as their altitudes.

Let AG, AL be two right parallelopipeds having the same base ABCD; then will they be to each other as their altitudes AE, AI.

Case first. When the altitudes are in the ratio of two whole numbers.

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Suppose the altitudes AE, AI are in the ratio of two whole numbers; for example, as seven to four. Divide AE into seven equal parts; AI will contain four of those parts. Through the several points of division, let planes be drawn parallel to the base; these planes will divide the solid AG into seven A small parallelopipeds, all equal to each other, having equal bases and equal altitudes. The bases are equal, because every section of a prism parallel to the base is equal to the base (Prop. II., Cor.); the altitudes are equal, for these altitudes are the equal divisions of the edge AE. But of these seven equal parallelopipeds, AL contains four; hence the solid AG is to the solid AL, as seven to four, or as the altitude AE is to the altitude AI.

Case second. When the altitudes are not in the ratio of two whole numbers.

Let AG, AL be two parallelopipeds whose altitudes have any ratio whatever; we shall still have the proportion

Solid AG solid AL:: AE: ÁI.

For if this proportion is not true, the first three terms remaining the same, the fourth term must be greater or less than AI. Suppose it to be greater, and that we have

Solid AG: solid AL:: AE: AO.

Divide AE into equal parts each less than OI; there will be at least one point of division between O and I. Designate that point by N. Suppose a parallelopiped to be constructed, having ABCD for its base, and AN for its altitude; and represent this parallelopiped by P. Then, because the altitudes AE, AN are in the ratio of two whole numbers, we shall have, by the preceding Case,

Solid AG: P::AE: AN.

But, by hypothesis, we have

Solid AG: solid AL:: AE: AO.

Hence (Prop. IV., Cor., B. II.),

Solid AL: P::AO:AN.

But AO is greater than AN; hence the solid AL must be greater than P (Def. 2, B. II.); on the contrary, it is less, which is absurd. Therefore the solid AG can not be to the solid AL, as the line AE to a line greater than AI.

In the same manner, it may be proved that the fourth term of the proportion can not be less than AI; hence it must be AI, and we have the proportion.

Solid AG: solid AL:: AE: AI.

Therefore, right parallelopipeds, &c.

PROPOSITION IX. THEOREM.

Right parallelopipeds, having the same altitude, are to each other as their bases.

Let AG, AN be two right parallelopipeds having the same altitude AE; then will they be to each other as their bases; that is,

Solid AG: solid AN :: base ABCD: base AIKL. Place the two solids so that their M surfaces may have the common angle BAE; produce the plane LKNO till it meets the plane DCGH in the line PQ; a third parallelopiped AQ will thus be formed, which may be compared with each of the paral- 1 lelopipeds AG, AN. The two solids AG, AQ, having the same base K AEHD, are to each other as their

altitudes AB, AL (Prop. VIII.); and

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the two solids AQ, AN, having the same base ALOE, are to each other as their altitudes AD, AI. Hence we have the two proportions

Solid AG: solid AQ :: AB: AL;

Solid AQ: solid AN :: AD: AI.

Hence (Prop. XI., Cor., B. II.),

Solid AG: solid AN:: ABXAD: ALXAI.

But ABXAD is the measure of the base ABCD (Prop. IV., Sch., B. IV.); and ALXAI is the measure of the base AIKL; hence

Solid AG: solid AN :: base ABCD: base AIKL. Therefore, right parallelopipeds, &c.

Any two right parallelopipeds are to each other as the prod ucts of their bases by their altitudes.

Let AG, AQ be two right parallelopipeds, of which the bases are the rectangles ABCD, AIKL, and the altitudes, the perpenaiculars AE, AP; then will the solid AG be to the solid AQ, as the product of ABCD by AE, is to the product of AIKL by AP.

Place the two solids so that their surfaces may have the common angle BAE; produce the planes necessary to form the third parallelo

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piped AN, having the same base with AQ, and the same altitude with AG. Then, by the last Proposition, we shall have Solid AG solid AN :: ABCD : AIKL.

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But the two parallelopipeds AN, AQ, having the same base Al L, are to each other as their altitudes AE, AP (Prop. VIII.); hence we have

Solid AN solid AQ:: AE: AP. Comparing these two proportions (Prop. XI., Cor., B. II.), we have

Solid AG solid AQ :: ABCD × AE: AIKL×AP.

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If instead of the base ABCD, we put its equal ABXAD, and instead of AIKL, we put its equal AIX AL, we shall have Solid AG: solid AQ :: AB× AD×AE: AI× AL×AP. Therefore, any two right parallelopipeds, &c.

Scholium. Hence a right parallelopiped is measured by the product of its base and altitude, or the product of its three dimensions.

It should be remembered, that by the product of two of more lines, we understand the product of the numbers which represent those lines; and these numbers depend upon the linear unit employed, which may be assumed at pleasure. If we take a foot as the unit of measure, then the number of feet in the length of the base, multiplied by the number of feet in its breadth, will give the number of square feet in the base. If we multiply this product by the number of feet in the altitude, it will give the number of cubic feet in the parallelopiped. If we take an inch as the unit of measure, we shall obtain in the same manner the number of cubic inches in the parallelopiped.

PROPOSITION XI. THEOREM.

The solidity of a prism is measured by the product of its base by its altitude.

For any parallelopiped is equivalent to a right parallelopiped, having the same altitude and an equivalent base (Prop. VII.). But the solidity of the latter, is measured by the product of its base by its altitude; therefore the solidity of the former is also measured by the product of its base by its altitude.

Now a triangular prism is half of a parallelopiped having the same altitude and a double base (Prop. V.). But the solidity of the latter is measured by the product of its base by its altitude; hence a triangular prism is measured by the product of its base by its altitude.

But any prism can be divided into as many triangular prisms of the same altitude, as there are triangles in the polygon which forms its base. Also, the solidity of each of these triangular prisms, is measured by the product of its base by its altitude; and since they all have the same altitude, the sum of these prisms will be measured by the sum of the triangles which form the bases, multiplied by the common altitude. Therefore, the solidity of any prism is measured by the product of its base by its altitude.

Cor. If two prisms have the same altitude, the products of the bases by the altitudes, will be as the bases (Prop. VIII., B. II.); hence prisms of the same altitude are to each other as their bases. For the same reason, prisms of the same base are to each other as their altitudes; and prisms generally are to each other as the products of their bases and altitudes.

PROPOSITION XII. THEOREM.

Similar prisms are to each other as the cubes of their homol ogous edges.

Let ABCDE-F, abcde-f be two similar prisms; then wil the prism AD-F be to the prism ad-f, as AB' to ab3, or as AF to af.

For the solids are to each other as the products of their bases and altitudes (Prop. XI., Cor.); that is, as ABCDEX AF, to abcde Xaf. But since the prisms are similar, the bases are similar figures, and are to each other as the squares of

their homologous sides; that is, as AB' to ab'. Therefore, we have

Solid FD: solid fd:: AB'x AF: ab xaf.

But since BF and bf are similar figures, their homologous sides are proportional; that is,

whence (Prop. X, B. II.),

Also

AF: af,

AB ab

AB: ab: AF: af".
AF af

AF af.

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Therefore (Prop. XI., B. II.),

ABX AF: ab xaf:: AF: af:: AB': ab'.

Hence (Prop. IV., B. II.), we have

Solid FD: solid fd: AB': ab3:: AF: af".

Therefore, similar prisms, &c.

PROPOSITION XIII. THEOREM.

If a pyramid be cut by a plane parallel to its base, 1st. The edges and the altitude will be divided proportionally. 2d. The section will be a polygon similar to the base.

Let A-BCDEF be a pyramid cut by a plane bcdef parallel to its base, and let AH be its altitude; then will the edges AB, AC, AD, &c., with the altitude AH, be divided proportionally in b, c, d, e, f, h; and the section bcdef will be similar to BCDEF.

First. Since the planes FBC, fbc are parallel, their sections FB, fb with a third F plane AFB are parallel (Prop. XII., B. VII.); therefore the triangles AFB, Afb are similar, and we have the proportion AF Af:: AB: Ab.

For the same reason,

AB: Ab:: AC: Ac,

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