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CONTENTS.

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PROPOSITION

PART III.-VOLUMES.

I. Two rectangular parallelopipeds having equal bases
are to each other as their altitudes.......

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III. Any two rectangular parallelopipeds are to each other as the products of their dimensions....

IV. Volume of rectangular parallelopiped..

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II. Two rectangular parallelopipeds having equal alti-
tudes, are to each other as their bases........

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EXERCISES (8)..............

V. Volume of a prism.................

VI. Volume of a right cylinder

VII. Volume of a ring...

EXERCISES (9).

VIII. Volume of triangular pyramid, or pyramid on any

base........

IX. Volume of frustum of pyramid..

X. Volume of cone....

XI. Volume of frustum of cone.

XII. Volume of wedge......

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EXERCISES (10)..........................

XIII. Lemma. If a triangle revolves round an axis situated in its plane, and passing through the vertex without crossing its surface, the volume generated is equal to the surface generated by the base multiplied by one-third of the altitude.....

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XIV. Volume generated by polygonal sector................

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XV. Volume of sector revolving round one of its sides....... 104

XVI. Volume of sphere.

105

XVII. Volume generated by a circular segment

107

XVIII. Volume of spherical segment...........

XIX. Determination of radius of a solid sphere...........

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MENSURATION

OF

LINES, SURFACES, AND VOLUMES.

MENSURATION is that branch of mathematical science which treats of the measurement of lines, surfaces, and volumes.

To measure a magnitude of any kind, is to find how often it contains another magnitude of the same kind called the unit; thus, to measure a surface, is to find how often it contains the unit of surface; similarly, to measure a volume, is to find how often it contains the unit of volume. In measuring such magnitudes, however, we do not do so by directly applying to the magnitude considered the unit magnitude; geometry affords us the means of deducing their measures, from the length of one or more lines connected with them.

The number which expresses how often a magnitude contains the unit, is called the numerical measure of that magnitude.

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