THEOREM X.

774. Tetrahedra (triangular pyramids) having equivalent bases and equal altitudes are equivalent.

Divide the equal altitudes a into n equal parts, and through each point of division pass a plane parallel to the base.

By 773, all sections in the first tetrahedron are triangles equivalent

to the corresponding sections in the second.

Beginning with the base of the first tetrahedron, construct on each

a

section, as lower base, a prism high, with lateral edges parallel to one

of the edges of the tetrahedron.

n

In the second, similarly construct prisms on each section, as upper base.

Since the first prism-sum is greater than the first tetrahedron, and the second prism-sum is less than the second tetrahedron, therefore the difference of the tetrahedra is less than the difference of the prism-sums.

But, by 771, each prism in the second tetrahedron is equivalent to the prism next above it on the first tetrahedron.

So the difference of the prism-sums is simply the lowest prism of the first series. As n increases, this decreases, and can be made as small

as we please by taking n sufficiently great; but it is always greater than the constant difference between the tetrahedra, and so that constant difference must be nought.

THEOREM XI.

775. A triangular pyramid is one-third of a triangular prism of the same base and altitude.

Let E ABC be a triangular pyramid. Through one edge of the base, as AC, pass a plane parallel to the opposite lateral edge, EB, and through the vertex E pass a plane parallel to the base. The prism ABC DEF has the same base and altitude as the given pyramid. The plane AFE cuts the part added, into two triangular pyramids, each equivalent to the given pyramid; for E ABC and A DEF have the same altitude as the prism, and its bottom and top respectively as bases; while E AFC and E AFD have the same altitude and equal bases.

BOOK XI.

MENSURATION, OR METRICAL GEOMETRY.

CHAPTER I.

THE METRIC SYSTEM. LENGTH, AREA.

776. In practical science, every quantity is expressed by a phrase consisting of two components, one a number, the other the name of a thing of the same kind as the quantity to be expressed, but agreed on among men as a standard or Unit. 777. The Measurement of a magnitude consists in finding this number.

778. Measurement, then, is the process of ascertaining approximately the ratio a magnitude bears to another chosen as the standard; and the measure of a magnitude is this ratio expressed approximately in numbers.

779. For the continuous-quantity space, the fundamental unit actually adopted is the METER, which is a bar of platinum preserved at Paris, the bar supposed to be taken at the temperature of melting ice.

780. This material meter is the ultimate standard universally chosen, because of the advantages of the metric system

of subsidiary units connected with it, which uses only decimal multiples and sub-multiples, being thus in harmony with the decimal nature of the notation of arithmetic.

781. The metric system designates multiples by prefixes derived from the Greek numerals, and sub-multiples by prefixes from the Latin numerals.

The abbreviation for meter is m.; hence km. for kilometer, mm. for millimeter.

782. The adoption of the meter gives the world one standard sect as fundamental unit.

783. The Length of any sect is its ratio to the meter expressed approximately in numbers.

Men of science often express their measurements in terms of a subsidiary unit, the Centimeter.

The length of a sect referred to the centimeter as unit is one hundred times as great as referred to the meter.

784. An accessible sect may be practically measured by the direct application of a known sect, such as the edge of a ruler suitably divided.

But because of incommensurability, any description of a sect in terms of the standard sect must be usually imperfect and merely approximate.

Moreover, in few physical measurements of any kind are more than six figures of such approximations accurate; and that degree of exactness is very seldom obtainable, even by the most delicate instruments.

785. For the measurement of surfaces the standard is the square on the linear unit.

786. The Area of any surface is its ratio to this square.

787. If the unit for length be a meter, the unit for area, a square on the meter, is called a square meter (m.2; better, m.2). In science the Square Centimeter (cm.) is adopted as the primary unit of surface.

PROOF. SPECIAL CASE: When the base and altitude of the rectangle are commensurable. In this case, there is always a sect which will divide both base and altitude exactly. If this sect be assumed as linear unit, the lengths a and b are integral