This section is from "The American Cyclopaedia", by George Ripley And Charles A. Dana. Also available from Amazon: The New American Cyclopędia. 16 volumes complete..

**Mensuration**, the art of measuring things which occupy space. This is the art which led to the formation of the science of geometry; and some schools of philosophy at the present day are inclined to limit the whole domain of mathematics to the field of mensuration, while extending this field so as to include time as well as space. The art is partly mechanical as well as mathematical, and even in its mathematical part is but the application or illustration of sciences that in their purity have no connection with material things. - There are three kinds of quantity in space, viz., length, surface, and solidity; and there are three di>tinct modes of measurement, viz., mechanical measurement, geometrical construc-tion, and algebraical calculation. For the last two modes arithmetical computation is a necessary adjunct; for the ratio to a unit quantity can be definitely stated in particular cases only as a numerical ratio. Lengths are measured on lines, and the measure of the length of a line is the numerical ratio which the line bears to a recognized unit of length, the inch, foot, or mile, determined in England and in this country by reference to metallic rods three feet long, kept by the governments as standards. The mechanical mode of determining lengths is called direct measurement.

Rods are directly compared with the standard, and accurately made of the same length, and those rods, "rules," or yard sticks, or else tapes and chains accurately graduated by direct comparison with such rules, are stretched side by side with the line to be measured, and the ratio observed. When the line is long and the rule is applied many times consecutively, the slight errors arising at the joining of the successive positions of the rod, being multiplied, become of serious practical importance. In geodesy, therefore, when base lines several miles long are to be accurately determined by direct measurement, an apparatus is used in which bars of different metals counteract each other's expansion and contraction. When the line is long, or when it is inaccessible, the length is usually measured by the second or third mode. - The measurement of a line by geometrical construction is effected by the direct measurement of accessible lines and angles in a figure of which the line to be measured forms a component part, and then drawing this figure upon paper, on a definite scale of a certain number of feet to the inch. The direct measurement of the unknown side upon the paper will evidently give the length of the line represented by it.

Thus, if one ship has sailed 50 miles E., and another from the same port 100 miles 30° E. of S., and we wish to know their distance apart, we may draw a line one inch long and a line half an inch long, making an angle of 60° with each other, and we shall find their extremities separated by .866 of an inch, showing the ships to be 86.6 miles asunder. "We do not include angles among quantities in space. Strictly an angle is a quantity, since it can be measured, and its measurement is necessary at times for the measurement of other quantities. But the measurement of angles is not, in the general use of language, included among the direct objects of mensuration. The measurement of a line by algebraic computation is effected as in geometrical construction, except that, instead of drawing the figure we calculate the length of the unknowm side from the known relations of the sides and angles of figures, and from tables giving numerical values for those relations in right triangles, into which all plane figures can be divided at pleasure.

In practice, it is easier to measure angles with great accuracy than long lines, and hence in geodesy only one base line is actually measured, while all the other distances of the survey are computed from the measurement of the angles in a network of triangles. - The second kind of quantity to be measured is surface. The area of a surface is its numerical ratio to a square surface whose side is a linear unit, that is, to a square foot, square inch, etc. This sort of measurement is never done directly or mechanically, but always by the measurement of lines, and generally by the use of the geometrical propositions, that all surfaces may be resolved into triangles; that all triangles are equivalent to the halves of rectangles having the same base and altitude; and that the area of a rectangle may be found by multiplying the number of units in its length by that in its breadth. The reduction of all surfaces to subjection to these propositions requires sometimes so much labor, that in surfaces of a more intricate form use is made of algebraical laws and of the differential calculus, according to the fundamental idea of fluxions, that a surface is generated by a moving line which constitutes, in two positions, two of the boundaries of the surface.

Thus a circle may geometrically be considered as composed of an unlimited number of triangles with their bases on the circumference and their vertices in the centre; or it may be considered algebraically as generated by a chord sweeping across it, beginning of no length, swelling to a diameter through the centre, and contracting again to zero. Either of these modes of viewing it leads to the same area of the circle, viz., the product of its circumference by half its radius, or, what is the same thing, '78539 of the square enclosing it. - The third species of quantity is solidity. The unit of measurement is here either a cube whose edge is a linear unit, or else it is an arbitrary number of cubic inches selected as a unit, such as the bushel of 2,150 inches, or the gallon of 231 inches. The direct or mechanical measurement of solidity is applied to liquids, or to solids separated into parts so small as to be handled somewhat in the manner of a liquid, as corn, for example, is poured from a basket. This direct measurement consists then in filling a vessel of known capacity with the article to be measured, repeatedly, until all is measured. The geometrical and algebraical modes of measuring solidity will be understood from the analogous modes of measuring lines and areas.

They are principally based on the doctrines, that the solidity of a right parallelopiped is found by multiplying the area of its base by its altitude; that a pyramid has one third the solidity of a parallelopiped of the same base and altitude; and that every solidity can be divided into pyramids and parallelopipeds. But in intricate cases it is easier to use fluxions, and consider the solid generated by the motion of a surface through it; a hemisphere, for example, might be considered as an unlimited number of pyramids with their apices at the centre, or as generated by the circular plane of its base, diminishing as it rose to the summit of the hemisphere, and there becoming a point. Mechanics use arithmetical rules or formulas derived from considerations such as we have here presented. The cask or barrel, for example, is treated as though one of several varieties of geometrical solids, and rules are given for discovering its solidity on those suppositions. The gauge rod is marked with the number of gallons which a cask of certain form would have if its diagonal distance from the centre of the bung to the inner end of the staves were the same as from the end of the rod to the spot where that number is engraved; and thus by thrusting the rod diagonally into the bung hole of an ordinary cask, the number of gallons it contains is readily determined.

The tonnage of ships is computed in the same way by assuming the figure of the ship to be of a certain model, and the tonnage is under- or over-estimated according to its departure from this average form. Many works have been published containing only practical rules without explanation, all essentially alike. In particular cases, ingenuity may devise particular modes for measuring the solidity or the area of very complicated figures; the earliest example is that of Archimedes determining the solidity of Hiero's crown by plunging it into water to discover how much of the fluid it displaced. Another example is Galileo's determination of the area included between a cycloid and its base by describing the cycloid upon a plate of metal, cutting it out, and comparing its weight with that of the generating circle cut out of a similar plate.

Continue to: