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# Superficial Solid

## length, surface, units and foot

SOLID, SUPERFICIAL, AND LINEAR DIMENSIONS. A solid, a surface, and a line, when they come to be the objects of arithmetic, are things as distinct as a weight and a time. That a surface is included by lines, or a solid by surfaces, makes no more of necessary connection between them than exists between weight and time, because the former can never be made sensible without the latter. Length only can measure length, a surface only a surface, a solid only a solid. Reasons of arithmetical convenience, net of necessity, make it advisable that whatever length may be chosen to measure length, the SQUARE on that length should be the surface by which fturface is measured, and the CUBE on that length the solid by which solidity is measured. Unfortunately, if a foot be the measure of 'length, the square on a foot and the cube on a foot have no other names than square foot and cubic foot. The farmer with his acres, and the distiller with his gallons, have an advantage which is denied to the young mathematician. Ask the first how many acres make a gallon, and the second how many gallons make an acre, and both would laugh at the question ; the third is allowed an indistinct conception of measuring surfaces and solids in feet or inches, as if they were lines, from the occurrence of the same word in all his measures.

Length is said to be a quantity of one dimension, surface of two, and solidity of three. The right line, the right surface or RECTANGLE, and the right solid or rectangular PARALLELOPIPED (the figure of a box, a die, a plank, a beam, &e.), are the implements of mensuration. Every surface must be reduced to the second form, and every solid to the third, before it can be measured. The rules (which tacitly contain these reductions) for measuring different superficial or solid figures will be found under the several beads : the two fundamental theorems by which measurement becomes practicable are as follows : 1. The numbers of linear units in the two sides of a rectangle being multiplied together, give the number of superficial units, square units, or squares on the linear unit, which the rectangle contains. Thus a 5 13 65 rectangle of 24 by 44 feet contains x or , or log square feet.

2. The numbers of linear units in the length, breadth, and thick ness of a right solid, being multiplied together, give the number of solid units, cubic units, or cubes on the linear unit, which the right solid contains. Thus a plank of 21 inches broad, inch thick, and 9 3 31 279 101 inches long, contains 7 x x 3, or or 34 cubic inches.