EXPLANATION OF CHARACTERS USED IN THIS WORK. There are various characters or marks used in Arithmetic, to denote several of the operations and propositions, the chief of which are as follows: = Equal, ...... The sign of Equality; as, 100 cents = $1, signifies that 100 cents are equal to one dollar. - Minus, or Less, The sign of Subtraction; as, 8-2 6: that is, 8, less 2, is equal to 6. + Plus, or More, The sign of Addition; as, 4+5=9: that is, 4, added to 5, is equal to 9. × Multiply by,.. The sign of Multiplication; as, 6x6 =36: that is, 6, multiplied by 6, is equal to 36. Divided by,... The sign of Division; as, 12÷3=4: that is, 12, divided by 3 is equal to 4. The signs of proportion; as, 2 :4 ::8:16: that is, as 2 is to 4, so is 8 to 16. : is to :: so is : to } ...... Shows that the difference between 7 and 2, added to 5, is equal to 10. Signifies that the square of 3 is required; as, 3×3=9. Signifies that the cube or third power of 4 is required; as, 4×4×4=64. Prefixed to any number, signifies that the square root of that number is required; as, 9=3. Prefixed to any number, signifies that the cube root of that number is required; as, 64=4. 3 MENSURATION OF SUPERFICIES AND SOLIDS. SECTION I. MENSURATION OF SURFACES. ART. 1. MENSURATION is that branch of Mathematics by which we ascertain the contents of superficial areas; the extension, solidities and capacities of bodies; and the lengths, breadths, &c. of various figures, either collectively or abstractly. The Mensuration of Solids is divided into two parts : II. The mensuration of their solidities. In order to form correct estimates of the extent of surfaces and solids, various rules have been adopted, most of which, the most valuable and useful in practice, will be found accompanying their respective problems in the following work. The surfaces or capacities of regular solids are readily calculated, but more intricacy attends the calculations of the surfaces and solid contents of many irregular bodies, as of frustrums, of pyramids, cones, &c. With the following treatise before him, the student or the mechanic may speedily perform all the calculations that ordinarily occur in the practical details of his business. Although Mensuration involves a knowledge of the elements of Geometry, yet it is not the object of this work to treat of that science at large. We shall therefore confine our exercises in this treatise to those measurements which will be most likely to prove beneficial to various classes of society, and especially to the operative mechanic, in the ordinary details of his business. DEFINITIONS. ART. 2. The following definitions, which are similar in substance to those found in Euclid, are here inserted for the convenience of reference, and to assist those who may be ignorant of Geometry in acquiring some knowledge of that science. I. Four-sided figures are variously named, according to their relative position and length of their sides. 2. A Rectangle, or Right Paral-D lelogram has its opposite sides equal, and all its angles right angles; as ABCD, (fig. 2.) C A 3. A Square is a figure whose sides are of equal length, and all its angles right Angles; as, ABCD, (fig. 3.) 4. A Rhomboid has its opposite sides equal, and its angles oblique; as, ABCD, (fig. 4.) II. When figures have more than four sides, they are classed under the head of Polygons. These again receive other particular names, according to the number of their sides or angles. A regular Polygon has all its sides and angles equal. A Pentagon is a regular Polygon of five sides; a Hexagon has six sides; a Heptagon has seven sides; an Octagon has eight sides; a Nonagon has nine sides; a Decagon has ten sides; an Undecagon has eleven sides; and a Dodecagon has twelve sides. III. A figure of three sides and angles is called a Triangle, and receives particular denominations from the relations of its sides and angles. 1. An equilateral triangle is that whose three sides are equal; as, ABC, (fig. 7.) 2. The height of a triangle is the length of a perpendicular drawn from one of the angles to the opposite side; as, Cp, (fig. 7.) The height of a four-sided figure is the perpendicular distance between two of its parallel sides; as, Dp, (fig. 1.) IV. The area, or superficial contents of any plane figure, is the measure of the space contained within the lines by which the figure is bounded. In calculating the area or the contents of any plane figure, some particular portion of surface is fixed upon as the measuring unit, with which the figure is to be compared. This is commonly a square, the side of which is the unit of length, being an inch, or a foct, or a yard, or any other fixed quantity, according to the measure peculiar to different artists. The same holds true, also, when the figure is a square. So, the area of the rhombus or rhomboides is equal to that of a parallelogram of the same base and altitude. Hence, the square foot, yard, &c., may be of any shape whatever, provided the foot contains 144 squares, each 1 inch square, and the yard 9 squares, each 1 foot square. And hence, the area or quantity of surface contained in a figure, is said to be so many square inches, square feet, or square yards. |