A superficies, or surface, is an extension of two dimensions, viz. length and breadth. A plane, or plane superficies, is that with wbich a right line may every way coincide. A plane superficies receives several denominations, according to the number and positions of the lines by which it is terminated; as follow: Fig. 1. A square is a right-angled equilateral parallelogram, whose four sides are equal, and its angles all right ones. A quadrangle is a figure made by four straight lines. Fig. 2. A parallelogram is a quadrangle whose opposite sides are parallel. An oblong, or rectangle, is longer than broad; but its op posite sides are equal, and all its angles right ones. A rhombus, or diamond figure, is a parallelogram whose sides are all equal, but its angles are not right angles. Fig. 3. A rhomboides is an oblique-angled parallelogram, whese opposite sides and angles only are equal. A triangle is a space included by three lines, and of conse quence has three angles; for every rectilineal plane figure has as many angles as sides. A right-angled triangle is that which has one right-angle, as Fig. in page 174, Fig. 4. An equilateral triangle is that whose three sides are all equal to each other. An isosceles triangle is that which has only two of its sides equal to one another. A scalene triangle is that which has all its sides un equal. An obtuse-angled triangle is that which has an obtuse angle. An acute-angled triangle is that which has every angle acute. Fig. 5. A trapezium is a quadrangle, whose opposite sides are not parallel. All right-lined figures, having more than four sides, are called polygons, and receive their names from the num ber of their sides or angles, Fig. 6. Having five sides or angles, is called a pentagon. A regular polygon is a figure with equal sides and equal angles. Fig. i. A circle is a plane figure bounded by a curve line called the circunference, every part whereof is equally distant from a point within, called the centre. A diameter, A B, of a circle, is a right line drawn through the centre, and terminated by the circum ference. The semi-diameter, A C, is called the radius. A semi-circle is a figure conained under a diameter, and that part of the circumference of a circle cut off by that diameter, as the line AB divides the circle into two semi-circles. Fig. 3. A segment is any part of a circle terminated by an arc, A DB, cut off by the line AB, called the chord. Fig. 9. A sector of' a circle is a part contained between two right lines or semi-diameters, and the intercepted arc of the circumference. Fig. 10. Represents the front of an arch built with stones of equal length, and is a segment of a sector, The hollow side, A B, of a curve, is called concave, and the raised side, C D, convex. Fig. 11. An ellipsis, or oval, is a figure bounded by a regular curve line, returning into itself, but its two axes cutting each other in the centre; one of which is longer (called the transverse axis) than the other (called the conjugate axis). A solid is that which bath length, breadth, an: thickness. Fig. 12. A cube is a solid bounded by sis equal squares. Fig. 13. A prism is a solid whose sides are parallelograms, and whose two ends are paraliei to each other. Fig. 1+. A cylinder is a round solid, like the rolling-stone of a bowling-green, whose two ends are equal and parallel circles. Fig. 15. A pyramid is a solid, whose base is a polygon, or right-lined figure, and whose sides, or triangles, meet in a point, C, called the vertex. Fig. 16. A cone is a round pyramid, or pyramid fiaving a circular base, in form like a sugar-loat. Fig. 17. 18. A frustum of a pyramid or cone is that part which remains, when any part nest the vertex is cut off by a plane parallel to the base. Fig. 19. A wedge is a solid, having a rectangular base, D.B, and two of the opposite sides ending in an acies or edges, EF. Fig. 20. A pavilion is a solid contained under five planes; the base is a rectangle or oblong, and the four sides terminate in a ridge, & F, parallel to a side of the base, AB, or CD, but unequal to it. Fig. 21. A prismoid is a solid contained under six planes; the bases, or ends, are parallel rectangles, and the four sides are quadrangles. Fig. 22. A sphere is a solid bounded by a convex surface, every point of which is equally distant from a point C, within, called the centre. The axis, or diameter of a sphere, is the right line A B. Fig. 23. A segment of a sphere is a part cut off by a plane A B. If the plane pass through the centre of the sphere, it will cut it equally in two, and each half is called a he misphere. Fig. 24. A spheroid is a solid resembling an egg, and is the body conceived to be generated by the revolution of an ellipse about its axis, and is denominated either prolate (oblong) or oblate, according as the revolution is made about the transverse axis or its conjugate. The axis about'which the revolution is made is the fixed axis, the other is the revolving axis. Fig. 25. A parabolic spindle is eight-fifteenths of its circum scribing cylinder. Fig. 26. Is the middle frustum of a spheroid. LXXV. SUPERFICIAL MEASURE. PROBLEM I. To multiply feet, inches, and parts, by feet, inches, and parts, which method is termed cross multiplication, but more properly duodecimals. RULE. Set the feet in the multiplier under the least denomination in the multiplicand, and the rest in order, beginning with the least denomination ; divide each product by 12, as you go on; place the first remainder under the multiplying figure, and the rest in order, adding each quotient to the next arising product, as in Sect. IX. and having finished the multiplication, the sum of all will be the product required. In general, thus; When feet are concerned, the produet is of the same denomination with the term multiplying feet. When feet are not concerned, the name of the product will be expressed by the sum of the indices of the two factors. EXAMPLES, (1) Multiply 17 feet, 7 inches, by 6 feet. (2) Multiply 47 feet, 8. inches, by 8 feet, 4 inches. (3) Multiply 7 feet, 10 inches, by 8 feet, 6 inches. (4) Multiply 64 feet, 7 inches, by 4 feet, 8 inches. (5) Multiply 12 feet, 8 inches, 9 parts, by 9. feet, 6 inches, parts. (6) Multiply 9 feet, 11 inches, 6 parts, by 11 feet, 8 inches. (7) Multiply 6+ feet, 10 parts, by 14 feet, 9 inches. (8) Multiplý 124 feet, 4 inches, by 42 feet, 9 seconds. (9) Multiply 16 feet, 7 inches, 10 parts, by 6 feet, 5 inches, 7 seconds. (10) Multiply 474 feet, 6 inches, 8 seconds, by 186 feet, 7 inches, 4 seconds. (11) Multiply 2 feet, 11', 8", 6", 7"", by 8 feet, 6', 7", (12) Multiply 46 feet, 6 in. 8', 1", by 6 feet, 4. in. 8', 6". PROBLEM II. To find the area of a parallelogram, whether it be a square, a rectangle, a rhombus, or a rhomboides. RULE. Multiply the length by the height or perpendicular breadth, and the product will be the area. If the area of a piece of ground in yards, be divided by 4840 (the number of square yards in one acre) the quotient will give the number of acres in the said piece.Org If the. area in links be divided by 100,000 (the number of square links in one acre), the quotient will give EXAMPLES. (1) What is the area in acres of a parallelogram whose length is 14,5 chains, and its breadth 9,75 chains ? yard, being in the form of a rhombus or rhomboides, PROBLEM III. 1 Case 1. To find the area of a triangle. Fig. 4. 1. Multiply one of its sides by the perpendicular let fall upon it half the product will be the 2. Multiply the base by half the perpendicular, or perpendicular area. a |