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A superficies, or surface, is an extension of two dimensions, viz. length and breadth.

A plane, or plane superficies, is that with which 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. 7. A circle is a plane figure bounded by a curve line

called the circumference, 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, AC, is called the radius. A semi-circle is a figure contained under a diameter, and

that part of the circumference of a circle cut off by that diaineter, as the line AB divides the circle into

two semi-circles. Fig. 8. A segment is any part of a circle terminated by an

arc, AD B, 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, AB, of a curve, is called concave, and the

raised side, CD, 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 wbich liath length, breadth, and 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 parallel to each other. Fig. 1t. A cylinder is a round solidl, 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 leaving 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, DB,

and two of the opposite sides ending in an acies or edge; EF.

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Fig. 20. A pavilion is a solid contained under fire planes;

the base is a rectangle or oblong, and the four sides terminate in a ridge, E F, parallel to a side of the base, AB, or C D, 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.

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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,

7 parts. (6) Multiply 9 feet, 11 inches, 6 parts, by 11 feet, 8 inches. (7) Multiply 64 feet, 10 parts, by 14 feet, 9 inches. (8) Multiply 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', 4", by 6 feet, 4 in. 8', 6".

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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

acres.

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EXAMPLES. (1) What is the area in acres of a parallelogram whose

length is 14,5 chains, and its breadth 9,75 chains ? (2) What is the area of a square whose side is 245 yards ? (3) How many square yards of paving are there in a court

yard, being in the form of a rhombus or rhomboides, whose length is 64 feet, 6 inches, and perpendicular breadth is 47 feet, 8 inches ?

PROBLEM III.

Case 1. To find the area of a triangle.
RULE.

Fig. 4. 1. Multiply one of its sides by the

perpendicular let fall upon it
from its opposite angle, and
half the product will be the

area.

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2. Multiply the base by half the

perpendicular, or perpendicular,
by half the base, and the pro- bil
dact gives the area.

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