4

APPENDIX,

CONTAINING

AN INTRODUCTION TO MENSURATION.*

DEFINITIONS.

I. AN ANGLE is the mutual inclination of two straight lines that meet one another in a point, which is called the VERTEX of the angle: or it is the degree of their opening or divergence.

Right Angle.

Right Angle.

II. When one straight line standing on another, makes with it two angles which are equal, each of these angles is called a RIGHT ANGLE; and the straight line which stands on the other is said to be PERPENDICULAR, or AT RIGHT ANGLES, to it, or to be A PERPENDICULAR to it. III. An OBTUSE ANGLE is greater than a right angle: an ACUTE ANGLE is less than a right angle.

B

Obtuse Angle,

A

Acute Angle.

An angle is usually named by three Dletters, the middle one being placed at the vertex, and the other two somewhere on the lines which contain the angle. Thus, the obtuse angle in the last diagram, may be called the angle ACB, or BCA, and the acute one the angle ACD, or DCA. When there is only one angle at the same point, it is best named by a single letter placed at that point.

*The small introductory tract on mensuration which is here given, is intended for the use of such pupils as may not have time or opportunity for studying a more extended course. For this reason, the easiest and most useful parts are selected, and the definitions and illustrations are delivered in plain and familiar terms, rather than with a view to mathematical precision. The rules are also given without demonstrations, as the pupils for whom this abstract is intended are not supposed to have read a preparatory course of mathematics. Those who may wish to prosecute the subject more extensively, may have recourse to any good modern treatise on the subject.

IV. PARALLEL STRAIGHT LINES are those which have everywhere equal perpendicular distances from each other.

V. A SURFACE, or SUPERFICIES, has length and breadth without thickness.

VI. A BODY, or, as it is often called, a SOLID, has length, breadth, and thickness, or depth.

VII. A FIGURE is a portion of space inclosed by one or more boundaries.

VIII. A FIGURE is said to be EQUILATERAL, if it have equal sides; and EQUIANGULAR, if it have equal angles.

IX. If a figure be contained by three lines, it is called a TRIANGLE; if by four, a QUADRILATERAL; if by more than four, a POLYGON.

X. An equilateral and equiangular polygon is often called a REGULAR POLYGON.

XI. Polygons of five, six, seven, eight, nine, ten, eleven, and twelve sides, are often called, respectively, PENTAGONS, HEXAGONS, HEPTAGONS, OCTAGONS, ENNEAGONS, or NONAGONS, DECAGONS, HENDECAGONS, and DODECAGONS.

XII. A quadrilateral which has its opposite sides parallel, is called a PARALLELOGRAM: and a parallelogram which has its angles right angles, is called a RECTangle.

XIII. A quadrilateral which has its sides equal, and its angles right angles, is termed a SQUARE; and a quadrilateral which has its sides equal, but its angles not right angles, is called a RHOMBUS.

XIV. A quadrilateral which has two sides parallel, and the other two not, is called a TRAPEZOID.

XV. Any quadrilateral, except a parallelogram or trapezoid, is termed a TRAPEZIUM.

XVI. A DIAGONAL of a figure is a straight line passing through two of its angles which are not adjacent to one another.

XVII. A CIRCLE is a figure contained, on a flat surface, by one line which is called the CIRCUMFERENCE; and is such that all straight lines drawn to the circumference from a certain point within the figure, called the CENTRE, are equal to each other. Any of those equal lines is called a RADIUS: and a line drawn through the centre, and terminated both ways by the circumference,

is called a DIAMETER.

double of a radius.

Hence, a diameter is evidently

XVIII. If the ends of a thread, FPf, be fastened to two pins, F, f, fixed at a less distance asunder than the length of the thread, and if the point of a pen or

D

P

A

F

C

E

B

pencil, P, be carried round in such a manner as to keep the thread constantly stretched, the curve line thus described, is called an ELLIPSE; the points F and f, where the pins are fixed, are called the FOCI (and each of them a FOCUS); the line AB, drawn through the foci, and terminated both ways

by the curve, is called the GREATER AXIS; and the line DE, drawn at right angles to this axis through its middle point, and terminated by the curve, is called the LESS AXIS.

XIX. MENSURATION is the method of determining by computation the comparative magnitudes of figures; and it is divided into two great branches, the mensuration of surfaces, and the mensuration of bodies or solids.

XX. The AREA of a surface is the space which it contains. In mensuration, the magnitude of this space is ascertained by the number of times that a given space, called the measuring unit, is contained in it.

XXI. The MEASURING UNIT which is adopted for surfaces, is a square whose side is some of the common measures of length, such as a square inch, a square foot, a square yard, &c.-(See the table of square measure, page 59.)

MENSURATION OF SURFACES.

RULE I. To find the area of a parallelogram: Multiply its length by its perpendicular breadth.

Hence, the area of a square is found simply by multiplying a side by itself.

A

B

Here the product of 36 and 15 is 540, the area in square feet.