ARITHMETIC, GEOMETRY, DRAWING, ETC

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**10.** The object of geometry is to measure the extent of bodies. A body has three dimensions, *length*, *breadth* and *thickness*, and one of these latter is sometimes termed *weight* or *depth*.

Either dimension taken by itself is measured by a straight line.

When the extent of a body is expressed by combining any two dimensions, it is termed *area* or *surface*, and when three are employed we obtain the solid measure or *volume*.

*Plane* geometry only takes cognizance of figures situated in one plane or surface, and therefore only possessing two dimensions; *solid* geometry, however, regards bodies as having all three dimensions.

Two lines are parallel when their distance apart is the same at all points. The same is also applicable to parallel planes.

Two lines or planes meeting each other will form an *angle*. The point at which they meet or intersect is termed the *apex* or *summit* of the angle.

A straight line is *perpendicular* or *at right angles* to another straight line, or to a plane, when all the angles which it makes with that line are equal.

A *circumference* of a circle is a curved line *lcdf* (fig. 1), such that all its points are equally distant from an internal point, *o*, termed the *center*. The *circle* is the space enclosed by the circumference.

It will be noticed that in geometry these two words are distinguished, although they are frequently referred to as identical. Thus, the rim of a wheel or balance is generally termed a circle.

Two circles (*lcdf* and *bra*, fig. 1), described from the same center are said to be *concentric*. When their centers do not coincide they are called *excentric* with regard to one another.

Any portion of a circumference, such as *fnd*, is termed an *arc* of the circumference, or, more commonly an *arc of a circle*.

A *chord* is a straight line, *fd*, which unites the two extremities of an arc. When the chord passes through the center of a circle it is termed a *diameter*.

The *radius* of a circle or circumference is a straight line drawn from the center to the circumference; and all the radii that can be thus drawn are equal. A diameter is, then, always double the radius, and conversely, the radius, is always half the diameter.

A *tangent* is a straight line that only touches a circumference at one point, as *gl* (fig. 1); whereas a secant cuts the circle, as *ij*.

A circumference is assumed to be divided into 360 equal parts, termed *degrees*. the degree is subdivided into sixty equal parts, or *minutes*, and the minute into 60 *seconds*. These are respectively symbolized by the marks ° ′ ″ placed at the right-hand top corner of the figure.

Such and expression as 18° 30′ 15.5″ would, then, be read 18 *degrees*, 30 *minutes*, and 15.5 *seconds*.

**11. Ratio of the circumference to the diameter.** The diameter of a circle is the circumference as 7:22; or, employing decimal fractions, as 1:3.14159 (a number which, in algebra, is always represented by the Greek letter π).

Knowing a diameter (D), the circumference, x, can be ascertained from the proportion:- 1:3.14159::D:x.

Knowing a circumference (c), the diameter, x, can be determined from the proportion:- 3.14159:1::c:x.

The latter proportion will also give the value of the radius, which is half the diameter.

**12.** The *superficial area of a circle* is equal to the circumference multiplied by half the radius, or to the square of the radius multiplied by 3.14159.

A *sector* is the circle enclosed between an arc and two radii bounding it as *bork* (fig. 1).

The area or surface of a sector can be ascertained by multiplying the length of the arc by half the radius.

A *segment* of a circle is the portion intercepted between an arc and its chord, as *fdn* (fig. 1).

The surface of a segment , as *bkrs*, can be obtained by subtracting from the area of the sector *obkr*, the area of the triangle, *bro* (**15**).

**13.** *Ring.* To determine the surface of a flat ring, the area of the inner circle must be subtracted from that of the outer circle; in other words, take the difference between the areas of the two circles that fix the inner and outer diameters of the ring.

The area of a flat ring can also be calculated by adding together the internal and external diameters; then multiplying the number so obtained by their difference and the decimal fraction 0.7854 (that is, 3.14159/4). The product will be the required area.

**14.** *Angles and their measurement.* When two lines meet one another, they form an angle, as we have already seen. If we take the apex as the center of a circle, the number of degrees intercepted between the two straight lines gives a measure of this angle.

The angle measured by a quarter of a circumference, or 90°, is termed a *right angle*.

An *obtuse* angle is greater than a right angle, and an angle that is less is termed an *acute* angle.

**15. Triangles, squares, etc., and their measurement.** The /triangle or plane area enclosed within three straight lines joined two and two together (A, B, C, fig. 2), is said to be *rectangular* when one of its angles is a right angle; it is *equilateral* when the three sides are equal, under which circumstances the three angles are also equal; and *isosceles* when only two sides are of equal length.

The sum of the three angles of a triangle is always equal to two right angles. If only two of the angles are known, it is thus easy to determine the third.

*Similar* triangles are characterized by the fact that their homologous sides (that is, the sides opposite to equal angles) are proportional.

*Peculiarity of the right-angled triangles.* The square described on the longest side, termed the *hypothenuse* (B, fig. 2), is equal to the sum of the squares described on the two other sides. Hence, it follows that, if the lengths of the two shorter sides are known, that of the hypothenuse can be ascertained by extracting the square root of the number formed by adding together the squares formed on these two sides (**5**).

If the hypothenuse is known and one of the shorter sides, the third can be determined by extracting the square root of the number formed by subtracting the square of the known side from the square of the hypothenuse.

The *surface of a triangle* is determined by multiplying one of the sides by half the perpendicular height of the angle opposite to this side.

**16.** The *surface of a square* or an oblong or *rectangle* (*abcd*, fig. 3) is equal to the product of the base multiplied by the height.

The sum of the squares described on the four sides is equal to twice the square described on a diagonal. This diagonal divides the rectangle into two equal rectangular triangles.

**17.** The *surface of a parallelogram or lozenge,* a plane figure with four sides, opposite pairs of which are parallel (*cfgd* and *cijd*, fig. 3),is equal to the product of one side multiplied by the perpendicular height of the figure.

The sum of the squares described on the four sides of a parallelogram is equal to the sum of the squares described on the two diagonals.

**18. Measures of various solid bodies.** The volume of a cube of parallelopiped (that is, a body bounded by six four-sided figures, every opposite two of which are parallel) is obtained by by multiplying the surface of the base by the height.

The *volume* of a straight cylinder is the product of the surface of the circle which forms its base into the height of the cylinder.

The *area* of the curved surface of a cylinder is obtained by multiplying the circumference of the circle forming its base by the height.

The *volume* of a tube or cylindrical ring of rectangular section, such as the arbor-nut of a barrel, or the rim of a circular balance, etc., is equal to the product of the plane surface of its base (**13**) into its height.

The *surface* of a *sphere* may be determined by multiplying the square of the diameter by 3.1416 (**11**).

The *volume* of a sphere is equal to this surface multiplied by a third of the radius.

Submitted by: Samuel Kirk (##)