CHAPTER I
DEFINITIONS OF TRIANGLES, CIRCLE, QUADRILATERALS, ETC.
Tue following has been carefully considered and selected to give the necessary information in a manner that will be easily understood, with- out previous instruction in the subject, as far as directly concerns Plumbers’ Work.
INSTRUMENTS
As there is a great benefit derived from the working of the various problems, and later, in the making of both working and finished drawings, it is perhaps advisable to give a list and a description of what is requisite as regards drawing instruments.
A good drawing board, 30 ins. by 22 ins., of yellow pine framed with hard wood, and with its surface perfectly level and the corners true right angles.
A T-square of suitable length, bound with a bevelled hard wood edge. Two set squares, having angles of 45° and 60° respectively— the first 6 ins. long and the latter 10 ins. long.
The compasses should have firmly fastened needle points. Buy just the instruments you require, i.c., a 44-in. half-set with knee joints and ink and pencil points, and a ruling pen for inking-in lines. It is prefer- able to use a pricker for marking off distances, ete. This can be bought, or a good one can be made by breaking a sewing needle and forcing the blunt end into the wood of a common penholder, leaving the point projecting half an inch. The foregoing instruments may be carried in a roll of wash leather.
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2 TRIANGLES, CIRCLE; QUADRILATERALS, ETC.
SCALES
These are indispensable. A set of paper scales can be obtained for a few pence.
The pencils should be of good quality, and for construction a hard HH or 3H is the best; for lining-in, an H_ pencil may be used. When sharpened, finish to a fine chisel edge with glasspaper.
A protractor (Fig. 1) is an instrument used for measuring angles already drawn, or for drawing an angle of a given degree (see Definitions). It consists either of a circular or semi-cir- cular disc, made in metal or celluloid. It may also be used as a flat rule. The semi-circular celluloid type is a convenient form; it is very thin, has easily read Pas degrees, and can be used with a fair amount of accuracy. Though perhaps not necessary at first, it is almost impossible to work without one in the later stages.
DRAWING PAPER
For ordinary pencil work smooth cartridge paper is the most suit- able; when the drawings have to be inked-in, a better quality is desirable, such as Whatman’s cartridge papers.
DEFINITIONS Axioms and Explanations of Terms, etc.
Def. 1.—A. Point denotes position only. It is shown by a dot, or a dot enclosed in a small circle.
Lines
Def. 2.—A Line has length and position, but neither breadth nor thick- ness. . It is indicated by letters placed at its extremities as A, B. Various methods of drawing lines are used, such as thick, thin, dotted, and chain lines.
Def. 3.—A Straight line is the shortest distance between two given points; it is also called a Right line.
Def. 4.—A Curved line is nowhere straight. There are different kinds of curved lines, such as (a) a simple curve, and (6) a compound curve.
Def. 5.—A Horizontal line is a level line, similar to the surface of still water.
Def. 6.—A Vertical or Perpendicular line is perfectly upright, like a plumb line.
LINES 3
Def. 7.—An Oblique line is neither horizontal nor vertical.
Def. 8.—Parallel lines are the same distance apart, and cannot meet, however far they may be produced.
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Angles
Def. 9.—An Angle is the inclination of two straight lines to each other which meet together; when two straight lines meet at a point AOB, the “corner” they form is termed “an angle.”
Def. 10.—When a line EO meets another CD at a point O, so that the adjacent angles on each side are equal, then that line is said to be perpendicular, and the angles formed are right angles.
‘A Right angle is divided into 90 degrees for the purposes of measuring, ete.
ca TRIANGLES, CIRCLE, QUADRILATERALS, ETC,
In Def. 10, if we continue the line EO below CD, then we have four right angles. Thus it will be seen (if a circle is described with centre O) that a circle contains four right angles, or 360°.
Def. 11.—An Obtuse angle is greater than a right angle. Def. 12.—An Acute angle is less than a right angle.
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The Circle
Def. 13.—A Cirele is a plane figure contained by one line called the “ Circumference,” and is such, that all lines drawn from a certain point within the figure to the circumference are equal to one another,
TRIANGLES 5
Def’ 14.—This point is called the “Centre” (Euc. I. Def. 15, 16).
Def, 15.—A line drawn through the centre and terminating at each end in the circumference, is the Diameter of the circle, and divides the circle into two equal parts, i.c., semi-circles.
Def. 16.—The Radius of a circle is the distance from the centre to any point in the circumference, and a line drawn at right angles to this line touching the circumference is called a Tangent to the circle.
Def. 17.—A Segment of a circle is the figure contained by a straight line and the portion of the circumference which it cuts off.
Def. 18.—An Are is a portion of the circumference of a circle.
Def. 19.—A. Sector is any portion enclosed by two radii and an are, ive, 4 circle =a Quadrant, }=a Sextant, and } circle = an Octant.
Triangles
Any figure formed by straight lines is termed rectilineal.
Def. 20.—Trilateral figures, or triangles, are those which are formed by three straight lines,
Triangles are named (1st) from the comparative lengths of their sides to each other :
Def. 21.—An Equilateral triangle has 3 equal sides,
Def. 22,—An Isosceles ” » 2 of its sides equal,
Def. 23.—A Scalene » 9 8 unequal sides ; and (2nd) from the magnitude of their angles :
Def. 24.—A Right-angled triangle has one of its angles a right angle, ie, 90°. The side opposite the right angle is called the “ Hypo- tenuse.” In Def. 24 AB is the hypotenuse.
Def. 25.—An Obtuse-angled triangle has one obtuse angle.
Def. 26.—An Acute-angled triangle has three acute angles (see Def. 22).
In all triangles the sum of the three angles always equals two right angles, or 180°. Thus, if two angles of a triangle are given, these together, subtracted from 180°, will give the third, or in the case of an isosceles triangle, if the angle at the apex or top be given, the result of this substracted from 180° and afterwards divided by 2 would give the angles at the base :—
180° — Angle at apex (say 40°) _ “SS ee
for if the three sides of a triangle are equal, the angles are equal; and if the two opposite sides are equal then the two opposite angles are also equal.
70° (each angle at base),
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6 TRIANGLES, CIRCLE, QUADRILATERALS, ETC.
Quadrilaterals
Def. 27.—A Quadrilateral figure or Quadrangle is bounded by four straight lines, the four angles equalling four right angles.
Def. 28.—A Payallelogram is a quadrilateral figure in which the opposite sidesare parallel. The Square, Rectangle or Oblong, the Rhombus, and the Rhomboid are parallelograms.
Def. 29.—A Square has all its sides equal and its angles right angles.
Def. 30.—A Rectangle or Oblong has its opposite sides equal and its angles right angles.
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Def. 31.—A Rhombus has all its sides equal, but its angles are not right angles.
Def. 32,—A Rhomboid has its opposite sides equal, but its angles are not right angles.
Def. 33.—A Trapezoid has only two sides parallel.
Def. 34.—A Trapezium (A) has none of its sides parallel, but may have two of its sides equal, as (B). It is then termed a Trapezium or kite,
Polygons
Def. 35.—A and B.A Polygon is a plane figure bounded by more than four straight lines.
If the sides are equal it is termed a regular Polygon. If the sides are unequal it is termed an irregular Polygon.
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POLYGONS 4
Polygons are named according to the number of their sides, viz. :—
A Pentagon has five sides. A’Nonagon has nine sides.
‘A Hexagon has six sides. ‘A Decagon has ten sides.
A Heptagon has seven sides. | An Undecagon has cleven sides.
‘An Octagon has eight sides. A Duodecagon has twelve sides. A
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