Scalene Triangles have all three sides of different lengths (age or aic), and for proof that the sides in those triangles are unequal, see Sect. 31 post, and in these triangles the angles are also unequal.

11. As regards the variation in angles.

Acute-Angled Triangles have every angle less than a right angle (ade, agc).

Right-Angled Triangles have one of the sides perpendicular to another side, thus giving a right angle (ahc).

Obtuse-Angled Triangles have one of the angles greater than a right angle (aci).

The right-angled triangle in the figure will also be scalene, the sides being of different lengths, but making sides ch and ca equal would make the triangle also isosceles, such a triangle only requiring two equal sides without regarding the extent of the included angle.

12. All the angles in any triangle are together equal to two right angles. (For demonstration see Div. C Sect. 10.) There cannot be more than one right or obtuse angle in a triangle.

13. In a right-angled triangle («hc, Fig. ▲ 1), the side opposed to the right angle (ah) is called the Hypothenuse, and the two other sides (ac and ch) the Base and Perpendicular.

In other triangles the point of the angle opposed to the side taken as base is termed the Apex or Summit of the triangle (bdgi).

14. A figure of four sides is called by the general term of a Quadrilateral, whatever may be the proportions of the

sides or the angles, and different quadrilaterals are thus distinguished.

15. Those which have both pairs of opposed sides parallel are included in the general term "Parallelograms," and include the following figures :—

Square. All four sides equal and four right angles (Fig. C 13).

Rectangle. Opposed sides (but not all four sides) equal, and four right angles (C 11).

Rhombus. Four equal sides, but not right angles (C 12, abcd).

Rhomboid. Opposed sides equal, but not right angles (C 12, bnmc).

Other four-sided figures include

Trapezoid. Only two parallel sides, with any angles (C 30, abnd).

Trapezium. Not any sides parallel, with any angles (C 30, abcd).

See Hutton's Mathematical Dictionary, v. 2, p. 610.

16. Plane figures of more than four sides (and perhaps, in strictness, any enclosed spaces) may be termed Polygons, and are named according to the number of their sides or angles, and are called regular if all the sides and angles are equal, and irregular if not so equal.

Pentagon, figure of five sides; Hexagon, six sides; Heptagon, seven sides; Octagon, eight sides; Nonagon, nine sides; Decagon, ten sides; Undecagon, eleven sides; Duodecagon, twelve sides, or with more sides, as a Polygon, with the named number of sides.

The several forms are given in the figures referred to in Div. B, where instructions for drawing the several figures are given (Sectns. 46 to 54).

17. Height or Altitude of a figure means the perpendi cular and not the slant height, unless otherwise expressed. Thus in Fig. 1 A od and not ad will be the height or altitude of triangle adc.

18. A Diagonal is a line drawn from one angle of a right-lined figure to another and opposed angle. Thus in Fig. C 9 and 10 the lines bd and ca are severally diagonals.

19. A Circle is a plane figure, bounded by a curved line called the Circumference, which is in all parts equidistant from a point within called the Centre. Fig. C 1, abcd being circumference of the smaller circle, and silp of the larger, and f the centre of both.

Diameter is a line passing through the centre, and bounded at both ends by the circumference (ac or sl). Radius is half diameter (af or sf).

Arc, any part of circumference (ik, il, &c.).

Chord, a right line connecting the two extremes of an arc (P p).

Segment, any part of a circle bounded by an arc and chord (P p P).

Semicircle, a segment of which the chord passes through the centre, and is thus a diameter (abca).

Sector, the space contained between two radii and their intercepted arc (kflk).

Quadrant, a sector containing one-fourth of a circle, being a right angle (ifli).

Sextant, a sector containing one-sixth of a circle.

Zone of a circle or any curvilineal figure, the space bounded by two parallel chords and their intercepted arcs (sljPs).

Concentric Circles have the same centres but different lengths of radii (abcd, silp).

Eccentric Circles have different centres (Fig. D 20, 21).

Circular Ring, the figure contained between two concentric circles (Fig. C 1, space between the larger and the smaller circle).

Lune, or Lunule, the figure formed by two arcs of circles having different radii (Fig. C 1, space between arc sil and dotted arc svl, radius of former being fs, and of latter ps, the common chord being in this case sl, diameter).

Tangent to a circle or any other curve is a right line which touches it, but only at one point (mn).

Secant, a line which cuts a circumference (Fig. D 23, ag, af, &c.)

Inscribed Figures, those in which the sides form chords in a circle (B 27, ghik).

Circumscribed Figures, those in which the sides form tangents of a circle (B 27, abdc).

20. For comparing the arc of an angle with a circumference, the latter is taken to be divided into 360 equal parts called Degrees, each degree into 60 equal parts called Minutes, and each minute into 60 equal parts called Seconds; a right angle containing 90 degrees, and a semicircle 180 degrees.

Degrees are marked °, minutes, seconds".

These divisions are, however, of little importance as regards ordinary geometrical operations, though of continuing use in trigonometry.

21. An Ellipse is a plane figure made by cutting a cone (a figure with a circular base, and terminating in a point) by a plane obliquely through the opposite sides of it, and when represented on a plane surface differs from a circle, in the contour or boundary line, being proportioned to cross axes of unequal length instead of the equal radii of a circle (see B, Fig. 49 to 51, and Sectns. 64, &c., of that Division).

An Oval differs from an ellipse in being shaped like an egg, and having one end more tapering than the other; and whilst the minor or perpendicular axis is equal in each case, the semi-major axis at the broader end is shorter, and at the narrower end is longer (B, Fig. 51, and Sectns. 67, &c.; F, Sect. 32).

22. Symbols or characters generally employed in geometrical operations:

+ (plus), or addition, indicates that the numbers between which it is placed are to be added together. Thus, 5+7 means that 5 and 7 are to be added together.

(minus), indicates that the number placed after it is to be subtracted from that placed before it; thus, 7 – 5 means that 5 must be subtracted from 7. The figures might be reversed, but the result would be a negative or minus quantity, being -2.

× indicates the product or multiplication of numbers; thus, 7 x 5 means 7 multiplied by 5. The same is generally understood with letters having a dot between them; thus, A.B, means A and B are to be multiplied together.

÷ indicates a division of the number which precedes it by that which follows; thus, 15÷8 means 15 divided by 8, and is often expressed by the fraction 15.