80. The Vertex of an angle, is its angular point; that is, the point where the legs of the angle meet. That of a figure, the uppermost angular point above the base.

81. When two lines cross each other, the opposite angles they make are called Vertical angles; as AED, and CEB. Prob. VIII.

82. Concentric figures are such as have the same

centre.

83.

Eccentric, or Non-concentric figures, are such as have different centres; hence if one circle be within another, the circumferences not being parallel, they are eccentric circles.

84. Equal plane figures are those whose areas are equal; thus, a circle is equal to a polygon, if the two have equal areas.

85.

E

Similar figures are such as have all the angles of the one equal to all the angles of the other, each to each, and the sides about the equal angles proportionals. The Irregular Hexagon ABCDEF, is similar to the corresponding irregular hexagon abcdef, and the triangles in the one figure

F

B

are respectively similar to the corresponding triangles in the other.

86. Homologous sides are the corresponding sides of similar figures, and are always proportional to one. another; thus, A B is homologous to a b, BC to bc, &c., because AB bears the same proportion to B C, as ab does to bc. In Arithmetic, homologous quantities are the antecedents or consequents of ratios, thus, 3 is to 4, as 6

is to 8. In this example, 3 and 6 are the antecedents, and 4 and 8 the consequents; therefore, 3 and 6 or 4 and 8 are homologous quantities.

Note. A Square is not similar to an oblong, although the figures are equiangular: neither is a rhombus similar to a rhomboid, if even they are equiangular.

The angles EDC and edc are called re-entering_angles. All the other angles are called salient angles. Vide Def. 72.

87. Identical figures are those which are both similar and equal.

88. An Ellipse is a section of a cone cut by a plane obliquely to its axis.

89. The Ellipse has two diameters, the longest of which is called its transverse, and the shortest, its conjugate. Vide Index for the word Ellipse.

90. One right-lined figure is said to be Circumscribed about another, or the latter is Inscribed in the former; when all the vertices of the inner figure, touch the sides of the outer one.

91. Aright-lined figure is Inscribed in a circle, or the circle Circumscribes it, when all the vertices of the former, touch the circumference of the circle.

92. A right-lined figure Circumscribes a circle, or the circle is Inscribed in it, when all the sides of the former are tangents to the circle.

93. If one figure be circumscribed about another, it is also said to be Described about it.

94. If a triangle be inscribed in a circle, either of its angles is In one segment, and stands On the other segment by which the said angle is subtended.

95. If two radii be drawn in a circle, the angle contained by them is called the angle at the Centre of the circle; and if from the points where these radii touch the circumference, two lines be drawn to any other point in the circumference, the angle contained by them is called the angle at the Circumference; and this angle contains exactly half the number of degrees of the angle at the centre.

96. The Sum of lines or planes is the quantity produced by addition. A line 3 inches long is equal to the sum of two other lines of 1 inch and 2 inches long; also a triangle whose area is three square inches, is equal to the sum of two other triangles, whose areas square inch and 2 square inches.

are 1

97. The Product of two lines is a rectangle, having one of the given lines for its base, and the other for its height.

98. A Multiple of a line or figure, is another line or figure that is exactly 2, 3, 4, or any other number of times as large as the first line or figure.

99. A Measure of a line or figure, is any line or figure, which, being applied to the first, would divide it into any number of parts, each equal to the said

measure.

100. To Transform a figure, is to change it into another figure of the same superficial or solid content; thus, if a square be made equal in area to a given triangle, the first is said to be a Transformation of the second. In Arithmetic, the word employed to express transformation, is Reduction.

101. The word Intersection is used when two lines cut each other, and the place where they cross is called the Point of intersection.

102. The word Bisect, signifies to divide into two equal parts, and Trisect to divide into three equal parts. 103. To Produce or Prolong a straight line is to lengthen it in the same straight line.

104. A Generatrix is that by which something is generated.

Thus, to give motion to a point, it becomes a generatrix, and a line is the result. In like manner a line may be said to generate a plane; and a plane, a solid. In these operations the generatrix or Generatrice is the generating element, and the thing generated is called the Generant.

105. A Directrix or Dirigent, is the line of motion, along which a line or plane, called a Describent, may be conceived to move, to describe a plane or solid.

The words describe, construct, make, and draw, are frequenty used in practical geometry, in one and the same sense.

The word equal is used when comparing two or more things whose magnitudes are of the same value, and in this sense it is employed in Euclid's Elements; but some eminent Mathematicians prefer the word equivalent, except where the things in question are precisely alike, in which cases only they employ the word equal.

When two lines or figures are said to coincide, it is meant that they agree in every respect.

The word without signifies outside.

A Problem is something proposed to be done.

A Geometrical representation of a building or other solid, when seen vertically, supposing the eye to be stationed at an infinite distance, is called a Plan. If the eye be conceived to be situated at an infinite distance horizontally, the drawing is then called an Elevation. If a solid be conceived to be cut by a plane passing through it in any direction, at right angles to the line of vision; a drawing of the internal parts thus exposed is called a Section.

The plural number of radius is radii; of superficies, superficies; of generatrix, generatrices; of directrix, directrices; of vertex, vertices; of rhombus, rhombi; of ellipse or ellipsis, ellipses; of focus, foci; and of trapezium, trapezia. The plural of other names is made by adding the letter s to the singular; as, tangent, tangents, &c.

A degree is divided into sixty equal parts, called minutes; these into sixty equal parts, called seconds; and these also into sixty equal parts, called thirds.

The Definitions and terms being understood, the student may turn his attention to drawing geometrical figures; but to do them neatly and accurately, mathematical instruments well finished are indispensable. To obtain these, application should be made to a respectable maker. Good second-hand instruments are sometimes to be met with at a cheap rate; but the novice should never purchase any, except at the recommendation of a competent judge. If badly filed, or if the points be not tempered steel, they will be of little or no service. The smallest number that can be available, must contain a pair of compasses with shifting leg; pen and pencil legs to fit in the compasses; and an ivory protractor, with scales engraved on it. A larger set contains, besides these, a pair of dividers, a long drawing pen, bow pen, and a bow pencil; the two latter for circles and arcs not exceeding an inch radius. Still larger sets contain hair compasses, lengthening bar, sector, and parallel ruler: these can be dispensed with, particularly the three latter. The sector is seldom used; and a "Triangle and Ruler," made by a good carpenter, answer all the purposes of a parallel ruler, and are not so likely as it to get out of repair.

Diagrams must not be drawn till the student is somewhat acquainted with his instruments. He must look at each one separately, and try to discover its use, which he will find little difficulty in doing. He may then draw, with a hard lead pencil, straight lines, and concentric circles, following the edge of the ruler for one, and taking care to press very gently on the compasses in describing the circles, lest the centre should be worn to a large hole. The lines must now be "inked in;" this is to be done with the best indian ink, rubbed up very black, and put into the drawing pens with a camel's hair pencil. When the ink is dry, the lines may be rubbed gently with indian rubber, to remove the lead pencil, and each line examined to ascertain whether the whole are equal, clear, and of uniform thickness; for to draw a good line is a great desideratum, and is moreover not an easy thing to do.

When continued lines of various thicknesses can be drawn at pleasure, dotted lines may be undertaken, both right and curved; but those lines intended to be dotted in ink, are not to be first dotted with lead pencil. Specimens of dotted lines may be seen in the diagrams accompanying the problems.