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number of times along the path of a given curve should, even approximately, make up and represent the length of that curve. In default of the means of direct measurement, however, we are enabled to infer with great accuracy, the lengths of all curves of regular outline, from a knowledge of the relation which, mathematical inquiry shows us, exists between the curve and straight lines drawn in connexion with it at certain points. Thus, the length of the circumference of a circle can always be derived or calculated from the known ratio which it bears to a straight line within it called the diameter ; the length of the periphery of an ellipse may be determined when the lengths of its transverse and conjugate axis are assigned, and so on.

(6.) Angles.-When two straight lines, which are not in the same direction, meet or cut one another at any degree of inclination, they form or include between them what is termed an angle. The lines themselves are called the sides of the angle, and their point of contact or intersection is called the vertex of the angle. A “plane rectilineal angle,” is the geometrical name for an angle described on a flat surface, and contained by straight lines. Angles may with propriety be considered as a species of magnitude, since they are as capable of being measured and their values expressed numerically as are the other kinds of magnitude-lines, surfaces, or solids. It should be particularly noted, however, that it is the amount of divergence of two straight lines proceeding in different directions from the same point, that constitutes an angle, and that the magnitude or value of an angle is not in any degree dependent on the length of either of its sides. Once a definite angle is created by the divergence of two straight lines from a point, it is wholly immaterial as respects the quality or magnitude of that angle, how far the original lines may be produced. Thus, it is evident that the opening immediately at the angular point A, in the annexed figure, remains the same, whether we take the sides of the angle to be represented by the lines A B, and A E, A C and A F, or A D and A G.


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It will greatly promote a clear apprehension of the nature of angular magnitudes in general, if the word “ angle” be regarded as meaning, not as it is loosely supposed to mean, a corner or something equivalent to a corner, but rather as signifying quantity of turning, that is, the successive positions which a straight line revolving on one of its extremities as a pivot or centre, might be made to assume relatively to another fixed line drawn from the same point.

For example, the longer hand of a common clock or watch describes in the course of each hour around the shorter hand, every possible variety of angle. A still better illustration would be afforded by laying a pair of compasses, with the legs closed, flat on a sheet of paper, and then causing the upper leg to perform, gradually, a complete revolution, while the under one is held stationary. In strictness, the hands of the clock, and the legs of the compasses must be supposed to partake of the character of mathematical lines, that is, to have no breadth or thickness what. ever. It is from this idea of revolution taking place in the formation of all angles, that the word vertex, or turning point, is derived.

Angles are usually designated, either by means of a single letter placed at the vertex, as the A in the last diagram ; or, when there are several angles at the same point which it is necessary to distinguish from each other, by means of three letters, arranged so that the middle letter shall always be that standing at the




vertex, while the first and third letters are stationed respectively at any distances
along the sides of the particular angle indicated.* Thus, we
speak of the upper of the two angles in the accompanying
figure, as the angle B A C, or C A B, and of the lower, as
the angle CAD, or D A C. In such a case as this, to A
speak only of the “ angle at A,” would leave it uncertain
whether we meant the angle B A C, or the angle D A C.

Instead of the word, angle, the sign is often conveniently employed, especially when there is occasion, in the course of geometrical reasoning, to refer to many different angles, or to the same angle repeatedly. Such an expression as < B A C, means the angle B A C.

Adjacent angles are those which lie side by side without the intervention of any other angle between them, such as the angles B A C and D A C.

When one straight line standing upon another straight line makes the adjacent angles equal to one another, each such angle is called a right angle, and either of the straight lines is said to be a perpendicular to the other. (Euclid, def. 10.)

Lines which are not mutually perpendicular, are said to be oblique to one another.
A right angle may also be defined to be the fourth

part of the angular space passed over by the nearer
extremity of a straight line revolving completely
around that extremity as an axis. If the line b for
example, be conceived to revolve around the point b,
until it attains the upright or perpendicular position,
6 d, it will evidently have performed one quarter of a
revolution, and when it coincides with b a, b e, and,
finally, with b c, its own original direction, it will have performed respectively, one-
half, three-quarters, and the whole of a revolution.

From this it follows, that all right angles are equal, and that the total angular space, or the sum of all the angles which can be formed by diverging lines, however numerous, about a common centre, must be equal to four right angles, since the angular space which they together fill, must be the same as that occupied by the four right angles surrounding that centre. It will also be manifest,-as demon. strated by Euclid, Prop. 13, Book I.,that the angles which one straight line makes with another upon one side of it, as the angles c b d and a b d, or a b e and cbe, in the figure above, are either two right angles, or are together equal to two right angles.

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An angle greater than a right angle is called an obtuse angle.

An angle less than a right angle is called an acute angle.

As the measurement of angular magnitudes does not necessarily enter into any part of the business of gauging, a description of the methods adopted for that pur

• Since the magnitude of an angle, as has just been shown in the text, does not depend on the length of either of its sides, it is immaterial at what distance from the vertex, the first and third of the letters naming the angle may be placed.


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pose may properly be omitted in the present work. The only species of angle with which officers are concerned is the right angle, and it will be shown in the article on Practical Geometry, how to draw lines perpendicular to each other, and also how it can be most readily ascertained whether a given angle is, or is not, a right angle.

(7.) Parallel Lines.-Two or more straight lines which lie in the same surface, and which do not incline or converge in any degree towards each other, but would always preserve the same distance apart, however far they might be produced in either direction, are called parallel lines. A B and C D represent parallel lines, and the fact of such lines being parallel may be readily determined by drawing perpendicular lines between them at any point, m and n, and observing whether these distances are equal. It is essential to the definition of parallelism that the lines should be in the same plane or surface. Two rulers, for instance, held one above the other, but pointing in different directions, could not properly be taken to represent parallel lines.

(8.) Plane Figures.-By a plane figure, as already explained, is meant a portion of space delineated on a plane or flat surface, and bounded by straight or curved lines.

A surface, or superficies, is defined in geometry to be “that which has length and breadth, but no thickness.” If á mathematical line be conceived to move in any manner through space, leaving behind it a trace or mark of its progress, it will describe a mathematical surface, for as the line itself has no thickness, the surface which it generates by its motion must also be devoid of that quality. The external limits of every solid body consist of such surfaces, and are, therefore, to be regarded strictly, as forming no portion the body which they enclose.

Surfaces, like lines, are divided into two general classes,-plane surfaces, and curved surfaces.

A plane, or plane surface, is that which lies with perfect evenness between its boundaries, so that no part of it is raised above or depressed below the level of any other part. It is the characteristic property of a plane surface, that if any two points be taken in it, and a straight line be drawn joining them, every point of that line or its continuation will be in contact with the surface, or as it is said, will lie in the surface; and this will be the case as respects every straight line whatever, that can be drawn, uniting any two points in a plane surface. In practice, the planeness or evenness of a surface is determined by placing a ruler which is constructed with a perfectly straight edge, in various positions along the surface submitted to examination, and observing whether every part of the edge of the ruler touches the surface.

A curved surface is that which does not lie evenly between its extremities. It is distinguished from a plane surface chiefly by the circumstance, that a straight line connecting any two points in a curved surface does not necessarily lie wholly in that surface.

Although there are certain curved surfaces in which it is possible so to select two points, that every part of the straight line joining them shall be in contact with the surface, yet the points which possess this property must be selected in a particular manner upon the surface, whereas, in the case of a plane, it is indifferent in what part of the surface the two points may be taken.

It is obvious that a straight line cannot be made to coincide, in any position, with the surface of a globe or sphere, except at one point,—that where it touches the globe ; but if a straight edge be applied lengthwise to a round column, it may be in contact with the surface at every point.

A plane surface of definite extent, bounded by straight lines, is called in geometry, a plane rectilinear figure.

No plane rectilinear figure can have a boundary of less than three lines, or in other words, two straight lines are not sufficient to enclose any portion of space. A single curved line, however, such as the circumference of a circle, may form the boundary of a figure.

Rectilinear figures of three sides are called triangles, a term which is derived from the inference, that if a figure have three sides, it must also have three angles, one formed by the junction of each pair of sides. There are several kinds of triangles, differing from each other, however, only in the relative value of their sides and angles. Thus,

An equilateral triangle has all its sides of equal length.
An isosceles triangle has two of its sides equal.
A scalene triangle has all its sides unequal.

A right-angled triangle has one of its angles a right angle; it cannot have moro than one, as it is proved in Euclid, Book I., Prop. 17, that « Any two angles of a triangle are together less than two right angles.

An obtuse-angled triangle has one of its angles an obtuse angle.
An acute-angled triangle has all its angles acute.
Plane figures of four sides are termed, generally, quadrilaterals.
A quadrilateral formed by two pairs of parallel sides is called a parallelogram.

The opposite sides and angles of a parallelogram are equal to one another. (Euclid, Book I., Prop. 34.)

When the sides of a parallelogram are joined at right angles, or are perpendicular to each other, the figure is called a rectangle. If one angle of a parallelogram be a right angle, it can be demonstrated that each of the other angles must also be a right angle. (Cor. to Euclid I., 34.)

A rectangle having all its sides of equal length is termed a square. This is the most important figure in plano geometry, since a square of definite dimensions is adopted as the measuring unit of every form of surface, or superficial magnitude.

A rhomboid is an oblique-angled parallelogram. It differs from the rectangle, therefore, in the fact of the sides not being perpendicular to each other.

A rhombus, or lozenge, is a rhomboid all the sides of which are equal. Thus, the rhomboid may be said to correspond to the rectangle, and the rhombus to the square.

A quadrilateral figure having only one pair of its opposite sides parallel, as a and b is called a trapezoid.

A quadrilateral figure which has none of its sides parallel is called a trapezium.


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The preceding comprehend all the possible varieties of four-sided rectilinear figures.

Plane rectilinear figures of more than four sides are named generally polygons, and, occasionally, multilateral figures, the former term having reference to the numerousness of the angles, and the latter to the numerousness of the sides. A polygon having four sides or angles is called a Pentagon. six


Heptagon. eight

Octagon and so on; the number of the sides, or, rather, of the included angles, conferring the denomination of each figure.

A regular polygon is one which has all its sides and angles equal. All other polygons are termed irregular.

Any straight line drawn across a quadrilateral figure a between two opposite angles, is termed a diagonal of that figure. Thus, A C and B D are the diagonals of the quadrilateral, A B C D.

It is plain that either of the diagonals resolves the figure into two triangles.

C Any side of a rectilinear figure may be considered, indifferently, its base, but, in strictness, the term should be applied only to the lowest or undermost side, that upon which the figure stands, or may be supposed to be constructed. The side towards which a perpendicular is drawn is invariably designated the base, with reference to such perpendicular, except in the case of a right-angled triangle, where the side opposite to the right angle is termed, distinctively, the hypotenuse, while either of the sides containing the right angle is called the base, and the other side, the perpendicular.

By the altitude or height of a triangle, or parallelogram, is meant the perpendicular distance from the base to the opposite angle or side. The angle to which the side taken as base lies opposite, is termed the vertical angle.

In each of the annexed trian. gles, let b c be the base; then bac will be the corresponding vertical angle. In the first triangle, a d is the altitude, and falls within the triangle. In the second, a c


6 is the altitude, and forms one of d the sides of the triangle. In the third, the dotted line, a d represents the altitude, and falls without the triangle, meeting the base produced at d. If either of the other sides were considered as the base, the vertical angle and altitude would vary accordingly.

The altitude of a rectangle may be either its length or breadth, as suits the occasion, or it may be a line drawn within the figure, parallel and equal to one of those dimensions. In other parallelograms, and in the trapezoid, the altitude is found in the same manner as in the case of a triangle, by drawing a perpendicular from the base to the opposite angle. A trapezium bas, properly speaking, no

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