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(21.) Another definition of parallel lines may be given as follows: Two lines are parallel when they have the same direction. If two lines, having the same direction, have also one point common, they will coincide and be identical.

The ordinary frames of windows and doors, and nearly all architectural framework, consist of systems of parallel lines at rightangles with each other. All fabrics produced in the loom consist of two systems of parallel threads, crossing each other at rightangles, so interlaced as to give strength and firmness to the cloth. The railway consists of two or more parallel lines of iron bars, called rails.

XIII. A plane figure is a limited portion of a plane. When it is limited by straight lines, the figure is called a rectilineal figure, or a polygon; and the limiting lines, taken together, form the contour or perimeter of the polygon.

(22.) The surfaces of level fields, bounded by straight fences, are polygonal figures. Floors of buildings are polygons, usually having four sides.

XIV. The simplest kind of polygon is one having only three sides, and is called a triangle. A polygon of four sides is called a quadrilateral; that of five sides is called a pentagon; that of six sides is called a heptagon; and so on for figures of a greater number of sides.

XV. A triangle having the three sides equal, is called an equilateral triangle; one having two sides equal, is called an isosceles triangle; and one having no two sides equal, is called a scalene triangle.

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XVI. A triangle having a right-angle, is called a right-angled triangle. The side opposite the right-angle is called the hypothenuse. Thus BAC is a right-angled triangle, right-angled at A; the side BC is the hypothenuse.

XVII. When the opposite sides of a quadrilateral are parallel, the figure is called a parallelogram.

XVIII. When the four angles of a parallelogram are right-angles, the figure is called a rectangle.

XIX. When the four sides of a rectangle are equal, the figure is called a square.

XX. When the four sides of a parallelogram are equal, and the angles not right, the figure is called a rhombus.

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XXI. When only two sides of a quadrilateral are parallel, the figure is called a trapezoid.

(23.) From the foregoing definitions, it will be seen that the quadrilateral includes the parallelogram, the rectangle, the square, the rhombus, and the trapezoid; the parallelogram includes the rectangle, the square, and the rhombus; and the rectangle includes the square.

Nearly all architectural structures, such as doors, windows, floors, and the sides of houses, are of the rectangular form. Among the different triangles employed in architecture and carpentry, the isosceles is most frequently to be found. It is the form usually given to the roofs of buildings, and to the pediment which surmounts and adorns porticos, doors and windows.

XXII. A diagonal of a polygon is a line joining the vertices of two angles, not adjacent.

(24.) From the above definitions, in connection with the diagrams, it will be readily seen that the triangle has no diagonal, the quadrilateral has two diagonals, the pentagon has five, and so on for polygons of a greater number of sides.

The number of diagonals of a polygon of n sides is given by this algebraic expression, §n (n —3). [See Elements of Algebra, Art. 178.]

XXIII. A circle is a plane figure bounded by one line, which is called the circumference; and is such that all straight lines drawn from a certain point within the circle to the circumference, are equal to one another.

This point is called the centre of the circle. One of the equal lines drawn from the centre of a circle to its circumference, is called a radius. The line passing through the centre, and terminating each way in the circumference, is called a diameter.

(25.) A circle might be defined as a plane figure bounded by a line, all the parts of which have the same degree of flexure.

DEFINITIONS OF TERMS.

1. An axiom is a self-evident proposition.

2. A theorem is a truth, which becomes evident by means of a train of reasoning called a demonstration. 3. A problem is a question proposed, which requires a solution.

4. A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem. 5. A corollary is an obvious consequence deduced from one or several propositions.

6. A scholium is a remark on one or several preceding propositions, which tends to point out their connection, their use, their restriction, or their extension.

7. A postulate is a problem, the method of solving which is obvious. It is therefore assumed or taken for granted by the geometer.

AXIOMS.

I. Things which are equal to the same thing, are equal to each other.

II. When equals are added to equals, the whole are equal.

III. When equals are taken from equals, the remainders are equal.

IV. When equals are added to unequals, the wholes are unequal.

V. When equals are taken from unequals, the remainders are unequal.

VI. Things which are double of the same or equal things, are equal.

VII. Things which are halves of the same thing, are equal.

VIII. Every whole is equal to all its parts taken together, and greater than any of them.

IX. Things which coincide, or fill the same space, are identical.

X. All right-angles are equal to one another.

POSTULATES.

I. To draw a straight line from any one point to any other point.

II. To produce a terminated straight line to any length.

III. To describe the circumference of a circle, from any centre, with any radius, or, in other words, at any distance from that centre.

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