(18.) From the above 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. (19.) 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. DEFINITION 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 wholes 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. PROPOSITIONS. PROPOSITION I. THEOREM. When a straight line meets another straight line, the sum of the two adjacent angles, thus formed, is equal to two right angles. Let the straight line AB be met by the straight line CD at the point D. Then will the two adjacent angles ADC, BDC be together equal to two right angles. A D F B Suppose the line DF to be at right angles to AB (Def. X). The angle ADC is composed of the two angles ADF and FDC therefore the sum of the two angles ADC, CDB is equal to the sum of the three angles ADF, FDC, CDB (Ax. II); of which the first ADF is a right angle (Def. X), and the sum of the other two FDC and CDB composes the right angle FDB: therefore the sum of the two angles ADC and BDC is equal to two right angles. Cor. 1. Hence, also, conversely, if the two angles ADC, BDC, on the same side of the line AB, make up together two right angles, then AD and DB will form a continued straight line AB. Cor. 2. Hence, all the angles which can be made at any point D, by any number of lines on the same side of AB, are together equal to two right angles. B Cor. 3. And as all the angles that can be made on the other side of AB are also equal to two right angles; therefore all the angles that can be made around the point D, by any number of lines, are equal to four right angles. (20.) The instrument called a square, which is extensively used in the arts for tracing lines at right angles to each other, consists of two flat rulers placed at right angles as in the adjoining figure. When much precision is required, great care should be taken by those purchasing this instrument, to test its accuracy. The preceding Proposition suggests a very simple and sure means of making such test. Suppose the straight line AB to be the edge of a board, or any other plane surface. Apply one side of the square so as to coincide with AC; then, along the other edge, upon the surface, draw the line CD. F D Now, reversing the square, apply the first A side so as to coincide with BC; and then, along the second side, trace upon the surface the line CF. If the square is perfectly accurate, it is obvious that the lines CD, CF will coincide. This method not only detects an error in the instrument, when it exists, but also shows the amount of error; that is, how much the angle of the square exceeds or falls short of being a right angle, or of containing just 90°. |