makes the adjacent angles equal, each of them is called a RIGHT ANGLE; and the straight line which stands upon the other is called a PERPENDICULAR to it. 10. An OBTUSE ANGLE is that which is greater than a right angle. 11. An ACUTE ANGLE is that which is less than a right angle. 12. PARALLEL LINES, or those straight lines which are in the same plane, and being produced ever so far both ways do not meet, are such as make equal alternate angles with a single third straight line which either meets or intersects them. 13. The distance between two points is the straight line which joins them; and the distance of a point from a straight line is the perpendicular from that point to the line. 14. A FIGURE is that which is enclosed by one or more boundaries. The space contained within the figure is called its area. 15 A PLANE RECTILINEAL FIGURE is that which is bounded by straight lines. The side on which the figure is supposed to stand is called its base. 16. A TRIANGLE is a figure bounded by three straight lines called its sides. Any side of a triangle may be called the base, and the angular point opposite to that side the vertex. 17. An EQUILATERAL triangle is that which has three equal sides. 18. An ISOSCELES triangle is that which has two equal sides. The third side is called the base. 19. A SCALENE triangle is that which has all its sides unequal. 20. A RIGHT-ANGLED triangle is that which has a right angle. The side opposite to the right angle is called the hypotenuse. 21. An OBTUSE-ANGLED triangle is that which has an obtuse angle. 22. An ACUTE-ANGLED triangle is that which has three acute angles. 23. A plane figure bounded by four straight lines is called a QUADRILATERAL. The straight line drawn across the figure and joining two of its opposite angles is called the DIAGONAL. 24. A TRAPEZIUM is a quadrilateral, none of the sides of which are parallel to another. 25. A TRAPEZOID is a quadrilateral which has two parallel sides. 26. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel. 27. A RHOMBOID is a parallelogram that has an acute angle. 28. A RHOMBUS is a rhomboid that has two adjacent sides equal. 29. A RECTANGLE is a parallelogram that has a right angle. 30. A SQUARE is a rectangle that has two adjacent sides equal. 31. Every plane figure bounded by more than four straight lines is called a POLYGON. Polygons receive particular names from the number of their sides and angles; thus a polygon of five sides is called a PENTAGON; of six sides, a HEXAGON; of seven sides, a HEPTAGON; of eight sides, an octagon, etc. 32. The ALTITUDE of a triangle or parallelogram is the perpendicular drawn from the opposite angle or side to the base. 33. A CIRCLE is a plane figure described by the revolution of a straight line about one extremity of it which remains fixed. The fixed point is called the CENTRE of the circle, the describing line the RADIANT, and the boundary traced by the remote end of the line the CIRCUMFERENCE. 34. The radius of a circle is a straight line drawn from the centre to the circumference. Cor. Hence all the radii of the same circle are equal, for each of them is equal to the radiant. a a 35. The DIAMETER of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 36. An ARC of a circle is any part of the circumference. 37. Magnitudes which coincide the whole of one with the whole of the other, or exactly fill the same space, are said to be equal in every respect. 38. When the several parts of one magnitude can be made to coincide with those of another, the magnitudes are equal only in area, but in other respects they may be very unequal. A PROPOSITION is a distinct portion of abstract science, the verbal statement of which is called its enunciation, and is either a theorem or a problem. A THEOREM is a proposition which requires to be established by a process of reasoning called a demonstration. A PROBLEM proposes some operation to be performed, and requires a solution. A LEMMA is a subsidiary theorem the truth of which must be established preparatory to the demonstration of a subsequent theorem or problem. An Axiom is a self-evident theorem, which cannot be made more certain by any proof. A POSTULATE is a request made by the geometer that he shall be allowed to perform a simple operation, evidently possible and absolutely necessary for the construction of the diagrams, or it is a self-evident problem. A COROLLARY is an obvious consequence that results from a demonstration. A direct demonstration proceeds by a regular series of deductions from the premises to the conclusion. An indirect demonstration establishes a proposition, by proving that every other possible supposition contrary to the truth enunciated leads to a conclusion that is absurd or contradictory, and has therefore been frequently called a reductio ad absurdum. a a An HYPOTHESIS is a supposition made in the enunciation of a proposition, or in the course of a demonstration, and may be either true or false. A scholium is a remark subjoined to a demonstration on the nature, the application, or the peculiarities of one or more preceding propositions. To join two points is a concise expression for drawing a straight line from the one of them to the other. POSTULATES. 1. Let it be granted that a straight line may be drawn from any one point to any other point. 2. That a straight line may be produced in a straight line to any required length. 3. That a circle may be described from any centre, with a radiant equal to any given straight line. AXIOMS. 1. Things which are equal to the same thing, or to equal things, are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double, triple, etc., of the same thing, or of equal things, are equal to one another. 7. Things which are each one half, one third, etc., of the same thing, or of equal things, are equal to one another. 8. The whole is greater than its part. 9. The whole is equal to all its parts taken together. When in a construction, or in the course of a demonstration, a a reference is made to a definition, a postulate, or an axiom, the contractions Def., Post., Ax., are used; and when a proposition already demonstrated is referred to, the number of the proposition is mentioned, as Prop. IV. B D From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line, it is required to draw from the point A a straight line equal to BC. From A as a centre, with a radiant equal to BC, describe (Post. 3) the circle DEF, and from A draw a straight line to any point in the circumference, as E, the line AE is equal to BC; A for the straight line AE being drawn from the centre A to the circumference is a radius (Def. 34) of the circle DEF, and therefore equal (Def. 34, Cor.) to the radiant; but the radiant was taken equal to BC, .wherefore AE and BC are each equal to the radiant; and things which are equal to the same thing are equal (Ax. 1) to one another, therefore AE is equal to BC. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, of which AB is the greater, it is required to cut off a part from AB equal to C the less. From A as a centre, with a radiant equal to C, describe (Post. 3) the circle DEF. Because the straight line AE extends from A -В the centre to the circumference it is a radius (Def. 34, Cor.) of the circle DEF, and there D F |