EXPLANATIONS OF TERMS. GEOMETRY proceeds by means of Definitions, Postulates, Axioms, and Propositions. A Definition is the explanation or meaning of a term of art; or describes a magnitude by enumerating its properties. A Postulate is a petition or demand, necessary to be granted, admitted as possible. An Axiom is a self-evident truth, which requires no proof to confirm it. A Proposition is a sentence in which something is affirmed or proposed to be done; it is either a theorem, or a problem. A Theorem is a proposition which proposes some truth to be proved; it wants demonstration. A Problem is a proposition which proposes something to be done or constructed; it requires solution. The Construction is the drawing of certain lines or figures, to enable us to demonstrate the theorem or to solve the problem. The Demonstration is the reasoning employed to show that the theorem is true, or that the problem is solved. A Corollary is a consequent truth, deduced from a foregoing demonstration. A Scholium is a remark made upon a foregoing proposition. A Lemma is a minor proposition, necessary to be demonstrated, previous to a more important one which follows it. In a theorem certain things are supposed and admitted, from which a conclusion is to be drawn; this supposition is called the hypothesis. Some propositions are proved directly by means of definitions, axioms, or propositions already demonstrated. Other propositions are proved indirectly, by showing that any other supposition or hypothesis than the one advanced would lead to an absurdity or impossibility. The sentence which expresses the substance of a proposition is called the Enunciation. DEFINITIONS. 1. A point is that which has no parts. 2. A line is length without breadth or thickness; as, the line A B. 3. The extremities of a line are points; as, the points A and B of the line A B. 4. A straight or right line is the shortest distance between two points, and lies evenly between them. 5. A superficies or surface is that which has length and breadth, but no thickness; as, s. 6. The extremities of a superficies are lines. 7. A plane superficies is that which lies evenly between its extreme lines. 8. A rectilineal angle is the opening or corner between two right lines, which meet in a point, but do not form one right line; as, the angle A. 9. The right lines which form an angle, are called the sides; and the point in which the sides me is called the vertex of the angle; as, the vertex A. 10. When several angles meet in one point, each angle is expressed by three letters, of which the middle letter is placed at the vertex of the angle, and another upon each of the sides which form the angle; thus, the angle formed by the lines C B and D B is called the angle C B D or DBC; the angle formed by the lines C B and E B is called the angle C B E or E B C. N.B. When there is only one angle at a point, it may be expressed by a single letter; as, the angle A or by three letters as, FAG. A 4 F B B S G C D E 11. When a right line standing upon another right line makes the adjacent angles equal to one another, each of them is called a right angle; and the right line which stands upon the other is called a perpendicular to it; thus, if the line AB, standing upon CD, makes the angles ABC and ABD equal, each of them is a right angle, and AB is perpendicular to C D. 12. One angle is said to be greater than another, when the lines which form the angle are farther from each other than the lines which form the other, measuring at equal distances from the vertex of each; thus, the angle EBC is greater than the angle ABC. 13. An angle greater than a right angle is called obtuse; as, the angle EBC. 14. An angle less than a right angle is called acute; as, the angle EBD. 15. A plane figure is a space inclosed by one or more lines. C 16. A circle is a plane figure bounded by one line called the circumference, and is of such a kind, that all right lines drawn from a certain point within the figure to the circumference are equal to one another; as, the circle DE F. 17. This point is called the centre of the E circle, and the right line drawn from the centre to the circumference is called the radius ; as, the radius AD drawn from the centre a : if more lines than one be drawn, they are called the radii. 18. A diameter of a circle is a right line passing through the centre, and terminated both ways by the circumference; as, the di ameter EF. 19. A semicircle is the figure contained by the diameter and half the circumference; as, H. 20. An arc is any part of the circumference; as, EL. L 21. A figure bounded by right lines is called a rectilineal figure; if it be bounded by three lines it is called a triangle; if bounded by four lines it is called a quadrilateral; if bounded by more than four lines it is called a polygon. 22. Triangles are of three kinds, when described according to the nature of their sides; namely, Equilateral, Isosceles, and Scalene. 23. An equilateral triangle is that which has three equal sides; as, the triangle ABC. 24. An isosceles triangle is that which has two equal sides; as, the triangle D E F. 25. A scalene triangle is that which has three unequal sides; as, the triangle GHI. 26. Triangles are also of three kinds, when described according to the nature of their angles; namely, Right-angled, Obtuse-angled, and Acute-angled. A D G 27. A right-angled triangle is that K which has one of its angles a right angle; o as, the triangle KLM. 28. An obtuse-angled triangle is that which has one of its angles an obtuse angle; as, the triangle N OP. 29. An acute-angled triangle is that which has all its angles acute; as, the triangle QRS. 30. Parallel right lines are such as are equidistant from each other, and if produced would never meet; as, AB and CD. 31. A square is a quadrilateral figure, having all its sides equal and all its angles right angles; as, H. 32. An oblong or rectangle is a quadrilateral figure, having all its angles right angles, but its length exceeds its breadth; as, K. 33. A rhombus is a quadrilateral figure having all its sides equal, but its angles are not right angles; as, L. 34. A rhomboid is a quadrilateral figure having the opposite sides equal, but its angles are not right angles; as, N. B E F H I R 35. All quadrilateral figures, whose opposite sides are parallel, are called parallelograms; and the line joining two of its opposite angles, is called the diagonal or diameter. POSTULATES. 1. Let it be granted that a right line may be drawn from any one point to any other point. 2. That a terminated right line may be produced to any length in a right line. 3. That a circle may be described from any centre, with any distance from that centre as radius. AXIOMS. 1. Things which are equal to the same, 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 of the same, are equal to one another. 7. Things which are halves of the same, are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. 9. The whole is greater than its part. 10. Two right lines cannot inclose a space. 11. All right angles are equal to one another. 12. If a right line meet two other right lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these two right lines being continually produced, shall at length meet on that side on which are the angles, which are less than two right angles. |