23. A quadrilateral figure is a plane figure enclosed by four straight lines. 24. A square is a four-sided figure which has all its sides equal and all its angles right angles. It is sufficient, if all the sides are equal, to have one angle a right angle, as it may then be proved that the other angles are right angles. 25. A rectangle, or oblong, is a four-sided figure which has not all its sides equal, but its angles are right angles. 26. A rhombus is a four-sided figure which has all its sides equal, but its angles are not right angles. 27. A rhomboid is a four-sided figure which has its opposite sides equal, but not all its sides equal, nor its angles right angles. 28. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. 29. A parallelogram is a four-sided figure which has its opposite sides parallel. 30. A trapezium is a quadrilateral which has two sides parallel Sometimes trapezium is used for any quadrilateral, and trapezoid for one having two sides parallel. 31. A polygon, or multilateral figure, is a plane figure enclosed by more than four straight lines. 32. A diagonal is a straight line joining two opposite angles of a quadrilateral, or any two angles of a polygon not consecutive. Postulates. Let it be granted 1. That a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced any length in a straight line. 3. Tbat a circle may be described from any centre, at any distance from that centre. These Postulates are Requests to be allowed to construct certain figures, and evidently require the aid of a ruler or straight-edge, not divided into parts, for drawing straight lines, and of compasses for describing circles. In Post. 3, Euclid tacitly makes this restriction, that the centre of the circle must be either extremity of the line taken as radius. Without this restriction Propositions I. 2 and I. 3 would be superfluous. But if we are allowed to transfer a figure from one place to another without change in the magnitude of its parts, surely we may transfer a straight line without altering its length ; and if so, then no such restriction is to be read into Post. 3, and Propositions I. 2 and I. 3 may be omitted. Axioms. 1. Things wbich are equal to the same thing 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 doubles of the same thing are equal to one another. 7. Things wbich are halves of the same thing are equal to one another. 8. The whole is greater than its part. 9. Magnitudes wbich coincide with one another, that is, which fill the same space, are equal to one another. 10. All right angles are equal to one another. 11. Two intersecting straight lines cannot be both parallel to the same straight line (called Playfair's Acciom). This form of the axiom is one that can be more readily accepted as an axiom than Euclid's, which is, “ If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles less than two right angles" (see I. 29, Cor.) The first eight of these axioms are General, or applicable to all kinds of Magnitude; the others are Geometrical. SECTION I. On the Properties of Angles and Triangles. PROPOSITION 1. PROBLEM. To describe an equilateral triangle on a given straight line. Given a st. line AB. AB. AB describe BCD (Post. 3). With centre B and distance BA describe O ACE. From the point C, in which the circles intersect, draw the st. lines CA, CB (Post. 1). Then will ABC be an equil. A. :: A is centre of O DCB, : AC = AB (Def. 11); and :: B is centre of O ACE, :: BC = AB. Now : AC, BC, each=AB, and things which are equal to the same thing are equal to one another (Ax. 1), :: AC BC. :. AC, BC, and AB are all equal; :: an equil. A ACB has been described on the given st. line AB. Q.E.F. EXERCISE 1. Show how the construction in Prop. 1 enables us to form two equilateral triangles on AB. 2. On a given straight line to construct an isosceles triangle having the equal sides each equal to a given line. 3. To construct a triangle having two sides each double of the base. For Euclid's Propositions 2 and 3, see Appendix to Book I. PROPOSITION 4. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by these sides equal to one another, their bases, or third sides, shall be equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite. Given AB = DE, AC = DF, and 2 A = D. and A ABC = A DEF. If A ABC be applied to A DEF so that B is on E and BA along ED, then point A falls on D, :: BA = ED (given); and AC falls on DF, : - A = _D (given); and point C falls on F,:: AC = DF (given). Then since B is on E and C on F, BC must coincide with EF (Def. 3). :: BC = EF; and LB coincides with and . is = LE, and 2 C coincides with and : is = LF, and A ABC wholly coincides with and :: is = A DEF in every respect. Q.E.D. Note.-The equality of the two triangles is demonstrated by the method of Superposition, which is the ultimate test of equality (Ax. 9). 4. Prove the above proposition when one of the triangles is placed in inverted, or in reversed, position. |