22. A quadrilateral is a plane figure bounded by four straight lines. The straight line which joins opposite angular points in a quadrilateral is called a diagonal. 23. A polygon is a plane figure bounded by more than four straight lines. 5 TRIANGLES. 24. An equilateral triangle is a triangle whose three sides are equal. 25. An isosceles triangle is a triangle two of whose sides are equal. 26. A scalene triangle is a triangle which has three unequal sides. 27. A right-angled triangle is a triangle which has a right angle. 6. The side opposite to the right angle in a right-angled triangle is called the hypotenuse. 28. An obtuse-angled triangle is a triangle which has an obtuse angle. 29. An acute-angled triangle is a triangle which has three acute angles. [It will be seen hereafter (Book I. Proposition 17) that every triangle must have at least two acute angles.] QUADRILATERALS. 6 30. A square is a four-sided figure which has all its sides equal and all its angles right angles. [It may be shewn that if a quadrilateral has all its sides equal and one angle a right angle, then all its angles will be rignt angles.] 31. An oblong is a four-sided figure which has all its angles right angles, but not all its sides equal. 32. A rhombus is a four-sided figure which has all its sides equal, but its angles are not right angles. 33. A rhomboid is a four-sided figure which has its opposite sides equal to one another, but all its sides are not equal nor its angles right angles. U 34. All other four-sided figures are called trapeziums. It is usual now to restrict the term trapezium to a quadrilateral which has two of its sides parallel. [See Def. 35.] 35. Parallel straight lines are such as, being in the same plane, do not meet, however far they are produced in either direction. 36. A Parallelogram is a four-sided figure which has its opposite sides parallel. 37. A rectangle is a parallelogram which has one of its angles a right angle. THE POSTULATES. Let it be granted, 1. That a straight line may be drawn from any one point to any other point. 2. That a finite, that is to say a terminated, straight line may be produced to any length in that straight line. 3. That a circle may be described from any centre, at any distance from that centre, that is, with a radius equal to any finite straight line drawn from the centre. NOTES ON THE POSTULATES. 1. In order to draw the diagrams required in Euclid's Geometry certain instruments are necessary. These are (ii) A pair of compasses with which to draw circles. In the Postulates, or requests, Euclid claims the use of these instruments, and assumes that they suffice for the purposes mentioned above. 2. It is important to notice that the Postulates include no means of direct measurement : hence the straight ruler is not supposed to be graduated ; and the compasses are not to be employed for transferring distances froin one part of a diagram to another. 3. When we draw a straight line from the point A to the point B, we are said to join AB. To produce a straight line means to prolong or lengthen it. The expression to describe is used in Geometry in the sense of to draw. ON THE AXIOMS. The science of Geometry is based upon certain simple statements, the truth of which is so evident that they are accepted without proof. These self-evident truths, called by Euclid Common Notions, are known as the Axioms. GENERAL AXIOMS. 1. Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 4. If equals be added to unequals, the wholes are unequal, the greater sum being that which includes the greater of the unequals. 5. If equals be taken from unequals, the remainders are unequal, the greater remainder being that which is left from the greater of the unequals. 6. Things which are double of the same thing, or of equal things, are equal to one another. 7. Things which are halves of the same thing, or of equal things, are equal to one another. 9.* The whole is greater than its part. * To preserve the classification of general and geometrical axioms, we have placed Euclid's ninth axiom before the eighth. GEOMETRICAL AXIOMS. 8. Magnitudes which can be made to coincide with one another, are equal. 10. Two straight lines cannot enclose a space. 11. All right angles are equal. 12. If a straight line meet two straight lines so as to make the interior angles on one side of it together less than two right angles, these straight lines will meet if continually produced on the side on which are the angles which are together less than two right angles. B That is to say, if the two straight A lines AB and CD are met by the straight line EH at F and G, in such a way that the angles BFG, DGF are together less than two right angles, it is asserted that AB and CĎ will meet if continually produced in the direction of B and D. 'H NOTES ON THE AXIOMS. 1. The necessary characteristics of an Axiom are (i) That it should be self-evident ; that is, that its truth should be immediately accepted without proof. (ii) That it should be fundamental; that is, that its truth should not be derivable from any other truth more simple than itself. (iii) That it should supply a basis for the establishment of further truths. These characteristics may be summed up in the following defini. tion. DEFINITION. An Axiom is a self-evident truth, which neither requires nor is capable of proof, but which serves as a foundation for future reasoning. 2. Euclid's Axioms may be classified as general and geometrical. General Axioms apply to magnitudes of all kinds. Geometrical Axioms refer specially to geometrical magnitudes, as lines, angles, and figures. 3. Axiom 8 is Euclid's test of the equality of two geometrical magnitudes. It implies that any line, angle, or figure, may be taken up from its position, and without change in size or form, laid down upon a second line, angle, or figure, for the purpose of comparison, and it states that two such magnitudes are equal when one can be exactly placed over the other without overlapping. This process is called superposition, and the first magnitude is said to be applied to the other. 4. Axiom 12 has been objected to on the double ground that it cannot be considered self-evident, and that its truth may be deduced from simpler principles. It is employed for the first time in the 29th Proposition of Book I., where a short discussion of the difficulty will be found. |