straight but uumarked ruler for drawing or producing straight lines, and that he may employ a pair of compasses so far as to describe circles with them : (3) In his third division he places his Axioms, or Common Notions, which are self-evident truths, that is, truths which are so simple as not to require proof, and they are for the most part concerned with the equality of magnitudes. Starting from these, Euclid proceeds to deduce less simple constructions and properties of geometrical figures in a series of separate propositions, those which explain new constructions being termed Problems, while those which elucidate new properties of figures are called Theorems. It is most important that the student should bear in mind that Euclid's full treatment of the subject is gradually evolved from the elementary principles ; each proposition does not stand by itself, but forms a part of the general structure ; it depends upon what has gone before it, and in its turn is itself the support of some that come after it. The following are the abbreviations we shall use in the Propositions : def. for definition. prop. for proposition. post. „ postulate. cons. » construction. ax. , axiom. hyp. hypothesis. Q.E.F. stands for Quod erat faciendum, or, which was to be done, and is placed at the conclusion of a problem. Q. E.D. stands for Quod erat demonstrandum, or, which was to be proved, and is placed at the conclusion of a theorem. T. D. PLANE GEOMETRY. DEFINITIONS. 1. A POINT is that which has no parts, or which has no magnitude. 2. A line is length without breadth. 4. A straight line is a line which lies evenly between its extreme points. 5. A superficies is that which has only length and breadth. 6. The extremities of a superficies are lines. 7. A plane superficies is that in which, any two points being taken, the straight line between them lies wholly in that superficies. 8. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together but are not in the same straight line. An angle is generally expressed by three letters, the middle one of which must be the letter at the point where the straight lines forming the angle meet, and of the other two letters one lies upon each of these straight lines; if, however, there be only one angle at the point, it may be expressed simply by the letter at that point : thus the angle contained by the straight lines BA, AC, may be expressed as BAC, or CAB, or simply as the angle A. o But none of the angles FEG, DEP, DEG, can be expressed as the angle E, because by this we should leave it uncertain which of the three angles was intended. 9. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of these angles is called a right angle; and the straight line which stands on the other is said to be perpendicular to it. 10. A plane figure is that which is inclosed by one or more lines. 11. A circle is a plane figure contained by one line (called the circumference), and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. 12. And this point is called the centre of the circle. 13. Rectilineal figures are those which are contained by straight lines. 14. Trilateral figures, or triangles, are contained by three straight lines. 15. An equilateral triangle has three equal sides. 16. An isosceles triangle has two sides equal. 17. A right-angled triangle is that which has a right angle ; and the side opposite to the right angle is called the hypothenuse. 18. A quadrilateral is a plane figure contained by four straight lines. 19. A square is a four-sided figure which has all ts sides equal, and all its angles right angles. 20. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet. The remaining Definitions (21 to 35) are placed by Euclid with the above, but they do not occur in the Propositions of the First Book, and some of them are not found again in his works. 21. A plane angle is the inclination of two lines to each other in a plane which meet together, but are not in the same straight line. 22. An obtuse angle is an angle greater than a right angle. 23. An acute angle is an angle less than a right angle. 24. A term or boundary is the extremity of anything; and a figure is that which is inclosed by one or more boundaries. 25. A diameter of a circle is a straight line drawn through the centre, and terminated both ways hy the circumference. 26. A semicircle is a figure contained by a diameter and the part of the circumference cut off by the diameter. 27. A segment of a circle is the figure contained by a straight line and the part of the circumference which it cuts off. 28. A scalene triangle is a triangle with three unequal sides. 29. An obtuse-angled triangle is a triangle which has an obtuse angle. triangle 30. An acute-angled triangle is a wbich has three acute angles. 31. An oblong is a four-sided figure which has all its angles right angles, but has 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. 34. A trapezium is any four-sided figure other than a square, an oblong, a rhombus, or a rhomboid. 35. A multilateral figure, or polygon, is a figure which is contained by more than four straight lines. |