Thus at the point A there are the angles CAH, HAF, FAB; CLASSIFICATION OF QUADRILATERALS. 31. A rhombus is a quadrilateral that has all its sides equal. Thus, if AB, BC, CD, DA are all equal, the quadrilateral ABCD is a rhombus. The rhombus ABCD is sometimes named by two letters placed at opposite corners, as AC or BD. Euclid defines a rhombus to be 'a B D quadrilateral that has all its sides equal, but its angles not right angles.' 32. A square is a quadrilateral that has all its sides equal, and all its angles right angles. Thus, if AB, BC, CD, DA are all equal, and the angles A, B, C, D right angles, the quadrilateral ABCD is a square. The square ABCD is sometimes named by two letters placed at opposite corners, as AC or BD; and it is said to be described on any one of its four sides. BL 33. A parallelogram is a quadrilateral whose opposite sides are parallel. A D Thus, if AB is parallel to CD, and AD parallel to BC, the quadrilateral ABCD is a parallelogram. The parallelogram ABCD is sometimes named by two letters placed at opposite corners, as AC or BD; and any one of its four sides may be called the base on which it stands. B4 34. A rectangle is a quadrilateral whose opposite sides are parallel, and whose angles are right angles. Thus, if AB is parallel to CD, AD A parallel to BC, and the angles A, B, C, D right angles, the quadrilateral ABCD is a rectangle. The rectangle ABCD is sometimes named by two letters placed at opposite corners, as AC or BD. In B books on mensuration, BC and AB would be called the length and the breadth of the rectangle. The definitions of a square and a rectangle are somewhat redundant—that is, more is said about a square and a rectangle than is absolutely necessary to distinguish them from other quadrilaterals. This will be seen later on. 35. A trapezium is a quadrilateral that has two sides. parallel. A D 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 to any length either way. 3. That a circle may be described with any centre, and at any distance from that centre. The three postulates may be considered as stating the only instruments we are allowed to use in elementary geometry. These are the ruler or straight-edge, for drawing straight lines, and the compasses, for describing circles. The ruler is not to be divided at its edge (or graduated), so as to enable us to measure off particular lengths; and the compasses are to be employed in describing circles only when the centre of the circle is at one given point, and the circumference must pass through another given point. Neither ruler nor compasses can be used to carry distances. If two points A and B are given, and we wish to draw a straight line from A to B, it is usual to say simply 'join AB.' To produce a straight line, means not to make a straight line when there is none, but when there is a straight line already, to make it longer. The third postulate is sometimes expressed, 'a circle may be described with any centre and any radius.' That, however, is not to be taken as meaning with a radius equal to any given straight line, but only with a radius equal to any given straight line drawn from the centre. [The restrictions imposed on the use of the ruler and the compasses, somewhat inconsistently on Euclid's part, are never adhered to in practice.] AXIOMS. 1. Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the sums are equal. 3. If equals be taken from equals, the remainders are equal. B 4. If equals be added to unequals, the sums are unequal, the greater sum being obtained from the greater unequal. 5. If equals be taken from unequals, the remainders are unequal, the greater remainder being obtained from the greater unequal. 6. Things which are doubles of the same thing are equal to one another. 7. Things which are halves of the same thing are equal to one another. 8. The whole is greater than its part, and equal to the sum of all its parts. 9. Magnitudes which coincide with one another are equal to one another. 10. All right angles are equal to one another. 11. Two straight lines which intersect one another cannot be both parallel to the same straight line. An axiom is a self-evident truth, or it is a statement the truth of which is admitted at once and without demonstration. Some of Euclid's axioms are general—that is, they apply to magnitudes of all kinds, and not to geometrical magnitudes only. The first axiom, which says that things which are equal to the same thing are equal to one another, applies not only to lines, angles, surfaces, and solids, but also, for example, to numbers, which are arithmetical, and to forces, which are physical, magnitudes. It will be seen that the first eight axioms are general, and that the last three are geometrical. It ought, perhaps, to be noted that some of the axioms are often applied, not in the general form in which they are stated, but in particular cases that come under the general form. For example, under the general form of Axiom 2 would come two particular cases: If equals be added to the same thing, the sums are equal; and If the same thing be added to equals, the sums are equal. Again, a particular case coming under the general form of Axiom 4 would be: If the same thing be added to unequals, the sums are unequal, the greater sum being obtained from the greater unequal. Axioms 6 and 7, on the other hand, are only particular cases of more general ones-namely, Things which are double of equals are equal, and Things which are halves of equals are equal; and these axioms again are only particular cases of still more general ones: Similar multiples of equals (or of the same thing) are equal, and Similar fractions of equals (or of the same thing) are equal. Axiom 9 is often called Euclid's definition or test of equality; and the method of ascertaining whether two magnitudes are equal by seeing whether they coincide—that is, by mentally applying the one to the other, is called the method of superposition. Two magnitudes (for example, two triangles) which coincide are said to be congruent; and this word, if it is thought desirable, may be used instead of the phrase, ‘equal in every respect.' Axiom 10 is, strictly speaking, a proposition capable of proof. The proof is not given here, as at this stage it would perhaps not be fully appreciated by the pupil. After he has read and understood the definitions of the third book, he will probably be able to prove it for himself. Axiom 11, frequently referred to as Playfair's axiom (though Playfair states that it is assumed by others, particularly by Ludlam in his Rudiments of Mathematics), has been substituted for that given by Euclid, which is proved as a corollary to Proposition 29. QUESTIONS ON THE DEFINITIONS, POSTULATES, AXIOMS. 1. How do we indicate a point? 2. What is the only thing that a point has? What has it not? 3. Could a number of geometrical points placed close to one another form a line? Why? 4. Draw two lines intersecting each other in two points. 5. Could two straight lines be drawn intersecting each other in two points? 6. What is Euclid's definition of a 'straight' line? 7. Could a number of geometrical lines placed close to one another form a surface? Why? 8. When two points are taken on a plane surface, and a straight line is drawn from the one to the other, where will the straight line lie? 9. If a straight line is drawn on a plane surface and then produced, where will the produced part lie? |