SECTION IV. GEOMETRICAL DRAWING. 60. DEF. A circle is a plane closed figure such that all straight lines drawn to it from a certain point within it are equal. 61. The certain point within the circle from which the equal lines are drawn is called the centre. 62. Each of the equal lines drawn from the centre to the circle is called a radius. 63. A finite straight line drawn through the centre of a circle and terminated both ways by the circle is called a diameter. 64. B The length of a circle is called its circumference. Thus in the figure ACBD is a circle; O is its centre; OA, OB, OC are each a radius; AOB is a diameter; the length of ACBDA is its circumference. 65. For the purpose of geometrical drawing, we must assume the following 'Postulates.' I. That we can draw a straight line from one given point to another given point. II. That we can produce a given finite straight line any distance either way. III. That we can describe a circle with any given point as centre, and with the line joining the centre to any other given point as radius. When we say that we must assume that we can draw a straight line etc., we mean that we are about to describe a system of theoretical geometrical drawing which we could actually carry out provided we could actually draw a straight line, produce it, and describe a circle. It is however obviously impossible to draw a straight line; a material straight line by itself does not exist. A near approach to drawing a straight line would be to colour part of a sheet of paper and make the boundary of the colour straight. But just as a fine straight rod does indicate a straight line (for we may agree that it shall indicate the theoretical line which is at its centre) so a penciled or an inked line on paper may indicate a straight line. We may agree that it shall indicate the line at its centre, or at one of its edges. It is however very difficult to carry this out in practice. Suppose for example there are many 'penciled lines' drawn through the same point; it will be very difficult to secure that their central lines shall all be exactly in the right position. In fact a line is not drawn; it is left to the eye to imagine or guess its exact position. Hence all practical geometrical drawing is not the exact theoretical drawing which is supposed in Euclid for the purposes of explanation. Practical geometrical drawing is always approximate. The figure drawn on paper is really a picture of the theoretical figure. With care and skill pictures can be drawn which are very accurate indeed; in fact pictures can be drawn which are accurate figures for all practical purposes. 66. There are certain phrases which are customary in Euclid which the student must understand. Join AB. This expression means that two points indicated by A and B are part of the diagram and the straight line joining A to B is to be drawn. 'A given finite straight line.' This expression means that the student is supposed to have already drawn on the paper on which he is to draw his diagram, a certain finite straight line. 'A given finite straight line AB.' This means that there is supposed to be given to the student already drawn on his paper a certain straight line of given length whose extremities are to be indicated by the letters A and B. 'A given straight line ACB and a given point C in it.' AB are two points on a given line of unlimited length (which may extend any length beyond A and beyond B) and C indicates a chosen point in the line AB. 'Produce CB.' This means that C and B are points, which are joined by a finite straight line; and this straight line is to be prolonged beyond B so that Produce BC' would mean that the straight line BC is to be prolonged in the direction beyond C. At the conclusion of the proof of a Geometrical Construction, or Problem, it is usual to put Q.E.F. i.e. Quod erat faciendum, 'which was to be done.' 67. We have assumed (i) that we can draw the straight line joining two given points, (ii) that a finite straight line which has been drawn can be extended or 'produced' to any desired extent either way, (iii) that with a given centre and at a given distance from that centre we can describe a circle. This last assumption may be stated thus:-Given the centre and one point on the circle, we can describe the circle. It is equivalent to saying that we can find the extremities of all lines which can be drawn from the centre of the same length as the radius of the circle. EXAMPLES VII. 1. AB, AC are two given straight lines terminated at A; AB is greater than AC; cut off a part from AB equal to AC. 2. AB is a given straight line and C is a given point; draw from A a straight line AD equal in length to AB and passing through C. 3. AB and AC are two given straight lines terminated at A; AB is less than AC; produce AB to D so that it shall be equal to AC. 4. AB is a given straight line; shew how to produce AB to C so that AC shall be (i) twice, (ii) three times AB. 5. AB and AC are two given straight lines; cut off from AB, produced as far as is necessary, a part equal to twice AC. 6. AB is a given straight line and C is a given point; draw through A and C a line AD whose length is three times AB. Proposition 1. 68. To describe an equilateral triangle upon a given finite straight line. Let AB be the given finite straight line, it is required to describe an equilateral triangle having AB as one of its sides. A E Construction. With A as centre and AB as a radius describe the circle BCD. With B as centre and BA as a radius describe the circle ACE. Let C be a point in which the two circles intersect. Join CA, CB. Then the figure ACB is an equilateral triangle such as is required. Proof. Because AC, AB are straight lines drawn from its centre A to the circle BCD, therefore AC is equal to AB. And because BA, BC are straight lines drawn from its centre B to the circle ACE, therefore BA is equal to BC. Therefore AC, BC and AB are equal to each other. Wherefore ABC is an equilateral triangle and it is described upon the given straight line AB. Q. E. F. |