drawn through them. How will you find a second point K? In the same way as H; or simply take the second intersection of the first pair of circles. Then join HK and we have the perpendicular bisector of AB (Fig. 10). If you had a line AB and wanted to bisect it, that is, to find its middle point, what would you do?-Draw the perpendicular bisector; the point O where it cuts AB is the middle point. Or we could measure AB; suppose it 8.31 cm.; from this find half the length, namely 4.15 cm., and measure off this distance from one end of the line. A Ο B FIG. 10. 20. Could you draw a perpendicular to a line CD from a point E that does not lie on the line?-We might fold the paper so that the crease passes through E, while one part of the line CD falls along another part. Then all the four angles where the crease crosses CD will fit if folded on one another, so that they are right angles and the crease -is the perpendicular we want. Could you draw the perpendicular without folding? Could you find on CD two points, Pand Q, such that Ewould IE Q/ FIG. 11. lie on the perpendicular bisector of PQ?-With E as centre, draw any circle that cuts CD in two points, and call these P and Q. Then E is equidistant from P and Q so that the perpendicular bisector of PQ passes through E (Fig. 11). How many points on this bisector do you need in order to draw it? -One in addition to E. Draw two equal circles, with centres at P and Q, and call one of the points where they cut one another H. Then EH is the perpendicular bisector of PQ, that is, it is the perpendicular from E on CD. How would you draw at E a perpendicular to CD when E lies on CD? Will the same method do ?—Yes, find P and Q equidistant from E, then find H not on CD and equidistant from P and Q; and EH is the perpendicular (Fig. 12). How would you use your protractor to draw a perpendicular to CD at a point E in it? Do so. Some protractors can be used to draw the perpendicular when E is not on CD. Can yours be so used? 21. Converse Propositions. We have found properties of circles so related that a condition and the conclusion of one become respectively the conclusion and a condition of the other. Thus we had the trio:-(1) The perpendicular bisector of a chord of a circle goes through the centre. (2) The line joining the middle point of the chord to the centre is at right angles to the chord. (3) The perpendicular from the centre on the chord bisects it. Any one of these is said to be the converse of any other. Another pair of converse propositions we had was :-(1) The line joining the centres of two circles bisects their common chord at right angles. (2) The perpendicular bisector of the common chord of two circles passes through the centres of the circles. rad. 6 cm. .B The following are statements related in the same way. Discuss which of them are true and which false. Draw a circle of radius 5 cm. and centre A and take a point B somewhere inside the circle (Fig. 13). Is it true that every point more than 5 cm. from A lies outside the circle; and that every point outside this circle is more than 5 cm. from A? Is it true that every point more than 5 cm. from B lies outside the circle; and that every point outside the circle is more than 5 cm. from B? Is it true that every point more than 10 cm. from A is outside the circle; and that every point outside the circle is more than 10 cm. from A? Is it true that every point more than 10 cm. from B is outside the circle; and that every point outside the circle is more than 10 cm. from B? FIG. 13. Does it appear whether, from the truth or falsity of a statement, you can tell whether a converse of the statement is true or false? EXERCISES. 1. The positions of four points are specified thus: from 0 to A is 1.6 miles north and 4.2 east; from A to B is 31 north and 2.3 west; from B to C is 2 north and 3.3 west; and from C to D is 3.7 south and 1.4 east. Show the points OABCD on the scale of 1 inch to one mile. A B FIG. 14. 2. In Fig. 14 the thick line AB represents a level railway which is carried partly through cuttings and partly on an embank ment. The irregular line represents the section of the ground. If the scale of horizontal distances is 1 inch to a mile, and the scale of vertical distances is 1 inch to 100 feet, draw up a table showing the height of the ground above the level of the railway at intervals of half a mile starting from A, and indicating points where the ground is below the level of the railway by prefixing the sign 3. Prick off Fig. 15; in which A and B are two forts. A body of soldiers at Y are just out of range of each fort and want to get as near as possible without going any nearer to either fort. Show the point they would go to, and draw the shortest path by which they could go. Draw the straight part of the path by laying your straight-edge against the circle and through their destination. 4. AB is a tube which revolves uniformly on a table about its middle point 0, that is, turning through the same angle every second. If a marble moves from one end of the tube to the other at a uniform speed (that is, travelling the same distance every second), whilst the tube makes a complete revolution, draw the path of the marble. Represent the tube by a line 5 inches long. 5. From a corner of a sheet of paper mark off 20 cm. along one edge to A and 13'4 cm. along the other to B. Fold over the corner along the line AB, and mark on the page the point where the corner falls. Open out the fold again and join C to the corner by a straight line. Prove shortly that this straight line is perpendicular to AB. CHAPTER II POSITION OF CHAIR ON SCHOOLROOM FLOOR 1. A CHAIR stands in a certain position on the schoolroom floor. What measurements would you take so that if the chair is moved it may be placed again in its former position? Discussion will soon show that this resolves itself into two questions, how to fix one leg of the chair, and how many legs must be fixed. The former question is the problem of the cache over again, and the same solution may be offered. The teacher accepts this and asks for another. A likely one is to measure the distances of the leg from the walls. Proceed to measure the distance from a wall, say the north wall. From what point of the wall will you measure? The suggestion may be made of the nearest point of the wall. Measure from various points and measure the angles these various distances make with the wall. Make a table of the various distances and the corresponding angles. To measure at right angles is thus suggested; or it may have been suggested at first. Suppose the leg of the chair known to be 4.6 feet from the north wall, measured at right angles. What locus does this give? on what line does the leg lie? Find by drawing; draw a number of lines 4.6 feet long perpendicular to the wall, either on the floor or to a scale of 1 cm. to 1 foot on paper. What is the locus of the ends of these perpendiculars? It looks like a straight line. Test it with your straight-edge. 2. Parallels. Fold a sheet of paper and call the crease you make A. Fold the crease A on itself in two different places, making creases B and C(Fig. 1). What angles do these creases B and C make with À?-Right angles, since the |