P; and so on, till you see what the relation is, and till you can prove it without any drawing at all.- -The sum of the distances must be greater than 9 cm. and the difference less than 9 cm. If you take any three points, and rule straight lines with your straight-edge to join them, the figure you make is called a triangle, and the three lines you have ruled are its sides. Make a triangle having sides 8 cm., 10 cm., and 13 cm. long, another with sides 8, 10, and 17 cm. long, and, if you can, four more with sides (1) 8, 10, 3 cm.; (2) 8, 10, 19 cm.; (3) 8, 10, 7 cm. ; (4) 8, 10, 1 cm. Try making other triangles with different sides till you can give a rule to distinguish the cases in which the triangle can be made; and justify your rule. The rule is important enough to be called a proposition, and may be stated thus-Any two sides of a triangle are together greater than the third. 5. Points of the Compass. The terms north and east have been used. How do you know which direction is north and which east? Point to the south. How do you know? It is the direction in which you must walk to go towards the sun at midday. Where does the sun rise and where set?-Roughly, it rises in the east and sets in the west. If you face south and then turn in the opposite direction, which way are you facing?-North. Does any one know the Pole Star and how it is used to fix directions? (The Pole Star is not necessary, and need not be followed up if no pupil knows it.) In If you face south and then turn halfway round to the north, which way are you facing ?-East or west. turning halfway round to the north you have turned through an angle called a right angle. Point out any right angles you see about the room, the tables, desks, &c. Through how many right angles do you turn in turning from south to north? And in turning from south back to south again? What fraction is a right angle of the whole angle you turn through in turning from south back to south again? What fraction is a right angle of the angle you turn through in turning round from any position till you again face in the same direction? it again so that A How many angles And are they all Then what Fold a sheet of paper, and call two points on the crease A and B. Open the sheet out and fold and B fall together. Open it out again. do the creases make ?Four. equal? Yes, because they fit together. angles do they make with one another?-Right angles. Take two creases that form a right angle and fold them on one another. What angle does the new crease make with the former two?-Half a right angle. By folding make an angle a quarter of a right angle. Note for the Teacher. If any pupil raises the question whether the size of an angle depends on the length of the arms, the question must be discussed. But the question should be left alone till a pupil raises it. It is a general principle not to point out logical difficulties to a pupil. It is a waste of time to explain a difficulty that the pupil does not feel. 6. Halves and quarters of a right angle have been mentioned. In shopping we avoid fractions of a shilling by using a smaller unit, the penny. In measuring we avoid fractions of a metre by using a smaller unit, the centimetre. So with angles it is convenient to have a smaller unit than the right angle. The smaller unit is called a degree, and A FIG. 1. B W S Draw half a 90 of them make a right angle. Look at your protractors; they are marked or graduated in degrees. dozen angles at random, and measure them in degrees. How many degrees are there in the first two angles together? Make an angle containing this number of degrees. Make another angle as big as the first three angles that you made. In turning from north to east, through how many degrees do you turn? And from north to south? And from north to north again? And in turning from north halfway to east? An angle may be indicated by a single letter or by three letters, as A, w, pqr, in Fig. 1. A straight line may be indicated by a single letter or by two, as S, BC. Care must be taken that the sign used singles out one angle or line. 7. The Circle. Draw a circle and fold the paper so that the crease cuts the circle. Open it out again. Call the two points of cutting C and D, and call the centre of the circle A. Draw the radii AC and AD, and fold again so that these radii lie along one another. How do C and D fall? Open out again and call the point where the creases cross 0. What are the angles at 0?-Right angles, for they all fit when folded together. What do you know of the lengths CO and OD?—They are equal because they fit together. 8. The straight line joining two points on a circle is called a chord. Draw any circle and lay in it a chord CD (Fig. 2). Bisect the chord at O by folding C on D. Join 0 to A, the centre of the circle, and measure the angles AOC and AOD. Repeat with several other chords. Again draw a circle and lay in it a chord CD. Draw a line bisecting the chord at right angles by folding Con D and creasing the paper. How does the centre A lie with respect FIG. 2. D to this perpendicular bisector? (When two lines meet at right angles they are said to be perpendicular to one another.) Repeat with several other chords. 9. Can you tell from your drawing whether the centre A lies within a millimetre of the perpendicular bisector of the chord? Within 0·1 of a mm.? Within 0.01 mm. ? And in the former case, when the centre A was joined to the middle point of the chord CD, can you tell whether the angle COA differs from a right angle by 1 degree? By 0·1 of a degree? There is a limit to the accuracy of a conclusion the centre of the circle. 10. The Isosceles Triangle. Draw any triangle ACD, having the sides AC and AD equal. A triangle with two sides equal is called an isosceles triangle, and the third side is called the base. Measure the angles at C and D. Repeat with three or four other isosceles triangles. What do you observe about the angles C and D in each case? Again, draw any straight line CD, and with your protractor draw lines from C and D making the same angle with CD and meeting in A, and so forming a triangle. Measure the lines AC and AD. Repeat with three or four other figures. What experimental result have you reached? 11. If in Art. 7 we began without a circle, but with only two equal lines AC and AD, would there be any difference in our conclusions? What do you know of the triangle ACD made by joining C and D?—It has two sides equal. Consider the triangle when AC and AD are folded together. What can you tell about it, and what of the angles at C and D?-They are equal because they fit together. State this as a proposition. The angles at the base of an isosceles triangle are equal. Now begin with a straight line CD (Fig. 3), and draw lines CE and DF from C and D on the same side of CD and making equal angles with it. Fold C on D. How do CE and DF fall?-Together, since the angles are equal. If, then, these lines are produced till they meet the crease, what happens?—They meet the crease in the same point. Call it G. Then we have a triangle F FIG. 3. GCD in which we made the angles C and D equal. What do we know of the sides ?GC is equal to GD because they fit. This equality may be written = GC = GD, the sign being a short way of writing 'is equal to'. State your result as a proposition. If a triangle has two of its angles equal, the two sides opposite to these angles are equal. 12. Let us return to the trees A and P, and the two circles with these as centres, at one of whose intersections the cache must lie. Call these intersections C and D (Fig. 4). Draw the figure and fold Con D, so as to get the crease that bisects CD at right angles. What do we know of this crease in connexion with the circle whose centre is A? And in connexion with the circle whose centre is P? We know that the crease passes through Aand A FIG. 4. through P. So that the perpendicular bisector of a chord that is common to two circles passes through the centres of the circles. 13. If we had found the middle point O of CD and joined |