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together they make a right angle or 90 degrees; so that each is 45 degrees.

Can you tell the size of the angle XOW?-It is 90 degrees, for the turn from OY to OX is 90 degrees and the turn from OY to OW is 180 degrees.

And what do you know of the sum of the angles BOX, XOW, and WOQ? It is 180 degrees.

And of these three angles you know two; what, then, is the third WOQ?It must be 45 degrees to make the sum 180 degrees.

10. To write this down in the ordinary way takes a long time, and some contractions or symbols are used to shorten the writing. Thus

[blocks in formation]

and by the help of these symbols our reasoning may be

stated thus

LXOY = 90°

.. LXOB = LBOY = 45°

LYOX + LXOW = 180° and LYOX = 90°

[blocks in formation]

and so on. But these contractions should be sparingly used at first as there is danger of concentrating the attention on the contractions instead of on the reasoning. The value of symbols is best realized if the pupils are allowed to do their writing in longhand till they begin to realize how tedious it is; they may then be encouraged to invent symbols for themselves and to use their own inventions for a time.

11. Now find the size of each angle of the figure by similar reasoning, and compare these results with the measured results. Can you now answer any of the questions at the beginning of article 9? Are all points on OQ equidistant from OZ and OW?—Yes; OQ bisects the angle ZOW and folding along OQ brings OZ and OW together, so that if any length is measured along OZ and OW, and perpendiculars drawn, they meet on OQ and have the same length; just as we found already for OB, OX and OY. So that we now know that all points on BOQ are equidistant from XOZ and YOW.

The question whether OQ bisects LWOZ is already answered.

Do all points equidistant from XOZ and YOW lie on the line BOQ? Find a number of such points by drawing, and then answer the question.-No, all such points do not lie on BOQ; some lie on the line that bisects the angles XOW and YOZ. Here, then, we have two converse propositions. Are they both true, or both untrue, or what? Can you sum up what we have found in a single proposition about a locus of points? A little discussion should lead to the statement that: The locus of points equidistant from two lines intersecting at right angles is the two lines that bisect the four angles made by the intersecting lines.

Ex. 2. Use the result of Article 8 to bisect a right angle by the help of ruler and compasses.

12. Suppose now that the lawn in Art. 7 is not rectangular, and that as before you know the cache is equidistant from two edges, the distances being measured perpendicular to the edges. Go over the work of Arts. 7-11, altering it to suit the altered condition.

13. Draw two straight lines AOB and COD crossing at 0, and measure all four angles. The angles AOC and BOD are called vertically opposite angles. The angles AOD and BOC also are called vertically opposite. You have met with several such pairs of angles. Can you state and prove any proposition about a pair of vertically opposite angles? The vertically opposite angles made by two intersecting lines are equal.


3. In article 12 it is found that the bisector OB of any angle XOY may be found by laying off equal lengths OA and OC along OX and OY and drawing perpendiculars at A and C to meet in B. Find out if it would do to join O to the middle point D of AC; would this line OD be the bisector of the angle?

4. Make a map of the schoolroom and the desks, &c., in it. Use a scale of 1 cm. to 1 ft., 1 in. to 1 ft., 1 cm. to 1 metre, or any other that is convenient.

5. How would you fix the position of a pencil that lies on your desk? A book? A triangle cut out of paper? An irregular sheet of paper?

6. How would you draw a line parallel to a line that is given you? How would you do it if the parallel was to go through a given point?

7. Two straight railroads, AOC and BOD, cross at O at right angles. A is 13 miles, and B is 1 miles distant from 0. Two trains pass A and B respectively, at noon, moving towards O at the uniform rates of 20 and 30 miles an hour. Find by measurement, and tabulate, the distances apart of the trains at noon, and at 1, 2, 3, 7 and 8 minutes after noon. By examination of your table of distances, state as accurately as you can the time when the trains were nearest to one another. Take O about the middle of the page, and work to a scale of 2 inches to a mile.

8. Draw on tracing-paper a straight line SH six inches long: mark on it a point B two inches distant from S. On a page of your book draw two straight lines XOP and OQ intersecting at right angles, XOP close to one side of the page and OQ across the middle of the page. Bring the tracing-paper over these lines, moving it about so that S passes up and down XOP and at the same time й moves to and fro along 0Q. Prick on the paper the points over which B passes and then draw a freehand smooth curve through these points.

9. A square ABCD, measuring 3 cm. in the side, rolls without sliding along a line XY, turning about B, C and D in succession. Draw (full size) the path of the point A from the position in which it is leaving XY till it again lies on XY. (A square has all its sides equal and all its angles right angles.)

10. Draw two circles of 1 inch radius with their centres 24 inches apart. On tracing-paper draw two straight lines each 1 inches long bisecting each other at right angles. By pricking through the tracing-paper and drawing through the points a smooth curve, determine the paths of the ends of one of the straight lines while

the ends of the other move along the circumferences of the two circles one end on each.

11. A point P is known to be at least 4.6 cm. from a point A of a straight line; shade the area in which P may not lie. P is also to be at least 4.6 cm. from another point B of the line; shade the additional prohibited area. Take more points C, D, . . . on the line, and suppose P to be at least 4.6 cm. from each, and shade the prohibited area. Finally, find experimentally and approximately the form of the prohibited area if P is to be at least 46 cm. from every point of the line.



1. THE determination of areas is a problem of great antiquity. Many races have passed through a stage of development in which the land of a community was periodically divided into a number of equal parcels; and the name Geometry itself means land-surveying.

Do you know any units in terms of which areas are measured?-Acre, hectare, square foot, square decimetre, square inch, square centimetre, and others. Draw as

many of these units as your paper is big enough to hold, and cut them out to use for measuring. Take a sheet of paper of any form, and see how many square inches you can mark out on it. Does this tell the area?- -Yes, except for the irregular pieces left at the edges. Are there no pieces left over except at the edges?- -We can mark out the inch-squares so that there will be no pieces left over except at the edges. Could you measure the playground in the same way with a square foot as unit ?—Yes, but it would be somewhat laborious. Let us see if we cannot find easier ways.

2. In measuring the sheet of paper, do you need to lay the unit-area on the sheet every time to mark out the units there? We could simply rule the sheet into squares, or we could place it on squared paper and trace its boundary on to the squared paper. Is there any choice as to where you mark out the first square?-We could use any straight side of the sheet as the side of squares, and if the sheet was rectangular* one square should be placed in a corner of the sheet.

* The sheet is rectangular when its four angles are all right angles.

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