'Adjacent' means lying next to, from the Latin ad, to, and jaceo, I lie. Adjacent angles have one of their bounding lines in common. Thus, each of the two adjacent angles BAC, BAD has BA for one boundary. Now, referring to Def. 9 once more, when BA stands upon CD in the way described in Def. 10, it cannot properly be said to be inclined at all to the line AD or AC. In other words, BAD and BAC are not angles as defined in Def. 9. The meaning of that definition is plain enough, and yet it would be better if we could find no objection to the words. But we must not linger on the point any more, except to point out to the student that he may derive some warning from it how much care is often necessary in order to ensure that our language shall really express our meaning.

Def. 10 is a double one, as it defines two things, that is, 'right angle' and 'perpendicular.' We may define 'right angle' alone by stopping at the semicolon. But, to define 'perpendicular' apart, we must say something like this::-"When a straight line stands on another so as to make the adjacent angles equal, either of the lines is called a perpendicular to the other."

The word 'perpendicular' is from the Latin perpendiculum, a plumbline.

EXAMINATION V.

1. Explain the word 'adjacent'; give its derivation, and also that of 'perpendicular.' 2. In the second figure of Art. 5 name an angle on each side of BAC adjacent to it. 3. Similarly for CAD in the same figure. 4. Write out the definition of a 'right angle.' 5. Define a 'perpendicular." 6. If BOD be a right angle, say whether AOD is obtuse or acute; also BOC, BOE, and COD respectively.

E

A

7. FIGURES.

13. A term or boundary is the extremity of anything.

14. A figure is that which is enclosed by one or more boundaries.

Definition 13 is merely an ordinary explanation of one word by another; such as any dictionary would give.

A figure might be more clearly described as a portion of space closed in by boundaries. This space, in "Plane Geometry," is also assumed to be plane space, or space in some plane surface real or imaginary. And therefore our 'figures' will always be 'plane figures,' or figures in a plane. The purpose of Plane Geometry is to prove by argument numerous important facts which have been discovered relating to plane figures, angles, and lines.

The boundaries of figures may be straight lines or curved lines, or some of one kind and some of the other. When there are more boundaries than one, each of them may be called a side.

If we wish to name a figure, we put letters along its boundary or boundaries, more especially at the corners, if there are any, and use three or more of these letters, taking care to use just enough to distinguish the figure from any other which may have some of the same letters as itself. Thus the circle in Diagram I., which has A, B, C, D, E, F, G, H, Kalong its circumference, may be called the circle ABC or BCD or ADH, or we may thus use any other three of the letters named. Sometimes we may have reason to use four or more of the same letters. It is the same with all other kinds of figures.

The word 'figure' is sometimes used for the boundary all round the figure, and then the space enclosed is called the area of the figure.

EXAMINATION VI.

B

1. How many figures are contained by the straight lines which form the capital letter A ? 2. How many figures respectively in B, D, 0, 8? 3. How many in the letters of the word 'dog? 4. Name the separate boundaries of the figure ABCDE in the diagram. 5. Also of the figure ABCDEF. 6. And of the figure ACDE. 7. What is the word 'area' used for, and what does 'figure' then mean?

E

8. CIRCLES.

15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.

16. And this point is called the centre of the circle.

17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

B

D

E

The most important of all figures is defined first, although bounded by a curved line (and not by "one straight line," as some beginners will inadvertently say!).

This is another double definition, defining 'circle' and also its 'circumference.'

Definition 16 might have been similarly incorporated with Def. 15, and by itself, of course, is no definition of 'centre of a circle.'

It would be well to learn, along with Defs. 15-17, the following one: 16 (a) Each of the equal straight lines which may be drawn from the centre to the circumference of a circle is called a radius.

The word 'circle' is often used instead of the longer word circumference,' when there is no danger of one's being misunderstood in doing so. For the derivations of some words now introduced, we have-Latin circulus, a ring; circum, around, and fero, I carry; radius, a spoke of a wheel; Greek dia, through, and metron, a measure.

EXAMINATION VII.

1. Give the derivations of 'circle,' 'circumference,' 'diameter,' 'radius,' and say which of these words is often used in place of another. 2. Distinguish between the circle ABCDE and the circumference ABCDE. 3. Define 'circumference' without saying more than is necessary. 4. Define 'centre of a circle,' and do not say too little. 5. If a straight line be drawn from a point in the circumference to the centre, what may it be called? 6. If the same straight line run through the centre until it meet the circumference again, what will then be its name? 7. Name all the radii in the figure of Defs. 15-17. 8. How many radii are equal to one diameter ?

9. PARTS OF CIRCLES.

18. A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter.

19. A segment of a circle is the figure contained by a straight line, and the part of the circumference which it cuts off.

B

Definition 18 is not used until we are nearly at the end of Book II., and could be dispensed with even there. As semi means 'half,' in using the name 'semicircle' we are assuming that the diameter of a circle cuts it into halves. If we follow the common practice of Euclid, this should not have been assumed, for the simple reason that we are able to prove it; as the learner will perhaps notice on reading Euclid's Book III.

Definition 19 is not used until Book III., and is repeated among the definitions of that book.

A portion of a circumference is often called an 'arc.'

For reasons just stated, the above definitions would have been better reserved for the Third Book, in which the circle is more especially studied.

10. RECTILINEAL FIGURES.

20. Rectilineal figures are those which are contained by straight lines.

KINDS OF RECTILINEAL FIGURES.

21. Trilateral figures, or triangles, by three straight lines. 22. Quadrilateral figures by four straight lines.

23. Multilateral figures, or polygons, by more than four straight lines.

'Trilateral,' from Latin tres, three, and latus, a side, means threesided; 'triangle' means three-angled.

'Quadrilateral' means four-sided; 'multilateral' many-sided; 'polygon' many-angled.

The Latin quatuor means four, multi, many or several; the Greek polus, many, and gonia, a corner.

EXAMINATION VIII.

1. What is the name given to the space between the straight line AC and the arc AEC in the figure of Art. 9? 2. Also to the space between

AOB and ADB in the same, O being centre? 3. Define an arc. 4. Connect the derivations and meanings in respect to 'trilateral,' 'quadrilateral,' 'multilateral,' 'polygon.' 5. State or write the name in letters of the triangle ABC in five other different ways. 6. Write down eight triangles which occur in the figure ABCD. 7. Write three quadrilaterals occurring in the figure ABCDEF. 8. Write two five-sided polygons in the same figure, and one six-sided polygon.

B

B

11. TRIANGLES NAMED ACCORDING TO THEIR SIDES,

Of three-sided figures

24. An equilateral triangle is that which has three equal sides.

25. An isosceles triangle is that which has two sides equal. 26. A scalene triangle is that which has three unequal sides.

да

The Latin æquus means equal; Greek isos, equal; skelos, leg; skalenos, limping, uneven.

12. TRIANGLES NAMED ACCORDING TO THEIR ANGLES.

27. A right-angled triangle is that which has a right angle. 28. An obtuse-angled triangle is that which has an obtuse angle.

29. An acute-angled triangle is that which has three acute angles.