AXIOMS.

14.

(1) Things equal to the same thing, or equal things, are equal to each other.

(2) If the same operation be performed on equals, the results will be equal.

(3) The whole is greater than any of its parts.

(4) The whole is equal to the sum of all its parts.

(5) If equals are added to unequals, the sums will be unequal in the same sense.

(6) If equals are subtracted from unequals, the remainders will be unequal in the same sense.

(7) Equals may be substituted for equals.

(8) Magnitudes whose boundaries coincide are equal. (9) Two points determine but one straight line.

(10) Two straight lines can intersect in but one point. (11) A straight line is the shortest distance between two points.

(12) Through the same point but one line can be drawn parallel to a given line.

POSTULATES.

15.

Let it be granted:

(1) That a straight line can be drawn joining any two given points.

(2) That a straight line can be produced to any extent in either direction.

(3) That a circle can be drawn with any point as the center and any finite straight line the radius.

(4) That on the greater of any two lines can be cut off a line equal to the less.

(5) That a figure can be moved unaltered to a new position.

(6) That two equal magnitudes can be made to coincide

SUGGESTIONS TO PUPILS.

The first step in the solution of a geometrical problem is to study it very carefully to understand the meaning of the language. The second step is the careful construction of a figure which shall afford a clear conception of what we have to do or prove. By constructing your figure carefully, relations between lines, angles, etc., are often suggested which might otherwise escape attention if the figure were carelessly constructed. Neatness is conducive to accuracy, while carelessness tends to inaccuracy. If your figure suggests certain relations, you are now ready to satisfy yourself whether they are real or apparent.

Your success and progress in solving geometrical problems will depend on your habit of watching for new properties that present themselves in various ways. The construction of a figure should be such that it shall not exhibit apparent relations not involved in the problem illustrated. That is, lines should not seem equal, or to be at right angles when they are not necessarily so. Triangles should not seem to be isosceles or right-angled unless the conditions of the problem require it. It is better to make a triangle whose angles are about 75°, 45°, 60°, for illustrating in general.

If quadrilaterals are spoken of in a problem, use the trapezium, and not the parallelogram. It is a good plan to draw the figure of the problem in heavy lines, and those used as helping lines more lightly, or in dotted or broken lines. Always letter every point of the figure which may be referred to as you proceed with your discussion. Keep the same figure as long as possible. Drawing new figures may distract the attention from a course of reasoning. After having exhausted the properties of the given figure, auxiliary lines may be drawn and resulting properties noted. The most useful auxiliary lines are obtained by

(1) Joining two given points.

(2) Drawing a line through a given point parallel to a given line.

(3) Drawing a line perpendicular to a given line at a given point within the line, or from a given point without the line.

(4) Producing a line its own length, or the length of another given line.

In preparing your lessons, write the statement of the proposition very carefully, as you have worked it out, to bring to class. After it has been corrected, then write it in your book.

PLANE GEOMETRY.

16.

The demonstration or proof of a theorem must be based upon definitions, axioms, postulates, and previously proved theorems. One of the simplest methods of proof of the equality of two figures is to show that when one is superposed upon the other their boundaries coincide, and the figures are consequently equal, by Axiom 8.

17.

Thus-suppose we wish to prove that
All straight angles are equal.

What is a straight angle? We know what the definition says, nothing more. The pupil must here review the definition until there is no doubt in his mind about what it states, and he can fully illustrate it.

(1) What is an angle in general?

(2) What is a straight angle?

After the definition has been mastered, let him draw two straight angles and attempt to prove them equal by showing that their sides must coincide when one is placed upon the other. Write the authority for each step in brackets after each statement. Compare your proof with that given below and see if yours fails in any essential point.

Given: A E B and M N O, any two straight angles. Required: To prove that angle A E B equals angle M NO.

Proof:

(1) Superpose

A E B upon / M N O so that point E will fall upon point N and side E B will take the direction and coincide with N O. [$15, Post. 5-Any figure can be moved unaltered to a new position.]

(2) Side E A will take the direction of side N M. [§11, 1A straight angle is an angle whose sides point in exactly opposite directions.]

(3) ... ZA E B = / M N O. [$14, 8-Magnitudes which can be made to coincide are equal.] But A E B and ZM NO were given any two straight angles; .. we con clude that all straight angles are equal.

18.

Cor. All right angles are equal.

(The proof follows directly from the definition of a right angle.)

The sections and exercises are numbered consecutively throughout the entire book.

N. B.- The proof of every section and exercise in the book is required of the pupil. When considered too difficult for the average pupil, partial proofs and suggestions are given to assist him. But he should never refer to these hints unless he has first exhausted his own resources to discover a proof of his own.