P.-Because the vertex of the one angle is, like wise, the vertex of the other. M.-What other angles are, here, vertical? P.—The angles a m c and b m d. M.-Compare them. P. They are equal. M.-Hence, the opposite or vertical angles formed by the intersection of two straight lines P.-Are equal to each other. M. This truth we have established; we have proved it to be true. If, then, you were called-upon to prove, that, "if two straight lines intersect each other, the opposite or vertical angles are equal," what would you do? P. We would do as we have just now done. M.-Well-the method by which a mathematical truth, such as this, is proved, we call its demonstration. M.-I shall rub out what I have written. Demonstrate, each of you, that, "if two straight lines intersect each other, the opposite or vertical angles are equal." The master will, here, have an opportunity of judging whether what precedes has been clearly understood. It is of importance that the written demonstration should be well performed, -the lines being neatly drawn, &c. The signs, which have been introduced, tend to render the expressions capable of being more quickly revised, and to facilitate the detection of errors. The following is a specimen of the manner of demonstration recommended : If two straight lines intersect each other, the opposite or vertical angles are equal. Let the straight lines ab and cd intersect each other in the point m; the vertical angles a m ca and b m d, and, also, cm b and am d, shall be equal. a m Because sa mc and c m b = 2rt. ▲s, b and, also, thes cm b and b m d = 2 rt. ≤s ; therefore, sam c and cm b = _s m b and b md: from each of these equals take-away ▲ cm b; there remains ▲ amc = bm d. from each of these equals take away there remains mbam d. 2 rt. Zs; sam cand am d: am c; M.On what truth does this demonstration chiefly depend? P. That the two angles which one straight line makes with another straight line are together equalto two right angles. M.-There are two other truths referred-to in your demonstration: what are they? P.-Angles which are equal to the same angles are equal to each other; and, if the same angle be taken from equal angles, the remaining angles are equal. M.-Do these last-mentioned truths require demonstration? P.-No; they are acknowledged at once. M.-Will it be readily admitted that, the two angles which one straight line makes with another straight line are equal to two right angles; and that, if two straight lines intersect each other, the vertical angles are equal? P.-No; these truths must be proved; they require demonstration. M.-Truths which are at once conceded, and, therefore, do not require demonstration, are called axioms. (Greek džíwua, from agios, worthy,—a proposition worthy of being believed.) M.—Instead of saying, "Angles which are equal to the same angles are equal to each other," we might say, in general, "Things which are equal to the same thing are equal to each other." Generalize the other axiom in a similar way. P.-If equals be taken from equals, the remainders are equal. M.-Endeavour to generalize a similar axiom. P.-If equals be added to equals, the sums are equal. M. From what obvious truth did you deduce this general axiom ? P.-If equal angles be added to equal angles, the sums are equal. The pupils will probably mention several axioms: if not, they may be led to discover some. It is of importance, that the substance of this section should be reduced to concise sentences, which ought to be written on the school-slate and committed to memory. SUBSTANCE OF SECTION I. 1. A line is length without breadth. 3. Two straight lines may either be parallel or not. 4. Parallel lines are such, that, if produced both ways ever so far, they do not meet. 5. If non-parallel lines be produced far enough, they meet or intersect each other. 6. If two straight lines meet, they may either form one angle or two angles. 7. An angle is the inclination, to one another, of two lines, which meet in one point. 8. That point is called the vertex of the angle; and the lines which contain the angle are called its legs. 9. Angles may either be equal or unequal. 10. If one straight line standing on another straight line makes the adjacent angles equal to each other, each of them is called a right angle. 11. All right angles are equal to each other. 12. An obtuse angle is that which is greater than a right angle. 13. An acute angle is that which is less than a right angle. 14. All obtuse angles are not equal to each other; nor are all acute angles equal to each other. 15. The two angles, which one straight line makes with another straight line, are either two right angles, or they are together equal to two right angles. 16. If two straight lines intersect each other, the four angles about the point of intersection are either four right angles, or they are, together, equal to four right angles. 17. If two straight lines intersect each other, the opposite or vertical angles are equal. The axioms should, in a similar manner, be written, and learnt by heart. Frequent repetition of what precedes is indispensably necessary before beginning the next section. The pupils must be able to write the preceding sentences in their order, whenever they are so required, and great strictness is to be observed with regard to the demonstration of No. 17. SECTION II.-THREE STRAIGHT LINES. you M.-Draw three straight lines, and state what observe respecting them, proceeding as you did with two straight lines. α с a f b с d 4th. They may all three be parallel. 5th. Two of them may be parallel, a and the third non-parallel. f. e |