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be used instead of circle'? (see Art. 8). 9. Can the real circle be properly spoken of as at a distance from its centre ? 10. What lines have their lengths equal to the distance of the circumference from the centre ? (see Art 8.) When the compasses are in use, what two points of the compasses are at a distance from each other equal to one of these lines ?

Note.-If the student does not possess some knowledge of Practical Geometry, let him use both straight-edge and compasses in drawing all figures until he arrives at the end of Proposition III. Afterwards he may proceed, as is the usual way, to illustrate his work by freehand diagrams.

16. SOME APPLICATIONS OF THE POSTULATES. Postulate 1.—Take two points at random; put A at one, and B at the other. By Postulate 1, we may join A to B by a straight line. This is usually expressed briefly by saying “join AB."

Postulate 3 is often expressed more briefly by using the word “radius,' thus : “A circle may be described with any centre and any radius.”

EXAMINATION XV. 1. What words are understood in the direction “join AB? 2. Take two points about half an inch apart on your paper, and call one A, the other B. Join AB. What postulate have you now applied ? 3. Lengthen out AB a little beyond B to some point, and call this other point F. What postulate have you now applied ? 4. With centre A and radius AB describe a circle (or circumference), and place the letters C, D along it. What may this circle be called, and what postulate has been applied in drawing it? 5. With the same centre A and a radius AF which is greater than AB describe another circle. Join AC, and produce it until it meets the other circle in some point; which call G. What postulates have been applied in these operations ? 6. In the same figure join AD; produce AD to meet the outer circle in a point H. Join BC, CD, FG, GH. How often has Postulate 1 been applied in these operations ?

Note.—The student may verify his construction by comparing it with the figure which accompanies the next Exercise.

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AXIOMS. 17. THE FOUNDATIONS OF REASONING IN GEOMETRY. Numerous facts are known relating to geometrical figures; and Euclid's object is to establish the most important of these by sound

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argument. In doing so, he uses some facts to lead us on to others; and from these he steps on to others again. In this process there must, of course, be a beginning—in other words, some facts must themselves be assumed as true; from which we may then proceed to deduce all the rest, which constitute the science of Geometry. Facts adopted in this way as true without proof are usually called axioms. It is clearly necessary that they should be chosen with special care and skill. Euclid performs this service for us in Geometry; and in selecting his axioms he appears to adopt as his rule that each one must be, firstly, required ; secondly, selfevident, that is, obvious to ordinary minds without proof; and thirdly, incapable of being proved.

18. COMMON NOTIONS. The first seven axioms apply to all kinds of magnitude whether geometrical or not; and Euclid himself calls them “common notions,” that is, notions which people in general have in their minds.

AXIOMS 1–7.

1. Things which are equal to the same thing are equal to one another.

2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal.

5. If equals be taken from unequals, the remainders are unequal.

6. Things which are double of the same, are equal to one another.

7. Things which are halves of the same, are equal to one another.

Although these are all so easy to understand, it will be to our future advantage to work a number of simple Exercises upon them.

When applying Axioms 4 and 5, we usually happen to know which is the greater of the pair of unequals; and for practical purposes these axioms might advantageously be stated as follows :

4. If equals be added to unequals, the wholes are unequal, that being the greater which contains the greater of the first unequals.

5. If equals be taken from unequals, the remainders are unequal, that being the greater which contains the greater of the first unequals.

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EXERCISE I.

H 1. If DH, in the diagram annexed, is equal to AB, prove that AC is also equal to DH.

The answer to this will be fully given as a model to assist in working some Exercises which follow it.

Proof. It is given that DH is equal to AB; and because A is the centre of the circle BCD, therefore AC is also equal to AB.

(Def. 15) But things which are equal to the same thing are equal to one another.

(Ax. 1) Therefore DH and AC are equal to one another.

Which was to be proved. 2. If CG is equal to AB, prove that AC is equal to CG.

(Let the proof be in little paragraphs like the one just given,

and give the references in parentheses.) 3. If AC is equal to CD, prove that AD is equal to CD.

Say what kind of triangle ACD must then be. 4. If AH is equal to HG, show that the triangle AHG must be equilateral

Nos. 5—9 refer to Diagram I.

5. Given that OA and AC are each equal to UV, prove that the triangle AOC must be equilateral. The answer will be given here as another model.

Proof.
It is granted that 0 A and AC are each equal to UV.
Therefore AC is equal to 0A.

(Ax. 1)
And since 0 is the centre of the circle ACE,
therefore OC is also equal to 0A.

(Def. 15) Therefore OC is equal to AC;

(Ax. 1) hence 04, OC, and AC are all equal to one another. Therefore the triangle 0 AC is equilateral.

(Def. 24) 6. Given that PQ is equal to OR; prove that the triangle UFQ is equilateral.

7. Given also EC is equal to 0A ; prove the triangle EOC equilateral.

8. Given that OPQ is an equilateral triangle'; show that PQ must be equal to os.

9. Given that CEO and CER are both equilateral; prove the quadrilateral OCRE equilateral.

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Nos. 10–17 refer to Diagram II., in which it must be remembered all the sides of the small triangles are equal. 10. Show that AQ is equal to AX.

Proof
Since AP is equal to AY,
and PQ is equal to YX;

therefore the whole which is formed by adding PQ to AP is equal to the whole formed by adding YX to AY;

(Ax. 2) that is, AQ is equal to AX. 11. Given AB is equal to AD; prove that BP is equal to DY. 12. Given that YZ is less than XT; show that HT is greater than EZ. 13. Given FL less than GM; prove RL less than SM. 14. Prove AQ equal to BP. 15. Given LF half of RE, and EZ half of FY; prove LF and EZ equal.

16. Given that quadrilateral APEY is equal to twice triangle PEY, and PQEY is double of the same triangle; prove that APEY is equal to PQEY.

17. If FE bisects the quadrilateral QFHE, and FH bisects EFGH; prove the triangles FQE and FGH are equal.

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19. MODIFIED USE OF SOME AXIOMS. We have already referred to modifications of Axioms 4 and 5, in which account is taken of the lesser and the greater in each pair of unequals. We may, and often do, apply other modifications of the axioms, such as the following; which are respectively related to axioms 2, 3, 4, 5, 6, 7:

If the same be added to equals, or if equals be added to the same, wholes are equal.

If the same be taken from equals, or equals from the same, the remainders are equal.

If the same be added to unequals, the wholes will be unequal, that one being the greater which contains the greater of the former unequals.

If the same be taken from unequals, the remainders will be unequal, &c.
Things which are doubles of equals are equal.
Things which are halves of equals are equal.

The learner is not advised to commit these to memory, but simply read them attentively, before working the next exercise.

The word · bisect' means to divide into two equal parts, that is, into halves.

One thing is said to be common' to two others when it belongs to both of them.

EXERCISE II. (Diagram II. is referred to throughout.) 1. Given the triangles APY, PEQ equal, show the quadrilaterals APEY, PQEY are equal.

Proof. It is given that the triangle APY is equal to the triangle PEQ; add to each the same triangle PEY;

then the whole APEY is equal to the whole PQEY. (Ax. 2) 2. Given the quadrilaterals PQEY, PEXY equal, prove the trapeziums AQEY, APEX are equal.

3. Given the quadrilaterals EFGH, FGVH equal, prove the triangles EFH, GVH equal.

4. Given the trapeziums СUGR, CVGS equal, prove the triangles GRS, GUV equal.

5. If GS bisects the quadrilateral GRSU, and SU bisects CUGS, prove the two quadrilaterals are equal.

6. The same being given, prove the triangles SCU, GRS equal.

7. If all the angles round the point E are equal to one another, prove the angles PEX, FEX are equal.

8. Also the angles PEX, HEY.

9. Given the angles FGV, HGU are equal, prove the angles FGH, VGU are also equal.

10. If all the angles of the triangle FEH are equal, also those of the triangle FGH, and also the angles FEH, FGH; prove the angles EFG, EHG are equal; and each double of the angle FEH.

20. SUPERPOSITION. Superposition means, literally, the placing of one magnitude upon another ; which we may do when we wish to compare their sizes or magnitudes. This is a very familiar process, and Euclid adopts it as his test of equality in our next axiom.

This axiom is very important. The three succeeding ones are somewhat related to it, and are given with it.

AXIOMS 8–11. 8. Magnitudes which coincide with one another, that is, which exactly occupy the same space or position, are equal to one another.

9. The whole is greater than its part. 10. Two straight lines cannot enclose a space. 11. All right angles are equal to one another.

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