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Hughes's Pupil Teachers' Examination Manuals.
A COMPLETE SET
MALE PUPIL TEACHERS'
QUESTIONS IN EUCLID,
TO SEPTEMBER 1879 (INCLUSIVE).
Compiled, Classified, and Graduated by
W. J. DICKINSON,
FORMERI.Y LECTURER ON GRAMMAR AND EUCLID AT THE BATTERSEA TRAINING COLLEGE;
PRACTICAL ENGLISH GRAMMAR AND ANALYSIS,' 'the difficulties of GRAMMAR
AND THE DIFFICULTIES OF EUCLID SIMPLARED
LONDON: JOSEPH HUGHES,
EUCLID, BOOK I.
Definitions, Axioms, Postulates, and General Questions, .
Propositions I.-XV. (inclusive),
Propositions XVI.-XXVI. (inclusive),
Propositions XXVII.-XLI. (inclusive),
NOTE. The deductions given at the end are deductions which, in the Examination Papers, were given independently, not being appended to any proposition. Along with the propositions, however, many deductions will be found as they were appended to the propositions, on which they were dependent, in the Examination Papers.
Although the propositions are grouped as above, for the sake of keeping the parts prescribed for the various years of apprenticeship separate (the parts formerly prescribed, as also those at present), the questions are arranged according to the propositions, and numbered accordingly.
Definitions, Axioms, Postulates, and General Questions. 1. Define superficies, right angle, semicircle, acute-angled triangle, and write out the postulates.
2. Define a plane rectilineal angle, a right angle, and a perpendicular, and write out Euclid's three postulates.
3. Define parallel straight lines and a parallelogram. Write out Euclid's twelfth axiom.
4. Define point, straight line, plane angle. A boy coming to the eleventh axiom says, 'How can right angles be equal to one another unless the lengths of the lines forming them are respectively equal?' Answer him.
5. Explain clearly why the following definition is insufficient: -'A plane rectilineal angle is the inclination of two straight lines to one another.' Define obtuse angle, isosceles triangle, rhomboid.
6. Define a plane superficies, a circle, an acute-angled triangle, parallel straight lines. What straight lines are there which though produced to any length never meet, yet are not parallel? 7. Define right-angled triangle, oblong, parallel straight lines, postulate, axiom, problem, theorem. When is one proposition said to be the converse of another?
8. Define point, plane, superficies, obtuse angle, semicircle, oblong, scalene triangle, parallelogram:
9. Draw examples of the different kinds of four-sided figures, plane rectilineal angles, and triangles. Write a definition of each. What is a postulate? Mention those of Euclid.
10. What is an axiom? Give as many of Euclid's axioms as you can.
11. Write out Euclid's definitions of parallel straight lines, a parallelogram, a right angle, and a circle.
12. Give Euclid's definitions of a circle, the centre of a circle, the radius of a circle, the diameter of a circle, a semicircle, and a segment of a circle.
13. Explain the following terms given in Euclid :-Proposition, problem, theorem, enunciation, hypothesis, axiom, postulate, and corollary.
14. Explain the difference between a problem and a theorem. What is a corollary? When is one proposition said to be the converse of another? Give examples from the First Book.
15. Distinguish between problems and theorems. Into what parts may every proposition be divided? What is an indirect demonstration? Define corollary with an example.
16. Define acute angle, figure, semicircle, parallelogram, and write out the axiom on which Euclid bases his reasonings on parallel lines, and also the axiom which is the definition of geometrical equality.
17. What figure is that of which the diagonals are equal, but do not intersect at right angles?
Propositions I.-XV. (inclusive).
1. Describe an equilateral triangle on a given finite straight line.
Same proposition. Is the demonstration direct or indirect? 2. Define a point and a straight line. From a given point draw a straight line equal to a given straight line.
Same proposition. What axioms are applied in this proposition? Same proposition. Draw the figure when the equilateral triangle is described on the side of the line opposite to that taken in your construction.
Same proposition. If you are able, prove this proposition by joining the point to the extremity of the given line which is farther removed from it, instead of the nearer one, as in Euclid.
3. Show how from the greater of two given straight lines to cut off a part equal to the less.
4. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to each other, they shall likewise have their bases or third sides equal, and the two triangles shall be equal, and their other angles shall be equal, each to each, viz. those to which the equal sides are opposite.
Same proposition. If a triangle ABC be turned over about its side AB, show that the line joining the two positions of C is perpendicular to AB.
5. What is a plane triangle, and how many different kinds of triangles does Euclid name? Prove that the angles at the base of an isosceles triangle are equal, and if the equal sides be produced the angles on the other side of the base shall be equal.
Show that every equilateral triangle is equiangular.