Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

21. Explain Cardan's Rule for solving cubic equations. Art. 308.

22. What are the roots of the equation x—7x2+14x=_V. Ex. 3. P. 381.

23. Give the Preliminaries to Horner's method. Art. 309. 24. Explain the Principles involved in Horner's method. Art. 300-314.

25. Find the roots of the equation x3-10x+6+1=0. Ex. 4. P. 400.

NOTE.-Many additional questions might have been proposed, but candidates who can answer satisfactorily the above questions need fcel no embarrassment in an examination in this science.

CHAPTER XIV.

Every Teacher of our First Class Schools is now expected to be well qualified in the Higher Mathematics. Hence a few questions are proposed in

GEOMETRY.

The references in the questions on Geometry are to "Davies' Legendre." B. stands for Book. D. for Definition. P. for Proposition. C. for Corotlary.

1. Define Geometry.

ANS.-Generically it means the art of measuring the earth; but as it is now used Geometry denotes the Science of magnitude in general,—the mensuration of lines, surfaces, solids, with their various relations.

[blocks in formation]

4. What is a Broken line? Curvea line? Surface? Plane? B. I. D. 8-11.

5. What is a Curved Surface? Plane Angle? Right-Angle? B. I. D. 12-14.

6. What are Oblique Angles? How many kinds? B. I. D. 15.

7. Define an Acute Angle. An Obtuse Angle. B. I. D. 15. 8. When are lines parallel? What is a plane figure? B. I. D. 16, 17.

9. What is a Polygon? Triangle? Hexagon? Octagon? B. I. D. 19.

10. Define an Equilateral polygon. Equiangular polygon. B. I. D. 20.

11. When are two polygons mutually equilateral and equiangular? B. I. D. 22.

12. How are Triangles classified? How many classes are there? B. I. D. 23.

13. Define a Scalene Triangle. An Isosceles triangle. B. I. D. 23.

14. Define Equilateral and Acute angled triangles. B. I. D. 23.

15. Define Right-angled triangles, and obtuse-angled triangles. B. I. D. 23.

16. What are Quadrilaterals? Divided into how many

B. I. D. 24.

classes?

17. Define Trapezium. Trapezoid. Parallelogram. B. I.

D. 24.

18. Into how many classes are parallelograms divided? B. I. D. 25.

19. Define a Rhombus. Rectangle. 20. What is a Diagonal? A Base?

Square. B. I. D. 25.
B. I. D. 26, 27.

B. I.

DEFINITION OF TERMS.

1. What is an axiom?

D. 27.

27.

[blocks in formation]

2. What is a Problem? Lemma? Proposition? B. I. D.

3. What is a Corollary? Scholium? Hypothesis? Postulate? B. I. D. 27.

EXPLANATION OF SIGNS.

REMARK. The explanation of the signs in Geometry is the same as is found in the questions on Algebra, which see.

AXIOMS.

1. How many axioms are there? B. I. P. 19.

2. Give them all accurately. B. I. 3. How many Postulates are there?

P. 19.

Give them. B. I. P.

THEOREMS.

1. Demonstrate Proposition I. Theorem. B. I.

2. Demonstrate Prop. IX. B. I. and P. XXV. B.I. Also Prop. XXVIII. B. I.

II.

[blocks in formation]

1. Define Proportion. Ratio. Antecedent. Consequent. B. D. 2.

2. How may the ratio of Magnitudes be expressed? B. II. D. 3.

3. When are magnitudes commensurable? When incommensurable? B. II. D. 3.

4. How will you illustrate the principles found in the last two questions? B. II. D. 4, 5.

5. When are four quantities in Proportion? B. II. D. 6. 6. When is a quantity a fourth proportional to the other three? B. II. D. 7.

7. When are three quantities in proportion? B. II. D. 8. 8. When are magnitudes in proportion by Alternation? B. II. D. 9.

9. When are magnitudes in proportion by Inversion? B. II.

D. 10.

10. When by Composition? When by Division? B. II. D. 11, 12.

11. What are Equimultiples of two quantities? B. II. D. 13.

12. When are two varying quantities reciprocally proportional? B. II. D. 14.

13. Demonstrate P. I, IX and XII. B. II.

OF THE CIRCLE.

B. III.

D. 1. B. III.

1. Define a Circle. Circumference.

2. What is the Radius? Diameter?

How do all the radii of

equal or the same circles, compare in magnitude? D. 2.

B. III.

3. What is an Arc? Chord? Sector? Segment? D. 3, 4,

5. B. III.

4. When is a straight line said to be inscribed in a circle? D. 6. B. III.

5. Define an inscribed triangle.

D. 7. B. III.

6. What is an inscribed polygon? D. 7. B. III.

7. Define a Secant line. A Tangent. D. 8, 9. B. III.

8. What is the point of contact?

9. Define the point of tangency. in a polygon? D. 11. B. III.

D. 9. B. III.

When is a circle inscribed

10. Demonstrate P. IV, VIII, XV and XVIII, in B. III. 11. Demonstrate Problems III, X, XIII and XV, in B. III.

BOOK IV.

1. Define Similar Polygons. D. 1. B. IV.

2. What are homologons, angles and sides? D. 2. B. IV. 3. What do you understand by area? Equivalent figures? D. 4. B. IV.

4. When are two sides of one polygon said to be reciprocally proportional to two sides of another? D. 5., B. IV.

5. What are similar Arcs, sectors, or segments? D. 6. B. IV.

6. What is the Altitude of a triangle? D. 7. B. IV.

7. What is the Altitude of a parallelogram? Of a Trapezoid? D. 8, 9. B. IV.

8. Demonstrate the following Proposition:

"The square described on the sum of two lines is equivalent to the sum of the squares described on the lines, together with twice the rectangle contained by the lines." P. VIII. B. IV.

"The square described on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides." P. XI. B. IV.

"In every quadrilateral inscribed in a circle, the rectangle of the two diagonals, is equivalent to the sum of the rectangles of the opposite sides taken two and two." P. XXXIII. B. IV.

9. Demonstrate Problems 10, 16 and 18. B. IV.

[merged small][ocr errors]

1. What is a regular polygon? D. 1. B. V.

2. How many sides may a regular polygon have? D. 2. B. V.

3. Demonstrate the following Proposition:

"To inscribe a square in a given circle." P. III. B. V.

4. And the following:

"In a given circle to inscribe a regular decagon." P. VI. B. V.

5. Also this Theorem:

"The arc of a circle is equal to the product of the radius by the circumference." P. XV. B. V.

BOOK VI.

1. When is a straight line perpendicular to a plane? D. 1, 3. B. VI.

2. When is a plane perpendicular to a line? D. 2. B. VI. 3. When are two planes parallel to each other? D. 3. B. VI. 4. Define a diedral angle, and the faces and edge of an angle. D. 4. B. VI.

5. What is the measure of a diedral angle? D. 4. B. VI. 6. Define a Polyedral angle. What is the face, edge and vertex of the Polyedral angle? D. 5. B. VI.

7. Demonstrate the following:

"Two planes which are perpendicular to the same straight line are parallel to each other." P. IX. B. VI.

8. "If two straight lines be cut by three parallel planes, they will be divided proportionally." P. XV. B. VI.

9. "The sum of either two of angle is greater than the third."

the plane angles which include a triedral P. XIX. B. VI.

BOOK VII.

1. Define a Polyedron Prism. Base of the prism. D. 1, 2. B. VII.

2. Describe the convex surface of a prism. D. 3. B. VII. 3. Define the altitude of a prism. What is a right prism? D. 5. B. VII.

4. What is a triangular prism? Parallelopipedon? B. VII.

5. What is a pyramid? Altitude of a pyramid? D. 9.

D. 7.

B.

« ΠροηγούμενηΣυνέχεια »