2. How does the chromatic minor differ from the diatonic

minor?

3. How comes it that music scales and methods should be liable to alteration?

4. Make a list of all diatonic and chromatic intervals found in the chromatic minor scale.

5. Give a simple rule for at once reading off in the minor any major tune.

III.

1. What do you understand by enharmonic?

2. On what instrument can the enharmonic scale be illustrated? On what instrument can it not? Give your reasons for this as fully as you can.

3. Whereabouts in the enharmonic scale does the diesis occur? 4. If Do is somewhat higher in pitch than Reb, and so on with the rest, what will be the natural arrangement of the intervals to secure the continuous ascending and then descending enharmonic scale?

5. Why do you omit E and Fb in the above series? In what other part of the scale is there a similar omission?

IV.

1. Give the etymology of transposition, and show how the term applies to music.

2. What transposition is exceedingly common?

3. To what extent could the National Anthem be transposed? and why should it or any other tune be so treated?

4. Take a few bars in each of the different tune measures, and transpose them twice.

5. A tenor song in Bb is required for a bass voice; point out the transposition necessary.

V.

cres

1. Give examples of the use of the signs "staccato," 99.66 cendo," "diminuendo," "fortissimo," "piano," "da capo," "dal segno,” and supply the abbreviations.

2. Which of the above affect the intensity of the music? What do you mean by intensity? and how is the required intensity of a note secured?

3. What signs are used to affect the rate of movement, the rhythm, the repetition?

4. If you have access to any of the following, state (1) what they are, and (2) what variety of compositions they contain :Mendelssohn's "Athalie."

5. Select convenient passages to transpose, and name intervals, chromatic and diatonic.

GEOMETRY.

EUCLID. BOOK II.

DEFINITIONS.

1. Every right-angled parallelogram is called a rectangle, and is said to be contained by any two of the straight lines which contain one of the right angles.

2. In every parallelogram, any one of the parallelogram about a diameter, together with the two complements, is called a Gnomon.

Thus the parallelogram H G, together with the complements A F, FC, is a gnomon, which is more briefly expressed by the letters AG K, or EH C, which are at the opposite angles of the parallelograms which make the gnomon.

Proposition I. Theorem.

If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line and the several parts of the divided line.

Let A and B C be two straight lines, and let B C be divided into any parts in the points D, E.

Then the rectangle contained by the straight lines A, B C shall be equal to the rectangle contained by A, B D, together with that contained by A, D E and that contained by A, E C.

From the point B draw B F at right angles to B C (I. 11). Make B G equal to A (I. 3).

Through G draw G H parallel to B C (I. 31),

and through D, E, C draw D K, E L, C H parallel to B G. Then the rectangle B H is equal to the rectangles B K, D L,

E H.

But B H is contained by A, B C,

for it is contained by G B, B C ; and G B is equal to A : B K is contained by A, B D,

for it is contained by G B, BD; and G B is equal to A: D L is contained by A, DE,

for it is contained by D K, DE; and D K is equal to B G,

or A :

and in like manner E H is contained by A, E C.

Therefore the rectangle contained by A, B C is equal to the rectangles contained by A, B D ; by A, D E ; and by A, E C. Therefore, if there be two straight lines, etc.

Proposition II. Theorem.

Q. E. D.

If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square on the whole line.

Let the straight line A B be divided into any two parts in the point C.

Then the rectangle A B, B C✶ together with the rectangle A B, A C shall be equal to the square on A B.

Upon A B describe the square A D E B (I. 46),
through C draw C F parallel to A D or B E (I. 31).
Then A E is equal to the rectangles A F, CE.
But A E is the square on A B

and A F is the rectangle A B, A C,

for it is contained by D A, A C, and D A is equal to A B : and CE is the rectangle A B, BC

for B E is equal to A B :

therefore the rectangle A B, BC together with the rectangle

А В, АС

is equal to the square on A B.

Therefore, if a straight line, etc.

Proposition III. Theorem.

Q. E. D.

If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts is equal to the rectangle contained by the two parts, together with the square on the aforesaid part.

Let the straight line A B be divided into any two parts in the point C.

Then the rectangle A B, B C shall be equal to the rectangle A C, C B, together with the square on B C.

*This is the usual mode of indicating the rectangle contained by two straight lines, as A B and B C, and is employed by us in the present and following propositions for the sake of brevity. The square on a line, as A B, may also be called the square of the line.