VI. 1. Write out the three forms of the diatonic minor in all the keys. 2. For what purpose are pieces written in minor mode at all? 3. Which note of the minor mode is the most effective? 4. If I sing "Do" sharp instead of "Do" in a minor phrase, what effect is produced? 5. Make a collection of minor mode phrases and comment on them. VII. 1. I find the signature of a piece of manuscript music to be two sharps placed on A and E; comment upon this. 2. In what major scale will E be sol? Give the signature. Do the same for keys in which E is fa, la, do, re, and si. 3. Take the above list of keys and state the relative minor of each. 4. Draw the great stave and write the following air in the treble staff. Then add the octave below to each note. Then bar it in compound common time with proper signature. A, G, F, Bb, A, C, D, E, C, B', F, D, Bb, G, C, C, G, A, F. The 4th, 9th, 14th, are crotchets. The last F is a dotted crotchet. All the rest are quavers. 5. Some tunes have been written in two sorts of time, duple and triple. What difference in effect will be noticed in contrasting them ? VIII. 1. What do you understand by intensity of sound? and what marks are used to show degrees of intensity in a piece of music? 2. The following signs are noticed in a musical exercise; explain the meaning of each: M=60; f; ff; p; pp; mf; cresc; dim; >; Largo ; Andante; Allegretto; DC; D$. 3. How are pauses provided for in written music? give a list of the necessary signs, GEOMETRY. EUCLID. BOOK I (27–48). Proposition XXVII. Theorem. If a straight line falling on two other straight lines makes the alternate angles equal to one another, these two straight lines shall be parallel. Let the straight line E F, which falls on the two straight lines A B, C D, make the alternate angles A E F, E F D equal to one another. Then A B shall be parallel to C D. For, if A B be not parallel to C D, A B and CD, being produced, will meet Let them be produced and meet towards B, D, in the point G. and its exterior angle A E F is greater than its interior But the angle A E F is equal to the angle E F G (hyp.) Therefore the angle A E F is both greater than, and equal to the angle E F G; which is impossible. Therefore A B, C D, being produced, do not meet towards B, D. In like manner it may be demonstrated that they do not meet when produced towards A, C. But those straight lines in the same plane, which meet neither way, though produced ever so far, are parallel to one another (Def. 35). Therefore A B is parallel to CD. Therefore, if a straight line falling on two other straight lines, etc. Proposition XXVIII. Theorem. Q. E. D. If a straight line, falling upon two other straight lines, makes the exterior angle equal to the interior and opposite angle upon the same side of the line, or makes the interior angles upon the same side together equal to two right angles, the two straight lines shall be parallel to one another. Let the straight line E F, which falls upon the two straight lines A B, C D, make the exterior angle E G B equal to the interior and opposite angle G H D upon the same side; or make the interior angles on the same side B G H, GHD together equal to two right angles. Then A B shall be parallel to C D. E G H Because the angle E G B is equal to the angle G H D, and the angle E G B is also equal to the angle A G H (I. 15), therefore the angle A G H is equal to the angle G H D (Ax. 1); and these are alternate angles; therefore A B is parallel to C D (I. 27). Again, because the angles B G H, G H D are together equal to two right angles, and that A G H, B G H are also equal to two right angles (I. 13), therefore the angles A G H, B G H are equal to the angles BGH, GHD (Ax. 1). Take away the common angle B G H ; therefore the remaining angle AGH is equal to the remaining angle G H D ; and these are alternate angles; therefore A B is parallel to C D (I. 27). Therefore, if a straight line, falling upon two other, etc. Proposition XXIX. Theorem. Q. E. D. If a straight line falls upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle upon the same side, and also the two interior angles upon the same side together equal to two right angles. Let the straight line E F fall upon the parallel straight lines A B, C D. Then, the alternate angles A GH, G H D shall be equal to one another, the exterior angle E G B shall be equal to the interior and opposite angle G H D upon the same side of E F, and the two interior angles B GH, GHD upon the same side of E F shall be together equal at two right angles. E A. C H For, if the angle A G H be not equal to the angle G HD, one of them must be greater than the other. Let A G H be greater than G H D. To each of these add the angle BGH, therefore the angles A GH, B G H are greater than the But the angles AGH, B G H are equal to two right angles (I. 13), therefore the angles B G H, G H D are less than two right angles. But if a straight line meets two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles that are less than two right angles (Ax. 12). Therefore the straight lines A B, C D, if produced far enough, will meet towards B and D. But they never meet, since they are parallel (hyp.) ; therefore the angle A G H is not unequal to the angle G H D, that is, the angle A G H is equal to the angle G H D. But the angle A G H is equal to the angle E GB (I. 15), therefore also the angle E G B is equal to the angle G H D (Ax. 1). Add to each of these the angle B G H, therefore the angles E G B, B G H are equal to the angles B GH, GHD (Ax. 2). But E G B, B G H are equal to two right angles (I. 13), therefore also BGH, G H D are equal to two right angles (Ax. 1). Therefore, if a straight line falls on two parallel straight lines, etc. Q. E. D. Proposition XXX. Theorem. Straight lines which are parallel to the same straight line are parallel to one another. Let the straight lines A B, C D be each of them parallel to E F. Then A B shall also be parallel to CD. |