Fugue (Prout)/Chapter 6

From Wikisource
Jump to navigation Jump to search

CHAPTER VI.


THE EXPOSITION AND COUNTER-EXPOSITION.


178. If the student clearly understands how to answer a fugue subject, and how to write a good countersubject against the answer, he will be ready to commence the next stage of his work—the composition of a complete exposition of a fugue. By the exposition, as mentioned in Chapter I. (§ 11), is meant that part of the fugue during which the voices make their first entries in succession, and which extends as far as the conclusion of the subject or answer (as the case may be) by the voice that enters last. If the student has mastered the preceding chapters of this volume, and has had sufficient practice in counterpoint to be able to add free parts to two parts that are in double counterpoint with one another (Double Counterpoint, Chapter VII.), he will find that the tasks now before him will offer him but little difficulty.

179. The first question to be considered is the order in which the different voices should enter in the exposition. The subject may in the first instance be announced by any voice, but the order of the subsequent entries will largely depend upon what voice has led.

180. In order to understand this clearly, it is needful to bear in mind the fact that the answer should always be at a distance of a fourth or fifth above or below the subject. It is also necessary to remember that the compass of the alto voice is about a fourth below that of the treble; that the tenor is an octave below the treble, and the bass an octave below the alto. We are speaking here of vocal music; but the parts in an instrumental fugue are mostly treated pretty much as if they were voice parts. For the purposes of fugal answer, we group the voices in pairs, the higher pair being the treble and tenor, and the lower the alto and bass. If the student remembers the directions for transposing a given subject which are given in Counterpoint, § 53, he will be aware that a subject given in the treble must be transposed a fourth or fifth lower for the alto, an octave lower for the tenor, and an eleventh or twelfth lower for the bass.

181. If we apply this principle to the matter now under consideration, we shall see that a subject announced by one of the higher pair of voices should be answered by one of the lower pair, and vice versa. This is the usual practice of fugue writers, though occasional exceptions, with which we need not now concern ourselves, are to be met with.

182. Our next question is, By which voice of the other pair (supposing the fugue to be for four voices) should the answer be given? If, for instance, the treble leads with the subject, should the alto or the bass have the answer? It is seldom difficult to decide this point, if we remember that it is generally best for the last entry in the exposition to be in an outer, rather than in a middle voice. The reason for this is that it is easier to distinguish the subject or answer when it is in an outer part, especially in fugues with four or five voices. In three-part fugues, owing to the thinner harmony, an entry in a middle part can be more clearly heard. In the forty-eight fugues of Bach's 'Wohltemperirtes Clavier,' the last entry is in an outer part in no fewer than forty-two—seven out of every eight.

183. In a two-part fugue, the exposition will be a very simple matter. One of the voices (it is indifferent which) leads with the subject; the other follows with the answer, which the leading voice accompanies with the counterpoint or countersubject, as the case may be, and the exposition is complete.

184. In a three-part fugue, the operation is somewhat longer. If one of the outer parts has the subject, the middle part usually has the answer, and the remaining outer part has the subject again. If the middle part commences, the answer may be equally well in either of the outer parts—it is quite immaterial which. Occasionally we find one outer part leading, and the other outer part following, the middle voice being the last to enter. This, however, is rare; out of twenty-six three-part fugues in the 'Wohltemperirtes Clavier' there are only two (Nos. 27 and 28) in which this method is adopted.

185. While the second entering voice gives the answer, the first continues with the countersubject, supposing there to be one; on the entry of the third voice with the subject, the second voice, which has just concluded the answer, continues with the countersubject, as the first voice did before. The countersubject, which previously accompanied the answer, must now be transposed a fourth or a fifth to serve as a counterpoint to the subject. It may also, in the case of a tonal fugue require some modification (§ 170). Meanwhile, the leading voice, having now completed the countersubject, adds a fresh counterpoint, which may also be a second countersubject; in this case the subject and the two countersubjects must be written in triple counterpoint. If the third part be a free part (as as mostly the case), triple counterpoint is not necessary.

186. Here the exposition of a three-part fugue may end; but, for a reason now to be shown, it is often continued a little further. It was said in the last chapter (§ 160) that a countersubject should be written in double counterpoint to the subject, so as to be able to be used both above and below it. But whenever a subject is announced by an outer part, and the other voices enter in regular ascending or descending order, a moment's thought will show the student that the countersubject, if given (as is best) to the voice which has just completed the subject or answer, will always occupy the same relative position to the subject, or answer. If the treble leads, the countersubject will always be above; and if the bass leads, it will always be below. In such cases, in order to show the countersubject during the exposition in both its aspects, as an upper and lower counterpoint, we frequently find one additional entry of subject or answer by the voice that first led, while the last entering voice has the countersubject. If the middle voice leads, it is evident that we shall have the countersubject in both positions without this additional entry, as it will be below the upper part and above the lower one. In such a case, as also when there is no regular countersubject, the additional entry is not needed. What has been said in this paragraph applies equally, under the same circumstances, to fugues with more than three voices.

187. To illustrate the matter now under notice, look at the example given in § 173. Here is an exposition of a three-part fugue, which, as it stands, might be considered quite complete. Before proceeding, however, to the first episode (see Chapter VII.) Bach introduces the answer in the treble (the voice which first led) in order to let the first countersubject (which has hitherto only been heard above the subject) appear below it in the bass—the voice which has just finished the subject.

188. Very frequently in the exposition of a fugue we meet with a codetta (§ 62) either before the first entry of the answer, or between the answer and the second entry of the subject. We must now show when such a codetta is expedient, and when, it is necessary.

189. Before the first entry of the answer a codetta will be necessary in the following cases. First, if the subject begin on the tonic with an accented note (e.g., on the first beat of a bar), and also end on the tonic, and the answer be below, it is clear that, as the answer will begin on the dominant, this note being a fourth below the tonic, cannot be sounded as a harmony note against it in two-part counterpoint.

\new Staff << \key g \major \time 4/4 \mark \markup \tiny { (\italic"a") }
  \new Voice \relative g' { \[ g2^"S" b4 a8 b | c4 e d c | b2 a | g \] }
  \new Voice \relative d' { \stemDown s1 s s d2_"A" } >>

Here it would be very bad to introduce the answer against the last note of the subject; it will be necessary to add a short codetta,

\new Staff << \key g \major \time 4/4 \mark \markup \tiny { (\italic"b") }
  \new Voice \relative g' { \[ g2^"S" b4 a8 b | c4 e d c | b2 a |
    g4 \] a8^\markup \tiny \italic "Codetta." b a4 g | \stemUp
    fis a d cis | b }
  \new Voice \relative d' { \stemDown s1 s s s d2 fis4 e8 fis | g4 } >>
&c.

thus deferring the entry of the answer till the following bar.

190. Another case in which a codetta is needed before the answer enters, is when the subject begins on an unaccented note near the end of a bar, and ends on an accented note at the beginning of a bar.

J. S. Bach. Wohltemperirtes Clavier, Fugue 36.

\new Staff { \override Score.Rest #'style = #'classical \key f \minor \time 2/4 \partial 8 \relative c'' <<
  { \[ c8^"S" | f, f f des'16 bes | e,8 e e16 f g aes |
    bes aes g aes bes des c bes |
    aes[ \] g^\markup \tiny \italic "Codetta." f g] aes bes c d |
    ees d c d ees g f ees } \\
  { r8 R2*3 | r4 r8 f,_"A" | c c c aes'16 f } >> }
&c.


191. A codetta is also sometimes introduced before the answer to avoid the collision of tonic and dominant harmony.

J. S. Bach. Wohltemperirtes Clavier, Fugue 7.

\new Staff { \key ees \major %correcting obvious error
\time 4/4 \mark \markup \tiny { (\italic"a") } \relative b' {
  \[ bes16^"S" g f g ees aes g aes c8 bes r a16 f |
    ees'8 d c4\trill bes16[ \] f' d bes] aes! f' d aes |
  << { g8 aes' g f ees16 c d ees f4 ~ } \\
     { ees,16 d c d bes ees d ees g8 f r d16 bes } >> } }
&c.

Here the implied harmony of the first half of the second bar is clearly that of a full cadence in the dominant. If we introduce the answer on the last note of the subject

\new Staff { \key ees \major \time 4/4 \mark \markup \tiny { (\italic"b") } \relative b' {
  bes16 g f g ees aes g aes c8 bes r a16 f |
  << { ees'8 d c4\trill bes8 aes' g f } \\
     { r2 ees,16 d c d bes ees d ees } >> } }
&c.

the effect, as everyone will feel, is simply atrocious.

192. We much more frequently find a codetta between the answer and the second entry of the subject. The reason in many cases is the same as that spoken of in our last paragraph—to avoid the collision of tonic and dominant harmony. If the subject begins and ends with a note of the tonic chord, such a codetta will mostly be desirable.

J. S. Bach. Wohltemperirtes Clavier, Fugue 8.

\new Staff { \key dis \minor \time 4/4 \mergeDifferentlyHeadedOn \mergeDifferentlyDottedOn \relative a' <<
  { R1 R1 | r2 ais4^"A" dis ~ | dis8 fis eis dis cis dis eis4 |
    ais, dis4. cis8 bis4 | ais8 } \\
  { \[ dis,4_"S" ais'4. b8 ais gis | fis gis ais4 dis, gis ~ |
    gis8 fis eis4 dis4. \] eis8 | fis4 gis ais4. gis8 |
    fis eis fis eis16 dis eis8 ais4 gisis8 | ais4. } >> }

The student will see at once that if the third voice enters here on the last note of the answer, there will be the same unpleasant effect as in example (b) of our last paragraph. Bach therefore inserts two bars of codetta before the entry of the next voice.

193. If the subject begins with the tonic and ends on the third or fifth of the tonic, a codetta is generally necessary, as the last note of the answer will be the third or fifth of the dominant, against both of which the tonic is a dissonance.

J. S. Bach. Wohltemperirtes Clavier, Fugue 22.

 \new Staff \relative b' { \key bes \minor \time 2/2 \mark \markup \tiny { (\italic"a") } \override Score.Rest #'style = #'classical <<
  { \[ bes2^"S" f | r4 ges' f ees | des \] c des ees |
    f e8 f g2 ~ | g4 } \\
  { R1 R1 | f,2_"A" bes, | r4 des' c bes | aes } >> }


J. S. Bach. Wohltemperirtes Clavier, Fugue 17.

 \new ChoirStaff << \override Score.Rest #'style = #'classical
  \new Staff \relative a { \key aes \major \time 4/4 \mark \markup \tiny { (\italic"b") }
    \[ r4^"S" aes8 ees' c aes f' des |
    ees4 ~ \] ees16 des c des ees f g ees aes bes c bes |
    aes g f aes g2 }
  \new Staff \relative e { \clef bass \key aes \major
    R1 | r4 \[ ees8^"A" aes g ees c' aes | bes[ \] ees, bes' g] s4 } >>

In both these examples the answer ends with the tonic chord of the dominant key and of this chord the original tonic forms no part.

194. We will now write an exposition of a fugue in three parts, taking a very simple and straightforward subject, and adding a regular countersubject.

\new ChoirStaff << \override Score.Rest #'style = #'classical \override Score.BarNumber #'break-visibility = #'#(#f #f #f)
  \new Staff \relative d' { \key d \major \time 4/4 \numericTimeSignature
    \[ d4^"S" g fis r8 b16 a |
    g8 fis e4 d8[ \] d16^\markup \tiny \italic "Codetta." e] fis8 gis
    \[ a8^"CS" cis4 b a16 gis a8 cis_\markup \tiny "(a)" |
    fis,16 gis a4 gis8 a[ \] b16 a] g8 a16 g |
    fis g a b cis d e8 a,4. d8 ~ | d a g4 fis8 r r4 |
    \[ a^"A" d cis r8 fis16 e | d8 cis b4 a8 \] }
  \new Staff \relative a { \clef alto \key d \major \numericTimeSignature
    R1 R1 \[ a4^"A" d cis r8 fis16 e |
    d8 cis b4 a8[ \] a b cis] |
    \[ d8^"CS" fis4 e d16 cis d8 fis |
    b,16 cis d4 cis8 d[ \] fis16^\markup \tiny "(b)" e] d8 e16 d |
    cis8 e fis16 e fis gis a8 e g4 ~ |
    g8 e ~ e16 fis e d cis8 }
  \new Staff \relative d { \clef bass \key d \major \numericTimeSignature
    R1*4 \[ d4^"S" g fis r8 b16 a |
    g8 fis e4 d8[ \] d16 e] fis8 gis |
    \[ a8^"CS" cis4 b a16 gis a8 cis^\markup \tiny "(c)" |
    fis,16 gis a4 gis8 a \] } >>

We have begun with the upper voice, to let all the parts enter in regular descending order. Notice first how, by means of syncopations, we have carefully contrasted the countersubject with the subject. At (a) will be seen the interval of a perfect fifth between the two voices. This is generally forbidden in two-part double counterpoint; it is possible here because its inversion, the perfect fourth (see (c) in the seventh bar), can be used as an accented auxiliary note.

195. We have considered the subject as ending on D, in the second bar. This necessitates a codetta (§ 189), which is imitated by each of the following voices. This is a case of very common occurrence. It would have been possible here to consider the subject as extending to the beginning of the third bar. In this case, the answer would have been tonal; we regard the subject as ending on the tonic, so as to illustrate the employment of the codetta.

196. On the entry of the bass with the subject, the alto, which had the answer, takes the countersubject, and the counterpoint of the treble is free. Little difficulty will be experienced in adding the free parts to the subject and countersubject by any student who has mastered Chapter VII. of Double Counterpoint.

197. The exposition might end at (b); but we have introduced the additional entry spoken of in § 186, to show the countersubject below as well as above the subject. This entry is made by the voice that began; but as this is a three-part fugue, it will be seen that the treble, which before had the subject, now has the answer. Notice also at (b) the rest before the entry in the treble. It is generally advisable, though not absolutely necessary, to let a rest precede the re-entry of the subject or answer.

198. The student should now take the subjects given at the end of Chapter IV., and write expositions on them in three parts. He will do well to write two or three expositions at least on the same subject, with different countersubjects, and altering each time the order of entry of the voices. He will be surprised to find how much variety he can obtain by this means in the treatment of the same subject.

199. In the exposition of a four-part fugue much greater variety is possible in the order of entry than with only three parts. But, for reasons already given (§§ 180–182), only a few of the possible twenty-four changes in the order are in actual use. It is doubtful whether, except in the cases of the irregular expositions, to be noticed later in this chapter, any instance can be found of an answer being given by the tenor to a subject announced in the treble, or by the alto to a subject announced in the bass. In consequence of the difference of pitch between the two pairs of voices, the answer should always be given by a voice of the other pair from that to which the voice belonged which had the subject. We said above, that it was generally best, especially in a four-part fugue, that the last entry be in an outer part. Consequently the orders of entry most frequently met with, and best, are the following four:—

  1. Treble, Alto, Tenor, Bass.
  2. Bass, Tenor, Alto, Treble.
  3. Alto, Tenor, Bass, Treble.
  4. Tenor, Alto, Treble, Bass.

The following are also possible, and sometimes to be met with, but are less good:—

  1. Treble, Bass, Tenor, Alto.
  2. Bass, Treble, Alto, Tenor.
  3. Alto, Treble, Bass, Tenor.
  4. Tenor, Bass, Treble, Alto.

200. In general, subject and answer should enter alternately throughout the exposition; and if the student examines the eight orders of entry just given he will see that in every case one pair of voices will have the subject, and the other the answer. This is by far the most common method, but there are occasional exceptions. In the 41st fugue of the 'Wohltemperirtes Clavier' the order of entry is Alto, Treble, Tenor, Bass, and the alto and tenor have the subject, the treble and bass the answer. Sometimes also another deviation from the regular practice is met with. In the first fugue of the same work, the order of entry is the same as that just noted—Alto, Treble, Tenor, Bass; but here the third voice (the tenor) has the answer, and the fourth (the bass) the subject. In two other fugues of the same work (Nos. 12 and 14) the fourth voice, instead of the answer, has an additional entry of the subject. Here, to avoid two immediate entries of the subject, the third and fourth entries are separated by a rather long codetta. We advise the student in his first attempts to adhere to the usual plan, introducing subject and answer alternately, and adopting one of the four preferable orders of entry given in the last paragraph.

201. In other respects an exposition for four voices resembles one for three, excepting that on the entry of the fourth voice two free parts must be added to the subject and countersubject, instead of only one. The additional entry spoken of above may be used at discretion if an outer part leads.

202. The exposition of a fugue may end in either the tonic or dominant key. If the subject end in the tonic, the answer will end in the dominant; in this case, if the number of voices engaged in the fugue be an even number (two, four, &c), the exposition will end in the dominant; if the number be odd (three or five), the exposition will end in the tonic. If the subject ends in the dominant, the case will be reversed; with an even number of voices the exposition will end in the tonic, and with an odd number in the dominant.

203. Exceptionally, cases are met with in which all the voices of a fugue do not take part in the exposition. The 26th fugue of the 'Wohltemperirtes Clavier' is an instance. The exposition, which occupies the first four bars, is for three voices only, and the fourth voice does not enter till the 19th bar, when it brings in the subject by augmentation, and the last ten bars of the fugue are for four voices. A similar example will be seen in Bach's Organ Fugue in C major—

\relative c' { \key c \major \time 2/2 c4 ~ c16 b c d e8 c a fis' | g8 }

of which the first 48 bars are a four-part fugue, without pedals. At the 49th bar the pedals enter with the subject in augmentation (as in the case last mentioned), and from this point to the end the fugue is in five parts.

204. We now give an exposition of a four-part fugue, taking the same subject as before, but adding an entirely different countersubject and beginning with a middle voice. We choose the alto, so as to retain the same key.

\new ChoirStaff << \override Score.BarNumber #'break-visibility = #'#(#f #f #f)
  \new Staff \relative a' { \key d \major \time 4/4 \numericTimeSignature
    R1*6 \[ a4^"A" d cis r8 fis16 e | d8 cis b4 a8 \] s }
  \new Staff \relative d' { \clef alto \key d \major \numericTimeSignature
    \[ d4^"S" g fis r8 b16 a |
    g8 fis e4 d8[ \] fis^\markup \tiny \italic "Codetta." e d] |
    \[ cis8^"CS" d16 e fis8 gis a e a4 ~ |
    a ~ a16 gis fis gis a8[ \] d,16 e] fis8 e |
    d d' cis b a4. g16 a | b8 a g a16 g fis8 d16 e fis e fis gis |
    a gis fis e d8 e ~ e fis16 g a8 fis ~ |
    fis16 gis a8 e b' e,16 fis e d_"&c." }
  \new Staff \relative a { \clef tenor \key d \major \numericTimeSignature
    R1*2 \[ a4^"A" d cis r8 fis16 e |
    d8 cis b4 a8[ \] b a g! |
    \[ fis^"CS" g16 a b8 cis d a d4 ~ |
    d ~ d16 cis b cis d8[ \] a gis b] |
    e, fis16 gis a8 b cis d16 e fis e d cis |
    b8 cis16 d e fis d e cis d cis b }
  \new Staff \relative d { \clef bass \key d \major \numericTimeSignature
    R1*4 \[ d4^"S" g fis r8 b16 a | g8 fis e4 d8[ \] fis e d] |
    \[ cis^"CS" d16 e fis8 gis a e a4 ~ | a ~ a16 gis fis gis a8 \] s } >>

Notice first that though in § 194 we gave the subject to the treble in the same key, we should have somewhat cramped ourselves in four parts by beginning so low when there were three other voices to come underneath it. The number of the parts in which a fugue is to be written should be taken into account in selecting the voice and pitch of the subject.

205. The student will by this time know enough of harmony and counterpoint to need no help in examining the above exposition. We will only remark that as the countersubject appears in both positions, the additional entry is here unnecessary; and that while our three-part exposition ended in the key of the tonic (not counting the redundant entry), this one ends in the key of the dominant (§ 202).

206. We sometimes meet with irregular expositions of fugues, in which the subject appears twice in succession before the answer is heard at all. The following is an example.

Handel. 'Solomon.'

\new ChoirStaff <<
  \new Staff \relative f' { \key f \major \time 2/2 <<
    { R1*4 r2 f | f e | a2. a4
      g2 g | f2. g4 | a bes c2 | f, f' | f e | f d | d c |
      d2. d4 | c1 ~ | c2 bes | bes a | bes f | f1 } \\
    { R1*14 bes,1 bes2 a | d2. d4 |
      c2 c | bes2. c4 | d ees f2_"&c." } >> }
  \new Staff \relative f { \key f \major \clef bass <<
    { r2 f | f e | a2. a4 | g2 g | f2. g4 | a bes c2 | c, f' |
      f e | f d | d c | d2. d4 | c1 | r2 bes4 c | d e f2 |
      f bes, | f'1 ~ f2 f | ees2. ees4 | d1 ~ | d2 c } \\
    { R1*18 r2 bes, | bes a } >> } >>


To save space we have given this passage in "short score." Here, as the subject is repeated, the two voices of the same pair follow each other; the alto and bass enter with the answer. Notice, in passing, that we have here another example of an answer in the subdominant (§71). A fugue beginning in this way is sometimes described as an "Octave Fugue" to distinguish it from the ordinary fugue, in which the second entry is at a distance of a fourth or fifth from the first. In the familiar chorus, "Fallen is the foe," in Handel's 'Judas,' will be seen a fugue in which all the four entries are in the octave.

207. In some fugues the exposition is followed, either immediately or after the first episode (which will be described in our next chapter), by what is called a Counter-exposition. This is really a second exposition in the same two keys (generally tonic and dominant) as the first, but with important differences. The chief of these are that in the counter-exposition the voices which before had the subject now have the answer, and vice versa; and that frequently the answer leads and the subject replies. In fugues 1 and 11 of the 'Wohltemperirtes Clavier' will be seen examples of the former, and in Nos. 26 and 33 of the same work, illustrations of the latter. Sometimes, as in the first fugue, the counter-exposition follows immediately on the close of the exposition; at others (as in Fugue 11) it is separated from it by an episode. To save space, we simply refer to these pieces without quoting them; we may fairly suppose that everyone who wishes to study fugal construction has a copy of the 'Forty-Eight' by him for reference. If not, the sooner he gets one, the better. It should be added that in the counter-exposition the entries of the voices are generally accompanied by free counterpoint in the other voices; it is seldom that the leading voice in the counter-exposition is found (as in the exposition) unaccompanied.

208. In many cases, when there is a counter-exposition, it is only partial; that is to say, not all the voices of the fugue take part in it. For example, in Fugue 38 of the 'Wohltemperirtes Clavier,' which is for three voices, the exposition, which ends at the 11th bar, is followed by the first episode. In bar 14 begins the counter-exposition. The bass, which before had the subject, now leads with the answer, which is now real, though it was at first tonal (§ 109). Before the completion of the answer by the bass, the treble, which before had the answer, enters with the subject (bar 16), but the close of the subject in the treble is followed by the second episode, the alto not entering in the counter-exposition with either the subject or answer, and having only free counterpoint throughout.

209. In the passage just referred to, we said that the treble entered with the subject "before the completion of the answer by the bass"; and this leads us to notice a feature very often to be met with when there is a counter-exposition. In many cases this portion of the work is used to introduce the first stretto,—that is to say, in the counter-exposition, the entries of the subject are closer together than in the first exposition. For instance, in the fugue just noticed, the answer enters in the exposition three bars later than the subject; but in the counter-exposition it is only two bars later. In the first fugue of the 'Forty-Eight' the answer in the counter-exposition follows the subject at only one crochet's distance, instead of a bar and a half; and in the 31st fugue of the same work the counter-exposition takes the form of a canon at one bar's distance first at the fifth below, between tenor and bass, and then at the fourth above, between alto and treble. Considerably more latitude is allowed as to the entries in the counter-exposition than in the exposition.

210. Occasionally we find a counter-exposition by inversion. In the fugue in G major, No. 15 of the 'Forty-Eight,' we shall find an example of this kind between the 20th and the 31st bars; and in the 46th fugue we shall find an elaborate counter-exposition, beginning at bar 42, in which not only the subject but the countersubject is inverted.

211. It must be clearly understood that the introduction of a counter-exposition into a fugue is purely optional. The larger number of fugues do not contain one at all. We only find one in thirteen out of the 48 fugues in the 'Wohltemperirtes Clavier'; and this is probably a fair average proportion. The instructions given in this chapter will sufficiently show the student how to write a counter-exposition, if he wishes to do so.

212. With the exposition or counter-exposition the first section of a fugue ends. The construction of the middle and final sections will be treated of in the following chapters. If the student can write a really good exposition, he is already well advanced on the road to fugal composition. Let him now take the subjects in Chapter IV., and write on them a series of fourpart expositions, as he has presumably already done in three-parts. He should also write expositions on subjects of his own invention. He must try to get as much variety as possible both in his countersubjects, and in the free added parts; he should also vary his order of entry in many different ways. He must keep his expositions when he has written them, to furnish him with material for the exercises he will have to write on the next chapter.