I have a manuscript with strings and harpsichord. Under the bottom staff of the harpsichord there is numbers, like 7. What are these numbers and what do they represent?
The numbers, as already noted, are "figured bass", where the keyboardist would improvise an accompaniment part. The figures indicate what intervals above the bass note should be represented in the accompaniment part. So, for example, a chord marked "6/5" requires a third, fifth, and sixth above the bass note (the third is typically implied). This is the first inversion of a seventh chord.
Speaking as one who plays a lot of basso continuo, there are some advantages:
The keyboardist can adjust the texture of the accompaniment part to the performance space, the size of the ensemble, and the preferences of the performers.
This offers the performers the opportunity to play the piece slightly differently each time, which appealed to the Baroque mindset.
If the composer didn't want to specify every single note, why bother? There are plenty of situations where they did write obbligato parts that specified every note (see, for example, the Bach viola da gamba sonatas). Remember that printed music was rare and most keyboardists knew how to do this, so at times the composer chose to let the performer sketch in the details.
For a really beautiful and aggravating example of Bach writing all kinds of counterpoint into a figured bass part, see the e minor sonata.
More about figured bass, in support of @KimFierens' answer:
The figured bass numbers and symbols on your scores were written by the original composer in the Baroque era, before about 1775. That was the only system they used back then for identifying chords. Our modern system of chord names had not been invented yet. That is why they don't write out modern chord names on those scores.
In the Baroque era, one finds two kinds of sheet music for harpsichord. In the first, usually for solo harpsichord, all the notes to be played are written out for the left and right hand on two staves of music, like on the modern piano. There may be some figured bass notation in the left hand of the harpsichord part, but in most cases there is not.
The second kind is notation for a musical ensemble, such as a singer, a string section, and harpsichord. In this example, the harpsichord is part of what is called basso continuo. (Today we call this the "rhythm section".) In this case the entire harpsichord part consists only of one staff of a bass clef with only a bass line (one note at a time) written out, together with the figured bass notation symbols.
Basso continuo notation is played like this: a bass instrument (upright bass, cello, or bassoon) plays the bass notes as written. A continuo instrument (harpsichord, or organ, theorbo, or lute) also plays the bass line in unison with the bass instrument, but the continuo player also improvises a chord accompaniment (in the right hand if you are talking about harpsichord or organ). The player is expected to figure out their own notes and chord voicings, observing and interpreting the figured bass symbols in order to play the correct chords. A good harpsichord/continuo player will never play the same piece exactly the same way twice, because it should be improvised. But the performer needs to understand all the rules about playing continuo from figured bass that were generally in use in the time period.
Because not many keyboardists these days learn how to improvise a continuo from figured bass symbols, there exists a modern type of sheet music called a realized continuo. For example, if you are hired to play harpsichord or organ in Handel's Messiah in an orchestra but you don't know how to improvise over a figured bass, you can purchase one or more printed scores where an editor (in the modern era) has written out every note for the left and right hands in their own version of a continuo. In other words, the editor has realized and improvised a full accompaniment and has then written out every note of the improvisation for the left and right hands. As a result you can play Handel's Messiah (and play it the same way every time) and achieve something musically acceptable for the orchestra, even though you are not really following the spirit of the Baroque because you aren't improvising a different performance every time you play a piece like a harpsichordist would in Handel's time.
There are many resources for reading and interpreting Baroque figured bass notation and using it. Kim Fierens has provided a link to one resource. There are also resources for the rules and practices for a keyboardist in improvising a continuo.
And if you are really curious, there are also resources for improvising continuo on the lute and the baroque guitar.
All modern Western music theory textbooks make extensive use of Baroque figured bass notation, adapting it to the study of chord progressions used in chorales. In modern parlance this is sometimes called bass position notation.
Figured bass and basso continuo notation continued to be used throughout the brief years of the style period we call the Classical period, and into the Romantic period as well, although it was in sharp decline by then. The last piece of historical music that I'm aware of that used figured bass and basso continuo is Beethoven's Mass in C, published in 1812.
What you're seeing is probably figured bass notation. The numbers indicate which chords should be played over the bass notes. A note with a 7 underneath it, for example, is shorthand for an uninverted seventh chord with that note as root.
See this link for a more detailed explanation: link.
If I may ask why don't they write out the chord as it is? We need to learn a whole set of notation. What did they earn by doing it this way?– NachmenDec 10, 2015 at 12:52
3Basso continuo playing almost always involved at least some degree of improvisation. In fact, this improvisatory "feel" was and is a major part of the aesthetic appeal of baroque music. The figures serve as a sort of guide, with which composers could make their harmonic intentions known, without having to spell everything out — leaving it to the performers to fill in the details while rehearsing/playing (this is called "realizing the bass"). Good keyboard and lute players were invariably adept at this art, often improvising quite elaborate realizations. Dec 10, 2015 at 13:54
1For an example of such florid continuo/figured bass playing, see: youtube.com/watch?v=RDvAOPKAUvs Dec 10, 2015 at 13:58
2@Nachmen Partly this is because musicians before the mid-eighteenth century did not generally think in terms of "chords." In JS Bach's book on playing continuo (which I recommend), he says simply that the continuo player plays the bass line with his left hand and plays consonances and dissonances above the bass with his right hand, according to the figures (=numbers). The figures just specify intervals above the bass. These happen to form what we may call "chords" but most musicians of that time thought in terms of contrapuntal lines (also known as "voice leading"). Dec 10, 2015 at 17:14