Astrologus sphaeram circumagebat, cum aqua subito fluere desiit.

Because their endings show their grammatical roles.

• astrologus is nominative singular, which is the case normally used for the subject.
• sphaeram is accusative singular, which is the case normally used for the direct object.

So:

• astrologus = the astrologer / astronomer
• sphaeram = the sphere

Even if the word order changed, the endings would still tell you who is doing the action and what is being acted on.

circumagebat comes from circumagere, a compound verb:

• circum- = around
• agere = to drive, move, set in motion

So circumagere means something like:

• to move around
• to rotate
• to whirl
• to turn around

In this sentence, sphaeram circumagebat means he was rotating the sphere or he kept turning the sphere around.

The imperfect tense often describes an action that was:

• ongoing,
• repeated,
• or in progress in the past.

So circumagebat means:

• was rotating
• kept rotating
• used to rotate
  depending on context

Here, the most natural sense is was rotating: it sets the background action that was happening when something else occurred.

That is very common in Latin narrative:

• imperfect = background, ongoing action
• perfect = a completed event that happens within that background

This is a very common contrast in Latin.

• circumagebat = an action in progress in the past
• desiit = a completed event, stopped

So the sentence presents:

1. an ongoing action: the astrologer was rotating the sphere
2. a sudden event: the water stopped flowing

Latin often uses this combination to show that one event interrupts or occurs during another.

So the tense contrast is meaningful, not random.

Because desinere often takes an infinitive in Latin.

• desiit = he/she/it stopped
• fluere = to flow

Together, fluere desiit means stopped flowing.

This is similar to English verbs like:

• began to flow
• continued to flow
• stopped flowing

So fluere is in the infinitive because it depends on desiit.

aqua is nominative singular.

It is the subject of desiit and of the infinitive idea fluere:

• aqua ... desiit = the water stopped
• aqua fluere desiit = the water stopped flowing

Even though fluere is an infinitive, the subject is still understood to be aqua.

Here cum most naturally means when.

So:

• Astrologus sphaeram circumagebat, cum aqua subito fluere desiit.
• The astrologer was rotating the sphere, when the water suddenly stopped flowing.

This is a temporal use of cum.

A learner should know that cum can have several meanings, including:

• when
• since
• although

The exact meaning depends on context and on the mood of the verb. Here, with desiit in the indicative, a straightforward temporal when is the best fit.

subito is an adverb meaning suddenly.

It modifies the verbal idea fluere desiit:

• aqua subito fluere desiit = the water suddenly stopped flowing

Latin adverbs are often placed quite flexibly. Here subito comes before fluere desiit, but it could appear in other positions too without changing the basic meaning.

Its main job is to emphasize the abruptness of the event.

Latin word order is much freer than English word order because Latin relies heavily on case endings.

English depends more on position:

• The astrologer turned the sphere

If you swap words in English, the meaning may change. In Latin, endings usually keep the roles clear.

So:

• Astrologus sphaeram circumagebat
• Sphaeram astrologus circumagebat
• Circumagebat astrologus sphaeram

all could mean basically the same thing, though the emphasis would differ.

In your sentence, the order is quite natural for Latin prose:

• subject
• object
• verb
• then a cum clause

Historically, it can be a little broader than the modern English word astrologer.

In many ancient contexts, the study of the stars did not divide neatly into modern astronomy and astrology. So astrologus may refer to someone who studies the heavens, and depending on context it may be translated as:

• astrologer
• astronomer
• star-gazer or student of the stars

If your provided meaning says one of these, follow that meaning for this sentence. But it is good to know that the Latin word can overlap with more than one English term.

Yes. sphaera means sphere, ball, or sometimes a celestial globe depending on context.

In a sentence with astrologus, it could easily suggest an astronomical model or globe. So sphaeram circumagebat may suggest not just any ball, but possibly a scientific or celestial sphere.

That said, grammatically it is just an ordinary accusative singular noun here: sphaeram.

Possibly, depending on how smooth you want the English to sound, but when is usually the most direct choice here.

Because the main verb is imperfect (circumagebat) and the cum clause has a completed event (desiit), the sense is:

• He was rotating the sphere when the water suddenly stopped flowing.

Using while in English can sometimes make it sound as if both actions were simply simultaneous and ongoing. But here one action is ongoing and the other is a sudden completed event, so when often captures the contrast better.

Here are the main forms:

• astrologus ← astrologus, astrologi = astrologer / astronomer
• sphaeram ← sphaera, sphaerae = sphere
• circumagebat ← circumago, circumagere, circumegi, circumactum = rotate, move around
• cum = when / since / although
• aqua ← aqua, aquae = water
• subito = suddenly
• fluere ← fluo, fluere, fluxi, fluxum = flow
• desiit ← desino, desinere, desii/desivi, desitum = stop, cease

Looking at the dictionary forms is often helpful because it lets you see what tense, case, or form changes have happened in the sentence.