In abaco pueri calculos movent, ut numerum celerius inveniant.

Because in can take either the accusative or the ablative, depending on the meaning.

• in + ablative = in/on a place, with no motion into it
• in + accusative = into/onto a place, showing motion toward it

Here, in abaco means on the counting board / on the abacus, so it describes location, not movement. That is why Latin uses the ablative abaco.

Abacus in Latin refers to a counting board or abacus. In this sentence, in abaco means something like on the abacus or on the counting board.

For a Roman context, this often meant a board on which counters were moved around to calculate.

Pueri is the nominative plural of puer, meaning boy. The nominative case is normally used for the subject of the sentence.

So:

• puer = boy
• pueri = boys

Since movent is they move, pueri fits as the subject: the boys move.

Because calculos is the direct object of movent.

The verb moveo, movere means to move, and the thing being moved goes in the accusative case.

So:

• calculi = counters / pebbles (nominative plural)
• calculos = counters / pebbles (accusative plural)

The boys are moving the counters, so calculos must be accusative.

Yes, originally it does. Calculus literally means a small stone or pebble. But in this context it means counting stones, counters, or tokens used for calculation.

This is a good example of how Latin often uses a concrete word in a specialized practical sense.

This is an ut-clause of purpose.

It means in order that they may find the number more quickly, or more naturally in English, so that they can find the number more quickly.

The structure is:

• ut = so that / in order that
• inveniant = subjunctive verb, because purpose clauses use the subjunctive

So the full idea is: The boys move counters on the abacus so that they may find the number more quickly.

Because after ut introducing a purpose clause, Latin normally uses the subjunctive.

So:

• inveniunt would mean they find (indicative, plain statement)
• inveniant means that they may find / so that they may find in this kind of clause

This is one of the most common uses of the subjunctive in Latin.

Because the main verb movent is in a primary tense (present), and Latin follows a sequence of tenses in subordinate clauses.

With a present main verb, a purpose clause usually takes the present subjunctive:

• movent = present
• inveniant = present subjunctive

If the main verb were in a past tense, you would normally expect the imperfect subjunctive instead.

Celerius means more quickly. It is the comparative adverb of celeriter (quickly).

So:

• celeriter = quickly
• celerius = more quickly

Latin often uses the comparative even where English might simply say more quickly without stating exactly what it is being compared to. The implied idea is more quickly than otherwise or faster.

Because the sentence is talking about the number or the total/result, not several numbers.

Numerum is singular accusative, the direct object of inveniant. The idea is that the boys use the counters to arrive at a numerical answer.

Depending on context, numerum could mean:

• a number
• the total
• the result of the calculation

Yes, inveniant literally means find, but in English we might translate it more naturally as:

• work out
• determine
• figure out

So although find is correct, the sense is mathematical: they move the counters in order to work out the number.

Latin word order is much more flexible than English word order because Latin uses case endings to show grammatical relationships.

English depends heavily on word order:

• The boys move the counters

Latin can move parts around more freely because the endings already show who is doing what.

This sentence has a very normal Latin feel:

• In abaco sets the scene first
• pueri gives the subject
• calculos movent gives the main action
• ut numerum celerius inveniant gives the purpose

So the order is natural for Latin, even though English would usually be less flexible.

Yes. Ut is a very common Latin word with several uses, including:

• as
• when
• so that / in order that
• that in some result clauses

But here, because it is followed by a subjunctive verb and expresses the goal of the action, it is clearly a purpose use: so that.

Yes: it describes what the boys are doing in a general or present-time sense.

The sentence is probably stating a customary or ongoing action:

• The boys move counters on the abacus...

It could be understood as:

• a general truth,
• a classroom description,
• or an action happening now.

The present tense works well for all of those in Latin.

Not really in this context. Grammatically, in with the ablative can mean in or on, and the best English choice depends on the object.

With a counting board or abacus, English usually prefers on the abacus or on the counting board, because the counters are being arranged on its surface or framework, not placed inside it in an ordinary spatial sense.

Here is a word-by-word breakdown:

• In — preposition, here taking the ablative
• abaco — ablative singular of abacus, on the abacus / counting board
• pueri — nominative plural of puer, the boys
• calculos — accusative plural of calculus, the counters / pebbles
• movent — 3rd person plural present active indicative of moveo, they move
• ut — so that / in order that
• numerum — accusative singular of numerus, the number / total
• celerius — comparative adverb, more quickly
• inveniant — 3rd person plural present active subjunctive of invenio, they may find / determine

So the sentence consists of:

1. a prepositional phrase: In abaco
2. a main clause: pueri calculos movent
3. a purpose clause: ut numerum celerius inveniant