Остроту зрения проверяют по таблице для проверки зрения.

Because остроту is the direct object of проверяют (check / test), so it is in the accusative case.

The dictionary form is острота (sharpness; acuity). Since it is a feminine noun ending in -а, its accusative singular becomes -у:

• острота → nominative
• остроту → accusative

So Остроту зрения проверяют literally means They check the acuity of vision.

Зрения is genitive singular of зрение (vision; eyesight).

It appears twice in the sentence, and both times it is genitive, but for slightly different reasons:

1. остроту зрения = the acuity of vision
   Here, зрения shows possession/relationship: acuity of what? → of vision

2. для проверки зрения = for the checking/testing of vision
   Here, проверка is a verbal noun (checking, testing), and the thing being checked often appears in the genitive: checking of what? → of vision

So in both places, English of vision / of eyesight corresponds to Russian зрения in the genitive.

This is a very common Russian pattern. Проверяют is 3rd person plural, but it does not always mean a specific they.

Russian often uses 3rd person plural to express an indefinite personal idea:

• Проверяют = they check / people check / it is checked

In English, we often translate this more naturally with a passive:

• Visual acuity is checked...

So the sentence does not necessarily mean some clearly identified group of people. It just describes the usual procedure.

Here по means something like:

• according to
• on the basis of
• using
• by means of

In medical or testing contexts, проверять по таблице means to test/check using a chart/table as the reference.

So:

• по таблице = using a chart / from a chart / according to the chart

This is idiomatic Russian. Even though English might prefer with a chart or using a chart, Russian naturally uses по here.

Because the preposition по usually requires the dative case in this meaning.

The noun is:

• таблица = chart, table

In the dative singular:

• таблица → таблице

So:

• по таблице = according to the chart / using the chart

This is a good phrase to memorize as a chunk: проверять по таблице.

Not in the furniture sense. Here таблица means a chart, table, or test chart.

In this sentence, it refers to the kind of chart used for eye testing, like an eye chart.

So although the basic meaning of таблица is often table, in context it is better understood as:

• chart
• eye chart
• test chart

It can sound a little repetitive in a word-for-word translation, but it is completely normal in Russian.

The sentence has:

• остроту зрения = visual acuity
• таблице для проверки зрения = a chart for testing vision

Russian often repeats nouns where English might avoid repetition. In English, you might naturally say:

• Visual acuity is checked using an eye chart.

Russian, however, is comfortable repeating зрение for clarity and precision.

Literally, it means:

• for the checking/testing of vision

Breakdown:

• для = for
• проверки = genitive singular of проверка (checking, test, examination)
• зрения = of vision

So таблица для проверки зрения is literally a chart for testing vision.

This is a very common Russian pattern:

• noun + для + verbal noun/genitive phrase

For example:

• прибор для измерения температуры = an instrument for measuring temperature

Because the sentence describes a general method or regular procedure, not one completed action.

• проверять / проверяют = imperfective → to check, to be checking, to check habitually
• проверить = perfective → to check once, to complete the check

Here the meaning is something like:

• This is how visual acuity is tested
• Visual acuity is checked by means of a chart

That kind of general statement usually takes the imperfective.

Yes. Russian word order is flexible, though different orders can sound more or less natural depending on emphasis.

The given sentence is neutral and clear:

• Остроту зрения проверяют по таблице для проверки зрения.

Possible variations are grammatically possible, for example:

• По таблице для проверки зрения проверяют остроту зрения.

That version puts more emphasis on the method first: Using a vision-testing chart, they check visual acuity.

So the original word order is natural, but not the only possible one.

Yes. A very literal translation would be:

• They check the acuity of vision according to a table for checking vision.

But more natural English would be:

• Visual acuity is tested using an eye chart.
• Eyesight is checked with an eye chart.

That is a good reminder that Russian and English often organize this kind of information differently.