Vapor per sudationem pervagatur.

The subject is vapor.

It is in the nominative singular, which is the case normally used for the subject of a sentence. The verb pervagatur is also 3rd person singular, so it matches vapor: the vapor spreads.

Because vapor is the subject, not a direct object.

• vapor = nominative singular
• vaporem = accusative singular

In this sentence, the vapor is the thing doing the action of pervagatur, so Latin uses the nominative: vapor.

Sudationem is accusative singular.

It is accusative because it follows the preposition per, and per takes the accusative case. So:

• per sudationem = through sweating / by means of perspiration

This is a very common Latin pattern:
per + accusative.

Because the sentence specifically uses the preposition per, which means through, by means of, or by way of.

Latin often uses the ablative without a preposition for means or instrument, but per + accusative can give a more concrete sense of movement or passage through something. Here, per sudationem suggests the vapor spreads through perspiration or by way of sweating.

So the choice is not random:

• sudatione would be a plain ablative expression
• per sudationem emphasizes the idea of through/by way of

Because pervagatur is a deponent verb form.

A deponent verb has:

• passive forms
• but an active meaning

So pervagatur looks like he/she/it is spread through if you judge only by the ending, but it actually means something like:

• wanders through
• spreads through
• diffuses through

This is very common in Latin, and it often surprises English speakers at first.

Pervagatur is:

• present tense
• indicative mood
• 3rd person singular
• deponent

So it means he/she/it spreads through, ranges through, or diffuses through.

Because the subject is vapor, the sense is the vapor spreads through ...

Yes, there is a historical and lexical connection, but they are not doing the same grammatical job in the sentence.

• per by itself is a preposition here: through
• pervagatur is a verb with the prefix per-

The prefix per- often adds the sense of:

• through
• thoroughly
• throughout

So pervagatur is stronger than plain vagatur. It suggests not just wandering, but spreading through or ranging throughout.

So in this sentence, you have:

• per sudationem = through sweating
• pervagatur = spreads through / diffuses through

That can feel repetitive in English, but it is perfectly normal in Latin.

Sudatio, sudationis is a noun meaning sweating, perspiration, or the act/process of sweating.

So per sudationem is not just through sweat as a substance, but more specifically through perspiration or through the process of sweating.

That distinction can matter:

• sudor often refers more directly to sweat
• sudatio refers more to sweating/perspiration

Because Latin does not have definite or indefinite articles like English the, a, or an.

So vapor can mean:

• vapor
• the vapor
• a vapor

The exact choice in English depends on the context and on the translation already given to the learner.

Yes. Latin word order is much more flexible than English word order because grammatical roles are shown mainly by case endings, not by position.

So these could still make sense:

• Per sudationem vapor pervagatur
• Vapor pervagatur per sudationem

The original order is perfectly normal, but Latin can move words around for emphasis, style, or rhythm.

A very literal way to feel the structure is:

• Vapor = the vapor
• per sudationem = through perspiration / by way of sweating
• pervagatur = spreads through / diffuses

So the sentence is built as:

The vapor spreads through perspiration.

Even if your given translation is smoother or more idiomatic, this literal breakdown helps you see how the Latin grammar works.

You would usually learn them as:

• vapor, vaporis m. = vapor, steam, heat
• sudatio, sudationis f. = sweating, perspiration

That means:

• vapor is a 3rd-declension masculine noun
• sudatio is a 3rd-declension feminine noun

In the sentence:

• vapor = nominative singular
• sudationem = accusative singular

So this sentence is also good practice in recognizing 3rd-declension endings.