Diametros sphaeram per medium secat.

You know mostly from case, not from word order.

• sphaeram ends in -am, a very common accusative singular ending for first-declension nouns, so it is the direct object.
• secat is a singular verb.
• That leaves diametros as the subject: the diameter.

So the basic structure is:

diametros = subject
sphaeram = object
secat = verb

Because diametros is a Greek loanword used in Latin mathematical language. Latin often keeps Greek-looking forms, especially in technical vocabulary.

So even though -os looks unusual from the point of view of basic Latin declensions, it is normal here for this borrowed word. A learner should simply recognize diametros as the noun meaning diameter.

Sphaeram is accusative singular.

It is accusative because it is the direct object of secat: the thing being cut. In other words, the diameter performs the action, and the sphere receives it.

Base form: sphaera = sphere
Here: sphaeram = sphere, as object

Secat is:

• 3rd person singular
• present tense
• active voice
• indicative mood

It comes from seco, secare, meaning to cut.

So secat means he/she/it cuts or simply cuts. Since the subject is diametros, the sense is the diameter cuts.

Because finite Latin verbs do not agree in gender.

Latin verbs like secat agree with their subject in person and number, but not in gender. So whether the subject is masculine, feminine, or neuter, the verb form can stay the same.

Gender agreement happens with things like:

• adjectives
• participles
• some pronouns

But not with a present indicative verb like secat.

Literally, per medium means through the middle.

In a geometric context, that often comes out naturally in English as:

• through the center
• through the middle
• sometimes effectively in half or into two equal parts, depending on context

So the phrase tells you how the sphere is cut: not just anywhere, but through its middle.

Because per takes the accusative case.

That is one of the first preposition rules Latin learners meet:

• per + accusative = through, by means of, and related ideas

So:

• per medium = through the middle

The case is required by the preposition per.

Because medium is not functioning here as an adjective directly modifying sphaeram.

Instead, it is being used more like a substantive expression: the middle. So the phrase means through the middle, not through the middle sphere or anything like that.

That is why you should read per medium as a unit.

Yes.

• per medium = through the middle
• in medio = in the middle

That is an important difference.
Here the idea is motion or passage through the middle, so per is the right preposition.

In a geometric sentence, per medium secat suggests cutting along a line that passes through the middle, not merely being located there.

Because Latin has no definite or indefinite articles.

Latin does not have separate words for:

• the
• a
• an

So diametros can mean diameter or the diameter, and sphaeram can mean a sphere or the sphere, depending on context. English has to supply the article when translating.

Less than in English.

Latin uses endings much more than word order to show grammatical function, so the sentence could be rearranged more freely than an English sentence. The current order is perfectly natural:

Diametros sphaeram per medium secat.

A reader still understands it because:

• diametros is the subject
• sphaeram is the object
• per medium is the prepositional phrase
• secat is the verb

Putting secat at the end is also very common in Latin prose.

Its basic meaning is cuts, but in this context it can easily carry the stronger geometric idea of cuts through the middle, and therefore effectively bisects.

So the exact English wording depends on how literal or how mathematical you want to be:

• more literal: cuts the sphere through the middle
• more mathematical: bisects the sphere

The Latin itself still uses the ordinary verb seco plus the phrase per medium to express that idea.