Neat! You can be more systematic than that if you want, as an aside

Hello!

Oh, 燃ゆ?

Umm

Is this a poem?

Um, do you know modern 燃える?

燃ゆ is the literary (文語) form of the same word

I think it was like 燃ゆ　→　燃ゆる　→　燃える

There were spelling changes. But the language itself changed over time too

燃ゆ was not just a funny spelling

Classical Japanese (文語 a.k.a. Early Modern Japanese a.k.a. "the literary langugae") had a different set of verb classes than Modern Japanese

燃ゆ was a 下二段活用動詞 "lower bigrade conjugation verb", and it had the 終止形 "sentence-final form" 「燃ゆ」 and the 連体形 "pre-noun form" 「燃ゆる」. But gradually Japanese speakers started using the 連体形 form to end sentences too, and the distinction between the two was lost, so 燃ゆる became the sentence-final form as well, but the vowel shifted from /u/ to /e/, and so ゆ became え (because there is no separate "ye")

Virtually all verbs lost their old 終止形 "sentence-final forms", although not all verbs underwent the same changes, and 死ぬ lost its old 連体形 "pre-noun form" instead

(That is, we say 死ぬ rather than 死ぬる, so it's an exception)

A literary language based on Early Modern Japanese was preserved and continued (and continues!) to be used many years after the language was no longer spoken, chiefly in writing and other literature (songs, etc.)

This continued for about 700 years (although it wasn't perfectly frozen, and did change over time) until the 言文一致 movement in the Meiji period, which brought the written and spoken languages together (basically, people started writing Japanese down more like Modern Japanese)

Even today the literary language survives in, well, literature!

Literary vocabulary, literary grammar

Of course, I'm glossing over a lot of details--obviously the history of Japanese literature and writing is complex, and there's a lot that I don't know

五段 verbs mostly come from Classical 四段 verbs

死ぬ is a singular exception

There are no other verbs like it in Modern Japanese

It forms a sub-class of its own

I've read that a couple dialects have regularized it to 死む

The reason 四段 verbs became 五段

Well, do you know what the numbers mean?

1-step, 2-step (monograde, bigrade) 4-step, 5-step (quadrigrade, quinquigrade)

You can tell by looking at the kana.

食べる is an 一段 verb because it only has the one surface stem form: 食べ

読む is a 五段 because it has five surface stem forms:

It used to be a 四段 verb. The last one, 読もう, used to be 読まう, so it only had four surface forms (ま・み・む・め, but not も)

But a regular sound change shifted /au/ → /oː/

So the six 活用形 "inflectional forms" of 食べる that 語尾 "(inflectional) word endings" attach to are simply 食べ・食べ・食べ・食べ・食べ・食べ

(Note that informally some people use 語尾 to mean something closer to 終助詞 "sentence-final particle", please don't be confused by this usage)

に indicates a relationship between the thing it marks and a following inflectable word (such as a verb)

So you could call it an ad-verbial particle

Because it specifies a relationship to verbs.

@Anthony Some linguists would say Japanese sentences are made up of two basic types of relationships: 連体 ad-nominal (relating to a following noun or noun equivalent) and 連用 ad-verbial (relating to a following verb or other inflecting word)

These correspond to the categories of 体言 and 用言 in Japanese grammar

And to the Japanese names of two of the 活用形 "inflected forms": 連用形 "ad-verbial form" and 連体形 "ad-nominal form"

I'm taking slight liberties in my glosses here

But people call these things by all sorts of names in English

I can think of six off the top of my head for 連用形: adverbial form, infinitive, continuative, conjunctive, Vmasu, "the stem". It's a mess of terminology!