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I was reviewing my Anki cards and it said that 減少 and 増加 are antonyms. Now,
for extra practice, I try to give at least one kunyomi reading to each kanji.
Then it occured to me that ふえる and へる sounded similar. This question may sound
crazy and naive, but could they be etymologically related, considering that
some words spelled with え now used to be spelled with へ?
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etymology false-etymology
Do ふえる and へる have a common etymology?
In Japanese, phonetic equivalence is generally a very poor predictor of common
etymology. Japanese is well known as having a preponderance of homophones,
mostly due to the accretion of possible readings for each kanji which has
developed over the centuries. But even among wago (和語), verbs ending in 'eru'
are extremely common. Still, it is always worth researching individual cases.

A search of the words on Daijirin doesn't reveal evidence of a common
etymology, at least from the definitions
[here](https://kotobank.jp/word/%E5%A2%97%E3%81%88%E3%82%8B%E3%83%BB%E6%AE%96%E3%81%88%E3%82%8B-371400)
and [here](https://kotobank.jp/word/%E6%B8%9B%E3%82%8B-625738). I also checked
Kojien (広辞苑第六版) - it lists 減る as appearing in the '[Genpei
Sesuiki](https://en.wikipedia.org/wiki/Genpei_J%C5%8Dsuiki)', an extended
version of the [Heike
Monogatari](https://en.wikipedia.org/wiki/The_Tale_of_the_Heike)
(approximately 12th century). According to Daijirin, 減る also appears in the
[Koyo Gunkan](https://en.wikipedia.org/wiki/K%C5%8Dy%C5%8D_Gunkan) (17th
century), so presumably, the word was in regular use during the interim
period. I couldn't find any contemporaneous examples of 増える, checking in the
same Kojien edition and Shinmeikai (新明解国語辞典 第五版). That doesn't mean it wasn't
in regular use, of course. But without solid examples of usage that provide
context for the words, I can't find any evidence to support your hypothesis of
a common etymology.

Nonetheless, it's an interesting hypothesis in light of the phonetic changes
you mention. However, I think we need more details on how those specific words
might have undergone change before we draw any solid conclusions about their
derivations.
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answer
My main reference is the 日本国語大辞典【にほんこくごだいじてん】 (NKD). This is one of the better
monolingual Japanese dictionaries for providing historical details and
etymologies.

### About ふえる

[Their entry for
ふえる](https://kotobank.jp/word/%E6%AE%96%E3%83%BB%E5%A2%97-2079199#E7.B2.BE.E9.81.B8.E7.89.88.20.E6.97.A5.E6.9C.AC.E5.9B.BD.E8.AA.9E.E5.A4.A7.E8.BE.9E.E5.85.B8)
shows that this comes from older form ふゆ, appearing some time in the middle of
the [Muromachi period](https://en.wikipedia.org/wiki/Muromachi_period)
(1336–1573). The verb root is _fuy-_.

Digging [further](https://kotobank.jp/word/%E3%82%86-650032), we find that
this ～ゆ ending might derive from the Old Japanese auxiliary or verb ending
that indicates spontaneous action that happens on its own, as well as
potential and passive senses. This underlying meaning of "happens on its own"
agrees with the fact that this verb is intransitive.

  * We can find a hint about the lurking "y" in the verb ふえる from its transitive counterpart ふやす. This ふやす can also be analyzed as the causative form of ふゆ: verb root _fuy-_ + linking vowel _-a-_ + causative / transitive ending _-su_.

The ～ゆ suffix had the so-called 下二段活用【しもにだんかつよう】 or "lower-bigrade
conjugation" pattern, where the conjugation forms ended in either _-u_ or
_-e_. This commonly shifted from ～ゆ to ～える as time passed, and this is the
same ～ゆ that we also see in classical 見【み】ゆ and 聞【き】こゆ, modern 見【み】える and
聞【き】こえる.

The core meaning of this verb has consistently been "to increase".

### About へる

Meanwhile, [the NKD entry for
へる](https://kotobank.jp/word/%E6%B8%9B%E3%83%BB%E8%80%97-2081413) shows that
this has pretty much always had the form へる, since it was first documented in
1170. The verb root is _her-_ , or in older stages of Japanese, _fer-_.

_(See also[this other
post](https://japanese.stackexchange.com/questions/82475/why-does-fu-changes-
to-pu-while-its-not-started-with-h/82477#82477) about the historical sound
shifts for the はひふへほ kana.)_

The core meaning of this verb has consistently been "to decrease".

### Comparing

  * The phonology (sound) suggests independent roots: _fuy-_ and _fer-_ are quite distinct, and there are no explainable sound shifts that could apply to suggest that these come from some shared origin.
  * The semantics (meanings) also suggest independent roots. While opposites sometimes have shared origins (consider English pairs like positive _terrific_ and negative _terrible_ , or _awesome_ and _awful_ ), we can usually trace the development of such meaning pairs. For _fuy-_ and _fer-_ , the two have always had opposite meanings, for as long as the terms have been used in written Japanese.

**Conclusion:** While it is possible that these two terms are cognates that
share a common ancestor root, and the sense and sound developments have simply
been lost to history, in consideration of the known development of both words,
it appears unlikely that these terms are related.
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