Greek:
Horizontal: Adj.
«οριζόντιος, -α, -ο» [o̞ɾiˈz̠o̞ŋ.d̠i.o̞s̠] (masc.), [o̞ɾiˈz̠o̞ŋ.d̠i.a] (fem.), [o̞ɾiˈz̠o̞ŋ.d̠i.o̞] (neut.) a MoGr Katharevousa construction (1766)
«ὁριζόντιος» [o̞ɾiˈz̠o̞ŋ.di.o̞s̠] a calque for the Fr.
horizontal, ultimately from the Ancient Greek 3rd declension masculine noun
«ὁρίζων-ὁρίζοντος» /hoˈri.s͡dɔːn/ (nom. sing.)-/hoˈri.s͡dontos/ (gen. sing.) --> initially
the adjective in the set phrase ὁρίζων κύκλος /hoˈri.s͡dɔːn ˈky.klos/ =
delineated circle, later,
the substantivization of the adjective, ὁρίζων (masc.) =
marking of boundaries, horizon, an Ancient Greek deverbal from the verb
«ὁρίζω» /hoˈri.s͡dɔː/ -->
to border, demarcate, separate, determine, define < Ancient Greek masc. noun
«ὅρος» /ˈho.ros/ -->
border, boundary, mark, term, limit, appointment, definition (the Proto-Hellenic *ϝόρϝoς /ˈwor.wos/ points towards IE origin, PIE *uo̯ru-o- < *ue̯ru-
to draw cf. Latin urvāre).
Horizontally: Adv.
«ὁριζοντίως» [o̞ɾiz̠o̞ŋˈdi.o̞s̠] (constructed in 1799), modern vernacular
«οριζόντια» [o̞ɾiˈz̠o̞ŋ.di.a].
Vertical: Adj.
«κάθετος, -τη, το» [ˈka.θe̞t̠o̞s̠] (masc.), [ˈka.θe̞t̠i] (fem.), [ˈka.θe̞t̠o̞] (neut.) < Ancient Greek adjective
«κάθετος, -τος, -τον» /ˈkɐ.thetos/ (masc. or fem.), /ˈkɐ.theton/ (neut.) -->
vertical, steep, sharp, a deverbal from the ancient athematic verb
«καθίημι» /kɐˈthi.ɛːmi/ -->
to let fall vertically, send down, drop a compound of the apocopic & elided preposition
«κατ'» /kɐt/ + athematic verb
«ἵημι» /ˈhi.ɛːmi/ -->
to send away, let go, throw, hurl (PIE *(H)ieh1-
to throw cf. Latin iacere).
Vertically: Ancient adverb
«καθέτως» /kɐˈthe.tɔːs/ or the modern vernacular
«κάθετα» [ˈka.θet̠a].
Perpendicular: Adj.
«κατακόρυφος, -φη, -φο» [kat̠aˈko̞.ɾifo̞s̠] (masc.), [kat̠aˈko̞.ɾifi] (fem.), [kat̠aˈko̞.ɾifo̞] (neut.) a modern Katharevousa construction (1843), a calque for the French
vertical, a compound of the prefix & preposition
«κατά» [kaˈt̠a] < Ancient Greek preposition
«κατά» /kɐˈtɐ/ + feminine noun
«κορυφή» [ko̞ɾiˈfi] < Ancient Greek feminine noun
«κορυφή» /koryˈphɛː/ -->
top, skull (it has been recognised as Pre-Greek due to the variants fem. κορυφή & masc. κόρυμβος /ˈko.rymbos/,
uppermost point of a ship, top of a mountain).
Perpendicularly: Adv.
«κατακορύφως» [kat̠ako̞ˈɾi.fo̞s̠] (a Katharevousa 1876 construction), or the modern vernacular
«κατακόρυφα» [kat̠aˈko̞.ɾifa].
Horizon: Masc. noun
«ορίζοντας» [o̞ˈɾi.z̠o̞ŋd̠as̠] which is the accusative of the ancient 3rd declension masc. noun
«ὁρίζων» (see 'horizontal').