What follows next in the sequence "unary, binary, ternary..."?

Solution 1:

The problem is that English uses two different kinds of adjectives to mean "first, second, etc". The ones in -ary without the -n- come from the Latin ordinals, "first, second, etc."; but they are different after 3. (An asterisk * indicates that the word is not found in (ordinary) English sources.)

1. Primus — primary "first"
2. Secundus — secondary "second"
3. Tertius — tertiary
4. Quartus — *quartary
5. Quintus — *quintary

...

The -arius suffix was also used in Latin with ordinals, and secundarius means something like "second, pertaining to two, second in rank", though it often comes very close to the simple ordinal secundus. It usually adds some connotation of ranks and order in a grand system. There is also secundanus, which I believe isn't much different.

The -n- ones come from Latin distributive adjectives, "one each, two each, etc."; they were always used in plural in Latin. They were sometimes also used in a sense roughly similar to the ordinals, which is probably why English uses them in an odd way.

1. Singuli — single/singular/singulary "one each"
2. Bini — binary "two each"
3. Terni/trini — ternary/*trinary
4. Quaterni — quaternary
5. Quini — quinary
6. Seni — senary
7. Septeni — septenary
8. Octoni — octonary
9. Noveni — *novenary
10. Deni — denary
11. Undeni — *undenary
12. Duodeni — duodenary
13. Terni/trini deni — *ternidenary/*tridenary

...

I believe the ones derived from ordinals were originally used to mean "second [in order]" in English, and the distributive -n- ones to mean "of two parts", or "characterised by the number 2". But then, because these meanings are related and often overlap, they got mixed up, resulting in the current defective lists, where the -n- forms serve both senses from 4 up.

The number one is the strangest exception of all, where a new word unary was made up, though no Latin equivalent exists (there is only unus, "one", but that is like using *duary from duo, "two"). Nonary is odd as well.

These are the Latin cardinal numbers for reference:

1. Unus/una/unum/etc. (depending on gender and case) — "one"
2. Duo/duorum/duarum/etc. (depending on case and gender) — "two"
3. Tres/trium/etc. (depending on case)
4. Quattuor
5. Quinque
6. Sex
7. Septem
8. Octo
9. Novem
10. Decem
11. Undecim
12. Duodecim
13. Tredecim

...

Solution 2:

The arity of a function or operation is the number of arguments or operands that the function takes.

N-ary:

• Nullary means 0-ary.

• Unary means 1-ary.

• Binary means 2-ary.

• Ternary means 3-ary.

• Quaternary means 4-ary.

• Quinary means 5-ary.

• Senary means 6-ary.

• Septenary means 7-ary.

• Octary means 8-ary.

• Nonary means 9-ary.

Hope this helps.

Solution 3:

I know I'm a little late here, but I thought it might be worth mentioning that Wikipedia has a great list of base systems, which goes all the way up to 16 (Hexadecimal, of course) without holes, and then on to 85 (Pentaoxagesimal). Here's a quick reproduction of part of it:

1. unary (not actually on the main list, but listed farther down as being used in tally marks)
2. binary
3. ternary
4. quarternary
5. quinary
6. senary
7. septenary (used in weeks)
8. octal
9. nonary
10. decimal (everybody's favorite!)
11. undecimal
12. duodecimal (used in hours, months)
13. tridecimal
14. tetradecimal 
15. pentadecimal
16. hexadecimal (Base16 encoding)

18 is octodecimal

20 is vigesimal

It's interesting to note that even our method of naming these systems reflects our attachment to the decimal system, as we begin to add prefixes after decimal. Also, if you want to form a higher number, it appears that you can use the following formula:

prefix for 2nd digit + prefix for 1st digit + gesimal

So, 27 is septemvigesimal. I invented this formula in answer to this question, but it appears to fit every case on the list.

Wikipedia also lists −2 as negabinary and −3 as negaternary. Theoretically, you can add the nega- prefix to anything, but I have no idea what you would use it for.

Solution 4:

As Cerberus notes, the first list you give is a combination of two lists, both on Latin roots: ordinals for 1–3, then arities (from distributive numbers). To keep these straight, and include the corresponding words from Ancient Greek, I’ve written two Wiktionary appendices:

• Appendix:English ordinal numbers
• Appendix:English arities and adicities

Solution 5:

Here comes another late arrival at the Latin ball. First let me say that I studied Latin from the age of about eight through to 23, and in all that time I scarcely encountered distributive, other thsan in Kennedy's Latin Primer (the Latin grammar book of the day). Outside that, and the sole example: bina castra. I can explain that this is because the word for 'camp', castra is plural. It is plural. The singulare, castrum, mean a fort. The Romans saw a camp as an accretion of small 'forts' (watchtowers). So duo castra for two camps would be ambiguous. Instead, to be clear, they used this distributive number. But it is difficult to find examples of it, outside the commentaries of Julius Caesar. (I have since encountered Plautus the writer of Roman comedy as a second source - see below)

The nearest is the Roman denarius nummus. The denarius is worth ten ases. And the first syllable is, of course, our old friend den-i - ten-(as)-sets. Otherwise, I have read quite a lot of Latin, prose and poetry, war and peace, but encountered scarcely any example. Consequently, clarity about its usage is extremely difficult.

There is a useful list of Latin numerals, including distributive numerals from the Later Latin Society: http://www.informalmusic.com/latinsoc/latnum.html. This includes a reasonable account of the meaning of distributive numbers, though the explanations leave at least me with the question "what the heck were they for?".

Such a list of numerals makes it look as if they are very widely used indeed. In fact, as far as I can tell, the binary numbers are constructed in accordance with a formative rule that allows us all the way to sets of infinity, which does not make them useful, and most can never have been used, possibly including undeni. The mathematics of bases might have given an opening, but too late, I fear!

I found a helpful explanation of distributive numbers from Allen and Greenough's Latin Grammar, cited by Dickinson College: http://dcc.dickinson.edu/grammar/latin/distributives. It provides a few other examples.

However, I have found an article in the Classical Review [Volume 21 Issue 7 Nov 1907 by J.P.Postgate - What follows next in the sequence "unary, binary, ternary..."?, which challenges the standard name and interpretation of the so-called distributives as a misnoma. They should be called, he argues, Collectives, giving useful citations: for example, Julius Caesar's account of the Britons he encountered in which groups of ten or twelve men (deni duodenique) shared wives.