How did "tot" come to mean a measure or ration?

How did tot,

A measure of spirits, especially rum.

get that meaning?

It seems to have come to mean a specific ration, as in the daily tot of rum given to a sailor in the Royal Navy (well, no longer daily; see Black Tot Day).

This is in almost complete opposition to the idea of a total, though I can see how perhaps you could tot up the tots to make a whole. Other possibilities might be an association with the small child meaning of tot (something tiny) or totter (stumbling).

(I'm also curious about teetotal, but it looks as though it came later, so it's unlikely to be a source for this meaning.)

The OED has for "tot" 4:

1a. A very small or tiny child.

'2. A very small drinking-vessel; a child's mug. (See also quot. 1845.) Chiefly dial.

'3. A minute quantity of anything, esp. of drink; a dram; also, anything very small.

[Quotes inserted because otherwise stackexchange replaces my '2' and '3' with '1' and '2'].

So the implication is that it comes from the "small child" meaning. The OED does not have an origin for that use.

From the Online Etymology Dictionary:

tot (2)
"to reckon up," 1760, from tot (n.), first recorded 1680s, short for total.

Etymology references from Memidex:

• New World Dictionary:
  tot | totted [past tense] | totting [present participle] Origin: probably from Scandinavian, as in Old Norse tuttr, small chap, tutta, little girl | contraction from "total"
  www.yourdictionary.com/tot

  tot Origin: probably from Scandinavian, as in Old Norse tuttr, small chap, tutta, little girl
  www.yourdictionary.com/totted

• American Heritage Dictionary:
  tot | totted [past tense] | totting [present participle] | tots [3rd-person singular present]
  Origin: unknown
  www.yourdictionary.com/tot

• Oxford Dictionary:
  tot [entry 1]
  First use: early 18th century
  Origin: (originally dialect): of unknown origin
  oxforddictionaries.com/definition/tot

  tot [entry 2] | totted [past tense] | totting [present participle] | tots [3rd-person singular present]
  First use: mid 18th century
  Origin: archaic tot "set of figures to be added up", abbreviation of "total" or of Latin totum "the whole"
  oxforddictionaries.com/definition/tot--2

  tot [entry 3] | totted [past tense] | totting [present participle] | tots [3rd-person singular present]
  First use: late 19th century
  Origin: slang tot "bone", of unknown origin
  oxforddictionaries.com/definition/tot--3

• Merriam-Webster:
  tot [entry 1, noun]
  First use: 1725
  Origin: unknown
  www.merriam-webster.com/dictionary/tot

  tot [entry 2, verb, abbreviation] | totted [past tense] | totting [present participle]
  First use: about 1772
  Origin: tot., abbreviation of total
  www.merriam-webster.com/dictionary/tot show=1

• Encarta Dictionary:
  tot [entry 2] | totted [past tense] | totting [present participle] | tots [3rd-person singular present]
  First use: Mid-18th century
  Origin: Shortening of total, or from Latin tot "this number, so many"
  encarta.msn.com/dictionary 701710519/definition.html