Should the word Boolean be capitalized?
Solution 1:
Wikipedia capitalizes Boolean, as does Wiktionary (both as an adjective and as a noun). Merriam-Webster and the American Heritage Dictionary of the English Language capitalize the adjective and don't have an entry for the noun.
What Wiktionary does not capitalize is the noun bool. M-W and AHD don't have an entry for bool.
A search in the British National Corpus returns 94 cites for Boolean, but sadly only 50 randomly selected ones are displayed at a time, so I just hit "reload" a few times. The results that I got each time showed the following distribution:
Boolean 30 28 30 26
boolean 19 21 18 22
BOOLEAN 1 1 2 2
A search for bool did not return a single result.
Solution 2:
There are many scientific nouns and adjectives that derive from their inventor's name, and which are still capitalized even though they are widely used. Examples include:
- the Gaussian function (or distribution)
- Coulombic interactions
- the Lagrangian and the Laplacian operators
- the Ohmic dissipation
- an Arrhenian behaviour (in chemistry)
- the trans-Planckian problem
The notable exceptions are chemical elements, whose name are never capitalized (e.g., “the symbol of einsteinium is Es”), and units of measurement (“a current of two amperes”).
Solution 3:
Capitalised Boolean for the two-state data items, lowercase bool for the C++ keyword derived from it.
(Google Ngram)
Your example is correct:
"Implement a variable using the the Boolean data type for the particular programming language that you are using."
Solution 4:
The usual convention in mathematics is to capitalize adjectives derived from proper names, but there are a few exceptions. The only exceptions that come to mind just now are "cartesian" and "abelian" (and perhaps "noetherian", though I've also seen that capitalized quite often). We (mathematicians) generally capitalize "Boolean," "Gaussian," "Euclidean," "Diophantine," "Artinian," "Eulerian," "Hamiltonian," "Pfaffian," etc. (Note that this answer is specifically about mathematical usage; I make no claims here about general English.)