I'm interested in some corner cases in notation related to equation, not sure what's the best classification of following expressions, are they identity, equation or formula?
a)$$1+\frac{1}{3}=1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...$$
b)$$x=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}...$$
c)$$x=\frac{\pi}{4}$$
d)$$3x={2^3}x-5x$$
e)$$x(x-1)=2x$$
By checking the expressions from 3 aspects "explicitness of the term", "explicitness of the expression", "counts of the roots", I can differentiate a) from the other 4 as it contains only "constant"s(or explicit terms) and "sign"s BUT the other 4 contain also variables(i.e. implicit terms).
The only differences I can sense among the other 4 are the number of root(s), or solution(s).
Especially for d) it has infinite roots, or with simplification to make it explicit, the final result gives the identity "x=x".
And although the b) and c) look so differently, in b), the right-hand-side expression to the equal sign may be implicit to some people, the root of them are exactly the same.
So my opinion of classification would be, since there's no sharp difference btw the implicit expression and explicit expression, e.g. in cases of b) and c), and the number of root(s) would be so dependent on the expressions itself, we shall only take the criteria "explicitness of the term" to define the "equation" and "formula", so all a)~e) are equations, and other than a) the other 4 are formulas.
However I'm so interested to cross check my understanding with the historical mathematicians. So I'd like to get some help here for the cites of historical mathematical works/manuscripts/books up to 1900s that making-use-of(or defining) terms "equation", "identity", "formula".