History of Mathematical Notation I would like to see a simple example which shows how mathematical notation were evolve in time and space.
Say, consider the formula
$$(x+2)^2=x^2+4{\cdot}x+4.$$
If I understand correctly, Franciscus Vieta would write something like this. (Feel free to correct me.)
$\overline{N+2}$ quadr. æqualia $Q+N\,4+4$.
($N$ stays for unknown and $Q$ for its square.)
Can you give me the other examples?
 A: "The big picture" that can be seen in Carlo Beenakker's example is 


Rhetorical (verbal);   Syncopated  (abbreviated words);       Symbolic. 


However, this well-known picture is very algebra-oriented and does not say anything about the use of notations in say geometry as discussed here "
The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History
" or as asked here Uppercase Point Labels in High-School Diagrams. 
Thus, at least for me, the interesting answers to OP questions would be those that are not related to algebra.  
A: You might also enjoy Earliest Uses of Symbols of Operation!!
A: 
Find a quantity such that $\varsigma\;\overline{\beta}$ squared equals
  $$\Delta^{\Upsilon}\;\varsigma\overline{\delta}\;{\stackrel{o}{M}}\overline{\delta}$$

(My translation into English from Diophantes $\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\eta$.)
A: For a quite extensive overview with many examples, you might want to check out The origins and development of mathematical notation.
I also enjoyed reading Stephen Wolfram's take on Mathematical Notation – past and future, with a great variety of illustrations.

The OP asks specifically for the evolution of one formula, "to see the big picture". Here is one example, taken from Math through the ages:

A: I think that the most complete source is still :


*

*Florian Cajori (1859-1939), A History of Mathematical Notations (1928 - Dover reprint).


We can easily find there examples regarding the evolution of "algebraic" symbolism through the modern era :


*

*§122 page 90 : Leonardo Piasano (Fibonacci) (c.1170 – c.1250), from Liber abaci

*§130 page 101 : Nicolas Chuquet (c.1445 – c.1488), from Triparty en la science des nombres

*§132 page 103 : Estienne de La Roche (1470–1530), from L'Arismetique (1520)

*§134-on page 106 : Luca Pacioli (c.1447–1517), from Summa de arithmetica, geometria, proportioni et proportionalita (1494)

*§140 page 117 : Gerolamo Cardano (1501 – 1576) ), from Artis Magnæ, Sive de Regulis Algebraicis Liber Unus (1545)

*§151 page 139 : Michael Stifel  (1487 – 1567), from Arithmetica integra (1544)

*§162 page 154 : Simon Stevin  (1548 – 1620), from De Thiende ('the art of tenths'), first published in Dutch in 1585 and translated into French as Disme

*§170 page 169 : Leonard (c.1515–c.1559) and Thomas Digges  (c.1546 – 1595) , from An Arithmetical Warlike Treatise Named Stratioticos (1579)

*§176 page 181 : François Viète  (1540 – 1603), from Zeteticorum libri quinque (1591)

*§191 page 205 : René Descartes (1596 - 1650), from La Géométrie (1637).
