History of Notation – Origin of the Abuse of Notation y=y(x)

ho.history-overviewnotation

It's quite common nowadays to name a function and the application of the function to its input with the same letter. (Possibly more so in applied areas. Certainly many calculus textbooks do this.)

When did this practice start?

In particular, did any of the old masters like Newton, Leibniz, Euler etc. ever write something like $y=y(x)$?

Clarification: The question is really about the history of this practice. With whom did it start?
I didn't want to discuss merits or demerits of this notation. If you want to provide a non standard interpretation of $y=y(x)$, please also back it up with historical references.

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Regarding the original question of who started literally writing $y=y(x)$ or something like it, which I understand Jacobi didn't do in the quoted 1840 paper: Cayley (1859, p. 3),

where $\Omega$ is regarded as a function of $r,v,y,$ or (as this may be expressed) where $\Omega = \Omega(r, v, y)$

sounds like an early example, in that he feels the need to explain the notation.

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One landmark could be the 1693 paper "An Instance of the Excellence of the Modern ALGEBRA, in the Resolution of the Problem of finding the Foci of Optick Glasses universally"(alt. link), where Edmond Halley (of comet fame) introduced the rules of signs in optics:

This Dioptrick Problem is that of finding the Focus of any sort of Lens, exposed either to Converging, Diverging or parallel Rays of Light, proceeding from, or tending to a given Point in the Axis of the Lens, be the ratio of Refraction what it will (..., p.961)

if $d$ be so small, as that $2r\rho$ exceed $dr+d\rho$, then will it be $-f$, or the focus will be Negative, which shows that the Beams after both Refractions still proceed Diverging. (..., p.963)

If the Lens be double Concave, the focus of converging Beams is negative, where it was affirmative in the case of diverging Beams on a double Convex (..., p.964)

A second use is to find what Convexity or Concavity is required, to make a vastly distant Object be represented at a given focus (...) in Glass $\smash[b]{\frac{rf}{2r-f}}=\rho$, whence if $f$ be greater than $2r$, $\rho$ becomes Negative, and $\frac{rf}{f-2r}$ is the Radius of the Concave sought. (..., p.967)

So here clearly $\rho$ can have "negative numbers as input", as you say.

According to Shapiro (1990, p.151) "An inherent limitation of Barrow (1669) and his contemporaries is that all line segments must be positive... Edmond Halley in his landmark paper (1693) overcame this drawback and derived a single equation that yielded the image point for all varieties of thick lenses."

According to Delambre (1816, p.X), reading this paper was what converted Lagrange to Mathematics, "and revealed to him his true destiny", no less.

Edit: From some more authorities' writings:

Struik (A Source Book in Mathematics, 1200-1800, p.168) says: "Newton's Enumeratio linearum tertii ordinis was first published together with his Opticks in London in 1704, but it was written much earlier, perhaps in or after 1676 (...) he freely used (a novelty at the time) positive and negative values of the coordinates."
Michel Serfati (Landmark Writings in Western Mathematics 1640-1940, p.18) says: "From 1655, the De sectionibus conicis of John Wallis used the Cartesian method for expressing algebraically the ancient geometric definitions and properties of the conics of Apollonius, systematically interpreting $x$ and $y$ as having any sign."
Kline (Mathematical Thought From Ancient to Modern Times, p.319) says: "John Wallis, in De Sectionibus Conicis (1655) (...) was also the first to consciously introduce negative abscissas and ordinates."
Boyer (History of Analytic Geometry, p.139) says: "Newton is sometimes given sole credit for the correct use of negative coordinates, but he had been anticipated to some extent by others, notably Wallis and Lahire. (...) the use of negative values of the coordinates were fairly well established before the middle of the [18th] century."

Mathematical Notation – The Letter \wp; Name & Origin

Apparently first introduced by Weierstrass in Winter 1862/63 lectures published by H. A. Schwarz (1881, 1885, 1892, 1893), §9:

Mit der Sigma-Function $\mathfrak Su$ ist die Pe-Function $\wp u=\wp(u\mid\omega,\omega')=\wp(u;g_2,g_3)$ durch die Gleichung $$ \wp u=-\frac{d^2}{du^2}\log\mathfrak S u=\frac{(\mathfrak S'u)^2-\mathfrak S u\mathfrak S''u}{\mathfrak S^2u} $$ verbunden. (...)

The letter and a reference to Schwarz's notes also appear on the first page of Weierstrass's paper Zur Theorie der elliptischen Functionen (1882). Attribution in e.g. (Schwarz student) H. Hancock's Lectures on the Theory of Elliptic Functions (1910), p. 309:

(...) the function which we thus have was called by Weierstrass the Pe-function and denoted by $$ \wp(u)\qquad\text{or more simply}\qquad\wp u $$

or R. Godement's Analysis I (2004), p. 181:

(...) the famous function $$ \wp(u)=1/u^2+\sum_{\omega\ne0}\left[1/(u-\omega)^2-1/\omega^2\right] $$ of Weierstrass (it already appeared in Eisenstein), with a $p$ which smacks of the gothic, of the italic and of the cursive, chosen by the inventor⁶⁵ and retained by posterity. (...)

⁶⁵ His biography in the DSB tells us that in the course of his fourteen years of high-school teaching he had to teach mathematics, physics, German, botany, geography, history, gymnastics “and even calligraphy”.

Note added: While I don’t know of a handwritten specimen by Weierstrass himself (asked about in comments by @NateEldredge and @ManfredWeis), there are a few in lecture notes of S. Pincherle who had studied with Weierstrass in Berlin: (1899-1900, Chap. XXII).

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Mathematical Notation – The Letter \wp; Name & Origin

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