Similar Functions

Summary: $Y$ and $Z$ are similar functions of $y$ and $z$ when both are built from their respective variables by the same formal expression. Euler introduces this as a bookkeeping notion used throughout the Introductio.

Sources: chapter1

Last updated: 2026-04-23

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Definition

If $Z$ is a function of $z$ and $Y$ a function of $y$ such that $Y$ is defined through $y$ and constants, in the same way as $Z$ is defined through $z$ and constants, then the functions $Y$ and $Z$ are said to be similar functions of $y$ and $z$ respectively. (source: chapter1, §26)

Operationally: replacing $z$ by $y$ in the expression for $Z$ yields the expression for $Y$.

Example

If $Z=a+bz+c{z}^2$ and $Y=a+by+c{y}^2$, then $Z$ and $Y$ are similar functions (source: chapter1, §26).

A common idiom is ”$Y$ is such a function of $y$ as $Z$ is of $z$.”

Use under substitution

Similarity is used even when the two variables are related. For instance, with $z=y+n$:

• “Such a function of $y$ as $ay+b{y}^3$” is similar to the function $a(y+n)+b(y+n{)}^3$ of $y+n$.

And with $y=1/z$:

• $\frac{a+bz+c{z}^2}{\alpha +\beta z+\gamma z^2}$ as a function of $z$ is similar to $\frac{a{z}^2+bz+c}{\alpha z^2+\beta z+\gamma }$ as a function of $1/z$ — after clearing, this is the same expression written in $1/z$.

Why this matters

The similar-function concept is the 18th-century precursor of a formula template or an abstract function symbol. It lets Euler talk about “the same function applied to different arguments” without the modern notation $f(\dots )$. He says it “is fruitfully used throughout all of higher analysis” (source: chapter1, §26).

Related pages

• function
• chapter-1-on-functions-in-general