Why "monoidal" transformation? In Carlo Beenakker's answer to this recent MO question, it turns out that the name "monoid" was first used in mathematics by Arthur Cayley for a surface of order $$ which has a multiple point of order $−1$.
On the other hand, a (old) name for the blow-up is "monoidal transformation". I suspect that there should be a link relating these two classical terminologies, but I was not able to find one. So let me ask the following

Question. Why was the blow-up classically known as "monoidal transformation"? Was this related to Cayley's "monoid"?  

 A: As I mentioned in the comments, Semple and Roth (Introduction to Algebraic Geometry) discuss this in "Examples on Chapter VIII" example 12 (p.187 of my version). As far as I can tell, this is what they are saying:
Homoloidal webs:
Let $V$ be a 4-dimensional vector space and suppose we're working in
the projective 3-space $PV$ with homogeneous coordinates
$[x_0:\cdots:x_3]$. Pick four homogeneous (degree $d$) polynomials
$F_0,\ldots,F_3\in Sym^d(V)$. We get a rational map $PV\to PV$
defined by
$$[x_0:\cdots:x_3]\mapsto[F_0(x):\cdots:F_3(x)]$$
How
should we think of this? We have a 3-parameter family of surfaces
$\sum_i\lambda_iF_i=0$ (parametrised by
$[\lambda_0:\cdots:\lambda_3]\in PV^*$, called a "homoloidal web" of
surfaces) and the equation $\sum_i\lambda_iF_i(x)=0$ defines a
hyperplane in $PV^*$ consisting of surfaces which contain $x$. Our
map sends $x$ to the point in $PV$ dual to this hyperplane.
Monoids:
Now suppose that our surfaces $F_i$ all have a singularity of order
$d-1$ at the point $p$. In other words, they're all monoids in the
sense of Cayley (because they have degree $d$). They call the
corresponding birational maps monoidal transformations. Certainly $p$ will be in the base locus of this system of surfaces, so you need to blow up at $p$ to get a morphism, but I'm still not sure why this is a reasonable definition.
They do discuss examples of this immediately afterwards, but I get lost in the notation.
