Definition of "centroid" before the invention of vectors or coordinate system?

Question

It is fairly easy to define the centroid of a finite set of points $v_1,...,v_n$ in $R^n$: $(v_1+...+v_n)/n$ will be it. But how do people define, or calculate the centroid of a set of point $v_1,..,v_n$ in 2D, or even 3D space, before the notion of vector or coordinate system is invented?

Answer 1

The centroid of a set of points points can be defined recursively as follows:

• the centroid of a single point is the point itself;

• the centroid $G$ of points $P_1,\ldots, P_n$ ($n>1$) is the unique point on the segment joining the centroid $G'$ of $P_2,\ldots, P_n$ with $P_1$ such that: $GG'/GP_1=1/(n-1)$.

Of course one should prove that this definition does not depend on the order of points $P_1,\ldots, P_n$, which can be done (I can provide details if necessary).

This also entails the following useful property:

• the centroid $G$ of points $P_1,\ldots, P_n$ and $Q_1,\ldots, Q_m$ is the unique point on the segment joining the centroid $G_n$ of $P_1,\ldots, P_n$ with the centroid $G_m$ of $Q_1,\ldots, Q_n$ such that: $GG_m/GG_n=n/m$.