In this paper we introduce the notion of Archimedes invariant partition of a curve and of a family of curves. Let \(f:\mathbb {R}\rightarrow \mathbb {R}\) be a strictly convex function and let n be an integer, \(n\ge 2\) . We say that an increasing sequence \(t=(t_0=0,t_1,\ldots ,t_n=1)\) of numbers of the interval [0; 1] defines Archimedes invariant partition of the curve \(\mathbb {R}\ni x \mapsto (x, f(x))\) if for every \(-\infty<A<B<+\infty \) the ratio \(P_f(A,B)/P_f(t,A,B)\) , where \(P_f(A,B)\) is the area of the domain \(\textbf{P}_f(A,B)\) enclosed between the graph of the function \(f\big |_{[A; B]}\) and the line segment with endpoints (A, f(A)) and (B, f(B)), and \(P_f(t,A,B)\) is the area of the convex polygon \(\textbf{P}_f(t;A,B)\) with the vertices \((x_k, f(x_k))\) , \(x_k=(1-t_k)A+t_kB\) , \(k\in \{0,1,\ldots ,n\}\) , is independent of the endpoints A and B. The sequence t defines the proportions on which we divide the interval [A; B], we call it a proportions sequence. Let \(\Gamma \) be a family of curves of above mentioned form. We say that a partition sequence t defines Archimedes invariant partition of the family \(\Gamma \) if it defines Archimedes invariant partition of each member of \(\Gamma \) . This idea comes from Archimedes’ theorem on the area of a parabolic segment which in fact says that the proportions sequence \((t_0=0, t_1=1/2, t_2=1)\) defines Archimedes invariant partition of the family of all parabolas. In the paper we show that for every integer n, \(n\ge 2\) , every proportions sequence t defines Archimedes independent partition of the family of all parabolas. We also fully characterize all proportions sequences t which define Archimedes invariant partitions of the family \(\Gamma :=\{\mathbb {R}\ni x \mapsto (x, p(x)):p\;\text {is a cubic polynomial}\}\) . In the paper we refine above mentioned invariance definitions in terms of a mean value property. This allows us to drop the strictly convexity restriction of f and to avoid division by zero.
