The Archimedean property of the real numbers says that there is no $\varepsilon>0$ such that
remains less than $1$ regardless of how small $\varepsilon$ is, as long as the finite cardinal number $n$ is large enough.
One explanation of why it's called "Archimedean" is that Archimedes of Syracuse (287 BC – c. 212 BC), a geometer, physicist, engineer, and inventor, said such quantities don't exist. Nonetheless, he used them brilliantly. He said his arguments using infinitesimals fall short of being complete proofs because infinitesimals don't exist. All of his arguments using infinitesimals are in one work, sometimes titled The Method.
Here is an example:
- Draw a secant line to a parabola with endpoints $A$ and $B$. (This need not be orthogonal to the axis, as one might think from some illustrations.)
- Through $B$, draw the tangent line; through $A$ draw a line parallel to the axis. These meet at $C$.
Archimedes claimed: One-third of the area of triangle $ABC$ is inside the curve.
To demonstrate this, he relied on the concept of center of gravity, which had first been introduced by him. He also relied on the concept of torque on a lever, also first introduced by him.
- Let $D$ be the midpoint between $A$ and $C$. Consider $D$ to be the fulcrum of a lever, which is the line $DB$.
- Let $E$ be on the line $DB$, just as far from the fulcrum $D$ as $B$ is, but in the opposite direction.
Archimedes showed that the center of gravity of the interior of the triangle is on this line $DB$, one-third of the way from $D$ to $B$. If one could let the whole weight of the triangle rest at that center of gravity, and a weight equal to that bounded by the curve and the secant line at $E$, then the lever is in equilibrium precisely of the proposition to be demonstrated is true.
To show that, he considered cross-sections parallel to the axis. Let the infinitesimal weight of each such cross section (proportional to its length) rest on the lever at the point where the cross-section intersects the lever. Archimedes claimed this would exert just as much torque on the lever as if the whole weight of the triangle rests at the center of gravity.
So let the infinitesimal weight one such cross-section rest on the lever $DB$ at the point where it crosses $DB$. And let a weight equal to that of the part of that cross-section that is inside the curve rest at $E$. If the lever is then in equilibrium, then we're done. But that is just what in modern language we would call the equation of the parabola. QUOD ERAT DEMONSTANDUM
Archimedes used the same method to show that the center of gravity of the interior of a hemisphere (i.e. half a sphere) is five-eights of the way from the pole to the center. And maybe more than a dozen other propositions; I don't remember exactly how many.