When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Introduction

Two points determine a straight line. There are few mathematical statements so clear and indisputable. Yet even this simple statement needs a little qualification. Although we would generally imagine the two points in question to be different from one another, we should probably make that explicit:

Postulate 1.  Two distinct points in the plane determine a unique straight line.

This is similar to the first postulate in Euclid's Elements: "to draw a straight line from any point to any point." His use of the linguistic construction "from … to" suggests that he also meant the two points to be distinct, but he didn't postulate the uniqueness of this line, even though he implicitly used uniqueness, for example in his proof of Proposition XI.3.

In analytic geometry, where we think in terms of algebraic equations instead of geometric constructions, we know how to calculate the equation of a straight line given two distinct pairs \((x_1,y_1)\) and \((x_2,y_2).\) If our concern is to find a linear function \(y=mx+b,\) then Postulate 1 needs the additional qualification that the two points have distinct \(x\)-coordinates.

How does this simple principle generalize to curved lines and larger numbers of points? This vague question can be answered in a number of ways: for example, three points can determine a circle or a parabola. More generally, five points usually determine a conic section; that is, either an ellipse, a parabola, or a hyperbola. Like the corresponding proposition for straight lines, this principle needs some qualification in order that the conic be uniquely determined. And like that result, it can be treated either as a problem in geometry or as a problem in algebra, involving second degree equations in two variables. Of course, this second approach only became possible in the 17th century, after the introduction of analytic geometry.

By the 18th century, mathematicians knew how to generalize this proposition about conic sections to curves of higher order. (We will define equations of degree two and higher later in this paper.) Colin Maclaurin (1698-1746), Gabriel Cramer (1704-1752), and Leonhard Euler (1707-1783) all independently discovered that \(\frac{n^2+3n}{2}\) points determine a curve of order \(n,\) subject to certain conditions. When \(n>2\) there are no geometric constructions; instead, Maclaurin, Cramer, and Euler used a counting argument that comes from considering the equations of such curves. Quite likely, other mathematicians of the time came to the same conclusion by themselves, but at least one of them – William Braikenridge (ca. 1700-1762) – thought that the correct number was actually \(n^2+1.\) We notice that this agrees with \(\frac{n^2+3n}{2}\) only in the two classical cases \(n=1\) and \(n=2.\) Instead of counting coefficients in an equation, Braikenridge's argument came from considering the way in which two lines or curves in the plane can intersect.