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

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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.
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.