In Einstein's "The Meaning of Relativity", https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1 I don't understand this passage:
"We shall now show briefly that there are geometrical entities which lead to the concept of tensors. Let $P_0$ be the centre of a surface of the second degree, $P$ any point on the surface, and $ξ_ν$ the projections of the interval $P_{0}P$ upon the co-ordinate axes. Then the equation of the surface is"
$a_{\mu \nu }\xi _{\mu }\xi _{\nu }=1$
The quantities a $\displaystyle a_{\mu \nu }$ determine the surface completely, for a given position of the centre, with respect to the chosen system of Cartesian co-ordinates. From the known law of transformation for the $\xi _{\nu }$, (3a) for linear orthogonal transformations, we easily find the law of transformation for the a $a_{\mu \nu }$"
I don't understand how $a_{\mu \nu }$ was brought in, nor the form of the equation, (and by the way 1 is the radius?)
Just because it is easy to draw in 2D consider an ellipse centered at $P_0$
$$ \left(\frac{\xi_1}{a}\right)^2 + \left(\frac{\xi_2}{b}\right)^2 = 1 \tag{1} $$
for some numbers $a$ and $b$, this is the red ellipse in the sketch below.
Note that this equation can also be written as
$$ a_{11}\xi_1\xi_1 + a_{22}\xi_2\xi_2 = 1 ~~\mbox{with}~~ a_{11} = 1/a^2, a_{22} = 1/b^2\tag{2a} $$
or as
$$ a_{\mu\nu}\xi_\mu\xi_\nu = 1 ~~\mbox{with}~~ a_{11} = 1/a^2, a_{22} = 1/b^2,a_{12} = a_{21} = 0\tag{2b} $$
A slightly more general case is the blue rotated ellipse on the sketch, for this one you can write the set of points that define it as
$$ A\xi_1 \xi_1 + B \xi_1 \xi_2 + C\xi_2 \xi_2 = 1 \tag{3a} $$
For some appropriate constants $A$, $B$ and $C$. As before you can rewrite it as
$$ a_{\mu\nu}\xi_\mu\xi_\nu = 1 \tag{3b} $$
which is again Eqn (2b). You can see the pattern here. You do not need to limit this to ellipses, you can also include hyperbolas. And this extends naturally to 3D, where you can include spheres, ellipsoids, hyperboloids of one and two sheets (...) The point is that if the surface is quadratic, you can always write as
$$ a_{\mu\nu}\xi_\mu\xi_\nu = 1 $$