I am trying to understand the following introductory passage in an early lecture by the philosopher/mathematician Gottlob Frege because I am interested in how Frege conceived of the role of geometric intuition in mathematical reasoning.



> 
> One of the most far-reaching advances made by analytic geometry in more recent times is that it regards not only points but also other forms (e.g., straight lines, planes, spheres) as elements of space and determines them b means of coordinates. In this way we arrive at geometries of more than three dimensions without leaving the firm ground of intuition. The geometry of straight lines for example is a four dimensional one, and so is the geometry of spheres. But there is a difference between the two, in that a sphere can always be determined unequivocally by four numbers, whereas it would seem that this is not possible in the case of straight lines. we determine a straight line by an equation between six quantities with a quadratic equation holding between them. We express this peculiarity of the geometry of straight lines by calling it a second order one, whereas the geometry of spheres is of the first order.
> 
> 
> The geometry of pairs of points in the plane, with which we will here be concerned, is four-dimensional and of the third order.
> 
> 
> 


How are we to understand "dimension" and "order" in this passage? I guessed that the dimension is the number of coordinates required (six for lines) minus the order of the equation required to hold between the coordinates (two for lines). To get a better grip on the basic idea I worked the algebra on the equations for lines in three dimensional space.


$x = x\_{0} + at$


$y = y\_{0} + bt$


$z = z\_{0} + ct$


So that:


$t = \frac{x - x\_{0}}{a} = \frac{y - y\_{0}}{b} = \frac{z - z\_{0}}{c}$


And:


$abct^{3} = xyz - xyz\_{0} - xy\_{0}z + xy\_{0}z\_{0} - x\_{0}yz + x\_{0}yz\_{0} + x\_{0}y\_{0}z - x\_{0}y\_{0}z\_{0}$


Substituting the original equations into that, and doing a bunch of algebra, gives a quadratic in t with coefficients expressed in $x\_{0}, y\_{0}, z\_{0}, a, b, c$. So I'm trying to make sense of Frege's description of the geometry of lines is dimension four and order two by thinking of the dimension as the number of coordinates minus the order of the equation required for those coordinates to characterize a line. But, I can't quite make that fit with what he says about the geometry of spheres.


Frege's early work in geometry is influenced by the analytic projective geometry in the style of Plucker, using homogeneous coordinates. I've consulted some internet sources and used the undergraduate textbook "Modern Geometries" to better understand homogeneous coordinates, but I have been unsuccessful finding definitions of dimension and order for this context. What does Frege mean by dimension and order in the above quotation?



Vocabulary of 19th Century analytic projective geometry: What are "order" and "dimension"?
As Francesco Polizzi remarks, the idea is to identify $\text{Sym}^2(\mathbb{P}^2)$ with a cubic hypersurface in $\mathbb{P}^5$. Frege's lecture (see link in comments) actually goes on to explain how this is done. The cubic hypersurface is (in the six variables $s\_1,s\_2,s\_3,t\_1,t\_2,t\_3$),


$\text{det} \begin{pmatrix}
s\_1 & t\_3 & t\_2 \cr
t\_3 & s\_2 & t\_1 \cr
t\_2 & t\_1 & s\_3
\end{pmatrix}=0$


and the identification of $\text{Sym}^2(\mathbb{P}^2)$ with this hypersurface is


$\{[x\_1:x\_2:x\_3],[y\_1:y\_2:y\_3]\}\mapsto [x\_1y\_1:x\_2y\_2:x\_3y\_3:\frac{1}{2}(x\_2y\_3+x\_3y\_2):\frac{1}{2}(x\_3y\_1+x\_1y\_3):\frac{1}{2}(x\_1y\_2+x\_2y\_1)]$.


Quite possibly, someone more knowledgeable than me can motivate this!


