# Explanation of the main points in Gauss's resultant calculus

After he read Mobius's 1827 treatise on the "barycentric calculus" (according to Gauss's own testimony, he read this treatise only in 1843), Gauss wrote down several unpublished notes on what he called "resultant calculus" (I don't know if this a general term or just Gauss's own term), and then he used this in the context of his studies of the "pentagramma myrificum". To make it easier to offer explanation of the main points of this calculus, I added a (google) translation of Gauss's fragment, which is very short:

[1] The barycentric calculus finds its counterpart in another (presumably still comprehensive) calculus, which one might call the resultant calculus. Just as the first deals with points in which one assumes heavy masses; so the latter would have as its object lines in which forces act. If $$a,b,c,d$$ and so on are such lines in which, in each of them in a certain sense, forces act which are proportional to the numbers $$\alpha,\beta,\gamma,\delta$$ and so on, then the equation becomes $$a\alpha+b\beta+c\gamma+d\delta+\cdots = 0$$, and it means that these forces keep each other in balance. [2] If $$B,C$$ are two points in $$a$$, one can quite appropriately put $$B-C = ai$$. [3] The main question now is what kind of force one has to apply, baycentrically to the angular points, or resultantically to the sides of the pentagon, so that the simplest possible relationships take place between them and their equivalents at the inner pentagon. [4] The co-linearity of two systems of points or straight lines can be best defined by the fact that one linear equation can always be divided between five points (lines) of one system and the five corresponding ones of the other. [5] The characteristic of colineation seems to be expressed most simply like this: let $$A,B,...$$ be straight lines, $$P,Q,...$$ points of the one system to which the lines $$a,b...$$ and the points $$p,q...$$ correspond in the other. If now one expresses the (vertical) distance of the point $$P$$ from the straight line $$A$$ using $$PA$$, then in general: $$\frac{PA\cdot QB\cdot pb \cdot qa}{pa \cdot qb\cdot PB \cdot QA}=1$$.

Now, I have many misunderstandings concerning this fragment:

• The basic idea - While the conception of "barycentric coordinates" is clear to me, the general conception of the "resultant calculus", as a kind of dual-equivalent to Mobius's calculus, is unclear to me. Therefore, I'd like to know if anyone can understand from the translation what is the basic idea behind it.
• Notation clarification - Does Gauss refer to complex sizes in what he denotes $$\alpha,\beta,\gamma,\delta$$ or just to real sizes? I ask because in note [2] Gauss uses $$i$$ (the imaginary unit).

I have several other questions on this fragment and its relation to the pentagramma myrificum, but as a start I just want to clarify the notation and the basic idea.

• What is meant by the forces keeping each other in balance? What are angular points? My first thought on ia was the imaginary axis (in the imaginary plane there are angles involved). Jul 19 at 21:44
• Where are the masses and how big are they? Jul 19 at 21:46
• I guess "Forces keeping each other in balance" means that the (vector) sum of the forces is zero. And the masses mentioned in note [1] are Mobius's physical metaphor for his barycentric coordinates (read the wikipedia article on "barycentric coordinates"). Jul 19 at 21:56
• Arre the forces metaphoric too? Jul 20 at 5:33
• So basically you ask what he means with his resultant calculus? Barycentric coordinates are clear. They are the coordinates of the point inside a triangle (3 points) at the intersection of three perpendicular lines going from a point to the other side. Im not sure if I see the connection with masses here. Are different masses put on the corners of an equilateral triangle? Yes, of course. Now Gauss replaced the points with masses for lines and forces. What has a pentagram to do with this? Jul 21 at 6:33