Section §4: In this section, entitled “the curious points of the triangle”, Geiser proves a few of the theorems from Chapter 16
without making use of transversals, but rather by means of points and sides of a triangle. He also defines the incircle and excircles of a triangle. Furthermore, he points out relationships between various points of the triangle, e.g. that the centroid S
5
Here, Geiser gives a different proof than in §2.
6
Namely, the theorems concerning the nature of the centroid, the circumcentre, the incentre, and the orthocentre. In order to prove that the angle bisectors meet at the incentre, Geiser introduces the notion of the distance of a point from a given straight line.
lies on the same line as the orthocentre H and the circumcentre M, with the relation HS = 2HM [15, p. 16].
Geiser then makes use of the theorems that he introduced so far to find ‘solutions of some simple problems’ [15, p. 19]. By solving, he really means proving by means of geometric construction; he explains how to derive certain results by drawing straight lines and circles, bisecting angles and employing the properties of particular points in a triangle. Thus, these proofs also serve as instructions. Examples of these problems are proving that for four triangles constructed by means of four straight lines the respective circumcircles intersect in a single point, and that the respective orthocentres of such triangles are collinear.
Section §5: In this section, Geiser introduces what he calls “körperliches Dreieck” or “Dreikant”. This translates to “solid triangle”, which I will use for want of a better English expression7. Geiser remarks that ‘as is commonly
known, spherical triangles do not differ significantly from solid triangles’ [15, p. 23]. He continues that:
We will assume the most fundamental terms with regard to solid and spherical triangles, from which we will derive a number of properties corresponding to various theorems regarding planar triangles, which were derived in the previous sections.
[15, p. 23]
Geiser proves these theorems for solid triangles, giving the analogous versions, or consequences, for spherical triangles after each proof. He remarks that he could have used spherical trigonometry instead, but does not go into
7
Geiser defines solid triangles as follows:
Three arbitrary planes in space, i.e. of which no two are parallel and all three of which do not intersect in a single line in particular, divide space into eight portions, each of which is a solid triangle […]. The angles defined by the planes are generally known as the angles of the solid triangle. Meanwhile, the angles created by the lines of intersection of the planes are called the sides of the triangles. [15, p. 23]
any detail [cf. 15, p. 29]. An example is the theorem that the great circle arcs through the corners and midpoints of the respective facing sides intersect in the centroid of the spherical triangle. In order to derive this theorem, Geiser proves that the planes through the edges and median lines of the respective facing sides in a solid triangle intersect in its centroidal axis [15, p. 23-24]. Using the definition of a cone of revolution containing the edges of a solid triangle8, he describes the nature of the in- and circumcentre of a spherical
triangle; he also shows that the analogous statement of the relationship proved in §4 [15, p. 16] holds true for solid triangles.
Lastly, Geiser shows that when a plane intersects the sides of a skew quadrilateral, resulting in eight sections, then the product of four non-adjacent sections equals the product of the other four sections [15, p. 30]. The proof of this theorem, and its converse, is again more algebraic than the previous proofs in this chapter.
Section §6: In this section, Geiser moves into three dimensions and considers analogous results of the theorems proved so far for the tetrahedron:
Elementary geometry consists of planar, spherical and spatial geometry, depending on the region in which its constructions are performed. The theory of the planar triangle in planimetrics corresponds to the theory of the spherical triangle in […] spherical geometry, and in spatial geometry we can consider it to correspond to the theory of the trilateral pyramid or tetrahedron. Holding on to this analogy, one can easily conjecture how certain planimetric theorems can be applied to space when looking at them more closely. However, it is not guaranteed that the conjecture will hold in every case.
[15, p. 31]
8
According to Geiser, the cone’s axis is the axis of the solid triangle’s edges, which in turn is the line of intersection of the median planes of a solid triangle.
Among the theorems that do hold in three dimensions, he proves that the bimedians (the line segments between the midpoints of two opposing edges) intersect in the tetrahedron’s centroid, and that the planes through the midpoints of the edges and perpendicular to them intersect in the circumcentre. As an example where the analogy does not hold in space, Geiser shows that in general, the altitudes of a tetrahedron do not intersect. Furthermore, he investigates what happens when one inscribes eight spheres in a tetrahedron and constructs their respective centres [15, p. 34-35], and derives results from the fact that the tangent plane of a sphere through the four vertices intersects the opposing triangular face in a straight line [15, p. 35- 37].
Section §7: Here, Geiser defines the n-gon, first in two and then in three dimensions, as well as the complete n-gon (where n-1 edges join in each vertex). Referring to figures, he gives some examples of different kinds of n- gons, e.g. concave and convex. Moreover, he derives the number of edges and diagonal vertices in complete polygons, polylaterals and polyhedra9.