Einige Gesetze über die Theilung der Ebene und des Raumes.

The article “Einige Gesetze über die Theilung der Ebene und des Raumes.” was published by J. Steiner in the very first volumn of the Journal für die reine und angewandte Mathematik in 1826. Journal für die reine und angewandte Mathematik is the oldest mathematics periodical in existence.†. My translation is meant to convey the ideas published by Steiner, and when presented with the choice between translating faithfully to the original text or clarity of his ideas, I admit to choosing the later. There are two footnotes original to the text, which appear with asterisks. Where helpful I included additional footnotes to clarify ideas in the article; these are denoted by daggers and do not appear in the original text. I welcome any corrections or improvements to this translation. Please contact me with suggestions at heavilin@usu.edu Geometry textbooks have primarily followed J. Pestalozzi’s methodology† to illustrate how many planer regions can be constructed from intersections of lines and circles. But this treatment does not reveal the underlying rules that permit a more general treatment of the topic. Moreover, the approach is even less effective in providing general rules for the formation of regions in space formed by intersections of arbitrary planes and spherical surfaces. Instead of the geometer’s question, “How many flat surfaces are necessary to create this or that body?”, we turn the question around and ask, “How many regions can be formed by a given number of planes?” We begin by observing at least four planes are needed to construct a bounded region in space. By extension conclude four planes can form at most one bounded region. This statement then begs the question, “How many regions can be formed by 4, 5, 6, 7, or more planes?” In this article we present general rules for dividing a plane into bounded regions by way of intersecting lines and circles. Thereafter we extend these ideas to space and present the rules determining the number of regions formed by intersecting planes and spherical surfaces. From this we can easily determine how many of these regions are bounded. Unlike the current treatment of this topic in geometry textbooks, we explain the relevant relationships and provide a context for answering these questions, and in doing so further the study of solid geometry. 1 Clearly a straight line∗ cut at n arbitrary points is broken into n + 1 regions∗∗ of which n−1 are finite and the two remaining regions are infinite, and that furthermore a closed-curve†† cut at n points is broken into n regions. 2 A straight line embedded in a plane cuts the plane into two regions; with the addition of a second line that cuts the first the number of regions of the plane increases by two. A third line that cuts the first lines, the number of regions increases to three, and with a fourth line, that cuts the first three in three points, to four, and so on. In fact each susequent line increases the number of regions in the plane by the number of existing regions through which the new line passes, therefore the plane with n lines will be cut into 2 + 2 + 3 + 4 + 5 + · · ·+ (n− 1) + n = 1 + n(n + 1) 2 = 1 + n + n(n− 1) 1 · 2 (1) †Johann Heinrich Pestalozzi (January 12, 1746 Ð February 17, 1827) was a Swiss pedagogue and educational reformer who exemplified Romanticism in his approach. Wikipedia ∗By straight line we mean an infinite line. ∗∗Also here we mean the following, when speaking of the regions of a plane or space, we mean only the single regions, and not regions constructed by adjoining more simple regions together. ††Here the author is referring to circles EINIGE GESETZE ÜBER DIE THEILUNG DER EBENE UND DES RAUMES. Math 3310 regions. If one simply wants to know the number of finite (bounded) regions of the plane, we note that the first three lines construct this piece. The fourth line increases the number of such regions by 2 and the fifth line increases by 3 and so on. Namely, for each additional line the number of bounded regions of the plane can increase by the number of intersections with boarders of the bounded regions that it interesects†, and therefore with n arbitrary lines there can be at most 0 + 0 + 1 + 2 + 3 + 4 + · · ·+ (n− 3) + (n− 2) = (n− 1)(n− 2) 2 = 1− n + n(n− 1) 1 · 2 (2) bounded regions of the plane. Substracting Eq. (2) from Eq. (1), the number of unbounded regions of the plane is