In his Newtonian studies Alexandre Koyré remarked that, from the area law of planetary motion, Kepler erroneously deduced that the planets move in their orbits with speeds inversely proportional to the distances from the sun. It was this assertion, which I suspected to be false, that led me to make a detailed study of the discovery of the area law. Quite independently and unknown to me, Curtis Wilson had been making a similar study of the discovery of the elliptical orbit. Since the arguments leading to the two Keplerian laws are inter-dependent, our researches may be described as variations on the same theme. What I wish to do this evening is to explain in simple terms our reconstruction of Kepler’s path to the elliptical orbit and the area law.

The following activity is one of a series developed for junior high school students (age 12-15). Each of the activities involves a historical discussion of a subject related to a certain curriculum topic, and takes the form of a teacher directed presentation/discussion with accompanying transparencies and worksheet.

The choice and development of the activities was guided by certain general principles. The connection between the regular curriculum and the selected subject is meant to give relevance to the history and also, to motivate and deepen student understanding of subject matter. The structure of the activity enables the teacher to change, omit, or to add if he wishes. Points for discussion and questions to be worked are deliberately inserted at intervals to activate students, because in previous trial versions which were closer to the lecture format we found that students lost the thread and hence, also interest. Reactions to the present format were much more encouraging.

The classical theorems of Ceva and Menelaus make assertions about the value of certain products of ratios of lengths in configurations in the affine plane. We shall use the term Ceva-type to describe any result of this general kind: one that specifies a configuration in affine space of n dimensions, defined only by incidences, about which one can make an assertion about a product of ratios of lengths, areas, etc. Several results of this kind are known. Apart from the classical results there are, for example, Ceva's and Menelaus' Theorems for n-gons, Hoehn's Theorem for pentagrams [1], and the Selftransitivity Theorem of [2].

The classical problem of the knight’s tour consists of moving a knight over a chess board in such a manner that it moves successively on to every possible square once and only once. If the initial and final squares of this tour are a knight’s move away from each other, then for obvious reasons the tour is termed re-entrant. The problem has a long and interesting history. Solutions due to De Moivre, Euler, Vandermonde, Warnsdorff and Roget, together with further references can be found in [1].

In Cundy and Rollett’s invaluable book on Mathematical Models, the authors begin their Chapter on Polyhedra by remarking that “The most suitable, and in many ways the most attractive, subject for an experiment in the construction of mathematical models is a set of polyhedra.” Various methods are in general use to produce finished models of polyhedra for the showcase, and similar methods form the basis of those considered suitable for constructing the simpler polyhedra in the classroom. That most commonly adopted is to draw the net of the required solid on a sheet of cardboard, allow for tabs as necessary, cut it out, score the creases half-through, fold up, and stick the tabs with some suitable quick-drying cement. For class-room use the major drawback to model making on these lines is the time-consuming and potentially messy process of sticking; and it is the object of this article to develop a method which dispenses with the use of paste or cement altogether. In fact I shall show how, using paper and scissors only, firm, neat models can be made, which in the case of the simpler solids could easily be produced in the classroom, and which in the case of the more complex polyhedra reveal by their construction in a striking manner some of the geometrical relationships of these solids.

1. It is well known that the solution of the ordinary linear differential equation with constant coefficients

(Dn + a1Dn−1 + + an−1D+an)y = F(x), .....(1)

where y, Dy, ..., Dn−1y take arbitrary values y0, y1, .., yn−1

when x = 0, ..... (2)

can be obtained by multiplying (1) by e−px, (p > 0) and integrating with regard to x from 0 to ∞

A selection process used by some compilers of big-prize competitions is to present a triangle (usually equilateral) divided into (say) nine congruent sub triangles by the lines joining corresponding trisection points of the sides, and to ask how many triangles there are in the figure. It is amusing to obtain the general result when each side of the original triangle A is divided into n equal parts and pairs of dividing points are joined by parallels to the sides in all possible ways.

If a series of points be pricked at random through a strip of paper, and the strip be held horizontally, certain of the points will appear as peaks, namely those which happen to be higher than either of their immediate neighbours. These we call maximal points. Similarly there are others, minimal points, which are lower than either of their immediate neighbours. Between each maximal point and an adjacent minimal point is a “run”, either a “run up” or a “run down”, the points in a run forming a steadily ascending (or descending) series. These runs will naturally be of various lengths, the length of a run being defined as the number of points in the run, including the first and the last. The shortest runs will thus be of length 2, others may be of length 3, 4, or some greater number, up to m, where m is the total number of points on the strip. Clearly where the points form a run of m there is only one run throughout the whole strip, the greatest possible number of runs is m-1, in which case each run is of length 2.