FIGURE 1a



The Arithmetic Mean



b is the arithmetic mean between a and c.



FIGURE 1b



The Geometric Mean



The length b is the geometric mean between lengths a and c. The area B is the geometric mean between areas A and C.



FIGURE 1c



Two Geometric Means Between Solids



There are two geometric means between a cube whose edge is 1 with volume 1, and a cube whose edge is 2 with volume 8. Proportionally, the there will be two geometric means between a cube of volume 1 and a cube of volume 2.







FIGURE 2a



The Proportions of a Parabola



The parabola is formed by moving right angle ABC so that vertex B moves along line OB while C moves along line OC. This forms the changing rectangle OBPC. Point P traces a parabola. By similar triangles, OA:OB::OB:OC or, OC=OB2.



FIGURE 2b



The Proportions of an Hyperbola



The hyperbola is formed by the corner B of rectangle OABC. As the sides of the rectangle change, the area stays constant. This maintains the proportion 1:OA::OA:OA x AB.



FIGURE 3



Menaechmus' Determination of Two Means by Conic Sections



The intersection of an hyperbola and a parabola determine the magnitudes that double the cube. The parabola is formed from OA=1 and right angle ABD. The hyperbola is formed from the green rectangle OBCD which has an area of 2. From the parabola, OA:OB::OB:OD, or 1:OB::OB:OC2. From the hyperbola, OB x BC = 2. Combining these two yields the proportion, 1:OB::OB:BC::BC:2. In other words, line OB will form a cube whose volume is 2 and BC will form a cube whose volume is 4.







FIGURE 4



Intersection of Cylinder and Torus



The curve formed by the intersection of a cylinder and a torus has the characteristic that Gauss called ``negative'' curvature.



FIGURE 5



Kepler's Projective Concept of Conic Sections



As the focus moves off to the left, the circle is transformed into an ellipse. At the boundary with the infinite, the ellipse becomes a parabola. The hyperbola is formed on ``other side'' of the infinite.



FIGURE 6



Equal Hyperbolic Areas



The areas between 1 and 2; 2 and 4; and 4 and 8, are all equal.







FIGURE 7



Leibniz's Construction of the Catenary



The catenary is formed as the arithmetic mean between two curves which Leibniz called ``logarithmic,'' and are today called exponential. In the figure, the blue lines are spaced equally along a horizontal axis. The ``logarithmic'' curve is formed by the vertical lengths which are in geometric proportion. OO=1; e'=OO2 and e=1/OO2; d'=OO3 and d=1/OO3, etc. The catenary is formed by adding length e to e' and dividing the combined length by two; then adding length d to d' and dividing the combined length by two, etc. The points of the catenary are equal to (OOn+1/OOn)/2.



FIGURE 8a



Projection of Equal Hyperbolic Areas



The points along the hyperbola that correspond to equal divisions of area are projected onto the axis, by drawing perpendicular lines from the axis to those points. This produces lengths, Ob,Oc,Od. Oa=1.



FIGURE 8b



Measuring the Lengths Along the Axis



When the perpendicular lines from the axis are extended to intersect the asymptote, they mark off the lengths 2n+1/2n. By inversion, the corresponding lengths along the axis are projections by a 45 degree angle of these lengths. Therefore, the lengths Ob, Oc, and Od are equal to (2n+1/2n)/2.



FIGURE 8c



The Relationship Between Hyperbola and Catenary



When lengths Oa, Ob, Oc, Od are set along a line at equal intervals, their endpoints form the catenary.