November 15th, 2004 • 8:00 PM

Animated Graphics for Lyndon LaRouche's

**[NOTE: Some of these graphics are large, and may take some minutes to display on your screen, especially if you have a low-speed Internet connection.]**

**FIGURE 1: Archytas Solution to the Doubling of the Cube, Take 1**

Archytas solved the problem of doubling the cube's volume, by finding a general method for finding *two* geometric means between *two* extremes. This was accomplished by forming a cylinder and a torus, by rotating one circle around a point on the circumference of another. When a 60-degree cone is produced to intersect both the torus and cylinder, the sought-for geometric means are found.

**Note:** To get a better view of this construction, you can change the viewing angle by clicking and dragging on the image.

**FIGURE 2:Archytas Solution to the Doubling of the Cube, Take 2**

Archytas's construction for doubling the cube was to find two geometric means between two extremes by the intersection of a torus, cylinder, and cone. The torus and the cylinder are formed by the unified action of rotating a semicircle in which a right angle is itself rotating. In the animation, the red circle is rotated about point O. This action sweeps out the torus. Simultaneaously, a cylinder is formed by the connected rotation of line PQ. This doubly-connected action causes point Q to move simultaneously on a line and a circle, while point P moves simultaneously on a circle and the "bold curve" formed by the intersection of the torus and cylinder. This action produces the manifold of all possible cases of two geometric means between two extremes. To find the specific case for the doubled cube, we require a 60-degree cone. The intesection of all three surfaces produces the desired magnitude.

**Note:** To see the animation from different viewpoints, hold down the left mouse button and move the mouse. To stop the animation double click on the image. To restart the animation, double click again.

**FIGURE 3: Retrograde Apparent Motion of Mars**

Here is a simulation of a time-lapse representation of the observed positions of Mars between June and December 2003. For part of this time Mars appears to be moving backward against the background of fixed stars.

**FIGURES 4 & 5: Apparent Motion of Mars in 2003 and 2004**

This is a simulation of the motion of Mars at two different parts of its orbit. Notice that in 2003, Mars moves a greater distance against the background of fixed stars than in 2004 in the same amount of time.

**FIGURE 6: Kepler's Hypothesis of Mars's Elliptical Orbit**

Kepler hypothesized that the non-uniform motion of Mars was the effect of an elliptical orbit in which the Sun was at one of the foci. In this orbit, Mars would move closer and farther from the Sun as it orbitted around. When Mars was farther from the Sun, it would move more slowly. When it was closer, it would move faster. Since its distance to the Sun was constantly changing, so was its velocity.

**FIGURE 7: Kepler's Discovery of the Equal Area, Equal Time Principle**

Kepler discovered that even though the motion of Mars was not constant, it was governed by a principle that equal portions of Mars's orbit produced equal areas sweeping out its motion.

**FIGURE 8: The Three Anomalies**

Kepler measured the non-uniform motion of Mars by three anomalies. The "true" anomaly which was measuered by the angle formed by a line connecting Mars to the Sun (seen here in blue). The "eccentric" anomaly was measured by projecting the planet's elliptical motion onto a circle and measuring the angle this projection made with the center of the circle (seen here in black). The "mean" anomaly measured the average, or constant motion (seen here in white).

**FIGURE 9: The Position of Ceres**

In January 1801, Guisseppi Piazzi discovered a new object in the sky which he observed for only 42 days. The object was later determined to be the asteroid Ceres. No one could determine the orbit of Ceres from these few positions—except for Carl Friedrich Gauss, who used Kepler's method, while all discredited authorities followed Isaac Newton.

**FIGURE 10: Motion of Ceres and Earth**

Gauss determined the orbit of Ceres by focussing on the relationship between the motion of Ceres and the Earth.

**FIGURE 11: Ceres Earth Areas**

Both the orbit of Ceres and the orbit of the Earth were following Kepler's principle of equal areas.

**FIGURE 12: Areas Swept Out by Ceres**

Gauss realized that since he knew the three positions of Ceres, and the time elapsed, he could determine the area swept out, and therefore the orbit, because the areas swept out was a function of the characteristic of the orbit as a whole. Gauss also understood that Kepler had forecast the existence of an exploded planet between Mars and Jupiter—precisely the region which Ceres was discovered to occupy.