Follow Johannes Kepler's brilliant analysis of the problem that stumped astronomers for two millenia. What are you looking at in the sky?
The Sun moves around a great circle called the ecliptic yearly. Thus the plane of the ecliptic always contains both the Earth and the Sun.
Northern hemisphere spring and summer last longer than fall and winter: The Sun takes longer to move the 180 degrees from March equinox to September equinox than to move the 180 degrees back to March equinox. These speed variations are identical every year. Simple, but not too simple.
Venus (and Mercury) oscillate from one side of the Sun to the other. The west-to-east leg (or morning to evening side of the Sun) takes much longer than the east-to-west leg. What seems to be happening?
Viewed from Earth, Venus certainly does not move around the Sun periodically. But is its orbit periodic with respect to the stars? If so, then when viewed from Venus, the Sun would follow a simple, periodic path through the stars.
You could have made the first yearly snapshot of Venus on any date. By starting from different dates, you can view the Venus's orbit from any direction in the ecliptic plane.
What would snapshots of the sky from Venus look like at these same twenty times? The Sun would be at twenty different points around Venus's orbital plane, and the Earth would appear at at twenty points around a copy of this Venus ring viewed from Earth, but with reversed sight lines.
To continue the analogy with Venus locations at Earth year intervals, build a ring of Mars locations at Mars year intervals. (The 687 day period of Mars was accurately known in antiquity.) The Sun is at different places now, so record those as well.
At each step, Earth, Sun, and Mars determine a plane, which cuts the sky in the great circle passing through the Sun and Mars. Since the Sun-Mars vector is the same at each step, all ten circles intersect at a single point.
Unfortunately, the intersection point of the Sun-Mars great circles is not an accurate way to find the Sun-Mars direction - in fact it fails completely when Mars crosses the ecliptic.
Thus, at opposition you know the absolute directions of all three sides of the Sun-Mars-Earth triangles at these ten times, so you know their shapes and orientations in 3D space! Furthermore, the Sun-Mars side is the same vector in all ten triangles.
Initially, you viewed all ten snapshots of the sky from the Earth, then from a fixed position relative to the Earth.
Comparing observations spaced by Mars years, you have surveyed points on Earth's orbit. The second part of Kepler's great insight was that by stepping Earth years you can survey new points on Mars's orbit.
You could continue stepping by Earth or Mars years to survey more points. But notice that your points on Earth's orbit are spaced by two Earth years minus one Mars year or 43.53 days.
Since you have triangulated fifteen 3D points on Mars's orbit in addition to ten on Earth's, you can view whole collection from anywhere.
Now that you have several points all around both orbits, can you guess how the Earth and Mars must move between your survey points?
To naked-eye precision, Earth's orbit is indistinguishable from a circle. However, Kepler noticed that the orbit of Mars is elongated by one part in 250 (0.4%) in the same direction as its center is displaced from the Sun. He also noticed an ellipse with 9% eccentricity has 0.4% elongation.
The slight speed variations in the motion of the Sun around the ecliptic exactly align with the direction the center of Earth's orbit is displaced from the Sun! The times of your survey points around Mars's much more eccentric orbit show the same pattern with much larger speed variations.
Kepler discovered an ingenious way to quantify the speeding up and slowing down of a planet as it moves closer and farther from the Sun - his Second Law: The radius vector from the Sun to a planet sweeps out equal areas in equal times.
Using Kepler's First and Second Laws, you can determine any orbit from only five observations! Thus Kepler's Laws represent a major practical advance, vastly reducing the time and effort required to survey an orbit.
Kepler discovered his first two laws by careful study of the motion of Mars relative to the Sun. Using those two laws, he found the orbital parameters for the other planets, and discoverd the quantitative relationship between the size and period of planetary orbits - his Third Law:
Today we remember Kepler for his laws of planetary motion. Eighty years later, Newton would recast Kepler's laws as his inverse square law of gravity - perhaps the greatest single breakthough in the history of science.
Here is everything you can see in the sky with your own eyes on a perfect night, except the Moon. (And including the Sun, even though you can't see where it is at night!) This wide angle view is 100° from side to side. Click the date to pause. Drag the sky to look around.
The tilt of the ecliptic relative to the equator (perpendicular to the Earth's spin axis) causes the seasons. The planets wander above and below the ecliptic, and also reverse direction sometimes - the motion of the Sun is much simpler. How would the Sun move viewed from another planet?
Day count: -
Interval between equinoxes:
March to September:
September to March:
Maybe you are watching Venus move around the Sun from a nearly edge-on vantage point. Venus moves around the Sun in the same direction as the Sun moves around Earth (apparently), but faster, passing behind the Sun on its west-to-east leg, then in front of the Sun (much nearer us) on its east-to-west leg.
To test this idea, record the positions of Venus spaced exactly one (sidereal) Earth year apart. At these intervals, your view of Venus's orbit is always from the same point on Earth's orbit, eliminating your changing vantage point. In twenty years, you begin to see the orbit of Venus take shape one point at a time!
Something remarkable happens: From two vantage points exactly 180° apart, all twenty snapshot points are perfectly colinear, and the Sun is on the line! You see that Venus's orbit lies in a plane containing the Sun - just like Earth - but inclined by about 3.4° to the ecliptic.
The point of this exercise is that Kepler used Mars instead of Venus. Instead of comparing observations one Earth year apart like this Venus ring, he began with observations one Mars year apart. Viewed from Mars, Earth would appear in a ring like the Venus ring you see from Earth.
Since you stepped by Mars year intervals, the ten Sun-Mars vectors you have recorded are identical both in direction and length. Thus, this Mars ring is a displaced copy of the ring of Sun points. In other words, you are looking at a copy of the Sun's (apparent) orbit relative to the Earth!
That intersection point is the common Sun-Mars direction. As with Venus, by choosing any other starting date for the ten snapshots, you can see the points on the orbit from any vantage point around Mars's orbit. Note that all ten points collapse into a line in the ecliptic at the two times Mars crosses it.
The one exception is when the Sun-Mars great circle is perpendicular to the ecliptic - called opposition. At opposition the Sun-Mars direction is very nearly the same as the Sun-Earth direction. Choose one of your ten snapshots exactly at opposition, and you will know the common Sun-Mars direction for all ten very accurately.
With this realization, Kepler became the first person to know what anything would look like from a vantage point other than Earth - his first great insight. Kepler could compute what these ten snapshots look like viewed from far above the ecliptic plane.
However, you are now free to move between times and view each triangle from a fixed position relative to the Sun if you choose! You see that you have surveyed ten points on Earth's orbit!
One Earth year after any of your ten survey observations, Earth is back at that survey point, while Mars advances around its orbit by some fixed amount. You can choose any two of your ten Earth points to triangulate this new position of Mars!
If you step by 43.53 days in either direction, you move from any of your known Earth points to an adjacent known point (unless you step off the end of the sequence). Thus, you can triangulate thirteen more points on Mars's orbit by taking a few such steps from the two points you know.
Just like Earth and Venus, Mars's orbit is confined to a plane, this time inclined about 1.8° from the ecliptic. Each planet orbits the Sun in its own plane, fixed relative to the stars. The slight inclination of these planes explains why we see planets wander out of the ecliptic.
You find that all the known points on either orbit very nearly lie on circles, but not exactly centered on the Sun. The eccentricity of Earth's orbit (off center distance as a fraction of radius) is about 1.7%, while the eccentricity of Mars's orbit is about 9%.
Thus if Mars's orbit is an ellipse, the Sun is at or near one focus. (Exaggerate the differences between your survey points and circles by a factor of 40 to see the elongation.) On this evidence, Kepler formulated his First Law: Planets orbit the Sun along ellipses with the Sun at one focus.
When you watch the motion at a true constant rate, it's hard to see these speed variations. Exaggerating the speed variations by a factor of fifteen for Earth, then by a factor of four for Mars, makes the non-uniformities obvious. In both cases, the planet moves faster when closer to the Sun.
This seems like a strange idea at first. However, your known survey points around both orbits are mostly spaced by equal times (43.53 days). You know the exact time Mars reaches those survey points, but not any others, so it's natural to study equal time sectors of its ellipse. Kepler noticed that within each orbit the sector areas match exactly.
For example, make five observations spaced equally in time to get five sight lines. Guess the distances along the first three sight lines to determine an ellipse, hence the other two distances. Adjust your three guesses to make the four sector areas equal, and you have the orbit.
The cube of the long axis of the ellipse is proportional to the square of the period. Unlike the first two, the Third Law relates the orbits of different planets. By putting another constraint on sector areas, the Third Law reduces the number of observations needed to determine an orbit to four.
However, Kepler's most original contribution is his use of periodicity to accurately survey 3D points on the orbits of the Earth and Mars. The greatest astronomers for two millenia all missed the picture - seeing the solar system may be the hardest problem anyone has ever solved.