# How did Kepler infer three-dimensional positions from Tycho Brahe's data?

This has bugged me for some time.

Tycho Brahe's data on planetary observations, presumably, consisted of the direction in which a planet was observed at a given date and time, but not the distance to the planet. What techniques did Kepler use to add a depth dimension to these observations, to create the three-dimensional data that one can start studying to arrive at his three laws?

• @rob I tried to calculate this my self from Keplers original data like I explain it in my answer. But it's a mess! So i Google'd a bit, and found out, that you do need the declination! To correct the mistake caused by the light bending in atmosphere. Kepler used a whole year to make these correction. I give up; I must say I cant calculate this like Kepler did; There is far more work needed than just to put the Tycho's data to excel and to get the results. Theoretically it's a simple calculation, but the devil is in details. Dec 14, 2015 at 18:53
• @Jokela You might want to re-post that under your answer - rob won't get notified here. Dec 14, 2015 at 19:11

And as the Mars is out side us, and rotates slower, it has an particular character that it even moves to "wrong direction" in the sky for a while. It must have been partially luck, that 5 of these observations is measures with enough accuracy this important point in orbit. (see link) Or maybe this was exactly the interesting "problem" they were laying their eyes at. Anyhow, this makes it quite easy to solve the distance with trigonometry. And this way Kepler had more then just the opposition positions of the orbit. This Picture explains how the rather simply trigonometry; aloud to solve also the distance through these observations.

This way you can solve the shape of the orbit. Ofcourse you might not have any true distance to anywhere, so you must decide that ie. the Distance from earth to sun is "1-something" and then you can start to calculate the rest, even though you can't scale the "1-Something" to meters. At 2012, over 400 years after Tycho's death, this "1-Something" is claimed to be accurately 149597870700 meters. The first definition was made allready by Archimedes. I am sure he claimed his result as accurate, as the present knowledge. I don't know when the name "AU" was given. Tycho used the very wrong value from Ptolemy, 1/20 from the true distance. Linear scaling obviously doesn't seem to have any influence to shapes. And the way Tycho did he's observations even eliminates the possibility to such a mistake. He took the distance to sun granted and measured practically only angles.

The way to get the exact positions through angles is called Triangulation. This method was invented by cartographer Gemma Frisus in 1533. Even today, with Theodolite is exact 3D positions measured through angles only. By setting up the machine, you only need to show two known points to be able to measure Anything in 3D-space. Depending on what is known, you may need to measure these angles from two positions, to get a distance. And you do need to have some fixed position;

Kepler used a fixed position of Mars defined by it's orbital period;

So to conclude this completely, the question was;

What techniques did Kepler use to add a depth dimension to these observations, to create the three-dimensional data that one can start studying to arrive at his three laws?

And the Mars provides practically two techniques, which are both introduced to this picture;

Technic "A"; The known Mars orbital period is used to fix the position of Mars(3), the angle is defined by measuring the directions to Mars at some Earth position (2) and then again exactly after 687 days, when Earth is in another position (1) because Earth has Orbitet 1.88 rounds (687/365). Two measurements made 2 years minus 43 days.

Technic "B"; The slower orbiting speed of Mars is used to produce quasi-fixed postion to Mars. The angle is first measured at Earth on 4 at the point when Mars "stops". Note that this position 6 is the Mars position "c" at the "retrogade motion"-picture. This Motion lasts 72 days, which means that 72/687 = 0.105 x 360 degrees must be removed from the movement of Earth, which is 72/360 = 0.2 x 360 degrees. This way the coordinates can be moved an the angle measured at Earth position 5 and Mars at 7/e can be used as a fixed point.

The measurement data of Tycho Brache aloud to define these distances 5 times with Methdod B,

And 16 times with Method A;

• 27.12.1582-13.11.1584,
• 21.12.1584-8.11.1586,
• 12.3.1585-28.1.1587,
• 15.4.1585-3.3.1587,
• 18.5.1585-5.4.1587,
• 27.3.1587-12.2.1589,
• 21.4.1587-9.3.1589,
• 23.9.1591-11.8.1593,
• 2.10.1591-18.8.1593,
• 10.11.1591- 26/28.9.1593,
• 23.1.1592-10.12.1593,
• 13.2.1592-30.12.1593,
• 17.2.1592-3.1.1594,
• 29/30.10.1593-15/18.9.1595,
• 26/27.11.1593-12/16.10.1595,
• 7-13.12.1593-25-30.10.1595,

if you reduce the accuracy to 2 days; it can be made atleast additional 8 times; 7.1.1585-23.11.1586, 14.1.1585-1.12.1586, 26.3.1585-10.1.1587, 7.5.1585-27.3.1587, 9.3.1589- 23.1.1591, 16.3.1589-3.2.1591, 4.4.1589- 19.2.1591, 3.2.1592-19.12.1593.

Especially 1593-1595 measurements provides really high accuracy. But it can be easily seen that such an high amount of measurements provides enough data to make solid conclusions.

Declination, Ascension Rob's Comment forces me to improve this aspect. As seen in the first picture the data includes the Declination, the other key information was the Time, when the Mars was seen in fixed direction. This time was first recorded with only 5-10 minute accuracy, and later with a minute accuracy. This time is of course Apparent solar time. Which practically means, that it's directly the angle to sun in an orbital plane. This means that the Declination is not needed at all, to calculate the distances with methods A and B. The declination is needed only to define the Retrogade Motion points e and c. It should be further noted that the Solar Mean time varies in order +/- 30 secs in one orbit, which means that Tycho's time measurements made in with 1 min accuracy, were as accurate as it can be. This fact simplifies the calculations to 2D.

Time Measurement It should not be left without notion, that Tycho Brahe was apparently the first person in Earth, who was even able to do these measurements. The first clock, able to even measure seconds, was build in 1579. In 1581 Tycho redesigned his clocks, so that they could display seconds. Yet, his four clocks were not accurate enough; disagreement was $+/- 4 s$. The more accurate Pendulum clock was invented and build first in 1644.

• A minor quibble on an excellent answer: your first plot shows variations in Mars's declination, but your top-down figures suggest changes in (mostly) right ascension. Is it safe to assume that Kepler was working in three dimensions and used both data sets?
– rob
Dec 10, 2015 at 21:30
• @rob Well, yes, this aspect is actually still missing; You of course need to use both, declination and Ascension, as the Earth is also rotating over it's own axle. But Tycho Didn't measure Ascension, instead he measured the accurate time when the Mars was at the same direction (ascension) with his "stonehenge-alike" measurement system. To get this "right" you have to use both, though Cos 20 is only ~0.965. I will fill up this gap some time soon Dec 11, 2015 at 5:31
• @rob ...After looking the Kepler's data I realized that you don't much need the Declination, as the Ascension was measured from Solar time instead of some mechanical angle measurement device, it doesn't make any mistake. Dec 12, 2015 at 18:53
• Hi, I am trying to replicate the first graph as mentioned in your answer, Was wondering where could I get the data of the 'RED LINE" -- The orbit as calculated by the modern methods. Oct 7, 2019 at 20:21
• Could you take a look at physics.stackexchange.com/questions/507041/… . Would really appreciate it Oct 8, 2019 at 11:04