Re: Cassinis triangulation of distance Earth-Mars in 1672

Hello, Paul!

You sound like someone both fascinated by and eager to learn about 
astronomy.  I will give some short answers to your direct questions, but 
before getting to that, let me give you a web reference that has a very wide
range of introductory astronomy topics at the university level:

     http://www.astro.
lsa.umich.edu/users/cowley/intro1.html
     http://www.astro.
lsa.umich.edu/users/cowley/intro2.html
     http://www.astro.
lsa.umich.edu/users/cowley/intro3.html

These are three very long web pages, which together comprise almost 40 
lectures of Professor Cowley's class at the University of Michigan  (and a 
hearty round of applause to the good professor for making this available... 
no small amount of work!). The subjects of parallax, etc, are all mentioned 
in there and you might want to read those parts germane to distance 
measurements, or--better yet--read all of it. It is a very wide-ranging 
class and well worth your time since you have an interest in the field.

Now to your questions:

>>>Roemer used the A.U. for his measurement of Lightspeed. Where did he get 
>>>it? 

    Roemer got it from the measurements by Cassini and others using the 
parallax of Mars to establish the Earth-Mars distance. That, combined with
the relative distances of all the planets (see below) from Kepler's 3rd Law,
allow you to solve for the actual distances.

>>>I found the reference to Cassini in your Article about parallaxes. 
>>>Since Earth and Mars are both constantly moving against each other, what 
>>>was Cassinis baseline? 

    Cassini's baseline was the diameter of the Earth, or some known part of 
it.  There are two possible ways to do this, and I must admit I don't know 
which one Cassini used. One is to have two observers widely separated on the 
Earth make simutaneous observations of Mars and background stars.  The other 
is to use the time span of the night to let one observer rotate around... 
this is especially effective if Mars is at or near opposition and thus 
observable almost all night, allowing almost the entire Earth diameter to be 
used. However, the second method is limited somewhat by difficulties in 
observation near the horizon due to atmospheric effects and by the fact that 
Mars will move some distance in 12 hours.

>>>I would like a sketch or diagram of the experiment.

    Your Mad Scientist admits to being a computer klutz and doesn't know
of a way to put diagrams into these answers.  Sorry!  However, there are
general diagrams of how parallax works in Prof. Cowley's class notes on the 
web (his diagram is for stars and uses the earth's orbital diameter as the
baseline, but the concept is very similar).

>>>(and the actual data itself??)

    This I don't have. If it still exists (Cassini's experiment was done in 
the 1600's), it is probably in a museum somewhere. There were no scientific 
journals in those days...the practitioners wrote letters to each other and 
passed them all over Europe so that everyone was kept informed of what was 
going on.

>>>One side and two angles means Cosinus Theorem?.

    I'm not familiar with your term "cosinus theorem" but with two angles 
and the side between them known, the triangle should be solvable.  The third
angle is immediately known (sum of all 3 is 180 degrees), and then the law 
of sines can be used:

     Sin A         Sin B         Sin C      where A, B, C are the angles
    -------   =   -------   =   -------       and a, b, c are the sides
       a             b             c          opposite those angles


>>>Your article says "it was known that Jupiter is about 5 times as 
>>>far as the Earth from the Sun" How was that known?? 

    This was known from Kepler's Third Law: the ratio of the cube of the 
semimajor axis of the orbital ellipse to the square of the orbital period is 
a constant.  Express the semimajor axis in Earth orbit radii (one earth 
orbit radius is defined to be 1 Astronomical Unit or AU) and the period in 
Earth years, and the constant is exactly 1.  Since the orbital periods of 
the planets can be observed directly, then the relative distance from the 
sun to any planet can be determined in AU.  For example, in round numbers, 
Jupiter takes about 12 years to complete an orbit. Square this, and take the 
cube root, and the answer is 5.24 AU for Jupiter's semi-major axis  (the 
actual numbers, from the Royal Astronomical Society of Canada's "Observer's 
Handbook", are 11.86 years for the orbital period and 5.203 AU for the 
axis).

>>> And Thanks!!!  Paul

And thank you for an interesting question.  

[Moderator's note: There is an archive of historical manuscripts here at the 
Royal Observatory in Greenwich (e-mail astroline@nmm.ac.uk). *If* Cassini 
corresponded, for example, with John Flamsteed, we may have the correspondance. 
Cambridge University Library or the archives of the Paris Observatory might be 
other places to look for historical manuscripts - Jim O'Donnell]