Lecture 7: Copernicus I (Copernican v. Ptolemaic Systems)
© Darrin Durant 2004
Copernicus v. Ptolemy
The Ptolemaic System (Claudius Ptolemy, 100-170)
The Eccentric Model: The earth is displaced from the
centre of the geometrical construction. One could also
put C in motion, either around the earth or around
another point, but in each case the deferent would have
as its centre either a fixed or a movable central-point.
E.g.: if the distance between C – E = 0.03 the radius of
the eccentric, the displaced circle will account for the 6
extra days the sun spends between the vernal and the
autumnal equinoxes.
Fig. 1: eccentric model
The Epicycle-Deferent Model: Accounts for the periodic
appearance of retrograde motion as seen from the
earth. In Fig. 2, the planet P is moving eastward with
the deferent and is at its maximum speed. If P were on
the inside of D (between D and E), then P would be
moving westward, against its deferent, and would be at
its slowest speed (and appearing to retrogress).
Planetary motions accounted for by varying the speed
and direction of the epicycle and the deferent.
Fig. 2: epicycle-deferent model
Fig. 3: equant model
Fig. 4: equant & epicycle
The Equant Model: The rate of rotation of a deferent (and so on) is required to be
uniform, not with respect to its own geometric centre, but with respect to an equant
point displaced from that centre. E.g.: in Fig. 3, P sweeps out equal angles in equal
times as measured at Q. Uniformity of angular motion (though not about the centre)
retained, but uniformity of linear motion about the circumference is given up.
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Lecture 7: Copernicus I (Copernican v. Ptolemaic Systems)
© Darrin Durant 2004
The geometrical equivalence of the Copernican and Ptolemaic systems
 Fig. 5: Explanations of the motion of the sun and the seasons
Earth
Geocentric
System
S3
S1
S2
Section of the
Starry Vault
*
*
3
*
*
*
*
*
2
*
*
*
*
*
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Sun
Heliocentric
System
E1
E3
E2
 Fig. 6: Explanations of the retrograde motion of the planets
Earth
Geocentric
System
P3
P2
P
P4
P
P5
*
Starry
Vault
*
*
5
*
2
3
4
*
1
*
P5
Heliocentric
System
P1
*
*
P4 P3 P2
P3
P1
E3
E4
E5
E2
E
E1
2
*
*
*
*
*
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Lecture 7: Copernicus I (Copernican v. Ptolemaic Systems)
© Darrin Durant 2004
How the sun ‘governs’ the planets in the Ptolemaic system
 Fig. 7:
1. The centre of the epicycles of Mercury and Venus are tied to the
earth-sun vector, keeping them roughly aligned with the sun.
2. The lines joining the epicycles of Mars, Jupiter and Saturn are
required to remain parallel to the earth-sun radius vector. This
means whenever a superior planet is at its point of maximum
retrograde motion, the sun will be opposite it in the heavens.
 Why the ordering?
 Ptolemy admits he cannot measure distances
 Mars, Jupiter, Saturn
Sun
(Earth), Moon, Mercury, Venus
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Lecture 7: Copernicus I (Copernican v. Ptolemaic Systems)
© Darrin Durant 2004
The Copernican System (1473-1543)
 Fig. 8:
Copernicus’ Heliocentric System:
is Fig. 8 accurate?
 ‘Mean Sun’
 Epicycles
 Dimensions?
 The nature of the orbs?
 Fig. 9:
The Copernican explanation of stellar parallax:
The line between a terrestrial observer and a
fixed star does not stay parallel to itself as the
earth moves in its orbit. Thus, the star’s
apparent position on the stellar sphere should
shift by a given angle (that between the two
lines, close to the intermediate star) during an
interval of six months.
 Fig. 10:
Copernicus’ explanation of the difficulty of
detecting stellar parallax: note that the
further away the sphere of the fixed stars,
the smaller the angle of parallax. An object
within the earth’s atmosphere would have a
large parallax, such as that of a comet. If
comets are objects out in the celestial realm,
then they would have a smaller parallax.
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Lecture 7: Copernicus I (Copernican v. Ptolemaic Systems)
© Darrin Durant 2004
Copernicus and Planetary Distances
 Fig. 11:
 Fig. 12:
SE2 P2=90
E2
Venus
P2
SE=10
SVE = 90
Earth
Sun
Sun
P1
E1
VES = 46
Fig.11: VS/SE = VS/10 = sin 46, but sin 46 = .72
Fig.12: obtain a right angle triangle by letting a planet move from opposition to 90
point. Using orbital times and times of motion (R), we get: cosE2SP2 = SE/SP =
10/R or R = 10/cosE2SP2
Copernicus and Ptolemy on ‘Bounded Elongation’
 Fig. 13: Ptolemy  the angle between S and P must be restricted (keep centre
of epicycle on ES vector)
Earth
Sun
Planet
 Fig. 14: Copernicus
28
Orbit of earth
Orbit of Venus
Earth
Orbit of Mercury
 The earth’s orbit contains the orbits
of the inferior planets
46
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Lecture 7: Copernicus I (Copernican v. Ptolemaic Systems)
© Darrin Durant 2004
Summary of the Copernican system
 3 motions:
1. Diurnal axial rotation (west to east)
2. Annual motion around sun
3. Conical motion of earth’s axis
 Problems:
Why don’t we all fly off the earth?
Why doesn’t the earth disintegrate?
If everything is going in a circle, why do bodies fall to earth?
Why can’t we detect stellar parallax?
Why does the moon orbit the earth?
Shouldn’t we see Phases of Venus?
If Copernicus and Ptolemy both use epicycles, then how do we choose
between them?
8. Scripture declares the universe to be geocentric
1.
2.
3.
4.
5.
6.
7.
 Issues:
1. Are the Copernican orbs ‘real’ (solid, material)
2. Is the system ‘heliocentric’ if there is a ‘mean sun’ centre?
3. Is the Copernican system is ‘simpler’ or ‘more harmonious”?
4. Which system was more ‘accurate’? Accurate according to what
criteria?
“Whether these eccentrics really exist in the spheres of the planets no
mortal knows, unless we are to declare them, and likewise epicycles, to
have been disclosed by some revelation of spirits (as some claim).”
Albertus de Brudzewo, 1495
“God the creator placed these bodies so far away from our senses that
that we are unable to produce principles of demonstration for them (as
we can in the [study] of other things) or to discover what is natural and
familiar, by means of which we may afterwards set out the causes of
particular appearances.”
Nicodemus Frischlin, De astronomicae artis… (1586)1
See Nicholas Jardine, “Epistemology of the Sciences”, in Schmitt et al, The Cambridge History of
Renaissance Philosophy (1988), 685-711. Albertus de Brudzewo taught mathematics at the University of
Cracow when Copernicus studied there.
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