Department of History
University of California, Irvine
Instructor:    Dr. Barbara J. Becker

Lecture 6.  Spheres and Harmonies.

Effect of the Humanist Movement on Astronomy
Re-examination of standard texts:

Claudius Ptolemy (2nd c CE) 

  • sought to establish a mathematical system from which positions of planets could be predicted in advance 
  • put together past and current theories and observations in a thirteen-book treatise called 

or The Mathematike Syntaxis
(The Mathematical Compilation)

excerpt from the Preface to Almagest (c. 150) by Claudius Ptolemy:

...And so, in general, we have to state:

  • that the heavens are spherical and move spherically; 
  • that the earth, in figure, is sensibly spherical also when taken as a whole;
  • [that the earth] in position, lies right in the middle of the heavens, like a geometrical center;
  • [that the earth] in magnitude and distance, has the ratio of a point with respect to the sphere of the fixed stars, having itself no local motion at all. 
And we shall go through each of these points briefly to bring them to mind.

The title of Ptolemy's treatise evolved to

or The Megiste Syntaxis
(The Greatest Compilation)

which the Arabs transliterated as al-Majisti.

Gerard of Cremona translated Ptolemy's work into Latin from Arabic (1175) and al-Magisti became known as Almagest.

Georg Peurbach (1426-1461) initiated the challenging task of translating the Almagest directly from Greek into Latin.  After Peurbach's untimely death, the project was carried out by his pupil, Johannes Müller of Königsberg (1436-1476), who was known by his latinized name, Johannes de Regiomontanus (King's Mountain).
  • Epitome of Ptolemy (1496)

Frontispiece from The Epitome of Ptolemy (1496) shows Ptolemy (left) reading from the Almagest while Regiomontanus listens attentively (right) and points to the well-ordered celestial scheme that Ptolemy's great work describes.

Gerard of Cremona's Latin translation of Almagest appeared in print in 1518.

The first edition of Almagest in the original Greek was published in 1538.

Celestial Motions to be Explained
  • daily motion of fixed stars (from east to west)
  • superimposed eastward motion of sun, moon, and planets against background of fixed stars
Closer observation reveals variations on basic motions:
  • heavenly bodies (most notably the sun) periodically appear to speed up and slow down
  • planets occasionally appear to reverse their normal course [retrograde motion]

Direct motion of Mars is interrupted by retrograde motion from October 13, 1978 to May 29, 1979.
  • points where celestial equator and ecliptic intersect appear to drift slowly westward [precession of the equinoxes]
Standard Greek Account
According to Aristotle:

All celestial bodies, by their nature, move in perfect circles at constant speeds.

  • How can we account for what we see and still adhere to the laws of physics?
  • How can we "save the appearances"?
In Ptolemy's system, retrograde motion is accounted for by introducing compound circular orbits. 

A planet (Mars) moves at constant speed on an epicycle.

The center of the epicycle moves around the earth at a constant speed on the deferent circle.

The combined motion of a planet on its epicycle and deferent circles results in the appearance of retrograde motion as seen from earth.


To further refine his system, Ptolemy included two additional mathematical devices:

The first, the eccentric circle, was already in use by Ptolemy's time.

Planet moves on a circle, but that circle is not centered on the earth.

Planet moves at constant speed, but will appear to move faster when it is closer to earth, slower when it is farther away.


The second mathematical device introduced by Ptolemy, the equant point, was his own invention.

The equant point (D) is located on the line connecting the earth (E) with the center of the planet's eccentric circle (F); DF = EF.

Center of the planet's epicycle (C) moves along the eccentric circle at a rate that would appear to be constant if viewed from the equant point.

The Epitome of Ptolemy by Regiomontanus (1496):
  • systematized and clarified Ptolemy's Almagest
  • became popular textbook; used by Columbus and Copernicus
  • didn't improve the predictive power of Ptolemy's system
  • didn't resolve problems with calendar
Minor annoying anomalies, or major unavoidable crisis?.....
Calendar Problems
Julian calendar year = 365.25 days, roughly equivalent to sidereal year (period of earth's revolution about the sun with respect to the stars).  Keeps civil time.

Tropical year (period of earth's revolution about the sun with respect to the vernal equinox) = 365.242199 daysKeeps seasonal time.

Tropical year is about 11 min 14 sec shorter than civil calendar year.

Difference adds up to roughly one day every 130 years:
40 BCE 360 CE 760 CE 1160 CE 1560 CE....
| | | | |
  | | | | |
day of
Mar 25 Mar 22 Mar 19 Mar 16 Mar 13....

Church calendar conflict: 

  • Christmas locked into civil calendar--December 25
  • Easter is a "moveable feast"--linked to vernal equinox (first Sunday after the first full moon after the vernal equinox)
  • If calendar problem remains unresolved, Easter will eventually occur in December!
  • prompted Copernicus to conclude that something was fundamentally wrong with Ptolemy's system


Postscript on calendar reform:  The Gregorian Calendar

In 1582, Pope Gregory XIII decreed that in the Catholic Christian world, October  4, 1582 would be followed by October 15.  He took this drastic action to bring the civil and seasonal calendars back into sync by realigning the vernal equinox with a part of the civil calendar that had been traditionally associated with the beginning of spring -- around the 20th of March. 

To prevent such disparities from building up in the future, he modified the rule for determining leap years.  In the Julian calendar, every year that is divisible by four is a leap year.  But that practice clearly inserts too many days.  How many?  Over a 400-year period, it will add three extra days.  So every 400 years, three days will have to be eliminated to keep the civil calendar in seasonal alignment.  How to choose those three days?  Gregory decreed that century years would no longer be designated as leap years -- unless they were divisible by 400!

Nicholas Copernicus (1473-1543)
Revolutionary, or Reactionary?
1473 born in Torun, Prussia (present-day Poland)
1491 enrolled at University of Cracow studied church law
1496 enrolled at University of Bologna studied Greek, philosophy, astronomy, medicine
1500 summoned to Rome for symposium on calendar reform
1501 enrolled at University of Padua; studied law and medicine
1503 returned to Torun; served as physician, cleric, scholar
1514 circulated treatise called Commentariolus (Little Commentary) among his friends

excerpts from
Commentariolus (1514)
by Nicholas Copernicus
The planetary theories of Ptolemy and most other astronomers, although consistent with the numerical data, seemed likewise to present no small difficulty.  For these theories were not adequate unless certain equants were also conceived ... a system of this sort seemed neither sufficiently absolute, nor sufficiently pleasing to the mind.

I considered whether there could be found a more reasonable arrangement of circles....

Let no one suppose that I have gratuitously asserted, with the Pythagoreans, the motion of the earth; strong proof will be found in my exposition of the circles.

For the principal arguments by which the natural philosophers attempt to establish the immobility of the earth rest for the most part on the appearances; it is particularly such arguments that collapse here, since I treat the earth's immobility as due to an appearance.

Copernicus' assumptions in Commentariolus:
  • earth is not center of universe
  • earth's distance from sun is small compared to distance to stars
  • apparent motions of heavens arise from motion of earth
  • retrograde motion is natural consequence of structure of system
1538 Georg Joachim Rheticus (1514-1575) arrived in Torun to study with Copernicus
1540 Rheticus published Narratio Prima (First Account)
1541 Rheticus encouraged Copernicus to prepare his treatise for publication
1542 (May) manuscript taken to printer in Nuremberg

Rheticus unable to oversee publication

task taken over by Andreas Osiander

1542 (Dec) Copernicus suffers cerebral hemorrhage
1543 (May) publication is completed

Copernicus dies shortly afterwards

Advantages to Copernicus' System

  • Explains variations in planetary brightness
  • Makes it possible to determine, with certainty, the proper ordering of celestial bodies in the heavens
Determining Relative Planetary Distances from the Sun in Copernicus' Heliocentric System

Distance of an inner planet (Venus) from the sun (sun-Earth distance = 1 AU)

Distance of an outer planet (Mars) from the sun (sun-Earth distance = 1 AU)

Distances from Planetary System Center
measured in Earth Radii, and (Astronomical Units)

430 (.4 AU)
9,600 (.4 AU)
820 (.75 AU)
17,000 (.72 AU)
1,100 (1 AU)
23,500 (1 AU)
1,700 (1.5 AU)
35,700 (1.5 AU)
6,000 (5.5 AU)
122,000 (5.2 AU)
10,500 (9.5 AU)
224,000 (9.5 AU)
too far to measure
6,300,000,000 to nearest star

Copernicus' system of the world from de Revolutionibus Orbium Coelestium (1543)

Disadvantages to Copernicus' System
  • No sensation of motion
  • Fixed stars show no signs of parallax shift
  • Still uses circles on circles; not really simpler
  • Violates principle of economy; why is there so much empty space between Saturn and the fixed stars?
  • No explanation for natural motions of terrestrial elements; why do things fall?
Which is Simpler??
The world system of Copernicus....

...or that of Ptolemy?

Why change?
Go to:
  • Book I of Mathematical Syntaxis, or Almagest (based on observations made from 127-151 CE) by Claudius Ptolemy (2nd c CE);
  • Astronomia Magna (1537) by a contemporary of Copernicus, and one of the more controversial figures in the history of science:  Theophrastus Bombastus von Hohenheim, a.k.a. Paracelsus (1493-1591);
  • the foreword and preface to the first edition of De Revolutionibus Orbium Coelestium [On the Revolutions of the Heavenly Spheres] (1543) by Nicholas Copernicus (1473-1543); and
  • the preface to Mysterium Cosmigraphicum [Cosmic Mystery] (1596) by Johannes Kepler (1571-1630).
Weekly Readings
Lecture Notes