The Mathematical Sciences in Antiquity

Influence of Greek Mathematics
- The Renaissance saw a revival of Greek mathematical works
- Important contributors include Archimedes and Apollonius of Perga.
- Apollonius’ focus on conic sections (ellipse, parabola, hyperbola) revolutionized geometry and influenced modern mathematical physics.

Primary Focus and Observations
- Early Greek astronomy centralized around star mapping and calendar development
- The challenge of synchronizing the solar year with lunar months led to various calendar systems

Solar Year vs. Lunar Month
- The solar year does not equate to an integer multiple of the lunar month, leading to discrepancies:
- A lunar calendar of twelve 30-day months is shorter by approximately five days per year.
- Various schemes were suggested to adjust this, culminating in the Metonic Cycle by Meton (fl. 425 B.C.)
- Meton’s Cycle relied on the observation that 19 years equal approximately 235 lunar months.
- This plan included 12 years of 12 months and 7 years of 13 months to correct the calendar with seasons.

Influential Figures
- Plato (427–348/47 B.C.) and Eudoxus of Cnidus (~390–~337 B.C.) introduced significant changes in astronomy
- Emphasis shifted from stars to planetary motion.

A geometrical model devised to explain celestial and planetary movements.
Concentric spheres:
- Celestial sphere: houses the stars and rotates daily to explain rising/setting.
- The earthly sphere remains stationary in the center.
Key Components
- Celestial Equator: Defined by Earth's equator projected into space.
- Ecliptic: The sun’s annual path inclined at approximately 23 degrees to the celestial equator, intersecting at equinoxes.
- Solstices: Points of maximum distance of the sun from the equator.
- Tropics: Defined by the paths of ecliptic through solstices (Tropic of Cancer and Capricorn).

Notable Characteristics
- The sun travels around the ecliptic yearly, while the moon completes a circuit monthly.
- Other planets (Mercury, Venus, Mars, Jupiter, Saturn):
- Follow the ecliptic, typically moving west to east but with speed variations.
- Example: Mars orbits in about 22 months (687 days) and exhibits retrograde motion, reversing its position every 26 months.
Retrograde Motion
- All planets except the sun and moon show this phenomenon, where a planet appears to reverse its direction while observed from Earth.
Mercury and Venus’ Spatial Constraints
- These planets never stray far from the sun:
- Maximum elongation: 23° for Mercury and 44° for Venus, likened to "dogs on a leash".

Nested Spheres Theory
- Proposed a solution to model each planet's irregular motion as a combination of uniform circular movements
- Mechanism:
- Each planet corresponds to a collection of concentric spheres with specific rotational patterns, addressing their complex movements.

Motion Visualization
- Inner spheres control speed, latitude changes, and retrograde motion, showcased by a “hippocampe” path (figure 5.5).
- For example, Mars utilizes 4 spheres to account for its apparent motion during observation.

Eudoxus’s model was mathematical, aiming for geometrical elegance without empirical precision.
No predictive precision was sought; qualitative correlations sufficed given the limited astronomical knowledge.

Callippus increased the sphere count for the sun and moon (four spheres each for planets).
Aristotle’s Synthesis
- Merged Eudoxus and Callippus’ spheres into a single cosmological model with physicality, locating aether as the material cause for motions.

Dual Motion Concept
- Each celestial body’s motion involves a natural rotation and imposed axial movement from the above sphere, guided by a desire for perfection corresponding with the “Unmoved Mover”.

Counteracting Sphere Models
- Introduced to produce simple daily rotation in planets by countering complex upper sphere motions.
- The mechanical transmission of motions amongst spheres remains ambiguous in Aristotle’s account.

Heraclides' Rotation Proposal
- Proposed that Earth rotates daily on its axis – gained partial acceptance yet often disregarded.
Aristarchus’ Heliocentric Proposition
- Suggests the sun as the static center of the cosmos with planets revolving around it, challenging traditional models.

Aristarchus’ Lunar Distance Calculations
- Developed a method for comparing distances of the sun and moon, though with measurement limitations (e.g., measurement inaccuracies in angle leading to significant errors).
Eratosthenes’ Measurement of Earth's Circumference
- Used gnomon shadow lengths between Alexandria and Syene, producing a remarkably accurate circumference estimation using geometry.
Hipparchus' Contributions
- Introduced analytical rigor to predictions, enhancing observational astronomy and correcting existing models.

Mathematical Syntaxis
- Ptolemy combined hipparchial methods with updated geometrical constructs, employing diverse models (eccentric, epicycle-on-deferent, equant) to match empirical observations once quality data was amassed.
Eccentric Model
- Planets move uniformly around a circle; gives the appearance of non-uniform motion relative to Earth’s positioning.
Epicycle-on-Deferent Model
- Accounts for discrepancies in appearance through additional circular motions (epicycles) combined with deferent motion, predicting periodic retrograde behavior effectively.
Equant Point Model
- Enhanced angular measurement techniques to maintain uniform circular motion perspective observed from a specific, fixed point, while speeds varied based on observations from Earth.
Interrelation of Models
- Ptolemy's comprehensive system allowed combination of various models to maximize predictive power, ensuring longer-term viability of these astronomical models across centuries.

The synthesis of geometric, physical, and empirical methods in ancient astronomy laid foundational principles for future scientific frameworks, highlighting the dynamic progress of astronomical thought towards a more mathematical and observational basis in the Hellenistic age.