On the Sizes and Distances of the Sun and Moon
The first geometric determination of cosmic distances — and the birthplace of heliocentrism
Tradition: Greek mathematical astronomy
The sun is far larger than the earth — the geometric proof that made heliocentrism thinkable
On the Sizes and Distances of the Sun and Moon is the only surviving work of Aristarchus of Samos. It applies rigorous Euclidean geometry to a single angular observation — the angle between sun and moon at the moment of half-moon (first or third quarter) — to derive upper and lower bounds on the relative distances and sizes of the sun and moon. Aristarchus's method is geometrically correct but limited by the precision of his angular measurement: he estimated the angle at 87 degrees (the true value is about 89.85 degrees), leading him to conclude that the sun is 18–20 times farther than the moon (the true ratio is about 390). Nevertheless, his conclusion that the sun is far larger than the earth was revolutionary and may have motivated his heliocentric hypothesis — reported by Archimedes in The Sand Reckoner — that the earth revolves around the sun. This hypothesis was rejected in antiquity but vindicated by Copernicus eighteen centuries later.
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Editions cited
- Aristarchus of Samos: The Ancient Copernicus (Thomas L. Heath, Oxford, 1913; Dover reprint)
- On the Sizes and Distances of the Sun and Moon (in Heath's Greek Astronomy, Dent, 1932)
- Berggren and Goldstein, "Aristarchus's On the Sizes and Distances" (Archive for History of Exact Sciences, 1987)
School Embodiments
The treatise is a masterpiece of deductive reasoning: from a single angular observation and geometric axioms, Aristarchus derives quantitative conclusions about cosmic distances.
"The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon." (Proposition 7)
The work belongs to the tradition of Greek mathematical astronomy: Eudoxus, Autolycus, and the programme of geometric cosmology.
The treatise follows the deductive format of Euclid — hypotheses, then propositions proved in order.
Aristarchus treats his geometric conclusions as describing the real physical arrangement of the cosmos, not mere computational devices.
Archimedes treats Aristarchus's heliocentric hypothesis as a physical claim about cosmic arrangement. (Sand Reckoner)
Cosmic structure is explained through geometry and observation without reference to mythology or theology.
The treatise contains no theological language; all arguments are geometric and observational.
Internal Tensions
The tension between the method's geometric validity and its observational limitation: the angular measurement was imprecise (87 degrees vs. the true 89.85 degrees), producing a quantitatively wrong result from a qualitatively correct method. The broader tension: a true heliocentric model was proposed, understood, and rationally rejected for lack of supporting evidence.
I. Time
Time is presupposed as the medium of celestial revolution but not theorised. The motions of sun and moon are regular and deterministic.
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II. Space
Space is the treatise's subject: Aristarchus measures cosmic distances geometrically. Space is vast, Euclidean, three-dimensional. The heliocentric hypothesis (reported by Archimedes) implies an enormously larger cosmos than the geocentric model.
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III. Matter
Sun and moon are physical bodies with definite sizes. The sun is far larger than the earth — possibly the reasoning behind heliocentrism.
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IV. Observer
The astronomer observes from the earth's surface with angular measurement. The revolutionary insight is that the observer's position is not the cosmic centre.
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V. Energy
Not addressed. The work is kinematic and geometric, not dynamical.
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VI. Information
Geometric-astronomical truths are universal and conserved. The method — combining observation with deduction — generates knowledge that transcends its historical moment.
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How On the Sizes and Distances of the Sun and Moon resolves each dilemma
27 resolved positions across 4 dimensions, including 3 distinctive where the majority of schools go the other way · 30 unaligned.
Each dimension is sorted so minority positions come first. Mainstream positions are folded into an expandable list.
Time · 9 dilemmas · 3 distinctive
Persistence, the future, and the direction of becoming.