Ibn al-Shatir's double-epicycle lunar model

An astronomical model can describe the Moon's direction yet produce an implausible variation in its distance from Earth. This is the problem Victor Roberts highlights in studying Ibn al-Shatir: Ptolemy's lunar theory exaggerated the changing distance, whereas Ibn al-Shatir's double-epicycle arrangement reduced that defect. The achievement concerns a particular mathematical relationship, not invention of the Moon, all astronomy or heliocentrism ¹.

The museum presents the model through Roberts's1957 study and the biographies by David A. King and Muammer Dizer actually read, not a claimed reading of the original manuscript. Similarity with Copernicus is a subject for comparison, but similar names and diagrams do not by themselves establish a documented transmission route between two astronomers ^{2 3}.

An epicycle is a small circle whose centre travels on another circle. Here the first small circle carries the centre of a second, on whose circumference the calculated Moon is located. Earth remains central to the overall description; combined circular motions represent the Moon's changing movement in Roberts's analysis. “Correction” does not mean adoption of gravitational dynamics or a modern elliptical orbit ⁴.

Direction and distance are different tests: two calculated directions can agree while their underlying models imply different distances. The accompanying image does not reconstruct the mechanism. It is a modern illustration of apogee, the farthest position, and perigee, the nearest, not the historical circles or a diagram of lunar phases ^{5 6}.

Ibn al-Shatir worked within a tradition using circles, calculation and tables to describe the heavens. Roberts identifies a specific exaggeration of lunar-distance variation in Ptolemaic theory, not an absence of earlier lunar observation or recognition of changing apparent speed. Success in some calculations did not necessarily mean consistency across every implication of a model ⁷.

King notes that works by Maragha scholars appear in the introduction to Ibn al-Shatir's handbook and inspired his non-Ptolemaic models. This establishes an intellectual setting, not permission to erase his contribution or assign every detail to one predecessor. Sharing a mathematical technique establishes neither a shared personal religious affiliation nor the same transmission path for every equation ⁸.

Ibn al-Shatir worked in Damascus as head timekeeper at the Umayyad Mosque, combining instruments, tables and theory. King distinguishes his importance in planetary theory from his timekeeping work, comparing the latter with his colleague al-Khalili's. Excellence in one subject does not require surpassing every contemporary in every other subject ⁹.

Roberts studied a microfilm of Bodleian Marsh139, reporting that it was copied in768AH, paired with1366. This attests to the book by that time, not an exact invention date. The record therefore uses an open-start bound ending in1366; it does not turn a witness's copying date into a day of composition, observation or discovery ⁷.

Sunni frame: King explicitly identifies Ibn al-Shatir himself as Muslim, providing an individual basis for Muslim identity. Sunni is the separately declared museum default, not this source's testimony to his personal legal school. The record does not infer religious identity from his name, city or employment alone ¹⁰.

Birth accounts remain distinct: the existing museum biography reports Ibn Hajar's Rabi al-Awwal704AH, Dizer gives Shaban705AH/March1306, and King writes circa1305. This article neither replaces the earlier text nor invents a calendrical reconciliation. It attributes the alternatives; understanding the lunar construction does not depend on resolving the birth date ^{11 12 13}.

Roberts gives a carrying-circle radius of60 normalized units and epicycle radii of6;35 and1;25. The semicolon separates the integer from sexagesimal fractions:6;35 means six plus thirty-five sixtieths, not decimal6.35. These are ratios within the model's presentation, not modern distances in kilometres ¹⁴.

In our teaching arithmetic, the small radii become395/60 and85/60. Their sum is480/60, or8; their difference is310/60, or5;10. Aligning or opposing two arrows changes their resultant length between those limits. This is a modern consistency check on printed numbers, not fractional notation claimed to come from the author's manuscript ⁵.

Roberts associates mean conjunction and opposition with an apparent epicycle radius of5;10 and quadratures with a limit of8. Conjunction and opposition concern relative solar and lunar longitudes; quadrature is a quarter-turn separation. This does not turn a dated true phase into an independent measurement of every circle: coordinated motions, not arrow lengths alone, govern the complete model ¹⁴.

Comparing60 with5;10 gives54;50 and65;10; comparing it with8 gives52 and68. The ratio68/52 is17/13, approximately1.30769. These are checks on published magnitudes, not a reimplementation of the tables or a certificate that the model's error against modern observations equals a particular number ⁵.

Dizer prints43;50 as the lower conjunction/opposition bound, whereas Roberts's printed page432, visually inspected, gives54;50. We retain the disagreement rather than silently correcting the source. Roberts also distinguishes the mean angular diameter taken from the text from the extremes he calculated himself; the modern scholar's entire calculation must not be reassigned to the historical author ^{15 2}.

King reports that Ibn al-Shatir's handbook was used, commented on and adapted in Damascus and Cairo for centuries, while instrument treatises remained popular in Syria, Egypt and Turkey. Circulation of tables or instrument manuals does not establish that all their users adopted the theoretical lunar construction or retested its distances. The source distinguishes practical calculation's reception from interest in the models ¹⁶.

European transmission remains attributed: Roberts in1957 leaves the question open and allows independent solutions; King later regards the reappearance of model details as evidence of some transmission. The museum does not combine these into an undocumented travel story. No named translator or particular Latin witness connecting this lunar theory with Copernicus has been established here ^{2 16}.

The model inherited Ptolemaic problems and the language of circular motions, but rearranged the circles rather than merely renaming them. King describes the apparent elimination of eccentric deferents and the equant through additional epicycles. That does not mean every one of Ibn al-Shatir's models offered the same benefit: the biography specifically associates reduced distance variation with the lunar case ³.

In the textual sequence, an observational work, Taliq al-arsad, preceded the surviving explanation of theory and described the author's procedures and parameters, according to King. No copy is known in that account. We cannot extract an imagined observing programme or recreate a missing notebook from a biographical sentence; knowing a title or reference is not reading the text ⁸.

Roberts finds Copernicus's lunar construction structurally identical apart from small parameter differences, and compares the two small radii in detail. Mathematical resemblance is a bounded claim that can be examined. It differs from asserting that every Copernican idea came from one predecessor or that both astronomers held the same overall picture of the universe ⁴.

Earth remains central in Ibn al-Shatir's model; lunar similarity does not erase that difference. Nor should Roberts's1957 statements about other planets be automatically replaced by a later generalisation about all models. This record counts one lunar contribution, not extra inventions obtained by renaming the circles or projecting its specific improvement onto the Sun and every planet ^{2 3}.

Prayer-time determination and astronomical calculation formed the astronomer's working environment, but the sources read here do not establish that this particular model changed every Damascene timetable or decided every crescent sighting. King separates tables of timekeeping functions from planetary theory; the author's occupation does not make every page of his work one interchangeable practical instruction ¹⁷.

Readers can nevertheless see the importance of a better question: not only where the Moon appears, but what a calculation implies for its distance and apparent size. This lesson follows Roberts's specific problem. It is not a historical census of beneficiaries or a claim that the model supplied measurements equivalent to those of the space age ¹⁸.

Modern study revealed that astronomy before Copernicus was not simply repeated copying of one unchanged construction. Roberts presents a particular surviving Arabic work, investigates its mathematics and states the limits of what he can conclude about influence. That historical value does not require an absolute-priority slogan or an imagined competition between modern nations ¹.

King dates the investigation of Ibn al-Shatir's planetary theory to the1950s and describes research into sources and influence as still preliminary in his account. The dates of scholarship should therefore be read alongside the dates of the models. A1957 conclusion and a later biographer's interpretation are not identical witnesses or a definitive inventory of everything subsequently discovered ¹⁹.

Bounded contribution: Roberts attributes this lunar configuration to Ibn al-Shatir and studies it in Nihayat al-sul. This does not claim that circles, observation or every astronomical correction began with him. Inherited components do not cancel the novelty of their arrangement; a specific innovation does not erase its predecessors ⁴.

Transmission readings: Roberts leaves independent development possible, whereas King favours transmission of some details. We present the dated, attributed interpretations without a museum verdict on a route not documented here. References to Maragha scholars establish intellectual context, not Ibn al-Shatir's personal sect or a single religious identity for all contributions ^{2 20}.

The model supports teaching sexagesimal numbers, vector addition and the distinction between angular agreement and distance consistency. Our paper examples test bounds and ratios only. Computing an actual position requires rates, a reference epoch, direction conventions and observational comparison; three lengths alone do not provide a ready-made lunar calendar ⁵.

The visible image is modern and generic: arrows join Earth to a near and a far point on a dashed path. Its proportions cannot yield Ibn al-Shatir's parameters, nor establish that his model followed that drawn path. Accurate captions are part of the historical explanation, not decoration added after the evidence is settled ⁶.

In1957 Roberts listed five witnesses: four at the Bodleian and one at Leiden, using Marsh139 for his study. This is an inventory dated to the article, not a present worldwide census or evidence that the museum inspected every copy. A manuscript copied within the author's lifetime, as described by Roberts, is not thereby an autograph ⁷.

King distinguishes the surviving theoretical treatise and new handbook from the observational work with no known copy. These are different layers of survival and transmission. A surviving table does not fill the gap of a missing observational book, and an original instrument by the same maker does not turn every mathematical theory into a diagram engraved on that instrument ²¹.

Roberts's title invites comparison with Copernicus and might lead a reader to expect a newly discovered European copy. The conclusion is more cautious. Reading the title together with the conclusion, numbers and figures prevents an arresting phrase becoming a transmission document that the author never claimed to possess ¹⁸.

Ibn al-Shatir should not be reduced to someone who preceded a European name. His biography combines craft, timekeeping, tables, instruments and theory. Here one lunar contribution is illuminated, while the complete quadrant remains in its separate museum record: a physical instrument and a mathematical construction are not the same kind of achievement ^{22 23}.

Make two paper arrows, one longer than the other, and place the second's beginning at the first's end. Pointing in the same direction gives their summed length; opposing them gives the difference. Intermediate positions change the distance to the final point. This teaches a resultant, not a historical machine or every aspect of lunar movement ¹⁴.

Choose lengths8 and3 in paper units: their sum is11 and difference5. Draw intermediate positions and notice that the endpoint is not always equally far away. Keep this miniature example separate from Roberts's published radii, and ask of each image whether it is historical evidence or modern explanation. No solar viewing or optical instrument is needed ^{5 6}.

• Roberts1957, complete article428–432; printed Figure2 and numbers431–432 visually inspected ²⁴.
• King, complete biography and bibliography: identity, works and reception ²⁵.
• Dizer2000, complete entry; numerical disagreement retained, not silently corrected ¹⁵.
• Earlier museum Ibn Hajar and Kashf al-zunun citations; originals not reread for this article ^{11 23}.
• Commons Scott Foresman public-domain context drawing; rights/history and image reinspected ⁶.