Sharaf al-Din al-Tusi's linear astrolabe

An ordinary plane astrolabe carries a representation of celestial circles on a surface. Sharaf al-Din al-Tusi is credited with a different arrangement: a graduated staff, used with threads, that performs some of the astrolabe's work. The marks are not simply a ruler's equal length divisions. They encode astronomical and angular information. This record concerns that particular linear adaptation, not the invention of all astrolabes, projection or celestial observation ^{2 3}.

Its interest lies in a deliberate exchange: less surface, but knowledge stored in scales and recovered through movable threads. The account also states limitations, especially in azimuth and meridian problems. A modest physical shape did not eliminate mathematical preparation or preserve every function equally well. Both ambition and limits belong in the story ^{4 5}.

MacTutor describes a wooden rod with graduations, a plumb line, a double cord and a perforated pointer, without sights. That is a general description, not a measured inventory of a medieval specimen inspected here. Carra de Vaux explains the threads as a means of obtaining chord lengths that are then interpreted as arcs through a scale on the staff ^{2 3}.

“Linear” concerns the arrangement, not a promise that every relationship becomes direct proportion. The editor compares its appearance and some uses with a calculating rule. This is his explanatory analogy, not evidence that Sharaf invented logarithms or that every later slide rule descended from his staff. Similar-looking carriers can embody quite different calculations ³.

The staff inherited a plane-astrolabe tradition in which the sphere and celestial circles were represented on a plane. The editor describes a further reduction: a line within that projected plane carries the representation. Earlier geometry is therefore part of the explanation, not something erased by an inventor's name. Sharaf was adapting an existing astronomical-computing practice ³.

The translated construction repeatedly refers to tables, a base line, horizon circles, shadows and stellar positions. Some knowledge was prepared before use and stored in the markings; other results emerged through geometrical manipulation. A wooden body did not make those tables unnecessary. Understanding what was inherited helps locate the contribution more precisely than calling it the first instrument to measure the heavens ⁶.

The biographies place Sharaf in an itinerant teaching world involving Damascus, Aleppo, Mosul and the Tus region. His pupil Kamal al-Din ibn Yunus reported studying Euclid and Ptolemy with him. This gives a context for mathematical teaching and transmission, not the address of the first instrument workshop or a dated manufacture receipt ⁷.

The detailed witness read here is later: Carra de Vaux's1895 edition of passages from al-Marrakushi, with French translation. The editor distinguishes construction from use passages and reports a missing ending in his manuscript. We read selected pages of that publication, not Sharaf's autograph, and did not collate its Arabic text against manuscripts. The record's twelfth-century horizon is an attribution period, not an exact invention date ^{8 7}.

This is Sharaf al-Din al-Muzaffar al-Tusi, not Nasir al-Din al-Tusi of the Maragha observatory. The1895 editor explicitly corrects an earlier addition of Nasir's name. Ibn Yunus's later teaching of Nasir does not transfer his own teacher's instrument to his pupil. A shared geographical surname is not an identity record ^{9 10}.

MacTutor identifies the individual as an Islamic mathematician. That supplies the Muslim identity basis rather than an inference from name, residence or patron. Sunni is separately the museum's declared default frame; no personal legal school was established by these readings. Ozel attributes improvement to Ibn Yunus and later description to al-Marrakushi, distinguishing invention, modification and exposition rather than absorbing every participant into one maker ^{11 4}.

Threads turn a geometrical configuration into lengths; scales translate suitable lengths back into angular information. The construction distinguishes functions for arcs, horizon-related quantities, stars and shadows. Colours and separate lines help distinguish marks. This is a distribution of calculation between a prepared object and an informed operator, not a staff that supplies an answer independently of its markings or the selected problem ^{3 12}.

A modern paper example illustrates only the chord principle. Two equal sides of length R enclosing angle θ have opposite side c=2R sin(θ/2). With R=10 units, θ=60 degrees gives c=10; θ=120 degrees gives about17.32, not20. Splitting the isosceles triangle into two right triangles proves the relation. This is independently derived teaching geometry, not a recovered historical calibration, a transcription of the staff's tables or its complete operating algorithm. A changing thread length or moving knot would change this idealized reading; no error measurements from an original instrument are available here.

Ozel supplies a bounded reception route: Ibn Yunus's improvement, al-Marrakushi's description, and manuscript copies listed in collections. That establishes reported textual and scholarly reception, not a count of manufactured instruments or widespread household ownership. A surviving treatise is not automatically a surviving wooden staff, and the listed manuscripts were not independently inspected in this research ^{4 13}.

The1895 publication creates another identifiable episode: French-language study of the instrument. Its editor considered, but doubted, derivation of Jacob's staff from it. The same word “staff” does not establish a manufacturing connection. We therefore do not draw a verified route from Sharaf's design to all European navigation staffs ¹⁴.

Tables are embedded in the design rather than merely consulted beside it. The translation describes horizon circles, their centres and radii, ascensions, chords and shadow quantities represented through marks. Some depend on the locality's latitude. Portability is consequently different from universal applicability: carrying a staff to another place does not prove that every local scale remains suitable there ¹⁵.

The text also discusses crowded markings and alternative locations for information about zodiac signs and stars. This is a reading-interface problem. A correctly engraved number can still be misread from the wrong scale. The museum explains that organization without supplying a working construction manual or inviting readers to aim an observing staff toward the Sun ¹⁶.

The pupil's modification and the later description belong to the idea's reception, as reported by Ozel. They do not prove that every surviving detail was present on the first staff. The1895 editor notes that al-Marrakushi sometimes depends on an unnamed author and does not confidently identify that author. We do not fill the gap with a guessed attribution ^{4 3}.

Likewise, a comparison with a calculating rule is not a documented genealogy of modern slide rules. It is useful to compare ways of distributing knowledge among scales and movable elements, but conceptual resemblance and historical transmission need different evidence. The accompanying chain represents antecedent practice, this adaptation and textual reception, not every later mechanical computer ¹.

The translated uses include altitude, day and night arcs, and time. They distinguish equal hours from hours connected with divisions of daylight, and relate the afternoon prayer-time problem to shadows and altitude. This places the staff within a practical astronomical culture, but does not provide a daily log from a market or household, or prove that ordinary people routinely owned it ¹⁷.

Ozel describes simplicity, low cost and fewer defects as intentions behind the instrument. Those are design aims, not measured superiority established here. We can describe a search for a less elaborate physical carrier without declaring wood always cheaper or threads always more accurate than metal. The elaborate explanation itself shows how much preparation could remain behind a simple exterior ^{4 18}.

Its historical value includes an alternative way of carrying astronomical computation: the object's shape need not resemble the full network of celestial circles to preserve some relationships. Yet this compression has limits. The translated witness explicitly calls its azimuth method incomplete and discusses difficulties near the equinoxes and in nighttime meridian determination. These qualifications are evidence, not modern blemishes to remove from a celebratory narrative ^{1 5}.

That is not a verdict that the whole design failed or succeeded universally. We can trace teaching, modification, description and later historical interest. The readings do not establish that it transformed world trade, solved ocean navigation or initiated a scientific revolution. A restrained account preserves its particular ingenuity without adding a global social consequence unsupported by the sources ^{4 19}.

The biographies credit this linear astrolabe to Sharaf, and the edition distinguishes him from Nasir. The structured attribution is an improvement within an astrolabe tradition, not universal priority in angular measurement or invention of the plumb line. “Improvement” states the relationship to the precursor; it does not deny the attribution of the particular staff arrangement to its maker ^{2 4 9}.

Jacob's staff remains a separate question. The editor describes its moving transverse piece and doubts the proposed derivation. We do not settle the entire history of both instruments here; the narrower conclusion is that this source does not warrant converting resemblance into certain ancestry. Nor should a pupil's improvements and a later expositor's details collapse into a single undated invention moment ^{20 4}.

The digitized edition offers evidence for studying instrument design, calculation and teaching. It does not certify that a modern replica matches an original or recommend the staff for navigation. This page's image is a modern public-domain teaching triangle. It supports discussion of lengths and angles, but is not a picture of the staff, its historical colour scheme or its original dimensions ^{1 21}.

The digital chain is likewise a conceptual diagram, not a manuscript illustration. The editor says his manuscript supplied no figure and that he did not draw one. A new picture must not acquire false authority merely by appearing beside a historical text. Our captions distinguish modern contextual geometry from an inspected artifact or a documented reconstruction ³.

What we verified is a published witness and selected pages, not an artifact. Its introduction identifies separate construction and use passages, a missing ending, and two tables retained in translation rather than repeated in the printed Arabic. Someone looking for a numerical table must know which part of the edition is being consulted; absence from one section does not imply absence from the publication ⁸.

Ozel lists treatises and copies in collections including Topkapi and the British Museum under the names used in that article. These are bibliographical leads, not claims that current shelfmarks or original manuscripts were checked. Nor do we declare that no medieval staff survives anywhere: not inspecting an original here is a much narrower statement than a worldwide inventory of loss ¹³.

“Staff of al-Tusi” preserves a personal association while showing how a name can mislead. The1895 correction distinguishes two Tusis; the teaching biography connects them through Ibn Yunus without merging them. Knowledge can travel through pupils, but responsibility for a particular work still requires evidence. Fame and a shared place-name do not automatically settle authorship ^{9 10}.

Stars, shadows and religious time appear together in the use chapters. Describing afternoon prayer-time, ascensions and zodiac signs records historical content; it is neither a religious ruling nor an endorsement of predictive astrology. Serving a practical religious problem also does not by itself establish the maker's personal denomination. Individual Muslim identification and the declared Sunni museum frame remain distinct ^{22 11}.

Imagine a map filled with circles and a designer asking which information could fit on a ruler. The answer is not simply to shrink the whole picture: selected marks and movable threads can preserve selected relationships. This is a teaching analogy for the editor's account, not a quotation of a conversation Sharaf actually had. Ask where a length appears, where an angle appears, and whether doubling one must double the other ³.

Our paper example gives a counterexample: equal sides10 units long enclose60 degrees and span10 units, but120 degrees spans only about17.32. Nothing needs to be pointed at the sky. No staff, cord near the face, telescope or solar observation is required. The lesson is that unequal spacings can store a relationship, and that a simple-looking object can carry extensive thought. This is modern independent geometry, not an archaeological experiment.

• ²³ O'Connor and Robertson,complete biography,July1999.
• ²⁴ Ahmet Ozel,biography and works,2012.
• ¹⁴ Editorial introduction; ¹⁸ selected French translation;no Arabic manuscript collation.
• ²¹ Modern public-domain geometry context.