Al-Kashi's plate of conjunctions

The plate of conjunctions is an astronomical computing instrument attributed to Ghiyath al-Din al-Kashi by Petra Schmidl's astronomical biography. Its task is to estimate the time of an expected conjunction through linear interpolation: moving between known endpoint information under an assumption of uniform change. It neither observes planets unaided nor extracts their motion from nothing, and it cannot turn an approximation into observational certainty ¹.

Schmidl distinguishes it from the “plate of heavens,” a broader planetary-position calculator. Sadettin Okten likewise identifies two instruments and assigns interpolation to the conjunction plate. This record therefore does not recount al-Zarqali's equatorium, al-Kashi's sine computation or his circumference calculation under a new name. The different computational question justifies the distinction ^{1 2}.

Conjunction does not mean physical contact. Here the calculation concerns the expected time at which the two longitudes being compared coincide; it does not imply collision, equal distance from Earth or visibility from every locality. Nor does an instrument's name establish a particular astrological use or validate predictions about human affairs ¹.

Schmidl calls it a simple mechanical application of elementary geometry. Mechanical does not mean powered, self-running or digital. Lengths and ratios can embody a computational operation, but the account read does not supply verified dimensions, materials or every engraved line. Our example explains interpolation in modern notation; it is not instructions for an authenticated historical replica ¹.

Celestial computation did not begin with al-Kashi. Schmidl describes his Khaqani Zij as a work engaging with and correcting aspects of the Ilkhani Zij, with trigonometric tables and spherical and calendrical computations. Iranica situates his work within an extended mathematical tradition rather than a sudden creation of all knowledge. Such a setting helps explain the need to fill a small computational gap between known values ^{3 4}.

Planetary calculators also formed a varied instrument tradition, including al-Zarqali's separately documented arrangement. Functional comparison does not establish a particular manuscript's route to al-Kashi, or give one predecessor authorship of every later geometrical application. The bounded claim here is an instrument within an older computational practice, not the world's first interpolation ^{1 2}.

Schmidl dates the first Nuzhat al-hadaiq text to10 February1416 at Kashan and its revision to June1426 at Samarkand; Okten gives the same years and distinguishes the locations. These are dates of a text and its description, not a workshop certificate recording metal cut or a machine first operated. The record uses an open latest documentary bound of1426, not an invented birthday for the design or a guessed Hijri equivalent ^{1 5}.

Al-Kashi worked under Ulugh Beg's patronage at Samarkand, participating in its observatory and calculations. Schmidl and Iranica differ in estimating his arrival; that difference need not be concealed to describe the plate. Samarkand in this record marks the documented revision context, not proof that the idea never existed at Kashan or that an observatory team jointly invented every detail ^{6 7}.

Both references attribute the instruments to al-Kashi, but this plate is not another count of his plate of heavens or calculation of pi. Generating planetary positions and estimating when two values coincide are related but distinct tasks. Authors, copyists and later commentators also have different roles; a later writer is not automatically a co-inventor of the earlier device ^{1 2}.

MacTutor explicitly identifies al-Kashi as an Islamic mathematician, the individual basis recorded in his biography. The museum's Sunni default is declared separately, not inferred as a personal school from name, city or patron. Iranica also cautions about self-praise in his letters to his father. The museum does not promote those descriptions into an objective ranking of every contemporary scholar ^{8 9}.

Schmidl describes the plate as a mechanical application of elementary geometry performing linear interpolation. Our modern explanation begins with two imaginary longitude differences: at hour6 the difference is plus2degrees, and at hour18 it is minus1degree. These are teaching choices, not planetary observations or numbers transcribed from al-Kashi's manuscript. We seek the zero between the two endpoints ¹.

Assume the difference changes linearly over the interval. Its total change is three degrees across twelve hours; reaching zero from plus2 requires two-thirds of that interval, eight hours. The estimate is hour14 on our imaginary clock. Modern notation expresses this as t=t0+(t1−t0)*(−d0)/(d1−d0). This follows from proportional parts; neither the Latin symbols nor this operation order is certified as the plate's engraving ¹⁰.

Interchange the two bodies and the differences become minus2 and plus1. The zero still occurs two-thirds through the interval. Rescaling both differences into a common new unit likewise leaves the result unchanged. Equal endpoint values, however, make the denominator zero: two nonzero equal values have no crossing in this model; two zero values fail to single out one instant. These are boundaries of our model, not recovered historical operating instructions ¹¹.

Angles also wrap around360degrees. Readings359 and1 may describe a two-degree forward movement rather than a backward358-degree sweep. A continuous angular representation must be selected before interpolation. Curved motion creates another limitation: an endpoint chord only approximates the intermediate change. Correct endpoints alone do not guarantee an exact intermediate event time. We are not claiming to recover the plate's wrap-handling procedure or measure its error ¹¹.

A first version and a revision establish a textual life, not a census of manufactured instruments or market users. Schmidl identifies an Arabic manuscript in London for the first version and a Tehran lithographic edition for the second. These are the reference's descriptions, not manuscripts personally inspected or collated by the museum ¹.

Okten mentions a later anonymous work dedicated to BayezidII and Kennedy's study of it. That is a valuable reception lead, not proof that every Ottoman plate reproduced this device exactly or that every detail reached Europe. We supply no translator, workshop route or global adoption figure unsupported by the accounts actually read ².

The conceptual component is a ratio between time change and numerical change within a linear model. Proportionality itself was not al-Kashi's invention; the specific attribution concerns a named instrument applying elementary geometry to conjunction timing. Distinguishing principle and design prevents both collapsing every calculator into one invention and denying innovation whenever its components are older ¹⁰.

The operation requires computed or assumed information before it starts. In the same account, the plate of heavens treats planetary positions through successive planar operations. The conjunction plate does not replace all tables, observations and theory. Our chain diagram therefore claims neither that al-Kashi invented every input nor that the modern triangle illustration was his historical source ¹.

The clearest reception evidence here is the textual and scholarly afterlife of the work and its instruments, including references to Edward Kennedy's study. We did not read his original book in this round; its presence in a bibliography is not permission to copy its figures or attribute measurements to pages we have not inspected. The instrument's function remains supported by the actual reference text ^{12 2}.

Modern computers also interpolate, but shared terminology does not create direct technological ancestry for particular software. That is a teaching comparison, not transmission of code or a production lineage. Nor does this instrument establish a new planetary theory: Iranica distinguishes al-Kashi's computational accomplishments from research into cosmological planetary models ¹³.

The described practical value lies in estimating a time geometrically rather than repeating every numerical step. The accounts read provide no timed benchmark, savings percentage or diary of daily use. We can describe a way to ease a computational task without claiming that it accelerated commerce, navigation or every resident's schedule ¹.

Conjunction time is not automatically naked-eye appearance or a local religious timetable. Horizon and location introduce other questions absent from our example. These limits make the instrument's task clearer, not less worthy: a specific solution need not be made responsible for all social life to deserve a place in a museum ¹⁴.

The plate illustrates the embodiment of a mathematical procedure and the division of astronomical work into tasks expressed through different media. Schmidl presents al-Kashi as a computational scientist and instrument-maker in an active scholarly environment. This does not mean every colleague agreed with him or used every design. Institutional history is richer than isolated-genius lists without erasing individual attribution ¹⁵.

Its historical interest is not a race against today's computer. It lies in understanding the question, the answer's assumptions, and the difference between accessible testimony and physical details not established here. No read reference demonstrates that every modern interpolation descends from this plate or compares its accuracy with spacecraft observations ^{1 16}.

Schmidl and Okten attribute the specific plate to al-Kashi, so the record treats this as an original named design, not absolute priority for interpolation. “Plate” does not merge it with the plate of heavens: the reference explicitly describes two instruments and different functions. The1426 revision is likewise not a fixed date of first conception, only our safe latest documentary bound ^{1 5}.

The material read supplies no documented attribution dispute requiring an invented contest between names. Not manufacturing controversy is different from proving global priority. “Original” is bounded by the adjacent written claim. A new witness could refine that claim through explicit evidence; the inventor label should never expand beyond what its sources support ¹.

Interpolation can be explained with a short table and paper while distinguishing the exercise from the historical plate. A computed zero outside the endpoints is extrapolation, not an internal estimate, and depends still more on extending the motion assumption. A numerical quotient is not necessarily a sound astronomical answer, and many decimal places do not turn approximation into measurement ¹¹.

Our images are modern. The right triangle provides ratio context, not planetary markings or plate divisions. The museum's generated chain diagram summarizes older computation, a particular device and later study; it does not copy a manuscript figure. This visual qualification matters because an attractive unlabelled picture can imply an artifact never inspected ^{17 1}.

Schmidl identifies dates and manuscript or publication descriptions for the Nuzha versions; Okten mentions the lithograph and later treatment. These are textual witnesses as reported by references, not a worldwide object survey. We neither identify an original plate displayed at a named museum nor declare that none survives anywhere without an appropriate investigation ^{1 5}.

Reading a modern article is not reading every manuscript it cites. Our research record separates actual reading from bibliography. A reconstruction or book figure cannot enter merely because a search engine shows it; permission and the distinction between original artifact and modern interpretation require separate verification ^{12 18}.

The Nuzha title combines literary language with a technical instrument description; its garden imagery is not evidence of an actual garden installation. Court association does not reduce a device to entertainment, and celestial conjunctions do not validate every astrological practice. What two read references establish is an explicit computational function ^{5 1}.

This case encourages distinguishing people, institutions and works. Al-Kashi made other contributions, each answering a different question. A simple plate need not become a footnote to famous pi, nor gain prestige by being called the father of every calculator. Specific, testable explanation offers visitors more than an inventory of unbounded superlatives ^{19 20}.

Draw a paper timeline from6 to18, write plus2 above the first endpoint and minus1 above the second. State clearly that these are imaginary numbers, not sky observations. With uniform change, where is zero? Split twelve into three equal pieces and take two; you arrive at14. This is our modern ratio exercise, not a transcription of original plate operations ¹⁰.

Reverse both signs and check that the time is unchanged. Then try two equal nonzero values: why can you not select a crossing? The exercise requires neither solar viewing nor going outside at night, and is not fortune-telling. Discovering the information's limits is as valuable as finding an answer; describing an old instrument differs from reproducing it on paper ¹¹.

• Petra Schmidl, Kashi, Biographical Encyclopedia of Astronomers,2007,613–615; complete actual ISMI text read ²¹.
• Sadettin Okten, KASI, TDV encyclopedia, instruments and work5 ²².
• Iranica and MacTutor for computational context, biography and inferential limits; cited original works not read here ^{23 8}.
• Modern triangle metadata, public-domain release and original2007 history ²⁴.