Habash al-Hasib's shadow tables

A shadow can be understood through a drawing, but the relationship between its length and a ray's angle can also be organized numerically. This is the contribution associated here with Habash al-Hasib al-Marwazi. Ihsan Fazlioglu connects him with shadow tables corresponding to what we now call tangent computation, attributing a priority claim to Sarton. We record a bounded computational improvement, not invention of shadows, sundials or all trigonometry ¹.

Knowing a geometric relationship and preparing a reusable numerical tool are different achievements. However, a modern source saying “first” is not an image of the old table. We have not read Sarton's original discussion or Habash's manuscript here. Priority therefore remains attributed, and the teaching values below are ours, not purported historical cells.

A table arranges values for lookup instead of repeating every geometric construction. A computational “shadow” is not merely a dark patch: it is a magnitude related to a gnomon's shadow under an agreed scale. MacTutor distinguishes the historical route of shadow and reciprocal-shadow quantities from the chord and sine route ².

Modern tangent and cotangent require a specified angle. If we choose a ray's elevation above the horizontal, horizontal shadow divided by vertical gnomon height equals the cotangent in our model. The reciprocal is the tangent. This is modern explanatory language, not a literal translation of every manuscript term or every gnomon orientation. Changing the angle convention changes which modern function represents a particular ratio.

Shadow measurement predates Habash by centuries. MacTutor discusses sundials and heights calculated from shadows, while describing Greek chords and Indian half-chords along another historical route. The world before this contribution was not an empty page; a numerical table was not humanity's first encounter with a changing shadow ³.

Francois Charette describes Habash learning Indian and Persian traditions, then working within Ptolemaic astronomy while revising observations. This does not make every element of his work derive from one source or emerge without earlier tables and instruments. The contribution belongs to a mixed inheritance rather than erasing that inheritance ⁴.

His work is better documented than his personal life. TDV reports Baghdad observations between829 and864, a long life and a possible death between864 and874. Charette discusses observations as late as868, places completion of the zij after869, and tentatively estimates a lifespan around796–894. We do not splice these estimates into certain birth and death dates ^{5 6}.

The record places the table contribution broadly in the ninth century. MacTutor's statement about Arabic shadow tables around860 is not an individual composition date. Court connections also do not establish formal membership in Ma'mun's observational team: Charette and TDV distinguish contact and benefit from participation that the evidence does not securely demonstrate ^{2 7}.

Charette explicitly calls Habash a Muslim astronomer in an authored scholarly biography. This is individual identity evidence, independent of Arabic name, city or patron's religion. Sunni is separately the museum's declared frame for the Abbasid setting, not a newly invented certificate of his personal legal school ⁸.

His role here is computational improver. We do not assign every contemporary astronomer as a table coauthor or confuse patrons and copyists with contributors. The maker's earlier source and original contributor attribution remain intact. This run adds sources and a fuller account without pretending to have read material used by an earlier contributor.

Draw a model on paper only: a vertical heighth, a horizontal lengths, and a hypotenuse representing a straight ray elevated by anglea above the horizontal. In the ideal right triangle, s/h=cot(a) and h/s=tan(a). Specifying angle and denominator prevents reversing the meaning when the drawing changes. These are our modern teaching equations, not symbols claimed to occur in Habash's zij.

At45degrees the two legs are equal, so the ratio is1. Choosing a teaching height of12units gives a shadow of12units. Twelve is our convenient choice, not a recovered historical gnomon height or table base. Doubling both lengths preserves their ratio: ratios distinguish shape from size.

At30degrees, s/h=sqrt(3); at60degrees, s/h=1/sqrt(3). For our height12 the corresponding lengths are 12sqrt(3) and 4sqrt(3). We retain exact radicals or explicitly label an approximation. Modern calculator places are not printed as purported manuscript entries. The accompanying modern triangle cannot establish the author's historical scaling convention.

A table makes numerical lookup possible, but equal angle increments do not produce equal shadow increments. From30to45degrees the ratio changes from square root of3 to1; from45to60 it changes from1 to the reciprocal of square root of3. These changes differ. This checks nonlinearity without borrowing unseen historical cells.

Solar altitude therefore does not directly equal hours since sunrise. Converting altitude into time requires astronomical geometry, including location, season and position on the daily path. We do not turn a descriptive modern paragraph into a linear law. At the ideal horizon the shadow ratio has no finite value; the overhead case has its own limit. Division by zero must not be presented as an ordinary table value.

Charette describes al-Biruni using Habash's zij early in his own career. TDV records studies of the zij by Ibn Iraq and al-Biruni, alongside Ibn Yunus's criticism of some Venus and Mercury latitude values. Reception involved use and criticism, not uninterrupted praise ^{9 10}.

Reception of the whole handbook does not prove that every reader copied the same shadow table unchanged. We therefore add no automatic transmission edge to every author or modern instrument. Following a particular cell through manuscripts requires evidence beyond the general history of a work's reception. A later copy's date is not the date of the original method.

The explanation rests on right-triangle geometry and earlier shadow measurement. Scholarship situates Habash within movement from Indian and Persian models to a Ptolemaic arrangement revised with observations. Charette emphasizes close adherence to many Ptolemaic models and procedures alongside improved parameters and computational methods. Following a tradition and improving it are not necessarily opposites ¹¹.

Sine, tangent and versed sine must remain distinct. TDV discusses sine tables and terminology for sine and versine, but this does not license copying damaged electronic formulas mixing a radius60 scale with normalization to1. We preserve the distinction between functions and give a consistently scaled modern example, rather than repairing historical mathematics by guessing ¹.

Charette describes significant auxiliary tables called jadwal al-taqwim and works concerning instruments, the sphere and qibla. These show the breadth of Habash's programme. They are not automatically alternate names for every shadow table or chapters to count mechanically as separate inventions. A numerical method's connection to an application needs identification, not merely the shared word “table” ¹².

Relationships between witnesses and methods remain historical questions. Our modern equation does not establish that a programmer calling a trigonometric function directly transmits Habash's text. A conceptual connection can be shown; a continuous technical genealogy requires documented traces, not similar calculator output.

MacTutor places shadow tables in the context of height measurement and sundials. Numerical organization makes a geometric relationship available for repeated calculation instead of requiring a fresh drawing each time. Our sources do not count Habash's users or quantify time saved by each surveyor, so we invent no social-adoption statistics ².

Calculation does not remove measurement conditions. Our model assumes a vertical gnomon, horizontal surface and defined angle. These assumptions explain why memorizing a value is insufficient. We ask visitors neither to conduct solar observations nor to use an incidental shadow for a professional or religious decision. The activity remains on paper.

This contribution belongs to the history of turning geometric relationships into numerical tools usable across problems. MacTutor gives a separate history of Latin shadow terms and later European tangent and cotangent names. The modern word tangent and its abbreviations therefore must not be attributed to Habash ¹³.

The work's value is not a compressed national priority contest. It formed part of a network of books, observations and languages, with revision built into the process. Visitors can appreciate Habash without believing he created every function from nothing, that Greek and Indian tables did not exist, or that later criticism cancels every earlier accomplishment.

TDV's strongest specific priority statement is attributed to Sarton, not a judgement we reached after comparing every surviving manuscript. We retain that attribution and use “improvement” in the record. MacTutor gives a general date around860 for Arabic shadow tables, not an individual author for every table in that paragraph ^{1 2}.

Three layers remain separate: older shadow computation, scholarship assigning tables to Habash, and a universal first-ever claim. This reading supports the middle layer sufficiently to explain the contribution and presents the third as a source's claim rather than transforming incomplete priority research into certainty.

A calculator can now supply a trigonometric function while concealing angle and unit conventions. Our paper example makes them visible: is the angle measured from horizontal or vertical? Is the answer a length or a ratio? Are inputs degrees or radians? These are modern teaching questions, not quotations from a zij introduction.

The image is a modern right triangle made by TheOtherJesse and released into the public domain in2007. It distinguishes opposite side, adjacent side and hypotenuse. It is not a surviving gnomon, Habash's table or an illustration of a Samarra workshop, and supplies no historical scale ¹⁴.

Charette distinguishes an Istanbul witness closer to the original from a Berlin recension containing later material, and argues that four titles denote one surviving Ptolemaic zij. TDV treats the names and the relationship between Dimashqi and Mumtahan in more differentiated discussion. We expose this interpretive difference instead of manufacturing four certain books from four names ^{15 16}.

Neither witness nor Debarnot's survey was inspected here. A statement written in2007 cannot certify the current publication situation. The zij record provides context for the tabular programme, not an original folio containing our teaching values. No folio number, row or gnomon height has been invented.

Charette explains al-Hasib as a title for mathematical-astronomical work, not a romantic claim that its bearer calculated everything mentally. TDV treats the meaning of Habash itself cautiously. A speculative nickname explanation must not become a personal appearance or ancestry assertion ^{8 5}.

The museum can tell a quieter story: a scholar reads inherited work, revises observations, organizes calculations, and is subsequently read and corrected by others. A memorable name or a modern instrument's precision cannot substitute for documenting a specific contribution.

Draw a right triangle on paper with equal legs. Their ratio is1 whether each measures2or4units. Enlarge the drawing without changing its shape and notice that size changes while ratio does not. This is our modern example, not a page from Habash.

Then read the source card: are we seeing an old table or a modern account of it? Both can teach us something, but they are different evidence. No outdoor gnomon or looking towards the Sun is needed. Paper is enough; solar observation stays outside this activity.

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