Ibn Muadh's twilight-based atmospheric height

Ibn Muadh al-Jayyani tried to infer the height of an atmospheric layer from the first dawn light and the last evening twilight. His twilight treatise connected something visible with a geometrical distance beyond direct reach. Calvo reports a solar depression of18degrees,parameters for Earth and Sun,and a result around52miles. That is a historical model's result,not an accurate measurement of the end of the entire atmosphere ¹.

The work was long associated with Ibn al-Haytham,but scholarly sources report Sabra's correction to Ibn Muadh. We distinguish authorship from the fame of another name,and formulating a quantitative question from the correctness of every assumption. This is not invention of air,twilight or a compulsory prayer-time rule ².

The problem is inverse:observe light,then infer a distance. The Sun need not stand above an observer's horizon to illuminate every place above that observer;its rays may reach a layer while Earth's shadow covers the eye's location. Turning this picture into a height requires assumptions about Earth,the Sun,and what becomes illuminated and visible ¹.

The treatise estimates a height;it does not describe a machine flying there. It does not directly measure air pressure or changing density. NASA now describes gas layers with different properties and says the exosphere has no definitive top altitude. A sharp assumed layer in a diagram therefore cannot be identified with the physical end of all atmospheric gas ^{3 4}.

Twilight knowledge did not begin from nothing. TDV reports different angles used by Habash,al-Nayrizi,al-Biruni and Ibn Yunus,including19,18and17degrees,and calculations distinguishing morning from evening. That list is not a set of identical measurements and does not establish Ibn Muadh as the first person to use18 ⁵.

Dawn and dusk mattered for astronomy and timekeeping,but the source distinguishes Ibn Muadh's concern with atmospheric height from determining prayer times. A natural-philosophical question does not become a ruling simply because its phenomenon also has religious uses. Different angles do not prove that every calculator used the same definition or observing conditions ⁶.

We place the treatise in the eleventh century,not on a specified day. Calvo's secure chronological anchor is the eclipse he described in1079;that does not make1079his death year or the date of the twilight text. Identification with the namesake born in989is qualified in the biography;Calvo instead reports scholarship suggesting an early-eleventh-century birth ^{7 8}.

The Arabic original is not known to survive in these accounts. Knowledge comes through Hebrew,Latin and Italian versions and scholarship comparing them. “Not known extant” is not proof that every copy was destroyed. A manuscript's appearance in a catalogue is also not a claim that the museum opened and read it. Survival itself is part of the history ^{7 3}.

MacTutor individually identifies al-Jayyani as an Islamic mathematician,and TDV explicitly calls Ibn Muadh a Muslim scholar. That supports individual identity;the museum's declared Sunni frame is separate,not inferred from a name or the model's validity. Nor does a name associated with Jaen establish the site of every observation ^{9 5}.

He belonged to a setting of judges and jurists and wrote on ratio,spherical trigonometry and astronomical tables. These are not automatically additional inventions inside this entry. A mechanical treatise attributed to him in some scholarship is not promoted as settled property:Smith's visible notes record researchers denying that attribution ^{1 10}.

Calvo lists four inputs:a solar depression of18degrees,a mean solar distance of1110Earth radii,a linear Sun/Earth size ratio of5.5:1,and Earth's circumference of24000miles. The reported result is around52miles. We must not remove the finite distance and size,then claim a full historical reproduction using only parallel rays ¹.

For understanding,we supply a simpler modern teaching model with explicit limits. Assume a circular Earth section of radiusR,straight parallel rays,a thin layer with a sharp upper boundary,and light arriving from an illuminated point on the observer's horizon. If solar depression isδ,thenα=δ/2arises between radii in the two right triangles. These are conditions of a diagram,not facts about the whole atmosphere.

The relation iscosα=R/(R+h),givingh/R=secα−1. Secant means reciprocal cosine,not another independently measured coefficient. Withδ=18,we haveα=9degrees. TakingR=3818length units givesh≈47.59185in the same units;atδ=19it gives≈53.08967. Both are museum recomputations of the simplified model. We do not equate47.59with the historical52.

TDV's web example switches from18to19degrees and contains a decimal-place error in its height expression. We do not copy0.1391instead of0.01391as a valid coefficient. Different inputs cannot be silently reconciled by erasing the source. Numerical checks test consequences of assumed geometry,not whether the real atmosphere has a sharp ceiling or whether refraction can actually be neglected ¹¹.

The simplified relation is not linear in angle:doubling the angle need not double height. A zero angle gives zero height in this geometry;scaling all lengths together at unchanged angle preserves ratios. These test the modern example's structure,not an additional ancient observation. They ask what the expression preserves and assumes,rather than accepting an answer because it has decimal places.

Calvo reports the Latin translation associated with Gerard of Cremona and a Hebrew translation by Samuel ben Jehuda. Smith's abstract counts one Hebrew,25Latin and one Latin-derived Italian exemplar;his critical text used comparative transcription of22Latin manuscripts. These figures serve different purposes and cannot simply be added as a readership census ^{7 3}.

Calvo describes52miles as standard in the Latin West until the late sixteenth century. Smith's abstract discusses seventeenth-century revision in light of refraction and barometric studies. This is reception followed by correction,not uninterrupted success or evidence that every later writer independently observed the phenomenon ^{12 3}.

The question depends on distinctions between an observer's horizon,Earth's form and its shadow. Smith's notes explain the conical shadow by assuming a Sun larger than Earth. We do not turn this into light emitted by the observer's eye or confuse it with the vision theory in Ibn al-Haytham's optical work,with which some manuscripts were bound ¹³.

An angle alone supplies no length unit:an Earth scale is required. In our modern example,doublingR doublesh while leaving the ratio unchanged. Silently replacing a historical mile with a kilometre and celebrating precision would therefore be misleading. Consistent parameters and declared units are essential even when the underlying physical model itself needs revision.

In Latin reception the figure became standard,then faced criticism connected with refraction. Calvo mentions Tycho Brahe's prominence in raising that issue and Kepler's subsequent reduction of the height,followed by later astronomers. These scholars are not co-authors of the earlier text. Revising an old number also does not erase the historical significance of formulating its problem ¹².

We draw no demonstrated line from the eleventh-century treatise to every current astronomical-twilight definition. The National Weather Service defines civil,nautical and astronomical twilight using the Sun's geometric centre at6,12and18degrees below the horizon. Matching18does not independently prove transmission or validate the height once inferred from it ¹⁴.

The story helps explain why light can remain after the Sun leaves your horizon. The sources read here,however,provide no calculation of money saved or ships rescued by this treatise. The documented effect is circulation of a text and numerical result in a scholarly tradition,not a daily-life benefit filled out with invented incidents ¹.

The connection between twilight and religious timekeeping remains context,with the author's aim distinguished. We supply neither hours nor a religious prescription,and do not ask visitors to look at the solar disc or follow it. Geometry can be explained on paper without turning the exhibit into unsafe observing instructions or a devotional ruling ⁶.

The striking act is extracting an indirect magnitude from a light phenomenon and communicating the model across languages. Its value lies in posing a quantitative problem,not in supplying the last atmospheric measurement. The reception of52miles supports a history of numerical influence;its revision shows that successful circulation alone does not establish truth ³.

Correcting the author's name reveals another scholarly task:examining witnesses and attributions rather than trusting a famous name in a later association. Smith discusses copies bound with the optics and forms ofAbhomadiappearing in several witnesses. That conclusion is reported from his study,not a claim that the museum collated those manuscripts itself ¹⁵.

**Sunni—declared frame:**individual Muslim identity is documented,and the frame does not settle physical validity. **Academic account:**TDV and Smith report Sabra's correction from Ibn al-Haytham to Ibn Muadh. Sabra's paper appears in references but was not independently read for this entry;the attribution relies on the scholarly accounts actually read ^{16 17 15}.

This correction does not authorize a claim to the first atmospheric theory in every civilization. The record preserves a historical attribution dispute without inventing an equally balanced modern contest between the two names. Nor does uncertainty about another work justify expanding the maker's inventions:each work requires its own evidence ¹⁸.

NASA describes layers differing in temperature,density and composition,and says the exosphere has no definitive top altitude. That is not our thin,sharply bounded teaching layer. Comparing52miles with a conventional space-boundary line and calling the numerical resemblance proof of medieval accuracy would misrepresent both quantities ¹⁹.

The modern image shows three blue twilight bands,day and night,and6/12/18degree marks. It depicts neither Ibn Muadh's rays nor Earth's conical shadow nor a measured height. Its role is distinguishing today's categories,not impersonating an eleventh-century figure. The bands are not a timetable promising equal durations ²⁰.

Calvo's short biography,MacTutor and TDV were read,as were Smith's public abstract and visible notes—not his entire critical Latin edition. Hebrew,Italian and Latin witnesses are reported by scholars,not copies we examined. Loss of the known Arabic original does not prevent attribution criticism using surviving versions ^{21 22}.

TWCarlson's7September2012diagram is available underCC BY-SA3.0. We inspected the actual image and rights and use the original SVG unmodified. Reproducing a scholarly page is unnecessary for clear explanation. The historical source supporting a claim remains distinct from the modern illustration helping readers distinguish categories ²³.

Names change in transmission between languages,and a text may acquire association with a famous author through its binding position. Smith notes name forms pointing to Ibn Muadh and limited explicit conflation with Ibn al-Haytham among the witnesses he consulted. A possible explanation for error is not evidence of intentional forgery by an unnamed copyist ¹⁵.

Inheritance includes both error and revision. A magnitude can remain authoritative for a time,then be reconsidered as understanding of light,pressure and air changes. That is no verdict on a people or faith;it is the history of one model,its circulation and its modification ^{12 3}.

Imagine on paper a circular Earth with a layer above it. Your location can enter shadow before everything overhead does. Ibn Muadh's question was whether the first light could reveal the layer's height. Answering required numbers and assumptions,not merely looking upward. The difference between seeing and inferring is the lesson ¹.

Then ask:what if the layer has no sharp roof? What if light changes direction? A calculation can correctly describe a drawing without describing the world as it is. Do not test this by looking at the Sun;the paper idea is enough to distinguish useful geometry from a physical conclusion needing further evidence ⁴.

• Calvo:model,parameters and reception ²¹.
• MacTutor/TDV:biography,identity and attribution ^{8 24}.
• Smith1992:public abstract/visible notes only ²⁵.
• NWS/NASA:separate modern comparison ^{14 4}.
• Commons:modern image/rights ²³.