Recovering Two Lost Treatises on Approximating the Sine of 1 ° from Qushjī’s Commentary on Zīj-i Ṣulṭānī
Bibliographic record
Abstract
The accuracy of medieval approximations of the Sine of 1° reached its peak with the works of the Persian mathematician and astronomer Ghiyāth al-Dīn Jamshīd al-Kāshī (d. 832/1429), and his patron Ulugh Beg (d. 853/1449), the Timurid ruler of Transoxiana and a mathematician and astronomer in his own right. Their works were written during the active phase of the Samarkand observatory, which was founded by Ulugh Beg and whose final product, Zīj-i Sulṭānī, was the most accurate zīj of the medieval period. Even though neither of their treatises on approximating the Sine of 1° has reached us, Kāshī’s and Ulugh Beg’s approximation methods were transmitted through the works of their colleagues at the observatory, namely Qāḍī-zāda al-Rūmī’s (d. after 844/1441) recension of Kāshī’s treatise and ʿAlāʾ al-Dīn ʿAlī al-Qūshjī’s (d. 879/1474) commentary on Zīj-i Sulṭānī. Unlike Qāḍī-zāda’s treatise, Qūshjī’s commentary has not received the attention it deserves from historians. Thus, it has not been noticed that what is presented in Qūshjī’s commentary under the rubric of “Ulugh Beg’s demonstrative method” is, in fact, a synthesis of Kāshī’s and Ulugh Beg’s approximation methods. The present article aims to fill this gap by offering an edition and English translation of the relevant passages of Qūshjī’s commentary, disti
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.007 | 0.021 |
| Scholarly communication | 0.005 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.004 | 0.016 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".