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
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
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 teacher head, 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".