This is a detailed examination of Chapter Five of the Third Treatise of Jamshīd Ghiyāth al-Dīn al-Kāshī (al-Kāshānī, d. 1429) from his book, The Key to Arithmetic (Miftāḥ al-Ḥisāb), also known as The Computers' Key (Miftāḥ al-Ḥussāb). A facsimile of the original manuscript is reproduced in Chapter Three of the text, along with an opposite-page translation and commentary. A sexagesimal multiplication table, useful to those engaged in computations from Babylonian, Greek, Arabic, and Byzantine scientific writings, is included in a back pocket of the book.
The Greek mathematician Autolykos wrote these treatises around 320 BC. They form the oldest material on Greek mathematical astronomy that has survived in a well preserved form. The books contain the complete Greek text and its translation into English, the latter published for the first time, while the Greek text is based on the critical edition of Joseph Mogenet. The Books of Autolykos deal with the simple geometry of points and large circles on a sphere, applying the axiomatic method systematically for the first time. These treatises give a clear insight into the status of Greek mathematical astronomy just before the initiation of the famous School of Alexandria.