How did an astrolabe turn the whole sky into a hand-held calculator?

It squashed the sky onto a disc you could hold. An astrolabe is a star chart and a physical model of the visible half-dome of the sky, and also an analog calculating device. Sight the Sun or a star along the arm on the back, hanging the instrument vertically, and you can read its altitude in degrees from the edge. Then the front does the work: it can identify stars or planets, work out latitude given the time (or the time given the latitude), and even survey or triangulate.

The front has two layers. A plate called a tympan is engraved for one specific latitude with circles of azimuth and altitude, showing the part of the sky above your local horizon. Above it spins the rete, a cut-out framework with pointers for the brightest stars and the ecliptic. Turn the rete once and you have gone through one day. All of it is a stereographic projection of the celestial sphere, normally made from the South Pole onto the plane of the Equator.

It was not invented in Baghdad: John Philoponus wrote the earliest surviving treatise in Greek around 550, and the story that Hypatia invented it is wrong. Muslim astronomers developed it further, adding angular scales and azimuth circles, and by the 10th century al-Sufi's text on it ran to 386 chapters describing more than 1,000 uses. It also had a religious use: determining prayer times and the qibla, the direction of Mecca. Centuries later, in England, Chaucer compiled a treatise on the astrolabe for his son, mainly based on a work by Messahalla or Ibn al-Saffar.