The above is not the name of the illest hip hop duo since OutKast. Instead, the torquetum and rectangulus are medieval astronomical instruments used for spherical trigonometry.

## The Torquetum of Jabir ibn Aflah

Also known as the turquet, the torquetum is designed to measure and convert three sets of coordinates—horizontal, equatorial, and ecliptic. First developed by Spanish Muslim astronomer and mathematician Jabir “Big Jabir” ibn Aflah in the late 12^{th} or early 13^{th} century CE, it is essentially an analog computer.

An example of a torquetum can be seen in Swiss artist Hans “Big Hans” Holbein the Younger’s *The Ambassadors* (below), above the elbow of the dude on the right.

## Wallingford’s Rectangulus

Approximately a century later in 1326, English Catholic monk and mathematician Richard “Big Dick” of Wallingford developed the rectangulus. Perpetually irked by the limitations of astrolabes, Big Dick designed the device to measure the angles between planets and other astronomical bodies, as the torquetum does; however, the rectangulus is calibrated via linear scales and read by plumb lines, and resolves polar measurements directly into their Cartesian components.

Technically, the rectangulus is a form of skeleton torquetum—a series of nested angular scales that allow measurement of azimuth and elevation in direct polar coordinates relative to the ecliptic. Prior to the invention of the rectangulus, converting these coordinates required the most advanced mathematics that had yet been developed. Ye olde rectangulus simplified these and further calculations.

The original rectangulus is lost to the sands of time, but extant examples date back to the 1600s. The device was comprised of a brass pillar with numerous linear scales hinged above it. Sights on the upper arm allowed it to be easily aimed at the intended astronomical object. Plumb bob lines descending from the scales intersected with linear scales, which were themselves marked on horizontal scales. This gave measurements in trigonometric ratios, rather than angles. Math, right?!

*Photo credit: irinaraquel via Foter.com / CC BY*