Math
Math is the underlying knowledge needed to make use of the angles and distances measured with other navigational instruments. Advancements in mathematics from the Ancient Greeks to the modern day have shaped our understanding of both navigation and the universe as a whole; the importance of mathematics in predictive efforts writ large is expanded upon in greater detail in the Path to Newton.
In Lost Without Longitude, we present only one example of a tool used for calculating mathematics  Galileo's compass.
Galileo's Compass
Description: Galieo's Geometic and Military Compass is a device that was, in some ways, a precursor to the slide rules used in by 20th century engineers. While other individuals such as Fabrizio Mordente and Thomas Hood created similar devices (all called "Sectors"), Galileo's is the best known, perhaps because of his fame in other areas.
Sectors were used by military officers and engineers for a variety of purposes: to measure distances and angles, to make a wide range of mathematical calculations, and even to determine how much gunpowder should be used for a cannonball of a particular type.
Related Instruments: Sector, Slide Rule
Usage Dates: Late 1500s to late 1800s
Images:
Video:
More:

Galileo's Compass  History of an Invention from the Museo Galileo. Includes a brief history of the device and multiple images.

Galileo's Instruments of Discovery at the Smithsonian. Includes photos of Galileo's other works.

A page on The Sector at the Galileo Project.

Galileo's own book on the operation of his compass, from the Internet Archive. Still in the original Italian.

"The Sector: its history, scales, and uses," a very good article by Erwin Tomash and Michael R. Williams that explains the provides an indepth explanation for how the Galileo Compass was used.