# Uraniborg Math Handbook

## English Translation

### Mathematics in the Sixteenth Century

Tycho Brahe was the greatest observer in astronomy prior to the invention of the telescope. Uraniborg was the name of his observatory in Denmark where he worked, along with many assistants, almost his whole life. This observatory was, in effect, a graduate school in astronomy where research was carried out. As such, a lot of teaching was done in addition to looking at the sky. The teaching was necessary because few of Tycho's assistants had his expertise in mathematics and, without that, Tycho would have had to do all of the calculations himself.

Unless you are an astronomer, this might not sound like much. However, an example included with this translation, taken from Tycho's logs, should convince you that “reducing” (extracting information from) just one observation could take you a long time even with a calculator. Back in the late sixteenth century, there were no calculators; there were not even any mathematical equations! Consequently, showing someone how to do a typical calculation was far from trivial. To make matters worse, these math problems were almost always in spherical trigonometry which is as hard as it sounds. So Tycho's chief assistants (his postdocs, as it were) took it upon themselves to produce a tutorial. The article linked below is a translation of this tutorial along with some additional, explanatory description.

It is not easy reading; there are difficulties of a different sort. For instance, one of the simplest algorithms, in English, instructs the student to proceed as follows:

 *** Planar Algorithm 2 *** Multiply the side adjacent to the right angle by the whole and divide by the side subtending the right angle to give the sine of the angle subtended by that side. The third angle is given by the complement of 90 of this arc. If you multiply the sine of this by the side subtending the right angle and divide by the whole, it gives the other side adjacent to the right angle that was sought. ******

Perfectly clear, yes? No?

It will be apparent, from the text, that another thing scientists did not have in those days was peer reviewers. There were a lot of typos and other errors that have been corrected here as much as possible.

### Description

• Introductory material (by the translator)
• Planar trigonometry algorithms
• Spherical trigonometry algorithms
• Numerical examples for the algorithms in modern notation
• An example from Tycho's logs (text and translation)