Match opposite pairs
Each side goes with the angle facing it: AB with the angle at C, AC with the angle at B. The baseline faces the far mark, which is why the third angle is needed.
40 / ANTIQUITY TO THE RENAISSANCE
Johann Müller · Regiomontanus
Stand on one bank of a river. Measure a baseline and two angles, and the sine rule tells you how far it is to a tower on the other side, without crossing.
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01 / THE SINE RULE
What can two angles tell you?
Slide station B along the bank and watch both angles change.
Walk 10 m along each sightline: your distance from the bank is 10 × the sine of the angle. Both sightlines reach the same height at the far mark.
Use the angle at the far mark and the measured baseline to find AC. Then move B and check that the answer stays the same.
Far mark:
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THE IDEAIn any triangle, each side divided by the sine of the angle opposite it gives the same number. Two angles and one known side are enough to find the other sides, without crossing the river.
The person behind the ideas
- Studied and taught in Vienna with Georg Peurbach.
- Completed On Triangles (De triangulis omnimodis) in 1464; it was printed in 1533.
- Set up a printing press in his house in Nuremberg around 1471 to publish scientific works.
More of the mathematics
From land to sky
The later books of On Triangles treat spherical triangles, whose sides are arcs of great circles on a sphere. These were the triangles astronomers needed.