German village of Unfinden, which is now part of Königsberg, Bavaria, and he is generally known by the Latinized

form of his birthplace, Regiomontanus.

Regiomontanus studied first at the University of Leipzig, and then under Georg von Peurbach in Vienna where he

completed the latter’s translation of Ptolemy’s Almagest, as well as making his own translations of works by

Archimedes, Apollonius and Heron.

Regiomontanus’ greatest work was his treatise

several years after his death.

treated independently of astronomy written by a European. It is divided into five books: the first two on plane

trigonometry give a systematic account of methods for solving triangles including the Law of Sines, and the other

three on spherical trigonometry. On many occasions he used rhetorical algebra to solve problems.

Another tract on trigonometry

Regiomontanus travelled widely in Germany and Italy, but later settled in Nuremberg where he established an

observatory and wrote some tracts on astronomy which he printed on his own press. He also made a mechanical

eagle that flapped its wings.

Regiomontanus died in suspicious circumstances in 1476 when working in Rome on the reformation of the

calendar. Some accounts say he was poisoned.

The sides AB and AC of a triangle ABC differ by 3 units. The perpendicular from A onto BC meets BC at D. AD is

10 units and the difference between BD and DC is 12 units.

Solve the triangle.

Regiomontanus used syncopated algebra and worked in Latin, but what follows using modern algebra in English is

essentially the method he followed.

Because the difference between BD and DC is four times the difference between AB and AC, it follows that the

sum of AB and AC is four times the sum of BD and DC.

Therefore, if the length of BC is x:

1. BD + DC = x

and, since the difference between BD and DC is 12 units, then they must be ½x + 6 and

½x – 6 respectively.

2. AB + AC = 4x

And, since the difference between AB and AC is 3, then they must be 2x + 1.5 and 2x – 1.5 respectively.

Hence the diagram looks like this:

Now use Pythagoras’ Theorem in one of the two right-angled triangles. It doesn’t matter which, so I will use triangle

ABD:

ABD:

This is the same result as that obtained by Regiomontanus.

Unfortunately this result gives a negative value for the length of DC, which means D should be outside the base BC.

It’s not clear whether or not Regiomontanus was aware of this.

Unfortunately this result gives a negative value for the length of DC, which means D should be outside the base BC.

It’s not clear whether or not Regiomontanus was aware of this.