Georg Mohr, Danish mathematician and theorist (b. 1640)

Jørgen Mohr: A Pioneer in Geometric Construction

Jørgen Mohr (Latinised as Georgius or Georg Mohr; born April 1, 1640, in Copenhagen – died January 26, 1697, in Kieslingswalde, now Koźlice in Poland) was a distinguished Danish mathematician whose profound contributions to geometry were, for a long time, surprisingly overlooked. He is primarily celebrated for being the first to rigorously prove what is now known as the Mohr–Mascheroni theorem. This fundamental theorem establishes a remarkable principle in classical Euclidean geometry: any geometric construction that can be performed using both a compass and a straightedge can, in fact, be achieved using a compass alone.

Mohr's groundbreaking work, titled "Euclides Danicus" (The Danish Euclid), was published in Amsterdam in 1672. In this treatise, he systematically demonstrated how all constructions typically attributed to Euclid – such as bisecting a line segment, constructing perpendicular lines, or copying an angle – could be accomplished solely with the aid of a compass. This was a significant revelation, as it showed that the straightedge, a tool considered indispensable for drawing lines, was theoretically redundant for a vast array of geometric problems if one possessed a compass capable of drawing circles of any desired radius.

Despite its revolutionary content, "Euclides Danicus" remained largely obscure for over a century after its publication. It was only in 1797, when the Italian mathematician Lorenzo Mascheroni independently published his work "Geometria del Compasso" (Geometry of the Compass), that the theorem gained prominence. Mascheroni's widely recognized work, which arrived at the same conclusions, initially overshadowed Mohr's earlier discovery. It wasn't until 1928, when a copy of Mohr's original "Euclides Danicus" was rediscovered in a bookstore in Copenhagen, that his rightful priority in proving this theorem was finally acknowledged, leading to its current designation as the Mohr–Mascheroni theorem.

The Significance of the Mohr-Mascheroni Theorem

The Mohr–Mascheroni theorem is more than just a mathematical curiosity; it has profound implications for the foundational understanding of geometric construction. It highlights the inherent power and versatility of the compass as a geometric tool. While a straightedge is limited to drawing straight lines, a compass can define points through the intersections of circles and can effectively 'transfer' lengths without explicitly drawing a line segment. For instance, to construct a perpendicular line to a given line through a point using only a compass, one would utilize intersecting arcs to define points that then allow the bisection of an arc, indirectly achieving the desired line.

This theorem essentially simplifies the set of fundamental tools required for classical Euclidean constructions, demonstrating that the ability to draw circles is sufficient for all constructions achievable with both circles and straight lines. It underscores the elegance and interconnectedness of geometric principles and continues to be a fascinating topic for mathematicians and educators exploring the boundaries of constructibility.

Frequently Asked Questions about Jørgen Mohr and the Mohr-Mascheroni Theorem

Who was Jørgen Mohr?

Jørgen Mohr was a Danish mathematician born in 1640, best known for being the first to prove the Mohr–Mascheroni theorem, a foundational result in geometric construction.

What is the Mohr–Mascheroni theorem?

The Mohr–Mascheroni theorem states that any geometric construction that can be performed using a compass and a straightedge can also be accomplished using only a compass.

When was the Mohr–Mascheroni theorem discovered?

Jørgen Mohr first proved and published the theorem in his book "Euclides Danicus" in 1672. It was independently rediscovered and popularized by Italian mathematician Lorenzo Mascheroni in 1797.

Why is the theorem named after two mathematicians?

It is named after both Jørgen Mohr and Lorenzo Mascheroni because Mohr was the first to prove it, but his work was largely unknown. Mascheroni independently rediscovered and published the theorem over a century later, bringing it to widespread mathematical attention. Mohr's priority was recognized only in the 20th century after his original text was rediscovered.

What is the significance of being able to construct with a compass alone?

The significance lies in demonstrating that the straightedge is not a necessary tool for performing classical Euclidean constructions. It reveals that the power of intersection points formed by circles is sufficient to define all points constructible with both tools, simplifying the theoretical basis of geometric constructions.