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In Mathematical Excursions to the World鈥檚 Great Buildings, Alexander J. Hahn puts historical landmarks to the test of algebra

IN THE 14th century, every decent European city needed a Gothic cathedral, and Milan was no exception. Several years after construction began, however, the masons noticed that the foundations of the church were unsound. They called in a German master builder who reviewed their plans and told them what the problem was 鈥� that the proportions of their Duomo were based on the triangle rather than the square, as convention dictated. He was correct, but not for the reasons he thought.

As Alexander J. Hahn explains in Mathematical Excursions to the World鈥檚 Great Buildings, 鈥渢he Gothic concept of structure relied on geometric and numerical relationships and not on the considerations of loads, thrusts, and stresses鈥�. Theologians thought of God as a geometer, and artists showed Him designing the cosmos with a compass. All that was necessary to engineer a sound building, architects believed, was the proper application of divine proportions.

鈥淭heologians thought of God as a geometer; artists showed Him designing the cosmos with a compass鈥�

Modern architects rely on algebra and calculus. Hahn turns these tools on historical structures from the Parthenon to the Hagia Sophia to St Paul鈥檚 Cathedral, revealing how they hold up and explaining the causes of visible contortions and cracks. Sometimes these insights shed light on aesthetic qualities, such as the unique arrangement of the Hagia Sophia鈥檚 domes. More engrossingly, Hahn employs mathematics to explore how architects have conceived of buildings through the ages.

In the case of Milan鈥檚 cathedral, Hahn鈥檚 discussion is especially rich because his maths plays out against a backdrop of detailed historical documentation, including the testimony of the German. What the master builder did not understand 鈥� and which Hahn explains 鈥� is that the reason square proportions tend to be safer is that the stability of Gothic structures depends on their height. Choose the wrong kind of triangle to determine a cathedral鈥檚 elevation, and your building will be too squat to keep the outward stresses manageable. The Milanese masons chose not to take the German鈥檚 advice, a choice which led to reinforcing buttresses later being added to the Duomo.

Unfortunately, although some of Hahn鈥檚 other explorations 鈥� notably his discussion of the Sydney Opera House 鈥� are equally nuanced, far too many are cursory and superficial. That, plus the dry textbook prose and his misplaced attempt to be architecturally comprehensive, makes these mathematical excursions feel more like a prepackaged bus tour than an intellectually stimulating holiday.