How Did Medieval Builders Construct Complex Cathedrals Before Advanced Mathematics? And More Questions From Our Readers

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Smithsonian magazine

September/October 2026

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How did Europeans build enormous, complex cathedrals before they had advanced mathematics?

Although medieval master builders could not do complex calculations, they had extensive experience in applied geometry and engineering. They used simple drafting tools such as the compass and the square to outline a cathedral’s proportions, relying on basic ratios to plan the relationship between the building’s different parts. They sketched the designs onto the floor of the mason’s lodge to guide workmen and used snaplines—taut cords—to lay out perfectly straight lines across large areas. Master builders used proprietary templates to guide finer carved shapes. They had in-depth knowledge from long apprenticeships in teams or guilds of stonemasons, carvers and sculptors who traveled widely from project to project. They applied what they’d learned about buttressing and vaulting from existing buildings and experimented with new ideas in the projects they supervised. Master builders examined the structural soundness of a building as construction proceeded and adjusted support elements—such as flying buttresses—to solve any issues they discovered along the way.—Stephen Clancy, art and architectural historian, Smithsonian Journeys

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Why did insects shrink in size over the past 300 million years? 

The scientific consensus involves several factors. The largest terrestrial arthropods arose during the Carboniferous and Permian periods, roughly 250 million to 350 million years ago. During this time, Earth’s atmosphere was up to 35 percent oxygen, compared with around 21 percent oxygen today. Insects with larger bodies could more easily and efficiently deliver oxygen to all their tissues because more oxygen was available in the air. These large insects also had fewer predators, more plentiful food supplies and many more ecological niches to occupy than later insects would have once other types of animals evolved. Over the past 300 million years, evolution began favoring insects that were more inconspicuous, needed less oxygen, and could live in tiny crevices and similar habitats that were too small for other creatures. —Floyd Shockley, collections manager of entomology, National Museum of Natural History

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Why do parrots live so much longer than dogs?

Unlike dogs, which live an average of 10 to 13 years, parrots can live 50 years or longer. A 2022 paper suggests that parrots’ unique intelligence plays a role in their longer life spans. Scientists looked at zoo records from around the world for 217 parrot species. They found a direct correlation between brain size and life expectancy. In the wild, intelligence helps parrots solve life-or-death problems like finding food sources. Larger brains also take more time to develop, so intelligent species have evolved longer life spans. In general, a parrot’s brain-to-body ratio is similar to that of primates, while dogs have a lower brain-to-body ratio. Birds also age differently from mammals. They have higher metabolic processes, which should degrade their cells faster. Yet birds suffer significantly less cellular damage than mammals over time. This marvel of bird biology is of great interest to scientists, as it may help us find ways to slow our own aging or reduce cellular damage from diseases. —Stacy Hill, assistant curator of the Bird House, National Zoo and Conservation Biology Institute