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Leonardo got so deeply immersed in these studies that he decided to begin an entire treatise on the anatomy of horses. Vasari claimed that it was actually completed, though that seems unlikely. As usual, Leonardo was easily distracted by related topics. While studying horses, he began plotting methods to make cleaner stables; over the years he would devise multiple systems for mangers with mechanisms to replenish feed bins through conduits from an attic and to remove manure using water sluices and inclined floors.
Leonardo Da Vinci
Walter Isaacson
SQUARING THE CIRCLE One topic related to the conservation of volume that particularly intrigued Leonardo, and would eventually obsess him, came from the ancient Greek mathematician Hippocrates. It involves a lune, a geometric shape that looks like the crescent of a quarter-moon. Hippocrates discovered a delightful mathematical fact: if you create a lune by overlapping a large half-circle with a smaller one, you can construct a right triangle inside the larger half-circle that has the exact same area as the lune. This was the first way discovered to calculate the exact area inside a curved shape, such as a circle or a lune, and to replicate that area in a straight-sided shape, such as a triangle or rectangle. That fascinated Leonardo. He filled his notebooks with shaded drawings in which he overlapped two half-circles and then created triangles and rectangles that had the same area as the resulting crescents. Year after year, he relentlessly pursued ways to create circular shapes with areas equivalent to triangles and rectangles, as if addicted to the game. Though he never gave the precise dates of any milestones he reached when making a painting, he treated these geometric studies as if each little success was a moment in history worthy of a notarial record. One night he wrote momentously, “Having for a long time searched to square the angle to two equal curves . . . now in the year 1509 on the eve of the Calends of May [April 30] I have found the solution at the 22nd hour on Sunday.”16 His pursuit of equivalent areas was aesthetic as well as intellectual. After a while, his experimental geometric shapes, such as his curved triangles, became artistic patterns. On one set of pages (fig. 59) he drew 180 diagrams of overlapping circular and straight-sided shapes, each one annotated with how the shaded and unshaded portions relate to each other in area.17 Fig. 59. Finding equivalent geometric areas. As usual, he decided to put together a treatise on the topic—De Ludo Geometrico, he called it (On the Game of Geometry)—and it filled page after page of his notebooks. Not surprisingly, it joined his other treatises in never being finished for publication.18 His choice of the word ludo is interesting; it implies a diversion or pastime that is engrossing but like a game. Indeed, the distraction that came from playing with lunes seemed to drive him to lunacy at times. But to him it was an enthralling mind game, one that he believed would get him closer to the secrets of nature’s beautiful patterns.
Leonardo Da Vinci
Walter Isaacson
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