the sphere is sliced by a plane passing through its center, the plane traces a circle on its interior surface. Between any two points, the circle describes a geodesic.
The surface of the earth is a model of spherical geometry. The Euclidean blackboard is flat, but the surface of the earth is curvedâand curved in the same way at every point. It is positively curved, a designation that unaccountably suggests a geometrical accomplishment on the order of overeating. In spherical geometry, straight lines are arcs and there are no parallel lines at all. Wandering geodesics intersect one another as they circumnavigate the globe. Lines of latitude are parallel, but save for the equator, they are not geodesics. The interior angles of a triangle add up to more than 180 degrees. Euclidean figures bulge as if bursting.
If Euclid is demoted on the surface of a sphere, he cannot be altogether denied. Geodesics on the surface of a sphere rest, after all, on its surface . The interior of a sphere remains a part of the general Euclidean background. This veil of indifference dropped, Euclidean straight lines return to prominence. The shortest distance between two sunny beaches is a straight line drilled through the earth from point to point.
The drill is a reminder. Having been cast away, Euclid has a tendency to return to any exercise in non-Euclidean geometry as an enveloping space, a contrasting structure, an astonishingly durable ghost. That ghostâis he lingering for any good reason? An ant may determine the non-Euclidean character of a sphere without any Euclidean contrast at her disposal. She may well conclude from purely local clues that the surface of a sphere is curved.
Clever ant.
But while the ant is looking at the surface of the sphere, who is looking at the ant, and from which perspective?
Enter ghost.
G AUSS HAD GRASPED the principles of non-Euclidean geometry; he had entertained his provocative thoughts in the silence of his study. It was left to the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky to do the rest.
No story in the history of mathematics is more romantic. Bolyaiâs father, Farkas, had been an amateur mathematician; he was often in correspondence with Gauss. Euclidâs parallel postulate obsessed him. Failure to establish the parallel postulate he regarded as âan eternal cloud on virgin truth.â The proofs that he sent eagerly to Gauss, Gauss promptly returned, the errors marked. Bolyaiâs son, János, was a prodigy and a polyglot, the master of nine difficult languages, a mathematician of distinction, a man of many gifts. High- strung and independent, he was consumed by duels, dances, and debts; he spent years in military service. And like his father, he was obsessed by the parallel postulate.
His father saw his son advancing toward the sinister, dark defile that had for so long obsessed him. He endeavored to warn him by means of letters that mingled the plan-gency and hysteria of a trainâs whistle: âDo not in any case have anything to do with the parallels. I know every twist and turn in this business and I have myself wandered in its fathomless night, which has extinguished every light and joy in my life. I beg you in the name of God. Leave the parallels in peace.â 2
There is the rattle of thunder in all the old Hungarian clouds, a flash of lightening, claptraps accumulating, father and son receding:
Ash on an old manâs sleeve
Is all the ash the burnt roses leave.
Dust in the air suspended
Marks the place where a story ended. 3
M UCH FURTHER TO the east, a Russian mathematician, Nikolai Ivanovitch Lobachevsky, was entering the same dark defile and finding it altogether to his taste. Like Bolyai, Lobachevsky was a man of intellectual powers that had been celebrated from his youth. He was original, determined, disciplined, and hardworking. When officials at the University of Kazan discovered that he
The surface of the earth is a model of spherical geometry. The Euclidean blackboard is flat, but the surface of the earth is curvedâand curved in the same way at every point. It is positively curved, a designation that unaccountably suggests a geometrical accomplishment on the order of overeating. In spherical geometry, straight lines are arcs and there are no parallel lines at all. Wandering geodesics intersect one another as they circumnavigate the globe. Lines of latitude are parallel, but save for the equator, they are not geodesics. The interior angles of a triangle add up to more than 180 degrees. Euclidean figures bulge as if bursting.
If Euclid is demoted on the surface of a sphere, he cannot be altogether denied. Geodesics on the surface of a sphere rest, after all, on its surface . The interior of a sphere remains a part of the general Euclidean background. This veil of indifference dropped, Euclidean straight lines return to prominence. The shortest distance between two sunny beaches is a straight line drilled through the earth from point to point.
The drill is a reminder. Having been cast away, Euclid has a tendency to return to any exercise in non-Euclidean geometry as an enveloping space, a contrasting structure, an astonishingly durable ghost. That ghostâis he lingering for any good reason? An ant may determine the non-Euclidean character of a sphere without any Euclidean contrast at her disposal. She may well conclude from purely local clues that the surface of a sphere is curved.
Clever ant.
But while the ant is looking at the surface of the sphere, who is looking at the ant, and from which perspective?
Enter ghost.
G AUSS HAD GRASPED the principles of non-Euclidean geometry; he had entertained his provocative thoughts in the silence of his study. It was left to the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky to do the rest.
No story in the history of mathematics is more romantic. Bolyaiâs father, Farkas, had been an amateur mathematician; he was often in correspondence with Gauss. Euclidâs parallel postulate obsessed him. Failure to establish the parallel postulate he regarded as âan eternal cloud on virgin truth.â The proofs that he sent eagerly to Gauss, Gauss promptly returned, the errors marked. Bolyaiâs son, János, was a prodigy and a polyglot, the master of nine difficult languages, a mathematician of distinction, a man of many gifts. High- strung and independent, he was consumed by duels, dances, and debts; he spent years in military service. And like his father, he was obsessed by the parallel postulate.
His father saw his son advancing toward the sinister, dark defile that had for so long obsessed him. He endeavored to warn him by means of letters that mingled the plan-gency and hysteria of a trainâs whistle: âDo not in any case have anything to do with the parallels. I know every twist and turn in this business and I have myself wandered in its fathomless night, which has extinguished every light and joy in my life. I beg you in the name of God. Leave the parallels in peace.â 2
There is the rattle of thunder in all the old Hungarian clouds, a flash of lightening, claptraps accumulating, father and son receding:
Ash on an old manâs sleeve
Is all the ash the burnt roses leave.
Dust in the air suspended
Marks the place where a story ended. 3
M UCH FURTHER TO the east, a Russian mathematician, Nikolai Ivanovitch Lobachevsky, was entering the same dark defile and finding it altogether to his taste. Like Bolyai, Lobachevsky was a man of intellectual powers that had been celebrated from his youth. He was original, determined, disciplined, and hardworking. When officials at the University of Kazan discovered that he