was prepared to do well any task assigned him, they assigned him every task and quickly took it for granted that he would do them well. They were not mistaken. Knowing little about architecture, Lobachevsky designed a stately and imposing university building. He became the administrative center of mathematical life, reading scholarship applications, searching out other mathematicians,placating and pleasing the faculty, and skillfully managing the bureaucracy by which the university lived, its endless accountants, survey takers, patronage peddlers, censors. He took charge of the university library. Until his tenure, it had been little used and poorly managed, the mice scampering through its shelves. He bound the old books and ordered new journals; he cleaned the place up.
L OBACHEVSKY â S REJECTION OF the parallel postulate, which he published in The Kazan Messenger in 1829, could not have been more severe. Instead of the Euclidean plane of old, there is the hyperbolic plane of new. Both planes are planes in that they are two-dimensional surfaces. There the similarity ends. Were the hyperbolic plane like the Euclidean plane, there would be no point in denying Euclidâs parallel postulate. In that way, lies madness.
Lobachevskyâs illustrations of the hyperbolic plane were contrived to fit into a small region of the Euclidean plane, but beyond the margins of the illustrations, the hyperbolic plane departs from flatness, turning itself over like an inside-out orange peel. The illustrations nonetheless convey what is unusual and strange. There is a straight line R, a point B marked on the line, a point P lying beyond R; there are straight lines x and y that pass through P ; and there is an angle θ ( Figure IX.1 ).
F IGURE IX.1. The hyperbolic plane
If θ is less than ninety degrees, straight lines passing through P will sooner or later intersect line R. If θ equals 90 degrees, a reversion to Euclid. The reversion marks the limits of the familiar.
But if θ is greater than 90 degrees? First, those straight lines are straight. Second, they are parallel to line R. And, third, there is more than one of them.
The theorems that follow are an imposition on common sense. The interior angles of a hyperbolic triangle sum to less than 180 degrees: A + B + C < Ï. Similar triangles are congruent. Lines that are parallel to a given line need not be parallel to one another. The circumference of a circle whose radius is R is greater than 2ÏR.
And Euclidâs parallel postulate is false.
W ITH A FEW simple straight lines, Nikolai Ivanovich Lobachevsky had placed the denial of Euclidâs parallel postulate squarely on the flat Euclidean page. The theorems that he derived in his masterpiece he derived properly. They followed impeccably from the axioms of neutral geometry and the denial of Euclidâs parallel postulate. The engine of inference purred without pause.
But if the denial of Euclidâs parallel postulate is to be convincing, it requires more than a derivation. It demands a model and so a way of being true. Illustrations have done what illustrations can do. If no model is forthcoming, there is no reason to suppose that the denial of the parallel postulate and the axioms of neutral geometry are consistent, mutually life-enhancing. If they are not, we are returned to the logical point from which non-Euclidean geometry represented a flight. An inconsistency would indicate a contradiction between the denial of the parallel postulate and the axioms of neutral geometry.
And this is precisely what no mathematician had been able to discover.
P ICTURES WHEN THEY appeared were for this reason welcomed. They revealed a world that might satisfy the axioms of non-Euclidean geometry. They were a source of inspiration and so a source of comfort.
It could be done. That is what the models suggested.
The Kazan Messenger had come and gone, when in 1868, the Italian mathematician Eugenio Beltrami published
L OBACHEVSKY â S REJECTION OF the parallel postulate, which he published in The Kazan Messenger in 1829, could not have been more severe. Instead of the Euclidean plane of old, there is the hyperbolic plane of new. Both planes are planes in that they are two-dimensional surfaces. There the similarity ends. Were the hyperbolic plane like the Euclidean plane, there would be no point in denying Euclidâs parallel postulate. In that way, lies madness.
Lobachevskyâs illustrations of the hyperbolic plane were contrived to fit into a small region of the Euclidean plane, but beyond the margins of the illustrations, the hyperbolic plane departs from flatness, turning itself over like an inside-out orange peel. The illustrations nonetheless convey what is unusual and strange. There is a straight line R, a point B marked on the line, a point P lying beyond R; there are straight lines x and y that pass through P ; and there is an angle θ ( Figure IX.1 ).
F IGURE IX.1. The hyperbolic plane
If θ is less than ninety degrees, straight lines passing through P will sooner or later intersect line R. If θ equals 90 degrees, a reversion to Euclid. The reversion marks the limits of the familiar.
But if θ is greater than 90 degrees? First, those straight lines are straight. Second, they are parallel to line R. And, third, there is more than one of them.
The theorems that follow are an imposition on common sense. The interior angles of a hyperbolic triangle sum to less than 180 degrees: A + B + C < Ï. Similar triangles are congruent. Lines that are parallel to a given line need not be parallel to one another. The circumference of a circle whose radius is R is greater than 2ÏR.
And Euclidâs parallel postulate is false.
W ITH A FEW simple straight lines, Nikolai Ivanovich Lobachevsky had placed the denial of Euclidâs parallel postulate squarely on the flat Euclidean page. The theorems that he derived in his masterpiece he derived properly. They followed impeccably from the axioms of neutral geometry and the denial of Euclidâs parallel postulate. The engine of inference purred without pause.
But if the denial of Euclidâs parallel postulate is to be convincing, it requires more than a derivation. It demands a model and so a way of being true. Illustrations have done what illustrations can do. If no model is forthcoming, there is no reason to suppose that the denial of the parallel postulate and the axioms of neutral geometry are consistent, mutually life-enhancing. If they are not, we are returned to the logical point from which non-Euclidean geometry represented a flight. An inconsistency would indicate a contradiction between the denial of the parallel postulate and the axioms of neutral geometry.
And this is precisely what no mathematician had been able to discover.
P ICTURES WHEN THEY appeared were for this reason welcomed. They revealed a world that might satisfy the axioms of non-Euclidean geometry. They were a source of inspiration and so a source of comfort.
It could be done. That is what the models suggested.
The Kazan Messenger had come and gone, when in 1868, the Italian mathematician Eugenio Beltrami published
Similar Books
