forward without ever being brought to a stop, and yet it will be finite the end will never be found, but the complete tour can be made. Their space will therefore be unbounded, since on a sphere one may always walk.
What they will call space will be the sphere on which they are confined, and on which take place all the phenomena with which they are acquainted.
In a word, their geometry will be spherical geometry. The straight line to them will be the shortest distance from one point on the sphere to another - that is to say, an are of a great circle. What kind of a geometry will they construct? In the first place, it is clear that they will attribute to space only two dimensions. But now suppose that these imaginary animals, while remaining without thickness, have the form of a spherical, and not of a plane figure, and are all on the same sphere, from which they cannot escape. In that case they will certainly attribute to space only two dimensions. Let us further admit that this world is sufficiently distant from other worlds to be withdrawn from their influence, and while we are making these hypotheses it will not cost us much to endow these beings with reasoning power, and to believe them capable of making a geometry. Let us imagine to ourselves a world only peopled with beings of no thickness, and suppose these "infinitely flat" animals are all in one and the same plane, from which they cannot emerge. Lobachevsky's propositions have no relation to those of Euclid, but they are none the less logically interconnected. If the circumference of a circle be divided into n equal parts, and tangents be drawn at the points of intersection, the n tangents will form a polygon if the radius of the circle is small enough, but if the radius is large enough they will never meet. It is impossible to construct a figure similar to a given figure but of different dimensions. For instance, the sum of the angles of a triangle is always less than two right angles, and the difference between that sum and two right angles is proportional to the area of the triangle. The theorems are very different, however, from those to which we are accustomed, and at first will be found a little disconcerting.
APPLIED MATH NON EUCLIDEAN GEOMETRY SERIES
From these hypotheses he deduces a series of theorems between which it is impossible to find any contradiction, and he constructs a geometry as impeccable in its logic as Euclidean geometry. He assumes at the outset that several parallels may be drawn through a point to a given straight line, and he retains all the other axioms of Euclid. Now, this is precisely what Lobachevsky has done. It would be, therefore, impossible to found on those premisses a coherent geometry. If it were possible to deduce Euclid's postulate from the several axioms, it is evident that by rejecting the postulate and retaining the other axioms we should be led to contradictory consequences. This little work has inspired most of the recent treatises to which I shall later on refer, and among which I may mention those of Beltrami and Helmholtz. But the question was not exhausted, and it was not long before a great step was taken by the celebrated memoir of Riemann, entitled: Über die Hypothesen welche der Geometrie zum Grunde liegen. They have nearly rid us of inventors of geometries without a postulate, and ever since the Académie des Sciences receives only about one or two new demonstrations a year.
Finally, at the beginning of the nineteenth century, and almost simultaneously, two scientists, a Russian and a Bulgarian, Lobachevsky and Bolyai, showed irrefutably that this proof is impossible. It is impossible to imagine the efforts that have been spent in pursuit of this chimera.
For a long time a proof of the third axiom known as Euclid's postulate was sought in vain. (3) Through one point only one parallel can be drawn to a given straight line.Īlthough we generally dispense with proving the second of these axioms, it would be possible to deduce it from the other two, and from those much more numerous axioms which are implicitly admitted without enunciation, as I shall explain further on. (2) A straight line is the shortest distance between two points (1) Only one line can pass through two points
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