A Brief History of Geometry

Leading to non-Euclidean Geometry

Much of the following is excerpted, with permission, from A Mathematics Sampler: Topics for the Liberal Arts, Fourth Edition by William P. Berlinghoff, Kerry E. Grant & Dale Skrien. New York: Ardsley House, 1996.

About 300 BC, Euclid organized all of the geometry and much of the arithmetic and number theory that was known (some 300 years of work by earlier Greek mathematicians) into a single, cohesive work, The Elements. Euclidean geometry is the geometry we learn informally as children and somewhat more formally in high school geometry; for most, it is just "geometry." Euclid's goal was to systematize the various relationships that had been observed among spatial figures, which he, like Plato and Aristotle, regarded as ideal representations of physical entities. Euclid's organizational scheme, the axiomatic method, was so ingenious that it has remained the paradigm for all of mathematics and much of science. And so persuasive and logically compelling was Euclid's geometry that it took over two thousand years before mathematicians began to suspect that there might be other ways of looking at geometry, that perhaps Euclid's geometry was not the geometry, but a geometry.


Euclid's work aimed to derive as many geometric statements as possible, by strictly logical argument, from a small number of initial statements, assumed as a foundation because they were so "obviously" true. The derived statements he called theorems; the initial statements were divided into two categories, axioms and postulates. In addition, Euclid started off most of the thirteen books of The Elements with a list of definitions, describing the technical terms that would appear in that book. For example, Book I has twenty-three definitions, covering such terms as point, line, angle, circle, triangle, quadrilateral, and parallel. For clarity in what follows, let us state just one of these:

Two lines are parallel if they are contained in the same plane, but do not meet, no matter how far extended in either direction.


The axioms, also known as common notions, were general statements about quantity and logic; Euclid asserted the following five:

  1. Things which are equal to the same thing are equal to each other.
  2. If equals are added to equals, the results are equal.
  3. If equals are subtracted from equals, the results are equal.
  4. Things which coincide with each other are equal.
  5. The whole is greater than any of its parts.


The postulates were specific geometric ideas that were assumed to be true. Euclid based all of his plane geometry on the following five postulates:

  1. A line segment can be drawn from any point to any other point.
  2. A line segment can be extended continuously in a straight line.
  3. A circle may be drawn with any center and distance (that is, radius).
  4. All right angles are equal to one another.
  5. Through a given point not on a given line exactly one line can be drawn parallel to a given line.


(It should be noted that postulate 5, as stated here, is an alternative and logically equivalent form of Euclid's fifth postulate; it is called Playfair's Postulate, named after the British mathematician John Playfair (1748-1819) who formulated it.)


The Elements then proceeds with literally hundreds of related assertions (the theorems) that can be logically derived from the postulates. And over the centuries that followed, thousands more were added by admiring followers of the axiomatic method. It is a tribute to Euclid's genius and organizational skill that so much can be proven from such meager beginnings. But almost from the very moment Euclid proposed his five postulates for geometry, there was serious suspicion that the fifth postulate, often called the parallel postulate, might actually be dependent upon (a logical consequence of) the other four. If a proof of this could be found, then plane geometry could be seen as derived from just four postulates rather than Euclid's five.

Even Euclid himself seemed to be bothered by the possible dependence of this postulate; he avoided using it until the proof of his twenty-ninth theorem, and subsequently did not use it again, though many later proofs depend upon it indirectly via that twenty-ninth theorem. The belief that the parallel postulate could be proved from the other four spread rapidly, and over the centuries many scholars established their mathematical reputations by constructing "proofs" of it in which the flaws were so subtle as to escape notice during their lifetimes. Even these flaws were uncovered eventually, however, and the independence of the parallel postulate remained an open and tantalizing question through the end of the eighteenth century.

In the early 1700s, an Italian teacher and scholar named Girolamo Saccheri made the first noteworthy attempt to attack the problem indirectly. He proposed to negate the parallel postulate and then to probe the resulting geometric tangle until he found a contradiction. The existence of that contradiction would establish the dependence of the postulate. Now, the parallel postulate specifies that there is exactly one line through a given point parallel to a given line; so its negation takes the form of two alternatives:

There is a line and a point not on it with the property that

  1. there are no lines parallel to this line throught this point; or
  2. there are at least two lines parallel to this line through this point.

Using the understanding that Euclid's second postulate implies straight lines to be infinitely long, Saccheri found a contradiction resulting from the first alternative. The second case was much more stubborn, though. But Saccheri was so convinced that his thread of logic must ultimately snag that he actually knotted it himself. After skillfully proving many valuable results, he ended by forcing a weak and vague conclusion about lines that merge at infinity, which he partially persuaded himself to be a logical contradiction. It apparently convinced almost no one, and even Saccheri himself was sufficiently skeptical to attempt another solution. However, his second effort was no more successful than the first.

Apparently, he and others were so convinced of the dependence of the parallel postulate that no attention was paid to the alternative possibility. This alternative would imply that the negation of the parallel postulate combined with Euclid's postulates 1--4 do not form an inconsistent system, but rather are the basis for a new, logically consistent geometry that is essentially different from the system Euclid described. Such a situation was not, it seems, within the realm of speculation for the scholars of the eighteenth century.

Almost a century later Saccheri's indirect approach was revived independently and almost simultaneously by four men. In the early years of the nineteenth century, the great German mathematician Carl Friedrich Gauss was the first to recognize that a new geometry is created if the parallel postulate is replaced by the assertion that through a given point there are at least two lines parallel to a given line. Oddly, he did not publish his findings. It is speculated that his reluctance stemmed from the dominance of Immanuel Kant's philosophy within the European intellectual community of that day. Kant's theory of metaphysics was based in part on the assertion that the human perception of space is necessarily Euclidean, and hence the assertion of an equally valid geometry with different properties would bring Gauss into direct conflict with Kant. Perhaps even Gauss did not want to pit his reputation as Germany's greatest mathematician against that of its greatest philosopher.

The first publication of this revolutionary possibility appeared in 1829, written by a Russian mathematician, Nicolai Lobachevsky, who devoted much of his life to developing this new geometry. Part of his work was anticipated by Janos Bolyai, a young Hungarian army officer; but Bolyai did not publish his results until 1832. Bolyai was also interested in developing what he called "absolute" geometry, a system based on Euclid's first four postulates alone. A geometry based on these four postulates and the assertion that there are no lines parallel to a given line appeared in 1854, when Bernhard Riemann (also of Germany) proved that Saccheri's contradiction in this case could be avoided. Riemann's insight was based on shifting from the commonly accepted understanding of Euclid's second postulate, which calls for the ability to continuously extend a line segment; if one understands this to imply that straight lines must be unlimited in extension, a very natural inference, then Saccheri's argument is correct. But if one suspends the usual visualization of a line (Think, for example of a circle instead.), then the requirement of continuous extendibility does not inescapably lead to infinitely long lines, and a consistent geometry with no parallels is possible.

The geometries developed by Bolyai/Lobachevsky and Riemann are called non-Euclidean geometries. They differ from Euclidean geometry only in their rejection of the parallel postulate. But this single alteration at the axiomatic foundation of the geometry has profound effects in its logical consequences. For example, in Lobachevskian geometry, parallel lines do not remain equidistant, but rather diverge from each other in one or both directions. Three non-collinear points may or may not lie on a circle. Every triangle has an angle sum strictly less than 180o, and the sum is not constant, but may be any number from 0o to 180o. The area of a polygon is not connected to the lengths of the sides (in any simple or proportional way), but is directly and simply connected to the size of the angles. There are no non-congruent similar figures, so angle-angle-angle is a valid triangle congruence theorem. The Pythagorean Theorem is not true. The ratio of the circumference of a circle to its diameter is greater than pi, and is different for different circles. Similar surprising results are true in Riemannian geometry, though the quantitative ones differ from the familiar Euclidean ones in the opposite direction compared to Lobachevskian.


Over the remainder of the century, the discoveries of Gauss, Bolyai, Lobachevsky, Riemann and others forced mathematicians and scientists to rethink what mathematics is and how it relates to other disciplines and to our understanding of the physical world. This was the beginning of the revolution in thinking that, for example, allowed Einstein to think of "curved space" and general relativity.

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