Mathematicians Reconsider Euclid's Parallel Postulate

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Mathematicians Reconsider Euclid's Parallel Postulate

Overview

Ever since the time of Euclid, mathematicians have felt that Euclid's fifth postulate, which lets only one straight line be drawn through a given point parallel to a given line, was a somewhat unnatural addition to the other, more intuitively appealing, postulates. Eighteenth-century mathematicians attempted to remove the problem either by deriving the postulate from the others, thus making it a theorem, or by replacing it with a simpler statement. Nineteenth-century mathematicians would change the postulate to generate logically consistent non-Euclidean geometries, which twentieth-century physicists would in turn propose as the true geometry of space and time.

Background

In his Elements of Geometry, the great Greek mathematician Euclid (335-270 b.c.) was forced to adopt a rather awkwardly worded fifth and final postulate:

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than the two right angles.

It is clear that Euclid was not entirely comfortable with this postulate, as he postponed its use in proving theorems as long as he could. Over the centuries mathematicians speculated that the postulate could actually be proved as a theorem from the other axioms and postulates or, failing that, that it could be reformulated in a far simpler way as befitted a self-evident truth about the nature of space.

The Greek historian Proclus (410-485) records that the first Ptolemy, ruler of Egypt at the time that Euclid was teaching in Alexandria, wrote a book on the fifth postulate including what he considered a proof from the other postulates. Proclus also identified the flaw in Ptolemy's argument. The Persian mathematician Nasir-Eddin (1201-1274), who translated Elements into Arabic also tried to prove the postulate, and his argument too had subtle flaws. The English mathematician John Wallis (1616-1703) offered a proof of his own in 1663, but this too was defective.

In 1733 Girolamo Saccheri (1667-1733), an Italian Jesuit priest and Professor of Mathematics at the University of Pavia, published a book entitled Euclid Free of Every Flaw. Saccheri, who had already published a book on logic, decided to prove the postulate using the logical technique of "reductio ad absurdum," in which one disproves a hypothesis by showing the preposterous results of carrying the proposition to its logical conclusion. Saccheri considered a quadrilateral with two adjacent right angles and two parallel sides each meeting the base at a right angle. If the fifth postulate were, in fact, a consequence of the other postulates, the remaining angles would have to be right angles. Saccheri began by assuming that one of the two other possibilities was true, that the angles were either obtuse or acute. There was no difficulty in proving that the assumption of obtuse angles led to a contradiction. The possibility of acute angles proved much more difficult to deal with. After much difficulty Saccheri was able to produce only a very subtle form of absurdity in which two parallel lines when extended to infinity would combine to form a single straight line with a common perpendicular. In fact, he had proved that the fifth postulate was independent of the others, and in the course of his investigation he had derived many of the important theorems of non-Euclidean geometry. Because he was not aware of the significance of his work, credit for the discovery of non-Euclidean geometry would go to others.

In a paper presented in 1763, Georg S. Klugel (1739-1812), a German professor of mathematics, suggested that the fifth postulate could not be proved. Klugel's suggestion was the stimulus for The Theory of Parallel Lines by the German mathematician Johann Heinrich Lambert (1728-1777), written in 1766 but not published during his lifetime. Lambert considered a quadrilateral with three right angles and investigated the consequences of assuming that the remaining angle was right, acute, or obtuse. Like Sacceri, Lambert obtained theorems that would be important in non-Euclidean geometry. He derived, for instance, formulas for the area of a triangle in terms of the difference of its angles and the sum of two right angles. Lambert also recognized that a geometry without a parallel postulate might apply to the surface of a sphere or to a saddle-like surface.

Joseph Fenn suggested a simple substitute for the fifth postulate in 1769. In the equivalent form proposed by John Playfair (1748-1819), it states:

Through a given point P not on a line l there is only one line in the plane of P and l which does not meet l.

It is in this form that the axiom appears in modern texts.

In his attempt to derive as much of geometry as possible without using the fifth postulate, the great French mathematician Joseph Louis Lagrange (1736-1813) was able to prove that if the sum of the angles in any one triangle was equal to two right angles, then the sum of the angles in every triangle would be the same. Without the fifth postulate, however, the sum of the angles in any triangle could not be determined.

Impact

Work on the parallel postulate in the eighteenth century set the stage for the development of non-Euclidean geometries in the nineteenth century. Then it was discovered that totally consistent geometries could be obtained by assuming that through a given point either no lines or multiple lines could be drawn parallel to a given line. Perhaps the first significant mathematician to work in the area of non-Euclidean geometry was the great German Mathematician Carl Friedrich Gauss (1777-1855), who may also have been the first individual to suggest that the geometry of physical space might not be Euclidean. To investigate the latter possibility, Gauss measured the sum of the angles of a triangle defined by three mountain peaks. The uncertainties inherent in physical measurements prevented a definitive conclusion. Credit for the invention of non-Euclidean geometry is generally considered to be shared by Gauss, the Russian mathematician Nikolai Ivanovich Lobachevsky (1793-1856), and the Hungarian mathematician Janos Bolyai (1802-1860). With the groundwork established in the eighteenth century, non-Euclidean geometry was ripe for exploration.

With the exception of the problematic nature of the parallel postulate, the majority of mathematicians in 1800 regarded mathematics as a grand body of established truths. Over the course of the nineteenth century the emergence of several versions of non-Euclidean geometry, together with new problems concerning the concept of number and paradoxes of set theory and logic, would force a reappraisal of the very nature of mathematics. To the philosopher Plato (c. 427-347 b.c.), mathematics had described an abstract reality of which the world as perceived by humans was an imperfect reflection. To Galileo (1564-1642) and Isaac Newton (1642-1727) it was an accurate description of the nature of space. To the philosopher Immanuel Kant (1724-1804), it was one of the fundamental categories of human reason, an order imposed on experience by the human mind. Now it seemed that there was an element of arbitrariness in mathematics. It could no longer be regarded as a reliable means of deriving truths from self-evident propositions. Mathematicians and philosophers would now be divided into different camps by what they believed to be the substance of mathematics. Formalists considered it to be the manipulation of symbols according to specified rules—a sort of game. Intuitionists considered it to be the exploration of ideas that could be clearly conceived in the mind. Many mathematicians, of course, chose to continue proving theorems and solving problems without becoming alarmed about the uncertain foundations of their subject.

Even within the context of Euclidean geometry, there were still additional problems with Euclid's original text. Euclid had made many tacit (or unwritten) assumptions that needed to be made explicit. Logicians had come to realize the futility of trying to define every term that a mathematician might use. A more rigorous geometry text was published in 1882 by Moritz Pasch (1843-1930), a German mathematician. Pasch selected the concepts of point, line, plane, and congruence as undefined terms, and introduced axioms, which are statements about the undefined terms to be accepted without proof. Pasch's axioms included one stating that when a line intersects one side of a triangle it also intersects one of the other two sides. Euclid had assumed this to be the case, but not explicitly.

Pasch's geometry book was followed by others also attempting to provide a complete and logically sound exposition of geometric ideas. Noteworthy are The Principles of Geometry, published in 1889 by the Italian mathematician Giuseppe Peano (1858-1932), and the Foundations of Geometry, also first published in 1899 by the German Mathematician David Hilbert (1862-1943). Hilbert's book would run through seven editions.

In 1912, in his general theory of relativity, the German-born physicist Albert Einstein (1879-1955) proposed that the gravitational attraction between masses could be understood as resulting from a curvature of space and time. Straight lines were defined by the paths that a light beam would take, and the sum of the angles of a triangle would depend on the distribution of matter in that region of space. Einstein's equations gave a relationship between the the curvature of space and the matter and energy nearby. Applied to the universe as a whole, Einstein's equations allowed for three possibilities: Euclidean (flat) geometry, a space of positive curvature, like the surface of a sphere, or a space of negative curvature, like the surface of a saddle. Which geometry in fact applies is still to be resolved by observation.

DONALD R. FRANCESCHETTI