# Geometry

# GEOMETRY

Until 1800, mathematics was divided into two great branches: *geometry* and *arithmetic*. Both were commonly regarded as the more obviously secure repositories of human knowledge. At this stage, geometry could be suitably defined as "the science which investigates the properties and relations of magnitudes in space, as lines, surfaces, and solids" (*Oxford English Dictionary* ). However, with the enormous enrichment of mathematics in the nineteenth century, the scope of geometry was greatly expanded and diversified, its content disrupted, and its epistemic standing called into question.

The word "geometry" comes from a Greek word that literally means measurement of the earth and was originally applied to the art of land surveying. But around 500 BCE or even earlier, the spatial properties and relations that had been codified by land surveyors in Mesopotamia and Egypt became in Greece the starting-point of inquiries of a more abstract sort that soon took leave of their down-to-earth origins. In this guise, geometry appeared to Plato as a testimony of the other-worldly origin of the human soul (*Meno* ) and was included by him as a compulsory item in the curriculum for would-be philosopher-kings (*Republic*, VII, 526c–528d). For more than twenty centuries, philosophers regarded the geometry created by Greek mathematicians from Eudoxus, through Euclid, to Archimedes and Apollonius as the standard of indubitable truth and cogent reasoning. As a result of later developments, geometry, with the rest of mathematics, came to be seen as a capital example of the loss of certainty that currently pervades most areas of civilized life (Kline 1980). In more than one sense, this enhances, rather than diminishes its philosophical significance.

This entry is divided into three sections. The first section touches on some philosophically noteworthy aspects of ancient geometry. The second section deals briefly with geometry and philosophy from 1600 to 1800. And the third section describes those episodes in the history of geometry since 1800 that had the greatest impact on twentieth-century philosophy.

## Topics in Ancient Geometry

### geometry in the middle east

According to Herodotus (2.109), the Greeks learned land surveying (*geōmētriē* ) from the Egyptians, who used it to reassess taxes on properties partially washed away by the Nile. It appears that this art was first cultivated in the Middle East to cope with the consequences of floods in southern Mesopotamia. Archaeological evidence from both regions displays applications of the so-called theorem of Pythagoras, and a clay tablet now at Yale University (YBC 7289) gives the length of the diagonal of a unit square as 1.41421296, the same approximation to √2 that Ptolemy used some 2,000 years later. The Old Babylonian scribe who calculated it probably knew that he could improve on this figure, but it is highly unlikely that he suspected that no algorithm could ever yield a perfectly accurate one. No extant document from ancient Egypt or Mesopotamia contains the general statement of a geometric theorem or anything that even remotely resembles a geometric proof.

### pythagoreans and irrationals

Thales of Miletus, "the first to philosophize," supposedly was also the first to prove a geometric theorem (namely, that a triangle with two equal sides also has two equal angles). The earliest proofs probably consisted of diagrams that plainly displayed the relations they were meant to prove (see Plato, *Meno*, 80d–86c). But Greek geometers soon produced purely discursive proofs (like the one given later in this paragraph). The Pythagoreans, intellectually and politically active in southern Italy throughout the fifth century BCE, worked intensely on mathematical problems, as they thought that numbers (i.e., the positive integers) are the principles of everything. This suggestive belief was supported by their discovery that musical chords are associated with simple numerical proportions. It broke down, however, when a member of the school, possibly Hippasus of Metapontum, showed that there are geometric magnitudes of the same kind whose relative sizes cannot be conveyed by numbers. Presumably, this was first demonstrated for the diagonal and the side of the regular pentagon; but it is proved more easily for the diagonal and the side of a square by the following argument transmitted in an appendix to Euclid's *Elements*.

Take the side of the square as the unit of length. Then, by Pythagoras's theorem, the length of the diagonal equals √2. But there are no two integers *a* and *b* such that (*a* /*b* )^{2} = 2. For suppose there are. Then, by simplification of the fraction *a* /*b* we should find two integers *p* and *q*, with no common divisor, such that (*p* /*q* )^{2} = 2. Then *p* ^{2} = 2*q* ^{2} and *p* is an even number, equal to 2*n*, say. (For the square of an odd number, say 2*n* + 1, is always odd, that is, 4*n* ^{2} + 4*n* + 1). But then 2*q* ^{2} = *p* ^{2} = 4*n* ^{2}, and *q* ^{2} = 2*n* ^{2}, so that *q* is also even. But this is impossible, for we assumed that *p* and *q* do not have a common divisor. Therefore, one cannot find two integers *a* and *b*, no matter how large, such that the diagonal of a square exactly equals *a* × 1/*b* of its side. Awareness of the existence of incommensurable lengths cut short dreams of grasping nature through numbers and opened a chasm between arithmetic and geometry.

### eudoxus's theory of proportions

Eudoxus of Cnidos (c. 390–c. 337 BCE) invented a method for representing the visible motion of each planet in the sky (including Sun and Moon) as the resultant of the combined uniform rotations of several geocentric spheres. Eudoxus's planetary models are the earliest extant example of geometrical representation of natural processes for the sake of predicting their future evolution. Their moderate predictive success may have motivated Plato's change of mind from his early view that real planetary motions are essentially irregular and unpredictable (*Republic* VII, 529d7–530b4) to his later commendation of mathematical astronomy as an efficient servant of theology (*Laws* VII, 822a4–c5; X, 897c4–9; XII, 966d6–967d2) and his endorsement of Eudoxus's program as the proper way of "saving the phenomena" of the sky (Simplicius, 7.492.30–35). Eudoxus also originated the method of exhaustion employed by Archimedes for calculating volumes enclosed by curved surfaces, which was the first step toward the creation of the integral calculus. But Eudoxus's chief contribution to geometry was his theory of proportions, preserved in book 5 of Euclid's *Elements*. With it, geometry recovered the computational powers it had lost when separated from arithmetic, and the road was opened for rigorously conceiving and handling physical quantities of all sorts.

Two magnitudes *a* and *b* are said to have a ratio *a* :*b* to one another if there are integers *m* and *n* such that *m* × *a* > *b* and *n* × *b* > *a*. (The assumption that any two lengths have a ratio to one another is known as the Archimedean postulate.) Eudoxus produced definitions by virtue of which ratios can be added and multiplied, yielding new ratios, and any two ratios *a* :*b* and *c* :*d* satisfy *trichotomy*, that is, either *a* :*b* = *c* :*d*, or *a* :*b* > *c* :*d*, or *c* :*d* > *a* :*b*. In this last case, there will always be an integer *n* such that *n* (*a* :*b* ) > *c* :*d*. Thus, it is natural to regard all Eudoxean ratios as magnitudes that have ratios to one another. This paves the way for setting up equations that combine magnitudes of very different kinds, for example, masses, distances, and times, or volumes, temperatures, and pressures (as represented by their respective ratios with the appropriate units). However, it is not apparent that anyone saw this before the seventeenth century.

### euclid's elements

In this, the most famous of mathematical textbooks, Euclid (c. 325–c. 265 BCE) organized the results and displayed the methods of fourth-century-BCE Greek geometry. It is usually taken for granted that the book is patterned after Aristotle's conception of a true science (*epistēmē* ). This must consist of a collection of universal statements (theorems) obtained by deductive inference from self-evident premises (axioms) and definitions using a few self-explanatory terms (primitives). However, Euclid's book, though prima facie it may seem to prove every theorem from five postulates and a short list of so-called common notions, often resorts to unspoken assumptions. Moreover, Euclid's deductions do not all fit into the narrow frame of Aristotle's logic, and use forms of inference first codified by George Boole (1815–1864), Augustus De Morgan (1806–1871), and Charles Sanders Peirce (1839–1914). Also, his primary definitions (e.g., "A *straight* is a line which lies evenly with the points in itself") would have to be further supplemented by axioms to be of use in deductions. It seems more likely, therefore, that Aristotle based his idea of a true science on his own grasp of what contemporary geometricians were doing (textbooks similar to Euclid's had been around since Aristotle was a student in Plato's Academy) but did not set a paradigm that they or their successors actually followed.

### euclid's postulates

The first three postulates are not statements, but requests to allow certain constructions. The third—"to describe a circle with any center and any radius"—would require an infinite drawing board, which is not self-evidently available. The fifth is a conditional existential statement: "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 produced indefinitely*, intersect on that side on which are the angles less than the two right angles." Obviously, the condition here printed in italics can only be met on an endless plane. So in the finite world of Aristotelian and medieval cosmology, this postulate is vacuously true, and its existential consequent may be false (there may well not be any such intersection). Still, if Euclid's other postulates and the Archimedean postulate are true, denial of the consequent implies that a quadrangle with three right angles has an acute angle at the remaining corner, so there can be no rectangles. It also implies that polygons with the same shape also have the same size, in which case Aristotle's suggestion (*Physics* 207b29–34) that all geometrical theorems can be demonstrated in his bounded cosmos by suitably scaling down the diagrams employed would simply be wrong.

These seemingly counterintuitive implications kept geometers throughout the centuries trying to prove the fifth postulate from other principles until, shortly after 1820, Nikolay Lobachevsky and Janos Bolyai dared to deny it and independently published essentially the same system of non-Euclidean geometry. It might be a sign of Euclid's genius that he did not gloss over the fact that this assumption (without which the theorem of Pythagoras will not stand) is not self-evident.

## Geometry and Philosophy at the Onset of Modernity

### nature geometrized

Aristotle taught that natural science, to adequately grasp its proper subject, must employ terms that connote the peculiar matter of each thing, for example, "snub," which only applies to fleshy noses, rather than "concave," which connotes merely a geometric shape (*Metaphysics* Ε, 1, 1025b30–1026a7; *Physics* II, 2, 194a2–27). Still, he agreed (*Metaphysics* Λ, 8) with the purely geometric description of astronomical phenomena proposed by Eudoxus, presumably because he believed that ether, the stuff that the heavens are made of, can change only by rotation about the center of the Earth, and this is properly described in geometric terms. Anyway, Aristotle's strictures on science did not deter Archimedes (c. 287–212 BCE) from dealing mathematically with the equilibrium and the flotation of bodies. In the meantime, astronomers from Apollonius (third cent. BCE) and Hipparchus (second cent. BCE), through Ptolemy (second cent. CE), to Copernicus (1473–1543) developed ever more complex geometric models of planetary motion, involving diverse circular motions about different centers (none of which coincides with that of the Earth).

After Galileo Galilei's telescope showed that there are mountains on the Moon and fleeting spots on the Sun, the distinction between celestial and terrestrial physics became pointless, and each took cues from the other. Thus Johannes Kepler (1571–1630) sought to explain the motion of planets (including the Earth) by *forces* exerted on them from the Sun, while Galileo (1564–1642) proposed a chronogeometrical model of free fall on the surface of the Earth, which he conceived as uniformly accelerated rectilinear motion. Lasting success was finally achieved by Isaac Newton (1642–1727), by dint of his mathematical genius and his consummate command of geometry. In the course of these efforts, Kepler (1609) had the words "God is always doing geometry" printed on the front page of his masterpiece, and Galileo wrote that the book of nature "is written in mathematical language, and its characters are triangles, circles and other geometrical figures, without which we cannot understand a word of it" (1623, sec. 6). René Descartes's contention that extension is the one and only clearly and distinctly conceivable attribute of bodies surely called for a comprehensive and thoroughgoing geometrization of physics and might have led to it had geometry been ripe enough to deal with its strenuous demands.

### descartes's revolution in geometry

Except for his first law of motion (the principle of inertia) and his work on the refraction of light, Descartes's direct contribution to physics, subjected to unsparing criticism by Christian Huygens, Gottfried Wilhelm Leibniz, and Isaac Newton, failed to gain admission into the classical canon. But modern mathematical physics would not have been possible without Descartes's *indirect* contribution to it, through his two great inventions in geometry: *coordinates* (independently introduced also by Pierre Fermat) and the *algebra of lengths*.

Coordinates are quantitative labels employed for identifying points in space. By means of them the relations among the points can be quantitatively represented and investigated. Nowadays geometric coordinates are drawn from the field of real numbers ℝ, which we regard as a natural extension of ℚ, the field of rationals, which, in turn, is constructed from the familiar integers. But this understanding of these matters was still far off in Descartes's time (although his geometric algebra was a decisive step toward it). To avoid anachronism, one must regard Descartes's original coordinates as oriented lengths or, more exactly, as Eudoxean ratios between such lengths and a conventionally chosen unit length. To assign so-called Cartesian coordinates to a point *P* in space, one takes the three distances *x, y*, and *z* from *P* to three mutually perpendicular planes (listed in a conventional order, the same for all points), and prefixes to each a plus sign or a minus sign, according to the side of the respective plane that faces *P* (again by convention). The Cartesian coordinates of *P* then form an ordered triple of oriented lengths, say 〈+*x*, −*y*, −*z* 〉. (There are other ways of defining coordinates: Oblique coordinates depend on three planes not at right angles to each other. Polar coordinates label a point *P* by its absolute distance from a fixed point *O*, the angle made by *OP* with a fixed plane Γ through *O*, and the angle made by the perpendicular projection of *OP* on Γ with a fixed line through *O* on that same plane.)

In Euclid's *Elements* (1956), segments are added to segments in an obvious way to obtain new segments; multiplying a segment *s* by an integer *n* amounts to adding *n* copies of *s* end to end; a straight segment or a length is never multiplied by another one. Until not too long ago, it was usually understood that such multiplications do occur in Euclid's book, but then the product of such a multiplication had to be an *area*. Descartes followed Euclid on the addition of lengths and defined the multiplication of a length *a* by a length *b* so as to yield still another length *ab*. Here is how. Draw two straight lines from a point *O*. Mark points *F* and *H* on one line so that *OF* has unit length and *OH* has length *a*. Mark point *G* on the second line so that *OG* has length *b*. Draw the straight line segment *FG*. Let the parallel to *FG* through *H* cut the second line at *K*. Clearly, then, *OH* /*OF* = *OK* /*OG*. Therefore, *OG* × *OH* = *OK* × *OF* ; in other words, *OK* has length *ab* (Figure 1).

By this procedure, entirely based on elementary geometrical knowledge available to Euclid, Descartes and his successors were able to represent all geometrical relations by equations or inequalities between given and unknown

quantities, and to solve geometrical problems algebraically. In the algebra all such quantities were handled in the same way as the positive integers and they were therefore called *numbers*. Newton explains: "By a number we do not understand a multitude of units, but rather the abstract ratio of any quantity to another quantity of the same kind, which is taken as a unit. There are three varieties of number: integers, rationals and irrationals" (1707, p. 2). Eventually, they were called *real* numbers, to distinguish them from the imaginary ones, that is, the multiples of √−1, which also turned up as solutions of algebraic equations.

The method of coordinates soon suggested the idea of a space with *n* dimensions, whose points would be labeled by *n* quantities. In particular, if 〈*x, y* 〉 denotes an arbitrary point on a plane, a straight line on that plane can now be defined as the set of points satisfying the linear equation *y* = *ax* + *b*, and a circle with radius *r* and center at 〈0, 0〉 as the set of points satisfying the quadratic equation *x* ^{2} + *y* ^{2} = *r* ^{2}. These two equations take care of all points on the plane that can be constructed with a ruler and a compass, which were the only points contemplated by Euclid. But after Descartes, mathematicians felt free to consider any curve defined by an algebraic equation or indeed by a convergent series, such as *y* = sin *x*, or *y* = *e ^{x}* (where

*e*is the base of the natural logarithms). Even though Euclid never countenanced the plethora of points obtainable in such ways and it does not follow from his postulates, what we normally call "Euclidean space" comprises them all.

### kant's philosophy of geometry

The overwhelming success of geometry in physics and astronomy induced some seventeenth-century philosophers to follow its example in ethics and metaphysics. The foremost instance of this is Benedict de Spinoza's *Ethica ordine geometrico demonstrata* (Ethics demonstrated in geometric order; 1677), but John Locke too believed that "if men would in the same method, and with the same indifferency, search after moral as they do mathematical truths," then "a great part of morality might be made out with that clearness, that could leave, to a considering man, no more reason to doubt, than he could have to doubt of the truth of propositions in mathematics, which have been demonstrated to him" (1690, IV.iii.20, xii.8).

Immanuel Kant, however, thought otherwise. Invidiously comparing geometry, as a science that "excels all others in certainty and distinctness," with metaphysics, which "has only just started out on the path to these goals" (1902–, 2: 168), he recommended, in 1763, that the latter stop imitating the former, in order to progress along that path. He soon went further. In his Latin dissertation of 1770, Kant taught that confusion and stagnation in metaphysics were due to the contamination of the human intellect with the sensuous notions of time and space. By thoroughly avoiding them, metaphysics will escape the temptations of materialism and determinism and become a secure science of God, freedom, and immortality. Yet in the *Critique of Pure Reason* (1781), Kant likened the purified understanding he advocated in 1770 to a bird that, tired by the resistance of the air, sets out to fly in a vacuum. In his mature view, the basic concepts of human thought—one and many, reality and negation, substance and cause—are not obtained from sensuous experience, but they can refer to objects only when applied to it, under the conditions of human sensibility, namely, space and time.

This decisive turnabout in the history of philosophy is closely related to Kant's reflections on geometry and its use in physics. In 1746 Kant spoke of a general or "supreme" geometry, adapted to a space with any number of dimensions. That the space we live in has only three dimensions is due to the empirical fact that all material particles are linked by forces governed by Newton's inverse-square law (1902–, 1: 34). But in 1768 he made a discovery that, he thought, put an end to all such explanations of space and spatial structure from the physical relations between bodies in space. No description of a shoe in terms of its different parts and the relations between them will allow us to tell a left shoe from the matching right shoe; the difference between the two shoes can be grasped only by considering their respective orientation in the space that embraces them. Kant understood this to imply that the bodily structure of bodies depends on that of space as a whole, which therefore is presupposed by them, rather than being only an expression of their interactions.

Kant was then faced with the following dilemma: Either (a) space itself is a substance, of which bodies are modes (a position that, according to Kant, results in Spinoza's unchristian and immoral deification of space), or (b) space must be thought of as possessing a novel, hitherto unheard of manner of existence (which implies a corresponding adjustment of the ontological standing of bodies as such). About 1769 Kant lighted on alternative (b), which he described around 1791 as one of the two hinges on which metaphysics must turn (the other one being the reality of freedom). He claims that "*space is not something objective and real*, neither a substance, nor an accident, nor a relation; it is rather a subjective and ideal scheme, so to speak—which issues from the nature of the mind according to a stable law—for coordinating everything that is sensed externally" (1902–, 2: 403). Or, in mantra form, *space is one of the forms of human sensibility* (time is the other one). As a consequence of this, things are bodies only insofar as they are actual or potential objects of our sense perception, but not as they are in themselves. (Indeed, Kant figured out in the early 1770s that the standard assumption that things in themselves are spatial would make it impossible to solve the contradictions regarding the limit of the physical world and the divisibility of its content, which he later set forth in the first two items of the Antinomy of pure reason.)

Kant's conception of space is the key to his philosophy of geometry (and is in turn reinforced by it). The epistemological problem of geometry lies in explaining how it can furnish us with precise quantitative information about things we have never met in real life and which anyway we could not measure accurately, for example, the exact size of the angles of a trillion-sided regular polygon. Plato proposed that this knowledge is remembered from another life in which we had direct access to the intelligible "form" of things. The fact that geometry contains such knowledge nourished similar hopes for metaphysics and ethics, which, however, were crushed by Kant's approach. In his view, geometry rests on our natural awareness of the conditions under which alone the manifold appearances displayed through our external senses "can be ordered into certain relations" (1787, B 34) and thus shaped into corporeal phenomena. Such awareness is not intellectual but intuitive, as we may gather from the example of the pair of shoes, described above, and also from the fact that geometrical proofs proceed by the "construction of concepts." Kant explains this expression somewhat intriguingly as follows: "To *construct* a concept means to exhibit a priori the intuition that corresponds to it: the construction of a concept therefore requires a *non-empirical* intuition which … as intuition is a *particular* object, but nevertheless, as the construction of a concept (a general idea), must convey universal validity for all possible intuitions that belong under the same concept" (1781/1787, A 713/B 742).

Anyone not put off by these opaque notions could well regard them as a proper explanation of the amazing success of geometry in physics. For, if geometry spells out the ordering that is required for us to grasp external phenomena, then it is no wonder that all external phenomena comply in every detail with the teachings of geometry. Soon, however, innovations in geometry moved the ground from under Kant's position and made it untenable. Before we turn to them, it should be emphasized that among these innovations, the best known one—the derivation of a consistent geometry from the denial of Euclid's fifth postulate—does not challenge Kant's view but somehow corroborates it. For Kant, geometry provides information, conveyed by what he called *synthetic* propositions, and this implies that any of its unproven principles can be denied without selfcontradiction.

## From Gauss to Hilbert and beyond

### non-euclidean geometry

The fact that Euclid's fifth postulate is not self-evident prompted several mathematicians to try to prove it. John Wallis (1616–1703) succeeded in inferring it from the assumption that for any given figure there is another one, similar to it, of any arbitrary size. This assumption is neither necessarily true nor empirically obvious, but it does provide a perspicuous characterization of Euclidean space.

Girolamo Saccheri (1733[1986]) sought to prove the postulate indirectly. He devised a quadrilateral thus constructed on a plane: Draw straight lines *m* and *n* through points *P* and *Q*, making right angles with the segment *PQ*. Mark points *A* and *B* on *m* so that *AP* = *PB*. Mark points *C* and *D* on *n* so that *CA* and *DB* are both perpendicular to *m* (See Figure 2).

If one assumes the Archimedean postulate (which Saccheri tacitly does), the fifth postulate will hold if and only if ∠ACQ and ∠BDQ are right angles. Saccheri assumed that these angles are obtuse and easily proved that, if so, two points can be joined by more than one line, which he considered absurd. He then assumed that both

angles are acute and derived from this hypothesis many surprising propositions that did not appear to be contradictory, until at last he reached one he pronounced "repugnant to the nature of the straight line."

The consequences that Saccheri drew from the acute-angle hypothesis reappeared in the nineteenth century in the private papers of Carl F. Gauss (1777–1855) and in independent publications by Nikolay I. Lobachevsky (1793–1856) and Janos Bolyai (1802–1860). These authors treated these consequences as theorems of an alternative system of geometry, based on the straightforward denial of Euclid's fifth postulate (with the others retained). This system has received various names, but by priority of publication, it should be called *Lobachevskian geometry*. In this geometry, the three interior angles of a triangle add up to less than two right angles, the difference being proportional to the area of the triangle. Therefore, similar triangles are congruent. Consider again the segment *PQ* perpendicular to straight line *m* at *P*. By the denial of the fifth postulate, there is a set *S* of straight lines through *Q* that form an acute angle with *PQ* on one or the other side of it and yet do not meet *m* on that side (let alone on the side where they form an obtuse angle with *PQ* ). Let α be the smallest of these angles. By symmetry, there are two lines in *S* that form angle α on either side of *PQ*. In Lobachevsky's terminology (independently adopted also by both Gauss and Bolyai), these two lines are called the *parallels* of *m* through *Q*, and α is the *angle of parallelism* for *PQ*. The size of α decreases as *PQ* grows. On any Lobachevskian plane, there is a unique length *h* such that the angle of parallelism for any segment of length *h* equals 45°. The length *h* provides an absolute standard of length for that plane. Kant's friend Johann Heinrich Lambert, who around 1766 worked on this subject along lines similar to Saccheri's, said there was "something alluring about this consequence which readily arouses the desire that the [acute angle] hypothesis be true!" (1786/1895, p. 162). Note that if, in the case discussed, *PQ* = *h*, the two parallels to *m* through *Q* are mutually perpendicular. Lambert thought this was an intolerable paradox.

The absence of contradiction in a long series of theorems inferred from the denial of the fifth postulate does not, of course, imply that Lobachevskian geometry is consistent. Lobachevsky proposed an argument for proving that his geometry is at least as tenable as Euclidean geometry. He showed that there is a logically formal correspondence between the equations of Lobachevskian trigonometry and the familiar equations of spherical trigonometry. By virtue of it, any contradiction derived from the former will be matched by one flowing from the latter. Such a contradiction would entail that the said standard trigonometric equations are false, and this in turn would entail the falsehood of the Euclidean principles from which these equations follow.

Lobachevsky also tried to ascertain whether his own geometry or Euclid's is true of physical space. He used astronomical data to calculate the sum of the internal angles of the triangle formed by three stars and concluded that the difference between the result obtained in a Lobachevskian space and the Euclidean value was well within the margin of observational error. Decades would pass before Hermann Lotze (1879, p. 774) pointed out that all such attempts are vain, for if astronomical measurements do not agree with Euclidean geometry, the disagreement can still be accounted for by a deviation of stellar light from its supposedly rectilinear trajectory.

### groups and invariants

Shortly before Lobachevsky's earliest publication on his geometry, Jean-Victor Poncelet's *Treatise on the Projective Properties of Figures* (1822) started a way of doing geometry that seemed more intuitive and tame but which ultimately was much more radical and would have deeper consequences than the denial of Euclid's fifth postulate. It is based on adding to each straight line *m* a "point at infinity" that *m* shares with every straight line parallel to it and treating all such "points" as belonging to a single "plane." This assumption enormously simplified the statement and the proof of geometric theorems concerning relations of incidence, collinearity, and coplanarity among points, lines, and planes. Metric features like distance and metric relations like congruence were totally ignored. Natural (initially tacit) assumptions regarding the neighborhood relations between the points at infinity and the standard points implied that ordinary space differed drastically from the new *projective space* in which it was now embedded. For example, a left shoe traveling indefinitely in a fixed direction would, after crossing the plane at infinity, return to its original location from the opposite side, in the guise of a right shoe. In this way, projective geometry disposed of Kant's claim about the irreducible difference between the two kinds of shoes, its ontological implications, and its intuitive roots.

Projective geometry grew in scope and sophistication at the hands of August Moebius (1790–1868), Julius Plücker (1801–1868), Karl Georg Christian von Staudt (1798–1867), Arthur Cayley (1821–1895), and others. Different sorts of numerical coordinates were introduced as sheer labeling devices, for in this metric-free context they plainly did not represent distances. The use of coordinates consisting of complex numbers made it possible to introduce more points, in addition to the familiar real points (those labeled by real numbers). These "complex points" are linked to real points and among themselves by relations of collinearity (if their coordinates satisfy the same linear equations) and vicinity (by dint of the neighborhood relations between real and nonreal numbers on the complex plane). The beautiful vistas opened by such developments inspired further flights of mathematical freedom leading to the creation of still other branches of geometry.

Moved by the confusing variety of geometrical methods and approaches, Felix Klein formulated his celebrated Erlangen Program, in which he seeks to unify all forms of geometry under a single overarching point of view. This is provided by the notion of *transformation group* and the related notion of *invariant*.

Let *S* be a set of points. A *transformation* (or *permutation* ) *T* of *S* assigns to each point *p* of *S* one and only one point *T* (*p* ) of *S*, in such a way that every point of *S* equals *T* (*p* ) for some *p*. In other words, a transformation of *S* is a one-to-one mapping of *S* onto itself. We say that *T* (*p* ) is the *value* of *T* at *p*. *T* is said to *send p* to *T* (*p* ). If *M* is a subset of *S, T* is said to *send M* to the set *T* (*M* ) = {*T* (*p* ): *p* ∈ *M* }. For every transformation *T*, there is an *inverse transformation T* ^{−1} that, for each *p* in *S*, sends *T* (*p* ) back to *p*. The identity transformation Id* _{S}* sends every

*p*in

*S*to itself. Given two transformations

*T*

_{1}and

*T*

_{2}, their

*product T*

_{2}

*T*

_{1}is the transformation that sends each

*p*in

*S*to the value of

*T*

_{2}at

*T*

_{1}(

*p*). The product of transformations is clearly

*associative*, that is, (

*T*

_{3}

*T*

_{2})

*T*

_{1}=

*T*

_{3}(

*T*

_{2}

*T*

_{1}) for any three transformations

*T*

_{1},

*T*

_{2}, and

*T*

_{3}. A set

*G*is a

*group of transformations*of

*S*if every element of

*G*is a transformation of

*S*and

*G*contains (1) the product of any two of its elements, (2) the inverse of every one of its elements, and (3) the identity transformation Id

*. Any subset of*

_{S}*G*that meets conditions (1) through (3) is said to be a

*subgroup*of

*G*.

Given a group *G* of transformations of a set *S*, let *R* be an *n* -adic predicate (*n* ≥ 1) such that, for any points *p* _{1}, …, *p _{n}* in

*S*and any transformation

*T*in

*G, R*(

*p*

_{1}, …,

*p*) implies that

_{n}*R*(

*T*(

*p*

_{1}), …,

*T*(

*p*)). We say then that

_{n}*R*is an

*invariant*of group

*G*or that

*R*is

*G-invariant*. Likewise, a function

*f*on

*S*is said to be

^{n}*G-invariant*if, for every

*n*-tuple 〈

*p*

_{1}, …,

*p*〉 of elements of

_{n}*S, f*(

*p*

_{1}, …,

*p*) =

_{n}*f*(

*T*(

*p*

_{1}), …,

*T*(

*p*)).

_{n}*G*is said to

*preserve*its invariants.

Klein's Erlangen Program for systematically ordering geometries is based on the following simple idea: Each geometry is the study of the invariants of a group, and the relations of inclusion between groups and their subgroups determine a hierarchy of geometries. Starting from the group of all possible transformations of an arbitrary set, whose sole invariant is the cardinality of the set, one descends, through multiple branches, right down to the trivial group, which is a subgroup of every group and preserves every property and relation, for it only comprises the identity transformation. In particular, projective geometry studies the invariants of the group of collineations, that is, the set of transformations that send straight lines to straight lines. This is a subgroup of the group of continuous transformations, whose invariants are the *topological* properties of projective space. Drawing on work by Arthur Cayley (1859), Klein (1871, 1873) found a way of defining different real-valued functions on point pairs that behaved, on well-defined regions of projective space, precisely like the distance functions of, respectively, Lobachevskian geometry (which he called *hyperbolic* ), Euclidean geometry (which he called *parabolic* ), and a third geometry (which he called *elliptic* ). Each of these functions was an invariant of a certain subgroup of the said group, comprising the collineations that map a specific quadric surface onto itself.

Klein's result led Russell (1897) to assert that the general "form of externality" is disclosed to us a priori in projective geometry, but its metric structure—which Russell wrongheadedly claimed can *only* be Lobachevskian, Euclidean, or elliptic—must be determined a posteriori by experiment. Poincaré took another view of this matter: If geometry is nothing but the study of a group, "one may say that the truth of the geometry of Euclid is not incompatible with the truth of the geometry of Lobachevsky, for the existence of a group is not incompatible with that of another group" (1887, p. 290). Euclidean geometry has seemed preferable only because the rotations and translations of the Euclidean group reflect, to a comfortable approximation, the motions of ordinary hard bodies in our environment.

### riemannian manifolds

In his lecture "On the Hypotheses Which Lie at the Foundation of Geometry" (1867), Bernhard Riemann took an approach to geometry that did not fit into Klein's Erlangen Program. He noted that traditional geometry rested on assumptions summed up in Pythagoras's theorem (by which the distance between a point with Cartesian coordinates 〈*x, y, z* 〉 and the origin 〈0, 0, 0〉 equals the positive square root of *x* ^{2} + *y* ^{2} + *z* ^{2}). These assumptions had been corroborated by using light rays to line up things and rigid bodies to measure the length of lines, and therefore were bound to break down on very small scales, where such physical objects are not available, and perhaps also on very large scales, where the errors of observation generated by using such instruments might become intolerable.

Riemann therefore proposed to proceed from more general assumptions toward a more flexible geometry that physicists could later resort to when they needed it. He took his cue from Carl Friedrich Gauss's work on the intrinsic geometry of surfaces (1828), which he extended to general spaces of *n* -dimensions (*n-manifolds*, for short). Though Riemann supposedly addressed his lecture to humanists (hence its meager use of mathematical symbolism), the meaning and reach of the lecture first became clear through its further elaboration by other outstanding mathematicians (e.g., Elwin Bruno Christoffel, Friedrich Schur, Wilhelm Killing, Élie Cartan, Hermann Weyl), and a whole century would pass before it was satisfactorily explained to undergraduate students (Spivak 1979). The following rough sketch of Riemann's breakthrough owes much to the light shed on it by such later developments.

An *n* -manifold *M* is furnished with coordinate systems or *charts*, by which different regions or *patches* of *M* are mapped continuously and one-to-one onto subsets of ℝ* ^{n}* (the set of all

*n*-tuples of real numbers, endowed with the neighborhood relations it inherits from the real-number field ℝ). Two charts defined on overlapping patches are said to be

*compatible*if the coordinate transformation between them is a smooth function from one open subset of ℝ

*onto another (possibly the same) one. An*

^{n}*atlas*of

*M*is a collection of compatible charts for

*M*such that every point of

*M*lies on the patch of at least one chart. Any atlas

*A*of

*M*determines a corresponding maximal atlas

*A*

_{max}comprising every conceivable chart of

*M*compatible with those in

*A*. The way the charts of

*A*

_{max}combine with each other in coordinate transformations reflects the overlapping and intertwining of the patches on which they are defined and thus specifies the global topology, the shape, of

*M*. (In the realm of 2-manifolds, or surfaces, the atlas of a pretzel differs from that of a donut or a bun.) The lengths of curves drawn in

*M*(which are best thought of as continuous mappings of an open interval of ℝ into

*M*) can then be defined in an endless variety of ways by rules that assign to each point

*p*of

*M*an appropriate function on the coordinate differentials at

*p*, in a manner that varies smoothly from point to point.

Riemann was aware that this approach gave the mathematician enormous freedom, and he proposed restricting the admissible functions, for the time being, to quadratic functions on coordinate differentials, which, on a small neighborhood of each point of *M*, would yield a definition of length in optimal agreement with the *n* -dimensional version of the Pythagorean theorem. To conform to the standard concept of length, he also required his quadratic functions to be positive-definite, that is, to take their values only among the nonnegative real numbers. This requirement was subsequently relaxed in the spacetime geometries of the theory of relativity, which, for this reason, are often called *semi-Riemannian*.

From the standpoint of twentieth-century mathematics, the characterization of an *n* -manifold *M* by means of an atlas of the kind described makes it possible to assign an *n* -dimensional real vector space to each point *p* of *M*, the *tangent space* at *p* (thus called by analogy with the plane tangent to a smooth surface at any point of it). A Riemannian metric on *M* assigns to each *p* of *M* a tensor of rank 2, that is, a bilinear function on its tangent space, that varies smoothly from point to point. Riemann's quadratic functions on the coordinate differentials at each point can be naturally obtained as appropriate representations, relative to one or another locally defined coordinate system, of such coordinate independent objects.

A given Riemannian metric μ on *M* determines a smoothly varying assignment, to each *p* of *M*, of a tensor of rank 4, that is, a quadrilinear function on its tangent space. Such an assignment is in effect a *field of tensors* (one at each point), but, for short, this assignment is called the *Riemann tensor* of the Riemannian manifold 〈*M*, μ〉. This is a natural generalization to *n* -manifolds of the analytically defined yet fairly intuitive concept of the Gaussian curvature of a surface (a real-valued function that is positive and constant on a sphere, variable but always positive on an ovoid, alternatively positive and negative on the surface of a saddle, and equal to 0 on a plane or a cylinder), and is therefore also called the *curvature tensor*, a term that mathematicians wield nonchalantly but that among philosophers has been a source of endless worries.

On this same analogy, *n* -manifolds of constant 0 curvature are said to be *flat*. In particular, Euclidean space is a flat 3-manifold, while Lobachevskian space is a 3-manifold of constant negative curvature, and Klein's elliptic space is a 3-manifold of constant positive curvature. But a generic Riemannian manifold 〈*M*, μ〉 has variable curvature, and therefore the only group of transformations of *M* that will preserve the metric μ is the trivial group consisting only of the identity transformation. Already for this reason, Riemannian geometry obviously cannot fall under the Erlangen Program. Another excluding reason is the fact that a geometric inquiry that considers general spaces of endlessly different shapes cannot be characterized by a group of transformations of one of these spaces onto itself. Still, there are Riemannian manifolds endowed with interesting symmetries, and group theory has been the tool of choice for studying them.

### hilbert's foundations

Euclid's putative program for logically inferring the truths of geometry from a sufficient list of unproven premises was fondly imitated by scientists and philosophers in the seventeenth century, but it was first properly carried out by Moritz Pasch (1882). He gathered what he regarded as the empirical foundation of geometry into a few undefined concepts concerning the shape, size, and reciprocal position of bodies and a few axioms that linked these concepts among themselves and with other concepts defined in terms of them. Pasch's axioms "state what has been observed in certain very simple diagrams" (p. 43). All other geometric statements should be proved from the axioms by the strictest deductive methods.

Pasch dealt with projective geometry. The first rigorous axiomatization of Euclidean geometry was given in David Hilbert's *Foundations of Geometry* (1899), a book that had a major influence on twentieth-century mathematics and philosophy. Hilbert invited the reader to consider three arbitrary sets of objects, which he called points, straights, and planes; three undefined relations of *incidence* between a point and a straight line, between a straight line and a plane, and among three points; and two undefined relations of *congruence* between two pairs of points (*segments*,) and between two equivalence classes of point triples (*angles* ). Hilbert linked these objects and relations through nineteen axioms, which—when supplemented with the "axiom of completeness" added in the second edition—are sufficient for characterizing the said objects and relations up to isomorphism. This means that if we have two threefold collections of points, straights, and planes having the prescribed relations of incidence and congruence in agreement with Hilbert's twenty axioms, there will always be a one-one mapping of the points, straights, and planes of one collection respectively onto the points, straights, and planes of the second that preserves all five sorts of relations. Such a structure-preserving mapping between structured sets is called an *isomorphism*. Evidently it can hold between two systems of intuitively very dissimilar objects.

Hilbert availed himself of this feature of axiomatic theories for studying the independence of some axioms from the rest. To prove such independence, he proposed actual instances (*models* ) of the structure determined by all the axioms but one, plus the negation of the omitted one. Gottlob Frege complained that the geometric axioms retained in these exercises could be applied to Hilbert's far-fetched models only by tampering with the natural meaning of words. Hilbert replied, on December 29, 1899, "Every theory is only a scaffolding or schema of concepts together with their necessary mutual relations, and the basic elements can be conceived in any way you wish. If I take for my points any system of things, for example, the system *love, law, chimney-sweep*, … and I just assume all my axioms as relations between these things, my theorems, for example, the theorem of Pythagoras, also hold of these things.… This feature of theories can never be a shortcoming and is in any case inevitable" (Frege 1967, p. 412). Hilbert's declaration of independence from sense experience and ordinary usage concisely expresses the modern view of mathematics as a universal "science of patterns" (Resnik 1997), in which geometry is barely distinguishable from its other branches, except on historical grounds.

### the geometry of the universe

This approach to pure geometry and mathematics gives physicists enormous freedom to choose the abstract structures they judge most suitable for representing (modeling) the phenomena under inquiry. Yet, as Albert Einstein (1921) pointed out, so long as physics remains unable to provide, from microphysical principles, an exact theoretical construction of the instruments it uses for measuring distances and times, it will continue to need a practical geometry, which must be suggested and corroborated by experience.

According to Einstein, the stability of sharp spectral lines justifies the postulate that any two ideal clocks, once running beside each other at the same rate, will always do so, no matter where and when they are brought together again for comparison. Under this postulate, and for spatiotemporal regions sufficiently small that gravity is practically homogeneous in them, experience has amply vindicated the validity of the flat semi-Riemannian geometry that Hermann Minkowski (1909) initially proposed for the whole world. For broader regions, Einstein's theory of general relativity assumes a semi-Riemannian spacetime geometry whose variable curvature reflects the variations in gravity.

By solving Einstein's equations of the gravitational field under cosmologically plausible special symmetry requirements, Alexander Friedmann (1922, 1924) produced big-bang models of the universe. These big-bang models were ready at hand to explain the systematic recession of the galaxies away from us when discovered by Vesto Slipher and Edwin Hubble before 1930 and the current low temperature of the background thermal radiation when discovered by Arno Penzias and Robert Wilson in 1964. It is worth emphasizing that the explanation of these phenomena from the Einstein field equations is purely geometrical—a consequence of the shape of the universe under the postulated symmetry requirements. Purely geometric arguments also support the proof by Stephen Hawking and Roger Penrose (1970) that, under physically very plausible assumptions, a generic relativistic space-time most likely contains black holes.

** See also ** Aristotle; Black Holes; Boole, George; Copernicus, Nicolas; De Morgan, Augustus; Descartes, René; Einstein, Albert; Frege, Gottlob; Galileo Galilei; Hilbert, David; Kant, Immanuel; Kepler, Johannes; Leibniz, Gottfried Wilhelm; Locke, John; Lotze, Rudolf Hermann; Mathematics, Foundations of; Newton, Isaac; Peirce, Charles Sanders; Plato; Poincaré, Jules Henri; Pythagoras and Pythagoreanism; Russell, Bertrand Arthur William; Space; Spinoza, Benedict (Baruch) de; Thales of Miletus; Weyl, (Claus Hugo) Hermann.

## Bibliography

### references

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Frege, Gottlob. *Kleine Schriften*, herausgegeben von I. Angelelli. Darmstadt, Germany: Wissenschaftliche Buchgesellschaft, 1967.

Friedmann, Alexander. "Über die Krümmung des Raumes." *Zeitschrift für Physik* 10 (1922): 377–386.

Friedmann, Alexander. "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes." *Zeitschrift für Physik* 21 (1924): 326–332.

Galileo Galilei. *Il saggiatore nel quale con bilancia esquisita e giusta si ponderano le cose contenute nella Libra astronomica e filosofica di Lotario Sarsi Sigensano*. Roma: Giacomo Mascardi, 1623. Partial English translation in Stillman Drake, *Discoveries and Opinions of Galileo*. New York: Doubleday, 1957 (*The Assayer* ).

Gauss, Carl Friedrich. *Disquisitiones generales circa superficies curvas*. Göttingen, Germany: Dieterich, 1828. English translation by A. Hiltebeitel and J. Morehead: *General Investigations of Curved Surfaces*. Hewlett, NY: Raven Press, 1965.

Hawking, Stephen W., and Roger Penrose. "The Singularities of Gravitational Collapse and Cosmology." *Royal Society of London Proceedings* A 314 (1970): 529–548.

*Herodotus*, with an English translation by A. D. Godley. 4 vols. Cambridge, MA: Harvard University Press, 1946.

Hilbert, David. "Die Grundlagen der Geometrie." In *Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals*. Leipzig, Germany: B.G. Teubner, 1899, pp. 3–92.

Kant, Immanuel. (A), *Critik der reinen Vernunft*. Riga: J.F. Hartknoch, 1781. Repeatedly translated into English as *Critique of Pure Reason*.

Kant, Immanuel. (B), *Critik der reinen Vernunft*. Zweyte hin und wieder verbesserte Auflage. Riga: J.F. Hartknoch, 1787. Second edition of Kant (A); variants normally included in English translations.

Kant, Immanuel. *Kant's gesammelte Schriften*, herausgegeben von der Preussischen Akademie der Wissenschaften (Bde. 1–22), der Deutschen Akademie der Wissenschaften zu Berlin (Bd. 23), und der Akademie der Wissenschaften zu Göttingen (Bde. 24, 25, 27–29). Berlin, 1902–. Page numbers of this edition are printed on the margin of *The Cambridge Edition of Kant's Works in Translation*.

Kepler, Johannes. *Astronomia nova aitiologetos seu physica coelestis, tradita commentariis de motibus stellæ Martis*. Pragæ, 1609. English translation by William H. Donahue: *New Astronomy*. Cambridge, U.K.: Cambridge University Press, 1992.

Klein, Felix. "Über die sogenannte Nicht-Euklidische Geometrie." *Mathematische Annalen* 4 (1871): 573–625.

Klein, Felix. "Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz)." *Mathematische Annalen* 6 (1873): 112–145.

Klein, Felix. *Vergleichende Betrachtungen über neuere geometrische Forschungen*. Erlangen, Germany: A. Duchert, 1872.

Kline, Morris. *Mathematics: The Loss of Certainty*. New York: Oxford University Press, 1980.

Lambert, Johann Heinrich. "Theorie der Parallellinien." *Magazin für die reine und angewandte Mathematik* 2 (1786): 137–164; 3 (1786): 325–358. Reprinted in P. Stäckel and F. Engel, *Die Theorie der Parallellinien von Euklid bis Gauss*. Leipzig, Germany: B.G. Teubner, 1895, pp. 152–207; our quotation refers to this edition.

Locke, John. *An Essay concerning Humane Understanding*. London: Thomas Basset, 1690.

Lotze, Hermann. *Metaphysik: Drei Bücher der Ontologie, Kosmologie und Psychologie*. Leipzig, Germany: S. Hirzel, 1879.

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Newton, Isaac. *Arithmetica universalis, sive De compositione et resolutione arithmetica liber*. Cambridge, England: Typis academicis, 1707.

Pasch, Moritz. *Vorlesungen über neueren Geometrie*. Leipzig, Germany: Teubner, 1882.

Plato. *Laches - Protagoras - Meno - Euthydemus*, with an English translation by W. R. M. Lamb. Cambridge MA: Harvard University Press, 1990.

Plato. *Laws*, with an English translation by R. G. Bury. 2 vols. Cambridge MA: Harvard University Press, 1984.

Plato. *Republic*, with an English translation by Paul Shorey. Cambridge MA: Harvard University Press, 1982.

Poincaré, Henri. "Sur les hipothèses fondamentales de la géométrie." *Bulletin de la Société mathématique de France* 15 (1887): 203–216.

Poncelet, Jean-Victor. *Traité des propriétés projectives des figures*. Paris: Bachelier, 1822.

Resnik, Michael D. *Mathematics as a Science of Patterns*. Oxford, U.K.: Clarendon Press, 1997.

Riemann, Bernhard. "Über die Hypothesen, welche der Geometrie zugrunde liegen." *Abhandlungen der Kgl. Gesellschaft der Wissenschaften zu Göttingen* 13 (1867): 133–152. For English translation, see under Spivak.

Russell, Bertrand. *An Essay on the Foundations of Geometry*. Cambridge, U.K.: Cambridge University Press, 1897. Unaltered reprint: New York, Dover, 1956.

Saccheri, Girolamo. *Euclides ab omni nævo vindicatus*. Mediolani: e typographia Pauli Antonii Montani, 1733. Reprinted as *Euclides vindicatus*, with facing English translation by G. B. Halsted, New York: Chelsea, 1986.

Simplicius. *In Aristotelis de Caelo Commentaria*, edidit I. L. Heiberg. Berlin: G. Reimer, 1894.

Spinoza, Benedictus. *Ethica ordine geometrico demonstrata*, in Opera posthuma. Amstelodami: J. Rieuwertsz, 1677.

Spivak, Michael. *A Comprehensive Introduction to Differential Geometry*. 2nd ed., Berkeley: Publish or Perish. 5 vols, 1979. Includes an excellent English translation, with mathematical commentary, of Riemann's lecture "On the hypotheses that lie at the foundation of geometry"; see Vol. 2, pp. 135ff.

### suggestions for further reading

Aaboe, Asger. *Episodes from the Early History of Mathematics*. New York: Random House, 1964.

Blumenthal, Leonard M. *A Modern View of Geometry*. San Francisco: W. H. Freeman, 1961.

Boi, Luciano. *Le problème mathématique de l'espace: Une quête de l'intelligible*. Berlin: Springer, 1995.

Bonola, Roberto. *Non-Euclidean Geometry: A Critical and Historical Study of Its Development*. English translation with additional appendices by H.S. Carslaw. New York: Dover, 1955. Includes English translations of works by Bolyai and Lobachevsky.

Buroker, Jill Vance. *Space and Incongruence: The Origin of Kant's Idealism*. Dordrecht, Netherlands: D. Reidel, 1981.

Coxeter, Harold S. M. *Introduction to Geometry*. 2nd ed. New York: John Wiley and Sons, 1969.

Descartes, René. *The Geometry*. Translated from the French and Latin by D. E. Smith and M. L. Latham, with a facsimile of the first edition of 1637. New York: Dover, 1956.

Ellis, George F. R., and Ruth M. Williams. *Flat and Curved Space-Times*. Oxford, U.K.: Clarendon Press, 1988.

Heath, T. L. *The Thirteen Books of Euclid's Elements*. Translated from the text of Heiberg with introduction and commentary. New York: Dover. 3 vols.

Heilbron, J. L. *Geometry Civilized: History, Culture, and Technique*. Oxford: Clarendon Press, 1998.

Hilbert, David and S. Cohn-Vossen. *Geometry and the Imagination*. Translated from the German book *Anschauliche Geometrie* by P. Nemenyi. New York: Chelsea, 1952.

Modenov, P. S. and Parkhomenko, A. S. *Geometric Transformations*. Translated and adapted from the first Russian edition by M. B. P. Slater. 2 vols. New York: Academic Press, 1965.

Nagel, Ernest. "The Formation of Modern Conceptions of Formal Logic in the Development of Geometry." *Osiris* 7 (1939): 142–224.

Neugebauer, Otto. *The Exact Sciences in Antiquity*. 2nd ed. New York: Dover, 1969.

Rosenfeld, B. A. *A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space*. Translated from the Russian by A. Shenitzer. New York: Springer, 1988.

Torretti, Roberto. *Philosophy of Geometry from Riemann to Poincaré*. Dordrecht, Netherlands: D. Reidel, 1978.

*Roberto Torretti (2005)*

# Geometry

# GEOMETRY

**GEOMETRY** . During the last two millennia bce, the period that produced most religious texts, geometry (lit., "earth measurement," from Greek *gaia, ge*, "the earth," and *metrein*, "to measure") was essentially a "geometrical algebra" with a focus on number. Problematic allusions to number and space, which abound in sacred texts, are presumably inspired by this early mathematical protoscience. During the much later development of Christianity and Islam, Euclidean geometry—based, so it seemed, on irrefutable deductive logic built from definitions, postulates, and theorems—became the rational paradigm for all sciences, including theology. The discovery of other geometries in the last two centuries has brought the realization that Euclidean geometry is merely a special case within a wider realm. Efforts to rid mathematics of its logical paradoxes have taught that perfect consistency and certainty are unattainable in rational thought. These developments, together with a new awareness of the complexity of physical space and a better understanding of how culture shapes perceptions, have dramatically altered philosophical dogmatism, making ancient and Eastern modes of thought more congenial to the modern West and contributing to the problems of contemporary religion.

## Neolithic Cultures (6000–3500 bce)

In southeastern Europe and the Near East, Neolithic peoples decorated the surfaces of cult objects with geometric motifs—circles, ovals, parallel lines, chevrons, triangles, squares, meanders, and spirals. These abstract designs came to abound in the folk arts of most of the cultures of the globe, seemingly irrespective

of time, of the degree of civilization attained, or of concomitant skill in the realistic depiction of natural objects such as animals, human faces, leaves, and landscapes. Creation myths inherited from Mesopotamia and Egypt of the third and second millennia bce and later from Palestine, China, and Greece, as well as those recorded in modern times in the Americas, Africa, and Oceania show that the act of divine creation is universally conceived as an ordering, a shaping and selection that brings a world, a cosmos, into being. During the Neolithic period, the abstract geometrical motifs that ornament dress, vessels, walls, and other artifacts found in the earliest shrines and villages were expressions of an intuitive identification of order with the sacred and a consequent mobilization of aesthetic feeling in control of design.

In *A History of Mathematics* (1968), Carl B. Boyer observes that pottery, weaving, and basketry, from the time of their Neolithic origins, "show instances of congruences and symmetry, which are in essence parts of elementary geometry." To Boyer, "simple sequences in design"—such as translations, rotations, and reflections (see figure 1)—"suggest a sort of applied group theory, as well as propositions in geometry and arithmetic." Formal propositions did not appear, however, until the Greeks initiated them in the fifth and fourth centuries bce. Group theory was not developed until the last two centuries; only recently has it been extended to cover the geometric symmetries in space that were already a concern in Palestinian stoneware, for instance, as early as 10,000 bce. The oldest mathematical texts, dating from circa 1900 to 1600 bce, show that geometry developed historically as "the science of dimensional order," in close alliance with arithmetic and algebra, although "the 'spaciness' of space and the 'numerosity' of number are essentially different things" (Alfred North Whitehead, *An Introduction to Mathematics*, 1911).

"There is a direct correlation between complexity of weaving and sophistication of arithmetic understanding,"

Walter A. Fairservis notes in *The Threshold of Civilization* (1975). In a settlement like that at Çatal Hüyük in central Turkey, occupied before 6500 bce, the frame posts of the houses are filled in with sunbaked bricks made from molds, furnishing strong economic motivation for knowing precisely "how many bricks were necessary for each wall"; hence, "counting and notation were very much a part of the cultural scene." Far cruder artifacts—sequences of notches incised in bone, studied by Alexander Marshack (1972)—suggest "systems of lunar and other notation" that push the origins of arithmetic far back into Paleolithic times, twenty-five thousand and more years ago. Modern archaeologists and anthropologists are thus producing alternative theories to those of Herodotos (fifth century bce), who believed geometry began in Egypt, motivated by the necessity of reestablishing boundaries after the annual Nile floods, and by Aristotle (fourth

century bce), who also assumed an origin in Egypt, but because "there the priestly caste was allowed to be at leisure."

## Ancient Egypt

Chief sources of knowledge of early Egyptian geometry are the Moscow Papyrus (c. 1890 bce) and the Rhind Papyrus (c. 1650 bce). The emphasis here is always on calculation, so that their geometry "turns out to have been mainly a branch of applied arithmetic" (Boyer, 1968). The concept of geometric similarity is applied to triangles, and there is a rudimentary trigonometry. There is a good approximation to π in the formula that computes the area of a circle by constructing a square on eight-ninths of its diameter (see figure 2). In addition, the Egyptians knew the formulas for elementary volumes and correctly calculated the volume of a truncated pyramid.

Modern scholars, however, are disappointed to find so little cause for the high estimation in which the Greeks later held Egyptian science. Respect for the organizational and engineering skills required for the building of palaces, canals, and pyramids, for example, is tempered by the realization that such civic projects entail little more then what Otto Neugebauer (1969) calls "elementary household arithmetic which no mathematician would call mathematics." Neugebauer concludes: "Ancient science was the product of a very few men; and these few happened not to be Egyptian." Of far greater interest is what was happening in Babylon.

## Babylon

Several hundred baked-clay tablets about the size of the palm of the hand, incised with neatly crowded rows of cuneiform inscriptions, provide more information about the mathematical sciences in Babylon circa 1900 to 1600 bce than exists for any other place or time preceding the *Elements* of Euclid, circa 320 bce. Standard tables of multiplication of reciprocals and a place value notation on base sixty facilitated computation at a level Neugebauer compares with that of Europe in the early Renaissance, more than three thousand years later.

Babylonian geometry, like that of Egypt, was still "applied algebra or arithmetic in which numbers are attached to figures" (see figure 3). The ratio between the side and diagonal of a square (i.e., the square root of two) was computed correctly to about one part in a million (see figure 4). Ratios between the areas of a pentagon, hexagon, and heptagon and those of squares built on one side were closely approximated, as was the value of π. A geometric concept of similarity is applied to circles, and perhaps also to triangles. An angle inscribed in a semicircle is known to be a right angle (see figure 5). The Pythagorean theorem (which holds that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides) was understood in all its generality a thousand years before Pythagoras. One tablet, known as Plimpton 322, develops a set of fifteen "Pythagorean triplets" (three numbers defining right triangles, such as 3,4,5) in a sequence in which acute angles vary by approximately one degree. This "prototrigonometry," unsuspected until the tablet was translated, is one of the most astonishing mathematical achievements of the ancient world. It demonstrates empirical knowledge of a general formula:

Assuming integers *p* and *q* with *p* > *q*, and taking *p* = 2 and *q* = 1, the 3,4,5 triplet emerges as follows:

SIDE | SIDE | HYPOTENUSE |

2 pq | p2 − q2 | p2 + q2 |

2 (2 × 1) = 4 | 4 − 1 = 3 | 4 + 1 = 5 |

It was from the Babylonians, rather than from the Egyptians, that the Greeks inherited the fund of empirical insights that they transformed into an exact science. The example just cited—in which the ratio of the musical octave, 2:1, is transformed into the Pythagorean triplet 3,4,5—is the foundation of Plato's cosmogony, and it may turn out to be one of the most important clues to the numerology of the ancient world.

## Greek Transformation of Egyptian and Babylonian Knowledge

Researchers are still trying to unravel the story of how the Greeks, in less than three centuries (600–300 bce), transformed geometry—inherited essentially as an art of making arithmetical relations visible—into a science based on definitions, postulates, and theorems (and appealing therefore to an invisible *logos* ). Material inherited from Egypt and Babylon through Thales (c. 585 bce), and Pythagoras (c. 550 bce) was riddled with confusion between exactness and approximation. A new "dialectical" spirit arose with Parmenides (c. 475 bce) and his followers. Precise definition, always elusive, was the new goal that Socrates (d. 399 bce) applied to moral and ethical questions by appealing to "harmonic" examples from the Pythagorean geometry of the vibrating string. Plato's affection for dialectics and his emphasis on abstraction bore unexpected fruit in Aristotle's rejection of his teacher's Pythagorean methods: "Dialectics is merely critical where philosophy claims to know" (*Metaphysics* 1004b). Aristotelian "first philosophy" developed a new syllogistic rigor in the generation after Theaetetus (c. 414–c. 369 bce) and Eudoxus (c. 400–c. 350 bce), who changed the method and enlarged the scope of mathematics. Another generation later, Euclid's *Elements* completed the mathematical transformation so successfully that the only traces of earlier Greek mathematics that survive must be gleaned from his work. Little else was deemed worth copying.

For current understanding of geometrical symbolism in ancient religions, the Greek transformation has almost no meaning. Although Christianity and Islam are historically young enough to have been affected by the transformation, their holy books show virtually no influence of the new Greek science. (A possible notable exception is the opening line of the *Gospel of John:* "In the beginning was the Logos.") Plato, the first author in history whose works have survived intact, made an extended commentary on the mathematical bias in his philosophy and offers the richest insight into the intentionality that inserted so many mathematical elements in ancient mythology. The musical geometry at the heart of his mathematics was common to both East and West; its simplest pragmatic formula is found in China.

## Harmonic Cosmology in China and Greece

The natural, or counting, numbers (1, 2, 3, …, infinity) are a primordial image of order. Developed systematically into Pythagorean triplets, they lead to a prototrigonometry of the plane. Applied systematically to the geometry of a vibrating string, they link the magical realm of tone with the numbers that measure the world. To use economical modern concepts, the octave ratio 1:2 becomes the cyclic module (Plato's matrix, or "universal mother") in which the even numbers are "modular activities" (doubling and halving merely produce further "octave identities") and the odd numbers are "modular residues" (meaning that they define new pitches within the octave matrix). To build a scale, the simplest procedure is to follow the old Chinese rule of adding or subtracting one-third (from any reference pipe or string length). This is the geometrical analogue of the musical procedure of tuning by ear: A subtraction of one-third correlates with the musical interval of an ascending perfect fifth (3:2); an addition of one-third correlates with the descending perfect fourth (4:3). To avoid fractions in the arithmetic, the reference length must contain one factor of three for every "tone child" to be generated. The Chinese pentatonic (five-tone) scale must therefore be generated from 3;s4 = 81.

Tone | C | G | D | A | E |

Number | 81 | 54 | 72 | 48 | 64 |

Operation | −1/3 | +1/3 | −1/3 | +1/3 |

Rearranged into scale order, this number sequence has a reciprocal "twin" that defines frequency ratios:

Tone | C | D | E | G | A |

Length ratio | 81 | 72 | 64 | 54 | 48 |

Frequency | 64 | 72 | 81 | 96 | 108 |

Both Chinese and Greek cosmology are projections from this tonal geometry, reducible to continued operations with the prime number 3. Note that the defining operation started on 34 = 92 = 81. Ancient China was conceived as 1/81 part of the whole world, that is, as 1/9 of one of the nine "great continents." China was also considered to be divided into nine provinces, so that each Chinese province was 1/729 of the whole world; now 36 = 93 = 729 is the base for the same tuning calculation when it is extended through seven tones for the complete diatonic scale, also standard in China in the fifth century bce. At about the same time, Philolaus, a Greek Pythagorean philosopher, conceived the year as made up of "729 days and nights," a number that would seem to come from nowhere but such a musical cosmology. Plato linked the seven tones in this set to the sun, moon, and five planets; later, Ptolemy (second century ce) linked the scale to the zodiac.

The numbers 64 and 81, on which the alternate scale progressions commence, and the number 108, largest in the set of pentatonic frequency values, are immortalized in various ways. In China there are 64 hexagrams in the *Yi jing* divination text. The numbers 82 = 64 and 92 = 81 have been the favored squares on which to construct the Hindu fire altar since Vedic times (c. 1500 bce). The number 108, upper limit in the set, is the number of beads in the Buddhist rosary. The tuning pattern itself has recently been discovered (but without numbers) on an Old Babylonian cuneiform tablet from circa 1800 bce.

The set of twelve consecutive tones generated by the above procedure constitutes a chromatic scale. In ancient China each twelve tones in turn became the tonic of the standard pentatonic scale for a particular calendric period. Throughout Chinese history the bureau of standards remained wedded to a tonal geometry: The length of a pitch pipe (an end-blown hollow tube) sounding the reference tone determined the standard for both weights and volumes. Each new regime established a new reference pitch; today there is the record of dozens of succesives changes in the bureau of standards as the reference pitch oscillated over the range of about the interval of a sixth.

Unless one knows the musical procedure, the Daoist formula for the creation of the creation of the world sounds mystical: "The Dao [the Way] produced one, the one produced two, the two produced three, and three produced the ten thousand things [everything]." The creation myth related by Plato in the *Timaeus* similary develops the world's harmonical soul and body from the numbers 1,2, 3. Pythagoreans frankly announced that, to them, "All is number," and Aristotle quotes them as saying, "The world and all that is in it is determined by the number of three." Plutarch describes planetary ditances in the Philoaus system as "a geometrical progression with three as the common ratio." For eight hundred years Greek astronomers toyed with varations on this planetary motion (c. 1600 ce) while still looking for the right tones to associate with each celestial body. East and West, the "geometry" of heaven and earth was musical and profoundly trintarian while astronomy was being gestated.

The Greek view of this tonal-planetary geometry points in the direction of a more abstract mathematical system. The Pythagorean "holy tetractys" was a pebble pattern symbolizing continuing geometric progression from a point through a line and a plane to the "solid" dimension (see figure 6). Plato takes advantage of the double meanings of integers (both as whole numbers and as reciprocal fractions) to generate the material for a seven-tone scale at the cube dimension (33=27; see figure 7). Nicomachus (fl. c. 100 ce), writing an introduction to Plato, simplifies the double view of the multiplication table 2 × 3 up to the limit of 36 = 93 = 729 (see figure 8.) Stones, musical tones, planets, numbers, and geometry are all part of one vast Pythagorean synthesis, replete with symbolic cross-reference and a supporting mythology. An exasperated Aristotle mocked it; Euclid made it obsolete.

The problem of "excess and deficiency" with which a musician wrestles in adjusting the geometry of the string by ear had its arithmetical analogue in the ancient problem of making approximations between the areas of a circle and square and the volumes of a sphere and cube. Thus harmonics was a paradigm for an ethics of moderation; "nothing too much," the Greek ideal, had its counterpart in Confucian concepts of morality and behavior. Thousands of years earlier, the Egyptians had conceived the scales of Maat as the "great balance" on which the heart of the deceased was figuratively weighed to test its fitness for immortality. Thus the wisdom literature of the ancient world shows remarkable parallels between cultures.

The historical record is so fragmented, however, that interpretation of geometric symbols remains speculative. Modern studies in the neurophysiology of vision and in related psychological inferences suggest that schematic, geometric relations play decisive roles. The universal acceptance of the octave ratio 1:2 is further evidence of human psychophysical norms that could generate correspondence between cultures that were never in contact. While they can neither fully document the paths of cultural diffusion nor even claim that diffusion is necessary, researchers cannot entirely rid themselves of suspicion that there was a great deal more diffusion than can be proven.

## Problems in Ancient Geometric Symbols

It is easy to imagine that the ancient stone circles that abound in Europe and America linked people to events in the sky—that the twenty-eight poles arranged in a circle for the lodge of the Arapaho Indians' Sun Dance, for instance, may correspond to twenty-eight lunar mansions; that the twelve sections of the Crow tribe's lodge for that ceremony may allude to the months of the year; and that other cultures possessed similar symbols of earth, sky, and calendar (Burckhardt, 1976). But many familiar symbols are more puzzling. Why, for instance, did the Pythagoreans take a five-pointed star (see figure 9) as their special symbol? Is it because each line cuts two others in "mean and extreme ratio" (meaning that the whole lines is to the longer segment as the longer is to the shorter) so that the figure symbolizes both "continuing geometric progression" (the world's "best bonds," for Plato) and a victory over the "darkness" of the irrational? Could the Hindu "drum of Śiva" (see figure 10)—with its inverted triangles and the interlocked triangles of the star-hexagon (see figure 11), prevalent in Indian and Semitic cultures—be related to the Pythagorean symbols in figures 6 and 7?

The so-called Pascal triangle (see figure 12), containing the coefficients for the expansion of the binomial (*a+b* ) 2,3,4,5,6 was known to Pingala (c. 200 bce) as Mount Meru, the Hindu-Buddhist holy mountain. Pingala interpreted this triangle as showing the possible variations of meter built from monosyllables, disyllables, trisyllables, and so on. Could this "Mount Meru" be related to the holy mountains of other Eastern religions? Is the Pythagorean tetractys simply the Greek form of older holy mountains? Is it significant that the Sumerian symbol for mountain is a triangular pile of bricks (see figure 13), aligned in a pattern Pythagoreans found useful for numbers? Is the hourglass shape of the later Buddhist holy mountain simply a geometric variation of the "drum of Śiva"?

Translation of the ancient Babylonian mathematical texts now makes clear that the computational sophistication achieved four thousand years ago was so great that the sacred texts of all peoples must be studied with new alertness for evidence of rational—and not merely poetic—inspiration. The old Sumerian-Babylonian gods possessed straightforward numerical "nicknames" (used for scribal shorthand) in sexagesimal (base sixty) notation; the three great gods—Anu-An, whose numerical epithet was 60 (written as a large 1), Ea-Enki, associated with 40, and Enlil, associated with 50—are functional equivalents of Plato's 3,4,5 Pythagorean genetic triad. Could some of the ancient religious mythology turn out to be mathematical allegory?

It seems curious that the ancient Greek altar at Delphi is built on cubic dimensions, as are the chapel of the Egyptian goddess Leto that Herodotos saw in the city of Buto, the Vedic fire altar, the Holy of Holies in Solomon's temple, and the ancient Sumerian ark (first to "rescue a remnant of mankind from the flood"). The name of the most sacred Islamic shrine, the Kaʿbah at Mecca, literally means "cube," and the city of New Jerusalem in the *Book of Revelation* is also measured in such dimensions. All of these cubic consonances between various religions suggest that poetic religious imagination has had a geometrical, "protoscientific" component for a very long time.

## Geometry since Euclid

The thirteen books of Euclid's *Elements* culminate in a treatment of the five regular, or Platonic solids—tetrahedon, cube, octahedron, dodecahedron, and icosahedron—each with uniform sides and angles and all capable, when replicated, of closely packing three-dimensional, abstract space. Euclid mastered the transformational symmetries of these "rigid" bodies.

In the third century bce Archimedes did important work on the area of the surface of a sphere and of a cylinder and on their respective volumes, and Apollonius carried the study of conic sections (ellipse, parabola, and hyperbola) to its highest development. Later Greek authors made further

advances in geometry, but the great wave of development that had begun scarcely four hundred years earlier was spent.

Later Hindu talent has been mainly arithmetic and algebraic; theorems on the areas and diagonals of quadrilaterals in a circle were contributed by Brahmagupta (c. 628 ce). Important Arab contributions to the solution of cubic equations by the method of intersecting conics were summarized by ʿOmar Khayyam (c. 1100 ce). European development has been rapid since Kepler's time; it was Kepler who introduced the concepts of the infinitely great and the infinitely small, which Euclid had carefully excised from consideration. The invention of analytic geometry by Pierre de Fermat and René Descartes in the seventeenth century led to a new integration of geometry and algebra.

The contributions of Euclid's *Elements* to the invention and development of the physical sciences during all these centuries is inestimable. His kind of logical, geometrical argument is the basis for Archimedes' formalization of the laws

of the lever; for the Greek development of astronomy as a physical science (by Hipparchus, Ptolemy, and others); for Galileo's work on the dynamics of the inclined plane; for Kepler's laws of planetary motion; for Newton's laws of planetary gravitational dynamics; and for an endless host of related physical sciences. Euclidean geometry, still being improved today, has thus been one of the Western world's most powerful engines of progress. The *Elements* has been "the most influential textbook of all times" (Boyer, 1968), and it was long assumed to be as certain a guide to geometry as the Bible to absolute truth. God, it was confidently asserted, is a geometer.

During the nineteenth century, which Boyer calls the "golden age of geometry," this almost-perfected world of traditional, Euclidean-inspired mathematical physics exploded with a creative energy, leading to a crisis in the very foundations

of mathematics. For example, one of the several new geometries that appeared, projective geometry, has nothing to do with measurement. Several "non-Euclidean" geometrics, by N. I. Lobachevskii, Wolfgang Bolyai, and G. F. B. Riemann, omit Euclid's famous sixth postulate concerning parallel lines, always somewhat suspect, to create equally logical geometries of even wider generality.

The favored status of Euclidean geometry has evaporated. There are no longer any assumptions that command universal assent, no systems of logic powerful enough to validate themselves even in mathematics. Mathematics has thus lost some of its certainty even while multiplying its powers. Einstein's notion of "space-time," with its curvatures and paradoxes, has relegated Euclidean geometry to the role of a convenient tool for certain modern intuitions within a severely limited local space. Space perception itself has been proved culturally biased; intuition can be taught new tricks. Space has also multiplied its hallowed three dimensions beyond any possibility of imagining; today the number of its dimensions is the number of independent variables in various formulas.

With the shattering of mathematical certainty and the ending of the idolatry of Euclidean rationalism, Western scientists in particular have felt a new attraction to the philosophical skepticism of Vedic poets and to Daoist and Buddhist feeling for how the world behaves. The ancient worldview was created in imagination; today imagination is still proving more powerful than logic. In the ferment of this present age, mathematics and physics are committed, perhaps more clearly than ever—but as an act of faith rather than of reason—to the primordial affection for symmetry, guided less by reason than by aesthetic feeling for elegance and beauty.

## See Also

Circle; Numbers; Quaternity; Sacred Space.

## Bibliography

For a basic textbook in Euclidean geometry and in the various non-Euclidean modern geometries, see H. S. M. Coxeter's *Introduction to Geometry* (1961; reprint, New York, 1969), notable for the elegant compression it achieves by rigorous pursuit of symmetry and of the "group" of transformations this includes. The history of geometry is set within the wider context of the whole of mathematics and the contributions of nations and individuals treated with great fairness in Carl B. Boyer's *A History of Mathematics* (New York, 1968). On the early period, Otto Neugebauer's *The Exact Sciences in Antiquity*, 2d ed. (New York, 1969), has become a classic. The transformation of Egyptian and Babylonian empirical knowledge into Greek science is studied in B. L. van der Waerden's *Science Awakening*, translated by Arnold Dresden (New York, 1963). Richard J. Gillings's *Mathematics in the Time of the Pharaohs* (Cambridge, Mass., 1972) is an effort to balance the somewhat negative views of mathematicians toward Egyptian science.

A splendidly illustrated study of Paleolithic stone and bone counting records is Alexander Marshack's *The Roots of Civilization* (New York, 1972). Neolithic symmetry in pottery and weaving designs is richly illustrated in Marija Gimbutas's *The Goddesses and Gods of Old Europe, 6500–3500 b.c.* (Berkeley, Calif., 1982). Modern concepts of symmetry are formalized, with great respect toward ancient craftsmen who explored it intuitively, in Hermann Weyl's *Symmetry* (Princeton, N.J., 1952).

Applications of traditional geometry to religious art and architecture can be found in Robert Lawlor's *Sacred Geometry: Philosophy and Practice* (New York, 1982) and Titus Burckhardt's *Art of Islam: Language and Meaning*, translated by J. Peter Hobson (London, 1976). A stone-by-stone analysis of the geometry employed by the several builders of a great cathedral is masterfully displayed in John James's *The Contractors of Chartres*, 2 vols. (Beckenham, U.K., 1980).

For a development of the tonal-geometrical symbolism hidden in ancient religious and philosophical texts, see my books *The Myth of Invariance* (New York, 1976), *The Pythagorean Plato* (Stony Brook, N.Y., 1978), and *Meditations through the Quran* (York Beach, Maine, 1981).

### New Sources

Gray, Jeremy. *Ideas of Space: Euclidean, Non-Euclidean and Relativistic.* New York, 1989.

Lachterman, David Rapport. *The Ethics of Geometry: A Genealogy of Morality*. New York, 1989.

Magnani, Lorenzo. *Philosophy and Geometry: Theoretical and Historical Issues*. Boston, 2001.

Mlodinow, Leonard. *Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace.* New York, 2001.

Romano, David Gilman. *Athletics and Mathematics in Corinth: The Origins of the Greek Stadion*. Philadelphia, 1993.

Szulakowski, Urszula. *The Alchemy of Light: Geometry and Optics in Late Renaissance Alchemical Illustration.* Symbola and Emblemata Studies in Renaissance and Baroque Symbolism, vol. 10. Leiden, 2000.

**Ernest G. McClain (1987)**

*Revised Bibliography*

# Geometry

# GEOMETRY.

While the origins of geometry are likely to remain a matter of pure speculation, the elaborate written cultures of ancient Egypt and Babylon provide a wealth of information about the uses of geometry. Area and volume measurements abound in work connected with taxation, the provision of cities, and large-scale building works. Sometimes the Babylonians' evidence (which survives because they wrote on durable clay tablets) spills over into purer matters, and reveals methods for finding the areas of circles, and an impressive calculation of the length of the diagonal of a unit square. The so-called Pythagorean Theorem for right-angled triangles was used to find sides and diagonals of rectangles, and approximate methods for finding square roots. Other tablets display a cut-and-paste method for dealing with questions that could be formulated as quadratic equations—the origins of the method of completing the square—that depends for its validity on a certain amount of elementary geometry.

## Antiquity and the Middle Ages

Unfortunately there is little evidence of the transmission of geometrical knowledge from either Egypt or Babylonia to the emerging Greek culture. Significantly, the Greeks seem to have been interested in proof, and the nature of mathematical knowledge, in a way that these other cultures were not. Plato's dialogues display these features in a dramatic way. In the *Meno,* for example, Plato has Socrates ask a slave boy about the diagonal of a square. What Socrates elicits is a comparison between the square of the diagonal and the square on the side; not a numerical answer, and not an approximation to 2, but an argument accompanied by a proof.

The advent of proof permitted an important discovery: 2 is what we would call an irrational number: there are no integers *p* and *q* such that 2 *p/q.* Historians used to present this discovery as momentous. Allegedly the mathematics of earlier, Pythagorean, times was based on the idea that everything could be counted, any two lengths could be regarded as multiples of a common measure. The incommensurability of the side and diagonal of a square put an end to that belief and caused a crisis in Greek mathematics. In the late twentieth century, however, historians retreated from this position. The only evidence for it is very late, and no contemporaneous evidence suggests a crisis. Rather, as Plato's dialogues suggest, there might have been surprise and excitement. The slave boy, after all, gave an acceptable answer. The existence of incommensurable pairs of lengths greatly complicated the theory of proportion, which is attributed to Eudoxus of Cnidus (c. 400–c. 350 b.c.e.) and presented in books 5 and 6 of Euclid's (fl. c. 300 b.c.e.) *Elements,* but again there is no suggestion of a crisis.

*The paradoxes of Zeno.*

Further evidence of the sophistication of Greek thought is found in Zeno of Elea's (c. 495–c. 430 b.c.e.) paradoxes, which survive only in the form of a somewhat hostile account by Aristotle (384–322 b.c.e.). Zeno aimed to show that the analysis of motion led inevitably to contradictions. Achilles may never catch a tortoise, because each time he runs to where the tortoise was it is still ahead, committing Zeno to an infinite chase. Indeed, by a somewhat similar argument he cannot get started. An arrow cannot move in an instant; therefore, it is at rest in every instant of its flight and therefore always at rest. Whatever Zeno's intention in proclaiming them, his paradoxes testify to a deep-seated interest in logical reasoning, and they continued to attract interest.

*The notion of proof.*

Much of Greek mathematics would be impossible without good notions of proof. The simplest form of proof was proof by showing, in which arrangements of pebbles were used to show such results as the sum of two odd numbers is even. Zeno's paradoxes display another form of reasoning, called *reductio ad absurdum,* in which a proposition is refuted by showing that it leads logically to a self-contradiction or other evident impossibility. This method was used extensively by Archimedes in his estimation of areas and volumes, and also earlier by Euclid in his *Elements,* for example when he showed that there are infinitely many prime numbers. For, if there are not, then there are only finitely many prime numbers, *p* _{1}, *p* _{2}, …, *p* _{n} say, in which case the number *p* _{1} *p* _{2} … *p* _{n} 1 is larger than any of these, so it cannot be prime, and yet it is divisible by none of them, so it must be prime.

Proofs in geometry turn approximate estimates based on a finite number of cases into certain knowledge. For example, the assumptions made at the start of Euclid's *Elements,* including the parallel postulate as described below, permitted Euclid to prove that the angle sum of a triangle is exactly two right angles by exhibiting a suitable pair of parallel lines, to prove Pythagoras's theorem by moving areas around, and, ultimately, to show that there are exactly five regular solids.

*Euclid's* Elements *and the axiomatization of geometry.*

The most impressive form of proof, however, in Greek mathematics is the axiomatic method, developed at length in Euclid's *Elements.* The aim, not perfectly honored but impressively so, was to state definitions of the fundamental terms, gives rules for what may be said about them, and then to derive truths successively from this base of assumptions (the axioms). The result is that later propositions in each book of the *Elements* depend in an elaborate, tree-like way, on the earlier ones, and confidence in these results depends on the transparency of the proofs and the quality of the original axioms.

*Apollonius and Archimedes.*

One of the intellectual high points of Greek mathematics is undoubtedly Apollonius of Perga's (c. 262–c. 190 b.c.e.) theory of conic sections. It is forbiddingly austere, but it goes a long way to creating a unified theory of all (nondegenerate) plane sections of a cone: the ellipse, parabola, and hyperbola. The names derive from the way their construction is shown to produce an area that falls short, is equal to, or exceeds another area (compare the terms for figures of speech: elliptical, or of few words; a parable is exact, hyperbole an exaggeration). The comparisons of areas yield a proportion, which is modernized as the equation of the curves, and Apollonius shows in some detail how the equation may be simplified by suitable geometric choices and how properties of the conic sections may be obtained, such as the focal properties of conics and the construction of tangents.

Archimedes (c. 287–212 b.c.e.), a near contemporary of Apollonius, has earned a reputation as the greatest of the Greek geometers not only for the brilliance of his achievements, but also perhaps because they are easier to admire. He found volumes of sections of cones and various solid figures, he was the first to show that the constant () that enters the formulas for the circumference and the area of a circle is in fact the same, and he also made a number of practical and mechanical discoveries. He also left a unique account, known as the Method, of how he came to some of his discoveries by heuristic means, regarding areas as made up of lines that could be moved around. A tenth-century copy of this account was discovered in 1906 in a monastery in Istanbul. It was then lost again, but reappeared in 1998, when it was put on auction and sold for the surprisingly small sum of $2.2 million.

*Arabic and Islamic work on geometry.*

Islamic scholars did much more than simply transmit Greek ideas to the later West, as some accounts have suggested. They far surpassed all previous cultures in geometric design. They also produced the most penetrating analyses of the single most obvious weakness in all of Euclid's *Elements* : the parallel postulate. Euclid had assumed that if two lines *m* and *n* cross a third, *k,* and the angles and the lines *m* and *n* make with *k* are less than two right angles on one side of the line (in the figure 2 right angles) then the lines will meet on that side of the line if they are produced sufficiently far (see fig. 1).

Unlike all Euclid's other assumptions, the parallel postulate makes claims about what happens arbitrarily far away and so could be false. However, very few theorems can be proved unless the parallel postulate is known, so mathematicians were in a quandary. Greek and still more Islamic commentators took the view that it would be better to drop the parallel postulate from the list of axioms, and to derive it instead from the other axioms as a theorem.

Remarkably, from Thabit ibn Qurrah (c. 836–901) to Nasir ad-Din at-Tusi (1201–1274), they all failed. To give just one example, Ibn al-Haytham (Alhazen; 965–1039) assumed that if a segment of fixed length and perpendicular to a given line moves with one endpoint on the line then the other end point draws a straight line, parallel to the given line. Certainly, the parallel postulate follows as a theorem if one may make this assumption, and the parallel postulate implies it, but this only invites the question: how can the assumption itself be proved, or is it merely an alternative assumption to the parallel postulate? Some years later, Omar Khayyám (1048?–?1131) objected to the assumption on just these grounds, arguing that it was an illegitimate use of motion in geometry to attempt to define a curve this way, still more to assume that the curve so produced was a straight line.

## Modern Era

Significant Western interest in mathematics ebbed for a long time during and after the Roman Empire, before flowing at times in the Middle Ages. Only in the sixteenth century did a continual process of growth begin, aided by the rediscovery of Greek and Arabic texts and the publication of editions of Euclid's *Elements* and the works of Apollonius and Archimedes. At the same time, the discovery of the method of single-focused perspective transformed first architecture and then the practice of painting, where it produced a dramatically heightened realism. The technique proved eminently teachable, although few painters apart from Piero della Francesca (c. 1420–1492) also understood the mathematics involved.

*Analytic and projective geometry.*

Girard Desargues (1591–1661) brought together projective ideas from both architecture and painting to create the first fully unified theory of conic sections (all nondegenerate conic sections are projections of a circle). This theory naturally highlights those aspects that are projective (such as tangency questions) and it led directly or indirectly to a number of novel discoveries over the next century before it petered out. It was then rediscovered by Gaspard Monge (1746–1818) and Jean-Victor Poncelet (1788–1867) at the time of the French Revolution. In the form of simple horizontal and vertical projections it became the core technique of descriptive geometry or engineering drawing, a mainstay of French mathematical education throughout the nineteenth century, and, of course, it is still in use in the early twenty-first century.

Poncelet's breakthrough at the start of the nineteenth century was to see that, for many geometric properties a curve is equivalent to any of its "shadows" (its images under central projection). His own way of doing this was not found to be acceptable by later mathematicians, but Michel Chasles (1793–1880) in France and August Ferdinand Möbius (1790–1868) and Julius Plücker (1801–1868) in Germany all independently found more rigorous ways of making his insight work, and the resulting subject of projective geometry became the fundamental geometry of the nineteenth century. Although the details remained obscure for some time, the key point was that projective geometry discussed geometric properties of figures that do not involve the concept of distance. Any theorem in projective geometry is true in Euclidean geometry, but not vice versa, and so projective geometry is more basic than Euclidean geometry.

Desargues's contemporary, René Descartes (1596–1650) was much more successful with a work that was ruthlessly modern in its approach, and entirely eclipsed earlier attempts. Descartes took contemporary algebra, rewrote it in simpler notation, and proceeded to solve geometric problems by recasting them in algebraic terms and solving them by algebraic means, then reinterpreting the solution in geometric terms. Typically, the algebraic solution will be a single equation between two unknowns. Descartes interpreted this as defining a curve, and gave an elaborate discussion of how, given an equation, the corresponding curve can be drawn. He published his ideas as an appendix (entitled *La géométrie* ) to a longer philosophical work in 1637.

Descartes did not explain the more elementary parts of his approach. This was done by a number of Dutch scholars who came after him, and the study of geometry by means of algebra (Cartesian, analytic, or coordinate geometry) was swiftly established. It took about a century for mathematicians to decide that the algebraic equation was an acceptable answer to a geometric problem, and to drop Descartes's search for geometric constructions, but the idea that geometric figures are naturally and fruitfully described by algebra has remained central to much of mathematics ever since.

*Differential geometry.*

Differential geometry, on the other hand, began as the study of curves and surfaces where the calculus is allowed. It is connected to such questions as: when a map of the earth's surface (assumed to be a sphere) is made on a plane, what geometric properties can be preserved? The decisive reformulation of the early nineteenth century came when Carl Friedrich Gauss (1777–1855) investigated the curvature of surfaces in space. The curvature of a surface at a point (and generally the curvature of a surface varies from point to point) is a measure of the best fitting sphere, plane, or saddle at that point (see figs. 2 and 3).

Gauss found that this quantity is intrinsic: it can be determined by measurements taken in the surface itself without reference to the ambient Euclidean space. This property was so unexpected Gauss called the result an exceptional theorem.

*Gaussian curvature and the emergence of non-Euclidean geometries.*

After Gauss's death it emerged that he, alone of the mathematicians of his time, had had some sympathy with efforts to show that Euclidean geometry was not the only possible geometry of space, and indeed his astronomer friends Friedrich Wilhelm Bessel (1784–1846) and Heinrich Wilhelm Matthäus Olbers (1758–1840) had also inclined that way. This leads back to the question of the parallel postulate.

Around 1830 János Bolyai (1802–1860) in present-day Hungary and Nicolai Ivanovich Lobachevsky (1792–1856) in remote Kazan in Russia, wrote down and published detailed accounts of what a geometry would look like in which the parallel postulate was false and the angle sum of a triangle is always less than two right angles (reprinted in English translation in Bonola, 1912). Although independent, their work is remarkably similar and can be described together. They studied geometry in two and three dimensions, and found formulas for triangles in the plane analogous to the formulas of spherical trigonometry for triangles on the sphere. These new formulas showed that small regions of the new geometry differed only slightly from small regions of the Euclidean plane, thus explaining why the new geometry had not been noticed hitherto, but they also showed what many a previous defence of the parallel postulate had hinted at—that the new geometry was different from Euclidean geometry in many respects.

Such was the novelty of Bolyai's and Lobachevsky's work that few read it and the published responses to it were extreme in their hostility. Most people instinctively found it incoherent; they "knew" it was wrong but were not willing to find out where. Gauss, however, wrote to Bolyai to say that he agreed with János's presentation but implying that he knew it all already (a claim for which there is little surviving evidence). János was so enraged he never published again. In 1840 Gauss nominated Lobachevsky to the Göttingen Academy of Sciences, but did nothing else to promote the new geometry. The result was that both men died without getting the acclaim their discoveries undoubtedly merited.

*Riemann's generalization for spaces of higher dimensions.*

The hegemony of Euclidean geometry came to an end not with the discoveries of Bolyai and Lobachevsky, but in stages, starting with Riemann's wholly novel approach to geometry that severely undercut it. Bernhard Riemann (1826–1866) was a student of mathematics at the University of Göttingen in the mid-1850s. In his postdoctoral thesis he set out the view that geometry was the study of any "space" of points upon which one could talk about lengths, and he indicated a variety of ways in which the techniques of the calculus could do such service. This is a rather natural and elementary idea, the problems come in spelling out the details in any useful way.

Riemann concentrated on intrinsic properties of the space, such as Gauss's idea of the curvature of a surface, and he noted that there would be different geometries on spaces with different intrinsic properties. That includes spaces of different dimensions, and also spaces of dimension two, say, but different curvatures. What it does not do is say that Euclidean space of some dimension is the source of geometric concepts, thus Euclidean geometry is overthrown.

Riemann's thesis was published posthumously in 1867, just in time to resolve the doubts of an Italian mathematician, Eugenio Beltrami (1835–1899), who had come to some of the same ideas. He immediately published his reworking of the geometry of Bolyai and Lobachevsky as the geometry of a surface of constant negative curvature, of which he had a description in a disc of unit (Euclidean) radius. Beltrami's map and Riemann's philosophy of geometry convinced mathematicians, but not all philosophers, of the validity of non-Euclidean geometry, as the Bolyai-Lobachevsky geometry became known.

## Twentieth Century

There, curiously, Riemann's ideas remained for more than a generation. There was some interest in novel three-dimensional geometries, almost none in geometries, in Riemann's sense, of higher dimensions, except to show that mechanics could be done in such a setting, and in simplifying the formidable algebraic complexity of the subject (today handled by means of the tensor calculus). The decisive change came with the work of Albert Einstein (1879–1955).

Einstein's special theory of relativity of 1905 was a thorough reworking of the mathematics of motion, and at first Einstein was unsympathetic to the geometrical reworking given to his ideas by Hermann Minkowski (1864–1909) in 1908. But when Einstein started to think of a general theory of relativity that would find an equivalence between forces and accelerations, he found the ideas of Riemannian differential geometry invaluable. The theory he came to in 1915 formulated gravitation as a change in the metric of space-time. On this theory, matter changes the shape of space, which is how it exerts its attractive force.

*Felix Klein's Erlangen program.*

By the 1870s, projective geometry had established itself as the fundamental geometry, with Euclidean geometry as a special case, along with some other geometries not described in this essay. The young German mathematician Felix Klein (1849–1925) then proposed to unify the subject, by treating projective geometry as the main geometry, and deriving the other geometries as special cases. Every geometry Klein was interested in, most strikingly non-Euclidean geometry, was defined on projective space or a subset of it, and the relevant geometric properties were those kept invariant by the action of a suitable subgroup of the group of all projective transformations. This view, called the Erlangen Program, after the university where Klein first published it, has remained the orthodoxy since the 1890s, when Klein republished it, but in its day the novelty was the explicit introduction of the then-novel concept of a group, and the shift of attention from properties of figures to the idea that these properties are geometric precisely because they are invariant under the appropriate group of transformations.

*Italians, Hilbert, and the axiomatization of geometry.*

Klein's geometries do not include many of the geometries Riemann had pointed toward. It included only those that have large groups of transformations, which, however, is most of those of interest in physics and much of mathematics for a long time, including, it was to transpire, Einstein's special theory of relativity. The first step beyond what Klein had done, and for that matter Riemann, was proposed by David Hilbert (1862–1943), starting in 1899, although a number of Italian geometers had had similar ideas in the decade before.

Hilbert was insistent that theorems in geometry should only use the properties of objects that entered into their definitions, and to this end he formulated elementary geometry carefully in terms of five families of axioms. What distinguished his work from his Italian predecessors was his insistence that there was an interesting new branch of mathematics to be explored, which studied axiom systems. It would aim at showing the independence of certain axioms from others, and establishing what sorts of axioms were needed to deduce particular results. Whereas the Italian mathematicians had mostly aimed at axiomatizing projective and perhaps Euclidean geometry once and for all, Hilbert saw the axiomatic method as both creative and of wide applicability. He even indicated ways in which it could work in mathematical physics.

By 1915, the axiomatization of geometry had begun to spread to other branches of mathematics as well, with a consequent improvement in the standards of formal proof, and Einstein's general theory of relativity had similarly improved the ideas about the applications of geometry.

*See also* ** Cosmology: Cosmology and Astronomy ** ;

**;**

*Greek Science***;**

*Islamic Science***;**

*Mathematics***;**

*Physics***.**

*Relativity*## bibliography

Berggren, J. Len. *Episodes in the Mathematics of Medieval Islam.* New York: Springer-Verlag, 1986.

Bonola, R. *History of Non-Euclidean Geometry.* Translated by H. S. Carslaw, preface by F. Enriques. Chicago: Open Court, 1912.

Bos, Henk. J. M. *Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction.* New York: Springer-Verlag, 2001.

Field, Judith V., and Jeremy J. Gray, eds. *The Geometrical Work of Girard Desargues.* New York: Springer-Verlag, 1987.

Fowler, David H. *The Mathematics of Plato's Academy: A New Reconstruction.* 2nd ed. Oxford: Clarendon, 1999.

Gray, Jeremy J. *The Hilbert Challenge.* Oxford: Oxford University Press, 2000.

——. *Ideas of Space: Euclidean, Non-Euclidean, and Relativistic.* 2nd edition. Oxford: Oxford University Press, 1989.

Hilbert, David. *Foundations of Geometry.* 10th English edition, translation of the second German edition by L. Unger. Chicago: Open Court, 1971.

Høyrup, Jens. *Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin.* New York: Springer-Verlag, 2002.

Jaouiche, K. *La théorie des parallèles en pays d'Islam: Contributions à la préhistoire des géométries non-euclidiennes.* Paris: Vrin, 1986.

Robson, Eleanor. "Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322." *Historia Mathematica* 28 (2001): 167–206.

Salmon, Wesley C., ed. *Zeno's Paradoxes.* Indianapolis: Bobbs-Merrill, 1970.

*Jeremy* *Gray*

# Geometry

# Geometry

*Rita Dove 1980*

This poem was published in Dove’s first complete book of poems, *The Yellow House on the Corner*, in 1980. “Geometry” like other poems in the same volume, explores the dynamic between knowledge and imagination. Through an evolving series of increasingly surprising and fantastic dramatic images, the poem takes the reader on a swift and fanciful excursion from indisputable knowledge (“I prove a theorem”) to the realm of imagination. The speaker of the poem seems to be suggesting that the very act of attempting to impose intellectual certainty results in the unleashing of a mysterious, and ultimately wonderful, transformative force. The geometrical “house” immediately “expands” from what is known and certain, and suddenly the speaker is no longer protected, but is “out in the open.” The windows, those framed devices through which the speaker observes the external world, “jerk free” and hinge “into butterflies” in a transformation from rational thought to imagination. The poet seems to be saying that where intellect and imagination “intersect” there is “sunlight,” or enlightenment, and that, in the end, it is the imagination that is “true and unproven.”

## Author Biography

Born in 1952 in Akron, Ohio, to well-educated parents, Dove is the daughter of Ray A. Dove, the first African-American chemist to break the racial barrier in the tire and rubber industry, and the former

Elvira Elizabeth Hord. An excellent student, Dove was invited to the White House in 1970 as a Presidential Scholar, ranking nationally among the best high school students of the graduating class of that year. She earned a bachelor’s degree from Miami University of Ohio in 1973—where she had enrolled as a National Achievement Scholar—and graduated summa cum laude. The following year, Dove studied at West Germany’s Tubingen University as a Fulbright scholar. This led to further studies at the Iowa Writers’ Workshop. There she met her husband, the German-born writer and journalist Fred Viebahn. In addition to her other achievements, which include fellowships from the National Endowment for the Arts, the Guggenheim Foundation, and the Andrew W. Mellon Foundation, Dove holds the distinction of having been the first African American, as well as the youngest individual, to hold the post of United States Poet Laureate, a position she held from 1993 to 1995. Dove lives with her husband and daughter in Charlottesville, Virginia, where she is professor of English at the University of Virginia Commonwealth.

### Media Adaptations

- Rita Dove, Maya Angelou, and S. E. Hinton are featured on a 1999 video from Films for the Humanities, entitled
*Great Woman Writers*. - Journalist Bill Moyers presents an in-depth look at Dove’s life and her writings in
*Poet Laureate Rita Dove*, a one-hour videocassette produced in 1994 and released by Films for the Humanities. It was originally broadcast on PBS as part of the*Bill Moyers’ Journals*series. - Rita Dove was the executive producer for
*Shine Up Your Words*, a 1994 television program meant to introduce students to poetry. It is available from Virginia Center for the Book, in Richmond, Virginia. *New Letters*magazine produced the radio series*New Letters on the Air*. This series is available on audiocassette, including #305,*Rita Dove*, which features the author reading and discussing her poems in 1985.

## Poem Text

I prove a theorem and the house expands:

the windows jerk free to hover near the ceiling,

the ceiling floats away with a sigh.

As the walls clear themselves of everything

but transparency, the scent of carnations 5

leaves with them. I am out in the open

and above the windows have hinged into

butterflies,

sunlight glinting where they’ve intersected.

They are going to some point true and unproven.

## Poem Summmary

### Line 1

In this first line, Dove sets the stage for the rest of the poem. The speaker asserts indisputable rational knowledge (“I prove a theorem”) and immediately a mysterious force is set in motion (“the house expands”).

### Lines 2–3

In these lines, inanimate objects, which are the product of rational thinking, take on living and even human characteristics: “the windows jerk free to hover near the ceiling,” and “the ceiling floats away with a sigh.” This attribution of human characteristics to inanimate objects is known as personification.

### Lines 4–6

In these lines, the mysterious force that dismantles everything that is known and certain continues. The walls disappear, “the scent of carnations leaves with them,” and suddenly the speaker is no longer protected: “I am out in the open.” The use of carnations may suggest a celebration of moving from one level of knowledge to another.

### Lines 7–9

In the final tercet, the transformation is complete: “above the windows have hinged into butterflies.” Windows, rationally constructed frames of perception, have been transformed into living creatures of the imaginative realm, “sunlight glinting where they’ve intersected.” The poet seems to be suggesting that where rational thought and imagination intersect there is enlightenment. These imaginatively transformed creatures “are going to some point true and unproven.”

## Themes

### Order and Disorder

Geometry is the branch of mathematics devoted to understanding physical space in terms of logical theorems. In Rita Dove’s poem “Geometry” human beings’ ability to understand the world in terms of logic is viewed as a mixed blessing. In the first stanza, the expansion of a house can be taken as a symbol that the intellect has conquered the limitations of the physical world, making what is there bigger and better. The poem’s speaker seems to control the dimensions of the house by understanding them. Up to this point, humanity’s ability to understand the principles of order that already exist in nature is presented as a marvelous skill because it has not only made the house possible but has also improved beyond its original sense of order, creating this expansion.

By the second stanza, however, the poem raises doubts about the overall worth of geometric order. It shows the things that are lost when there is too much importance placed on logical understanding. The walls “clear themselves,” presumably of art works that have been hung on them, which relies

### Topics for Further Study

- Rewrite an existing geometric proof, explaining all of the steps in the proof in your own words.
- Dove has said in interviews that poetry is the meeting of words and music. Explore the relationship between music and geometry and explain it in a poem.
- Research the ways in which butterflies have changed and relocated in the past thirty or forty years to adapt to the growth of the human population. Report on their fate: Are they becoming extinct or “going to some point true and unproven”?
- Of all flowers, Rita Dove chose to note that it was “the scent of carnations” that disappeared when the theorem was proven. What are the associations that people have with carnations? Talk to a florist and then make a chart of the characteristics of carnations that might be ruined by excessive logic.
- Find an event that occurs in everyday life that you find mysterious and develop your own theorem to explain it. Try to follow the form of a mathematical proof in your explanation.

on a sense of disorder that has no place in logical theorems. Flowers then lose their scent because their fragrance does not fit into geometric equations. The joys of life are the disorderly and illogical ones, which cannot be appreciated when humans focus strictly on their ability to create order.

In the end, the poem finds a peaceful compromise between order and disorder by observing that the untidy elements that give life pleasure can never be completely deadened by theorems but will always be able to escape them. The windows, made by humans with the help of geometry, have some element to them that makes them as natural and free as butterflies, with the sunlight shining off them in a way that is aesthetically pleasing but not measurable by geometry. The last line refers to “some point true and unproven,” expressing the confidence that the natural world has its own order that exists independently of the geometric sense of order.

### Beginning and Ending

In a world that thinks that logic is the only really important thing, the proof of a geometric theorem might be considered an end unto itself. The proof may be the start of a different road of scientific inquiry, as scientists and mathematicians apply the information from the theorem to some practical use, but that one particular theorem has been proven, marking an end to a line of inquiry. In this poem, though, Dove presents the proof of the theorem as the beginning of the physical world’s independence. Abstract thought, such as geometry, has been seen as confining the essence of nature in the past, but this poem shows that nature’s essence can never be captured in a theorem. It is a never-ending resource. As many times as humans can create logical models of the physical world, the world has even more mysteries that go beyond all logic. Just when it might seem that geometry has made the pleasures of art and flowers vanish, as depicted in the poem, the physical world asserts itself again.

In this poem, man-made windows are no more contained by logic than are butterflies. Both have “unproven” qualities about them that go beyond their mathematical qualities, which is why the poem presents them, in the end, as escaping. As Dove presents it, no one logical proof can offer complete understanding of the physical world, but instead it represents the start of a new line of inquiry in the quest for knowledge about reality, which is constantly elusive.

### Absurdity

This poem presents a struggle against the constraints of logic. It is a warning that the clearly defined view of the world that is sought by mathematics is too limited, because it only presents a small segment of reality. To make her readers think about reality in ways that go beyond logic, Dove presents them with a sense of reality that is unfamiliar and unexpected. By weaving absurd notions throughout the poem, she is able to counter the human predisposition for logic with the equally strong tendency toward imagination.

Of course, it is absurd to state that a mental act like proving a theorem can cause a physical result like making a room expand, but it is exactly the absurdity of such a statement that forces readers to reconsider the situation being described. Describing a natural and predictable physical reaction would not pique readers’ curiosity: when Dove describes things that could not happen, she challenges her readers’ assumptions about what they do and do not know. Mathematical equations do not make windows float or walls turn transparent, but the poem does raise the issue of how these imaginary consequences resemble the actual goals of geometric proofs.

## Style

“Geometry” is a contemporary American narrative poem. It is like traditional, formalist poetry only in its organization into stanzas. The stanzas are of equal length of three lines each known as tercets; this organization conveys a sense of geometrical symmetry even though three is an uneven number. The poem employs no formal rhyme scheme. It is written in free verse, which means it uses no set pattern of meter, but contains its own unique accents and rhythms. The poet chooses consciously where to break the lines, and does so to produce the sounds that make its ultimate rhythm.

## Historical Context

### Euclidean Geometry

Most principles of geometry upon which mathematicians base their work today—and for the past twenty-three centuries—are related to the theories and methods first recorded around 300 B.C.E. by the Greek writer Euclid. His comprehensive work on mathematical theory, *The Elements*, was probably heavily based on the work of his predecessor Eudoxus, who had been a student of the philosopher Plato. Euclid refined Eudoxus’s theories, along with geometric principles that were the results of generations of mathematicians. His *Elements*, written in Egyptian Alexandria, has been a central influence for twenty-three centuries, from the Hellenistic world after the conquest of Alexander the Great to the Roman Empire, to the Byzantine Empire, the Islamic Empire, into the medieval world and on to today.

*The Elements* is a comprehensive treatise that brings together geometry, proportion, and number theory, tying them all into one complete theory for the first time. It is divided into thirteen books. The first six are about geometry. At the heart of Euclid’s geometry are five postulates. A postulate is a rule that is assumed to be true and does not have to be proved, as opposed to a theorem, which needs

### Compare & Contrast

**1980:**The United States Department of Education is developed, comprised of a staff of seventeen thousand full-time employees.**Today:**Some people feel that the centralized Department of Education should be disbanded because it cannot adequately understand local issues that affect schools’ environments.**1980:**A study by UCLA and the American Council on Education finds that college freshmen express more interest in money and power than at any time in the past fifteen years. It is the beginning of a period that came to be known as The “Me” Decade.**Today:**After a long period of economic stability in the 1990s, many students take economic stability for granted. Colleges are seeing renewed interest in careers that are not focused on accumulating wealth, such as mathematics and poetry.**1980:**Humanity’s understanding of the universe expands with the findings of Voyager I, an unmanned space craft that made new discoveries about Saturn’s moons as part of its three-year, 1.3 billion-mile journey.**Today:**Plans are underway to send two unmanned space crafts to Pluto, the farthest planet in our solar system.**1980:**The United States Supreme Court finds, in the case of*Diamond v. Chakrabarty*that a man-made life form—specifically, a microorganism that could eat petroleum in cases of spills—can be patented.**Today:**Biotechnology and genetic technology are growing scientific fields and lucrative sectors of the stock market.

proving. Euclid’s first three postulates have to do with construction. For instance, the first one states that it is possible to draw a straight line between any two points. The second and third postulates deal with defining straight lines and circles. The fourth postulate states that all right angles are equal. The fifth postulate was to become a challenge to the mathematical community for centuries to come. It states that two lines are parallel if they are intersected by a third one with identical interior angles. This postulate assumed many facts about parallel lines continuing on for infinity. Euclid himself was said to be uncomfortable with the absolute truth of this statement and declared it to be a given truth only after some hesitation. Its acceptance was a factor that defined a set of geometric theories as Euclidean geometry.

### Non-Euclidean Geometry

For centuries, mathematicians tried either to prove Euclid’s fifth postulate right once and for all or to find the overlooked element that proved it to be wrong. In 1482, the first printed edition translating Euclid’s work from Arabic to Latin appeared, stimulating the progress. During the 1600s, various mathematicians rewrote the fifth postulate in ways that helped redefine such concepts as “acute angle” or “parallel” in new ways. By 1767, the French writer Jean Le Rond d’Alembert referred to the problem of parallel lines as “the scandal of elementary geometry.”

In the early nineteenth century, there arose various schools of geometry that rewrote the assumptions, creating whole systems of understanding space without having to accept the fifth postulate. Collectively, these schools of thought came to be known as non-Euclidean geometries. There are two different types of non-Euclidean geometry, each relying on a different understanding of the concept of parallelism. Those that assume that there is no such thing as a “parallel” line that will fail to eventually meet the original one are called “elliptic geometries”; those that assume that there can be multiple lines passing through a point that will parallel the original line without touching it are referred to as “hyperbolic geometries.”

Three mathematicians, working independently of one another, came up with systems of geometry (almost at the same time) in the beginning of the 1800s, all of which left out Euclid’s problematic fifth postulate. Carl Frederich Gauss is credited with being the first of them. Gauss disliked controversy and was unwilling to disagree with the prevailing view that Euclid’s geometry was the inevitable, indisputable truth, so he devised his system in private and did not publish his findings. In 1823, Gauss read the works of Janos Bolyai, a Rumanian whose non-Euclidean theories were hidden in his introduction to a book by his father, who was also a famous mathematician. Though Bolyai could not have known of Gauss’s results, his theories were similar. In 1829, a Russian, Nikolai Lobachevsky, who was himself unfamiliar with the work of Gauss and Bolyai, published his own work of non-Euclidian geometry. These three gave rise to a new way of conceiving of space, changing the assumptions that had been put into place by Euclid more than two thousand years earlier. It is just this sort of advancement of knowledge, of restructuring assumptions that were previously taken to be indisputable truth, that Rita Dove considers in her poem “Geometry.”

## Critical Overview

Critic Nelson Hathcock, writing *Critical Survey of Poetry*, says that while Dove “can exult in the freedom that imagination makes possible,” she also demonstrates in her poems that such imaginative liberty has its costs and dangers. He writes about “Geometry”: “Dove parallels the study of points, lines, and planes in space with the work of the poet…. Barriers and boundaries disappear in the imagination’s manipulation of them, but that manipulation has its methodology or aesthetic.” For example, in “Geometry,” the voice of the poem tells us: “I prove a theorem.” Critic Robert McDowell, writing in *Callaloo* about *Yellow House on the Corner*, praises Dove’s “storyteller’s instinct,” her “powerful images,” and “her determination to reveal what is magical in our contemporary lives.”

Well-known critic Helen Vendler, in a 1991 article in *Parnassus: Poetry in Review*, says that Dove “looks for a hard, angular surface to her poems,” and that “She is an expert in the disjunctive.” By this, Vendler means Dove is an expert in disunity or, or that she is very good at expressing an opposition between the meanings of words.

## Criticism

### David Kelly

*Kelly is an adjunct professor of creative writing and literature at Oakton Community College and an associate professor of literature and creative writing at College of Lake County and has written extensively for academic publishers. In this essay, Kelly examines reasons why it would be a mistake to include Dove’s poem in the tradition of anti-scientific poetry.*

It would be very easy for readers to oversimplify the message that can be found in Rita Dove’s poem “Geometry,” taking the poem to be nothing more than yet another burlesque of humanity’s endless fascination with intellectual order. Read lightly, the poem does in fact seem to suggest that the drive to make order out of chaos is a vain and hopeless one that is doomed to failure. It begins with a blunt, triumphant declaration of success, as the speaker announces proof of a theorem. After that, the poem does not portray geometry as any sort of mastery of the world, but instead things go haywire: the house expands, the ceiling fades away, the odors of nature vanish.

These are not the results that are expected to follow proving a theorem, and their illogical nature must be particularly offensive to the mathematician who tried to find some sense of order with the initial proof. Predictability is the point of geometry; when chaos results, it can seem like the poem’s speaker, and mathematicians in general, are doomed to fail. This interpretation is supported by a long-standing tradition that the arts have of presenting rational thought as an affront to nature, creating some sort of battle zone between the natural and the rational.

It is one of the most basic questions about being human, and Dove handles it with such sublime grace that readers could easily miss the overall significance of what she says. Philosophers have long divided human essence into two parts, recognizing the distinction between our mammalian bodies that make us part of the physical world even as the purely human capacity to reason separates us from the physical. In recent centuries, poets have tended to side with nature, presenting reason as a form of corruption that alienates humanity from the rest of the natural world. Just because this has been the trend, though, and even though the poem does approach serious thought playfully, still there is not enough evidence for reading “Geometry” as an assault on the weakness of logic.

The ancient Greeks, whose ideas have formed the basis of Western thought, recognized this basic duality in the human condition, representing it in the forms of Apollo, the god of (among other things) light and therefore of logic and truth, and Dionysus, the god of fertility and of wine, whose followers celebrated irrationality. Their concept of humanity’s divided essence has come down through time to the present day, when it is still thought that “too much” logic will lead to an orderly but sterile, emotionless existence, whereas the absence of logic leaves one in the realm of animal instinct, at the mercy of unexpected violence and unforeseen occurrences. The Greeks may have worshipped Apollo and Dionysus equally, but the fear of favoring one too heavily over the other has caused supporters to divide rigidly into two camps.

In general, most fields of human endeavor can be seen as drawing on both their intellectual achievement and their physical contact with the natural world. Architects, for instance, cannot design purely theoretical buildings without any recognition of the terrain and the atmospheric conditions that those buildings will be housed in; even physicists, who deal with concepts that are too minute, grand, or old for human experience, find that their theories are pointless if they cannot be supported by some real-world evidence. Geometry is one of the most abstract of cerebral pursuits, with only the thinnest relationship to immediate reality. Poetry was once a field of abstract thought, although it has become increasingly focused on the world’s physical nature.

This is, to a large extent, the legacy of the romantic movement that began at the end of the eighteenth century. It followed on the heels of the Enlightenment, when the intellectual world focused on applying scientific methods to understanding human behavior. The French and American Revolutions, for example, were Enlightenment byproducts, and one can see in them the shift from political order based on tradition to political order based on rational principles, such as the rule of the majority. As with most intellectual movements, romanticism is marked by its movement in the direction opposite from the movement that came before—in this case, from intellectualism to physicality. The romantic response to the Enlightenment

*“Proving a theorem should provide a sense of completeness, but in this line there is less a sense of liberation than of vulnerability.”*

was to focus attention on humanity’s relationship to nature. If logic is a set of ideas that can be transferred from one situation to another, the romantics turned away from shared knowledge to focus on the subjective experience of the individual writer; if logic is used to find ways to channel water, build bridges, and traverse mountains, romanticism focused on appreciating but not controlling the natural world. The common use of the word “romantic,” referring to love within a personal relationship, offers insight into the nature of romanticism; romance emphasizes personal experience and is generally accepted to be beyond of the rules of logic. To apply geometric theorems to romantic love would strike most people as heartless and cynical. In its extreme, romanticism would reject the intrusion of any and all such mental designs.

The age of romanticism has long since faded, but its most enduring legacy is the bond forged between poetry and nature. Poetry is, of course, a cerebral event, built of words, not of flesh or earth. Though some poetry can be instructive or contemplative, most poetry offers straight, unexplained description, or at least relies heavily on the physical evidence that humans know from their five senses. There is a basic distrust, in modern poetry, of ideas that the poet spoon-feeds to the reader, and so poetry instead moves to capture the physical experience with words. Some poets have extended their distrust of theorizing to a deep resentment and suspicion of all logic. From Whitman to Eliot to Neruda, there is a clear path of poets who have been resistant to order, with an assumption that logic and creativity cannot exist at the same time and that one must therefore give way to the other. Since this has been the prevailing trend for the past century or two, it is understandable that readers might assume “Geometry” to be an attack on the insufficiency of words.

### What Do I Read Next?

- The best poetry of the early part of Dove’s career is collected in
*Selected Poems*, an anthology of works from*The House on the Yellow Corner, Museum*, and*Thomas and Beulah*, which won her the Pulitzer Prize for 1987.*Selected Poems*was published by Pantheon Books in 1993. - Dove is also a novelist. Her book
*Through the Ivory Gates*, published by Vintage Books in 1993, tells the fictional tale of a young black artist, whose life is much like the author’s, who returns to her home in Akron to run an artists-in-schools program. - W. S. Merwin’s poem “The Horizons of Rooms” is similar to “Geometry” in the way that it contemplates the ways that humans surround themselves with logical constructs of their own making, forgetting about the independent world of nature that goes beyond human order. It is found in Merwin’s 1988 collection
*The Rain in the Trees*, published by Knopf. - Walt Whitman’s poem “When I Heard the Learn’d Astronomer” expresses sorrow at the ways that scientific knowledge narrows one’s experience of the world. It can be found in
*Walt Whitman: The Complete Poems*, edited by Francis Murphy, published by Viking Press in 1990. - Other poems like “Geometry” can be found in
*Against Infinity: An Anthology of Contemporary Mathematical Poetry*, initiated, collected, and edited by Ernest Robson and Jet Wimp, published by Primary Press in 1979. - “Ode to Numbers,” by the Chilean poet Pablo Neruda, is a short poem that looks at math in the same spirit that informed Dove. It can be found in the anthology
*Selected Odes of Pablo Neruda*, translated by Margaret Sayers Peden, published in 1990 by University of California Press. - Linda Pastan’s poem “Arithmetic Lesson: Infinity” is included in her collection
*Carnival Evenings: New and Selected Poems, 1968–1998*, published in 1998 by W. W. Norton Company.

In fact, the imagery Dove uses in the poem does lend itself to be interpreted as being anti-logical. Though the first stanza presents the proof of a theorem as an uplifting experience, with the windows and ceiling floating up as if all of the weight of the physical world had been rendered irrelevant, the second is clouded with hints of the theorem’s unintended side effects. Walls are cleared of paint, paper, or anything else that may have adorned them; flowers lose their fragrance. The second stanza ends with “I am out in the open.” Proving a theorem should provide a sense of completeness, but in this line there is less a sense of liberation than of vulnerability. Readers who see this poem as another example of art rejecting science will focus on the second stanza, with the implication of the danger it carries.

It does not help that the poem’s stance toward geometry is not cleared up in the final stanza, which is, if anything, more ambiguous than the previous two. The physical room that the speaker describes *does* experience an uplifting sense of freedom from the same proof that took the walls away. Does this mean that finding the proof is a good thing because it has liberated the physical world (giving man-made windows the independence and beauty of natural butterflies, for example) or that the proof is bad because life is only tolerable in the places that have escaped the deadening confines of geometry? The poem does not explicitly say, but it does have several aspects that should lead readers to accept intellectualism and not treat it, as so many poets have, as the enemy.

One clue is that this final stanza, though open to interpretation in several ways, clearly is meant to evoke a mood of hope and optimism. The dominant images are of sunlight and truth, and the poem does not say that either has suffered from the proving of the theorem. If reality is escaping from geometry here, it isn’t being aggressively pursued, indicating that its escape is part of the overall plan. In fact, the final word, “unproven,” loops the process back to the opening salvo, “I prove a theorem,” indicating that even something as intellectual as a geometric proof is a part of the cycle of nature.

A minor point, but one still worth mentioning, is the poem’s structure. It does not follow any strict rhythm or rhyme scheme, but it does have a geometric symmetry, with three stanzas of three lines each. Such a structure could be meant to parody the rigors of geometry, but if this were the case, Dove could have made the case better by using a sing-song pattern to mock the lack of inspiration in formal thought. Instead, the limited use of regular structure implies that order can, in a limited way, be of some good.

It is too simple to say that logic and instinct are mutually exclusive, that the world only has room for nature or rationality, but not both. Obviously, both can come together: The combination of reason with physicality is what defines humanity. Readers who have become accustomed to seeing poets and other writers take sides in this conflict are used to reading the works of extremists, who either warn that humans might become unfeeling machines if mathematical order prevails or that barbaric destruction will rule if mathematical precision is forgotten. Usually, poets tend to favor instinct over reason. It is the self-expressive thing to do. Rita Dove is too intelligent to deal in half-truths, however, and “Geometry,” a poem that seems simple and light, refuses to take the easy way out. This poem is too intelligent either to embrace or to reject logic blindly but instead establishes its place in the vast strangeness of the universe.

**Source:** David Kelly, Critical Essay on “Geometry,” in *Poetry for Students*, The Gale Group, 2002.

### Judi Ketteler

*Ketteler has taught literature and composition. In this essay, Ketteler discusses the way in which Rita Dove makes a comparison between geometry and poetic form.*

The poem “Geometry,” by Rita Dove, is a poem about ideas and space and the way in which ideas and space represent possibility and liberation. A mathematical science, the discipline of geometry revolves around precision and around measurements that add up to an organic whole to prove a scientific truth. The human mind has the capability to create such precision and order, to make sense of what would otherwise be chaos.

By titling the poem “Geometry,” Dove alerts the reader as to the subject of the poem. Unlike a “riddle poem” (such as Emily Dickinson’s “A Narrow Fellow in the Grass”), this poem makes its metaphor explicit—in this case, the comparison of geometry and poetry. The reader then begins reading this poem thinking about the science of geometry and brings with him- or herself ideas about geometry and what it means. Simply defined, geometry is the branch of mathematics that deals with the relations and measurements of lines, angles, surfaces, and solids. Most students study geometry at some point in their schooling and, as

*“Nothing in all Emerson’s writings is more eloquent and popular than some bits of his patriotic verse.”*

part of their learning, have to memorize theorems, proofs, and formulas. Geometry is exact; a measurement is what it is; an angle is what it is—there are no grey areas. Whether the reader likes or dislikes geometry, these are some of the perceptions he or she may bring to this poem upon glancing at the title.

If, by chance, the reader has forgotten his or her high school geometry, the first line brings it all back: “I prove a theorem and the house expands.” The word “theorem” is very much associated with geometry, and proving theorems is a main tenet. The speaker immediately takes ownership of the poem, as well as the action of doing geometry. The first line highlights the setting of the poem nicely as well. The reader can imagine a school-age girl, inside of her house, working on her geometry problems. Dove is deliberate in her choice of verbs for the first line. She doesn’t equivocate or say “I study” or “I grasp”; instead, she says, “I prove”— a strong statement. The speaker is clearly both confident and competent in her geometry skills.

The second part of the first line is even more interesting: “the house expands.” There is a causal connection; the house expands because the speaker proved a theorem. The house even takes on human characteristics. The last two lines of the first stanza showcase this personification: “the windows jerk free to hover near the ceiling, / the ceiling floats away with a sigh.” The mood is one of lightness. The softness of “hover” and “with a sigh” suggests this is a peaceful transformation. The house is expanding beyond its walls. The walls are, in fact, ceasing to exist. And the liberating force is the theorem, which the speaker has proven to be true.

There is a way in which the house in this poem stands in for the mind, especially in the way that it expands. Literary critic Therese Steffen writes in her book *Crossing Color: Transcultural Space and Place in Rita Dove’s Poetry, Fiction and Drama:*

Two slightly different readings are imaginable: Either the house metaphorically portrays the mind, or the mind-blowing expansion blasts the house apart.” In any case, it is the mental powers at work that cause the shift from solid to soft. What was once a stable structure is drifting apart. In the same way, what was once a stable knowledge base is drifting too— expanding outwards and upwards.

The reader might think about childhood drawings of a house—very angular, consisting of a box with a triangle roof, a rectangle door and two box windows, usually crisscrossed with a “t.” As a physical space, the house is very much a center of geometrical shapes—walls, doors, windows, floors, ceilings, and furniture. But it is also a center of ideas; in other words, of cultural space. Therese Steffen reads the difference between physical space and cultural space in this way: “Cultural space, as distinguished from place and location, is a space that has been seized upon and transmuted by imagination, knowledge, or experience.” This is a useful distinction because it helps us separate the metaphorical from the actual. If we speak in literal terms, we know the actual house isn’t actually expanding; rather, the cultural space the house represents is expanding—namely the mind of the speaker.

As the poem progresses to the second stanza, the structure of the house continues to destabilize. “As the walls clear themselves of everything / but transparency, the scent of carnations / leaves with them. I am out in the open.” That last line of the second stanza is very powerful—why is the speaker “out in the open”? Why has this geometry caused her to lose her grounding? Even sensory perception has faded away with the carnations. Critic Helen Vendler, in her book *The Given and the Made*, reads this as an experience of “pure mentality”: “As the windows jerk free and the ceiling floats away, sense experience is suspended; during pure mentality, even the immaterial scent of carnations departs.” The speaker is one with her mind—outside forces do not seem to matter. Her surroundings have become “transparent,” leaving nowhere to hide. One way to read “openness” is that the speaker’s foundation of knowledge has been so altered, the “walls” around her mind so shaken, that all of the limits of imagination and understanding that previously held her back have now vanished. There is great liberation in the transparency because it allows her to see beyond what she previously thought were the limits.

Though some readers may love geometry and see unlimited possibilities in mathematical science, to claim that theorems and geometry problems are inherently beautiful and liberating is still a hard sell for many math-fearing readers. Dove isn’t speaking strictly of geometry, though. Just as the house can be read as a metaphor for the mind, geometry itself has a metaphorical quality, especially as it relates to Dove’s true love: poetry. Vendler understands the poem “Geometry” in this way:

It is a poem of perfect wonder, showing Dove as a young girl in her parents’ house doing her lessons, mastering geometry, seeing for the first time the coherence and beauty of the logical principals of spatial form. The poem ‘Geometry’ is really about what geometry and poetic form have in common.

Both geometry and poetry concern space. Simply speaking, geometry takes a logical approach and studies the relationship of objects to the space around them. Poetry takes a more fluid, less tangible approach in that it “studies” the inner space of the mind and the mind’s relationship to thoughts and ideas. Poetry and geometry are alike in that they both seek truth. Geometry is guided by logical principles: If *x* and *y* are true, then we can make a statement about *z*, and it must be true as well. While this is a mathematical way of thinking, it is also highly poetic. There is poetry in the thought process and in the belief that the truth is important in that it helps us to organize our world and understand our place in it. Theorems are as much about shapes and angles as they are about human beings. The speaker in the poem “Geometry” is swept away by these thoughts and connections, and her world is altered as a result.

The experience of “pure mentality” continues through the third and final stanza. There is a sense of a great transformation in the final lines: “and above the windows have hinged into butterflies, / sunlight glinting where they’ve intersected. / They are going to some point true and unproven.” Butterflies are often symbolic of beauty, wonder, and freedom. Here, the windows have actually transformed into butterflies. Solid materials like wood, brick, and glass have changed into brightly colored, delicate wings. Steffen remarks: “This liberating move from the initial “prove” to the final unproven … metamorphoses the wallbound window-frames like earthbound caterpillars into butterflies.” The windows of the house, which provide only a limited view on the world, are exchanged for a more expansive view through the eyes of butterflies. They are flying away, as the speaker says, “to some point true and unproven.”

The last line of the poem suggests that there is much more still to be discovered. The speaker begins the poem by “proving” a theorem. This sets the initial outward movement into action. This new knowledge leaves the speaker left out in the open, without solid walls to shield and limit her. Her entire relationship to the world has shifted by the last stanza. The solid windows have “intersected” with the liberated butterflies—an intersection of an old way of thinking and the new way of thinking and looking at the world. Another way to read the line “sunlight glinting where they’ve intersected,” is to understand it as the intersection of geometry and poetry—the meeting point of mathematical science and emotional introspection. In that intersection is true liberation, which causes the curious, well-rounded mind to continue searching for truth in the world.

This is a poem about poetry and about the beauty of ideas and human thought processes. But it does not exist in a vacuum. Thinking about the larger social implications for this poem enriches the reading of it. The speaker in “Geometry” experiences a liberation brought on by learning something new about herself and the world around her. The saying “knowledge is power” comes to mind. There is great power in the implications of this poem— the sense of wonderment increases with each stanza, as boundaries disappear and possibilities loom. By proving the theorem, a whole new world opens up to the speaker, and it is a world where windows can transform into butterflies.

In short, education is the real stimulus behind the speaker’s transformation. And Dove, a highly educated woman, not to mention former poet laureate of the United States, certainly knows the value of education. Much of Dove’s poetry speaks to the African-American experience. This poem does not so much speak to that experience as it does to the value of education, which is certainly relevant to the African-American experience. Education is wonderful in that it brings about personal enlightenment, but it is also the way out of poverty and despair. Poetry and abstract ideas about space and people’s relationship to the world may seem far removed from the social and cultural realities of everyday working people, particularly poor people who are more concerned with basic needs. However, as the final lines of “Geometry” suggest, there is a key intersection—whether it be the intersection of rational thought and emotion, of thought and action, or of old and new—that can lead to liberation.

**Source:** Judi Ketteler, Critical Essay on “Geometry,” in *Poetry for Students*, The Gale Group, 2002.

## Sources

Hathcock, Nelson, *Critical Survey of Poetry*, Magill, 1991, pp. 954–61.

McDowell, Robert, “The Assembling Visions of Rita Dove,” in *Callaloo*, Vol. 9, No. 1, Winter 1986, pp. 52–60.

Steffen, Therese, *Crossing Color: Transcultural Space and Place in Rita Dove’s Poetry, Fiction and Drama*, Oxford University Press, 2001.

Vendler, Helen, “A Dissonant Triad,” in *Parnassus: Poetry in Review*, Vol. 16, No. 2, 1991, pp. 391–404.

———,*The Given and the Made: Strategies of Poetic Redefinition*, Harvard University Press, 1995.

## Further Reading

Bachelard, Gaston, *The Poetics of Space*, Beacon Press, 1994.

This renowned modern philosophical text, first published in 1964, explores poetry’s relationship to the measurable, physical world.

Dove, Rita, *The Poet’s World*, Library of Congress, 1995.

This publication actually consists of the texts of two addresses that Dove made to the Library of Congress while she was poet laureate. Her view of poetry’s overall significance and her goals as an individual poet are emphasized.

Mlodinow, Leonard, *Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace*, Free Press, 2001.

Dove’s poem assumes that its reader has a sense of what geometry is about. In this book, Mlodinow traces the history of geometry by discussing the major figures who have shaped modern thought, giving a funny, spry spin to a topic that students can sometimes find dull and dense.

Steffen, Therese, *Crossing Color: Transcultural Space and Place in Rita Doves’ Poetry, Fiction and Drama*, Oxford University Press, 2001.

In one of the only books analyzing Dove’s overall career, this recent publication looks at the issues of spatial concept that are raised in “Geometry.”

# Geometry

# Geometry

Perpendicular lines and planes

Geometry, the study of points, lines, and other figures in space, is a very old branch of mathematics. Its ideas were undoubtedly used, intuitively if not formally, from earliest times. Walking along a straight line toward a particular destination is the shortest way

to get there; lining an arrow up with the target is the way to hit it; sitting in a circle around a fire is the most equitable way to share the warmth. Early humans need not have been students of formal geometry to know and to use these ideas that all relate to geometry.

As early as 2,600 years ago the Greeks had not only discovered a large number of geometric properties, they had begun to see them as abstract ideas to be studied in their own right. By the third century BC, they had created a formal system of geometry. Their system began with the simplest ideas and, with these ideas as a foundation, went well beyond much of what is taught in schools today.

## Proof

Typically one learns arithmetic and algebra by experiment or by being told how to do it. Geometry, however, is taught logically. Its ideas are established by means of proof. One starts with definitions, postulates, and primitive terms; then proves his or her way through the course.

The reason for this procedure goes back to the forenamed Greeks, and in particular to Greek mathematician Euclid of Alexandra (c.325–c.265 BC). Twenty-three hundred years ago he wrote a book called the *Elements*. This book contains no exercises, no experiments, no applications, no questions—just proofs, the proof of one proposition after another.

For centuries the *Elements* was the basic text in geometry. British mathematician and ancient Greek historian Sir Thomas Little Heath (1861–1940), in his 1925 translation of the *Elements*, quotes Indian-British mathematician Augustus De Morgan (1806-1871): “There never has been. . .a system of geometry worthy of the name, which has any material departures. . .from the plan laid down by Euclid.” Nowadays the *Elements* word ’behold’ has been replaced with texts that have exercises, problems, and applications, but the emphasis on proof remains. Even the most obvious fact, such as the fact that the opposite sides of a parallelogram are equal, is supposed to go unnoticed, or at least unused, until it has been proved. Whether or not this makes sense, the reader will have to decide for himself or herself, but sensible or not, proof is and will probably continue to be a dominant component of a course in geometry.

Proofs can vary in formality. They can be as formal as the two-column proofs used in textbooks in which each statement is identified as an assumption, a definition, or the consequence of a previously proved property. They can be informal with much left for the reader to fill in. They can be almost devoid of explanation, as in the ingenious proof of the Pythagorean theorem given by Hindu mathematician and astronomer Bhaskara Acharya (1114–1185) in the twelfth century. His proof consisted of a single word ’behold’ and a drawing.

## Constructions

Another lasting influence of Euclid’s *Elements* is the emphasis which is placed on constructions. Three of the five postulates on which Euclid based his geometry describe simple drawings and the conditions under which they can be made. One such drawing (construction) is the circle. It can be drawn if one knows where its center and one point on it are. Another construction is drawing a line segment between two given points. A third is extending a given line segment. These are the so-called ruler-andcompass constructions upon which Euclidean geometry is based.

Modern courses in geometry are frequently based on other postulates. Some, for example, permit one to use a protractor to draw and measure angles; some allow the use of a scale to measure distances. Even so, the traditional limitations that the Euclidean postulates placed on constructions are often observed. Protractors, scales, and other drawing tools, which would be easier and more accurate to use, are forbidden. Constructions become puzzles, intriguing but separate from the logical structure of the course, and not overly practical.

## Points, lines, and planes

Points, lines, and planes are primitive terms; no attempt is made to define them. They do have properties, however, which can be explicitly described. Among the most important of these properties are the following.

Two distinct points determine exactly one line. That line is the shortest path between the two points. Bricklayers use these properties when they stretch a string from corner to corner to guide them in laying bricks.

Two points also determine a ray, a segment, and a distance, symbolized for points A and B by AB (or BA when B is the endpoint), AB, and AB respectively. (Some authors use AB to symbolize all of these, leaving it to the reader to know which is meant.) Three non-collinear points determine one and only one plane.

The photographer’s tripod exploits this to hold the camera steady; while the chair on an uneven floor rocks back and forth between two different planes determined by two different combinations of the four legs.

If two points of a line lie in a plane, the entire line lies in the plane. It is this property that makes the plane flat. Two distinct lines intersect in at most one point; two distinct planes intersect in at most one line. If two *coplanar* lines do not intersect, they are parallel. Two lines that are not coplanar cannot intersect and are called skew lines. Two planes which do not intersect are parallel.

A line which does not lie in a plane either intersects that plane in a single point, or is parallel to the plane.

## Angles

An angle in geometry is the union of two rays with a common endpoint. The common endpoint is called the vertex and the rays are called the sides. Angle ABC is the union of BA and BC. When there is no danger of confusion, an angle can be named by its vertex alone. It is also handy from time to time to name an angle with a letter or number written in the interior of the angle near the vertex. Thus angles ABC, B, and x are all the same angle.

When the two sides of an angle form a line, the angle is called a straight angle. Straight angles have a measure of 180°. Angles that are not straight angles have a measure between 180° and0°. The reflex angles, whose measures exceed 180°, encountered in other branches of mathematics are not ordinarily used in geometry. If the sum of the measures of two angles is 180°, the angles are said to be supplementary. Right angles have a measure of 90°. Lines that form right angles are also said to be perpendicular. If the sum of the measures of two angles is 90°, the angles are called complementary. Angles that are smaller than a right angle are called acute. Those angles that are larger than a right angle but smaller than a straight angle are called obtuse. When two lines intersect, they form two pairs of opposite or vertical angles. Vertical angles are equal.

A ray that divides an angle into two equal angles is called an angle bisector. Points on an angle bisector are equidistant from the sides of the angle.

## Parallel lines and planes

Given a line and a point not on the line, there is exactly one line through the point parallel to the line.

Two coplanar (lines on the same plane) lines l1 and l2 in Figure 1, cut by a transversal t, are parallel if and only if:

1) Alternate interior angles (e.g., d and e) are equal.

2) Corresponding angles (e.g., b and f) are equal.

3) Interior angles on the same side of the transversal are supplementary (see Figure 1).

These principles are used in a variety of ways. A draftsperson uses 2) to rule a set of parallel lines. Number 1) is used to show that the sum of the angles of a triangle is equal to a straight angle.

If a set of parallel lines cuts off equal segments on one transversal, it cuts off equal segments on any other transversal (see Figure 2) a draftsperson finds this idea useful when he or she needs to subdivide a segment into parts that are not readily measured, such as thirds. If transversal AC in Figure 2 is slanted so that AC is three units, then the parallel lines through the unit points will divide AB into thirds as well.

If a set of parallel planes is cut by a plane, the lines of intersection are parallel. This property and its converse are used when one builds a bookcase. The set of shelves are, one hopes, parallel, and they are supported by parallel grooves routed into the sides.

## Perpendicular lines and planes

If A is a given point and CD a given line, then there is exactly one line running through A that is perpendicular to CD. If B is the point on line CD that also resides on the line running perpendicular to

CD, then that line, AB, is the shortest distance from point A to line CD.

In a plane, if CD is a line and B a point on CD, then there is exactly one line through B perpendicular to CD. If B happens to be the midpoint of CD, then AB is called the perpendicular bisector of CD. Every point on AB is equidistant from C and D.

If a line QP is perpendicular to a plane at a point P, then it is perpendicular to every line in the plane that passes through P. Carpenters use this property when they make sure that a doorframe is perpendicular to the floor. Otherwise the door will rub on the floor, as someone who lives in an old house is likely to know.

A line will be perpendicular to a plane if it is perpendicular to two lines in the plane. The carpenter, in setting up the door frame, need not check every line with his or her square; two lines will do.

If perpendiculars are not confined to a single plane, there will be an infinitude of lines through B perpendicular to CD, all lying in the plane that is perpendicular to CD. If B is a midpoint, this plane will be the perpendicular-bisector plane of CD, and every point on this plane will be equidistant from C and D.

Two planes are perpendicular if one of the planes contains a line that is perpendicular to the other plane. The panels of folding screens, for example, stay perpendicular to the floor because the hinge lines are perpendicular to the floor.

## Triangles

Triangles are plane figures determined by three non-collinear points called vertices. They are made up of the segments, called sides, which join them. Although the sides are segments rather than rays, each pair of them makes up one of the triangle’s angles.

Triangles may be classified by the size of their angles or by the lengths of their sides. Triangles whose angles are all less than right angles are called acute. Those with one right angle are right triangles. Those with one angle larger than a right angle are obtuse. (In a right triangle, the side opposite the right angle is called the hypotenuse and the other two sides legs.) Triangles with no equal sides are scalene triangles. Those with two equal sides are isosceles. Those with three equal sides are equilateral. There is no direct connection between the size of the angles of a triangle and the lengths of its sides. The longest side, however, will be opposite the largest angle; and the shortest side, opposite the smallest angle. Equal sides will be opposite equal angles.

In comparing triangles it is useful to set up a correspondence between them and to name corresponding vertices in the same order. If CXY and PST are two such triangles, then angles C and P correspond; sides CY and PT correspond; and so on.

Two triangles are congruent when their six corresponding parts are equal. Congruent triangles have the same size and shape, although one may be the mirror image of the other. Triangles ABC and FDE are congruent provided that the sides and angles which appear to be equal are in fact equal.

One can show that two triangles are congruent without establishing the equality of all six parts. Two triangles will be congruent whenever:

1) Two sides and the included angle of one are equal to two sides and the included angle of the other (SAS congruence).

2) Two angles and the included side of one are equal to two angles and the included side of the other (ASA congruence).

3) Three sides of one are equal to three sides of the other (SSS congruence).

Triangle congruence applies not only to two different triangles. It also applies to one triangle at two different times or to one triangle looked at in two different ways. For example, when the girders of a bridge are strengthened with triangular braces, each triangle stays congruent to itself over a period of time, and does so by virtue of SSS congruence.

Two triangles can also be similar. Similar triangles have the same shape, but not necessarily the same size. They are alike in the way that a snapshot and an enlargement of it are alike. When two triangles are similar, corresponding angles are equal and corresponding sides are proportional.

One can show that two triangles are similar without showing that all the angles are equal and all the sides proportional. Two triangles will be similar when:

1) Two sides of one triangle are proportional to two sides of another triangle and the included angles are equal (SAS similarity).

2) Two angles of one triangle are equal to two angles of another triangle (AA similarity).

3) Three sides of one triangle are proportional to three sides of another triangle (SSS similarity).

The properties of similar triangles are widely used. Artists, for example, use them in making smaller or larger versions of a picture. Map makers use them in drawing maps; and users, in reading them.

Figure 3 shows a right triangle in which an altitude BD has been drawn to the hypotenuse AC by AA similarity, the triangles ABC, ADB, and BDC are similar to one another.

By virtue of these similarities, one can write AC/BC = BC/DC and AC/AB = AB/AD. Then, using AD + DC = AC and a little algebra, one ends up with (AB)2 + (BC)2 = (AC)2, or the Pythagorean theorem: “In a right triangle the sum of the squares on the legs is equal to the square on the hypotenuse.” This neat proof was discovered by Bhaskara, as mentioned earlier.

The altitude BD in Figure 3 is also the mean proportional between AD and DC. That is, AD/BD = BD/DC.

In triangle ABC, if DE is a line drawn parallel to AC, it creates a triangle similar to ABC. It therefore divides AB and BC proportionally. Conversely, a line that divides two sides of a triangle proportionally is parallel to the third side. A special case of this is a

segment joining the midpoints of two sides of a triangle. It is parallel to the third side and half its length.

Each triangle has four sets of lines associated with it: medians, altitudes, angle bisectors, and perpendicular bisectors of the sides. In each set, the three lines are, remarkably, concurrent; that is, they all pass through a single point. In the case of the medians, which are lines from a vertex to the midpoint of the opposite side, the point of concurrency is the centroid, the center of gravity. The angle bisectors are concurrent at the incenter, the center of a circle tangent to the three sides. The perpendicular bisectors of the sides are concurrent at the circumcenter, the center of a circle passing through all three vertices. The altitudes, which are lines from a vertex perpendicular to the opposite side, are concurrent at the orthocenter.

## Quadrilaterals

Quadrilaterals are four-sided plane figures. Various special quadrilaterals are defined in various ways. The following are typical:

Trapezoid: A quadrilateral with one pair of parallel sides.

Parallelogram: A quadrilateral with two pairs of parallel sides.

Rhombus: A parallelogram with four equal sides.

Kite: A quadrilateral with two pairs of equal adjacent sides.

Rectangle: A parallelogram with four right angles.

Square: A rectangle with four equal sides. It is a special kind of rhombus.

Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle.

In any quadrilateral, the sum of the angles is 360°. In a cyclic quadrilateral opposite angles are supplementary.

The diagonals of any parallelograms bisect each other. The diagonals of kites and any rhombus are perpendicular to each other.

Opposite sides of parallelograms are equal.

## Circles

A circle is a set of points in a plane which are a fixed distance from a point called the center, C (see Figure 4) a chord is a segment, DE, joining two points on the circle; a radius is segment, CA, joining the center and a point on the circle; a diameter is a chord, DB, through the center. A tangent, DF, is a line touching the circle in a single point. The words radius and diameter can also refer to the lengths of these segments.

An arc is the portion of the circle between two points on the circle, including the points. A major arc is the longer of the two arcs so determined; a minor arc, the shorter. When an arc is named it is usually the minor arc that is meant, but when there is danger of confusion, a third letter can be used, e.g., arc DAB.

All circles are similar, and because of this fact the ratio of the circumference to the diameter is the same for all circles. This ratio, called pi or *p,* was shown to be smaller than 22/7 and larger than 223/71 by ancient Greek mathematician Archimedes (287-212 BC) about 240 BC.

An arc can be measured by its length or by the central angle that it subtends. A central angle is one whose vertex is the center of the circle.

An inscribed angle is one whose vertex is on the circle and whose sides are two chords or one chord and a tangent. Angles EDB and BDF are inscribed angles. The measure of an inscribed angle is one half that of its intercepted arc. Any inscribed angle that intercepts a semicircle is a right angle; so is the angle between a tangent and a radius drawn to the point of tangency.

If X is a point inside a circle and AB and CD any two chords through X, then X divides the chords into segments whose products are equal. That is, (AX)(XB) = (CX)(XD).

## Area

Areas are expressed in terms of squares such as square inches, meters, miles, etc. Formulas for the areas of various plane figures are based upon the formula for the area of a rectangle, lw, where l is the length and w the width. The area of a parallelogram is bh, where b is the base and h the height (altitude), measured along a line perpendicular to the base. The area of a triangle is half that of a parallelogram with the same base and height, bh/2. When the triangle is equilateral, h = \/3 b/2 so the area is \/3b 2 /4. A trapezoid whose parallel sides are b1 and b2 and whose height is h can be divided into two triangles with those bases and altitudes. Its area is (b1 + b2)h/2.

The area of a quadrilateral with sides a, b, c, and d depends not only on the lengths of the sides but on the size of its angles. When the quadrilateral is cyclic (all four endpoints are on in a circle), its area is given by a remarkable formula discovered by Hindu mathematician and astronomer Brahmagupta (598-668) in the seventh century:

where s is the semi-perimeter (a + b + c + d)/2. This formula includes Heron’s formula, discovered in the first century, for the area of a triangle,

as a special case. By letting d = 0, the quadrilateral becomes a triangle, which is always cyclic.

The area of a circle can be approximated by the area of an inscribed regular polygon. As the number of sides of this polygon increases without limit, its area approaches cr/2, where c is the circumference of the circle and r the radius. Since c = 2πr, the area of the circle is π^{2}.

The surface area of a sphere of radius r is four times the area of a circle of the same radius, 4πr^{2}.

The lateral surface of a right circular cone can be unrolled to form a sector of a circle (see Figure 5) its

area is πrs, where r the radius of its base and s is the *slant* height of the cone.

## Volumes

The volumes of geometric solids are expressed in terms of cubes that are one unit on a side, such as cubic centimeters or cubic yards. The volume of a rectangular solid (box) whose length, width, and height are l, w, and h is lwh. The volume of a prism or a cylinder is Bh, where B is the area of its base and h its height measured along a line perpendicular to the base. The volume of a pyramid or cone is one-third that of a prism or cylinder with the same base and height, that is Bh/3. The volume of a sphere of radius r is 4πr^{3}/3.

It is interesting to note that the volumes of a cylinder, a hemisphere, and a cone having the same base and height are in the simple ratio 3:2:1.

## Other geometries

The foregoing is a summary of Euclidean geometry, based on Euclid’s postulates. Euclid’s fifth postulate is equivalent to assuming that through a given point not on a given line, there is exactly one line parallel to the given line. When one assumes that there is no such line, elliptical geometry emerges. When one assumes that there is more than one such line, the result is hyperbolic geometry. These geometries are called non-Euclidean. Non-Euclidean geometries are as correct and consistent as Euclidean, but describe special spaces. Geometry can also be extended to more than three dimensions. Other special geometries include projective geometry, affine geometry, and topology.

## Resources

### BOOKS

Burton, David M. *The History of Mathematics: An Introduction*. New York: McGraw-Hill, 2007.

### KEY TERMS

**Proposition—** A statement which can be proved as a logical consequence of postulates, definitions, or previously proved propositions.

Euclid. *Elements*. Translated by Sir Thomas L. Heath, New York: Dover Publishing Co., 1956.

Henle, Michael. *Modern Geometries: Non-Euclidean, Projective, and Discrete*. Upper Saddle River, NJ: Prentice Hall, 2001.

Noronha, Maria Helena. *Euclidean and Non-Euclidean Geometries*. Upper Saddle River, NJ: Prentice Hall, 2002.

Setek, William M. *Fundamentals of Mathematics*. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

Silvester, John R. *Geometry: Ancient and Modern*. Oxford, UK, and New York: Oxford University Press, 2001.

Slavin, Stephen L. *Geometry: A Self-teaching Guide*. Hoboken, NJ: John Wiley & Sons, 2005.

Weisstein, Eric W. *The CRC Concise Encyclopedia of* Mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2003.

J. Paul Moulton

# Geometry

# Geometry

Geometry, the study of points, lines, and other figures in **space** , is a very old branch of **mathematics** . Its ideas were undoubtedly used, intuitively if not formally, from earliest times. Walking along a straight line toward a particular destination is the shortest way to get there; lining an arrow up with the target is the way to hit it; sitting in a **circle** around a fire is the most equitable way to share the warmth. Early humans need not have been students of formal geometry to know and to use these ideas.

As early as 2,600 years ago the Greeks had not only discovered a large number of geometric properties, they had begun to see them as abstract ideas to be studied in their own right. By the third century B.C., they had created a formal system of geometry. Their system began with the simplest ideas and, with these ideas as a foundation, went well beyond much of what is taught in schools today.

## Proof

Typically one learns **arithmetic** and **algebra** by experiment or by being told how to do it. Geometry, however, is taught logically. Its ideas are established by means of "proof." One starts with definitions, postulates, and primitive terms; then proves his or her way through the course.

The reason for this goes back to the forenamed Greeks, and in particular to Euclid. Twenty-three hundred years ago he wrote a beautiful book called the *Elements*. This book contains no exercises, no experiments, no applications, no questions—just proofs, the **proof** of one proposition after another.

For centuries the *Elements* was the basic text in geometry. Heath, in his 1925 translation of the *Elements*, quotes De Morgan: "There never has been...a system of geometry worthy of the name, which has any material departures...from the plan laid down by Euclid." Nowadays the *Elements* has been replaced with texts which do have exercises, problems, and applications, but the emphasis on proof remains. Even the most obvious fact, such as the fact that the opposite sides of a **parallelogram** are equal, is supposed to go unnoticed, or at least unused, until it has been proved. Whether or not this makes sense, the reader will have to decide for himself or herself, but sensible or not, proof is and will probably continue to be a dominant component of a course in geometry.

Proofs can vary in formality. They can be as formal as the two-column proofs used in text-books in which each statement is identified as an assumption, a definition, or the consequence of a previously proved property; they can be informal with much left for the reader to fill in; or they can be almost devoid of explanation, as in the ingenious proof of the **Pythagorean theorem** given by the Hindu mathematician Bhaskara in the twelfth century. His proof consisted of a single word "behold" and a drawing.

## Constructions

Another lasting influence of Euclid's *Elements* is the emphasis which is placed on **constructions** . Three of the five postulates on which Euclid based his geometry describe simple drawings and the conditions under which they can be made. One such drawing (construction) is the circle. It can be drawn if one knows where its center and one point on it are. Another construction is drawing a line segment between two given points. A third is extending a given line segment. These are the so-called ruler-and-compass constructions upon which Euclidean geometry is based.

Modern courses in geometry are frequently based on other postulates. Some, for example, permit one to use a protractor to draw and measure angles; some allow the use of a scale to measure distances. Even so, the traditional limitations which the Euclidean postulates placed on constructions are often observed. Protractors, scales, and other drawing tools which would be easier and more accurate to use are forbidden. Constructions become puzzles, intriguing but separate from the logical structure of the course, and not overly practical.

## Points, lines, and planes

Points, lines, and planes are primitive terms; no attempt is made to define them. They do have properties, however, which can be explicitly described. Among the most important of these properties are the following:

Two distinct points determine exactly one line. That line is the shortest path between the two points. Bricklayers use these properties when they stretch a string from corner to corner to guide them in laying bricks.

Two points also determine a ray, a segment, and a **distance** , symbolized for points A and B by AB (or BA when B is the endpoint), AB, and AB respectively. (Some authors use AB to symbolize all of these, leaving it to the reader to know which is meant.) Three non-collinear points determine one and only one **plane** .

The photographer's tripod exploits this to hold the camera steady; the chair on an uneven floor **rocks** back and forth between two different planes determined by two different combinations of the four legs.

If two points of a line lie in a plane, the entire line lies in the plane. It is this property which makes the plane "flat." Two distinct lines intersect in at most one point; two distinct planes intersect in at most one line. If two *coplanar* lines do not intersect, they are **parallel** . Two lines which are not coplanar cannot intersect and are called "skew" lines. Two planes which do not intersect are parallel.

A line which does not lie in a plane either intersects that plane in a single point, or is parallel to the plane.

## Angles

An **angle** in geometry is the union of two rays with a common endpoint. The common endpoint is called the "vertex" and the rays are called the "sides." Angle ABC is the union of BA and BC. When there is no danger of confusion, an angle can be named by its vertex alone. It is also handy from time to time to name an angle with a letter or number written in the interior of the angle near the vertex. Thus angles ABC, B, and x are all the same angle.

When the two sides of an angle form a line, the angle is called a "straight angle." Straight angles have a measure of 180°. Angles which are not straight angles have a measure between 180° and 0°. The "reflex" angles, whose measures exceed 180°, encountered in other branches of mathematics are not ordinarily used in geometry. If the sum of the measures of two angles is 180°, the angles are said to be "supplementary." "Right" angles have a measure of 90°. Lines which form right angles are also said to be **perpendicular** . If the sum of the measures of two angles is 90°, the angles are called "complementary." Angles which are smaller than a right angle are called "acute." Those which are bigger than a right angle but smaller than a straight angle are called "obtuse." When two lines intersect, they form two pairs of "opposite" or "vertical" angles. Vertical angles are equal.

A ray which divides an angle into two equal angles is called an angle "bisector." Points on an angle bisector are equidistant from the sides of the angle.

## Parallel lines and planes

Given a line and a point not on the line, there is exactly one line through the point parallel to the line.

Two *coplanar* lines l1 and l2, cut by a transversal t are parallel if and only if

- Alternate interior angles (e.g., d and e) are equal.
- Corresponding angles (e.g., b and f) are equal.
- Interior angles on the same side of the transversal are supplementary (see Figure 1).

These principles are used in a variety of ways. A draftsman uses 2) to rule a set of parallel lines. Number 1) is used to show that the sum of the angles of a triangle is equal to a straight angle.

If a set of parallel lines cuts off equal segments on one transversal, it cuts off equal segments on any other

transversal (see Figure 2). A draftsman finds this useful when he or she needs to subdivide a segment into parts which are not readily measured, such as thirds. If transversal AC in Figure 2 is slanted so that AC is three units, then the parallel lines through the unit points will divide AB into thirds as well.

If a set of parallel planes is cut by a plane, the lines of intersection are parallel. This property and its converse are used when one builds a bookcase. The set of shelves are, one hopes, parallel, and they are supported by parallel grooves routed into the sides.

## Perpendicular lines and planes

If A is a given point and CD a given line, then there is exactly one line running through A that is perpendicular to CD. If B is the point on line CD that also resides on the line running perpedicular to CD, then that line, AB, is the shortest distance from point A to line CD.

In a plane, if CD is a line and B a point on CD, then there is exactly one line through B perpendicular to CD. If B happens to be the midpoint of CD, then AB is called the perpendicular bisector of CD. Every point on AB is equidistant from C and D.

If a line QP is perpendicular to a plane at a point P, then it is perpendicular to every line in the plane which passes through P. This property is used by carpenters when they make sure that a door frame is perpendicular to the floor. Otherwise the door will rub on the floor, as someone who lives in an old house is likely to know.

A line will be perpendicular to a plane if it is perpendicular to two lines in the plane. The carpenter, in setting up the door frame, need not check every line with his or her square; two will do.

If perpendiculars are not confined to a single plane, there will be an infinitude of lines through B perpendicular to CD, all lying in the plane which is perpendicular to

CD. If B is a midpoint, this plane will be the perpendicular-bisector plane of CD, and every point on this plane will be equidistant from C and D.

Two planes are perpendicular if one of the planes contains a line which is perpendicular to the other plane. The panels of folding screens, for example, stay perpendicular to the floor because the hinge lines are perpendicular to the floor.

## Triangles

Triangles are plane figures determined by three non-collinear points called "vertices." They are made up of the segments, called sides, which join them. Although the sides are segments rather than rays, each pair of them makes up one of the triangle's angles.

Triangles may be classified by the size of their angles or by the lengths of their sides. Triangles whose angles are all less than right angles are called "acute." Those with one right angle are "right" triangles. Those with one angle larger than a right angle are "obtuse." (In a right triangle the side opposite the right angle is called the "hy potenuse" and the other two sides "legs.") Triangles with no equal sides are "scalene" triangles. Those with two equal sides are "isosceles." Those with three equal sides are "equilateral." There is no direct connection between the size of the angles of a triangle and the lengths of its sides. The longest side, however, will be opposite the largest angle; and the shortest side, opposite the smallest angle. Equal sides will be opposite equal angles.

In comparing triangles it is useful to set up a correspondence between them and to name corresponding vertices in the same order. If CXY and PST are two such triangles, then angles C and P correspond; sides CY and PT correspond; and so on.

Two triangles are "congruent" when their six corresponding parts are equal. Congruent triangles have the same size and shape, although one may be the mirror image of the other. Triangles ABC and FDE are congruent provided that the sides and angles which appear to be equal are in fact equal.

One can show that two triangles are congruent without establishing the equality of all six parts. Two triangles will be congruent whenever

- Two sides and the included angle of one are equal to two sides and the included angle of the other (SAS congruence).
- Two angles and the included side of one are equal to two angles and the included side of the other (ASA congruence).
- Three sides of one are equal to three sides of the other (SSS congruence).

Triangle congruence applies not only to two different triangles. It also applies to one triangle at two different times or to one triangle looked at in two different ways. For example, when the girders of a bridge are strengthened with triangular braces, each triangle stays congruent to itself over a period of time, and does so by virtue of SSS congruence.

Two triangles can also be similar. Similar triangles have the same shape, but not necessarily the same size. They are alike in the way that a snapshot and an enlargement of it are alike. When two triangles are similar, corresponding angles are equal and corresponding sides are proportional.

One can show that two triangles are similar without showing that all the angles are equal and all the sides proportional. Two triangles will be similar when

- Two sides of one triangle are proportional to two sides of another triangle and the included angles are equal (SAS similarity).
- Two angles of one triangle are equal to two angles of another triangle (AA similarity).
- Three sides of one triangle are proportional to three sides of another triangle (SSS similarity).

The properties of similar triangles are widely used. Artists, for example, use them in making smaller or larger versions of a picture. **Map** makers use them in drawing maps; and users, in reading them.

Figure 3 shows a right triangle in which an altitude BD has been drawn to the hypotenuse AC. By AA similarity, the triangles ABC, ADB, and BDC are similar to one another.

By virtue of these similarities one can write AC/BC = BC/DC and AC/AB = AB/AD. Then, using AD + DC = AC and a little algebra, one ends up with (AB)2 + (BC)2 = (AC)2, or the Pythagorean **theorem** : "In a right triangle the sum of the squares on the legs is equal to the square on the hypotenuse." This neat proof was discovered by Bhaskara, mentioned earlier.

The altitude BD in Figure 3 is also the **mean** proportional between AD and DC. That is, AD/BD = BD/DC.

In triangle ABC, if DE is a line drawn parallel to AC, it creates a triangle similar to ABC. It therefore divides AB and BC proportionally. Conversely, a line which divides two sides of a triangle proportionally is parallel to the third side. A special case of this is a segment joining the midpoints of two sides of a triangle. It is parallel to the third side and half its length.

Each triangle has four sets of lines associated with it: medians, altitudes, angle bisectors, and perpendicular bisectors of the sides. In each set, the three lines are, remarkably, concurrent, that is, they all pass through a single point. In the case of the medians, which are lines from a vertex to the midpoint of the opposite side, the point of concurrency is the "centroid," the center of gravity. The angle bisectors are concurrent at the "incenter," the center of a circle tangent to the three sides. The perpendicular bisectors of the sides are concurrent at the "circumcenter," the center of a circle passing through all three vertices. The altitudes, which are lines from a vertex perpendicular to the opposite side, are concurrent at the "orthocenter."

## Quadrilaterals

Quadrilaterals are four-sided plane figures. Various special quadrilaterals are defined in various ways. The following are typical:

**Trapezoid** :A **quadrilateral** with one pair of parallel sides.

Parallelogram: A quadrilateral with two pairs of parallel sides.

Rhombus: A parallelogram with four equal sides.

Kite: A quadrilateral with two pairs of equal adjacent sides.

**Rectangle** : A parallelogram with four right angles.

Square: A rectangle with four equal sides. It is a special kind of rhombus.

Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle.

In any quadrilateral the sum of the angles is 360o. In a cyclic quadrilateral opposite angles are supplementary.

The diagonals of any parallelograms bisect each other. The diagonals of kites and any rhombus are perpendicular to each other.

Opposite sides of parallelograms are equal.

## Circles

A circle is a set of points in a plane which are a fixed distance from a point called the center, C (see Figure 4). A "chord" is a segment, DE, joining two points on the circle; a radius is segment, CA, joining the center and a point on the circle; a diameter is a chord, DB, through the center. A "tangent," DF, is a line touching the circle in a single point. The words "radius" and "diameter" can also refer to the lengths of these segments.

An "arc" is the portion of the circle between two points on the circle, including the points. A major **arc** is the longer of the two arcs so determined; a minor arc, the shorter. When an arc is named it is usually the minor arc that is meant, but when there is danger of confusion, a third letter can be used, e. g. arc DAB.

All circles are similar, and because of this the **ratio** of the circumference to the diameter is the same for all circles. This ratio, called **pi** or π, was shown to be smaller than 22/7 and larger than 223/71 by the mathematician Archimedes about 240 B.C.

An arc can be measured by its length or by the central angle which it subtends. A central angle is one whose vertex is the center of the circle.

An inscribed angle is one whose vertex is on the circle and whose sides are two chords or one chord and a tangent. Angles EDB and BDF are inscribed angles. The measure of an inscribed angle is one half that of its intercepted arc. Any inscribed angle that intercepts a semicircle is a right angle; so is the angle between a tangent and a radius drawn to the point of tangency.

If X is a point inside a circle and AB and CD any two chords through X, then X divides the chords into segments whose products are equal. That is, AX XB = CX XD.

## Area

Areas are expressed in terms of squares such as square inches, meters, miles, etc. Formulas for the areas of various plane figures are based upon the formula for the area of a rectangle, lw, where l is the length and w the width. The area of a parallelogram is bh, where b is the base and h the height (altitude), measured along a line perpendicular to the base. The area of a triangle is half that of a parallelogram with the same base and height, bh/2. When the triangle is equilateral, h = √3 b/2 so the area is √3 b2/4. A trapezoid whose parallel sides are b1 and b2 and whose height is h can be divided into two triangles with those bases and altitudes. Its area is (b1 + b2)h/2.

The area of a quadrilateral with sides a, b, c, and d depends not only on the lengths of the sides but on the size of its angles. When the quadrilateral is cyclic (all four endpoints are on in a circle), its area is given by a remarkable formula discovered by the Hindu mathematician Brahmagupta in the seventh century:

where s is the semi-perimeter (a + b +c + d)/2. This formula includes Heron's formula, discovered in the first century, for the area of a triangle,

as a special case. By letting d = 0, the quadrilateral becomes a triangle, which is always cyclic.

The area of a circle can be approximated by the area of an inscribed regular polygon. As the number of sides of this polygon increases without **limit** , its area approaches cr/2, where c is the circumference of the circle and r the radius. Since c = 2πr, the area of the circle is πr2.

The surface area of a **sphere** of radius r is four times the area of a circle of the same radius, 4πr2.

The lateral surface of a right circular cone can be unrolled to form a sector of a circle (see Figure 5). Its

area is πrs, where s is the *slant* height of the cone and r the radius of its base.

## Volumes

The volumes of geometric solids are expressed in terms of cubes which are one unit on a side, such as cubic centimeters or cubic yards. The **volume** of a rectangular solid (box) whose length, width, and height are l, w, and h is lwh. The volume of a **prism** or a cylinder is Bh, where B is the area of its base and h its height measured along a line perpendicular to the base. The volume of a **pyramid** or cone is one third that of a prism or cylinder with the same base and height, that is Bh/3. The volume of a sphere of radius r is 4πr3/3.

It is interesting to note that the volumes of a cylinder, a hemisphere, and a cone having the same base and height are in the simple ratio 3:2:1.

## Other geometries

The foregoing is a summary of Euclidean geometry, based on Euclid's postulates. Euclid's fifth **postulate** is equivalent to assuming that through a given point not on a given line, there is exactly one line parallel to the given line. When one assumes that there is no such line, elliptical geometry emerges. When one assumes that there is more than one such line, the result is hyperbolic geometry. These geometries are called "non-Euclidean." Non-Euclidean geometries are as correct and consistent as Euclidean, but describe special spaces. Geometry can also be extended to more than three dimensions. Other special geometries include **projective geometry** , affine geometry, and **topology** .

## Resources

### books

Euclid. *Elements.* translated by Heath, Sir Thomas L., New York: Dover Publishing Co., 1956.

Gullberg, Jan, and Peter Hilton. *Mathematics: From the Birth of Numbers.* W.W. Norton & Company, 1997.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* New York: CRC Press, 1998.

J. Paul Moulton

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Proposition**—A statement which can be proved as a logical consequence of postulates, definitions, or previously proved propositions.

# Geometry

# Geometry

The term geometry is derived from the Greek word *geometria,* meaning "to measure the Earth." In its most basic sense, then, geometry was a branch of mathematics originally developed and used to measure common features of Earth. Most people today know what those features are: lines, circles, angles, triangles, squares, trapezoids, spheres, cones, cylinders, and the like.

Humans have probably used concepts from geometry as long as civilization has existed. But the subject did not become a real science until about the sixth century b.c. At that point, Greek philosophers began to express the principles of geometry in formal terms. The one person whose name is most closely associated with the development of geometry is Euclid (c. 325–270 b.c.), who wrote a book called *Elements.* This work was the standard textbook in the field for more than 2,000 years, and the basic ideas of geometry are still referred to as Euclidean geometry.

## Elements of geometry

**Statements.** Statements in geometry take one of two forms: axioms and propositions. An axiom is a statement that mathematicians accept as being true without demanding proof. An axiom is also called a postulate. Actually, mathematicians prefer not to accept any statement without proof. But one has to start somewhere, and Euclid began by listing certain statements as axioms because they seemed so obvious to him that he couldn't see how anyone would disagree.

One axiom is that a single straight line, and only one, can be drawn through two points. Another axiom is that two parallel lines (lines running next to each other like train tracks) will never meet, no matter how far they are extended into space. Indeed, mathematicians accepted these statements as true without trying to prove them for 2,000 years. Statements such as these form the basis of Euclidean geometry.

However, the vast majority of statements in geometry are not axioms but propositions. A proposition is a statement that can be proved or disproved. In fact, it is not too much of a stretch to say that geometry is a branch of mathematics committed to proving propositions.

**Proofs.** A proof in geometry requires a series of steps. That series may consist of only one step, or it may contain hundreds or thousands of steps. In every case, the proof begins with an axiom or with some proposition that has already been proved. The mathematician then proceeds from the known fact by a series of logical steps to show that the given proposition is true (or not true).

**Constructions.** A fundamental part of geometric proofs involves constructions. A construction in geometry is a drawing that can be made with the simplest of tools. Euclid permitted the use of a straight edge and a compass only. An example of a straight edge would be a meter stick that contained no markings on it. A compass is permitted in order to determine the size of angles used in a construction.

Many propositions in geometry can be proved by making certain kinds of constructions. For example, Euclid's first proposition was to show that, given a line segment AB, one can construct an equilateral triangle ABC. (An equilateral triangle is one with three equal angles.)

## Plane

A plane is a geometric figure with only two dimensions: width and length. It has no thickness. The flatness of a plane can be expressed mathematically by thinking about a straight line drawn on the plane's surface. Such a line will lie entirely within the plane with none of its points outside of the plane.

A plane extends forever in both directions. Planes encountered in everyday life (such as a flat piece of paper with certain definite dimensions) and in mathematics often have a specific size. But such planes are only certain segments of the infinite plane itself.

## Plane and solid geometry

Euclidean geometry dealt originally with two general kinds of figures: those that can be represented in two dimensions (plane geometry) and those that can be represented in three dimensions (solid geometry). The simplest geometric figure of all is the point. A point is a figure with no dimensions at all. The points we draw on a piece of paper while studying geometry do have a dimension, of course, but that condition is due to the fact that the point must be made with a pencil, whose tip has real dimensions. From a mathematical standpoint, however, the point has no measurable size.

Perhaps the next simplest geometric figure is a line. A line is a series of points. It has dimensions in one direction (length) but in no other. A line can also be defined as the shortest distance between two points. Lines are used to construct all other figures in plane geometry, including angles, triangles, squares, trapezoids, circles, and so on. Since a line has no beginning or end, most of the "lines" one deals with in geometry are actually line segments—portions of a line that do have a limited length.

In general, lines can have one of three relationships to each other. They can be parallel, perpendicular, or at an angle to each other. According to Euclidean geometry, two lines are parallel to each other if they never meet, no matter how far they are extended. Perpendicular lines are lines that form an angle of 90 degrees (a right angle, as in a square or aT) to each other. And two lines that cross each other at any angle other than 90 degrees are simply said to form an angle with each other.

**Closed figures.** Lines also form closed figures, such as circles, triangles, and quadrilaterals. A circle is a closed figure in which every part of the figure is equidistant (at an equal distance) from some given point called the center of the circle. A triangle is a closed figure consisting of three lines. Triangles are classified according to the sizes of the angles formed by the three lines. A quadrilateral is a figure with four sides. Some common quadrilaterals are the square (in which all four sides are equal), the trapezoid (which has two parallel sides), the parallelogram (which has two pairs of parallel sides), the rhombus (a parallelogram with four equal sides), and the rectangle (a parallelogram with four right- or 90-degree angles).

**Solid figures.** The basic figures in solid geometry can be visualized as plane figures being rotated through space. Imagine that a circle is caused to rotate around its center. The figure produced is a sphere. Or imagine that a right triangle is rotated around its right angle. The figure produced is a cone.

## Area and volume

The fundamental principles of geometry involve statements about the properties of points, lines, and other figures. But one can go beyond those fundamental principles to express certain measurements about such figures. The most common measurements are the length of a line, the area of a plane figure, or the volume of a solid figure. In the real world, length can be determined using a meter stick or yard stick. However, the field of analytic geometry provides a way to determine the length of a line by using principles adapted from geometry.

Mathematical formulas are available for determining the area of any figures in geometry, such as rectangles, squares, various kinds of triangles, and circles. For example, the area of a rectangle is given by the formula A = l · h, where l is the length of the rectangle and h is its height. One can find the areas of portions of solid figures as well. For example, the base of a cone is a circle. The area of the base, then, is A = π · r^{2}, where π is a constant whose value is approximately 3.1416 and r is the radius of the base. (Pi [π] is the ratio of the circumference of a circle to its diameter, and it is always the same, no matter the size of the circle. The circumference of a circle is its total length around; its diameter is the length of a line segment that passes through the center of the circle from one side to the other. A radius is a line from the center to any point on the circle.)

## Words to Know

**Axiom:** A mathematical statement accepted as true without being proved.

**Construction:** A geometric drawing that can be made with simple tools, such as a straight edge and a compass.

**Euclidean geometry:** A type of geometry based on certain axioms originally stated by Greek mathematician Euclid.

**Line:** A collection of points with one dimension only—that of length.

**Line segment:** A portion of a line.

**Non-Euclidean geometry:** A type of geometry based on axioms other than those first proposed by Euclid.

**Plane geometry:** The study of geometric figures that can be represented in two dimensions only.

**Point:** A figure with no dimensions.

**Proposition:** A mathematical statement that can be proved or disproved.

**Proof:** A mathematical statement that has been demonstrated logically to be correct.

**Solid geometry:** The study of geometric figures that can be represented in three dimensions.

Formulas for the volume of geometric figures also are available. For example, the volume of a cube (a three-dimensional square) is given by the formula V = s^{3}, where s is equal to the length of one side of the cube.

## Other geometries

With the growth of the modern science of mathematics, scholars began to ask whether Euclid's initial axioms were necessarily true. That is, would it be possible to imagine a world in which more than one straight line could be drawn through two points. Such ideas often sound bizarre at first. For example, can you imagine two parallel lines that do eventually meet at some point far in the distance? If so, what does the term parallel really mean?

Yet, such ideas have turned out to be very productive for the study of certain special kinds of spaces. They have been given the name non-Euclidean geometries and are used to study certain kinds of mathematical, scientific, and technical problems.

# geometry

**geometry** Branch of mathematics concerned with shapes. Euclidean geometry deals with simple plane and solid figures. *Analytic geometry* (co-ordinate geometry), introduced by Descartes (1637), applies algebra to geometry and allows the study of more complex curves. *Projective geometry*, introduced by Jean-Victor Poncelet (1822), concerns itself with projection of shapes and with properties that are independent of such changes. More abstraction occurred in the early 19th century with formulations of non-Euclidean geometry by Janos Bolyai and N. I. Lobachevsky, and differential geometry, based on the application of calculus. See also topology

# geometry

ge·om·e·try / jēˈämətrē/ •
n. the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. ∎ (pl. -tries) a particular mathematical system describing such properties: *non-Euclidean geometries.* ∎ [in sing.] the shape and relative arrangement of the parts of something: *the geometry of spiders' webs.*

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**geometry**