Encyclopedia
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern
mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry.
Early geometry
The earliest recorded beginnings of geometry can be traced to
ancient Egypt , the
ancient Indus Valley , and ancient Babylonia from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in
surveying,
construction,
astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of
calculus. For example, both the Egyptians and the Babylonians were aware of versions of the
Pythagorean theorem about 1500 years before
Pythagoras; the Egyptians had a correct formula for the volume of a
frustum of a square pyramid; the Babylonians had a
trigonometry table.
Ancient Indian geometry
Harappan geometry
The geometry used in the
Indus Valley Civilization of
North India and
Pakistan from around 3000 BC was just as advanced as its contemporaries in Egypt and Mesopotamia, and mostly developed as a result of advanced
urban planning, which is evident from the perfect grid pattern of
Harappa and
Mohenjo-daro where streets were laid out in perfect right angles. The geometry used by this early
Harappan civilization was for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced
brick technology, which utilised ratios. The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding. Brick sizes were in a perfect ratio of 4:2:1. Decimal weights were based on ratios of 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with each unit weighing approximately 28 grams, similar to the English ounce or Greek uncia.
Many of the weights uncovered have been produced in definite geometrical shapes which present knowledge of basic geometry, including the circle. This culture also produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles.
Further to the use of circles in decorative design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. Some historians believe this points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of
p.
In
Lothal, a thick ring-like shell object found with four slits each in two margins served as a
compass to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees. Such shell instruments were probably invented to measure 8–12 whole sections of the horizon and sky, explaining the slits on the lower and upper margins. Archaeologists consider this as evidence the Lothal experts had achieved something 2,000 years before the Greeks are credited with doing: an 8–12 fold division of horizon and sky, as well as an instrument to measure angles and perhaps the position of stars, and for navigation purposes. Lothal contributes one of three measurement scales that are integrated and linear . An ivory scale from Lothal has the smallest-known decimal divisions in Indus civilization. The scale is 6 mm thick, 15 mm broad and the available length is 128 mm, but only 27 graduations are visible over 146 mm, the distance between graduation lines being 1.704 mm . The sum total of ten graduations from Lothal is approximate to the
angula in the
Arthashastra. The Lothal craftsmen took care to ensure durability and accuracy of stone weights by blunting edges before polishing. The Lothal weight of 12.184 gm is almost equal to the Egyptian
Oedet of 13.792 gm.
Vedic geometry
During the Vedic period of Indian mathematics , many rules and developments of geometry are found in Vedic works as a result of the mathematics required for the construction of religious
altars. These include the use of geometric shapes, including triangles, rectangles, squares, trapezia and circles, equivalence through numbers and area,
squaring the circle and vice versa, the
Pythagorean theorem and a list of
Pythagorean triples discovered algebraically, and computations of
p .
As a result of the mathematics required for the construction of these altars, many rules and developments of geometry are found in Vedic works. These include:
- Use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.
- Equivalence through numbers and area.
- Squaring the circle and vice versa.
- Pythagorean triples discovered algebraically.
- Statements of the Pythagorean theorem and a numerical proof.
- Computations of p, with the closest being correct to 2 decimal places.
Lagadha was probably the earliest known mathematician to have used geometry and
trigonometry for
astronomy.
Yajnavalkya composed the
Shatapatha Brahmana, which contains geometric aspects, including several computations of p, with the closest being correct to 2 decimal places , and gives a rule implying knowledge of the Pythagorean theorem.
The
Sulba Sutras , which is another name for geometry, were composed between 800 BC and 500 BC and were appendices to the Vedas giving rules for the construction of religious altars. The
Sulba Sutras contain the first use of irrational numbers,
quadratic equations of the form a x
2 = c and ax
2 + bx = c, the use of the
Pythagorean theorem and a list of
Pythagorean triples discovered algebraically
predating Pythagoras, geometric solutions of linear equations, and a number of geometrical proofs. These discoveries are mostly a result of altar construction, which also led to the first known calculations for the
square root of 2, which were correct to a remarkable 5 decimal places.
Baudhayana composed the
Baudhayana Sulba Sutra, which contains a statement of the Pythagorean theorem, geometric solutions of a linear equation in a single unknown, several approximations of
p , along with the first use of irrational numbers and quadratic equations of the forms ax
2 = c and ax
2 + bx = c, and a computation for the square root of 2, which was correct to a remarkable five decimal places.
Manava composed the
Manava Sulba Sutra, which contains approximate constructions of circles from rectangles, and squares from circles, which give approximate values of
p, with the closest value being 3.125.
Apastamba composed the
Apastamba Sulba Sutra, which contains the method of
squaring the circle, considers the problem of dividing a segment into 7 equal parts, calculates the square root of 2 correct to five decimal places, solves the general linear equation, and also contains a numerical proof of the
Pythagorean theorem, using an area computation. The historian Albert Burk claims this was the original proof of the theorem which
Pythagoras copied on his visit to India.
Classical Greek geometry
For the ancient
Greek mathematicians, geometry was the crown jewel of their sciences, probably developing independently of Indian geometry, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of an "axiomatic theory", which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.
Thales and Pythagoras
Thales of
Miletus , was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived.
Pythagoras of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to
Babylon and
Egypt. The theorem that bears his name was not his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.
Plato
Plato , the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked
ruler or a
protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possible
compass and straightedge constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system.
Aristotle , Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs which was not substantially improved upon until the 19th century.
Hellenistic geometry
Euclid
Euclid , of
Alexandria, probably a student of one of Plato’s students, wrote a treatise in 13 books , titled
The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as
Euclidean geometry. The treatise is not a compendium of all that the
Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by
Ptolemy I, King of Egypt.
The Elements began with definitions of terms, fundamental geometric principles , and general quantitative principles from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.
- Any two points can be joined by a straight line.
- Any finite straight line can be extended in a straight line.
- A circle can be drawn with any center and any radius.
- All right angles are equal to each other.
- If two straight lines in a plane are crossed by another straight line , and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect .
It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks.
The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.
Archimedes
Archimedes , of
Syracuse,
Sicily, when it was a Greek city-state, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time . Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.
Archimedes had followed Eudoxian methods to write out geometric solutions. One solution to the area and volume of a parabola used unit fractions, a form of rigorous arithmetic notation that was created by Egyptians 1,700 years earlier. A unit fraction link between Archimedes' method of slicing the parabola into small pieces, creating the first form of calculus, as given by the proof
- 4A/3 = A + A/3 + A/12
and, its 1/4th geometric infinite series form
- 4A/3 = A + A/4 + A/16 + A/64 + ... A/ + ...
The Moscow Mathematical Papyrus, dating to 2,000 BCE also sliced the area of a truncated pyramid, exactly finding its area, as Archimedes later applied by following the Eudoxian 1/4th geometric series, and proving his result by unit fraction arithmetic.
After Archimedes
After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus , author of
Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.
The great
Library of Alexandria was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port , but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.
Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.
Islamic geometry
The
Islamic
Caliphate established across the
Middle East,
North Africa,
Spain,
Portugal,
Afghanistan and parts of
Pakistan, began around 640 CE.
Islamic mathematics during this period was primarily algebraic rather than geometric, though there were important works on geometry. Scholarship in Europe declined and eventually the
Hellenistic works of
antiquity were lost to them, and survived only in the Islamic centers of learning.
Although the Muslim mathematicians are most famed for their work on
algebra, number theory and number systems, they also made considerable contributions to geometry,
trigonometry and mathematical
astronomy, and were responsible for the development of algebraic geometry. Geometrical magnitudes were treated as "algebraic objects" by most Muslim mathematicians however.
The successors of Mu?ammad ibn Musa al-?warizmi undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
Al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.
Thabit ibn Qurra
Although Thabit ibn Qurra contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to real numbers,
integral calculus, theorems in
spherical trigonometry, analytic geometry, and
non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the
Ptolemaic system, and in mechanics he was a founder of statics.
An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept.
In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments.
After Thabit ibn Qurra
Ibrahim ibn Sinan , who introduced a method of integration more general than that of
Archimedes, and al-Quhi were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular al-Haytham, studied optics and investigated the optical properties of mirrors made from
conic sections.
Astronomy, time-keeping and
geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials.
Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy.
Omar Khayyám
Omar Khayyám was a
Persian mathematician, as well as a poet. Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general method of solving
cubic equations by intersecting a parabola with a circle. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of
non-Euclidean geometry. Omar Khayyam also combined the use of trigonometry and
approximation theory to provide methods of solving algebraic equations by geometrical means. He was mostly responsible for the development of algebraic geometry.
In a paper written by Khayyam before his famous algebra text
Treatise on Demonstration of Problems of Algebra, he considers the problem:
Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal. Khayyam shows that this problem is equivalent to solving a second problem:
Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem in turn led Khayyam to solve the cubic equation x
3 + 200x = 20x
2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by compass and straightedge, a result which would not be proved for another 750 years.
His
Treatise on Demonstration of Problems of Algebra contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations . However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations.
In
Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-Euclidean geometry, although this was not his intention. In trying to prove the parallel postulate he accidentally proved properties of figures in non-Euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered.
Sharafeddin Tusi
Persian mathematician Sharafeddin Tusi did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the study of algebraic geometry.
The 17th century
When Europe began to emerge from its
Dark Ages, the
Hellenistic and
Islamic texts on geometry found in Islamic libraries were translated from
Arabic into
Latin. The rigorous deductive methods of geometry found in Euclid’s
Elements of Geometry were relearned, and further development of geometry in the styles of both Euclid and Khayyam continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.
In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by
Rene Descartes and
Pierre de Fermat . This was a necessary precursor to the development of
calculus and a precise quantitative science of
physics. The second geometric development of this period was the systematic study of
projective geometry by Girard Desargues . Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by
Hellenistic geometers, notably Pappus . The greatest flowering of the field occurred with
Jean-Victor Poncelet .
In the late 17th century,
calculus was developed independently and almost simultaneously by
Isaac Newton and
Gottfried Wilhelm von Leibniz . This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.
The 18th and 19th centuries
Non-Euclidean geometry
The old problem of proving Euclid’s Fifth Postulate, the "
Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. Though Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of Euclid's theories of parallels and his proof of properties of figures in non-Euclidean geometries contributed to the eventual development of
non-Euclidean geometry. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri,
Lambert, and
Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century,
Gauss,
Johann Bolyai, and
Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first
non-Euclidean geometry. By 1854,
Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for
Einstein's
theory of relativity.
It remained to be proven mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.
While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense distances. With the development of relativity theory in physics, this question became vastly more complicated.
Introduction of mathematical rigor
All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by
David Hilbert in 1894 in his dissertation
Grundlagen der Geometrie . Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.
Analysis situs, or topology
In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as
topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.
The 20th century
Developments in algebraic geometry included the study of curves and surfaces over finite fields, rather than the real or complex numbers.
Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and
cryptography. For some properties of one of the smallest finite spaces, the 3-dimensional projective space over the two-element field, see the . With the advent of the computer, new disciplines such as computational geometry or digital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth.
See also
- List of geometry topics
- Important publications in geometry.
- Interactive geometry software
- Book written by " A2 " about two and three-dimensional space, to understand the concept of four dimensions
External links
- at cut-the-knot
- An online tool to compute lines, surfaces and volumes of the main plane and solid figures, through direct and indirect formulas.
- Software for learning and teaching mathematics and geometry. The standard in Education.
- Kig is a Free Software program for exploring geometric constructions.
- A free dynamic geometry tool, useful for exploring geometry.
- Free software for performing compass and straightedge constructions in the manner of the Ancient Greeks.
- by Antonio Gutierrez.
- at cut-the-knot
-
- Stanford Encyclopedia of Philosophy:
- by Elmer G. Wiens
-
-
- An online resource for problems
- All lessons introduce mathematical concepts, step by step, with animations of text, points, lines and figures in general. Solution of problems is also given step by step. Colors are used to give hints and clues to follow the concept or the solution of the problems.
