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## Two millennia of mathematics : from Archimedes to Gauss - Colby College Libraries

Seller Inventory B More information about this seller Contact this seller 3. Language: English. Brand new Book. A collection of inter-connected topics in areas of mathematics which particularly interest the author, ranging over the two millennia from the work of Archimedes to the "Werke" of Gauss. The book is intended for those who love mathematics, including undergraduate students of mathematics, more experienced students and the vast unseen host of amateur mathematicians.

It is equally a useful source of material for those who teach mathematics. Softcover reprint of the original 1st ed. Seller Inventory AAV More information about this seller Contact this seller 4. Seller Inventory LIE More information about this seller Contact this seller 5. More information about this seller Contact this seller 6. Seller Inventory AAZ More information about this seller Contact this seller 7.

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Former Library book. Great condition for a used book! Minimal wear. Seller Inventory GRP More information about this seller Contact this seller Condition: Good. Shows some signs of wear, and may have some markings on the inside. Condition: Used: Good. About this Item: Springer. Ships with Tracking Number! May not contain Access Codes or Supplements. May be ex-library. Buy with confidence, excellent customer service!. Seller Inventory q. The evidence for that destruction is the most definitive and secure.

Caesar's invasion may well have led to the loss of some 40,, scrolls in a warehouse adjacent to the port as Luciano Canfora argues, they were likely copies produced by the Library intended for export , 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 4th century.

The Serapeum was certainly destroyed by Theophilus in , and the Museum and Library may have fallen victim to the same campaign. In the Bakhshali manuscript , there is a handful of geometric problems including problems about volumes of irregular solids. The Bakhshali manuscript also "employs a decimal place value system with a dot for zero. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" including cube roots, fractions, ratio and proportion, and barter and "practical mathematics" including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain.

Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

### Two millennia of mathematics : from Archimedes to Gauss, George M. Phillips

Chapter 12 also included a formula for the area of a cyclic quadrilateral a generalization of Heron's formula , as well as a complete description of rational triangles i. Brahmagupta's formula: The area, A , of a cyclic quadrilateral with sides of lengths a , b , c , d , respectively, is given by. The first definitive work or at least oldest existent on geometry in China was the Mo Jing , the Mohist canon of the early philosopher Mozi BC. It was compiled years after his death by his followers around the year BC. In addition, the Mo Jing presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.

The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts i.

As to its invisibility there is nothing similar to it. The mathematician, inventor, and astronomer Zhang Heng AD used geometrical formulas to solve mathematical problems. Zu Chongzhi AD improved the accuracy of the approximation of pi to between 3. The Nine Chapters on the Mathematical Art , the title of which first appeared by AD on a bronze inscription, was edited and commented on by the 3rd century mathematician Liu Hui from the Kingdom of Cao Wei.

This book included many problems where geometry was applied, such as finding surface areas for squares and circles, the volumes of solids in various three-dimensional shapes, and included the use of the Pythagorean theorem. The book provided illustrated proof for the Pythagorean theorem, [29] contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomon , the circle and square, as well as measurements of heights and distances.

This was more accurate than Liu Hui's contemporary Wang Fan , a mathematician and astronomer from Eastern Wu , would render pi as 3. In terms of solid geometry, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. Areas for the [33]. Volumes for the [32]. Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician Shen Kuo CE , Yang Hui who discovered Pascal's Triangle , Xu Guangqi , and many others.

By the beginning of the 9th century, the " Islamic Golden Age " flourished, the establishment of the House of Wisdom in Baghdad marking a separate tradition of science in the medieval Islamic world , building not only Hellenistic but also on Indian sources. Although the Islamic 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.

Al-Mahani born conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji born completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today. 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. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem , which he extended from special right triangles to all triangles in general, along with a general proof. Ibrahim ibn Sinan ibn Thabit born , who introduced a method of integration more general than that of Archimedes , and al-Quhi born were leading figures in a revival and continuation of Greek higher geometry in the Islamic world.

These mathematicians, and in particular Ibn 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. A paper in the journal Science suggested that girih tiles possessed properties consistent with self-similar fractal quasicrystalline tilings such as the Penrose tilings. The transmission of the Greek Classics to medieval Europe via the Arabic literature of the 9th to 10th century " Islamic Golden Age " began in the 10th century and culminated in the Latin translations of the 12th century.

### Convergence testing

An anonymous student at Salerno travelled to Sicily and translated the Almagest as well as several works by Euclid from Greek to Latin. Eugenius of Palermo d. Advances in the treatment of perspective were made in Renaissance art of the 14th to 15th century which went beyond what had been achieved in antiquity. In Renaissance architecture of the Quattrocento , concepts of architectural order were explored and rules were formulated. Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings, [40] notably Masolino da Panicale and Donatello.

Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several. As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood with help from his friend the mathematician Toscanelli , [41] but did not publish, the mathematics behind perspective.

Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen's Optics'. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective.

## Pi goes on forever

Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective.

Perspective remained, for a while, the domain of Florence. Jan van Eyck , among others, was unable to create a consistent structure for the converging lines in paintings, as in London's The Arnolfini Portrait , because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world.

The culmination of these Renaissance traditions finds its ultimate synthesis in the research of the architect, geometer, and optician Girard Desargues on perspective, optics and projective geometry. The Vitruvian Man by Leonardo da Vinci c. The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. In the early 17th century, there were two important developments in geometry.

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 c.

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 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.

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. By 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 , Bernhard Riemann , a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic self-contained 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 proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 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? With the development of relativity theory in physics, this question became vastly more complicated. 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, now known as Hilbert's axioms , were given by David Hilbert in in his dissertation Grundlagen der Geometrie Foundations of Geometry.

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. In the midth 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. Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. 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.

From Wikipedia, the free encyclopedia. Projecting a sphere to a plane. Outline History. Concepts Features. Line segment ray Length. Volume Cube cuboid Cylinder Pyramid Sphere. Tesseract Hypersphere. Main article: Egyptian geometry. Main article: Babylonian mathematics. See also: Greek mathematics.