# Bibliography of Primary Sources

# Bibliography of Primary Sources

Apollonius of Perga. *Conics* (c. 200 b.c.). This work consisted of 8 books with some 400 theorems. In this great treatise, he set forth a new method for subdividing a cone to produce circles, and discussed ellipses, parabolas, and hyperbolas—shapes he was the first to identify and name. In place of the concentric spheres used by Eudoxus, Apollonius presented epicircles, epicycles, and eccentrics, concepts that later influenced Ptolemy's cosmology. Even more significant was his departure from the Pythagorean tendency to avoid infinites and infinitesimals: by opening up mathematicians' minds to these extremes, Apollonius helped make possible the development of the infinitesimal calculus two millennia later. The most important factor in this monumental work, however, was not any one problem, but Apollonius's overall approach, which opened mathematicians' minds to the idea of deriving conic sections by approaching the cone from a variety of angles. By applying the *latus transversum* and *latus erectum,* lines perpendicular and intersecting, Apollonius prefigured the coordinate system later applied in analytic geometry.

Archimedes. *On the Equilibrium of Planes* (c. 240 b.c.). Here Archimedes considered the mechanics of levers and the importance of the center of gravity in balancing equal weights.

Archimedes. *On the Sphere and Cylinder* (c. 240 b.c.). In this work Archimedes built on the previous work of Euclid to reach conclusions about spheres, cones, and cylinders. As described in *The Scientific 100* (Simmons 1996): "He showed that if these figures have the same base and height—imagine a cone inscribed in a hemisphere which itself is inscribed within a cylinder—the ratio of their volumes will be 1:2:3. In addition, the surface of the sphere is equivalent to two-thirds of
the surface of the cylinder which encloses it." Archimedes was immensely pleased with this discovery, even requesting that his family have a sphere and cylinder engraved on his tombstone.

Archimedes. *On Floating Bodies* (c. 240 b.c.). Archimedes used *On Floating Bodies* to recount his theory regarding water displacement and help found the science of hydrostatics. In this book, he demonstrated that when an object of any shape and weight is floated in water, its vertical, buoyant force is equal to the weight of the water it displaces. One legend of Archimedes holds that he first understood this connection between the weight of a floating object and the resulting increase in water level while watching bath water rise as he sunk his body into a tub. He was said to have been so excited by this insight that he jumped from the tub and ran stark naked through the streets proclaiming his discovery.

Aristaeus the Elder. *Five Books Concerning Solid Loci* (c. 390 b.c.). The curves, lines, and points of cones were the subject matter of this book, which Euclid later credited as the source for much of his own writing on conics in Book XIII of his *Elements.*

Aristotle. *Organon* (c. 340 b.c.). A series of works, including the *Prior* and *Posterior Analytics,* in which Aristotle established many of the fundamental rules of logic, which were applied to philosophical inquiry, mathematics, and many other branches of science. In particular, he delineated the rules of deductive argument and developed symbolic notation to express such arguments.

Aryabhata. *Aryabhatiya* (a.d. 499). Bringing together teachings from ancient Greek and Indian astronomers, as well as new ideas from Aryabhata himself, this work developed various rules for arithmetic and trigonometric calculations. It also contained a number of important "firsts" or near-firsts, including one of the first recorded uses of algebra. Furthermore, it was one of the first texts to include the idea of number position or place value (i.e., tens, hundreds, thousands, etc.). These concepts would have enormous impact as they moved westward, as would another idea implemented by Aryabhata in his text: the Hindu numeral system.

Autolycus of Pitane. *On the Moving Sphere* and *On Risings and Settings* (c. 310 b.c.). The two earliest surviving mathematical works. *On the Moving Sphere* is a study of spherical geometry with a clear astronomical application, and *On Risings and Settings* a work on astronomical observations. It is possible that the first of these was based on a now-lost geometry textbook by Eudoxus of Cnidus, whose theory of homocentric spheres Autolycus supported.

Boethius. *Arithmetic* (c. a.d. 520). This work would later become medieval scholars' principal source regarding Pythagorean number theory.

Brahmagupta. *Brahmasphutasiddhanta* (The opening of the universe, a.d. 628). This work defined zero as the result obtained when a number is subtracted from itself—by far the best definition of zero up to that time. Brahmagupta also provided rules for "fortunes" and "debts" (positive and negative numbers), and used a place-value system much like that which exists today. In addition, the work offered an algorithm for computing square roots, a method for solving quadratic equations, and rudimentary forms of algebraic notation.

Chang Ts'ang. *Chiu-chang Suan-shu* or *Jiuzhang Suanshu* (Nine chapters of mathematical art, c. 160 b.c.). Chang's authorship of this work is in not certain. In any case, the work, the oldest known Chinese mathematical text, contained 246 problems, which as the title suggested were presented in nine chapters. Some four centuries after Chang Ts'ang, Liu Hui wrote a famous commentary on the "Nine Chapters."

Cleomedes. *On the Circular Motions of the Celestial Bodies* (first century a.d.). This work provides valuable information regarding the work of Cleomedes's more distinguished predecessors. *On the Circular Motions* is made up almost entirely of ideas taken from others, most notably Posidonius, and contains a mixture of accurate and wildly inaccurate data. Of particular interest is the fact that the text serves as the principal source regarding the methodology used by Eratosthenes in making his famous measurement of Earth's circumference.

Dharmakirti. *Seven Treatises* (seventh century a.d.). Presented a system of syllogistic reasoning that became highly influential in the East. In this work, Dharmakirti outlined a precise form of syllogism that, like its Western counterpart, consisted of three parts; however, the purpose of those three parts—and indeed the methodology governing their use—was radically different.

Diocles. *On Burning Mirrors* (c. first century b.c.). Diocles is remembered almost entirely for this fragmentary manuscript. In it, he discussed not only the physical problem referred to in the title, but such subjects as cutting a sphere with a plane, as well as the famous Delian problem of doubling the cube. *On Burning Mirrors* may actually have been a collection of three separate short works, combined under a single title that does not reflect the whole. In any case, the book consisted of 16 geometric propositions, most of which involved conics.

Diophantus of Alexandria. *Arithmetica.* (c. a.d. 250). Considered one of the greatest mathematics works of its era and the first coherent work in number theory in the history of mathematics. It apparently consisted of 13 volumes, though only 6 have survived, in which Diophantus describes several propositions regarding number theory, including one that inspired Pierre de Fermat to pose his famous "Last Theorem." Though earlier mathematicians, including the Egyptians and Babylonians, had explored many of the number theory problems posed by Diophantus, they had never before been assembled together in a single work. It was this compilation of problems in number theory that helped launch it as an independent segment of mathematics.

Eratosthenes. *Geography.* A lost work in which Eratosthenes published his theories and calculations. The title reflects the first-known use of the term *geography, *which means "writing about the Earth." Although his calculations were disputed in his own time, they allowed the development of maps and globes that remained among the most accurate produced for over a thousand years. This, in turn, sparked interest in geography and geodesy, and emboldened regional seafaring exploration using only the most primitive navigational
instruments. Eratosthenes's work, moreover, helped solidify belief in a round Earth, and promoted an early theory that the relative warmth or coolness of a locations climate was determined by its distance from the equator. *Geography* also supported the concept of antipodes—undiscovered lands and peoples on the "other side" of the world.

Eratosthenes. *On the Measurement of the Earth* (c. 225 b.c.). Now lost, this work marked the foundation of geodesy, the branch of mathematics that deals with determination of Earth's size and shape, and the location of points on its surface. Among the topics of study within geodesy is the system of latitude and longitude, which Eratosthenes seems to have pioneered in his maps, the most accurate in the world at the time.

Euclid. *Elements* (c. 300 b.c.). Considered the bible of geometry for 2,000 years, this work remains the most influential textbook in history, and one of the essential works of human civilization. The book is primarily a summation of mathematical knowledge passed down from Pythagoras onward, and its genius lies in its cogent explanation of basic principles, as well as its clear and thorough explication of geometric proofs. Consisting of 13 books in which Euclid elaborated on some 450 propositions, the *Elements* begins with a definition of points, lines, planes, angles, circles, triangles, quadrilaterals, and parallel lines. In Book II, Euclid addressed rectangles and squares; in Book III, circles; and in Book IV polygons. He continued with a discussion of proportion and area (Book V), followed by an application of this theory to plane geometry (Book VI). Book VII covers arithmetic, Book VIII irrational numbers, and Book IX rational numbers, while the remainder of the volume is devoted to three-dimensional, or solid, geometry. Euclid's five postulates are among the most important aspects of his work; the first three of these focus on construction with the straight edge and circle or compass, the only tools of Euclidean geometry, while the fourth states that all right angles are equal. Most controversial, however, was the fifth postulate, which discussed the relationship between two straight lines placed side by side.

Eudemus of Rhodes. *History of Geometry* and *History of**Astronomy* (fourth century b.c.). Though these books have been lost, much of what they contained was passed on to other ancient writers, and collectively they constitute a principal source of information on numerous ancient thinkers and their achievements.

Eudoxus of Cnidus. *On Speeds* (c. 375 b.c.). Here the author presented a new theory of the motion made by the Sun, Moon, and planets. Given the spherical shape of Earth, Eudoxus imagined a series of concentric spheres around it, and eventually developed a description of 27 spheres necessary for picturing the movement of all known bodies.

Geminus. *Theory of Mathematics* (c. 100 b.c.). Now lost, the work presented an overview of geometry. The latter was by then a long-established discipline among the Greeks, and this gave Geminus a certain perspective that would have been beyond the reach of his predecessors. Thus he undertook to define mathematics as a whole, and to classifying it within the context of the sciences. It constituted a valuable early attempt to give shape to the mathematical discipline, and to place it within the context of scientific study.

Hero of Alexandria. *Metrica* (c. first century a.d.). An important work on geometry that was lost and not rediscovered until 1896. It contains formulas to compute the areas of things like triangles, cones, and pyramids. The area of the triangle is often attributed to Hero, but it is likely he borrowed it from Archimedes or the Babylonians.

Hippocrates of Chios. *Elements of Geometry* (c. 460 b.c.). Now lost, this was a mathematical textbook. The first work of its kind, it would have an enormous impact on another book of a similar title, the highly influential *Elements* of Euclid. In his work, known through the writings of Aristotle, Proclus, and others, Hippocrates became the first mathematician to adopt scientifically precise and logical methodology for developing geometrical theorems from axioms and postulates. It is also likely that *Elements of Geometry *contained the first written explanation of Pythagorean principles, since the Pythagoreans who preceded him did not believe in committing their ideas to writing.

Hypsicles of Alexandria. *Elements, Book XIV* (c. 150 b.c.). In "Book XIV," often mistakenly included with the original writings of Euclid, Hypsicles improved on Apollonius's approach to problems involving a dodecahedron and an icosahedron inscribed in the same sphere. Hypsicles is also credited with works on polygonal numbers, regular polyhedra, and arithmetic progressions. The latter appears in his *On the Ascension of Stars,* the first astronomical text to divide the Zodiac into 360 degrees.

Li Ch'un-feng. *Ten Classics of Mathematics* (c. a.d. 650). Li Ch'un-feng led a group of scholars who created commentaries on the "Nine Chapters of Mathematical Art," the "Mathematical Classic of the Gnomon of Chou," and other works. These came to be known collectively as the *Ten Classics of Mathematics,* which remained the definitive mathematical text in China for at least four centuries.

Liu Hui. *Commentary on "Nine Chapters of Mathematical**Art"* (c. a.d. 263). The "Nine Chapters" was a text of unknown authorship dating back to the first century b.c. The oldest known Chinese mathematical text, it contained 246 problems, which as the title suggested were presented in nine chapters. The first of these concerned arithmetic and the fundamentals of geometry, and included a discussion of the counting rods.

Manava. *Sulbasutras* (c. 750 b.c.). These were early Hindu mathematical texts, of which Manava was the author of one. Mathematics in ancient India primarily served the purposes of priestly rites, and Manava's *Sulbasutra* concerns accurate construction of altars for making sacrifices. Long before Greek mathematicians began trying to square the circle, the Manava *Sulbasutra* offered information on converting squares or rectangles to circles. Among the various values for π given in the text was the figure of 25/8 or 3.125.

Nicomachus. *Arithmetike eisagoge* (Introduction to arithmetic, c. a.d. 100). A highly influential if rather unusual mathematical text. Here, Nicomachus examined odd, even, prime, composite, and perfect numbers. He also presented an interesting theorem in which he showed that by adding consecutive odd numbers, successively including one additional number, it was possible
to produce a series of the sums of all cubed numbers. The book also shows the strange, highly unscientific, side of Pythagorean mathematics, which for instance assigned personalities to numbers. Pappus and other mathematicians of the late ancient world despised Nicomachus's book, while Boethius perhaps revealed himself as a true medieval with his admiration of it. He turned it into a schoolbook, and despite—or perhaps because of—its peculiarities, the work became the standard arithmetic text of the Middle Ages. Not until the Crusades (1095-1291), when western Europeans increasingly became exposed to Arab versions of more significant ancient works, was it replaced.

Nicomedes. *On Conchoid Lines* (c. 250 b.c.). This work, the author's most notable—and perhaps only—written work, has been lost. Portions of it that have survived in the writings of others, however, provide historians with knowledge of the conchoid and lemma. The first of these looks rather like what a modern person would describe as a very flat Bell curve, but to the Greeks it appeared like a sea creature; hence the derivation of the name from *konche,* or mussel shell. Below this curve was a line, and beneath that a point parallel to the apogee of the curve. By determining the length of a segment from the apogee to the lower point, it was possible to find segments of equal length—both of which also intersected the curve—on either side of that one. This in turn yielded a trisected angle, providing a solution of sorts to one of the great problems of antiquity. The lemma was a minor theorem he discovered in the course of working on the duplication of the cube problem.

Pappus of Alexandria. *Synagoge* (Collection, fourth century a.d.). While serving as a handbook of geometry, the *Synagoge* also incorporated the work of earlier mathematicians, and in many cases is the only existing source for these ideas. It also contains influential work on astronomical and projective geometry, and both René Descartes and Isaac Newton used Pappus's work.

Panini. *Astadhyayi* (c. 400 b.c.). In this work, Panini gave some 4,000 aphoristic rules for Sanskrit, which—in part thanks to his systematization—remained largely unchanged for the next two millennia. Panini's linguistic formulae have often been likened to mathematical functions, and it has been suggested that the Hindu number system and mathematical reasoning are linked to the structure of the Sanskrit language. Plato. *The Republic* (c. 380 b.c.). Here Plato wrote that plane and solid geometry were two of the five subjects necessary to the education of the philosopher-king. Mastering geometry trained the student in logical argument and taught him to search for and love truth, which Plato represented with eternal and ideal geometric forms. In other words, knowledge of geometry and logic became prerequisite to the study of philosophy. Plato also recorded the research efforts of Theodorus and Theatetus to geometrically demonstrate the existence of the irrational numbers between √3 and √17.

Theodosius of Bithynia. *Sphaerics* (second century b.c.). Intended to provide a mathematical grounding for astronomy, the *Sphaerics* expands on the *Elements* of Euclid with regard to spheres.

Wang Hs'iao-t'ung. *Ch'i-ku Suan-ching* (c. a.d. 625). Includes the first known use of cubic equations in a Chinese text. The work offers 20 problems involving mensuration, but does not provide any rule for solving cubic equations. Foremost mathematician in China during the seventh century, Wang Hs'iao-t'ung was regarded as an expert on the calendar.

Won-wang. *I-ching* (c. 1150 b.c.). This work includes some mathematical information. The book discusses the magic square, a matrix in which the sum of all figures added along any straight line is the same. The magic square, which the text attributes to the semi-legendary Emperor Yu, is in turn related to the faces of the dice.

Zeno of Elea. *Epicheiremata* (c. 450 b.c.). Apparently contained the author's famous paradoxes, of which Plato claimed there were 40 or more. Both it and the majority of the paradoxes have long since disappeared, though it is likely that the other problems in principle resembled the four that survive. These four failed in their attempt to prove that motion was impossible, but they did impress philosophers with the importance of logic itself. Through use of logic, Zeno seemingly created a series of statements that could not possibly be true. Thus was born the scientific study of the dialectic, which has captured the imagination of philosophers and mathematicians ever since.

**NEIL SCHLAGER**

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