## Similar

This book is a critical, in-depth study of Kahn’s philosophy of education and his unique pedagogy. It is the first extensive and comprehensive investigation of the Kahn Master’s Class as seen through the eyes of his graduate students at Penn.

This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams.

Get down to the basics — get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals

Conquer proofs with confidence — follow easy-to-grasp instructions for understanding the components of a formal geometry proof

Take triangles in strides — learn how to take in a triangle's sides, analyze its angles, work through an SAS proof, and apply the Pythagorean Theorem

Polish up on polygons — get the lowdown on quadrilaterals and other polygons: their angles, areas, properties, perimeters, and much more

Open the book and find:

Plain-English explanations of geometry terms

Tips for tackling geometry proofs

The seven members of the quadrilateral family

Straight talk on circles

Essential triangle formulas

The lowdown on 3-D: spheres, cylinders, prisms, and pyramids

Ten things to use as reasons in geometry proofs

Learn to:

Core concepts about the geometry of shapes and geometry proofs

Critical theorems, postulates, and definitions

The principles and formulas you need to know

Fortunately, there's Schaum's. This all-in-one-package includes more than 650 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 25 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible.

More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.

This Schaum's Outline gives you

665 fully solved problems Concise explanations of all geometry concepts Support for all major textbooks for geometry coursesFully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores!

Fortunately, there's Schaum's. This all-in-one-package includes more than 600 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible.

More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.

This Schaum's Outline gives you

618 fully solved problems to reinforce knowledge Concise explanations of all trigonometry concepts Updates that reflect the latest course scope and sequences, with coverage of periodic functions and curve graphing.Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum’s to shorten your study time--and get your best test scores!

Schaum's Outlines--Problem Solved.

Hollywood actress and math whiz Danica McKellar has completely shattered the “math nerd” stereotype. For years, she’s been showing girls how to feel confident and ace their math classes – with style! With Girls Get Curves, she applies her winning techniques to high school geometry, giving readers the tools they need to feel great and totally “get” everything from congruent triangles to theorems, and more. Inside you’ll find:

· Time-saving tips and tricks for homework and tests

· Illuminating practice problems (and proofs!) with detailed solutions

· Totally relatable real-world examples

· True stories from Danica’s own life as an actress and math student

· A Troubleshooting Guide, for getting unstuck during even the trickiest proofs!

With Danica as a coach, girls everywhere can stop hiding from their homework and watch their scores rise!

Does the thought of geometry make you jittery? You're not alone. Fortunately, this down-to-earth guide helps you approach it from a new angle, making it easier than ever to conquer your fears and score your highest in geometry. From getting started with geometry basics to making friends with lines and angles, you'll be proving triangles congruent, calculating circumference, using formulas, and serving up pi in no time.

Geometry is a subject full of mathematical richness and beauty. But it's a subject that bewilders many students because it's so unlike the math they've done before—it requires the use of deductive logic in formal proofs. If you're having a hard time wrapping your mind around what that even means, you've come to the right place! Inside, you'll find out how a proof's chain of logic works and even discover some secrets for getting past rough spots along the way. You don't have to be a math genius to grasp geometry, and this book helps you get un-stumped in a hurry! Find out how to decode complex geometry proofs Learn to reason deductively and inductively Make sense of angles, arcs, area, and more Improve your chances of scoring higher in your geometry class

There's no reason to let your nerves get jangled over geometry—your understanding will take new shape with the help of Geometry For Dummies.

About the Book

Established as a successful practical workbook series with more than 20 titles in the language learning category, Practice Makes Perfect now provides the same clear, concise approach and extensive exercises to key fields within mathematics. The key to the Practice Makes Perfect series is the extensive exercises that provide learners with all the practice they need for mastery.

Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with angles, circles, triangles, graphs, functions, the laws of sines and cosines, and more!

100s of Problems!

* Step-by-step answer sets clearly identify where you went wrong (or right) with a problem

* Get the inside scoop on graphing trig functions

* Know where to begin and how to solve the most common equations

* Use trig in practical applications with confidence

As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity--and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex.

Moving and enlightening, A Certain Ambiguity is a story about what it means to face the extent--and the limits--of human knowledge.

This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Can geometry be exciting?

It can for those who can appreciate the beauty of numbers and their relationships.

This book contains a meticulous geometric investigation of the 5 Platonic Solids and 5 other important polyhedra, as well as reference charts for each solid.

Poly (many) hedron (face) means "many faces." Polyhedra are 3 dimensional figures with 4 or more faces, or sides. These polyhedra are reflections of Nature herself, and a study of them provides insight into the way the world is structured. Nature is not only beautiful, but highly intelligent. As you explore the polyhedra in this book, this will become apparent over and over again.

The book contains a geometric explanation of the Phi Ratio and Fibonacci series, and a detailed analysis of the pentagon, which forms the basis for many of these solids. The pentagon is composed entirely of Phi relationships, and is integral to a proper understanding of sacred geometry.

With over 140 full-color illustrations, this book is perfect for teachers and students of geometry alike. It is a must for those who are serious about sacred geometry.

This book is dedicated to those who can appreciate the logic of numbers and the beauty of nature, for they are both aspects of the same unifying principle.

Prerequisites: knowledge of simple algebra and elementary trigonometry. No brainiac math skills required! Only the ability to appreciate nature's own logic.

Book #1 in the Geometric Explorations Series

From Loving Healing Press (www.LovingHealing.com)

From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map.

Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.

This book presents the state of the art of geometrical tolerancing, covers the latest ISO and ANSI/ASME standards and is a comprehensive reference and guide for all professional engineers, designers, CAD users, quality managers and anyone involved in the creation or interpretation of CAD plans or engineering designs and specifications.

* For all design and manufacturing engineers working with these internationally required design standards

* Covers ISO and ANSI geometrical tolerance standards, including the 2005 revisions to the ISO standard

* Geometrical tolerancing is used in the preparation and interpretation of the design for any manufactured component or item: essential information for designers, engineers and CAD professionals

* The only desktop geometrical tolerancing reference

* For all CAD users, engineers, designers, drafting professionals and anyone who needs to specify or interpret product specifications to international standards

* Simple and quick to use, visually indexed, large format presentation for ease of use

"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."

(David Parrott, Australian Mathematical Society)

"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community."

(European Mathematical Society)

"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."

(Denis Bonheure, Bulletin of the Belgian Society)

This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises.

A no-nonsense practical guide to trigonometry, providing concise summaries, clear model examples, and plenty of practice, making this workbook the ideal complement to class study or self-study, preparation for exams or a brush-up on rusty skills.

About the Book

Established as a successful practical workbook series with over 30 titles in the language learning category, Practice Makes Perfect now provides the same clear, concise approach and extensive exercises to key fields within mathematics.

The key to the Practice Makes Perfect series is the extensive exercises that provide learners with all the practice they need for mastery.

Market / Audience

For students who need to review and practice trigonometry, whether to keep up with class work or to prepare for a test or exam (such as SAT and ACT in the US, or GCSE in the UK).

International suitability: High

Benefit to the Customer

Workbook is not exam specific, yet it provides thorough coverage of the trigonometry skills required in most math tests.

About the Authors

Carolyn Wheater ( Hawthorne, NJ) teaches middle school and upper school mathematics at the Nightingale-Bamford School in New York City. Educated at Marymount Manhattan College and the University of Massachusetts, Amherst, she has taught math and computer technology for 30 years to students from preschool through college. She is a member of National Council of Teachers of Mathematics (NCTM) and the Association of Teachers in Independent Schools.

The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.

"In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."

Mathematics Teaching

"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."

Christoph Bandt, Mathematical Reviews

"...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."

H.Haase, Zentralblatt

About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.

Key features:

* Gradual progression in problem difficulty builds and strengthens mathematical skills and techniques

* Basic topics include trigonometric formulas and identities, their applications in the geometry of the triangle, trigonometric equations and inequalities, and substitutions involving trigonometric functions

* Problem-solving tactics and strategies, along with practical test-taking techniques, provide in-depth enrichment and preparation for possible participation in various mathematical competitions

* Comprehensive introduction (first chapter) to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry expose advanced students to college level material

103 Trigonometry Problems is a cogent problem-solving resource for advanced high school students, undergraduates, and mathematics teachers engaged in competition training.

Other books by the authors include 102 Combinatorial Problems: From the Training of the USA IMO Team (0-8176-4317-6, 2003) and A Path to Combinatorics for Undergraduates: Counting Strategies (0-8176-4288-9, 2004).

The next chapter is an introduction to real and complex manifolds. It contains an exposition of the theorem of Frobenius, the lemmata of Poincaré and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thom's transversality theorem.

Chapter 3 includes characterizations of linear differentiable operators, due to Peetre and Hormander. The inequalities of Garding and of Friedrichs on elliptic operators are proved and are used to prove the regularity of weak solutions of elliptic equations. The chapter ends with the approximation theorem of Malgrange-Lax and its application to the proof of the Runge theorem on open Riemann surfaces due to Behnke and Stein.

Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure.

The book is arranged in four sections, devoted to realizing the universal principle force equals curvature:

Part I: The Euclidean Manifold as a Paradigm

Part II: Ariadne's Thread in Gauge Theory

Part III: Einstein's Theory of Special Relativity

Part IV: Ariadne's Thread in Cohomology

For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum.

Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos).

Elements of Topology provides a basic introduction to point-set topology and algebraic topology. It is intended for advanced undergraduate and beginning graduate students with working knowledge of analysis and algebra. Topics discussed include the theory of convergence, function spaces, topological transformation groups, fundamental groups, and covering spaces.

The author makes the subject accessible by providing more than 250 worked examples and counterexamples with applications. The text also includes numerous end-of-section exercises to put the material into context.

This book is composed of 12 chapters that review the principles, concepts, and analytic proofs of geometric theorems, families of lines, the normal equation of the line, and related matters. Other chapters highlight the application of graphing, foci, directrices, eccentricity, and conic-related topics. The remaining chapters deal with the concept polar and rectangular coordinates, surfaces and curves, and planes.

This book will prove useful to undergraduate trigonometric students.

The University of Toronto Undergraduate Competition was founded to provide additional competition experience for undergraduates preparing for the Putnam competition, and is particularly useful for the freshman or sophomore undergraduate. Lecturers, instructors, and coaches for mathematics competitions will find this presentation useful. Many of the problems are of intermediate difficulty and relate to the first two years of the undergraduate curriculum. The problems presented may be particularly useful for regular class assignments. Moreover, this text contains problems that lie outside the regular syllabus and may interest students who are eager to learn beyond the classroom.

While containing all the key results of basic topology, Essential Topology never allows itself to get mired in details. Instead, the focus throughout is on providing interesting examples that clarify the ideas and motivate the student, reflecting the fact that these are often the key examples behind current research.

With chapters on:

continuity and topological spaces deconstructionist topology the Euler number homotopy groups including the fundamental group simplicial and singular homology, and fibre bundlesEssential Topology contains enough material for two semester-long courses, and offers a one-stop-shop for undergraduate-level topology, leaving students motivated for postgraduate study in the field, and well-prepared for it.

The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.

The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.

The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics.

Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.

Learn and review on the go! Use Quick Review Geometry/Trigonometry Study

Notes to help you learn or brush up on the subject quickly. You can use the

review notes as a reference, to understand the subject better and improve your

grades. Easy to remember facts to help you perform better.

Meet JJ, an unusual character with a unique vantage position from which he can measure and monitor humanity’s progress. Armed with a device that compels all around it to tell the truth, JJ offers a satirical evaluation of our attitudes to numeracy and logic, touching upon several aspects of life on Earth along the way, from the criminal justice system and people’s use of language to highway driving and modern art.

A collection of mathematically-flavored stories and jokes, interlaced with puzzles, paradoxes and problems, fuse together in an entertaining, free-flowing narrative that will engage and amuse anyone with an interest in the issues confronting society today. JJ demonstrates how a lack of elementary mathematical knowledge can taint our work and general thinking and reflects upon the importance of what is arguably our most valuable weapon against ignorance: a sound mathematical education.

What is JJ’s prognosis for our future? There’s only one way to find out...

"Numbers, logic, human behavior and aliens: this unique book blends them all into a captivating narrative of serious talk and satire, where wit and scholarly details are counterpointed by instructive puzzles and mathematical fun. A ‘must read’ for anybody who appreciates humor and culture."

Stanislav Potapenko, Department of Civil and Environmental Engineering, University of Waterloo, Canada

"... a real delight. Constanda has managed to intertwine stories, puzzles, logic and some very rich mathematics concepts into a very readable, enjoyable novel... I believe this book should be in the personal library of every high school mathematics teacher."

Tom Becvar, St Louis University High School, USA

"...a highly readable, unique and fascinating combination of humor, mathematics and social commentary that is factual, educational and, more importantly, understandable. Given what is taking place in today’s society, J.J. Moon brings mathematics back to earth! I’ll never go through the car buying process again without thinking about SCAM 16!"

Jerry Hoopert, VP / Chief Administrative Officer, Tulsa Teachers Credit Union, USA

Key features:

* Begins with a brief survey of basic notions in algebraic and differential geometry, Lie groups and Lie algebras

* Examines how, in a new chapter, Clifford algebra is relevant to robot kinematics and Euclidean geometry in 3D

* Introduces mathematical concepts and methods using examples from robotics

* Solves substantial problems in the design and control of robots via new methods

* Provides solutions to well-known enumerative problems in robot kinematics using intersection theory on the group of rigid body motions

* Extends dynamics, in another new chapter, to robots with end-effector constraints, which lead to equations of motion for parallel manipulators

Geometric Fundamentals of Robotics serves a wide audience of graduate students as well as researchers in a variety of areas, notably mechanical engineering, computer science, and applied mathematics. It is also an invaluable reference text.

-----

From a Review of the First Edition:

"The majority of textbooks dealing with this subject cover various topics in kinematics, dynamics, control, sensing, and planning for robot manipulators. The distinguishing feature of this book is that it introduces mathematical tools, especially geometric ones, for solving problems in robotics. In particular, Lie groups and allied algebraic and geometric concepts are presented in a comprehensive manner to an audience interested in robotics. The aim of the author is to show the power and elegance of these methods as they apply to problems in robotics."

--MathSciNet

Key features of Putnam and Beyond

* Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants.

* Each chapter systematically presents a single subject within which problems are clustered in every section according to the specific topic.

* The exposition is driven by more than 1100 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors.

* Complete solutions to all problems are given at the end of the book. The source, author, and historical background are cited whenever possible.

This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for self-study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to to expand their mathematical horizons.

The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems.

Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem- solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops.

This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated developement. As Professor Morris Kline of New York University noted, "This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels."

Beginning with an overview and analysis of mathematics, the first of three major divisions of the book progresses to an exploration of analytic geometry, algebra, and ordinary differential equations. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. It furthur examines prime numbers, the theory of probability, approximations, and the role of computers in mathematics. The theory of functions of a real variable opens the final section, followed by discussions of linear algebra and nonEuclidian geometry, topology, functional analysis, and groups and other algebraic systems.

Thorough, coherent explanations of each topic are further augumented by numerous illustrative figures, and every chapter concludes with a suggested reading list. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference.

"This is a masterful English translation of a stupendous and formidable mathematical masterpiece . . ." — Social Science