In this post we will see a wonderful book on mathematics  titled Straight Lines and Curves by N. Vasilyev and V. Gutenmacher.

About the book:

The authors N. B. Vasilyev and V. L. Gutenmacher are professors of mathematics at Moscow University. N. B. Vasilyev works in the field of the application of mathematical methods to biology, while V. I. Gutenmacher works in the field of mathematical methods used in the analysis of economic models.

In addition to their scientific work, they have both written many articles and books for high school and university students, and have worked with the Correspondence Mathematics School, which draws its pupils from all over the Soviet Union. They have worked on the committee organizing the “mathematical olympiad” problem competitions, which have greatly stimulated interest in mathematics among young people in the Soviet Union. They regularly contribute to the magazine “Kvant” (“Quantum”), a remarkable educational magazine devoted to mathematics and physics. This book contains a wealth of material usually found in geometry courses, and takes a new look at some of the usual theorems.

It deals with paths traced out by moving points, sets of points satisfying given geometrical conditions, and problems on finding maxima and minima. The book contains more than 200 problems which lead the reader towards some important areas of modern mathematics, and will interest a wide range of readers whether they be high school or university students, teachers, or simply lovers of mathematics.

The book was translated from the Russian by Anjan Kundu ans was published by Mir Publishers in 1980. It was also republished by Birkhauser in 2004. The link below is for the Mir Edition.

For those who really want to explore the world of mathematics, I would highly recommend using dynamic mathematics software GeoGebra. It is a Free and Open Source Software. Exploring the problems and exercises in this particular book (and of course otherwise also) with help of GeoGebra, has been for me, an exceptionally rewarding experience. This will reveal to you many subtle points and lead you to mathematical depths which you may have not thought of (or could not) diving into.

And I couldn’t but resist to add this quote:

… the feeling for beauty in mathematics is infectious.
It is caught not taught. It affects those with a flair for the subject.

– H. E. Huntley from The Divine Proportion

I hope you too get infected and infect others as well.

PDF | 11.3 MB | OCR | Cover | 600 dpi | Bookmarked | Paginated

(Some parameters on IA maybe different.)

You can get the book here (IA) and here (filecloud).

Password, if needed: mirtitles

Table of Contents

Preface (7)

INTRODUCTION (9)

Introductory Problems (9)
Copernicus’ Theorem (13)

1. SET OF POINTS (17)

A Family of Lines and Motion (23)
Construction Problems (25)
Set of Problems (30)

2. THE ALPHABET (35)

A Circle and a Pair of Arcs (38)
Squares of Distances (42)
Distances from Straight Lines (51)
The Entire Alphabet (57)

3. LOGICAL COMBINATIONS (60)

Through a Single Point (60)
Intersection and Union (67)
The “Cheese” Problem (74)

4. MAXIMUM AND MINIMUM (78)

Where to Put the Point (82)
The “Motor-Boat” Problem (84)

5. LEVEL CURVES (90)

The “Bus” Problem (90)
Functions on a Plane (93)
Level Curves (94)
Graph of a Function (94)
The Map of a Function (100)
Boundary Lines (101)
Extrema of Functions (103)

6. QUADRATIC CURVES (108)

Ellipses, Hyperbolas, Parabolas (108)
Foci and Tangents (113)
Focal Property of a Parabola (117)
Curves as the Envelopes of Straight Lines (121)
Equations of Curves (124)
The Elimination of the Radicals (129)
The End of Our Alphabet (130)
Algebraic Curves (138)

7. ROTATIONS AND TRAJECTORIES (140)

The Cardioid (141)
Addition of Rotations (142)
A Theorem on Two Circles (153)
Velocities and Tangents (157)
Parametric Equations (166)
Conclusion (170)

ANSWERS, HINTS, SOLUTIONS (172)
APPENDIX I. Method of Coordinates (181)
APPENDIX II. A Few Facts from School Geometry (183)
APPENDIX III. A Dozen Assignments (187)

Notation (196)