Customer Reviews

Plane And Solid Geometry - Wentworth-Smith Mathematical Series

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Wentworth's "Geometry" was written around 1900 and served as a standard for teaching students Euclidean geometry in American secondary schools. The subject treated is synthetic geometry; absolutely no reference is made to co-ordinates, groups, or graphs and everything is defined and derived geometrically in a classical way. Throughout the book, definitions, theorems, and proofs are presented in a consistent and organized manner. Diagrams and figures are found throughout and are easily referenced. All theorems are proved and the treatment is fully rigorous. The volume really consists of two books in one and can be thought of as a modern rendition of Euclid's Elements.

The first book treats plane geometry and compass constructions as well as loci. This material is more likely to be familiar to most students. Here a complete treatment of parallel lines, perpendicular lines, methods of proof, congruence, polygons, triangles, quadrilaterals, circles, chords, similarity, proportion, Pythagorean theorem, area of polygons, regular polygons, concurrence, inscribing, circumscribing, circumference and the area of a circle are all covered.

The second book treats solid geometry. This is three dimensional geometry and includes planes, dihedral angles, polyhedral angles, surface area, volume, polyhedrons, prisms, parallelepipeds, pyramids, 5 regular polyhedra, cylinders, cones, and spheres.

Several appendices introduce symmetry, extrema, and spherical segments. There is also a "recreations" section which provides exercises in fallacious proofs. A brief history of geometry and a table of formals is also provided.

The exercises range from simple to quite difficult. No answers or examples are provided. A few of the questions are worded obscurely and can be difficult to understand. However, most of the book should be accessible to high school students once they understand basic algebra (up to factoring and the quadratic formula). I would say the book can be complete in about 10 months by spending two days for every exercise set and doing a fair amount of the questions (please do not waste your time doing ALL the exercises, and opt for something like 33%). The theorems unfortunately are rarely named. That is, a theorem will be called Proposition XV but never something like "S.A.S congruence" which aids in memorization. Overall, this book is THE bible for geometry and makes for an excellent reference and learning guide.


PS.
Unfortunately, most schools have abandoned teaching the subject in this manner in favor of a more modern treatment that includes vectors and co-ordinates. I believe this is partly responsible for the declining performance of students in mathematical areas across North America. Nothing can replace the intuitive nature of geometry in training a mathematician in proofs and deductive reasoning. Instead, our students are rushed into analytic geometry and calculus and first encounter axioms and proofs in linear algebra during the first two years of college. Do you want to know why the angles of a triangle add up to 180? Can you estimate square roots using a compass and ruler? Do you know how to derive the formulas for the area of a circle or the volume of a sphere? Do you know the different ways of constructing a plane? Did you know there are only 5 regular polyhedra and that constructing more is impossible? If not, you are seriously missing out on some good mathematics! Strongly recommended for quantitative enthusiasts and contest competitors.

Other Recommendations:
For a gentler approach to plane geometry I recommend Harold Jacobs "Geometry". The best feature is that theorems are named and writing proofs is more explicative. His algebra book is also great for elementary algebra.

For a complete treatment of proofs and how to write them get a copy of Velleman's "How to Prove It". It is difficult to learn proofs from geometry books alone so this makes an excellent supplement to any budding mathematician. Will also train you in logic.

For a collection of challenging plane geometry problems in preparation for competitions or to boost problem solving skills check out Posamentier/Salkind "Challenging Problems in Geometry".

For a more challenging exposition of the same subject matter with an excellent set of questions the Russian adaptation of Kiselev's two geometry books "Planimetry and Stereometry" make for good reading. Very terse and rigorous, not for the faint of heart! I suggest you tackle it after working through Wentworth.