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This book arises from mature experience allied with an appreciation of contemporary trends of thought, it has been tested under classro om conditions, it requires high standards of thought on the part of both pupil and teacher, and it has a number of novel aspects without breaking with traditional methods.

In recent years, the trend in our syllabuses up to the Ordinary Level in the G.C.E. has been to reduce the number of formal theorems asked. The teacher has had little to learn as his ability in rider work has so greatly transcended the ability of his pupils: he has perhaps allowed himself to be bored. This may account for the fact that the attention of reformers has been drawn towards facets of work in algebra, as being a more obvious field in which change would be acceptable, by challenging our teaching skill in the presentation of new material.

In spite of the American style which inclines to put a number to every postulate, definition and theorem, Dr. Lewis's text does much to bring a happy marriage between so called "modern mathematics" and the content of our geometry syllabuses up to the standard of our fifth forms and a little beyond. The author investigates the need for definitions and postulates, first in everyday situations and then in mathematics generally. Point, line and set then appear as undefined terms whose properties need to be made clear by examples: there follow the ideas of betweenness, line segment, ray and angle. The postulates needed for the four elementary operations are followed by those of the reflexive, symmetric and transitive properties of equality. Congruence appears as a one to one correspondence between the vertices of two polygons such that (a) all the corresponding sides are equal and (b) all the corresponding angles are equal. Ratio is defined as the quotient of the measure of two quantities, when the quantities are expressed in the same unit. A locus of points is the set of those points and only those points, that satisfy given conditions.

The elements of set theory, particularly intersection and union first appear in connection with the building of figures, and later along with loci and cartesian geometry. There is a careful examination by Venn diagrams of the relations between the statements (i) If p then q, (ii) If q then p (iii) If not p then not q (iv) If not q then not p This comes first with general and then with geometrical situations. The relationships are later established by truth tables. The set theory and the logic are developed on formal lines but only so far as they are needed for practical purposes in the book.

The text itself is very readable and smooth, in many places quite informal without being familiar, and without the slightest trace of looseness, triviality or "writing down to a level". The preface mentions grades 10 and 11 in the U.S.A., corresponding to the 15 and 16 year old grammar school pupils in this country, so the level of writing would be a little difficult for younger pupils. On the other hand, the author has increased the pace at which a pupil may move by his deliberate policy of providing readymade diagrams for the first half of the riders in each exercise.

The less usual parts include chapters on coordinate geometry, in which set notation is used skilfully but not obtrusively, some formal solid geometry and an eight page incursion into non-euclidean geometry with three theorems and eight tangible riders.

The format of the book is large being 9.5 in. by 6.3 in., the printing, layout and symbolism are clear and most attractive, the book is profusely illustrated and two colour printing is used for both diagrams and text. A good index is provided, but there do not appear to be teachers' copies with answers. In my opinion, this book represents an important contribution to the teaching apparatus of any school and copies should be widely available for use by school staffs, if not for rotation amongst classes.

Dr. Harry Lewis was principal of Arts High School, Newark, New Jersey. He was formerly the chairman of the Mathematics Department of Ease Side High School of Newark, having taught mathematics for many years in the Newark Public School System. He is the coauthor of textbooks on business mathematics and has taught at the New York University School of Education.

Preface
Contents
List of Symbols
1. Definitions and Their Place in a Proof
Need for definitions
Who determines the definitions of words?
Constructing a definition
Need for undefined terms
The language of geometry
Test
2. Definitions of Geometric Terms
The measure of a line segment
The measure of an angle
Drawing a conclusion based on the reverse of a definition
Drawing conclusions on the basis of definitions and the reverse of definitions
Test
3. Assumptions and Their Place in a Proof
How do the blind draw conclusions?
Postulates in geometry
The sum and difference of two line segments
The sum and difference of two angles
The addition postulate
The subtraction postulate
The multiplication and division postulates
The postulates of equality
Applications of the postulates of geometry
Test
4. The "Simple" Theorems
Theorem on right angles
Theorem on straight angles
Theorems on supplementary and complementary angles
Vertical angles
Test
5. Congruence of Triangles
Correspondence
Correspondence related to polygons
Congruent polygons
Postulates for proving triangles congruent
Applications of the postulates on congruent to formal proofs
Proving line segments or angles congruent through congruent triangles
Further conclusions that can be drawn on the basis of congruent triangles
Overlapping triangles
The isosceles triangles
The S.S.S. theorem
The hypotenuse-leg method of congruence
Problems involving congruence of more than one pair of triangles
Test and review
Try this for fun
6. Perpendicularity
Meaning of distance and its relation to perpendicular lines
Conditional and categorical statements
Test and review
7. Perpendicularity in Space Geometry
The meaning of determine
Further conditions under which a plane is determined
Methods of determining a plane
Perpendicularity between a line and a plane
Test and review
8. The Indirect Proof and Parallelism
Nonintersecting lines and the indirect proof
Parallelism─Section I
Parallelism─Section II
Parallelism─Section III
Uniqueness and existence
The parallelogram─Part I
The parallelogram─Part II
Test and review
Try this for fun
9. Parallelism in Space
Dihedral angles
Test and review
Try this for fun
10. The Angles of a Polygon
The angles of a polygon
A brief journey into non-Euclidean geometry
Test and review
11. Similar Triangles
Ratios and proportion
Theorems basic to the proofs of similarity
Similar triangles
Proving ratios equal and products equal
The right triangle
The theorem of Pythagoras
Test and review
Try this for fun
12. Coordinate Geometry─An Introductions
Plotting points
Distance between two points and dividing a line segment into any given ratio
Parallelism and perpendicularity
Test and review
Try this for fun
13. Coordinate Geometry─The Graph
The straight line
Intersection of two sets
Analytic proofs of problems from synthetic geometry
The graphs of inequalities
Locus of points
The circle
Test and review
Try this for fun
14. The Circle
Chords equidistant from the center of a circle
Tangents and secants
The sphere
The relation between angles and arcs
Applications of the theorems on angle measurement
Chords, tangent segments, and secant segments
Test and review
Try this for fun
15. Locus─Synthetic Geometry
Theorem, converse, inverse, and contrapositive
Locus theorems
Compound loci in synthetic geometry
Straightedge and compass constructions
More about construction with straightedge and compass
Test and review
Try these for fun
16. Inequalities
Inequalities
Test and Review
Try this for fun
17. Areas of Polygons and Circles
Area of the parallelogram, the triangle, and the trapezoid
Areas of similar triangles
Areas of regular polygons
Circumference of a circle
Area of a circle
Test and review
Try this for fun
18. Volumes
Volume of a prism
Volume of a pyramid
Surface area and volume of a cylinder and a cone
Volume and surface area of a sphere
Test and review
Try this for fun
19. Trigonometry
The trigonometric ratios
Some relationships between the trigonometric ratios
Test and review
Table of Squares and Square Roots
Index
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