PRINT LINKED RELATED

MacLane's Postulates

    In the 1960's Saunders MacLane proposed a new set of axioms that are more like Birkhoff's than Euclid's or Hilbert's axioms. Like Birkhoff's axioms, MacLane's axioms also use the real numbers; and so has fewer axioms than Hilbert's axioms did. MacLane's approach also does not generate any new theorems in Euclidean geometry but rather simplifies previous attempts at erecting the foundations of Euclidean geometry. The main differences are that MacLane uses a function to measure distance between points (called a metric function) and the so called Continuity Axiom, which incorporates the Crossbar Proposition into Euclidean geometry as an axiom. This topic states MacLane's axioms, from which all of Euclidean geometry can be proven.

Comment (Undefined Terms) Point, distance, line, and angle measure.

Axiom (Distance Axioms)
    (i) There are at least two points.
    (ii) If and are points, is a nonnegative number.
    (iii) For points and if and only if
    (iv) If and are points,

Axiom (Line Axioms)
    (i) A line is a set of points containing more than one point.
    (ii) Through two distinct points there is one and only one line.
    (iii) Three distinct points lie on a line if one of them is between the other two.
    (iv) On each ray from a point and to each positive real number there is a point with

Axiom (Angle Axioms)
    (i) If   and are rays from the same point, is a real number modulo
    (ii) If and are three rays from the same point, then
    (iii) If is a ray from and is a real number (modulo 360), then there is a ray from such that
    (iv) If then if and only if

Axiom (Similiarity Axiom) If two ordered triangles and have and (for positive) they are similiar.

Axiom (Continuity Axiom) Let be proper. If is between and then Conversely, if then the ray meets the interval

Geometry Books

Calculus (With Analytic Geometry)(8th edition) List Price: $209.95 Buy Used: $70.00 You Save: $139.95 (67%) New (56) Used (166) from $70.00Designed for the three-semester calculus course for math and science majors, Calculus continues to offer instructors and students new and innovative teaching and learning resources. This was the first (more)

The Misbehavior of Markets: A Fractal View of Financial Turbulence List Price: $17.95 Buy New: $10.24 You Save: $7.71 (43%) New (7) Used (5) from $7.48From the inventor/founder of fractal geometry, the award-winning book that turns modern financial theory on its head Mathematical superstar and inventor of fractal geometry, Benoit Mandelbrot, has spent (more)

The Fractal Geometry of Nature Buy New: $31.00 New (7) Used (15) from $27.90Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line. The complexity of nature's shapes differs in kind, not merely degree, from that of the shapes of ordinary (more)

The Golden Ratio: The Story of PHI, the World's Most Astonishing Number List Price: $14.95 Buy Used: $4.53 You Save: $10.42 (70%) New (38) Used (57) Collectible (1) from $4.53Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale (more)

Trigonometry (9th Edition) List Price: $138.67 Buy New: $94.00 You Save: $44.67 (32%) New (28) Used (20) from $92.99Over the years, the text has been shaped and adapted to meet the changing needs of both students and educators. As always, special care was taken to respond to the specific suggestions of users and reviewers (more)

Geometry List Price: $94.88 Buy Used: $6.95 You Save: $87.93 (93%) New (29) Used (114) from $6.95Geometry (more)

Geometry Workbook For Dummies (For Dummies (Math & Science)) List Price: $16.99 Buy New: $3.83 You Save: $13.16 (77%) New (38) Used (20) from $3.83Geometry is one of the oldest mathematical subjects in history. Unfortunately, few geometry study guides offer clear explanations, causing many people to get tripped up or lost when trying to solve a proof?even (more)

Geometry For Dummies (For Dummies (Math & Science)) List Price: $19.99 Buy New: $8.41 You Save: $11.58 (58%) New (52) Used (11) from $8.40Learning geometry doesn?t have to hurt. With a little bit of friendly guidance, it can even be fun! Geometry For Dummies, 2nd Edition, helps you make friends with lines, angles, theorems and postulates. (more)

Euclid's Elements List Price: $29.95 Buy New: $24.45 You Save: $5.50 (18%) New (1) Used (1) from $24.45Green Lion Press has prepared a new one-volume edition of T.L. Heath's translation of the thirteen books of Euclid's "Elements" In keeping with Green Lion's design commitment, diagrams have been placed (more)

College Algebra with Modeling and Visualization (3rd Edition) (Rockswold Series) List Price: $138.67 Buy Used: $18.00 You Save: $120.67 (87%) New (24) Used (124) from $18.00Gary Rockswold focuses on teaching algebra in context, answering the question, "Why am I learning this?" and ultimately motivating the students to succeed in this class. In addition, the author's understanding (more)