Every now and then comes a book that squeezes a lot of informative material into a highly readable and fun package. Jim Tolpin and George Walker's latest effort, From Truth to Tools, is just such a book. Sure, it's loaded with the geometric principles that underlie many of the tools we use in carpentry and cabinetmaking. It offers up a bit of history on the ancient uses of these truths, the basic geometric principles, to accomplish great works without modern equipment. And it schools us gently in the roots of our terminology that lie in their Greek and Latin origins. Sound boring? Not at all. This is a fun read that's easy on the mind and informative at the same time.

Everything geometric begins with a single point, without which nothing else is possible. The awl is the mechanical manifestation of the point. Attach two awls and you end up with dividers and can now mark off distances. From these, standard units of measure were developed—cubits in ancient Egypt defined by the length of the forearm of whomever happened to be Pharaoh at the time, the foot in European application and now the meter. Story sticks, folding rules and tape measures are the linear measurement tools deriving from this truth.

Add a pencil to dividers and create a compass. Now you can draw circles. On a longer stick, the compass becomes a trammel beam. As Tolpin and Walker explain, circles are the foundation for much valuable geometry and many of the remaining principles are based on work with circles. Draw two circles having their center points on the rims of each other and you've created a condition of symmetry. The line connecting the centers of these circles equals their radii and leads to the concept of the line segment. In practical terms, this shows up as a string line and, better, a chalk line. Draw a line through the center of symmetrical circles and you've created a condition of bilateral symmetry, the truth underlying a perfect straightedge.

Then there are marking tools: the marking awl, birdcage or brad awl, and long-shafted scribing awl. Other line-creating tools are markers and knives, and Tolpin and Walker discuss when and how best to use them in woodworking.

When you connect the center points of symmetrical circles with their intersection you form a triangle. This is the foundation of surfaces called planes and to the handplane, which is essentially a rectangle with a bit of iron set in a triangular position. Chisels and saw teeth employ the same triangular shape.

Draw lines through the origins of the symmetrical circles that connect with the intersections of their rims and you've created a set of parallel lines. These are the truth that underlies yet more tools. Winding sticks are one example, as are marking gauges and table saw rip fences. A taper gauge lays out lines parallel to the edge of a tapered board. Boat builders use another variant called a diminishing stick. Deriving from the truth of parallelism are yet other tools: the miter block, bevel gauge and the transfer gauge used to determine where to cut baseboard for a perfect fit, among others.

Returning to the circle, Tolpin and Walker show how a square is defined by constructing triangles inside a circle encompassing a pair of symmetrical circles. This defines right angles that lead to the square and try square, framing square, pinch sticks and the resultant angle square developed by Peter Galbert to lay out the rake and splay of chair legs.

Add a bit of lead to a string and you have a plumb line, the basis for yet other tools such as the mason's level and center finder.

Tolpin and Walker return to the sector, discussed in their earlier books, a proportional calculator and forerunner to the slide rule used until the end of the 18th century to measure whole number relationships. Proportional dividers, used to scale up or down drawings, derive from the same principle as sectors.

Angles can be set using a scale of chords or a protractor, and they describe how to construct one.

Not all tools are for small-scale use. In laying out large structures, such as bridges and, yes, pyramids, various tools have been used in the field. The triplet uses a 3-4-5 relationship in the legs of a triangle as the foundation for other tools—the carpenter's ten-foot pole and the thirteen-knot cord that are used to set right angles on large scale projects. The groma and merchet used plumb lines and line-of-sight to establish right angles over a distance, such as across a river.

The chorobate was a table used by the Romans that incorporated sight lines and a water trough for measuring levels. This ancient tool is succeeded by today's spirit level.

They explain how geometry is also used by loggers, using a felling gauge that is essentially a quadrant, to judge where a tree's top will land when the tree is felled.

The final section explains how Eratosthanes calculated the earth's circumference, or geometron, 2300 years ago by employing several of the truths laid out in the book.

An appendix includes references to downloadable templates for a scale of chords, sector, and Peter Galbert's resultant angle square, and to a resource for further exploration of intuitive math.

This book is charmingly illustrated throughout and comes in a hardbound edition with full color cover. You don't have to be a mathematician or a history scholar to fully appreciate the book. It's accessibly written and laid out in logical, easy-to- digest bites. Any woodworker with an interest in understanding the geometric underpinnings of carpentry and cabinetmaking will both appreciate and enjoy it.

J. Norman Reid is a woodworker, writer, and woodworking instructor living in the Blue Ridge Mountains with his wife, a woodshop full of power and hand tools and four cats who think they are cabinetmaker's assistants. He is the author of Choosing and Using Handplanes. He can be reached by email at nreid@fcc.net.