So what is the idea, and why is it so powerful? Well the idea is in a word: symmetry. It's very useful for many reason to have objects that have a circular symmetry. Think of the wheel and what an important discovery that was and you'll have some idea of the power of a circle. On top of being useful, circularly symmetric objects have a certain beauty and elegance which cannot be denied.
Throughout history we see evidence of round pillars etc. in architecture and art, and the lathe was one of the first instruments to allow for easily reproducible circular objects. Can you imagine the difficulty in trying to carve something into a perfect cylinder manually, it would require constant measuring an readjustment, however a wood lathe, will do this simply by the nature of how a piece is carved.
As long as your spindle is well centered at both ends, you can't help but produce a piece that is perfectly round. This is the power of the wood lathe. It allows for automatic rounding. The definition of a circle is a shape where every point is exactly the same distance from the center, and this is exactly what a lathe does. By keeping your lathe cutting tool a fixed distance from the center of your turning spindle, you carve the piece of wood such that every point on it's surface is an equal distance from the center.
A lathe by nature is not a complicated tool, although some modern lathes certainly appear to be, with computer controls and variable speeds. At it's simplest, and in it's earliest forms a lathe was little more than one person turning a piece of wood, maybe using some string, and another person carving it with a sharp tool. This simple design manages to get the job done however, and subsequent designs may have improved the speed and reliability of the turning process but have kept the same fundamental idea.
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