There are several ways that one might produce a line equal to a given line.
A carpenter might use a tape measure or a story-stick. A story-stick is an unmarked object of equivalent length to another. Its advantage lies in that it eliminates measurement. There are no mistakes of calculation. One simply uses the concrete object as a guide in reproducing equivalent lengths.
Both the tape measure and the story-stick have something in common, they relieve the woodworker of the need to comprehend the objects themselves. The woodworker relies upon his tools. In a sense, the tools know the measurements in a way that the woodworker does not.
These tools allow one to make highly precise measurements which would be nearly impossible by the eye alone. But they are also the reason why this aspect of woodworking is a craft and not a science.
Let’s think about how Euclidean Geometry deals with this. It is often stated that classical geometry is the geometry of the compass and straightedge. This is only partly true. If we mean that classical geometry seeks to produce that which can be constructed with these tools alone we speak the truth (that is until we enter into solid geometry). But if we mean that the compass and straightedge are the measures by which classical geometry attains to its status as a mathematical science, this is a misunderstanding which must be corrected.
Book I, Proposition 2 seeks to reproduce a given line at a given point. Euclid by no means suggests we use a compass to measure or reproduce this point. In fact, the Elements, his classic textbook, nowhere mentions this tool.
Instead, the construction is a purely mental act which can be aided by imagery made with compass and straightedge, but is in no way limited to or dependent upon such tools.
What is the significance of this fact?
It means that unlike the woodworker, unlike someone using tools to aid the mind or hands, the geometer actually intellectually grasps the equality of the two lines in the production directly by means of the intellect alone. There is no mediator between the geometer and the objects of geometry.
While a mechanical art is a knowledge of how to do something, Geometry, and all apodictic or demonstrative science is knowledge of things themsleves.
This is no small difference. It is the difference between someone who knows that x² is the formula for the area of a square, and someone who knows why. It is the difference between someone who listens to their guide who tells them which plants are poisonous, and someone who can identify them of her own. It is the difference between someone who gets a hug every time they are kind to a sibling, but knows not why, and someone who knows that his happiness is loving God and his neighbor.
Aristotelian or classical science means we have actual insight into the ‘why’ of things. Geometry is an amazing example of such a science. It provides those ‘aha’ moments because it directly involves us in the act of inductive insight (or mathematical deduction).
Perhaps we can hear with new respect the fact that the Latin word science (as its Greek counterpart) simply means ‘knowledge’. To possess a science in this sense was to have direct knowledge of the world.
Perhaps we can also sense how modern experimental or instrumental science fails in this regard to give the same kind of direct access to nature, and thus cannot satisfy the intellect in the same way. For all its power and efficiency, modern science often fails to fully engage us in this manner. Modern science can examine the cloth of a jacket, but it has forgotten how to look in the face the man who wears it.
In studying Euclid, we simply remember that knowledge and our encounter with it is ultimately meant to be intimate, to be personal, and that now and again, our longing to encounter reality face to face truly can be satisfied.