Law of average (rule of the two diameters)

The Law of average (rule of the two diameters) turns a piece of field measurement into an ethical analogy: the crossing point of Walden Pond’s greatest length and breadth, where its greatest depth lies, becomes a model for locating character’s depth amid the varied contours of a life. The image gives Thoreau a way to connect observed geography with moral interpretation, while keeping the passage’s starting point in a specific act of surveying rather than abstract speculation.

Surveying the pond

Desiring to recover Walden Pond’s long-lost bottom, Thoreau surveyed it before the ice broke up in early 1846, using a compass, chain, and sounding line. He mapped the pond at a scale of ten rods to an inch and entered more than a hundred soundings. The procedure mattered: the analogy emerges from measurements made across an irregular body of water, not from a diagram imposed on a perfectly symmetrical lake.

On the map, the greatest depth appeared near the center. Thoreau laid a rule lengthwise and then breadthwise, finding that the line of greatest length crossed the line of greatest breadth exactly at the point of greatest depth. The coincidence surprised him because the pond’s middle was nearly level, its outline far from regular, and its extreme length and breadth had been obtained by measuring into coves. The result is striking precisely because the shape is not a simple oval whose center would make the answer obvious.

The finding prompts him to wonder whether the same rule might lead toward the deepest part of the ocean, or even help explain the height of mountains as the opposite of valleys. He notes that a hill is not highest at its narrowest part. The language remains exploratory: the pond suggests a possible general principle, and the map supplies an instance to think with, rather than proof that every landscape obeys a universal formula.

From geography to character

Thoreau names the inference drawn from the pond the law of average. Its two intersecting measures, he proposes, can guide thought toward the sun in the system and the heart in a person. For human character, the lines are not literal lengths on a map: they pass through the aggregate of particular daily behaviors and the waves of a life. Their intersection indicates a person’s height or depth, while the life’s coves and inlets keep the figure from becoming a simple, featureless outline.

This method treats conduct over time as evidence. A single deed or inclination is only one contour; the broader pattern of a person’s days supplies the length and breadth from which character can be inferred. Thoreau suggests that a person’s shores and adjacent country—the circumstances around them—may help disclose a concealed bottom. The analogy therefore resists judging character from an isolated surface impression: interpretation requires attention to the pattern and the conditions that shape it.

The surrounding country can suggest depth without mechanically determining it. Mountainous circumstances and an Achillean shore, with peaks overshadowing and reflected in the person’s bosom, imply a corresponding depth; by contrast, a low, smooth shore indicates shallowness on that side. Thoreau also compares a bold, projecting brow with a corresponding depth of thought. These are proposed correspondences in his moral geography: outward form and circumstance become clues to inward life, but the passage presents them as interpretive analogies, not as measurements that settle every judgment.

Inclinations as coves

The image of depth grows more particular when Thoreau turns from a person’s whole character to individual inclinations. Every cove, he says, has a bar across its entrance; each inclination is a harbor in which a person may be detained and partially land-locked for a season. The figure captures both attraction and confinement. A disposition offers a sheltered place for a thought, but the same enclosure can keep that thought from moving freely back into the wider life.

Inclinations are not usually whimsical in this account. Their form, size, and direction are shaped by the shore’s promontories and by ancient axes of elevation: features accumulated in the landscape stand for older formations in a person’s nature. The metaphor makes an inclination intelligible as part of a larger structure, not merely a passing preference. What becomes a harbor has a history in the contours that surround it.

The bar can change. Storms, tides, and currents may gradually raise it, or the water may subside until the bar reaches the surface. Then what began as an inclination in the shore, a place where a thought was harbored, becomes an individual lake, cut off from the ocean. The thought acquires its own conditions: it may change from salt to fresh, becoming a sweet sea, a dead sea, or a marsh. The sequence gives the analogy a dynamic edge. Character is not only read from a settled map; inclinations can deepen into separated worlds as conditions alter their connection to the wider current of life.

Method and significance

The rule’s force comes from the movement between observation and inference. Thoreau first records a real pond by scale and soundings, notices an unexpected intersection, and then tests the finding as a way of thinking about character. His language shifts openly from the measured lake to ethics. The result is neither a detached natural-history report nor a claim that moral life can be reduced to geometry: the survey furnishes an image through which daily behavior, circumstance, and inward depth can be considered together.

The analogy also holds two ideas in tension. A person’s character has a depth that is not immediately visible, yet that depth may be approached through the pattern of conduct and the shape of a life’s circumstances. At the same time, coves, bars, and changing waters complicate any simple reading: people can be temporarily sheltered, partially enclosed, or transformed as their inclinations acquire new boundaries. The map offers a rule for looking, not a complete account of everything hidden beneath the surface.

In this sense, the pond’s irregular outline is essential to the moral comparison. The greatest depth does not announce itself through a perfectly regular boundary; it appears where the measured dimensions cross despite the nearly level middle and the coves that complicate the extremes. Thoreau makes that coincidence a prompt to examine a life in its breadth and length, rather than to mistake its visible outline for its whole measure.