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While not required, it is recommended you have a good grasp on these concepts if you want to get the most from this page:
G is for Geometry - Basics of the working tools
The Sky Was Never Moving - The motions of the Earth in the cosmos
The Astrolabe - View the night sky from one perspective.
Read the summary shown below or jump into the Square & Compass Tool
First, Let's Talk About the Symbols
First, Let's Talk About the Symbols
Symbols function as compressed carriers of meaning, allowing complex structures—mechanical, moral, or cosmological—to be preserved in simplified form. An allegory of symbols operates by mapping observable forms to underlying processes: a circle may stand in for cyclic recurrence, a square for constraint or boundary, a compass for measured expansion. The surface image is not the message itself but a stable encoding of relationships that can be reconstructed by a trained observer. This makes symbols durable across time, especially in contexts where direct technical language is unavailable or likely to be lost. The limitation, however, is ambiguity—without a shared framework, the same symbol can drift into purely decorative or mythic interpretation, losing its operational meaning. The allegory, then, is not just symbolic storytelling; it is a method of preserving instruction in a form resilient to cultural and linguistic decay.
As an example, I'll paraphrase "Born in Blood" by John J. Robinson. Take a look at the American Flag. You see 13 stripes representing the colonies, 50 stars for the states, and when you reminder yourself of these things you suddenly remember the story that made that flag. Without this reference, the image is open to interpretation that is not there.
Set your frame of reference.
Set your frame of reference.
Look South, stretch your arms along East and West, and then lean back until you see the pole star (Polaris). This axis is your plumb line.
Hold a construction square against your chest, with one leg protruding outward, like your nose. This defines your local right angle.
From this position, the sky will appear to rotate across your field of view, from left to right.
To simplify what we see, make a few useful assumptions:
1. The Earth is still.
Your brain favors relative motion over absolute motion. For our purposes, we remain fixed and let the sky move.
2. Coordinates are assigned, not inherent.
Objects do not move in X, Y, and Z directions by themselves—the observer defines the frame, the cosmos moves its direction regardless how we decide to define it.
3. The sky is projected onto a surface.
We treat the cosmos as if it is projected onto a curved surface and then flattened into a 2D view.
From a single point—your eye—everything is defined by angle and distance.
With the right frame and these simplifying assumptions, we can begin to measure the Earth, the Moon, and the Sun—and take the first steps into geometry.
Set the Frame
Set the Frame
Find your bearings
Before you measure anything, you need a stable reference.
Use the horizon, the Sun’s path, and Polaris to establish east, west, and true north. This defines your frame—everything else depends on it.
Find Your Position
Find Your Position
Locate yourself from the sky
Once your frame is set, the sky tells you where you are.
By measuring the height of Polaris and the Sun at noon, you can determine your latitude from a single observation point.
Read the Cycle
Read the Cycle
Track time through motion
The Sun’s path changes every day.
Watch how the horizon spread and noon height shift over time to determine the length of the day and your position within the seasonal cycle.
Square and Compass
Square and Compass
Understand the system
The same structure you used to measure the sky is captured in the square and compass.
A fixed axis and a moving arc—together they describe orientation, position, and time as one unified system.
And the Moon, too
And the Moon, too
By measuring how the Moon moves relative to the Sun, you begin to uncover the structure of its orbit.
The Moon’s horizon spread and meridian height reveal that its path is slightly tilted compared to the Sun, giving its orbital inclination. Over longer periods, you’ll notice those extremes slowly expand and contract, exposing the 18.6-year nodal cycle. Tracking how the Moon shifts against the Sun day by day reveals its period, while comparing its position over time introduces parallax-like effects—your first step toward measuring distance.
Angular Velocity (Time Derivatives)
Angular Velocity (Time Derivatives)
Once you move from position to change over time, the system reveals deeper behavior.
By tracking how the Sun’s spread and noon height shift day to day, you begin to see that its motion isn’t uniform—seasons progress at different rates, revealing Earth’s elliptical orbit. That same non-uniform motion leads to the Equation of Time, showing that solar time drifts from clock time and requires correction.
a Second Location to determine distance
a Second Location to determine distance
This is where scale enters the system.
When two observers measure the same Moon at the same time from different locations, the difference in their viewing angles reveals more than just motion—it reveals size and distance. From that single comparison, you can determine the Earth’s radius, estimate the Moon’s distance through parallax, and begin to establish the absolute scale of the entire system.
The Earth's Curve
The Earth's Curve
This is the simplest real-world step.
By measuring a stick’s height and the length of its shadow, you can directly calculate the Sun’s altitude from your location. Repeat the same measurement at different locations, and the change in angle begins to reveal the curvature of the Earth itself.
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