Discuss with respect to celestial navigation the formulation of Nories Nautical Ex­ meridian tables

Excellent question. Discussing the formulation of Norie’s Ex-Meridian tables takes us to the very heart of the practical genius behind celestial navigation. It’s a classic example of how complex spherical trigonometry was simplified into a straightforward, tabular method for the working mariner.

Let’s break down the discussion into several key parts:

1. The Navigational Problem: What are Ex-Meridian Tables for?
2. The Theoretical Foundation: The Spherical Trigonometry behind the sight.
3. The Simplification: How the complex formula is made practical.
4. The Formulation of Norie’s Tables: How the tables are structured to solve the simplified formula.
5. The Second Correction: An additional refinement for greater accuracy.

1. The Navigational Problem: The Need for Ex-Meridian Sights

The simplest and most reliable method for finding latitude at sea is the Meridian Passage or “Noon Sight.” At the exact moment the sun (or any celestial body) crosses the observer’s meridian, it reaches its maximum altitude. At this instant, the calculation for latitude is simple arithmetic:

Latitude = Declination ± (90° - Maximum Altitude)

The problem is that it’s easy to miss the exact moment of meridian passage. Clouds might obscure the sun, the ship’s motion could make a sextant reading impossible, or the navigator might be occupied with other duties.

An Ex-Meridian (or “near-meridian”) observation is a sight taken a short time before or after meridian passage. The altitude observed will be slightly less than the maximum altitude at meridian passage. The purpose of Ex-Meridian tables is to calculate the small correction that must be added to this observed altitude to find what the altitude would have been at meridian passage.

2. The Theoretical Foundation: The PZX Triangle

All celestial navigation calculations are based on solving the PZX (Pole-Zenith-Celestial Body) spherical triangle.

• P: The elevated celestial pole (North or South).
• Z: The observer’s zenith (the point directly overhead).
• X: The position of the celestial body (e.g., the Sun).

The sides and angles of this triangle are:

• Side PZ: Co-Latitude = 90° - Latitude
• Side PX: Polar Distance = 90° ± Declination
• Side ZX: Zenith Distance = 90° - Altitude
• Angle at P: Hour Angle (t or LHA)

The relationship between these elements is given by the Spherical Cosine Rule: cos(ZX) = cos(PZ) * cos(PX) + sin(PZ) * sin(PX) * cos(t)

Substituting the navigational terms: sin(Altitude) = sin(Latitude) * sin(Declination) + cos(Latitude) * cos(Declination) * cos(t)

This is the fundamental formula.

3. The Simplification: From Cosine Rule to Practical Correction

Our goal is to find the difference between the observed altitude (Alt_obs) at a small hour angle t, and the meridian altitude (Alt_m) where t = 0.

Let the correction be x, such that Alt_m = Alt_obs + x.

From the cosine rule:

• sin(Alt_obs) = sin(Lat)sin(Dec) + cos(Lat)cos(Dec)cos(t)
• At meridian passage, t=0 and cos(t)=1. Therefore: sin(Alt_m) = sin(Lat)sin(Dec) + cos(Lat)cos(Dec)

The difference is: sin(Alt_m) - sin(Alt_obs) = cos(Lat)cos(Dec) * (1 - cos(t))

This is the key relationship. The correction we need is related to Latitude, Declination, and the Hour Angle (t). For the small angles involved in an ex-meridian sight, two mathematical approximations are crucial:

1. Small Angle Approximation for Sine: For a small correction x (in radians), sin(Alt_m) - sin(Alt_obs) ≈ x * cos(Alt_m).
2. Half-Angle Formula: (1 - cos(t)) = 2 * sin²(t/2). For a small angle t (in radians), this is approximately t²/2.

Combining these, we get: x * cos(Alt_m) ≈ cos(Lat)cos(Dec) * (t²/2)

Solving for the correction x: x ≈ (t²/2) * [cos(Lat) * cos(Dec) / cos(Alt_m)]

Since Alt_m is related to Lat and Dec (Alt_m = 90 - (Lat ~ Dec)), this can be rewritten as: x ≈ (t²/2) * [cos(Lat) * cos(Dec) / sin(Lat ~ Dec)] (Note: Lat ~ Dec means Lat - Dec if they are the same name, and Lat + Dec if contrary name).

This is the formula that Norie’s tables are designed to solve. The correction is proportional to the square of the time from meridian passage (t²).

4. The Formulation of Norie’s Nautical Ex-Meridian Tables

Instead of forcing the mariner to calculate t² and cos(Lat), cos(Dec), etc., Norie’s brilliance was to break the formula down into pre-computed parts. The tables are typically structured in two main parts.

Let’s rewrite the correction x in a form suitable for tables:

Correction (x) = [A] * [B]

Where:

• [A] is a factor depending only on the Hour Angle (t).
• [B] is a factor depending on the Latitude and Declination.

Table I: The Time Factor (The “A” Value)

• Argument (Input): Hour Angle (t), expressed in minutes and seconds of time from meridian passage.
• Formulation: This table calculates a value proportional to t². The constant of proportionality is chosen to make the final units come out as minutes of arc (') for the correction. The table essentially pre-computes (t²/2) scaled by a constant. Because the correction varies as the square of the time, doubling the time from noon quadruples the correction.

Table II: The Latitude & Declination Factor (The “B” Value or Multiplier)

• Arguments (Inputs): Observer’s DR Latitude and the Sun’s Declination.
• Formulation: This table pre-computes the term cos(Lat) * cos(Dec) / sin(Lat ~ Dec). The user enters the table with their approximate latitude and the body’s declination to find a single multiplier.

The Practical Process:

1. Navigator determines the hour angle (t) in minutes of time from local noon.
2. Enter Table I with t to find a value (let’s call it the “At” factor).
3. Enter Table II with DR Latitude and Declination to find a multiplier.
4. Correction = (Value from Table I) × (Multiplier from Table II).
5. This correction is then added to the observed altitude (Ho) to get the computed Meridian Altitude.
6. Latitude is then found using the standard Meridian Passage formula.

5. The Second Correction

The t² approximation works very well for hour angles up to about 15-20 minutes. For larger hour angles, or in high latitudes where the sun’s altitude changes more slowly, the approximation begins to fail.

Norie’s tables therefore include a Second Correction.

• Formulation: This correction accounts for the next term in the Taylor series expansion of the cosine function. It is proportional to t⁴ and also depends on the altitude itself. In practice, it’s calculated using a simpler formulation: Second Correction = C × (First Correction)², where C is a factor found in a small table, usually entered with the tangent of the meridian altitude.
• Application: The second correction is almost always subtracted if the first correction is large. It is a small refinement to the much larger first correction.

Summary of Formulation

In essence, the formulation of Norie’s Ex-Meridian tables is a masterful piece of practical science. It:

1. Starts with the exact Spherical Cosine Rule.
2. Uses mathematical approximations (t²) valid for the specific conditions of a near-meridian sight.
3. Deconstructs the resulting formula x ≈ (t²/2) * [cos(Lat)cos(Dec) / sin(Lat~Dec)] into two independent parts.
4. Tabulates these parts separately (a time-dependent part and a latitude/declination-dependent part).
5. Allows the navigator to solve a complex trigonometric problem by looking up two numbers in tables and performing a single multiplication, making the process fast, simple, and robust in a pre-electronic era.