NATARAJAThe sailings — rhumb lines, meridional parts & the day's work
Learning outcomes & permitted supports
- Work a Mercator sailing both ways: course from tan C = DLong/DMP, distance from DLat·sec C — and run it back to the start
- Say what meridional parts ARE (stretched latitude minutes) and why they make the rhumb line straight on the chart
- Choose parallel, mid-latitude or Mercator sailing for the leg in hand, and state what the mid-lat method approximates
- Explain why the great circle is shorter than the rhumb it crosses — and by how much it matters on an ocean run
- Keep a day's work: courses and distances → DR position, the paper habit behind every GPS cross-check
Supports in this lab: Calculator and the meridional-parts idea; the engine uses the spherical MP formula (7915.7·log tan(45°+φ/2)) and says so where Norie's spheroidal table would differ. Golden 878/878 — every sailing runs forward and back to machine precision.
Glossary: rhumb line · meridional parts · departure · DLat & DLong · great circle · dead reckoning
Two roads between the same ports
On a Mercator chart the rhumb line draws itself straight — because the chart stretches each latitude by its meridional parts precisely so that a constant course IS a straight line. The great circle is shorter, but its course changes every mile. Drag the ports around and watch who wins by how much.
| method | course | distance | note |
|---|---|---|---|
| Mercator (exact rhumb) | 297.746° | 3222.0 nm | tan C = DLong/DMP · one course the whole way |
| Mid-latitude (plane) | 297.492° | 3249.4 nm | gap vs Mercator: 27.43 nm — grows with the latitude span |
| Great circle | 306.480° initial | 3203.9 nm | saves 18.1 nm, but the course keeps changing |
| DLat · DLong (shortest way) | 1500.0′ N · 3120.0′ W |
|---|---|
| Meridional parts A → B | 603.1 → 2244.3 (DMP 1641.2) |
| Departure (east–west miles made good) | 2851.5 nm W |
Meridional parts here are the SPHERICAL form (MP = 7915.7·log₁₀ tan(45°+φ/2)); Norie's printed tables add the Earth's ellipticity — at 45° the table reads 3013.4 against our 3029.9. Same idea, stated spheroid. The exam accepts the tables; the physics is this.
The day's work — dead reckoning, leg by leg
Course steered and distance run, three legs from position A: each leg is a little sailing problem forward. The engine runs each rhumb exactly, so the DR closes back through the Mercator relation — no mid-lat drift stacking up.
| leg | course | distance | arrives at |
|---|---|---|---|
| 1 | ° | nm | 9.3°N 70.1°E |
| 2 | ° | nm | 8.0°N 69.3°E |
| 3 | ° | nm | 7.8°N 66.8°E |
DR after the day's run: 7.8°N 66.8°E — this is the position every celestial fix in OP-15 is CORRECTING.