Brief description

The cross-staff (Jacob’s staff) is an early navigational and astronomical instrument used to measure the angular distance between two visible objects, such as the horizon and a celestial body (the Sun or Polaris).

Use / Function

• Latitude Determination: Measuring the elevation angle of Polaris or the noon Sun above the sea horizon to determine latitude.
• Surveying: Measuring horizontal and vertical angles between land landmarks, buildings, or mountain peaks.
• Scale: Personal and portable field instrument.

Operating principle

The cross-staff operates on right-triangle trigonometry. A cross-piece (transom) slides along a long, square main shaft (staff). By holding the end of the staff close to the eye and sliding the transom until its top edge aligns with the upper target (e.g., a star) and its bottom edge aligns with the lower target (e.g., the horizon), similar triangles are created. The distance along the graduated scale on the main staff directly corresponds to the subtended angle:

$$\tan\left(\frac{\theta}{2}\right) = \frac{\text{Transom Length} / 2}{\text{Distance from Eye}}$$

How to create it

1. Shape Main Staff: Craft a straight, square wooden beam (approx. 1 meter long) from hard, warp-resistant timber like oak or boxwood.
2. Fabricate Transoms: Cut one or more perpendicular wooden cross-pieces of known precise lengths.
3. Fit Sliding Sleeve: Create a tight central brass or wooden collar on the transom so it slides smoothly along the staff without wobbling.
4. Graduate Scales: Calculate trigonometric angles and engrave scale markings on each of the four faces of the main staff, calibrated for specific transom lengths.
5. Attach Sight/Vanes (Optional): Add small brass end sights or pinholes to improve alignment precision.

Materials needed

• Essential materials:
  • Wood: Dense, well-seasoned wood (boxwood, mahogany, oak) for shaft and cross-pieces.
• Tools:
  • Chisel, Plane, and Saw: For shaping timber accurately.
  • Carving gouge or scribing tool: For marking scale graduations.
• Possible substitutes:
  • Brass or Bone: For sliding sleeve bushings to prevent wear on wooden contact surfaces.

Variants and improvements

• Multi-Transom Cross-Staff: Uses three or four interchangeable transoms of different lengths to measure different angle ranges (e.g., 5°–15°, 10°–30°, 20°–60°).
• Backstaff (Davis Quadrant): Developed later to allow observers to measure solar elevation with their back to the Sun, avoiding direct eye strain or blindness.

Limits and risks

• Eye Hazard: Direct observation of the Sun can cause severe eye injury or blindness unless dark filter glass or indirect shadow technique is used.
• Parallax and Eye Position: Variations in eye placement at the end of the staff introduce measurement errors.
• Ship Motion: Difficult to maintain steady dual alignment on a rolling vessel at sea.
• Scale Compression: Accuracy drops significantly at high altitude angles (>60°) due to geometric scale compression.