(Note: this section requires familiarity with the tangent function. See "The Tangent" in the math refresher section.)

    The picture on the left is meant to represent the astronomer Claudius Ptolemy, who lived around 150 AD. It is an old picture, though not old enough for the artist to have actually known what Ptolemy looked like. But what is that gentleman holding?

    No, it isn't a religious symbol--the proportions are not right, and the marks on the stick do not seem appropriate. It is actually a cross staff (or "Jacob's staff"), a tool widely used by astronomers and navigators before the invention of the telescope, and for a while afterwards. It consists of a main staff with a perpendicular crosspiece, attached at its middle to the staff and able to slide up and down along it.

    The device was apparently invented by Rabbi Levi ben Gershon (1288-1344), a Jewish scholar who lived in Provence, in southern France, also referred to as "Gersonides." It is described in his Hebrew "Book of the Wars of the LORD," also translated to Latin; the name is borrowed from an ancient book whose only surviving trace is a casual reference in the bible, Book of Numbers, ch. 21, v. 14. Claudius Ptolemy lived more than 1000 years earlier, so the drawing on top indeed takes considerable artistic license.

    Astronomers used the cross-staff for measuring the angle between the directions of two stars. Other, older instruments for this purpose existed, used by scholars such as Hipparchos and Ptolemy, but none was as portable, which made the cross-staff eminently suitable for navigation at sea. Ships' officers used it to measure the elevation angle of the noontime Sun above the horizon, which allowed them to estimate their latitude (see section on navigation). The problem of getting dazzled by the Sun later led to the invention of the backstaff (pictured below on the left) where the sunlight fell onto a target, not into the eye. Columbus may well have used one. This was greatly improved around 1594 by Captain John Davis, so perhaps the "Mayflower" used the upgraded design.

    To measure the angle between two stars, an astronomer would place the staff just below one eye (drawing) and slide the cross-piece up and down. The cross-piece would have a pair of open sights sticking out perpendicular to the drawing at symmetric locations such as B and B' (often several pairs of sights, some spaced further apart than others). The astronomer would slide the cross-piece up and down, until sight B covered one of the stars and sight B' the other. For use at night, slits make convenient sights (see below).

    After that was achieved, the instrument would be lowered and the distance AC would be measured. Then if A was the angle between the staff and the direction of one star, from the definition of the tangent

    The distance BC between the sight and the stick was already known to the astronomer--so, using a table of tangents, the angle denoted by A could be calculated. Since the instrument was symmetric, the angle between the directions of the stars was 2A.

The materials:

• A wooden yardstick
• A wooden 1-foot ruler
• A piece of wood to serve as cross-piece,
          about the size of the ruler
          (a 2nd ruler is OK too).
• A thick paper (manila) folder, which will be cut up.
• (Optional) An empty cardboard box from
          cereal, frozen food, etc., with a glossy
          side. If not available, cardboard from
          the manila folder may be used.
• 4-6 rubber bands
• A stapler
• Two paper clips, and some paper.

(1)     Using the cardboard and the rubber band, make a "slider" on the yardstick, as follows. With a side of a cereal box, cut a strip 2" wide and 5" long, and wrap it around the yardstick, with the glossy side next to the wood; the cross-section should look approximately like Figure (1a). If you are using manila-folder material, make the strip about 1" longer and fold the end a few times as in Figure (1b), to make it press harder against the wood.
      Move the slider near the end of the yardstick and slip over it 3 turns of the rubber band--more turns for a longer band. None of the rubber should touch the wood, otherwise the slider won't move easily. The bands should fit fairly tightly, so that the slider is easily moved, but does not shift once placed in any position.
      Finally spread two turns of the rubber band to form an "X", as shown in Figure (2).

(2)       Carefully lifting the rubber loops, push the cross-piece through the "X", as shown in Figure (3). The crossing of the X should be at about the center of the cross-piece, which should be perpendicular to the yardstick. (However, no great accuracy is needed in meeting these conditions.)

(3)       Make two more sliders and place them on the cross-piece--one to the left of the yardstick, one to its right. These sliders should be narrower, only about 1.25" wide, and their rubber bands should also form an "X" shape.

(4)       From the cereal-box material, cut two strips 5/8" wide and 4" long (you may later shorten them). If you are using the manila folder material, make each strip 1 1/4" wide and then fold it to get double thickness.
      Slip each strip through the "X" in a direction perpendicular to the cross-piece, so that its middle is under the "X", then crease and bend them as in Figure (4), to support one of the cards you will make in the next step.

(5)       From the manila folder cut two cards, 5" wide and 2.5" high. In each card cut two slots about 1.25" from the edges, each about 1/8" wide (1/4" wide for rough measurements) and 1.75" long, as drawn in Figure 5.
      Insert each card between a pair of supports, as shown, making sure the slits are perpendicular to the cross-piece, as drawn. When everything is lined up, use the stapler to tie the card to its supports (the diagonal line in Fig. 5 is the staple). If the supports are too long, you may first cut off their ends--but their length does not affect performance.
   
Your cross-staff now should resemble the figure given at the beginning.



(6)       Always use either the inner pair or the outer pair of slits.
        Ignore the slits you do not use.

      To use the cross-staff, start by sliding the cross-piece to about the middle of the yardstick. Sight the yardstick along the line halfway between the stars whose separation you want to measure, then place the two sliders on the cross-piece so that the slits are approximately where the stars are located. Finally, adjust the yardstick slider until you see both stars flash simultaneously in both slits.

If that is too hard, try working with wider slits, such as 1/4" wide. Like all astronomy measurements, these too are best performed under a dark sky, with your eye used to the darkness. With a sky lit up by city lights or moonlight, you might find that you can only conduct measurements on bright stars.

      When you are satisfied that you have got both stars simultaneously in the slits, take down the cross staff and measure the distance B'B using your ruler.

      In the drawing at the start of this web page, the stick is exactly in the middle (CB = CB'), but do not worry if the distances do not turn out equal. Just measure the distance BB' and assume the stick is right in the middle.

    If the stars are very close, use the inner pair of slits and/or move the cross-piece further away on the stick. If they are widely apart, use the outer pair and perhaps go closer to the eye. Of course, use just one eye and close the other.

      The rest is just a calculation with tangents.