## The Cross-Staff

The cross-staff was first described, as an astronomical tool, by the Medieval Jewish scholar Levi ben Gersohn about 1342. In order to accommodate a broad range of angles he used a long staff, and was probably aided by an assistant to move the cross-piece in taking measurements. These early cross-staffs had only a single scale. For the book on the cross-staff see The Cross-Staff by Morser-Bruyns.

The cross-staff above is made of teak. The staff is 1/2" square stock cut from a plank and planed smooth by hand. The cross-pieces are of 1/8" x 1 1/2" sheet cut from 1" stock with a table saw and planed. They are glued and nailed to square bolsters with the edges ogee'd with a router. The 1/2" square holes were cut with a mortising machine after assembly. The staff is 32" long and graduated on the four sides from 5° - 16°, 10° - 30°, 20° - 60°, and 40° - 90°. The graduations are cut into the wood, while the figures are stamped. The graduations were determined by calculation to correspond to the cross pieces. The cross pieces are 2 1/2", 5", 10", and 20" long, corresponding to the respective scales on the staff. The eye-end of the staff is blunt while the far end is pointed, in accordance with navigational tradition (probably to enable ease of use at night).

Note that for the cross-staff the staff can be calibrated in any linear scale. The only important information is the ratio of the length of the cross-piece to the length measured on the staff! Thus one could measure both in, say, rice grains, little finger widths, fly's eyes, or whatever! Trigonometry can then be used to determine the angles at leisure. (A leisurely pace would not be uncommon in astronomy, however, it might be more trouble for rangefinding for artillery!)

For most instruments of course, one does the calculations in advance, and graduates the staff directly in degrees (note that in this case the scales will not be linear). Thus to determine the angular distance between two objects one slides the cross-piece along the staff until the two lines-of-sight just graze the ends, as shown in the illustration. The angle would then be read directly off the graduations on the staff where the cross piece intersects it.

Each line of sight forms the hypotenuse of a right triangle with the staff and cross piece as the adjacent and opposite sides, respectively. To layout the Staff then we can look at one of these triangles (which now encompass half of the observed angle) as seen below:

Trigonometric Calculation Method: The tangent of angle A is then Tan = opposite side/adjacent side = a/b. For a cross piece of given length we then have length = 2a, and the angle measured = 2A, with the length along the staff equal to b. Thus b = a/(tan A). For example, to find the length to mark off along the staff for the angle 20° with a 6 inch cross-piece, we have: b = (6/2) / (tan 20°/2) = 3 / (tan 10°). For the modern staff-maker this is an ideal place to utilize a spread sheet. For illustration you may check out my sample table of angles for 5-90° by 5°. If you are interested in setting up your own spread sheet I have provided a table of sample calculations based on Excel. If you want to make a number of cross-staffs then it is more efficient to make a layout board rather than measure the distances individually. Thus for the workshop described below I used the calculated distances at 5° intervals to make a layout board for 3" and 9" cross pieces. The board itself is 5" wide and 4' long (only about half is shown):

Geometric Method: A second way of determining the distances along the staff is the Geometric method. In this method a protractor is used to draw the various angles one is interested in on a board or card the length of the staff and the width on one half of the cross piece. The distances are then determined by the intersections of the angle lines with the edge of the board. One may now lay the staff along the board and mark off the distances for the various angles. For the workshop described below I made a board marked at 10° intervals for cross pieces of three different lengths, 3", 6", and 9". Thus there are three lines above and parallel to the center line:

The board itself is 10" wide and 4' long (only half is shown in the illustration). Both edges may be used in layout. For the 3" cross piece one looks at the intersection of the angle with the line closest to the center line (1 1/2" out = 3 x 0.5).

Copy Method: A third traditional method of calibration is to simply copy another staff. In this case one lines up the two staffs, and using a square and marking tool transfers the markings from the finished staff to the new staff. The cross pieces of the two staffs must also match.

Making a Simple Cross-Staff

This cross-staff was one of the projects for my 1998 workshop, "Medieval Scientific and Philosophical Instruments."

Materials (provided at the workshop):
• Hardwood dowel, 1/2" diameter x 4' long (sight along the dowel before using it to make sure it is straight). A three foot dowel would be more realistic for use by a single individual. the four foot instrument requires an assistant for effective use of the full length, but it does allow a wider range of angle measurement.
• Cross pieces, 3/4" x 1 1/2" x 3" and 9" long, respectively. A 1/2" hole is drilled in the center of each, and the ends are chamfered for better sighting, as illustrated below:

Construction:

Lightly sand the cross pieces to remove splinters and provide a good finish. Caution do not change the length of the cross piece while sanding!

• Check to make sure that your cross pieces slide down the entire usable length of your dowel. A tight fit is OK - it can be waxed after graduation.
• Choose whether you are going to use the geometrical or calculation method.
• Line up the end of your staff with the origin of your rule or geometrical layout board. (The layout boards at the workshop have stops at the origin end - just butt the end of the staff against this ledge.)
• Take a small square and, for the calculation method line it up at the proper graduation on the scale:

Or, for the geometrical method line it up at the inter section of the angle line and cross piece line.

• Finally, mark off the distance with a ballpoint pen. If you are using two different cross pieces you may want to use different colors for the two sets of graduations.
• Once the staff is graduated you can wax it to make it operate more smoothly.
• For navigation staffs it was traditional to point the end away from the eye. This makes it easier to use at night.

 Instruments Medieval Science & Scientific Instruments References