Simple

Quadrants

The quadrant is one of the earliest and simplest of measuring instruments for astronomy, navigation, and surveying. In operation one sights an object, such as a star, through the two sighting vanes along the 90° line (right edge in picture above), while holding the quadrant in one hand, and then clamps the string against the scale with the other hand. When sighting the Sun, one lines the quadrant up such that the image of the Sun formed by the upper pinnule falls on the lower pinnule. One then clamps the string as above and reads the angle on the scale.

Large Wood Quadrant

I constructed the instrument shown above in 1986 of 1/2" Ash plank recovered from pieces of an old broken cabinet I found. It has a radius from tip to edge of 9 3/8". The straight edges were cut and squared on a table saw, then the arc was rough-cut with a band saw. The straight edges were then finished with a Jack-plane, and the arc finished by hand with a Stanley Compass-plane. (I really enjoy using hand tools, but I don't have the time or patience to use them exclusively, so most of my rough work involves power tools.) To provide a better surface for drawing and writing a sheet of "parchment" paper was glued onto the wood base. The graduations (initially penciled in using a protractor), lettering etc. were then done with India ink with drafting and calligraphy pens.

The bob was hand turned on a wood lathe from 3/8" brass round stock with a file (careful, if it grabs you can be damaged). There is a small hole in the top for the thread, with a cross hole through the grooved part to tie it. The sites are fabricated from 12 gauge brass sheet stock with carefully centered holes. The sites have an inverted "J" profile, with the stem in the wood and the curve overhanging the parchment. This allows the holes to align over the 90° line and the thread of the plumb bob to also line up at 90°. This quadrant has a shadow square (discussed below) as well as the graduated arc for aid in solving

5" Copper Alloy Quadrant (front)

5" Copper Alloy Quadrant (back)

5" Copper Alloy Quadrant

In 2002 I found some US Coast Guard 4" diameter pipe at our local metal salvage company (Arcata Salvage). It turned out to be a 5% Ni-Copper alloy of a lovely bronze color with walls about 1/8" thick. I immediately purchased a short length as a source of both metal rings and flatstock. To make the quadrant I cut off a length of pipe 5" long, slit one side and opened it up and flattened it with hammers and a heavy steel flat "anvil." After cutting an arc to make a quarter circle I sanded one side smooths to provide an attractive surface for graduation (an image of the back unfinished side can be seen above). I used a punch to prick a mark near the vertex to place one leg of my large dividers to lay out the four arcs seen on the photo of the finished quadrant above. With dividers still set to the largest arc, the first divisions of the arc were scribed as described in my photo essay on graduation of a quadrant. I also drilled a small diameter (1/16") hole at the same punched mark (center of the arc) for the plummet line.

After graduation two slots were made along one edge of the quadrant with a file, carefully cutting them so they just touch the scribed line (90° or 0°) delineating the arc. Small brass sites with identically aligned holes were then fitted and soldered in place with "easy" silver solder. Next, figure punches were used to add the ten-degree graduation numbers. Finally, a small plummet, shaped with files on a lathe, was made and hung from the vertex with a short piece of string.

Making a six inch simple quadrant

This quadrant was one of the projects developed for my 1998 workshop, "Medieval Scientific and Philosophical Instruments." The workshop has been repeated a number of times subsequently using different types of plywood. For a professional workshop for the Scientific Instrument Society at Harvard University in 2007, brass quarter circles were provided along with simple metal working tools.

quarter circle, spade lugs, string and weight

Materials (provided at the workshop):
6 3/8 inch square of 1/4" plywood, cut with a band or jig saw to a quarter of a circle of 6 3/8 inch radius. Two holes are drilled along on edge to accommodate the sights, carefully aligned with the zero line of the quadrant. A third, small hole, is drilled at the vertex of the quadrant, 1/4 inch in from each edge.
Two male spade wire connectors (with central holes) for sites. (The enlarged portion of the sleeve on the brand provided was cut off, as shown in the upper connecter in the illustration below.)
A short (7-8 inches) length of fine cord or heavy thread to suspend the bob.
A symmetrical fishing weight for a plumb bob (a small, #1, worms head weight was used, but heavier weights will serve better with wind etc.).

Construction :

Lightly sand the edges and surfaces of the plywood quarter circle as necessary to remove the rough edges and splinters. The surface should be smooth enough to write on with a ball point pen (felt pens may also be used if the wood is sealed before-hand).
Check the two straight edges of your quarter circle for square. If they are at exactly 90°, then draw straight lines parallel to each edge passing through the center of the small hole at the vertex (they should be about 1/4" in from each edge). If the two straight edges are not at 90°, draw one and then draw the second at 90° to the first. Again both should pass through the center of the vertex hole. These will be the 0° and 90° line for your quadrant. (If you wish to use classical. geometrical methods of dividing, then only draw one line at this time. The second will be determined with a dividers in the process of dividion.)
Next layout the arcs defining the scales. In the design here four arcs are drawn. Beginning from the outer edge, the first three are drawn using the three different beam compasses (6", 5 3/4", and 5 1/4") shown below.

The compasses are made from hardwood, with a sharpened finishing nail for the point (it must be sharpened round, so no edges remain) inserted through a predrilled hole 3/8" from one end. The other end was made into a pen clamp by drilling two holes (one pen size about 1/2" in from the end and the other slightly larger about 3/8" further in) and then cutting a slit in from the end through the pen hole ant into the other. A body hole for a wood screw is then drilled through the side up to the slot, and then a pilot hole through the remainder. A waxed wood screw is then used a a clamping screw.

beam compass with arcs drawn on quadrant

- First, a 6" arc using the largest beam compass is drawn. To use this compass, place a pen in it and line up the tip so that it sticks out about 1/16" less than the nail point, and clamp it in. Now put the point into the vertex hole, and holding the beam in one hand with the pen perpendicular to the plywood, slowly turn the plywood blank under the pen to make an arc from between the 0° and 90° lines. Next, using the other two beam compasses, make the 5 3/4", and 5 1/4" arcs:
- Finally, a 3" radius arc is drawn using the edge of the 6" protractor as guide:

You are now ready to layout the graduations. (At this point, you may wish to consider alternate methods of graduating an arc discussed elsewhere on my site.) I find that doing so in stages reduces my chance of error, thus:
- First draw the graduations at 10° intervals. Using a shape pencil and your protractor make light marks every 10° (10, 20, 30, etc.) on the 3" arc. Next take your straight-edge and line it up with your 10° mark and the center of the vertex hole. Draw a line with the pen from the 3" to the 6" arc. (You may want to practice on a piece of paper or scrap wood to get the pen line to pass through the center of the marks. An old trick is to place the pen point on the mark then move the ruler up into contact with it.) Repeat with each of the other marks.
- Beginning with the 90° line (adjacent to the site holes), label the graduation line. I wrote mine above the 5 1/4" arc straddling each graduation.
- Next draw the graduations at 5° intervals. Again begin by making light pencil marks on the 3" arc. Then align your ruler and draw lines with the pen between the 5 1/4" and 6" arcs.
- Finally, draw the 1° graduations. This may be readily accomplished in two ways. 1) Proceed as above, graduating between the 5 3/4" and 6" arcs. 2) Use a dividers to mark four equally spaced intervals on the 6" arc, and draw graduations between the 5 3/4" and 6" arcs. Any easy way to set your dividers is to line it up at 2° intervals on the 3" arc of the protractor - this will give 1° of arc on the 6" circle.

quadrant with arcs and shadow square drawn

Adding a Shadow Square. The shadow square is used to find the opposite side of a triangle when the adjacent side is known, or vice-versa. In other words it solves simple trigonometric problems based on the tangent function. If you want to layout a shadow square using angle measurements you must calculate the angles for given ratios using the arctangent function. I have provided a table of sample values and formulae below. Of course the easiest way to layout a shadow square is to base it on similar triangles. Quite simply, you decide on how many divisions you want, divide the length of the side of your square by the number of divisions, and then set a dividers for that distance. You now use the dividers to lay off the required divisions. Finally, line up a straight-edge between the vertex of the quadrant or shadow square and your divisions and draw line segments. You can add a shadow square to your six inch quadrant as follows. (Shadow squares with 12 divisions seem to be the most common. I have chosen 9 divisions for this quadrant due to the ease of layout with a mm scale.) Layout a 45° line in pencil from the vertex of the quadrant to the innermost (3") arc. Using a square or ruler draw lines to the intersection of the line with the arc perpendicular to the 0° and 90° lines. Next draw a second pair of lines parallel to and above the first pair between the 0° and 90° lines and the 45° layout line at positions where they are exactly 45 mm long. Now mark off 5 mm intervals along these lines. If you now draw lines between the parallel lines which intersect the vertex and the intervals you will create 9 trapazoidal spaces on each side. Traditionally the spaces are alternately left open and filled in, as seen in the figure.

Calculation of a Shadow Square

Spread Sheet (Excel) Formulae and Table of Angles

Note that in column C, the Angle in Radians, I am using ATAN2, which in Excel is the arctangent of x & y coordinates, where x = number in column A and y = number in column B. Column D then converts the angle into degrees, since there are 2 pi radians in a circle (pi radians = 180°).

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