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.
The instrument shown above is constructed of 1/2" Ash
plank with 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 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, 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 a the graduated arc for aid in solving surveying
type problems.
Making a six inch simple
quadrant
This quadrant was one of the projects for my 1998 workshop,
"Medieval Scientific and Philosophical Instruments."
Materials (provided at the workshop - illustration
below):
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.
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.) 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.
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.
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.