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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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A chord based octant or quadrant
From: John Brown
Date: 2012 Nov 2, 10:31 -0700

As a strictly backyard / armchair navigator I am interested in taking sun and star sights using simple homemade instruments and have been experimenting with chord based octants / quadrants for a number of years.

I first got the idea when I realized that a hacksaw blade makes the ideal radius of a circle. The distance between the center of the two holes at each end is exactly 300mm. If two blades were joined at one end it should be possible to find a simple way to calculate the angle between them when one blade is horizontal and the other is aimed directly at the sun. It took a few days to come up with the theorem needed. "The line drawn from the center of a circle perpendicular to a chord bisects the chord" seemed to fit the bill and a short while later I was able to work out that the formula required was "angle = 2 x ASN ( Chord / 2R ) " where the chord is the distance measured between the center of the holes at the free ends of the blades and R is the 300mm radius of an imaginary circle. Note that 2R has nothing to do with the diameter of the circle.

The resulting instrument was used as follows - the lower arm was held in a horizontal position whilst the upper arm was rotated until the two pins of the simple sight were in alignment when projected onto a piece of card. The nut and bolt holding the two blades together were kept slightly tight so that the blades would hold their position when aligned with the sun. After noting the time, the ends of the instrument were then placed on a strip of paper and the center of each hole marked. These are labelled A and A and a second sight done and labelled B and B. Any number of sights can be done before measuring starts. Measuring the lengths of the chords is done by accurately measuring the distances of A - A , B - B etc to the nearest tenth of a mm and matching them up with the times that each sight was taken. Altitudes are calculated using the above formula. As the blades measure 300mm from the centers of the holes then R = 300 and therefore 2R = 600.

This instrument was fun to use even though a bit cumbersome. Large angles were made difficult by the increasing length of the chords involved .As might be expected, the results were not very accurate but it did generally give results to within half a degree or so. Most importantly it showed that the theory of measuring angles without a scale was possible in practice.

The next idea was to imitate a normal quadrant and to replace one of the hacksaw blades with a line and reference point drawn on a backboard. The line was drawn just in from the left hand edge and a right angles to the top edge . A 160mm steel ruler replaced the second hacksaw blade. It was hung from a nail near the top of the reference line and a reference point was marked where the center of the end of the ruler met the line. All chords were measured from this point.. The radius of the swinging arm was measured from the reference point to the point where the ruler swings on the nail. This worked out at 166.3mm

The instrument operates in the same manner as a normal quadrant except that the cord and weight are replaced by a free swinging ruler. When the top edge of the instrument is pointing at the sun and the ruler is hanging vertically, the instrument is carefully tipped on its back so as not to move the ruler. A mark is made at the center of the end of the ruler. The distance from this point to the reference point gives the length of the chord. Rather conveniently the plumb bob ruler can be used to measure the chord. Any number of sights can be taken before measurements are done as long as they are numbered. If all is well a procession of dots will show the outline of the circumference of a circle. For each set of sights a short length of painter's masking tape is placed where the end of the ruler falls so that the markings can be simply erased by removing the tape when they are no longer required. If a large piece of masking tape is used the time of each observation can be recorded on the face of the instrument. A watch set to UTC can be attached with blue tack.

I had three or four of these instruments and the only difference between them was the sights. All I had to do was transfer the ruler from one to the other and, by averaging the sights of all the instruments, intercepts of about 5nm or better were possible on a good day.

The final development has been a plumb bob consisting of a washer suspended on a length of cotton thread. This method is so simple that a working instrument can be made in just a few minutes - see photo of the emergency card. The end of the chord is marked at the center of the washer.

I have also tried a version with a mirror for taking star sights but unfortunately I'm at that stage of life when near and far won't focus together and glasses only make things worse. I am sure that it should work.

I wonder if the idea of a chord based quadrant has been used before? Recent discussions on NavList suggest that there is nothing new but only old ideas recycled. All the books that I have read on navigation assume that for emergency navigation there will be a scale of some sort available and David Burch in his books even gives instructions to make one if required. Perhaps this is an idea that has been tried before and rejected and everyone knows except me. Naturally the photos attached only show some of the better results along the way and none of the disappointments but as an emergency tool I think that the chord based quadrant might have a place. I hope that this has been of interest.

The attachments show the following.....

Quad 1 The original idea for measuring altitudes without a scale.

Quad 2 One of the original instruments using a ruler in place of a hacksaw blade. The idea for the sight was that the shadow from the front block should exactly cover the back block when aimed directly at the sun but this did not work out. This is reflected on the notes on the masking tape. Note that sights 2 and 5 had the same chord lengths.

Quad 3 The smaller instrument is an octant with a pin and nut sight which I had hoped to use on stars by clamping the instrument to a large L bracket but unfortunately eyesight limitations did not permit me to test it properly. The larger instrument was the first on which I used the cotton thread and washer plumb bob. It is converted from octant to quadrant by simply shortening the length of the plumb bob and marking a new reference point. The sight consists of a piece of plastic which gives the thinnest shadow when pointed directly at the sun. The two sights shown on the masking tape were taken a few minutes apart and averaged out quite well.

Quad 4 An instrument made from a cutting board costing two dollars. The cotton thread is held at the back of the instrument with blue tack which itself is covered by a piece of card to stop the instrument from sticking when it is put down. The results shown are not unusual. This was one of the first times that I had used a cardboard sight as Lego didn't take to being glued to this type of plastic with the type of glue that I had available.

Quad 5 A quadrant made in about 5 minutes. The plumb bob hangs from the center of the blue band and the radius is measured from there to the apex of the angle formed by the V at the bottom. R = 260mm. The intercept of 3.2 NM was rather unexpected. A small magnet attached to the washer adds extra weight.

John Brown,

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