Programming and verifying the Horizontal Instrument

Oughtred was essentially a contemporary of Shakespeare, the bane of the English literature courses of my youth.  Many of the words he used have aged out of the language, others he used now have slightly different meanings. Since much of the terminology and notation we now use for mathematics were still being developed, so like others of his day, Oughtred invented his own when he needed to.    Spelling was both different and not really standardized.    

The part of his book devoted to this Instrument includes very detailed instructions on how to construct one using basic geometric methods.     It also explains the many uses of the device.   Until I read it, I had the uninformed opinion that sundials are essentially no more than clocks.   This particular example, and probably many other styles of dial had a lot more going on.   

The Horizontal Instrument is a computer,  designed to do calculations related to the position of the sun at any time of day during the year.   I hadn't actually closely read his examples until the end of developing the paper replica.   What struck me when I did, is that most of the computations didn't require one to be out in the sun.

Programming the paper replica

I actually did the first version of the Horizontal Instrument before doing the Circles of Proportion.

When I started the development of the replica Horizontal Instrument, I tried to follow along with Oughtreds detailed instructions to generate a stereographic projection.   My thinking was that I could translate the geometry back into modern trigonometry and then program from that.     One of the things about this projection is that all of the curves are circles, so it should be easy to make....all you need is their centre points and radii.

To my enduring dismay, I was simply unable to follow along with Oughtred.   His Elizabethan prose and unusual terminology and steps confounded my attempts to perform the projection.    After my third failure, I went online and pulled the basic math from online sources.      

Oughtred's ecliptic curves are peculiar to the Instrument, so I figured those out with trig.   The top and sides of those curves give 3 points of the circles that comprise them, and the radius and centre points could be determined from that.     I was able to puzzle out the location of the Pole of the world, I still don't know how to place the 2 Poles of the Ecliptics.   For those, I'll go back to Oughtred, and see if I can figure them out.

Verifying the replica

I  started by making a replica of the Davenport Instrument and used it to ensure the replica agreed with the original.   Chiefly, I looked for intersections between the hour and declination curves on the circumference, because they are like fingerprints,  the positions are uniquely determined by the latitude. I  also used images for Instruments made by Elias Allen, but that didn't offer a lot of different latitudes to test with,  so I also used an online Stereographic projection calculator; that gave me a few more latitudes to evaluate.

Once I was convinced my replica progrAM was producing what seemed to be accurate projections,  I made a  quick version for my latitude and did a couple of computations out in the sun, and others at the desk.