Oughtred's Circles of Proportions

The Circles of Proportions  

I became interested in the Circles of Proportion because of the unusual nocturnal scales on the front side and the horizontal instrument on the reverse.     Later calculators moved away from celestial computing to more earthbound concerns, like timber calculations and liquor taxes.   And, of course, I am interested in learning how to render scales using period-authentic lettering, and Oughtred's Circles offer a snapshot of how most of the examples would have been made in the 17th and 18th centuries.

The Computing Scales

Oughtred's original book dated 1632 The Circles of Proportion and the Horizontal Instrument described a calculator featuring 8 scales.  Three of these were Tangent scales (T), two were sine scales (S), one was a logarithmic number (N), and one was an equally divided linear scale (E).     

The book described how the scales were to be rendered, gives a brief introduction to how to set up a computation, and follows this with numerous examples of calculations where the calculator would be useful.

There was an update to the slide rule mentioned in the 'An Addition Unto the Use of the Instrument Called the Circles of Proportion...', dated 1633.   

Here, Oughtred increased the number of scales by 2 to 10 scales, by adding a Sine and a Tangent scale.   

The body of this book concerns computations related to navigation and that is the reason given for the extended scale set modification.    I've not studied this book extensively, so I'm not sure how the improved scale sets would have helped with navigation-related problems.

Oughtred also had a section on a linear slide rule developed near the end of 'An Addition Unto the Use of the Instrument Called the Circles of Proportion'.   

The Nocturnal scales

In addition to the mathematical computing scales, Oughtred had some scales near the center of the instrument that could be used to determine the time at night. 

A nocturnal was an instrument used to find the time at night, these were used especially by navigators until sea-worthy chronometers were widely available.   There are 3 scales supporting two types of nocturnals on the front face of the Circles of Proportion,  called the 'outer' and 'inner' nocturnal

 A traditional nocturnal has a fixed month scale, a movable hour scale (a disc), and a movable pointer.   A good explanation of the traditional nocturnal can be found here.     One of the two nocturnals on this instrument (the outer nocturnal) functions as this type of nocturnal with the exception that they had a movable disc with the hour scale.    

The Circles of Proportion do not have a movable hour disc, which is puzzling because it makes the time determination calculation more difficult.    The nocturnal scales on this instrument, all fixed, are a clock (hour/quarter hour) scale, a calendar (day and month) scale, and the constellation (zodiac) scale.

With the traditional nocturnal, you first set a pointer on the hour dial to the day of the month, and sight through a hole in the center of a nocturnal to locate the Pole Star.   You move the arm of the calculator to line up with Alioth,  a star in the handle of the Big Dipper  (In some nocturnals one or more stars other than Alioth can be used).    When the arm is set to Alioth, you can directly read the time on the hour scale.

On Oughtred's instrument, the 2 arms are held in place by a hollow rivet which provides the peephole by which Polaris is seen.   You use the right cursor arm to find Alioth, and read the hour amount off the scale.    You add that to the setting one arm to the current month and day.   The second step provides the same offset that the rotation of a movable hour-circle.     

While the computation is more involved than a standard nocturnal, Oughtred probably just thought this a trivial matter.    One thing about it is that you must do most of the math back in your house because you need light to see the hour-circle.   Note that  Nocturnals were usually fashioned to enable working in the dark.   There would be finger holds or points extending from the device's movable hour dial, so you could count off hours with your fingers in the dark.

Oughtred's other Nocturnal scale  (the inner nocturnal) was based on using the "meridian" at your location.   The meridian is the line of longitude you are standing on; it intersects the horizon due South of your location (assuming you are in the northern hemisphere).      The meridian method may have been widely understood and understood by Oughtred's contemporaries but I've not located any instruments called nocturnals that use this technique.  

For this computation, you have to know in advance where the meridian hits the horizon at your location.    

With that known, you look southward for any of the stars representing the constellations that are engraved on the calculator.   Specifically, you pick the one that is closest to this meridian and manipulate the instrument to determine what the time is; the method is similar to the previous one - that you need to add two different hour-circle amounts.