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A schema for horizontal dials is a set of instructions used to construct horizontal sundials using compass and straightedge construction techniques, which were widely used in Europe from the late fifteen century to the late nineteen century. The common horizontal sundial is a geometric projection of an equatorial sundial onto a horizontal plane.
The special properties of the polar-pointing gnomon (axial gnomon) were first known to the Moorish astronomer Abdul Hassan Ali in the early thirteenth century ^{[1]} and this led the way to the dial-plates, with which we are familiar, dial plates where the style and hour lines have a common root.
Through the centuries artisans have used different methods to markup the hour lines sundials using the methods that were familiar to them, in addition the topic has fascinated mathematicians and become a topic of study. Graphical projection was once commonly taught, though this has been superseded by trigonometry, logarithms, sliderules and computers which made arithmetical calculations increasingly trivial/ Graphical projection was once the mainstream method for laying out a sundial but has been sidelined and is now only of academic interest.
The first known document in English describing a schema for graphical projection was published in Scotland in 1440, leading to a series of distinct schema for horizontal dials each with characteristics that suited the target latitude and construction method of the time.
The art of sundial design is to produce a dial that accurately displays local time. Sundial designers have also been fascinated by the mathematics of the dial and possible new ways of displaying the information. Modern dialling started in the tenth century when Arab astronomers made the great discovery that a gnomon parallel to the earths axis will produce sundials whose hour lines show equal hours or legal hours on any day of the year: the dial of Ibn al-Shatir in the Umayyad Mosque in Damascus is the oldest dial of this type. ^{[1]} Dials of this type appeared in Austria and Germany in the 1440s.^{[2]}
A dial plate can be laid out, by a pragmatic approach, observing and marking a shadow at regular intervals throughout the day on each day of the year. If the latitude is known the dial plate can be laid out using geometrical construction techniques which rely on projection geometry, or by calculation using the known formulas and trigonometric tables usually using logarithms, or slide rules or more recently computers or mobile phones. Linear algebra has provided a useful language to describe the transformations.
A sundial schema uses a compass and a straight edge to firstly to derive the essential angles for that latitude, then to use this to draw the hourlines on the dial plate. In modern terminology this would mean that graphical techniques were used to derive \sin x and m\tan y and from it \sin x .\tan y . ^{[2]}
Setting out a dial for 52°N. The three initial lines.
Marking the latitude, laying out length \sin\phi , and copying to G on the vertical.
A tangent:Laying out the length \tan\theta
A tangent of a sine: Laying out lines of length \sin\phi\tan h , where h is an integer 0 .. 5
Such geometric constructions were well known and remained part of the high school (UK grammar school) curriculum until the New Maths revolution in the 1970s.^{[3]}
The schema shown above was used in 1525 (from an earlier work 1440) by Dürer is still used today. The simpler schema were more suitable for dials designed for the lower latitudes, requiring a narrow sheet of paper for the construction, than those intended for the higher latitudes. This prompted the quest for other constructions.
The first part of the process is common to many methods. It establishes a point on the north south line that is sin φ from the meridian line.
The significant problem is the width of the paper needed in the higher latitudes. ^{[5]}
Benedetti, an impoverished nobleman worked as a mathematician at the court of Savola. His book which describes this method was De gnomonum umbrarumque solarium usu published in 1574. It describes a method for displaying the legal hours, that is equal hours as we use today, while most people still used unequal hours which divided the hours of daylight into 12 equal hours- but they would change as the year progressed. Benedettis method divides the quadrant into 15° segments. Two construction are made: a parallel horizontal line that defines the tan h distances, and a gnomonic polar line GT which represents sin φ.
Benedetti included instructions for drawing a point gnomon so unequal hours could be plotted. ^{[6]}
Quadrant with 15° segments.
constructing the rays.
Finding the origin.
Adding the hour lines.
Dial face.
(Fabica et usus instrumenti ad horologiorum descriptionem.) Rome Italy.
The Clavius method looks at a quarter of the dial. It views the horizontal and the perpendicular plane to the polar axis as two rectangles hinged around the top edge of both dials. the polar axis will be at φ degrees to the polar axis, and the hour lines will be equispaced on the polar plane an equatorial dial. (15°). Hour points on the polar plane will connect to the matching point on the horizontal plane. The horizontal hour lines are plotted to the origin.
^{[7]}
^{[5]}
From G casting \tan h \sin\phi on the horizontal.
The actual hour lines 9, 10, 11, 12, 1, 2, 3.
Construction lines removed.
Finding the angle for 4 and 5.
Drawing 4 and 5.
Construction lines removed. The completed dial plate for 52°N. Stirrup (1652)
The Jesuit Mario Bettini penned a method which was posthumously published in the book Recreationum Mathematicarum Apiaria Novissima 1660.
^{[8]}
Gnomonic triangle
Circle marked each 30°
Construction Lines
Hour lines drawn to the origin
The dial plate
William Leybourn published his "Art of Dialling"^{[4]} in 1669, a with it a six-stage method. His description relies heavily on the term line of chords, for which a modern diallist substitutes a protractor. The line of chords was a scale found on the sector which was used in conjunction with a set of dividers or compasses. It was still used by navigators up to the end of the 19th century.^{[5]}
This method requires a far smaller piece of paper,^{[5]} a great advantage for higher latitudes.
This method uses the properties of chords to establish distance m. \sin \theta in the top quadrant, and then transfers this distance into the bottom quadrant so that \sin \phi \sin \theta is established. Again, a transfer of this measure to the chords in the top quadrant. The final lines establish the formula \tan \kappa = \sin\theta\sin\phi\over\cos\theta = \tan\theta\sin\phi
This is then transferred by symmetry to all quadrants. It was used in the Encyclopedia Britannica First Edition 1771, Sixth Edition 1823^{[11]}
Draw the gnomon and diameters at the target angle.
Mark the top quadrants at 15° angles, and connect with chords
Transfer the half-chord length to the lower radius, and draw across.
Raise verticals
Draw hour line through the intersections.
Resulting dial for 52°.
The Dom Francois Bedos de Celles method (1760) ^{[13]} otherwise known as the Waugh method (1973) ^{[14]}^{[5]}
Constructing the 7, 8, 4, 5 lines
Marking the 7, 8, 4, 5 lines
The completed dial plate for 52°N. Bedos de Celles (1790)
This method first appeared in Peter Nicholsons A popular Course of Pure and Mixed Mathematics in 1825. It was copied by School World in Jun 1903, then in Kenneth Lynch's, Sundial and Spheres 1971. ^{[15]} It starts by drawing the well known triangle, and takes the vertices to draw two circles at radius (OB) sin φ and (AB) tan φ. The 15° lines are drawn, intersecting these circles. Lines are taken horizontally, and vertically from these circles and their intersection point (OB sin t,AB cos t) is on the hour line. That is tan κ = OB sin t/ AB cos t which resolves to sin φ. tan t.
^{[15]}
The basic triangle
The circles
The 15° measure
The intersection points
The completed dial
This was an early and convenient method to use if you had access to an astrolabe as many astrologers and mathematicians of the time would have had. The method involved copying the projections of the celestial sphere onto a plane surface. A vertical line was drawn with a line at the angle of the latitude drawn on the bisection of the vertical with the celestial sphere. ^{[16]}
London dial
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