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Altitude sundials for seasonal and equalhoursAllan A. Mills aa Astronomy Group, University of Leicester, Leicester, LE1 7RH, UKVersion of record first published: 18 Sep 2006.

To cite this article: Allan A. Mills (1996): Altitude sundials for seasonal and equal hours, Annals ofScience, 53:1, 75-84

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ANNALS OF SCIENCE, 53 (1996), 75-84 75

Note

Altitude Sundials for Seasonal and Equal Hours

ALLAN A. MILLS

Astronomy Group, University of Leicester, Leicester LE1 7RH, UK

Received 21 January 1995

Summary Altitude sundials do not require knowledge of the N - S direction to quantify t ime in terms of ei ther seasonal or equal hours. Methods lbr determining the corresponding dial patterns are reviewed, and accurately computed data for a latitude of 51 ~ presented as horizontal, vertical, rectangular, and pillar dials on both t imekeeping systems.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2. Methods for generat ing the dial pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3. Equal-hour altitude dials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4. Seasonal-hour altitude dials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

1. Introduction Like all sundials, altitude dials may be calibrated according to seasonal or equal hour

timekeeping systems. Deciding on one of these systems, and the latitude at which the dial is to be used, leaves the remaining variables of solar altitude, azimuth, and declination; and as we wish to employ the first as the timetelling element, we cannot eliminate its effect by sloping the gnomon as in the common directional dial. All we can do is:

(1) employ a point as the shadow-casting index; (2) annul changes is azimuth by manually turning the gnomon to be in the vertical

plane containing the sun; and (3) allow for changes in declination by date-labelling the calibration curves for the

shadow thrown by the point.

The major advantage of altitude dials is that it is not necessary to know the N-S direction when using them, but another benefit is that, unlike vertical directional dials, they are applicable to the early morning and late evening hours of midsummer. They therefore lend themselves to portable instruments, although still being latitude dependent. Calculation of the dial pattern is considerably more difficult than for equal-hour azimuth sundials, which no doubt accounts for the comparative scarcity of this family.

0003 3790/96 $12.00 i() 1996 Taylor & Francis Ltd.

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76 Allan A. Mills

2. Methods for generating the dial pattern Fundamentally, this involves converting the universal right ascension (RA) and

declination (Dec) coordinates of the Sun (the latter available from tables ~) to specific local altitude and azimuth coordinates, given the latitude of the site and the number and type of hour divisions to be marked. This operation has long been vital in quantitative positional astronomy, so a number of techniques have been developed over the centuries to cope with it.

2.1. Direct observation

Marking the shadow's position at intervals throughout a sunny day would define the curve traced out on that date, and the same procedure carried out on other days would demonstrate their changing curvature and displacement over the year. However, some independent timekeeper or method is required to enable the hour divisions to be marked thereon.

2.2. Graphical methods

A favourite technique for drawing standard dials, the geometric construction becomes complicated for altitude dials. The best account is given by Fuller, 2 who was endeavouring to rediscover a method that could have been used in ancient times.

2.3. The torquetum Accounts of this elegant instrument are commonly vague, merely saying that it was

'used by astronomers'. This tends to suggest that it was either for taking direct observation of heavenly bodies, or was a teaching device to illustrate their apparent motions. 3 The torquetum had a much more important function than these: it is clearly a mechanical analogue computer for transforming between equatorial (RA and Dec), ecliptic (celestial latitude and longitude) and local (altitude and azimuth) coordinates. It was indeed used by astronomers--but in the study and at leisure--to transform measurements initially obtained by more convenient methods. Thus, for example, a certain RA and Dec could be set on the protractors, the wedge adjusted to a given latitude, and the sights used to place a mark on the wall. The wedge would then be removed, the sights realigned on the mark, and the corresponding altazimuth coordinates read off.

Such an instrument is just what is needed by the practical sundial designer too (even today it would not be obsolete), and a workmanlike example is included in the collection of dials and associated apparatus pictured in Holbein's painting The Ambassadors

(National Gallery, London).

2.4. Calculation by spherical trigonometry

Plane trigonometry was known to ancient Greek philosophers, and spherical trigonometry was developed from it by medieval Arab astronomers with the accurate computation of the astrolabe and the direction to Mecca as major goals. An excellent introduction is given by H. R. Mills, and a more complete exposition by Smart. 4 The

i For example, those in A. E. Waugh, Sundials." Their Theor), and Construction (New York, 1973). -~A. W. Fuller, 'Universal Rectilinear Dials', Mathematical Gazette, 41 (1957), 9-24. 3 j. A. Bennett, The DivMed Circle (Oxford, 1987). H. Michel, Scientific Instruments" in Art and History

(London, 1967), gives a colour reproduction of a fine torquetum by Erasmus Habermel, c. 1600. 4 H. R. Mills, PositionalAstronomy andAstro-Navigation Made Easy (New York, 1978); W. M. Smart,

Textbook on Spherical Astronomy, 6th edn (Cambridge, 1977).

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Altitude Sundials 77

modern diallist has a huge advantage over earlier practitioners in the availability of the pocket electronic calculator.

2.5. Tables Finally, precomputed sets of table do exist, s

3. Equal-hour altitude dials For calculation, a convenient formula of spherical tr igonometry is

sin A = sin ~b sin 6 + cos ~b cos 6 cos (HA), (1)

where A is altitude o f the Sun, ~b is latitude of a place, 6 is declination of the Sun, and (HA) is hour angle of the Sun (all measurement are in degrees).

The latitude is fixed by the location at which the dial is to be used, and the equal hour system means that the hourly divisions of time are represented by 15 ~ increments of hour angle either side of noon. With the declination of the Sun on a given date obtained from tables, the above formula enables the altitude of the Sun at the chosen site to be calculated for any hour of any day of the year. The symmetry of the situation halves the work required, but the user of the dial is expected to know the approximate date and whether it is morning or afternoon.

Mills 6 gives a useful table of altitudes computed at monthly intervals for 5 I~ so this latitude was adopted for all the calculations below. This numerical data may be presented and utilized in many ways, always remembering that the shadow of a definite point is essential.

3.1. The horizontal disc dial

The length f of the shadow thrown by a vertical gnomon of height h is given by

= hco t A.

These lengths are plotted radially versus date in Figure 1. A cardboard disc bearing this pattern may be placed over a vertical nail of appropriate scale height, and rotated until the shadow of the nail points to the date of the observation. The tip of the shadow will then indicate the time of day whenever the Sun is visible. Although simple to construct this is a poor dial because the hour lines are close together in summer and the shadow long and fuzzy in winter.

3.2. The vertical disc dial

The length ( of the shadow thrown downwards upon a vertical dial by a horizontal gnomon of projection h, turned to be in the vertical plane containing the Sun, is given by

# = htan A.

These lengths are plotted radially versus date in Figure 2. A working sundial can be made by driving a nail through a square o f wood until it protrudes a convenient length, and then pushing a card dial scaled from Figure 2 on it. Hang the device up by a cord secured at one corner, and revolve the dial until the date is vertically below the nail.

5 Tables of Computed Altitude and Azimuth, US Hydrographic Publication no. 214. Each volume covers 10 ~ of latitude.

6 Mills (note 4).

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78 Al lan A. Mi l l s

Figure 1. The horizontal equal-hour altitude dial, disc type.

Figure 2. The vertical equal-hour altitude dial, disc type.

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Altitude Sundials 79

Then turn the whole thing so that the nail is in the plane of the Sun. (This can be done most accurately ifa plumb-line is hung from the shank of the nail.) The tip of the shadow then indicates local apparent solar time. Longitude and equation of time corrections would have to be applied to make this agree with clock time.

Improved variants of this simple portable dial are described by Mills v and Higgins. 8

3.3. Rectangular coordinates In the past, it was usual to plot the data for the vertical dial on rectangular

coordinates, giving the curves shown in Figure 3. Note how some of these cut the upper zero axis, being applicable only to the long days of summer. With this construction the gnomon must be slid along the top of the dial to the correct date before being turned into the Sun.

3.4. The flag dial A more compact design is achieved by utilizing the bilateral symmetry of the

pattern. It is cut along the middle and the resulting broad edge pivoted on a vertical post to allow it to be turned towards the S u n . 9 Such a 'flag dial' is illustrated in Figure 4.

3.5. The pillar dial An even more compact form is achieved by rolling the whole- -or one-half----of the

plot into a cylinder. The gnomon must then be rotated about the axis of the latter to the

am,6pm # ~

Length of gnomon ~-

I I APR I I oCT 21 22 18 21 21 24 21 21 22 23 24 1

Capricorn Aquarius Aries Taurus G Cancer Leo Virgo Libra Scorpio Sag rius v~ T t5 o:D Q ~ -,-- rq

I

Figure 3.

0

10

20

- 30

-40

-50 ,-&

s5 ~=

- 60

65

Data for the vertical equal-hour altitude dial plotted on rectangular coordinates.

7 Ibid. 8 K. Higgins, 'The Classification of Sundials', Annals of Science, 9 (1953), 342-8. 9 Bedos de Celles, La Gnomonique Pratique (Paris, 1760 and 1774); R. R. J. Rohr, Sundials: History,

Theory and Practice (Toronto, 1970).

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80 Allan A. Mills

Figure 4. An eighteenth-century 'flag dial' for equal hours. Courtesy of the Mus6e de la Vie Wallone, Libge.

date o f observation before turning the whole instrument towards the Sun. This design constitutes the pillar dial, shepherd 's dial, or chilinder.t~ An example is included in the Holbein painting mentioned above, while a beautifully made French model in the Whipple Museum, Cambridge, is illustrated in Turner. ~j

Old and ornate pillar dials frequently include the sign of the zodiac in which the Sun is traditionally placed during a given calendrical period. 'Tradit ionally ' because (in spite of the claims of astrologers) precession has moved the entire zodiac through nearly two signs since the dates were laid down by reference to the vernal equinox.

v0 C. Kren, 'The Traveller's Dial in the Middle Ages: The Chilinder', Technology and Culture, 18 (1977), 419-35.

v l G. L'E. Turner, Nineteenth-century Scientific Instruments (London, 1983), pit. opposite p. 48.

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Altitude Sundials 81

3.6. Other designs Ring, 12 navicula, 13 Reg iomontanus ] 4 and capuchin t5 dials are described in the

references given.

4. Seasonal-hour altitude dials Computat ion of the dial pattern for these instruments requires the prel iminary step

of obtaining the hour angle (HA) at which the Sun rises and sets on chosen dates through the year. Employing spherical trigonometry, this is obtained from the formula

cos (HA) = - tan ~b tan 6, (2)

where the symbols have the same meanings as before. This arc, when divided by 6, gives the degrees of HA corresponding to a seasonal hour on the date to which the declination applies. This figure, and its multiples, is then substituted in formula 1 to find the corresponding solar altitude A for a chosen latitude (here 51 ~ This data is shown in Table 1, and may be presented in the same forms as already given for the equal-hour analogues.

4.1. Horizontal and vertical disc dials

These dial patterns are given in Figures 5 and 6. An ingenious Roman portable dial in the Museum of the History of Science, Oxford,

is related to the latter. Its combined gnomon/scale is adjustable for date, and divided into six un-numbered portions, each subtending 15 ~ Stebbins and Field 16 have shown

Table 1. The Sun' s altitude at ends of the seasonal hours through the year, for a latitude of 51 ~

Sun's altitude (~

Sun's Dec 6th 5th 4th 3rd 2nd 1 st Sunrise Date (~ (Noon) 7th 8th 9th 10th 11 th Sunset

21 June + 23.44 62.44 58.24 48.26 35-96 23-17 10-90 0

21 July + 20.6 59.60 55-93 46.87 35.33 23.04 11-01 0 24 May

22 August + 11.8 50-71 48.20 41.50 32.21 21.64 10.66 0 21 April

23 September 0 39.00 37.43 33.02 26-42 18-34 9-37 0 21 March 24 October - 11.7 27.30 26-35 23.62 1 9 . 3 1 13.74 7.21 0 18 February 22 November 19.9 19.10 18.49 16.70 13.82 9.98 5.33 0 22 January 21 December - 23-44 15.56 15.08 13.66 11-35 8-24 4.43 0

12j. Moore, 'Ring Dials', Bulletin of the British Sundial Society, no. 2 (1994), 11-14. ~3 Fuller (note 2); D. J. Price, 'The "Little Ship of Venice": A Middle English Instrument Tract', Journal

fi)r the History of Medicine and Allied Sciences, 15 (1960), 399-407; J. Kragten, The Little Ship of Venice (Eindhoven, 1989).

14 Fuller (note 2). ~SWaugh (note 1); Mills (note 4); F. W. Sawyer III, 'A Universal Nomographic Sundial', Bulletin of

the British Sundial Society, no. 3 (1994), 14-17. i~ F. A. Stebbins, 'A Roman Sun-dial', Journal of the Royal Astronomical Society of Canada, 52 (1958),

250-4; J. V. Field, 'Some Roman and Byzantine Portable Sundials and the London Sundial-Calendar', in Histor3' of Technology, Xll, edited by G. Hollister-Short and F. James (London, 1990), 103-35.

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82 Allan A. Mills

Figure 5. The horizontal seasonal-hour altitude dial, disc type.

Figure 6. The vertical seasonal-hour altitude dial, disc type.

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Alti tude Sundials

Sunrise & Suhse~

83

2rid & 11m

4th & 9th

5~h & 8th

Noen

] _ l 4 I_ I i I t _ l JAN FEB MAR APR MAY JUN OOk AUG SEP OCT NOV DEC

21 22 18 21 21 24 21 21 22 23 24 22 21

Cap~ricom I Aquarius I Pi . . . . I Abes T Ta . . . . I Gemini I C . . . . . I Leo Virgo Libra Scorpio Sagittarius

Figure 7.

10

20

3O

- 4 O

== 50

5 s

�9 6 0

6 5

"Data for the vertical seasonal-hour altitude dial plotted on rectangular coordinates.

Figure 8. Drawing of a pillar dial from a manuscript dated to before AD 1350 (Bodleian Library, MS Ashmole 1522, f. 181v). The Latin inscription beneath the gnomon is believed to be an abbreviated form of Polus 12 punctos umbra. Courtesy of the Bodleian Library, Oxford.

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84 Alti tude Sundials

that its construction and geometry of use would give satisfactory approximations to the seasonal hours, the maximum error being _+ 15 minutes of time.

4.2. The pi l lar dial

The plot on rectangular coordinates is shown in Figure 7. Note that it is easily distinguished from that for equal hours (Figure 3) because none of the curves intersects the upper horizontal zero axis.

Establishing this pattern took a couple of days with an electronic calculator, so it was a surprise to find it carefully represented on a medieval dial. Gunther Iv reproduces (apparently without realizing its seasonal hour nature) a drawing of a pillar dial from a manuscript ~s dated to belbre AD 1350 (Figure 8). The Latin inscription written upside down around the base gives the months and their zodiacal equivalents to which the gnomon must be set before turning the instrument into the plane o f the Sun. I am inclined to think a torquetum must have been used to lay out this grid. With all pillar dials, the angle between the vertical and a line joining the tip o f the gnomon to the noon line at the summer solstice is equal to the latitude of the site for which the dial was designed less 23.5 ~ This construction gives about 4 8 ~ south German loca t ion- - for the dial, but the nature of the drawing does not permit any firm conclusion.

J7R. T. Gunther, Early Science in Oxford, II (Oxford, 1923), 122. ISBodleian Library, Oxford, MS Ashmole no. 1522, f. 181v. Before AD 1350.

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