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Seasonal-hour sundials on vertical andhorizontal planes, with an explanation ofthe scratch dialAllan A. Mills aa Astronomy Group , University of Leicester , Leicester, LE1 7RH,U.K.Published online: 18 Sep 2006.

To cite this article: Allan A. Mills (1993) Seasonal-hour sundials on vertical and horizontalplanes, with an explanation of the scratch dial, Annals of Science, 50:1, 83-93, DOI:10.1080/00033799300200131

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ANNALS OF SCIENCE, 50 (1993), 83-93

Scasonal-Hour Sundials on Vertical and Horizontal Planes, with an Explanation of the Scratch Dial

ALLAN A. MILLS

Astronomy G r o u p , University of Leicester, Leicester LE1 7RH, U.K.

Received 6 November 1991

Summary The true form of the seasonal-hour sundial, in both vertical and horizontal planes, has been calculated by spherical trigonometry and displayed with the aid of computer graphics. These grids are presented, and show that the hour lines are in fact shallow curves at sites not on the equator. The curvature becomes very apparent at latitudes exo~vding 50 ~ The true seasonal-hour pattern for a vertical dial at a latitude of 52"6~ is compared with the equiangular scratch dial.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2. The common sundial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3. Ancient timekeeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.1. Problems with the ancient system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2. The vertical plane seasonal-hour sundial at the equator . . . . . . . . . . . . . 85 3.3. Greek and Roman scaphe dials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4. Greek and Roman planar dials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4. Saxon dials and scratch dials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5. True seasonal-hour planar dials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1. Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2. Comparison between true and equiangular dial patterns . . . . . . . . . . . . 93

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

1. Introduction The familiar sundial with its sloping gnomon is, as sundials go, a comparatively

recent innovation, for it delineates equal hours throughout the year and appears to have become common following the introduction and spread of the mechanical clock in the thirteenth and fourteenth centuries.

Before that time all Western cultures had continued the Mesopotamian system of dividing the sunlit port ion of the day into twelve equal parts. These intervals will consequently vary in their absolute duration with the time of year, hence they are called 'seasonal hours'. Ancient sundials were therefore required to mark the passage of time in terms of a varying unit, and must rank among the earliest sophisticated scientific instruments.

Means for accurately achieving this twelve-fold division of the illuminated period appear to have been first achieved by the Greeks, using three-dimensional stone dials incorporating a complete or partial hemisphere. Only the tip of the gnomon was the operative index, so it could be fixed in any convenient position; horizontal was common. However, these massive dials were difficult and expensive to construct and

0003-3790/93 $10-00 �9 1993 Taylor & Francis Ltd.

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84 A.A. Mills

there was a demand for more easily made two-dimensional dials in the vertical plane, with matching horizontal pointed gnomons. Although apparently simple, these projections are in fact mathematically complex: only on the equator will the hour markings of a vertical dial take the form of straight radial lines at 15 ~ intervals. Ancient civilizations never fully solved the problem, but employed approximations that would have been satisfactory in latitudes below 45~

However, when the Romans moved into northern Europe, taking their dials with them, the traditional planar form became grossly in error. The descendants of these forms are the scratch dials to be found on many of the older parish churches in England. Nevertheless, as long as everyone employed the same pattern of dial, then they would be perfectly satisfactory for ensuring synchronicity. Scratch dials should be thought of as event markers rather than as defective timetellers.

2. The common sundial The sundial, with its characteristic triangular gnomon projecting from a supporting

wall or pedestal, 1 represents for most people one of the earliest scientific instruments. It is therefore a surprise to many to learn that this type of sundial is not represented

among extant ancient sundials. 2 The earliest surviving sloping-gnomon equal-hour portable dial is dated 1451, but similar examples dating from 1453-63 are conserved in a number of museums. 3 One of the earliest (and certainly the best known) monumental dial of this nature in Europe is the sculptural 'Man with a Sundial' in Strasbourg Cathedral, 4 which bears the date 1493.

These dials were a response to the spread of the mechanical clock in the preceding two centuries: ancient dials were expected to delineate the passage of time according to a completely different system.

3. Ancient timekeeping The simplest division of the sunlit portion of the day is into two equal parts, the

forenoon and afternoon, centred about the midday point we now call noon. This is when the sun achieves its highest elevation and lies on the meridian of the site, so that the shadow thrown by a nail driven perpendicularly into a south-facing wall coincides with a plumb-line hanging from its base. This constitutes an early form of noon mark.

However, division into two parts is very coarse and long ago the Mesopotamian cultures (which had a base-twelve arithmetic) divided the entire sunlit period into twelve equal parts. This system proved eminently suitable for agricultural communities with no good source of artificial illumination; it was so successful that it was copied by Egypt, Greece, and Rome, and remains fossilized in our own (2 • 12) or 24-hour timekeeping.

1 R. N. Mayal! and M. L. Mayall, Sundials: How to Know, Use and Make Them (Boston, 1962). 2 Although what appears to be a remarkable hybrid dial, originally ihcorporating a polar gnomon but

graduat~l to read in seasonal hours, was excavated in Afghanistan in 1975 and associated with Alexander's campaign of 329 n.c. See R. R.-J. Rohr, 'A Unique Greek Sundial', Bulletin of the British Sundial Society, no. 3 (1990), 14-17.

3 F. A. B. Ward, 'The Earliest String-Gnomon Sundials', Antiquarian Horolooy, 9 (1976), 800-1. *R. R.-J. Rohr, Les Cadrans Solaires (Paris, 1965). English edition as Sundials: History, Theory and

Practice (Toronto, 1970), plate 7.

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Notes and Discussions 85

3.1. Problems with the ancient system The fundamental difficulty with the ancient timekeeping system follows from the

66.5 ~ tilt of the Earth's spin axis with respect to the plane of its orbit around the sun, resulting in the illuminated period being of constant duration throughout the year only for a site on the equator. Elsewhere, the sun is above the horizon for a longer period in summer, reaching a maximum on the day of the summer solstice. Only at the equinoxes will the sunlit period become equal to the night, so that all the 'equinoctial' hours become equal at 1/24 of the entire day.

The variable duration of the ancient 'hour' is emphasized by referring to them as seasonal, unequal, s temporary, temporal, planetary, bibfical, Jewish, or antique hours. The term 'seasonal hour' will be used here.

3.2. The vertical plane seasonal-hour sundial at the equator A 'nail in a southerly wall' noon mark erected at an imaginary site on the equator

easily becomes the progenitor of a sundial (Figure 1). A horizontal shadow of the nail appears on the left at sunrise, and rotates as shown at a uniform velocity about the base of the nail as the day progresses. At midday it crosses the vertical noon line, and finally fades when once again horizontal to the right at sunset. (The shadow can never fall above the horizontal in any vertical dial.) This behaviour is repeated exactly on every sunny day throughout the year, so a dial keeping 'hours' could be readily and accurately constructed by simply inscribing an equiangular (15 ~ grid of radial straight fines converging on the base of the gnomon.

The numeration that follows from the construction is included in Figure 1. It will be seen that:

Sunrise -- Beginning of the first hour Mid-morning = End of the third hour Midday = End of the sixth hour Mid-afternoon = End of the ninth hour Sunset = End of the twelfth hour

r - - -SUNRISE , ~ . ~ SUNSET

\ 3rd

Figure 1. A south-facing vertical dial at a site on the equator. The entire shadow of a horizontal peg-gnomon moves in an anti-clockwise direction at a uniform rate of 15 ~ per hour.

5 However, they are equal throughout any given day.

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86 A.A. Mills

Figure 2. Fragment of a planar Roman dial found at Borcovicus station on Hadrian's Wall. Now in Chesters Museum, Northumberland.

3.3. Greek and Roman scaphe dials The best Greek and Roman seasonal-hour sundials were quite large three-

dimensional stone forms based on a partial sphere or 'scaphe': only the shadow of the tip of the gnomon 6 was the time-telling index. These dials have been analysed elsewhere, 7 so will not be elaborated on here. They could in theory tell time accurately if carved to a true sphere and correctly calibrated for a given site, but many extant examples s appear to be sadly deficient in these respects.

3.4. Greek and Roman planar dials Nevertheless, the amount of stone and skilled work required no doubt meant that a

scaphe dial was an expensive object, affordable only by prosperous citizens for their villas or as donations for erection in the town forum. There would have been a need for cheaper dials, particularly if they could be made by oneself or by an ordinary labourer. A projection on the vertical plane, as could be visualized by looking from a distance directly along the horizontal pointed gnomon of a scaphe dial, would be ideal.

This fiat dial would certainly have been easier to make, but unfortunately it was by no means easy to calibrate. Comparison with a well-made scaphe could suggest that the hour lines should bunch upwards in pairs, but it is obviously only too easy for oral

No ancient sundial has survived complete with its entire original gnomon: metal was much too precious to be discarded.

~A. A. Mills, 'Seasonal-Hour Sundials', Antiquarian Horology, 19 (1990), 147-70. Some errors were corrected in the next issue of this journal; 19 (1990), 315.

s S. L. Gibbs, Greek and Roman Sundials (New Haven, 1976).

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Notes and Discussions 87

instruction or personal memory to deteriorate down the easy path towards equian- gular straight hour lines converging on the base of the gnomon, although we have seen that this is correct only for a site on the equator. An example of such a 'debased diar 9 is provided by a fragment 1~ recovered from the Borcovicus station on Hadrian's Wall in northern England (Figure 2).

4. Saxon dials and scratch dials Spread among the older parish churches of England (and known too on the

Continent) are a few skilfully carved 'Saxon' dials 11 and a multitude of deteriorating ' s c r a t c h d ia l s ' . 12 In general, these are based on a semicircle divided radially into eight parts 13 (Figure 3) or twelve parts (Figure 4), although errors such as arcs and lines above the horizontal abound. Emphasis may be placed upon the two '45 ~ lines and the noon line by incising them more deeply or by adding short cross-bars.

The monastic rule stipulated prayer five times a day: at dawn, halfway through the morning, midday, halfway through the afternoon, and finally at sunset. It would therefore appear reasonable to suggest that the accentuated lines mark the traditional times (canonical hours) at which the incumbent should hold private or public prayer. We know that in England an equiangular dial pattern must be wildly inaccurate as a timeteller, but for the purpose and community that such patterns were intended to serve this did not matter. Anyone in the district who wished to join his parish priest and fellow Christians in body or in spirit at prayer had only to scratch a similar patttern around a nail in any convenient south-facing fence or wall and watch the shadow. In other words, the scratch dial was an event marker rather than a defective timeteller. It admirably fulfilled the criteria of simplicity and practicality and was, in its unpreten- tious way, a highly successful instrument; otherwise there would not have been so many of them! t4

5. True seasonal-hour planar dials Ancient peoples never completely solved the problem of projecting the three-

dimensional scaphe dial upon a vertical or horizontal plane, although skilled geometers could achieve close approximations that would have been perfectly adequate south of the Alps.15

It is therefore relevant to enquire what the correct patterns should be. This problem was addressed by a number of distinguished nineteenth-century mathematicians, each of whom presumably solved it to his own personal satisfaction. Unfortunately, their publications 16 are so complex, long-winded, and obscure that they were virtually

9 However, see the point made below concerning 'event markers'. l~ B. Brown, The Arts in Early En0/and 6 vols (London, 1921), v, 171-5. 11H. F. K. Eden and E. Lloyd, The Book of Sundials (London, 1900). 12 For a bibfiography of fiterature on scratch dials see the paper referred to in footnote 7; to these should

be added 'Sundial Literature Published by T. W. Cole', collated by C. K. Aked, Bulletin of the British Sundial Society, no. 3 (1991), 5-12.

13 For a discussion of the so-called 'octaval' or 'tidal' system of time measurement attributed to the Vikings, see A. J. Turner, 'Anglo-Saxon Sundials and the "Tidal" or "Octaval" System of Time Measurement', Antiquarian Horology, 15 (1984), 76-7.

14 Fortunately, the British Sundial Society is now recording details of scratch dials in Britain. The Secretary is Mr D. A. Young, Brook Cottage, 112 Whitehall Road, Chingford, London E4 6DW.

15 See, for example, M. Archinard, 'The Diagram of Unequal Hours', Annals of Science, 47 (1990), 173-90, and references cited therein.

16An extensive list is given in the paper referenced in footnote 7.

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88 A . A . Mills

(a)

(b)

Figure 3. Scratch dial on a southerly wall of the parish church at Boothby Pagnell, Lincolnshire. The lower picture shows the appearance produced when a pencil stub was temporarily inserted in the central hole.

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Notes and Discussions 89

Figure 4. Scratch dial in the porch of North Witham parish church, Lincolnshire.

inaccessible to antiquaries and--i t would appear from the replication of effort----even to their own (unacknowledged) colleagues. Part of the problem would appear to stem from a desire to obtain general equations for all the hour lines at any latitude and orientation, rather than presenting a few calculated dial patterns. The spherical trigonometry required for the latter endeavour is really quite basic, and the calculations tedious rather than difficult.

The availability of computer-generated graphics has, of course, completely altered the situation. A Fortran program was written for the VAX computer at the University of Leicester that enabled vertical or horizontal dials to be plotted for any latitude. The results are summarized in Figures 5 and 6. The patterns for latitudes 50 ~ to 60 ~ at 2 ~ intervals have been published on a larger scale. 17

5.1. Characteristics

1. Common to dials in all planes (a) The dials indicate seasonal hours, although these become identical with equal

hours at the equinoxes. (b) Only the tip of the gnomon is the effective shadow-casting element. Its

supporting stalk may therefore be placed in any position, although per- pendicular to the dial surface was usual.

17 A. A. Mills, 'Seasonal-hour Sundials for the British Isles', Bulletin of the British Sundial Society, no. 3 (1990), 15-21.

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90 A.A. Mills

ii

0 ~

+ I

10 ~ I

"1-

2 0 ~

§

4 0 "

3 0 "

5 0 "

6 0 ~ 6 5 "

Figure 5. Computed seasonal-hour vertical dials for an upright south-facing wall at the given latitudes. The tip of the gnomon is always 1/20 of the long axis of each box above the dial plane. The short dashes mark the equinoctial line.

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0 ~

2 0 ~

6 0 ~

4.

/ t~,m~tt t

4 0 "

10"

30"

/ t,,,,,,I \ 50"

Notes and Discussions 91

65"

///i Figure 6. Computed seasonal-hour horizontal dials for the given latitudes. The vertical

shadow-casting pillar is situated at the cross, and its tip is always 1/20 of the long axis of each box above the dial plane. The short dashes mark the equinoctial line,

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92 A.A. Mills

(c) A point was probably the most common form of termination of the gnomon, but a ball could have been preferred occasionally.

(d) The hour lines are really very shallow curves. However, the departure from linearity becomes noticeable only at latitudes greater than 45 ~

2. Additional characteristics of vertical dials (a) The upper ends of the hour lines curl in at an increasing rate so as to converge on

the base of a horizontal gnomon. (b) The midday shadow falls on the vertical line marking the end ofthe sixth hour. ts (c) The straight line approximations to the hour lines obtained by extending the

lower portions of the true curves meet in pairs on the extension of the midday axis above the horizontal. This is an area where the shadow of the tip of the gnomon can never fall. The angles between these straight line approximations decrease upwards- - in the opposite direction to the equal hour dial.

(d) The dial is symmetrical if described upon a truly south-facing wall, but becomes asymmetric about the midday vertical in declining dials.

1 = ' �9 . . . . . . . . . . . . : : ~ . . . . . . . . . . . . . . . . . . . 12th

Figure 7. An accurate seasonal-hour vertical dial for Leicester (52.6~ The tip of the south- facing gnomon is 1/5 the length of the long axis of box above the plane of the dial. The dotted lines show a superimposed scratch dial with its equiangular hour lines converging on the base of the gnomon.

is Therefore, the easiest way to determine whether a vertical plane dial is intended to delineate seasonal or equal hours is to see whether the vertical midday line is marked '6' or 'IT respectively. This criterion is particularly useful when examining folding diptych dials with multiple pin-gnomons. Cases are known where the dial pattern is an incorrect hybrid of the two systems: see, for example, A. A. Mills, 'Isaac Newton's Sundials', Antiquarian Horology, 20 (1992), 126-39.

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Notes and Discussions 93

. Additional characteristics of horizontal dials (a) Display curvature of the hour lines more strongly than vertical dials for the same

latitude. (b) Become very large in area compared with the height of the shadow-generating

pillar, especially above 450.19

5.2. Comparison between true and equiangular dial patterns Figure 7 shows in full lines the computed seasonal-hour south-facing vertical dial

for the middle of England (Leicester; latitude 52.6~ Upon it, in dotted lines, is superimposed an equiangular 'scratch' dial. It will be seen that the end of the third hour as imputed by the latter is, in fact, always.3/4 to 1 seasonal hour late for the true mid- point of the morning. Similarly, the mid-afternoon conclusion of the ninth hour is always indicated 3/4 to 1 seasonal hour too early. It has already been remarked that erroneous calibration does not really matter if everyone is using the same pattern of dial but, rather than being a problem, the hidden errors could have certain advantages. Noon is always correct.

6. Conclusions Unlike equal-hour sundials, seasonal-hour dials employ only the tip of their

gnomon as the shadow-casting index. Its shadow registers on the slightly curved hour lines converging upon a point perpendicularly below the index. In the vertical dial the spacing between the hour hncs tends to decrease upwards. Ancient dials, although intended to delineate seasonal hours, were often no better than approximations. However, the equiangular 'Saxon' and 'scratch' dials should be considered practical and effective 'event markers' rather than defective timetellers.

19 E. Buchner, Die Sonnenuhr des Auffustus (Mainz, 1982).

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