A Volvelle for the Equation of Daylight

Dave Abercrombie, abe@rahul.net


This volvelle is from Galluci's Theatro del Mundo ye tiempo Granada published 1657 in Venice. It is based on the Equation of Daylight, and is useful for calculations involving "seasonal hours" (aka "unequal hours"). This page outlines the mathematics, use, and significance of the volvelle.

Description

This volvelle was published as page 189 of a 1657 edition of Galluci's Theatro del Mundo ye tiempo Granada. The pointer arm is attached to the page with a strand of red silk. The arm is marked with signs of the Zodiac, with symbols for Aries and Libra at the end, and with Virgo and Pisces near the center. The innner circular scale on the page is marked with numbers from 0 through 29, representing degrees within a sign. The arm can be rotated to provide data at any desired combination of sign and degrees. This example is a slightly weak impression. The volvelle has a diameter of about 135 mm.

I am not sure what language is used for the text, but I think it is Spanish. I suspect that the text on the reverse side is not a discussion of this volvelle.

Numeric data transcribed from volvelle

The following data have been transcribed directly from the volvelle. The numbers are in base 60 (i.e., "sexagesimal") as is commonly used for angles (e.g., degrees and arcminutes) or time (hours and minutes). For example, at 6 degrees Leo, the volvelle reads 12 degrees and 25 minutes.

degrees Aries, Libra Taurus, Scorpius Gemini, Sagittarius Cancer, Capricorn Leo, Aquarius Virgo, Pisces
0 15 - 00 13 - 27 12 - 12 11 - 43 12 - 12 13 - 27
1 14 - 55 13 - 24 12 - 10 11 - 43 12 - 14 13 - 30
2 14 - 52 13 - 22 12 - 09 11 - 43 12 - 16 13 - 35
3 14 - 50 13 - 20 12 - 07 11 - 43 12 - 18 13 - 37
4 14 - 47 13 - 18 12 - 06 11 - 43 12 - 20 13 - 40
5 14 - 43 13 - 15 12 - 05 11 - 43 12 - 23 13 - 42
6 14 - 40 13 - 13 12 - 04 11 - 43 12 - 25 13 - 45
7 14 - 37 13 - 10 12 - 02 11 - 44 12 - 27 13 - 49
8 14 - 35 13 - 06 12 - 00 11 - 45 12 - 30 13 - 52
9 14 - 32 13 - 02 11 - 58 11 - 45 12 - 32 13 - 55
10 14 - 30 13 - 00 11 - 56 11 - 46 12 - 35 13 - 58
11 14 - 26 12 - 57 11 - 55 11 - 47 12 - 37 14 - 02
12 14 - 23 12 - 55 11 - 53 11 - 48 12 - 40 14 - 05
13 14 - 20 12 - 52 11 - 52 11 - 49 12 - 42 14 - 07
14 14 - 16 12 - 50 11 - 50 11 - 49 12 - 45 14 - 10
15 14 - 13 12 - 47 11 - 50 11 - 50 12 - 47 14 - 13
16 14 - 10 12 - 45 11 - 49 11 - 51 12 - 50 14 - 16
17 14 - 07 12 - 42 11 - 49 11 - 52 12 - 52 14 - 20
18 14 - 05 12 - 40 11 - 48 11 - 53 12 - 55 14 - 23
19 14 - 02 12 - 37 11 - 47 11 - 55 12 - 57 14 - 26
20 13 - 58 12 - 35 11 - 46 11 - 56 13 - 00 14 - 30
21 13 - 55 12 - 32 11 - 45 11 - 58 13 - 02 14 - 32
22 13 - 52 12 - 30 11 - 45 12 - 00 13 - 06 14 - 35
23 13 - 49 12 - 27 11 - 44 12 - 02 13 - 10 14 - 37
24 13 - 45 12 - 25 11 - 43 12 - 04 13 - 13 14 - 40
25 13 - 42 12 - 22 11 - 43 12 - 04 13 - 15 14 - 43
26 13 - 40 12 - 20 11 - 43 12 - 06 13 - 18 14 - 47
27 13 - 37 12 - 18 11 - 43 12 - 07 13 - 20 14 - 50
28 13 - 35 12 - 16 11 - 43 12 - 09 13 - 22 14 - 52
29 13 - 30 12 - 14 11 - 43 12 - 10 13 - 24 14 - 55

Based on the Equation of Daylight

The numeric data on this volvelle can be generated with the "Equation of Daylight". First, I define some terms that I'll be using below:

lambda  = longitude of Sun (geocentric)
epsilon = obliquity of ecliptic (here 23 deg 28 min)
delta   = declination of Sun
phi     = observer's Latitude (here 38 degrees)
n       = Equation of Daylight
length  = length of daytime, etc. depending on context

One first needs to calculate the Sun's declination (delta) from the longitude of the Sun (lamda) and the obliquity of the ecliptic (epsilon) as shown below. I used a value of 23 deg 28 min, although any value near this works just as well (see below for details). The formula for declination has symmetry over 180 degrees, so one need only calculate values for half of the year (as long as you pay attention to the plus/minus sign).

sin(delta) = sin(lambda)sin(epsilon)

Once we have the Sun's declination (delta), we can use the observer's latitude (phi) to calculate the Equation of Daylight (n) as shown here. A latitude of 38 degrees is used by the volvelle, and it is correct only at that latitude.

sin(n) = tan(phi)tan(delta)

The Equation of Daylight (n) from above is usually converted into more useful units. For example, if you double it then add to 180 degrees, you obtain the angle that the sky rotates in it dinural motion between sunrise and sunset. Since the sky rotates 15 degrees per hour, you can divide the angular Equation of Daylight by 15 to obtain in hours the length of day or night time. Both of these versions are shown below.

length = 180 + 2(n) gives degrees of Sun rotation per daylight
length = (180 + 2(n))/15 gives hours of daylight
length = 12 + (n/7.5) also gives hours of daylight

However, this volvelle is based on the conversion shown below which exhibits two differences from the conversion shown above. First, a negative sign is used instead of a plus sign. This shifts the focus of the data from daytime to nighttime (that is for Aries through Virgo). Secondly, 12 is used as the denominator rather than 15.

length = 180 - 2(n) gives degrees of Sun rotation per nighttime
length = (180 - 2(n))/12 gives degrees of sky rotation per 1/12 night
length = 15 - (n/6) gives degrees of sky rotation per seasonal hour

The Equation of Daylight was also known in the Middle Ages as the "Ascensional Differences".

The values of Equation of Daylight calculated via spreadsheet do not match exactly the values transcribed from the volvelle. At least part of the "error" is due to rounding, and there might be other types of error in the volvelle. However, the fit was good: the standard deviation of error was 52 seconds (i.e., two-thirds of the volvelle values were within 52 seconds of the values calculated by spreadsheet).

Significance and Use

This volvelle indicates how many degrees the sky rotates in its dinural motion during a "seasonal hour" (more on that term below). For example, when the Sun is 8 degrees of Leo (end of July), the volvelle shows that the sky will rotate 12 degrees 30 minutes during a seasonal nighttime hour.

The form of the Equation of Daylight presented by this volvelle makes sense particularly if one is interested in twelfths of a night (or day). Before the advent of mechanical clocks, it was common to divide daytime and nighttime each into 12 portions. During summer, the daytime portions would be longer that the nighttime portions, since the daytime was longer than the nighttime (although all 12 of the daytime portions would have the same duration, and all 12 of the nighttime portions would also be equal to each other but shorter than the daytime portions).

These portions have various names including "seasonal hours" and "unequal hours". Since the length of day and night varies with the Earth's seasons, the length of the "seasonal hours" also varies with season. And since, in general, the length of the daylight was not equal to the length of nighttime, the duration of a daytime "unequal hour" is not equal to the duration of a nighttime "unequal hour".

The Equation of Daylight is also useful when working with "Italian hours". These hours were obtained by dividing the complete dinural rotational period into 24 equal portions, just like we do. However, the counting of the Italian Hours would begin at sunset (or a half hour after sunset), rather than midnight (like us now and most of Northern Europe then). Knowing the length of night would help in the conversion between Italian Hours and modern-style hours. However, I would have expected the form using a denominator of 15 to be more efficient for this calculation that the denominator of 12 used by this volvelle.

Although this volvelle was published in Venice at a latitude above 45 degrees, it was calculated for a latitude of 38 degrees. This lower latitude of the volvelle is appropriate for use in much of the Mediterranean, and it is practically exact for places like Sicily and Athens.

Interestingly, the circular format of this volvelle seems to have few, if any, advantages over tabular presentation of the data. There are in fact no functional, geometric, or clarity benefits from the circular style of data presentation of this vovelle. It is certainly compact and attractive, but no loss in function would result from a simple, rectangular, tabular display (as shown above). Note that this is in contrast to, say, an astrolabe, where the circular design has geometric properties that are essential to its use.

Galluci apparently flourished 1569-1597. A Latin version of the "Theatro del Mundo..." , published in 1586 contains 51 different volvelles. So although Galluci's maps are rare, the volvelles are not quite so. That there would be so many in one volume leads me to think that Galluci considered volvelles as routine, rather than as the exception. I initially thought otherwise, assuming that the extra effort and expense to publish them might limit their use to only the most essential data or issues. But with as many as 51 different volvelles in one book, some are bound to address obscure or less common problems.

Limitations

Strictly speaking, the Equation of Daylight does not really provide the exact duration of day or night. It ignores issues such as refraction, height of eye, apparent diameter of the Sun's disk, and movement of the Sun during the day. It actually indicates the difference in Right Ascension between an object on the horizon and another (fictional) object with the same declination on the opposite horizon. However, ignoring these details does not significantly impair the usefulness of the Equation of Daylight.

References

These data and equations can be downloaded in this spreadsheet.

Ptolemy's Almagest, Book II, Chapter 9 discusses many useful calculations one can perform with the Equation of Daylight. Ptolemy also specifically compares division by 12 to division by 15. However, Ptolemy did not use the name "Equation of Daylight", and lacking modern trigonometry, he did not use the formulas described above

Otto Neugebauer's A History of Ancient Mathematical Astronomy (Springer-Verlag, 1975) discusses the Equation of Daylight on page 36, equation 2.

N.M. Swerdlow and Otto Newgebauer's Mathematical Astronomy in Copernicus's De Revolutionibus (Springer-Verlag, 1984) discusses's 16th century estimates of the obliquity at page 104.

J.L. Heilbron's The Sun in the Church : Cathedrals As Solar Observatories (Harvard University Press, 1999) provides a history of estimates of the obliquity in the 17th century.

Jean Meeus's Astronomical Algorithms (Willmann-Bell, 1991) provides many modern astronomical formulae, including historical values of the obliquity and basic coordinate conversion such as calculation of declination.

The Bologna Observatory has a description of Italian hours. The Dutch Sundial Society provides construction details and an image of a "hemsipherium" sundial that displays Italian Hours

Glossary


© Dave Abercrombie, February 2002