 Keith's Home page Java applet page: Keith's Astrolabes Astrolabe information index

# Back of astrolabe

The back of a medieval astrolabe usually showed several interesting features, many of which can be displayed with this program.

To see a display of the back of a typical astrolabe, you can press the 'f8' keyboard function key, or use the menu:

Menu: Astrolabe /show reverse of Astrolabe
The outer area almost invariably displayed three circular scales, and in the central area were drawn diagrams, each usually occupying either one or two quarters of the available space, each quarter being called a quadrant.

## Scales

The outer area of the back of almost every medieval astrolabe shows three circular scales. The outer one is divided into degrees. Adjacent to it is usually a Zodiac scale, and close to that is a calendar scale. Occasionally, particularly on Islamic astrolabes, more scales were drawn having a variety of uses.

The degree scale was used with an alidade to find the height of the Sun or a star.

To use a medieval astrolabe you need to know the apparent position of the Sun around the ecliptic circle for any day of the year. The ancients solved the problem of the apparent varying motion of the Sun around the celestial sphere (which we now know to be due to the elliptical orbit of the Earth around the Sun) by using either

a linearly divided calendar scale whose centre is slightly offset from the centre of the Zodiac scale or
a non-linearly divided calendar scale whose centre was concentric with the centre of the Zodiac scale.
With either type of calendar scale they were able to find the position of the Sun on the Zodiac scale for any day of the year.

The ecliptic circle on the rete of the astrolabe is also marked with the Zodiac scale, so the current position of the Sun on the rete could then be found.

## The degree scale and the alidade

The degree scale engraved around the outside edge of the back of an astrolabe was used with an alidade.

An alidade is a pointer which can be rotated about the centre of the back of an astrolabe, and its angular position can be found on the degree scale. Each end of the alidade has a small plate projecting away from the astrolabe in which are holes and/or notches which are used as sights.

To use an alidade, the astrolabe is first suspended freely using the suspension system attached to the throne. The alidade is then rotated to pointed towards the Sun or a star. When the rays of the Sun passing through the upper sight can be seen to fall over the lower sight, the height of the Sun can be read from the position of the alidade over the degree scale.

Notches allowed the alidade to be aligned more easily with stars, whose altitude could then be found on the degree scale.

## The Zodiac scale

The Zodiac scale on the back of an astrolabe is a circular scale showing the 12 signs of the Zodiac, each sign occupying 30 degrees, the scale having divisions to allow the position of the Sun to be found as it moves across each Zodiac sign.

The Zodiac scale was used with the adjacent calendar scale to find the position of the Sun in the Zodiac on any day of the year.

The signs of the Zodiac, together with the ecliptic longitude at the beginning of each sign, are:

Aries (0°), Taurus (30°), Gemini (60°),
Cancer (90°), Leo (120°), Virgo (150°),
Libra (180°), Scorpio (210°), Sagittarius (240°),
Capricorn (270°), Aquarius (300°), Pisces (330°).
Thus, Scorpio ranges from 210 to 240 degrees of ecliptic longitude. If a planet is stated to be at a position of 25 degrees into Scorpio, it has an ecliptic longitude of 235 degrees.

However, note that the Zodiac names are also the names of constellations. Astronomers commonly say that a planet is in a certain constellation, and the constellations do not have the same ecliptic longitudes as the signs of the Zodiac.

## Calendar scales

The ancients recognised that there was a problem in predicting the precise position of the Sun throughout the year in relation to the positions of the stars on the celestial sphere. They considered this to be due to the varying speed of the Sun around the celestial sphere We now know it to be due to the elliptical orbit of the Earth around the Sun and the varying speed of the Earth around its orbit.

They knew that on two days of the year the Sun rose when due East and set when due West. These were the days of the Spring and Autumn equinoxes. They knew that there was a different number of days between the former and the latter and the latter and the former. By arranging that the date of the first of these corresponded with the start of Aries (the First Point of Aries or FPoA) and that the date of the second of these corresponded with the start of Libra, they had a correlation between the Zodiac calendar scale and the calendar scale which closely corresponded throughout the year. Leap Years were a minor problem, best ignored. The result was a pair of linearly divided circular scales, one divided into days/months and the other into degrees, whose centres were slightly offset one from the other.

By extending the divisions of the offset calendar scale to the inside edge of the Zodiac scale, it was practical to produce a concentric but non-linear calendar scale. This concentric scale had the advantage that it took less room, but there was no difference in the inherent accuracy.

Because the years of the turn of centuries were not leap years in the Julian calendar, the date of the Spring equinox varies. Consequently, the date on the calendar scale which corresponds with the First Point of Aries has sometimes been used as a guide to the date of manufacture of medieval astrolabes. However, Henri Mitchell warns against pitfalls when using this dating method.

When you are displaying the back of the astrolabe, you can select which of the two types of calendar scale to display, or you can choose to display both. Probably the easiest method of selection is to use the '6' key on the keyboard, which cycles through displays of

a non-concentric calendar scale,
a concentric calendar scale, or
both.

You can also use the menu to select each of these options:
Menu: Astrolabe /Reverse - Calendars /Offset centre
Menu: Astrolabe /Reverse - Calendars /Concentric
Menu: Astrolabe /Reverse - Calendars /Offset + Concent
The graduations on the concentric scale displayed here are plotted using an algorithm, and will be precise for a couple of millennium on either side of the year 2000. You will see that the difference between the concentric and offset scales (that is, the error inherent in the offset scale) is small enough to be ignored.

You will see that the direction of the offset of the offset calendar scale (its closest point to the Zodiac scale) alters by a small amount as you change the Centuries. This is due to the changing perihelion of the earth, the perihelion rotating around the Sun by 360 degrees in 20930 years.

To change the year in 100 year steps, you can use the +++ and --- buttons within the 'set year' box in the panel, and watch the offset direction gradually change, indicated by the position of the minimum size of the gap between it and the adjacent Zodiac scale, and by the tiny arrow in the centre of the back. The positions of the graduations along the concentric calendar scale also change as the date is changed. (After several thousand years, the two scales differ due to the imprecise use of the year in the formula used for the concentric scale.)

It is interesting to observe that the direction of the offset is nominally vertical from about 700AD to about 1800AD.

The central area within the outer scales was commonly used for diagrams. Each diagram was usually drawn either in one quadrant or in a pair of quadrants. You can select which of several diagrams you want to display in each of the four quadrants.

The quickest way to view the options is to use keyboard keys, the '7', '8', '9' and '0' keys allowing you to cycle through the options:

'7' - the upper left quadrant,
'8' - the upper right quadrant,
'9' - the lower left quadrant and
'0' - the lower right quadrant.
In fact, the two upper quadrants have the same options, as have the two lower quadrants, although the displays in the two quadrants are left-to-right reversed from each other.

The upper quadrants can show:

Unequal hours curves
Unequal hours arcs
Sigma curves - the current latitude
Sigma curves - 25 to 65 degrees in tens, + latitude
Sigma curves - 30 to 60 degrees in tens, + latitude
The lower quadrants can show:
Shadow square - 12 divisions
Shadow square - 10 divisions
Shadow square -   7 divisions
These options can also be selected for each quadrant using the menu:
Menu: Astrolabe /Reverse - upper left quadrant
Menu: Astrolabe /Reverse - upper right quadrant
Menu: Astrolabe /Reverse - lower left quadrant
Menu: Astrolabe /Reverse - lower right quadrant

There is another page which deals with unequal hours. On that page you will find explanations of the unequal hours arcs and curves shown on the backs of astrolabes.

The Sigma curves allow you to determine the maximum height of the Sun throughout the Zodiac year, this depending upon latitude. Sigma curves are therefore displayed for the latitude which is shown in the panel on the left of the astrolabe display and also for odd or even groups of latitudes.

The sigma diagrams allow the Sun's Zodiacal position for specified latitudes to be found. The arcs indicate the divisions of the Zodiac, the Zodiac names being indicated by their initial letters across the bottom and side of the diagram. A line from the centre of the astrolabe through the Sun's current Zodiacal position shows the height of the midday Sun on the outer edge of the diagram.

The shadow squares were most commonly used to calculate the heights of objects by ratios. It was common to use ratios of  7 and 12, but we would now use ratios of 10. That is, a ratio would be found of so many parts in 7, or 12, or 10. The ratio for which the shadow square was engraved is the number which appears in the corner. Thus, if the number is 7, there will be 7 main divisions along the bottom of the shadow square as well as down the side.

The lower edge of the shadow square was named 'umbra recta' and the side was named 'umbra versa'.

Suppose the shadow square was calibrated for ratios of 10. If the distance of an object from the astrolabe was 8 units, and the alidade pointing to the top of the object crossed the bottom of the shadow square at the fourth division (measured from the vertical line down the centre of the astrolabe) then the height of the object would be 10 x 8 units divided by 4 units, which is 20 units.

Perhaps the shadow square was calibrated for ratios of 7, and the distance of an object from the astrolabe was 8 units. If the alidade pointing to the top of the object crossed the bottom of the shadow square at the sixth division (measured from the vertical line down the centre of the astrolabe) then the object would be 7 x 8 divided by 6 units tall which is a little more than 9 units.

However, with a square for ratios of 7 and a distance of 6 units, if the alidade crossed the line numbered 5 below the horizontal line, the height of the object would be 5 x 6 divided by 7 units tall which is 4 and a bit units.

With all of the above, the height you obtain is the height above that of the observer's eye level, of course. By holding the astrolabe on its side, you can obviously use the same technique to find the distance across a river using suitable objects on both banks.

It is also possible to use the shadow square to find the tangents of angles. Thus, with a square for ratios of 12, a pointer through the third division from the vertical takes you to the 14th degree from the vertical on the outside scale. Hence the tangent of 14° is 3/12. Similarly, a pointer through the 5th division down from the horizontal takes you to the 67.5th degree from the vertical. Hence, tan 67.5° is 12/5.

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Copyright Keith Powell 1999-2002