Drafting the Astrolabe 10: The Shadow Squares

The next task is to finish off the rear of the astrolabe. Two scales remain to be drawn, the Unequal hours and the shadow squares.

Drafting the shadow squares is straight-forward: Two square boxes in the bottom half of the astrolabe back, each side divided into a set number of sections. The only trick bit will be finding the correct size for the boxes. I want the shadow squares to be as big as possible, as larger means more accurate to draw and to use, but they also need to fit into the remaining space inside the calendar ring. Because the calendar ring is offset, I have to take that offset into account

The largest dimension of the shadow square is its 45 degree diagonal, so if I draw a line at 45 degrees from the center to the right bottom side of the astrolabe, stopping at the inside edge of the calendar ring, I can use that point to erect the rest of the square.

Step 1: Draw the 45 degree diagonal from the center down and right to where it touches the inside of the calendar ring on the bottom right side of the astrolabe back. Mentally label this point “A” (See Figure A).

Step 2: Erect a vertical line from point A to the horizontal center line of the astrolabe (see Figure B).

Step 3: Erect a line parallel to the horizontal center line of the astrolabe, running from point A to the left side of the astrolabe (see Figure C).

Step 4: Draw the 45 degree diagonal from the center down and left to where it touches the line drawn in the previous step. Mentally label this point “B” (see Figure C).

Step 5: Erect a vertical line from point B to the horizontal center line of the astrolabe as you did in step 2 (see Figure D).

Step 6: Draw a scale line inside the box you just drew (see Figure E).

Step 7: Divide and label the Shadow Squares.

Drafting the Astrolabe 9: The Calendar Ring Completed


Completed calendar ring

Drafting the Astrolabe: Concerning Accuracy

When drawing arcs to bisect lines or angles it is best to set the compass radius so that the construction arcs cross as close to 90 degrees as possible. The shallower the angle of incidence, the less defined the intersection, and the greater chance of the line drawn between the intersections being off by a (small, admittedly) amount.
Angle for bisecting an angle

Bisecting an angle

When using the Compass to measure distances, it is easier to transfer large distances than small ones with accuracy. If I need to draw a line and mark a point 1/32 of the way along it, I can do this with good accuracy by repeatedly dividing the line in half by construction. But if I want to transfer that 1/32 distance, it is easier and more accurate to transfer the 31/32 measurement. The mark ends up in the same place, but adjusting the compass accurately is easier.

When dividing a circle it is better to work each stage in the division over the whole circle then move to the next. If I need to divide a circle into 24 sections, I should first quarter the whole circle, then trisect all four quarters, then bisect all twelve sections. I could try to save time by trisecting one quarter, and bisecting one section of that; then use the compass to copy the arc of that 1/24 section 24 times. But I find that small errors in measurement compound if you do it that way. Working the whole circle at once tends to cancel out small errors.

Drafting the Astrolabe 9: The Calendar Ring

Well… That was a bit more intense than I expected.

I was expecting that drawing the calendar ring on the back of my astrolabe would be rather involved, but actually doing it took more patience than I expected. This was definitely the hardest part so far.

Drawing the offset ring was straight-forward. As described previously computing the direction and distance of the offset is fairly simple, once you have the equations, and drawing the lines presents no problem. Where the difficulty lies is dividing the calendar ring. As described earlier, I need to mark off a section equal to the equivalent 5.25 days and then divide the remainder of the circle into 360 parts. Because I’m no longer working with a full circle, I can’t just quarter and then trisect like i did with the zodiac ring. Here I have to bisect each angle, and fudge “trisection” with the compass. Fudging the trisection by adjusting the compass to mark off 1/3 of the arc by eye is full of potential for small errors. Moreover, the first bisection that has to be made is to divide a huge arc – It was harder to do that accurately than I expected. I had to invent techniques to make it easier to divide accurately. After much cussing and erasing, I have a result that is fairly accurate, I need to ink and label it and then I will post a scan of the result.

Drafting the Astrolabe 8: On Tools

While I’m working on this project, I’m also researching the tools and techniques used by the instrument makers who made these astrolabes. The two period manuals I am working from assume a certain shared level of knowledge, and do not mention tools or engraving techniques, except in passing.

I am finding the art of the period a better resource; in particular, I found a very nice portrait by Hans Holbein the Younger:


Nicholas Kratzer by Hans Holbein the Younger

Nicholas Kratzer by Hans Holbein the Younger

(Note: WikiMedia is listing this image as Public Domain, if anyone claims the rights to the photograph, please contact me at the email to the right and we can discuss it)

The original image can be found here.

The subject of the portrait is shown with several tools and a partially completed sundial compendium similar to those described in Hartmann. The tools for designing and laying out the lines of a sundial are the same ones that would be needed in designing and laying out an astrolabe.

Detail of tools

Detail of tools

Detail of tools

Detail of tools

Identifiable in the portrait are two straight-edges (one on the table, one hanging), two scribing compasses (on the table and hanging), a set of dividers (by the hammer on the right), a scriber (in front, right of center), some scissors and some other tools whose use is not immediately evident.

My friend Miguel, of Spanish Peacock was nice enough to make me a very nice straight-edge out of hardwood, based on the one in the painting:

Tools for Astrolabe Construction

Tools for Astrolabe Construction

A bit later, I stumbled across a two pairs of scribing dividers at a Harbor Freight sale.

I’m currently looking into buying a good scriber, and I will be ready to start playing with brass. I have neither the time nor skill to build a complete astrolabe from scratch, but I’m thinking a brass version of one of the climate plates might be instructive to attempt.

The Future: Moving the Astrolabe Generator away from Flex

I like Flex, but the tools to use it are not cheap. I can’t afford $250+ for the next version of Flash Builder and from what I’m seeing even Adobe is jumping on the HTML5 bandwagon. If Adobe offered a free, noncommercial version of Flash Builder, I might be convinced to stay with the platform. but I’m not seeing a future for Flex/Flash and the open-source community is not embracing it.

I’ve looked at HTML5, and it has potential, but it suffers from the same problem all the previous versions of HTML/JavaScript suffered from: uneven browser support. I don’t want to have to rewrite my interface code for every browser version that comes along. That got old years ago. I looked at other options as well, but Java is still the best choice I can find: It is object oriented, strongly typed, open source, platform independent, and has a massive list of free and open source tools to work with it.

The conversion will take time, But I’m planning on releasing the java version as 4.0 in the Fall. If anyone wants to jump on the bandwagon, contact me and I’ll send you my notes on the overhaul. As always, the code is checked in at SourceForge[http://sourceforge.net/projects/astrolabegenera/].

While I was digging through the process of planning the new version, I set up and ran the original java project:

Version .5 of the Astrolabe Generator

I’ve made a bit of progress since then.

The Future: Version 3 of the Astrolabe Generator

I spent most of the last weekend coding like crazy. I now have three new optional scales to add to the back of the astrolabe:Preview of Version 3 of the Astrolabe Generator

The Lunar Mansions. A calendar/astrology scale. This is visible inside the shadow squares, center bottom.

Two views of the Arcs of the signs: Equal sized and projected(top left shows the projected view of the arcs, with the Noon lines and the Qibla lines)

These are still not ready for use(note the missing labels…), but I’m planning on finishing them for the 3.0 release later this spring.

Drafting the Astrolabe 7: The Calendar Scale.

The next step in drafting the astrolabe is constructing the calendar scale.

Up until now the process of creating scales has been straightforward. The time scale on the front and the altitude and zodiac scales on the back are simple in concept and execution, requiring no math to draw. The calendar scale, on the other hand, needs to be aligned to the zodiac scale so that when the alidade pointer is rotated to a given date, the pointer crosses the zodiac scale at the Sun’s ecliptic position on that date. This is not straightforward, as the Earth’s orbit is elliptical, and the Sun appears to move along the ecliptic at different speeds at different times of the year; in other words, it covers a slightly wider arc per day in winter than in summer.

Traditionally, there are two ways to allow for the Sun’s uneven motion. The first is the concentric method, where the calendar circles are drawn concentrically with the zodiac circles. In this case the days are drawn on the scale so that the space between the day marks varies depending on where on the scale they are. To accomplish this, the astrolabe maker must determine the exact location of the Sun for each day. This method requires a lot of careful calculation and measurement, and would be very time-consuming, unless reference tables are available. I could use this method, as I have access to an accurate ephemeris. But the drawing of the day lines would be more finicky that I would like to attempt for a first try.

The second, simpler method is the one I’m going to use. This second method, the eccentric method, offsets the center of the calendar scale from the center of the zodiac scale, modeling the offset of Earth’s orbit. This method is actually very accurate, well within the overall accuracy of the astrolabe itself [Morrison: 111] and was used extensively by astrolabe makers. To accomplish the drafting this scale accurately, three factors must be known in advance: The direction of the offset, the distance of the offset, and the rotation of the calendar.

First, direction; from Stoeffler:

Find the Apogee of the Sun, corresponding to the time of construction of our astrolabe through the Alfonsine Tables, or through other tables. For example, in the Year of Christ 1510, with exact calculations, it is around 1 degree 16 minutes of Cancer. Therefore, get ready to place it on this point of Cancer, as this is its position in the year mentioned, that is, at the 16th minute of the second degree of Cancer.” [Stoeffler: 66]

The Alfonsine Tables {http://en.wikipedia.org/wiki/Alfonsine_tables} and other tables mentioned are tables of astronomical data that would have been expected to be available to someone designing astronomical instruments of the period.

Lacking such tables, I will have to do some computation. Morrison goes deeply into the math in his section on drawing the eccentric calendar; so I won’t try to duplicate his work in detail here, if you are interested I recommend his book very much. Here is a summary of the process:

Step 1. Compute the Julian Century (T)

The Julian Day for January 1, 2000 at noon UT is 2451545.0 [after Morrison: 337].

The current Julian Day (For the time of writing (November 14, 2011), is 2455879.5).

The Julian Century is computed as follows:

T = (2455879.5 – 2451545.0) / 36525 = 0.118672142 [after Morrison: 337].

Step 2. Find the angle of perihelion (Q)

Q = 102.937348 + (1.7195269*T) + (0.00045962*T^2) + (0.000000499*T^3) = 103.1414144 [after Morrison: 112].

Or 103 degrees rounded off. Counting around counterclockwise from Aries 0, this gives me an angle of Cancer 13.

Next, I need to know the magnitude of the offset. Stoeffler gives an offset of 1/32nd of the inner radius of the zodiac ring (0.03125 x R) [Stoeffler: 67]; where Morrison computes it to be 0.0334 x R [Morrison: 112]. Morrison is definitely the more accurate; but as the difference is about a hundredth of an inch for the circle I’ll be drawing, I’ll go with Stoeffler, as it is easier to draft.

With these first two factors known I can draw the rings for the calendar. The procedure is as follows (See the figure to the right):

A. Find the center of the zodiac ring. This is, of course already marked; I used it to draw the zodiac circles.

B. Draw a construction line (to be erased later) from the center to the inside edge of the zodiac ring at an angle of 103 degrees from Aries 0 (to Cancer 13).

C. Divide this line repeatedly until you have a mark 1/32 of the way from the center to the edge (exaggerated here).

D. Using that point as a center, draw the circles for the calendar ring.

Finally, I need to know where to start drawing the calendar; that is, how to rotate the calendar so that it is properly aligned with the zodiac.

Stoeffler calls for aligning the beginning of January with Capricorn 20:

Make the subdivisions of the days and of the months in this way. Set the rule on the center E and on the 20th degree of Capricorn and draw a line segment crossing all the eccentric circles. It will point out the beginning of January and be marked with a G. Starting from it, in the opposite direction to the succession of the Signs, count around 5 degrees 20 minutes. Set the rule between this point and the center E. Draw a line from the first eccentric circle to the second, which will be H. The remaining arc must now be divided (this small arc excepted) into 360 equal parts…” [Stoeffler: 67]

The sky has shifted a bit in 500 years. Using the equations in Morrison’s book, I come up with a rotation figure of -79.4757 (call it -79.5 degrees). Starting from Aries 0 and counting clockwise (remember, the sun moves counterclockwise through the zodiac ring, and we are rotating backwards) that puts the line for the beginning of January at Capricorn 10.5.

I now have all the figures I need to begin drafting the Calendar Ring.

Drafting the Astrolabe 6: The Zodiac Scale

OK, yes, that was tedious. But now it is done.

I have started work on the back of the astrolabe: The first scale, the zodiac ring, is finished.

Drawing this used a combination of geometric construction and fudging with the compass. First, draw the outside edge of the astrolabe (set the compass using the finished front drawing). When this is done draw 4 more circles of diminishing radius to make the scale. Then trisect the four quadrants. This divides the scale into the twelve astrological signs. Fudge with the compass to divide each sign into three parts, giving you the ten-degree marks. Then next, bisect the 10-degree angles to give the five-degree marks. Finally manually divide each five degree section into five one-degree ticks.

The Back of the Astrolabe

The scales should be marked as follows: The top half of the outermost ring should be marked 0-90 degrees on each side (zero for the horizontal and counting up to 90 for the vertical). This is the elevation scale, used to measure elevation above the horizon.

The innermost scale needs to be marked with the zodiac (I used the symbols, but spelling the names out is also used): Starting with Ares on the right side above the horizontal line and working counter-clockwise through the rest of the zodiac in order. Finally, each zodiac section has to have the 10 and 20 degree ticks marked (note that these are marked counter-clockwise as well).

Next up: Adding the Calendar.

Drafting the Astrolabe 5: A Recipe for Fudge(ing)

The next big chore will be creating the major scales for the back of the astrolabe. This will involve drawing one set of rings representing the Zodiac and another set representing the Calendar.
Drafting the zodiac is straight-forward, if tedious. The circle needs to divided into 360  one-degree ticks. The process is the same one I used to create my protractor (link).
The next step will be -um- interesting. The calendar scale sits inside the zodiac ring and is oriented such that when placed on a date, the adidade pointer will also point to the position of the sun on the zodiac for that date. Unfortunately, where 360 divides up well, 365 does not. in fact, the only divisors other the 1 and itself are 5 and 73. Now I can divide the circle into 5 equal parts (quintrants?)by cheating with the compass, but after that 73 is prime and not divisible equally. In addition the year is not 365 days…. it is 365.25.
So I looked at my sources. Stoeffler has this to say:
Hummm. Clever.
So I am to remove the angle equivalent of 5.25 days from the circle, and divide the remaining arc into 360 ticks. Much easier. Or rather somewhat easier; as the 360-day section of the circle is NOT 360 degrees, I will not be able to quarter it and trisect etc. like I will the zodiac. I will need to bisect repeatedly and do some fudging with the compass when I need to divide an angle other than in half.
Question: Am I dividing into 365 or 365.25?
It occurs to me that at the finished diameter of my astrolabe, the .25 day tick will be approximately the same as the width of the line used to mark it. If I remove the .25 and use 365 ticks, each tick will be thrown off a bit: By approximately .0007 of the width of the space. This is not going to be measurable with this instrument.
An examination of the backs of several period astrolabes shows that leaving out the .25 is a regular practice. In fact, going back to Stoeffler: “The GH arc will be divided into 5 parts and 1/4, if it is held to precision” (Stoeffler:67). So for this project I’ll go with 365 days.