Shadows and Waves - the Land of Trigonometry

Front of astrolabe. The top right is the sine quadrant. The lower left is the shadow (umbra) square.

1. Height of an Object

To find the height of an object like a tree or a building, the easy way is to walk to a place where the top of the object is at 45 degrees as measured by the alidade. Gradually move closer or farther so that it is at exactly 45 degrees. Now the distance you are from the object is the same as the height of the object. If you have previously calculated the length of your normal stride, you can now walk to the object, counting your steps, multiply that number by the length of your stride (*), and you will know the height of the object. If it's a relatively small object you might have to add your height (up to eye level) since the measurement is made from eye level.

* To know the length of your stride, make a mark on the ground, walk at least ten paces and make another mark, then measure the distance with a tape measure. Divide the distance by the number of steps.

If you can't put yourself in a position where you can sight the object at 45 degrees you'll need to use the UMBRA RECTA or the UMBRA VERSA on the front of the astrolabe. The names refer to shadows ("umbra"s) cast by upright objects. These two scales correspond to the modern trigonometric concepts of tangent (umbra recta) and cotangent (umbra versa). These functions are ratios of two of the sides of a right triangle. In the drawing point A is where you're standing, Point B is the top of the object you're measuring, a is the distance to the object and b is the height. When you sight thru the alidade you are measuring angle x. The tangent of angle x is the same as the ratio of b to a. For example, if you measure angle x as 60 degrees, the tangent (umbra recta) is 1.7.

So     b/a=1.7    b=a(1.7)

The height of the object is 1.7 times the distance to it. Walk off the distance, multiply, and you have the answer.

The 45degree method worked because the tangent of 45 degrees is 1.

2. Distance to an Object

To find the distance to a far object we just lay that same triangle flat on the ground and start at point C. With the astrolabe lying flat, line the alidade up with the distant object. Keeping the astrolabe in exactly the same position, turn the alidade 90 degrees to the left. Have someone walk off a pretty good distance in that direction as you watch them thru the alidade. When the helper stops he's at point A (wherever he stops that's it, as long as he's along the 90 degree line). Put a marker at point C that will be visible from point A. Now bring the astrolabe to point A, lay it flat and look thru the alidade first at point C, then at point B. Read the angle between them and make the same calculation as before.

3. Waves of Light

The top of the front of the some astrolabes contains a grid of little squares. Using these in combination with the Rule on the alidade we can read the sine of each angle. The sine is another trigonometric function which, on the astrolabe, was used as a quick calculator of the amount of daylight at any time of year.

For many of you we may already be too deep into mathematics. For some others it's time for a little challenge. The basic physical facts are these. If we plot the days of the year along the x axis and the hours of daylight along the y axis we'll produce a sine wave which, at 40 degrees latitude goes from 15 hours on June 22 to 9 hours on December 22 and then back, passing thru midpoints of 12 hours on the equinoxes of September 22 and March 22. The amount of light changes very little in the month before and after the solstices (June and December), but changes very quickly around March and September.

At 60 degrees Latitude we see the same 12 hour midpoints, but the extremes are 18 and 6 hours. And at 20 degrees the extremes are 14 and 10 hours. How can we calculate the number of hours of daylight corresponding to these facts? (Hint - use the degree markings as calendar days, counting 0-90 days before or after the solstice. We'll need to calculate how many hours more than twelve or less than twelve.)


The Ishango bone - 20,000 BC

Geoffrey Chaucer never finished the Treatise on the Astrolabe (tho he lived about nine years after leaving off in 1391). Some scholars think the boy Lowys may have died. Then again, he may have simply gone off to school, and they had much less contact, or Chaucer may have been too busy to finish.

He did draw some "conclusions" about the use of the astrolabe in astrology. After 600 years it shouldn't seem strange that I should see astrology differently. Even the progression of the nodes has left the constellations out of their houses. Remember that I said that the Earth's motions bring us back to ALMOST the same place ? From year to year it isn't even noticeable. But over the course of centuries we notice that the lion's head is now in the house of Virgo. No problem. The virgin's not home; she's next door. Likewise with the other constellations.

I don't look upon my ancestors as simple-minded creatures. We see the past from the privileged point of cumulative culture. But there's nothing that our minds posess that wasn't put there by the step by step effort of our ancestors. And in the house of Science three of the great ancestors were Mythology, Astrology and Alchemy. Science and Literature are their children. From generations upon generations of stargazers came the eventual child of mankind who scratched the number of days of moon cycles on a cave wall. From the wonder of seeing the wandering planets in the sky came the stories of mythology, still the backbone of 90% of our literature.

From my personal observations of a few hundred people over the forty or so years that I've been aware of the Eastern and Western Astrologies, these systems have had very little value in predicting how any individual might act at any moment, or even in describing a personality in general.

An individual is an individual. And the individual I have aspired to be is like that child of numbers. Thirty days is thirty days. Twenty-nine is twenty-nine. It won't be thirty just because I want it to be thirty, or because I think it should be thirty. Oh, look ! The next one IS thirty. And then a couple twenty-nines. And then over time I can almost see a pattern. I can't quite figure it out, but I know it's not exactly thirty or twenty-nine. Oh, beautiful moon !

And tho, on rare occasions, I do play the lottery, I have no illusions about my chances of winning.

Horse and dots, Lascaux, France, 18,000 BC


      Epilogue: The Horse's Tale