The Universe of Discourse

How long is the banana?

Mark Dominus:

One of the types of problems that al-Khwarizmi treats extensively in his book is the problem of computing inheritances under Islamic law.

Well, these examples are getting sillier and sillier, but

... but maybe they're not silly enough:

A rope over the top of a fence has the same length on each side and weighs one-third of a pound per foot. On one end of the rope hangs a monkey holding a banana, and on the other end a weight equal to the weight of the monkey. The banana weighs 2 ounces per inch. The length of the rope in feet is the same as the age of the monkey, and the weight of the monkey in ounces is as much as the age of the monkey's mother. The combined ages of the monkey and its mother is 30 years. One-half the weight of the monkey plus the weight of the banana is one-fourth the sum of the weights of the rope and the weight. The monkey's mother is one-half as old as the monkey will be when it is three times as old as its mother was when she was one-half as old as the monkey will be when it is as old as its mother will be when she is four times as old as the monkey was when it was twice as old as its mother was when she was one-third as old as the monkey was when it was as old as its mother was when she was three times as old as the monkey was when it was one-fourth as old as its is now. How long is the banana?

(Spoiler solution using, guess what, Linogram....)

        define object {
          param number density;
          number length, weight;
          constraints { density * length = weight; }
          draw { &dump_all; }
        }

        object banana(density = 24/16), rope(density = 1/3);
        number monkey_age, mother_age;
        number monkey_weight, weight_weight;

        constraints { 
          monkey_weight * 16 = mother_age;
          monkey_age + mother_age = 30;
          rope.length = monkey_age;
          monkey_weight / 2 + banana.weight  = 
                1/4 * (rope.weight + weight_weight);
          mother_age = 1/2 * 3 * 1/2 * 4 * 2 * 1/3 * 3 * 1/4 * monkey_age;
          monkey_weight = weight_weight;
        }

        draw { banana; }

        __END__

        use Data::Dumper;
        sub dump_all {
          my $h = shift;
          print Dumper($h);
        #  for my $var (sort keys %$h) {
        #    print "$var = $h->{$var}\n";
        #  }
        }

OK, seriously, how often do you get to write a program that includes the directive draw { banana; }?

The output is:

        $VAR1 = bless( {
                         'length' => '0.479166666666667',
                         'weight' => '0.71875',
                         'density' => '1.5'
                       }, 'Environment' );

So the length of the banana is 0.479166666666667 feet, which is 5.75 inches.

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The dot product in 1580?

Buy Voyages and Discoveries: Selections from Hakluyt's Principal Navigations from Bookshop.org (with kickback) (without kickback)

The Principal Navigations, Voyages, Traffiques, & Discoveries of the English Nation was published in 1589, a collection of essays, letters, and journals written mostly by English persons about their experiences in the great sea voyages of discovery of the latter half of the 16th century.

One important concern of the English at this time was to find an alternate route to Asia and the spice islands, since the Portuguese monopolized the sea route around the coast of Africa. So many of the selections concern the search for a Northwest or Northeast passage, routes around North America or Siberia, respectively. Other items concern military battles, including the defeat of the Spanish Armada; proper outfitting of whaling ships, and an account of a sailor who was shipwrecked in the West Indies and made his way home at last sixteen years later.

One item, titled "Experiences and reasons of the Sphere, to proove all partes of the worlde habitable, and thereby to confute the position of the five Zones," contains the following sentence:

First you are to understand that the Sunne doeth worke his more or lesse heat in these lower parts by two meanes, the one is by the kinde of Angle that the Sunne beames doe make with the earth, as in all Torrida Zona it maketh perpendicularly right Angles in some place or other at noone, and towards the two Poles very oblique and uneven Angles.

This explanation is quite correct. (The second explanation, which I omitted, is that the sun might spend more or less time above the horizon, and is also correct.) This was the point at which I happened to set down the book before I went to sleep.

But over the next couple of days I realized that there was something deeply puzzling about it: This explanation should not be accessible to an Englishman of 1580, when this item was written.

In 2006, I would explain that the sun's rays are a directed radiant energy field in direction E, and that the energy received by a surface S is the dot product of the energy vector E and the surface normal vector n. If E and n are parallel, you get the greatest amount of energy; as E and n become perpendicular, less and less energy is incident on S.

Englishmen in 1580 do not have a notion of the dot product of two vectors, or of vectors themselves for that matter. Analytic geometry will not be invented until 1637. You can explain the weakness of oblique rays without invoking vectors, by appeal to the law of conservation of energy, but the Englishmen do not have the idea of an energy field, or of conservation of energy, or of energy at all. They do not have any notion of the amount of radiant energy per unit area. Galileo's idea of mathematical expression of physical law will not be published until 1638.

So how do they have the idea that perpendicular sun is more intense than oblique? How did they think this up? And what is their argument for it?

(Try to guess before you read the answer.)

In fact, the author is completely wrong about the reason. Here's what he says the answer is:

... the perpendicular beames reflect and reverberate in themselves, so that the heat is doubled, every beam striking twice, & by uniting are multiplied, and continued strong in forme of a Columne. But in our Latitude of 50. and 60. degrees, the Sunne beames descend oblique and slanting wise, and so strike but once and depart, and therefore our heat is the lesse for any effect that the Angle of the Sunne beames make.

Did you get that? Perpendicular sun is warmer because the beams get you twice, once on the way down and once on the way back up. But oblique beams "strike but once and depart."

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