On our way, I perused a small volume of mathematical conundrums. "Holmes, here's a good puzzle. If a man is in the center of a lake and a fog descends, what is the shortest path he can choose to reach land? Apparently, no one knows. But they think the shortest path is one that goes straight for a while, makes a sharp left turn, goes straight a bit, curves around and then goes straight again." "Fascinating," Holmes said, his sar- casm barely concealed. "Look. We've arrived."
The duke looked haggard. "Holmes, I fear that the Ghastleigh Goat has been stolen." The animal in question was a family heirloom, some three feet long and made of bronze. It was worth vir- tually nothing but contained a secret drawer full of sensitive documents. "Donnell," Holmes muttered. "Quick- ly, show me where you kept the goat."
The duke led us to a small, drafty cel- lar. "there," he said, pointing to a large safe in one corner. Holmes studied the drain, looked at he ventilation grille and inspected the locks on the cellar door and on the safe. Falling to his knees, he found a paper wrapper. He sniffed the air and stood up.
"The story is clear, Your Grace. The thief made his entrance and exit through the safe and removed the goat. The goat wouldn't fit through the grille, so he attached it to an inflatable rubber tube and dropped it into the drains to float away and be collected outside the grounds.
"Instead the tube must have suffered a puncture. Before the goat could float away, it sank and blocked the drain, as you can smell. The documents are in the dr@s. But we cannot locate them from this end; the drain is too deep. We must break into the drainage system at a more convenient point." "There's a drain that cuts across the front lawn and comes up in the cellars," tlw duke suggested. "You can tell where it is in the summer because the grass changes color above it. Now it's covered in snow. If I recall correctly, it runs within 100 yards of the statue of the water nymph"
"We must dig a trench to this drain, determine its outlet and retrieve the goat before Dunnett does." "We must dig quickly!" the duke urged. "And in the right direction, too," I put in. "Othenvise we might miss the drain altogether."
"What we need to know," Holmes said, "is the shortest trench guaranteed to meet every straight line passing with- in 100 yards of the statue of the water nymph [see top illustration on opposite page]." "We could dig a circular trench with a 100-yard radius," the duke proposed. "With a length of 2007C yards, or about 628 yards," Holmes rapidly culated. "Can we do better?" "How about a straight line, 200 yards long, cutting across the duke's circle?" I asked. "Excellent, Watson, except that such a trench misses many possible drain positions." "Okay. Two such lines then, at right angles, for a total length of 400 yards?"
"Same problem, Watson. Mathematically, we are looldng for the shortest curve that meets every chord of a circle having a radius of lOOyards - a chord being any straight line that meets the circle at two points."
"Holmes, just think of how many chords there are to consider!" "Yes, it would be nice to simplify things. Ali. I've got it. In fact, we need consider only the tangents to the circle, Watson. These lines meet the circle at one point along its edge. After all, a curve that meets all tangents to a circle necessarily meets all the chords. "Choose any chord and consider the two tangents parallel to it [see bottom illustration on this page]. The curve meets one tangent at point B and the other at C. By way of continuity, the part of the part of the curve that joins B to C must cut the chord." He rubbed Ns chin. "I almost have it, but there is a gap in my reasoning." "We don't have time to worry about details, Holmes. What's your general idea?" "Well, there is a class of curves that automatically meet every tangent: those that start and end on opposite sides of the same tangent, wrapping around the circle, remaining outside or on it. Call them straps, since they strap a tangent to the circle."
"Go on, go on."
"Finding the shortest strap is simple. First, observe that the strap must touch the circle somewhere. If not, it could be tightened down until it did and would then be shorter. Suppose that it first meets the circle at a point B and last meets it at some point C. Then AB and CD must be straight lines. Otherwise the strap could be shortened by straightening those segments. Moreover, BC must be a single arc of the circle, for similar reasons."
"I think the lines AB and CD must be tangents to the circle," I said. "If they weren't, the curve could be shortened even more by moving B and C to positions where they were tangents." "Of course. But where should the points A and D be located? I think that both AB and CD need to be perpendicular to the tangent AD. if they were not at right angles, we could swing the strap around until they were, and then again the strap would be shorter." "Yes," I cried. "The arc BC is a semicirde. We have found the shortest curve!" "Unfortunately, we have found only the shortest strap," Holmes said, frowning. "But I do find it hard to see how any other curve meeting our requirements could be shorter." We stood in silence for a few minutes. "All may not be lost," I finally said. "How long is this shortest strap anyway?"
"(2 + pi r) r, or about 514 yards in this case." "That saves 89 yards compared with my plan," the duke exclaimed. "There is no time to lose. I'll summon the men!" While they dug, Holmes and I continued to seek even shorter curves but found none. "Holmes! Do you remember my book?" I pulled it from my pocket. "Listen. "The shortest path that meets every chord of a circle of radius r consists of two parallel straight segments of length r and a semicircle.""
"Just as I deduced. I confess, Watson, I misjudged the utility of your slim volume. Out of curiosity, how do we know that the path proposed is truly the shortest?" "It has been proved beyond any doubt by several different mathematicians. These proofs are terribly complex, though It would be of great interest if anyone could find a short, simple proof" "Perhaps I might-" Holmes began, when suddenly the duke let out a yell. His diggers had located the drain.
Holmes sighted its length. "We shall find the outlet and the thief - beyond the thicket in the distance." We secreted ourselves among the trees and settled down to wait. No sooner had the sun set than we heard footsteps. A masked figure came into view, and Holmes seized him "Now we shall see, he declared. "As I deduced at the start, it is - who are you?" "My God, it's Lucinda, the maid," the duke said. "What are you doing here?"
"Please, Your Grace. Yesterday I had to go to the cellar. The door was locked, so I climbed in through the grille. The safe was open, and I saw a funny old goat inside. I pulled it out to take a !ook, but it was really heavy, and I accidentally dropped it down the drain. I panicked. So I shut the safe and left. I intended to return it once I'd- Anyway, I was about to crawl up the drain to look for it when this gentleman" - she gave Holmes a winsome smile - "leaped on me." "So the Ghastleigh Goat is at the bottom of the drain," the duke mused, "I have a SOO-yard trench in my lawn, and Dunnett remains at large." He gave Holmes a hard stare. "It is a question of logical deduction," Holmes said. "When you have eliminated the impossible, then whatever remains, however improbable-"
"Yes, Holmes, yes! Go on!" I begged. "-remains improbable," he finished. "But don't quote me on that." "My lips are sealed." But my notebook is not. After all, a biographer has to make a living.