|A tiny bit of mathematics is used by Sherlock Holmes
to solve this mystery. In it, he ties together the disappearance of a
housemaid, the discovery of the dead body of the chief butler and a strange
poem which the head of each generation is required to pass on to his son by
family tradition. As it turns out, the poem is the key to finding a hidden
treasure that one of the Musgrave anscestors sought to protect and
geometry (see note below) is required to work out the location that the tip of a tree's
shadow would have occupied at the time the ritual was written.
Your description says that Holmes used trigonometry to work out the location of the tip of the shadow. Not so. He used geometry. By measuring the length of the shadow of a known length, he determined what would have been the length of the shadow of a tree that had been lost to a lightning strike. He used the theorem of similar triangles. He never measured or calculated an angle.
Ah, you are right! I stand corrected, as is the comment to which you referred. Thank you, Mr. O'Brien. In fact, to avoid any further confusion, the exact passage in question from the original story is reproduced below for your perusing pleasure:
|(quoted from Musgrave Ritual)|
"This was excellent news, Watson, for it showed me that I was on the right road. I looked up at the sun. It was low in the heavens, and I calculated that in less than an hour it would lie just above the topmost branches of the old oak. One condition mentioned in the Ritual would then be fulfilled. And the shadow of the elm must mean the farther end of the shadow, otherwise the trunk would have been chosen as the guide. I had, then, to find where the far end of the shadow would fall when the sun was just clear of the oak."
"That must have been difficult, Holmes, when the elm was no longer there."
"Well, at least I knew that if Brunton could do it, I could also. Besides, there was no real difficulty. I went with Musgrave to his study and whittled myself this peg, to which I tied this long string with a knot at each yard. Then I took two lengths of a fishing-rod, which came to just six feet, and I went back with my client to where the elm had been. The sun was just grazing the top of the oak. I fastened the rod on end, marked out the direction of the shadow, and measured it. It was nine feet in length.
"Of course the calculation now was a simple one. If a rod of six feet threw a shadow of nine, a tree of sixty-four feet would throw one of ninety-six, and the line of the one would of course be the line of the other. I measured out the distance, which brought me almost to the wall of the house, and I thrust a peg into the spot. You can imagine my exultation, Watson, when within two inches of my peg I saw a conical depression in the ground. I knew that it was the mark made by Brunton in his measurements, and that I was still upon his trail.
Holmes was doing this from the heights of the oak and elm trees (and the length of the shadows thrown at a specific time) in around the year 1890 (10 years earlier in the case of the elm that was measured by Musgrave before it was struck by lightning and cut down 10 years previously. But the ritual was written in 1648 or 1649. This would make both the oak and the elm around 200 years older (and thus not as tall) than when Holmes did his calculation in 1890. Elm and Oak trees grow at different rates, so surely Holmes and Brunton would have to know what the heights of both the two trees were 200 years previously, when the ritual was written, not the heights as they were in the mid 1800s.
Surely, the length of the tree shadows in 1890 could not be used to indicate the lengths they would have been in 1650?
Peter, I agree that the height of the elm when the “ritual” was created would be what was needed, not its height at the time it was cut down. So, if the height changed in that time, then the ritual would no longer point to the right spot. I'm not an arborist…but I am pretty sure that you are correct that a tree's height can change drastically over centuries.
I've been in touch with the Sherlock Holmes Society in London and they agree that there are anomalies in the story of "The Musgrave Ritual" and this has been debated in the society for many years.