“ ‘When my old tutor used to give me an exercise in trigonometry, it always took the shape of measuring heights. When I was a lad I worked out every tree and building in the estate.’

“This was an unexpected piece of luck. My data were coming more quickly than I could have reasonably hoped.

“ ‘Tell me,’ I asked, ‘did your butler ever ask you such a question?’

“Reginald Musgrave looked at me in astonishment. ‘Now that you call it to my mind,’ he answered, ‘Brunton did ask me about the height of the tree some months ago in connection with some little argument with the groom.’

“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.

“From this starting-point I proceeded to step, having first taken the cardinal points by my pocket-compass. Ten steps with each foot took me along parallel with the wall of the house, and again I marked my spot with a peg. Then I carefully paced off five to the east and two to the south. It brought me to the very threshold of the old door. Two steps to the west meant now that I was to go two paces down the stone-flagged passage, and this was the place indicated by the Ritual.

“Never have I felt such a cold chill of disappointment, Watson. For a moment it seemed to me that there must be some radical mistake in my calculations. The setting sun shone full upon the passage floor, and I could see that the old, foot-worn gray stones with which it was paved were firmly cemented together, and had certainly not been moved for many a long year. Brunton had not been at work here. I tapped upon the floor, but it sounded the same all over, and there was no sign of any crack or crevice. But, fortunately, Musgrave, who had begun to appreciate the meaning of my proceedings, and who was now as excited as myself, took out his manuscript to check my calculations.

“ ‘And under,’ he cried. ‘You have omitted the “and under.”’

“I had thought that it meant that we were to dig, but now, of course, I saw at once that I was wrong. ‘There is a cellar under this then?’ I cried.

“ ‘Yes, and as old as the house. Down here, through this door.’