It was now necessary to complete the observations of the evening before by measuring the height of the cliff above the level of the sea.

“Shall you not need an instrument similar to the one which you used yesterday?” said Herbert to the engineer.

“No, my boy,” replied the latter, “we are going to proceed differently, but in as precise a way.”

Herbert, wishing to learn everything he could, followed the engineer to the beach.  Pencroft, Neb, and the reporter remained behind and occupied themselves in different ways.

Cyrus Harding had provided himself with a straight stick, twelve feet long, which he had measured as exactly as possible by comparing it with his own height, which he knew to a hair.  Herbert carried a plumb-line which Harding had given him, that is to say, a simple stone fastened to the end of a flexible fiber.  Having reached a spot about twenty feet from the edge of the beach, and nearly five hundred feet from the cliff, which rose perpendicularly, Harding thrust the pole two feet into the sand, and wedging it up carefully, he managed, by means of the plumb-line, to erect it perpendicularly with the plane of the horizon.

That done, he retired the necessary distance, when, lying on the sand, his eye glanced at the same time at the top of the pole and the crest of the cliff.  He carefully marked the place with a little stick.

Then addressing Herbert—­“Do you know the first principles of geometry?” he asked.

“Slightly, captain,” replied Herbert, who did not wish to put himself forward.

“You remember what are the properties of two similar triangles?”

“Yes,” replied Herbert; “their homologous sides are proportional.”

“Well, my boy, I have just constructed two similar right-angled triangles; the first, the smallest, has for its sides the perpendicular pole, the distance which separates the little stick from the foot of the pole and my visual ray for hypothenuse; the second has for its sides the perpendicular cliff, the height of which we wish to measure, the distance which separates the little stick from the bottom of the cliff, and my visual ray also forms its hypothenuse, which proves to be prolongation of that of the first triangle.”

“Ah, captain, I understand!” cried Herbert.  “As the distance from the stick to the pole is to the distance from the stick to the base of the cliff, so is the height of the pole to the height of the cliff.”

“Just so, Herbert,” replied the engineer; “and when we have measured the two first distances, knowing the height of the pole, we shall only have a sum in proportion to do, which will give us the height of the cliff, and will save us the trouble of measuring it directly.”

The two horizontal distances were found out by means of the pole, whose length above the sand was exactly ten feet.

The first distance was fifteen feet between the stick and the place where the pole was thrust into the sand.