# Works of Jules Verne/The Mysterious Island/Dropped from the Clouds/Chapter 14

CHAPTER XIV A PROBLEM IN TRIANGLES
The next day, the 16th of April, and Easter Sunday, the settlers issued from the Chimneys at daybreak, and proceeded to wash their linen. The engineer intended to manufacture soap as soon as he could procure the necessary materials—soda or potash, fat or oil. The important question of renewing their wardrobe would be treated of in the proper time and place At any rate their clothes would last at least six months longer, for they were strong, and could resist the wear of manual labor. But all would depend on the situation of the island with regard to inhabited land. This would be settled to-day if the weather permitted.
The sun rising above a clear horizon, announced a magnificent day, one of those beautiful autumn days which are like the last farewells of the warm season. 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 is the 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. The second distance between the stick and the bottom of the cliff was five hundred feet.
These measurements finished, Cyrus Harding and the lad returned to the Chimneys. The engineer then took a flat stone which he had brought back from one of his previous excursions, a sort of slate, on which it was easy to trace figures with a sharp shell. He then developed the following proportions: 15:500::10:x. From which it was proved that the granite cliff measured 333 feet in height.
Cyrus Harding then took the instrument which he had made the evening before, the space between its two legs giving the angular distance between the star Alpha and the horizon. He measured, very exactly, the opening of this angle on a circumference which he divided into 360 equal parts. Now, this angle, by adding to it the twenty-seven degrees which separated Alpha from the antarctic pole, and by reducing to the level of the sea the height of the cliff on which the observation had been made, was found to be fifty-three degrees. These fifty-three degrees being subtracted from ninety degrees—the distance from the pole to the equator—there remained thirty-seven degrees. Cyrus Harding concluded, therefore, that Lincoln Island was situated on the thirty-seventh degree of southern latitude, or taking into consideration through the imperfection of the performance, an error of five degrees, that it must be situated between the thirty-fifth and the fortieth parallel.
There was only the longitude to be obtained, and the position of the island would be determined. The engineer hoped to attempt this the same day, at twelve o'clock, at which moment the sun would pass the meridian.
It was decided that Sunday should be spent in a walk, or rather an exploring expedition, to that side of the island ​between the north of the lake and Shark Gulf, and if there was time they would push their discoveries to the northern side of Cape South Mandible. They would breakfast on the downs, and not return till evening.
At half-past eight the little band was following the edge of the channel. On the other side, on Safety Islet, numerous birds were gravely strutting. They were divers, easily recognized by their cry, which much resembles the braying of a donkey. Pencroft only considered them in an eatable point of view, and learnt with some satisfaction that their flesh, though blackish, is not bad food.
Great amphibious creatures could also be seen crawling on the sand; seals, doubtless, who appeared to have chosen the islet for a place of refuge. It was impossible to think of those animals in an alimentary point of view, for their oily flesh is detestable; however, Cyrus Harding observed them attentively, and without making known his idea, he announced to his companions that very soon they would pay a visit to the islet. The beach was strewn with innumerable shells, some of which would have rejoiced the heart of a conchologist; there were, among others, the phasianella, the terebratula, etc. But what would be of more use, was the discovery, by Neb, at low tide, of a large oyster-bed, among the rocks, nearly five miles from the Chimneys.
"Neb will not have lost his day," cried Pencroft, looking at the spacious oyster-bed.
"It is really a fortunate discovery," said the reporter, "and as it is said that each oyster produces yearly from fifty to sixty thousand eggs, we shall have an inexhaustible supply there."
"Only I believe that the oyster is not very nourishing," said Herbert.
"No," replied Harding. "The oyster contains very little nitrogen, and if a man lived exclusively on them, he would have to eat not less than fifteen to sixteen dozen a day."
"Capital!" replied Pencroft. "We might swallow dozens and dozens without exhausting the bed. Shall we take some for breakfast?" And without waiting for a reply to his proposal, knowing that it would be approved of, the sailor and Neb detached a quantity of the mollusks. ​They put them in a sort of net of hibiscus fiber, which Neb had manufactured, and which already contained food; they then continued to climb the coast between the downs and the sea.
From time to time Harding consulted his watch, so as to be prepared in time for the solar observation, which had to be made exactly at mid-day.
All that part of the island was very barren as far as the point which closed Union Bay, and which had received the name of Cape South Mandible. Nothing could be seen there but sand and shells, mingled with débris of lava. A few sea-birds frequented this desolate coast, gulls, great albatrosses, as well as wild duck, for which Pencroft had a great fancy. He tried to knock some over with an arrow, but without result, for they seldom perched, and he could not hit them on the wing.
This led the sailor to repeat to the engineer, "You see, captain, so long as we have no fowling-pieces, we shall get nothing!"
"Doubtless, Pencroft," said the reporter, "but it depends on you. Procure us some iron for the barrels, steel for the hammers, saltpeter, coal, and sulphur for powder, mercury and nitric acid for the fulminate, and lead for the shot, and the captain will make us first-rate guns."
"Oh!" replied the engineer, "we might, no doubt, find all these substances on the island, but a gun is a delicate instrument, and needs very particular tools. However, we shall see later!"
"Why," cried Pencroft, "were we obliged to throw over-board all the weapons we had with us in the car, all our implements, even our pocket-knives?"
"But if we had not thrown them away, Pencroft, the balloon would have thrown us to the bottom of the sea!" said Herbert.
"What you say is true, my boy," replied the sailor. Then passing to another idea, "Think," said he, "how astounded Jonathan Forster and his companions must have been when, next morning, they found the place empty, and the machine flown away!"
"I am utterly indifferent about knowing what they may have thought," said the reporter.
"It was all my idea, that!" said Pencroft.
​"A splendid idea, Pencroft!" replied Gideon Spilett, laughing, "and which has placed us where we are."
"I would rather be here than in the hands of the Southerners," cried the sailor, "especially since the captain has been kind enough to come and join us again."
"So would I, truly!" replied the reporter. "Besides, what do we want? Nothing."
"If that is not—everything!" replied Pencroft, laughing, and shrugging his shoulders. "But, some day or other we shall find means of going away!"
"Sooner, perhaps, than you imagine, my friends," remarked the engineer, "if Lincoln Island is but a medium distance from an inhabited island, or from a continent. We shall know in an hour. I have not a map of the Pacific, but my memory has preserved a very clear recollection of its southern part. The latitude which I obtained yesterday placed New Zealand to the west of Lincoln Island, and the coast of Chili to the east. But between these two countries, there is a distance of at least six thousand miles. It has, therefore, to be determined what point in this great space the island occupies, and this the longitude will give us presently, with a sufficient approximation, I hope."
"Is not the archipelago of the Austral Islands the nearest point to us in latitude?" asked Herbert.
"Yes," replied the engineer, "but the distance which separates us from it is more than twelve hundred miles."
"And that way?" asked Neb, who followed the conversation with extreme interest, pointing to the south.
"That way, nothing," replied Pencroft.
"Nothing, indeed," added the engineer.
"Well, Cyrus," asked the reporter, "if Lincoln's Island is not more than two or three hundred miles from New Zealand or Chili?"
"Well," replied the engineer, "instead of building a house we will build a boat, and Master Pencroft shall be put in command——"
"Well then," cried the sailor, "I am quite ready to be captain—as soon as you can make a craft that's able to keep at sea!"
"We shall do it, if it is necessary," replied Harding.
But whilst these men, who really hesitated at nothing, were talking, the hour approached at which the observation ​was to be made. What Cyrus Harding was to do to ascertain the passage of the sun at the meridian without an instrument of any sort, Herbert could not guess.
The observers were then about six miles from the Chimneys, not far from that part of the downs in which the engineer had been found after his enigmatical preservation. They halted at this place and prepared for lunch, for it was half-past eleven. Herbert went for some fresh water from a stream which ran near, and brought it back in a jug which Neb had provided.
During these preparations Harding arranged everything for his astronomical observation. He chose a clear place on the shore, which the ebbing tide had left perfectly level. This bed of fine sand was as smooth as ice, not a grain out of place. It was of little importance whether it was horizontal or not, and it did not matter much whether the stick, six feet high, which was planted there, rose perpendicularly. On the contrary, the engineer inclined it towards the south, that is to say, in the direction of the coast opposite to the sun, for it must not be forgotten that the settlers in Lincoln Island, as the island was situated in the southern hemisphere, saw the radiant planet describe its diurnal arc above the northern, and not above the southern horizon.
Herbert now understood how the engineer was going to proceed to ascertain the culmination of the sun, that is to say its passing the meridian of the island or, in other terms, the south of the place. It was by means of the shadow cast on the sand by the stick, a way which, for want of an instrument, would give him a suitable approach to the result which he wished to obtain.
In fact, the moment when this shadow would reach its minimum of length would be exactly twelve o'clock, and it would be enough to watch the extremity of the shadow, so as to ascertain the instant when, after having successively diminished, it began to lengthen. By inclining his stick to the side opposite to the sun, Cyrus Harding made the shadow longer, and consequently its modifications would be more easily ascertained. In fact, the longer the needle of a dial is, the more easily can the movement of its point be followed. The shadow of the stick was nothing but the needle of a dial.
When he thought the moment had come, Cyrus Harding ​knelt on the sand, and with little wooden pegs, which he stuck into the sand, he began to mark the successive diminutions of the stick's shadow. His companions, bending over him, watched the operation with extreme interest. The reporter held his chronometer in his hand, ready to tell the hour which it marked when the shadow would be at its shortest. Moreover, as Cyrus Harding was working on the 16th of April, the day on which the true and the average time are identical, the hour given by Gideon Spilett would be the true hour then at Washington, which would simplify the calculation. Meanwhile as the sun slowly advanced, the shadow slowly diminished, and when it appeared to Cyrus Harding that it was beginning to increase, he asked, "What o'clock is it?"
"One minute past five," replied Gideon Spilett directly. They had now only to calculate the operation. Nothing could be easier. It could be seen that there existed, in round numbers, a difference of five hours between the meridian of Washington and that of Lincoln Island, that is to say, it was mid-day in Lincoln Island when it was already five o'clock in the evening in Washington. Now the sun, in its apparent movement round the earth, traverses one degree in four minutes, or fifteen degrees one hour. Fifteen degrees multiplied by five hours give seventy-five degrees.
Then, since Washington is 77° 3′ 11″, as much as to say seventy-seven degrees counted from the meridian of Greenwich—which the Americans take for their starting-point for longitudes concurrently with the English—it followed that the island must be situated seventy-seven and seventy-five degrees west of the meridian of Greenwich, that is to say, on the hundred and fifty-second degree of west longitude.
Cyrus Harding announced this result to his companions, and taking into consideration errors of observation, as he had done for the latitude, he believed he could positively affirm that the position of Lincoln Island was between the thirty-fifth and the fortieth parallel, and between the hundred and fiftieth and the hundred and fifty-fifth meridian observation was, it may be seen, of five degrees on both
The possible fault which he attributed to errors in the sides, which, at sixty miles to a degree, would give an error ​of three hundred miles in latitude and longitude for the exact position.
But this error would not influence the determination which it was necessary to take. It was very evident that Lincoln Island was at such a distance from every country or island that it would be too hazardous to attempt to reach one in a frail boat.
In fact this calculation placed it at least twelve hundred miles from Tahiti and the islands of the Austral archipelago, more than eighteen hundred miles from New Zealand, and more than four thousand five hundred miles from the American coast!
And when Cyrus Harding consulted his memory, he could not remember in any way that such an island occupied, in that part of the Pacific, the situation assigned to Lincoln Island.
TO BE CONTINUED IN VOLUME VI
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