Phyfe remained near the site as Underwood and Terry set their crew to the routine task of weighing, measuring, and photographing the object, while Underwood considered what else to do.

"You know, this thing has got me stymied, Terry. Since it can't be touched by an Atom Stream, that means there isn't a single analytical procedure to which it will respond—that I know of, anyway. Does your knowledge of the Stroids and their ways of doing things suggest any identification of it?"

Terry shook his head as he stood by the port of the laboratory ship watching the crews at work outside. "Not a thing, but that's no criterion. We know so little about the Stroids that almost everything we find has a function we never heard of before. And of course we've found many objects with totally unknown functions. I've been thinking—what if this should turn out to be merely a natural gem from the interior of the planet, maybe formed at the time of its destruction, but at least an entirely natural object rather than an artifact?"

"It would be the largest crystal formation ever encountered, and the most perfect. I'd say the chances of its natural formation are negligible."

"But maybe this is the one in a hundred billion billion or whatever number chance it may be."

"If so, its value ought to be enough to balance the Terrestrial budget. I'm still convinced that it must be an artifact, though its material and use are beyond me. We can start with a radiation analysis. Perhaps it will respond in some way that will give us a clue."

When the crew had finished the routine check, Underwood directed his men to set up the various types of radiation equipment contained within the ship. It was possible to generate radiation through almost the complete spectrum from single cycle sound waves to hard cosmic rays.

The work was arduous and detailed. Each radiator was slowly driven through its range, then removed and higher frequency equipment used. At each fraction of an octave, the object was carefully photographed to record its response.

After watching the work for two days, Terry wearied of the seemingly non-productive labor. "I suppose you know what you're doing, Del," he said. "But is it getting you anywhere at all?"

Underwood shook his head. "Here's the batch of photographs. You'll probably want them to illustrate your report. The surfaces of the object are mathematically exact to a thousandth of a millimeter. Believe me, that's some tolerance on an object of this size. The surfaces are of number fifteen smoothness, which means they are plane within a hundred-thousandth of a millimeter. The implications are obvious. The builders who constructed that were mechanical geniuses."

"Did you get any radioactive dating?"

"Rather doubtfully, but the indications are around half a million years."

"That checks with what we know about the Stroids."

"It would appear that their culture is about on a par with our own."

"Personally, I think they were ahead of us," said Terry. "And do you see what that means to us archeologists? It's the first time in the history of the science that we've had to deal with the remains of a civilization either equal or superior to our own. The problems are multiplied a thousand times when you try to take a step up instead of a step down."

"Any idea of what the Stroids looked like?"

"We haven't found any bodies, skeletons, or even pictures, but we think they were at least roughly anthropomorphic. They were farther from the Sun than we, but it was younger then and probably gave them about the same amount of heat. Their planet was larger and the Stroids appear to have been somewhat larger as individuals than we, judging from the artifacts we've discovered. But they seem to have had a suitable atmosphere of oxygen diluted with appropriate inert gases."

They were interrupted by the sudden appearance of a laboratory technician who brought in a dry photographic print still warm from the developing box.

He laid it on the desk before Underwood. "I thought you might be interested in this."

Underwood and Terry glanced at it. The picture was of the huge, gemlike artifact, but a number of the facets seemed to be covered with intricate markings of short, wavy lines.

Underwood stared closer at the thing. "What the devil are those? We took pictures of every facet previously and there was nothing like this. Get me an enlargement of these."

"I already have." The assistant laid another photo on the desk, showing the pattern of markings as if at close range. They were clearly discernible now.

"What do you make of it?" asked Underwood.

"I'd say it looked like writing," Terry said. "But it's not like any of the other Stroid characters I've seen—which doesn't mean much, of course, because there could be thousands that I've never seen. Only how come these characters are there now, and we never noticed them before?"

"Let's go out and have a look," said Underwood. He grasped the photograph and noted the numbers of the facets on which the characters appeared.

In a few moments the two men were speeding toward the surface of their discovery astride scooters. They jockeyed above the facets shown on the photographs, and stared in vain.

"Something's the matter," said Terry. "I don't see anything here."

"Let's go all the way around on the scooters. Those guys may have bungled the job of numbering the photos."

They began a slow circuit, making certain they glimpsed all the facets from a height of only ten feet.

"It's not here," Underwood agreed at last. "Let's talk to the crew that took the shots."

They headed towards the equipment platform, floating in free space, from which Mason, one of the Senior Physicists, was directing operations. Mason signaled for the radiations to be cut off as the men approached.

"Find any clues, Chief?" he asked Underwood. "We've done our best to fry this apple, but nothing happens."

"Something did happen. Did you see it?" Underwood extended the photograph with the mechanical fingers of the spacesuit. Mason held it in a light and stared at it. "We didn't see a thing like that. And we couldn't have missed it." He turned to the members of the crew. "Anyone see this writing on the thing?"

They looked at the picture and shook their heads.

"What were you shooting on it at the time?"

Mason glanced at his records. "About a hundred and fifty angstroms."

"So there must be something that becomes visible only in a field of radiation of about that wave length," said Underwood. "Keep going and see if anything else turns up, or if this proves to be permanent after exposure to that frequency."

Back in the laboratory, they sat down at the desk and went through the file of hundreds of photographs that were now pouring out of the darkroom.

"Not a thing except that one," said Terry. "It looks like a message intended only for someone who knew what frequency would make it visible."

Underwood shook his head. "That sounds a little too melodramatic for me. Yet it is possible that this thing is some kind of repository, and we've found the key to it. But what a key! It looks as if we've got to decipher the language of the Stroids in order to use the key."

"The best men in the field have been trying to do that for only about seventy-five years. If that's what it takes, we may as well quit right now."

"You said that this was nothing like any other Stroid characters that you had seen. Maybe this belongs to a different cultural stratum. It might prove easier to crack. Who's the best man in the field on this stuff?"

"Dreyer at the semantics lab. He won't touch it any more. He says he's wasted fifteen years of his life on the Stroid inscriptions."

"I'll bet he will tackle this, if it's as new as you think it is. I've seen some of those antiquarians before. We'll get Phyfe to transmit some copies of this to him. Who's the next best man?"

"Probably Phyfe himself."

"It won't be hard to get him started on it, I'll bet."

It wasn't. The old scientist was ecstatic over the discovery of the inscriptions upon the huge gem. He took copies of the pictures into his study and spent two full days comparing them with the known records.

"It's an entirely new set of characters," he said after completing the preliminary examination. "We already have three sets of characters that seem to be in no way related. This is the fourth."

"You sent copies to Dreyer?"

"Only because you requested it. Dreyer admitted long ago that he was licked."

During the week of Phyfe's study, the work of radiation analysis had been completed. It proved completely negative with the single exception of the 150 A. radiation which rendered visible the characters on the gem. No secondary effects of any significance whatever had been noted. The material reflected almost completely nearly every frequency imposed upon it.

Thus, Underwood found himself again at the end of his resources. It was impossible to analyze material that refused to react, which was refractive to every force applied.

Underwood told Terry at the conclusion of a series of chemical tests, "If you want to keep that thing out here any longer, I'm afraid you've got to think of some more effective way of examining it than I have been able to do. From a physical standpoint this artifact is in about the same position as the language of the Stroids had been semantically—completely intractable."

"I'm not afraid of its being sent back to the museum now. Papa Phyfe's got his teeth into it and he won't let go until he cracks the key to this lingo."

Underwood didn't believe that it would ever be solved, unless by some lucky chance they came upon a sort of Rosetta Stone which would bridge the gap between the human mind and that of the alien Stroids. Even if the Stroids were somewhat anthropomorphic in makeup as the archeologists believed, there was no indication that their minds would not be so utterly alien that no bridge would even be possible.

Underwood felt seriously inclined to abandon the problem. While completely fascinating, it was hardly more soluble than was the problem of the composition of the stars in the days before the spectroscope was invented. Neither the archeologists, the semanticists, nor the physicists yet had the tools to crack the problem of the Stroids. Until the tools became available, the problem would simply have to go by the boards. The only exception was the remote possibility of a deliberate clue left by the Stroids themselves, but Underwood did not believe in miracles.

His final conviction came when word came back from Dreyer, who said, "Congratulations, Phyfe," and returned the copies of the Stroid characters with a short note.

"Well, that does it," said Underwood.

Phyfe was dismayed by Dreyer's reply. "The man's simply trying to uphold a decaying reputation by claiming the problem can't be solved. Send it to the museum and let them begin work on it. I'll give it my entire time. You will help me, if you will, Doctor Bernard."

Terry himself was becoming somewhat dismayed by the magnitude of the mystery they had uncovered. He knew Phyfe's bulldog tenacity when he tackled something and he didn't want to be tied to semantics for the rest of the term of the expedition.

Underwood, however, had become immersed in X-ray work, attempting to determine the molecular structure of the artifact from a crystallographic standpoint, to find out if it could be found it might be possible to disrupt the pattern.

After he had been at it for about a week, Terry came into the lab in a disgruntled mood at the completion of a work period.

"You look as if Papa gave you a spanking," said Underwood. "Why the downcast mood?"

"I think I'll resign and go back to the museum. It's useless to work on this puzzle any longer."

"How do you know?"

"Because it doesn't follow the laws of semantics with respect to language."

"Maybe the laws need changing."

"You know better than that. Look, you are as familiar with Carnovan's law as I am. It states that in any language there is bound to be a certain constant frequency of semantic conceptions. It's like the old frequency laws that used to be used in cryptographic analysis except a thousand times more complex. Anyway, we've made thousands of substitutions into Carnovan's frequency scale and nothing comes out. Not a thing. No concept of ego, identity, perfection, retrogression, or intercourse shows up. The only thing that registers in the slightest degree is the concept of motion, but it doesn't yield a single key word. It's almost as if it weren't even a language."

"Maybe it isn't."

"What else could it be?"

"Well, maybe this thing we've found is a monument of some kind and the inscriptions are ritualistic tributes to dead heroes or something. Maybe there's no trick at all about the radiation business. Maybe they used that frequency for common illumination and the inscription was arranged to show up just at night. The trouble with you strict semanticists is that you don't use any imagination."

"Like to try a hand at a few sessions with Papa Phyfe?"

"No, thanks, but I do think there are other possibilities that you are overlooking. I make no claim to being anything but a strictly ham semanticist, but suppose, for example, that the inscriptions are not language at all in the common sense."

"They must represent transfer of thought in some form."

"True, but look at the varied forms of thought. You are bound down to the conception of language held as far back as Korzybski. At least to the conception held by those who didn't fully understand Korzybski. You haven't considered the concept of music. It's a very real possibility, but one which would remain meaningless without the instrument. Consider also—Wait a minute, Terry! We've all been a bunch of thoroughbred dopes!"

"What is it?"

"Look at the geometrical and mechanical perfection of the artifact. That implies mathematical knowledge of a high order. The inscriptions could be mathematical measurements of some kind. That would explain the breakdown of Carnovan's principles. They don't apply to math."

"But what kind of math would be inscribed on a thing like that?"

"Who knows? We can give it a try."

It was the beginning of their sleeping period, but Terry was fired with Underwood's sudden enthusiasm. He brought in a complete copy of all the inscriptions found upon the facets of the black gem. Underwood placed them on a large table in continuous order as they appeared around the circumference.

"It's mud to me," said Terry. "I'm the world's worst mathematician."

"Look!" exclaimed Underwood. "Here's the beginning of it." He suddenly moved some of the sheets so that one previously in the middle formed the beginning of the sequence. "What does it look like to you?"

"I've seen that until I dream of it. It's one Phyfe tried to make the most of in his frequency determinations. It looks like nothing more than some widgets alongside a triangle."

"That's exactly what it is, and no wonder Phyfe found it had a high frequency. That is nothing more nor less than an explanation of the Stroid concept of the differential. This widget over here must be the sign of the derivative corresponding to our dy/dx."

Hastily, Underwood scrawled some symbols on a scratch pad, using combinations of "x"s and "y"s and the strange, unknown symbols of the Stroids.

"It checks. They're showing us how to differentiate! Not only that, we have the key to their numerical system in the exponentials, because they've given us the differentiation of a whole series of power expressions here. Now, somewhere we ought to find an integral expression which we could check back with differentiation. Here it is!"

Terry, left behind now, went to the galley and brewed a steaming pot of coffee and brought it back. He found Underwood staring unseeingly ahead of him into the dark, empty corners of the lab.

"What is it?" Terry exclaimed. "What have you found?"

"I'm not sure. Do you know what the end product of all this math is?"

"What?"

"A set of wave equations, but such wave equations as any physicist would be thought crazy to dream up. Yet, in light of some new manipulations introduced by the Stroids, they seem feasible."

"What can we do with them?"

"We can build a generator and see what kind of stuff comes out of it when we operate it according to this math. The Stroids obviously intended that someone find this and learn to produce the radiation described. For what purpose we can only guess—but we might find out."

"Do we have enough equipment aboard to build such a generator?"

"I think so. We could cannibalize enough from equipment we already have on hand. Let's try it."

Terry hesitated. "I'm not quite sure, but—well, this stuff comes about as near as anything I ever saw to giving me what is commonly known as the creeps. Somehow these Stroids seem too—too anxious. That sounds crazy, I know, but there's such alienness here."

"Nuts. Let's build their generator and see what they're trying to tell us."