Napoleon’s visit to the pyramids was a grander excursion than the visit I’d made earlier with Talma and Jomard. More than a hundred officers, escorting soldiers, guides, servants, and scientists crossed the Nile and hiked up to the Giza plateau. It was like a holiday outing, a train of donkeys bearing French wives, mistresses, and a cornucopia of fruits, sweets, meats, and wine. Parasols were held in the sun. Carpets were spread on the sand. We would dine next to eternity.
Conspicuous by his absence was Silano, who I was told was conducting his own investigations in Cairo. I was glad I’d tucked Astiza safely out of the way.
As we trudged up the slope I reported Talma’s hideous death to Bonaparte, to gauge his reaction and plant doubt in his mind about my rival. Unfortunately, my news seemed to annoy our commander more than shock him. “The journalist had barely started my biography! He shouldn’t have wandered off before the country is pacified.”
“My friend disappeared when Silano arrived, General. Is that coincidence? I fear the count may be involved. Or Bin Sadr, that Bedouin marauder.”
“That marauder is our ally, Monsieur Gage. As is the count, an agent of Talleyrand himself. He assures me he knows nothing about Talma, and in any event he has no motive. Does he?”
“He said he wanted the medallion.”
“Which you said you lost. In a nation of a million restive natives, why do you suspect only the people who are on our side?”
“But are they on our side?”
“They are on my side! As you will be, when you begin to solve the mysteries we brought you here for! First you lose your medallion and calendar, and now you make accusations against our colleagues! Talma died! Men do in war!”
“They don’t have their heads delivered in a jar.”
“I have seen parts worse than that delivered. Listen. You saw the defeat of our fleet. Our success is imperiled. We are cut off from France. Rebel Mamelukes are gathering in the south. The population is not yet resigned to its new situation. Insurgents commit atrocities precisely to sow the kind of terror and confusion you’re exhibiting. Stand fast, Gage! You were brought to solve mysteries, not create them.”
“General, I’m doing my best, but Talma’s head was clearly a message…”
“A message that time is of the essence. I cannot afford sympathy, because sympathy is weakness, and any weakness on my part invites our destruction. Gage, I tolerated an American’s presence because I was told you might be useful in investigating the ancient Egyptians. Can you make sense of the pyramids or not?”
“I am trying, General.”
“Succeed. Because the moment you are of no use to me, I can have you jailed.” He looked past me, the admonition given. “Ah. They are big, aren’t they?”
The same awe that I’d felt on my initial visit was experienced by others as they came within view of the Sphinx and the pyramids behind. Customary chatter went silent as we clustered on the sand like ants, the depth of time palpable. Their shadows on the sand were as distinct as the pyramids themselves. It was not the ghosts of the long-vanished workmen and pharaohs I experienced, but rather the serene spirit of the structures themselves.
Napoleon, however, scrutinized the monuments like a quartermaster. “As simple as a child might build, but they certainly have size. Look at that volume of stone, Monge! Building this big one here would be like marshalling an army. What are the dimensions, Jomard?”
“We’re still digging, trying to find the base and the corners,” the officer replied. “The Great Pyramid is at least seven hundred and fifty feet on each side and more than four hundred and fifty feet high. The base covers thirteen acres, and while the building stones are huge, I calculate there are at least two and a half million of them. The volume is large enough to easily contain any of the cathedrals in Europe. It is the largest structure in the world.”
“So much stone,” Napoleon murmured. He asked the dimensions of the other two pyramids as well and, using a Conte pencil, began jotting calculations of his own. He played with mathematics in the way other men might doodle. “Where do you think they got the stone, Dolomieu?” he asked as he worked.
“Somewhere nearby,” the geologist replied. “Those blocks are limestone, the same as the bedrock of the plateau. That’s why they appear eroded. Limestone isn’t very hard, and wears easily from water. In fact, formations of limestone are frequently perforated with caves. We might expect caves here, but I must assume this plateau is solid, given the aridity. Reportedly there is also granite inside the pyramid, and that must have come from many miles away. I suspect the facing limestone also came from a separate quarry of finer rock.”
Napoleon displayed his calculations. “Look, it is absurd. With the stone in these pyramids you could build a wall two meters high and one meter thick around all of France.”
“I hope you don’t expect us to do so, General,” Monge joked. “It would weigh millions of tons to take home.”
“Indeed.” He laughed. “At last I have found a ruler who eclipses my own ambition! Khufu, you dwarf me! Yet why not simply tunnel into a mountain? Is it true the Arab tomb robbers didn’t find a corpse inside?”
“There is no evidence anyone was ever buried here,” Jomard said. “The main passage was blocked by enormous granite plugs that seem to have guarded…nothing.”
“So we are presented with another mystery.”
“Perhaps. Or perhaps the pyramids serve some other purposes, which is my own theory. For example, the pyramid’s placement, near the thirtieth parallel, is intriguing. It is almost exactly one-third of the way from the equator to the North Pole. As I was explaining to Gage here, the ancients hint that the Egyptians might have understood the nature and size of our planet.”
“If so, they are ahead of half the officers in my army,” Bonaparte said.
“Equally striking, the Great Pyramid and its companions are oriented in the cardinal directions of north, south, east, and west more precisely than modern surveyors typically achieve. If you draw a line from the pyramid’s center to the Mediterranean, it exactly bisects the Nile Delta. If you draw diagonal lines from one pyramid corner to the opposite and extend those, one going northeastward and one northwestward, they form a triangle that perfectly encloses the delta. This location was no accident, General.”
“Intriguing. A symbolic location to tie upper and lower Egypt together, perhaps. The pyramid is a political statement, do you think?”
Jomard was encouraged by this attention to his theories, which other officers had jeered at. “It is also interesting to consider the pyramid’s apothem,” he said enthusiastically.
“What’s an apothem?” I interrupted.
“If you drew a line down the middle of one face of the pyramid,” the mathematician Monge explained, “from point to base, so that you divide its triangle in two, that line is the apothem.”
“Ah.”
“The apothem,” Jomard went on, “appears to be exactly six hundred feet, or the length of the Greek stadia. That’s a common measurement found throughout the ancient world. Could the pyramid be a standard of measurement, or be built to a standard that long predates the Greeks?”
“Possibly,” Bonaparte said. “Yet using this as a measuring stick seems an even more absurd excuse for such a monument than a tomb.”
“As you know, General, there are sixty minutes in each degree of latitude or longitude. That apothem also happens to be one-tenth of one minute of one degree. Is this mere coincidence? Even odder, the perimeter of the pyramid’s base equals half a minute, and two circuits a full minute. Moreover, the perimeter of the pyramid’s base appears to be equal to the circumference of a circle whose radius is the pyramid’s height. It’s as if the pyramid was sized to encode the dimensions of our planet.”
“But dividing the earth into three hundred sixty degrees is a modern convention, is it not?”
“On the contrary, that number can be traced to Babylon and Egypt. The ancients picked three hundred sixty because it signifies the days of the year.”
“But the year is three hundred sixty-five,” I objected. “And a quarter.”
“The Egyptians added five holy days when that became apparent,” Jomard said, “just as we revolutionaries have added holidays to our thirty-six ten-day weeks. My theory is that the people who built this structure knew the size and shape of the earth and incorporated those dimensions into this structure so they’d not be lost, should learning decline in the future. They anticipated, perhaps, the Dark Ages.”
Napoleon looked impatient. “But why?”
Jomard shrugged. “Perhaps to reeducate mankind. Perhaps simply to prove that they knew. We build monuments to God and military victory. Perhaps they built monuments to mathematics and science.”
It seemed improbable to me that people so long ago could know so much, and yet again there was something fundamentally right about the pyramid, as if it were trying to convey eternal truths. Franklin had mentioned a similar rightness to the dimensions of Greek temples, and I remember that Jomard had tied everything to that strange Fibonacci number sequence. Again I wondered if these games of arithmetic had anything to do with the secret of my medallion. Mathematics made my mind fog.
Bonaparte turned to me. “And what does our American friend think? What is the view from the New World?”
“Americans believe things should be done for a purpose,” I said, trying to sound wiser than I was. “We’re practical, as you said. So what is the practical use of this monument? Perhaps Jomard has a point that this is more than a tomb.”
Napoleon was not fooled by my rambling. “Well, the pyramid has a point, at least.” We dutifully laughed. “Come. I want to look inside.”
While most of our party was content to picnic, a handful of us entered the dark hole on the pyramid’s north face. There was a limestone portal marking the pyramid’s original entrance that had been constructed by the ancient Egyptians. This entry, Jomard explained, was only revealed when Muslims stripped off the pyramid’s casing for stone to build Cairo; in ancient times it had been disguised by a cleverly hidden hinged door of stone. No one had known precisely where it lay. So before it was revealed, medieval Arabs made an attempt to plunder the pyramid by simply starting their own entry. In 820, Caliph Abdullah al-Mamun, knowing that historians recorded a northern entrance, had a band of engineers and stone masons chew their own tunnel into the pyramid in hopes of striking the structure’s corridors and shafts. As luck would have it, he began below the earlier door. It was this excavation we entered.
While their guess of the entry’s placement was off, the tunneling Arabs soon struck a narrow shaft inside the pyramid that had been built by the Egyptians. Just under four feet high, this shaft descended from the original entrance at an angle that Jomard had calculated was twenty-three degrees. Crawling upward, the Arabs found the original entrance to the outside and a second shaft ascending into the pyramid at the same slope the first descended. Such an upward shaft had never been mentioned in ancient chronicles, and it was blocked with granite plugs too hard to chisel through. Sensing he had found a secret pathway to treasure, Al Mamun ordered his men to tunnel around the plugs through the softer surrounding limestone blocks. It was hot, dirty, noxious work. The first granite plug was succeeded by another, and then a third. After great effort they rejoined the upward shaft, but found it was now plugged with limestone. Determined, they excavated even that. Finally, they broke through, and found…
“Nothing,” Jomard said. “And yet something, which you will see today.”
Under the geographer’s direction, we reconnoitered this architectural confusion of entrances and junctions, and then stoop-walked to peer down the descending shaft that the Arabs had first encountered. The blackness at its end was total.
“Why a slope and not steps?” Napoleon wondered.
“To slide things, perhaps,” Jomard said. “Or maybe this is not an entry at all but serves some other function, such as a pipe, or a telescope pointed at a particular star.”
“The biggest monument in the world,” Bonaparte said, “and it makes no sense. There’s something here we are missing.”
With the help of torches carried by local guides we cautiously made our way downward about one hundred meters, stepping sideways for purchase. Carved blocks gave way to a smooth shaft through limestone bedrock, and then the shaft ended in a cavelike room with a pit and an uneven floor. It seemed unfinished.
“As you can see, this shaft seems to lead nowhere,” Jomard said. “We’ve found nothing of interest.”
“Then what are we doing here?” Bonaparte asked.
“The lack of obvious purpose is what is intriguing, don’t you think? Why did they dig down here? And wait, it gets better. Let’s go up again.”
We did so, panting, and sweaty. Dust and bat guano stained our clothes. The air in the pyramid was warm, moist, and musty.
Back at the junction of tunnel and shafts, we now ascended above our original entry point and entered the ascending shaft so laboriously excavated by al-Mamun’s men. This one rose at the same angle the initial one descended, and again, it was too low to stand upright. There were no steps and it was awkward to climb. After sixty meters, hot and panting, we emerged at another junction. Ahead, running level, was a low passage that led to a largely featureless room with gabled roof that the Arabs had dubbed the Queen’s Chamber, even though our guides told us there was no evidence any queen had ever been buried there. We crawled to it and stood up. There was an alcove at one end, possibly for a statue or upright coffin, but it was empty. The room was remarkable only in its plainness. Its granite blocks were absolutely featureless, each weighing many tons and so finely jointed that I couldn’t slip a piece of paper between them.
“The gabled roof might deflect some of the pyramid’s weight to the chamber walls,” Jomard said.
Napoleon, impatient at the dirty indignities we were enduring, curtly ordered us back out to the junction where the shaft continued to ascend. He wanted to see the King’s Chamber above.
Now the cramped, dwarflike passageway changed to one for giants. The ascending passage broadened and rose, forming an inclined gallery that climaxed in a corbelled roof almost thirty feet above our heads. Again there were no steps; it was like climbing a slide. Fortunately, guides had fixed a rope. Once more, the stonework was as perfect as it was plain. This section’s height seemed as inexplicable as the dwarf-sized passage before.
Had humans really built this?
An Arab guide held his torch high and pointed at the ceiling. I could see dark clots up there, marring the perfect symmetries, but I didn’t know what they were.
“Bats,” Jomard whispered.
Wings twitched and rustled in the shadows.
“Let’s hurry up,” Napoleon commanded. “I’m hot and half suffocated.” The torch smoke stung.
The gallery was forty-seven meters long, Jomard announced after unreeling a tape, and again had no obvious purpose. Then the climb ended and we had to stoop to advance horizontally again. Finally we entered the pyramid’s biggest room, built a third of the way up the structure’s mass.
This King’s Chamber was a featureless rectangle built of colossal red granite blocks. Again, the simplicity was odd. The roof was flat and the floor and walls barren. There was no sacred book or bird-headed god. The only object was a lidless black granite sarcophagus set at the far end, as empty as the room itself. At about seven feet long, three and a half feet wide, and three feet high, it was too big to have fit through the tight entry we’d just crawled through, and must have been put in place as the pyramid was built. But Napoleon for the first time seemed intrigued, inspecting the rock casket closely.
“How could they have hollowed this out?” he asked.
“The room’s dimensions are also interesting, General,” Jomard said. “I measure thirty-four feet long, seventeen wide. The chamber floor represents a double square.”
“Imagine that,” I said, more mocking than I meant.
“He means its length is twice its width,” Monge explained. “Pythagoras and the Greeks were interested in the harmony of such perfect rectangles.”
“The chamber’s height is half the length of the room’s diagonal,” Jomard added, “or nineteen feet. Gage, help me here and I’ll show you something else. Hold this end of my tape in that corner.”
I did so. Jomard extended his tape diagonally to the opposite wall, exactly halfway along its length. Then, as I held the tape in my corner, he walked his end across the room until what had been a diagonal now lay alongside the wall I occupied. “Voilà!” he cried, his voice echoing in the rock room.
Once more I did not display the anticipated excitement.
“Don’t you recognize it? It’s what we talked about at the pyramid’s summit! The golden number, or golden mean!”
Now I saw it. If you divided this rectangular room into two squares, measured the diagonal of one of those squares, and laid that line on the long side of the chamber, the ratio between that length and what was left was the supposedly magical 1.618.
“You’re saying this room incorporates Fibonacci numbers in the same way the pyramid itself does,” I said, trying to sound casual.
Monge’s eyebrows raised. “Fibonacci numbers? Gage, you’re more of a mathematician than I would have guessed.”
“Oh, I’ve just been picking it up here and there.”
“So what’s the practical use of these dimensions?” Napoleon asked.
“It represents nature,” I ventured.
“And it encodes the Egyptian kingdom’s basic units of measurement,” Jomard said. “In its length and proportions, I think it lays out a system of cubits, just as we might design the metric system into the proportions of a museum.”
“Interesting,” the general said. “Still, to build so much—it’s a puzzle. Or a lens, perhaps, like a lens to focus light.”
“That’s what I feel,” Jomard said. “Any thought you think, any prayer you make, seems amplified by the dimensions of this pyramid. Listen to this.” He began a low hum, then a thrumming chant. The sound echoed weirdly, seeming to vibrate through our bodies. It was like striking a note of music that lingered in the air.
Our general shook his head. “Except that this focuses—what? Electricity?” He turned to me.
If I’d grandly said yes, he probably would have given me a reward. Instead, I looked vacant as an idiot.
“The granite coffer is also interesting,” Jomard said, to fill the awkward silence. “Its interior volume is exactly half its exterior volume. While it seems sized for a man or a casket, I suspect its precise dimensions are no accident.”
“Boxes within boxes,” Monge said. “First this chamber, then the outside of the sarcophagus, then the inside…for what? We have a host of theories, but no one answer I feel is conclusive.”
I looked up. It felt like millions of tons were pressing down toward us, threatening at any moment to obliterate our existence. For a moment I had the illusion the ceiling was descending! But no, I blinked, and the chamber was as before.
“Leave me,” Bonaparte suddenly commanded.
“What?”
“Jomard is right. I feel power here. Don’t you feel it?”
“It feels oppressive and yet alive,” I offered. “Like a grave, and yet you feel light, insubstantial.”
“I want to spend some time in here alone,” the general told us. “I want to see if I can feel the spirit of this dead pharaoh. Perhaps his body is gone but his soul remains. Perhaps Silano and his magic are real. Perhaps I can feel Gage’s electricity. Leave me with an unlit torch in the dark. I’ll come down when I’m ready.”
Monge looked concerned. “Perhaps if one of us remained as guard…”
“No.” He climbed over the lip of the black sarcophagus and lay down, staring at the ceiling. We looked down at him and he smiled slightly. “It’s more comfortable than you might think. The stone is neither too cold nor hot. Nor am I too tall, are you surprised?” He smiled at his little joke. “Not that I plan to remain here forever.”
Jomard looked troubled. “There are accounts of panic…”
“Never question my courage.”
He bowed. “To the contrary, I salute you, my general.”
So we dutifully filed out, each torch in turn disappearing through the low entryway until our commander was left alone in the dark. We worked our way down the Grand Gallery, letting ourselves down by the rope. A bat took flight and flapped down toward us, but an Arab waved a torch and the blind creature veered away from the heat, settling again on the ceiling. By the time we got down to the smaller shaft that led down to the pyramid entrance, I was soaked with sweat.
“I’ll wait for him here,” Jomard said. “The rest of you file outside.”
I needed no encouragement. The day seemed lit with a thousand suns when we finally emerged on the outside of the pyramid’s sand-and-rubble slope, clouds of dust puffing off our now-filthy clothes. My throat was parched, my head aching. We found shade on the east side of the structure and sat to wait, sipping water. The party members who had remained outside had scattered over the ruins. Some were circuiting the other two pyramids. Some had erected little awnings and were having lunch. A few had climbed partway up the structure above us, and others competed to see how high up the pyramid’s side they could hurl a rock.
I mopped my brow, acutely conscious that I seemed no closer to solving the medallion’s mystery. “All this great pile for three little rooms?”
“It doesn’t make sense, does it?” agreed Monge.
“I feel like there’s something obvious we’re not seeing.”
“I’m guessing we’re to see numbers, as Jomard said. It may be a puzzle meant to occupy humankind for centuries.” The mathematician took out paper and began his own calculations.
Bonaparte was absent for a full hour. Finally there was a shout and we went back to meet him. Like us he emerged dirty and blinking, skidding down the rubble to the sand below. But when we ran up I saw he was also unusually pale, his eyes having the unfocused, haunted look of a man emerging from a vivid dream.
“What took you so long?” Monge asked.
“Was it long?”
“An hour, at least.”
“Really? Time disappeared.”
“And?”
“I crossed my arms in the sarcophagus, like those mummies we’ve seen.”
“Mon dieu, General.”
“I heard and saw…” He shook his head as if to clear it. “Or did I?” He swayed.
The mathematician grasped his arm to hold him up. “Heard and saw what?”
He blinked. “I had a picture of my life, or I think it was my life. I’m not even sure if it was the future or the past.” He looked around, whether to be evasive or to tease us, I know not.
“What kind of picture?”
“I…it was very strange. I won’t speak of this, I think. I won’t…” Then his eyes fell on me. “Where’s the medallion?” he abruptly demanded.
He took me by surprise. “It’s lost, remember?”
“No. You’re mistaken.” His gray eyes were intent.
“It went down with L’Orient, General.”
“No.” He said it with such conviction that we looked at each other uneasily.
“Would you have some water?” Monge asked worriedly.
Napoleon shook his head as if to clear it. “I will not go in there again.”
“But, General, what did you see?” the mathematician pressed.
“We will not speak of this again.”
All of us were uncomfortable. I realized how much the expedition relied on Bonaparte’s precision and energy, now that I’d seen him dazed. He was imperfect as a man and a leader, but so commanding, so dominant in purpose and intellect, that all of us had unconsciously surrendered to him. He was the expedition’s spark and its compass. Without him, none of this would be happening.
The pyramid seemed to be looking down on us mockingly, the perfect peak.
“I must rest,” Napoleon said. “Wine, not water.” He snapped his finger and an aide ran to fetch a flask. Then he turned to me. “What are you doing here?”
Had he lost all his senses? “What?” I was confused by his confusion.
“You came with a medallion and a promise to make sense of this. You’ve claimed to have lost the one and haven’t fulfilled the other. What is it I felt in there? Is it electricity?”
“Possibly, General, but I have no instrument to tell. I’m as baffled as anyone.”
“And I am baffled by you, a suspected murderer and an American, who comes on our expedition and seems to be of no use and yet is everywhere! I’m beginning to not trust you, Gage, and it is not comfortable being a man I don’t trust.”
“General Bonaparte, I have been working to earn your trust, on the battlefield and here! It does no good to make wild guesses. Give me time to work on these theories. Jomard’s ideas are intriguing, but I’ve had no time to evaluate them.”
“Then you will sit here in the sand until you do.” He took the flask and drank.
“What? No! I have studies in Cairo!”
“You’re not to return to Cairo until you can come back and tell me something useful about this pyramid. Not old stories, but what it is for and how it can be harnessed. There’s power here, and I want to know how to tap it.”
“I want nothing less! But how am I to do that?”
“You are a savant, supposedly. Discover it. Use the medallion you pretend to have lost.” Then he stalked away.
Our little group watched him in stupefaction.
“What the devil happened to him in there?” Jomard said.
“I think he hallucinated in the dark,” Monge said. “Lord knows I wouldn’t stay in there alone. Our Corsican has guts.”
“Why did he focus on me?” His antagonism had shaken me.
“Because you were at Abukir,” the mathematician said. “I think the defeat is gnawing on him more than he will admit. Our strategic future is not good.”
“And I’m to camp out here staring at this structure until it is?”
“He’ll forget about you in a day or two.”
“Not that his curiosity isn’t warranted,” Jomard said. “I need to read the ancient sources again. The more I learn of this structure, the more fascinating it seems.”
“And pointless,” I grumbled.
“Is it, Gage?” asked Monge. “There’s far too much precision for pointlessness, I think. Not only too much labor, but too much thought. In doing more calculations just now, another correlation occurred to me. This pyramid is indeed a mathematical plaything.”
“What do you mean?”
“I will need to check my guess against Jomard’s figures, but if we extrapolate the pyramid’s slope to its original peak, a bit higher than it is now, and compare its height to the length of two of its sides, I believe we arrive at one of the most fundamental numbers in all mathematics: pi.”
“Pi?”
“The ratio of a circle’s diameter to its circumference, Gage, is considered by many cultures to be sacred. It’s about twenty-two parts to seven, or 3.1415…the number has never been completely computed. Still, every culture has tried to come as close as they can. The ancient Egyptians came up with 3.160. The pyramid’s ratio of height to two of its sides appears to come very close to that number.”
“The pyramid stands for pi?”
“It was built, perhaps, to conform to the Egyptian value of that number.”
“But again, why?”
“Once more we butt up against ancient mysteries. But it’s interesting, is it not, that your medallion included a diameter inside a circle? Too bad you lost it. Or did you?”
Interesting? It was a revelation. For weeks I’d been journeying blindly. Now I felt like I knew definitely what the medallion was pointing to: the pyramid behind me.