The coastal dawn revealed the great scaphe in all its pale glory. After eleven days of brutal effort, what had once seemed like a rudimentary excavation in the sand now rested as a large limestone basin, smooth and white under the first light of morning. The fire had transformed dead shells into quicklime; the sand, the water, and the mortar had done the rest.
In the exact center of the basin, the wooden gnomon stood perfectly vertical. Bhāskara paced back and forth on the edge of the scaphe, unable to hide his anticipation.
Tribal observed in silence. Although he knew the deep laws of matter, there was something fascinating in discovering whether that mathematical logic could be transformed into a concrete measurement of the world.
Alaya, beside him, kept her arms crossed.
"If you keep walking like that, you'll dig another scaphe," she commented.
Bhāskara did not even seem to hear her. The Sun climbed. The shadow of the gnomon, long in the early morning, advanced slowly across the clear surface of the basin. Bhāskara finally entered the scaphe. He knelt next to the dark mark and began to measure.
"One thumb... two palms..." he murmured.
He checked the height of the rod again. Checked the shadow. Repeated the measurement. Tribal approached.
"Is it right?"
Bhāskara raised a finger, asking for silence. The Sun continued to rise. The shadow shrank. When it finally reached its shortest length, Bhāskara marked the spot carefully. Then he compared the height of the gnomon with the length of the shadow. His eyes narrowed. He made some calculations in the sand. Then he smiled.
"Fifty parts."
Tribal tilted his head.
"Fifty parts of what?"
Bhāskara stood up.
"Of a circle."
He walked to a clean area of the sand and drew a large circle.
"Observe. A complete turn has three hundred and sixty degrees."
He divided the circle with the stick.
"The shadow reveals the inclination of the Sun's rays. In this case, approximately seven and two-tenths degrees." He wrote:
360 ÷ 7.2 = 50
Alaya uncrossed her arms.
"So this shadow represents one part in fifty?"
"Exactly."
Bhāskara pointed to the gnomon.
"We do not need to know the size of the world to measure it."
Tribal remained looking at the circle drawn in the sand.
"Then how do you measure that which we cannot see in its entirety?"
Bhāskara turned his eyes to the shadow.
"Like this."
For a few moments, no one said anything. The wind passed over the white surface of the scaphe. Bhāskara then began to explain.
"The Sun is so distant from the planet that its rays reach us practically parallel. If the surface of the world were perfectly flat, two identical rods would receive the light at the same angle. But the surface is not flat."
He ran his hand over the circle.
"It curves."
Tribal understood before Bhāskara finished.
"And the shadow reveals that curvature."
The astronomer smiled broadly.
"Exactly!"
Alaya looked at the line drawn in the sand.
"But a shadow only measures the place where we are. We still lack the second point."
Bhāskara's smile disappeared.
"Yes."
He looked to the south.
"Suryashan."
The golden city remained far beyond the horizon.
"On the solstice, when the Sun reaches the zenith over Suryashan, a perfectly vertical rod will cast no shadow at all. The light will descend directly upon it."
"Here we have an angle. In Suryashan we will have the point corresponding to zero shadow. Between the two exists an arc of the world's surface."
He wrote again in the sand:
7.2° → 1/50 of the circle
"If we discover the exact distance between this place and Suryashan, we will have a known part of the planet's circumference."
Tribal looked toward the horizon.
"And then?"
Bhāskara pointed to the number.
"We will multiply that distance by fifty."
The simplicity of the conclusion seemed greater than the calculation itself. Tribal smiled.
"So we don't need to walk all the way around the planet."
"No!"
Bhāskara opened his arms.
"We only need to measure a small part of the world to discover the size of the whole!"
Alaya observed the two of them for a few seconds.
"And how much will we have to walk?"
Bhāskara fell silent. He looked south. Then at the rope they had prepared.
"To Suryashan."
The next morning, they departed. The measuring rope was stretched between Bhāskara's ankles, keeping the span of each stride always equal. Tribal checked the tension before the first step. One stride. Then another. And another. The counting began.
Bhāskara recorded every measurement. At the end of each day, he transferred the numbers to the parchment. The waves were left behind. The sand gave way to dry earth. And the line they had drawn toward the south became the rule of the journey. Every step had a value. Every distance needed to be preserved. Any deviation could compromise the measurement.
During the first few days, the march was peaceful. Then, the terrain began to change. The plain became irregular. The air gained moisture. A deep sound began to grow ahead of them. Bhāskara stopped. The rope between his ankles grew taut. Ahead, the earth simply ended. A massive canyon tore through the continent. At the bottom of the rift, a gigantic river surged between the rocks, dark and furious. The current dragged logs and stones, disappearing into the distance. There was no bridge. There was no shallow crossing.
Bhāskara pulled out the parchment. The line indicating Suryashan pointed exactly straight ahead. He looked at the map. Then at the river. Then back at the map. Tribal approached.
"We can go around it."
Bhāskara did not answer. Alaya looked at the line drawn on the parchment.
"If we go around, we will lose the straight line."
Bhāskara squeezed the parchment between his fingers. Mathematics had given them a direction. But the earth did not seem willing to obey. He took a step to the edge of the canyon. Down below, the river roared.
For the first time since the journey began, Bhāskara did not have an equation for what lay before him. He looked at Suryashan on the map. Then at the abyss. And he understood: knowing the path did not mean knowing the journey.