Chương 23: Chapter 1.23: Absolute Units

Tiro bowed to Favius and gestured to the household slave behind him, who carried a heavy wooden box.

"Ice from the cellars, alongside tools to break it, just as the young master ordered." Favius cracked the box open and felt the cold, slightly dirty ice, cut out of a mountain lake in winter and transported down here. Funny how that is the solution to ice in most cultures before refrigeration. He smiled and turned to Tiro.

"Very well. Have it brought to the lecture hall." Tiro nodded, and the carrier left.

"Does the young master have another request?" Tiro's smile was carefully measured. Favius schooled his face into a thoughtful expression and stroked his chin. All my requests until now were practical. Well, maybe the ice was excessive? But given that he passes everything along to Hesnia, I wonder what she is making out of the collection of items I've requested. Maybe I should order him to get me a pregnant goat next. Just to see if he can do it within a day. Favius sighed and shook his head.

"That will be all for now. You've done well, Tiro." As he left for the gymnasium, the cold morning air on his bare chest, Favius noticed Tiro following him.

"Young master, may I ask—" Favius made a cutting gesture with his hand.

"You may not. I make my requests for my own reasons. And I believe none of them were unreasonable." From the corner of his eye, he saw several expressions cross the man's face, none of them exactly happy, before he bowed again and left. After a morning of physical training and cultivation exercises—and no less than three of Caliga's attempts to assassinate Brutus—they walked down the corridor to the lecture hall, where Favius had stockpiled his props. Caliga danced around him.

"So, today you're going to show us something practical. But what is it?" She poked him in the side of the stomach, and he gasped.

"Stop that!" He tried to slap her hand away, but she evaded nimbly and grinned.

"Tell me something, and I'm going to stop it." She tried to poke him again, but this time he was on guard, blocked, and counterattacked in one smooth move.

"Ow!" she hissed, dancing to the side.

"You can't do that!" She spun around to Urania, who walked beside Brutus.

"Urania, tell your brother he's not allowed to poke me." Urania's eyes widened as she was pulled out of a conversation about cultivation with the warrior. She looked between the two of them and shook her head. Caliga grasped her chest and gasped.

"Betrayal!" Favius chuckled and shook his head, trying to focus on the lesson he was about to teach. As they entered the lecture hall, Serran Vale was already waiting. All four of his students settled at their desks while Favius walked to the front of the room. He surveyed the collection of props—wooden sticks, tools, a piece of leather, clay, an already simmering pot. Yes, everything seems to be here. He nodded to everyone present and smiled.

"We've talked about calculations, but almost as important as how to calculate is what to calculate. And the mathematics we care about—" he gestured to the majordomo "—tend to concern themselves with physical quantities. We need to measure the goods that flow through our estate." Favius picked up a short wooden stick. It's the closest thing to a decimeter I could make, so… He held up the wooden stick and met the eyes of his not-quite-official students.

"Behold, one of the two elements fundamental to making a system to measure everything: the decimeter." The word came out strange: tenth measure. He repeated it slowly, and Serran narrowed his eyes at him.

"Is it just supposed to represent an arbitrary length, or is there anything special about it?" Favius nodded eagerly.

"Yes. You see, the measurements that are so common—our collection of everyday units: fingers, steps, forearms—might seem practical in their contexts. Yet a functional system needs to be able to seamlessly convert between them." He was met with a mix of frowns and furrowed foreheads. Favius shifted his weight from foot to foot. Seems like they haven't been struck by an epiphany about the glorious metric system yet. He held out his decimeter stick.

"So—this is the base unit of measurement. If you take it ten times, you get the next unit: the meter. If you take a tenth of it, we gain the next smallest unit: the centimeter. And each time we multiply by ten, we get another useful unit." Urania was nodding along, while Caliga's head was tilted. Brutus had a thoughtful expression, yet the majordomo shook his head. Unauthorized use: this story is on Amazon without permission from the author. Report any sightings.

"I can see the advantage of a system where all units are related by simple ratios, but I do not understand the weird names." Favius waved his hand.

"The names are irrelevant. What matters is that a lot of costly mistakes are being made if the conversion between the units is too complicated and they are not properly related to each other." He waved the man's concern away. The classical example is the incident where NASA had crashed a Mars rover onto the red planet because of a conversion error. But I can't use that here, obviously. But Serran wasn't convinced.

"This 'tenth measure' you propose—it's nearly the same length as a handspan. Why not call it a handspan, then a ten-handspan, and a tenth-handspan, and so on?" Favius met the man's eye and took a deep breath before replying. He's not wrong. My conflict with Cassian and his traditional thinking… Maybe if I take a local unit's name and define it to my own standard, that would make my system more digestible.

He hummed and recalled the mess of city-based norms, or units that were defined and used exclusively within a single collegium of craftsmen. This world had hundreds of local units in the Republic alone. He nodded to the man.

"You make a good point, majordomo." Soon, Brutus interjected, and they discussed which names to use. In the end, it resulted in a mixture of colloquial names and technical names. While everything was defined from the span, the centimeter-equivalent unit would also be called a finger, and the meter-equivalent an arm. At some point, Caliga cleared her throat and pointed to the other props.

"You haven't brought all this just to talk about a stick, right? Will we do more than debate how to name it?" Favius chuckled.

"Impatient as always, Caliga. But—" He ran a hand through his hair.

"You're right. To define any sophisticated system of measurement, one needs a fundamental unit of length and a common fluid." He grasped a thick-walled jug of water and raised it, pouring a little onto the floor.

"Water." He moved the jug to Caliga's desk before bringing a wooden box, clay, and a piece of thick leather as well.

"Caliga, Urania," he addressed his sisters, "we need to make a hollow cube from the clay, all whose sides are as long as this span-long piece of wood. Or rather—" he looked at the thickness of the leather "—slightly longer. Can you do it?" The two of them nodded and began forming the clay into the wooden box. While the girls were engaged with pottery, he considered his plan again. Introducing the metric system is necessary for everything in the future, but out of the seven base units…

well, ampere and mole only become relevant once we mess about with electricity and chemistry. But candela? He shook his head. Candela was always the weirdest of the standard units. Light intensity relative to the sensitivity of the human eye. I honestly don't have any idea how to do that in this world. Especially with shifters like me. While his sisters finished the liter cube, he walked to the desk and set up a scale and several jars of water.

He balanced an empty jug against several stone weights until the scale was even. When he was done, he took all the stones he had used and put them into a leather sack. Serran observed with a slightly bemused expression as Favius returned to his sisters' pottery project and stuffed the leather sheet into the cube. As close to a perfect cubic decimeter as I'm going to get. He studied the uneven surfaces. Maybe I should have done this with wood, but this should do for a demonstration.

None of this is going to be highly precise. But as long as we are within a ten percent error margin, we can refine things later. His sisters cleaned the clay off their hands, and Favius held up the jug of water.

"Behold." He poured the water into the vessel until it was filled almost exactly to the top.

"Brutus," he turned to his protector and gestured at the vessel, "would you care to venture a guess about what I've done here? Why is this important?" The warrior scratched his beard and narrowed his eyes.

"You have constructed volume from length, which we've now filled with water." He glanced over at the scale.

"I assume that there is something special about this particular volume which you are going to relate to the idea of weight." Favius clapped his hands.

"Very good. You see—" He picked up the ungainly liter cube and poured the water into the jug whose weight he had balanced out on the scales before. He placed it back on the now uneven scale, then gestured at Caliga.

"Would you mind balancing the scales using those rocks and then putting them into a bag?" Caliga nodded and did as she was told. But Serran cleared his throat.

"So if I understand what you're doing correctly, you want to use distance and the weightiness of water to define weight?"

"Yes!" Favius snapped his fingers and smiled before frowning.

"Weightiness… not quite the word." If the locals don't have an idea about the relationship between volume and weight, it makes sense that they have no concept of density. Still strange. Brutus scratched his beard and pointed to the icebox and the kettle.

"It's a clever idea, but water expands when it is heated, so how can you get a consistent measurement from that?" Favius made a sour expression and nodded.

"That is a good catch. For our current demonstration, this is sufficient. But for the book—well, I ought to make it clear that it should be water collected from freshly molten ice." The two older men hummed. As Caliga returned with a leather sack filled with stones, Favius took it from her and held it up like a relic.

"This is—" he wanted to say kilogram, but recalled the previous discussion "—the unit that we shall call a stone. The same rules we discussed regarding the span—how you can multiply and divide it by ten to get to other units of weight—apply here as well." Serran nodded.

"Consistency between volume and weight. I see the value of that. It is certainly going to make the conversion between the different amphora sizes of cities and foreign traders easier." Brutus tossed the sack several times and then rotated it by its string, smiling at Favius. The boy shared a knowing smile with him. At first, I'm just going to spread measurements. Newtonian physics… I need to think about the potential dangers of spreading that particular tidbit too early.

Apart from all the rotation and momentum stuff I shared with Brutus. Favius shook his head and turned back to the desk.

"As we've already used water to define weight, we may also use it to define temperature." Favius tilted his head. For now, I'll have to use mass and weight interchangeably, as much as that hurts. Not that that has hindered the old imperial system from just that. He took out a piece of ice and gestured to the steam over the pot.

"Water that has just frozen is at zero degrees." He paused, considering the word, but in this case, degrees seemed the best choice.

"It is a somewhat arbitrary scale, but it's mathematically convenient if we divide the interval between ice and steam into one hundred steps." Everyone nodded, but Favius himself shook his head. Might seem as if I'm just reinventing the units used on Earth, but I have no idea about air pressure and gravity here. So what I'm calling one hundred degrees is the boiling point on this world. But it could be anything between eighty and one hundred-something degrees back on Earth.

Urania gestured around at the experiments.

"I can see how the temperature system is practical. Anyone will be able to reinvent it from instructions. But the stone and the span depend on a length that you've decided upon arbitrarily." Favius nodded, but it was Brutus who replied.

"There is nothing that is consistently the same length all over the world. But—" he gestured behind him at the stack of wax tablets "—your brother is writing a book."

"So he's just going to spread a standard length in the book?" Favius nodded.

"With instructions to any copyist to copy the exact length."

"Sometimes," he smiled at Urania, "an idea must take precedence over the physical realities." While the others discussed the current lessons, he walked back to his props. Preparing for the last unit of measurement, he took out a string and a weight, bound them together, and began swinging the construct. He smiled.

"Lastly, let us talk about how one can use the span to define the central unit of time."