Outside the house.
Watching Little Ox hurry back inside, Xu Yun vaguely realized what was going on and quickly followed him.
"Bang—"
As soon as he entered, Xu Yun heard the sound of something heavy striking.
He looked over and saw Little Ox standing beside the desk with a frustrated expression, his left hand clenched into a fist, his knuckles pressing hard against the tabletop.
Clearly, Little Ox had just thrown a deliberate punch at the desk.
Xu Yun walked over and asked,
"Mr. Newton, what's the matter?"
"You wouldn't understand."
Little Ox waved him off irritably, but a few seconds later, something seemed to occur to him.
"Fat Fish, do you—or that Sir Han Li—know anything about mathematical tools?"
Xu Yun looked at him again with feigned confusion and asked,
"Mathematical tools? Do you mean a ruler? Or a compass?"
At those words, Little Ox's heart sank halfway. But he'd already started explaining, so he couldn't just stop there. He continued,
"Not physical tools, but a theory for calculating rates of change.
"Take the dispersion phenomenon we just discussed. It's a kind of instantaneous rate of change, and it might even involve particles too small for the naked eye to see.
"To calculate this rate of change, we need another tool that can continuously add things together, to calculate the product of the refraction angles.
"For example, multiplying n copies of a+b means taking the product of a or b from each a+b. Like (a+b)^2=a^2+2ab+b^2. Never mind, I doubt you'd understand anyway."
Xu Yun gave him a knowing look and said,
"I understand. Yang Hui's Triangle."
"Right, so we'll get ready to go see Uncle William in a bit. Wait—what did you say?"
Little Ox had been following his own train of thought, but when he heard Xu Yun's words, he froze. Then he abruptly looked up and stared at him.
"'Yang Hui's Triangle'? What's that?"
Xu Yun thought for a moment, then held out his hand to Little Ox.
"Could you pass me the pen, Mr. Newton?"
If this had happened a day earlier, when Little Ox had first met Xu Yun, he would have refused his request outright.
He might even have added, "Who do you think you are?"
But after they had worked through the dispersion phenomenon not long ago, Little Ox had begun to feel a faint interest in and respect for Xu Yun—or rather, for the Sir Han Li behind him.
Otherwise, he wouldn't have gone to such lengths to explain things to Xu Yun just now.
So, faced with Xu Yun's request, Little Ox did something rare: he handed him the pen.
Xu Yun took it and quickly drew a diagram on the paper:
1
11
121
1331 (Please ignore the ellipsis; without it, the site automatically indents the lines. This is driving me crazy.)
Xu Yun drew eight rows. The two numbers at the ends of each row were both 1, forming an equilateral triangle.
Anyone familiar with this diagram would recognize it as the famous Yang Hui's Triangle, also called Pascal's Triangle. The latter name was more widely accepted in the international mathematical community.
But in fact, Yang Hui discovered the triangle more than four hundred years before Pascal.
Yang Hui lived during the Southern Song dynasty. In 1261, he preserved a precious diagram in Detailed Analysis of the Nine Chapters on the Mathematical Art: the "Method of Finding Roots" diagram. It was the oldest surviving triangular arrangement of numbers for which there was reliable evidence.
However, for certain well-known reasons, Pascal's Triangle became much more widely circulated. Some people didn't even recognize the name Yang Hui's Triangle.
So despite Yang Hui's original written record, the mathematical triangle was still called Pascal's Triangle.
But there was something worth noting.
Pascal studied this triangular diagram in 1654, and officially published it in late November 1665. From the present day, that was—
a whole month away!
That was why Xu Yun had started with the dispersion phenomenon.
Dispersion was a classic differential model, even more so than Universal Gravitation. Both the angle of deflection and its "seven-in-one" appearance pointed directly to the tools of calculus.
The concept of 1/7 was even directly linked to expressing exponents as fractions.
If Little Ox, after encountering the dispersion phenomenon, didn't think of his current quandary—the "theory of fluxions"—then he might as well wash up and go to bed.
Little Ox encountered the dispersion phenomenon—Little Ox grew curious—Little Ox measured the data—Little Ox thought of the theory of fluxions—Xu Yun introduced Yang Hui's Triangle.
It was a perfect logical progression, a trap leading from physics to mathematics.
As for why Xu Yun had drawn the diagram, the reason was simple:
Yang Hui's Triangle was a thorn in the heart of every mathematician!
Our ancestors had invented Yang Hui's Triangle first, and there was conclusive evidence to prove it. Why should it be credited to someone else just because of the humiliations of modern history?
He couldn't do anything about the original timeline, nor did he have the ability to change it. But in this time and space, Xu Yun wouldn't let Yang Hui's Triangle share its name with Pascal's!
With Old Master Newton as his witness, Yang Hui's Triangle would be Yang Hui's Triangle.
A term that belonged to China alone!
Xu Yun silently let out a breath, then continued drawing a few lines on the paper.
"Mr. Newton, look. The two sloping sides of this triangle are made up of the number 1, and every other number is the sum of the two numbers above it.
"As the diagram shows, any number C(n,r) is equal to the sum of the two numbers above it, C(n-1,r-1) and C(n-1,r)."
As he spoke, Xu Yun wrote down a formula:
C(n,r)=C(n-1,r-1)+C(n-1,r)(n=1,2,3,··)
And
(a + b)^2= a^2 + 2ab + b^2
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 6ab^3 + b^4
(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
When Xu Yun wrote out the cubic expansion, Little Ox's expression gradually turned serious.
By the time Xu Yun had written the sixth-power expansion, Little Ox could no longer sit still.
He abruptly stood up, snatched the pen from Xu Yun, and started writing himself.
(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + a^6!
It was clear.
The nth row of Yang Hui's Triangle had n terms, and the sum of those terms was 2 to the power of n-1. The coefficients in the expansion of (a+b) to the nth power corresponded, in order, to the terms in the (n+1)th row of Yang Hui's Triangle!
Although this expansion was child's play for Little Ox—indeed, it was a basic operation in the binomial expansion—
this was the first time anyone had so clearly represented the coefficients of a root expression with a diagram!
More importantly, the mth number in the nth row of Yang Hui's Triangle could be expressed as C(n-1,m-1), the number of ways to choose m-1 elements from n-1 distinct elements.
That would be an enormous help with the subsequent binomial derivation Little Ox was working on!
But—
Little Ox's brows gradually furrowed again.
The appearance of Yang Hui's Triangle had opened up a new line of thought, but it did little to help with the problem he was stuck on: expanding (P+PQ)m/n.
That was because Yang Hui's Triangle dealt with coefficients, while the problem giving Little Ox a headache involved exponents.
Little Ox was like an experienced cyclist.
He'd rounded a bend in the mountain road and suddenly seen a wide-open stretch a hundred meters ahead, with breathtaking scenery, but a massive pile of fallen rocks just over ten meters in front of him blocked the way.
Just as Little Ox was hesitating, Xu Yun slowly added,
"By the way, Mr. Newton, Sir Han Li also studied Yang Hui's Triangle.
"Later, he discovered that the exponent in a binomial expression didn't necessarily have to be an integer. Fractions—and even negative numbers—seemed to work too."
"He didn't explain how to prove the case for negative numbers, but he did leave a proof for fractional exponents."
"He called it—"
"Han Li Expansion!"
Before you continue