A more outrageous bet would be on which concubine the Emperor favored today—sometimes the Emperor himself was backing these bets, can you believe it?

Basically, other than the succession of the imperial throne, anything could become the subject of a wager.

Therefore.

Something quite miraculous happened:

In the Northern Song Dynasty, until before the year 1023, the name of each person who hit the jackpot annually was recorded.

In the Seventh Year of Yuan You, which is 1092 AD.

A certain Emperor of Europe in Bianjing won more than seven hundred guans of money, and the registered name was Han Gonglian.

Thus, some people in later generations of the mathematical community firmly believe that this Han Gonglian was indeed the mathematician, that they were the same person.

After all, the name Han Gonglian is fairly rare, and the odds of coincidence aren't high.

However, another group of people denied this for lack of accurate records.

Although officially it's for the sake of so-called rigor, in reality, Xu Yun was more inclined to think it was out of the anger of those who never win...

As the focus returned to the original subject.

After introductions were made, Xu Yun briefly reiterated the issue at hand.

After a while.

Several mathematicians, at least top-ranked ones of the current era, formally began their calculations.

Just look at this setup:

Jia Xian, Han Gonglian, and Liu Yi, these three mathematicians are recorded in history books.

The other three might not be famous, with little historical documentation.

But from simple exchanges, it was evident that they had significant mathematical talent, just overlooked because they weren't recognized as mathematicians.

It could even be said.

In this era, in the year 1100 AD.

These six were the strongest math troupe in the world!

A truly exclusive lineup.

From a future perspective.

The problem posed by Xu Yun wasn't particularly difficult:

It was a gateway problem in Fresnel approximation, technically a part of geometric optics, with multiple solution methods available.

The simplest one, of course, was the geometric optics diagram method.

However, as simple as it is, the diagram method provides very limited information; it can only detail the imaging properties of a lens with a known focal length.

It can't link the focal length with the lens's inherent properties, representing the simplest mathematical approach.

Going further, one could employ the fundamentals of geometric optics, namely Fermat's Principle.

With Fermat's Principle, one could determine the influence of lens shape and material on imaging under geometrical optics approximation, being mathematically more complex.

The third stage was the Huygens-Fresnel principle, the scalar wave diffraction theory of light.

Using this theory for image analysis could provide more information—such as the influence of lens aperture, which is why larger apertures are preferred for astronomical telescopes.

The most rigorous, naturally, would be Maxwell's equations, solving the wave equation under given boundary conditions.

But this last method is exceedingly complex.

To give the most straightforward example:

Everyone has seen the blackboards in university lecture halls, right?

If you used the fourth method, at least six of those blackboards would be needed—and even then, you might not derive an analytical solution.

So unless the earlier approximation theories aren't applicable, this method is generally avoided.

For this reason, Xu Yun intended to follow the third approach.

Although this third method is much more complex theoretically, evaluating a lens requires performing double integrals twice.

But firstly, it produces the best practical results, and in an era where the theoretical framework is severely lagging, the importance of practical results is paramount.

Secondly...

Old Jia, he was actually the real inventor of Yang Hui's Triangle.

Yang Hui's Triangle is one of the most appropriate ancient tools for solving integrals, so theoretically having Old Jia take that step is quite practical.

Of course.

The step referred to here isn't about inventing calculus, but a temporary conceptual application.

After all, a single Yang Hui's Triangle alone can't conjure calculus; it requires a certain mathematical foundation.

The most critical aspect is.

This mathematical foundation refers not just to individual accumulation but to the entire mathematical community and era's accumulation, a qualitative transformation.

Therefore, Xu Yun wasn't planning to rush into solving everything at once, especially as his relationship with Little Niu was quite good, being drinking buddies after all.

Returning to the original focus.

Discovering a new field, Old Jia, Han Gonglian, and the others showed considerable enthusiasm.

After all, in these times, teamwork for overcoming challenges was rare.

They were seen either discussing their thoughts or directly engaging in data measurement.

For example, in Liu Yi's hands, there appeared a primitive tool:

A ruler.

Speaking of rulers, one must first introduce another concept:

Angle.

In their long technological practice, the ancient Huaxia formed an abstract concept of angles quite early—this 'early' can be traced back three to four thousand years.

Unfortunately.

They did not develop precise angle measurement from this.

Note.

It's about precise measurement.

This situation persisted until the Ming Dynasty, when the concept of angles brought by the missionary Matteo Ricci broke this impasse:

The "Elements of Geometry", which he translated with Xu Guangqi, defined angles, their classifications, various situations, ways to represent angles, and how to compare angles.

Since then, the division of angles into 360 degrees formally entered the recognition of the Huaxia mathematicians.

Before that.

Huaxia only had two rough ways to measure angles.

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