Newton and the Birth of Fluxions

From falling apples to a question

Cambridge, 1665. A young Isaac Newton notices that a falling apple speeds up. He asks: what is the speed at a single instant?

He models the position as $s(t) = 4.9t^2$ and compares average speeds over shrinking intervals:

$$\frac{s(t+\Delta t)-s(t)}{\Delta t} = 9.8t + 4.9\Delta t.$$

Letting $\Delta t \to 0$ gives $9.8t$, the instantaneous speed.

Fluxions

Newton calls instantaneous rates fluxions and introduces dot notation:

• $\dot{s}$: fluxion of position (instantaneous velocity)
• $\dot{t}$: fluxion of time (equals $1$)

For the apple, $\dot{s} = 9.8t$ and differentiating again yields constant acceleration $9.8\ \text{m/s}^2$.

Geometric view

Plotting $s(t)$ as a parabola:

• Average speed = secant slope.
• Instantaneous speed = tangent slope.
• Shrinking the interval turns the secant into the tangent.

General definition

For any $y = f(x)$,

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}.$$

Fluxions thus capture motion, change, and tangents in one limit.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\dot{s}$	符号	s dot	位置对时间的瞬时变化率（速度）
$\Delta t$	数学符号	Delta t	有限时间增量

中英对照

中文术语	英文术语	音标	说明
流数	fluxion	/ˈflʌkʃən/	牛顿用于瞬时变化率的术语
切线斜率	tangent slope	/ˈtændʒənt sləʊp/	曲线在一点的瞬时变化率
割线斜率	secant slope	/ˈsiːkənt sləʊp/	连接两点的平均变化率

NextLeibniz and the Rise of DifferentialsLeibniz’s geometric route to the derivative and the differential notation we still use.