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Our federal income tax law defines the tax y to be paid in terms of the income x; it does so in a clumsy enough way by pasting several linear functions together, each valid in another interval or bracket of income. An archeologist who, five thousand years from now, shall unearth some of our income tax returns together with relics of engineering works and mathematical books, will probably date them a couple of centuries earlier, certainly before Galileo and Vieta.

The Mathematical Way of Thinking, an address given at the Bicentennial Conference at the University of Pennsylvania, 1940.

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# A Disquisition on the Square Root of Three

The classical Greek ladder for $$\sqrt{3}$$ to six-place accuracy begins like this:
 $$1$$ $$1$$ $$\frac{1}{1}=1.000000$$ $$1$$ $$2$$ $$\frac{2}{1}=2.000000$$ $$3$$ $$5$$ $$\frac{5}{3}\approx 1.666667$$ $$4$$ $$7$$ $$\frac{7}{4}=1.750000$$ $$11$$ $$19$$ $$\frac{19}{11}\approx 1.272727$$ where each rung $$\langle a\quad b\rangle$$ is $$15$$ $$26$$ $$\frac{26}{15}\approx 1.733333$$ followed by $$\langle a+b\quad 3a+b\rangle,$$ $$41$$ $$71$$ $$\frac{71}{41}\approx 1.731707$$ written in reduced form, $$56$$ $$97$$ $$\frac{97}{56}\approx 1.732143$$ with $$\sqrt{3}$$ approximated by $$\frac{b}{a}.$$ $$153$$ $$265$$ $$\frac{265}{153}\approx 1.732026$$ $$209$$ $$362$$ $$\frac{362}{209}\approx 1.732057$$ $$571$$ $$989$$ $$\frac{989}{571}\approx 1.732049$$ $$780$$ $$1351$$ $$\frac{1351}{780}\approx 1.732051$$
While the ladder could begin with any pair of nonnegative integers, not both zero, the rung $$\left\langle 1\quad 1\right\rangle$$ was used here because it yields the “classical” Greek ladder. The ladder stops where it did because that's where it yields the six-place accuracy that was presented at the outset of this paper. The seven-place denominator of $$1000000$$ has been beaten by the three-place $$780$$ – quite an improvement.