In Pursuit of the Unknown Read Online Free PDF Page A

metric. And it is here that the tale comes full circle like the snake Orobouros and swallows its own tail, because the metric contains visible remnants of Pythagoras.
    Suppose, for example, that the manifold has three dimensions. Let the coordinates of a point be ( x , y , z ), and let ( x + d x , y + d y , z + d z ) be a nearby point, where the d means ‘a little bit of’. If the space is Euclidean, with zero curvature, the distance d s between these two points satisfies the equation
    d s 2 = d x 2 + d y 2 + d z 2
    and this is just Pythagoras, restricted to points that are close together. If the space is curved, with variable curvature from point to point, the analogous formula, the metric, looks like this:
    d s 2 = X d x 2 + Y d y 2 + Z d z 2 + 2 U d x d y + 2 V d x d z + 2 W d y d z
    Here X, Y, Z, U, V, W can depend on x, y and z. It may seem a bit of a mouthful, but like Pythagoras’s equation it involves sums of squares (and closely related products of two quantities like dx dy ) plus a few bells and whistles. The 2s occur because the formula can be packaged as a 3 × 3 table, or matrix:

    where X, Y, Z appear once, but U, V, W appear twice. The table is symmetric about its diagonal; in the language of differential geometry it is a symmetric tensor. Riemann’s generalisation of Gauss’s remarkable theorem is a formula for the curvature of the manifold, at any given point, in terms of this tensor. In the special case when Pythagoras applies, the curvature turns out to be zero. So the validity of Pythagoras’s equation is a test for the absence of curvature.
    Like Gauss’s formula, Riemann’s expression for curvature depends only on the manifold’s metric. An ant confined to the manifold could observe the metric by measuring tiny triangles and computing the curvature. Curvature is an intrinsic property of a manifold, independent of any surrounding space. Indeed, the metric already determines the geometry, so no surrounding space is required. In particular, we human ants can askwhat shape our vast and mysterious universe is, and hope to answer it by making observations that do not require us to step outside the universe. Which is just as well, because we can’t.
    Riemann found his formula by using forces to define geometry. Fifty years later, Einstein turned Riemann’s idea on its head, using geometry to define the force of gravity in his general theory of relativity, and inspiring new ideas about the shape of the universe: see Chapter 13 . It’s an astonishing progression of events. Pythagoras’s equation first came into being around 3500 years ago to measure a farmer’s land. Its extension to triangles without right angles, and triangles on a sphere, allowed us to map our continents and measure our planet. And a remarkable generalisation lets us measure the shape of the universe. Big ideas have small beginnings.

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Shortening the proceedings
    Logarithms

What does it tell us?
    How to multiply numbers by adding related numbers instead.
Why is that important?
    Addition is much simpler than multiplication.
What did it lead to?
    Efficient methods for calculating astronomical phenomena such as eclipses and planetary orbits. Quick ways to perform scientific calculations. The engineers’ faithful companion, the slide rule. Radioactive decay and the psychophysics of human perception.

N umbers originated in practical problems: recording property, such as animals or land, and financial transactions, such as taxation and keeping accounts. The earliest known number notation, aside from simple tallying marks like ||||, is found on the outside of clay envelopes. In 8000 BC Mesopotamian accountants kept records using small clay tokens of various shapes. The archaeologist Denise Schmandt-Besserat realised that each shape represented a basic commodity – a sphere for grain, an egg for a jar of oil, and so on. For security, the tokens were sealed in clay wrappings. But it