Where did That Number Come From?: Chronological Histories and Derivations of Numbers Important in Science

Where did That Number Come From?

Chronological Histories and Derivations of Numbers Important in Science

Brian Stedjee

Copyright © 2021 Brian Stedjee

All rights reserved

First Edition

NEWMAN SPRINGS PUBLISHING

320 Broad Street

Red Bank, NJ 07701

First originally published by Newman Springs Publishing 2021

ISBN 978-1-63692-834-0 (Paperback)

ISBN 978-1-63692-835-7 (Digital)

Printed in the United States of America

Table of Contents

Preface

The Length of a Year

The Size of the Earth

The Diameter of the Moon

Orbital Periods of the Planets

Relative Distances to the Planets

Pi

The Speed of Sound

The Mass of the Atmosphere

e: The base of the Natural Logarithm

The Speed of Light

Distances to the Sun and Planets

0 Degrees, 32 Degrees, and 212 Degrees Fahrenheit

The Mass of the Earth

The Mass of the Sun

Masses of Planets

Absolute Zero

Distancing to Nearby Stars

Atomic and Molecular Weights

The Gas Constant R

The Mass of the Moon

The Rydberg Constant

Planck's Constant, the Stefan-Boltzmann Constant, Wien's Displacement Constant, and Boltzmann's Constant

The Surface Temperature of the Sun

Avogadro's Number

Charge and Mass of an Electron

The Size of Atoms

Sizes of Nuclei and the Proton

Atomic Numbers

The Fine Structure Constant

The Age of the Earth

The Size and Age of the Universe

The Temperature of the Earth's Core

Bibliography

About the Author

Preface

Where did that number come from? Almost everyone has, at one time or another, seen a reported figure for the speed of light or the mass of the sun in an almanac, encyclopedia, or handbook. Those numbers are critical components of the foundations on which the pure and applied sciences, such as chemistry, physics, astronomy, earth sciences, biology, and engineering, are built. Yet, most people probably take them as matters of fact without ever considering their sources.

Over the centuries, some known and countless unknown researchers, driven by curiosity about and fascination with the Earth and what lies beyond it, used their creativity and imaginations to find the special numbers that are such critical components of today's complex technologies. The results of their work are seen everywhere in our lives today; yet most people have never heard their names.

This book presents, in chronological order, a series of brief historical sketches that, hopefully, will help explain to the educated layperson where those amazing numbers come from. It is possible that some people simply don't believe in the reasoning behind the discoveries and may ask, How could anybody figure something like that out anyway?

If a person wanted to answer that question—really wanted to find out where a particular number came from—he could simply call the local college and ask a teacher, couldn't he? He could call, but he might not get an answer. Many teachers simply do not know the origins of the numbers they memorize and solve problems with, nor do all teachers realize that the numbers change over time as more and more sensitive experiments are performed.

It was the realization of my own ignorance of the histories of many (if not most) of the numbers we accept, without questioning that prompted the research into the discovery and, in many cases, evolution of the numbers over time with generations of researchers basing their work on the results of experiments and analyses done by their predecessors.

For example, most of the values for masses and distances of objects are based on experimentally determined constants. The mass of the Earth calculation depends on the universal gravitational constant G. The mass of an electron calculation requires the elementary charge constant e. And to find the distance to Mars, you need the speed of light C.

The values for these constants are not the same in the newer handbooks as they are in the older ones because the experiments have improved. This is important because mass and distance values cannot be known to any greater accuracy than that of the constants they are based on.

Here are some examples of numbers whose accuracy has improved over time as researchers built on and learned from the work done by others, often centuries earlier:

The degree of accuracy of some of those constants has improved considerably over the years.

*This is the elementary charge constant, which should not be confused with e—the natural logarithm.

The Length of a Year

No one knows who first determined the length of a year. We do know that the people who built Stonehenge (ca. 2000 BC) obviously knew it, as well as a number of other astronomical facts. Because the Earth is tilted on its axis, the Sun appears to rise and set farther and farther to the north (in the northern hemisphere) until June 21 each year.

Figure 1-1

Then it begins rising and setting farther south. Given two set points to sight the sunrise, it's possible to determine June 21 each year.

Figure 1-2

The length of a year is 365.256 days.

The Size of the Earth

The Greeks knew that the Earth is spherical and one of them, Eratosthenes (276–195 BC), measured it. Eratosthenes noticed that on June 22 in Syene, Egypt (near modern day Aswan), a sundial cast no shadow. Another version of this story tells us that Eratosthenes noticed that sunlight struck the bottom of a vertical well at that time of day. At Alexandrea, Egypt (500 miles north of Syene), sunlight arrives 7 1/5 degrees from the vertical at noon on June 22. Eratosthenes reasoned that if the sun's rays were parallel and the Earth was spherical, then the ratio between 7 1/5 degrees and the number of degrees in a full circle (360) multiplied by the distance between Alexandria and Syene should give the circumference of Earth.

Figure 2-1

The circumference of Earth at the equator is, in fact, 24,902 miles or 40,077 kilometers.

The Diameter of the Moon

Sometime between 160 and 130 BC, the Greek astronomer, Hipparchus, successfully determined the size of the Moon and its distance from Earth. He based the calculation on the size of Earth's shadow on the Moon during a lunar eclipse. Because the sun is the same angular width as the Moon, Earth's shadow on the Moon is