## The Mathematics of Measurement: A Critical HistoryThe Mathematics of Measurement is a historical survey of the introduction of mathematics to physics and of the branches of mathematics that were developed specifically for handling measurements, including dimensional analysis, error analysis, and the calculus of quantities. |

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### Contents

THE EMERGENCE OF THE PRACTICAL CONCEPTION OF NUMBER | 9 |

Number words and recitation counting | 11 |

Written numerals | 13 |

The early working concept of number | 14 |

Abstract numbers | 16 |

Informal beginnings of conventional and formal numbers | 17 |

The interpretation of irrationals negatives and imaginary numbers | 18 |

EARLY SCIENTIFIC METROLOGY AND QUANTITATIVE LAWS | 22 |

Symbolic algebra and the calculus in kinematics | 110 |

Symbolic algebra in geometrical optics and astronomy | 112 |

The language of Newtons Principia | 113 |

MATHEMATICS IN EIGHTEENTHCENTURY EXACT SCIENCE | 117 |

Quantity of matter | 118 |

The concept of inertia | 122 |

Newtons laws of motion and his law of gravity | 124 |

Abbreviated ratio equations units and functional equations | 126 |

Early weights and measures | 23 |

Limitations of the technology of measurement | 26 |

Angle and time measurements in astronomy | 27 |

Quantification without metrology | 30 |

Quantitative laws in antiquity and in the middle ages | 31 |

EARLY APPLICATIONS OF GEOMETRY AND PROPORTION | 32 |

The character of greek theoretical geometry | 34 |

Euclidean geometry | 35 |

Abstract points lines and surfaces | 36 |

Applying idealized geometry to nature | 38 |

Configurations for physical quantities and relations | 39 |

Proportion | 41 |

Homogeneous and heterogeneous proportions | 42 |

Physical laws and compound ratios | 44 |

Proportion in Medieval exact science | 45 |

Ratio reduced to a single number | 46 |

Features of laws stated as proportions | 47 |

Applications of early trigonometry | 48 |

Trigonometry during the Renaissance | 50 |

THE PROGRESS OF QUANTIFICATION | 51 |

The basic reference quantities of measurement | 52 |

Applications of measurement and mathematics from the Renaissance | 56 |

Sensitivity to observational error | 57 |

Improvements in clockwork | 58 |

Balances to the eighteenth century | 59 |

The discovery of quantitative laws | 61 |

Galilean abstraction | 62 |

The quantification of electric charge tension and current | 63 |

ARITHMETICAL ALGEBRA AND SYMBOLIC GEOMETRY | 66 |

Verbal geometrical algebra | 68 |

Applications of arithmetical algebra to natural philosophy | 70 |

Analytical trigonometry | 74 |

The growing competence of number | 75 |

The promotion of heterogeneous ratios | 77 |

Opposition to algebra | 78 |

Justifications of algebra | 80 |

The infinitesimal calculus | 82 |

Algebra in physics in the seventeenth century | 86 |

The influence of geometry | 87 |

Comparative measurement | 89 |

Points of entry of algebra into technology and physics | 90 |

Thomas Harriot | 94 |

The concept of the moment of a force | 96 |

The algebraic representation of the moment of a force | 98 |

Varieties of force | 99 |

Components of a force | 100 |

Dynamical momentum | 105 |

Mechanical work and vis viva | 108 |

Varignons analytical kinematics and dynamics | 127 |

Algebra in scientific engineering | 131 |

Euler to Lagrange | 132 |

Celestial mechanics | 136 |

Algebra in magnetism and electricity | 138 |

The beginnings of the analytical theory of heat | 142 |

THE METRIC REFORM AND THE UNITS OF PHYSICS | 145 |

The centigrade scale of temperature | 147 |

The impact of the metric movement | 148 |

Absolute units and equations in the theory of heat | 149 |

Biots Traité de physique | 154 |

Absolute units in mechanics | 155 |

Work kinetic energy and thermodynamics | 159 |

Absolute units in dynamical astronomy | 160 |

ABSOLUTE QUANTIFICATION IN ELECTRICITY AND MAGNETISM | 163 |

Poisson and magnetic units | 164 |

Gauss and absolute units in magnetism | 166 |

Ampère and analytical electrodynamics | 168 |

Biot Savart Laplace and analytical electromagnetism | 170 |

Wilhelm Weber and absolute electric current | 171 |

Absolute units for electrical resistance and for electromotive force | 172 |

The concept of energy in electricity | 175 |

The proliferation of unit systems in electromagnetism | 177 |

The British Association standards | 179 |

The practical system of electrical units | 183 |

The Giorgi system and the Systèms International | 184 |

DIMENSIONAL ANALYSIS AND THE QUANTITY CALCULUS | 188 |

The dimensional analysis of Fourier | 191 |

Gauge invariance conversion factors dimensions | 193 |

The quantity calculus | 197 |

Webers base calculus | 200 |

Maxwells dimensional analysis | 202 |

Multiple ambiguity | 205 |

DIMENSIONAL EXPLORATION | 208 |

The optimum number of base quantities | 214 |

Dimensionless quantities | 215 |

The status of dimensional theory | 217 |

INTERPRETING THE ALGEBRA OF PHYSICS | 219 |

Formal algebra and formal arithmetic in modern physics | 221 |

Debates over the quantity calculus | 224 |

A symbolic mimicry of nature? | 229 |

Conclusion | 236 |

UNITS IN THE SYSTÈME INTERNATIONAL | 237 |

A coherent system of measurement | 238 |

Notes | 240 |

282 | |

308 | |

### Common terms and phrases

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