Chapter 91
Quantity, magnitude, measure and number as properties of things.
114 passages · 17 principal · Covers the 49 volumes of primary texts
Quantity is the property by which things are larger or smaller, more or fewer, heavier or lighter. It covers magnitude, which is continuous like a line or a stretch of time; number, which is discrete like a flock or a census; and measure, which joins the two by fixing a unit and counting how often it goes into a thing. In the Harvard Classics the idea appears in very different settings. Philosophers puzzle over what it means for one thing to exceed another. A critic asks why vast objects move us. Economists and surveyors argue over the standard bushel. Physiologists, chemists and physicists treat a measured number as decisive evidence. That range makes it a great idea for these books. Quantity seems at first the plainest of properties, yet the authors keep asking whether size belongs to things or to observers, whether every degree can be reckoned, and what is gained when a question is turned into a calculation.
The first question is what makes one thing greater than another. In the *Phaedo*, Socrates recalls that he once thought he understood why one man is taller than another “by a head,” or why ten exceeds eight by two1. He came to see that such answers explain nothing. A head cannot be the cause of greatness, since the same head would make the other man smaller. Excess must instead be explained by magnitude and number themselves2. Pascal turns to the formal side. He observes that number, space, motion and time can always be made greater or less than any given magnitude3. He also recalls Euclid’s definition of homogeneous magnitudes: two magnitudes are of the same kind when one, multiplied, can exceed the other4. Hume grants that the relations of quantity and number are intricate but traceable, through equality and inequality5. He draws back, however, from Pascal’s endless divisibility. To him, a real quantity infinitely less than any finite quantity is a prodigious and unsupportable notion6. The mathematician and the empiricist thus disagree at the very point where quantity meets the infinite, and the texts leave that disagreement standing.
Other writers treat quantity less as a puzzle than as a sign of order. Dante’s heavens are arranged so that each orb’s extent is proportioned to the virtue spread through its parts7, so that size there expresses spiritual power. Emerson takes the opposite tone of plain fact. Things act according to their quality and quantity, and a pound of water weighs the same in a tempest as in a pond8. Emerson’s quantity is stubborn and indifferent to circumstance. Berkeley’s is not. Philonous argues that the same object appears great to one perceiver and small to another, and from this he concludes that it cannot have one true size in itself9. The fable of the frog who swells to rival the ox makes comparison of size its whole subject10. It shows how readily magnitude becomes a matter of rivalry and point of view.
Some things seem to admit of more and less without admitting exact measure. Taine treats a people’s wants and faculties as quantities of degree, like pressure or weight, yet he concedes they cannot be measured exactly11. Darwin finds that the fertility of crosses runs by imperceptible degrees from none to perfect, and in some cases beyond12. There is a scale here, but no clean boundary. Cicero warns against the opposite mistake, which is to measure where measure does not belong. Friendship, he says, should not be kept like an account between debtor and creditor13. Between them these passages mark the outer edge of the idea, where degree remains real but counting would distort.
Burke shows how magnitude bears on feeling. Greatness of dimension, he holds, is a powerful cause of the sublime. Height affects us less than depth, both affect us more than length, and extreme littleness can move us in a similar way14. Greatness also has limits. Excessive length defeats itself through perspective, so a mean between the excessive and the short works best15. The effect of a great object depends on its uniformity as well as its size16. Burke offers a physical account of the effect, in which great dimensions work by an accumulation of impressions on the eye17. Beauty runs the other way. Size is the first property Burke examines in beautiful things, and he finds them comparatively small18. Great dimensions therefore draw an object away from mere beauty19. He also rejects the older view that beauty consists in proportion. Proportion, he argues, is only the measure of relative quantity, a matter of mensuration to which the mind is indifferent20. Yet he grants that magnitude is judged against a species. A thing that greatly exceeds the usual size of its kind is great rather than beautiful21. This half-concession echoes Berkeley’s relativity, though Burke places the standard in the species rather than in the single perceiver.
When quantity enters practice, the question becomes one of units. Herodotus measures Egypt in fathoms, furlongs, parasangs and schoines. He notes that peoples poor in land measure it by smaller units and peoples rich in land by larger ones, and he uses these units to reckon the length of the coast22. He makes Egyptian distances vivid by setting them beside the road from Athens to Pisa23. He gives the circuit and depth of a lake, the height of pyramids24, and the cubits of a monolith and its statues25. The tale in the *Thousand and One Nights* uses the most bodily unit of all: Sindbad paces round the dome and finds it fifty steps in circuit26. Such local and personal measures serve description well, but they fail commerce and science. Harrison complains that Elizabethan England has varying bushels, pounds and quarters, and he urges one standard for the whole realm27. Centuries later Kelvin presses the same complaint against the foot and the inch and argues for adopting the French metric system28.
Adam Smith shows how measure and value are bound together. Coin names first stood for definite weights of metal, the pennyweight being a fixed fraction of the pound29. A commodity that itself varies cannot measure others accurately, any more than a foot or a handful that kept changing could measure length30. Measures, coin included, become uncertain when they drift from their standard. Money-price must therefore be understood as a quantity of pure metal31, and a change in the ratio between gold and silver alters the quantities owed in payment32. Smith extends the reckoning to the whole economy. He argues that the money needed to circulate a given annual produce is a determinate quantity33. He guesses its proportion at between a fifth and a thirtieth of that produce, while admitting that it is impossible to determine34. He holds that paper currency is limited by the specie it replaces35. He also notes that people grasp a quantity of goods more readily than the abstract quantity of labour36. Like Taine, Smith reasons with quantities he cannot measure exactly. Unlike Taine, he presses on with figures anyway.
In the natural sciences of these volumes, measurement becomes the means of proof. Harvey’s argument for the circulation of the blood rests on simple arithmetic. He estimates what the heart expels at each beat and multiplies by the beats in half an hour. The total far exceeds all the blood in the body37 and more than food could supply38. He applies the same reckoning to the blood passing through the arms, limbs and neck39, and multiplies it by a thousand to show the scale40. Pasteur does the same at the scale of milligrams. He calculates the oxygen available in a flask41 and shows that less than a milligram remains in the boiled liquid42. He compares the weight of yeast with the sugar it ferments43 and measures the oxygen absorbed per gram of yeast. From this he computes how much oxygen yeast would need if it could not live without air44,45. He tabulates gases and weights46,47 and proves assimilation by recording the yeast’s gain in weight48. Holmes fights his medical battle with mortality figures per thousand births49. Faraday tallies the carbon burned each day by a man, a horse and the whole of London50.
Faraday’s lectures teach that matter keeps fixed ratios. Oxygen and hydrogen combine only by weight, eight to one51. Their proportions by volume differ from their proportions by weight52,53,54. Carbon, oxygen and lime combine in stated quantities as well55. He shows that equal weights may occupy very unequal bulks, so that a little platinum weighs as much as half a pint of water56,57. Water expands about seventeen hundred times on becoming steam58,59. He sets the gases side by side in tables of weight per pint and per cubic foot60,61,62. He weighs the air, a cubic foot of it coming to about an ounce and a fifth63, and the air of a room to over a ton64. His shadows show proportion at its simplest: at twice the distance the shadow is four times as large, and at three times it is nine65.
Helmholtz takes up the problem Harrison and Kelvin raised about standards, this time for force. He wants a precise measure of work in place of the vague feeling of exertion66. He finds it in weight multiplied by height of fall, the foot-pound, which applies to every machine67. Four pounds raised one inch equal one pound raised four inches68, and equal products show equal expenditure69. The conservation of force then becomes a claim about an unchanging quantity70. It can be tested because heat and work, measured independently, agree closely71, and because the equivalents found from different gases agree as well72,73. Kelvin defines mass by the motion equal forces produce in equal times74. He measures sound and light in the same spirit. He calculates the lengths and periods of organ pipes75, gives the wave-lengths and frequencies of red and violet light76, and notes the doubling at each octave77. With a grating he derives the wave-length of sodium light by simple proportion78, and the result agrees with exact measurement to within one per cent79,80.
The same habit reaches the earth and the sky, where it meets magnitudes beyond imagination. Lyell holds that errors in estimating time are fatal to any rational view of earlier ages81. He shows how repeated elevations of three feet might raise a mountain chain82. Darwin calculates that even the slow-breeding elephant would number nineteen million within about 750 years83. He adds strata thickness and fault displacements in feet and miles84,85, estimates areas by weighing cut-out paper maps86, and sounds coral reefs87,88. He measures beds of gravel89 and stems of kelp90, and he counts species island by island91. He even tests the reported precision of the bee’s cells against measurement92. Geikie measures river sediment to reckon how many years would wear down a continent93. Newcomb estimates the extent of space from the number of stars, much as one might estimate a field from its grains of wheat94. He admits his figures are rough95 but still gives average stellar speeds of about twenty miles a second96.
Vast numbers raise a new question: can they really be understood? Geikie sets 180,000 cubic miles of sediment beside a mountain ridge97. Helmholtz measures glaciers against Heidelberg and the Rhine valley98, and Darwin lays South American distances over Europe99. These comparisons assume that great magnitudes must be translated into familiar ones before the mind can grasp them. Kelvin denies this. A million million, he insists, is as intelligible as a small number100,101. Yet he too compares frequencies to make them graspable102, and he shows how a force a four-millionth of gravity would barely move a plummet103. His position answers Burke and Berkeley from the side of mathematics. For the senses, size remains relative and overwhelming. For calculation, it is simply number.
Numbers enter human affairs as argument. Burke counts inhabitants, voters, contributions and deputies to expose the inequality of France’s new representation104,105. Voltaire averages the lengths of English and French reigns106. Haskell estimates the Gettysburg dead from burial reports and a ratio of about one killed to five wounded107,108. Smith shows that capital lent can far exceed the money that conveys it109 and judges prosperity by comparing produce with consumption110. Pascal measures the stake in a wager against the chances of gain and loss111. Cervantes mocks the urge to count. Sancho makes the tally of goats ferried over a river essential to his tale, so that losing count ends the story112.
The questions these books open remain open. Philosophers still disagree, as Pascal and Hume do, over whether there can be a quantity smaller than any finite one. Berkeley’s relativity sits uneasily beside Faraday’s fixed ratios, though Burke’s species standard offers a partial bridge between them. Kelvin trusts that large numbers can be understood, while Geikie and Darwin keep reaching for comparisons. Taine and Cicero still ask whether every degree in life can rightly be reckoned. What the moderns add to Plato’s question about greater and less is a new kind of trust in standard units, measured equivalents and calculated proof. Their own admissions of rough estimates and things impossible to determine show that this trust has limits.
Introductory essay written by Claude Opus 5.5 from the outline and the notes on every passage below; quotations are checked against this edition.
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