Linz, November 1613 · Kepler, Nova Stereometria Doliorum Vinariorum, 1615

Imperceptible at First

In November 1613 a wine seller in Linz measured Johannes Kepler's new casks with a single rod pushed through the bunghole, whatever their shape. Kepler doubted it would work. In the book he printed in 1615 he showed why it nearly does: the Austrian cask sits on the top of a curve, and there a cooper's error costs almost nothing. Slide a cask along the rod's arc, then let an Austrian and a Rhenish cooper miss by the same amount.

What the seller did

Kepler had remarried that autumn and was stocking his house with wine. He tells the story in the dedication of the book, dated at Linz on 17 December 1613:

When some casks had been brought into the house and stored, four days later the seller came with a measuring rod, and with that one and the same rod he examined all the casks alike, without distinction, without regard to their shape, without reasoning or calculation. He put the bronze tip of the rod into the filling-hole of the full cask, slantwise, down to the heel of each of the two wooden discs (which in everyday speech we call the bottoms), and when this length from the top of the belly to the lowest point of each round board came out equal on both sides, he announced, from the number stamped on the rod at the place where this length ended, how many measures the cask held; and the price was reckoned by that number.

“Demissa enim acie virgæ aeneâ in orificium infusorium pleni cadi transversim ad calcem vtriusque orbis lignei, quos fundos vernaculo vsu dictitamus …”

Kepler, Nova Stereometria Doliorum Vinariorum (Linz, 1615), dedication, sig. A2r to A3r. Translation ours, from the 1615 print.

His objection was geometric, not personal. He wondered (mirari) how one slanting line through half a cask could be evidence of its capacity, and he doubted (dubitare) the method, because a very short cask with slightly wider heads could show the same length from the hole to the bottom of either head. Then he learned that this use of the rod was established here by public authority and that its gaugers were sworn to it, and he decided that working out whether it could be trusted was a fit first task for a new husband.

The bunghole is fixed and the rod is fixed, so the far corner of the head can only sit on the dashed arc. Every cask on that arc gets the same number from the rod. They do not hold the same wine. A cask as long as it is wide holds 93% of the best one; a cask twice as long as it is wide, 92%. But near the top of the curve the shape can wander a long way before anything shows: every shape from 1.25 to 1.60 times as long as wide holds within 1% of the best.

The top of the curve

Kepler modelled half the cask, from the bung to one head, as a cylinder whose diagonal is the rod. His fifth theorem of Part II finds the best one:

Of all cylinders having the same diagonal, the greatest and most capacious is the one whose base diameter is to its height in the ratio semidupla [√2 to 1] …

“Omnium Cylindrorum, diagonium eandem habentium, maximus & capacissimus est is, cujus diameter Basis, est ad altitudinem in proportione semidupla …”

Part II, Theorema V, sig. I3r. Translation ours.

The height there is the half-cask, so the whole cask should be √2 ≈ 1.41 times as long as it is wide. Then he looked at what the Austrian coopers actually did. Their rule, he says, was to take a third of the stave's length for the radius of the head: staves one and a half times the diameter. That is longer than 1.41, and he put the difference down to the stave ends, which run on past the grooves that hold the heads. Either way the cask lands on the flat part of the curve (at 1.5, the tool above says 99.8% of the best), and the corollary that follows is the sentence this page is named for:

Hence it is clear that it was by some good and geometrical genius that the Austrian coopers keep this rule for building a cask … For the other figures, ending at points close to G on this side and beyond, vary the capacity very little, because the capacity of the figure AGC is a maximum; and around the maximum, on both sides, the decrements are imperceptible at first.

“circa maximam verò utrinq; circumstantes decrementa habent initio insensilia.”

Part II, Theorema V, Corollarium II, sig. I4v. The print reads insensilia, not the insensibilia often quoted. Translation ours.

A little further down the same page he asks who would deny that Nature teaches geometry by instinct alone, since the coopers, “led only by their eyes and the beauty of the form” (solis oculis & speciei pulchritudine ducti), had learned to build the most capacious shape. He had wondered whether some excellent geometer had once taught them, and ruled it out: no trace of such a proof survived in the books, and the Rhineland and other wine country did not build this way; there, he says, they mostly make longer casks.

Kepler ran exactly this experiment in the German edition of the book (1616), with the same rod of 20. Every figure below is his, printed in chapter 76; the third column is recomputed on this page from (400 − h²) × h, and they agree to the unit. (The Deutsches Textarchiv's transcription reads the last one as 1625; the page image shows 2625, and his own difference of 203 from 2828 requires it.)

An Austrian cask half a part too long holds 3072 where the best holds 3080: short by one part in 385, which Kepler put as “von 10 Emmern kaum ein Achtering weniger” (hardly one Achtering less in ten Eimer). (He measured against 3080, the best rounded to the unit; against the unrounded 3079.2 the loss is one in 428, close to the instrument's reading at +0.45; the exact miss from 11.547 to 12 is 0.453.) One part too short, one in 280. Put the Austrian rod on a Rhenish cask of the usual shape and the loss is one in about 12: the rod says five and a half Eimer, and the cask holds five (“sechßthalbe Em̃er halten solte/ nur fünff Emmer hat”). Give the Rhenish cask its own rod and it is still exposed. Kepler shortened its half-length by a seventh of a part (14 instead of 14.14) and it gained one part in 101; lengthened to 15, it lost 203, which he called the 14th Eimer and “ein merckliches”, a noticeable amount. His conclusion for the Austrian cask: too little on both sides, “aber vmb ein vnkenliches vnnd schier gar nichts” (but by an unrecognisable, almost nothing).

Why the top is flat

At the top of a smooth hill, the ground is level. Walk a small step either way and you drop by an amount proportional to the square of the step, not the step itself. For the rod's arc the arithmetic is short: with the rod fixed, the fraction of the best volume that a cask of shape k holds is, near the top, about exp(−⅔ u²), where u is the natural log of k/√2. A cask 10% longer than the best loses 0.6%; 20% longer, 2.3% (the square rule estimates 0.6% and 2.2%). A Rhenish cask is not at a top. Its volume changes in proportion to the error itself, so in the second instrument a miss of half a part costs the Rhenish cask more than ten times what it costs the Austrian one.

Compare an error that is not protected this way. The rod's scale is cubic (content goes as the cube of the reading, as the next section shows), so a gauger who misreads the rod itself by 1% gets the content wrong by about 3%. The cask's shape, which the whole method ignores, matters far less than the one thing it measures.

This is the observation that would later become the rule that a derivative is zero at a maximum. Kepler states it about circles and figures, with no derivative anywhere; he returns to it twice more in the Latin book, for the belly (recalling that near the best shape the loss “non est observabilis inter initia”, is not observable at first, he adds that in this cask “nihil ferè … variat venter amplus an strictus”, a wide belly or a narrow one changes almost nothing) and for a cask whose two heads differ slightly (“insensibilis erit capacitatum in vtraque medietate differentia”, the difference will be imperceptible). Part III closes the argument about fraud: the Austrian rod is safe because the cask is built to this shape, and Kepler advises that the rule for building it be enforced by the magistrates, or else that the rod's authority be revoked by public decree for casks of abnormal shape. He also takes a swing at the careful computers of other methods, who, he says, strain out the gnat of the tiniest fractions and swallow the camel of errors (culicem excolantes fractionum minutissimarum, sed camelum errorum deglutientes).

The rod itself

A rod that reads volume from length has to be graduated in cubes. Kepler states the rule for casks of similar shape (Part II, Theorema XXVI: capacity goes as the cube of the length from the filling-hole to the bottom of a head) and describes the rod's layout: at the end of the first part the number 1, at 2 parts 8, at 3 parts 27, at 4 parts 64, at 5 parts 125, with unequal divisions between. The German edition calls it a rod “von vngleichen Cubischen Thailungen” calibrated to the Austrian Eimer, which the gaugers had to swear to use and no other.

For Linz he gives numbers. A rod reading of one and a half Linz feet means a small Eimer of 40 Achtering; three feet, 8 Eimer; six feet, 64. The Linz inch is divided into 19 puncten, 100 puncten on the rod mean 1 Achtering, and an Achtering is 614,110 cubic puncten (chapter 93). Put together, the rod says a cask holds 0.6141 times the cube of the reading. The best plain cylinder on the rod's arc holds 0.6046 times the cube, so the Linz rod credits the cask with 1.6% more than the best cylinder could hold. That is our arithmetic, not Kepler's. The most likely reading is that real Austrian casks have bellies and a bellied cask holds more than the cylinder between its heads, but we did not find where the rod's constant came from, and Kepler's modern editor warns against turning his printed scales into modern units, because the paper shrank in printing.

What the story does not say

The episode is a favourite of calculus textbooks and has grown in the retelling. Checked against the 1615 dedication:

• The dedication does not say it was wedding wine. The wedding, to Susanna Reuttinger, was at Eferding on 30 October 1613 (we have not established whether that date is Old or New Style). The dedication says only that in November, having brought home a new bride, he laid in wine for the household, and that the seller came to gauge it four days after delivery. MacTutor's biography of Kepler has him noticing the rod “at the wedding celebrations”; the dedication does not say so.
• Nobody was accused of cheating. Kepler doubted the method, not the seller. The one mention of fraud, in Part III, is about casks of abnormal shape, and his conclusion is that the Austrian rod is sound.
• He was not outraged. The Latin verbs are mirari and dubitare: he wondered, and he doubted.
• It was several casks (dolijs aliquot), not one barrel.
• He did not invent calculus over it. Part I of the book sums solids in the manner of Archimedes, cutting them into infinitely many thin pieces, and is a real ancestor of the integral calculus. The maximum in Part II is argued geometrically. The first draft, a few pages long, was finished with the dedication in December 1613 and then sat unpublished; all of Part II, including the maximum, was written in a six-week rewrite in the summer of 1615, which Kepler undertook partly to give Linz's first printer, Hans Plank, a first job (Hammer's account in the Gesammelte Werke).

Sources

1. Johannes Kepler, Nova Stereometria Doliorum Vinariorum, Linz: Plank, 1615. Scan of the EPFL copy: archive.org/details/chepfl-lipr-AXB124. Dedication sig. A2r to A4v; Theorema V sig. I3r; Corollaria I and II sig. I4v; Theorema XXVI sig. N1v; Part III sig. N3r to N3v.
2. Johannes Kepler, Außzug auß der Uralten Messe-Kunst Archimedis, Linz, 1616. Deutsches Textarchiv: deutschestextarchiv.de/book/show/kepler_messekunst_1616. Chapter 2 (the Austrian rod), chapter 76, pp. 64 to 65 (the worked casks), chapter 93, p. 99 (the Linz rod).
3. Franz Hammer (ed.), Johannes Kepler Gesammelte Werke, vol. IX, Mathematische Schriften, Munich: Beck, 1960, with the history of the text and a German translation of the dedication. Open access: publikationen.badw.de/de/002334745.
4. The wedding date and place: Biobibliographisches Handbuch der Kalendermacher, entry “Kepler, Johannes”, Universität Bremen; and the town of Eferding's 2013 anniversary, eferdingerland.at.
5. J. J. O'Connor and E. F. Robertson, “Johannes Kepler”, MacTutor History of Mathematics, mathshistory.st-andrews.ac.uk/Biographies/Kepler/.