A ball rolling down a parchment groove is not a falling body, and a stone sphere on a workbench is not the earth, so anyone arguing from one to the other owes an account of why the artificial case tells you about the natural one.
The previous lesson watched Galileo build a situation that does not occur in nature in order to measure something that does. That move is more important than any of his results, and it was being made at the same time by people with no interest in astronomy. This lesson takes three of them: a physician who built a model of the earth, a physician who did arithmetic on the heart, and a Lord Chancellor who tried to write down the rules. Between them they establish what an experiment has to be like before its result can be believed about anything outside the room it was done in.
A little earth on the bench
William Gilbert was president of the College of Physicians and physician to Elizabeth I, and he spent seventeen years and much of his own money on magnets. De Magnete, published in 1600, is the first book of its kind: a long, sober report of experiments, most of them his own, with the failures included.
The puzzle he attacked was the compass. A magnetised needle points roughly north, which had been known and used for four centuries and explained by an attraction to the pole star, or to a magnetic mountain in the far north, or to a point in the heavens. In 1576 the London instrument maker Robert Norman had added a second fact: a needle balanced so that it can turn in a vertical plane does not lie flat but dips downwards, in London by about , nearly three quarters of the way to vertical. No attraction from a distant point in the sky explains that.
Gilbert's answer was to build a model. He had a lodestone turned into a sphere, called it a terrella, a little earth, and mapped the behaviour of a small pivoted needle over its surface. The needle points along the meridian, towards the pole of the sphere. Placed at the equator of the terrella it lies flat; nearer a pole it dips, and at the pole it stands upright. Break the sphere in two and each piece is a complete magnet with two poles. Heat it past a certain point and it loses its virtue.
Every one of those behaviours matches what a compass does on the earth. Gilbert's conclusion is the flat sentence that the earth itself is a great magnet, that its directive power comes from its own body rather than from the sky, and that the compass is responding to the ground under the sailor's feet.
Example. For a uniformly magnetised sphere, the dip angle at latitude satisfies . What dip does that predict for London, and how does it compare with Norman's measurement?
London is at latitude , so and , giving
Norman measured , so the model is degrees low. That is a substantial error by the standards of Tycho's astronomy and a small one for a first physical theory of a planetary property. Turned round, the measured dip corresponds to a latitude of on a uniform sphere, five degrees north of where London actually is.
Now you. What should be concluded from the discrepancy?
Answer
Not that the model is wrong, and not that it is right, but that the earth is not a uniformly magnetised sphere. The field has irregular contributions from the crust and from the fluid interior, and the axis of the main dipole is tilted away from the axis of rotation, so any formula in latitude alone must fail by degrees somewhere. The decisive evidence came in 1634, when Henry Gellibrand compared measurements of magnetic declination at London: about degrees east in 1580, about 6 degrees east in 1622, and east in his own observation, a drift of roughly a tenth of a degree a year, always in the same direction. A property of a permanently magnetised stone cannot wander like that. The field went on swinging west to about by 1820 and has since come back through zero. Gilbert's terrella captures the dominant term of something that is real, and everything that departs from it is information about the interior of the earth rather than a defect in the experiment. Notice that Gellibrand's discovery was only possible because three people had recorded a number, with a date, in a form somebody else could compare against.
Where the model stops working
Gilbert did not stop where the evidence did. He argued that because a magnetic sphere has an axis it must rotate, treating the earth's daily rotation as a magnetic consequence, and he suggested that magnetic virtue holds the parts of the cosmos in their relations. Kepler picked that up and built his wrong physics of the planets on it. The same method, the same man, the same book: one conclusion that stands and one that does not, and nothing internal to the method separates them. What separates them is that the first was checked against measurements made by other people in other places, and the second was not checkable at all.
Half an hour of arithmetic
The received account of the blood, from Galen in the second century, was that blood is made in the liver out of digested food, is carried outwards through the veins to the parts of the body, and is there consumed. A small quantity seeps through invisible pores in the wall between the two sides of the heart, mixes with air from the lungs, and is distributed by the arteries. There is no circuit. Blood is manufactured, delivered and used up.
William Harvey, physician to James I and then Charles I, had studied at Padua under Fabricius, who had described the valves in the veins in 1603 without working out what they were for. In De Motu Cordis, published in 1628, Harvey killed the received account with a calculation that anybody could have done at any point in the preceding fourteen centuries.
Example. Take the left ventricle to hold about 2 ounces, suppose that only an eighth of its contents is expelled at each beat, and take a slow pulse of 1000 beats in half an hour. How much blood leaves the heart in that half hour, and what follows?
An eighth of 2 ounces is ounces, so the output is ounces, which at 16 ounces to the pound is
A man of 150 lb contains something like 11 lb of blood in total. So on deliberately pessimistic assumptions, the heart pushes out more than the body's entire content of blood in half an hour, and several times a man's weight in a day. It cannot all be manufactured from food, since nobody eats that; and it cannot all be consumed at the periphery, since there is nothing there to consume it. The only possibility left is that the same blood goes round again.
Now you. Redo the argument with modern figures: a stroke volume of 70 mL, a pulse of 72 a minute, and a blood volume of 5 litres.
Answer
The output is mL a minute, or litres, so the entire blood volume passes through the heart in almost exactly one minute. Over half an hour that is litres, thirty times the body's content. Harvey's conservative figures understate the true rate by a factor of about nine and the conclusion survives anyway, which is the mark of a well-built quantitative argument: it is designed so that the answer does not depend on the accuracy of the inputs. He could have been wrong about the ventricle's capacity by a factor of five in the direction that weakens his case and still have won it. Compare Kepler's eight arcminutes two lessons ago, which depended on the data being right to an arcminute. Both are legitimate, but an argument that is robust to its own uncertainties is much harder to resist.
The route the blood takes
The arithmetic says blood must return; it does not say by what route. Harvey supplied the route with experiments on the arm that anyone can repeat. Bind a limb tightly and the arteries are compressed, the limb below goes pale, and the veins do not fill. Slacken the ligature so that the deeper arteries are open but the surface veins are still pressed, and the limb swells and the veins below the binding stand out, because blood is arriving through the arteries and cannot get back. Press a finger on a raised vein and slide a second finger along it towards the heart: the vein empties and stays empty, and it refills only from the far end, never from the heart's side. The valves Fabricius had drawn all face the same way, towards the heart. Blood goes out by arteries, back by veins, and only in that direction.
The honest limit is that Harvey never saw the join. Nothing in his account explains how blood crosses from the smallest artery to the smallest vein, and he was reduced to speaking of porosities in the flesh. Marcello Malpighi saw the capillaries in a frog's lung with a microscope in 1661, four years after Harvey's death. The circulation was accepted for thirty years on the strength of an argument whose last step was missing, because the alternative had been arithmetically destroyed.
Bacon's programme, and the ledger on it
Francis Bacon, Lord Chancellor of England until his conviction for taking bribes in 1621, wrote no experimental treatise of any consequence and discovered nothing. His importance is that he tried to write down what the new practice should be before there was much of it.
The Novum Organum of 1620 sets out to replace Aristotle's logic. Its diagnosis is that the mind is not a clean mirror: Bacon's four idols name the systematic errors of human beings in general, of each person's own temperament and training, of the words we inherit, and of the philosophical systems we are taught. Its remedy is to gather instances in an organised way. For any property under investigation, compile a table of cases where it is present, a table of the nearest comparable cases where it is absent, and a table of cases where it varies in degree, and then work upwards to what is always present when the property is and always absent when it is not. Where two explanations survive, look for an instantia crucis, an instance at a fork in the road, a case where the two accounts predict opposite things.
Set against that, what Bacon actually got wrong is a long list. He rejected the Copernican system. He rejected Gilbert's magnetic earth. He had no use for mathematics, which was already the most productive tool in the field, and he thought the labour of collecting instances could be distributed among ordinary men of no special ability. The method as literally specified does not work: there is no theory-free collection of instances, since deciding which cases are the comparable ones is already a theoretical act, and no amount of listing would have produced the law of fall or the ellipse.
What survives is worth more than the method. The idea that knowledge of nature is a collective enterprise rather than a personal accomplishment, that it accumulates, that it requires organisation and funding and a division of labour, that negative instances must be sought rather than avoided, and that the point of it is use: all of that is Bacon, and the next lesson shows an entire institution built on his prospectus by men who had read him carefully and quietly dropped his logic.
What makes a contrived case count
Pull the three together and the conditions under which an artificial situation licenses a claim about nature start to be visible.
The contrivance has to isolate the factor of interest while leaving it recognisably itself. Galileo's incline slows fall without changing what fall is; Gilbert's terrella is a sphere of magnetic material like the one he claims the earth to be. The step is defensible exactly to the extent that the resemblance is argued rather than assumed, which is where Gilbert's rotation argument fails: nothing about the terrella spins.
The result has to be checkable against something you did not arrange. Harvey's arithmetic predicts consequences in the intact living arm, and those consequences can be produced in front of witnesses. Gilbert's dip law predicts a number a navigator can measure at sea.
And the phenomenon has to survive other hands. This is the condition none of the three could yet guarantee, because there was no reliable way for a report to travel, no agreed way to describe an apparatus so that someone else could build it, and no forum in which a contested result could be tried.
Example. Harvey's ligature demonstration and Gilbert's terrella both persuade, but not in the same way. What is the difference?
Harvey's ligature is a demonstration on the very thing in question: the arm is a real arm, the blood is real blood, and the audience watches the actual system misbehave when interfered with. No inference from model to original is needed, and the only question is whether the interference has been correctly described. Gilbert's terrella is an analogy, and it carries an unavoidable extra premise: that the earth is relevantly like this stone. That premise is supported by the number of independent behaviours the model reproduces, direction, dip, the two poles, the effect of breaking the stone, but it is a premise, and it is exactly where Gilbert went wrong when he pushed on to rotation. The general lesson is that a model earns credibility by the count of independent things it gets right, and loses it the moment it is used for a property the resemblance was never tested on.
Now you. Bacon's crucial instance is meant to decide between two surviving explanations at a stroke. Why is a single such instance rarely as decisive in practice as it sounds?
Answer
Because the two explanations never make their opposite predictions on their own. Each needs supporting assumptions about the apparatus, the materials and what else is going on, so a failed prediction can always be blamed on one of those instead. The telescope lesson had a live example: an observation that contradicted the received view could be attributed to a defect of the instrument, and there was no independent way to settle that. The way out is not logical but practical, and it is the whole business of the following lessons: build the instrument so that its behaviour is separately checkable, repeat the trial in other hands, and vary the supporting conditions until the result is the only thing that stays fixed. A crucial instance is real, and the vacuum provides the best one of the century, but it becomes crucial only after a great deal of work has closed off the escape routes.
That vacuum experiment is where this goes next. The question of whether nature abhors a vacuum had been argued from texts for four hundred years, and it was closed in four, by a glass tube full of mercury and a walk up a mountain.