Counting the Little Animals

How small? How many?
How van Leeuwenhoek calibrated and counted the multitude of tiny things he discovered.

It was an extraordinary claim, and no one believed it. Yet the man making the claim, Antony van Leeuwenhoek, trusted his eyes, amplified by his lenses. These lenses were so powerful that they let him break past the limits of the macroscopic world, the world of everyday things that we can see and touch. The first human to cross this threshold, he entered the microscopic world, a world that no one in his time, the 17th century, even suspected.

The more Leeuwenhoek looked, the more he was surprised and delighted. In his letter of October 9, 1676, he says that his sighting of thousands of animalcules in a drop of water ... (my translation):

... dit was voor mijn onder alle de wonderheden, die ik inde natuer heb ontdekt, het alderverwonderendste, en ik moet seggen, datter voor mij, tot nog toe geen grooter vermaek in mijn oog is geweest, als dese gesigten, van soo veel duisenden, van levende schepsels, in een kleijn droppeltje water te sien, door malcanderen bewegen, ijder bijsonder schepsels, sijn bijsonder bewegingh hebbende.

... this was for me among all the marvels, that I have discovered in nature, the most marvellous,  and I must say, that for me, up to now has been no greater pleasure in my eye, as these sights, of so many thousands, of living creatures, to see in a small drop of water, moving through each other, each special creature, having its special motion.

video of vorticella through Leeuwenhoek lens by candlelight

courtesy of Lesley Robertson, TU Delft

The claims

The claims by Galileo -- that it was the earth moving, not the sun -- violated the most obvious common sense, that the earth is standing still. Leeuwenhoek's claims also violated the most obvious common sense. He claimed that:

  • there was a world we could not see composed of whole societies of intricately formed little animals
  • the different kinds of animals were very, very tiny
  • there were millions, even billions of them in a glassful of water or the guts of an animal, all animals, including humans

The trick, of course, was that you could see this strange world only if you looked through this little instrument, this little lens. Even then, you had to hold the lens very close to your eye in just the right way. After that, you had to be very patient. And only one person had this special little lens.

Were the little animals really there? Was it a magic lens?

"Figments of the imagination"

On November 12, 1680, as a newly elected Fellow of the Royal Society, Leeuwenhoek wrote to Robert Hooke, by then its co-secretary and curator of experiments (Dutch from Ondervindingen 1684, p. 28, and my translation):

... na de maal het my te meermaal is te voren gekomen, en veel tegen sprekens lyde, dat ik omtrent de dierkens maar verdigtselen voor den dag bragt, en dat'er Heeren in Frankrijk syn, die derven seggen, dat het geen levende dierkens sijn die ik vertoon ...

so heb ik nogtans het contrarie aen verscheyde vermaarde Heeren vertoont, en derf seggen, dat sodanige Heeren, so verre nog niet gekomen syn, datse goede observatien konnen te weeg brengen.

... it has come to the fore many times, and I endure many speaking against me, that I brought forth the little animals as but figments of the imagination, and that there are gentlemen in France, that venture to say, that they are not living animals that I show ...

so I have nonetheless showed the contrary to various renowned gentlemen, and dare say, that such gentlemen, have yet to come so far, that they can make good observations.

So after he figured out how to make the lenses, then how to make the microscopes, and finally how to patiently see what he called "little animals" (diertgens) and other tiny structures, Leeuwenhoek had two other questions to answer.

  • How little were these animals? They had no names, though we now call them bacteria, protozoa, and cells.
  • How many of them were swimming around?

Having trained himself to be a surveyor, van Leeuwenhoek knew enough arithmetic to be able to answer both questions. The question he didn't answer, or even ask: what were the little animals for? if God created everything, what was their purpose? If not a purpose, then what was their function?

The numbers did not have to be "inconceivable" for others to doubt. In 1710, Leeuwenhoek was kind enough to spend some time with Zacharias von Uffenbach, a 27-year-old German jurist, who wrote (Dobell translation):

Mr. Leeuwenhoek showed us further the eye of a fly, which appeared very remarkable under the microscopium. ...

He is of opinion that a fly, according to his view, has more than a hundred, nay, more than a thousand, eyes; which is only one of this good man's extraordinary notions, which seem wont to have more of ingenuity than foundation.

You can see why Leeuwenhoek would be less than patient with visitors. He would show them something that they could see with their own eyes, and then they would call what they saw an "opinion", an "extraordinary notion".

How he measured and counted

Leeuwenhoek lived before micrometers, before standards of measurement for small things. Such standards like the inches and miles had long been used by surveyors for large things. Birch's History recounts the November 8, 1677, meeting of the Royal Society at which, for the second week in a row, Hooke was unable to replicate van Leeuwenhoek's results. Birch continues that Hooke:

then shewed a way of measuring the bigness of any object seen through the microscope, which was by opening the other eye, and seeing some other object with the left eye, whilst the right eye sees the object through the microscope:

and it was evident, that a pipe not bigger than a pig's bristle appeared a cylinder of about three inches diameter.

Given these constraints and crude solutions, Leeuwenhoek compared his tiny animals to common but still visible things: grains of sand, millet seeds, and hair. The claims from October 1676 that Hooke was trying to replicate had been followed by a letter earlier in 1677 titled in Philosophical Transactions as "Wherein Some Account is Given of the Manner of His Observing So Great a Number of Little Animals in Divers Sorts of Water".

How great a number? In a drop of water, Leeuwenhoek computed 2,730,000 (see image), and underestimating on purpose, claimed only "1000000 living Creatures in one drop of water". The details are discussed on the "No Longer Any Doubt" page.

A few years later, he was still patiently explaining his methods. His letter of November 12, 1680, continues:

As they'll say 'tis not credible that so great a many of these little animalcules can be comprehended in the compass of a sand-grain, ... I have figured out their proportions thus, in order to exhibit them yet more clearly to the eye:

He enclosed this drawing with his letter and then explained it.

Let me suppose, for example, that I see a sand-grain but as big as the spherical body ABGC and that I see, besides, a little animal as big as D, swimming, or running on the sand-grain; and measuring it by my eye, I judge the axis of the little animal D to be the twelfth part of the axis of the supposed sand-grain AG; consequently, according to the ordinary rules, the volume of the sphere ABGC is 1728 times greater than the volume of D.

He then compared objects of progressively smaller size.

Now suppose I see, among the rest, a second sort of little animals, which I likewise measure by my eye ...; and I judge its axis to be the fifth part, though I shall here allow it to be but the fourth part (as Fig E), of the axis of the first animacule D; and so, consequently, the volume of Fig D is 64 times greater than the volume of Fig E.

This last number, multiplied by the first number 1728 comes then to 110,592, the number of the little animals like Fig E, which are as big (supposing their bodies to be round) as the sphere ABGC.

His comparison progressed one more level.

But now I perceived a third sort of little animalcule, like the point F, whereof I judge the axis to be only a tenth part of that of the supposed animalcule E; wherefore 1000 animalcules such as F are as big as one animalcule like E.

This number, multiplied by the one foregoing, then makes more than 110 million little animals [like F] as big as a sand-grain.

Using simple yet inexorable arithmetic, Leeuwenhoek calculated that a drop of water could easily contain millions of moving, eating, reproducing animals, far more tiny animalcules than there were people in the Dutch Republic. A glass of water had more animalcules than there were people on Earth.

On September 17, 1683, he wrote:

I have had several gentlewomen in my house, who were keen on seeing the little eels in vinegar: but some of 'em were so disgusted at the spectacle, that they vowed they'd ne'er use vinegar again.

But what if one should tell such people in future that there are more animals living in the scum on the teeth in a man's mouth, than there are men in a whole kingdom? ... All the people living in our United Netherlands are not as many as the living animals that I carry in my own mouth this very day.

Scientist or conjurer?

This claim defied common sense as much as Galileo's claims that the earth moved and, relative to it, the sun stayed still.

Unfortunately, Galileo's claims challenged Church doctrine as well as a literal reading of the Bible. Galileo's confrontation with authority got him ex-communicated. The equally revolutionary claims of van Leeuwenhoek challenged only academic doctrine. His claims brought him honors and still made him, like Galileo, one of the most famous scientists of his time.

But was he a scientist? Or a conjurer?

In 1677, having just begun his observations, Leeuwenhoek wrote:

I'm well aware that these my writings will not be accepted by some, as they judge it to be impossible to make such discoveries: but I do not bother about such contradictions. Among the ignorant, they're still saying about me that I'm a conjuror, and that I show people what does not exist: but they're to be forgiven, they know no better. I well know there are whole Universities that won't believe there are living creatures in the male seed: but such things don't worry me, I know I'm in the right.

Early in his career, van Leeuwenhoek had to calculate these specific numbers to satisfy other scientists. Later, having established himself and become trusted, he used words like menigte, multitude, as an indeterminate number.

But as late as July 1684, in a letter addressed to Francis Aston, secretary of the Royal Society, he was still working through his calculations so that others could understand that his huge numbers were more than his imagination. In this case, he was examining the brains of a sheep and the tiny vessels that it contained.

While I viewed the incomprehensible number, of these small Vessells; It was very delightfull to me, to contemplate how every one of them, spread it self into severall branches. ...

For the better representing the inconceivable thinness of the blood Vessells, I made the following Calculations. viz. 100 red Globules lyeing side by side, do not equall the Axe of a [grain of] Sand: let then a Million of them be equall to its solid content.

There are blood Vessells in the brain, which I judge 1/64 part of a blood Globule would be too big to pass thro'; so that the Diameter of the Vessell, is to that of the Globule, as 1 to 4, and if a coarse [grain of] Sand be divided into 64 Millions of parts, 1 of the parts (if it be stiff and unplyable,) will not pass thro' one of the smallest Vessells of the brain.

While a grain of sand would not satisfy today's scientists as a standard unit of measurement, it was the best van Leeuwenhoek had. It was enough to make his point about the relative size of the creatures he observed.