Counting the Little Animals

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

More little animals in a drop of water than people on the face of the Earth?

It was an extraordinary claim, and few 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):

... 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 there has been no greater pleasure in my eye as these sights of so many thousands of living creatures in a small drop of water, moving through each other, each special creature having its special motion.

The claims

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

  • there was a world no human had ever seen that was composed of whole societies of intricately formed little animals
  • the different kinds of animals were very, very tiny
  • there were millions, even billions of the little animals in a glassful of water or the guts of an animal that humans could see, all animals, including humans

The trick, of course, was that Leeuwenhoek's friends and colleagues could see this strange world if they looked through only this little instrument, through only this little lens. Even then, they had to hold the lens very close to their eye in just the right way. After that, they had to be very patient. They had to learn to see. And only one person had this special little lens the equal of which no one had every made before, anywhere in the world, except this ex-haberdasher and minor city official in Delft.

Leeuwenhoek's friends and colleagues had some reasonable questions. Were the little animals really there? Was it a magic lens?

"Figments of the imagination"

In Letter 33 of November 12, 1680 (AB 65), Leeuwenhoek, the recently elected fellow of the Royal Society, wrote to Robert Hooke, by then its co-secretary and curator of experiments (Dutch from Ondervindingen, 1684, p. 28, my translation):

... 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 who 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. He had questions that he would spend the next four decades exploring: what were the little animals for? if God created everything, what was their purpose? If not a purpose, then what was their function?

However, there were two questions that Leeuwenhoek could answer right away. Having trained himself to be a surveyor and wine guager, he knew enough arithmetic to answer both.

  • How little were these animals?
  • How many of them were swimming around?

The numbers did not have to be "inconceivable" for others to doubt. Thirty years later, in 1710, the 78-year-old, world-renowned Leeuwenhoek was kind enough to spend some time with Zacharias von Uffenbach (left), 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.

Indeed, Leeuwenhoek could 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 micrometry was well developed, before standards of measurement for small things. (The term "micrometry" entered the language in the 1840's.) Such standards as inches and miles had long been used by surveyors for large things. Birch's History recounted the November 8, 1677, meeting of the Royal Society at which, for the second week in a row, Hooke was unable to replicate Leeuwenhoek's results. Birch continued 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 Leeuwenhoek's Letter 19 of March 23, 1677 (AB 31) to Henry Oldenburg, 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". (For details, see Hooke's Three Tries under Learn more below.)

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

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 16, 1683, he wrote to Anthonie Heinsius:

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 never 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 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 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 vessels; It was very delightfull to me, to contemplate how every one of them, spread it self into several 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 equal the axe of a [grain of] sand: let then a million of them be equal to its solid content.

There are blood vessels 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 vessels of the brain.

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