Wicksell and the Corpuscle Problem

April 9, 2012

Again, I strongly disagree with your statements and conclusions. You still fail to understand the importance of these works in relation to current stereology practice. Instead, you seem rather obsessed with parsing words. <Irrelevant demands redacted>

So you disagree. Fine. How do you explain the following huge mistakes.

 Wicksell was a Swedish physician interested in quantification of thyroid globules. He realized that counting the number of intersections per unit (Na) on random 2-D planes through a matrix containing 3-D objects provides a biased estimate of the number of 3-D objects per unit volume (Nv). The sources of this bias, which Wicksell referred to as the Corpuscle Problem, are the object’s size, shape, and orientation within the matrix. Though this relationship had been known for centuries, Wicksell was one of the first to make a thorough analysis. For more information on this topic, see my book published by the <removed>

  1. Wicksell was not a physician.
  2. Wicksell did not assist in the study of the thyroid

Wicksell was not interested in the number of profiles as representing the number of objects. A rather cursory read of the article would have told you this.

What I suspect is that you have not read ANY of the relevant material. Nothing. From the outset Wicksell approach the problem from a 3-dimensional perspective.

You claim that a bias, which Wicksell did not consider, or address, was named the corpuscle problem. You might want to take the time to learn what the corpuscle problem is all about. It’s not that complicated.

Here is a hint. In the 1925 paper Wicksell only considers balls. Thus the shape and orientation are moot claims on your part. As I have stated before Wicksell was interested in determining the number of balls regardless of the size distribution.

Please take the time to read Wicksell’s paper. The math is reasonable and interesting. You’ll learn who Wicksell was and what organ he investigated for a researcher at the university.


Comment on Buffon and response

April 9, 2012

The needle problem presented to the RAS by Buffon provides an empirical demonstration of stochastic geometry and probability theory. The probability of an intersection between the lines on the floor and the needle is directly related to (a) the length of the needles; and (b) the distance between the lines on the floor. If one only knows a or b, and does the simple experiment, i.e., determines the probability of an intersection based number of intersections (I) for a given number of tosses (N), the resulting probability (I/N) can be used to estimate the unknown quantity. The approach is unbiased because the more sampling that is done, hat is, the more tosses of the needle, the more closely the estimate converges on the true or expected value. Thus, Buffon’s Needle Problem as illustrated by the figure in question provides a prototypical example for the use of a probe with known parameters to estimate a unknown parameter by bringing the probe and object in random contact. Would the blogger like to suggest a better way to illustrate this concept? For more information on this topic, see<title withheld>,


The needle problem was one of at least problems solved correctly by Buffon. He provided a number of other calculations that were incorrect such as the 2 sets of orthogonal lines problem.


Buffon’s work assumes that a < b.


There is nothing about Buffon’s work that is unbiased since he did not sample, nor did he consider sampling. What distinguishes Buffon’s work is that it is the first probability problem solved for a non-discrete situation.

The idea of sampling with a needle comes from LaPlace. That comes well after Buffon’s death.

The image shows someone tossing needles. There is good reason that Buffon did toss coins to verify his derivation for Franc Carreau and he may have used bread sticks to test his needle computation, but he did this to verify the probabilities he derived. He was not sampling. He was not estimating. He did not employ a probe.

I believe the first known sampling using Buffon’s needle comes in 1901 with a faked math paper.

Comments and Response

April 9, 2012

You seem to be intentionally trying to muddle the situation. As a measure of area fraction (Aoil/ Arock), Delesse cut out and weighted outlines of oil phases traced onto paper (Aoil), and then did the same for the reference area (Arock). He then used a chemical method to determine volume of oil in the rock (Voil), and Archimedes method to measure the volume of the rock (Vrock). The Delesse principle simply show that for a random cut through a volume of rock that contains an oil phase, the area fraction is equivalent to the volume fraction (Aoil/ Arock = (Voil/ Vrock). Rosival and Glagolev used the same approach to show that a similar relationship with volume fraction holds for line fraction (Loil/ Lrock) and point fraction (Poil/ Prock), respectively. Would the blog writer care to suggest a better figure to illustrate this principle? For more information on this topic, see my book published by the <title withheld>

There are quite a few glaring mistakes here:
1.Delesse did not weigh paper.
2.Delesse was studying ores, not oil in rocks.
3.Delesse did not use the Archimedes method in his analysis.
4.It is not known if Delesse used chemical analyses to compare to his geometrical method
5.Delesse did not use random sections. In fact the argument was over what preferential orientation to use
6.He claimed that the area fraction was the same as the volume fraction, but he did not prove it
7.The spelling is Rosiwal
8.Rosiwal did not use a similar approach
9.Glagolev did not show anything about the method he used
10.Glagolev built a device to implement an existing method The method had been in use for years in Europe before Glagolev built his device.

As far as the figure being misleading, the Delesse principle was only applied to macroscopic analysis. The Rosiwal method was only applied to microscopic analysis. Glagolev built a device for a microscope. Glagolev’s device used a single point for a field of view.

I have seen the book and the book repeats these same mistakes. Time to correct these mistakes.

Delesse weighed metal, not paper.
Delesse did not use a reference volume.
Neither Delesse or Rosiwal or Glagolev ever proved anything.

There is not a single paper published using Delesse’s method. Not until Shand published was there a published paper using a geometrical method for modal analysis. That’s over 60 years after Delesse published.

Wicksell and scimitars

April 6, 2012

In an earlier form of the write up the following was stated.

The Corpuscle Problem arises from the fact that not all arbitrary-shaped 3-D objects have the same probability of being sampled by a 2-D sampling probe (knife blade).

This is not true either. The corpuscle problem did not address arbitrary shaped objects. It addressed balls. That’s all. The goal of the corpuscle problem was to count the number of ball shaped objects in tissue knowing only the profiles observed on slices. This is not a trivial problem. The balls can have a size distribution that is unknown.  The probability is not whether or not the blade samples an object, but how the blade samples a population of balls. Wicksell already knew the probability of sampling a ball. That was rather obvious to him. He was working on a much more complicated problem: how is the population sampled?

An experiment was done with potatoes and scimitars … actually scimitars were not available. Potatoes were sliced with a blade to produce a macroscopic data set that was employed in a corpuscle problem.

Since Wicksell’s 1926 paper showed that the problem was intractable for triaxial ellipsoids a number of different methods have been tried that give reasonable results for sections in opaque media. These include rocks, cement, asphalts, and other materials.

Wicksell and the Corpuscle problem

April 6, 2012
The following has been claimed about Wicksell’s paper published in 1925.
The work of S.D. Wicksell in the early 20th century (Wicksell, 1925) demonstrated the Corpuscle Problem — the number of profiles per unit area in 2-D observed on histological sections does not equal the number of objects per unit volume in 3-D; i.e., NA ≠ NV.

Is that what Wicksell published? No.

Wicksell was asked to assist in a problem where some anatomical shapes were seen as profiles on sections. What Wicksell solved was how to relate discs to the balls from which they were sliced. He related what was seen on slices to the original objects. The term ball is used here since that is the structure, which is not a sphere. Also, the profiles were observed to be discs, not circles.

In doing this he makes a number of important statements concerning the applicability of the mathematics to the problem at hand. He covers issues such as the size distribution of balls. He covers the issue of the position distribution.

In 1926 he reports that the solution he provides only works for a set of spheres. It does not work for ellipsoidal forms.

The corpuscle problem relates the distribution and numbers of profiles to the numbers of density of objects in the original material. The corpuscle problem is not useful in biology, but is used  in other fields such as geology, metallurgy, and studies of cements and roads.

To relate the corpuscle problem to the issue of the numerical density per unit area being not the same as the numerical density per unit volume simply shows a lack of understanding of Wicksell and his work.

Images that do not reflect the material

March 26, 2012

I was recently chastised by a <name removed on request> about my posts pointing out that the posted online history of stereology is muddled. It is as I have shown in the last 3 posts.

I also noticed that the artwork that goes with these historical posts is also quite off the mark.

Consider the image posted with the Cavalieri comments. It shows an object sectioned uniformly at a distance T. Cavalieri never did that. What is missing that is an essential part of the Cavalieri process is the height of the object shown next to an object of the same height.

Next to Delesse are 3 images. The images are labeled Delesse, Rosiwal, and Glagolev. The lines are straight. Rosiwal used paths and some of them were curved. The lines intersect. Rosiwal thought that his method required that the lines had to be far enough apart to avoid sampling any rock particle by more than 1 line. Finally Glagolev used a single point for his sampling. Glagolev did not invent point counting. He invented a machine to do point counting. In fact, his microscope point counting machine allowed counting rates close to what modern motorized stereological systems can offer.

The Buffon image shows an interesting situation. Buffon was prone to giving solutions without showing the derivation. This parquet floor is one of the problems where Buffon was wrong. His solution for the orthogonal lines is not correct. The correct solution was given by LaPlace in 1814, well after the death of Buffon. LaPlace did not name the origin of the problem so Buffon is not mentioned.

Good images support the story, not add more mud to the waters.

Continued exploration of the muddling of stereological history

March 26, 2012

Here is another part of the muddled history of stereology.

In 1847, the French mining engineer and geologist, Auguste Delesse, demonstrated that the expected valuefor[sic] volume fraction of an object varies in direct proportion to the observed area fraction of the object’s profile on a random section through the material containing the objects of interest. Today, by point counting, a derivative of the Delesse Principle, provides the basis for accurate and efficient estimates for object and regions volumes.

The first obvious muddled issue is the name. The first name of Delesse is Achille, not Auguste. The name Auguste is the first name of Rosiwal. Is this part of the confusion?

Delesse was  interested in what is known as modal analysis. He wanted to know what fraction of the rocks were of a particular type of mineral. He developed a technique so arduous to accomplish that he only performed it a few times and no one else did it.

Delesse did not use random sections. One of the big questions was the orientation that should be used. Those same issues would be discussed well into the 1950s by Chayes.

Is modern point counting a derivative of the Delesse principle. Not really. Point counting had an independent origin. The work had nothing to do with modal analysis. The original work was not even related to volumes. The mathematics might look similar and there is a mathematical relation, but the fact is that point counting was not derived from either Delesse’s work, or Rosiwal’s work.


March 23, 2012

Buffon is a nickname given to George Leclerc. As a young man he stupefied his family by declaring he wanted to be a scientist. It was a terrible blow to the family.

He saw only 1 way to really make it as a scientist and that was to become a member of the RAS, The Royal Society in France. As a young man he went and lived with a chemist. It was assumed that he would give a presentation on chemistry to gain acceptance into the RAS. Instead in 1733 he gave a presentation on gambling and probability. These sorts of problems were not new, but Buffon’s was. Unlike other solved problems which involved discrete items such as number of cards or faces to a die Buffon’s problem involved continuous quantities such as the position of a coin on a floor of tiles. He was not admitted into the RAS until 1734 in part due to the interest in verifying his material.

His presentation was recorded in the annals of the RAS but his original paper was lost. He wrote 2 other papers on his geometrical probability problems, but they too are lost.

So why do we see claims such as the following:

“Count George Leclerc Buffon presented the Needle Problem to the Royal Academy of Sciences in Paris, France.”

The work would have been permanently lost save for Buffon’s publishing mania. At one point he had a best seller in France on Natural History. It was the rage to own, not necessary read the many volumes. The appendices in one of the later volumes had a curious entry about these probability problems. That’s right. A book on natural history had an appendix in it having nothing to do with natural history. That was the norm for Buffon. He tossed in all sorts of odds and ends into his books. It is believed that the included paper was not the original content of his 1733 work, but more likely a combination of his earlier papers.

Others such as LaPlace commented on  his problems, but did not name them. In 1814 he published a correction to a mistake made by Buffon, but did not name Buffon.

About 100 years ago one of the pioneers of stochastic geometry found the geometric probabilities paper in the back of Buffon’s natural history book. The book was published in 1777 leading to the mistaken claim that Buffon presented this material in 1777.

After his 1736 paper on the subject Buffon never did anything in the field again. He had turned his attention to natural history. Buffon died in 1788 and left behind a legacy in natural history and certainly left an influence on one of proteges LaMarck.

Digits of precision and software

March 22, 2012

One of the basic notions that kids are taught in school is digits of precision. Even going back to elementary school kids are taught as early as grade 2 about approximate numbers. This is done using rulers that are marked out in units of inches or centimeters. The student has to pick the closest line on their ruler. Later on reading dials covers telling time. The distance and time issues are extended to measurements of volume and even surface area.

The idea is covered again and again especially as the arithmetic is advanced. Computations involving measurements are reduced to the significant digits.

Somewhere along the line this concept is lost as computers are used to perform the calculations. It is unfortunately all too easy to find journal articles which report numbers with 5 or 6 digits of precision. The numbers are copied out of whatever software is being used and pasted into the article being written.

Most stereological work is limited to 2 digits of precision. There might be 3 digits at times.

The measurements themselves are limited to 3 digits. If the measurements are taken from a computer screen, then the screen is often limited to about 1000 pixels. That limits the precision to 3 digits. Due to imaging issues the number of significant digits might be less than 3. Slices and sections cut with a microtome are limited to 3 digits at best. The thickness measurement is limited.

Stereological estimates are often done to a CE of .05. That limits the precision of the results to less than 2 digits.

Report values properly. Do not copy and paste from the software.

Cavalieri and the reason his name erroneously appears in stereology

March 20, 2012

Cavalieri was interested in mathematics and worked on a problem he called indivisibles.  The purpose of his work was to compute areas and volumes. He worked at a time before calculus had been invented. He was not the only one worked on this idea. It had been tossed around in one form or another for quite a while. In fact, Galileo had his own version of this idea. Robertval also worked on this idea.

The general idea is that an area can  be decomposed into line segments. The lines are 1 dimension less than the quantity being computed. A volume can be decomposed  into planar sections. Cavalieri’s theorem states that if 2 objects have the same height and the length of the line segments (areas of the planar sections) is the same for all positions from top to bottom, then the 2 areas (volumes) are the same.

The method here is a comparison. One object is compared to another. Some measure of the 2 objects is determined to be the same.

In 1902 some geologists examining crystals did something else yet claimed that the Cavalieri theorem applied. That is the first time that the name Cavalieri was associated with a volume estimation technique. Were this an actual application of the Cavalieri theorem then 2 objects would have been compared or an object would have been constructed that had the same volume as the crystals. Neither was done. Instead volumes were determined using a model-based methodology. This mistake was pointed out a long time ago.

Despite that initial error the name Cavalieri has stuck. Today we have the following statement posted all over the internet:

“the volume of non-classically shaped objects could be estimated in an accurate manner from the sum of areas on the cut surfaces of the objects”

This is wrong in several ways.

1. Cavalieri’s work dealt with the solutions to problems of shapes bounded by mathematical equations. He needed to be able to determine the areas and he used what was referred to as quadrature. This is like integration in calculus.

2. Cavalieri did  not estimate anything. His theorem and his work required the use of all sections.

3. Cavalieri’s work was not accurate. It was exact.

4. Cavalieri did not work with cut surface except to double check his work. One of the methods of integration was to cut shapes out of heavy paper stock and to weight them. This provides an independent check of the mathematics.

What is done today in stereology has nothing to do with Cavalieri. Cavalieri never estimated. Cavalieri never used point counting. Point counting is not developed until the 1920s.

This mistake or rather misnomer in stereology has led to people making reference to Cavalieri’s books. For a while it appeared to be fashionable to add the reference to Cavalieri. That fad appears to be over.