by Colin Pask
The concept of circulation, rather than continuous blood production by the liver, is central to Harvey's theory, and he needs all available evidence to convince people to give up the ancient and respected description of this vital bodily function. It is here that his calculation is introduced. Again, we can follow Harvey's own words from his chapter 9:
Let us assume, either arbitrarily or from experiment, the quantity of blood which the left ventricle of the heart will contain when distended to be, say two ounces, three ounces, one ounce and a half—in the dead body I have found it to hold upwards of two ounces. Let us assume further how much less the heart will hold in the contracted than in the dilated state; and how much blood it will project into the aorta upon each contraction—and all the world allows that with the systole something is always projected, a necessary consequence demonstrated in the third chapter, and obvious from the structure of the valves; and let us suppose as approaching the truth that the fourth, or fifth, or sixth, or even but the eighth part of its charge is thrown into the artery at each contraction; this would give either half an ounce, or three drachms, or one drachm of blood as propelled by the heart at each pulse into the aorta; which quantity by reason of the valves at the root of the vessel, can by no means return into the ventricle. Now in the course of half an hour, the heart will have made more than one thousand beats, in some as many as two, three, and even four thousand. Multiplying the number of drachms propelled by the number of pulses, we will have either one thousand half ounces, or one thousand times three drachms, or a like quantity of blood, according to the amount which we assume as propelled with each stroke of the heart, sent from this organ into the artery, a larger quantity in every case than is contained in the whole body!4
The calculation is simple: the amount of blood flowing as sent out by the heart in a certain time is given by the amount at each contraction multiplied by the number of heartbeats in that time. The exact details are not important; all of the assumptions show that there is more blood in this flow than could possibly be made by the liver. Galen's revered theory is discredited, and a revolution has begun in medicine. Calculation 26, Harvey establishes blood circulation is almost trivially simple, but its impact means that it must be on my list of important calculations. It helped to overturn over a thousand years of medical practice and ushered in our present-day understanding of how animal bodies function.
8.1.2 The Scientific Setting
The scientific revolution relied on three important approaches in science: the use of experiments (rather than relying on old written accounts), the use of a mechanical description and discrete or atomic theories of matter, and the use of mathematics. The work of Galileo, Boyle, Newton, and their contemporaries is often taken as instigating the scientific revolution. It might seem that Harvey was the great pioneer in medical science and biology, however, the real story is more involved (see Gregory).
Certainly, Harvey says the following of “true philosophers”:
nor are they so narrow-minded as to imagine any of the arts or sciences transmitted to us by the ancients, in such a state of forwardness or completeness, that nothing is left for the ingenuity and industry of others.5
But Harvey does display great respect for those ancients, referring to the “divine Galen” and making many references to Aristotle. The calculation, which to us seems so conclusive, is a tiny part of Harvey's book. In the final paragraph, he mentions the many supporting points “in the course of dissections” that confirm his theory, rightly emphasizing the importance of the great range of experiments he conducted, but with no mention of calculations. One of Harvey's critics was the German professor Caspar Hoffman, and, according to Andrew Gregory,
Hoffman accused Harvey of “abandoning anatomy for logistics,” in making calculations about the heart and blood flow, and said that: “Truly, Harvey, you are pursuing the incalculable, the inexplicable, the unknowable.”6
This comment seems a little unfair. Harvey clearly did respect the ancients, but his work surely inspired those who saw that it was time to challenge and overthrow some of the old ideas and prejudices. The use of a simple but irrefutable calculation was a powerful factor in that challenge.
8.2 COLLECTING AND ANALYZING DATA ON LIFE AND DEATH
In the previous section, we saw how Harvey discovered the blood flow mechanism operating in all animals. Although the bodily function mechanisms might be the same in all cases, no two humans are exactly the same, and the variations are of great interest. Biology is based on samples of organisms, and collections of data allow us to identify general trends, study the range of variations, and draw some general conclusions. The work discussed in this section is some of the earliest of this type relating to human populations. Another example involving a spread of species will be given later.
We are all born and we all die. But how long we live and how we die vary enormously. The first in-depth study of these variations is often taken to be the work of John Graunt (1620–1674). Although he was a draper, Graunt developed an interest in the London Bills of Mortality and the information that might be extracted from them. The Bills of Mortality listed the number of people dying from certain causes over a given week and also gave information about christenings, funerals, and other events (an example is given by Lewin and De Valois). Graunt's very well-received small book, Natural and Political Observations Made upon the Bills of Mortality, was published in 1662, and it led Charles II to have Graunt made a member of the Royal Society—quite an achievement for a “haberdasher of small-wares.”7 Graunt tried to estimate both the population and its breakdown (males/females, married/single, and different age groups). Of course, one aim of the bills was to provide information about the occurrence of the plague and movements of people.
The data in the Bills of Mortality is interesting if you want to know how many people died from “Griping in the Guts” or how many people were “Murthered” (in fact, surprisingly few), but it was not easy to extract precise data on population trends. Some things called for an explanation; in his conclusion, Graunt said he wished to know “why the Burials in London exceed the Christenings, when the contrary is visible in the Country.”8 London was a city of comings and goings, and the population did not have the underlying stability required for demographic studies.
8.2.1 The Breslau Data Becomes Available
The registers of births and deaths were very detailed and carefully kept in the city of Breslau in Silesia. Evangelical pastor Caspar Neumann (1648–1715) used them to fight popular superstitions such as the idea that health depended on the phases of the moon. Neumann passed his data on to Gottfried Wilhelm von Leibniz, from whom it went to France and then on to London where its value was finally recognized. Most importantly, the data for the years 1687–1691 were relatively stable; the city of Breslau did not have the continuous influx of people that so confused the data for London. The person who recognized the true value of the Breslau data was Edmond Halley—yes, he of the comet and transit of Venus fame!
In1693, Halley published a groundbreaking paper, “An Estimate of the Degrees of the Mortality of Mankind, Drawn from curious Tables of the Births and Funerals at the City of Breslaw,” in the Royal Society's Philosophical Transactions. (Excerpts from both Graunt's book and Halley's paper are available in Newman's The World of Mathematics.) In this paper, Halley demonstrated how to organize data into a useful table, how to make immediate observations and deductions, and how to make use of this data in calculations. After suitably organizing the data (rounding the population aged under one to 1,000, for example) Halley produced the table shown in figure 8.1. This table shows how the population of 34,000 was distributed over the different age levels, both year by year and in groups of seven years.
Having organized the data into a suitable form, Halley could extract information from his table “whose uses are manifold.”9 His first use was to find “the proportion of men able to bear arms.” (Graunt tried a similar calculation for London.) After defining this group as me
n aged between 18 and 56, it was a simple matter of adding up the numbers in the table and assuming roughly half were males. His result was “about 9000, or 9/34, or somewhat more than a quarter of the Number of Souls, which may perhaps pass for a Rule for all other places.” Halley is making an important point: deductions using the Breslau data might be valid for other populations.
Figure 8.1. Halley's table of population distribution in Breslau. From Edmond Halley, “An Estimate of the Degrees of the Mortality of Mankind, Drawn from Curious Tables of the Births and Funerals at the City of Breslaw, with an Attempt to Ascertain the Price of Annuities upon Lives,” Philosophical Transactions 17 (1693).
Halley's second use was to show the “Vitality in all Ages,” by which he means the chances of someone at one age living to be another. For example, he shows that the chance of a person age 25 not dying before turning 26 is 560 to 7, or 80 to 1, because his tables show that the 567 persons alive at 25 have reduced to 560 at age 26. Similarly he shows that “it is 377 to 68, or 5½ to 1, that a man of 40 does live 7 years.” Similarly, in his third use, Halley shows how to calculate the odds of someone dying or life expectancy. In his example, he shows that “a man of 30 may reasonably expect to live between 27 and 28 years.”
8.2.2 The Mathematics of Insurance
Halley's first uses of his Breslau table simply calculate numbers revealing how the population develops and ages. In the next uses, Halley turns to how the results may be applied, and in doing so, he begins a branch of mathematics that has been of commercial importance ever since. Halley makes clear his intentions for his fourth use, which is worth quoting in full:
By what has been said, the Price of Insurance upon Lives ought to be regulated, and the difference is discovered between the price of ensuring the life of a man of 20 and 50, for example: it being 100 to 1 that a man of 20 dies not in a year, and but 38 to 1 for a man of 50 years of age.10
His fifth use begins: “On this depends the Valuation of Annuities,” and it is here that Halley establishes what is often called the cornerstone of actuarial science. (In fact the full title of Halley's paper continues from that given above with the words with an Attempt to ascertain the Price of Annuities upon Lives.) There has always been a desire and a need to arrange funds for a secure and comfortable life in old age. One way is to purchase an annuity, whereby a sum of money is paid by a person to a government or other organization so that later a steady income is guaranteed after some defined year (often the year of the person's retirement) and usually until death. It sounds like a dry and boring topic, but annuities go back to at least Roman times. (Their surprising history can be found in the works of Ciecka, Hald, Kopf, and Lewin and De Valois.)
The basic question is: How much should one pay for an annuity? Halley has been considering the chances of people living beyond any given age, and he now wants this to be used in answering that question:
for it is plain that the purchaser ought to pay for only such a part of the value of the Annuity, as he has chances that he is living; and this ought to be computed yearly, and the sum of all those yearly values being added together, will amount to the value of the annuity for the Life of the Person proposed.
There is also the point that the organization providing the annuity will invest the funds paid to generate the annuity at a rate of interest p, and this is compounded to add to the deposited amount. Thus the money available to the provider is actually the sum paid plus the interest to be earned on that sum. Obviously the person buying the annuity should be aware that the sum paid need not be as big as at first expected when those extra earned interest contributions are taken into account. Halley showed how to evaluate the cost of annuities when these points are properly considered. He used the example of interest at 6 percent, so p = 0.06.
Halley explains his method in prose rather than in mathematical notation making it hard to follow. It amounts to using the following formula (see Ciecka or Hald) for the cost of an annuity paying unit amount per year after year n:
The quantities Lj (used in the above equation with j equal to n and n + m) are given by the entries in Halley's table (figure 8.1) for those living in year j, and their ratios give the chances of surviving into future years. The upper limit on the sum means carry on until there are no survivors for that year. Unless a computer is used, “this will without doubt appear to be a most laborious calculation,” as Halley puts it. However, “after a not ordinary number of Arithmetical Operations” he produces a table of annuity values for five-year intervals for ages 1 to 70.
In uses six and seven, Halley considers the cases where more than one life is involved. This is a complicated problem, but he is still able to use his stated calculating principles.
Selling annuities was a way for governments to raise money, and this was happening in England during Halley's time. Clearly the government had no idea about the mathematics involved or how to properly cost an annuity; while they were instantly gaining money (to use for running the country), they were going to lose out when the time came to pay the annuities. Halley saw the opportunity; using the results in his table, he says, “this shows the great advantage of putting money into the present fund lately granted to their Majesties.” I am not sure when a government official finally caught up with the theory and realized they were running an overgenerous insurance scheme!
Edmond Halley may be best known for his comet, but his clever and innovative pioneering work in actuarial science makes calculation 27, Halley values annuities, a worthy addition to my list of important calculations. (Work immediately following on from Halley's is described by Hald, and Lewin and De Valois. The use of information from large databases and extensive surveys is now a standard part of the biological and social sciences; see Cohen for a gentle introduction.)
8.3 AN EARLY FUNDAMENTAL ADVANCE IN GENETICS
The nineteenth century saw the development of Mendelian genetics and Darwin's theory of evolution by natural selection. In the twentieth century, they came together to give our modern framework for biology, and the mathematical basis was built up by R. A. Fisher, J. B. S. Haldane, and Sewall Wright. The result was the subject of population genetics, and it is the derivation of a first significant result in this field that is the subject for this section. (An easily accessible and readable introduction is given by Samir Okasha.) The result incorporates Mendel's discoveries and tackles one of Darwin's worries: How can variability be maintained in breeding populations that he saw as blending the characteristics of the individuals?
8.3.1 Some Background
We consider a large population of sexually reproducing organisms. The organisms are taken to be diploids: each cell contains two copies of each chromosome, one inherited from each parent. In order to reproduce, the parents produce gametes (which fuse together in sexual reproduction), and these gametes are haploid: they contain only one of each chromosome pair. The fusion process gives a new cell or zygote, which is again diploid, and this gives rise to the new organism. This is the life cycle followed by most multicelled animals and many plants.
The most basic problem arises when one particular chromosome locus or slot (or gene as it has now become), has two possible forms, known as alleles, which I denote by A and a. In the organism or phenotype, they will be responsible for some particular characteristic. For example, it could be the gene controlling eye color, with A giving brown eyes and a giving blue eyes. Or it could be the smooth or wrinkled seeds, the large or small plant heights, or the white or purple flower colors, in Mendel's pea-breeding experiments.
Over a large population, the alleles will occur with particular frequencies f(A) and f(a). (You may think of the fractions of each found in a large survey.) I will let
For example, if 80 percent of the time we find the A allele and 20 percent the a allele,
Clearly those will also be the frequencies of the alleles occurring in the gametes produced by the parents for sexual fusion. However, the zygotes so produced will have a pair of chromosomes with the particular
chromosome locus (gene) under consideration being either A or a according to the gametes used to produce them. Thus we can now work out the frequencies of finding the various pairs of alleles—f(AA), f(Aa), and f(aa)—in the zygotes (and hence in the new organism as it develops after the sexual reproduction). Using the frequencies for the alleles themselves in the gametes we are led to
The frequencies should add up to one, which is correct if we note equation (8.1):
p2 + 2pq + q2 = (p + q)2 = 12 = 1.
For the numerical example in equation (8.2),
f(AA) = 0.64 f(Aa) = 0.32 f(aa) = 0.04 and 0.64 + 0.32 + 0.04 = 1.
We now know how the alleles are distributed in the new generation of organisms. For example, if we check the pair of chromosomes having the slot or locus under discussion, we will find one has the A allele and the other the a allele in 32 percent of the cases. They both have the a allele in 4 percent of the cases.
8.3.2 Analysis and Concerns
Suppose that when the genes are expressed, allele A is dominant and a is recessive. For example, brown eyes must follow when we find the AA or Aa pairs, but blue eyes only follow from the aa pair. Thus in the new generation, for our numerical example, we expect to find 4 percent of individuals with blue eyes. Going back to the parent population, the allele for blue eyes actually occurred in 20 percent of cases according to equation (8.2). It appears that the dominant allele has led to a drastic reduction in blue eyes and the natural feeling is that if the process repeats to give the next generation, blue eyes will be even further reduced in number. The apparent conclusion is that the dominant form of the gene will quickly rule the whole population.
