by Colin Pask
This implies a strong and constantly operating check on population from the difficulty of subsistence. This difficulty must fall somewhere; and must necessarily be severely felt by a large portion of mankind.
I label this calculation 1, Malthus on population growth. By means of his simple calculation Malthus has identified a problem (the failure to match resources to increasing population) that has troubled mankind in almost all civilizations and continues to cause misery in much of the world today. The value of the calculation is in dramatically highlighting the problem.
1.2.1 Observations
Notice that we have now seen a common pathway for calculations with the following steps:
State the problem.
Identify the defining elements.
Translate the problem into a mathematical form.
Determine how to carry out the calculation.
Obtain the required result.
Analyze the result to see what it is telling us.
Malthus's example also illustrates another property of calculations: sometimes the objective is to identify a trend or type of behavior; in this case the way population growth outstrips the availability of resources needed to support it. The actual detailed numbers (population doubling in 25 years for Malthus) are not of particular interest; it is the pattern they reveal that is of importance.
Of course, some people will always want a more realistic assumption, as Charles Babbage amusingly demonstrated.9 After reading the couplet “Every minute dies a man, Every minute one is born,” in Alfred Tennyson's Vision of Sin, Babbage wrote to Tennyson suggesting that it be changed to “Every minute dies a man, And one and a sixteenth is born.” But even that was not enough as he continued “I may add that the exact figures are 1.167, but something must, of course, be conceded to the laws of metre.” At least Babbage wished to build in the trend of an increasing population, rather than the static one as Tennyson would have it.
Malthus's calculation is a good example of one that produces a set of numbers. However, sometimes a calculation may result in a formula. For example, if we denote the population in time period n by pn, then Malthus tells us that pn = 2n. The formula may be one to be tested against experimental results, and its accuracy may be a test of the validity of the underlying theory. On other occasions, the formula may be deduced by finding the appropriate mathematical expression to describe data from natural observations or from carefully designed experiments.
It is a feature of many good calculations that they lead us to ask questions and to build in other characteristics of the problem. In chapter 12 we will see how Malthus's work leads into whole new areas of both population modeling and mathematics.
It may be that an original calculation fails to take account of some essential aspects of the situation under investigation, perhaps something unknown at the time when it was made. This would invalidate any conclusions to be drawn from the calculation. We will see some famous examples in later chapters.
The results of a calculation might also lead us to different but related questions; in Malthus's case: How many people can the earth support? In fact, this question had been considered by Antonie van Leeuwenhoek (1632–1723), a man better known as a pioneer in microscopy. (Details are given by Cohen, who explains how Leeuwenhoek came up with a maximum population of 13,385,000,000.)
Finally, we should note that Darwin was aware of the conclusions to be drawn from Malthus's work when he considered how the essential competition for scarce or limited resources might be incorporated into his theory of evolution. Calculations may clearly have an influence far beyond their immediate context.
The actual calculations made by Malthus may have been extremely simple, but they have led to some vitally important developments in science and in the affairs of mankind.
1.3 OUTLINE FOR THE BOOK
I have grouped my chosen calculations into broad subject areas, and within each group, I use chronological ordering so we can see how calculations sometimes mark a major milestone in the history of science. Sometimes I will refer to a group of calculations. Emphasis is placed on the physical sciences, since it is there that mathematical descriptions and calculations have long been accepted as part of the subject, but examples from the biological sciences are also given. However, first, in chapters 2 and 3, I present some examples of calculations that are of purely mathematical interest, although in almost every case there is also a link to the applied mathematical world.
Chapter 4 shows how calculations have helped us to get to know our home, the earth. Calculations are becoming ever more important (think weather forecasting and climate change), and I will touch on some issues involved in a discussion section.
Chapters 5, 6, and 7 examine the part played by calculations when we consider our earth as a planet in a larger framework, first in the solar system and then in the universe as a whole. It is here that calculations become of necessity and exceptional importance.
Chapter 8 moves from the physical to the biological (having already seen Malthus's calculations earlier in this chapter). The physical world seems to demand the use of mathematics when we study it, but the triumphs of such an approach are fewer—but still vitally important—in the biological domain, and I will speculate a little about that.
The next three chapters deal with the basic and most fundamental parts of our physical world: light and the building blocks of matter. Discussion of the latter subject is divided into two chapters since nuclear physics provides some of the most spectacular (and perhaps disturbing) examples of the impact of calculations.
Chapter 12 examines how calculations have allowed us to discover the nature of motion in its various forms and to see how general mathematical descriptions were developed. It is here that we shall see the impact of modern computers as a vital ingredient in the business of calculation.
Finally comes a chapter of summary and evaluation. Can we say how, and to what extent, calculations have molded and supported the progress of science? Even more contentiously, can we pick out those calculations that are of supreme importance, those worthy of the label “great calculations”? It is an evaluation of their history, context, and impact that contributes to an assessment of a particular calculation, and here opinions are bound to differ. Please note that the title of this book is Great Calculations and not THE Great Calculations, or some other title suggesting that there is a definitive list. This is surely a personal matter, and many readers may have a quite different list. That is part of the fun, and I ask you not to be too irate if your personal favorite is missing.
References for all chapters are given in the notes section and also in the bibliography, which contains details of all mentioned books and papers. I have given examples of suitable references so that the interested reader may further explore each calculation.
If you cannot wait for the story to unfold, but want to see my list of important calculations right now, please go to the beginning of chapter 13!
in which we look at two outstanding examples of calculations made a long time ago.
Studies in cognitive science have revealed that humans (and some other animals) have certain innate mathematical abilities. At the simplest level, we readily distinguish the difference between groups of small numbers of objects (the power of numerosity) and recognize primitive geometric shapes and properties. This is linked to the natural world and our need to understand it in order to survive and prosper. Further development leads to arithmetic and simple ideas of measurement and design. Humans have also developed mathematical concepts and procedures that go beyond that link to the natural world. We speak of pure mathematics, with its emphasis on the mathematical formalism itself, as opposed to applied mathematics, where that formalism is exploited in order to describe the physical world. However, there is a continual interplay between pure and applied mathematics as problems and techniques are suggested and improved. Many people find it quite mysterious that abstract mathematical work later turns out to be useful in science.
This chapter and the next describe some calculations that are basically in the realm of pure mathematics, but that back-and-forth dialog between such calculations and their applications is ever present and will be nicely illustrated. I have chosen two examples from the work of ancient mathematicians for discussion in this chapter.
2.1 HISTORY BEGINS: TO MESOPOTAMIA FOR CALCULATION 2
In the period from around 3000 BCE, civilizations prospered in Mesopotamia, the region between the Tigris and Euphrates Rivers that we know today as Iraq. The region is sometimes called the cradle of civilization. People gathered in large towns, and city-states evolved. Major building and construction projects were undertaken, and farming using irrigation systems allowed large populations to be fed. There was development of ideas and practices in many areas: administration, planning, legal, military, trading, and religion. Naturally there was also the development of methods and materials for keeping records of all these things.
In one of the most significant of all steps in civilization, the early Sumerians (and later the Babylonians, Assyrians, and Hittites) developed cuneiform writing. A reed stylus was pressed into clay tablets to produce records that, after some process of drying or baking, became permanent. (The book by Robinson gives an introduction to writing and its importance.) Thousands upon thousands of ancient Mesopotamian clay tablets survive today. So it is that prehistory first moves over to history, as writing allows knowledge of life in those times to be obtained directly from the records left by the people themselves.
Many of the tablets are concerned with matters of everyday life such as administration, commerce, and scribal teaching and practice work. But tablets also give us what is surely the first great work of literature, The Epic of Gilgamesh. Even today the story of the tyrant king is fascinating to read. In Andrew George's translation of the work, you can see many examples of cuneiform tablets and an appendix explaining the translation process. Legal codes were developed, culminating in the 282 laws set down by Hammurabi around 1750 BCE. The Code of Hammurabi may be read in surviving cuneiform inscriptions, and we can appreciate how justice was meted out fairly, if somewhat harshly.
According to Jerry Brotton in his A History of the World in Twelve Maps, the very first map of the world may be viewed on a tablet from Sippar in southern Iraq. The tablet is now held in the British Museum (see bibliography). We can see how geometrical figures—circles, triangles, and rectangles—are carefully drawn. There is a hole presumably left by a pair of compasses used to draw the large circles. The cuneiform text explains the map; for example, beyond the outer circle is a “salt sea” representing the ocean encircling the inhabitable world.
Naturally the worlds of commerce, engineering, and administration, as well as the extensive investigations of astronomy and astrology initiated in Mesopotamia, led to the invention of mathematical ideas and their use in problem solving. (Otto Neugebauer is the classic reference.) Many thousands of tablets tell us that these people were skilled in mathematics, and they produced tables of all kinds. For numerical work, they counted to ten and then used multiples of ten as we do today. However, when they reached sixty they changed over to use 60 as the base for their numbers. As an example of the sexagesimal system, where we write:
1565 = 1 × 103 + 5 × 102 + 6 × 10 + 5,
those ancient mathematicians wrote:
425 = 4 × 602 + 2 × 60 + 5.
I have used the double underline to emphasize that they used cuneiform symbols, not the 4, 2, and 5 that we use today. Similarly their fractions were considered in terms of 60 rather than our decimal ten. For example, while we might use things like 7/100, they worked with expressions like 7/60 or 11/360. Today we still retain this ancient use of base 60 in our measurement of angles in terms of degrees, minutes, and seconds.
Tablets reveal how ancient peoples frequently used mathematics in practical applications and in all sorts of problems and puzzles, much as we still do today. It is possible to see how mathematics (as well as other disciplines) was taught, and how many problem sets, as well as worked examples and useful tables, were recorded. Enough tablets were discovered in one ruined building in Nippur to suggest it was a school. (The interested reader is referred to Eleanor Robson's highly informative and entertaining article, “Mathematics Education in an Old Babylonian Scribal School.”)
There is one clay tablet that has achieved great fame, and I have chosen the mathematics it records as my second noteworthy calculation.
2.1.1 Plimpton 322
Around 1923, George Plimpton purchased the clay tablet shown in figure 2.1. It is now labeled as “Plimpton 322” in the collection at Columbia University. Physically, it is approximately 12.5 cm by 8.8 cm, and the left-hand edge indicates that a portion has broken off and been lost. There are headings and fifteen rows of numbers set out in four columns. The reverse side is blank, but may have been reserved for an extension of the given table (see below). The tablet was probably written around 1800 BCE in the ancient city of Larsa in present-day Iraq. (There is a large literature based on Plimpton 322, with early work by Neugebauer, later work by Eleanor Robson, and the recent major review by Britton, Proust, and Shnider, and references therein, providing a wealth of detail and discussion.)
Figure 2.1. The clay tablet known as Plimpton 322. Courtesy of the Plimpton Collection, Rare Book and Manuscript Library, Columbia University.
The table on Plimpton 322 concerns Pythagorean triples. (I shall continue to use this term even though the work in question predates Pythagoras by a very long time.) Recall that three integers (a, b, c) form a Pythagorean triple if
Therefore, choosing the smallest possible numbers gives (3, 4, 5) since 32 + 42 = 25 = 52. Are there any more Pythagorean triples? After a little experimentation you might come up with (5, 12, 13) since 52 + 122 = 169 = 132. But how far would simple experimentation take you beyond that? Would you be likely to find the triple (4601, 4800, 6649), for example?
Obviously the name comes from the fact that a, b, and c could be the lengths of the sides of a right-angled triangle with hypotenuse of length c. Equally well, a and b could be the side lengths of a rectangle with diagonal of length c, and it appears that the ancient Mesopotamian mathematicians used that interpretation. Table 2.1 sets out the relevant Plimpton 322 examples.
Table 2.1. Numbers related to the Plimpton 322 tablet expressed in decimal form.
The four columns on the Plimpton 322 tablet give numbers corresponding to those shown in table 2.1 but with a left out. The tablet numbers, written in cuneiform and using base 60, have been converted here to decimal form. Six errors (two of which appear to be simple copying errors) have been corrected, and obvious values for broken-off parts of the tablet have been inserted.
We see that the ancient mathematicians in Mesopotamia produced a remarkable table of Pythagorean triples. For me it represents an outstanding pioneering mathematical achievement, and without hesitation I have taken it as calculation 2, Mesopotamian Pythagorean triples in my list of significant calculations.
2.1.2 Discussion
A brief glance at the examples in table 2.1 tells us that this was not just some haphazard search for Pythagorean triples. Even with an electronic calculator, the reader will find it is a tedious business finding these sets of numbers, and a search starting with the smallest numbers in (3, 4, 5) will not be very helpful. Some of the numbers in table 2.1 are quite large: 18541 in row 4, for example.
Just looking at the triples on the tablet might suggest a jumble of results, maybe just stumbled upon. But how likely is it that anyone would just happen across
22912 + 27002 = 35412?
The numbers in column one suggest a different story; they steadily decrease and tell us that this table is not a random set of examples but instead represents the results of a systematic procedure. Some people note that, if a geometric interpretation is considered, the figures in column one give the square of the cosecant of the angle in the appropriate right-angle triangle. (The cosecant of an angle in a r
ight-angled triangle is found by dividing the length of the hypotenuse by the length of the side opposite that angle.) However, this interpretation, while certainly valid, does not correspond to any such things in Mesopotamian mathematics. I return to this construction problem in the next section.
There has also been great debate about why Plimpton 322 was produced. It could have been as a table of results for use in solving problems of various kinds, including those based on the geometry of rectangles. It could be a table of examples for teachers to use. The interested reader will find discussions noted in the references.
2.1.3 Constructing the Table
Now, some details for those who wish to consider how the table on Plimpton 322 was constructed. There is an extensive literature on this topic, but I will present the well supported argument propounded by Britton, Proust, and Shnider. There are two points to discuss: How can Pythagorean triples be calculated? And what input should be used to produce a systematic list using that method?
If we take a number p and from it find the two numbers x and y according to
a little algebra shows that
Thus we have found the triple (y, 1, x). For example, choosing p = 2 gives (3/4, 1, 5/4). This is not a true Pythagorean triple since integers are required. However, multiplying all numbers in a triple gives another triple (simply scaled, we might say), and multiplying here by 4 gives us the (3, 4, 5) Pythagorean triple.
The above procedure uses a number p and its inverse 1/p. The Mesopotamian mathematicians made extensive use of inverses and produced tables of them for use in division (dividing by p is equivalent to multiplying by 1/p) and other calculations. (See the article by Eleanor Robson.) They also had methods for the scaling process. Thus using the idea expressed in equations (2.2) and (2.3) seems like a natural approach for them to use when seeking triples.
