Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  3.1.2 The Liber Abbaci

  We are lucky to have the modern translation of Liber Abbaci by Laurence Sigler. Three things become apparent when looking at the book. First, it is an enormous work, about 600 pages in Sigler's translation. This is not just a simple textbook or mathematical record; it is a compendium of mathematical results. Second, the title translates as “Book of Calculation,” and the arithmetic is much as we know it today; this is not a book about the abacus or other calculating devices. Third, this is a book for use by teachers and merchants wishing to learn the new approach to numbers. Liber Abbaci contains an enormous number of examples, both serious and entertaining.

  Liber Abbaci comprises fifteen chapters, with the first seven devoted to the basics of numbers and arithmetic. Five chapters present applications that are of importance for merchants, accountants, and so on. The final three chapters deal with mathematical techniques, with the usual array of innovative examples.

  Chapter 1 opens with the key to the whole book and the progress in mathematics that it produced:

  The nine Indian figures are

  9 8 7 6 5 4 3 2 1.

  With these nine figures and the sign 0 which the Arabs call zephyr, any number whatsoever is written, as is demonstrated below. A number is a sum of units, or a collection of units, and through the addition of them the numbers increase by steps without end.

  He goes on to explain about tens, hundreds, and so on. There are examples of how those awkward Roman numerals could be neatly handled:

  write 2023 for MMXXIII and write 4301 for MMMMCCCI.

  The chapter concludes with simple addition and multiplication tables, just as we learn them in primary school today. Tables for manipulating fractions are given in later chapters.

  Chapters 2–7 develop arithmetic using those “Indian figures.” For example, chapter 2 teaches us how to multiply, and like any good teacher, Fibonacci builds up a series of ever more complicated examples such as:

  7 × 308 = 2156, 70 × 81 = 5670, 123 × 456 = 56088,

  and finally 12345678 × 87654321 = 1082152022374638.

  A measure of Fibonacci's thoroughness is gained by noting that he suggests checking answers using the old technique called “casting out nines” and (of course) detailed examples are given.

  The next four chapters show how to use the new arithmetic in the banking, trading, and the business world:

  8. On Finding the Value of Merchandise by the Principal Method

  9. On the Barter of Merchandise and Similar Things

  10. On Companies and their Members

  11. On the Alloying of Monies

  Next is the large (187 pages in Sigler's translation) and untitled chapter 12. Fibonacci illustrates many mathematical procedures (including the sum of an arithmetic progression and the sum of a series of squares) using a large collection of what today we might call recreational problems (see below, sections 3.1.3 and 3.1.4).

  Finally there are three chapters on more advanced mathematical ideas. Fibonacci shows how to find square and cube roots using approximation techniques, including the idea of building up a series of ever more accurate answers (as we saw Archimedes doing for bounds on π). There is a great deal of algebra here (and in earlier chapters) but the modern reader may struggle to appreciate that as it is given in the rhetorical form, where words but no symbols are used, which was prevalent before Viete introduced our modern symbolic form in 1591.

  In summary, in Liber Abbaci, Fibonacci introduced part of the world to arithmetic easily performed using Hindu numerals and set out a breathtaking array of examples of all kinds.

  3.1.3 A Recreational Mathematics Example

  “Purse problems” have a long history, and Fibonacci included his own versions. Here is one from chapter 12:

  Two men who had denari found a purse with denari in it; thus found, the first man said to the second, if I take these denari of the purse then with the denari I have I shall have three times as many as you have. Alternately, the other man responded, and if I shall have the denari of the purse with my denari, then I shall have four times as many as you have. It is sought how many denari each has, and how many denari they found in the purse.

  Today we solve this problem using a little linear algebra. Let the two men have x1 and x2 denari, and the purse contain p denari. Then the problem tells us that

  p + x1 = 3x2 and p + x2 = 4x1,

  which we easily solve to get . In the second of his approaches to the problem, Fibonacci does some similar algebra except that for him, in the rhetorical form, an unknown is “the thing” (whereas we would use a symbol like x). We notice that there are three unknowns (p, x1, and x2), and the two conditions in the problem only allow us to find two unknowns, which are therefore given in terms of the third. The next step is to notice (as Fibonacci effectively does) that we are only interested in whole numbers, so we will say there are 11 denari in the purse (p = 11), and then the men have 4 and 5 denari. (Of course, we could use any multiple of 11 as p.) Fibonacci is cleverly using the fact that his problem requires a solution in whole numbers not fractions.

  3.1.4 But What about the Rabbits?

  Of all the 600 pages in Liber Abbaci, the content of one particular page in the chapter on recreational mathematics has become extremely famous:

  How Many Pairs of Rabbits are Created by One Pair in One Year?

  A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.

  Fibonacci shows how to proceed month by month and create what we now call the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and 377. Thus there will be 377 pairs of rabbits in twelve months. (I return to the modeling of population growth in chapter 12.) Fibonacci is demonstrating the explosive nature of the growth of rabbit populations. Obviously the early settlers in Australia had not read Liber Abbaci because in 1859 they released twenty-four rabbits into the wild, and the resulting plague continues to cause trouble for farmers to this very day!

  The Fibonacci numbers are now known to occur in a great range of biological problems, and through one simple example, he created a whole research field. Pure mathematicians have discovered many interesting properties of the whole sequence of Fibonacci numbers.

  3.1.5 Evaluation

  Fibonacci wrote other books, including Liber Quadratorum in which he demonstrated his talents as a mathematician by presenting a series of increasingly involved propositions about the squares of numbers (now a venerable topic in number theory). However, in Liber Abbaci, Fibonacci is gathering together a magnificent array of mathematical techniques and examples, which he collected from a variety of sources and which he supplements with his own ideas and methods. It is a very approachable, comprehensive treatise, and it changed the way people did mathematics for the next three hundred years. For this reason, I record calculation 4, Fibonacci's presentation in Liber Abbaci. A fitting summary is given by Keith Devlin in his biography of Fibonacci:

  The greatness of Liber Abbaci is due to its quality, its comprehensive nature, and its timeliness. It was good, it provided merchants, bankers, business people, and scholars with everything they need to know about the new arithmetic methods, and it was the first to do so. Though there were to be many smaller, derivative texts that would explain practical arithmetic, it would be almost three hundred years before a book of comparable depth and comprehensiveness would be written.1

  3.2 CALCULATION 5 TO MAKE LIFE EASIER

  Which would you rather do: multiply 512 by 128, or add 9 and 7? No doubt you settled for the addition. If the numbers were larger, it would make your choice even more emphatic—or would it? It may be that you respond by saying the question seems a little silly and you do not much care; after all it is only the choice of which button to press on a calculator. However, in the days before calculating machines were readily available, additi
on was comparatively simple, but a complicated multiplication would be an agonizing and time consuming task. So it was in 1614 that John Napier came to write:

  Seeing as there is nothing (right well beloved students in the Mathematics) that is so troublesome to Mathematical practice, nor that doth more molest and hinder Calculators, than the Multiplications, Divisions, square and cubical Extractions of great numbers, which besides the tedious expense of time, are for the most part subject to many slippery errors. I began therefore to consider in my mind, by what certain and ready Art I might remove those hinderances.2

  The stage was set for the invention of an invaluable new mathematical technique.

  3.2.1 The Basic Idea

  Take a look at table 3.1.

  Table 3.1. Numbers n in an arithmetic progression and a corresponding geometric progression.

  The columns giving the n values form an arithmetic progression while the corresponding 2n columns form a geometric progression (just as we saw in section 1.2). In the first case, additions are involved as we move down the columns, whereas it is multiplications in the second case. I also remind you about (or introduce you to) the exponent laws:

  From the table we can see that 512 × 128 is the same as 29 × 27 and the exponent law tells us that is 29+7 = 216. Consulting the table gives 216 as 65,536 so we conclude that 512 × 128 = 65,536. Thus we have seen that writing numbers as 2 to some power allows us to use addition of exponents to carry out tedious multiplications. You can show how divisions are carried out by finding that 512/128 is 29–7 = 22 = 4. Because 65,636 is 216 its square root must be 28 = 256.

  While we have seen a wonderfully simple way to carry out complicated calculations just using additions or subtractions, the table of numbers on which we can use those ideas is extremely limited. However, for any given number y it is possible to find an exponent x so that

  y = 2x and x is called the logarithm of y, written as log2(y) = x.

  Naturally, x is no longer a simple integer for most cases. Log is short for logarithm and the subscript 2 on the log symbol is to remind us that we are using 2 as the base for our calculations. In fact we can use any number as the base (obviously requiring new tables like 3.1), and we shall see that base 10 has become a standard. Table 3.1 is just a table of logarithms with base 2; in the heading we could replace 2n with y and n by log2(y). For example, for 256 the table gives 8 as log2(256).

  The exponent laws can now be written as laws for manipulating logarithms:

  Using table 3.1 we find

  log(512 × 128) = log(512) + log(128) = 9+ 7 = 16 = log(65,536)

  and so we deduce that 512 × 128 = 65,536.

  3.2.2 John Napier

  The basic ideas about logarithms and the recognition of their value were first explained by John Napier (1550–1617). He was the eighth Laird of Merchiston and after 1608, lived in Merchiston Castle near Edinburgh. He was a very active and inventive man whose writings covered many subjects including agriculture, engineering, and military matters. He also fathered twelve children. (See the book by Calinger for further details.) Napier was a staunch Scottish Presbyterian and enemy of Catholics, and his best-selling work, A Plaine Discovery of the Whole Revelation of Saint John: Set Down in Two Treatises, ran through twenty-one editions in English and had many translations.

  John Napier seems to be an unlikely mathematical innovator, but the quote given in the first paragraph of section 3.2 shows that he well understood the perils of calculation; his great discovery was a method for enormously reducing those perils. Napier introduced logarithms into mathematics, but not in the form given above. Napier hit upon the essential point—the use of the correspondence between the terms in an arithmetic series and those in a geometric series—but he dealt with it in terms of the motion of points along two lines, in one case with uniform speed (so distance traveled forms an arithmetic progression) and in the other with variable speed (to give effects like a geometric progression). (See the book by Goldstine and the articles by Bruce, Carslaw, and Jagger for details.) Using clever mathematical methods and a great deal of time (which only someone like a nobleman might have), Napier produced tables of logarithms, and his acclaimed Mirifici Logarithmorum Canonis Descriptio was published in 1614. According to historian Ronald Calinger “his [Napier's] logarithms were the cardinal computational accomplishment of the Reformation.”3

  3.2.3 Henry Briggs

  Naturally, many people became interested in logarithms, but the major figure in their development was Henry Briggs (1561–1631). He was the first professor of geometry in the new Gresham College in London and later held the same position at Merton College in Oxford. Briggs recognized the magnitude of Napier's achievements, and, in 1615, he made the arduous journey to Edinburgh to consult Napier. Briggs stayed for a month, and the two men came up with the ideas that led to logarithms as we know them today.

  One major change saw the adoption of base ten for future logarithms. (The construction method used by Napier results in a complicated structure as explained in the references cited above.) Briggs used a number of ingenious mathematical methods to calculate logarithms—see Goldstine and Bruce's aptly named paper “The Agony and the Ecstasy—The Development of Logarithms by Henry Briggs.” In his 1624 Arithmetica Logarithmetica, Briggs gave the base ten logarithms for all the integers from 1 to 20,000 and from 90,000 to 100,000 with instructions on how to find the other obviously missing cases. He gave the results to fourteen-figure accuracy! The second edition, published in 1828 with the help of the Dutch mathematician Adriaan Vlacq, covered all integers up to 100,000 but reduced to ten-figure accuracy. Briggs also published extensive tables of sines and tangents of angles and their logarithms in his Arithmetica Logarithmic, sometimes working to fifteen-figure accuracy. These were tables needed in many applications, and it was centuries before there was any improvement over them.

  These tables revolutionized the business of calculation and were soon seized upon by scientists and engineers, particularly those needing to manipulate expressions involving trigonometry.

  3.2.4 Impact

  The invention of logarithms and tables of them were a boon for people such as astronomers (like Kepler) and surveyors. (The cited books by Calinger and Jagger tell the story of the widespread use and importance of logarithms.) The situation was famously summed up by the most eminent of all French scientists, Pierre-Simon Laplace, when he said that logarithms, “by shortening the labors, doubled the life of the astronomers.”4 Napier himself seems to have been in no doubt about the value of his invention because his Mirifici Logarithmorum Canonis Descriptio opens with a poem which, translated from the original Latin, reads:

  This book is small if you consider just the words,

  but if you consider its use, Dear Reader, this book is huge.

  Study it and you will learn that you owe as much to this little book

  as to a thousand large volumes.5

  As a person who was at school and college in the pre-electronic-calculator era, I know just what Napier meant! I have no hesitation in adding to the list calculation 5, production of tables of logarithms.

  Napier invented a wooden device (“Napier's bones”) for mechanically using logarithms. William Oughtred and others made advances in that line, and eventually it led to the “modern” slide-rule, which was an essential part of the toolkit of engineers until the era of electronic calculators. (For more information, see the book by M. R. Williams, and it is also worth seeking out the old Handbook of the Napier Tercentenary Exhibition edited by E. M. Horsburgh.)

  Finally, logarithms give a fine example of the way mathematicians can sometimes transform the nature of a problem using a mathematical technique; in this case, multiplications turned into additions by moving into the world of logarithms. Similar advances were made later by introducing Fourier transforms and Laplace transforms.

  3.3 THE MASTER GIVES US CALCULATION 6

  In mathematics, we often create a sequence of numbers or answers. We have seen Malthus calculati
ng values for a population and its resources as they develop over a set of time intervals, Archimedes finding a sequence of ever-better approximations for π, and Fibonacci showing us how to find the number of pairs of rabbits as their population explodes. The next step in some cases is to sum the sequence of terms, and we call that a mathematical series. The sum is denoted by s with a subscript counting how many terms ai are to be included:

  s4 = a1 + a2 + a3 + a4 and generally sn = a1 + a2 + a3 +…+ an.

  For example, using terms as in Malthus's arithmetic progression we find

  1 + 2 + 3 + 4 + 5 = 15.

  In his Liber Abbaci, Fibonacci gave the general formula for summing an arithmetic progression which starts with the number a and increases by d each term:

  and Fibonacci wrote sn = × the number of terms × the sum of the first and last terms.

  Fibonacci also gave the sum of squares:

  Malthus used a geometric progression, and the sum for a general problem of this type (starting with a and multiplying by r to get each new term) is also well known:

  A problem on the ancient Egyptian Rhind papyrus (see section 2.1.4) called for this sum with a = 1 and r = 7 taken to 5 terms and got the answer 2,801. The problem is often repeated in various similar forms; for us it is the old puzzle “as I was going to St. Ives…,” while for Fibonacci in the Liber Abbaci, it was “seven old men go to Rome…”

  But now, a word of caution: the above results are special, and it is not always easy to find a formula for the sum in a series. For example, finding sn for the so called “harmonic series,”

  is notoriously difficult.

  3.3.1 The Next Level

  Mathematics often involves the infinite, and it is natural to ask what happens when we let the series continue on forever. We can ask for the value of sn as n becomes as large as we like (“in the limit as n approaches infinity”). Clearly, for an arithmetic progression such as in equation (3.1), the answer becomes infinitely large and we say the series diverges. For a geometric progression as in equation (3.2), the answer depends on the multiplier r; when r >