In addition to its treatment of Hindu-Arabic arithmetic, Liber abbaci covers the beginnings of algebra and some applied mathematics. Some of the methods Leonardo described may have been his own invention, but he obtained much from existing sources, primarily Arabic texts or Latin translations thereof, and from discussions with the Arabic mathematicians he encountered on his travels. In all cases, he provided rigorous proofs to justify the methods, in the fashion of the ancient Greeks, and illustrated everything with copious worked examples designed to provide exercises in using the new methods.
Leonardo divided the book into fifteen chapters, the titles of which vary from manuscript to manuscript, suggesting that the scribes who made copies felt free to make what they felt were clarifying improvements. The titles in Sigler’s English translation are:
Dedication and prologue
1. On the recognition of the nine Indian figures and how all numbers are written with them; and how the numbers must be held in the hands, and on the introduction to calculations
2. On the multiplication of whole numbers
3. On the addition of them, one to the other
4. On the subtraction of lesser numbers from greater numbers
5. On the divisions of integral numbers
6. On the multiplication of integral numbers with fractions
7. On the addition and subtraction and division of numbers with fractions and the reduction of several parts to a single part
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
12. On the solutions to many posed problems
13. On the method elchataym and how with it nearly all problems of mathematics are solved
14. On finding square and cubic roots, and on the multiplication, division, and subtraction of them, and on the treatment of binomials and apotomes and their roots
15. On pertinent geometric rules and on problems of algebra and almuchabala.
His opening chapter describes how to write—and read—whole numbers in the Hindus’ decimal system. Leonardo began: “These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.” He went on to explain the principles of place value, describing the forms of the numerals, and showing how to write large numbers (either by putting a dot—adcentare—above each hundred, and below each thousand, or by linking groups of three numerals with a small curved stroke called a virgula). When his chapter title promised to explain “how the numbers must be held in the hands,” he meant that quite literally. He described a procedure to calculate on the fingers, widely used by medieval scholars and traders, which was regarded as the easiest and quickest way of performing calculations. Manuscript copies of Liber abbaci and those of many other arithmetic books that were to follow often included a drawing showing the various finger positions used to represent different whole numbers. Leonardo also provided addition and multiplication tables to be referred to—or memorized—in order to facilitate computations. In all, he devoted several pages to this introductory description of the numerals, which would have been his readers’ first encounter with modern numbers.
The approach Leonardo took to multiplication in chapter 2 differs little from the one used today to teach children how to multiply two whole numbers together. He began with the multiplication of pairs of two-digit numbers and of multidigit numbers by a one-place number and then worked up to more complicated examples. He described various methods for checking the answers.24 It is interesting to note that Leonardo described multiplication before covering addition in chapter 3 and subtraction in chapter 4.25
Whole number division and simple fractions are described in chapter 5. An “integral number”, or “integer”, is a technical term for a whole number, positive, negative, or zero.
The topic of chapters 6 and 7 is what are today called mixed numbers, numbers that comprise both a whole number and a fractional part. Leonardo explained that you calculate with them by first changing them to fractional form (what we would today call “improper fractions”), computing with them, and then converting the answer back to mixed form.
Having described the basic methods of Hindu-Arabic arithmetic in the first seven chapters, Leonardo devoted most of the remainder of the book to practical problems. Chapters 8 and 9 provide dozens of worked examples on buying, selling, and pricing merchandise, using what we would today call reasoning by proportions—the math we use to check the best deal in the supermarket. For example, Leonardo asked: if two pounds of barley cost five solidi, how much do seven pounds cost, showing how to work out the answer. In chapter 10, he explained how to use similar methods to manage investments and profits of companies and their members, and showed how to decide who should be paid what.
Chapter 11, “On the alloying of monies”, met an important need at that time. Italy had the highest concentration of different currencies in the world, with twenty-eight different cities issuing coins during the course of the Middle Ages, seven in Tuscany alone. Their relative value and the metallic composition of their coinage varied considerably, from one city to the next and over time. This state of affairs meant good business for money changers, and Liber abbaci provided them with plenty of examples on problems of that nature. Also, with governments regularly revaluing their currencies, gold and silver coins provided a more stable base, and since most silver coinage of the time was alloyed with copper, problems about minting and alloying of money were important.
The lengthy chapter 12 presents 259 worked examples, some requiring only a few lines to solve, others spread over several densely packed pages. In modern terminology, the main focus is essentially “algebra”—not the symbolic reasoning we associate with the word today, rather algebraic reasoning expressed in ordinary language, often referred to as “rhetorical algebra”. Symbolic algebra did not appear until 1591, when the French amateur mathematician and astronomer François Viėte published his book In artem analyticam isagoge (Introduction to the analytic art), explaining how to formulate and solve an equation in symbolic form, much as we do today. Mathematicians in Leonardo’s time used a variey of methods to solve problems that today would be handled using symbolic algebra.
The term elchataym that appears in the title of chapter 13 refers to a rule known also as “Double False Position”. The word is Leonardo’s Latin transliteration of the Arabic al-khata’ayn, which means “the two errors”. The name reflects the fact that you start with two approximations to the sought-after answer, ideally one too low, the other too high, and then reason to adjust them to give the correct answer. Leonardo formulated his worked problems in several ingenious ways, in terms of snakes, four-legged animals, eggs, business ventures, ships, vats full of liquid which empty through holes, how a group of men should share out the proceeds when they find a purse or purses, subject to various conditions, how a group of men should each contribute to the cost of buying a horse, again under various conditions, as well as some problems in pure number terms.
Though most of the problems in chapters 12 and 13 are of a practical nature, Leonardo included some that have a whimsical flavor. For example, if a spider climbs a wall of a cistern, advancing a certain number of feet each day, but slips back so many feet each night, how long will it take to climb out? For some of these entertaining problems he presented clever solutions that may have been of his own devising.
In his penultimate chapter, Leonardo’s main focus was on methods for handling roots. He used the classifications given by Euclid in book 10 of Elements for the sums and differences of unlike roots—known as binomials and apotomes respectively. The discovery that √2 is irrational led the ancient Greeks to a study of what they called “incommensurable magnitudes”. Euclid’s term for a sum of two incommensurables, such as √2 + 1, was binomial (a “two-name” magnitude), and a difference, such as √2 – 1, he called an apotome. Handling incommensurables by means of what we would now regard as algebraic expressions was a common feature of Greek and medieval mathematics, and was the way they dealt with algebraic binomials such as “two things less a dirham” (d – 2 in modern algebraic notation). This is one reason why many medieval Italian treatments of algebra began with a long chapter on binomials and apotomes.
Chapter 15, the final chapter, deals with algebraic equations—again expressed in rhetorical fashion—using the methods descibed in al-Khwārizmī’s Algebra. Leonardo also fulfilled a promise he made in the preface, to deal with things “pertaining to geometry.” This is not a treatment of geometry as such, but rather an investigation of the connection between geometry and arithmetic, particularly the use of algebraic and geometric ideas to solve problems in arithmetic.
For the twenty-first-century reader who picks up the English translation of Liber abbaci1 and starts to read it, its most striking feature is the enormous number of worked examples that fill the book’s six hundred pages. A beginning math student today does not require, and would not tolerate, such a deluge of examples on basic arithmetic and algebra, but Leonardo was writing for a thirteenth-century audience. To most people at the time, symbolic numbers and their arithmetic were alien. Arithmetic was something practiced by the merchants, who used finger reckoning or an abacus.26
Many of the problems Leonardo presented involved weights. This was an important but complicated subject, since units of weight differed from one city to another and therefore frequently had to be converted. For example, one worked problem in chapter 8 is titled: “On finding the worth of Florentine rolls27 when the worth of those of Genoa is known”.2 A typical worked problem in this section of the book starts:
If one hundredweight of linen or some other merchandise is sold near Syria or Alexandria for 4 Saracen bezants, and you will wish to know how much 37 rolls are worth, then …3
In chapter 10, “On companies and their members”, Leonardo demonstrated valuable methods for solving problems such as determining the payouts in the following scenario:
Three men made a company in which the first man put 17 pounds, the second 29 pounds, the third 42 pounds, and the profit was 100 pounds.4
Toward the end of chapter 11, he gave a curious problem that became quite well known to mathematicians. It is called “Fibonacci’s Problem of the Birds”:
ON A MAN WHO BUYS THIRTY BIRDS OF THREE KINDS FOR 30 DENARI
A certain man buys 30 birds which are partridges, pigeons, and sparrows, for 30 denari. A partridge he buys for 3 denari, a pigeon for 2 denari, and 2 sparrows for 1 denaro, namely 1 sparrow for 1⁄2 denaro. It is sought how many birds he buys of each kind.5
What makes this problem particularly intriguing is that apparently there is not enough information to solve it. If you let x be the number of partridges, y the number of pigeons, and z the number of sparrows, then the information you are given leads to two equations:
x + y + z = 30 (the number of birds bought equals 30)
3x + 2y + 1⁄2 z = 30 (the total price paid equals 30)
But in general three equations are needed to find three unknowns. What makes this problem different is the availability of one crucial additional piece of information that enables the solution: the values of the three unknowns must all be positive whole numbers. (The man buys three kinds of birds, so none of the unknowns can be zero, and he surely does not buy fractions of birds.) Start by doubling every term in the second equation to get rid of the fraction:
x + y + z = 30
6x + 4y + z = 60
Subtract the first equation from the second to eliminate z:
5x + 3y = 30
Since 5 divides the first term and the third, it must also divide y. So y is one of 5, 10, 15, et cetera. But y cannot be 10 or anything bigger, since then it could not satisfy that last equation. Thus y = 5. It follows that x = 3 and z = 22. Leonardo, as usual, presented the solution in words, not symbols, but apart from that, this is his solution.
Like mathematics teachers and authors before him and since, Leonardo clearly knew that many of the people who sought to learn from him would have little interest in theoretical, abstract problems. Though mathematicians feel entirely at home in a mental world of abstract ideas, most people prefer the more concrete and familiar. And so, in order to explain how to use the new methods he learned during his visit to North Africa, Leonardo looked for ways to dress up the abstractions in familiar, everyday clothing. The result is a class of problems that today go under the name “recreational mathematics”.
For instance, Leonardo presented a series of “purse problems” to try to put into everyday terms the mathematical challenge associated with dividing up an amount of money—or anything else that people may have wanted to share—according to certain rules. The first was:
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 denari as you have. It is sought how many denari each has, and how many denari they found in the purse.6
Students today would be expected to solve this problem using elementary algebra (equations), and this takes at most a few lines. But restricted to the methods available at the time, Leonardo’s solution filled almost half a parchment page. (At heart, it is the same as the modern answer by algebra, but without symbolic equations it takes a lot more effort, and a great deal more space on the page, to reach the answer.)
More complicated variations followed, totaling eighteen purse problems in all, including a purse found by three men, a purse found by four men, and finally a purse found by five men. Each problem took a full parchment sheet to solve. Adding still more complexity, Leonardo presented a particularly challenging problem in which four men with denari find four purses of denari, the solution of which took several pages.
Although many of the variants to the purse problem Leonardo presented seem to be of his own devising, the original problem predated him by at least four hundred years. In his book Ganita Sara Sangraha, the ninth-century Jain mathematician Mahavira (ca. 800–870) presented his readers with this problem:
Three merchants find a purse lying in the road. The first asserts that the discovery would make him twice as wealthy as the other two combined. The second claims his wealth would triple if he kept the purse, and the third claims his wealth would increase five fold.
The reader has to determine how much each merchant has and how much is in the purse. This is precisely Leonardo’s first purse problem in chapter 12 of Liber abbaci. Presumably Leonardo came across the puzzle by way of an Arab text.
Leonardo’s purse problems involved divisions that required only whole numbers. To explain how to proceed when fractions are involved, he used a different scenario his readers could relate to: buying horses. The first horse problem reads:
Two men having bezants found a horse for sale; as they wished to buy him, the first said to the second, If you will give me 1⁄3 of your bezants, then I shall have the price of the horse. And the other man proposed to have similarly the price of the horse if he takes 1⁄4 of the first’s bezants. The price of the horse and the bezants of each man are sought.7
Again, today we would solve this problem using symbolic algebraic equations. Leonardo solved it thus:
You put 1⁄4 1⁄3 in order, and you subtract the 1 which is over the 3 from the 3 itself; there remains 2 that you multiply by the 4; there will be 8 bezants, and the first has this many. Also the 1 which is over the 4 is subtracted from the 4; there remains 3 that you multiply by the 3; there remains 9 bezants, and the other man has this many. Again you multiply the 3 by the 4; there will be 12 from which you take the 1 that comes out of the multiplication of the 1 which is over the 3 by the 1 which is over the 4; there remain 11 bezants for the price of the horse; this method proceeds from the rule of proportion, namely from the finding of the proportion of the bezants of one man to the bezants of the other; the proportion is found thus.8
This definitely does not read like a modern-day math textbook. It is far more reminiscent of a cooking recipe written for beginners and therefore leaving nothing to chance. Leonardo explained step-by-step what digits had to be written where, and what to do to them.
Among a total of twenty-nine horse-type problems Leonardo presented is one in which five men buy five horses9 and another, which he titled “A Problem Proposed to Us by a Most Learned Master of a Constantinople Mosque”,10 in which five men buy not a horse but a ship, and still another where seven men buy a horse.11
One of Leonardo’s problems leads to the particularly nasty answer that a certain businessman walks away from a partnership in Constantinople with a profit of
When Europeans in Leonardo’s time learned the Hindu-Arabic number system, they wrote fractions before whole numbers, and built those fractions up from right to left, with each new fraction representing that part of what is to the right. For example,
The right-to-left ordering may simply be a carryover from the writing of Arabic, although for the most part Arabic texts expressed Hindu-Arabic numbers rhetorically, using words instead of symbols. Leonardo would have articulated the above fraction as the Arabic mathematicians would both write and speak it: “Four fifths, and two thirds of a fifth, and one half of a third of a fifth.”
Decimal expansions are a special case of this notation when the denominators are all 10. For example, Leonardo would have written today’s decimal number 3.14159 as
Though decimal representation seems far simpler to us today, there was little need for it in Leonardo’s time, as no one counted anything in tenths. In fact, the method they used to represent fractions was particularly well suited for calculations involving money. In the monetary system used in medieval Pisa, 12 denari equaled 1 solidus and 20 solidi equaled 1 libra, so 2 librae, 7 solidi, and 3 denari would be written
Units of weight and measure could be even more complex. According to Leonardo, Pisan hundredweights “have in themselves one hundred parts each of which is called a roll, and each roll contains 12 ounces, and each of which weighs 1⁄2 39 pennyweights; and each pennyweight contains 6 carobs and a carob is 4 grains of corn.”12
Interestingly, an Arabic arithmetic text written by al-Uqlidisi in Damascus in 952 did in fact use place-value decimals to the right of a decimal point, but no one saw any particular reason to adopt it, and so the idea died, not to reappear again for five hundred years, when Arab scholars picked up the idea once more. Decimal fractions were not used in Europe until the sixteenth century.
Fractions written after the whole number part in Leonardo’s time denoted multiplication. For example, 1⁄2 of 3.14159 would have been written
Though Leonardo divided up his book in terms of different kinds of applications, underlying his solutions were a surprisingly small number of different techniques. The most notable were “the Rule of Three”, “the Rule of False Position”, “the Rule of Double False Position”, and a nonsymbolic form of “Algebra”, in particular a method he called regula recta (Direct Method), which we would describe today as rhetorical algebra restricted to linear equations.
The Rule of Three was an ancient method for solving proportions problems, such as:
If 10 men can dig a trench in 4 days, how long will 7 men take to dig a similar trench?
Today we solve such problems fairly easily. One approach is to calculate the amount of time it would take one man to dig a trench, namely 4 × 10 = 40 days (since 10 men would dig ten times as fast), and then divide by 7 to get the time it would take 7 men (who would progress seven times as fast), namely 40⁄7 = 55⁄7 days.
In general, the Rule of Three can be applied to problems that can be described in terms of proportions, like this:
Given three numbers, find a fourth in such proportion to the third as the second is to the first.
And the rule then says:
First place the numbers in such order that the first and third be of one kind, and the second the same as the number required. Multiply the second and third numbers together, and divide the product by the first, the quotient will be the answer to the question.
In modern algebraic terms, given numbers a, b, c, we have to find a number x such that x is to c as b is to a, i.e., so that
x : c = b : a
To solve this problem, you multiply both sides by c to get
x = b·c / a
The equation a·x = b·c that you get by multiplying the original proportion by a·c was sometimes referred to as “the product of the means equals the product of the extremes”.
In the case of the trench problem, a = 7, b = 4, c = 10 (so the first and the third are of one kind, namely number of men, and the second is of the same kind as the number required, namely number of days), and the rule says the answer is given by multiplying 4 by 10, to give 40, and then dividing by 7, to give 55⁄7 days.
The Rule of Three was known in China as early as the first century. Indian texts discussed it from the fifth century onward—it appeared in the Āryabhatīya and was extended and elaborated upon in Bhāskara’s commentaries, in which he applied it to problems quite similar to those analyzed by Leonardo. It was introduced into the Islamic world in about the eighth century. Renaissance Europe called it the Golden Rule, presumably because of the importance in commerce of solving relative proportions problems (which arise all the time in pricing and exchange) and the consequent utility of having a simple mechanical procedure that anyone could use. It was the basis for two of the other rules that were in common use at the time: the Rule of False Position and the Rule of Double False Position.
In modern terms, the Rule of False Position solves simple linear equations of the type Ax = B. Used by the ancient Egyptians, it consists of making a guess at the solution and then finding the correct answer by forming a proportion with the desired result:
result from guess : target value = guess : solution
For example, we can use the method of False Position to solve the following problem from the famous Rhind Papyrus, currently housed in the British Museum:
Find a quantity such that when its seventh part is added to it gives 19.
In this problem, the target value is 19. We must start with a guess for the answer. It doesn’t really matter what we choose, but it is best to select something that makes the arithmetic easier. Since the problem involves the seventh part of the answer, a good guess is 7. If we make that guess, then adding a seventh to it gives 7 + 1 = 8. That is the result from the guess. So, according to the rule:
8 : 19 = 7 : solution
By the Rule of Three, you get solution by multiplying 19 by 7 and dividing by 8, namely 133⁄8, or 165⁄8.
Leonardo devoted most of his lengthy chapter 12 to solving problems using the Rule of False Position. He divided it into nine parts, each devoted to problems of a particular kind. The third part was titled “Problems of trees, and other similar problems for which solutions are found”, and contains almost a hundred solved examples. The first five problems established what Leonardo meant by a tree problem: One wants to know the total length of a tree when given the proportion that lies beneath the ground. Presenting such problems in terms of trees was simply a convenient artifact; many practical problems that arise in commerce require finding the total amount based on a known proportion. It is because Leonardo solved such problems using the Rule of False Position that he called the method the “Tree Rule”. The first problem says:
There is a tree 1⁄3 and ¼ of which lies under the ground, and it is 21 palms. It is sought what is the length of the tree.13
The expression “1⁄3 and ¼” means “1⁄3 + ¼”; expressing fractional parts using sums of reciprocals was common practice at the time. Leonardo first solved the problem using the Rule of Three he discussed earlier:
Because ¼ and 1⁄3 are found in 12, you understand the tree itself to be divided into 12 equal parts, of which a third and a fourth, that is 7 parts, are 21 palms. Therefore proportionally, as the 7 is to the 21, so the 12 parts to the length of the tree. And because whenever four numbers are proportional, the product of the first by the fourth is equal to the product of the second by the third, therefore if you multiply the second 21 by the known third 12, and you divide by the first number similarly, namely by the 7, there will come up 36 for the fourth unknown number, namely for the length of that tree.
Next he introduced his Tree Rule (i.e., the Method of False Position):
There is indeed another method which we use, namely that you put for the unknown thing some arbitrary known number which is divided evenly by the fractions which are put in the problem itself. And according to the posing of that problem, with that put number you strive to discover the proportion occurring in the solution of that problem.
Leonardo then used this rule to obtain his alternative solution:
For example, the sought number of this problem is the length of the tree. Therefore you put that to be 12, since it is divided evenly by the 3 and the 4 which are under the fraction lines. And because it is said 1⁄3 and ¼ of the tree are 21, you take 1⁄3 + 1⁄4 of the put 12. There will be 7, which if it were by chance 21 we would certainly have what was proposed, namely that the tree be 12 palms. But because 7 is not 21, it turns out therefore, proportionally as the 7 is to the 21 so the put tree to the sought one, namely the 12 to the 36. Therefore one is in the habit of saying, “For the 12 which I put, 7 results. What should I put so that 21 results?” And if it is said this way, the outermost numbers should be multiplied together, namely the 12 by the 21, and the result should be divided by the remaining number.
In terms of the arithmetic needed, Leonardo’s two methods are the same. But they are conceptually different, and Leonardo drew a clear distinction between them. The first method, the Rule of Three, uses a convenient subdivision of the unknown; the second method, the Tree Rule, involves guessing a value for the unknown and then correcting it by a proportion calculation.
The Rule of Double False Position, the focus of Leonardo’s chapter 13, seems to have its origins in China sometime before 100 C.E., where it was called Ying pu tsu (too much and not enough) or sometimes the thia nu rule, thia meaning “the latest appearance of the waning moon”, nu “the earliest appearance of the waxing moon”. It was known to the Arabic mathematicians, who called it hisab al-khata’ayn, and was described in the works of al-Khwārizmī in the ninth century. Medieval European scholars called it elchataym or el cataym. It can be used to solve linear equations not only of the form Ax = B, for which the Method of False Position can be used, but also the more general form Ax + B = C.
The rule begins with two guesses to the answer and then uses a systematic method (or perhaps inspired guesswork) to refine them. A simple application would be to solve a problem like this:
A, B, and C built a house which cost $500, of which A paid a certain sum, B paid $10 more than A, and C paid as much as A and B both; how much did each man pay?
With such simple numbers, we could solve this problem by guesswork. A solution by the Method of Double False Position would go like this: We begin with a low guess, say A pays $80. Then B pays $90 and C pays $170. This means that altogether the three pay $340. This answer has an error of 500 – 340 = 160 dollars. Now we start with a high guess, say A pays $150. Then B pays $160 and C pays $310, giving a total of $620 for all three. This time the error is 500 – 650 = –120 dollars. Expressed in modern terms as a formula, the Rule of Double False Position says that the correct answer for A is given by
This gives:
Thus A pays $120, B pays $130, and C pays $250.
A modern solution would use algebra, formulating three equations that represent the information given:
A + B + C = 500, B = A + 10, C = A + B
Three equations, three unknowns, which you solve by substitution to give A = $120, B = $130, C = $250.
Leonardo used the rule to solve the following problem in chapter 13 of Liber abbaci:
A certain man went on business to Lucca, next to Florence, and then back to Pisa, and he made double in each city, and in each city he spent 12 denari, and in the end nothing was left for him. It is sought how much he had at the beginning.14
He calculated as follows:
You indeed put it that he has 12 denari of which he made double in the first trip, and thus he had 24 denari from which he spent 12 denari and there remained for him another 12 denari, of this he made double in the remaining two trips and he spent in each 12 denari; there remained for him at the end 12 denari. Therefore in the position I erred in value by plus 12; therefore you put it that he had 11 denari from which, as he made double in the three trips and spent in each 12 denari, there remained for him at the end 4 denari, namely 8 fewer than in the first position. And therefore this position is too large. Whence you say: for the 1 which I decreased in the capital I approximated more closely by 8; how many shall I decrease again so that the approximation is decreased by 4 further? You therefore multiply the 4 by the 1, and you divide by the 8, the quotient will be ½ of one denaro; this subtracted from the 11 denari leaves 10½ denari for the capital.
In modern terminology, Leonardo’s first guess was 12 with the resulting first error 12, and his second guess was 11 with the second error being 4, so the formula says the correct answer is
The formula—that mysterious-looking rule—is simply a modern way to describe the method. Medieval mathematicians would set up a proportion between the two incorrect answers and use the Rule of Three, which is what Leonardo was doing. The simplest solution is the modern one using symbolic algebra. You let A be the amount the man starts out with, then calculate that after visiting Lucca he has 2A – 12, after Florence 2(2A – 12) – 12 = 4A – 36, and after returning to Pisa 2(4A – 36) – 12 = 8A – 84. This final amount must be 0, so 8A = 84, giving A = 10½.
The algebraic method was known to Leonardo, indeed to the Arab scholars centuries before him, and was described by al-Khwārizmī around 830 in his algebra book. Without symbolic notation, algebraic solutions were not significantly shorter than the other methods he described; nevertheless, starting in the third section of chapter 12 of Liber abbaci, Leonardo gave many examples of how to solve problems using algebra. He introduced his alternative approach with a puzzle about money:
[I]t is proposed that one man takes 7 denari from the other, and he will have five times the second man. And the second man takes 5 denari from the first, and he will have seven times the denari of the first.15
The problem is to determine how much each man starts off with. First, Leonardo solved it by the Tree Method, getting the answer that the first man has 72⁄17 denari and the second 914⁄17. He then said he would solve it by an alternative approach—what he called the “Direct Method” (regula recta): One begins by calling the sought-after quantity a “thing” (res) and then forms an equation (expressed in words) that is solved step-by-step to give the answer. Expressed symbolically, this is precisely the modern algebraic method. Without modern symbolic notation, Leonardo’s solution looks complicated:
In solving problems there is a certain method called direct that is used by the Arabs, and the method is a laudable and valuable method, for by it many problems are solved. If you wish to use the method in this problem, then you put it that the second man has the thing and the 7 denari which the first man takes, and you understand that the thing is unknown, and you wish to find it, and because the first man, having the 7 denari, has five times as much as the second man, it follows necessarily that the first man has five things minus 7 denari because he will have 7 of the denari of the second; thus he will have five whole things, and to the second will remain one thing, and this the first will have five times it; therefore if from the first man’s portion is added 5 to the second’s that he takes, then the second will certainly have 12 denari and the thing, and to the first will remain five things minus 12 denari, and thus the second has sevenfold the first; that is because one thing and 12 denari are sevenfold five things minus 12 denari; therefore five things minus twelve denari are multiplied by the 7, yielding 35 things less 7 solidi28 that is equal to one thing and one solidi; therefore if to both parts are added 7 solidi, then there will be thirty-five things equal to one thing and 8 solidi because if equals are added to equals, then the results will be equal. Again if equals are subtracted from equals, then those which remain will be equal; if from the above written two parts are subtracted one thing, then there will remain 34 things equal to 8 solidi; therefore if you will divide the 8 solidi by the 34, then you will have 214⁄17 denari for each thing; therefore the second has 914⁄17 denari, as he has one thing and 7 denari. Similarly, if from five things, namely from the product of the 214⁄17 by 5, are subtracted 7 denari, then there will remain 72⁄17 denari for the first man, as we found above.16
To a modern reader, this does not look much like algebra. But Leonardo’s phrases “if equals are added to equals, then the results will be equal” and “if equals are subtracted from equals, then those which remain will be equal” provide a clue to what he was doing. When Leonardo wrote “thing” (in Latin res or cosa), he meant “the unknown quantity”, or more familiarly, x. In essence, Leonardo was formulating two linear equations and then solving them by adding and subtracting equals to each side and by substitution.
Avoiding the introduction of solidi (Leonardo replaced 12 denari by 1 solidus) and working throughout with denari, a present-day solution would look like this:
Let A be the amount the first man has, B the amount of the second. The conditions of the problem can be expressed as the equations
A + 7 = 5(B – 7)
B + 5 = 7(A – 5)
These can be rearranged to give
A = 5B – 42
B = 7A – 40
Use the second equation to replace B in the first:
A = 5(7A – 40) – 42
A = 35A – 200 – 42
34A = 242
Substituting for A in the earlier equation for B:
This is not exactly a translation of Leonardo’s calculation into symbolic form, since he had only one unknown, whereas today we find it more natural to start with two, one for each man in the problem. But this is just a symbolic technicality, though it does raise the interesting question—to which we have no answer—of how he came up with his particular “thing”.
Liber abbaci ends very abruptly with the passage:
And let us say to you, I multiplied 30-fold of a census by 30, and that which resulted was equal to the sum of 30 denari and 30-fold the same census; you put the thing for the census, and you multiply the 30 things by the 30 yielding 900 things that are equal to 30 things plus 30 denari; you take away the 30 things from both parts; there will remain 870 things equal to 30 denari; you therefore divide the 30 by the 870 yielding 1⁄29 denaro for the amount of the thing.
Leonardo was discussing an arithmetic problem solved by algebra. “Census” is the Latin translation of the Arabic mal (amount of money, or fund), and is used in many Arabic arithmetic problems, no matter what method is used to solve it. “Mal” (and “census”) is also the name of the second degree unknown in medieval algebra.
To a modern reader, this ending may seem a little strange. There is no conclusion, no summary, no reflection on what has been accomplished, no suggestion of new work to be done or of new things to try. Leonardo simply completed his description of the solution to yet another problem and then stopped writing. In fact, many medieval mathematical texts ended in this manner. The most an author might add was a brief statement about the copyist and the date, perhaps with thanks to God. Authors did not write conclusions the way they do today.
But what for Leonardo was the end of a project marked the beginning of an arithmetic revolution that would sweep through Europe. To be sure, one should not overdramatize the effect of just one man and one book. Western Europe, and Italy in particular, had entered a period of significant change before Leonardo was born, in commerce, finance, and education. Many of those changes depended on arithmetic, which provided a highly receptive environment for Liber abbaci to have an impact. There is good reason to believe Leonardo was very aware of that situation and wrote his book precisely to meet that need. Certainly, by the time he wrote his second edition—the only one we can read today—he knew of the importance of his work, as is reflected in the way he presented it.
Hindu-Arabic arithmetic was slowly finding its way across the Mediterranean by conduits other than Liber abbaci and would surely have come to dominate European trade, commerce, and finance eventually, even if Leonardo had not written his book.
The greatness of Liber abbaci is due to its quality, its comprehensive nature, and its timeliness: It was good, it provided merchants, bankers, businesspeople, and scholars with everything they needed 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 (Luca Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalità, published in Venice in 1494).