Spanish Colonial Mathematics: A Window
on the Past
Edward Sandifer
Ed Sandifer ([email protected]) is professor of
mathematics at Western Connecticut State University. His
Ph.D. in mathematics is from the University of
Massachusetts in Amherst. An initial interest in ring theory
and combinatorics evolved into an interest in the history of
mathematics under the influence of the Institute for the
History of Mathematics and Its Use in Teaching. He is
especially interested in Spanish Colonial Mathematics and
in the work of Leonhard Euler. He amuses himself and his
neighbors by running, and has run the Boston Marathon
every year since Nixon was president.
Introduction
If mathematics is a window on the world, then the history of mathematics is a window
on the past. For the English colonies in America, that window is largely closed to us.
By the time James Hodder’s arithmetic book was published in Boston in 1719, the
Colonial Period was two-thirds over. It may surprise a North American reader to know
that by that time, at least eleven works of mathematics had been written and published
in the Spanish colonies.
For our definition of “works of mathematics”, we use the authority of L. C. Karpinski’s 1940 Bibliography of Mathematical Works Printed in America Through 1850 [2].
For early works, Karpinski considered a work to be mathematics if it contained a chapter or more on mathematics. For later works, he was more stringent. Recent work by
Bruce Burdick of Roger Williams University in Rhode Island, as yet unpublished, has
turned up some Spanish colonial books containing symbolic logic. However, since
logic was a topic in a law curriculum at the time, and not of a mathematical, scientific
or commercial curriculum, Karpinski did not include those books. Burdick has also
come across some Spanish colonial books on astronomy and navigation with some
mathematical content that perhaps Karpinski overlooked. His is still work in progress
so, for us, a book is a work of mathematics if Karpinski says it is one.
Using seven of these eleven books as guides, we are going to take a 150-year tour
of the Spanish colonial world. We will look at the mathematics in those books and
how the colonists adapted it to fit the problems of the New World. We will see how
and why the Spanish colonies developed differently from other American colonies,
especially the English colonies. Through all this, we will get a glimpse of colonial life
in America four hundred years ago.
A bit of history
There may be readers who already know or still remember what they learned of colonial history. We ask them please bear with us as we review some of the points important
to our story.
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1440s Gutenberg’s printing press
1492 Columbus first visits America. Permanent colonies established within five
years.
1519 Cortez arrives in Mexico
1536 First printing press set up in Mexico City
1556 First Spanish colonial mathematics book, written and published in Mexico
1607 English settle in Jamestown, Virginia
1719 Hodder’s Arithmetick, written in England, first printed in Boston
The Spanish and the English had clearly different purposes in holding colonies in
America. The Spanish used their American colonies as a source of gold and silver,
and they regarded it as their responsibility to convert the indigenous people to Christianity. The English, on the other hand, used their American colonies as a source of
raw materials for manufacturing and as a market for manufactured goods. The English
avoided allowing manufacturing technology to be brought to America, since goods
manufactured in America might reduce demand for goods manufactured in England.
The Spanish, though, had no qualms about sending then-modern technology to the
New World, so Spanish colonies published mathematics a century and a half before
the English colonies did.
For comparison, we give a table showing the first New World mathematics books
in each of the nine languages listed in Karpinski.
1556
1696
17031
1730
1742
1775
1813
1833
1835
Spanish
Latin (published in Peru)
English
Dutch
German
French
Portuguese
Hawaiian
Choctaw
Among Cortez’s company when he arrived in Mexico was a friar, Juan Diez. Brother
Juan, besides being responsible for the spiritual care of the Conquistadores, served as a
kind of secretary and treasurer for the colony. He addressed hundreds of documents to
the King of Spain to report the state of the Spanish colonies in the New World. He was
also assigned the task of converting the Aztec emperor Montezuma to Catholicism.
In 1556, thirty-seven years after Cortez’s arrival, twenty years after the printing
press was set up, the first mathematics book rolled off the presses. It was written by
Brother Juan Diez, probably the same Juan Diez who arrived with Cortez. Some people
think that the Juan Diez who arrived with Cortez would have been too old by 1556,
so it might be a different man. But I like the idea of an aging friar, realizing that his
own imminent death would leave the new Spanish colony without anybody skilled in
1 According to Karpinski [2], the first mathematics published in the English colonies was in 1703, John Hill’s
The Young Secretary’s Guide, a reprint of an English work with only a small amount of mathematics. The first
book of mathematics was Hodder’s Arithmetick, in 1719, also written in England. The first one both written
and published in the English colonies was Isaac Greenwood’s 1726 work A course of philosophical lectures . . .
illustrating and confirming Sir Isaac Newton’s laws of matter and motion. This ambiguity about what to count as
“first” does not apply to mathematical works in the other American colonies.
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mathematics, who wrote a book to try to preserve his skills and to pass them on to the
next generation.
The Sumario Compendioso
Following the custom of the times, Diez gave his book a very long title, Sumario
Compendioso de las quentas de plata y oro que in los reynos del Piru son necessarias a
los mercadores: y todo genero de tratantes. Los algunas reglas tocantes al Arithmetica.
Fecho por Juan Diez freyle. (Compendious summary of the counting of silver and gold
that are necessary in the kingdoms of Peru to merchants and all kinds of traders. The
other rules touching on Arithmetic. Made by Juan Diez, friar.) Readers who know
Spanish will note that the Spanish of Brother Juan’s day was somewhat different than
the modern language.
The Sumario is one of the rarest books in the world. Only three copies are known
to exist today, one in the British Museum, one in the Escorial in Spain, and one in the
Huntington Library in San Marino, California. There are, however, at least two facsimile editions of the Sumario [4] and there have been articles about it in the American
Mathematical Monthly [3] and in the Mathematics Teacher [1]. To avoid repeating too
much of what is easily available elsewhere, here we will mention only a couple of the
high points from the Sumario.
The Sumario in the Huntington Library consists of 103 folios, sheets numbered on
one side but printed on both. The other copies have 105 folios. Most of these involve
pages and pages of tables in Roman numerals. The tables helped in calculations for
assaying gold and silver and for assessing taxes. By using them, a person could be an
assayer or a tax collector who knew only how to add and subtract and without knowing
how to multiply or divide.
Beyond the tables, the remaining pages are mostly simple one-variable algebra
problems, including quadratics, and some rules for converting among the many money
systems in use at the time. They also include an example of how to multiply. Diez
apparently did not have a very good sense of sequencing, for his multiplication example comes after a number of algebra problems that require multiplication for their
solution.
Diez’s method of multiplication is interesting, and it is worth a closer look. Figure 1
shows how Diez performs the calculations to multiply 875 by 978. We will leave to
the reader most of the exercise of figuring out how the method works, but here are a
few hints. In the first row below the first line, 72 is 8 times 9. The next calculation is
8 times 7. The 5 in 56 goes in the second row under the 2, and the 6 goes in the first
row. The next calculation is 8 times 8, and we leave the rest to the reader.
Figure 1.
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This method of multiplication, as well as Diez’s examples of algebra and quadratics
resemble some Italian mathematics in the late 1400s. This suggests that perhaps Diez
or his teachers may have studied in Italy.
Diálogos militares
The second Spanish colonial mathematics book followed the Sumario by almost thirty
years, coming in 1583. Also published in Mexico City, Diálogos militares, de la formación è información de personas, instrumentos, y cosas necessarias para el buen uso
dela guerra (Military dialogues on the formation and information about personnel, instruments, and things necessary for the good conduct of war.) The book is in the form
of a dialog between “M. Montañes” and “V. Viscayno”, which translate roughly as
“Country Bumpkin” and “Wise Gentleman” respectively. Montañes asks advice about
being a good soldier and officer, and somehow manages to ask exactly the right questions to lead the discussion along.
Diego Garcia de Palacia, author of Diálogos militares and its sequel Instructión
náuthica, which we will discuss below, seems to have been quite a character. He was
born in Spain about 1530, and nothing else is known of the first forty-five years of his
life. By 1576, he was Auditor of Guatemala, a high bureaucratic position. In 1578 he
offered to conquer and pacify the Philippine Islands on behalf of the King, at his own
expense, but the King apparently did not accept the offer. Apparently, being Auditor
was a lucrative career. In 1581 he was sent to Mexico City where he was awarded a law
degree and made Rector of the University. Later, he commanded an expedition against
Sir Francis Drake and the English in the South Sea, but no details of that expedition
seem to remain. Somehow, he found the time to write two books on military science,
both with some mathematical content. Nobody seems to have recorded for posterity
when, where, or how he died, but he is reported to have died disgraced and broke.
In the 192 folios of Diálogos militares de Palacia gives us four chapters. The first
chapter, “Qualities of a soldier and officer” is an essay on ethics and morals. The
second chapter, “Advice on going out to war” is an essay on logistics and strategy.
Chapter 3, “Nature and composition of gunpowder, rules of perspective, instruments”
is what the title says. The fourth chapter is titled “Formation of squadrons”.
Chapter 3 includes ten different gunpowder recipes. Five are designated for the use
of artillery and five for arcabuzes or rifles. Eight of the ten recipes are given as ratios
of the ingredients saltpeter, sulfur, and carbon. The ratios given for rifle gunpowder
are 7:2:1, 6:2:1 and 18:2:3. The second of these, de Palacia tells us, is the recipe of the
“miracle worker Albertus Magnus”. The five ratios for artillery gunpowder are 9:2:3,
100:10:30, 100:20:38, 10:2:3 and 9:1:2. He doesn’t tell us whether the measures are
to be taken by volume or by weight. Either way, that is quite a range of ratios, and it
is no wonder that gunpowder was often unreliable. We leave it to the reader to decide
if there is a substantial difference between gunpowder for artillery and that for rifles.
For reasons of safety, we suggest analysis rather than experimentation.
One of the two remaining recipes for rifle gunpowder is to mix five parts saltpeter
and one part each of sulfur, carbon, and a substance I have not been able to identify,
carcoma. The other recipe is the only one that uses units instead of ratios: 7 pounds of
saltpeter, 11 ounces of sulfur and 1 pound of carbon “made from orange peels.”
Now that we understand gunpowder, de Palacia explains why it is necessary to
aim above a target in order to hit it. Figure 2 shows how he thought the path of the
bullet has to be above the line of sight, so that, when the bullet runs out of speed, it
will fall onto the target. It would be another twenty years before Galileo, some 4000
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Figure 2.
miles away, would show that trajectories were parabolic, and so de Palacia’s analysis
would not be accepted today. On the other hand, his trajectories were not the triangular
trajectories described by the ancient Greeks. Rather, they more resemble the curved,
but not parabolic, trajectories described by Tartaglia in Italy in the 1400s. We moderns
might scoff that de Palacia’s theory was incorrect, but we should not forget that the
theory worked for him. His bullets hit their targets.
Also in Chapter 3 we learn about some instruments for measuring distance and
elevation and how to use those measurements in aiming cannons. Figure 3 shows two
of the illustrations from this chapter. The first shows a device for measuring the angle
at which a cannon is aimed. One of its arms is placed in the cannon barrel, and the
plumb line indicates the angle of elevation in “points”. The second illustration is one
of a series of similar illustrations, this one showing the cannon elevated one point.
Figure 3.
It is the fourth chapter, “The formation of squadrons” that has the most actual mathematical content. Suppose, for example, that your squadron has 25 men, and you want
to have them march in the form of a square. He shows us that you should march your
men in five rows of five men each.
For his second example, he shows that you can have your 25-man squadron march
in the form of a triangle by having one man in the first row, three in the second, then
five, seven, and nine men. This takes several pages, and climaxes with an illustration
showing how to have a 100-man squadron march in the form of a triangle.
If triangles aren’t enough, we learn how to march 500 men in the form of a cross
(Figure 4), a crown or an octagon, each carefully illustrated with a diagram of typeset os. Lest we think that de Palacia was concerned with showing off with fancy marching formations, he does give military reasons for each of the formations. The cross, for
example, allowed him to put riflemen at the eight outside corners where they would not
interfere with each other, and the octagon did not leave the individuals at the corners
so exposed to the enemy.
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Figure 4.
The Pragmatica
In 1584, the year after the publication of the Diálogos militares, a printing press was
set up in Lima, Peru. The first item printed on the new printing press was a four-page
edict of the Spanish King Philip, Pragmática sobre los diez días del año (Practical
things about the ten days of the year). The Catholic world was in the process of converting from the old Julian calendar to the new Gregorian calendar, and that required
skipping ten days on the calendar. The Pragmática commanded that the date to follow
October 4, 1583 would be October 15, 1583. The news was almost a year late in reaching the New World! The Pragmática also told how certain practical things like rent,
taxes, salaries, executions and, most importantly, saints’ days, were to be handled.
The King was fair in his proclamation: most things were to be prorated to account
for the shortened month. For things like executions, feast days, and saints’ days, the
ten days were “double days”. That is, on October 15, all the celebrations and events
scheduled for October 5 or October 15 would be held. Likewise, on October 16, they
would hold the celebrations both for October 16 and October 6, and so forth. No
saint would miss his or her day, no feast days would be skipped, and no condemned
people would go unexecuted. It is hard to know how useful the proclamation actually
was, arriving a year late.
It is not clear why Karpinski chose to call the Pragmática a work of mathematics.
It is interesting, and only one copy survives, but it contains no actual mathematics.
Perhaps he included it because it was the culmination of many years of mathematical
and astronomical controversy that had involved many of Europe’s most prominent
mathematicians.
Instrución náuthica
Four years after the success of the Diálogos militares, de Palacia published a sequel,
Instructión náuthica: para el buen uso, y regimiento de los naos, su traça y y govierno
conforme à la altura de México (Nautical instruction: for the good use and arrangement
of ships, their motion and operation adapted to the latitude of Mexico). Somebody,
probably a book dealer about a hundred years ago, has written in pencil on the inside
cover of the copy of Instructión náuthica at Brown University “Very Rare £45 only
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one other copy known”. If the inscription is accurate, then prices of rare books gone
up considerably in the last century, and the book is as rare as possible without being
unique.
Despite its rarity, Instructión náuthica was an influential and fairly well known
book, and it has been translated into English. The main reason for its notoriety is a 39page Vocabulario at the end of Chapter 4 detailing “the words and phrases for talking
with people of the sea”. People who do reconstructions of old ships like the ones
Columbus used, and nautical historians in general, think that de Palacia’s Vocabulario
is a gold mine of technical detail.
Again, de Palacia writes in four books. He tries to continue the dialog between
Montañes and Viscayno, but the form does not work so well for him this time. Most
of Viscayno’s answers are chapter-length, and Montañes has hardly anything to say.
In the first book, Viscayno explains that the earth is a sphere. Contrary to what you
may have learned in high school, this was known centuries before Columbus, but it
became useful information only when people began sailing across significant pieces of
the globe. We learn about the declination of the sun, the Biseisto or Leap Year, and tide
tables. He tells us how to use a quadrant and an astrolabe and how to calculate latitude.
He concludes the book with lessons on how to use the North Star or the Southern Cross
to determine what time it is at night.
The second book is mostly devoted to finding what was then called el Aureo numero,
or the “Golden Number”. This is in no way related to the number we sometimes call
the “golden ratio”. Explaining the Golden Number will take a bit of astronomy. As we
all know, the moon orbits around the earth approximately once a month in an orbit is
not a perfect circle, but closer to an ellipse. The moon’s closest point to the earth is
called perigee, and is approximately 360,000 km. The most distant point is apogee and
is about 405,000 km. The moon’s orbit is not always in the same position around the
earth. That is to say, the positions of apogee and perigee drift slowly over the years,
in a 19-year cycle. That 19-year cycle is one of the keys to predicting eclipses. To
16th century navigators it was important to know not only the phase of the moon (full,
crescent, new, etc.) and its position in the Zodiac, but also its position in its apogeeperigee cycle. To keep track of this cycle, each year had its Golden Number, a number
from 1 to 19 telling where the year is in the apogee-perigee cycle. Our Viscayno tells
us how to find the year’s Golden Number: essentially, add one to the year number,
modulo 19.
Since de Palacia couldn’t expect that all of his readers would know how to divide,
he shows us how to calculate the Golden Number with tricks involving the digits of
the year number. Then he shows us some more tricks for counting on your knuckles
for finding the Golden Number for years from 1500 clear up to 2300. He hoped that
his book would still be useful for almost 800 years!
The third chapter covers some basic astrology and some tables of the moon. The last
chapter describes some of the different ships in use at the time and the responsibilities
of the various crew members, before concluding with that famous Vocabulario.
Libro de plata reduzida
We are now at the sixth book on Karpinski’s list, published by Francisco Juan Garreguilla in 1607 with the title Libro de plata reduzida que trata de leyes baias desde
20 Marcos, hasta 120. Con sus Abezedarios al margen. Con una tabla general a la
postre. (Book of the reduction of money which treats the basic laws from 20 points up
to 120. With its indexes in the margin. With a general table at the end.) Although this
is mostly a big book of tables, there are a few interesting things about it.
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Figure 5.
There is a bit of irony in the title, where the phrase “reduction of money” is used
where today we might call it “taxation”.
These tables supplement the tables in the Sumario, and another set of tables published in 1597 by Joan de Belveder. Garreguilla was Contador or Treasurer in Lima.
He mentions both Juan Diez and Joan de Belveder in his preface, highlighting the fact
that these books were not written as museum pieces. Rather, they were written because
people needed them and used them.
Like most books of the time, the Libro de plata reduzida has a fairly long dedication
to one of the author’s patrons, or at least somebody the author hopes will be his patron.
Most books also have a few words written by somebody else in praise of the author.
This book features three sonnets telling us the wonderful deeds and character of the
author. One of them, “Sonnet to the author of this useful and excellent work, under
the license of Pedro de Oña.”, is given in Figure 5. It continues “from the first one in
your lineage, which gave life to Mercury and strength to Mars,” and finally “May our
sun, Garreguilla, never set.” Those folks really knew how to write a strong letter of
recommendation.
The tables in the Libro de plata reduzida told the tax assessor how much tax to
charge on various amounts of money and for many different tax rates, called “points.”
They are more extensive and more general than the tables in the Sumario compendioso,
which were designed primarily for assessing taxes on mined gold and silver. With the
Libro de plata reduzida, taxes could be levied on almost anything.
The Libro de plata reduzida features notched pages, like the notched tabs in some
dictionaries. These are the Abezedarios mentioned in the title. The user could find the
interest rate he needed marked on one of the notches and open the book directly to
that section. The word abezedarios is an interesting one. It has the same origins as
the word “alphabet”. The initial “a” is from the letter A in the Spanish alphabet. The
“be” is the letter B, and the “zed” is the letter Z, or maybe a corruption of the letter C.
The syllables “-arios” play the same role in Spanish as the endings “-ites” or “-ness”
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play in English, making the rest of the word into a noun. So, the word abezedarios
could be translated as “alphabet thingies”. The word does not seem to be in many
modern Spanish-English dictionaries, and it was a bit of a challenge to figure out how
to translate it.
Finally, perhaps this book should be considered two books instead of just one.
About three quarters of the way through the volume, there is another title page, almost identical to the first, but with a slightly different title, Libro de plata reduzida
que trata desde treynta Marcos, hasta ciento y veynte y neuve de toda ley, de dos mil
trezientos y ochenta. Con su Abezedario al margen. Con tres tablas ala postre (Book
of the reduction of money, which covers from 20 points up to 129, and for each law
to 2380 [points]. With its index in the margin. With three more tables at the end.) In
those days, publishing anything required permission from the King, and this section
has its own separate licencia. The other materials, dedication, poems in praise of the
author, are all the same as for the first part, but they are printed a second time, as if the
section of the volume were intended to stand alone. Finally, a careful reading of the
title shows that this part of the book has its own Abezedario, in the singular, setting it
apart from the rest of the book.
Breve aritmética
We jump to 1675 now, not because there were no Spanish colonial mathematics books
between 1607 and 1675, but because we haven’t had a chance to see any of the three
works published in that period. Benito Fernandez de Belo’s book is Breve aritmética
por el mas sucinto modo, que hasta oy se ha visto. Trata en las quentas que se pueden
ofrecer para formar campos y esquadrones. (Brief arithmetic for the most succinct
method which has been seen up to today. Treating calculations that one can do for the
formation of camps and squadrons.) It really is brief, eleven folios of text preceded by
four folios of dedications. There are two poems proclaiming the value of the book and
the generosity and heroism of its author.
The purpose of this short book seems to be to amplify on the last chapter of Diálogos
militares, which was by then almost a hundred years old. That last chapter was about
the formation of squadrons, and it required division and the extraction of square roots,
calculations that had not yet been described in any Spanish colonial mathematics
books. The Breve aritmética seems designed to fill that gap. The first of its four chapters explains an algorithm for division. The method is now called the “galley method”,
and, though it seems strange to modern times, it was the most common algorithm of the
time. The second chapter is about areas of circles. It uses the popular approximation
of 22/7 for π.
Chapter 3 is one of the oddest chapters anywhere. Its title is “Example and proof of
the first squadron arranged in a square”, and it consists only of 576 os arranged in a
24-by-24 square. There are no words in the chapter.
Chapter 3 motivates Chapter 4, which begins with a description of square numbers. Belo explains that, if you know how many men are in your squadron, then you
have to take a square root in order to find how big a square they will make. Then he
demonstrates, using 576 as an example, how to extract square roots.
Not all of Belo’s examples are rigged to come out even. He asks us to find the
dimensions of a squadron in terreno formation and containing 800 men. For some
now obscure reason, a formation is terreno if it is 7/3 times as long as it is wide.
To solve the problem, he first multiplies 800 by 7, then divides that by 3, giving 1866.
Then he takes the square root of 1866 and gets 43, the number of rows in the squadron,
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and divides that into 800 to get 18, the number of men in each row. Belo works out all
these calculations in detail. He also gives us another 43-by-18 array of os, which are
supposed to prove the solution is correct. It doesn’t seem to bother him at all that he
has 26 men left over.
After a few similar examples, Belo turns to triangles, showing that 324 men can
march in 18 rows of lengths 1, 3, 5, . . . , 35.
Now we come to what Belo clearly thinks is the climax of his little book, forming
squadrons into pentagons. He tells us that pentagon shapes are also known as “Bonete,
ò mitra”, a bonnet or mitre. Since the formation already has names, Belo may not
have been the first one to think of marching in pentagons. His analysis depends on the
dissection of the pentagon into two triangles and a rhombus as shown in Figure 6. To
find the dimensions of the pentagon, he tells us to divide the number in our squadron
by 5, then double that and take the square root. With luck, we get a fraction. Round that
fraction down to get the isosceles sides of the two triangles, and round it up to get the
sides of the rhombus. In his example of 278 men in the squadron, his calculation goes
278/5 = 55 (ignoring the fraction), doubled gives 110. The root of this is between 10
and 11.
Figure 6.
Belo checks his work by finding the number of men in each piece. The big triangle
has 1 + 3 + 5 + · · · + 19 = 100 men. The little triangle has 1 + 2 + 3 + · · · + 10 = 55
men, and the rhombus has 11 · 11 = 121 men. This adds up to 276, and Belo notes that
there are two people left over.
In another example, Belo forms an octagon with 600 people in it, leaving 15 left
over. We leave it to the reader to discover Belo’s dissection and analysis for the octagon.
Unlike the Diálogos militares, Belo gives us no military reasons for forming octagons or pentagons with our squadrons. It seems he just thinks they are interesting.
Bound in at the end of the book is a beautiful 10 by 13 fold-out woodcut illustration (Figure 7) that can only be described as a doodle. This illustration is not
mentioned in the text. It contains sketches of fortifications and doodles of cannons,
with a big, curly signature “Belo” in the middle. It looks like every copy of the Breve
aritmética was an autographed copy.
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Figure 7.
Cubus et sphaera
The last Spanish colonial mathematics book we consider is also the last one published
before the English colonies joined in publishing mathematics books. It is the eleventh
book in Karpinski, published in Peru in 1696. It is the first one written in Latin.
In Cubus, et sphaera geometrice duplicata, Juan Ramón Coninkius gives what he
believes is a ruler and compass construction that doubles the volume of a given cube
or sphere. We now know that such a construction is impossible, so there must be some
mistake in Coninkius’s work. Now we call people who attempt cube doubling and its
sister problems, circle squaring and angle trisecting “mathematical cranks”. In 1696
they were still legitimate and popular problems and most important mathematicians
at least tried them. The better mathematicians were good enough to find their own
mistakes before they published their efforts.
Coninkius was the Cosmographer of Peru and held a chair in mathematics at the
University there. He had come to the New World from Belgium after receiving a solid
mathematical education in Europe. He certainly loved the New World. His introduction
is a flowery praise of New World ideas and opportunities, and he offers his solution
to the centuries-old problem as proof that civilization, which he thought had become
stale and stagnant in Europe, was about to rise to new heights in the Americas.
History suggests that Coninkius’s ideas about the future might have been a little
wrong, and we know there was something wrong with his mathematics as well. His
analysis begins with doubling a square. In the course of that analysis, he tells us that
the square root of 200 is exactly 144/26, a value that is approximately 0.01 too large.
He then tries to use this to derive a cube root of 200. Though we know he must have
made more mistakes, it is nearly impossible to find them. What Karpinski says is the
only remaining copy of his book, the one at Yale University, is missing its fold-out
illustrations. Without them, the “proof” is impenetrable.
The end of the era
We have looked at seven of the first eleven Spanish colonial mathematics books, all
published before 1700 and all published before any English colonial mathematics
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books. By the time the twelfth Spanish colonial mathematics book was published in
Guatemala in 1732, the English had published thirteen and the Dutch had published
one. The silver and gold mines in Mexico and South America were depleted, and the
English colonies were growing wealthier. The roles of the American colonies in their
respective empires had changed, and that change was reflected in their mathematics
books. The Spanish colonies continued to produce new mathematics books at about
the same rate through 1800. By then, though, the English colonies had become the
United States of America, and they were producing a dozen new mathematics books
each year. The era of Spanish colonial mathematics books had ended.
The reader may wonder about the four Spanish colonial mathematics books not
discussed here. I omit them because I haven’t seen them. The first one I missed is
number five in Karpinski, Joan de Belveder’s 1597 book Libro general de reduciones
de plata, . . . . I also missed Karpinski’s number six, Libro de quentas, y reducciones de
plata y oro, . . . , by D. Alvaro Fuentes y de la Cerda, published in Guatemala in 1615.
No copies of this second book remain. Both are books of tables.
Karpinski’s seventh book is by Pedro de Paz, Arte para aprender todo el menor del
arithmética, sin maestro (The art of learning all of the minor art of arithmetic, without
a teacher), published in 1623 in Mexico.
The last one is the ninth one in Karpinski, Anatasius Reaton’s 1649 book Arte menor
de arismética, y modo de formar campos. (Minor art of arithmetic and methods for
forming camps.)
Some interesting points emerge when we look at all eleven Spanish colonial mathematics books at once. First, they are all written in America, to address issues in the
Spanish colonies. They are not reprints or copies of European works. Second, they
mostly fit together with each other, with the exception of the last one about doubling
cubes, and the Pragmática. The four books of tables build one upon the other. Four
books follow the thread begun by the Diálogos militares in a presentation that is more
and more mathematical. Paz’s book is apparently just arithmetic, but it fits in with the
military sequence. The two odd publications, the Pragmatica and the Cubus et sphaera
stand pretty much by themselves.
The reader who has made it clear to the end of this article might want to look farther
into this subject. Alas, there is not much available, and what is available is mostly about
the Sumario compendioso. There are translations of the Sumario, Diálogos militares,
Instrución nauthica and Breve aritmética, but, again except for the Sumario, they are
hard to find. Perhaps this article will inspire interest in the subject. Maybe somebody
will look more deeply into some of these old books and open a little wider this window
on the past.
Recent developments. Bruce Burdick has recently discovered that there is a fourth
copy of the Sumaria Compendioso in the archives at Duke University. He has also
found a second copy of Cubus et sphaera, complete with illustrations, in the British
Museum.
Acknowledgements. I used the copy of the Sumario Compendioso at the Huntington Library
in San Marino, California. The copy of Cubus et sphaera is at the Beineke Rare Book Library
on the Yale campus in New Haven, Connecticut. The other five books are at the John Carter
Brown Library on the Brown campus in Providence, Rhode Island. I sincerely thank these
libraries and their librarians for so generously sharing these rare treasures. I also thank Bruce
Burdick, Marjorie Ewall, Kim Plofker, Danny Otero, and an anonymous referee for their help
and advice.
VOL. 33, NO. 4, SEPTEMBER 2002 THE COLLEGE MATHEMATICS JOURNAL
277
References
1. Shirley B Gray, and C. Edward Sandifer, The Sumario Compendioso: The New World’s first mathematics
book, Mathematics Teacher 94 (2001) #2 98–105.
2. Louis C. Karpinski, Bibliography of Mathematical Works Printed in America through 1850, University of
Michigan Press, 1940.
3. David Eugene Smith, The first work in mathematics printed in the New World, American Mathematical
Monthly 28 (1921) 10–15.
4. David Eugene Smith, The Sumario compendioso of Brother Juan Diez, the earliest mathematical work of the
New World, Ginn, Boston, 1921. This is a facsimile edition and translation of the Sumario.
Mathematics Without Words
Sidney H. Kung (University of North Florida, [email protected]) has an integral
transform that makes at least one definite integral easy:
b
f (x) d x =
a
b
a
=
π/4
ln(1 + tan x) d x =
0
π/8
0
=
π/8
0
=
0
278
(a+b)/2
f (a + b − x) d x =
π/8
( f (x) + f (a + b − x)) d x
a
b
(a+b)/2
( f (x) + f (a + b − x)) d x
(ln(1 + tan x) + ln(1 + tan(π/4 − x)) d x
1 − tan x
ln(1 + tan x) + ln 1 +
1 + tan x
ln 2 d x =
dx
π
ln 2.
8
c THE MATHEMATICAL ASSOCIATION OF AMERICA