Ricardo Manapat - Mathematical Ideas in Early Philippine Society

philippine studies Ateneo de Manila University • Loyola Heights, Quezon City • 1108 Philippines

Mathematical Ideas in Early Philippine Society

Ricardo Manapat Philippine Studies vol. 59 no. 3 (2011): 291–336 Copyright © Ateneo de Manila University Philippine Studies is published by the Ateneo de Manila University. Contents may not be copied or sent via email or other means to multiple sites and posted to a listserv without the copyright holder’s written permission. Users may download and print articles for individual, noncommercial use only. However, unless prior permission has been obtained, you may not download an entire issue of a journal, or download multiple copies of articles. Please contact the publisher for any further use of this work at [emailprotected].

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R i c ard o M anapat

Mathematical Ideas in Early Philippine Society Posthumous Essay

This essay is a preliminary effort in outlining a history of mathematics in the Philippines. It calls attention to the existence of a highly developed enumeration and arithmetical system prior to the Spanish conquest, and argues that this enumeration system had unique characteristics that distinguished it from other Southeast Asian societies. Other mathematical and scientific ideas, such as the use of geometric concepts and astronomical tools, in the preconquest Philippines are also discussed and presented. Keywords: ethnomathematics • preconquest societies • indigenous culture • historiography

PHILIPPINE STUDIES 59, no. 3 (2011) 291–336

I represent a people that is little known to you. Today we

Status Questionis

are lost to civilization in the far reaches of the eastern seas.

The greatest difficulty in attempting a history of mathematics or of mathematical thinking in the Philippines is the absence of sources. There is nothing written on the topic. The accepted standard texts of the history of mathematics such as those by Kline,1 Eves,2 and others, while providing generous space to mathematical developments in the “non–Western” world, do not make even the slightest mention of the Philippines. On the other hand, the thousands of texts written on Philippine history since the 16th century are analogously deficient in that they concentrate on political, economic, social, institutional, or regional history, completely neglecting the history of mathematics or even of science in the Philippines as a separate and important area of concern. This gaping lacuna leads one to almost fall into the temptation to classify the Philippines as one of the histories or cultures which Morris Kline, the dean of the history of mathematics, describes as non-mathematical:

We have no government of our own, we have no flag—but we have a soul, a proud cultural heritage of our ancient Tagala race, and even now after three centuries of Spanish assimilation it is struggling for light and expression. – Juan Luna, 1897

Filipino reading Sir Isaac Newton’s Principia mathematica (1934, 440) experiences a most pleasant surprise upon encountering an explicit reference to the Philippines in this seventeenth-century classic of mathematics and science: [The waters in the Gulf of Tonkin] flow and ebb, not twice, as in other ports, but once only every day; . . . There are two inlets to this port and the neighboring channels, one from the seas of China, between the continent and the island of Leuconia; the other from the Indian sea, between the continent and the island of Borneo . . .

The quote is found in Proposition XXIV, Theorem XIX of his Principia and forms part of one of Newton’s many elaborations of the theory of gravitation. The context of the quote is Newton’s development of the observations of Edmund Halley, the famous astronomer and benefactor of Newton, concerning the effects of the gravitational pull of the moon on the ebb and flow of the tides along the equator and its significance for the theory of wave interference. The reference to the Philippines is done through the mention of Leuconia, the ancient Ptolemaic name for the Philippines. While the mention of the Philippines was through the indirectness of a mere obiter, clearly given to merely illustrate a scientific theory, the citation is still intriguing enough to lead one to the historical obverse and to inquire into the state of mathematics and the sciences in the Philippines while Newton was writing his magnum opus and developing the calculus and classical mechanics. This essay then concerns itself not with the mechanics of the rise and fall of the tides but with the historical ebb and flow of ideas on the side of the globe farthest from Newton as he was hewing his celestial mechanics within the ivied halls of Trinity College in Cambridge.

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As we examine the early civilizations, one remarkable fact emerges immediately. Though there have been hundreds of civilizations, many with great art, literature, philosophy, religion, and social institutions, very few possessed any mathematics worth talking about. Most of the civilizations hardly got past the stage of being able to count to five or ten.3

The temptation easily becomes a sin of commission when one accepts at face value some of the judgments made by Spanish friars about the Filipino’s lack of capacity for mathematics and science in the chronicles of early Philippine society. Fray Gaspar de San Agustin, for example, publishing his Compendio del arte de la lengua tagala (Compendium of the Art of the Tagalog Language) in 1703 wrote that “. . . los tagalos son poco aritméticos”4—the Tagalogs are little suited for mathematics, following this judgment with a harsher evaluation two pages later, “Pero los tagalos en el contar son varios y malos aritméticos”—Tagalogs in counting are unreliable and bad mathematicians.”5 One Fray Eladio Zamora, another Agustinian friar like Gaspar de San Agustin, supports the view of his predecessor by making a similar claim about the seventeenth- and eighteenth-century Filipino in his survey of education in the eighteenth-century Philippines, citing “. . .

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the small capacity of the [indio] for the sciences . . .”6 The Spanish friar chroniclers, moreover, were not alone in this view but were concurred with by many Spanish secular historians who also believed in the then-prevailing caricature that the indio was not only indolent and vice-ridden but was also a beast of burden who possessed the most minimal of intellectual skills. One such opinion directly relating to our present concern is the view expressed by Vicente Barrantes who stated in 1869 that “The indios learn to reckon with great difficulty. They generally take shells or stones to help them, which they heap up and count.”7 It is clear that an attempt to do a history of mathematics and science in the Philippines, especially one that concentrates on the period where traditional Philippine society experiences a profound transformation as it interacts with Spanish colonialism, cannot productively proceed from these premises. The Algebra of the Weaving Patterns, Gong Music, and Kinship System of the Kankana-ey of Mountain Province (1996), a short yet most important book published by a group of mathematicians from the University of the Philippines in Baguio, proceeds from a different perspective about mathematics and society and provides a set of premises which permits us to explore the history of mathematics in the Philippines in a more productive manner. The book focuses on three areas in the life of the Kankana-ey, one of seven principal linguistic groups in the Cordillera region in Northern Philippines—traditional weaving, indigenous gong music, and customary kinship patterns—and successfully shows that abstract mathematical ideas and principles such as geometric transformations and algebraic structures like frieze groups are “imbedded” in these indigenous practices. The present study starts from the premise that mathematical principles are indeed imbedded in the practices of society and that the absence of a history of mathematics in the Philippines is less attributable to the inherent incapacity of a people for mathematical and scientific abstraction than to the negligence of mathematicians and historians in abstracting, formalizing, and documenting these principles. The present work is a preliminary effort in outlining a history of mathematics in the Philippines. It attempts to call attention to the presence of a relatively developed set of mathematical ideas in early Philippine society. It will establish that a highly developed enumeration and arithmetical system was already developed by the time of the Spanish conquest and that this enumeration system had unique characteristics which distinguished it from its other Southeast Asian neighbors. Other mathematical

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and scientific ideas such as the use of geometric concepts and astronomical tools in early Philippine society will also be discussed and presented.

The Nebulous Roots Traces of mathematical ideas can be found even in the earliest moments of Philippine society which have been hitherto recorded. Basic geometric ideas, albeit in rudimentary form, are found in periods as early as the Angono Petroglyphs, the set of prehistoric rock and cave drawings found in the hills of Angono, a mountainous area south of Manila which juts from the Cordillera mountain range and extends to Laguna de Bay, the largest lake in Asia. Anthropologists who have studied the Angono Petroglyphs have found it difficult to date the rock drawings with precision but conjecture that they probably date to the late Neolithic Period or 3,000 years B.C. since the artifacts excavated from the area come from that time. Jesus Peralta, the most seasoned anthropologist from the National Museum, describes the petroglyphs: As a general rule the drawings are of human figures, consisting of line incisions of circular or domelike heads with or without necks set on a rectangular or V–shaped body. The arms, sometimes with digits, and the legs are also lineally executed, and are usually flexed. An inventory of the drawings produced a total of 127 figures clearly discernible as integral units. This count excludes other incisions that comprise slashes, naturally occurring holes, scratches, pits, pockmarks and other surface alterations on the rockwall. Some incisions on the rockwall can be recognized: triangles, rectangles, and circles. There is a high degree of probability that the triangles drawn singly have sexual connotations. These triangles are more or less equilateral, standing on an angle with a short line bisecting this angle. There is a complex of 4 triangles forming a parallelogram. One rectangle stands on one short side. Other nonfigurative cuts appear on the wall. A set of four parallel horizontal lines are unequal in length. There is also a set of five lines radiating from a common center.8

The prehistoric figures, furthermore, demonstrate that the Neolithic artists intuitively knew how to work with the notions of symmetry and

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proportion since the rock and cave drawings show a respect for the basic mathematical and aesthetical ideas of symmetry and proportion, as well as the more complicated idea of mathematical scaling, as seen in the successful resizing of the stone etchings from the actual, bigger figures of men and animals they represent. The drawings, more significantly, evidence the important capability of abstraction. The Angono Petroglyphs use simple lines to draw their figures, implying a more abstract approach to the subject matter, in contrast to their counterparts in European prehistoric lithic art, as for example the drawings in Altamira, where the animals drawn on the cave walls have a more realistic character. The Neolithic artists of Angono used lines to draw the figures which represented themselves and other members of their community, implying at least three different levels of abstraction: • • •

the abstraction necessary to draw and properly utilize a line; the abstraction shown in using a line to depict a figure; and the abstraction required to see that the figures drawn represent the artists and the members of their community.

simulation, but in an actual three-dimensional wood carving, the most complicated contortions possible of Moebius strips and other advanced ideas of topology. Another equally impressive demonstration of geometric thinking in ancient Philippine society can be found in the practice of shipbuilding. Not only were the mastery of the concepts of convexity, concavity, and the proper proportion between ship breadth and length to ensure sailing efficiency demonstrated, but more significant to note was the practice of constructing ships and boats to fit inside each other, with as much as twelve ships all fitting inside each other, exactly in the same manner as Russian dolls contain each other, an impressive demonstration, in gigantic three dimensional wooden models, of the mathematical ideas of sets, subsets, the measurement of volumes, and ordinality. Fr. Francisco Colin, a Jesuit priest who was amongst the first Spanish religious chroniclers who wrote about early Philippine society, recorded the practice in Catanduanes in his 1663 Labor evangelica: They were shipbuilders by profession. They made a great quantity of very light craft, which they took for sale throughout the region in

This tendency to represent artistic ideas in relatively abstract terms can still be found in the art of many actual indigenous Filipino communities. The different tribes in the northern Cordillera region, for example, have preserved this tendency towards abstraction, as seen for example in their depiction of their religious icons such as anitos and bululs which are carved in wood with an abstract or even “modern” character, even if the artistic and religious traditions date back to the prehistoric and almost nebulous past. Much of the art from these very same northern Cordillera communities also exhibit this tendency towards the abstract, as seen for example in the foldable stools and three-legged tables they produce from a single piece of wood. In the case of the table, three legs are fashioned from a single piece of log but are cut in such a way that they still form a single piece of wood. The legs can be made to open as an expandable tripod when necessary but can be collapsed back into the single log when desired, the joins between the three legs executed through a series of intricate holes to permit the three legs to collapse and expand as needed, all carefully carved so that the three legs do not dissociate from the original log. The design, presumably of a most ancient origin, employs, not in Escher-like drawings or as computer

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a very curious way, very much like the nests of boxes they make in Flanders. They built a large vessel, undecked, without using either nails or futtock timbers; then they built a smaller vessel which fitted exactly inside the first; then a third which fitted exactly inside the second; and so on, so that a large biroco might in the end have ten or twelve other vessels inside it of four specific types which they called biroco, virey, barangay and binitan. When they reach a port where they hope to make a sale—and they go as far as Calilaya, Balayan, Mindoro and other places more than a hundred leagues from their shipyards— they take out the smallest vessel and then the rest in order, so that he who saw but one ship enter the harbor would in an hour be puzzled to see ten or more craft in the water.9

The ancient Filipino institution of debt and usury reveals yet another instance of mathematical thinking. The universally accepted means of exchange and store of value in ancient Philippine society was not money but rice, the staple food. Debts were therefore incurred through this commodity. Repayment of the debt was expected to be accompanied with interest

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since rice was not only immediate food but was also seed, or capital, used in planting to produce more rice. The payment of interest represented the alternative cost of rice or the benefits forgone during the period the rice was on loan. The Tagalog term for interest, tubo, or growth, captures precisely the roots of the practice in the system of planting and harvesting rice. The amount of interest or tubo to be charged is directly related to the potential harvest. Miguel de Loarca, one of the earliest Spanish colonizers, describes the exact manner how interest was computed: If one lends another rice and a year passes without the debt being

of writing descended from different variants of Sanskrit, which was in wide use throughout Southeast Asia, extending from Bali through Thailand and Vietnam, during the tenth century, forming the basis of the different writing systems of these societies. The LCI is significant for many reasons,11 but for our present purposes it is important to point out that the document uses the Saka calendar system, permitting us to count and mark the years, and also shows a precise measurement for gold, implying the use of a standard system of weights and measures. The LCI also refers to the phases of the moon to fix the precise day within the month, implying familiarity with basic concepts of astronomy. A less technical translation of the LCI reads

paid, since rice is something that is planted, if it is not repaid in the first year of sowing, double the amount of the loan must be paid in the

In the Saka-year 882 (A.D. 900) in the month of March-April on the

second year, and four times the third, and so on at this rate. This alone

fourth day of the dark half of the moon which is a Monday, Lady

is their way of taking interest. Some indeed give a different account of

Angkatan, with her child Bukah, she the wife of His Honor Namwran,

it, but they have not well understood the matter. . .

appeared before the Chief and Commander of Tundun (Tondo) and

Scribe. Upon the instruction of the Chief and Commander of Tundun,

Interest grew not through simple arithmetical accretion but was doubled every harvest time. It was conceived of as a function of the productive value of rice used as capital or seed. An implicit distinction was made in this practice between simple arithmetical growth and exponential growth, the latter explicitly related to organic growth and the core insight behind what we now know as the exponential function. While admittedly we do not have here specific mathematical techniques of the exponential function that permit us to describe intricacies of beta particle decay or the dynamics of population growth, the idea that something grows not just arithmetically but geometrically, by compounding interest over the different harvest seasons, implicitly seeing a relationship between organic growth and interest payments, demonstrates an ability to perceive abstract mathematical relationships and utilize these patterns in everyday life. The recently discovered Laguna Copperplate Inscription (LCI), hitherto the earliest document or artefact relating to early Philippine society which, as a precise date, also reveals the use of mathematics in the ancient history of the country. The LCI, recorded during the equivalent of the last year of ninth century A.D., is a formal legal document engraved on a copper plate absolving a nobleman and his family of debts. The LCI, written in an ancient Malay language connected to Sanskrit, Old Malay and Old Tagalog, was inscribed on the plate using the Old Kawi script, the ancient system

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Jayadewa, a former chief of Pailah, a legal document was recorded clearing Namwran of a debt in gold amounting to 1 kati and 8 suwarna (around 926.4 grams). The debt was owed the Chief of Dewata representing the Chief of Mdang. Witnessing the legal ceremony were the Leader of Puliran (Pulilan), Kasumuran; the Leader of Pailah, Ganasakti, and the Leader of Binwangan, Bisruta. . . .12

The Cosmology of the Plebe The system of reckoning the years and the related astronomical work can be assumed to have continued into the early sixteenth century since Spanish conquistadores and chroniclers have recorded aspects of such practices when they arrived, with some accounts suffering from the attempt to read the Western calendar method in the peculiar manner early Filipinos practiced astronomy and counted time. One such misinterpretation was made by Fr. Francisco lgnacio Alcina, a Spanish missionary writing during the second half of the seventeenth century, who characterized the early Visayans with the most colorful description of plebe imperita or plebians without skill since the cosmological and scientific practices of these ancient Filipinos did not exactly coincide with Alcina’s own. Juan Francisco de San Antonio, a Franciscan missionary, provides a more sympathetic account in his Cronicas, written in 1738, where he describes the

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difference in the approach of the early Filipinos in the way they marked time and fashioned their cosmology:

It was exceptionally original and interesting how our people divided time before the present civilization. Not yet knowing the watch, they managed time through what they observed in the stars, in what they

It is not known whether these natives divided the time in hours, days,

saw in the plants and animals, and what they noted in their natural

weeks, months, or years, or made any other division of time. As this

movements.15

was necessary to them for the reckoning of their commerce, trade, and contracts (in which they all engaged), they used for reckoning their times of payment, and for other transactions and business of their government—for the hours, the state of the sun in the sky, the crowing of the cock, and the laying time of the hens, and several other enigmas which are still employed in the Tagálog speech. To keep account of the changing of seasons, they knew when it was winter or summer by the trees, and their leaves and fruit. They knew of the division into

Let us then take a closer look at how time was conceived and reckoned in ancient Philippine society. Some amount of astronomy was already known and practiced, as would be expected of a race which depended much on traversing the high seas and eking out an existence highly dependent on agriculture and the vagaries of the seasons. Juan de Plasencia, a Franciscan, thus recorded in his Customs of the Tagalogs, written in Manila in 1589:

months or years by moons. Consequently, in order to designate the date of payment, they said “in so many moons, in so many harvests, or

They worshiped, too, the moon, especially when it was new, at which

in so many fruiting’s of such and such a tree.” These were the methods

time they held great rejoicings, adoring it and bidding it welcome.

employed in their trading and government.

Some of them also adored the stars, although they did not know them

by their names, as the Spaniards and other nations know the planets–

This cosmology was certainly different from the paradigm used by Tycho Brahe, Johannes Kepler, and Newton in fixing the position of the stars and developing celestial mechanics. It was an approach more akin to the ancient Greek concept of kairos which viewed time in terms of subjective moments rather than kronos or the strictly measureable time which can be sliced into the most minute part with the help of a digital watch.14 This approach to the marking of time of course was not necessarily less valid than one based on the positivistic attitudes in Europe during the eighteenth and nineteenth centuries, for this cosmology of the plebe imperita also had its notions of regularity and periodicity and had served adequately the needs of its users. The eminent Filipino scholar Pedro Serrano Laktaw, writing in his posthumously published classic Estudios gramaticales sobre la lengua tagálog, has only but praises for ancient Filipino methods of telling time:

The appearance of the Pleiades signified the beginning of the agricultural season, the fundamental unit through which early Filipinos organized their experience of change and of time. Miguel de Loarca in his 1582 account notes that it is when the Pleiades appear that a new agricultural cycle is deemed to start and preparations for the new planting season begin. William Henry Scott, a specialist in early Philippine society, informs us that, apart from the Pleiades, other heavenly constellations were also used by ancient Filipinos to mark the changing seasons:

Es sobremanera original y curiosa la distribución del tiempo que

The agricultural cycle began, as Loarca noted, with the appearance of

practicaba nuestro pueblo antes de llegar a la actual civilización. No

certain stars. Most often these were the Pleiades in the constellation

conociendo aún el reloj, se gobemaba en esto por lo que en los astros

of Taurus which can first be seen in June locally called Moroporo,

observaba, en los animales y plantas veía, y notaba en sus mismas

meaning either “the boiling lights” or a flock of birds. Swiddens were

naturales acciones.

prepared at that time, and seeds were sown in September when they

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with the exception of the morning star, which they called Tala. They knew, too, the “seven little goats” [the Pleiades–as we call them–and, consequently, the change of seasons, which they call Mapolon; and Balatic which is our Greater Bear. . . .16

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This basic unit of change was given the name taon by the early Filipinos. The felicitously perceptive Franciscan Juan Francisco de San Antonio, already cited earlier, records for posterity the special meaning this term had in ancient times:

harvest, such as, “Taon na didto dile or It’s already harvest in their place,” and that old people were those who had seen many harvests (Sánchez 1617, 504v). Tuig also meant harvest, as in tinutuigan or what is ready for harvest, but it has the added connotation of anything periodic or recurring, such as the coming of the rains, panuigan sang olan sang habagat or sang amihan, either from the south or north, or the process of menstruation, as in tinuig na siya. The second meaning, it should be pointed out, demonstrates its affinity with the Tagalog tuwing which means every time or every occasion. Dag-on is when “everything is in bloom,” such as the flowering of trees and plants, “panog-on sa manga kakahuyan or when everything is in bloom, indicating the alternation of the seasons.”19 The Franciscan Plasencia unfortunately fails to appreciate the qualitatively different way ancient Filipinos conceived of change and of time, implicitly judging the ancient ways as inadequate practices which are fortunately going to be superceded with the continuing spread of Christianity:

They expressed “the year” in their old speech by the word taòn. It is

These natives had no established division of years, months, and days;

metaphorical, for it really means “the assembling of many,” and that

these are determined by the cultivation of the soil, counted by the

they have joined together months to make one year. They had a word

moons, and the different effect produced upon the trees when yielding

to signify seasons and climates, namely panahòn. But they never knew

flowers, fruits, and leaves: all this helps them in making up the year. . . .

were directly overhead at sunset, though the exact time depended on local climatic conditions. Indeed, because the rainy season varied from island to island, in some places farmers made use of the Big Dipper (Ursa Major), which they called Losong (rice mortar) [from where the term Luzon comes] or Balatik (ballista), though in Panay Balatik was what they called the two bright stars in Gemini. Still others planted when the Southern Cross was upright at sunset, a constellation that looked to the Visayans it looked like [sic] a coconut palm, Lubi, or blowfish, Butete. Similarly, the constellation Aries, the Ram, they called Alimango, the Crab.17

the word “time” [tiempo], in its general sense, and there is no proper Tagalog word for it; but they use the Spanish word only, corrupted

It seems, however, that now since they have become Christians, the

after their manner, for they make it tiyempo.

seasons are not quite the same, for at Christmas it gets somewhat

cooler. The years, since the advent of the Spaniards, have been

Taon, thus, did not exactly mean the year as the twelve calendar months corresponding to 365.25 days but referred to the larger phenomenon, encompassing cosmological, environmental, agricultural, and even religious elements, where and when everything got together—“the assembling of many”—to mark the start of a new season. The different uses of the term taon in present Tagalog and other Philippine languages give a glimpse of the original richness of the concept: pagkakataon, opportunity; nagkataon, by chance; nataon, to occur at the same time; mataon, to occur at the same time by chance; itinaon, to set or to schedule; maitaon, to be able to set or to schedule at the same time. Scott reminds us that Visayans have three different words corresponding to “year”—taon, tuig, and dag-on, each having its special connotations. Scott, citing the Sanchez Visayan dictionary of 1617, says that taon means

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determined by the latter, and the seasons have been given their proper names, and they have been divided into weeks.20

These changes Plasencia refers to involved not merely changing one name for another but more fundamental transformations involving worldviews as well as important semantic shifts in meaning. Taon, in due time, shifted meaning from its poetically-rich original connected with cosmic and agricultural cycles to the abstract, calendar-based meaning of 365.25 days, the present sense it has for modern Filipinos. Serrano Laktaw calls attention to this historical shift: Con la civilización que trajeron los españoles, conocieron el taón o año de 12 meses, adoptando los mismos nombres con que los nombran

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los españoles los suyos, diciendo: Enero, Febrero, Marzo, Abril, Mayo,

is when they weed their fields. Another they call [Cabuy: crossed out

Junio, Julio, Agosto, Setiembre, Octubre, Noviembre y Diciembre.21

in MS.] Yrarapun; it is the time when they begin to harvest the rice. Another they call Manalulsul, in which the harvesting is completed. As

With the civilization brought by the Spaniards, they came to know the

for the remaining months, they pay little attention to them, because in

year of 12 months, adopting the same names used by the Spaniards,

those months there is no work in the fields.23 24

saying: January, February, March, April, May, June, July, August, September, October, November, and December.

But the original difference between the Filipino taon and the year or año was clear, as Alcina himself reminds us:

The moon, which waxed and waned more often than the corning and going of the stars, then acquired a magical meaning for early Filipinos, as Scott summarizes for us: Visayans also believed that just as the moon times the human menstrual

[Taon or tuig is a word] with which they also counted the years, but

cycle, so its phases controlled all biological growth. Starfish were said

without computing or numbering the months, which from harvest to

to increase and decrease in size as the moon waxed and waned; crab

harvest they would count as eleven or twelve distinct and past, and

shells hardened and softened to the same rhythm; yellow turtles only

which they called tuig, and although they now confuse it with the

grew at nighttime when there was a moon, black ones during the dark

year, it was not a single year but an indefinite time because that word

of the moon, and white ones during the daytime. So too, coconut trees

means to them the same as “time” does to us.

were thought to produce one new sprout each new moon; the silklike

fibers of the ulango palm had to be gathered at quarter moon; and

In the same way the taon was conceived in terms of the periodic appearance of the Pleaides and other stars, the next unit used to organize time was the buan, literally the moon, where the taon was divided in terms of the different times the moon waxed and waned during the period. The month therefore in this ancient system was a lunar month, similar to Chinese practice, but quite different from the European calendar where calendar months do not coincide with the movements of the moon. Miguel de Loarca describes the practice:

stems of boats made from dao roots could be expected to outlast the vessel but only if cut during the waning moon. Furthermore, Visayans had a prescription for which phase of the moon was best for gathering any of a dozen varieties of abaca, though the most commonly planted variety was one that could be cut at any time. The dark of the moon was considered sinister because it was the favorite of witches and aswang, who fled at the first sight of the crescent moon showing its horns. Fieldwork and weaving were

They divide the year into twelve months, although only seven [sc.

accordingly forbidden the following day as a precaution against illness

eight] of these have names; they are lunar months, because they are

during the coming month, and a one- to three-day holiday was taken

reckoned. The second is called Dagancahuy, the time when the trees

to celebrate the full moon because the diwata came to earth at that

are felled in order to sow the land. Another month they call Daganenan

time. Nobody doubted that an eclipse (bakunawa) was caused by a

bulan; it by moons. The first month is that in which the Pleiades

huge sawa, python, trying to swallow the moon, and that it had to

appear, which they call Ulalen comes when the wood of those trees

be frightened away by noisy pounding on mortars and house floors,

is collected from the fields. Another is called Elquilin [Elkilin], and

followed by another holiday. . . .25 26

is the time when they bum over the fields. Another month they call Ynabuyan, which comes when the bonanças [or the fair winds when the monsoon is changing] blow. Another they call Cavay [Kaway]; it

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The week did not correspond to any celestial cycle27 so it did not form part of ancient Filipino system of reckoning time. There was no need to

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resort to the artifice of the week, since days within the buan or month could be precisely counted, as we shall soon see. But increasing Spanish inroads into Philippine culture assured that the concept of week would eventually be accepted. By the eighteenth century at least the concept of week was already in use, as recorded by San Antonio Francisco:

dies just before a son or grandson is born. The fifth or sixth night of waning was parik, to level or flatten, because it then rose so late the witches had many hours of darkness in which to beat down the earth by the stomping of their feet during their dances. Katin was the third quarter, so it had crossed this second barrier (lakad na an magsaguli) by the twenty-fourth or twenty-fifth night, and then got ready

[Days are counted] and so on until they have the difference of weeks,

for new moon again (malasumbang) about the twenty-ninth. This

which they call by the name Domingo, saying “so many Domingos.”

was the dark of the moon, or what the Spaniards called conjunción

[i.e., Sunday, Domingo being the Spanish word; evidence that this

(meaning the conjunction of the sun and the moon) when the moon

method of styling the week was evolved after the conquest].28

disappeared for a night or two. To the Visayans, it was then dead, lost, or gone hunting.

If the second basic unit of time was the buan, defined by the waxing and waning of the moon, then the system for measuring how far the current buan has elapsed was dependent on the moon’s evolving shape in the night sky. The days or nights from one moon to a new one could be easily ascertained. Scott relates the detailed manner in which the progression of the month is monitored through a description of the moon’s changing shape: The new moon was subang the first night it could be seen, or more colorfully, kilat-kilat, a little lightning flash. When it appeared as a full crescent the next night or two, it seemed to have opened its eyes

These phases of the moon were common time markers known to all. They would say, for example, “Duldulman an bulan [The moon begins to wane today]” or “Paodtononta an bulan [Let’s wait for the quarter moon]” (Sánchez 1617, 37, 188). And nasubang nga tao was a newcomer or upstart.29

The Tagalogs, however, had a slightly different method, preferring to count days rather than the Visayan practice of marking nights. As San Antonio records:

(gimata) or, alternately, closed its mouth (ungut)—like a baby’s on a mother’s breast. Then came a “three-day moon” or high new moon,

The days were reckoned by the name of the sun, namely, árao. Thus

hitaas na an subang, followed by balirig, the fourth or fifth night, and

the Tagalogs now reckon ysang árao “one day;” dalauang árao, “two

next it was “near the zenith” (odto). When it appeared as an exact half

[days].”30

disk—what western calendars call the first quarter moon—it was directly overhead at sunset, and therefore odto na an bulan. Then as it continued to wax, it “passed the barrier” (lakad), and when it was lopsided both before and after full moon, it looked like a crab shell (maalimangona). The full moon was greeted with a variety of names—paghipono, takdul, ugsar—but most significantly as dayaw, perfect or praiseworthy, fit

The next units of organizing time based on natural periodic occurences in nature were, alternatively, day and night. “The night is called gab-i; and the day arao, from the name of the sun.”31 It cannot be ascertained for the moment when the European clock was introduced to the Philippines. But the Cronicas of San Antonio records the use of the clock and the acceptance of the concept of the hour by the eighteenth century:

recognition of its spectacular shape and sunset-to-sunrise brilliance.

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And as it began to wane—that is, darken (madulumdulum)—a night

Only there are no terms to indicate the hours of the clock [in their

or two later, it set on the western horizon just before dawn and so

speech]; and now the Castilian [names of] hours are Tagalized, in

was called banolor, to exchange or take by mistake—like a man who

order to indicate the hours of time. They call the clock horasàn, that

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is, “a thing in which one sees the hours;” whether in its place or in the

katanhalíän, mediodía; tanhálinğ tapat, tanhálinğ tírik, saulo anğ

instrument made for it.32

áraw, a medio día en punto, porque habiendo llegado el sol al cenit proyecta su luz directamente sobre la cabeza; likid na, kunğ kiling nğ,

Prior to the use of the concept of hour to subdivide the day, ancient Filipinos determined the time of day through the movement and position of the sun, with both the Visayans and Tagalogs using picturesque descriptions of the passing of time during the day. Scott summarizes the manner in which Visayans marked time during the day:

a las doce y media o la una; lipás na, a las dos; mababá na, cuando el sol va cayendo, como a las cuatro; lúlúnod na, cuando está para ponerse el sol; nalúnod na, ya se puso el sol.34 The moments of the day are divided in the following manner pagsikat nanğ tálang baquero, around 3:30 in the morning; pagsikat nang tálangbatúgan, when the light comes out; madaling a’raw, the

The Visayans divided the daylight hours into a dozen or more specific

dawn; pagliwayway, kung bukangliwayway, when dawn breaks;

times according to the position of the sun. Between dawn and noon,

pagbabanang manok, when the cocks come down, already a clear

they reckoned nasirakna, shining, and nabahadna, climbing, and then

sky but still without the sun; namimitak na anğ áraw kung umááraw

iguritlogna, time for hens to lay, and makalululu, when your bracelets

na the appearance of the sun; áraw na kung umaga na, it is already

slid down your raised arm if you pointed at the sun. High noon was

day; hampás tikin anğ áraw, around seven in the morning, when they

odto na an adlaw; followed by two points of descent in the afternoon,

say that the sun is within the grasp of one un tikin: máaga pa, it is

palisna and ligasna; until midway to setting, tungana. Natupongna

still early, around seven thirty or eight in the morning; mataas na

sa lubi was when the sun sank to the height of the palm trees seen

ang araw, the sun is already night, around ten until twelve; tanhali,

against the horizon; and sunset was apuna; or natorma, when the sun

katanhalián, midday; tanháling tapat, tanhaling tirik, saulo anğ áraw,

finally disappeared. Day ended with igsirinto, when it was too dark to

high–noon, because the sun, having reached its height, now projects

recognize other people.

the light directly above the head; likid na, kung kilinğ na, twelve–

thirty or one; lipás na, two o’clock; mababa na, when the sun starts

The Tagalogs, on the other hand, had their own system and terminology, summarized for us by Serrano Laktaw. The day was divided in the following manner:

to go down, around four o’clock; lulunod na, when the sun starts to disappear; when the sun already has set.

The night, on the other hand, had less divisions: Los momentos del día las distribuían así: pagsíkat nanğ tálang baquero, como las tres y media de la mañana; pagsíkat nanğ tálanğ batúgan,

A la noche llaman gabí. Y sus momentos las reparten así: silim na

al salir del lucero; maralinğ áraw, el amanecer; pagliwayway, kunğ

kunğ sumísilim na, va oscureciendo; takipsilim, entre dos luces;

bukanğliwayway, al romper el alba; pagbabâ nang manok, al bajar

malálim na anğ gabí, muy avanzada la noche como a las diez o las

los gallos, estando ya bastante claro, pero aún sin sol; namímitak na

once; hátinğgabí, media noche.35

anğ áraw kunğ umááraw na, la salida del sol; áraw na kunğ umaga

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na, ya es de día; hampás tikín anğ áraw, como las siete de la mañana,

The night is called gabí. And it is divided in the following manner: silim

que es cuando dicen estar el sol al alance de un tikin; maaga pa, aún es

na kunğ sumísilim na, when it is getting dark; takipsilim, between two

temprano, como las siete y media o las ocho de la mañana; mataás na

lights; malálim na anğ gabí, when the night is already very advanced,

anğ áraw, ya está alto el sol, como a las diez hasta las doce; tanhali,

like ten or eleven in the evening; hátinğgabi, midnight.

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Serrano Laktaw notes that with the spread of the watch this system disappeared, eventually replaced by the European method of telling time: “Con la vulgarización del reloj va desapareciendo todo esto, y se cuenta ya al estilo europeo” (With the spread of the watch all of this started to disappear, and now they count time in the European style).

The Scales and the Confessional Commerce within the islands as well as trade with its Southeast Asian neighbors required the development of a system of weights and other measures. It was almost natural that the system of dry measures would revolve principally around the staple food, rice, but the system was also used to measure salt, mongo, and others. The most common measure would have been the dakot or handful since this would have been the most convenient way to handle the grain. But quantities in this system would of course have varied from hand to hand so a more uniform system had to be evolved for trade. Serrano Laktaw records the old system of the measurement of volume that evolved in the trading of these commodities: Measure

Definition

Metric Equivalent

Kabán

25 gantas or salop

75 liters

Kalahatian

half a cavan

37.5 liters

Ganta or salop

8 chupas

3 liters

Kágitnáan (or kalahating salop or kalahating ganta)

4 chupas

1.5 liters

Chupa, gátañg or gahenan

4 apatan

0.375 liters

and practical sense since this measure was never meant to be used with this digital exactness. The better approach to appreciating this ancient method of measuring the volume of grains and similar items is to view the kabán as the measure used for wholesale or bulk transactions, while ganta, salop, and chupa were used for retail purposes. These senses of the terms are still partly preserved in present-day Tagalog where kabán-kabán is used to denote, sometimes even in figurative speech, great quantities or bulk deliveries, while one can still hear, at least in the mind’s ear, a mother’s instructions to buy kalahating salop ng bigas or half a salop of rice from the neighborhood store when the household readies itself for lunch, a practice observed at least up until relatively recent times before the metric system completely totally took over. The ganta, salop, and chupa were defined in terms of each other through multiples of eight and four, most probably because the way they were measured and compared with each other was through a continuous division by two, since dividing by halving was the most convenient and comparatively more accurate way of dividing a quantity of grain. It might appear at first that the system is inconvenient since the kabán was defined as a multiple of 25 of the ganta or salop, making direct comparisons with the chupa and apatan, which were related to the ganta or salop by multiples of two, difficult. But reflection tells us that this would have not been much of a practical problem if we accept the observation made earlier that the kabán was a wholesale unit, while the rest were for retail transactions, thus making direct calculations between the chupa and the kabán for example, unnecessary. Serrano Laktaw instructs how counting using these measures was conducted: Y la fraseología era: isanğ gátanğ na bigás o isanğ gahénanğ bigás, una chupa de arroz limpio hasta 7 solamente, porque 8 chupas ya forman sanğsalop, una ganta, hasta 24 salop o ganta, porque 25 de

Apatan

estos forman ya sangkabán, un caván. Desde aquí con labí sa hasta

The three more important measures are printed in bold. Serrano Laktaw leaves apatan undefined and without a metric equivalent in his Estudios gramaticales. While the previous definitions provide us with enough information to calculate the metric equivalent of apatan to be 0.09375 liters, this arithmetical exercise appears to have little mathematical

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dos cavanes, dalawang kabán; labí sa kabán sanğsalop, un caván y una ganta, etc.36 The phrasing was isanğ gátanğ na bigás or isanğ gahénanğ bigás, one chupa of clean rice until 7 only, because 8 chupas already form

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sanğsalop, one ganta, until 24 salop or ganta, because 25 of these already form sangkabán, one cavan. From here with labí sa until two cavans, dalawanğ kabán; labí sa kabán sangsalop, one cavan and one ganta, etc.

The verb takal signified the act of measuring, usually through the salop or ganta, the traditional metering container, while the progressive form of the verb, tinatakal, connotes either the act of measuring or the act of slowly transferring from one container to the other. The word is explained in the Cronicas of Fray San Antonio: “The act of measuring in this manner is expressed by the word ‘tacal’ among the Tagalogs.”37 This ancient method of measurement lasted, surprisingly, until very late in Philippine history, when in the 1970s the use of the metric system was dictated by presidential decree. Prior to that, rice retailers had to register their wooden measuring boxes with the proper government offices to ensure that they complied with the standard volume for the ganta, salop, and chupa, with the wooden boxes duly stamped to ensure proper compliance and ensure that the public was not defrauded by unscrupulous merchants. But even when the proper scales were used, some amount of fraud was still possible, as when an experienced hand could vary the amount of rice by pouring it only very lightly into the measuring box, equally careful in shaving the top of the box with the traditional ruler-like wooden stick, resulting in less rice filling the purportedly standard measuring box. These unscrupulous practices, which the introduction of the metric system in the 1970s attempted to remedy through the measurement of metric weight rather than traditional volume, were, amusingly, apparently already practiced in early Philippine society, as gleamed through Fray Sebastian de Totanes’s 1745 Manual tagalog para la administracion de los sacramentos.38 Manual tagalog, published together with Totanes’s Arte de la lengua tagala, was a manual to help Spanish priests, especially those who were still struggling to learn Tagalog, to administer the sacrament of confession, by providing a set of ready questions for the penitent. The questions, quite detailed and even overly suggestive, were divided into two, depending on whether the father confessor wanted to administer a regular, lengthy confession or a shorter one. The regular confession, or La buena confesión, comprised 429 questions, at least in our count, covering items number 124 through number 553, from pages 56 through 143, while the shortened version, or Confesionario breve,

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had 82 questions, found on pages 143 through 154 of the Manual tagalog, with both sets consisting of quite detailed, truly embarrassing personal questions of the penitent. The questions were organized around the different commandments the penitent might have transgressed. For our present purposes, Question # 456 of Totanes’s Manual tagalog, based on the seventh and tenth commandments, provides extremely amusing historical color to our discussion of ancient Filipino weights and measures. Totanes first states the seventh and tenth commandments in Old Tagalog and in Spanish: Ang icapito, at icapolong otos nang P. Dios Houag cang magnacao. Houag cang magnasa nang di mo ari. El séptimo, y décimo mandamientos de Dios Ntro. Señor No hurtes. No desees la hacienda agena.39

Then forming part of the examination of conscience of the penitent for these commandments is the following question: 456. Has usado de dos gantas, ó medidas una grande para comprar, y otra pequeña para vender á otro? Y lo mismo te pregunto en cuanto á pesar, y medir con vara, braza, etc. 456. Nagdalaua cang salop caya sa pagtacal; isang malaqui sa pagbili mo, at isang munti sa pagbibilimo sa iba? At gayon ding itinatanong co sa iyo tongcol sa pagtitimbang, at sa pagsucat nang balanga?40 456. Have you used two measures, either in gantas or salop, a big one for buying and another, a smaller one, for selling to others? I also ask the same concerning weighing and measuring. . . etc.

The question is doubly amusing, since it shows that the practice of merchants defrauding their clients through a false system of weights and measures was already in practice as early as the time of Totanes, and also since the question, no matter how innocent the intention was in asking,

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was actually opening the possibility of the crime to the penitent who might otherwise not have thought up the deed. But if there was any fiddling around with the system of measuring volume, the most grievous would not have been the occasional, petty peccadillo of mischievous individuals. Fray San Antonio hints at how the Spanish crown defined and employed the weights and measures used so that the friars could maximize what they received: When the king issues orders for rice, it is reckoned by cabáns of twentyfour gantas apiece; and now it is known that it is of pálay rice, which is rice with the husk and uncleaned. When vouchers are issued for the stipends and the support of the religious ministers, the reckoning is by

Number of salop and gantas

Labíngisá o labingisahan

eleven

Dalawang̃puoán

twenty

The measurement of weights, on the other hand, took two forms, using two different types of scales, depending upon the quantity to be measured and the accuracy required. Items of high value, which needed accurate measurement, were weighed through the talaró or timbangan, the latter term still the popular name for the weighing scale. This is the usual balance with two weighing plates on each side, with some designed to be small enough to be carried in person.

fanegas, at the rate of two cabáns of twenty-four gantas each, of the

Those metals were employed in their trading only by the weight, which

said pálay rice uncleaned. And because his Majesty chooses that they

was used alone for silver and gold; and that weight they called talàro,

give it to us very clean, it is now ruled in the royal accountancy that

and was indicated by balances, like ours. They reckoned and divided

forty-eight gantas of the fanega of pálay is equivalent to a basket of

by this.42

twenty gantas of bigas, which is the name for cleaned rice. Hence the king in his charity, in order to give us our sustenance in the rice without waste, gives valuation to the measure at his own pleasure, for the rice with husk, so that the quantity may be doubled. The estimation of the king in this is not the same as looking into the hollow measure in its strict capacity, as has been already explained.41

The salop or ganta, it appears, was extended to also serve as a system of measuring liquids such as wine, vinegar, and oils, all products from thriving industries. The accepted unit was the tinaja, equivalent to 16 gantas, while the smallest was the bukohan, súbok salop, or salopan, all equal to one ganta, the first expression related to the quantity which might fit within a coconut shell, hence the term bukohan, while the last clearly relates to salop, which explains the consequent identification of meanings between the terms ganta and salop in the system of dry measures. Serrano Laktaw lists the other measuring cups which were known to have existed: Number of salop and gantas

Animan

six

Pitohan

seven

Walohan

eight

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The unit used for gold and other items where precise measurements were especially desired was called tahel, with the tinga, sapaha, and ama forming the other units for progressively lower quantities. The gold, which they call guinto, was also reckoned by weight. The largest weight is the tahel, which is the weight of ten reals of silver— or, as we say, of one escudo. The half-tahel is called tinga, which is the weight of five reals. The fourth part is called sapaha, which is two and one-half reals.43

Serrano Laktaw and Scott provide us this table of weights:44 1 táhel

2 ting̃a [or paningan]

1 ting̃á

2 sapahà

1 sapahà

2 amas

1 amas

2 balabato

1 balabato

2 kupang

1 kupang

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This means that, for example, four kupang was equivalent to one mas, while one tahel was equal to 16 mas. Alternative spellings and pronunciations for tahel are tahil, tael or tae; amas was also mas, from the Malay emas. An alternative manner of weighing small quantities, perhaps yet more ancient than the foregoing, is to compare the gold with beans or rice:

Serrano gives the two tables for the system of weights, the second providing the Spanish equivalents used during the early period of colonization:48 Unit

Equivalent

1 pico

10 sinantan

1 sinantan

2 banal or 10 kate

They also used other metaphorical terms (as the Spanish do the term

1 banal*

5 kate

granos), and said sangsága, which is the weight of one red kidney-

1 cate

2 soco or 16 tahel

bean [frixolillo] with a white spot in the middle.45

1 soco*

8 táhel

1 táhel

16 amas o adarmes

This little seed was called saga or sangsága and served as the basic unit of weight in this system. Gold and gold dust, used as a means of payment, therefore were measured as multiples of these beans, with the following other units used:46 1 balay

3 bahay

1 bahay

3 saga

1 saga

Tagalogs treated one saga as the equivalent of three palay (grain of rice) seeds. Scott therefore observes that the term sumasaga was extended to mean buying cheap items. Such precise measurements then required accurate scales. Scales which gave the fair and exact weight were called matapat na taiarô while the untrusty ones were referred to as may kaná. The measurement of heavier items required the use of another type of weighing scale, called the sinantanan or sinantan, what the Spaniards term the romana, where there is only one weighing plate and the balance is achieved through a system of weights on the lever, such as the system used in the infirmary scales to measure the weights of patients before digital and spring systems were adopted: In order to weigh bulkier things, such as wax, silk, meat, etc., they had steelyards, which they called sinantan, which was equivalent to

Spanish Unit

1 quintal

4 arrobas

1 arroba

25 libras

1 libra

16 onzas

1 onza

16 adarmes

There appears to have been an attempt to place a uniform system of weights during the early eighteenth century. San Antonio, writing in his Cronicas, records this attempt and the resulting equivalents: Consequently, these old weights having been adjusted to the Spanish weights by the regulations of the year 1727, one cate is equivalent to one libra, six onzas; one chinanta to thirteen libras, and twelve onzas; hence one quintal, of eighty of the old cates, corresponds to four arrobas and ten libras of our weight. A pico of one hundred cates is equivalent to five arrobas, twelve and one-half libras, in the new arrangement. As in the case of gold, one tahel must weigh one and one-fourth onzas in our weight.49

Very little of these ancient practices survive. Since the Spanish colonial administration continually used the European system in their measurements, the traditional, even poetic, methods of counting had to give way to the metric system. Serrano Laktaw thus laments:

ten cates, of twenty onzas [i.e., ounce] apiece. The half of that they called banál, which was five cates; and the half of the cate they called soco.47

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La adopción del sistema métrico–decimal abolió poco a poco estas nativas antigüedades, hasta el punto de que en la actualidad ni noticia de ellas tiene la presente generación.50

The adoption of the metric–decimal system little by little abolished these native ancient practices, until we have reach the point that the present generation does not even have news of them.

Counting and the Last Number We discuss in this section the counting system and the different kinds of numbers ancient Filipinos used.51 Cardinal numbers in Old Tagalog can be classified into either simple or complex. Simple numbers are those which consist of one single meaningful element, while complex numbers are those which are made up of more than one meaningful element.52 Simple cardinal numbers in Old Tagalog are the counting numbers from one through ten: 1–10 *Isá

*Dalawá, o dalwá

*Tatló

*Ápat

*Limá

*Ánim

*Pitó

*Waló

*Siyam

*Puló or Puô

11–19 *Labi-ng-isá

1 more than 10

*Labi-ng-dalawá

2 more than 10

*Labi-ng-tatló

3 more than 10

*Labi-ng-ápat

4 more than 10

*Labi-ng-limá

5 more than 10

*Labi-ng-ánim

6 more than 10

*Labi-ng-pitó

7 more than 10

*Labi-ng-waló

8 more than 10

*Labi-ng-siyam

9 more than 10

Numbers 11 through 19 are formed by adding the prefix labing to the simple root number. The prefix is formed from labi or more than plus the linker –ng. Labingtatlo or 13, for example, means three more than 10.54 This method implies the use of Base 10. This system also implies the process of addition and the mathematical notion of greater than (>). In contrast, European languages, English included, generally use the equivalent of the conjunction “and” to form the number—for example, 22 would be generally expressed as twenty and two. Multiples of 10

The old system of counting in Tagalog made a distinction between puló and puô, both of which signified ten. Puló was used when counting consecutively, as from one through ten.53 Isa, dalawa, . . . puló. The origin may have come from punô, meaning full, indicating that the fingers of the two hands were already full in the counting process. Puô, on the other hand, was a contraction from puló and was used when the quantity ten was used

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by itself, adding the prefix isang or sang, producing sangpuô. Modern usage merely uses the latter, sangpuô, usually further contracting the word from sangpuô to sampu. Complex numbers in Tagalog are formed by using one of the simple cardinal numbers and combining it with another meaningful element. The numbers 11 through 19 are formed in this way:

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*Dalawang̃puô

*Tatlong̃puô

*Apat na puô

*Limang̃puô

*Anim na puô

*Pitong̃puô

*Walong̃puô

*Siyam na puô

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The number 20 is expressed as a multiple of 10, as are all multiples of 10, until 90. Dalawangpuô, tatlongpuô, . . . siyam na puô. Linguists who have studied the language used to express the multiples of ten believe that there is an implied process of multiplication involved: “pu times ten.”55 Thus, dalawangpuô is really dalawa na puô—or two times 10; tatlongpuô is tatlong na puô—or three times 10. The interesting implication here is that for double digit numbers, with the exception of multiples of 10, the implied operation of addition is involved—or more precisely, the relation greater than (>)—where the number in the one’s place is added to the number in the tens place or compared as a relation. But when it comes to the multiples of ten, the implied process is multiplication, where a number is multiplied by puô or 10. 21–100

The conceptual framework and the corresponding grammatical rules undergo a significant change starting with the number 21 in Old Tagalog. Twenty-one is expressed in old Tagalog as maykatlongisa. Numbers from 21 through 99, with the exception of the multiples of ten which follow their own rules, are formed by combining four semantic elements: • • • •

the prefix may– or mey–; the second prefix or infix –ika– or –ka–; the multiple of ten towards which one is counting; the number in the one’s place.

The rule for forming numbers starting with 21 is to use the prefix may–, signifying to have or to have in existence, then add the infix –ka–, a contraction of –ika–, then use the multiple of 10 towards one is heading, such as 30 when one is expressing 21, then the actual quantity in the one’s place, in this case isa or one. Thus 21 is expressed as may–ika–katlong–isá or simply maykatlongisa. Serrano Laktaw gives us this example from the counting of kabans: De dalawanğ kaban, en adelante, con may, como queda dicho en los

From two kabans onwards, with –may–, just in the same way as in the cardinal numbers: Maykatlonğ kalahátinğ kaban, two kabans and a half.

We have two authorities from the early eighteenth century who explain to us the use and the conceptual significance of this method of counting. Gaspar de San Agustin, writing in his 1703 Compendio del arte de la lengua tagala, explains that writing 21 as maykatlongisá is really in effect saying the quantity one towards the number 30: “—May catlong isa—, veinte y uno, esto es uno para treinta, etc.”57 Fray Sebastian de Totanes, whose Confesionario we already cited earlier, has basically the same explanation, when he says that we are really indicating the multiple of ten towards which we are walking or heading, and that this is the manner of counting until we reach 100: Para proseguir contando desde 20, se hace con –mey–, que significa tener, tomando el numero del diez á que se camina (que desde 20. Vg.: es –tatlo–, porque es el 30, ó tercer diez) anteponiendoles –ca–, y despues el número intermedio ligado con el mismo diez, con sola –n–, si este acabase en vocal, ó sin ligazon alguna, si acaba en consonante; y asi se prosigue hasta 100 que es –daan–, pero para nombrarle solo, se le antepone –sang– (segun el citado no. 359) Y dirá en rigor tagalog, tengo para 30. Vg.: Tantos que en nuestro castellano son veinte y tantos, Vg.58 To continue counting from 20, one does it with –mey–, which signifies “to have,” taking the multiple of ten towards one is heading (which from 20 is three, because it is 30, or the third ten), placing before it –ca–, and then a single –n– after the intermediate number connected with the same ten, if this number ends with the vowel ‘o’, without any addition if it ends with a consonant. In this way one heads towards 100, which is –daan–, but to name it by itself, one places –sang–. And in Tagalog one will say, “I have so much towards 30, which in our Spanish is twenty and something.”

cardinales: Maykatlonğ kalahátinğ kabán, dos cavanes y medio.56

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Here are the examples from Serrano Laktaw: 20–100 Maykatlong̃isá

Maykatlong̃dalawá

Maykatlong̃tatló

Maykápat na isa

Maykápat na dalawá

Maykápat na tatló

Later Tagalog counts these numbers differently. In contrast to the ancient method, it now merely uses the coordinating conjunction “at” to imply the addition of the number in the one’s place to the number in the ten’s place, much in the same way that Western systems do, a clear indication of the later Spanish influence on the numeration system. Thus, 21 is no longer maykatlongisá but dalawangpu at isa or dalawampút isa, losing the implicit image of counting or heading towards a fixed amount in favor of the implied process of addition. This is now how the modern Tagalog speaker would say the number. Más tarde se adoptó la manera de contar española: isanğ kabán at

Maykalimang̃isá

kalahati, caván y medio; sanğdaanğ kabán, ápat na salop at limanğ

Maykalimang̃dalawá

gátanğ o gahenan, cien cavanes, cuatro gantas y cinco chupas, etc.59

Maykalimang̃ápat

Maykánim na isá

Maykánim na tatló

Maykánim na limá

Maykapitong̃isá

Maykapitonánim

Maykapitong̃pitó

Maykawalong̃isá

Maykawalong̃tatló

Maykawalong̃ápat

Maykasiyam na isá

Maykasiyam na pitó

Maykasiyam na waló

Maykaraang̃isá

Maykaraang̃ápat

Labi sa raan isa

101

Maykaraang̃siyam

Labi sa raan sang̃puô

110

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Much later, the Spanish system of counting was adopted: isanğ kabán at kalahati, one and a half caván sanğdaanğ kaban, ápat na salop at limanğ gatanğ o gahenan, one hundred cavanes, four gantas and five chupas, etc. 100–199

The formation of numbers from 101 through 199 in the ancient system was formed in a way similar to the formation of numbers below 100, where they were designated by simply using the prefix labi–sa. This time the prefix labi–sa is placed before the term daan, while the quantity over a hundred, occupying the one’s and the ten’s place, is placed after it. Thus, in the old system, then number 101 is labi sa raan isa. (The modem method again follows the Spanish where the conjuction “at” or “and” is used—for example, isangdaan at dalawangpuo for 120, which in the ancient method would be labi sa raan dalawangpu.) The set of numbers 101 through 199 analogously follows the rules for forming numbers 11 through 19, implying the relation of more than or greater than (>). Here are Serrano Laktaw’s examples:

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Labi sa raan labing̃ isá

111

Maykatlong̃daang̃isá

201

Labi sa raan tatlong̃puô

130

Maykatlong̃daang̃ sangpuó

210

Labi sa raan maykápat na isa

131

Maykatlong̃daang̃labingisá

211

Labi sa raan apat na puô

140

Maykatlong̃daang̃dalawang̃puô

220

Labi sa raan maykalimang̃dalawá

142

Maykatlong̃daang̃ maykatlong̃isá

221

Labi sa raan limang̃puô

150

Maykatlong̃daan maykápat ná dalawá

232

Labi sa raan maykánim na ápat

154

Labi sa raan ánim na puô

160

Maykápat na raang̃tatló

303

Labi sa raan maykapitong̃limá

165

Maykápat na raang̃ maykalimang̃tatló

343

Labi sa raan pitong̃puô

170

Labi sa raan maykawalong̃ánim

176

Maykalimang̃daang̃ dalawá

402

Labi sa raan walong̃puô

180

Maykalimang̃daang̃ maykápat na ápat

434

Labi sa raan maykasiyam na pitó

187

Maykalimang̃daang̃ maykápat na waló

438

Labi sa raan maykaraang̃tatló

193 Maykánim na raang̃ápat

504

Maykánim na raang̃ maykánim na limá

555

Maykánim na raang̃ maykapitong̃ápat

564

Maykapitong̃daang̃ isá

601

Maykapitong̃daang̃ maykalimang̃ limá

645

Maykapitong̃daang̃ maykapitong̃ pitó

667

Maykawalong̃daang̃labing̃tatló

713

Maykawalong̃daang̃maykatlong̃limá

725

Maykawalong̃daang̃maykawalong̃pitó

777

Maykasiyam na raang̃pitó

807

Maykasiyam na raang̃maykápat na limá

835

Maykasiyam na raang̃ maykasiyam na waló

888

Maykalíbong̃waló

908

Maykalíbong̃ siyam na puô

990

Maykalíbong̃ maykaraang̃ siyam

999

Multiples of 100 are formed in a fashion analogous to the formation of the earlier multiples of 10. An implied process of multiplication is involved in counting by the hundreds, where daan signifies “times one hundred” or multiplication by 100. Thus, isangdaan or sangdaan (100) is one multiplied by a hundred; dalawangdaan (200) is two multiplied by a hundred; and so on. Multiples of 100 *Sang̃daan

100

*Dalawang̃daan

200

*Tatlong̃daan

300

*Ápat na raan

400

*Limang̃daan

500

*Ánim na raan

600

*Pitong̃daan

700

*Walong̃daan

800

*Siyam na raan

900

201–999

Counting from 201 through 999 is done with the prefix may–, in a manner analogous to counting from 21 through 99:

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1000 *Sang̃libo

1,000

Isang libo is 1,000, or one times one thousand, again libo signifies multiplication by a thousand, analogous to the multiples of ten and hundred. From 1,001 through 1,999, the prefix labi sa is used, exactly in the same way that numbers were formed from 101 through 199: Labí sa libo isa: 1,001. 2001

From 2,001 through 9,999, the prefix mayka– is used, exactly in the same manner that numbers were formed from 201 through 999, as for example: Maykatlonglibong̃–isá: 2,001.

Labi sa laksâ ápat na puô

10,040

Labi sa laksâ ánim na puô

10,060

Dalawang̃laksâ

20,000

From 20,001 through 99,999, they are formed with the prefix mayka– in the same manner as 2,001 through 9,999: Maykatlong̃laksang̃ maykalíbong̃libo

29,000

Maykatlong̃laksang̃ maykalíbong̃libo maykawalong̃daan~may kawalongwaló

29,778

Maykalimang̃laksang̃maykatlong̃libong̃ maykasiyam na raan maykápat na dalawa

42,832

100,000

Maykatlonglibong̃tatló Maykatlonglibotatlong̃daan

2,300

*Sang̃yuta

100,000

(for greater clarity, the second form for 2,300 is used) Maykasiyam na líbomaykawalongdaanglimá

8,705

Maykalibong̃libomaykasiyam na raang̃ápat

9,804

Maykalibong̃libomaykasiyam na raang̃labing̃waló

9,818

10,000 *Sanglaksa

10,000

Ten thousand was termed isang laksa. Again, as in the previous multiples of 10, a process of multiplication, this time by 10,000, was implied by the use of the term laksa, such that dalawang laksa or 20,000 was two multiplied by 10,000. The term comes from the Sanskrit –laksha– which means 100,000, but the meaning changed into the lesser quantity of 10,000 as it passed on to Old Malay and Old Tagalog.60 The term still exists in ordinary Tagalog, albeit with phonetic changes, as dagsa, signifying a great amount, as in dagsa-dagsa or nagdagsaan ang mga tao sa EDSA—an uncountable or great number of people went to EDSA. From 10,001 through 19,999, the numbers are formed in the same way as they were formed from 1,001 through 1,999:

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A hundred thousand or 100,000 was termed as yuta, yota, or sangyuta. The use of yuta again implied, as in the previous multiples of 10, the process of multiplication, such that dalawang yuta meant two times 100,000 or 200,000. Pardo de Tavera writes that ayuta originally meant 10,000 in Sanskrit, implying that there has been a semantic interchange of meaning between yuta and laksa as it passed into Old Tagalog.61 100,001–199,999

The formation of numbers from 100,001 through 199,999 is through the previous artifice of the prefix labi sa, exactly in the same way as one counts from 10,000 through 19,999: Labí sa yutà sang̃puó

100,010

Labí sa yutà maykatlong̃daan labing̃siyam

100,219

Labí sa yutà maykawalong̃daang̃ maykapitong̃ limá

100,765

200,001–999,999

From 200,001 through 999,999, the numbers are formed with the prefix mayka–, in the same way that they were formed when counting from 20,001 through 99,999:

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Maykatlong̃yutà tatlong̃puô

200,030

Maykápat na yutà maykanim na pitó

300,057

Maykawalong̃yutà labí sa laksâ maykatlong̃libo maykapitong̃daan maykawalong̃ánim

712,676

1,000,000 *Sang̃ang̃awáng̃aw

1,000,000

One million is sang–angaw–angaw or one times 1,000,000, in the same way that the previous multiples of 10 denoted multiplication. It can also be expressed in the following alternative ways:

example of the name of a number which preserved, upon its passing into Tagalog, the same meaning it had in Sanskrit).63 It should be clear that this system of counting was an extremely sophisticated one. Spanish friars who studied Tagalog grammar expressed great amazement at the consistency and efficiency with which the language formed words to express numbers. Fray Sebastian de Totanes records these observations: Este es el rigoroso modo de contar el tagalog. Para comprender bien el ingenioso artificio, con que cuentan, nótese en lo dicho lo uniforme, que procede en todas sus mutaciones. Desde el primer 10, hasta el segundo (que es el 20,) cuenta con labi; pues lo mismo observa desde

Angawangaw Isangangawangaw sang̃ang̃awáng̃aw (or sang̃puong̃yutà)

el primer 100, hasta el segundo (que es el 200,) y desde el primer

1,000,000

diez mil, hasta el segundo (que es el veinte mil,) y desde el primer cien mil, hasta el segundo, que es el doscientos mil. Desde el segundo diez, (que es el veinte) hasta el décimo diez, (que es el ciento,)

Expressing one million as sangpuongyuta or ten 100,000, implies, as in the previous cases, a clear understanding of place value.

cuenta con mey; pues lo mismo hace desde el segundo ciento, (que

10,000,000

es el cien mil,) y desde el segundo cien mil, (que es el doscientos mil,)

Ten million in Old Tagalog is –cati–or –kati–. Pardo de Tavera gives us enough information to construct the following table to track the changes in meaning as the words passed from Sanskrit to Old Malay, to Old Tagalog:62

hasta el décimo cien mil, que es el millon. Todo se verá practicado en

es el doscientos), hasta el décimo ciento, (que es el mil,) y desde el segundo diez mil, (que es el veinte mil,) hasta el décimo diez mil, (que

lo dicho, si se reflexiona para comprender el artificio. Aunque ya con la comunicacion de los españoles, muchos cuentan como nosotros, y asi dicen: Dalawangpuo at isá, veinte y uno. Sangdáan at limá, ciento y cinco. Limánğ daang dalauánğpouó at limá quinientos y veinte y cinco, y asi de los demás números.64

Yuta

Laksa

Kati

Sanskrit

10,000 (ayuta)

100,000 (laksha)

10,000,000 (kôti)

Old Malay

1,000,000 (yuta, djuta)

10,000 (laksa)

100,000 (keti)

counts. In order to understand well the ingenious artifice through

Old Tagalog

100,000

10,000

10,000,000

changes. From the first 10 until the second (which is the number 20),

With regard to the number –cati– or –kati–, or 10,000,000, Pardo de Tavera observes that it is perhaps the only word for a number which has preserved its original Sanskrit meaning as it passed into Old Tagalog: “Este es quizás al único ejemplo de un nombre de cantidad que conserva, al pasar al Tag., la misma significación que en el Sans” (This is perhaps the only

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This therefore is the rigorous method through which the Tagalog which they count, note how uniform it proceeds throughout all of its it counts with labi; well, the same is seen from the first 100 until the second (which is 200), and from the first ten thousand until the second (which is twenty thousand), and from the first one hundred thousand until the second (which is two hundred thousand). From the second ten (which is twenty), until the tenth ten (which is one hundred) it counts with –mey–, which is the same from the second one hundred (which

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is two hundred), until the tenth one hundred (which is a thousand), and from the second ten thousand (which is twenty thousand), until the tenth ten thousand (which is 100,000) and from the second one hundred thousand (which is two hundred thousand), until the tenth one hundred thousand (which is one million). Everything is seen as done in the same manner, if one takes time to understand the method. Although, because of interaction with the Spaniards, many count as we do, and therefore they say: dalawangpuó at isa, twenty and one. Sangdaan at lima, one hundred and five. Limangdaan dalawang puo at lima, five hundred and twenty and five, similarly with the other numbers.

The observations of Totanes are important because they call attention not only to the exact mathematical pattern but also to the aesthetic beauty involved in the way this enumeration system was conceived and practiced. The basic building block was the simple numbers from one through nine. Complex numbers are built upon these simple numbers with strict mathematical and grammatical consistency. Numbers from 11 through 19 have their rules based on the prefix labi–. This pattern of constructing numbers through the prefix labi– is consistently carried out for 101, 1,001, 10,000, etc. As soon as twenty is reached, another set of rules for the construction of numbers take over, the rule of using the prefix may–, as well as a different way of looking at the counting process. Counting is no longer merely adding to the previous number but is a process of heading towards another quantity, which in this case is the nearest multiple of ten, as in maykatlongisá is one towards the number thirty. The process of counting with the prefix may– is repeated for the multiples of 20, 200, 2,000, etc. Another way of forming numbers, representing yet a third method and mental framework, is the counting of the different multiples of ten. Whereas the prefixes labi– and may– respectively represent the notions of greater than (>) and the idea of heading towards a greater quantity, in this case the nearest multiple of ten, the multiples of ten themselves represent a different idea of counting and involves a different mental process. As pointed out earlier, both the simple and the succeeding multiples of ten imply a process of multiplication. We thus have three different mental processes working within this enumeration system.

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A graphic analogy may be proposed to further describe this ancient enumeration system. Imagine a figure of concentric circles of ever increasing radii drawn in three different colors in an ever-repeating pattern. At the center of the circles is also the point of origin of the number line. As one continues to extend the number line and counts through to ever higher numbers, one also passes through the different three–colored concentric circles, where the three–color pattern represents the three different rules for constructing numbers. Counting numbers is not merely going through a straight line but is also an exercise in constructing a system of three–colored concentric circles whose colors vary periodically. While most other enumeration systems would merely have straightforward repetition of patterns, the ancient Tagalog system of counting is quantum arithmetic. No other enumeration system has such sophistication. This was an enumeration system which was complete, as it could express any number it desired; mathematically and grammatically consistent, as mathematical patterns and grammatical structure strictly followed rules; efficient, since it made full and efficient use of repeating patterns; and aesthetically impressive because of its level of sophistication. Furthermore, the implied processes of addition and multiplication contained in the formation of the counting words are also totally consistent with our modern understanding of the idea of place value in arithmetic.

The Last Number: lsang Bahala Did the counting process ever end? Apparently Old Tagalog, as well as Old Malay, did not subscribe to the notion of mathematical infinity but instead had what is called “limit numbers” or numbers beyond which one stops at counting. For ancient Tagalog number crunchers, this limit number was a thousand yuta, 100,000,000, or one hundred million. Beyond this number was an inconceivable mathematical void which no one crosses. Fray Francisco Blancas de San Jose fortunately was able to record and preserve this mathematical notion in his 1610 Arte y reglas de la lengua Tagala: . . . fang libong yota y millares de yota no fe conoce: fino dizen fang bahala, que es dezir un que fe yo, ycao na ang bahala, echa por effos trigos de Dios: que ya no fe puede pensar.65

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. . . one thousand yota and thousands of yota is not known. Instead they say sang bahala, which means “What do I know? I leave it up to you [Bahala ka]. What can I do? Of these things one can no longer conceive.”

1903b, vol. 7, pp. 188–89. 17 William Henry Scott, Barangay: Sixteenth-Century Philippine Culture and Society. Quezon City: Ateneo de Manila University Press, 1994, pp. 123–24. 18 San Antonio, “Cronicas,” pp. 359–60.

Notes

19 Mateo Sanchez, Vocabulario de la lengua Bisaya, Manila, 1711, as cited in Scott, Barangay, p. 123.

Morris Kline, Mathematical Thought from Ancient to Modern Times. Oxford: Oxford University

Howard Eves, An Introduction to the History of Mathematics. New York: Holt, Rinehart, and

Press, 1972.

20 Juan de Plasencia, “Customs of the Tagalogs,” pp. 189–90. 21 Serrano Laktaw, Estudios gramaticales, p. 360. 22 Francisco Ignacio Alcina, Historia de las islas e indios de Bisayas, 1668, as cited in Scott,

Winston, 1961. 3

Morris Kline, Mathematics for the Nonmathematician. New York: Dover Publications, 1967, p. 11.

Gaspar de San Agustin, Compendio del arte de la lengua tagala. Manila: lmprenta de “Amigos del Pais,” Calle de Anda, 1703, 1787, 1879, p. 115.

Barangay, p. 123. 23 Miguel de Loarca, “Relation of the Filipinas Islands,” 1582–1583, reprinted in Blair and Robertson, 1903a, vol. 5, p. 165. 24 Scott, Barangay, p. 123, cautions us and qualifies Loarca’s account: “What Loarca called months

San Agustin, Compendio, p. 117.

Eladio Zamora, O.S.A. Las corporaciones religiosas en Filipinas, Valladolid: Imprenta y Libresia

were seasonal events connected with swidden farming: they do not appear in early Visayan

Religiosa de Andrés Martin, 1901, reprinted in Emma Blair and James Robertson, The Philippine Islands, 1493–1898, 1907b, vol. 46, p. 347. 7

16 Juan de Plasencia, “Customs of the Tagalogs,” Manila, 1589, reproduced in Blair and Robertson,

Vicente Barrantes, Apuntes interesantes, Madrid: Impr. de El Pueblo, 1869, reprinted in Blair and Robertson, 1907a, vol. 45, p. 292.

dictionaries as the names of months, nor do any other names.” 25 Ibid., p. 124. 26 Scott reminds us that the system, like other calendar solutions proposed by other civilizations, has some significant problems: “. . . the Visayan month was a lunar month—29-and-a-half days and 43 minutes, to be exact—so twelve of them did not add up to a year, but only 354 days. They

Jesus T. Peralta, “Petroglyphs and Petrographs,” in Kasaysayan, 1998, vol. 2, p. 135.

were therefore not the equivalent of months in the western calendar, which are arbitrary divisions

Francisco Colin, Labor evangelica de los Obreros de la Compañia de Jesus en las lslas Filipinas.

unrelated to the moon, approximately one-twelfth of a solar year of 365 days.” Ibid., p. 122.

Madrid: 1663, vol. 1, reprinted 1904, p. 25.

27 Ibid., p. 121.

10 Miguel de Loarca, “Tratado de las islas Filipinas,” written around 1580, reproduced in Juan Jose Delgado, Historia general sacro-profana, politica y natural de las islas del poniente llamadas Filipinas (Manila: Juan Atayde, 1892), p. 383. 11 The detailed discussion of the historical and anthropological significance of the LCI is found in

28 San Antonio, “Cronicas,” p. 359. 29 Scott, Barangay, p. 121. 30 San Antonio, “Cronicas,” p. 359.

Antoon Postma, “The Laguna Copper-Plate Inscription (LCI): A Valuable Philippine Document,”

31 Ibid.

National Museum Papers vol. 2, no. 1 (1991), pp. 1–25. Also cited in E. P. Patanñe, The Philippines

32 Ibid.

in the 6th to the 16th Centuries, pp. 83–103. Quezon City: LSA Press, 1996.

33 Scott, Barangay, p. 121.

12 For the present purposes we use the translation provided by Patanñe, The Philippines, p. 85. Postma, the first translator of the LCI, provides four different translations of the document, reflecting his growing understanding of the text. 13 Juan Francisco de San Antonio, “Cronicas de la Provincia de San Gregorio Magno,” chap. 45, reproduced in Blair and Robertson, 1906, vol. 40, pp. 358–59. 14 The notorious practice of so-called “Filipino time” or habitual tardiness, now slowly disappearing, might in fact be traced or at least related to this ancient manner of telling time: I am not late for

34 Serrano Laktaw, Estudios gramaticales, p. 361. 35 Ibid. 36 Serrano Laktaw, Estudios gramaticales, p. 357. 37 San Antonio, “Cronicas,” p. 363. 38 Sebastian de Totanes, Arte de la lengua tagala y Manual tagalog para la administracion de los sacramentos. Binondo: lmprenta de Miguel Sanchez y Ca., Sampaloc, 1745.

a meeting, despite having arrived past the agreed hour, because the meeting had not yet started

39 Totanes, Arte de la lengua tagala, p. 118.

when I arrived.

40 Ibid., p. 121.

15 Pedro Serrano Laktaw, Estudios gramaticales sobre la lengua tagalog. Santa Cruz, Manila:

41 San Antonio, “Cronicas,” p. 363.

Imprenta de Juan Fajardo, 1929, p. 360.

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References

42 Ibid., p. 361. 43 Ibid.

Alcina, Francisco Ignacio. 1668. Historia de las islas e indios de Bisayas. Part 1, Books 1–4, Victor

44 The first three rows are from Serrano Laktaw’s tables (Estudios gramaticales, p. 358), while the last three are reconstructed from data provided by Scott. 45 San Antonio, “Cronicas,” pp. 361–62.

Baltazar transcription, 1962. Chicago: University of Chicago Philippine Studies Program. Barrantes, Vicente. 1907. Apuntes interesantes, 1870. In The Philippine Islands, 1493–1898, vol. 45: 1736, ed. Emma Blair and James Robertson, 286–95. Cleveland: Arthur H. Clark.

46 Table constructed from data provided by Scott, Barangay.

Blair, Emma Helen and James Robertson, eds. 1903a. The Philippine Islands, 1493–1898, vol. 5: 1582–1583. Cleveland: Arthur H. Clark.

47 San Antonio, “Cronicas,” p. 358. 48 The basic table is from Serrano Laktaw (Estudios gramaticales, p. 358), but the two items with asterisks are additions to the Serrano Laktaw table from data provided by San Antonio.

———. 1903b. The Philippine Islands, 1493–1898, vol. 7: 1588–1591. Cleveland: Arthur H. Clark. ———. 1906. The Philippine Islands, 1493–1898, vol. 40: 1690–1691. Cleveland: Arthur H. Clark.

49 San Antonio, “Cronicas,” p. 362.

———. 1907a. The Philippine Islands, 1493–1898, vol. 45: 1736. Cleveland: Arthur H. Clark.

50 Serrano Laktaw, Estudios gramaticales, p. 358.

———. 1907b. The Philippine Islands, 1493–1898, vol. 46: 1721–1739. Cleveland: Arthur H. Clark.

51 This section reproduces all of the examples found in Serrano Laktaw’s Estudios gramaticales,

Blancas de San Jose, Francisco. 1610. Arte y reglas de la lengua Tagala. Bataan.

pp. 352–56, since this important study methodically describes the old system but yet has never been given the attention it deserves. Other sources apart from Serrano Laktaw are indicated accordingly. We follow Serrano Laktaw’s practice of marking with an asterisk ancient usage that

Colin, Francisco. 1663/1904. Labor evangelica , ministerios apostolicos de los Obreros de la Compañia de Jesus en las lslas Filipinas, vol. 1, ed. Pablo Pastells, S.J. Reprint, Barcelona: Henrich. Eves, Howard. 1961. An introduction to the history of mathematics. New York: Holt, Rinehart, and

is still conserved in present practice. 52 Paul Schachter and Fe Otanes, Tagalog Reference Grammar. Berkeley: University of California

Winston. Kline, Morris. 1967. Mathematics for the nonmathematician. New York: Dover Publications.

Press, 1972, p. 200. 53 Serrano Laktaw, Estudios gramaticales, p. 352.

———. 1972. Mathematical thought from ancient to modern times. Oxford: Oxford University Press.

54 The linker –ng undergoes phonetic changes and becomes labindalawa, labing-anim, labimpito

Loarca, Miguel de. 1892. Tratado de las islas Filipinas, written around 1580. In Historia general sacro–

depending on whether the first consonant of the following simple number is bilabial, dental/

profana, politica y natural de las islas del poniente llamadas Filipinas, Juan Jose Delgado. Manila:

alveolar, or velar/glottal; see Schachter and Otanes, Tagalog Reference Grammar, pp. 200–201,

Juan Atayde. Also reprinted in The Philippine Islands (1903a), 1493–1898, vol. 5: 1582–1583, ed.

for the linguistic rules governing these changes.

Emma Blair and James Robertson, 34–187. Cleveland: Arthur H. Clark.

55 Schachter and Otanes, Tagalog Reference Grammar, p. 200.

Newton, Isaac. 1934. Sir Isaac Newton’s mathematical principles of natural philosophy and his system of the world, trans. Andrew Motte. Berkeley: University of California Press.

56 Serrano Laktaw, Estudios gramaticales, p. 357. 57 Gaspar de San Agustin, Compendio del arte de la lengua tagala. Manila: Imprenta de “Amigo del Pais,” Calle de Anda, 1703, 1787, 1879, p. 115. 58 Sebastian de Totanes, Arte de la lengua tagala y Manual tagalog para la administracion de los sacramentos. Binondo: Imprenta de Miguel Sanchez y Ca., Sampaloc, 1745.

Pardo de Tavera, T. H. 1887. El sanscrito en la lengua tagalog. Paris: Imprimerie de la faculté de médecine. Patanñe, E. P. 1996. The Philippines in the 6th to 16th centuries. Quezon City: LSA Press. Peralta, Jesus T. 1998. Petroglyphs and petrographs. In Kasaysayan: The Story of the Filipino People, vol. 2: The earliest Filipinos, 134–43. [Manila]: Asia Publishing.

59 Serrano Laktaw, Estudios gramaticales, p. 357. 60 T. H. Pardo de Tavera, El sanscrito en la lengua tagalog. Paris: lmprimerie de la faculté de médecine, 1887, p. 34.

Plasencia, Juan de. 1903. Customs of the Tagalogs. In The Philippine Islands, vol. 7: 1588–1591, ed. Emma Blair and James Robertson, 173–96. Cleveland: Arthur H. Clark. Postma, Antoon. 1991. The Laguna copper-plate inscription (LCI). National Museum Papers 2(1):

61 Ibid., p. 55.

1–25.

62 Ibid.

San Agustin, Gaspar de. 1703/1787/1879. Compendio del arte de la lengua tagala. Manila: lmprenta

63 Ibid., p. 26.

de “Amigos del Pais,” Calle de Anda.

64 Totanes, Arte de la lengua tagala, p. 103.

San Antonio, Juan Francisco de. 1906. Cronicas de la Provincia de San Gregorio Magno, chap. 45. In

65 Francisco Blancas de San Jose, Arte y reglas de la lenqua Tagala. Bataan, 1610, p. 266.

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Sanchez, Mateo. 1617/1711. Vocabulario de la lengua Bisaya, compuesto por el R.P. Matheo Sanchez de la Sagrada Compañia de Jesus y avmentada por otros PP. de la misma Compañia, para el vso y como Manila: Colegio de la Sagrada Compañia de Jesus por D. Gaspar Aquino de Belen. Microfilm Schachter, Paul and Fe Otanes. 1972. Tagalog reference grammar. Berkeley: University of California Press. Scott, William Henry. 1994. Barangay: Sixteenth-century Philippine culture and society. Quezon City: Ateneo de Manila University Press. Serrano Laktaw, Pedro. 1929. Estudios gramaticales sobre la lengua tagalog. Manila: lmprenta de Juan Fajardo. Totanes, Sebastian de. 1745. Arte de la lengua tagala y Manual tagalog para la administracion de los sacramentos. Binondo: lmprenta de Miguel Sanchez y Ca., Sampaloc. University of the Philippines College Baguio, Discipline of Mathematics Faculty. 1996. The algebra of the weaving patterns, gong music, and kinship system of the Kankana-ey of Mountain Province. [Baguio City]: Discipline of Mathematics Faculty, University of the Philippines College Baguio. Zamora, Eladio, O.S.A. 1907b. Las corporaciones religiosas en Filipinas, 1901. In The Philippine Islands, 1493–1898, vo1. 46: 1721–1739, ed. Emma Blair and James Robertson, 319–63. Cleveland: Arthur H. Clark.

Editor’s Note This essay was prepared for a graduate course on the History of Mathematics offered by Dr. Mari-Jo P. Ruiz, which the late Mr. Ricardo Manapat took in the second semester of academic year 2000–2001. Dr. Ruiz, who had kept the original essay, broached its possible publication. Philippine Studies is very grateful to Mrs. Angelita L. Manapat and her children, Maria Teresa Manapat Hayakawa, Maria Lourdes Manapat de Pala, Maria Cristina Manapat-Sims, and Jose Alfredo Manapat, for permission to publish it in this journal. Except for minor editorial changes, such as ensuring consistency of the author’s style and verifying the exactness of cited extracts, this posthumous publication reproduces the original paper in the manner it was composed, for which reason the journal’s house style has been suspended. The reference list, however, has been reformatted to conform to the journal’s style.

Ricardo Manapat

was director of the Records Management and Archives Office (National

Archives of the Philippines), from 1996–1998 and 2002–2008. In 1976 he obtained his A.B. Philosophy degree and graduated with Departmental Honors from the Ateneo de Manila University, where he became an Instructor in the Philosophy Department from 1977–1979. He completed his M.A. in Spanish in Rizal Studies at the University of the Philippines in 2004. In 2005 he began his PhD studies at La Trobe University, where the following year he was awarded the Commonwealth International Student Scholarship as well as the DM Myer Medal for most outstanding graduate student. He was in the midst of doctoral research when he died of myocardial infarction on 24 December 2008. He was 55 years old. He authored Some are Smarter than Others: The History of Marcos’ Crony Capitalism (New York: Aletheia Publications, 1991) and was editor-in-chief of the “Smart File,” Smart File Magazine Animal Farm Series.

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