Yupana


A yupana is an abacus used to perform arithmetic operations dating back to the time of the Incas.

Types

The term yupana refers to two distinct classes of objects:
Although very different from each other, most of the scholars who have dealt with table-yupana, have then extended its reasoning and theories to the yupana of Poma de Ayala and vice versa, perhaps in an attempt to find a unifying thread or a common method. The Nueva Coronica was discovered only in 1916 in the library of Copenhagen and that part of the studies on it were based on previous studies and theories regarding table-yupanas.

History

Several chroniclers of the Indies described, unfortunately approximately, the Incan abacus and its operation.

Felipe Guaman Poma de Ayala

The first was Guaman Poma de Ayala that in 1615 approximately, wrote:
In addition to providing this brief description, Poma de Ayala draws a picture of the yupana: a board of five rows and four columns in which are designed a series of white and black circles.

José de Acosta

The father Jesuit José de Acosta wrote:

Juan de Velasco

Father Juan de Velasco wrote:

Table-yupana

Chordeleg

The earliest known example of a table-yupana was found in 1869 in Chordeleg, Azuay Province, Ecuador. It is a rectangular table of wood consisting of 17 compartments, of which 14 are square, 2 are rectangular, and one is octagonal. On two edges of the table there are other square compartments raised and symmetrically arranged one another, to which two square platforms, are overlapped. These structures are called towers. The table presents a symmetry of the compartments with respect to the diagonal of the rectangle. The four sides of the board are also engraved with figures of human heads and a crocodile. As a result of this discovery, Charles Wiener began in 1877 a systematic study of these objects. Wiener came to the conclusion that the table-yupanas served to calculate the taxes that farmers paid to the Incan empire.

Caraz

Found at Caraz in 1878 - 1879, this table-yupana is different from that of Chordeleg as the material of construction is the stone and the central compartment of octagonal shape is replaced with a rectangular one; towers also have three shelves instead of two.

Callejón de Huaylas

A series of table-yupanas much different from the first, was described by Erland Nordenskiöld in 1931. These yupana, made of stone, present a series of rectangular and square compartments. The tower is composed of two rectangular compartments. The compartments are arranged symmetrically with respect to the axis of the smaller side of the table.

Triangular yupana

These yupana, made of stone, have 18 compartments of triangular shape, arranged around the table. On one side there is a rectangular tower with only one floor and three triangular compartments. In the central part there are four square compartments, coupled between them.

Chan Chan

Identical to the yupana of Chordeleg, both for the material and the arrangement of the compartments, this table-yupana was found in the archaeological complex of Chan Chan in Peru in 1967.

Cárhua de la Bahía

Discovered in the province of Pisco, these table-yupanas are two tables in clay and bone. The first is rectangular, has 22 square and three rectangular compartments, and has no towers. The second is rectangular containing 22 square compartments, two L-shaped and three rectangular in the center. The compartments are arranged symmetrically with respect to the axis of the longer side.

Huancarcuchu

Discovered in the upper Ecuador by Max Uhle in 1922, this yupana is made of stone and its bins are drawn. It has the shape of a scale consisting of 10 overlapping rectangles: four on the first floor, three on the second, two in the third and one in the fourth. This yupana is the one that is closest to the picture by Poma de Ayala in Nueva Coronica, while having a line less and being half drawn.

Florio

C. Florio presents a study
which does not identify a yupana in these archaeological findings, but an object whose name is unknown and which has been forgotten. Instead, this object is to connect to the tocapu called “llave inca” and to the yanantin-masintin philosophy. The scholar reaches this conclusion starting from the lack of objective evidences which recognize a yupana in this object, a belief that consolidated over years only for the repeat of this hypothesis never demonstrated, and by crossing data from the Miccinelli Documents and the tocapu catalogued by Victoria de la Jara.
Supposing to colour the different compartments of the table-yupana, C. Florio identifies a drawing very similar to a really existing tocapu and catalogued by Victoria de la Jara. In addition, in the tocapu reported in figure D, also catalogued by V. de la Jara, Florio identifies a stylization of the tocapu C and the departure point for creating the tocapu “llave inca”. She finds the relation between the table-yupana and the Inca key also in their connection with the concept of duality: the table-yupana structure is clearly dual and Blas Valera in “Exul Immeritus Blas Valera populo suo” describes the tocapu we call Inca key as representing the concept of the “opposite forces” and the “number 2”, both strictly linked to the concept of duality.
According to C. Florio, the real yupana used by the Incas is that of Guáman Poma, but with more columns and rows. Guáman Poma would have represented just the part of the yupana useful for carrying out a specific calculation, which Florio identifies to be a multiplication.

Theories of Yupana Poma de Ayala

Henry Wassen

In 1931, Henry Wassen studied the yupana of Poma de Ayala, proposing for the first time a possible representation of the numbers on the board and the operations of addition and multiplication. He interpreted the white circles as gaps, carved into yupana in which to insert the seeds described by chroniclers: so the white circles correspond to empty gaps, while the blacks circles correspond to the same gaps filled with a black seed.
The numbering system at the base of the abacus was positional notation in base 10.
The representation of the numbers, then followed a vertical progression such that the units were positioned in the first row from the bottom, in the second the tens, hundreds in the third, and so on.
Wassen proposed a progression of values of the seeds that depends on their position in the table: 1, 5, 15, 30, respectively, depending on who occupy a gap in the first, second, third and fourth columns. Only a maximum of five seeds could be included in a box belonging to the first column, so that the maximum value of said box was 5, multiplied by the power of the corresponding line. These seeds could be replaced with one seed of the next column, useful during arithmetic operations. According to the theory of Wassen, therefore, the operations of sum and product were carried out horizontally.
This theory received a lot of criticism due to the high complexity of the calculations and was therefore considered inadequate and soon abandoned.
By way of example, the following table shows the number 13457.

This first interpretation of the yupana of Poma de Ayala was the starting point for the theories developed by subsequent authors, up to the present day. In particular, no one ever moved away from the positional numbering system until 2008.

Emilio Mendizabal

was the first to propose in 1976 that the Inca were using, as well as the decimal representation, also a representation based on the progression 1,2,3,5. Mendizabal in the same publication pointed out that the series of numbers 1,2,3 and 5, in the drawing of Poma de Ayala, are part of the Fibonacci sequence, and stressed the importance of "magic" that had the number 5 for civilization the north of Peru, and the number 8 for the civilizations of the south of Peru.

Radicati di Primeglio

In 1979, Carlos Radicati di Primeglio emphasized the difference of table-yupana from that of Poma de Ayala, describing the state of the art of the research and theories advanced so far. He also proposed the algorithms for calculating the four basic arithmetic operations for yupana of Poma de Ayala, according to a new interpretation for which it was possible to have up to nine seeds in each box with vertical progression for powers of ten. The choice of Radicati was to associate to each gap a value of 1.
In the following table is represented the number 13457

William Burns Glynn

In 1981, the English textile engineer William Burns Glynn proposed a positional base 10 solution for the yupana of Poma de Ayala.
Glynn, as Radicati, adopted the same Wassen's idea of full and empty gaps, as well as a vertical progression of the powers of ten, but proposed an architecture that allowed to greatly simplify the arithmetic operations.
The horizontal progression of the values of the seeds in its representation is 1, 1, 1 for the first three columns, so that in each row is possible to deposit a maximum of ten seeds. Ten seeds of any row is correspond to a single seed of the upper line.
The last column is dedicated to the memory, which is a place where you can drop momentarily ten seeds, waiting to move them to the upper line. According to the author, this is very useful during arithmetic operations in order to reduce the possibility of error.
The solution of Glynn has been adopted in various teaching projects all over the world, and even today some of its variants are used in some schools of South America.
In the following table is represented the number 13457

Nicolino de Pasquale

The Italian engineer Nicolino de Pasquale in 2001 proposed a positional solution in base 40 of the yupana of Poma de Ayala, taking the representation theory of Fibonacci already proposed by Emilio Mendizabal and developing it for the four operations.
De Pasquale also adopts a vertical progression to represent numbers by powers of 40. The representation of the numbers is based on the fact that the sum of the values of the circles in each row gives as total 39, if each circle takes the value 5 in the first column, 3 in the second column, 2 in the third and 1 in the fourth one; it is thus possible to represent 39 numbers, united to neutral element ; this forms the basis of 40 symbols necessary for the numbering system.
One of the possible representations of the number 13457 in the yupana by De Pasquale is shown in the following table:

The theory of De Pasquale opened, in the years after his birth, great controversy among researchers who divided mainly into two groups: one supporting the base 10 theory and another supporting the base 40 one. The Spanish chronicles of the time of the conquest of the Americas indicated that the Incas used a decimal system and that since 2003 the base 10 has been proposed as the basis for calculating both with the abacus and the quipu
De Pasquale has recently proposed the use of yupana as astronomical calendar running in mixed base 36/40 and provided its own interpretation of the Quechua word huno, translating it as 0.1. This interpretation diverges from all the chroniclers of the Indies, starting from Domingo de Santo Tomas which in 1560 translated huno with chunga guaranga.

Cinzia Florio

In 2008 Cinzia Florio proposes an alternative and revolutionary approach in respect to all the theories proposed so far. For the first time we deviate from the positional numbering system and we adopt the additive, or sign-value notation.
Relying exclusively on the design of Poma de Ayala, the author explains the arrangement of white and black circles and interprets the use of the abacus as a board for making multiplications, in which the multiplicand is represented in the right column, the multiplier in the two central columns and the result is shown in the left column. See the following table.

The theory differs from all the previous in several aspects: first, the white and black circles would not be gaps that may be filled with a seed, but rather different colors of seeds, representing respectively tens and units.
Secondly, the multiplicand is entered in the first column respecting the sign-value notation: so, the seeds can be entered in any order and the number is given by the sum of the values of these seeds.
The multiplier is represented as the sum of two factors, since the procedure for obtaining the product is based on the distributive property of multiplication over addition.
The table multiplier drawn by Poma de Ayala with that provision of the seeds, represent according to the author, the calculation: 32 x 5, where the multiplier 5 is decomposed into 3 + 2. The sequence of numbers 1,2,3,5 would be casual, contingent to the calculation done and not related to the Fibonacci series.
ProductMultiplicatorMultiplicatorMultiplicand
3X2X
◦◦◦••◦◦•••◦
◦◦◦◦•◦◦•◦◦•
•••••◦◦◦◦•◦
◦◦◦◦•◦◦•◦•◦
◦◦◦•••••◦◦•
151966432

Key: ◦ = 10; • = 1; The operation represented is: 32 x 5 = 32 x = + = 64 + 96 = 160
The numbers represented in the columns are, from left to right: 32, 64 = 32 x 2 and 32 x 3 = 96 and finally 151. In this issue are based all possible criticisms of this interpretation, since 151 is obviously not the sum of 96 and 64. Florio, however, notes that a mistake of Poma de Ayala, in designing a black circle instead of a white one, would have been possible. In this case, changing just a black circle with a white one in the last column, we obtain the number 160, which is exactly the product sought as the sum of the quantities present in the central columns.
With a yupana as the one designed by Poma de Ayala can not be represented every multiplicands, but it is necessary to extend the yupana vertically to represent numbers whose sum of digits exceeds 5. The same thing goes for the multipliers: to represent all the numbers is necessary to extend the number of columns. Apart from the supposed error calculation , is the only one that identifies in the yupana of Poma de Ayala a mathematical and consistent message and not a series of random numbers as in other interpretations.

Chroniclers of the Indies

  • Theory by Wassen and table-Yupana

  • Theory by Glynn Burns and school projects

  • Theory by De Pasquale

*

  • Theory by C. Florio


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