Juan Pérez de Moya became known in the Iberian Peninsula because of the publication of his

Juan Pérez de Moya, conocido en la Península Ibérica por la publicación de la

Juan Pérez de Moya was born in Santisteban del Puerto, in 1513, and died in Granada in 1596. He was not a mathematician of first magnitude, because his works do not contain original contributions; nevertheless, Valladares Reguero (Valladares Reguero,

Moya began his mathematical production in 1554 with the

The three works just mentioned were then reordered originating the

Having collected in the

Five years later, Moya revisited all the mathematical material that he had already produced, and after improving (correcting some existing lapses) and expanding it, he publishes it in a voluminous work, the

Based on the themes developed in the

The first printed geometrical treatise in the Spanish literature is Juan de Ortega’s

In sixteenth century Europe there are some unavoidable names in the study of Geometry and its applications, such as Finé, Bovelles, Ringelberg, Peletier, Apiano, Dürer, and others, whose works were directed to specific areas of that science. One of the most important works, both for the diversity of geometrical themes presented and for the relations established between practical Geometry and its theoretical foundations, is Tartaglia’s

The first notions of geometry given by Moya are included in the

Going back to Moya, we now present the statement and resolution of a problem concerning a round field the perimeter of which measures 44 staffs and the diameter is asked. He notes that the ratio between the circumference and diameter of any circle is 3 1/7 (

Es una tierra redonda, la qual tiene de circunferencia 44 varas, demando: ¿que tendra de diametro? Para saber esta, y sus semejantes, tendras por regla general que la proporcion de la circunferencia a su diametro es tripla sexquiseptima, y al contrario, del diametro a su circunferencia es subtripla sexquiseptima (Pérez de Moya,

The procedures used by the cited authors are described in abacus’s treatises (Simi,

Despite the similarity observed between Moya’s

On the other hand, some of the problems in

The first volume of

The work is organized in forty-four chapters; the first forty providing useful tools to understand the remaining four. It starts such as

Y porque con mayor fundamento se pueda disputar y dar razon de esta arte [geometría], pondremos primero tres generos de principios, siguiendo la orden de Euclides, que son diffiniciones, peticiones, y communes sentencias (Pérez de Moya,

But while in Euclid’s

The last chapters of

Medir una cosa no es otro, sino saber quantas medidas famosas contiene la cosa que se mide. Medida famosa dizen a una qualquier medida usada y notoria acerca de algunas o de muchas gentes (Pérez de Moya,

His terms are very similar to those used by Tartaglia:

Misurare alcuna quantita, non vuol inferir altro, che un voler trovar quante volte si ritrovi in quella alcuna famosa quantita, over qual parte, over quante parti sia di detta famosa quantita. Per famose quantita si debbe intendere per quelle specie di misure communamente usitate per le provincie, over citta, in tai misurationi (Tartaglia,

In the context of altimetry, the text shows how to determine heights, depths and inaccessible distances, using “regla status” and “two staffs”. These instruments were not mentioned in

The questions about squaring the circle and duplicating the cube, which were under the attentive eyes of the mathematicians of the Renaissance, also took the interest of Moya. In the first case, he presented three approximate procedures (Pérez de Moya,

Ma per non lasciar di narrar quanto che nella quadratura del detto cerchio ho trovato scritto, & massime di quele, che nella practica sono di qualche cõmodita, over utilita, Carlo Bouile in fin de l’opera sua, da un brevíssimo modo, over regola da ridure un quadrato in un cerchio (…) (Tartaglia,

Tartaglia does not mention the title of Bovelles’s work, but we can check that this subject is studied in the last folia of the

The importance of doubling (obviously approximately) the cube is also clear in this work of Moya. For the construction, he recommends the lecture of the fourth book of Dürer’s

Alberto Durero enel quarto libro de su Geometria pone la orden practicalmente que se ha de tener para saber doblar, o tresdoblar el cubo, o cuerpo quadrado a forma de dado, porque es regla necessaria para muchas cosas mechanicas la quise poner aquí (Pérez de Moya,

The mention of Dürer confirms the influence that this author had in Spain, mainly due to the practical features of his work (Vasconcelos,

The influence of Tartaglia’s

The

The volume is structured in four Books (

Moya uses a big part of the

There are some important differences between the

Moya defines geometry similarly to Tartaglia, (Pérez de Moya,

Moya follows Tartaglia when he describes the division of a line segment in mean and extreme ratio (or, as they say,

Moya also follows Tartaglia in many other occasions: for example in the definition of angle

The subjects related to circumscriptions between polygons and circles are introduced by means of seven definitions similarly to Tartaglia (Tartaglia,

Campano |
Tartaglia |
Moya |
---|---|---|

Elementos IV, 1 |
Cap. IX, 1 |
---- |

Elementos IV, 2 |
Cap. IX, 2 |
Cap. 32 |

Elementos IV, 3 |
Cap. IX, 3 |
Cap. 31 |

Elementos IV, 4 |
Cap. IX, 4 |
Cap. 27 |

Elementos IV, 5 |
Cap. IX, 5, 6 |
Cap. 23 |

Elementos IV, 6 |
Cap. IX, 7 |
Cap. 34 |

Elementos IV, 7 |
Cap. IX, 8 |
Cap. 33 |

Elementos IV, 8 |
Cap. IX, 9 |
Cap. 28 |

Elementos IV, 9 |
Cap. IX, 10 |
Cap. 24 |

Elementos IV, 10 |
Cap. IX, 11 |
Cap. 16. art. 5 |

Elementos IV, 11 |
Cap. IX, 12 |
Cap. 36 |

Elementos IV, 12 |
Cap. IX, 13 |
Cap. 35 |

Elementos IV, 13 |
Cap. IX, 14 |
Cap. 29 |

Elementos IV, 14 |
Cap. IX, 15 |
Cap. 25 |

Elementos IV, 15 |
Cap. IX, 16 |
Cap. 38 |

Elementos IV, 16 |
Cap. IX, 17 |
----- |

There are two other topics in Moya’s studies reflecting the great influence of Tartaglia’s work. One is focused on the doubling of the cube and the other on the construction of an isosceles triangle with both angles of the base double of the angle at the vertex (Pérez de Moya,

For the doubling of the cube, Moya explains the method of the “very ingenious” (

(…) si tirando una linea de un punto a otro [o sea, de K a L], passar por el punto, o angulo D auras acertado, y si passare baxo, abre mas el compas y señala con el de la misma manera, y si passare alto, cierra el compas hasta tanto que se haga, que la linea que del un punto al otro se echare passe justamente por el punto D, como haze la linea KL (Pérez de Moya,

To prove that this construction satisfies the required, Moya notes that the line segments FK and CL are the mean proportionals between

And he uses Euclid’s

Lo qual todo se demuestra por la duodecima diffinicion del quinto de Euclides, y por la treynta y seys del onzeno, que infieren en substancia, que siendo quatro lineas continuas proporcionales, la proporcion del cubo de la primera linea al cubo de la segunda, sera como la proporcion de la primera a la quarta de las dichas líneas (Pérez de Moya,

These arguments are analogous to those of Tartaglia:

Tutto questo si approva, & dimostra per la duodecima diffinitione del quinto, & per la trentesimasesta del undecimo, che in sostanza inferiscono, che essendo quattro linee continue proportionali, la proportione del cubo della prima, al cubo della seconda sara, come la proportione della prima alla quarta di dette linee (Tartaglia,

As we know, the major works of practical geometry of the Renaissance included the study of measuring instruments, since this subject is closely linked with the resolution of problems of architecture, fortification, engineering, astronomy, cosmography and navigation

As Maroto and Piñero note, there are important similarities between Moya and Finé concerning the study of altimetry instruments and its applications (Maroto and Piñero,

The division of the circle into equal parts only using the ruler and the compass was studied by the authors of the Renaissance. Sometimes, they included in their texts approximate constructions, even when exact constructions were possible (Simi,

Moya knew that, with an opening of the compass equal to the radius of the circumference, he could determine the sixth part of the circumference and that this would allow him to divide it in twelve and in twenty-four equal parts. A similar reasoning would have enabled him to divide the circumference in 8, 16 and 32 equal parts, justifying it by Euclid’s

Our author does not explain why he uses approximate procedures when there are exact constructions for the same purposes; but we stress that in all of them the compass has a fixed opening which is equal to the radius of the circumference that he wants to divide, and that they provide very good approximations (the error is nearly one thousandth part of the radius of the circle considered). Furthermore, Moya also indicates a construction (obviously approximate) for the division of the circle in 36 equal parts, using the fourth division of the circumference into three equal parts (Pérez de Moya,

The use of the compass with a fixed opening in the constructions was a real challenge in the mid-sixteenth century. Tartaglia devotes to this subject a chapter of the Fifth Part of

Moya’s

Para medir triángulos, ay tantos modos, y primores que dezir que quererlos referir aqui seria confundir los entendimientos de algunos medidores, con los muchos preceptos, los quales por averlos puesto en otro volumen, solo pondre una regla general para medir cualquier triangulo de qualquiera suerte y genero que sea, con solo la noticia de sus lados (Pérez de Moya,

In this work, Moya keeps a similar style to

One topic discussed is the division of the circle. Moya gives the exact construction for dividing the circle into three equal parts and constructions for the approximate division of a circle in 5, 7 and 11 parts, which efficiently work; he takes from Dürer the division in 7 equal parts (Pedoe,

Moya refers to some of his sources. Euclid is the most cited author and while he does not always clarify the name and the version of the

Since the writing of his first geometrical text until the printing of

The geometrical works of the sixteenth century that mainly influenced him were from French and Italian authors, such as Peletier, Finé, Bovelles, Cardano and Tartaglia, but to this list should be added (and it is not exhaustive!) the names of Dürer, Ringelberg, Frisius, and the Spanish Aguilera. Sometimes, Moya indicates incompletely the titles of the texts consulted. For example, he simply says Forcius’s

We are sure that among the geometrical works of the sixteenth century his main source is Tartaglia’s

The interest that Euclid’s

As it is known, the first printed edition of Euclid’s

I am very grateful to Antoni Malet and to Carlos Sá for their thoughtful and detailed suggestions.

Research funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT – Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.

See the letter addressed by Moya to readers (Pérez de Moya,

When Moya published

The printing of the first book was delayed by the large number of its figures, and therefore the second was printed before it.

See the letter wrote in December of 1567, in (Pérez de Moya, “El Bachiller Ivan Perez de Moya a los lectores”).

This is a rare book of which only a copy is known. It is kept in the National Library of Portugal (Res 6553P). A brief study of this work is given in (Silva and Malet,

See (Silva,

See (Silva,

See (Ortega,

According to (Smith,

See, for example, (Pacioli,

Pacioli dedicates 76 folios of the second part of his

The example given for the first conception involves three segments and the example given for the sixth conception is numerical (Pérez de Moya,

Moya referes to

For example to the area of the regular pentagon he gives A=

It says: “Continued greatness cannot be infinitely increased but may be infinitely decreased” (Pérez de Moya,

Cardano says the following: «Pro mensurando trigono æquialtero quadrabis latus eius, & productum multiplicabis per 13, & divide per 30. (…) si autem velles præcisius multiplica per 433. & divide per 1000.» (Cardano,

Moya says: “I took all the major part of this article from Cardano” (

For biographical notes about Ringelberg see (Pereira,

The

Moya mentions the italian geometer but he does not indicate the name of the work that he consulted (Pérez de Moya,

See (Pérez de Moya,

Also in (Rojas,

See (Pérez de Moya,

See how Tartaglia deals with this subject in (Tartaglia,

For the different instruments used in architecture in the sixteenth century, see (Gessner,

On the texts that include problems of altimetry we mention: Juan Martín Población’s

This is not the first time that Moya cites Aguilera, as shows (Pérez de Moya,

Moya refers to proposition 46 from Euclid’s

Moya claims to Campano’s

In (Pérez de Moya,

In 1553, Giovanni Battista Benedetti (Venezia, 1530 – Turim, 1590) publishes in Venice a work entirely devoted to this topic, entitled

The title is “Principles of Geometry from which the students of Liberal Arts can take advantage, and also for every man whose profession requires the use of the ruler and the compass. Containing the procedures for measuring and dividing lands” (

This work has by five books which were published together for the first time in 1584, in Venice, under the title

See (Pérez de Moya,