The first notions of geometry given by Moya are included in the Fourth Book of his Arithmetica practica, y speculativa, and developed in a section consisting of three short chapters. The main objective of this text is to introduce the rules of land surveying, as the title of the last chapter shows (Muestra la orden de medir tierras). It was usual for the first Iberian printed treatises of arithmetics to include a brief section with some geometrical topics. It is the case, for example, of Juan de Ortega’s Compusicion de la arte dela arismetica y Juntamente de Geometria (Ortega, 1512), of Gaspar Nicolas’s Tratado da Pratica Darismetyca (Nicolas, 1519), and of Joan Ventallol’s Pràtica mercantívol (Ventallol, 1521). Apart from the form, also the contents and the methods used are similar in these texts. The contents consist, in general terms, on the resolution of practical problems related with the calculation of the perimeter and the area of ​​a plane figure. The problems posed in the case of the circle provide the approximate value 3 1/7 (figure 1) of π, which was also adopted by the authors mentioned above9. This value of π was already specified in the earlier texts of arithmetic and practical geometry (Smith, 1958, vol.2, p. 307) in particular in Pacioli’s Summa (Pacioli, 1494, f. 30r)10. However, this was not the only option for the value of π. For example, the anonymous author of Libro de Arismética que es dicho alguarismo proposes to multiply the diameter of a circle by 3 1/3 to obtain its perimeter (Caunedo y Córdoba, 2000, p. 207). 


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 (proporcion tripla sexquiseptima):


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, 1562, p. 310).


The procedures used by the cited authors are described in abacus’s treatises (Simi, 2000, p. 156). They involve rules, such as the rule of three, the arithmetic concepts of square root and proportion, and simple geometric knowledges, such as the perimeter and the area of plane figures, and the theorems of Pythagoras and of Heron; for example Gaspar Nicolas proposes and solves forty-two geometric problems in such conditions (Nicolas, 1519, ff. 80r-94v).


Despite the similarity observed between Moya’s Fourth Book and the other mentioned works, there is a significant difference in the introductory part of their texts. This part reveals the attention paid by our author to the definitions of the geometric objects with which he works, even when the text is directed for land surveyors, to whom the more theoretical issues possibly would not be important. This brief introduction to the Euclidean geometry gives an idea of the importance that Moya puts on the formulations and rigor. Euclid and Aristotle are authors mentioned in this Fourth Book. We stress that Moya was familiar with Ortega’s Compusicion de la arte dela arismetica y Juntamente de Geometria, as is shown by the similarity between the two texts: (Pérez de Moya, 1562, p. 310) and (Ortega, 1512, ff. 196r-197r). But Moya mentions neither this title nor its author. As Sesiano already observed there is a work by Moya printed in 1563 with a reprint of the Tratado subtilíssimo by Juan Ortega (Sesiano, 1988, p. 38); we could verify that it is the Sixth Book in (Ortega, 1563, pp. 241v-254v).


On the other hand, some of the problems in Fourth Book do not appear in the works of the Iberian authors referred, but they are included in Pacioli’s Tractatus Geometrie11. The similarities with Pacioli concern the type of elementary utensils used in the calculation of distances, heights and depths, and on the way of introducing geometry (Pacioli, 1494, f. 50v) and (Pérez de Moya, 1562, p. 316). Moya gives analogous definitions to those of Pacioli’s Summa, and uses analogous terms to designate the same concepts; for example, helmuayn and helmuariphe in (Pacioli, 1494, ff. 1r-1v) and (Pérez de Moya, 1562, pp. 308-309). Nevertheless Pacioli’s text is much more detailed and extensive12.


The first volume of Fragmentos Mathematicos is a text on geometry with different characteristics from the Fourth Book of Arithmetica practica, y speculativa (1562). It is much more extensive, entirely devoted to geometry, and related to problems that can be solved with the ruler and compass. According to Moya’s letter to the readers (Pérez de Moya, 1568, “El Bachiller Ivan Perez de Moya a los lectores”), this work must constitute a fundamental set of tools to those who study other subjects. 


The work is organized in forty-four chapters; the first forty providing useful tools to understand the remaining four. It starts such as Fourth Book with a set of definitions, postulates and ordinary sentences, or concepts, which are essential as a support of geometry:


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, 1568, pp. 2-3).


But while in Euclid’s Elements the text is presented without further explanation, Moya clarifies the contents with geometrical or numerical examples13. And he uses Euclid’s Elements to justify some practical constructions. In the Fragmentos Mathematicos we can find elementary subjects, such as the division of a line segment in equal parts by means of a ruled paper, but also some elaborate ones, such as justifications involving propositions of Euclid’s Elements14. The constructions include the circumscription of a triangle by a circle, the quadrature of polygons and of the circle, the duplication of the cube, and a set of units of measures useful to cosmographers and geometers. Some of these units are already given in the works of Roman authors from the Classical Antiquity, which Moya cites: Columela, Vitruvius and Pliny. Moya mentions in particular the fifth book of Columela’s Res Rustica, the third book (first chapter) of Vitruvius, and the second book (chapter 23) of Pliny. The titles of Vitruvius and Pliny’s works are not mentioned, but it is highly probable that Moya was referring to De architectura and to Historiae Naturalis, because these are emblematic works of those authors.


The last chapters of Fragmentos Mathematicos are dedicated to issues of measurement: altimetry, planimetry and stereometry (for Moya these involves the measurement of corporeal things whatever the material of which they consist). Here the concept of measure is made explicit by Moya for the first time, referring it to the comparison with some known unit which he calls “famous measure”:


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, 1568, p. 1).


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, 1560a, p. 1).


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 Fourth Book. In planimetry, Moya teaches how to measure areas of regular polygons, the circle and the oval, and indicates rules to determine volumes (the “mass”) and surface areas of solids (platonic and other non-regular solids). These studies have different applications, which Moya exemplifies in some practical problems. Sometimes, the rules proposed do not lead to exact values, and Moya knows it, but possibly their use would not require greater accuracy15.


Fragmentos Mathematicos differ from Fourth Book by the diversity of authors cited in the former. Nevertheless, the references are not exhaustive and we can see some significant similarities with other texts which Moya does not denounce. We give three examples: the contents of the last common sentence are paraphrases from Aristotle’s Physics (book III)16; the two formulas for determining the approximate area of an equilateral triangle when it side is known, A=13xl2/30, and A=433xl2/1000, are both from Cardano (Cardano, 1663, p. 117)17; the procedure for the division of a line segment into equal parts is already described in Ringelberg’ Ad mathematicen (Ringelberg, 1531, pp. 481-482) and in Peletier’s De L’usage of Geometrie (Peletier, 1573, p. 22). We have no doubt that Moya owes to Cardano a great part of the study on measurement of areas, mainly to polygons with more than four sides. Besides, Moya takes some applications of altimetry from Frisius, referring to a short treatise which is published together with the cosmography of Apiano (Pérez de Moya, 1568, p. 166)18. Moya does not indicate its title, but we can easily understand that it is Libellus de Locorum describendorum ratione, et de eorum distantiis inveniendis, nunquam antehac visus (Frisio, 1548, ff. 55r- 68v). 


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, 1568, pp. 69-71) − two of them adapted from Archimedes’s Measurement of the Circle (propositions 1 and 2) (Dijkdteerhuis, 1987, p. 222), and the third one (Pérez de Moya, 1568, pp. 77-79) from Ringelberg19– but Moya does not mention the name of this author. The last process mentioned is based on a very simple construction given in the General Trattato (Tartaglia, 1560b, ff. 17v-22v) which Tartaglia associates to Bovelles (Carlo Bouile):


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, 1560b, p. 22r).


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 Géométrie en françoys (Bovelles, 1511, ff. 39v-40r)20. The construction, also presented in Ringelberg’ Ad mathematicen (Ringelberg, 1531, p. 485), consists in dividing the diameter of the circle in eight equal parts and taking the square whose diagonal is equal to ten of those parts (figure 2). The area of this square is approximately equal to the area of the circle.


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 Geometry, stressing their efficacy:


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, 1568, p. 85).


The mention of Dürer confirms the influence that this author had in Spain, mainly due to the practical features of his work (Vasconcelos, 1929, p. 74).


The influence of Tartaglia’s General Trattato is clear in several passages of Fragmentos Mathematicos. For example, Moya proposes four procedures to divide a line segment in equal parts, two of which are given in the General Trattato; and in the case of the ovals, both the description and the construction of the figures are from Tartaglia (Tartaglia, 1560c, ff. 22v-23v) whose name Moya mentions (Pérez de Moya, 1568, p. 61)21.


The Geometria Practica y Speculativa is the second volume of Tratado de Mathematicas. It is the most important geometrical text written by our author. Moya reorders the chapters of Fragmentos Mathematicos and, occasionally, reorganizes the subjects.


The volume is structured in four Books (Libros), divided in chapters, some of them consisting of articles. It begins with a summary indicating the contents in detail. 


Moya uses a big part of the Tratado to explore practical applications of geometry (Books II-IV) reserving the first book for the theoretical foundation of the issues contained in the other three. The Tratado de Geometria includes all the contents of the Fragmentos Mathematicos that Moya believes to be essential, which makes this work more extensive and varied than the previous ones. Furthermore, it provides much more references to the sources, which are also more accurate. The division of the circle, the construction of polygons, and the excess between the inscribed and circumscribed polygons to the same circle are topics treated in Tratado, and not included in Fragmentos Mathematicos. But the opposite also happens: for example the construction of circles with area multiple or submultiple of a given circle presented in Fragmentos Mathematicos is not described in Tratado de Geometria (Pérez de Moya, 1568, pp. 72-77). It is true that Tratado de Geometria is more complex than the previous geometrical texts of our author, but its language is equally simple; however Moya recognizes that not all mathematical proofs are accessible to all its readers22. 


There are some important differences between the Fragmentos Mathematicos and the Tratado de Geometria. The most significant ones are probably the emphasis put in proofs of the Tratado, which are systematically founded on Euclid’s Elements, and the distinction concerning practical and theoretical geometry. Moya identifies practical geometry with the use of the ruler and the compass, describing the latter’s movements in detail and he associates theoretical geometry with Euclid’s Elements. Furthermore, the Tratado de Geometria includes alternative strategies that minimize the construction work in some particular cases, and Moya stresses the importance of the generalization. For example, in Fragmentos Mathematicos the construction of the perpendicular to a line segment only concerns the midpoint of the line segment, but in Tratado de Geometria all possibilities are considered (Pérez de Moya, 1573, pp. 23-24). Moya explores the concept of perpendicularity in two distinct ways: one using a procedure justified by Euclid’s Elements (Proposition I, 11) and the other offering a completely intuitive and practical construction, in which he simply uses a square. This last construction is analogous to the one which Tartaglia ascribes to the mathematically uneducated (simplice natural) (Tartaglia, 1560c, f. 3v). 


Moya defines geometry similarly to Tartaglia, (Pérez de Moya, 1573, p. 5) (Tartaglia, 1560a, p. 1); and shares his opinion about the advantage of associating theoretical and practical geometrical knowledge. He begins his geometrical texts with definitions, postulates and common sentences, similarly to Euclid’s Elements, following Campano’s terminology. In Tratado he admits the existence of two types of points, corresponding to the interpretations of the matematico and the natural. He stresses that to the matematico the point is an abstract entity, indivisible and invisible, and to the natural the point is a very small sign that can be seen and divided into parts (Pérez de Moya, 1573, p. 6). This differentiation is adopted by Tartaglia, and other authors of the Renaissance such as Piero de la Francesca (Davis, 1977, pp. 41- 42) and Leonardo da Vinci (Arrighi, 2004, pp. 368-369), but we attest that Moya uses Tartaglia’s terms to express these notions23.


Moya follows Tartaglia when he describes the division of a line segment in mean and extreme ratio (or, as they say, the division of a line segment according to the proportion that has the means and two extremes). He explains the construction similarly to Tartaglia and uses the same figure (figure 3).


Moya also follows Tartaglia in many other occasions: for example in the definition of angle24; in the definition of the oval – a figure that attracted great interest among designers, painters, architects and craftsmen − commonly used in illuminations. In Tratado de Mathematicas he gives a construction of the oval (Pérez de Moya, 1573, p. 57), invented by Tartaglia in 1547 (Tartaglia, 1560a, ff. 11v-12r), which is more general than those of previous designers and architects. Moya refers to the difference between problem and theorem (Pérez de Moya, 1573, p.50), following Tartaglia (Tartaglia, 1560b, f. 1r), and stresses that all propositions dealing with triangles presented in the corresponding chapter of Tratado are problems. 


The subjects related to circumscriptions between polygons and circles are introduced by means of seven definitions similarly to Tartaglia (Tartaglia, 1545, pp. 1-2) (Campano only gives two in Euclid’s Elements Book IV); and Moya uses the ruler and the compass to solve the problems proposed in General Trattato, which the Italian mathematician solves in a way that he claims to be “more expeditious and more brief than that of Euclid”. Nevertheless, our author gives a personal touch to this study by adopting an ordering of the subject different from the one of Tartaglia (see table I).


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, 1573, p. 53) (Tartaglia, 1560a, f. 14v). It is surprising that Moya does not allude to the relationship between such an isosceles triangle and a regular pentagon; he merely reproduces (Pérez de Moya, 1573, p. 56) an approximate construction of the regular pentagon that Bovelles exhibits in his Geometrie practique (Bovelles, 1551, f. 20r). 


For the doubling of the cube, Moya explains the method of the “very ingenious” (ingeniosissimo) Tartaglia, believing that he, better than any other, shows a general rule to address this issue25. Moya starts with a cube of edge a, and then constructs the rectangle CDFE (figure 4) with sides a and 2a, the diagonals of which intersect at G. He then traces the lines EI and EH, containing the sides EF and EC, and looks for the points K, belonging to EI, and L, belonging to EH, such that K, L and D are aligned, and K and L are both equally distant from G. The line segment FK thus obtained is equal to the edge of the required cube. The points K and L are discovered by trial; he draws a circumference centered in G and intersecting the lines EI and EH in K and L. Then, he gives suggestions to facilitate the final task, stressing that the points K and L are well determined only if the line segment KL contains D. If D falls above KL it is necessary to increase the opening of the compass, and if D falls below KL it is necessary to reduce that opening:


(…) 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, 1573, p. 248). 


To prove that this construction satisfies the required, Moya notes that the line segments FK and CL are the mean proportionals between a (= EC) and 2a (= CD), in other words:


And he uses Euclid’s Elements, saying the following:


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, 1573, p. 248).


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, 1560c, f. 21r). 


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 navigation26. We find measuring contents in Dürer, Peletier, Apiano, Frísio and Finé, for example, and it is not surprising that Moya knew their works, because they were very well represented in the Spanish libraries. In Aspectos de la Ciencia aplicada en la España del Siglo de Oro, Maroto and Piñero say that at the beginning of the sixteenth century the study of instruments constituted a significant part of the work of some Spanish authors27. But we stress that the only Spanish author cited by Moya on this subject, is “doctor” (doctor) Aguilera (Pérez de Moya, 1573, p.33), referring to his work Canons astrolabii universalis published in 155428.


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, 2006, pp. 341-349). But we can add that there are also significant differences. Actually, for example, Moya graduates the geometric quadrant from 1 to 12 (Pérez de Moya, 1573, p. 94), while Finé graduates it from 1 to 60 (Finé, 1587, f. 29r) which allows more accurate measurements. Furthermore Moya generally uses in the applications different values ​​from Finé and he refers to different versions of Euclid’s Elements. Moya is consulting Campano’s version, and Finé probably follows Zambert’s version29.


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, 2002, p. 14). Moya exposes the subject for the first time in his Tratado de Geometria explaining how to divide the circumference in 2, 6, 8, 12, 16, 24, 32 and 36 equal parts. The study begins with the construction of the perpendicular bisector of a cord and uses the procedure and the figure presented in the General Trattato (Tartaglia, 1560c, ff. 10r-10v); the only difference is in the version of Euclid’s Elements used, because Moya follows Campano while Tartaglia follows his own Euclid’s Elements version (Tartaglia, 1545, f. 65v)30.


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 Elements book IV. However, he used a different technique beginning with the division of the quadrant in three parts, which leads to an approximate division of the circumference.


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, 1573, p. 43). In the second volume of Fragmentos Mathematicos (1567) a procedure was already referred, which gives an approximate division of a circle in 36 equal parts, using its division in nine parts (Pérez de Moya, 1567, p. 436). Moya ascribed it to Rodrigo Zamorano and considered it a good invention. Nevertheless, in Tratado de Geometria (1573) he indicates another construction, reproduced from Porres de Osorio (probably from his Nuevas Proposiciones Geometricas), which leads to a better approximation than the former31. 


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 General Trattato (Tartaglia, 1560c, ff. 22r- 22v), using this technique in the solving of several problems of Euclid’s Elements32; and the emphasis given by Moya to this kind of constructions reflects, once again, the influence he has had from Tartaglia.


Moya’s Principios de Geometria consists of two “Books” (Libros). The first book begins with definitions, petitions and ordinary sentences as in his previous geometric works. And it includes fundamental knowledge as, for example, the procedure for dividing a line segment in equal parts and for constructing different kinds of triangles, a square, a regular pentagon, a circle, an oval. It also teaches the division of the circle in equal parts and the squaring of the circle. The title of the second book is “It concerns things belonging to the kind of measure called Planimetry which belong to measuring and dividing land” (Trata de cosas pertenecientes al genero de medida, que dicen Planimetria que pertenece al medir e dividir tierras). The complete title of the book points to its recipients: men who study liberal arts and surveyors33. Moya advises the reader that its content is neither original nor comprehensive and that it is a part of his Tratado de Geometria. 


Principios de Geometria has a reduced size, allowing easy transport and handling, and its content is more practical than the previous Tratado de Geometria. The authors’ main concern is to indicate the simplest process for solving each proposed question, as shown in the following remark:


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, 1584, vol.2, chapter X).


In this work, Moya keeps a similar style to Fragmentos Mathematicos and Tratado de Geometria, but he is now more interested in the applications of geometry to the solving of practical problems and in the details related to the construction of rudimentary tools of measurement. Here, the problem follows immediately the corresponding topics, showing its practical use; Moya does not dedicate to them a special section in the end of the work, as he had done in his other geometrical texts.


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, 1976, p. 68).


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 Elements that he uses, he usually mentions the book and the proposition used. The first book of the Elements is the most often cited but there are also allusions to books III, IV and VI (in the first book of Principios de Geometria) and to books II and V (in the second book). The only new references are Serlio (1475-1554) who is the author of Tratado de Architettura34, and Vignola (referred by Pinola in the text) who is author of Le due regole della prospettiva. Moya citesVignola in the context of architecture. On the other hand, many of the references mentioned in his other works are not included here, certainly because Moya did not think that they were useful to the recipients of this work.