114 UDC 514.11(091)(450) 371.3::514 Teaching Innovations, 2014, Volume 27, Issue 3, pp. 114–126 Mª Rosa Massa Esteve1, PhD Universitat Politècnica de Catalunya, Centre de Recerca per a la Història de la Tècnica, Barcelona, Spain Original Paper Historical activities in the mathematics cla- ssroom: Tartaglia’s Nova Scientia (1537)2 Abstract: Th e History of Mathematics can be developed both implicitly and explicitly in the classroom. Learning about the history of mathematics can therefore contribute to improving the integral education and training of students. Th e aim of this paper is to analyze the proposal of an historical activity based on the work Nova Scientia (1537) by Tartaglia for use in the mathematics classroom. Th is analysis will show the use of a Renaissance mathematical instrument for measuring the height of a mountain in order to motivate the study of trigonometry in the mathematics classroom, as well as to show students the explanatory role of mathematics in regard to the natural world. Key words: History of mathematics, teaching, Niccolò Tartaglia, Nova Scientia, geometry. Introduction Th e history of mathematics shows how math- ematics has frequently been used to solve problems concerning human activity as well as for helping to understand the world that surrounds us. Th e study of historical processes enables us to see how the dif- ferent aspects of mathematics have been combined together in a repeated interaction of application and development. Th us, for instance, geometry, which emerged as a means of measure, has evolved along- side the problems of measurement (Stilwell, 2010); trigonometry has developed in order to solve prob- 1 m.rosa.massa@upc.edu 2 Th is research is included in the project: HAR2013-44643-R. lems of both astronomy and navigation (Zeller, 1944), while algebra, which came more to the fore in problem-solving, especially in mercantile arith- metic during the Renaissance, was later to become an indispensable tool for solving problems in geometry and number theory (Bashmakova & Smirnova, 2000; Massa Esteve, 2005a). All this knowledge will un- doubtedly enrich the mathematical background and training of teachers, some references to which can be found in the historiography (Calinger, 1996; Fauvel &Maanen, 2000; Demattè, 2006; Massa Esteve et al., 2011; Lawrence, 2012). In Catalonia, the implementation of the his- tory of mathematics in the classroom has for twen- ty years inspired some individual initiatives among Received: 20 July 2014 Accepted: 10 September 2014 115 Historical activities in the mathematics classroom: Tartaglia’s Nova Scientia (1537) teachers (Romero & Massa, 2003; Guevara et al., 2008; Roca-Rosell, 2011; Massa-Esteve, 2012). Th e academic year 2009-2010 saw the inauguration of a new course for training pre-service teachers of mathematics in secondary education. Th e syllabus of this Master’s degree launched at the universi- ties includes a compulsory section on the history of mathematics and its use in the classroom. One of the subjects of this course concerns engineers-artists in the Renaissance, and a proposal of an historical ac- tivity on this subject in the mathematics classroom has been presented to pre-service teachers. Th e aim of this paper3 is to analyze the pro- posal of the implementation of this historical activ- ity, and also to discuss whether these kinds of activi- ties can show students how mathematics may play an explanatory role in regard to the natural world. Furthermore, the paper considers whether working with instruments and following the procedures rec- ommended to their users in the past off er students today a valuable appreciation of mathematical prac- tices (Heering, 2012). Usefulness of the history of mathematics in the classroom Th e usefulness of the history of mathematics in the classroom is described through our theoreti- cal and practical approach, with the aim of persuad- ing people about the need for this type of training. Knowledge of the history of mathematics can assist in the enrichment of teaching tasks in two ways: by providing students with a diff erent vision of mathe- matics, and by improving the learning process (Katz, 2000; Jankvist, 2009; Panagiotou, 2011). A diff erent vision of mathematics Teachers with knowledge of the history of mathematics will have at their command the tools for 3 A fi rst version of this paper was presented in the First Euro- pean Autumn School of History of Science and Education, 15-16 November 2013 in Barcelona. conveying to students a perception of this discipline as a useful, dynamic, human, interdisciplinary and heuristic science (Massa Esteve, 2003, 2010). Teach- ers in possession of such knowledge are able to show students a further relevant feature of mathematics – that it can be understood as a cultural activity. His- tory shows that societies develop as a result of the scientifi c activity undertaken by successive genera- tions, and that mathematics is a fundamental part of this process. Mathematics can be presented as an intellectual activity for solving problems in each pe- riod. Th e societal and cultural infl uences on the his- torical development of mathematics provide teach- ers with a view of mathematics as a subject depend- ent on time and space and thereby add an additional value to the discipline (Katz & Tzanakis, 2011). It is also worth pointing out that not only as teachers, but also as mathematicians, the history of mathematics enables us to arrive at a greater compre- hension of the foundations and nature of this disci- pline. Th e history of mathematics provides the devo- tees of this science with a deeper approach to an un- derstanding of the mathematical techniques and con- cepts used every day in the classroom. Knowing his- tory of our discipline helps us explain how and why the diff erent branches of mathematics have taken shape: analysis, algebra and geometry, their diff erent interrelations and their relations with other sciences. An improvement in the learning process Th e history of mathematics as a didactic re- source can provide tools to enable students to un- derstand mathematical concepts better. Th e history of mathematics can be employed in the mathematics classroom as an implicit and explicit didactic resource (Jahnke et al., 1996). Th e history of mathematics as an implicit re- source can be employed by teachers in the design phase by choosing contexts, by preparing activities (problems and auxiliary sources) and also by draw- ing up the teaching syllabus for a concept or an idea. In addition to its importance as an implicit tool for 116 Mª Rosa Massa Esteve improving the learning of mathematics, the history of mathematics can also be used explicitly in the class- room for the teaching of mathematics. Although by no means an exhaustive list, four areas may be men- tioned where the history of mathematics can be em- ployed explicitly in Catalonia: 1) for proposing and directing research work at baccalaureate level using historical material; 2) for designing and imparting elective subjects involving the history of mathemat- ics; 3) for holding workshops, anniversary celebra- tions and conferences, and 4) for implementing sig- nifi cant historical texts in order to improve under- standing of mathematical concepts (Massa Esteve, 2005b; Romero et al., 2007, 2009; Massa & Romero, 2009). Th is paper is focused on the last point, which is, presenting an historical text involving mathemat- ical instruments employed in the Renaissance. Historical activities in the mathematics classroom Historical texts can be used throughout the diff erent steps in the teaching and learning process: to introduce a mathematical concept; to carry out an exploration of it more deeply; to provide an ex- planation of the diff erences between two contexts; to motivate study of a particular type of problem or to clarify a process of reasoning. In order to use historical texts properly, teach- ers are required to present historical fi gures in con- text, both in terms of their own objectives and the concerns of their period. Situating authors chron- ologically enables us to enrich the training of stu- dents. Th us, students learn diff erent aspects of the science and culture of the period in question in an interdisciplinary way. It is important not to fall into the trap of the amusing anecdote or the biographical detail without any mathematical content. It is also a positive idea to have a map available in the class- room to situate the text both geographically and his- torically. Teachers should clarify the relationship be- tween the original source and the mathematical concept under study, so that the analysis of the sig- nifi cant proof should be integrated into the mathe- matical ideas one wishes to convey. Th e mathemati- cal reasoning behind the proofs should be analyzed and contextualized within the mathematical sylla- bus by associating it with the mathematical ideas studied on the course so that students may see clear- ly that it forms an integral part of a body of knowl- edge. In addition, addressing the same result from diff erent mathematical perspectives enriches stu- dents’ knowledge and mathematical understanding (Massa Esteve, 2014). Th e aims of the implementation of the histor- ical activity in the mathematics classroom are: a) To learn about the sources on which knowl- edge of mathematics in the past is based; b) To recognize the most signifi cant changes in the discipline of Mathematics; those which have infl uenced its structure and classifi cation, its meth- ods, its fundamental concepts and its relation to other sciences; c) To show students the socio-cultural rela- tions of mathematics with politics, religion, philoso- phy and culture in each period, as well as with other spheres; d) To encourage students to refl ect on the de- velopment of mathematical thought and the trans- formations of natural philosophy. Case study: Historical activity based on Tartaglia’s Nova Scientia (1537) Th e following historical activity deals with the work Nova Scientia (1537) by Niccolò Fontana Tart- aglia (1499/1500-1557). In order to implement the activity in the classroom, it is recommendable to be- gin with a brief presentation of the epoch, the Italian Renaissance, and Tartaglia himself. Th e aims of the author as well as the features of the work would then 117 Historical activities in the mathematics classroom: Tartaglia’s Nova Scientia (1537) be analyzed, and fi nally students are encouraged to construct an instrument for measuring degrees and to follow the reasoning of a signifi cant proof, in or- der to acquire new mathematical ideas and perspec- tives. Th is classroom activity would be implement- ed in the last cycle of compulsory education (14-16 year olds) with the aim of introducing and motivat- ing the study of trigonometry. Th e context: Th e Italian Renaissance Th e period from the mid-14th century to the beginning of the 17th century was the age of the Re- naissance, so called because it represented the re- birth of interest in the Greece and Rome of Classical antiquity (Rose, 1975; Hall, 1981). Artists, writers, scientists, and even the more refi ned craft smen looked to the past for inspiration and examples on which to model their own work. Latin and Greek were the indispensable keys to style, knowledge, and good taste, assuming a foun- dational signifi cance in education that they were to retain for centuries. Th is was the period of the great voyages of discovery which enlarged the horizons of the Western civilization, as did the invention of printing, with its incalculable eff ects upon human communication and the spread of information. Th e stream of wealth from the New World helped to de- velop the already growing economies of Europe. Th e major infl uence Renaissance had on technology was in the fi eld of architecture. Th e abandonment of Gothic forms by the Italian architect Filippo Bru- nelleschi (1377-1446) and his successors, and the gradual spread of the Neo-Classical Palladian style of building from Italy over the whole of Europe in- volved changes in building techniques. Teachers could argue that the inventions of the modern world demonstrated its technologi- cal superiority: this was especially the lesson of Jan Stradanus’ Nova Reperta (1570), a volume of splen- did engravings also produced near the end of the 16th century. We can use this image of Nova Reperta to show all these advances to students (see Figure 1). Figure 1. Nova Reperta 118 Mª Rosa Massa Esteve With this image the teacher can discuss with the students how the ancients had not mastered the “super-natural” force of gunpowder, nor discovered how to multiply books and pictures by printing. Nei- ther had they found the direct route to the East, nor the New World to the West; they remained ignorant of the use of the magnetic compass and of other nav- igational aids which had made the 15th century voy- ages of discovery possible. Th e ancients also lacked windmills, iron-shod horses, the art of making spec- tacles, mechanical clocks and iron-founding. In terms of the basic inventions and improve- ments made in the middle Ages, the Renaissance did little more than increase their size and scope. Ma- chines became larger and more intricate and pro- duction increased. Th ere were three major innova- tions during the Renaissance: gunpowder, the com- pass and printing. However, the Renaissance gave rise to a frame of mind which was increasingly re- ceptive to further technological development. Th e historical author: Tartaglia Tartaglia, an engineer and scientist of the Re- naissance, was taught fi rst in abacus school and then further taught himself mathematics. Tartaglia be- longed to that group of engineers and mathemati- cians who looked upon Archimedes as their role- model. Th eory, practice, and knowledge and its ap- plication were all part of the goal of scientifi c knowl- edge of a mathematician. Hence, Tartaglia took a new role and presented a new image of the science of mathematics, which encompassed all these fi elds of study and action (Bennett & Johnston, 1996). Some works by Tartaglia are: Nova Scientia (1537, 2nd edi- tion 1558), Quesiti et Inventioni Diverse (1546), Ge- neral Trattato di numeri et misure (1556-1560) and Euclid’s Elements (1543). Tartaglia is deemed as a great mathemati- cian of this period because of his use of geometry, and for his invention and development of a proce- dure for solving the cubic equation, but which Car- dano later published claiming it as his own (Giusti, 2010; Gavagna, 2010). Tartaglia embodied the im- age of the engineer mathematician that appeared in Italy in the Cinquecento and whose aim was “to solve the problems of his professions and to practice the art of invention”. Th e historical work: Nova Scientia In his work Nova Scientia (1537), Tartaglia introduced a new science: ballistics. In this work he tried to determine the form taken by the trajectory of a cannonball (Valleriani, 2013; Tartaglia, 1998). In the frontispiece of the work dealing with the theory of ballistic phenomena, Tartaglia pre- sents an image that seems to go back to the Pla- tonic idea, according to which mathematics consti- tutes the key to the door of science and philosophy. Th e image depicts two fortresses: One is situated on a top of mountain or hill entitled Philosophy, and fl anked by Plato and Aristotle; the other is situat- ed at the bottom, and called the Quadrivium, which of course consists of Music, Arithmetic, Geometry and Astronomy but to which is added a new science: Perspective. Tartaglia is seen at the center as Master of Ceremonies, presenting the principles of science that constituted ballistics. To enter this Sancta San- torum of Knowledge one must pass through a door guarded by Euclid. Euclid’s Elements in the Cinque- cento period were not only the foundation but the paradigm or manner to attain all wisdom (propae- deutic function) (see Figure 2). Tartaglia’s book is not a treatise on motion in a medieval sense, that is to say, he does not analyze the nature of motion (Tartaglia, 1998). He states that he will address the study of the movement of a pro- jectile ejected from a cannon or by whatever “artifi - cial machine or matter that will be appropriated to throw violently a body equally weighty into the air.” (Defi nition XIII). Th is current of thinking in, which including the artifi cial machine into the theoretical investigations, came to the fore in the middle of the XV century (Gessner, 2010). Th e practice was estab- lished by engineers and others trained in the atel- 119 Historical activities in the mathematics classroom: Tartaglia’s Nova Scientia (1537) iers of craft smen from the north of Italy and Germa- ny. Th e machine and its artifi ces were regarded as a way of conducting research into the world. In fact, the mechanics of the Cinquecento may be regarded as a science of machines. Th e theoretical analysis of the functioning of machines and their eff ects is pre- dominant in other subjects in mechanics, but can use geometry because machines such as the balance, the lever and the pulley are simple to analyze by geo- metric methods. Figure 2. Frontispiece of the Nova Scientia. Tartaglia, 1537. At that time, the principal problems in the analysis of movement of a cannonball were the questions of what happened when the ball was in the air. Th ese were: By how many degrees should the cannon be inclined to the horizontal so that the ball could hit a target located at a particular distance? At what inclination must a ball be fi red so that the ex- pected distance would be the maximum of possible distances? Tartaglia, who was at the time professor of mathematics at Venice, gave the fi rst answers to these questions. He asserted that the maximum dis- tance of a ball fi red from a cannon could be obtained by inclining the cannon 45º on the horizontal. Fur- thermore, he provided another answer that was even more surprising; Tartaglia claimed that the trajec- tory that the ball described through the air consist- ed of a curve. Th is claim contradicted the Aristote- lian doctrine of movement, according to which the movement that the ball must follow will be a straight line until it reaches its maximum height, aft er which it will fall vertically to the center of the earth (Hen- ninger-Voss, 2002). Th us, Aristotle’s doctrine pro- vided for no curved movement. However, the tra- jectory of a cannonball according to Tartaglia was composed of three parts; one rectilinear, one curve that follows an arc of circumference, both represent- ing the trajectory’s violent motion and, fi nally, one rectilinear of natural motion (see Figure 3). Figure 3. Tartaglia’s movement. Tartaglia, 1537. Th is work by Tartaglia enjoyed considerable success. By 1583, the text in Italian had reached sev- en editions and had been translated into many lan- guages. Tartaglia, an expert on the matter, subse- 120 Mª Rosa Massa Esteve quently returned to the problem of movement in his work Quesiti et invention diverse (1546). Th e text: the Euclidian way Nova Scientia consists of 3 books. Th e fi rst book contains 14 defi nitions, 5 suppositions, 4 com- mune sentences and 6 propositions with some cor- ollaries. Book two contains 14 defi nitions, 4 sup- positions and 9 propositions with some corollar- ies, and the third book contains 5 defi nitions and 12 propositions. In looking at Tartaglia’s Nova Sci- entia, teachers in the classroom may comment with the students on the Euclidian way of presenting this new practical science (Ekholm, 2010). In Proposition I of the fi rst book, Tartaglia states his results on movement, including the fol- lowing words: “Proposition First. All bodies of equal weight with natural movement, the further they move away from the beginning and approach their end point, the more they gain in speed.”4 In the fourth supposition of the second book, he again addresses the inclination of the cannon on the horizontal to achieve the maximum distance. Teachers can discuss with students the inclination at which the expected distance would be the maxi- mum (see Figure 4). In addition, in the Proposition VIII of the sec- ond book he proves that the inclination of the can- non must be 45º, basing his reasoning on the propo- sition VII, where he uses geometry in similar trian- gles (see Figure 5): “Proposition VIII. If the same moving power ejects or throws equally heavy bodies, which are similar and equal to each other, violently through the air but in diff erent manners, the one [equally heavy body] that accomplishes its 4 Th e translations are adapted from the English version by Val- leriani of Tartaglia’s edition of the 1558 (Valleriani, 2013).“Prop- ositione Prima. Ogni corpo egualmente grave nel moto natura- le, quanto piu el se andara aluntanando dal suo principio, over appropinquando al suo fi ne, tanto piu andara veloce.” (Tartaglia, 1537). transit at an elevation of 45 degrees above the horizon produces its eff ect farther away from its beginning and above the plane of the horizon than [if it were] elevated in any other way.”5 Figure 4. Tartaglia, 1537. Figure 5. Figure of the Proposition VII. Tartaglia, 1537. 5 „Propositione VIII. Se una medema possanza movente eiet- tara, over tirara corpi egualmente gravi simili, et eguali in diver- si modi violentemente per aere, Quello que fara il suo transito elevato a 45 gradi sopra a l’orizonte fara etià il suo eff eto piu lon- tan dal suo principio sopra il pian de l’orizonte che in qualunque altro modo elevato” (Tartaglia, 1537) 121 Historical activities in the mathematics classroom: Tartaglia’s Nova Scientia (1537) Th e mathematical instrument Tartaglia constructs two gunner’s quadrants, one with a graduate arc to measure the inclination of the cannonball, and the other instrument for solving the problem of measuring the distances and height of an inaccessible object. He off ers an explanation of the fi rst instrument at the beginning of the book in the dedicatory letter, as well as examining its con- struction accurately. He also gives examples with cannon (see Figure 6). Figure 6. Dedication letter. Tartaglia, 1537. In the third book, from the Proposition I to the Proposition IV, he provides a description of the material required for constructing the second gunner’s quadrant: the rule and the setsquare, and checks its angles in the following propositions. Fi- nally in the Proposition VI of the third book, Tarta- glia constructs this gunner’s quadrant (see Figure 7). Th is gunner’s quadrant is used by Tartaglia for measuring the height of inaccessible objects in the propositions of the third book, as shown below. Figure 7. The gunner’s quadrant. Tartaglia, 1537. Th e signifi cant proof: Proposition VIII of the third book Th e proposition we will look into more detail I fi nd a good example to be employed in the class- room because Tartaglia uses the gunner’s quadrant, while at the same time using geometry in similar tri- angles in the proof to determine the distances and height of an inaccessible object. In the classroom implementation, students could be prompted to re- produce the reasoning of this proof with the geome- try of triangles before introducing the trigonometry. In the Proposition VIII of the third book Tartaglia proves how to obtain the height of a vis- ible, but inaccessible object. He claims: “I would like to investigate the height of a visible object that one can move to the level of the base, and at the same time I would like to determine the distance through the hypotenuse or diameter of the height.” 6 6 “Propositione VIII. Voglio investigare l’altezza de una cosa ap- parente che si poscia andaré alla basa, over fondamento di quella, etiam tutto a un tempo voglio comprehendere la distantia ypo- thumissale, over diametrale di tal altezza”. (Tartaglia, 1537). 122 Mª Rosa Massa Esteve Th e image of this proposition clarifi es the ge- ometric reasoning (see Figure 8): Figure 8. Figure of the Proposition VIII. Tartaglia, 1537. Aft er providing accurately an explanation of the construction of gunner’s quadrant, together with the students the teacher could follow the reasoning of the proof using the similarity of triangles. For ex- ample, they can draw a fi gure with triangles that re- produces the geometric problem (see Figure 9). Together with the students, the teacher can reproduce the geometrical proof using similar tri- angles, Pythagoras’ theorem and Th ales’ theorem.7 Th e teacher can also show the use of this fi gure to solve other problems in the classroom; for instance, the height of a house, or a distance of an object. In fact, these kinds of problems are solved today by trigonometry, and furthermore this historical activ- 7 In this case I am referring to Elements VI. 2: “If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the tri- angle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle.”(Heath, 1956). ity also justifi es the introduction of the teaching of trigonometry. Figure 9. Reproduction of the mathematical problem Concluding remarks In order to transmit to the students the idea that mathematics is a science in a continuous state of evolution, and that it is the result of the joint and on- going work of many people rather than knowledge amassed by independent contributions arising from fl ashes of inspiration, it is recommendable to pre- sent historical activities in the classroom. Th is his- torical activity shows the process from geometry to trigonometry and also how a mathematical instru- ment can be used in the mathematics classroom, the gunner’s quadrant, for instance, or an instrument for measuring degrees, in order to obtain the height of inaccessible objects such as trees or mountains. As regards to the question posed about whether working with instruments and following the procedures recommended to their users in the past can provide students today with a valuable ap- preciation of the past practices, it should be taken into account that when working with instruments 123 Historical activities in the mathematics classroom: Tartaglia’s Nova Scientia (1537) in the classroom, it may not be appropriate to fol- low exactly the instructions of the users in the past. Th e function and effi cacy of these instructions of the past practices sometimes are not connected to the nowadays world of students. However, the replica- tion of such procedures in the construction of in- struments could inspire ideas for constructing simi- lar instruments to reproduce this practice with stu- dents today. Students can learn about how mathe- matical instruments were used in the past to solve real problems. Actually, in the Renaissance, technological developments in military and artistic spheres, as well as in scientifi c instruments, were made through the study of mathematics, which became increasing- ly regarded as a universal tool for solving problems. Th us, the question whether mathematics acquired an explanatory role in regard to the natural world gives rise to further questions about how and why this was so, and leads to discussions on the nature of math- ematics. One may consider that this historical activ- ity clearly shows the explanatory role of mathemat- ics for solving problems of practical geometry in the Renaissance, such as the problem of the inclination of a cannon when one wishes to achieve the maximum distance, as well as the problem of fi nding the height of inaccessible objects. Th e mathematical ideas used for the proofs of these propositions can be found in Euclid’s Elements: Pythagoras’ theorem, Th ales’ theo- rem, and in the principles relating to incommensu- rable lengths. In fact, history of mathematics shows that mathematics is used to address a natural phe- nomenon, and in this sense it shows the usefulness of mathematics for revealing the natural world. Th e originality of this historical activity resides in the use of a text, which does not consist entirely of pure math- ematics, to give an explanation for the movement of the military projectiles, which could be described as geometrization of real-life problems. 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Историју математике, као имплицитни извор, наставници могу да користе да би осмислили фазу часа користећи различите контексте, припремајући наставне активности (проблемске ситуације и помоћне изворе за сазнавање) и креирајући наставни силабус у функцији формирања појмова или идеја. Осим као имплицитном средству за побољшање учења математике, историја математике може да се користи експлицитно у разреду ради поучавања математике. Имплементација важних историјских текстова може да обезбеди средства која ће ученицима омогућити да боље разумеју математички појам. Циљеви имплементације историјске активности на часовима математике су: а) учење о изворима на којима се заснива знање математике у прошлости; б) препознавање најзначајнијих промена у математичким дисциплинама − оне које су утицале на структуру и класификацију, на њене методе, основне појмове и везу са другим наукама; в) указивање ученицима на социокултурну везу математике и политике, религије, филозофије и културе, у сваком периоду, као и везе са осталим сферама, и коначно, што је најважније, подстицање ученика да се изразе у вези са математичком мишљу и трансформацијом природне филозофије. Циљ овог рада је анализа студије случаја предлога историјске активности, базиране на раду „Nova Sci- entia“ (1537) Никола Фонтане Тартаље (Niccolò Fontana Tartaglia (1499/1500–1557)), за коришћење на часовима математике. Ова анализа ће показати употребу ренесансног математичког инструмента за мерење висине планине да би се мотивисало проучавање тригонометрије на часовима математике, као и показивање улоге математике у објашњавању природног света. Штавише, у раду разматрамо да ли рад на инструментима и мерења помоћу њих, препоручиваних корисницима у прошлости, омогућавају ученицима у садашњости адекватно вредновање мерењем инструментима из прошлости. Кључне речи: историја математике, поучавање, Николо Тартаља, „Nova Scientia“, геометрија.