## The Should I Become a Mathematician? Thread

Euler III
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Phases in rhetoric development of treatises on algebra - The abacus tradition

"While the medieval tradition of riddles or problems with standard recipes was carried through to sixteenth-century arithmetic books, a new tradition of algebraic problem solving emerged in Renaissance Italy. The Catalogue by Warren van Egmond (1980) provides ample evidence of a continuous thriving of algebraic practice from the fourteenth till the sixteenth century."

"Over two hundred manuscripts provides an insight in the practice of teaching the basics of arithmetic and algebra to sons of merchants in the abacus schools of major towns in Renaissance Italy. The more skilled of these abacus masters drafted treatises on algebraic problem solving in the vernacular."

"These consist typically of a short introduction on the basic operations on polynomials and the rules for solving problems (resolving equations). The larger part of these treatises is devoted to the algebraic solution of problems. We can state that the algebraic practice of the abacus tradition is the rhetorical formulation of problems using an unknown. The solution typically depends on the reformulation of the problems in terms of the hypothetical unknown. The right choice of the unknown is half of the solution to the problem. Once the several unknown quantities are expressed in the rhetorical unknown, the analytic method consists of manipulating the polynomials and applying the rules of algebra (resolution of equations) to the point of the resolution of a value for the unknown."

As an example of the rhetoric of algebraic problem solving let us look at the major abacus master of the fourteenth century, Antonio de’ Mazzinghi (Problem 9, Arrighi 1967):

Italian: Truova 2 numeri che, multiplichato l’uno per l’altro, faccino 8 e i loro quadrati sieno 27

English: Find two numbers which, multiplying one with the other gives 8, and [adding] their squares gives 27.

After the problem text is given, the solution typically starts with the hypothetical definition of an unknown: “Suppose that the first quantity is one cosa”. The skill of abacus master and the elegance of the problem-solving method depends mostly on the clever choice of the unknown. Maestro Antonio not only was skilful in this, he also was the very first to introduce multiple unknowns for solving difficult problems in an elegant way.

"Ma per aguagliamenti dell’algibra anchora possiamo fare; e questo è che porremo che lla prima quantità sia una chosa meno la radice d’alchuna quantità, l’altra sia una chosa più la radice d’alchuna quantità. Ora multiplicherai la prima quantità in sè et la seconda quantità in sè et agugnerai insieme et araj 2 censi et una quantità non chonosciuta, la quale quantità non chonosciuta è quel che è da 2 censj infino in 27, che v’è 27 meno 2 censj, dove la multiplichatione di quella quantità è 13 1/2 meno i censo."

------

Instead of using the cosa for one of the numbers, or two unknowns for the two numbers, Maestro Antonio here uses

x-sqr(y) and x+sqr(y).

Squaring these two numbers gives

x^2-2x*sqr(y)+y and x^2+2x*sqr(y)+y respectively

Adding them together results in 2x^2+2y, which is equal to 27.

The auxillary unknown thus is 13 1/2 - x^2.

-------

"This text fragment from the end of the fourteenth century is exemplary for the abacus tradition. Algebraic practice consists of analytical problem solving. The rhetorical structure depends on the reformulation of the given problem in terms of the cosa and applying the analytical method to arrive at a value for the unknown. The unknown quantities can then easily be determined. A test subsitituting the values of the quantities in the original problem provides proof of the validity of the solution."

[More painful word problems anyone?]
 Euler IV The Beginning of Algebraic Theory: from Pacioli to Cardano "By the end of the fifteenth century we observe a change in the rhetorical structure of algebra treatises. While the solution to problems still remains the major focus of the texts, authors pay more attention to the introductory part. While a typical abacus text on algebra was limited to thirty or forty carta, the new treatises easily fill hundred folio’s. Two trends contribute to more comprehensive approach: the use of the algorism as a rhetorical basis for an introductory theory and the extraction of general principles from practice." The amalgamation of the algorism with the abacus text "The algorism, as grown from the first Latin translations of Arab adoptions of Hindu reckoning, describes the Hindu-Arabic numerals and the basic operations of addition, subtraction, multiplication and division. In later texts we also find doubling and root extraction as separate operations. These operations are applied to natural numbers, fractions and occasionally also sexadecimal numbers. Through the influence of Boetian arithmetic, some algorisms also include sections on proportions and progressions. Whereas we find this structure also in abacus texts on arithmetic, the treatises on algebra have a different character." "The introductory part extends on early Arab algebra with the six rules for solving quadratic problems, lengthened by some derived rules. By the end of the fifteenth century algebraic treatises also incorporate the basic operations on arithmetic and broaden the discussion on whole numbers and fractions with irrational binomials and cossic numbers. We witness this evolution in Italy as well as in Germany. The culmination of this evolution is reflected in the Practica Arithmeticae of Cardano (1539). Cardano begins his book with the numeration of whole numbers, fractions, and surds (irrational numbers) as in the algorisms. He then adds de numeratione denominationum placing expressions in an unknown in the same league with other numbers, which is completely new." "In doing so he shows that the expansion of the number concept has progressed to the point of accepting polynomial expressions as one of the four basic types of numbers. He further discusses the basic operations in separate chapters and applies each operation to the four types. Also, he applies root extraction to powers of an unknown in the same way as done for whole numbers (chapter 21). He continues by constructing aggregates of cossic numbers with whole numbers, fractions or surds (chapter 33 to 36). As an example of the aggregation of cossic numbers with surds, he shows how sqr(3) multiplied with 4x^2+5x gives sqr(48x^4+120x^3+75x^2). Though Cardano was not the first, his Practica Arithmeticae is a prime example of the adoption of the algorism for the rhetorical structure of the new text books on algebra, and functioned as a model for later authors. Cossic numbers were in this way fully integrated with the numeration of the species of number and presented as the culmination of the application of the operations of arithmetic. --------- Extracting general principles from algebraic practice "For a second trend in the amplification of an introductory theory in algebraic treatises we can turn to Pacioli. It has long been suspected that Pacioli based his Summa de arithmetica geometria proportioni et proportionalita of 1494 on several manuscripts from the abacus tradition." "These claims have been substantiated during the past decades for large parts of the Geometry. Ettore Picutti has shown that “all the ‘geometria’ of the Summa, from the beginning on page 59v. (119 folios), is the transcription of the first 241 folios of the Codex Palatino 577”, (cited in Simi and Rigatelli 1993). Margaret Daly Davis (1977) has shown that 27 of the problems on regular bodies in Pacioli’s Summa are reproduced from Pierro’s Trattao d’abaco almost literally. Franci and Rigatelli (1985) claim that a detailed study of the sources of the Summa would yield many surprises. Yet, for the part dealing with algebra, no hard evidence for plagiarism has been given. While studying the history of problems involving numbers in geometric progression (GP), I found that a complete section of the Summa is based on the Trattato di Fioretti of Maestro Antonio. Interestingly, this provides us with a rare insight in Pacioli’s restructuring of old texts, and as such, in the shift in rhetorics of algebra books." Pacioli: Famme de 13 tre parti continue proportionali che multiplicata la prima in laltre dui, la seconda in laltre dui, la terça in laltre dui, e queste multiplicationi gionti asiemi facino 78. Maestro Antonio: Fa’ di 19, 3 parti nella proportionalità chontinua che, multiplichato la prima chontro all’altre 2 e lla sechonda parte multiplichato all’altre 2 e lla terza parte multiplichante all’altre 2, e quelle 3 somme agunte insieme faccino 228. Adimandasi qualj sono le dette parti. In modern notation, the general structure of the problem is as follows: x/y = y/z x+y+z=a x(y+z)+y(x+z)+z(x+y)=b Maestro Antonio is the first to treat this problem and uses values a=19 and b=228. Expanding the products and summing the terms gives: 2xy+2xz+2yz=228, but as y^2=xz we can write this also as 2xy+2y^2+2yz=228, or 2y(x+y+z)=228 Given the sum of 19 for the three terms, this results in 6 for the middle term. Antonio then proceeds to find the other terms with the procedure of dividing a number into two extremes such that their product is equal to the square of the middle term. Pacioli solves the problem in exactly the same way. However, the rhetorical structure is quite different. Maestro Antonio performs an algebraic derivation on a particular case. Instead, Pacioli justifies the same step as an application of a more general principle, defined as a general key.... "The restructuring of material and the shift in rhetoric is in itself an important aspect in the development of sixteenth-century textbooks on algebra. Pacioli raised the testimonies of algebraic problem solving from the abacus masters to the next level of scientific discourse, the textbook. When composing the Summa, Pacioli had almost twenty years of experience in teaching mathematics at universities all over Italy. His restructuring of abacus problem solving methods is undoubtedly inspired by this teaching experience. Cardano’s Practica Arithmeticae continues to build on this evolution and the two works together will shape the structure of future treatises on algebra." ----- [now doesn't that look like part of the algebra book puzzler that mathwonk tossed at us this summer?] [With that x(y+z)+y(x+z)+z(x+y)=b fragment!]
 Euler V -------- Algebra as a model for method and demonstration "The two decades following Cardano’s Practica Arithmeticae were the most productive in the development towards a symbolic algebra. Cardano (1545) himself secured his fame by publishing the rules for solving the cubic equation in his Ars Magna and introduced operations with two equations. In Germany, Michael Stifel (1544) produce his Arithmetica Integra which serves as a model of clarity and method for many authors during the following two centuries." "Stifel also provided significant improvements in algebraic symbolism, which have been essential during the sixteenth century. He was followed by a Johannes Scheubel (1550) who included an influential introduction to algebra in his edition of the first six books on Euclid’s Elements. This introduction was published separately in the subsequent year in Paris as the Algebrae compendiosa (Scheubel, 1551) and reissued two more times. In France, Jacques Peletier (1554) published the first French work entirely devoted to algebra, heralding a new wave of French algebraists after the neglected Chuquet (1484) and de la Roche (1520)." "Johannes Buteo (1559) built further on Cardano, Stifel and Peletier to develop a method for solving simultaneous linear equations, later perfected by Guillaume Gosselin (1577). In 1560, an anonymous short Latin work on algebra was published in Paris. It appeared to be of the hand of Petrus Ramus and was later edited and republished by Schoner (1586, 1592). The work depended on Scheubel’s book to such a measure that Ramus refrained from publishing it under his own name. In Flanders, Valentin Mennher published a series of books between 1550 and 1565, showing great skill in the application of algebra for solving practical problems." "England saw the publication of the first book treating algebra by Robert Recorde (1557). This Whetstone of witte was based on the German books of Stifel and more importantly Scheubel. It introduced the equation sign as a result of the completion of the concept of an equation. It would take too long to review all these books. Only some general trends and changes in the rhetorical structure of the sixteenth-century algebra textbook will be discussed." "Giovanna Cifoletti (1993) is one of the few who wrote on the rhetoric of algebra and specifically on this period. She attributes a high importance to Peletier’s restructuring of the algebra textbook. However, we have shown that the merger of the algorism with the practical treatises of the abacus tradition was initiated by the end of the fifteenth century, culminating in Cardano (1539)." "This trend cannot be attributed to Peletier, as proposed by Cifoletti. On the other hand, Peletier was an active participant in the humanist reform program which aimed not only at language and literature but also at science publications. His works on arithmetic (1549), algebra (1554) and geometry (1557) make explicit references to this program and reflections on the rhetoric of mathematics teaching. Cifoletti (1993) demonstrates how Peletier intentionally evokes the context of the author as the classical Orator in order to approach a textbook from the point of view of rhetoric. He rebukes on the demonstration of mathematical facts by his predecessors, explicitly referring to Stifel and Cardano. His ideal model for mathematical demonstration is exemplified by the rules of logic represented under the form of a syllogism. In his introduction to Euclid’s Elements he considers the application of syllogisms in mathematical proof as analogous with that of an lawyer at the court house, the rules of rhetoric: "Que si quelqu’un recherche curieusement, pourquoi en la démonstration des propositions ne se fait voir la forme du syllogisme, mais seulement y apparoissent quelques membres concis du syllogisme, que celui là sache, que ce seroit contre la dignité de la science, si quand on la traite à bon escient, il falloit suivre ric à ric les formules observées aux écoles. Car l’advocat, quand il va au barreau, il ne met pas sur ses doigts ce que le Professeur en rhétorique lui a dicté: mais il s’étudie tant qu’il peut, encore qu’il soit fort bien recours des preceptes de rhétorique, de faire entendre qu’il ne pense rien moins qu’à la rhétorique." [it's interesting how you can skim through it pretty easily seeing three words stick out: advocat/syllogisme/rhetorique] ------- So, how did Peletier apply his understanding of rhetoric in his Algebre? Cifoletti (1993) points at the contamination of the rhetorical notion of quaestio and the algebraic notion of problems, initiated by Ramus and Peletier, and fully apparent in the Regulae of Descartes. She goes as far as to identify the algebraic equation with the rhetorical quaestio (Cifoletti 1993): "But I also think that from the point of view of the history of algebra, so crucial for later theoreticians of Method, 'quaestio' has played a fundamental role because it has allowed consideration of the process of putting mathematical matters into the form of equations in a rhetorical mode.' "In Cicero’s writings, the quaestio is an important part of rhetorical theory. He distinguishes between the 'quaestio finita', related to time and people, and the 'quaestio infinita', as a question which is not constrained. The quaestio finita is also called causa, and the alternative name for quaestio infinita is propositum, related to the aristotelian notion of thesis. Cicero discerns the two types of propositum, the first of which is propositum cognitionis, theoretical, and the second is propositum actions, practical. Both these types of quaestio infinita have their role in algebra as the art addresses both theoretical and practical problems." -------- "I believe the rhetorical function of algebra recognized by the authors cited above, is contained more in the development of algebraic symbolism, than in the changing role of quaestio. I have argued elsewhere that the period between Cardano (1539) and Buteo (1559) has been crucial for the development of the concept of the symbolic equation." "The improved symbolism of Viète, and symbols in general, are the result, rather than the start, of symbolic reasoning. It is precisely Cardano, Stifel, Peletier and Buteo who shaped the concept of the symbolic equation by defining the combinatorial operations which are possible on an equation. The process of representing a problem in a symbolic mode and applying the rules of algebra to arrive at a certain solution, have reinforced the belief in a mathesis universalis. Such a universal mathesis allows us not only to address numerical problems but possibly to solve all problems which we can formulate." "The thought originates within the Ramist tradition as part of a broader philosophical discussion on the function and method of mathematics, but the term turns up first in the writings of Adriaan Van Roomen (1597). The idea will flourish in the seventeenth century with Descartes and Leibniz. A mathesis universalis is inseparably connected with the newly invented symbolism. As Archimedes only needed the right lever to be able to lift the world, so did the new algebraist only need to formulate a problem in the right symbolism to solve it. Nullum non problema solvere, or “leave no problem unsolved” as Viète would zealously write at the end of the century. Much has been written on the precise interpretation of Descartes’ use of the term. The changing rhetoric of algebra textbooks at the second half of the sixteenth century gives support to the interpretation of Chikara Sasaki, in which mathesis universalis can be considered as algebra applied as a model for the normative discipline of arriving at certain knowledge. This is the function Descartes describes in Rule IV of his Regulae. Later, Wallis (1657) uses Mathesis Universalis as the title for his treatise on algebra and includes a large historical section discussing the uses of symbols in different languages and cultures. As a consequence, the study of algebra delivers us also a tool for reasoning in general." -------
 Euler VI --------- The generalization of problems to propositions "For the modern reader of sixteenth-century algebras, it is difficult to understand why it took so long before algebraic problems became formulated in more general terms. Many of the textbooks of mid-sixteenth century contained hundreds of problems often of similar types intentionally dispersed over the pages. It is evident that someone who can solve the general case, can all individual problems belonging to that case. What is more, the need for generality was duly recognized. For example Cardano (1545) writes “We have used this variety of examples so that you may understand that the same can be done in other cases” (Witmer 1968)." "There is a specific historical reason for the lack of generality. By 1560, the algebraic symbolism was developed to a point where multiple equations of higher degree could be simultaneously formulated without ambiguities." "One crucial aspect was missing: the tools for the generalization of the values of the coefficients. This required the generalization of the concept of an equation to a general structure which can be approached under different circumstances. It was Viète who initiated the shift from the solution of problems to the study of the structure of equations and transformations of equations." "Let us look at one example as an illustration of the importance of the new symbolism for coefficients. In the In Artem Analyticem Isagoge, Viète (1591) studies several problems with numbers in GP, as did Cardano and Stifel before him. The latter two construct equations in order to solve specific instances of problems with numbers in GP. On the other hand, Viète is interested in the relationship between the properties of numbers in GP and the structure of the quadratic and cubic equation. He investigates the circumstances in which one can be transformed into the other." .... "Before Viète this crucial property of this quadratic equation could not be represented. Viète therefore introduced the use of the vowels A, E, I, O and U to represent unknowns, and the use of consonants for the constants and coefficients of an equation." "However, others after Viète show the inclination to reformulate classic problems in more general terms. Christopher Clavius, the great reformer of mathematics teaching, published his own Algebra in Rome in 1608. Unexpectedly, he ignores most of the achievements and improvements in symbolism of the second half of the sixteenth century and goes back to Stifel’s Arithmetica Integra as a model for structure and for most of his large problem collection." "This method of generalization is completed by Jacques de Billy (1643) who treats no less than 270 problems on numbers in GP in his Nova Geometriae Clavis Algebra. For each problem, he gives a general formulation, a construction method, an algebraic derivation and a general canon. de Billy abandons the terms ‘problem’ and quaestio and instead uses propositio. The general formulation of problems thus allows him to dispose of problems altogether and move to general propositions which constitute a new body of mathematical theory." "The generalization of problems thus achieved more than one had hoped for. Not only did it provide a solution method to all problems of that type. Also it constituted a body of mathematical knowledge that could be referred to in a rhetorical exposition, strengthening its persuasive power." ----------- An attempt at an axiomatic theory "The method of de Billy, of generalizing problems and turning their solution into canons which are universally applicable and to be used in the derivation of other propositions, was taken over by a new wave of algebraists in England. Despite of the fact that he published only a concise introduction to algebra in French (1637) and the Latin treatise on numbers in GP (1643), de Billy was well appreciated in England." "The books were not issued again in France. In England however, William Leybourn (1660) added a translation of de Billy’s 'Abrégé des préceptes d'algèbre' as the fourth part of his Arithmetic, first published in 1657. This popular work was reprinted several times up to the eighteenth century. But it is de Billy’s other work which influenced the rhetoric of English algebra textbooks in the later half of the seventeenth century. In England, the need for rigor in the demonstration of algebraic reasoning was felt more directly. The prime model for truthful reasoning was, without doubt, Euclidian geometry, constructing theorems which follow from axioms by deductive reasoning." "Before the seventeenth century, algebra was considered a practice, performed by those skilled in the art. It required experience and knowledge of many rules, which had their own name such as the regula alligationis. The idea of a universal mathesis rendered knowledge of such rules superfluous." "Algebra was basically not different from geometry or arithmetic (Wallis 1657). Algebra starts from simple facts which can be formulated as axioms. All other knowledge about algebraic theorems can be derived from these axioms by deduction. John Wallis introduced the term axioms in relation to algebra in an early work, called Mathesis Universalis, included in his Operum mathematicorum (1657). With specific reference to Euclid’s Elements, he gives nine Axiomata, called communes notationes, referring to the function of symbolic rewriting. 1 Due eidem sunt aequalia, sunt et inter se aequalia if A = C and B = C then A = C 2 Si aequalibus aequalia addantur, tota sunt if A = B then A + C = B + C 6 Quae eiusdem sunt dupliciae sunt inter se aequalia 2A = A + A 7 Quae eiusdem sunt dimidia, sunt aequalia inter se A/2 = A – A/2 etc etc... "Some years later, John Kersey (1673, Book IV) expanded on this and formulated 29 axioms “or common notions, upon which the force of inferences or conclusions, about the equality, majority and minority of quantities compared to one another, doth chiefly depend”. Although using many more axioms, he basically reformulates those from Wallis. The method of constructing theorems or canons and the belief in the infallibility of the chain of reasoning becomes apparent from Kersey’s explication of the difference between the analytic and the synthetic approach in the introduction (Kersey 1673): 'Algebra which first assumes the quantity sought, whether it be a number or a line in a question, as if it were known, and then, with the help of one or more quantities given, proceeds by undeniable consequences, until that quantity which at first was but assumed or supposed to be known, is found to some quantity certainly known, and is therefore known also.' 'Which analytical way of reasoning produceth in conclusion, either a theorem declaring some property, proportion or equality, justly inferred from things given or granted in a proposition, or else a canon directing infallibly how that may be found out or done which is desired; and discovers demonstrations of the certainty of the resulting theorem or canon, in the synthetical method, or way of composition, by steps of the analysis, or resolution.' "The quote is an excellent example to illustrate how the rhetoric of the algebra textbooks in the second half of the seventeenth century adopts of the Euclidian style of demonstration." ------ "The attempt to grasp the foundations of algebraic reasoning in basic axioms, was pursued until the early eighteenth century. Before Euler in Germany, the most influential writer of textbooks on mathematics was Christian Wolff (1713-1715). His Elementa matheseos universae was originally issued in two volumes. The first one treats the traditional disciplines arithmetic, algebra, geometry and trigonometry. A later addition added a wide variety of practical mathematics, from optics and astronomy to fortification and pyrotechnics. With the Basel edition this standard textbook was enlarged to five volumes, reprinted and adapted several times in the eighteenth century (Wolff, 1732). Immanuel Kant owned a copy of the first edition and was intimitaly acquainted with Wolff’s work (Warda 1922).The book had an important influence on Kant’s conception of the synthetic a priori in his Critique of Pure Reason (Shabel, 2003). Especially Kant’s view on the role of algebra in symbolic construction, as based on the manipulation of geometrically constructible objects, is strongly influenced by the way Wolff conceived algebra." The part on algebra in the Elementa was also published separately in a Compendium (Wolff, 1742) and translated into English (Wolff, 1739) and German. Wolff starts his Compendium with an introduction to the methodo mathematica describing the axiomatic method. In the introduction to arithmetic, preceding the algebra, he gives eight axioms “on which the general way of calculation is founded”, corresponding with these of Wallis (1657). He adds (Wolff, 1739): 'The delivering of these may seem superfluous, but it will be found that they are of great help to the understanding of Algebra, giving a clear idea of the way of reasoning that is used therein.' "While the axioms define the basic properties of quantities and, as such, belong to the realm of arithmetic, they are considered functional for the study of algebra. Wolff deals with many problems, always formulated in the general way, leading to a general solution and illustrated by a numerical example. The solution is often presented as a theorem." "While the axiomatic approach was abandoned in the most common textbooks after Euler, the attempts by Wallis, Kersey and Wolff extented into the nineteenth century through some lesser-know works. Perkins (1842) lists ‘four axioms used in solving equations’. Ingrid Hupp (1998) studied a tradition of three university professors teaching mathematics at the university of Wurzburg. Franz Huberti (1762), Franz Trentel (1774) and Andreas Metz (1804) all continued Wolff’s approach to express the essentials of algebra and arithmetic by axioms." "Their motivation may have been more didactical than in pursuance of a mathesis universalis. The axiomatic method brings rigor, clarity and brevity to the mathematical discipline, all too much inundated by numerous individual rules and recipes. Metz uses these properties of the axiomatic structure of algebra explictly as an argument to include it in an elementary textbook on arithmetic (Metz 1804)." "While Wolff defined axioms but never used them in his Algebra, Huberti and to a larger extent Trentel and Metz, occasionally apply the axioms in derivations (Hupp 1998)." "Though the axiomatic method, found in algebra textbooks until the early nineteenth century does not match the standards of mathematical logic emerging in the late nineteenth century, the axiomatic model of Euclidian geometry is used rhetorically to arrive at “undeniable consequences”. The purpose of algebra moves from the solution of numerical problems to the construction of a body of certain mathematical knowledge formulated by means of theorems and derived by rigorous deduction. Importantly, problems are the main instrument in this rhetorical transition. The whole body of knowledge, in the form of theorems, is derived from generalized problems. The changing role of problems has facilitated the rhetorical transition of algebra textbooks." -------
 Euler VIII --------- The generalization of problems to propositions "For the modern reader of sixteenth-century algebras, it is difficult to understand why it took so long before algebraic problems became formulated in more general terms. Many of the textbooks of mid-sixteenth century contained hundreds of problems often of similar types intentionally dispersed over the pages. It is evident that someone who can solve the general case, can all individual problems belonging to that case. What is more, the need for generality was duly recognized. For example Cardano (1545) writes “We have used this variety of examples so that you may understand that the same can be done in other cases” (Witmer 1968)." "There is a specific historical reason for the lack of generality. By 1560, the algebraic symbolism was developed to a point where multiple equations of higher degree could be simultaneously formulated without ambiguities." "One crucial aspect was missing: the tools for the generalization of the values of the coefficients. This required the generalization of the concept of an equation to a general structure which can be approached under different circumstances. It was Viète who initiated the shift from the solution of problems to the study of the structure of equations and transformations of equations." "Let us look at one example as an illustration of the importance of the new symbolism for coefficients. In the In Artem Analyticem Isagoge, Viète (1591) studies several problems with numbers in GP, as did Cardano and Stifel before him. The latter two construct equations in order to solve specific instances of problems with numbers in GP. On the other hand, Viète is interested in the relationship between the properties of numbers in GP and the structure of the quadratic and cubic equation. He investigates the circumstances in which one can be transformed into the other." .... "Before Viète this crucial property of this quadratic equation could not be represented. Viète therefore introduced the use of the vowels A, E, I, O and U to represent unknowns, and the use of consonants for the constants and coefficients of an equation." "However, others after Viète show the inclination to reformulate classic problems in more general terms. Christopher Clavius, the great reformer of mathematics teaching, published his own Algebra in Rome in 1608. Unexpectedly, he ignores most of the achievements and improvements in symbolism of the second half of the sixteenth century and goes back to Stifel’s Arithmetica Integra as a model for structure and for most of his large problem collection." "This method of generalization is completed by Jacques de Billy (1643) who treats no less than 270 problems on numbers in GP in his Nova Geometriae Clavis Algebra. For each problem, he gives a general formulation, a construction method, an algebraic derivation and a general canon. de Billy abandons the terms ‘problem’ and quaestio and instead uses propositio. The general formulation of problems thus allows him to dispose of problems altogether and move to general propositions which constitute a new body of mathematical theory." "The generalization of problems thus achieved more than one had hoped for. Not only did it provide a solution method to all problems of that type. Also it constituted a body of mathematical knowledge that could be referred to in a rhetorical exposition, strengthening its persuasive power." ----------- An attempt at an axiomatic theory "The method of de Billy, of generalizing problems and turning their solution into canons which are universally applicable and to be used in the derivation of other propositions, was taken over by a new wave of algebraists in England. Despite of the fact that he published only a concise introduction to algebra in French (1637) and the Latin treatise on numbers in GP (1643), de Billy was well appreciated in England." "The books were not issued again in France. In England however, William Leybourn (1660) added a translation of de Billy’s 'Abrégé des préceptes d'algèbre' as the fourth part of his Arithmetic, first published in 1657. This popular work was reprinted several times up to the eighteenth century. But it is de Billy’s other work which influenced the rhetoric of English algebra textbooks in the later half of the seventeenth century. In England, the need for rigor in the demonstration of algebraic reasoning was felt more directly. The prime model for truthful reasoning was, without doubt, Euclidian geometry, constructing theorems which follow from axioms by deductive reasoning." "Before the seventeenth century, algebra was considered a practice, performed by those skilled in the art. It required experience and knowledge of many rules, which had their own name such as the regula alligationis. The idea of a universal mathesis rendered knowledge of such rules superfluous." "Algebra was basically not different from geometry or arithmetic (Wallis 1657). Algebra starts from simple facts which can be formulated as axioms. All other knowledge about algebraic theorems can be derived from these axioms by deduction. John Wallis introduced the term axioms in relation to algebra in an early work, called Mathesis Universalis, included in his Operum mathematicorum (1657). With specific reference to Euclid’s Elements, he gives nine Axiomata, called communes notationes, referring to the function of symbolic rewriting. 1 Due eidem sunt aequalia, sunt et inter se aequalia if A = C and B = C then A = C 2 Si aequalibus aequalia addantur, tota sunt if A = B then A + C = B + C 6 Quae eiusdem sunt dupliciae sunt inter se aequalia 2A = A + A 7 Quae eiusdem sunt dimidia, sunt aequalia inter se A/2 = A – A/2 etc etc... "Some years later, John Kersey (1673, Book IV) expanded on this and formulated 29 axioms “or common notions, upon which the force of inferences or conclusions, about the equality, majority and minority of quantities compared to one another, doth chiefly depend”. Although using many more axioms, he basically reformulates those from Wallis. The method of constructing theorems or canons and the belief in the infallibility of the chain of reasoning becomes apparent from Kersey’s explication of the difference between the analytic and the synthetic approach in the introduction (Kersey 1673): 'Algebra which first assumes the quantity sought, whether it be a number or a line in a question, as if it were known, and then, with the help of one or more quantities given, proceeds by undeniable consequences, until that quantity which at first was but assumed or supposed to be known, is found to some quantity certainly known, and is therefore known also.' 'Which analytical way of reasoning produceth in conclusion, either a theorem declaring some property, proportion or equality, justly inferred from things given or granted in a proposition, or else a canon directing infallibly how that may be found out or done which is desired; and discovers demonstrations of the certainty of the resulting theorem or canon, in the synthetical method, or way of composition, by steps of the analysis, or resolution.' "The quote is an excellent example to illustrate how the rhetoric of the algebra textbooks in the second half of the seventeenth century adopts of the Euclidian style of demonstration." ------ "The attempt to grasp the foundations of algebraic reasoning in basic axioms, was pursued until the early eighteenth century. Before Euler in Germany, the most influential writer of textbooks on mathematics was Christian Wolff (1713-1715). His Elementa matheseos universae was originally issued in two volumes. The first one treats the traditional disciplines arithmetic, algebra, geometry and trigonometry. A later addition added a wide variety of practical mathematics, from optics and astronomy to fortification and pyrotechnics. With the Basel edition this standard textbook was enlarged to five volumes, reprinted and adapted several times in the eighteenth century (Wolff, 1732). Immanuel Kant owned a copy of the first edition and was intimitaly acquainted with Wolff’s work (Warda 1922).The book had an important influence on Kant’s conception of the synthetic a priori in his Critique of Pure Reason (Shabel, 2003). Especially Kant’s view on the role of algebra in symbolic construction, as based on the manipulation of geometrically constructible objects, is strongly influenced by the way Wolff conceived algebra." The part on algebra in the Elementa was also published separately in a Compendium (Wolff, 1742) and translated into English (Wolff, 1739) and German. Wolff starts his Compendium with an introduction to the methodo mathematica describing the axiomatic method. In the introduction to arithmetic, preceding the algebra, he gives eight axioms “on which the general way of calculation is founded”, corresponding with these of Wallis (1657). He adds (Wolff, 1739): 'The delivering of these may seem superfluous, but it will be found that they are of great help to the understanding of Algebra, giving a clear idea of the way of reasoning that is used therein.' "While the axioms define the basic properties of quantities and, as such, belong to the realm of arithmetic, they are considered functional for the study of algebra. Wolff deals with many problems, always formulated in the general way, leading to a general solution and illustrated by a numerical example. The solution is often presented as a theorem." "While the axiomatic approach was abandoned in the most common textbooks after Euler, the attempts by Wallis, Kersey and Wolff extented into the nineteenth century through some lesser-know works. Perkins (1842) lists ‘four axioms used in solving equations’. Ingrid Hupp (1998) studied a tradition of three university professors teaching mathematics at the university of Wurzburg. Franz Huberti (1762), Franz Trentel (1774) and Andreas Metz (1804) all continued Wolff’s approach to express the essentials of algebra and arithmetic by axioms." "Their motivation may have been more didactical than in pursuance of a mathesis universalis. The axiomatic method brings rigor, clarity and brevity to the mathematical discipline, all too much inundated by numerous individual rules and recipes. Metz uses these properties of the axiomatic structure of algebra explictly as an argument to include it in an elementary textbook on arithmetic (Metz 1804)." "While Wolff defined axioms but never used them in his Algebra, Huberti and to a larger extent Trentel and Metz, occasionally apply the axioms in derivations (Hupp 1998)." "Though the axiomatic method, found in algebra textbooks until the early nineteenth century does not match the standards of mathematical logic emerging in the late nineteenth century, the axiomatic model of Euclidian geometry is used rhetorically to arrive at “undeniable consequences”. The purpose of algebra moves from the solution of numerical problems to the construction of a body of certain mathematical knowledge formulated by means of theorems and derived by rigorous deduction. Importantly, problems are the main instrument in this rhetorical transition. The whole body of knowledge, in the form of theorems, is derived from generalized problems. The changing role of problems has facilitated the rhetorical transition of algebra textbooks." -------

 Quote by mathwonk i am puzzled. the copy of euler i have linked contains hundreds of exercises.
Well, maybe there are enough exercises then. I guess I just got a little suspicious of the fact that there are no "Questions for Practice" after several topics. I probably have to take a deeper look into the material.

RJinkies, thanks for posting. A related topic might be the approach some writers of mathematical learning materials have today; problem based material rather than a rigorous exposition of the subject by the writer himself/herself. For instance, "Polynomials" by Barbaue (which is a book I have looked into and maybe aspire to get some time in the future to learn from) is an algebra book that basically gives you a bunch of exercises/problems, and it seems to me that the idea is that you more or less are expected to discover and conceive the general theory yourself. I also think that "Algebra" by Gelfand has a similar approach.