##
**From the Géométrie to calculus: the problem of the tangents and the origins of infinitesimal calculus.**
*(Italian.
English summary)*
Zbl 1441.01005

As the title clarifies, this paper faces some aspects of the development of mathematics between the publication of Descartes’ Géométrie (1637) and the beginning of the 18th century connected with the problem of the tangents, of the quadratures and of the origin of infinitesimal calculus. The author identifies two research programmes from which the infinitesimal calculus arose: 1) the problem to draw the tangent to a curve; 2) Cavalieri’s Geometria indivisibilium to which the quadrature question is connected. This question is, in its turn, related to Newton’s inquires and results on the infinite series.

Content of the paper: The article begins explaining the method to trace the tangent to a curve expounded by Descartes in his Géométrie. The author of the article under review clarifies that, given a point \(P\equiv(x_0,y_0)\) of the curve, Descartes determined first the equation of the circumference tangent to the curve at \(P\) and its radius. Afterwards, he determined the tangent as the perpendicular on the radius at \(P\). The author explains that this method is based on the possibility to write the variable \(y\) as a function of \(x\) in the equation of the curve and to replace such a value in the equation of the circumference, so to obtain a polynomial \(Q(x)=(x-x_0)^2R(x)\) the degree of which is \(2n\) if the degree of the curve is \(n\). By equating in \(Q(x)\) the coefficients of the terms having the same degree, you get a system in which the number of the unknowns is the same as the number of the equations, so that the centre and the radius of the circumference can be determined and, thus, the tangent (p. 210). The author claims that in principle the tangent problem is solved, but that, as a matter of fact, the calculations are very complicated in easy cases, too. He offers the example of the parabola (p. 210).

Afterwards, the author analyses the method invented by Descartes’ rival, Pierre Fermat, to determine maxima and minima and to trace the tangents. As a matter of fact, the author clarifies that it is appropriate to speak of two Fermat’s methods: the former devised by Fermat for the problem of maxima and minima and based on the idea that, if \(M\) is a maximum of the curve \(f\), two values \(A\) and \(E\) exist, one on the left and the other on the right of the abscissa corresponding to the maximum \(M\), such that \(f(A)=f(E)=Z\), with \(Z<M\). Considering the ratio \(\frac{f(A)-f(E)}{A-E}=0\), when the points \(A\) and \(E\) get closer and closer until coinciding with the abscissa of the maximum, and you pose \(A=E\) in the previous equation, the abscissa of the maximum will be determined and, hence, the maximum itself. The author also presents an example offered by Fermat in which the method is applied. He explains why the application of this method to the tangent problem is involved in a difficulty, overcome by Fermat by a slight change, but a change that, in fact, makes the procedure far more malleable, insofar as it breaks the symmetry between the points \(A\) and \(E\) and makes it suitable to solve the tangent problem. This modification is so important that the author considers it remarkable enough to identify a new method. Thus, the author explains the use made by Fermat of his method to reach the adequatio, based on a quantity – indicate by \(E\) – which, at all effects, is an infinitesimal (p. 212). It is shown how Fermat applied his method to find the tangent to a parabola, to the cissoids of Diocles – curiously, the author writes cissoids of Nicomedes – and to the cycloid (pp. 213–216).

In the following section, the quarrel between Fermat and Descartes as to the best tangent method is analysed. Two objections moved by Descartes against Fermat’s methods are presented. The former depends on a pretentious interpretation by Descartes of Fermat’s method. The latter is more founded, but can, in any case, be overcome (pp. 216–218).

The following chapter concerns the improvement of Descartes’ method made by Florimond de Beaune, Hudde and Sluse. These mathematicians worked on three aspects: 1) to determine the tangent without passing through the tangent circumference, which was useful to reduce the degree of the equation determining the tangent from \(2n\) to \(n\), if \(n\) is the degree of the curve; 2) the problem of the elimination of a variable and the connected question of the necessity to solve a complicate system of equations; 3) the fact that Descartes’ method was not applicable to transcendental curves (p. 219). The merit of the idea to consider directly the tangent \(y=y_0+m(x-x_0)\) at a point was by de Beaune, but Hudde had the great merit to guess that the resort to the polynomial \(R(x)\) could be avoided so that the whole procedure becomes far easier (p. 219). For, he grasped that a polynomial \(Q(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n\) has a double root in \(x_0\) if and only if the equation \(Q_1(x_0)=0\) is also verified, where \(Q_1(x)=k_0a_0+k_1a_1x+k_2a_2x^2+\dots +k_na_nx^n\) and the \(k_i\) belong to an arithmetical progression. Working on these bases, Hudde was able to avoid the resort to the auxiliary polynomial \(R(x)\). The author refers to the example of the parabola \(py=x^2\). Replacing \(y_0+m(x-x_0)\) instead of \(y\), the two polynomials \(Q(x)=x^2-py_0-pm(x-x_0)\) and \(Q_1(x)=2x-pm\) are obtained from which it is easy to write the equation of the tangent. Some years later, Hudde arrived at a complete automation of his method. Sluse reached results analogous to Hudde’s (pp. 219–220). However, neither Hudde nor Sluse offered the proof that the method worked. The merit of this demonstration is due to Huygens (1667, but published in 1693). This interesting demonstration, which is explained in a clear manner, is based on the introduction of an infinitesimal quantity \(e\), which resembles closely the infinitesimal quantity Fermat had introduced in his tangent method (p. 220). Newton had also arrived at the Sluse rule for the tangents. Finally, the author expounds a further and more general rule (also applicable to a class of transcendental curves) to draw the tangents invented by Tschirnhaus (p. 222).

The author comments that at the end of the 1670’s there were several methods to solve the tangent problem, some of them deriving from Descartes’ ideas, others from Fermat’s. Those derived from Fermat’s train of thoughts also allowed the mathematicians to draw the tangents to transcendental curves. These remarkable successes notwithstanding, some classes of curves still escaped the capabilities of such methods, in particular, the curves whose equation included many radicals (p. 222). As a matter of fact, Fermat had taught how to eliminate the radicals from the equation of the tangents, but in many cases the resulting polynomial was of so high degree that it was impossible to work it (p. 223). Here, the author makes an important consideration: despite the cleverness and broad applicability of Fermat’s method, it has a feature: it has a global character in the sense that the adequatio and the derived equation are treated as a sole mathematical object. To be clearer: when nowadays we obtain the tangent or the maxima and minima of a function \(f(x)\) through the derivative we operate in two steps: 1) determination of the derivative \(f'(x)\); 2) solution of the equation \(f'(x)=0\), whereas Fermat joins these two steps in the determination of the adequatio. This is a remarkable difference because, through the differentiation rules, it is possible to simplify the problem and to solve it in many cases which were precluded to Fermat’s method, because it is possible to operate on the single elements composing the function. Whereas, an equation is always global, in the sense that if the function \(f(x)=g(x)+h(x)\) is the sum two functions, the equation \(f(x)=0\) has nothing to do with \(g(x)=0; h(x)=0\), it is not possible to trace back the two functions \(g(x), h(x)\) within the sum. Thence, Fermat had not yet developed the concept of differential and the rules governing the operation with such concept. Even the less, it is possible to speak of derivate in relation to Fermat’s method (p. 223). Another problem with these tangent methods is that all of them try to determine the tangent through the sub-tangent, which is a natural choice from a geometrical point of view, but it can involve a complicate algebraic operation (p. 224).

In the following section, the author explains the way in which Leibniz, in a very famous paper published in 1684, introduced his concept of differential as well as the rules of differentiation (p. 224–225). The rules of differentiation allowed Leibniz to solve completely the tangent problem for algebraic curves and to reduce it to the determination of the differential for transcendental curves. The author stresses that, in Leibniz, the differentials – instead of the sub-tangent – become the principal parameters to determine the tangent (p. 225). He also addresses the problem of how Leibniz considered his infinitesimal quantities. For, Leibniz had always denied the mathematical existence of infinitesimal quantities and had considered them as a device, a “fiction” as the imaginary and the negative numbers, useful in the calculations, but deprived of a real mathematical existence (p. 226). The author points out with convincing argumentations that, when Leibniz spoke of the “inverse problem of tangents”, he was not referring either to the sole problem of the quadratures or to the fundamental theorem of the integral calculus, but to the question concerning the general solution of differential equations, on which most of the problems regarding geometry and physics relied.

Afterwards, the author gives a general picture of the theory of indivisibles, without entering, in this case, in many details. He refers to the acquisitions of this theory, but also to the conceptual and logical problems involved in it through a glance on the works of Cavalieri, Torricelli and Angeli. A very interesting application by Pascal of the method of indivisibles is expounded, where Pascal overcame many logical difficulties connected with Cavalieri’s formulation – in particular insofar as the problematic question of the retto e obliquo transito is concerned –, so as to arrive close to the concept of integral (pp. 228–229). The author points out that the lack of an enough elaborated concept of function did not allow then mathematicians to develop the theory of the quadratures to the same level as the tangents’ (p. 229). Because of this reason, starting from the 1650’s, the propulsive force of the indivisibles method was exhausting. In its place, the infinite series become a powerful method used to achieve the quadrature. Under this respect, the very master of the infinite series was Isaac Newton. The author enters several details of the method of fluxions expounded by Newton in a series of works written in the second half of the 1760’s but published many years afterwards, at the beginning of the 18th century. He also poses a comparison between Newton’s fluxions and Leibniz’s differentials claiming that Newton’s calculus appears as a natural consequence of his kinematic standpoint and that, hence, it results less innovative than Leibniz’s differentials. The author states that the real fundamental element of Newton’s calculus is the use of the infinite series, through which Newton, as a matter of fact, solved all the quadrature problems debated at that time, by reducing it to calculate an integral of powers (p. 231). The author shows the way in which Newton solved by series the differential equation \[ \dot{y}=1-3x+y+x^2+xy \] (p. 232) to conclude that, in this manner, Newton overcame Leibniz’s results expounded in 1684 and seems to give a definitive answer to the problems of mathematical analysis.

Afterwards, a section on the priority dispute Newton-Leibniz for the invention of the infinitesimal calculus follows (pp. 233–237). I do not enter this section because the history is well known. Rather, it is interesting what the author claims in the last section of his work entitled “Unicuique suum”. Here, he argues that, given the situation at the end of the 17th century, Newton’s results were far superior than Leibniz’s, but that for the successive research, Leibniz’s concepts were more fruitful than Newton’s. Interestingly, the author claims that this depends on the fact that Newton’s method of approximation through the infinite series closes the problems of quadratures and solution of differential equations, but does not exhaust them. In substance, such a method is scarcely susceptible of an evolution. In contrast to this, it is exactly the evolution of Leibniz’s concepts which allowed the mathematicians from continental Europe to solve a series of problems (see the quotation by Johannes Bernoulli, 1713, referred to by the author, p. 238) which were precluded to English mathematicians, always faithful to the infinite series method.

Commentary: This paper is very good. It offers a broad and clear picture of the evolution of the most important mathematical concepts between 1637 and the beginning of the 18th century. The mathematical details are well explained and framed within their context. The historical analyses are perspicuous and refined. I recommend this article to any reader. On the other hand, I stress some little differences in my interpretations in respect to those offered in the paper: 1) it seems to me that Fermat’s method for the tangent is by far superior than Descartes’. Fermat guessed that infinitesimal quantities were necessary to solve this problem and, though with the limitations underlined, he was very skilful in managing his infinitesimal quantities. In my opinion, in the comparison Descartes-Fermat the author has a too neutral approach; 2) it is profoundly true, as the author claims, that in the 17th century the concept of function did not exist. Nonetheless, it seems to me that some authors arrived closer at the concept of function – though in an intuitive and implicit way – more than the author is available to recognize. For example, it seems to me that Newton arrived close because, in the Methodus fluxionum et serierum infinitarum, all the variables are functions of the time \(t\). Therefore, he considered the variable \(t\) as independent and the others as dependent from \(t\). Obviously, you do not find a definition of function. However, I am expounding a different interpretation not as a critics to the excellent article I am reviewing, but because it is only natural that different historians of mathematics can offer different interpretations.

Content of the paper: The article begins explaining the method to trace the tangent to a curve expounded by Descartes in his Géométrie. The author of the article under review clarifies that, given a point \(P\equiv(x_0,y_0)\) of the curve, Descartes determined first the equation of the circumference tangent to the curve at \(P\) and its radius. Afterwards, he determined the tangent as the perpendicular on the radius at \(P\). The author explains that this method is based on the possibility to write the variable \(y\) as a function of \(x\) in the equation of the curve and to replace such a value in the equation of the circumference, so to obtain a polynomial \(Q(x)=(x-x_0)^2R(x)\) the degree of which is \(2n\) if the degree of the curve is \(n\). By equating in \(Q(x)\) the coefficients of the terms having the same degree, you get a system in which the number of the unknowns is the same as the number of the equations, so that the centre and the radius of the circumference can be determined and, thus, the tangent (p. 210). The author claims that in principle the tangent problem is solved, but that, as a matter of fact, the calculations are very complicated in easy cases, too. He offers the example of the parabola (p. 210).

Afterwards, the author analyses the method invented by Descartes’ rival, Pierre Fermat, to determine maxima and minima and to trace the tangents. As a matter of fact, the author clarifies that it is appropriate to speak of two Fermat’s methods: the former devised by Fermat for the problem of maxima and minima and based on the idea that, if \(M\) is a maximum of the curve \(f\), two values \(A\) and \(E\) exist, one on the left and the other on the right of the abscissa corresponding to the maximum \(M\), such that \(f(A)=f(E)=Z\), with \(Z<M\). Considering the ratio \(\frac{f(A)-f(E)}{A-E}=0\), when the points \(A\) and \(E\) get closer and closer until coinciding with the abscissa of the maximum, and you pose \(A=E\) in the previous equation, the abscissa of the maximum will be determined and, hence, the maximum itself. The author also presents an example offered by Fermat in which the method is applied. He explains why the application of this method to the tangent problem is involved in a difficulty, overcome by Fermat by a slight change, but a change that, in fact, makes the procedure far more malleable, insofar as it breaks the symmetry between the points \(A\) and \(E\) and makes it suitable to solve the tangent problem. This modification is so important that the author considers it remarkable enough to identify a new method. Thus, the author explains the use made by Fermat of his method to reach the adequatio, based on a quantity – indicate by \(E\) – which, at all effects, is an infinitesimal (p. 212). It is shown how Fermat applied his method to find the tangent to a parabola, to the cissoids of Diocles – curiously, the author writes cissoids of Nicomedes – and to the cycloid (pp. 213–216).

In the following section, the quarrel between Fermat and Descartes as to the best tangent method is analysed. Two objections moved by Descartes against Fermat’s methods are presented. The former depends on a pretentious interpretation by Descartes of Fermat’s method. The latter is more founded, but can, in any case, be overcome (pp. 216–218).

The following chapter concerns the improvement of Descartes’ method made by Florimond de Beaune, Hudde and Sluse. These mathematicians worked on three aspects: 1) to determine the tangent without passing through the tangent circumference, which was useful to reduce the degree of the equation determining the tangent from \(2n\) to \(n\), if \(n\) is the degree of the curve; 2) the problem of the elimination of a variable and the connected question of the necessity to solve a complicate system of equations; 3) the fact that Descartes’ method was not applicable to transcendental curves (p. 219). The merit of the idea to consider directly the tangent \(y=y_0+m(x-x_0)\) at a point was by de Beaune, but Hudde had the great merit to guess that the resort to the polynomial \(R(x)\) could be avoided so that the whole procedure becomes far easier (p. 219). For, he grasped that a polynomial \(Q(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n\) has a double root in \(x_0\) if and only if the equation \(Q_1(x_0)=0\) is also verified, where \(Q_1(x)=k_0a_0+k_1a_1x+k_2a_2x^2+\dots +k_na_nx^n\) and the \(k_i\) belong to an arithmetical progression. Working on these bases, Hudde was able to avoid the resort to the auxiliary polynomial \(R(x)\). The author refers to the example of the parabola \(py=x^2\). Replacing \(y_0+m(x-x_0)\) instead of \(y\), the two polynomials \(Q(x)=x^2-py_0-pm(x-x_0)\) and \(Q_1(x)=2x-pm\) are obtained from which it is easy to write the equation of the tangent. Some years later, Hudde arrived at a complete automation of his method. Sluse reached results analogous to Hudde’s (pp. 219–220). However, neither Hudde nor Sluse offered the proof that the method worked. The merit of this demonstration is due to Huygens (1667, but published in 1693). This interesting demonstration, which is explained in a clear manner, is based on the introduction of an infinitesimal quantity \(e\), which resembles closely the infinitesimal quantity Fermat had introduced in his tangent method (p. 220). Newton had also arrived at the Sluse rule for the tangents. Finally, the author expounds a further and more general rule (also applicable to a class of transcendental curves) to draw the tangents invented by Tschirnhaus (p. 222).

The author comments that at the end of the 1670’s there were several methods to solve the tangent problem, some of them deriving from Descartes’ ideas, others from Fermat’s. Those derived from Fermat’s train of thoughts also allowed the mathematicians to draw the tangents to transcendental curves. These remarkable successes notwithstanding, some classes of curves still escaped the capabilities of such methods, in particular, the curves whose equation included many radicals (p. 222). As a matter of fact, Fermat had taught how to eliminate the radicals from the equation of the tangents, but in many cases the resulting polynomial was of so high degree that it was impossible to work it (p. 223). Here, the author makes an important consideration: despite the cleverness and broad applicability of Fermat’s method, it has a feature: it has a global character in the sense that the adequatio and the derived equation are treated as a sole mathematical object. To be clearer: when nowadays we obtain the tangent or the maxima and minima of a function \(f(x)\) through the derivative we operate in two steps: 1) determination of the derivative \(f'(x)\); 2) solution of the equation \(f'(x)=0\), whereas Fermat joins these two steps in the determination of the adequatio. This is a remarkable difference because, through the differentiation rules, it is possible to simplify the problem and to solve it in many cases which were precluded to Fermat’s method, because it is possible to operate on the single elements composing the function. Whereas, an equation is always global, in the sense that if the function \(f(x)=g(x)+h(x)\) is the sum two functions, the equation \(f(x)=0\) has nothing to do with \(g(x)=0; h(x)=0\), it is not possible to trace back the two functions \(g(x), h(x)\) within the sum. Thence, Fermat had not yet developed the concept of differential and the rules governing the operation with such concept. Even the less, it is possible to speak of derivate in relation to Fermat’s method (p. 223). Another problem with these tangent methods is that all of them try to determine the tangent through the sub-tangent, which is a natural choice from a geometrical point of view, but it can involve a complicate algebraic operation (p. 224).

In the following section, the author explains the way in which Leibniz, in a very famous paper published in 1684, introduced his concept of differential as well as the rules of differentiation (p. 224–225). The rules of differentiation allowed Leibniz to solve completely the tangent problem for algebraic curves and to reduce it to the determination of the differential for transcendental curves. The author stresses that, in Leibniz, the differentials – instead of the sub-tangent – become the principal parameters to determine the tangent (p. 225). He also addresses the problem of how Leibniz considered his infinitesimal quantities. For, Leibniz had always denied the mathematical existence of infinitesimal quantities and had considered them as a device, a “fiction” as the imaginary and the negative numbers, useful in the calculations, but deprived of a real mathematical existence (p. 226). The author points out with convincing argumentations that, when Leibniz spoke of the “inverse problem of tangents”, he was not referring either to the sole problem of the quadratures or to the fundamental theorem of the integral calculus, but to the question concerning the general solution of differential equations, on which most of the problems regarding geometry and physics relied.

Afterwards, the author gives a general picture of the theory of indivisibles, without entering, in this case, in many details. He refers to the acquisitions of this theory, but also to the conceptual and logical problems involved in it through a glance on the works of Cavalieri, Torricelli and Angeli. A very interesting application by Pascal of the method of indivisibles is expounded, where Pascal overcame many logical difficulties connected with Cavalieri’s formulation – in particular insofar as the problematic question of the retto e obliquo transito is concerned –, so as to arrive close to the concept of integral (pp. 228–229). The author points out that the lack of an enough elaborated concept of function did not allow then mathematicians to develop the theory of the quadratures to the same level as the tangents’ (p. 229). Because of this reason, starting from the 1650’s, the propulsive force of the indivisibles method was exhausting. In its place, the infinite series become a powerful method used to achieve the quadrature. Under this respect, the very master of the infinite series was Isaac Newton. The author enters several details of the method of fluxions expounded by Newton in a series of works written in the second half of the 1760’s but published many years afterwards, at the beginning of the 18th century. He also poses a comparison between Newton’s fluxions and Leibniz’s differentials claiming that Newton’s calculus appears as a natural consequence of his kinematic standpoint and that, hence, it results less innovative than Leibniz’s differentials. The author states that the real fundamental element of Newton’s calculus is the use of the infinite series, through which Newton, as a matter of fact, solved all the quadrature problems debated at that time, by reducing it to calculate an integral of powers (p. 231). The author shows the way in which Newton solved by series the differential equation \[ \dot{y}=1-3x+y+x^2+xy \] (p. 232) to conclude that, in this manner, Newton overcame Leibniz’s results expounded in 1684 and seems to give a definitive answer to the problems of mathematical analysis.

Afterwards, a section on the priority dispute Newton-Leibniz for the invention of the infinitesimal calculus follows (pp. 233–237). I do not enter this section because the history is well known. Rather, it is interesting what the author claims in the last section of his work entitled “Unicuique suum”. Here, he argues that, given the situation at the end of the 17th century, Newton’s results were far superior than Leibniz’s, but that for the successive research, Leibniz’s concepts were more fruitful than Newton’s. Interestingly, the author claims that this depends on the fact that Newton’s method of approximation through the infinite series closes the problems of quadratures and solution of differential equations, but does not exhaust them. In substance, such a method is scarcely susceptible of an evolution. In contrast to this, it is exactly the evolution of Leibniz’s concepts which allowed the mathematicians from continental Europe to solve a series of problems (see the quotation by Johannes Bernoulli, 1713, referred to by the author, p. 238) which were precluded to English mathematicians, always faithful to the infinite series method.

Commentary: This paper is very good. It offers a broad and clear picture of the evolution of the most important mathematical concepts between 1637 and the beginning of the 18th century. The mathematical details are well explained and framed within their context. The historical analyses are perspicuous and refined. I recommend this article to any reader. On the other hand, I stress some little differences in my interpretations in respect to those offered in the paper: 1) it seems to me that Fermat’s method for the tangent is by far superior than Descartes’. Fermat guessed that infinitesimal quantities were necessary to solve this problem and, though with the limitations underlined, he was very skilful in managing his infinitesimal quantities. In my opinion, in the comparison Descartes-Fermat the author has a too neutral approach; 2) it is profoundly true, as the author claims, that in the 17th century the concept of function did not exist. Nonetheless, it seems to me that some authors arrived closer at the concept of function – though in an intuitive and implicit way – more than the author is available to recognize. For example, it seems to me that Newton arrived close because, in the Methodus fluxionum et serierum infinitarum, all the variables are functions of the time \(t\). Therefore, he considered the variable \(t\) as independent and the others as dependent from \(t\). Obviously, you do not find a definition of function. However, I am expounding a different interpretation not as a critics to the excellent article I am reviewing, but because it is only natural that different historians of mathematics can offer different interpretations.

Reviewer: Paolo Bussotti (Udine)

### MSC:

01A45 | History of mathematics in the 17th century |

01A50 | History of mathematics in the 18th century |

26-03 | History of real functions |

26A06 | One-variable calculus |