Guide The Origins of Infinitesimal Calculus

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Collins's requests to distinguished mathematicians for help in solving practical problems recur in his correspondence. He asked for numerical methods that facilitated the computation of logarithmic tables and of annuities. To solve the latter problem Collins received from Gregorie as well as from Newton series specifically derived for it.

On 24 March Collins sent to James Gregorie the first four terms of one of Newton's series for a zone of the circle. Two of the practical problems that concerned the mathematical practitioners were of extreme mathematical difficulty, and both were related to navigation. The second problem is the rectification of the logarithmic spiral, defined by the condition that to angular increments in arithmetical progression correspond exponential increments in the radii or distances to the spiral pole. Such an spiral has the property that the angle between it and its radii is always the same, and is therefore closely related to the loxodromes.

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The problem itself is theoretically connected to the determination of longitude and to the general problem of finding rhumb lines. Only Gregorie provided an answer, including its demonstration, which is a real tour de force involving the numerical quadrature of the logarithmic curve. It is not obvious which, if any, was the real practical value of the solutions to such problems, at least to the nautical ones.

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The solutions Gregorie provided by means of powerful theoretical methods were probably too accurate for the technical means then available either to pilots or to map makers. On the other hand, the mathematical interest of some of the problems Collins posed is huge, and they reverberate through the mathematical literature up to the first decades of the 18th century. The work of James Gregorie is an excellent exemple of the status of mathematical knowledge within the Royal Society and also of the ways in which the Society impinged on the mathematical research agenda.

History of calculus - Wikipedia

The problems itself, however, and their solutions, were not as prominent in the institutional life of the Society as were problems of a more philosophical and gentle nature. Indeed, almost none of the deep mathematical results Collins received in answer to his queries did find a place in the Philosophical Transactions. However the fact remains that Gregorie and Newton, among others, provided original solutions to non-trivial, sometimes very difficult mathematical problems coming from the world of the mathematical practitioners.

Perhaps we may find here a partial explanation for the fact that the 17th-century methods of series expansion, which in that century were of interest mostly for their numerical applications, are an essentially English contribution. One of the first publications in which series expansions played a prominent role in solving practical problems came from the pen of Nicolaus Mercator Kaufmann , an early fellow of the Royal Society of Danish origin who settled in London for many years.

Among them, Mercator offered a brilliant if apparently simple idea. Since it was known from the work of Gregoire of St.

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Belonging to a mathematical genre sometimes called infinitesimal geometrical analysis, they are probably the most rich and deep 17th-century mathematical books antedating the infinitesimal calculus literature. Both make a clever use of the so-called differential or characteristic triangle, on which they ground the quadratures and rectifications of many difficult curves.

Methodologically, they provide geometrical results about geometrical objects, i. As we shall see, Barrow offered a theoretical discussion that justified the use of such mathematical objects.

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Although many results demonstrated in these books were previously known to Torricelli, Mengoli, Gregoire de St Vincent, Tacquet, and others, most of them were here for the first time set forth in as general a form as possible, convincingly proved, and included in a consistent general framework. The relationship between tangents and quadratures was not particularly emphasized during most of the seventeenth century, not even when leading mathematicians came upon results that unmistakably revealed them.

Before the calculus techniques of Newton and Leibniz, however, mathematicians did not reach a truly general method that applied to all kinds of curves then known. After his studies in Aberdeen and publishing a book on optics , he started on his European tour He spend a few years in Padua, probably studying with Stephano degli Angeli — , a competent mathematician who had been a student of Torricelli and was teaching mathematics at Padua. Angeli published many books dealing with the quadratures of different kinds of spirals, parabolas, and hyperbolas.

In Padua Gregorie published Vera circuli et hyperbolae quadratura By an involved argument, Gregorie concluded that the limit could not be expressed as a rational function of a0 and 3 S. It got him in an ugly quarrel with Huygens, then probably the most prestigious and respected mathematician in Europe. Figure 1 In , also in Padua and just before returning to London, Gregorie published Geometriae pars universalis. In fact the title of the work makes direct reference to the set of "universal" theorems that opens the book. It may be divided in two parts. The first one contains 11 general theorems about the relationships between tangent, arc, and area in any monotonic curve Prop.

Given a curve of ordinates y upon the axis x, whose quadrature is known for all the values of y, it shows how to find a curve, u, upon the same axis, whose arc length at every point is given by the area of the curve y. Translating to our notions, this is to say that z whose integral is u is equal to u divided by the subtangent of u, which as indicated above corresponds to what we call the derivative of u.

Rather to the contrary, we want to emphasize that mathematical results cannot be easily translated from a given language and theoretical framework to another one. Those are general methods for transforming curves that to put it anachronistically are most conveniently expressed in polar coordinates such as spirals or sectors of conic sections into curves that are usually expressed in rectangular coordinates. Then he demonstrates that the area 0 of the evolute is twice the area of the involute. And, finally, that at corresponding points I on the involute and J on the evolute , that is points such that the radius of I equals the ordinate of J, the tangents to the curves make the same angles with the radius and with the ordinate.

Next, the Geometriae pars universalis deals with volumes, surfaces, and centers of gravity of solids, where Gregorie is interested in giving relations between the solid of revolution and the generating figure prop. Newton's own copy of the book, still extant, displays Newton's "dog-ears", or signs of dog-earing, marking places that interested him.


Whiteside has shown, many problems solved in Newton's Method of Fluxions originate in theorems of the Universal Part of Geometry. Just upon reading it, Gregorie wrote the Exercitationes geometricae, a small mathematical tract consisting of four different sections. There is, first of all, an answer to Huygens's criticism to the Vera quadratura, in which Gregorie included some rather insulting remarks. Then comes an elegant geometrical proof of Mercator's series expansion for the hyperbolic area Mercator had provided none , and an improved series expansion for the same.

Through the Mercator example Gregorie understood the power of series expansions and turned to explore them. To calculate them, Gregorie did not worked either with fluxions or with differentials, but apparently the evidence we have is highly fragmentary worked with the quotient between a variable and its subtangent which corresponds to our derivative.

go He also found analytical rules to find the new subtangent when a given curve was modified so that the ordinates were now a function of a new variable. With this arsenal of new algorithms, Gregorie could solve the difficult problems Collins sent to him see previous section , as well as series expansions for tanx, secx, arcsecx. Isaac Barrow Barrow's best-known scientific works are two sets of lectures he prepared between and in fulfillment of his duties as Lucasian professor of mathematics, Geometrical Lectures and Optical Lectures.

Barrow also wrote Mathematical Lectures, which contain a substantial analysis of the object and method of the mathematical sciences. Barrow publicly delivered them in Cambridge in three consecutive years, from to He produced updated editions of Euclid, Apollonius, and Archimedes in which he introduced algebraic language and somewhat modified the demonstrations. Barrow graduated from Cambridge around Under the influence of the famous Cambridge Platonist, Henry More, he studied natural philosophy and produced a thesis on the insufficiency of the Cartesian hypothesis.

Finally, in he became Chaplain of his Majesty and devoted himself to theology.

Contemporary accounts of his life and character strongly convey the impression that he grew skeptical about the fruitfulness and worth of natural philosophy, and as his skepticism grew he took refuge in geometrical optics and mathematics, and came to advocate a strongly mathematized natural philosophy. Then, in the last years of his life, Barrow abandoned mathematics to prolifically write sermons and religious texts—over two thousand pages of them, highly regarded by scholars as well as by the Anglican hierarchy.

The Mathematical Lectures This unfairly neglected book analyses the nature and status of the mathematical sciences; it discusses the nature of space and extended geometrical magnitude; why they are not to be identified; how the latter grounds the notion of number; and what is the relationship between arithmetic and geometry. The final set of lectures deals with ratio and proportionality. Other important topics are dealt with rather quickly, like the nature of infinitesimals or the status of algebra.

Since the science that deals with magnitude in general is geometry, Barrow concludes that geometry subsumes all mathematics. To measure magnitudes means to chose as unit one homogeneous magnitude that is particularly well-known and compare to it the other magnitudes in order to find the ratio that they keep to the unit. This ratio measures or makes known the quantity of that specific magnitude.

Lines are usually taken to represent the measures of res quanta, but they in fact are only "symbols" or "indexes" of the things measured—they are only a commodious way to visualize the quantity of some magnitude, i.

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That ratio or proportion is in itself not reducible to anything. It s not a number, or a res quanta, it is just an abstract relationship. Numbers in themselves have no distinct meaning. A given quanta say, a line may be called 1, or 2, or whatever number you like. And conversely, any given number can be applied to whatever line or magnitude we like.

Left by themselves, numbers cannot be operated. If they are, it is done only on the tacit understanding that they represent homogeneous magnitudes measured with the same unit.