15 Dec 2008

Shattering Geometrical Fetters: Euler's Analysis Revolution

Scott Wollschleger asks in the comments to the "Euler in the History of the Calculus" entry:

what were the "geometrical fetters"?

which he asks in reference to this quote in Boyer:

Euler’s formalist approach to calculus freed it from all “geometrical fetters. It also made more acceptable the arithmetic interpretation which was later to clarify the calculus through the limit concept which Euler himself neglected” (246b).

Hegel articulates the logical problem the early methods of calculus encountered, despite their techniques producing accurate results; he writes in Science of Logic § 586

there is a return of the finite determinateness of quantity and the operation cannot dispense with the conception of a quantum which is merely relatively small. The calculus makes it necessary to subject the so-called infinitesimals to ordinary arithmetical operations of addition and so on, which are based on the nature of finite magnitudes, and therefore to regard them momentarily as finite magnitudes and to treat them as such. It is for the calculus to justify its procedure in which it first brings them down into this sphere and treats them as increments or differences, and then neglects them as quanta after it had just applied forms and laws of finite magnitudes to them.

When differentiating, finite values are treated as zero when convenient for the operation, so certain magnitudes are treated both as something and nothing in two phases of one same process, which is illogical and calls for a more solid theoretical explanation. We see this part of the operation carried out here [by MIT's David Jerison], at the very end of the algebra work:


This paradox was resolved with the limit concept, whose development took many hundreds of years after the advent of the calculus. Until Cauchy's work of the early 19th century, the limit concept lacked precision of formulation, even though it's development began with the Greek method of exhaustion and was expressed in Newton's Principia. The cause for the retarded development was the long history of the limit concept being conceived through geometrical intuition, for as well the quantitative values in arithmetic and algebra were considered in terms of geometrical magnitudes. Moreover, those who invented calculus regarded it as an instrument for determining relationships between quantities in geometrical problems. (Boyer 271-272)

When illustrating the limit, it was common to evoke the definition of a circle as the limit of a polygon.

But certain geometrical concerns cause us to misconceive what in fact the limit is; for we might wonder,

Is it the approach to coincidence of the sides of the polygon with the points representing the circle? Does the polygon ever become the circle? Are the properties of the polygon and the circle the same? It was questions such as these that retarded the acceptance of the limit idea, for they were similar to those of Zeno in demanding some sort of visualization of the passage from the one to the other by which the properties of the first figure merge into those of the second. (Boyer 272c)

Although Euler did not succeed in breaking entirely from geometrical limitations, he did make great steps in that direction. Boyer obtains the "geometrical fetters" phrase from John Meez's A History of European Thought in the Nineteenth Century, footnote 1, page 103:

1 See on this point the opinion of an authority, Hermann Hankel, in his highly interesting and suggestive lecture, ' Die Entwickelung der Mathematik in den letzten Jahrhunderten ' (Tubingen, 1869, republished by P. du Bois-Reymond, 1884). Speaking of the age of Leibniz he says : “Though on the Continent mathematicians were not so conservative as in England, where a purely geometrical exposition was considered to be the only one worthy of mathematics, yet the whole spirit of that age was directed to the solution of problems in geometrical clothing, and the result of the calculus had mostly to be retranslated into geometrical forms. It is the inestimable merit of the great mathematician of Basel, Leonhard Euler, to have freed the analytical calculus from all geometrical fetters, and thus to have established analysis as an independent science. Analysis places at its entrance the conception of a function, in order to express the mutual dependence of two variable quantities. . . . The abstract theory of functions is the higher analysis. . . . The conception of a function has been slowly and hesitatingly evolved out of special and subordinate conceptions. It was Euler who first established it, making it the foundation of the entire analysis, and hereby he inaugurated a new period in mathematics " (p. 12, &c. ).

Video from MIT OpenCourseWare. Creative Commons Licence. Prof. David Jerison's 18.01 Single Variable Calculus, Video Lecture 1.

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.

Meez, John. A History of European Thought in the Nineteenth Century. London: Adamant Media Corporation, 2004.

Text available online at


image from
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.287a.

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