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A Primer on Early Calculus
L'Hospital

From the Preface of his 1696 volume of "Analyse des infiniment petits Pour l'intelligence des lignes courbes"  and its 1730 English translation:

"I must own my self very much obliged to the Labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groenengen, as having made free with their Discoveries as well as those of Mr. Leibnitz
(sic):  So that whatever they please to claim as their own, I frankly return them."

Guillaume François Antoine, Marquis de l'Hôpital
(1661 – 1704)

So what is l'Hospital's written explanation of what today we call his "Rule"?

He writes, "the infinitely small part by which a quantity is continually increased or diminished, called the Difference be considered as the assemblage of an infinity of straight lines, each infinitely small, or as a polygon having an infinite number of sides."

Today we denote the "Difference" of an x quantity by dx.  Clearly l'Hospital considers a segment of a line as part of the curve.  Thus, we encounter the concept of a curve, the concept of tangent, the concept of slope and the problem of what method to use when handling indeterminate forms, i.e., 0/0 and ∞/∞.

While we call his method l'Hospital's Rule, l'Hospital himself only used the word "rule" for his first five propositions.  Moreover, all of these propositions were based on
Leibniz's six brief pages published in Acta Eruditorum, (1684, pp. 467-473)The French word "regles" appears on the frontispiece (see the background below and to the right) and on five propositions, none of which treat the indeterminate form.  Expanding the usage of the word "regles", i.e., rule, appears to be a much later attachment to his name.  Yes, we in mathematics like to name problems, propositions, theorems, etc. with a person's name.  But seldom is this initiated by the individual himself.  After all, modesty and decorum are at stake.

So where does the indeterminant form first appear?  For clarity, we turn to the 1730 translation.
On page 191 of Section IX,  l'Hospital introduces the following:

and Fig. 130.

L'Hospital on Leibniz
L'Hospital on Leibniz:

"I must here in justice own, (as Mr. Leibnitz (sic) himself has done in Journal des Scavans for August, 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis,
as appears by his excellent Principia, published first in the year 1687 which almost wholly depends upon the use of the said Calculus.

But the method of Mr. Leibnitz's is much more easy and expeditious, on account of the notation he uses, not to mention the wonderful assistance it affords on many occasions."

Marquis de l'Hôspital
(1661 - 1704)

Johann van Waveren Hudde
(1628 - 1704)
 Starting with the study of tangents to curves, all of the men represented on this web page made significant contributions to the initial formulation of what today we call  The Calculus. See the Acta Eruditorum of 1697.

 Galileo Galilei (1564-1642) René Descartes (1596 - 1650) Pierre de Fermat (1601 - 1665) Blaise Pascal (1623 - 1662) Leibniz (1646 - 1716) Jacob Bernoulli (1654 - 1705) Editor of  Acta Eruditorum Leonhard Euler (1707 - 1783) Sir Isaac Newton (1642 - 1727)

We find at least two famous disputes in play when writing about this crucial chapter in the history of mathematics. First, there is the legendary dispute over the priorty between Newton and Leibniz. Who grasped the basic concepts of calculus first? To this controversy readers should also know Johann Bernoulli became outraged when l'Hospital published Analyse with his name as the sole author. Bernoulli had tutored l'Hospital over several months both in Paris and in Basel on this topic. Thus Bernoulli felt his ideas, particularly the explanations, were stolen.

Interestingly, viewers of this web page can see the cover page (frontispiece) in the background.  No name appears!  Moreover, no name appears in print throughout the entire volume.  Students always ask about the correct spelling of the name.  We prefer the usage of the 1730 translation - l'Hospital.  (Use an "L" at the beginning of a sentence.)

No matter what one's opinions on priority and credit for scientific work, scholars agree that the Analyse is the first textbook on differential calculus. Newton's work was in notebooks or published late. Leibniz's six pages of propositions were too terse to be understood without great effort on the part of the most advanced mathematicians. Thus, l'Hospital's Analyse  with good examples spread over 181 pages of explanations was the crucial elementary introduction for the calculus so needed to advance mathematics.  Moreover, in the preface he gave credit for prior work on l'intelligence des lignes courbes  - curved lines, the genesis on our National Curve Bank Project: A MATH Archive - to no less than the following mathematicians:

Archimedes, Viète, Descartes, Pappus, Pascal, Barrow, Leibniz, Newton, two Bernoullis, Craig, Huygens, Tschirnhausen and Hudde.
Reading the original source, the lasting impression is l'Hospital wrote a great book and was more than willing to express his debt to others.
Shirley B. Gray, December 29, 2009
The Analyse des infiniment (1696) images are reproduced with permission of The Huntington Library, San Marino, CA.  Students of mathematics are most grateful for the opportunity to view the original  sources.

The English translation (1730) of Analyse des infiniment was prepared by E. Stone and printed by William Innys.