E212 -- Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum
(Foundations of Differential Calculus, with Applications to Finite Analysis and Series)
Summary:
This is a volume on differential calculus. It is divided into two books, one of which has 9 chapters and other 18. The Euler Archive has provided a 1787 printing of the Latin text (Ticino: Petri Galeatii), shown below.
- Front Matter
- Part I
- Chapter 1: De differentiis
finitis.
- Chapter 2: De usu
differentiarum in doctrina serierum.
- Chapter 3: De infinitis
atque infinite parvis.
- Chapter 4: De
differentialium cujusque ordinis natura.
- Chapter 5: De
differentiatione functionum algebraicarum unicam variabilem involventium.
- Chapter 6: De
differentiatione functionum transcendentium.
- Chapter 7: De
differentiatione functionum duas pluresve variabiles involventium.
- Chapter 8: De formularum
differentialium ulteriori differentiatione.
- Chapter 9: De
aequationibus differentialibus.
- Part II
- Chapter 1: De
transformatione serierum.
- Chapter 2: De
investigatione serierum summabilium.
- Chapter 3: De inventione
differentiarum finitarum.
- Chapter 4: De conversione
functionum in series.
- Chapter 5: Investigatio
summae serierum ex termino generali.
- Chapter 6: De summatione
progressionum per series infinitas.
- Chapter 7: Methodus
summandi superior ulterius promota.
- Chapter 8: De usu calculi
differentialis in formandis seriebus.
- Chapter 9: De usu calculi
differentialis in aequationibus resolvendis.
- Chapter 10: De maximis et
minimis.
- Chapter 11: De maximis et
minimis functionum multiformium pluresque variabiles complectentium.
- Chapter 12: De usu
differentialium in investigandis radicibus realibus aequationum.
- Chapter 13: De criteriis
radicum imaginariarum.
- Chapter 14: De
differentialibus functionum in certis tantum casibus.
- Chapter 15: De valoribus
functionum, qui certis casibus videntur indeterminati.
- Chapter 16: De
differentiatione functionum inexplicabilium.
- Chapter 17: De
interpolatione serierum.
- Chapter 18: De usu
calculi differentialis in resolutione fractionum.
- Supplementary Material
Publication:
- Originally published as a book in 1755
- Opera Omnia: Series 1, Volume 10
Documents Available:
- Book I was translated into English by John Blanton, and published by Springer in 2000. Selections are available via Google Books: E212.
- Much of Chapters 5 and 6 of Book II have been translated into English by David Pengelley, and are available, along with many other excellent resources, at his Teaching with Original Historical Sources in Mathematics web page in pdf and dvi formats.
- E212 is discussed in Ed Sandifer's How Euler Did It September 2005 column published online by the MAA.
- Alexander Aycock (aaycock at students dot uni dash mainz dot de) is in the process of translating Book II into English. Drafts of his work are provided here; please send your comments or feedback to him at the email address indicated above.
- Chapter 1: On the transformation of series
- Chapter 2: On the investigation of summable series
- Chapter 3: On finding finite differences
- Chapter 4: On the conversion of functions into series
- Chapter 5: Investigation of the sum of series from the general term
- Chapter 6: On the summation of progressions by means of infinite series
- Chapter 7: The superior method of summing further promoted
- Chapter 8: On the Use of differential calculus in the formation of series
- Chapter 9: On the use of differential calculus in the resolution of equations
- Chapter 10: On maxima and minima
- Chapter 11: On maxima and minima of multiform functions and such containing several variables
- Chapter 12: On the use of differentials in the investigation of the real roots of equations
- Chapter 13: On criteria for imaginary roots
- Chapter 14: On differentials of functions in only certain cases
- Chapter 15: On the values of functions which in certain cases seem to be undetermined
- Chapter 16: On the differentiation of inexplicable functions
- Chapter 17: On the interpolation of series
- Chapter 18: On the use of differential calculus in the resolution of fractions
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E212 include:
- Ferraro G., “Analytical symbols and geometrical figures in eighteenth-century calculus.” Studies in History and Philosophy of Science, 32A (3), 535-555 (Sep 2001).
- Ferraro G., “Differentials and differential coefficients in the Eulerian foundations of the calculus.” Historia Mathematica, 31 (1), pp. 34-61 (Feb 2004).
- Gould HW., “Euler formula for nth differences of powers.” American Mathematical Monthly, 85 (6), pp. 450-467 (1978).
- Grabiner JV., “The changing concept of change - the derivative from Fermat to Weierstrass.” Mathematics Magazine, 56 (4), pp. 195-206 (1983).
- Ruthing D., “Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N..” Mathematical Intellingencer, 6 (4), pp. 72-77 (1984).
- Volkert K., “History of pathological functions - on the origins of mathematical methodology.” Archive for History of Exact Sciences, 37 (3), pp. 193-232 (1987).
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