**History and ****Hermeneutics**** for Mathematics Education**

**Storia
ed Ermeneutica per la Didattica della Matematica**

**A treatise by Carmichael (1855)**

**Un trattato di Carmichael (1855)**

Carmichael, R. (1855), *A Treatise on the Calculus of Operations*,
Longman, Brown, Green and Longmans,

**Contents**

Chapter I. Introduction
(pp. 1-7)

Chapter II. Elementary Principles (pp. 8-13)

Chapter III. Application to the Integration of Linear Total Differential
Equations

Section II. Application
of preceding Theorems (pp. 21-28)

Chapter IV. Application to the Integration of Linear Partial Differential
Equations

Section II. Application
of preceding Theorems (pp. 34-48)

Chapter V. Integration of various additional Classes of Differential
Equations, Total and Partial (pp. 49-66)

Chapter VI.

Section II. Evaluation
and Extension of Definite Integrals (pp. 88-91)

Chapter VII. Interpretation of Symbols of Operation (pp. 92-102)

Chapter VIII. Application to Analytic Geometry (pp. 103-124)

Chapter IX. Micellaneous Applications in the Differential and Integral
Calculus (pp. 125-136)

Chapter X. Application to the Calculus of Finite Differences (pp. 137-152)

Appendix A. On the Calculus of Variations (pp. 153-160)

Appendix B. On the Quadrature of Surfaces, and the Rectification of Curves (pp.
161-167)

Appendix C. Additional Applications to Integration (pp. 168-170)

“**Introduction.**

The Calculus of Operations, in the
greatest extension of the phrase, may be regarded as that science which treats
of bthe combinations of symbols of operation, conformably to certain given
laws, and of the relations by which these symbols are connected with the
subjects on which they operate.

As the principal object of the present work
is to reduce and simplify the labours of the student, and as the great
practical utility of the Calculus of Operations, at present, arises from its
bearing upon the Differential and Integral Calculus, and the Calculus of Finite
Differences, the symbols of operation employed in illustration, and the laws by
which they are governed, and those belonging to the branches of analysis named,
and are consequently bfamiliar to the reader” (*A Treatise on the Calculus of Operations*, p. 1).

“**Elementary principles.**

Two symbols F and G are said to be *commutative*
when, *u* being the subject on which they operate,

FG
*u* = GF *u*

A symbol F is said to be *distributive*
when, *u* and *v* being two distinct subjects,

F
(*u*+*v*) = F *u* + F *v*

A symbol F is said to be *iterative*,
or to follow the law of indices, when

F* ^{m}
*F

It is to be observed that this third
formula is not to be regarded as a law of symbolic combination in the same
sense as the first, nor as a law of symbolic operation in the same sense as the
second. In fact, F^{m}*u* is rather to be regarded as a
mere abbreviated notation for the result of the operation F performed *m*
times successively upon *u*,

F
F F ... *u*

than as equivalent to the result of any *operation
raised to the power m*, performed on *u*.

The laws stated being the principal ones
which occour in the practical employment of the Calculus of Operations, we
shall for the present confine our attention to them. They may be called,
respectively,

II. the Law of Distribution

III. the Law of Indices

We may at once observe, that whatever
theorem is true for any one symbol which satisfies these laws is true for every
symbol which satisfies them.

Now the symbols of *numbers* satisfy
them; indeed all algebraical equations may be considered as having the same
subject, unity, and the constants, as denoting sums of operations performed on
unity. Again the symbols of *differentiation* satisfy those laws; for if *u*
be a function of the two independent variables *x* and *y*, it is
known that

D* _{x}*D

that, if *u* be a function of *x*
only,

D* _{x}*
(

and that

D_{x}^{m}* *D_{x}^{n}*u* = D_{x}^{m}^{+n} *u*

Hence we deduce the important
consequence, that every theorem in Algebra, which depends on those laws, has an
analogue in the Differential Calculus” (*A
Treatise on the Calculus of Operations*, pp. 8-9).

See moreover:

Si veda inoltre:

L’Hospital, G. de (1716), *Analyse
des infiniment petits*, Papillon, Paris (II ed.).

Newton, I. (1740), *Le methode des fluxions et
des suites infinites*, Debure, Paris (I ed.: 1736).

Riccati,
V. (1752), *De
usu motus tractorii in constructione Aequationum Differentialium Commentarius*,
Lelio della Volpe, Bologna.

Paulini
a S. Josepho (P. Chelucci) (1755), *Institutiones analyticæ earumque usus in
Geometria*, Gessari, Napoli.

Euler,
L. (1787) *Institutiones
Calculi Differentialis cum eius usu in Analysi Finitorum ac Doctrina Serierum*,
I, II, Galeati, Pavia (II ed.; I ed.: 1755).

Euler,
L. (1796), *Introduction
a l’Analyse Infinitésimale*, I, II, Barrois, Paris (I ed. in French).

Brunacci,
V. (1804), *Corso
di Matematica sublime*, I, II, Allegrini, Firenze.

Lagrange, J.L. (1813), *Théorie
des fonctions analytiques*, Courcier, Paris.

Cauchy, A.L. (1836), *Vorlesungen uber die Differenzialrechung*,
Meyer, Braunschweig.

Lacroix, S.F. (1837), *Traité
elementaire du Calcul Différentiel et du Calcul Intégral*, Bachelier, Paris
(V ed.).

De Morgan, A. (1842), *The
differential and integral Calculus*, Baldwin and Cradock,

Sturm, Ch. (1868), *Cours d’Analyse*, I, II,
Gauthier-Villars, Paris.

Carnot, L.N.M. (1881), *Réflections
sur la métaphisique du Calcul Infinitésimal*, Gauthier-Villars, Paris (I
ed.: 1797).

Laurent, H. (1885-1887-1888), *Traité
d’Analyse*, I, II, III, Gauthier-Villars, Paris.

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**History and ****Hermeneutics**** for Mathematics Education**

**(Giorgio T. Bagni, Editor)**

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