Colin Maclaurin's "A Treatise of Algebra in Three Parts" [1] contains:
I. The fundamental rules and operations,
II. The composition and resolution of equations of all degrees,
and the different affections of their roots,
III. The application of algebra and geometry to each other,
to which an appendix concerning the general properties of geometrical
lines is also added.
According to W.W. Rouse Ball, this treatise is "[a]mong Maclaurin's minor
works". Despite the fact that it is certainly influenced by Newton's
_Universal Arithmetic_, I would still deem it an important publication.
Ball goes on to remark:
"It contains the results of some early papers of Maclaurin; notably
of two, written in 1726 and 1729, on the number of imaginary roots
of an equation, suggested by Newton's theorem; and of one, written
in 1729, containing the well-known rule for finding equal roots by
means of the derived equation. In this book negative quantities are
treated as being not less real than positive quantities. To this
work a treatise, entitled _De linearum geometricarum proprietatibus
generalibus_, was added as an appendix; besides a paper of 1720
above alluded to, it contains some additional and elegant theorems.
Maclaurin also produced in 1728 an exposition of the Newtonian
philosophy, which is incorporated in the posthumous work printed
in 1748. Almost the last paper he wrote was one printed in the
_Philosophical Transactions_ for 1743 in which he discussed from a
mathematical point of view the form of a bee's cell."
And, as far as the query is concerned, I would like to recall also that
Maclaurin's _Algebra_ (Part I, "The fundamental rules and operations",
p. 86 ff) contains enough ingredients of the method today known as the
so-called "Cramer's rule". Obviously enough, Maclaurin does not dress
his formulas with 'Determinants' sauce!
I append below "Theorem I" (paragraph 86) so that listmembers may gather
precisely Colin's notation and style.
T H E O R E M I
§86. Suppose that two Equations are given, involving two
unknown Quantities, as,
/
| ax + by = c
<
| dx + ey = f
\
af - dc
then shall y = ---------
ae - db
Where the Numerator is the difference of the Products of
the opposite Coefficients in the Orders in which 'y' is not
found, and the Denominator is the Difference of the Products
of the opposite Coefficients taken from the Orders that
involve the two unknowns Quantities.
For, from the first Equation, it is plain that
c - by
ax = c - by .. and x = --------
a
f - ey
from the 2d, dx = f - ey .. and x = --------
d
c - by f - ey
therefore -------- = --------, and cd - dby = af - aey
a d
whence aey - dby = af - cd
af - cd
and y = ---------
ae - db
ce - bf
after the same Manner, x = --------- . [p. 82]
ae - db
In a similar fashion, Maclaurin deals with the system
/
( ax + by + cz = m
|
< dx + ey + fz = n
|
( gx + hy + kz = p
\
and with elementary algebra proves, for instance, that:
aep - ahn + dhm - dbp + gbn - gem
z = -----------------------------------
aek - ahf + dhc - dbk + gbf - gec
The Swiss mathematician Gabriel Cramer gave the general rule for n x n
systems in his celebrated treatise "Introduction a\ l'analyse des lignes
courbes alge/briques" [2], published in 1750. There he states how these
terms can be computed as products of certain coefficients in the equations,
and how the sign is determined. Cramer also explains how the n numerators
of the fractions can be found by replacing certain coefficients in this
calculation by constant terms of the system. This rule, without proof,
appears in the Appendix of [2], and arises out of the need to find the
equation of a plane curve passing through a number of given points. My
quick and *rough* translation of a few lines may give an idea of Cramer's
wording:
"...
The examination of these formulas provides this general rule.
Let be n equations & n unknowns. One will find the value of each
unknown by forming n fractions of which the common denominator has
as many terms as there are permutations of n different things.
Each term is composed of letters ZYXV &c. always written in the
same order, but to which one distributes, as exponents, the first
n numbers arranged in all possible ways. Thus, when one has three
unknowns, the denominator has [1 x 2 x 3] 6 terms, composed of the
three letters ZYX, which receive successively exponents 123, 132,
213, 231, 312, 321. One adds to these terms the signs + or -,
according to the following rule. When an exponent in the same term
is followed, mediately or immediately, by an exponent smaller I
will called that a disturbance.
..."
For the benefit of many readers who may not have an easy access to
Cramer's masterpiece, I have also taken the 'pains' to waddle to our
classic's library. I have typed the passage of Cramer's Appendix which
is relevant here, converted it to a PDF file, and uploaded it to the Net.
The URL is:
http://members.tripod.com/~Historia_Matematica/Cramer.pdf
Surely, Seki Kowa (1683) and Gottfried W. Leibniz (for instance, a letter
to l'Hospital on April 28, 1693) should be mentioned in connection with
the first traces (proto-history) of Cramer's rule.
W.W. Rouse Ball finally remarked upon Maclaurin's influence as follows:
Maclaurin was one of the most able mathematicians of the 18th
century, but his influence on the progress of British mathematics
was on the whole unfortunate. By himself abandoning the use both
of analysis and of the infinitesimal calculus, he induced Newton's
countrymen to confine themselves to Newton's methods, and it was
not until about 1820, when the differential calculus was introduced
into the Cambridge curriculum, that English mathematicians made any
general use of the more powerful methods of modern analysis.
Although this passage has been quoted and approved by prestigious
historians, I am wondering what others on this forum think about Ball's
remark -- Judith?, Piers?, V. Frederick?, Victor?, Christopher? ...
La gracia esta si *todos* aportamos un granito de arena, don't you think so?
[1] Maclaurin, Colin (1698-1746):
_A treatise of Algebra_, London: Printed for A. Millar & J. Nourse, 1748.
[2] Cramer, Gabriel (1704-1752):
"Introduction a\ l'analyse des lignes courbes alge/briques", Gene\ve:
Fre\res Cramer, 1750.
Greetings from sunny Montevideo,
Julio Gonzalez Cabillon
On Tue, 18 Nov 1998, Joao Bosco Pitombeira wrote:
| Gostaria de receber informacoes sobre a historia da "regra de Cramer"
|
| Grato
|
| Joao Bosco Pitombeira
| jbpit@saci.mat.puc-rio.br
| Departamento de matematica
| PUC-Rio
| Rio de Janeiro
| Brasil