Lagrange J. L. Лекции по элементарной математике (Чикаго, 1898) М. , Lagrange J.-L. Lectures on elementary mathematics (Chicago, 1898)(T)(175s)_M

Lagrange J. L. Лекции по элементарной математике (Чикаго, 1898) М.

Lagrange J.-L. Lectures on elementary mathematics (Chicago, 1898)(T)(175s)_M_.djvu

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Date Jul 15, 2005

Cites: Then making,
for the sake of simplicity,
the two roots A and В of the reduced equation assume
the form
^T, B=f-gV-T...
Therefore, if the value of л'3 be increased by
insensible degrees from 8/ to infinity, the value of g*
will also augment by insensible and corresponding
degrees from zero to infinity...
Now, from the nature of a
circle the same chord с corresponds not only to the
arc s but (calling the entire circumference u) also to
the arcs
и — s, 2u-\-s, 3 и — s, ...
The last equa-
equation gives ab = —f—pfrom which I conclude that ab
a-\- b
also is necessarily a real quantity...
The following is another method of reaching the
same formulae, less direct than that which has already
been expounded to you, but which, on the other hand
has the advantage of being analogous to the method
of Cardan for equations of the third degree...
The preceding remark should be added to article 777 of Euler'i Algebra
and to article 37 of the author's Note XIII of the ТгаЦё de la resolution des
equations nuniiriques...
The more that value approaches to zero, the more
will the value of x which has produced it approach to
a root of the equation...
If, now, we consider the equation of the curve, it
is plain in the first place, that by making x=-0 we
shall have y~u; and consequently that the sign of
the ordinatej/ will be the same as that of the quantity
u, the last term of the proposed equation...
Consequently,
y—^7 will be a quantity smaller than the smallest
positive value of/; and in like manner we shall find
a quantity smaller than the smallest negative value
of /...
Similarly, let h be the greatest coefficient of the terms
having a contrary sign to the first term after x has
been changed into —x; and let m — ri be the expo-
exponent of x in the first term having a contrary sign to
the first term of the equation as thus altered...
The equation in у is reduced, by the vanishing of its
last term, to the (?n — 2)th degree,—being divisible
by je2...
Then taking the unknown
quantity for the abscissa x, and the function of the
unknown quantity, or the quantity compounded of
that quantity and the known quantities, which forms
one side of the equation, for the ordinatejc, the curve
described by these co-ordinates x and у will give by
its intersections with the axis those values of x which
are the required roots of the equation...
Hence, whatever be the value of the quantity A,
it is plain that the values of у will necessarily pass General
from positive to negative, both for x negative and for S0lutl0n-
x positive and greater than a...
We shall likewise find that if
b is positive and с is negative, the equation will have
two real roots, one greater and one less than —a...
Whence it follows that every point A taken upon the
straight line PA will in general give eight upon the
straight line PD, all of which must be separately and
successively considered to obtain all the possible so-
solutions...
We have al-
already illustrated their employment in resolving equa-
equations, and their consideration is always useful in the
approximate description of curves, for the reason that
a curve of this kind can always be made to pass
THE EMPLOYMENT OF CURVES...
But the last expression forjc (equation 2)
is preferable, partly because of the simplicity of the
I48 THE EMPLOYMENT OF CURVES...
But, apart from the intrinsic
improbability of this view which is at variance with
the truth that science is nearly always gradual and
organic in growth, modern historical researches have
traced the germs and beginnings of algebra to a much
remoter date, even in the line of European historical
continuity...