Opening the text

Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8Various

History

Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8

Name: Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8
Author: Various

Various

Encyclopedias and dictionaries

For purely geometrical questions the only general method available is
practically that employed by Fermat. If a quantity depends on the
position of some point P on a curve, and if its value is equal at two
neighbouring points P and P´, then at some position between P and P´ it
attains a maximum or minimum, and this position may be found by making P
and P´ approach each other indefinitely. Take for instance the problem
of Regiomontanus "to find a point on a given straight line which
subtends a maximum angle at two given points A and B." Let P and P´ be
two near points on the given straight line such that the angles APB and
AP´B are equal. Then ABPP´ lie on a circle. By making P and P´ approach
each other we see that for a maximum or minimum value of the angle APB,
P is a point in which a circle drawn through AB touches the given
straight line. There are two such points, and unless the given straight
line is at right angles to AB the two angles obtained are not the same.
It is easily seen that both angles are maxima, one for points on the
given straight line on one side of its intersection with AB, the other
for points on the other side. For further examples of this method
together with most other geometrical problems on maxima and minima of
any interest or importance the reader may consult such a book as J. W.
Russell's _A Sequel lo Elementary Geometry_ (Oxford, 1907).

  The method of the differential calculus is theoretically very simple.
  Let u be a function of several variables x1, x2, x3 ... x_n, supposed
  for the present independent; if u is a maximum or minimum for the set
  of values x1, x2, x3, ... x_n, and u becomes u + [delta]u, when x1,
  x2, x3 ... x_n receive small increments [delta]x1, [delta]x2, ...
  [delta]x_n; then [delta]u must have the same sign for all possible
  values of [delta]x1, [delta]2 ... [delta]x_n.

  Now
                                            _                                                                   _
                __ [delta]u                |   __ [delta]²u       __      [delta]³u                              |
    [delta]u = \   --------- [delta]x1 + ½ |  \   ---------- + 2 \   ------------------- [delta]x1 [delta]x2 ... | + ...
               /__ [delta]x1               |_ /__ [delta]x1²     /__ [delta]x1 [delta]x2                        _|

Public-domain text, read in full here on John Shaqi.

Opening the text

Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8Various

History

Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8

Various

Encyclopedias and dictionaries

For purely geometrical questions the only general method available is
practically that employed by Fermat. If a quantity depends on the
position of some point P on a curve, and if its value is equal at two
neighbouring points P and P´, then at some position between P and P´ it
attains a maximum or minimum, and this position may be found by making P
and P´ approach each other indefinitely. Take for instance the problem
of Regiomontanus "to find a point on a given straight line which
subtends a maximum angle at two given points A and B." Let P and P´ be
two near points on the given straight line such that the angles APB and
AP´B are equal. Then ABPP´ lie on a circle. By making P and P´ approach
each other we see that for a maximum or minimum value of the angle APB,
P is a point in which a circle drawn through AB touches the given
straight line. There are two such points, and unless the given straight
line is at right angles to AB the two angles obtained are not the same.
It is easily seen that both angles are maxima, one for points on the
given straight line on one side of its intersection with AB, the other
for points on the other side. For further examples of this method
together with most other geometrical problems on maxima and minima of
any interest or importance the reader may consult such a book as J. W.
Russell's _A Sequel lo Elementary Geometry_ (Oxford, 1907).

  The method of the differential calculus is theoretically very simple.
  Let u be a function of several variables x1, x2, x3 ... x_n, supposed
  for the present independent; if u is a maximum or minimum for the set
  of values x1, x2, x3, ... x_n, and u becomes u + [delta]u, when x1,
  x2, x3 ... x_n receive small increments [delta]x1, [delta]x2, ...
  [delta]x_n; then [delta]u must have the same sign for all possible
  values of [delta]x1, [delta]2 ... [delta]x_n.

  Now
                                            _                                                                   _
                __ [delta]u                |   __ [delta]²u       __      [delta]³u                              |
    [delta]u = \   --------- [delta]x1 + ½ |  \   ---------- + 2 \   ------------------- [delta]x1 [delta]x2 ... | + ...
               /__ [delta]x1               |_ /__ [delta]x1²     /__ [delta]x1 [delta]x2                        _|

Public-domain text, read in full here on John Shaqi.

Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8

Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg": Volume 17, Slice 8

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