Encyclopaedia Britannica > Volume 15, PLA-RAM

P O R
[ 399 ]
P O R
Porifm. pofition, and alfo a point R, to find a point D in one of
—v the eiven lines, fo that DE and DF being drawn per¬
pendicular to BC, AC, and DR, joined ; DE -f-DF
may have to DR a given ratio. It is plain, that ha-
vincr found G, the problem would be nothing more than
to find D, fuch that the ratio of GD2 to DR% and
therefore that of GD to D R, might be given, the point
D being in the circumference of a given circle, as is
well known to geometers.
The fame porifm alfo afiifts in the folution of
another problem. For if it were required to find D
fuch that DE2 + DF2 might be a given fpace ; having
found G, DG7 would have to DE2 + DF2 a given ra¬
tio, and DG would therefore be given ; whence the fo¬
lution is obvious. _ .
The connection of this porifm with the impoihble
cafe of the problem is evident; the point D being that
from which, if perpendiculars be drawn to AC and CB,
the fum of their fquares is the. leaft pofiible. for iince
DFH M1 : DGZ :: LOz + LM= : LG"‘; and finee
LG is lefs than DG, LOJ+LMl muft be lefs than
LF1 -f DE\ It is evident from what has now appear¬
ed, that in fome inftances at leaft there is a clofe con-
neftion between thefe propofitions and the maxima or
minimat and of confequence the impofiible cafes of pro¬
blems. The nature of this connexion requires to be
farther inveftigated, and is the more interefting becaufe
the tranfition from the indefinite to the impollible cafe
feems to be made with wonderful rapidity. I hus in
the firft propolition, though there be not properly
fpeaking an impoffible cafe, but only one where the
point to be found goes off ad injimtumr it may be re¬
marked, that if the given point F be anywhere out of
the line HD (fig. i.), the problem of drawing GB
equal to GF is always pofiible, and admits of juft one
folution ; but if F be in DH, the problem admits of
no folution at all, the point being then at an infinite
diftance, and therefore impofiible to be afligned. There
is, however, this exception, that if the given point be
at K in this fame line, DH is determined by making
DK equal t© DL. Then every point in the line DE
gives a folution, and may be taken for the point G.
Here therefore the cafe of numberlefs folutions, and of
no folution at all, are as it were conterminal, and fo clofe
to one another, that if the given point be at K the
problem is indefinite; but if it remove ever fo little from
K, remaining at the fame time in the line DH, the
problem cannot be refolved. I his affinity might ha\e
been determined a prion : for it is, as we have feen, a
general principle, that a problem is converted into a po¬
rifm when one or when two of the conditions of it ne-
ceflarily involve in them fome one of the reft. Sup-
pofe, then, that two of the conditions are exa&ly in
that ft ate which determines the third; then while they
remain fixed or given, fhould that third one vary or
differ ever fo little from the ftate required by the other
two, a contradiftion will enfue: therefore if, in the hy-
pothefis of a problem, the conditions be fo related to one
another as to render it indeterminate, a porifm is pro¬
duced ; but if, of the conditions thus related to one ano¬
ther, fame one be fuppofed to vary, while the others con¬
tinue the fame, an abfurdity follows, and the problem
becomes impoffible. Wherever, therefore, any problem
admits both of an indeterminate and an impoffible cafe,
it is certain, that thele cafes are nearly related to one
Plate
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another, and that fome of the conditions by which they
are produced are common to both.” It is fuppofed -
above, that two of the conditions of a problem involve
in them a third; and wherever that happens, the con-
clufion which has been deduced will invariably take
place. But a porifm may in fome cafes be fo fimple as
to arife from the mere coincidence of one condition
with another, though in no cafe whatever any incon-
fiftency can take place between them. There are,
however, comparatively few porifms fo fimple in their
origin, or that arife from problems where the conditions
are but little complicated ; for it ufually happens that
a problem which can become indefinite may alfo become
impofiible; and if fo, the conne&ion already explained
never fails to take place.
Another fpecies of impofiibility may frequently arife
from the porifmatic cafe of a problem which will affedl
in fome meafure the application of geometry to aftrono-
my, or any of the fciences depending on experiment or
obfervation. For when a problem is to be refolvcd by
help of data furnifiied by experiment or obfervation,
the firft thing to be confidered is, whether the data fo
obtained be fufficient for determining the thing fought;
and in this a veiy erroneous judgment may be formed,
if we reft fatisfied with a general view of the fubjeft; for
tho’ the problem may in general be refolved from the data
with which we are provided, yet thefe data may be fo
related to one another in the cafe under confideration,
that the problem will become indeterminate, and inftead
of one folution will admit of an indefinite number. This
we have abeady found to be the cafe in the foregoing pro¬
pofitions. Such cafes may not indeed occur in any of the
practical applications of geometry ; but there is one o£
the fame kind which has actually occurred in aftronomy.
Sir Ifaac Newton, in his Principia, has confidered a
fmall part of the orbit of a comet as a ftraight line de-
feribed with an uniform motion. From this hypothefis,
by means of four obfervations made at proper intervals
of time, the determination of the path of the comet is
reduced to this geometrical problem : Four ftraight
lines being given in pofition, it is required, to draw a
fifth line acrofs them, fo as to be cut by them into
three parts, having given ratios to one another. Now
this problem hud been conftrudted by Dr Wallis and
Sir Chriftopher Wren, and alfo in three different ways
by Sir Ifaac himfelf in different parts of his works; yet
none of thefe geometers obferved that there was a par¬
ticular fituation of the lines in which the problem ad¬
mitted of innumerable folutions : and this happens to
be the very cafe in which the problem is applicable to
the determination of the comet’s path, as was firft dil-
covered by the Abbe Bofcovieh, who was led to it
by finding, that in this way he could never deter¬
mine the path of a comet with any degree of cer¬
tainty.
Befides the geometrical there is alfo an algebraical
analyfis belonging to porifms ; which, however, does not
belong to this place, becaufe we give this account of
them merely as an article of ancient geometry; and the
ancients never employed algebra in their inveftigations.
Mr Playfair, profeffor of mathematics in the univerfity
of Edinburgh, has written a paper on the origin and.
geometrical inveftigation of potifms, which is publifhed
in the third volume of the Tranfailions of the Royal
Society of Edinburgh, from which this account ot the
5 £ubje&
Porifm.