Encyclopaedia Britannica > Volume 15, PLA-RAM

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Porifm. therefore Fermat’s idea of porifms, founded upon this
circumftance, could not fail to be imperfeft.
To confirm the truth of the preceding theory, it may
be added, that profelfor Dr Stewart, in a paper read
a confiderable time ago before the Philofophical Society
of Edinburgh, defines a porifm to be “ A propofition
affirming the poffibility of finding one or more condi¬
tions of an indeterminate theorem;” where, by an in¬
determinate theorem, he meant one which expreiTes a re¬
lation between certain quantities that are determinate
and certain others that are indeterminate ; a definition
which evidently agrees with the explanations which have
been here given.
If the idea which we have given of thefe propofitions
be juft, it follows, that they are to be difeovered by
confidering thofe cafes in which the conftruftion of a
problem fails, in confequence of the lines which by
their interfeftion, or. the points which by their pofi-
tion, were to determine the problem required, happen¬
ing to coincide with one another. A porifm may
therefore be deduced from the problem to which it be-
longs, juft as propofitions concerning the maxima and
minima of quantities are deduced from the problems of
which they form limitations; and fuch is the moft natu¬
ral and obvious analyfis of which this clafs of propofi-
tions admits.
The following porifm is the firft of Euclid’s, and the
firft alfo which was reftored. It is given here to exem¬
plify the advantage which, in inveftigations of this kind,
may be derived from employing the laiv of continuity
in its utmoft extent, and purfuing porifms to thole ex¬
treme cafes where the indeterminate magnitudes increafe
ad infinitum.
muft pafs through the point to be found M ; for if not,
it may be demonftrated juft as above, that AE" does
not pafs through H, contrary to the fuppofition. 1 he
point to be found is therefore in the line E'B, which is
given in pofition. Now if from E there be drawn EP
parallel to AE', and ES parallel to BE', BS : SE :: BL
SExBL ^ a P • PF • • A F ?
Purine.
LN='
BS
therefore FG : LN :
, and AP: PE :: AF : FG=-
PEXAF.SEXBL
AP~ : BS
PExAF
XBS : SExBLxAP; wherefore the ratio of FG to
LN is compounded of the ratios of AF to BL, PE to
ES, and BS to AP ; but PE : SE :: AE': BE', and
BS :AP: : DB : DA for DB : BS : : DE': E'E ::
DA : AP; therefore the ratio of FG to LN is com*
pounded of the ratios of AF to BL, AE' to BE', and
DB to DA. In like manner, becaufe E" if a point in
the line DE and AE", BE" are inflefted to it, the
ratio of FH to LM is compounded of the fame ratios
of AF to BL, AE' to BE', and DB to DA ; there¬
fore FH : LM : : FG: NL (and confequently) : : PIG
: MN; but the ratio of HG to MN is given, being the
fame as that of a to /3; the ratio of P H to LM is
therefore alfo given, and FH being given, LM is given
in magnitude. Now LM is parallel to BE', a line
given in pofition ; therefore M is in a line QM, parallel
to AB, and given in poiition ; therefore the point M,
and alfo the line KLM, drawn through it parallel to
BE', are.given in pofition, which were to be found.
Plate
Hence this conftnnftion : From A draw AE' parallel
to FK, fa as to meet DE in E'; join BE', and take in
it BQ^ fo that « : 6 : : HF : BQj-and through Qdraw
Thu'poiifm may be confidcred as having occurred in QM parallel to AB. Let HA be drawn, and produ-
the folutuon of the followmg problem: Two points A, B, tdl « meet DE m E , and draw BE', meeting QN*
(fig. 4.) and alfo three ftraight lines DE, FK, KL, be-
CCCCXLH ing given in pofition, together with two points H and M
in two of thefe lines, to infleft from A and B to a point
in-the third, two lines that ffiall cut off from KF and
KL two fegments, adjacent to the given, points PI and
M, having to one another the given ratio of to n.
Now, to find whether a porifm be connected with this
problem, fuppofe that there is, and that the following
propofition is true. Two points A and B, and two
ilraight lines DE, FK, being given in pofition, and
in M ; through M draw KML parallel to BE, then is
KML the line and M the point which were to be
found. There are two lines which will anfwer the con*
ditions of this porifm ; for if in QB, produced on the
other fide of B, there be taken B</=BQ_, and if qm
be drawn parallel to AB, cutting MB in m; and if m k
be drawn parallel to BQ^, the part m n, cut oft by EB
produced, will be-equal to MN, and have to HG the
ratio required. It is plain, that whatever be the ratio
of a to t1, and whatever be the magnitude of FH, if the
dfol po nt H in one of ihem,°a ifne LK may be found, other thing, given remain the fame, the lines found w.ii
, ^ at be all parallel to BE'. But if the ratio of « to d re-
and alfo a point in it M, both given in pofition, fuch
that AE and BE inflefted from the points A and B
to any point whatever of the line DE, fhafl cut off fiom
the other lines FK and LK fegments HG and MN
adjacent to the given points H and M, having to one
another the given ratio of a to f.
Firft, let AE, BE', be infleaed to the point E', fo
that AE' may be parallel to FK, then fhall E B be pa¬
rallel to KL, the line to be found ; for if it be not pa¬
rallel to KL, the point of their interfeftion muft be at
a finite diftance from the point M, and' therefore ma¬
king as /3 to a; fo this diftance to a fourthproportional,
the diftaneg from H at which AE' interfeas 1 K, will
be equal to that fourth proportional. But AE does
not interfea FK, for they are parallel by eonftruaion ;
therefore BE' cannot interfea KL, which is therefore
parallel to BE', a line given in pofition. Again, let
AE", BE", be infleaed to E", fo that AE' may pafs
through the given point H : then it is plain that BE"
be all parallel
main the fame likewife, and if only the point H van',
the pofition of KL will remain the fame, and tiie point
M will vary.
Another general remark which may be made on the
analyfis of porifms is, that it often happens, as in the
laft example, that the magnitudes required may all, or
a part of them, be found by confidering the extreme
cafes; but for the difeovery of the relation between
them, and the indefinite magnitudes, we muft have re-
courfe* to the hypothefis of the porifm in its moft gene¬
ral or indefinite form; and muft endeavour fo to con¬
duit the reafoning, that the indefinite magnitudes may
at length totally difappear, and leave a propofition af-
futing the relation between determinate magnitu es
only.
For this purpofe Dr Simfcn frequently employs two
ftatementa of the general hypothefis, which he comparts
together. As for iuitar.ee, in his analyfis of the laft po-
^ niin.