Page images

Fig. 1

[blocks in formation]
[merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][graphic][subsumed][subsumed][subsumed][subsumed][merged small]


Poriam. tio, the point F is in the circumference of a given circle, problem, suppose that there is, and that the following Porism,

we bave a locus. But when conversely it is said, if a proposition is true. Two points A and B, and two
circle ABC, of which the centre is o, be given by po- straight lines DE, FK, being given in position, and
sition, as also a point E; and if D be taken in the line also a point H in one of them, a line LK

may be found,
EO, 60 that EO XOD=AOʻ; and if from E and D and also a point in it M, both given in position, such
the licea EF, DF be inflected to any point of the cir. that AE and BE inflected from the points A and B
comference ABC, the ratio of EF to DF will be given, to any point whatever of the line DE, shall cut off from
viz. the same with that of EA to AD, we have a local the other lines FK and LK segments HG and MN ad-

jacent to the given points H and M, having to one an-
Lastly, when it is said, if a circle ABC be given by other the given ratio of a to B.
position, and also a point E, a point D may be found First, let AE', BE', be inflected to the point E', so
such that if EF, FD be inflected from E and D to any that AE' may be parallel to FK, then shall E'B be pa-
point F in the circumference ABC, these lines shall rallel to KL, the line to be found; for if it be not pa-
have a given ratio to one another, the proposition be- rallel to KL, the point of their intersection must be at
comes a porism, and is the same that has just now been a finite distance from the point M, and therefore mak-

ing as ß to , so this distance to a fourth proportional,
Hence it is evident, that the local theorem is changed the distance from H at wbich AE' intersects FK, will
into a porism, by leaving out what relates to the deter be equal to that fourth proportional. But AE' does
mination of D, and of the given ratio. But though all not intersect FK, for they are parallel by construction;
propositions formed in this way from the conversion of therefore BE' cannot intersect KL, which is therefore
Joci, are porisms, yet all porisms are not formed from parallel to BE',

parallel to BE', a line given in position. Again, let the conversion of loci; the first, for instance of the pre AE", BE", be inflected to E“, so that AE" may pass ceding cannot by conversion be changed into a locus ; through the given point H: then it is plain that BE" therefore Fermat's idea of porisms, founded upon this must pass through the point to be found M; for if not, circumstance, could not fail to be imperfect.

it may be demonstrated just as above, that AE“ does To confirm the truth of the preceding theory, it may not pass through H, contrary to the supposition. The be added, that Professor Dugald Stewart, in a paper read point to be found is therefore in the line E"B, which is a considerable time ago before the Philosophical Society given in position. Now if from E there be drawn EP of Edinburgh, defines a porism to be “A proposition parallel to AE', and ES parallel to BE', BS : SE :: BL affirming the possibility of finding one or more condi


PE=AF tiong of an indeterminate theorem ;" where, by an in- :LN=

and AP: PE :: AF:FG=

BS determinate theorem, he means one which expresses a

PEX AF SEX BL relation between certain quantities that are determinate therefore FG: LN ::

and certain others that are indeterminate ; a definition

which evidently agrees with the explanation which has XBS: SEX BLXAP; wherefore the ratio of FG to
been here given:

LN is compounded of the ratios of AF to BL, PE to
If the idea which we have given of these propositions Es, and BS to AP; but PE : SE :: AE': BE', and
be just, it follows, that they are to be discovered by BS : AP :: DB : DA, for DB : BS :: DE': E'E ::
considering those cases in which the construction of a DA : AP; therefore the ratio of FG to LN is com-
problein fails, in consequence of the lines which by pounded of the ratios of AF to BL, AE' to BE', and
their intersection, or the points wbich by their posi DB to DA. In like manner, because E" is a point ja
tion, were to determine the problem required, happen- the line DE and AF", BE" are inflected to it, thu
ing to coincide with one another. A porism may there ratio of FH to LM is compounded of the same ratios
fore be deduced from the problem to which it belongs, of AF to BL, AE to BE', and DB to DA ; there-
just as propositions concerning the maxima and minima fore FH : LM:: FG: NL (and consequently) :: HG
of quantities are deduced from the problems of which : MN; but the ratio of HG to MN is given, being by
they form limitations; and such is the most natural and supposition the same as that of « to B; the ratio of FH
ohvious analysis of which this class of propositions ad- to LM is therefore also given, and FH being given,

LM is given in magnitude. Now LM is parallel to The following porism is the first of Euclid's, and the BE', a line given in position; therefore M is in a line first also which was restored. It is given here to ex QM, parallel to AB, and given in position; therefore emplify the advantage which, in investigations of this the point M, and also the line KLM, drawn through it kind, may be derived from employing the law of conti parallel to BE', are given in position, which were to be nuity in its utmost extent, and pursuing porisms to found. Hence this construction : From A draw AE those extreme cases where the indeterminate magnitudes parallel to FK, so as to meet DE in E'; join BE', and increase ad infinitum.

take in it BQ, so that a: B :: HF : BQ, and through This porism may be considered as having occurred in Q draw QM parallel to AB. Let HA be drawn, and

the solution of the following problem: Two points A, B, produced till it meet DE in E", and draw BE“, meetig.4. (fig. 4.) and also three straight lines DE, FK, KL, being QM in M; through M draw KML parallel to

ing given in position, together with two points H and M BË', then is KML the line and M the point which
in two of these lines, to inflect from A and B to a point were to be found. There are two lines which will an-
in the third line, two lines that shall cut off from KF swer the conditions of this porism; for if in QB, produ-
and KL two segments, adjacent to the given points H ced on the other side of B, there be taken B g=BQ,
and M, having to one another the given ratio of « to B. and if q m be drawn parallel to AB, cutting MB in m;
Now, to find whether a porism be connected with this and if ma be drawn parallel to BQ, the part m n, cut
VOL. XVII. Part I.






· LB: +

Porism. off by EB produced, will be equal to MN, and bave the first, second, and tifth of these propositions, it is ma- Poriss

to HG the ratio required. It is plain, that whatever nifest that
be the ratio of a to B, and whatever be the magnitude LB LA AB
of FH, if the other things given remain the same, the + CL·LB'+

lines found will be all parallel to BE. But if the ratio
of a to B remain the same likewise, and if only the point Again, because
H vary, the position of KL will remain the same, and CL:LA ::(LB : LE::DB : DG::)DB:: DBX DG,
the point M will vary.

Another general remark which may be made on the

therefore DB X DG=ADB'.
analysis of porisms is, that it often happens, as in the

CL last example, that the magnitudes required may all, or

And because a part of them, be found by considering the extreme CL: LB::(LA: LE :: DA: DH::)DA': DAXDH, cases; but for the discovery of the relation between

LB them, and the indefinite magnitudes, we must have re therefore DA


-DA'. From the result of

course to the hypothesis of the porism in its most gene-
ral or indefinite form ; and must endeavour so to con these two last propositions we have
duct the reasoning, that the indefinite magnitudes may LB

at length totally disappear, and leave a proposition as.


serting the relation between determinate magnitudes

but DAXDII= twice trian. ADH, and DBX DG= For this purpose Dr Simson frequently employs two twice trian. BDG, and therefore DAX DH+DBX statements of the general hypothesis

, which he compares DG=2 (trian. A DH+trian. BDG)= 2 (trian. AEB together. As for instance, in his analysis of the last po. +trian. AEG)=AB XLE+EKXHG. Now it has he only in

LB but also another point o, anywhere in the same line, been proved, that D.1XDII+DBX DG=DA'

CL to both of which he supposes lines to be inflected from


LB the points A, B. This double statement, however,

AB -= cannot be made without rendering the investigation long

CL and complicated; nor is it even necessary, for it may be


avoided by having recourse to simple porisms, or to loci, "LA'+

--AD +

LD’, therefore

or to propositions of the data. The following porism is


given as an example where this is done with some difii-


-BD’culty, but with considerable advantage both with re CL


CL gard to the simplicity and shortness of the demonstration.

was to be demonstrated. It will be proper to premise the following lemma. Let

Por ism. Let there be three straight lines AB, AC, AB (fig. 7.) be a straight line, and D, L any two points CB given in position (fig. 5.); and from any point in it, one of which D is between A and B; also let

whatever in one of them, a: D, let perpendiculars be CL be any straight line. Then shall

drawn to the other twn, as DF, DE, a point G may be LB LA LB LA


found, such, that if GD be dirawn from it to the point


the square of that line shall have a given ratio to the

sum of the squares of the perpendiculars DF and DE, For place CL perpendicular to AB, and through the which ratio is to be found. points A, C, B describe a circle, and let CL meet the Draw AH, BK perpendicular to BC and AC; and circle again in E, and join AE, BE. Also draw DG

Also draw DG in AB take L, so that AL : LB :: AH' : BK: :: parallel to CE, meeting AE and BE in H and G, and AC: CB*. The point L is therefore given; and if draw EK parallel to AB. Then, from the elements of a line N be taken, so as to have to AL the same ratio geometry,

that AB' has to AH', N will be given in magnitude. CL : LB :: (LA: LE ::) LA: LAXLE,

Also, since All': BK: ::AL: LB, and AH': AB':

AL : N, ex equo, BK*: AB: :: LB : N. Draw LO, LB and hence LBXLE= -·LA.

LM perpendicular to AC, CB; LO, LM are thereCL fore given in magnitude. Now, because AB' : BK ::

Also CL : LA :: (LB : LE ::) LB: : LB X LE, AD' : DF, N : LB :: AD': DF, and DF=T

and hence LB x LE= ·LB?.

•AD?; and for the same reason DE:= BD'; but,

Now CL : LB :: LA : LE :: EK or LD : KH,



AD'+ -.BD'and CL : LA :: LB : LE :: EK or LD : KG, by the preceding lemma,


N therefore, (Geom. Sect. III. Theor. 8.)


•AL’+ * BL+ -DL"; that is, DE+DFCL: AB :: (LD: GH ::) LD': EK XGH,


and hence EK X GHS -·LD.

LOʻ+LA+W.DL. Join LG, then by hspoele-

sis LO'+LM has to LG', the same ratio as DF+ From the three equations which we liare deduced from DE' bas to DG’; let it be that of R to N, then LOʻ

--AL' +

[ocr errors]

Fig. 7.



[ocr errors]





The same porism also assists in the solution of another Pori-m. Prism LM'=LGʻ; and therefore DE'+DF=NIG?+ problem. For if it were required to find D such that DE:

AF DL?; but DP'+Dr=DG?; therefore, would have to DE +DI- a given ratio, and DG

would therefore be given ; whence the solution is obvi-

DL'S DG’and ·DL=-(DG-

The connection of this porism with the impossible case
LGʻ); therefore DGʻ-LGʻ has to DL a constant of the problem is evident; the point L being that from
satio, viz. that of AB to R. The angle DLG is there- which, if perpendiculars be drawn to AC and CB, the
fore a right angle, and the ratio of AB to R that of sum of their squares is the Icast possible. For since
equality, otherwise LD would be given in magnitude, DF +DE : DG:: LOʻ+LM': LGʻ; and since
contrary to the supposition. LG is therefore given in LG is less than DG, LO:+LM must be less than
position: and since R:N:: AB:N:: LOʻ+LM: DF+DE'.
LG'; therefore t'e square of LG, and consequently It is evident from what has now appeared, that in
LG, is given in magnitude. The point G is there some instances at least there is a close connection be-
fore givell, and also the ratio of DE+DF to DG, tween these propositions and the maxima or minima, and
wirich is the same with that of AB to N.

of consequence the impossible cases of problems. The
The construction easily follows from the analysis, but nature of this connection requires to be farther investi-
it may be rendered more simple ; for since All?: A B* gated, and is the more interesting because the transition
:: AL: N, and BK?: AB:: BL:N; therefore. AH from the indefinite to the impossible case seems to be
+BK?: AB: :: AB : N. Likewise, if AG, BG, be made with wonderful rapidity. Thus in the first propo-
joined, AB:N:: AHY : AG', and AB:N:: BK: sition, though there be not properly speaking an impos.
BG?; wherefore AB:N:: AH+BK: AGʻ+BG , sible case, but only one where the point to be found
but it was proved that AB:N: ALP+BK: AB', goes off ad infinitum, it may be remarked, that if the
sherefore AG? +BG--=ABP; therefore the angle AGB given point F be anywhere out of the line HD (fig. 1.),
is a right one, and AL : LG :: LG: LB. If there the problem of drawing GB equal to GF is always pos-

fore AB be divided in I, so that AL : LB :: AH': sible, and admits of just one solution ; but if I be in 1:

BK”; and if LG, a mean proportional between AL DH, the problem admits of no solution at all, the point
aod LB, be placed perpendicular to AB, G will be the being then at an infinite distance, and therefore impossi-
point required.

ble to be assigned. There is, however, this exception,
The step in the analysis, by which a second intro- that if the given point be at K in this same line, DH is
duction of the general hypothesis is avoided, is that in determined by making DK equal to DL. Then every
which the angle GLD is concluded to be a right angle; point in the line DE gives a solution, and may be taken
which follows from DGʻ_GL' having a given ratio to for the point G. Here therefore the case of numberless
LD', at the same time that LD is of no determinate solutions, and of no solution at all, are as it were conter-
magnitude. For, if possible, let GLD be obtuse (fig. 6.), minal, and so close to one another, that if the given
and let the perpendicular from G to AB meet it in V, point be at K the problem is indefinite; but if it re-
therefore V is given : and since GD-LG=LD'+ move ever so little from K, remaining at the same time
2DL XLV; therefore, by the supposition, LD'+2DL in the line DH, the problem cannot be resolved. This
XLV must have a given ratio to LD'; therefore the affinity might have been determined à priori: for it is,
ratio of LD: to DLXVL, that is, of LD to VL, is as we have seen, a general principle, that a problem is
given, so that VL being given in magnitude, LD is al converted into a porism when one or when two of the
so given. But this is contrary to the supposition ; for conditions of it necessarily involve in them some one of
LD is indefinite by hypothesis, and therefore GLD the rest. Suppose, then, that two of the conditions are
cannot be 'obtuse, nor any other than a right angle. exactly in that state which determines the third ; then
The conclusion that is here drawn immediately from the while they remain fixed or given, should that third one
indetermination of LD would be deduced, according to vary or differ ever so little from the state required by
Dr Simson's method, by assuming another point D' the other two, à contradiction will ensue : therefore if,
any how, and from the supposition that GD'_GL: in the hypothesis of a problem, the conditions be so re-
LD" :: GD'-GL': LD", it would easily appear that lated to one another as to render it indeterminate, a po-
GLD must be a right angle, and the ratio that of equa- rism is produced; but if, of the conditions thus related

to one another, some one be supposed to vary, while the
These porisms facilitate the solution of the general others cantinue the same, an absurdity follows, and the
problems from which they are derived. For example, problem becomes impossible. Wherever, therefore, any
let three straight lines AB, AC, BC (fig. 5.), be given problem admits both of an indeterminate and an impossi-
in position, and also a point R, to find a point D in one ble case, it is certain, that these cases are nearly relat-
of the given lines, so that DE and DF being drawn ed to one another, and that some of the conditions by
perpendicular to BC, AC, and DR, joined; DE+DF: which they are produced are common to both.
may have to DR’ a given ratio. It is plain, that hav It is supposed above, that two of the conditions of a
ing found G, the problem would be nothing more than problem involve in them a third; and wherever that
to find D, such that the ratio of GD' to DR2, and happens, the conclusion which has been deduced will
therefore that of GD to DR, might be given, from invariably take place. But a porism may in some cases
which it would follow, that the point D is in the cir be so simple as to arise from the mere coincidence of one
cumference of a given circle, as is well known to geo condition with another, though in no case whatever any

inconsistency can take place between them. There are,

Dd 2


« PreviousContinue »