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Porentru, PORENTRU, is a town of Swifferland, in Elfgaw, restoration of the porisms of Euclid, which has all the Porism.
Porism. and capital of the territory of the bishop of Barle. It has appearance of being juft. It precisely corresponds to

a good castle, where he resides. It has in it, however, Pappus's description of them. ` All the lemmas which
nothing else worth taking notice of, except the cathe- Pappus has given for the better understanding of Eu-
dral. I'he bishop is a prince of the empire. It is seat- clid's propositions are equally applicable to those of
ed on the river Halle, near mount Jura, 22 miles south Dr Simson, which are found to differ from local theo-
of Bale. E. Long. 7. 2. N. Lat. 47- 34.

rems precisely as Pappus affirms those of Euclid to
PORISM, in geometry, is a name given by the have done. They require a particular mode of analysis,
ancient geometers to two classes of mathematical propo- and are of immense service in geometrical investigation ;
fitions. Euclid gives this name to propositions which on which account they may juftly claim our atten-
are involved in others which he is professedly investiga- tion.
ting, and which, although not his principal object, are While Dr Simson was employed in this inquiry, he
yet obtained along with it, as is e, refled by their name carried on a correspondence upon the subject with the
porifmatu, “ acquifitions.” Sucli propositions are now late Dr M. Stewart, professor of mathematics in the
called corollaries. But he gives the same name, by way university of Edinburgh ; who, besides entering into Dr
of eminence, to a particular class of propofitions which Simson's views, and communicating to him many curi-
he collected in the course of his researches, and selected ous porisms, pursued the same subject in a new and very
from among many others on account of their great sub- different dire&tion. He published the result of his in-
ferviency to the business of geometrical investigation in quiries in 1746, under the title of General Theorems,
general. These propofitions were so named by him, not caring to give them any other name, left he might
either from the way in which he discovered them, while appear to anticipate the labours of his friend and for-
he was investigating something else, by which means mer preceptor. The greater part of the propositions
they might be considered as gains or acquisitions, or from contained in that work are porisms, but without de-
their utility in acquiring farther knowledge as steps in monstrations; therefore, whoever wishes to investigate
the investigation. In this sense they are porismata; for one of the most curious subjects in geometry, will there
a opet w fignifies both to investigate and to acquire by in. find abundance of materials, and an ample field for dif-
veftigation. These propofitions formed a collection, cuffion.
which was familiarly known to the ancient geometers Dr Simfon defines a porism to be “a propofition, in
by the name of Euclid's porisms ; and Pappus of Alex, which it is proposed to demonstrate, that one or more
andria says, that it was a most ingenious collection of things are given, between which, and every one of in-
many things conducive to the analysis or solution of the numerable other things not given, but assumed accord-
most difficult problems, and which afforded great delighting to a given law, a certain relation described in the
to those who were able to understand and to investigate proposition is shown to take place.”
them.

This definition is not a little obscure, but will be Unfortunately for mathematical science, however, this plainer if expressed thus : “ A porism is a proposition valuable collection is now loft, and it still remains a doubt. affirming the possibility of finding such conditions as ful question in what manner the ancients conducted their will render a certain problem indeterminate, or capable researches upon this curious fubject. We have, however, of innumerable solutions.” This definition agrees

with reason to believe that their method was excellent both Pappus’s idea of these propositions, so far at least as in principle and extent, for their analyfis led them to they can be understood from the fragment already menmany profound discoveries, and was rettricted by the fe- tioned; for the propositions here defined, like those verest logic. The only account we have of this class which he describes, are, ftri&tly speaking, neither theoof geometrical propositions, is in a fragment of Pappus, rems nor problems, but of an intermediate nature bein which he attempts a general definition of them as a tween both; for they neither fimply enunciate a truth fet of mathematical propositions distinguishable in kind to be demonstrated, nor propose a question to be resol. from all others; but of this distinction nothing remains, ved, but are affirmations of a truth in which the deterexcept a criticism on a definition of them given by some mination of an unknown quantity is involved. In as geometers, and with which he finds fault, as defining far, therefore, as they assert that a certain problem may them only by an accidental circumstance, “ Pori/ma become indeterminate, they are of the nature of theoeft quod deficit hypothefi a theoremate locali.

rems; and, in as far as they seek to discover the condi. Pappus then proceeds to give an account of Euclid's tions by which that is brought about, they are of the porisms; but the enunciations are so extremely defec- nature of problems. tive, at the same time that they refer to a figure now We shall endeavour to make our readers understand lost, that Dr Halley confeffes the fragment in question this subject distinctly, by confidering them in the to be beyond his comprehension.

way in which it is probable they occurred to the anThe high encomiums given by Pappus to these pro- cient geometers in the course of their researches: this positions have excited the curiosity of the greatest geo- will at the same time show the nature of the analysis pemeters of modern times, who have attempted to dif- culiar to them, and their great use in the solution of cover their nature and manner of investigation. M. problems. Fermat, a French mathematician of the last century, It appears to be certain, that it has been the solution of attaching himself to the definition which Pappus cri- problems which, in all states of the mathematical sciticises, published an introduction (for this is its modest enccs, has led to the discovery of geometrical truths : title) to this subject, which many others tried to eluci- the first mathematical inquiries, in particular, must have date in vain. At length Dr Simson of Glasgow, by occurred in the form of questions, where something was patient inquiry and some lucky thoughts, obtained given, and something required to be done; and by the

reasoning

a

a

Porism. reasoning necessary to answer thefe questions, or to dif- ing the given circle ABC in B, let H be its centre, join Porism.

cover the relation between the things given and those HB, and let HD be perpendicular to DE. From D
to be found, many truths were suggested, which came draw DL, touching the circle ABC in L, and join HL;
afterwards to be the subject of leparate demonstra. also from the centre G, with the distance GB or GF,
tions.

describe the circle BKF, meeting HD in the points K
The number of these was the greater, because the an- and K. Then HD and DL are given in position and
cient geometers always undertook the solution of pro- magnitude ; and because GB touches the circle ABC,
blems, with a scrupulous and minute attention, into. HŘG is a right angle ; and since G is the centre of the
much that they would scarcely suffer any of the collate- circle BKF, therefore HB touches the circle BKF, and
ral truths to escape their observation.

HB:= the rectangle K HK ; which rectangle +DK
Now, as this cautious manner of proceeding gave an =HD', because K'K is bisected in D, therefore
opportunity of laying hold of every collateral truth con. HL'+ KD-=DH'=HL’and =LD'; therefore DK?
nected with the main object of inquiry, these geometers =DL’,

and DK=DL; and since DL is given in mag-
foon perceived, that there were many problems which in nitude, DK is also given, and K is a given point : for the
certain cafes would admit of no folution whatever, in same reason K' is a given point, and the point F being
consequence of a particular relation taking place among given by hypothesis, the circle BKP is given by position.
the quantities which were given. Such problems were The point G, the centre of the circle, is therefore gi-
said to become impoflible : and it was soon perceived, ven, which was to be found. Hence this construction :
that this always happened when one of the conditions Having drawn HD perpendicular to DE, and DL
of the problem was inconsistent with the rett. Thus, touching the circle ABC, make DK and DK' each
when it was required to divide a line, so that the rec- equal to DL, and find G the centre of the circle de.
tangle contained by its segments might be equal to a scribed through the points KFK; that is, let FK be
given space, it is evident that this was possible only when joined and bifected at right angles by MN, which meets
the given space was less than the square of half the line; DE in G, G will be the point required; that is, if
for when it was otherwise, the two conditions dehining, GB be drawn touching the circle ABC, and GF to the
the one the magnitude of the line, and the other the given point, GB is equal to GF.
rectangle of its segments, were inconsistent with each The synthetical demonftration is easily derived from
other. Such cases would occur in the solution of the the preceding analysis; but it must be remarked, that in
most simple problems ; but if they were more compli- some cases this construction fails. For, first, if F fall
cated, it mult have been remarked, that the construc- anywhere in DH, as at F, the line MN becomes paral-
tions would sometimes fail, for a reason directly contra- lel to DE, and the point G is nowhere to be found; or,
ry to that just now assigned. Cases would occur, where in other words, it is at an infinite distance from D.-
the lines, which by their intersection were to determine This is true in general ; but if the given point F coin-
the thing fought, instead of interfecting each other as cides with K, then MN evidently coincides with DE;
they did commonly, or of not meeting at all as in the so that, agreeable to a remark already made, every point
above mentioned case of impossibility, would coincide of the line DE may be taken for G, and will satisfy
with one another entirely, and of course leave the pro- the conditions of the problem ; that is to say, GB will
blem unresolved. It would appear to geometers upon equal to GK, wherever the point G be taken in the.
a little reflection, that fince, in the case of determinate line DE: the fame is true if F coincide with K. Thus
problems, the thing required was determined by the in- we have an instance of a problem, and that too a very
tersection of the two lines already mentioned, that is, fimple one, which, in general, admits but of one folu-
by the points common to both; so in the case of their tion; but which, in one particular case, when a certain
coincidence, as all their parts were in common, every relation takes place among the things given, becomes
one of these points muft give a solution, or, in other indetinite, and admits of innumerable solutions. The
words, the folutions must be indefinite in number, propofition which results from this case of the problem

Upon inquiry, it would be found that this proceed. is a forism, and may be thus enunciated :
ed from some condition of the problem having been in- A circle ABC being given by position, and also a
volved in another, so that, in fact, there was but one, straight line DE, which does not cut the circle, a point
which did not leave a sufficient number of independent K may be found, fuch, that if G be any point what-
conditions to limit the problem to a fingle or to any

de

ever in DE, tie straight line drawn from G to the
terminate number of solutions. It would soon be per- point K Thall be equal to the straight line drawn from
ceived, that these cases formed very curious propofitions G touching the given circle ABC.”
of an intermediate nature between problems and theo. The problem which follows appears to have led to
rems, and that they admitted of being enunciated in a the discovery of many porisms.
manner peculiarly elegant and concise. It was to such A circle ABC ("g. 2.) and two points D, E, in a
propofitions that the ancients gave the name of pori/ms. diameter of it being given, to find a point F in the cir-
This deduction requires to be illustrated by an example: cumference of the given circle ; from which, if straight
suppose, therefore, that it were required to resolve the lines be drawn to the given points E, D, these straight
following problem.

lines shall have to one another the given ratio of to fi, Plate A circle ABC (fig. 1.), a straight line DE, and a which is supposed to be that of a greater to a less.CCCCXII point F, being given in positien, to find a point G in the Suppose the problem resolved, and that F is found, so

Itraight line DE such, that GF, the line drawn from that FE has to FD the given ratio of 2 to ng produce
it to the given point, shall be equal to GB, the line EF towards B, bisect the angle EFD by FL, and
drawn from it touching the given circle.

DFB by FM: therefore EL.LD::EF: FD, that
Suppole G to be found, and GB to be drawn touch- is in a given ratio, and since ED is given, each of the seg

3 D 2

ments

لب

a

Porifm. ments EL, LD, is given, and the point L is also given; is equal to AOH, and therefore the angle FOB to Porifin.

because DFB is bisected by FM, EM:MD:: EF:FD, HOG, that is, the arch FB to the arch HG. This
that is, in a given ratio, and therefore M is given. propofition appears to have been the last but one in
Since DFL is half of DFE, and DFM half of DFB, the third book of Euclid's Porisms, and the manner of
therefore LFM is half of (DFE+DFB), therefore its enunciation in the porismatic form is obvious.
LFM is a right angle; and since the points L, M, are The preceding proposition also affords an illustration
given, the point F is in the circumference of a circle of the remark, that the conditions of a problem-are in
described

upon LM as a diameter, and therefore given volved in one another in the porismatic or indefinite case;
in position. Now the point F is also in the circumfe- for here several independent conditions are laid down,
rence of the given circle ABC, therefore it is in the in- by the help of which the problem is to be resolved.
tersection of the two given circumferences, and there- Two points D and E are given, from which two lines
fore is found. Hence this construction : Divide ED are to be inflected, and a circumference ABC, in which
in L, so that EL may be to LD in the given ratio of these lines are to meet, as also a ratio which these lines
a to B, and produce ED also to M, so that EM may are to have to each other. Now these conditions are
be to MD in the same given ratio of a to b; bisect all independent on one another, so that any one may be
LM in N, and from the centre N, with the distance changed without any change whatever in the rest. This
NL, defcribe the semicircle LFM; and the point F, in is true in general; but yet in one case, viz. when the .
which it intersects the circle ABC, is the point ie- points are so related to one another that their rectangle
quired.

under their distances from the centre is equal to the
The synthetical demonstration is easily derived from square of the radius of the circle ; it follows from the
the preceding analysis. It must, however, be remark- preceding analysis, that the ratio of the inflected lines :
ed, that the construction fails when the circle LFM is no longer a matter of choice, but a necessary conse -
falls either wholly within or wholly without the circle quence of this disposition of the points..
ABC, so that the circumferences do not intersect; and From what has been already said, we may trace the
in these cases the problem cannot be solved. It is also imperfect definition of a porism which Pappus ascribes
obvious that the construction will fail in another case, to the later geometers, viz. that it differs from a local
viz. when the two circumferences LFM, ABC, entire. theorem, by wanting the hypothesis assumed in that
ly coincide. In this case, it is farther evident, that theorem. – Now, to understand this, it must be observed,
every point in the circumference ABC will answer the that if we take one of the propositions called loci, and
conditions of the problem, which is therefore capable make the conftruction of the figure a part of the hypo-
of numberless solutions, and may, as in the former, in- thesis, we get what was called by the ancient geo-
stances, be converted into a porism. We now inquire, meters a local theorem. If, again, in the enunciation of
therefore, in what circumstances the point L will coin- the theorem, that part of the hypothesis which contains
cide with A, and also the point M with C, and of con- the construction be suppressed, the propofition thence
fequence the circumference LFM with ABC. If we arising will be a porism, for it will enunciate a truth,
suppose that they coincide EA:AD::2:3 :: EC: and will require to the full understanding and investiga-
CD, and EA : EC :: AD:CD, or by conversion EA tion of that truth, that something should be found, viza.
: AC :: AD:CD-AD::AD:2DO, O being the the circumstances in the construction supposed to be on
centre of the circle ABC ; therefore, also, EA:AO:: mitted.
AD: DO, and by composition EO:A0:: AO: DO, Thus, when we say, if from two given points E, D,

therefore EOXOD=A0'. Hence, if the given points (fig. 3.) two straight lines EF, FD, are inflected to a Pla'e E and D (Ag. 3.) be fo ftuated, that EOXOD= third point F, fo as to be to one another in a given raCCCCXII AO', and at the fame time a:ß:: EA: AD::EC: tio, 'the point F is in the circumference of a given

CD, the problem admits of numberless folutions; and circle, we have a locus. But when conversely it is
if either of the points D or E be given, the other point, said, if a circle ABC, of which the centre is O.,
and also the ratio which will render the proble. inde- be given by position, as also a point E; and if D be ta-
terminate, may be found. Hence we have this po- ken in the line EO, so that EOXOD=A0'; and if
rism :

from E and D the lines EF, DF be inflected to any
“A circle ABC, and also a point D being given, point of the circumference ABC, the ratio of EF to
another point E may be found, such that the two lines DF will be given, viz, the same with that of EA to
inflected from those points to any point in the circum- AD, we have a local theorem.
ference ABC, shall have to each other a given ratio, Lastly, when it is said, if a circle ABC be given by
which ratio is also to be found.” Hence also we have position, and also a point E, a point D may be found,
an example of the derivation of porisms from one ano. such that if EF, FD be inflected from E and D to any
ther, for the circle ABC, and the points D and E re point F in the circumference ABC, these lines thall
maining as before (fig. 3.), if, through D, we draw have a given ratio to one another, the propofition be-
any line whatever HD B, meeting the circle in B and comes a porism, and is the same that has just now been
H; and if the lines EB, EH, be also drawn, these lines investigated.
will cut off equal circumferences BF, HG. Let FC Hence it is evident, that the local theorem is changed
be drawn, and it is plain from the foregoing analysis, into a porism, by leaving out what relates to the deter-
that the angles DFC, CFB, are equal; therefore if mination of D, and of the given ratio. But though all
OG, OB, be drawn, the angles BOC, COG, are also propositions formed in this way from the conversion of
equal ; and consequently the angles DOB, DOG. In loci, are porisms, yet all porisms are not formed froin
the same manner, by joining AB, the angle DBE be the conversion of loci ; the first, for instance, of the pre.
úg biíccted by BA, it is evident that the angle AOF ceding cannot by conversion be changed into a locus

therefore

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Porifnr. therefore Fermat's idea of porisms, founded upon this must pass through the point to be found M; for if not, Porism., 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 Dr Stewart, in a paper read point to be found is therefore in the line E B, which is a confiderable time ago before the Philosophical Society given in pofition. 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 poflibility of finding one or more condi

SEXBL

PEXAF tions of an indeterminate theorem;" where, by an in

:LN

and AP:PE:: AF: FG= determinate theorem, he meant one which expresses a re

BS

AP lation between certain quantities that are determinate therefore FG:LN::

PEXAF SEXBL

:: PEXAF
and certain others that are indeterminate; a definition

AP BS
which evidently agrees with the explanations which have XBS: SEXBLXAP; 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 juft, 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 comproblem fails, in confequence of the lines which by pounded of the ratios of AF to BL, AE to BE, and their intersection, or the points which by their polía DB to DA. In like manner, because E" is a point in tion, were to determine the problem required, happen the line DE and AE”, BE" are inflected to it, the ing to coincide with one another.

À porism may

ratio of FH to LM is compounded of the same ratios therefore be deduced from the problem to which it bé. of AF to BL, AE' to BE', and DB to DA; therelongs, just as propositions concerning the maxima and fore FH:LM::FG:NL (and consequently) :: HG minima of quantities are deduced from the problems of : MN; but the ratio of HG to MN is given, being the which they form limitations; and such is the most natu- same as that of a to B; the ratio of FH to LM is ral and obvious analysis of which this class of propofi- therefore also given, and FH being given, IM is given tions admits.

in

magnitude. Now LM is parallel to BE, a line The following poriím is the first of Euclid's, and the given in position; therefore M is in a line QM, parallel first also which was restored. It is given here to exem

to AB, and given in position-; therefore the point M, plify the advantage which, in investigations of this kind, and also the line KLM, drawn through it parallel to may be derived from employing the law of continuity BE, are given in pofition, which were to be found. in its utmost extent, and pursuing porisms to those ex

Hence this construction : From A draw. AE paralick treme cases where the indeterminate magnitudes increase to FK, so as to meet DE in E'; join BE', and take in ad infinitum.

it BQ, so that a : 8::HF: BQ, and through draw This porism may be considered as having occurred in QM parallel to AB: Let HA be drawn, and produ

the solution of the following problem: Two points A, B, ced till it meet DE in E", and draw BE", meeting QM Plate

(fig. 4.) and also three straight lines DE, FK, KL, be in M; through M draw KML parallel to BE, then is CCCCXilling given in position, together with two points Hand M KML the line and M the point which were to be

in two of these lines, to inflect from A and B to a point found. . There are two lines which will answer the con-
in the third, two lines that shall cut off from KP and ditions of this porism ; for if in QB, produced on the
KL two segments, adjacent to the given points H and other fide of B, there be taken Bq=BQ, and if qm
M, having to one another the given ratio of 4 to l. be drawn parallel to AB, cutting MB in m; and if
Now, to find whether a porism be connected with this be drawn parallel to BQ, the part mn, cut off by EB
problem, suppose that there is, and that the following produced, will be equal to MN, and have to HG the
proposition is true. Two points A and B, and two ratio required. It is plain, that whatever be the ratio
itraight lines DE, FK, being given in position, and of a to !, and whatever be the magnitude of FH, if the
also à point H in one of them, a line LK may be found, other things given remain the fame, the lines found will
and also a point in it M, both given in position, such be all parallel to BE. But if the ratio of a to B re-
that AE and BE. inflected from the points A and B main the same likewise, and if only the point H vary,
to any point whatever of the line DE, shall cut off from the position of KL will remain the same, and the point
the other lines FK and LK segments HG and MN M will vary.
adjacent to the given points. H and M, having to one Another general remark which may be made on the
another the giren ratio of a to .

analysis of porisms is, that it often happens, as in the
First, let AE', BE', be.infected to the point E', so last example, that the magnitudes required may all, or
that AE' may be parallel to FK, then shall E'B be pa- a part of them, be found by considering the extreme
rallel to KL, the line to be found; for if it be not pa. cafes; but for the discovery of the relation between
rallel to KL, the point of their interfection must be at them, and the indefinite magnitudes, we must have res
a fipite distance from the point M, and therefore ma course to the hypothesis of the porism in its molt genea
king as s to a; fo this distance to a fourth proportionał, ral or indefinite form; and mult endeavour fo to con-
the distance from H at which AE' interfects FK, wilí duct the reasoning, that the indefinite magnitudes may
be equal to that fourth proportional. But AE does at length totally disappear, and leave a proposition ai-
not interfect FK, for they are parallel by construction' ; serting the relation between determinate magnitujes
therefore BE caunot intersect KL, which is therefore only.
parallel to BE', a line given in position. Again, let For this purpose Dr Simson frequently employs two
AE", BE“, be infected to E", so that AE' may pass statements of the general hypothesis, which he compares
through the given point H: then it is plain that BE" together. As for instance, in his analysis of the lat po- .

5

rilin,

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ALE

L.BD"; but,

the points A, B This double flatement, however, by the preceding lenina, WP.AD:+44BD

'

R

N

Plate

AB

R

R

AB.DL'=(DG'

R

a

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Po iím rism, he affumes not only E, any point in the line DE, AD'; and for the same reason DE'=

Porifm.

' but also another point O, anywhere in the same line,

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

LB

LB ,

,

' N

N

N

. and complicated ; nor is it even heceffary, for it may be ·AL:+41-BL’+AB-DL"; that is, DE*+DF-2

?'' = N

N having to simpler to loci, or to propofitions of the data. The following porism Loʻ+LM'+ AB DL'. Join LG then by hypothes ,

is given as an example where this is done with some , difficulty, but with conliderable advantage both with fis LO'+LM', as to LG’, the same ratio as DF + regard to the simplicity and fhortness of the demonftra- DE- has to DG'; let it be that of R to N, then LO * tion. It will be proper to premise the following lemma. LM=LG“; and therefore DE+DF=LG+

RLG*; and therefore DE+DP•=LG+

N
Let AB (fig. 7.) be a Itraight line, and D, L any
CCCCXIII
two points in it, one of which D is between A and B; 2.DL”; but DE'+DF=RDGʻ; therefore,

:

, N

N

N let CL be any straight line. LB.AD'+LA.BD'=

.
CL CL

LG!

BA.DL'DG,and AB.
АВ

N

N
LB. AL'PLABLAB-DL. Place CL perpen- LG“); therefore DG-LG has to DL-a constant

:+
CL CL CL

';
dicular to AB, and through the points A, C, B, de- ratio, viz. that of AB to R. The angle DLG is there-
scribe a circle; and let CL meet it again in E, and join fore a right angle, and the ratio of AB to R that of
AE, BE. Draw DG parallel to CE, meeting AE equality, otherwise LD would be given in magnitude,
and BE in H and G. Draw EK parallel to AB. contrary to the fuppofition. LG is therefore given in

LB pofitionR:: :
CL: LB ::(LA: LE::) LA': LAXLE=1B LA: LG*; therefore the fquare of LG, and confequently

= ?
CL

LG, is given in magnitude. The point G is there-
CL:LA::(LB: LE::) LB':LBXLE= LA.BL’. fore given, and also the ratio of DE +DF to DG’,

CL

which is the same with that of AB to N. Now CL : LB :: LA: LE::(EK) LD: KH, and

The construction easily follows from the analysis, but CL:LA:: LB:LE:: (EK) LD: KG; therefore, it may be rendered more simple ; for fince AH?: AB: (V. 24.) CL:AB :: (LD: GH ::) LD" : EKXGH: :AL:N, and BK*: AB?: : BL: N; therefore AH

:: AB.LD"; therefore LB.LA'+

LA

AB '

CE

+BK?: AB2 : : AB:N. Likewise, if AG, BG, be CL

CL

CL: CL joined, AB:N:: AH' : AGʻ, and AB:N::BK2 : •LD'=ABXLE+EKXGH. Again, CL:LA:: BG?; wherefore AB:N:: AK2+ BK?: AG+BG{: : Ꭰ: : (LB : LE :: DB : DG : :) DB : DBXDG=LA and AGʻ+BG=AB2; therefore the angle AGB is a

:

CL right one, and AL:LG::LG: LB. If therefore AB •DD', and CL : LB :: (LA: LE::DA : DH::) be divided in L, so that AL:LB :: AH’: BK2; and DA”: DAXDH=LEDA; therefore CIDA'+ placed perpendicular to AB, G will be the point reLB

LB

, ,
';
CL

CL

quired.
LADB'=ADxDH+DBXDG=ABXLE+EKX
CL

1 he step in the analysis, by which a second intro

duction of the general hypothesis is avoided, is that in LB LA

LB
GH; wherefore

DA? +
-DB'= ULLA’t which the angle GLD is concluded to be a right angle;

'
CL
CL CL

which follows from DG-GL, having a given ratio to
AB
B'+41 LD'. 2. E. D.

LD’, at the same timethat LD is of no determinate magCL CL

nitude. For, if poffible, let GLD be obtufe (fig. 6.), Let there be three straight lines AB, AC, CB and let the perpendicular from G to AB meet it in V, given in position (fig. 5.); and from any point what. therefore V is given : and since GD-LG-=LDP+ ever in one of them, as Ó, let perpendiculars be drawn 2DLXLV; therefore, by the supposition, LD + 2DL to the other two, as DF, DE, a point G may be found, XLV must have a given ratio to LDP, therefore the fuch, that if GD be drawn from it to the point D, the ratio of LD- to DLXVL, that is, of LD to VL, is square of that line shall have a given ratio to the sum of given, so that VL being given in magnitude, LD is althe squares of the perpendiculars DF and DE, which

fo given. But this is contrary to the supposition ; for ratio is to be found.

LD is indefinite by hypothesis, and therefore GLD Draw AH, BK perpendicular to BC and AC; and

cannot be obtuse, nor any other than a right angle. in AB take L, so that AL:LB :: AH’: BK’::

The conclusion here drawn immediately from the indeAC':CB'. The point L is therefore given ; and if termination of LD would be deduced, according to N be taken, so as to have to AL the same ratio that Dr Simfon's method, by assuming another point D' AB: has to AH, N will be given in magnitude. Al any how, and from the supposition that GD-GL’: fo, fince AH': BK'::AL:LB, and XH': AB':: LD::GD-GL’: LD, it would easily appear that

?AH
AL:N, ex equo BK?: AB':: LB : N. Draw LO, GLD must be a right angle, and the ratio that of equa-
LM perpendicular to AC, CB; LO, LM are there. lity.
fore given in magnitude. Now, because AB?: BK?:: These porisms facilitate the solution of the general
AD': DF, N: LB :: AD': DF, and DF'=

LB problems from which they are derived. For example, let
N three straight lines AB, AC, BC (fig. 5.), be given in

poli

:

:

:

LA LB' +

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