Forifm. reafoning neceffary to answer thefe queftions, or to difcover the relation between the things given and thofe to be found, many truths were fuggefted, which came afterwards to be the fubject of separate demonftrations.

The number of thefe was the greater, because the ancient geometers always undertook the folution of problems, with a fcrupulous and minute attention, info much that they would scarcely fuffer any of the collateral truths to escape their obfervation.

Now, as this cautious manner of proceeding gave an opportunity of laying hold of every collateral truth connected with the main object of inquiry, thefe geometers foon perceived, that there were many problems which in certain cafes would admit of no folution whatever, in confequence of a particular relation taking place among the quantities which were given. Such problems were faid to become impoffible and it was foon perceived, that this always happened when one of the conditions of the problem was inconfiftent with the reft. Thus, when it was required to divide a line, fo that the rectangle contained by its fegments might be equal to a given space, it is evident that this was poffible only when the given space was lefs than the square of half the line; for when it was otherwise, the two conditions defining, the one the magnitude of the line, and the other the rectangle of its fegments, were inconfiftent with each other. Such cafes would occur in the folution of the moft fimple problems; but if they were more complicated, it must have been remarked, that the conftructions would sometimes fail, for a reason directly contrary to that just now affigned. Cafes would occur, where the lines, which by their intersection were to determine the thing fought, instead of intersecting each other as they did commonly, or of not meeting at all as in the above mentioned cafe of impoffibility, would coincide with one another entirely, and of course leave the problem unrefolved. It would appear to geometers upon a little reflection, that fince, in the cafe of determinate problems, the thing required was determined by the interfection of the two lines already mentioned, that is, by the points common to both; fo in the cafe of their coincidence, as all their parts were in common, every one of thefe points must give a folution, or, in other words, the folutions must be indefinite in number.

Upon inquiry, it would be found that this proceed ed from fome condition of the problem having been involved in another, so that, in fact, there was but one, which did not leave a fufficient number of independent conditions to limit the problem to a fingle or to any determinate number of folutions. It would foon be perceived, that these cases formed very curious propofitions of an intermediate nature between problems and theorems; and that they admitted of being enunciated in a manner peculiarly elegant and concife. It was to fuch propofitions that the ancients gave the name of porifms. This deduction requires to be illustrated by an example: fuppofe, therefore, that it were required to refolve the following problem.

Plate A circle ABC (fig. 1.), a ftraight line DE, and a CCCCXII point F, being given in pofition, to and a point G in the ftraight line DE fuch, that GF, the line drawn from it to the given point, fhall be equal to GB, the line drawn from it touching the given circle.

Suppole G to be found, and GB to be drawn touch

ing the given circle ABC in B, let H be its centre, join Porifni. HB, and let HD be perpendicular to DE. From D draw DL, touching the circle ABC in L, and join HL; alfo from the centre G, with the distance GB or GF, defcribe the circle BKF, meeting HD in the points K and K. Then HD and DL are given in position and magnitude; and because GB touches the circle ABC, HBG is a right angle; and fince G is the centre of the circle BKF, therefore HB touches the circle BKF, and HB' the rectangle K HK; which rectangle +DK' =HD', because K'K is bisected in D, therefore HL+KD DH'HL'and = LD'; therefore DK =DL', and DK=DL; and fince DL is given in magnitude, DK is alfo given, and K is a given point: for the fame reafon K' is a given point, and the point F being given by hypothefis, the circle BKP is given by position. The point G, the centre of the circle, is therefore given, which was to be found. Hence this construction :

Having drawn HD perpendicular to DE, and DL touching the circle ABC, make DK and DK' each equal to DL, and find G the centre of the circle defcribed through the points K FK; that is, let FK be joined and bifected at right angles by MN, which meets DE in G, G will be the point required; that is, if GB be drawn touching the circle ABC, and GF to the given point, GB is equal to GF.

The fynthetical demonftration is easily derived from the preceding analysis; but it must be remarked, that in fome cafes this conftruction fails. For, firft, if F fall anywhere in DH, as at F', the line MN becomes parallel to DE, and the point G is nowhere to be found; or, in other words, it is at an infinite distance from D.— This is true in general; but if the given point F coincides with K, then MN evidently coincides with DE; fo that, agreeable to a remark already made, every point of the line DE may be taken for G, and will fatisfy the conditions of the problem; that is to say, GB will be equal to GK, wherever the point G be taken in the line DE: the fame is true if F coincide with K. Thus we have an instance of a problem, and that too a very fimple one, which, in general, admits but of one folution; but which, in one particular case, when a certain relation takes place among the things given, becomes indefinite, and admits of innumerable folutions. The propofition which refults from this cafe of the problem is a porifm, and may be thus enunciated:

"A circle ABC being given by pofition, and alfo at ftraight line DE, which does not cut the circle, a point K may be found, fuch, that if G be any point whatever in DE, the ftraight line drawn from G to the point K fhall be equal to the straight line drawn from G touching the given circle ABC."

The problem which follows appears to have led to the discovery of many porifms.

A circle ABC (g. 2.) and two points D, E, in a diameter of it being given, to find a point F in the circumference of the given circle; from which, if straight lines be drawn to the given points E, D, thefe ftraight lines shall have to one another the given ratio of to f, which is fuppofed to be that of a greater to a lefs.Suppose the problem refolved, and that F is found, fo that FE has to FD the given ratio of to, produce EF towards B, bifect the angle EFD by FL, and DFB by FM: therefore EL: LD :: EF: FD, that is in a given ratio, and fince ED is given, each of the seg 3 D 2

ments

به

is equal to AOH, and therefore the angle FOB to Pótifier. HOG, that is, the arch FB to the arch HG. This propofition appears to have been the laft but one in the third book of Euclid's Porifms, and the manner of its enunciation in the porifmatic form is obvious.

Porifnt. ments EL, LD, is given, and the point L is alfo given; becaufe DFB is bifected by FM, EM:MD:: EF: FD, that is, in a given ratio, and therefore M is given. Since DFL is half of DFE, and DFM half of DFB, therefore LFM is half of (DFE+DFB), therefore LFM is a right angle; and fince the points L, M, are given, the point F is in the circumference of a circle defcribed upon LM as a diameter, and therefore given in pofition. Now the point F is alfo in the circumference of the given circle ABC, therefore it is in the interfection of the two given circumferences, and therefore is found. Hence this conftruction: Divide ED in L, fo that EL may be to LD in the given ratio of a to, and produce ED alfo to M, so that EM may be to MD in the fame given ratio of a to ; bifect LM in N, and from the centre N, with the distance NL, defcribe the femicircle LFM; and the point F, in which it interfects the circle ABC, is the point required.

The fynthetical demonstration is easily derived from the preceding analysis. It muft, however, be remarked, that the conftruction fails when the circle LFM falls either wholly within or wholly without the circle ABC, fo that the circumferences do not interfect; and in these cafes the problem cannot be folved. It is alfo obvious that the construction will fail in another cafe, viz. when the two circumferences LFM, ABC, entire ly coincide. In this cafe, it is farther evident, that every point in the circumference ABC will answer the conditions of the problem, which is therefore capable of numberless folutions, and may, as in the former inftances, be converted into a porifm. We now inquire, therefore, in what circumstances the point L will coincide with A, and alfo the point M with C, and of confequence the circumference LFM with ABC. If we fuppofe that they coincide EA: AD:::::EC: CD, and EA: EC:: AD: CD, or by converfion EA : AC::AD: CD-AD:: AD: 2DO, O being the centre of the circle ABC; therefore, alfo, EA : AO :: AD: DO, and by compofition EO: AO:: AO: DO, therefore EOXOD=AO. Hence, if the given points E and D (fig. 3.) be fo fituated, that EOXOD= CCCCXIII AO3, and at the fame time a ::: EA: AD:: EC: CD, the problem admits of numberlefs folutions; and if either of the points D or E be given, the other point, and also the ratio which will render the proble. indeterminate, may be found. Hence we have this porifm:

Pla'e

"A circle ABC, and alfo a point D being given, another point E may be found, fuch that the two lines inflected from thefe points to any point in the circumference ABC, fhall have to each other a given ratio, which ratio is also to be found." Hence alfo we have an example of the derivation of porisms from one another, for the circle ABC, and the points D and E remaining as before (fig. 3.), if, through D, we draw any line whatever HDB, meeting the circle in B and H; and if the lines EB, EH, be alfo drawn, thefe lines will cut off equal circumferences BF, HG. Let FC be drawn, and it is plain from the foregoing analysis, that the angles DFC, CFB, are equal; therefore if OG, OB, be drawn, the angles BOC, COG, are alfo equal; and confequently the angles DOB, DOG. In the fame manner, by joining AB, the angle DBE beng bifected by BA, it is evident that the angle AOF

The preceding propofition alfo affords an illuftration of the remark, that the conditions of a problem are in volved in one another in the porifmatic or indefinite cafe; for here feveral independent conditions are laid down, by the help of which the problem is to be refolved. Two points D and E are given, from which two lines are to be inflected, and a circumference ABC, in which thefe lines are to meet, as also a ratio which these lines are to have to each other. Now these conditions are all independent on one another, fo that any one may be changed without any change whatever in the reft. This is true in general; but yet in one cafe, viz. when the points are fo related to one another that their rectangle under their distances from the centre is equal to the square of the radius of the circle; it follows from the preceding analyfis, that the ratio of the inflected lines is no longer a matter of choice, but a necessary confe quence of this difpofition of the points..

From what has been already faid, we may trace the imperfect definition of a porifm which Pappus afcribes to the later geometers, viz. that it differs from a local theorem, by wanting the hypothefis affumed in that theorem.- Now, to understand this, it must be observed, that if we take one of the propofitions called loci, and make the conftruction of the figure a part of the hypothefis, we get what was called by the ancient geometers a local theorem. If, again, in the enunciation of the theorem, that part of the hypothefis which contains the conftruction be fuppreffed, the propofition thence arifing will be a porifm, for it will enunciate a truth, and will require to the full understanding and inveftiga tion of that truth, that fomething should be found, viz. the circumstances in the construction supposed to be o mitted.

Thus, when we fay, if from two given points E, D, (fig. 3.) two ftraight lines EF, FD, are inflected to a third point F, fo as to be to one another in a given ratio, the point F is in the circumference of a given circle, we have a locus. But when converfely it is. faid, if a circle ABC, of which the centre is Q, be given by pofition, as alfo a point E; and if D be taken in the line EO, so that EOXOD=AO1; and if from E and D the lines EF, DF be inflected to any point of the circumference ABC, the ratio of EF to DF will be given, viz, the fame with that of EA to AD, we have a local theorem.

Laftly, when it is said, if a circle ABC be given by pofition, and alfo a point E, a point D may be found, fuch that if EF, FD be inflected from E and D to any point F in the circumference ABC, these lines fhall have a given ratio to one another, the propofition becomes a porism, and is the fame that has just now been inveftigated.

Hence it is evident, that the local theorem is changed into a porifm, by leaving out what relates to the determination of D, and of the given ratio. But though all propofitions formed in this way from the converfion of loci, are porifms, yet all porifms are not formed from the converfion of loci ; the first, for instance, of the pre. ceding cannot by conversion be changed into a locus

therefore

Plate CCCCXII;

To confirm the truth of the preceding theory, it may be added, that profeffor 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 conditions of an indeterminate theorem;" where, by an indeterminate theorem, he meant one which expresses a relation 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 these propofitions be juft, it follows, that they are to be discovered by confidering those cafes in which the construction of a problem fails, in confequence of the lines which by their interfection, or the points which by their pofition, were to determine the problem required, happening to coincide with one another. A porifm may therefore be deduced from the problem to which it belongs, juft as propofitions concerning the maxima and minima of quantities are deduced from the problems of which they form limitations; and fuch is the most natural and obvious analysis of which this clafs of propofi

tions admits.

The following porism is the first of Euclid's, and the firft alfo which was restored. It is given here to exemplify the advantage which, in inveftigations of this kind, may be derived from employing the law of continuity in its utmost extent, and purfuing porifms to thofe extreme cafes where the indeterminate magnitudes increase ad infinitum.

This porifm may be confidered as having occurred in the folution of the following problem: Two points A, B, (fig. 4.) and alfo three ftraight lines DE, FK, KL, being given in pofition, together with two points H and M in two of these lines, to inflect from A and B to a point in the third, two lines that fhall cut off from KF and KL two fegments, adjacent to the given points H and M, having to one another the given ratio of to . 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 traight lines DE, FK, being given in pofition, and alfo a point H in one of them, a line LK may be found, and alfo a point in it M, both given in pofition, fuch that AE and BE inflected from the points A and B to any point whatever of the line DE, fhall cut off from 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 A.

First, let AE', BE', be inflected to the point E', fo that AE' may be parallel to FK, then fhall E'B be раrallel to KL, the line to be found; for if it be not parallel to KL, the point of their interfection must be at a finite diftance from the point M, and therefore making as s to a; fo this distance to a fourth proportional, the distance from H at which AE interfects FK, will be equal to that fourth proportional. But AE does not interfect FK, for they are parallel by construction; therefore BE' cannot interfect KL, which is therefore parallel to BE', a line given in pofition. Again, let AE", BE", be inflected to E", fo that AE' may pafs through the given point H: then it is plain that BE"

:LN=

SEXBL

BS therefore FG: LN::

PEXAF SEXBL AP BS

PEXAF AP :: 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 compounded of the ratios of AF to BL, AE to BE, and DB to DA. In like manner, because E" is a point in the line DE and AE", BE" are inflected 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; therefore FH: LM:: FG: NL (and confequently) :: HG MN; but the ratio of HG to MN is given, being the fame as that of a to 6; the ratio of FH 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 pofition; therefore the point M, and alfo the line KLM, drawn through it parallel to BE', are given in pofition, which were to be found. Hence this conftruction: From A draw AE parallel' to FK, fo as to meet DE in E'; join BE', and take in it BQ, fo that a:6:: HF: BQ, and through Q draw QM parallel to AB. Let HA be drawn, and produced till it meet DE in E", and draw BE", meeting QM 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 answer the conditions of this porifm; for if in QB, produced on the other fide of B, there be taken Bq=BQ, and if q m be drawn parallel to AB, cutting MB in m; and if ma be drawn parallel to BQ, the part mn, cut off 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, and whatever be the magnitude of FH, if the other things given remain the fame, the lines found will be all parallel to BE'. But if the ratio of a to remain the fame likewife, and if only the point H vary, the pofition of KL will remain the fame, and the point M will vary.

α

Another general remark which may be made on the analysis of porifmns 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 difcovery of the relation between them, and the indefinite magnitudes, we must have res courfe to the hypothefts of the porism in its most general or indefinite form; and muít endeavour fo to conduet the reasoning, that the indefinite magnitudes may at length totally difappear, and leave a propofition alferting the relation between determinate magnitudes only.

For this purpose Dr Simfor frequently employs two ftatements of the general hypothefis, which he compares together. As for inftance, in his analysis of the laft po 5

rifin,

Po ifm

rifm, he affumes not only E, any point in the line DE, but also another point O, anywhere in the fame line, to both of which he fuppofes lines to be inflected from the points A, B. This double flatement, however, cannot be made without rendering the investigation long and complicated; nor is it even neceffary, for it may be avoided by having recourfe to fimpler porifms, or to loci, or to propofitions of the data. The following porifm is given as an example where this is done with fome difficulty, but with confiderable advantage both with regard to the fimplicity and fhortness of the demonftration. It will be proper to premife the following lemma. Let AB (fg. 7.) be a straight line, and D, L any CCCCXIII two points in it, one of which D is between A and B; let CL be any straight line. ·AD'+LA・BD'=

Plate

LB

CL

CL

-·BL" DL'. Place CL perpen

CL dicular to AB, and through the points A, C, B, deferibe a circle; and let CL meet it again in E, and join AE, BE. Draw DG parallel to CE, meeting AE and BE in H and G. Draw EK parallel to AB. CL: LB::(LA: LE::) LA': LA×LE=

LB CL

LA

LA.BL.

CL

CL: LA:: (LB: LE::) LB': LBXLE=
LA::(LB:
Now CL: LB::LA:LE:: (EK) LD: KH, and
CL: LA:: LB:LE:: (EK) LD: KG; therefore,
(V. 24.) CL: AB :: (LD: GH ::) LD': EKXGH.

AB CL

LB

CL

LA

AB

LD; therefore -·LA'+ CL CL •LD=ABXLE+EKXGH. Again, CL: LA:: (LB: LE::DB: DG : :) DE': DBXDG-LA CL DD', and CL: LB:: (LA: LE::DA: DH::) DA':

LA

CL

LB.

DA'; therefore D

DB2=AD×DH+DB×DG=AB-LE+EK×

ᏞᏴ

GH; wherefore DA'+ADB'=LLA'

LA

CL

LB CL

CL

CL

Let there be three ftraight lines AB, AC, CB given in pofition (fig. 5.); and from any point whatever in one of them, as D, let perpendiculars be drawn to the other two, as DF, DE, a point G may be found, fuch, that if GD be drawn from it to the point D, the fquare of that line fhall have a given ratio to the fum of the fquares of the perpendiculars DF and DE, which ratio is to be found.

Draw AH, BK perpendicular to BC and AC; and in AB take L, fo that AL: LB:: AH': BK':: AC: CB. The point L is therefore given; and if N be taken, fo as to have to AL the fame ratio that AB has to AH, N will be given in magnitude. Alfo, fince AH': BK':: AL: LB, and AH': AB' : : AL: N, ex equo BK': AB':: LB: N. Draw LO, LM perpendicular to AC, CB; LO, LM are therefore given in magnitude. Now, because AB: BK:: AD': DF, N: LB:: AD': DF', and DF'= LB

N

LG'); therefore DG-LG has to DL a conftant

ratio, viz. that of AB to R. The angle DLG is therefore a right angle, and the ratio of AB to R that of equality, otherwife LD would be given in magnitude, contrary to the fuppofition. LG is therefore given in pofition and fince R: N:: AB:N:: LO' +LM': LG; therefore the fquare of LG, and confequently LG, is given in magnitude. The point G is therefore given, and alfo the ratio of DE+DF to DG', which is the fame with that of AB to N.

The conftruction eafily follows from the analyfis, but it may be rendered more fimple; for fince AH: AB2 ::AL: N, and BK: AB2:: BL: N; therefore AH2 +BK: AB:: AB: N. Likewife, if AG, BG, be joined, AB: N:: AH': AG, and AB:N:: BK2: BG; wherefore AB: N:: AK2+ BK: AG+ BG" and AG+BGAB; therefore the angle AGB is a right one, and AL: LG:: LG: LB. If therefore AB be divided in L, fo that AL: LB:: AH2: BK2; and if LG, a mean proportional between AL and LB, be placed perpendicular to AB, G will be the point required.

The ftep in the analyfis, by which a fecond introduction of the general hypothefis is avoided, is that in which the angle GLD is concluded to be a right angle;

which follows from DG-GL, having a given ratio to LD', at the fame time that LD is of no determinate magnitude. For, if poffible, let GLD be obtufe (fig. 6.), and let the perpendicular from G to AB meet it in V, therefore V is given: and fince GD-LG LD2+ 2DLXLV; therefore, by the fuppofition, LD+2Dİ XLV must have a given ratio to LD2; therefore the ratio of LD to DLXVL, that is, of LD to VL, is given, fo that VL being given in magnitude, LD is alfo given. But this is contrary to the fuppofition; for LD is indefinite by hypothefis, and therefore GLD cannot be obtufe, nor any other than a right angle. The conclufion here drawn immediately from the indetermination of LD would be deduced, according to Dr Simfon's method, by affuming another point D' any how, and from the fuppofition that GD-GL': LD" : : GD2-GL': LD, it would easily appear that GLD must be a right angle, and the ratio that of equality.

Thefe porifms facilitate the folution of the general problems from which they are derived. For example, let three ftraight lines AB, AC, BC (fig. 5.), be given in