PORISM

PORISM. Let there be three straight lines AB, AC, CB given in position; (fig. 5) and from any point whatever in one of them, as D, let perpendiculars be drawn to the other two, as DE, DF, a point G may be found, such, that if GD be drawn from it to the point D, the square of that line shall have a given ratio to the sum of the squares 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, so that $AL\; :\; LB\; ::\; AH^2\; :\; BK^2\; ::\; AC^2\; :\; CB^2$. The point L is therefore given; and if a line N be taken, so as to have to AL the same ratio that $AB^2$ has to $AH^2$, N will be given in magnitude. Also, since $AH^2\; :\; BK^2\; ::\; AL\; :\; LB$, and $AH^2\; :\; AB^2\; ::\; AL\; :\; N$, ex equo, $BK^2\; :\; AB^2\; ::\; LB\; :\; N$. Draw LO, LM perpendicular to AC, CB. LO, LM are therefore given in magnitude. Now, because $AB^2\; :\; BK^2\; ::\; AD^2\; :\; DF^2$, $N\; :\; LB\; ::\; AD^2\; :\; DF^2$, and $DF^2\; =\; \backslash frac\{LB\}\{N\}\; \backslash cdot\; AD^2$. For the same reason $DE^2\; =\; \backslash frac\{AL\}\{N\}\; \backslash cdot\; BD^2$.

But, by the preceding lemma, $\backslash frac\{LB\}\{N\}\; \backslash cdot\; AD^2\; +\; \backslash frac\{AL\}\{N\}\; \backslash cdot\; BD^2\; =$

thesis $LO^2\; +\; LM^2$ has to $LG^2$, the same ratio as $DE^2\; +\; DF^2$ has to $DG^2$; let it be that of R to N, then $LO^2\; +\; LM^2\; =\; \backslash frac\{R\}\{N\}\; \backslash cdot\; LG^2$; and therefore $DE^2\; +\; DF^2\; =\; \backslash frac\{R\}\{N\}\; \backslash cdot\; LG^2\; +$

therefore $DG^2\; -\; LG^2$ has to $DL^2$ a constant 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, otherwise LD would be given in magnitude, contrary to the supposition. LG is therefore given in position; and since $R\; :\; N\; ::\; AB\; :\; N\; ::\; LO^2\; +\; LM^2\; :\; LG^2$; therefore the square of LG, and consequently LG, is given in magnitude. The point G is therefore given, and also the ratio of $DE^2\; =\; DF^2$ to $DG^2$, which is the same with that of AB to N.

The construction easily follows from the analysis, but it may be rendered more simple; for since $AH^2\; :\; AB^2\; ::\; AL\; :\; N$, and $BK^2\; :\; AB^2\; ::\; BL\; :\; N$; therefore $AH^2\; +\; BK^2\; ::\; AB^2\; ::\; AB\; :\; N$. Likewise, if AG, BG, be joined, $AB\; :\; N\; ::\; AH^2\; :\; AG^2$, and $AB\; :\; N\; ::\; BK^2\; :\; BG^2$; wherefore, $AB\; :\; N\; ::\; AH^2\; +\; BK^2\; :\; AG^2\; +\; BG^2$, but it was proved that $AB\; :\; N\; ::\; AH^2\; +\; BK^2\; :\; AB^2$, therefore $AG^2\; +\; BG^2\; =\; AB^2$; therefore the angle AGB is a right angle, and $AL\; :\; LG\; ::\; LG\; :\; LB$. If therefore AB be divided in L, so that $AL\; :\; LB\; ::\; AH^2\; :\; BK^2$; and if LG, a mean proportional between AL and LB, be placed perpendicular to AB, G will be the point required.

The step in the analysis, by which a second introduction of the general hypothesis is avoided, is that in which the angle GLD is concluded to be a right angle; which follows from $DG^2\; -\; LG^2$ having a given ratio to $LD^2$, at the same time that LD is of no determinate magnitude. For, if possible, let GLD be obtuse, (fig. 6.) and let the perpendicular from G to AB meet in V, therefore V is given; and since $GD^2\; -\; LG^2\; =\; LD^2\; \backslash times\; 2DL\; \backslash times\; LV$; therefore, by the supposition, $LD^2\; +\; 2DL\; \backslash times\; LV$ must have a given ratio to $LD^2$; therefore the ratio of $LD^2$ to $DL\; \backslash times\; VL$, that is, of LD to VL, is given, so that VL being given in magnitude, LD is also given. But this is contrary to the supposition; for LD is infinite by hypothesis, and therefore GLD cannot be obtuse,

Porism. nor any other than a right angle. The conclusion that is here drawn immediately from the indetermination of LD would be deduced, according to Dr. Simson's method, by assuming another point D' any how, and from the supposition that $GD^2\; :\; GL^2\; ::\; LD^2\; :\; LD\text{'}^2$, it would easily appear that GLD must be a right angle, and the ratio that of equality.

These porisms facilitate the solution of the general problems from which they are derived. For example, let three straight lines AB, AC, BC, (fig. 5), be given in position, and also a point R, to find a point D in one of the given lines, so that DE and DF being drawn perpendicular to BC, AC, and DR, joined; $DE^2\; +\; DF^2$ may have to DR² a given ratio. It is plain, that having found G, the problem would be nothing more than to find D, such that the ratio of GD² to DR², and therefore that of GD to DR, might be given, from which it would follow, that the point D is in the circumference of a given circle, as is well known to geometers.

The same porism also assists in the solution of another problem. For if it were required to find D such that $DE^2\; +\; DF^2$ might be a given space; having found G, DG² would have to $DE^2\; +\; DF^2$ a given ratio, and DG would therefore be given; whence the solution is obvious.

The connection of this porism with the impossible case of the problem is evident; the point L, being that from which, if perpendiculars be drawn to AC and CB, the sum of their squares is the least possible. For since $DF^2\; +\; DE^2\; :\; DG^2\; ::\; LO^2\; +\; LM^2\; :\; LG^2$; and since LG is less than DG, $LO^2\; +\; LM^2$ must be less than $DF^2\; +\; DE^2$.

It is evident from what has now appeared, that in some instances at least there is a close connection between these propositions and the maxima or minima, and of consequence the impossible cases of problems. The nature of this connection requires to be farther investigated, and is the more interesting because the transition from the indefinite to the impossible case seems to be made with wonderful rapidity. Thus in the first proposition, though there be not properly speaking an impossible case, but only one where the point to be found goes off in infinitum, it may be remarked, that if the given point F (fig. 1), be anywhere out of the line HD, the problem of drawing GB equal to GF is always possible, and admits of just one solution; but if F be in DH, the problem admits of no solution at all, the point being then at an infinite distance, and therefore impossible to be assigned. There is, however, this exception, that if the given point be at K in this same line, DH is determined by making DK equal to DL. Then every point in the line DE gives a solution, and may be taken for the point G. Here therefore the case of numberless solutions, and of no solution at all, are as it were conterminal, and so close to one another, that if the given point be at K the problem is indefinite; but if it remove ever so little from K, remaining at the same time in the line DH, the problem cannot be resolved. This affinity might have been determined a priori; for, as we have seen, it is a general principle, that a problem is converted into a porism when one or two of the conditions of it necessarily involve in them some one of the rest. Suppose, then, that two of the conditions are exactly in that state which determines the third; then whilst they remain fixed or given, should that third one vary or differ ever so little from the state required by the other two, a contradiction will ensue. Therefore if, in the hypothesis of a problem, the conditions be so related to one another as to render it indeterminate, a porism is produced; but if, of the conditions thus related to one another, some one be supposed to vary, whilst the others continue the same, an absurdity follows, and the problem becomes impossible. Wherever, therefore, any problem admits both of an indeterminate and an impossible case, it is certain, that these cases are nearly related to one another,

and that some of the conditions by which they are produced Porism. are common to both.

It is supposed above, that two of the conditions of a problem involve in them a third; and wherever that happens, the conclusion which has been deduced will invariably take place. But a porism may in some cases be so simple as to arise from the mere coincidence of one condition with another, though in no case whatever can any inconsistency take place between them. There are, however, comparatively few porisms so simple in their origin, or that arise from problems where the conditions are but little complicated; for it usually happens that a problem which can become indefinite may also become impossible; and if so, the connection already explained never fails to take place.

Another species of impossibility may frequently arise from the porismatic case of a problem which will affect in some measure the application of geometry to astronomy, or any of the sciences depending upon experiment or observation. For when a problem is to be resolved by means of data furnished by experiment or observation, the first thing to be considered is, whether the data so obtained be sufficient for determining the thing sought; and in this a very erroneous judgment may be formed, if we rest satisfied with a general view of the subject. For though the problem may in general be resolved from the data with which we are provided, yet these data may be so related to one another in the case under consideration, that the problem will become indeterminate, and instead of one solution will admit of an indefinite number. This we have already found to be the case in the foregoing propositions. Such cases may not indeed occur in any of the practical applications of geometry; but there is one of the same kind which has actually occurred in astronomy. Sir Isaac Newton, in his Principia, has considered a small part of the orbit of a comet as a straight line described with an uniform motion. From this hypothesis, by means of four observations made at proper intervals of time, the determination of the path of the comet is reduced to this geometrical problem. Four straight lines being in position, it is required to draw a fifth line across them, so as to be cut by them into three parts, having given ratios to one another. Now this problem had been constructed by Dr. Wallis and Sir Christopher Wren, and also in three different ways by Sir Isaac himself in different parts of his works; yet none of these geometers observed that there was a particular situation of the lines in which the problem admitted of innumerable solutions; and this happens to be the very case in which the problem is applicable to the determination of the comet's path, as was first discovered by Boscovich, who was led to it by finding, that in this way he could never determine the path of a comet with any degree of certainty.

The preceding account of this interesting branch of the ancient geometry, is taken from a very elegant and elaborate paper, On the Origin and Investigation of Porisms, by the late Professor Playfair, published in the third volume of the Transactions of the Royal Society of Edinburgh, and also in his collected Works, (vol. iii. p. 178.) For further particulars on the subject, the reader may consult the original paper of Professor Playfair; Dr. Simson's treatise, De Porismatibus, contained in his Opera Reliqua; a paper in the fourth volume of the Edinburgh Transactions, by Professor Wallace, entitled, Some Geometrical Porisms, with examples of their application to the Solution of Problems; Dr. Traill's Account of the Life and Writings of Robert Simson, M.D.; and Professor Leslie's Geometrical Analysis. At the end of Dr. Stewart's General Theorems, above mentioned, five very remarkable porisms are enunciated, the demonstration of the first of which is given in Leybourn's Mathematical Repository, (vol. i.) and of the remaining four in the fifth volume of the same work published

Pork in 1830. The algebraic analysis may frequently be applied with great advantage in the investigation of porisms, but this manner of treating the subject does not come within the scope of the present article.