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 DF, DE, 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 $AB^2\; :\; BK^2\; ::$

sis $LO^2\; +\; LM^2$ has to $LG^2$, the same ratio as $DF^2\; +\; DE^2$ has to $DG^2$; let it be that of R to N, then $LO^2\; +\; LM^2$

Porism. $LM^2\; =\; \backslash frac\{R\}\{N\}\; \backslash cdot\; LG^2$; and therefore $DE^2\; +\; DF^2\; =\; \backslash frac\{R\}\{N\}\; \backslash cdot\; LG^2\; +\; \backslash frac\{AB\}\{N\}\; \backslash cdot\; DL^2$; but $DE^2\; +\; DF^2\; =\; \backslash frac\{R\}\{N\}\; \backslash cdot\; DG^2$; therefore, $\backslash frac\{R\}\{N\}\; \backslash cdot\; LG^2\; +\; \backslash frac\{AB\}\{N\}\; \backslash cdot\; DL^2\; =\; \backslash frac\{R\}\{N\}\; \backslash cdot\; DG^2$ and $\backslash frac\{AB\}\{N\}\; \backslash cdot\; DL^2\; =\; \backslash frac\{R\}\{N\}\; (DG^2\; -\; LG^2)$;

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 $AL^2\; :\; AB^2\; ::\; AL\; :\; N$, 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 one, 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 $DG^2\; :\; GL^2$ 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 it in $V$, therefore $V$ is given: and since $GD^2\; -\; LG^2\; =\; LD^2\; +\; 2DL\; \backslash times\; LV$; 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 indefinite by hypothesis, and therefore $GLD$ cannot be obtuse, 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 $D\text{'}$ any how, and from the supposition that $GD\text{'}^2\; -\; GL^2\; :\; LD\text{'}^2\; ::\; GD^2\; -\; GL^2\; :\; LD^2$, 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 $DR^2$ 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 $GD^2$ to $DR^2$, 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 $G$, $DG^2$ 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 $DE^2\; +\; DF^2\; :\; DG^2\; ::\; LO^2\; +\; LM^2\; :\; LG^2$; and since $LG$ is less than $DG$, $LO^2\; +\; LM^2$ must be less than $DE^2\; +\; DF^2$.

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 further 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 ad infinitum, it may be remarked, that if the given point $F$ be anywhere out of the line $HD$ (fig. 1.), 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 it is, as we have seen, a general principle, that a problem is converted into a porism when one or when 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 while 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, while 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 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 any inconsistency can take place between them. There are,

Porisms. 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 on 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 the abbé 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.

Besides the geometrical there is also an algebraical analysis belonging to porisms; which, however, does not belong to this place, because we give this account of them merely as an article of ancient geometry; and the ancients never employed algebra in their investigations. Mr Playfair, formerly professor of mathematics, and now of natural philosophy in the university of Edinburgh, has written a paper on the origin and geometrical investigation of porisms, which is published in the third volume of the Transactions of the Royal Society of Edinburgh, from which this account of the subject is taken. He has there promised a second part to his paper, in which the algebraical investigation of porisms is to be considered. This will no doubt throw considerable light upon the subject, as we may readily judge from that gentleman's known abilities, and from the specimen he has already given us in the first part.

Such as are desirous of knowing more of this subject may consult Dr Simson's treatise De Porismatibus, which

is contained in his Opera Reliqua, published after his death at the sole expence of the earl of Stanhope. We have already mentioned Dr Stewart's General Theorems, which contain many beautiful porisms, but without demonstrations. A considerable number of them, however, have been demonstrated by the late Dr R. Small, of Dundee, in the Trans. R. S. Edin. vol. ii. There is also a paper upon the subject of porisms by Mr W. Wallace, now of the Royal Military College, in the fourth volume of the same work, entitled Some Geometrical Porisms, with examples of their application to the Solution of Problems.