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Porism. tio, the point F is in the circumference of a given circle, we have a locus. But when conversely it is said, if a circle ABC, of which the centre is O, be given by position, as also a point E; and if D be taken in the line EO, so that EOXOD=AO'; 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 same with that of EA to AD, we have a local theorem.

Fig. 4.

Lastly, when it is said, if a circle ABC be given by position, and also a point E, a point D may be found such that if EF, FD be inflected from E and D to any point F in the circumference ABC, these lines shall have a given ratio to one another, the proposition becomes a porism, and is the same that has just now been investigated.

Hence it is evident, that the local theorem is changed into a porism, by leaving out what relates to the determination of D, and of the given ratio. But though all propositions formed in this way from the conversion of loci, are porisms, yet all porisms are not formed from the conversion of loci; the first, for instance of the preceding cannot by conversion be changed into a locus; therefore Fermat's idea of porisms, founded upon this circumstance, could not fail to be imperfect.

To confirm the truth of the preceding theory, it may be added, that Professor Dugald Stewart, in a paper read a considerable time ago before the Philosophical Society of Edinburgh, defines a porism to be "A proposition affirming the possibility of finding one or more conditions of an indeterminate theorem ;" where, by an indeterminate theorem, he means one which expresses a relation between certain quantities that are determinate and certain others that are indeterminate; a definition which evidently agrees with the explanation which has been here given.

If the idea which we have given of these propositions be just, it follows, that they are to be discovered by considering those cases in which the construction of a problem fails, in consequence of the lines which by their intersection, or the points which by their position, were to determine the problem required, happening to coincide with one another. A porism may therefore be deduced from the problem to which it belongs, just as propositions concerning the maxima and minima of quantities are deduced from the problems of which they form limitations; and such is the most natural and obvious analysis of which this class of propositions ad

mits.

The following porism is the first of Euclid's, and the first also which was restored. It is given here to exemplify the advantage which, in investigations of this kind, may be derived from employing the law of continuity in its utmost extent, and pursuing porisms to those extreme cases where the indeterminate magnitudes increase ad infinitum.

This porism may be considered as having occurred in the solution of the following problem: Two points A, B, (fig. 4.) and also three straight lines DE, FK, KL, being given in position, together with two points H and M in two of these lines, to inflect from A and B to a point in the third line, two lines that shall cut off from KF and KL two segments, adjacent to the given points H and M, having to one another the given ratio of a to B. Now, to find whether a porism be connected with this VOL. XVII. Part I.

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problem, suppose that there is, and that the following Porism, proposition is true. Two points A and B, and two straight lines DE, FK, being given in position, and also a point H in one of them, a line LK may be found, and also a point in it M, both given in position, such that AE and BE inflected from the points A and B to any point whatever of the line DE, shall cut off from the other lines FK and LK segments HG and MN adjacent to the given points H and M, having to one another the given ratio of a to s.

:LN=

First, let AF, BE', be inflected to the point E', so that AE' may be parallel to FK, then shall E'B be parallel to KL, the line to be found; for if it be not parallel to KL, the point of their intersection must be at a finite distance from the point M, and therefore making as ẞ to a, so this distance to a fourth proportional, the distance from H at which AE' intersects FK, will be equal to that fourth proportional. But AE' does not intersect FK, for they are parallel by construction; therefore BE' cannot intersect KL, which is therefore parallel to BE', a line given in position. Again, let AE", BE", be inflected to E", so that AE" may pass through the given point H: then it is plain that BE" must pass through the point to be found M; for if not, it may be demonstrated just as above, that AE" does not pass through H, contrary to the supposition. The point to be found is therefore in the line E"B, which is given in position. Now if from E there be drawn EP parallel to AE', and ES parallel to BE', BS: SE :: BL SEX BL PE AF and AP: PE::AF: FG-BS AP therefore FG: LN:: PEXAF SEX BL :: PEXAF AP BS XBS: SEX BLX AP; 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 same ratios of AF to BL, AE' to BE', and DB to DA; therefore FH: LM :: FG: NL (and consequently) :: HG : MN; but the ratio of HG to MN is given, being by supposition the same as that of a to ß; the ratio of FH to LM is therefore also given, and FH being given, LM is given in magnitude. Now LM is parallel to BE', a line given in position; therefore M is in a line QM, parallel to AB, and given in position; therefore the point M, and also the line KLM, drawn through it parallel to BE', are given in position, which were to be found. Hence this construction: From A draw AE' parallel to FK, so as to meet DE in E'; join BE', and take in it BQ, so that a ß:: HF: BQ, and through Q draw QM parallel to AB. 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 porism; for if in QB, produced on the other side 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 m n, cut Dd

off

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Poriam. tio, the point F is in the circumference of a given circle, we bave a locus. But when conversely it is said, if a circle ABC, of which the centre is O, be given by position, as also a point E; and if D be taken in the line EO, so that EO×OD=AO'; and if from E and D the lizce EF, DF be inflected to any point of the circumference ABC, the ratio of EF to DF will be given, viz. the same with that of EA to AD, we have a local theorem.

Fig. 4.

Lastly, when it is said, if a circle ABC be given by position, and also a point E, a point D may be found Buch that if EF, FD be inflected from E and D to any point F in the circumference ABC, these lines shall have a given ratio to one another, the proposition becomes a porism, and is the same that has just now been investigated.

Hence it is evident, that the local theorem is changed into a porism, by leaving out what relates to the determination of D, and of the given ratio. But though all propositions formed in this way from the conversion of loci, are porisms, yet all porisms are not formed from the conversion of loci; the first, for instance of the preceding cannot by conversion be changed into a locus; therefore Fermat's idea of porisms, founded upon this circumstance, could not fail to be imperfect.

To confirm the truth of the preceding theory, it may be added, that Professor Dugald Stewart, in a paper read a considerable time ago before the Philosophical Society of Edinburgh, defines a porism to be "A proposition affirming the possibility of finding one or more conditions of an indeterminate theorem ;" where, by an indeterminate theorem, he means one which expresses a relation between certain quantities that are determinate and certain others that are indeterminate; a definition which evidently agrees with the explanation which has been here given:

If the idea which we have given of these propositions be just, it follows, that they are to be discovered by considering those cases in which the construction of a problem fails, in consequence of the lines which by their intersection, or the points which by their posítion, were to determine the problem required, happening to coincide with one another. A porism may therefore be deduced from the problem to which it belongs, just as propositions concerning the maxima and minima of quantities are deduced from the problems of which they form limitations; and such is the most natural and obvious analysis of which this class of propositions ad

mits.

The following porism is the first of Euclid's, and the first also which was restored. It is given here to exemplify the advantage which, in investigations of this kind, may be derived from employing the law of continuity in its utmost extent, and pursuing porisms to those extreme cases where the indeterminate magnitudes increase ad infinitum.

This porism may be considered as having occurred in the solution of the following problem: Two points A, B, (fig. 4.) and also three straight lines DE, FK, KL, being given in position, together with two points H and M in two of these lines, to inflect from A and B to a point in the third line, two lines that shall cut off from KF and KL two segments, adjacent to the given points H and M, having to one another the given ratio of a to B. Now, to find whether a porism be connected with this VOL. XVII. Part I.

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problem, suppose that there is, and that the following Porism, proposition is true. Two points A and B, and two straight lines DE, FK, being given in position, and also a point H in one of them, a line LK may be found, and also a point in it M, both given in position, such that AE and BE inflected from the points A and B to any point whatever of the line DE, shall cut off from the other lines FK and LK segments HG and MN adjacent to the given points H and M, having to one another the given ratio of a to ß.

:LN=

First, let AF, BE', be inflected to the point E', so that AE' may be parallel to FK, then shall E'B be parallel to KL, the line to be found; for if it be not parallel to KL, the point of their intersection must be at a finite distance from the point M, and therefore making as ẞ to a, so this distance to a fourth proportional, the distance from H at which AE' intersects FK, will be equal to that fourth proportional. But AE' does not intersect FK, for they are parallel by construction; therefore BE' cannot intersect KL, which is therefore parallel to BE', a line given in position. Again, let AE", BE", be inflected to E", so that AE" may pass through the given point H: then it is plain that BE" must pass through the point to be found M; for if not, it may be demonstrated just as above, that AE" does not pass through H, contrary to the supposition. The point to be found is therefore in the line E"B, which is given in position. Now if from E there be drawn EP parallel to AE', and ES parallel to BE', BS: SE :: BL SEX BL PE=AF and AP: PE::AF: FG9 BS AP PEXAF SEX BL therefore FG: LN :: :: PEXAF AP BS XBS: SEX BLX AP; 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 AF", BE" are inflected to it, the ratio of FH to LM is compounded of the same ratios of AF to BL, AE' to BE', and DB to DA; therefore FH: LM:: FG: NL (and consequently): HG : MN; but the ratio of HG to MN is given, being by supposition the same as that of a to ß; the ratio of FH to LM is therefore also given, and FH being given, LM is given in magnitude. Now LM is parallel to BE', a line given in position; therefore M is in a line QM, parallel to AB, and given in position; therefore the point M, and also the line KLM, drawn through it parallel to BE', are given in position, which were to be found. Hence this construction: From A draw AE' parallel to FK, so as to meet DE in E'; join BE', and take in it BQ, so that a: :: 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 porism; for if in QB, produced on the other side of B, there be taken B q=BQ, and if q m be drawn parallel to AB, cutting MB in m; and if mλ be drawn parallel to BQ, the part m n, cut Dd

off

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