Descriptio autem quam tradit (Pappus) porismaturn adeo brevis est et obscure, et injuria temporis aut Miter vitiate, ut nisi Deus benigne animum et vires dederit in ea petinaciter inquirere, in perpetuum forsan geometric latu isset." Simsoni Opera Reliqua, p. 513.
Dr. Trail, in his life of Simson, gives the following ac count of the discovery.
" Dr. Simson maintained for some time his resolution of abstaining from all attempts at the rediscovery of po risms ; but happening one day to be walking with some friends on the banks of the river Clyde at Glasgow, and by accident being left behind his company, he inadvertently fell into a reverie respecting porisms.
" Some new ideas struck his mind, and with his chalk having drawn some lines on an adjoining tree, at that mo ment, for the first time, he acquired a just notion of one of Euclid's porisms." § The first publication of Simson on this subject, was a paper inserted in the Philosophical 7'ransactions for the year 1723; it was not, however, until after his death, that the whole of his investigations were made public in the posthumous edition of his works, for which the mathema tical world is indebted to the munificence of the late Earl Stanhope. Some few years after, this subject attracted the attention of Mr. Mayfair, who has given a most philo sophical account of the origin of this class of propositions, and has removed whatever obscurity remained attached to them. The paper in which his views are explained, is in deed a model of that peculiarly beautiful style of writing for which he was so justly celebrated, and which is unfor tunately so rarely met with in the literary productions of mathematicians.
In the geometrical explanation of porisms, we shall avail ourselves of the light which he has thrown on the subject, and then endeavour to supply those observations which he promised respecting their algebraical investigation.
The definition of porisms which Simson has given, is unquetionably rather obscure ; and without an example of one of these propositions, it is by no means easy to com prehend its meaning : it appears therefore preferable to postpone the explanation of the term until the reader is made acquainted with the thing. The ancient geometers examined every problem on which they bestowed their at tention with the most minute scrutiny : unacquainted with the comprehensive generalization which is introduced into every geometrical problem by the application of algebra, they carefully inquired into every separate case that could cause any change in the magnitude or relative position of the data, fearful lest that mode of solution they had con trived for it in one case, might not equally apply to others.
Such a laborious course of inquiry, although adverse to rapid advancement, was well calculated to make them per fectly acquainted with every thing remarkable which the solution of the problem could present ; and it must soon have occurred to them, that in many cases the general con struction would fail, and no solution be obtained, in conse quence of some peculiar relation between the data. Such is the case if we attempt to divide a given line into two parts, whose rectangle is equal to a given square. When the given square is greater than that described on half the given line, no solution can be obtained. In such cases, the problem became impossible, and it was always found that some two at least of the data were contradictory to each other. In the illustration, we have chosen the two conditions, defining the magnitude of the line and that of the rectangle of its segments are incompatible.
When a problem contained an impossible case, another question presented itself; to determine the limits amongst the relations of the data, so that it shall just remain possi ble ; and with respect to the problem itself, to construct it so that a certain quantity, instead of being given,.shall be the greatest or least possible. The elegant construc tions to which this gave rise, under the name of maxima and minima, are well known to geometers.
These circumstances would occur when the data were but few, and the problem simple; but in the consideration of questions a little less elementary, it must have been ob served, that besides this method, by which the construc tion became useless, another of quite an opposite nature was sometimes introduced. It might happen that two lines or two circles by whose intersection the point ought to be determined, instead of cutting each other as in the general case, or not intersecting each other at all, as in the impossible one, should wholly coincide. The true interpretation of this circumstance could not long remain unnoticed. Since that point, which was common to the two lines, determined the point to be found, and in the case of two circles intersecting, there were two points in common, and therefore equally fulfilling the condition, it was natural to conclude, that when the lines or circles co incided, all points being in common, all would equally satisfy the problem. Here, then, an affinity of solutions appeared, yet they were all connected by a certain law. The reason of such a singular result, must soon have been found in the coincidence between two of the data, and thus a less number of data being given than were sufficient, the problem became indeterminate.