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Poriím. pofition, and also a point R, to find a point D in one of another, and that some of the conditions by which they Porifin. the given lines, so that DE and DF being drawn per- are produced are common to both.”
It is supposed
pendicular to BC, AC, and DR, joined ; DE+DF above, that two of the conditions of a problem involve
may have to DR’ a given ratio. It is plain, that ha. in them a third ; and wherever that happens, the con.
ving found G, the problem would be nothing more than clusion which has been deduced will invariably take
to find D, such that the ratio of GD' to DR', and place. But a porism may in son cases be so simple as
therefore that of GD to DR, might be given, the point to arise from the mere coincidence of one condition
D being in the circumference of a given circle, as is with another, though in no case whatever any incon-
well known to geometers.
sistency can take place between them.
The same porism also assists in the solution of however, comparatively few porisms so simple in their
another problem. For if it were required to find D origin, or that arise from problems where the conditions
such that DEP+DF might be a given space ; having are but little complicated; for it usually happens that
found G, DG’ would have to DE+DF a given ra- a problem which can become indefinite may also become
tio, and DG would therefore be given ; whence the so- imposible; and if so, the connection already explained
lution is obvious.
never fails to take place.
The connection of this porism with the impoffible Another species of impoffibility may frequently arise
case of the problem is evident ; the point L being that from the porismatic case of a problem which will affect
from which, if perpendiculars be drawn to AC and CB, in some measure the application of geometry to astrono-
the sum of their squares is the least possible. For fince my, or any of the sciences depending on experiment or
DF+DE': DG* :: LO'+LM: LG'; and fince observation. For when a problem is to be resolved by
LG is less than DG, LO:+LM' must be less than help of data furnished by experiment or observation,
DF’+DE”. It is evident from what has now appear- the first thing to be considered is, whether the data so
ed, that in some instances at least there is a close con-
obtained be sufficient for determining the thing fought, nection between these propofitions and the maxima or and in this a very erroneous judgment may be formed, minima, and of confequence the impossible cases of pro- if we reft satisfied with a general view of the subject; for blems. The nature of this connection requires to be tho' the problem may in general be resolved from the data farther investigated, and is the more interesting because with which we are provided, yet these data may be so: the transition from the indefinite to the impossible case related to one another in the case under consideration, seems to be made with wonderful rapidity. Thus in that the problem will become indeterminate, and instead the first proposition, though there be not properly of one folution will admit of an indefinite number. This speaking an impossible case, but only one where the we have already found to be the case in the foregoing propoint to be found goes off ad infinitum, it may be re- positions. Such cases may not indeed occur in any of the
marked, that if the given point F be anywhere out of practical applications of geometry; but there is one of Plate the line HD (fig. 1.), the problem of drawing GB the same kind which has actually occurred in astronomy. ccccx!!! equal to GF is always possible, and admits of just one Sir Ifaac Newton, in his Principia, has considered a
solution; but if F be in DH, the problem admits of small part of the orbit of a comet as a straight line de.
no folution at all, the point being then at an infinitescribed with an uniform motion. From this hypothetis,
distance, and therefore impossible to be assigned. There by means of four observations made at proper intervals
is, however, this exception, that if the given point be of time, the determination of the path of the comet is
at K in this same line, DH is determined by making reduced to this geometrical problem : Four straight
DK equal to DL. Then every point in the line DE lines being given in position, it is required to draw a
gives a solution, and may be taken for the point G. fifth line across them, so as to be cut by them into
Here therefore the case of numberless solutions, and of three parts, having given ratios to one another. Now
no solution at all, are as it were conterminal, and so close this problem had been constructed by Dr Wallis and
to one another, that if the given point be at Kthe Sir Christopher Wren, and also in three different ways
problem is indefinite; but if it remove ever so little from by Sir Ifaac hirself in different parts of his works; yet
K, remaining at the same time in the line DH, the none of these geometers observed that there was a par-
problem cannot be resolved. This affinity might have ticular situation of the lines in which the problem ad-
been determined à priori : for it is, as we have seen, a mitted of innumerable solutions : and this happens to
general principle, that a problem is converted into a po- be the very cafe in which the problem is applicable to
rism when one or when two of the conditions of it ne- the determination of the comet's path, as was first dis-
cefiarily involve in them some one of the rest. Sup. covered by the Abbé Boscovich, who was led to it
pose, then, that two of the conditions are exactly in by finding, that in this way he could never deter-
that state which determines the third; then while they mine the path of a comet with any degree of cere
remain fixed or given, should that third one vary or tainty.
differ ever so little from the state required by the other Besides the geometrical there is also an algebraical
two, a contradiction will ensue: therefore if, in the hy- analysis belonging to porisms; which, however, does not
pothesis of a problem, the conditions be so related to one belong to this place, because we give this account of
another as to render it indeterminate, a porism is pro- them merely as an article of ancient geometry; and thie
duced; but if, of the conditions thus related to one ano- ancients never employed algebra in their inveftigations.
ther, some one be supposed to vary, while the others con- Mr Playfair, professor of mathematics in the univerfity
tinue the same, an absurdity follows, and the problem of Edinburgh, has written a paper on the origin and.
becomes impoffible. Wherever, therefore, any problem geometrical investigation of porilms, which is published.
admits both of an indeterminate and an impossible case, in the third volume of the Transactions of the Royal
it is certain, that these cases are nearly related to one Society of Edinburgh, from which this account of the
Pork, subject is taken. He has there promised a second part Most of the roads and fields are so steep, that no carriages Pore to his paper, in which the algebraicai investigation of any kind can be used; all the crops are therefore
n of porisms is to be considered. This will no doubt carried in with crooks on horses, and the manure in
Porphyry throw considerable light upon the subject, as we may wooden pots called defels. Many of the poor are ema teadily judge from that gentleman's known abilities, and ployed in spinning yarn for the Dunfter manufactory. from the fpecimen he has already given us in the firit part. W. Long. 3. 32. N. Lat. 51. 14.
PORK, the ficth of swine killed for the purposes of PORO. See CALAURIA. food. See Sus.
PORPESSE, in ichthyology. See DELPHINUS.
The hog is the only domestic animal that we know PORPHYRIUS, a famous Platonic philofopher, was
of no use to man when alive, and therefore seems pro- born at Tyre in 233, in the reign of Alexander Seve-
perly designed for food. Besides, as loathsome and ugly rus. He was the disciple of Longinus, and became
to every human eye, it is killed without reluctance. the ornament of his school at Athens ; from thence he
The Pythagoreans, whether to preserve health, or on went to Rome, and attended Plotinus, with whom he
account of compassion, generaliy forbade the use of ani. lived fix years. After Plotinus's death he taught phi.
mal food; and yet it is alleged that Pythagoras reserved lofophy at Rome with great applause ; and became well
the use of hog's flesh for himself. The Jews, the skilled in polite literature, geography, astronomy, and
Egyptians, &c. and other inhabitants of warm countries, music. He lived till the end of the third century, and
and all the Mahometans at present, reject the use of died in the reign of Dioclefian. There are still extant
pork. It is difficult to find a satisfactory reason for this, his book on the Categories of Aristotle ; a Treatise on
or for the precept given to the Jews respecting it, tho' Abstinence from Flesh; and several other pieces in
unquestionably there was some good one for it
. The Greek. He also composed a large treatise against the
Greeks gave great commendations to this food; and Galen, Christian religion, which is lost. That work was an-
though indeed that is suspected to be from a particular fwered by Methodius bilaop of Tyre, and also by Eu-
fondness, is everywhere full of it. The Romans confi- febius, Apollinarius, St Augustin, St Jerome, St Cyril,
dered it as one of their delicacies ; and if some of the and Theodoret. The emperor Theodofius the Great
inhabitants of the northern climates have taken an aver- caused Porphyrius's book to be burned in 338. Those
fion to it, that probably arose from the uncultivated of his works that are still extant were printed at Cam-
state of their country not being able to rear it. Pork bridge in 1655, 8vo, with a Latin verfion.
is of a very tender structure; increased perhaps from a “ Porphyrius (lays Dr Enfield) was, it must be own-
peculiarity in its economy, viz. taking on fat more ed, a writer of deep erudition; and had his judgment and
readily than any other animal
. Pork is a white meat integrity been equal to his learning, he would have defer-
even in its adult state, and then gives out a jelly in very ved a dittinguished place among the ancients. But neither
great quantity. On account of its little perspirability the splendor of his diction, nor the variety of his reading,
and tenderness it is very nutritious, and was given for can atone for the credulity or the dishonefty which till.
that intention to the athletæ. With regard to its alka- ed the narrative parts of his works with so many extra-
lescency, no proper experiments have yet been made ; vagant tales, or interest the judicious reader in the ab-
but as it is of a gelatinous and fucculent nature, it is ftrufe subtelties and myftical flights of his philofophical Cullen's probably less so than many others. Upon the whole, writingg." Mat. Med. it appears
valuable nutriment; and the rea- PORPHYRY, a genus of stones belcnging to the son is not very obvious why it was in some countries order of faxa. It is found of several different colours, forbid. It is said that this animal is apt to be diseased; as green, deep.red, purple, black, dark-brown, and but why were not inconveniences felt on that account grey. Under the name of porphyry, Mr Kirwan and in Greece ? Again, it has been alleged, that as Palestine M. de Saussure include those stones which contain eiwould not rear these animals, and as the Jews had ther felt-spar, fchoerl, quartz, or mica, with other species learned the use of them in Egypt, it was necessary they of crystallized stone on a filiceous or calcareous ground. fould have a precept to avoid them. But the Egyp. There are a great many different kinds. M. Ferber des tians themselves did not use this meat ; and this reli- fcribes 20 varieties under four species, but in general it gious precept, indeed, as well as many others, seems is considered with relation to its ground, which is met to have been borrowed from them. Possibly, as pork with of the colours already mentioned. When the is not very perspirable, it might increase the leprosy, ground is of jasper, the porphyry is commonly very which was said to be epidemic in Palestine ; though hard ; the red generally contains felt-spar in small white .this is far from being certain.
dots or specke ; and frequently, together with these, PORLOCK, in the county of Somerset in England, is black spots of schoerl
. The green is often magnetic,
a small sea-port town fix miles west from Minehead. This and is either a jasper or schoer), with spots of quartz.
whole parish, including hamlets, contains about 110 Sometimes a porphyry of one colour contains a frag-
houses, and nearly 6oo inhabitants. The situation of the ment of another of a different colour. Those that have
very romantic, being nearly surrounded on all chert for their ground are fusible per se. The calca-
fides, except toward the sea, by fteep and lofty hills, inter- reous porphyry consists of quartz, felt-spar, and mica,
sected by deep vales and hollow glens. Some of the hills in separate grains, united by a calcareous cement ; and,
are beautifully wooded, and contain numbers of wild deer. lastly, the micaceous porphyry consists of a greenith
The valleys are very deep and picturesque; the sides be- grey micaceous ground, in which red felt-fpar and
ing steep, scarred with wild rocks,and patched with woods greenish soap-rock are inserted.
and foreit l rubs. Some of them are well cultivated The porphyry of the ancients is a moft elegant mass
and ftudd : with villages or single farms and cottages, of an extremely firm and compact structure, remarkably
Jtbough agriculture here is very imperfectly understood. heavy, and of a fine strong purple, variegated inore or