041 



PORISM. 



POROSITY. 



612 



cathedrals, both of which advance out very considerably. Some of the 

 porches in our larger parish churches have a room above. [PARVISE.] 

 Wooden porches are common in the smaller churches of every period 

 of English pointed Gothic architecture. 



In our ancient domestic architecture the porch, where it occurs at 

 all, forms a marked, though not always a central feature, in the 

 principal front. When it projects from the main structure, it is 

 usually carried up so as to have a room, or else what forms a bay in a 

 room, over it ; and it is not unfrequently carried up higher than the 

 rest, so as to form a kind of tower ; or else the porch is recessed 

 within the building, and presents externally merely an open arch. In 

 many Elizabethan buildings, the porch, though forming a narrow com- 

 partment of the whole front, is profusely ornamented, even where the 

 rest is quite plain. Kirby, in Northamptonshire, the seat of Lord- 

 Chancellor Hatton, offers a most elaborate, not to say extravagant, 

 example of the kind. 



PORISM (nApiaim). An intermediate class of propositions, between 

 problems and theorems, was, as we are informed by Pappus, distin- 

 guished by the ancient geometers under the name of porisms. Unfor- 

 tunately, however, the only notices of them by the ancients themselves, 

 which are found in their remaining works, occur in the ' Collectiones 

 Mathematicie ' of Pappus Alexandrinus, and the commentaries of 

 Proclus on the Elements of Euclid, hi both places so very imperfectly, 

 that till of late years mathematicians were not agreed on their exact 

 interpretation. The description of porisms by Pappus, which he gives 

 in the preface to the seventh book of his above-mentioned work, in an 

 account of Euclid's work on the subject, is, in all the manuscripts 

 which have been examined, extremely mutilated, and every attempt to 

 restore them, before the masterly hand of Robert Simson took up the 

 subject, had completely failed. The first part of the description, 

 which seems to be entire, is calculated only to excite curiosity, being 

 too general for conveying any precise notion of these propositions, or 

 for giving any effectual assistance for the recovery of them ; and the 

 remainder, containing a detail of the contents of Euclid's work, is 

 through the whole so corrupt that all endeavours to explain it were 

 nugatory. Several celebrated geometers indeed flattered themselves 

 that they had obtained possession of the secret ; but even Dr. Halley, 

 with all his acuteness, relinquished the task, and adds, after giving the 

 original, "hactenus porismatum descriptio ncc mini intellects nee 

 lectori profutura." The definition which Pappus quotes from the 

 ancienU is too general to be useful, and perhaps implied more than 

 our acquaintance with the language in which ke wrote can enable us 

 to determine. He says that a "theorem is something requiring 

 demonstration, a problem in which something U proposed to be con- 

 structed ; but a porism, that which requires investigation ; " and 

 though this definition certainly does corresiwnd to the nature of these 

 propositions, yet it is deficient in discrimination, and of itself neither 

 i . nvi-ys any precise notion of Euclid's porisms, nor gives assistance 

 in the investigation of any individual proposition. Dr. Simson's 

 restored definition is as follows, literally translated : " A purism ia a 

 proposition in which it is proposed to demonstrate that some one thing 

 or more things are given, to which, as also to each of innumerable 

 other things not given, but which have the same relation to those 

 which are given, it is to be shown that there belongs some common 

 affection described in the proposition." The following less literal 

 translation may probably be better understood : " A porism is a pro- 

 position in which it U proposed to demonstrate that one or more things 

 are given, between which and every one of innumerable other things 

 not given, but aisumed according to a given law, a certain relation 

 described in the proposition is to be shown to take place." Dr. Simson 

 illustrates the propriety and accuracy of this definition by many 

 examples, and it is so framed as to correspond with all the intimations 

 of Pappus respecting porisms, and also with the character of the few 

 individual porisms of Euclid which Dr. Simson had discovered. It 

 may therefore justly be considered as expressive of the notions on this 

 subject entertained by the ancients, although probably, as in the cases 

 of theorem and problem, no precise definition was given of porism. It 

 has been objected to Simson's definition, that it may be inferred from 

 it that a porism partaken more of the nature of a problem than a 

 theorem, and consequently is inconsistent with the "intermediate 

 nature " mentioned by Pappus. In his enunciation it is affirmed that 

 certain things may be found which shall have the relations or properties 

 Hi described. Now were it simply proposed to investigate certain 

 things which would have the properties expressed in the porism, it 

 may be regarded as a problem ; but if these things are found by a con- 

 struction described in the enunciation, the proposition becomes a 

 theorem affirming the truth of the properties asserted; and then a 

 demonstration only i required, without any investigation, in the 

 manner which appears to liave been practised by the later mathemati- 

 cians alluded to by Pappus. The enunciation of a porism as a problem 

 is not consistent with the usual character of such propositions. 

 Problems usually, whatever difficulty may attend their solution, .ire 

 almost immediately recognised, by those having some knowledge of 

 geometry, as either |>"--il>lr in certain circumstances of the data, or as 

 altogether impossible ; and it would be unusual to propose as a problem 

 " to find thing* with certain properties, respecting the possibility of which 

 no judgment can be formed without an analysis, or such consideration as 

 is equivalent to an analysis." For example, if it had been proposed as a 



ARTS AND SCt. DIV. VOL. VI. 



problem in the time of Apollonius, to find in a given parabola a point 

 having the property of the focus, that point being then unknown, such 

 a proposition would not have been considered as a proper problem, but 

 would in reality have been a porism. To take another example : 

 Proclus, in his commentaries on the Elements, mentions the first pro- 

 position of the third book, " to find the centre of a circle," as a porism, 

 being in some measure between a problem and a theorem. But 

 Proclus, however distinguished as a philosopher, was no mathematician, 

 and as a circle, from Euclid's definition of it, must have a centre, the 

 proposition to find that centre seems to be a proper problem. Had the 

 circle been defined from another of its properties, as, for instance, from 

 its being produced by the extremity of a straight line moving at right 

 angles to another straight line, given in magnitude and position, and in 

 the same plane, so that the square of the moving line be always equal 

 to the rectangle by the segments into which it divides the given line ; 

 then the finding of the centre would be a proper porism, and might be 

 enunciated thus : " within a given circle (defined in the manner just 

 mentioned) a point may be found from which all straight lines drawn 

 to the circumference will be equal." 



Having thus placed before our readers the most probable restoration 

 of the ancient meaning of the term porism, we proceed to notice briefly 

 what modern geometers have given us on the subject. First in im- 

 portance stands the admirable paper on porisms by Professor Playfair, 

 in the first volume of the ' Transactions of the Royal Society of 

 Edinburgh,' which was read before that body in July, 1784. He im- 

 proves on Simson's definition, and substitutes the following : " A 

 porism is a proposition affirming the possibility of finding such con- 

 ditions as will render a certain problem indeterminate, or capable of 

 innumerable solutions." This, it must be confessed, is an important 

 and elegant simplification, and fully conveys every idea contained in 

 the more prolix definition of Simson ; but at the same time we agree 

 with Dr. Trail in thinking that Dr. Simson's is expressed more nearly 

 in the language and manner of the ancient geometers : " Though I 

 admire the ingenuity and fully admit the soundness of this definition, 

 and also the utility of the principle on which it is founded in the 

 discovery of porisms, I must acknowledge my doubt of that particular 

 notion of a porism having ever been adopted, or even proposed, among 

 the ancient geometricians." (Trail's ' Life of Simson,' pp. 50, 51.) A 

 paper on porisms, containing some examples in the higher geometry, 

 by Lord Brougham, was inserted in the ' Philosophical Transactions of 

 the Royal Society,' in 1798. Fryer has given a popular history of the 

 discovery of porisms, in the last edition of Simson's Geometry. Lastly, 

 the most complete exposition of them that has yet appeared may be 

 found in the ' Apcrcu Historique sur 1'Origine et le Ddveloppement 

 des Me'thodes en Gdomctrie,' 4to, BruB., 1837, by M. Chasles, of the 

 French Institute : as well as in his recently published work, ' Les trois 

 livres de Porismes d'Euclide, retablies pour la premiere fois,' Paris, 

 1860, 8vo. 



Porism was also used by the Greek geometers to denote a corollary 

 to a proposition, and the frequent use of the word hi this sense, as well 

 as in the other, by Pappus and Proclus, has occasioned much con- 

 fusion. Proclus says that " corollary is one of the geometrical appel- 

 lations, but it has a twofold signification," and he proceeds to describe, 

 in a very obscure manner, the difference between the two meanings of 

 the term. 



(See Proclui in Eudidem, edit. Hervagii, fol. Basil., 1533, fol. 18. 

 We refer the reader also to Henry Savile's Prcckctiona in Eudidm, 

 4to, Oxon, 1621, p. 18 ; and Trail's Life of Simson, p. 92.) 



POROSITY is that condition of material bodies which consists in 

 the discontinuity of their molecules, the intervals between these being 

 called pores (from Tripos, a passage). Porosity is a property common 

 to all bodies in nature, at least we know none in which the particles 

 are contiguous to one another. By pores however we do not mean the 

 cantict as in sponge and cork, which'.are visible to the eye, and scarcely 

 those of other bodies which may be rendered so by the aid of a micro- 

 scope. In bodies whose pores are not manifest, the existence of the 

 intervals between the molecules is proved by various circumstances. 

 Thus many of the metals become more compact by hammering, and 

 all of them contract in bulk by a reduction of temperature. We may 

 also refer to the Florentine experiment, for determining whether or 

 not water is compressible, the fluid was by pressure forced through 

 the pores of the vessel of gold in which it was contained. Again, the 

 porosity of bodies is inferred from their elasticity and the sounds 

 which are heard'when the molecules are in a state of vibration : also, 

 in transparent bodies it is inferred from the fact that the particles of 

 light pass through them, or that the vibrations of an tetherial fluid 

 take place among the molecules. 



When salt .is dissolved in water, the particles of the salt seem to 

 introduce themselves between those of the water, so that the volume 

 of the mixture is less than the sum of the volumes of the separate 

 substances ; and the like may be said of the mixture of alcohol with 

 water ; in which cases the particles of one of the kinds of substance 

 appear to enter and occupy the spaces between the particles of the 

 other. The intervals between the particles of gaseous substances are 

 very great; and though, in some cases, the volume of a mixture is 

 equal to the um of the volumes of the separate gases, yet, in others, 

 it is equal to not more than 4, J, J, or J of the sum of the separate 

 volumes. A body of aqueous vapour composed of a volume rcpre* 



T T 



