P A R P A R 255 may be illustrated by the story which is told of Lagrange. It is said that towards the end of his life lie wrote and actually took to the Institute a paper dealing with the theory of parallels. He had begun to read it ; but, before he had proceeded very far, something struck him. He stopped reading, muttered "II faut que j y songe encore," and put the paper in his pocket (De Morgan, Budget of Paradoxes, p. 173). There appears to be no doubt that the true theory first presented itself to the mind of Gauss. The history of the matter is interesting, and deserves to be more generally known than it appears to be. In his earlier days, before his career in life was determined, when he had to consider the possibility of his becoming a teacher of mathematics, he drew up a paper in which he gave a philosophical development of the elements of mathematics. It was probably in the course of this discussion (about 1792) that he first came across the difficulty of the parallel axiom. He arrived at the conclusion that geometry became a logically consistent structure only after the parallel axiom was given as part of its foundation; and he convinced himself that this axiom could not be proved, although from experience (for example, from the sum of the angles of the geodesic triangle Brocken, Hohenhagen, Inselberg) we know that it is at least very approximately true. If, on the other hand, this axiom be not granted, there follows another kind of geometry, which he developed to a considerable extent and called the antieuclidian geometry. 1 Writing to Bessel on the 27th January 1829, he says " In leisure hours now and then I have again been reflecting on a subject which with me is now nearly forty years old ; I mean the first principles of geometry; I do not know if I have ever told you my views on that matter. Here too I have carried many things to farther consolidation, and my conviction that we cannot lay the foundation of geometry completely a priori lias become if possible firmer than before. Meantime it will be long before I bring myself to work out my very extensive researches on this subject for publication, perhaps I shall never do so during my lifetime ; for I fear the outcry of the Boeotians, were I to speak out my views on the question." Bessel entered heartily into the ideas of Gauss, and urged him to publish them regardless of the Boeotians. Concerning the generality of mathematicians in his day, Gauss probably judged rightly, however, for his intimate correspondent Schumacher was, as we learn from their correspondence in 1831, unable to follow the new idea. One of the letters (Gauss to Schumacher, 12th July 1831) is of great interest because it shows us that Gauss was then in full possession of the most important propositions of what is now called hyperbolic geometry. In particular he states that in hyperbolic space the circumference of a circle of radius r is Trk( e * _ e * J, where k is a constant, which we know from experience to be infinitely great compared with any length that we can measure (supposing, he means, the space of our experience to be hyperbolic), and which in Euclid s geometry is infinite. Gauss never published these researches ; and no traces of them seem to have been found among his papers after his death. Our first knowledge of the hyperbolic geomctiy dates from the publication of the works of N. Lobatschewsky and W. Bolyai. Lobatschewsky s views were first published in a lecture before the Faculty of Mathematics and Physics in Kasan, 12th February 1826. See Frischanf, Elcmente dcr dbsolutcn Geometric, Leipsic, 1876, page 33. Speaking of a German edition of Lobatschewsky s work, which he had seen published at Berlin in 18-10, Gauss says that he finds nothing in it which is materially new to him, but that Lobatschewsky s method of development is different from his own, and is a masterly performance carried out in the true geometric spirit. The theory received its complement in the famous Habilitationsschrift of Riemann, in which the elliptic geometry for the first time appears. Beltrami, Helmholtz, Cayley, Klein, and others have greatly developed the subject ; but it is 1 Sartorius von Waltershausen, Gauss zum Ged&chtniss, Leipsic, 1856, p. 81. unnecessary to pursue its later history here, since all essential details will be found in the article MEASUREMENT, vol. xv. p. 6f>9. All that we need do is to call the attention of those who busy themselves with mental philosophy to this generalization of geometry, as one of the results of modern mathematical research which they cannot afford to. overlook. (G. CH.) PARALYSIS, 2 or PALSY, the loss of the power of muscular action due to some interruption to the nervous mechanism by means of which such action is excited (see " Nervous System " in PHYSIOLOGY). In its strict sense the term might include the loss of the influence of the nervous system or any of the bodily functions, the loss of common sensation or of any of the special senses ; but other terms have come to be associated with these latter conditions, and the word " paralysis " in medical nomenclature is usually restricted to the loss or impairment of voluntary muscular power. Paralysis is to be regarded rather as a symptom than a disease^?- se, and is generally connected with some well-marked lesion of some portion of the nervous system. According to the locality and extent of the nervous system affected, so will be the form and character of the paralysis. It is usual to regard paralysis as depending on disease either of the brain, of the spinal cord, or of the nerves distributed to parts and organs ; and hence the terms cerebral, spinal, and peripheral paralysis respectively. The distribution of the paralytic condition may be very extensive, tending to involve in greater or less measure all the functions of the body, as in the general paralysis of the insane (see INSANITY) ; or again, one half of the body may be affected, or one or more extremities, or it may be only a certain group of muscles in a part sup plied by a particular nerve. Reference can be made here only to the more common varieties of paralysis, and that merely in general terms. 1. Paralysis due to Brain Disease. Of this by far the most common form is palsy affecting one side of the body, or Hemipleyia. It usually arises from disease of the hemi sphere of the brain opposite to the side of the body affected, such disease being in the form of haemorrhage into the brain substance, or the occlusion of blood-vessels, and consequent arrest of the blood supply to an area of the brain; or again it may be due to the effect of an injury, or to a tumour or mor bid growth in the tissues of the brain. The character of the seizure and the amount of paralysis vary according to the situation of the disease or injury, its extent, and its sudden or gradual occurrence. The attack may come on as a fit of apoplexy, in which the patient becomes suddenly unconscious, and loses completely the power of motion of one side of the body ; or a like result may arise more gradually and without loss of consciousness. In either case of "complete" hemiplegia the paralysis affects more or less the muscles of the tongue, face, trunk, and extremities. Speech is thick and indistinct, and the tongue, when pro truded, points towards the paralysed side owing to the unopposed action of its muscles on the unaffected side. The muscles of the face implicated are chiefly those of mastication. The paralysed side hangs loose, and the corner of the mouth is depressed, but the muscles clos ing the eye are as a rule unimpaired, so that the eye can be shut, unlike what occurs in another form of facial paralysis (Bell s palsy). The muscles of respiration on the affected side, although weakened, are seldom wholly paralysed, but those of the arm and leg are completely powerless. Sensation may at the first be impaired, but as a rule returns soon, unless the portion of the brain affected be that which is connected with this function. Rigidity of the paralysed members is occasionally present as an early or a late symptom. In many cases of even complete hemiplegia improvement takes place after the lapse of - From irapavf.iv, to relax. Wickliffe lias palesy, and another old form of the word is parlesy .
