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Inner self.

special sensations and enters, though little suspected, into all our
higher feelings. If, as sometimes happens in serious nervous
affections, the whole body or any part of it should lose common
sensibility, the whole body or that part is at once regarded as
strange and even as hostile. In some forms of hypochondria, in
which this extreme somatic insensibility and absence of zest leave
the intellect and memory unaffected, the individual doubts his
own existence or denies it altogether. Ribot cites the case of such
a patient who, declaring that he had been dead for two years,
thus expressed his perplexity :-"J'existe, mais en dehors de la
vie réelle, matérielle, et, malgré moi, rien ne m'ayant donné la
mort. Tout est mécanique chez moi et se fait inconsciemment."1
It is not because they accompany physiological functions essential
to the efficiency of the organism as an organism, but simply
because they are the most immediate and most constant sources of
feeling, that these massive but ill-defined organic sensations are
from the first the objects of the directest and most unreflecting
interest. Other objects have at the outset but a mediate interest
through subjective selection in relation to these, and never become
so instinctively and inseparably identified with self, never have the
same inwardness. This brings us to a new point. As soon as
definite perception begins, the body as an extended thing is dis-
tinguished from other bodies, and such organic sensations as can be
localized at all are localized within it. At the same time the
actions of other bodies upon it are accompanied by pleasures and
pains, while their action upon each other is not. The body also is
the only thing directly set in motion by the reactions of these
feelings, the purpose of such movements being to bring near to it
the things for which there is appetite and to remove from it those
towards which there is aversion. It is thus not merely the type of
occupied space and the centre from which all positions are reckoned,
but it affords us an unfailing and ever-present intuition of the actu-
ally felt and living self, to which all other things are external, more
or less distant, and at times absent altogether. The body then first
of all gives to self a certain measure of individuality, permanence,
and inwardness.

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such and such an aim in life. The main instrument in the forma-
tion of this concept, as of others, is language, and especially the
social intercourse that language makes possible. Up to this point
the presentation of self has shaped that of not-self, that is to say,
external things have been comprehended by the projection of its
characteristics. But now the order is in a sense reversed: the indi-
vidual advances to a fuller self-knowledge by comparing the self
within with what is first discernible in other persons without. So
far avant l'homme est la société ; it is through the "us" that we learn
of the "me" (comp. p. 75 note 1). Collective action for common ends
is of the essence of society, and in taking counsel together for the
good of his tribe each one learns also to take counsel with himself
for his own good on the whole; with the idea of the common weal
arises the idea of happiness as distinct from momentary gratification.
The extra-regarding impulses are now confronted by a reasonable
self-love, and in the deliberations that thus ensue activity attains
to its highest forms, those of thought and volition. In the first
we have a distinctly active manipulation of ideas as compared
with the more passive spectacle of memory and imagination.
Thereby emerges a contrast between the thinker and these objects
of his thought, including among them the mere generic image of
self, from which is now formed this conception of self as a person.
A similar, even sharper, contrast also accompanies the exercise of
what is very misleadingly termed "self-control," i.e., control by this
personal self of "the various natural affections," to use Butler's
phrase, which often hinder it as external objects hindered them. It
is doubtful whether the reasoning, regulating self is commonly
regarded as definitely localized. The effort of thinking and concen-
trating attention upon ideas is no doubt referred to the brain, but
this is only comparable with the localization of other efforts in the
limbs; when we think we commonly feel also, and the emotional
basis is of all the most subjective and inalienable. If we speak of
this latest phase of self as par excellence "the inner self" such
language is then mainly figurative, inasmuch as the contrasts just
described are contrasts into which spatial relations do not enter.

But with the development of ideation there arises within this
what we may call an inner zone of self, having still more unity
and permanence. We have at this stage not only an intuition of
the bodily self doing or suffering here and now, but also memories
of what it has been and done under varied circumstances in the
past. External impressions have by this time lost in novelty and
become less absorbing, while the train of ideas, largely increased in
number, distinctness, and mobility, diverts attention and often shuts
out the things of sense altogether. In all such reminiscence or
reverie a generic image of self is the centre, and every new image
as it arises derives all its interest from relation to this; and so
apart from bodily appetites new desires may be quickened and old
emotions stirred again when all that is actually present is dull and
unexciting. But desires and emotions, it must be remembered,
though awakened by what is only imaginary, invariably entail actual
organic perturbations, and with these the generic image of self comes
to be intimately combined. Hence arises a contrast between the
inner self, which the natural man locates in his breast or pphy, the
chief seat of these emotional disturbances, and the whole visible
and tangible body besides. Although from their nature they do
not admit of much ideal representation, yet, when actually present,
these organic sensations exert a powerful and often irresistible in-
fluence over other ideas; they have each their appropriate train,
and so heighten in the very complex and loosely compacted idea
of self those traits they originally wrought into it, suppressing to
an equal extent all the rest. Normally there is a certain equilibrium
to which they return, and which, we may suppose, determines the
so-called temperament, naturel, or disposition, thus securing some
tolerable uniformity and continuity in the presentation of self.
But even within the limits of sanity great and sudden changes of
mood are possible, as, e.g., in hysterical persons or those of a
"mercurial temperament,' or among the lower animals at the
onset of parental or migratory instincts. Beyond those limits—as
the concomitant apparently of serious visceral derangements or the
altered nutrition of parts of the nervous system itself-complete
"alienation" may ensue. A new self may arise, not only distinct
from the old and devoid of all save the most elementary know-
ledge and skill that the old possessed, but diametrically opposed
to it in tastes and disposition,-obscenity, it may be, taking the
place of modesty and cupidity or cowardice succeeding to generosity
or courage. The most convincing illustrations of the psycho-
logical growth and structure of the presentation of self on the lower
levels of sensation and ideation are furnished by these melancholy
spectacles of minds diseased; but it is impossible to refer to them
in detail here.

Self as a

Passing to the higher level of intellection, we come at length person. upon the concept which every intelligent being more or less distinctly forms of himself as a person, M. or N., having such and such a character, tastes, and convictions, such and such a history, and

1 Bases affectives de la Personnalité," in Revue philosophique, xviii. p. 149.

The term "reflexion" or internal perception is applied to that state Self-conof mind in which some particular presentation or group of presenta-scioustions (x or y) is not simply in the field of consciousness but there as ness. consciously related to self, which is also presented at the same time. Self here may be symbolized by M, to emphasize the fact that it is in like manner an object in the field of consciousness. The relation of the two is commonly expressed by saying, "This (2 or y) is my (M's) percept, idea, or volition; I (M) it is that perceive, think, will it." Self-consciousness, in the narrowest sense, as when we say, "I know myself, I am conscious that I am," &c., is but a special, though the most important, instance of this internal perception: here self (M) is presented in relation to self (with a difference, M'); the subject itself-at least so we say-is or appears as its own object.

"2

It has been often maintained that the difference between consciousness and reflexion is not a real difference, that to know and to know that you know are "the same thing considered in different aspects. But different aspects of the same thing are not the same thing, for psychology at least. Not only is it not the same thing to feel and to know that you feel; but it might even be held to be a different thing still to know that you feel and to know that you know that you feel,-such being the difference perhaps between ordinary reflexion and psychological introspection. The difficulty of apprehending these facts and keeping them distinct seems obviously due to the necessary presence of the earlier along with the later; that is to say, we can never know that we feel without feeling. But the converse need not be true. How distinct the two states are is shown in one way by their notorious incompatibility, the direct consequence of the limitation of attention : whatever we have to do that is not altogether mechanical is ill done unless we lose ourselves in the doing of it. This mutual exclusiveness receives a further explanation from the fact so often used to discredit psychology, viz., that the so-called introspection and indeed all reflexion are really retrospective. It is not while we are angry or lost in reverie that we take note of such states, but afterwards, or by momentary side glances intercepting the main interest, if this be not too absorbing.

But we require an exacter analysis of the essential fact in this retrospect the relation of the presentation a or y to that of self or M. What we have to deal with, it will be observed, is, implicitly at least, a judgment. First of all, then, it is noteworthy that we are never prompted to such judgments by every-day occurrences or acts of routine, but only by matters of interest, and, as said, gener

192.

2 So-misled possibly by the confusions incident to a special faculty of reflexion, which they controvert-James Mill, Analysis, i. p. 224 sq. (corrected, however, by both his editors, pp. 227 and 230), and also Hamilton, Lect., i. p. 3 It has been thought a fatal objection to this view that it implies the possi bility of an indefinite regress; but why should it not? We reach the limit of our experience in reflexion or at most in deliberate introspection, just as in space of three dimensions we reach the limit of our experience in another respect. But there is no absurdity in supposing a consciousness more evolved and explicit than our self-consciousness and advancing on it as it advances on that of the unreflecting brutes.

1

ally when these are over or have ceased to be all-engrossing. Now in such cases it will be found that some effect of the preceding state of objective absorption persists, like wounds received in battle unnoticed till the fight is over,-such, e.g., as the weariness of muscular exertion or of long concentration of attention; some pleasurable or painful after-sensation passively experienced, or an emotional wave subsiding but not yet spent ; "the jar of interrupted expectation," or the relief of sudden attainment after arduous striving, making prominent the contrast of contentment and want in that particular; or, finally, the quiet retrospect and mental rumination in which we note what time has wrought upon us and either regret or approve what we were and did. All such presentations are of the class out of which, as we have seen, the presentation of self is built up, and so form in each case the concrete bond connecting the generic image of self with its object. In this way and in this respect each is a concrete instance of what we call a state, act, affection, &c., and the judgments in which such relations to the standing presentation of self are recognized are the original and the type of all real predications (comp. p. 81). The opportunities for reflexion are at first few, the materials being as it were thrust upon attention, and the resulting "percepts " are but vague. By the time, however, that a clear conception of self has been attained the exigencies of life make it a frequent object of contemplation, and as the abstract of a series of instances of such definite self-consciousness we reach the purely formal notion of a subject or pure ego. For empirical psychology this notion is ultimate; its speculative treatment falls altogether-usually under the heading "rational psychology"—to metaphysics. Conduct. The growth of intellection and self-consciousness reacts powerfully upon the emotional and active side of mind. To describe the various sources of feeling and of desire that thus arise-esthetic, social and religious sentiments, pride, ambition, selfishness, sympathy, &c.-is beyond the scope of systematic psychology and certainly quite beyond the limits of an article like the present.2 But at least a general résumé of the characteristics of activity on this highest or rational level is indispensable. If we are to gain any oversight in a matter of such complexity it is of the first importance to keep steadily in view, as a fundamental principle, that as the causes of feeling become more complex, internal, and representative the consequent actions change in like manner. We have noted this connexion already in the case of the emergence of desires, and seen that desire in prompting to the search for means to its end is the primum movens of intellection (pp. 73-75). But intellect does much more than devise and contrive in unquestioning subservience to the impulse of the moment, like some demon of Eastern fable; even the brutes, whose cunning is on the whole of this sort, are not without traces of self-control. As motives conflict and the evils of hasty action recur to mind, deliberation succeeds to mere invention and design. In moments of leisure, the more imperious cravings being stilled, besides the rehearsal of failures or successes in the past, come longer and longer flights of imagination into the future. Both furnish material for intellectual rumination, and so we have at length (1) conceptions of general and distant ends, as wealth, power, knowledge, and-self-consciousness having arisenthe conception also of the happiness or perfection of self, and (2) maxims or practical generalizations as to the best means to these ends. Instead of actions determined by the vis a tergo of blind passion we have conduct shaped by what is literally prudence or foresight, the pursuit of ends that are not esteemed desirable till they are judged to be good. The good, it is truly urged, is not to be identified with the pleasant, for the one implies a standard and a judgment and the other nothing but a bare fact of feeling; thus the good is often not pleasant and the pleasant not good; in talking of the good, in short, we are passing out of the region of nature into that of character. It is so, and yet this progress is itself so far natural as to admit of psychological explication. As already urged (p. 72), the causes of feeling change as the constituents of consciousness change and depend more upon the form of that consciousness as that increases in complexity. When we can deliberately range to and fro in time and circumstances, the good that is not directly pleasant may indeed be preferred to what is only pleasant while attention is confined to the seen and sensible; but then the choice of such good is itself pleasant,-pleasanter than its rejection would have been. Freedom of will in the sense of absolute arbitrariness or "causeless volition," then, is at least without support from experience. The immediate affirmation of self-consciousness

They have thus a certain analogy to the presentative element in external perception, the re-presentative elements being furnished by the rest of the generic image of self. But, as this generic image is combined with and prim. anly sustained by a continuous stream of organic sensations, the analogy is

not very exact.

The paychology of a century or so ago, like the biology of the same period, was largely of the "natural history" type and was much occupied with such desorptions: writers like Dugald Stewart, Brown, and Abercrombie, e.g., draw freely from biography (and even from fiction) illustrations of the popularly received mental faculties and affections. A very complete and competent handling of the various emotions and springs of action will be found in Bain, The Emotions and the Will; Nahlowsky, Das Gefühlsleben, 2d ed., 1884, is also good.

that in the moment of action we are free must be admitted indeed, but it does not prove what it is supposed to prove the existenco of a liberum arbitrium indifferentia-but only that the relation of the end approved to the empirical self as then presented was the determining motive. This freedom of this empirical self is in all cases a relative freedom; hence at a later time we often come to see that in some past act of choice we were not our true selves, not really free. Or perhaps we hold that we were free and could have acted otherwise; and this also is true if we suppose the place of the purely formal and abstract conception of self had been occupied by some other mood of that empirical self which is continuously, but at no one moment completely, presented. It must, however, be admitted that psychological analysis in such cases is not only actually incomplete but in one respect must necessarily always remain so; and that for the simple reason that all we discern by reflexion must ever be less than all we are. That empirical self that the subject sees and even fashions is after all only its object and workmanship, not itself. If this be so, the indeterminist position, that particular acts are not fully determined by aught in consciousness, can neither be certainly established nor finally overthrown on scientific grounds; but the presumption is against it. In another sense, however, it may be allowed that freedom is possible, if not actual, viz., as synonymous with self-rule or autonomy. Freedom applies not to the ultimate source of an activity but to execution; that man is free "externally" who can do what he pleases, and when we talk of internal freedom the same meaning holds.3

ence.

BIBLIOGRAPHY.-A. Historical.-There are few good works on the history of psychology; the only one in English (R. Blakey, History of the Philosophy of Mind from the Earliest Period to the Present Time, London, 1848) is said to be worthless. F. A. Carus's Geschichte der Psychologie (Leipsic, 1808) is at least useful for referA work bearing the same title by H. Siebeck, of which only the first part has yet appeared (consisting of two divisions-(i.) Die Psychologic von Aristoteles, (ii.) Die Psychologie von Aristoteles bis zu Thomas von Aquino, Gotha, 1880 and 1884) is thoroughly and carefully done. Die Philosophie in ihrer Geschichte (I. Psychologie), by the late Professor Harms (Berlin, 1878), is also good. Ribot's La Psychologic Anglaise contemporaine (2d ed., Paris, 1875) and La Psychologie Allemande contemporaine (Paris, 1879) are lucid and concise in style, though the latter work in places is superficial and inaccurate.

B. Positive.--The most useful and complete work as an introduction, and for the English reader, is Mr Sully's well-arranged and well-written Outlines of Psychology (2d ed., London, 1885). Of more advanced text-books the late Professor Volkmann's Lehrbuch der Psychologic (2 vols., 3d ed., Köthen, 1885, edited by Cornelius) is a monument worthy the lifelong labours it entailed. Written in the main from a Herbartian standpoint, it is still the work of one who not only had read and thought over all that was worth reading by psychologists of every school but was unusually gifted with the qualities that make a good investigator and a good expositor. The importance of the Herbartian psychology to English students has been too long overlooked; while it has much in common with the English preference for empirical methods, it is in aim, if not in attainment, greatly in advance of English writers in exactness and system. Other excellent works of the same school are M. W. H. Drobisch's Empirische Psychologie (Leipsic, 1842), T. Waitz's Lehrbuch der Psychologic als Naturwissenschaft (Brunswick, 1849), and Steinthal's Einleitung in die Psychologie und Sprachwissenschaft (Berlin, 1871). To the honoured name of Lotze belongs a distinguished place in any enumeration of recent productions in philosophy; his Medicinische Psychologie (Göttingen, 1852) is still valuable; but it is out of print and scarce. A large part of his Mikrokosmos (3 vols., 3d ed., 1876-80; translated into English, 2 vols., 1885) and one book of his Metaphysik (2d ed., 1884; also translated into English) are, however, devoted to psychology. The close connexion between the study of mind and the study of the organism has been more and more recognized as the present century has advanced, and the doctrine of evolution in particular has been as fruitful in this study as in other sciences that deal with life. In this respect Mr Herbert Spencer's Principles of Psychology (2 vols., 2d ed., 1870) and Data of Ethics (1879) occupy a foremost place. Dr Bain's standard volumes, The Senses and the Intellect (3d ed., 1873) and The Emotions and the Will (3d ed., 1875), contain a good deal of "physiological psychology," but no adequate recognition of the importance of the modern theory of development; still, with the exception of Locke, perhaps no English writer has made equally important contributions to the science of mind. It is very questionable whether the time has yet come for a systematic treatment of the connexions of mind and body. Wundt's Physio logische Psychologie (2 vols., 2d ed., 1880) is rather a physiology added to a psychology than an attempt at such a systematic treatment. It is, however, a thoroughly able work by one who is both a good psychologist and a good physiologist. (J. W*.)

PSYCHOPHYSICS. See WEBER'S LAW.

3 See ETHICS, to which these questions more fitly belong.

PTARMIGAN. See GROUSE, vol. xi. p. 222. PTERODACTYLE. The extinct flying reptiles known as "pterodactyles" are among the most aberrant forms of animals, either living or extinct. Since the beginning of this century, when Blumenbach and Cuvier first described the remains of these curious creatures, they have occupied the attention of naturalists, and various opinions have been expressed as to their natural affinities. The general proportions of their bodies (excepting the larger head and neck) and the modification of the forelimb, to support a membrane for flight, remind one strongly of the bats, but the resemblance is only superficial; a closer inspection shows that their affinities are rather with reptiles and birds.

In all pterodactyles the head, neck, and forelimb are large in proportion to the other parts of the body (fig. 1). The skull is remarkably avian, and even the teeth, which

tremity of the vertebra next in front of it. The eight or nine cervical vertebræ are always large, and are succeeded by about fourteen or sixteen which bear ribs. Probably there are no vertebra which can be called lumbar. The sacrum consists of from three to six vertebræ. The tail is short in some genera and very long in others. The sternum has a distinct median crest, and the scapula and coracoid are also much like those of carinate birds. The humerus has a strong ridge for the attachment of the pectoral muscle, and the radius and ulna are separate bones. There are four distinct metacarpals; passing from the inner or radial side, the first three of these bear respectively two, three, and four phalanges, the terminal ones having had

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FIG. 1.-Pterodactylus spectabilis, Von Meyer, natural size, from the lithographic slate., humerus; ru, radius and ulna; mc, metacarpals; pt, pteroid bone 2, 3, 4, digits with claws; 5, elongated digit for support of wing membrane; st, sternum, crest not shown; is, ischium; pp, prepubis. The teeth are not

shown.

most of them possess, and which seem so unbird-like, are paralleled in the Cretaceous toothed birds of North America. Judging from the form of the skull, the brain was small, but rounded and more like that of a bird than that of a reptile. The position of the occipital condyle, beneath and not at the back of the skull, is another character pointing in the same direction. The nasal opening is not far in advance of the large orbit, and in some forms there is a lachrymo-nasal fossa between them. The premaxillæ are large, while the maxillæ are slender. In certain species the extremities of the upper and lower jaws seem to have been covered with horn, and some forms at least had bony plates around the eye. The union of the post-frontal bone with the squamosal to form a supra-temporal fossa is a reptilian character. Both jaws are usually provided with long slender teeth, but they are not always present. The vertebral column may be divided into cervical, dorsal, sacral, and caudal regions. The centra of the vertebræ are procœlous,—that is, the front of each centrum is cup-like and receives the ball-like hinder ex

FIG. 2.-Rhamphorhynchus phyllurus, Marsh, from the Solenhofen slates, onefourth natural size, with the greater part of the wing membranes preserved. x, caudal membrane; st, sternum; h, humerus; sc, scapula and coracoid; wm, wing membrane." claws. The phalanges of the outermost digit are much elongated, and except in one doubtful form are always four in number. It is the extreme elongation of this outer digit, for the support of the patagium, which is the most characteristic feature of the pterodactyle's organization. A slender bone called the "pteroid" is sometimes seen extending from the carpal region in the direction of the upper part of the humerus. Some naturalists look upon the pteroid merely as an ossification of a tendon, corre sponding with one which is found in this position in birds, while others are inclined to regard it rather as a rudimentary first digit, modified to support the edge of the patagium. The pelvis is small. In form the ilia resemble rather the ornithic than the reptilian type; but the other portions of the pelvis are more like those of the crocodiles. The hind

limb is small, and the fibula seems to have been feebly | born at Ptolemais Hermii, a Grecian city of the Thebaid. developed and fixed to the tibia. The hind foot has five It is certain that he observed at Alexandria during the digits in some forms, but only four in others. In the reigns of Hadrian and Antoninus Pius, and that he surlatter case the number of phalanges to each digit, counting vived Antoninus. Olympiodorus, a philosopher of the from the tibial side, is two, three, four, five respectively. Neoplatonic school who lived in the reign of the emperor The long bones and vertebræ, as well as some parts of the Justinian, relates in his scholia on the Phado of Plato skull, contained large pneumatic cavities similar to those that Ptolemy devoted his life to astronomy and lived for found in birds. There can be little doubt that the ptero- forty years in the so-called Пrepà тoû Kavúßov, probably dactyles had the power of sustained flight. The large size elevated terraces of the temple of Serapis at Canopus near of the sternal crest indicates a similar development of the Alexandria, where they raised pillars with the results of pectoral muscles and a corresponding strength in the arms. his astronomical discoveries engraved upon them. This The form of the forelimb, especially its outer digit, indi- statement is probably correct; we have indeed the direct cates in no uncertain manner that it supported a flying evidence of Ptolemy himself that he made astronomical obmembrane; but within the last few years this has been servations during a long series of years; his first recorded more clearly demonstrated by the discovery of a specimen observation was made in the eleventh year of Hadrian, in the Solenhofen slates with the membrane preserved 127 A.D., and his last in the fourteenth year of Antoninus, (fig. 2). The occurrence of pterodactyle remains in marine 151 A.D. Ptolemy, moreover, says, "We make our obserdeposits would seem to indicate that they frequented the vations in the parallel of Alexandria." St Isidore of Seville seashore; and it is tolerably certain that those forms with asserts that he was of the royal race of the Ptolemies, and long and slender teeth were, in part at least, fish-eaters. even calls him king of Alexandria; this assertion has been Seeing, however, that the armature of the jaws varies followed by others, but there is no ground for their opinion. considerably in the different genera, it is most likely that Indeed Fabricius shows by numerous instances that the their diet varied accordingly. name Ptolemy was common in Egypt. Weidler, from whom this is taken, also tells us that according to Arabian tradition Ptolemy lived to the age of seventy-eight years; from the same source some description of his personal appearance has been handed down, which is generally considered as not trustworthy, but which may be seen in Weidler, Historia Astronomiæ, p. 177, or in the preface to Halma's edition of the Almagest, p. lxi. Ptolemy's work as a geographer is treated of below (p. 91 sq.), and an account of the discoveries in astronomy of Hipparchus and Ptolemy has been given in the article ASTRONOMY. Their contributions to pure mathematics have not yet been noticed in the present work. Of these the chief is the foundation of trigonometry, plane and spherical, including the formation of a table of chords, which served the same purpose as our table of sines. This branch of mathematics was created by Hipparchus for the use of astronomers, and its exposi tion was given by Ptolemy in a form so perfect that for be compared with the doctrine as to the motion of the 1400 years it was not surpassed. In this respect it may heavenly bodies so well known as the Ptolemaic system, which was paramount for about the same period of time. maic system was then overthrown, the theorems of HipThere is, however, this difference, that, whereas the Ptoleparchus and Ptolemy, on the other hand, will be, as Delambre says, for ever the basis of trigonometry. The astronomical and trigonometrical systems are contained in μαθηματικὴ σύνταξις, the great work of Ptolemy 'H panatiky σivτagis, or, as Fabricius after Syncellus writes it, Μεγάλη σύνταξις τῆς ἀστρονομίας; and in like manner Suidas says οὗτος [Πτολ.] "ypaye rov péyav doтpoνóμov To σúvтagi. The Syntaxis τὸν μέγαν ἀστρονόμον ἤτοι σύνταξιν. of Ptolemy was called Ὁ μέγας ἀστρονόμος to distinguish it from another collection called Ὁ μικρὸς ἀστρονόμος, also highly esteemed by the Alexandrian school, which contained some works of Autolycus, Euclid, Aristarchus, Theodosius of Tripolis, Hypsicles, and Menelaus. To designate the great work of Ptolemy the Arabs used the superlative μεγίστη, from which, the article al being prefixed, the known, is derived. hybrid name Almagest, by which it is now universally

Pterodactyles present so many avian peculiarities that it has been proposed to place them in a special group, to be called Ornithosauria, which would hold a position intermediate between Aves and Reptilia. On the other hand, pterodactyles are thought by most authorities to have a closer relationship with the reptiles, and the different genera are placed in a separate order of the Reptilia called Pterosauria. The most important genera are five. (1) Pterodactylus; these have the jaws pointed and toothed to their extremities, and the tail very short. (2) Rhamphorhynchus (fig. 2); this genus has the jaws provided with slender teeth, but the extremities of both mandible and upper jaw are produced into toothless beaks, which were probably covered with horn; the tail is extremely long. (3) Dimorphodon; in this form the anterior teeth in both upper and lower jaws are long, but those at the hinder part of the jaws are short; the tail is extremely long. (4) Pteranodon; similar in most respects to Pterodactylus, but the jaws are devoid of teeth. In these four genera the outer digit of the manus has four phalanges. (5) Ornithopterus; this form is said to have only two phalanges in the outer digit of the manus; the genus, however, is very imperfectly known, and it has been suggested that it may perhaps be a true bird. The Pterosauria are only known to have lived during the Mesozoic period. They are first met with in the Lower Lias, the DimorPhoton macronys from Lyme Regis being perhaps the earliest known species. The Jurassic slates of Solenhofen have yielded a large number of beautifully preserved examples of Pterodactylus and Rhamphorhynchus, and remains of the same genera have been found in England in the Stonesfield slate. have also been obtained in some abundance from the Cretaceous Bones of pterodactyles phosphatic deposits near Cambridge; and their remains have been met with occasionally in the Wealden and Chalk of Kent. The Peranodon is only known from the Upper Cretaceous rocks of North America. The Pterosauria were for the most part of moderate or small size (see fig. 1), but some attained to very considerable dimensions; for instance, Rhamphorhynchus Bucklandi from the Stonesfield slate probably measured 7 feet between the Rhamphorhynchus Buckland; wing-tips. But the largest forms existed apparently towards the close Ce of the Mesozoic period, the pterodactyles of the British Cretaceous rocks and the American Pteranodon being of still larger size: some of them, it is calculated, must have had wings at least

20 feet in extent.

See Buckland, Bridgewater Treatise, 1836; Cuvier, Ossements fossiles, vol. v. pp. 350 (1824); Huxley, "On Rhamphorhynchus Bucklandi," in Quart. Journ. Gen. Soc., vol. xv. p. 658 (1859), and Anatomy of Vertebrated Animals (1871), 26; Marsh, "Notice of New Sub-order of Pterosauria (Pteranodon)," Amer. Journ. Sei, and Art, vol. xi. p. 507 (1876), and on the "Wings of Pterodac tyles in Amer. Journ, Science, vol. xxiii. p. 251 (1882): Owen, Palarontogion Society (1851, 1859, 1860); Seeley, Ornithosauria (1870); Von Meyer, Reptilien aus dem lithograph. Schiefer (Fauna der Vorwelt) (1859), and Palæontographion, vol. x. p. 1 (issi). (E. T. N.)

PTOLEMIES, the Macedonian dynasty of sovereigns of Egypt. See EGYPT, vol. vii. pp. 745-748, and MACEDONIAN EMPIRE, vol. xv. p. 144.

PTOLEMY (CLAUDIUS PTOLEMEUS), celebrated as a mathematician, astronomer, and geographer. He was a native of Egypt, but there is an uncertainty as to the place of his birth; some ancient manuscripts of his works describe

We proceed now to consider the trigonometrical work of Hipparchus and Ptolemy. In the ninth chapter of the first book of the Almagest Ptolemy shows how to form a table of chords. He supposes the circumference divided into 360 equal parts (Tμhμata), and then bisects each of these parts. Further, he divides the diameter

1 Weidler and Halma give the ninth year; in the account of the eclipse of the moon in that year Ptolemy, however, does not say, as

him as of Pelusium, but Theodorus Meliteniota, a Greek in other similar cases, he had observed, but it had been observed

writer on astronomy of the 12th century, says that he was

(Almagest, iv. 9).

into 120 equal parts, and then for the subdivisions of these he employs the sexagesimal method as most convenient in practice, i.e., he divides each of the sixty parts of the radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts. In the Latin translation these subdivisions become "partes minutæ prima" and "partes minuta secundæ," whence our "minutes" and "seconds" have arisen. It must not be supposed, however, that these sexagesimal divisions are due to Ptolemy; they must have been familiar to his predecessors, and were handed down from the Chaldæans. Nor did the formation of the table of chords originate with Ptolemy; indeed, Theon of Alexandria, the father of Hypatia, who lived in the reign of Theodosius, in his commentary on the Almagest says expressly that Hipparchus had already given the doctrine of chords inscribed in a circle in twelve books, and that Menelaus had done the same in six books, but, he continues, every one must be astonished at the ease with which Ptolemy, by means of a few simple theorems, has found their values; hence it is inferred that the method of calculation in the Almagest is Ptolemy's own.

As starting-point the values of certain chords in terms of the diameter were already known, or could be easily found by means of the Elements of Euclid. Thus the side of the hexagon, or the chord of 60°, is equal to the radius, and therefore contains sixty parts. The side of the decagon, or the chord of 36°, is the greater segment of the radius cut in extreme and mean ratio, and therefore contains approximately 37P 4′ 55′′ parts, of which the diameter contains 120 parts. Further, the square on the side of the regular pentagon is equal to the sum of the squares on the sides of the regular hexagon and of the regular decagon, all being inscribed in the same circle (Eucl. XIII. 10); the chord of 72° can therefore be calculated, and contains approximately 70° 32′ 3′′. In like manner, the square on the chord of 90°, which is the side of the inscribed square, is twice the square on the radius; and the square on the chord of 120°, or the side of the equilateral triangle, is three times the square on the radius; these chords can thus be calculated approximately. Further, from the values of all these chords we can calculate at once the chords of the arcs which are their supplements.

This being laid down, we now proceed to give Ptolemy's exposition of the mode of obtaining his table of chords, which is a piece of geometry of great elegance, and is indeed, as De Morgan says, of the most beautiful in the Greek writers."

one

He takes as basis and sets forth as a lemma the well-known theorem, which is called after him, concerning a quadrilateral inscribed in a circle: The rectangle under the diagonals is equal to the sum of the rectangles under the opposite sides. By means of this theorem the chord of the sum or of the difference of two arcs whose chords are given can be easily found, for we have only to draw a diameter from the common vertex of the two arcs the chord of whose sum or difference is required, and complete the quadrilateral; in one case a diagonal, in the other one of the sides is a diameter of the circle. The relations thus obtained are equivalent to the fundamental formulæ of our trigonometry

sin (A+B)=sin A cos B+cos A sin B, sin (AB)=sin A cos B-cos A sin B, which can therefore be established in this simple way. Ptolemy then gives a geometrical construction for finding the chord of half an arc from the chord of the are itself. By means of the foregoing theorems, since we know the chords of 72° and of 60°, we can find the chord of 12°; we can then find the chords of 6o, 3°, 110, and three-fourths of 1°, and lastly, the chords of 4, 7, 9°, 10, &c.,—all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3. Performing the calculations, he finds that the chord of 10 contains approximately 1P 34' 55", and the chord of three-fourths of 1° contains OP 47' 8". A table of chords of arcs increasing by 13° can thus be formed; but this is not sufficient for Ptolemy's purpose, which was to frame a table of chords increasing by half a degree. This could be effected if he knew the chord of one-half of 1°; but, since this chord cannot be found geometrically from the chord of 13°, inasmuch as that would come to the trisection of an angle, he proceeds to seek in the first place the chord of 1o, which he finds approximately by means of a lemma of great elegance, due probably to Apollonius. It is as follows: If two unequal chords be inscribed in a circle, the greater will be to the less in a less ratio than the are described on the greater will be to the arc described on the less. Having proved this theorem, he proceeds to employ it in order to find approximately the chord of 1°, which he does in the following manner

chord 60' 60 chord 45'

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4

4

45,2.e.,<,..chord 1°< chord 45';

3

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as the approximate value of the chord of 1°. The chord of 1° being thus known, he finds the chord of one-half of a degree, the approximate value of which is OP 31′ 25′′, and he is at once in a position to complete his table of chords for arcs increasing by half a degree. Ptolemy then gives his table of chords, which is arranged in three columns; in the first he has entered the arcs, increasing by half-degrees, from 0° to 180°; in the second he gives the values of the chords of these arcs in parts of which the diameter contains 120, the subdivisions being sexagesimal; and in the third he has inserted the thirtieth parts of the differences of these chords for each half-degree, in order that the chords of the intermediate arcs, which do not occur in the table, may be calculated, it being assumed that the increment of the chords of arcs within the table for each interval of 30' is proportional to the increment of the arc.1

For brevity we use modern notation. It has been shown that the chord of 45' is OP 47' 8" q.p., and the chord of 90' is 1o 34′ 15′′ q.p. ; hence it follows that approximately

chord 1° <1P 2′ 50′′ 40′′ and > 1o 2′ 50′′.

Since these values agree as far as the seconds, Ptolemy takes 1o 2′ 50′′

Trigonometry, we have seen, was created by Hipparchus for the use of astronomers. Now, since spherical trigonometry is directly applicable to astronomy, it is not surprising that its development was prior to that of plane trigonometry. It is the subject-matter of the eleventh chapter of the Almagest, whilst the solution of plane triangles is not treated separately in that work.

To resolve a plane triangle the Greeks supposed it to be inscribed in a circle; they must therefore have known the theorem-which is the basis of this branch of trigonometry--The sides of a triangle are proportional to the chords of the double arcs which measure the angles opposite to those sides. In the case of a right-angled triangle this theorem, together with Eucl. I. 32 and 47, gives the complete solution. Other triangles were resolved into rightangled triangles by drawing the perpendicular from a vertex on the opposite side. In one place (Alm., vi. c. 7; vol. i. p. 422, ed. Halma) Ptolemy solves a triangle in which the three sides are given by finding the segments of a side made by the perpendicular on it from the opposite vertex. It should be noticed also that the eleventh chapter of the first book of the Almagest contains incidentally some theorems and problems in plane trigonometry. The problems which are met with correspond to the following: Divide a given arc into two parts so that the chords of the doubles of those arcs shall have a given ratio; the same problem for external section. Lastly, it may be mentioned that Ptolemy (Alm., vi. 7; vol. i 8 30 p. 421, ed. Halma) takes 3o 8′ 30′′, i.e., 3+. + =3.1416, as 60 3600 the value of the ratio of the circumference to the diameter of a circle, and adds that, as had been shown by Archimedes, it lies between 3 and 341.

The foundation of spherical trigonometry is laid in chapter xi. on a few simple and useful lemmas. The starting-point is the wellknown theorem of plane geometry concerning the segments of the sides of a triangle made by a transversal: The segments of any side are in a ratio compounded of the ratios of the segments of the other two sides. This theorem, as well as that concerning the inscribed quadrilateral, was called after Ptolemy-naturally, indeed, since no reference to its source occurs in the Almagest. This error was corrected by Mersenne, who showed that it was known to Menelaus, an astronomer and geometer who lived in the reign of the emperor Trajan. The theorem now bears the name of Menelaus, though most probably it came down from Hipparchus; Chasles, indeed, thinks that Hipparchus deduced the property of the spherical triangle from that of the plane triangle, but throws the origin of the latter further back and attributes it to Euclid, suggesting that it was given in his Porisms. Carnot made this theorem the basis of his theory of transversals in his essay on that subject. It should be noticed that the theorem is not given in the Almagest in the general manner stated above; Ptolemy considers two cases only of the theorem, and Theon, in his commentary on the Almagest, has added two more cases. The proofs, however, are general. Ptolemy then lays down two lemmas: If the chord of an arc of a circle be cut in any ratio and a diameter be drawn through the point of section, the diameter will cut the arc into two parts the chords of whose doubles are in the same ratio as the segments of the chord; and a similar theorem in the case when the chord is cut externally in any ratio. By means of these two lemmas Ptolemy deduces in an ingenious manner-easy to follow, but difficult to discover from the theorem of Menelaus for a plane triangle the corresponding theorem for a spherical triangle: If the sides of a spherical triangle be cut by an arc of a great circle, the chords of the doubles of the segments of any one side will be to each other in a ratio compounded of the ratios of the chords of the doubles of the segments of the other two sides. Here, too, the theorem is not stated generally; two cases only are considered, corresponding to The the two cases given in plano. Theon has added two cases. proofs are general. By means of this theorem four of Napier's formulæ for the solution of right-angled spherical triangles can be easily

1 Ideler has examined the degree of accuracy of the numbers in these tables and finds that they are correct to five places of decimals.

2 On the theorem of Menelaus and the rule of six quantities, see Chasles, Aperçu Historique sur l'Origine et Développement des Méthodes en Géométrie, note vi. p. 291.

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