Showing posts with label Ancient Science. Show all posts
Showing posts with label Ancient Science. Show all posts

Sunday, May 3, 2009

The Power of Soul - The Meditations of Marcus Aurelius - Unpublished Selections Explained, Med. XI.16



Meditation XI.16 - The Power of Soul - Translated by George Long and rewritten by Russell McNeil


As to living in the best way, this power is in the soul, if it be indifferent to things which are indifferent.1 And it will be indifferent, if it looks on each of these things separately and all together,2 and if it remembers that not one of them produces in us an opinion about itself, nor comes to us;3 but these things remain immovable, and it is we ourselves who produce the judgments about them,4 and, as we may say, write them in ourselves, it being in our power not to write them,5 and it being in our power, if by necessity these judgments have imperceptibly got admission to our minds, to wipe them out;6 and if we remember also that such attention will only be for a short time, and then life will be at an end.7 Besides, what trouble is there at all in doing this?8 For if these things are according to nature, rejoice in them, and they will be easy to you:9 but if contrary to nature, seek what is conformable to your own nature, and strive towards this,10 even if it bring no reputation;11 for each of us is allowed to seek our own good.12

Explanation

(1) The Stoic soul or psyche is not the obscure intangible and indefinable object of religious traditions. Even in the ancient world the Stoics understood the soul as a materially defined and localized entity. If we take Stoic principles literally, the soul or its properties should therefore be accessible and measurable using the techniques of physics. The ancient Stoics would marvel at modern electroencephalography (EEG) and magnetoencephalography (MEG) with their detection of alpha, beta, delta, and theta waves, and other measurables of brain activity. These electromagnetically generated signals, they would conclude, are actually analogs of active mind-based human rhythms - in effect, they are direct measurements of the soul at work. These responses of the soul can be triggered by many things. The notion of "indifference" in this meditation applies to those areas of living that ought to trigger indifferent cognitive responses. This - the Stoic will claim - is a matter of choice and attitude. The pleasure-pain and fight-flight responses, in particular, are important, but only for bodily survival, and ought (in Stoicism) to trigger the appropriate cognitive responses (panic and addiction are examples of what the Stoics would say would be inappropriate cognitive responses). But these areas have nothing to do with "living in the best way," or living harmoniously with nature, or being virtuous.

(2) The Stoic would take a detached clinical perspective around areas of indifference. See indifferent things as they are, and understand why they exist, and for what purposes.

(3) The acquisition of an expensive material possession might drive us to extremes. We might work for years to save up for a fast sports car. But this object, the car, once we have it, has no "opinion" about itself. A similar argument can be made about all of our physical pleasures, or the pains that we spend an inordinate amount of time, money and energy trying to avoid.

(4) The view that maximizing pleasure and minimizing pain is the reason we exist is a judgment call. None of the pleasures or pains of life are in any sense important in terms of their influences over what really matters in life. They have no opinions about themselves.

(5) We have the power to see the difference between what is truly vital, and what is indifferent. We can "write" those differences onto our souls.

(6) We also have the power to "wipe out" those judgments that have in the past obscured our real reasons for living.

(7) This approach - to remember that it is easy to shift a long-held attitude, if we look at the brevity of life - has been subsumed into a variety of twelve-step approaches to addiction recovery. In the words of those programs, we can resolve these sometimes difficult addictions and our attitudes toward addictions by rewriting our personal scripts on a daily basis, and doing this "one day at a time."

(8) The "one day at a time" approach does work. It is a manageable and "trouble" free way to deal with seemingly intractable problems.

(9) Following nature will be painless and be a cause for rejoicing because the objects in this approach do give opinions about themselves, and those objects are both beautiful and good for those who choose to examine them.

(10) If things are contrary to nature, we ought to be indifferent toward them. We will be indifferent if we truly see them as they are, and notice that these things are indeed not conformable to our true nature.

(11) One of the difficulties we have in dealing with addictions, for example, is the isolation that our abandonment of those addictions brings. We lose the community of peers that are so often held together by the addiction. This feels like a loss of "reputation." But reputation in the Stoic scheme is nothing other than a form of fame - something we seek to bring comfort to our ego. It is really self-serving, and in opposition to nature. Our true goal in life must be other-serving. This is what will bring us real and lasting happiness.

(12) We have the power to act in ways that are unique to our own gifts and abilities. The virtues we have, and the ways we may express these define our individual paths.

Russell McNeil, PhD, is the author of The Meditations of Marcus Aurelius: Selections Annotated and Explained by Skylight Paths Publishing. The unpublished selections presented in this Blog are provided as supplemental material to the published selections which are annotated and explained in the book. The published selections are referenced in this Blog by page number and section.

Saturday, May 2, 2009

The Five Elements - The Meditations of Marcus Aurelius - Unpublished Selections Explained, Med. XI.20



Meditation XI.20 - The Five Elements - Translated by George Long and rewritten by Russell McNeil

Your aerial part1 and all the fiery parts2 which are mingled in you,3 though by nature they have an upward tendency,4 still in obedience to the disposition of the universe they are overpowered here in the compound mass (the body).5 And also the whole of the earthy part6 in you and the watery,7 though their tendency is downward,8 still are raised up and occupy a position which is not their natural one.9 In this manner then the elemental parts obey the universal,10 for when they have been fixed in any place, by necessity they remain there until again the universal shall sound the signal for dissolution.11 Is it not then strange that your intelligent part12 only should be disobedient and discontented with its own place?13 And yet no force is imposed on it,14 but only those things which are conformable to its nature:15 still it does not submit, but is carried in the opposite direction.16 For the movement towards injustice and intemperance and to anger and grief and fear is nothing else than the act of one who deviates from nature.17 And also when the ruling faculty is discontented with anything that happens,18 then too it deserts its post:19 for it is constituted for piety and reverence towards the gods20 no less than for justice. For these qualities also are comprehended under the generic term of contentment with the constitution of things, and indeed they are prior to acts of justice.21


Explanation

(1) In Aristotelian and Stoic physics all passive matter was presumed to be a mixture, in varying proportions, of the four primary substances or elements, earth, water, air (aerial) and fire. A fifth unchanging element referred to as aether (also ether), or quintessence, was presumed to exist above the realm of the Earth and the principle element in the composition of the stars and planets. The Stoics modified this scheme by referring to aetherial (or etherial) substance as pneuma. Pneuma, unlike aether, was universally distributed and co-mingled with the other four elements in varying degree, and associated with the active principle of nature or Logos. Like aether, pneuma was considered indestructible. Unlike the passive elements, pneuma did not dissolve with the death of the body.

(2) The fiery element co-mingles with the body and generates the body's heat.

(3) The dissolution of the body at death begins with the release of the fiery part leaving the body cold.

(4) The natural motions of the four elements (when unrestrained) were upward (air and fire) and downward (earth and water).

(5) In Stoic physics, the overpowering of the natural motions of the four elements in the compound mass of the body would be regulated through the superior agency of the fifth element, the active element, pneuma. In Aristotelian physics the fifth element (aether) was not present in the sublunary realm (below the orbit of the moon) and consequently offered no equivalent conjecture as to why the four basic elements were overpowered in the body without the intervention of a fifth element.

(6) The earthy part would be the principle element of bone for example.

(7) The chief element in the composition of bodily fluids.

(8) Water flows downward, but fluids in the body can move upward.

(9) This argument would also apply to the animation of the entire body which, although composed mainly of earthy and watery elements, can overpower those natural tendencies when in motion, or when leaping into the air, or climbing a tree.

(10) The natural elements obey their universal tendencies, when not constrained by a superior power.

(11) The dissolution of the elements of the body will occur at death - an operation of nature signaled and predetermined by the disposition of the active principle of nature.

(12) The intelligent part (or active aspect of our nature) is distinct from the four elements, and never of necessity constrained by those four elements.

(13) We can disobey our natural intelligence and allow discontent into our lives because human beings are free to act in opposition to nature, but such attitudes are in opposition to the natural movement of the intelligent part which is otherwise unconstrained. In other words we have the free will to override our intelligence (see also Meditation XI.36).

(14) The only clear reason Stoics offer for overriding our natural intelligence is ignorance of the law of nature - although Stoicism does not disallow willful disobedience.

(15) The things that conform to the nature of intelligence are the same things that are conformable to divinity, such as justice and wisdom and the other virtues.

(16) The "opposite direction" in this case is the direction away from virtue.

(17) In other words the direction of vice.

(18) These estrangements away from nature form the basis for a wide spectrum of mental pathologies with significant modern relevance. These are discussed in detail in Meditation XI.19 (published in the book).

(19) The ruling faculty can "desert its post" because we are always free to do so. Again, we have free will.

(20) The ruling faculty is constituted to do what is best for the universe (or the community). See Meditation XII.14 for a discussion of what the Stoics mean by the "gods."

(21) In saying that these things are "prior to acts of justice," Marcus reminds us that the universe is beautiful and inherently good. Human beings need to understand this is so, and can see that this is so by examining nature. When we realize this existential truth, we will act justly. Absent this knowledge, human beings are predisposed to act in contrary ways, or in response to those things that their animal and emotional natures tell them are good.

Russell McNeil, PhD, is the author of The Meditations of Marcus Aurelius: Selections Annotated and Explained by Skylight Paths Publishing. The unpublished selections presented in this Blog are provided as supplemental material to the published selections which are annotated and explained in the book. The published selections are referenced in this Blog by page number and section.

Sunday, April 26, 2009

Meditation on a Star -The Meditations of Marcus Aurelius - Unpublished Selections Explained, Med. XI.27



Meditation XI.27 - Meditation on a Star - Translated by George Long and rewritten by Russell McNeil

The Pythagoreans1 bid us in the morning look to the heavens that we may be reminded of those bodies which continually do the same things and in the same manner perform their work,2 and also be reminded of their purity and nudity. For there is no veil over a star.3


Explanation

(1) Pythagoras of Samos (ca. 576 - ca. 495 BCE) was a Greek mathematician-philosopher and founder of a mystical religious movement based on his ideas. The Pythagoreans responded to the puzzle of existence through the window of numbers - and were convinced that the mystery of reality could be decoded through a close examination of numerical relationships.

(2) The Pythagoreans had access to (or were aware of) astronomical observational records extending back thousands of years. That evidence indicated that the cycles of the stars, constellations and planets were thoroughly predictable, and that the bodies themselves were unchanging. This invariance led many to believe that the stars and planets were divine and eternal.

(3) From the Stoic perspective the stars and planets offered direct observational evidence of pure divinity. This divinity would not have the transcendent character of the more mystical Pythagoreans, but provided direct evidence of the flawless character of the active aspect of nature, divorced as it was from the ever-changing passive character of the human body, or of the Earth itself, with its dynamic mixtures of earth, air, water and fire. Unlike the Pythagoreans, the Stoics would have associated these celestial displays with Logos, or the universal intelligence. The Stoics uniquely understood human nature as distilled from this same universal reason, which was cloaked within (and often confused by) the body. There was, however, no such cloak, or "veil," over a star. The contemplation of a star therefore offered the Stoic a singular meditative opportunity to observe nature in its purest form.

Russell McNeil, PhD, is the author of The Meditations of Marcus Aurelius: Selections Annotated and Explained by Skylight Paths Publishing. The unpublished selections presented in this Blog are provided as supplemental material to the published selections which are annotated and explained in the book. The published selections are referenced in this Blog by page number and section.

Wednesday, April 22, 2009

Stoics Must Walk the Walk - The Meditations of Marcus Aurelius - Unpublished Selections Explained, Med. XI.29




Meditation XI.29 – Stoics Must Walk the Walk - Translated by George Long and rewritten by Russell McNeil


Neither in writing nor in reading will you be able to lay down rules for others before you shall have first learned to obey rules yourself.1 Much more is this so in life.2

Explanation

(1) Marcus makes frequent reference to reading and writing in his meditations. Stoic methodology is based on objective inquiry - in many respects it anticipates the modern hypothetico-deductive model of the scientific method, with one notable difference. The goal of Stoic inquiry is to discover the rules of engagement in life through an objective examination of nature. Scientific method, in contrast, is detached from moral inquiry. In Stoicism the laws of nature are abstract templates for virtue. They are the "rules" Marcus refers to in the meditation. because we have free will, it is possible, of course, to acquire an intellectual understanding of these rules, and then to ignore them. But it is only in acting in accord with those rules that a Stoic can truly understand what living rightly means. Living rightly in Stoic terms confers enlightenment (serenity) on the actor. This is a state of being that is the goal of all Stoics. Without this experience it is not possible to effectively guide or advise others. To attempt to do so would be false and inauthentic.

(2) A sage Stoic, qua Stoic, must walk the walk before attempting to lead. Marcus must certainly be reflecting about himself - about his writing, his reading, his teaching, and about his leadership - in this deeply personal reflection.

Russell McNeil, PhD, is the author of The Meditations of Marcus Aurelius: Selections Annotated and Explained by Skylight Paths Publishing. The unpublished selections presented in this Blog are provided as supplemental material to the published selections which are annotated and explained in the book. The published selections are referenced in this Blog by page number and section.

Tuesday, September 25, 2007

Nicomachus of Gerasa (c 100 CE)


Nicomachus of Gethisrasa is mentioned in a small number of sources and we can date him fairly accurately from the information given. Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower limits on his dates. On the other hand Apuleius, the Platonic philosopher, rhetorician and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Latin so this gives an upper limit on his dates. One of the most interesting references is by Lucian, the rhetorician, pamphleteer and satirist who was born about 120 AD, who makes one of his characters say: You calculate like Nicomachus. Clearly Nicomachus had achieved fame for his arithmetical work! Nicomachus was a Pythagorean. This is obvious from his writings on numbers and music, but we are also told this by Porphyry who says that he was one of the leading members of the Pythagoreans School. Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry. Unlike Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples. However Introduction to Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs. Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation. Sometimes Nicomachus stated a result that is simply false and then illustrated it with an example that happens to have the properties described in the result. We must deduce from this that some of the results are merely guesses based on the evidence of the numerical examples (and in some cases perhaps even based on one example!). - from Malaspina Biography

Books from Alibris: Nicomachus

Monday, September 10, 2007

Lucretius (c 95 BCE-55 BCE)

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Quotation

Pleasant it is, when over a great sea the winds trouble the waters, to gaze from shore upon another's great tribulation; not because any man's troubles are a delectable joy, but because to perceive you are free of them yourself is pleasant.

Books

Please browse our Amazon list of titles about Lucretius. For rare and hard to find works we recommend our Alibris list of titles about Lucretius.

AlibrisResearch

COPAC UK: Lucretius
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Biographical

The great Latin didactic poet. Our sole information concerning his life is found in the brief summary of Jerome, written more than four centuries after the poet's death. Jerome followed, often carelessly, the accounts contained in the lost work of Suetonius De Viris Illustribus, written about two centuries after the death of Lucretius; and, although it is likely that Suetonius used the information transmitted by earlier grammarians, there is nothing to guide us to the original sources. According to this account the poet was born in 95 B.C. He became mad in consequence of the administration of a love-philtre; and after composing several books in his lucid intervals, which were subsequently corrected by Cicero, he died by his own hand in the forty-fourth year of his age. Donatus states in his Life of Virgil, a work also based on the lost work of Suetonius, that Lucretius died on. the same day on which Virgil assumed the toga virus, that is, in the seventeenth year of Virgil's life, and on the very day on which he was born, and adds that the consuls were the same, that is Cn. Pompeius Magnus and M. Licinius Crassus, consuls in 70 and again in 55. The statements cannot be perfectly reconciled; but we may say with certainty that Lucretius was born between 98 and 95 BCE, and died in 55 or 54. A single mention of his poem, the De Rerum Natura (which from the condition in which it has reached us may be assumed to have been published posthumously) in a letter of Cicero's to his brother Quintus, written early in 54 BCE, confirms the date given by Donatus as that of the poet's death. The statements of Jerome have been questioned or disbelieved on the ground of their intrinsic improbability. They have been regarded as a fiction invented later by the enemies of Epicureanism, with the view of discrediting the most powerful work ever produced by any disciple of that sect. It is more in conformity with ancient credulity than with modern science to attribute a permanent tendency to dcrangejnent to the accidental administration of any drug, however potent. A work characterized by such strength, consistency and continuity of thought is not likely to have been composed "in the intervals of madness" as Jerome says. Donatus, in mentioning the poet's death, gives no hint of the act of suicide. The poets of the Augustan age, who were deeply interested both in his philosophy and in his poetry, are entirely silent about the tragical story of his life. Cicero, by his professed antagonism to the doctrines of Epicurus, by his inadequate appreciation of Lucretius himself and by the indifference which he shows to other contemporary poets, seems to have been neither fitted for the task of correcting the unfinished work of a writer whose genius was so distinct from his own, nor likely to have cordially undertaken such a task.

Yet these considerations do not lead to the absolute rejection of the story. The evidence afforded by the poem rather leads to the conclusion that the tradition contains some germ of fact. It is remarkable that in more than one passage of his poem Lucretius writes with extraordinary vividness of the impression produced both by dreams and by waking visions. It is true that the philosophy of Epicurus put great stress on these, as affording the explanation of the origin of supernatural beliefs. But the insistence with which Lucretius returns to the subject, and the horror with which he recalls the effects of such abnormal phenomena, suggest that he himself may have been liable to such hallucinations, which are said to be consistent with perfect sanity, though they may be the precursors either of madness or of a state of despair and melancholy. Other passages, wher he describes himself as ever engaged, even in his dreams, on his task of inquiry and composition, produce the impression of ar unrelieved strain of mind and feeling, which may have ended in some extreme reaction of spirit, or in some failure of intellectual power, that may have led him to commit suicide. But th strongest confirmation of the tradition is the unfinished condition in which the poem has reached us. The subject appears to have been fully treated in accordance with the plan sketched out in the introduction to the first book. But that book is thr only one which is finished in style and in the arrangement o its matter. In all the others, and especially in the last three the continuity of the argument is frequently broken by passages which must have been inserted after the first draft of the arguments was written out. Thus, for instance, in his account of the transition from savage to civilized life, be assumes at v.1011 the discovery of the use of skins, fire, &c., and the first beginning of civil society, and proceeds at 1028 to explain the origin of language, and then again returns, from 1090 to 1160, to speculate upon the first use of fire and the earliest stages of political life. These breaks in continuity show what might also be inferred from frequent repetitions of lines which have appeared earlier in the poem, and from the rough workmanship of passages in the later books, that the poem could not have received the final revision of the author. Nor is there any great difficulty in believing that Cicero edited it; the word "emendavit," need not mean more than what we call "preparing for press."

From the absence of any claim on the part of any other district of Italy to the honour of having given birth to Lucretius it is inferred that he was of purely Roman origin. No writer certainly is more purely Roman in personal character and in strength of understanding. His silence on the subject of Roman greatness and glory as contrasted with the prominence of these subjects in the poetry of men of provincial birth such as Ennius, Virgil and Horace, may be explained by the principle that familiarity had made the subject one of less wonder and novelty to him. The Lucretian gens to which he belonged was one of the oldest of the great Roman houses, nor do we hear of the name, as we do of other great family names, as being diffused over other parts of Italy, or as designating men of obscure or servile origin. It may well be assumed that Lucretius was a member of the Roman aristocracy, belonging either to a senatorian or to one of the great equestrian families. If the Roman aristocracy of his time had lost much of the virtue and of the governing qualities of their ancestors, they showed in the last years before the establishment of monarchy a taste for intellectual culture which might have made Rome as great in literature as in arms and law. A new taste for philosophy had developed among members of the governing class during the youth of Lucretius, and eminent Greek teachers of the Epicurean sect settled at Rome at the same time, and lived on terms of intimacy with them. The inference that Lucretius belonged to this class is confirmed by the tone in which he addresses Gaius Memmius, a man of an eminent senatorian family, to whom the poem is dedicated. His tone is quite unlike that in which Virgil or even Horace addresses Maecenas. He addresses him as an equal he expresses sympathy with the prominent part he played in public life, and admiration for his varied accomplishments, but on his own subject claims to speak to him with authority.

Although our conception of the poet's life is necessarily vague and meagre, yet his personal force is so remarkable and so vividly impressed on his poem, that we seem able to form a consistent idea of his qualities and characteristics. We know, for example that the choice of a contemplative life was not the result of indifference to the fate of the world, or of any natural coldness or even calmness of temperament. In the opening lines of the second and third books we can mark the recoil of a humam and sensitive spirit from the horrors of the reign of terror which he witnessed in his youth, and from the anarchy and confusion which prevailed at Rome during his later years. We may also infer that he had not been through his whole career so much estranged from the social life of his day as he seems to have been in his later years. Passages in his poem attest his familiarity with the pomp and luxury of city life, with the attractions o the public games and with the pageantry of great military spectacles. But much the greater mass of the illustrations of his philosophy indicate that, while engaged on his poem he must have passed much of his time in the open air, exercising at one the keen observation of a naturalist and the contemplative vision of a poet. He seems to have found a pleasure, more congenial to the modern than to the ancient temperament, in ascending mountains or wandering, among their solitudes. References to companionship in these wanderings, and the well-known description of the charm of a rustic meal speak of kindly sociality rather than of any austere separation from his fellows.

Other expressions in his poem imply that he was also a student of books. Foremost among these were the writings of Epicurus; but he had also an intimate knowledge of the philosophical poem of Empedocles, and at least an acquaintance with the works of Democritus, Anaxagoras, Heraclitus, Plato and the Stoical writers. Of other Greek prose writers he knew Thucydides and Hippocrates; while of the poets he expresses in more than one passage the highest admiration of Homer, whom he imitated in several places. Next to Homer, Euripides is most frequently reproduced by him. But his poetical sympathy was not limited to the poets of Greece. For his own countryman Ennius he expresses an affectionate admiration; and he imitates his language, his rhythm and his manner in many places. The fragments of the old tragedian Pacuvius and of the satirist Lucilius show that Lucretius had made use of their expressions and materials. In his studies he was attracted by the older writers, both Greek and Roman, in whose masculine temperament and understanding he recognized an affinity with his own.

His devotion to Epicurus seems at first sight more difficult to explain than his enthusiasm for Empedocles or Ennius. Probably he found in his calmness of temperament, even in his want of imagination, a sense of rest and of exemption from the disturbing influences of life; while in his physical philosophy he found both an answer to the questions which perplexed him and an inexhaustible stimulus to his intellectual curiosity. The combative energy, the sense of superiority, the spirit of satire, characteristic of him as a Roman, unite with his loyalty to Epicurus to render him not only polemical but intolerant and contemptuous in his tone toward the great antagonists of his system, the Stoics, whom, while constantly referring to them, he does not condescend even to name. With his admiration of the genius of others he combines a strong sense of his own power. He is quite conscious of the great importance ahd of the difficulty of his task; but he feels his own ability to cope with it.

It is more difficult to infer the moral than the intellectual characteristics of a great writer from the personal impress left by him on his work. Yet it is not too much to say that there is no work in any literature that produces a profounder impression of sincerity. No writer shows a juster scorn of all mere rhetoric and exaggeration. No one shows truer courage, not marred by irreverence, in confronting the great problems of human destiny, or greater strength in triumphing over human weakness. No one shows a truer humanity and a more tender sympathy with natural sorrow.

The peculiarity of the poem of Lucretius, that which makes it unique in literature, is that it is a reasoned system of philosophy, written in verse. The prosaic title De Rerum Natura, implies the subordination of the artistic to a speculative motive. As in the case of nearly all the great works of Roman literary genius, the form of the poem was borrowed from the Greeks. The rise of speculative philosophy in Greece was coincident with the beginning of prose composition, and many of the earliest philosophers wrote in the prose of the Ionic dialect; others, however, and especially the writers of the Greek colonies in Italy and Sicily, expounded their systems in continuous poems composed in the epic hexameter. Most famous in connexion with this kind of poetry are Xenophanes and Parmenides, the Eleatics and Empedocles of Agrigentum. The last was less important as a philosopher, but greater than the others both as a poet and a physicist. On both of these grounds he had a greater attraction to Lucretius. The fragments of the poem of Empedocles show that the Roman poet regarded that work as his model. In accordance with this model he has given to his own poem the form of a personal address, he has developed his argument systematically, and has applied the sustained impetus of epic poetry to the treatment of some of the driest and abstrusest topics. Many ideas and expressions of the Sicilian have been reproduced by the Roman poet; and the same tone of impassioned solemnity and melancholy seems to have pervaded both works. But Lucretius, if less original as a thinker, was probably a much greater poet than Empedocles. What chiefly distinguishes him from his Greek prototypes is that his purpose is rather ethical than purely speculative; the zeal of a teacher and reformer is more strong in him than even the intellectual passion of a thinker. His speculative ideas, his moral teaching and his poetical power are indeed interdependent on one another, and this interdependence is what mainly constitutes their power and interest. But of the three claims which he makes to immortality, the importance of his subject, his desire to liberate the mind from the bonds of superstition and the charm and lucidity of his poetry - that which he himself regarded as supreme was the second. The main idea of the poem is the irreconcilable opposition between the truth of the laws of nature and the falsehood of the old superstitions. But, further, the happiness and the dignity of life are regarded by him as absolutely dependent on the acceptance of the true and the rejection of the false doctrine. In the Epicurean system of philosophy he believed that he had found the weapons by which this war of liberation could be most effectually waged. Following Epicurus he sets before himself the aim of finally crushing that fear of the gods and that fear of death resulting from it which he regards as the source of all the human ills. Incidentally he desires also to purify the heart from other violent passions which corrupt it and mar its peace. But the source even of these - the passions of ambition and avarice - he finds in the fear of death; and that fear he resolves into the fear of eternal punishment after death.

The selection of his subject and the order in which it is treated are determined by this motive. Although the title of the poem implies that it is a treatise on the "whole nature of things," the aim of Lucretius is to treat only those branches of science which are necessary to clear the mind from the fear of the gods and the terrors of a future state. In the two earliest books, accordingly, he lays down and largely illustrates the first principles of being with the view of showing that the world is not governed by capricious agency, but has come into existence, continues in existence, and will ultimately pass away in accordance with the primary conditions of the elemental atoms which, along with empty space, are the only eternal and immutable substances. These atoms are themselves infinite in number but limited in their varieties, and by their ceaseless movement and combinations during infinite time and through infinite space the whole process of creation is maintained. In the third book he applies the principles of the atomic philosophy to explain the nature of the mind and vital principle, with the view of showing that the soul perishes with the body. In the fourth book he discusses the Epicurean doctrine of the images, which are cast from all bodies, and which act either on the senses or immediately on the mind, in dreams or waking visions, as affording the explanation of the belief in the continued existence of the spirits of the departed. The fifth book, which has the most general interest, professes to explain the process by which the earth, the sea, the sky, the sun, moon and stars, were formed, the origin of life, and the gradual advance of man from the most savage to the most civilized condition. All these topics are treated with the view of showing that the world is not itself divine nor directed by divine agency. The sixth book is devoted to the explanation, in accordance with natural causes, of some of the more abnormal phenomena, such as thunderstorms, volcanoes, earthquakes, &c., which are special causes of supernatural terrors.

The consecutive study of the argument produces on most readers a mixed feeling of dissatisfaction and admiration. They are repelled by the dryness of much of the matter, the unsuitableness of many of the topics discussed for poetic treatment, the arbitrary assumption of premises, the entire failure to establish the connexion between the concrete phenomena which the author professes to explain and these assumptions, and the erroneousness of many of the doctrines which are stated with dogmatic confidence. On the other hand, they are constantly impressed by his power of reasoning both deductively and inductively, by the subtlety and fertility of invention with which he applies analogies, by the clearness and keenness of his observation, by the fullness of matter with which his mind is stored, and by the consecutive force, the precision and distinctness of his style, when employed in the processes of scientific exposition. The first two books enable us better than anything else in ancient literature to appreciate the boldness and, on the whole, the reasonableness of the ancient mind in forming hypotheses on great matters that still occupy the investigations of physical science. The third and fourth books give evidence of acuteness in psychological analysis; the fourth and sixth of the most active and varied observation of natural phenomena; the fifth of original insight and strong common sense in conceiving the origin of society and the progressive advance of man to civilization. But the chief value of Lucretius as a thinker lies in his firm grasp of speculative ideas, and in his application of them to the interpretation of human life and nature. All phenomena, moral as well as material, are contemplated by him in their relation to one great organic whole, which he acknowledges under the name of "Natura daedala rerum," and the most beneficent manifestations of which he seems to symbolize and almost to deify in the "Alma Venus," whom, in apparent contradiction to his denial of a divine interference with human affairs, he invokes with prayer in the opening lines of the poem. In this conception of nature are united the conceptions of law and order, of ever-changing life and interdependence, of immensity, individuality, and all-pervading subtlety, under which the universe is apprehended both by his intelligence and his imagination.

Nothing can be more unlike the religious and moral attitude of Lucretius than the old popular conception of him as an atheist and a preacher of the doctrine of pleasure. It is true that he denies the doctrines of a supernatural government of the world and of a future life. But his arguments against the first are real1y only valid against the limited and unworthy conceptions of divine agency involved in the ancient religions; his denial of the second is prompted by his vital realization of all that is meant by the arbitrary infliction of eternal torment after death. His war with the popular beliefs of his time is waged, not in the interests of licence, but in vindication of the sanctity of human feeling. The cardinal line of the poem, Tantum religio potuit suadere malorum, is elicited from him as his protest against the sacrifice of Iphigenia by her father. But in his very denial of a cruel, limited and capricious agency oi the gods, and in his imaginative recognition of an orderly, all-pervading, all-regulating power, we find at least a nearer approach to the higher conceptions of modern theism than in any of the other imaginative conceptions of ancient poetry and art. But his conception even of the ancient gods and of their indirect influence on human life is more worthy than the popular one. He conceives of them as living a life of eternal peace and exemption from passion, in a world of their own; and the highest ideal of man is, through the exercise of his reason, to realize an image of this life. Although they are conceived of as unconcerned with the interest of our world, yet influences are supposed to emanate from them which the human heart is capable of receiving and assimilating. The effect of unworthy conceptions of the divine nature is that they render a man incapable of visiting the temples of the gods in a calm spirit, or of receiving the emanations that "announce the divine peace" in peaceful tranquillity. The supposed "atheism" of Lucretius proceeds from a more deeply reverential spirit than that of the majority of professed believers in all times.

His moral attitude is also far removed from that of ordinary ancient Epicureanism or of modern materialism. Though he acknowledges pleasure to be the law of life, yet he is far from regarding its attainment as the end of life. What man needs is not enjoyment, but "peace and a pure heart." The victory to be won by man is the triumph over fear, ambition, passion, luxury. With the conquest over these nature herself supplies all that is needed for happiness. Self-control and renunciation are the lessons which he preaches.

It has been doubted whether Cicero, in his short criticism in the letter already referred to, concedes to Lucretius both the gifts of genius and the accomplishment of art or only one of them. Readers of a later time, who could compare his work with the finished works of the Augustan age, would certainly disparage his art rather than his power. But with Cicero it was different. He greatly admired, or professed to admire, the genius of the early Roman poets, while he shows indifference to the poetical genius of his younger contemporaries. Yet he could not have been insensible to the immense superiority in rhythmical smoothness which the hexameter of Lucretius has over that of Ennius and Lucilius. And no reader of Lucretius can doubt that he attached the greatest importance to artistic execution, and that he took a great pleasure, not only in "the long roll of his hexameter, "but also in producing the effects of alliteration, assonance, &c., which are so marked a peculiarity in the style of Plautus and the earlier Roman poets. He allows his taste for these tricks of style to degenerate into mannerism. And this is the only drawback to the impression of absolute spontaneity which his style produces. He was unfortunate in living before the natural rudeness of Latin art had been successfully grappled with. His only important precursors in serious poetry were Ennius and Lucilius, and, though he derived from the first of these an impulse to shape the Latin tongue into a fitting vehicle for the expression of elevated emotion and imaginative conception, he could find in. neither a guide to follow in the task he set before himself. The difficulty and novelty of his task enhances our sense of his power. His finest passages are thus characterized by a freshness of feeling and enthusiasm of discovery. But the result of these conditions and of his own inadequate conception of the proper limits of his art is that his best poetry is clogged with a great mass of alien matter, which no treatment in the world could have made poetically endurable. [Adapted from Encyclopedia Britannica (1911)]

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Wednesday, August 29, 2007

Hippocrates of Cos (c 460-377 BCE)

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Make a habit of two things: to help; or at least to do no harm.

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Hippocrates of Cos -- about 460 BC - 377 BC. Ancient Greek physician, commonly regarded one of the most outstanding figures in medicine of all times. He is often called "the father of medicine". He was the leader of a medical school of Cos and the author of most of writings of the school. He had a great impact on succeeding generations of practitioners of medicine and some general rules still apply. His work and writings rejected the superstition and magic of primitive "medicine" and laid the foundations of medicine as a branch of science. The whole collection of works of the Hippocratic medical school were gathered as the Hippocratic Corpus. The best known of the Hippocratic writings is the Hippocratic Oath.

Hippocratic Oath

I swear by Apollo the physician, by Aesculapius, Hygeia, and Panacea, and I take to witness all the gods, all the goddesses, to keep according to my ability and my judgement, the following Oath:

To consider dear to me as my parents him who taught me this art; to live in common with him and if necessary to share my goods with him; to look upon his children as my own brothers, to teach them this art if they so desire without fee or written promise; to impart to my sons and the sons of the master who taught me and the disciples who have enrolled themselves and have agreed to the rules of the profession, but to these alone the precepts and the instruction. I will prescribe regimen for the good of my patients according to my ability and my judgement and never do harm to anyone. To please no one will I prescribe a deadly drug nor give advice which may cause his death. Nor will I give a woman a pessary to procure abortion. But I will preserve the purity of my life and my art. I will not cut for stone, even for patients in whom the disease is manifest; I will leave this operation to be performed by practitioners, specialists in this art. In every house where I come I will enter only for the good of my patients, keeping myself far from all intentional ill-doing and all seduction and especially from the pleasures of love with women or with men, be they free or slaves. All that may come to my knowledge in the exercise of my profession or in daily commerce with men, which ought not to be spread abroad, I will keep secret and will never reveal. If I keep this oath faithfully, may I enjoy my life and practice my art, respected by all men and in all times; but if I swerve from it or violate it, may the reverse be my lot.


Another famous, time-honoured medical rule ascribed to Hippocrates is : "Primum non nocere" (First, do no harm). [This article is licensed under the GNU Free Documentation License and uses material adapted in whole or in part from the Wikipedia article on Hippocrates.]

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Hipparchus (190-125 BCE)



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Never deceive a friend.

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Hipparchus (Greek: Hipparcos), Greek astronomer, mathematician and geographer, born: 190 B.C., Antigoneia since the year 30 Nicaea (Greek: Nikaia) when Cisimah gave its name, ancient district Bithynia, (modern-day Iznik) in province Bursa, in modern day Turkey, died: 120 B.C., probably the island of Rhodes. The exact dates of his life are not known for sure, but he is believed to have observed from 162 to 126 B.C. Date of his birth was calculated by J. B. J. Delambre, based on clues in his work. We don't know anything about his youth either. Most of what is known about Hipparchus is from Strabo's Geographica (Geography), from Pliny the Elder's Naturalis historia (Natural sciences) and from Ptolemy's Almagest. He probably studied in Alexandria. His main original works are lost. His only preserved work is the Commentary on Aratus, a commentary on a poem by Aratus which describes the constellations and the stars which comprise them. This work contains many measurements of stellar positions. For his accession he holds the place of originator and father of scientific astronomy. He is believed to be the greatest Greek astronomer observer and he is at the same time entitled the greatest astronomer of ancient times, although Cicero still though about Aristarchus of Samos. Some put on this place also Ptolemy of Alexandria.

Hipparchus had ranked stars after their brightness in six magnitude classes, what we, as magnitudes , still use today since Ptolemy. He arranged value of 1 to 20 brightest stars, to weaker ones value of 2 and so forth to the stars with a class of 6, which can be barely seen. Later astronomers with telescopes and photographic plates and with other measuring devices for the light had extended a luminosity with a density of light current j of a star on the Earth. Observations with measuring devices for the light had shown that the density of light current of a star with a magnitude 1m is hundred times greater of a star with a magnitude 6m.

Hipparchus had made a lot of astronomical instruments, which were used for a long time with naked-eye observations. About 150 B.C. he made first astrolabe, which was improved in 3rd century by Arab astronomers and brought by them in Europe in 10th century. With astrolabe Hipparchus was able to measure among the first the geographical latitude and time. Gnomon was changed during his time. They put it in a metallic hemisphere, which was devided inside in concentric circles and it used as a portable instrument, named scaphion, for determination of geographical coordinates from measured solar altitudes. With this instrument Eratosthenes of Cyrene 220 B.C. had measured the length of Earth's meridian and after that they used this instrument to make smaller maps. Hipparchus had proposed to determine the geographical longitudes of several cities at solar eclipses. In fact eclipse doesn't arise simultaneously in all points of the center of the lunar shadow, but his method would give the most accurate data as would any previous one, if it would be correctly carried out. But his method unfortunately never saw its proper usage and for this reason maps were rather inaccurate.

It is thought that Hipparchus compiled the first catalog of stars, and also compiled the first trigonometry tables. He tabulated values for the chord function, which gave the length of the chord for each angle. In modern terms, the chord of an angle equals twice the sine of half of the angle, e.g., chord of A = 2sin(A/2). He had a method of solving spherical triangles. Theorem in plane geometry called Ptolemy's theorem was developed by Hipparchus. This theorem was elaborated on by Lazare Carnot.

Hipparchus is perhaps most famous for having been the first to measure the precession of the equinoxes (There is some suggestion that the Babylonians may have known about precession but it appears that Hipparchus was to first to really understand it and measure it). According to al-Battani Chaldean astronomers had distinguished the tropical and siderical year. He stated they had around 330 B.C. an estimation for the length of siderical year to be Sk = 365.2576388d = 365d 6h 11m with an error of 110s. This phenomenon was probably also known to Kidinnu around 314 B.C. A. Biot and Delambre attribute the discovery of precession also to old Chinese astronomers. Hipparchus had used almost the basic astronomical instruments gnomon, atrolabe, armiral sphere and so. Before him Meton and his students had determined 440 B.C. the two points of the solstice. Hipparchus on his own in Alexandria 146 B.C. determined the equinoctial point. He used Archimedes' observations of solstices. Year after 145 B.C. also on his own he determined the length of tropical year to be:

TH = 365.24653...d = 365d5h55m12s (TH = 365.24653...d = 365d5h55h elsewhere), which differs from today's T = 365.24219...d = 365d5h48m45s for only 6m27s (6m15s). Before him the Chaldean astronomers knew the lengths of seasons are not equal. Hipparchus fully measured the length of winter and spring to be 184 1/2 days, summer and autumn 180 1/2 days. In his geocentrical view, which he preferred, he explained this fact with adoption the Earth is not in the centre of Sun's orbit around it, but it lies eccentrically for 1/24 r. With his estimation of the length of seasons he tried to determine, as of today, linear eccentricity of Earth's orbit and according to J. L. E. Dreyer he got the incorrect value e = 0.04166. The question is if he is really author of this estimation? After that he lived from 141 B.C. to 126 B.C. mostly on the island of Rhodes, again in Alexandria and in Syracuse and around 130 B.C. in Babylon made a lot of precise and lasting observations. When he measured the length of gnomen shadow at solstice he determined the length of tropical year and he was finding times of the known bright star sunsets and times of sunrises. From all of these measurements he 134 B.C. found the length of siderical year to be SH = 365.2569444...d = 365d6h10m, which differs from today's S = 365.2563657...d = 365d6h9m10s for 50s. Hipparchus had measurements of the times of solstices from Aristarchus dating from 279 B.C. and from the school of Meton and Euctemon dating from 431 B.C.. This was a long enough period of time to allow him to calculate the difference between the length of the sidereal year and the tropical year, and led him to the discovery of precession. When he compared both lengths, he saw the tropical year is shorter for about 20 minutes from siderical. And as first in the history he correctly explained this with retrogradical movement of vernal point V over the ecliptic for about 45" or 46" (36" or 3/4' according to Ptolemy) per annum (today's value is 50,387", 50,26") and he showed the Earth's axis is not fixed in space. After that he 135 B.C. enthusiastic of nova star in constellation of Scorpion with equatorial armiral sphere measured ecliptic coordinates of about 850 (1600 or 1080, which is false quoted many times elsewhere) and till 129 B.C. he made first big star catalogue. This map served him to find any changes on the sky and for great sadness it is not preserved today. His star map was thoroughly modified as late as 1000 years after 964 by A. Ali Sufi and 1500 years after 1437 by M. T. ibn Sh. ibn T. Ulugh Beg. Later, Edmond Halley would use his star catalog to discover proper motions as well. His work speaks for itself. And another sad fact is that we do not know almost nothing from his life, what was already stressed by Fred Hoyle. In his star map Hipparchus drew position of every star on the basis of its celestial latitude, (its angular distance from the celestial equator) and its celestial longitude (its angular distance from an arbitrary point, for instance as is custom in astronomy from vernal equinox). This system was also transferred to maps for Earth. Before him longitudes and latitudes were used by Dicaearchus of Messana, but they got their meanings in coordinate net not until Hipparchus. By comparing his own measurements of the position of the equinoxes to the star Spica with those of Euclid's contemporaries Timocharis of Alexandria and Aristil 150 years earlier, and the records of Chaldean astronomers and specially Kidinnu's records he still later observed that the equinox had moved 2o relative to Spica. He also noticed this motion in other stars. He obtained a value of not less than 1o in a century. The modern value is 1o in 72 years. He also knew the works Phainomena (Phenomena) and Enoptron (Mirror of Nature) of Eudoxus of Cnidus, who had near Cyzicus on the southern coast of the Sea of Marmara his school and through Aratus' astronomical epic poem Phenomena Eudoxus' sphere, which was made from metal or stone and where there were marked constellations, brightest stars, tropic of Cancer and tropic of Capricorn. These comparisons embarrassed him because he couldn't put together Eudoxus' detailed statements with his own observations and observations of that time. From all this he found that coordinates of stars and Sun had systematically changed. Their celestial latitudes L remained unchanged, but their celestial longitudes B had reduced as would equinoctial points, intersections of ecliptic and celestial equator, move with progressive velocity every year for 1/100'. After him many Greek and Arab astronomers had confirmed this phenomenon. Ptolemy compared his catalogue with those of Aristil, Timocharis, Hipparchus and the observations of Agrippa and Menelaus of Alexandria from the early 1st century and he finally confirmed Hipparchus empirical fact that poles of celestial equator in one Platonic year or approximately in 25,777 years encircle ecliptical pole. The diameter of these circles is equal to the inclination of ecliptic. The equinoctial points in this time traverse the whole ecliptic and they move for 1o in a century. This velocity is equal to Hipparchus' one. Because of these accordances Delambre, P. Tannery and other French historians of astronomy had wrongly jumped to conclusions that Ptolemy recorded his star catalogue from Hipparchus' one with an ordinary extrapolation. This was not known until 1898 when Marcel Boll and the others had found that Ptolemy's catalogue differs from Hipparchus' one not only in the number of stars but otherwise.

This phenomenon was named by Ptolemy just because the vernal point V leads the Sun. In Latin praecesse means to overtake or to outpass and today means to twist or to turn too. Its own name shows this phenomenon was discovered practically before its theoretical explanation, otherwise would be named with a better term. Many later astronomers, physicists and mathematicians had occupied themselves with this problem, practically and theoretically. The phenomenon itself had opened many new promising solutions in several branches of celestial mechanics: Thabit's theory of trepidation and oscillation of equinoctial points, Newton's general gravitational law, which had explained it in full, Euler's kinematic equations and Lagrange's equations of motion, D'Alembert's dynamical theory of the movement of the rigid body, some algebraic solutions for special cases of precession, Flamsteed's and Bradley's difficulties in making of precise telescopic star catalogues, Bessel's and Newcomb's measurements of precession and finally the precession of perihelion in Einstein's General Theory of Relativity. Luni-solar precession causes the motion of point V by the ecliptic in the opposite direction of apparent solar year's movement and the circulation of celestial pole. This circle becomes a spiral because of additional ascendancy of the planets. This is planetary precession where the ecliptical plane swings from its central position for + or - 4o in 60,000 years. The angle between ecliptic and celestial equator e = 23o 26' is reducing for 0.47" per annum. Besides the point V slides by equator for p=0.108" per annum now in the same direction as Sun. The sum of precessions gives an annual general precession in longitude P = 50,288" which causes the origination of tropical year.

Hipparchus described the motion of the Sun and obtained a value for the eccentricity. It was known that the seasons were of unequal length, not something that would be expected if the sun moved around the earth in a circle at uniform speed (of course today we know that the planets move in ellipses, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609). His solution was to place the earth not at the center of the sun's motion, but at distance from the center. This model of the sun's motion described the actual motion of the sun fairly well.

Hipparchus also studied the motion of the moon and obtained more accurate measurements of some periods of the motion than existed previously, and undertook to find the distances and sizes of the sun and moon. About 139 B.C. he determined the length of synodic month to 23/50s. He discovered the irregularity in lunar movement, which changes medium lunar longitude and today is called equalization of the center with a value:

I = 377' sin m + 13' sin 2m,

where m is medium anomaly of the Moon. [This article is licensed under the GNU Free Documentation License and uses material adapted in whole or in part from the Wikipedia article on Hipparchus.]

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Tuesday, August 28, 2007

Herophilus (335 BCE-280 BCE)

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Quotation

When health is absent, wisdom cannot reveal itself, art cannot manifest, strength cannot fight, wealth becomes useless, and intelligence cannot be applied.

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Herophilus was a Greek physician who became an anatomist in the Museum at Alexandria. Unfortunately all of his writings have been lost. He dissected human bodies following death to ascertain the "nature of the fatal malady." He was quoted frequently by Galen, Dioscorides, Pliny and Plutarch. [Adapted from Ohio State University History of Horticulture]

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Saturday, August 18, 2007

Claudius Galen (c 130 BCE-200 BCE)

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Quotation

Employment is nature's physician, and is essential to human happiness.

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Claudius Galen of Pergamum (131-201 AD) was a Greek physician. He performed extensive dissections and vivisections on animals. He is the author of On the Natural Faculties. he was also the personal physician to Roman emperor and Stoic philosopher Marcus Aurelius.

Many of his views about human anatomy were false since he had performed his dissections on pigs, apes, and dogs. [The material in this paragraph is licensed under the GNU Free Documentation License and uses material adapted in whole or in part from the Wikipedia article on Claudius Galen.]

Detailed Biography

Called Gallien by Chaucer and other writers of the middle ages, the most celebrated of ancient medical writers, was born at Pergamus, in Mysia, about A.D. 130. His father Nicon, from whom he received his early education, is described as remarkable both for excellence of natural disposition and for mental culture; his mother, on the other hand, appears to have been a second Xanthippe. In 146 Galen began the study of medicine, and in about his twentieth year he left Pergamus for Smyrna, in order to place himself under the instruction of the anatomist and physician Pelops, and of the peripatetic philosopher Albinus. He subsequently visited other cities, and in 158 returned from Alexandria to Pergamus. A few years later he went for the first time to Rome. There he healed Eudemus, a celebrated peripatetic philosopher, and other persons of distinction; and ere long, by his learning and unparalleled success as a physician, earned for himself the titles of Paradoxologus, the wonder-speaker, and Paradoxopoeus, the wonder-worker, thereby incurring the jealousy and envy of his fellow-practitioners. Leaving Rome in 168, he repaired to his native city, whence he was soon sent for to Aquileia, in Venetia, by the emperors Lucius Verus and Marcus Aurelius. In 170 he returned to Rome with the latter, who, on departing thence to conduct the war on the Danube, having with difficulty been persuaded to dispense with his personal attendance, appointed him medical guardian of his son Commodus. In Rome Galen remained for some years, greatly extending his reputation as a physician, and writing some of his most important treatises. It would appear that he eventually betook himself to Pergamus, after spending some time at the island of Lemnos, where he learned the method of preparing a certain popular medicine, the terra lemnia or sigillata. Whether he ever revisited Rome is uncertain, as also are the time and place of his death. According to Suidas, he died at the age of seventy, or in the year 200, in the reign of Septimius Severus. If, however, we are to trust the testimony of Abul-faraj, his decease took place in Sicily, when he was in his eightieth year. Galen was one of the most versatile and accomplished writers of his age. He composed, it is said, nearly 500 treatises on various subjects, including logic, ethics and grammar. Of the published works attributed to him, 83 are recognized as genuine, 19 are of doubtful authenticity, 45 are confessedly spurious, 19 are fragments, and 15 are notes on the writings of Hippocrates.

Galen, who in his youth was carefully trained in the Stoic philosophy, was an unusually prolific writer on logic. Of the numerous commentaries and original treatises, only one has come down to us, the treatise on Fallacies in dictione. Many points of logical theory, however, are discussed in his medical and scientific writings. His name is perhaps best known in the history of logic in connexion with the fourth syllogistic figure, the first distinct statement of which was ascribed to him by Averroes. There is no evidence from Galen's own works that he did make this addition to the doctrines of syllogism, and the remarkable passage quoted by Minoides Minas from a Greek commentator on the Analytics, referring the fourth figure to Galen, clearly shows that the addition did not, as generally supposed, rest on a new principle, but was merely an amplification or alteration of the indirect moods of the first figure already noted by Theophrastus and the earlier Peripatetics. [Adapted from Encyclopedia Britannica (1911)]

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Sunday, August 12, 2007

Euclid (c 323 - c 283 BCE)

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Quotation

A line is length without breadth.

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Euclid was a Greek mathematician who lived in the 3rd century B.C.

His most famous work is the Elements, a book in which he deduces the properties of geometrical objects and integers from a set of axioms, thereby anticipating the axiomatic method of modern mathematics.

Although many of the results in the Elements originated with earlier mathematicians, one of Euclid's major accomplishments was to present them in a single logically coherent framework.

His fifth postulate, called the Parallel Postulate, states that for any line and any point not on that line, there exists a unique line passing through the point and never intersecting the line. It was long assumed to follow from the other axioms, but in the 19th century, Janos Bolyai (and probably Carl Friedrich Gauss before him) realized that its negation leads to consistent non-euclidean geometries, which were later developed by Lobatchevsky and Riemann. In addition to a treatment of geometry, Euclid's book also contains the beginnings of elementary number theory, such as the notion of divisibility, the greatest common divisor and the Euclidean algorithm to determine it, and the infinity of prime numbers. While the Elements was still used in the 20th century as a geometry text book and has been considered a fine example of the formally precise axiomatic method, Euclid's treatment does not hold up to modern standards and some logically necessary axioms are missing. The first correct axiomatic treatment of geometry was provided by Hilbert in 1899. [This article is licensed under the GNU Free Documentation License and uses material adapted in whole or in part from the Wikipedia article on Euclid .]

Books from Alibris: Euclid