In ancient Greece the continued existence of the dead depended on their constant remembrance by the living. The after-life, for the ancient Greeks, consisted of a grey and dreary world in the time of Homer (8th century BCE) and, most famously, we have the scene from Homer's Odyssey in which Odysseus meets the spirit of the great warrior Achilles in the nether-world where Achilles tells him he would rather be a landless slave on earth than a king in the underworld. By the time of Plato, however (4th century BCE) the after-life had changed in character so that souls were better rewarded for their pains once they had left the earth; but only in so much as the living kept their memory alive.
The Land of the Dead
The afterlife was known as Hades and was a grey world ruled by the Lord of the Dead, also known as Hades. Within this misty realm, however, were different planes of existence the dead could inhabit. If they had lived a good life and were remembered by the living they could enjoy the sunny pleasures of Elysium; if they were wicked then they fell into the darker pits of Tartarus while, if they were forgotten, they wandered eternally in the bleakness of the land of Hades. While both Elysium and Tartarus existed in the time of the writer Hesiod (contemporary of Homer) they were not understood then in the same way they came to be.
If people had lived a good life & were remembered by the living they could enjoy the sunny pleasures of Elysium.
In Plato's dialogue of The Phaedo, Socrates delineates the various plateaus of the after-life and makes it clear that the soul who, in life, devotes itself to the Good is rewarded in the beyond with a much more pleasant existence than those who indulged their appetites and lived only for the pleasures the world has to offer. As most people, then as now, viewed their lost loved ones as paragons of human virtue (whether they were or not, in fact) it was considered one's duty to the dead to remember them well, regardless of the life they had lived, the mistakes they had made, and, thereby, provide them with continued existence in Elysium. This remembrance was not considered a matter of personal choice but, rather, an important part of what the Greeks knew as Eusebia.
Piety in Ancient Greece
We translate the Greek word `Eusebia' today as `piety' but eusebia was much more than that: it was one's duty to oneself, others and the gods which kept society on track and made clear one's place in the community. Socrates, for example, was executed by the city-state of Athens after having been convicted of impiety for allegedly corrupting the youth of Athens and speaking against the established gods. However unjust we may see Socrates' end today he would, in fact, have been guilty of impiety in that he encouraged the youth of Athens, by his own example, to question their elders and social superiors. This behavior would have been considered impious in that the youth were not acting in accordance with eusebia, i.e. they were forgetting their place and obligations in society.
Eusebia & the After-life
In the same way that one had to remember one's duty toward others in one's life, one also had to remember one's duty to those who had departed life. If one forgot to honor and remember the dead one was considered impious and, while this particular breach of social conduct was not punished as severely as Socrates' breach, it was certainly frowned upon severely. Today, should one consider the tombstones of the ancient Greeks - whether in a museum or just below the Acropolis in Athens - one finds stones with comfortable, common scenes depicted: a husband sitting at table as his wife brings him his evening meal, a man being greeted by his dogs upon returning home. These simple scenes were not merely depictions of moments the deceased enjoyed in life; they were meant to remind the living viscerally of who that person was in life, of who that person still was now in death, and to spark the light of continued remembrance in order that the `dead' should live in bliss eternally. In ancient Greece death was defeated, not by the gods, but by the human agency of memory.
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Contributor's Note: This article first published on the website Suite 101. C. 2008 Joshua J. Mark
Ancient Greeks: Everyday Life, Beliefs and Myths
When someone died in Ancient Greece, they would be washed. A coin would be placed in their mouth, to pay the ferrymen who took the dead across the rivers in the different parts of the Underworld. When the Greeks conquered Egypt, they adopted the Egyptian tradition of mummification. They used simple boxes for burying their dead or the deceased would be burned, and their ashes buried in a special pot.
Tombs and Gravestones
Entrances to tombs, where the dead were laid to rest, were made of marble. Heads of Gorgons were carved on to the tomb doors to ward off evil. The tombs were made to stop the dead being forgotten and sometimes they were carved with pictures, showing the deceased with people they knew in life.
Inside the tomb the family of the deceased person placed valuable objects with their body, like pottery, jewellery and coins. It was believed that they would be able to use these objects in the Underworld. Every year families visited the tombs of their dead relatives, making offerings and decorating the tomb.
Living the Ancient Greek Death
The first rite of passage, or prothesis, means laying out of the body. (Image: Walters Art Museum/Public domain)
Putting Oneself in the Sandals of a Dying Greek
The ancient Greeks held certain ideas about death. One of the most characteristic motifs that people find on ancient Greek tombstones is the handshake between the living and dead. Both figures invariably exhibit a dignified calm. That’s what Greek tragedy is all about—looking death squarely in the eye. As a Greek, they knew that terrible things happen and they knew, too, that by confronting them head-on, they’d be able to deal with them and get on with life. One could posit that the Greeks got it just right.
But one needs to put oneself in the sandals of a dying Greek to understand it. It’s an unpleasant thought, but there’s no escaping it if one wants to fully experience the other side of history.
The Role of a Physician in Death
Let’s assume one is dying in one’s home, surrounded by one’s relatives, including young children. There won’t be any physician at hand to give painkillers.
A physician may have offered treatment in the earlier stages of sickness, but once it became inevitable that there could only be one outcome, the medical profession had nothing to offer anymore.
It’s also extremely unlikely that a physician would be called in to put one out of one’s misery by euthanasia, a coined word of Greek etymology meaning ‘good death’, but which has no ancient Greek equivalent. In fact, the Hippocratic Oath, which was probably widely adopted, enjoined upon those physicians who took it “not to administer a poison to anybody who asked for one and not to propose such a course”. So let’s hope that one’s final illness is short and painless.
This is a transcript from the video series The Other Side of History: Daily Life in the Ancient World. Watch it now, Wondrium.
The Role of Gods in Death
The poet Keats has a wonderful line in Ode to a Nightingale: “I have been half in love with easeful death”. The Greeks conceived of easeful death in the form of the God Apollo, who came to strike them down with his so-called ‘gentle arrows’. That’s the best that he or any other of the gods had to offer. They certainly didn’t have any consolation to give someone.
In Euripides’ play the Hippolytus, when Hippolytus is dying, the goddess Artemis, to whom he has devoted himself exclusively all his life and with whom he’s had a very close relationship, bids him farewell. She explains to him that it’s not lawful for a deity to be present at the death because the pollution that a corpse releases would taint her.
The one god who may have taken some slight interest in the fate of the dying is the healing God Asclepius. When Socrates passes from this world to the next in Plato’s dialogue the Crito, he has this to say, “I owe a cock to Asclepius. See that it’s paid.” Cocks were sacrificed to Asclepius. Socrates may be indicating that Asclepius eased his passing, although it’s possible, too, that he’s merely suggesting philosophically that death is a ‘cure’ for life.
The First Rite of Passage: Prothesis
in ancient Greece, as soon as one died, the women in one’s family began keening and ululating so that everyone in the neighborhood knew of the individual’s demise. It was the women, too, who took charge of one’s body and prepared it for burial. They closed one’s mouth and eyes, tied a chin strap around one’s head and chin to prevent the jaw from sagging they washed the whole body, anointed it with olive oil they clothed the body and wrapped it in a winding sheet, leaving only one’s head exposed.
Then they laid the body on a couch with one’s head propped up on a pillow and one’s feet facing the door. After getting all this done, they sang dirges in one’s honor.
This is the scene that is depicted on the very earliest Greek vases with figurative decoration. It’s called the prothesis, which literally means the laying out of the body. It represents the first stage in the process that will take one from this world to the next, ‘from here to there’, as the Greeks put it. Meanwhile, relatives and friends would call at the house and join in the grieving.
The Second Rite of Passage: Ekphora
The second rite of passage is the ekphora. Ekphora means literally ‘the carrying out of one’s body’—specifically from one’s home to one’s place of burial. According to Athenian law, the ekphora had to take place within three days of one’s death, although in hot weather it’s likely that it would have taken place much sooner. The ekphora had to take place before sunrise so that it wouldn’t create a public nuisance.
If one was wealthy, one’s body would be transported in a cart or carriage drawn by horses. This scene is also depicted on the earliest vases with figurative decoration. Professional undertakers might also be employed to bear the corpse and break up the ground for burial. These professionals were known as ‘ladder men’ klimakophoroi, because they’d lay one’s body on a ladder, which they carried horizontally.
If professional undertakers were employed, they wouldn’t have any physical contact with the family members before this phase. The Greeks would have been shocked and appalled by the idea of handing over one’s body to professionals to prepare it for burial.
The Third Rite of Passage: BurialPottery was one of the most
common grave gifts for the dead. (Image: British Museum/Public domain)
It was one’s relatives who conducted the burial service. No priests were present either. Priests were debarred for exactly the same reason that Artemis absented herself from the dying Hippolytus, so as not to incur pollution. Because if they incurred pollution, they might transmit it to the gods.
Absolutely nothing is known about the details of the burial service. Truth be told, it’s not even known if there was a burial service as such. If any traditional words were spoken, they were not recorded. Both inhumation and cremation were practiced, although cremation, being more costly, was seen as more prestigious. If one was cremated, then one’s relatives would gather the ashes and place them in an urn, which they then would bury along with the grave gifts.
The commonest grave gift was pottery. In fact, that’s why so many high-quality Greek vases have survived intact—because they were placed intact in the ground.
Over time, however, the Greeks became more stingy. Chances are, if one died in the 4th century B.C., all one would get is a couple of oil flasks known as lêkythoi filled with olive oil—olive oil was regarded as a luxury item. Some Greeks, however, were so stingy that they purchased lêkythoi with a smaller internal container to save them the expense of filling the whole vase with oil. Supposedly, they thought the dead wouldn’t notice.
As soon as the filling of the grave was done, they’d erect a grave marker over it. After completing the third and final rite of passage, all the mourners would return to the grieving home for a commemorative banquet.
The Burial Laws
Since one’s corpse was regarded as a source of pollution—the Greek word for the pollution is miasma, which means much the same in English—one had to be buried outside the city walls. In the ancient Greece, burial within a settlement was extremely rare after the 8th century B.C. The same was true of Rome. The earliest Roman law code, the Law of the Twelve Tables, dated 450 B.C., contains the provision, “The dead shall not be buried or burnt inside the city.”
It is not certain, but the origins of the belief in pollution may be connected with a kind of primitive sense of hygiene. Dead one’s relatives and anyone else who had come into contact with the corpse were debarred from participation in any activities outside the home until the corpse had undergone purification.
Reintegration into the community for mourners didn’t take place until several weeks after the funeral. One’s relatives also had to take measures to prevent the polluting effect of one’s corpse from seeping into the community. That included providing a bowl of water brought from outside the house so that visitors could purify themselves on leaving.
Common Questions About Living the Ancient Greek Death
The three stages are the laying out or the prothesis , the funeral procession or the ekphora, and the burial or the Interment.
Greeks honored the dead by following the three rites of passage , by building the tombs in Ceramicus, the Potter’s Quarter, and by offering the grave goods.
Greeks prepared for the afterlife by following the three rites of passage and offering the grave goods.
According to the burial law in ancient Greece, one had to be buried outside the city walls.
Roman Beliefs About Life After Death
The funerals of the dead took place in quite an organized manner. This was mainly done by the professionals. The professional provided with the mourning by the women, some forms of dances and music also accompanied this. There was a difference as to how funeral took place for the poor and for the wealthy people.
For the poor people, the funeral was something that took place in a very simple manner and for the wealthy people, the funeral happened on a large scale and it was quite a fantastic ceremony.
There were people who wore masks and they were the ones that rode the chariot. Romans did either burial or cremation. In case of cremation, the dead were cremated on a pyre. The gifts and the personal belongings of the person were kept with him in his tomb.
And in the case of in humanization, the bodies were protected. This protection was done either with the help of a sack, a wood-like structure, etc.
The belief in life after death, which is maintained by each of the Abrahamic religions, raises the metaphysical question of how the human person is to be defined. Some form of mind-body dualism, whether Platonic or Cartesian, in which the mind or soul survives the death of the…
…provides an argument for a life after death in which the injustices and inequities of the present life are remedied.
American Indian religions
The beliefs of the Aztec concerning the other world and life after death showed the same syncretism. The old paradise of the rain god Tlaloc, depicted in the Teotihuacán frescoes, opened its gardens to those who died by drowning, lightning, or as a result…
…most groups believed in an afterlife. It was generally thought that the souls of the recently deceased would hover around the community and try to induce close friends and relatives to join them in their journey to eternity thus, the elaborate funerary rites and the extensive taboos associated with death…
Ancient European religions
They believed in a life after death, for they buried food, weapons, and ornaments with the dead. The druids, the early Celtic priesthood, taught the doctrine of transmigration of souls and discussed the nature and power of the gods. The Irish believed in an otherworld, imagined sometimes as underground…
…and imaginative picture of the afterlife. The living were perpetually obsessed by their care for the dead, expressed in elaborate, magnificently equipped and decorated tombs and lavish sacrifices. For, in spite of beliefs in an underworld, or Hades, there was also a conviction that the individuality of the dead somehow…
…alluded to the kind of afterlife that was expected for the deceased. The Elysium-like concept of the afterlife prevailed in the Archaic period, but in the ensuing centuries one finds a growing emphasis on the darker realm of the underworld. Frescoes show its ruler, Hades (Etruscan Aita), wearing a wolf-skin…
No unified conception of the afterlife is known. Some may have believed that fallen warriors would go to Valhalla to live happily with Odin until the Ragnarök, but it is unlikely that this belief was widespread. Others seemed to believe that there was no afterlife. According to the “Hávamál,” any…
…it was believed, to the realm of Hades by Hermes but the way was barred, according to popular accounts, by the marshy river Styx. Across this, Charon ferried all who had received at least token burial, and coins were placed in the mouths of corpses to pay the fare.
…last things, especially death and afterlife) with their discoveries, they invested music, geometry, and astronomy with religious values. According to their doctrine, the original home of the soul was in the stars. From there it fell down to earth and associated with the body. Thus, man was a stranger on…
…most Romans’ ideas of the afterlife, unless they believed in the promises of the mystery religions, were vague. Such ideas often amounted to a cautious hope or fear that the spirit in some sense lived on, and this was sometimes combined with an anxiety that the ghosts of the dead,…
Ancient Near and Middle Eastern religions
…for the tomb and the next world. Egyptian kings are commonly called pharaohs, following the usage of the Bible. The term pharaoh, however, is derived from the Egyptian per ʿaa (“great estate”) and dates to the designation of the royal palace as an institution. This term for palace was used…
Belief in an afterlife and a passage to it is evident in predynastic burials, which are oriented to the west, the domain of the dead, and which include pottery grave goods as well as personal possessions of the deceased. The most striking development of later mortuary practice was…
…common Indo-European notion of the hereafter, pictured as a pastureland with grazing cattle “for which the dead king sets out.” This suggests that the Indo-European forebears of the later speakers of Hittite, Palaic, and Luwian, as well as those of minor members of this group, entered Anatolia together, following a…
…of the personal continuance of life after death. Many baptized early Christians were convinced they would not die at all but would still experience the advent of Christ in their lifetimes and would go directly into the Kingdom of God without death. Others were convinced they would go through the…
…the ability to destroy and bring back to life all creatures, who are limited and are, therefore, subject to God’s limitless power.
…arose a belief in an afterlife, for which the dead would be resurrected and undergo divine judgment. Before that time, the individual had to be content that his posterity continued within the holy nation. But, even after the emergence of belief in the resurrection of the dead, the essentially ethnic…
…may have continued to exist, but it was not to be understood any longer as life. The existence of the dead in sheol, the netherworld, was not living but the shadow or echo of living. For most biblical writers this existence was without experience, either of God or of anything…
…extremely subtle position that equated immortality with the cleaving of the human intellect to the active intellect of the universe, thus limiting it to philosophers or to those who accepted a suitable philosophical theology on faith. Little or no consensus was evident in the modern period, though the language of…
…belief, every person after his death becomes a kami, a supernatural being who continues to have a part in the life of the community, nation, and family. Good men become good and beneficial kamis, bad men become pernicious ones. Being elevated to the status of a divine being is not…
…fate awaiting individuals in the afterlife. Each act, speech, and thought is viewed as being related to an existence after death. The earthly state is connected with a state beyond, in which the Wise Lord will reward the good act, speech, and thought and punish the bad. This motive for…
…the gift of immortality this afterlife was first sought by the pharaohs and then by millions of ordinary people. The second was the concept of a postmortem judgment, in which the quality of the deceased’s life would influence his ultimate fate. Egyptian society, it has been said, consisted of the…
…soul with personal survival or continuity after death, there is an equally ancient view that emphasizes the continuity of life. This view, to which the Dutch anthropologist Albertus Christiaan Kruyt gave the term soul-stuff (a term he contrasted with the postmortem soul), is chiefly found among the rice cultivators of…
…that death is followed by everlasting life elsewhere—in Sheol, hell, or heaven—and that eventually there will be a universal physical resurrection. Others (e.g., Buddhists, Orphics, Pythagoreans, and Plato) have held that people are reborn in the time flow of life on earth
To give the dead new life beyond the grave, mourners may allow life-giving blood to fall upon the corpse sacramentally. In this cycle of sacramental ideas and practices, the giving, conservation, and promotion of life, together with the establishment of a bond of union with the sacred order, are…
…is to appeal to a life after death the hardships of this life, whether caused by natural evil or by moral evil, are as nothing compared with the rewards to come, and they are a necessary factor in preparing one for the afterlife through moral training and maturation. This line…
The Theater in Ancient Greece
Theater in ancient Athens was performed at the agora. Later, the theatrical events became so big that they were moved to an open-air auditorium below the Athenian Acropolis. Open-air auditoriums were built in most Greek cities, some holding as many as 15,000 spectators.
Theater performances became part of the religious festival to Dionysus, the god of wine. The festival lasted five days and had as many as three full dramas performed in one day. The dramas were judged contests, and the winning actors and playwrights received prizes. The dramas were sponsored by rich citizens know as choregoi.
Three types of plays were developed in ancient Greece, the tragedy, the comedy, and the satire. A tragedy was about Greek heroes and gods. The plots often showed conflicts between men and gods, and the endings were often bad for the main characters. Comedies were often politically-based stories, or they featured conflicts between men and women. They were meant to be lighthearted fun. Satires were often witty, cutting, and ironic stories that mocked human vice and folly.
Early plays were performed with just one actor, but casts were later expanded to include three actors. Actors wore masks that signaled to the audience the identity and possibly the mood of the character at a certain moment or scene in the play. One actor played multiple parts, changing masks to portray different characters. Actor&rsquos costumes signaled the mood and characteristics of the character. Darker clothing was associated with the tragic character, and light clothing was associated with happy or funny roles.
The History of Gambling in Ancient Greece
Modern forms of gambling can be traced back to many ancient cultures, from China to Egypt and beyond.
Yet, the truth is that ancient Greece played a bigger part in the development of modern forms of gambling than most places.
A Look at the Origins of Gambling in Greece
You wouldn’t expect casinos with the highest payout slot machines , but Ancient Greece had their own means of placing wagers.
Gambling games based on throwing dice and tossing coins have been mentioned in some ancient Greek books and stories. Some sources suggest the game of poker may also have begun here, although others think that it was first played in China or Persia.
Ancient Greek icosahedronal Dice
What can’t be denied is that gambling was hugely popular in this culture, with special places where gamblers could go to place some bets. We can see it in sculptures and paintings too, with people betting on fights and races.
Interestingly, the Gods Hermes and Pan are both said to have placed wagers, while Zeus, Poseidon and Hades decided how to split up the world by drawing straws. Yet, some Greek philosophers were against gambling and thought that it would damage civilization if left unchecked.
Some of the Most Popular Games
One of the games that is often mentioned as being popular in the olden days in Greece is Heads and Tails. This was first played with shells, before the introduction of coins made it easier to gamble on which side would end up facing upwards. Pitch and Toss was a game that involved throwing coins at a wall.
Perhaps the simplest game of all was the one called Par Impar Ludere. One player would hold a bunch of small items in one hand and the other person had to guess whether the total number of objects was odd or even. The Greeks would bet on the outcome, and it also became popular in the Roman empire. Gambling is also believed to have been a huge factor in the early Olympic Games
It is claimed the Palamedes invented dice when Troy was under siege and that this led to his dice being used in a Temple of Fortune in Corinth. However, this appears to just be a legend, as the first mention of dice in Greece can be traced all the way back to 6000BCE.
A theory that runs through the ancient Greek’s love of gambling is that the Gods controlled the games. Even the outcome of a game of pure luck like throwing dice was considered to be in the lap of the gods.
Ancient Greek astragali used for playing gambling games
Modern Gambling in Greece
If we fast forward in time to the present day, we can see that gambling in Greece is legal in land-based establishments. The big cities all tend to have a few casinos in them, while the islands that are popular with tourists also offer casinos to visitors.
Among the most famous casinos in the country is the Mont Parnes Regency Casino of Athens. It dates from the 1960s and is located in the National Forest of Parnitha. A luxurious landmark in the capital, this is a stylish casino with lots of different ways of gambling.
Apparently, the oldest casino in Greece was built in Loutraki in the early part of the 20th century. Most modern casinos here are refined and exclusive establishments where players can bet in comfort.
The Greek Gambling Commission controls betting in the country, while players in Greece can easily and safely access a large range of online casinos and sports betting sites from foreign operators. This means that currently people can wager on the likes of soccer, tennis and basketball betting online.
It is still something of a gray area, though, as the Greek regulators and the European courts have produced different opinions on the legality of online gambling in Greece.
Therefore, it is worth keeping an eye out for any future changes to the legislation in this fast-moving industry that could have an effect on Greek players.
Greek Life As Depicted in Homer's Epic: The Odyssey
In Homer’s epic, The Odyssey, various aspects of the ancient Greeks are revealed through the actions, characters, plot, and wording. Homer uses his skill as a playwright, poet, and philosopher to inform the audience of the history, prides, and achievements of the ancient Greeks, and, also, to tell of the many values and the multi-faceted culture of the ancient Greek caste. The Greeks had numerous values and customs, of which the primary principles are the mental characteristics of an individual, the physical characteristics of an individual, the recreations and pastimes the Greeks enjoyed, the way in which a host treats a guest, the religious aspects, and finally, the Greeks’ view on life, revealed in The Odyssey which shows and defines their culture
One of the most prominent of the mental characteristics the ancient Greeks valued was the cleverness and the wit of an individual. This can be discerned from The Odyssey because of many instances and events in which Odysseus uses his brain’s wit and other tricks to get himself out of a risky situation. Examples of this are when he tells Polyphemos the Cyclopes that his name is Nobody, when he overcomes Circe’s magic with the help of moly, when he fills his men’s ears with wax and ties himself to a post so that he and his men can get by the Sirens safely, and when he disguises himself as a beggar and reveals his true identity to few. Odysseus is by “far the best of mortal men for counsel and stories” (Bk. XIII, 297 – 298). Also, Odysseus is said to be able to match a god in wits and trickery (Bk. XIII, 291 – 295). Penelope, Odysseus’ wife also uses her wit and trickery to get herself out of situations. An example of this is when she pretends to be weaving a shroud for Laertes, but actually undoes at night as much as she had done in the morning. Athene, the goddess of wisdom, provides another example of the usage of wit and tricks. Athene disguises Odysseus as a beggar and also surrounds him with a mist numerous times so that his former acquaintances will not see or recognize him.
Other significant mental characteristics that the Greeks valued are faithfulness and loyalty. There are many, many examples of loyalty and faithfulness in The Odyssey. The four most significant examples are Penelope, Eumaios, Philoitois, and Argos. Penelope is Odysseus faithful wife who never slept with anyone else besides Odysseus, even though she was tempted. She also keeps herself hoping that Odysseus is still alive and will someday come home. Eumaios is the loyal swineherd who helps Odysseus overcome the suitors. Philoitois is the loyal ox herd who also helps Odysseus overcome the suitors. Argos is the “patient-hearted … dog” (Bk. XVII, 292) of Odysseus. Odysseus tests these individuals (except the dog) to decide whether he can trust them or not. He also tests other individuals, such as the servants, to find if they are loyal to him or not.
Physical characteristics were just as important to the Greeks as mental characteristics. Strength was one of the more dominantly looked upon of the physical characteristics. Strength was a common test and was used to gauge a man’s place in the real world. Penelope used strength as a test for the competition for the suitors. The competition was to be able to string Odysseus’ bow and shoot it accurately, the prize (marriage of Penelope) going to “the one who takes the bow in his hands, strings it with greatest ease, and sends an arrow clean through all twelve axes” (Bk. XXI, 75 – 76). Strength was also a part of the Phaiakian’s competition. Strength was needed for the discus throwing (which Odysseus excelled in), wrestling, and boxing. Also, the Greeks loved competition, proven by the fact that they urged on Odysseus and Iros to fight. And when they finally saw blood, they went crazy, laughing and cheering like it was the most exciting thing in the world.
The Greeks enjoyed many recreations and pastimes, of which dancing, singing, and storytelling were dominant. The Phaiakians were known for their terpsichorean skills, and as Odysseus said, wonder and awe cane to him when he watched the dancing (Bk. VIII, 382 – 384). Singing was also a well-loved recreation. Singers were well known and loved by all. As Odysseus said to Demodokos, “Demodokos, above all mortals beside I prize you” (Bk. VIII, 487). The only survivor of those who had plotted against Odysseus was Phemios, the singer of the suitors. He survives because Odysseus allows him to live because of his gift of voice from the gods. As Telemachus says about the suitors, “This is all they think about, the lyre and the singing” (Bk. 1, 159). Storytelling is yet another virtue and is prized by Greeks. Menelaos tells of his adventures to Telemachus, Odysseus tells of his adventures to the Phaiakians, and Odysseus tells of his false adventures to Eumaios. Another pastime that the Greeks enjoyed very much is feasting, or in crude terms, eating and drinking. The suitors always eat and provide plenty, even though they eat Odysseus cattle and drink Odysseus’ wine. They have many drinking contests to see who can drink the most, and usually, at the end the contestants usually become bacchanalian. The suitors always have a “desire for eating and drinking” (Bk. 1, 150) according to Telemachus.
The treatment of a guest was very important in the times of the ancient Greeks. It defined your social class, and it also helped you in favour with Zeus, who is the god of travellers and guests. A wide range of things can be classified as hospitality, but the general idea is always the same and cannot change. Hospitality was giving any stranger food, warmth, shelter and comfort before asking questions such as their name, heritage, or means of transportation. Hospitality also meant an ear for every word and respect for every word as well. Also, the host is responsible for being the aegis of the guest while the guest resides at his home. Telemachus feels that he cannot provide this for his father (in guise of a beggar), and is therefore ashamed. “How can I take and entertain a stranger guest in my house? I myself am young and have no faith in my hands’ strength to defend a man, if anyone else picks a quarrel with him (Bk. XVI, 69 – 72). Examples of good hospitality are abundant throughout The Odyssey, such as when Athene goes to Telemachus in Ithaca, when Telemachus goes to Nestor, when Telemachus goes to Menelaos, when Odysseus goes to the Phaiakians, and when Odysseus goes to Eumaios. Presents at arrival are expected, but presents at departure are not always present. However, in case of a wealthy, generous, or friendly host, presents, even those with incalculable and immense values can be exchanged.
The religious beliefs and aspects of the ancient Greek culture are very defined and strict. The Greeks believed that the world was watched over by Zeus and other Olympian gods, and that these gods decided their future. They also believed that the gods’ wills could be turned with sacrifices. This is why Odysseus, Telemachus, and many, many other characters made so many sacrifices to the gods. These characters also pray to the gods so that the gods can hear them and fulfil their wishes. The Greeks also believed in “life” after death in the underworld with Hades. Another religious aspect of the Greek culture was prophecies. Prophecies and prophet were abundant, but the supply of accurate prophecies and prophets were much less abundant, and the demands for these were high, making them scarce. The two main prophets in The Odyssey were Teiresias and Theoklymenos. Teiresias was a dead prophet who Odysseus went to consult in the underworld. He prophesized most aspects of Odysseus’ journey accurately and because of him, Odysseus was able to survive his wanderings. Theoklymenos was a prophet from a family of prophets. He could prophesize quite accurately from bird auguries, as shown when he prophesizes that Telemachus “shall have lordly power forever” (Bk. XV, 534). Homer uses quite a few bird auguries in The Odyssey, one in the beginning to warn the suitors of Odysseus’ homecoming (Bk. II, 146 – 154), and two near the end, both to symbolize Odysseus’ triumph over the suitors.
The ancient Greeks had an optimistic view on life, a view that makes nice, happy endings, but are unfortunately not very realistic. The Greeks believed that in the end of any hardship or endurance, justice would emerge and show its victorious smile to the victim. They believed that persistence and determination would come through in the end. The Greeks also believed that in a battle between good and evil, good will triumph in the end. The view that good triumphs versus evil can be seen in the epic when Odysseus (good) kills all of the suitors (bad) against virtually impossible odds. The view that justice will emerge in the end is shown in The Odyssey when all unfaithful servants and maids are killed. The view of persistence and determination succeeding is proved by the fact that Odysseus “who, after much suffering, came at lest in the twentieth year back to his own country” (Bk. XXIII, 101 – 102) survived all of his shipwrecks, attacks, and other hindrances and ultimately succeeds in coming back home.
Throughout The Odyssey, Greek values and the Greek culture are constantly shaped by the flow of the author’s pen, which narrates a story with an intricate plot. The epic allows the modern-day public know about the times when men fought with their hands and their heads, when the gods dominated cultures, and when love and faithfulness meant something. The Odyssey is a great work of a great poet, Homer, who not only captures the essence of the ancient Greek spirit and culture, but also tells a story that can be passed down from generation to generation, without any fear of growing old.
Conic Sections in Ancient Greece
The knowledge of conic sections can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C. it is reported that he used them in his two solutions to the problem of "doubling the cube". Following the work of Menaechmus, these curves were investigated by Aristaeus and of Euclid. The next major contribution to the growth of conic section theory was made by the great Archimedes. Though he obtained many theorems concerning the conics, it does not appear that he published any work devoted solely to them. Apollonius, on the other hand, is known as the "Great Geometer" on the basis of his text Conic Sections , an eight-"book" (or in modern terms, "chapter") series on the subject. The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely.
In the years following Apollonius the Greek geometric tradition started to decline, though there were developments in astronomy, trigonometry, and algebra (Eves, 1990, p. 182). Pappus, who lived about 300 A.D., furthered the study of conic sections somewhat in minor ways. After Pappus, however, conic sections were nearly forgotten for 12 centuries. It was not until the sixteenth century, in part as a consequence of the invention of printing and the resulting dissemination of Apollonius' work, that any significant progress in the theory or applications of conic sections occurred but when it did occur, in the work of Kepler, it was as part of one of the major advances in the history of science.
This paper will investigate the history of conic sections in ancient Greece. We will examine the work of the aforementioned mathematicians relevant to conic sections, with specific attention given to Apollonius's text on Conic Sections .
Pappus and Proclus
It may seem odd to begin with these late figures, but the significance of Pappus and Proclus needs to be established early. While Pappus of Alexandria was a competent mathematician and geometer, we are interested here in his work as a mathematical commentator and historian of mathematics. Residing mainly in Alexandria, roughly 500 years after the likes of Euclid, Archimedes, and Apollonius graced the intellectual scene, Pappus wrote several commentaries on the works of many great mathematicians of the past (that is, of his past!). One of his most important contributions was his Mathematical Collection , an eight book series that included commentary and historical notes, as well as several original propositions and extensions of existing works. In Book VII, he discusses twelve treatises of the past which included Apollonius' Conic Sections , Euclid's Surface Loci , and Aristaeus' Solid Loci (Eves, 1990, p. 183-4). Pappus gives us great insight into the lives and works of Greek geometers. He had access to works which are now lost, and in addition to being a skilled mathematician in his own right, he provides a valuable link to ancient Greek geometry.
Proclus, who lived during the fifth century A.D., was also a notable mathematical historian. Like Pappus, he had access to original documentation of the mathematics of the Classical and Hellenistic eras that is no longer available. His Eudemian Summary is an invaluable source of information about early Greek mathematical work up to Euclid (Eves, 1990, pp. 74- 75). His authority will be called upon in this paper, particularly when examining the influence of Aristaeus and Euclid.
According to tradition, the idea of the conic sections arose out of the exploration of the problem of "doubling the cube". This problem, and the accompanying story, is presented in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes, which has come down to us as quoted by Eutocius in his commentary on Archimedes' On the Sphere and Cylinder , which appears in Heath. Eratosthenes told the king that the legendary King Minos wished to build a tomb for Glaucus and felt that its current dimensions - one hundred feet on each side - were inadequate.
- Too small thy plan to bound a royal tomb. Let it be double yet of its fair form Fail not, but haste to double every side.
Clearly, doubling every side will increase the volume by a factor of eight, not by the desired factor of two. Mathematicians worked diligently on this problem, but were having tremendous difficulty in solving it. A breakthrough of a kind occurred when Hippocrates of Chios reduced the problem to the equivalent problem of "two mean proportionals", though this formulation turned out to be no easier to handle than the previous one (Heath, 1961, p. xviii). Eratosthenes continued by mentioning the Delians, who had an interest in exactly the same problem of "doubling a cube". When they called upon geometers at Plato's Academy in Athens for a solution, two geometers found answers to the equivalent mean proportions problem. Archytas of Tarentum used "half- cylinders", and Eudoxus used "curved lines". These solutions, however, only gave demonstrations of the existence of the desired number as a geometrical quantity, but they could not actually construct the mean proportion mechanically, so they did not reach the point of practical application until Menaechmus, who he achieved it with considerable difficulty (Heath, 1961, pp. xvii-xviii).
The mention above of Hippocrates' mean proportions is of interest. What this means is that, given two lengths a and b, we find x and y such that a:x::x:y and x:y::y::b, or in modern notation a/x=x/y=y/b if we denote this ratio by r, then r^3 = (a/x)(x/y)(y/b)=a/b, and as Hippocrates noted, that if the segment a is twice as long as the segment b, then the doubling of the cube would be solved using the length r. Needless to say, he did not have an algebraic notation able to support the argument in the form we have given it, and had to argue directly.
Menaechmus was a pupil of Eudoxus, a contemporary of Plato (Heath, 1921, p. 251). Much of what we know about Menaechmus's work comes to us from the commentaries of Eutocius, a Greek scholar who discussed the works of many mathematicians of his and earlier times, including Menaechmus, Archimedes, and Apollonius. In his solutions, Menaechmus essentially finds the intersection of (ii) and (iii) (see Solution 1, below), and then, alternatively, the intersection of (i) and (ii) (see Solution 2, below). Menaechmus's proof deals with the general case of the mean proportions. Once we have this, we can take the special case a=2b to double the cube. Before giving these two solutions, it should be noted that Menaechmus did not use the terms "parabola" and "hyperbola" - these terms are due to Apollonius. Instead, he called a parabola a "section of a right-angled cone", and a hyperbola a "section of an obtuse-angled cone" (Heath, 1921, p. 111).
- Solution 1:
- Let AO, AB be two given straight lines such that AO > AB and let them form a right angle at O.
- Suppose the problem solved and let the two mean proportionals be OM measured along BO produced and ON measured along AO produced. (Heath, 1921, p. 253).
- Complete the rectangle OMPN.
- Because AO : OM = OM : ON = ON : OB, we have by cross multiplication the following relations:
- (1) OB.OM = ON² = PM² [the "." refers to multiplication], such that P lies on a parabola which has O for its vertex, OM for its axis, and OB for its latus rectum.
- (2) AO.OB = OM.ON = PN.PM, such that P lies on a hyperbola with O as its center, and OM and ON as its asymptotes.
- To find the point P, we must construct the parabola in (1) and the hyperbola in (2), and once we do so, the intersection of the two solves the problem, for AO : PN = PN : PM = PM : OB.
- Solution 2:
- Suppose AO and AB are given and the problem to be solved as in the first two steps to Solution 1.
- Again, we have AO : OM = OM : ON = ON : OB, giving us
- (1) as in solution 1, the relation OB.OM = ON² = PM ², such that P lies on a parabola which has O for its vertex, OM for its axis, and OB for its latus rectum.
- (2) the relation AO.ON = OM² = PN², such that P lies on a parabola which has O for its vertex, ON for its axis, and OA for its latus rectum.
- To find the point P, we must construct the two parabolas described in (1) and in (2). The intersection gives us the point P such that AO PN = PN : PM = PM : OB
While it is apparent that Menaechmus utilized what later became known as conic sections, did he really have a construction involving a cone in mind when he solved the problem of doubling the cube? Heath argues that he did, for the following reason. In the same letter from Eratosthenes to Ptolemy mentioned above, Eratosthenes stated, in connection with a discussion of his own solution to the problem, that there is no need to resort to "cutting the cone in the triads of Menaechmus" (Heath, 1961, xviii). In addition to this quotation appearing in Eutocius' commentary on Archimedes, Proclus confirms that conics were discovered by Menaechemus (Heath, 1961, xix) .
Now that we have seen how Menaechmus first applied the conic sections, one might wonder, "How did he think of obtaining these curves from a cone?". Though there is virtually no information on this question itself, intuition tells us that the keen observational skills of Greek mathematicians would be attracted to such shapes. It is likely that the first conic section noticed in nature would have been an ellipse. If one cuts a cylinder at an angle other than a right angle to its axis, the result is an ellipse. In fact, Euclid notes in his Phaenomena that a cone or cylinder cut by a plane not parallel to the base results in a section of an acute-angled cone which is "similar to a [shield]" (Heath, 1921, 125). A natural extension of this phenomena would by the cutting of a cone in a similar fashion. Then perhaps they moved the cutting plane so that it does not cut the cone completely. What types of curves result? How are each of their properties similar to the other sections? How are they different? This is a possible, and probably simplified, discussion of the flowing of ideas that led to the study of conic sections.
Neugebauer suggests that the origin of the concept is in the theory of sundials, since the sheaf of light rays involved in the design of sundials is a cone which is cut by the plane of the horizon in a hyperbola, and a portion of that hyperbola is then marked out on the sundial.
According to Geminus, the ancients revolved a right triangle about one of its legs to determine a cone. Additionally, only right cones were known. Of these right-angled cones, there are three types. Evidently the vertical angle at the top of the cone could be less than ninety degrees, more than ninety degrees, or exactly ninety degrees (Heath, 1921, p. 111). We will see later when we study Apollonius, that there is a fundamental difference in the types of cones he considers. The segment connecting the "top point" of the cone to the center of the circular base is always a right angle. Apollonius considers a more general form of the cone do not assume the right angle (Heath, 1961, p. 1). Returning the specialized cones from the account of Geminus, these cones were called acute-angled, obtuse-angled, and right-angled cones (not to be confused with right cones, which refer to the revolution of a right triangle). In addition to the two names for hyperbola and parabola given previously, an ellipse was known as a "section of an acute-angled cone" (Heath, 1921, p. 111).
Nothing is known of the methods used by Menaechmus's to deal with these curves (Cajori, 1924, p. 27). Heath discusses what he calls his "probable" method, based on the assumption that Menaechmus's constructions of his curves would likely be rather simple and direct, but instructive enough to demonstrate the salient properties. This will not be discussed any further. Fortunately we have extensive documentation of the treatises of later geometers, notably Appolonius, on the subject of conic sections.
Aristaeus and Euclid
We next come to the (again, lost) works of Aristaeus `the elder' and of the celebrated Euclid on conic sections. Since we do not have the original works by these two men on conic sections, our knowledge of them is derived from the comments of Pappus, whose writings are discussed in Heath, using a translation by Hultsch:
The four books of Euclid's conics were completed by Apollonius, who added four more and produced eight books of conics. Aristaeus, who wrote the still extant five books of solid loci connected with the conics, called one of the conic sections the section of an acute-angled cone, another the section of a right-angled cone and the third the section of an obtuse-angled cone. Apollonius says in his third book that the `locus with respect to three or four lines' had not been completely investigated by Euclid, and in fact neither Apollonius himself nor any one else could have added in the least to what was written by Euclid with the help of those properties of conics only which had been proved up to Euclid's time Apollonius himself is evidence for this fact when he says that the theory of that locus could not be completed without the propositions which he had been obliged to work out for himself. Now Euclid-regarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system, being moreover in no wise contentious and, though exact, yet no braggart like the other-wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations . (Heath, 1961, pp. xxi-xxii)
Before discussing the implications of Pappus' words, we turn to Proclus to give us some insight into the concept of a "solid locus". He defines a locus to be "a position of a line or surface involving one and the same property" (Heath, 1961, p. xxxii). Loci are divided into two classes, "line-loci", and "surface-loci". Within the line loci are "plane- loci" and "solid-loci". Plane-loci are generated in a plane, like the straight line. Solid-loci are generated from a section of a solid figure, i.e. the cylindrical helix and the conic sections. Pappus makes a division of what Proclus calls the solid-loci. He breaks this category into "linear-loci" and "solid-loci", not to be confused with what Proclus calls solid-loci. Solid-loci, to Pappus, are sections of cones (parabolas, ellipses, and hyperbolas), and linear-loci are more complicated lines than straight lines, circles, and conic sections (Heath, 1961, p. xxxiii).
With this information, along with Pappus' passage, Heath drew several conclusions concerning the works of Euclid and Aristaeus concerning conic sections. First, Aristaeus' treatment of solid loci concentrated on parabolas, ellipses, and hyperbolas, i.e. he considered conics to be loci. Second, Aristaeus' treatise on solid loci came first, and contained more original ideas and theorems than did Euclid's. Pappus says that Euclid wrote about the basic theory of conic sections, targeting his propositions to prepare readers to analyze the solid loci of Aristaeus (Heath, 1961, p. xxxii). Along these same lines, Heath remarks that "Euclid's Conics was a compilation and rearrangement of the geometry of the conics so far as known in his time, whereas the work of Aristaeus was more specialized and more original" (Heath, 1921, pp. 116-7). Third, Aristaeus used the terms "section of right-angled, acute-angled, and obtuse-angled cone", the accepted names for these curves until Apollonius. Finally, The Conics of Euclid was superseded by Conic Sections by Apollonius.
In addition to the ideas above, a key to pull from the work of Aristaeus and Euclid is that they were a source upon which mathematicians based their work upon, or at least consulted. We will see this in action as we continue our discussion with Archimedes and Apollonius.
"No survey of the history of conic sections could be complete without a tolerably exhaustive account of everything bearing on the subject which can be found in the extant works of Archimedes" (Heath, 1961, p. xli). There is no substantiated evidence that he ever wrote an entire work devoted to conic sections, but his knowledge of the subject is obvious in the works we do have. Among the treatise Archimedes published was Quadrature of the Parabola, Conoids and Spheroids, Floating Bodies, and Plane Equilibrium. These works share a common thread-they require the extensive use of the properties of parabolas, Archimedes' specialty amongst the conic sections (Heath, 1921, p. 124).
Heath says that Euclid's Conics is the probable source from which Archimedes adopts basic principles of conics that he assumes without proof (Heath, 1921, p. 122). He uses the "old", pre-Apollonius names for the conic sections (i.e. section of an acute-angled cone = ellipse) (Heath, 1961, p. xlii). Before going on it is important to clarify his vocabulary. Diameters are what we consider to be the axes of the ellipse (both the major and minor). These two diameters are conjugate. The axis of a parabola is also called a diameter, and the other diameters are called "lines parallel to the diameter". The diameter of a hyperbola is the portion of what we consider the axis within the single-branched hyperbola (Archimedes consider the second branch to be part of the same curve). The center of the hyperbola was called the point in which the "nearest lines to the section of an obtuse-angled cone" (asymptotes) meet (Heath, 1921, p. 122).
Heath cites several assumptions Archimedes made on the basis of previous works by the likes of Euclid and Aristaeus. With reference to the central conics:
- The straight line drawn from the center of an ellipse, or the point of intersection of the asymptotes of a hyperbola, through the point of contact of any tangent, bisects all chords parallel to the tangent In the ellipse, the tangents at the extremities of either of two conjugate diameters are both parallel to the other diameter. If a cone, right or oblique, be cut by a plane meeting all the generators, the section is either a circle or an ellipse. If a line between the asymptotes meets a hyperbola and is bisected at the point of concourse, it will touch the hyperbola If x, y are straight lines drawn, in fixed directions respectively, from a point on a hyperbola to meet the asymptotes, rectangle xy is constant. With reference to parabolas in particular, Parallel chords are bisected by one straight line parallel to the axis, which passes through the point of contact of the tangent parallel to the chords. If the tangent at Q meet the diameter PV in T, and QV be the ordinate to the diameter, PV = PT [see Apollonius for definition of ordinate]. All parabolas are similar (Heath, 1921, pp. 123-24)
The nature of Archimedes' writings seems to be such that he only proves what is not reasonably obvious to a trained mathematician. What was obvious to Archimedes, however, does not always coincide with what is obvious to most people! By the same argument, the propositions that Archimedes does prove tend to be very difficult. Archimedes seemed to be less concerned with developing a complete, systematic treatment of the conics (which in any case was accessible in the now lost works of others), but rather with using what was already established and/or easily proved develop deep and challenging theorems. For this reason, this paper, while it has given a basic background of the assumptions and basic trends of Archimedes' study, will not examine the original proofs he gave.
Along with Euclid and Archimedes, Apollonius is the third member of the trio of great geometric minds of Ancient Greece. "It is no exaggeration to say that almost every significant subsequent geometrical geometric, right up to and including the present time, finds its origin in some work of these three great scholars" (Eves, 1963, 25). Only a small amount of information is known about the life of Apollonius. He was born in the city of Perga, in Pamphylia, which was is located in southern Asia Minor, now Turkey. The date of his birth again is agreed upon by both Eves and Heath to be approximately 262 B.C., that is, approximately 25 years after the birth of Archimedes. As a young man he traveled to Alexandria to study with the successors of Euclid. He flourished during the reign of Ptolemy Euergetes ("The Benefactor", 247-222 B.C.). He continued to be a recognized scholar during the reign of Ptolemy Philopator (222-205 B.C.). (Heath, 1921, 126). It is also known that he visited Pergamum, where he met Eudemus, to whom he dedicated the first two books of his Conic Sections (Heath, 1921, 126). The third through seventh books (and possible the eighth, which is lost) were dedicated to King Attalus I (241-197 B.C.), a fact which has helped historians estimate the years of his lifetime.
Four of Apollonius' eight books have come down to us in Greek. The eighth book is completely lost - we do not even have any knowledge of its contents. Books V-VII have reached us in an Arabic translation, whose date is debatable. Eves and Heath consider it to be a ninth century translation (Eves, 1990, p. 171). Cajori on the other hand writes of a 1250 translation, without any mention of the ninth century one (Cajori, 1924, 38). Two brothers from the family Muh, Ahmad and al-Hasan, first contemplated translating Conic Sections into Arabic during the ninth century. They almost lost interest due to the poor condition of their manuscripts. Ahmad received a copy of Eutocius' edition of Books I-IV and had them translated by Abi Hilal al-Himsi (died 883/4). He then gave a different manuscript of books V-VII to Thabit ibn Qurra (lived 826-901) to translate. Confirming Cajori's mention of the 1250 translation, Heath reports that in 1248, another translation was made by Nasir ad-Din (Heath, 1921, p. 127).
Apollonius opens each of his surviving books with a preface. The preface to Book I, which serves as a general preface for the whole series, and to Book V have been included in Appendix A. From the general preface we learn that the first four books of Conic Sections completed and formalized the previous work known to Apollonius at the time. According to Heath, Apollonius never claims the material covered in the first four books to be original, except for certain theorems in Book III, and the investigations in Book IV. What he does contend, however, is that his treatise is more complete and rigorous than previous works on the subject, which agrees with the comments of Pappus (Heath, 1961, p. lxxvii). Unlike most of the first four books, books five through seven covered new concepts which went beyond the "essentials". Heath states, The real distinction between the first four books and the fifth consists rather in the fact that the former contain a connected and scientific exposition of the general theory of conic sections as the indispensable basis for further extensions of the subject in certain special directions, while the fifth Book is an instance of such specialization the same is true of the sixth and seventh books (Heath, 1961, p. lxxvi).
Before we examine individual propositions from Conic Sections , it might be appropriate to mention the origin of the names of the conic sections as we know them to be today. According to Eves, the terms "ellipse", "parabola", and "hyperbola" were adopted from early Pythagorean vernacular referring to the "application of areas" (the form of "geometric algebra" recorded in Euclid's Elements , Book II. When applying a rectangle to a line segment [by aligning one edge of the rectangle to the segment with one corner of the rectangle matching up with one endpoint], the "other" corner of the rectangle either fell short of, met exactly, or exceeded the end of the segment. These three cases were respectively called "ellipsis", "parabole", or "hyperbole". Eves shows how these terms were applied in a similar spirit to the conic sections by Apollonius in the following manner:
- Let AB be the principal axis of a conic. Let P be any point on the conic. Let Q be the foot of the perpendicular to AB. Mark off a distance AR, perpendicular to AB by a distance now known as the latus rectum or parameter of the curve. Apply to the segment AR, a rectangle having for one side AQ and an area equal to (PQ)². If the rectangle exceeds the segment AR, then the conic is a hyperbola. If the rectangle coincides with the segment AR, then the conic is a parabola. If the rectangle falls short of the segment AR, then the conic is an ellipse. (Eves, 1963, pp. 30-1)
This argument alone does not seem to be a proof, or even a definition. As it is written, it certainly does not appear in Apollonius' Conic Sections, though, later, when his propositions are discussed, a similarity to these will be evident. Eves statements, however, do seem to check out when one follows the steps. The first three statements are clear, and common to all three cases. Not explicitly stated, let F be a focus of the given conic section, and K an endpoint of the latus rectum. Here are examples (non-Greek) of each of the three cases:
Before we go into Apollonius' method for proving these relationships, it would only be appropriate to start, as he did, by defining the relevant terms.
If a straight line indefinite in length, and passing always through a fixed point, be made to move round the circumference of a circle which is not in the same plane with the point, so as to pass successively through every point of that circumference, the moving straight line will trace out the surface of a double cone, or two similar cones lying in opposite directions and meeting in the fixed point, which is the apex of each cone.
The circle about which the straight line moves is called the base of the cone lying between the said circle and the fixed point, and the axis is defined as the straight line drawn from the fixed point or the apex to the center of the circle forming the base.
The cone so described is a scalene or oblique cone except in the particular case where the axis is perpendicular to the base. In this latter case the cone is a right cone.
If a cone be cut by a plane passing through the apex, the resulting section is a triangle, two sides being straight lines lying on the surface of the cone and the third side being the straight line which is the intersection of the cutting plane and the plane of the base.
Let there be a cone whose apex is A and whose base is the circle BC, and let O be the center of the circle, so that AO is the axis of the cone. Suppose now that the cone is cut by any plane parallel to the plane of the base BC, and DE, and let the axis AO meet the plane DE in o. Let p be any point on the intersection of the plane DE and the surface of the cone. Join Ap and produce it to meet the circumference of the circle BC in P. Join OP, op.
Then, since the plane passing though the straight lines AO, AP cuts the two parallel planes BC, DE in the straight lines OP, op respectively, OP, op are parallel.
And, BPC being a circle, OP remains constant for all positions of p on the curve DpE, and the ratio Ao : Ao is also constant.
Therefore, op is constant for all points on the section of the surface by the plane DE. In other words, that section is a circle.
Hence all sections of the cone which are parallel to the circular base are circles (Heath, 1961, pp. 1-2).
Conic Sections continues to define a diameter to be a straight line bisecting each of a series of parallel chords of a section of a cone. In each of the examples below, PP' is a diameter:
In the figures above, if QQ' is bisected by diameter PP' at V, then PV is called an ordinate, or a straight line drawn ordinate-wise. The length PV cut off from the diameter by any ordinate QV is called the abscissa of QV (Heath, 1961, pp. 7-8).
We now turn to Apollonius' definitions of the conic sections as we attempt to connect them to the definition Eves gave above. The case of the parabola will be given as an example of Apollonius' developments:
First let the diameter PM of the section be parallel to one of the sides of the axial triangle as AC, and let QV be any ordinate to the diameter PM. Then if a straight line PL (supposed to be drawn perpendicular to PM in the plane of the section) be taken of such a length that PL:PA = BC² : BA.AC , it is to be proven that QV² = PL.PV
Let HK be drawn through V parallel to BC . Then, since QV is also parallel to DE , it follows that the plane through H, Q, K is parallel to the base of the cone and therefore produces a circular section whose diameter is HK . Also QV is at right angles to HK.
Now, by similar triangles and parallels,
HV : PV = BC : AC and VK : PA = BC : BA.
Hence, QV² : PV.PA = PL : PA = PL.PV : PV.PA
It follows that the square on ay ordinate to the fixed diameter PM is equal to a rectangle applied to the fixed straight line PL drawn at right angles to PM with altitude equal to the corresponding abscissa PV. Hence the section is called a Parabola .
The fixed straight line PL is called the latus rectum , or the parameter of the ordinates.
This parameter, corresponding to the diameter PM , will be denoted by the symbol p below. Thus,
This proof differs from that given above, for the earlier exercise assumed the focus to be known. Apollonius chooses PL in such a way that it represents the latus rectum, or focal width of the curve. Due to the earlier development, that any plane parallel to the base and cutting the cone completely is a circle. Through the use of the sets of parallel lines QV and DE, HK and BC, and through the similar triangles HKA and BCA, it follows rather directly as Apollonius states. Just like in the previous demonstration (Eves), the square of the ordinate (QV²) is equal to the length of the latus rectum (PL) times the abscissa of QV (PV).
Apollonius' definitions of the hyperbola and ellipse follow along a similar line. For the hyperbola, the area of the rectangle (set equal to the square of the ordinate) overlaps the fixed latus rectum. For the ellipse, the area of the rectangle falls short of the fixed latus rectum. Reiterating from before, Heath suggests that these definitions indicate that the names come from the Pythagorean terms relating to the application of areas to segments.
The final topic of Apollonius' Conic Sections to be considered is his treatment of tangents. He develops this topic in both Book I and Book V. Book V introduces the idea of "maximum" and "minimum" lines to refer to tangents and normals, respectively. This book, considered by Eves to be "the most remarkable and original" of the seven we have today, quickly becomes very difficult to read and follow. The propositions and relationships it proves, which today are more easily shown using differential calculus, are rigorously explored in the classic Greek geometric fashion (Heath, 1961, pp. lxxv-lxxvi). Preliminary theorems, however, are not terribly difficult to follow. First we will look at two propositions from the first book concerning tangents (one will be stated and discussed, the other formally proved), and then we will look at one Book V theorem.
Proposition 11 states, If a straight line be drawn through the extremity of the diameter of any conic parallel to the ordinates to that diameter, the straight line will touch the conic, and no other straight line can fall between it and the conic (Heath, 1961, p. 22). That is, no straight line can fit between a tangent line and the curve to which it is tangent. This seems like a reasonable statement, related to the definition of tangent line used later in the development of the calculus (though, among other things, too "global" in scope).
Apollonius proves this in two cases, one for a parabola, and one for the ellipse, hyperbola, and circle [interesting that he would include the circle].
Proposition 12: If a point T be taken on the diameter of a parabola outside the curve and suh that TP = PV, where V is the foot of the ordinate from Q to the diameter PV, he line TQ will touch the parabola.
We have to prove that the straight line TQ or TQ produced does not fall within the curve on either side of Q.
For, if possible, let K, a point on TQ or TQ produced, fall within the curve, and through K draw Q'KV' parallel to an ordinate and meeting the diameter in V' and the curve in Q'.
Then Q'V'² : QV² > KV'² : QV², by hypothesis, > TV'² : TV²
Hence, 4TP.PV' : 4TP.PV > TV'² : TV²
But, since by hypothesis TV' is not bisected in P,
which is absurd. Therefore, TQ does not at any point fall within the curve, and is therefore a tangent.
The figure for this proof by contradiction can be redrawn to show what is being assumed, that there exists a point K on TQ such that K lies inside the parabola. We then construct KQ'V' parallel to the ordinate QV.
Then, using our assumption that Q'V' > KV', the given TP = PV, and the similar triangles TVP and TV'Q', we arrive at the contradiction.
We now move ahead to Book V to get a feel for Apollonius' idea of minimum with a simple case of the concept:
Proposition 82 In a parabola, if E be a point on the axis such that AE is equal to half the latus rectum, then the minimum straight line from E to the curve is AE and, if P be any other point on the curve, PE increases as P moves further from A on either side. Also, for any point:
Let AL be the parameter or latus rectum. Then, PN² = AL.AN = 2AE.AN
Adding EN², we have, EN² = 2AE.AN + EN² = 2AE.AN + (AE - AN)² = AE² + AN
Thus, PE² > AE² and increase with AN, i.e. as P moves further and further from A. Also the minimum value of PE is AE, or AE is the shortest straight line from E to the curve.
[In this proposition, as well as many others in Book V, Apollonius considers three cases, where N is between A and E, where N coincides with E and PE (perpendicular to axis), and where AN is greater than AE-we will only consider this one case for brevity's sake]
The proof starts by stating the general relationship between the ordinate, abscissa, and latus rectum of a parabola. This is a special case of the parabola in which E is chosen on the diameter such that AE is half the latus rectum, which is reflected in the rewriting of the original relationship. Because PN is perpendicular to PE, EN² is added to both sides of the equation, and due to the Pythagorean Theorem, the left-hand-side of the equation reduces to PE². The rest of the proof follows easily.
This paper has attempted to provide a systematic introduction to the work of the Greek geometers involved in the development of conic section theory. It started with the work of Menaechmus, who first used conics to solve the doubling of the cube. It is unknown how many properties of the conics he knew, though it is generally accepted he did know they came from the cutting of a cone. After Menaechmus, Aristaeus and Euclid formalized and expanded upon the conics (Aristaeus was more original). Then came the great Archimedes, who used the elementary theory of conic sections to develop important concepts about parabolas, and extended that far beyond the scope of this paper. The culmination of the subject came at the hands of Apollonius, who, in eight volumes, rigorously developed all that was known about conic sections before him, and added a multitude of propositions that were original (we believe) to him, so much in fact that Eves notes, "The treatise is considerably more complete than the usual present-day college course in the subject".
After the era of these great mathematicians, there was a lull in the growth of conic sections until Pappus. He expanded upon much of what was known, and also proved to be a valuable source to modern math historians trying to learn about the Greek methods. With the passing of Pappus and perhaps Proclus, conics disappeared for over 1000 years until being re-born in the 15th and 16th centuries. Though the work of scientists and mathematicians, like Kepler who was both, conics evolved from a novel intellectual exercise in Ancient Greece, to a powerful modeling tool for explaining the physical laws of the universe.
Selected Prefaces to Conic Sectons (Translated by Halley, Printed in Heath)
Apollonius to Eudemus, greeting.
If you are in good health and circumstances are in other respects as you wish, it is well I too am tolerably well. When I was with you in Pergamum, I observed that you were eager to become acquainted with my work in conics therefore I send you the first book which I have corrected, and the remaining books I will forward when I have finished them to my satisfaction. I daresay you have not forgotten my telling you that I undertook the investigation of this subject at the request of Naucrates the geometer at the time when he came to Alexandria and stayed with me, and that, after working it out in eight books, I communicated them to him at once, somewhat too huuriedly, without a thorough revision (as he was on the point of sailing), but putting down all that occurred to me, with the intention of returning to them later. Wherefore I now take the opportunity of publishing each portion from time to time, as it is gradually corrected. But, since it has chanced that some other persons also who have been with me have got the first and second books before they were corrected, do not be surprised if you find them in a different shape.
Now of the eight books the first four form an elementary introduction the first contains the modes of producing the three sections and the opposite branches [of the hyperbola-Heath] and their fundamental properties worked out more fully and generally than in the writings of other authors the second treats the properties of the diameters and axes of the sections as well as the asymptotes and other things of general importance and necessary for determining limits of possibility, and what I mean by diameters and axes you will learn from this book. The third book contains many remarkable theorems useful for the synthesis of solid loci and determinations of limits the most and prettiest of these theorems are new, and, when I discovered them, I observed that Euclid had not worked out the synthesis of the locus with respect to three and four lines, but only a chance portion of it and that not successfully: for it was not possible that the synthesis could have been completed without my additional discoveries. The fourth book shows in how many ways the sections of cones meet one another and the circumference of a circle it contains other matters in addition, none of which has been discussed by earlier writers, concerning the number of points in which a section of cone or the circumference of a circle meets [the opposite branches of a hyperbola-Heath].
The rest [of the books-Heath] are more by way of suplusage [`more advanced' but literally implies extensions of the subject beyond the mere essentials-Heath in the form of a footnote]: one of them deals somewhat fully with minima and maxima, one with equal and similar sections of cones, one with theorems involving determination of limits, and the last with determinate conic problems.
When all the books are published it will of course be open to those who read them to judge them as they individually please. Farewell.
Apollonius to Attalus, greeting.
In this fifth book I have laid down propositions relating to maximum and minimum straight lines. You must know that our predecessors and contemporaries have only superficially touched upon the investigation of the shortest lines, and have only proved what straight lines touch the sections and, conversely, what properties they have in virtue of which they are tangents. For my part, I have proved these properties in the first book (without however making any use, in the proofs, of the doctrine of the shortest lines) inasmuch as I wished to place them in close connection with that part of the subject in which I treated of the production of the three conic sections, in order to show at the same time that in each of the three sections numberless properties and necessary results appear, as they do with reference to the original (transverse) diameter. The propositions in which I discuss the shortest lines I have separated into classes, and dealt with each individual case by careful demonstration I have also connected the investigation of them with the investigation of the greatest lines above mentioned, because I considered that those who cultivate this science needed them for obtaining a knowledge of the analyis and determination of problems as well as for their synthesis, irrespective of the fact that the subject of one of those which seem worthy of study for their own sake. Farewell.
How the Greeks Changed the Idea of the Afterlife
Their secret cults help shape the way we think of what happens after death.
The world of ancient Greece was filled with gods, led by the towering Olympians—Zeus, Hera, Apollo, Poseidon, Athena, and other giants of mythology. Alongside worship of these divine inhabitants of Olympus were hundreds of cults focused on local deities and heroes.
People prayed to these gods for the same reasons we pray today: for health and safety, for prosperity, for a good harvest, for safety at sea. Mostly they prayed as communities, and through offerings and sacrifice they sought to please the inscrutable deities who they believed controlled their lives.
But what happens after death? In this, the ancients looked to Hades, god of the underworld, brother of Zeus and Poseidon. But Hades gave no reassurance. Wrapped in misty darkness, cut by the dread River Styx, the realm of Hades (“the unseen”) was, the poet Homer tells us, a place of “moldering horror” where ordinary people—and even heroes—went after they died.
Sympathetic interest in the human condition eventually led the Greeks to adopt new forms of religion and new cults. No longer seen as a joyless fate, the afterlife became more of a personal quest. Mystery cults, shrouded in secrecy, promised guidance for what would come after death. The mystery rites were intensely emotional and staged like elaborate theater. Those of the great gods on the Greek island of Samothrace took place at night, with flickering torch fire pointing the way for initiates. Guarded on pain of death, the rituals remain mysterious to this day.
By the fourth century B.C., cults had emerged that claimed to offer purification by cleansing initiates of the stain of humanity. The foundations for new religions were falling into place. And when Christianity swept the ancient world, it carried with it, along with guidance from a single deity, remnants of the old beliefs: the washing away of human corruption through mystic rites, the different fates awaiting the initiated and uninitiated, and the reverence for sacred texts.