But Eudoxus the Cnidian, who was somewhat junior to Leon, and the companion of Plato, first of all rendered the multitude of those theorems which are called universals more abundant; and to three proportions added three others; and things relative to a section, which received their commencement from Plato, he diffused into a richer multitude, employing also resolutions in the prosecution of these.

It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantaneously destroyed and shall remain forever exposed to the play of the eternal waves.

Again, Amyclas the Heracleotean, one of Plato's familiars, and Menæchmus, the disciple, indeed, of Eudoxus, but conversant with Plato, and his brother Dinostratus, rendered the whole of geometry as yet more perfect. But Theudius, the Magnian, appears to have excelled, as well in mathematical disciplines, as in the rest of philosophy. For he constructed elements egregiously, and rendered many particulars more universal. Besides, Cyzicinus the Athenian, flourished at the same period, and became illustrious in other mathematical disciplines, but especially in geometry. These, therefore, resorted by turns to the Academy, and employed themselves in proposing common questions.

Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.

But Hermotimus, the Colophonian, rendered more abundant what was formerly published by Eudoxus and Theætetus, and invented a multitude of elements, and wrote concerning some geometrical places. But Philippus the Mendæan, a disciple of Plato, and by him inflamed in the mathematical disciplines, both composed questions, according to the institutions of Plato, and proposed as the object of his enquiry whatever he thought conduced to the Platonic philosophy.

This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth.

For this, to draw a right line from every point, to every point, follows the definition, which says, that a line is the flux of a point, and a right line an indeclinable and inflexible flow.

The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular.

If two right lines cut one another, they will form the angles at the vertex equal. ...This... is what the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales...

A transition, therefore, is not undeservedly made from sense to consideration, and from this to the nobler energies of intellect. Hence, as the certain knowledge of numbers received its origin among the Phœnicians, on account of merchandise and commerce, so geometry was found out among the Egyptians from the distribution of land. When Thales, therefore, first went into Egypt, he transferred this knowledge from thence into Greece: and he invented many things himself, and communicated to his successors the principles of many. Some of which were, indeed, more universal, but others extended to sensibles.

But after these, Pythagoras changed that philosophy, which is conversant about geometry itself, into the form of a liberal doctrine, considering its principles in a more exalted manner; and investigating its theorems immaterially and intellectually; who likewise invented a treatise of such things as cannot be explained in geometry, and discovered the constitution of the mundane figures.

Animals are rational; in most of them logos is imperfect, but it is certainly not wholly lacking. So if, as our opponents say, justice applies to rational beings, why should not justice, for us, also apply to animals?

Things essentially incorporeal, because they are more excellent than all body and place, are every where, not with interval, but impartibly. Things essentially incorporeal are not locally present with bodies but are present with them when they please; by verging towards them so far as they are naturally adapted so to verge. They are not, however, present with them locally, but through habitude, proximity, and alliance. Things essentially incorporeal, are not present with bodies, by hypostasis and essence; for they are not mingled with bodies. But they impart a certain power which is proximate to bodies, through verging towards them. For tendency constitutes a certain secondary power proximate to bodies.

The Pythagoreans made kindness to beasts a training in humanity and pity.

Soul, indeed, is a certain medium between an impartible essence, and an essence which is divisible about bodies. But intellect is an impartible essence alone. And qualities and material forms are divisible about bodies. Not everything which acts on another, effects that which it does effect by approximation and contact; but those natures which effect any thing by approximation and contact, use approximation accidentally.

So people should abstain from other animals just as they should from the human.

Incorporeal hypostases, in descending, are distributed into parts, and multiplied about individuals with a diminution of power; but when they ascend by their energies beyond bodies, they become united, and proceed into a simultaneous subsistence, through exuberance of power.

The fleshless diet contributes to health and to a suitable endurance of hard work in philosophy.

I think that when friendship and perception of kinship ruled everything, no one killed any creature, because people thought the other animals were related to them.

The utility of a science which enables men to take cognizance of the travellers on the mind's highway, and excludes those disorderly interlopers, verbal fallacies, needs but small attestation. Its searching penetration by definition alone, before which even mathematical precision fails, would especially commend it to those whom the abstruseness of the study does not terrify, and who recognise the valuable results which must attend discipline of mind. Like a medicine, though not a panacea for every ill, it has the health of the mind for its aim, but requires the determination of a powerful will to imbibe its nauseating yet wholesome influence: it is no wonder therefore that puny intellects, like weak stomachs, abhor and reject it.

Every good thing is gentle and consistent, progressing in good order and not going beyond what is right.

Every body is in place; but nothing essentially incorporeal, or any thing of this kind, has any locality.

Cicero said loud-bawling orators were driven by their weakness to noise, as lame men to take horse.

When one asked him what boys should learn, "That," said he, "which they shall use when men."

When Eudæmonidas heard a philosopher arguing that only a wise man can be a good general, "This is a wonderful speech," said he; "but he that saith it never heard the sound of trumpets."

Diogenes the Cynic, when a little before his death he fell into a slumber, and his physician rousing him out of it asked him whether anything ailed him, wisely answered, "Nothing, sir; only one brother anticipates another,-Sleep before Death."

After the battle in Pharsalia, when Pompey was fled, one Nonius said they had seven eagles left still, and advised to try what they would do. "Your advice," said Cicero, "were good if we were to fight jackdaws."

Agesilaus was very fond of his children; and it is reported that once toying with them he got astride upon a reed as upon a horse, and rode about the room; and being seen by one of his friends, he desired him not to speak of it till he had children of his own.

A soldier told Pelopidas, "We are fallen among the enemies." Said he, "How are we fallen among them more than they among us?"

About Pontus there are some creatures of such an extempore being that the whole term of their life is confined within the space of a day; for they are brought forth in the morning, are in the prime of their existence at noon, grow old at night, and then die.

After he routed Pharnaces Ponticus at the first assault, he wrote thus to his friends: "I came, I saw, I conquered."

When Demaratus was asked whether he held his tongue because he was a fool or for want of words, he replied, "A fool cannot hold his tongue."

Cato the elder wondered how that city was preserved wherein a fish was sold for more than an ox.

The measure of a man's life is the well spending of it, and not the length.

As Cæsar was at supper the discourse was of death,-which sort was the best. "That," said he, "which is unexpected."

Lysander, when Dionysius sent him two gowns, and bade him choose which he would carry to his daughter, said, "She can choose best," and so took both away with him.

Cato instigated the magistrates to punish all offenders, saying that they that did not prevent crimes when they might, encouraged them. Of young men, he liked them that blushed better than those who looked pale.

For many, as Cranton tells us, and those very wise men, not now but long ago, have deplored the condition of human nature, esteeming life a punishment, and to be born a man the highest pitch of calamity; this, Aristotle tells us, Silenus declared when he was brought captive to Midas.

As Athenodorus was taking his leave of Cæsar, "Remember," said he, "Cæsar, whenever you are angry, to say or do nothing before you have repeated the four-and-twenty letters to yourself."

A physician, after he had felt the pulse of Pausanias, and considered his constitution, saying, "He ails nothing," "It is because, sir," he replied, "I use none of your physic."

Cato requested old men not to add the disgrace of wickedness to old age, which was accompanied with many other evils.

There are two sentences inscribed upon the Delphic oracle, hugely accommodated to the usages of man's life: "Know thyself," and "Nothing too much;" and upon these all other precepts depend.

"Young men," said Cæsar, "hear an old man to whom old men hearkened when he was young."

And when the physician said, "Sir, you are an old man," "That happens," replied Pausanias, "because you never were my doctor."

He said they that were serious in ridiculous matters would be ridiculous in serious affairs.

To one commending an orator for his skill in amplifying petty matters, Agesilaus said, "I do not think that shoemaker a good workman that makes a great shoe for a little foot."

Socrates thought that if all our misfortunes were laid in one common heap, whence every one must take an equal portion, most persons would be contented to take their own and depart.

When one told Plistarchus that a notorious railer spoke well of him, "I 'll lay my life," said he, "somebody hath told him I am dead, for he can speak well of no man living."

Dionysius the Elder, being asked whether he was at leisure, he replied, "God forbid that it should ever befall me!"

Antagoras the poet was boiling a conger, and Antigonus, coming behind him as he was stirring his skillet, said, "Do you think, Antagoras, that Homer boiled congers when he wrote the deeds of Agamemnon?" Antagoras replied, "Do you think, O king, that Agamemnon, when he did such exploits, was a peeping in his army to see who boiled congers?"