Parmenides of Elea 504/500 B.C.
Zeno of Elea 464/460 B.C.
Melissus of Samos 441 B.C. (Samos in Asia Minor, the same as Pythagoras)

Those philosophers were active in the City State of Elea in Southern Italy. Both Parmenides and Zeno were supposed to be born in Elea, too.
Elea was established in 540 B.C. for whose founding, Xenophanes wrote a poem of two thousand words to celebrate it.
It is said that this so-called Eleatic philosophy was founded by Xenophanes, who was associated with the founding of Elea itself (See the above). Copleston takes the same view about the Eleatic school.
Aristotle also said that Parmenides was Xenophanes' disciple. These dates, except Melissus whose sea battle with the Athenian navy was recorded independently, are based on Xenophanes' date of his poem to celebrate the founding of Elea in 540 B.C. We subtracted 40 years from it for Parmenides and further 20 (altogether 60) years for Zeno, for we have no clue for the dating of the Eleatic philosophers. However, in conjunction with Melissus's acmé, they are not that far way from the truth. Nevertheless, many recently argue against these dates regarding their exactitude.
For example, Plato's later dialogues, Parmenides, is often quoted as being not consistent with the above datings. In Parmenides, Plato portrays that Socrates who was at about 20 years of age met Parmenides, when Parmenides and Zeno visited Athens for the first time. Plato describes that Parmenides who looked like God with white beard was 65 years of age, while Zeno was 40 years old. Although Socrates' first encounter with Parmenides and Zeno in this dialogue particularly regarding its exact content may very well be a product of Plato's creative imagination, the relative ages of these three philosophers could very well be correct. When Socrates was 20 years old, that is 450 B.C., thus Zeno's acme is then 450 B.C. and in consequence Parmenides' is 475 B.C.
Further, from these datings if they were correct, we may bring down the date of Heracleitus' acme to 475 B.C., too.(See the section on Heracleitus in the Ionian-Italian Migrant Philosophers.)
Melissus was born in Samos, the same birth place as Pythagoras. No doubt, therefore, Melissus grew up with the Pythagorean philosophy, together with the tradition of the Ionian natural philosophies, and yet it is interesting that he came all the way from the Asia Minor to Elea in Southern Italy to study under Parmenides. This may indicate that the Eleatic school of philosophy was quite widely known in the Magna Graeca because of its originality and its revolutionary content.
At that time communication networks must work quite well, since it is said that Parmenides had been influenced by the philosophy of Samos through Xenophanes and Ameinias and yet Parmenides' philosophy was reexported back to Samos by Melissus. It also indicates that around the time when Melissus wanted to further his study in philosophy, Parmenides and the Eleatic School was so widely known that Melissus was able to choose Parmenides as his teacher by going all the way from Samos, Asia Minor, to the west side of Southern Italy.
Melissus was the Admiral of Samos' navy and revolted against Athens. Pericles lead the unbeatable Athenian Navy and tried to attack the Samosian Navy. Nevertheless, because of a still unknown reason, Pericles withdrew a portion of his ships, the circumstance of which favored Melissus and he won the ship battle with the Athenians. This was so unusual that it was recorded in history.
On the one hand, the Eleatic philosophy seems so abstract that it may appear obsolete in the practical matter, Melissus mastered it well. On the other, Melissus showed himself as a man of practical affairs in that he was politically active, was further a Navy Admiral and won the battle against the world-well known strong Athenian navy. This is an impressive event. Therefore, philosophy was often considered as not being practical, and yet it is, once applied to a particular situation by a given philosopher, indeed very useful and effective. Knowledge is Power!

Parmenides of Elea (500 B.C.)

The Miletian (Ionian) philosophers sought the principle of the universe in the concrete material substance which is perceivable by the senses (e.g., Water or Air). It is a material substance ("hé ousia" [substance] is "that which becomes the subject of a proposition and does not become the predicate"(Aristotle's definition of "substance"based on the structure of language = "the being which exists by itself and does not need anything else for its existence"‹the ontological, Cartesian definition.)
First of all, such a principle as water or air is a material substance(=substratum) and is primarily referred by the subject term of the judgement (i.e., something definite and particular) with its unique properties (particular qualities).
Secondly, it is by our sense experience that such a material substance is given, i.e., is known. Experience as the means and source of knowing (such a material principle) is by nature not infallible, therefore is finite, and uncertain. Thus experience is not certain as a way of knowing. In other words, experience can not validate the reality of its object at all.
The philosophers of the Eleatic school sought the knowledge which is infallible, necessary, indubitable and universally valid. If they were alive in the post Renaissance like Descartes, they would have searched this apodeicticity in the way or kind of knowing itself. Parmenides however sought in the special kind or the unique way of Being which should provide such apodeictic knowledge. If we may say in the Aristotelian fashion, Parmenides and his school looked for the immutable, self-identical principle not in a material substance as a substratum(the subject of the proposition), but in the predicate of the judgement. It is not just a predicate, but the widest, the most encompassing, the most fundamental predicate of all, namely "Being"(to eon).
Thus when we make a subject out of this Being, its predicate of the proposition is none but Being itself: Being is Being. This is tautologous and is apodeictically true, i.e., absolutely necessarily and universally true. On the other hand, nothing (=Non-being, to mé on) does not exist. This, too, is apodeictically true. For anything exists (i.e., is) at all, it must not come into Being, nor perish into Non-being. It must have existed and exist and will exist, is absolutely immutable and no change whether this is locomotion or alteration. It is also true that ex nihilo nihil fit (out of nothing nothing comes into Being).

Being exists;

Non-being does not (exists).

These two are the self evident, thus absolutely indubitably true axioms.
From these two axioms, several propositions(theorems) derive immediately:

1) Being is one and self identical with itself (in unity, therefore, not many).

For if many Beings should exist, then Non-being must also exist between these many Beings. This is contradictory to the basic axioms.

2) Being is qualitatively one and the same everywhere(neither dense nor thin, too).
For if it were, there must be Non-being mixed with Being, which is contradictory to the
basic axioms.

3) No alternation, nor locomotion, i.e., no change, does not exist.
For if change should exist, it presupposes that Non-being exists, which is contradictory to the axioms.

Therefore, our universe as reality is one, self same, thus does not change.

Furthermore, Being as being perfect which is spherical, Being is a sphere in shape.

This Being cannot be known by our experience, but only by reason (to noein or hé noesis) alone. Thus what really Is is authentically known (by reason), and what is authentically known (by Reason or Thought) is Being.

Therefore, according to Parmenides:

Being (To Eon) is Rational (=known by Reason‹Noésis‹) and what is Rational (=known by Reason‹Noésis‹) is Being (To Eon).

This thesis of identity of Being and knowledge became one of the most important principles that the philosopher must to attack to solve. Probably the most successful attempt to solve this question to our satisfaction was later made by Hegel in his dialectic thinking at the culmination of the Western Philosophy. In case of Hegel, the formulation was:
Was vernunftig ist wirklich, und was wirklich ist vernunftig.
(What is rational is real, and what is real is rational.)

What we know through our sense experience is thus to be viewed by Parmenides as a Mere Illusion.
Our common sense purports what is known through the senses is real. Against this conception of Reality known by sense experience, for the first time in the history of the Western philosophy, Parmenides was supposed (We pointed out the possibility of Heracleitus to have done this earlier) to make an attempt to transcend beyond the empirical phenomena to the ultimate Reality. This ultimate Reality can be known only through reason, unmixed with the senses or experience, i.e., by means of pure Thought alone.
To briefly elaborate this identity of Being and Thought:

Only that which is known by Pure Thought, or is grasped by Reason, does genuinely and immutably exists. In other words, it is impossible for us to think of that which is not (=Non-being), but we can only think and comprehend by Reason or Thought that which is (=Being) alone.

Thus Being, according to Parmenides, is the sole Reality which is one, continuous, indivisible, qualitatively equal, of no generation nor corruption, no locomotion, no condensation, no rarefication. This Being is logical and solely comprehensible by Reason. While our senses are fallible and correspond to the opinion(he doxa), our Reason being infallible correlates itself to genuine Knowledge(epistémé).
In consequence, to Parmenides, who might be concerned about the radical nature of his own thought, Being (to eon) is not an object of our mundane experience, but is transcendent from it. Therefore, Parmenides must have only Goddess reveal and prophesize this Reality as Truth in the form of poetry (although it is not uncommon to express philosophical thought in verse rather than in prose at that time).

Zeno of Elea (464/460 B.C.)

Zeno are well known for his invention of dialectic (hé dialektiké techné) and many paradoxes which have his name on. He was remembered as the master of dispute to demonstrate that, once we accepted the assumptions of our common sense, we inevitably had to accept the logical consequences of the absurdity and contradiction in common sense in order to show that Parmenides' Being (to eon) is the genuine reality. We know very little of his life. As the most faithful student of Parmenides, Zeno accompanied Parmenides everywhere and travelled many city states. However, it must be more likely that through his effective argumentation, Zeno contributed to make clear the unfeasibility of Ionian natural philosophy, which presupposed the motion of generation and corruption (condensation and rarefication) and "one and many".
Zeno attempted to show the contradictions and inconsistencies in our common sense knowledge.According to our common sense, the world of experience be the sole reality. By demonstrating our common sense knowledge as paradoxical, Zeno exerted himself in proving that Parmenides' Being is the only genuine Reality. Zeno not only was responsible for devising the socalled "indirect proof", but also he has been universally considered as the founder of the "dialectic".
For example, in stead of arguing that Being is one, Zeno argued that, if Being were many, as our common sense dictates, many contradictions would logically follow, e.g. the existence of Non-being, or the non-existence of motion from that premiss, for example. The latter cannot be, therefore, Being (=Reality) must be one. Or e.g., if we agree that Being (=Reality) is many, then we must conclude from this that Being is both finite and infinite at the same time. For in order to distinguish one Being from the other Being, there must be another Being inbetween and ad infinitum.
Aristotle held that Zeno was the inventor of dialectics. What does this mean? Zeno formulates the proposition (B) which is the negation of (A) to be demonstrated to be true. Then he tries to draw a consequence(C) from (B) which turns out to be self contradictory. Thus since (C) as self contradictory resulted from the premiss (B), in order for (C) to be true, (A) in stead of (B),i.e., the negation of (A), must be postulated as its premiss.
Zeno consciously employed the logical method for demonstration of the point at stake, which Parmenides implicitly conceived of. By means of Zeno's endeavor, logic became a conscious, deliberate activity of perusing truth and knowledge.
For example, Aristotle reproduced Zeno's argument for the impossibility of motion in his Physics 6-9:

In order to go from A to B, one has to go through infinite numbers of the intermediaries, i.e., within a ceratin finite time, we have to go through a infinite number of points. This is impossible.

An argument that Achilles with twice the speed can never pass the tortoise. The argument of the standstill of a flying arrow. These are only two of the paradoxes resulted from the assumption of the infinite divisibility of a finite line and the specialization of time.

In the history of philosophy, there is always a leap in the development of thinking and this is considered as one of the prime examples for this. Although the Eleatic thinking was not perfect, it was a courageous leap to go and follow from the apodeictic true axioms wherever the arguments might lead, even far beyond the limit of human experience and how contrary the conclusion of the arguments may be to experience.
However, this was at the same time the beginning of the human overconfidence in the European Reason. In the History of the Western Philosophy, we must wait for Nietzsche until this dominance of and blind trust in Reason was critically confronted.
While Zeno negatively denies the mundane doxa (opinion) of the mortal beings that is solely based on our experience, Melissus positively affirms Being itself.

Melissus of Samos (441 B.C.)

Melissus developed Parmenides' thought a step further from the proposition, "Being is, while Non-being is not." In stead of following his detailed arguments, here we will attempt to comprehend how Parmenides' and Melissus' thoughts differ.
One of Parmenides' descriptions of Being states that Being is finite (i.e., not indeterminate). According to Parmenides, that Being is atelés (infinite, limitless) means that Being is incomplete (atelés means purposeless or endless) and something lacking. Thus Being is shaped as spherical, thus limited, and qualitatively equal everywhere, and is finite.
Against Parmenides' thesis that Being (to eon) is finite, Melissus held that Being is infinite. For since there is no generation, nor corruption possible for Being, Being must have neither the For if Being is finite at least in the temporal sense, Being must have the beginning and the end in time. This, according to Melissus, does not make sense. Therefore,Being must be infinite at least in the temporal sense, i.e., at least temporally infinite, thus timeless.
The second point Melissus added to Parmenides' thought was that, in addition to Nonbeing, there exists no "void" (to kenon), while Parmenides had implicitly considered Non-being be void. From this, it must be more rigorously concluded that there is no motion (whether it is locomotion, or condensation, or rarefication).
Thirdly, Melissus was also to have said that Being does not suffer pain, or enjoys pleasure, for, according to Melissus, nothing can be added or subtracted from Being.

The most important contribution by the Eleatic School to the history of the Western philosophy is the clear establishment of the superiority of Reason (ho nous) both as the principle of Being and of cognition at the same time, while Parmenides, Zeno and Melissus made knowledge of sense experience not only inferior (to that of Reason) but not even worth the name of knowledge at all, because sense experience provides us mere illusion, not truth. This was the explicit beginning of the long tradition of Rationalism in Western philosophy.
It is also worth noting that, according to the Eleatic philosophy, truth is for most parts covered up by illusion, prejudices and pre-conceived ideas. In order to "uncover" truth, we must free ourselves from our common sense and its conviction. Thus it was established that common sense is not conducive to philosophical inquires, but rather something which is to be critically questioned and from which we must liberate ourselves in order to pursue philosophy and philosophical knowledge.
The Eleatic philosophy established the two value logic which even absolved dialectic until the end of German Idealism.
The Eleatic philosophy further demonstrated that we as a philosopher must have the courage and trust in logical inference whatever its consequence may be, as long as we start with truth as premisses.
The Eleatic philosophers had a very clear conception of Being, which is not material, but an abstract concept derived from the "predicate term concepts."
Many of the prominent historians of the Western philosophy in the past contend that Parmenides' Being is material and consider him a materialist, who holds that the ultimate Being is matter or material substance. This contention is not only wrong, but also is utterly misleading and misses the most important contribution of Parmenides' philosophical endeavor to the history of the Western philosophy. Parmenides was well aware of how unconventional and anti-common sense belief his philosophical wisdom was. As mentioned previously, that was why Parmenides had to use epic, not prose, to express his philosophical ideas and felt even the necessity to have the Goddess speak about his philosophical ideas as her prophecies.
It is as erroneous to call Parmenides a materialist as Anaximander. It is wrong to underestimate such great geniuses' contributions which were way ahead of their times.
The Being as an entity, that which is, and the Being which as a principle makes an entity exist are not yet articulated at the time of the Eleatic School.