These private notes belong to Lenin’s “wartime notebooks” from 1915, and were first published in 1925 in the magazine Bolshevik, No. 5-6.
As the First World War raged on, and in the lead-up to the October Revolution of 1917, Lenin spent some time reviewing and summarizing the work of various philosophers. Amid producing a “conspectus” on Heraclitus and Aristotle, he jotted down what is known to be his tersest and most definitive exposition on the question of dialectics.
The Marxists Internet Archive provides an edition of this text that is very focused on remaining faithful to Lenin’s original script, attempting to preserve stenciled letters and underlines and parentheses and side-notes. The editorial choices made here are aimed at improving readability. Please refer to the original editions if citing this document.
The splitting of a single whole and the cognition of its contradictory parts is the essence — one of the “essentials,” one of the principal (if not the principal) characteristics or features — of dialectics. 
The correctness of this aspect of the content of dialectics must be tested by the history of science. This aspect of dialectics usually receives inadequate attention from the likes of Plekhanov; the identity of opposites is taken as the sum-total of examples (“for example, a seed,” “for example, primitive communism”),  and not as a law of cognition (and as a law of the objective world):
- In mathematics: addition and subtraction, differential and integral.
- In mechanics: action and reaction.
- In physics: positive and negative charge.
- In chemistry: the combination and dissociation of atoms.
- In social science: the class struggle.
The identity of opposites is the recognition or discovery of the contradictory, mutually exclusive, opposite tendencies in all phenomena and processes of nature, including mind and society.  In order to understand any process of the world in its spontaneous development, in its real existence, in its “self-movement,” we must understand it as a unity of opposites. Development is the “struggle” of opposites. The two basic — possible, historically observable — conceptions of development or evolution are: development as a cycle of increase and decrease, and development as a unity of opposites.
In the first conception of motion the driving force or source of self-movement remains uninterrogated (or it is made external — God, etc.). In the second conception the chief attention is directed precisely to seeking the source of this self-movement.
The first conception is lifeless, pale and dry. The second is living. The second alone furnishes the key to the “self-movement” of everything existing; it alone furnishes the key to “leaps” and “breaks in continuity,” to the “transformation into the opposite,” to the destruction of the old and the emergence of the new.
The unity (coincidence, identity, equilibrium) of opposites is conditional, temporary, transient, relational; whereas the struggle of mutually exclusive opposites is as absolute as development and motion are absolute. 
In his Capital, Marx first analyses the simplest, most common, most ordinary and fundamental everyday relation of bourgeois society, a relation encountered billions of times: the exchange of commodities. In this very simple phenomenon, in this “cell” of bourgeois society, his analysis reveals all the contradictions — or the germs of all the contradictions — of modern society. The subsequent exposition shows us the development (both growth and movement) of these contradictions and of this society in the sum of its individual parts, from its beginning to its end.
This should be the method of study and presentation of dialectics in general, for Marx’s dialectics of bourgeois society is only a particular case of dialectics. Begin with what is the simplest and most ordinary or common, with any proposition: “the leaves of a tree are green,” “John is a man,” “Fido is a dog,” etc. Already here we have dialectics, as Hegel’s genius recognised: the individual is the universal. 
Thus, the opposites — the individual opposed to the universal — form an identity: the individual does not exist outside of the connection that leads to the universal; the universal exists only in the individual and through the individual; every individual is, one way or another, a universal; every universal is a fragment, or an aspect, or the essence of an individual; every universal only approximately embraces all the individual objects; every individual enters incompletely into the universal; every individual is connected by thousands of transitions to other kinds of individuals (things, phenomena, processes); etc.
Here already we have the foundations for asserting that these relations are necessary, that connectedness in nature is objective and independent of observers, etc. Already here we have what is necessary and what is contingent, what is essential and what is phenomenal. For in saying “John is a man,” “Fido is a dog,” “this is a leaf of a tree,” etc., we disregard a number of attributes as contingent; we separate the individual from the universal, and counterpose one against the other.
Thus, in any proposition we can (and must) identify the germs of all the elements of dialectics — as in the nucleus of a cell — and thereby show that dialectics is a property of all human cognition in general. Natural science shows us (though it must be demonstrated in every particular instance) an objective nature with the same qualities: the transformation of the individual into the universal, of the contingent into the necessary, transitions and modulations, and the reciprocal connection of opposites. Dialectics is the theory of knowledge of Hegel, and of Marxism. This is the “aspect” of the matter (not an aspect but the essence of the matter) to which Plekhanov — not to speak of other Marxists — paid no attention.
Knowledge is represented as a series of circles both by Hegel and by the modern “epistemologist” of natural science, the philosophical vulgarizer and foe of Hegelianism (which he did not understand!), Paul Volkmann. 
“Circles” in philosophy: 
- Ancient: from Democritus to Plato and the dialectics of Heraclitus.
- Renaissance: Descartes vs. Gassendi (Spinoza?).
- Modern: Holbach — Hegel (via Berkeley, Hume, Kant). Hegel — Feuerbach — Marx.
Dialectics as living, many-sided knowledge (with the number of sides eternally increasing) — with an infinite number of shades of every approach and approximation to reality, with a philosophical system growing into a whole out of each shade — is immeasurably richer than “metaphysical” materialism, whose main problem is its inability to apply dialectics to the Bildertheorie, to the process and development of knowledge.
Philosophical idealism is only nonsense from the standpoint of crude, simple, metaphysical materialism. From the standpoint of dialectical materialism, on the other hand, philosophical idealism is a one-sided and exaggerated  development, inflating and distending one of the aspects or facets of knowledge into an absolute — divorced from matter and nature, deified. It is true that idealism is clerical obscurantism, but philosophical idealism is also, more correctly, a road to clerical obscurantism through one of the sides of the infinitely complex dialectical knowledge of man. 
Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles or a spiral. Any segment of this curve can be transformed, one-sidedly, into an independent and complete straight line, which, if one does not see the forest for the trees, leads one into the quagmire of clerical obscurantism, where it is anchored by the class interests of the ruling classes. Rectilinearity and one-sidedness, woodenness and petrification, subjectivism and subjective blindness — these are the epistemological roots of idealism. And clerical obscurantism — philosophical idealism — has, of course, epistemological roots. It is not groundless; it is a sterile flower, undoubtedly, but a sterile flower that grows on the living tree of living, fertile, genuine, powerful, omnipotent, objective, absolute human knowledge.
See the quotation from Philo on Heraclitus at the beginning of Section III, “On Cognition,” in Lasalle’s book on Heraclitus. That is precisely how Hegel, too, puts the matter — Aristotle in his Metaphysics continually grapples with it and combats Heraclitus and Heraclitean ideas. ↩
The same is true in Engels, “in the interests of popularisation…” ↩
It would be more correct, perhaps, to say “unity”? Although the difference between the terms identity and unity is not particularly important here. In a certain sense both are correct. “Unity of opposites” is the division of a unity into mutually exclusive opposites together with the reciprocal relation. ↩
NB: The distinction between subjectivism (skepticism, sophistry, etc.) and dialectics, incidentally, is that in objective dialectics the difference between the relative and the absolute is itself relative. For objective dialectics there is an absolute within the relative. For subjectivism and sophistry the relative is only relative, and excludes the absolute. ↩
See Aristoteles, Metaphysics, Book 3, Part 4, Section 8-9: “We could not suppose that there is a house besides the particular houses.” [web] ↩
See Hegel, Logic and P. Volkmann, Epistemological Foundations of the Natural Sciences. ↩
Is a chronology of persons essential? No! ↩
J. Dietzgen, “überschwengliches.” ↩
NB: This is aphorism. ↩