A resemblance of relations; an agreement or likeness between things in some circumstances or effects, when the things are otherwise entirely different. Thus, learning enlightens the mind, because it is to the mind what light is to the eye, enabling it to discover things before hidden.
Karslake
(Biological) A relation or correspondence in function, between organs or parts which are decidedly different.
(Geometrical) Proportion; equality of ratios.
(Grammatical) Conformity of words to the genius, structure, or general rules of a language; similarity of origin, inflection, or principle of pronunciation, and the like, as opposed to anomaly. Johnson
Homology:
The quality of being homologous; correspondence; relation; as, the homology of similar polygons.
(Biological) Correspondence or relation in type of structure in contradistinction to similarity of function; as, the relation in structure between the leg and arm of a man; or that between the arm of a man, the fore leg of a horse, the wing of a bird, and the fin of a fish, all these organs being modifications of one type of structure.
(Chemical) The correspondence or resemblance of substances belonging to the same type or series; a similarity of composition verying by a small, regular difference, and usually attended by a regular variation in pysical properties; as, there is an homology between methane, CH4, ethane, C2H6, propane, C3H8, etc., all members of the paraffin series. In an extended sense, the term is applied to the relation between chamical elements of the same group; as, chlorine, bromind, and iodine are said to be in homology with each other. Cf. Heterology.
General homology(Biological), the higher relation which a series of parts, or a single part, bears to the fundamental or general type on which the group is constituted. Owen
Serial homology(Biological), representative or repetitive relation in the segments of the same organism, - as in the loster, where the parts follow each other in a straight line or series. Owen Cf. Homotypy.
Special homology(Biological), the correspondence of a part or organ with those of a different animal, as determined by relative postition and connection. Owen
Metaphor:
(Rhetorical) The transference of the relation between one set of objects to another set for the purpose of brief eplanation; a compressed simile; e.g., the ship ploughs the sea. Abbott and Seeley. 'All the world's a stage' Shakespeare
Analysis:
A resolution of anything, whether an object of the senses or of the intellect, into its constituent or original elements; an examination of the component parts of a subject, each separately, as the words which compose a sentence, the tones of a tune, or the simple propositions which enter into an argument. it is opposed to synthesis.
(Chemical) The separation of a compound sustance by chemical processes, into its constituents, with a view to ascertain either (a) what elements it contains, or (b) how much of each element is present. The former is called qualitative, and the latter quantitative analysis.
(Logic) The tracing of things to their source, and the resolving of knowledge into its original principles.
(Mathematics) The resolving of problems by reducing the conditions that are in them to equations.
"This confusion between qualities of animals and qualities of human society is an example of the problem of homology and analogy. By homologous traits, biologists mean those properties of organisms that are shared by different species because they have a common biological origin and some common biological genetic ancestry, and they derive from common fetures of anatomy and development. Even though they look very different and are used for very differnet purposes, the bones of a human arm and of a bat's wing are homologous because they are anatomically derived from the same structures and influenced by the same genes. On the other hand, a bat's wing and an insect's wing are only anagous. That is, they look superficially alike and they seem to serve the same function, but they have no origin in common at the genetic or morphological level. But analogy is in the eye of the observer. How do we decide that slavery in ants and ant queens are like human slavery and like human royal families? How do we decide that the coyness we see in people is the same as the behaviour in animals called coyness? What happens is that human categories are laid on animals by analogy, partly as a matter of convenience of language, and then these traits are 'discovered' in animals and laid back on humans as if they had a common origin."
Lewontin, R.C., The Doctrine of DNA: Biology as Ideology, 1991
Analogy is one of a human being's most powerful tools. In one sense, any description of anything else is an analogy, and by extension, any thought (being a 'transcription' of the 'real' thing is also an analogy. Alternatively, these could be described as homologous (that is, 'the same as' rather than 'like'). There's potentially an element of tautology here (we can't perceive in any other way), but it is also clear that the perception of a thing or an event is not that thing or event, and that different people have different perceptions of the same event - sometimes quite radically different.
A fascinating part of this is the different ways in which different ideas can be more or less analogous to each other, often very much in accordance with an individual's perception.