Etymology
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infrequent (adj.)
1530s, "little used" (now obsolete); 1610s, "not occurring often," from Latin infrequentem (nominative infrequens) "occurring seldom, unusual; not crowded, absent," from in- "not" (see in- (1)) + frequens "repeated, regular, constant, often" (see frequent). Related: Infrequently.
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infrequency (n.)
1670s, fact of being infrequent," from Latin infrequentia "a small number, thinness, scantiness," abstract noun from infrequentem (nominative infrequens) "occurring seldom, unusual; not crowded, absent," from in- "not" (see in- (1)) + frequens (see frequent). Older in this sense is infrequence (1640s). Earlier infrequency was used in a now-obsolete sense of "state of being unfrequented" (c. 1600).
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rare (adj.1)

[thin, few, unusual] late 14c., "thin, airy, porous" (opposed to dense); mid-15c., "few in number and widely separated, sparsely distributed, seldom found, very infrequent;" from Old French rer, rere "sparse" (14c.) and directly from Latin rarus "thinly sown, having a loose texture; not thick; having intervals between, full of empty spaces" (antonym of densus). Sometimes reconstructed to be from a PIE root *ere- "to separate; adjoin."

"Having the particles not close together," hence "few in number," hence, "unusual." Sense of "remarkable from uncommonness," especially "uncommonly good" is from late 15c. (Caxton). Related: Rareness. In chemistry, rare earth is from 1818.

Rare implies that only few of the kind exist : as, perfect diamonds are rare. Scarce properly implies a previous or usual condition of greater abundance. Rare means that there are much fewer of a kind to be found than may be found where scarce would apply. [Century Dictionary, 1895]
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egomania (n.)

"obsessive self-centeredness," 1825 (in a letter of English critic William Sidney Walker, published in 1852), from ego + mania. Not in common use before 1890s, where it translated German Ich-Sucht.

[The egomaniac, as opposed to the megalomaniac,] does not regard it as necessary to dream of himself as occupying some invented social position. He does not require the world or its appreciation to justify in his own eyes himself as the sole object of his own interest. He does not see the world at all. Other people simply do not exist for him. The whole 'non-Ego' appears in his consciousness merely as a vague shadow or a thin cloud. The idea does not even occur to him that he is something out of the common, that he is superior to other people, and for this reason either admired or hated ; he is alone in the world ; more than that, he alone is the world and everything else, men, animals, things are unimportant accessories, not worth thinking about. [Max Nordau, "Degeneration," English translation, 1895]

Nordau's book was much-read, debated, and cited at the time and the word was associated with him (e.g. The Agora, July 1895).

Walker's use aside, its infrequent print appearance before 1895 seems to have been largely in the side of medicine that dealt with psychological matters:

The most frequent, yet the most extraordinary of these perversions of temper, are seen in young females. It is a species of aberration of the intellect, but short of insanity, real enough, but exaggerated, fictitious, factitious, and real at the same time. It frequently has its origin in dyspepsia, hysteria, or other malady, and in emotion of various kinds, such as disappointment, vexation, &c. Its object is frequently to excite and to maintain a state of active sympathy and attention, for which there, is as it were, a perpetual, morbid, and jealous thirst. It was rather aptly designated, by the clever relative of one patient, an ego-mania. [Marshall Hall, M.D., "Practical Observations and Suggestions in Medicine," London, 1845]
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