The word 'rivulet' entered English in the late sixteenth century, most likely through Italian 'rivoletto' (a tiny stream), the diminutive of 'rivolo' (a small stream), itself from Latin 'rīvulus' (a small stream), the diminutive of 'rīvus' (a stream, a brook). The PIE root is *h₁reyH- (to flow, to stream), which also gave, through Latin 'rīvus,' the English words 'river,' 'rival,' 'derive,' and 'arrive,' as well as the place name 'Riviera.'
A rivulet is the smallest unit of flowing water that has a name. It is smaller than a brook, which is smaller than a creek, which is smaller than a stream, which is smaller than a river — though the precise thresholds between these terms are not standardized and vary by region and usage. What distinguishes a rivulet is its tininess: it is a trickle, a thread of water finding its way downhill, too small to carry anything but itself.
The word is notable for its double diminution. Latin 'rīvus' (stream) was already modest — not a great river but an ordinary brook. Latin 'rīvulus' made it smaller. Italian 'rivoletto' made it smaller still. English 'rivulet' preserves this accumulated smallness. The word sounds small: the soft vowels and liquid consonants create an aural impression of delicacy that matches the thing described.
The Latin root 'rīvus' (stream) generated a remarkable family of English words, several of which have traveled so far from their watery origin that the connection is invisible without etymological knowledge.
'River' itself comes from Latin 'rīpārius' (of the riverbank), from 'rīpa' (bank), rather than directly from 'rīvus' — but the association is close, and some scholars have proposed that 'rīpa' and 'rīvus' share the same prehistoric root.
'Rival' comes from Latin 'rīvālis' (one who uses the same stream), from 'rīvus.' In Roman law and custom, people who shared access to an irrigation stream were 'rīvālēs' — co-users of the same water source. Since water rights were among the most contentious issues in agricultural communities, these stream-sharers were frequently in competition and dispute. The semantic shift from 'co-user of a stream' to 'competitor' was natural and apparently irresistible. The English word 'rival' preserves this ancient connection between shared water and conflict.
'Derive' comes from Latin 'dērīvāre' (to draw off water from a stream, to divert a channel), from 'dē-' (from, away) + 'rīvus.' To derive something was originally to channel it away from its source, like diverting water from a stream. The intellectual sense — to derive a conclusion, to derive a word's meaning — preserves the image of drawing something from a source and channeling it in a new direction.
'Arrive' comes from Latin 'ad-' (to, toward) + 'rīpa' (riverbank), via Old French 'ariver' (to come to shore, to reach land). To arrive was to reach the bank — to come to shore after a journey by water. The word's maritime origin is forgotten in modern usage, where arriving by car or plane or foot has replaced arriving by boat, but the riverbank is still embedded in the word.
'Riviera' — used for the coastal regions of southern France and northwestern Italy — comes from Italian 'riviera' (coast, shore), from Latin 'rīpāria' (riverbank region). The French and Italian Rivieras are famous for their beauty, but the word itself simply means 'the shore.'
The rivulet, then, is the humblest member of a distinguished family. It is the diminutive of the diminutive, the smallest expression of the PIE root for flowing water. Yet from this same root came words for competition, derivation, arrival, and some of the most celebrated coastlines in the world. The rivulet trickles, but the root from which it springs has flowed through the entire history of the English language.