kʷetwr̥-plo-
“fourfold, quadrupled”quadruple, quarter, squad
Compound yielding Latin quadrus, English quadruple, quarter, squad, squadron.
Discussion
The PIE form *kʷetwr̥-plo- (fourfold, quadrupled) is a multiplicative compound of *kʷetwóres (four) with the fold-suffix *-plo-. The literal meaning is "four-layered" — the fourth member of the multiplicative series that includes *dwi-plo- (twofold/double) and *tri-plo- (threefold/triple).
Latin quadruplus (fourfold) continues the form with the expected Italic treatment of the labiovelar: PIE *kʷ > Latin qu. The derivatives are institutional and mathematical: quadruple (four times as much), quarter (one fourth), quartet (a group of four), quadrant (a fourth part of a circle), quadrilateral (four-sided), and squad (from Italian squadra, originally a square formation of soldiers — four-sided military geometry). The word square itself may be distantly related through the same "four" base.
The numeral *kʷetwóres (four) is independently one of PIE's most stable reconstructions: English four, Latin quattuor, Greek téttares, Sanskrit catvā́ras, Lithuanian keturì, Old Irish cethir, and Gothic fidwor all continue the same form. The labiovelar *kʷ surfaces differently in each branch — Latin qu-, Greek t-, Sanskrit c-, Germanic f- (by Grimm's Law) — making the cognate set a textbook demonstration of regular sound correspondences.
The English word four descends through Old English fēower from PGmc *fedwōr, with Grimm's Law converting PIE *kʷ to Germanic *f. The ordinal fourth (OE fēorþa) and the fraction quarter (via French from Latin quartus) coexist in English as Germanic native and Latin borrowing respectively — the same numeral inherited twice through different channels.
The multiplicative suffix *-plo- itself derives from PIE *pel- (to fold), making all the -ple/-fold words literally "foldings": double is two-fold, triple is three-fold, quadruple is four-fold. The suffix survived into Latin as -plus/-plex (simplex, duplex, complex, multiplex) and into Germanic as -fold (twofold, manifold). The physical image underlying all mathematical multiplication is the folding of cloth or paper — to multiply by four is to fold four times.
The cultural significance of four in PIE thought is reflected in the word's productivity. Four-part divisions appear throughout IE cultural vocabulary: the four seasons, the four cardinal directions, the four elements (a Greek elaboration of an older pattern), and the quadripartite social structures reconstructed for some PIE-descendant societies. Whether PIE itself had a fourfold cosmological schema remains debated, but the linguistic productivity of *kʷetwóres suggests the number held structural importance beyond mere counting.