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Almost, pRth/prath means to extend, increase, become famous, etc., and pRthvI/pRthivI, feminine of pRthu, literally spread, does mean earth, the entire world. pRtha is a spread palm (of hand), an unit of measure; and pRthA, the feminine of that, is the name of the daughter of the yAdava shUra/vasu who was adopted daughter of kunti/kuntibhoja, and was therefore also called kuntI, daughter of kunti. To the best of my knowledge pRthA was not used synonymously with pRthvI.

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pArtha does mean son of pRthA. It could also mean son of pRthi, or short of pArthiva, a prince, or short of pArthona, the Sanskritized transliteration of the Greek name for the zodiac sign Virgo; but the famous usage was as son of pRthA.