Tolkien introduced the root ᴹ√THA in Quendian & Common Eldarin Verbal Structure (EVS1) of the 1930s to serve as the basis for the (Old) Noldorin future suffix -thā. Tolkien said: “This thā is probably in origin a defining adverb = then, next, since with time-reference the pronominal stem √THA in Noldorin referred forward” (PE22/97). Tolkien described a similar system for Sindarin in Common Eldarin: Verb Structure (EVS2) of the early 1950s: “The element thā is adverbial, and meant originally ‘then, next’. The pronominal stem the/tha when used in Sindarin with time-reference pointed forward to the future” (PE22/131).
In notes on Words, Phrases and Passages in the Lord of the Rings from the late 1950s or early 1960s, Tolkien assigned a disinct meaning for √ATH as “ease, comfort, heal”, forming the basis for the name of the healing plant S. athelas or Q. asëa aranion (PE17/49). In Definitive Linguistic Notes (DLN) from 1959 Tolkien gave √ATHA as the basis for a number of words such as verb Q. asya-/S. eitha- “to ease, assist” and prefix Q. as(a)-/S. ath- “easily” (PE17/148).
In Late Notes on Verbs (LVS) from 1969, Tolkien connected √THA and √ATHA as a “mono-consonantal reversible” root ÞĀ̆/AÞA combining these two senses. As Tolkien described it:
> Its basic sense was probably “be helpful”, be willing to assist, in any work etc., agree, consent ... this sense is obscured in the uses of aþa- with other verbs, it may be supposed to be the earliest sense, because of derivatives made from it, such as Q aþea (< apaya) name (as = “beneficial” ?) of a herb (PE22/165).
From this, Tolkien described a variety of derivatives, such as Q. aþumo “good companion”, S. natha- (< an-aþa-) “bring help to, save, rescue” and Q. þá or aþa “I agree”, or in longer form aþanye “I will (do so)”. In Sindarin, this verb developed into a sort of future inflection:
> In S. the verb aþa, atha had become agglutinated to the verb stem, and formed a kind of “future”, expressing the intention of the subject, closely resembling in sense and uses English will (when not mere future): I will (I’ll) go, he will (he’ll) go, espec. in the 1st and 3rd persons. In the second person the implication of “will” of the subject is clearest in questions or negation (PE22/167).
As described in this document, the Sindarin verbal suffix -atha was not a true future, but was rather an expression of intent. Nevertheless, this document connected both prior senses of the roots √THA “future” and √ATHA “ease, assist, comfort” via a more basic sense of “be helpful”, with atha- “agree” > “I will” > (in Sindarin only) a future sense resembling English “I will [I intend to] go”. In this same document, however, Tolkien considered splitting out the medical senses of √ATHA into a distinct root √HATHA “treat kindly/make easy, (help to) cure”; see that entry for details.