The more I look at it, the more complicated the stemma of the Mathers-Crowley Goëtia gets. That is:
1. The original 17th-century redaction depends on the 1665 Robert Turner translation of the Heptameron (and thus the Harley Lemegeton, which was further heavily redacted from the text represented by the Sloane copies, can. not. possibly. derive from the work of someone who died in 1656). There are two citations in the Heptameron's Exorcismus Spirituum Aërorum that were omitted in Turner's translation, though included in his typeset of the Latin (addressing the spirits as vos qui, vestra culpa, de coelo eiecti fuistis usque ad infernum locum, "you who, for your sin, were cast out of heaven even unto the infernal region" -- and exorcising them per eam quæ Ecclesia Dei nominatur, "by she who is called the Church of God") and were similarly skipped over in the corresponding sections of the second conjuration of the Ars Goëtia, and two instances of sedem ("seat") being rendered "seal," that are perpetuated into the Goëtia, as well as the questionable translation of "nona cohorte" as "in the ninth Legion" in the "Beralanensis" conjuration. All these points differ, for example, in the other known 17th-century English translation of the Heptameron, (BL Sloane MS. 3851 fol. 61r-74r -- see specifically fol. 64v, 65v).
2. The redactor of the Ars Goëtia (going on the Sloane MS. 3825 version), while perpetuating the aforementioned errors and omissions, besides the major rearrangements (e.g. displacing the concluding section of the Exorcism / Vinculum into "The Constraint," and moving the bulk of the "Beralanensis" conjuration into the first conjuration), freely rephrased citations in the second conjuration and made minor rearrangements in their order. Other passages remain, barring vagaries of spelling, capitalisation and punctuation, verbatim from Turner.
Since the Vinculum Spirituum was a floating text, extant exemplars of which show a great deal of variation, that was incorporated into the Lucidarium prior to that being turned into the Heptameron, it is of course possible that the redactor of the Goëtia had a variant copy in Latin or English, probably not under that title (else why, as previously remarked, make up something else to meet the demand for the "Spirits' Chain"?): on the other hand, there are no citations in the second conjuration of the Goëtia that are not in the Heptameron version of the Vinculum; there is no mention, for instance, of the great and terrible name Gin which Noah heard and spake and was delivered from the Flood, which appears in other English texts of the period deriving from the Vinculum (Sloane MS. 3824 fol. 14r, 144v; see also Sloane MS. 3847 fol. 14r).
3. Mathers' redaction, while retaining the major rearrangements, reverts some minor rearrangements and is generally phrased much closer to Turner's translation than to the Sloane Lemegeton MSS. It also has major variations that are not attested by any of the BL Lemegeton MSS. and, in so far as they relate to the general rearrangement of the conjurations between the Heptameron and Ars Goëtia, are not simply derived from Turner; this may have been based on an otherwise unknown Lemegeton text, or may have been invented by Mathers. For instance, the first conjuration in Sloane MS. 3825 runs as followeth:
I Invocate and conjure you spirit N. & being with power armed from the supreame Majesty I throughly [sic] command you by Beralanensis, Baldachiensis, Paumachiæ & Apologiæ Sedes, and the most powerfull princes, Genio Liachidi ministers of the Tartarean seat, cheeefe princes of <of> the seat of Apologia, in the Ninth Legion; I exorcize & powerfully command you spirit N, in and by him that said the word and it was done; and by all the holy and most glorious names of the most holy and true God, and by these his most holy names Adonai, El, Elohim, Elohe, Zebeoth, Elion, Escerchie, Jah, Tetragrammaton, Saday, that you forthwith appeare and shew yourselves here unto me before this Circle, in a fair and humane shape, without any deformity or ugly shew, and without delay, doe ye come; from all parts of the world to make <make> rationall answeres unto all Things which I shall ask of you, and come yee peaceably visibly and afably without delay, manifesting what I desire, being conjured by the name of the Eternall living and true God, Helioren, I conjure you by the especiall and true name of your God that you are obedient unto and by the name of the King, which beareth rule over you, that forthwith you come without Tarring and fullfill my desires, and command, and persist unto the End, & according to my Intentions and I conjure you by him by whome all Creatures are obedient and by this ineffable name Tetragrammaton Jehovah, which being heard, the Elements are overthrown, the aire is shaken, the sea runneth back, the fire is quenched, the Earth trembleth, and all the hosts of Celestialls, Terrestialls & Infernalls doe tremble, and are troubled and confounded together, that you visibly and affelby speak unto me with a Clear voice Intelligible, and without any ambiguity, therefore come ye in the name Adonai Zebeoth, Adonai Amiorem, com com why stay you hasten Adonay, Saday, the Kinge of kings commandeth you.
The other Sloane Lemegeton texts have only minor variations with this. Whereas Mathers' text runs:
I do invocate and conjure thee, O Spirit, N., and being with power armed from the Supreme Majesty, I do strongly command thee, by Beralanensis, Baldachiensis, Paumachia, and Apologiæ Sedes; by the most Powerful Princes, Genii, Liachidæ, and Ministers of the Tartarean Abode; and by the Chief Prince of the Seat of Apologia in the Ninth Legion, I do invoke thee, and by invocating conjure thee. And being armed with power from the Supreme Majesty, I do strongly command thee, by Him Who spake and it was done, and unto whom all creatures be obedient. Also I, being made after the image of God, endued with power from God, and created according to His will, do exorcise thee by that most mighty and powerful name of God, El, strong and wonderful; O thou Spirit N. And I command thee by Him who spake the Word and his Fiat was accomplished, and by all the names of God. Also be the names Adonai, El, Elohim, Ehohi, Ehyeh Asher Ehyeh, Zabaoth, Elion, Iah, Tetragrammaton, Shaddäi, Lord God Most High, O thou spirit N., that thou dost forthwith appear unto me here before this Circle in a fair human shape, without any deformity or tortuosity. [&c., &c., &c. -- see the CP Goëtia for the rest.]
"I, being made after [...] O thou Spirit N," is displaced from the start of the "Exorcism of the Spirits of the Air" and does not appear anywhere in the Sloane 3825 Goëtia.
4. Mathers had access to one or more Lemegeton texts in unspecified private collections. At least one of these was separate from the Frederick Hockley transmission line (an incomplete exemplar of which is in Wellcome MS. 3203): Mathers gives four versions of the introductory description of the work detailing the separate books, three of which include the Ars Notoria: Hockley's source (which he described as "a MS. in two volumes, about the date of 1680 to 1700") only had the first four "books" (see his copy of the introductory description in Wellcome MS. 4665); and of the four BL Lemegeton MSS., Sloane MS. 2731 omits the Ars Notoria and Harley MS. 6483 omits the outline description. Hockley's version of the first conjuration is also closer to the Sloane copies than to the Mathers redaction (the copy in Wellcome 3203 omits some phrases, possibly by copyist eye-skip). The copy which was later acquired by Professor E.M. Butler (who reproduced an illustration from it in her Ritual Magic) also apparently lacked the Ars Notoria (she characterises it as being similar in content to Sloane 2731).
5. Mathers, or someone in an otherwise unknown MS. tradition, while generally modernising spelling and standardising the orthography of names of God of Hebrew origin, introduced archaic English inflectional forms that weren't in the 17th-century versions.
I have uploaded a tabulation of the parallel passages to Scribd.
