This is a very rough translation of Aquinas's article in the Summa on the Filioque. The Latin is here; the Dominican Fathers translation is here.
To the second [article] we proceed in this way.
 It seems that the Holy Spirit does not proceed from the Son. For according to Dionysius, we must not dare to say something about the divine substance, save according what divinity expresses to us through sacred oracles [ex sacris eloquiis]. But sacred Scripture does not express that the Holy Spirit proceeds from teh Son, but only that He proceeds from the FAther; as appears in John 15, "Spirit of truth, who proceeds from the Father." Therefore the Holy Spirit does not proceed from the Son.
 Further, in the Symbol of the Council of Constantinople we read this: "We believe in the Holy Spirit, Lord and Life-giver, proceeding from the Father, with the Father and the Son to be adored and glorified." Therefore it should not be added in our Symbol that the Holy Spirit proceeds from the Son, but it seems that they who added this are to be anathematized.
 Further, Damascene says, "The Holy Spirit we say is from the Father, and we name Him Spirit of the Father, but we do not say the Holy Spirit is from the Son, although we name Him the Spirit of the Son." Therefore the Holy Spirit does not proceed from the Son.
 Nothing proceeds from that in which it rests. But the Holy Spirit rests in the Son. For it is said in the legend of the Blessed Andrew, "Peace be to you, and to all who believe in one God the Father, and in His one Son, our only Lord Jesus Christ, and in one Holy Spirit, proceeding from the Father, and remaining in the Son." Therefore the Holy Spirit does not proceed from the Son.
 Further, the Son proceeds as Word. But our spirit in us does not seem to proceed from our word. Therefore neither does the Holy Spirit proceed from the Son.
 Further, the Holy Spirit proceeds completely [perfecte] from the Father. Therefore it is superfluous to say that He proceeds from the Son.
 Further, in things perpetual there is no difference between actual being [esse] and possibility [posse], as is said Physics III; and much les in divine things. But the Holy Spirit is able to be distinguished from the Son, even if He did not proceed from Him. For Anselm says, in the book on the procession of the Holy Spirit, "The Son and Holy Spirit have their being from the Father, but in diverse ways, the one by being born , the other by proceeding, so they are distinct from each other." And later on he says, "For even if the Son and the Holy Spirit were not through distinct [plures] for any other reason, through this alone they would be different [diversi]. Therefore the Holy Spirit is distinguished from the Son, from whom He does not have His being [ab eo non existens].
But on the contrary is what Athanasius [i.e., the Quicunque Vult attributed to Athanasius] says, "The Holy Spirit from the Father and the Son, not made, not born, but proceeding."
I respond that it must be said taht it is necessary to say that the Spirit is from the Son. For if He is not from Him, in no way is He personally distinguished from Him, as appears from what has been said above. For it is not possible to say that the divine persons are distinguished from each other according to something absolute, because it would follow that there would not be three in one essence; for whatever is said of the divine absolutely pertains to the unity of the essence. Therefore it must be conceded that the divine persons can be distinguished from each other only relatively. But the persons cannot be distinguished relatively, save insofar as they are opposed; which appears in this, that the Father has two relations, according to one of which He is referred to the Son, and according to the other to the Holy Spirit, but as these are not opposed, they do not constitute two persons, but pertain only to the one person of the Father. If therefore there be found in the Son and in the Spirit only two relations which refer each to the Father, these relations would not be opposed to each other, just as the two relations the Father has to them are not. Therefore, since the Father is one person, it would follow that the Son and the Holy Spirit would be one person having two relatednesses opposed to the two relations of the Father. But this is heretical, for it takes away faith in the Trinity. Therefore it is fitting that the Son and the Holy Spirit be refered to each other through opposed relations. But there cannot be in the divine any opposed relations save relations of origin, as is proved above. But opposed relations of origin are attributed according to principle and according to what is from the principle. Therefore it must be conceded that it is necessary to say that either the Son is from the Holy Spirit, which nobody says, or the Holy Spirit is from the Son, which we confess.
And this is consonant with the notion of the procession of each. For it is said above that the Son proceeds by way of intellect, as Word; but the Holy Spirit by way of will, as Love. But it is necessary that love proceed from the word, for we do not love something save according to the mental conception we apprehend. Therefore it is also clear according to this that the Holy Spirit proceeds from the Son.
The same is also shown from the order of things. For we nowhere find this, that different things proceed from one another without order, save in those things that differ materially; as one smith produces many knives that are materially distinct from each other, but are not ordered to each other. But in things in which there is not only a distinction of material, we always find in a mutlitude of products some order. Hence also in the order of produced creatures, the beauty [decor] of divine wisdom is manifested. If therefore there proceeds from the one person of the Father two persons, namely, the Father and the Holy Spirit, it is fitting that there be some order of one to the other. Nor can any other order be assigned, save the order of nature, whereby one is from the other. Therefore it is not possible to say that the Son and the Holy Spirit proceed from the Father such that neither proceeds from the other, unless we post in them a material distinction, which is impossible.
Hence also the Greeks themselves understand the procession of the Holy Spirit to have some order to the Son. For the concede that the Holy Spirit is the Spirit of the Son, and is from the Father through the Son. Some of them are also said to concede that He is from the Son, or that He flows forth from Him, but not that He proceeds; which seems to be either from ignorance or from stubbornness. For if one rightly considers it, one discovers that the word 'procession' most commonly pertains to all that denotes origin of any kind. For it is used to designate any kind of origin; as a line proceeding from a point, a ray from the son, a river from a spring; and similarly with everything else. As it pertains to any origin, one can conclude that the Holy Spirit proceeds from the Son.
Therefore to the first it must be said that we ought not to say of God what is not found in sacred Scripture, either by word or by sense. But although we do not find by word in sacred Scripture that the Holy Spirit proceeds from the Son, we find it according to the sense; and especially where the Son says (John 16), speaking of the Holy Spirit, "He will glorify me, because He receives of me." For by a rule of sacred Scripture it must be held that what is said of the Father is fittingly understood of the Son, although there be added an exclusive saying, save only in those things in which the Father and the Son are distinguished according to opposed relations. For when the Lord says (Matthew 11), "Nobody knows the Son save the Father," this does not exclude that the Son knows Himself. Thus when it is said that the Holy Spirit proceeds from the Father, even though it be added that He proceeds from the Father alone, this does not at all exclude the Son, because the Father and the Son are not opposed according to one's being the principle of the Holy Spirit; but only according to this, that this is Father and that Son.
To the second it must be said that in every Council there has been instituted some Symbol, according to some error which is condemned in the Council. Hence subsequent Councils did not make any other Sybmol than the first, but what was implicitly contained in the first Symbol, is explained by some addition against insurgent heresies. Hence in the determination of the Council of Chalcedon it is said, that those who were gathered together in the Council of Constantinople, handed down the doctrine of the Holy Spirit, not implying that there was less in the preceeding Council (which was gathered at Nicaea); but declaring against the heretics what they had left understood. Therefore because in the ancient times of the Council there had not yet arisen the error that says that the Holy Spirit does not proceed from teh Son; it was not necessary to propose it explicitly. But afterward, certain errors arising, in another Council gathered in the West, it was expressed by the authority of the Roman pontiff; by whose authority even the ancient Councils were gathered together and confirmed. Nonetheless it was contained implicitly in that according to which it is said that the Holy Spirit proceeds from the Father.
To the third it must be said that the Holy Spirit's not proceeding from the Son was first introduced by the Nestorians, as appears in the Nestorian Sybmol condemned in the council of Ephesus. And this error was held by the Nestorian Theodoret, and many after him; among whom was also the Damascene. Hence in this his sentence does not stand. Although it is said by some that while the Damascene, did not confess that the Holy Spirit is from the Father, he also does not deny it with these words.
To the fourth it must be said that when the Holy Spirit is said to rest or stay in the Son, it is not at all excluded that He proceeds from Him, because the Son also is said to say in the Father, when He nonetheless proceeds from the Father. Also it is said that the Holy Spirit rests in the Son either as lover of the lover rests in the beloved, or according to the human nature of Christ, about which it is written (John 1), "On whom you shall see the Spirit descending, and staying on Him, He it is who baptizes."
To the fifth it must be said that word in the divine is not attributed according to the likness of the vocal word, from which spirit does not proceed, for it would then be said only metaphorically; but according to the likeness of the mental word, from which love proceeds.
To the sixth it must be said that from this, that the Holy Spirit completely proceeds from the Father, not only is it not superfluous to say that the Holy Spirit proceeds from the Son, but it is altogether necessary. For one power is in the Father and the Son; and whatever is from the Father, necessarily is from the Son, unless it is repugnant to the property of filiation; for the Son is not from Himself, although He is from the Father.
To the seventh it must be said that the Holy Spirit is personally distinguished from the Son in this, that the origin of one is distinguished from the origin of the other. But the difference of origin itself is because of this, that the Son is from the Father alone, whereas the Holy Spirit is from the Father and the Son. For otherwise the processions would not be distinguished from each other as is shown above.