Would it be correct to think of God as a great body having different limbs only three of which we (mankind) have perceived?
No this would be totally against the Trinity. We know and have proof that God is Three in One and One in Three. Nothing in the Holy Bible implies otherwise.
Our Lord Jesus Christ commissioned His disciples to go and preached His Words to all saying, "All authority has been given to Me in heaven and on earth. Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all things that I have commanded you" (Mt 28:18-20). Every believer is baptized on this doctrine. It is through the baptism in the name of the Trinity that we are regenerated from death to life. If there was any other "limb or person' as you mentioned Our Lord surely would have mentioned since this is the declaration of faith.
Having, then, this full assurance, we are baptized as we have been commanded, and we believe as we are baptized, and we hold on as we believe; so that with one accord our baptism, our faith, and our ascription of praise are to the Father, and to the Son, and to the Holy Spirit.
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