The Brazen Serpent, Jacob's Ladder, and the Cross
Old Testament: Numbers 21:8-9
And the Lord said to him: Make a brazen serpent, and set it up for a sign: whosoever being struck shall look on it, shall live. Moses therefore made a brazen serpent, and set it up for a sign: which when they that were bitten looked upon, they were healed.
Old Testament, also: Genesis 28:12-13
And he saw in his sleep a ladder standing upon the earth, and the top thereof touching heaven: the angels also of God ascending and descending by it, And the Lord leaning upon the ladder, saying to him: I am the Lord God of Abraham thy father, and the God of Isaac; the land, wherein thou sleepest, I will give to thee and to thy seed.
New Testament: John 1:51; John 3:14-15
... Amen, amen I say to you, you shall see the heaven opened, and the angels of God ascending and descending upon the Son of man. ... And as Moses lifted up the serpent in the desert, so must the Son of man be lifted up: That whosoever believeth in him, may not perish; but may have life everlasting.
The connection
Christ Himself draws this typology explicitly. The serpent, an image of the very thing that caused death, is lifted up so that looking on it brings life. Christ, made sin for us though sinless, is lifted up on the Cross so that looking to Him in faith heals the wound of the fall. Christ claims a second image alongside it in the very same Gospel: Jacob's ladder, "the angels of God ascending and descending upon the Son of man" (John 1:51), reaching from Jacob's stone pillow to the gate of heaven (Genesis 28:12-13). The two images share one axis between earth and heaven at a moment of crisis, but where the ladder shows glory and communion, the serpent shows the curse itself lifted up and borne. Read together, the Cross is both: the curse-bearing is the gate-opening, the very thing raised up to heal is what throws heaven's gate wide.
Moral sense
Stop trying to cure yourself; look, in bare faith, at what shames you until it saves you. But the gaze must pass through the sign to what it signifies: even this brazen serpent, made at God's own command, later had to be destroyed by King Ezechias when Israel began burning incense to it instead of looking through it (4 Kings 18:4).
Anagogical sense
Seeing the glorified Christ face to face, no longer faith's look at a hidden God, but sight. The ladder's promise, heaven opened and the angels passing freely between, is that horizon completed: no longer a vision granted to one sleeper at Bethel, but the gate standing open to all who look to the Son of man lifted up.
The Fathers
St John Chrysostom, Homily 27 on John, sets the figure beside its fulfilment: "there the hanging serpent healed the bites of serpents, here the Crucified Jesus cured the wounds inflicted by the spiritual dragon." St Justin Martyr, First Apology 60, says Moses took brass "and made it into the figure of a cross."
Warrant
Checked in the Greek: ophis is the same noun in both, the serpent set on the pole at Numbers 21:8-9 in the Septuagint and the serpent Christ names at John 3:14, where the reference to Moses is explicit in any case. One more thing the Greek holds, and it is the plainest kind of warrant there is: the name. Mouses stands at Numbers 21:8-9 in the Septuagint and at John 3:14, where Christ says "as Moses lifted up the serpent in the desert". The Gospel names the man as well as the sign.
Catechism of the Catholic Church, paragraph 2130.
The verses
Numbers 21:8 · Numbers 21:9 · Genesis 28:12 · Genesis 28:13 · John 1:51 · John 3:14 · John 3:15