Bull
The bull surfaces in the umbrella material almost entirely as a sacrificial animal — paired with the goat, paired with the ashes of the heifer, and weighed against what its blood can finally accomplish for sin. The narrow scope here belongs in the wider treatment of bullocks, heifers, and the burnt-offering system; see Offerings for the procedural detail.
The blood of bulls and goats
Hebrews carries the umbrella's full weight. The first verse states the cleansing achieved within the priestly service of the older covenant: "For if the blood of goats and bulls, and the ashes of a heifer sprinkling those who have been defiled, sanctify to the cleanness of the flesh:" (He 9:13). Bull-blood and goat-blood are paired with the burnt remains of the heifer; together they reach a cleanness "of the flesh" — outward purification of the body.
The second verse states the limit of that same blood: "For it is impossible that the blood of bulls and goats should take away sins" (He 10:4). The pairing now bears the negative — the same animals whose blood cleansed the flesh cannot remove sin. The bull, in the umbrella's frame, names both the legitimate sacrificial victim and the precise reason a further offering is needed.