Introduction
The midrash (Shemot Rabbah 38:2) identifies the following expressions for atonement of the sin of the golden calf. Each term represents a different aspect of the sin and a corresponding atonement.
| Term | Translation | Verse | Function |
| פר | Bull | Exodus 29:1, Leviticus 4:3 | Sin offering |
| פרה | Cow | Numbers 19:2 | Purification |
| עגל | Calf | Leviticus 9:2-3 | Sin offering |
Animals
Bull פר
The first term refers to Aaron’s involvement, as a leader, with the golden calf and subsequent atonement. In Leviticus 4:3 the Torah commands a high priest who erred in a matter of judgement, involving a sin which would lead to excision if committed intentionally, to bring a bull offering for atonement. Due to the severity of the sin, the meat of this sin offering is burnt outside the temple. Similarly the meat of the sin offering of Exodus 29:1, presented during the seven days of inauguration of the tabernacle (משכן) is burnt outside the camp.
Cow פרה
The cow, as a mother, refers to the impact of the sin over generations in terms of death. Rashi quoting Midrash Tanchuma 8 on Parshat Chukat explains let the mother (i.e. cow) clean the mess made by its child (i.e. golden calf). The ashes of this cow, after slaughter, are mixed with water to purify someone who has had contact with the dead either by touching, carrying, or being covered by the same roof. The Talmud (Avodah Zara 5a) states that if not for the sin of the golden calf the Israelites would be free from death. After receiving the Torah the Israelites would have returned to the state of Adam before the sin of eating from the tree of knowledge. (Another opinion states that death was decreed from Adam but the sin of the golden calf made the Israelites susceptible to persecution from the other nations.) In any event the cow refers to purification from death brought by the sin of the golden calf. In effect the mother (i.e. cow) provides purification from death caused by her son (i.e. calf).
Calf עגל
The calf refers to the actual sin of the golden calf and Hashem’s forgiveness to Aaron, allowing him to serve as high priest (Midrash Tanchuma Shmini 4). Aaron offered this calf as a sin offering on the eighth day of inauguration (first of Nissan). Due to the severity of the sin, the meat of this sin offering is also burnt outside the camp (Leviticus 9:11). It is interesting to note that besides this offering, the term calf is never specifically mentioned in the Torah in reference to an offering even though one can offer a calf as a voluntary offering. The Torah is sensitive to the experience of the Israelites and does not want to recount this sin when mentioning offerings which are a mandatory means to coming close to Hashem.
Conclusion
The Torah uses different terms for offerings to atone for the golden calf, balancing the need for atonement while sensitively maintaining respect for the Israelites.