How does Saint Joachim fit into Saint Luke’s Genealogy

The Sacrifice of St Joachim by Fra Giotto
Today is the feast day of Saint Joachim, father of the Virgin Mary, in the Extraordinary Calendar (1962). The genealogy of Saint Matthew is read for the feast. Oddly enough, Saint Joachim is not mentioned. Neither does the name appear in the genealogy of Saint Luke. This has led some to doubt the Catholic conviction that Saint Joachim is the historic father of the Blessed Virgin Mary.
Tradition holds that Saint Luke’s genealogy is that of the Blessed Virgin Mary. When Saint Luke writes “Joseph, [ ] of Heli”, without the word “son” being present in the Greek, the Evangelist indicates that “Joseph, of Heli” is to be read “Joseph, [son-in-law] of Heli.” Joseph is not the biological son of Heli, but the son-in-law of Heli. Cornelius a Lapide holds that Joachim/Heli was the full brother of Jacob, the father of Saint Joseph. This would make Joseph and Mary first cousins.
Early Church Fathers note that “Heli” who is mentioned as the “father of Joseph” in Saint Luke’s genealogy is one and the same as Saint Joachim. 
Heli is short for Heli-achim (meaning “God prepares”), a Hebrew alternate form for the Hebrew Jeho-achim (meaning “Jehovah prepares”).
All this being said, it would seem more appropriate to have Saint Luke’s genealogy, not Saint Matthew’s, to be read at Holy Mass for Saint Joachim’s feast day. If any one has the reason why it’s Matthew and not Luke, please leave a comment below.
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