Around this time, the heredity of San hakase grew in importance, and being part of a hereditary family joined passing Hoshi as a requirement for assuming the role of San hakase; initially, in the early Heian period, the Iehara and Okura clans began to make it into the heredity, but this did not last; then, after the ruin of the preceding clans, the Ozuki and Miyoshi clans began to utilize heredity, made their own Sando into hereditary learning and secret teachings, and excluded other clans by adopting capable disciples to inherit the family name.
この頃になると、算博士の世襲化が進み算道によって奉試に合格するとともに、譜第の家系であることが算博士就任の要件とされるようになり、当初は家原氏・大蔵氏が平安時代前期には世襲化も兆候を見せるものの長続きせず、両氏の没落後は小槻氏・三善氏の両氏が世襲するようになり、自己の算道を家学・秘伝化し、また有能な門人を養子として家名を継承させることで他氏を排除するようになった。