On the planet Euclid, there are 10 days in each month and 18 months in each year. Once every seven years, however, there is a leap year and one day is added to the third month. Each week contains five days which are (in order) ArchimedesDay, BernoulliDay, CavalieriDay, DesCartesDay, and EulerDay.

Fermet was born on a DescartesDay on the first day of the fourth month in a leap year. On what day of the week will Fermet’s 25th birthday fall?

The year's length is a multiple of 5 days, so the day of the week will advance by one every 7 years.

25 years is 7*3 + 4, so he will have lived through 3 leap years.
That means that in years 21-27 his birthday will be on Dday+3=Bday (4+3 mod 5)