Johnnie and math Ph.D. here. It's a bad, or at least slow, way to prepare for math or math-heavy graduate school as you'll need to take additional coursework elsewhere in order to be competitive. On the other hand, freshman math at St. John's (Euclid, a bit Apollonius, start into Ptolemy) was for a significant fraction of students in my year the first math class they had ever enjoyed. Euclid's Elements, despite its many inefficiencies, and faults, does not heavily rely on mastery and/or acceptance of earlier curricular material, nor is it designed in service to later curricular material that most students are in no position to anticipate or appreciate.
On the third hand, reading Newton is a really bad way to learn calculus; the college uses a lab manual which more closely resembles a modern calculus course.
It appears to be a liberal arts program. Is this substantially different, with respect to the rigor of mathematics, than most other comparable programs? They might cover more calculus (maybe at a theoretical level?) than most liberal arts programs.
I'd expect the rigor is fine, but the particulars that are learned differ.
I doubt the distinction matters at all for the vast majority of grads, especially ones who don't intend to become mathematicians. Learning how to math is probably more important than the specific material, outside a handful of things. You can pick up the rest as-needed, and for the vast majority of people, "the rest" that is in fact ever needed for the entire rest of their lives, will be very little. Especially if they're pursuing a classics-based liberal arts degree.
I doubt many of their grads are planning to become actual computer-scientists or mathematicians or mech. engineers or any of that. Lawyer, maybe doctor, maybe writer, maybe an ordinary computer programmer, that sort of thing. As long as you're not afraid of math, you'll be fine in any of those not having had a typical PDE class or whatever.
Yes, it is substantially different with respect to content than standard undergraduate mathematics programs. It covers a few historically important texts and does not teach (if those texts are any indication) most of what is usually taught in an undergraduate math degree. (A poster above writes: "Freshman math was almost entirely the study of Euclid and Nicomachus.")
So this is the books used in an undergraduate liberal arts degree (your degree is IN liberal arts). These are the math tagged books in a quirky bachelors in philosophy degree, essentially. They do not have a math degree (or any degrees aside from bachelors in liberal arts?).
I see - I understood "liberal arts program" above to mean a liberal arts college in general (typically offering a mathematics major). I agree that this reading list is better suited for something like "history of math for humanities students."
On the third hand, reading Newton is a really bad way to learn calculus; the college uses a lab manual which more closely resembles a modern calculus course.