Curves and equations, also known as the Cartesian Connection, is an important and fundamental concept in plane analytic geometry, but it has been shown that students lack a rigorous understanding. Focusing on the concept of curves and equations, we investigated 84 American and British analytic geometry textbooks published from 1826 to 1963. The study found that three types of non-rigorous definitions were prevalent in early textbooks published before the 20th century. The first type merely recognized a connection between curves and equations but did not express the nature of the strict correspondence. The second type, although intentionally emphasizing that each point needs to be satisfied, examined only one aspect of the correspondence, neglecting the bidirectional nature inherent in the relationship between curves and equations. The third category of non-rigorous definitions confused the concepts of curves with functions. Definitions of curves and equations have become increasingly rigorous in post-20th century textbooks, and these rigorous definitions can be divided into descriptive definitions, definitions based on the concept of set, and definitions based on sufficient and necessary conditions. It is noteworthy that the second category of non-rigorous definitions was not eliminated until the middle of the 20th century. Epistemological barriers in the historical development of the curves and equations concepts can become cognitive barriers for students in the classroom as well. Instructional strategies can be developed to guide students in recognizing non-rigorous definitions, leading them through a historical reconstruction of how concepts related to curves and equations have evolved. This approach facilitates a natural progression from qualitative to rigorous understanding. In addition, early textbooks also provide a variety of methods for verifying the Cartesian Connection, which provide abundant materials for teaching.