# How did definitions of a limit vary before the epsilon-delta definition?

My understanding is that before the epsilon-delta definition of a limit, the rigor and soundness of the definition of a limit was not good enough.

So, how did the definitions of a limit vary before the epsilon-delta definition?

I've read the Wikipedia page for the definition, so I understand past mathematicians couldn't define an infinitesimal. But, I was just hoping for further explanation and elaboration on what they did and didn't know compared to now. Was there no use of formal statements, quantifiers, conditionals back then, like seen in the epsilon-delta definition?

I was kind of hoping to see some notation and statements, equations, etc.

• There was no use of formal statements, quantifiers and conditionals until the late 19th century, or even verbal equivalents of those. Limits and infinitesimals were treated informally based on kinematic intuitions and/or algebraic tricks just as they are today outside of "foundational" passages in textbooks and mathematical logic. – Conifold Jan 2 at 7:57
• Since virtually all of this older literature is now digitized and freely available (at least where I'm at), you can look yourself --- google books search for {"calculus" AND "infinitesimal"} for 1700-1825. I didn't include limit because often this word and even the process was implicit, but you can include it if you want. I think you'll find that your concern with "formal statements", "quantifiers", "conditionals", etc. is a bit ahistorical. – Dave L Renfro Jan 2 at 7:57
• There were no precise definition of limit; there were an intuitive notion of of continuously changing quantity. – Mauro ALLEGRANZA Jan 2 at 10:26
• Could one say, "It was not until the definition of a limit was formally stated using quantifiers and conditionals that a precise definition was found."? I'm not very familiar with real numbers but I assume there is some connection there with that, too. So, basically, what I'm asking is what makes the epsilon-delta definition sound as compared to earlier notions. – watchy Jan 2 at 13:09
• A precise definition was proposed by Weierstrass about the same time that precise constructions of real numbers by him, Dedekind and Cantor appeared. But precision and formalism have little to do with soundness, as the word is commonly understood. Soundness means faithfulness to some pre-existing concept, and the pre-existing concept of a limit was too vague to talk about soundness here. – Conifold Jan 3 at 9:03