Peirce's existential graphs are significant in predicate calculus for several reasons:

1. Visual Representation: Peirce developed existential graphs as a way to visually represent logical propositions and their relationships. This graphical approach allows for intuitive manipulation and understanding of logical statements, making complex logical relationships clearer.

2. Expressiveness: The graphs can express quantifiers such as "there exists" and "for all" in a way that aligns closely with natural language reasoning. This expressiveness is crucial in predicate calculus as it allows for the formulation of statements that include existential and universal quantifications.

3. Formal System: Peirce's existential graphs provide a formal system that can be used to derive logical conclusions. They maintain the same level of rigor as traditional predicate calculus while offering a different medium of expression that can simplify the reasoning process.

4. Connections to Modern Logic: Peirce's work foreshadowed many developments in modern logic and computer science, particularly in areas like graph theory and visual programming. The existential graphs can be seen as an early attempt to bridge formal logic with visual reasoning, influencing later developments in logical systems.

5. Reduction of Syllogistic Logic: Peirce aimed to reduce syllogistic reasoning to a graphical form, allowing for a more efficient means of reasoning that could potentially replace or enhance traditional symbolic logic.

In summary, Peirce's existential graphs serve as a powerful tool in predicate calculus, enhancing the understanding and manipulation of logical statements through visual means, thereby contributing to the broader field of logic and reasoning.