I'm reading SL Loney's plane trigonometry book and I arrived at a theorem saying : "the length of the circumference of a circle always bears a constant ratio to its diameter."
Now, in this proof he uses two propositions from the book VI of Euclid. Now, am I supposed to be reading these proposition and all the definitions,axioms,etc. or I can take them for granted and continue reading the theorem without them ? How is it done, in your opinion?(I'm new to the domain of proofs ^^)
Thank you!
It would suffice to say that all circles are "similar" to each other. That's why circumference of a circle always bears a constant ratio to its diameter.
In my opinion: Even if you don't understand it, it doesn't make much of a difference.
Whether you accept certain theorems at face value, or whether you choose to investigate those results more deeply, is up to you, and it depends on the level of understanding you wish to have.
Certainly, if you want a deep understanding of the fundamental material in a topic, you should read and understand all of the prior results leading up to certain ideas. But it is not always necessary to do so, and it is easy to get lost in reading proofs, alternative proofs, equivalent definitions, etc.
Ultimately, deciding what you should focus your efforts on is easily determined by answering the following questions:
Short answer: Look at the proofs.
Long answer: If your goal is to understand and use mathematics, then you can skip or skim over proofs. However, proofs are important for two reasons: (a) They force you to think logically and not jump to conclusions, (b) the method employed in the proofs often lead to answers to other questions.
If you are starting out on learning proofs, I can't think of a better introduction than the first two books of Euclid (later books may not be that easy to follow).
