Dido’s argument is from the 2nd century bc.
Euclid’s axioms are cool ig.
In Euclid’s actual language of the postulates, there is an intense focus on doing, and producing.
The fifth postulate is not necessarily true. You can make mathematical systems that lack the fifth postulate, like in curved geometry with hyperbolas.
Philosophical and mathematical proofs may be similar, but geometric proofs certainly are quite different.
Renaissance painters usually used mathematical ratios in order to regularise the sort of paintings they made. These were often called city of God, or the ideal city, paintings.
The discovery of linear perspective:
The greek sources were rediscoverd, but mathematically they didn’t add anything to greek geometry to create it, there was just a new application for a new kind of depiction.
Rather, it had to do with optics. With how we view things.
What is interesting is that no matter what stance one took on whether the lines come from the eyes or to the eyes, the elaboration of the lines of perspective are still perfectly straight and unparallel.
Octagons were important in early perspective drawings, as they used 45 degree angles.
Galile and Newton ost crucially thought that we can apply mathematical laws to the heavens.
Conclusion: mathematics was important to art.
The pythagoreans thought that the rational numbers were the only real numbers, whilst now a days we call a different set of numbers real, against the imaginary (complex) numbers.
The reals are usually thought of as a continuous quantity that can be depicted on a number line.
An identity is an equation that is always true, despite the values of the variables.
eg. a + b = b + a
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Parentheses
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standard functions, such as logarithms, trigonometric functions
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exponents and roots
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multiplication and division
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addition and subtraction
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in the case of operations of the same level of precedence, evalutation proceeds from left to right.
Before, it was not obvious to have a 90 degree angle grid system in cartesian graphs.