“If there were no calculus, there would be no cell phones, computers, or microwave ovens.
Moreover, there will be no radio, no television, no ultrasound images for expectant mothers, and no GPS for lost travelers.
We would never have split the atom, discovered the human genome, or put a man on the moon.
“Even the American Declaration of Independence might not have come out.”
--- p.11

“This book looks at calculus from a much broader perspective than usual.
It covers many cousins ​​and derivatives of calculus that are spread across mathematics and adjacent fields.
Because this big tent perspective is unorthodox, I hope my approach doesn't cause any confusion.
For example, when I said earlier that without calculus there would be no computers, cell phones, etc., that doesn't mean that calculus alone created all these wonderful inventions.
Not at all.
Science and technology are inseparable partners and are the stars of this amazing show.
My point is that calculus played a significant (albeit often supporting) role in creating the world we know today.”
--- p.16

“Calculus began as a byproduct of geometry.
In ancient Greece, around 250 BC, there was a group of passionate mathematicians who tried to solve the riddle of curves.
They had an ambitious plan to use infinity to bridge the gap between curves and straight lines.
Once the bridge was completed, it was hoped that the methods and techniques of rectilinear geometry could be carried over the bridge and used to solve the riddle of curves.
I thought that with the help of infinity, I could solve all the existing unsolved problems.
At least that was the plan.
At the time, this plan must have seemed reckless.
Infinity had a questionable reputation.
Infinity was known as a useless and terrifying being.
It even had ambiguous and confusing properties.
"What exactly is infinity? Is it a number? A place? A concept?"
--- p.42

“Archimedes may have been the designer of marvelous war machines, and he was undoubtedly a brilliant scientist and engineer, but his real claim to immortality lies in his contributions to mathematics.
He laid the foundations of integral calculus.
Although this profound concept is clearly evident in his work, the world did not take notice of it until nearly 2,000 years later.
It would be too polite to simply say that Archimedes was ahead of his time.
Has anyone in history been more ahead of their time than Archimedes?
Two strategies appear repeatedly in Archimedes's work.
The first strategy was to make enthusiastic use of the principle of infinity.
Archimedes, in his quest to explore the mysteries of circles, spheres, and other curved shapes, always approximated them with many straight lines, planes, and three-dimensional shapes with faceted surfaces like jewels.
By imagining more and more pieces and making them smaller and smaller, we get closer and closer to reality through the limits of an infinite number of pieces, approaching the exact value.
To use this strategy properly, you had to be a master of math and puzzles, as it required combining many numbers or pieces to arrive at a conclusion.
Another surprising strategy was to combine mathematics and physics, that is, the ideal and the real.
Specifically, it combined geometry, the field that studies form, with mechanics, the field that studies motion and force.
Sometimes geometry was used to make mechanics easier, and sometimes the opposite was done by gaining insight into pure form from mechanics.
By skillfully using these two strategies, Archimedes was able to penetrate deeply into the mystery of curves.”
--- p.80

“Galileo was the first person to put the scientific method into practice.
Rather than simply quoting authority figures or sitting around the table theorizing, he sought to gain information by interrogating nature using careful observation, creative experimentation, and elegant mathematical modeling.
And this approach has led to many surprising discoveries.
The simplest and most surprising discovery of all was the secret of the law of falling bodies hidden in the odd numbers 1, 3, 5, 7…
Aristotle argued that heavy objects fall because they seek their natural position at the center of the universe.
Galileo thought this was nothing more than empty words.
And instead of thinking about 'why' objects fall, we tried to quantify 'how' they fall.
To do this, we had to find a way to measure the falling object and track its position at every moment.
It was not an easy task.
Anyone who has ever dropped a stone from a bridge knows how fast it falls.
Measuring a falling stone would require a very accurate clock, but Galileo didn't have one.
“Also, tracking a rapidly falling stone moment by moment would require several very good video cameras, something that was not readily available in the early 17th century.”

--- p.135