Took a while to get back to this.

The term also denotes indivisible things. How many points are in a line of one meter? A billion billion? Those are very small bits, but even a billion billionth of a meter can be divided into a half a billion billionth, and those halves into quarters. So, because there is no end to dividing any positive amount, that being a finite number seems hard to accept. But, if we assume that there's an infinite number ( which is what we learn in school) and we say that each point has a size, the line is now infinitely long. So, we can say that each point has no size. But that would seem to mean that each point is the size of 0, and 0+0=0....which would seem to mean our line , is, well, not a line.

The ancient Greeks knew about this problem. Zeno the Eleatic listed a number of paradoxes based on it, maybe the most famous one being Achilles and the Tortoise ( I won't get into them but there's a nice article on it here ). But they moved away from thinking about this problem, and Euclid focused on much more intuitive geometry, with reasonable propositions. But Archimedes did step cautiously into the use of infinitesimals, showing that they could be used to calculate the volumes enclosed by circles, or of spheres and cylinders. In the Renaissance, some mathematicians , taking their lead from Archimedes, began to use them as well.

One important part of the Counter Reformation, when the Catholic Church moved strongly to combat the spread of Protestantism, were the Jesuits. They were created as an army to argue and convince, to lead people away from heresy and to solidly stand in the security of Church teachings. Accordingly, the Jesuits created many schools- in fact, in Catholic countries they dominated the education systems. Their schools were strict, and highly respected. But they were strictly hierarchical, and with a very strong commitment to Catholic dogma being superior to everything else, even experience and discovery ( Jesuits famously stipulating that, if the Church said black was white, it would become white) . Mathematics was a part of the education they provided, but subordinate to theology. Indivisibles were an affront to the idea of an orderly world- why would God allow such paradoxes? Their mathematicians ( like Clavius, who was greatly responsible for the reform creating the Gregorian calendar) for the most part stayed away from them, embracing the more intuitively reasonable Euclid. However, it became hard to avoid the subject. In the 17th c. one, Cavalieri, showed rather innocuously how they could work- while publicly denying he was doing so. He was criticized nonetheless. By the time of the ban in 1632, the use of indivisibles had been reviewed a few times already. Under some years of the reign of Pope Urban XV, the Jesuits were out of favor and the subject was ignored. But by 1632 the Catholic church was again in a war against heresy, all across Europe, with the Thirty Years War being pushed along by the Emperor Frederick II ( also a graduate of the Jesuit schools) . It was in no mood to debate, and reflect. It was the period in which the simple miller Menocchio ( the subject of Gian Carlo Ginzburg's classic The Cheese and the Worms) would be burnt for too much theological musing.. It was the time in which Galileo was condemned to house arrest for eight years, after showing evidence of the earth moving around the sun. Cavalieri, ironically, would move in with the house-imprisoned Galileo and help him with his writings for the last three months of his life.

The result was that mathematics simply moved on without the Jesuits. John Wallis would advance the field of infinitesimals, Newton and Leibniz would discover calculus, Leonhard Euler would dominate the field and explore algorithms n the 18th c..Plenty more difficult things appeared than infinitesimals. Things called imaginary numbers ( like the square root of -1, very important for working out electrical resonance ) would be found to be, well, just very very useful. ..and the Jesuits in Italy would try very hard to ignore it all.

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Amir Alexander: How a Dangerous Mathematical Theory Shaped the Modern World