16

There is indeed a difference between 18th century and modern concepts of algebra. For example, Lagrange and his contemporaries did not define algebraic structures into existence by specifying axioms for operations on abstract sets. Most of algebra was about integers and polynomials, which were assumed to pre-exist, and they were not thought as manifestations ...


14

The problem can be seen as part of more general attempts to extend the domain of operations defined initially for naturals (integers, or rationals), only. A prominent natural example would be taking an $n$-th power (another would be the binomial coefficients, and related Newtonian series). The problem is, thus, not an isolated one, but rather part of a ...


8

The answer depends on what you consider the "modern concepts". In modern algebra, they usually do not consider polynomials or rational functions as functions of the variable $x$, but exactly as you say in $1$, as elements of the field $C(y)$. So Lagrange's writing is not more symbolic/abstract than modern algebra, but more abstract than modern "high school ...


6

Aristotle's idea of earthquakes caused by "winds within the Earth" had much currency for centuries, if they were ascribed to natural causes at all. Here is from Agnew's History of Seismology on the prevailing thought in the 17-18th centuries: "With the decline of Aristotelian thought in early modern Europe, other ideas were put forward, ...


6

Euler's motivation R. Hilfer on pg. 18 of "Threefold Introduction to Fractional Derivatives" states, "Derivatives of non-integer (fractional) order motivated Euler to introduce the Gamma function ...." Euler introduced in the same reference given by Hilfer essentially $$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=\frac{x^{\alpha-\...


5

I don't know, but Images of earth from outside had been made for more than 500 years. A globe of the Earth would seem to count as a "depiction of any kind". The sphericity of the Earth was established by Greek astronomy in the 3rd century BC, and the earliest terrestrial globe appeared from that period. The earliest known example is the one ...


4

The reasons are the same as extending the binomial formula to non-positive exponents (Newton binomial). It involves generalized binomial coefficients which cannot be expressed in factorials. Binomial formula with non-integer exponent arises, for example in the computation of the arc length of an ellipse, and in other natural problems. The first part of the ...


4

(Partial answer - and a bit long winded) Regarding comparable telescopes - source : The Herschel Objects and How to Observe Them, by James Mullaney. Herschel's telescopes far surpassed in both quality and size any other telescope in the world at the time. After comparative trials at a number of observatories in England including Greenwich, he stated ...


3

The best answer to that question is the following book: The steam engine of Thomas Newcomen by L. T. C. Rolt, J. S. Allen 1977 Moorland Pub. Co. ; New York : Science History Publications, To summarize what I learned from this book. The boilers did not generate enough steam pressure to move pistons. You needed to use the weight of the atmosphere for the ...


3

"suspect" is hard to track... François Viète in Variorum de rebus responsorum mathematics liber VIII (1593) discovered the first infinite product in the history of mathematics by giving an expression of $\pi$ with what is now called Viète's formula. John Wallis, like Viète, expressed $\pi$ in the form of an infinite formula, but involving only ...


3

One can easily name one main intrinsic reason: invention of Calculus in the very end of the previous century, and "invention of mathematical physics" by Newton. It happened in the very beginning of 18-s century that sufficiently many people suddenly realized that mathematics can effectively explain the world. But of course, there were also outside reasons, ...


1

The progress in 18th century followed two important breakthroughs in 17th. Algebraic notation which was essentially modern appeared in Descartes's Geometry published already in 1637, and the calculus was invented by Newton and Leibniz by 1680s. Some methods of the calculus of variations were also developed in 17th century, in connection with the famous ...


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