I'm writing a period novel set in 17th century Europe where Leibniz and Newton were newly publishing their works on Calculus (or Newton's Principia).
And there is one part that Leibniz had to to explain his concept in Calculus to my main character, so I wanted to ask about a part of his 1686 article.
According to Leibniz's article in 1686, written in Latin, There was a differential equation written $$pdy = xdx$$ Since $$d \frac{1}{2} xx = xdx$$ and since $$\int xdx = \frac{1}{2} xx,$$ $$\int pdy = \frac{1}{2} xx .$$
I still don't get why he invented integral symbol for "sum".
What does the "sum" actually mean in that article?
And why was there no "+C" ?
I cannot afford to buy the source of mathematics (1200-1800) which has the translated version of it, so I decided to ask here.
Here is the translation of the section above, from Struik: A Source Book in Mathematics, 1200-1800.
I proceed to this subject in the following way. Let the ordinate be $x$, the abscissa $y$, and the interval between perpendicular and ordinate, described before, $p$. Then according to my method it follows immediately that $p\,dy = x\,dx$, as Dr. Craig has also found. When we now subject this differential equation to summation we obtain $\int p\,dy = \int x\,dx$ (like powers and roots in ordinary calculations, so here sum and difference, or $\int$ and $d$, are each other’s converse). Hence we have $\int p\,dy = {1\over 2}xx$, which was to be demonstrated. Now I prefer to use $dx$ and similar symbols, rather than special letters, since this $dx$ is a certain modification of the $x$ and by virtue of this it happens that — when necessary — only the letter $x$ with its powers and differentials enters into the calculus, and transcendental relations are expressed between $x$ and some other quantity. Transcendental curves can therefore be expressed by an equation, for example, if $a$ is an arc, and the versed sine $x$, then $a = \int dx : \sqrt{2x-xx}$ and if the ordinate of a cycloid is $y$, then $y = \sqrt{2x-xx} + \int dx : \sqrt{2x-xx}$, which equation perfectly expresses the relation between the ordinate $y$ and the abscissa $x$. From it all properties of the cycloid can be demonstrated. The analytic calculus is thus extended to those curves that hitherto have been excluded for no better reason than that they were thought to be unsuited to it. Wallis’s interpolations and innumerable other questions can be derived from this.