Apographum schediasmatis a Newtono olim scripti

Probleme

An equation being given expressing the Relation of two or more lines x, y & z described in the same time by two or more moving bodies A, B, C, & to find the relation of their velocities p, q, r, &c

Solution

Set all y^e terms on one side of the Equation that they become equal to nothing. Multiply each term by so many times $\frac{\mathrm{p}}{\mathrm{x}}$ as x hath dimensions in that term. Secondly multiply each term by so many times $\frac{\mathrm{q}}{\mathrm{y}}$ as y hath dimensions in it. Thirdly multiply each term by so many times $\frac{\mathrm{r}}{\mathrm{z}}$ as z hath dimensions in it &c. The summ of all these products shall be equal to nothing. Which Equation gives the relation of p, q, r, &c.

Or more generally thus.

Order the Equation according to the dimensions of x, & putting a & b for any two numbers whether rational or not, multiply the thers thereof by any part of this Progression $\frac{\mathrm{a}\mathrm{p}-3\mathrm{b}\mathrm{p}}{\mathrm{x}}.\frac{\mathrm{a}\mathrm{p}-2\mathrm{b}\mathrm{p}}{\mathrm{x}}.\frac{\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{p}}{\mathrm{x}}.\frac{\mathrm{a}\mathrm{p}}{\mathrm{x}}.\frac{\mathrm{a}\mathrm{p}+\mathrm{b}\mathrm{p}}{\mathrm{x}}.\frac{\mathrm{a}\mathrm{p}+2\mathrm{b}\mathrm{p}}{\mathrm{x}}.\frac{\mathrm{a}\mathrm{p}+3\mathrm{b}\mathrm{p}}{\mathrm{x}}.\mathrm{\&c.}$ Also order the Equation according to the dimensions of y & multiply the terms of it by this Progression $\frac{\mathrm{a}\mathrm{q}-3\mathrm{b}\mathrm{q}}{\mathrm{y}}.\frac{\mathrm{a}\mathrm{q}-2\mathrm{b}\mathrm{q}}{\mathrm{y}}.\frac{\mathrm{a}\mathrm{q}-\mathrm{b}\mathrm{q}}{\mathrm{y}}.\frac{\mathrm{a}\mathrm{q}}{\mathrm{y}}.\frac{\mathrm{a}\mathrm{q}+\mathrm{b}\mathrm{q}}{\mathrm{y}}.\frac{\mathrm{a}\mathrm{q}+2\mathrm{b}\mathrm{q}}{\mathrm{y}}.\frac{\mathrm{a}\mathrm{q}+3\mathrm{b}\mathrm{q}}{\mathrm{y}}.\mathrm{\&c}$ Also order it according to the dimensions of z & multiply its terms by this Progression $\frac{\mathrm{a}\mathrm{r}-3\mathrm{b}\mathrm{r}}{\mathrm{z}}.\frac{\mathrm{a}\mathrm{r}-2\mathrm{b}\mathrm{r}}{\mathrm{z}}.\frac{\mathrm{a}\mathrm{r}-\mathrm{b}\mathrm{r}}{\mathrm{z}}.\frac{\mathrm{a}\mathrm{r}}{\mathrm{z}}.\frac{\mathrm{a}\mathrm{r}+\mathrm{b}\mathrm{r}}{\mathrm{z}}.\frac{\mathrm{a}\mathrm{r}+2\mathrm{b}\mathrm{r}}{\mathrm{z}}.\frac{\mathrm{a}\mathrm{r}+3\mathrm{b}\mathrm{r}}{\mathrm{z}}$. &c. The summ of these products shall bee equal to nothing. Which equation gives the relation of p, q, r, &c.

Examples.

Example 1. If the propounded Equation be ${\mathrm{x}}^3-2\mathrm{x}\mathrm{x}\mathrm{y}+4\mathrm{x}\mathrm{x}-7\mathrm{x}\mathrm{y}\mathrm{y}-{\mathrm{y}}^3-103=0$. By the precedent Rule the first operation will produce $3\mathrm{x}\mathrm{x}\mathrm{p}-4\mathrm{x}\mathrm{y}\mathrm{p}+8\mathrm{x}\mathrm{p}+7\mathrm{y}\mathrm{y}\mathrm{p}$. The second produceth $\mathrm{-2}\mathrm{x}\mathrm{x}\mathrm{q}+14\mathrm{x}\mathrm{y}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{q}$. And these two added together make $3\mathrm{x}\mathrm{x}\mathrm{p}-4\mathrm{x}\mathrm{y}\mathrm{p}+8\mathrm{x}\mathrm{p}+7\mathrm{y}\mathrm{y}\mathrm{p}-2\mathrm{x}\mathrm{x}\mathrm{q}+14\mathrm{x}\mathrm{y}\mathrm{q}-3\mathrm{y}\mathrm{y}\mathrm{q}=0$.

Example 2. If the Equation be ${\mathrm{x}}^3-2\mathrm{a}\mathrm{a}\mathrm{y}+\mathrm{z}\mathrm{z}\mathrm{x}-\mathrm{y}\mathrm{y}\mathrm{x}+\mathrm{z}\mathrm{y}\mathrm{y}-{\mathrm{z}}^3=0$ the first operation will produce $3\mathrm{p}\mathrm{x}\mathrm{x}+\mathrm{p}\mathrm{z}\mathrm{z}-\mathrm{p}\mathrm{y}\mathrm{y}$, the second $\mathrm{-2}\mathrm{a}\mathrm{a}\mathrm{q}-2\mathrm{y}\mathrm{x}\mathrm{q}+2\mathrm{z}\mathrm{y}\mathrm{q}$, the third $\mathrm{+2}\mathrm{z}\mathrm{x}\mathrm{r}-\mathrm{y}\mathrm{y}\mathrm{r}-3\mathrm{z}\mathrm{z}\mathrm{r}$. The summ of w^ch is $3\mathrm{p}\mathrm{x}\mathrm{x}+\mathrm{p}\mathrm{z}\mathrm{z}-\mathrm{p}\mathrm{y}\mathrm{y}-2\mathrm{a}\mathrm{a}\mathrm{q}-2\mathrm{y}\mathrm{x}\mathrm{q}+2\mathrm{z}\mathrm{y}\mathrm{q}+2\mathrm{z}\mathrm{x}\mathrm{r}-\mathrm{y}\mathrm{y}\mathrm{r}-3\mathrm{z}\mathrm{z}\mathrm{r}=0$.

Demonstration

If two bodies A & B move uniformly, the one successively \succesively/ from a to c, e, g &c the other in the same times from b to d, f, h &c then are the lines ac & bd, ce & df, eg & fh &c in the same proportion to one another as are the velocities p & q wherewith the bodies describe them <2r> And tho they move not uniformly yet are the infinitely little lines w^ch each moment they describe, as the velocities are which they have while they describe them. As if the body A with the velocity p describe the infinitely little line po in one moment, the body B with the velocity q will describe the line {illeg}qo in the same moment For p.q:::po. {illeg}qo. So that if the described lines be x & y in one moment they will be x+po & y+{illeg}qo in the next moment.

Now if the equation expressing the relation between the lines x & y be for instance $\mathrm{r}\mathrm{x}+\mathrm{x}\mathrm{x}-\mathrm{y}\mathrm{y}=0$