II. EQƲATION.
HAving thus given each quantity its Name, and making no further distinction between the quantities given and those sought, but treating them all promiscuously, and as already known, you must carefully search into and discuss all the Cir∣cumstances of the Question, and making various Comparisons of the quantities, by adding, substracting, multiplying, and dividing them, &c. 'till at length, which is the chief aim and design of it, you can express one and the same quantity two ways, which is that we call an Equation: And you must find as many of these Equations, or Equalities of literal quantities, (as expressing the same thing) as there are several unknown quantities in the Question, independent on each other, and consequently denominated by so many different Letters, z, y, x, &c. But if so many Equations cannot be found, after hav∣ing exhausted all the Circumstances of the Questions by one or two Equations; that is a sign the other unknown quanti∣ties may be assumed at pleasure: Which the Examples we shall hereafter bring will more fully shew.
But as here also (as likewise in all this Art) Ingenuity and Use do more than Rules and Precepts; yet we will here shew the principal Fountains, for the sake of young Beginners, whence Equations, according to circumstances obvious in the Questi∣on, are usually had. These are partly Axioms self evident, E. g.
That the whole is equal to all its parts taken together.
That those quantities which are equal to one third, are equal among themselves.
That the Products or Rectangles under the Parts or Segments, are equal to the Product of the whole.
Partly some universal Theorems that are certain and already demonstrated, as,