Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

2. In simple Quadratick Equations.

1.

• IF xx=ab
• or y{powerof2}=1c
• or z{powerof2}=¾dd

you'l have

• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉

that is to a mean propor∣tional between

• a and b
• 1 and c
• ¾d and d

and so the Construction will be had from n. 3. Schol. 2. Prop. 34. (see Fig. 3.)

• 2. If y{powerof2}=fg+kl
• or x{powerof2}=fg−kl

you'l have

• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉

make there∣fore on the one side the Right-angled Triangle ABC (Fig. 4.) whose side

AB is = to a mean Proportional between f and g,

BC is = to a mean Proportional between k and l;

On the other a Right-angled ▵ (Fig. 5.) whose side AB is = to a mean Proportional between f and g, and the side BC = to a mean Proportional between k and l; and on the one hand the Hypothenusa, on the other the side

• AC will be the va∣lue of y
• AC will be the va∣lue of x

sought.

And all by vertue of the Pythag. Theor. and according to Schol. 5. Prop. 34. or the Consectarys of Prop. 44. See Fig 4. and 5.

3. If z{powerof2}=〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉 extracting the Roots on both sides, z will =〈 math 〉〈 math 〉, and so be the second case of simple Equations.

4. If y{powerof2} be =〈 math 〉〈 math 〉, make first as l to f, so g to a fourth which call n, and by putting ln for fg, you'l have y{powerof2}=〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉. Make Secondly, as m to n, so h to a fourth, which call p; and by putting mp for nh, you'l have y{powerof2}=

〈 math 〉〈 math 〉 i. e. pk, and so the first case of the present Equations.

5. If 〈 math 〉〈 math 〉 in the first place the Rectangles fg and lm being turned into Squares, and collected into one Sum, make them =nn. Then (since cc and cd are multiplyed by qb+bd) in like manner qb and bd added make pp; and you'l have 〈 math 〉〈 math 〉. Thirdly (since pp is already multiplyed by cc+cd) having added cc+cd into one Sum that they may e. g. make rr; x{powerof2} will =〈 math 〉〈 math 〉, and so be the third case of the present Equations.