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

SCHOLIƲM.

HEnce, lastly, we have a new genesis of the hyperbola in Plano about its given diameters from the speculations of(β) 1.1De Witt; if, viz. having drawn the lines AB and EF cross one another at pleasure (Fig. 132) to the angle BCF you conform the move∣able angle BCD (acd being to be delineated in the opposite hyperbola equal to the contiguous ACD) one of whose legs is conceived to be indefinitely extended, but the o∣ther CD of any arbitrary length; and to the end of it D ap∣ply the slit of a moveable ruler GD about the point G at any arbitrary interval GD (but yet parallel to the leg CB in this first station) and so carrying together along with it the moveable angle BCD about the line ECF, but so that the leg CD may always remain fast to it, and the other CB be inter∣sected in its progress by the ruler GDH, e. g. in b or β This point of intersection, thus continually moved on, will describe the curve bGβ, which we thus prove to be an hyperbola: Be∣cause the ruler GDH turning about the pole G, and carried from D e. g. to d or δ cuts the leg of the moveable angle CB

brought to the situation cb or γβ, and in the mean while remain∣ing always parallel to it self; and having drawn from the points of intersection b and β and G the lines GI, bK and βη parallel to the ruler CF, because e. g. in the second station, having ta∣ken the common quantity cD from the equal ones CD and cd, the remainders Cc and Dd are equal, and by reason of the si∣militude of the ▵ ▵ dcb and dDG,

• as Dd
• i. e. Cc
• or bK

to DG,

• so dc
• i. e. DC
• or GI

to cb; the rectangle of Kb into bc will be = □ of DG into GI, by Prop. 18. lib. 1. and in like manner, when in the third station having added the common line Dγ to the equal ones CD and γδ, the whole lines Dδ and Cγ are equal, and, by reason of the similitude of the ▵ ▵ βγδ and GDδ

• as Dδ
• i. e. Cγ
• or βη

is to DG so

• is γδ
• i. e. DC
• or GI

to γβ; the □ of ηβ into βγ=□ of DG into GI, by the same 18. Prop. Wherefore the three points b, G, β, (and so all the o∣thers that may be determined the same way) are in the hyper∣bola, whose asymptotes are CB and CF and its centre C, &c. by the present Prop. Consect. 6. Q. E. D.

You may also determine innumerable points of this curve separately without the motion we have now prescrib'd, viz. as the point a in the opposite hyperbola, if thro' any assumed point c in the asymptote CE you draw a parallel to the other asymptote CA, and having made cd equal to CD, from G thro' d draw Gda, and so in others.

Notes

• (β) 1.1

  De Witt Elem. Curv. lib. I. cap. 2. prop. 3.