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

SHCOLIƲM II.

HEnce you have the common mechanical ways of describ∣ing the ellipsis and hyperbola about their given axes; and the ellipsis, if the Foci N, N, (Fig. 129. n. 1.) are gi∣ven, or found according to Consect. 3. Prop. 7. and having therein stuck or fixed two pins, put over them a thread NFn tyed both ends together precisely of the length you design the greater ax DE to be of, and having put your pencil or pen in that ▵-string draw it round, always keeping it equally extended or tight. Now because the parts or portions of the thread re∣main always equal to the whole ax DE, what we proposed is evident by the present Prop. which may also be very elegantly described by a certain sort of Compasses, a description where∣of Swenterus gives us in his Delic. Physico-Math. Part. 2. Prop. 20. which may be also done by a sort of organical Mechanism, by the help of two rulers moveable in the Foci GN and Hn (n 2.) and equal to the transverse ax DE, and fastned a∣bove by a transverse ruler GH equal to the distance of the Fo∣ci, as may appear from the Figure. For if the style F be moved round within the fissures of the cross rulers Hn and GN the curve thereby described will be an ellipsis from the pro∣perty we have just now demonstrated of it, which it hath in every point F. For the triangles HGN and NHn, which have one common side HN, and the others equal by construction, are equal one to another, and consequently the angles FHN and FNH equal, so also the legs HF and FN, and so likewise FN and Fn together are equal to Hn = DE; which is the very property of the ellipse we are now treating of. But Van Schoo∣ten, who taught us this delineation, hints, that, if thro' the middle point •• of the line HN you draw the line IFL, it will touch the ellipsis in the point F; for since the angles IFH and IFN are equal, by what we have just now said, the vertical angle LFn of the one IFH, will be necessarily equal to the o∣ther IFN: But this equality of the angles, made by the line KL drawn thro' F, with both those drawn from the centres, is here a sign of contact, as is in the circle the equality of the angles with a line drawn from its one centre. So that after this way you may draw a tangent thro' any given point F of

[illustration]

Pag. 192.

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the ellipsis without this organical apparatus of Rulers; viz. if, having drawn from both the Focus's thro' the given point F the right lines nH, NG equal to the Latus Transversum DE, you bisect HN in I and draw IFL: Or if the line that connects the extremes GH be produced to K, and you draw thence KFL, viz. in that case where GH and Nn are not parallel; other∣wise a line drawn thro' the point F parallel to them would be the tangent sought.

As to the hyperbola, there is a mechanick method of draw∣ing that also, not unlike the others, from a like property in that, communicated by the same Van Schooten, viz. If ha∣ving found the Focus's N and n (Fig. 129. n. 3.) you tye a thread NFO in the Focus N and at the end of the ruler nO of the length of the transverse ax DE; then putting in a pen or the moveable leg of a pair of compasses (nor would it be dif∣ficult to accommodate the practice we before made use of to this also) draw or move it within the thread NFO from O to E, so that the part of the thread NO may always keep close to the ruler as if it were glued to it. For if we call the length of the thread X, and the transverse ax ob as above, the ruler nO will be, by the Hypoth. = X + ob. Make now the part of the thread OF = ½ X, the remainder or other part will be NF = ½ X and nF = ½ X + ob, and the difference between FN and Fn, = ob. Make OF = ¾ X, then will FN be ½ X and Fn ¼ X + ob, the difference still remaining ob and so ad infinitum. In short, since the difference of the whole thread and of the whole ruler is ob, and in drawing them, the same OF is taken from both, there will always be the same difference of the re∣mainders. Hence also assuming at pleasure the points N and n you may describe hyperbola's so, the thread NFO be short∣er than the ruler nFO: For if it were equal there would be described a right line perpendicular to Nn, thro' the middle point C.

There yet remains one method of describing hyperbola's and ellipses in Plano, by finding the several points without the help or Apparatus of any threads or instruments, viz. in the ellipsis, having given or assumed the transverse axis DE and the Foci N and n (Fig. 130. n. 1.) if from N at any arbi∣trary distance, but not greater than half the transverse ax NF, you make an arch, and keeping the same opening of the com∣passes

you cut off, from the transverse ax, EG, and then, taking the remaining interval GD, from n you make another arch•• cutting the former in F, and so you will have one point of the ellipse, and after the same way you may have innume∣rable others, f, f, f, &c.

In like manner to delineate the hyperbola, having given o•• assumed the transverse ax DE and the Focus's N and n (n. 2.) if from N at any arbitrary distance NF you strike an arch, and keeping the same aperture of the compasses from the dia∣meter continued, you cut off EG, and then at the interval GD from n make another arch cutting the former in F, you will have one point of the hyperbola, and after the same way innumerable others, f, f, &c.