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

CHAP. I. Of the chief Properties of the Conick Sections.

Proposition I.

IN the Parabola (GKEH Fig. 114) the(α) 1.1 square of the semi-ordinate (IK) is equal to the rectangle IL made by the Latus Rectum EL and the abscissa EI.

Demonstration.

MAke the sides of the cone that is supposed to be cut, AB =a, BC=b, and moreover EB=oa, and EI=eb, and AC=c; therefore NI will be = ec, by reason of the si∣militude of the ▵ ▵ BCA and EIN; and EP or IO=oc, by reason of the similitude of the ▵ ▵ ABC and EBP. There∣fore ▭ NIO=oecc=□ IK by the Schol. of Prop. 34 (n. 3) and Prop 17. Lib. 1. Now if a line be sought which with the abscissa EI shall make the ▭ IL=□ IK you will

have it by dividing the said square by the Abscissa EI. viz. 〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉 = EL. And this is called the Latus Rectum, viz. in relation to the Abscissa EI with which it makes that rectangle, which, it's evident, is = □ IK, and from this equality the section has the name of Parabola, in Apollo∣nius.

CONSECTARYS.

I. THis Latus Rectum, expressed by the quantity 〈 math 〉〈 math 〉, may be found out after a shorter way, if you make as b to c (the side of the cone parallel to the section BC at the Diame∣ter of the base AC) so oc (the side EP called by some the Latus Primarium) to a fourth.

II. But if any one, with Apollonius, had rather express this by meer data in the cone it self as cut (because oc or that La∣tus Primarium EP is not a line belonging to the cone it self) he may easily perceive, if the quantity of the Latus Rectum found above, be multiplyed by the other side of the cone a, there will be produc'd the equivalent 〈 math 〉〈 math 〉 which instead of the proportion above will furnish us with this other,

• as ab− to cc− so oa
• □ of AB into BC−□ AC−EB

to a fourth; which is the very proportion of Apollonius in Prop. 9. Lib. 1. and confirms our former.

SCHOLIƲM I.

HEnce you have an easie and plain way of describing a Parabola, having the top of the ax and the Latus Re∣ctum given, viz. by drawing several semiordinates whose extreme points connected together will exhibit the curvity of the Parabola. But you may find as many semiordinates as you please, if having cut off as many parts of the Ax as you please, you find as many mean proportionals between the

Latus Rectum and each of those parts or Abscissa's. See n. 2. and 3. Fig. 47. Introduct. to Specious Analysis.

SCHOLIƲM II.

HEnce also we have a new genesis of the parabola in Plan•• from the spculations of De Witte, viz. if the rectiline∣ar angle HBG (Fig. 115.) conceived to be moveable about the fixed point B be conceived so to move out of its first situ∣ation with its other leg BH along the immoveable rule EF, that it may at the same time move also the ruler HG, from its first situation DK, all along parallel to it self, and with the other leg BG let it all along cut the said ruler HG, and with this point of its intersection continually moving from B to∣wards G it will describe a curve. That this curve will be the parabola of the antients is hence manifest, because it will have this same first property of the parabola. For, 1. if the an∣gle HBG (n. 1.) be supposed to be a right one, and BD or HI=a, BI or KG=b (viz. in that station of the angle and rule HG by which they denote the point G in the inter∣section) you'l have by reason of the right angle at B, BI, i. e. b a mean proportional Between HI i. e. a, and IG or BK, and so this as an abscissa = 〈 math 〉〈 math 〉. Wherefore if BK i. e. 〈 math 〉〈 math 〉 be multiplyed by BD=a, the rectangle DBK will be = bb =□KG; which is the first property of the parabola: So that it follows, since the same inference may be made of any other point in this curve, that this curve will be the parabo∣la, BD or HI its Latus Rectum, KG a semiordinate, and BK its axis, &c. 2. If the angle HBG be an oblique one (num. 2) it may be easily shewn from what we have supposed that the ▵ ▵ DBH and BKG will be equiangular: Therefore as BD (i. e. a) to DH sc. BI (i. e. b) so KG sc BI (i. e. b) to BK (i. e. 〈 math 〉〈 math 〉) Therefore again the □ DBK=bb□KG. QED.

Consect. 3. It is also evident in this second case, that BK drawn parallel to the ax, but not thro' the middle of the pa∣rabola,

will be a diameter which will have for its vertex B, its Latus Rectum BD, and semiordinate GK, &c.

Consect. 4. Therefore you may find the Latus Rectum in a given parabola geometrically, if you draw any semiordinate whatsoever IK (Fig. 116.) and make the abscissa EF equal to it, and from F draw a parallel to the semiordinate IK, and from E draw the right line EK thro' K cutting off FH the Latus Rectum sought; since as EI to IK so is EF (i. e. IK) to FH by Prop 34. lib. 1. wherefore having the ab∣scissa and semiordinate given arithmetically, the Latus Re∣ctum will be a third proportional.

Consect. 5. Since therefore the Latus Rectum found above is 〈 math 〉〈 math 〉, if you conceive it to be applyed to the parabola in LM, so that N shall be that point which is called the Focus, LN will be 〈 math 〉〈 math 〉 and its square 〈 math 〉〈 math 〉 and this divided by the Latus Rectum 〈 math 〉〈 math 〉 will give occ〈 math 〉〈 math 〉 for the abscissa EN: So that the distance of the Focus from the Vertex will be ¼ of the Latus Re∣ctum.

Consect. 6. Since therefore EN is = 〈 math 〉〈 math 〉 if for EF you put ib, NF will be = 〈 math 〉〈 math 〉, whose square will be found to be 〈 math 〉〈 math 〉, to which if there be added □ GF=oicc, by Prop. 1. the square of NG will be = 〈 math 〉〈 math 〉 whose root (as the extraction of it and, without that, the ana∣logy of the square NF with the square NG manifestly shews) will be 〈 math 〉〈 math 〉; so that a right line drawn from the Focus to the end of the ordinate, will always be equal to the abscissa EF

+EN i. e. (if EO be made equal to EN) to the compounded line FO.

SCHOLIƲM III.

HEnce you have an easier way of describing the parabola in Plano from the given Focus and Vertex, viz. (Fig: 117.) the axis being prolonged thro' the vertex E to O, so that EO shall = EN, if a ruler HI be so moved by the hand G, according to PQ, from OF to HI, that putting in a style or pin, it shall always keep the part of the Thred NGI (which must be of the same length with the rule HI) as fast as if it were glued to it (which perhaps might also be done with the Compasses by an artifice which we will hereafter al∣so accommodate to the hyperbola) and at the same time it will describe in Plano the part of the line EGR. That this will be a parabola is evident from the foregoing Consect. because as the whole thred is always = to the ruler IH; so the part GN is always necessarily equal to the part GH, i. e. to the line FO. Moreover from the same sixth Consect. and Fig. 116. may be drawn another easie way of describing the para∣bola in Plano from the Focus and Vertex given thro' innume∣rable points G to be found after the same way: viz. If from any assumed point in the ax F you draw to the ax a perpendi∣cular, and at the interval FO from the Focus N you make an intersection in G. Which innumerable points G will be de∣termined with the same facility, having given only, or assu∣med the axis and Latus Rectum, by vertue of the present Proposition. For if, having assumed at pleasure the point F in the axis, you find a mean proportional between the Latus Rectum and the abscissa EF, a semiordinate FG made e∣qual to it, will denote or mark the point G in the parabola sought.

Proposition II.

IN the hyperbola (GKEH Fig 118)(α) 1.2 the square of the semiordinate (IK) is equal to the rectangle (IL) made of the Latus Rectum (EL) and the abscissa (EI) together with ano∣ther rectangle LS of the said abscissa (EI or LR) and RS a fourth proportional to DE the Latus Transver∣sum, (EL) the Latus Rectum, and EI the Abscissa.

Demonstration.

Suppose the side of the cone AB here also = a, and BM parallel to the section = b, and the intercepted line AM=c, and EI=eb; all according to the analogy we have observed in the parabola; and NI will be as there = ec. Making moreover MC=d and the Latus Transversum DE=ob, so that DI shall be = ob+eb; then will (by reason of the simi∣litude of the ▵ ▵ BMC, DEP, and DIO) EP be = od, and IO=od+ed, and so QO=ed. Therefore □ NIO will be = oecd+eecd=□IK But this square divided by the Abscissa EI=eb gives 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉 for the line IS which with the abscissa would make the rectangle ES = to the said square ••K. Now therefore, if here also we call a Line the Latus Rectum found after the same way as in the parabola, viz by making as b− to c− so od to a fourth 〈 math 〉〈 math 〉 (as a line parallel to the section — to the intercepted diam. so the Latus Primarium, but that the other part 〈 math 〉〈 math 〉 will be a fourth proportional to bc and ed or to eb, ec▪ and ed, or (to speak with Apollonius as we have done in the Prop.) to ob, 〈 math 〉〈 math 〉 and eb (for in these three cases you I have the same fourth 〈 math 〉〈 math 〉) Wherefore now it is evident that the square of the se∣miordinate

oecd+eecd is equal to the rectangle IL (made by the Latus Rectum 〈 math 〉〈 math 〉 into the abscissa eb=oecd) together with ▭ LS of this fourth proportional 〈 math 〉〈 math 〉 into the same ab∣scissa Eb, which is = eecd. Which was to be found and de∣monstrated.

CONSECTARYS.

I. HEnce you have in the first place the reason why Apol∣lonius called this Section an Hyperbola; viz. because the square of the ordinate IK exceeds or is greater than the rectangle of the Latus Rectum and the Abscissa.

II. Since therefore the Latus Rectum here also as well as in the parabola is found by making as b to c so od to 〈 math 〉〈 math 〉 (i. e. as the parallel to the section BM is to the intercepted Diam. AM so is the Latus Primarium EP to a fourth EL.) If any one had rather express this Latus Rectum after Apollonius's way, he will easily perceive, this quantity being found and multi∣plyed both Numerator and Denominator by b the parallel to the section, there will come out the equivalent quantity 〈 math 〉〈 math 〉 which gives us instead of the former proportion this other,

• as bb− to cd− so ob to a fourth;
• □BM−▭AMC—Latus Transversum to a fourth;

which is that of Apollonius in Prop. 12. Lib. 1. and consequent∣ly herein confirms our former.

III. You may also have this Latus Rectum geometrically, by finding a third proportional (as we have done in the pa∣rabola Consect. 4. Prop. 1.) to the abscissa EI (Fig. 119.) and the semiordinate IK (= EF;) and then find a fourth pro∣portional EL to DI (the sum of the Latus Transversum and abscissa) and FH already found, or IS equal to it, and DE (the Latus Transversum) and that will be the Latus Rectum sought.

[illustration]

Pag. 172.

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126

126

127

128

SHCOLIƲM.

FRom this third Consectary, we may reciprocally from the Latus Rectum and transverse given, find out and apply as many semiordinates to the ax as you please, and so describe the hyperbola thro' their (ends or) infinite points: viz. if assuming any part of the abscissa EI, you make as DE to EL so DI to IS; and then find a mean proportional IK between IS and the abscissa EI, and that will be the semiordinate sought: And both this praxis and the Consect. may be abun∣dantly proved by setting it down in, and making use of, the literal Calculus.

Proposition III.

IN the Ellipsis (KDEK, Fig. 120.) the(α) 1.3 square of the semiordinate (IK) is equal to the rectangle (IL) of the Latus Rectum (EL) and the abscissa (EI) (less or) taking first out another rectangle (LS) of the same abscissa (EI or LR) and RS a fourth proportional to (DE) the Latus Transversum (EL) the Latus Rectum and (EI) the abscissa.

Demonstration.

Suppose the side of the cone to be AB here also = a and BM parallel to the section = b and the intercepted AM=c, and EI=eb; and NI will be again = ec, all as in the hyperbola. And mak••ng also here as in the hyper∣bola MC=d, and the Latus Transversum DE=ob, so that DI will be ob−eb; then will (by reason of the simili∣tude of the ▵ ▵ BMC, DEP and DIO) EP be=od, and IO=od−ed. Therefore ▭ of NIO will be = oecd−eecd=□IK. But this square divided by the abscissa EI=eb gives 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉 for that line IS which with the abscissa would make the rectangle ES= to the said square IK. Now therefore if we call the Latus Rectum a right line found after the same way as in the parabola, by making ac∣cording to Cons. 1. Prop. 1.

as b to c− so od− to a fourth 〈 math 〉〈 math 〉 i. e. as the line pa∣rallel to the section — to the intercepted diameter — so the Latus Primarium, &c. It is manifest that the Latus Re∣ctum is one part of the line just now found; and the other part 〈 math 〉〈 math 〉 is a fourth proportional to b, c and ed, or (to speak with Apollonius as we have done in the Prop.) to 〈 math 〉〈 math 〉 and eb (for there will come out the same quantity 〈 math 〉〈 math 〉) where∣fore now it is evident that the □ of the semiordinate IK is e∣qual to the ▭ IL (of the Latus Rectum 〈 math 〉〈 math 〉 and the ab∣scissa eb=oecd) having first taken out thence the ▭ LS, or eecd out of that fourth proportional 〈 math 〉〈 math 〉 by the same abscissa eb; which was to be found and demonstrated.

CONSECTARYS.

I. HEnce yov have first of all the reason of the name of the Ellipse, which Apollonius gave to this section; viz. because the square of the semiordinate IK is defective of, or less than the rectangle of the Latus Rectum and the abscis∣sa.

II. Since therefore the Latus Rectum here also as well as in the parabola and hyperbola, is found by making as b to c so od to 〈 math 〉〈 math 〉 (i. e. as BM parallel to the section is to the inter∣cept. diam. AM so the Latus Primarium EP to a fourth EL) now if any one had rather express this Latus Rectum after A∣pollonius's way, he will easily see that the quantity above found being multiplyed both Numerator and Denominator by b, that there will come out an equivalent one 〈 math 〉〈 math 〉, which instead of the former proportion will give this other,

as

• bb− to cd− so ob to a fourth;
• BM−▭AMC—Latus Transvers. to a fourth;

which is the same with that we have also found in the hyper∣bola, and which also Apollonius has Prop. 13. Lib. 1.

III. This Latus Rectum may also be had geometrically, if you find, 1. in the hyperbola a third proportional FH to the abscissa EI (Fig. 121.) and semiordinate IK (= EF.) 2. But EL a fourth proportional to DI (the difference of the Latus Transversum and the abscissa) and the found FH, or IS equal to it, and the Latus Transversum DE, is the Latus Rectum sought.

SCHOLIƲM.

FRom this third Consect. we may reciprocally, having the Latus Rectum and Transversum given, apply as many semiordinates to the ax as you please, and so draw the ellipsis thro' as many points given as you please, viz. if, taking any ab∣scissa EI, you make as DE to EL so DI to a fourth IS; then between this IS and the abscissa EI find a mean proportional IK, and that will be the semiordinate sought: And this Prax∣is also and the third Consect. may be abundantly proved by making use of a literal Calculus. For e. g. here a fourth proportional to ob, 〈 math 〉〈 math 〉 and ob−eb will be 〈 math 〉〈 math 〉 and a mean proportional between this fourth and eb will be 〈 math 〉〈 math 〉, &c.

Proposition IV.

IN a Parabola(α) 1.4 the squares of the ordinates are to one another as the ab∣scissa's.

Demonstration.

For if EF (Fig. 122.) be called ib, as above EI was cal∣led eb, since the Latus Rectum is 〈 math 〉〈 math 〉 the square of FG wil be = oicc. Therefore □ IK will be to □ FG as oecc—oicc

• e to i or
• eb to ib.

Q. E. D.

CONSECTARY.

HEence having drawn LO parallel to the ax or diameter EF, if it be cut by the transverse line EG in M and by the curve of the parabola in K; then will OL, ML, and KL be continual proportionals. For EF is to EN as FG to NM or IK, by reason of the similitude of the ▵ ▵ EFG and ENM. But the squares FG and IK are in duplicate proportion of EF to EN by Prop. 35. Lib. 1. and are also in the same propor∣tion as the abscissa's EF and EI by the pres. Therefore EF to EI is also in duplicate proportion of EF to EN i. e.

• EF is to EN as EN to EI Q. E. D.
• OL is to ML as ML to KL Q. E. D.

Proposition V.

IN the hyperbola and Ellipsis(α) 1.5 the squares of the Ordinates are as the rectangles contained under the lines which are intercepted between them, and the Vertex's of the Latus Transversum's.

Demonstration.

For, if EF (Fig. 118. and 120.) be called ib, as EI was above called eb, then will according to Prop. 2. and the 3d. dedu∣ction.

GF=oicd+eicd in the Hyperb.

oicd−eicd in the Ellips.

and the ▭ DFE=oibb+iibb in the Hyperb.

oibb−iibb in the Ellips.

Therefore the □ KI is to the square GF as oecd±eecd to oicd ± iicd i. e. as oe±ee to oi±ii.

and ▭ DIE is to the ▭ DFE, as oebb±eebb to oibb±iibb i. e. in like manner as

oe±ee to oi±ii. Q. E. D.

CONSECTARY I.

IN the Ellipsis this may be more commodiously expressed apart thus; the squares of the ordinates (KI and GF) are as the rectangles contained under the segments of the Diame∣ter (viz. DIE and DFE) in which sense this property is also common to the circle, as in which the squares of the ordinates are always equal to the rectangles of the segments.

CONSECTARY II.

THerefore, if the Latus Rectum be conceived to be apply∣ed in the hyperbola, so that N shall be the Focus; (see Fig. 123.) then will LN=〈 math 〉〈 math 〉, and its square be 〈 math 〉〈 math 〉. But as the □ KI to the square LN, so is the ▭ DIE to the ▭ DNE i. e. oecd+eecd to 〈 math 〉〈 math 〉 so is oebb+eebb to 〈 math 〉〈 math 〉. But now the ▭ of the whole DE and the part added EN into the part added EN, i. e. ▭ DNE (=〈 math 〉〈 math 〉 together with the square of half CE (=〈 math 〉〈 math 〉) is = □ compounded of half and the part added CN= 〈 math 〉〈 math 〉 by Prop. 9. lib. 1. Wherefore CN the distance of the Focus from the centre is = 〈 math 〉〈 math 〉. But 〈 math 〉〈 math 〉 is the fourth part of the ▭ of the Latus Transversum ob and the Latus Rectum 〈 math 〉〈 math 〉 (or the

fourth part of the figure, as Apollonius calls it) and 〈 math 〉〈 math 〉 is the □ of 〈 math 〉〈 math 〉 i. e. of half the Latus Transversum. Wherefore we have found the following Rule of determining the Focus in an hyperbola: If a fourth part of the figure (or the rectangle of the Latus Rectum into the Transversum) be added to the square of half the Latus Transversum, and from the sum you extract the square root; that will be the distance of the Focus from the center CN: And hence substracting half the Latus Transversum CE, you will have distance of the Focus from the Vertex EN.

CONSECTARY III.

IN like manner in the Ellipsis having drawn the ordinate LM (Fig. 124.) that the Focus may be in N, the □ LN would be 〈 math 〉〈 math 〉 as above, and by a like inference □ DNE=〈 math 〉〈 math 〉. But now □ DNE together with the square of the dif∣ference CN is equal to the □ of half CE by Propos. 8. lib. 1. and consequently the □ CN is = □ CE−□DNE, that is, 〈 math 〉〈 math 〉. Wherefore CN the distance of the Focus from the centre is = 〈 math 〉〈 math 〉. Wherefore we have found the following Rule to determine the Focus in the El∣lipse. If the fourth part of the figure (or the rectangle of the Latus Rectum into the Latus Transversum) be substracted from the square of half the Latus Transversum, and from the remain∣der you substract the square root; that will be the distance of the Focus from the Centre CN: And taking hence half the La∣tus Transversum CE, you'l have the distance of the Focus from the Vertex EN.

SCHOLIƲM I.

BOth the Rules are easie in the practice, for since 〈 math 〉〈 math 〉 is nothing but the square of CE, and 〈 math 〉〈 math 〉 nothing but the re∣ctangle of ¼ DE into LM; if between LM and ¼ DE or MO (Fig. 125.) you find a mean proportional MN, (and so whose □ is equal to that □) and in the hyperbola join to it at right angles MC=CE, the hypothenusa CN will be the distance sought of the Focus from the centre: And the same may be had in the Ellipsis, if (n. 2.) having described a semi-circle upon CM=CE you draw or apply the mean found MN, and draw CN.

SCHOLIƲM II.

HEnce also we have(α) 1.6 a new genesis of the Ellipse in Plano about the diameters given, from the speculations of Monsieur de Witt; viz. If about the rectilinear angle DCB (Fig. 126. n. 1. and 2.) consider'd as immove∣able, the rule NLK (which all of it will equal the greater semidiameter CB, and with the prominent part LK the lesser CD) be so moved that N going from C to D, and L from B to C may perpetually glide along the sides of the angle, the extreme point in the K in the mean while describing the curve BKE (and in a like application the other quadrants) and that this curve thus described will be the ellipsis of the ancients is hence manifest, because it has the second property of the ellipse just now described. For, 1. if the angle DCB or NCB be supposed to be a right one (as in Fig. 126. num. 1.) and the rule KN in the same station, it marked out the point K, and having apply'd the semiordinate KI, and drawn the perpendicular LM, from the square KL and the square CE (as being equal) substract mentally the equal squares LM and CI, and there will remain by virtue of the Pythagorick Theor. on the one hand □ KM and on the other by

Prop. 8. lib. 1. □ DIE equal among themselves. But now the square of KI is to the square of KM (i e. to the □ DIE) as the square of KN to the square of KL (i. e. as the square of CB to the square of CE) by reason of the similitude of the ▵ ▵ KLM and KNI; and since the same may be demonstra∣ted after the same manner of any other semiordinate Ki viz. that its □ Ki is to the □ DiE as the square CB to the square CE. It also follows, that the □ KI is to the □ DIE as □ Ki to the □ DiE, and alternatively, the square KI will be to the square Ki as the □ DIE to the □ DiE; which is the se∣cond property of the ellipse. 2. If the angle NCE be not a right one (as in Fig. 126. n. 2. and the like cases) having drawn NO and KP parallel to the rule nlB in the first station, [in which station the angle NCE, to which the flexible ruler is to be made, is determined, viz. by letting fall the perpendicular Bl from the extremity of one diameter upon the other, and moreover by adding or substracting the difference of the semi-diameters ln] having also drawn the Ordinate KIM, and PI parallel to CN; which being done the ▵ ▵ CBl and IKF, and also CBn and IKP will be similar. Where∣fore having joined NP, from the parallelism of the lines IP and NC and the similitude of the aforesoid ▵ ▵, as also of NCO and nCl, it will be easie to conclude that NCIP is a parallelogram. Wherefore since KN is = CE and □ KN = □ CE, having substracted the squares of the equal lines NP and CI, there will remain on the one hand □ KP on the other the □ DIE equal among themselves as above. Therefore the square of KI will be to the square of KP (i. e. to the □ DIE) as the square of BC to the square of Bn (i. e. to the square of CE) as in the former case: And since here also the same may be demonstrated after the same manner of any o∣ther semiordinate Ki; we may infer as above, that the □ □ KI and Ki are to one another as the rectangles DIE and DiE, &c.

But after what way the same ellipses may be described by these right lined angles without any of these rulers thro' infi∣nite points given, will be be manifest from the same figures to any attentive Person. For having once determined the angle NCE or nCD (num. 2. e. g.) if NL or nl be applyed where you please by help of a pair of compasses, and continued to

K, so that LK or lk shall be equal to lB, you will have eve∣ry where the point K, &c.

CONSECTARY IV.

SInce in the hyperbola (Fig. 123.) the □CN−□CE=□DNE, and in the ellipsis (Fig. 124.) □CE−□CN=□DNE, by vertue of Prop. 9 and 8. lib. 1 if for CN you put on both sides for brevity's sake m, then will the □ DNE in the hyperbola be rightly expressed in these terms 〈 math 〉〈 math 〉, and in the ellipsis in these 〈 math 〉〈 math 〉.

Proposition VI.

IN the parabola(α) 1.7 the Latus Rectum is to the sum of two semiordinates (e. g. IK+FG i. e. HO in Fig. 122.) as their difference (OG) to the difference of the abscissa's (IF or (KO.)

Demonstration.

For if the greater abscissa EF be made = ib, and the less EI=eb, the semiordinates answering to them FG and IK will be √ooic and √oecc as is deduc'd from Prop. 1. Where∣fore if you set in the same series

1	2	3
The Latus R.	−sum of the semiord.	−their diff.
〈 math 〉〈 math 〉	−√oicc+√oecc	−√oicc−√oecc
	4
	−diff. of the absciss.
	−ib−eb

and multiply the extremes and means, you'l have on both sides the same product oicc−oecc, which will prove by vertue of Prop. 19. lib. 1. the proportionality of the said quantities. Q. E. D.

SCHOLIƲM.

THis is that property of the parabola, whereon the Clavis Geometrica Catholica of Mr. Thomas Baker is founded, which as unknown to the ancients, nor yet taken notice of by Des Cartes, he thinks was the reason why that incomparable Wit could not hit upon those universal rules for solving all Equations howsoever affected. Concerning which we shall speak further in its place. We will only further here note, that Baker was not the first Inventor of this property, but had it, as he himself ingeniously confesses, out of a Manuscript communicated to him by Tho. Strode of Maperton, Esquire.

Proposition VII.

IN the hyperbola and ellipsis(α) 1.8 the Latus Rectum is to the Latus Transversum, 〈◊〉〈◊〉 the square of any semiordinate (e. g. IK in Fig. 118▪ and 120.) to the rectangle (DIE) con∣tained under the lines intercepted between it and the Vertex's of the Latus Transversum.

Demonstration.

For the Latus Rectum is on both sides 〈 math 〉〈 math 〉, the Latus Trans∣versum ob, &c. Wherefore if you make in the same series as the Latus R. to the Lat. Transv. so the □ IK to ▭ DIE

• 〈 math 〉〈 math 〉 in hyperb. 〈 math 〉〈 math 〉
• in ellips. 〈 math 〉〈 math 〉

The rectangles of the extremes and means will both be ooebcd±oeebcd, and so will prove the proportionality of the said quantities, by Prop. 19. lib. 1. Q. E. D.

CONSECTARY I.

HEnce having given in the ellipsis (see Fig. 124) the La∣tus Rectum and the transverse ax, you may easily obtain the second ax or diameter, if you make as the Lat. Transv. to the Lat. Rect. so the □ DCE to □AC 〈 math 〉〈 math 〉.

CONSECTARY II.

THerefore the □ of the whole AB will be = oocd=□ of the Latus Rectum into the Lat. Transv. (which Apllo∣nius calls the Figure) so that the second Ax (and any second Diameter) will be a mean proportional between the Latus Re∣ctum and the Latus Transversum. Hence in the hyperbola also the second or conjugate diameter may be called a mean proportional between the Latus Rectum and Transversum, i. e. √oocd or a line which is equal in power to the Figure, as A∣pollonius speaks.

SCHOLIƲM I.

HEnce may be derived another and more simple way of delineating organically the ellipsis in Plano about the given axes AB, DE (Fig 127.) which Schooten has given us▪ viz. by the help of two equal rulers CG and GK move∣able about the points G and G: If, viz. the portions CF and HK are equal to half the lesser ax AC, but taken with both the augments (viz. CF+FG+GH) may = ½ the greater ax CD or CI; and the point K moving along the produced line DE the point H may describe the curve EHAD. That this will be an ellipsis will be evident by vertue of this seventh Prop. from a property that agrees to this curve in all its points H. For having drawn circles about each diameter, and the lines IHN, FO perpendicular to CE; and having made the Latus Rectum EL, which is a third proportional to DE and AB by the second Consect. of this Prop. &c. by reason of the similitude of the triangles CFO, CIN, FO will be to FC as

IN to IC, and alternatively FO to IN as FC to IC i. e. as AC to CE or AB to DE. Therefore also the square of FO (or HN) will be to the square of IN, as the square of AB to the square of DE, by Prop. 22. lib. 1. i. e. as EL the Latus Rectum to ED the Latus Transversum, by Prop. 35. lib. 1. But the □ IN is = DNE from the proporty of the circle. Therefore □ FO (or of the semiordinate HN) is to the ▭ DNE as EL the Latus Rectum to ED the Latus Transv. therefore by ver∣tue of the pres. Prop. the point H is in the Ellipsis, and so any other, &c. Q. E. D.

CONSECTARY III.

NOW if in the ellipsis the □ of AC the second Ax (= 〈 math 〉〈 math 〉 by Consect. 1.) and □ CN the distance of the Fo∣cus from the centre (= 〈 math 〉〈 math 〉 by Censect 3. Prop. 5. the figure whereof you may see n. 124.) be joined in one sum; the □ AN will be = 〈 math 〉〈 math 〉, and so the line AN=〈 math 〉〈 math 〉 i. e. to half the Latus Transversum: So that hence having the axes given you may find the Foci, if from A at the interval CD you cut the transverse ax in N and N.

CONSECTARY IV.

NOW if, on the contrary, in an hyperbola (Fig. 123.) the □ AC or EF=〈 math 〉〈 math 〉 be substracted from the □ CF or CN=〈 math 〉〈 math 〉 by vertue of Consect. 2. Prop. 5. there will remain 〈 math 〉〈 math 〉 and its root 〈 math 〉〈 math 〉, i. e. half the Latus Trans∣versum CD: So that here also, the axes being given, you may find the Focus's, if from the vertex E you make EF a per∣pendicular to the ax = to the second Ax AC, and at the in∣terval CF from the centre C you cut the Latus Transversum continued in N and N.

SCHOLIƲM II.

BUT now, that the right lines KN and KN drawn from any other point (e. g. K) to the Foci, when taken toge∣ther in the ellipsis, but when substracted the one from the o∣ther in the hyperbola, are equal to the Latus Transversum DE, we will a little after demonstrate more universally, and also shew an easie and plain Praxis of delineating the ellipsis and hyperbola in Plano, having the axes and consequently the Fo∣ci given.

CONSECTARY V.

SInce we have before demonstrated Consect. 2. and 3. Prop. 5. that the □ DNE in the hyperbola and also in the el∣lipsis is = 〈 math 〉〈 math 〉; and here in Consect. 1. the □ of the second semi-diameter AC is also = 〈 math 〉〈 math 〉; it is evident that this □ AC is equal to the □ DNE.

CONSECTARY VI.

IT is hence moreover evident, if the square of half the trans∣verse diameter GE=〈 math 〉〈 math 〉 be compared with the square of half the second diameter AC or EF=〈 math 〉〈 math 〉, multiplying both sides by 4 and dividing by o; they will be to one another as obb to ocd i. e. further dividing both sides by b, as ob to 〈 math 〉〈 math 〉 the Latus Transversum to the Latus Rectum.

CONSECTARY VII.

BUT since also the □ DIE is to the □ IK as the Latus Transversum to the Latus Rectum, by vertue of the pre∣sent 7. Prop. the square of CE the transverse semidiam. will be

to the square of AC the second semidiam. (or by the 5th. Con∣sect. of this, to the □ DNE) as the □ DIE to the square of IK.

CONSECTARY VIII.

YOU may also now have the □ IK (which otherwise in the hyperb. is eocd+eecd, in the ellipse oecd−eecd, by vertue of Prop. 2. and 3.) in other terms, if you make as □ CE to the □ DNE so the □ DIE to a fourth; i. e.

as 〈 math 〉〈 math 〉 to 〈 math 〉〈 math 〉 by vertue of Consect. 4. Prop. 5.

(so oebb+eebb in the hyperb.

as 〈 math 〉〈 math 〉 to 〈 math 〉〈 math 〉 so oebb−eebb in the ellipsis.

For hence by the Golden Rule the square IK may be infer'd as a fourth proportional.

In the hyperbola 〈 math 〉〈 math 〉;

In the ellipsis 〈 math 〉〈 math 〉:

The use of which quantities will presently appear.

Proposition VIII.

THE

• Aggregate in the ellipse
• Difference in the hyperb.

of the right lines(α) 1.9 KN and Kn (Fig. 128.) drawn from the same point K to both the Focus's is equal to the transverse ax DE.

An Ocular Demonstration.

WHich consists wholly in this to find the lines KN and Kn by help of the right-angled triangles IKN and IKn (sc. the hypothenuses having the sides given) and afterwards see if the sum of both in the ellipse, and difference in the hyperbola be = ob, i. e. to the transverse ax DE.

I. In the Ellipsis.

Putting for CN (which above Prop. 5. Cons. 3. was found to be 〈 math 〉〈 math 〉) I say putting for it m, you'l have

IN=CI+CN=½ob−eb+m

In=Cn−CI=m−½ob+eb

Therefore □IN= 〈 math 〉〈 math 〉 □In= 〈 math 〉〈 math 〉

Add to each □ IK which was found in Prop. 7. Consect. 8. in the ellipsis = 〈 math 〉〈 math 〉 and you'l have □KN= 〈 math 〉〈 math 〉 and by extracting the roots (which is easie) you'l have

〈 math 〉〈 math 〉 and

〈 math 〉〈 math 〉;

Sum ob. Q. E. D.

II. In the Hyperbola.

Putting again m for CN (which above Cons. 2. Prop. 5. was found to be 〈 math 〉〈 math 〉) and you'l have

IN =CI+CN=½ob+eb+m

In=CI−Cn=½ob+eb−m. Therefore □IN= 〈 math 〉〈 math 〉 and □In= 〈 math 〉〈 math 〉

Add to both the □ IK which was found in Prop. 7. Consect. 8. in the hyperbola 〈 math 〉〈 math 〉 and you'l have □KN= 〈 math 〉〈 math 〉

□Kn= 〈 math 〉〈 math 〉 and extracting the roots out of these (which is easie) you'l have KN= 〈 math 〉〈 math 〉.

Kn= 〈 math 〉〈 math 〉 (which is a false or impossible root, for it would be CE−CN and moreover — another quantity.

Or Kn= 〈 math 〉〈 math 〉; which is a true and possible root.

The difference therefore of the true roots is = ob. Q. E. D.

SCHOLIƲM I.

WE first of all tried to make a literal Demonstration by using the quantity of the square IK as you have it expressed Prop. 2. and 3. and the quantity IN as it was com∣pounded of CI=ob+eb+CN= 〈 math 〉〈 math 〉, &c. but we found it very tedious in making only the squares of IN and In. Then for the surd quantity CN we substituted another, viz. m, and we produced the squares of IN and In as above, but we added the square of IK in its first value; and thus we obtain'd the squares KN and Kn, but in such terms, that the exact roots could not be extracted, but must be exhibited as surd quantities, and consequently we must make use of the rules belonging to them to find their sum or difference, which we laid down Cons. 3. Prop. 7. and Consect. Prop. 10. Lib. 1. which tho' it would succeed, yet would be full of trouble and tediousness. Therefore at length when we came to use those other terms which express the square IK, the business succeed∣ed as easie as we could wish, and that in a plain and easie way and no less pleasant, which I doubt not but will also be the opinion of the Reader, who shall compare this with other demonstrations of the same thing, which only lead indirectly to this truth, or with them, which de Witte has given us in E∣lem▪ Curvar. lin. p. m. 293. and 302. and which he thinks

easie and short enough in respect of others both of the ancients and moderns, and which we have reduced into this yet more distinct form, and accommodated to our schemes.

Preparation for the Hyperbola.

Make as

• CD to CN
• EE−CN

so CI to CM so that the □ of

• DCM
• MCE

will be = □

• NCI
• nCI.

Because therefore it will be by Consect. 7. Prop. 7. as □ CD to □ DNE, so the □ DIE to the □ IK.

And also by composition, as □ CD to

• □CD+□DNE
• i. e. □ CN per 9. lib I.

so DIE to DIE + □ IK.

Therefore by a Syllepsis, as □ CD to the □ CN so

• □ CD + DIE
• i. e. □ CI

to □ CN + DIE + □ IK.

But also by the Hypothesis. as the □ CD to the □ CN so □ CI to □ CM. Therefore □ CM is = □ CN + DIE + □ IK.

Demonstration.

Since therefore it is certain that the difference between DM and EM is the transverse ax DE; if it be demonstrated that DM is = KN and EM = Kn, the business will be done, be∣cause the difference between KN and Kn is also the transverse ax DE.

Resolve the □ KN.

It is certain that NIq + IKq=KNq.

Substitute for NIq, by the 7. Lib. 1. CIq+CNq+2NCI.

Preparation for the Ellipsis.

Make as CD to CN so CI to CM.

So that the □ DCM is = □ NCI.

Because therefore by Consect. 7. Prop. 7. as □ CD to □ DNE so □ DIE to the □ IK;

Then also by dividing, as □ CD to

• □ CD−DNE
• i. e. □ CN, by 8. l. 1.

so DIE to DIE−□IK.

Therefore by a Dialepsis, as □ CD to □ CN, so

• □CD−□DIE
• i. e. CI □ by 8. cit.

to □ CN−DIE+□IK.

But also by the Hypothesis, as □ CD to □ CN, so □ CI to □ CM: Therefore □ CM is = □ CN−DIE + □ IK.

Demonstration.

Since therefore it is certain that the sum of DM and EM is the transverse ax DE; if it be demonstrated that DM is = KN and EM = Kn, the business will be done, because the sum of KN and Kn is also equal to the transverse ax DE.

Resolve the □ KN.

It is certain that NIq+••Kq=KNq.

Substitute for NIq, by the 7. lib. 1. CIq+CNq+2NC••

Then will CIq+CNq+2NC••+IKq=KNq.

Substitute for C••q, by the 9. lib. 1. CDq+DIE; then will CDq+DIE+CNq+2NC••+Kq=KNq.

Resolve also □ DM.

It is certain that CMq+CDq+

• 2DCM
• 2NCI

= DM, by the 7. lib. 1.

Substitute for CMq its value by the Preparation, and you'l have CNq−D••E+••Kq+CDq+2NCI=DMq:

Which were before = KNq.

Therefore KN=DM; which is one.

In like manner resolve □ Kn.

It is certain that nlq+••Kq=Knq.

Substitute for nIq, by Consect. 1. Prop. 10. Lib. 1. C••q+CNq−2nCI, and you'l have

CIq+CNq−2nCI+IKq=Knq.

Substitute for CIq, by the 9. lib. 1. CDq+D••E, and you'l have 〈 math 〉〈 math 〉=Knq.

Resolve also the □ EM.

It is certain that 2CDq+2CMq−DMq=EMq per 13. lib. 1.

Then will CIq+CNq+2NCI+IKq=KNq.

Substitute for CIq by the 8. lib. 1. CDq−DIE; then will 〈 math 〉〈 math 〉.

Resolve also the □ DM.

It is certain that CMq

• 2DCM
• 2NCI

= DMq per 7. lib. 1.

Substitute for CMq its value from the preparation, and you'l have CNq−DIE+IKq+CDq+2NCI=DMq:

Which before were = KNq.

Therefore KN = DM; which is one.

In like manner resolve the □ Kn.

It is certain that nIq+IKq=Knq.

Substitute for nIq by Consect. 1. Prop. 10. lib. 1.

CIq+CNq−2NCI, and you'l have CIq+CNq−2NCI+IKq=Knq.

Substitute for CIq per 8. lib. 1. CDq−DIE, and you'l have CDq−DIE + CNq−2NCI+IKq=Knq.

Resolve also □ EM.

It is certain that 2CDq+2CMq−DMq=EMq per 13.1.

Substitute the value of DMq first found above, and you'l have CDq+CMq−2nCI=EMq.

Substiute for CMq the value as in the preparation, and you'l have CDq+CNq+DIE−2nCI+IKq=EMq: Which were before = Knq.

Therefore Kn=EM; which is the other.

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.

129

130

131

132

133

134

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.

Proposition IX.

IF the secondary ax, or conjugate diameter of the hyperbola AB (Fig. 131.) be applied parallel to the vertex E, so that it may touch the hyperbola, and OE, EP are equal like BC and AC, and from the centre C you draw thro' O and P right lines running on ad insinitum, and lastly QR parallel to the Tangent OP; you'l have the following

CONSECTARYS.

I. THE parts QG and HR intercepted between the curve and those right lines CQ, CR will be equal; for by reason of the similitude of the ▵ ▵ CEP and CFR as also CEO, CFQ as CE is to EO (and EP) so will CF be to FQ and FR, and consequently these will be equal; and so taking away the semiordinates FG and FH which are also equal, the remainders GQ and HR will be also equal, and consequently the □ □ QGR, GRH, &c. all equal among themselves: Which we had already deduced before in Consect. 2. and 3. Def. 7.

II. The rectangle QGR will be = □ EO or EP = 〈 math 〉〈 math 〉 i. e. (as Apollonius speaks) to the fourth part of the figure: For by

reason of the similitude of the ▵ ▵ CEO, CFQ, CE will be to EO as CF to FQ: i. e. as the □ CE to the □ EO i. e. as (by Cons 2. 7.) the Lat. Transv. to the Lat. Rect.

• so the □ CF
• to th □ FQ

i. e. as (by the 7. Prop.) as the □ DFE to the □ FG

But now if from the □ CF you take the □ DFE, there will remain the □ CE, by Prop. 9. lib. 1. and if from the □ FQ you take the □ FG there will remain the □ QGR, by Prop. 8. lib. 1. wherefore that remaining □ CE to this remain∣ing □ QGR will be, as was the whole square CF to the whole square FQ, by Prop. 26. lib. 1. i. e. as was the □ CE to the □ EO; consequently the □ QGR and the square EO (to which the same square CE bears the same proportion) will be equal among themselves.

III. Since this is also after the same manner certain of any other rectangle ggr or grh, &c. it follows that all such rect∣angles are equal among themselves.

IV. Wherefore it is most evident, since the lines FR, fr, &c. and so GR and gr grow so much the longer, by how much the more remote they are from the vertex E; that on the contrary the lines QG and qg must necessarily so much the more decrease and grow shorter, and consequently the right line CQ approach so much nearer and nearer to the curve EG.

V. But that they can never meet or coincide altho' produ∣ced ad infinitum will thus appear; if it were possible there could be any concourse or meeting, so that the point G and Q or g and q could any where coincide, it would follow from Consect. 2. that as the □ DFE to the square FG so the square CF to the square FQ i e▪ to the same square FG; and so that the □ DFE would be = □ CF; which is absurd by Prop. 9 lib. 1. so that now it is evident that the lines COQ and CPR drawn according to Consect. 1. are really Asymptotes (i. e. they will never(α) 1.10 coincide (viz. with the curve of the hyperbola) as Apollonius has named them.

VI. Having drawn the right lines from G and g parallel to both the asymptotes, viz. GS and gs and likewise GT and gt, the rectangles TGS and tgs will be(α) 1.11 e∣qual among themselves. For by reason of the similitude of the ▵ ▵ TQG and tqg, first, TG will be to QG as tg to qg; and, by reason of the equality of the □ □ QGR and qgr, secondly, QG will be to gr reci∣procally as qg to GR, by Prop 19▪ lib. 1. and by reason of the similitude of the ▵ ▵ SGR and sgr, thirdly, as gr to gs so GR to GS, wherefore (since in two series

1. 2. 3. as TG to QG to gr to gs so tg to qg to GR to GS) you'l have ex aequo or by proportion of equality as TG to gs so tg to GS, by Prop. 24 lib. 1. Therefore, by Prop. 17. of the same, the □ of TG into GS=□ of tg into gs. Q. E. D.

SCHOLIƲM.

HEnce, lastly, we have a new genesis of the hyperbola in Plano about its given diameters from the speculations of(β) 1.12De 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

• (α) 1.1

  I. Pro∣perty of the Pa∣rab. Apoll. Prop. 11. Lib. 1.
• (α) 1.2

  I. Pro∣perty of the Hy∣perb. Apoll. Prop. 12. Lib. 1.
• (α) 1.3

  I. Pro∣pert. of the El∣lipse of Apoll. Prop. 13. Lib. 1.
• (α) 1.4

  2. Property of the ••arab. 20. Prop. Apoll. Lib. 1. Con.
• (α) 1.5

  Proper∣ty of the Hy∣perbola and Ellips. Apol. 21. Lib. I.
• (α) 1.6

  De Witt Elem. Curv. Lib. 1. Cap. 3. Prop. 13.
• (α) 1.7

  3. Pro∣perty of the Pa∣rab.
• (α) 1.8

  The 3. Property of the Hyperb. and El∣lips. Apollon. lib. I. Prop. 21. pars prior.
• (α) 1.9

  Apollon. Lib. 3. Prop. 51. and 52.
• (α) 1.10

  Apoll. Prop. 1. lib. 2.
• (α) 1.11

  Apollon. Prop. 12. lib. 2.
• (β) 1.12

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