of KD and KM (13. 1 Sup.), and therefore in a constant ratio. Case 5. Lastly, let one of the right lines PQ, and LR, meeting each other in L, as LR, be a tangent, touching the section in a vertex R of the principal axis F, and of course perpendicular to that axis (Cor. 1. 11. 1 Sup.), and therefore parallel to the directrix DG, the other PQ being a secant, meeting the directrix in D, and the section or sections in and Q. The rectangle PLQ is to the square of LR, in a constant ratio.

Let fall the perpendiculars FU and LH on the directrix DX, draw FO parallel to LR meeting LH in O; and, because of the parallelograms HR and OR, the right lines HL, OL and OF are severally equal to UR, FR and LR (34. 1 Eu.); therefore OL is to LH, as FR to RU, and therefore in the determining ratio (6. 1 Sup.); whence, a right line being supposed to be drawn from F to L, the rectangle PLQ is to the difference of the squares of FL and LO, or, because of the right angled triangle FOL and parallelogram OR, to the square of OF (47. 1 Eu.), or, which is equal (34. 1 Eu.), LR, as the square of LD is to the difference of the squares of LD and LO (13. 1 Sup.), and therefore, as before, in a constant ratio.

Scholium. That the truth of this proposition may appear more clearly, in the different positions of the point K, another secant skt is exhibited in the figures to the three first cases, meeting the secant DPQ, in a different situation, with respect to the section, as internally, instead of externally; the reasoning in the demonstration of the proposition applying, by substituting the small letters s, k, t, g and x, for their respective capitals.

Cor. 1. If any right line (KZ, see fig. 1 of this prop.), touching a conick section, meet two parallel right lines (GT, gt), cutting the section or opposite sections; the rectangles under the segments of the secants, between the tangent and the section or sections, are to each other, as the squares of the segments of the tangent, between the parallels and the contact.

For the ratios of the rectangles under these segments of the secants, to the squares of the respective segments of the tangent which they meet, being by this prop. equal, by alternating these rectangles are to each other, as the same squares.

Cor. 2. Or if any right line, touching a conick section, meet two parallel right lines, touching the section or opposite sections; it may in the same manner be proved, that the squares the segments of the parallel tangents between their contacts and the tangent which they meet, are to each other, as the squares of

of

the segments of that tangent, between the parallels and its con

tact.

Cor. 3. Or if any right line, touching a conick section, meet two parallel right lines, whereof one is a tangent, and the other a secant; the square of the segment of the tangent, and rectangle under the segments of the secant, between the tangent which they meet and the section, are to each other, as the squares of the segments of that tangent between the parallels and its contact.

Cor. 4. Or if any right line, cutting a conick section or opposite sections in two points, meet two parallel right lines cutting in like manner the same section or sections; the rectangles under the segments of the parallels between the section or sections, and the right line which they meet, are to each other, as the rectangles under the segments of the right line which they so meet, between the parallels and the section or sections.

Cor. 5. Or if any right line, cutting a conick section or opposite sections, meet two parallel right lines touching the same section or opposite sections; the squares of the segments of the parallel tangents between their contacts, and the secant which they meet, are to each other, as the rectangles under the segments of that secant, between the parallels, and the section or sections.

Cor. 6. Or if any right line, cutting in two points a conick section or opposite sections, meet two parallel right lines, whereof one is a tangent, and the other a secant; the square of the segment of the tangent, and the rectangle under the segments of the secant, between the section or sections and the secant which they meet, are to each other, as the rectangles under the segments of that secant between the parallels, and the section or sections.

1

PROP. XV. THEOR.

A right line (PH), drawn from any point (P) of a hyperbola (BP), to the adjacent directrix (DX), parallel to the adjacent asymptote (CK), is equal to the distance (PF) of the same point, from the adjacent focus (F).

Let E be other focus, and AB the principal axis; draw AK at right angles to AB, meeting the asymptote CK in K, and PX at right angles to DX.

BF

E A CD

K

Because AK is equal to the second semiaxis (Def. 19. 1 Sup.), its square is equal to the difference of the squares of CE and CA (2. 1 Sup.), and the square of AK is also equal to the difference of the squares of CK and CA (47. 1 Eu.); whence, the difference of the squares of CE and CA, and of CK and CA, being cach equal to the square of AK, are equal to each other (Ax. 1. 1), adding to each of these differences the square of CA, the squares of CE and C are equal, and so CE is equal to CK; and PF is to PX, as CE, or, which has been just proved equal to it, CK, is to CA (Schol. 6. 1 Sup.), or, because of the equiangular triangles ACK and XHP, as PH is to PX (4. 6 Eu.); therefore PF and PH, having the same ratio to PX, are equal (9. 5 Eu).

Scholium.-Hence, a focus F, the adjacent directrix DX, and an asymptote CK of a hyperbola being given, the section may be described. Let XHZ be an instrument similar to a square, but with one side HZ moveable about H, so as to make the angle ZHX equal to a given one. Let one side of it HX be applied to the directrix DX, and the other side HZ, being toward the part on which is F, be so inclined to the side HX, that it may be parallel to CK; and to the extremity Z, of the side HZ, let one extremity of a thread of the same length as HZ be fastened, and let its other extremity, the thread going round a pin in the side HZ, at the point, be fastened at the point F, and because the thread FZ is equal to HZ, taking PZ from each, F remains equal to PH; let the side HX of the instrument be moved along the line DX, and, the thread remain

ing extended, let the pin, affixed to the side HZ of the instrument, describe the line BP, which is the hyperbola required, as is manifest from this proposition.

Hence appears further, the close analogy, which exists between the parabola and hyperbola, seeing that, if the side of the square, to which the thread and pin, are applied, deviate ever so little either way from a right angle with the other side, the figure becomes a hyperbola.

PROP. XVI. THEOR.

If, in a right line parallel to an asymptote of a hyperbola, any point be taken, and also a finite right line, which is to the distance of that point from the directrix, adjacent to the hyperbola which it meets, in the determining ratio; the rectangle under the distances of the point, wherein the parallel right line meets the hyperbola, from that wherein it meets the directrix, and from the assumed point, is to the difference of the squares of the distance of that point, from the focus adjacent to the same directrix, and the assumed finite right line, as the square of the segment of the same parallel, between the directrix and hyperbola, to the square of a right line, joining the focus, to the point, in which the parallel meets the directrix.

The rectangle DPK is to the difference of the squares of KF and KM, as the square of PD is to the square of FD.

Let fall the perpendicular PH on DX, and from K draw KN parallel to PF, meeting DF, produced if necessary, in N.

Because of the equiangular triangles DKN and DPF, KN is to KD, as PF is to PD (4. 6 Eu.); whence, PF being equal to PD (15.1 Sup.), KN is equal to KD. And since KN is to KD, as PF is to PD, and, because of the equiangular triangles DKX and DPH, DK is to KX, as DP to PH (4. 6 Eu.), by equality, KN is to KX, as PF to PH (22. 5 Eu.), or in the determining ratio; therefore KN or KD is equal to KM (Hyp. and 9.5 Eu).

From the centre K, at the distance KD, KN or KM, let the circle DNM be described; and, because of the parallels PF and KN, the rectangle DPK is to the rectangle DFN, or, which is equal (Cor. 3. 36. 3 Eu.), the difference of the squares of KF and KM, as the square of DP is to the square of DF (20. 6 and Cor. 3. 22.5 Eu).

Cor. The segment (KD), of a right line (DM) parallel to an asymptote (CZ) of a hyperbola, between any point (K) in the parallel, and the directrix, is to a perpendicular (KX) let fall from the same point on the directrix, in the determining ratio.

For, because of the equiangular triangles DKX and DPH, KD is to KX, as PD, or its equal (15. 1 Sup.) PF, is to PH (4. 6 Eu.),or in the determining ratio.

Scholium. The reasoning in this proposition and corollary applies, whether the point taken in the parallel, be within or without the hyperbola, as the point k, a right line being supposed to be drawn from k to F, by substituting the small letters k, x, m and n for their respective capitals.

PROP. XVII. THEOR.

If two right lines, parallel to each other, both touch or both cut in two points, or one of them touch, and the other so cut, a hyperbola or opposite hyperbolas, and meet a right line parallel to an asymptote; the squares of the segments of the tangents, or rectan gles under the segments of the secants, between the right line parallel to the asymptote, and the point or points wherein they meet the hyperbola or hyperbolas, are to each other, as the seg ments of the right line parallel to the asymptote, between the parallels, and the concourse of that right line, with the hyperbola, which it meets.