the segments of that tangent, between the parallels and its contact.

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.

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.

H

E A CD B.F

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 each 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 CK 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 P, be fastened at the point F, and because the thread FPZ is equal to HZ, taking PZ from each, PF remains equal to PH; let the side HX of the instrument be moved along the line DX, and, the thread romain

ing extended, let the pin, affixed to the side HZ of the instrument, describe the line BP, which is the hyperbola required, asis 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.

In a right line DK, parallel to an asymptote CZ of a hyperbola, let any point whatever K be taken; let P and D be the points in which DK meets the hyperbola and directrix DX adjacent thereto, and F the adjacent focus; let KM be taken on DH, having to a perpendicular KX, let fall from K on the directrix DX, the determining ratio, and let DF, PF and FK be joined.

k

F

N

P

X

M

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 rectangles 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 segments of the right line parallel to the asymptote, between the parallels, and the concourse of that right line, with the hyperbola, which it meets.

Let, in fig. 1, RL and rl be two tangents, parallel to each other, touching opposite hyperbolas in R and r, meeting a directrix DG, in H and h, and DK parallel to the asymptote CZ in L and 1; and let secants ST and st meet a hyperbola, as in fig. 1, or opposite hyperbolas, as in fig. 2, in S and T, s and t, the directrix DG in G and g, and the parallel to the asymptote in K and k. The squares of LR and Ir in fig. 1, and the rectangle SKT and skt in fig. 1 and 2, are to each other, as the right lines PL and Pl, PK and Pk.

Join KF and DF; and KD and kD in fig. 1 and 2, and LD and ID in fig. 1, have to perpendiculars let fall from the points K and k, L and 1, on the directrix DG, the determining ratio (Cor. 16. 1 Sup.), and let, first, the parallel right lines which meet DK, be, in fig. 1, the tangent LR and the secant ST; and since the rectangle SKT is to the difference of the squares of KF and KD, as the square of KG is to the difference of the squares of KG and KD (13. 1 and Cor. 16. 1 Sup.), and the difference of the squares of KF and KD is to the rectangle DPK, as the square of DF is to the square of DP (16. 1 Sup. and Theor. 3. 15. 5. Eu.), by compounding these ratios, the rectangle SKT is to the rectangle DPK, in a ratio compounded of the ratios of the rectangle SKT to the difference of the squares of KF and KD, and of the same difference to the rectangle DPK, (Def. 13. 5 Eu.), or, which has been just shewn to be equal, of the ratios of the square of KG to the difference of the squares of KG and KD, and of the square of DF to the square of DP.