Let EG, eg, be two conjugate diameters parallel to the sides of the parallelogram, and dividing it into four less and equal parallelograms. Also, draw the ordinates DE, de, and cÊ perpendicular to PQ; and let the axis produced meet the sides of the parallelograms, produced, if necessary, in T and t. Then, by theor. 7, CT: CA :: CA: CD,

The Difference of the Squares of every Pair of Conjugate Diameters, is equal to the same constant Quantity, namely, the Difference of the Squares of the two Axes.

For, draw the ordinatés ED, ed.

Then, by cor. to theor. 10, ca2 = CD2 - cd2,

ca2 = de2
cacD2 + DE2 ·
CE2 CD2 + DE2,

ce' cd2 + de3 ;

ce2 = CD2 + DE3 cd3 - de3,

CA

2

2

AB ab2 = EG2

CE

THEOREM XII.

All the Parallelograms are equal which are formed between the Asymptotes and Curve, by Lines drawn Parallel to the Asymptotes.

For, let A be the vertex of the curve, or extremity of the semi-transverse axis ac, perp. to which draw AL or al, which will be equal to the semi-conjugate, by definition 19. Also, draw HDeh parallel to L,

Then, by theor. 2, ca2: AL2 :: CD3 and, by parallels,

theref. by subtract.

CA: AL: CD2: DH';
CA3: AL:: CA2: DH'
rect. HE. Eh ;

conseq. the square AL2 = the rect. HE
But, by sim. tri. PA: AL:: GE : EH,
and, by the same, QA : al :: EK : Eh ;

theref. by comp. PA. AQ ALGE. EK: HE . Eh; and because

AL HE. Eh, theref. PA. AQ GE. EK.

But the parallelograms CGEK, CPAQ, being equiangular, are as the rectangles GE. EK and PA. AQ.

Therefore the parallelogram GK = the paral. PQ. That is, all the inscribed parallelograms are equal to one another.

Q. E. D.

Corol. 1. Because the rectangle GEK or CGE is constant, therefore GE is reciprocally as CG, or ce: CP:: PAGE. And hence the asymptote continually approaches towards the curve, but never meets it for GE decreases continually

as co increases; and it is always of some magnitude, except when to is supposed to be infinitely great, for then GE is infinitely small, or nothing. So that the asymptote co may be considered as a tangent to the curve at a point infinitely distant from c.

H I

K

D E

Corol. 2. If the abscisses CD, C, CG, &c. taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates DH, EI, GK, &c. parallel to the other asymptote, be a decreasing geometrical progression, having the same ratio. For, all the rectangles CDH, CEI, CGK, &c. being equal, the ordinates DH, EI, GK, &c, are reciprocally as the abscisses CD, CE, CG, &c. which are geometricals. And the reciprocals of geometricals are also geometricals, and in the same ratio, but decreasing, or in converse order.

C

THEOREM XIII.

The three following Spaces between the Asymptotes and the Curve, are equal; namely, the Sector or Trilinear Space contained by an Arc of the Curve and two Radii, or Lines drawn from its Extremities to the Centre; and each of the two Quadrilaterals, contained by the said Arc, and two Lines drawn from its Extremities parallel to one Asymptote, and the intercepted Part of the other Asymp

For, by theor. 12, CPAQ = CGEK;
subtract the common space CGIQ,

there remains the paral. PI the par. IK ;
To each add the trilineal IAE, then
the sum is the quadr. PAEG = QAEK.
Again, from the quadrilateral CAEK
take the equal triangles, CAQ, CEK,
and there remains the sector CAE= QAEK.
Therefore CAE = QAEK = PAEG.

Q. E. D.

SCHOLIUM.

In the figure to theorem 12, cor. 2, if cD = 1, and CE, CG, &c. be any numbers, the hyperbolic spaces HDEi, iegk, &c. are analogous to the logarithms of those numbers. For, whilst the numbers CD, CE, CG, &c. proceed in geometrical progression, the correspondent spaces proceed in arithmetical progression; and therefore, from the nature of logarithmns are respectively proportional to the logarithms of those numbers. If the angle c were a right angle, and CD = DH=1; then if CE were = 10, the space DEIH would be 2-30258509, &c.; if co were 100, then the space DGKн would be 4-60517018: these being the Napierean logarithms to 10 and 100 respectively. Intermediate arears corresponding to intermediate abscisse would be the appropriate logarithms. These are usually called Hyperbolic logarithms; but the term is improper: for by drawing other hyperbolic curves between HIK and its asymptotes, other systems of logarithms would be obtained. Or, by changing the angle between the asymptotes, the same thing may be effected. Thus, when the angle c is a right angle, or has its sine = 1, the hyperbolic spaces indicate the Napierean logarithms; but when the angle is 25° 44′ 27′′, whose sine is 43429448, &c. the modulus to the common, or Briggs's, logarithms, the spaces DEIH, &c. measure those logarithms. In both cases, if spaces to the right of DH are regarded as positive, those to the left will be negative; whence it follows that the logarithms of numbers less than 1 are negative also.

THEOREM XIV.

The sum or difference of the semi-transverse and a line drawn from the focus to any point in the curve is equal to a fourth proportional to the semi-transverse, the distance from the centre to the focus, and the distance from the centre to the ordinate belonging to that point of the curve.

That is,

FE+ACCI, or FE=AI;
ACCI, or fe=BI.

and fE

Where CA: CF :: CD: cr the 4th propor. to ca, CF, CD.

AFDI

For, draw AG parallel and equal to ca the semi-conjugate; and join co meeting the ordinate DE produced in н.

Then, by theor. 2, ca2: AG2 :: CD2-ca2 : DE2;
and, by sim. As, CA2: AG2 :: CD2-ca2 : DH2—AG2;
consequently DE2=DH2—AG2=DI2 —ca3.

Also FD=CF CD, and FD2=CF-2CF. CD+CD2;
but, by right-angled triangles, Fd2+de2=fe2;
therefore FE-cr-ca-2cF. CD+CD2+ DH3.

and, by supposition, 2cF. CD=2ca . ci; theref. FECA2-2cs. CI+CD2+DH3.

But, by supposition, ca2 : CD2 :: cF2 or ca2+AG2 : cr2;

and, by sim. As,

therefore

consequently

CA: CD2: CA2+AG2: CD2+H2;
CI2=CD2+DH2==ch2;

FE CA-2CA. CI+Ci2.

And the root or side of this square is FE CI—CA⇒AI.
In the same manner is found fE=CI+CA=BI.

Q. E. D.

Corol. 1. Hence CH=C1 is a 4th propor. to ca, CF, CD.

Corol. 2. And fE+FE=2CH or 2c1; or FF, CH, ƒ are in continued arithmetical progression, the common difference being ca the semi transverse.

=

Corol. 3. From the demonstration it appears, that DE2 = DI12 · AG3 = DH-ca. Consequently DH is every where greater than DE; and so the asymptote CGH never meets the curve, though they be ever so far produced: but DH and DE approach nearer and nearer to a ratio of equality as they recede farther from the vertex, till at an infinite distance they become equal, and the asymptote is a tangent to the curve at an infinite distance from the vertex.

THEOREM XV.

If a line be drawn from either focus, perpendicular to a tangent to any point of the curve; the distance of their intersection from the centre will be equal to the semi-trans. verse axis.