From any point N in the hyperbola, draw NM at right angles to the axis, and let it meet the asymptote in K; then (from similar triangles and 22.6),

But (Art. 123),

As EA2: AH2 :: EM2: MK2.

As EA2: ED2 (AH3) :: AM.MB : MN2;

and EM is greater than AM.MB (6.2); therefore, MK2 is greater than MN, and MK greater than MN.

ART. 133. Retaining the construction in the last article, produce KM to meet the asymptote EQ in L; then shall KN.NL = ED2.

As in last article, we have,

EA2: ED2:: EM2 : MK2 :: AM.MB: MN2 :: (19.5) EM2-AM.MB: MK2-MN2 : : (6.2 and 5.2) AE2 : KN.NL. KN.NL ED2 = HA.AI.

Hence,

Cor. Hence, OPQ being drawn parallel to KL, the rectangle OP.PQ = KN.NL; and, therefore (16.6),

As KN OP :: PQ : NL.

ART. 134. The asymptotes continually approach to the hyperbola.

Taking KL and OQ as in the last article, it is evident that, ER being greater than EM, PQ is greater than NL; but, KN: OP: PQ : NL;

hence, OP is less than KN.

ART. 135. Through the vertex A, and any other point N of the hyperbola, let the lines AS, NT be drawn parallel to one of the asymptotes EQ, meeting the other in S and T; then,

As ES: ET :: TN : SA.

Draw AW, NV parallel to EO; then the triangles IAW, LNV, are similar; as are also HAS, KNT; hence the following analogies:

As AW: NV:: AI: NL :: (Art. 133) KN: HA :: TN: AS. As ES ET :: TN: SA.

Therefore,

Cor. If PU be drawn parallel to EQ, then

EU: ET:: TN: UP.

Scholium. From the property demonstrated in Art. 135 is deduced a relation between logarithms and the areas contained between the hyperbolic curve and its asymptote. Let EA ED, and consequently ES = SA, and SEW a right angle. The hyperbola is then called an equilateral or rectangular one. If in that case we assume ES = 1; and of course the square SW also = 1; then ET being estimated in units of ES, and the area ASTN in units of SW, it is proved by writers on differentials that ASTN is the logarithm of

ET, provided the modulus (Art. 15) m = 1. Hence, those logarithms are termed hyperbolic.

=

It is, however, observable, that these hyperbolic areas may be made to express logarithms of other kinds. For, if the relation between the axes is such that, ES being 1, the area of the parallelogram SW shall be expressed by the modulus, the area of ASTN will be the logarithm of ET, according to the system to which that value of m belongs. But the demonstration of these properties would lead further into the differential and integral calculus, than the design of this work admits.

SECTION V.

SPHERICAL PROJECTIONS.

ARTICLE 136. The business of Spherical Projections is to represent by lines, drawn or described on a plane given in position, the circles which are described on the surface of a sphere. The lines thus drawn or described on the plane, are called the projections of the circles which they represent; and are so framed that, to an eye properly located, every circle on the sphere will appear coincident with its representative.

Def. 1. The plane on which the circles of the sphere are represented, is called the plane of projection; and the point where the eye is supposed to be located, the projecting point. A right line drawn from the projecting point to any point on the sphere, and extended to meet the plane of projection, is termed a projecting line.

Def. 2. Every circle on the sphere is called an original circle; and the figure which represents it on the plane of projection, a projected circle.

Def. 3. In the orthographic and stereographic projections, the plane of projection is supposed to pass through the centre of the sphere. Then the common section of this plane and the spherical surface is a circle, which is called the primitive circle. This circle is evidently a great one (Art. 45); and, being both on the sphere and plane of projection, may be considered as an original circle, projected into itself.

Def. 4. In the orthographic projection, the projecting point is in the axis of the primitive circle; but so remote, that all the

projecting lines drawn to the different points of the sphere may be considered as parallel.

Def. 5. In the stereographic projection, the projecting point is at one of the poles of the primitive circle.

Def. 6. The line of measures of any circle which is to be projected, is the common section of the plane of projection, and another plane which passes through the axes, both of the primitive circle and of the circle to be projected.

Def. 7. The semitangent of an arc is the tangent of half the arc; not half the tangent of the arc.

Of the Orthographic Projection.

ART. 137. If a right line AB be projected orthographically upon a plane, the projection will be a right line; and the original line will be to its projection, as radius to the cosine of the inclination of the original to the plane of projection.

in a, b. Conceive a plane to pass through Aa, Bb; this plane will include AB, and be at right angles to the plane of projection (def. 4 and 17.2 sup). The common section of these planes will evidently be the projection of AB; but this section is a straight line (3.2 sup.) contained between Aa and Bb; that is, it is the line ab.

Through A draw AD parallel to ab; then is DAB the inclination of AB to the plane of projection. And (Art. 28)

radius cosine DAB :: AB: AD or ab.