12°. The same conditions as in 10°, except that the centre of the lower wheel is 4 feet from the plumb line dropped from the centre of the wheel above the floor.

13°. It is required to construct three equal friction wheels to run tangent to each other and to an axle two inches in diameter. What must be their common radius, and what the radius of the circular bed cut for them in the centre of a wheel?

14°. If the top-masts of two ships, having elevations of 90 and 100 feet above the level of the sea, are seen from each other at the distance of 25'7 miles, what is the diameter of the earth?

15°. How far can the Peak of Teneriffe be seen at sea?

16°. How far will a water level fall away from a horizontal line, sighted at one end in a distance of one mile, the diameter of the earth being estimated at 7,960 miles?

17°.* If AC, one of the sides of an equilateral triangle ABC, be produced to E, so that CE shall be equal to AC; and if EB be drawn and produced till it meets in D, a line drawn from A at right angles to AC; then DB will be equal to the radius of the circle described about the triangle.

18°. If an angle B of any triangle ABC, be bisected by the straight line BD, which also cuts the side AC in D, and if from the centre A with the radius AD, a circle be described, cutting BD or BD produced in E; then BE: BD :: AD : CD.

19°. Let ABC be a triangle right angled at B; from A draw AD parallel to BC, and meeting in D, a line drawn from B at right angles to AC; about the triangle ADC describe a circle, and let E be the point in which its circumference cuts the line AB or AB produced; then AD, AB, BC, AE, are in continued proportion.

20°. Let ABC be a circle, whose diameter is AB; and from D any point in AB produced, draw DC touching the circle in C, and DEF any line cutting it in E and F; again, draw from C a perpendicular to AB, cutting EF in H; then,

ED2 CD2 :: EH : FH.

21°. Let ABC be a circle, and from D, a point without it, let three straight lines be drawn in the following manner: DA touching the circle in A, DBC cutting it in B and C, and DEF cutting it in E and F; bisect the chord BC in H, draw AH, and produce it till it meets the circumference in K; draw also KE and KF cutting BC in G and L. The lines HG and HL are equal.

* "Prize Problems," Yale College, 1840

SECTION SECOND.

The Ellipse.

Def. 1. An ellipse is a plane curve described by the intersection of two radii, varying in such manner as to preserve in sum the same constant quantity, while they revolve about two fixed points

Take the middle

F, F, distant from each other by the line 2c. point O between F, F', for the origin of rectangular coördinates, the line passing through F, F', being the axis of X.

There

Now, it is obvious that, if the curve cuts the axis of x, y for that point will be reduced to nothing; therefore, if we make y = 0, and what x becomes for this value of y, we have (200) a2 • 02+(a2 — c2)x_。 = a2(a2 — c2);

denote by x,

=0

Cor. 1. The ellipse cuts the axis of abscissas at equal (201) distances on the right and left of the origin, which ¿x'

distance = a.

Fig. 432.

When x = 0, the curve cuts the axis of y, but this condition gives (200)

a2yo+(a2 — c2) • 02 = a2 (a2 — c2) ;
Yz_o=±(a2 — c2)‡ ; hence,

Y

Cor. 2. The ellipse cuts the axis of ordinates at equal (202) distances above and below the origin, which distance, denoted by

Def. 2. The line MON = 2a, is denominated the Major Axis or the Conjugate Diameter, and passes through the Foci,* F, F'; the line N POQ 2b, is the Minor Axis or the Trans

=

verse Diameter, being perpendicular to the former. [b can never > than a.]

If in (200) we substitute b2 for (a2 — c2), the equation of the ellipse becomes

22

a2y2+ b2x2 = a2b2, or

y2
+
b2 a2

where the constants are the semimajor and semiminor axes. Cor. 3. The ellipse is symmetrical in reference to both (204) axes; since,

For every value of x, whether + or —,

Cor. 4. The major axis bisects all chords parallel to the (205) minor, and the minor axis bisects all chords parallel to the major. Cor. 5. The origin bisects all chords drawn through it, (206)

* Foci, plural of focus, fire-place.

and is, consequently, the Centre of the Ellipse; HOT therefore these chords are Diameters.

Fig. 437.

Cor. 6. Diameters, equally inclined to the major axis, (207) are equal; and the converse.

Cor. 7. Any ordinate of the ellipse is to the correspond- (208) ing ordinate of the circle, described on the major axis, as the semiminor axis is to the semimajor.

For let y, Y, be corresponding ordinates of the ellipse and circle; we have (203)

Cor. 8. The circle, described on the major axis, circum- (209) scribes the ellipse. Hence,

Cor. 9. The angle embraced by chords, drawn from any (210) point of the ellipse to the extremities of the major axis,

is obtuse.

[How?]

Fig. 439.

Cor. 10. Any abscissa of the ellipse is to the correspond- (211) ing abscissa of the circle described on the minor axis, as the semimajor axis to the semiminor.

Cor. 11. The circle described on the minor axis is in- (212) scribed in the ellipse.

Cor. 12. The angle embraced by chords drawn from any (213) point of the ellipse to the extremities of the minor axis, is acute.

Def. 3. The double ordinate drawn through the focus, is denominated the Parameter of the major axis, and sometimes the Latus Rectum.

To find the parameter we have only to make x = c in (200); whence there results

Cor. 13. The Parameter is a third proportional to the (214)

major and minor axes.

Putting the parameter = p, and substituting in (203) we get

y2 =

=

P 2a

(a2 — x2),

(215)

for the equation of the ellipse, in terms of the parameter and semimajor axis.

Def. 4. We call e the Eccentricity because it expresses the ratio of the distance, c, of the focus from the centre to the semimajor axis, and thus determines the form of the ellipse, as round or flat. When the eccentricity is = 0, the ellipse becomes a circle. Equation (216) is that of the ellipse, referred to its eccentricity and semimajor axis, and is convenient in astronomy.

It is sometimes desirable to have the equation of the ellipse, when the left hand extremity of the major axis is made the origin of abscissas. In order to this, we have only to substitute x a instead of x in (203), as the new x exceeds the old by a, y remaining the same; which done, there results,

Fig. 4311.

(218)

The origin might be transported to any other point, either in the curve or elsewhere, by changing the value of y as well as that

of x.

Scholium. It is easy to show that any equation referred to rectangular coordinates, and of the form

py2+qx2 = r,

is the equation of an ellipse; for we have