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So that the three angles are as follow, viz. A 27° 4′; L B 37° 20′; C 115° 36. 1053. THEOREM IV. If the triangle be right-angled, any unknown part may be found by the following proportion : ·

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As radius

Is to either leg of the triangle,

So is tangent of its adjacent angle

To the other leg;

And so is secant of the same angle
To the hypothenuse.

For AB being the given leg in the right-angled triangle ABC, from the centre A with any assumed radius AD describe an arc DE, and draw DF perpendicular to AB, or parallel to BC. Now, from the definitions, DF is the tangent and AF the secant of the arc DE, or of the angle A, which is measured by that are to the radius AD. Then, because of the parallels BC, DF, we have AD: AB:: DF: BC, and ::AF: AC, which is the same as the theorem expresses in words.

Note. Radius is equal to the sine of 90°, or the tangent of 45°, and is A expressed by 1 in a table of natural sines, or by 10 in logarithmic sines. Example 1. In the right-angled triangle ABC,

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10.221848

2.431363

Note. There is another mode for right-angled triangles, which is as follows:
:-

ABC being such a triangle, make a leg AB radius; or, in other words, from the centre A and distance AB describe an arc BF. It is evident that the other

leg BC will represent the tangent and the hypothenuse AC the secant of the are BF or of the angle A.

In like manner, if BC be taken for radius, the other leg AB represents the tangent, and the hypothenuse AC the secant of the arc BG or angle C.

If the hypothenuse be made radius, then each leg will represent the sine of its opposite angle; namely, the leg AB the sine of the arc AE or angle C, and the leg BC the sine of the arc CD or angle A.

C

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Then the general rule for all such cases is, that the sides of the triangle bear to each other the same proportion as the parts which they represent. This method is called making every side radius.

1054. If two sides of a right-angled triangle are given to find the third side, that may be found by the property of the squares of the sides (Geom. Prop. 32. ; viz. That the square of the bypothenuse or longest side is equal to both the squares of the two other sides together). Thus, if the longest side be sought, it is equal to the square root of the sum of the two shorter sides; and to find one of the shorter sides, subtract one square from the other, and extract the square root of the remainder.

1055. The application of the foregoing theorems in the cases of measuring heights and distances will be obvious. It is, however, to be observed, that where we have to find the length of inaccessible lines, we must employ a line or base which can be measured, and, by means of angles, which will be furnished by the use of instruments, calculate the lengths of the other lines.

SECT. V.

CONIC SECTIONS.

1056. The conic sections, in geometry, are those lines formed by the intersections of a plane with the surface of a cone, and which assume different forms and acquire different properties, according to the several directions of such plane in respect of the axis of the cone. Their species are five in number.

1057. DEFINITIONS.-1. A plane passing through the vertex of a cone meeting the plane of the base or of the base produced is

called the directing plane. The plane VRX (fig. 404.) is the directing plane, 2. The line in which the directing plane meets the plane of the base or the plane of the base produced is called the directrix. The line RX is the directrix. 3. If a cone be cut by a plane parallel to the directing plane, the section is called a conic section, as AMB or AHI (fig. 405.)

4. If the plane of a conic section be cut by R another plane at right angles passing along the axis of the cone, the common section of the two planes is called the line of the axis.

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5. The point or points in which the line of the axis is cut by the conic surface is or are called the vertex or vertices of the conic section. Thus the points A and B (figs. 404. and 405.) are both vertices, as is the point A or vertex (fig. 406.).

6. If the line of the axis be cut in two points by the conic surface, or by the surfaces of the two opposite cones, the portion of the line thus intercepted is called the primary axis. The line AB (figs. 404. and 405.) and AH (fig. 406.) is called the primary axis.

7. If a straight line be drawn in a conic section perpendicular to the line of the axis so as to meet the curve, such straight line is called an ordinate, as PM in the above figures.

8. The abscissa of an ordinate is that portion of the line of axis contained between the vertex and an ordinate to that line of axis.

B

D

H

Fig. 406.

Thus in figs. 404, 405, and 406. the parts AP, BP of the line of axis are the abscissas AP.

9. If the primary axis be bisected, the bisecting point is called the centre of the conic section.

10. If the directrix fall without the base of the cone, the section made by the cutting plane is called an ellipse. Thus, in fig. 404., the section AMB is an ellipse. It is evident that, since the plane of section will cut every straight line drawn from the vertex of the cone to any point in the circumference of the base, every straight line drawn within the figure will be limited by the conic surface. Hence the axis, the ordinates, and abscissas will be terminated by the curve.

11. If the directrix fall within the base of the cone, the section made by the cutting plane is called an hyperbola. Hence it is evident, that since the directing plane passes alike through both cones, the plane of section will cut each of them, and therefore two sections will be formed. And as every straight line on the surface of the cone and on the same side of the directing plane cannot meet the cutting plane, neither figure can be enclosed.

12. If the directrix touch the curve forming the base of the cone, the section made by the cutting plane is a parabola.

OF THE ELLIPSIS.

1058. The primary axis of an ellipsis is called the major axis, as AB (fig. 407.); and a straight line DE drawn through its centre perpendicular to it, and terminated at each extremity by the curve, is called the minor axis.

1059. A straight line VQ drawn through the centre and terminated at each extremity by the curve is called a diameter. Hence the two axes are also diameters.

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1060. The extremities of a diameter which terminate in the curve are called the vertices of that diameter. Thus the points V and Q are the vertices of the diameter VQ.

1061. A straight line drawn from any point of a diameter parallel to a tangent at either extremity of the diameter to meet the curves is called an ordinate to the two abscissas. Thus PM, being parallel to a tangent at V, is an ordinate to the two abscissas VP, PQ. 1062. If a diameter be drawn through the centre parallel to a tangent at the extremity of another diameter, these two diameters are called conjugate diameters. Thus VQ and RS are conjugate diameters.

1063. A third proportional to any diameter and its conjugate is called the parameter or latus rectum.

1064. The points in the axis where the ordinate is equal to the semi-parameter are called the foci.

1065. THEOREM I. In the ellipsis the squares of the ordinates of an axis are to each other as the rectangles of their abscissas.

Let AVB (fig. 408.) be a plane passing through the axis of the cone, and AEB another section of the cone perpendicular to the plane of the former ; AB the axis of the elliptic section, and PM, HI ordinates perpendicular to it; then it will be

PM2: HI2:: APx PB: AH × HB.

For through the ordinates PM, HI draw the circular sections KML, MIN parallel to the base of the cone, having KL, MN for their diameters, to which PM, HI are ordinates as well as to the axis of the ellipse. Now, in the similar triangles APL, AHN,

AP PL:: AH: HN,

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Coroll. 1. If C be the centre of the figure, AP × PB-CAo- CP2, and AH × HB= CA2-CH2.

Therefore PM2: HI2:: CA2-CP2: CA- CH2. For AP-CA-CP, and PB= CA+ CP: consequently AP x PB=(CA-CP) (CA+CP)= CA-CP2; and in the same manner it is evident that AH × HB=(CA+CH)(CA-CH) = CA2 — CH2. Coroll. 2. If the point P coincide with the middle point C of the semi-major axis, PM will become equal to CE, and CP will vanish; we shall therefore have

PM2: HI2:: CA2-CP2: CA2-CH2

Now CE2: HI2:: CA2; CA2-CH2, or CA2 × HI2= CE2(CA2— CH2).

1066. THEOREM II. In every ellipsis the square of the major axis is to the square of the minor axis as the rectangle of the abscissas is to the square of their ordinate.

Let AB (fig. 409.) be the major axis, DE the minor axis, C the centre, PM and HI ordinates to the axis AB; then will

CA2: CE2:: AP x PB: PM2.

For since by Theor. L., PM2; HI2:: AP × PB: AH x HB; and if A the point H be in the centre, then AH and HB become each equal to CA, and HI becomes equal to CE; therefore

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M

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Coroll. 1. Hence, if we divide the two first terms of the analogy by AC, it will be CE CA: AP PB: PM2. But by the definition of parameter, AB: DE:: DE: pa

2CE2

rameter, or CA: CE::2CE: parameter=CA let us call P; then

2CE2
CA

Therefore is the parameter, which

AB: P:: AP x PB: PM2. Coroll. 2. Hence CA: CE2:: CA2- CP2: PM2. ^A+CP)=(AP × PB).

For CA-CP2 = (CA-CP)

1067. THEOREM III. In every ellipsis, the square of the minor axis is to the square of the major axis as the difference of the squares of half the minor axis and the distance of an ordinate from the centre on the minor axis to the square of that ordinate.

E

M

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Draw MQ (fig. 410.) parallel to AB, meeting CE in Q; then will

A

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CE2 CA2:: CE- CQ2: QM2;

For by Cor. 2. Theor. II., CA2; CAo-CP2:: CE2: PM2;
Therefore, by division, CA2: CP2::CE: CE2- PM2.

G

Fig. 410.

Therefore, since CQ=PM and CP=QM; CA2: QM2:: CE2: CE÷- CQ2.

G

E

Coroll. 1. If a circle be described on each axis as a diameter, one being inscribed within the ellipse, and the other circumscribed about it, then an ordinate in the circle will be to the corresponding ordinate in the ellipsis as the axis belonging to this ordinate is to the axis belonging to the other; that is,

CA: CE:: PG: PM,

and CE CA::pg: pM;

and since CA2: CE2:: AP x PB: PM2,

and because AP ×

or CA

PB = PG2; CA2: CE2:: PG2: PM2,
CE:: PG: PM.

Fig. 411.

B

In the same manner it may be shown that CE: CA::pg pM, or, alternately, CA: CE::PM: pg; therefore, by equality, PG: PM::pM: pg, or PG: Cp::CP: pg : therefore CgG is a continued straight line.

Coroll. 2. Hence, also, as the ellipsis and circle are made up of the same number of corresponding ordinates, which are all in the same proportion as the two axes, it follows that the area of the whole circle and of the ellipsis, as also of any like parts of them, are in the same ratio, or as the square of the diameter to the rectangle of the two axes; that is, the area of the two circles and of the ellipsis are as the square of each axis and the rectangle of the two; and therefore the ellipsis is a mean proportional between the two circles.

Coroll. 3. Draw MQ parallel to GC, meeting ED in Q; then will QM=CG=CA; and let R be the point where QM cuts AB; then, because RMGC is a parallelogram, RM is equal to CG=CE; and therefore, since QM is equal to CA, half the major axis and RM=CE, half the minor axis QR is the difference of the two semi-axes, and hence we have a method of describing the ellipsis. This is the principle of the trammel, so well known among workmen.

If we conceive it to move in the line DE, and the point R in the line AB, while the point M is carried from A, towards E, B, D, until it return to A, the point M will in its progress describe the curve of an ellipsis.

1068. THEOREM IV. The square of the distance of the foci from the centre of an ellipsis is equal to the difference of the square of the semi-axes.

E

G

Let AB (fig. 412.) be the major axis, C the centre, F the focus, and FG the semi-parameter; then will CE-CA2-CF2. For draw CE perpendicular to AB, and join FE. By Cor. 2. Th. II., CA2: CE2::CA2CF2 FG2, and the parameter FG is a third proportional to CA, CE; therefore CA2: CE2:: CE2: FG2, and as in the two analogies the first, second, and fourth terms are identical, the third terms are equal; consequently

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Fig. 412.

B

Coroll. 2. The two semi-axes and the distance of the focus from the centre are the sides of a right-angled triangle CFE, and the distance FE from the focus to the extremity of the minor axis is equal to CA or CB, or to half the major axis.

Coroll. 3. The minor axis CE is a mean proportional between the two segments of the axis on each side of the focus. For CE-CA2- CF2=(CA + CF) × (CA —- CF). 1069. THEOREM V. In an ellipsis, the sum of the lines drawn from the foci to any point in the curve is equal to the major axis.

Let the points F, f (fig. 413.) be the two foci, and M a point
in the curve; join FM and ƒM, then will AB=2CA=FM+ƒM.
By Cor. 2. Th. II., CA: CE2:: CA2-CP2: PM2,
But by Th. IV., CE CA2-CF2;

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And by taking the rectangle of the extremes and means, and dividing the equation by CA2, the result is

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In the same manner it may be shown that FM=CA+ these is FM+ƒM=2CA.

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Coroll. 1. A line drawn from a focus to a point in the curve is called a radius vector, and the difference between either radius vector and half the major axis is equal to half the difference between the radius vectors. For, since

Coroll. 2. Because CF::CP: CA-ƒM.

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Coroll. 3. Hence the difference between the major axis and one of the radius vectors gives the other radius vector. For, since FM + Mƒ=2CA;

Therefore FM=2CA-Mf.

H

E

h

Af

B

G

C

Coroll. 4. Hence is derived the common method of describing an ellipsis mechanically, by a thread or by points, thus:- - Find the foci Ff (fig. 414.), and in the axis AB assume any point G; then with the radius AG from the point F as a centre describe two arcs H, H, one on each side of the axis; and with the same radius from the point ƒ describe two other arcs h, h, one on each side of the major axis. Again, with the distance GB from the point ƒ describe two arcs, one on each side of the axis, intersecting the arcs HH in the points HH; and with the same radius from the point ƒ describe two other arcs, one on each side of the axis, intersecting the arcs described at h, h in the point h, h. find as many points as we please; and a sufficient number being found, the curve will be formed by tracing it through all the points so determined.

H

h

Fig. 414.

In this manner we may

1070. THEOREM VI. The square of half the major axis is to the square of half the minor axis as the difference of the squares of the distances of any two ordinates from the centre to the difference of the squares of the ordinates them

selves.

Let PM and HI (fig. 415.) be ordinates to the major axis AB; draw MN parallel to AB, meeting HI in the point N; then will PM=HN, and MN=PH, and the property to be demonstrated is thus expressed—

CA2: CE2:: CP-CH2: HI2-HNo.

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Or by producing HI to meet the curve in the point K, and making CQ= CP, the property to be proved will be

By Cor. 2. Theor. II.
Therefore

But, by division,
Alternately,

And, since we have above,
Therefore, by equality,

But since

And since

Therefore

CA2: CE2:: PH x HQ: KN.
CA2: CE::CA2-CP2: PM2,
CA2: CE:: CA2-CH2: HIo.

CA2- CH2: CA2-CP2:: HI2: PM2 or HN2;
CA2-CH: CP2-CH2::HI2: HI-HN2.
CA2-CH2 HI2:: CP2-CH2; HI2-HN2;
CA2-CH2: HI2:: CA2: CE,

CA2: CE2:: CP2-CH2: HIo- HN2;
CP2_CH%=(CP−CH)(CP+CH)=PH × QH,
HI2-HN2=(HI−HN)(HI+HN)= NI × KN,
CA2: CE2:: PH x HQ: NI × NK.

X

Coroll. 1. Hence half the major axis is to half the minor axis, or the major axis is to the minor axis, as the difference of the squares of any two ordinates from the centre is to the rectangle of the two parts of the double ordinate, which is the greatest made of the sum and difference of the two semiordinates. For KN=HK+HN=HI+HN, which is the sum of the two ordinates, and NI-HI-HN, which is the difference of the two ordinates. Coroll. 2. Hence, because CP2-CH2=(CP-CH)(CP + CH), and since HI-HNo = (HI-HN)(HI+HN), and because CP-CH= PH and HI-HN=NI; therefore

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