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its tangent set off from G, given F the centre; through G describe the c.rele GI from the centre F.

Cor. Hence, a great circle perpendicular to the primitive, is a right-hne CDE drawn throngb the centre perpendicular to the line of measures.

N.B. When the centre F lies at too great a distance, draw EG io cut AB in H; or lay the semi-tangent of DG from C to H; and, throngi tie thuec points G, H, I, draw a circle (which may be done, if only a small portion is required, by an instrument called a cylographi, invented by the Editor of this work).

Prop. 13. To describe an oblique circle, at a given distance from a given pole.

Draw the line of measures AB through the point p, if it is given, and draw DE perpendicular to it; also draw EpP; or, ii the pointp is not given, set off the height of

PB the pole above the primitive from B to P, then from P set off PH=PI equal the distance of the circle from

E its pole, and draw EH and EI to intersect AB in F and G. About the diameter FG describe the required circle.

The same by the Scale. If the point P is given, apply Cp to the semi tangent, and it will give the distance of the pole from D, the pole of projection opposite to the projecting point. This distance being obtained, it will be easy to find the greatest and nearest distances of the circle from the pole of the primitive opposite to the projecting point; take the semi-tangent of these distances, and set thein off from C to G and F, both the same way if the circle lie all on one side; but each its own way, if on different sides of D.

And then FG is the diameter of the circle required.

Cor. t. If F be the pole of a great circle as of DLE, draw EFH through the pole F; make HK equal 90° ; draw EK cutting the line of measures wu L, through the three points D, L, E, draw the required great circle.

Cur. 2. Hence it will be easy to draw one circle parallel to another.

Prop. 14. Through two given points A and B, to draw a gieat circle. Through one of the points A, and

T through the centre draw a line ACG, and EF perpendicular to it; draw AE and EG perpendicular to AE ; then, through the three points A, B, G, draw G the required circle. Or thas, from E, found as before, draw

1 EH, HCI, and EIG, which gives a third point G, throngh which the circle must pass.

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The same by the Scale. Draw ACG, and apply AC to the semi tangents; find the degrees; set off the semi-tangent of its supplement from C to G for a third point

Or tons, apply AC to the tangents, and set off the tangent of its complement from C to G; and, through the three points A, B, G, describe the required circle.

For, since HEI, or AEG, is a right-angle, A and G represent opposite points of the sphere, and whence all circles, passing through A and G, are great sircles.

Prop. 15. About a given pole, and through a given point, to describe the representation of a circle.

Let P be the pole, and B the given point; through P and B de- IL scribe the great circle AD, by Prop.

B 14, the centre of which is E; through the centre C draw CPH ;

А and, from the centre E, draw EB

C

D and BF perpendicular to it. With the centre F, and radius FB, decribe the required circle BHA,

E

Prop. 16. To find the poles of any circle FNG.
Through its centre

D draw the line of mea

H sures AG, and DE

K perpendicular to it : draw EFH, and set off its distance from H to P, and draw f 1 ΑΙ

P" B Epp, then p is the pole.

Or thos, draw EFH ak EIG, and bisect HI in P, and draw EpP, and p is the internal pole. Lastly, draw PCQ, EQq, and q is the external pole.

In a great circle DLE, draw ELK, and make DHAK, or KH=AD, and draw EFH, and F is the pole.

L

C F

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The same by the Scale. Apply CF to the semi-tangents, and note the degrees. Take the sum of the degrees, and of the distance of the circle from its pole, if the circle be all one side, but their difference if it encompass the pole of projection, set off the semi-tangent of this sum, or difference, from C to the internal pole p, and the semi-tangent of its supplement Cp, will give the external pole g.

Or thus, apply CF and CG to the semi-tangents, set off the semi-tangent of half the sum of the degrees, if the circle lies ali one way, or of half the difference if it encompass the pole of projection, from C to the pole p, and the semi tangent of the supplement, Co gives the external pole .

In a great circle, as DLE, draw the line of measnres AB perpendicular to DE, and set the tangent and co-tangent of half its inclination from the centre C, different ways, to F and f.

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Prop. 17. To draw a great circle at any given inclination above the primitive ; or, making any given angle with it at a given point.

Draw the line of measures AB, and DCE perpendicular to it;

H make EK=2HD equal twice the complement of the inclination of

CVL BE the circles; or DK = 2AH = twice the inclination ; and draw EKF, then F is the centre of RGD, the circle required.

E Or thus, draw De and AB perpendicular to it, and let D be the poiot given, make AH equal to the inclinatiou, and draw EGH and HCN, and ÉNO to cut AB in O. Then bisect GO in F, for the centre of the required circle.'

The same by the Scale. Set off the tangent of the inclination on the line of measures from C to F, then F is the centre. Set off the semi-tangent of the complement from C to G, then GP or DF is the radius.

Or the secant of the inclination being set off from G or D to F, will give the centre.

Prop. 18. Through a given point P, to draw a great circle, to make a given angle with the primitive.

D Through the point given P, and the

K centre C, draw the line AB and DE per

L pendicular to it; set off the given angle HA from A to H and from H to K, and draw BGK; with the radius CG, and centre AH

B

C C, describe the circle GIF; and, with the radius BG, and centre P, cross that circle in F; then, with the radius FP,

M and centre F, describe the circle LPM, required.

E The same by the Scale. With the tangent of the given angle, and one foot in C, describe the arc FG with the secant of the given angle, and une foot in the given point P. cross that arc at F; from the centre F describe a circle througli the point P.'

Prop. 19. To draw a great circle to make a given angle with a given oblique circle FPR, at a given point P in that circle. Through the centre C, and the

given

FG point P, draw the right-line DE, and

D AB perpendicular to it; draw APG, and make BM=2DG ; and draw AM to cut DE in I; draw IQ perpendicular to DE, then IQ is the line

II. wherein the centres of all circles are found, which pass through the point P. Find N, the centre of the given circle FPR, and make the angle NPL equal to the given angle, da L is the centre of the circle HPK, as

P

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The same by the Soole. Through P and C draw DE; apply CP to the semi-tangents, and set off the tangent of its complement from C to I, or the secant from P 10 I; on DI erect the perpendicular IQ; find the centre N of FPR, and make the angle NPL eqnal the angle given, and L is the centre.

Cor. If one circle is to be drawn perpendicular to another, it must be drawn through its poles.

Erop. 20. To draw a great circle through a given point P, to make a given angle with a given great circle DE. About the given point P, as a pole,

D

T by Prop. 13, Cor. 1, describe the

к great circle FG; find I, the pole of

L the given circle DE, and by Prop. 16;

P aboutthe pole I, by Prop. 13, describe the small circle HKL, at a distance equal to the given angle, to intersect FG in H; about the pole H de

H scribe, by Prop. 13, the great circle APB as required.

E Prop. 21. To draw a great circle to cut two given great circles abd, elf, at given angles. Find the poles s, r of

d the two given circles, by Prop. 16, about which draw two parallels phk and pnk, at the distances respectively equal to the angles given, by Prop. 13, the point of intersection p, is the pole of the required circle moq.

Cor. Hence, to draw a right circle, to make with an oblique circle abd, any given angle.

Draw a parallel hk at a distance from the pole of the oblique circle equal to

pildo the given angle. Its intersection f with the primitive, gives the pole of the right circle gct, as required.

Prop. 22. To lay any number of degrees on a great circle, or to measure any arc of it. Let AFI be the primitive ; find

R the internal pole P of the given cir. cle DEH, by Prop. 16, lay the de

F grees on the primitive from A to F,

OI and draw PA, PF, intercepting the

ADCP part DE required.

Or, to measure De, draw PEF and PDA, and AF is its measure, which, applied to thu line of chords, shows how many degrees it contains.adid

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Or thus, find the external pole p of the given circle, set off the given degrees from I to K, and draw pl, på, intercepting the part DE required; or, to measure DE, through D and É draw pl, PK, then KI is the measure of DE.

Or this, through the internal pole P draw the lines DPG and EPL, setting off the given degrees from G to L on the circle GL; then DE is the arc required. Or, if it be required to measure DE, the degrees in the arc GL is the measure of DE,

Or thus, set off the given degrees from G to H on the circle GL, and, from the external pole P draw pg, pH, intercepting DE, the arc required; or, to measure DE, draw pDG, PEH, then the degrees on GH are equal to DE.

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By the Scule for Right Circles. See fig. Prop. 22. Let CA be the right circle, take the number of degrees from the semi-tangents, and set them off from C to D for the arc CD.

Or, if the given degrees are to be set off from A, then take the degrees off from the semi-tangents from 90° towards the beginning, and set them off from A to D, and if CD was to be measured, apply it to the beginning of the semitangents; and, to measure AD, apply it from 90° backwards, and the degrees intercepted gives its measure.

N.B. The primitive is measured by the line of chords, or else it is actually divided into degrees.

Prop. 23. To set off any number of degrees on a lesser circle, or to measure any arc of it.

Fig. 1. Let the lesser circle be DEH; find its internal pole P, by Prop. 16, describe the circle AFK parallel to the primitive, by Prop. 11, and as far from the projecting point, as the given circle DE is from its internal pole P, set off the given degrees from A to F, and draw PA and PF intersecting the given circle in D and E ; then DE is the arc required. Or, to measure DE, draw PDA and PEF, and AF exhibits the degrees on DE.

Or thus, find the external pole p, of the given circle, by Prop. 16, describe the lesser circle AFK as far from the projecting point, as DE the given circle is from its pole p, by Prop 11 ; set off the degrees from I to K, and draw pDI and pEK, then De represents the given number of degrees ; or to measure DE, draw pdi, and pĖK, aud KI is the measure of DĚ.

Or thos, let O be the centre of the given circle DEH; through the internal pole P, draw the lines DPG and EPL, divide the quadrant GQ into ninety equal degrees, and if the degrees be set off from G to L, then DE will repre sent these degrees. Or the degrees in GL will measore DE.

Or thus, divide the quadrant GR into ninety equal parts, or degrees, and set off the given degrees from G to H, and draw pDÅ, and pEH, from the external pole p; then DE will represent the given degrees.

Or, through D and E, draw pDG, and rEH, then the number of equal de grees op GH is the measure of DE.

Scholium. Any circle parallel to the primitive is divided or measured, by drawing lines from the centre to the divisions of the primitive.

Prop. 24. To measure an angle:

By Cor. 1, Prop. 13, about the angular point as a pole, describe a great circle, and note where it intersects the legs of the angle; through these points of intersection, and the angular point, draw two right lines to cut the primitive; the arc of the primitive, intercepted between them, is the measure of the angle. This needs no example.

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