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Therefore, the nearer any place is to the equator, the less will its arctic and antarctic circles be; and on the contrary, the farther any place is from the equator, the greater they are. So that,

At the poles, the equator may be considered as both an arctic and antarctic circle, because its plane is coincident with that of the horizon.

But at the equator (that is, in a right sphere) there is neither arctic nor antarctic circle.

They who live under the northern polar circle, have the tropic of Cancer for their arctic, and that of Capricorn for their antarctic circle.

And they who live on either tropic, have one of the polar circles for their arctic, and the other for their antarctic circles.

Hence, whether these circles fall within or without the tropics, their distance from the zenith of any place is ever equal to the difference between the pole's elevation and that of the equator above the horizon of that place.

From what has been said, it is plain there may be as many arctic and antarctic circles, as there are individual points upon any one meridian between the north and south poles of the earth.

Many authors have mistaken these mutable circles, arid have given their names to the immutable polar circles, which last are arctic and antarctic circles, in one particular case only, as has been shewn.

Problem xix. "lo find the circle or parallel of perpetual apparition, or occupation of a fixed star,

in a given latitude.

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By rectifying the globe to the latitude of the place, and turning it round on its axis, it will be immediately evident, that the circle of perpetual apparition is that parallel of declination which is equal, to the complement of the given lalitu?^ northward; and for the perpetual occupation, it is the same parallel southward ; that is to say, in other words, all those stars whose declinations exceed the co-latitude, will always be visible, or above the horizon; and all those in the opposite hemisphere, whose declination, exceeds the co-latitude, never rise above the horizon.

For instance; in the latitude of London 51 deg. 30 min. whose co-latitude is 38 deg. 30 min. gives the parallels desired; for all those stars which are within the circle, towards the north pole, never descend below our horizon; and all those stars which are within the same circle, about the south pole, can never be seen in the latitude of London, as they never ascend above its horizon.

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OF PROBLEMS RELATING TO THE AZIMUTH, &C. OF THE SUN ANLI STARS.

Problem xx. 7 he latitude of the place and the sun's place being given, to find the suns amplitude.

That degree from east to west in the horizon, wherein any object rises or sets, is called the amplitude.

Rectify the globe, and bring the sun's place to the eastern side of the meridian, and the arch of the horizon intercepted between that point and the eastern point will be the sun's amplitude at rising.

If the same point be brought to the western side of the horizon, the arch of the horizon intercepted between that point and the western point, will be the sun's amplitude at setting.

Thus, on the 24th of May the sun rises at four, with 36 degrees of eastern amplitude, that is, 36 degrees from the east towards the north, and sets at eight, with 36 degrees of western amplitude.

The amplitude of the sun at rising and setting increases with the latitude of the place; and in very high northern latitudes, the sun scarce sets before he rises again. Homer had heard something of this, though it is not true of the Laestrygones, to whom he applies it:

Six days and nights a doubiful course we steer; J

The next, proud Lamps' lofty towers appear, >

And Liestrygonia's gates arise distinct in air. )

The shepherd quitting here at night (he plain,

Calls, tc :ucceed his cares, the watchful twain.

I'ut he (hat scorns the chairs of sleep (o wear,

And adds the herdsman's to the shepherd's tare,

So near the pastures and so short the way, 7

13 :s double toils to ay claim a double pay, *

A.n<l join the labours of the night and ilay,

Problem xxi. To find the sun's altitude at arty given time of the day.

Set the centre of the artificial sun to his place in the ecliptic upon the globe, and rectify it to the latitude and zenith; bring the centre of the artificial sun under the strong brass meridian, and set the hour index to that XII which is most elevated; turn the globe to the given hour, and move the graduated edge of the quadrant to the centre of the artificial sun; and that degree on the quadrant, which is cut by the sun's centre, is the sun's height at that time.

The artificial sun being brought under the strong brass meridian, and the quadrant laid upon its centre, will shewits meridian, or greatest altitude forthat day.

If the sun be in the equator, his greatest or meridian altitude is equal to the elevation of the equator, which is always equal to the co-latitude of the place.

Thus, on the 24th of May, at nine o'clock, the sun has 44 deg. altitude, and at six in the afternoon 2O degrees.

OF THE AZIMUTHAL OR VERTICAL CIRCLES.

The vertical point, that is, the uppermost point of the celestial globe, represents a point in the heavens, directly over our heads, which is called our zenith.

From this point circular lines may be conceived crossing the horizon at right angles.

These are called azimuth or vertical circles. That one which crosses the horizon at 10 degrees distance from the meridian on either side is called an azimuth circle of 1O deg.; that which crosses at 20 is called an azimuth of 2O degrees.

The azimuth of 90 deg. is called the prime vertical: it crosses the horizon at the eastern and western points.

Any azimuth circle may be represented by the graduated edge of the brass quadrant of altitude when the centre upon which it turns is screwed to that point of the strong brass meridian which answers to the latitude of the place, and the place is brought into the zenith.

If the said graduated edge should lie over the sun's centre or place, at any given time, it will represent the sun's azimuth at that time.

If the graduated edge be fixed at any point, so as to represent any particular azimuth, and the sun's place be brought there, the horary index will shew at what time of that day the sun will be in that particular azimuth.

Here it may be observed, that the amplitude and azimuth are much the same.

The amplitude shewing the bearing of any object when it rises or sets, from the east and west points of the horizon.

The azimuth the bearing of any object when it is above the horizon, either from the north or south points thereof. 1 hese descriptions and illustrations being understood we may proceed to'

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Problem xxn. Tojind at what time the sun is due east, the day and the latitude being givtn.

Rectify the globe; then if the latitude and declination are of pne kind, bring the quadrant of alti

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