A TABLE,

Showing how many geographical and American miles make a degree of longitude in every degree of latitude.

Deg. Geo. Am. Deg. Geo. Am. Deg. Geo. Am.

Lat. M'ls. M'ls.

Lat. M'ls. M'ls. Lat. M'ls. M'ls.

The above table is thus calculated:

Multiply the cosine of the latitude by 60, and you will find the length of one degree on that latitude in geographical miles; then the geographical miles being multiplied by 1.152, will give the American miles.

For instance, to find how many geographical and American miles make one degree in the parallel of 84. In the first place, the cosine of $4 degrees, taken from a table of natural sines and cosines, is .104528 to radius unity; now .104528 multiplied by 60, gives 6.27168, the number of geographical miles in one degree on that parallel of latitude; but 6.27, which is the number in the table, will answer our present purpose. Again, 6.27 multiplied by 1.152 will give 7.22304, the number of American in one degree on the same parallel of latitude; three decimal places are rejected, and only 7.22 inserted in the table.

The reason of the preceding calculation is evident from this principle, that the circumferences of circles are to each other as their radii; and that the radius of_any parallel of latitude is equal to the cosine of that latitude; hence, if the radius of the equator be taken equal to unity, it follows that unity, or 1, is to cosine of any latitude, so is 60 geographical miles, the length of a degree on the equator, to the number of geographical in one degree of longitude on that parallel of latitude; and, consequently, the cosine of any latitude multiplied by 60, will give the length of one degree of longitude in that parallel of latitude.

The intelligent student, who is curious to make the calculation of the preceding table, will find a correct table of natural sines and cosines in my edition of Gibson's Surveying.

PROBLEM XVIII.

To find at what rate per hour the inhabitants of any given place are carried, from west to east, by the revolution of the earth on its axis.

RULE. Find how many miles make a degree of longitude in the latitude of the given place, (by the preceding Prob. or the annexed table,) which multiply by 15 for the answer.

The reason of this rule is obvious, for if m be the number of miles contained in a degree, we have 24 hours : 360o multiplied by m:: 1 h.: the answer; or, which amounts to the same thing, 1:15×m::1:the answer; therefore, the number of geographical, or American miles in a degree of longitude in any given latitude, multiplied by 15, will duce the answer in geographical, or American miles.

The above rule is on a supposition that the earth revolves on its axis, from west to east, in 24 hours; but it has been already observed, (Chap. VII. Art. 6,) that the earth makes one complete revolution on its axis in 23 hours, 56 minutes, 4.1 seconds; hence, where greater accuracy is required, we must multiply the number of geographical miles by 15.041 for the answer.

EXAMPLES.

1. At what rate per hour are the inhabitants of Pekin carried from west to east by the revolution of the earth on its axis ?

Answer. The latitude of Pekin is 400, in which parallel a degree of longitude is equal to 46 geographical, or 53 American miles. (See Ex. 1. Prob. XVII.) Now, 46 multiplied by 15, produces 690, and 53 multiplied by 15 produces 795; hence, the inhabitants of Pekin are carried 690 geographical, or 795 American miles per hour.

By the table. In latitude 400 a degree of longitude is equal to 45.96 geographical miles, and 52.94 American miles. Now, 45.96 multiplied by 15, produces 689.4; and 52.94 multiplied by 15 will give 794.1: Hence, the inhabitants in this parallel are carried 689.4 geographical, or 794.1 American miles per hour, by the earth's revolution on its axis; which result is more correct than the former. And, if we multiply 45.96 by 15.041, and also 52.94 by 15.041, the answer will be found still more correctly.

2. At what rate per hour are the inhabitants of the following places carried, from west to east, by the revolution of the earth on its axis: Truxillo, a town in Peru; Sofala, a town in Africa, and capital of a country of the same name; Lahore, a city of Asia, and the capital of a province of the same name, several times the capital of Hindoostan and the residence of the great Moguls; Kiev, a city in European Russia, situated on the right bank of the Dnieper; and Christiana, the most beautiful city in Norway, situated in a bay or gulf, about 25 miles from the sea.

PROBLEM XIX.

The hour of the day at any particular place being given, to find what hour it is in any other place.

RULE. Bring the place at which the time is given to the brass meridian, and set the index of the hour circle to the given hour at that place: then, turn the globe till that place for which the time is required be brought to the meridian, and the index will show the hour at that place.

If the place where the hour is sought lie to the east of that wherein the time is given, turn the globe westward; but if it lie to the west, the globe must be turned eastward.

Or, bring the given place to the meridian, and set the index of the hour circle to 12; turn the globe (as before) till the other place comes to the meridian, and the hours passed over by the index will be the difference of time between the two places.

If the place where the hour is sought, lie to the east of that wherein the hour is given, the difference of time must be added to the given time; but if to the west, subtract the difference of time: Thus, a place 15 degrees to the eastward of another, has the sun on its meridian an hour earlier than the latter place; therefore, when it is 12 o'clock in the former place it is but 11 o'clock in the latter; and 12 o'clock in the latter place corresponds to 1 o'clock in the former, &c.

Or, without the hour circle, find the difference of longitude between the two places, (by Prob. VI.) and convert it into time by allowing 15 degrees to an hour, or 4 minutes of time to one degree. The difference of longitude in time, will be the difference of time between the two places, with which proceed as in the last rule.

To convert degrees, minutes, and seconds into time, at the of 360 degrees for 24 hours, and the contrary,一

Say as 360°: 24h. or as 150: 1h.:: any number of degrees, &c.: the time required.

The converse of this rule will give the degrees. Hence, degrees of longitude may be converted into time by multiplying by 4, observing that minutes or miles of longitude multiplied by 4, produce seconds of time, and degrees of longitude, when multiplied by 4, correspond to minutes of time: and, on the contrary, minutes of time divided by 4, will give degrees of longitude: if there be a remainder after dividing by 4, multiply it by 60, and divide the product by 4, or, which amounts to the same thing, multiply the remainder by 15, the quotient in the former case, or the product in the latter, will be minutes of a degree, or miles of longitude.

EXAMPLES.

1. When it is 9 o'clock in the morning at NewYork, what hour is it at Dieppe, a sea-port of France, in the English Channel?

By the first method. Bring New-York to the meridian, and set the index of the hour circle to 9 o'clock; then, by turning the globe westward till Dieppe comes to the meridian, the index will point to 2 o'clock nearly, which is the hour at that place; hence, as Dieppe lies to the east of NewYork, when it is nine in the morning at the latter place, it is two in the afternoon at the former.

By the second method. Bring New-York to the meridian and set the index to 12 o'clock, then, by turning the globe, as before, till Dieppe be brought to the meridian, the hours passed over by the index will be five, which is the difference of time between both places. And, because Dieppe lies to the east of New-York, this difference of time must be added to the given time; that is, 5 hours added to 9 hours will give 14 hours; consequently, it is 2 hours past noon, or 2 o'clock in the afternoon at Dieppe.

By the third method. The difference of longitude between both places is found (by Prob. VI.) to be 75°5′. Now 75 degrees, divided by 15, will produce 5, and 5' multiplied by 4 will give 20; hence, the difference of time corresponding to the difference of longitude, is equal to 5 hours, 20 seconds, with which proceed as in the last method, and you will find the time at Dieppe to be 2 hours and 20 seconds past 12 o'clock, when it is nine in the morning at New-York, which is nearly the sume as before.