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PROBLEMS TO ILLUSTRATE THE USE OF THE TABLE.

PROBLEM.

To find the Natural Versed Sine, Natural Versed Sine Supplement,

Natural Co-versed Sine, Natural Sine, and Natural Cosine, of any given Arch, expressed in Degrees, Minutes, and Seconds.

RULE.

Enter the Table, and find the natural versed sine, versed sine supplement, co-versed sine, natural sine, or natural cosine, answering to the given degree, minute, and next less tenth second; to which add the proportional part answering to the odd seconds, found at the bottom of the page, if a natural versed sine, co-versed sine above 90., or natural sine be wanted; but subtract the proportional part, if a versed sine supplement, co-versed sine under 90., or natural cosine, be required : and the sum, or remainder, will be the natural versed sine, natural sine, natural versed sine supplement, co-versed sine, or natural cosine, of the given arch.

Example 1.

Required the natural versed sine, versed sine supplement, co-versed sine, natural sine, and natural cosine, answering to 42:12:36??

To find the Natural Versed Sine:

Natural versed sine to 42:12:30 =
Proportional part to

259293

20

6". =

Add

Given Arch 42:12:36". Natural versed sine = 259313

To find the Versed Sine Supplement :-
Versed sine supplement to 42:12.30”.
Proportional part to

6.

Subtract

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1.740707

20

.

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Required the natural versed sine, versed sine supplement, co-versed sine, natural sine, and natural cosine, answering to 109:53:45”?

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To find the Versed Sine Supplement :-
Versed sine sup. to 109:53:40
Proportional part to

5"

Subtract

659712

23

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- To find the Natural Sine:

Natural sine to 70:6:10. Sup. to 109:53:50"
Proportional part to 5"

Add

940305

8

Supplement 70:6:15. to given arch, nat, sine = 940313

To find the Natural Cosine:

Natural cosine to 70:6:10" Sup. to 109:53:50.
Proportional part to

5"

Subtract

340334

23

Supplement 70:6:15"to given arch, nat, cosine=340311

Remark. Since the natural sines and natural cosines are not extended beyond 90 degrees, therefore, when the given arch exceeds that quantity, its supplement, or what it wants of 180 degrees, is to be taken, as in the above example. And when the given arch is expressed in degrees and minutes, the corresponding versed sine supplement, co-versed sine under 90 degrees, and natural cosine, are to be taken out agreeably to the note in page 57, which see,

PROBLEM II.

To find the Arch corresponding to a given Natural Versed Sine, Versed

Sine Supplement, Co-versed Sine, Natural Sine, and Natural Cosine.

RULE.

Enter the Table, and find the arch answering to the next less natural versed sine, or natural sine, but to the next greater versed sine supplement, co-versed sine, or natural cosine ; the difference between which and that given, being found in the bottom of the page, will give a number of seconds, which, being added to the arch found as above, will give the required arch,

Example 1.

Required the arch answering to the natural versed sine 363985 ?

Solution. The next less natural versed sine is 363959, corresponding to which is 50:30'10"; the difference between 363959 and the given natural versed sine, is 26; corresponding to which, at the bottom of the Table, is 7", which, being added to the above-found arch, gives 50:30:17", the required arch.

Note.—The arch corresponding to a given natural sine is obtained precisely in the same manner.

Example 2.

Required the arch corresponding to the natural versed sine supplement 1.464138?

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Solution. The next greater natural versed sine supplement is 1.464155; corresponding to which is 62:20:40"; the difference between 1.464155 and the given natural versed sine supplement, is 17; answering to which, at the bottom of the Table. is 4", which, being added to the above-found arch, gives 62:20:44", the required arch.

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Note.-The arch corresponding to a given co-versed sine, or natural cosine, is obtained in a similar manner.

Remark 1.

The logarithmic versed sine of an arch may be found by taking out the common logarithm of the product of the natural versed sine of such arch by 10000000000; as thus :

Required the logarithmic versed sine of 78:30:45”.? The natural versed sine of 78.30:45" is . 800846, which, being multiplied by 10000000000, gives 8008460000; the common log. of this is 9.903549; which, therefore, is the logarithmic versed sine of the given arch, as required.

Remark 2.

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The Table of Logarithmic Rising may be readily deduced from the Natural versed sines; as thus :

Reduce the meridian distance to degrees, by Table I., and find the natural versed sine corresponding thereto; now, let this be esteemed as an integral number, and its corresponding common log. will be the loga

rithmic sising.

Example.

Required the logarithmic rising answering to 4:50":45?

4.50:45! = 72:41:15, the natural versed sine of which is 702417; the common log. of this is 5. 846595, which, therefore, is the logarithmic rising required.

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Logarithms are a series of numbers invented, and first published in 1614, by Lord Napier, Baron of Merchiston in Scotland, for the purpose of facilitating troublesome calculations in plane and spherical trigonometry, These numbers are so contrived, and adapted to other numbers, that the sums and differences of the former shall correspond to, and show, the products and quotients of the latter.

Logarithms may be defined to be the numerical exponents of ratios, or a series of numbers in arithmetical progression, answering to another series of numbers in geometrical progression; as,

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0. 1. - 2. 3. 4. 5. 6.

7.

8. ind. or log. 1. 10. 100. 1000. 10000. 100000 1000000 10000000. 100000000

ge.pro.

Whence it is evident, that the same indices serve equally for any geometrical series; and, consequently, there may be an endless variety of systems of logarithms to the same common number, by only changing the second term 2. 3. or 10. &c. of the geometrical series of whole numbers.

In these series it is obvious, that if any two indices be added together,

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