Required the natural versed sine, versed sine supplement, co-versed sine, natural sine, and natural cosine, answering to 109:53:45?? Given arch. . 109:53:45 Nat. versed sine 1.340311 To find the Natural Sine : Natural sine to . 70:6:10 Sup. to 109:53:50" 940305 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. 340334 5" . Subtract 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? 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. 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. 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 logarithmic rising. Example. Required the logarithmic rising answering to 4:5045!? 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. TABLE XXVIII. Logarithms of Numbers. 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, 5. 6. 7. 0. 1. 2. 3. 4. 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, their sum will be the index of that number which is equal to the product of the two terms, in the geometrical progression to which those indices belong: thus, the indices 2. and 6. being added together, make 8; and the corresponding terms 4. and 64. to those indices (in the first series), being multiplied together, produce 256, which is the number corresponding to the index 8. It is also obvious, that if any one index be subtracted from another, the difference will be the index of that number which is equal to the quotient of the two corresponding terms: thus, the index 8. minus the index 3 = 5; and the terms corresponding to these indices are 256 and 8, the quotient of which, viz., 32, is the number corresponding to the index 5, in the first series. And, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power; thus, the index, or logarithm of 16, in the first series, is 4; now, if this be multiplied by 2, the product will be 8, which is the logarithm of 256, or the square of 16. Again,-if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm of that root: thus, the index or logarithm of 256 is 8; now, 8 divided by 2 gives 4; which is the logarithm of 16, or the square root of 256, according to the first series. The logarithms most convenient for practice are such as are adapted to a geometrical series increasing in a tenfold ratio, as in the last of the foregoing series; being those which are generally found in most mathematical works, and which are usually called common logarithms, in order to distin guish them from other species of logarithms. In this system of logarithms, the index or logarithm of 1, is 0; that of 10, is 1; that of 100, is 2; that of 1000, is 3; that of 10000, is 4, &c. &c.; whence it is manifest, that the logarithms of the intermediate numbers between 1 and 10, must be 0, and some fractional parts; that of a number between 10 and 100, must be 1, and some fractional parts; and so on for any other number: those fractional parts may be computed by the following Rule. To the geometrical series 1. 10. 100. 1000. 10000. &c., apply the arithmetical series 0. 1. 2. 3. 4. &c., as logarithms. Find a geometrical mean between 1 and 10, or between 10 and 100, or any other two adjacent terms of the series between which the proposed number lies. Between the mean thus found and the nearest extreme, find another geometrical mean in the same manner, and so on till you arrive at the number whose logarithm is sought. Find as many arithmetical means, according to the order in which the geometrical ones were found, and they will be the logarithms |