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;3999.17, &c. (equal to Half the Side of a Decagon inscribed in the Circle) let the Co-fine of 3", the Difference between 18° and 15°, be found *; from which the Co-fine of 45' will be had, by two Bisections only: Whence the Sines of all the Arches in the Progression 1° 30', 2° 15', 3°oo', 3° 45', &c. may be determined (by Theor. 1. p. 13.) and that to any assigned Degree of Exačtness.
The Sines of all the Terms of the Progression 45", 1° 30', 2° 15', &c. up to 60°, being thus derived, the next thing is to find, by Help of these, the Sines of all the intermediate Arches, to every single Minute.
This, if you desire no more than the 4 or 5 first Places of each (which is exact enough where nothing less than Degrees and Minutes is regarded), may be effected by barely taking the proportional Parts of the Differences.
But if a greater Degree of Accuracy be insisted on, and you would have a Table carried on to 7 or 8 Places, each Number (which is sufficient to give the Value of an Angle to Seconds, and even to Thirds, in most Cases), then the Opperation may be as follows:
3°. To the lesser Extreme add the foremention’d Excess; and, to the Sum, add the first Remainder 3 to this Sum add the next Remainder, and so on continually: Then the several Sums thus arising will respectively exhibit the Sines of all the intermediate Arches, to every single Minute, exclusive of the last; which, if the Work be right, will agree with the greater Extreme itself, and therefore will be of Use in proving the Operation.
But to illustrate the Matter more clearly, let it be proposed to find the Sines of all the intermediate Arches between 3° oo' and 3° 45', to every single Minute, those of the Extremes being given, from the foregoing Method, equal to Soś233595 and ,0654.0312 respectively. Here, the Sum of the Sines of the Extremes being multiplied by ,ooooooo.423, the first Produćt will be ,OOOOOOOO498, &c. or ,oooooooo.50, hearly, (which is sufficiently exačt for the present Purpose); and this, again, multiplied by 22, gives soooooo 1 1. for a 2d Produćt; which added to ,ooo2903815, * Part of the Difference of the two given Extremes, will be ,ooo2904915, the Excess of the Sine of 3° ol' above that of 3° oo'. From whence, by proceeding according to the 2d and 3d Rules, the Sines of all the other intermediate Arches are had, by Addition and Subtraction only. See the Operation,
Again, as a second Example, let it be required to find the sines of all the Arches, to every Mi
nute, between 59° I two Extremes being
After the same Manner the Sines of all the intermediate Arches between any other two proposed Extremes may be derived, even up to 90 Degrees;
C 2 but
but those of above 60° are best found from those below, as has been shewn elsewhere. The Reasons upon which the foregoing Operations are founded, depend upon Principles too foreign from the main Design of this Treatise, to be explained here (even would Room permit); however, as to the Correctness and Utility of the Method itself, I will venture to affirm, that, Whoever has the Inclination, either to calculate new Tables, or to examine those already extant, will not find -One §. of the Trouble, this Way, as he un3.
avoidably must according to the common Methods.