« ForrigeFortsett »
But notwithstanding all these Artificies and Comendiums, a Method (similar to that in Page 16.) for finding the Logarithms of large Numbers, one from another, by Addition and Subtraction, only, . still seems wanting in the Calculation of Tables. I shall, therefore, here subjoin such a Method.
1. Let A, B and C denote any three Numbers in arithmetical Progression, not less than Ioooo each, whereof the common Difference is Ioo.
2. From twice the Logarithm of B, subtract the Sum of the Logarithms of A and C, and let the Remainder be divided by Ioooo.
3. Multiply the Quotient by 49,5, and to the Produćt add or a Part of the Difference of the Logarithms of A and B; then the Sum will be the Excess of the Logarithm of A + 1 above that of A.
Thus, let it be proposed to find the Logarithms of all the whole Numbers between 17900 and 1810o; those of the two Extremes 1790o and 18100, and that of the mean (18oco) being given.
soooooooo.134 (see Rule 2.) which multiplied by 49,5, and the Produćt added to Log B - Log. A IOO gives soooo.24261 of for the Excess of the Logarithm of A + 1 above that of A (by Rule 3.) From whence the Work, being continued according to Rule 4 and 5, will stand as follows.
* . - Note,
The Logarithms found according to this Method, in Numbers between IOOoo and 20000, are true to 8 or 9 Places of Figures: Those of Numbers between 20000 and 5oooo err only in the 9" or io" Place; and those of above 50ooo are trug to Io Places, at least.
Again, for AB, it will be, as Radius: Sine of C (54°40').; : AC (17919) : A B (by Theorem 2.) Whence, by adding the Logarithms of the second and third Terms together and subtraćting that of the first (as above), we have AB = 14611. See the Operation. -
C Moreover, in the oblique plane Triangle ABC, let there be given AB = 75, AC = 6o, and the included Angle A = 48°; to find the other two An
A B gles. Then (by Theort
- - 5.) it will be