By thefe two laft Examples the Logarithm of 10 may be found; thus 3XL2 L23L8= L&= 2.0794415416797 2231435513142 Nap. Log. of 8× = Nap. Log. of 10=2.3025850929941 3. Let it be required to find the Value of the Index n for making Briggs's Logarithms, his Logarithm of 10 being 1. The Question propos'd is the fame with this; viz; 10000 &c. indefinitely Quere n. 12 × Naper's Logarithm of 10 is 1. SOLUTION. 10000 &c. indefinitely 72 X × 2.302585092994 &c. is 1 (by the preceeding ); confequently (by multiplying each part by 1) 10000 &c. indefinitely x 2.302585, &c. = n = 2302585092994 &c. indefinitely 2302585092994045684 017991454684364207601101488628772976033328 &c. indefinitely as computed by others; therefore 10000 &c. indefinitely = B. 43429448190325182765112 891891660508229439700580366656611445, &c; and confequently 2 ß = b = .8685889638065036 &c. Note, Note, The Logarithms of 2 and 3 may be expeditiously had by finding the Logarithms of and (byRule 2), whofe Sum is the Logarithm of x = 2; And this Logarithm 2 added to that of is the Logarithm of x2 = 3: But they along with the Logarithm of 10 may be fooner had by finding the Logarithms of, and ; thus Firft for Naper's Logarithm of Herer 15, S = 16, and 16-15 =3= aa 16+15 961) 1. And, by Rule 2. OPERATION. = 2x=ba=.0645161290322 | 1)ba=.0645161290322 961)baba' 6713436943)ba3 961) baba- 6985885)bas= 961)ba-ba 7267)ba?= 223781231 139718104 2x ba=.0408163265306 ba=.0408163265306 2401)baba3 1699971953)ba3= 56665732 2401)babas 2401)ba-ba 708025)bas: = 14160 4 Naper's Logarithm of 4.0408219945202 And next for Naper's Logarithm of 8 I Here r = 80, S = 81, and a = 81-80 81+80=; and aa 25921) 1. Now by Rule 2. OPE OPERATION. 2×18 ba=.0124223602484 | ba=.0124223602484 1597464 .6931471805599. th 1st 5th 6th L 6th 3 x L 2 10th L 2 L2 = L 31.0986122886681. L 82.0794415416798. 5th-10th 11hL10=2.302 5850929940 . Note, Naper's Logarithm of 10, and of any other Number being known, Briggs's Logarithm of the faid Number may be found by this Proportion. As Naper's Logarithm of 102.30258 &c. Is to Briggs's Logarithm of 10 = 1 So is Naper's Logarithm of any other Number To Briggs's Logarithm of the faid Number. But the Index b being known, the best way is to find Briggs's Logarithm a-rew: Thus, Ex. I. Let it be required to find Briggs's Logarithm of 2 to eleven Places. Note, The Index b must have a Figure, at least, more than the intended Logarithm is to have; therefore in this Ex. it must have 12 or 13 Figures in it; viz. b. 868588963806. The most expeditious Method of finding Briggs's Logarithm of 2, that I know, is thus, 21° 1024; and, putting 1000 and s 1024, we r 24 have LL12, which (when found by the following Operation) being added to the Logarithm of r which is 3, will give the Logarithm of xr, or of s, viz. of 21; °; and Ex. II. Let it be required to find Briggs's Logarithm of 3 to ten or eleven Places. Since Briggs's Logarithm of 2 is known, his Logarithm of 3 may be foon had by finding his Logarithm of 1, by Rule II, and adding it to the Logarithm of 2, which Sum will give the Logarithm of 1 x 23: But it may be fooner found by Rule IV. Thus: z being=3, and d=2, s will be 4, and r2; and therefore dd+2rs=y=1+16=17, and y y 289. Ex. III. Let it be required to find Briggs's Logarithm of 20001, by Ru'e V. ++ Here t =20001; and Briggs's Logarithm of ± t- -I L300 L 1000L1.0001 is fuppofed to be known, it being.00004342727687=u: Ex. IV. Let it be required to find Briggs's Logarithm of the prime Number 17. 1. Since Briggs's Logarithm of the adjoining Number 16 (24) which is, by Ex. I. 4x. 3010299956639= 1.2041199826559., is known, that of 17 is foon found by Rule II. from the Logarithm of 7; for this added to the Logarithm of 16 gives the Logarithm of 17. Herer 16, s=17; ==a; and 1089) Iaa. 17-16 therefore 17+16=3 OPERATION... .86858&c.xba=.0263208776911 | ba.02632087769£1 1089)baba3 1089)ba-ba 1089)ba-ba 2416976833)ba 3 221944 5ba5 203 7)ba7 80565894 44389 29 Sof 1.0263289387223 Briggs's Logarithm of 16-1.2041199826559. Briggs's Logarithm of 171.2304489213782 |