Sidebilder
PDF
ePub

Rx (=,8635 &c.) =,006839283 &c.

127

[blocks in formation]

-S=

7.

,006839424 &c. and 2 Log. 8-Log. 9-S ,845098040 &c. the common Logarithm of required. But the fame Conclufion may be brought out by fewer Terms of the Series, if the Logarithms of the three firft Primes 2, 3 and 5 be fuppofed known; because those of 48 and 50 (which are compofed of them) will likewife be known: From whence the Logarithm of 7 (= Log. 49= 14 Log. 48 +Log.50, +S' will come out,845098040

4

will

&c. (as before) Which Value will be true to 1 Places of Figures by taking the firft Term of the Series, only.

Again, let the common Logarithm of the next Prime Number, which is 11, be required. Here a may be taken = 10, 11 and c = 12; but fewer Terms of the Series will fuffice, if other three Numbers, compofed of 11 and the inferior Primes, be taken, whereof the common Difference is an Unit. Thus, becaufe 982 × 7×7,99 3 x 11 (9 × 11), and 100 = 2 × 2 × 5 × 5 (or 10 × 10), let there be taken a 98, b99, and c=100; and then, by the first Term of the Series only, the Log. of

=

3x

99 will be found true to 14 Places; whence that of 11 (Log. 99-Log. 9) is also known.

But

But notwithstanding all these Artificies and Compendiums, a Method (fimilar to that in Page 16.) for finding the Logarithms of large Numbers, one from another, by Addition and Subtraction, only, ftill feems wanting in the Calculation of Tables. I fhall, therefore, here fubjoin fuch a Method.

1. Let A, B and C denote any three Numbers in arithmetical Progreffion, not less than 10000 each, whereof the common Difference is 100.

2. From twice the Logarithm of B, fubtract the Sum of the Logarithms of A and C, and let the Remainder be divided by 10000.

3. Multiply the Quotient by 49,5, and to the Product add Part of the Difference of the Logarithms of A and B; then the Sum will be the Excels of the Logarithm of A+ 1 above that of A.

4. From this Excess let the Quotient (found by Rule 2.) be continually fubtracted, that is, first from the Excess itself, then from the Remainder, then from the next Remainder &c. &c.

5. To the Logarithm of A add the said Excess, and to the Sum add the firft of the Remainders; to the last Sum add the next Remainder &c. &c. then the feveral Sums, thus arifing, will exhibit the Logarithms of A + 1, A + 2, A+ 3 &c. refpectively.

Thus, let it be propofed to find the Logarithms of all the whole Numbers between 17900 and 18100; thofe of the two Extremes 17900 and 18100, and that of the mean (18000) being given.

Then,

[blocks in formation]

,00000000134 (fee Rule 2.) which multiplied by A 49,5, and the Product added to Log. B-Log. Á

100

gives ,00002426107 for the Excefs 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.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

The Logarithms found according to this Method, in Numbers between 10000 and 20000, are true to 8 or 9 Places of Figures: Thofe of Numbers between 20000 and 50000 err only in the 9th or 10th Place; and thofe of above 50000 are true to ro Places, at least.

Having explained the Manner of constructing a Table of Logarithms, and that by various Methods, I now come to fhew the Ufe of fuch a Table in the Business of Trigonometry.

Firft, in the rightangled plane Triangle ABC, let there be given the Hypothenufe AC= 17910 Feet, and the Angle A 35° 20′; to find the Perpendicular BC and the Base AB.

[blocks in formation]

Here, because Radius: Sine 35° 20′ :: 17910: BC (by Theor. 2. p. 4.) we have BC= Sine 35° 20'x17910

Radius

Therefore, because the Addition and Subtraction of Logarithms, answers to the Multiplication and Divifion of the natural Numbers (fee p. 36, 37.) we have Log, BC Log. Sine 35° 20′ + Log. 17910— Log. Radius.

But, by the Tables of artificial, or logarithmic, Sines*, the Log. Sine of 35° 20′ will appear to be 9,7621775; to which add 4,2530956, the Log. of 17910, and from the Sum (14,0152731) take 10, the Log. of Radius, and there refults 4,0152731 the Log. of BC; which, in the Tables, anfwers to 10358, the Length of BC required.

*A Table of Artificial Sines is nothing more than a Table of the Logarithms of the Numbers expreffing the Natural Sines, to the Radius 10000.00000; whofe Logarithm is 10.

E

Again,

Again, for AB, it will be, as Radius : Sine of C (54° 40′): AC (17910) AB (by Theorem 2.) Whence, by adding the Logarithms of the fecond and third Terms together and fubtracting that of the first (as above), we have AB=14611. See the Operation.

[blocks in formation]

Log. Sine C (54° 40′) 9,9115844

Log. AC (17910)

Log. AB 14611

A

As AB+ AC (135)

[ocr errors][merged small][merged small][merged small][merged small]

4,2530956

4,1646800

Moreover, in the oblique plane Triangle ABC, let there be given A B = 75, AC 60, and the included Angle A= 48°; to find the other two AnB gles. Then (by Theorem 5.) it will be

its Log. 2,1303338

its Log. 1,1760913

10,3514169

11,5275082

C-B

to the T.

= 14° 00′

9,3971744

2

Which 14°, added to 66°, the half Sum of the Angles C and B, gives the greater C

80°; and

fubtracted therefrom, leaves the leffer B 52°.

Laftly,

« ForrigeFortsett »