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Cafe zd. When the feet in the multiplicand are more than 12, and those in the multiplier under 12, work with the feet as in cafe the firft.

And for the inches take parts of the multiplicand, viz. 6 inches, 4 inches, 34, 2 82 I z &c. 12

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Cafe 3d. When the feet in both factors exceed 12; multiply the feet, by the feet, and for the inches in the multiplier, take parts in the multiplicand as in cafe zd.

And for the inches in the multiplicand take parts from the feet in the multiplier only, and the fum of all the parts, and product in feet, is the answer.

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THE

YOUNG STUDENT's

POCKET COMPANION.

PART II.

Of LOGARITHMS, GEOMETRY,

And TRIGONOMETRY.

The INTRODUCTION.

OGARITHMS are the noble invention of JOHN NEPER, Baron, of Marchiston in Scotland, the use of them was originally written in Latin by the Author, but fince tranflated and illuftrated, with his approbation, by our countryman HENRY BRIGGS. Their ufe extends to Arithmetic, Trigonometry, Geography, Dialing, Navigation, and Aftronomy.

The name is derived of Logos, reason, and Arithmos, numbers, whence they are rational or proportional numbers, being an artificial fet fo contrived, as to render the tedious operations of Arithmetic and Trigonometry, fuch as multiplication, divifion, proportion, and the extraction of roots, more easy.

M 3

Thefe

I

Thefe logarithms are claffed in tables to all natural numbers from 1 to 10.000. but fuch tables may be extended to much greater numbers, and alfo to decimals and mixed numbers. Since the logarithms of all numbers under 10 have their indices o; all numbers from 10 to 100 have 1 for their index; from 100 to 1000 2 is the index; fo the index of the logarithm of any number is less than the number of its digits, thus the index cfthe logarithm of 768472 will be 5. alfo if the index of a logarithm in operation fhould be 4 the number anfwering thereto confifts of 5 places or tens of thoufands.

The index of a decimal is negative in fuch fort that: the index of a decimal confifting of tenth parts is -9. but if the tens place be a cypher, the index of its logarithm is 8 &c, as in the fubjoined table.

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The indices, of mixed numbers are governed by the integral part thereof entirely.

In addition of logarithms, if the indices be pofitive or integral, add them and you have the true fum. If the indices be all decimal or negative, add them and if the fum be under 10. add 10 thereto. If juft 10 add 1; if above 10 cast 10 away, the fum or remainder will be the true index required.

If the indices be fome pofitive, and fome negative, add them alfo; if the fum be 10 or more caft 10 away; and the remaining index is pofitive and the number correfponding is integral; if on of the indices be under 10 that fum is a f and its numerical reprefentative is a decimal.

e index,

In fubtraction of logarithms, if the indices be all integral or pofitive, then fubtract as ufual, if the index of the dividend be greater. If one or both the indices be negative, and that of the dividend fmaller than that of the divifor, add 10 to the lefs; and if the index of the dividend, be pofitive, the remaining index will be pofitive, if not it is negative, and the number correfponding thereto is a decimal.

In multiplication of logarithms if the index be negative you caft away the tens and the remainder is a negative index.

N. B. That as many tens as you caft away in multiplying the logarithm for the fquare, cube, &c. fo many tens you must borrow to the index in dividing for the fquare, cube root, &c. in order to obtain the true. negative index.

To find a logarithm to a number of 5 6 7 or 8 places by means of a table of logarithms to 4 places only. As to find the logarithm of 7842964; first the logarithm of 7842000 is6.89443; and the logarithm of 7843000 is 6.89448; here the difference of the numbers is 1000, and the difference of the logarithms is 5. • 1000 5 964: 4.8 whence 6.89443 +4.8 6.89447.8 is the logarithm which was required.

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If a logarithm be given to find the corresponding number of 5 6 7 or 8 &c. places. As for example let the number be found to the logarithm 6.47284.

The nearest number in the table is 2970, and its logarithm 3.47276, the next is 2971, and its logarithm 3.47290; here the difference of the numbers is 1, and the difference of the logarithms is 14, alfo the difference of the next lefs in the table and propofed logarithm is 8, fetting afide the index 6, by which it appears the number fought muft confift of 7 places. 14 1000 :: 8: 571 hence 2970000 + 571 = 2970571 the number required.

In working proportions by logarithms, take the compliment arithmetic of the firft term, beginning at the left hand, fubating every figure from 9 and the laft from 10 and the the fum of the logarithms of the 1st. 2d. and 3d. terms are the logarithm of the 4th.

Multiplication by Lagarithms.

1

RULE. The fum of the logarithms of the factors is the logarithm of the product.

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1.649724 S.

Pro. 1728 3.237543 S. Pro. 44.64

Divifion by Logarithms.

RULE. The difference of the logarithms of

the factors is the logarithm of the quotient.

Numb. Log.

Numb, Leg

Di. 1728 3.237543

Di. 44.64

1.649724

By 144 2.158362 S By

3.6

0.556302 S

1.093422 Diff.

Quo. 12 1.079181 Diff. Quo. 12.4

Involution by Logarithms.

RULE. Multiply the logarithm of the number to be involved by the index of the power viz. by2 for the fquare, 3 for the cube, 4 for the biquadrate, &c. and the product is the logarithm of the power.

Let 16 be fquared Let 12 be cubed

Numb. Log.

16 1.204120

Numb. Log.

12 1.079181

For the fquare. by 2 For the cube X by

Anf. 256

3.

2.408240 Anf. 1728 3.237543

Evolution by Logarithms.

RULE. Divide the logarithm of the power by the index of the root, viz. for the fare root divide by 2, for the cube root by 3, for the quadrate by 4, and the quotient is the logarithm of the root.

The

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