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By a similar train of reasoning, as was used in deducing the rule for the cube root, we determine, in general, for any root, the following

- RULE. I. Point the number off into periods of as many figures each as there are units in the index, denoting the root. · II. Find by trial, the figure of the first period, which will be the first figure of the root ; place this figure to the left, in a col. umn called the FIRST COLUMN. Then multiply it by itself, and place the product for the first term of the SECOND COLUMN.

This, multiplied by the same figure, will give the first term of the THIRD COLUMN. Thus continue until the number of columns is one less than the units in the index, denoting the root.

Multiply the term in the LAST COLUMN by the same figure, and subtract the product from the first period, and to the re. mainder bring down the next period, and it will form the FIRST DIVIDEND.

Again, add this same figure to the term of the FIRST COLUMN, multiply the sum by the same figure, and add the product to the term of SECOND COLUMN; which, in turn, must be multiplied by the same figure, and added to the term of THIRD COLUMN, and 80 on till we reach the LAST COLUMN, the term of which will form the FIRST TRIAL DIVISOR.

Again, beginning with the FIRST COLUMN, repeat the above process until we reach the column next to the last. And so con. tinue to do until we obtain as many terms in the FIRST COLUMN as there are units in the index, denoting the root ; observing in each successive operation to terminate on the column of the next inferior order.

III. Seek how many times the FIRST TRIAL DIVISOR, when there are annexed to it as many ciphers, less one, as there are units in the index, is contained in the FIRST DIVIDEND, the quotient figure will be the second figure of the root. Then proceed with this figure the same as was done with the first figure; observing to advance the terms of the different columns as many places to the right, as the number expressing the order of the column ; that is, advancing the terms of the FIRST COLUMN one place, those of the SECOND COLUMN two places, and so for the succeeding columns.

After completing the requisite number of terms in the differ. ent columns, by means of this second figure of the root, then proceed to obtain the third figure of the root, in the same way as the second figure was obtained; and in this way the operation can be continued until all the periods are brought down. If there is still a remainder, the process can be extended by form. ing periods of ciphers.

Examples. 1. What is the fifth root of 36936242722257 ?

OPERATION. 1st 2D 3D 4TH

root. COL. COL. COL. COL.

5 25 125 625 3693'62427'22357(517 10 75.500 3125

3125 15 150 1250 32525251 56862427 20 250 1275251 33826005 32525251 251 25251 1300754 347673946051 2433717622357 252 25503 1326510

2433717622357 253 25756 1344842293 254 26010 2557 2618899

2. What is the 7th root of 2? Ans. 1.10409, nearly. 3. What is the Ith root of 11? Ans. 1.24357, nearly. 4. What is the 5th root of 5?'

5. What is the 7th root of 1231171548132409344?

OPERATION. .

1st 2D 3D 4TH 5тн

6TH
COL. COL. COL. COL.

COL.
COL.

12311'7154813'2409344)384
27. i 81

243
729

2187
6 27 108 405

1458
5103

101247154813
9 54 270 1215

5103 - 11568197824 92545582592
12 90 540 2835

808149728 21076554688

87015722212409344
15 135 945 37231216. 1188544608 21753930553102336 87015722212409344
18 189 1110152 47549360 1663938528
218 20644 1289768 59424240 169343966275584
226 22452 1484360 72979760
234 24324 1694440 737528368896 .
242 26260 1920520
250 28260 1932692224
258 30324
2664 3043056

193 CHAPTER XII.

DUODECIMALS.

82. In decimals, we have seen that the figures decrease in in a tenfold ratio, from the left towards the right.

In DUODECIMALS, this decrement goes on in a twelve fold ratio.

The different denominations are the foot, (f.,) the prime, or inch, (",) the second, (",) the third, (",) the fourth, ('''') the fifth, ('''',) and so on.

Thus, 7f.,6', 3", 4'", 5'1", is read, 7 feet, 6 primes, 3 seconds, 4 thirds, 5 fourths.

The accents, used to distinguish the denominations below feet, are called indices. Taking the foot for the unit, we have the following relations :

l'=1 of 1 foot,
1"=t of t of 1 foot =iia of 1 foot,
1''=is of 1 of the of 1 foot =ite of 1 foot,
l''=is of il of is of is of 1 foot = otso of 1 foot,

&c. &c. &c. Addition and subtraction of duodecimals, are performed like addition and subtraction of other compound numbers, remembering that 12 of any denomination make one of the next greater denomination.

MULTIPLICATION OF DUODECIMALS.

83. Suppose we wish to multiply 14f. 7' by 2f. 3', we should proceed as follows:

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2f. 3 3f. 71 gr

29f. 2 Ans. 32f. 9' g=32f.+of a foot+Išo of a foot.

EXPLANATION. We began on the right-hand, and multiplied the multiplicand through, first by the primes of the multiplier, then by the feet of the multiplier, thus: 3'x75x 1=1 of a ft. which is 21"=1' 9''; we write down the 9", and carry the l' to the next product; again, 14f. x 3'=14X =1 of a foot, which is 42'; now adding in the l', which was to carry from the last product, we have 43'=3f. 7', which we write down, thus finishing the first line of products.

Again, we have 2f.x7=2x1=14 of a foot, which is 14=lf. 2'; we write the 2' under the seconds of the last line, and carry 1f. to the next product ; 2f. x 140.=28f.to which, adding in the 1f., which was to carry from the last product, we have 29f., which we place underneath the feet of the last line. Taking the sum, we find 32f. 9'' 9'', for the answer.

From the above we infer, that if we consider the index of the feet to be 0, then the denominalion of the product will be denoted by the sum of the indices, representing the factors.

Thus, feet by feet, produces feet; feet by primes, produces primes ; primes by primes, produces seconds, &c., &c.

Hence, to multiply a number consisting of feet, inches, seconds, &c., by another number consisting of like qualities, we have this

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