25.6 francs is exchanged for £1? What is the arbitrated price between London and Paris, when 3 francs=480 rees, 400 rees=34s. Flemish, and 358. Flemish = £1 ?

9. A person in London owes another in Petersburg a debt of 460 rubles, which must be remitted through Paris. He pays the requisite sum to his broker, at a time when the exchange between London and Paris is 23 francs for £1, and between Paris and Petersburg 2 francs for one ruble. The remittance is delayed until the rates of exchange are 24 francs for £1., and 3 francs for 2 rubles. What does the broker gain or lose by the transaction?

10. A trader in London owes a debt of 508 pistoles to one in Cadiz : is it more advantageous to him to remit directly to Cadiz, or circuitously through France? the exchanges being £1=254 francs, 19 francs 1 Spanish pistole, 4 Spanish pistoles = £3.

SQUARE ROOT.

=

178. The SQUARE of a given number is the product of that number multiplied by itself. Thus 36 is the square of 6.

The square of a number is frequently denoted by placing the figure 2 above the number, a little to the right. Thus 62 denotes the square of 6, so that 6236.

179. The SQUARE Root of a given number is a number, which, when multiplied by itself, will produce the given number.

The square root of a number is sometimes denoted by placing the sign before the number, or by placing the fraction above the number a little to the right. Thus √36 or (36)* denotes the square root of 36; so that /36 or (36)+= 6.

180. The number of figures in the integral part of the Square Root of any whole number may readily be known from the following considerations:

Hence it follows that the square root of any number between 1 and 100

must lie between 1 and 10, that is, will have one figure for its integral part; of any number between 100 and 10000, must lie between 10 and 100, that is, will have two figures in its integral part; of any number between 10000 and 1000000, must lie between 100 and 1000, that is, must have three figures in its integral part; and so on. Wherefore, if a point be placed over the units' place of the number, and thence over every second figure to the left of that place, the points will shew the number of figures in the integral part of the root. Thus the square root of 99 consists, so far as it is integral, of one figure; that of 198 of two figures; that of 176432 of three figures; that of 1764321 of four figures; and so on.

181. The following Rule may be laid down for extracting the square root of a whole number.

RULE. "Place a point or dot over the units' place of the given number, and thence over every second figure to the left of that place, thus dividing the whole number into several periods. The number of points will shew the number of figures in the required root (Art. 180).

Find the greatest number whose square is contained in the first period at the left; this is the first figure in the root, which place in the form of a quotient to the right of the given number. Subtract its square from the first period, and to the remainder bring down the second period. Divide the number thus formed, omitting the last figure, by twice the part of the root already obtained, and annex the result to the root and also to the divisor. Then multiply the divisor, as it now stands, by the part of the root last obtained, and subtract the product from the number formed, as above mentioned, by the first remainder and second period. If there be more periods to be brought down, the operation must be repeated."

After pointing, according to the Rule, we take the first period, or 13, and find the greatest number whose square is contained in it. Since the square of 3 is 9, and that of 4 is 16, it is clear that 3 is the greatest num

ber whose square is contained in 13; therefore place 3 in the form of a quotient to the right of the given number. Square this number, and put down the square under the 13; subtract it from the 13, and to the remainder 4 affix the next period 69, thus forming the number 469. Take 2 × 3, or 6, for a divisor; divide the 469, omitting the last figure, that is, divide the 46 by the 6, and we obtain 7. Annex the 7 to the 3 before obtained, and to the divisor 6; then multiplying the 67 by the 7 we obtain 469, which being subtracted from the 469 before formed, leaves no remainder; therefore 37 is the square root of 1369.

Reason for the above process.

Since (37)2=1369, and therefore 37 is the square root of 1369; we have to investigate the proper Rule by which the 37, or 30+7, may be obtained from the 1369.

Now 1369=900+469=900+49 +420

= (30)2+72 + 2 × 30 × 7
=(30)2+2 x 30 × 7+72

where we see that the 1369 is separated into parts in which the 30 and the 7, together constituting the square root, or 37, are made distinctly apparent. Treating then the number 1369 in the following form, viz.

(30)+2×30 × 7+72

we observe that the square root of the first part or of (30), is 30; which is one part of the required root. Subtract the square of the 30 from the whole quantity (30)3 + 2 × 30 × 7 +72, and we have 2 × 30 × 7 + 72 remaining. Multiply the 30 before obtained by 2, and we see that the product is contained 7 times in the first part of the remainder, or in 2 × 30 × 7; and adding the 7 to the 2 × 30, thus making 2 × 30+7 or 67, this latter quantity is contained 7 times exactly in the remainder 2 × 30 × 7+ 7 or 469; so that by this division we shall gain the 7, the remaining part of the root. If we had found that the 2 × 30+7 or 67, when multipled by the 7, had produced a larger number than the 469, the 7 would have been too large, and we should have had to try a smaller number, as 6, in its place. The process will be shewn as follows;

This operation is clearly equivalent to the following: 900+420 +49 (30+7

which is the mode of operation pointed out in the Rule.

Note 1. The reasoning will be better understood when the Student has made some progress in Algebra.

Note 2. The divisor obtained by doubling the part of the root already obtained, is often called a trial divisor, because the quotient first obtained from it by the Rule in (Art. 181), will sometimes be too large. It will be readily found, in the process, whether this is the case or not, for when, according to our Rule, we have annexed the quotient to the trial divisor, and multiplied the divisor as it then stands by that quotient, the resulting number should not be greater than the number from which it ought to be subtracted. If it be, the quotient is too large, and the number next smaller should be tried in its place.

Note 3. If at any point of the operation, the number to be divided by the trial divisor be less than it; we then affix a cypher to the root, and also to the trial divisor, bring down the next period, and proceed according to the Rule.

Ex. 2. Find the square root of 74684164.

Ex. 3. Find the square root of 71690512350625.

it appears, that in extracting the square root of decimals, the decimal places must first of all be made even in number, by affixing a cypher to the right, if this be necessary; and then if points be placed over every second figure to the right, beginning as before from the units' place of whole numbers, the number of such points will shew the number of decimal places in the root.

183. If there be no whole number, or integral part in the given number, we must, in pointing begin with the second figure from that which would be the units' place, if there were a whole number, and mark successively over every second figure to the right. If there be a whole number as well as a decimal, it will be the safest method to begin at the units' place and point over every second figure to the right and left of it. The number of points over the whole numbers and decimals will shew respectively the numbers of figures in the integral and decimal parts of the root. Thus if the given number were 6115'23, place the first point over the 5, and mark from it to the right and left, thus 6115.23. If the given number were 58.432, first make the decimal places even in number thus, 58·4320, and then point thus 58·4320.