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illustration. In multiplying a number of two or more digits * by itself or some other similar number, according to the ordinary process, the product of the local values of the members of each factor is formed in so compact a manner-the various stops are so closely combined, as to leave no distinct clue to the nature of the several independent operations which really take place. In order to see how the successive results are derived and expressed as a whole, it is necessary to examine the details of some simple case of involution, such as the following :

87 = 80 +7 (multiply the two parts 87 80 + 7 together, each by each). 609

80% + 80 ? Add. 696

+ 80 X 7 + 72 ) 7569 = 80% + 2 (80 x 7) + 7 From this it appears, that the square of the sum of two numbers (80 and 7) is equal to the square of each of those numbers added to twice their product; and it will be found that the same property holds good whatever number be taken, or whatever be the two parts into which it may be supposed to be divided. If, instead of 80 and 7, in the present example, the parts were 77 and 10, 68 and 19, 50 and 37, or any other two numbers making up 87, the sum of their squares, together with twice their product, would give a result equal to the square of their sum, namely, 87.+

Knowing the method of formation and the composition of a square number, we are enabled wben such a number presents itself, to recover its square root merely by undoing in proper order those processes which we have seen to be requisite for the production of the square. Thus, roverting to the expression

80% + 2 (80 x 7) + 7 = 7569, We observe, on separating 7569 into periods, that its root must consist of two figures, and lie between 100 and 10,000. (See page 146.) As 80° = 6400, and 90% = 8100, it is evident also that the first or leading root figure must be 8, and consequently that the root itself must be between 80 and 90. If then we subtract 80 or 6400 from 7569, the remainder will be 1169, and this, as above shown, is made up of 2 (80 x 7) + 7", or twice the product of the two figures of the root added to the square of the second figure. Divide 1169 by 2 X 80 or 160, that is, double the first root figure ; the quotient is 7, with the remainder 49 = 7, and we infer accordingly that 7 is the second figure of the root, or that the entire root is 87, since 160 + 7 or 167 is contained exactly 7 times in 1169. The details of the operation are exhibited below under three different forms. 80% + 2 (80 x 7) + 7% (80 + 7 6400 + 1120 + 49 (80 +7 75|69 (87

6400

64

802

160 + 7)

2 x 80 + 7) 2 (80 x 7) + 78

2 (80 x 7) +7°

1120 + 49
1120 + 49

167)1169

1169

• The figures of a number, when considered independently of the value which they possess in the arithmetical scale, are termed digits. Thus the number 428 has for its digits, 4, 2, and 8, but the local significance of the first of these is 400, and of the second, 20.

| See Euclid, Prop., 4, Book 2, for a demonstration of this property as regards magnitudes in general, and therefore as regards numbers, which are a particular species of magnitude

The second process is merely a development or an expansion of the first. The third, which is performed as the rule directs, is an abridgment of the second,-all the numbers which there stand in the same line being collected into one sum, and the writing down of final ciphers being omitted for the sake of expedition. There will now be little difficulty in understanding why the first root-figure is doubled to form the first trial-divisor, nor how it is the last figure of the dividend is excluded in applying that divisor, and the latter rendered complete by adding to it the rootfigure it has just enabled us to discover.

When a square number consists of more than three or four figures, and its root, therefore, of more than two figures, the process of evolution is carried on by a succession of uniform steps, each precisely similar to that which is taken to arrive at the second figure of a root consisting only of two figures. In the extraction of a root of three places of figures, for instance, the hundreds and tens having been determined as before by the method applicable to a root of no greater value than tens and units, let the hundreds and tens together be regarded as tens, thus reducing the number of figures in the root apparently to two, then the units may be found in the same manner as the tens were previously discovered, that is, by repeating the operation for the discovery of the second figure of the root; and whatever be the magnitude of a square number, the several parts of its root may be derived one after another on a like supposition as to the number of its digits.

A full and satisfactory explanation of this subject, however, cannot be given without the aid of algebra.

Application of the square root.—The most important use of the rule for the extraction of the square root is its enabling us to find the mean proportional * between two given numbers.

Several questions in mensuration, gauging, and other branches of practical mathematics require the computation of mean proportionals, as may be seen on referring to subsequent parts of this work. An example or two, therefore, will suffice in the present place.

It should be remarked that the mean proportional between two numbers, has to the smaller of those numbers, the same ratio or relation as the larger number has to it (the mean proportional). Thus 6, the mean proportional between 4 and 9, is once and a half as great as 4, and 9 is once and a half as great as 6-that is, 4 : 6::6:9, or multiplying together means and extremes respectively, 6' = 4 x 9, whence 6 =1/4 x 9;. and this property is evidently true of numbers in general, namely, that the mean proportional between any two numbers is the square root of their product.

Example (1.) It is proved in geometry, that the area of an ellipse is equal to the area of a circle, the diameter of which is a mean proportional between the axis of the ellipse. What, then, is the diameter of a circle equal in area to an ellipse, the longer axis of which is 24 and the shorter 18 inches ?

24 x 18 = 432, and 1 432 = 20.78.

Answer. 20.78 inches. Example (2.) A substance placed in one scale of a balance having arms of

* See page 109 for a definition of this term.

unequal length, is observed to weigh 130lbs., and when placed in the other scale only 123 lbs. What is the true weight of the substance ?

Answer. The mean proportional between the two observed

weights, or 123 x 130 = 126.45 lbs. The reason of this answer may be inferred from the properties of the lever. *

Besides its application in the computing of mean proportionals, the extraction of the square root is required in the solution of various practical problems, such as the following :

To find the side of a square equal in area to a given surface of any form.
To find the diameter of a circle from its area.
To find the hypothenuse of a right-angled triangle, when the base and

perpendicular are given.
To find the area of a triangle, the three sides being given,
To find gauge points,” or the square roots of constant divisors, for squares,

circles, &c., &c., &c. For the mode of obtaining these results, officers are referred to the chapters on Mensuration.

Extraction of the cube root.-It has been already remarked that the ordinary process of extracting any root higher than the square or second root is extremely tedious.t Horner's method is much more simple and involves less labour, but for most purposes the subjoined rule, laid down by Dr. Hutton, will afford a sufficient approximation to the cube root, when a table of logarithms is not at hand. I

RULE.—Ascertain by trial, in the manner below indicated, a number nearly equal to the required root, take the cube of this number, and call it the assumed cube.

Then say, by the Rule of Three, As the sum of the given number and twice the assumed cube is to the sum of the assumed cube and twice the given number. 80 is the root of the assumed cube to the root required, nearly. Again, by employing the cube of the root just found as a new assumed cube, and proceeding as before, & close approach may be had to the value of the required root, and so on any number of times, always using the cube of the root last determined as a new assumed cube. But it will rarely be necessary to incur the trouble of making a second approximation.

In order to judge with facility between what numbers the required root lies, the given number should be divided into periods of three figures each, beginning at the units' place ; for it is evident, that since the cube root of 1000 is 10, of 1000000 is 100, &c., the cube root of any number less than 1000 must consist of one figure ; of any number between 1000 and 1000000, of two figures, &c., and accordingly the number of periods of three figures each, with any smaller period remaining, will indicate the number of figures in the cube root required.

It will also be of assistance to refer to the annexed table of the cubes of the first nine numbers :

28 = 8:38 = 27 : 4' = 64 : 5o = 125 : 6' = 216 : 78 = 343 : 89 — 512 : 98 = 729.

* Soe any elemontary treatise on Natural Philosophy or Mechanics.

See page 146.
The reason of this rule cannot be properly explained without having recourso to algebra

Example (1.) What is the oube root of 672 ?

Here it is at once seen that the required root lies between 88 and 9', and also that it is much nearer to the latter. Let 729 be the assumed cube. Then by the ruleAssumed cube 729 Given No. 672

2

1344

1458

Root of
Given No. 672 Assumed cube 729

- assumed cube.
2130 :

2073 : : 9 : 8.7592 This is precisely the same result as would be got from a table of five-figure logarithms. The exact process gives as the root, 8.7590, &o.

Example (2.) Extract the cube root of 21035.8.

As there are two periods in this number, the root consists of two figures, and lies, as may easily be inferred from the table above, between 20 and 30, the latter being the nearer limit. Taking 30° or 27000 for the assumed cube, we have

27000 21035.8

54000

42071.6 21035.8

27000. 75035.8 : 69071.6 : : 30 : 27.615 If greater correctness be desired, a cube still nearer to the given one may be assumed by saying, As the difference between 208 and 30° is to 10, the difference of their roots, so is the difference between the given number and 20% to the difference of their roots-that is, 27000

21035.8
8000
20

8000 19000 : 10 :: 13035.8 : 7 And, adding 7 to 20, we have 27° or 19683 as an assumed cube. Operating as the rule directs, we now find the required root to be 27.605, which agrees with the answer worked out by the regular process.

To extract the cube root of a decimal fraction, annex one or two ciphers, if necessary, so that the number of figures shall be three, or some multiple of three. Then treat the decimal as a whole number, extract its root by the preceding rule, and divide the result by 10, 100, &c., according as there are one, two, or more periods of three figures in the given decimal, including the added ciphers, if any. Thus, the cube root, of .0076 is the same as the cube root of .007600, and this equals the cube root of 1078000 199 = .1966.

The cube root of a vulgar fraction is best obtained by reducing the fraction to a decimal, and proceeding as explained in the last paragraph.

To extract the fourth root of a number, first extract the square root, and then the square root of this ; but logarithms, as before observed, afford the readiest means of extracting roots of every degree, from the square root upwards.

Applications of the cube root.—Example (1.) To find the length of the side of & cubical * vessel that shall be equal in content to any given vessel of a different shape. Rule-Extract the cube root of the content of the given vessel, which will be the length required.

A cylindrical vessel has the capacity of 33273 cubic inches. What is the side of a cubical vessel of equal capacity ?

Answer. W/33273 — 32.16 inches.

Example (2.) The length of a rectangular vessel is 216, its breadth 125, and its depth 64 inches. What are the corresponding dimensions of a vessel of the same shape that shall contain 8 times as much ? Rule-Multiply the cubes of the given dimensions respectively by the given ratio, and extract the cube roots of the products.

Answer. Length, 432 inches.

Breadth, 250
Depth, 128 ,

MISCELLANEOUS QUESTIONS FOR EXERCISE. The ensuing questions are intended as a supplement to those which have been given at the end of each article, and are for the most part of a very simple character, although, in some instances, the right method of solution may not at once suggest itself to persons unaccustomed to reason on numerical data.

(1.) £663 4s. 14d, is paid as malt duty. The charges had all arisen from coucht gauges. Required the net and gross bushels of malt respectively.

Answer. 4890 bush. net ; 6000 bush. gross. (2.) The 5 per cent. additional duty on a quantity of malt amounts to £6 78. 23d. What is the quantity ?

Answer. 985 bushels. (3.) A cubic inch of distilled water, at the temperature 62°F., weighs 252.458 grains ; there are 7,000 grains in a lb. Required the number of cubic inches in a pint, weighing 14 lbs.

Answer. 34.695. (4.) 17 gallons of spirits at 108. 6d. per gallon, are mixed with 7 gallons at a different price. What is that price, if 20 per cent. be gained by selling the compound at 138, per gallon ?

Answer. 11s. 79d. (5.) If oranges are bought at the rate of 3 a penny, how must they be sold to gain 25 per cent. ?

Answer. At the rate of 12 for 5d. (6.) 120 oranges are bought at the rate of 5 a penny, and 120 others at the rate of 6 a penny ; they are sold at the rate of 11 for 2d. What is the gain or loss per cent. ?

Answer. A loss of 100 per cent. . That is, a vessel the length, breadth, and depth of which are all equal.

+ It is supposed that every officer understands the technical terms used in questions of this kind, and knows the rate of the various dutios.

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