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7. Required the two mean proportionals between 3 and 375 Ans. 15 and 75.

8. The earth revolves round the sun in 3654 days, at 95 millions of miles distance from him; required the distance of Jupiter from the sun, supposing he revolves about the sun in 4332; days a ? Ans. 494.109000 miles nearly.

8. In the temple of Apollo, in the island of Delos, there was a cubical altar, 3 feet each way, made of the horns of animals forcibly bent and entwined together, said to be the work of Apollo in his infancy; required the side of a cube double, and of another, half of the same 2 Ans. side of the double, 3.77976 feet; side of the half, 2,3811 feet.

By Approximation e.

Rule I. Call the given number whose root is required to be found, the number.

II. Find by trials a power nearly equal to the number, and call its root, the assumed root.

III. Add 1 to the index of the power, and call the result, the

the said root for the first mean; and this product by the root for the second:

thus in the example * v: × 2 = the first mean, and this product multiplied

by "v o = the second mean,

d The cubes of the distances of any two planets from the sun, are as the squares of the times in which they each revolve round him; in this example therefore 355;}* : 4332].” : 95 milliono the cube of Jupiter's distance, the cube root of which is the answer: in the same manner the distances of all the other planets may be found, their periodic times, with that of the earth, and its distance from the sun, being known.

• Approximation, (from the Latin ad to, and proarimus nearest,) is a continual approach, still nearer and nearer, to the quantity sought ; by this method the roots of numbers are found, not exactly, but to any assigned degree of nearness, short of absolute exactness: the rule is in substance the same as that first given by Dr. Hutton, in the first volume of his Mathematical Tracts. There are rules by which the roots of earact powers may be accurately determined; but for the roots of high powers, the operations require too much time and labour to be of any real use in practice. An universal investigation of the above rule will be given, when we treat of the resolution of the higher equations in Algebra

sum ; subtract 1 from the index, and call the result, the difference. IV. Multiply the power by the sum, and the number by the difference, and add both products together for the first term. V. Multiply the number by the sum, and the power by the difference, and add both products together for the second term. VI. Make a rule of three stating, thus; say as the first term: is to the second term ; ; so is the assumed root: to a fourth number, (found by the rule of three,) which will be the root of the given number nearly. VII. Involve the root found to the given power, and if the power and given number are nearly equal, the work is finished; but if not, the operation must be repeated, thus; VIII. Let the root found be called the assumed root, and its power the power, and proceed with these and the given number, sum, and difference, as before, whence a root will be obtained still nearer the truth. In this manner the operation may be repeated at pleasure, observing always to use the last found root, and its power, for the assumed root and power.

ExAMPLEs. 1. Required the cube root of 520. Here 520 = the number. 3 = the inder. 3 + 1 = 4 = the sum. 3–1 = 2 = the difference. I find by trials that 8 is nearly equal to the cube root of 520; therefore 8 = the assumed root, and S) = 512 = the power. Then 512 x 4 = 2048 = the power multiplied by the sum. 520 × 2 = 1040 = the number multiplied by the diffe

- [rence. their sum = 3088 = the first term.

And 520 x 4 = 2080 = the number multiplied by the sum. 512 × 2 = 1024 = the power multiplied by the diffe[rence,

their sum 3104 = the second term.

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2. Required the 5th root of 40. Here 40 = the number. 5 = the index. 5 + 1 = 6 = the

sum, 5–1 = 4 = the difference. Let the root found by trials be

2 = the assumed root; then 2' = 32 = the power.

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To repeat the operation. . Here 2,0909 = the root assumed. 2,0909)* = 39.963757, &c.

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Wherefore 399.782543 : 399.855029 :: 2.0909 : 399.855029 x 2.0909 399.782543 otear. 3. What is the cube root of 17.54? Root 2.5982, &c. 4. What is the 4th root of 94.75853 : Root 3.1196, &c. 5. What is the 5th root of 3124? Root 4.9996, &c. 6. Extract the 6th root of 48. Root 1.90636, &c. 7. Required the 7th root of 581. Root 2.4824, &c. 8. Required the 8th root of 72138957.88, Root 9,5999, &c.

= 2.091279108543 = the root extremely 9. What is the 9th root of 2 Root 1.080059, &c. 10. Extract the second, third, fourth, fifth, sixth, seventh, eighth, and minth roots of one hundred, and find their sum. Ans. 27.84716, &c.

284. In all the foregoing examples the index of the root is a fraction, having 1 for its numerator; examples however sometimes occur, in which the numerator of the index is greater than l; in this case the root is extracted by the following:

Rule I. Involve the given number to that power which i denoted by the numerator of the index, from whence extract the root denoted by the denominator: or,

II. First extract the root denoted by the denominator, then involve this root to the power denoted by the numerator'.

11. Find the value of 83. Thus by Rule I. 8]*=64, and "vö4 = 4, the root required. By Rule II. * v8 = 2, and 2* = 4, the root as before. 12. Required the value of IO):. Thus io) = 10000 and * y10000 = 6.3006, &c. the root : or * v10 = 1.5849, &c. and 1.5849]*= 6.3096, the root as before. 13. Required the value of 24. Ans. 1.68179, &c. 14. Required the value of 1023 #. Ans. 15.993, &c. 15. Required the value of 1064S4. Ans. 484. 16. Required the sum of the values of gli, 33, als, and 5.

Ans. 10.72196, &c. .” PROGRESSION. 285. When several numbers or terms are placed in regular succession, the whole is called a series. *

286. If the terms of a series successively increase or decrease, according to some given law, the series is said to be in progression.

287. Progression is of two kinds, Arithmetical and Geometri

f In this rule both Involution and Evolution are employed; the numerator of the fractional index denoting a power, and the denominator a root, thus in ex. 11.8% denotes either the cube root of the square of 8, or the square of the cube root of 8: on this principle the rule depends.

cal, arising from the manner in which the successive increase or decrease is made ; namely, either by addition or subtraction, or by multiplication or division.


288. A series of numbers is said to be in Arithmetical Progression, when the terms successively increase or decrease by the constant addition or subtraction of a number, called the common differences. There are five particulars belonging to questions in arithmetical progression; viz.: 1. The least term, 2. The greatest term, 3. The number of terms. 4. The common difference. 5. The sum of all the terms. Any three of these five being given, the remaining two may be found, as is shewn by the rules and examples following. 289. The least term, the greatest term, and the number of terms, being given, to find the sum of all the terms. Rule. Add the least and greatest terms together, multiply the sum by half the number of terms, and the product will be the sum required.

} called the extremes.

- ExAMPLEs. 1. The least term is 3, the greatest 17, and the number of terms 8, in an arithmetical progression; required the sum of the term S. Thus 3 + 17 = 20 = sum of the extremes. And 4 (or half 8) = half the number of terms. Then 20 x 4 = 80 = the sum required.

s when the progression consists of three or four terms only, it is usually called an arithmetical proportion ; and the middle terms are called arithmetical means.

The essential property of an arithmetical progression is this ; namely, “The sum of the two extreme terms is equal to the sum of every two mean “terms equally distant from the extremes;” from this property many others, some of which are the subject of the following rules, are easily deduced; but as this cannot be conveniently done without Algebra, it was thought best to refer to the Algebraic part of the work for proof of the rules here given. The word progression is derived from the Latin progredior, to go forward,

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