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α

Whence we have THEOREM 1. pR-a, THEOR. 2. P,

α

THEOR. 3.

t

R. THEOR. 4.

Ρ

log.a-log.p
log. R

Rt

t; the three latter of which follow immediately from the first; the fourth cannot be conveniently exhibited in numbers without the aid of logarithms.

By means of these four theorems, all questions of compound interest may be solved.

EXAMPLES.-1. What is the amount of 1250l. 10s. 6d. for 5 years, at 4 per cent. per annum, compound interest?

Here p (1250l. 10s. 6d.=) 1250.525, t=5, R=1.04. Then theor. 1.(pRt=) 1250.525 x 1.0451250.525 x 1.2166 =1521.388715-1521l. 7s. 94d.-a.

2. What principal will amount to 2001. in 3 years, at 4 per cent. per annum ?

Here a=200, R=1.04, t=3, and theor. 2. (

200

1.124864

-=177.7992=1771. 15s. 114d.=p.

α 200

R

1.043

3. At what rate per cent. per annum will 500l. amount to 5781. 16s. 3d. in 3 years?

Here p=500, a=(5781. 16s. 3d.=) 578.8125, t=3;

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and,

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Ρ

P. 3. Art. 63.=) —×5

-x 5.25 1.05 R: wherefore, (since R-1

=r,) we have R-1=.05=r, viz. 5 per cent. per annum.

4. In how many years will 2251. require to remain at interest, at 5 per cent. per annum, to amount to 260l. 9s. 3 d.?

Here p=225, R=1.05, a=(260l. 9s. 34d.=) 260. 465625: log. a-log. p_log. 260.465625-log. 225

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5. Any number which measures two or more numbers, is called their common measure; and the greatest number that will measure them, is called their greatest common measure.

Thus 1, 2, 3, and 6, are the common measures of 12 and 18; and 6 is their greatest common measure.

COR. Hence the greatest common measure of several numbers cannot be greater than the least of those numbers; and when the least number is not a common measure, the greatest common measure cannot be greater than half the least. Def. 2.

cor.

6. An even number is that which can be divided into two equal whole numbers.

Thus 6 is an even number, being divisible into two equal whole numbers, 3 and 3, &c.

7. An odd number is that which cannot be divided into two equal whole numbers; or, which differs from an even number by unity. Thus, 1, 3, 5, 7, &c. are odd numbers.

COR. Hence any even number may be represented by 2 a, and any odd number by 2a+1, or 2 a—1.

8. A prime number is that which can be measured by itself and unity only'.

bers.

Thus, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, &c. are prime num

1 Hence it appears, that no even number except 2 can be a prime, or that all primes except 2 are odd numbers; but it does not follow that all the odd numbers are primes: every power of an odd number is odd, consequently the powers of all odd numbers greater than 1, after the first power, will be composite numbers.

Several eminent mathematicians, of both ancient and modern times, have made fruitless attempts to discover some general expression for finding the prime numbers: if n be made to represent any of the numbers 1, 2, 3, 4, &c. then will all the values of 6n+1 and 6 n-1 constitute a series, including all the primes above 3; but this series will have some of its terms composite numbers thus, let n=1, then 6n+1=7 and 6n-1-5, both primes; if n=2, then 6n+1=13, and 6 n−1=11, both primes; if n=3, then 6n+1 =19, and 6n-1=17, both primes, &c. Let n=6, then 6n+1=37 a prime, but 6n—1=35 (=5×7) a composite number; also if n=8, then 6 n+1= 49 a composite number, and 6-147 a prime, &c. For a table of all the prime numbers, and all the odd composite numbers, under 10,000, see Dr. Hutton's Mathematical Dictionary, 1795. Vol. II. p. 276, 278.

9. Numbers are said to be prime to each other, when unity is their greatest common measure m.

Thus, 11 and 26 are prime to each other, for no number greater than 1 will divide both without remainder.

10. A composite number is that which is measured by any number greater than unity.

Thus, 6 is a composite number, for 2 and 3 will each measure it.

COR. Hence every composite number will be measured by two numbers: if one of these numbers be known, the other will be the quotient arising from the division of the composite number, by the known measure.

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11. The component parts of any number, are the numbers (each greater than unity) which multiplied together, produce that number exactly.

Thus, 2 and 3 are the component parts of 6, for 2×3=6; 3, 4, and 5 are the component parts of 60, for 3×4x5=60, &c. 12. A perfect number" is that which is equal to the sum of all its aliquot parts.

m Numbers which are prime to one another, are not necessarily primes in the sense of def. 8. thus 4 and 15 are composite numbers according to def. 10. but they are prime to each other, since unity only will divide both. Hence two even numbers cannot be prime to each other.

In the Scholar's Guide to Arithmetic, 7th Ed. p. 104. 9. it is asserted, that "If a number cannot be divided by some number less than the square root thereof, that number is a prime." Now this cannot be true; for neither of the square numbers 9, 25, 49, &c. &c. can be measured by any number less than its square root, and yet these numbers are not primes: a slight alteration in the wording will however make it perfectly correct; thus, "If a number which is not a square, cannot be divided by some number less than the square root thereof, that number is a prime." This interpretation was undoubtedly intended by the learned author, although his words do not seem to warrant it. "The following table is said to contain all the perfect numbers at present known.

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These numbers were extracted from the Acts of the Petersburg Academy, in several of the volumes of which, Tracts on the subject may be found.

Thus, 6 is a perfect number, for its aliquot parts are 1(=~~

1

of 6) 2 (=—=—= of 6) and 3 (=— of 6) and 1+2+3=6.

13. An imperfect number is that which is greater or less than the sum of its aliquot parts; in the former case it is called an abundant number, in the latter, a defective number.

Thus, 8 and 12 are imperfect numbers; the former (viz. 8) is an abundant number, its aliquot parts being 1, 2 and 4, the sum of which 1+2+4=7, is less than the given number 8. The latter (viz. 12) is a defective number, its aliquot parts being 1, 2, 3, 4, and 6, the sum of which, viz. 16, is greater than the given number 12.

14. A pronic number is that which is equal to the sum of a square number and its root

Thus, 6, 12, 20, 30, &c. are pronic numbers; for 6=(4+ √4=) 4+2; 12=(9+√/9=) 9+3; 20=(16+√/16=) 16 +4; 30=(25+ √/25=) 25+5, &c.

Property 1. The sum, difference, or product of any two whole numbers, is a whole number. This evidently follows from the nature of whole numbers, for it is plain that fractions cannot enter in either case.

COR. Hence the product of any two proper fractions is a fraction.

2. The sum of any number of even numbers is an even number. Thus, let 2 a, 2 b, 2 c, &c. be even numbers. (See def. 7. cor.) Then 2 a+2b+2c+, &c.=their sum; but this sum is evidently divisible by 2, it is therefore an even number; def. 6.

COR. Hence if an even number be multiplied by any number whatever, the product will be even.

3. The sum of any even number of odd numbers is an even number.

Thus, (def. 7. cor.) let 2a+1, 2b+1, 2c+1, and 2 d+1, be an even number of odd numbers.

Then will their sum 2a+2b+2c+2d+1+1+1+1, be an even number; for the former part 2a+2b+2c+2d is even, by def. 6. and the latter consisting of an even number of units is likewise even; wherefore the sum of both will be even, by property 2. COR. Hence if an odd number be added to an even, the sum will be odd.

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