And (y3 +u3=) 72=x2z+xz2, and things that are equal to the same are equal; therefore 9z-z2.9=72, or 9z-z2=8, or z2-9z=-8; whence by completing the square, &c. z=8, x= (9-2) 1, y= (3‚√x2z=) 2, u= (3√/xz2=) 4.

5. Of four numbers in geometrical progression, the product of the two least is 8, and of the two greatest 128; what are the numbers?

Let x, y, u, and z, be the numbers.

-.

y น

32

Therefore (8 x 128=) 1024=u2y2, or uy=32, and u=—.

32

y

y::y: where multiplying

y

Y

6. The sum of 3 numbers in geometrical progression is 14, and the greater extreme exceeds the less by 6; what are the numbers? Ans. 2, 4, and 8.

125. Def. Compound Interest is that which is paid for the use, not only of the principal or sum lent, but for both principal and interest, as the latter becomes due at the end of the year, half-year, quarter, or other stated time.

To investigate the rules of Compound Interest.

Let p the principal, r=the rate per cent. t=the time, R= (1+r) the amount of 11. for a year, called the ratio of the rate per cent. a=the amount.

:

Then since 1 pound is to its amount for any given time and rate: so are any number of pounds: to their amount for the same time and rate; therefore as

1

PPR the first,.

PR PR

1: R::

second,
third,

year's amount.

=

PR: : pR'=th,

PR: PR fourth,

α

Whence we have THEOREM 1. pR'=α, THEOR. 2. P,

α

log.a-log.p

THEOR. 3.

t

R. THEOR. 4.

Ρ

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. (pR*=) 1250.525 × 1.04 5: 1250.525 x 1.2166 =1521.388715=15217. 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. (

α 200

R

1.043

200

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

1.124864

3. At what rate per cent. per annum will 500l. amount to 578l. 16s. 3d. in 3 years?

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

P. 3. Art. 63. =) ——×5 ×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 225l. require to remain at interest, at 5 per cent. per annum, to amount to 260l. 9s. 34d. ?

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

whence, theor. 4. (

log. R

log. 1.05

=

=3 years=t.

0.0211893

2.4157506-2.3521825 0.0635681

0.0211893

=)

5. What sum will 500l. amount to in 3 years, at 5

per annum? Ans. 5781. 16s. 3d.

per

cent.

6. What principal will amount to 15211. 7s. 94d. in 5 years,

at 4 per cent. per annum? Ans. 1250l. 10s. 6d.

7. At what rate per cent. will 7211. amount to 1642l. 19s. 94d. in 21 years? Ans. 4 per cent.

8. In how many years will 7217. be at interest at 4 per cent. to amount to 1642l. 19s. 94d. Ans. 21 years.

If the interest be payable half-yearly, make t=the number of half-years, that is twice the number of years, and r=half the rate per cent. but if the interest be payable quarterly, let t=the number of quarter-years, viz. 4 times the number of years, and r=one-fourth of the rate per cent. and let R=r+1 in both cases, as before *.

126. To determine some of the most useful properties of

numbers.

Def. 1. One number is said to be a multiple of another, when the former contains the latter some number of times exactly, without remainder.

Thus 12 is a multiple of 1, 2, 3, 4, and 6.

COR. Hence every whole number is either unity, or a multiple of unity.

2. One number is said to be an aliquot part of another, when the former is contained some number of times exactly in the latter.

Thus 1, 2, 3, 4, and 6, are aliquot parts of 12, for 1 is TT, 2 is t, 3 is 4, 4 is, and 6 is 4 of 12.

COR. Hence no number which is greater than half of another number, can be an aliquot part of the latter.

3. One number is said to measure another number, when it will divide the latter without remainder.

Thus each of the numbers 1, 2, 4, 5, 10, and 20, measures 20. 4. One number is said to be measured by another, when the latter will divide the former without remainder.

Thus 20 is measured by 1, 2, 4, 5, 10, and 20.

COR. Hence every aliquot part of a number measures that number, and every number is measured by each of its aliquot parts, and by itself.

k It was at first intended to investigate and apply every rule in arithmetic, but want of room obliges us to omit Equation of Payments, Loss and Gain, Barter, Fellowship, and Exchange; these will be easily understood from the doctrine of proportion, of which we have amply treated.

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 2 a+1, or 2 a—1.

S. A prime number is that which can be measured by itself and unity only 1.

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 62-1 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 6 n—1=35 (=5 × 7) a composite number; also if n=8, then 6 n+1: 49 a composite number, and 6 n-1-47 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 TM.

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.

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 2x3=6; 3, 4, and 5 are the component parts of 60, for 3 × 4×5=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. n The following table is said to contain all the perfect numbers at present known.

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.