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and a ar,
Thus, 2, 6, 9, 3, and a, ar, br, b, are inversely propor
I 1 tional; because 2, 6,
br' 7 , are directly proportional.
GEOMETRICAL PROGRESSION, is when a series of quantities have the same constant ratio; or which increase, or decrease, by a common multiplier, or divisor.
Thus, 2, 4, 8, 16, 32, 64, &c., and a, ar, ara, am, art, &c., are series in geometrical progression.
The most useful properties of geometrical proportion and progression are contained in the following theorems :
1. If three quantities be in geometrical proportion, the product of the two extremes will be equal to the square of the mean.
Thus, if the proportionals be 2, 4, 8, or a, b, c, then will 2 X 8= 42, and a xc= .
2. Hence, a geometrical mean proportional, between any two quantities, is equal to the square root of their product.
Thus, a geometric mean between 4 and 9 is = V 36 = 6. And a geometric mean between a and b is = v ab.*
3. If four quantities be in geometrical proportion, the product of the two extremes will be equal to that of the means.
Thus, if the proportionals be 2, 4, 6, 12, or a, b, c, d; then will 2 x 12 = 4 X 6, and a xd=b X c.
4. Hence, the product of the means of four proportional quantities, divided by either of the extremes, will give the other extreme ; and the product of the extremes, divided by either of the means, will give the other mean. Thus, if the proportionals be 3, 9, 5, 15, or a, b, c, d; 9 x 5
3 x 15 then will
* If two or more geometrical means between any two quantities be required, they may be expressed as below:
Vazb and Vab2 = two geometrical means between a and b.
, cabeynti,ca" Ko) n+1 = any number (n) of geometrical
means between a and b. Where
is the ratio; so that if a be multiplied by this, it will give the first of these means; and this last being again multiplied by the same, will give the second ; and so on.
5. Also, if any two products be equal to each other, either of the terms of one of them, will be to either of the terms of the other, as the remaining term of the last is to the remaining term of the first.
Thus, if ad = bc, or 2 x 15 = 6 x 5, then will any of the following forms of these quantities be proportional:
Directly, a:b::c:d, or 2 : 6::5:15.
In all of which cases, the product of the two extremes is equal to that of the two means.
6. In any continued geometrical series, the product of the two extremes is equal to the product of any two means that are equally distant from them; or to the square of the mean, when the number of terms is odd. Thus, if the series be 2, 4, 8, 16, 32; then will
2 X 32 = 4 X 16 = 82. 7. In any geometrical series, the last term is equal to the product arising from multiplying the first term by such a power of the ratio as is denoted by the number of terms less one.
Thus, in the series 2, 6, 18, 54, 162, we shall have 2 x 34 2 X 81 = 162.
And in the series, a, ar, ara, ar), art, &c., continued to n terms, the last term will be
1= ar-). 8. The sum of any series of quantities in geometrical progression, either increasing or decreasing, is found by multiplying the last term by the ratio, and then dividing the difference of this product and the first term by the difference between the ratio and unity. Thus, in the series 2, 4, 8, 16, 32, 64, 128, 256, 512, we
512 X 2 - 2 shall have
1024 - 2 2= 1022, the sum of the 2-1
Or the same rule, without considering the last term, may be expressed thus :
Find such a power of the ratio as is denoted by the number of terms of the series ; then divide the difference between this power and unity, by the difference between the ratio and unity, and the result, multiplied by the first term, will be the sum of the series.
Thus, in the series a tar vara + ar3 + art, &c., to ar we shall have
Where it is to be observed, that if the ratio, or common multiplier, r, in this last series, be a proper fraction, and consequently the series a decreasing one, we shall have, in that case, a +ar+ara + ar} + ar, &c., ad infinitum
1 9. Three quantities are said to be in harmonical proportion, when the first is to the third, as the difference between the first and second is to the difference between the second and third.
Thus, a, b, c, are harmonically proportional, when a : 6 ::a-6:
5 c, or a:0::b-a:C b. And c is a third harmonical proportion to a and b, when
ab 2a-6 10. Four quantities are in harmonical proportion, when the first is to the fourth, as the difference between the first and second is to the difference between the third and fourth.
Thus, a, b, c, d, are in harmonical proportion, when a:d::a b: d, or a:d::b-a:d- C. And d is a
fourth harmonical proportional to a, b, c, when d=
2ain each of which cases it is obvious, that twice the first term must be greater than the second, or otherwise the proportionality will not subsist.
11. Any number of quantities, a, b, c, d, e, &c., are in harmonical progression, if a:c::a b:b b:d::b bC :6--
dicie::c-d:d-e; &c. 12. The reciprocals of quantities in harmonical progression, are in arithmetical progression.
Thus, if a, b, c, d, e, &c., are in harmonical progression, 1 1 1 1 1
&c., will be in arithmetical progression. a'
b'c' d' e
13. An harmonical mean between any two quantities, is equal to twice their product divided by their sum.
2ab Thus = an harmonical mean between a and 5.*
* In addition to what is here said, it may be observed, that the ratio of two squares is frequently called duplicate ratio ; of two square roots, sub-duplicate ratio ; of two cubes, triplicate ratio ; and of two cube roots, sub-triplicate ratio ; &c.
1. The first term of a geometrical series is 1, the ratio 2, and the number of terms 10; what is the sum of the series? Here 1 x 2 =1 X 512 = 512, the last term.
512 X 2-1 1024 - 1 And
1023, the sum required. 2-1 1
1 2. The first term of a geometrical series is the ratio
3' and the number of terms 5; required the sum of the series. 1
1 1 1
the last term.
2 81 162' 1-TIX}_1-TO 121 3 121 And
243 2 162 3. Required the sum of 1, 2, 4, 8, 16, 32, &c., continued to 20 terms.
1 1 1 1 1 4. Required the sum of 1,
&c., continued 2' 4' 8' 16' 32'
127 to 8 terms.
128" 1 1 1 1 5. Required the sum of 1,
&c., continued to 3' 9' 27' 81'
9841 10 terms.
19683 6. A person being asked to dispose of a fine horse, said he would sell him on condition of having a farthing for the first nail in his shoes, a halfpenny for the second, a penny for the third, twopence for the fourth, and so on, doubling the price of every nail. to 32, the number of nails in his four shoes; what would the horse be sold for at that rate ?
Ans. 44739241. 55. 3 d.
THE DOCTRINE OF EQUATIONS, is that branch of algebra which treats of the methods of determining the values of unknown quantities by means of their relations to others which are known.
This is done by making certain algebraic expressions equal to each other (which formula, in that case, is called an equation), and then working by the rules of the art, till the quantity sought, is found equal to some given quantity, and consequently becomes known.
The terms of an equation are the quantities of which it is =
composed; and the parts that stand on the right and left of the sign =, are called the two members, or sides, of the equation.
Thus, if x= a + b, the terms are a, a, and b; and the meaning of the expression is, that some quantity x, standing on the lefthand side of the equation, is equal to the sum of the quantities a and b on the righthand side.
A simple equation is that which contains only the first power of the unknown quantity: as,
x + a = 3b, or ax bc, or 2x + 3a = 562 ; Where x denotes ihe unknown quantity, and the other letters, or numbers, the known quantities.
A compound equation is that which contains two or more different powers of the unknown quantity : as,
2a + ax = b, or 203 4x2 + 3x = 25. Equations are also divided into different orders, or receive particular names, according to the highest power of the unknown quantity contained in any one of their terms: as quadratic equations, cubic equations, biquadratic equations, &c.
Thus, a quadratic equation is that in which the unknown quantity is of two dimensions, or which rises to the second power : as,
2a = 20; 202 + ax=b, or 3x2 + 10x = 100. . A cubic equation is that in which the unknown quantity is of three dimensions, or which rises to the third power: as,
202 = 27; 22.3 3x = 35; or 203 ana to bx = c. A biquadratic equation is that in which the unknown quantity is of four dimensions, or which rises to the fourth power: as 24 = 25; 5.x4 - 4x=6; or 24 ax: + baca
d. And so on for equations of the 5th, 6th, and other higher orders, which are all denominated according to the highest power of the unknown quantity contained in any one of their
The root of an equation is such a number or quantity, as, being substituted for the unknown quantity, will make both sides of the equation vanish, or become equal to each other.
A simple equation can have only one root; but every compound equation has as many roots as it contains dimensions, or as is denoted by the index of the highest power of the unknown quantity, in that equation.
Thus, in the quadratic equation 2 + 2x = 15, the root, or value of x, is either + 3 or 5; and, in the cubic equation 203 9x2 + 26x = 24, the roots are 2, 3, and 4, as will be found by substituting each of these numbers for x.
In an equation of an odd number of dimensions, one of its