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Letter to Editor
March 25, 1737
The Virginia Gazette
Richmond, Williamsburg, Richmond County, Virginia
What is this article about?
A letter to Mr. Parks solves a prior question on four numbers in continued proportion, identifying them as 384, 768, 1536, and 3072 via algebraic method, and proposes a new question on five such numbers with given sums.
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Mr. Parks,
I have here answer'd the Question propos'd in your last; second Question.
The Four Numbers sought, are 384, 768, 1536, and 3072;
which I investigate in this Manner:
Put s = the Sum of the Extremes;
s = the Sum of the Means,
a = one of the Means,
Then, x - a, will be = the other.
x - a, will be one Extreme, and
x^2 - 2xa + a^2 the other.
The Proportion then will stand thus:
= 2 x a +
required a?
2 x a a
= s; the Sum of the Extremes:
x - 3xa + 3a x
- 3x . a 3xa = s x a - s
= s x a - s a + 3x a - 3x a
x t
s x s
, t s
Whence by substituting the Values of x, and s, in this Equation, a, will be equal to 1536.
2x - 2x 2 . 2
x - 4 = 768
= 384 and
= 3072, the
above Numbers.
QUESTION.
It is required to find Five Numbers in continued Proportion, the Sum of whose Extremes, is 1989; and the Sum of the Squares of the Second and fourth Terms, is 930,852?
I have here answer'd the Question propos'd in your last; second Question.
The Four Numbers sought, are 384, 768, 1536, and 3072;
which I investigate in this Manner:
Put s = the Sum of the Extremes;
s = the Sum of the Means,
a = one of the Means,
Then, x - a, will be = the other.
x - a, will be one Extreme, and
x^2 - 2xa + a^2 the other.
The Proportion then will stand thus:
= 2 x a +
required a?
2 x a a
= s; the Sum of the Extremes:
x - 3xa + 3a x
- 3x . a 3xa = s x a - s
= s x a - s a + 3x a - 3x a
x t
s x s
, t s
Whence by substituting the Values of x, and s, in this Equation, a, will be equal to 1536.
2x - 2x 2 . 2
x - 4 = 768
= 384 and
= 3072, the
above Numbers.
QUESTION.
It is required to find Five Numbers in continued Proportion, the Sum of whose Extremes, is 1989; and the Sum of the Squares of the Second and fourth Terms, is 930,852?
What sub-type of article is it?
Informative
What themes does it cover?
Education
Science Nature
What keywords are associated?
Continued Proportion
Mathematical Puzzle
Algebraic Solution
Numbers In Proportion
Reader Question
What entities or persons were involved?
Mr. Parks
Letter to Editor Details
Recipient
Mr. Parks
Main Argument
solves the second question from the previous issue with four numbers in continued proportion: 384, 768, 1536, 3072, using algebraic substitution; proposes a new question on five such numbers.
Notable Details
Algebraic Method Using Sums Of Extremes And Means
Equation Substitution Yields A = 1536