2. If the radius be 10, and PD 4.

Then CD-DP1046CP; and /CA-CP2

And 2CD16+PD≈20, or CD—10,

PROBLEM IX.

To find the diameter and circumference of a circle, either from

the other.

RULE I.*

As 7 is to 22, so is the diameter to the circumference. so is the circumference to the diameter.

As 22 is to 7,

RULE

The ratio of the diameter of a circle to its circumference has never yet been exactly attained. Nor can a square, or any other right-lined figure, be found, that shall be equal to a given circle. This is the famous problem, called the squaring of the circle, which has exercised the abilities of the greatest mathematicians for ages, and been the occasion of so many endless disputes.Several persons of considerable eminence have, at different times, pretended, that they had discovered the exact quadrature; but their errors have soon been detected, and it is now generally looked upon as a thing impossible to be done.

But though the relation between the diameter and circumference cannot be expressed in known measure, it may yet be approximated to any assigned degree of exactness. And thus that incomparable Geometer, the great Archimedes, about two thousand

years

RULE 2.

As 113 is to 355, so is the diameter to the circuinference.

As 355 ameter.

is to 113, so is the circumference to the di

RULE

years ago, had discovered this ratio to be nearly as 7 is to 22, which is the same as our first rule.

By inscribing and circumscribing polygons of 96 sides, he found the ratio to be less than 34, but greater than 3 to 1; and thence inferred the ratio above mentioned, as may be seen in his book de dimensione circuli. And in this manner was the problem more anciently performed by Philo Gedarensis, and by Apollonius Pergaus, in a work, not come to our hands, called Ocyteoboos, as we are are informed by Eutocius, in his commentary on Archimedes.

the

The ratio of Vieta and Metius is that of 113 to 355, which is something more exact than the former, and is the same as second rule.

But the first, who ascertained this ratio to any great degree of exactness, was Van Ceulen, a Dutchman, in his book de Circulo et Adscriptis. He found, that if the diameter of a circle was 1, the circumference would be 314159265358979323846264338 3279502884 nearly. And this is exactly true to 36 places of decimals, and was effected by means of the continual bisection of an arc of a circle, which was so exceedingly troublesome and laborious, that it must have cost him incredible pains. It is said. to have been thought so curious a performance, that the numbers were cut on his tomb stone, in St. Peter's church yard at Leyden. This last number was not only confirmed, but extended to double the number of places, by the ingenious Mr. Abraham Sharp.

But since the invention of Fluxions, and the Summation of Infinite Series, there have been several methods found out for doing

the

RULE 3.

As I is to 31416, so is the diameter to the circum ference.

As 3'1416 is to I, so is the circumference to the di ameter.

EXAMPLES.

the same thing with less labour and trouble, and far more expedition. Mr. John Machin, Professor of Astronomy in Gresham College, has, by these means, given a quadrature of the circle, which is true to 100 places of decimals; and M. De Lagny and M. Euler have carried it still farther. And these last expressions are so extremely near the truth, that, except the ratio could be completely obtained, we need not wish for a greater degree of

accuracy.

The method of obtaining this proportion from the doctrine of fluxions may be shewn as follows:

Take Ac any arc of a circle, and let cr be an indefinitely small tangent at the point c.

Then draw the lines as in the figure, and put OA=r, Ab=x, bey, ATt, and Acz; and for the fluxion of a simple quantity put a point over it.

Now, since the triangles ren, clO, and TAO are similar, wę

rx

fluxion of the arc, in terms of the versed sine.

And

EXAMPLES.

1. To find the circumference of a circle, whose diame ter AB is 10.

By.

And also, Ob (—y2): Oc (r) : ; nr -(9): cr

ry

2

fluxion of the arc, in terms of the sine.

And, in like manner, AT (t); OT (√r2+t3) :: cn

; but OT (√r2+12) : OA (r) :: Oc (r).

and therefore Ab (x) =

3

Now, from any of the three forms of fluxions here found, their fluents, or the value of the arc itself, will become known.

But the third form, expressed in terms of the tangent, will be the most convenient, because it is entirely free from radical quan

Then, by taking Acto any given arc, whose tangent can be found in terms of the radius, the series will become known; and being repeated as often as Ac is contained in the whole circumference, we shall have the length of the circumference in terms of the diameter.

Thus, if the radius be 1, and Ac be part of the circumfer ence, or 45°, its tangent will be equal to the radius, and the se

ries will become I ÷+子 + · of 45°, and 8 x : 1 − ÷ +} − ÷ + ÷ whole circumference.

X

This series is the simplest form, that can possibly be obtained, but in order to get another, that will converge faster, we must

take