IV.

THE HYPERBOLA AS CONNECTED WITH THE LOGARITHMS.

I.

WHILE a straight line PM (Fig. XVII.) moves parallel to itself along the indefinite straight line CPD with a velocity always proportional to the distance of its extremity P from a fixed point C, let its other extremity M approach to or recede from P, fo that PM may defcribe equal spaces in equal times: The point P will defcribe a part PP' or Pp' of the straight line CD, while the point M describes a correfponding part MM' or Mm of the curve m'SM'.

2. If the motion is fuppofed to have begun at P, the area PM M'P' or PM m'p' is the logarithm of the abfcifs CP' or Cp'.

3. In order that equal spaces may be described in equal times, it is evident that the greater or fmaller the abfcifs CP' or Cp' becomes with regard to CP, the smaller or greater must the ordinate P'M' or p'm' become with regard to PM; Therefore CP': CP:: PM: P'M', or Cp': CP :: PM: p'm'; Therefore the product of any abfcifs by the correfpondent ordinate is a conftant quantity: Therefore

4. The curve m'SM' is a hyperbola having CD for one of its affymptotes, and Cs, parallel to the ordinates, for the other.

C:

5. From this manner of conceiving the generation of the hyperbola might be deduced the properties of that curve and of the logarithms. That CD and C, for inftance, touch the curve at an infinite diftance from C appears from this: When the abfcifs is infinite, the ordinate must be zero, and when the abfcifs is zero, the ordinate must be infinite, in order that their product may equal the finite quantity PM x CP: And that the logarithm of CP is zero appears from this; PM is length without breadth and therefore no space.

6. Let CP=a, PM, PP'x and P'M'=y; we have (3) y=, or, developing the fraction in the manner first taught by Nicolas Mercator*,

7. It is evident that the space PMM`P' is equal to the fum of all the ordinates y+y"+"+ &c. on the abfcifs x. If the abfcifs is fuppofed to be divided into an infinite number of infinitely finall and equal the abfciffæ corresponding to the ordinates y', y",y'", &c. parts, called 1, 2, 3, &c: therefore (6)

may

be

Now, as was first demonftrated by Wallis, the fum 1+2"+3"+ &c. continued to infinity, that is to x" in this cafe, being equal to "+!

; we

formed by the af

fymtotes and the diftance MN=m of the point M of the curve from

rm

the assymptote CD; as is evident from its value = Sin' where r de

notes the radius of the circle.

Arith. Infinit.

SECTION.