# Page:Philosophical Transactions - Volume 003.djvu/15

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thereto, and finds the Logarithms of all Primitive Numbers under 1000 by one Multiplication, two Divisions, and the Extraction of the Square Root, but for Prime Numbers greater, much more easily.

Concerning the construction of Logarithms Mr. Nicholas Mercater hath a Treatise, intituled Logarithmotechnia, likewise at the Press, from which the Reader may receive further satisfaction. And as for Primitive Numbers, and whether any odd number proposed less than 100000 be such, the Reader will meet with a satisfactory Table at the end of a Book of Algebra, written in High Dutch by John Henry Rohn, now translated and enriched, and near ready for publick view.

The Area of an Hyperbola not being yet given by any Man, we think fit a little to explain the Author's meaning.

In Figure 1. Let the Curve DIL represent an Hyperbola, whose Asymptotes AO, AK, make the Right Angle OAK, the Author propounds to find the Hyperbolick space ILNK, contained by the Hyperbolical Line IL, the Asymptote KM, and the two Right Lines IK, LM, which are parallel to the other Asymptote AO.

 He puts the Lines IK = 1 000 000 000 000 LM = 1 000 000 000 000 0 AM = 1 000 000 000 000 Hence KM = 9 000 000 000 000

Whence he finds the space LIKM

 to be ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 230 258 509 299 404 562 401 78681 too little. 230 258 509 299 404 562 401 78704 too great.

Note: If IK be put for an Unit, then LM may represent 10, and HG 1000, and FE 1024: And, by what is demonstrated by Gregory of St. Vincent, it holds,

As the space IBLMKI, Is to the Logarithm of LM, to wit, of 10: So is the space IBEFKI, To the Logarithm of the Number represented by the Line EF, to wit, of 1024

The Author by the same method finds the Area of the space GEFH to be 237 165 266 173 160 421 183 067, and the space LIKM abovesaid being taken for the Logarithm of 10, and tripled, is the Logarithm of 1000, the which added to the space now found, makes the sum 69314718055994529141719170, and 1024, being the 10th Power of 2, the 10th part of this number is the Hyperbolical Logarithm of the Numb. 2, to wit, 6931471805599452914171917. And it holds by proportion,

As 23025850929940456249178700, the Logarithm of 10, To 6931471805599452914171917, the correspondent Logarithm of 2: So 1 000 000 000 000 000 000 000 000 0, the Logarithm of 10 in the

Tables,