go PHILOSOPHICAL TRANSACTIONS. [aNNO 1730. 



tween the corresponding rays of the spiral. — For those rays are a series of terms 

 in a continued geometric progression; and the parts of the circumference form 

 a series of terms in arithmetic progression. Now the terms of the arithmetic 

 series being taken as the exponents of the corresponding terms in the geometric 

 series, there will be the same relation between each geometric term and its corre- 

 lative, as between numbers and their logarithms. And hence the proportional 

 spiral is also called the logarithmic spiral. 



3. That proportional spiral, which intersects its radii at angles of 43 degrees, 

 produces logarithms that are of Napier's kind. — For, if the ditterence betweeiv 

 the first and second terms in the geometric series was indefinitely small, and the 

 first division of the circumference was of the same magnitude; then may that 

 part of the spiral, intercepted between the first and second radii, be taken as 

 the diagonal of a square, two of whose sides are parts of those radii ; therefore 

 the spiral which cuts its rays at angles of 45°, has a kind of logarithms belong- 

 ing to it, so related to their corresponding numbers, that the smallest variation 

 between the first and second terms in the geometric series, is equal to the loga- 

 rithm of the second term, a cypher being taken for the logarithm of the first. 

 But of this kind are the hyperbolical logarithms, or those first made by their 

 inventor the Lord Napier: consequently the logarithms to that spiral which cuts 

 its rays at angles of 43°, are of the Napierian kind. 



4. The rhumb-lines on the globe arc analogous to the logarithmic spiral. — 

 For every oblique rhumb cuts the meridian at equal angles; and it is a property 

 in stereographic projections, that the lines in it intersecting each other, form 

 angles equal to those which they represent on the sphere. Therefore a projection 

 of the sphere being made on the plane of the equator, the meridians will become 

 the radii of the equator, and the rhumbs intersecting them at equal angles, will 

 become the proportional spiral. Hence, the arcs of the equator, or the differ- 

 ences of longitude reckoned from the same meridian, are as the logarithms of 

 those parts of the corresponding meridians, intercepted between the centre and 

 rhumb-line. 



) 5. A sea chart being constructed, in which the meridians are parallel to each 

 other, and the lengths of the degrees of latitude increase in the same proportion as 

 the meridional distances decrease on the globes, will constitute a Mercator's chart, 

 in which, besides the positions of places having the same proportions to each 

 other, as on the globes, the rhumb lines will be represented by right lines. — 

 For none but right lines can cut at equal angles several parallel right lines. 



6. The divisions of the meridi:m line on a Mercator's chart, are the same as a 

 table of the differences of longitude answering to each minute, or small differ- 

 ence of latitude on the rhumb line making angles of 45° with the meridians. — 

 For, in such a chart, the parallels of latitude are equal to the equator, and are 

 at right angles to the meridians; and therefore a rhumb of 45° cuts the meri- 



