VOL. XIX.] PHILOSOPHICAL TKANSACTIONS. 6Q 



long train of consequences and complication of proportions, by which the evi- 

 dence of the demonstration is in a great measure lost, and the reader wearied 

 before he attain it. Nor with less labour and apparatus has Dr. Barrow, in his 

 Geometrical Lectures, lect. xi, app. i, proved, that the sum of all the secants 

 of any arch is analogous to the logarithm of the ratio of radius -f- sine to radius 

 — sine, or, which is all one, that the meridional parts answering to any degree 

 of latitude, are as the logarithms of the rationes of the versed sines of the 

 distances from both the poles. Since which. Dr. Wallis, on occasion of a para- 

 logism committed by one Mr. Norris in this matter, has more fully and clearly 

 handled this argument, as may be seen in N° 176 of these Transactions. But 

 neither Dr. Walhs nor Dr. Barrow, in their said treatises, have any where 

 touched on the aforesaid relation of the meridian line to the logarithmic tan- 

 gent ; nor has any one, that I know of, yet discovered the rule for computing, 

 independently, the interval of the meridional parts answering to any two given 

 latitudes. 



Wherefore, having attained, as I conceive, a very easy and natural demon- 

 stration of the said analogy, and having found out the rule for exhibiting the 

 difference of meridional parts, between any two parallels of latitude, without 

 finding both the numbers of which they are the difference ; I hope I may be 

 entitled to a share in the improvements of this useful part of geometry. De- 

 siring no other favour of some mathematical pretenders, than that they think 

 fit to be so just, as neither to attribute my desire to please the honourable the 

 Royal Society in these exercises to any kind of vanity or love of applause in me, 

 (who too well know how very few these things oblige, and how small reward 

 they procure,) nor yet to complain coram non judice, that I arrogate to myself 

 the inventions of others, and on that pretext to depreciate what I do, unless at 

 the same time they can produce the author I wrong, to prove their assertions. 

 Suchdisingenuity, as I have always most carefully avoided, so I wish I had not 

 too much experience of it in the very same persons, who make it their business 

 to detract from that little share o f reputation I have in these things. But to 

 return to the matter in hand, let us demonstrate the following proposition. 



The meridian line is a scale of logarithmic tangents of the half comple- 

 ments of the latitudes. — For this demonstration, it is requisite to premise these 

 four lemmata. 



Lemma I. — In the stereographic projection of the sphere on the plane of the 

 equinoctial, the distances from the centre, which in this case is the pole, are 

 laid down by the tangents of half those distances, that is, of half the compie- 

 ments of the latitudes. This is evident from Eucl. 3, 20. 



Lemma 11. — In the stereographic projection, the angles, under which the 



