VOL. XXIX.J PHILOSOPHICAL TRANSACTIONS. 185 



are all parallel lines, not divided equally, as in the common plain chart, which 

 is therefore erroneous, but the minutes and degrees, or strictly the fluxions of 

 the meridian, at every several latitude, are proportional to their respective 

 secants. Or a degree in the projected meridian, at any latitude, is to a degree 

 of longitude in the equator, as the secant of the same latitude is to radius. 



Tlie reason of which enlargement of the elements of latitude is, to counter- 

 balance the enlargement of the degrees of longitude. For in this projection 

 the meridians being all parallel, a degree of longitude at, suppose, 6o° latitude, 

 is become equal to a degree in the equator, whereas it really is, on the globe's 

 surface, only half as much, the radius of the parallel of 6o°, that is its cosine, 

 being but half the radius of the equator. Therefore to proportion the degrees 

 of latitude to those of longitude, a degree, or elemental particle, in the meri- 

 dian, is to be as much greater than a degree, or like particle, in the equator, 

 as the radius of the equator is greater than the radius of the parallel of latitude, 

 viz. its cosine. 



In fig. 8, pi. 4, let the radius cd represent half the equator, dm an arc of 

 the meridian; ms its sine, ce its secant; then is cs equal to its cosine; and cs 

 : CM :: cd (= cm) : ce, that is, as cosine : to radius :: so is radius : to secant. 

 The cosines being then, in this projection, supposed all equal to radius, or, 

 which comes to the same, the parallels of latitude being all made equal to the 

 equator, the radius of the globe, at every point of latitude, by the precedent 

 analogy, is supposed equal to the secant of latitude; and consequently the 

 elements, minutes, &c. of the meridian must be proportional to their respective 

 secants. 



The way Mr. Wright takes for making his table of meridional parts, is by a 

 continual addition of natural secants, beginning at 1 minute, and so proceeding 

 to 89 deg. Dr. Wallis, in Philos. Trans. N° 176, finds the meridional part 

 belonging to any latitude by this series, putting s for its natural sine, viz. s + 

 ^^ -f- 4-*^ -}- -1^^ + -i^^ &c. which gives the meridional part required. How to 

 find the same mechanically by means of an easily constructed curve line, is what 

 I shall now show. 



1. Prepare a ruler ab, fig. 9, of a convenient length, in which let bo be 

 equal to the radius of the intended projection. To the point o as a centre, on 

 the narrower edge of the ruler, fasten a little plate-wheel wh tight to the ruler, 

 and of a diameter a little more than the thickness of the ruler. Let kr, fig. 8, 

 represent another long ruler, to which ar is a perpendicular line. Place the 

 ruler ab on the line ar, with the centre of the wheel at a. Then with one 

 hand holding fast the ruler kr, with the other hand slide the end b of the ruler 



VOL. VI. B B 



