[ I02 ] 



concave arch, at which a tangent plane would be parallel to the 

 upper furface, and which point muft be the centre of the bubble: 



alfo 



b3th; and let tlie centre of the bubble reft, not at the point A but at D, making the 

 angle ^CZ) the error of the inftruinent, whofe fine is BD, and verfcd fine BA: 

 the fame line B A will be the verfed fine of the angle of aberration in all levels of 

 fimilar conftvudion and regular fliape, whatever be the degree of curvature of their 

 tubes : for becaufe the force with which fluids move in any direftion through a pipe 

 or orifice is q. p. equal to the weight of the fuperincumbent fluid, which in this cafe 

 is that of .1 column of liquor, whofe altitude will he A B (if while the centre of 

 the bubble is at D, its extremity extends to ^j but lefs, and equally fo in both 

 tubes, if the extremity refts at a diftance from A on either fide) ; fo the weight and 

 height of this column muft be in both levels the fame, becaufe it is counterpoifed by 

 the attraction of cohefion (I fpeak not of eledfric attraflion) arrefting the liquor, 

 and preventing the extremities of the bubble from fettling in an horizontal plane ; 

 which attra£lion is the fame and invariable in tubes of equal diameters whatever be 

 their curvature, and therefore in both tubes A F, AG, the centre of the bubble 

 will relt in the line B D : let it reft in the tube of lefler curvature or greater radius 

 at the point E, and in the other tube A D in D ; then the arch A E, meafuring the 

 angle of aberration in the former tube will be to the arch AD the aberration in the 

 latter in the inverfe ratio of the line B E to the line B D : thefe lines are the fines 

 of the angles; but the fine or ordinate of a circle is a mean proportional between 

 the verfed fine or abfcifle and its complement to the diameter ; for if lines be drawn 

 from the extremes of the diameter to any point in the circumference, as for inftance, 

 to D in the circle A D, they conftitute an angle in a femicircle, which muft be a 

 right one, (El. 3. 31), from whofe vertex the fine DB falls perpendicular 

 on the bafe -, fo that the verfed fine AB is to the fine B D ^s this BD 

 is to the complement of the diameter B H (El. 6. 8.) ; i. e. the reiSangle 

 A B, BH IS equal to the fquare of BD (El. 6. 16.); alfo in the circle 

 AE the reaangle under the fame abfcifle A B, and the complement to its 

 diameter is equal to the fquare of B E : but thefe tv/o rearangles, fince one of their 

 fides AB \i common, muft be in the ratio of the other fides; i. e (becaufe A B li 

 fo very fmall that it may be negleded), in the ratio of the whole diameters, or of 

 the radii which are as the diameters; which ritio is that of the fquare oi B D 

 to the fquare of BE; confequently when the fines are as i to 2, the diameters or 



radii 



