4o6 THE MATHEMATICAL PRINCIPLES [BOOK 111. the motion of the sun in the interval between the same conjunctions. But as this motion is to the annual motion 360°, so is the motion of the node but just now found 1S° 19' 5" 53'" to its annual motion, which will there fore be 19° IS' I" 23'" ; and this is the mean motion of the nodes in the sidereal year. By astronomical tables, it is 19° 21' 21" 50'". The dif ference is less than 3- j^- part of the whole motion, and seems to arise from the eccentricity of the moon's orbit, and its inclination to the plane of the ecliptic. By the eccentricity of this orbit the motion of the nodes is too much accelerated ; and, on the other hand, by the inclination of the orbit, the motion of the nodes is something retarded, and reduced to its just velocity. PROPOSITION XXXIII. PROBLEM XIV. To find the true motion, of the nodes of the moon. In the time which is as the area NTA— NrfZ (in the preceding Fig.) that motion is as the area NAe, and is thence given ; but because the cal culus is too difficult, it will be better to use the following construction of the Problem. About the centre C, with any interval CD, describe the circle BEFD ; produce DC to A so as AB may be to AC as the mean motion to half the mean true motion when the nodes are in their quadratures (that is, as 19° 18' I" 23'" to 19° 49' 3" 55'" ; and therefore BC to AC as the difference of those motions 0G Jl' 2" 32'" to the latter motion 19° 49' 3" 55'", that is, as 1 to 38T\). Then through the point D draw the indefinite line Gg, touching the circle in. I) ; and if we take the angle BCE, or BCF, equal to the double distance of the sun from the place of the node, as found by the mean motion, and drawing AE or AF cutting the perpendicular DG in G, we take another angle which shall be to the whole motion of the node in the interval be tween its syzygies (that is, to 9° IV 3") as the tangent DG to the whole circumference of the circle BED, and add this last angle (for which the angle DAG may be used) to the mean motion of the nodes, while they are passing from the quadratures to the syzygies, and subtract it from their mean motion while they are passing from the syzygies to the quadratures, we shall have their true motion ; for the true motion so found will nearly agree with the true motion which comes out from assuming the times as the area NTA — NrfZ, and the motion of the node as the area NAe ; as whoever will please to examine and make the computations will find : and this is the semi -menstrual equation of the motion of the nodes. But there is also a menstrual equation, but which is by no means necessary for find