Table for computing the Moon’s motion for 12 hours at any intermediate time.

h.	m.	(x.)	1st diff.	(y.)	1st diff.	(z.)	1st diff.
0.	0.	–,5000000.	+.	+,3333333.	–.	+,0833333.	–.
			20833.		20805.		1750.
	3.	–,4979167.		+,3312528.		+,0831583.
			20834.		20746.		1779.
	6.	–,4958333.		+,3291782.		+,0829804.
			20833.		20688.		1808.
	9.	–,4937500.		+,3271094.		+,0827996.
			20833.		20631.		1837.
	12.	–,4916667.		+,3250463.		+,0826159.
			20834.		20573.		1864.
	15	–,4895833.		+,3229890.		+,0824295
			20833.		20515.		1893.
	18.	–,4875000.		+,3209375.		+,0822402.
			20833.		20457.		1920.
	21.	–,4854167.		+,3188918.		+,0820482.
			20834.		20399.		1948.
	24	–,4833333.		+,3168519.		+,0818534.
			20833.		20342.		1976.
	27.	–,4812500.		+,3148177.		+,0816558.
			20833.		20284.		2003.
	30.	–,4791667.		+,3127893.		+,0814555.
			20834.		20225.		2030.
	33.	–,4770833.		+,3107668.		+,0812525.
			20833.		20168.		2056.
	36.	–,4750000.		+,3087500.		+,0810469.
			20833.		20110.		2084.
	39.	–,4729167.		+,3067390.		+,0808385.
			20834.		20052.		2110.
	42	–,4708333.		+,3047338		+,0806275.
			20833.		19994.		2137.
	45	–,4687500.		+,3027344		+,0804138.
			20833.		19937.		2163.
	48.	–,4666667.		+,3007407.		+,0801975.
			20834.		19878.		2189.
	51.	–,4645833.		+,2987529.		+,0799786.
			20833.		19821.		2215
	54.	–,4625000.		+,2967708.		+,0797571.
			20833.		19763.		2240.
	57.	–,4604167.		+,2947945.		+,0795331.
			20834.		19704.		2266.
1.	0.	–,4583333.		+,2928241.		+,0793065.
			20833.		19647.		2291.
	3.	–,4562500.		+,2908594.		+,0790774.
			20833.		19589.		2317.
	6.	–,4541667.		+,2889005.		+,0788457.
			20834.		19532.		2341.
	9.	–,4520833.		+,2869473.		+,0786116.
			20833.		19473.		2366.
	12.	–,4500000.		+,2850000.		+,0783750.
			20833.		19416.		2391
	15.	–,4479167.		+,2830584.		+,0781359.
			20834.		19357.		2415
	18.	–,4458333.		+,2811227.		+,0778944.
			20833.		19300.		2439.
	21.	–,4437500.		+,2791927.		+,0776505.
			20833.		19242.		2464.
	24.	–,4416667.		+,2772685.		+,0774041.
			20834.		19184.		2487.
	27.	–,4395833.		+,2753501.		+,0771554.
			20833.		19126.		2511.
	30.	–,4375000.		+,2734375.		+,0769043.
			20833.		19069.		2535.
	33.	–,4354167.		+,2715306.		+,0766508.
			20834.		19010.		2558.
	36.	–,4333333.		+,2696296.		+,0763950.
			20833.		18952.		2581.
	39.	–,4312500.		+,2677344.		+,0761369.
			20833.		18895.		2603.
	42.	–,4291667.		+,2658449.		+,0758766.
			20834.		18837.		2627.
	45.	–,4270833.		+,2639612.		+,0756139.
			20833.		18779.		2650.
	48.	–,4250000.		+,2620833.		+,0753489.
			20833.		18721.		2672.
	51.	–,4229167.		+,2602112.		+,0750817.
			20834.		18663.		2693.
	54.	–,4208333.		+,2583449.		+,0748124.
			20833.		18605.		2717.
	57.	–,4187500.		+,2564844.		+,0745407.
			20833.		18548.		2738.
2.	0.	–,4166667.		+,2546296.		+,0742669.
			20834.		18490.		2759.
	3.	–,4145833.		+,2527806.		+,0739910.
			20833.		18431.		2781.
	6.	–,4125000.		+,2509375		+,0737129.
			20833.		18374.		2803.
	9.	–,4104167.		+,2491001.		+,0734326.
			20834.		18316.		2823.
	12.	–,4083333.		+,2472685		+,0731503.
			20833.		18258.		2845.
	15	–,4062500.		+,2454427.		+,0728658.
			20833.		18200.		2866.
	18	–,4041667.		+,2436227.		+,0725792.
			20834.		18143.		2886.
	21.	–,4020833.		+,2418084.		+,0722906.
			20833.		18084.		2906
	24.	–,4000000.		+,2400000.		+,0720000.
			20833.		18027.		2927.
	27.	–,3979167.		+,2381973.		+,0717073.
			20834.		17969.		2947.
	30.	–,3958333.		+,2364004.		+,0714126.
			20833.		17910.		2966.
	33.	–,3937500.		+,2346094.		+,0711160.
			20833.		17853.		2987.
	36.	–,3916667.		+,2328241.		+,0708173.
			20834.		17796.		3006.
	39.	–,3895833.		+,2310445.		+,0705167.
			20833.		17737.		3025.
	42.	–,3875000.		+,2292708.		+,0702142.
			20833.		17679.		3045.
	45.	–,3854167.		+,2275029.		+,0699097.
			20834.		17622.		3063.
	48.	–,3833333.		+,2257407.		+,0696034.
			20833.		17563.		3082.
	51.	–,3812500.		+,2239844.		+,0692952
			20833.		17506.		3101.
	54.	–,3791667.		+,2222338.		+,0689851.
			20834.		17448.		3120.
	57.	–,3770833.		+,2204890.		+,0686731.
			20833.		17390.		3137.
3.	0.	–,3750000.		+,2187500.		+,0683594.
			20833.		17332.		3156.
	3.	–,3729167.		+,2170168.		+,0680438.
			20834.		17275.		3174.
	6.	–,3708333.		+,2152893.		+,0677264.
			20833.		17216.		3191.
	9.	–,3687500.		+,2135677.		+,0674073.
			20833.		17159.		3209.
	12.	–,3666667.		+,2118518.		+,0670864.
			20834.		17100.		3226.
	15.	–,3645833.		+,2101418.		+,0667638.
			20833.		17043.		3244.
	18.	–,3625000.		+,2084375.		+,0664394.
			20833.		16985.		3260.
	21.	–,3604167.		+,2067390.		+,0661134.
			20834.		16927.		3277.
	24.	–,3583333.		+,2050463.		+,0657857.
			20833.		16869.		3294.
	27.	–,3562500.		+,2033594.		+,0654563.
			20833.		16812.		3311.
	30.	–,3541667.		+,2016782.		+,0651252
			20834.		16753		3326.
	33	–,3520833.		+,2000029.		+,0647926.
			20833.		16696.		3343.
	36.	–,3500000.		+,1983333.		+,0644583.
			20833.		16638.		3358.
	39.	–,3479167.		+,1966695.		+,0641225
			20834.		16579.		3375.
	42.	–,3458333.		+,1950116.		+,0637850.
			20833.		16522		3390
	45.	–,3437500.		+,1933594.		+,0634460.
			20833.		16465.		3405.
	48.	–,3416667.		+,1917129.		+,0631055.
			20834.		16406.		3421.
	51.	–,3395833		+,1900723.		+,0627634.
			20833.		16348		3435.
	54.	–,3375000.		+,1884375.		+,0624199.
			20833.		16291		3450.
	57.	–,3354167.		+,1868084.		+,0620749.
			20834.		16232.		3465.
4.	0.	–,3333333.		+,1851852.		+,0617284.
			20833.		16175.		3480.
	3.	–,3312500.		+,1835677.		+,0613804.
			20833.		16117.		3493.
	6.	–,3291667.		+,1819560.		+,0610311.
			20834.		16059.		3508.
	9.	–,3270833.		+,1803501.		+,0606803.
			20833.		16001.		3522.
	12.	–,3250000.		+,1787500.		+,0603281.
			20833.		15943.		3536.
	15.	–,3229167.		+,1771557.		+,0599745.
			20834.		15886.		3549.
	18.	–,3208333.		+,1755671.		+,0596196.
			20833.		15827.		3562.
	21.	–,3187500.		+,1739844		+,0592634.
			20833.		15770.		3576.
	24.	–,3166667.		+,1724074.		+,0589058.
			20834.		15712.		3589.
	27:	–,3145833.		+,1708362.		+,0585469.
			20833.		15654.		3601.
	30	–,3125000.		+,1692708.		+,0581868.
			20833.		15596.		3614.
	33.	–,3104167.		+,1677112.		+,0578254.
			20834.		15538		3627.
	36.	–,3083333.		+,1661574.		+,0574627.
			20833.		15480.		3638.
	39.	–,3062500		+,1646094.		+,0570989.
			20833.		15423.		3651.
	42	–,3041667.		+,1630671.		+,0567338.
			20834.		15365.		3663.
	45.	–,3020833.		+,1615306.		+,0563675.
			20833.		15306.		3675.
	48.	–,3000000.		+,1600000.		+,0560000.
			20833.		15249.		3686.
	51.	–,2979167.		+,1584751.		+,0556314.
			20834.		15191.		3698.
	54.	–,2958333.		+,1569560.		+,0552616.
			20833.		15133.		3709.
	57.	–,2937500.		+,1554427.		+,0548907.
			20833.		15075		3720.
5.	0.	–,2916667.		+,1539352.		+,0545187.
			20834.		15018.		3731.
	3	–,2895833		+,1524334.		+,0541456.
			20833.		14959.		3741.
	6.	–,2875000.		+,1509375		+,0537715.
			20833.		14902.		3752.
	9.	–,2854167.		+,1494473.		+,0533963.
			20834.		14844.		3762.
	12.	–,2833333.		+,1479629.		+,0530201.
			20833.		14785.		3773.
	15.	–,2812500.		+,1464344.		+,0526428.
			20833.		14728.		3782.
	18.	–,2791667.		+,1450116.		+,0522646.
			20834.		14671.		3792.
	21.	–,2770833.		+,1435445.		+,0518854.
			20833.		14612.		3802.
	24.	–,2750000.		+,1420833.		+,0515052.
			20833.		14554.		3811.
	27.	–,2729167.		+,1406279.		+,0511241.
			20834.		14497.		3820.
	30.	–,2708333.		+,1391782.		+,0507421.
			20833.		14438.		3830.
	33	–,2687500.		+,1377344.		+,0503591.
			20833.		14381.		3838.
	36	–,2666667.		+,1362963.		+,0499753.
			20834.		14323.		3847.
	39.	–,2645833.		+,1348640.		+,0495906.
			20833.		14265.		3855.
	42.	–,2625000.		+,1334375.		+,0492051.
			20833.		14208.		3864.
	45.	–,2604167.		+,1320167.		+,0488187.
			20834.		14149.		3872.
	48.	–,2583333.		+,1306018.		+,0484315.
			20833.		14091.		3880.
	51.	–,2562500.		+,1291927.		+,0480435.
			20833.		14034.		3887.
	54	–,2541667		+,1277893.		+,0476548.
			20834		13975.		3896.
	57.	–,2520833		+,1263918.		+,0472652.
			20833.		13918.		3902.
6.	0.	–,2500000.		+,1250000.		+,0468750.
			20833.		13860.		3910.
	3.	–,2479167.		+,1236140.		+,0464840.
			20834.		13802.		3917.
	6.	–,2458333.		+,1222338.		+,0460923.
			20833.		13745.		3924.
	9.	–,2437500.		+,1208593.		+,0456999.
			20833.		13686.		3930.
	12.	–,2416667.		+,1194907.		+,0453069.
			20834.		13628.		3937.
	15.	–,2395833.		+,1181279.		+,0449132.
			20833.		13571.		3943.
	18.	–,2375000.		+,1167708.		+,0445189.
			20833.		13513.		3950.
	21.	–,2354167.		+,1154195.		+,0441239.
			20834.		13454.		3955.
	24.	–,2333333.		+,1140741.		+,0437284.
			20833.		13397.		3961.
	27.	–,2312500.		+,1127344.		+,0433323.
			20833.		13340.		3968.
	30.	–,2291667.		+,1114004.		+,0429355.
			20834.		13281.		3972.
	33.	–,2270833.		+,1100723.		+,0425383.
			20833.		13223.		3977.
	36.	–,2250000.		+,1087500.		+,0421406.
			20833.		13166.		3982.
	39.	–,2229167.		+,1074334.		+,0417424.
			20834.		13107.		3988.
	42.	–,2208333.		+,1061227.		+,0413436.
			20833.		13050.		3992.
	45	–,2187500.		+,1048177.		+,0409444.
			20833.		12992.		3997.
	48	–,2166667.		+,1035185		+,0405447.
			20834.		12934.		4001.
	51.	–,2145833.		+,1022251.		+,0401446.
			20833.		12876.		4005.
	54.	–,2125000.		+,1009375.		+,0397441.
			20833.		12819.		4009.
	57.	–,2104167.		+,0996556		+,0393432
			20834.		12760.		4012.
7.	0.	–,2083333.		+,0983796		+,0389419.
			20833.		12702.		4016.
	3.	–,2062500.		+,0971094.		+,0385403.
			20833.		12645.		4020.
	6.	–,2041667.		+,0958449.		+,0381383.
			20834.		12587.		4024.
	9.	–,2020833.		+,0945862.		+,0377359.
			20833.		12529.		4026.
	12.	–,2000000.		+,0933333.		+,0373333.
			20833.		12471.		4032.
	15.	–,1979167.		+,0920862.		+,0369304.
			20834.		12413.		4032.
	18.	–,1958333.		+,0908449.		+,0365272.
			20833.		12355.		4034.
	21.	–,1937500.		+,0896094.		+,0361238.
			20833.		12298.		4037.
	24.	–,1916667.		+,0883796.		+,0357201.
			20834.		12240.		4039.
	27.	–,1895833.		+,0871556.		+,0353162.
			20833.		12181.		4041.
	30.	–,1875000.		+,0859375.		+,0349121.
			20833.		12124.		4043.
	33.	–,1854167.		+,0847251.		+,0345078.
			20834.		12066.		4044.
	36.	–,1833333.		+,0835185.		+,0341034.
			20833.		12008.		4046.
	39.	–,1812500.		+,0823177.		+,0336988.
			20833.		11950.		4047.
	42.	–,1791667.		+,0811227.		+,0332941.
			20834.		11893.		4048.
	45.	–,1770833.		+,0799334.		+,0328893.
			20833.		11834.		4049.
	48.	–,1750000.		+,0787500.		+,0324844.
			20833.		11777.		4050.
	51.	–,1729167.		+,0775723.		+,0320794.
			20834.		11719.		4050.
	54.	–,1708333.		+,0764004.		+,0316744.
			20833.		11660.		4051.
	57.	–,1687500.		+,0752344.		+,0312693.
			20833.		11603.		4051.
8.	0.	–,1666667.		+,0740741.		+,0308642.
			20834.		11546.		4051.
	3.	–,1645833.		+,0729195.		+,0304591.
			20833.		11487.		4051.
	6.	–,1625000.		+,0717708.		+,0300540.
			20833.		11429.		4051.
	9.	–,1604167.		+,0706279.		+,0296489.
			20834.		11372.		4049.
	12.	–,1583333.		+,0694907.		+,0292440.
			20833.		11313.		4048.
	15.	–,1562500.		+,0683594.		+,0288391.
			20833.		11256.		4048.
	18.	–,1541667.		+,0672338.		+,0284343.
			20834.		11198.		4047.
	21.	–,1520833.		+,0661140.		+,0280296.
			20833.		11140.		4046
	24.	–,1500000.		+,0650000.		+,0276250.
			20833.		11082.		4045
	27.	–,1479167.		+,0638918.		+,0272205.
			20834.		11025.		4042
	30.	–,1458333.		+,0627893.		+,0268163.
			20833.		10966.		4041.
	33.	–,1437500.		+,0616927.		+,0264122.
			20833.		10909.		4039.
	36.	–,1416667.		+,0606012.		+,0260083.
			20834.		10850.		4037.
	39.	–,1395833.		+,0595168.		+,0256046.
			20833.		10793.		4034.
	42.	–,1375000.		+,0584375.		+,0252012.
			20833.		10735.		4032.
	45.	–,1354167.		+,0573640.		+,0247980.
			20834.		10677.		4030.
	48.	–,1333333.		+,0562963.		+,0243950.
			20833.		10619.		4026.
	51.	–,1312500.		+,0552344.		+,0239924.
			20833.		10562.		4023.
	54.	–,1291667.		+,0541782		+,0235901.
			20834.		10503.		4020.
	57.	–,1270833.		+,0531279.		+,0231881.
			20833.		10446.		4017.
9.	0.	–,1250000.		+,0520833.		+,0227864
			20833.		10388.		4013.
	3.	–,1229167.		+,0510445.		+,0223851.
			20834.		10330.		4009.
	6.	–,1208333.		+,0500115.		+,0219842.
			20833.		10271.		4005.
	9.	–,1187500.		+,0489844.		+,0215837.
			20833.		10214.		4001.
	12.	–,1166667.		+,0479630.		+,0211836.
			20834.		10157.		3997.
	15.	–,1145833.		+,0469473.		+,0207839.
			20833.		10098.		3992.
	18.	–,1125000.		+,0459375.		+,0203847.
			20833.		10041.		3987.
	21.	–,1104167.		+,0449334.		+,0199860.
			20834.		9982.		3982.
	24.	–,1083333.		+,0439352.		+,0195878.
			20833.		9925.		3977.
	27.	–,1062500.		+,0429427.		+,0191901.
			20833.		9867.		3973.
	30.	–,1041667.		+,0419560.		+,0187928.
			20834.		9809.		3967.
	33.	–,1020833.		+,0409751.		+,0183961.
			20833.		9751.		3961.
	36.	–,1000000.		+,0400000.		+,0180000.
			20833.		9693.		3956.
	39.	–,0979167.		+,0390307.		+,0176044.
			20834.		9636.		3949.
	42.	–,0958333.		+,0380671.		+,0172095.
			20833.		9577.		3943.
	45.	–,0937500.		+,0371094.		+,0168152.
			20833.		9520.		3937.
	48.	–,0916667		+,0361574.		+,0164215.
			20834.		9462.		3931.
	51.	–,0895833.		+,0352112.		+,0160284.
			20833.		9404.		3923.
	54.	–,0875000.		+,0342708.		+,0156361.
			20833.		9346.		3917.
	57.	–,0854167.		+,0333362.		+,0152444.
			20834.		9288.		3910.
10.	0.	–,0833333.		+,0324074.		+,0148534.
			20833.		9230.		3903.
	3.	–,0812500.		+,0314844.		+,0144631.
			20833.		9173.		3895.
	6.	–,0791667.		+,0305671.		+,0140736.
			20834.		9114.		3888.
	9.	–,0770833.		+,0296557.		+,0136848.
			20833.		9057.		3879.
	12.	–,0750000.		+,0287500.		+,0132969.
			20833.		8999.		3872.
	15.	–,0729167.		+,0278501.		+,0129097.
			20834.		8941.		3864
	18.	–,0708333.		+,0269560.		+,0125233.
			20833.		8883.		3855.
	21.	–,0687500.		+,0260677.		+,0121378.
			20833.		8825.		3847.
	24.	–,0666667.		+,0251852.		+,0117531.
			20834.		8768.		3839.
	27.	–,0645833.		+,0243084.		+,0113692.
			20833.		8709.		3829.
	30.	–,0625000.		+,0234375.		+,0109863.
			20833.		8652.		3820.
	33.	–,0604167.		+,0225723.		+,0106043.
			20834.		8594.		3811.
	36.	–,0583333.		+,0217129.		+,0102232.
			20833.		8536.		3801.
	39.	–,0562500.		+,0208593.		+,0098431.
			20833.		8478.		3793.
	42.	–,0541667.		+,0200115.		+,0094638.
			20834.		8420.		3783.
	45.	–,0520833.		+,0191695.		+,0090855.
			20833.		8362.		3772.
	48.	–,0500000.		+,0183333.		+,0087083.
			20833.		8304.		3762.
	51.	–,0479167.		+,0175029.		+,0083321.
			20834.		8247.		3752.
	54.	–,0458333		+,0166782.		+,0079569.
			20833.		8189.		3741.
	57.	–,0437500.		+,0158593.		+,0075828.
			20833.		8130.		3731.
11.	0.	–,0416667.		+,0150463.		+,0072097.
			20834.		8073.		3720.
	3.	–,0395833.		+,0142390.		+,0068377.
			20833.		8015.		3709.
	6.	–,0375000		+,0134375.		+,0064668.
			20833.		7957.		3698.
	9.	–,0354167.		+,0126418.		+,0060970.
			20834.		7900.		3686.
	12.	–,0333333.		+,0118518.		+,0057284.
			20833.		7841.		3675.
	15.	–,0312500.		+,0110677.		+,0053609.
			20833.		7783.		3663.
	18.	–,0291667.		+,0102894.		+,0049946.
			20834.		7726.		3651.
	21.	–,0270833.		+,0095168.		+,0046295.
			20833.		7668.		3639.
	24.	–,0250000.		+,0087500.		+,0042656.
			20833.		7610.		3627.
	27.	–,0229167.		+,0079890.		+,0039029.
			20834.		7552.		3614.
	30.	–,0208333.		+,0072338.		+,0035415.
			20833.		7494.		3601.
	33.	–,0187500.		+,0064844.		+,0031814.
			20833.		7437.		3589.
	36.	–,0166667.		+,0057407.		+,0028225.
			20834.		7378.		3576.
	39.	–,0145833		+,0050029.		+,0024649.
			20833.		7321.		3562.
	42.	–,0125000.		+,0042708.		+,0021087.
			20833.		7263.		3549.
	45	–,0104167.		+,0035445.		+,0017538.
			20834.		7204.		3535.
	48.	–,0083333.		+,0028241.		+,0014003.
			20833.		7147.		3522.
	51.	–,0062500.		+,0021094.		+,0010481.
			20833.		7089.		3508.
	54.	–,0041667.		+,0014005.		+,0006973.
			20834.		7032.		3494.
	57.	–,0020833.		+,0006973.		+,0003479.
			20833.		6973.		3479.
12.	0.	–,0000000.		+,0000000.		+,0000000.

Construction of the Table.

When three positions of the Moon at noon and midnight are taken from the Nautical Almanac immediately preceding, and two following the intermediate1 time required, the series for the formation of the decimal fractions (x)(y)(z) will be as follows:—

Let	r,	represent the intermediate time equal to or less than 12 hours:
	t,	12 hours, or 720 minutes.
	C,	the Moon’s third position at noon or midnight.

Then, C, + r.⁄t. + r. × r – 1.⁄2t. + r × r – 2.⁄3t. + r. × r + 1.⁄4t..

The three last terms of the series reduced, will give the respective numerators and denominators of (x)(y)(z) which have been brought into their corresponding decimals throughout the table: or,

Let	r,	represent the intermediate time equal to, or less than 12 hours as before.
	t,	1 hour, or 60 minutes: then

– 12. + r.⁄24, = (x.)

+ 288, – 36r. + rr.⁄864. = (y)

+ 3456, – 144r. – 24rr. + rrr.⁄41472. = (z.)

The numbers or decimal fractions on the table, answering to one hour, are required by the first series.

1⁄12

11⁄24

23⁄36

13⁄48.

11⁄24, the value of (x), reduced, is	= ,4583333.
11⁄24 × 23⁄36 = 253⁄864, the value of (y)	= ,2928241.
11⁄24 × 23⁄36 × 13⁄48 = 3289⁄41472, the value of (z)	= ,0793065.

By the 2d series.

– 12 + 1.⁄24. = –11⁄24 (x)

+ 288 – 36 + 1 × 1.⁄864. = 253.⁄864. (y)

+ 3456 – 144 – 24 × 1 × 1. + 1 × 1 × 1⁄41472. = 3289⁄41472., (z)

Equal to the fractions found by the first series,—and so of any other

intermediate time from 0 to 12 hours.

Application.

 to find the Moon’s motion for 12 hours; also, the correct place in longitude latitude,

 right ascension or declination, at any intermediate time between 0 and 12 hours.

Required the motion for 12 hours, and the Moon’s correct place in longitude, September 20th 1809, at 4 h. 48 m. apparent time for the meridian of Greenwich.

Take three positions next before, and two following the given time, out of the Nautical Almanac, and set them down, with their successive differences, as follow:—

1809.			s.	°	′	″
Sep.	19	Noon,	10.	1.	50.	23.	A.		°	′	″
								+	7.	0.	44	a1.		′	″
	〃	Midn:	10.	8.	51.	7.	B.						–	2.	3.	a2.		′	″
								+	6.	58.	41.	b1.					–	0.	23	a3.
	20.	Noon,	10.	15.	49.	48.	C.						–	2.	26	b2.					0.	a4.
		>						+	6.	56.	15.	c1.					–	0.	23	b3
	〃	Midn:	10.	22.	46.	3.	D.						–	2.	49.	c2.
								+	6.	53.	26.	d1
	21.	Noon,	10.	29.	39.	29.	E.

The third differences being uniform, the fourth difference vanishes, or becomes nothing in this example.

In all cases, c1, c2. b3, and a4. are the differences to be used in the application

of the foregoing table.

		°	′	″	dec.
	c1 ″	6.	56.	15.	000.
(x) –,3000000. ×	c2. – 169	+		50.	700.
(y) +,1600000. ×	b3.  – 23	–		3.	680.
(z) +,0560000  ×	a4. 0			0.	000.
Moon’s motion for 12 hours, at 4 h. 48 m		6.	57.	2.	020.

As 12 h. to 6.° 57.′ 2.″ 020. dec. so is 4 h. 48 m. to 2.° 46.′ 48.″ 808 dec., which added to C, gives 10. s. 18.° 36.′ 36.″ 808. dec. the Moon’s true place at the required time, admitting the positions at noon and midnight stated in the Nautical Almanac to be strictly correct.

The reverse of the proposition, viz. what time at Greenwich on the 20th September, 1809, will the Moon’s longitude be 10. s 18.° 36.′ 36.″ 808 dec., may be thus found.

	s.	°	′	″
Given longitude	10.	18.	36.	36.	808.
September 20th Noon, (C)	10.	15.	49.	48	–
difference,	–	2.	46.	48.	808.

As c1. 6.° 56.′ 15.″ to 12 h. so 2.° 46.′ 48.″ 808 dec., to 4 h. 48 m. 32 sec. 53 dec. the approximated time

The numbers in the table at the approximated time, are

(x) –,2996238. (y) +,1597200, and (z) +,0559336.

			°	′	″
		c1.	6.	56.	15.	000.
(x) –,2996238. × c2.	–	169.	+		50.	636.
(y) +,1597200. × b3.	–	23	–		3.	673.
(z) +,0559336  × a4		0			0.	000
Moon’s motion for 12 hours, at the approx: time			6.	57.	1.	963.

To find, by interpolation, the numbers or decimal fractions (y) and (z) at any other time not expressed in the table.

Take two numbers immediately preceding, and two next following the time required, find their first and second differences, to which the proper signs + or –, are to be prefixed. Multiply, by algebraical process, a mean of the second differences—

	m.	sec.
For	0.	10	by	–¹⁷⁄₆₄₈		the product added to, or subtracted from the proportional part of the first difference, as the signs direct, will give the correction to be applied to the decimal fraction next preceding the required time. ☞. The first differences of (x) being uniform or as nearly so as possible, simple proportion is only necessary.
		20		–⁴⁄₈₁.
		30		–⁵⁄₇₂
		40		–⁷⁄₈₁
		50		–⁶⁵⁄₆₄₈
	1.	0		–¹⁄₉
		10		–⁷⁷⁄₆₄₈
		20		–¹⁰⁄₈₁.
		30		–⅛
		40		–¹⁰⁄₈₁
		50		–⁷⁷.⁄₆₄₈
	2.	0		–¹⁄₉
		10		–⁶⁵⁄₆₄₈
		20		–⁷⁄₈₁
		30		–⁵.⁄₇₂.
		40		–⁴⁄₈₁
		50		–¹⁷⁄₆₄₈,

---

Let the decimal fractions (y) and (z) answering to 4 h. 25 m. 40 sec. be required.

h.	m.
4.	21	+,1739844.			+,0592634.
			–15770			–3576
	24.	,1724074.		+58.	,0589058.		–13
			–15712.			–3589.			–12½ mean.
	27.	,1708362		+58.	,0585469.		–12
			–15654			–3601.
	30	,1692708.			,0581868.
As 3 m: to 15712. so 1. m. 40. sec. to –8729, nearly, the proportional part of the first difference.The second difference, +58, multiplied by –¹⁰⁄₈₁, is –,7, added to 8729, makes 8736, the correction, which subtracted from ,1724074, leaves +,1715338, the decimal fraction (y.)					As 3. m to 3589. so 1. m. 40. sec to 1994, nearly. The second diff: –12.5, multiplied by –¹⁰⁄₈₁, gives +1, taken from 1994, gives –1993, the correction, which subtracted from ,0589058, leaves +,0587065, the decimal fraction (z)

If (x) be sought for the above time, the first difference for 3 minutes is 20834, and the proportion for 1 m. 40 sec. = 11574, which subtracted from –,3166667, leaves –,3155093, at the time required.

---

As 6.° 57.′ 1.″ 963 dec. to 12 h. so 2.° 46.′ 48.″ 808. dec to 4 h. 48 m. 0 sec. the true time at Greenwich, when the Moon had that longitude, without a necessity of repeating the operation.

---

But if the Moon’s motion for 12 hours between A and B, be required, the series so far as it respects the fourth difference (z) will be different, and of course, the value of the equation arising from that difference will not be the same:—the first series, according to that arrangement, will be—

A. + r.⁄t + r. × r – 1.⁄2t. + r. × r – 2⁄3t. + r. × r – 3.⁄4t.

The three last terms to be applied in the manner before directed.

The second series will be—

– 12. + r.⁄24., = (x)

+ 288 – 36r. + rr.⁄864. = (y.)

– 10368. + 1584r. – 72rr. + rrr.⁄41472. = (z.)

(x.) and (y.) are the same as before; but (z) will at 1 h. intermediate time, be

By Series 1st

1⁄12

11⁄24

23⁄36

35⁄48

–11⁄24 × 23.⁄36. × 35.⁄48., = 8855⁄41472, reduced, –,2135175, the sign being negative. in this case, and affirmative in the other.

By the second series.

– 10368 + 1584 × 1 – 72 × 1 × 1. + 1 × 1 × 1 × 1.⁄41472 = 8855.⁄41472., the same as above.

The differences a1, a2, a3. and a4. are to be used, when one position before, and four following the intermediate time are taken, but this being a different principle of construction from that on which the table was formed, no example need be given.

---

Example II.

 Required the time of full Moon at the Capitol, in the city of Washington, in the month of November, 1809.

Subtract the Sun’s longitude from that of the Moon, and set down three positions next preceding, and two following the opposition, as follow:—

		☽’s long. – ⊙’s long.
1809.			s.	°	′	″		1st diff:
Novem:	20th	Midnight	5.	17.	35.	30½.		°	′	″			2d diff.
							+	5.	32.	31½	a1					3d diff:
	21.	Noon,	5.	23.	8.	2.						–	1.	34½	a2.		″			4th dif
							+	5.	30.	57.	b1					+	9½	a3		″	dec
	〃	Midnight,	5.	28.	38.	59.						–	1.	25.	b2				–	0.	5	a4
							+	5.	29.	32.	c1.					+	9.	b3.
	22.	Noon,	6.	4.	8.	31.						–	1.	16	c2
							+	5.	28.	16.	d1.
	〃	Midn:	6.	9.	36.	47.

	s.	°	′	″	dec
From	6.	0.	0.	0.	000.
Subtract position midnight of 21st	5.	28.	38.	59
diff.		1.	21.	1.	000.

As c1. 5.° 29.′ 32.″ to 12 h. so 1.° 21.′ 1.″ to 2 h. 57 m. 0 sec. the approximated time at Greenwich past midnight.

The numbers in the table at 2. h. 57. m. are

(x) –,3770833. (y) +,2204890. (z) +,0686731.

		°	′	″	dec.
	C1,	5.	29.	32.	000
	(x) –,3770833. × c2. – 76″	+	0.	28.	658.
	(y) +,2204890. × b3 + 9	+	0.	1.	984.
	(z) +,0686731. × a4. – 0.5	–	0.	0.	034.
diff:	⊙ and ☽’s Long. for 12 h. at the approx. time	5.	30.	2.	608.

As 5.° 30.′ 2.″ 608. dec to 12 h. so is 1.° 21.′ 1.″ to 2. h. 56. m. 32. sec nearly, or 14. h. 56. m. 32. Sec. according to the astronomical method of reckoning.

	D.	H.	M.	S.
Time of opposition at Greenwich,	21.	14.	56.	32.
Longitude in time (West)	–	5.	7.	38.
Full Moon in Novemr at Washn	21.	9.	48.	54.

---

Many other examples relating to the Moon’s latitude, right ascension and declination, might be given; but the preceding are supposed to be sufficient to shew the usefulness of the table; great care has been taken to compute the numbers or decimal fractions correctly, which may be verified by taking the second differences: those differences are nearly uniform in (y); and in z, they gradually decrease to about the 8th hour, after which, the sign changes from – to +, and they increase for the remaining hours contained in the table.

---