On July 1, at 8 a.m. by my watch, it was 8h. 4m. by my clock. I took the watch to Greenwich, and, when it said 'noon', the true time was 12h. 5m. That evening, when the watch said '6h.', the clock said '5h. 59m.'
On July 30, at 9 a.m. by my watch, it was 8 h. 57m. by my clock. At Greenwich, when the watch said '12h. 10m.', the true time was 12h. 5m. That evening, when the watch said '7h.', the clock said '6h. 58m.'
My watch is only wound up for each journey, and goes uniformly during any one day: the clock is always going, and goes uniformly.
How am I to know when it is true noon on July 31?
When the clock says '12h. 2m. 29
On July 1, watch gained on clock 5m. in 10 h.; i.e. ½m. per hour; i.e. 2m. in 4h. Hence, when watch said ‘noon’, clock said ‘12h. 2m.’; i.e. clock was 3m slow of true time, when true time was 12h. 5m.
On July 30, watch lost on clock 1m. in 10h.; i.e. 6 sec. per hour; i.e. 19 sec. in 3h. 10m. Hence, when watch said ‘12h. 10m.’, clock said ‘12h. 7m. 19sec.’; i.e. clock was 2m. 19sec. fast of true time, when true time was 12h. 5m.
Hence clock gains, on true time, 5m. 19sec. in 29 days; i.e. 319 sec. in 29 days; i.e. 11 sec. per day; i.e.
|(24 × 12)|
hen cent, while true time goes 5m., watch goes 5m.
Now, when true time is 12h. 5m. on July 31, clock is (2m. 19sec. + 11sec.) fast of it; i.e. says ‘12h. 7½m.’ Hence, if true time be put 5m. back, clock must be put 5m.
sec. back; i.e. must be put back to 12h. 2m. 29
Hence, on July 31, when clock indicates this time, it is true noon.