Easy answer: you're confusing time with linear measurement.

You don't get seconds of time until you have a device capable of measuring seconds of time. The Egyptians didn't use the Babylonian 60-unit measurement, and thus didn't have 60 "seconds" in a minute... in fact, they didn't measure time that precisely.

And your measurement of the second may be off. The official definition is:
"The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom at a temperature of 0 Kelvin."

And prior to 1967, it was:
"the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time."
[en.wikipedia.org]

And an arc-second is rather larger than the pyramid.