Meton's heliotropion (a device showing the turning point of the sun, i.e. the solstice) was set up on the Pnyx in the archonship of Apseudes (433/2 BC). The essential part of it was a fixed gnomon to cast a shadow onto a fixed surface. What is thought to be its base has been found on the platform behind the Bema.
If the path of the sun is charted at the same time of day at regular intervals through the year, either by noting the position of the shadow on the ground (as done in antiquity) or by taking a photo of the sun in the sky (as in the image below, from A Robinson's book The Story of Measurement), an elongated figure-8 results, called an analemma.
Shortest day (the winter solstice) corresponds to the bottom of the figure here described, and longest day (the summer solstice) to the top.
Few people realize how much our calendar owes to the Romans. Let me illustrate: If some of them had been more popular emperors, our calendar might run 'January February March Neroneus Claudius Germanicus July August Domitian November December'. Domitian was following the precedent set by Julius Caesar (= July) and Octavian Augustus (= August), but people refused to honour him thus. Julius and Augustus, however, were appreciated enough for their names to stick as those of the months in high summer. Further, the shifting of the start of the year back from March to January shunted what was the seventh month into ninth position (Sept = seven), the eighth into tenth (Oct = eight), the ninth into eleventh (Nov = nine) and the tenth into twelfth (Dec = ten).
Also, because Augustus was not inferior to Julius, his month had to have the same number of days as Julius', to wit, 31, even if it wrecked what was hitherto an entirely rational system, and that in turn required us all to learn silly verses like '30 days hath September' etc. in order to remember the new irrational scheme.
March 31 April 30 May 31 June 30
September 30 October 31 November 30 December 31
- do you see the pattern? It was originally probably a month of 31 days followed by a month of 30 days repeated from January to December. This does not of course add up to 365, never mind the quarter.
Alternating 31-30 day months would give
6 months of 31 days = 186, + 6 x 30 =180; total = 366 = too long.
The Roman calendar was some 90 days adrift of the seasons when Julius Caesar took it upon himself to sort it all out; January was falling in late autumn instead of mid winter. For the calculations, he employed a Greek astronomer called Sosigenes. Sosigenes ignored Hipparkhos' measurements on the length of the year made more than a hundred years earlier and as a result the Julian calendar had to be reformed by Pope Gregory in the C16; if he'd used Hipparkhos' figures we would still be using the Julian calendar today and the people of Pope Gregory's time would not have 'lost' 10 days...
When in due course a month was named after Julius, and it was decided to make that month the one after June, that month was in the 31 day position. Hence July has 31 days.
It would have been a dishonour to Augustus to give him a shorter month than Julius', so August was going to be a 31 day month. If they had positioned it in what we now call October, the substitution might have gone smoothly. But they decided to give him high summer. That would have given three 31-day months in sequence. To avoid that, the day quotas per month from September to the end of the year were reversed. The calendar now had seven 31 day months - and it had been too long when it only had six 31 day months!
(7 x 31 =) 217 + (5 x 30 =) 150 = 367.
This is of course 2 days too long. It was decided to take them out of February, and that's why February ordinarily has 28 days, and 29 in a leap year, when the four quarters of a day ignored over the past three years now make up a whole day.
References: Suetonius Domitian 13. For further reading see R Hannah Greek and Roman Calendars Duckworth 2005.