I am the one who is best informed of the time of this prayer, i.e. the night prayer. The Messenger of Allah ﷺ used to offer it at the hour when the moon went down on its third night.  

أَنَا أَعْلَمُ النَّاسِ بِوَقْتِ هَذِهِ الصَّلاَةِ صَلاَةِ الْعِشَاءِ الآخِرَةِ كَانَ رَسُولُ اللَّهِ ﷺ يُصَلِّيهَا لِسُقُوطِ الْقَمَرِ لِثَالِثَةٍ  

We remained one night waiting for the Messenger of Allah ﷺ to offer the Isha prayer. He came out to us when one-third of the night has passed or even after it. We did not know whether anything kept him occupied or there was some other matter. When he came out, he said: Are you waiting for this prayer ? Were it not that it would impose a burden on my people, I would normally pray with them at this time. He then gave orders to the mu'adhdhin who declared that the time of the prayer had come.  

مَكَثْنَا ذَاتَ لَيْلَةٍ نَنْتَظِرُ رَسُولَ اللَّهِ ﷺ لِصَلاَةِ الْعِشَاءِ فَخَرَجَ إِلَيْنَا حِينَ ذَهَبَ ثُلُثُ اللَّيْلِ أَوْ بَعْدَهُ فَلاَ نَدْرِي أَشَىْءٌ شَغَلَهُ أَمْ غَيْرُ ذَلِكَ فَقَالَ حِينَ خَرَجَ أَتَنْتَظِرُونَ هَذِهِ الصَّلاَةَ لَوْلاَ أَنْ تَثْقُلَ عَلَى أُمَّتِي لَصَلَّيْتُ بِهِمْ هَذِهِ السَّاعَةَ ثُمَّ أَمَرَ الْمُؤَذِّنَ فَأَقَامَ الصَّلاَةَ  

We waited for the Prophet ﷺ to offer the night prayer. He delayed until people thought that he would not come out and some of us said that he had offered the prayer. At the moment when we were in this condition the Prophet ﷺ came out. People said to him as they were already saying. He said: Observe this prayer when it is dark, for by it you have been made superior to all the peoples, no people having observed it before you.  

أَنَّهُ سَمِعَ مُعَاذَ بْنَ جَبَلٍ يَقُولُ ارْتَقَبْنَا النَّبِيَّ ﷺ فِي صَلاَةِ الْعَتَمَةِ فَأَخَّرَ حَتَّى ظَنَّ الظَّانُّ أَنَّهُ لَيْسَ بِخَارِجٍ وَالْقَائِلُ مِنَّا يَقُولُ صَلَّى فَإِنَّا لَكَذَلِكَ حَتَّى خَرَجَ النَّبِيُّ ﷺ فَقَالُوا لَهُ كَمَا قَالُوا فَقَالَ لَهُمْ أَعْتِمُوا بِهَذِهِ الصَّلاَةِ فَإِنَّكُمْ قَدْ فُضِّلْتُمْ بِهَا عَلَى سَائِرِ الأُمَمِ وَلَمْ تُصَلِّهَا أُمَّةٌ قَبْلَكُمْ  

We observed the prayer after nightfall with the Messenger of Allah ﷺ, and he did not come out till about half the night had passed. He then said: Take your places. We then took our places. Then he said: The people have prayed and gone to bed, but you are still engaged in prayer as long as you wait for the prayer. Were it not for the weakness of the weak and for the sickness of the sick. I would delay this prayer till half the night had gone.  

صَلَّيْنَا مَعَ رَسُولِ اللَّهِ ﷺ صَلاَةَ الْعَتَمَةِ فَلَمْ يَخْرُجْ حَتَّى مَضَى نَحْوٌ مِنْ شَطْرِ اللَّيْلِ فَقَالَ خُذُوا مَقَاعِدَكُمْ فَأَخَذْنَا مَقَاعِدَنَا فَقَالَ إِنَّ النَّاسَ قَدْ صَلَّوْا وَأَخَذُوا مَضَاجِعَهُمْ وَإِنَّكُمْ لَنْ تَزَالُوا فِي صَلاَةٍ مَا انْتَظَرْتُمُ الصَّلاَةَ وَلَوْلاَ ضَعْفُ الضَّعِيفِ وَسَقَمُ السَّقِيمِ لأَخَّرْتُ هَذِهِ الصَّلاَةَ إِلَى شَطْرِ اللَّيْلِ