Doctors say there has been little change and his condition continues to be described as serious but stable.
Mr Mandela, 94, had been ill for some days before being transferred in the early hours of Saturday to a hospital in Pretoria.
It is the third time this year he has been admitted to hospital.
The BBC’s Karen Allen in Pretoria says there is a sense of calm across South Africa, and also a quiet hope that the man who led the fight against apartheid may regain his strength once again.
Mr Mandela’s wife, Graca Machel, cancelled a scheduled appearance in London to remain at her husband’s bedside.
On Saturday, presidency spokesman Mac Maharaj said that although Mr Mandela was again suffering from pneumonia, he was breathing on his own – “a positive sign”.
The former president’s close friend Archbishop Desmond Tutu has led calls for South Africans to pray for him and wish him a speedy recovery.
Meanwhile, South African President Jacob Zuma has continued to hold engagements away from the capital.
He has indicated that he will visit Mr Mandela if doctors advise him to do so.
Nelson Mandela served as president from 1994 to 1999.
He was previously imprisoned for 27 years, and is believed to have suffered damaged lungs while working in a prison quarry.
He contracted tuberculosis in the 1980s while being held in jail on the windswept Robben Island.
He retired from public life in 2004 and has been rarely seen in public since.
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