Police searching for the 75-year-old, who taught at Fettes College in the Capital, made the discovery on Sunday morning on land close to the A696 near Kirkwhelpington in Northumberland.
Last week, Detective Superintendent Andrew Patrick revealed police had begun scouring land along the road, and said they had narrowed down the search after “extensive enquiries”.
Officers were pictured combing woodland in the area near the village – more than 80 miles from the pensioner’s home in the South Learmonth Gardens near Stockbridge.
The body has yet to be identified but the family of Dr Coshan have been informed.
Detective Superintendent Patrick said: “Our thoughts remain with Peter’s family at this incredibly difficult time and we are continuing to provide them with support.
“I would like to take the opportunity to pass on our thanks to everyone who has come forward with information which has helped our investigation.”
Dr Coshan was reported missing on August 12 after having been seen in Seafield Road, Edinburgh, shortly before midnight on the previous day.
At the same time, they said two men had been arrested and charged in connection.
Paul Black, 63, and Paul McNaughton,27, appeared at Edinburgh Sheriff Court on August 18. They made no plea to a charge that they murdered Dr Coshan.
They also made no plea to charges of theft and attempting to defeat the ends of justice.
The investigation later centred on the Scotland-England border and police made an appeal for help to find a old-style dark coloured Vauxhall Vectra with a 57 registration plate, which had been seen in the area.
The former biology teacher taught at Fettes, of which Sir Tony Blair was once a pupil, from 1972 until his retirement in 2005.
He also lead the school’s Duke of Edinburgh’s Award scheme for 33 years.
A spokesperson for the college said: “We are shocked and deeply saddened to hear of the tragic developments and our thoughts are with his family and friends at this difficult time.”
Helen Harrison, head of Fettes College, also paid tribute to the pensioner, who she described as “an inspirational biology teacher”.
“He will be remembered fondly by many, not least by those who were taught by him, tutored by him and introduced to the hills through his enthusiasm for the Duke of Edinburgh Award Scheme, which he ran for 33 years,” she said.