He died a few months after they arrived from the weavers in Brussels at the end of 1519.
But it is certain he never saw them together with the cartoons he had drawn four years earlier as the basis for the tapestries.
No one ever has.
Until now.
Today staff at the Victoria and Albert Museum were putting the finishing touches to an exhibition of the tapestries, displayed for the first time alongside the cartoons which have been on display at the museum since 1865, and are among its foremost treasures.
Curators and workmen were raising four of the heavy 494-year-old tapestries, woven in wool, silk and gilt-metal thread, which have been loaned by the Vatican - at its own suggestion - as an accompaniment toto mark this month's visit to Britain by Pope Benedict XVI.
The cartoons and tapestries will be together on show for just six weeks.
"No one has ever seen them together before," said Clare Browne, the museum's curator of textiles, who described the tapestries as "among the most extraordinary productions of their era".
Watching with barely suppressed excitement was the V&A's expert on Raphael, Mark Evans, who has been studying the works since he was an undergraduate. "When the Vatican rang up in February and offered to loan us the tapestries for an exhibition, to say that my jaw dropped would be an understatement," he said.
"I had always thought the logistical difficulties and political support necessary would make it impossible. This is my opportunity to realise something that has been at the fringes of my thinking for 40 years. I know I will never do anything like this again."
The series of 10 tapestries, of events taken from the Acts of the Apostles, was commissioned by Pope Leo X in 1515.
Raphael, a busy artist already at work on frescoes for the papal apartments, completed the designs, painted as full-size cartoons, within a year and it was from these that the finest weavers in Europe, based 1,000 miles away in Brussels, wove their work.
It is estimated that each tapestry would have taken one loom a year to complete and at least seven were delivered to Rome in December 1519. So highly were they prized, that when Leo X died within two years, deep in debt, some of the tapestries were pawned. Only four of the surviving tapestries are robust enough to have been driven to London.
The cartoons themselves, which had been cut into strips so they could be placed under the loom and copied, remained in Brussels until they were bought a century later by Charles I. They have been owned by the royal family ever since and placed on loan to the museum for the last 145 years.
Also in the exhibition are Raphael's preliminary preliminary drawings for the designs, loaned from the royal collection.
Had Raphael been able to compare his paintings with the finished designs, he would have spotted some intriguing differences.
The robe Christ is wearing in the work depicting his Charge to St Peter is a plain white, or pastel colour in the cartoon, but was decorated with gold stars in the tapestry.
They went even further in the tapestry known as the Sacrifice at Lystra, showing the aftermath of the healing of a lame man by St Paul; probably believing the man did not figure strongly enough in the crowd, the weavers replaced the crutch Raphael had painted on the ground beside him with a wooden leg instead: the victim apparently having grown a new limb instead of merely having his own restored to muscled health.
The Flemish tapestries have greater vivacity than one of the Mortlake copies, also in the exhibition, from 100 years later, which is evidently the product of reverent imitation rather than exuberant flair.
Evans said: "You can see the sheer visual intelligence of Raphael in these designs. It is very common today to be snooty these days about brainpower in comparison to emotions, but the painter's great narrative shines through. It is supercharged."
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