When is a Botticelli not a Botticelli? When, for that matter, is a Holbein not a Holbein? And how did a portrait of a Bavarian town clerk turn into one of Martin Luther by Holbein, and then back again? And how come a brunette, peeping from behind a curtain, is not a demure young Victorian lady after all, but a rather more seductive Renaissance blonde?

These are some of the questions posed in Close Examination: Fakes, Mistakes and Discoveries, on view until September 12. It is the first major exhibition of its kind, and the first under the gallery’s new policy to show its own collections each summer in free, focused exhibitions.

There are more than 40 paintings to see. Among them are works by Dürer, Raphael, Rembrandt, Verrocchio, Corot, Courbet, Friedrich — or not as the case may be — each telling the story behind the painting, revealing the contribution of science to the modern understanding of artworks. It celebrates the collaboration of the gallery’s conservators, art historians and world-leading Scientific Department, founded in 1934, working to unravel secrets locked up in an artwork’s paint and varnish, the wood of a panel or frame, correct misconceptions about authorship or authenticity, and distinguish between workshop collaborations and period copies or modern forgeries.

The six rooms represent challenges faced: Deception and Deceit; Mistakes: Attributions Downgraded; Transformation and Modifications; Secrets and Conundrums; Redemption and Recovery; and one room devoted to Botticelli.

In 1874, the Gallery acquired two works by Sandro Botticelli showing Venus. One was Venus and Mars, painted around 1485. The other, as described in the Christie’s catalogue, was “Venus reclining in a Landscape with three amorini pelting her with roses”. The sale of these and others from the collection of Alexander Barker, the son of a fashionable boot maker, caused quite a stir. Many eulogised over the paintings, saying “these represent the classical side of Botticelli’s genius”. But some were less happy. A letter from the director’s wife expresses her displeasure at the purchase of two “second-rate” Botticellis. Today, while Venus and Mars is regarded as an authentic Botticelli, the other is simply labelled An Allegory, Italian, Florentine, about 1500. Yet, curators say it does have some connections with Botticelli, possibly being made by a follower.

The first Botticelli you see is in the opening room. It looks typical, a pretty young veiled Madonna, child in her arms, painted in tempera (and other media) on panel. But though it was praised when it first surfaced as a Botticelli in 1930, some had doubts. Kenneth Clark (Director of the National Gallery from 1934) suggested it had “something of the silent cinema star about it”. Yes, it does look as if Gloria Swanson could have modelled for it.

Close examination during conservation work post Second World War cast further doubts: some pigments weren’t right; the Madonna’s robe was painted using Prussian blue, a pigment invented in the early 18th century. In 1994, paint analysis by energy-dispersive X-ray (EDX) spectrometer confirmed the diagnosis: synthetic pigments unavailable until the 19th century were present, including cobalt blue and zinc chromate. In addition, signs of ageing typical of a Renaissance work, such as surface craquelure (the network of cracks developing as paint layers age), paint losses and worm holes in the wood panel, were all artificial.

It was painted by one of the best forgers of the late 19th century, Umberto Giunti, a teacher at the Institute of Fine Art in Siena who produced many a ‘Renaissance’ painting for avid collectors.

The Portrait Group in this same room, a Renaissance profile portrait, may also have been his work. Now labelled as Italian, early 20th Century, pigments again gave the game away, as did the use of shellac, used to simulate an aged appearance.The children in the group do look very 20th century.

As for the ‘Holbein’, that’s another story: an 18th-century makeover of an existing portrait by an unknown German artist of Alexander Mornauer, town clerk of the Bavarian city of Landshut (between 1464 and 1488), was altered between 1720 and the end of the century into ‘Martin Luther’. His tall hat was shortened and rounded, and the background changed to a vivid blue to make it look like a work by Holbein. The deceit was discovered when Ashok Roy, Director of Scientific Research, ran his finger over the blue and felt it to be far too smooth; azurite, which would have been used, ground by hand in the 15th century, is generally coarser; the too smooth Prussian blue only became widely available from the 1720s.

Woman at a Window is a repainting job done to suit Victorian tastes. Routine restoration in 1978 revealed the painting’s secrets: changed hair colour, softened jaw line, breasts veiled more discreetly, eyes far less sultry. Scholars had wondered about the work for some time. Was it a portrait by Palma Vecchio of his daughter? Or Lorenzo Lotto? Now, it is simply attributed to an unknown Italian artist.

This is an enjoyable, informative exhibition. Seeing 40 case studies takes time: paintings plus pictures of the originals or alterations, details of the scientific analysis, but there’s a good link to exhibition on the National Gallery’s website that gives extra details: www.nationalgallery.org.uk