My eureka moment came late. Late in life, late in thought, late in realisation. It crept up on me, all unawares, and surprised me, after the fact.

Allow me to explain.

Somewhere near the heart of what I've been trying to do these past few years is to understand what it is about a few daubs of paint on canvas that is so meaningful for us, and so consequential; how the painting itself, that small thing within the frame, can become something more than a possession - a familiar, a fetish, an emblem, a kind of moral spirit level, an agent of equilibration, as Seamus Heaney says, "an upright, resistant, and self-bracing entity within the general flux and flex"; how the artist is not merely a world-mirror but a world-maker; how a painter of the humblest objects (pots and pans, apples and pears) contrives to show us what human beings are capable of; and how one human being can become an exemplar for another, a process beautifully described by Paul Valery: "To become for someone else an example of the dedicated life, being secretly invoked, pictured, and placed by a stranger in a sanctum of his thoughts, so as to serve him as a witness, a judge, a father, a hallowed mentor."

Henri Matisse is a textbook case. In 1899 he felt impelled to buy a small Cezanne - a daring thing to do at the time - Three Bathers (c 1879-82). Matisse was then struggling to survive as an artist; he was chronically impoverished and already married. The down payment of 500 francs was far beyond his means. He raised it by pawning a magnificent emerald ring that was a wedding present - and one of his wife's cherished possessions.

Over the next three decades, through thick and thin, this Cezanne became something of a touchstone for the whole family; even in the worst of times, it was unthinkable to part with it. For Matisse himself, it was the object of veneration.

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In 1936, he decided to give it to the Museum of the City of Paris. On the day he dispatched it, he composed a letter to the director of the museum.

"In the thirty-seven years I have owned this canvas, I have come to know it quite well, though not entirely, I hope; it has sustained me morally in the critical moments of my venture as an artist; I have drawn from it my faith and my perseverance; for this reason, allow me to request that it be placed so that it may be seen to its best advantage. For this it needs both light and adequate space. It is rich in colour and surface, and seen at a distance it is possible to appreciate the sweep of its lines and the exceptional sobriety of its relationships.

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"I know that I do not have to tell you this, but nevertheless I think it is my duty to do so; please accept these remarks as the excusable testimony of my admiration for this work, which has grown increasingly greater ever since I have owned it. Allow me to thank you for the care that you will give it, for I hand it over to you with complete confidence ... ".

Matisse was not the only one. Pablo Picasso was not much given to veneration, but Cezanne was an exception. "He was my one and only master!" he told his sympathetic friend Brassai. "Don't you think I've looked at his paintings? I've spent years studying them. Cezanne! He was like a father to us all. He was the one who protected us." Picasso owned four Cezannes, including Aix Cathedral Seen from the Atelier des Lauves (1902-04), a watercolour on white paper. The glory of this work is its radical lack of finish - it is the white paper that appears to organise the watercolour. The pathos of the paper was one of Cezanne's great discoveries. "As soon as he touched the canvas, the picture was already there," marvelled Picasso, consciously or unconsciously echoing Renoir: "How does he do it? He has only to put two dabs of colour on a canvas and it's already something."

The third man of modern art, Georges Braque, felt the same. Braque, too, owned a small Cezanne, Bouquet of Peonies in a Green Jar (c 1898). It hung in his bedroom, together with a landscape by his father and one of his own skull paintings. It was possessed of tutelary properties. First of all, it was unvarnished. Varnish was anathema to Braque. Healthy paintings did not need it. His Cezanne came straight from the studio, he would point out, straight from the hands of the master. The sense of transmission was palpable. Better still, it was unfinished, or so people thought. Braque dissented, and delighted in it. He delighted also in the metamorphic confusions of its identity. In the original catalogue raisonne of Cezanne's work, the green jar is a blue vase; and according to his dealer's stockbook, the peonies are roses.

Floral sensations rather than botanical specimens, they explode out of the jar like a rocket out of a bottle. Braque never tired of studying them. The fascination lay in what was painted and what was left unpainted - the fireworks and the blanks in between - not the thing, but the effect it produces, as Mallarme wrote. What Braque glimpsed in the Bouquet of Peonies was the life with which he passionately identified. "Cezanne! He swept away the idea of mastery in painting. He was not a rebel, Cezanne, but one of the greatest revolutionaries; this will never be sufficiently emphasised. He gave us a taste for risk. His personality is always in play, with his weaknesses and his strengths. With him, we're poles apart from decorum. He melds his life in the work, the work in his life."

These men went on to exert an influence of their own. Braque in particular became an exemplary figure for his generation and the generations that followed. By the 1940s, anyone who was anyone had a small Braque. For many it had almost talismanic significance.

The writer and adventurer Andre Malraux carried one with him on his travels. Into his bag, on his ceaseless questing, went his entourage of fetishes - two or three of the usual kind, and the Braque, a beachscape, with an empty boat, an uncanny blue, and the night drawing in.

Jean Paulhan, the editor of the Nouvelle Revue Francaise and an active resistant, had one on the wall of the little office to which he had been relegated during the Occupation. One day, a young firebrand by the name of Claude Roy came to see him, to seek his advice. Paulhan sat him down and made him welcome, discoursing capaciously in his cultivated way, dodging and weaving, joking and provoking. "Defeat is certain. One cannot make war with excessive regard for humanity. Some had no wish to fight for the King of England. Others had no wish to fight for the Committee of Ironworks. In fact no one wished to fight for anyone." Roy interjected feebly and in vain. Paulhan was reaching his peroration. "Who knows which is worse? To be convinced that one is right, or (too quickly) that one is wrong. Victory and defeat whisper the same thing: what is done, is well done."

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Roy's head was spinning. He asked what he should do. Without a word, Paulhan steered him gently towards the painting on the wall, The Kitchen Table (1941). The two men looked at it in silence. After a while Paulhan spoke again, of a lemon whose yellow was exactly what it wanted.

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Roy took his leave. That evening he received a characteristic note from Paulhan. It read: "I can scarcely see how we can avoid a long war. All the same, one would wish it short. I feel one should set aside a few acres, a small corner where the air is free, where no one lies (even with the very best of intentions). I don't ask for a big corner, by any means."

Braque fortified that small corner. The poet Francis Ponge was sustained during the Occupation by The Banjo, also known as Mandolin and Score (1941). Wherever he laid his head - Ponge was another active resistant - he pinned up an illustration of that painting, torn out of a cheap picture book, "a little like my flag, or my reasons for living (and struggling)". It was the colours he remembered, and their application, "very bold, but properly arranged in all their variety, with an especially violent mauve". This tattered Braque held, and Ponge with it. It furnished his world and nourished his imagination: "That's why I could live. Happily. That's the society (of friends) for which I fought ... In sight of that, during my rare moments of leisure, guided by the Latin alphabet and the roots of our French words, I wrote."

Researching a biography of Braque, I went to see the distinguished American abstract artist Ellsworth Kelly (born 1923), who was known to be an admirer. For some it was the Braque-world in its totality that seized the imagination. For others it was a part of the whole, an element, an intriguing detail. For Kelly, it was the shapes. The shape of a light fitting he appropriated from the corner of a Braque still life became Sneaker or Brooklyn Bridge (1955) - metamorphism worthy of the master - and thereafter a whole series of Brooklyn Bridges. He was hooked. When I went to see him, some 50 years after that revelation, he had recently bought himself a small Braque, a late landscape, not so much painted as excavated, pigment by pigment, as if the old artist had followed his own advice and dug down to the root, "the foundation of all art: clay".

Kelly was entranced. He had positioned it not on the wall, but on a little lectern on his desk. He loved to study it - reading it slowly, repeatedly, sometimes with a magnifying glass; examining the pigments, as if taking soil samples; revelling in the communion with the unfettered Braque, just as Braque had revelled in the unvarnished Cezanne. Sitting next to him at the desk, I thought I caught something of what one painter means to another.

That was not the moment. When the biography appeared, in 2005, I was faced with a rather different question: why Braque? By this stage, it may be that most authors have a good idea, or at any rate a good answer. Disconcertingly, I did not. I had various answers - all too many answers - none of which seemed to me to be entirely satisfactory. In search of something better, I borrowed from Jean Paulhan, who had asked himself the same question. Paulhan's answer effectively if rather enigmatically turned the tables: "He chose me."

This answer at least had the virtue of succinctness. Even more disconcerting, however, it left me with an inchoate feeling of expressing a deeper truth, the nature of which I could not grasp. Brooding on Braque, and on the mysteries of transmission, I couldn't help regretting that we didn't have one at home to worship, like Ellsworth Kelly and all the rest.

There were no paintings in the house when I was growing up, I fell to ruminating, other than occasional pieces by my mother, a talented but distracted amateur. There were a few scattered reproductions. One of these dwelt in the Stygian gloom at the top of the stairs, hung at a height for the cat to contemplate on its nocturnal rambles. The cat, I expect, gave it as much attention as I ever did.

Yet it was treasured by my mother, and I still had it. I got it down and looked at it, as if for the first time. It is a postcard-size illustration mounted on a card and framed. It is a little faded, but the signature is unmistakable, and the date is clear. It is Still Life with Bread and Jug (1938), by Georges Braque. It is my very own small Braque. Eureka!

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