The library, like the school, was a place for which Louis Kahn felt the deepest reverence. Books were Kahn's most treasured possessions, for 'the world is put before you through the books', and Kahn felt that books were literally priceless: 'A book is tremendously important. Nobody ever paid for the price of a book, they only paid for the printing.' Therefore, Kahn believed that the library should be a sacred place: 'The book is an offering ... The library tells you of this offering.' In 1956, almost ten years before he was commissioned to design the library at the Phillips Exeter Academy in New Hampshire, Kahn had already begun to question the typical programme of the library in his competition entry for Washington University: 'The spaces and their constituent form as a building should originate from broad interpretations of use rather than the satisfaction of a program for a specific system of operation.' Kahn held that the usual library programme led to two quite distinct and separated spaces, 'one for people, one for books', yet he strongly believed that 'books and the reader do not relate in a static way'.

Though not evident in the submitted scheme, it was during the design of the Washington University Library that Kahn first spoke about the pivotal idea of the individual reading carrel, and its generative potential, stating that it was his desire to find a space construction system in which the carrels were inherent in the support which harboured them. Reading within a cloistered space with natural light in nearness to the building surfaces seemed good. Revealing his historical inspiration, Kahn then quoted from a historical description of the medieval monastic library at Durham, England, with its cloister colonnade glazed from floor to ceiling, and carrels equipped with desks set into every window niche, while, on the other side of the cloister, against the wall of the church and away from the sunlight, were placed great wooden cabinets full of books. This source would have been reinforced by Kahn's memories of the 'window' seats, overlooking the central courtyard, built into the upper level of the monastery cloister of Bramante's S. Maria della Pace in Rome, which Kahn had visited.

For Kahn, the architecture of the library naturally evolved from this inspiring beginning: 'Then from the smallest characteristic space harboured in the construction itself, the larger and still larger spaces would unfold ... Wall-bearing masonry construction with its niches and vaults has the appealing structural order to provide naturally such spaces.' While it would be another ten years before Kahn actually designed a building where, as he said, the carrel is the niche which could be the beginning of the space order and its structure, the concept was so compelling that Kahn never ceased considering its implications. A year later, in 1957, Kahn arrived at his second pivotal insight into the nature of the library: 'A man with a book goes to the light. A library begins that way. He will not go fifty feet away to an electric light.' Related directly to the concept of reading carrels at the periphery, this empathetic understanding of the nature of the individual act of reading was complemented by Kahn's third pivotal insight - the collective expression of the library as an institution, embodied in the great central room which, upon entry, presents us with the books.

Charged by Richard Day, Phillips Exeter Academy's new principal, with finding an architect capable of giving the school a significant work of modern architecture (as opposed to the neo-Georgian style heretofore characteristic of the campus), the library building committee interviewed a number of the leading architects of the day, including I. M. Pei, Paul Rudolph, Philip Johnson and Edward Barnes. Yet the committee was immediately struck by Kahn's profound and richly nuanced conception of the library as a modern institution, and he was awarded the commission in November 1965.
The previous year, Kahn had made the astonishing statement: 'You plan a library as though no library ever existed', indicating his intention to return to the beginnings, to the original inspiration for the library as a place, rather than to accept the prevailing programmatic definition. Kahn's definitive concept for the Exeter Library, developed in 1966, evolved from his three insights into the nature of the library as an institution, and resulted in a literal inversion of the traditional library programme and plan type. The habitual separation of the central reading room from the peripheral book stacks was turned inside out, so that the reading rooms were now at the outer edge, as carrels with natural light; the book stacks were within, protected from the natural light; 'and again the emergence of light in the center', as Kahn said, in the great top-lit central hall where one sees the books upon entry

From the very beginning of the design process, Kahn conceived of these three types of spaces as if they were three buildings, constructed of different materials and at different scales, buildings-within-buildings - a large-scale interpretation of his concrete 'sunshields' wrapped around glass rooms at the Salk Institute Meeting House - yet here each layer was to be habitable. At Exeter Library, the outermost building layer, housing the double-height reading carrels, was to be load-bearing brick; the inner building layer, housing the single-storey book stacks, was to be reinforced concrete; and the central room, wrapped by the outer two layers, was to reach the full height of the building. In this way, as he had done with his 'composite order in the Indian Institute of Management, Kahn again engaged both archaic and modern methods of construction in the same building: 'The brick structure was made in an old-fashioned way, and the interior structure was done in today's techniques.' Kahn meant 'old-fashioned' structure quite literally, for in his early studies he indicated that the brick outer building was to have stacked sets of semi-circular masonry arches, reminiscent of ancient Roman theatres and arenas, while the central room was to have giant circular and semi-circular masonry arches.

The plan of Exeter Library was from the very beginning heavily indebted to Wright's Unity Temple, not only in its cruciform-in-square plan but also in its location of the stairs in the corners; in its double-square plan mezzanine floors overlooking the central, full-height, top-lit, square space; and - perhaps most telling - in the plan dimensions of the central room: a 32 foot (9.5 metre) square, exactly matching the sanctuary of Unity Temple. Kahn's Exeter Library plan is in fact presaged remarkably closely by Rudolph Schindler's unrealized competition design for the Bergen Public Library of 1920-29 which was also clearly based upon Wright's Unity Temple plan. That Kahn was very probably entirely unaware of this unpublished library design by Schindler further illustrates the degree to which these ordering principles are part of a shared tradition in modern architecture.

Once this three-layer (reading carrels, book stacks and central hall) cruciform-in-square plan was set, Kahn's early designs for Exeter Library involved an exhaustive exploration of the possibilities of the four outer corners, which at various times were proposed as freestanding, projecting towers or retracted, re-entrant corners; as housing the stairs or as seminar rooms; and as trapezoidal, triangular, square, semi-circular or circular in form. Of these early designs, the scheme proposing circular corners, housing the stairs set on a 45 degree diagonal to the square main building, is the most monumental, with the grid of windows of the five double-height floors of reading rooms framed by the solid cylindrical corner towers, nearly 100 feet (30 metres) tall, opened only on the diagonal where a vertical slot runs from top to bottom.

The final scheme emerged with Kahn's recognition of the primacy of the reading carrels, which he made stand free as 'brick buildings', forming the library's four elevations, and his concomitant removal of the corner towers to create re-entrant corners, which present on the exterior the 16 foot (4.9 metre) depth of the load-bearing brick structure -measured from the outer brick face to the concrete columns of the book stacks within, precisely one-half of the 32 foot (9.5 metre) dimension of the central room. Each structural bay in the 'brick buildings' is 20 l/2 feet (6.2 metres) wide and the brick structure is 121/2 feet (3.8 metres) deep, a 'golden section' in proportion. The 'concrete building', housing the book stacks, measures 20 feet ( 6 metres) deep and 40 feet ( 12 metres) wide, a double square flanking the square central space - exactly as in Wright's Unity Temple. The height of the four 'middle' floors of the library, which house the book stacks, is 35 feet (10.5 metres), and the width of the central space, measured from outside face to outside face of the concrete wall, is also 35 feet (10.5 metres)- together making a square in elevation, into which Kahn inscribed a circular opening 30 feet (9 metres) in diameter. That these proportions were of significance for Kahn is most clearly indicated by the fact that the height of the central room, measured from the floor to the bottom of the roof structure, which had been set at the beginning of the design in 1966 at 50 feet (15 metres), was changed by Kahn in 1968 to 52 feet (16 metres). This was a seemingly small change, but one of the greatest importance, for, taken together with the 32-foot (9.5 metre) plan dimension, it resulted in the section of the central hall being a perfect 'golden section' proportion (1:1.618), as we see it today.

Seen across the grass lawn of the Phillips Exeter Academy campus, the Library is a massive, cubic brick block, 111 feet (334 metres) wide and 80 feet (24 metres) tall, its re-entrant corners stepping back to reveal the four 88 foot (2.7 metre) wide 'brick buildings' housing the carrels. Each facade extends beyond the last perpendicular brick pier at either end, as well as the recessed 45 degree wall at each corner, thus appearing to be a freestanding plane or screen. In the four facades, which are aligned with the cardinal directions, brick piers are spanned by flat 'jack' arches at the floor lines; a single storey at the ground floor, with four double-height floors above. As the building rises, the brick piers decrease in width; the window openings between the piers increase in width; and the flat arches (the angled masonry of which affects this transition in the piers' width) increase in depth at each floor. The whole forms a 'statically hierarchical' expression of the load-bearing brick walls - the piers thicker at the bottom where the load is greatest, and thinner at the top where the load is least, allowing us to see 'the way they bring the weight down' to the ground. Kahn intended that the inhabitant empathetically read the building's structure, embodied in the brick piers of the facade, perceiving through the changes in their widths the way in which the piers at the top 'are dancing like angels', as compared to 'the bottom, where they are grunting'.
A deeply shadowed arcade runs around the Library at the ground, while the top floor of the 'brick building' is opened as a pergola, through which we can see the sky beyond- dark below, light above. In the three double-height floors between, the openings are glazed, with a large sheet of glass recessed in the depth of the brick wall placed above teak-wood-clad volumes set flush to the outside face of the brick wall, and in which are typically opened small double windows, lighting the pairs of carrels within. At the transition between the recessed upper window and the teak-wood carrel is a bent stainless-steel drip - the only element Kahn allowed to project forward of the brick wall – which makes a sharp shadow line. Red-coloured sandstone coping (cap-beams) are set above the open, 6 foot (8 metre) brick balustrades at each corner balcony and at the rooftop pergola. At ground level the brick walls stand upon a band of black stone, expressing the concrete foundation beneath, and the heavy, open arcade anchors the building to the ground; as Kahn stated, 'The arcade is a landscape thing. It belongs to the building, certainly, but it also belongs to the entrance and belongs to the grounds.'

We may enter the Library, as Kahn said, from any direction, through the arcade: 'From all sides there is an entrance. If you are scurrying in the rain to get to the building, you can come in at any point and find your entrance. It's a continuous, campus-type entrance'. Once within the low brick-floored and brick-walled arcade we make our way to the north side of the building, where a glass-walled entry vestibule is found. Entering we are in a double-height space, in the centre of which rises a superb double staircase, its tall cylindrical outer walls framing two curved stair sets, left and  right, which meet in the middle at the landing above a in Michelangelo’s Laurentian Library (1525) in Florence, we find we have entered at the lower service level, and must climb this magnificent stair to reach the primary floor, the piano nobile. The outer edge of Kahn's stair forms a 32 foot (9.5 metre) diameter circle - the exact width of the Library's central room - centred precisely midway between the outer edge of the arcade below and the edge of the central room above. The concrete structure of the stair is exposed on its inner curve, but all surfaces that we touch as we climb are clad in travertine marble - the outer walls, the stair risers and treads, and the handrails. Above our heads, a pair of full-floor-deep concrete transfer beams, with triangular buttress-like ends and tension tie beams, span over the stair from left to right, framing large openings through which we see the entry hall.

At the centre of the Exeter Library, Kahn brings us up into the entry hall - a square space that rises the full height of the building, its giant circular concrete openings revealing the books on all four sides, celebrating the purpose of the building; 'So you feel the invitation of the books', as Kahn said. Inevitably, our eyes are first drawn up the lines of the corner piers, over the square concrete walls pierced by the circular openings, to the deep concrete beams which cross at the ceiling overhead, their bottom edges forming an enormous dark X-shape against the brightness above, where the light from clerestory windows opened on all sides strikes the sides of the beams. This great entry hall, composed of the primary geometric forms of squares, circles and triangles, an astonishing space absolutely critical to the experience and function of the building, was in fact absent from the written programme of spaces given to Kahn at the start of the project- a room with 'no name' without which the Library, as it stands today, is simply inconceivable. In this room, Kahn realized his ideal: 'A glorious central and single space, the walls and their light left in faceted planes, the shapes of the record of their making, intermingled with the serenity of light from above.'

A grand piano was placed in the entry hall some time after Exeter Library opened, which would certainly have pleased Kahn, for the sound of music in this great room is truly wonderful. Kahn would also have been pleased by the table which has also been placed in the otherwise empty entry hall, for in 1964 he had imagined such a table in the library, 'upon which the books lie, and these books are open. They are planned very, very cleverly by the librarian to open to pages ... with marvellous drawings.' Thus, even before receiving the Exeter commission, Kahn already believed that the library is not only, or even primarily, a place where 'you are thumbing through the files and catalogues' to fulfil assigned homework research, but rather a place that gives each person who enters what was for Kahn the joyful experience of 'discovering a book'.

After taking in this first impression, we begin to examine this room more carefully. The floor of the entry hall is made of light-coloured travertine, as is the stair, and our eye goes first to its corners, where stand four concrete piers, 18 inches (46 centimetres) thick and 6 feet (1.8 metres) deep, which are turned on the diagonal so we see their narrow ends, illuminated in the light from above. Unexpectedly, from the dark corners behind these piers we see thin slices of sunlight, for narrow windows are opened in the outside corner of the two stairs, at the inside of the building's re-entrant corners. This important detail, quite similar to the narrow slit-windows that Wright opened in the shadowed corner stairs of Unity Temple, was arrived at quite late in the design, which had previously shown L-shaped piers at the main floor (with X-shaped piers above), as well as elevators at the centre of the stairs, both of which would have blocked  this glimpse of light from the corners. The other two corner also are given narrow windows at their outside corners, but, as these are the lavatories, their inner corners are not opened to the central room.

Across the central hall from the entry stair is the reference desk, to the Ieft and right are the card catalogue and periodicals, all three positioned around the central space, under the book stacks. Each of the 'concrete buildings' housing the books is framed at either end by solid concrete walls, the 16 foot (4.9 metre) depth of the book stacks, with four concrete columns at third-points between, all supporting a thickened concrete slab upon which are carried the metal book cases, illuminated by fluorescent lights (whose light is not deleterious for books, as is sunlight). At the mezzanine level just above the main floor, the full floor-deep concrete truss-beams receive the load of the pairs of columns supporting the book stacks above, transferring this load to the solid concrete walls at either side. The 40 foot (12 metre) wide space in the depth of the transfer beam is left open at the entry stair, as we have seen, to 'dramatize the support', as Kahn said, and is enclosed with wooden walls on the other three sides of the entry hall. Above these transfer beams at the mezzanine,  and reaching to the underside of the diagonal beams at the ceiling, are concrete bracing beams spanning between the four corner piers, each a square opened with a 30 foot (9 metre) diameter circle, through which the four levels of wood-panel faced book stacks are visible from the floor of the entry hall.

At the outer edge of the Library are placed the double-height reading rooms, framed by the tall brick piers marching down the 80 foot (24 metre) long space at 20 foot (6 metre) intervals. The reading rooms and the book stacks are connected by the concrete floor slab, which spans between the 'concrete buildings' and the 'brick buildings’. Carpet covers the concrete floor in both the book stacks and the reading rooms, where the carpet is divided by bands of black slate that connect the pairs of brick piers. As Kahn had understood, it is the most natural thing to find a book in the protective darkness of the stacks and carry it the short distance to these sunlit reading rooms. The mezzanine book stack level above is framed at its edge by a low wood-panelled wall and built into the thickness of the depth of the brick pier are bookcases and places to sit and read, overlooking the reading room below. In the larger upper portion of each opening in the facade, a wooden frame holds a single window, set flush to the inside face of the brick wall, so that the sunlight floods into the reading room from above. Under each of these large windows, a double study carrel is set into the depth of the brick wall; as Kahn said, 'The name carrel implies something which is in the construction itself, which you find as a good place to read'

Each carrel is an elegantly detailed piece of oak furniture, divided at its midpoint by a privacy panel allowing two readers to sit facing without disturbing each other, and providing each reader with an L-shaped desk and return, with book shelves beneath. The inside edge of each carrel is closed by an angled privacy wall, while the outside edge, at the exterior of the building, has a sliding panel with which the small window may be closed and the sunlight modulated to suit the reader: 'The carrel is the room within a room', as Kahn described them. A double set of fluorescent lights provides night-time illumination, the upper one recessed into the concrete slab, lighting the reading room, and the lower one suspended directly over the carrels.