Saint Francis Receives the Stigmata (detail, 1904), Maurice Denis (1870–1943)

 

"Eight Hundred Years On, Brother Sun Still Rises Every Morning"

October 2026 Magnificat

On Sunday, October 4, 2026, we celebrate the eight-hundredth anniversary of the death of Saint Francis of Assisi. This is why, on the cover of your Magnificat, we offer for your contemplation a monumental work by Maurice Denis: Saint Francis Receiving the Stigmata. Painted in 1904, when the “Nabi of the beautiful icons” was leaving behind the audacity of his youth for a more interior, more classical, more prayerful path, this work preserves the ­influence of Gauguin and Sérusier in a taste for flat areas of color, for rhythm, for chromatic order; but it now makes them subservient to a life choice. The artist is thirty-four; he does not renounce modernity: he baptizes it.

This large panel is one single breath, a breath of primordial light. Look how this dawn tears open the heavens, and how, in the morning of the world, everything is ablaze! Here is the earth offering itself to its rays: the hills, the trees, the rocks, the grasses, the distance; everything receives, everything responds. Everything calls forth thanksgiving.

And here is Francis, so small in this incandescent immensity. Yet small and meaningful, like a simple three-letter word: “Yes.” There he stands, on the very edge of the cliff of La Verna, as though at the end of the visible universe, offered to the Invisible, his arms outstretched in the form of a cross. And creation as a whole seems gathered behind him in order, with him, to become an offering. Francis has walked through this creation barefoot, almost for fear of wounding it. For he loved it as brothers and sisters love one ­another, finding his dearest confidants in fire and water, in the bird, in the wolf and the lamb. Creation, for him, is not an environment but a kinship. It is not matter for predation; it is God’s gift, an appeal to gratitude, a summons to praise.

This is what Maurice Denis renders so admirably: under his brush, our Mother Earth becomes a canticle and calls for our ­voices. And this both gently and powerfully challenges us. Praise and thanksgiving are the first act of justice towards reality. One who, in ­thanksgiving, calls all creatures his brothers and sisters, and the earth his mother—how could he devastate them? One who receives the world as a gift—how could he treat it as prey?

Yet let us follow Maurice Denis further still. For this light is not only cosmic: it is crucified. Brother Sun is not the pagan idol of a self-sufficient world; he is the sign of Another. His brightness ­descends, crosses space, and reaches the body of the poor man of Assisi. Francis no longer contemplates only the beauty of God in creatures; he receives in his own flesh the very form of Christ. The singer of nature becomes the Alter Christus, configured to him in whom all things were created, in heaven and on earth. Francis, who had embraced poverty, is now embraced by the cross. Thus, the stigmata are the burning signature of Christ at the bottom of a life made poor by having been entirely given.

I live, now not I, but Christ lives in me (Galatians 2:20). Jesus ­countersigns upon Francis what Francis had long since allowed Jesus to write in his deeds: poverty, gentleness, obedience, fraternity, mercy, self-gift to the very end. Before the crucified Jesus conformed Francis’ hands, feet, and side to his own, Francis had already conformed his whole life to that of Jesus.

Thus the two Francises become one: the poet of Brother Sun and the crucified one of Mount La Verna; the brother of all ­creatures and the brother of the suffering Christ; he who sings creation and he who bears in his flesh the new and eternal creation. The ­artist brings them both to light in a single dawn: nature receives its ­splendor from God, and man receives his splendor from Christ.

So let us allow this light to reach us, to bathe us, to transfigure us. Let us ask Francis to teach us the gaze that blesses, the poverty that sets free, the gentleness that disarms, the likeness to Christ that alone saves our lives and the world. Eight hundred years have passed, yet Brother Sun still rises. And holiness remains that reality beyond our strength, impossible to conquer, and yet so simple to receive and to offer: to become poor enough, fraternal enough, loving enough for God to sign, authenticate, and make effective his real presence in our lives.

Pierre-Marie Dumont

Saint Francis Receives the Stigmata (detail, 1904), Maurice Denis (1870–1943)

The Monk by the Sea (1808–1810), Caspar David Friedrich (1774–1840)

 

"The Monk by the Sea" (1808–1810), Caspar David Friedrich (1774–1840)

October 2026 Magnificat

From the relative safety of shore, a solitary figure surveys white-capped waters, brooding clouds, and a boundless sky. The horizon stretches unbroken before him; the heavens dominate his diminutive form. Though his religious garb suggests faith, his head and heart are level with the restless waves. He is a question—an image of existential longing. He is Caspar David Friedrich’s Monk by the Sea.

Viewing Friedrich’s art is like “eavesdropping on someone’s prayers” (Helmut Börsch-Supan). The artist’s devout Lutheran faith and ascetic lifestyle informed his landscapes—often populated by lone, faceless figures—with a reverential mood. Friedrich stood at the vanguard of German Romanticism, and his oeuvre echoes the movement’s firm conviction: “Nature is the living, ­visible garment of God” (Goethe, 1808). When criticized for eschewing traditional religious iconography in favor of landscape, the artist asserted the religious possibilities of the genre and his distinctively Christian approach.

From a tender age, Friedrich was captivated by the natural beauty of Rügen, an island near his hometown of Greifswald on the Baltic Sea. It was here that he first heard the shore sermons of Lutheran pastor Ludwig Gotthard Kosegarten, whose plein-air preaching to seasonal fishermen was accompanied by the “trumpets of the sea and the many-voiced pipe organ of the storm” (Kosegarten, Jucunde, 1803). The island’s austere coast—with its icy waters, bracing winds, and barren dunes—is the setting for Monk by the Sea.

Genesis

No topographical markers betray this location. Friedrich empties the canvas, distilling “into the most elemental pictorial language an experience of the natural world that…approaches the universal and the timeless” (Hokanson & Seidenstein, The Soul of Nature). This dearth of detail allows for a symbolic, spiritual reading of land, sea, and sky. Divorced from the particularities of place and time, these natural forces evoke the Genesis narrative—a recurring theme in Friedrich’s art.

Standing on the dry land (Genesis 1:9), the monk feels the mighty wind move over the darkened surface of the deep as leaden clouds remit to light (Genesis 1:2-3). He gazes out over God’s creation as if contemplating its origin: In the beginning, when God created the heavens and the earth—and the earth was without form or shape, with darkness over the abyss and a mighty wind sweeping over the waters—Then God said: Let there be light, and there was light (Genesis 1:1-3).

The watery chaos (Gn 1:2)—tĕhôm, in Hebrew—is associated with confusion and disturbance, ­potentiality and formlessness. A return to this ­primordial state implies movement from intelligibility to ­incoherence, form to fragmentation, life to death. The tĕhôm is a common Biblical metaphor for extreme affliction: Save me, God, for the waters have reached my neck. I have sunk into the mire of the deep, where there is no foothold. I have gone down to the watery depths; the flood overwhelms me (Psalm 69:1-2).

Existential angst

There is a profound, melancholic resonance between the monk and the sea. Hemmed in by a low horizon, the monk’s figure is framed by its turbulent waves—his robe shares their inky hue. Out of the depths (Psalm 130:1) he invokes the infinite, curved like a question mark below the vast sky.

Monk by the Sea was originally exhibited with its counterpart, Abbey in the Oakwood, a depiction of the monk’s burial. Silhouetted in the waning light, Gothic ruins tower over a cemetery, where oaks spread gnarled branches toward a crescent moon. The abbey, the trees, the moon—each more ancient than the last—highlight the transience of the monk’s life. In a similar vein, as Friedrich himself comments, any traces of the monk’s brief presence on the beach will be quickly erased by the wind.

In both paintings, the absence of perspectival markers creates an almost disorienting sense of incalculable space. Together, the pair suggests the monk’s feelings of insignificance amidst the immensity of space and time. His mortality seems to threaten a descent into the void. Existential angst wars with faith within.

The sublime

God’s response to existential distress often looks like a kind of re-creation. As Peter sinks in the Sea of Galilee, Christ repeats the pattern of Genesis, pulling the “rock”—the dry land—from the waters. Job—crying out, beset by suffering—receives an enigmatic reply from the midst of the whirlwind: Where were you when I founded the earth? (Job 38:4).

Here too, the monk’s attention is directed toward the grandeur of the created order—toward a perspective as high and as wide as the vast expanse of sky. He must see creation anew with the eyes of faith, and discern the divine within the very vastness that disturbs his spirit.

A quintessential example of the Romantic sublime, Monk by the Sea depicts an experience of the incomprehensible, even terrifying, force and immensity of nature. From the Latin sub—“up to”—and limen—“threshold”—it describes man’s fleeting brush with the far side of reason. Often eliciting wonder and awe, this intense confrontation can also lead to despair: feelings of personal insignificance and mortal dread. Approached in faith, however, it offers a thrilling glimpse of the Creator, whose intricate management of the universe overwhelms man’s faculties. This “sacred intuition,” says Friedrich, is “seen and recognized only in belief.”

While the monk’s frame is dwarfed by the heavens, so too is the narrow strip of sea, suggesting divine dominion over the forces of chaos that beset his soul. Isolated on desolate dunes, the monk anxiously considers his place within the cosmic order. Yet—far above him—calm light shines.

Amy Giuliano

Holds degrees in art history from Yale and theology from the Angelicum, Rome.

The Monk by the Sea (1808–1810), Caspar David Friedrich (1774–1840), Alte Nationalgalerie, Berlin. 

Saint Jerome in His Study (c. 1435), Jan van Eyck (c. 1390–1441)

"Saint Jerome in His Study" (c. 1435), Jan van Eyck (c. 1390–1441)

For centuries the Church has remained indebted to Saint Jerome, the 4th-century scholar and Doctor of the Church whose translation work made it possible for “the treasures of the Bible to be opened more lavishly, so that richer fare may be provided for the faithful at the table of God’s Word” (Sacrosanctum Concilium, 51). In his early-15th-century masterpiece Saint Jerome in His Study, the gifted Flemish painter Jan van Eyck offers a striking visual homily. A prominent figure in the Early Netherlandish painting school, van Eyck uses vibrant color, warm light, and rich symbolic details to evoke the power of God’s Word in the life and work of Saint Jerome.

From a young age, Jerome (c. 347–420) was immersed in the study of literature and languages. Not long after his baptism in 366, he placed his significant linguistic and literary talents at the service of the Word of God. He lived an ascetic life of study and prayer in a continuous lectio divina, reading and contemplating God’s Word. The fruit of his long years of study was the collection of original and revised Latin translations known as the Vulgate, which became the official Latin text of Scripture.

Feasting at the table of God’s word

Jan van Eyck places Saint Jerome in the corner of a warm interior space where he is seated at an ornate desk covered with a deep green tablecloth. The delicately carved high-back seat suggests a cathedra, a bishop’s liturgical chair. Jerome is clothed in a vibrant red robe and a red hat or galero, a medieval headdress worn by cardinals. The saintly scholar’s head is inclined in deep contemplation of God’s Word.

With his right hand, Jerome gently touches the pages of Scripture. One can feel the movement of the sacred text as the living Word of God. In a blue curtained niche behind him are recessed shelves with large and small reference books and an astrolabe, an astronomical instrument used to chart the stars. Green and red bead tassels indicating his cardinal rank hang nearby. And on top of a white jar labeled “TYRIRICA,” thought to be an antidote to physical or spiritual poisons, is a pomegranate, a symbol of Christ’s Resurrection. The dark glass bottle near the sacred text reflects light streaming in from the high upper left window. This subtle glimmer of reflected light evokes the light of the Holy Spirit who inspires Jerome’s study and contemplation of God’s Word.

A large hourglass on the table in front of Saint Jerome suggests the gradual passage of time and human mortality. On the folded letter, in plain sight, is a clear salutation, addressed to an unnamed cardinal. Historians suggest it refers to Cardinal Niccolò Albergati, patron and friend of van Eyck, who may have commissioned and owned this exquisite painting. In the lower right corner, a crouched lion alludes to the legend that Saint Jerome extracted a thorn from the paw of the animal, who became his lifelong companion.

Jesus, the living and incarnate Word

“Ignorance of the Scriptures is ignorance of Christ.” These words attributed to Saint Jerome direct our gaze to Jesus Christ, who stands at the heart of Sacred Scripture. And what may we learn from Saint Jerome today? “This above all: to love the Word of God in Sacred Scripture…. It is important that every Christian live in contact and in personal dialogue with the Word of God…. On the one hand, it must be a truly personal dialogue because God speaks with each one of us through Sacred Scripture and it has a message for each one of us. We must not read Sacred Scripture as a word of the past, but as the Word of God that is also addressed to us” (Pope Benedict XVI).

Moreover, Pope Benedict continues, “although it is a personal Word, it is also a Word that builds community, builds the Church. We must therefore read it in communion with the living Church.” Saint Jerome reminds us that “human opinions come and go. What is very modern today will be very antiquated tomorrow. On the other hand, the Word of God is the Word of eternal life, it bears within it eternity and is valid for ever. By carrying the Word of God within us, we therefore carry within us eternity, eternal life.”

Jem Sullivan, Ph.D.

Associate professor in the School of Theology and Religious Studies at The Catholic University of America and the author of Way of Beauty: Rekindling Eucharistic Amazement with Visio Divina (Our Sunday Visitor).

 

Saint Jerome in His Study (c. 1435), Jan van Eyck (c. 1390–1441), Detroit Institute of Arts. ©  Detroit Institute of Arts / Bridgeman Images.

The Creation (c. 1480–1490)

"The Little Rabbit That Chose the Better Part"

Magnificat September 2026

From September 1 to October 4, the feast of Saint Francis of Assisi—in 2026 we mark the eighth centenary of his death—the Church leads us into the Season of Creation. In the spirit of this season, we contemplate this illumination painted around 1480, which comes to us from the close of the Middle Ages, when the image, with its poetic and symbolic language, sets us before the very foreground of the creative act.

And now, behold

See how everything begins, not in matter, but in glory. There is not a primordial chaos; there is gold. This burning sheet, silent, sovereign, does not represent a thing, but the glory of Being. Gold, here, is the radiance of glory; it is light before it has encountered any obstacle; it is the first morning of the world, even before the dawn has received a name. God said, “Let there be light!” ( Genesis 1:3 )

The gold background does not merely fill the image; it grounds it. It does not decorate the world; it proclaims its source, its life, and its fulfilment. It is the signature of uncreated light. It is the first day that endures beneath all days. It is the Let there be light that never ceases to resound. It is the brightness before the sun, the brightness of God, the brightness of the light that does not wane, the radiance of divine glory, by which creation is a transfiguration.

And here, at the threshold of the world, is the uncreated Light, ­begotten of uncreated Light, the first-born before all creation. Look: all things are created through him and for him (Col 1:15-17). See how, even before the creation of man and woman, it is a human figure that, rightly, the ­illuminator gives us to contemplate. Yet it is not the venerable old man representing the Father in so many images: it is a young man. Yes, it is indeed he: it is already Christ—before Bethlehem, before Nazareth, ­before the Jordan, before the cross, before the empty tomb—Christ in his creative eternity: the Word.

For when Scripture narrates the work of the six days, it is marked by this decisive formula: God said. God speaks, and it is done. God creates by his Word. Now the Word of God is not a breath lost in the air: it is Someone. He is the Son, the Only-Begotten, the Image of the invisible God, the One in whom all things were created. (Colossians 1:15–16 )

The hand that speaks

Thus the illuminator has clothed him with all his signs. First, his youth: not the passing testimony of his age, but the eternal newness of the Son. His blue halo rayed with gold: the blue of the heavenly heights, the gold of glory, of the light that does not dim. His crown consecrates him King of the universe, but with a disconcerting kingship: he does not ascend the throne to dominate others; he will descend from it in order to raise them to it; he does not have subjects under his feet, but brothers and sisters whose feet he will wash; he does not weigh upon humanity with his divine omnipotence; out of love, he will become man so that man may become God.

And precisely his red mantle, which covers his heavenly blue tunic, speaks of his ardent Love, of his life given, of his blood ­already ­mysteriously poured out in the very dawn of creation. And his hands! The right hand is the hand of the Pantocrator, the hand of Christ the Almighty: the hand that speaks, the creative hand that says Fiat and it is done. The left hand is open: it also speaks, it says: Father, thy will be done. (Matthew 6:10)

Thus, through the image, the artist makes us see the Word as he unfolds his creative power. In the upper right, here is the ­firmament of the second day, spread out like a great blue mantle in which the clouds dispense the freshness of the first morning. In this sky are kindled the lights of the fourth day. That they have a face is not naïveté, it is a poetic language:

Praise to you, my Lord, for all creation,

Most specially our noble Brother Sun,

Bringing the day by which you grant us light.

 

Pierre-Marie Dumont

The Creation (c. 1480–1490), illumination from Ms 101, fol. 7, Getty Museum, Los Angeles. Image: Courtesy of Getty’s Open Content Program.

American Paleobotanist 🇺🇸 👨‍🔬 Ralph Works Chaney (August 24, 1890 – March 3, 1971)

 

From McGraw-Hill encyclopedia of science and technology. McGraw-Hill Modern Men of Science; 426 Leading Contemporary Scientists Presented by the Editors of the McGraw-Hill Encyclopedia of Science and Technology. McGraw-Hill, 1966.

CHANEY, Ralph Works

American paleobotanist

Born August 24, 1890, Chicago, Illinois, USA

Died March 3, 1971, Berkeley, California, USA

In his study of Tertiary forests, Chaney developed a dynamic approach based upon close resemblances between plants of the past and those now living.  He considered plant fossils as representatives of the vegetation of their day, in contrast to their study as individual specimens whose structure or phylogeny is the primary concern of many paleobotanists.

Chaney's familiarity with living plants influenced his emphasis on similarities rather than on differences in designating fossil species.

But he did not use modern specific names for plants older than Pleistocene, arguing that the incompleteness of their record did not justify assuming that there had been no changes of specific rank since Tertiary time. Firmly believing that taxonomy should be a tool rather than a burden, he set up stratigraphic species where differences in size or other minor characters could be detected in rocks of different ages. In like manner he tended to establish geographic species for similar plants separated by ocean or climatic barriers. Recognition of elements whose fossils had close living equivalents in major areas did much to clarify areal and systematic relationships of fossil plants to each other and to those that have survived in modern forests.

Chaney's collecting procedure required handling of large numbers of specimens, so that variations within designated fossil species could be compared with those of related living plants. A census taken in the field showed which species were abundant near sites of deposition, and perhaps suggested more remote habitats in adjacent uplands for sparsely represented plants.

A standard section was set up in the John Day Basin of eastern Oregon, where abundant and well-preserved fossils occurred in stratigraphic sequence. Successive floras showed differences in composition, and Chaney noted a progressive reduction in size and texture of leaves. This provided a basis for dating the floras of other areas, and for their assignment to the same stage in the Tertiary sequence if they occurred in the same general latitude.

Quantitative appraisal of leaf characters provided a basis for estimating Tertiary climates. Large, thick, camptodrome-veined leaves now characterize the evergreen forests of the tropics, while trees of higher latitudes are largely deciduous, with smaller, thinner leaves, and craspedodrime venation. The change in leaf characters from Lower to Upper Tertiary shows a response to progressive changes toward the cooler and drier climate of our day.

Chaney pointed out that the temperate Eocene plants of Alaska did not appear in Oregon until Oligocene time, and that subtropical Eocene plants of Oregon have survived only in low latitudes. He suggested the term "geoflora" for groups of plants in mass migration; the Arcto-Tertiary Geoflora has maintained itself with only minor changes in composition since the Eocene, during which time its distribution has been shifted from Alaska southward across the United States. The Neotropical-Tertiary Geoflora has moved from Washington and Oregon into Mexico and Central America.  Recognition of a deciduous conifer, Metasequoia, as a fossil by S. Miki in 1941, and its almost contemporary discovery as a living tree by T. Wang in central China, has done much to confirm the concept of geofloras. Chaney had previously predicted that a fossil "redwood" might be discovered, with a deciduous habit suited to occurrence at high latitudes. He visited the natural occurrence of Metasequoia in 1948 to determine that its associates were members of the Arcto-Tertiary Geoflora, and revised the fossil records of several members of the Taxodiaceae.

Most fossil floras contain plants that lived at low to middle altitudes. Chaney suggested that since the lapse rate corresponds to successively lower temperatures at higher latitudes, an Eocene flora living near  sea level in Alaska may have had a counterpart in the mountains of Oregon; its absence or scant representation in the Oregon Eocene was to be expected in view  of its remoteness from sites of deposition, but we could reconstruct such a montane flora by studying floras of similar age in deposits to the north. An early elaboration of this concept led to his suggestion that angiosperms may have had their origin at high altitudes during the Jurassic period, although they first appeared in the fossil record in Cretaceous time.

While there has been well-defined latitudinal control of forest distribution during the Tertiary period, Chaney noted significant departures. A given flora lived farther south with increasing distance from the Pacific Coast; it ranged far to the north across ocean basins. Plotting the occurrence of Eocene floras across North America and Eurasia, he showed that lines connecting similar floras (isoflors) followed the paths of modern isotherms and of major forest types. He concluded that Eocene position of continents and oceans in relation to each other and to the axis of rotation was essentially the same as it is today. This placed the burden of proof on advocates of continental drift and polar migrations during later geologic time.

A descendant of pioneer Illinois farmers, Chaney majored successively in zoology, botany, and geology at the University of Chicago, where he received his B.S. in geology in 1912 and his Ph.D. in paleontology in 1919. Several years of high-school and university teaching (State University of Iowa) preceded his appointment in 1922 as research associate of the Carnegie Institution of Washington; he retired in 1957.  From 1930 until 1957, he was also professor of paleontology at the University of California, Berkeley.  He carried on field work widely in the Americas and Asia. During World War II he set aside his paleobotanical work to become assistant director of the University's Radiation Laboratory in an administrative capacity. He was elected to the National Academy of Sciences in 1947.

For background information see Paleobotany in the McGraw-Hill Encyclopedia of Science and Technologu.

American mathematician Salomon Bochner 👨‍🏫 🇺🇸 (20 August 1899 – 2 May 1982)

From McGraw-Hill encyclopedia of science and technology. McGraw-Hill Modern Men of Science; 426 Leading Contemporary Scientists Presented by the Editors of the McGraw-Hill Encyclopedia of Science and Technology. McGraw-Hill, 1966.

BOCHNER, Salomon

American mathematician

Born August 20, 1899, Cracow, Poland (then Austria-Hungary)

Died 2 May 1982 Houston, Texas  

Bochner's achievements were mainly in the field of harmonic or Fourier analysis, that is, in the theory of representation of general functions by trigonometric series, trigonometric integrals, and more comprehensive expansions. But he was also active in the theory of functions of several complex variables, especially on Cauchy formulas, and on functions in tubular domains, which latter topic has applications to hyperbolic differential equations and to problems in quantum field theory. In differential geometry he introduced a general topic that goes under the name "curvature and Betti numbers."

In physics and technology many phenomena are viewed as "waves" (which subsumes "vibrations" or "oscillations"); there are waves in acoustics, hydronamics, electrodynamics, optics, and quantum mechanics, among others. Invariably, a "general" wave is an additive, finite or infinite, superposition of "simple" waves, a simple wave being one with a specific single "frequency" is associable; also invariably, in each context there is a certain "typical" case, in which all occurring frequencies are integer-values multiples of a single one, which is then the smallest one. This skeletal "sameness" within physical contexts of different provenance, especially the sameness of the "typical" case, is brought about by the fact that, in each case, a certain function f(x), the wave function (say on an interval of length 2π), is represented as a series, with constant coefficients, of the "simple" functions { sin nx, cos nx } or, in the complex version, of the "simple" functions  { e^imx }. This mathematical expansion is of course independent of the physical interpretation in which the function f(x) may be involved.

Another "typical" case arises if, as J.B. Fourier did in the early 19th century, we vary the interval over which f(x) is defined, and then let the length of the interval tend to infinity. This replaces the trigonometric series by a trigonometric integral, that is, the Fourier coefficients by a certain function, say 𝝋(ɑ), which is called the Fourier transform of f(x). Now the relation between between f(x) and its transform 𝝋(ɑ) is a "dual" or "reciprocal" one, meaning that f(x) can be reobtained from 𝝋(ɑ) in more or less the same manner as 𝝋(ɑ) was obtained from f(x). After lying dormant for the better part of a century, this duality began to be studied in earnest at the beginning of the 20th century. The principle underlying it is so general and comprehensive that the awareness and presence of it was felt in ever wider and ever more numerous areas of mathematics and physics. For instance, if formulated in a suitably comprehensive version, this duality subsumes the de Broglie duality between waves and corpuscles.

A new event occurred in the early 1920s when the mathematician Harold Bohr (younger brother of the physicist Niels Bohr) introduced, on -∞ < x < ∞, so-called almost periodic functions (which had been foreshadowed as far back as in the mechanics of Lagrange), each of which has a "discrete" Fourier expansion, in which however the "frequencies" λ , which occur in the "simple" functions { e^iλx }, can be arbitrary real numbers, with no requirement at all of commensurateness between any two of them.

Bochner soon introduced an algorithmic summability process, the so-called Bochner-Fejer process, which makes it just as easy to handle the Fourier expansion of almost periodic functions-for any type of almost periodicity, however general-as the customary case of periodic functions. Furthermore, Bochner gave an entirely different characterization of the class of almost periodic function on the line, namely, by a certain topological property of compactness. It was this alternate definition that afterwards enabled John von Neumann to extend almost periodicity from the euclidean line to other group spaces.

In the field of Fourier integrals, Bochner found a much applied criterion for a continuous complex-valued function 𝝋(ɑ) to be representable as Fourier-Stieltjes integral 

𝝋(ɑ) = ∫ e^iaxdF(x)

in which dF ≥ 0. For this to be possible it is not only  necessary but also sufficient that 𝝋(ɑ) be positive-definite, meaning that for any finitely many points ɑ₁,…, ɑ_n , and complex constants  c₁,…, c_n , there is:

𝚺   (p,q=1…n) c_pc_q𝝋 (ɑ_p-ɑ_q ) ≥ 0

This criterion has several applications in the theory of probability; it is also usable for the derivation of the spectral representation of a self-adjoint operator in Hilbert space; and it has been generalized and applied to functions on topological group spaces.

Bochner was also a precursor in the theory of the so-called Schwartz distributions in that he introduced generalized Fourier transforms for functions that do not grow faster at infinity than a power of x.

After attending high school in Berlin, Bochner studied mathematics at the University of Berlin, terminating with a Ph.D. in 1921. In 1924-26 he studied, partly as a fellow of the International Education Board, with Harold Bohr in Copenhagen, and with G. H. Hardy and J. E. Littlewood in Oxford and Cambridge. In 1926-33 he was lecturer at the University of Munich, where he wrote his first book on Fourier integrals. In 1933 he joined the mathematics department of Princeton University and in 1951 became Henry Burchardt Fine Professor there. He was elected to the National Academy of Sciences in 1950.

Bochner wrote "Several Complex Variables" (1948), "Fourier Transforms" (1949), "Curvature and Betti Numbers" (1953), "Harmonic Analysis and the Theory of Probability" (1955), "Fourier Integrals" (1959), and "The Role of Mathematics in the Rise of Science" (1966).

For background information see "Analysis of Variance"; "Fourier Series and Integrals"; "Integral Transform" in the McGraw-Hill Encyclopedia of Science and Technology.

"The Well-Stocked Kitchen" (1566), Joachim Beuckelaer (c. 1534–1576)

 

 "The Well-Stocked Kitchen" (1566), Joachim Beuckelaer (c. 1534–1576)

Magnificat July 2026

 

 What would you like for dinner tonight? In this vivid scene by the Flemish painter Joachim Beuckelaer, there are plenty of options. Fowls in various stages of defeathering, listless rabbits, and succulent slabs of meat await the hands of a pair of kitchen maids busily skewering and plucking the corpses under their care. Vegetarians may even be catered for, given the range of verdant vegetables and rich, juicy fruits. Fine, choice wines seem to be on offer, as a contented-looking man by the fire seems to be ensuring that the vintage is still good. Amid the bustle of the well-stocked kitchen, several other figures are enjoying the commotion, perhaps hoping to serve as impromptu taste testers for the feast.

In the background of this rather active still-life, another scene unfolds. Framed within a classical arch and partially covered with a flowing canopy, a man with outstretched arms speaks to a group of women who listen attentively at his feet. Another woman has come upon the group, and seems to make a modest protest. Several men observe the scene with interest. In contrast to the vivid colors of the foreground fowl and fauna, these figures are painted in muted tones, their blue and green cloaks contrasting with the vivid reds that frame the kitchen scene.

In these juxtaposed scenes, Beuckelaer draws on an artistic tradition of monumental genre scenes developed by his teacher Pieter Aertsen (c. 1507–1575). In a 1552 painting now in Vienna, Aertsen contrasts a foreground still-life of a table full of luscious food with a depiction of Jesus talking with two women. An inscription on the mantlepiece makes the occasion clear: Mary has chosen the better part (Luke 10:42). Aertsen juxtaposes the kitchen of earthly delights with the biblical story of Mary and Martha recounted in Luke 10:38-42 and proclaimed in the liturgy on the feast of Saints Martha, Mary, and Lazarus (July 29):

Jesus entered a village where a woman whose name was Martha welcomed him. She had a sister named Mary who sat beside the Lord at his feet listening to him speak. Martha, burdened with much serving, came to him and said, “Lord, do you not care that my sister has left me by myself to do the serving? Tell her to help me.” The Lord said to her in reply, “Martha, Martha, you are anxious and worried about many things. There is need of only one thing. Mary has chosen the better part and it will not be taken from her.”

Following in the brush strokes of his master, Beuckelaer’s "Well-Stocked Kitchen" juxtaposes the outward goods of earthly food and drink and the inward mystery of Christ’s comforting and challenging words. In Beuckelaer’s depiction, the outstretched arms of Christ seem to distinguish between the outward cares of Martha and the inward concern of Mary while encircling both women in his magnanimous embrace.

In the view of some art historians, Aertsen’s and Beuckelaer’s paintings are meant to contrast the self-love of those who follow worldly desires and the divine love of those who strive for the Kingdom of Heaven. While there is certainly truth to this perspective, there is also an element of synthesis between the outward and the inward in these paintings. Christ has truly come into this world to live among us. From the wedding feast at Cana onwards, Jesus truly dwelt among his people, dining and drinking with them. Jesus does not come to destroy our human nature, but to restore and elevate it.

Jesus does not deny the need for outward things—in his Incarnation, the invisible Son of the Father has truly taken on a human body, like us in all things but sin, that needs to be nourished—but he insists on the right ordering of the external and the internal. He has taken on our flesh not only to dwell among us, but to give us the chance of dwelling with him. Through his visible form, he lifts us up to the love of things invisible: “For in the mystery of the Word made flesh a new light of your glory has shone upon the eyes of our mind, so that, as we recognize in him God made visible, we may be caught up through him in love of things invisible” (Preface I of the Nativity of the Lord).

Jesus knows that we need to eat and drink, but invites us to rightly order these earthly needs: Do not worry and say, “What are we to eat?” or “What are we to drink?” or “What are we to wear?” All these things the pagans seek. Your heavenly Father knows that you need them all. But seek first the kingdom [of God] and his righteousness, and all these things will be given you besides (Matthew 6:31-33).

Even in the liturgy, we need to seek a right ordering of the external and the internal. As Cardinal Ratzinger once preached, “In the Liturgy we should also give the service of Martha: offering a sacred space to the Lord, offering him our preparations, well-planned ceremonies, and the chant, offering him the gifts of the earth, the bread and the wine. All of this is very necessary, and it is equally necessary to do it well. Yet nonetheless, if the ‘Marian’ dimension is not also present in the liturgy—the contemplative dimension in which we simply sit at the Lord’s feet—the most essential thing has gone missing.”

In the sacred banquet of the Eucharist, Jesus unites the outward nourishing of bodily food with the inward nourishing of the soul. He offers not merely the bounties of a well-stocked kitchen, but something far more satisfying: Whoever eats my flesh and drinks my blood has eternal life, and I will raise him on the last day (John 6:54). The same Jesus who visited the home of Martha and Mary wants to enter our homes as well: Behold, I stand at the door and knock. If anyone hears my voice and opens the door, [then] I will enter his house and dine with him, and he with me (Revelation 3:20).

 

Father Innocent Smith, O.P.

Dominican friar of the Province of Saint Joseph and professor at the University of Notre Dame. He is the author of Bible Missals and the Medieval Dominican Liturgy.

 
The Well-Stocked Kitchen (1566), Joachim Beuckelaer (c. 1534–1576), Rijskmuseum, Amsterdam. Photo : Rijksmuseum, Amsterdam.

 

The Sower (c. 1865), Jean-François Millet (1814–1875)

 

 

 "Blessed Are the Arable Hearts"

Magnificat July 2026 

A sower went out to sow. Thus begins the world, thus begins the Gospel (Luke 8:5) , thus begins this painting…

The arm is flung forward, the hand opens, the seeds take flight to fall like a rain that makes the earth fruitful. And in this gesture lies the entire parable—the good soil and the harvest.

As far as the eye can see, the arable land unfolds. The earth is open, the land that the hand of man has freed from the stones, purged of brambles, torn from the curse like a promise about to be fulfilled. As far as the eye can see, here is the prepared land. Not the stony ground where the seeds burn, not the brambles where the young shoot is strangled, but that vast fertile field like a womb waiting to be made fruitful.

As far as the eye can see, Lord, here is this fallow land, the fruit of men’s labor. Land shaped by sweat, sanctified by it, land where every plowing is an act of faith, where every furrow opened is an act of hope, where the preparation of soil is an act of charity.

Now, Jean-François Millet (1814–1875) has placed us, the beholders of his work, in a position to become its actors: we are set so low, almost at ground level, that we ourselves become one of the furrows. Before this painting, we are part of the good soil that is about to be fertilized by the august gesture of the Sower. For this land, the fruit of human labor, is our heart. Our heart turned, broken, plowed, opened, offered: freed from the hardness of the stones, released from the grip of the brambles. Blessed are the arable hearts…

Millet painted a peasant sowing. He shows him to us from below: we look up to him, and this raises and exalts him. His silhouette is outlined by the light of the setting sun piercing through the clouds, a technique used in ancient bas-reliefs to designate men of providence. His gesture is broad, theatrical, arm outstretched: a movement that belongs more to the monuments raised in our public squares to honor great men than to simple narrative painting.  

The mystery of mysteries   

This monumentality bestowed by Millet upon his pictorial expression is not in vain: here the peasant sows as Abel the righteous sowed at the dawn of the world, and as Jesus of Nazareth sowed his word when the fullness of time had come.  Millet painted what he himself confessed: “The peasant at his work is like a priest at his Mass.” Then everything becomes clear: the step is a procession, the hand is a blessing, the land is an altar.  For to sow is to begin to accomplish the mystery wherein the grain, fallen into the earth, dies in order to live, is lost in order to multiply, disappears in order to rise again. Mystery of mysteries where God explains nothing, yet in the end accomplishes everything in us, for us.

Van Gogh considered Millet “the sower of modern art.” And indeed he was, for he showed that art needs neither gold nor marble nor grand style to be an expression of the sacred, but only a peasant advancing, an arm opening, a grain falling into the earth. He looked upon the humble man and in him discerned a man raised to the rank of the divine; he looked upon the earth and there discerned the table of the Last Supper; he looked upon labor and discerned a liturgy. He contemplated the earth and the work of men, and from plowing to harvest he discerned the history of salvation. And since then, every artist who seeks truth rather than appearance—a presence that speaks to the soul, rather than affectation that speaks to pride—sows in the furrow opened by Jean-François Millet, shepherd in his childhood, plowman in his adolescence, who before his easel became an expert in parables.

 

Pierre-Marie Dumont

 
The Sower (c. 1865), Jean-François Millet (1814–1875), The Clark Art Institute, Williamstown, Massachusetts. © 2011 Sterling and Francine Clark Art Institute / photo Michael Agee